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ON THE SPLITTING PRINCIPLE OF BENIAMINO SEGRE 18 Oct 2021 Camilla Felisetti Claudio Fontanari ON THE SPLITTING PRINCIPLE OF BENIAMINO SEGRE 18 Oct 2021arXiv:2105.00892v3 [math.AG] We state and prove in modern terms a Splitting Principle first claimed by Beniamino Segre in 1938, which should be regarded as a strong form of the classical Principle of Connectedness. Introduction The memoir [S38], appeared in the Annali di Matematica Pura e Applicata in 1938, is the last paper published in Italy by Beniamino Segre before the end of Second World War, as a consequence of the odious racial laws promulgated by the Italian Fascist Regime in that same year. The title Un teorema fondamentale della geometria sulle superficie algebriche ed il principio di spezzamento alludes to a Splitting Principle stated there for the first time (albeit with a sloppy formulation and an incomplete proof). Francesco Severi, who praised that work as the best one of his former student (see [BC98], p. 233), extensively reconsidered it in his treatise [S59], published in 1959. Such Splitting Principle should be regarded as a strong form of the classical Principle of Connectedness, attributed by Segre to Enriques and by Severi to Noether, stating that if the general member of a flat family {X t } of closed subschemes of P k parameterized by an irreducible curve T of finite type is connected, then X t is connected for all t ∈ T (see for instance [H77], III, Exercise 11.4 on p. 281, or [S09], Proposition 6.5). In modern terms, we state it as follows (see [S38], p. 111, and [S59], pp. 81-82): Theorem 1.1 (Splitting Principle). Let {E} be a flat family over a normal base of nodal curves on a smooth surface F of geometric genus p g . Suppose that the general element E of {E} is irreducible and that a special element E 0 of {E} splits as E 0 = C + D with C, D irreducible. Let Γ := C ∩D = Γ 1 ⊔Γ 2 , where Γ 1 is the set of points which are limits of nodes of the general curve in {E} and Γ 2 is its complement in Γ. Assume that |D |D | is non-empty and C is sufficiently general with respect to D, in particular that |C(−Γ 1 )| has no base points on D. If c i is the cardinality of Γ i then we have c 2 ≥ p g + 1, unless the points in Γ 2 are linearly dependent with respect to K F . The assumptions that all curves in {E} are nodal and that C is general with respect to D are both missing from Segre's statement in [S38] and are added by Severi in [S59], p. 81. We point out that the splitting of Γ into the disjoint union of Γ 1 and Γ 2 is well-defined only if E 0 is assumed to have double points as singularities (hence it is implicit in Segre's argument, see in particular [S38], §10, p. 122: "I punti di Γ si potranno allora distinguere in due categorie, secondochè provengono o meno come limiti da punti doppi di E, ossia rispettivamente a seconda che non risultano oppure risultano punti di collegamento tra C e D; denotiamo ordinatamente con Γ 1 , Γ 2 i gruppi costituiti dai punti del primo o del secondo tipo, (...) talchè sarà Γ = Γ 1 + Γ 2 "). Furthermore, Severi's statement in [S59], p. 81, assumes that the curve D is ordinaria, which in particular implies that the characteristic series E |D is complete on D. Severi in [S59], p. 197, comments: "(...) abbiamo stabilito nel n. 23 il notevole principio di spezzamento This research project was partially supported by GNSAGA of INdAM and by PRIN 2017 "Moduli Theory and Birational Classification". 2020 Mathematics Subject Classification. 14-03, 14D05, 14D06. Keywords and phrases. Splitting principle, Connectedness principle, Algebraic system, Flat family, Nodal curve. di B. Segre contenuto nella Memoria degli Annali di Matematica, 1938, però sotto l'ipotesi, qui aggiunta, che una componente del limite della curva che tende a spezzarsi sia una curva ordinaria irriducibile (appartenente cioè totalmente ad un sistema irriducibile avente su di essa la serie caratteristica completa). Siccome questo principio fa entrare in giuoco soltanto il genere geometrico p g della superficie F e non l'irregolarità,è ragionevole supporre (n. 23) che il suo fondamento topologico sia in relazione soltanto ai cicli bidimensionali della riemanniana di F ; epperò, siccome essoè vero, qualunque sia p g , sopra le superficie regolari, e per p g = 0 su ogni superficie irregolare apparisce naturale di supporre che il principio stesso sia vero sempre." Finally, in [S59], p. 81, the curve D is assumed to be nonsingular ("priva di nodi "), but our modern proof shows that also this assumption is unnecessary. Indeed, the main ingredients are Riemann-Roch theorem, Serre duality and adjunction formula, which hold also in the singular case up to replacing the canonical bundle K D with the dualizing sheaf ω D (see for instance [BHPV04], p. 62). We are going to apply these formulas to restrictions to D of divisors on the smooth surface F , hence to Cartier divisors on the nodal curve D. The proofs Our proof of Theorem 1.1 relies on a couple of crucial remarks. Lemma 2.1. We have Γ 2 = ∅. Proof. If Γ 2 = ∅ then all nodes of E 0 = C + D belong to Γ 1 , i.e. they are limits of nodes of E. Hence by [T76], Théorème 1 on p. 73, locally around E 0 we may resolve simultaneously all singularities of E. In this way we would obtain a family of irreducible curves degenerating to a disconnected one, contradicting the classical Principle of Connectedness (see for instance [H77], III, Exercise 11.4 on p. 281). Lemma 2.2. There is at least one point P ∈ Γ 2 which is not a base point of the complete linear series |E 0|D − Γ 1 | on D. Proof. Assume by contradiction that (1) h 0 (D, E 0|D − Γ 1 ) = h 0 (D, E 0|D − Γ 1 − Γ 2 ) = h 0 (D, E 0|D − Γ). On the other hand, since E 0 = C + D we have |E 0|D − Γ 1 | ⊇ |C |D − Γ 1 | + |D |D |. Moreover, since C |D − Γ 1 = Γ 2 = ∅ by Lemma 2.1 and |C(−Γ 1 )| has no base points on D by assumption, we have h 0 (D, C |D − Γ 1 ) ≥ 2. Hence we deduce h 0 (D, E 0|D − Γ 1 ) ≥ h 0 (D, C |D − Γ 1 ) + h 0 (D, D |D ) − 1 ≥ h 0 (D, D |D ) + 1 = h 0 (D, E 0|D − C |D ) + 1 = h 0 (D, E 0|D − Γ) + 1, contradicting (1), so the claim is established. Proof of Theorem 1.1. We follow Segre's approach in [S38]. Let d := h 1 (D, D |D ). We have two possibilities: (i) c 2 ≥ d + 1 or (ii) c 2 ≤ d. (i) Suppose c 2 ≥ d + 1. Let i := h 2 (F, D) . We first prove that (2) d ≥ p g − i. Indeed, by adjunction K F |D = ω D − D |D and by Serre duality d = h 1 (D, D |D ) = h 0 (D, ω D − D |D ) = h 0 (D, K F |D ). The short exact sequence on F 0 → K F (−D) → K F → K F |D → 0 yields a long exact sequence 0 → H 0 (K F (−D)) → H 0 (K F ) → H 0 (K F |D ) → . . . hence p g ≤ i + d. If i = 0, we immediately get c 2 ≥ d + 1 ≥ p g + 1. If instead i > 0, then the points in Γ 2 are dependent with respect to K F , i.e. h 0 (F, K F (−Γ 2 )) > p g − c 2 . Indeed, on the one hand by (2) we have (3) p g − c 2 ≤ p g − d − 1 ≤ p g − p g + i − 1 = i − 1. On the other hand, since any global section of K F which vanishes on D vanishes in particular on Γ 2 , we have h 0 (F, K F (−Γ 2 )) ≥ h 0 (F, K F (−D)) = h 2 (F, D) = i. By (3) we conclude that h 0 (F, K F (−Γ 2 )) ≥ i > p g − c 2 . (ii) Suppose c 2 ≤ d. Let P ∈ Γ 2 and set Γ * 2 := Γ 2 \ P . 1 Observe first that the linear series |D |D + Γ * 2 | on D is special. In fact, h 1 (D, D |D + Γ * 2 ) = h 0 (D, ω D − D |D − Γ * 2 ) ≥ h 0 (D, ω D − D |D ) − c 2 + 1 = h 1 (D, D |D ) − c 2 + 1 = d − c 2 + 1 ≥ 1. In particular, by adjunction we have that H 0 (D, ω D − D |D − Γ * 2 ) ∼ = H 0 (D, K F |D − Γ * 2 ) is non-zero. We are going to prove that the natural inclusion H 0 (F, K F − Γ 2 ) ⊆ H 0 (F, K F − Γ * 2 ) is an isomorphism for some choice of P ∈ Γ 2 , i.e. that the points in Γ 2 are dependent with respect to K F . Indeed, by Lemma 2.2, there exists at least one P ∈ Γ 2 such that the complete linear series |E 0|D − Γ 1 | = |C |D + D |D − Γ 1 | = |D |D + Γ − Γ 1 | = |D |D + Γ 2 | on D does not admit P as a base point. On the other hand, by the Riemann-Roch theorem h 0 (D |D + Γ 2 ) = h 1 (D |D + Γ 2 ) + deg(D |D + Γ 2 ) + 1 − p a (D) h 0 (D |D + Γ * 2 ) = h 1 (D |D + Γ * 2 ) + deg(D |D + Γ 2 ) − 1 + 1 − p a (D). Since h 0 (D |D + Γ * 2 ) = h 0 (D |D + Γ 2 ) − 1 then h 1 (D |D + Γ 2 ) = h 1 (D |D + Γ * 2 ) and by Serre duality (4) h 0 (D, ω D − D |D − Γ * 2 ) = h 0 (D, ω D − D |D − Γ 2 ). Suppose now by contradiction that the inclusion H 0 (F, K F − Γ 2 ) ⊆ H 0 (F, K F − Γ * 2 ) is strict, i.e. that there exists an effective divisor A in PH 0 (F, K F − Γ * 2 ) not passing through P . Note that A ∩ D = D, since P ∈ D \ A. Now, if Γ * 2 is not empty, then A ∩ D = ∅, since ∅ = Γ * 2 ⊂ A ∩ D, and A |D is a nontrivial effective divisor on D lying in PH 0 (D, K F |D − Γ * 2 ) \ PH 0 (D, K F |D − Γ 2 ), contradicting (4). The same conclusion holds if Γ * 2 is empty but A ∩ D = ∅. On the other hand, if Γ * 2 is empty and A |D = 0, then K 1 Note that Γ 2 is non empty by Lemma 2.1, but Γ * 2 might be. F |D ∼ = O D and by adjunction we have ω D = D |D . Hence (4) implies 1 = h 0 (D, O D ) = h 0 (D, O D (−P )) = 0 and this contradiction ends the proof. . W Barth, K Hulek, C Peters, A Van De Ven, Springer-VerlagBerlin HeidelbergCompact complex surfacesW. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact complex surfaces, Springer-Verlag Berlin Heidelberg (2004). A Brigaglia, C Ciliberto, Geometria Algebrica, La matematica italiana dopo l'unità. Marcos Y Marcos; MilanoA. Brigaglia, C. Ciliberto, Geometria algebrica, in: La matematica italiana dopo l'unità, Marcos Y Marcos, Milano (1998), 185-320. Algebraic Geometry. R Hartshorne, Springer-VerlagNew YorkR. Hartshorne, Algebraic Geometry, Springer-Verlag New York (1977). Un teorema fondamentale della geometria sulle superficie algebriche ed il principio di spezzamento. B Segre, Ann. Mat. Pura Appl. 171B. Segre, Un teorema fondamentale della geometria sulle superficie algebriche ed il principio di spezza- mento, Ann. Mat. Pura Appl. 17 (1938), no. 1, 107-126. A smoothing criterion for families of curves. E Sernesi, preprintE. Sernesi, A smoothing criterion for families of curves, preprint February 2009, available online at http://www.mat.uniroma3.it/users/sernesi/smoothcrit.pdf. Geometria dei sistemi algebrici sopra una superficie e sopra una varietà. algebrica. Edizioni Cremonese. F Severi, RomaF. Severi, Geometria dei sistemi algebrici sopra una superficie e sopra una varietà. algebrica. Edizioni Cremonese, Roma 1959. Résolution simultanée. I. II., in: Séminaire sur les singularités des surfaces. B Teissier, Lecture Notes in Mathematics. 777Springer-VerlagCent. Math.Éc. Polytech.B. Teissier, Résolution simultanée. I. II., in: Séminaire sur les singularités des surfaces (Cent. Math.Éc. Polytech., Palaiseau 1976-77), Lecture Notes in Mathematics 777, Springer-Verlag, Berlin (1980), 71-146. E-mail: camilla.felisetti@unitn.it. Trento, ItalyTrento, Italy. E-mail: camilla.felisetti@unitn.it Claudio Fontanari Dipartimento di Matematica Università di Trento Via Sommarive 14. Claudio Fontanari Dipartimento di Matematica Università di Trento Via Sommarive 14 E-mail: claudio.fontanari@unitn.it. Trento, ItalyTrento, Italy. E-mail: claudio.fontanari@unitn.it

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{'abstract': 'We state and prove in modern terms a Splitting Principle first claimed by Beniamino Segre in 1938, which should be regarded as a strong form of the classical Principle of Connectedness.', 'arxivid': '2105.00892', 'author': ['Camilla Felisetti ', 'Claudio Fontanari '], 'authoraffiliation': [], 'corpusid': 233481657, 'doi': '10.1007/s10231-021-01171-w', 'github_urls': [], 'n_tokens_mistral': 4322, 'n_tokens_neox': 3789, 'n_words': 2183, 'pdfsha': '8ebbbecaed6e4a815d4764ee1e529dfa5a4a119e', 'pdfurls': ['https://arxiv.org/pdf/2105.00892v3.pdf'], 'title': ['ON THE SPLITTING PRINCIPLE OF BENIAMINO SEGRE', 'ON THE SPLITTING PRINCIPLE OF BENIAMINO SEGRE'], 'venue': []}

arxiv

Calibrating AI Models for Wireless Communications via Conformal Prediction 15 Dec 2022 Student Member, IEEEKfir M Cohen kfir.cohen@kcl.ac.uk Viterbi Faculty of Electrical and Computing Engineering Technion-Israel Institute of Technology WC2R 2LS, 3200003London, HaifaU.K., Israel Member, IEEESangwoo Park sangwoo.park@kcl.ac.uk Viterbi Faculty of Electrical and Computing Engineering Technion-Israel Institute of Technology WC2R 2LS, 3200003London, HaifaU.K., Israel Fellow, IEEEOsvaldo Simeone osvaldo.simeone@kcl.ac.uk. Viterbi Faculty of Electrical and Computing Engineering Technion-Israel Institute of Technology WC2R 2LS, 3200003London, HaifaU.K., Israel Shlomo Shamai Viterbi Faculty of Electrical and Computing Engineering Technion-Israel Institute of Technology WC2R 2LS, 3200003London, HaifaU.K., Israel Life Fellow, IEEEShitz Viterbi Faculty of Electrical and Computing Engineering Technion-Israel Institute of Technology WC2R 2LS, 3200003London, HaifaU.K., Israel Shitz)Shlomo Shamai Viterbi Faculty of Electrical and Computing Engineering Technion-Israel Institute of Technology WC2R 2LS, 3200003London, HaifaU.K., Israel Calibrating AI Models for Wireless Communications via Conformal Prediction 15 Dec 202210.18742/rnvf-m0761 The authors acknowledge the use of King's Computational Research, Engineering and Technology Environment (CREATE). Retrieved November 21, 2022, from https://doi.org/10.18742/rnvf-m076. Kfir M. Cohen, Sangwoo Park, and Osvaldo Simeone are with King's Communication, Learning, & Information Processing (KCLIP) lab, Department of Engineering, King's College London, 2Index Terms Calibrationset predictionreliabilityconformal predictioncross-validationBayesian learningwireless commu- nications When used in complex engineered systems, such as communication networks, artificial intelligence (AI) models should be not only as accurate as possible, but also well calibrated. A well-calibrated AI model is one that can reliably quantify the uncertainty of its decisions, assigning high confidence levels to decisions that are likely to be correct and low confidence levels to decisions that are likely to be erroneous. This paper investigates the application of conformal prediction as a general framework to obtain AI models that produce decisions with formal calibration guarantees. Conformal prediction transforms probabilistic predictors into set predictors that are guaranteed to contain the correct answer with a probability chosen by the designer. Such formal calibration guarantees hold irrespective of the true, unknown, distribution underlying the generation of the variables of interest, and can be defined in terms of ensemble or time-averaged probabilities. In this paper, conformal prediction is applied for the first time to the design of AI for communication systems in conjunction to both frequentist and Bayesian learning, focusing on demodulation, modulation classification, and channel prediction.A. MotivationHow reliable is your artificial intelligence (AI)-based model? The most common metric to design an AI model and to gauge its performance is the average accuracy. However, in applications in which AI decisions are used within a larger system, AI models should not only be as accurate as possible, but they should also be able to reliably quantify the uncertainty of their decisions. As an example, consider an unlicensed link that uses AI tools to predict the best channel to access out of four possible channels. A predictor that assigns the probability vector of [90%, 2%, 5%, 3%] to the possible channels predicts the same best channel -the first -as a predictor that outputs the probability vector [30%, 20%, 25%, 25%]. However, the latter predictor is less certain of its decision, and it may be preferable for the unlicensed link to refrain from accessing the channel when acting on less confident predictions, e.g., to avoid excessive interference to licensed links [1], [2].As in the example above, AI models typically report a confidence measure associated with each prediction, which reflects the model's self-evaluation of the accuracy of a decision. Notably, neural network models implement probabilistic predictors that produce a probability distribution across all possible values of the output variable.The self-reported model confidence, however, may not be a reliable measure of the true, unknown, accuracy of a prediction. In such situations, the AI model is said to be poorly calibrated.As illustrated in the example inFig. 1, accuracy and calibration are distinct criteria, with neither criterion implying the other. It is, for instance, possible to have an accurate predictor that consistently underestimates the accuracy of its decisions, and/or that is overconfident where making incorrect decisions (see fourth column inFig. 1). Conversely, one can have inaccurate predictions that estimate correctly their uncertainty (see fifth column inFig. 1).Deep learning models tend to produce either overconfident decisions [3], or calibration levels that rely on strong assumptions about the ground-truth, unknown, data generation mechanism [4]-[9]. This paper investigates the use of conformal prediction (CP) [10]-[12] as a framework to design provably well-calibrated AI predictors, with distribution-free calibration guarantees that do not require making any assumption about the ground-truth data generation mechanism.B. Conformal Prediction for AI-Based Wireless SystemsCP leverages probabilistic predictors to construct well-calibrated set predictors. Instead of producing a probability vector, as in the examples inFig. 1, a set predictor outputs a subset of the output space, as exemplified inFig. 2. A set predictor is well calibrated if it contains the correct output with a pre-defined coverage probability selected by the system designer. For a well-calibrated set predictor, the size of the prediction set for a given input provides a measure of the uncertainty of the decision. Set predictors with smaller average prediction size are said to be more efficient [10]. This paper investigates CP as a general mechanism to obtain AI models with formal calibration guarantees for communication systems. The calibration guarantees of CP hold irrespective of the true, unknown, distribution underlying the generation of the variables of interest, and are defined either in terms of ensemble averages[10]or (a) Examples of probabilistic predictors for two inputs x 1 and x 2 : As compared to the ground-truth distribution in the second column, the first predictor (third column) is accurate, assigning the largest probability to the optimal decision (indicated as "opt" in the second column) and also well calibrated, reproducing the true accuracy of the decision; the second predictor (fourth column) is still accurate, but it is underconfident on the correct decision (for input x 1 ) and overconfident on the correct decision (for input x 2 ); the third predictor (fifth column) is not accurate, producing a uniform distribution across all output values, but is well calibrated if the data set is balanced [13]; and the last predictor (sixth column) is both inaccurate and poorly calibrated, providing overconfident decisions. (b) Confidence versus accuracy for the decisions made by the corresponding predictors. A well-calibrated set predictor can be inefficient if it returns excessively large set predictions (forth column). In contrast, a poorly-calibrated set predictor (fifth column) returns set predictions that include the true value of the label with a probability smaller than 1 − α. in terms of long-term averages [14]. CP is applied in conjunction to both frequentist and Bayesian learning, and specific applications are discussed to demodulation, modulation classification, and channel prediction. C. Related Work Most work on AI for communications relies on conventional frequentist learning tools (see, e.g., the review papers [15]- [18]). Frequentist learning is based on the minimization of the (regularized) training loss, which is interpreted as an estimate of the ground-truth population loss. When data is scarce, this estimate is unreliable, and hence the focus on a single, optimized, model parameter vector often yields probabilistic predictors that are poorly calibrated, producing overconfident decisions [3], [19]- [21]. Bayesian learning offers a principled way to address this problem [22], [23]. This is done by producing as the output of the learning process not a single model parameter vector, but rather a distribution in the model parameter space, which quantifies the model's epistemic uncertainty caused by limited access to data. A model trained via Bayesian learning produces probabilistic predictions that are averaged over the trained model parameter distribution. This ensembling approach to prediction ensures that disagreements among models that fit the training data (almost) equally well are accounted for, substantially improving model calibration [24], [25]. In practice, Bayesian learning is implemented via approximations such as variational inference (VI) or Monte Carlo (MC) sampling, yielding scalable learning solutions [23]. VI methods approximate the exact Bayesian posterior distribution with a tractable variational density [26]- [29], while MC techniques obtain approximate samples from the Bayesian posterior distribution [30]- [32]. Among other applications to communications systems, Bayesian learning was studied for network allocation in [33]- [35], for massive MIMO detection in [36]- [38], for channel estimation in [39]- [41], for user identification in [42], and for multi user detection in [43], [44]. Extensions to Bayesian meta-learning have been investigated in [20]. Exact Bayesian learning offers formal guarantees of calibration only under the assumption that the assumed model is well specified [4], [5]. In practice, this means that the assumed neural network models should have sufficient capacity to represent the ground-truth data generation mechanism, and that the predictive uncertainty should be unimodal for continuous outputs (since conventional likelihoods are unimodal, e.g., Gaussian) [5], [23], [24]. These assumptions are easily violated in practice, especially in communication systems in which lower-complexity models must be implemented on edge devices, and access to data for specific network configurations is limited. Specific examples are provided in [21] for applications including modulation classification [45], [46] and localization [47], [48]. Robustified versions of Bayesian learning that are based on the optimization of a modified free energy criterion were shown empirically to partly address the problem of model misspecification [4], [5], with implications for communication systems presented in [21]. However, robust Bayesian learning solutions do not have formal guarantees of calibration in the presence of misspecified models. Another family of methods that aim at enhancing the calibration of probabilistic models implement a validationbased post-processing phase. Platt scaling [49] and temperature scaling [3] find a fixed parametric mapping of the trained model output that minimizes the validation loss, while isotonic regression [50] applies a non-parametric binning approach. These recalibration-based approaches cannot guarantee calibration, as they may overfit the validation data set [51] and they are sensitive to the inaccuracy of the starting model [52]. Conformal prediction is a general framework for the design of set predictors that satisfy formal, distribution-free, guarantees of calibration [10], [11]. Given a desired miscoverage probability α, CP returns set predictions that include the correct output value with probability at least 1 − α under the only assumption that the data distribution is exchangeable. This condition is weaker that the standard assumption of "i.i.d." data made in the design of most machine learning systems. The original work on CP, [10], introduced validation-based CP and full CP. Since then, progress has been made on reducing computational complexity, minimizing the size of the prediction sets, and further alleviating the assumptions of exchangeability. Cross-validation-based CP was proposed in [53] to reduce the computational complexity as compared to full CP, while improving the efficiency of validation-based CP. The authors of [54], [55] proposed the optimization of a CP-aware loss to improve the efficiency of validation-based CP, while avoiding the larger computational cost of cross-validation. The work [56] proposed reweighting as a means to handle distribution shifts between the examples in the data set and the test point. Other research directions include improvements in the training algorithms [57], [58], and the introduction of novel calibration metrics [59], [60]. Finally, online CP, presented in [14], [61], was shown to achieve long-term calibration over time without requiring statistical assumptions on the data generation. D. Main Contributions To the best of our knowledge, with the exception of the conference version [62] of this paper, this is the first work to investigate the application of CP to the design of AI models for communication systems. The main contributions of this paper are as follows. • We provide a self-contained introduction to CP by focusing on validation-based CP [10], cross-validation-based CP [53], and online conformal prediction [61]. The presentation details connections to conventional probabilistic predictors, as well as the performance metrics used to assess calibration and efficiency. • We propose the application of offline CP to the problems of symbol demodulation and modulation classification. The experimental results validate the theoretical property of CP methods of providing well-calibrated decisions. Furthermore, they demonstrate that naïve predictors that only rely on the output of either frequentist or Bayesian learning tools often result in poor calibration. • Finally, we study the application of online CP to the problem of predicting received signal strength for over-the-air measured signals [63]. We demonstrate that online CP can obtain the predefined target long-term coverage rate at the cost of negligible increase in the prediction interval as compared to naïve predictors. The conference version [62] of this work presented results only for symbol demodulation, while not providing background material on CP and not considering online CP. In contrast, this work is self-contained, presenting CP from first principles and including also online CP. Furthermore, this work investigates applications of CP to modulation classification and to channel prediction by leveraging real-world data sets [63], [64]. For reproducibility purposes, we have made our code publicly available 1 . The rest of this paper is organized as follows. In Sec. II, we define set predictors, and introduce the relevant performance metrics. Then, in Sec. III, naïve set predictors are introduced that do not provide guarantees in terms of calibration. Sec. IV describes conformal prediction, a general methodology to obtain well-calibrated set predictors. Sec. V details online conformal prediction, which is well suited for time-varying data. Applications to wireless communications are investigated in the following sections: Symbol demodulation is studied in Sec. VI; modulation classification in Sec. VII; and channel prediction in Sec. VIII. Sec. IX concludes the paper. II. PROBLEM DEFINITION This section introduces set predictors, along with key performance metrics of coverage and inefficiency. To this end, we start by describing the data-generation model and reviewing probabilistic predictors. A. Data-Generation Model We consider the standard supervised learning setting in which the learner is given a data set D = {z[i]} N i=1 of N examples of input-output pairs z[i] = (x[i], y[i]) for i = 1, . . . , N , and is tasked with producing a prediction on a test input x with unknown output y. Writing z = (x, y) for the test pair, data set D and test point z follow the unknown ground-truth, or population, distribution p 0 (D, z). Apart from Sec. V, we further assume throughout that the population distribution p 0 (D, z) is exchangeable -a condition that includes as a special case the traditional independent and identically distributed (i.i.d.) data-generation setting. Note that we will not make explicit the distinction between random variables and their realizations, which will be clear from the context. p 0 (D, z|c) = p 0 (z|c) N i=1 p 0 z[i] c(1) for some ground-truth sampling distribution p 0 (z|c) given the variable c, under the exchangeability assumption, the joint distribution can be expressed as p 0 (D, z) = E p0(c) p 0 (D, z|c) ,(2) where E p(x) [·] denotes the expectation with respect to distribution p(x). The vector c in (2) can be interpreted as including context variables that determine the specific learning task. For instance, in a wireless communication setting, the vector c may encode information about channel conditions. In Sec. V, we will consider a more general setting in which no assumptions are made on the distribution of the data. B. Probabilistic Predictors Before introducing set predictors, we briefly review conventional probabilistic predictors. Probabilistic predictors implement a parametric conditional distribution model p(y|x, φ) on the output y ∈ Y given the input x ∈ X , where φ ∈ Φ is a vector of model parameters. Given the training data set D, frequentist learning produces an optimized single vector φ * D , while Bayesian learning returns a distribution q * (φ|D) on the model parameter space Φ [23], [24]. In either case, we will denote as p(y|x, D) the resulting optimized predictive distribution p(y|x, D) =      p(y|x, φ * D ) for frequentist learning E q * (φ|D) [p(y|x, φ)] for Bayesian learning. Note that the predictive distribution for Bayesian learning is obtained by averaging, or ensembling, over the optimized distribution q * (φ|D). We refer to Appendix A for basic background on frequentist and Bayesian learning. From (3), one can obtain a point predictionŷ for output y given input x as the probability-maximizing output aŝ y(x|D) = argmax y ∈Y p(y |x, D).(4) In the case of a discrete set Y, the hard predictor (4) minimizes the probability of detection error under the model p(y|x, D). The probabilistic prediction p(y|x, D) also provides a measure of predictive uncertainty for all possible outputs y ∈ Y. In particular, for the point predictionŷ(x|D) in (4), we have the predictive, self-reported, confidence level conf(x|D) = max y ∈Y p(y |x, D) = p ŷ(x|D) x, D .(5) As illustrated in Fig. 1, the performance of a probabilistic predictor can be evaluated in terms of both accuracy and calibration, with the latter quantifying the quality of uncertainty quantification via the confidence level (5) [3]. Specifically, a probabilistic predictor p(y|x, D) is said to be well calibrated [3] if the probability that the hard predictorŷ =ŷ(x|D) equals the true label matches its confidence level π for all possible values of probability π ∈ [0, 1]. Mathematically, calibration is defined by the condition P y =ŷ p(ŷ|x, D) = π = π, for all π ∈ [0, 1](6) where the probability P(·) follows the ground-truth distribution p 0 (x, y). Stronger definitions, like that introduced in [66], require the predictive distribution to match the ground-truth distribution also for values of y that are distinct from (4). C. Set Predictors A set predictor is defined as a set-valued function Γ(·|D) : X → 2 Y that maps an input x to a subset of the output domain Y based on data set D. We denote the size of the set predictor for input x as |Γ(x|D)|. As illustrated in the example of Fig. 2, the set size |Γ(x|D)| generally depends on input x, and it can be taken as a measure of the uncertainty of the set predictor. The performance of a set predictor is evaluated in terms of calibration, or coverage, as well as of inefficiency. Coverage refers to the probability that the true label is included in the predicted set; while inefficiency refers to the average size |Γ(x|D)| of the predicted set. There is clearly a trade-off between two metrics. A conservative set predictor that always produces the entire output space, i.e., Γ(x|D) = Y, would trivially yield a coverage probability equal to 1, but at the cost of exhibiting the worst possible inefficiency of |Y|. Conversely, a set predictor that always produces an empty set, i.e., Γ(x|D) = ∅, would achieve the best possible inefficiency, equal to zero, while also presenting the worst possible coverage probability equal to zero. Let us denote a set predictor Γ(·|·) for short as Γ. Formally, the coverage level of set predictor Γ is the probability that the true output y is included in the prediction set Γ(x|D) for a test pair z = (x, y). This can be expressed as coverage(Γ) = P y ∈ Γ(x|D) , where the probability P(·) is taken over the ground-truth joint distribution (a) (b) Γ x[i]|{z[j]} i−1 j=1 Γ(x|D) y Y y[i] Y timep 0 (D, (x, y)) in (2). The set predictor Γ is said to be (1 − α)-valid if it satisfies the inequality coverage(Γ) = P y ∈ Γ(x|D) ≥ 1 − α.(7) When the desired coverage level 1 − α is fixed by the predetermined target miscoverage level α ∈ [0, 1], we will also refer to set predictors satisfying (7) as being well calibrated. Following the discussion in the previous paragraph, it is straightforward to design a valid, or well-calibrated, set predictor, even for the restrictive case of miscoverage level α = 0. This can be, in fact, achieved by producing the full set Γ(x|D) = Y for all inputs x. One should, therefore, also consider the inefficiency of predictor Γ. The inefficiency of set predictor Γ is defined as the average prediction set size inefficiency(Γ) = E Γ(x|D) ,(8) where the average is taken over the data set D and the test pair (x, y) following their exchangeable joint distribution p 0 (D, (x, y)). In practice, the coverage condition (7) is relevant if the learner produces multiple predictions using independent data set D, and is tested on multiple pairs (x, y). In fact, in this case, the probability in (7) can be interpreted as the fraction of predictions for which the set predictor Γ(x|D) includes the correct output. This situation, illustrated in Fig. 3(a), is quite common in communication systems, particularly at the lower layers of the protocol stack. For instance, the data D may correspond to pilots received in a frame, and the test point z to a symbol within the payload part of the frame (see Sec. VI). While the coverage condition (7) is defined under the assumption of a fixed ground-truth distribution p 0 (D, z), in Sec. V we will allow for temporal distributional shifts and we will focus on validity metrics defined as long-term time averages (see Fig. 3 Fig. 4. A naïve probabilistic-based (NPB) set predictor uses a pre-trained probabilistic predictor to include all output values to which the probabilistic predictor assigns the largest probabilities that reach the coverage target 1 − α. This naïve scheme has no formal guarantee of calibration, i.e., it does not guarantee the coverage condition (7), unless the original probabilistic predictor is well calibrated. (b)). D train predict x model {p (y |x, D)} y ∈Y order probabilities threshold { } 0 1 1 − α Γ NPB (x|D) III. NAÏVE SET PREDICTORS Before describing CP in the next section, in this section we review two naïve , but natural and commonly used, approaches to produce set predictors, that fail to satisfy the coverage condition (7). A. Naïve Set Predictors from Probabilistic Predictors Given a probabilistic predictor p(y|x, D) as in (3), one could construct a set predictor by relying on the confidence levels reported by the model. Specifically, aiming at satisfying the coverage condition (7), given an input x, one could construct the smallest subset of the output domain Y that covers a fraction 1 − α of the probability designed by model p(y|x, D). Mathematically, the resulting naïve probabilistic-based (NPB) set predictor is defined as Γ NPB (x|D) =argmin Γ∈2 Y |Γ| (9) s.t. y ∈Γ p(y |x, D) ≥ 1 − α for the case of a discrete set, and an analogous definition applies in the case of a continuous domain Y. Fig. 4 illustrates the NPB for a prediction problem with output domain size |Y| = 4. Given that, as mentioned in Sec. I, probabilistic predictors are typically poorly calibrated, the naïve set predictor (9) does not satisfy condition (7) for the given desired miscoverage level α, and hence it is not well calibrated. For example, in the typical case in which the probabilistic predictor is overconfident [3], the predicted sets (9) tend to be too small to satisfy the coverage condition (7). B. Naïve Set Predictors from Quantile Predictors While the naïve probabilistic-based set predictor (9) applies to both discrete and continuous target variables, we now focus on the important special case in which Y is a real number, i.e., Y = R. This corresponds to scalar regression problems, such as for channel prediction (see Sec. VIII). Under this assumption, one can construct a naïve set predictor based on estimates of the α/2and (1 − α/2)-quantiles y α/2 (x) and y 1−α/2 (x) of the ground-truth distribution p 0 (y|x) (obtained from the joint distribution p 0 (D, z)). In fact, writing as y q (x) = inf y ∈ R : y −∞ p 0 (y |x) dy ≤ q(10) the q-quantile, with q ∈ [0, 1], of the ground-truth distribution p 0 (y|x), the interval y α/2 (x), y 1−α/2 (x) contains the true value y with probability 1 − α. Defining the pinball loss as [67] q (y,ŷ) = max − (1 − q)(y −ŷ), q(y −ŷ) (11) for q ∈ [0, 1], the quantile y q (x) in (10) can be obtained as [68] y q (x) = argmin y∈R E p0(y|x) q (y,ŷ) .(12) Therefore, given a parametrized predictive modelŷ(x|φ), the quantile y q (x) can be estimated asŷ(x|φ D,q ) with optimized parameter vector φ D,q = argmin φ 1 N (x,y)∈D q y,ŷ(x|φ) .(13) With the estimateŷ(x|φ D,α/2 ) of quantile y α/2 (x) and estimateŷ(x|φ D,1−α/2 ) of quantile y 1−α/2 (x), we finally obtain the naïve quantile-based (NQB) predictor Γ NQB (x|D) = ŷ(x|φ D,α/2 ),ŷ(x|φ D,1−α/2 ) .(14) The naïve set prediction in (14) fails to satisfy the condition (7), since the empirical quantilesŷ q (x) generally differ from the ground-truth quantiles y q (x). IV. CONFORMAL PREDICTION In this section, we review CP-based set predictors, which have the key property of guaranteeing the (1 − α)-validity condition (7) for any predetermined miscoverage level α, irrespective of the ground-truth distribution p 0 (D, z) of the data. We specifically focus on validation-based CP [10] and cross-validation-based CP [53], which are more practical variants of full CP [10], [69]. In Sec. V, we cover online CP [14], [61]. A. Validation-Based CP (VB-CP) In this subsection, we describe validation-based CP (VB-CP), which partitions the available set D = D tr ∪ D val into a training set D tr with N tr samples and a validation set D val with N val = N − N tr samples ( Fig. 5(a)). This class of methods is also known as inductive CP [10] or split CP [53]. VB-CP operates on any pre-trained probabilistic model p(y|x, D tr ) obtained using the training set D tr as per (3). At test time, given an input x, VB-CP relies on a validation set to determine which labels y ∈ Y should be included in the predicted set. Specifically, for any given test input x, a label y ∈ Y is included in set Γ VB (x|D) depending on the extent to which the candidate pair (x, y ) "conforms" with the examples in the validation set. This "conformity" test for a candidate pair is based on a nonconformity (NC) score. An NC score for VB-CP can be obtained as the log-loss NC(z = (x, y)|D tr ) = − log p(y|x, D tr )(15) or as any other score function that measures the loss of the probabilistic predictor p(y|x, D tr ) on example (x, y). It is also possible to define NC scores for quantile-based predictors as in (14), and we refer to [61] for details. VB-CP consists of a training phase (Fig. 5(a)-(d)) and of a test phase (Fig. 5(e)). During training, the data set D tr is used to obtain a probabilistic predictor p(y|x, D tr ) as in (3) (Fig. 5(b)). Then, NC scores NC(z val [i]|D tr ), as in (15), are evaluated on all points z val [i], i = 1, . . . , N val in the validation set D val (Fig. 5(c)). Finally, the real line of NC scores is partitioned into a "keep" region and a "discard" region ( Fig. 5(d)), choosing as a threshold the split D D tr D val train D tr NC D val NC z val [1]|D tr NC z val [2]|D tr NC z val N val |D tr . . . . . . . . . (a) (b) (c) model model NC (d) (e) 0-quantile (1 − α)-quantile keep discard NC {(x, y )} y ∈Y NC ((x, y )|D(1 − α)-empirical quantile of the N val NC scores {NC(z val [i]|D tr )} N val i=1 . Accordingly, we "keep" the labels y with NC scores that are smaller than the (1 − α)-empirical quantile of the validation NC scores, and "discard" larger NC scores. During testing (Fig. 5(e)), given a test input x, |Y| NC scores are evaluated, one for each of the candidate labels y ∈ Y, using the same trained model p(y|x, D tr ). All candidate labels y for which the NC score NC((x, y )|D tr ) falls within the "keep" region are included in the predicted set of VB-CP. Mathematically, the VB-CP set predictor is obtained as Γ VB (x|D) = y ∈ Y NC((x, y )|D tr )(16)≤ Q α {NC(z val [i]|D tr )} N val i=1 , where the empirical quantile from the top for a set of N real values {r[i]} N i=1 is defined as Q α {r[i]} N i=1 = (1 − α)(N + 1) th smallest value of the set {r[i]} N i=1 ∪ {+∞}.(17) B. Cross-Validation-Based CP (CV-CP) VB-CP has the computational advantage of requiring the training of a single model, but the split into training and validation data causes the available data to be used in an inefficient way. This data inefficiency generally yields set predictors with a large average size (8). Unlike VB-CP, cross-validation-based CP (CV-CP) [53] trains multiple models, each using a subset of the available data set D. As detailed next and summarized in Fig. 6 Specifically, as illustrated in Fig. 6, K-fold CV-CP [53], referred here as K-CV-CP, first partitions the data set Fig. 6(a)), for a predefined integer K ∈ {2, . . . , N } such that the ratio N/K is an integer. D into K disjoint folds {S k } K k=1 , each with N/K points, i.e., ∪ K k=1 S k = D ( During training, the K subsets D \ S k are used to train K probabilistic predictors p(y|x, D \ S k ) defined as in (3) ( Fig. 6(b)). Each trained model p(y|x, D \ S k ) is used to evaluate the |S k | = N/K NC scores NC z k D \ S k for all validation data points z k ∈ S k that were not used for training the model (Fig. 6(c)). Unlike VB-CP, K-CV-CP requires keeping in memory all the N validation scores for testing. These points are illustrated as crosses in Fig. 6(c). During testing, for a given test input x and for any candidate label y ∈ Y, CV-CP evaluates K NC scores, one for each of the K trained models. Each such NC score NC (x, y ) D \ S k is compared with the N/K validation scores obtained on fold S k . We then count how many of the N/K validation scores are larger than NC (x, y ) D \ S k . If the sum of all such counts, across the K folds {S k } K k=1 , is larger than a fraction α of all N data points, then the candidate label y is included in the prediction set ( Fig. 6(d)). This criterion follows the same principle of VB-CP of including all candidate labels y that "conform" well with a sufficiently large fraction of validation points. Mathematically, K-CV-CP is defined as Γ K-CV (x|D) = y ∈ Y K k=1 z k ∈S k 1 NC (x, y ) D \ S k } (18) ≤ NC z k D \ S k ≥ α(N + 1) , where 1(·) is the indicator function (1(true) = 1 and 1(false) = 0). The left-hand side of the inequality in (18) implements the sums, shown in Fig. 6(d), over counts of validation NC scores that are larger than the corresponding NC score for the candidate pair (x, y ). K-CV-CP increases the computational complexity K-fold as compared to VB-CP, while generally reducing the inefficiency [53]. The special case of K = N , known as jackknife+ [53], is referred here as CV-CP. In this case, each of the N folds S k , k = 1, . . . , N uses a single cross validation point. In general, CV-CP is the most efficient form of K-CV-CP, but it may be impractical for large data set sizes due to need to train N models. The number of folds K should strike a balance between computational complexity, as K models are trained, and inefficiency. Specifically, for frequentist learning, the optimization algorithm producing the parameter vector φ * D in (3) must be permutation-invariant. This is the case for standard methods such as full-batch gradient descent (GD), or for non-parametric techniques such as Gaussian processes. For Bayesian learning, the distribution q * (φ|D ) in (3) must also be permutation-invariant, which is true for the exact posterior distribution [23], as well as for approximations obtained via MC methods such as Langevin MC [23], [31]. K-fold split D train D\S1 NC S1 {NC(z1|D\S1)} z1∈S1 (a) (b) (c)NC ((x, y ) |D\Sk) NC ((x, y ) |D\SK) (x, y ) y ∈ Y y = y = . . . x Γ K-CV (x|D) = { } # # # α(N + 1) The requirement on permutation-invariance can be alleviated by allowing for probabilistic training algorithms such as stochastic gradient descent (SGD) [70]. With probabilistic training algorithms, the only requirement is that the distribution of the (random) output models is permutation-invariant. This is, for instance, the case if SGD is implemented by taking mini-batches uniformly at random within the training set D [70]- [72]. With probabilistic training algorithms, however, the validity condition (7) of CV-CP is only guaranteed on average with respect to the random outputs of the algorithms. Specifically, under the discussed assumption of permutation-invariance of the NC scores, by [53, Theorems 1 and 4], CV-CP satisfies the inequality P y ∈ Γ CV (x|D) ≥ 1 − 2α,(19) while K-CV-CP satisfies the inequality P y ∈ Γ K-CV (x|D) ≥1 − 2α − min 2(1−1/K) N/K+1 , 1−K/N K+1 ≥1 − 2α − 2/N .(20) Therefore, validity for both cross-validation schemes is guaranteed for the larger miscoverage level of 2α. Accordingly, one can achieve miscoverage level of α, satisfying (7), by considering the CV-CP set predictor Γ CV (x|D) with α/2 in lieu of α in (18). That said, in the experiments, we will follow the recommendation in [53] and [71] to use α in (18). V. ONLINE CONFORMAL PREDICTION In this section, we turn to online CP. Unlike the CP schemes presented in the previous section, online CP makes no assumptions about the probabilistic model underlying data generation [14], [61]. In the offline version of CP reviewed in the previous section, all N samples of the data set D are assumed to be available upfront (see Fig. 3(a)). In contrast, in online CP, a set predictor Γ i for time index i is produced for each new input x[i] over time i = 1, 2, . . . Specifically, given the past observations {z[j]} i−1 j=1 , the set predictor Γ i x[i] {z[j]} i−1 j=1 outputs a subset of the output space Y. Given a target miscoverage level α ∈ [0, 1], an online set predictor is said to be (1 − α)-long-term valid if the following limit holds lim I→∞ 1 I I i=1 1 y[i] ∈ Γ i x[i] {z[j]} i−1 j=1 = 1 − α(21) for all possible sequences z[i] with i = 1, 2, . . . Note that the condition (21), unlike (7), does not involve any ensemble averaging with respect to the data distribution. We will take (21) as the relevant definition of calibration for online learning. Rolling conformal inference (RCI) [61] adapts in an online fashion a calibration parameter θ[i] across the time index i as a function of the instantaneous error variable err[i] = 1 y[i] / ∈ Γ i (x[i]) ,(22) which equals 1 if the correct output value is not included in the prediction set Γ i (x[i]), and 0 otherwise. This is done using the update rule θ[i + 1] ← θ[i] + γ err[i] − α ,(23) where γ > 0 is a learning rate. Accordingly, the parameter θ is increased by γ(1 − α) if an error occurs at time i, and is decreased by γα otherwise. Intuitively, a large positive parameter θ [i] indicates that the set predictor should be more inclusive in order to meet the validity constraint (21); and vice versa, a large negative value of θ[i] suggests that the set predictor can reduce the size of the prediction sets without affecting the long-term validity constraint (21). Following [61], we elaborate on the use of the calibration parameter θ[i] in order to ensure condition (21) for an online version of the naïve quantile-based predictor (14) for scalar regression. A similar approach applies more broadly (see [14], [73], and [74] Γ RCI i x[i] D[i] (24) = ŷ(x|φ D[i],α/2 ) − ϕ(θ[i]),ŷ(x|φ D[i],1−α/2 ) + ϕ(θ[i]) , where ϕ(θ) = sign(θ) exp |θ| −1 (25) is the so-called stretching function, a fixed monotonically increasing mapping. The set predictor RCI (24) "corrects" the NQB set predictor (14) VI. SYMBOL DEMODULATION In this section, we focus on the application of offline CP, as described in Sec. IV, to the problem of symbol demodulation in the presence of transmitter hardware imperfections. This problem was also considered in [20], [75] by focusing on frequentist and Bayesian learning. Unlike [20], [75], we investigate the use of CP as a means to obtain set predictors satisfying the validity condition (7). A. Problem Formulation The problem of interest consists of the demodulation of symbols from a discrete constellation based on received baseband signals subject to hardware imperfections, noise, and fading. The goal is to design set demodulators that output a subset of all possible constellation points with the guarantee that the subset includes the true transmitted signal with the desired target probability 1 − α. In the context of channel decoding, this type of receiver is referred to as a list decoder [76]. To keep the notation consistent with the previous sections, we write as y[i] the i-th transmitted symbols, and as x[i] the corresponding received signal. Each transmitted symbol y[i] is drawn uniformly at random from a given constellation Y. We model I/Q imbalance at the transmitter and phase fading as in [62]. Accordingly, the ground-truth channel law connecting symbols y[i] into received samples x[i] is described by the equality x[i] = e ψ f IQ (y[i]) + v[i],(26) for a random phase ψ ∼ U[0, 2π), where the additive noise is v[i] ∼ CN (0, SNR −1 ) for signal-to-noise ratio level SNR. Furthermore, the I/Q imbalance function [77] is defined as f IQ (y[i]) =ȳ I [i] + ȳ Q [i],(27) where  ȳ I [i] y Q [i]   =   1 + 0 0 1 −     cos δ − sin δ − sin δ cos δ     y I [i] y Q [i]   ,(28) B. Implementation As in [20], [75], demodulation is implemented via a neural network probabilistic model p(y|x, φ) consisting of a fully connected network with real inputs x[i] of dimension 2 as per (26), followed by three hidden layers with 10, 30, and 30 neurons having ReLU activations in each layer. The last layer implements a softmax classification for the |Y| possible constellation points. We adopt the standard NC score (15), where the trained model φ D for frequentist learning is obtained via I = 120 GD update steps for the minimization of the cross-entropy training loss with learning rate η = 0.2; while for Bayesian learning we implement a gradient-based MC method, namely Langevin MC, with burn-in period of R min = 100, ensemble size R = 20, learning rate η = 0.2, and temperature parameter T = 20. We assume standard Gaussian distribution for the prior distribution [31]. Details on Langevin MC can be found in Appendix A. We compare the naïve set predictor (9), also studied in [20], [75], which provides no formal coverage guarantees, with the CP set prediction methods reviewed in Sec. IV. VB-CP uses equal set sizes for the training and validation sets. We target the miscoverage level as α = 0.1. The coverage level is set to 1 − α = 0.9, and each numerical evaluation is averaged over 50 independent trials (new channel state c) with N te = 100 test points. C. Results We consider the Amplitude-Phase-Shift-Keying (APSK) modulation with |Y| = 8. The SNR level is set to SNR = 5 dB. The amplitude and phase imbalance parameters are independent and distributed as ∼ Beta( /0.15|5, 2) and δ ∼ Beta(δ/15 • |5, 2), respectively [75]. and Fig. 8 shows the empirical inefficiency = 1 N te N te j=1 Γ(x te [j]|D) ,(30) both evaluated on a test set D te = {(x te [j], y te [j])} N te j=1 with N te = 100, as a function of the size of the available data set D. We average the results for 50 independent trials, each corresponding to independent draws of the variables {D, D te } from the ground truth distribution. This way, the metrics (29)- (30) provide an estimate of the coverage (7) and of the inefficiency (8), respectively [53]. From Fig. 7, we first observe that the naïve set predictor, with both frequentist and Bayesian learning, does not meet the desired coverage level in the regime of a small number N of available samples. In contrast, confirming the theoretical calibration guarantees presented in Sec. IV, all CP methods provide coverage guarantees, achieving coverage rates above 1 − α. Furthermore, as seen in Fig. 8, coverage guarantees are achieved by suitably increasing the size of prediction sets, which is reflected by the larger inefficiency. The size of the prediction sets, and hence the inefficiency, decreases as the data set size, N , increases. In this regard, due to their more efficient use of the available data, CV-CP and K-CV-CP predictors have a lower inefficiency as compared to VB predictors, with CV-CP offering the best performance. Finally, Bayesian NC scores are generally seen to yield set predictors with lower inefficiency, confirming the merits of Bayesian learning in terms of calibration. VII. MODULATION CLASSIFICATION In this section, we propose and evaluate the application of offline CP to the problem of modulation classification [45], [46]. A. Problem Formulation Due to the scarcity of frequency bands, electromagnetic spectrum sharing among licensed and unlicensed users is of special interest to improve the efficiency of spectrum utilization. In sensing-based spectrum sharing, a transmitter scans the prospective frequency bands to identify, for each band, if the spectrum is occupied, and, if so, if the signal is from a licensed user or not. A key enabler for this operation is the ability to classify the modulation of the received signal [78]. The modulation classification task is made challenging by the dimensionality of the baseband input signal and by the distortions caused by the propagation channel. Data-driven solutions [79] have shown to be effective for this problem in terms of accuracy, while the focus here is on calibration performance. Accordingly, we aim at designing set modulation classifiers that output a subset of the set of all possible modulation schemes with the property that the true modulation scheme is contained in the subset with a desired probability level 1 − α. To this end, we adopt the data set provided by [80], which has approximately 2.5 × 10 6 baseband signals of 1024 I/Q samples, each produced using one out of 24 possible digital and analog modulations across different SNR values and channel models. We focus only on the high SNR regime (≥ 6 dB). This data set D is made out of approximately 1.28 × 10 6 (x, y) pairs, where x is the channel output signal of dimension 2048 and y is the index of one of the |Y| = 24 possible modulations. The SNR value itself is not available to the classifier. B. Implementation We use a neural network architecture similar to the one used in [80], which has 7 one-dimensional convolutional layers with kernel size 3 and 64 channels for all layers, except for the first layer with has 2 channels. The convolution layers are followed by 3 fully-connected linear layers. A scaled exponential linear unit (SELU) is used for all inner layers, and a softmax is used at the last, fully connected, layer. We assume availability of N = 4800 pairs (x, y) for the data set D, while gauging the empirical inefficiency and coverage level with N te = 1000 held-out pairs. A total number of I = 4000 GD steps with fixed learning rate of 0.02 are carried out, and the target miscoverage rate is set to α = 0.1. VB partitions its available data into equal sets for training and validation. C. Results In this problem, due to computational cost, we exclude CV-CP and we focus on K-CV-CP with a moderate number of folds, namely K = 6 and K = 12. In Fig. 9, box plots show the quartiles of the empirical coverage (29) and of the empirical inefficiency (30) As also noted in the previous section, VB-CP suffers from larger predicted set size as compared to K-CV-CP, due to poor sample efficiency. A small number of folds, as low as K = 6, is sufficient for K-CV-CP to outperform VB-CP. This improvement in efficiency comes at the computational cost of training six models, as compared to the single model trained by VB-CP. VIII. ONLINE CHANNEL PREDICTION In this section, we investigate the use of online CP, as described in Sec. V, for the problem of channel prediction. We specifically focus on the prediction of the received signal strength (RSS), which is a key primitive at the physical layer, supporting important functionalities such as resource allocation [81], [82]. A. Problem Formulation Consider a receiver that has access to a sequence of RSS samples from a given device. We aim at designing a predictor that, given a sequence of past samples from the RSS sequence, produces an interval of values for the next RSS sample. To meet calibration requirements, the interval must contain the correct future RSS value with the desired rate level 1 − α. Unlike the previous applications, here the rate of coverage is evaluated based on the time average 1 t t i=1 1 y[i] ∈ Γ i x[i] {z[j]} i−1 j=1 .(31) B. Implementation We build the CP set predictor by leveraging the probabilistic neural network used in [61] as the model class for the quantile predictors in (13)- (14). Each quantile predictor consists of a multi-layer neural network that pre-processes the most recent K pairs {z[i − K], . . . , z[i − 1]}; of a stacked long short-term memory (LSTM) [83] with two layers; and of a post-processing neural network, which maps the last LSTM hidden vector into a scalar that estimates the quantile used in (14). For details of the implementation, we refer to Appendix C. Fig. 11 report the time-average coverage = 1 C. Results Fig. 10 and I I i=1 1 y[i] ∈ Γ i x[i] {z[j]} i−1 j=1(32) and the time-average inefficiency = 1 for online CP (24), compared to a baseline of the naïve quantile-based predictor (14), as a function of the time window size I for data sets [63] and [63], respectively. We have discarded 1000 samples for a warm-up period for both metrics (32) and (33). I I i=1 Γ i x[i] {z[j]} i−1 j=1(33) In both cases, the naïve predictor is seen to fail to satisfy the coverage condition (21) min φ L D (φ)=− 1 N (x,y)∈D log p(y|x, φ) (34) =E p D (x,y) − log p(y|x, φ) , with empirical distribution p D (x, y) defined by the data set D. Bayesian learning addresses epistemic uncertainty by treating the model parameter vector as a random vector φ with prior distribution φ ∼ p(φ). Ideally, Bayesian learning updates the prior p(φ) to produce the posterior distribution p(φ|D) as p(φ|D) ∝ p(φ) N i=1 p y[i] x[i], φ(35) and obtains the ensemble predictor for the test point (x, y) by averaging over multiple models, i.e., p(y|x, D) = E p(φ|D) [p(y|x, φ)].(36) In practice, as the true posterior distribution is generally intractable due to the normalizing factor in (35), approximate Bayesian approaches are considered via VI or MC techniques (see, e.g., [23]). In the experiments, we adopted Langevin MC to approximate the Bayesian posterior [23], [31]. Langevin MC adds Gaussian noise to each standard GD update for frequentist learning (see, e.g., [23,Sec. 4.10]). The noise has power 2η/T , where η is the GD learning rate and T > 0 is a temperature parameter. Langevin MC produces R model parameters {φ[r]} R r=1 across R consecutive iterations. We specifically retain only the last R samples, discarding an initial burn-in period of R min iterations. The temperature parameter T is typically chosen to be larger than 1 [86], [87]. With the R samples, the expectation term in (36) is approximated as the empirical average 1 R R r=1 p(y|x, φ[r]). We observe that Langevin MC is a probabilistic training algorithm, and that it satisfies the permutation-invariance property in terms of the distribution of the random output models discussed in Sec. IV-C. APPENDIX B ALGORITHMIC DETAILS FOR ROLLING CONFORMAL INFERENCE The RCI algorithm is reproduced from [61] in Algorithm 1. The architecture of the set predictor is inspired by [61], and made out of three artificial neural networks. The first, [c 1 k [i], h 1 k [i]] = f LSTM w k [i], c 1 k−1 [i], h 1 k−1 [i] φ 1 LSTM [i](37)[c 2 k [i], h 2 k [i]] = f LSTM h 1 k [i], c 2 k−1 [i], h 2 k−1 [i] φ 2 LSTM [i] .(38) The The miscoverage rate was set to α = 0.1, the learning rate to η = 0.01, and we chose γ = 0.03 for the calibration parameter θ in (23). Fig. 1. (a) Examples of probabilistic predictors for two inputs x 1 and x 2 : As compared to the ground-truth distribution in the second column, the Fig. 2 . 2Set predictors produce subsets of the range of the output variable (here { } ) for each input. Calibration is measured with respect to a desired coverage level 1 − α: A set predictor is well calibrated if the true label is included in the prediction set with probability at least 1 − α. Mathematically, exchangeability requires that the joint distribution p 0 (D, z) does not depend on the ordering of the N + 1 variables {z[1], . . . , z[N ], z}. Equivalently, by de Finetti's theorem [65], there exists a latent random vector c with distribution p 0 (c) such that, conditioned on c, the variables {z[1], . . . , z[N ], z} are i.i.d. Writing the conditional i.i.d. distribution as Fig. 3 . 3(a) The validity condition (7) assumed in offline CP is relevant if one is interested in the average performance with respect to realizations (D, z) ∼ p 0 (D, z) of training set D and test variable z = (x, y). Input variable x is not explicitly shown in the figure, and the horizontal axis runs over the training examples in D and the test example z. (b) In online CP, the set predictor Γ i uses its input x[i] and all previously observed pairs z[1], . . . , z[i − 1] with z[i] = (x[i], y[i]) to produce a prediction set. The long-term validity (21) assumed by online CP is defined as the empirical time-average rate at which the predictor Γ i includes the true target variable y[i]. Fig. 5 . 5Validation-based conformal prediction (VB-CP): (a) The data set is split into training and validation set; (b) A single model is trained over the training data set; (c)-(d) Post-hoc calibration is done by evaluating the NC scores on the validation set (c) and by identifying the(1 − α)-quantile of the validation NC scores. This divides the axis of NC scores into a "keep" region of NC scores smaller than the threshold, and into a complementary "discard" region (d). (e) For each test input x, VB-CP includes in the prediction set all labels y ∈ Y for which the NC score of the pair (x, y ) is within the "keep" region. , during the training phase, each data point z[i] in the validation set is assigned an NC score based on a model trained using a subset of the data set D that excludes z[i], with i ∈ {1, ..., N }. Then, for testing, the inclusion of a label y in the prediction set for an input x is based on a comparison of NC scores evaluated for the pair (x, y ) with all the N validation NC scores. Fig. 6 . 6K-fold cross-validation-based conformal prediction (K-CV-CP): (a) The N data pairs of data set D are split into K-folds each with |S k | = N/K samples; (b) K models are trained, each using a leave-fold-out data set of |D \ S k | = N − N/K pairs; (c) NC scores are computed on the N/K holdout data points for each fold S k ; (d) For each test input x, all labels y ∈ Y for which the number of "higher-NC" validation points exceeds a fraction α of the total N points are considered in the prediction set. CV-CP is the special case with K = N .C. Calibration GuaranteesVB-CP (16) satisfies the coverage condition (7)[10] under the only assumption of exchangeability (see Sec. II-A).The validity of CV-CP requires a technical assumption on the NC score. While in VB-CP the NC score is an arbitrary score function evaluated based on any pre-trained probabilistic model, for CV-CP, the NC score NC(z|D ) must satisfy the additional property of being invariant to permutations of the data set D used to train the underlying probabilistic model. Consider the log-loss NC(z = (x, y)|D ) = − log p(y|x, D ) (15), or any other score function based on the trained model p(y|x, D ), as the NC score. CV-CP requires that the training algorithm used to produce model p(y|x, D ) provides outputs that are invariant to permutations of the training set D . Rather, it models the observations as a deterministic stream of input-output pairs z[i] = (x[i], y[i]) over time index i = 1, 2, . . . ; and it targets a coverage condition defined in terms of the empirical rate at which the prediction set Γ i at time i covers the correct output y[i]. , via the additive stretching function ϕ(θ[i]) based on the calibration parameter θ[i]. As the time index i rolls, the calibration parameter θ[i] adaptively inflates and deflates according to (23). Upon each observation of new label y[i], the quantile predictor model parameters φ D[i],α/2 and φ D[i],1−α/2 can also be updated, without affecting the long-term validity condition (21) [61, Theorem 1]. We refer to Appendix B for further details on online CP. with y I [i] and y Q [i] being the real and imaginary parts of the modulated symbol y[i]; andȳ I [i] andȳ Q [i] standing for the real and imaginary parts of the transmitted symbol f IQ (y[i]). In (28), the channel state c consists of the tuple c = (ψ, , δ) encompassing the complex phase ψ and the I/Q imbalance parameters ( , δ). Fig. 7 . 7Coverage for naïve set predictor (9), VB-CP (16), CV-CP, and K-CV-CP (18) with K = 4, for symbol demodulation problem (Section VI). For every set predictors, the NC scores are evaluated either using frequentist learning (dashed lines) or Bayesian learning (solid lines). Fig. 8 . 8Average set prediction size (inefficiency) for the same setting ofFig. 7. Fig. 7 7te [j] ∈ Γ(x te [j]|D) , Fig. 9 . 9Coverage and inefficiency for NPB (9), VB-CP (16), and K-CV-CP (18) with K = 6 and K = 12, for the modulation classification problem (implementation details in Section VII-B). The boxes represent the 25% (lower edge), 50% (solid line within the box), and 75% (upper edge) percentiles of the empirical performance metrics evaluated over 32 different experiments, with average value shown by the dashed line. from 32 independent runs, with different realizations of data set and test examples. The lower edge of the box represents the 0.25-quantile; the solid line within the box the median; the dashed line within the box the average; and the upper edge of the box the 0.75-quantile. As can be seen in the figure, the naïve set predictor is invalid (see average shown as dashed line), and it exhibits a wide spread of the coverage rates across the trials. On the other hand, all CP set predictors are valid, meeting the predetermined coverage level 1 − α = 0.9, and have less spread-out coverage rates. Fig. 10 . 10CP for time-series Outdoor of [64]: (top) Time-average coverage (32) of naïve set prediction and online CP; (bottom) Time-averaged inefficiency (33) of naïve set prediction and online CP. Fig. 11 . 11CP for time-series NLOS_Head_Indoor_1khz in [63]: (top) Time-average coverage (32) of naïve set prediction and online CP;(bottom) Time-average inefficiency (33) of naïve set prediction and online CP. f pre (·), is a multi-layer perceptron (MLP) network with hidden layers of 16, 32 neurons each, parametrized by vector φ pre [i]. It is meant to apply a pre-process over the most recent observed K = 20 pairs {z[i − K], . . . , z[i − 1]} to be transformed element-wise into a length-K vector w[i] = w 1 [i], . . . , w K [i] , in which the k-th element (k = 1, . . . , K) is w k [i] = f pre z[i − K + k − 1] φ pre [i] .Effectively, this will serve as a temporal sliding K-lengthwindow, with a time-evolving pre-processing function. The second neural network, f LSTM (·) has two layers with model parameter vectors φ 1 LSTM [i] (first layer) and φ 2 LSTM [i] (second layer), which retains a memory via the hidden state vectors h and c, initialized at every time index i as c 1 0 [i] = c 2 0 [i] = h 1 0 [i] = h 2 0 [i] = 0. By accessing the previous K pairs via the vector w[i], this recurrent neural network extracts temporal patterns by sequentially transferring information via LSTM cells (with shared parameter vectors) in the image of hidden and cell state vectors c k [i], h k [i] via the LSTM cells. These vectors flow along the LSTM by concatenating k = 1, . . . , K cells, and forming vectors of length 32 each third and last network is a post-processing MLP f post (·) with one hidden layer of 32 neurons, and with parameter vector φ post [i], which maps the last LSTM hidden 64-length vector h K[i] = [h 1 K [i], h 2 K [i]] into a scalar f post x[i], h K [i] φ post [i] ∈ Rthat estimates the quantile for the output y[i]. Accordingly, the time evolving model parameter is the tupleφ[i] = (φ pre [i], φ 1 LSTM [i], φ 2 LSTM [i], φ post [i]).(39)This model is instantiated twice for the regression problem: one for the α/2 lower quantile and the other for 1 − α/2 upper quantile. For every time instant i, after the new output y[i] is observed, continual learning of the models is taken place by training the models with corresponding pinball losses (11) using the new pair (x[i], y[i]), while initializing the models as the previous models at time instant i − 1. ). Denote the data set D[i] = {z[j]} i−1 j=1 as having all previously observed labeled data set up till time i − 1. The key idea behind RCI is to extend the naïve prediction interval (14) depending on the calibration parameter θ[i] as channel ID, which determines the carrier frequency used at time i out of the 16 possible bands, as the input x[i]. At time i, we observe a sequence of RSS samples z[1], . . . , z[i − 1] with z[i] = (x[i], y[i]), and the goal is to predict the next RSS sample y[i] via the online set predictor Γ RCIThe second data set[63] reports samples y[i], measured in dBm, on a 5.8 GHz device-to-device link without additional input. Hence, in this case, we predict the next RSS sample y[i] using the previous RSS samples y[1], . . . , y[i − 1]. Note that the prior works[63],[64] adopted standard probabilistic predictors, while here we focus on set predictors that produce a prediction interval Γ RCIThis is computed as the fraction previous time instants i ∈ {1, ..., t} at which the set predictor Γ i includes the true RSS value y[i]. We consider two data sets of RSS sequences. The first data set records RSS samples y[i] in logarithmic scale for an IEEE 802.15.4 radio over time index i [64]. We further use the available side information on the time-variant i (24). i x[i] {z[j]} i−1 j=1 . for both data sets, while online CP converges to the target level 1 − α = 0.9. This result is obtained by online CP with a modest increase of around 8% for both data sets in terms of inefficiency.IX. CONCLUSIONSAI in communication engineering should not only target accuracy, but also calibration, ensuring a reliable and safe adoption of machine learning within the overall telecommunication ecosystem. In this paper, we have proposed the adoption of a general framework, known as conformal prediction (CP), to transform any existing AI model into a well-calibrated model via post-hoc calibration for communication engineering. Depending on the situation of interest, post-hoc calibration leverages either an held-out (cross) validation set or previous samples. Unlike calibration approaches that do not formally guarantee reliability, such as Bayesian learning or temperature scaling, CP provides formal guarantees of calibration, defined either in terms of ensemble averages or long-term time averages. Calibration is retained irrespective of the accuracy of the trained models, with more accurate models producing smaller set predictions.To validate the reliability of CP-based set predictors, we have provided extensive comparisons with conventional methods based on Bayesian or frequentist learning. Focusing on demodulation, modulation classification, and channel prediction, we have demonstrated that AI models calibrated by CP provide formal guarantees of reliability, which are practically essential to ensure calibration in the regime of limited data availability.Future work may consider applications of CP to other use cases in wireless communication systems, as well as extension of involving training-based calibration[55],[84] and/or meta-learning[85].FREQUENTIST AND BAYESIAN LEARNING Given access to training data set D = z[i] = (x[i], y[i]) N i=1 with N examples, frequentist learning finds a model parameter vector φ * D by tackling the following empirical risk minimization (ERM) problemAPPENDIX A Algorithm 1 : 1Rolling Conformal Inference (for Regression)[61] Inputs : α = long-term target miscoverage levelθ[1] = initial calibration parameter φ lo [1], φ hi [1] = initial modelsParameters : I = number of online iterations γ = learning rate for calibration parameter η = learning rate for model updatesOutput : {Γ RCI i x[i] {z[j]} i−1 j=1 } I i=1 = predicted sets for {x[i]} I i=1 1 for i = 1, . . . , I time instants do Retrieve a new data sample (x[i], y[i])Set prediction of new inputCalculate set Γ RCI i x[i] {z[j]} i−1 j=1 using 5 ŷ x[i] φ lo [i] − ϕ(θ[i]),ŷ x[i] φ hi [i] + ϕ(θ[i])Check if prediction is unsuccessfulerr[i] ← 1 y[i] / θ[i + 1] ← θ[i] + γerr[i] − α Update models using new sample 11 φ lo [i + 1] ← φ lo [i] − η∇ φ α/2 y[i],ŷ x[i] φ lo [i] 12 φ hi [i + 1] ← φ hi [i] − η∇ φ 1−α/2 y[i],ŷ x[i] φ hi [i] 13 return predicted sets {Γ RCI i x[i] {z[j]} i−1 j=1 } I IMPLEMENTATION OF ONLINE CHANNEL PREDICTION2 3 4 6 7 ∈ Γ RCI i x[i] {z[j]} i−1 j=1 8 Update calibration parameter 9 10 i=1 APPENDIX C https://github.com/kclip/cp4wireless Advances in cognitive Radio Networks: A survey. 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{'abstract': "When used in complex engineered systems, such as communication networks, artificial intelligence (AI) models should be not only as accurate as possible, but also well calibrated. A well-calibrated AI model is one that can reliably quantify the uncertainty of its decisions, assigning high confidence levels to decisions that are likely to be correct and low confidence levels to decisions that are likely to be erroneous. This paper investigates the application of conformal prediction as a general framework to obtain AI models that produce decisions with formal calibration guarantees. Conformal prediction transforms probabilistic predictors into set predictors that are guaranteed to contain the correct answer with a probability chosen by the designer. Such formal calibration guarantees hold irrespective of the true, unknown, distribution underlying the generation of the variables of interest, and can be defined in terms of ensemble or time-averaged probabilities. In this paper, conformal prediction is applied for the first time to the design of AI for communication systems in conjunction to both frequentist and Bayesian learning, focusing on demodulation, modulation classification, and channel prediction.A. MotivationHow reliable is your artificial intelligence (AI)-based model? The most common metric to design an AI model and to gauge its performance is the average accuracy. However, in applications in which AI decisions are used within a larger system, AI models should not only be as accurate as possible, but they should also be able to reliably quantify the uncertainty of their decisions. As an example, consider an unlicensed link that uses AI tools to predict the best channel to access out of four possible channels. A predictor that assigns the probability vector of [90%, 2%, 5%, 3%] to the possible channels predicts the same best channel -the first -as a predictor that outputs the probability vector [30%, 20%, 25%, 25%]. However, the latter predictor is less certain of its decision, and it may be preferable for the unlicensed link to refrain from accessing the channel when acting on less confident predictions, e.g., to avoid excessive interference to licensed links [1], [2].As in the example above, AI models typically report a confidence measure associated with each prediction, which reflects the model's self-evaluation of the accuracy of a decision. Notably, neural network models implement probabilistic predictors that produce a probability distribution across all possible values of the output variable.The self-reported model confidence, however, may not be a reliable measure of the true, unknown, accuracy of a prediction. In such situations, the AI model is said to be poorly calibrated.As illustrated in the example inFig. 1, accuracy and calibration are distinct criteria, with neither criterion implying the other. It is, for instance, possible to have an accurate predictor that consistently underestimates the accuracy of its decisions, and/or that is overconfident where making incorrect decisions (see fourth column inFig. 1). Conversely, one can have inaccurate predictions that estimate correctly their uncertainty (see fifth column inFig. 1).Deep learning models tend to produce either overconfident decisions [3], or calibration levels that rely on strong assumptions about the ground-truth, unknown, data generation mechanism [4]-[9]. This paper investigates the use of conformal prediction (CP) [10]-[12] as a framework to design provably well-calibrated AI predictors, with distribution-free calibration guarantees that do not require making any assumption about the ground-truth data generation mechanism.B. Conformal Prediction for AI-Based Wireless SystemsCP leverages probabilistic predictors to construct well-calibrated set predictors. Instead of producing a probability vector, as in the examples inFig. 1, a set predictor outputs a subset of the output space, as exemplified inFig. 2. A set predictor is well calibrated if it contains the correct output with a pre-defined coverage probability selected by the system designer. For a well-calibrated set predictor, the size of the prediction set for a given input provides a measure of the uncertainty of the decision. Set predictors with smaller average prediction size are said to be more efficient [10]. This paper investigates CP as a general mechanism to obtain AI models with formal calibration guarantees for communication systems. The calibration guarantees of CP hold irrespective of the true, unknown, distribution underlying the generation of the variables of interest, and are defined either in terms of ensemble averages[10]or", 'arxivid': '2212.07775', 'author': ['Student Member, IEEEKfir M Cohen kfir.cohen@kcl.ac.uk \nViterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel\n', 'Member, IEEESangwoo Park sangwoo.park@kcl.ac.uk \nViterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel\n', 'Fellow, IEEEOsvaldo Simeone osvaldo.simeone@kcl.ac.uk. \nViterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel\n', 'Shlomo Shamai \nViterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel\n', 'Life Fellow, IEEEShitz \nViterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel\n', 'Shitz)Shlomo Shamai \nViterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel\n'], 'authoraffiliation': ['Viterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel', 'Viterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel', 'Viterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel', 'Viterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel', 'Viterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel', 'Viterbi Faculty of Electrical and Computing Engineering\nTechnion-Israel Institute of Technology\nWC2R 2LS, 3200003London, HaifaU.K., Israel'], 'corpusid': 254685775, 'doi': '10.48550/arxiv.2212.07775', 'github_urls': ['https://github.com/kclip/cp4wireless'], 'n_tokens_mistral': 28619, 'n_tokens_neox': 24679, 'n_words': 14567, 'pdfsha': 'e1b9537f31657163227d033d1f602e2b61dde429', 'pdfurls': ['https://export.arxiv.org/pdf/2212.07775v1.pdf'], 'title': ['Calibrating AI Models for Wireless Communications via Conformal Prediction', 'Calibrating AI Models for Wireless Communications via Conformal Prediction'], 'venue': []}

arxiv

Bayesian Forecasting in the 21st Century: A Modern Review 7 Dec 2022 December 8, 2022 Gael M Martin David T Frazier Worapree Maneesoonthorn Rubén Loaiza-Maya Florian Huber Gary Koop Didier NibberingJohn Maheu Anastasios Panagiotelis Gael M Martin Monash University Australia Monash University Australia; Worapreee (Ole) Maneesoonthorn University of Melbourne Australia; Rubén Loaiza-Maya Monash University Australia; Florian Huber Gary Koop University of Salzburg Austria University of Strathclyde UK John Maheu McMaster University Canada; Anastasios Panagiotelis Didier Nibbering University of Sydney Australia Monash University Australia Bayesian Forecasting in the 21st Century: A Modern Review 7 Dec 2022 December 8, 2022Bayesian predictionmacroeconomicsfinancemarketingelectricity demandBayesian computational methodsloss-based Bayesian prediction The Bayesian statistical paradigm provides a principled and coherent approach to probabilistic forecasting. Uncertainty about all unknowns that characterize any forecasting problem -model, parameters, latent states -is factored into the forecast distribution, with forecasts conditioned only on what is known or observed. Allied with the elegance of the method, Bayesian forecasting is now underpinned by the burgeoning field of Bayesian computation, which enables Bayesian forecasts to be produced for virtually any problem, no matter how large, or complex. The current state of play in Bayesian forecasting is the subject of this review. The aim is to provide readers with an overview of modern approaches to the field, set in some historical context. Whilst our primary focus is on applications in the fields of economics and finance, and their allied disciplines, sufficient general details about implementation are provided to aid and inform all investigators. Introduction Why Bayesian forecasting? The Bayesian statistical paradigm uses the rules and language of probability to quantify uncertainty about all unknown aspects of phenomena that generate observed data. This core characteristic of the paradigm makes it particularly suitable for forecasting, with uncertainty about the unknown values of future observations automatically expressed in terms of a probability distribution. Moreover, Bayesian methods allow a user to seamlessly, and systematically, yield probabilistic forecasts that reflect uncertainty about all unknowns, and that condition only on known past events, or 'data'. To paraphrase Geweke and Whiteman (2006), Bayesian methods prevail in forecasting because they explicitly employ the principle of relevant conditioning: the use of forecasts that condition only on observed data, with all uncertainty about the model, parameters, latent variables, etc., correctly accounted for in the production of said forecasts. This stands in stark contrast to forecasting approaches based on frequentist methods, which typically condition on a given model (or set of models), and on point estimates of model unknowns, plus adopt ad hoc methods (if any) to quantify uncertainty regarding unknown quantities. Indeed, the ability of Bayesian forecasters to incorporate the uncertainty associated with the production of forecasts, while utilizing all available information -both a priori and sample information -in a principled manner, led Granger et al. (1986) to conclude that: "In terms of forecasting accuracy a good Bayesian will beat a non-Bayesian, who will do better than a bad Bayesian." Echoing these sentiments, in our opinion, the power of the Bayesian forecasting paradigm is a product of the paradigm's requirement that all elements of the statistical problem necessary to produce forecastsfuture observations, past observations, parameters, latent variables, models -are treated as arguments of a joint probability distribution. The express probabilistic formulation of these elements, in turn, allows a Bayesian to invoke the standard rules of probability to produce a distribution for an unknown future value that is conditioned on the known past data, and is marginal of all other arguments that are inherently unknown. While this ability to marginalize all unknowns through probability calculus is the hallmark of the Bayesian approach, the benefits of the paradigm, and what ultimately in our opinion defines a 'good Bayesian', is the attention to detail necessary to successfully implement Bayesian methods. In Bayesian forecasting, before we ever attempt to produce a forecast, we must first carefully enumerate all possible sources of uncertainty; and construct reasonable prior beliefs for these quantities, which often include (possibly several layers of) latent variables that have a specific and delicate interaction with the observed data; always taking great care to ensure that these prior beliefs do not conflict with the observed data. Then and only then can we 'turn the Bayesian crank' to produce the joint posterior distribution over all unknown quantities (including future values), and ultimately integrate out the quantities we are not interested in to obtain the (posterior) predictive distribution for the future values of our random variables of interest. The attention to detail necessary to produce Bayesian forecasts means that very few (if any) implicit assumptions are maintained, and what explicit assumptions are maintained can often be rationalized/tested against the data. Consistent with the internal coherence of the Bayesian statistical paradigm, the basic manner in which all Bayesian forecasting problems are framed is the same. What differs however, from case to case, is the way in which the problem is solved -i.e. the way in which the forecast distribution is accessed. To understand why this is so, it is sufficient to recognize that virtually all Bayesian quantities of interest, including forecast distributions, can be expressed as expectations of some sort. For most models that are used to predict empirically relevant data these expectations are not available in closed form. Hence, in any practical problem, implementation of Bayesian forecasting is both model-and data-dependent, and relies on advanced computational tools. Different forecasting problems -defined by different forms and 'sizes' of models and data sets -require, in turn, different approaches to computation. The evolution of the practice of Bayesian forecasting has, as a consequence, gone hand-in-hand with developments on the computational front; with increasingly large and complex models rendered amenable to a Bayesian forecasting approach via access to modern techniques of computation. The purview of this review In this review, we give a modern take on the current landscape of Bayesian forecasting. Whilst excellent textbook treatments of Bayesian forecasting are given in Geweke (2005) and West and Harrison (2006), and with Geweke and Whiteman (2006) reviewing specific aspects of Bayesian forecasting in a slew of practical settings, the field has advanced by leaps and bounds in the last twenty years. Therefore, we believe the time is ripe to consider a review of the subject that touches on many of the novel and exciting areas now being explored. The methodological advances we review have general applicability to all discipline areas; nevertheless, due to our own interests, expertise and experience -and to keep the scope of the paper manageable -we have chosen to focus primarily on applications in the economic sciences. Whilst the paper is not designed to be a treatise on Bayesian computation, sufficient details are provided to enable the practitioner to understand why numerical tools are needed in most forecasting settings, and how they are used. The general structure of the paper is as follows. In Section 2 we provide a short tutorial on Bayesian forecasting. This begins with an outline of the Bayesian forecasting method, followed by an overview of the computational techniques used to implement the method. In Section 3 we then take the reader on a potted chronological tour of Bayesian forecasting, up to the present day. We begin by giving a snapshot of the forecasting problems tackled during the last few decades of the 20th century, and the computational solutions that were adopted then -most notably, Markov chain Monte Carlo (MCMC) algorithms. We then look at the types of 'intractable' forecasting problems that are increasingly encountered in the 21st century, and provide an overview of the new computational solutions that have been proposed to tackle such problems. We also outline very recent developments in which misspecification of the forecasting model is explicitly acknowledged, and conventional likelihood-based Bayesian forecasting eschewed as a consequence; with problem-specific measures of forecast accuracy (or forecast loss) used, instead, to drive the production of forecast distributions. Section 4 then provides the reader with more detailed reviews of modern Bayesian forecasting in the following four broad fields: macroeconomics, finance, marketing, and electricity pricing and demand. Section 5 closes the paper with a brief summary of the current state of play. Before proceeding further, we make a note about scope and language. To render the scope of the paper manageable we focus primarily on Bayesian forecasting in 'time series models' -i.e. models for random variables that are indexed by time -and on using such models to say something about the values that these random variables will assume in the future. These future values may be informed only by past observations on the variable, or may also depend on the known values of covariates, or regressors. We also follow the convention in the Bayesian literature by using the terms 'forecast' and 'prediction' (and all of their various grammatical derivations) synonymously and interchangeably in this case, for the sake of linguistic variety. The fundamental principles of Bayesian prediction apply equally to data indexed by something other than time. The term 'forecast' is not used in this case as it is a term reserved for temporal settings. The main exceptions to our focus on time series models, and forecasting per se, occur in Section 4.3, in which models for cross-sectional data are used to predict customer choice in marketing settings, and Section 4.4, in which models for electricity demand that have a spatial dimension are referenced. A Tutorial on Bayesian Forecasting The Bayesian forecasting method For the sake of illustration, we assume a scalar random variable y t , and define the (T × 1) vector of observations on y t as y 1:T = (y 1 , y 2 , ..., y T ) ′ . We assume (for the moment) that y 1:T has been generated from some parametric model with likelihood p(y 1:T |θ), with θ = (θ 1 , θ 2 , ..., θ p ) ′ ∈ Θ ⊆ R p a p-dimensional vector of unknown parameters, and where we possess prior beliefs on θ specified by p(θ). Using the same symbol y 1:T to denote both the vector of observed data and the T -dimensional vector random variable, we define the joint distribution over y 1:T and θ as p(y 1:T , θ). Application of the standard rules of probability to p(y 1:T , θ) yields Bayes theorem (or Bayes rule), p(θ|y 1:T ) = p(y 1:T |θ)p(θ) p(y 1:T ) , where p(y 1:T ) = Θ p(y 1:T |θ)p(θ)dθ. Bayes theorem provides a representation for the posterior probability density function (pdf) for θ, p(θ|y 1:T ), as proportional to the product of the likelihood function and the prior. The term p(y 1:T ) defines the marginal likelihood, and the scale factor [p(y 1:T )] −1 in (1) ensures that p(θ|y 1:T ) integrates to one. Now, define y T +1 as the (one-step-ahead) future random variable, where we focus on one-step-ahead forecasting in Sections 2 and 3 merely to simplify the exposition. Assuming y T +1 to be a continuous random variable (again, for illustration), standard probability manipulations lead to the following expression for the forecast (or predictive) pdf for y T +1 : p(y T +1 |y 1:T ) = Θ p(y T +1 |θ, y 1:T )p(θ|y 1:T )dθ. When no confusion arises, we also refer to p(y T +1 |y 1:T ), albeit loosely, as the forecast (or predictive) distribution, or simply as the 'predictive'. 1 The density p(y T +1 |y 1:T ) summarizes all uncertainty about y T +1 , conditional on the assumed model -which underpins the structure of both the conditional predictive, p(y T +1 |θ, y 1:T ), and the posterior itself -and the prior beliefs that inform p(θ|y 1:T ). Point and interval predictions of y T +1 , and indeed any other distributional summary, can be extracted from (2). In the case where the model itself is uncertain, and a finite set of parametric models, M 1 , M 2 ,...,M K , is assumed to span the model space, a 'model-averaged' predictive, p M A (y T +1 |y 1:T ), is produced as p M A (y T +1 |y 1:T ) = K k=1 p(y T +1 |y 1:T , M k )p(M k |y 1:T ),(3) where p(y T +1 |y 1:T , M k ) denotes the density in (2), but now conditioned explicitly on the kth model in the set. The kth posterior model probability, p(M k |y 1:T ), k = 1, 2, ..., K, is computed via a further application of Bayes theorem in which the (initial) joint distribution of interest is defined over both the model space and the space for the parameters of each of the K models. Standard manipulations lead to p(M k |y 1:T ) ∝ p(y 1:T |M k )p(M k ),(4) where p(y 1: T |M k ) = Θ k p(y 1:T |θ k , M k )p(θ k |M k )dθ k ,(5) for each k = 1, 2, ..., K, with θ k denoting the parameter set for the kth model. As is clear, analytical evaluation of p(y T +1 |y 1:T ) in (2) requires, at the very least, a closed-form expression for p(θ|y 1:T ). Typically, however, such an expression is not available, with most posteriors being known only up to a constant of proportionality, as p(θ|y 1:T ) ∝ p(y 1:T |θ)p(θ). The main exceptions to this occur when p(y 1:T |θ) is from the exponential family, and either a natural conjugate, or convenient noninformative prior is adopted; specifications which may be suitable for some simple (and low-dimensional) empirical problems, but are certainly not broadly applicable in practice. Analytical evaluation of p M A (y T +1 |y 1:T ) in (3) also requires a closed-form expression for each p(y 1:T |M k ) (with normalization of p(M k |y 1:T ) then straightforward); once again a rare thing beyond the exponential family (and standard prior) setting. Hence the need for numerical computation to implement Bayesian forecasting in virtually all realistic empirical problems. 2 An overview of computation The form of (2) makes it clear that the Bayesian predictive pdf, p(y T +1 |y 1:T ), is nothing more than the posterior expectation of the predictive conditional on θ. Hence, accessing p(y T +1 |y 1:T ) amounts to the 1 We note that p(yT +1|y1:T ) is sometimes referred to as a 'posterior' predictive in the literature, given that it is produced by averaging the conditional predictive, p(yT +1|θ, y 1:T ), with respect to the posterior density, p(θ|y 1:T ). We do not adopt this expression, leaving it to the context to make it clear as to whether the term 'predictive' is being used to refer to the distribution that is marginal of θ, p(yT +1|y1:T ), or that which is conditioned on θ, p(yT +1|θ, y 1:T ). We also streamline the exposition by not using explicit notation for any observed covariates on which the model for yt may depend, and on which the predictive for yT +1 would condition, unless this is essential. 2 Numerous textbook illustrations of the material in this section can be found. In addition to the references Geweke (2005) and West and Harrison (2006) cited earlier, some examples are Zellner (1971), Koop (2003) and Robert (2007). evaluation of an expectation. This insight is helpful, as it enables us to see many of the computational methods that are used to access p(y T +1 |y 1:T ) -in cases where it is not available in closed form -simply as different ways of numerically estimating an expectation. It is convenient to group Bayesian computational methods into three categories: 1) Deterministic integration (or quadrature) methods (Davis and Rabinowitz, 1975;Naylor and Smith, 1982); 2) Exact simulation methods; and 3) Approximate methods. Given that the production of p(y T +1 |y 1:T ) involves integration over θ, only in very low-dimensional models is 1) a feasible computational method on its own, due to the well-known 'curse of dimensionality' that characterizes numerical quadrature. Hence, the computational methods in 2) and 3) are those most commonly adopted, and will be our focus here; noting that quadrature may play still a limited role within these alternative computational frameworks. The methods in 2) use simulation to produce M draws of θ, θ (i) , i = 1, 2, ..., M , from the posterior p(θ|y 1:T ), which, in turn, define M conditional predictives, p(y T +1 |θ (i) , y 1:T ), i = 1, 2, ..., M , the mean of which is used to estimate (2). Alternatively, if it is easier to simulate from p(y T +1 |θ (i) , y 1:T ) than to evaluate it at any point in the support of y T +1 , M draws of y T +1 , y (i) T +1 , i = 1, 2, ..., M , are taken, one for each draw θ (i) , and kernel density estimation methods used to produce an estimate of p(y T +1 |y 1:T ). Different simulation methods are distinguished by the way in which the posterior draws are produced. Methods in 2) include Monte Carlo sampling (Metropolis and Ulam, 1949), importance sampling (IS) (Hammersley and Handscomb, 1964;Kloek and van Dijk, 1978;Geweke, 1989) and MCMC samplingincluding Gibbs sampling (Geman and Geman, 1984;Gelfand and Smith, 1990) and Metropolis-Hastings (MH) algorithms (Metropolis et al., 1953;Hastings, 1970) -with MCMC being by far the most common simulation method used to compute forecast distributions in practice. The term 'exact' arises from the fact that, under appropriate conditions (including convergence of the Markov chain to p(θ|y 1:T ) in the case of the MCMC algorithms), such methods all produce a √ M -consistent estimate of the ordinate p(y T +1 |y 1:T ), at any point in the support of the random variable y T +1 ; this estimate can thus be rendered arbitrarily accurate, for large enough M. We refer the reader to: Chib and Greenberg (1996) and Geyer (2011) for reviews of MCMC sampling; Casella and George (1992) and Chib and Greenberg (1995) for descriptions of the Gibbs and MH algorithms (respectively) that are useful for practitioners; and Andrieu et al. (2004), Robert and Casella (2011) and Martin et al. (2022b) for historical accounts of MCMC sampling. Geweke and Whiteman (2006) also serves as an excellent reference on the use of these computational methods in a forecasting context. Given the critical role played by MCMC methods in the production of Bayesian forecasts, the basic principles of the algorithms are also outlined below in Section 3.1; with more recent developments of both IS and MCMC -most notably sequential Monte Carlo (SMC) (Gordon et al., 1993;Chopin and Papaspiliopoulos, 2020) and pseudo-marginal MCMC (Beaumont, 2003;Andrieu and Roberts, 2009;Andrieu et al., 2011) -discussed briefly in Section 3.2. The methods in 3) replace p(θ|y 1:T ) in the integrand of (2) with an approximation of some sort, and evaluate the resultant integral. In so doing, such methods do not aim to estimate p(y T +1 |y 1:T ) itself, but some representation of it, defined as the expectation of p(y T +1 |θ, y 1:T ) with respect to the relevant posterior approximation. The methods in 3) have been based on the principles of approximate Bayesian (Price et al., 2018), variational Bayes (VB) , and integrated nested Laplace approximation (INLA) , and produce what are termed 'approximate' forecast, or predictive distributions. Suffice to say that the principle adopted for estimating the 'approximate predictive' so defined is typically one and the same: draws of θ from the approximate posterior (however produced) are used to produce either a sample mean of conditional predictives, or M draws of y T +1 from p(y T +1 |θ, y 1:T ), with kernel density estimation then applied. Production of (3) requires the computation of each model-specific predictive, plus the computation of each (5). The first set of K computations would proceed via the sorts of steps outlined above. Computation of the K marginal likelihoods could also be performed via one of the three broad methods listed above (in particular 2) or 3)); however, the fact that each (5) 3 Bayesian Forecasting: A Chronological Tour 3.1 The late 20th century: The advent of MCMC As is clear from the brief synopsis above, it is simulation that is key to computing forecast distributions when they are not available in closed form. While the use of simulation to compute statistical quantities of interest was known by the 1970s (Metropolis and Ulam, 1949;Metropolis et al., 1953;Hammersley and Handscomb, 1964;Hastings, 1970), the technology required to perform simulation in a convenient and timely fashion was not yet available, and simulation-based computation thus remained largely out of reach. To quote Geweke and Whiteman (2006): "In the beginning, there was diffuseness, conjugacy and analytical work!" In the latter part of the 20th century, things changed. The increased speed and availability of desktop machines (Ceruzzi, 2003), allied with critical advances in simulation methodology, led to a proliferation of methods for accessing p(y T +1 |y 1:T ) via the simulation of draws from p(θ|y 1:T ). To this end, we give a brief outline of the pre-eminent posterior simulation algorithms of the 1990s (and into the early 2000s): Gibbs sampling (Section 3.1.1), MH-within-Gibbs sampling (Section 3.1.2), and (MH-within-) Gibbs sampling allied with data augmentation (Section 3.1.3); touching on the types of forecasting models that were able to be treated via such methods, including the ubiquitous state space models that underpin much modern Bayesian forecasting. To keep the exposition concise, we place all algorithmic details in Appendix A, and reference specific algorithms from Appendix A at suitable points in the text. Gibbs sampling As a general rule, if p(θ|y 1:T ) does not have a closed-form representation, it is also not amenable to Monte Carlo sampling, as the latter requires that p(θ|y 1:T ) can be decomposed into recognizable densities, from which computer simulation is feasible. IS (Kloek and van Dijk, 1978;Geweke, 1989), via use of an 'importance' or 'proposal' density, q(θ|y 1:T ), that matches p(θ|y 1:T ) well and which can be drawn from, is a possible solution in some cases. However, the algorithm can fail to produce representative draws from p(θ|y 1:T ) when the dimension of θ is large, due to the difficulty of finding a q(θ|y 1:T ) that is a 'good match' to p(θ|y 1:T ) in high dimensions. In contrast, under certain conditions, a Gibbs sampler is able to produce a (dependent) set of draws from the joint posterior via iterative sampling from lower dimensional, and often standard, conditional posteriors. In other words, a Gibbs sampler takes advantage of the fact that, while joint and marginal posterior distributions are usually complex in form and unable to be simulated from directly, conditional posteriors are often standard and amenable to simulation. Given the satisfaction of the required convergence conditions (Geyer, 2011), draws θ (i) , i = 1, 2, ..., M , produced via iterative sampling from the full conditionals converge in distribution to p(θ|y 1:T ) as M → ∞, and can be used to produce a √ M -consistent estimate of the ordinates of p(y T +1 |y 1:T ) across the support of y T +1 in the manner described in Section 2.2. Decisions about how to partition, or 'block' θ need to be made (Liu et al., 1994;Roberts and Sahu, 1997), with a view to increasing the 'efficiency' of the chain which, in effect, amounts to ensuring an accurate estimate of p(y T +1 |y 1:T ) for an given number of draws, M. (See Algorithm 1 in Appendix A.1.) Chib (1993) and McCulloch and Tsay (1994) are the earliest examples of using Gibbs algorithms for Bayesian estimation and prediction in time series settings. Whilst Chib (1993) looks expressly at an autoregressive (AR) structure in the error term of a linear regression model, the proposed Gibbs scheme also nests the steps required for sampling from an observed AR model. (See also Section 8.1, Koop, 2003.) The key points to note are that the conditional posteriors are shown to retain a closed form even when the specification allows for i) multiple lags of y t , ii) an informative prior on θ, iii) a prior that imposes stationarity on the model, and iv) a Student t (rather than a Gaussian) distribution; with any one of these modifications precluding analytical treatment of p(θ|y 1:T ) and p(y T +1 |y 1:T ). McCulloch and Tsay (1994) entertain even more complex versions of an AR model, to cater for random level shifts, outliers and missing values, and with an informative prior invoked for all unknowns. Once again, despite such model and prior specifications implying that closed forms for p(θ|y 1:T ) and p(y T +1 |y 1:T ) do not exist, the conditional posteriors all retain a closed form and, hence, allow for a straightforward application of a Gibbs scheme. As one would anticipate however, a 'pure' Gibbs algorithm based on a full set of standard conditionals is not always possible, with the more typical situation being one in which one or more of the conditionals -associated with any given partitioning of the parameter space -are not available in closed form. The following section describes how to adapt a Gibbs algorithm in cases where certain conditional components are not known in closed form, and, in so doing, illustrates a powerful simulation-based algorithm for accessing p(y T +1 |y 1:T ) in more complex settings. MH-within-Gibbs sampling The Gibbs sampler is only one example of an MCMC algorithm. The first such example -the 'Metropolis' algorithm -appeared in a paper that has assumed an important status in the history of statistics: Metropolis et al. (1953) 3 . The 'Metropolis' algorithm was subsequently generalized by Hastings (1970), and it is this 'MH' version of the method that is typically referenced. For the purpose of this review, the key purpose of the MH algorithm is to enable sampling from non-standard conditionals within a Gibbs algorithm, in particular when the dimension of the conditionals precludes (say) the exclusive use of inverse cumulative distribution function (ICDF) sampling. 4 Under regularity, a Markov chain that converges to p(θ|y 1:T ) can be produced by embedding an MH algorithm (or MH algorithms) within an outer Gibbs loop. In short, an MH-within-Gibb algorithm proceeds by drawing from any non-standard conditional indirectly, via a 'candidate', or 'proposal' distribution that is deemed to be a good match to the inaccessible conditional, and accepting the draw with a given probability. Critically, the formula that defines the acceptance probability involves evaluation of the non-standard conditional only up to its integrating constant; hence the conditional need not be known in its entirety. 5 Again, under appropriate regularity, the draws θ (i) , i = 1, 2, ..., M , from the MHwithin-Gibbs algorithm converge in distribution to p(θ|y 1:T ) as M → ∞, and can be used to produce a √ M -consistent estimate of the ordinates of p(y T +1 |y 1:T ). (See Algorithm 2 in Appendix A.2.) As will become evident in the subsequent empirical review sections, MH-within-Gibbs algorithms remain the dominant form of method used to sample from posteriors -and to estimate predictive distributions -for time series models for which a convenient partitioning of the parameter space is available, and for which the conditional posteriors are known up to their integrating constants. Hence, we reserve further elaboration on the use of such algorithms in practice until the appropriate points in Section 4. MCMC, data augmentation, and state space models For many empirical problems in economics and related fields, a suitable model can be partitioned into two sets: static unknowns θ, which are fixed throughout time, and latent data, z 1:T = (z 1 , z 2 , ..., z T ) ′ , that are a function of time. The latent states may be intrinsic to the model -as in a state space model -or may be auxiliary variables introduced purely for the purpose of facilitating posterior sampling. Application of a Gibbs-based MCMC scheme to the joint, or 'augmented' set of unknowns {θ, z 1:T } is often referred to as 'data augmentation', in the spirit of Tanner and Wong (1987), and such schemes have enabled the Bayesian analysis of large classes of time series models that would otherwise have been inaccessible. We illustrate here the basic principles of the approach using a state space model governed by a measurement density for the observed scalar random variable, y t , and a Markov transition density for a scalar state variable, z t , p(y t |z t , θ) (7) p(z t |z t−1 , θ).(8) Using the generic notation in (7) and (8), the augmented posterior is p(θ, z 1:T |y 1:T ) ∝ p(y 1:T |z 1:T , θ)p(z 1:T |θ)p(θ). In certain cases, the model structure is such that a pure Gibbs scheme can be used to produce draws from p(θ, z 1:T |y 1:T ) and, thus, from p(θ|y 1:T ); an insight obtained independently by Carter and Kohn (1994) and Frühwirth-Schnatter (1994) for the case of the linear Gaussian state space model, for example. However, implementation of such a scheme will, by definition, require both p(θ|z 1:T , y 1:T ) and p(z 1:T |θ, y 1:T ) to have recognizable forms. In more general cases, in which either the measurement or state equation has non-linear and/or non-Gaussian features, the resulting conditionals will not necessarily have a known closed form, which necessitates the addition of MH steps within the outer Gibbs loop. To conclude, and once again using the generic notation in (7) and (8), once draws have been produced from p(θ, z 1:T |y 1:T ), the predictive pdf, p(y T +1 |y 1:T ) = z T +1 z 1:T Θ p(y T +1 |z T +1 , θ, y 1:T )p(z T +1 |z T , θ)p(θ, z 1:T |y 1:T )dθdz 1:T dz T +1 ,(10) can be estimated is the usual way, using subsequent draws from p(z T +1 |z T , θ) and p(y T +1 |z T +1 , θ, y 1:T ), or by averaging the conditional predictives over all draws of z T +1 and θ. The 21st Century: Intractable forecasting models What do we mean by 'intractable' ? The MCMC methods that evolved during the late 20th century continue to serve as the 'bread and butter' of Bayesian forecasting, as will be made evident in Section 4. Nevertheless, more ambitious forecasting problems are now being tackled, and this has tested the mettle of some of the early algorithms! As a consequence, Bayesian forecasters have begun to exploit more modern computational techniques, and it is those techniques that we touch on briefly in this section. It is convenient to characterize these newer computational developments as different types of solutions to so-called 'intractable' forecasting problems, by which we mean: 1) Forecasts based on models with data generating processes (DGPs) that cannot be readily expressed as a pdf, or probability mass function (pmf); 2) Forecasts based on high-dimensional models, with a very large number of unknowns; 3) Forecasts produced using extremely large data sets. Problems that feature problem 1) are referred to as doubly intractable problems, as not only is p(θ|y 1:T ) not available in its entirety (as is typical), but the DGP itself is also not able to be expressed analytically. With reference to 1), the MCMC methods referenced so far entail the evaluation of the DGP as a pd(/m)f, either in the calculation of the acceptance probability in any MH sub-step, or in the specification of full conditionals in any 'pure' Gibbs step. Hence, they are infeasible when DGPs do not admit such a representation. Many such DGPs exist (see, for example, Martin et al., 2022a, for a list of examples); however, particularly pertinent ones to mention here are continuous time models in finance with unknown transition densities (Gallant and Tauchen, 1996), α-stable models for financial returns (and/or their volatility) (Peters et al., 2012;Martin et al., 2019), and stochastic dynamic equilibrium models in economics (Calvet and Czellar, 2015). With regard to 2), whilst, in principle ( The methods in the following sections have been designed to solve one or more of these instances of intractability. The techniques in Section 3.2.2 do so whilst preserving the 'exact' nature of the estimate of p(y T +1 |y 1:T ), whilst those in Section 3.2.3 aim to produce an approximation of p(y T +1 |y 1:T ) only. Exact computational solutions The first two decades of the 21st century have witnessed a wealth of advances in both MCMC and IS-based algorithms. The goal of the newer MCMC algorithms -at their heart -is to explore the high mass region of the joint posterior more efficiently, in particular when the dimension of the parameter space is large. This, in turn, enables a more accurate estimate of p(y T +1 |y 1:T ) to be produced for a given computational budget. This goal has been achieved via a variety of means, which (in the spirit of Robert et al., 2018, andMartin et al., 2022b) can be summarized as: i) the use of more geometric information about the target posterior, most notably the use of 'Hamiltonian' up-dates (Neal, 2011b;Hoffman and Gelman, 2014); ii) the use of better MH candidate, or proposal distributions, including those that 'adapt' to previous draws (Nott and Kohn, 2005;Roberts and Rosenthal, 2009 Whilst not designed expressly to deal with problems of scale, sequential Monte Carlo (SMC) methods -which exploit the principles of IS -have developed in parallel to the expansion of the MCMC stable. Developed initially for the sequential analysis of state space models, via methods of 'particle filtering' (Gordon et al., 1993), SMC methods have evolved into a larger suite of methods used to perform both sequential and non-sequential tasks (Naesseth et al., 2019;Chopin and Papaspiliopoulos, 2020). For the purpose of this review, the most pertinent development is the melding of particle filtering with MCMC in state space settings to produce a particle marginal MH (PMMH) algorithm (Andrieu et al., 2011;Flury and Shephard, 2011;Pitt et al., 2012;Doucet et al., 2015;Deligiannidis et al., 2018). Such algorithms tackle intractability type 1) in the dichotomy of the previous section, by replacing an 'unavailable' likelihood function by an unbiased estimate -produced via the particle filter -in an MH algorithm which, under regularity, retains the posterior p(θ|y 1:T ) as its invariant distribution. Given the increasingly important role played by PMMH, a brief algorithmic description of it is included in Algorithm 3 in Appendix A.4. 7 Approximate computational solutions In situations in which the dimension, or structure of the forecasting model, or the size of the data set, still precludes the use of an MCMC or a PMMH approach, an approximate method may be the only computational option. The cost of adopting such a solution is that these methods no longer directly target the exact predictive, p(y T +1 |y 1:T ); instead, an approximation of p(y T +1 |y 1:T ) becomes the goal. The spirit of these methods is to approximate p(y T +1 |y 1:T ) via some feasible approximation to the posterior p(θ|y 1:T ). Denoting the posterior approximation generically by g(θ|y 1:T ), the resultant approximate predictive can be expressed as g(y T +1 |y 1:T ) = Θ p(y T +1 |θ, y 1:T )g(θ|y 1:T )dθ,(11) in the case where there are only static unknowns. When the model features both static parameters and time-varying latent parameters, and exploiting the Markov property of the state process in (8), the approximate predictive can be represented as g(y T +1 |y 1:T ) = z T +1 z T Θ p(y T +1 |z T +1 , θ, y 1:T )p(z T +1 |z T , θ)p(z T |θ, y 1:T )g(θ|y 1:T )dθdz T dz T +1 . Given draws of θ from g(θ|y 1:T ), and given an appropriate forward-filtering algorithm to draw from p(z T |θ, y 1:T ) when needed, a simulation-based estimate of g(y T +1 |y 1:T ) can be produced in the usual way, either as a sample mean of the conditional predictives defined by the draws of θ (and z T +1 ), or by applying kernel density techniques to the draws of y T +1 from the conditional predictive. With reference to the taxonomy of intractable problems delineated in Section 3.2, the different methods of producing g(θ|y 1:T ) (and, hence, g(y T +1 |y 1:T )) can be categorized according to whether they are being used to obviate 1) or to tackle a problem of scale: 2) and/or 3). Both ABC and BSL avoid the need to evaluate the DGP and, hence, are feasible methods in the doubly intractable settings of category 1). In brief, both methods require only simulation, not evaluation, of the DGP. The approximation of p(θ|y 1:T ) arises, primarily, from the fact that both methods -in different ways -degrade the information in the full data set, y 1:T , to the information contained in a set of summary statistics, η(y 1:T ). As such, the target becomes the so-called 'partial' posterior for θ, which conditions on η(y 1:T ), rather than y 1:T . The quality of the approximation is thus dependent on the informativeness of the summaries, as well as on other forms of approximation invoked in the implementation of the methods. Vanilla versions of both algorithms are provided in Algorithms 4 (Appendix A.5) and 5 (Appendix A.6) respectively. In contrast to ABC and BSL, VB and INLA still target the exact posterior p(θ|y 1:T ), but provide approximations that can be computationally convenient when the scale of the empirical problem is large in some sense (so problem 2) and/or problem 3)), often as a consequence of the specification of a high number of latent, or 'local', parameters in the model, in addition to the (usually) smaller set of 'global' parameters (θ in our notation). Adopting the technique of the calculus of variations, VB produces an approximation of p(θ|y 1:T ) that is 'closest' to p(θ|y 1:T ) within a chosen variational family, We refer the interested reader to Martin et al. (2022a) for an extensive review of all of these approximate Bayesian methods, as well as more complete coverage of the existing literature, including references to in-depth reviews of specific methods. Martin et al. also includes discussion of 'hybrid' methods that mix and match features of more than one computational technique, with the aim of tackling multiple instances of 'intractability' simultaneously. Regardless of which approximation method is used, the hope is that the resulting approximate predictive g(y T +1 |y 1:T ) performs well relative to the inaccessible exact predictive, and that issue is addressed in certain work cited in the empirical reviews in Section 4. The 21st Century: Misspecified forecasting models The role of model specification in Bayesian forecasting Inherent in the conventional Bayesian approach to forecasting is the assumption that the process that has generated the observed data tallies with the particular model that underpins the likelihood function. Bayesian model averaging (BMA) -and the resultant predictive in (3) -has evolved as a principled way of catering for uncertainty about the predictive model, and BMA remains a very important technique in the Bayesian arsenal. Nevertheless, underpinning BMA is still the assumption that the true process is spanned by the set of models over which one averages -i.e. that the so-called M-closed view of the world (Bernardo and Smith, 1994) prevails. In response to these perceived limitations of the conventional approach, attention has recently been given to producing predictions that are 'fit for purpose', by focusing the Bayesian machinery on the specific goals of the predictive analysis at hand. In the following sections we briefly summarize three such approaches, all of which move beyond the conventional likelihood-based Bayesian up-date, and Mclosed paradigm: seeking to produce accurate predictions without recourse to the assumption of correct model specification. Focused, or 'loss-based' Bayesian prediction Loaiza-Maya et al. (2021) propose an approach to Bayesian prediction expressly designed for the context of misspecification. In brief, rather than a correct predictive model being assumed, a prior is placed over a class of plausible predictive models. The prior is then updated to a posterior via a sample criterion function that is constructed using a scoring rule that rewards the type of predictive accuracy (e.g. accurate prediction of extreme values) that is important for the particular empirical problem being tackled. With a criterion function that explicitly captures predictive accuracy replacing the likelihood function in the conventional Bayesian up-date, the explicit need for correct model specification is avoided. Following , and using generic notation, for P a convex class of predictive distributions on (Ω, F), the predictive accuracy of P ∈ P can be assessed using a scoring rule S : P × Ω → R. If the value y eventuates, then the positively-oriented 'score' of the predictive P , is S(P, y). The expected score under the true unknown predictive P 0 is defined as S(·, P 0 ) := y∈Ω S(·, y)dP 0 (y). A scoring rule is said to be proper relative to P if, for all P, G ∈ P, S(G, G) ≥ S(P, G), and is strictly proper, relative to P, if S(G, G) = S(P, G) ⇐⇒ P = G. Scoring results are important mechanisms as they elicit truth telling within the forecasting exercise: if the true predictive P 0 were known, then in terms of forecasting accuracy as measured by the scoring rule S(·, ·) it would be optimal to use P 0 . Different scoring rules rewards different forms of predictive accuracy (see , Opschoor et al., 2017, and Martin et al., 2022c for expositions); hence the motivation to drive the update by the score that 'matters'. Since P 0 and the expected score S(·, P 0 ) are unattainable in practice, an estimate based on y 1:T is used to define the sample criterion, S T (θ) := T −1 t=0 S[p(y t+1 |θ, y 1:t ), y t+1 ], where p(y t+1 |θ, y 1:t ) is the pdf associated with a given P. Adopting the exponential updating rule proposed by Bissiri et al. (2016) (see also Giummolè et al., 2017, Holmes andWalker, 2017, Guedj, π w (θ|y 1:T ) = exp [wS T (θ)] π(θ) Θ exp [wS T (θ)] π(θ)dθ ,(13) for some learning rate ω ≥ 0, calibrated in a preliminary step. This posterior explicitly places high weight on -or focuses on -values of θ that yield high predictive accuracy in the scoring rule S(·, ·). As such, the process of building a Bayesian predictive as: p w (y T +1 |y 1:T ) = Θ p(y T +1 |θ, y 1:T )π w (θ|y 1:T )dθ,(14) is termed focused Bayesian prediction (FBP) by the authors. By construction, when the predictive model, p(y T +1 |θ, y 1:T ), is misspecified, (14) will -out-of-sample -often outperform, in the chosen rule S(·, ·), the likelihood (or log-score)-based predictive in (2), and this is demonstrated in Loaiza-Maya et al. both theoretically and in extensive numerical illustrations. Since a positively-oriented score can, equivalently, be viewed as the negative of a measure of predictive loss, FBP can also be referred to as 'loss-based' prediction. Such terminology is indeed adopted in Frazier et al. (2021), in which the principles delineated here are extended to high-dimensional models, and approximations to both π w (θ|y 1:T ) and p w (y T +1 |y 1:T ) based on VB proposed, and validated. Bayesian predictive combinations: Beyond BMA The predictive distributions within the 'plausible class' referenced above may characterize a single dynamic structure depending on a vector of unknown parameters, θ, or may constitute weighted combinations of predictives from distinct models, in which case θ comprises both the model-specific parameters and the weights. As such, FBP provides a coherent Bayesian method for estimating weighted combinations of predictives via predictive accuracy criteria, and without the need to assume that the true model is spanned by the set of constituent predictives -an assumption that underpins BMA, as we have noted. A similar motivation underlies other contributions to the extensive Bayesian literature on estimating combinations of predictives that has now developed (and which rivals the large frequentist literature on forecast combinations that has also evolved 8 ), with predictive performance -quantified by a range of userspecified loss measures -driving the posterior up-dating of the weights. Indeed, the Bayesian literature, having access as it does to powerful computational tools, has been able to invoke more complex weighting schemes than can be tackled via frequentist (optimization) methods. Notable contributions, including some also driven by the criterion of predictive calibration (Dawid, 1982;Dawid, 1985;, include Billio world that accords with the reality of misspecification. Bayesian predictive decision synthesis A third approach that seeks to produce Bayesian predictions without relying explicitly on correct model specification is Bayesian predictive synthesis (BPS) (Johnson, 2017;McAlinn and West, 2019; (BPDS) by Tallman and West (2022). In particular, BPDS provides a sound decision-theoretic framework for constructing forecast combinations, and can be shown to encompass several commonly-suggested Bayesian forecasting approaches. The starting point of BPDS is the production is a prior distribution over the m-dimensional unknown outcome y -implicitly indexed by T + 1 in a time series forecasting application -and the information set H, encoded via the J predictive models {h j (y|x j ) : 1 ≤ j ≤ J}, where x = (x 1 , . . . , x J ) denotes the collection of vectors of (possibly latent) dummy variables associated with a decision. The decision maker then constructs a predictive by integrating out x using a 'synthesis function' α(y|x): p(y|H) = X α(y|x) J j=1 h j (y|x j )dx 1 . . . dx J . The choice of the synthesis function α(·|x) can be used to drive the analysis. For instance, in the case of forecast combinations, we can take h j (y|x) = p j (y|x, M j ), for some model M j , and then any set of synthesis functions α j (·|x) such that the combination density p(y|H) = X J j=1 ω j α j (y, x j | x) p j (y | x, M j ) J k=1 ω k α k (y, x k | x) dx 1 . . . dx J . is a valid density, for given weights 0 < ω j < 1, J j=1 ω j = 1. Specific choices of α j (y, x j | x) then produce different forecast combination methods (see, Johnson, 2017, for a discussion); for example, in In an attempt to 'focus' the BPDS approach towards decisions that are tailored to a specific loss function, Tallman and West (2022) propose taking as their synthesis function, α j (y, x j |x) = exp{τ ′ (x)S j (y, x j )}, where the score S j (y, x j ) is a k-dimensional vector that measures the utility one receives from realizing outcome y under decision x j , and τ (x) is a vector that weights the directional relevance of the scores S j (y, x j ). While the BPS framework, as a whole, can set the tenor of the predictions towards dynamic forecast updates that produce predictions tailored towards a loss function of interest, via the choice of synthesis function α(·|·), BPS is ultimately tied to a 'likelihood-type' framework, or at least a log-loss function, due to the presence of the latent variables x, which must be integrated out via assumed predictive models, p j (y|x, M j ), and the production of these individual predictives using exact (i.e., likelihoodbased) Bayesian methods. 9 While the BPDS approach can somewhat circumvent the reliance on the likelihood, due to its ability to focus on specific scores, this approach appears to be distinct from methods that entirely replace the likelihood function for the up-date. Therefore, a very interesting research path would involved combining the methods based on generalized posteriors discussed in Section 3.3.2 with the BPS framework. Selective Reviews of Bayesian Forecasting: Discipline-Specific Examples Having established the necessary details regarding the production of Bayesian forecasts in general contexts, we now review how this general probabilistic mechanism is employed to produce Bayesian forecasts in several important empirical fields. In order to produce a comprehensive and up-to-date review of each area, a range of discipline experts have been invited to write the various sections, with the authorship flagged in the section headings. This means that the style of coverage differs somewhat across sections, as suits the topic, and as fits with the perspective of the authors; however, we have aimed to retain notation that -as far as possible -is consistent both across sections and with the notation used in the earlier parts of the paper, and in the technical appendix; and to ensure that the basic layout of all sections is the same. As noted earlier, other than in Section 4.3 -in which cross-sectional consumer choice data is modelled -and in Section 4.4, in which spatial models are briefly referenced, time series problems and forecasting are the primary focus. Macroeconomics (Florian Huber and Gary Koop) Central banks and other policy institutions routinely collect vast amounts of time series data on key macroeconomic outcomes. One stylized fact is that these data sets often display substantial co-movements and this calls for modeling all these series jointly to produce accurate point and density forecasts. This, however, leads to large-scale models that are prone to overfitting, ultimately resulting in weak out-ofsample forecasting performance. This helps explain the popularity of Bayesian methods for macroeconomic forecasting. They can easily handle many parameters and, through appropriate prior choice, deal effectively with questions related to model and specification uncertainty in macroeconomic models. At a high level of generality, there are two modelling approaches used by macroeconomic forecasters. The first uses reduced-from models and imposes relatively little economic structure on the data. The second uses structural models such as dynamic stochastic general equilibrium (DSGE) models that are often estimated through Bayesian techniques; see, among many others, Smets and Wouters (2007), Adolfson et al. (2007) and Del Negro et al. (2016). However, reduced-form approaches have proved more popular and, in this section, our focus will be on them. As stated above, macroeconomists are typically interested in modeling the joint evolution of a set of macroeconomic quantities. To set up a general framework for understanding the types of models used for forecasting, assume that an M -dimensional vector y t is related to a K-dimensional vector of explanatory variables x t through y t = g(x t ) + ε t ,(15) where g : R K → R M is a function and ε t is N (0 M , Σ t ). 10 This general specification nests most important reduced-form models commonly used in macroeconomics and can be used to explain the main issues that arise. For instance, if x t = (y ′ t−1 , . . . , y ′ t−p ) ′ contains p lags of y t , g(x t ) = Ax t is a linear function with M × K(= M p) coefficient matrix A, and Σ t = Σ is constant over time we have a standard vector autoregressive (VAR) model. If we set x t = f t with f t denoting a set of Q ≪ M latent factors and g(f t ) = Λf t is linear with Λ being an M × Q matrix of factor loadings and f t evolves according to some stochastic process (such as a VAR) we end up with a dynamic factor model (DFM, see Stock and Watson, 2011). Factor augmented VARs (Bernanke et al., 2005) combine a VAR with a DFM. The dependent variables in the VAR part of the model are a subset of y t plus a small number of factors. Traditionally, VARs and factor models have been linear and homoskedastic. But there is a great amount of empirical evidence in most macroeconomic data sets of parameter change, both in the conditional mean and the conditional variance. This can be accommodated through particular choices for g and Σ t . For the latter, stochastic volatility processes have proved particularly popular. For the former, various parametric forms for g lead to time-varying parameter VARs (TVP-VARs) which assume that the coefficients of the VAR evolve according to a random walk. But it is also worth noting that there is an increasing literature which assumes g is unknown and uses Bayesian nonparametric methods to uncover its form (see, for example, Kalli and Griffin, 2018, Adrian et al., 2021, and Huber et al., 2020. If we set M = 1 we obtain single-equation time series regressions which are particularly popular in inflation forecasting (e.g. based on the Phillips curve). If we additionally set x t = 1 and allow for time varying parameters, we can obtain models such as the unobserved components stochastic volatility (UCSV) model of Stock and Watson (2007) which is commonly used to forecast inflation (for recent applications, see Chan et al., 2013, Stock and Watson, 2016, and Huber and Pfarrhofer, 2021. This general framework defines a class of likelihood functions. As per the outline in Section 2.1, Bayesian forecasting involves multiplying a chosen likelihood function by an appropriate prior to produce a posterior which can be used to produce the predictive density. The choice of prior and computational method used for posterior and predictive inference will be case specific and we will have more to say about some interesting cases below. But a few general comments are worth noting here. First, the choice of prior matters much more in models such as the large VAR, which have a large number of parameters relative to the number observations, than in models with fewer parameters such as the UCSV model or the DFM. Second, for linear homoskedastic models with conjugate priors analytical formulae for the posterior and the one-step-ahead predictive density are available. For all other cases, MCMC methods are available. These take the general form outlined in Section 3.1. However, as noted in Section 3.2, MCMC methods typically do not scale well and can be computationally slow in models involving large numbers of parameters (such as large VARs) or large numbers of latent states (such as TVP-VARs). Thus, the focus of many recent papers has been on developing either improved MCMC algorithms or approximate VB methods for speeding up computation. Thirdly, our discussion so far focuses on forecasting with a single model. In practice, it is common to find that forecasts improve if many models are combined. Thus, either BMA or, alternatively, the methods outlined in Section 3.3.3 are commonly used by macroeconomic forecasters. With this general framework established, it is worthwhile to offer some additional detail about some of the most important 21st century developments and a discussion of how they have led to improvements in macroeconomic forecasting. Large VARs Going back to early work such as Doan et al. (1984a), Bayesian VARs have been used successfully in a variety of macroeconomic forecasting applications. Recently, they have enjoyed even greater popularity due to the rise of the large VAR. The pioneering large VAR paper was Bańbura et al. (2010). Subsequently, dozens of papers have used large VARs for macroeconomic forecasting (see, among many others, Carriero et al., 2009, Koop, 2013b, Carriero et al., 2015, Giannone et al., 2015, and Hauzenberger et al., 2021b. Large VARs, involving dozens or even hundreds of dependent variables, have been found to forecast well and improve upon single-equation techniques and DFMs. Large VARs are heavily over-parameterized and, thus, Bayesian prior shrinkage has been essential in ensuring their forecasting success. We will discuss priors shortly, but at this point we highlight the fact that the use of large Bayesian VARs has been one of the major recent developments in macroeconomic forecasting. Prior shrinkage in VARs a j ∼ N (0, ψ j λ), ψ j ∼ f 1 , λ ∼ f 2 , where a j is the j th VAR coefficient, λ controls global shrinkage since it is common to all coefficients and Adding stochastic volatility (SV) The other main development that has had a tremendous impact on applied macroeconomic forecasting in the 21st century is the development of models such as VARs that incorporate parameter change and nonlinearity. Put simply, the macroeconomic world is rarely linear and homoskedastic and models that relax these assumptions have been found to improve macroeconomic forecasting. These improvements lie not only in point forecasts, but more importantly in density forecasts. Given the increasing interest, by central banks and academics alike, in issues such as forecast uncertainty and tail risk, the fact that these new models produce more accurate predictive densities increases their value. A popular specification for VARs with SV involves factorizing the error variance-covariance matrix as Σ t = A 0 H t A ′ 0 with A 0 being a lower triangular matrix with unit diagonals 13 and H t = diag(e h 1t , . . . , e h M t ) being a diagonal matrix with log-volatilities evolving according to simple stochastic processes such as a independent random walks or AR(1) processes. In an important contribution, Clark (2011) considers a VAR-SV and finds it to produce accurate point and density forecasts relative to homoskedastic models, with gains being particularly pronounced using forecast metrics involving the entire predictive density. Building on this insight, several other researchers have analyzed the role of heteroskedasticity in macroeconomic forecasting in VARs (see, for example, Clark andRavazzolo, 2015, andChiu et al., 2017) and confirm the result that using SV pays off when the focus is on obtaining accurate density forecasts. However, a problem with the standard SV specification is that the computational burden relative to homoskedastic VARs is increased enormously. This makes it difficult to do Bayesian forecasting with large VARs with SV. As a remedy, Carriero et al. (2016) propose a simple common stochastic volatility (CSV) specification that assumes the shock variances to be driven by a single common volatility factor, maintaining conjugacy and thus leading to computationally efficient MCMC algorithms. They acknowledge that this model is simplistic but show that it yields much more accurate forecasts than homoskedastic VARs in a standard US macroeconomic forecasting application. To gain more flexibility, researchers have developed algorithms that allow for estimating large VARs with M independent SV processes. Carriero et al. (2019) propose techniques that permit equationby-equation estimation of such VARs and thus render estimation of larger models with SV feasible. Modified versions of this algorithm form the basis of several recent papers that combine large data sets with SV for macroeconomic forecasting (see, among others, Huber and Feldkircher, 2019, Chan, 2021. Adding time variation in the VAR coefficients The previous discussion has emphasized that capturing changing error variances is key for obtaining precise forecasts. However, it may also be important to allow for structural change in the VAR coefficients themselves. One popular multivariate model that captures both changes in the VAR coefficients and error variances is the TVP-VAR-SV model proposed in Primiceri (2005), which assumes that the VAR coefficients β t = vec(A t ) are time-varying and evolve according to a multivariate random walk while Σ t is a multivariate SV process. This model is a multivariate state space model which can be estimated using adaptations of the techniques outlined in Section 3.1.3. The innovations to the states govern the amount of time variation in the parameters. Various shrinkage priors (often based on the global-local shrinkage priors discussed above) have been proposed that allow for data-based selection of whether time variation in a corresponding coefficient is 13 A0 can also be time varying. necessary or not. These priors are typically elicited on the non-centered parameterization of the state space model (see Frühwirth-Schnatter and Wagner, 2010) and can help minimize overfitting concerns and produce improved forecasts. D' Agostino et al. (2013) is an important early contribution to the macroeconomic forecasting literature using TVP models. This paper uses a small TVP-VAR with SV and shows that it produces more accurate point predictions, outperforming simpler univariate benchmarks and constant parameter VARs. One key shortcoming of this model, however, is that it only uses a small information set. This has led to several researchers proposing new methods that can be used in higher dimensions. Various approaches are possible, including models that restrict the TVP process (e.g. by imposing a factor structure which allows for time variation in a large number of parameters to be driven by a low number of factors, see Chan et al., 2020). As mentioned above, shrinkage priors are used to keep the curse of dimensionality in check. These priors are typically used after transforming the model to allow for equation-by-equation estimation. Such approaches mean fairly high-dimensional TVP-VARs can be estimated without risk of over-fitting, and in a reasonable amount of time. MCMC-based forecasting with large TVP-VARs and regressions is also an active field of research and different shrinkage methods and advances in computation have led to improvements in the forecasting performance of TVP models (see, among many others, Hauzenberger et al., 2021a). However, it is worth noting that if computation does become a concern, approximate methods (e.g. using the VB methods outlined in Section 3.2.3) can be used. Approaches which avoid the need for MCMC are developed in Koop and Korobilis (2013) and Koop and Korobilis (2018). The former paper proposes large approximate TVP-VARs based on forgetting factors whereas the latter one uses VB techniques to forecast inflation with large TVP regression models. Bayesian nonparametric VARs Up to this point we have assumed that the conditional mean function g takes a known form. However, it could be that the functional form is unknown. Bayesian nonparametric techniques, such as Bayesian additive regression trees (BART, see Chipman et al., 2010), Gaussian processes and kernel regressions (Adrian et al., 2021) or infinite mixtures (Kalli and Griffin, 2018), allow the researcher to uncover such unknown functional forms and produce precise macroeconomic forecasts. In general, they have had great success, but they have been found to be particularly useful in studies that focus on the tails of predictive distributions or on the handling of outliers such as the ones experienced during the pandemic (see, for example, Huber et al., 2020, andClark et al., 2022b). Kalli and Griffin (2018) propose a nonparametric VAR that builds on an infinite mixture model with the mixture weights being driven by the lagged endogenous variables. They show, using US and UK data, that their model yields competitive forecasts, with accuracy gains in terms of point and density predictions increasing sharply for higher forecast horizons. Clark et al. (2022a) use BART-based VARs to perform tail forecasting of US output, unemployment and inflation in real time, finding that nonparametric techniques work well in the tails and for higher-order forecasts. With a particular focus on predictive accuracy during the pandemic, Huber et al. (2020) develop mixed frequency nonparametric VARs and show that these models yield substantially more precise nowcasts during the Covid-19 period. Conclusions and further directions We have outlined how Bayesian methods have been used successfully for macroeconomic forecasting. Most of the discussion related to VARs, which are a class of models where Bayesian methods have proved particularly popular. But it is worth noting that empirically-relevant extensions (e.g. SV or TVP) can be added to other multivariate time series models such as DFMs or FAVARs, as can the VAR prior shrinkage methods (e.g. global-local shrinkage methods) we have discussed. It is also worth noting that we have focused on models that do not restrict the coefficients. However, restricted VARs are often used for forecasting. For instance, vector error correction models (which impose cointegrating restrictions) or multi-country VARs such as global VARs are restricted VARs. We have also focused on forecasting as opposed to the closely related field of nowcasting. Mixed frequency VARs, which jointly model quickly-released, high-frequency variables (e.g. monthly variables such as surveys, employment and inflation) and slowly-released, low-frequency variables (e.g. quarterly variables such as GDP), have proved very popular with nowcasters. Bayesian methods are typically used with such models (see, for example, Schorfheide and Song, 2015, McCracken et al., 2021, Huber et al., 2020 and, in real-time nowcasting exercises they tend to perform well. Finance (John Maheu, Worapree Maneesoonthorn and Gael Martin) A pertinent question in financial analysis is whether the risks associated with financial assets -and the prices of those risks -are predictable in ways that are useful in applications such as portfolio allocation, risk management and derivative pricing. With risk factors typically being represented as latent distributional features of observable financial variables, it follows that two key goals in the statistical analysis of financial problems are: i) The accurate prediction of latent distributional features; and ii) The development of complex, non-linear state space models to underpin this prediction. Both of these goals lend themselves naturally to a Bayesian treatment given, in turn, the automatic production of predictive distributions via the Bayesian paradigm, and the swathe of computational methods available to estimate complex models -most notably those with a latent variable structure. In particular, the growth in financial derivatives markets from the 1990s onwards has generated the need to model the underlying asset as a continuous time process, almost always augmented with continuous time processes for the asset volatility, and often via jump diffusions. Such models -whilst 'convenient' in the sense of allowing for closed-form solutions for derivative prices -are challenging from a statistical point of view, given that they typically need to be treated as a (discretized) non-linear state space model, and may require multiple sources of data to enable separate identification of model parameters and risk premia. Estimation of and forecasting with such models is nevertheless computationally feasible via Bayesian methods, with MCMC algorithms of one form or another forming the backbone of the early treatments (Eraker, 2001;Eraker et al., 2003;Eraker, 2004;Forbes et al., 2007;Johannes et al., 2009). We refer the reader to Jacquier and Polson (2011) and Johannes and Polson (2010) for comprehensive reviews of the application of Bayesian methods in finance up to the first decade of the 21st century. The coverage includes, in short, Bayesian approaches to: portfolio allocation, return predictability, asset pricing, volatility, covariance, 'beta' and 'value at risk' prediction, continuous time models (and discretized versions thereof), interest rate modelling, and derivative (e.g. option) pricing. Our goal in the current review is to outline the more recent advances that have evolved over the last decade, in particular those that have exploited (in one way or another) new methodological advances, new sources of data, and modern computational techniques. In order, we shall briefly review: the use of diverse data sets, including derivative prices and high-frequency measures of financial quantities; the treatment of DGPs that are unavailable in closed form; the analysis of high-dimensional models; and the application of non-parametric modelling. Multiple sources of financial data It is now a well-established fact that the constant volatility feature of a geometric diffusion process for a financial asset price is inconsistent with both the observed dynamics in return volatility and the excess kurtosis and skewness that characterizes the typical empirical return distribution; see Bollerslev et al. (1992) for an early review. The option pricing literature supports this finding, with certain empirical regularities, such as 'implied volatility smiles', seen as evidence that asset prices deviate from the geometric Brownian motion assumption that underlies the Black and Scholes (1973) option price (Bakshi et al., 1997;Hafner and Herwartz, 2001;Lim et al., 2005). Hence, the 21st century has seen the proliferation of many alternative specifications for asset prices, and associated theoretical derivative prices, most of which are nested in a general framework of (discretized) bivariate jump diffusion models for the asset itself and its volatility. Allied with these developments has been the growth in access to transaction-level 'high-frequency' data -in both the spot and options markets -which, in itself, has spawned new approaches to inference and forecasting in the financial sphere. The Bayesian literature has brought to bear on this problem the power of computational methods -both established, and more recent -to enable the multivariate state space models that have emerged from this literature to be estimated, and probabilistic predictions of all dynamic variables -the return itself, volatility, random jumps (in either the return or the volatility, or both), and various risk premia -to be produced. With reference to the generic notation for a state space model in (7) and (8), Bayesian approaches over the last decade can be categorized according to the specification adopted for the (multivariate) measurement at time t, y t and, hence, for the (multivariate) state, z t , being modelled and forecast. Some work exploits data from both the spot and options market to predict volatility and its risk premia (Maneesoonthorn et al., 2012), and theoretical option prices (Yu et al., 2011;Carverhill and Luo, 2022 14 ); other work combines 'low-frequency' daily observations on returns with high-frequency measures of volatility and/or price jumps to predict (in some combination) returns, volatility, and the size and occurrence of price jumps (Jin and Maheu, 2013;Maneesoonthorn et al., 2017;Frazier et al., 2019); whilst further work combines daily returns with futures prices in predicting various financial quantities of interest (Fileccia and Sgarra, 2018;Gonzato and Sgarra, 2021). Financial models that are 'unavailable' All but one of the papers cited in the previous paragraph share a common feature -namely, a DGP that can be expressed as a probability density (or mass) function. With reference to (9), it is the availability of a closed form for p(y 1:T , z 1:T |θ) = p(y 1:T |z 1:T , θ)p(z 1:T |θ), that renders feasible the MCMC methods used in the said works. In contrast, Frazier et al. (2019) adopt a process for the latent log-volatility that is driven by an α-stable innovation, such that p(z 1:T |θ) is unavailable, and MCMC infeasible as a consequence. Instead, ABC is adopted for inference, and an approximate predictive of the form of (11) produced instead. In addition to providing theoretical validation of the approach, the authors demonstrate, in range of different simulation settings, that despite inaccuracy at the posterior level, the approximate predictive is always a very close match to the exact predictive. Related work in which an ABC method is used to conduct forecasting appears in Canale and Ruggiero (2016) For other recent Bayesian treatments of intractable models of this sort that continue to exploit MCMC principles (with or without an ABC component), see Vankov et al. (2019) and Müller and Uhl (2021). 15 Large financial models Thus far, we have reviewed Bayesian treatments of models for single financial assets. That is, the models may have specified multiple latent components, and potentially multiple measurements, but they still aim to explain (and forecast) quantities related to a single asset. Models for multiple assets are also critically important in financial applications, with the relationship between financial assets determining the extent to which diversification can be achieved, as well as how risks permeate the various sectors of the financial market. Indeed, Bayesian methods are particularly suitable for dealing with such multivariate models, since the dimensionality of z 1:T is typically much larger than that of y 1:T and, hence, challenging to deal with via any other means. Chib et al. (2009) provide an early review of the Bayesian analysis of multivariate SV models, with all work up to this point utilizing traditional MCMC techniques, and the statistical and predictive analysis limited to relatively low-dimensional systems (up to ten assets). Subsequent work has focused on the development of more flexible multivariate distributions (Nakajima, 2017), and the use of sparse factor structures and shrinkage priors in constructing larger-dimensional models (Zhou et al., 2014;Kastner et al., 2017;. More recently, with the advances made in VB methods, inference and prediction in very large-dimensional financial models is now possible (Gunawan et al., 2021;Chan and Yu, 2022;Frazier et al., 2022;Quiroz et al., 2022). There is also a growing interest in the prediction of co-movements of various sorts, with: Bernardi et al. Bayesian nonparametric modelling in finance As noted, simple parametric assumptions such as additive Gaussian innovations are inconsistent with the stylized features of financial data. Whilst more suitable non-Gaussian/non-linear models can be built (as highlighted above), Bayesian nonparametric modelling allows for further flexibility via the incorporation of Dirichlet process mixture (DPM) structures. Such an approach has been shown to provide robustness to distributional assumptions and can improve point forecasts, but the main gain has been significant improvements in the accuracy of predictive densities, and of risk measures derived from those densities. The advancement of the literature in this direction has been aided by the stick-breaking representation (Sethuraman, 1994) and the introduction of the slice sampler (Walker, 2007;Kalli et al., 2011). Jensen and Maheu (2010) introduce an extension to a standard SV model to capture the unknown return innovation distribution via a DPM. The DPM specification has also been inserted into other popular models in finance, with: Jensen and Maheu (2014) adopting a DPM to jointly model the return and future log-volatility distribution; Delatola and Griffin (2013) capturing the so-called leverage effect; Ausín et al. (2014) applying a DPM to univariate GARCH models; and Kalli and Griffin (2015) using Bayesian nonparametric modelling to aggregate autoregressive processes to produce a SV model with long-range dependence. Extensions to multivariate financial models have also occurred: in a multivariate GARCH setting in Jensen and Maheu (2013); and in a Cholesky-type multivariate SV model in Zaharieva et al. (2020). Markov switching models (Chib, 1996) feasible in the IHMM. The IHMM structure has been used to model the univariate GARCH distribution (Dufays, 2016), and the multivariate GARCH distribution (Li, 2022); and to provide a nonparametric model for realized measures, including realized covariance matrices (Jin and Maheu, 2016;Liu and Maheu, 2018;Jin et al., 2019), with all papers documenting very large improvements in density forecast accuracy from the IHMM. Other applications of the IHMM include Shi and Song (2016), who use the IHMM to date and forecast speculative bubbles, and who also adopt a version with GARCH effects; Yang (2019), who studies the relationship between stock returns and real growth with a multivariate IHMM model; and more recently Jin et al. (2022), who employ the DPM prior in the infinite Markov pooling of predictive distributions, with forecasting applications to interest rates, realized covariances and asset returns. Other approaches to time dependence in Bayesian nonparametrics for finance include Griffin and Steel (2011), who introduce a time-dependent stick breaking process in a general setting and develop a SV model for returns. More recently, Sun et al. (2020) use a weighted DPM to forecast return distributions, while Zamenjani (2021) allows for lagged covariates to impact the weights in the DPM model through a probit stick-breaking process. Marketing (Rubén Loaiza Maya and Didier Nibering) Bayesian methods are applied to a wide range of marketing problems; see Rossi and Allenby (2003) for a review of the early literature. More recently, these methods have been increasingly used for the purpose of prediction, for instance in customer choice behaviour (Toubia et al., 2019;Araya et al., 2022), customer demand (Posch et al., 2022), customer satisfaction (Mittal et al., 2021), dynamic pricing (Bastani et al., 2022), advertising effectiveness (Danaher et al., 2020;Loaiza-Maya et al., 2022) and recommender systems (Ansari et al., 2018). Given the large variety of marketing applications, we focus in this section on the modelling of customer choice to illustrate the key principles of Bayesian prediction in marketing problems. A common problem in marketing is that of setting the price level of a set of products so that total profits are maximized. To estimate these optimal prices, predictions of how customers will react to price changes are crucial. Predictions of customer choices under different marketing environments can be constructed by choice models. These models are estimated using data about the product choices of customers in the marketplace, a survey, an experiment, etc. (Rossi et al., 2012). Figure 1 shows a prediction of interest in this context: the predicted purchase probability of a customer for product j (y-axis) as a function of the price of product k (x-axis). The predicted purchase probability can be constructed for a customer for which only a few choices are observed, or for a new customer for which we do not observe choices in the data. Although this section, as noted earlier, focuses on cross-sectional data, choice models can also be applied to the forecasting of future choice probabilities by using panel data (Gilbride and Allenby, 2004;Terui et al., 2011). extending the multinomial logit model to a nested logit model (Poirier, 1996;Lahiri and Gao, 2002) or a random parameter logit model (Train, 2009). On the other hand, the multinomial probit model does not impose the IIA property, and as such is commonly used in the analysis of economic choice behaviour, where complementary and substitution effects are important. For instance, the multinomial probit model has been recently used in the analysis of car choices (Karmakar et al., 2021), grocery brand choices (Miyazaki et al., 2021), employment choices (Mishkin, 2021), and car parking choices (Paleti, 2018). This section presents a review of Bayesian prediction with the multinomial probit model. Multinomial probit model specification The variable of interest is y i ∈ {0, 1, 2, . . . , J}, which indicates the choice made by individual i among a set of J + 1 choice alternatives. This choice is modeled to be conditional on a set of J latent utilities z i = (z i1 , . . . , z iJ ) ′ , so that p(y i |z i ) = I [z iy i = max(z i )] if max(z i ) > 0, I[y i = 0] if max(z i ) ≤ 0,(16) where I[A] is one if statement A is true and zero otherwise. The base category j = 0 is one of the choice alternatives, which is selected a priori. The base category is observed whenever all the latent utilities are less than zero. The utilities are expressed in terms of r predictors via a linear Gaussian model, p(z i |X i , θ) = φ J (z i ; X i β, Σ) ,(17) where φ J (z; µ, C) denotes a J-variate normal density with mean µ and covariance matrix C, X i a J × r matrix of predictor values, β an r−dimensional vector of coefficients, and Σ a covariance matrix that captures complementary and substitution effects between the choice alternatives. Combined, (16) and (17) give rise to the augmented likelihood function of the multinomial probit model p(y, z|θ, X) = n i=1 p(y i |z i )p(z i |X i , θ),(18)where θ = {β, Σ}, y = {y i } n i=1 , z = {z i } n i=1 , and X = {X i } n i=1 , with n the total number of individuals. For a given prior distribution p(θ), the augmented posterior distribution of the model is given as p(θ, z|y, X) ∝ p(y, z|θ, X)p(θ). Albert and Chib (1993) were the first to propose the use of data augmentation (see Section 3.1.3 herein) for conducting Bayesian analysis of the multinomial probit model. The predictive distribution Consider now an individual s, with predictor values X s , whose choice behaviour we would like to predict. The predictive for individual s, can be written as p(y s |X s , y, X) = Θ zs p(y s |z s )p(z s |θ, X s )dz s z p(θ, z|y, X)dz dθ,(20) from which the predictive choice probabilities P r(y s = j) = p(j|X s , y, X), such as those in Figure 1, can be constructed. The specification and computation of the predictive distribution in (20) . Conditional on the draws for the latent utilities, sampling β from its full conditional is straightforward. Generating from the conditional distribution of Σ is nonstandard as the scale restrictions on Σ have to be taken into account. Scalable Bayesian prediction In addition to the challenges delineated above, it is difficult to scale p(y s |X s , y, X) to problems with large choice sets or a large number of observations. Recent advances in the computation of the predictive have focused on tackling the scalability issues in J and n, as we discuss below. When considering a full covariance matrix specification for Σ, the total number of parameters increases quadratically with J. For problems with large choice sets and small samples, this implies that the ratio of total number of parameters to total number of observations is large, making it difficult to construct accurate predictions. Loaiza-Maya and Nibbering (2022b) propose a spherical transformation of the covariance matrix of the latent utilities that imposes a parsimonious factor structure and a trace restriction. As a result, the total number of parameters grows only linearly with J. The authors demonstrate that this parsimonious structure leads to improved predictive performance over full covariance matrix specifications. Additionally, as noted above, the construction of the posterior predictive entails evaluation of the integral over the latent utilities z. Although MCMC is able to solve this integral, it does so by generating the utility vector for each individual from a multivariate truncated normal, which is a computationally costly exercise (McCulloch and Botev, 2017). This renders MCMC algorithms impractical for problems where a large n is considered. VB can be employed to tackle problems with large n. Adapting the generic descriptions of VB in Section 3.2.3 and Appendix A.7, the application of VB in this setting considers the class of approximating densities Q with elements q λ (θ, z) ∈ Q, indexed by the variational parameters λ. The exact augmented posterior is approximated by qλ(θ, z) with an optimal variational parameter equal tô λ = arg min λ∈Λ KL [q λ (θ, z)||p(θ, z|y, X)] ,(21) where KL denotes the Kullback-Leibler divergence. Then the variational predictive is constructed aŝ p λ (y s |X s , y, X) = Θ zs p(y s |z s )p(z s |θ, X s )dz s z qλ(θ, z)dz dθ. Calibration of the variational approximation requires a scale-identified expression for p(y, z|θ, X). To achieve this, Girolami and Rogers (2006) consider an identity matrix covariance structure, while Fasano and Durante (2022) fix Σ at predetermined values. Loaiza-Maya and Nibbering (2022a) propose a method for a multinomial probit model with a factor covariance structure. This method uses the hybrid variational approximation q λ (θ, z) = q λ (θ)p(z|y, θ, X) introduced by Loaiza-Maya et al. (2022). Electricity Pricing and Demand (Anastasios Panagiotelis) Forecasting in electricity markets is critical for efficient day-to-day operation of power grids, long-term planning of infrastructure and increasingly, at a disaggregated level, for the management of smart grids. The scope of this subsection will cover forecasting electricity prices, electricity load/demand and generation by source of power, primarily wind and solar. Hereafter these problems will collectively be referred to as 'electricity forecasting'. Motivations for electricity forecasting can be found in general reviews such as Weron (2014) however notwithstanding this, Bayesian methods have found success in the field. There are very few instances of Bayesian forecasting in electricity markets that predate the early 2000s, although we now cover some notable exceptions. Bunn (1980) consider the case of updating load forecasts in an online fashion by computing a Bayesian model average of load profiles of a cloudy and sunny day. Meanwhile, Bayesian VARs have been used by Gunel (1987), Beck and Solow (1994) and Joutz et al. (1995) to forecast energy demand, nuclear power generation and demand prices and consumption respectively. A Bayesian VAR shrinks autoregressive coefficients to either a random walk or white noise depending on whether data are stationary or non-stationary and was popularized in macroeconomics by Doan et al. (1984b) (see also Section 4.1). The performance of Bayesian VARs in early electricity forecasting applications is mixed; Beck and Solow (1994) find evidence in favour of Bayesian autoregression, Joutz et al. (1995) find that Bayesian VARs are effective for forecasting demand, but not price, while Gunel (1987) does not find any improvement at all from using Bayesian VARs rather than conventional ARIMA models. With the advent and popularization of MCMC methods, Bayesian forecasting has begun to find greater success in the field of electricity forecasting. In the literature of roughly the past two decades, there are three common major motivations for using Bayesian forecasting, two of which have antecedents in the earlier literature. The first, is the use of 'Bayesian models' 16 , which have now grown well beyond Bayesian VARs to include models with latent volatilities, models with a spatial dimension, and Bayesian neural networks. The second is the use of BMA for forecast combination. The third is the production of full probabilistic forecasts via Bayesian computation. These are now each discussed in turn. Bayesian models The structure inherent in many electricity forecasting problems provides a motivation for the innovative use of priors to improve forecasting accuracy. Although the early literature cited before found somewhat ambiguous results when comparing Bayesian VARs to classical alternatives, more recent work finds evidence in favour of a Bayesian approach; see Raviv et al. (2015) for point forecasts and Gianfreda et al. (2020) for both point and density forecasts. An important aspect of this work is the exploitation of the intraday nature of the data, since typically hourly prices are stacked in a VAR model. The intraday structure lends itself to priors that shrink parameters corresponding to consecutive hours of the day that are close to one another. An early application of this approach can be seen in Cottet and Smith (2003). Since electricity data are increasingly available not only at a high temporal frequency but also at a variables. Examples in electricity forecasting include a latent jump process for price spikes (Chan et al., 2014) and SV models (Smith, 2010;Kostrzewski and Kostrzewska, 2019). Also, in recent years, Bayesian analysis of machine learning models has become increasingly popular. This includes neural network models (Brusaferri et al., 2019;Ghayekhloo et al., 2019;Capone et al., 2020), where VB is typically used. Also, Bayesian regression trees (BART) have been applied to electricity forecasting by Nateghi et al. (2011) and Alipour et al. (2019), who find that they outperform non-Bayesian counterparts. Finally, there is an extensive literature on using Bayesian networks for forecasting in energy; see Adedipe et al. (2020) for a review of these methods in forecasting wind generation. Bayesian model averaging (BMA) As noted earlier, the importance of forecast combination is widely appreciated in the forecasting literature. Whilst, as highlighted in Section 3.3.3, many different Bayesian approaches to forecast combination have now been explored, BMA remains a very important method in the sphere of electricity forecasting. As described in Section 2.1, BMA uses posterior model probabilities as combination weights. Whenever the choice of model is parameterized, the predictive density has an interpretation as a forecast combination. Examples include Smith (2000) who combines forecasts from regression models that include different predictor sets, and Panagiotelis and Smith (2008) who average over models with different combinations of skew and symmetric marginal distributions. It is also common in the electricity forecasting literature to produce point forecasts from different models and then combine these using BMA as a post-processing step. This approach grew out of research combining ensembles of forecasts from numerical weather predictions (NWPs) (Raftery et al., 2005;Sloughter et al., 2010). In the NWP setting, forecasts are the outputs of deterministic physical models. Statistical models are then formed by assuming that for k = 1, . . . , K, p(y t |a k , b k , f k , σ 2 , M k ) ∼ N (a k +b k f k , σ 2 ), where f k is the k th NWP and a k , b k and σ 2 are additional parameters. These statistical models are then combined using the usual BMA machinery described by (3). Uncertainty over a k , b k and σ 2 is integrated out in the usual way, and there are no additional parameters since the f k are obtained deterministically. This approach has been used in energy forecasting by Coelho et al. (2006), who motivate forecasting rainfall as a input into forecasting generation from hydroelectric dams, and Du (2018) who use wind forecasts to predict generation from wind farms. The work of Raftery et al. (2005) has been subsequently extended to the case where the forecasts Probabilistic forecasting A common motivation for taking a Bayesian approach is the ease with which the computational machinery of MCMC or approximate methods produce a full predictive density rather than only point forecasts. Key operational decisions in electricity forecasting depend on quantities other than the predicted mean; see Nowotarski and Weron (2018) and references therein for discussion. While the importance of probabilistic forecasting is often highlighted in Bayesian papers it is not always the case that forecasts are evaluated in a way that assesses the quality of the full predictive distribution 18 . For example, often probabilistic forecasts are summarized by prediction intervals, and the empirical coverage of these intervals used as a means of checking model quality; for an early example see Pezzulli et al. (2006), and more recently Wang et al. (2017) and Kostrzewski and Kostrzewska (2019), where the latter show that Bayesian methods compare favourably to non-Bayesian alternatives for forecasting electricity prices. Kostrzewski However, the use of scoring rules and, hence, the explicit recognition of the distributional form of the forecasts, is becoming increasingly popular as a means of evaluating predictive distribution in both Bayesian and non-Bayesian electricity forecasting. The CRPS is particularly amenable to Bayesian inference since it is usually approximated using a Monte Carlo sample from the predictive density. For an early example of its use in Bayesian electricity forecasting see 17 The same point does not apply when combining ensembles from NWPs since the forecasting models are deterministic. 18 We note that in some cases, this is challenging; for example for long-run forecasts as in Da Silva et al. (2019). Panagiotelis and Smith (2008); for later examples, see Bracale and De Falco (2015), Brusaferri et al. (2019) and Gianfreda et al. (2020). Other scoring rules are less commonly used in the Bayesian electricity forecasting literature although Ohtsuka et al. (2010), where the log score is used, is a notable exception. In Summary Bayesian forecasting is underpinned by a single core principle: uncertainty about the future value of a random variable is expressed using a probability distribution that is, ultimately, conditioned only on observable, or known, information about the random variable. Nothing, surely, could be more natural as a way of framing the forecasting exercise, and the Bayesian approach to forecasting is -arguablyone of the most compelling features of the paradigm. 19 In large measure, the challenge has, potentially, been in the implementation of Bayesian forecasting: computing the expectation that defines the predictive distribution; most particularly when accessing (draws from) the posterior itself is difficult. And as models have become larger and more challenging, and as the data sets have grown 'bigger', this problem of accessing the exact posterior has only increased. However, as this review has demonstrated, the expansion of the forecasting problems being tackled has gone hand-in-hand with the development of new and improved computational methods designed expressly to access challenging posteriors, and in a reasonable computing time. Notably, when it comes to accurate forecasting, somewhat crude approximations of the posterior have been found to still yield accurate predictions; meaning that Bayesian forecasting remains viable for large and complex models for which approximate computation of posteriors is the only feasible approach. The more fundamental problem of model mis-specification can also be managed, by moving away from the conventional likelihood-based Bayesian up-dating and allowing forecast accuracy itself -and its link to the future decisions that depend on that accuracy -to drive the up-dating. This, in turn, ensures that forecasts are 'fit for purpose', despite the inevitable mis-specification of the forecasting model. Allied with the computational power that now drives the Bayesian engine, this ability to generalize the paradigm beyond its traditional links with the likelihood principle is a potent, if not yet fully realized, force in forecasting. Algorithm 5. Note that, for a given θ, the draws z (j)1:T ∼ i.i.d. p(·|θ), j = 1, . . . , m, are used to estimate µ(θ) and Σ(θ) as µ m (θ) = 1 m m j=1 η(z (j)1:T ) and Σ m (θ) = 1 m−1 m j=1 (η(z (j)1:T ) − µ m (θ))(η(z (j)1:T ) − µ m (θ)) ′ . The M draws of θ are used to produce an estimate of p(θ|η(y 1:T )) via kernel density methods. Algorithm 5 BSL-MCMC Algorithm for i = 1, . . . , M do Draw θ * ∼ q(θ|θ (i−1) ) Produce µ m (θ) and Σ m (θ) using j = 1, . . . , m independent model simulations at θ * Compute the synthetic likelihood L * = N [η(y); µ m (θ * ), Σ m (θ * )] and L (i−1) defined in a corresponding manner Compute the Metropolis-Hastings ratio: r = L * π(θ * )q(θ (i−1) |θ * ) L (i−1) π(θ (i−1) )q(θ * |θ (i−1) ) if U (0, 1) < r then Set θ (i) = θ * , µ m (θ (i) ) = µ m (θ * ) and Σ m (θ (i) ) = Σ m (θ * ) else Set θ i = θ (i−1) , µ m (θ (i) ) = µ m (θ (i−1) ) and Σ m (θ (i) ) = Σ m (θ (i−1) ) end if end for A.7 Variational Bayes (VB) VB seeks the best approximation to p(θ|y 1:T ) over a 'variational family' of densities Q, with generic element q(θ). Typically this proceeds by minimizing the Kullback-Leibler (KL) divergence between q(θ) and p(θ|y 1:T ), which produces the variational approximation as where KL [q(θ)|p(θ|y 1:T )] = E q [log(q(θ))] − E q [log(p(θ, y 1:T ))] + log(p(y 1:T )) and p(θ, y 1:T ) = p(y 1:T |θ)p(θ). Given that the unknown normalizing constant log(p(y 1:T )) in (23) does not depend on q, the (infeasible) optimization problem in (22) with the so-called evidence lower bound (ELBO) defined as: ELBO[q(θ)] := E q [log(p(θ, y 1:T ))] − E q [log(q(θ))]. The usefulness of VB is that, for certain models, p(y 1:T |θ), and certain choices of Q, the optimization problem in (24) can be solved efficiently using various numerical algorithms. Most notably, for problems in which θ, and possibly y 1:T also, are high-dimensional, the production of q * (θ) is much faster is a prior, rather than a posterior expectation does have implications for precise manner in which computation is implemented. (SeeArdia et al., 2012, and Llorente et al., 2021, for details). Such a treatment was the method of attack for large classes of models in the 1990s and 2000s. Relevant contributions here, which include specific treatments of the ubiquitous stochastic volatility (SV) model, are Polson et al. (1992), Jacquier et al. (1994), Shephard and Pitt (1997), Kim et al. (1998), Chib et al. (2002), Stroud et al. (2003), Chib et al. (2006), Strickland et al. (2006), Omori et al. (2007)andStrickland et al. (2008). The reviews ofFearnhead (2011) andGiordani et al. (2011) provide more detailed accounts and extensive referencing of this earlier literature. 6 (See also Appendix A.3.) and under appropriate regularity), a convergent MCMC chain can be constructed for any model, the exploration of a very highdimensional parameter space via an MCMC algorithm can be prohibitively slow(Tavaré et al., 1997;Rue et al., 2009;Braun and McAuliffe, 2010;Lintusaari et al., 2017;Betancourt, 2018;Johndrow et al., 2019). Hence, in models with a very large number of unknowns -including those with multiple sets of high-dimensional latent variables -the production of an accurate MCMC-based estimate of p(y T +1 |y 1:T ) in a practical amount of time, may not be possible. Finally, regarding 3) MCMC schemes require pointwise (i.e. for each y t ) evaluation of p(y 1:T |θ) at each draw of θ, thereby inducing an O(n) computational burden at each iteration in an MCMC chain. Such schemes can thus struggle when confronted with 'big data'(Bardenet et al., 2017). ); iii) various types of combinations of multiple chains(Jacob et al., 2011;Neal, 2011a;Neiswanger et al., 2013;Glynn and Rhee, 2014;Huber, 2016;Jacob et al., 2020); or iv) the use of ex-post variance reduction methods(Craiu and Meng, 2005;Douc and Robert, 2011;Owen, 2017;Baker et al., 2019). We refer the reader to Green et al.(2015), Robert et al. (2018) and Dunson and Johndrow(2019)for detailed reviews of modern developments in MCMC, and toJahan et al. (2020) for an overview of the way in which certain of the newer methods manage the problem of scale -in terms of either the unknowns or the data, or both. whilst INLA applies a series of nested Laplace approximations(Laplace, 1774;Tierney and Kadane, 1986;Tierney et al., 1989) to a high-dimensional latent Gaussian model to produce an approximation of p(θ|y 1:T ). Both VB and INLA exploit state-of-the-art optimization techniques, for the purpose of minimizing the 'distance' between p(θ|y 1:T ) and the variational approximation in the case of VB, and for the purpose of producing the mode of the high-dimensional vector of latent states in the case of INLA. The basic principles of VB and INLA are provided in Appendices A.7 and A.8 respectively. 2019, Lyddon et al., 2019, and Syring and Martin, 2019), Loaiza-Maya et al. (2021) define the generalized (or Gibbs) posterior: et al. (2013), Casarin et al. (2015a), Casarin et al. (2015b), Casarin et al. (2016), Pettenuzzo and Ravazzolo (2016), Aastveit et al. (2018), Bassetti et al. (2018), Baştürk et al. (2019) and Casarin et al. (2019). Once again adopting the language of Bernardo and Smith (2009), this literature seeks to move Bayesian predictive combinations beyond the M-closed world of BMA to the M-open the case of McAlinn and West (2019) and McAlinn et al. (2020), the synthesis function is taken to be the density of a (possibly multivariate) dynamic linear factor model. Many different priors have been used with VARs. Traditionally, natural conjugate priors in the Minnesota tradition were used since these allowed for analytical posterior and one-step-ahead predictive inference. Definitions of these priors and discussions of their properties are available in standard sources such as Koop and Korobilis (2010) and Dieppe et al. (2016). These priors are subjective and require the user to select prior hyperparameters, most importantly those relating to the strength of prior shrinkage. In recent years, a range of alternative priors have been proposed which are more automatic, requiring fewer subjective prior choices by the researcher. For instance, Giannone et al. ( 2015 ) 2015develop methods for estimating shrinkage parameters in conjugate priors, thus avoiding the need for their subjective elicitation. Chan (2022) also uses a conjugate prior and develops methods for selecting shrinkage parameters using a prior which relaxes some of the restrictive assumptions of the Minnesota prior. There are also a range of methods which automatically decide on the optimal degree of shrinkage for each VAR coefficient. These are the global-local shrinkage priors which are widely used with regressions and in machine learning applications and increasingly used with VARs. 11 Global-local shrinkage priors have the form ψ j controls local shrinkage since it is specific to the j th coefficient. The densities f 1 and f 2 are mixing densities and a large range of choices of them have been proposed. One choice leads to stochastic search variable selection, used with VARs inGeorge et al. (2008),Koop (2013a) andKorobilis (2013) and many others. Other choices lead to the Dirichlet-Laplace prior used with VARs byKastner and Huber (2021), or the normal-gamma and horseshoe priors used inHuber and Feldkircher (2019) andCross et al. (2020), and there are many others. Since these priors are Gaussian at the first layer of the hierarchy, textbook MCMC algorithms for all the VAR parameters can be easily implemented. 12 , Kon KamKing et al. (2019),Virbickaitė et al. (2020) andPesonen et al. (2022). ABC treatment of a conditional likelihood for a time series of financial returns, p(y 1:T |z 1:T , θ), that is unavailable in closed form is also investigated inCreel and Kristensen (2015),Martin et al. (2019) andChakraborty et al. (2022), withChakraborty et al. (2022) proposing a modularized version of ABC. (2015) predicting the interdependence between U.S. stocks with Bayesian time-varying quantile regressions; Geraci and Gnabo (2018) capturing and predicting the interconnectedness of financial institutions through Bayesian time-varying VARs; and Alexopoulos et al. (2022) modelling and predicting common jump factors in a large panel of financial returns. A potential drawback of the DPM model is that it neglects time dependence in the unknown distribu-tion. An important extension of the DPM prior is the hierarchical Dirichlet process of Teh et al. (2006), which allows for the construction of a prior for an infinite hidden Markov model (IHMM), which allows for time dependence in a flexible manner. The introduction of the beam sampler of Van Gael et al. (2008), which extends the slice sampler, renders conventional posterior sampling methods for finite-state Figure 1 : 1PredictedThe two most popular models used to predict choice behaviour are the multinomial logit and multinomial probit models. The multinomial logit model imposes the independence of irrelevant alternatives (IIA) property(McFadden, 1989), which means that it cannot capture general substitution patterns among choice alternatives. The IIA property of this model can be relaxed under certain assumptions by poses three key challenges. First, p(y s |z s ) requires a choice of base category. This choice affects the prior predictive choice probabilities, and hence the (posterior) predictive choice probabilities can be sensitive to the choice of base category; see Burgette and Nordheim (2012). Burgette et al. (2021) propose a symmetric prior specification to address this problem. The parameters θ are not identified under this prior, but this does not affect the predicted probabilities. Second, the parameters θ lack scale identification, as p(y i |z i ) = p(y i |cz i ) for any positive scalar c. Different solutions have been proposed to fix the scale, all based on a constraint on the specification of Σ. For instance, McCulloch et al. (2000) fix the first leading element of Σ to unity. This approach is sensitive to the ordering of the choice categories in the model. Burgette and Nordheim (2012) fix the trace of Σ, which is invariant to the way the choice categories enter the model. Third, the computation of p(y s |X s , y, X) involves the evaluation of the integrals over the latent utilities in z s and z. Since no analytical solution for these integrals is available, they are solved with MCMC sampling steps. The latent utility of each choice category is sampled from a univariate truncated normal, conditional on the latent utilities for all the other choice alternatives, for each individual (McCulloch and for price forecasting, Lindberg et al. (2019) for load forecasting, Antonanzas et al. (2016) for solar power forecasting and Giebel and Kariniotakis (2017) for wind power forecasting. These reviews indicate that the majority of work in electricity forecasting does not employ a Bayesian approach; high spatial resolution, there are further examples in the literature of using priors to exploit neighbourhood structure. Examples include Ohtsuka et al. (2010) who use spatial ARMA processes to predict electricity load in nine Japanese regions, and Gilanifar et al. (2019) who use spatio-temporal Gaussian processes to forecast residential-level electricity demand. Even where spatial information is unavailable, hierarchical models estimated using Bayesian methods have been used to produce disaggregate energy demand forecasts; examples can be found in Mori and Nakano (2014) and Wang et al. (2017) who use Gaussian processes, and Grillone et al. (2021) who use regression. Informative hierarchical priors have been used in instances where data sets are small in size, or unavailable; for example, Pezzulli et al.(2006) elicit priors for future trajectories of temperature in the winter using past observations, and Launay et al. (2015) elicit priors for the electricity demand of 'non-metered' households using data on 'metered' households. While the aforementioned examples take a Bayesian approach to exploit the use of priors in novel ways, another strain of the Bayesian forecasting literature is based on estimating models with latent f k are not the outputs of deterministic physical models but are point forecasts from statistical models, each with their own unknown parameters. For exampleNowotarski et al. (2014) adopt the approach ofRaftery et al. (2005) but where the f k are obtained from statistical time series models with parameters estimated using frequentist techniques. This approach is not fully Bayesian (despite being referred to asBMA in the literature), since although the model average integrates over the uncertainty in a k , b k and σ 2 it does not integrate over uncertainty in the parameters of the underlying time series models used to generate the point forecasts f k . 17 This approach is found to perform poorly relative to frequentist forecast combination schemes. In a similar vein, Hassan et al. (2015) and Raza et al. (2017) combine electricity load forecasts from different neural networks. and Kostrzewska (2019) also evaluate quantile forecasts using the pinball loss, as do Yang et al. (2019) and Sun et al. (2019) both for forecasting residential-level load (net of solar PV generation in the latter case). q(θ)|p(θ|y 1:T )] , , is replaced by the equivalent (and feasible) optimization problem:q * (θ) := arg max q(θ)∈Q {E q [log(p(θ, y 1:T ))] − E q [log(q(θ))]} , For example,Dongarra and Sullivan (2000) rank the 'Metropolis' algorithm proposed inMetropolis et al. as one of the 10 algorithms "with the greatest influence on the development and practice of science and engineering in the 20th century". 4 Any non-standard probability distribution can, in principle, be drawn from using ICDF sampling. The term 'Griddy Gibbs' sampling was first used byRitter and Tanner (1992) to refer to the use of ICDF sampling to draw from non-standard conditionals in a Gibbs scheme. Given that the method amounts to the use of numerical quadrature, it suffers from the curse of dimensionality, and is thus infeasible for drawing from anything other than very low-dimensional conditionals. SeeBauwens and Lubrano (1998) for the application of the Griddy-Gibbs sampler to a generalized autoregressive conditionally heteroscedastic (GARCH) model for financial returns.5 Moreover, and in contrast to IS, the requirement to find a well-matched proposal distribution is facilitated by the dimension reduction invoked by the breaking down of the high-dimensional joint posterior into the lower dimensional conditionals, before any proposal distribution needs to be specified. We also note here the work ofChib and Greenberg (1994), in which the state space representation of an autoregressive moving average (ARMA(p,q)) model(Harvey, 1981) was exploited, and the principle of data augmentation invoked, in order to enable a MH-within-Gibbs scheme to be applied. PMMH is actually a special case of the general pseudo-marginal MH technique (also sometimes denoted by the abbreviation 'PMMH'), in which a 'pseudo' likelihood, produced -in some manner or another -as an unbiased estimator of the true likelihood, is used within an MH algorithm. See, for example, the subsampling methods based on pseudo-marginal MCMC (Bardenet et al., 2017; Quiroz et al., 2018; Quiroz et al., 2019) used expressly to improve the performance of MCMC in the case of a large-dimensional y1:T (i.e. intractability type 3)). See Hall and Mitchell (2007),Ranjan and Gneiting (2010),Geweke and Amisano (2011) andGneiting and Ranjan (2013) for early contributions to the frequentist forecast combination literature, andWang et al. (2022) for a recent review. We note that whilst Geweke and Amisano(2011)is not explicitly Bayesian, in terms of estimating the optimal predictive combination, it provides important insights into the connection between the 'optimal linear pool' and BMA, and also uses Bayesian numerical methods in the production of some of the constituent forecast distributions. For an alternative combination approach based on dynamic weights and non-linear filtering, we refer toBillio et al. (2013), as cited earlier. Note that the Gaussianity assumption is not essential; mixtures of Gaussian distributions, for example, can be used to produce flexible error distributions if deemed necessary (see, for example,Clark et al., 2022a, andPrimiceri, 2022). They are also used with DFMs to select the number of factors. 12 In large VARs with global-local shrinkage priors, MCMC methods can nevertheless be very slow, with much faster VB methods developed inGefang et al. (2022). We note that whilst a time series model is constructed in the case of these two references, the (out-of-sample) prediction of option prices is across the cross section of strike prices and maturities. We also make note ofFulop and Li (2019) who exploit spot and options data to produce filtered estimates (as opposed to strictly out-of-sample predictions) of latent volatility and price jump intensity. The citation of Creel and Kristensen (2015), Martin et al. (2019), Vankov et al. (2019) and Müller and Uhl(2021)is relevant to this review, despite these references not having an explicit component on forecasting. By a 'Bayesian model' we generally mean a model with a prior and likelihood estimated by Bayesian inference. Bayesian methods for finding tuning parameters such as the automatic relevance determination (seeHippert and Taylor, 2010, for an example in electricity forecasting), and Bayesian optimisation lie beyond the scope of this section. The notation z1:T used in this section and in Section A.6 below is not to be confused with the use of z1:T to denote a vector of latent variables elsewhere in the paper. A Further Computational DetailsA.1 Gibbs samplingUnder the required regularity conditions (seeTierney, 1994)the Gibbs sampler yields a Markov chain with invariant distribution, p(θ|y 1:T ), via a transition kernel that is defined as the product of full conditional posteriors associated with the joint. For the case of θ partitioned into B mutually exclusive blocks, θ = (θ 1 , θ 2 , ..., θ b , ..., θ B ) ′ , the steps of the Gibbs algorithm are given in Algorithm 1.Algorithm 1 Gibbs Sampling AlgorithmSpecify an initial value θ (0) and partition the parameter set into B mutually exclusive blocks., |y 1:T ) end for end for Return a sample of draws from p(θ|y 1:T ).A.2 MH-within-Gibbs samplingIn Algorithm 2 we provide the generic steps of the so-called 'MH-within-Gibbs' algorithm, for the case of θ partitioned into B mutually exclusive blocks, θ = (θ 1 , θ 2 , ..., θ b , ..., θ B ) ′ . The symbol p * b represents (the ordinate of) a kernel of the corresponding conditional p b (·|·).Algorithm 2 MH-within-Gibbs AlgorithmSpecify an initial value θ (0) , a partition of the parameter set into B mutually exclusive blocks, and a proposal distribution q b (θ b |y 1:T ) for b ∈ {1, . . . , B}.end if end for end for Return a sample of draws from p(θ|y 1:T ).The b th candidate density q b (θ b |y 1:T ) may be chosen to deliberately target the form of the b th condi-, y 1:T ), in which case the algorithm may be referred toA.5 ABC based on summary statisticsThe simplest (accept/reject) form of the ABC algorithm, as based on a chosen vector of summaries, η(y 1:T ), proceeds via the steps in Algorithm 4, with the accepted draws of θ used to produce an estimate of p(θ|η(y 1:T )), via kernel density methods. That is, ABC targets only the partial posterior, p ε (θ|η(y 1:T )). This partial posterior is equivalent to p(θ|y 1:T ) if and only if η(y 1:T ) is sufficient for conducting inference on θ, and for ε → 0. Clearly, the very problems for which ABC is required imply that sufficient statistics are not available, and the requirement that ε → 0 is infeasible in practice; so inference via ABC is only ever intrinsically approximate. 20 A.6 BSL based on summary statistics BSL mimics ABC in targeting a posterior for θ that conditions on a vector of summaries η(y 1:T ), rather than the full data set y 1:T ; however the summaries play a different role in the algorithm. Once again with reference to the simplest version of the algorithm, the steps of the BSL-MCMC algorithm are as given in Algorithm 3 PMMH AlgorithmStep 1: Initialization, i = 0 (a) Set θ (0) arbitrarily and (b) Run an SMC algorithm targeting p(z 1:T |y 1:T , θ (0) ), sample z1:T ∼p(z 1:T |y 1:T , θ (0) ) and let p(y 1:T |θ (0) ) denote the marginal likelihood estimate.Step 2: for i = 1, . . . , M do (a) Draw θ c ∼ q(θ|y 1:T , θ (i−1) ), (b) Run an SMC algorithm targeting p(z 1:T |y 1:T , θ c ), sample z c 1:T ∼p(z 1:T |y 1:T , θ c ) and let p(y 1:T |θ c ) denote the marginal likelihood estimate.(c) Compute the Metropolis-Hastings ratio:1:T = z c 1:T ,p(y 1:T |θ (i) ) =p(y 1:T |θ c ) elseSet θ (i) = θ (i−1) , z1:T = z (i−1) 1:T ,p(y 1:T |θ (i) ) =p(y 1:T |θ (i−1) ) end if end for Return a sample of draws from p(θ|y).Algorithm 4 Accept/Reject ABC Algorithm Based on Summary Statisticsfor i = 1, . . . , M do Simulate θ (i) , i = 1, 2, ..., M , from p(θ), and artificial data z (i) 1:T from p(·|θ (i) ); Accept θ (i) if d{η(z (i) 1:T ), η(y 1:T )} ≤ ε, where d{·, ·} denotes a generic metric and ε > 0 a pre-specified tolerance parameter. end for Quantifying time-varying forecast uncertainty and risk for the real price of oil. 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MH-within-Gibbs sampling in state space models The application of MH-within-Gibbs sampling within a state space setting is qualitatively the same as described in Algorithm 2, except that the joint set of unknowns is augmented to (θ, z 1:T ), and decisions about partitioning need to be made for both θ|z 1:T and z 1:T |θ. Decisions about the 'blocking' of z 1:T are particularly important, given both the dimension of z 1:T and the time-series dependence embedded in the state process, as are matters of parameterizing the state space model. We refer the reader to. A Strickland, Shephard and Pittfor) for illustrations of state blocking in which the block sizes are selected randomly. and to Frühwirth-SchnatterA.3 MH-within-Gibbs sampling in state space models The application of MH-within-Gibbs sampling within a state space setting is qualitatively the same as described in Algorithm 2, except that the joint set of unknowns is augmented to (θ, z 1:T ), and decisions about partitioning need to be made for both θ|z 1:T and z 1:T |θ. Decisions about the 'blocking' of z 1:T are particularly important, given both the dimension of z 1:T and the time-series dependence embedded in the state process, as are matters of parameterizing the state space model. We refer the reader to: Shephard and Pitt (1997) and Strickland et al. (2006) for illustrations of state blocking in which the block sizes are selected randomly; and to Frühwirth-Schnatter (2004) and Strickland et al. (2008) for Frühwirth-Schnatter, 1994) to be used to produce a candidate draw of (any particular block of) z 1:T , conditional on θ. As noted in Section 3.2.2 (and in the review. T (e.G, Kim, Early Bayesian treatments of non-linear state space models often exploited a linear Gaussian approximation at some point, for the purpose of defining candidate densities for. Andrieu et al.Braun and McAuliffe1more recent approaches to such models have exploited PMMH principles instead. Algorithm 3 reproduces the algorithm in. Section 2.4.2, adapted slightly to match the notation of the current paper. To simplify the exposition, the algorithm of p(θ|y 1:T ) (and any associated quantities. including predictives) via simulationEarly Bayesian treatments of non-linear state space models often exploited a linear Gaussian approxima- tion at some point, for the purpose of defining candidate densities for (blocks of) z 1:T (e.g., Kim et al., 1998; Stroud et al., 2003; Strickland et al., 2006), thereby enabling a Kalman filter-based 'forward fil- tering, backward sampling' algorithm (Carter and Kohn, 1994; Frühwirth-Schnatter, 1994) to be used to produce a candidate draw of (any particular block of) z 1:T , conditional on θ. As noted in Section 3.2.2 (and in the review, Giordani et al., 2011), more recent approaches to such models have exploited PMMH principles instead. Algorithm 3 reproduces the algorithm in Andrieu et al. (2011), Section 2.4.2, adapted slightly to match the notation of the current paper. To simplify the exposition, the algorithm of p(θ|y 1:T ) (and any associated quanti- ties, including predictives) via simulation (Braun and McAuliffe, 2010; Kabisa et al., 2016; Wand, 2017; The relationship between (23) and (25), plus the fact that KL. Korobilis Koop, also means ELBO[q * (θ)] is a lower bound on the logarithm of the 'evidence', or marginal likelihood, p(y 1:TKoop and Korobilis, 2018). The relationship between (23) and (25), plus the fact that KL[·] ≥ 0, also means ELBO[q * (θ)] is a lower bound on the logarithm of the 'evidence', or marginal likelihood, p(y 1:T ); Different VB methods are defined by both the choice of Q and the manner in which the optimization is implemented, and we refer the reader to. Blei, for reviews, including algorithmic details for specific VB methods. Different VB methods are defined by both the choice of Q and the manner in which the optimization is implemented, and we refer the reader to Ormerod and Wand (2010), Blei et al. (2017), and Zhang et al. (2018), for reviews, including algorithmic details for specific VB methods. A.8 Integrated nested Laplace approximation (INLA). A.8 Integrated nested Laplace approximation (INLA) adapted the very early approximation method of Laplace (1774) to approximate posteriors (and associated quantities) in the latent Gaussian model class, which encompasses a large range of -potentially high-dimensional -models, including the non-Gaussian state space models that feature heavily in economics and finance. In brief, Rue et al. use a series of nested Laplace approximations, allied with low-dimensional numerical integration, denoting their method by integrated nested Laplace approximation (INLA) as a result. As with VB, INLA eschews simulation for optimization, exploiting bespoke numerical algorithms designed for the specific (albeit broad) model class. We refer the reader to Rue. Rue, Martino and Rieblerand Wood (2019) for implementation detailsRue et al. (2009) adapted the very early approximation method of Laplace (1774) to approximate pos- teriors (and associated quantities) in the latent Gaussian model class, which encompasses a large range of -potentially high-dimensional -models, including the non-Gaussian state space models that feature heavily in economics and finance. In brief, Rue et al. use a series of nested Laplace approximations, allied with low-dimensional numerical integration, denoting their method by integrated nested Laplace approximation (INLA) as a result. As with VB, INLA eschews simulation for optimization, exploiting bespoke numerical algorithms designed for the specific (albeit broad) model class. We refer the reader to Rue et al. (2009), Rue et al. (2017), Martino and Riebler (2019), and Wood (2019) for implementation details.

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{'abstract': 'The Bayesian statistical paradigm provides a principled and coherent approach to probabilistic forecasting. Uncertainty about all unknowns that characterize any forecasting problem -model, parameters, latent states -is factored into the forecast distribution, with forecasts conditioned only on what is known or observed. Allied with the elegance of the method, Bayesian forecasting is now underpinned by the burgeoning field of Bayesian computation, which enables Bayesian forecasts to be produced for virtually any problem, no matter how large, or complex. The current state of play in Bayesian forecasting is the subject of this review. The aim is to provide readers with an overview of modern approaches to the field, set in some historical context. Whilst our primary focus is on applications in the fields of economics and finance, and their allied disciplines, sufficient general details about implementation are provided to aid and inform all investigators.', 'arxivid': '2212.03471', 'author': ['Gael M Martin ', 'David T Frazier ', 'Worapree Maneesoonthorn ', 'Rubén Loaiza-Maya ', 'Florian Huber ', 'Gary Koop ', 'Didier NibberingJohn Maheu ', 'Anastasios Panagiotelis ', 'Gael M Martin ', '\nMonash University\nAustralia\n', '\nMonash University\nAustralia; Worapreee (Ole) Maneesoonthorn\n', '\nUniversity of Melbourne\nAustralia; Rubén Loaiza-Maya\n', '\nMonash University\nAustralia; Florian Huber\n', '\nGary Koop\nUniversity of Salzburg\nAustria\n', '\nUniversity of Strathclyde\nUK\n', '\nJohn Maheu\nMcMaster University\nCanada; Anastasios Panagiotelis\n', '\nDidier Nibbering\nUniversity of Sydney\nAustralia\n', '\nMonash University\nAustralia\n'], 'authoraffiliation': ['Monash University\nAustralia', 'Monash University\nAustralia; Worapreee (Ole) Maneesoonthorn', 'University of Melbourne\nAustralia; Rubén Loaiza-Maya', 'Monash University\nAustralia; Florian Huber', 'Gary Koop\nUniversity of Salzburg\nAustria', 'University of Strathclyde\nUK', 'John Maheu\nMcMaster University\nCanada; Anastasios Panagiotelis', 'Didier Nibbering\nUniversity of Sydney\nAustralia', 'Monash University\nAustralia'], 'corpusid': 254366353, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 65679, 'n_tokens_neox': 55574, 'n_words': 32767, 'pdfsha': 'ff2c392cbc42d601e8712a12d3324e19a3c21307', 'pdfurls': ['https://export.arxiv.org/pdf/2212.03471v1.pdf'], 'title': ['Bayesian Forecasting in the 21st Century: A Modern Review', 'Bayesian Forecasting in the 21st Century: A Modern Review'], 'venue': []}

arxiv

Numerical Methods for Detecting Symmetries and Commutant Algebras Sanjay Moudgalya Department of Physics Institute for Quantum Information and Matter California Institute of Technology 91125PasadenaCaliforniaUSA Walter Burke Institute for Theoretical Physics California Institute of Technology 91125PasadenaCaliforniaUSA Olexei I Motrunich Department of Physics Institute for Quantum Information and Matter California Institute of Technology 91125PasadenaCaliforniaUSA Numerical Methods for Detecting Symmetries and Commutant Algebras (Dated: June 5, 2023) For families of Hamiltonians defined by parts that are local, the most general definition of a symmetry algebra is the commutant algebra, i.e., the algebra of operators that commute with each local part. Thinking about symmetry algebras as commutant algebras allows for the treatment of conventional symmetries and unconventional symmetries (e.g., those responsible for weak ergodicity breaking phenomena) on equal algebraic footing. In this work, we discuss two methods for numerically constructing this commutant algebra starting from a family of Hamiltonians. First, we use the equivalence of this problem to that of simultaneous block-diagonalization of a given set of local operators, and discuss a probabilistic method that has been found to work with probability 1 for both Abelian and non-Abelian symmetries or commutant algebras. Second, we map this problem onto the problem of determining frustration-free ground states of certain Hamiltonians, and we use ideas from tensor network algorithms to efficiently solve this problem in one dimension. These numerical methods are useful in detecting standard and non-standard conserved quantities in families of Hamiltonians, which includes examples of regular symmetries, Hilbert space fragmentation, and quantum many-body scars, and we show many such examples. In addition, they are necessary for verifying several conjectures on the structure of the commutant algebras in these cases, which we have put forward in earlier works[1][2][3]. Finally, we also discuss similar methods for the inverse problem of determining local operators with a given symmetry or commutant algebra, which connects to existing methods in the literature. A special case of this construction reduces to well-known "Eigenstate to Hamiltonian" methods for constructing Hermitian local operators that have a given state as an eigenstate. The study of symmetries is at the heart of many areas of physics. For example, understanding of symmetries leads to classifications of various phases of matter and transitions between them [4][5][6], definitions of Gibbs ensembles that describe thermodynamics of systems [7,8], etc. Hence, given the Hamiltonian for a quantum system, determining its symmetries is of utmost importance and is usually the first step in approaching any problem. Most symmetries studied in quantum many-body physics, such as Z 2 parity symmetry, U (1) particle number conservation, or SU (2) spin conservation, are examples of "onsite" symmetries represented by global unitary operators that are tensor products of single-site unitary operators that form representations of the corresponding groups; and symmetries of this form are usually easy to "guess" by quick direct inspection of the Hamiltonian. However, a relatively recent realization is that not all symmetries that appear in natural Hamiltonians are of this type, and the exploration beyond such "conventional" on-site global symmetries has only recently been initiated in various contexts [1][2][3][9][10][11][12][13][14][15]. Particularly in the context of dynamics, invoking such unconventional symmetries has also been shown to be necessary to understand phenomena of weak ergodicity breaking [16][17][18][19][20] that have now been observed in multiple experiments [21][22][23][24][25]. The study of unconventional symmetries first requires a clear general definition of a symmetry, which is not immediately obvious. Allowing arbitrary operators that commute with the given Hamiltonian to be valid conserved quantities implies that any Hamiltonian has exponentially many conserved quantities, namely, its eigenstate projectors are all trivially conserved. In [1][2][3] we introduced and developed one framework that provides a non-trivial definition of symmetries and conserved quantities, which naturally captures both conventional symmetries in standard Hamiltonians [2] as well as unconventional symmetries that explain the weak ergodicity breaking phenomena of quantum many-body scars [3] and Hilbert space fragmentation [1]. In particular, we focus on families of Hamiltonians that are composed of local terms, and the bond or local algebra A generated by the local terms [1][2][3]26]. The commutant algebra C is the centralizer of the local algebra, i.e., the algebra of all conserved quantities of the said family of Hamiltonians, which we will also refer to as the symmetry algebra. Hence we can think of any symmetry as being characterized by a pair of algebras (A, C), which are simply the algebras of symmetric operators and the symmetry algebra respectively. This definition of conserved quantities starting from an arbitrary local algebra A removes the restriction to onsite unitary and other conventional symmetries, since the commutant C can include non-on-site or even non-local operators, and moreover it need not have any simple underlying group structure, as illustrated through several examples in [1,3]. In addition, this language allows a systematic counting of the number and the sizes of dynamically disconnected blocks or Krylov subspaces in terms of the dimensions of the commutant, or more precisely the dimensions of the irreducible representations of the local algebra and the commutant. This in turn leads to a broad classification of systems in terms of the scaling of the number of Krylov subspaces with system size [1] (or equivalently their Coherence Generating Power [27,28]), as well as to a precise definition of Hilbert space fragmentation [1]. Moreover, a complete determination of the full commutant C corresponding to a local algebra A enables an exhaustive characterization of all symmetric local operators. That is, the von Neumann Double Commutant Theorem (DCT) guarantees that any operator that commutes with all operators in C should be in the local algebra A, hence it should be expressible in terms of the known local generators of A, which enables systematic construction and characterization of all local operators with a given symmetry. We demonstrated this principle in action by constraining the forms of operators symmetric under several regular symmetries [2], and also by constructing local Hamiltonians that possess some set of Quantum Many-Body Scars [3]. However, unlike simple on-site symmetries, determining the full commutant C corresponding to a local algebra A or determining the dimensions of their irreducible representations is far from straightforward in practice. Moreover, we need these properties for many of the results we discuss in previous works, e.g., the Hilbert space decomposition and the application of the DCT require an explicit proof that a particular set of operators generates the full commutant, and not just some subalgebra of the true commutant. As evident from examples in [2,3], we were able to analytically establish the commutants only in some simple cases, and in many cases we also needed to invoke some reasonable assumptions. Hence in this work, we discuss some numerical methods necessary to construct the commutant C corresponding to a local algebra A, which can either be used to determine and numerically construct operators in C, or to quickly verify key properties, e.g., the dimension of the C. One approach we discuss is to view this as problem of simultaneous block diagonalization of the (noncommuting) generators of the local algebra A. The problem of simultaneous block diagonalization appears in several other contexts [29][30][31][32][33][34][35][36], notably in the study of "noise commutants" and "noiseless subsystems" in the quantum error correction literature [29][30][31]. However, some of these methods are not always directly applicable or completely general, e.g., the methods proposed to detect symmetries in [32,33] only work for Abelian symmetries, hence we build upon them and adapt them to the quantum many-body problem we have in hand, including with non-Abelian symmetries. In another approach, we also exploit the locality of the generators of A and map the problem of finding the commutant to determining the ground state of a frustrationfree Hamiltonian, which is a superoperator acting on the space of all operators. This method is applicable is any dimension, but the brute force implementation is computationally more expensive than the previous method, since it required the diagonalization of a superoperator. However, for one-dimensional systems, this mapping enables us to use ideas from Matrix Product State (MPS) algorithms to obtain an efficient method to determine the commutant C, building on methods previously used for related problems [37][38][39]. Finally, for the sake of completeness, we also discuss the "inverse" of our method, which constructs local sym-metric operators corresponding to a given commutant algebra. Remarkably, this coincides with and also unifies many of the existing methods used in the literature for constructing local operators with a given symmetry [32,40] or those that possess a particular eigenstate [41][42][43][44][45][46][47][48][49]. Since such local operators are in the algebra A corresponding to a given commutant C, such methods are also useful in determining a set of local generators of A. In addition, these methods are useful in systematically constructing two distinct types of symmetric operators, which we referred to as type I and type II symmetric operators in an earlier work [3], and we discuss such applications. This paper is organized as follows. In Sec. II, we briefly review the concepts of bond, local, and commutant algebras and Hilbert space decomposition that we use in the rest of this work. In Sec. III, we discuss a method for simultaneous block diagonalization of operators in the local algebra and using it to construct operators in the commutant. In Sec. IV, we discuss a method for constructing the commutant as a tensor network, and we illustrate efficient method to obtain the commutant in one dimension. Finally, in Sec. V, we discuss numerical methods that construct local operators in the local algebra given its commutant, and we conclude with open questions in Sec. VI. II. BOND, LOCAL, AND COMMUTANT ALGEBRAS We now briefly review the concepts of local and commutant algebras required for this work, and we refer to our previous works [1][2][3] for a detailed discussion of their properties. Given a set of generically non-commuting operators { H α } on a Hilbert space H, we can define its commutant to be the set of all operators O that satisfy [ H α , O] = 0 ∀α.(1) These operators O form an associative algebra C, which we refer to as the commutant algebra, which can also be viewed as the centralizer of the algebra generated by the operators { H α }, denoted by A := ⟨⟨{ H α }⟩⟩, where the notation implicitly assumes inclusion of the identity operator 1 and also closure under Hermitian conjugation. A and C are hence examples of von Neumann algebras [50][51][52], and are centralizers of each other as a consequence of the DCT. As a consequence, the centers of the algebras A and C (i.e., the subalgebra that commutes with all the elements in the algebra) coincide, and can be written as Z = A ∩ C; and when C or A is Abelian, we obtain that Z = C ⊆ A or Z = A ⊆ C respectively. In quantum matter applications, we are interested in cases where the operators { H α } are local, which can refer to either strictly local operators with a support over a few nearby sites, or extensive local operators that are sums of strictly local terms throughout the system. Consequently, we refer to the algebra A generated by these terms as a bond algebra if { H α } contains only strictly local terms, or more generally as a local algebra if it contains some extensive local terms. The commutant algebra C is then naturally the symmetry algebra of families of Hamiltonians [1,2] of the form H = α J α H α ,(2) where {J α } is an arbitrary set of coefficients. As we discuss in other works, the language of commutant algebras can be used to understand a variety of phenomena such as Hilbert space fragmentation [1], conventional symmetries [2], and quantum many-body scars [3]. In all of these cases, C and A are respectively the "symmetry" algebra (i.e., all conserved quantities) and the algebra of "symmetric" operators (i.e., all operators that commute with the conserved quantities in C). Note that the commutant C should be interpreted as the symmetry algebra for generic families of Hamiltonians constructed from the set { H α }. We usually start with a set { H α } that is "homogeneously" distributed over the lattice, which is physically reasonable and captures physically relevant symmetries. Breaking homogeneity and excluding some operators from the set might lead to a larger commutant, which, while mathematically correct, is not physically relevant. For example, in some cases this exclusion might effectively "split" the system into some commuting parts, giving rise to several "unwanted" symmetries. However, with judicious choices, it is easy to avoid such situations. An important property that we heavily use in this work is the decomposition of the Hilbert space as a consequence of the existence of algebras A and C that are centralizers of each other [1][2][3][53][54][55][56], given by H = λ H (A) λ ⊗ H (C) λ ,(3) where H (A) λ and H (C) λ respectively denote D λ -and d λdimensional irreps of A and C. Equation (3) can be simply viewed as a tensored basis in which all the operators in A are simultaneously (maximally) block-diagonal. Operationally this means that any operators h A ∈ A, h C ∈ C, and h Z ∈ Z can be unitarily transformed as W † h A W = λ (M λ ( h A ) ⊗ 1 d λ ),(4)W † h C W = λ (1 D λ ⊗ N λ ( h C )),(5)W † h Z W = λ (c λ ( h Z )1 D λ ⊗ 1 d λ ),(6) where M λ ( h A ) and N λ ( h C ) are some D λ -dimensional and d λ -dimensional matrices respectively, c λ ( h Z ) is a cnumber, and W is a fixed unitary matrix. Equation (4) precisely denotes the simultaneous block-diagonalization of the operators in (including the generators of) A, which is equivalent to the simultaneously block-diagonalization of operators in (including the generators of) C, shown in Eq. (5). Since the Hamiltonians we are interested in belong to the local algebra A, this decomposition can be used to precisely define dynamically disconnected quantum number sectors or "Krylov subspaces" of the Hamiltonian [1]. In particular, there are blocks labelled by different λ's in Eq. (3) which can be uniquely specified by eigenvalues under a minimal set of generators of the center Z. Further, for each λ, there are d λ number of identical D λdimensional Krylov subspaces, which can be uniquely labelled by eigenvalues under a minimal set of generators of any maximal Abelian subalgebra of C [1]. It is also possible that D λ = 1 for some λ, which correspond to "singlets" of the algebra A [2], i.e., simultaneous eigenstates of all the operators in the algebra A, including the family of Hamiltonians we are interested in. The singlets can either be degenerate or non-degenerate depending on whether the corresponding d λ is greater than 1 or equal to 1 respectively, and they play an important role in the study of quantum many-body scars [3]. All singlets have the property that any "ket-bra" operator of the form |ψ λ,α ⟩⟨ψ λ,β | for any two degenerate singlets |ψ λ,α ⟩ and |ψ λ,β ⟩, not necessarily distinct, are a part of the commutant algebra C. With this background, in the following sections, we address the following question: Given a family of Hamiltonians of the form of Eq. (2) or a local algebra A = ⟨⟨{ H α }⟩⟩, how does one numerically determine the exhaustive list or number of operators in its commutant C, construct the decomposition of Eq. (4), and determine the associated dimensions {D λ } and {d λ }? We discuss two numerical methods that answer (parts of) this question, and the best method depends on the system being studied. III. SIMULTANEOUS BLOCK DIAGONALIZATION We first discuss a method to determine the blocks or Krylov subspaces of a given family of Hamiltonians by simultaneously block-diagonalizing the operators { H α }. This effectively implements the unitary transformation of Eqs. (4)-(6), which gives a direct access to the dimensions {D λ } and {d λ } and also to the operators in the commutant C. Similar methods have been used in the literature to "detect" symmetries in one-parameter families of Hamiltonians [32] or unitary operators [33], to construct noise commutants in the context of quantum error correction [29][30][31], or more generally, to simultaneously block-diagonalize certain sets of operators [34][35][36]. To determine the blocks, it is typically sufficient to work with two randomly chosen Hermitian operators from the family (i.e., from the algebra A = ⟨⟨{ H α }⟩⟩) which we refer to as H (1) = α J (1) α H α , H (2) = α J (2) α H α .(7) The rationale for working with just two operators is that these two "randomly chosen" operators typically generate the full algebra A, i.e., A = ⟨⟨H (1) , H (2) ⟩⟩. However, we emphasize that H α here must be "generic enough" for this to happen and for the procedure we outline below to work flawlessly, and we do not want them to have any other degeneracies apart from those due to the structure in Eq. (4). An example of such "accidental" degeneracies occurs when all the { H α } are non-interacting particlehole symmetric terms (e.g., those shown in #2 in Tab. II or III in [2]), in which case H (1) and H (2) have additional degeneracies due to particle-hole symmetry in the single-particle spectrum, which are not explained by the commutant language. We can circumvent such issues by including ( We start with extracting operators in the center Z, and to do this, it is sufficient to identify blocks labelled by different λ. It is clear from Eq. (6) that the projectors onto these blocks span the center Z. Since all operators in the center Z commute with H (1) , the eigenstates of H (1) can be labelled by (i.e., have definite) eigenvalues under operators in (or generators of) Z. 1 Since all operators in Z also commute with H (2) , its matrix elements between eigenstates of H (1) that differ in eigenvalues under some operators in Z vanish. Using the form of the operators in Z of Eq. (6), we can conclude that the matrix of H (2) in the eigenbasis of H (1) is certainly block-diagonal with blocks of sizes given by {D λ d λ }, which are all labelled by different λ, but this does not rule out the existence of smaller blocks within each of these blocks. Nevertheless, for generic choices of H (1) and H (2) , we can rule out the existence of smaller blocks within blocks in H (2) that are two-dimensional or more. As evident 1 Since we assume that the generic (random) H (1) does not show any "accidental" degeneracies, i.e., the corresponding matrices {M λ (H (1) )} are some generic matrices unrelated to each other for different λ's, any set of degenerate eigenvectors of H (1) must belong to precisely one D λ d λ -dimensional block λ and hence these states must be eigenvectors of all operators in Z. from Eq. (4), when restricted to a block labelled by λ, there exists a unitary operator that transforms H (1) and (1) ) and M (2) in the eigenbasis of H (1) restricted to the block labelled by λ) is not expected to have any smaller block-diagonal structure, allowing us to determine these blocks by expressing H (2) in the eigenbasis of H (1) . H (2) into M (1) D λ ⊗ 1 d λ and M (2) D λ ⊗ 1 d λ respectively [where M (1) D λ := M λ (HD λ ⊗1 d λ in a "random" eigenbasis 2 of M (1) D λ ⊗ 1 d λ (hence the matrix of H On the other hand, this does not identify the blocks where D λ = 1 (i.e., blocks composed of singlets of the algebra A), since both H (1) and H (2) restricted to such blocks are proportional to the identity (1 D λ =1 ⊗ 1 d λ ) = 1 d λ , hence H (2) is diagonal in the eigenbasis of H (1) . However, the singlet blocks labelled by different λ are generically non-degenerate under eigenvalues of H (1) and H (2) , hence each such degenerate subspace of H (1) and H (2) corresponds to a block labelled by λ consisting of d λ degenerate singlets of A. This completes the identification of the blocks labelled by distinct λ in Eq. (3), and an orthogonal basis (with respect to the Frobenius inner product) for the center Z is given by the projectors onto these blocks. When C is Abelian, these projectors span the full commutant since C = Z. Hence this procedure is sufficient for constructing Abelian commutants and identifying a subset of Abelian symmetries of a given family of systems, which was demonstrated in [32,33]. B. Extracting {D λ } and {d λ } Once the blocks labelled by λ are identified, the dimensions {D λ } and {d λ } can directly be extracted as follows. For random choices of the coefficients, the eigenvalues of H (1) and H (2) restricted to a block labelled by λ appear in multiplets of degeneracies given by {d λ }, since they can both be expressed as shown in Eq. (4). These multiplicities of the eigenvalues of H (2) within each D λ d λ -dimensional block can be used to determine d λ corresponding to that block, which can in turn be used to deduce D λ . This allows us to completely determine the dimensions {D λ } and {d λ }, starting from two generic choices of operators H (1) and H (2) . Quantities such as the number of Krylov subspaces, the dimensions of (i.e., 2 By this we mean that eigenvectors within degenerate subspaces are "randomly" chosen. In practice, this is not guaranteed to be the case for black-box eigensolvers, i.e., the eigenvectors within degenerate subspaces might have some hidden structure (e.g., they are sometimes organized according to the total spin in the computational basis). Hence it is better to explicitly act with a random unitary to randomize the eigenvectors within each degenerate subspace to ensure that this method works. number of linearly independent operators in) the commutant and the local algebra can then be computed directly: they are given by K = λ d λ , dim(C) = λ d 2 λ , and dim(A) = λ D 2 λ respectively [1]. Note that while this extraction of the data {D λ }, {d λ } works in general, we do not yet have the information about the basis that gives the subblock-diagonal structure within each block λ, Eqs. (4)-(5). C. Extracting C for Non-Abelian Commutants While the method described in Sec. III A completes the simultaneous block diagonalization and the construction of the full commutant when C is Abelian, there are additional operators in the commutant when C is non-Abelian, and more work is needed. Let us denote the tensored basis inside the block λ implied in the Hilbert space decompositions in Eqs. (3)-(6) as {|ψ αβ ⟩}, 1 ≤ α ≤ D λ , 1 ≤ β ≤ d λ . For a D λ -dimensional Krylov subspace K β = span α {|ψ αβ ⟩}, we can construct the projectors onto the Krylov subspaces Π β,β := α |ψ αβ ⟩⟨ψ αβ |. In addition, for degenerate Krylov subspaces K β and K β ′ for β ̸ = β ′ [i.e., distinct blocks labelled by the same λ in Eq. (4)], we can construct the operators Π β,β ′ := α |ψ αβ ⟩⟨ψ αβ ′ |. It is then easy to show that the operators {Π β,β ′ } span the full commutant [1]. In App. A we discuss a method for constructing the operators of the commutant using the matrix elements of H (2) in the eigenbasis of H (1) restricted to a block labelled by λ with D λ ≥ 2. In particular, we use the fact that within such a block λ, H (1) and H (2) can be unitarily transformed into M D λ for a random choice of H (1) . This allows us to express the operators in the commutant restricted to the block λ in terms of known quantities [see Eq. (A14)] and hence construct the full commutant. For singlet blocks with D λ = 1, the operators in the commutant are simply the "ket-bra" operators of the degenerate singlets (e.g., |ψ β ⟩⟨ψ β ′ | for degenerate singlets |ψ β ⟩ and |ψ β ′ ⟩) which can be constructed directly from the corresponding eigenvectors of H (2) (which are also eigenvectors of H (1) ). Note that this method requires a full exact diagonalization of H (1) , hence its time complexity grows exponentially with system size [as O(d 3L loc ) for a system of size L and local Hilbert space dimension d loc ], and its application is practical only for small system sizes. D. Examples We now discuss some examples of this method being applied to extract values of {D λ } and {d λ } in various types of systems. We separately consider models with conventional symmetries, those with Hilbert space fragmentation, and those with QMBS, and we depict all the results in Figs. 1(a)-(c). [57,58] as degenerate QMBS [3]. A few one-dimensional blocks along with one outstanding large block indicate the presence of QMBS. Note that for the sake of clarity any "degeneracies" in (D λ , d λ ) have been resolved by introducing a small horizontal separation. D λ (b) L =4 L =5 L =6 L =7 L =8 2 4 6 d λ 10 0 10 1 10 2 10 3 10 4 D λ (c) L =4 L =5 L =6 L =7 L =8 Conventional Symmetries We start with an example of conventional symmetries in the case of the family of one-dimensional spin-1/2 Heisenberg models, given by the family of Hamiltonians H = L j=1 J j ( ⃗ S j · ⃗ S j+1 ), where ⃗ S j = (S x j , S y j , S z j ) are the spin-1/2 operators on site j. The boundary conditions for these can be either periodic or open, and the symmetry algebra does not depend on this choice. This family of systems has been discussed using the commutant algebra framework in [1,2], and the local algebra and commutant algebra pair is given by A SU (2) = ⟨⟨{ ⃗ S j · ⃗ S j+1 }⟩⟩, C SU (2) = ⟨⟨S x tot , S y tot , S z tot ⟩⟩,(8) where S α tot := j S α j , α ∈ {x, y, z}, is the total spin operator in the directionα. The dimensions of irreducible representations for this pair of algebras, i.e., the numbers {D λ } and {d λ }, obtained using the simultaneous block-diagonalization method are shown in Fig. 1(a). For this pair of algebras, the distinct λ in the decompositions of Eqs. (4)-(6) correspond to the total spin quantum number which for even system size L takes values λ = 0, 1, . . . , L/2 and for odd system size takes λ = 1/2, 3/2, . . . , L/2 [namely, the eigenvalues of ⃗ S 2 tot are given by λ(λ + 1)]. The horizontal axis in the figure indicates d λ = 2λ + 1 which is the familiar degeneracy of spin-λ sector. In particular, d λ=0 = 1 marks the space of spin-singlets (i.e., states that have eigenvalue 0 under S 2 tot ), while d λ=L/2 = L+1 marks the ferromagnetic manifold. Note that all ferromagnetic states are degenerate singlets of the algebra A SU (2) generated by the individ-ual Heisenberg terms, hence D λ=L/2 = 1. In general, the sizes of each of the quantum number sectors of spin-λ, given by D λ , are known exactly [1,59] D λ = L L/2 + λ − L L/2 + λ + 1 .(9) As we depict in Fig. 1(a), the simultaneous block diagonalization procedure reproduced all these numbers correctly. While the blocks labelled by different values of λ, hence the operators in the center Z SU (2) := A SU (2) ∩ C SU (2) , were directly obtained using the method described in Sec. III A, extracting the full non-Abelian commutant C SU (2) required the implementation of the procedure described in Sec. III C. To ensure that these procedures work as intended, we found that the explicit randomization of eigenvectors, described in footnote 2 was particularly important. Hilbert Space Fragmentation We move on to an example of this method applied to models exhibiting Hilbert space fragmentation. We focus on the spin-1 Temperley-Lieb models, given by the family of Hamiltonians H T L = L−1 j=1 J j ( ⃗ S j · ⃗ S j+1 ) 2 , where ⃗ S j = (S x j , S y j , S z j ) here are the spin-1 matrices on site j, and we have used open boundary conditions. The local algebra in this case is given by A TL = ⟨⟨{( ⃗ S j · ⃗ S j+1 ) 2 }⟩⟩,(10) and the corresponding commutant C TL was computed explicitly in [1,59]; we are not aware of a compact expres-sion for C T L . The values of {d λ } and {D λ } for this pair of algebras is shown in Fig. 1(b). These numbers were also analytically computed in [59], and while the number of distinct blocks labelled by distinct λ and the values D λ are identical to the spin-1/2 Heisenberg model case in the previous section, the degeneracies {d λ } are given by d λ = [2λ + 1] q := q 2λ+1 − q −(2λ+1) q − q −1 , q = 3 + √ 5 2 .(11) For example, the block with the largest d λ , given by the block labelled by λ = L/2, also has D λ=L/2 = 1 and contains "ferromagnetic" product states |α 1 , α 2 , . . . , α L ⟩ with α j ̸ = α j+1 , j = 1, . . . , L − 1 for OBC along with their full SU (3) lowered multiplets, see [59] and references therein for the details; hence d λ=L/2 grows exponentially with system size L. All our numerical results shown in Fig. 1(b) are consistent with the above analytical results [59] and provide an independent check of these predictions. Quantum Many-Body Scars Finally, we provide an example of this method applied to a model of QMBS. As a non-trivial example, we consider spin-1 models on a periodic chain that realize the exact tower of scars found in the AKLT model [57,58,[60][61][62] as exact degenerate scars [3]. The local algebra for this case is given by [3] A (p) scar = ⟨⟨{Π [j,j+2] h [j,j+2] Π [j,j+2] }⟩⟩,(12) where h [j,j+2] is a sufficiently generic three-site operator, and Π [j,j+2] are three-site projectors chosen such that they vanish on the AKLT scar tower, see App. D of [3] for details on their precise construction. As conjectured in [3], the commutant algebra C (p) scar is expected to be spanned by ket-bra operators of the N scar states in the common kernel of the projectors {Π [j,j+2] }, where N scar depends on the system size L. Equivalently, we expect the Hilbert space decomposition in Eqs. (4)-(6) to simply consists of two distinct λ's, one corresponding to the thermal block and one to the scar block, which we denote by λ = t and λ = s respectively. We then expect the scar block to have (D λ=s , d λ=s ) = (1, N scar ), and the thermal block to have (D λ=t , d λ=t ) = (dim(H) − N scar , 1), where dim(H) = 3 L is the total dimension of the Hilbert space. The common kernel of these projectors for various system sizes has been conjectured in [3] (see Sec. V C and App. D there), and we briefly summarize the results here. For L odd, the common kernel is spanned by two statesthe AKLT ground state and the spin-polarized ferromagnetic state, hence d λ=s = 2. For L = 2 × odd, the kernel is spanned by the L/2 QMBS tower of states, denoted by {|ψ n ⟩} and defined in Eq. (22) of [3], as well as the ferromagnetic state, which is not a part of the tower of QMBS for these system sizes [60]; hence the total number of states in the kernel is L/2 + 1. On the other hand, for L = 2 × even, in addition to the L/2 + 1 QMBS tower of states {|ψ n ⟩} (which now includes the ferromagnetic state), there are two additional states in the kernel, denoted by 1 k=±π/2 in [3,57], which add up to L/2 + 3 states in the kernel. These explain all the values of {D λ } and {d λ } shown in Fig. 1(c) for various system sizes. Note that these results were only conjectured and not proven in our previous work [3], hence the results here illustrate non-trivial discovery/validation of the full commutant algebras. IV. LIOUVILLIAN APPROACH We now discuss an alternate method to construct operators in and determine the dimension of the commutant, which, in certain cases, allows us to determine the commutant for much larger system sizes. We start by in- This common kernel can be expressed as the null subspace of the positive semi-definite (p.s.d.) superoperator terpreting operators O as vectors | O), hence we obtain [ H α , O] = 0 ⇐⇒ L Hα := H α ⊗ 1 − 1 ⊗ H T α | O) = 0,(13)P := α P Hα := L † Hα L Hα , P| O) = 0 ⇐⇒ L Hα | O) = 0 ∀α,(14) where the second condition follows due the p.s.d. property of all P Hα . The dimension of the commutant, dim(C), is simply the dimension of this nullspace. The commutant can be numerically computed straightforwardly using this approach, although the timecomplexity is worse than the previous simultaneous block-diagonalization method since it involves the diagonalization of the superoperator P and scales as O(d 6L loc ). Hence a direct application of this method is usually limited to extremely small system sizes, although efficient methods for determining the kernel (e.g., Lanczos algorithm) can be used to improve its performance. Nevertheless, further progress can be made in some cases by noting that the operators { H α } are local terms, either strictly local or extensive local terms. When they are strictly local, so are the Hermitian p.s.d. superoperators P Hα := L † Hα L Hα . 3 Hence operators in the commutant are the frustration-free "ground states" of a local superoperator "Hamiltonian" P. B. Efficient Method in One Dimension Efficient numerical methods to obtain the frustrationfree ground states of a one-dimensional Hamiltonian are known in the literature [37][38][39], and they can be directly adapted to be applied to this problem. We discuss one such method in detail in App. B and provide a summary here. Note that this method borrows ideas from existing numerical algorithms based on Matrix Product States (MPS), particularly the Density Matrix Renormalization Group (DMRG) [64]. However, unlike DMRG, this algorithm is "exact", i.e., it does not use any variational optimizations or approximations (hence its accuracy is limited only by accumulations of machine round-off errors). Strictly local generators with OBC We start with the simplest case, namely, obtaining commutants of bond algebras generated by nearestneighbor terms, say { H j,j+1 }, with open boundary conditions (OBC). We can then define the nearest-neighbor Liouvillians { L j,j+1 } and correspondingly { P j,j+1 } using Eqs. (13) and (14) respectively. For a system of size L, we start by envisioning the χ L := dim(C L ) operators in the commutant C L , spanned by an orthonormal basis {|C (L) µ L )} χ L µ L =1 , as a Matrix Prod- uct State (MPS) consisting of tensors {A N } L N =1 with an open auxiliary index on the right. We wish to construct these tensors such that this property is true for any system of size 2 ≤ N ≤ L, i.e., the N -site MPS with an open auxiliary index on the right [see Eq. (B4) and Fig. 2(a)] spans the N -site commutant C N (i.e., the centralizer of the operators { H j,j+1 } N −1 j=1 ). Hence the MPS is not a standard one, and its bond dimension can grow from left to right, and the N -th bond from the left, N = 2, · · · , L, 3 Concretely, we can interpret the two tensored copies of the Hilbert space in Eq. (13) as two legs of a ladder (or two layers of a bilayer in higher dimensions), as common in the study of superoperators, e.g., in the Lindblad master equation [63]. The superoperators and operators in the original Hilbert space are operators and states in the doubled (ladder) Hilbert space respectively. It is then clear that if Hα is strictly local in the original Hilbert space, so are L Hα and also P Hα in the doubled (ladder) Hilbert space. has a dimension χ N := dim(C N ). We then solve for the tensors of the MPS recursively from the left using ideas similar to DMRG [64]. In particular, we solve for the two tensors A 1 and A 2 on the left by requiring that the MPS restricted to two sites is annihilated by P 1,2 [see Eq. (B5) and Fig. 2(b)]. We then determine the rest of the tensors {A N } L N =3 recursively, i.e., we can use the form of the (N − 1)-th tensor to determine the N -th by requiring that P N −1,N annihilates the N -site MPS [see Fig. 2(c)]; in particular, this condition can be rephrased in terms of determining the nullspace of an d 2 loc χ N −1 -dimensional matrix M [see Eq. (B9) and Fig. 2(d)]. The dimension of the auxiliary index χ N for the N -th tensor A N (which is also the dimension of the N -site commutant) is simply the dimension of the nullspace of that matrix. Hence the dimension of the commutant for a system of L sites can be determined using this recursive process. As we discuss in App. B, this method can be generalized in many directions. For bond algebras in OBC that are generated by r-site strictly local terms, the Nth tensor A N can be recursively determined using the previous (r − 1) tensors {A N −r+1 , · · · , A N −1 }, and it requires a diagonalization of an d 2 loc χ N −1 -dimensional matrix. Hence the time-complexity of this method for such a bond algebra and large systems of size L naively scales as O(Ld 6 loc χ 3 L ), although in practice there can be steps in the computation that are comparable in timecomplexity, e.g., the construction of the aforementioned matrix itself takes O(d 4(r−1) loc χ 3 L ). Nevertheless, these estimates make it clear that the efficiency of this method depends on the scaling of the commutant dimension with system size. If the commutant dimension stays constant or grows at most polynomially with the system size, e.g., in systems with conventional symmetries such as U (1) or SU (2) [1,2] or systems exhibiting quantum manybody scars [3], this method can be highly efficient, as we show in Figs. 3(a) and 3(c). However, if the commutant dimension grows exponentially with the system size, e.g., for systems exhibiting Hilbert space fragmentation [1], the accessible system sizes are limited, as we show in Fig. 3(b). In such cases, it can in practice be more efficient to work with the simultaneous blockdiagonalization method discussed in Sec. III. Strictly local generators with PBC or extensive local generators While this MPS construction works rather neatly for commutants of bond algebras generated by strictly local terms with OBC, additional work needs to be done to obtain commutants of algebras generated with strictly local terms with periodic boundary conditions (PBC) or with extensive local operators. In these cases, we first express Eq. (14) as P = P obc + P str + P ext , where (i) P obc , (ii) P str , and (iii) P ext respectively contain the P Hα 's corresponding to the generators H α that are (i) strictly local and can be viewed as generating the OBC bond algebra, (ii) strictly local and straddle the PBC "boundary", and (iii) extensive local. We then proceed by first determining the χ L -dimensional OBC commutant, i.e., the kernel of P obc , and we denote this subspace as C L := {|C | C (N) μ N ) = … μ N A N-1 A N A 1 A 2 A 1 A 2 1,2 = 0 A N A N-1 N-1 A * N A * N-1,N = 0 μ N-1 (a) (b) (c) A N-1 N-1 A *L N-1,N ℒ † N-1,N m N m′ N μ′ N-1 A N-1 N-1 A *̂ N-1,N μ N-1 μ′ N-1 m′ N m N = = (d) M [m′ N , m N ](L) µ L )} χ L µ L =1 . We then compute T str/ext , the χ L -dimensional matrices that are the restrictions of P str and P ext in the subspace C L ; their matrix elements are given by ( T str/ext ) µ L ,µ ′ L := (C (L) µ L | P str/ext |C (L) µ ′ L ).(15) The kernel of the χ L -dimensional matrix T str + T ext is then the commutant of the full local algebra, including all the generators. 4 As we discuss in Apps. B 3 and B 4, the restricted matrices T str and T ext can be efficiently computed using the MPS form for C L and using transfer matrices of its MPS tensors and MPO forms of P Hα when H α is extensive local [see Eqs. (B12) and (B19)]. Note that this method involves the construction of Υχ 2 N −1 × Υχ 2 N transfer matrices for every N ≤ L (where Υ = 1 for straddling operators and Υ ∼ O(1) for extensive local operators, 4 Note that a zero eigenvector of T str/ext , i.e., P str/ext restricted to the space C L , is always a zero eigenvector of P str/ext . This is because for any such vector | O 0 ) ∈ C L , we have by definition ( O 0 | P str/ext | O 0 ) = 0. Since P str/ext is positive semi-definite, this means that P str/ext | O 0 ) = 0. related to the bond dimension of the MPO of the corresponding P Hα ), their multiplication, and then the diagonalization a χ L -dimensional matrix. Hence the full timecomplexity of this method for large system sizes naively scales as O(LΥ 2 χ 4 L ), although a better scaling might be possible with an efficient tensor contraction ordering [65]. Although this scaling is polynomial in system size when the OBC commutant dimension scales polynomially, it is worse than the OBC problem, and the large exponents that show up in typical cases of interest (e.g., if χ L ∼ L 2 , the time-complexity scales as ∼ L 9 ) can be a significant hinderance in practice. Also note that the time-complexity depends on the scaling of the dimension of the OBC commutant, which might be larger than the scaling of the full commutant when the PBC or extensive local operators are included. Hence this method might not be efficient even if we expect the full commutant to have a small dimension. C. Restricting the form of the conserved quantity Finally, we show that this method can also be directly extended to search for conserved quantities in the commutant that are of a particular form, e.g., strictly local or extensive local operators of a fixed range. We denote the vector space spanned by operators of this form as V, and a linearly independent basis for V by {|V µ )}, and construct the operator P V , defined as the restriction of P to V. That is, the matrix elements of P V read ( P V ) µ,µ ′ = (V µ | P|V µ ′ ).(16) Then using the p.s.d. property of P we can show that the nullspace of P V is spanned by operators in C that are in V, i.e., P V | O) = 0 ⇐⇒ P| O) = 0 & | O) ∈ V.(17) D. Examples We now provide examples of computations of the dimension of the commutant algebra, i.e., the number of linearly independent operators in the algebra, using the efficient Liouvillian MPS method. We again illustrate this separately for cases with regular symmetries, fragmented systems, and QMBS systems, and we depict the results in Fig. 3(a)-(c). As discussed in Sec. IV B, this method is powerful in practice compared to the simultaneous blockdiagonalization method only for local algebras generated by strictly local terms with open boundary conditions. Note that the dimension of the commutant is related to the dimensions of the irreducible representations as dim(C) = λ d 2 λ [1], which provides an alternate way to compute it if the d λ 's are known. Conventional symmetries As examples of systems with conventional symmetries, we consider spin-1/2 systems with Z 2 , U (1), and SU (2) symmetries. The local and commutant algebras for these are discussed in [2], see Tab. I there. To summarize, the corresponding algebra pairs are given by A Z2 = ⟨⟨{S x j S x j+1 }, {S z j }⟩⟩, C Z2 = ⟨⟨ j S z j ⟩⟩, A U (1) = ⟨⟨{S x j S x j+1 + S y j S y j+1 }, {S z j }⟩⟩, C U (1) = ⟨⟨S z tot ⟩⟩, A SU (2) = ⟨⟨{ ⃗ S j · ⃗ S j+1 }⟩⟩, C SU (2) = ⟨⟨S x tot , S y tot , S z tot ⟩⟩,(18) where S α j , α ∈ {x, y, z} are the spin-1/2 operators on site j, and S α tot = j S α j . For the Z 2 and U (1) symmetries, the analytically known dimensions of the commutants can be directly obtained by counting the number of linearly independent operators, whereas in the case of SU (2) it is easier to compute dim(C) using the known values of {d λ }, the dimensions of the irreducible representations. In particular, we obtain dim(C Z2 ) = 2, dim(C U (1) ) = L + 1, and dim(C SU (2) ) = L+3 3 , and we refer readers to [1] for the details of the calculations. Our numerical results in Fig. 3(a) are consistent with these analytical predictions. Note that the system sizes accessible in the case of SU (2) are much smaller than the others due to the faster scaling of dim(C SU (2) ) with the system size L. Hilbert space fragmentation For fragmented systems, we apply this method to the t − J z models [66,67], spin-1 dipole conserving models [68], and spin-1 TL models [59], which have been studied in the commutant algebra framework in [1]. As discussed there, the t − J z models and the spin-1 dipole conserving models are examples of "classical fragmentation", i.e., the commutants in these cases, which we denote by C t−Jz and C dip , are completely spanned by operators that are diagonal in the product states basis. Hence they are Abelian, and their dimension is simply the number of distinct blocks in the product state basis, i.e., the number of "Krylov subspaces" [69]. The structure of the Krylov subspaces in these cases has been discussed in detail in [1], and we obtain dim(C t−Jz,obc ) = 2 L+1 − 1 and dim(C dip,obc ) = 2P L+1 − 1, where P L+1 is the (L + 1)th Pell number, which grows exponentially with L. On the other hand, the spin-1 Temperley-Lieb model exhibits "quantum fragmentation", and the dimension of the commutant can be computed directly using the dimensions of the irreducible representations, known from earlier literature [59], also shown in Eq. (11). All of these numbers are consistent with the dimensions obtained from the efficient MPS Liouvillian method, shown in Fig. 3(b). Note that the system sizes accessible here are much smaller than in the case of conventional symmetries due to the exponential scaling of the commutant dimensions. Quantum Many-Body Scars Finally, we apply this method to several models exhibiting QMBS studied in [3]. Note that there is a distinction between systems with degenerate QMBS and non-degenerate QMBS, in particular the local and commutant algebras in these cases are different. From the examples studied in [3], it appears that to obtain commutant algebras corresponding to non-degenerate QMBS, it is necessary to include an extensive local operator in the generators of the local algebra, whereas that is not the case for commutant algebras corresponding to degenerate QMBS. Since the efficiency of the MPS method is truly evident only for commutants of local algebras generated only by strictly local operators with OBC, we will only consider examples of degenerate QMBS throughout this section. First, we consider the case where the four OBC AKLT ground state are degenerate QMBS -Hamiltonians with this feature can be systematically constructed by an explicit "embedding" of these states into the spectrum [70]. The bond algebra in this case is given by A AKLT scar = ⟨⟨{P AKLT j,j+1 h j,j+1 P AKLT j,j+1 }⟩⟩,(19) where P AKLT j,j+1 are the spin-1 AKLT projectors, and h j,j+1 is a sufficiently generic two-site operator, see [3] for the details. The commutant of A AKLT scar , with OBC, is spanned by the identity operator, along with all ket-bra operators formed by states in the kernel of {P AKLT j,j+1 }, which are the four OBC AKLT ground states; hence the dimension of the commutant is given by dim( C AKLT scar ) = 17. Second, we consider models with the OBC AKLT tower of QMBS [57] as degenerate QMBS, where the bond algebra is given by [3] A (o) scar = ⟨⟨Π where {h [j,j+2] } and {h L−1,L }, along with the identity operator. Hence we expect its dimension to grow as N 2 scar + 1, where N scar is the number of states in this kernel. As discussed in [3] [see Eq. (D20) there], we expect N scar = L/2 + 1 when L is even, and N scar = (L + 1)/2 when L is odd, which then gives the expected dimension of the commutant in these cases. Finally, we consider models with the π-bimagnon tower in the OBC one-dimensional spin-1 XY model as degenerate QMBS [20,71,72], which resembles the η-pairing tower of states in the Hubbard model [72][73][74][75][76]. We consider the case where the commutant retains a U (1) spin conservation symmetry, and in [2] we conjectured the lo-cal and commutant algebra pair A (XY) scar = ⟨⟨{S x j S x j+1 + S y j S y j+1 }, {(S z j ) 2 }, {(S z j + S z j+1 )(1 − S z j S z j+1 )}⟩⟩, C (XY) scar = ⟨⟨{|Φ m ⟩⟨Φ n |}, S z tot ⟩⟩,(21) where S α j are the spin-1 operators on site j, {|Φ n ⟩} are the L + 1 QMBS states of the spin-1 XY model [see Eq. (29) there for the definition], and S z tot is the total spin operator. The dimension of C (XY) scar commutant can be obtained straightforwardly by counting the number of linearly independent operators, and we obtain dim( C (XY) scar ) = L(L + 4). 5 . In each of these cases, the commutant dimensions we expect are consistent with the results in Fig. 3(c), and these also provide additional verifications to the commutants conjectured but not proven in [3]. Note that in Fig. 3(c), the system sizes studied in the spin-1 XY case are larger than for the AKLT tower of states since the bond algebra in the former case is generated by two-site terms as opposed to three-site terms. For the sake of completeness, we discuss the inverse problem in the same language, i.e., given the generators of the commutant algebra C, the task is to construct local operators in the algebra A, which are by definition symmetric operators that possess the symmetry algebra C. As we discuss in Sec. V B, this inverse method is completely equivalent to methods used in several previous works for the same purpose [40][41][42][43]. In addition, there is a larger body of literature that focuses on recovering parent Hamiltonians for a general state, but their approaches are distinct from ours, e.g., they might assume additional structure of the state or resort to variational optimizations over some parameter space [44][45][46][47][48][49]. A. Inverse Method We first note that determining the space of all symmetric operators is straightforward using the methods discussed in Secs. III and IV. That is, the "symmetry" (i.e., the Double Commutant Theorem) between A and C in Eqs. (3)- (5) implies that the construction of the full algebra A from the generators of C can be done in exact analogy to the construction of C from the generators of A. Focusing on the Liouvillian method, given a set of generators of the commutant algebra { Q α }, we can define superoperators as L Qα := Q α ⊗ 1 − 1 ⊗ Q T α .(22) Analogous to Sec. IV, the common kernel of { L Qα }, or equivalently, the ground states of the superoperator P ′ := α L † Qα L Qα(23) are the linearly independent operators that span A. However, for quantum matter applications, we are usually interested in constructing symmetric local Hamiltonians, hence we wish to obtain the set of local operators in A. Since any set of local operators with a finite range of at most r max form a vector space, say V loc , this can be achieved using a direct analogy to the discussion in Sec. IV C. In particular, we can compute P ′ V loc , the restriction of the matrix P ′ to V loc ; this is defined analogous to Eq. (16). P ′ V loc has properties analogous to Eq. (17), i.e., its ground states are operators in the vector space A ∩ V loc , which in turn are all the local operators in A with range at most r max . B. Connection to Previous Methods We now elaborate on the precise relations of this inverse method to some of the methods introduced earlier in the literature for similar purposes. First, in [32,40], such symmetric local Hamiltonians were understood as zero modes of an appropriately constructed "commutant matrix," which is precisely equivalent to the matrix P ′ V loc . Second, [41] introduced a "correlation matrix" method for obtaining Hermitian local operators that have a given state |ψ⟩ as an eigenstate, which is equivalent to obtaining local operators that commute with |ψ⟩⟨ψ|. Applying our method to this problem, this is simply set of "ground states" of the matrix P ′ V loc , where V loc := span{|V µ )} is a vector space spanned by local operators {V µ } of interest. The matrix elements of P ′ V loc are then given by ( P ′ V loc ) µ,µ ′ = (V µ | L † |ψ⟩⟨ψ| L |ψ⟩⟨ψ| |V µ ′ ) = Tr [|ψ⟩⟨ψ| , V µ ] † [|ψ⟩⟨ψ| , V µ ′ ] = ⟨ψ| {V µ , V µ ′ } |ψ⟩ − 2 ⟨ψ| V µ |ψ⟩ ⟨ψ| V µ ′ |ψ⟩ ,(24) where V µ are assumed to be Hermitian operators. This is precisely the "correlation matrix" defined in [41] [see Eqs. (1.1) and (2.7) there], and they too determine the local operators by determining the zero eigenvectors of the correlation matrix; hence these methods are completely equivalent when applied to determine Hermitian operators that have individual states |ψ⟩ as eigenstates. Finally, [42] independently introduced a "covariance matrix" method where the ground states of the matrix are local operators that have a given state |ψ⟩ as an eigenstate. This covariance matrix differs from P ′ V loc or the correlation matrix of Eq. (24), and is defined as ( P ′′ V loc ) µ,µ ′ := ⟨ψ| V µ V µ ′ |ψ⟩ − ⟨ψ| V µ |ψ⟩ ⟨ψ| V µ ′ |ψ⟩ ,(25) where {V µ } is a linearly independent set of Hermitian operators that spans V loc . Any ground state of ( P ′′ V loc ) (i.e., with eigenvalue 0) corresponds to an operator in V loc that has |ψ⟩ as an eigenstate. While such an operator might be non-Hermitian in general, Hermitian operators can be obtained by restricting to real eigenvectors in the ground state space. Using the fact that P ′ V loc = P ′′ V loc + ( P ′′ V loc ) * and that P ′′ V loc and P ′ V loc are both positive-semidefinite Hermitian matrices, it is easy to show that the subspace spanned by all the ground states of P ′ V loc exactly coincides with the subspace spanned by real ground states of P ′′ V loc . Hence, if we are interested in Hermitian local operators, and we interpret the procedure of [42] as solving for the real eigenvectors, these methods are equivalent. When applied to states with an MPS representation, these methods are also closely related to the tensor network method discussed in [43], which determines strictly local or extensive local (emergent) symmetry operators given the MPS ground state of a certain Hamiltonian. C. Construction of Type I and Type II Symmetric Operators We now discuss how to apply this method to distinguish two types of symmetric operators that can be constructed out of a bond algebra A := ⟨⟨{ H α }⟩⟩, i.e., an algebra generated by a set of strictly local operators { H α }, where none of the H α are extensive local. To recap, in [3] we found qualitatively new types of symmetric Hamiltonians (i.e., extensive local operators) corresponding to symmetry algebras that are "unconventional", for example, commutant algebras that explain QMBS. One obvious class of symmetric extensive local operators are those that can be expressed as a sum of symmetric strictly local operators, and we refer to these as Type I symmetric Hamiltonians. Type II symmetric Hamiltonians are then those that cannot be expressed as a sum of symmetric strictly local operators, i.e., they necessarily involve highly non-local expressions in terms of the strictly local generators of A. In a previous work [2], we showed that for commutants generated by on-site unitary operators, all symmetric Hamiltonians are of type I, whereas in another previous work [3] we showed in the case of QMBS that there are Hamiltonians that are type II. Further, we introduced the notion of equivalence classes of type II symmetric operators, where two type II operators are equivalent if they differ by the addition of a type I symmetric operator. Since type I symmetric operators of a given range form a vector space that is a subspace of the vector space of all symmetric operators of that range, the set of equivalence classes of type II operators has an appropriate quotient space structure. These equivalence classes of type II operators of range at most r max can be directly extracted numerically using the inverse methods discussed in this section, as we now discuss. Note that similar ideas were used to discover various Hamiltonians with various examples of QMBS in [62,72,77]. Concretely, we start by considering the set of all clusters of sites on the lattice of range r max , which we denote by {R}, and the associated vector spaces {V R } of strictly local operators with support strictly within the respective clusters. We can then apply the method of Sec. V A using V loc = R V R to compute the vector space O loc is the space of all symmetric local operators, both strictly local and extensive local, of range at most r max , and it includes both type I and type II symmetric operators. (Here and below, the dependence of the discussed operator spaces on r max is understood implicitly.) To separate the type I and type II operators, we can then apply the method of Sec. V A restricting to each such cluster R, i.e., by using V loc = V R . This yields the vector space of operators in A that have support only on the cluster R, which we denote by O , whose dimension is given by the difference N II := dim(O (A) /O (A) I ) = dim(O (A) ) − dim(O (A) I ). Numerically applying the procedure to standard examples of on-site unitary symmetries, we then recover that N II = 0 for all choices r max ≪ L, consistent with the proof that there are no Type II symmetric operators in such cases [2]. On the other hand, applying this method to unconventional symmetries such as QMBS, we find in certain cases that N II can increase with r max , which shows that several independent type II symmetric operators can exist, as discussed in [3]. VI. CONCLUSIONS AND OUTLOOK In this work, we provided two methods to numerically construct commutant algebras corresponding to families of Hamiltonians. One of these involves simultaneous block diagonalization of two randomly chosen operators in the family and builds on earlier works of similar nature [32,33]. The other method maps this onto a problem of determining the frustration-free ground state of a Liouvillian superoperator, which can be efficiently solved in one dimension using MPS-based techniques discussed in [39]. These methods are useful in determining all the symmetries or dynamically disconnected "Krylov subspaces" of a particular family of systems, and we demonstrate this by applying these methods to several examples where we detect the presence of conventional symmetries [2], Hilbert space fragmentation [1], or Quantum Many-Body Scars [3]. In addition, they allow us to conjecture and corroborate commutants corresponding to local algebras we study in cases where we are not able to provide a proof, and could also be useful in other contexts, e.g., in quickly checking if a family of Hamiltonians has some unexpected symmetries. Finally, we also discussed inverse methods to determine local symmetric operators given the set of generators of a symmetry algebra. These can be useful in determining the exhaustive set of generators for a local algebra corresponding to a given symmetry algebra, or for identifying distinct types of local symmetric operators. While in this work we have adapted a "proof-ofprinciple" approach to demonstrate the numerical methods, we believe there are many avenues to make these methods much more efficient and hence more widely applicable to physically relevant families of Hamiltonians. Moreover, as evident from some of the examples, even the efficient Liouvillian method of Sec. IV B in one dimension works best in practice only when the generators of the local algebra are strictly local and with open boundary conditions. The addition of terms with periodic boundary conditions or extensive local terms is naively a significant hindrance, and it would be interesting to explore tricks that might make those cases computationally more tractable. It would also be interesting to generalize this method to two dimensions, where many Hamiltonians of physical interest lie, and perhaps ideas from the theory of Projected Entangled Pair States (PEPS) [78] might be useful. Numerical methods for determining the commutant are also useful for scanning through physically relevant models looking for "unconventional symmetries," which includes scars and fragmentation. Indeed, in an upcoming work [79], we apply such methods to discover examples of Strong Zero Modes (SZM) [80,81] that can be understood within the commutant algebra framework. This in turn allows us to construct non-integrable models with SZM, settling the debate of whether SZM can only occur in integrable models. There are multiple questions on the theoretical front too. First, it would be interesting to connect the methods discussed here to other methods introduced in the literature for identifying unconventional symmetries such as quantum many-body scars and Hilbert space fragmentation. For example, [82] used integer factorizations of characteristic polynomials of Hamiltonians to detect Hilbert space fragmentation and quantum many-body scars, while [83] used machine learning methods to detect quantum many-body scars. Furthermore, the Liouvillian method here shows that symmetry algebras can be understood as frustration-free ground state manifolds of local superoperators. This suggests that this problem might be analytically tractable, and indeed we find examples of such cases, which leads to several insights on symmetric systems with locality, and we will report these results elsewhere [84]. Also, given that several types of conventional and unconventional symmetries can be understood within the commutant algebra framework [1][2][3], it is natural to wonder if the frustration-free ground state property introduces some general constrains on the kind of operators that are allowed to be symmetries, e.g., do symmetry operators necessarily have low operator entanglement? We defer explorations of such questions to future work. Inserting decompositions of identity in terms of these states in Eq. (A5), we obtain Π β,β ′ = α,γ,δ,γ ′ ,δ ′ |ϕ γδ ⟩⟨ϕ γδ | U (α0α) |ϕ αβ ⟩⟨ϕ αβ ′ | ( U (α0α) ) † |ϕ γ ′ δ ′ ⟩⟨ϕ γ ′ δ ′ |. (A7) Using Eqs. (A6) and (A3), we obtain ⟨ϕ γδ | U (α0α) |ϕ αβ ⟩ = δ γα ⟨y (γ) δ |U (α0) (U (α) ) † |y (α) β ⟩ = δ γα ⟨v δ |(U (α) ) † U (α0) |v β ⟩ := δ γα V (αα0) δβ ⟨ϕ αβ ′ | ( U (α0α) ) † |ϕ γ ′ δ ′ ⟩ = δ αγ ′ ⟨y (α) β ′ |U (α) (U (α0) ) † |y (γ ′ ) δ ′ ⟩ = δ αγ ′ ⟨v β ′ |(U (α0) ) † U (α) |v δ ′ ⟩ = δ αγ ′ V (α0α) β ′ δ ′ ,(A8) where we have defined V (α ′ α) β ′ β := ⟨v β ′ | V (α ′ α) |v β ⟩ and V (α ′ α) := (U (α ′ ) ) † U (α) , a d λ -dimensional unitary matrix. Substituting Eq. (A8) in (A7), we obtain Π β,β ′ = α,δ,δ ′ |ϕ αδ ⟩ V (αα0) δβ V (α0α) β ′ δ ′ ⟨ϕ αδ ′ | = α,δ,δ ′ (V (α0α) βδ ) * V (α0α) β ′ δ ′ |ϕ αδ ⟩⟨ϕ αδ ′ |.(A9) We now show that we can relate {V (α ′ α) β ′ β } to the matrix elements of H (2) in the {|ϕ αβ ⟩} basis, i.e., the eigenbasis of H (1) . As a consequence of Eqs. (A1) and (A3), the matrix elements of H (2) read ⟨ϕ α ′ β ′ | H (2) |ϕ αβ ⟩ = ⟨x α ′ | M (2) D λ |x α ⟩ ⟨y (α ′ ) β ′ |y (α) β ⟩ = ⟨x α ′ | M (2) D λ |x α ⟩ ⟨v β ′ | (U (α ′ ) ) † U (α) |v β ⟩ = ⟨x α ′ | M (2) D λ |x α ⟩ V (α ′ α) β ′ β . (A10) The ratios of the matrix elements of V (α ′ α) can then be written in terms of matrix elements of H (2) , which completely determines the matrix V (α ′ α) up to a single non-zero element. That is, V (α ′ α) = c α ′ α G (α ′ α) , G (α ′ α) β ′ β := ⟨ϕ α ′ β ′ | H (2) |ϕ αβ ⟩ , c α ′ α := [⟨x α ′ | M (2) D λ |x α ⟩] -1 . (A11) The absolute value of c α ′ α can be obtained by imposing the unitarity of V (α ′ α) , (G (α ′ α) ) † G (α ′ α) = 1 |c α ′ α | 2 1 =⇒ |c α ′ α | 2 = det G (α ′ α) − 2 d λ .(A12) Plugging this into Eq. (A11), the matrix elements V (α ′ α) β ′ β can be expressed as V (α ′ α) β ′ β = 1 det G (α ′ α) 1 d λ ⟨ϕ α ′ β ′ | H (2) |ϕ αβ ⟩ ,(A13) where we have ignored the phase factor, which can be arbitrary and does not enter into the expression for the operators in the commutant. We can then rewrite Eq. (A9) as Π β,β ′ = α,δ,δ ′ |c α0α | 2 (G (α0α) βδ ) * G (α0α) β ′ δ ′ |ϕ αδ ⟩⟨ϕ αδ ′ | = α,δ,δ ′ det G (α0α) − 2 d λ (G (α0α) βδ ) * G (α0α) β ′ δ ′ |ϕ αδ ⟩⟨ϕ αδ ′ |,(A14) which are all in terms of "known" quantities. This allows us to construct C λ = span β,β ′ { Π β,β ′ }. Repeating this procedure for all the blocks labelled by different λ's allows us to construct the full commutant C. Bond algebras generated by nearest-neighbor terms with OBC We first illustrate this method for bond algebras generated by nearest-neighbor terms with OBC. In this case, the algebra generators { H α } and the superoperators { P Hα } discussed in Sec. IV are strictly local nearest-neighbor terms, which we denote by { H j,j+1 } L−1 j=1 and { P j,j+1 } L−1 j=1 , where L is the system size. Our aim is to construct the commutant recursively, i.e., to obtain the commutant of an N -site system from the commutant of an (N − 1)-site system. In the following, we denote the commutant of a system of size n as C n := span{|C (n) µn )} χn µn=1 , χ n := dim(C n ), (C (n) α |C (n) β ) = δ α,β ,(B1)µ N −1 )} as |C (N ) µ N ) = d 2 loc m N =1 χ N −1 µ N −1 =1 |C (N −1) µ N −1 ) ⊗ |m N ) [A [m N ] N ] µ N −1 µ N , (B2) where [A [m N ] N ] µ N −1 µ N can be viewed as elements of some tensor A N with a d 2 loc -dimensional physical index labelled by m N , and two auxiliary indices of dimensions χ N −1 and χ N labelled by µ N −1 and µ N respectively. We can then repeatedly apply Eq. (B2), e.g., applying twice we obtain |C (N ) µ N ) = µ N −2 ,µ N −1 m N −1 ,m N |C (N −2) µ N −2 ) ⊗ |m N −1 m N ) [A [m N −1 ] N −1 ] µ N −2 µ N −1 [A [m N ] N ] µ N −1 µ N ,(B3) and applying N times, we obtain the MPS form of |C (N ) µ N ): |C (N ) µ N ) = {mj } N j=1 ,{µj } N −1 j=1 [A [m1] 1 ] µ1 [A [m2] 2 ] µ1µ2 [A [m3] 3 ] µ2µ3 · · · [A [m N ] N ] µ N −1 µ N |m 1 m 2 · · · m N ),(B4) which we show pictorially in Fig. 2(a). Note that it is convenient to view {A [mj ] j } N j=2 as a χ j−1 × χ j matrix over the auxiliary indices, and the leftmost tensor A [m1] 1 as a χ 1 -dimensional vector with a single auxiliary index, although we will sometimes implicitly assign a dummy auxiliary index to A 1 and treat it as a χ 0 × χ 1 matrix where χ 0 := 1. To construct the commutant C N , we hence need to solve for the tensors {A j } N j=1 . We start with N = 2, and directly solve for the tensors A 1 and A 2 as follows. The vectors {|C (2) µ2 )} are defined as the orthonormal span of the kernel of P 1,2 , which can be obtained by a direct diagonalization. To construct the individual tensors, we can perform a Schmidt decomposition similar to Eq. (B2) on the vectors |C (2) µ2 ) to obtain |C (2) µ2 ) = µ1,m2 [A [m2] 2 ] µ1µ2 |C (1) µ1 ) ⊗ |m 2 ), |C (1) µ1 ) = d 2 loc m1=1 [A [m1] 1 ] µ1 |m 1 ),(B5) where |C (1) µ1 )} in the computational basis gives us the tensor A 1 . The condition that P 1,2 satisfies is shown pictorially in Fig. 2(b). Given the initial tensors, we can solve for the remaining tensors recursively, using ideas similar to those used in Density Matrix Renormalization Group (DMRG) algorithms [64]. Suppose we know the operators |C (N −1) µ N −1 ) in the (N − 1)-site commutant C N −1 , i.e., we have the tensors {A j } N −1 j=1 . We can solve for the tensor A N by requiring P N −1,N |C (N ) α ) = 0 ⇐⇒ (C (N ) α ′ | P N −1,N |C (N ) α ) = 0, ∀α, α ′ .(B6) Note that the implication uses the positive semi-definite property of P N −1,N , where it is easy to show that (C (N ) α | P N −1,N |C (N ) α ) =⇒ P N −1,N |C (N ) α ) = 0. Moreover, since P N −1,N is a two-site operator, it is convenient to express {|C (N ) µ N )} as in Eq. (B3), in which case Eq. (B6) can be written as (relabelling α, Fig. 2(c). Since the tensor A N −1 is known from the previous step of the recursion, we can use Eq. (B7) to solve for the tensor A N . To do so, we note that Eq. (B7) is equivalent to α ′ → µ N , µ ′ N ) µ µ ′ N −1 ,µ N −1 m N −1 ,m N m ′ N −1 ,m ′ N (m ′ N −1 m ′ N | P N −1,N |m N −1 m N )[A [m ′ N −1 ] N −1 ] * µµ ′ N −1 [A [m ′ N ] N ] * µ ′ N −1 µ ′ N [A [m N −1 ] N −1 ] µµ N −1 [A [m N ] N ] µ N −1 µ N = 0, ∀ µ N , µ ′ N ,(B7)µ ′ N −1 ,m ′ N µ N −1 ,m N [A [m ′ N ] N ] * µ ′ N −1 µ ′ N M [m ′ N ,m N ] µ ′ N −1 ,µ N −1 [A [m N ] N ] µ N −1 µ N = 0, ∀µ N , µ ′ N ,(B8) where we have defined M [m ′ N ,m N ] µ ′ N −1 ,µ N −1 := µ m N −1 ,m ′ N −1 (m ′ N −1 m ′ N | P N −1,N |m N −1 m N )[A [m ′ N −1 ] N −1 ] * µµ ′ N −1 [A [m N −1 ] N −1 ] µµ N −1 ,(B9) which is pictorially shown in Fig. 2 . This method can also be extended to cases where { P Hα } consist of multiple types of strictly local terms of various ranges. We can then "absorb" the smaller range terms into the longer range ones while ensuring that all terms in { P Hα } have been included, and apply the same procedure. For example, given two terms of P (1) [j,j+r1−1] and P (2) [j,j+r2 −1] of ranges r 1 and r 2 where r 1 ≥ r 2 , we can replace these terms by a new term of range r 1 , e.g., P (1,2) [j,j+r1−1] := P (1) [j,j+r1−1] + P (2) [j,j+r2 −1] . The kernel of P (1,2) [j,j+r1−1] is guaranteed to be the common kernel of P (1) [j,j+r1−1] and P (2) [j,j+r2 −1] since they are positive semi-definite operators. This procedure is also useful in cases where the ranges of { P Hα } vary throughout the system, e.g., when the generators include additional shorter range terms on the boundaries of the system. P L-1 P L P 1 P 2 b r b l = A j j A * μ′ j-1 μ j-1 μ j μ′ j μ j -1 μ′ j -1 [E] μ j μ′ j = A j j A *Ô μ j-1 μ′ j-1 μ j μ′ j μ j -1 μ′ j -1 [EÔ ] μ j μ′ j = A j j A * P μ j μ′ j μ j-1 μ′ j-1 ν j-1 ν j μ j -1 ν j -1 μ′ j -1 μ′ j ν j μ j [Ẽ P ] = (C (L) μ′ L |̂ β 1̂ α L | C (L) μ L ) = … μ L A L-1 A L A 1 A 2 … μ′ L 1 A * 2 A * L -1 A * L A *̂ α β (C (L) μ′ L |̂ ext | C (L) μ L ) = … μ L A L -1 A L A 1 A 2 … μ′ L 1 A * 2 A * L -1 A * L A * … P L -1 P L P 1 P 2 b r b l (a) (b) (d) (f) (e) Bond algebras with strictly local generators and PBC We now consider the case where the bond algebra generators { H α } consist of strictly local terms with PBC, e.g., terms such as H L,1 straddling the "boundary" in addition to { H j,j+1 } L−1 j=1 . In many cases, the addition of this straddling term does not give rise to a new bond algebra, e.g., in the free-fermion and Hubbard algebras discussed in [2], although this might not be evident a priori. In the following we refer to the commutant of the bond algebra ⟨⟨{ H j,j+1 } L−1 j=1 ⟩⟩ as the OBC commutant and the commutant of the bond algebra ⟨⟨{ H j,j+1 } L−1 j=1 , H L,1 ⟩⟩ as the PBC commutant. For simplicity, we restrict ourselves to bond algebras generated by nearest neighbor terms, the generalization to other types of terms is straightforward. For a system of size L we first compute the OBC commutant C L using the methods in App. B 1, and its basis vectors {|C µ L )} that are annihilated by the straddling superoperators P str , e.g., P L,1 . These linear combinations are in the kernel of the χ L -dimensional Hermitian and positive semi-definite matrix T str , the restriction of P str to C L , whose elements are given by ( T str ) µ ′ L µ L := (C (L) µ ′ L | P L,1 |C (L) µ L ).(B10) While it is memory intensive to compute the vectors {|C (L) µ L )}, the matrix elements can nevertheless be computed efficiently using the MPS form for {|C (L) µ L )} using the "transfer matrices" {E j } and {E O j } corresponding to the tensors {A j }, defined as [E j ] µ ′ j−1 µj−1 µ ′ j µj := d 2 loc mj =1 [A [mj ] j ] * µ ′ j−1 µ ′ j [A [mj ] j ] µj−1µj , [E O j ] µ ′ j−1 µj−1 µ ′ j µj := d 2 loc mj ,m ′ j =1 [A [m ′ j ] j ] * µ ′ j−1 µ ′ j (m ′ j | O|m j )[A [mj ] j ] µj−1µj ,(B11) where O is an operator acting on the site j. Note that these transfer matrices can be viewed as χ 2 j−1 × χ 2 j matrices by introducing composite indices (µ ′ j−1 , µ j−1 ) and (µ ′ j , µ j ), and they are shown diagrammatically in Figs. 4(a,c). Decomposing the straddling operator as P L,1 = α,β O α L O β 1 , it is easy to see using Eqs. (B4) and (B11) that the matrix elements of T str of Eq. (B10) can be expressed as ( T str ) µ ′ L ,µ L = α,β (C (L) µ ′ L | O β 1 O α L |C (L) µ L ) = α,β [E O β 1 E 2 E 3 · · · E L−1 E O α L ] µ ′ 0 ,µ0 µ ′ L ,µ L .(B12) where the equality of the matrix element on the L.H.S. to the transfer matrix expression on the R.H.S. is evident from Fig. 4(b) (we have indicated µ ′ 0 , µ 0 for clarity, but these are fixed µ ′ 0 = µ 0 = 1 and can be dropped, the convention being [A ] µ1 ). Note that if the operators O α L and O β L are fermionic, i.e., are odd under fermion parity, one would need to introduce a Jordan-Wigner string that runs throughout the system that enters into the transfer matrix expression in Eq. (B12). The PBC commutant can then be expressed in terms of the vectors {|C (L) µ L )} by diagonalizing the χ L -dimensional matrix T str . This method can also be applied to bond algebras with multiple terms straddling the boundary, e.g., in the case of bond algebras generated by n-site terms for n > 2. The T str matrices similar to Eq. (B10) can be constructed separately for each type of straddling term, and the full PBC commutant is the kernel of the sum of the matrices, which is guaranteed to be the common kernel of the individual matrices since they are positive semi-definite. Local algebras with some extensive local generators Finally, we can also apply similar ideas to local algebras where the list of generators includes an extensive local term, say H ext . The idea is again to first compute the OBC commutant C L and the MPS form of its basis vectors {|C (L) µ L )}. We then compute the matrix elements of the P ext superoperator of Eq. (14) corresponding to the extensive local operator H ext between the vectors that span the OBC commutant: ( T ext ) µ ′ L µ L := (C (L) µ ′ L | P ext |C (L) µ L ).(B13) To compute this overlap, it is convenient to represent P ext , which is translation-invariant if H ext is translationinvariant, as a Matrix Product Operator (MPO) [85], i.e., P ext = {m ′ j },{mj } [b l T P [m ′ 1 m1] P [m ′ 2 m2] . . . P [m ′ L m L ] b r ]|m ′ 1 · · · m ′ L )(m 1 · · · m L |,(B14) where the {P [m ′ k ,m k ] } are Υ-dimensional matrices, and b l , b r are Υ-dimensional vectors whose elements can be chosen to be some fixed numbers. We pictorially show the MPO form of Eq. (B14) in Fig. 4(d). Note that since P ext := L † ext L ext , This shows that L ext is a sum of range-r strictly local superoperators, and its MPO can be constructed using several standard techniques known in the literature [58,86,87]. For example, if H ext = S z tot = j S z j , which is the extensive local operator we are interested in several examples, it is easy to show that L ext is an MPO of bond dimension 2, i.e., L ext = {m ′ j },{mj } [b l Q T Q [m ′ 1 m1] Q [m ′ 2 m2] . . . Q [m ′ L m L ] b r Q ]|m ′ 1 · · · m ′ L )(m 1 · · · m L |,(B16) where Q is the MPO tensor which can be viewed as a 2-dimensional matrix with elements as single-site superoperators, and b l Q , b r Q are 2-dimensional vectors; their expressions read Q = 1 ⊗ 1 S z ⊗ 1 − 1 ⊗ S z 0 1 ⊗ 1 , b l Q = 1 0 , b r Q = 0 1 .(B17) Coming back to the computation of Eq. (B13), we can define generalized transfer matrices as [ E P j ] µ ′ j−1 νj−1µj−1 µ ′ j νj µj := d 2 loc m ′ j ,mj =1 [A [m ′ j ] j ] * µ ′ j−1 µ ′ j [P [m ′ j ,mj ] ] νj−1,νj [A [mj ] j ] µj−1µj ,(B18) which is shown in Fig. 4(e) and can be viewed as a (χ 2 j−1 Υ × χ 2 j Υ) matrix. We can then express the matrix elements of T ext of Eq. (B13) as ( T ext ) µ ′ L ,µ L = ν0,ν L (b l ) ν0 [ E 1 E 2 · · · E L−1 E L ] µ ′ 0 ,ν0,µ0 µ ′ L ,ν L ,µ L (b r ) ν L ,(B19) where the boundary vectors b l and b r only carry the auxiliary indices of the MPO, as shown in Fig. 4(f) [and µ ′ 0 = µ 0 = 1 are fixed and can be dropped as explained after Eq. (B12)]. The commutant of this local algebra is then given by the kernel of the Hermitian positive semi-definite matrix T ext . ⊗ 1 d λ , and we assume that there are no degeneracies in the spectrum of M FIG. 1 . 1(Color online) The sizes and degeneracies of the blocks (Krylov subspaces) for several types of Hamiltonians with non-trivial commutant algebras, extracted using the Simultaneous Block Diagonalization method for various system sizes L. {D λ } denote the sizes of the blocks and {d λ } denote their degeneracies. For details in each case, see Sec. III D. (a) Conventional Symmetries: Spin-1/2 Heisenberg model. The blocks are the SU (2) total spin quantum number sectors. (b) Hilbert Space Fragmentation: Spin-1 Temperley-Lieb Models [1]. The small sizes of blocks and large degeneracies indicate the presence of Quantum Hilbert Space Fragmentation. (c) Quantum Many-Body Scars (QMBS): Spin-1 models realizing the periodic boundary condition spin-1 AKLT tower of scars where L Hα is the Liouvillian corresponding to the term H α , i.e., it represents the adjoint action of the Hamiltonian, hence L Hα |•) := [ H α , •]. Using Eq. (13) and the definition of the commutant in Eq. (1), the commutant is the common kernel of the Liouvillian superoperators { L Hα }. A. Mapping onto a Frustration-Free Ground State Problem μ′ N-1 μ N- 1 ,FIG. 2 . 12Liouvillian approach for the efficient construction of commutants of nearest-neighbor bond algebras in one dimension with open boundary conditions. (a) MPS representation for the basis vectors of the commutant on N sites. (b) Equation for solving for the leftmost two unknown tensors A1 and A2 by diagonalizing the known matrix P1,2. (c) Equation for recursively solving for the unknown tensor AN in terms of the known tensor AN−1 and the known matrix PN−1,N . (d) Matrix M for which the tensor AN is an eigenvector with eigenvalue 0. online) The dimension of the commutant algebras for several families of Hamiltonians, extracted using the efficient Liouvillian method. As discussed in Sec. IV B, this works best in cases where the local algebra is generated by strictly local terms with open boundary conditions, and we have only demonstrated such cases. For details in each case, see Sec. IV D. (a) Conventional Symmetries: Families of models with Z2, U (1), and SU (2) symmetries; (b) Hilbert Space Fragmentation: t − Jz models, spin-1 Temperley-Lieb (TL) models, spin-1 dipole conserving models; (c) Quantum Many-Body Scars: OBCAKLT ground states as scars, AKLT scar tower as degenerate scars, spin-1 XY π-bimagnon scar tower as degenerate scars. Note that dim(C) is a limiting factor in this method, hence smaller system sizes are accessible for systems with larger dim(C). , {Π [j,j+2] h [j,j+2] Π [j,j+2] L } are sufficiently generic operators in the bulk and on the boundary respectively, and {Π [j,j+2] } and {Π L } are three-site and two-site bulk and boundary projectors respectively; see App. D of [3] for the details of their construction. The commutant of this algebra, C (o) scar , was conjectured to be fully spanned by ket-bra operators of the common kernel of the projectors {Π [j,j+2] } and {Π 5 There are 2L + 1 operators of the form {(S z tot ) m , m = 0, 1, . . . , 2L} and (L + 1) 2 operators of the form {|Φm⟩⟨Φn|}, of which 2 (|Φ 0 ⟩⟨Φ 0 | and |Φ L ⟩⟨Φ L |) can be expressed as linear combinations of {(S z tot ) m } V. INVERSE PROBLEM: CONSTRUCTING SYMMETRIC OPERATORS FROM THE SYMMETRY 6 6 6A quick proof is as follows. We wish to show ( P ′V loc )µ,ν xν = 0 ⇐⇒ ( P ′′ V loc )µ,ν xν = 0, where repeated indices are summed over and xν is assumed to be real in the R.H.S. Starting with the L.H.S., the relation between P ′′ V loc and P ′ V loc shows that ( P ′′ V loc )µ,ν xν = −( P ′′ V loc ) * µ,ν xν =⇒ xµ( P ′′ V loc )µ,ν xν = −xµ( P ′′ V loc ) * µ,ν xν =⇒ xµ( P ′′ V loc )µ,ν xν = 0, where we have used matrix, we have xµ = x * µ (all its eigenspaces can be chosen to be real), hence x * µ ( P ′′ V loc )µ,ν xν = 0, which is equivalent to the R.H.S. since P ′′ V loc is p.s.d. For the other direction, we start with the R.H.S. and assume xµ = x * µ to obtain. ( P ′′ V loc ) * µ,ν xν = 0. Using the expression for P ′ V loc , the L.H.S. follows. all operators in A that are linear combinations of strictly local operators with support on any one of the clusters {R} (operators with different supports among {R} can appear in the sum). Note that O (A) vector space of all type I symmetric operators of range at most r max , which includes both strictly local and extensive local operators. With these vector spaces O (A) and O (A) I , we directly obtain the equivalence classes of type II symmetric operators, i.e., the quotient space O (A) where (O 1 1|O 2 ) := 1 D Tr(O † 1 O 2 )is the usual Hilbert-Schmidt overlap of two operators, and D is the dimension of the Hilbert space in which the operators O 1 and O 2 act. Note that {|C (n) µn )} are operators on the Hilbert space of n sites, hence they are linear combinations of computational basis operators |m 1 · · · m n ), 1 ≤ m j ≤ d 2 loc , where d loc is the on-site Hilbert space dimension assumed for simplicity to be the same on all sites.To construct the commutant recursively, it is convenient to think of {|C(N ) µ N )} as a Matrix Product State (MPS), see Fig. 2a. In principle, an MPS can be understood via successive Schmidt decompositions of the "vector" |C (N ) µ N ). To begin, the decomposition w.r.t. a bipartition of the chain into the left (N − 1) sites and the rightmost N -th site is of the form |C (N ) µ N ) = α |L α 1,··· ,N −1 ) ⊗ |R α N ), where |L α 1,··· ,N −1 ) and |R α N ) are Schmidt vectors with supports on the left and right partitions respectively. Since { P j,j+1 } N −1 j=1 vanish on |C (N ) µ N ) by definition, they must also vanish on the corresponding left Schmidt vectors {|L α 1,··· ,N −1 )}. Hence the left Schmidt vectors {|L α 1,··· ,N −1 )} are in the commutant C N −1 and can be expressed as linear combinations of |C (N −1) µ N −1 ), and the right Schmidt vectors {|R α N )} are a linear combinations of the computational basis vectors {|m j )}. With this in mind, |C (N ) µ N ) can always be expressed in terms of {|C (N −1) ( 1 ) 1µ1 ) is a vector (which does not have any interpretation as a basis vector of a commutant) with support only on the first site. Note that in Eq. (B5), the simultaneous Schmidt decomposition of {|C (2) µ2 )} (achieved using the Singular Value Decomposition of properly reshaped amplitudes lumping in the index µ 2 ) gives us the tensor A where we have used the orthonormalization conditions of Eq. (B1) for the vectors {|C (N −2) µ N −2 )}. The condition of Eq. (B7) is shown diagrammatically in (d). Note that we can introduce composite indices (m ′ N , µ ′ N −1 ) and (m N , µ N −1 ) in Eqs. (B8) and (B9) and view A N and M as matrices of dimensions (d 2 loc χ N −1 ) × χ N and (d 2 loc χ N −1 ) × (d 2 loc χ N −1 ) respectively. Further, it is easy to check using Eq. (B9) that M[m ′ N ,m N ] µ ′ N −1 ,µ N −1 = M [m N ,m ′ N ] µ N −1 ,µ ′ N −1 * , hence M with thesecomposite indices is a Hermitian matrix. Moreover, since P N −1,N = L † N −1,N L N −1,N (explicitly given in our setting but also true for any Hermitian positive semi-definite P N −1,N ), it is easy to see that the matrix M can be expressed as M = G † G for an appropriately defined matrix G of shape χ N −2 d 4 loc × d 2 loc χ N −1 ; this is also evident fromFig. 2(d). Hence M is a Hermitian positive semi-definite matrix, and the tensor A N in Eq. (B8) is the nullspace of M , since A N with composite indices is a (d 2 loc χ N −1 ) × χ N matrix where the columns are vectors that make up the nullspace of the matrix M . The tensor A N can thus be constructed by diagonalizing a (d 2 loc χ N −1 )-dimensional matrix. The dimension of the kernel of the matrix M determines the χ N , which is also the dimension of the commutant C N . Note that in this method, the tensor A N can be obtained with only the knowledge of the tensor A N −1 and the strictly local term H N −1,N . [Note that the assumed orthonormalization of the prior set {|C(N −2)µ N −2 )} is automatically ensured in the recursive construction starting from Eq. (B5) and subsequent diagonalizations of Hermitian matrices M at each step.] The full commutant C N can be constructed from the tensors {A j } N j=1 using Eq. (B4), although storing it explicitly is typically memory intensive. 2 . 2Bond algebras generated by strictly local r-site terms with OBC The method presented in App. B 1 can be generalized straightforwardly to bond algebras generated by r-site terms with OBC. In this case, the generators of the algebra { H α } and the superoperators { P Hα } can be denoted by { H [j,j+r−1] } L−r+1 j=1 and { P [j,j+r−1] } L−r+1 j=1 . The tensors {A j } r j=1 can be obtained by solving for the kernel of P [1,r] and performing successive Schmidt decompositions similar to the r = 2 case shown in Eq. (B5). The remaining tensors can be obtained recursively by imposing the conditions of Eq. (B6) for (C (N ) α ′ | P [N −r+1,N ] |C (N ) α ) = 0. In particular, expressing |C (N ) µ N ) in terms of {|C (N −r) µ N −r )} and the tensors {A j } N j=N −r+1 [similar to Eq. (B3) for r = 2], this condition ext … FIG. 4 . 4Diagrammatic representation of various tensors required to efficiently construct commutants for bond algebras in one dimension with periodic boundary conditions and for local algebras in one dimension. can be written in terms of the matrix elements of P [N −r+1,N ] and the tensors {A j } N j=N −r+1 [similar to Eq. (B7) for r = 2]. Going through steps similar to Eqs. (B7) to (B9), we can express A N in terms of the kernel of a Hermitian positive semi-definite (d 2 loc χ N −1 )-dimensional matrix M , and the dimension of this kernel is given by χ N , the dimension of the commutant C N . Hence the tensor A N can be determined with the knowledge of the previous (r − 1) tensors {A j } N −1 j=N −r+1 and the term P [N −r+1,N ] , and the full commutant C N can be constructed from the tensors {A j } N j=1 [ m1 ] 1 ] m11µ0,µ1 = [A ] 1,µ1 ≡ [A[m1] 1 [m1] 1 the MPO matrices {P [m ′ k ,m k ] } can be constructed from the MPO matrices for L ext . This can in turn be constructed directly from the expression of the terms in H ext . For example, if H ext = j h [j,j+r] , where { h [j,j+r] } are some range-r strictly local operators, L ext is given byL ext = H ext ⊗ 1 − 1 ⊗ H T ext = j ( h [j,j+r] ⊗ 1 − 1 ⊗ h [j,j+r] ). (B15) ACKNOWLEDGEMENTSWe particularly thank Pengfei Zhang for useful discussions and for sharing unpublished notes on[39]. We also thank Bryan Clark, Nick O'Dea, Laimei Nie, and Frank Pollmann for useful discussions. This work was supported by the Walter Burke Institute for Theoretical Physics at Caltech; the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant PHY-1733907); and the National Science Foundation through grant DMR-2001186. S.M. acknowledges the hospitality of the Centro de Ciencias de Benasque Pedro Pascual, where a part of this work was completed.Appendix A: Constructing Non-Abelian Commutants Using Block-DiagonalizationIn this appendix, we give details on constructing non-Abelian commutants using the simultaneous block diagonalization method discussed in Sec. III. According to Eq. (4) the Hamiltonians H(1)and H(2)can be unitarily transformed into a basis where they are simultaneously block-diagonal, i.e.,In the following, we assume that the blocks labelled by λ have been "resolved" using methods discussed in Sec. III A, and that the numbers d λ and D λ have already been extracted using the methods described in Sec. III B. The tensor product form used to write the expected finer block-diagonal structure inside the block λ in the R.H.S.'s of the above equations assumes a formal factoring of the corresponding D λ d λ -dimensional space as Hλ . However, this tensor factoring of the space is not initially known to us other than that it exists. According to Eq. (5), the unitarily-transformed commutant W † CW restricted to the block labelled by λ is spanned by operators of the formis a complete orthonormal basis in this space. Denoting this restricted commutant in the original computational basis by C λ , it can be written aswhere in the last form U d λ can be any fixed unitary acting in Hλ . While these operators in C λ can be directly constructed if the W or the block-diagonal basis is known, we do not have a direct access to such information. What we have instead is how the instances H(1)and H (2) act in our computational basis, in particular we can diagonalize H (1) and evaluate expectations values of H(2)in the eigenstates of H(1). In particular, diagonalizing H (1) of the form of Eq. (A1) and assuming there are no degeneracies in the spectrum of M (1) D λ , we obtain its eigenstates and d λ -fold degenerate eigenvalues {m α }: λ since diagonalizing H (1) only gives us an arbitrary basis in the degenerate space corresponding to its eigenvalue m α . Our task is to construct C λ using only the information we have, which we now develop.Since |ywhere we have used that {|x α ⟩} form a complete orthonormal basis in H (A) λ . Choosing the fixed unitary U d λ in Eq. 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{'abstract': 'For families of Hamiltonians defined by parts that are local, the most general definition of a symmetry algebra is the commutant algebra, i.e., the algebra of operators that commute with each local part. Thinking about symmetry algebras as commutant algebras allows for the treatment of conventional symmetries and unconventional symmetries (e.g., those responsible for weak ergodicity breaking phenomena) on equal algebraic footing. In this work, we discuss two methods for numerically constructing this commutant algebra starting from a family of Hamiltonians. First, we use the equivalence of this problem to that of simultaneous block-diagonalization of a given set of local operators, and discuss a probabilistic method that has been found to work with probability 1 for both Abelian and non-Abelian symmetries or commutant algebras. Second, we map this problem onto the problem of determining frustration-free ground states of certain Hamiltonians, and we use ideas from tensor network algorithms to efficiently solve this problem in one dimension. These numerical methods are useful in detecting standard and non-standard conserved quantities in families of Hamiltonians, which includes examples of regular symmetries, Hilbert space fragmentation, and quantum many-body scars, and we show many such examples. In addition, they are necessary for verifying several conjectures on the structure of the commutant algebras in these cases, which we have put forward in earlier works[1][2][3]. Finally, we also discuss similar methods for the inverse problem of determining local operators with a given symmetry or commutant algebra, which connects to existing methods in the literature. A special case of this construction reduces to well-known "Eigenstate to Hamiltonian" methods for constructing Hermitian local operators that have a given state as an eigenstate.', 'arxivid': '2302.03028', 'author': ['Sanjay Moudgalya \nDepartment of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA\n\nWalter Burke Institute for Theoretical Physics\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA\n', 'Olexei I Motrunich \nDepartment of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA\n'], 'authoraffiliation': ['Department of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA', 'Walter Burke Institute for Theoretical Physics\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA', 'Department of Physics\nInstitute for Quantum Information and Matter\nCalifornia Institute of Technology\n91125PasadenaCaliforniaUSA'], 'corpusid': 256616021, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 38410, 'n_tokens_neox': 33701, 'n_words': 20069, 'pdfsha': '3fb054d344fd3ccad437af9ec7a427749c3ef39b', 'pdfurls': ['https://export.arxiv.org/pdf/2302.03028v2.pdf'], 'title': ['Numerical Methods for Detecting Symmetries and Commutant Algebras', 'Numerical Methods for Detecting Symmetries and Commutant Algebras'], 'venue': []}

arxiv

OGNIAN KASSABOV 1982 OGNIAN KASSABOV THE AXIOM OF COHOLOMORPHIC (2n + 1)-SPHERES IN THE ALMOST HERMITIAN GEOMETRY 1 SERDICA Bulgaricae mathematicae publicationes 819821 Center for mathematics and mechanics In his book on Riemannian geometry [1] E. C a r t a n proved a characterization of a real-space-form, using the axiom of planes. There are many results in this direction also for a Kaehler manifold. B.-Y. C h e n and K. O g i u e [4] have proved that a Kaehler manifold, which satisfies the axiom of coholomorphic 3-spheres is flat. In this paper we prove a generalization of this theorem for an almost Hermitian manifold. 1. Introduction. Let N be an n-dimensional submanifold of an 2m-dimensional almost Hermitian manifold M with Riemannian metric g and almost complex structure J. Let∇ and ∇ be the covariant differentiations on M and N, respectively. It is well known, that the equation α(X, Y ) = ∇ X Y − ∇ X Y, where X, Y ∈ XN, defines a normal-bundlevalued symmetric tensor field, called the second fundamental form of the immertion. The submanifold N is said to be totally umbilical, if α(X, Y ) = g(X, Y )H for all X, Y ∈ XN, where H = (1/n)trace α is the mean curvature vector of N in M. In particular, if α vanishes identically, N is called a totally geodesic submanifold of M. For X ∈ XN, ξ ∈ XN ⊥ we write ∇ X ξ = −A ξ X + D X ξ, where −A ξ X (respectively, D X ξ) denotes the tangential (resp. the normal) component of ∇ X ξ. A normal vector field ξ is said to be parallel, if D X ξ = 0 for each X ∈ XN. By an n-plane we mean an n-dimensional linear subspace of a tangent space. A 2n-plane (respectively an n-plane) α where 1 ≤ n ≤ m is said to be holomorphic (respectively, antiholomorphic) if Jα = α (respectively Jα ⊥ α). A (2n + 1)-plane α is called coholomorphic if it contains a holomorphic 2n-plane. An almost Hermitian manifold M is said to satisfy the axiom of holomorphic 2n-planes (respectively 2n-spheres) if for each point p ∈ M and for any 2n-dimensional holomorphic plane π in T p M there exists a totally geodesic submanifold N (respectively a totally umbilical submanifold N with nonzero parallel mean curvature vector) containing p, such that T p N = π, where n is a fixed number, 1 ≤ n < m. An almost Hermitian manifold M is said to satisfy the axiom of antiholomorphic nplanes (respectively n-spheres) if for each point p ∈ M and for any n-dimensional antiholomorphic plane π in T p M there exists a totally geodesic submanifold N (respectively a totally umbilical submanifold N with nonzero parallel mean curvature vector) containing p, such that T p N = π, where n is a fixed number, 1 < n ≤ m. An almost Hermitian manifold is called an RK-manifold, if R(X, Y, Z, U) = R(JX, JY, JZ, JU) for all X, Y, Z, U ∈ T p M, p ∈ M. We have proved in [5]: T h e o r e m A. Let M be a 2m-dimensional almost Hermitian manifold, m ≥ 2. If M satisfies the axiom of holomorphic 2n-planes or the axiom of holomorphic 2n-spheres for some n, 1 ≤ n < m, then M is an RK-manifold with pointwise constant holomorphic sectional curvature. T h e o r e m B. Let M be a 2m-dimensional almost Hermitian manifold, m ≥ 2. If M satisfies the axiom of antiholomorphic n-planes or the axiom of antiholomorphic nspheres for some n, 1 < n ≤ m, then M is an RK-manifold with pointwise constant holomorphic sectional curvature and with pointwise constant antiholomorphic sectional curvature. Consequently, if m ≥ 3, then M is one of the following: 1) a real-space-form, or 2) a complex-space-form. These theorems generalize some results in [3,6,9]. It is not difficult to see that if n > 1 then the holomorphic analogue of Theorem B holds. Following B.-Y. C h e n and K. O g i u e [4], L. V a n h e c k e formulates the following axiom of coholomorphic (2n + 1)-spheres [8]: For each point p ∈ M and for each coholomorphic (2n + 1)-plane π in T p M, there exists a (2n + 1)-dimensional totally umbilical submanifold N of M containing p, such that T p N = π, where n is a fixed integer, 1 ≤ n < m. We shall prove the following theorem. T h e o r e m. Let M be a 2m-dimensional almost Hermitian manifold, m ≥ 2. If M satisfies the axiom of coholomorphic (2n + 1)-spheres for some n, then M is conformal flat. Hence, using [7] we have C o r o l l a r y 1. Let M be a 2m-dimensional connected Kaehler manifold, m ≥ 2. If M satisfies the axiom of coholomorphic (2n + 1)-spheres for some n, then either M is flat or M is locally a product of two 2-dimensional Kaehler manifolds with constant curvature K and −K, respectively, K > 0. The case m ≥ 3 in corollary 1 is treated in [4]. An almost Hermitian manifold M which satisfies ( ∇ X J)X = 0 for all X ∈ XM is said to be an NK-manifold. Using the classification in [7] we have also C o r o l l a r y 2. Let M be a 2m-dimensional NK-manifold, m ≥ 2. If M satisfies the axiom of coholomorphic (2n + 1)-spheres for some n, then M is one of the following: 1) a flat Kaehler manifold, 2) locally a product M 1 × M 2 , where M 1 (respectively M 2 ) is a 2-dimensional Kaehler manifold with constant curvature K (respectively −K), 3) a 6-dimensional manifold of constant curvature K > 0, 4) locally a product M 3 × M 2 , where M 3 is a 6-dimensional NK-manifold of constant curvature K > 0. An almost Hermitian manifold M is said to be of pointwise constant type α, provided that for each point p ∈ M and for each X ∈ T p M we have α(p)g(X, X) = λ(X, Y ) = λ(X, Z) with λ(X, Y ) = R(X, Y, Y, X) − R(X, Y, JY, JX) whenever the planes defined by X, Y and X, Z are antiholomorphic and g(Y, Y ) = g(Z, Z) = 1. If for X, Y ∈ X(M) with g(JX, Y ) = g(X, Y ) = 0, λ(X, Y ) is a constant whenever g(X, X) = g(Y, Y ) = 1, then M is said to have global constant type. C o r o l l a r y 3. Let M be an almost Hermitian manifold with pointwise constant type α. If M satisfies the axiom of coholomorphic (2n + 1)-spheres for some n and if dimM ≥ 6, then M is a space of constant curvature α and M has global constant type. Corollary 3 is proved in [8] for an RK-manifold. Preliminaries. Let M be an 2m-dimensional almost Hermitian manifold with Riemannian metric g, almost complex structure J and covariant differentiation ∇. The curvature tensor R, associated with ∇ has the following properties: 1) R(X, Y )Z = −R(Y, X)Z 2) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0 3) R(X, Y, Z, U) = −R(X, Y, U, Z) for all X, Y, Z, U ∈ T p M, p ∈ M, where R(X, Y, Z, U) = g(R(X, Y )Z, U). The Weil conformal curvature tensor C is defined by C(X, Y, Z, U) = R(X, Y, Z, U) − (1/(2m − 2)){g(X, U)S(Y, Z) −g(X, Z)S(Y, U) + g(Y, Z)S(X, U) − g(Y, U)S(X, Z)} +(S(p)/((2m − 1)(2m − 2))){g(X, U)g(Y, Z) − g(X, Z)g(Y, U)}, where S and S(p) are the Ricci tensor and the scalar curvature of M, respectively. Now, let N be a submanifold of M, as in section 1. The normal component of R(X, Y )Z, where X, Y, Z ∈ XN is given by (2.1) (R(X, Y )Z) ⊥ = (∇ X α)(Y, Z) − (∇ Y α)(X, Z) , where (∇ X α)(Y, Z) = D X α(Y, Z) − α(∇ X Y, Z) − α(Y, ∇ X Z) and if N is totally umbilical submanifold of M, (2.1) reduces to (2.2) (R(X, Y )Z) ⊥ = g(Y, Z)D X H − g(X, Z)D Y H . 3. Proof of the Theorem. Let X, Y be arbitrary unit vectors in T p M, p ∈ M, such that X is perpendicular to Y, JY . Applying the axiom of coholomorphic (2n + 1)-spheres for a coholomorphic plane, which contains X, JX, JY and is perpenducular to Y and using (2.2) we obtain If m > 2, we take a unit vector Z, perpenducular to X, JX, Y, JY . Using again the axiom of coholomorphic (2n + 1)-spheres and (2.2) we find Hence, by the properties of the curvature tensor R we obtain (3.8) R(X, Y, Z, U) = 0 . Making use of (3.1)-(3.8) it is not difficult to prove that R(X, Y, Z, U) = 0 for an arbitrary orthogonal quadriple X, Y, Z, U ∈ T p M. According to a well known theorem of Schouten [2] the Weil conformal curvature tensor of M vanishes. R e m a r k. If a Riemannian manifold M of dimension n > 3 is conformal flat, then there exists a totally umbilical submanifold N of dimension n < m through every point of M and in every n-dimensional direction of that point (see [2]). Consequently, if M is a conformal flat 2m-dimensional almost Hermitian manifold, m ≥ 2, then M satisfies the axiom of coholomorphic (2n + 1)-spheres for every n, 1 ≤ n < m. X, JX, JX, Y ) = g(D X H, Y ) , R(X, JY, JY, Y ) = g(D X H, Y ) . Hence (3.3) R(X, JX, JX, Y ) = R(X, JY, JY, Y ) . From (3.2) we have R(Y + JY, JX, X, Y − JY ) = 0 and consequently (3.4) R(X, Y, Y, JX) = R(X, JY, JY, JX) . X, Y, Y, JX) = R(X, Z, Z, JX) . If m ≥ 4, let U be a unit vector in T p M, perpenducular to X, JX, Y, JY, Z, JZ. From (3.6) we have 2R(X, JX, JX, U) = R(X, Y +Z, Y +Z, U) = 0, which gives R(X, Y, Z, U) = −R(X, Z, Y, U). Leçons sur la géometrie des espaces de Riemann. ParisE. C a r t a n. Leçons sur la géometrie des espaces de Riemann. Paris, 1946. Some characterizations of complex space forms. K B. -Y. C H E N, O G I U E, Duke Math. J. 40B. -Y. C h e n, K. O g i u e. Some characterizations of complex space forms. Duke Math. J., 40, 1973, 797-799. . K B. -Y. C H E N, O G I U E, Two theorems on Kaehler manifolds. Michigan Math. J. 21B. -Y. C h e n, K. O g i u e. Two theorems on Kaehler manifolds. Michigan Math. J., 21, 1974, 225-229. On the axiom of planes and the axiom of spheres in the almost Hermitian geometry. Serdica, 8O. K a s s a b o v. On the axiom of planes and the axiom of spheres in the almost Hermi- tian geometry. Serdica, 8, 1982, 109-114. Conditions for constancy of the holomorphic sectional curvature. K N O M I Z U, J. Diff. Geom. 8K. N o m i z u. Conditions for constancy of the holomorphic sectional curvature. J. Diff. Geom., 8, 1973, 335-339. 4-dimensional conformally flat Kaehler manifolds. Tohoku Math. J. 24S. T a n n o. 4-dimensional conformally flat Kaehler manifolds. Tohoku Math. J., 24, 1972, 501-504. The axiom of coholomorphic (2p+1)-spheres for some almost Hermitian manifolds. 30L. V a n h e c k e. The axiom of coholomorphic (2p+1)-spheres for some almost Hermit- ian manifolds. Tensor (N.S.), 30, 1976, 275-281. On real representation of Kaehler manifolds. K Y A N O, I , Ann. Math. 61K. Y a n o, I. M o g i. On real representation of Kaehler manifolds. Ann. Math., 61, 1955, 170-189.

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{'abstract': 'In his book on Riemannian geometry [1] E. C a r t a n proved a characterization of a real-space-form, using the axiom of planes. There are many results in this direction also for a Kaehler manifold. B.-Y. C h e n and K. O g i u e [4] have proved that a Kaehler manifold, which satisfies the axiom of coholomorphic 3-spheres is flat. In this paper we prove a generalization of this theorem for an almost Hermitian manifold.', 'arxivid': '1004.3855', 'author': [], 'authoraffiliation': [], 'corpusid': 119310276, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3973, 'n_tokens_neox': 3434, 'n_words': 1991, 'pdfsha': 'e378909ad72f655f9152bb5b4ee5960e0d5841ce', 'pdfurls': ['https://arxiv.org/pdf/1004.3855v1.pdf'], 'title': ['OGNIAN KASSABOV', 'OGNIAN KASSABOV'], 'venue': ['THE AXIOM OF COHOLOMORPHIC (2n + 1)-SPHERES IN THE ALMOST HERMITIAN GEOMETRY 1 SERDICA Bulgaricae mathematicae publicationes']}

arxiv

* )( * * ), A. Cerica( 1 ), C. DAuria( 1 ) V Agostini ; F Casaburo ; G De Bonis ; F Di Mauro ; L Martone ; G Organtini ; F Piacentini Dipartimento di Fisica Università di Roma -Roma Italy ( ) INFN Dipartimento di Fisica, Università di Genova and INFN-Sezione di Genova Sezione di Roma -RomaItaly * )( * * ), A. Cerica( 1 ), C. DAuria( 1 ) Measurement of the cosmic ray flux by an ArduSiPM-based muon telescope in the framework of the Lab2Go project.Summary. -Within Istituto Nazionale di Fisica Nucleare (INFN) outreach activities, the Lab2Go project is of great significance. Its goal is involving high school teachers and students in several laboratory activities, aiming at increasing the weight of experimental contents in teaching and learning. In this article we present the measurement, carried out in the framework of the Lab2Go project, of the cosmic muon flux made by an ArduSiPM-based muon telescope. -Introduction At the end of the XVIII century, the spontaneous discharge of electroscopes was observed [1]. After Röntgen's X-rays discovery [2], it was proposed that the leakage of electric charge was due to the ionization caused by radioactivity coming from the underground. This hypothesis was dismissed by Pacini, who showed that the radiation under the sea was significantly lower than at the surface [3]. In 1912, during a balloon flight up to an altitude of 5200 m, Hess observed an increasing of the radiation with the altitude [4], proving that it comes from the space. Later, Millikan named this radiation Cosmic Rays (CRs) [5]. When CRs enter the atmosphere, they interact producing a particle showers called Extensive Air Showers (EAS), that can be divided in three components: hadronic, electromagnetic and muonic [6]. The latter, due to the low interaction between muons and the atmosphere, can be detected at sea level. Here, muon flux is approximately of 1 µ ± / min /cm 2 [7], but it depends on the Zenith angle θ and, for θ < 75 • , it is: F (θ) = F 0 cos 2 θ counts / s /m 2(1) being F 0 the flux at θ = 0 [8]. Experimentally, we can measure it rotating the detector, and thus verifying the expected formal relation between measured flux and angle. In this article, we present the measurement of the cosmic muon flux made by an ArduSiPM-based muon telescope in the framework of the Lab2Go project [9] at Liceo "L. Pietrobono" in Alatri (Italy). Despite our experimental setup doesn't allow to estimate the electron contamination, due to the higher interaction of electrons in atmosphere, it is negligible for our educational purpose. -Experimental setup and procedure The experimental setup ( Fig. 1a) consists of a muon telescope made of two Ar-duSiPMs. ArduSiPM is a transportable particle detector [10] created and developed by INFN Roma, constituted by an electronic shield connected to an Arduino DUE [11], a 5 cm × 5 cm × 5 mm scintillator and a Silicon PhotoMultiplier (SiPM). To build up the muon telescope, the two ArduSiPMs are placed on a mechanical structure that hosts the scinitllators as well, and is free to rotate. The distance between scintillators is 7.8 cm, corresponding to an acceptance of 0.16 [12]. To automatize the angle measurement, the inclination is derived by g components measured by an accelerometer at reast connected to the rotating mechanical structure. The number of events measured by each one of the ArduSiPMs and by both of them (time width= 10 µs), the acquisition time and the angle are read by an M5Stack (Fig. 1b) [13]. -Data analysis and results The number of coincidences per unit time was measured at different angles as N (θ) T , being N (θ) the number of events detected by both the ArduSiPM and T the acquisition time. Due to the low rate of detected muons (it depends on the acceptance), long acquisition time was needed for each angle, from approximately 1h at θ = 0 up to approximately 2h at θ = 60 • . The uncertainty on N (θ) has been evaluated assuming a Gaussian uncertainty when counts were > 30, and a Poisson uncertainty otherwise. Being A the area of the scintillator, values of F (θ) = N (θ) T A for several angles have been interpolated (Fig. 2) with the function Eq. 1, resulting in F 0esti = (2.49 ± 0.18) · 10 −2 counts / min /cm 2 as derived from the fit. Despite the large uncertainties due to poor statistics (few tens of detected muons), the relation in Eq. 1 is qualitatively verified as shown in Fig. 2. In addition, to further stress out the need for a quantitative validation and to allow the students to acquire the concept of agreement between measurements, we evaluated a test function. In particular, we started assuming no-correlation between the estimated F 0esti and measured F 0meas = (2.68 ± 0.45) · 10 −2 counts / min /cm 2 flux parameters (zerohypothesis) and we consider t = |F0 esti −F0 meas | σ 2 F 0 esti +σ 2 F 0meas , whose result (t = 0.4) has been used to estimate a one-sided gaussian p-value. We got p-value=0.7 (> 0.05, then verified at 95% C.L.) and an agreement within 1σ. -Students' satisfaction At the end of the Lab2Go course, an anonymous satisfaction questionnaire has been proposed to each one of 17 students (10 male and 7 female) attending Lab2go, in order to evaluate their level of satisfaction concerning the lecture topics, their engagement, the quality of teaching and the overall project. For each of the 10 questions, students (attending the last three years of Liceo Scientifico) could express their impression, from a minimum of 1 (unsatisfactory) to a maximum of 4 (very satisfactory). The results of the total 170 answers (17 students * 10 questions) have been summarised in an histogram (Fig. 3) resulting in a mean grade µ = 3.435 ± 0.048. -Conclusions In the framework of Lab2Go, the measurement of the cosmic muon flux by an ArduSiPMbased muon telescope has been proposed to students of Liceo "L. Pietrobono" in Alatri. Thanks to this experiment we had the possibility to introduce many topics, as CRs, particle physics, muons, antimatter, special relativity, particle detectors and time coincidence, going far beyond the topics commonly taught in physics lectures at high schools. Moreover, students learned how to process, interpolate and show data, taking into account the uncertainties and estimating the agreement between measurements. A satisfaction questionnaire proposed to students at the end of the course reported a positive evaluation of the proposed activity and, in general, of Lab2Go overall. * * * Fig. 1 . 1: (a) Muon telescope during the measurement at Liceo "L. Pietrobono". (b) Data read by the M5Stack. Fig. 2 . 2: Flux interpolation. Fig. 3 . 3: Results of satisfaction questionnaires. The authors acknowledge Mauro Mancini, Francesco Safai Therani, and Comitato di Coordinamento III missione (CC3M)-INFN. Two centenaries: The discovery of cosmic rays and the birth of Lajos Jánossy. P Király, Journal of Physics: Conference Series. 40912001P. Király, "Two centenaries: The discovery of cosmic rays and the birth of Lajos Jánossy," Journal of Physics: Conference Series, vol. 409, p. 012001, feb 2013. Centennial of röntgen's discovery of x-rays. R I Frankel, The Western journal of medicine. 1648764624pmidR. I. Frankel, "Centennial of röntgen's discovery of x-rays," The Western journal of medicine, vol. 164, pp. 497-501, Jun 1996. 8764624[pmid]. Penetrating radiation at the surface of and in water. D Pacini, D. Pacini, "Penetrating radiation at the surface of and in water." https://arxiv.org/abs/1002.1810, 2010. On the observations of the penetrating radiation during seven balloon flights. V Hess, V. Hess, "On the observations of the penetrating radiation during seven balloon flights." https://arxiv.org/abs/1808.02927, 2018. The birth and development of coincidence methods in cosmic-ray physics. L Bonolis, Walther Bothe, Bruno Rossi, American Journal of Physics. 79L. Bonolis, "Walther Bothe and Bruno Rossi: The birth and development of coincidence methods in cosmic-ray physics," American Journal of Physics, vol. 79, pp. 1133-1150, nov 2011. Energy spectrum and mass composition of high-energy cosmic rays. A Haungs, H Rebel, M Roth, Reports on Progress in Physics. 661145A. Haungs, H. Rebel, and M. Roth, "Energy spectrum and mass composition of high-energy cosmic rays," Reports on Progress in Physics, vol. 66, p. 1145, jun 2003. Characterization of atmospheric muons at sea level using a cosmic ray telescope. J Autran, D Munteanu, T Saad, S Saoud, Moindjie, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 903J. Autran, D. Munteanu, T. Saad Saoud, and S. Moindjie, "Characterization of atmospheric muons at sea level using a cosmic ray telescope," Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 903, pp. 77-84, 2018. Investigation of the zenith angle dependence of cosmic-ray muons at sea level. M Bektasoglu, H Arslan, Pramana. 80M. Bektasoglu and H. Arslan, "Investigation of the zenith angle dependence of cosmic-ray muons at sea level," Pramana, vol. 80, pp. 837-846, 2013. Il progetto Lab2Go per la diffusione della pratica laboratoriale nelle scuole. M Andreotti, La Fisica nella Scuola. M. Andreotti and et al., "Il progetto Lab2Go per la diffusione della pratica laboratoriale nelle scuole," La Fisica nella Scuola, vol. 3-4, 2020. The ArduSiPM a compact trasportable software/hardware data acquisition system for sipm detector. V Bocci, G Chiodi, F Iacoangeli, M Nuccetelli, L Recchia, 2014 IEEE Nuclear Science Symposium and Medical Imaging Conference. V. Bocci, G. Chiodi, F. Iacoangeli, M. Nuccetelli, and L. Recchia, "The ArduSiPM a compact trasportable software/hardware data acquisition system for sipm detector," in 2014 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), pp. 1-5, 2014. Realizzazione di un telescopio a basso costo per lo studio della distribuzione angolare di muoni con i rivelatori ArduSiPM. C Chiarini, C. Chiarini, "Realizzazione di un telescopio a basso costo per lo studio della distribuzione angolare di muoni con i rivelatori ArduSiPM." https://sites.google.com/view/particle- detectors/thesis?pli=1, 2019.

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{'abstract': 'Measurement of the cosmic ray flux by an ArduSiPM-based muon telescope in the framework of the Lab2Go project.Summary. -Within Istituto Nazionale di Fisica Nucleare (INFN) outreach activities, the Lab2Go project is of great significance. Its goal is involving high school teachers and students in several laboratory activities, aiming at increasing the weight of experimental contents in teaching and learning. In this article we present the measurement, carried out in the framework of the Lab2Go project, of the cosmic muon flux made by an ArduSiPM-based muon telescope.', 'arxivid': '2301.12948', 'author': ['V Agostini ', '; ', 'F Casaburo ', '; ', 'G De Bonis ', '; ', 'F Di Mauro ', '; ', 'L Martone ', '; ', 'G Organtini ', '; F Piacentini ', '\nDipartimento di Fisica\nUniversità di Roma -Roma\nItaly (\n', '\n) INFN\nDipartimento di Fisica, Università di Genova and INFN-Sezione di Genova\nSezione di Roma -RomaItaly\n'], 'authoraffiliation': ['Dipartimento di Fisica\nUniversità di Roma -Roma\nItaly (', ') INFN\nDipartimento di Fisica, Università di Genova and INFN-Sezione di Genova\nSezione di Roma -RomaItaly'], 'corpusid': 256389645, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3147, 'n_tokens_neox': 2741, 'n_words': 1607, 'pdfsha': '3edd62bb225859561ca9dba6b0aac82e5734b3e2', 'pdfurls': ['https://export.arxiv.org/pdf/2301.12948v2.pdf'], 'title': ['* )( * * ), A. Cerica( 1 ), C. DAuria( 1 )', '* )( * * ), A. Cerica( 1 ), C. DAuria( 1 )'], 'venue': []}

arxiv

Q-ball candidates for self-interacting dark matter Jun 2001 (April, 2001) Alexander Kusenko Department of Physics and Astronomy UCLA 90095-1547Los AngelesCA RIKEN BNL Research Center Brookhaven National Laboratory Upton11973NY Paul J Steinhardt Department of Physics Princeton University 08540PrincetonNJ Q-ball candidates for self-interacting dark matter Jun 2001 (April, 2001)arXiv:astro-ph/0106008v1 1 We show that non-topological solitons, known as Q-balls, are promising candidates for selfinteracting dark matter. They can satisfy the cross-section requirements for a broad range of masses. Unlike previously considered examples, Q-balls can stick together after collision, reducing the effective self-interaction rate to a negligible value after a few collisions per particle. This feature modifies predictions for halo formation. We also discuss the possibility that Q-balls have large interaction cross-sections with ordinary matter.PACS numbers: 95.35.+d, 98.80.Cq The standard cold dark matter model based on nonrelativistic, collisionless particles successfully predicts the formation of structure on large scales exceeding a megaparsec, but appears to make problematic predictions about structure on galactic and sub-galactic scales. The dark matter density profile in the cores of galaxies, the number of satellites, the thickenings of disks, the density of low mass objects, gravitational lens statistics, and the asphericity of cluster cores found in numerical simulations appear to be at variance with observations [1,2]. The difficulties suggest either that dark matter is not cold [3] or that dark matter is not collisionless [1]. In either case, the conventionally preferred candidates for dark matter, weakly interacting massive particles (WIMPS) or axions, would be ruled out. In this paper, we propose non-topological solitions known as Q-balls as candidates for collisional dark matter. Q-balls occur in a wide range of particle physics models, can be produced copiously in the early universe, and can be stable. As candidates for self-interacting dark matter, they possess particularly advantageous and interesting features, as shown in this paper. First, Q-balls can satisfy the requisite cross-section conditions for a much wider range of masses than ordinary point-like particles. Second, depending on the detailed interactions of the fields from which they are generated, Q-balls can scatter inelastically, leading to modifications of halo evolution compared to elastically scattering collisional dark matter. A particularly interesting limit is where they collide and stick. In this case, the population evolves and the crosssection and mass relations change as scattering proceeds. The effect is that scattering becomes insignificant as time proceeds. The initial scatterings smooth out halo cores, but heat conduction ceases afterwards and gravothermal collapse is avoided. A third feature of Q-balls is that they can have significant interactions with ordinary matter (although this is not required). A large range of parameter space is ruled out by current experimental bounds, but significant unconstrained range remains, suggesting new directions for dark matter searches. Basic Properties of Q-balls Q-balls are the ground state configurations for fixed charge Q in theories with interacting scalar fields φ that carry some conserved global U(1) charge [4][5][6]. If the field configuration is written in the form φ(x, t) = e iωtφ (x),(1) its charge is Q = 1 2i φ * ↔ ∂ t φ d 3 x = ω φ 2 d 3 x.(2) The form ofφ(x) is determined minimizing the energy E = d 3 x 1 2 |φ| 2 + 1 2 |∇φ| 2 + U (φ) ,(3) where the potential U (φ) has a minimum at φ = 0 and is invariant under the global U(1) transformation φ → e iθ φ. In the thin-wall limitφ(x) = φ 0 is nearly constant in the interior (r < R) and drops rapidly to zero for r > R. Using (2), one can write the energy (3) as E ≈ Q 2 2V φ 2 0 + V U (φ 0 ),(4) where V is the Q-ball volume. The minimum of energy in eq. (4) with respect to V is E = µQ, where µ = 2U (φ 0 )/φ 2 0 . In the thin-wall limit, the minimum of E with respect to the value φ 0 corresponds to µ → µ 0 = min 2U (φ) φ 2 .(5) Depending on the potential, µ 0 in eq. (5) can be finite (Type I) or infinite (Type II). The mass of a Type I Q-ball is M (Q) = µQ For large Q-balls (Q → ∞), µ → µ 0 and φ → φ 0 ("thin-wall" limit) [4][5][6]. For smaller values of Q, µ can be computed in a "thick-wall" approximation [7]. For Q < 10 radiative corrections become important [8]. In any case, µ is less than the mass of the φ particle as a consequence of condition (5). The radius of the Type I Q-ball is R Q ≈ 3 4π 1/3 Q 1/3 (µφ 2 0 ) 1/3 .(6) Type II Q-balls occur if the scalar potential grows slower than the second power of φ. Then the Q-ball never reaches the thin-wall regime, even if Q is large. The value of φ inside the soliton extends to a value as large as the gradient terms allow, and the mass of a Qball is proportional to Q p , p < 1. In particular, if the scalar potential has a flat plateau U (φ) ∼ m 4 flat at large φ, then the mass of a Q-ball is [9] M (Q) ∼ m flat Q 3/4 and the size is R Q ∼ m −1 flat Q 1/4 . Q-balls as self-interacting dark matter: As in generic examples of self-interacting dark matter, Q-ball scattering in regions of high density facilitates heat exchange in dark matter cores that smooths out their distribution and, also, enhances the stripping of dark matter from satellites that accelerates their tidal destruction. Both effects serve to resolve the problems of cold, collisionless dark matter. For these purposes, the basic requirement is that the ratio of self-interaction cross section σ DD to particle mass M must be in the range [ 1] S = σ DD M = 8 × 10 −25 − 1 × 10 −23 cm 2 GeV −1 = 0.5 − 6 cm 2 g −1 ,(7) For point particles whose dominant scattering is s-wave, Hui has shown that unitarity implies a cross-section bounded above by σ DD ∼ 1/(M v rel ) 2 , where v rel ≈ 300 km/s is the typical velocity of the dark matter particles. In this case, the maximal mass for a point particle is M ≈ 10 GeV [10]. Q-balls are extended objects which can evade this bound. Higher partial waves contribute to their scattering such that their cross-section is essentially geometric (except in the limit of very small coupling), σ DD = πR 2 Q . Then, S for Type I Q-balls is σ DD M ≈ 9π 16 1/3 M (Q) φ 0 4/3 Q 4/3 M (Q) 3 ,(8) and for Type II Q-balls is σ DD M ∼ m −3 flat Q −1/4 ∼ Q 2 M (Q) 3 .(9) Note that both expressions for S can greatly exceed the unitarity bound for large Q > 10 5 . Hence, it is possible to have Q-ball candidates that satisfy the requirements on S for a range of masses much greater than 10 GeV (the unitarity limit). If Q-balls are of Type I and no restriction is placed on the relative magnitude of φ 0 and µ, the mass of the φ particle can range from below a keV to well beyond the electroweak scale. If the mass of φ > M Z , such a scalar field could make extremely heavy, strongly interacting Q-balls (cf. Ref. [12]). Naturalness arguments, while not rigorous, suggest potentials in which φ 0 ∼ µ. In this case, for Type I Q-balls (M (Q) ∼ µQ), a satisfactory choice of parameters is in the range around µ ∼ φ 0 ∼ 20MeV, Q ∼ 10 − 10 3 . For Type II Q-balls (with M (Q) ∼ m flat Q 3/4 ), the analogous relations are m flat ∼ 20 MeV and Q ∼ 10 4 − 10 5 . Note that, if the global U(1) symmetry of the Q-balls is associated with baryon number, as in most examples considered previously [11], empirical constraints specific to baryonic processes do not permit the requisite large cross-sections. However, there is no problem with more general U(1) symmetries. Q-ball production in the early universe: Several mechanisms could lead to a formation of Q-balls in the early universe. First, they can be produced in the course of a phase transition [13]. Second, solitosynthesis, a process of gradual charge accretion similar to nucleosynthesis, can take place [14][15][16]. Finally, Q-balls can emerge from fragmentation of a scalar condensate [11] formed at the end of inflation. Solitosynthesis occurs through an accretion of charge. It requires some universal asymmetry η Q of the global charge Q, similar to baryon asymmetry of the universe. When the temperature drops below some critical value T c ∼ µ/| ln η Q | [15], a Q-ball minimizes both the energy and the free energy of the system, and a rapid coalescence of global charge into Q-balls occurs [14,15]. The number of Q-balls and their mass density in the universe depends on the value of Q-asymmetry, η Q , and is largely unconstrained. Fragmentation of a coherent scalar condensate can lead to a copious production of Q-balls [11]. At the end of inflation, scalar fields develop large expectation values along those directions in the potential that have small masses or flat plateaus [17]. The subsequent rolling of the condensate can encounter an instability, as a result of which the scalar condensate can break up into Q-balls [11]. This process has been studied both analytically [11,18] and numerically [19,20] and was shown to produce a sharply peaked distribution of sizes of Qballs. There is also some evidence that Q-balls and anti-Q-balls can form from the same condensate while the overall charge asymmetry η Q [20] is small or zero. The number density of Q-balls formed in this way depends on the shape of the potential at large φ and the horizon size at the time of formation. The only strict constraint is that the separation between Q-balls should be of the order of their size at the time of formation. For us this translates into a red shift at which Q-balls are formed. Q-ball scattering and sticking: Whereas previous studies of collisional dark matter assumed that they scatter elastically, Q-balls can either merge or split after a collision depending on whether energy can be dissipated [21]. The merger of Q-balls requires the kinetic energy to dissipate quickly on the time scale of the collision. In the absence of additional interactions, emission of φ particles is the only channel for such dissipation. There are many modes of oscillations of an excited Q-ball: volume, surface, etc. The basic mode for a Type I Q-ball deforms the entire Q-ball and has a frequency Ω ∼ φ 2/3 0 µ 1/3 Q −1/3 , which decreases with Q. Production of φ particles in these oscillations is very efficient if Ω > m φ . When Ω is close to m φ , production of φ particles is enhanced by parametric resonance. For Ω > m φ , there are several other resonant bands. If, however, Ω < m φ , particles are produced very inefficiently, and it is unlikely that the Q-ball will dissipate any energy at all on the time scale of a typical collision. Therefore, if Ω > m φ , Q-balls can merge, but if Ω < m φ , they are more likely to fragment. Additional interactions of φ with other light states can enhance the dissipation and increase the probability of merging. For Type-II Q-balls, there is a strong energetic bias toward merging as opposed to fragmentation. The subject of Q-ball collisions, clearly, deserves further studies. Merger or sticking together of dark matter can lead to novel dynamics of the halo compared to the standard case of elastic self-interactions. As perturbations begin to grow, the density of Q-balls is too low for there to be significant scattering. As the halo density profile becomes steeper and denser, Q-ball collision and merger takes place. Mergers replace two particles with charge Q with a single particle of charge 2Q. According to Eqs. (8) and (9), the ratio S ≡ σ DD /M decreases as Q increases and the mean velocity v rel decreases by two. Both effects decrease the interaction rate until, ultimately, selfinteraction ceases. Thereby, a stable population of Qballs is reached with lower central density than occurs if there are no collisions. A modest amount of kinetic energy is lost, but the fraction of particles that scatter (halo particles scattering off particles in the central core) is a tiny fraction of the total halo. Because the collisions self-adjust the population from interacting to noninteracting, heat conduction ceases and gravothermal collapse is avoided. Hence, it is interesting to compare predictions for small halos formed early in the universe when the density is high. For collisionless dark matter, there are many such halos and they are extremely cuspy and dense. For elastically scattering dark matter, collisions smooth the core distribution but, then, gravothermal collapse causes the central density to rise again. In the case of merging Q-balls, the core density is reduced and it remains that way. In the case of splitting, binding energy can be converted into kinetic energy. Since the binding energy can exceed the gravitational binding to the halo, splitting can lead to conversion of two similar-size Q-balls into one large Q-ball that remains gravitationally bound to the halo and one small fragment that escapes. It is possible to imagine that both merger and splitting play a role. Suppose S ≡ σ DD /M is initially large. In a small young halo with a dense core, collision and merger transforms the population into large Q-balls with small S. Large Q-balls are more likely to split and have energy escape. The halo structure will be influenced by both effects. What kinds of interaction are possible between Q-balls and baryons? If the field φ has only gravitational interactions with matter, Q-balls cannot be detected directly. Condition (7) for φ 0 ∼ µ (the naturalness condition) suggests that m φ must be close to 0.02GeV. But, if φ has a mass below 50 MeV, it cannot have strong interactions with matter because of a combination of collider bounds, in particular because a neutral pion could decay into φφ pairs. Likewise, a light φ field cannot have weak interactions because it would violate the precision measurements at LEP of the Z width. Q-balls can interact strongly with ordinary matter if m φ > 50 MeV. Alternatively, there is the the possibility of a significant cross-section with ordinary matter if the interactions are mediated by some new physics. In a number of Grand Unified and string-inspired models, light fields are accompanied by additional interactions. For example, an interesting possibility is that an additional gauge U(1)', unrelated to either the global U Q (1) or the Standard Model gauge group, is spontaneously broken at some high scale [22]. We will use this model to illustrate possible interactions that the field φ can have with matter, which can ultimately make Q-balls detectable. Let us suppose that φ interactions with matter are mediated by some vector boson with mass M Z ′ . Then the cross section for Q-ball interactions with a nucleon is, roughly, σ Qp ∼ F g 2 Q 2 M 2 Z ′ ,(10) where g is some coupling constant and F is a form factor. The Q 2 -dependence occurs because of the coherent scattering of a nucleus off the φ quanta in the condensate, and the form factor F accounts for a fraction of Q-matter that scatters coherently. If the size of the Qball is smaller than that of a nucleus, F is of the order of 1. If the Q-ball is much larger than the nucleus, F ∼ (R n /R Q ) 3 , where R n is the size of the nucleus. The resulting dark matter (Q-ball)/proton crossections, σ Qp , are large enough to be detected for a broad range of parameters. Fig. 1 summarizes the current limits on σ Qp and M based on existing searches [2]. Superposed are the predictions for Q-balls. A large range of parameters is already ruled out, but there remain unexplored regimes. One consists of Q-balls with large crosssections and masses larger than a TeV. Since the local Empirical constraints (shades of grey) on Q-ball/proton cross section and mass assuming φ interacts through an intermediate boson Z ′ with M Z ′ ≈ 1 TeV and g ≈ 0.1. Experiments are described in Ref. [23]. The most likely range for Q-balls is enclosed by hatched boundary (either Type I with φ0/µ within a few orders of magnitude of unity or Type II). White regions are currently unconstrained experimentally. If no restriction is placed on φ0/µ0, the predicted range for Q-balls expands to the dot-dashed boundary, including even more untested territory. dark matter density is 0.4 GeV/cm 3 and the mean velocity is 300 km/s, the flux is of order 10 5 cm −2 s −1 . Another possibility is relatively light Q-balls with masses about 1 GeV and a weak cross section, below 10 −31 cm 2 . The flux of these particles would be high, of order 10 10 cm −2 s −1 . Different strategies would have to be adopted to search in the two regimes, but both are feasible, as will be discussed in a future paper. To summarize, a new scalar field can, in the form of Qballs, be the cold dark matter consistent with all present observations. The self-interactions of Q-balls are characterized by a large cross section due to their extended geometry, a property that can naturally explain the flattened density profiles of dark matter halos. If Q-balls scatter inelastically and merge, scattering may cease after the profile is flattened. It is conceivable that Q-balls have significant interactions with ordinary matter, either strong interactions or interactions mediated by a heavy Z ′ boson. In this case, the Q-balls can be detected in near-future experiments. We thank D. Spergel and J.P. Ostriker for many useful remarks, G. Gelmini and S. Nussinov for discussions of Q-ball interactions with matter, and P. McGuire for aid in determining existing constraints in the Figure. FIG. 1. Empirical constraints (shades of grey) on Q-ball/proton cross section and mass assuming φ interacts through an intermediate boson Z ′ with M Z ′ ≈ 1 TeV and g ≈ 0.1. Experiments are described in Ref. [23]. 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{'abstract': 'We show that non-topological solitons, known as Q-balls, are promising candidates for selfinteracting dark matter. They can satisfy the cross-section requirements for a broad range of masses. Unlike previously considered examples, Q-balls can stick together after collision, reducing the effective self-interaction rate to a negligible value after a few collisions per particle. This feature modifies predictions for halo formation. We also discuss the possibility that Q-balls have large interaction cross-sections with ordinary matter.PACS numbers: 95.35.+d, 98.80.Cq', 'arxivid': 'astro-ph/0106008', 'author': ['Alexander Kusenko \nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA\n\nRIKEN BNL Research Center\nBrookhaven National Laboratory\nUpton11973NY\n', 'Paul J Steinhardt \nDepartment of Physics\nPrinceton University\n08540PrincetonNJ\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCA', 'RIKEN BNL Research Center\nBrookhaven National Laboratory\nUpton11973NY', 'Department of Physics\nPrinceton University\n08540PrincetonNJ'], 'corpusid': 6996584, 'doi': '10.1103/physrevlett.87.141301', 'github_urls': [], 'n_tokens_mistral': 7478, 'n_tokens_neox': 6161, 'n_words': 3822, 'pdfsha': '473a4d5e86f7752a6481f0aff00ba3a45d5d27ad', 'pdfurls': ['https://export.arxiv.org/pdf/astro-ph/0106008v1.pdf'], 'title': ['Q-ball candidates for self-interacting dark matter', 'Q-ball candidates for self-interacting dark matter'], 'venue': []}

arxiv

Beam Dynamics in Independent Phased Cavities 2022 Y K Batygin Los Alamos National Laboratory 87545Los AlamosNMUSA Beam Dynamics in Independent Phased Cavities Nuclear Instruments and Methods in Physics Research 10401671922022 Linear accelerators containing the sequence of independently phased cavities with constant geometrical velocity along each structure are widely used in practice. The chain of cavities with identical cell lengths is utilized within a certain beam velocity range, with subsequent transformation to the next chain with higher cavity velocity. Design and analysis of beam dynamics in this type of accelerator are usually performed using numerical simulations. A full theoretical description of particle acceleration in an array of independent phased cavities has not been developed. In the present paper, we provide an analytical treatment of beam dynamics in such linacs employing Hamiltonian formalism. We begin our analysis with an examination of beam dynamics in an equivalent traveling wave of a single cavity, propagating within accelerating section with constant phase velocity. We then consider beam dynamics in arrays of cavities, utilizing an effective traveling wave propagating along with the whole accelerator with the velocity of synchronous (reference) particle. The analysis concluded with the determination of the matched beam conditions. Finally, we present a beam dynamics study in 805 MHz Coupled Cavity Linac of the LANSCE accelerator facility. Introduction Most of the modern accelerators contain sequences of accelerating sections with equidistant cells. Wide application of superconducting RF technology resulted in the development of accelerating structures, where the utilization of identical cells in a single structure is unavoidable due to technological reasons. In the linac comprising of independently phased cavities, it is possible to change the velocity profile of synchronous particle, and, therefore, to change the output energy. This option is especially important in heavy ion linacs accelerating particles with various mass-to-charge ratios. Analysis of beam dynamics and design of accelerator sections with identical cells are typically performed using numerical methods due to the absence of synchronism between particle and RF field, and unavoidable phase slippage in RF field. Analysis of beam dynamics in the array of structures requires consideration of beam dynamics around the reference particle, which velocity, in the general case, might be significantly different from the geometrical phase velocity in each individual cavity. Analytical expressions provide a significant advantage in the analysis of beam dynamics versus numerical simulations. Various aspects of beam dynamics in such structures in the closed analytical form are presented in References [1][2][3][4][5][6]. In this paper, we develop an analytical treatment of longitudinal beam dynamics in an array of cavities with multiple identical cells, using Hamiltonian formalism. Beam Dynamics in Accelerating Section with Equidistant Cells Consider longitudinal beam dynamics in a structure with identical cells (see Fig. 1). Most of such structures in ion accelerators are -structures with cell length , where is the geometrical velocity and is the RF wavelength. Acceleration of particles in such field can be considered as dynamics in an equivalent traveling wave propagating along with the structure with constant phase velocity and with amplitude , where is the average field per accelerating gap, is the transit time factor (see Appendix A), and is the phase of a particle in traveling wave [7]: , where is the wave number. The phase is also a phase of a particle in the standing wave at the moment of time when the particle crosses the center of the accelerating gap. Differentiation of Eq. (1) along the longitudinal coordinate together with the equation for particle energy gain provides a set of equations for on-axis particle dynamics in traveling wave [1]: π β g λ / 2 β g λ = 2π / ω β g E = E o T (β ) E o T (β )(1), (2) ,(3) where m and q are mass and charge of particle, and is the relativistic factor (normalized particle energy). Because the phase velocity of the accelerating wave is constant, , the phase slippage is inevitable in this type of accelerating structure. Equations (2), (3) can be derived from Hamiltonian , where Hamiltonian equations are , . In the standing wave structure with identical cells, the average field per cell is constant, = const, and variation of particle velocity along the cavity is typically small, , therefore, the amplitude of accelerating field can be approximated to be constant . Because the geometrical velocity is also a constant, = const, the Hamiltonian, Eq. (4), is a constant of motion. From Hamiltonian, Eq. (4), the integral of particle motion in such field, , is (5) is a nonlinear equation, which determines the beam energy in the structure, , as a function of RF phase of particle , (6) where and are initial phase and energy, correspondingly. . (5) k z = 2π / (β g λ) ϕ z dϕ dz = 2π λ ( 1 β − 1 β g ) dγ dz = qE mc 2 cosϕ γ = (1− β 2 ) −1/2 β g = const H = 2π λ ( γ 2 − 1 − γ β g ) − qE mc 2 sinϕ dγ / dz = − ∂H / ∂ϕ dϕ / dz = ∂H / ∂γ E o Δβ / β << 1 E = E o T (β ) ≈ const β g C = Hλ / (2π ) γ 2 − 1 − γ β g − qEλ 2π mc 2 sinϕ = C Equation (b) Figure 2: (a) phase space trajectory of a particle in an RF structure with equidistant cells, (b) equivalent traveling wave. In the accelerating section with , the synchronous phase in each individual accelerator structure is , and acceleration is achieved as a rotation in phase space around the synchronous phase (see Fig. 2). To find the value of beam energy in closed form, let us express the constant in Eq. (5) through the value of RF phase , at which the particle velocity is equal to geometrical velocity, : γ ϕ γ 2 − 1 − γ β g = γ o 2 − 1 − γ o β g + qEλ 2π mc 2 (sinϕ − sinϕ o ) ϕ o γ o β g < 1 ϕ s = −90 o C ϕ m β = β g ,(7) where the energy corresponding to geometrical velocity of the cavity is . Using expansion , the Eq. (5) becomes . Equation (9) explicitly connects particle energy along accelerating structure, , with the phase of a particle in RF field, . The value of is determined from Eq. (9) by the initial value of beam phase , and initial energy : . Equation (9) determines two values of particle energy for each phase, depending on the cavity length: larger, , and smaller, , than that determined by the geometry of the structure, . The values of the final energy, , corresponding to the final phase are: , (11) where the negative sign is taken when , while the positive sign is taken when . Energy gain in accelerator structure of length can be expressed as , where is the effective phase of the particle in RF field of the cavity defined by: . To determine the phase slippage of particle in cavity, let us rewrite Eq. (2) as , where we took into account that . From Equations (9), (13), the derivative of RF phase of the particle over dimensionless time, , is C = β g γ g − γ g β g − qEλ 2π mc 2 sinϕ m γ g = (1− β g 2 ) −1/2 βγ ≈ β g γ g + (γ − γ g ) β g − 1 2 (γ − γ g ) 2 (β g γ g ) 3 (γ − γ g ) 2 (β g γ g ) 3 = qEλ π mc 2 (sinϕ m − sinϕ ) γ ϕ ϕ m ϕ o γ o sinϕ m = sinϕ o + π (β g γ g ) 3 mc 2 qEλ (γ g − γ o ) 2 γ f ≥ γ g γ f ≤ γ g γ g γ f ϕ f γ f = γ g ± qEλ(β g γ g ) 3 π mc 2 sinϕ m − sinϕ f γ f < γ g γ f > γ g L n ΔW = qE o T (β )L n cosϕ eff ϕ eff cosϕ eff = mc 2 (γ f − γ o ) qE o T (β ) L n dϕ dt = ω (1− β β g ) ≈ −ω (γ − γ g ) β g 2 γ g 3 dβ = dγ / (βγ 3 ) ωt .(14) The time of particle acceleration in the structure is determined by the integration of Equation (14): . Expanding RF phase of particle around as , the integral, Eq. (15), can be approximated as . (16) Equation (16) connects the dimensionless time of particle acceleration in the cavity, , with the phase slippage in RF field from to . The right hand of Equation (16) has a positive sign for , and a negative sign for . In case the particle trajectory in phase space passes the value of , like that illustrated in Figure 2, the time, , should be calculated as a sum of that required for phase variation from the initial value of to , and then from to final value : . For accelerating structures working on -mode, the number of accelerating cells is: , and the length of the cavity is . In the accelerating section with , ions are always slower than the phase velocity of the accelerating wave. The minimal energy for particles to be accepted into a continuous unlimited acceleration in such a structure (see Appendix B): . is too large for usual values of accelerating gradients and wavelength in ion accelerators. For example, for = 5 MeV/m, =1 m, the minimal required kinetic energy of protons should be W min = 275 GeV. However, dϕ d(ωt) = qEλ π mc 2 β g γ g 3 (sinϕ m − sinϕ ) Δ(ωt) = πβ g γ g 3 ( mc 2 qEλ ) dϕ sinϕ m − sinϕ ϕ o ϕ f ∫ ϕ ϕ m sinϕ ≈ sinϕ m + (ϕ − ϕ m )cosϕ m − 0.5(ϕ − ϕ m ) 2 sinϕ m Δ(ωt) ≈ 2π β g γ g 3 mc 2 qEλ sinϕ m {arcsin[1+ (ϕ m − ϕ f )tanϕ m ] − arcsin[1+ (ϕ m − ϕ o )tanϕ m ]} Δ(ωt) ϕ o ϕ f ϕ f > ϕ o ϕ f < ϕ o ϕ m Δ(ωt) ϕ o ϕ m ϕ m ϕ f Δ(ωt) = Δ(ωt) ϕ o ϕ m + Δ(ωt) ϕ m ϕ f π N cell ! Δ(ωt) π L n = N cell β g λ 2 β g = 1 γ min = 1+ ( qEλ π mc 2 ) 2 2 qEλ π mc 2 E λ ion beams can be accelerated in finite-length sections with , while particle RF phase is varied within (see Fig. 10). From Eq. (6), the final energy of the particle in the accelerator section with is , which is a unique function of , , , in contrast with the section with . The effective phase of the particle in RF field is determined by Eq. (12). From Eq. (2) for sections with , , the time of particle acceleration in the section is . (22) The particle velocity along the section as a function of RF phase is determined by Eq. (B-2): . Integration of Eq. (22) connects the time of particle acceleration within the cavity with phase slippage in RF field: . For a typical case of ion acceleration below the minimal energy , the traveling time can be estimated as , where is the average velocity of a particle in the cavity. Beam Dynamics in Array of Cavities The dynamics of the beam in an array of accelerating cavities can be described in classical terms of particle oscillations around the synchronous phase of reference (synchronous) particle, which velocity is equal to that of the effective traveling wave . The dynamics of the reference particle is determined by the geometry of the accelerating channel and shifts of RF phases between cavities. While the reference particle travels from the center of the last cell of cavity ( ) to the center of the first cell of cavity ( ) separated by the distance (see Fig. 1), the phase of RF field is changed in each cavity by β g = 1 −π / 2 < ϕ < π / 2 β g = 1 γ f = 1 2 [γ o − γ o 2 − 1 + ( qEλ 2π mc 2 )(sinϕ o − sinϕ f ) + 1 γ o − γ o 2 − 1 + ( qEλ 2π mc 2 )(sinϕ o − sinϕ f ) ] ϕ o ϕ f γ o β g < 1 β g = 1 dϕ / dt = ω (1− β ) Δ(ωt) = dϕ 1− β ϕ o ϕ f ∫ β(ϕ) = 1− [γ o − γ o 2 − 1 + ( qEλ 2π mc 2 )(sinϕ o − sinϕ )] 2 1+ [γ o − γ o 2 − 1 + ( qEλ 2π mc 2 )(sinϕ o − sinϕ )] 2 Δ(ωt) = ϕ f − ϕ o 2 + 1 2 dϕ [γ o − γ o 2 − 1 + ( qEλ 2π mc 2 )(sinϕ o − sinϕ )] 2 ϕ o ϕ f ∫ γ min Δ(ωt) ≈ ϕ f − ϕ o 1− β β ϕ s (z) β s (z) n n + 1 ! d n the value , where . From that condition, the velocity of reference particle after cavity ( ) is ,(26) where is the difference in RF phases of cavities, which includes the integer number of RF periods and a fractional part : From Eq. (26), the profile of velocity of reference particle along accelerator does not depend on the amplitude of RF field in cavities [2]. The effective synchronous phase of the linac is determined by the rate of increase of velocity of the reference particle along with the machine. From Eq. (3), taking into account that , the expression for synchronous phase is , where is the amplitude of equivalent traveling wave propagating along the linac. Within the cavity, the velocity of reference particles can be approximated as . The amplitude is the ratio of the cavity voltage to the effective length occupied by the cavity, , which includes the cavity length, and the halves of drift spaces between cavities (see Fig. 1 ) ,(30) where and are the average fields in RF gaps and the transit time factor in cavity ( ), correspondingly. The velocity of the reference particle is changing within the cavity from to , therefore, the rate of increase of velocity of the reference particle in the cavity with the number ( ) is . Combining Eqs. (28), (29), (31), the synchronous phase of the linac at the cavity ( ) is determined as: . (32) φ = ωt d t d = ! d n / (β s c) n β n = 2π ! d n λ (ϕ n − ϕ n+1 ) ϕ n − ϕ n+1 Δϕ n ϕ n − ϕ n+1 = 2π m − Δϕ n m = 0, 1, 2,.. dγ = βγ 3 dβ cosϕ s (z ) = 1 qE mc 2 β s γ s 3 dβ s dz E β s _ n = β n−1 + β n 2 E U n = E o _ n L n L n + 0.5(d n + d n+1 ) E = E o _ n T n (β s _ n ) L n L n + 0.5(d n + d n+1 ) E o _ n T n (β s ) n β n−1 β n n dβ s dz ≈ β n − β n−1 L n + 0.5(d n + d n+1 ) n cosϕ s _ n = mc 2 qE o _ n T n L n β s _ n γ s _ n 3 (β n − β n−1 ) The values of , , define the dynamics of the reference particle in the equivalent traveling wave and are entirely determined by accelerator channel. The beam velocity and effective phase , Eq. (12), do not nesseseraly coincide with , , creating a mismatch between the beam and accelerating wave. The performed analysis allows us to determine normalized acceptance of accelerator and matched conditions for the beam in linac. The separatrix of longitudinal particle motion in a traveling wave with amplitude , phase velocity , and synchronous phase is defined as [7] , where is the deviation of particle momentum from that of synchronous particle, and is the phase deviation from synchronous phase (see Fig. 3). The phase width of separatrix is , and the half-width of separatrix in momentum, , is [7] , where is the dimensionless frequency of small-amplitude oscillation around synchronous particle: . (35) β s (z) ϕ s (z) E(z) ϕ eff β s (z) ϕ s (z) E β s ϕ s p ζ 2 2mγ 3 + qEβ s λ 2π [sin(ϕ s +ψ )-ψ cosϕ s +sinϕ s -2ϕ s cosϕ s ] = 0 p ζ = p z − p s ψ = ϕ s − ϕ Φ s ≈ 3 ϕ s p ζ sep p ζ sep mc = 2β s γ s 3 Ω ω 1- ϕ s tgϕ s Ω / ω Ω ω = qEλ mc 2 sinϕ s 2πβ s γ s 3 p ζ = p z − p s ζ = z − z s H = p ζ 2 2mγ 3 + mγ 3 Ω 2 ζ 2 2(36) with the maximum value of determined by Eq. (34) (see Fig. 3). The normalized longitudinal acceptance, , is specified as . For small absolute values of synchronous phase, , one can use expansion , and the normalized longitudinal acceptance can be approximated as . ( Within this approximation, the effective phase length of separatrix is estimated as , which gives a realistic evaluation of the phase width of the separatrix, because the tail of the separatrix is rarely populated by particles. According to Eq. (39), the longitudinal acceptance increases with energy as . Equation (36) determines zero-intensity averaged matched beam with given longitudinal emittance , where longitudinal beam radius , and beam half-momentum spread, are (see Fig. 3): , . The more exact matched conditions with varied beam ellipse along accelerator should be specified using numerical methods [8]. In presence of strong space charge forces, the matched conditions are modified taking into account coupling between transverse and longitudinal particle oscillations due to space charge forces [9]. Beam Dynamics in LANSCE 805 MHz Linac Los Alamos Linear Accelerator H = const ζ sep mγ 3 Ω 2 ζ sep 2 2 = p ζ sep 2 2mγ 3 ζ sep = 2 β s c ω 1− ϕ s tanϕ s ε acc = ζ sep p sep / (mc) ε acc = 2 π λ β 2 γ 3 ( Ω ω )(1− ϕ s tanϕ s ) ϕ s tanϕ s ≈ ϕ s + ϕ s 3 / 3 ε acc ≈ 2 3π β 2 γ 3 ( Ω ω ) ϕ s 2 λ = 2 3π 3/2 (β γ λ) 3/2 ϕ s 5/2 qE mc 2 Φ s eff = 2π (2ζ sep ) βλ = 4 1− ϕ s tgϕ s ≈ 4 ϕ s 3 = 2.31ϕ s ε acc~( β γ ) 3/2 ε z R z p ζ (R z ) matched = ε z λ 2πγ 3 ( ω Ω ) ( p ζ mc ) matched = 2π γ 3 ε z λ ( Ω ω ) The structure of CCL accelerator is illustrated in Fig. 4, 5, and the parameters of the machine are presented in Table 1 Ω / ω = 9.3⋅10 −3 p ζ sep / mc = 3.8 ⋅10 −3 ε acc = 7.46 π cm mrad Experimental determination of longitudinal beam emittance in the accelerator is performed through measurement of the longitudinal beam size after Tank 3 in the DTL at a beam energy of 70 MeV [12], and measurement of momentum spread of the 800-MeV beam in a high-dispersive point of high-energy beam transport [13]. The typical value of the phase length of the bunch at 70 MeV is 7 o on 201.25 MHz scale, and that of the 800 MeV beam momentum spread is . Due to adiabatic damping of phase oscillations in a linear accelerator, the momentum spread is changing as [7] . ( . The initial phase length of the matched beam is , while for the mismatched beam it was selected to be . Longitudinal oscillations of the matched beam remain close to linear within separatrix, resulting in a small emittance growth (see Fig. 7a). The larger longitudinal size of the mismatched beam causes stronger nonlinear dependence of longitudinal oscillation frequency on amplitude, resulting in creation of tails in longitudinal phase space, and to a noticeable beam emittance growth (see Fig. 7b). Summary The beam dynamics in independent phased cavities was analyzed using the Hamiltonian approach. The analysis is based on the representation of the accelerating field as an equivalent traveling wave. Analytical expressions for energy gain and phase slippage in individual accelerating cavities were evaluated. The particle dynamics in a sequence of cavities with equidistant cells was analyzed as a dynamics around synchronous (reference) particle in an effective accelerating wave of the whole machine. Longitudinal acceptance and matched beam conditions are determined. Developed analysis was applied to the dynamics of the beam in LANSCE 805 MHz Coupled Cavity Linac. Δp / p ≈ 10 −3 Δp p ∼ 1 β 5/4 γ 1/4 ε z = 4ε z _ rms R z = p ζ / mc = 1.26 ⋅10 −3 µ t / µ s µ z / µ zo ϕ o − ϕ m ≈ 10 o ϕ s = −30 o Φ s eff ≈ 70 o 25 o 43.75 o Δε z / ε z = 0.058 Δε z / ε z = 0.28 Appendix A. Transit Time Factor in Sections with Identical Cells Consider -type accelerating structure based on a combination of identical cells of length , where is a constant value of geometrical particle velocity. Structure containing cells has a total length of . During acceleration, particle velocity becomes unavoidably different from the constant value of the geometrical velocity of the structure, , resulting in reduction of the transit time factor [1]. In a structure with an odd number of cells, the center of the structure, , is located in the geometrical center of the middle gap (see Fig. 8 a). The transit time factor of such a structure is [1] , (A-1) where is the on-axis distribution of accelerating field. Integration of Eq. (A-1) is performed within the structure length . Expression (A-1) is a product of two values , (A-2) where is the transit time factor for particle with , , (A-3) and is the normalized factor, which represents reduction of transit time factor due to difference in design and actual particle velocities : . (A-4) For harmonic-type distribution of the field along the axis, , typical for superconducting structures, the value of [1], and the normalized factor is for odd . (A-5) π β g λ / 2 β g N L s = Nβ g λ / 2 β ≠ β g z = 0 T = ∫ − L s /2 L s /2 E g (z )cos( 2π z βλ )dz ∫ − L s /2 L s /2 E g (z )dz = [ ∫ − L s /2 L s /2 E g (z )cos( 2π z β g λ )dz ∫ − L s /2 L s /2 E g (z )dz ] [ ∫ − L s /2 L s /2 E g (z )cos( 2π z βλ )dz ∫ − L s /2 L s /2 E g (z ) cos( 2π z β g λ )dz ] E g (z ) − N β g λ / 4 ≤ z ≤ Nβ g λ / 4 T = T o ⋅T s (N ,β / β g ) T o β = β g T o = ∫ − L s /2 L s /2 E g (z )cos( 2π z β g λ )dz ∫ − L s /2 L s /2 E g (z )dz T s (N ,β / β g ) β ≠ β g T s (N ,β / β g ) = ∫ − L s /2 L s /2 E g (z ) cos( 2π z βλ )dz ∫ − L s /2 L s /2 E g (z ) cos( 2π z β g λ )dz E g (z ) = E max cos(2π z / β g λ) T o = π / 4 T s (N ,β / β g ) = − L s /2 L s /2 ∫ cos( 2π z β g λ ) cos( 2π z βλ )dz − L s /2 L s /2 ∫ cos 2 ( 2π z β g λ )dz N In structures with even number of cells, the center of the structure is located at the null of the longitudinal electric field (see Fig. 8 b). For such a structure, the time of arrival of the particle to the point with the coordinate is , (A-6) where is the phase of RF field at the time of particle arrival to the center of the accelerating gap. Energy gain per structure is given by . (A-7) Because the filed distribution is odd function of z, the second integral in expression (A-7) is equal to zero and the transit time factor for such structure is expressed as [1] . Combining Eqs. (A-2), (A-5), (A-9), the normalized factor in a structure with arbitrary number of cells is [14]: . (A-10) Figure 9 illustrates the value of normalized factor for structures with various cell numbers. Appendix B. Dynamics in Accelerating Sections with βg = 1 Analysis of beam dynamics in accelerating sections with is well developed for electron linacs [15]. In accelerating sections with all particles are slower than the phase velocity of the accelerating wave, and there is no synchronous particle. The Hamiltonian, Eq. (4), describing particle dynamics in this structure, is z t(z) = 1 ω (ϕ + π 2 ) + dz β(z)c 0 z ∫ ϕ ΔW = −q cosϕ [ E g (z) sin(k z z)dz − L s /2 L s /2 ∫ + tanϕ E g (z) cos(k z z)dz − L s /2 L s /2 ∫ ] E g (z) T = ∫ − L s /2 L s /2 E g (z )sin( 2π z βλ )dz ∫ − L s /2 L s /2 E g (z )dz T s (N ,β / β g ) = − L s /2 L s /2 ∫ sin( 2π z β g λ ) sin( 2π z βλ )dz − L s /2 L s /2 ∫ sin 2 ( 2π z β g λ )dz N T s (N ,β / β g ) = sin π N 2 ( β g β − 1) π N 2 ( β g β − 1) − (−1) N sin π N 2 ( β g β + 1) π N 2 ( β g β + 1) T s β g = 1 β g = 1 . (B-1) The integral of motion, Eq. (5), can be expressed through particle velocity or particle energy: , or (B-2) . (B-3) For particles, which energy continuously increases in the field of accelerating wave, the particle velocity asymptotically approaches the value of unit, Particles with energy cannot be captured into continuous unlimited acceleration, however, they can be accelerated in the limited-length sections while particle RF phase is varied within the interval . H = 2π λ (βγ − γ ) − qE mc 2 sinϕ 1− β 1+ β + qEλ 2π mc 2 sinϕ = C γ − γ 2 − 1 + qEλ 2π mc 2 sinϕ = C β ∞ → 1 qEλ 2π mc 2 sinϕ ∞ = C ϕ ∞ C ≤ qEλ / (2π mc 2 ) γ o ≥ 1 2 [ qEλ 2π mc 2 (1− sinϕ o ) + Figure 1 : 1Accelerating structure of independently phased cavities. Figure 3 : 3Phase space trajectories, (dotted) elliptical approximation of separatrix, (bold) normalized longitudinal emittance of matched beam. The separatrix, Eq. (33), can be approximated in the phase space of canonical-conjugate variables , by the Hamiltonian , [10] consists of 201.25 MHz Drift Tube Linac accelerating particles from 0.75 MeV to 100 MeV, and 805 MHz Side-Coupled Linac (CCL), accelerating particles from 100 MeV to 800 MeV. The CCL consists of 104 tanks, which are grouped into 44 accelerating modules (modules 5-48). . Energy gain per module is changing from 13 MeV at the beginning of the linac to 16 MeV at the end of linac. The linac has a length ~ 700 m with an average real estate gradient of = 1 MeV/m, and average accelerating field = 1.15 MV/m, providing acceleration with the synchronous phase .(43)The smallest value of longitudinal acceptance is at the beginning of CCL acceleration, where particle momentum has a value of . Equations (34), (35), (38) provide the following parameters for the longitudinal phase space area of accelerator: dimensionless longitudinal oscillation frequency , half -width of separatrix in momentum , and longitudinal acceptance . Figure 4 : 4Accelerating tanks of 805 MHz Coupled-Cavity linac separated by quadrupole doublets. Figure 5 : 5Layout of 805 MHz LANSCE linac[11]. of the beam size and momentum spread gives an estimate of the longitudinal normalized beam emittance at 70 MeV as = 0.7 π-cm-mrad. Equations (41), (42) give the values of matched beam size 5.9 mm and half-momentum spread at the beginning of linac.Space charge depression parameters of transverse and longitudinal oscillations are correspondingly, and weakly affect zero-intensity matching parameters.Figures 6 and 7illustrate the longitudinal dynamics of the beam in a sequence of 805 MHz CCL tanks of matched and mismatched beams. In the accelerator, particles are injected into each tank with momentum, lower than the geometrical value, and extracted with momentum, larger than the geometrical value. The initial and final RF phases are approximately the same. A typical value of phase slippage within the tank is . According to Eq. (40), the effective separatrix length for a structure with is Figure 6 : 6Dynamics in 805 MHz linac sections with constant : (a) matched beam, (b) mismatched beam. Figure 7 : 7Longitudinal phase-space distribution of the beam in LANSCE 805 MHz linac: (a) matched beam, (b) mismatched beam. Eq. (A-2), the transit time factor, Eq. (A-8), can be represented as a product of two terms with Figure 10 10the asymptotic value of RF phase. Therefore, for particles captured into continuous unlimited acceleration, the value of constant, Eq. (B-4), is restricted as . From Eqs. (B-3), (B-4), the energy of injected particles to be captured into continuous unlimited acceleration has to be .(B-5)The equity sign in Eq. (B-5) determines boundary phase space trajectory (separatrix) illustrates phase space trajectories of particles in structure with . Only particles whose initial energy is larger than that, determined by separatrix equation, , can be captured into continuous unlimited acceleration. The equation defines the phase , where the initial particle energy required for unlimited acceleration, has the minimal possible value: γFigure 8 : 8o (ϕ ) ≥ γ sep (ϕ )∂γ sep / ∂ϕ = 0 < γ min −π / 2 < ϕ < π / 2 Field distribution in a -structure with (a) odd number of cells, (b) even number of cells. Figure 9 : 9Normalized transit time factor for -type accelerating structure with constant phase velocity for different values of cell numbers. Figure 10 . 10Longitudinal particle trajectories in the field with β s = 1, bold lines are separatrices. The area of separatrix in variablesis the normalized longitudinal acceptance of accelerator. The value of Hamiltonian, Eq. (36), is constant along the elliptical trajectory, , therefore, the longitudinal half-size of separatrix, , is given by , or . Table 1 . 1Parameters of LANSCE 805-MHz linac.Module Energy (MeV) Length (m) UT (MV) 5 113.02200 11.79010 16.09360 6 125.92400 11.68780 15.84787 7 139.39500 12.20490 16.44505 8 153.41299 12.70280 17.00951 9 167.95799 13.18170 17.54443 10 182.10600 12.82690 16.96636 11 196.68500 13.21910 17.38345 12 211.67900 13.59650 17.77822 13 226.37399 13.29720 17.32801 14 240.91701 13.16380 17.05645 15 255.77499 13.45020 17.33382 15 270.93399 13.72600 17.59341 17 286.38599 13.99150 17.84242 18 302.11700 14.24700 18.16461 19 317.56900 13.99770 17.84242 20 333.26401 14.21880 18.12303 21 349.19299 14.43180 18.39320 22 365.34699 14.63700 18.65302 23 381.71701 14.83470 18.90247 24 397.70902 14.49470 18.46597 25 413.88901 14.66560 18.68304 26 430.24899 14.83040 18.89088 27 446.78400 14.98940 19.09297 28 463.48700 15.14270 19.28696 29 480.35199 15.29070 19.47401 30 496.75000 14.86970 18.93479 31 513.28802 14.99770 19.09646 32 529.96301 15.12140 19.25461 33 546.76001 15.24080 19.39550 34 563.69898 15.35640 19.55943 35 580.75299 15.46800 19.69228 36 597.27399 14.98620 19.07680 37 613.90002 15.08280 19.19809 38 630.63000 15.17650 19.31811 39 647.45801 15.26710 19.43130 40 664.38300 15.35500 19.54329 41 681.40100 15.44000 19.65070 42 698.50897 15.52240 19.75458 43 715.70502 15.60220 19.85628 L a T Wangler, RF Linear Accelerators. New YorkWileyT. Wangler, RF Linear Accelerators, Wiley, New York, 2008. Beam Dynamics Basics in RF Linacs. N Pichoff, CERN Accelerator School. N. Pichoff, "Beam Dynamics Basics in RF Linacs", CERN Accelerator School, Zeegse, The Netherlands, 24 May -2 June 2005, Editor D. Brandt, CERN-2006-012 (2006). Linear Accelerators. M Vretenar, CERN-20154-009CAS-CERN Accelerator School; Advanced Accelerator Physics. W. HerrM. Vretenar, "Linear Accelerators", CAS-CERN Accelerator School; Advanced Accelerator Physics, Trondheim, Norway, 19-29 August 2013, Edited by W. Herr, CERN-20154-009 (2014). The Effect of Phase Slippage in Multicell Cavities on Longitudinal Beam Dynamics. F Gerigk, CERN-NUFACT-NOTE 2001-072F. Gerigk, "The Effect of Phase Slippage in Multicell Cavities on Longitudinal Beam Dynamics", PS/RF Note 2001-009, CERN-NUFACT-NOTE 2001-072 (2001). Superconducting, energy variable heavy ion linac with constant , multicell cavities of CH-type. S Minaev, Phys. Rev. Special Topics -Accelerators and Beams. 12120101S. Minaev, et al "Superconducting, energy variable heavy ion linac with constant , multicell cavities of CH-type", Phys. Rev. Special Topics -Accelerators and Beams 12, 120101 (2009). Multi-Charge Beam Dynamics in an Ion Linac. P N Ostroumov, K W Shepard, Phys. Rev. Special Topics -Accelerators and Beams. 330101P.N. Ostroumov and K.W. Shepard, "Multi-Charge Beam Dynamics in an Ion Linac", Phys. Rev. Special Topics -Accelerators and Beams 3, 030101 (2000). I M Kapchinskiy, Theory of Resonance Linear Accelerators. HarwoodI.M. Kapchinskiy, Theory of Resonance Linear Accelerators, Harwood, 1985. Advanced Approach for Beam Matching along the Multi-Cavity SC CW Linac at GSI. S Yaramyshev, J. Phys. Conf. Series. 106752005S. Yaramyshev et al "Advanced Approach for Beam Matching along the Multi-Cavity SC CW Linac at GSI", J. Phys. Conf. Series 1067 052005 (2018). Six-dimensional matching of intense beam with linear accelerating structure. Y K Batygin, Nuclear Inst. and Methods in Physics Research. 995165074AY.K. Batygin, "Six-dimensional matching of intense beam with linear accelerating structure", Nuclear Inst. and Methods in Physics Research, A 995 (2021) 165074. LANSCE High Power Operations and Maintenance Experience. K W Jones, P W Lisowski, Proceedings of the Sixth International Topical Meeting on Nuclear Applications of Accelerator Technology. the Sixth International Topical Meeting on Nuclear Applications of Accelerator TechnologySan Diego, CaliforniaK.W. Jones and P.W. Lisowski, "LANSCE High Power Operations and Maintenance Experience," in Proceedings of the Sixth International Topical Meeting on Nuclear Applications of Accelerator Technology, San Diego, California, 1-5 June 2003, pp. 372-375. . H Takeda, J H Billen, LA-UR-98-4478H. Takeda, J.H. Billen, LA-UR-98-4478 (1998). Bunch Shape Monitor Measurement at the LANSCE Linac. I Draganic, Proc. of NAPAC 2016. of NAPAC 2016Chicago, IL USAI. Draganic et al, "Bunch Shape Monitor Measurement at the LANSCE Linac", Proc. of NAPAC 2016, Chicago, IL USA, MOA3CO03 (2016). Experimental study of passive compensation of space charge at the Los Alamos National Laboratory Proton Storage Ring. M A Plum, Phys. Rev. Special Topics -Accelerators and Beams. 264201M. A. Plum et al, "Experimental study of passive compensation of space charge at the Los Alamos National Laboratory Proton Storage Ring", Phys. Rev. Special Topics -Accelerators and Beams 2, 064201 (1999). Transit Time Factor of a Multi-Cell Standing Wave Cavity. J.-F Ostiguy, FermilabReportJ.-F.Ostiguy, "Transit Time Factor of a Multi-Cell Standing Wave Cavity", Fermilab Report (2017). Formulae and Procedures Useful for the Design of Linear Accelerators. P Lapostolle, M Weiss, CERN-PS-2000-001P. Lapostolle and M. Weiss, "Formulae and Procedures Useful for the Design of Linear Accelerators", CERN-PS-2000-001 (2000).

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{'abstract': 'Linear accelerators containing the sequence of independently phased cavities with constant geometrical velocity along each structure are widely used in practice. The chain of cavities with identical cell lengths is utilized within a certain beam velocity range, with subsequent transformation to the next chain with higher cavity velocity. Design and analysis of beam dynamics in this type of accelerator are usually performed using numerical simulations. A full theoretical description of particle acceleration in an array of independent phased cavities has not been developed. In the present paper, we provide an analytical treatment of beam dynamics in such linacs employing Hamiltonian formalism. We begin our analysis with an examination of beam dynamics in an equivalent traveling wave of a single cavity, propagating within accelerating section with constant phase velocity. We then consider beam dynamics in arrays of cavities, utilizing an effective traveling wave propagating along with the whole accelerator with the velocity of synchronous (reference) particle. The analysis concluded with the determination of the matched beam conditions. Finally, we present a beam dynamics study in 805 MHz Coupled Cavity Linac of the LANSCE accelerator facility.', 'arxivid': '2302.09431', 'author': ['Y K Batygin \nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n'], 'authoraffiliation': ['Los Alamos National Laboratory\n87545Los AlamosNMUSA'], 'corpusid': 250512685, 'doi': '10.1016/j.nima.2022.167192', 'github_urls': [], 'n_tokens_mistral': 11478, 'n_tokens_neox': 9271, 'n_words': 5723, 'pdfsha': '6be41a38457f4c9a4045fa319fa3a7676025cf28', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09431v1.pdf'], 'title': ['Beam Dynamics in Independent Phased Cavities', 'Beam Dynamics in Independent Phased Cavities'], 'venue': ['Nuclear Instruments and Methods in Physics Research']}

arxiv

AN EFFICIENT THRESHOLD DYNAMICS METHOD FOR TOPOLOGY OPTIMIZATION FOR FLUIDS Huangxin Chen Haitao Leng ANDDong Wang Xiao-Ping Wang AN EFFICIENT THRESHOLD DYNAMICS METHOD FOR TOPOLOGY OPTIMIZATION FOR FLUIDS We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model. 2010 Mathematics Subject Classification. 35K93, 35K05, 65M12, 35Q35, 49Q10, 65M60, 76S05. Introduction Topology optimization in fluid mechanics has become a significant problem due to its application in many industrial problems such as the optimization of transport vehicles and biomechanical structure. The process of topology optimization allows the introduction of new boundaries as part of the solution and is thus more flexible than shape optimization, which requires that the topology be predetermined. The method of topology optimization was originally developed for the optimal design in structural mechanics [4,5] and has been applied in a variety of physical fields such as acoustics, electromagnetics, fluid flow, and thermal problems [5,7,13,45,34,11]. Topology optimization was first applied to fluid mechanics by Borrvall and Petersson [7] by adopting the concept of density methods to Stokes flows. In [7], the domain with fluidsolid regions was treated as the porous medium, the Brinkman flow was introduced to obtain a well-posed problem to minimize the total dissipation power, and the discrete optimization problem was further solved with the method of moving asymptotes (MMA) [46] to obtain the optimal designed regions for fluids and solids. Topology optimization in fluid mechanics has since been extended to the Darcy-Stokes flow model [23,53], Navier-Stokes flow [21,37,55,12,19,49], and non-Newtonian flow [40], and it has also been applied in the design of more complicated fluidic devices [2,35,36]. Several successful methods have also been recently introduced to improve the performance of topology optimization in fluid mechanics. For instance, the level set method was applied to fluidic topology optimization (cf. [55,8,49] and the references therein), and the fluid-solid interface is described by the zero-level set of a level set function. In [49], the authors further studied the fluidic topology optimization framework by combining the level set method and the extended finite-element method. Phase field-based topology optimization for fluids was considered in [19], in which the gradient flow method was used to find the optimal topology. Among these methods, a critical step is to update the fluid-solid regions by solving the Hamilton-Jacobi equations in the level set method [55], by solving a parameter optimization problem via a nonlinear programming method [49], or by solving the Cahn-Hilliard or Allen-Cahn system via the phase field approach [19]. The threshold dynamics method developed by Merriman, Bence, and Osher (MBO) [30,31,32] is an efficient method for simulation of the motion of the interface driven by the mean curvature. To be more precise, let D ⊂ R d be a domain whose boundary Γ = ∂D is to be evolved via motion by mean curvature. The MBO method is an iterative method, and at each time step, it generates a new interface, Γ new (or equivalently, D new ) via the following two steps: Step 1. Solve the Cauchy initial value problem for the heat diffusion equation until time t = τ , u t = ∆u, u(t = 0, ·) = χ D , where χ D is the indicator function of domain D. Letũ(x) = u(τ, x). Step 2. Obtain a new domain D new with boundary Γ new = ∂D new by D new = x :ũ(x) ≥ 1 2 . The MBO method has been shown to converge to continuous motion by mean curvature [3,9,18,47]. Esedoglu and Otto gave a variational formulation for the original MBO scheme and successfully generalized this type of method to multiphase problems with arbitrary surface tensions [15]. The method has attracted considerable attention due to its simplicity and unconditional stability. It has since been extended to deal with many other applications, including the problem of area-preserving or volume-preserving interface motion [44], image processing [51,17,29], problems of anisotropic interface motion [33,43,6,14], the wetting problem on solid surfaces [54], the generation of quadrilateral meshes [48], graph partitioning and data clustering [20], and auction dynamics [25]. Various algorithms and rigorous error analysis have been introduced to refine and extend the original MBO method and related methods for these problems (see, e.g., [16,24,32,42,41,50]). Adaptive methods have also been used to accelerate this type of method [26] based on nonuniform fast Fourier transform. Laux et al. [27,28] rigorously proved the convergence of the method proposed by [15], and a generalized manifold-valued threshold dynamics method was developed by [38,52,39]. In this paper, we introduce an efficient and simple strategy based on the threshold dynamics method to update the topology of fluid-solid regions. In our approach, the total energy consists of the dissipation power in the fluid and the perimeter regularization and is subject to a fluid volume constraint and an incompressibility condition. The perimeter term is based on convolution of the heat kernel with the characteristic functions of regions. Based on minimization of an approximate total energy, an efficient threshold dynamics method is derived for topology optimization for fluids. The porous medium approach is used in our algorithm, and we introduce the Brinkman equation, which "interpolates" between the Stokes equation in the flow region and some Darcy flow through a porous medium (a weakened nonfluid region). We then solve the Brinkman equation for the whole domain by the standard mixed finite-element method and update the fluid-solid regions by convolution and with a simple thresholding step. In particular, the convolutions can be efficiently computed on a uniform grid by fast Fourier transform (FFT) with the an optimal complexity of O(N log N ). The proposed algorithm is very simple and easy to implement. Extensive numerical results show that the proposed algorithm converges at many fewer iterations than the method given by [7], which indicates the high efficiency of the proposed algorithm. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters. We also show that the method has the total energy decaying property. The paper is organized as follows. In Section 2, we show the mathematical model. In Section 3, we introduce an approximate energy to the total energy and derive an efficient threshold dynamics method. The unconditional stability of the threshold dynamics method (i.e., the energy decaying property) is proved in Section 4. We discuss the numerical implementation in Section 5 and verify the efficiency and the energy decaying property of the algorithm in Section 6. We make conclusions, and discuss some ideas for future work in Section 7. Mathematical model In this section, we consider the mathematical model for topology optimization for fluids in Stokes flow. Denote Ω ∈ R d (d = 2, 3) as the computational domain, which is fixed throughout optimization, and assume that Ω is a bounded Lipschitz domain with an outer unit normal n such that R d \Ω is connected. Furthermore, we denote Ω 0 ⊂ Ω as the domain of the fluid, which is a Caccioppoli set 1 and Ω \ Ω 0 ∈ Ω as the solid domain. Throughout the paper, we use the standard notations and definitions for Sobolev spaces (cf. [1]). Our goal is to determine an optimal shape of Ω 0 that minimizes the following objective functional consisting of the total potential power and a perimeter regularization term, min (Ω0,u) J 0 (Ω 0 , u) = Ω µ 2 |Du| 2 − u · f dx + γ|Γ| (2.1) subject to ∇ · u = 0, in Ω, (2.2a) ∇p − ∇ · (µ∇u) = f , in Ω 0 , (2.2b) u = 0, in Ω \ Ω 0 , (2.2c) u| ∂Ω = u D , on ∂Ω, (2.2d) |Ω 0 | = β|Ω| with a fixed parameter β ∈ (0, 1). (2.2e) Here, u : Ω → R d , Du is the distributional derivative of u, µ is the dynamic viscosity of the fluid, p is the pressure, u D : ∂Ω → R d is a given function, f : Ω → R d is a given external force, |Γ| is the perimeter of the boundary of Γ = ∂Ω 0 , and γ > 0 is a weighting parameter. Derivation of the algorithm In this section, we develop an efficient threshold dynamics method for the topology optimization problem discussed in (2.1) and (2.2) for fluids in Stokes flow. Note that the goal is to determine the optimal interface between liquid and solid that minimizes functional (2.1) subject to constraints (2.2). Motivated by the idea from the threshold dynamics methods developed by [15], [54], [51], we use the indicator functions for the fluid region and the solid region to implicitly represent the interface. 3.1. Approximate energy. Define an admissible set B as follows: B :={(v 1 , v 2 ) ∈ BV (Ω) | v i (x) = {0, 1}, v 1 (x) + v 2 (x) = 1 a.e. in Ω, and Ω v 1 dx = V 0 }, (3.1) where BV (Ω) is the vector space of functions with bounded variation in Ω, and V 0 is the fixed volume of the fluid region. We introduce χ 1 (x) to denote the indicator function of the fluid region Ω 0 , i.e., χ 1 (x) := 1, if x ∈ Ω 0 , 0, otherwise, and χ 2 (x) as the indicator function of Ω \ Ω 0 , i.e., χ 2 (x) = 1 − χ 1 (x). The interface Γ is then implicitly represented by χ 1 and χ 2 . Let χ = (χ 1 , χ 2 ) and we have χ ∈ B. It is well known that the perimeter of the interface Γ can be approximated by, |Γ| ≈ π τ Ω χ 1 G τ * χ 2 dx, (3.2) where G τ (x) = 1 (4πτ ) d 2 exp − |x| 2 4τ is the Gaussian kernel (See [15]). We solve the optimization problem (2.2) by iteration. At each iteration, one must solve the Stokes equation in the fluid domain, which is changing in the iteration. It is more convenient numerically to use the porous medium approach as in [19,10]. The idea is to "interpolate" between the Stokes equation in the fluid domain (i.e., {x| χ 1 (x) = 1}) and u = 0 in the solid domain (i.e., {x| χ 2 (x) = 1}) by introducing an additional penalization term, ∇ · u = 0, in Ω, (3.3a) ∇p − ∇ · (µ∇u) + α(x)u = f , in Ω, (3.3b) u| ∂Ω = u D , on ∂Ω. (3.3c) Here, α(x) is a smooth function that varies between 0 andᾱ τ through a thin interface layer Γ, andᾱ −1 τ is the permeability. In the current representation of the interface, we use the 0.5 level set of φ = G τ * χ 2 to approximate the position of the interface Γ. It is well known that such φ is a smooth function between [0, 1] and admits a change from 0 to 1 in an O( √ τ ) thin layer. Hence, α is given by α =ᾱ τ φ =ᾱ τ G τ * χ 2 . (3.4) In the limiting model (i.e., τ 0),ᾱ τ should be set as +∞ to make the constraints {u = 0 in Ω \ Ω 0 } satisfy. Also, to ensure that the velocity vanishes outside the fluid domain when τ 0, we add a penalty termᾱ τ 2 G τ * χ 2 |u| 2 to the objective functional. In subsequent calculations, for numerical consideration, we fixᾱ τ as a sufficiently large constant,ᾱ. In this porous media approach, the system (3.3) is solved for a fixed domain Ω. Finally, combining (2.1), (3.2), (3.4), and the penalty term, we arrive at the following approximate objective functional J τ (χ, u) = Ω µ 2 |Du| 2 +ᾱ 2 |u| 2 G τ * χ 2 − u · f + γ π τ χ 1 G τ * χ 2 dx. (3.5) Remark 3.1. For simplicity, we use the same τ in the second and the fourth terms of the above approximate energy. Indeed, one can also use different values of τ in the two terms and the property of the algorithm will be similar. Now, we consider the following approximate formulation of the problem by min (χ,u) J τ (χ, u), subject to χ = (χ 1 , χ 2 ) ∈ B and u satisfy (3.3). (3.6) In the following, we give the derivation of the threshold dynamics scheme to solve (3.6). 3.2. Derivation of the scheme. In this section, we use a coordinate descent algorithm to minimize the approximate energy (3.5) with constraints (3.3). A similar idea has been applied in the design of a threshold dynamics method of image segmentation [51]. Given an initial guess χ 0 = (χ 0 1 , χ 0 2 ), we compute a series of minimizers (3.8) for k = 0, 1, 2, · · · . Here, the admissible set S is defined as u 0 , χ 1 , u 1 , χ 2 , · · · , u k , χ k+1 , · · · such that u k = arg min u∈S J τ (χ k , u), (3.7) χ k+1 = arg min χ∈B J τ (χ, u k ),S := u ∈ H 1 u D (Ω, R d ) | ∇ · u = 0 where H 1 u D (Ω, R d ) = {u ∈ H 1 (Ω, R d ) | u| ∂Ω = u D }, and B is defined in (3.1) . Given the k-th iteration χ k , we first solve (3.7) to get the u k . It is easy to see that the constraint minimization problem is equivalent to the following u k = arg min u∈H 1 u D (Ω,R d ) J τ (χ k , u) + Ω p∇ · u dx with p as a Lagrangian multiplier. Variation of the above functional leads to the following Brinkman equation. That is, u k can be obtained by solving      ∇ · u = 0, in Ω ∇p − ∇ · (µ∇u) + α(χ k )u = f , in Ω u| ∂Ω = u D (3.9) where α(χ k ) =ᾱ 2 G τ * χ k 2 . Because J τ (χ k , u) is convex in u, the solution (u k , p k ) of (3.9) is a minimizer of J τ (χ k , u). The following lemma shows the existence of u for the system (3.9) for a given χ ∈ B. Lemma 3.1 ([19, 22]). For every χ ∈ B, some u ∈ H 1 u D (Ω, R d ) exist that satisfy ∇ · u = 0 such that Ω µ∇u · ∇v + α(χ)u · v dx = Ω f · v dx, ∀ v ∈ V, (3.10) where V := {v ∈ H 1 0 (Ω, R d ) | ∇ · v = 0} . Given u k , we now rewrite the objective functional J τ (χ, u) intoJ τ,k (χ) as follows: J τ,k (χ) := J τ (χ, u k ) = Ωᾱ 2 χ 2 G τ * |u k | 2 dx + γ π τ Ω χ 1 G τ * χ 2 dx + Ω µ 2 |Du k | 2 − u k · f dx. (3.11) The next step is to find χ k+1 such that χ k+1 = arg min χ∈BJ τ,k (χ). (3.12) It is the minimization of a concave functional on a nonconvex admissible set B. However, we can relax it to a problem defined on a convex admissible set by finding r k+1 such that r k+1 = arg min r∈HJ τ,k (r), (3.13) where H is the convex hull of B defined as follows: H :={(v 1 , v 2 ) ∈ BV (Ω) | v i (x) ∈ [0, 1], i = 1, 2, and v 1 (x) + v 2 (x) = 1 a.e. in Ω, Ω v 1 dx = V 0 }, (3. 14) The following lemma shows that the relaxed problem (3.13) is equivalent to the original problem (3.12). Therefore, we can solve the relaxed problem (3.13) instead. Therefore, we need only prove thatr ∈ B. Lemma 3.2. Let u ∈ H 1 u D (Ω, R d ) We prove by contradiction. Ifr ∈ B, there is a set A ∈ Ω and a constant 0 < C 0 < 1 2 , such that |A| > 0 and 0 < C 0 <r 1 (x),r 2 (x) < 1 − C 0 , for all x ∈ A. We divide A into two sets A = A 1 ∪ A 2 such that A 1 ∩ A 2 = ∅ and |A 1 | = |A 2 | = |A|/2. Denote r t = (r t 1 , r t 2 ) where r t 1 =r 1 + tχ A1 − tχ A2 and r t 2 =r 2 − tχ A1 + tχ A2 with χ A1 and χ A2 being the indicator functions of the domain A 1 and A 2 , respectively. When 0 < t < C 0 , we have 0 < r t 1 , r t 2 < 1 and r t 1 + r t 2 =r 1 +r 2 = 1, and Ω r t 1 dx = Ωr 1 dx = V 0 . This implies that r t ∈ H. Furthermore, direct computations give, d 2 dt 2J τ,k (r) = 2γ √ π √ τ Ω d dt r t 1 G τ * d dt r t 2 dx = 2γ √ π √ τ Ω (χ A1 − χ A2 )G τ * (χ A2 − χ A1 ) dx = −2γ √ π √ τ Ω (χ A1 − χ A2 )G τ * (χ A1 − χ A2 ) dx = −2γ √ π √ τ Ω G τ /2 * (χ A1 − χ A2 ) G τ /2 * (χ A1 − χ A2 ) dx ≤ 0. The penultimate step comes from the fact that the heat kernel is a self-adjoint operator and forms a semigroup with various values of τ . From the above inequality, the functional is concave on the pointr. Thus,r cannot be a minimizer of the functional. This contradicts the assumption. Now, we show that (3.13) can be solved with a simple threshold dynamics method. BecauseJ τ,k (r) is quadratic in r, we first linearize the energyJ τ,k (r) at r k bỹ J τ,k (r) ≈J τ,k (r k ) + L τ,k r k (r − r k ), (3.16) where L τ,k r k (r) = Ω γ π τ r 1 G τ * r k 2 + γ π τ r 2 G τ * r k 1 + r 2ᾱ 2 G τ * |u k | 2 dx (3.17) = Ω (r 1 φ 1 + r 2 φ 2 ) dx. Here φ 1 = γ π τ G τ * r k 2 and φ 2 =ᾱ 2 G τ * |u k | 2 + γ π τ G τ * r k 1 . Then (3.13) can be approximately reformulated into (3.18) χ k+1 = arg min r∈H L τ,k r k (r) = arg min r∈H Ω (r 1 φ 1 + r 2 φ 2 ) dx. The following lemma, in particular, (3.21) shows that (3.18) can be solved in a pointwise manner by χ k+1 1 (x) = 1 and χ k+1 2 (x) = 0, if φ 1 (x) < φ 2 (x) + δ, χ k+1 1 (x) = 0 and χ k+1 2 (x) = 1, otherwise, (3.19) where δ is chosen as a constant such that Ω χ k+1 1 dx = V 0 . Lemma 3.3. Let φ 1 = γ π τ G τ * χ k 2 , φ 2 =ᾱ 2 G τ * |u| 2 + γ π τ G τ * χ k 1 and (3.20) D k+1 1 = {x ∈ Ω| φ 1 − φ 2 < δ} for some δ such that |D k+1 1 | = V 0 . Then for χ k+1 = (χ k+1 1 , χ k+1 2 ) with χ k+1 1 = χ D k+1 1 and χ k+1 2 = 1 − χ k+1 1 , we have L τ,k χ k (χ k+1 ) ≤ L τ,k χ k (χ k ) for all τ > 0. Proof. Because L τ,k χ k is a linear functional, we only need to prove that there holds L τ,k χ k (χ k+1 ) ≤ L τ,k χ k (χ) (3.21) for all χ = (χ 1 , χ 2 ) ∈ B. For each (χ 1 , χ 2 ) ∈ B, we know χ 1 = χD 1 and χ 2 = χD 2 for some open setsD 1 ,D 2 in Ω, such that D 1 ∩D 2 = ∅,D 1 ∪D 2 = Ω and |D 1 | = V 0 . Let A 1 =D 1 \D k+1 1 = D k+1 2 \D 2 and A 2 =D 2 \D k+1 2 = D k+1 1 \D 1 . We must have |A 1 | = |A 2 | due to the volume conservation property. Because A 1 ⊂ D k+1 2 , we have φ 1 (x) − φ 2 (x) ≥ δ, χ k+1 1 (x) − χ 1 (x) = −1, ∀x ∈ A 1 . Similarly, because A 2 ∈ D k+1 1 , we have φ 1 (x) − φ 2 (x) < δ, χ k+1 1 (x) − χ 1 (x) = 1, ∀x ∈ A 2 . Therefore, using χ k+1 1 − χ 1 + χ k+1 2 − χ 2 = 0, we have L τ,k χ k (χ k+1 ) − L τ,k χ k (χ) =γ π τ Ω (χ k+1 1 − χ 1 )φ 1 + (χ k+1 2 − χ 2 )φ 2 dx =γ π τ Ω (χ k+1 1 − χ 1 )(φ 1 − φ 2 ) dx =γ π τ Ω (χ A2 (φ 1 − φ 2 ) − χ A1 (φ 1 − φ 2 )) dx ≤γ π τ Ω (χ A2 δ − χ A1 δ) dx = γ π τ δ(|A 2 | − |A 1 |) = 0. To determine the value of δ, one can treat Ω χ k+1 1 dx − V 0 as a function of δ and use an iteration method (e.g., bisection method or Newton's method) to find the root of Ω χ k+1 1 dx − V 0 = 0. For the uniform discretization of Ω, a more efficient method is the quick-sort technique proposed in [54]. Assume we have a uniform discretization of Ω with grid size h, we can approximate Ω χ k+1 1 dx by M h 2 , we then sort the values of φ 1 − φ 2 in an ascending order and simply set χ k+1 1 = 1 on the first M points and χ k+1 2 = 1 on the other points. Remark 3.2. In many implementations, one may solve Stokes equation on nonuniform grid points. To preserve the volume for the discretization on nonuniform grids, although the volume cannot be simply approximated by the number of grid points times the size of each cell, a similar technique can be applied. One can still sort the values of φ 1 − φ 2 in ascending order, save the index into S, calculate the integrating weight at each grid point into V, and set V = 0 and i = 0. Then, δ can be simply found by: + 1)). Now, we are led to a threshold dynamics algorithm for topology optimization problem (3.6) for fluids in Stokes flow in the following. while V < V 0 ; i ← i + 1; V = V + V(S(i)); end; δ = φ 1 (S(i + 1)) − φ 2 (S(i Algorithm 1. Discretize Ω uniformly into a grid T h with grid size h and set M = V 0 /h d . Step 1. Input: Set τ > 0,ᾱ > 0, k = 0, a tolerance parameter tol > 0 and give the initial guess χ 0 ∈ B. Step 2. Iterative solution: 1. Update u. Solve the Brinkman flow equations 3. Compute e k χ = χ k+1 1 − χ k 1 2 . If e k χ ≤ tol, stop the iteration and go to the output step. Otherwise, let k + 1 → k and continue the iteration.      ∇ · u = 0, in Ω ∇p − ∇ · (µ∇u) + α(χ k )u = f , in Ω u| ∂Ω = u D by mixed finite-element method to get u k , where α(χ k ) =ᾱG τ * χ k 2 . 2. Update χ. Evaluate φ 1 = γ π τ G τ * χ k 2 , φ 2 =ᾱ 2 G τ * |u| 2 + γ π τ G τ * χ k 1 . Step 3. Output: A function χ ∈ B that approximately solves (3.6). Remark 3.3. We note that in the original MBO method, on one hand, the algorithm can be easily stuck when τ is very small because, in the discretized space, τ is so small that no point can switch from one phase to another (i.e., χ 1 changes from 0 to 1 or 1 to 0) at one iteration step. On the other hand, with a large τ , the interface can easily move but creates large error. Hence, we apply the adaptive in time technique [54] in numerical experiments by modifying Algorithm 1 into an adaptive algorithm by adjusting τ during the iterations. Indeed, we set a threshold value τ t and a given tolerance e t , if e k χ ≤ e t , let τ new = ητ with η ∈ (0, 1) and update τ := τ new in the next iteration unless τ ≤ τ t . Otherwise, τ will not be updated, and the iteration will continue with the same τ . We use this adaptive strategy for the choice of τ in the numerical experiments. Stability Analysis In this section, we prove the unconditional stability property of the proposed algorithm. Specifically, for the series of minimizers u 0 , χ 1 , u 1 , χ 2 , · · · , u k , χ k+1 , · · · , computed by Algorithm 1, we prove J τ (χ k+1 , u k+1 ) ≤ J τ (χ k , u k ) for all τ > 0. We first introduce Lemma 4.1 which leads us to J τ (χ k+1 , u k ) ≤ J τ (χ k , u k ) for all τ > 0. Lemma 4.1. For a fixed u k , let χ k+1 be the k + 1-th iteration derived from Algorithm 1, we have J τ (χ k+1 , u k ) ≤ J τ (χ k , u k ) for all τ > 0. Proof. From the linearization ofJ τ,k (χ k ) in (3.16), we have J τ (χ k , u k ) =L τ,k χ k (χ k ) − γ π τ Ω χ k 1 G τ * χ k 2 dx + Ω µ 2 |Du k | 2 − u k · f dx, J τ (χ k+1 , u k ) =L τ,k χ k (χ k+1 ) − γ π τ Ω χ k+1 1 G τ * χ k 2 + χ k+1 2 G τ * χ k 1 − χ k+1 1 G τ * χ k+1 2 dx + Ω µ 2 |Du k | 2 − u k · f dx. Then, we calculate J τ (χ k+1 , u k ) − J τ (χ k , u k ) =L τ,k χ k (χ k+1 ) − L τ,k χ k (χ k ) + γ π τ Ω (χ k+1 1 − χ k 1 )G τ * (χ k+1 2 − χ k 2 ) dx =L τ,k χ k (χ k+1 ) − L τ,k χ k (χ k ) − γ π τ Ω (χ k+1 1 − χ k 1 )G τ * (χ k+1 1 − χ k 1 ) dx =L τ,k χ k (χ k+1 ) − L τ,k χ k (χ k ) − γ π τ Ω G τ /2 * (χ k+1 1 − χ k 1 ) 2 dx ≤L τ,k χ k (χ k+1 ) − L τ,k χ k (χ k ). Because we have L τ,k χ k (χ k+1 ) − L τ,k χ k (χ k ) ≤ 0 from Lemma 3.3, we are led to J τ (χ k+1 , u k ) − J τ (χ k , u k ) ≤ 0 for all τ > 0. We are now led to the following theorem which proves the total energy decaying property Theorem 4.2. For the series of minimizers u 0 , χ 1 , u 1 , χ 2 , · · · , u k , χ k+1 , · · · , calculated with Algorithm 1, we have J τ (χ k+1 , u k+1 ) ≤ J τ (χ k , u k ) (4.1) for all τ > 0. Proof. For all τ > 0, from (3.7) , we have J τ (χ k+1 , u k+1 ) ≤ J τ (χ k+1 , u k ). From Lemma 4.1, we have J τ (χ k+1 , u k ) ≤ J τ (χ k , u k ). Thus, combining the above together gives the stability estimate (4.1). Remark 4.1. We remark here that, as we proved, the energy is decaying for any given τ . If τ changes from τ 1 to τ 2 at the k th iteration with τ 1 > τ 2 in our adaptive in time strategy, for example, χ k is generated by τ 1 and χ k+1 is generated by τ 2 . The energy is decaying in the sense that J τ2 (χ k+1 , u k+1 ) ≤ J τ2 (χ k , u k ) where the energy J at two iterations χ k and χ k+1 are approximated by the same τ 2 . Numerical Implementation In this section, we illustrate the implementation of Algorithm 1, with a focus on Step 2. The Brinkman equations (3.3a-3.3c) are solved with the mixed finite-element method, and the Taylor-Hood finite-element space is used for discretization, which satisfies the discrete inf-sup condition [22]. Let T h be a uniform triangulation of the domain Ω, and N h is the set of all vertices of T h . For a given χ h = (χ h 1 , χ h 2 ) ∈ B h where B h is the discrete version of B defined on N h . For the uniform regular triangulation of the domain, all values are evaluated on uniform quad grid points. Thus, we can use FFT for efficient evaluation of the discretized convolutions. We introduce the Taylor-Hood finite-element space V h := {v ∈ H 1 (Ω, R d ) | v| K ∈ [P 2 (K)] d , K ∈ T h }, Q h := {q ∈ L 2 (Ω, R) | Ω q dx = 0, q| K ∈ P 1 (K), K ∈ T h }. Let V D h := {v ∈ V h | v| ∂Ω = u h D }, where u h D is the a suitable approximation of the Dirichlet boundary condition u D on the boundary edges/faces of T h . For the solution of (3.3a-3.3c), find (u h , p h ) ∈ V D h × Q h such that −(p h , ∇ · v h ) + (µ∇u h , ∇v h ) + (α(χ h )u h , v h ) = (f , v h ), ∀ v h ∈ V 0 h , (∇ · u h , q h ) = 0, ∀ q h ∈ Q h . The above bilinear form can be easily extended to the Brinkman equations both with Dirichlet boundary Γ D and Neumann boundary Γ N , where Γ D ∩ Γ N = ∅, Γ D ∪ Γ N = ∂Ω, and (µ∇u − pI) · n| Γ N = g. When u h is obtained, we proceed to use the FFT to evaluate (φ h 1 , φ h 2 ) on each node of N h as follows: φ h 1 = γ π τ G τ * χ h 2 , φ h 2 =ᾱ 2 G τ * |u h | 2 + γ π τ G τ * χ h 1 . Following Algorithm 1, we can now use (φ h 1 , φ h 2 ) to update the indicator function χ h by the approach stated in Algorithm 1. Numerical experiments In this section, we perform extensive numerical testing to demonstrate the efficiency of Algorithm 1 with an adaptive strategy for the choice of τ . We choose η = 0.5 in the update of τ . If no confusion is possible, we still denote by τ as its initialization in the following. 6.1. Two dimensional results. We firstly test the proposed algorithm for the two dimensional problems. For most of examples in this subsection, we assume that the Dirichlet boundary condition with a parabolic profile and the magnitude of the velocity is set as |u D | = g(1 − (2t/l) 2 ) with t ∈ [−l/2, l/2], where l is the length of the section of the boundary at which the inflow/outflow velocity is imposed. The direction of the inflow/outflow velocity is illustrated in the following examples. Example 6.1. The first example shown in Figure 6.1 is the optimal design of a diffuser that was tested for topology optimization for fluids using MMA in [7]. Here, we apply Algorithm 1 to obtain the optimal design of the diffuser. Let g = 1 and 3 for the inflow and outflow velocities, respectively. We set the fluid region fraction as β = 0.5 and test the problem on a 128 × 128 grid. We first perform the simulations withᾱ = 2.5 × 10 4 , τ = 0.01, γ = 0.1 and with two types of initial distribution of χ 1 , as shown in Figure 6.2; that is, the initial fluid region is restricted in the middle of the domain in the left graph of Figure 6.2 (Case 1), and the initial fluid region satisfies a random distribution in the right graph of Figure 6.2 (Case 2). In both cases, we always arrive at the same optimal design result shown in the left graph of Figure 6.3, which also shows the quiver plot of the approximate velocity in the fluid region. The optimal design result seems similar to the result obtained by MMA in [7]. The energy decaying property can be observed in the right graph of Figure 6.3 which shows the energy curves for the above two cases of the initial distribution of χ 1 . The iteration converges in about 25 steps in both cases. In the next example, we increaseᾱ = 2.5 × 10 5 . Again, we use the initial fluid region of Case 1 with τ = 0.001, γ = 0.01. The optimal design of the diffuser and the approximate velocity in the fluid region are shown in the left graph of Figure 6.5. It seems that the fluid region at the left boundary reaches top and bottom boundaries in this case. The energy decaying property is also observed in Figure 6.5. The iteration converges even more quickly at about 10 steps. We also test the problem with the same inflow Dirichlet boundary condition as above, but we replace the outflow Dirichlet boundary condition with a homogeneous Neumann boundary. A similar optimal design of diffuser is then obtained as above for the cases ofᾱ = 2.5 × 10 4 andᾱ = 2.5 × 10 5 . For the case d = 0.5, we choose a random initial distribution χ 1 , as shown in the left graph of Figure 6.7. We remark that γ can also be set to zero in Algorithm 1. For fixed τ = 0.001, we test γ = 0.01, 0.001, 0. The optimal design result is nearly the same for the three choices of γ, as shown in the middle graph of Figure 6.7, and the energy decaying property is observed from the energy curves in the right graph of Figure 6.7. For the case d = 1.5, we choose an initial distribution χ 1 with the fluid region located in the middle of the domain as Case 1 of Example 6.1. We set τ = 0.01 and γ = 0.0001. The optimal design result and the approximate velocity are shown in the left graph of Figure 6.8, and the energy decaying property is also observed from the energy curve in the right graph of Figure 6.8. Compared with the computational cost used by MMA in [7], we find that our algorithm converges more quickly to the optimal result (cf. Table 2). Example 6.3. We consider another example studied in [7] that includes a body fluid force term imposed in the local circular region with center [1/2, 1/3] and radius r = 1/12. We show the design domain in Figure 6.9. The inflow and outflow Dirichlet boundaries are located with centers [0, 2/3] and [1, 2/3] respectively. Let g = 1 for the inflow and outflow velocities, and let the fluid region fraction be β = 1/4. We test the problem with various choices of body fluid force on a 128 × 128 grid, and we always chooseᾱ = 2.5 × 10 4 , τ = 0.01, γ = 0.0001 in this example. We test the cases for three different force terms f = [−1125, 0], [562.5, 0], [1687.5, 0]. We choose the initial distribution χ 1 with the fluid region located in a circular region with center [1/2, 1/2] and radius 1/ √ 3π. The optimal results and energy curves are plotted in Figures 6.10 to 6.12 for various values for force f , and the new algorithm also converges more quickly to the optimal results than the MMA shown in [7]. One can observe that for f = [−1125, 0] the fluid flow is in a clockwise direction near the center roundabout (left An interesting phenomenon observed in this example was the appearance of a tiny local solid at the center of the roundabout for the two cases of f = [−1125, 0], [1687. 5,0], and the tiny local solid is clearer when the grid is finer (cf. Figure 6.13). Example 6.4. Finally, we consider optimal design for a three-terminal device shown in Figure 6.14. The inflow and outflow Dirichlet boundaries are located with centers [0, 0.3] and [1, 0.7], and the homogeneous Neumann boundary is located on the left boundary with center [0, 1.1]. Let g = 0.5 for the inflow velocity and the fluid region fraction be β = 0.3. We chooseᾱ = 2.5 × 10 4 , τ = 0.01, γ = 0.0001 in this example and test the problem on a grid 80 × 112. |u D | =ḡ 1 − (y − a) 2 + (z − b) 2 l 2 , whereḡ is the prescribed velocity at the center of the flow profile at which the inflow/outflow velocity is imposed, l is the radius of the flow profile, (y, z) are Cartesian coordinates on a x-plane, and (a, b) are the center of a circle on a x-plane. Example 6.5. The design domain of this example is shown in Figure 6.16. For the inflow, we letḡ = 1, l = 1 2 , and (a, b) = ( 1 2 , 1 2 ) on x = 0 plane. For the objective of mass conservation, we letḡ = 9, l = 1 6 , and (a, b) = ( 1 2 , 1 2 ) on x = 1 plane. We set the fluid region fraction is β = 0. 35. This example was already tested by the level set method in [8]. Here we apply our new Algorithm 1 to obtain the optimal diffuser. Throughout this example, we choose the initial distribution χ 1 with fluid domain in a region of {(x, y, z) : x ∈ (0, 1), y ∈ (0, 1), z ∈ ( 7 20 , 7 10 )}. Firstly, we test the case withᾱ = 2.5 × 10 4 , τ = 0.05, and γ = 0.01 on 32 × 32 × 32 and 64 × 64 × 64 grids. In the following, the interface between solid and fluid regions for the optimal design is shown, and the fluid Next, the energy decay properties of the Algorithm 1 with different parameters τ and γ for this problem are shown for the same case ofᾱ = 2.5 × 10 4 in Figure 6.20. We note that the optimal design results for different parameters τ and γ are similar to that in the left graphs of Figure 6.17 and Figure 6.18. From the two graphs of Figure 6.20, we find that the energy converges to almost the same value when τ or γ is fixed. We test this problem based on the Algorithm 1 withᾱ = 2.5 × 10 4 , τ = 0.05, and γ = 0.01 on 32 × 32 × 32 and 64 × 64 × 64 grids. The initial distribution χ 1 with fluid domain is located in a region of {(x, y, z) : x ∈ (0, 1), y ∈ (0, 1), z ∈ ( 1 2 , 3 4 )}. The corresponding optimal design result is shown in the left graphs of Figure 6.22 and Figure 6.23. From the left graphs of Figure 6.22 and Figure 6.23, we can see that the interface between solid and fluid regions is more smooth when the simulation is performed on the fine grid. From the right graphs of Figure 6.22 and Figure 6.23, the energy decaying property is also observed. The iteration converges in about 50 steps and 70 steps on coarse and fine grids respectively. In Figure 6.24, we present the slice of optimal design result at z = 25/64 on a 32 × 32 × 32 grid, and the approximate velocity in the fluid region is also included. 6.3. Discussions on the robustness and efficiency of our algorithm. The numerical results in the previous subsections demonstrated the robustness and efficiency of our algorithm. First, the final optimal design result seems to be insensitive to the initial distribution of χ 1 . As shown in the first and second two dimensional examples for the case with α = 2.5 × 10 4 , even with a random initial distribution of χ 1 , we always get the same final optimal diffuser (cf. Figures 6.2-6.3). From the viewpoint of energy stability, the energy decaying property is proved mathematically and observed numerically for the problem with different initial distributions of χ 1 . Moreover, from the numerical results in Figure 6.4 for the Example 6.1 and Figure 6.20 for the Example 6.5, we can see that our algorithm is also robust for the different choices of parameters used in the algorithm. Next, we compare some of the numerical results above with some existing methods for topology optimization of fluids in Stokes flow in the literature. We mainly compare the numerical results of our algorithm with the results using MMA in [7] and the level set approach in [8]. Table 4. Comparison of different methods for Example 6.5. The parameters used in our algorithm are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. In our algorithm, only a Brinkman problem is solved without the need to solve adjoint problem at each iteration step, and the indicator functions of fluid-solid regions are easily updated based on simple convolutions followed by a thresholding step. Therefore, the computational cost at each iteration is less than that in MMA [7] or in the level set approach [8]. Thus, our algorithm is much simpler and easier to implement than those methods. Tables 1 and 2 list the number of iterations of our algorithm, the MMA, and the level set approach for two examples, Table 3 shows the number of iterations of the MMA and our algorithm for Example 6.3, and Table 4 presents the number of iterations of the level set approach and our algorithm for Example 6.5. We can see that our algorithm converges in many fewer steps. Discussion and conclusions In this paper, we introduce a new efficient threshold dynamics method for topology optimization for fluids in Stokes flow. We aim to minimize a total energy functional that consists of the dissipation power and the perimeter approximated by nonlocal energy. During the iterations of the algorithm, only a Brinkman equation requires solution by a mixed finite-element method, and the indicator functions of fluid-solid regions are updated by a thresholding step that is based on the convolutions computed by the FFT. A simple adaptive in time strategy is used to accelerate the convergence of the algorithm. The total energy decaying property of the proposed algorithm is rigorously proved and observed numerically. Several numerical examples are tested to verify the efficiency of the new algorithm, and we show that the new algorithm converges more rapidly for most the examples than the MMA used in [7]. Compared to existing methods for topology optimization for fluids, we believe that the proposed algorithm is simple and easy to implement. For the numerical experiments that we have performed thus far, the proposed method always finds an optimal topology and the numerical results are insensitive to the initial guess and parameters. We believe that our algorithm can also be extended to topology optimization for fluids in Navier-Stokes flow. be a given function and r = (r 1 , r 2 ).Proof. Letr = (r 1 ,r 2 ) ∈ H be a minimizer of the functionalJ τ,k (r) on H. Because B ⊂ H, we havẽ J τ,k (r) = min r∈HJ τ,k (r) ≤ min r∈BJ τ,k (r). Sort the values of φ 1 − φ 2 in an ascending order, and set χ k+1 1 = 1 on the first M points and χ k+1 2 = 1 on the other points. Figure 6 . 61. (Example 6.1) Design domain for the diffuser example. Figure 6 . 62. (Example 6.1) Left (Case 1): Initial distribution of χ 1 . Right (Case 2): Initial distribution of χ 1 . Figure 6 . 63. (Example 6.1) Left: Optimal diffuser for the caseᾱ = 2.5 × 10 4 and the approximate velocity in the fluid region. Right: Plot of energy curves for two cases of distribution of χ 1 . In this case, the parameters are set asᾱ = 2.5 × 10 4 , τ = 0.01, γ = 0.1.Next, we test the case (initial fluid region of Case 1) for various parameters. We first fixᾱ = 2.5 × 10 4 , τ = 0.01 and vary γ = 0.01, 0.005, 0.001. We then test the cases for fixed γ = 0.001 and various choices of τ = 0.05, 0.01, 0.001. The optimal design of the diffuser is similar to the result in the left graph ofFigure 6.3. Figure 6 . 64. (Example 6.1) Plot of energy curves for case 1 of distribution of χ 1 withᾱ = 2.5 × 10 4 . Left: For fixed τ = 0.01, energy curves for the cases of γ = 0.01, 0.005, 0.001. Right: For fixed γ = 0.01, energy curves for the cases of τ = 0.05, 0.01, 0.001. Figure 6 . 64 shows the energy decaying property for each of these cases. In all cases, the iteration converges in fewer than 25 steps. Figure 6 . 65. (Example 6.1) Left: Associated optimal diffuser and approximate velocity in the fluid region. Right: Plot of energy curve for Case 1 of distribution of χ 1 . In this case, the parameters are set as α = 2.5 × 10 5 , τ = 0.001, γ = 0.01. Example 6 . 2 . 62In this example, we test the double pipes problem shown inFigure 6.6. The inflow and outflow Dirichlet boundaries are located with centers [g = 1 for the inflow and outflow velocities, respectively, and let the fluid region fraction be β = 1/3. We test the problem withᾱ = 2.5 × 10 4 on a 128 × 256 grid for d = 0.5 and on a 192 × 128 grid for d = 1.5. Figure 6 . 6 . 66(Example 6.2) Design domain for the double pipes example. Figure 6 . 7 . 67(Example 6.2) For the case d = 0.5. Left: Initial distribution of χ 1 . Middle: Optimal double pipes and approximate velocity in the fluid region. Right: For fixed τ = 0.001, energy curves for the cases of γ = 0.01, 0.001, 0. Figure 6 . 8 . 68(Example 6.2) For the case d = 1.5, the parameters are set as τ = 0.01 and γ = 0.0001. Left: Optimal double pipes and approximate velocity in fluid region. Right: Energy curve. Figure 6 . 9 . 69(Example 6.3) Design domain for the example with a force term. Figure 6 . 610. (Example 6.3) For the example with a force term f = [−1125, 0] on a grid 128 × 128. Left: Optimal design result and approximate velocity in the fluid region. Right: Energy curve. Figure 6 . 611. (Example 6.3) For the example with a force term [562.5, 0] on a grid 128 × 128. Left: Optimal design result and approximate velocity in the fluid region. Right: Energy curve. Figure 6 . 612. (Example 6.3) For the example with a force term [1687.5, 0] on a grid 128 × 128. Left: Optimal design result and approximate velocity in the fluid region. Right: Energy curve. Figure 6 . 613. (Example 6.3) Optimal design results for for example with force term f = [1687.5, 0]. Left: Optimal design result on a coarse grid 128 × 128. Right: Optimal design result on a fine grid 256 × 256. graphs inFigure 6.10), while for f = [1687.5, 0] it is in a counterclockwise direction (left graph ofFigure 6.12). Figure 6 . 614. (Example 6.4) Design domain for the example with a force term. Figure 6 . 615. (Example 6.4) Left: Optimal design result for example of three-terminal device and approximate velocity in the fluid region. Right: Energy curve. We choose the initial distribution χ 1 , with the fluid region located in double parallel pipes [0, 1] × [13/60, 23/60] ∪ [0, 1] × [37/60, 47/60]. The optimal result was obtained after 29 iterations. The optimal design result and the approximate velocity are shown in the left graph of Figure 6.15. The energy decaying property is also observed from the energy curve in the right graph of Figure 6.15. 6.2. Three dimensional results. We now present the numerical examples in three dimensions. For the Dirichlet boundary condition in the following examples, we always assume that the magnitude of the velocity is set as Figure 6 . 616. (Example 6.5) Design domain. Figure 6 . 617. (Example 6.5) Left: Optimal design result on a 32 × 32 × 32 grid. Right: Energy curve.In this case the parameters are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. Figure 6 . 618. (Example 6.5) Left: Optimal design result on a 64 × 64 × 64 grid. Right: Energy curve.In this case the parameters are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. region locates in the interior of subdomain surrounded by the interface. The optimal diffusers are presented in the left graphs ofFigure 6.17 andFigure 6.18 and the energy decaying property can be observed in the right graphs ofFigure 6.17 andFigure 6.18. The optimal design results seem to be similar to that in[8].The iteration converges in about 25 steps and 35 steps on coarse and fine grids respectively. Additionally, the slice of optimal design result at y = 0.5 on 32 × 32 × 32 grid and the approximate velocity in the fluid domain are provided inFigure 6.19. Figure 6 . 619. (Example 6.5) The slice of optimal design result and the approximate velocity in fluid region at y = 0.5 on a 32 × 32 × 32 grid. The parameters are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. Example 6 . 6 . 66In this example we assume that there are four flow profiles on the inflow boundary and one flow profile on the outflow boundary. The design domain is shown in Figure 6.21. For the four inflow profiles, we letḡ = 1, the radius is set as l = 1 8 and the centers of circles are ( 1 4 , 1 4 ), ( 1 4 , 3 4 ), ( 3 4 , 1 4 ) and ( 3 4 , 3 4 ) on the x = 0 plane respectively. For the outflow profile, we letḡ = 1, l = 1 4 and (a, b) = ( 1 2 , 1 2 ) on the x = 1 plane. We set the fluid region fraction as β = 1 4 . Figure 6 . 620. (Example 6.5) Plot of energy curves forᾱ = 2.5 × 10 4 on 32 × 32 × 32 grid. Left: For fixed τ = 0.05, energy curves for the cases of γ = 0.1, 0.01, 0.001. Right: For fixed γ = 0.01, energy curves for the cases of τ = 0.05, 0.01, 0.001. Figure 6 . 621. (Example 6.6) Design domain. Figure 6 . 622. (Example 6.6) Left: Optimal design result on a 32 × 32 × 32 grid. Right: Energy curve.In this case the parameters are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. Figure 6 . 623. (Example 6.6) Left: Optimal design result on a 64 × 64 × 64 grid. Right: Energy curve.In this case the parameters are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. Figure 6 . 624. (Example 6.6) The slice of optimal design result and the approximate velocity in fluid region at z = 25/64 on a 32 × 32 × 32 grid. The parameters are set asᾱ = 2.5 × 10 4 , τ = 0.05, γ = 0.01. Table 1. Comparison of the number of iterations of different methods to obtain the optimal design result for Example 6.1 with α = 2.5 × 10 4 . The parameters used in our algorithm are set as τ = 0.001, γ = 0.01.Methods Grid Number of iterations MMA 100 × 100 33 Level set 96 × 96 197 Our algorithm 128 × 128 21 Methods Grid Number of iterations MMA 150 × 100 236 Level set 216 × 144 681 Our algorithm 192 × 128 35 Table 2 . 2Comparison of the number of iterations of different methods to obtain the optimal design result for Example 6.2 with d = 1.5. The parameters used in our algorithm are set as τ = 0.01, γ = 0.0001.Methods Surface force density Number of iterations −1125 229 MMA 562.5 66 1687.5 69 −1125 21 Our algorithm 562.5 18 1687.5 28 Table 3 . 3Comparison of the number of iterations of the MMA and our algorithm to obtain the optimal design results for Example 6.3. 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Osting, A diffusion generated method for computing Dirichlet partitions, arXiv preprint arXiv:1802.02682, 2018. Topology optimization of regions of Darcy and Stokes flow. N Wiker, A Klarbring, T Borrvall, Int. J. Numer. Meth. Engrg. 69N. Wiker, A. Klarbring, and T. Borrvall, Topology optimization of regions of Darcy and Stokes flow, Int. J. Numer. Meth. Engrg., 69 (2007), pp. 1374-1404. An efficient threshold dynamics method for wetting on rough surfaces. X Xu, D Wang, X P Wang, J. Comput. Phys. 330X. Xu, D. Wang, and X. P. Wang, An efficient threshold dynamics method for wetting on rough surfaces, J. Comput. Phys., 330 (2017), pp. 510-528. A variational level set method for the topology optimization of steady-state Navier-Stokes flow. S Zhou, Q Li, J. Comput. Phys. 227S. Zhou and Q. Li, A variational level set method for the topology optimization of steady-state Navier-Stokes flow, J. Comput. Phys., 227 (2008), 10178-10195.

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{'abstract': 'We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model. 2010 Mathematics Subject Classification. 35K93, 35K05, 65M12, 35Q35, 49Q10, 65M60, 76S05.', 'arxivid': '1812.09437', 'author': ['Huangxin Chen ', 'Haitao Leng ', 'ANDDong Wang ', 'Xiao-Ping Wang '], 'authoraffiliation': [], 'corpusid': 119705857, 'doi': '10.4208/csiam-am.so-2021-0007', 'github_urls': [], 'n_tokens_mistral': 21491, 'n_tokens_neox': 17765, 'n_words': 11152, 'pdfsha': '3db6ab38c4848f50144c31056a7266790e2a381a', 'pdfurls': ['https://arxiv.org/pdf/1812.09437v1.pdf'], 'title': ['AN EFFICIENT THRESHOLD DYNAMICS METHOD FOR TOPOLOGY OPTIMIZATION FOR FLUIDS', 'AN EFFICIENT THRESHOLD DYNAMICS METHOD FOR TOPOLOGY OPTIMIZATION FOR FLUIDS'], 'venue': []}

arxiv

On the MISO Channel with Feedback: Can Infinitely Massive Antennas Achieve Infinite Capacity? Jinyuan Chen On the MISO Channel with Feedback: Can Infinitely Massive Antennas Achieve Infinite Capacity? 1 We consider communication over a multiple-input single-output (MISO) block fading channel in the presence of an independent noiseless feedback link. We assume that the transmitter and receiver have no prior knowledge of the channel state realizations, but the transmitter and receiver can acquire the channel state information (CSIT/CSIR) via downlink training and feedback. For this channel, we show that increasing the number of transmit antennas to infinity will not achieve an infinite capacity, for a finite channel coherence and a finite input constraint on the second or fourth moment. This insight follows from our new capacity bounds that hold for any linear and nonlinear coding strategies, and any channel training schemes. In addition to the channel capacity bounds, we also provide a characterization on the beamforming gain that is also known as array gain or power gain, at the regime with large number of antennas. I. INTRODUCTION Motivated by the increasing demand for higher data rates in wireless communication systems, a significant effort is being made to study the use of massive multiple-input multiple-output (massive MIMO) systems [1]- [3]. As equipped with a large number of antennas, the massive MIMO system has a potential to boost the channel's beamforming gain that is also known as array gain or power gain (cf. [4], [5]). In the massive MIMO channels, for example in a massive multiple-input single-output (MISO) channel, the capacity may increase logarithmically with the number of antennas (cf. [4]- [7]), which implies that infinitely massive antennas may allow us to achieve an infinite capacity, even with a finite power constraint at the transmitter. However, the above exciting result is based on the key assumption that the instantaneous fading coefficients are perfectly known to the receiver/transmitter (perfect CSIR/CSIT). In general, CSIT and CSIR entail channel training and feedback. In a typical system with frequency-division-duplex (FDD) mode, CSIT comes from channel training and feedback operating over the downlink channel and feedback channel respectively. The overhead of the training and feedback may in turn affect the channel capacity. Therefore, it remains open if a massive MIMO system could still provide a significant capacity benefit as we expected. Specifically, we might ask the following question: Can infinitely massive antennas always achieve an infinite capacity in a massive MIMO channel? In this work, we study this question by focusing on a massive MISO block fading channel with output feedback. We assume that the transmitter and receiver have no prior knowledge of the channel state realizations, but the transmitter and receiver can acquire the channel state information via downlink training and feedback. We begin with a simple case where the coherence of the channel block is T c = 2 and the input signals are limited by a finite second-moment constraint that is also known as long-term average power constraint. Since the coherence is T c = 2, the transmitter could use the first and the second channel uses of each channel block for channel training and data transmission, respectively. Based on this scheme, one might tentatively expect an infinite rate for the case with infinite number of transmit antennas, because a little channel state information might be very useful for this case. However, we show that in this setting increasing the transmit-antenna number to infinity will not yield an infinite capacity. The result holds for any linear and nonlinear coding strategies, and any channel training schemes. This result is in sharp Jinyuan Chen is with Louisiana Tech University, Department of Electrical Engineering, Ruston, USA (email: jinyuan@latech.edu). A short version of this work has been submitted to ISIT2017. arXiv:1703.01287v1 [cs.IT] 3 Mar 2017 contrast to the result of the setting with perfect CSIT/CSIR (e.g. through a genie-aided training and feedback), in which the capacity will go to infinity as the antenna number grows to infinity (cf. [7]). As a main contribution of this work, we derive both capacity upper bound and lower bound for the MISO channel with feedback under the second moment and the fourth moment input constraints, respectively. For the case with a finite channel coherence and a finite input constraint on the fourth moment, the result reveals again that increasing the transmit-antenna number to infinity will not yield an infinite capacity. In addition to the capacity bounds, this work also provides a characterization on the channel's beamforming gain at the regime with large number of antennas. Similarly to the degrees-of-freedom metric (cf. [8]) that usually captures the prelog factor of capacity at the high power regime, beamforming gain is used in this work to capture the prelog factor of capacity at the high antenna-number regime. The capacity of the channels with feedback, or with imperfect CSIT/CSIR, has been studied extensively in the literature for varying settings, e.g., the point-to-point channels [9]- [21] and the broadcast channel [22]- [26]. However, a common assumption in those works above is that imperfect CSIT and CSIR were acquired without considering the overhead in channel training. The channel training overhead can not be negligible when the number of channel parameters to be estimated is large and the channel coherence is relatively small. This work is categorized in the line of works studying the multiple-antenna networks where CSIT and CSIR were acquired via channel training with overhead, such as [27]- [32]. To the best of our knowledge, the previous works in this direction usually considered some specific assumptions, e.g., linear coding strategy, dedicated channel training (a certain time is dedicated specifically for channel training) and short-term input constraint. In this work, we study a broad setting that considers any linear and nonlinear coding strategies, any channel training schemes, any short-term and long-term input constraints. The remainder of this work is organized as follows. Section II describes the system model. Section III provides the main results of this work. The converse and achievability proofs are described in Sections IV, V, VI and the appendices. The work is concluded in Section VII. Throughout this work, (•) T , (•) * , (•) H and (•) −1 denote the transpose, conjugate, conjugate transpose and inverse operations respectively. || • || denotes the Euclidean norm, det(•) denotes the determinant, tr(•) denotes the trace, and | • | denotes the magnitude. We use A 0 to denote that matrix A is Hermitian positive semidefinite, and use A B to mean that B − A 0. Logarithms are in base 2. We let e j i = (e i , e i+1 , · · · , e j ) if i ≤ j else let e j i denote an empty term, and let e j = e j 1 . I(•), H(•) and h(•) denote the mutual information, entropy and differential entropy, respectively. • denotes the largest integer not greater than the argument and • denotes the smallest integer not less than the argument. Z and C denote the sets of integers and complex numbers, respectively. [a mod m] denotes the modulo operation, i.e., [a mod m] = r if the number of a can be represented as a = m + r for ∈ Z and |r| < |m|. u ∼ CN (u 0 , Ω 0 ) denotes that the random vector u is proper complex Gaussian distributed with mean u 0 and covariance Ω 0 . u is said to be proper if E[(u − E[u])(u − E[u]) T ] = 0. When a complex Gaussian vector is proper and with zero mean, it is said to be circularly symmetric complex Gaussian. u ∼ X 2 (k) denotes that u is a chi-squared random variable that is defined as the sum of of squares of k independent and identically distributed (i.i.d.) standard normal N (0, 1) random variables. Unless for some specific parameters, the random matrix, random variable and random vector are usually denoted by the bold italic uppercase symbol (e.g., U ), bold italic lowercase symbol (e.g., u) and bold italic lowercase symbol with underline (e.g., u) respectively, while the corresponding realizations are non-bold (e.g., U , u and u). II. SYSTEM MODEL We consider a MISO channel where a transmitter with M (M ≥ 2) antennas sends information to a single-antenna user, as illustrated in Fig. 1. The signal received by the user at time t is given as y t = h T t x t + z t ,(1) t = 1, 2, · · · , n, where x t denotes the transmitted signal vector at time t, z t ∼ CN (0, 1) denotes the additive white Gaussian noise channel vector at time t, and h t,m denotes a channel coefficient of the mth transmit antenna at time t. We assume a block fading model (cf. [28], [33]), in which the channel coefficients remain constant during a coherence block of T c channel uses and change independently from one block to the next, i.e., (AWGN), h t [h t,1 , h t,2 , · · · , h t,M ] T ∼ CN (0, I M ) denotes the M × 1 User TX Feedback Downlink channelh Tc+1 = h Tc+2 = · · · = h Tc+Tc = h ( +1)Tc+1 for = 0, 1, · · · , L − 1 and L = n/T c , where n, L, T c are assumed to be integers. We assume that the channel coefficients in each block are initially unknown to the transmitter and the user. At the end of each time t, the user can feed back the channel outputs to the transmitter over an independent feedback link. For simplicity we assume that the feedback link is noiseless (error-free) and with a unit time delay, i.e., at the beginning of time t + 1, the transmitter knows y t (y 1 , y 2 , · · · , y t ). For this feedback communication of total n channel uses, the transmitter wishes to send the user a message index w that is uniformly distributed over {1, 2, · · · , 2 nR }. We specify a (2 nR , n) feedback code with encoding maps x t : {1, 2, · · · , 2 nR } × C t−1 → C M , t = 1, 2, · · · , n(2) that result in codewords (or code functions, more precisely) x n (w, y n−1 ) = x 1 (w), x 2 (w, y 1 ), · · · , x n (w, y n−1 ) . Then the user decodes the message with decoding mapŝ w n : C n → {1, 2, · · · , 2 nR }. We consider two cases of constraints on the input signals. At first we consider the second moment input constraint such that 1 n n t=1 E x t (w, y t−1 ) 2 ≤ P(5) where the expectation is over all possible noise and fading sequences as well as the message w. This second moment constraint is also known as the average power constraint. We then consider the fourth moment input constraint such that 1 n n t=1 E x t (w, y t−1 ) 4 ≤ κ 2 P 2(6) where κ is a positive constant. The fourth moment input constraint has been introduced in several communication scenarios (cf. [34]- [38]). For some certain cases, imposing the fourth moment constraint is identical to imposing a limitation on the kurtosis that is a measure of peakedness of the signal (cf. [35]- [37]). The probability of error P (n) e is defined as P (n) e 1 2 nR 2 nR w=1 Pr ŵ n (y n ) = w|w = w . A rate R (bits per channel use) is said to be achievable if there exists a sequence of (2 nR , n) codes with P (n) e → 0 as n → ∞. The capacity of this channel C is defined as the supremum of all achievable rates. In this work, we specifically focus on the capacity effect of the channel with large number of antennas, which may be captured by the metric of beamforming gain. For the capacity effect of the channel with high power, one might consider the metric of degrees-of-freedom that is beyond of the scope of this work. For a finite input constraint P (or κP ) and for a fixed α log T c log M , α ≥ 0 that is the ratio between coherence and antenna number in a logarithmic scale, or equivalently T c M α , the beamforming gain of the channel is defined as b(α) lim sup M →∞ C(α, P, M ) log M . Similarly to the definition of generalized degrees-of-freedom (GDoF, see [39]), the beamforming gain b(α) captures the capacity prelog factor for a class of channels with a fixed α, at the regime with a large number of antennas. In this setting b = 0 means zero beamforming gain, while b = 1 denotes a full beamforming gain. For the ideal case with perfect CSIT and CSIR (e.g., through a genie-aided method) one might achieve a full beamforming gain. However, for this setting where CSIR and CSIT are acquired via downlink training and feedback, the beamforming gain is generally unknown so far. In the following we seek to characterize the beamforming gain of this setting. III. MAIN RESULTS This section provides the main results for a MISO channel with feedback defined in Section II. The proofs are shown in Sections IV, V, VI and the appendices. Before showing the main results of this work, let us first revisit the ideal case of MISO channel with perfect CSIT and CSIR, and with a second moment input constraint. For this ideal case, the work in [7] characterized the channel capacity in a closed form C ideal = max P (γ): γP (γ)fγ (γ)dγ=P γ log 1 +P (γ) · γ f γ (γ)dγ(7) where γ h t 2 , f γ (γ) is the probability density function of γ, and the optimal power allocation of P (γ) is based on a water-filling algorithm (cf. [7]). When the antenna-number M is large, the capacity in (7) tends to log(1 + P M ), which is summarized in the following proposition. Proposition 1 (Ideal case). For the MISO channel with perfect CSIT and CSIR, and with a finite secondmoment input constraint P , the channel capacity tends to log(1 + P M ) for a large M . Proposition 1 follows from the capacity expression in (7) and the asymptotic analysis that is provided in Appendix A. Proposition 1 reveals that the capacity of a MISO channel with perfect CSIT and CSIR will go to infinity as the antenna-number M grows to infinity, even with a finite input constraint P . Let us now go back to the MISO channel with feedback defined in Section II, where the transmitter and receiver have no prior knowledge of the channel state realizations, but the transmitter and receiver can acquire the CSIT/CSIR via downlink training and feedback. We begin with a specific case of coherence, T c = 2, and provide a capacity result in the following theorem. Theorem 1 (Capacity upper bound). For the MISO channel with feedback defined in Section II and given a coherence T c = 2, the capacity is upper bounded as C ≤ 1.5 log(1 + P ) + 0.5 under the second moment input constraint in (5). The proof of Theorem 1 is provided in Section IV. Theorem 1 reveals that under a finite secondmoment input constraint P and T c = 2, the capacity of this MISO channel with feedback will not go to infinity as the antenna-number M grows to infinity. This result in sharp contrast to the result of the perfect CSIT/CSIR case, in which the capacity will be infinite as M grows to infinity (see Proposition 1). In Theorem 1 we just consider the specific case of T c = 2. When T c = 1 it is reduced to the fast fading case without CSIT -in this case increasing the antenna number M to infinity will not improve too much on the channel capacity even with perfect CSIR (cf. [6]). Note that for the case without CSIT but with perfect CSIR, the channel capacity tends to log(1 + P ) for large M , under the second moment input constraint (cf. [6]). When T c ≥ 3, we conjecture that the channel capacity will still not go to infinity as M grows to infinity, for a finite P and a finite T c . The following result summarizes the capacity upper bound for the case with a fourth moment input constraint. Theorem 2 (Capacity upper bound). For the MISO channel with feedback defined in Section II, the capacity is upper bounded by C ≤ log 1 + min M + 2, √ 2(T c + 1) · κP under the fourth moment input constraint in (6). The proof of Theorem 2 is provided in Section V and the appendices. Theorem 2 reveals that given a finite fourth-moment input constraint κP and a finite T c , again, the capacity will not go to infinity as the antenna number M grows to infinity. The following result summarizes the capacity lower bound. Theorem 3 (Capacity lower bound). For the MISO channel with feedback defined in Section II, the capacity is lower bounded by C ≥ T c −T τ T c · log 1 + P · max{(T τ −1), 1/2} 2 + 1 P − 1 max{T τ , 2} under the second moment input constraint; while under the fourth moment input constraint the capacity is lower bounded by C ≥ T c −T τ T c · log 1 + P o · max{(T τ −1), 1/2} 2 + 1 Po − 1 max{T τ , 2} where P o κP √ 3 and T τ min{M,Tc} log max{4, min{M,Tc}} . The capacity lower bound in Theorem 3 is a lower bound on the achievable rate of a proposed scheme (see Section VI). The proposed scheme is a simple scheme that uses only T τ number of transmit antennas (T τ ≤ M ). The scheme consists of a downlink training phase and a data transmission phase for each coherence block of the channel. Specifically, for each coherence block with T c channel uses, the duration of downlink training phase is T τ number of channel uses, while the duration of data transmission phase is T c − T τ number of channel uses. The choice of T τ is critical to the scheme performance, because with too small T τ there is not enough time for the channel training, while with too large T τ there is not enough time for the data transmission. In our scheme we set T τ = min{M,Tc} log max{4,min{M,Tc}} which yields the achievable rate in Theorem 3. Note that the rate in Theorem 3 can be further improved since we just focus on the simple scheme. In this work, we specifically focus on the channel capacity effect of the system with large number of antennas, which may be captured by the metric of beamforming gain. The following results summarize the beamforming gain of the channel under two input constraints respectively. Proposition 2 follows directly from Theorem 1. Proposition 2 reveals that there is no beamforming gain at a specific case of T c = 2. In this case, increasing the antenna number M to infinity will not improve too much on the capacity. Theorem 4 (Beamforming gain). For the MISO channel with feedback defined in Section II, the beamforming gain is characterized as b(α) = min{α, 1} under the fourth moment input constraint. Theorem 4 follows from the capacity bounds in Theorems 2 and 3 under the fourth moment input constraint (see Appendix D for the proof). As illustrated in Fig. 2, a full beamforming gain, b = 1, is achievable when α ≥ 1. Intuitively, for the case with large α, α 1, the coherence is very large and consequently the channel can be considered as a static channel, in which a full beamforming gain could be achieved easily via sufficiently long downlink training, as the time overhead of downlink training can be negligible in this static case. Theorem 4 reveals an interesting insight that, instead of α 1, α = 1 is sufficient for achieving a full beamforming gain. When 0 ≤ α ≤ 1 (or, equivalently, T c ≤ M ) the beamforming gain grows linearly with α, which implies that in this case the capacity grows logarithmically with the coherence T c in an asymptotic sense. IV. CONVERSE: THE CASE WITH A SECOND MOMENT INPUT CONSTRAINT For the channel model defined in Section II, we will show that the rate is upper bounded by R ≤ 1.5 log(1 + P ) + 0.5 (bits/channel use) under the second moment input constraint in (5) and given a fixed coherence T c = 2. The result reveals that, given a fixed coherence T c = 2 and a finite P , increasing the antenna number M to infinity will not yield an infinite capacity. The result holds for any linear and nonlinear schemes, and holds for any channel training schemes. Our proof is mainly based on minimum mean square error (MMSE) estimation techniques and MIMO techniques. In the proof we will use Lemmas 1 -3 that are shown at the end of this section. At first we bound the rate as follows: nR = H(w) = I(w; y n ) + H(w|y n ) ≤ I(w; y n ) + n n (8) = n t=1 h(y t y t−1 ) − h(y t w, y t−1 ) + n n (9) ≤ n t=1 h(y t y t−1 ) − h(y t w, y t−1 , h t , x t ) + n n (10) = n t=1 h(y t y t−1 ) − log(πe) + n n (11) = n t=1 E y t−1 h y t y t−1 = y t−1 − n log(πe) + n n(12) where (8) follows from Fano's inequality and n → 0 as n → ∞; (9) results from chain rule; (10) uses the fact that conditioning reduces differential entropy; (11) is from that h(y t w, y t−1 , h t , x t ) = h(y t − h T t x t w, y t−1 , h t , x t ) = h(z t ) = log(πe) . We proceed to upper bound the differential entropy h y t y t−1 = y t−1 in (12). Note that the average power of y t given ( y t−1 = y t−1 ) is E |y t | 2 y t−1 = y t−1 = E |h T t x t + z t | 2 y t−1 = y t−1 = 1 + E |h T t x t | 2 y t−1 = y t−1 . Since differential entropy is maximized by a circularly symmetric complex Gaussian distribution with the same average power, we have h y t y t−1 = y t−1 ≤ log πe 1 + E |h T t x t | 2 y t−1 = y t−1 .(13) Then, by plugging (13) into (12) it gives the following bound on the rate: nR − n n ≤ n t=1 E log πe 1 + E |h T t x t | 2 y t−1 − n log(πe) = n t=1 E log 1 + E |h T t x t | 2 y t−1 .(14) Let us now focus on the expectation term E |h T t x t | 2 y t−1 in (14). Note that x t is a function of (y t−1 , w), which implies that x t and h t can be correlated. Based on this fact, computing the value of E |h T t x t | 2 y t−1 could be challenging in general. We also note that the value of E |h T t x t | 2 y t−1 could be a function of M , depending on the correlation of x t and h t . In what follows we seek to bound the value of E |h T t x t | 2 y t−1 for the specific case of T c = 2. We will use the notations ofĥ t ,h t and Ω t given aŝ h t 0 if t ∈ {2 + 1} L−1 =0 x * t−1 y t−1 x t−1 2 +1 if t ∈ {2 + 2} L−1 =0(15)Ω t I M if t ∈ {2 + 1} L−1 =0 I M − x * t−1 x T t−1 x t−1 2 +1 if t ∈ {2 + 2} L−1 =0 (16) h t (y t−1 , w) ∼ CN (0, Ω t ) (see Lemma 1 below); (23) follows from the identity that tr(AB) ≤ λ max (A)tr(B), where λ max (A) corresponds to the maximum eigenvalue of matrix A, for positive semidefinite m × m Hermitian matrices A, B; (24) results from the facts that λ max (ĥ t * ĥ t T ) = ĥ t 2 , tr(x t x H t ) = x t 2 , and λ max (Ω t ) ≤ 1 (see Lemma 2 below). At this point, by plugging (24) into (14) we can bound the rate as nR − n n ≤ n t=1 E log 1 + E ĥ t 2 · x t 2 y t−1 + E x t 2 y t−1(25) Note that bothĥ t and x t are the functions of (y t−1 , w), which implies that computing E ĥ t 2 · x t 2 y t−1 in (25) would be challenging. In the following we will bound the value of ĥ t 2 given y t−1 for the specific case of T c = 2. When T c = 2, the channel changes every two channel uses. In this case, h t can not be estimated from the knowledge of (y t−1 , w) if t is an odd number. If t is an even number, h t can be estimated from the knowledge of (y t−1 , w) at a certain degree of precision. From (15) we note thatĥ t is the MMSE estimate of h t given (y t−1 , w). Given the expression ofĥ t in (15), now we compute ĥ t 2 as ĥ t 2 = 0 if t ∈ {2 + 1} L−1 =0 x t−1 2 ·|y t−1 | 2 ( x t−1 2 +1) 2 if t ∈ {2 + 2} L−1 =0(26) When t is an even number, ĥ t 2 can be bounded as ĥ t 2 = x t−1 2 · |y t−1 | 2 ( x t−1 2 + 1) 2 ≤ |y t−1 | 2 x t−1 2 + 1 ≤ |y t−1 | 2 , if t ∈ {2 + 2} L−1 =0(27) By plugging (26) and (27) into (25) it gives nR − n n ≤ L−1 =0 E log 1 + E ĥ 2 +1 2 · x 2 +1 2 y 2 + E x 2 +1 2 y 2 + L−1 =0 E log 1 + E ĥ 2 +2 2 · x 2 +2 2 y 2 +1 + E x 2 +2 2 y 2 +1 (28) ≤ L−1 =0 E log 1 + E x 2 +1 2 y 2 + L−1 =0 E log 1 + E |y 2 +1 | 2 · x 2 +2 2 y 2 +1 + E x 2 +2 2 y 2 +1 (29) = L−1 =0 E log 1 + E x 2 +1 2 y 2 + L−1 =0 E log 1 + (|y 2 +1 | 2 + 1) · E x 2 +2 2 y 2 +1 ≤ L−1 =0 log 1 + E x 2 +1 2 + L−1 =0 E log |y 2 +1 | 2 + 1 · 1 + E x 2 +2 2 y 2 +1 (30) = L−1 =0 log 1 + E x 2 +1 2 + L−1 =0 E log 1 + E x 2 +2 2 y 2 +1 + L−1 =0 E log 1 + |y 2 +1 | 2 ≤ L−1 =0 log 1 + E x 2 +1 2 + L−1 =0 log 1 + E x 2 +2 2 + L−1 =0 log 1 + E |y 2 +1 | 2(31) where n is assume to be an even number and L n/2 for this case of T c = 2; (28) results from splitting one summation term in (25) into two summation terms; (29) is from (26) and (27). In the first summation term in (30) the outer expectation is moved inside the logarithmic function, which will not reduce the value. (30) also follows from the fact that (1 + a 1 (1 + a 2 )) ≤ (1 + a 1 )(1 + a 2 ) for a 1 , a 2 ≥ 0. In the second and third summation terms in (31) the outer expectations are moved inside the logarithmic functions, respectively, which again will not reduce the values. In the next step we will bound the expectation term E |y 2 +1 | 2 in (31). In this case of T c = 2, the channel changes independently at time t = 2 + 1, which implies that the input signal x 2 +1 should be independent of the channel h 2 +1 . Therefore, E |y 2 +1 | 2 =1 + E |h T 2 +1 x 2 +1 | 2 =1 + E tr(h T 2 +1 x 2 +1 x H 2 +1 h * 2 +1 ) =1 + E tr(h * 2 +1 h T 2 +1 x 2 +1 x H 2 +1 ) =1 + tr E h * 2 +1 h T 2 +1 · E x 2 +1 x H 2 +1 (32) =1 + tr I M · E x 2 +1 x H 2 +1 (33) =1 + E x 2 +1 2 (34) where (32) follows from the independence between x 2 +1 and h 2 +1 ; (33) is from that h 2 +1 ∼ CN (0, I M ). By plugging (34) into (31) it then yields nR − n n ≤ L−1 =0 log 1 + E x 2 +1 2 + L−1 =0 log 1 + E x 2 +2 2 + L−1 =0 log 2 + E x 2 +1 2 (35) = n t=1 log 1 + E x t 2 + L−1 =0 log 2 + E x 2 +1 2 (36) ≤ max n t=1 E[ xt 2 ]≤nP n t=1 log 1 + E x t 2 + L−1 =0 log 2 + E x 2 +1 2 (37) ≤ max n t=1 E[ xt 2 ]≤nP n t=1 log 1 + E x t 2 + max n/2−1 =0 E[ x 2 +1 2 ]≤nP n/2−1 =0 log 2 + E x 2 +1 2 (38) =n log(1 + P ) + n 2 log(2 + 2P ) (39) =1.5n log(1 + P ) + 0.5n(40) where (36) uses the definition that n = 2L for this case of T c = 2; (37) follows from that the right-handside (RHS) of (36) is maximized by the optimal power allocation subject to the power constraint in (5); in (38) the maximization is moved into two summation terms, which will not reduce the value; (39) follows from Lemma 3 (see below). Finally, by taking n → ∞ it gives the bound R ≤ 1.5 log(1 + P ) + 0.5 which completes the proof. Now we provide some lemmas used in our proofs. The following lemma describes the well-known MMSE estimate result (see, for example, [40,Chapter 15.8]). Lemma 1. [40, Chapter 15.8] Consider two independent random vectors u ∈ C M ×1 , z ∈ C N ×1 , with two vectors being complex Gaussian, that is, z ∼ CN (0, I N ) and u ∼ CN (û 1 , Ω 1 ) for some fixedû 1 and fixed Hermitian positive semidefinite Ω 1 . Let A ∈ C N ×M be a fixed matrix and let y = Au + z then the conditional density of u given y is u|y ∼ CN (û, Ω) whereû =û 1 + Ω 1 A H (AΩ 1 A H + I N ) −1 (y − Aû 1 ), Ω = Ω 1 − Ω 1 A H (AΩ 1 A H + I N ) −1 AΩ 1 . Furthermore, the two random vectorsû and v u −û are independent, and the conditional density of v given y is v|y ∼ CN (0, Ω). Note that in Lemma 1,û and v are two jointly proper complex Gaussian vectors and the covariance matrix of those two vectors vanishes, which implies thatû and v are independent -the lack of correlation implies independence for two jointly proper Gaussian vectors [41]. In our setting, we consider the case of y t = x T t h t + z t , where x t a deterministic function of (y t−1 , w) given the encoding maps as in (2). When T c = 2 and t is an even number, Lemma 1 reveals that (15) and (16). Lemma 2. Consider any vectors e i ∈ C M ×1 , i ∈ Z, and let h t = x * t−1 y t−1 x t−1 2 +1 is the MMSE estimate of h t given (y t−1 , w) and h t |(y t−1 , w) ∼ CN (ĥ t , Ω t ), whereĥ t and Ω t are defined inK t I M − t−1 i=1 K i e * i e T i K i e T i K i e * i + 1 , t = 2, 3, 4, · · ·(41) and K 1 I M , then we have 0 K t I M , ∀t ∈ {1, 2, 3, · · · }. Proof. The proof is shown in Appendix B-A. Lemma 3. The solution for the following maximization problem maximize n t=1 log(1 + s t ) subject to n t=1 s t ≤ m s t ≥ 0, t = 1, 2, · · · , n is s 1 = s 2 = · · · = s n = m/n, for a positive constant m > 0. Proof. The proof is shown in Appendix B-B. V. CONVERSE: THE CASE WITH A FOURTH MOMENT INPUT CONSTRAINT For the channel model defined as in Section II, we will show that the rate is upper bounded by R ≤ log 1 + min M + 2, √ 2(T c + 1) · κP under the fourth moment input constraint in (6). The result reveals that, given a finite coherence T c and a finite κP on the fourth moment input constraint, then increasing the antenna-number M to infinity will not yield an infinite capacity. The result holds for any linear and nonlinear schemes, and holds for any channel training schemes. Similarly to the proof in Section IV for the case with a second moment input constraint, the proof for this case with a fourth moment input constraint is based on MMSE estimation techniques, MIMO techniques, as well as Cauchy-Schwarz inequality. In the proof we will use Lemmas 1 -6 shown in Section IV and in this section (see below). Following the steps (8)-(11), we bound the rate as follows: nR ≤ n t=1 h(y t y t−1 ) − log(πe) + n n (42) ≤ n t=1 h(y t ) − n log(πe) + n n(43) where (42) is from (8)-(11), (43) results from the fact that conditioning reduces differential entropy. We proceed to upper bound the differential entropy h y t in (43). Note that the average power of y t is E |y t | 2 = E |h T t x t + z t | 2 = 1 + E |h T t x t | 2 . Again, by using the fact that differential entropy is maximized by a circularly symmetric complex Gaussian distribution with the same average power, we have h y t ≤ log πe 1 + E |h T t x t | 2 .(44) Then, by combining (44) and (43) it yields the following bound on the rate: nR − n n ≤ n t=1 log πe 1 + E |h T t x t | 2 − n log(πe) = n t=1 log 1 + E |h T t x t | 2 .(45) Let us now focus on the expectation term E |h T t x t | 2 in (45). Similarly to the case with a second moment input constraint, x t is a function of (y t−1 , w), which implies that x t and h t can be correlated and consequently computing the value of E |h T t x t | 2 could be challenging in general. In what follows we seek to bound the value of E |h T t x t | 2 . We will use the notations ofĥ t ,h t and Ω t given aŝ h t t−1 i=Tc t−1 Tc +1 Ω i x * i (y i − x T iĥi ) x T i Ω i x * i + 1 for t = T c + 1,(46)Ω t I M − t−1 i=Tc t−1 Tc +1 Ω i x * i x T i Ω i x T i Ω i x * i + 1 for t = T c + 1,(47)h t h t −ĥ t ∀t ∈ {1, 2, · · · , n}(48) andĥ Tc+1 = 0, Ω Tc+1 = I M , ∀ ∈ {0, 1, · · · , L − 1}. From Lemma 4 (see below) we note thatĥ t is the MMSE estimate of h t given (y t−1 , w). By following the similar steps in (18) -(24), now we bound the value of E |h T t x t | 2 as E |h T t x t | 2 = E E |h T t x t | 2 y t−1 , w = E E tr h * t h T t x t x H t y t−1 , w = E tr E h * t h T t y t−1 , w · x t x H t (49) = E tr E (ĥ t +h t ) * (ĥ t +h t ) T y t−1 , w · x t x H t (50) = E tr E ĥ t * ĥ t T +h t * h t T +ĥ t * h t T +h t * ĥ t T y t−1 , w · x t x H t = E tr ĥ t * ĥ t T + Ω t · x t x H t (51) = E tr ĥ t * ĥ t T x t x H t + tr Ω t x t x H t ≤ E λ max (ĥ t * ĥ t T ) · tr(x t x H t ) + λ max (Ω t ) · tr(x t x H t ) (52) ≤ E ĥ t 2 · x t 2 + x t 2(53) where (49) stems from the fact that x t is a deterministic function of (y t−1 , w) given the encoding maps as in (2); in (50) we just replace h t withĥ t +h t , whereĥ t andh t are defined in (46)-(48); (51) results from the fact thatĥ t is a deterministic function of (y t−1 , w) given the encoding maps as in (2), and the fact thath t (y t−1 , w) ∼ CN (0, Ω t ) (cf. Lemma 4); (52) follows from the identity that tr(AB) ≤ λ max (A)tr(B) for positive semidefinite m × m Hermitian matrices A, B; (53) results from the facts that λ max (ĥ t * ĥ t T ) = ĥ t 2 , tr(x t x H t ) = x t 2 , and λ max (Ω t ) ≤ 1 (see Lemma 2 in Section IV). At this point, by combining (53) and (45) we bound the rate as nR − n n ≤ n t=1 log 1 + E ĥ t 2 · x t 2 + E x t 2 = n t=1 log 1 + E ( ĥ t 2 + 1) · x t 2 .(54) Sinceĥ t and x t are two functions of (y t−1 , w), it implies that computing (54) is also challenging. In order to bound E ( ĥ t 2 + 1) · x t 2 in (54), we use Cauchy-Schwarz inequality, that is, E[ab] ≤ E[|a| 2 ] · E[|b| 2 ] for any two random variables a and b. Then, it yields E ĥ t 2 · x t 2 or E ( ĥ t 2 + 1) · x t 2 inE ( ĥ t 2 + 1) · x t 2 ≤ E ( ĥ t 2 + 1) 2 · E x t 4 which, together with (54), gives the following bound on the rate nR − n n ≤ n t=1 log 1 + E ( ĥ t 2 + 1) 2 · E x t 4 (55) ≤ n t=1 log 1 + min M 2 +4M +1, 2[(t − 1) mod T c ] 2 + 7[(t − 1) mod T c ] + 1 · E x t 4 (56) ≤ n t=1 log 1 + min M + 2, √ 2(T c + 1) · E x t 4 (57) ≤ max n =1 E[ x 4 ]≤nκ 2 P 2 n t=1 log 1 + min M + 2, √ 2(T c + 1) · E x t 4(58) = n log 1 + min M + 2, √ 2(T c + 1) · κP(59) where (55) results from (54) and Cauchy-Schwarz inequality; (56) follows from the (66) in Lemma 5 (see below); [t mod T c ] denotes a modulo operation; (57) stems from that M 2 + 4M + 1 < (M + 2) 2 and that 2[(t − 1) mod T c ] 2 + 7[(t − 1) mod T c ] + 1 ≤ 2(T c − 1) 2 + 7(T c − 1) + 1 < 2(T c + 1) 2 ; (58) results from maximizing the RHS of (57) under a fourth moment constraint (cf. (6)); where (59) follows from Lemma 6. At this point, as n → ∞, we have the bound R ≤ log 1 + min M + 2, √ 2(T c + 1) · κP and complete the proof. Now we provide some lemmas used in our proofs. The following lemma is the extension of Lemma 1. Lemma 4. Consider independent random vectors u ∈ C M ×1 and z t ∈ C N ×1 , t = 1, 2, · · · , T , with each vector being complex Gaussian, that is, z t ∼ CN (0, I N ) and u ∼ CN (û 1 , Ω 1 ) for some fixedû 1 and some fixed Hermitian positive semidefinite Ω 1 . Let y t = A t u + z t , t = 1, 2, · · · , T, where A t ∈ C N ×M is a deterministic function of (y t−1 , w); w is a fixed parameter (or a set of fixed parameters). Then, the conditional density of u given (y t−1 , w) is u (y t−1 , w) ∼ CN (û t , Ω t ) whereû t û 1 + t−1 i=1 Ω i A H i (A i Ω i A H i + I N ) −1 (y i − A iû i ) (60) Ω t Ω 1 − t−1 i=1 Ω i A H i (A i Ω i A H i + I N ) −1 A i Ω i(61) for t = 2, 3, · · · , T . Furthermore,û t and v t u −û t are conditionally independent given (y t−2 , w) for t = 2, 3, · · · , T , and the conditional density of v t given ( y t−1 , w) is v t (y t−1 , w) ∼ CN (0, Ω t ). Proof. The proof is shown in Appendix B-D. Lemma 5. Forĥ t defined in (46), we have E ĥ t 2 ≤ [(t − 1) mod T c ] (62) E ĥ t 2 ≤ M (63) E ĥ t 4 ≤ 2[(t − 1) mod T c ] 2 + 5[(t − 1) mod T c ] (64) E ĥ t 4 ≤ M 2 + 2M (65) E ( ĥ t 2 + 1) 2 ≤ min M 2 + 4M + 1, 2[(t − 1) mod T c ] 2 + 7[t mod T c ] + 1(66) for t = 1, 2, · · · , n. [t mod T c ] denotes a modulo operation. Proof. The proof is shown in Appendix B-E. Proof. The proof is shown in Appendix B-C. VI. ACHIEVABILITY In this section we provide an achievability scheme for the MISO channel with feedback. To this end, the proposed scheme can achieve a rate R (bits/channel use) that is lower bounded by R ≥ T c − T τ T c · log 1 + P · max{(T τ − 1), 1/2} 2 + 1 P − 1 max{T τ , 2}(67) under the second moment input constraint (cf. (5) P o , the proposed scheme will satisfy the fourth moment input constraint and achieve the declared rate. In the following we will just describe the scheme for the case with a second moment input constraint. The proposed scheme is a simple scheme that uses no more than T c number of transmit-antennas. The scheme consists of a downlink training phase and a data transmission phase for each coherence block of the channel. The choice of phase duration is critical to the scheme performance, because with too small duration for training phase there is not enough time for the channel training, while with too large duration for training phase there is not enough time for the data transmission. In this scheme we set the durations of the training phase and data transmission phase as T τ = min{M, T c } log max{4, min{M, T c }} , T d = T c − T τ(68) respectively, as illustrated in Fig 3. Without loss of generality we focus on the scheme description for the first channel block, corresponding to the time index t ∈ {1, 2, · · · , T c }. Note that h 1 = h 2 = · · · = h Tc and h 1 = [h 1,1 , h 1,2 , · · · , h 1,M ] T . A. Downlink training The goal of the downlink training phase with feedback is to allow both user and transmitter to learn the channel state information. At time t, t = 1, 2, · · · , T τ , the downlink training is operated over the tth transmit-antenna in order to estimate the channel h 1,t , where h 1,t denotes the channel coefficient between the tth transmit antenna and the user during the first channel block. By setting the pilot signal as x t = √ P [0, 0, · · · , 0, 1, 0, · · · , 0] T , where the nonzero value is placed at the tth element, then the received signal of user at time t is given as y t = √ P h 1,t + z t , t = 1, 2, · · · , T τ .(69) As a result, the user observes T τ channel training outputs that can be written in a vector form: y τ = √ P h τ + z τ ,(70) where y τ [y 1 , y 2 , · · · , y Tτ ] T , h τ [h 1,1 , h 1,2 · · · , h 1,Tτ ] T and z τ [z 1 , z 2 , · · · , z Tτ ] T . After receiving the channel training outputs, the user can estimate channel h τ with MMSE estimator: h τ = √ P P + 1 y τ .(71) The MMSE estimateĥ τ and estimation errorh τ h τ −ĥ τ are two independent complex Gaussian vectors, whereĥ τ ∼ CN (0, P P +1 I) andh τ ∼ CN (0, 1 P +1 I). After MMSE estimation, the user feeds back the value ofĥ τ to the transmitter over an independent feedback link (the transmitter can also obtain the MMSE estimateĥ τ if the user feeds back the channel outputs to the transmitter). B. Data transmission After obtaining the channel state information ofĥ τ (CSIT) the transmitter sends the data information with linear precoding: x t = √ Pĥ * τ ĥ τ s t , t = T τ + 1, T τ + 2, · · · , T c (focusing on the first channel block), where s t denotes the information symbol with unit average power. The corresponding signal received at the user is given as: y t = h T τ x t + z t = √ P (ĥ τ +h τ ) Tĥ * τ ĥ τ s t + z t = √ P ĥ τ s t + √ Ph T τĥ * τ ĥ τ s t + z t , t = T τ + 1, T τ + 2, · · · , T c(72) (again, focusing on the first channel block). The channel input-output relationship in (72) can be further expressed in a vector form: y d = √ P ĥ τ s d + √ Ph T τĥ * τ ĥ τ s d + z d (73) where y d [y Tτ +1 , y Tτ +2 , · · · , y Tc ] T , s d [s Tτ +1 , s Tτ +2 · · · , s Tc ] T and z d [z Tτ +1 , z Tτ +2 , · · · , z Tc ] T . Note that the conditional distribution ofh T τĥ * τ ĥ τ givenĥ τ is a Gaussian distribution, that is,h T τĥ * τ ĥ τ ĥ τ ∼ CN (0, 1 P +1 ). Rate analysis: We now analyze the achievable rate of the proposed scheme. At first we assume that the input symbol s t , ∀t, is circularly symmetric complex Gaussian distributed, i.e., s t ∼ CN (0, 1), and is independent ofĥ τ and h τ . The following proposition provides a lower bound on the achievable ergodic rate. Proposition 3. The achievable ergodic rate for the scheme with Gaussian input, training and feedback, and data transmission as described in Sections VI-A,VI-B is bounded as R ≥ T c − T τ T c · log 1 + P · max{(T τ − 1), 1/2} 2 + 1 P − 1 max{T τ , 2} under the second moment input constraint (cf. (5)), where T τ = min{M,Tc} log max{4,min{M,Tc}} . Proof. The proof is shown in Appendix C. VII. CONCLUSION In this work we provided capacity bounds for the MISO block fading channel with a noiseless feedback link, under the second and fourth moment input constraints respectively. The result holds for any linear and nonlinear coding strategies, any channel training schemes, any long-term and short-term input constraints. The result reveals that, increasing the transmit-antenna number M to infinity will not yield an infinite capacity, for the case with a finite second-moment input constraint and T c = 2, and for the case with a finite fourth-moment input constraint and a finite coherence T c . In addition to the capacity bounds, this work also provided a characterization on the channel's beamforming gain for some cases. Specifically, for the case with a finite fourth-moment input constraint and T c = M α , the result reveals that α = 1 is sufficient for achieving a full beamforming gain. When 0 ≤ α ≤ 1, the beamforming gain increases linearly with α. The result has provided some practical insights for the massive MIMO system operating with FDD mode. APPENDIX A PROOFS OF PROPOSITION 1 In this section we provide the proof of Proposition 1, for the ideal case of MISO channel with perfect CSIT and CSIR, and with a second moment input constraint. For this ideal case, the channel capacity is characterized in [7] in a closed form C ideal = max P (γ): γP (γ)fγ (γ)dγ=P γ log 1 +P (γ) · γ f γ (γ)dγ(74) where γ h t 2 , f γ (γ) is the probability density function of γ, and the optimal power allocation of P (γ) is based on a water-filling algorithm (cf. [7]). We here focus on the asymptotic analysis when the antenna-number M is large. For the capacity C ideal expressed in (74), it can be upper bounded as: C ideal = max P (γ): Eγ [P (γ)]=P E γ log 1 +P (γ) · γ ≤ max P (γ): Eγ [P (γ)]=P E γ log 1 +P (γ) + E γ log 1 + γ (75) ≤ max P (γ): Eγ [P (γ)]=P log 1 + E γ [P (γ)] + log 1 + E γ [γ](76) = log 1 + P + log 1 + M (77) = log 1 + P M + P + M where (75) results from the identity that log(1 + a 1 a 2 ) ≤ log(1 + a 1 ) + log(1 + a 2 ) for any a 1 ≥ 0 and a 2 ≥ 0; (76) stems from Jensen's inequality; (77) follows from the fact that E γ [γ] = E[ h t 2 ] = M . Let us now focus on the lower bound on C ideal expressed in (74). Since C ideal is determined by the optimal power allocation ofP (γ) over all possible power allocation strategies. Clearly, settingP (γ) = P , ∀γ (equal power allocation) gives a lower bound on C ideal . Therefore, C ideal = max P (γ): Eγ [P (γ)]=P E γ log 1 +P (γ) · γ ≥ E γ log 1 + P · γ (79) ≥ E γ log P · γ + (80) = E γ log 2γ + log P 2 + ≥ log max{2M − 2, 1} + log P 2 + (81) = log max{(M − 1)P, P/2} + ≥ log(1 + (M − 1)P ) − 1(82) where (79) uses a suboptimal power allocation, i.e.,P (γ) = P , ∀γ, which will not increase the value of C ideal ; (80) uses the notation of (•) + = max{•, 0}; (81) stems from Lemma 7 (see below), that is, E γ log 2γ ≥ log max{2M − 2, 1}; note that 2γ = 2 h t 2 ∼ X 2 (2M ); (82) follows from the identity that log x + ≥ log(1 + x) − 1 for a positive x. Therefore, combing the upper bound and lower bound in Lemma 7. If u ∼ X 2 (k) is a chi-square random variable with k ≥ 2 degrees of freedom, k is an even number, then E[log u] ≥ log max{k − 2, 1}. Proof. If u is a chi-square random variable with k ≥ 2 degrees of freedom, its probability density function is given by f X (u) = u k/2−1 e −u/2 2 k/2 Γ(k/2) u > 0 0 else (84) where Γ(•) is a Gamma function (cf. [42]). When k ≥ 2 and k is an even number, we have E[ln u] = ψ(k/2) + ln 2 (see 4.352-1 in [43]), where ψ(x) is the digamma function. Note that ψ(1) = −γ o , where γ o ≈ 0.57721566 is Euler's constant, and for any integer x > 1 the digamma function ψ(x) can be expressed as ψ(x) = −γ o + x−1 p=1 1 p (cf. [44], [45]). Therefore, when k > 2 and k is an even number, we have APPENDIX B PROOFS OF LEMMAS 2 -6 In this section we provide the proofs of Lemmas 2 -6. A. Proof of Lemma 2 We will prove that, for any vectors e i ∈ C M ×1 for i ∈ Z, and for K t I M − t−1 i=1 K i e * i e T i K i e T i K i e * i + 1 , t = 2, 3, 4, · · ·(88) and K 1 I M , then 0 K t I M , ∀t ∈ {2, 3, · · · }. From the definition in (88), we have K t+1 = K t − K t e * t e T t K t e T t K t e * t + 1 , t ∈ {1, 2, 3, · · · }.(89) One can easily check from (89) that, if K t is a Hermitian matrix, then K t+1 is also a Hermitian matrix for t ∈ {1, 2, 3, · · · }. Since K 1 I M is a Hermitian matrix, then from the above recursive argument it is true that K t is a Hermitian matrix for t ∈ {1, 2, 3, · · · }. In the second step, we will prove that if the Hermitian matrix K t is positive semidefinite, then the Hermitian matrix K t+1 is also positive semidefinite for t ∈ {1, 2, 3, · · · }. Specifically, if the Hermitian matrix K t is positive semidefinite, t ∈ {1, 2, 3, · · · }, then for any vector x ∈ C M ×1 we have x H K t+1 x = x H K t − K t e * t e T t K t e T t K t e * t + 1 x (90) = x H K t x − |x H K t e * t | 2 e T t K t e * t + 1 = b H b − |b H c| 2 c H c + 1 = b 2 + b 2 c 2 − |b H c| 2 c 2 + 1 ≥ b 2 + b 2 c 2 − b 2 c 2 c 2 + 1 (91) ≥ 0 (92) where (90) is from the definition in (89); the Hermitian positive semidefinite matrix K t is decomposed as K t U ΛU H = U Λ 1/2 U H U Λ 1/2 U H using singular value decomposition method, where U and Λ are the unitary matrix and diagonal matrix respectively, b U Λ 1/2 U H x and c U Λ 1/2 U H e * ; where (91) results from Cauchy-Schwarz inequality, i.e., |b H c| 2 ≤ b 2 c 2 . Since the Hermitian matrix K 1 is positive semidefinite, then from the above recursive argument it is true that the Hermitian matrix K t is positive semidefinite, t ∈ {1, 2, 3, · · · }. From the above steps we have proved that the matrix K t is Hermitian positive semidefinite, t ∈ {1, 2, 3, · · · }, which means that 0 K t , ∀t ∈ {1, 2, 3, · · · } In the next step we will prove that K t I M , ∀t ∈ {1, 2, 3, · · · } From the definition in (88), we have I M − K t t−1 i=1 K i e * i e T i K i e T i K i e * i + 1 , t = 2, 3, 4, · · ·(93) Since matrix K t is Hermitian positive semidefinite, t ∈ {1, 2, 3, · · · }, it holds true that K t e * t e T t K t e T t K t e * t + 1 0, t = 1, 2, 3, · · · (94) because for any vector x ∈ C M ×1 we have x H Kte * t e T t Kt e T t Kte * t +1 x = |x H Kte * t | 2 e T t Kte * t +1 ≥ 0. Then combing (94) and (93) it gives I M − K t t−1 i=1 K i e * i e T i K i e T i K i e * i + 1 0, t = 2, 3, 4, · · · which implies that I M K t , t = 2, 3, 4, · · · At this point, we completes the proof. B. Proof of Lemma 3 This subsection shows that the solution for the following minimization problem minimize − n t=1 log(1 + s t ) subject to n t=1 s t ≤ m − s t ≤ 0, t = 1, 2, · · · , n is s 1 = s 2 = · · · = s n = m/n, for a positive constant m > 0. Note that the Lagrangian of this convex optimization problem is given as L(s 1 , s 2 , · · · , s n , β, µ 1 , µ 2 , · · · , µ n ) = − n t=1 log(1 + s t ) + β n t=1 s t − m + n t=1 µ t · (−s t ), where β, µ 1 , µ 2 , · · · , µ n are Lagrangian parameters. By solving the following Karush-Kuhn-Tucker (KKT) conditions: ∂L(s 1 , s 2 , · · · , s n , β, µ 1 , µ 2 , · · · , µ n ) ∂s t = − 1 (1 + s t ) · ln 2 + β − µ t = 0, t = 1, 2, · · · , n,(95)β( n t=1 s t − m) = 0,(96)−µ t s t = 0, t = 1, 2, · · · , n (97) n t=1 s t − m ≤ 0,(98) −s t ≤ 0, t = 1, 2, · · · , n (99) β ≥ 0, µ t ≥ 0, t = 1, 2, · · · , n then it gives the optimal solution of s 1 = s 2 = · · · = s n = m/n. Note that s 1 = s 2 = · · · = s n = m/n, β = n (n+m)·ln 2 , and µ 1 = µ 2 = · · · = µ n = 0 satisfy the above KKT conditions. At this point we completes the proof. Note that one could also use the symmetric optimization method to prove the result of this lemma. C. Proof of Lemma 6 This subsection shows that the solution for the following minimization problem minimize − n t=1 log(1 + c √ s t ) subject to n t=1 s t ≤ m − s t ≤ 0, t = 1, 2, · · · , n is s 1 = s 2 = · · · = s n = m/n, for positive constants m > 0 and c > 0. The proof is similar to that of Lemma 3. Note that the Lagrangian of this convex optimization problem is given as L(s 1 , s 2 , · · · , s n , β, µ 1 , µ 2 , · · · , µ n ) = − n t=1 log(1 + c √ s t ) + β n t=1 s t − m + n t=1 µ t · (−s t ), where β, µ 1 , µ 2 , · · · , µ n are Lagrangian parameters. By solving the following KKT conditions: ∂L(s 1 , s 2 , · · · , s n , β, µ 1 , µ 2 , · · · , µ n ) ∂s t = − c 2 · s −1/2 t (1 + c √ s t ) · ln 2 + β − µ t = 0, t = 1, 2, · · · , n,(101)β( n t=1 s t − m) = 0,(102)−µ t s t = 0, t = 1, 2, · · · , n (103) n t=1 s t − m ≤ 0,(104) −s t ≤ 0, t = 1, 2, · · · , n (105) β ≥ 0, µ t ≥ 0, t = 1, 2, · · · , n then it gives the optimal solution of s 1 = s 2 = · · · = s n = m/n. Note that s 1 = s 2 = · · · = s n = m/n, β = 1 ( m n + 1 c · √ m n )·2 ln 2 , and µ 1 = µ 2 = · · · = µ n = 0 satisfy the above KKT conditions. D. Proof of Lemma 4 We will prove the following statement. Consider independent random vectors u ∈ C M ×1 and z t ∈ C N ×1 , t = 1, 2, · · · , T , with each vector being complex Gaussian, that is, z t ∼ CN (0, I N ) and u ∼ CN (û 1 , Ω 1 ) for some fixedû 1 and Hermitian positive semidefinite Ω 1 . Let y t = A t u + z t , t = 1, 2, · · · , T, where A t ∈ C N ×M is a deterministic function of (y t−1 , w); w is a fixed parameter (or a set of fixed parameters). y denotes an empty term if ≤ 0. Then, the conditional density of u given (y t−1 , w) is u (y t−1 , w) ∼ CN (û t , Ω t ) whereû t û t−1 +û t−1,t , Ω t Ω t−1 − Ω t−1,t(107)andû t−1,t Ω t−1 A H t−1 (A t−1 Ω t−1 A H t−1 + I N ) −1 (y t−1 − A t−1û t−1 ) Ω t−1,t Ω t−1 A H t−1 (A t−1 Ω t−1 A H t−1 + I N ) −1 A t−1 Ω t−1 for t = 2, 3, · · · , T . Let v t u −û t thenû t and v t are conditionally independent given (y t−2 , w), and the conditional density of v t given (y t−1 , w) is v t (y t−1 , w) ∼ CN (0, Ω t ) for t = 2, 3, · · · , T . This lemma is the extension of the well-known MMSE estimate result that is expressed in Lemma 1. Now we proceed with the proof. We first consider the simple case of t = 2. From Lemma 1 we conclude that the conditional density of u given (y 1 , w) is u|(y 1 , w) ∼ CN (û 2 , Ω 2 )(108)whereû 2 =û 1 + Ω 1 A H 1 (A 1 Ω 1 A H 1 + I N ) −1 (y 1 − A 1û 1 )(109) and Ω 2 = Ω 1 − Ω 1 A H 1 (A 1 Ω 1 A H 1 + I N ) −1 A 1 Ω 1(110) where A 1 is a deterministic function of w by definition. It follows from Lemma 1 thatû 2 and v 2 u−û 2 are independent; the conditional density of v 2 given (y 1 , w) is v 2 |(y 1 , w) ∼ CN (0, Ω 2 ). We then consider the case of t = 3 (T ≥ 3). By using the result in (108), that is, u|(y 1 , w) ∼ CN (û 2 , Ω 2 ), it yields the following conclusion: z 2 u (y 1 , w) ∼ CN 0 u 2 , I N 0 N ×M 0 M ×N Ω 2 .(111) Let us now look at the following vector y 2 u = I N A 2 0 M ×N I M z 2 u (112) where A 2 is a deterministic function of (y 1 , w). It is well known that the affine transformation of a complex proper Gaussian vector also yields a complex proper Gaussian vector, that is, if e ∈ C q×1 ∼ CN (µ, Q), then it holds true that Be ∼ CN (Bµ, BQB H ) for fixed µ ∈ C q×1 , B ∈ C p×q and Q ∈ C q×q (see, e.g., [6], [41]). Therefore, by combining (111) and (112) it gives y 2 u (y 1 , w) ∼ CN A 2û 2 u 2 , K 1,1 K 1,2 K 2,1 K 2,2 K(113) where K 2,2 = Ω 2 , K 2,1 = Ω 2 A H 2 , K 1,2 = A 2 Ω 2 , K 1,1 = A 2 Ω 2 A H 2 + I N .(114) Let us consider a new vector obtained from the following affine transformation: y 2 u−K 2,1 K −1 1,1 y 2 = B y 2 u where B = I N 0 N ×M −K 2,1 K −1 1,1 I M . As mentioned, affine transformation of a complex proper Gaussian vector also yields a complex proper Gaussian vector. Therefore, y 2 u−K 2,1 K −1 1,1 y 2 (y 1 , w) ∼ CN A 2û 2 u 2 − K 2,1 K −1 1,1 A 2û 2 , K 1,1 0 N ×M 0 M ×N K 2,2 −K 2,1 K −1 1,1 K 1,2 . (115) The result in (115) implies that u−K 2,1 K −1 1,1 y 2 (y 1 , w) ∼ CN û 2 − K 2,1 K −1 1,1 A 2û 2 , K 2,2 − K 2,1 K −1 1,1 K 1,2 .(116) The result in (115) also implies that the two vectors y 2 and u−K 2,1 K −1 1,1 y 2 are conditionally independent given (y 1 , w) because their conditional cross-covariance vanishes. Based on this independence and (116), it gives u − K 2,1 K −1 1,1 y 2 (y 2 , y 1 , w) ∼ CN û 2 − K 2,1 K −1 1,1 A 2û 2 , K 2,2 − K 2,1 K −1 1,1 K 1,2(117) and u−K 2,1 K −1 1,1 y 2 +K 2,1 K −1 1,1 y 2 u (y 2 , y 1 , w) ∼ CN û 2 +K 2,1 K −1 1,1 (y 2 −A 2û 2 ) û 3 , K 2,2 −K 2,1 K −1 1,1 K 1,2 Ω 3 .(118) Finally, plugging (114) into (118) leads to the following conclusion: u|(y 2 , y 1 , w) ∼ CN (û 3 , Ω 3 ) whereû 3 =û 2 + Ω 2 A H 2 (A 2 Ω 2 A H 2 + I N ) −1 (y 2 − A 2û 2 ) and Ω 3 = Ω 2 − Ω 2 A H 2 (A 2 Ω 2 A H 2 + I N ) −1 A 2 Ω 2 , as defined in (107). Let v 3 u −û 3 . Then, the conditional density of v 3 given (y 2 , y 1 w) is v 3 |(y 2 , y 1 , w) ∼ CN (0, Ω 3 ). Note thatû 3 is conditionally independent of v 3 given (y 1 , w), since Cov(û 3 , v 3 y 1 , w) E û 3 − E[û 3 |y 1 , w] v 3 − E[v 3 |y 1 , w] H y 1 , w = 0 and the vectorsû 3 and v 3 are two jointly proper Gaussian vectors given (y 1 , w). The lack of correlation implies independence for two jointly proper Gaussian vectors. For the general case when t = 4, 5, · · · , T , the proof is similar to the previous case. At this point we complete the proof. E. Proof of Lemma 5 Forĥ t defined as in (46) and (47), we will prove the following bounds E ĥ t 2 ≤ [(t − 1) mod T c ] (119) E ĥ t 2 ≤ M (120) E ĥ t 4 ≤ 2[(t − 1) mod T c ] 2 + 5[(t − 1) mod T c ](121)E ĥ t 4 ≤ M 2 + 2M (122) E ( ĥ t 2 + 1) 2 ≤ min M 2 + 4M + 1, 2[(t − 1) mod T c ] 2 + 7[(t − 1) mod T c ] + 1(123) for t = 1, 2, · · · , n. Let us provide some lemmas that will be used in our proof. At first we rewrite the definitions ofĥ t and Ω t in (46) and (47) aŝ h t+1 ĥ t +ĥ t,t+1 ,ĥ t,t+1 Ω t x * t (y t − x T tĥt ) x T t Ω t x * t + 1 for t + 1 = T c + 1 (124) Ω t+1 Ω t − Ω t x * t x T t Ω H t x T t Ω t x * t + 1 for t + 1 = T c + 1(125) andĥ Tc+1 = 0, Ω Tc+1 = I M , ∀ ∈ {0, 1, · · · , L − 1}. Lemma 8. Forĥ t+1 andĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, we have E ||ĥ 1 +ĥ 1,2 +ĥ 2,3 + · · · +ĥ t,t+1 || 2 = E ||ĥ 1 || 2 + ||ĥ 1,2 || 2 + ||ĥ 2,3 || 2 + · · · + ||ĥ t,t+1 || 2 . Proof. See Appendix B-F. Lemma 9. Forĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, the following bounds hold E ||ĥ t,t+1 || 2 w, y t−1 ≤ 1,(126)E[||ĥ t,t+1 || 2 ] ≤ 1.(127) Proof. See Appendix B-G. Lemma 10. Forĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, the following inequalities hold E ||ĥ t,t+1 || 4 w, y t−1 ≤ 3,(128)E[||ĥ t,t+1 || 4 ] ≤ 3.(129) Proof. See Appendix B-H. Lemma 12. Forĥ t+1 =ĥ t +ĥ t,t+1 defined in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, we have E ||ĥ t+1 || 4 ≤ E ĥ t 4 + 4 · E ĥ t 2 + 3. Proof. See Appendix B-I. Now we are ready to prove Lemma 5. At first we focus on the case of t ∈ {1, 2, · · · , T c } and prove (62) in Lemma 5 (or equivalently, (119)): E ĥ t 2 = E ||ĥ 1,2 || 2 + ||ĥ 2,3 || 2 + · · · + ||ĥ t−1,t || 2 (130) ≤ 1 + · · · + 1 (131) = t − 1(132) where (130) results from Lemma 8; (131) follows from Lemma 9. For the general case of t ∈ {1, 2, · · · , n}, we note thatĥ t is a function of (x t−1 Tc t−1 Tc +1 , y t−1 Tc t−1 Tc +1 ) (cf. (46), (47)), where y t−1 Tc t−1 Tc +1 corresponds to the channel outputs (up to time t − 1) within the current channel block associated with time t. We also note that the previous results in (132) only depends on the number of channel outputs within the current channel block. Therefore, one can easily follow the previous steps and show that E ĥ t 2 = t−1 i=Tc t−1 Tc +1 E[||ĥ i,i+1 || 2 ] ≤ 1 + 1 + · · · + 1 (133) = [(t − 1) mod T c ], t ∈ {1, 2, · · · , n}(134) whereĥ i,i+1 Ω i x * i (y i −x T iĥi ) x T i Ω i x * i +1 for T c t−1 Tc (46) and (47); (133) is again from Lemma 9. + 1 ≤ i ≤ t − 1; whereĥ i and Ω i are defined in We now prove (63) in Lemma 5 (or (120)): E ĥ t 2 ≤ E ĥ t 2 + E h t 2 = E ĥ t +h t 2 − E ĥ H tht =0 − E h H tĥt =0 (135) = E h t 2 (136) = M(137) whereh t h t −ĥ t ; (135) is from the identity that a + b 2 = a 2 + b 2 + a H b + b H a for any two vectors a, b ∈ C M ×1 ; (136) follows from the fact that E ĥ H tht = E E ĥ H tht w, y t−1 = E[0] = 0 by using the result thath t (w, y t−1 ) ∼ CN (0, Ω t ) and thatĥ t is deterministic given (w, y t−1 ); similarly, E h H tĥt = 0; (137) is from the assumption that h t ∼ CN (0, I M ) . We focus on the case of t ∈ {1, 2, · · · , T c } and prove (64) in Lemma 5 (or (121)): E ĥ t 4 ≤ E ĥ t−1 4 + 4 · E ĥ t−1 2 + 3 (138) ≤ E ĥ t−1 4 + 4(t − 1) + 3 (139) ≤ E ĥ 1 4 + 4(1 − 1) + 4(2 − 1) + · · · + 4(t − 1) + 3(t − 1) (140) = 2(t − 1) 2 + 5(t − 1)(141) where (138) follows from Lemma 12; (139) is from the result in (132); (140) follows by repeating the steps of (138) and (139); (141) uses the definition thatĥ 1 = 0. For the general case of t ∈ {1, 2, · · · , n}, we again note thatĥ t is a function of (x t−1 Tc t−1 Tc +1 , y t−1 Tc t−1 Tc +1 ). Therefore, one can easily follow the previous steps and show that E ĥ t 4 ≤ E ĥ Tc t−1 Tc +1 4 + [(t−1)modTc] k=0 4k + 3([t mod T c ] − 1) (142) = 2[(t − 1) mod T c ] 2 + 5[(t − 1) mod T c ](143) where (143) uses the definition ofĥ Tc t−1 Tc +1 = 0. We now prove (65) in Lemma 5 (or (122)): E ĥ t 4 ≤E ĥ t 4 + E h t 4 ≥0 + E 2 ĥ t 2 h t 2 ≥0 + E 4Re 2 (ĥ H tht ) ≥0 + E 4( ĥ t 2 + h t 2 ) · Re(ĥ H tht ) =0 (144) =E ĥ t +h t 4 (145) =E h t 4 whereh t h t −ĥ t ; (145) stems from the identity that a + b 4 = a 4 + b 4 + 2 a 2 b 2 + 4Re 2 (a H b) + 4( a 2 + b 2 ) · Re(a H b) for any two vectors a, b ∈ C M ×1 , where Re(•) denotes the real part of the argument; (146) follows from Lemma 11; (144) results from the fact that E 4( ĥ t 2 + h t 2 ) · Re(ĥ H tht ) =E E 4( ĥ t 2 + h t 2 ) · Re(ĥ H tht ) (w, y t−1 ) =E 4 ĥ t 2 · E Re(ĥ H tht ) (w, y t−1 ) =0 + E 4 h t 2 · Re(ĥ H tht ) (w, y t−1 ) =0 (147) =E[0 + 0] (148) =0 where (147) results from the fact thatĥ t is deterministic given (w, y t−1 ); (148) follows from the identities that E[Re(a H b)] = Re(E[a H b]) = 0 and E[ b 2 · Re(a H b)] = Re(E[a H b · b 2 ]) = 0 for a fixed vector a and a Gaussian vector b ∼ CN (0, K); note that the odd-order moments of a complex proper Gaussian vector are zeros (see, e.g., [48] (62) and (64) gives E ( ĥ t 2 + 1) 2 = E[ ĥ t 4 ] + 2E[ ĥ t 2 ] + 1 ≤ 2[(t − 1) mod T c ] 2 + 7[(t − 1) mod T c ] + 1. At this point we complete the proof of Lemma 5. F. Proof of Lemma 8 We here prove that, forĥ t+1 andĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, we have E ||ĥ 1 +ĥ 1,2 +ĥ 2,3 + · · · +ĥ t,t+1 || 2 = E ||ĥ 1 || 2 + ||ĥ 1,2 || 2 + ||ĥ 2,3 || 2 + · · · + ||ĥ t,t+1 || 2 . For the case of t ∈ {1, 2, · · · , T c − 1}, we have E ||ĥ 1 +ĥ 1,2 +ĥ 2,3 + · · · +ĥ t,t+1 || 2 = E ||ĥ t +ĥ t,t+1 || 2 (149) = E E ||ĥ t +ĥ t,t+1 || 2 w, y t−1 (150) = E E ||ĥ t || 2 + ||ĥ t,t+1 || 2 +ĥ H tĥt,t+1 +ĥ H t,t+1ĥt w, y t−1 = E ||ĥ t || 2 + E ||ĥ t,t+1 || 2 w, y t−1 + 0 + 0 (151) = E ||ĥ t || 2 + E E ||ĥ t,t+1 || 2 w, y t− = E ||ĥ t || 2 + E ||ĥ t,t+1 || 2 = E ||ĥ t−1 +ĥ t−1,t || 2 + E ||ĥ t,t+1 || 2 (152) = E ||ĥ t−1 || 2 + E ||ĥ t−1,t || 2 + E ||ĥ t, where (149) uses the definitions ofĥ t+1 andĥ t,t+1 as in (124) and (125); (150) is from the identity that E[a] = E[E[a|b]] for random a and b; (151) follows from the fact thatĥ t,t+1 is a complex Gaussian vector with zero mean given (w, y t−1 ) (cf. Lemma 4) and the fact thatĥ t is a deterministic function of (w, y t−1 ) given the encoding maps as in (2); (152) use the definitions ofĥ t−1 andĥ t−1,t as in (124) G. Proof of Lemma 9 We will prove that, forĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c −1}, the following bounds hold E ||ĥ t,t+1 || 2 w, y t−1 ≤ 1, E[||ĥ t,t+1 || 2 ] ≤ 1. We will just prove the first inequality, as the second inequality follows immediately from the first inequality and the identity that E[||ĥ t,t+1 || 2 ] = E E ||ĥ t,t+1 || 2 w, y t−1 . Given thatĥ t,t+1 = H. Proof of Lemma 10 We will prove that, forĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, the following inequalities hold E ||ĥ t,t+1 || 4 w, y t−1 ≤ 3,(160)E[||ĥ t,t+1 || 4 ] ≤ 3.(161) We will just prove the first inequality in (160), as the second inequality in (161) follows immediately from (160) and the identity that E[||ĥ t,t+1 || 4 ] = E E ||ĥ t,t+1 || 4 w, y t−1 . The proof of (160) follows from the proof steps of Lemma 9. Forĥ t,t+1 defined as in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, we have E ||ĥ t,t+1 || 4 w, y t−1 = E Ω t x * t (y t −x T tĥt ) x T t Ω t x * t + 1 H Ω t x * t (y t −x T tĥt ) x T t Ω t x * t + 1 2 w, y t−1 = E Ω t x * t (x T th t + z t ) H Ω t x * t (x T th t + z t ) 2 x T t Ω t x * t + 1 4 w, y t−1 (162) = E x T t Ω t Ω t x * t 2 · x T th t + z t 4 x T t Ω t x * t + 1 4 w, y t−1 = x T t Ω t Ω t x * t 2 x T t Ω t x * t + 1 4 · E x T th t + z t 4 w, y t−1(163) where (162) use the definition ofh t h t −ĥ t ; (163) follows from the fact that x t is a deterministic function of (y t−1 , w) and Ω t is a deterministic function of (y t−2 , w) given the encoding maps as in (2). Let us focus on the inner expectation term in (163). Note that, for two complex numbers a and b, we have |(a + b)| 4 = |a| 4 + |b| 4 + 2|a| 2 |b| 2 + 4Re 2 (ab * ) + 4(|a| 2 + |b| 2 ) · Re(ab * ). In the following, we will replace a and b with x T th t and z t respectively and compute E x T th t + z t 4 w, y t−1 . At first we note that given z t ∼ CN (0, 1) andh t |(y t−1 , w) ∼ CN (0, Ω t ), the following equalities holds true: E[z t ] = 0 (165) E[|z t | 2 ] = 1 (166) E[z t z t ] = 0 (167) E[z t · |z t | 2 ] = 0 (168) E[|z t | 4 ] = 3(169) E[x T th t |w, y t−1 ] = 0 (170) E[|x T th t | 2 |w, y t−1 ] = x T t Ω t x * t (171) E[|x T th t | 4 |w, y t−1 ] = 3(x T t Ω t x * t ) 2(172) APPENDIX C PROOF OF PROPOSITION 3 In this section we provide the proof of Proposition 3. Note that our rate analysis is closely inspired by [9], [28]. For the proposed scheme with Gaussian input, training and feedback described in Sections VI-A,VI-B, the scheme achieves the following ergodic rate R = 1 T c I(s d ; y τ , y d ) by encoding the message over sufficiently large number of channel blocks, where the relationship between s d , y τ and y d are given in (70) and (73). The achievable rate can be lower bounded as: T c R =I(s d ; y τ , y d ) =I(s d ;ĥ τ , y τ , y d )(193)≥I(s d ;ĥ τ , y d )(194)=I(s d ;ĥ τ ) + I(s d ; y d ĥ τ ) =I(s d ; y d ĥ τ ) (195) =I(s d , √ P ĥ τ s d ; y d ĥ τ ) (196) ≥I( √ P ĥ τ s d ; y d ĥ τ ) (197) =h( √ P ĥ τ s d ĥ τ ) − h( √ P ĥ τ s d y d ,ĥ τ ) =T d · E[log(πeP ĥ τ 2 )] − h( √ P ĥ τ s d y d ,ĥ τ )(198) where (193) results from the fact that ĥ τ is a function of y τ ; where (194) and (197) are from the fact that adding more information will not reduce the mutual information; (195) is from our input assumption that s d and ĥ τ are independent; (196) uses the fact that √ P ĥ τ s d is a function of s d andĥ τ ; (198) follows from the fact that s d ∼ CN (0, I T d ), where T d = T c − T τ (cf. (68)). Let us focus on the second term in (198): h( √ P ĥ τ s d y d ,ĥ τ ) ≤ Tc t=Tτ +1 h( √ P ĥ τ s t y t ,ĥ τ ) (199) = Tc t=Tτ +1 h( √ P ĥ τ s t − β t y t y t ,ĥ τ ) (200) ≤ Tc t=Tτ +1 h( √ P ĥ τ s t − β t y t ĥ τ ) (201) ≤ Tc t=Tτ +1 E log πe · E √ P ĥ τ s t − β t y t 2 ĥ τ(202) where (199) is from chain rule and the fact that conditioning reduces differential entropy, where y d [y Tτ +1 , y Tτ +2 , · · · , y Tc ] T , s d [s Tτ +1 , s Tτ +2 · · · , s Tc ] T and y t = √ P ĥ τ s t + √ Ph T τĥ * τ ĥ τ s t + z t(203) (cf. 72); (200) results from that h( √ P ĥ τ s t y t ,ĥ τ ) = h( √ P ĥ τ s t −β t y t y t ,ĥ τ ) for any deterministic function β t of y t andĥ τ ; (201) is from the fact that conditioning reduces differential entropy; (202) uses the fact that Gaussian distribution is the differential entropy maximizer given the same second moment of E √ P ĥ τ s t − β t y t 2 ĥ τ . In the next step we will focus on a single term inside the summation in (202). Specifically, we will choose a proper β t to minimizes E | √ P ĥ τ s t − β t y t | 2 ĥ τ , which will in turn tighten the bound in (202), where y t is expressed in (203). This is equivalent to the MMSE estimation problem. For the MMSE estimation problem, the optimal c to minimize E |u − cv| 2 ] is c = E[uv * ] E[|v| 2 ] and in this case E |u − c v] = E[|u| 2 ] − |E[uv * ]| 2 E[|v| 2 ] , for two random variables u and v with zero mean. Therefore, the optimal β t can be chosen as β t = E √ P ĥ τ s t y * t ĥ τ E |y t | 2 ĥ τ = P ĥ τ 2 P ĥ τ 2 + P σ 2 + 1(204) where σ 2 1 P + 1 corresponding to the variance ofh T τĥ * τ ĥ τ givenĥ τ . Remind thatĥ τ andh τ are independent with each other,ĥ τ ∼ CN (0, P P +1 I) andh τ ∼ CN (0, 1 P +1 I). By setting β t as in (204), we have E | √ P ĥ τ s t − β t y t | 2 ĥ τ = E | √ P ĥ τ s t | 2 ĥ τ − E[ √ P ĥ τ s t y * t ĥ τ ] 2 E[|y t | 2 ĥ τ ] = P ĥ τ 2 − P ĥ τ 2 2 P ĥ τ 2 + P σ 2 + 1 = P ĥ τ 2 · (P σ 2 + 1) P ĥ τ 2 + P σ 2 + 1 . By plugging (205) and (202) into (198), we have: T c R ≥ T d · E log πeP ĥ τ 2 − T d · E log πe · P ĥ τ 2 · (P σ 2 + 1) P ĥ τ 2 + P σ 2 + 1 = T d · E log 1 + P ĥ τ 2 P σ 2 + 1(206) Note thatĥ τ ∼ CN (0, P P +1 I) and δĥ τ ∼ CN (0, 2I), for δ 2(P + 1) P . It then implies that δĥ τ 2 is chi-squared distributed with 2T τ degrees of freedom, that is, δĥ τ 2 ∼ X 2 (2T τ ). If u is a chi-square random variable with k ≥ 2 degrees of freedom, its probability density function is given by (84) and its probability density function is zero when u ≤ 0. Therefore, without loss of generality we consider δĥ τ 2 as a positive chi-squared random variable with 2T τ degrees of freedom. Then, from (206) we further have T c R ≥ T d · E log 1 + P δ 2 δĥ τ 2 P σ 2 + 1 = T d · E log( δĥ τ 2 ) + T d · E log 1 δĥ τ 2 + P/δ 2 P σ 2 + 1 ≥ T d · E log( δĥ τ 2 ) + T d · log 1 E δĥ τ 2 + P/δ 2 P σ 2 + 1 (207) = T d · E log( δĥ τ 2 ) + T d · log 1 2T τ + P/δ 2 P σ 2 + 1(208) where (207) follows from the fact that g(x) = log( 1 x + c) is a convex function since ∂ 2 g(x) ∂x 2 ≥ 0 for any x > 0, where c > is a constant; (208) results from that E δĥ τ 2 = 2T τ , since δĥ τ 2 ∼ X 2 (2T τ ). Let us now we focus on the first term in (208). From Lemma 7 described in Appendix A, we note that if u ∼ X 2 (k) is a chi-square random variable with k ≥ 2 degrees of freedom, k is an even number, then E[log u] ≥ log max{k − 2, 1} which, together with the fact that δĥ τ 2 ∼ X 2 (2T τ ), implies that E[log( δĥ τ 2 )] ≥ log(max{2(T τ − 1), 1}).(209) Finally, by plugging (209) into (208) we have: T c R ≥ T d · log max{2(T τ − 1), 1} + T d · log 1 2T τ + P/δ 2 P σ 2 + 1 = T d · log max{2(T τ − 1), 1} 2T τ + P δ 2 · max{2(T τ − 1), 1} P σ 2 + 1 = T d · log 1 − 1 max{T τ , 2} + P · max{2(T τ − 1), 1} P σ 2 δ 2 + δ 2 = (T c − T τ ) · log 1 − 1 max{T τ , 2} + P · max{(T τ − 1), 1/2} 2 + 1 P where δ 2 2(P +1) P , σ 2 1 P +1 and T d (T c − T τ ). By dividing both two sides of (210) with T c , it gives the final lower bound on the achievable rate of the proposed scheme. At this point we complete the proof. Fig. 1 . 1MISO channel with a feedback link. Fig. 2 . 2Beamforming gain b vs. α for the MISO channel with feedback, under the fourth moment input constraint. Proposition 2 (Beamforming gain). For the MISO channel with feedback defined in Section II and given a coherence T c = 2, the beamforming gain is b = 0 under the second moment input constraint. Lemma 6 .≥ 0, t = 1, 2 , 62The · · · , n is s 1 = s 2 = · · · = s n = m/n, for positive constants m > 0 and c > 0. Fig. 3 . 3The model of downlink training and data transmission, where the downlink training and data transmission are operated over Tτ and T d channel uses of each channel block. ( 78 ) 78and (82) leads to the following conclusion: log(1 + (M − 1)P ) − 1 ≤ C ideal ≤ log 1 + P M + P + M For a finite P , we have lim M →∞ log(1 + (M − 1)P ) − 1 log(1 + P M ) = 1 and lim M →∞ log 1 + P M + P + M log(capacity C ideal tends to log(1 + P M ) for a large M . At this point, we complete the proof. p ≥ ln m + γ o for any positive natural number m (cf.[46]). When k = 2, thenE[ln u] = ψ(1) + ln 2 = −γ o + ln 2 ≥ 0.(87)Finally, by combing (86) and (87), we have E[log u] = 1 ln 2 E[ln u] ≥ 1 ln 2 ln(max{k−2, 1}) = log(max{k− 2, 1}). Lemma 11 . 11[47, Theorem 6] Let u ∈ C M ×1 ∼ CN (0, Ω). For a fixed Hermitian matrix A ∈ C M ×M , then E[(u H Au) 2 ] = 2tr(AΩAΩ) + (tr(AΩ)) 2 . and (125); (153) follows from the previous steps in (149)-(152); (154) follows from the same step in (153). Note thatĥ 1 = 0. based on the capacity bounds in Theorems 2 and 3. Theorem 2 reveals that the capacity of a MISO channel with feedback is upper bounded as C ≤ log 1 + min M + 2, √ 2(T c + 1) · κP under the fourth moment input constraint. By replacing T c with M α , then the beamforming gain is upper bounded as b(α) ≤ lim M →∞ log 1 + min M + 2, √ 2(M α + 1) · κP log M = lim M →∞ log 1 + min M, M α log M ,M α } ) · log(1 + min{M,M α } log min{M,M α } ) log M = lim M →∞ log(1 + min{M, M α }) log M = min{1, α} which matches the upper bound. Note that when α = 0, the upper and lower bounds on the beamforming gain are matched immediately. At this point we complete the proof. Tc}} . For the case with a fourth moment input constraint (cf. (6)), the proposed scheme achieves the similar rate R with difference being that in the latter case P is replaced with P o κP √ 3 . Note that by replacing the input power P with), where T τ = min{M,Tc} log max{4,min{M,Downlink Training Data Transmission Channel block 1 T τ T d · · · Downlink Training Data Transmission Channel block 2 T τ T d = E[||ĥ 1 || 2 ] + E[||ĥ 1,2 || 2 ] + E[||ĥ 2,3 || 2 ] + · · · + E[||ĥ t,t+1 || 2 ]t+1 || 2 (153) . . . t x t | 2 = tr(h T t x t x H t h * t ) = tr(h * t h T t x t x H t )by using the identity of tr(AB) = tr(BA) for any matrices A ∈ C m×q , B ∈ C q×m ; (20) stems from the fact that x t is a deterministic function of (y t−1 , w) given the encoding maps as in(2); in (21) we just replace h t withĥ t +h t , whereĥ t andh t are defined in (15)-(17);(22) results from the fact thatĥ t is a deterministic function of (y t−1 , w) given the encoding maps as in(2), and the fact that the conditional density ofh t given (y t−1 , w) is t Ω t x * t − x T t Ω t Ω t x * t x T t U ΛU H x * t − x T t U ΛU H U ΛU H x * t = x T t U (Λ − Λ 2 )U H x * t ≥ 0by using the singular value decomposition of Ω t U ΛU H , where U and Λ are the unitary matrix and diagonal matrix respectively. Note that if 0 Ω t I M , then U (Λ − Λ 2 )U H 0. At this point we completes the proof. ACKNOWLEDGEMENTWe wish to thank AyferÖzgür and Andrea Goldsmith for helpful comments during the early stage of this work.whereĥ t,t+1 is defined in (124) and (125); where (155) use the definition ofh t h t −ĥ t and the fact that y t − x T tĥt = x T th t + z t ; (156) results from the facts that x t is a deterministic function of (y t−1 , w) and Ω t is a deterministic function of (y t−2 , w) given the encoding maps as in (2); (157) follows from the facts thath t |(y t−1 , w) ∼ CN (0, Ω t ) and that z t is independent ofh t ; (158) follows from thatwhere the first inequality follows from that 0 Ω t I M (cf.Lemma 2)and that x T where (172) follows from Lemma 11 in Appendix B-E, i.e., E[|x T tht | 4 |w,(169) also follows from Lemma 11. By using (164)-(172), we havewhere(173)is from (164); (174) follows from (165)-(172) as well as the fact that z t is independent of x t andh t ; (175) stems from the following conclusion for two independent complex random variables a and b, b167)). In the above we replace a and b with x T th t and z t respectively. The last step in (176) follows from (171).By plugging(176)into(163), we havewhere (178) uses the fact that x T t Ω t x * t ≥ 0 since Ω t 0 (cf. Lemma 2); (179) follows from the same step in (158), i.e.,At this point we completes the proof.I. Proof of Lemma 12We will prove that, forĥ t+1 =ĥ t +ĥ t,t+1 defined in (124) and (125), t ∈ {1, 2, · · · , T c − 1}, we haveWe will at first focus on the upper bound of E ĥ t +ĥ t,t+1 4 w, y t−1 . Remind thatĥ t,t+1 =and thath t |(y t−1 , w) ∼ CN (0, Ω t ) (cf. Lemma 4), whereh t h t −ĥ t . Thus, one can easily conclude that).(181)30Note that, for any two vectors a, b ∈ C M ×1 , a + b 4 can be expanded as in (164). Then, by replacing a and b withĥ t andĥ t,t+1 respectively, we havewhere (182) results from (164); (183) follows from the fact thatĥ t is deterministic given (w, y t−1 ), and the identities thatfor a fixed vector a and a Gaussian vector b ∼ CN (0, K); note that the odd-order moments of a complex proper Gaussian vector are zeros (see, e.g.,[48]); (184) results from Lemma 9 and Lemma 10, i.e., E ĥ t,t+1 4 w, y t−1 ≤ 3 and E ĥ t,t+1 2 w, y t−1 ≤ 1; (185) follows from thatwhere (186) follows from the identity thatfor two vectors a and b with the same dimension; (187) stems from (181), i.e., h t,t+1 |(y t−1 , w) ∼ CN (0, (190) follows from the same step in (158), i.e., x T t Ω t Ω t x * t ≤ x T t Ω t x * t + 1. 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Available. 62C.-P. Chen and F. Qi, "The best lower and upper bounds of harmonic sequence," RGMIA Res. Rep. Coll. Available online at http://rgmia.org/v6n2.php, vol. 6, no. 2, 2003. Moments of the complex multivariate normal distribution. S A Sultan, D S Tracy, Linear Algebra and its Applications. 237S. A. Sultan and D. S. Tracy, "Moments of the complex multivariate normal distribution," Linear Algebra and its Applications, vol. 237, pp. 191 -204, 1996. Moments and cumulants of the multivariate real and complex Gaussian distributions. K Triantafyllopoulos, Department of Mathematics, University of BristolK. Triantafyllopoulos, "Moments and cumulants of the multivariate real and complex Gaussian distributions," 2002, Department of Mathematics, University of Bristol.

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{'abstract': 'We consider communication over a multiple-input single-output (MISO) block fading channel in the presence of an independent noiseless feedback link. We assume that the transmitter and receiver have no prior knowledge of the channel state realizations, but the transmitter and receiver can acquire the channel state information (CSIT/CSIR) via downlink training and feedback. For this channel, we show that increasing the number of transmit antennas to infinity will not achieve an infinite capacity, for a finite channel coherence and a finite input constraint on the second or fourth moment. This insight follows from our new capacity bounds that hold for any linear and nonlinear coding strategies, and any channel training schemes. In addition to the channel capacity bounds, we also provide a characterization on the beamforming gain that is also known as array gain or power gain, at the regime with large number of antennas.', 'arxivid': '1703.01287', 'author': ['Jinyuan Chen '], 'authoraffiliation': [], 'corpusid': 3918747, 'doi': '10.1109/tit.2018.2851784', 'github_urls': [], 'n_tokens_mistral': 33900, 'n_tokens_neox': 29124, 'n_words': 18293, 'pdfsha': '9ed69d3566653a01e424a21ea810e336d253082f', 'pdfurls': ['https://arxiv.org/pdf/1703.01287v1.pdf'], 'title': ['On the MISO Channel with Feedback: Can Infinitely Massive Antennas Achieve Infinite Capacity?', 'On the MISO Channel with Feedback: Can Infinitely Massive Antennas Achieve Infinite Capacity?'], 'venue': []}

arxiv

Learning from Multiple Independent Advisors in Multi-agent Reinforcement Learning 2023. May 29 -June 2, 2023 Sriram Ganapathi Subramanian sriram.subramanian@vectorinstitute.ai Vector Institute TorontoCanada Matthew E Taylor matthew.e.taylor@ualberta.ca University of Waterloo WaterlooCanada Kate Larson kate.larson@uwaterloo.ca Alberta Machine Intelligence Institute University of Alberta Edmonton, EdmontonCanada, Canada Mark Crowley mcrowley@uwaterloo.ca University of Waterloo WaterlooCanada Sriram Ganapathi Subramanian University of Waterloo WaterlooCanada Matthew E Taylor University of Waterloo WaterlooCanada Kate Larson University of Waterloo WaterlooCanada Mark Crowley University of Waterloo WaterlooCanada Learning from Multiple Independent Advisors in Multi-agent Reinforcement Learning IFAAMAS . of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023)London, United Kingdom272023. May 29 -June 2, 2023 Multi-agent reinforcement learning typically suffers from the problem of sample inefficiency, where learning suitable policies involves the use of many data samples. Learning from external demonstrators is a possible solution that mitigates this problem. However, most prior approaches in this area assume the presence of a single demonstrator. Leveraging multiple knowledge sources (i.e., advisors) with expertise in distinct aspects of the environment could substantially speed up learning in complex environments. This paper considers the problem of simultaneously learning from multiple independent advisors in multi-agent reinforcement learning. The approach leverages a two-level -learning architecture, and extends this framework from single-agent to multi-agent settings. We provide principled algorithms that incorporate a set of advisors by both evaluating the advisors at each state and subsequently using the advisors to guide action selection. We also provide theoretical convergence and sample complexity guarantees. Experimentally, we validate our approach in three different test-beds and show that our algorithms give better performances than baselines, can effectively integrate the combined expertise of different advisors, and learn to ignore bad advice. INTRODUCTION Reinforcement learning (RL) has been successful in obtaining superhuman performances in a wide range of challenges such as Atari games [26], Go [35], and simple robotic tasks like opening doors and learning visuomotor policies [20]. However, it has not been straightforward to replicate these successes in complex real-world problems. One reason is that these problems often have a multiagent structure, where more than one learning agent participates at the same time, resulting in complicated dynamics. Despite research advances in multi-agent reinforcement learning (MARL) [11], poor sample efficiency in existing algorithms is one issue that still causes significant hurdles in applying MARL to complex problems [33]. Using external sources of knowledge that help in accelerating MARL training is one solution [2], which has extensive support in literature [33]. However, most prior work include two limiting assumptions. First, all demonstrations need to come from a single demonstrator [4]. In complex MARL environments, since agents learn policies that meet the twin goals of responding to changing opponent(s) and environments [22], a learner can likely benefit from multiple knowledge sources that have expertise in different parts of the environment or different aspects of the task. Second, all demonstrations are near-optimal (i.e., from an "expert") [29]. In practice, these knowledge sources are typically sub-optimal, and we broadly refer to them as advisors (to differentiate from experts). In this paper, we provide an approach that simultaneously leverages multiple different (sub-optimal) advisors for MARL training. Since the advisors may provide conflicting advice in different states, an algorithm needs to resolve such conflicts to take advantage of all the advisors effectively. We propose a two-level learning architecture and formulate a -learning algorithm for simultaneously incorporating multiple advisors in MARL, improving upon the previous work of Li et al. [21] in single-agent RL. This architecture uses one level to evaluate advisors and the other learns values for actions. Further, we extend our approach to an actor-critic variant that applies to the centralized training and decentralized execution (CTDE) setting [24]. Since RL is a fixed point iterative method [40], we provide convergence results, proving that our -learning algorithm converges to a Nash equilibrium [27] (under common assumptions). Additionally, we provide a detailed finite-time analysis of our -learning algorithm under two different types of learning rates. Finally, we experimentally study our approach in three different multi-agent test-beds, in relation to standard baselines. Since we relax the two limiting assumptions regarding learning from demonstrators in MARL, our hope is that this approach will spur successes in real-world applications, such as autonomous driving [10] and fighting wildfires [15], where MARL methods could use existing (sub-optimal) solutions as advisors to accelerate training. RELATED WORK This work is most related to the approach of reinforcement learning from expert demonstrations (RLED) [29]. A well-known RLED technique is deep -learning from demonstrations (DQfD) [12], which combines a temporal difference (TD) loss, an L2 regularization loss, and a classification loss that encourages actions to be close to that of the demonstrator. Another method, normalized actorcritic (NAC) [16], drops the classification loss and is more robust under imperfect demonstrations. However, NAC is prone to weaker performances than DQfD under good demonstrations due to the absence of classification loss. A different approach, human agent transfer (HAT) [42], extracts information from limited demonstrations using a classifier, while confidence-based human-agent transfer (CHAT) [47] improves HAT by using a confidence measurement to safeguard against sub-optimal demonstrations. A related approach is the teacher-student framework [44], where a pretrained policy (teacher) can be used to provide limited advice to a learning agent (student). Subsequent works expand this framework towards interactive learning [1], however, almost all works in this area assume a moderate level expertise for the teacher. Moreover, these are all independent methods primarily suited for single-agent environments, and may not be directly applicable in MARL context. Furthermore, external knowledge sources have also been used in MARL [33], where prior works often assume near optimal experts [30,49] or are only applicable to restrictive settings, such as fully cooperative or zero-sum competitive games [28,34,35,46,52]. Leno et al. [34] introduced a framework where an agent can learn from its peers in a shared learning environment, in addition to learning from the environmental rewards. Here the peers can be sub-optimal, however this work only applies to cooperative environments. Other works have provided a cooperative teaching framework for hierarchical learning [18,50]. For multi-agent general-sum environments, advising multiple intelligent reinforcement agents -decision making (ADMIRAL-DM) [38] is a -learning approach that incorporates real-time information from a single online sub-optimal advisor. One limitation of many prior works is the assumption of a single source of demonstration. In MARL, it may be possible to obtain advisors from different sources of knowledge that provide conflicting advice. For single-agent settings, Li et al. [21] provides the two-level -learning (TLQL) algorithm that incorporates multiple advisors in RL. The TLQL maintains two -networks, where the firstnetwork (high-level) keeps track of each advisor's performance and the second -network (low-level) learns the quality of each action. We improve upon TLQL and make it applicable to MARL settings. BACKGROUND Stochastic Games: A -player stochastic game is represented by a tuple ⟨ , 1 , . . . , , 1 , . . . , , , ⟩, where is the state space, is the action space of the agent ∈ {1, . . . , }, and : × 1 × · · · × − → R is the reward function of . Also, : × 1 × · · · − → Ω( ) is the transition probability that determines the next state given the current state and the joint action of all agents, where Ω is a probability distribution. Finally, ∈ [0, 1) is the discount factor. At each time , all agents observe the global state and take a local action [32]. The joint action = { 1 , . . . , } determines the immediate reward for and the next state of the system ′ . Each agent learns a suitable policy that gives the best responses to its opponent(s). The policy is denoted by : − → Ω( ). Let ≜ ( 1 , . . . , ) be the joint policy of all agents. At a state , the value function of under the joint policy is ( ) = ∞ =0 E , [ | 0 = , ] . This represents the expected discounted future reward of , when all agents follow the policy from the state . Related to the value function, is the action-value function or the -function. The -function of agent , under the policy , is given by, ( , ) = ( , ) + E ′ ∼ [ ( ′ )]. The setting we consider is general-sum stochastic games, where the reward functions of the different agents can be related in any arbitrary fashion. In this setting, the Nash equilibrium is typically considered as the solution concept [13], where the joint policy represents the joint policy of all agents except . In a Nash equilibrium, each agent plays the best response to the other agents and any deviation from this response is guaranteed to be worse off. Further, Hu and Wellman [13] proved that the -updates of an agent , using the Nash payoff at each stage eventually converges to its Nash value ( * ), which is the action-value obtained by the agent when all agents follow the joint Nash equilibrium policy for infinite periods. Two-level -learning: The TLQL algorithm [21] enables singleagent learning under the simultaneous presence of multiple advisors providing conflicting demonstrations. Here, the challenge is to determine which advisor to trust in a given state. In this regard, the TLQL contains two -tables, a high-level -table (abbreviated as high-) and a low-level -table (abbreviated as low-). The highstores the value of the ⟨ , ⟩ pair, where ∈ represents an advisor (with representing the set of all advisors). The highalso stores the value of following the RL policy in addition to each advisor. The low-maintains the value of each state-action pair. At each time step, the agent observes the state and selects an advisor (or the RL policy) from the high-using the -greedy strategy. If the high-returns an advisor, then the advisor's recommended action is performed. If the RL policy is returned, then an action is executed from the low-based on the -greedy strategy [39]. The low-is updated using the vanilla -learning Bellman update [48]. Subsequently, the high-is updated using a synchronization step. In this step, when an advisor's action is performed, the value of the advisor in the high-is simply assigned the value of that action from the low-. Finally, the high-of the RL policy is updated using the relation ℎ ℎ ( , ) = max ( , ). This synchronization update of high-preserves the convergence guarantees, due to the policy improvement guarantee in single-agent -learning [39]. There are two important limitations of TLQL. First, the highthat represents the value of the advisors also depends on the RL policy through the synchronization step. This value represents the value of taking the action suggested by the advisor at the current state and then following the RL policy from the next state onward. This definition is problematic since at the beginning of training, the RL policy is sub-optimal, and the objective is to accelerate learning by relying on external advisors and avoid using the RL policy at all. As advisors are evaluated at each state using the RL policy, it is likely that the most effective advisor among the set of advisors is not being followed until the RL policy improves. At this stage, it might be possible to simply follow the RL policy itself, defeating the purpose of learning from advisors. Second, the advisors have not been evaluated at the beginning of learning. Hence, it is impossible to find the most suitable advisor to follow, from the available advisors. While TLQL simply follows an -greedy exploration strategy, this approach could take many data samples to figure out the right advisor. We address both these limitations. TWO-LEVEL ARCHITECTURE IN MARL We consider a general-sum stochastic game, where there are a set of agents that are learning a policy with an objective of providing a best response to the other agents as well as the environment. Each agent can access a finite set of (possibly sub-optimal) independent advisors . We use to represent an advisor of , where ∈ { 1 , . . . , | | }. Each advisor can be queried by to obtain an action recommendation at each state of the stochastic game. These online advisors provide real-time action advice to the agent, which helps in learning to dynamically adapt to opponents. We consider a centralized training setting and assume 1) the advisors are fixed and do not learn, 2) the communication between agents and their advisors is free, 3) there is no communication directly between learning agents, 4) the environment is fully observable (i.e., an agent can observe the global state, all actions, and all rewards), and 5) the state and action spaces are discrete. Though we require these assumptions for theoretical guarantees, we will show that it is possible to relax a number of these assumptions in practice. To make TLQL applicable to multi-agent settings, we parameterize both the -functions with the joint actions, as is common in practice [22]. Also, we do not maintain the RL policy in the high-table and do not perform a synchronization step. These steps are no longer needed to preserve the convergence results in multi-agent settings, since we do not have a policy improvement guarantee (unlike in single-agent settings) [41]. Instead, we choose to use the probabilistic policy reuse (PPR) technique [6], where a hyperparameter ( ′ ∈ [0, 1]) decides the probability of following any advisor(s) (i.e., using the high-) or the agents' own policy (i.e., using the low-) for action selection, at each time step during training. This hyperparameter starts with a high value (maximum dependence on the available advisor(s)) at the beginning of training and is decayed (linearly) over time. After some finite time step during training, the value of this hyperparameter goes to 0 (no further dependence on any advisor(s)) and the agent only uses its low-(own policy) for action selection. This helps in two ways: 1) in the time limit ( − → ∞), a learning agent has the possibility of recovering from poor advising (by learning from the environment), and 2) eventually the trained agent can be independently deployed (with no requirement of having access to any advisor(s)). The general structure of our proposed Multi-Agent Two-Level -Learning (MA-TLQL) algorithm is given in Figure 1. Since we are in a fully observable setting, like [13], we specify that each agent maintains copies of the -tables of other agents from which it can obtain the joint actions of other agents for the current state. If such copies cannot be maintained, agents could use the previously observed actions of other agents for the joint action as done in prior works [38,51]. We use the two-level architecture, where each agent will maintain a high-as well as a low-. The high-provides a value for the ⟨ , − , ⟩ tuples, where − = { 1 , . . . , −1 , +1 , . . . , } is the joint action of all agents except the agent . This high-is a value estimate for the advisor as estimated by the agent at the state and joint action − . The high-estimates are updated with an evaluation update given by ℎ ℎ +1 ( , − , ) = ℎ ℎ ( , − , ) + ( + ℎ ℎ ( ′ , ′− , ) − ℎ ℎ ( , − , ) ,(1) where and ′ are the states at and + 1, and is the learning rate. Also, − and ′− are joint actions at and ′ , respectively. As described previously, a hyperparameter is used to decide between choosing to follow an advisor or the RL policy. If the agent follows an advisor, the high-is used to select an advisor using an ensemble selection technique. Let us denote, Q = {ℎ ℎ ( , − , 1 ), . . . , ℎ ℎ ( , − , )}, to represent the highestimates of a set of advisors (with | | = ) advising the same action to an agent . Here, represents an advisor ∈ {1, . . . , } of . Then the value of vote for action , at the state and the joint action − , denoted by V ( , − , ), is calculated as V ( , − , ) = max Q + =1, ≠arg max ℎ ℎ ( , − , ) 1 ( ) ℎ ℎ ( , − , ). (2) Here, ( ) is the number of times the agent has visited the state . In Eq. 2, if an action is recommended by more than one advisor, the value of its vote is a weighted sum of all high-estimates of advisors recommending that action. Each high-estimate (except the best high-estimate) is weighted by the reciprocal of the number of times the respective state is visited. In this way, when a state is visited many times, the advisor with the best high-estimate is likely to be followed (wisdom of individual). When a state is visited only a few times, then the action suggested by a majority of advisors is likely to be selected (wisdom of crowd). From Eq. 2, if an action is recommended by only one advisor, then the value of vote for will be equal to the high-estimate of that advisor. After the value of votes for all actions are calculated, the action with the maximum value of vote is executed, and the high-estimate of the advisor recommending this action is updated by the agent using Eq. 1. If the agent decides to use its RL policy, it uses its low-, which contains a value for the ⟨ , − , ⟩ tuples (value for each action). At each step, the low-is updated using a control update as follows: +1 ( , − , ) = ( , − , ) + + max ′ ( ′ , ′− , ′ ) − ( , − , ) . (3) Now we describe how MA-TLQL addresses the two limitations of TLQL. The first was the dependence of high-on the RL policy in TLQL. Note, the high-in MA-TLQL maintains the values of the advisor themselves, i.e., the value of following the advisor's policy from the current state onward (see Eq. 1). Thus, the coupling between the advisor values and the RL policy is removed (no synchronization). The second was the difficulty in picking the right advisor in TLQL. MA-TLQL uses an ensemble technique to choose the advisor during the early stages of learning. In later stages, it switches to following the best advisor according to the high-estimates, which addresses this limitation of TLQL. In Appendix J, we present a toy example that illustrates the limitations of TLQL. We provide the complete pseudocode for a tabular implementation of the MA-TLQL algorithm in Appendix A (Algorithm 1). Further, we extend this approach to large state-action environments using a neural network based implementation (Algorithm 2), which uses a target network and a replay buffer, as in the Deep -learning (DQN) algorithm [26]. We also provide an actor-critic implementation (Algorithm 3) which is suitable for CTDE [24]. We will refer to this algorithm as multi-agent two-level actor-critic (MA-TLAC). In MA-TLAC, each agent has two actors and two critics (high-level and low-level), where the respective -functions serve as the critic and the corresponding policies serve as the actors. In this CTDE method, agents can obtain global information (including actions and rewards of other agents) during training, however, the agents only require access to its local observation during execution. This makes our method applicable to partially observable environments as in Lowe et al. [24]. MA-TLAC applies to continuous state space environments as well (refer to Appendix A for more details). THEORETICAL RESULTS We present a convergence guarantee for tabular MA-TLQL and characterize the convergence rate. For these results, we build on some prior works that provide several fundamental results on the nature of stochastic iterative functions [3,5]. We apply these to MA-TLQL in general-sum stochastic games using three assumptions from Hu and Wellman [13], where the first two are standard [40]. Assumption 1. Every ∈ S and ∈ , for every agent are visited infinitely often, and the reward function (∀ ) stays bounded. Assumption 2. For all , , and , 0 ≤ ( , ) < 1, ∞ =0 ( , ) = ∞, ∞ =0 [ ( , )] 2 < ∞. Assumption 3. The Nash equilibrium is a global optimum or saddle point in every stage game of the stochastic game. The third assumption is a restriction on the nature of the stochastic game. Several prior works note that this assumption is restrictive but needed to theoretically prove the convergence of -learning methods in general-sum stochastic games with two or more agents. In practice, however, it is still possible to observe convergence of -learning methods when this assumption is violated [13,38,51]. Now we prove our theoretical results. All theorem statements are provided here, while the proofs can be found in Appendices B -D. First, we provide the convergence guarantee for the low-. Recall, the PPR technique guarantees that the MA-TLQL dependence on high-is only until a finite time step during training. After this step, the agent only uses its low-for action selection. As the convergence result in Theorem 1 is provided in the time limit ( − → ∞), the influence of high-can be neglected for this result. Theorem 1. Given Assumptions 1, 2, 3, the low-values of an agent converges to its Nash value in the limit ( − → ∞). Next, we provide sample complexity bounds for the MA-TLQL algorithm. Instead of explicitly considering the high-values, we specify that the underlying joint policy has a covering time of . The covering time specifies an upper bound on the number of time steps needed for all state-joint action pairs to be visited at least once starting from any state-joint action pair. Further, since the action selection is only based on the low-values in the limit ( − → ∞), we are most interested in the sample complexity of low-, where the dependence on the high-is effectively represented by . Regarding sample complexity, as is done in [5], we distinguish between two kinds of learning rates. Consider the following equation for the low-(rewriting Eq. 3 and dropping for simplicity), +1 ( , ) = 1 − ( , ))( ( , ) + ( , ) + max ( +1 , +1 ) .(4) The value of ( , ) = 1 [#( , , ) ] , where #( , , ) is the number of times until that the joint action is performed at . Here, we consider ∈ (1/2, 1]. The learning rate is linear if = 1, and the learning rate is polynomial if ∈ (1/2, 1). The next theorem provides a lower bound on the number of time steps needed for convergence in the case of a polynomial learning rate. From Assumption 1, let us specify that all rewards for the agent are bounded by max . We consider a variable max , which denotes the maximum possible low-value for the agent , which is bounded by max = max /(1− ). Additionally, we also use another variable = (1 − )/2 to present our upcoming results concisely. Theorem 2. Let us specify that with probability at least 1 − , for an agent , || − * || ∞ ≤ . The bound on the rate of convergence of low-, , with a polynomial learning rate of factor is given by (with * as the Nash -value of the agent ) = Ω 1+3 2, max ln( | |Π | | max ) 2 2 1− / + ( ln max + 1)/2 1 1− .(5) Assuming the same action spaces for all agents (i.e. | 1 | = | 2 | = · · · = | | = | |), we note that the dependence on the number of agents is ln | | = ln | |. Overall this results in a sub-linear dependence on the number of agents based on the value of , which is far superior to recent works that report an exponential dependence on the number of agents when learning in generalsum stochastic game environments (with an arbitrary number of agents) for convergence to a Nash equilibrium [23,36]. Further, the dependence on the state space and action space in Theorem 2 is sub-linear (ln | |), and the dependence on the covering time is Ω( 2 −3 2 + 1/1− ), which is a polynomial dependence. The next theorem considers the linear learning rate case. Theorem 3. Let us specify that with probability at least 1 − , for an agent , || − * || ∞ ≤ . The bound on the rate of convergence of low-, , with a linear learning rate is given by = Ω ( + + 1) 1 ln max 2, max ln( | |Π | | max ) 2 2 2 ,(6) where is a small arbitrary positive constant satisfying ≤ 0.712. Theorem 3 shows that the bound is linear in the number of agents and sub-linear in the state and action spaces. This linear dependence on the number of agents is also superior to prior results [23,36]. Note, the dependence on the covering time in Theorem 3 could be much worse than that of Theorem 2, depending on the value of max and . Since the value of is small, the dependence is certainly worse than that obtained for the polynomial learning rate case. Also, the dependence on max is exponential as opposed to a polynomial dependence for Theorem 2. The last two theorems illustrate the performance benefit in using a polynomial learning rate as opposed to a linear learning rate in our algorithm. EXPERIMENTS AND RESULTS We consider three different experimental domains, one each for competitive, cooperative, and mixed settings, where each agent has access to a set of four advisors. We use neural network implementations of MA-TLQL and MA-TLAC, along with 5 other baselines: DQN [26], DQfD [12], CHAT [47], ADMIRAL-DM [38], and TLQL [21]. In Appendix F, we tabulate the characteristics of these baselines and provide further details regarding our choices. Since CHAT and ADMIRAL-DM assume the presence of a single advisor, we use a weighted random policy approach for implementing these two algorithms in the multiple-advisor setting, as in Li et al. [21]. If different advisors provide different actions at the same state, each action is weighted based on the number of advisors suggesting that action. For DQfD, during pre-training [12], we populate the replay buffer using advisor demonstrations from all the available advisors. For all our experiments, we will describe the critical details here, while the complete description is in Appendix K. All the experiments are repeated 30 times, with averages and standard deviations reported. For statistical significance we use the unpaired 2-sided t-test and report -values, where < 0.05 is considered significant. The tests compare the highest performing algorithm (typically MA-TLQL) with the second-best baseline and best/average advisor performance. We conduct a total of seven experiments. The code for all experiments is open-sourced [37]. Appendix K tabulates all our experimental settings. Appendix L provides the hyperparameter details and Appendix M contains the wall clock times. Experiments 1-4 use the competitive, two-agent version of Pommerman [31]. The environment is complex, with each state containing roughly 200 elements related to agent position and special features (e.g., bombs). The reward function is sparse: agents only receive a terminal reward of {−1, 0, +1}. Experiments are conducted in two phases. In the first phase (training), our algorithms and the baselines train against a standard DQN opponent for 50,000 episodes, where we plot the cumulative rewards. During this phase, algorithms can use advisors to accelerate training. In the second phase (execution), we test the performance of the trained policies against DQN for 1000 episodes, where we plot the win rate (fraction of games won) for each algorithm. During this phase, agents cannot access advisors, take no exploratory actions, and do not learn. All advisors pertaining to these four experiments are rule-based agents. Experiment 1: Our first experiment uses a set of four advisors ranked in terms of quality from Advisor 1 to Advisor 4. Here, Advisor 1 is the best advisor, capable of teaching the agent all skills needed to win the game of Pommerman, and Advisor 4 only suggests random actions. In Pommerman, there is a fixed set of six skills that an agent needs to master to be able to win [31]. Since this set of advisors can teach all these skills, we say the agent has access to a sufficient set of advisors. We plot the training and execution performances in Figure 2(a) and (b) respectively, including the performance of the best and average advisors (average of all Advisors 1-4) against DQN. MA-TLQL gives the best performance ( < 0.01) and is the only algorithm providing a better performance than the best advisor ( < 0.11) in both training and execution. MA-TLAC performs better than the average advisor ( < 0.04). None of the others show better performances than the average advisor. CHAT and ADMIRAL-DM are not capable of leveraging and distinguishing amongst a set of advisors. DQfD uses pre-training, which is not very effective in the non-stationary multi-agent context. Learning from online advising is preferable in MARL. Also, DQfD and CHAT are independent techniques that are not actively tracking the opponent's performance. While TLQL is capable of learning from multiple advisors, its independent nature in addition to coupling of advisor values with the RL policy reduces its effectiveness in multi-agent environments. MA-TLQL gives a better performance than MA-TLAC in both training and execution ( < 0.01). As noted previously, the -learning family of algorithms tends to induce a positive bias while using the maximum action value, which leads to providing the best possible response [45]. This explains the superior performance of MA-TLQL. We conclude that MA-TLQL is capable of leveraging a set of good and bad advisors. Further, the training results in Figure 2(a) show that MA-TLQL is able to learn a better policy faster than the baselines by using advisors ( < 0.01). The evaluation results in Figure 2(b) show that amongst all algorithms trained for the same number of episodes, MA-TLQL provides the best performance, when deployed without any advisors ( < 0.01). Both observations point to better sample efficiency in MA-TLQL. Supplementary experiments in Appendix E show that MA-TLQL comes to relying more on good advisors than poor advisors, as compared to the baselines, illustrating its superiority. Experiment 2: We use the same domain as in Experiment 1, but with a different set of advisors. Now, all four advisors can teach strictly different Pommerman skills. For example, Advisor 1 can teach how to escape the enemy (and nothing else), and Advisor 2 can teach how to obtain necessary power-ups (and nothing else -full details are in Appendix K). These advisors provide psuedorandom action advice in states outside their expertise. This set of advisors is also a sufficient set. Now, learning agents must decide what advisor to listen to in the current state. From the training and execution results in Figure 3(a) and (b), we see that MA-TLQL gives the best overall performance ( < 0.02), exceeding the average performance of the four advisors ( < 0.05). Since all four advisors have similar quality, we only choose to use the average performance of the four advisors in this experiment for comparison. We conclude that MA-TLQL is capable of leveraging the combined knowledge of a set of advisors with different individual expertise, during learning. Experiment 3: We use the same domain as in Experiment 1 but with a different set of four advisors. These advisors are similar to the set of advisors in our first experiment, where Advisor 1 gives the best advice throughout the domain, and Advisor 4 is random. However, this set of advisors is not capable of teaching all the strategies (i.e, Pommerman skills) needed to win in Pommerman, and compose an insufficient set (more details in Appendix K). It is critical for agents to learn from the environment in addition to the advisors. Training and execution results in Figure 4 shows the superior performance of MA-TLQL, the only algorithm that outperforms the best advisor ( < 0.05) and all baselines ( < 0.02). Surprisingly, TLQL performs better than MA-TLAC ( < 0.02), likely due to the positive bias of -learning. This experiment reinforces the observation that MA-TLQL is capable of learning from good advisors and avoids bad advisors (also see Appendix E). Since MA-TLQL outperforms the best advisor, this experiment demonstrates that MA-TLQL can learn from both, advisors and through direct interactions with the environment, hence having a much improved sample efficiency as compared to other algorithms that learn only from the environment. This is observed during both training and execution. Experiment 4: This is similar to the Experiment 2: four advisors have similar quality, but each understands a different Pommerman skill. However, our set of advisors in this experiment are insufficient to teach all the skills in Pommerman, and the agent must also learn from the environment. The results in Figure 5 shows that MA-TLQL is capable of leveraging the combined expertise of the advisors and learning from the environment to obtain the best performance, as compared to the baselines ( < 0.04) and advisors ( < 0.05). This makes MA-TLQL more sample efficient than the prior algorithms. Experiment 5: We now switch to a four-agent version of Pommerman, which is two vs. two. This is a mixed setting as agents need to learn cooperative as well as competitive skills. Overall, this is a more complex domain with a larger state space. We consider four sufficient advisors of different quality, similar to Experiment 1. We conduct two phases -training (for 50,000 episodes) and execution (for 1000 episodes). The training and execution results in show that MA-TLQL provides the best performance compared to the baselines ( < 0.04) but does not perform better than the best available advisor. Since this is a more complex domain, MA-TLQL needs a larger training period for learning good policies. However, MA-TLQL still performs better than the average performance of the four advisors ( < 0.03). We conclude that although MA-TLQL's performance suffers in the more difficult mixed setting, it still outperforms all the other baselines and is capable of distinguishing between good and bad advisors (see also Appendix E). From both training and execution results in Figure 6, we note that MA-TLQL has a superior sample efficiency as compared to the other baselines. [8]. There are eight pursuer learning agents that learn to capture a set of 30 randomly moving targets (evaders) (details in Appendix K). We use four pre-trained DQN networks as the advisors, learning for 500, 1000, 1500, and 2000 episodes, respectively. We again have two phases -training and execution. During training, all algorithms are trained for 2000 episodes. The trained networks are then used in the execution phase for 100 episodes with no further training or influence from advisors. Figure 7(a) plots the episodic rewards obtained during training and the Figure 7(b) plots the number of targets captured in the execution phase, where MA-TLQL shows the best performance ( < 0.03). Hence, MA-TLQL can outperform all baselines in a cooperative environment as well. Experiment 7: This final experiment considers a mixed cooperative competitive Predator-Prey environment which is a part of the Multi Particle Environment (MPE) suite [24]. Our implementation uses a discrete action space and a continuous state space (more details in Appendix K). There are a total of eight predators trying to capture eight prey (prey are not removed, but respawned upon capture). In our experiment, each algorithm trains the predators while the prey is trained using a standard DQN opponent. The experiments have two phases of training and execution, which is modelled as a CTDE setting. Here each agent obtains information about the actions and rewards of all other agents during training, but only has local observation during execution. Since this environment requires decentralization during execution, we omit the fully centralized MA-TLQL and ADMIRAL-DM. We also omit DQfD since it gave poor performances previously. As in Experiment 6, we use four pretrained DQN (predator) networks as advisors (trained for 1000, 2000, 7000, and 12000 episodes). Training is conducted for 12000 episodes and execution is conducted for 100 episodes. The training results in Figure 8(a) (plot of episodic rewards) show that MA-TLAC is the most sample efficient compared to other algorithms as it is able to leverage the available advisors better than others, thus outperforming them ( < 0.04). The execution results in Figure 8(b) plots the average prey captured by each algorithm. MA-TLAC outperforms others during execution as well ( < 0.03). From all the p-values across the seven experiments, we note that most of our observations are statistically significant. Despite observing MA-TLQL outperforming the best advisor in many of the experiments, some of these comparisons are not statistically significant (i.e., ≥ 0.05). While the main experiments of the paper consider fixed advisors, our algorithms can also be implemented with learning/changing advisors (see Appendix G). In Appendix I we study performances under different numbers of advisors. Also, our algorithms can be used along with opponent modelling techniques as done by prior works [9] (more details in Appendix H). ABLATION STUDY In this section, we run an ablation study on the three components of MA-TLQL that differ from the previously introduced TLQL algorithm by Li et al. [21]. To recall these three components are: i) joint action (JA) updates, ii) ensemble method (EM), and iii) advisor evaluation (AE). For this ablation study we will consider the twoagent version of Pommerman with four sufficient advisors having different (Experiment 1) and similar quality (Experiment 2). The ablation results corresponding to Experiment 1 are given in Figure 9, where we plot the performances of TLQL and MA-TLQL in addition to TLQL with each of the three components. In Figure 9(a) and (b), the performance of TLQL with each of the three components is better than vanilla TLQL. TLQL using the ensemble method (i.e., TLQL+EM) is able to perform better than vanilla TLQL, since at the beginning of training the -values of the advisors are not accurate, and the ensemble technique chooses the advisor action that is agreed upon by most advisors in the given set (in line with our discussions in Section 4). Recall that the set of four different advisors had four advisors of decreasing quality, with the first three advisors capable of teaching some useful Pommerman skills and the last advisor being just random (see Appendix K). Using the ensemble prevents the use of the random advisor, as the first three advisors are more likely to agree upon an action, increasing the possibility of the agent choosing that action. Further, we see that TLQL highly benefits from using the joint action update (i.e., TLQL+JA) instead of an independent update seen in vanilla TLQL. The joint action update explicitly considers the strategies of other agent(s) and helps in providing stronger best responses as compared to an independent update in the multi-agent environments. Finally, TLQL using advisor evaluation in the hightable (i.e., TLQL+AE) provides the best benefit compared to the other components. As discussed in Section 3 and Section 4, the highdefinition in vanilla TLQL is limiting since the advisor evaluation through the high-is coupled with the inaccurate RL policy (and AE addresses this limitation). Further, from Figure 9, we see that MA-TLQL (integrating all the three components) shows the best performance as compared to vanilla TLQL and individual TLQL implementations with each of the three components ( < 0.05). Thus, MA-TLQL is able to seamlessly integrate the advantages of each of the individual components of TLQL, demonstrating its superiority. We also consider a similar ablation study using Experiment 2 (see Figure 10). As in Figure 9, we see that TLQL with each of the three components performs better than vanilla TLQL. Since we have four advisors of similar quality where each advisor is good at a different Pommerman skill, their agreement on an action is expected to be small. Hence, the ensemble technique (i.e., TLQL+EM) provides only a small improvement over vanilla TLQL. However, the other two components (i.e., TLQL+JA and TLQL+AE) provides a good performance benefit over TLQL. Finally, MA-TLQL, that integrates all the three components, provides the best performance ( < 0.03). CONCLUSION This paper provided a principled approach for learning from multiple independent advisors in MARL. Inspired by Li et al. [21], we present a two-level architecture for multi-agent environments. We discuss two limitations in TLQL and address these limitations in our approach. Also, we provide a fixed point guarantee and sample complexity bounds regarding the learning of MA-TLQL. Additionally, we provided an actor-critic implementation that can work in the CTDE paradigm. Further, we performed an extensive experimental analysis of MA-TLQL and MA-TLAC in cooperative, competitive, and mixed settings, where we show that these algorithms are capable of suitably leveraging a set of advisors, and perform better than baselines. As future work, we would like to consider human advisors and further explore some avenues in the real-world context. A ALGORITHM PSEUDOCODES A complete pseudocode of a tabular implementation of ourlearning based algorithm (MA-TLQL) is given in Algorithm 1. All agents initialize a low-table and a high-table in line 2. Then at each state, all agents choose to perform an action in lines 8-20. This action can come from the advisor or the RL policy as described in Section 4. Then the action is executed, and the next state and reward are observed in line 21. Finally, the values for the lowas well as the high-are updated (line 22 and line 23) according to equations presented in Section 4. The value of ′ is linearly decayed from a high-value to a value close to zero during training (line 24). To make Algorithm 1 applicable to high dimensional state and action spaces, we provide a function approximation-based implementation of MA-TLQL in Algorithm 2. Here neural networks are used as the function approximator, and the algorithm uses a separate target network and a replay buffer for training, as introduced in the well-known DQN algorithm [26]. The agent maintains a high-network and two low-networks (evaluation and target networks) and updates these networks using the temporal difference (T.D.) errors with the update equations presented in Section 4. If the full state of the stochastic game is not available, the agent can simply use its observation instead of the state, as applicable in most function approximation-based RL methods. We also extend Algorithm 2 to an actor-critic implementation described in Algorithm 3. This algorithm is called multi-agent twolevel actor-critic (MA-TLAC). This algorithm uses the policy as the actor and the -values as the critic, consistent with prior work [19]. We maintain two actors, and two critics to reflect the two-level (high and low) nature of our algorithm. The high-level actor determines an advisor and the high-level critic helps train the high-level actor, using the T.D. errors. Similarly, the low-level actor determines the appropriate action, with the low-level critic providing the T.D. errors for training. In MA-TLAC, since we use a separate actor network for advisor selection, we do not use the ensemble technique from Eq. 2. Instead, the high-level actor directly chooses one amongst the given advisors for the current state. The advantage of this algorithm is that it can be implemented using the popular CTDE paradigm [24], since the actors do not require the actions of other agents for action/advisor selection. In CTDE, global information (i.e., information from other agents) is available during training time but not available during execution. This CTDE based implementation also allows our method to be used in partially observable domains, since the actors can use the local observations for action/advisor selection while the critic can use the joint actions and states during training, as described in Lowe et al. [24]. Also, since the ensemble technique (Eq. 2) is not used in MA-TLAC, it is also applicable to continuous state space environments as well (unlike MA-TLQL, which is only applicable to environments with discrete state spaces). All the algorithm pseudocode provided in this section assume that all agents are using the same algorithmic steps for learning where it can maintain copies of updates of other agents, as done in prior work [13]. If this is not possible, the agents would directly use the observed previous actions of other agents for its updates. For each agent , get the current state 7: For each agent , get the joint actions of other agents − at state using the respective copies and previous actions of all agents 8: For each agent , let be a uniform random number between 0 and 1 9: if < ′ then 10: Let ′ be a uniform random number between 0 and 1 11: if ′ < then 12: Choose an advisor using the high-values of agent for the current state and joint action of other agents from Eq. 2 and use its action as the current action 13: else 14: Set the advisor as a random advisor from and use its action as the current action . 15: end if 16: else if > ′ and < then 17: Set the action as a random action from the action space 18: Update value of low-for the agent using Eq. 3. Obtain the next actions for other agents ′− from the respective copies and previous actions of other agents 23: If an advisor was chosen, update value of high-of the advisor for the agent using Eq. 1 24: At the end of each episode, linearly decay ′ 25: end while For each agent , get the current state 6: For each agent , get the joint actions of other agents − at state using the respective copies and previous actions of all agents 7: For each agent , let be a uniform random number between 0 and 1 8: if < ′ then 9: Let ′ be a uniform random number between 0 and 1 10: if ′ < then 11: Choose an advisor using the high-values of agent from Eq. 2 for the current state and joint action of other agents using the high-, , and use its action as the current action 12: else 13: Set the advisor as a random advisor from and use its action as the current action . 14: end if 15: else if > ′ and < then 16: Set the action as a random action from the action space 17: If an advisor was used, for each agent , store ⟨ , , , ′ , ′ , ⟩ in replay buffer D ′ , where is the advisor 23: Set the next state ′ as the current state 24: At the end of each episode, linearly decay ′ For each agent , get the current state 7: For each agent , let be a uniform random number between 0 and 1 8: if < ′ then 9: Let ′ be a uniform random number between 0 and 1 10: if ′ < then 11: Choose an advisor using the high-level actor , for the agent , for the current state , and use its action as the current action 12: else 13: Set the advisor as a random advisor from and use its action as the current action 14: end if 15: else if > ′ and < then 16: Set the action as a random action from the action space 17: If an agent used an advisor , then update the advisor's -estimate. 27: For each , set = + ( ′ , ′− , ) according to Eq. 1 28: Obtain the next actions for other agents ′− from the respective copies 29: For each , update the high-level critic by minimizing the loss L ( ) = ( − ( , − , )) 2 where is the advisor chosen by the agent 30: For each , calculate the advantage estimate using the relation ( , − , ) = − − ( | ) ( , − ,) 31: For each , update the high-level actor using the log loss J ( − ) = log − ( | ) ( , − ,) 32: Set the next state as the current state = ′ 33: At the end of each episode, linearly decay ′ 34: end while B PROOF OF THEOREM 1 Theorem 1. Given Assumptions 1, 2, 3, the low-values of an agent converges to its Nash value in the limit ( − → ∞). Proof. Our proof will be along the lines of Theorem 3 in Subramanian et al. [38]. Let us consider a lemma from prior work. Lemma 1. A random iterative process Δ +1 ( ) = (1 − ( ))Δ ( ) + ( ) ( )(7) where ∈ , = 0, 1, . . . , ∞, converges to zero with probability one (w. p. 1) if the following properties hold: 1. The set of possible states is finite. 2. 0 ≤ ( ) ≤ 1, ( ) = ∞, 2 ( ) < ∞ w. p. 1 , where the probability is over the learning rates . 3. || E{ ( )|P }|| ≤ K ||Δ || + , where K ∈ [0, 1) and converges to zero w. p. 1. 4. var{ ( )|P } ≤ (1 + ||Δ || ) 2 , where is some constant. Here P is an increasing sequence of -fields that includes the past of the process. In particular, we assume that , Δ , −1 ∈ P . The notation || · || refers to some (fixed) weighted maximum norm and the notation var refers to the variance. Proof. Refer to Theorem 1 in Jaakola et al. [14] for proof. □ Across this section, since we are only focusing on the lowvalues, with a small abuse of notation, we will use to denote the low-values. Now, we define a Nash operator , using the following equation, ( , ) = E ′ ∼ [ ( , ) + 1 * ( ′ ) · · · * ( ′ ) ( ′ )](8) where ′ is the state at time + 1, ( 1 * ( ′ ), . . . , * ( ′ )) is the Nash equilibrium solution for the stage game ( 1 ( ′ ), . . . , ( ′ )), and is the transition function. denotes the -value of a representative agent . Lemma 2. Under Assumption 3, the Nash operator as defined in Eq. 8 forms a contraction mapping with the fixed point being the Nash -value of the game. Proof. See Theorem 17 of Hu and Wellman [13]. □ Now, since the operator forms a contraction mapping, || − * || ≤ || − * ||, is satisfied for some ∈ [0, 1) and all . Here * is the Nash -value of the agent . The objective is to apply Lemma 1 to show that the low-in MATLQL converge to the Nash values. The first two conditions of Lemma 1 are satisfied from the Assumption 1 and Assumption 2. Now, comparing Eq. 7 and Eq. 3 we get that can be associated with the state joint action pairs ( , ) and Δ ( , ) can be associated with ( , ) − * ( , ). Here, * ( , ) is the Nash value of the agent . Now we get Δ +1 ( ) = (1 − ( ))Δ ( ) + ( ) ( ),(9) where ( ) = + ℎ, ( +1 ) − * ( , ) + [max ( +1 , +1 ) − ℎ, ( +1 )] = Δ + ℎ, ( +1 ) − * ( , ) + ( , ) = Δ , ( , ) + ( , )(10) The Nash value function ℎ, ( ) of an agent is defined as the expected cumulative discounted future rewards obtained by the agent , given that all agents follow the Nash policy * . Here, we set, ( , ) = , ( , ) = ( , ) = 0 if ( , ) ≠ ( , ) . Let the -field generated by all the random variables provided by ( , , 1 , . . . , , −1 , . . . , 1 , 1 , 1 , 0 ), be represented by P . Now, all the -values are P measurable which makes Δ and , P measurable and this satisfies the measurability condition of Lemma 1. Hu and Wellman [13] proved that the result, ℎ, ( +1 ) ≜ ( ′ , 1 * , . . . , * ) = 1 * ( ′ ) · · · * ( ′ ) * ( ′ ) holds (see the proof in Lemma 10 of [13]). Hence, from Lemma 2, we can show that the E[ ] forms a contraction mapping. This can be done using the fact that E( * ) = * (refer to Lemma 11 in [13]). Here, the norm is the maximum norm on the joint action. Now, we have the following for all , From the definition of and the Assumption 3, it can be shown that the value of converges to 0 in the limit of time (see Theorem 3 in Subramanian et al. [38]). || E[ ,( Thus, it follows from Lemma 1 that the process Δ converges to 0 and hence, low-value for an agent , converges to Nash value * . □ C PROOF OF THEOREM 2 In this section, we give the proof for Theorem 2. For providing this bound we use the notion of covering time . The covering time means that within steps from any start state, all state-joint action pairs are performed at least once by all agents. Similar to Dar and Mansour [5], we clarify that we do not need to assume that the state-joint action pairs are being generated by any particular strategy. Same as in Appendix B, across this section, we will use to denote the low-values. For a representative agent , we will focus on the value of = || − * || and the aim is to bound the time until ≤ . Here the norm denotes the maximum difference of the values across all states and joint actions. The proof of this theorem follows the Theorem 4 in Dar and Mansour [5]. While the work of Dar and Mansour was restricted to single-agent MDPs, our result extends the analysis of Dar and Mansour to the general-sum stochastic game setting. In line with the Eq. 4, let us consider a stochastic iterative process of the form, +1 ( ) = (1 − ( )) ( ) + ( )(( )( ) + ( )). (13) As mentioned in Section 5, let us specify = 1− 2 . Also, consider a constant max where the max denotes the maximum lowvalue possible to be obtained in the stochastic game by the agent . Hence, the relation || 0 || ≤ max holds. Further, let us consider a sequence , with 1 = max and +1 = (1 − ) for all ≥ 1. By the nature of this construction, the sequence is guaranteed to converge to 0, since at each step the value of is being continuously multiplied by a fractional value. Now we can prove the following result. Lemma 3. For every , there exists a time such that, for any ≥ we have || || ≤ . Proof. See Theorem 8 in Dar and Mansour [5]. □ The Lemma 3 guarantees that at time ≥ , for any the value of || ( )|| is in the interval [− , ]. We can state the following lemma, where we bound the number of iterations until ≤ . Lemma 4. For ≥ 1 ln( max / ) we have ≤ . Proof. Since we have that 1 = max and = (1 − ) −1 , we will need a that satisfies = max (1 − ) ≤ . By taking a logarithm on both sides, we get ln ( max )(1 − ) ≤ ln( ) ln( max ) + ln(1 − ) ≤ ln( ) =⇒ ln( max ) − ln( ) ≤ − ln(1 − ) =⇒ ln( max / ) ≤ ∞ =0 / =⇒ 1/ ln( max / ) ≤ .(14) In the last step we omit the higher powers since is a small fraction. This proves the result. □ Now, we define a sequence of times , with reference to the MA-TLQL low-level -updates with a polynomial learning rate. Let us define +1 = + . Here the term denotes the decay factor of the learning rate with = 1 ( +1) for ∈ (1/2, 1). The term specifies the number of steps needed to update each state joint action pair at least times. The time between the and +1 is denoted as the th iteration. Now we provide a definition for the number of times a state joint action pair is visited. Definition 1. Let ( , , 1 , 2 ) be the number of times that the state joint action pair ( , ) was performed in the time step interval [ 1 , 2 ]. Before providing the suitable bounds, we would like to provide some equations that relate the stochastic iterative technique given in Eq. 13 with the -update in Eq 4. First we provide a formal definition of a stochastic game, that will be useful for our further analysis. Consider a stochastic game that can be defined as follows: Towards the same, we define an operator that can be represented as (17) From this construction is bounded by max for all and has zero expectation. Further we will define two other sequences ; and ; where represents some initial time. These are given by the following equations. ( )( , ) = | | =0 ( )( ( , ) + max ∈ ( , )).(15)+1; ( , ) = (1 − ( )) ; ( , ) + ( ) ( , )(18) where ; ( , ) = 0. The value of ; bounds the contributions of ( , ), to the value of , starting from an arbitrary . Now we have +1; ( , ) = (1 − ( )) ; ( , ) + ( , )(19) where ; = . Next, we can state a lemma that will bound the -functions w.r.t the sequences ; and ; . Proof. See Lemma 4.4 in Bertsekas and Tsitsiklis [3]. □ From the Lemma 5 we see that a bound on the difference depends on the bound for ; and ; . So we can bound ; and ; separately and two bounds together will provide a bound for . We first provide a result on the nature of the sequence ; . Lemma 6. The sequence ; is a monotonically decreasing sequence. Proof. From Eq. 19 we can write (subtract from both sides), Proof. For each state-joint action pair , we are assured that ( , , , +1 ) ≥ , since the covering time is and the underlying policy has made steps (since we have the relation +1 = + ). ( +1; ( , ) − ) = (1 − ( )) ; ( , ) + (1 − ( , )) = (1 − ( ))( ; ( , ) + ).(21) where +1 = (1 − ). We aim to show that after time +1 = + , for any ≥ for any ≥ +1 we have ≤ 2 . By definition, we can rewrite as, = (1 − ) Π − =1 (1 − + ) = 2 Π − =1 (1 − + ) = 2 Π − =1 (1 − 1 | ( , , +1 , ) | )(23) the last identity follows from the definition of . Since the 's are monotonically decreasing, ≤ 2 (1 − 1 ) − .(24) For ≥ + we have, ≤ 2 1 − 1 ≤ 2 .(25) The last step is obtained from the fact that lim − →∞ (1 − (1/ )) = 1/ . Hence, ; ( , ) ≤ ( + 2 ) . □ From Lemma 7 we have provided a bound for the term ; for = +1 which will automatically hold for all ≥ +1 , since the term ; is monotonically decreasing (from Lemma 6) and deterministic (from Eq. 19 and Lemma 5). Next For bounding the sequence ; we consider the interval ∈ [ +1 , +2 ]. First we provide a lemma that bounds the coefficients in this interval and bounds the influence of ( , ) in this interval. Proof. Since Here, the * is from the fact that in a time interval of , each statejoint action pair is performed at-least / times by the definition of covering time. Hence, we get the relation that , , + ( , ) ≤ ( / ) . Now, consider the expectation of˜+ ( , ). By definition, we have that, + ( , ) has zero mean and is bounded by max for any history and state-joint action pair, hence, E[˜+ ( , )] = E[ , , + ( , ) + ( , )] = , , + ( , ) E[ + ( , )] = 0.(28) Next we can prove that it is bounded as well, |˜+ ( , )| = | , , ( , ) + ( , )| ≤ | , , ( , ) max | ≤ ( / ) max (29) □ Now, let us define , ; ( , ) = =1˜+ ( , ). The objective is to prove that this is a martingale difference sequence having bounded differences. ( , )| ≤ ( / ) max (30) Proof. We first note that the term , ( , ) is a martingale sequence, since Let = ( , , , ), then for any ∈ [ +1 , +2 ] we have that ≤ +2 − = +1 + +1 − = + + ( + ) − = + + 2 ≤ Θ( 1+ ).(35) By Lemma 9 we can apply Azuma's inequality to − +1, ; ( , ) with = ( / ) max . Therefore, we can derive that | ; ( , )| ≥˜| ∈ [ +1 , +2 ] ≤ 2 −˜2 2 = +1 2 ≤ 2 −˜2 2 2 ≤ 2 −˜2 2 2 2, max ≤ 2 −˜2 1+3 2, max(36) for some constant > 0. We can set˜= 2 −˜2 ≥ 1 − +2 = +1 [∀ , ∀ , | ; ( , )| ≥˜] ≥ 1 − +2 = +1˜| |Π | | ≥ 1 − ( +2 − +1 )˜| |Π | |(37) We would like to set a level of probability of 1 − , for each state-joint action pair. From the above equation we get, 1 − ( +2 − +1 )˜| |Π | | = 1 − / =⇒ ( +2 − +1 )˜| |Π | | = / . =⇒˜= /( +2 − +1 ) | |Π | | .(38) Thus taking˜= ( +2 − +1 ) | | | | assures 1 − for each statejoint action pair. As a result we have, = Θ 1+3 2, max ln( | |Π | | / ) 2 = Θ 1+3 2, max ln( | |Π | | max /(˜)) 2(39)Setting˜= (1 − 2/ ) gives the desired bound. □ Now that we have bounded for each iteration the time needed to achieve the desired probability 1 − . The next lemma provides a bound for error in all iterations. Lemma 11. Consider the low-update given in Eq. 4, with a polynomial learning rate. With probability 1 − , for every iteration ∈ [1, ] and time ∈ [ +1 , +2 ] we have | ; ( , )| ≤ (1 − 2 ) , i.e., ∀ ∈ [1, ], ∀ ∈ [ +1 , +2 ], ∀ , : | ; ( , )| ≤ (1 − 2 ) ≥ 1 −(40) given that 0 = Θ 1+3 2, max ln( max | |Π | | /( )) 2 2 1/(41) Proof. From Lemma 10 we have that ∀ ∈ [ +1 , +2 ] : | ; | ≥ (1 − 2 ) ≤(42) Using the union bound we have that [∀ ≤ , ∀ ∈ [ +1 , +2 ]| ; | ≥˜] ≤ =1 [∀ ∈ [ +1 , +2 ]| ; | ≥˜] ≤ (43) where˜= (1 − 2 ) . □ The next lemma solves the recurrence Proof. Let us define the following series +1 = =0 + 0(45) with an initial condition 0 = Proof. The proof of Theorem 2 follows from Lemmas 7, 11, 4, and 12. Specifically, from the relation in Lemma 12 substitute the value of 0 from the Lemma 11 (value of 0 ), and value of from Lemma 4 (lower bound for ). From the Lemma 7 and Lemma 11, we see that the condition required in Lemma 5 is satisfied to provide a lower bound. □ D PROOF OF THEOREM 3 In this section, we aim to show that the size of the th iteration is (1 + ) for some positive constant ≤ 0.712. The covering time property guarantees that in (1 + ) steps, each pair of state-joint actions are performed at least (1 + ) times. The sequence of times in this case is +1 = + (1 + ) . As in the last section, we will first bound ; and then bound ; . As in Appendix B, we will use to denote the low-values across this section as well. The proof of this theorem follows the Theorem 5 in Dar and Mansour [5]. While the work of Dar and Mansour was restricted to single-agent MDPs, our result extends the analysis of Dar and Mansour to the general-sum stochastic game setting. Lemma 13. Consider the low-update given in Eq. 4, with a linear learning rate. Assume that for ≥ we have that ; ( , ) ≤ . Then for any ≥ + (1 + ) = +1 we have that ; ( , ) ≤ ( + 2 2+ ) . Proof. For each state-joint action pair, we are assured that ( , , , +1 ) ≥ (1 + ) , since in an interval of (1 + ) steps, each state-joint action pair is visited at least (1 + ) times by the definition of covering time. Let , ( , ) = + , where = (1 − ) . We now have, = (1 − )Π − =1 (1 − + ) = 2 Π − =1 (1 − + ) = 2 Π − =1 (1 − 1 | ( , , +1 , ) | ),(48) where the last identity follows from the fact that = 1 ( , ,0, ) . Since the 's are monotonically decreasing, using = +(1+ ) , we get (using < 0.712), ≤ 2 (1 − 1 (1+ ) ) − ≤ 2 (1 − 1 (1+ ) ) 1+ ≤ 2 (1 − 1 (1+ ) ) 1+ ≤ 2 ≤ 2 2+ (49) Hence, ; ( , ) ≤ ( + 2 2+ ) . □ The following lemma enables the use of Azuma's inequality. Lemma 14. For any ≥ and 1 ≤ ≤ we have that , ; ( , ) is a martingale sequence, which satisfies, | , ; ( , ) − −1, ; ( , )| ≤ max ( , , 0, )(50) Proof. ). Proof. By Lemma 14 we can apply Azuma's inequality on the term, − +1, ; (note that − +1, ; = ; ), and with the expression = Θ max ( , ,0, ) for any ≥ +1 . Therefore, we derive that, [| ; ≥˜|] ≤ 2 −2˜2 = 2 ≤ 2 −˜2 ( , , , ) 2, max(54) for some positive constant c. Let us define, ( , ) = 1, if ( , ) ≠ 0 ( , ) = 0, otherwise .(55) Using the union bound and the property that, in an interval of length (1 + ) , each state-joint action pair is visited at least (1 + ) times, we get Proof. From Lemma 15, we know that ∀ ∈ [ +1 , +2 ] : | ; ( , )| ≥˜ ≤ [∀ ≥ ((1 + ) + 1) : | ; ( , )| ≥˜] ≤ ∞ =( (1+ ) +1) | ; ( , )| ≥˜ ≤ ∞ =( (1+ ) +1) ( , )2 −˜2 ( , ,0, ) 2, max ≤ 2 −˜2 ( (1+ ) ) 2, max ∞ =0 −˜2 2, max = 2 −˜2 ( (1+ ) ) 2, max 1− −˜2 2, max = Θ( 2, max − ′˜2 2,max∀ ∈ [ +1 , +2 ] : | ; | ≥ 2+ ≤ .(58) Using the union bound, we have that, ∀ ≤ , ∀ ∈ [ +1 , +2 ] : | ; | ≥ 2+ ≤ =1 ∀ ∈ [ +1 , +2 ]| ; | ≥ 2+ ≤ . (59) □ Theorem 3. Let us specify that with probability at least 1 − , we have for an agent , || − * || ∞ ≤ . The bound on the rate of convergence of low-, , with a linear learning rate is given by = Ω ( + + 1) 1 ln max 2, max ln( | |Π | | max ) 2 2 2 , where is a small arbitrary positive constant satisfying ≤ 0.712 Proof. The Theorem 3 follows from Lemmas 16, 13, 4, and the fact that +1 = + (1 + ) = 0 ((1 + ) + 1) . Specifically, substitute the value of from Lemma 4 (lower bound for ), value of 0 from Lemma 16 (value of 0 ), and see that Lemma 13 and Lemma 16 satisfy the condition for the lower bound in Lemma 5. □ E FREQUENCY OF LISTENING TO ADVISORS This section plots the frequency of listening to each advisor in some of our experiments. We would like to show that the MA-TLQL listens more to the good advisor and avoids the bad advisor more than other related baselines. For these experiments, we consider the TLQL [21] and ADMIRAL-DM [47] algorithms for comparison. Since the CHAT implementation uses the same method to choose advisors as ADMIRAL-DM (weighted random policy approach), we will omit CHAT for these results (performance is similar to ADMIRAL-DM). DQfD uses pretraining and does not choose advisors in an online fashion; hence we omit DQfD for these experiments as well. We consider our first experiment in Section 6 (Experiment 1), where we had a set of four sufficient advisors of different quality. The first advisor (Advisor 1) had a better quality than the others, and the agents must listen more to this advisor. On the other hand, Advisor 4 only suggested random actions and the agents are expected to avoid listening to this advisor. In Figure 11(a), we plot a curve that corresponds to the percentage of time steps an algorithm listened to Advisor 1 out of all the time steps the algorithm had an oppourtunity to listen to one of the available advisors. From the plots, we see that MA-TLQL listens more (compared to the other baselines) to this advisor (Advisor 1) from the beginning until the end of training. Since the MA-TLQL uses an ensemble technique to choose an advisor, this gives it a distinct advantage in the early stages of training. Further, since MA-TLQL performs an explicit evaluation of the advisors independent of the RL policy, it manages to listen more to the correct advisor as compared to other baselines. MA-TLAC also listens more to the good advisor as compared to the other baselines (TLQL, ADMIRAL-DM). Since TLQL couples the advisor evaluation with the RL policy, it listens a lot less to the good advisor as compared to MA-TLQL. Also, TLQL considers the RL policy as part of the high-level table, which makes it less reliant on advisors. This could be a problem when good advisors are available. In Figure 11(b), we plot the percentage of each algorithm listening to the bad advisor (Advisor 4). We see that MA-TLQL has the least dependence on this advisor as compared to all other algorithms. This reinforces our observation that MA-TLQL is most likely to choose to listen to the correct advisors in this multi-agent setting. We plot the percentage of listening to Advisor 1 and Advisor 4 in Experiment 3 from Section 6 that had an insufficient set of four advisors of decreasing quality. The results in Figure 12(a) and (b) show that MA-TLQL listens more to the good advisor and less to the bad advisor, same as our observations for Experiment 1. Similarly, we plot the percentages of listening to the good and bad advisor in the Pommerman team environment (mixed setting) used in Experiment 5 in Figure 13(a) and (b). Here we plot the results for one of the two pommerman agents playing in the same team (the other agent's results are similar). Once again, we note that MA-TLQL listens more to the correct advisor than the other algorithms and better avoids the bad advisors compared to the other algorithms. F NATURE OF ALGORITHMS CONSIDERED In Table 1, we tabulate all the algorithms considered in this paper. The differences between the algorithms stem from the nature of the algorithm (independent or multi-agent), ability to naturally support learning from conflict demonstrations (i.e., more than one advisor), and type of demonstrations that they naturally support (offline vs. online). Offline demonstrations are demonstrations that are collected (in a memory buffer) from an advisor before the "training phase" where the algorithm is trained using interactions with the environment. These demonstrations are typically used to train the algorithm in a "pre-training" phase before regular training, as done in several prior works [7,12,17]. Alternatively, online demonstrations are obtained in real-time during the training phase (and not pre-collected). Here, an agent can actively obtain action recommendations directly from an advisor for the current game context. In general, in all the experiments considered in this paper, we found that algorithms that consider actions of other agents to provide best-responses (non-independent) performed better than independent algorithms which consider all other agents to be part of the environment. One reason for this behaviour could be the fact that independent algorithms break the Markovian assumptions in reinforcement learning methods [41]. Additionally, we found that algorithms that learn from online advising perform better as compared to algorithms that learn from offline advising. If algorithms Multi-agent One Online TLQL [21] Independent More than one Online MA-TLQL (ours) Multi-agent More than one Online MA-TLAC (ours) Multi-agent More than one Online Table 1: Description of all algorithms considered in this paper learn from online advising, then these algorithms can exploit the knowledge of advisors in response to dynamically changing other agent(s), in real-time. Since other agent(s)/opponent(s) are not typically available before training in multi-agent environments, offline demonstrations are not very successful in multi-agent training, as opposed to single-agent training where they were quite successful [12]. In this paper, we considered environments where the advising come from multiple sources of independent knowledge. In many states, the different advisors provide conflicting recommendations. Hence, algorithms that can effectively resolve conflicting information from the different advisors were successful as compared to other algorithms that are not capable of naturally supporting learning under multiple conflicting advisors. As seen from Table 1, the only two algorithms that have all the three desirable properties (i.e, support multi-agent update, learn from conflicting advisors, support offline demonstrations) are MA-TLQL and MA-TLAC which gave the best performance in most of the seven experiments considered in Section 6. In our paper, we do not consider HMAT [18] and LeCTR [28] as baselines, though these can also be classified as action advising methods. We do not consider these algorithms as appropriate benchmarks due to two important reasons. First, they are both restricted to two-agent cooperative settings while we are interested in more general domains including those with more than two agents and both competitive and mixed-motive environments. Second, we are interested in independent learning from a set of external advisors, while both HMAT and LeCTR focus on peer-to-peer learning (partly due to their focus on teaching teams, as opposed to our work which is on more general multi-agent learning problems). G MA-TLQL WITH LEARNING ADVISORS In all the experiments in this paper we considered advisors that are fixed and non-changing, which we specified in Section 4. If the advisors are fixed, their -values can be determined using the highupdates in MA-TLQL. The fixed nature of advisors allows us to provide theoretical guarantees of convergence of the high-values (under the assumption of infinite updates in the limit) using similar arguments as in Theorem 1. However, experimentally we can still consider other types of advisors which are actively learning during the training stage and hence are updating the policies actively (though using such advisors do not have any theoretical guarantees of convergence). In this section, we will revisit Experiment 6 and Experiment 7 from Section 6 and study the performances of MA-TLQL and MA-TLAC under learning advisors. For statistical significance we use the unpaired 2-sided t-test and report -values, as in the earlier experiments. First we consider Experiment 6, where we used the Pursuit cooperative environment along with four pre-trained networks of DQN as the advisor. Now, we will consider the same set of four advisors and label them as "fixed advisors". Additionally, we will let the same four pre-trained networks of DQN continue training while it is being actively used for action advising. We label this set of four advisors as "changing advisors". In Figure 14(a) we plot the performances (plot of episodic rewards) of MA-TLQL and MA-TLAC along with the fixed as well as the changing advisor set. While using the changing advisor set, we see that MA-TLQL and MA-TLAC outperform their counterparts using the fixed advisors ( < 0.04). As the advisors are actively learning and changing their strategies during the training stage, they are able to provide better action recommendations to MA-TLQL and MA-TLAC at the different parts of the environment. This is reflected in their superior performances. Similarly, we revisit Experiment 7 with the Predator-Prey environment and consider two different sets of advisors. In Experiment 7, we used four pre-trained DQN networks as advisors. We will reuse the same set of four advisors and label them as the "fixed advisors". A continuously training counterpart is labelled as the "changing advisors". We plot the performances of MA-TLAC in the Predator-Prey experiment in Figure 14(b), and once again we notice that MA-TLAC using the changing advisor set outperforms MA-TLAC using the fixed advisor set ( < 0.03). Hence, experimentally we see that MA-TLQL and MA-TLAC can still be used with changing advisors which provide good empirical performances. However, the non-stationary nature of these advisors makes it impossible to provide theoretical guarantees of convergence. This is similar to using independent algorithms in multi-agent environments, which do not have any theoretical guarantees of convergence to either a local or global optimum, yet empirically, several prior works have noted that these algorithms perform well in various multi-agent environments [11,25]. H MA-TLQL WITH OPPONENT MODELLING In all the experiments in this paper, the MA-TLQL used a multiagent update where it simply tracked opponent actions (by considering the previous action) and did not perform any active opponent modelling. The core focus of this paper is learning from advisors, and we chose not to perform any particular opponent modelling technique to keep the algorithm simple. However, in this section, we will implement MA-TLQL along with an opponent modelling technique that uses a separate neural network (2 Relu layers of 50 neurons and an output layer) to predict the action of the opponent. The network uses the state and previous action of the opponent as the input and predicts the action of the opponent. This predicted action of the opponent is used by the agent to calculate best responses. Finally, the actual observed action of the opponent is used to define a cross-entropy loss function that is used to train the network. We revisit Experiment 3 and Experiment 4 in Section 6 where we used four insufficient advisors of different and similar quality. Now, we will use the same experimental procedure (with the same set of advisors) and consider the performances of MA-TLQL with and without active opponent modelling. For statistical significance we use the unpaired 2-sided t-test and report -values. We plot the performances in Figure 15. From both Figure 15(a) and (b) we see that MA-TLQL with opponent modelling performs better than the MA-TLQL algorithm that does not perform any active opponent modelling. In the Figure 15(a) we see that opponent modelling only provides a small increase in performance ( < 0.2). One reason for this is the fact that MA-TLQL without opponent modelling is already showing a high performance in this experiment by learning efficiently from the advisors. Alternatively, Figure 15(b) shows a considerable increase in performance while MA-TLQL is using opponent modelling ( < 0.04). Here, the performance of MA-TLQL without opponent modelling is not as good is the performance in Figure 15(a) and opponent modelling shows a marked improvement in performance. I MA-TLQL WITH DIFFERENT NUMBERS OF ADVISORS In this section we train MA-TLQL with different number of advisors in the Pursuit environment considered in Experiment 6 of Section 6. Recall that in Experiment 6 we considered four pre-trained networks of DQN as the advisor (pre-trained for 500, 1000, 1500, and 2000 episodes). Labelling these advisors, we will denote Advisor 1 as the advisor pre-trained for 500 episodes, Advisor 2 as the advisor pre-trained for 1000 episodes, Advisor 3 as the advisor pre-trained for 1500 episodes, and Advisor 4 as the advisor pre-trained for 2000 episodes. In this experiment, we initially train MA-TLQL with no advisor (denoted as MA-QL), and subsequently train MA-TLQL with the addition of one advisor from the set of advisors. MA-QL always chooses actions from the low-(since there are no advisors, the high-is not maintained). The objective is to study the performance of MA-TLQL under the presence of different numbers of advisors. For statistical significance we use the unpaired 2-sided t-test and report -values. The results are given in Figure 16. We see that MA-QL trained without any advisor gives the least performance. MA-TLQL trained using one advisor (Advisor 1) gives a better performance than MA-QL ( < 0.06). Next, we train MA-TLQL with two advisors (Advisor 1 and Advisor 2), which gives a better performance than MA-QL and MA-TLQL trained with one advisor ( < 0.1). Similarly, MA-TLQL with three advisors (Advisor 1, Advisor 2, and Advisor 3) gives a better performance than the case with two advisors ( < 0.09) and the best performance is given by MA-TLQL with all four advisors ( < 0.04). This result shows that MA-TLQL performance can keep improving with the addition of better advisors (than those available in the current set), which shows that MA-TLQL is capable of identifying the right advisor from the available set and exploiting the expertise of different available advisors in a multi-agent environment. From the -values, we note that the observation of the best performance of MA-TLQL with all four advisors is statistically significant. While we observe constantly improving performances with each advisor, some of these comparisons are not statistically significant as provided by the -values. J ILLUSTRATIVE EXAMPLE In this section we would like to show a toy example where the TLQL updates as provided by Li et al. [21] takes a longer time to figure out the best advisor from a set of advisors, as compared to the MA-TLQL updates, we introduced in this paper. We will just use a single agent based grid-world environment as given in Figure 17, instead of multi-agent environments. The TLQL updates will use the Bellman update for the low-updates and a subsequent synchronization step to update its high-values as proposed in Li et al. [21]. In MA-TLQL, the low-values will use the control update as given in Eq. 3 albeit in a single-agent fashion (joint actions need not be considered), since this environment only contains one agent. The high-will use the evaluation update as given in Eq. 1 in a singleagent fashion. Also, for simplicity, we do not use the ensemble selection strategy in Eq. 2 for the MA-TLQL action selection in this example. Rather we only use the high-and low-updates for advisor and action evaluation at the given state. Here we have a set of 6 states { 1, 2, 3, 4, 5, } with the agent starting at state 1 and trying to reach the goal state . The states 3, 4 and are the terminal states where the agent receives a reward of -1 in states 3 and 4, and a reward of +1 in state . The state 5 only exists for symmetry (cannot be reached in practice). The agent can take one of the two actions { , } (to denote right and down respectively) at each state. In this environment, it can be seen that the agent needs to take action in states 1 and 2 to obtain the maximum rewards. The agent has access to two advisors 1 and 2, where 1 is an optimal advisor providing the correct action (right) at every state and 2 is a sub-optimal advisor which provides action with probability 0.5 and the action with probability 0.5. All transitions in this domain are deterministic. Also, we specify that the learning rate ( ) is 0.1 and the discount factor ( ) is 0.9. Now we consider the -updates (high-level and low-level) pertaining to both MA-TLQL and TLQL. At the beginning, we initialize all the -values to 0 arbitrarily. At the initial time step ( = 0) let us assume that the agent starts at the initial state 1. Let both the advisors suggest action . Then the agent takes this action and first updates its low-using the equation, 1 ( 1, ) = 0 ( 1, ) + + max 0 ( 2, ) − 0 ( 1, ) 1 ( 1, ) = 0 + 0.1 0 + 0.9 × 0 − 0 1 ( 1, ) = 0. (60) Now, TLQL will set the value of the high-of both advisors to be 0 (synchronization). The MA-TLQL updates will also yield a value of 0 for both the advisors. In Table 2 we tabulate the -values for the non-terminal states ( 1 and 2). At the next time step = 2, the agent is at state 2. Again let both advisors suggest the right action. Now the low-is updated as, 2 ( 2, ) = 1 ( 2, ) + + max 1 ( , ) − 1 ( 2, ) 2 ( 2, ) = 0 + 0.1 1 + 0.9 × 0 − 0 2 ( 2, ) = 0.1 (61) We specify that when the next state is terminal, the temporal difference (T.D.) target is the reward itself. Both the algorithms, TLQL and MA-TLQL, will have the same high-values in this case as well. The -values for time = 2 are tabulated in Table 3. Now, we move to time = 3. Since the goal has been reached at the previous time step, the agent resets back to the initial state 1. Now, let us assume that both advisors specify the right action again at state 1. = 2 ( 1, ) + + max 2 ( 2, ) − 2 ( 2, ) 3 ( 1, ) = 0 + 0.1 0 + 0.9 × 0.1 − 0 3 ( 1, ) = 0.009(62) Again, high-values for MA-TLQL and TLQL will be the same as the low-values and are tabulated in Table 4. Next, the agent moves to state 2. We are at time = 4. Let the advisor 2 specify action at this state (Advisor 1 always specifies ). Also let us assume that the agent chooses to listen to 2 at this state ( -values of both advisors are the same, so the agent is indifferent between the two advisors) and hence it performs action (down) from 2. Now the Q-value for the low-can be updated as, 4 ( 2, ) = 4 ( 2, ) + + max 4 ( 4, ) − 4 ( 2, ) 4 ( 2, ) = −0.1(ℎ ℎ 4 ( 2, 2) = ℎ ℎ 3 ( 2, 2) + + ℎ ℎ 3 ( 4, 2) − ℎ ℎ 3 ( 2, 2) ℎ ℎ 4 ( 2, 2) = 0.1 + 0.1(−1 − 0.1) ℎ ℎ 4 ( 2, 2) = −0.01(64) All -values for time = 4 are tabulated in Table 5. Since the state 4 was a terminal state, the agent is back to state 1. We are at time = 5. At this state, let us assume that both the advisors specify action . Now the agent chooses to perform this action and updates its low-using, 5 Since both the advisors specified action , both advisors are assigned the same -values in the high-in the TLQL update. Now, the high-estimate of advisor 1 will be the same as the low-value in the MA-TLQL update as well. However, the highestimate of the advisor 2 of the MA-TLQL update will be as follows, Table 6. Comparing the high-estimate of TLQL and MA-TLQL, we see that in the TLQL updates (Table 6b), the agent is indifferent between following advisor 1 or advisor 2 in state 1 (same -values), while it would decide to choose advisor 1 at the state 2. Even after 5 update steps TLQL has not been able to determine the right advisor (as advisor 1 is better than 2) for both states. In contrast, the MA-TLQL updates found in Table 6c clearly show a highervalue for the advisor 1 than the advisor 2 for both the states. This example presents a situation where the MA-TLQL distinguishes between a good and a bad advisor faster than the vanilla TLQL update as introduced by Li et al. [21]. ℎ ℎ 5 ( 1, 2) = ℎ ℎ 4 ( 1, 2) + + ℎ ℎ 4 ( 2, 2) − ℎ ℎ 4 ( 1, 2) K EXPERIMENTAL DETAILS This section provides the complete details of all of our experimental domains, including details about the reward function and the advisors used. For the Pommerman and Pursuit environments, we assume that all the actions of other agents are either directly observable (fully observable), shared amongst agents, or provided by the game engine to perform centralized updates. For the MPE environment, the actions of other agents are observable only during training and not during execution (CTDE). Table 7 contains a summary of all of our experimental settings along with the associated configuration of advisors. K.1 Pommerman In Pommerman, the complete set of skills needed to be learned in order to win games include 1) escaping from the enemy, 2) obtaining power ups (bombs/life), 3) killing the enemy, 4) blasting walls to open routes, and 5) coordinating with a teammate (in the team version) [31]. In our experiments, we consider two Pommerman domains. The first four experiments use the two-agent version of Pommerman and the fifth experiment uses the four-agent team version of Pommerman. Each episode in our training and execution experiments corresponds to a full Pommerman game with a randomized board and a maximum of 800 steps before completion. The game ends either when all the steps are completed or when one of the two Pommerman agents dies (two-agent version). In the team version, the game ends either when all the steps are complete (800 steps maximum) or when one of the two teams completely dies. The first experiment (Experiment 1) uses four sufficient advisors of varying quality. The first advisor (Advisor 1) can teach all the strategies (i.e., various Pommerman skills as mentioned in Section 6) needed to win the Pommerman game. Advisor 2 can teach moves associated with killing the opponent if the opponent is very close to the agent and defensive strategies that help avoid the enemy. Hence, all the agents should follow the first advisor as far as possible, and the fourth advisor must be avoided entirely. In the second experiment (Experiment 2), we use a new set of four advisors. Here, the first advisor teaches only defensive skills (escaping the enemy), the second advisor can only teach aggressive skills (killing the enemy), the third advisor can only teach strategies that enable obtaining the power ups, and the fourth advisor teaches ways to seek and blast open wooden walls which opens up various paths in the game. It can be seen here that no one advisor is can teach all the strategies needed to win in Pommerman. However, together, all four advisors can teach the requisite strategies, and they need to be leveraged appropriately. In Experiment 3, we use a set of four advisors of decreasing quality as in the first experiment; however, none of the advisors are can teach strategies that seek enemies and kill them (insufficient set). Also, none of the advisors are capable of teaching the skills needed to seek wooden walls to blast open. Hence, this set of advisors is insufficient for winning the Pommerman game. The first advisor (Advisor 1) can teach strategies to escape from an enemy, kill the enemy if the enemy is right next to the agent, and obtaining the power-ups. The second advisor (Advisor 2) can only teach strategies that help in escaping the enemy or killing an enemy close to the agent. Advisor 3 can only teach strategies that pertain to escaping the enemy, and Advisor 4 only provides random suggestions. In Experiment 4, we have a set of advisors similar to the second experiment; however, the advisors are incapable of teaching sufficient skills needed to win in Pommerman. The first advisor (Advisor 1) teaches only defensive skills to escape an enemy. The second advisor (Advisor 2) helps in learning a strategy that can kill an enemy right next to the agent. The third advisor (Advisor 3) can teach strategies that lead to obtaining the power-ups, and the fourth advisor (Advisor 4) can teach skills needed to blast open wooden walls, if the agent is very close to the wall. The fifth experiment (Experiment 5) with the team domain uses the same set of advisors as Experiment 1. The Pommerman environment was released by Resnick et al. [31] under the Apache2 license. K.2 Pursuit domain The pursuit domain was first introduced by Gupta et al. [8], and we use the implementation provided by the Petting Zoo environment [43] (released under MIT license). The game has a set of 8 pursuer agents cooperating with each other to capture a set of 30 evaders in the environment. We use the same reward function and environmental parameters as in Terry et al. [43] with some minor modifications. In our setting, the agents get a reward of +1 for hitting (tagging) the evaders and a reward of +30 for catching an evader. An evader is decided to be caught if it is surrounded by a group of at least two pursuer agents. The captured evaders are removed from the environment. All agents get an urgency penalty of -0.1 at each time step of the game. Each episode has a maximum of 500 steps. The episode terminates either when all the evaders are captured or when all the steps are completed. All the pursuers receive the same reward at all time steps (global reward structure), where the rewards are distributed amongst all agents. Each pursuer observes a 7 × 7 grid around itself (the entire grid is 16 × 16), which means that the pursuer can get full information about other pursuers and evaders within this observable grid and no information outside this grid. K.3 Multi Particle Environment (Predator-Prey) The Multi Particle Environment (MPE) was originally released by Lowe et al. [24] as a set of testbeds for the purpose of testing algorithms that pertain to cooperative and mixed cooperativecompetitive settings with characteristics of communication and obstacle interactions. From this suite of testbeds we use the Simple Tag environment that pertains to a Predator-Prey setting where a set of three predators try to capture a single prey. We use the environment defined as a part of the Petting Zoo library [43] (released under MIT license). Here all agents have a discrete action space with a set of 5 actions (four cardinal directions and one action that signifies no movement). Both the predators and prey have a continuous observation space that corresponds to the velocity and position of all agents including the agent itself. The predators get a reward of +10 for hitting (colliding/tagging) prey and the prey get a punishment of -10 for being hit by any predator (the prey is not removed from the environment). The prey also receives a small additional penalty for exiting the field of play (see Terry et al. [43] for more details). All the predators receive the same reward at all time steps (global reward structure), where the rewards are distributed amongst all predators. Our environment contains eight predators and eight prey, in addition to five obstacles that block the path of the predators and prey. Each game contains 500 steps of training or execution. We model this domain as a CTDE setting, where the actions taken, and the rewards obtained by all agents are available to each agent during training, but not available during execution. L HYPERPARAMETERS AND IMPLEMENTATION DETAILS All the hyperparameters in our implementation of baselines are either the same or closely match the values recommended by the respective papers that introduced these algorithms. Our algorithms also use similar hyperparameters as the baseline algorithms, with a few exceptions (for performance and computational efficiency reasons). The DQN [26], CHAT [47], TLQL [21], ADMIRAL-DM [38], and MA-TLQL implementations use almost the same hyperparameters. These algorithms use a learning rate of 0.01, a discount factor of 0.9, a replay memory size of 2 × 10 6 , and a fixed exploration rate of 0.9. The target network is replaced every 10 learning iterations using the hard replacement strategy. The evaluation and target networks use 3 fully connected layers (2 ReLU layers of 50 neurons and an output layer). We use a batch size of 32. For the CHAT implementation, we use the neural network based confidence variant (NNHAT) from Wang and Taylor [47]. We set a confidence threshold of 0.6 and use 3 fully connected layers (2 layers using ReLU as the activation function with 50 neurons and an additional output layer). The advisors are directly used in CHAT instead of preparing decision based rules from classifier models as done in [47] due to performance reasons and also because the advisors used in our experiments are either rule-based agents or pretrained networks and not human advisors as designed in CHAT. Since these advisors are considered to be extracted rules in CHAT (not actual demonstrators/advisors), we allow CHAT dependence on advisors in the execution phase as well. We use the PPR technique in TLQL for our experiments, though this is not used in Li et al. [21]. This is done due to two reasons. First, unlike in single-agent settings, independent -learning methods do not have a policy improvement guarantee in multi-agent settings (as discussed in Section 4) [41], hence, the vanilla TLQL is not guaranteed to stop depending on advisors as motivated by Li et al. [21]. Second, without PPR, the experimental TLQL training performance is very good (since it has unlimited dependence on advisors), while the execution performances are very poor (since advisors are not available during execution). The best execution performances for TLQL is obtained while using PPR. Regarding the implementation of the PPR technique, the TLQL, ADMIRAL-DM, MA-TLAC, and MA-TLQL implementations start with a value of ′ = 1, which is linearly decayed to 0 at the end of training. There is no influence of advisors during execution, and hence, ′ = 0 during execution. To stay consistent with the description in Li et al. [21], the TLQL implementation uses a total of 3 networks (evaluation and target networks for low-in addition to high-). The high-and lownetworks use fully connected layers with the same architecture as described for the DQN. The MA-TLQL implementation uses four networks (two evaluation and target networks for the lowand high-respectively), also having the same configuration as described for the DQN. Further, the MA-TLQL uses a second replay buffer for the high-in addition to the replay buffer of the low-. Both buffers use the same memory size of 2 × 10 6 . Regarding DQfD [12], we set 1 × 10 6 as the demo buffer size and perform 50,000 mini-batch updates for pretraining. The replay buffer size is twice the size of the demo buffer. The N-step return weight is 1.0, the supervised loss weight is 1.0 and the L2 regularization weight is 10 −5 . The epsilon greedy exploration is 0.9. The discount factor is 0.99 and the learning rate is 0.002. The network architecture uses 3 fully connected layers (2 ReLU layers of 24 neurons and an output layer). The pretraining for DQfD comes from a data buffer related to a series of games where the advisors compete against each other. The MA-TLAC uses two actor networks and two critic networks. The network architecture is the same as described for MA-TLQL. The actor networks use a learning rate of 10 −6 , and the critic networks use a learning rate of 10 −3 . For all training experiments, we use a set of 30 random seeds (1 -30). We use a new set of 30 random seeds (31 -60) for the execution experiments. M WALL CLOCK TIMES All the training for the experiments were conducted on a virtual machine having 2 Nvidia A100 GPUs with a GPU memory of 40 GB. The CPUs use the AMD EPYC processors with a memory of 125 GB. The Pommerman experiments took an average of 12 hours wall clock time to complete, and the Pursuit experiments took an average of 15 hours wall clock time to complete. * = [ 1 * , . . . , * ] for all ∈ and all satisfies ( ; * , , . . . , * ] Figure 1 : 1Structure of MA-TLQL, for a representative agent having access to a set of AD advisor(s) Figure 2 : 2Two agent Pommerman with four sufficient advisors of different quality (Experiment 1) Figure 3 : 3Two-agent Pommerman with four sufficient advisors of similar quality (Experiment 2) Figure 4 : 4Two-agent Pommerman with four insufficient advisors of different quality (Experiment 3) Figure 6 6 Figure 5 : 5Two-agent Pommerman with four insufficient advisors of similar quality ( Figure 6 : 6Team (mixed) Pommerman (Experiment 5) : This experiment switches to the cooperative Pursuit domain Figure 8 : 8Mixed Predator-Prey setting (Experiment 7) Figure 9 : 9Ablation results using Experiment 1 Figure 10 : 10Ablation results using Experiment 2 all ∈ 1, . . . , , ∈ , and ∈ : ( , , − ) ← − 0 where − = [ 1 , . . . , −1 , +1 , . . . , ] 3: For all ∈ , ∈ , and for all ∈ 1, . . . , : ℎ ℎ ( , − , ) ← − 0 4: Initialize a value for hyperparameters and ′ and 5: while training is not finished do 6: joint action , observe joint reward and the next state ′ , where = [ 1 , . . . , ] and = [ 1 , . . . , ] 22: joint action , observe joint reward and the next state ′ , where = [ 1 , . . . , ] and = [ 1 , . . . , ] 21: For each agent , store ⟨ , , , ′ , ′ ⟩ in replay buffer D , where = [ 1 , . . . , ], ′ = [ ′1 , . . . , ′ ]. Obtain the next actions for other agents ′− from the respective copies and previous actions of other agents 22: minibatch of K experiences ⟨ , , , ′ , minibatch of K experiences ⟨ , , , ′ , ′ , parameters of the target network for each agent by copying over the evaluation network every T steps: − ← − and − ← − 34: high-level critic and actor networks for all ∈ {1, . . . , } 4: Initialize a value for hyperparameters and ′ and 5: while training is not finished do 6: joint action , observe joint reward and the next state ′ , where = [ 1 , . . . , ] and = [ 1 , . . . , ] 21: For each agent , obtain the joint actions of other agents − (current observed actions of other agents) at state 22: Set = + max ′ ( ′ , ′− , ′ ) according to Eq. 3 23: For each , update the low-level critic by minimizing the loss L ( ) = ( − ( , − , )) 2 24:For each , calculate the advantage estimate using the relation ( , − , ) , update the low-level actor using the log loss J ( − ) = log − ( | ) ( , − , )26: , )|P ]|| ≤ || ( , ) − * ( , )|| = ||Δ || (11) Now from Eq. 10, || E[ ( , )|P ]|| ≤ || E[ , ( , )|P ]|| + || E[ ( , )|P ] || ≤ ||Δ || + || E[ ( , )|P ]|| (12) This satisfies the third condition of Lemma 1, provided that = || E[ ( , )|P ]|| converges to 0 with probability 1 (w. p. 1.). Definition 2 . 2A stochastic game is defined as ⟨S, , A, , R, ⟩ where S is a finite set of states, is the finite set of agents, | | = , and A = 1 × . . . × is the set of joint actions, where is the finite action set of an agent , and = ( 1 , . . . , ) ∈ A is the joint action where an agent takes action ∈ . Furthermore, , ( ) : × A × ↦ → [0, 1] is the transition function that provides the probability of reaching state from state when all agents are performing the joint action ∈ A in state , ( , ) = { 1 ( , ), . . . , ( , )} is the set of reward functions, where ( , ) : × A ↦ → R is the reward function of the agent , and is the discount factor satisfying 0 ≤ < 1. Here the = { , − }, where − denotes the joint action of all agents except the agent . Rewriting the -function with we get, +1 ( , ) = (1 − ( , )) ( , ) + ( , )(( )( , ) + ( , )). (16) Let be the state that is reached by performing joint action at time in state and ( , ) be the reward observed by the agent at state ; then ( , ) = ( , ) + max ∈ ( , ) + max ∈ ( , ) . Lemma 5 . 5For every state and joint action and time , we have − ; ( , ) + ; ( , ) ≤ * ( , ) − ( , ) ≤ ; ( , ) + ; ( , ) Now, the convergence of || +1; ( , ) − || follows since lim − →∞ Π = (1 − ( , )) = 0. This shows that the sequence ( +1; ( , ) − ) monotonically decreases to 0 and hence the sequence ; monotonically decreases to . This proves our result. □ Next, we provide a bound on the value of ; . Lemma 7. Consider the low-update given in Eq. 4, with a polynomial learning rate and assume that for any ≥ we have ; ( , ) ≤ . Then for any ≥ + = +1 we have ; ( , ) ≤ ( + 2 ). Let , ( , ) = + , where = (1 − ) . Now we have the following expression, +1, ( , ) = (1 − ) ; ( , ) + = + (1 − ) we bound the term ; by (1 − 2 ) . The sum of the bounds for ; ( , ) and ; ( , ), would be ( + ) = (1 − ) = +1 , as desired. Now we state a definition for a sequence , , ( , ) and , ; . Definition 3. Let ; ( , ) = (1 − ( , )) −1; ( , ) + ( , ) ( , ) , ) = + ( , )Π = + +1 (1 − ( , )). Lemma 8 . 8Let˜, + ( , ) = , , + ( , ) + ( , ), then for any ∈ [ +1 , +2 ] the random variable˜+ ( , ) has zero mean and bounded by ( / ) max . + into two parts, the first + and the second = Π = + +1 (1 − ). Since, is bounded from above by 1, we have, , , + ( , ) ≤ + ( , ) = 1 |#( , , + ) | * ≤ ( + ) ≤ ( ) . Lemma 9 . 9For any ∈ [ +1 , +2 ] and 1 ≤ ≤ we have that , ; ( , ) is a martingale sequence that satisfies, . variable + −1 denotes all previous values of . Also, by Lemma 8 we can show that˜+ ( , ) is bounded by ( / ) Consider the low-update given in Eq. 4, with a polynomial learning rate. With probability at least 1 − we have that, for every state-joint action pair | ; ( , )| ≤ (1 − 2 ) for any ∈ [ +1 , +2 ], i.e. ∀ , ∀ ∈ [ +1 , +2 ] : | ; ( , )| ≤ (1 − 2 ) ≥ 1 − For each state-joint action pair comparing , ; ( , ) and ; ( , ) we note that ; ( , ) = − +1, ;( , ). Using the union bound we get, [∀ , ∀ , ∀ ∈ [ +1 , +2 ] : , ( , , ∀ , ; ( , ) ≥˜] . Let us specify that with probability at least 1 − , for an agent , || − * || ∞ ≤ . The bound on the rate of convergence of low-, , with a polynomial learning rate of factor is given by (with * as the Nash -value of ) ,;. ( , ) is a martingale difference sequence since, Consider the low-update given in Eq. 4, with a linear learning rate. With probability at least 1 − we have that for every state-joint action pair | ; ( , )| ≤ 2+ , for any > +1 and any positive constant ≤ 0.712, i.e., ∀ ∈ [ +1 , +2 ] : ; ( , ) ≤ 2+ ≥ 1 − (53) given that ≥ Θ( 2, max ln( max | |Π | | )/( ) that for every ≥ +1 (and as a result for any ∈ [ +1 , +2 ]), with probability at least 1 − the statement holds at every state-joint action pair.□ We have bounded for each iteration the time needed to achieve the desired probability of 1 − . The following lemma provides a bound for the error in all the iterations. Lemma 16. Consider the low-update given in Eq. 4, with a linear learning rate. With probability 1 − , for every iteration ∈ [1, ], time ∈ [ +1 , +2 ], and any positive constant ≤ 0.712, we have | ; | ≤ 2+ , i.e., ∀ ∈ [1, ], ∀ ∈ [ +1 , +2 ] : | ; | ≤ 2+ ≥ 1 − (57) given that 0 = Θ 2, max ln( max | | | | /( )) Figure 11 : 11Frequency of listening to advisors in the twoagent Pommerman experiment with four sufficient advisors of different quality (Experiment 1) Figure 12 : 12Frequency of listening to advisors in the twoagent Pommerman experiment with four insufficient advisors of different quality (Experiment 3) Figure 13 : 13Frequency of listening to advisors in the team Pommerman experiment with four sufficient advisors of different quality (Experiment 5) Figure 14 : 14Performances of MA-TLQL and MA-TLAC under changing (learning) advisors and fixed advisors. Here (a) corresponds to the setting in Experiment 6 and (b) corresponds to the setting in Experiment 7, described in Section 6. Figure 15 : 15vs. One Pommerman: Insufficient different quality advisors (b) One vs. One Pommerman: Insufficient similar quality advisors Performances of MA-TLQL with and without opponent modelling using four insufficient advisors. Here (a) corresponds to the setting in Experiment 3 and (b) corresponds to the setting in Experiment 4, described in Section 6. Figure 16 : 16MA-TLQL with different numbers of advisors in the Pursuit environment (setting used in Experiment 6, described in Section 6) Figure 17 : 17Toy environment to compare updates of TLQL and MA-TLQL MA-TLQL high-at time = 1 The low-values are updated as, MA-TLQL high-at time = 2 5 ( 1, ) = 0.009 + 0.1 0 + 0.9 × 0.1 − 0.009 5 ( 1, ) = 0.0171. ℎ ℎ 5 ( 1, 2 ) 2= 0.009 + 0.1 0 + 0.9 × −0.01 − 0.009 ℎ ℎ 5 ( 1, 2) = 0.009 − 0.0018 = 0.0072 (66) At this stage (time = 5) all the -values are tabulated in Algorithm 2 AlgorithmMA-TLQL Neural Network Method1: Let denote a set of advisors available to the agent 2: Initialize , − for all ∈ 1, . . . , (to denote low-). Ini- tialize , − for all ∈ 1, . . . , (to denote high-) 3: Initialize a value for hyperparameters and ′ and 4: while training is not finished do 5: Table 2 : 2TLQL and MA-TLQL updates at time = 1. The columns refer to the actions and the rows refer to the states. Table 3 : 3TLQL and MA-TLQL updates at time = 2 (b) TLQL high-at time = 3 (ST./AC.) Adv. 1 Adv. 2 (c) MA-TLQL high-at time = 3(ST./AC.) Right Down S1 0.009 0 S2 0.1 0 (a) Low-values at time = 3 (ST./AC.) Adv. 1 Adv. 2 S1 0.009 0.009 S2 0.1 0.1 S1 0.009 0.009 S2 0.1 0.1 Table 4 : 4TLQL and MA-TLQL updates at time = 33 ( 1, ) 63) The high-values for the MA-TLQL update is given by, (ST./AC.) Right DownS1 0.009 0 S2 0.1 -0.1 (a) Low-values at time = 4 (ST./AC.) Adv. 1 Adv. 2 S1 0.009 0.009 S2 0.1 -0.1 (b) TLQL high-at time = 4 (ST./AC.) Adv. 1 Adv. 2 S1 0.009 0.009 S2 0.1 -0.01 (c) MA-TLQL high-at time = 4 Table 5 : 5TLQL and MA-TLQL updates at time = 4 (ST./AC.) Right DownS1 0.0171 0 S2 0.1 -0.1 (a) Low-values at time = 5 (ST./AC.) Adv. 1 Adv. 2 S1 0.0171 0.0171 S2 0.1 -0.1 (b) TLQL high-at time = 5 (ST./AC.) Adv. 1 Adv. 2 S1 0.0171 0.0072 S2 0.1 -0.01 (c) MA-TLQL high-at time = 5 Table 6 : 6TLQL and MA-TLQL updates at time = 5 The third advisor (Advisor 3) can only teach defensive strategies that Exp. DomainType Advisors # of train- ing agents 1 Two- agent Pommer- man Competitive 4 sufficient advi- sors with differ- ent quality 2 2 Two- agent Pommer- man Competitive 4 sufficient advi- sors with similar quality 2 3 Two- agent Pommer- man Competitive 4 insufficient ad- visors with dif- ferent quality 2 4 Two- agent Pommer- man Competitive 4 insufficient ad- visors with simi- lar quality 2 5 Four- agent Pommer- man Mixed 4 sufficient advi- sors with differ- ent quality 4 6 Pursuit SISL Cooperative 4 insufficient ad- visors with dif- ferent quality 8 7 Predator- Prey MPE Mixed (CTDE) 4 insufficient ad- visors with dif- ferent quality 8 Table 7 : 7Description of experimental settings help in escaping the enemy, and cannot teach aggressive strategies needed to kill the enemy. 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{'abstract': 'Multi-agent reinforcement learning typically suffers from the problem of sample inefficiency, where learning suitable policies involves the use of many data samples. Learning from external demonstrators is a possible solution that mitigates this problem. However, most prior approaches in this area assume the presence of a single demonstrator. Leveraging multiple knowledge sources (i.e., advisors) with expertise in distinct aspects of the environment could substantially speed up learning in complex environments. This paper considers the problem of simultaneously learning from multiple independent advisors in multi-agent reinforcement learning. The approach leverages a two-level -learning architecture, and extends this framework from single-agent to multi-agent settings. We provide principled algorithms that incorporate a set of advisors by both evaluating the advisors at each state and subsequently using the advisors to guide action selection. We also provide theoretical convergence and sample complexity guarantees. Experimentally, we validate our approach in three different test-beds and show that our algorithms give better performances than baselines, can effectively integrate the combined expertise of different advisors, and learn to ignore bad advice.', 'arxivid': '2301.11153', 'author': ['Sriram Ganapathi Subramanian sriram.subramanian@vectorinstitute.ai \nVector Institute\nTorontoCanada\n', 'Matthew E Taylor matthew.e.taylor@ualberta.ca \nUniversity of Waterloo\nWaterlooCanada\n', 'Kate Larson kate.larson@uwaterloo.ca \nAlberta Machine Intelligence Institute\nUniversity of Alberta\nEdmonton, EdmontonCanada, Canada\n', 'Mark Crowley mcrowley@uwaterloo.ca \nUniversity of Waterloo\nWaterlooCanada\n', 'Sriram Ganapathi Subramanian \nUniversity of Waterloo\nWaterlooCanada\n', 'Matthew E Taylor \nUniversity of Waterloo\nWaterlooCanada\n', 'Kate Larson \nUniversity of Waterloo\nWaterlooCanada\n', 'Mark Crowley \nUniversity of Waterloo\nWaterlooCanada\n'], 'authoraffiliation': ['Vector Institute\nTorontoCanada', 'University of Waterloo\nWaterlooCanada', 'Alberta Machine Intelligence Institute\nUniversity of Alberta\nEdmonton, EdmontonCanada, Canada', 'University of Waterloo\nWaterlooCanada', 'University of Waterloo\nWaterlooCanada', 'University of Waterloo\nWaterlooCanada', 'University of Waterloo\nWaterlooCanada', 'University of Waterloo\nWaterlooCanada'], 'corpusid': 256274681, 'doi': '10.48550/arxiv.2301.11153', 'github_urls': ['https://github.com/Sriram94/'], 'n_tokens_mistral': 36662, 'n_tokens_neox': 32366, 'n_words': 20122, 'pdfsha': '96369f983029317a1727e2fcf1573f7c497b1ea2', 'pdfurls': ['https://export.arxiv.org/pdf/2301.11153v2.pdf'], 'title': ['Learning from Multiple Independent Advisors in Multi-agent Reinforcement Learning', 'Learning from Multiple Independent Advisors in Multi-agent Reinforcement Learning'], 'venue': ['IFAAMAS']}

arxiv

Hidden Gems: 4D Radar Scene Flow Learning Using Cross-Modal Supervision Fangqiang Ding fding@ed.ac.uk University of Edinburgh Andras Palffy a.palffy@tudelft.nl Delft University of Technology Dariu M Gavrila d.m.gavrila@tudelft.nl Delft University of Technology Chris Xiaoxuan Lu University of Edinburgh Hidden Gems: 4D Radar Scene Flow Learning Using Cross-Modal Supervision This work proposes a novel approach to 4D radar-based scene flow estimation via cross-modal learning. Our approach is motivated by the co-located sensing redundancy in modern autonomous vehicles. Such redundancy implicitly provides various forms of supervision cues to the radar scene flow estimation. Specifically, we introduce a multitask model architecture for the identified cross-modal learning problem and propose loss functions to opportunistically engage scene flow estimation using multiple cross-modal constraints for effective model training. Extensive experiments show the state-of-the-art performance of our method and demonstrate the effectiveness of cross-modal supervised learning to infer more accurate 4D radar scene flow. We also show its usefulness to two subtasks -motion segmentation and ego-motion estimation. Our source code will be available on https://github.com/Toytiny/CMFlow. Introduction Scene flow estimation is to obtain a 3D motion vector field of the static and dynamic environment relative to an ego-agent. In the context of self-driving, scene flow is a key enabler to navigational safety in dynamic environments by providing holistic motion cues to multiple subtasks, such as ego-motion estimation, motion segmentation, point cloud accumulation, multi-object tracking, etc. Driven by the recent successes of deep neural networks in point cloud processing [22,35,36,40,51,54], predominant approaches to scene flow estimation from point clouds adopt either fully- [12,27,34,49,52] or weakly- [10,11] supervised learning, or only rely on self-supervised signals [2,9,21,24,31]. For supervised ones, the acquisition of scene flow annotations is costly and requires tedious and intensive human labour. In contrast, self-supervised learning methods require no annotations and can exploit the inherent spatio-temporal relationship and constraints in the input data to bootstrap scene flow learning. Neverthe- Figure 1. Cross-modal supervision cues are retrieved from colocated odometer, LiDAR and camera sensors to benefit 4D radar scene flow learning. The source point cloud (red) is warped with our estimated scene flow and gets closer to the target one (blue). less, due to the implicit supervision signals, self-supervised learning performance is often secondary to the supervised ones [10,23,49], and they fail to provide sufficiently reliable results for safety-critical autonomous driving scenarios. These challenges become more prominent when it comes to 4D radar scene flow learning. 4D automotive radars receive increasing attention recently due to their robustness against adverse weather and poor lighting conditions (vs. camera), availability of object velocity measurements (vs. camera and LiDAR) and relatively low cost (vs. Li-DAR) [5,30,32,41]. However, 4D radar point clouds are significantly sparser than LiDARs' and suffer from nonnegligible multi-path noise. Such low-fidelity data significantly complicates the point-level scene flow annotation for supervised learning and makes it difficult to rely exclusively on self-supervised training-based methods [9] for performance and safety reasons. To find a more effective framework for 4D radar scene flow learning, this work aims to exploit cross-modal supervision signals in autonomous vehicles. Our motivation is based on the fact that autonomous vehicles today are equipped with multiple heterogeneous sensors, e.g., Li-DARs, cameras and GPS/INS, which can provide complementary sensing and redundant perception results for each other, jointly safeguarding the vehicle operation in complex urban traffic. This co-located perception redundancy can be leveraged to provision multiple supervision cues that bootstrap radar scene flow learning. For example, the static points identified by radar scene flow can be used to estimate the vehicle odometry. The consistency between this estimated odometry and the observed odometry from the colocated GPS/INS on the vehicle forms a natural constraint and can be used to supervise scene flow estimation. Similar consistency constraints can be also found for the optical flow estimated from co-located cameras and the projected radar scene flow onto the image plane. While the aforementioned examples are intuitive, retrieving accurate supervision signals from co-located sensors is non-trivial. With the same example of optical and scene flow consistency, minimizing flow errors on the image plane suffers from the depth-unaware perspective projection, potentially incurring weaker constraints to the scene flow of far points. This motivates the following research question: How to retrieve the cross-modal supervision signals from co-located sensors on a vehicle and apply them collectively to bootstrap radar scene flow learning. Towards answering it, in this work we consider opportunistically exploiting useful supervision signals from three commonly co-existent sensors with the 4D radar on a vehicle: odometer (GPS/INS), LiDAR, and RGB camera (See Fig. 1). This cross-modal supervision is expected to help us realize radar scene flow learning without human annotation. In our setting, the multi-modal data are only available in the training phase, and only 4D radar is used during the inference stage. Our contributions can be summarized as follows: • Our work is the first 4D radar scene flow learning using cross-modal supervision from co-located heterogeneous sensors on an autonomous vehicle. • We introduce a multi-task model architecture for the identified cross-modal learning problem and propose loss functions to effectively engage scene flow estimation using multiple cross-modal constraints for model training. • We demonstrate the state-of-the-art performance of the proposed CMFlow method on a public dataset and show its effectiveness in downstream tasks as well. Related Work Scene flow. Scene flow was first defined in [44] as a 3D uplift of optical flow that describes the 3D displacement of points in the scene. Traditional approaches resolve pixel-wise scene flow from either RGB or RGB-D images based on prior knowledge assumptions [6, 13-15, 38, 39, 45, 48] or by training deep networks in a supervised [17,18,28,29,37] or unsupervised [16,26,53,55] way. In contrast, some other methods directly infer pointwise scene flow from 3D sparse point clouds. Among them, some methods rely on online optimization to solve scene flow [8,25,33]. Recently, inspired by the success of point cloud feature learning [35,36,40,51], deep learning-based methods [3,12,27,52] have been dominant for point cloudbased scene flow estimation. Deep scene flow on point clouds. Current point cloudbased scene flow estimation methods [2, 3, 10-12, 21, 24, 27, 31, 34, 46, 47, 49, 52] established state-of-the-art performance by leveraging large amount of data for training. Many of them [3,34,46,47,49] learn scene flow estimation in a fully-supervised manner with ground truth flow. These methods, albeit showing promising results, demand scene flow annotations, of which the acquisition is labour-intensive and costly. Another option is to leverage a simulated dataset [29] for training, yet this may result in poor generalization when applied to the real data. To avoid both human labour and the pitfalls of synthetic data, some methods design self-supervised learning frameworks [2,21,24,31,52] that exploit supervision signals from the input data. Compared with their supervised counterparts, these methods require no annotated labels and thus can train their models on unannotated datasets. However, the performance of these self-supervised learning methods is limited [21,52] by the fact that no real labels are used to supervise their models. Some recent methods [10,11] try to seek a trade-off between annotation efforts and performance by combining the ego-motion and manually annotated background segmentation labels. Although pseudo ground truth ego-motion can be easily accessed from onboard odometry sensors (GPS/INS), annotating background mask labels is still expensive and need human experts to identify foreground entities from complex scenarios. Radar scene flow. Previous works mostly estimate scene flow on dense point clouds captured by LiDAR or rendered from stereo images. Thus, they cannot be directly extended to the sparse and noisy radar point clouds. To fill the gap, a recent work [9] proposes a self-supervised pipeline for radar scene flow estimation. However, just like in other self-supervised methods, the lack of real supervision signals limits its scene flow estimation performance and thus hinders its application to more downstream tasks. Unlike the aforementioned works, we propose to retrieve supervision signals from co-located sensors in an automatic manner without resorting to any human intervention during training. Note that we use cross-modal data to provide all supervision at once only in the training stage and do not require other modalities during inference. Method Problem Definition Scene flow estimation aims to solve a motion field that describes the non-rigid transformations induced both by the motion of the ego-vehicle and the dynamic objects in the scene. For point cloud-based scene flow, the inputs are two consecutive point clouds, the source one P s = {p s i = {c s i , x s i }} N i=1 and the target one P t = {p t i = {c t i , x t i }} M i=1 , where c s i , c t i ∈ R 3 are the 3D coordinates of each point, and x s i , x t i ∈ R C are their associated raw point features. The outputs are point-wise 3D motion vectors F = {f i ∈ R 3 } N i=1 that align each point in P s to its corresponding position c ′ i = c s i + f i in the target frame. Note that P s and P t do not necessarily have the same number of points and there is no strict point-to-point correspondence between them under real conditions. Therefore, the corresponding location c ′ i is not required to coincide with any points in the target point cloud P t . In our case of 4D radar, the raw point features x s i , x t i ∈ R 2 include the relative radial velocity (RRV) and the radar cross-section (RCS) measurements [1]. RRV measurements, resolved by analyzing Doppler shift in radar frequency data, contain partial motion information of points. RCS can be seen as the proxy reflectivity of each point, which is mainly affected by the reflection property of the target and the incident angle of the beams. Overview The overall pipeline of our proposed method is depicted in Fig. 2. To bootstrap cross-modal supervised learning, we apply a multi-task model that predicts radar scene flow in two stages. The first stage starts by extracting base features with two input point clouds, which are then forwarded to two independent heads to infer per-point moving probabilities and initial point-wise scene flow vectors. On top of them, the second stage first infers a rigid transformation respective to radar ego-motion and outputs a binary motion segmentation mask. Then, the final scene flow is obtained by refining the flow vectors of identified static points with the derived rigid transformation. In summary, our multitask model's outputs include a rigid transformation (i.e. the ego-motion), a motion segmentation mask (i.e. which targets are static or dynamic), and the final refined scene flow. To supervise these predictions, we extract corresponding supervision signals from co-located modalities and train the entire model end-to-end by minimizing a loss L composed of three terms: L = L • ego + L • seg + L • f low .(1) Here, L • ego is the ego-motion error that supervises the rigid transformation estimation using the odometry information. Our motion segmentation error L • seg constrains the predicted moving probabilities with a point-wise pseudo motion segmentation label, which is obtained by fusing information from the odometer and LiDAR. We further supervise the final scene flow with signals given by LiDAR and RGB camera in L • f low . In the following section (Sec. 3.3), we first briefly introduce each module of our model. We then explain how we extract signals from co-located modal-ities to supervise our outputs in the training phase (Sec. 3.4). More details on our method can be found in the supplementary materials. Model Architecture Similar to [2,[9][10][11]43], our model is designed in a twostage fashion, with a rough initial flow derived in the first stage and refined in the second stage to obtain the final estimate of scene flow, as shown in Fig. 2. A description of all components of our model can be found below. Backbone. Following [9,10,27,52], our backbone network directly operates on unordered point clouds P s and P t to encode point-wise latent features. In particular, we first apply the set conv layers [27] to robustly extract multi-scale local features for individual point clouds and propagate features from P t to P s using the cost volume layer [52] for feature correlation. We then concatenate multi-stage features of P s (including the correlated features) and forward them into another multi-scale set conv layers to generate the base backbone features E ∈ R N ×Ce . Specifically, the maxpooling operation is used along the channel axis to restore the global scene features after each set conv layer, which are then concatenated to per-point local features. Initial flow and motion segmentation head. Given the base backbone features E that represents the inter-frame motion and intra-frame local-global information for each point p s i ∈ P s , we apply two task-specific heads for decoding. The first head is to produce an initial scene floŵ F init = {f init i ∈ R 3 } N i=1 , while another is for generating a probability mapŜ = {ŝ i ∈ [0, 1]} N i=1 that denotes the moving probabilities of points in P s (w.r.t. the world frame). We implement both heads with a multi-layer perceptron and map the output to probabilities with Sigmoid(·) in the motion segmentation head. Ego-motion head. With the natural correspondences {c s i , c s i +f init i } N i=1 formed by the initial scene flowF init between two frames and the probability mapŜ 1 , we restore a rigid transformationT ∈ R 4×4 that describes the radar ego-motion using the differentiable weighted Kabsch algorithm [19]. To mitigate the impact of the flow vectors from moving points, we compute 1 −Ŝ as our weights and normalize the sum of them to 1. Besides, we generate a binary motion segmentation mask by thresholdingŜ with a fixed value η b to indicate the moving status of each point. We use this binary mask as our motion segmentation output and to identify stationary points for flow refinement below. Refinement layer. As the flow vectors of static points are only caused by the radar's ego-motion, we can regularize their initial predictions with the more reliable rigid trans-formationT. The refinement operation is simply replacing the initial flow vectorf init i of identified stationary points with the flow vector induced by radar's ego-motion, which is derived as [f i 1] ⊤ = (T − I 4 )[c s i 1] ⊤ . The final scene flow is attained asF = {f i ∈ R 3 } N i=1 . Temporal update module. Apart from the aforementioned basic modules, we also propose an optional module that can be embedded into the backbone to propagate previous latent information to the current frame. More specifically, we apply a GRU network [7] that treats the global features in the backbone as the hidden state and update it temporally across frames. During training, we first split long training sequences into mini-clips with a fixed length T and train with mini-batches of them. The hidden state is initialized as a zero vector for the first frame of each clip. When evaluating on test sequences, we emulate the training conditions by re-initializing the hidden state after T frames. In general, our model can deliver solutions to three different tasks, i.e. scene flow estimation, ego-motion estimation and motion segmentation, with 4D radar point clouds. Specifically, these outputs are compactly correlated to each other. For example, accurate motion segmentation results will benefit ego-motion estimation, which further largely determines the quality of the final scene flow. Cross-Modal Supervision Retrieving A key step in our proposed architecture is to retrieve the cross-modal supervision signals from three co-located sensors on autonomous vehicles, i.e. odometer, LiDAR and camera, to support model training without human annotation. This essentially leads to a multi-task learning problem. Of course, the supervision signals from individual modality-specific tasks (e.g., optical flow) are inevitably noisier than human annotation. However, we argue that if these noisy supervision signals are well combined, then the overall noise in supervision can be suppressed and give rise to effective training anyway. In the following, we detail how we extract cross-modal supervision signals and subtly combine them to formulate a multi-task learning problem. Ego-motion loss. To supervise the rigid transformationT derived in our ego-motion head, it is intuitive to leverage the odometry information from the odometer (GPS/INS). As a key sensor for mobile autonomy, the odometer can output high-frequency ego-vehicle poses, which can be used to compute the pseudo ground truth radar ego-motion transformation O ∈ R 4×4 between two frames. The ground truth rigid transformation T = O −1 can then be derived to summarize the rigid flow component F r = {f r i } N i=1 induced by the ground truth radar ego-motion, where [f r i 1] ⊤ = (T − I 4 )[c s i 1] ⊤ . Our ego-motion loss is formulated as: L • ego = 1 N N i=1 (T − T)[c s i 1] ⊤ 2 ,(2) where we supervise the estimatedT by encouraging its associated rigid flow components to be close to the ground truth ones. By supervisingT, we can implicitly constrain the initial scene flowf init i for static points. More importantly, the static flow vectors in the final flowF can also be supervised as the refinement is based onT. Motion segmentation loss. Unlike ego-motion estimation, supervising motion segmentation with cross-modal data is not straightforward as no sensors provide such information. To utilize the odometry information, we generate a pseudo motion segmentation label with the rigid flow component F r given by the odometer and the radar RRV measurements {v i } N i=1 . More specifically, we first approximate the RRV component ascribed to the radar ego-motion by v r i ≈ u ⊤ i f r i /∆t. Here, u i is the unit vector with its direction pointing from the sensor to the point c s i and ∆t is time duration between two frames. Then, we compensate the radar ego-motion and get the object absolute radial ve- locity ∆v i = abs(v i − v r i ). With per-point ∆v i , the pseudo motion segmentation label S v = {s v i ∈ {0, 1}} N i=1 can be derived by thresholding, where 1 denotes moving points. More details on our thresholding strategy can be found in the supplementary materials. Note that one shortcoming is that tangentially moving targets are not distinguished. Besides the odometry information, we also leverage the LiDAR data to generate a pseudo foreground (movable objects) segmentation label S f g . To this end, we first feed LiDAR point clouds into an off-the-shelf 3D multi-object tracking (MOT) pretrained model [50]. Then we divide radar points from P s into the foreground and background using the bounding boxes (e.g. pedestrian, car, cyclist) produced by 3D MOT to create S f g . Besides, we can also assign pseudo scene flow vector label to foreground points by: a) retrieving the ID for each bounding box from the first frame, b) computing the inter-frame transformation for them, c) deriving the translation vector for each inbox point based on the assumption that all points belonging to the same object share a universal rigid transformation. The resulting pseudo scene flow label is denoted as F f g = {f f g i } N i=1 , where we leave the label empty for all identified background points. Given the two segmentation labels S v and S f g , directly fusing them is impeded by their domain discrepancy, i.e., not all foreground points are moving. Therefore, we propose to distill the moving points from S f g by discarding foreground points that either keep still or move very slowly. We implement this by removing the rigid flow component F r from F f g and get the non-rigid flow component of all foreground points. Then we obtain a new pseudo motion segmentation label S l = {s l i ∈ {0, 1}} N i=1 by classifying points with apparent non-rigid flow as dynamic. A more reliable pseudo motion segmentation S = {s i } N i=1 can be consequently obtained by fusing S l and S v . For points classified as moving in S l , we have high confidence about their status and thus label these points as moving in S. For the rest of the points, we label their s i according to S v . Finally, as seen in Fig. 2, our motion segmentation loss can be formulated by encouraging the estimated moving probabilitieŝ S to be close to the pseudo motion segmentation label S: L • seg = 1 2 ( N i=1 (1 − si)log(1 −ŝi) N i=1 (1 − si) + N i=1 silog(ŝi) N i=1 si ) . (3) Here, we use the average loss of static and moving losses to address the class imbalance issue. Scene flow loss. In the final scene flow outputF, the flow vectors of static points have been implicitly supervised by the ego-motion loss (c.f. Eq. (2)). In order to further constrain the flow vectors of moving points, we formulate two new loss functions. The first one is L • mot , which is based on the pseudo scene flow label F f g derived from 3D MOT results. In this loss, we only constrain the flow vectors of those moving points identified in S l through: L • mot = 1 N i=1 s l i N i=1 s l i (fi − f f g i ) 2 .(4) In addition to utilizing the odometer and LiDAR for crossmodal supervision, we also propose to extract supervision signals from the RGB camera. Specifically, we formulate a loss L • opt using pseudo optical flow labels W = {w i ∈ R 2 } N i=1 . To get this pseudo label, we first feed synchronized RGB images into a pretrained optical flow estimation model [42] to produce an optical flow image. Then, we project the coordinate c s i of each point onto the image plane and draw the optical flow vector at the corresponding pixel m i of each point. Given W, it is intuitive to construct the supervision for our scene flow prediction by projecting it on the image plane. However, minimizing the flow divergence in pixel scale has less impact for far radar points due to the depth-unawareness during perspective projection. Instead, we directly take the point-to-ray distance as the training objective, which is more insensitive to points at different ranges. The loss function can be written as: L • opt = 1 N i=1 si N i=1 siD(c s i +fi, mi + wi, θ)(5) where D denotes the operation that computes the distance between the warped point c s i +f i and the ray traced from the warped pixel m i + w i . θ denotes sensor calibration parameters. Note that we only consider the scene flow of moving points identified in S here. Apart from the above two loss functions used to constrain the final scene flow, we also employ the self-supervised loss L • self in [9] to complement our cross-modal supervision. See the supplementary for more details. The overall scene flow loss is formulated as: L • f low = L • mot + λoptL • opt + L • self ,(6) where we set the weight λ opt = 0.1 in all our experiments. Experiments Experimental Setup Dataset. For our experiments, we use the View-of-Delft (VoD) dataset [32], which provides synchronized and calibrated data captured by co-located sensors, including a 64-beam LiDAR, an RGB camera, RTK-GPS/IMU based odometer and a 4D radar sensor. As is often the case with datasets focused on object recognition, the test set annotations of the official VoD dataset are withheld for benchmarking. However, our task requires custom scene flow metrics for which we need annotations to generate ground truth labels. Therefore, we divide new splits ourselves from the official sets (i.e., Test, Val, Train) to support our evaluation. Given sequences of data frames, we form pairs of consecutive radar point clouds as our scene flow samples for training and inference. We only generate ground truth scene flow and motion segmentation labels for samples from our Val, Test sets and leave the Train set unlabelled as our method has no need for ground truth scene flow annotations for training. Please see the supplementary details on our dataset separation and labelling process. Metrics. We use three standard metrics [27,31,52] to evaluate different methods on scene flow estimation, including a) EPE [m]: average end-point-error (L 2 distance) between ground truth scene flow vectors and predictions, b) AccS/AccR: the ratio of points that meet a strict/relaxed condition, i.e. EPE < 0.05/0.1 m or the relative error < 5%/10%. We also use the c) RNE [m] metric [9] that computes resolution-normalized EPE by dividing EPE by the ratio of 4D radar and LiDAR resolution. This can induce sensor-specific consideration and maintain a fair comparison between different sensors. Besides, we compute the RNE [m] for moving points and static points respectively, and denote them as MRNE [m] and SRNE [m]. Baselines. For overall comparison, we apply seven stateof-the-art methods as our baselines, including five selfsupervised learning-based methods [2,9,21,31,52] and two non-learning-based methods [4,33], as seen in Tab. 1. To keep a fair comparison, we use their default hyperparameter settings. We retrain scene flow models offline on our VoD Train set for learning-based methods and directly optimize scene flow results online for non-learning-based ones. Implementation details. We use the Adam optimizer [20] to train all models in our experiments. The learning rate is initially set as 0.001 and exponentially decays by 0.9 per epoch. The Val set is used for both model selection during training and for determining the values of our hyperparameters. We set our classification threshold η b = 0.5 (c.f. Sec. 3.3) for all experiments. When activating our temporal update module, the length of mini-clips T is set as 5. Please refer to the supplementary materials for our hyperparameter searching process. Table 2. Ablation experiments on combing supervision signals from diverse modalities. Abbreviations: odometer (O), LiDAR (L), camera (C). Note that we disable the temporal update scheme in this study to highlight the impact of modalities for training. Scene Flow Evaluation Overall results. We quantitatively compare the performance of our methods to baselines on the Test set, as shown in Tab. 1. Compared with both non-learning-based and self-supervised counterparts, CMFlow shows remarkable improvement on all metrics by leveraging cross-modal supervision signals for training. Our method outperforms the second-best approach [9] by 37.6% on EPE, implying drastically more reliability. Our performance is further improved when adding the temporal update scheme (c.f. Sec. 3.3) in the backbone, i.e. CMFlow (T). We also observe that the performance slightly degrades on the AccS metric when activating the temporal update. This suggests us that introducing temporal information will somewhat disturb the original features and thus reduce the fraction of already fine-grained (e.g. EPE<0.05 m) predictions. Impact of modalities. A key mission of this work is to investigate how supervision signals from different modalities help our radar scene flow estimation. To analyze their impact, we conduct ablation studies and show the breakdown results in Tab to a further improvement on all metrics, as seen in row (c)-(e). These results validate that our method can effectively exploit useful supervision signals from each modality to facilitate better scene flow estimation. However, compared to that from the odometer, the gains brought by these two sensors are smaller, especially for the camera, which only increases the AccS by 1.3%. In our opinion, the reason for this is two-fold. First, in Eq. (4) and Eq. (5), the pseudo scene/optical flow labels only constrain the flow vectors of identified moving points, which are significantly fewer than static ones. Thus, they have limited influence on the overall performance. Second, the supervision signals extracted from these sensor modalities could be noisy, as exhibited in Fig. 3. The dashboard reflections on the car windscreen severely disturb the optical flow estimation on images, which further results in erroneous supervision in Eq. (5). As for LiDAR point clouds, both inaccurate object localization and false negative detections occur in the 3D MOT results, which not only affect the generation of pseudo motion segmentation labels (Eq. (3)) but bring incorrect constraints to scene flow in Eq. (4). Impact of the amount of unannotated data. Being able to use more unannotated data for training is an inherent advantage of our cross-modal supervised method compared to fully-supervised ones as no additional annotation efforts Table 3. Motion segmentation evaluation. For row (a), we report the results of the self-supervised baseline [9]. In row (b), we replace S by S v provided by odometer. A.D. denotes adding extra unannotated data (c.f. Fig. 5) for training. Label S v Label S l A.D. mIoU (%) Gain (%) (a) 46.9 - (b) ✓ 52.8 +5.9 (c) ✓ ✓ 54.1 +1.3 (d) ✓ ✓ ✓ 57.1 +3.0 are required. Here, we are interested in if the performance of CMFlow could surpass that of fully-supervised methods when more unannotated data is available for training. To this end, we further mix in an extra amount of unlabeled data provided by the VoD dataset [32] 2 in our cross-modal training sets. As for fully-supervised methods, we select the state-of-the-art method, PV-RAFT [49], for comparison. As PV-RAFT needs ground truth scene flow annotations for training, we utilize all available annotated samples from the Train set for it. The analysis of the correlation between the percentage of added unannotated training data and the performance is shown in Fig. 5. As we can see, the performance of CMFlow improves by a large margin on both two metrics by using extra training data. In particular, after adding only 20% of the extra unannotated training samples (∼140% more than the number of training samples used for PV-RAFT), CMFlow can already outperform PV-RAFT [49] trained with less annotated samples. This implies the promise of our method for utilizing a large amount of unannotated data in the wild. Qualitative results. In Fig. 4 source frame by its estimated 3D motion vector), both the static background and multiple objects with different motion patterns are aligned well between two frames. It can also be observed that our method can give accurate scene flow predictions, close to the ground truth one. Subtask Evaluation Motion segmentation evaluation. Apart from scene flow estimation, our multi-task model can additionally predict a motion segmentation mask that represents the real moving status of each radar point (c.f. Sec. 3.3). Here, we evaluate this prediction of CMFlow and analyze the impact of its performance in Tab. 3. Since this is a binary classification task for each point, the mean intersection over union (mIoU) is computed by taking the IoU of moving and static classes and averaging them. As the two ingredients to form the final pseudo motion segmentation label S used to supervise our output in Eq. (3), both S v and S l contributes to our performance improvement on the motion segmentation task (row (a)-(c)). Moreover, our mIoU further increases with extra training data to learn motion segmentation on 4D radar point clouds. We also visualize some qualitative motion segmentation results of row (d) in Fig. 6, where our method can segment moving points belonging to multiple dynamic objects accurately in complicated scenarios. Ego-motion estimation evaluation. One important feature of our method is that we can estimate a rigid transformation that represents relative radar ego-motion transform between two consecutive frames in dynamic scenes (c.f. Sec. 3.3). To demonstrate this feature, we evaluate this output on the Test set and show the ablation study results in Tab. 4 with two metrics: the relative translation error (RTE) and the relative angular error (RAE). With the cross-modal supervision from the odometry in row (b), we can directly constrain our ego-motion estimation in Eq. (2) and thus improve the performance by a large margin. Using LiDAR and camera supervision (c.f. row (c)) can also help as they lead to better motion segmentation and scene flow outputs, which further benefit the compactly associated ego-motion Table 4. Evaluation of ego-motion estimation between two frames. T denotes we activate the temporal update module. Ours Ground Truth ICP Figure 7. Odometry results as a byproduct of our scene flow estimation. The ground truth is generated using the RTK-GPS/IMU measurements. We plot the results on two challenging test sequences. Please refer to more results in supplementary videos. estimation. We also activate the temporal update module in the backbone, which also increases the overall performance. With our high-level accuracy on ego-motion estimation between consecutive frames, we are also interested in whether our results can be used for the more challenging long-term odometry task. We accumulate the interframe transformation estimations and plot two ego-vehicle trajectories in Fig. 7. Without any global optimization, our method can provide accurate long-term trajectory estimation in dynamic scenes by only estimating inter-frame transformations and remarkably outperform the ICP baseline [4]. Conclusion In this paper, we presented a novel cross-modal supervised approach, CMFlow, for estimating 4D radar scene flows. CMFlow is unique in that it does not require manual annotation for training. Instead, it uses complementary supervision signals retrieved from co-located heterogeneous sensors, such as odometer, LiDAR and camera, to constrain the outputs from our multi-task model. Our experiments show that CMFlow outperforms our baseline methods in all metrics and can surpass the fully-supervised method when sufficient unannotated samples are used in our training. CMFlow can also improve two downstream tasks, i.e., motion segmentation and ego-motion estimation. We hope our work will inspire further investigation of crossmodal supervision for scene flow estimation and its application to more downstream tasks, such as multi-object tracking and point cloud accumulation. Figure 2 . 2Cross-modal supervised learning pipeline for 4D radar scene flow estimation. The model architecture (c.f. Sec. 3.3) is composed of two stages (blue/orange block colours for stage 1/2) and outputs the final scene flow together with the motion segmentation and a rigid ego-motion transformation. Cross-modal supervision signals are utilized to constrain outputs with various loss functions (c.f. Sec. 3.4). Figure 3 . 3Illustration of the causes of noisy supervision signals from camera and LiDAR. The top row shows an example of the noisy optical flow estimation on RGB images. The bottom row exhibits unreliable object recognition on LiDAR point clouds. We enlarge regions of interest and mark them with amber circles. Figure 4 . 4Qualitative scene flow results in two scenes. From left to right: 1) radar points from the source frame projected to the corresponding RGB image (points are coloured by distance from the sensor), 2) two input radar point clouds, the source one (pink) and the target one (green), 3) the source point cloud warped by our predicted scene flow and the target radar point cloud, 4) the source point cloud warped by ground truth scene flow and the target one. We mark dynamic objects in amber and apply the zooming-in operation for them. added unannotated training data[%] Figure 5 . 5Analysis of the performance when adding more unannotated training data. For the training of CMFlow, we retain the samples from the Train set and add different percentages of data from the extra unannotated VoD part, which provides ∼ 28.5k more training samples. Figure 6 . 6, we showcase example scene flow results of CMFlow (trained with the extra training samples) compared to the ground truth scene flow. By applying the estimated scene flow (i.e. moving each point in the Visualization of motion segmentation. The left column shows the corresponding image with radar points (coloured by range) projected onto it. In the middle and right columns, moving points are shown in orange while static points are shown in blue. 4D Radar Point CloudsInitial Flow Head Motion Seg. 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{'abstract': 'This work proposes a novel approach to 4D radar-based scene flow estimation via cross-modal learning. Our approach is motivated by the co-located sensing redundancy in modern autonomous vehicles. Such redundancy implicitly provides various forms of supervision cues to the radar scene flow estimation. Specifically, we introduce a multitask model architecture for the identified cross-modal learning problem and propose loss functions to opportunistically engage scene flow estimation using multiple cross-modal constraints for effective model training. Extensive experiments show the state-of-the-art performance of our method and demonstrate the effectiveness of cross-modal supervised learning to infer more accurate 4D radar scene flow. We also show its usefulness to two subtasks -motion segmentation and ego-motion estimation. Our source code will be available on https://github.com/Toytiny/CMFlow.', 'arxivid': '2303.00462', 'author': ['Fangqiang Ding fding@ed.ac.uk \nUniversity of Edinburgh\n\n', 'Andras Palffy a.palffy@tudelft.nl \nDelft University of Technology\n\n', 'Dariu M Gavrila d.m.gavrila@tudelft.nl \nDelft University of Technology\n\n', 'Chris Xiaoxuan Lu \nUniversity of Edinburgh\n\n'], 'authoraffiliation': ['University of Edinburgh\n', 'Delft University of Technology\n', 'Delft University of Technology\n', 'University of Edinburgh\n'], 'corpusid': 257255545, 'doi': '10.48550/arxiv.2303.00462', 'github_urls': ['https://github.com/Toytiny/CMFlow.'], 'n_tokens_mistral': 15507, 'n_tokens_neox': 13602, 'n_words': 8413, 'pdfsha': '1d99fb26a6497ffa62dd5afe296e18b0a05639b1', 'pdfurls': ['https://export.arxiv.org/pdf/2303.00462v3.pdf'], 'title': ['Hidden Gems: 4D Radar Scene Flow Learning Using Cross-Modal Supervision', 'Hidden Gems: 4D Radar Scene Flow Learning Using Cross-Modal Supervision'], 'venue': []}

arxiv

Analytic description and explicit parametrization of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure 3 Aug 2010 P A Djondjorov Institute of Mechanics Bulgarian Academy of Sciences Acad. G. Bonchev St Block 41113SofiaBulgaria V M Vassilev Institute of Mechanics Bulgarian Academy of Sciences Acad. G. Bonchev St Block 41113SofiaBulgaria I M Mladenov Institute of Biophysics Bulgarian Academy of Sciences Acad. G. Bonchev St Block 211113SofiaBulgaria Analytic description and explicit parametrization of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure 3 Aug 2010Preprint submitted to International Journal of Mechanical Sciences August 4, 2010Elastic ring (tube)Hydrostatic pressureEquilibrium shapesParametric equationsTube conductivitySimilarity law * Corresponding author; tel The parametric equations of the plane curves determining the equilibrium shapes that a uniform inextensible elastic ring or tube could take subject to a uniform hydrostatic pressure are presented in an explicit analytic form. The determination of the equilibrium shape of such a structure corresponding to a given pressure is reduced to the solution of two transcendental equations. The shapes with points of contact and the corresponding (contact) pressures are determined by the solutions of three transcendental equations. The analytic results presented here confirm many of the previous numerical results on this subject but the results concerning the shapes with lines of contact reported up to now are revised. (P.A. Djondjorov), vasilvas@imbm.bas.bg (V.M. Vassilev), mladenov@obzor.bio21.bas.bg (I.M. Mladenov) Introduction In the present paper, the problem for determination of the equilibrium shapes of a circular inextensible elastic ring subject to a uniformly distributed external force that acts normally to the ring in the ring plane is addressed. This problem is also referred to as the stability problem or buckling of the circular shape of the ring and the other equilibrium shapes are called buckled [1,2,3]. It is also known (see, e.g., [4,5,6]), that if a cylindrical elastic shell of circular cross section (i.e., a tube) is subject to a uniform external pressure, which is normal to its middle surface, then the typical cross section of the deformed tube takes the same shapes as the axis of a deformed elastic ring does provided that the latter is a simple curve (i.e., a curve without intersections). Therefore, here the term "ring" will be used to indicate both a proper ring and a tube. It should be noted also that in the majority of the works in this field, the distributed force acting on a ring is called pressure as in the case of a shell. Following this tradition, we will use the same term in the present study remembering that pressure means force per unit length in the case of a ring and force per unit area in the case of a shell. Maurice Lévy [7] was the first who stated and studied the problem under consideration and reduced the determination of the foregoing equilibrium shapes in polar coordinates to two elliptic integrals for the arclength and polar angle regarded as functions of the squared radial coordinate. He found also several remarkable properties of the equilibrium ring shapes and obtained that if the pressure p is such that p < (9/4)(D/ρ 3 ), where D and ρ are the ring bending rigidity and radius of the undeformed shape, respectively, then the ring possesses only the circular equilibrium shape. Later on, Halphen [8] and Greenhill [9] derived exact solutions to this problem in terms of Weierstrass elliptic functions on the ground of complicated analyses of the properties of the aforementioned elliptic integrals. Halphen (see [8, p. 235]) found out that non-circular shapes with n ≥ 2 axes of symmetry are possible only for pressures greater than p n = (n 2 −1)(D/ρ 3 ). Halphen [8] and Greenhill [9] presented also several examples of non-circular equilibrium ring shapes. It should be noted, however, that the exact solutions reported in [8,9], representing the polar angle as a function of the radius, appeared to be intractable and many researchers continued searching exact solutions [1,10,11,12,13,14,15], while others used various approximations [2,4,6,16] on the way to determine the equilibrium shapes of the ring. Carrier [1] was the first who reconsidered the foregoing problem for the buckling of an elastic ring about half a century after the works by Lévy, Halphen and Greenhill. He expressed the curvature of the deformed ring in terms of Jacobi cosine function [17] involving several unknown parameters to be determined by a system of algebraic equations. However, he succeeded to find approximate solutions to this system only for small deflections from the undeformed circular ring shape (see the exhaustive analysis provided recently by Adams [11] who has criticized and developed Carrier's work [1]). Tadjbakhsh and Odeh [2] studied the boundary-value problem describing the buckled shapes of the ring and the associated variational problem. They proved the existence of solutions to the boundary-value problem in the case of small deflections from the undeformed state and the existence of solutions of the associated variational problem (week solutions to the foregoing boundaryvalue problem) describing buckled shapes of an arbitrary deflection. Watanabe and Takagi [12] thoroughly analyzed the variational problem for determination of the ring shapes stated by Tadjbakhsh and Odeh [2] and obtained analytic expressions for the curvature of the ring (in terms of Jacobi elliptic functions and Carrier's parameters [1]) and formulas for the slope angle at the points where the curvature has extrema. They also proved that non-circular shapes with n ≥ 2 axes of symmetry exist for pressures greater than p n = (n 2 − 1)(D/ρ 3 ), thus extending Halphen's result, and found out, moreover, that each such shape in unique. This paper completes the branch of analysis of the considered problem that is based more or less on the approach suggested more than half a century ago by Carrier [1]. In the recent papers [13,14,15], the present authors have studied the differential equation for the curvature of the ring with the aim to achieve an analytic description of the cylindrical equilibrium shapes of lipid bilayer membranes. The determination of the analytic solutions of this equation reported in [15] does not follow Carrier's approach [1]. Instead of this, the explicit formulas for the curvature of the buckled shapes are obtained in forms similar to those suggested by Zhang [10] for lipid bilayer membranes and by Fukumoto [18] in the context of fluid mechanics. In [15], the parametric equations of the directrices of the considered cylindrical surfaces are expressed in an explicit analytic form, the necessary and sufficient conditions for such a surface to be closed are derived and several sufficient conditions for its directrix to be simple or self-intersecting are given. The equilibrium shapes of closed planar elastic loops subject to the constraints of fixed length and enclosed area are studied also in the works by Arreaga et al. [19], Capovilla et al. [20] and Guven [21]. Flaherty et al. [4] presented a numerical determination of the equilibrium shapes of elastic rings or tubes subject to uniform external pressure. They suggested a scenario for the evolution of the equilibrium shape as the pressure increases, however, some stages of this scenario are not confirmed here. Recently, Wu et al. [16] presented approximate solutions of the considered problem by means of a special Fourier expansion of the curvature. They developed a linearized algebraic system for the unknown coefficients of this Fourier expansion and thus obtained an approximation of the ring shape. The aim of the present paper is to provide an analytic description of the equilibrium states of an elastic ring or tube subject to a uniform hydrostatic pressure going as far as possible, to develop efficient computational procedures completing this analytic description and to reexamine some of the most important results obtained in this field. The layout of the paper is as follows. The statement of the problem is given in Section 2. A concise derivation of the most important Lévy's results is given in Section 3 where the equation for the ring curvature and the parametric equations of the equilibrium shapes are derived as well. Using these results, in Section 4, the symmetry of the equilibrium shapes is justified. All periodic solutions to the equation for the ring curvature and the expressions for the corresponding slope angles are presented in explicit analytic form in Section 5. Two systems of transcendental equations are derived in Sections 6 and 7 allowing to determine the equilibrium shapes corresponding to a given pressure and to calculate the values of the pressure at which the equilibrium shapes possess points of contact. The results concerning the equilibrium shapes with lines of contact obtained in [4] are reexamined in Section 8. Differential equations for the equilibrium states Let us consider a ring made of a homogeneous isotropic linearly elastic material and assume that it is represented by its middle axis. Suppose now that the ring is subject to a uniform external pressure p acting along the normal vector to its stress-free configuration. The analysis carried out in the present work is based on the following three assumptions: (i ) the ring axis is inextensible; (ii ) the pressure p preserves its magnitude and always acts as an external uniformly distributed force along the inward normal vector to the deformed ring axis, i.e., it is a uniform (simple) hydrostatic pressure; (iii ) the deformed ring axis is a regular closed plane curve Γ parametrized by its arclength s. Next, let the curve Γ be given by means of the coordinates x(s), y(s) of its position vector r(s) with respect to a certain rectangular Cartesian coordinate frame in the Euclidean plane, i.e., r(s) = x(s) i + y(s) j, where i and j are the unit vectors along the coordinate axes x and y, respectively. Consequently, the unit tangent vector t(s) and the unit inward normal vector n(s) to the curve Γ are given as follows t(s) = x ′ (s) i + y ′ (s) j, n(s) = −y ′ (s) i + x ′ (s) j.(1) Here and throughout this paper primes denote derivatives with respect to the arclength s. Recall that the foregoing unit tangent and normal vectors are related to the curvature κ(s) of the curve Γ through the Frenet-Serret formulas [22] t ′ (s) = κ(s)n(s), n ′ (s) = −κ(s)t(s). Finally, let M(s), N(s) and Q(s) denote the bending moment and the components of the stress resultant force F(s) along the tangent and normal vectors to the curve Γ, i.e., F(s) = N(s)t(s) + Q(s)n(s).(3) Then, the particular constitutive equation relating the moment M(s) with the curvature κ(s) and the form of the stress-free configuration of the ring together with the system of differential equations F ′ (s) = −p n(s),(4)M ′ (s) = −F(s) · n(s),(5) representing the local balances of the force and moment, respectively, in accordance with the assumption (ii ), and the closure conditions following from the assumption (iii ), which, without loss of generality, may be written in the form r(L) = r(0), t(L) = t(0),(6) where L is the length of the deformed ring, determine entirely the equilibrium state of the ring under consideration (see, e.g., [5,Ch. 4)]). Here and throughout this paper the dot stands for dot (scalar) product of two vectors. Let us remark also that using Eq. (3) and the Frenet-Serret formulas (2) one can represent the system of differential equations (4), (5) in the scalar form N ′ (s) = Q(s)κ(s),(7)Q ′ (s) = −N(s)κ(s) − p,(8) M ′ (s) = −Q(s). Parametric equations for the equilibrium shapes The aim of this Section is to prove the most important facts concerning the problem under consideration established by Maurice Lévy in his memoir [7] and to derive the parametric equations of the equilibrium ring shapes. Using the expression for the normal vector, see Eqs. (1), one can integrate the equilibrium condition (4) and express the force vector in the form F(s) = py(s) i − px(s) j,(10) Next, taking the dot product of both sides of Eq. (10) with the normal vector one gets, bearing in mind the second one of equations (1), the relation F(s) · n(s) = −p[x(s)x ′ (s) + y(s)y ′ (s)], which allows to integrate the balance of moment equation (5) and to obtain M(s) = p 2 (r 2 (s) + c),(12) where c is an arbitrary constant of integration, cf. [7, 3 • (p. 9)]. It is noteworthy that the relations (10) -(12) hold regardless of the particular material properties of the ring and the form of its stress-free configuration. In terms of the slope angle ϕ(s) one has the expressions x ′ (s) = cos ϕ(s), y ′ (s) = sin ϕ(s),(13)κ(s) = ϕ ′ (s),(14) and using Eqs. (13) can rewrite Eqs. (1) in the form t(s) = cos ϕ(s) i + sin ϕ(s) j,(15)n(s) = − sin ϕ(s) i + cos ϕ(s) j.(16) Then, combining Eqs. (3), (10), (15) and (16) one obtains the parametric equations of the deformed ring shape in the form x(s) = − 1 p Q(s) cos ϕ(s) − 1 p N(s) sin ϕ(s), y(s) = − 1 p Q(s) sin ϕ(s) + 1 p N(s) cos ϕ(s).(17) Evidently, in view of Eq. (15), the second one of the closure conditions (6) implies that the rotation number of the deformed ring axis is 2mπ, where m is an integer, i.e., ϕ(L) = ϕ(0) + 2mπ,(18) whereas the first one of them transforms, on account of Eqs. (12), (17) and (18), into the obvious conditions for periodicity of the forces and moment N(L) = N(0), Q(L) = Q(0), M(L) = M(0).(19) Again, neither the form of the parametric equations (17) nor the forms of the boundary conditions (18) and (19) depend on the particular material properties or the stress-free configuration of the ring. Let us now assume that the constitutive equation of the ring is M(s) = D(κ(s) − κ • ),(20) where D is its bending rigidity and κ • = 1/ρ is the curvature of its stress-free configuration, which is supposed to be a circle of radius ρ. Using Eq. (20) and the Frenet-Serret formulas (2) we can immediately integrate the system of differential equations (7) -(9) obtaining the following expressions for the tangent and normal components of the stress resultant force F(s) N(s) = − D 2 (κ 2 (s) − 2µ), Q(s) = −Dκ ′ (s),(21) and the single ordinary differential equation for the ring curvature κ ′′ (s) + 1 2 κ 3 (s) − µκ(s) − σ = 0,(22) where σ = p/D and µ is an arbitrary constant of integration. On the other hand, combining equations (20), (11) and (12) we obtain the relation N 2 (s) + Q 2 (s) = 2pD(κ(s) − κ • ) − p 2 c, which, in view of Eqs. (21), implies κ ′ (s) 2 = P (κ(s)),(23) where P (κ) is a fourth-order polynomial of the curvature κ of the form P (κ) = − 1 4 κ 4 + µκ 2 + 2σκ + ε (24) whose free term ε = −2σκ • − σ 2 c − µ 2 incorporatesx(s) = 1 σ κ ′ (s) cos ϕ(s) + 1 2σ (κ 2 (s) − 2µ) sin ϕ(s), y(s) = 1 σ κ ′ (s) sin ϕ(s) − 1 2σ (κ 2 (s) − 2µ) cos ϕ(s),(25) obtained by substituting Eqs. (21) in the general formulas (17). However, the parametric equations (25) describe a shape that a ring of bending rigidity (18) and (19) hold for L = 2πρ; note that the latter equality follows from the assumption (i ). If this is the case, then the respective solution κ(s), its first derivative κ ′ (s) and its indefinite integral ϕ(s) = κ(s)ds,(26) cf. Eq. (14), determine entirely the equilibrium state of the pressurized ring. Indeed, the shape of the ring is determined explicitly by Eqs. (25) and the values of the moment and forces acting along the ring axes are given by the formulas (20) and (21), respectively. It should be remarked that each such solution κ(s) is necessarily a periodic function with period L = 2πρ, due to the condition (19), and if T is its least period, then L = nT , where n is a positive integer. Since ϕ(nT ) = nϕ(T ), as follows by formula (26), the closure condition (18) takes the form ϕ(T ) = 1 n ϕ(0) + 2mπ n ·(27) Symmetry of the ring shapes The symmetry of the ring shapes is discussed, usually without going into much detail, by almost all authors that contributed to the solution of the considered problem. A concise justification of this property is given below. Let Γ be a curve whose curvature κ(s) is a periodic solution of Eq. (23) with least period T . For each integer i denote by Γ (−) it and Γ (+) it the parts of the curve Γ corresponding to s ∈ [T i − t,T i ] and s ∈ [T i ,T i + t], respectively, whereT i = i(T /2) and t ∈ R. Since the relations t(T i ) · r(T i + s) = −t(T i ) · r(T i − s) (28) n(T i ) · r(T i + s) = n(T i ) · r(T i − s)(29) hold for each couple of numbers i ∈ N and s ∈ R, the curves Γ It should be noted that relations (28) and (29) are easily verified using the parametric equations (25) and the formulas κ(T i + s) = κ(T i − s), κ ′ (T i + s) = −κ ′ (T i − s)(30) and ϕ(T i + s) = ϕ(T i ) + ϕ(s)(31) that follow from κ(−s) = κ(s) and the definition (26) of the slope angle ϕ(s). In other words, the curve Γ can be thought of as generated by successive reflections of its part Γ (−) 1t , t =T 1 , about the axes directed by the normal vectors n(T i ), i ∈ N. Apparently, if the curve Γ (−) nt ∪ Γ (+) nt get closed smoothly for some positive integer n, which may happened only for t =T n , then the closed curve Γ = Γ (−) nt ∪ Γ (+) nt has n distinct axes of symmetry and is said to have n-fold symmetry or to be of n mode. Analytic expressions for the ring curvatures and slope angles As it was mentioned in the Introduction, Eq. (23) was studied by the present authors with the aim to describe the cylindrical equilibrium shapes of fluid lipid bilayer membranes (see [13,14,15]). Explicit analytic expressions for all periodic solutions of Eq. (23) and for the corresponding slope angles (26) are presented in [15]. These results are outlined in this Section. Depending on the values of the parameters σ, µ and ε, there exist two cases in which the polynomial P (κ) attains positive values and hence Eq. (23) has real-valued solutions: (I) the polynomial P (κ) has two simple real roots α, β ∈ R, α < β, and a pair of complex conjugate roots γ, δ ∈ C, δ =γ; (II) the polynomial P (κ) has four simple real roots α < β < γ < δ ∈ R. In the first case, the polynomial P (κ) is nonnegative in the interval α ≤ κ ≤ β, while in the second one, it is nonnegative in the intervals α ≤ κ ≤ β and γ ≤ κ ≤ δ. It should be noted that the roots α, β, γ and δ of the polynomial P (κ) can be expressed explicitly through its coefficients µ, σ and ε and vice versa. Indeed, after some standard algebraic manipulations (see, e.g., [23]), one can find the following expressions for the roots of the polynomial P (κ) − √ ω ± 2µ − 2σ 1 ω − ω, √ ω ± 2µ + 2σ 1 ω − ω, where ω = (2µ + ζ) 2 − 2 2 3ε 6ζ , ζ = 3 2 2 3 (3 2 σ 2 + √ χ) − 2 3 µ (µ 2 + 3 2 ε) , χ = 2 2 3ε µ 2 + ε 2 − 3 2 µσ 2 − 3σ 2 2 2 µ 3 − 3 3 σ 2 . Then, one can denote properly each of the above expressions for the roots in accordance with the notation introduced in the cases (I) and (II), respectively. Simultaneously, by Vieta's formulas one obtains α + β + γ + δ = 0,(32) due to the absence of a term with κ 3 in the polynomial P (κ), see Eq. (24), and µ = 1 4 α 2 + β 2 + γ 2 + αβ + αγ + βγ ,(33)σ = − 1 8 (α + β) (α + γ) (β + γ) ,(34)ε = 1 4 αβγ(α + β + γ). Case (I) Let the parameters µ, σ and ε be such that the polynomial P (κ) has roots as in case (I), namely, two of them are real (α < β) and the other two constitute a complex conjugate pair which, in view of relation (32), can be written in the form γ = − α + β 2 + iη, δ = − α + β 2 − iη,(35) where η is a nonnegative real number. In this case, Eq. (23) Let η = 0 and hence the roots of the polynomial P (κ) are simple. Denote λ 1 = 1 4 √ AB, k 1 = 1 2 − 4η 2 + (3α + β) (α + 3β) 2AB ,(36) where A = 4η 2 + (3α + β) 2 , B = 4η 2 + (α + 3β) 2 .(37) Evidently, A > 0, B > 0, λ 1 > 0 and 0 < k 1 < 1. In this case, each solution of Eq. (23) can be expressed by the function κ 1 (s) = (Aβ + Bα) − (Aβ − Bα) cn(λ 1 s, k 1 ) (A + B) − (A − B) cn(λ 1 s, k 1 ) ,(38) which takes real values for each s ∈ R and is periodic with least period T 1 = (4/λ 1 )K(k 1 ) due to the periodicity of the Jacobi function cn(λ 1 s, k 1 ). Here, K(·) denotes the complete elliptic integral of the first kind. The corresponding slope angle (26) can be written in the form ϕ 1 (s) = Aβ − Bα A − B s + α − β 2λ 1 k 2 1 + C arctan k 2 1 + C sn (λ 1 s, k 1 ) dn (λ 1 s, k 1 ) + (A + B) (α − β) 2λ 1 (A − B) Π (−C, am(λ 1 s, k 1 ), k 1 ) ,(39) where Π(·, ·, ·) denotes the incomplete elliptic integral of the third kind and C = (A − B) 2 4AB ·(40) Now, let η = 0 and (3α + β)(α + 3β) > 0. Then, the polynomial P (κ) has one double and two simple real roots. The curvature and the slope angle (26) are expressed in terms of elementary functions as follows κ 2 (s) = (Aβ + Bα) − (Aβ − Bα) cos(λ 1 s) (A + B) − (A − B) cos(λ 1 s) , ϕ 2 (s) = Aβ − Bα A − B s + 8(α − β) A − B arctan A B tan 1 2 λ 1 s · where λ 1 , A and B remain defined by formulas (36) and (37), respectively. Case (II) Let the parameters µ, σ and ε be such that the polynomial P (κ) has roots as in case (II), that is α < β < γ < δ ∈ R. Denote λ 2 = 1 4 (γ − α) (δ − β), k 2 = (β − α) (δ − γ) (γ − α) (δ − β) · Since γ − α > β − α > 0 and δ − β > δ − γ > 0, it is seen that λ 2 > 0 and 0 < k 2 < 1. In this case, each solution of Eq. (23) can be expressed by one of the following functions κ 3 (s) = δ − (δ − α)(δ − β) (δ − β) + (β − α)sn 2 (λ 2 s, k 2 ) , κ 4 (s) = β + (γ − β)(δ − β) (δ − β) − (δ − γ)sn 2 (λ 2 s, k 2 ) , see [15,Theorem 2]. The functions κ 3 (s) and κ 4 (s) take real values for each s ∈ R and are periodic with least period T 2 = (2/λ 2 )K(k 2 ) because of the periodicity of the function sn 2 (λ 2 s, k 2 ). Their indefinite integrals (26) can be written as ϕ 3 (s) = δs − δ − α λ 2 Π β − α β − δ , am(λ 2 s, k 2 ), k 2 , ϕ 4 (s) = βs − β − γ λ 2 Π δ − γ δ − β , am(λ 2 s, k 2 ), k 2 , respectively. It should be remarked, that the indefinite integrals ϕ j (s), j = 1, . . . , 4, of the foregoing solutions κ j (s) of Eq. (23) are chosen so that ϕ j (0) = 0. Moreover, κ j (0) always coincides with a certain root of the polynomial P (κ) (actually, κ j (0) = α for j = 1, 2, 3 and κ 4 (0) = γ) and hence κ ′ j (0) = 0, according to Eq. (23). It is important to note also that the functions κ j (s) are strictly increasing for s ∈ [0, T /2] where T is the respective least period. Determination of the equilibrium shapes In the present study, our primary interest is in the determination of the equilibrium ring shapes that are curves without intersections, i.e., simple curves. It is established in [15] that only the solutions of Eq. (23) that fall under the case (I) with η = 0 and α + β = 0 may give rise to simple curves. Therefore, hereafter we will restrict our analysis to the regular closed curves Γ n of curvatures κ(s) = κ 1 (s) given by formula (38), which have n axis of symmetry and meet all the necessary conditions for that to be simple. Thus, the closure condition (27) for such a curve Γ n reads ϕ 1 (T 1 ) = ± 2π n ,(41) as ϕ 1 (0) = 0 (see the remark at the end of Section 5), n ≥ 2 due to the four vertex theorem (see, e.g., [24]) and m = ±1 since the rotation number of a simple regular closed curve must be ±2π, see [25]. Note, however, that there exist regular closed curve with rotation number ±2π, which are not simple. Then, substituting the expression T 1 = (4/λ 1 )K(k 1 ) for the least period of the solutions (38) of Eq. (23) in the general formula (39) for the corresponding slope angle ϕ 1 (s) one can rewrite the closure condition (41) in the form (A + B)(α − β) 2λ 1 (A − B) Π(−C, k 1 ) + Aβ − Bα λ 1 (A − B) K(k 1 ) = ± π 2n ·(42) Finally, substituting the same expression for the period T 1 in the relation L = nT 1 in order to take into account that the length of the ring L is fixed and does not change upon deformation, see assumption (i ), one obtains 1 λ 1 K(k 1 ) = π 2n ·(43) after setting for simplicity, without loss of generality, L = 2π, i.e., κ • = ρ = 1. (34) and (35), in terms of the positive parameters σ, η and q as follows α = 4σ η 2 + q 2 − q, β = 4σ η 2 + q 2 + q ·(44) Thus, given an integer n ≥ 2 and a pressure p by means of the parameter σ (called hereafter simply "pressure") the problem for the determination of the foregoing equilibrium shapes of the ring corresponding to this pressure is reduced to the computation of the solutions η and q of the transcendental equations (42) and (43). It is important to notice that this problem has no nontrivial solution if 0 < σ ≤ σ bn and has a unique nontrivial solution if σ > σ bn , see [12, Theorem 2]. Here, σ bn = n 2 − 1 is the so-called buckling pressure and by a trivial solution we mean the one, which corresponds to the ring shape that is a circle of radius ρ = 1. Here, the transcendental equations (42) and (43) Three examples of such shapes, which confirm the results presented in [4] are given in Fig. 2. a b c Equilibrium shapes with points of contact In [2,4], it is established that for each mode n = 2, 3, 4 there is a value of the pressure σ, called contact pressure and denoted by σ cn , at which some points of the respective buckled ring shape of n-fold symmetry come into contact. In the aforementioned works, it is also observed that if the applied pressure σ is such that σ bn < σ < σ cn , then the corresponding buckled shape of n mode is simple. It should be noted, that the values for the contact pressures reported in [2,4] are obtained solving numerically a rather complicated nonlinear boundary-value problem. In the present study, the determination of the non-circular equilibrium ring shapes with points of contact and the respective contact pressures is reexamined being reduced to the computation of the common solutions of the transcendental equations (42), (43) and one more algebraic, when n = 2, or transcendental, when n > 2, equation in the way to be described below. Proceeding to the examination of this problem, let us first clarify, slightly extending the definition used in [4], that an n-mode equilibrium ring shape Γ n is said to have a point of contact if it is not self-intersecting, but there is at least one couple of values s 1 and s 2 of the archlength s such that 0 < s 1 < s 2 < L and r(s 2 ) = r(s 1 ), t(s 2 ) = ±t(s 1 ). This means that at the point of contact r(s 2 ) = r(s 1 ) the curve Γ n is tangent to itself. Such a double point on a curve is called a cusp or tacnode, see [26]. The objective now is to reformulate the above conditions in a form suitable for the developing of an efficient procedure for computation of the contact pressures corresponding to the foregoing equilibrium ring shapes. For that purpose, it is convenient to use the relations κ(s) = σ 2 r 2 (s) − µ 2 + ε 2σ(46) and r(s) · t(s) = 1 σ κ ′ (s), r(s) · n(s) = − 1 2σ (κ 2 (s) − 2µ), which follow from Eqs. (23) - (25) and allow, taking into account Eqs. (15) and (16), conditions (45) to be cast in the form κ 1 (s 2 ) = κ 1 (s 1 ), κ ′ 1 (s 2 ) = ±κ ′ 1 (s 1 ),(47)κ 2 1 (s 2 ) − 2µ = ±(κ 2 1 (s 1 ) − 2µ),(48) and ϕ 1 (s 2 ) = ϕ 1 (s 1 ) + 2lπ,(49) if the sign in the second one of Eqs. (45) is plus, or ϕ 1 (s 2 ) = ϕ 1 (s 1 ) + (2l + 1)π,(50) if the foregoing sign is minus. Here, l is an integer. In the case when t(s 2 ) = t(s 1 ), conditions (47) imply s 2 = s 1 +jT 1 , where j is a positive integer and hence, according to Eqs. (26) and (41), we have ϕ 1 (s 2 ) = ϕ 1 (s 1 ) ± j 2π n ·(51) Now, combining Eqs. (49) and (51) we obtain j = ±ln, which means that s 2 = s 1 ± lnT 1 = s 1 ± lL and therefore the assumptions t(s 2 ) = t(s 1 ) and 0 < s 1 < s 2 < L turn out to be incompatible. That is to say that a noncircular equilibrium ring shape can not have cusps of this type. In the remaining case t(s 2 ) = −t(s 1 ), conditions (47) and (48) read Hence, in view of Eqs. (26) and (41), condition (50) takes the form κ 1 (s 2 ) = κ 1 (s 1 ), κ ′ 1 (s 2 ) = −κ ′ 1 (s 1 ),(52)κ 2 1 (s 2 ) − 2µ = κ 2 1 (s 1 ) − 2µ = 0.(53)ϕ 1 (s 1 ) = ± π n − π 2 − lπ,(54) if s 2 = T 1 − s 1 , or ϕ 1 (s 1 ) = ±π − π 2 − lπ,(55) if s 2 = nT 1 − s 1 . Next, suppose that the curve Γ n is such that condition (53) κ 1 (s − 1 ) = − 2µ, κ 1 (s + 1 ) = 2µ .(56) According to Eqs. (56), there is some s 0 ∈ [0, T 1 /2] for which κ(s 0 ) = 0 and therefore, in the light of the above considerations, s − 1 < s 0 < s + 1 and κ 1 (0) ≤ − √ 2µ < 0 < √ 2µ ≤ κ 1 (T 1 /2) because the curvature κ 1 (s) is strictly increasing for s ∈ [0, T 1 /2] (see the remarks at the end of Section 5). Hence, κ 2 1 (0) − 2µ ≥ 0, κ 2 1 (T 1 /2) − 2µ ≥ 0 .(57) Taking into account that κ ′ 1 (0) = κ ′ 1 (T 1 /2) = 0 and ϕ 1 (0) = 0, the first of parametric equations (25) implies x(T 1 /2) = 1 2σ κ 2 1 (T 1 /2) − 2µ sin ϕ 1 (T 1 /2).(58) The property (31) of the slope angle allows the closure condition (41) to be recast in the form It is clear that the slope angle ϕ 1 (s) has a minimum at s = s 0 since ϕ ′ 1 (s 0 ) = κ 1 (s 0 ) = 0 and ϕ ′′ 1 (s 0 ) = κ ′ 1 (s 0 ) > 0. Actually, this is the only local extremum of this function in the interval (0, T 1 /2) since s 0 is the only point in the foregoing interval where the curvature is equal to zero. Therefore, ϕ 1 (s) < π/n for s ∈ (0, T 1 /2) since ϕ(0) = 0 and ϕ(T 1 /2) = π/n. ϕ 1 (T 1 /2) = ± π n ,(59) One can show also that ϕ(s) > −π for s ∈ (0, T 1 /2). Indeed, if ϕ(ŝ) = −π for someŝ ∈ (0, T 1 /2), then r ′ (ŝ) = x(ŝ) cos ϕ(ŝ) + y(ŝ) sin ϕ(ŝ) r(ŝ) = − x(ŝ) r(ŝ) ≤ 0 since x(s) ≥ 0 for each s in the considered interval. However, this inequality contradicts the fact that r(s) is strictly increasing for s ∈ (0, T 1 /2) as follows from Eq. (46), and hence, ϕ(s) > −π in this interval. In view of the inequalities −π < ϕ(s) ≤ π/n, conditions (54) and (55) take the form ϕ 1 (s 1 ) = π n − π 2 , ϕ 1 (s 1 ) = − π 2 , respectively. Apparently, ϕ ′ 1 (s − 1 ) < 0 and hence ϕ 1 (s) is decreasing in the neighbourhood of s = s − 1 , while ϕ ′ 1 (s + 1 ) > 0 and hence ϕ 1 (s) is increasing at s = s − 1 . Therefore, contact points (if exist) should be such that ϕ 1 (s − 1 ) = π n − π 2 ,(60) or ϕ 1 (s + 1 ) = − π 2 ,(61) otherwise the respective curve Γ n is self-intersecting. Finally, taking the inverse s = 1 λ 1 F arccos Aβ + Bα − κ(A + B) Aβ − Bα − κ(A − B) , k 1 of the function κ = κ 1 (s), which is readily achieved by Eq. (38) and well defined for each s ∈ [0, T 1 /2], one, bearing in mind that s − 1 , s + 1 ∈ [0, T 1 /2] and Eqs. (56), obtains s − 1 = 1 λ 1 F arccos Aβ + Bα + √ 2µ(A + B) Aβ − Bα + √ 2µ(A − B) , k 1 ,(62)s + 1 = 1 λ 1 F arccos Aβ + Bα − √ 2µ(A + B) Aβ − Bα − √ 2µ(A − B) , k 1 ,(63) where F(·, ·) is the incomplete elliptic integral of the first kind. Thus, two triples of transcendental equations (42), (43) and (60) or (61) arise for the determination of the n-mode equilibrium ring shapes with points of contact. In both cases, the respective transcendental equations involve as unknowns only the four parameters σ, η, q and n since the archlengths s − 1 and s + 1 , which may correspond to points of contact are determined explicitly in terms of these parameters by formulas (62) The obtained values for the respective contact pressures σ cn are presented in Table 1. The equilibrium ring shapes with points of contact corresponding to the contact pressures σ c2 , σ c3 and σ c4 are depicted in Fig. 3. It is worth noting that in the special case n = 2 the transcendental equation (60) may be replaced by the algebraic relation σ = (η 2 + q 2 ) 2 16q ·(64) Indeed, in this case, expression (60) simplifies to ϕ 1 (s − 1 ) = 0 meaning that s − 1 = 0. In fact, there exist two values of the archlength within the interval [0, T 1 /2] in which ϕ 1 (s) = 0 but the other one is necessarily greater than s 0 and therefore it is disregarded. Then, the first of Eqs. In [4], it is claimed that for each mode n beyond the contact pressure σ cn there exists a continuous range of pressures, from σ cn up to a certain pressure denoted by σ 0n , such that for each σ cn ≤ σ ≤ σ 0n the respective ring shape exhibits contacts at isolated points only. The pressure σ 0n is set in [4] to be the one for which the curvature at the corresponding contact point is zero. If so, however, then in view of Eqs. (66) µ = 0 and s − 1 = s + 1 = s 0 . Therefore, the equation κ(s)−2µ = 0 has exactly one solution for s ∈ [0, T 1 /2] and hence, according to [15,Theorem 3], the corresponding ring shape is self-intersecting. Thus, in this respect the results of Flaherty et al. [4] turn out to be inaccurate in spite of the fact that they are widely accepted and even confirmed numerically by other authors (see, e.g., [27]). Actually, for all modes n in the range 2 ≤ n ≤ 15 our computations based on the procedure described in Section 6 show that if the applied pressure σ is such that σ bn < σ < σ cn , then the corresponding buckled shape of n mode is simple, while for σ > σ cn this shape always has points of self-intersection. Our conjecture is that this behaviour is inherent to all modes. For n = 2, 3, 4, Flaherty et al. [4] affirm that for each pressure σ such that Equilibrium shapes with lines (areas) of contact Apparently, for rings of finite thickness self-intersecting shapes are not possible because they are not planar, nevertheless for a very thin ring such a shape may be considered as a good approximation of its equilibrium state. A tube evidently can not take a self-intersecting shape, but there is a good reason to expect that tubes subject to sufficiently high pressure posses equilibrium shapes with areas of contact (lines of contact of their cross sections). Flaherty et al. [4] suggest similarity transformations to be used for the determination of such shapes, but realize this idea in a very complicated way. Moreover, the construction developed in [4] for the said purpose makes use of the curves Γ 0n corresponding to the pressures σ 0n , which are wrongly regarded as curves with isolated points of contact as it was noted and discussed above. Below, an alternative approach is presented for constructing equilibrium ring (tube) shapes with lines (areas) of contact based on the same "similarity" idea that actually arises out of the following property of Eqs. (22) and (23). Under the transformation (s, κ) −→ (s/λ, λκ), where λ is an arbitrary real number, each equation of form (22) corresponding to certain constants µ and σ transforms into an equation of the same form but with new coefficients: µ −→ λ 2 µ, σ −→ λ 3 σ. The same holds true for equation (23) if ε −→ λ 4 ε in addition. In other words, equations (22) and (23) are invariant with respect to the similarity transformation Λ : (s, κ; µ, σ, ε) −→ (s/λ, λκ; λ 2 µ, λ 3 σ, λ 4 ε). Consequently, the parametric equations (25) imply that the shapes whose parameters are related by such a transformation Λ are similar, the respective scaling factor being 1/λ. Accordingly, if a closed curve Γ is scaled in this way, then its length L and area A change to L/λ and A/λ 2 , respectively. Thus, given n ≥ 2, let the curve Γ cn of length L cn = 2π be the equilibrium shape with points of contact corresponding to the contact pressure σ cn and letΓ be the shape (of the same length) with lines of contact corresponding to a pressureσ > σ cn . The curveΓ is constructed in two steps. First, scaling the curve Γ cn with a factor (σ/σ cn ) 1/3 one obtains another curveΓ cn which has the same number of contact points because it is similar to the curve Γ cn but corresponds to the pressureσ and its length isL cn = 2π(σ cn /σ) 1/3 < L cn . Then Consequently, the moment and force also suffer jumps at the aforementioned points since their limit values from the bent parts of the curve are M b = −D 2µ + κ • , N b = 0, Q b = ±D P − 2µ ,(65) while along each line of contact the resultant pressure is zero and M l = −Dκ • , N l = 0, Q l = 0.(66) Equations (65) In our opinion, this property of the constructed curvesΓ allows this shapes to be regarded as equilibrium ring (tube) shapes with lines (areas) of contact at least in the week sense discussed above. Γ M ′ (s)ds = − Γ F(s) · n(s)ds,(67) In the light of the results presented in this Section, it should be remarked that the similarity law (5.4) obtained in [4,Section 5], which concerns the conductivity of a buckled tube conveying an incompressible viscous fluid, has to be revised. Actually, this law, which expresses the conductivity of a tube with areas of contact through that of a tube whose cross sections have just points of contact, should be replaced by the following one C(σ) = σ cn σ 4/3 C(σ cn ), σ > σ cn ,(69) where C(σ) denotes the conductivity of a tube subject to pressure σ. It is necessary to do so because σ cn is the unique pressure for which there exists an equilibrium tube shape of n-fold symmetry whose cross sections have only isolated points of contact. Concluding remarks In the present paper, the problem for determination of the equilibrium shapes of a circular inextensible elastic ring (tube) subject to a uniform hydrostatic pressure is reexamined. For the first time, more than a century ago, this problem was stated and studied by Maurice Lévy in his memoir [7]. Here, a concise derivation of the most important facts established in [7], see 1 • -3 • (p. 9), concerning the existence of a "centre of the elastic forces", following by Eq. (10), and the properties reflected by Eqs. (11) and (12) is given in Section 3. Then, the parametric equations of the equilibrium shapes are expressed through the forces and slope angle, see Eqs. (17). It is noteworthy that neither the relations (10) - (12) nor the forms of the parametric equations (17) or boundary conditions (18) and (19) depend on the particular material properties or stress-free configuration of the ring. A concise justification of the symmetry of the ring shapes discussed by many authors (usually without going into much detail) is given in Section 4. It is shown analytically that each ring shape is symmetric and can be obtained by successive reflections of its part corresponding to the first half period of the curvature with respect to its symmetry axes. Further, assuming that the stress-free configuration of the ring is a circle of radius ρ, the case in which the linear constitutive equation (20) holds is considered. In this case, the equilibrium state of the ring is determined by the periodic solutions of the nonlinear ordinary differential equation (23) for the ring curvature, which are such that the closure condition (27) holds. In fact, the shape of the ring is determined explicitly by the parametric equations (25) and the values of the moment and forces acting along the ring axes are given by Eqs. (20) and (21), respectively. Explicit analytic expressions for all periodic solutions of Eq. (23) and for the corresponding slope angles are presented in Section 5. In Section 6, the determination of the equilibrium shapes corresponding to a given pressure σ is reduced to the computation of the common solutions of two transcendental equations (42) and (43). Shapes with isolated points of contact are studied in Section 7. It is shown that the pressures at which such a shape is attained can be obtained The most important results achieved in Sections 6 and 7 are as follows. In contrast to the assertion in [4] that for each mode n there exists a range of pressures for which the respective ring shape has only isolated points of contact, we found, solving numerically Eqs. (42), (43) and (64) or (60), that for each mode 2 ≤ n ≤ 15 there is a unique such pressure, namely σ cn . Moreover, for all modes in the range 2 ≤ n ≤ 15 our computations based on the procedure described in Section 6 show that if the applied pressure σ is such that σ bn < σ < σ cn , then the corresponding buckled shape of n mode is simple, while for σ > σ cn this shape always has points of self-intersection. Our conjecture is that this behaviour is inherent to all modes. Section 8 concerns the equilibrium ring (tube) shapes with lines (areas) of contact that are expected to occur for pressures greater than the respective contact pressure instead of the self-intersecting shapes (unnatural for tubes and planar rings) predicted by the considered model. Here, the construction of these shapes is based on the similarity properties of Eqs. (23) and (25) following in general outline the idea suggested by Flaherty et al. in [4], but the uniqueness of the contact pressures σ cn is taken into account. The shapes obtained in this way are shown to satisfy the total balances (67) and (68) of the respective forces and moments, which is good reason to consider them as equilibrium shapes. Finally, the expression for the so-called similarity law obtained in [4] is revised, see Eq. (69). The interested reader can find the Mathematica notebooks developed for the solution of the transcendental equations of Sections 6 and 7 as well as the notebooks developed for the construction of the shapes described in Section 8 at http://www.bio21.bas.bg/ibf/dpb files/mfiles/. omitting the constant of integration since it is always possible to eliminate it by choosing the origin of the coordinate frame at a certain privileged point which, actually, is the one that Lévy called "centre of the elastic forces", cf. [7, 1 • (p. 9)]. Eq. (10) implies that at each point of the deformed ring configuration the magnitude of the force vector F (s) = |F(s)| is proportional to the magnitude of the position vector r(s) = |r(s)|, the magnitude of the pressure |p| being the coefficient of proportionality (cf. [7, 2 • (p. 9)]), i.e., F (s) = |p| r(s). all the constants of integrations introduced so far. Actually, Eq. (23) is a first integral of Eq. (22) (see [15, Sec. 2] for more details). In this context, ε is viewed as an arbitrary constant of integration. Each sufficiently smooth real-valued solution κ(s) of an equation of form (23) corresponding to a certain triple of given values of the parameters µ, ε and σ = 0 generates, up to a rigid motion in the plane, a unique plane curve Γ of curvature κ(s). The components of the position vector of this curve can by expressed in the form D could take being subject to pressure p = σD if and only if the regarded solution κ(s) of the respective equation of form (23) is such that the closure conditions are symmetric with respect to the axis directed by the normal vectors n(T i ). Fig. 1 1illustrates the simplest case i = 1, t =T 1 . Figure 1 : 1Position vectors r(T /2 ± s), tangent t(T /2) and normal n(T /2) vectors to a curve; the dashed line represents the axis of symmetry. has periodic solutions if η = 0 or η = 0 and (3α + β)(α + 3β) > 0, see [15, Theorem 1]. The left-hand sides of Eqs. (42) and (43) are functions of the parameters α, β and η, see Eqs. (36), (37) and (40). However, for the aims of the present study it is convenient to express the parameters α and β, using formulas corresponding to given integer n ≥ 2 and pressure σ > σ bn are solved numerically in two steps using Mathematica . First, the two curves in the (η, q) plane defined by Eqs. (42) and (43) are plotted using the routine ContourPlot in order to identify roughly the values of the coordinates of their intersection point. Then, these values are put as starting values in the routine FindRoot, which is employed to obtain the solutions η and q of the system of equations (42), (43) with a sufficient accuracy. Once such a solution is determined, formulas (44), (38), (39), (40) and the parametric equations (25) allow to depict the corresponding equilibrium ring shapes using the routine ParametricPlot. Figure 2 : 2Equilibrium ring shapes corresponding to: (a) σ = 4.75 (2-fold symmetry); (b) σ = 16.25 (3-fold symmetry); (c) σ = 35.25 (4-fold symmetry). Now, taking into account the symmetry of the considered type of ring shapes, the invariance of Eq. (22) under the translation of its independent variable s and the particular properties of its solutions reflected by the formulas (30), without loss of generality one may assume that s 1 ∈ [0, T 1 /2] and can easily enumerate all the values s 2 of the arclength for which conditions (52) hold and the part of the curve Γ n corresponding to s ∈ [s 1 , s 2 ] may have double points of the considered type only, namely: s 2 = T 1 − s 1 and s 2 = nT 1 − s 1 . which, in view of the second of inequalities (57) and the fact that σ > 0, means that the sign of x(T 1 /2) coincides with that of the right hand side of Eq. (59). If the latter sign is minus, then x(T 1 /2) < 0 as implied by Eq. (58) and the second of inequalities (57). On the other hand, x(0) = 0 and x ′ (s) = cos ϕ(s) meaning that x(s) attains positive values in the interval (0, T 1 /2). Therefore, the respective curve Γ n is self-intersecting in this case because it intersects one of its symmetry axis -the one directed by the normal vector n(0) = −j. In this way, we arrive at the conclusion that a contact point may occur only if the sign of the right hand side of closure condition Eq. (59) is plus. and (63), and the same holds true for the left hand sides of Eqs. (60) and (61) in view of the general expression (39) for the slope angle. Of course, one should remember that the sign of the right hand side of Eq. (42) is plus in this context. To summarize: given an integer n ≥ 2, each solution of any one of the aforementioned two triples of transcendental equations gives the value of the contact pressure σ cn and the values of the parameters η and q determining in this way an equilibrium ring shape of n-fold symmetry with points of contact. Solving numerically the foregoing two systems of transcendental equations using the routine FindRoot in Mathematica we have found that the system consisting of Eqs. (42), (43) and (61) does not have solutions for 2 ≤ n ≤ 15, but the system of equations (42), (43) and (60) has a unique solution for each such mode n. Our conjecture is that this happens for all modes. Figure 3 : 3(56) reads α 2 −2µ = 0. Substituting here expressions (44) 1 and (33) for the root α and the parameter µ, respectively, and accounting for Eqs. (35) and (44) 2 one obtains Eq. Ring shapes with points of contact: (a) σ = 5.247; (b) σ = 21.65; (c) σ = 51.844. The values of contact pressures σ cn obtained for n = 2, 3, 4 confirm exactly the results presented in [4, formulas (2.11)], see also Fig. 3, but the latter are interpreted therein as the lowest values of the pressures at which an isolated point of contact occurs. 12 ,Figure 4 :Figure 5 :Figure 6 : 124565.247 ≤ σ ≤ 10.34, 21.65 ≤ σ ≤ 81.81 or 51.84 ≤ σ ≤ 207.2, respectively, the corresponding equilibrium ring shapes have only isolated points of contact without being self-intersecting. Our results presented in Figs. 4, 5 and 6 show that this is not the case. Let us recall that for each σ > σ bn the corresponding equilibrium ring shape of n-fold symmetry is unique, see [Theorem Ring shapes corresponding to: (a) σ = 6.48; (b) σ = 9.24; (c) σ = 10.34. Ring shapes corresponding to: (a) σ = 28.56; (b) σ = 56.09; (c) σ = 81.81. Ring shapes corresponding to: (a) σ = 70.7; (b) σ = 140; (c) σ = 207.2. Figure 7 : 7, the curveΓ is obtained by substituting each point of contact of the curveΓ cn by a line segment of length 2π(1−(σ cn /σ) 1/3 )/n along the respective symmetry axis of the curveΓ cn so as its total length to become 2π. Examples of shapes with lines of contact are presented in Figs. Shapes with lines of contact corresponding to: (a) σ = 10.34 (2-fold symmetry); (b) σ = 81.81 (3-fold symmetry); (c) σ = 207.2 (4-fold symmetry). Figure 8 : 8Shapes with lines of contact corresponding to: (a) σ = 400 (6-fold symmetry); (b) σ = 800 (9-fold symmetry); (c) σ = 1500 (12-fold symmetry). It is clear that the tangent, normal and position vectors of a shapeΓ with lines of contact constructed in the foregoing way are continuous at each point of the curveΓ. However, its curvature suffers jumps at the end points of the line segments used to substitute the contact points of the respective auxiliary curveΓ cn because the limit values of the curvature from the bent parts of the curve and from the line segments are − √ 2µ = 0 and zero, respectively. and (66) follow by the constitutive equation (20), the general solution (21) of Eqs. (7) -(9) and Eqn. (23). Thus, the local balances (4) and (5) of the force and moment are violated for the shapes with lines of contact. Fortunately, however, the total balances Γ F ′ (s)ds = − Γ p n(s)ds, of these quantities are satisfied. Indeed, Eqs. (67) and (68) hold on the curvê Γ cn since it corresponds to an equilibrium shape without jump discontinuities of the force and moment. On the other hand,Γ =Γ cn ∪ {Line segments} and the integrals in Eqs. (67) and (68) taken along the line segments are equal to zero because here p = 0, M ′ (s) = 0 and F(s) = 0. computing the common solutions of Eqs. (42), (43) and (64) in the case n = 2, or Eqs. (42), (43) and (60) for n > 2. Table 1 . 1Contact pressures.n 2 3 4 5 6 7 8 σ cn 5.247 21.650 51.844 97.834 161.077 242.682 343.517 n 9 10 11 12 13 14 15 σ cn 464.276 605.522 767.719 951.253 1156.450 1383.580 1632.890 AcknowledgementsThis research is supported by the contract # 35/2009 between the Bulgarian and Polish Academies of Sciences. On the buckling of elastic rings. G F Carrier, J. Math. and Phys. 26G.F. Carrier, On the buckling of elastic rings, J. Math. and Phys. 26 (1947) 94-103. Equilibrium states of elastic rings. I Tadjbakhsh, F Odeh, J. Math. Anal. Appl. 18I. Tadjbakhsh, F. Odeh, Equilibrium states of elastic rings, J. Math. Anal. Appl. 18 (1967) 59-74. Bifurcation theory applied to buckling states of a cylindrical shell. J Chaskalovic, S Naili, ZAMP. 46J. Chaskalovic, S. Naili, Bifurcation theory applied to buckling states of a cylindrical shell, ZAMP 46 (1995) 149-155. Post buckling behaviour of elastic tubes and rings with opposite sides in contact. J E Flaherty, J B Keller, S I Rubinow, SIAM J. Appl. Math. A. 23J.E. Flaherty, J.B. Keller, S.I. Rubinow, Post buckling behaviour of elastic tubes and rings with opposite sides in contact, SIAM J. Appl. Math. A 23 (1972) 446-455. Nonlinear problems of elasticity. S S Antman, Applied Mathematical Sciences. 107Springer-VerlagS.S. Antman, Nonlinear problems of elasticity, Applied Mathematical Sciences, Vol. 107, Springer-Verlag, New York, 1995. Buckling and collapse of open and closed cylindrical shells. C Pozrikidis, J. Eng. Math. 42C. Pozrikidis, Buckling and collapse of open and closed cylindrical shells, J. Eng. Math. 42 (2002) 157-180. Mémoire sur un nouveau cas intégrable du problème de l'élastique et l'une de ses applications. M M Lévy, Journal de Mathematiques Pures et Appliquees série 3 X (1884). M.M. Lévy, Mémoire sur un nouveau cas intégrable du problème de l'élastique et l'une de ses applications, Journal de Mathematiques Pures et Appliquees série 3 X (1884) 5-42. Deuxième partie: Applications a la mécanique, a la physique, a la géodésie, a la géométrie et au calcul intégral, Gauthier-Villars et fils. G H Halphen, 1888ParisLa corbeélastique plane sous pression uniforme, Chap. V in Traité des fonctions elliptiques et de leurs applicationsG.H. Halphen, La corbeélastique plane sous pression uniforme, Chap. V in Traité des fonctions elliptiques et de leurs applications. Deuxième partie: Applications a la mécanique, a la physique, a la géodésie, a la géométrie et au calcul intégral, Gauthier-Villars et fils, Paris, 1888. The elastic curve under uniform normal pressure. A G Greenhill, Mathematische Annalen LII (1889). A.G. Greenhill, The elastic curve under uniform normal pressure, Math- ematische Annalen LII (1889) 465-500. A complete classification of closed shapes for cylindrical vesicles. S G Zhang, Acta Phys. Sin. (Overseas Edn). 6S.G. Zhang, A complete classification of closed shapes for cylindrical vesicles, Acta Phys. Sin. (Overseas Edn) 6 (1997) 641-655. Postbuckling of circular rings: An analytical solution. A J Adams, J. Math. Phys. 4932902A.J. Adams, Postbuckling of circular rings: An analytical solution, J. Math. Phys. 49 (2008) 032902. Representation formula for the critical points of the Tadjbakhsh-Odeh functional and its application. K Watanabe, I Takagi, Japan J. Indust. Appl. Math. 25K. Watanabe, I. Takagi, Representation formula for the critical points of the Tadjbakhsh-Odeh functional and its application, Japan J. Indust. Appl. Math. 25 (2008) 331-372. Plane curves associated with integrable dynamical systems of the Frenet-Seret type. P A Djondjorov, V M Vassilev, I M Mladenov, Proc. 9th International Workshop on Complex Structures, Integrability and Vector Fields. 9th International Workshop on Complex Structures, Integrability and Vector FieldsSingaporeWorld Scientific Publishing CoP.A. Djondjorov, V.M. Vassilev, I.M. Mladenov, Plane curves associated with integrable dynamical systems of the Frenet-Seret type, In: Proc. 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia 25-29 August 2008, World Scientific Publishing Co., Singapore, 2009, pp. 56-63. On the translationallyinvariant solutions of the membrane shape equation. V Vassilev, P A Djondjorov, I M Mladenov, Geometry, Integrability and Quantization. I. Mladenov and M. de LeónSOFTEX, Sofia8V.M Vassilev, P.A. Djondjorov, I.M. Mladenov, On the translationally- invariant solutions of the membrane shape equation, In: Geometry, In- tegrability and Quantization, Vol. 8 (Eds. I. Mladenov and M. de León), SOFTEX, Sofia, 2007, pp. 312-321. Cylindrical equilibrium shapes of fluid membranes. V M Vassilev, P A Djondjorov, I M Mladenov, J. Phys. A: Math. Theor. 41435201V.M. Vassilev, P.A. Djondjorov, I.M. Mladenov, Cylindrical equilibrium shapes of fluid membranes. J. Phys. A: Math. Theor. 41 (2008) 435201. Analytical approximations to large postbuckling deformation of elastic rings under uniform hydrostatic pressure. B S Wu, Y P Yu, Z G Li, Int. J. Mech. Sci. 49B.S. Wu, Y.P. Yu, Z.G. Li, Analytical approximations to large post- buckling deformation of elastic rings under uniform hydrostatic pressure, Int. J. Mech. Sci. 49 (2007) 661-668. H Hancock ; E. Jahnke, F Emde, F Lösch, The definitions and properties of the elliptic functions and integrals can be found. Dover, New York; Teubner, Stuttgart; New York; New YorkSpringer-VerlagElliptic Functions and ApplicationsThe definitions and properties of the elliptic functions and integrals can be found, e.g., in: H. Hancock, Elliptic Integrals, Dover, New York, 1958; E. Jahnke, F. Emde, F. Lösch, Tafeln Höherer Funktionen, Teub- ner, Stuttgart, 1960; K. Chandrasekharan, Elliptic Functions, Springer- Verlag, New York, 1985; D. Lawden, Elliptic Functions and Applica- tions, Springer-Verlag, New York, 1989. Stationary configurations of a vortex filament in background flows. Y Fukumoto, Proc. R. Soc. Lond. A. R. Soc. Lond. A453Y. Fukumoto, Stationary configurations of a vortex filament in back- ground flows, Proc. R. Soc. Lond. A 453 (1997) 1205-1232. Areaconstrained planar elastica. G Arreaga, R Capovilla, C Chryssomalakos, J Guven, Phys. Rev. E. 6531801G. Arreaga, R. Capovilla, C. Chryssomalakos, J. Guven, Area- constrained planar elastica, Phys. Rev. E 65 (2002) 031801. Elastica hypoarealis. R Capovilla, C Chryssomalakos, J Guven, Eur. Phys. J. B. 29R. Capovilla, C. Chryssomalakos, J. Guven, Elastica hypoarealis, Eur. Phys. J. B 29 (2002) 163-166. Laplace pressure as a surface stress in fluid vesicles. J Guven, J. Phys. A: Math. Gen. 39J. Guven, Laplace pressure as a surface stress in fluid vesicles, J. Phys. A: Math. Gen. 39 (2006) 3771-3785. Introduction to Geometry. H S M Coxeter, WileyNew Yorksecond ed.H.S.M. Coxeter, Introduction to Geometry, second ed., Wiley, New York, 1969. Mathematical Handbook for Scientists and Engineers: Definition, Theorems, and Formulas for Reference and Review. G A Korn, T M Korn, McGraw-HillNew YorkG.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and En- gineers: Definition, Theorems, and Formulas for Reference and Review, McGraw-Hill, New York, 1968. The four vertex theorem and its converse. D Deturck, H Gluck, D Pomerleano, D S Vick, Notices Amer. Math. Soc. 54D. DeTurck, H. Gluck, D. Pomerleano, D.S. Vick, The four vertex the- orem and its converse, Notices Amer. Math. Soc. 54 (2007) 192-207. Über die Drehung der Tangenten und Sehnen ebener Kurven. H Hopf, Compositio Math. 2H. Hopf,Über die Drehung der Tangenten und Sehnen ebener Kurven, Compositio Math. 2 (1935) 50-62. Academic Press Dictionary of Science and Technology. C G Morris, Academic Press IncSan DiegoC.G. Morris (Ed.), Academic Press Dictionary of Science and Techno- logy, Academic Press Inc., San Diego, 1992. Buckling and collapse of heavy tubes resting on a horizontal or inclined plane. M G Blyth, C Pozrikidis, European Journal of Mechanics A/Solids. 21M.G. Blyth, C. Pozrikidis, Buckling and collapse of heavy tubes rest- ing on a horizontal or inclined plane, European Journal of Mechanics A/Solids 21 (2002) 831-843.

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{'abstract': 'The parametric equations of the plane curves determining the equilibrium shapes that a uniform inextensible elastic ring or tube could take subject to a uniform hydrostatic pressure are presented in an explicit analytic form. The determination of the equilibrium shape of such a structure corresponding to a given pressure is reduced to the solution of two transcendental equations. The shapes with points of contact and the corresponding (contact) pressures are determined by the solutions of three transcendental equations. The analytic results presented here confirm many of the previous numerical results on this subject but the results concerning the shapes with lines of contact reported up to now are revised. (P.A. Djondjorov), vasilvas@imbm.bas.bg (V.M. Vassilev), mladenov@obzor.bio21.bas.bg (I.M. Mladenov)', 'arxivid': '1008.0533', 'author': ['P A Djondjorov \nInstitute of Mechanics\nBulgarian Academy of Sciences Acad. G. Bonchev St\nBlock 41113SofiaBulgaria\n', 'V M Vassilev \nInstitute of Mechanics\nBulgarian Academy of Sciences Acad. G. Bonchev St\nBlock 41113SofiaBulgaria\n', 'I M Mladenov \nInstitute of Biophysics\nBulgarian Academy of Sciences Acad. G. Bonchev St\nBlock 211113SofiaBulgaria\n'], 'authoraffiliation': ['Institute of Mechanics\nBulgarian Academy of Sciences Acad. G. Bonchev St\nBlock 41113SofiaBulgaria', 'Institute of Mechanics\nBulgarian Academy of Sciences Acad. G. Bonchev St\nBlock 41113SofiaBulgaria', 'Institute of Biophysics\nBulgarian Academy of Sciences Acad. G. Bonchev St\nBlock 211113SofiaBulgaria'], 'corpusid': 119144041, 'doi': '10.1016/j.ijmecsci.2011.02.005', 'github_urls': [], 'n_tokens_mistral': 18999, 'n_tokens_neox': 16341, 'n_words': 10343, 'pdfsha': 'cb2416d6053f1ca10b6e5e4130ac09588fa28c2f', 'pdfurls': ['https://arxiv.org/pdf/1008.0533v1.pdf'], 'title': ['Analytic description and explicit parametrization of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure', 'Analytic description and explicit parametrization of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure'], 'venue': []}

arxiv

Ferromagnetism and magneto-dielectric effect in insulating LaBiMn 4/3 Co 2/3 O 6 thin films R Ranjith Asish K Kundu M Filippi B Kundys W Prellier B Raveau J Laverdière M P Singh S Jandl Laboratoire CRISMAT UMR 6508 CNRS ENSICAEN 6 Boulevard Maréchal JuinCaen-14050France Department de physique & Regroupement québécois sur les materiaux de pointe Université de Sherbrooke J1K2R1SherbrookeQuébecCanada Ferromagnetism and magneto-dielectric effect in insulating LaBiMn 4/3 Co 2/3 O 6 thin films 1 High quality epitaxial thin films of LaBiMn 4/3 Co 2/3 O 6 perovskite were fabricated on (001)-oriented SrTiO 3 and LaAlO 3 substrates by the pulsed laser deposition technique. Magnetization measurements reveal a strong magnetic anisotropy and a ferromagnetic behavior that is in agreement with a super-exchange interaction between Mn 4+ and Co 2+ ions, which are randomly distributed in the B-site. A distinct anomaly is observed in the dielectric measurements at 130K corresponding to the onset of the magnetic ordering, suggesting a coupling. Above this temperature, the extrinsic Maxwell-Wagner effect is dominating. Theses results are explained using the Raman spectroscopic studies indicating a weak spin-lattice interaction around this magnetic transition. Multifunctional materials (multiferroics, magneto dielectric, spintronics etc.) are of interest owing to their potential device applications, such as non-volatile memories, magnetic read heads, tunnel junction spin filtering etc 1 . Materials that exhibit such behavior possess a coupling between electronic and magnetic properties, which is reflected in their respective order parameters. Magnetic semiconductors or insulators have been identified as single phase materials that offer the potential to exhibit simultaneously electronic and magnetic ordering, and until now, very few compounds of this type have been reported (e.g. BiMnO 3 , 2,3 CdCr 2 Se 4 , 4 La 2 Mn(Co/Ni)O 6 , 5,6, Bi 2 MnNiO 6 , 7,8 and diluted magnetic semiconductors 9 ). In the metal oxide members containing bismuth, ferromagnetism originates from superexchange interactions between adjacent cations through the oxygen and ferroelectricity likely originates from the lone pair electrons of the Bi 3+ ion, which induce structural distortions. As a result, the investigation of ferromagnetic insulators containing bismuth offers the potential to generate magnetoelectric properties. In this respect, the insulating LaBiMn 4/3 Co 2/3 O 6 (LBMCO) perovskite is attractive since at low temperature it is a hard ferromagnet (useful for memory device). 10 This material has a higher ferromagnetic Curie temperature (130K) than BiMnO 3 (T C~1 05K) and can be prepared at normal pressure. LBMCO possesses a high value of thermoelectric power at room temperature and p-type polaronic conductivity. Nevertheless, no detailed ferroelectric studies have been reported for this compound. However, recent investigations of the magneto-dielectric 11 or spin filtering, 12 for these Bi-based materials in the form of thin films have enhanced the possibility of device applications. LBMCO films (1500Å) were deposited on (001)-oriented SrTiO 3 (STO) and LaAlO 3 (LAO) substrates supplied by CrysTec (GmbH, Germany) by the pulsed laser 3 deposition (PLD). A KrF excimer laser (λ = 248nm, 3Hz) was focused on the dense ceramic target of LBMCO, prepared through conventional sol-gel method. 10 The deposition was carried out at a substrate temperature of 650 o C under 20 mTorr of oxygen. The crystalline quality and the epitaxial nature of the films were confirmed by x-ray diffraction (λ=1.5406Å) using normal θ-2θ configuration (Seifert) and Φscans (Phillips Xpert). The cationic ordering and the spin-lattice interactions were investigated by polarized Raman spectroscopy using a Labram-800 microscope spectrometer equipped with a He:Ne laser(λ=632.8nm) and nitrogen cooled CCD The diffraction pattern reveals that the films are highly oriented and no alternate phases or extra orientations were detected. Bulk LBMCO crystallizes in a Pnma space group with lattice parameters of a bulk =5.529 Å, b bulk =7.801 Å and c bulk =5.513 Å. 10 Consequently, the lattice parameters of the film would have the following the relationships: a P =c P =a bulk /√2=3.905 Ǻ, b=2a P , 13 To further investigate the presence of weak spin -lattice coupling, the local structure of the thin films was investigated by polarized Raman spectroscopy. 17,18 Temperature dependence Raman spectra were measured on film samples grown on both STO and LAO surfaces in the x'x', xx, xy, and x'y' polarization configurations. A typical Raman spectra collected at 300K is shown in Figure 4. 19 , which occurs in the vicinity of the magnetic transition temperature. In the present case, however, the magnetic transition temperature is about 123K for bulk (see Fig. 2), which is much lower than the observed phonon peak shifts at 230 K (see Fig. 4). It is also important to note that this temperature behavior correlates well with the temperature dependence dielectric properties of these films, which also peak around the same temperature. The same ambiguity has been also observed in other magnetodielectric materials, such as DyMnO 3 20 . detector. The magnetization (M) and the capacitance (C) measurements were investigated as a function of temperature (T) and magnetic field (H) by a Quantum Design AC SQUID and physical property measurement system (PPMS) coupled with an impedance analyzer (Agilent technologies-4284A), respectively. Figure 1 1shows the Θ−2Θ x-ray diffraction pattern of the LBMCO thin film. where a P refers to the lattice parameter of the cubic perovskite sub-cell (close to 3.9Å). Based on this, the films would have either a [010] or a [101]-orientation with respect to the substrate plane. Such an orientation does not affect the properties and the micro-structural characterizations will be published elsewhere. On LAO, the out-of-plane lattice parameter is calculated to be 3.907Å. Such a value matches well with STO substrate (3.905 Ǻ), as can be seen from the overlap of the diffraction peaks. The out-of-plane lattice parameter is slightly different than the bulk lattice parameter by +0.2%, when 4 assuming as a [010]-orientation, and -0.08%, if a [101]-orientation of the film is assumed. The Φ-scan recorded around (103) reflection of the film is shown in the inset of Figure 1. The presence of the 4 peaks, 90°-separated confirms the cube-oncube growth of the thin film with respect to the substrate, indicating the epitaxial nature of the film. Figure 2 10 Figure 3 2103shows the temperature dependent zero-field-cooled (ZFC) and fieldcooled (FC) magnetization of the LBMCO film on LAO substrate. Magnetic field was applied both parallel (H||(100) S ) and perpendicular (H||(001) S ) to the substrate surface. At 10K, thin films exhibited a higher magnetization (2.21µ B /f.u (FC)) for the parallel applied magnetic field parallel to substrate (film) surface (H||(100) S ) in comparison to the perpendicular applied magnetic field (H||(001) S ) (0.68µ B /f.u.(FC)). A strong magnetic anisotropy (along the directions parallel and perpendicular of the film plane) can be associated with a single domain or epitaxial orientation of the LBMCO films (on both STO and LAO substrates). The inset of Figure 2 shows the magnetic hysteresis loop, (M-H), for the LBMCO films recorded at 10 and 100K. At 10K, a remnant magnetization (M r ) value of 2.26 µ B /f.u. and 2.43 µ B /f.u. and a coercive field (H C ) value of 8.0 and 7.1 kOe are observed for LBMCO films deposited on LAO and STO, respectively. As expected, the high temperature (M-H) behavior is linear similar to paramagnetic phase and dominated by the diamagnetic behavior of the substrate. The obtained (M-H) plots and the strong divergence between ZFC-FC magnetization at low temperature are similar to those of the bulk material reported earlier. On the other hand, the H C values for the LBMCO thin films are greater than those of the bulk phase. 10 The small variation in the H C as well as in the Curie temperature (T C ) values observed in LBMCO films on different substrates may be due to the tensile strain of the films. The ZFC and FC magnetization data of LBMCO films exhibit a clear 5 transition from a paramagnetic state to a ferromagnetic state at T C . The T C valueswere calculated from the minimum position of the dM FC /dT versus temperature. A T C value of ~123K and 115K is obtained for the film deposited on LAO and STO substrates, respectively, which is in close agreement with the bulk(130K).10 A large divergence between ZFC and FC data at low temperature was observed for both substrates. The T irr (the temperature at which, ZFC and FC magnetization diverges) is also higher for LAO film (T irr ~100K) whereas for STO film the T irr value is 85K.Aforementioned, low temperature magnetic properties prove the existence of ferromagnetism in LBMCO film, and are in close agreement with the possibility of FM interaction between Mn 4+ and Co 2+ cations via super-exchange mechanism. 4-6 The spin-glass behavior of the low temperature magnetic phase has been reported previously in the bulk. shows the dielectric measurements carried out at 1 MHz in the temperature range of 10 to 300K, both for bulk and thin film samples. In contrast to the film, the bulk exhibits a huge rise in the dielectric constant above 150K owing to an extrinsic Maxwell-Wagner kind relaxation, which arises from the grain boundaries present in the system 14 . Above 250K, the dielectric constant further increases in the bulk samples, which can be attributed to the space charge effect with an increase in temperature, owing to the semiconducting nature of the system. Around 200K, the curve of dielectric constant, measured at 1MHz, exhibits an anomaly in both the thin film and the bulk. These features are frequency dependent (not shown) as expected from extrinsic effect. 14 . Another dielectric anomaly is observed around the magnetic transition temperature (123K), for thin film samples. This anomaly argues for the presence of a weak spin lattice interaction in the thin films around the magnetic transition. The onset of interactive ordered magnetic clusters could give rise to spin 6 lattice coupling, which distorts the lattice, and alters the dielectric constant 15 . The presence of lone pair electrons of bismuth can facilitate this lattice distortion at the onset of magnetic ordering. Nevertheless, the specific details of the origin and the nature of the observed weak spin-lattice interaction remain ambiguous, and are currently under study in the case of both bulk and thin films.The dielectric anomaly around the magnetic transition observed in thin film samples is dominated by the extrinsic Maxwell-Wagner effect at low frequencies (< 500kHz). The imaginary dielectric constant (ε") was observed to be in the range of 245 -150 within the observed temperatures of 10-300K respectively. The inset ofFigure 3shows the isothermal magnetic field-dependence of the dielectric constant measured on thin film at 10K, under 1MHz. A positive magnetodielectric effect ∆ε (where ∆ε is calculated using the following formula [(ε(H)-ε(0))/ε(0)]x100 %) of 0.7% was observed at 10K. This small effect was observed up to 100K, which is close to the magnetic transition temperature of 123K, without the overlap of other extrinsic effects. The weak magnetodielectric effect (+0.7%) observed could arise from the intrinsic spin-lattice coupling.16 In orthorhombic perovskite crystal symmetry, 24 Raman modes are allowed, but only 4 Raman modes were observed in these samples. These modes appear at 270 cm -1 (xx and x'x' configurations), 518 cm -1 (not seen in x'y'), 626 cm -1 (xy and x'x') and 635 cm -1 (xx and x'x'). Detailed studies further show that the Raman peaks are characterized by a very large full-width at half maxima (FWHM) value. For example, the FWHM value for the 635 cm -1 phonon mode (A g -mode) is about 60 cm -1 . Such broadness in the phonon peaks indicates that the cations are randomly distributed, leading to large cationic disorder among the Mn 4+ and Co 2+ . Moreover, the apparent lack of polarization dependence further attests to the lack of a cation ordering. To observe the spin-lattice interactions, a temperature dependent shift of the 633 cm -1 Raman mode was plotted for films grown on STO and LAO substrates (see inset of Fig. 4). This plots reveals that as the temperature decreases, the shift in frequency first peaks up around 230K for STO and 250 K for LAO, respectively, and further decreases until 80K, after which it stays constant. The observed softening trend is consistent with what has been reported for other double perovskite manganites. The modulation of the super-exchange integral by the phonons is generally observed as softening of phonons Figure captions Figure captions Figure 1 . 1XRD pattern of LBMCO thin films on STO and LAO substrates. Inset shows the Φ-scan around (103) reflection of the thin film. Figure 2 . 2(color online) Temperature dependent ZFC (open symbol) and FC (solid symbol) magnetization (M), of LaBiMn 4/3 Co 2/3 O 6 thin film on LaAlO 3 (001) substrates (H = 1000 Oe, applied parallel to film surface H||(100) S , and perpendicular to the film surface, H||(001) S . The insets show the magnetic hysteresis curves at different temperatures (H||(100) S ). Contribution from the substrate was substracted. Figure 3 . 3(color online) Variation of the relative permittivity (ε r ) with temperature for LBMCO compound (left for bulk, and right for thin film). The inset shows the variation of ∆ε % {[ε (H)-ε (0)/ ε (0)] x 100} with magnetic field at 10K for LBMCO/LAO film. Fig. 4 . 4(color online) Raman spectrum of the LBMCO/LAO in both HH and HV geometry. Inset shows the softening of a Raman mode with temperature in both substrates. In summary, highly oriented epitaxial LaBiMn 4/3 Co 2/3 O 6 thin films were grown. The films deposited, on STO and LAO substrates, exhibited a clear ferromagnetic behavior with a Curie temperature of 123K and 115K, respectively, and a saturation magnetization of 2.21µ B /f.u. for the applied magnetic field parallel to the film surface and 0.68µ B /f.u. for the applied magnetic field perpendicular to the film surface. The high frequency anomaly observed in the dielectric measurements of the 8 thin films around the onset of magnetic ordering temperature, suggests the presence of a weak spin-lattice interaction. A weak magneto-dielectric effect of 0.7% at 10K arises from the existing spin-lattice interaction was also confirmed by observation of softening of phonon modes in polarized Raman spectroscopic studies.AKK thanks the French Ministry of Education and Research for a fellowship award. This work was carried out in the frame of the NoE FAME (FP6-5001159-1), the . W Prellier, M P Singh, P Murugavel, J. Phys. Cond. Matter. 17803W. Prellier, M.P. Singh and P. 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N S Rogado, J Li, A W Sleight, M A Subramanian, Adv. Mater. 172225N.S. Rogado, J. Li, A.W. Sleight, and M.A. Subramanian, Adv. Mater. 17, 2225 (2005). . W Prellier, A Fouchet, B Mercey, J. Phys. Cond. Matter. 151583W. Prellier, A. Fouchet and B. Mercey, J. Phys. Cond. Matter 15, R1583 (2003). . A K Kundu, V Pralong, V Caignaert, C N R Rao, B Raveau, J. Mater. Chem. 173347A. K. Kundu, V. Pralong, V. Caignaert, C. N. R. Rao, and B. Raveau, J. Mater. Chem. 17, 3347 (2007). . C H Yang, S H Lee, T Y Koo, Y H Jeong, Phys. Rev. B. 75140104C. H. Yang, S. H. Lee, T. Y. Koo, and Y. H. Jeong, Phys. Rev. B 75, 140104(R) (2007). . M Gajek, M Bibes, S Fusil, K Bouzehouane, J Fontcuberta, A Barthelemy, A Fert, Nat. Mater. 6296M. Gajek, M. Bibes, S. Fusil, K. Bouzehouane, J. Fontcuberta, A. Barthelemy, and A. Fert, Nat. Mater., 6, 296 (2007). . B Mercey, J Wolfman, W Prellier, M Hervieu, Ch Simon, B Raveau, Chem. Mater. 122858B. Mercey, J. Wolfman, W. Prellier, M. Hervieu, Ch. Simon, B. Raveau, Chem. Mater. 12, 2858 (2000). . G Catalan, Appl. Phys. Lett. 88102902G.Catalan, Appl. Phys. Lett., 88, 102902 (2006). . J F Scott, Journal of Materials Research. 222053J.F.Scott, Journal of Materials Research, 22, 2053 (2007). . J F Scott, Phys. Rev. B. 162329J.F. Scott, Phys. Rev. B 16, 2329 (1977). . K D Troung, J Laverdiere, M P Singh, S Jandl, P Fournier, Phys. Rev. B. 76132413K.D. Troung, J. Laverdiere, M.P.Singh, S. Jandl and P. Fournier, Phys. Rev. B, 76, 132413 (2007). . M N Illiev, M V Abrashev, A P Litvinchuk, V G Hajiev, H Guo, A Gupta, Phys.Rev.B. 75104118M.N. Illiev, M.V.Abrashev, A.P.Litvinchuk, V.G.Hajiev, H.Guo and A.Gupta, Phys.Rev.B, 75, 104118 (2007). . E Granado, A Garcia, J A Sanjurjo, C Rettori, I Torriani, E Prado, R D Sanchez, A Canerio, S B Oseroff, Phys. Rev. B. 6011879E.Granado, A.Garcia, J.A. Sanjurjo, C.Rettori, I.Torriani, E.Prado, R.D. Sanchez, A.Canerio and S.B.Oseroff, Phys. Rev. B., 60, 11879 (1999). . 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{'abstract': 'High quality epitaxial thin films of LaBiMn 4/3 Co 2/3 O 6 perovskite were fabricated on (001)-oriented SrTiO 3 and LaAlO 3 substrates by the pulsed laser deposition technique. Magnetization measurements reveal a strong magnetic anisotropy and a ferromagnetic behavior that is in agreement with a super-exchange interaction between Mn 4+ and Co 2+ ions, which are randomly distributed in the B-site. A distinct anomaly is observed in the dielectric measurements at 130K corresponding to the onset of the magnetic ordering, suggesting a coupling. Above this temperature, the extrinsic Maxwell-Wagner effect is dominating. Theses results are explained using the Raman spectroscopic studies indicating a weak spin-lattice interaction around this magnetic transition.', 'arxivid': '0801.3321', 'author': ['R Ranjith ', 'Asish K Kundu ', 'M Filippi ', 'B Kundys ', 'W Prellier ', 'B Raveau ', 'J Laverdière ', 'M P Singh ', 'S Jandl ', '\nLaboratoire CRISMAT\nUMR 6508\nCNRS\nENSICAEN\n6 Boulevard Maréchal JuinCaen-14050France\n', '\nDepartment de physique & Regroupement québécois sur les materiaux de pointe\nUniversité de Sherbrooke\nJ1K2R1SherbrookeQuébecCanada\n'], 'authoraffiliation': ['Laboratoire CRISMAT\nUMR 6508\nCNRS\nENSICAEN\n6 Boulevard Maréchal JuinCaen-14050France', 'Department de physique & Regroupement québécois sur les materiaux de pointe\nUniversité de Sherbrooke\nJ1K2R1SherbrookeQuébecCanada'], 'corpusid': 117506408, 'doi': '10.1063/1.2842409', 'github_urls': [], 'n_tokens_mistral': 5970, 'n_tokens_neox': 4974, 'n_words': 2922, 'pdfsha': '5fa2d3a2f7b4c4a07aa54c2e208d1b55545301de', 'pdfurls': ['https://arxiv.org/pdf/0801.3321v1.pdf'], 'title': ['Ferromagnetism and magneto-dielectric effect in insulating LaBiMn 4/3 Co 2/3 O 6 thin films', 'Ferromagnetism and magneto-dielectric effect in insulating LaBiMn 4/3 Co 2/3 O 6 thin films'], 'venue': []}

arxiv

Neural Forecasting at Scale Philippe Chatigny University of Sherbrooke SherbrookeQCCanada Shengrui Wang University of Sherbrooke SherbrookeQCCanada Jean-Marc Patenaude Laplace Insights SherbrookeQCCanada Boris N Oreshkin Unity Technologies Labs, MontrealQCCanada Neural Forecasting at Scale Univariate time series forecastingdeep neural networksN-BEATSensemble models We study the problem of efficiently scaling ensemble-based deep neural networks for multi-step time series (TS) forecasting on a large set of time series. Current state-of-the-art deep ensemble models have high memory and computational requirements, hampering their use to forecast millions of TS in practical scenarios. We propose N-BEATS(P), a global parallel variant of the N-BEATS model designed to allow simultaneous training of multiple univariate TS forecasting models. Our model addresses the practical limitations of related models, reducing the training time by half and memory requirement by a factor of 5, while keeping the same level of accuracy in all TS forecasting settings. We have performed multiple experiments detailing the various ways to train our model and have obtained results that demonstrate its capacity to generalize in various forecasting conditions and setups. Introduction In the past few years, abundant evidence has emerged suggesting that deep neural networks (DNN) constitute an effective modeling framework for solving time series (TS) forecasting problems. DNN models have been shown to produce state-of-the-art forecasts when large homogeneous datasets with multiple observations are available [1]. The success of DNN is largely accounted for by two factors: (i) the cross-learning on multiple time-series 1 [3,4,5] and (ii) the use of over-specified large capacity ensemble models 2 . However, the high computational requirements of such models in comparison to statistical models have raised concerns regarding their applicability in practical scenarios [3]. Indeed, the deployment of a reliable DNN with an automatic training procedure is far more challenging because of this cost and other factors such as optimal architecture and hyperparameter tuning which various authors discused in previous studies [10,11]. These factors can be summarized by the following. The need to render these methods more efficient has been pointed out multiple times [3,12] and is one of the core challenges that must be solved to democratize their use. Currently, they require much time, specialized hardware and energy to train and deploy. Besides their model size, which can render their use cumbersome, re-training these models at every forecast for different TS is not viable for most organizations given their training time. Reducing memory requirements and computational cost, as well as offering model that are "ready-to-use" once trained are key aspects to improve upon to make these model more accessible to smaller organizations who neither have the money nor the data to support frequent retraining. Only recently has some work been done to evaluate how to generalize these models to multiple types of TS when trained on public datasets that cover various TS settings while maintaining an acceptable level of accuracy within a zero-shot regime [13], i.e. to train a neural network on a source TS dataset and deploy it on a different target TS dataset without retraining, which provides a more efficient and reliable solution to forecast at scale than its predecessor even in difficult forecasting conditions, or in a few-shot learning regime, i.e., by fine-tuning the model to the target dataset of interest [14,15]. In the ensemble case, the cost of runing these models is amplified since most of the current top-performing models rely on independent training of ensemble 1 Cross-learning is the approach were a single model is trained on multiple TS. The model assumes that all TS follow the same process and that each TS are independent samples of this process. A notable instance of such model is the FFORMA model [2]. 2 Over-specified large capacity ensembles refers to ensemble of models (DNN here) where each model have high number of parameters, perhaps larger than what would be needed to get a good training error. We refer the reader to recent empirical [6,7] and theoretical [8,9] evidences that indicates that larger networks may indeed be easier to train to achieve better results. members. Producing forecasts with a small ensemble size without affecting accuracy is of great interest for smaller organization. On the other hand, there are plentiful examples of successful deployment of neural networks in large-scale TS forecasting. It appears that the benefits of using such models in practice definitely outweigh the associated costs and difficulties [16]. First, such models are scalable: a neural TS forecasting model usually performs better as the scale of the data used to train it increases. This has been observed in various TS competitions [3,17,18]. Second, they can be reusable: we can reuse a model to produce forecasts over multiple TS [13] not observed during training faster and produce forecast in real-time. They also offer flexibility: It is typicaly easier to adjust a DNN-based model for handling missing values [19], adjusting its parameters based on custom business/scientific objectives [20] or considering multi-modal (various source and representation of data such as text, video, etc.) within a single end-to-end model [21]. These benefits made neural TS forecasting models popular and even mainstream in various settings. In fact, the prevalent use of neural networks manifests a paradigm shift in data-driven forecasting techniques, with fullyautomated models being the de-facto standard in organizations that that can afford DNN based forecasting workflows. Many examples of DNN for TS exist. Some of the largest online retail platforms are using neural networks to forecast product demand for millions of retail items [22,23]. AutoML solutions with heavy use of DNNs like [24] are being used in various settings and have been demonstrated to be very competitive [3] with almost no human involvement. Some companies that need to allocate a large pool of resources in different environments are using neural networks to anticipate required resources for different periods of the day [25,26]. Large capital markets companies are using neural networks to predict the future movement of assets [27] via a process that links the trade-generating strategies with notifications and trade automation from these forecasts. 3 However, these approaches are often not accessible to smaller organization because of their cost to opperate. We propose to tackle the problems within a single approach to facilitate the use of DNN for TS forecasting at scale. Related Work TS forecasting models: Traditional local, univariate models for TS forecasting include the autoregressive integrated moving average (ARIMA) model [30], exponential smoothing methods (HOLT, ETS, DAMPED, SES) [31,4] decomposition-based approaches, including the THETA model [32], and autoregressive (AR) models with time-varying coefficients as in [33,34]. Global univariate TS models that rely on deep neural networks (DNNs) have been proposed recently as alternatives to these models such as DEEP-STATE [35], DEEP-AR [22] and more recently Transformer -based models [36,37]. In contrast to the traditional approaches, they can be trained with multiple independent TS simultaneously and handle non-stationary TS without preprocessing steps. One of the key difference between these two class of model is are how they approach the forecasting problem. Traditonal models typically learn from TS locally, by considering each TS as a separate regression task and fitting a function to each (local model) whereas DNNs do so by fitting a single function to multiple TS (global model) [16]. Some concerns have been raised regarding machine learning (ML) publications claiming satisfactory accuracy without adequate comaprison with the well-established statistical baselines and using inappropriate criteria often leading to misleading results [38]. It is inspiring to see that recent ML publications such as [35,22,39] have largely solved these problems by following more rigorous evaluation protocols and baseline comparisons. Ensemble methods: Combining multiple models is often a more straightforward strategy to produce accurate forecasts than finding the best parameterization for one particular model [40,41]. Recently, both M4 and M5 forecasting competitions have empirically confirmed the accuracy of ensemble methods [17,3]. Notable instances of model for univariate TS forecasting include that use this method FFORMA [2] (second entry in M4), ES-RNN [42] (first entry in M4) and subsequently N-BEATS 4 [39]. Because they use en-sembling, these models, especially N-BEATS, have high computational and memory complexities, which require specialized infrastructure to accelerate their training and store the trained models [4]. For example, the full N-BEATS models consists of 180 individual models. It takes around 11'755 hours to train on the full M4 dataset using 1 NVIDIA GTX 2080Ti GPU. Furthermore, the total size of the models in ensemble is 160 GB, which, depending on the number of training logs and saved snapshots of the model, can increase to over 450 GB. In comparison, the Theta method takes around 7 min to do the same. N-BEATS: The overall N-BEAT model [39] is designed to apply signal decomposition of the original TS similar to the "seasonality-trend-level" approach of [43] but using a fully connected neural networks organized into a set of blocks. Each blocks applies a decomposition of the signal it is given as input and make a forecast from this signal and pass the reminder of the signal to the other block. Beside the parmaterization of each block, one as to specify the number of past observations all blocks must consider which we refer as the lookback windows. When trained on large datasets, N-BEATS is trained with a bagging proeceduce [44] on various lookback windows, losses, and subpopulations to produce an ensemble of models. All of these models are independently trained and inflate the parameter size of the ensemble and thus the time to train the complete model. Hence, the major issues of using these model at scale come down to parameter size of the model, time to train the model and whether or not we can offset the operating costs of DNN based forecasting workflows for ensemble models. This paper seek to reduce the computational complexity gap between classical and neural TS models by proposing a more memory-and computation -efficient version of the N-BEATS model [39]. Our approach achieves this by re-formulating the original fully-connected N-BEATS architecture as a single kernel convolution, which allows for training multiple models, each with different lookback windows, in parallel on the same GPU while sharing most of the parameters in the network. This leads to reduced ensemble training time and memory footprint as well as reduced ensemble model size, which positively affects the costs of training, querying and storing the resulting ensemble without compromising its accuracy. Our contributions can be summarized as follows: [1] We introduce N-BEATS(P), a multi-head parallelizable N-BEATS architecture that permits the simultaneous training of multiple global TS models. Our model is twice as fast as N-BEATS, has 5 times fewer parameters than its predecessor, and performs at the same level of accuracy on M4 dataset than N-BEATS and generalize well in other TS forecast condition. [2] Our model is faster to train and more accurate than the top-scoring models of the M4 competition (ES-RNN [42] FFORMA [2]). The remainder of this paper is organized as follows. Section 3 describes the univariate TS forecasting problem. Section 4 presents our modeling approach. Section 5 outlines empirical evaluation setup and our results. Finally, Section 6 presents our conclusions. Problem Statement We consider the univariate point forecasting problem in discrete time where we have a training dataset of N TS, D train = {X T i +1:T i +H } N i=1 . The task is to forecast future values of the series, Y (i) T i +1:H ∈ R H , given a regularly-sampled sequence of past observations, X (i) 1:T i ∈ R T (i) . We use the bold notation to define vectors, matrix and tensor. To solve the task, we define a forecasting function f θ : R l → R H , parameterized with a set of learnable parameters θ ∈ Θ ⊂ R M where l ≤ T i The parameters of the forecasting function can be learned using an empirical risk minimization framework based on the appropriate samples of forecasting function inputs, Z in ∈ R l , and outputs, Z out ∈ R H , taken from the training set: θ = arg min θ∈Θ Z in ,Zout∈D train L(Z out , f θ (Z in ))(1) A few remarks are in order regarding the selection of the model input window size l. The optimal choice of l is highly data-dependent. In terms of general guidelines, TS with a swiftly changing generating process [45] will favor small values of l, as historical information quickly becomes outdated. TS with long seasonality periods will favor larger l, as observing at least one and maybe a few seasonality periods may be beneficial for making a more informed forecast. Obviously, several conflicting factors can be at play here and finding a universally optimal solution for all TS does not seem viable. Therefore, l can be treated as a hyperparameter selected on a TS-specific validation set. A more productive and accurate solution would entail using an ensemble of several models, each trained with its own l, as in [39]. However, this solution tends to inflate the ensemble size, and that is the problem we aim to address in this paper. In general, increasing the diversity of an ensemble [46] with different forecasting models usually results in the inflation of the ensemble size and computational costs. Therefore, we focus on providing a solution to more effectively parallelize training of the N-BEATS ensemble, which is obviously applicable to situations other than using multi-l ensembles. Model Model architecture The basic building block of the proposed model has a multi-head architecture and is depicted in Fig. 1 (left). Each -th block can take as input up to W input signals x ( ) lw ; w ∈ {1, . . . , W } of the same TS with different lookback windows l = [l 1 , · · · l W ], and generates two output vectors for each of the input signals provided: the backcast signalx ( ) lw of length l w and the forecast signalỹ ( ) lw of length equals to the forecast horizon H. We set each l w to a multiple of H ranging from 2H to 7H.x ( ) lw is fed to the next block for its input andỹ ( ) lw is added to the previous forecast from the previous block. Internally, the basic building block is divided into four parts. The first part consists of W independent FC input layers that project the signal into a fixed higher-dimensional representation z ( ) lw ∈ R + . This is done by mapping the w-th model with φ w : R lw → R + such as z ( ) lw = FC lw (x ( ) lw ). To achieve parallelization in practice, we do this mapping with φ w : R L → R + where L = max(l) instead and pad missing values of the W − 1 signals with 0 that have smaller lookback windows. We set the padding to 0 for the missing values of the lookback and and make sure the FC doesn't have bias to guarantee obtaining the same result as mapping W times the input of each models sequentialy with their respective φ w . Figure 1: Illustration of the proposed model. The basic block consists of multi-head and multi-output fully connected (FC) layers with ReLU non-linear activations, where some layers are shared between the W models. Each block input X l ∈ R N ×L×W contains the same input signal at different lookback windows l 1 · · · l W , where for each of the W representations of the signal, missing values are padded with 0. The multi-output part of the block consists of W independent layers (represented by the blue cube in the figure) that predict basis expansion coefficients both forward θ f lw (Forecast) and backward θ b lw (Backcast) for each of the W models. A stack can have layers with shared g b lw and g f lw . Forecasts are aggregated by summing over all partial forecasts of each block, enabling us to retrieve which block had the most impact in making the forecast. Parallelization is achieved by forcing head layers of each block to have the same input size, by using mask layers in the input layer to consider only the T w first observations of input signals and reshaping the tensor to force computation in parallel instead of sequentially applying computation in a loop for each of the W models. The second part consists of a shared FC stack that takes as input the TS representation produced in the first part and outputs forward θ This approach allows us to parallelize the computation of the forecast by considering an input X ∈ R N ×L×W and producing outputỸ ∈ R N ×H×W and obtain W forecasts for N TS simulatenously for each of the W lookback windows. These opperations are repeated iteratively over all blocks across all stacks of the model. Thus, the computation of the forecast and backcast for the -th block given the w-th signal, is described by the following equations: z ( ) lw = FC(FC(· · · (FC ,lw (x ( ) lw ) (2) θ f,( ) lw = FC f lw (z ( ) lw ) (3) θ b,( ) Tw = FC b lw (z ( ) lw ) (4) y ( ) lw = g b,( ) lw (θ f,( ) lw ) = dim θ f,( ) lw i=1 θ f,( ) i,lw v f,( ) i,lw(5) x ( ) lw = g f,( ) lw (θ b,( ) Tw ) = dim θ b,( ) lw i=1 θ b,( ) i,Tw v b,( ) i,Tw(6) Here, FC corresponds to a fully connected layer with ReLU non-linearity activation [47], and v f,( ) i,lw and v b,( ) i,lw are forecast and backcast basis vectors for the -th block. These vectors can either be chosen to be learnable parameters or can be set to specific functional forms that are fixed prior to training the model. In Eq. 6, the number of time FC is applied is based on the number of layer and is part of the specification of the model. Eqs.2-6 are then repeated iteratively for all blocks, following the same architecture topology as N-BEATS [39]. The individual blocks are stacked using two residual branches. The first branch, illustrated in Fig. 1 (middle), runs over the backcast signal produced by each block and iteratively decomposes the initial TS signal such that the subsequent block consider the residual of its preceding block. The second branch, illustrated in Fig. 1 (right), aggregates the partial forecast of each block. These operations are described by the following equations: x ( +1) lw = x ( ) lw −x ( ) lw (7) y lw = ỹ ( ) lw(8) Generic and Interpretable Model Version Multiple versions of this approach can be provided to parameterize each of the W models. For instance, both the generic and interpretable versions of N-BEATS proposed in [39] are compatible with our model. We will briefly describe these two extensions; we refer the reader to the original paper for more details [39]. The generic architecture: in this version, g b lw and g f lw are specified as a linear projection of the previous layer output such that the outputs of the -th block are described as follows: The interpretable architecture: Similar to the traditional TS decomposition into trend and seasonality found in [43,48], trend and seasonality decomposition can be enforced in V f,( ) lw and V b,( ) lw . [39] proposed to do this by conceptually separating the set blocks into two stacks such that one stack of blocks is parameterized with a trend model (T ) and the other with a seasonal model (S). All block in a stack shared the same parameters. The trend model consists of constraining the basis function to modelize a trend signal, i.e., using a function polynomial of small degree p as follows: y ( ) lw = V f,( ) lw θ f,( ) lw + B f,( ) lwx ( ) lw = V b,( ) lw θ b,( ) lw + B b,( ) lw (9) where V f,( ) lw ∈ R H×dim θ f,( ) lw , B b,( ) lw ∈ R H and V b,( ) lw ∈ R L×dim θ b,( ) lw , B b,( ) lwy ( ) lw = g f,( ) lw,trend (θ f,( ) lw ) = Tθ f,( ) lw ; T = [1, t, · · · t p ](10) where T is a matrix of powers of p. Thus the waveform extracted will follow a monotonic or a slowly varying function. The seasonal model constrains the basis function to modelize periodic functions, i.e, g f,( ) Tw (θ f,( ) lw ; V f,( ) t,lw ), using Fourier series as follows: y ( ) lw = g f,( ) lw,seas. (θ f,( ) lw ) = Sθ f,( ) lw ;(11)S = [1, cos(2πt, · · · , cos(2π H/2 − 1 t), sin(2π H/2 − 1 t)] Thus, by first (1) applying the trend model and then (2) lw . In any configuration of the model, estimating the parameters of the model problem is done by maximum likelihood estimation (MLE). To simplify the notation, we consider eq. 12 as the function that establishes the forecast, where θ NBEATS is the set of all parameters of each block and x i lw is the i-th TS considered with input size of length l w . y lw Y i = NBEATS(x i lw ; θ NBEATS )(12) Thus, optimizing the model consists of optimizing eq. 13. We use a stochastic gradient descent optimization with Adam [49] over a fixed set of itterations and a three-steps learning rate schedule. Here L(N BEAT S(x n Tw ; θ NBEATS ), y (n) ) corresponds to some metric function that measures the quality of the forecast to the ground truth Y. Note that we combine the losses of the forecasts of all models, using the mean values to promote cooperation between the different models. Following the same training framework as [39], we used the MAPE, MASE and SMAPE losses to build the ensemble, all of which are detailed in the following section. We refer the reader to [39] for design choice of the model and a exhaustive discussion on the parameter choice of this model. In our work we reuse the same set of parameters and do not apply hyper-parameters search at the exception of the yearly TS where our model converge to a stable results earlier (10k itterations instead of 15k). θ * NBEATS = argmin θ * NBEATS 1 N N i=0 1 W W w=1 L(NBEATS(x i lw ; θ NBEATS ), y i )(13) Experimental setup We conducted the experimental evaluation of the forecasting methods on 6 datasets which include a total of 105'968 unique TS when combined and over 2.5 million forecasts to produce on these TS. We report the accuracy of our model on the first and dataset and consider the rest to assess the model ability to generalize in other settings. We details all datasets here and report our generalization results on zero-shot forecasting in Appendix C. The datasets are the following: (1) (public) M4: 100'000 heterogeneous TS from multiple sectors that include economic, finance, demographics and other industry used in the M4 TS competition [17,4]. (2) (public) M3: 3003 heterogeneous TS from derived from mostly from financial and economic domains [50]. (3) (public) Tourism: 1311 TS of indicators related to tourism activities sampled monthly, quarterly and yearly [51,52]. (4) (public) Electricity: 370 TS of the hourly electricity usage of 370 customers over three years [53,54]. S. financial markets, each covering different types of asset classes including stocks, bonds, commodities, currencies and market indexes, or a proxy for a market index covering a larger set of financial asset than the dataset used in [55]. For the M4, M3 and Tourism datasets, target TS trajectories were specified by the competition's organizers with each subpopulation of TS with the same frequency (Hourly, Quarterly, etc..) having its own horizon ( We compared the forecast accuracy of our approaches with the reported accuracy of other TS models in the M4 TS competition including FFORMA [2] and ES-RNN [42]. In reporting the accuracy of these models, we relied upon the accuracy and the pre-computed forecasts reported in their respective original paper. The statistical models were produced on R using the forecast package [56] and we measured the training time to train and produce each forecast of our model as well as the Theta method. We also relied upon the reported running time of the implementation provided in [4]. Finally, all models were compared on a naive forecast, i.e., a random walk model or a seasonally adjusted random walk, that assumes all future values will be the same as the last known one(s). This was done to assess whether the forecasts of these models are accurate in the first place. M AP E(x, x) = 100 H H i=1 |x T +i − x T +i | x T +i (14) M ASE(x, x) = 1 H H i=1 |x T +i −x T +i | 1 T −m T t=m+1 |x t − x t−m | (15) SM AP E(x, x) = 200 H H i=1 |x T +i −x T +i | |x j |+|x T +i | (16) OW A(x, x) = 1 2 SM AP E SM AP E N AIV E2 + M ASE M ASE N AIV E2 (17) N D(x, x) = H i=1 |x T +i − x T +i | H i=1 |x T +i | (18) M DA(x, x) = 1 H H i=1 sign(x T +i − x T ) = sign(x T +i − x T )(19) We evaluated the forecast accuracy using 8 standard TS metrics: the mean absolute percentage error (MAPE) used in the Tourism compeition [51], the mean absolute scaled error (MASE) [57], the scaled mean absolute percentage error (SMAPE) used in the M3 competition [50], the normalized deviation (ND) used in [22] and the mean directional accuracy (MDA). Additionally for the M4 competition, we evaluated the model on the overall weighted average (OWA) between the SMAPE and the MASE such that a seasonally-adjusted naive (NAIVE2) forecasting model obtains a score of 1.0 [4]. For instance, an OWA of 0.90 means that the forecast is on average 10% better than a NAIVE2 model with respect to both the SMAPE and MASE metrics. The MDA measures the model's ability to produce forecasts where the trajectory follows the actual change of the TS relative to the last known value: the higher the MDA is, the better a model predicts the trend of a TS at any given time. For all other metrics, the lower the value, the better a model predicts the TS. For the M4 dataset, we only consider the OWA, the MASE and the MDA. The other metrics were used in order to compare ourselve with other methods and other datasets as detailed in Appendix C. Eq. 14-19 describes how these metrics are computed.x is the forecast, x is the ground truth and m is the time interval between successive observations considered by the organizers for each data frequency, i.e., 12 for monthly, four for quarterly, 24 for hourly and one for yearly, weekly and daily data. Without loss of generality to previous equatios T is the number of point in-sample observed to make the forecastushe and H is the forecast horizon. We present the results of the baseline and benchmark accuracies for the M4 dataset in Table 1. The table gives the reported accuracy of N-BEATS reported in the original papers [39], the replicated results using the publicly accessible implementation provided by the original authors along with their scaled versions [13] based upon their implementation and our model NBEATS(P). Three main conclusions can be drawn: Baseline and Benchmark (1) Scaling TS to allow generalization on other datasets for the N-BEATS model, as presented in [13], adds a penalty on the OWA metrics for the M4 dataset, which suggests that there is a trade-off between accuracy and generalization on other datasets for DNN-based models. (2) Figure 2 details how sensemble size has an impact on computational time to train. It can be seen that applying a bagging procedure [44] 3 to 4 times is sufficient to get an accurate ensemble for both the NBEATS and NBEATS(P) model but NBEATS(P) is more efficient the larger the ensemble size is. (3) The top-performing models do not differ significantly with respect to the coverage of the TS forecasted and the mean directional accuracy (MDA). This provides an argument that if one is mainly interested in predicting the TS variation from the forecast origin, relying on the fastest implementation of the top-performing models for a first initial prediction is a cost-effective solution. . Regarding (2), we illustrate this phenomenon in fig. 3 by plotting the TSNE embedding of each series of M4 computed from the same set of features used in the FFORMA [2] by comparing the top performing model with the N-BEATS(i) model by coloring each TS with its individual MASE accuracy. We refer the reader to table.A.5 and [2,59] for a detailed overview of the 42 features used and their interpretation. Appendix A details the distribution of these features over all datasets we considered. Note that there are no substantial differences between the approaches, despite some subtle regions of the graph where we can observe N-BEATS(i) performing better overall than ES-RNN. Table 2 presents the time to train each model accuracy as well as the average pairwise absolute percentage error correlation of the forecast residuals between ensemble members, in a way to similar to the experimental evaluation of M4 submission performed in [64]. At first glance, we see that the training time of these models be as long as multiple weeks. N-BEATS models timing are reported with a single GPU. In comparison to N-BEATS, N-BEATS(P) takes less time regardless of the variant used but the gain is observed especially Table 1: Averaged forecasting results of the M4 competition for the evaluated models. The OWA metric is presented for each seasonal pattern observed. Forecasts from models in italics were pre-computed except for the N-BEATS models. We replicate the results with the implementation provided by the authors, e.g: N-BEATS (I) (original) vs N-BEATS (I) (our). MLP ' and RNN ' models are appended with "'" to signify that these model were trained per TS using a seasonal and trend decomposition with manual pre-and post processing steps [4]. We also considered a coverage indicator which measures the number of series that a model forecasts better than an arbitrary MASE accuracy threshold of τ = 1.0. We also added the MDA of the forecast. Training Time and Number of Parameters for the generic architecture. Both the original approach and ours are at the same level of correlation and, while ours is slightly less diverse, it is roughly twice as efficient and achieves the same level of accuracy. We can observe that using the scaled version has little impact in terms of diversity. In our preliminary result, we also observed that there was no significant difference in terms of the TS samplers used to train N-BEATS(P) where for instance, different TS were sampled for the W model. In practice, computational time remains more than significant for training these models on a single GPU. However, we can speed up the training by training model simulatenously. If we consider N-BEATS(G) vs N-BEATS(P, G), each ensemble member trained on a single TS frequency takes on average 12 & 27 minutes, and at worst, 19 & 57 minutes respectively. N-BEATS(G)'s time is for a single lookback and thus requires 1080 (6 lookback × 3 losses × 6 frequencies × 10 repeats) independent models to be trained whereas N-BEATS(P, G) requires only (3 losses×6 frequencies×10 repeats) independent models. Thus, if one would have access to 1080 GPUS, the total training time of N-BEATS(G) could be done in 20 min, but this is an unrealistic amount of ressources for most organization. Our approach cuts down that cost: given 10 GPUs, the N-BEATS(G) and a greedy schedudling of model training, it would take roughly 35h to train whereas our would only take 16h with all 10 repeats. Using 20 gpus, our model would achieve it in 8 hours whereas the previous model would take 18. When forecasting larger amount of TS, say 1 billions monthly distinct TS, estimating the cost can be difficult. We can make a reasonable assumption that the computational time required to train a model scales linearly with the number of time series to forecast altought it takes roughly the same to train on different subpopulations or the other based on the number of itterations. Hence assuming we are using the same 3 losses and bagging procedure and it takes a single model to train the monthly TS where the number of itterations required linearly scaled from the one used in M4 (see B.9) and require 75k itterations instead, it would take 1878h and 1551h for the generic and interetable version of our model on 30 GPUS. Altough a larger dataset may benefit from a deeper & wider model further inflating the cost, the number of itterations might not need to be this high. However, our approach will result in similar gain in the scenario of deeper & wider model. Regardless, training these model for everyday usage requires a lot of computational ressouce. In order for the cost to be kept low at this scale, training would need to be done less frequently and models would have to remain outdated to some extent as recent trends and structural changes in the data wouldn't be used to update the model parameters. To further show the performance of our model, we show in Table. 3 that one could have achieved the same average accuracy as the top M4 competitions entries by training our for approximatively 2h on a single GPU without bagging. Thus, even with minimal amount of ressources, smaller organizations can train 54 DNN-based models, each on TS of different freqencies, losses and lookback windows very fast making our model far more accessible to small organization who doesn't have dozens of gpus available. Since the training procedure of our approach takes roughly a fixed amount of time to train regardless of the number of TS to forecast 5 , forecasting more TS might requires more itterations and/or more parameters for the model to capture the dynamics of these additonal TS. Therefore the training time is expected to increase the more TS we want to forecast in the training regime. However, the overall training time doesn't increase linearly with the number of TS to forecast as increasing the number of itteration is a fixed cost and the number of itterations to train Quarterly (24K TS) or Monthly (48K TS) was the same in our setting. This leaves, the total number of itterations to train all models, the forecast horizons, the number of parameters and the number of models trained simultanously to have effect on training time. The difference in improvement factor between parallelized generic and interpretable versions of N-BEATS(P) is due to the hidden layer sizes between the two versions. Having a higher number of hidden neurons reduce the computational gain of training multiple models conjointly as it saturate GPU usage. If we have a sufficiently expressive model without requiring too many hidden neurons, N-BEATS(P) is expected to produce accurate forecasts at a fraction of the cost. Otherwise the gain will be diminished. Regardless, these results show that ensemble diversity and accurate forecasts could have been achieved with reduction in resources and computation time. Given the increasing trend of top-performing models requiring ever more Conclusion We proposed an efficient novel architecture for training multiple TS models conjointly for univariate TS forecasting. We empirically validated the flexibility of our approach on the M4 TS datasets as well as assessing its generalizability to other domains of application, using 5 other datasets which, combined, cover over 2.5 million forecasts. We provided forecasts in various TS settings at the same level of accuracy as current state-of-the-art models with a model that is twice as fast while requiring 5 times fewer parameters than the top performing model. We highlighted both stylized facts and limitations of the performance of the model studied, in an effort to provide insights to TS practitioners for operating DNN-based models at scale. Our results suggest that training global univariate models conjointly by sharing parts of their parameterizations yield competitive forecasts in a fraction of the time and does not significantly impair either forecast accuracy or ensemble diversity. and Traffic datasets exhibit multiple seasonal patterns, whereas datasets like Finance exhibit large difference from other datasets in terms of high order autocorrelation (x_acf10), autoregressive conditional heteroscedasticity (archlm, garch_r2), strength of trend (trend) and high variance of the mean of observation from non-overlapping windows (stability). All sampled TS from all datasets are summarized in Fig. A.5 using the T-SNE algorithm [69]. Each point of this graph correspond to the 2-dimensional embedded space of a single TS computed from the same set of endogenous statistical features [2] . One can observe the heterogeneity of these datasets and note that the subpopulations of TS within a dataset can have high variance in their statistical properties while being similar to other subpopulations of other datasets. When considering the Finance dataset, we can see how the [59]. Default values when test failed was 0. The M4 7 dataset is a publicly accessible dataset that contains a large set of 100'000 heterogenous TS sampled from the ForeDeCk database for the M4 competition [17]. The database is compiled at the National Technical University of Athens and is built from multiple diverse and publicly accessible sources. It includes TS frequently encountered in business domains such as industries, services, tourism, imports/exports, demographics, education, labor & wage, government, households, bonds, stocks, insurances, loans, real estate, transportation, and natural resources & environment. TS were sampled at different frequencies [Yearly, Quarterly, Monthly, Weekly, Daily and Hourly] each with different forecast horizons, i,e, [6,8,18,13,14,48] according to the competition organizer. Table A.6 outlines the composition of the M4 dataset across domains and forecast horizons. All TS were provided with a prepossessing scaling procedure to ensure positive observed values at all time-steps with minimum observed values greater than or equal to 10. The scaling was applied only to sampled TS whose minimum oberved value was smaller than 10 by adding a per-TS constant to all TS to ensure that the minimal values was positive. All other TS were unaltered by any preprocessing step. The dataset was subdivized into a training and a test dataset by the M4 TS competition organizers. For further details on this dataset, we refer the reader to the following: [4,17]. We relied on the pre-computed forecasts and PI available at https: The Tourism 9 dataset is a publicly accessible dataset that contains TS collected by [51] from tourism government agencies and academics who had used them in previous tourism forecasting studies. The TS of this dataset are highly variable in length. It includes yearly, quarterly and monthly TS. Table. A.8 details the proportion of TS from each frequency. For further detail on this dataset, we refer the reader to [51]. This dataset was considered for zero-shot forecasting, to examine a case where the target dataset comes from domains that are not present in the M4 dataset. / Appendix A.4. Electricity and Traffic Datasets Details Electricity 10 and Traffic [70] are two publicly available datasets from the Univeristy of California Irvine Machine Learning repository. The Electricity dataset contains the hourly electricity usage monitoring of 370 customers over three years, with some clients being added during the the observation periods creating cold-start conditions for producing some forecasts. The Traffic dataset contains TS of the hourly occupancy rates, scaled in the (0,1) range for 963 lanes of freeways in the San Francisco Bay area over a period of slightly more than a year. Both of these dataset exhibit strong seasonal patterns due to their nature and are mostly homogeneous. These two TS datasets are used de facto to evaluate the quality of DNN-based TS models as in [22,35,71]. We included these two datasets as a sanity check for zero-shot forecasting, to ensure that zero-shot forecasts were accurate in a setting where it is relatively easy to produce accurate forecasts. Appendix A.5. Finance dataset The Finance dataset contains daily closing prices of U.S. MFs and ETFs observed between 2005-07-01 and 2020-10-16 and traded on U.S. financial markets, each covering different types of asset classes including stocks, bonds, commodities, currencies and market indexes, or a proxy for a market index. The dataset was obtained through three data providers: (1) Fasttrack 11 , a professional-grade data provider for financial TS, (2) Yahoo Finance API and (3) the Federal Reserve of Saint-Louis (FRED) database. Part of this dataset is proprietary, so we do not have permission to share that part publicly. However, the list of securities is given in Table. D.13 to help interested readers reconstruct the dataset from public data sources. We considered this dataset in our zero-shot experiments by sampling the TS at three different frequencies [dDaily, weekly and monthly] and specifying the same forecast horizon as that of the M4 dataset. We used this dataset to present a worst-case scenario for zero-shot. First this is a case where the forecasting application is notorious for its forecasting difficulty. Moreover, the source dataset on which we train our model has at most 10K TS to train from and at worst 164 TS, which force zero-shot generalization with very few training data. Also, by sampling the TS at large scale, we emulated how zero-shot could be applied on the whole history of the TS, similar to the procedure carried out by portfolio managers and quantitative analyst to backtest the validity of their investment strategies. The TS were split into We used the same overall training framework as [39] including the stratified uniform sampling of TS in the source dataset to train the model. Training N-BEATS and N-BEATS(P) was done by first segmenting the training dataset into non-overlapping subsets based on the TS frequency they were observed in. Then, independent training instances were trained, one each group by specifying the forecast horizon of each instance based on the common forecast horizon of the subset. Table B.9 presents the HP settings used to train all N-BEATS and N-BEATS(P) models on the different subsets of M4. Except for the number of iterations on the yearly TS, all other HPs are the same. We did not proceed with an exhaustive parameter search since this was not the focus of our work. We were interested in whether or not we could make the N-BEATS model model faster and more usable in practical scenarios. For zero-shot application, we relied on the scaled version of each model, i.e. where the TS is scaled based on its maximum observed value over its lookback periods. With one exception, the model trained on a given frequency split of a source dataset is used to forecast the same frequency split on the target dataset. The only exception is follows: when transferring from M4 to M3, the Other subpopulation of M3 is forecast with the model trained on the Quarterly subpopulation of M4. Table B.10 describe the different zero-shot training regimes on which the model was trained on the source dataset. The source code to replicate the experiments for both traditional forecasting regime and zero-shot forecasting of Appendix C is available at: https://anonymous.4open.science/r/actm-7F90. Appendix C. Zero-shot forecasting To test whether our model can generalize to other datasets, we evalate its capacity to support zero-shot TS forecasting, i.e., to train a neural network on a source TS dataset and deploy it on a different target TS dataset without retraining, which provides a more efficient and reliable solution to forecast at scale than its predecessor. We present a flowchart of the zero-shot forecasting regime in fig. C.6. In this setting a single model is trained once on a source datasets and can be used to forecast multiple target datasets without retraining as in [13]. We demonstrate that N-BEATS(P) has comparable level of accuracy than N-BEATS for zero-shot generalization ability in various settings. It can operate on various domains of applications and on target datasets that are out-of-distribution of the source dataset it was trained on, i.e. on dataset from other dommains, settings and/or that have different statistical properties than the dataset it was trained on. We evaluated their performance in the zero-shot regime on all other datasets (M3, Tourism, Electricty, Traffic, Finance) by training models on the M4 dataset only using scaled TS as in [13]. The reason for this preprocessing step was to prevent catastrophic failure when the target dataset scale is different from that of the source dataset. We tested 3 different setups for zero-shot forecasting, which we denote by R O , R SH,LT and R SH . R O is a setup where we use the same model to produce results on M4 (Table 1) and apply it on the target dataset. This required us to truncate the forecast or apply the model iteratively on the basis of previous forecasts to ensure the forecast size is the same as the target dataset. The model was not trained to operate when this condition occurs. R SH is a setup where the model is trained with the same number of iterations as R O but we specified the model's forecast horizon to be the same that of the target datasets. R SH,LT is the same training regime as R SH , but we allowed the model to consider TS samples from further in the past during training and trained the model with more iterations. The rationale of testing these training setup is to evaluate the impact of training the model longer for generalization and to to test the model in forecast condition it wasn't trained to do (e.g. in R O when the forecast horizon of the target dataset exceed or is inferior to the forecast horizon of the target dataset). When training the model, we consider an hyper-parameter L H which is a coefficient defining the length of training history immediately preceding the last point in the train part of the TS that is used to generate training sample. This coefficient multiplied by the forecast horizon determined the maximum number of most recent points in the train dataset for each TS to generate training sample. The higher that coeffcient, the further in the past we consider TS observation to train this model. To produce forecasts, we used the subset of the ensemble models trained on the same TS frequency to produce the multiple forecasts and combined them by median aggregation. Detailed explanations of this aggregation, selection of L H and the ensemble parameters used are given in Appendix B. We compared against statistical baselines and other ML models such as DEEP-STATE [35], N-BEATS [39,13], DEEP-AR [22], FFORMA [2], ES-RNN [42], Deep Factors [72] and many others including statistical baselines already evaluated on theses datasets. In reporting the accuracy of these models, we relied upon the accuracy and the pre-computed forecasts reported in their respective original paper. (Table. 1), which required to truncation of the forecast or applying the model iteratively at the basis of previous forecasts to ensure the forecast size was the same that of the target dataset. R SH is trained in the same fashion as R O but we specified the model's forecast horizon to be the same as that of the target datasets. R SH,LT is the same training regime as R SH except that the model is allowed to consider TS samples from further in the past while training: See Table. B.10 for more detail. Results for models with * appended to their names are replicated from the original papers and signifies an anonymous submission for which we do not know the methodology. (2) Comparing with [13], where a different training regime was used, the difference between their results and ours highlights the importance of the optimization procedure to facilitate transfer to another dataset. In certain cases, some datasets (e.g., Tourism, will benefit from a longer training, to the detriment of the forecast accuracy on the source dataset. In other cases, like the Electricity dataset, no adjustments are required between the source and the target dataset. (3) The case of the Tourism dataset highlights the importance of ensuring that the forecast horizon of the source dataset used to train the model is longer than or equal of the target dataset; this is a key factor in producing reliable zero-shot forecasts. Considering that the M4 dataset includes a large number of heterogenous TS that contain at least some TS with similar statistical properties to those present in the target dataset, zero-shot forecasting can be easily deployed with a pre-trained DNN model and can produce initial forecasts that are on par with the level of accuracy of multiple baselines, and sometimes benchmarks, very quickly at a minimal cost. However, not all settings will benefit equally from this approach. The Finance dataset is a prime example of a setting where zero-shot forecasting produce mixed results. In this setting, the source TS dataset has very few TS to train from in comparison to the test sets. Also, these TS are very difficult to forecast in a univariate setting since they are almost all non-ergodic, heteroscedastic, and have high noise-to-signal ratio. Despite these added difficulties, both N-BEATS and N-BEATS(P) can produce forecasts at a comparable level to a single statistical model in term of MDA when using the R O training setup. However, the zero-shot regime achieved forecasts better than a naive one only by sampling these TS at a monthly frequency, which coincidentally is the largest pool of TS in M4 (48'000.) In comparison, the daily (3594) and weekly (227) subsets contain fewer TS. Hence, even under poor conditions of application for zero-shot N-BEATS(P), we can still produce preliminary forecasts quickly. These results highlight the importance of selecting a good source dataset but even in subpar conditions, our approach can still generalize well with respect to the MDA metric. T i } N i=1and a test dataset of future values of these TS D eval = {Y (i) predictors of expansion coefficients for each of the W lookback periods.The third part consists of the W independent backward g ∈ R L are basis vectors learned by the model, which can be taught as waveforms. Because no additional constraints are imposed on the form of V f,( ) lw the waveforms learned do not have inherent structure on how they should look. applying the seasonal model within the doubly residual stacking topology of the model, we obtain a model that applies TS component decomposition in a similar way to than traditional decomposition approaches. Basis functions are a generalization of linear regression where we replace each input with a function of the input. Here, the polynomial and the Fourier series are functions that model uses the trend and seasonality and take as input the embedding computed from the TS at each block ant not the raw TS values. In the case of the generic version, the basis functions for the forecasts and the backast are respectively the vectors V f,( ) lw and V b,( ) ( 5 ) 5(public) Traffic: 963 TS of the hourly occupancy rates on the San Francisco Bay Area freeways scaled between 0 and 1[53,54].(6) (proprietary) Finance: 321 TS observed between 2005-07-01 and 2020-10-16 of the adjusted daily closing price of various U.S. mutual funds and exchange traded funds traded on U. Figure 2 : 2OWA metric (left) and Time (right) in minutes for the different N-BEATS models configurations as a function of ensemble size. Figure 3 : 3MASE coverage for the ES-RNN (left) and N-BEATS(i) (right) over the M4 dataset. Each point on the graphs corresponds to a single TS and the darker its color, the better the given model according to the MASE. Horizontal and vertical coordinates represent the value of the two-dimensional embedding computed with TSNE[58] from the statistical features of the series which we detail in Appendix A. All TS with MASE values over 3 where assigned the same color to facilitate visualization. 85 ± 0.02 N-BEATS(G) scaled 11755 0.84 ± 0.04 N-BEATS(I) [39] 7437 0.89 ± 0.02 N-BEATS(I) scaled 6607 0.85 ± 0.03 N-BEASTS(I+G) [39] 19211 0.84 ± 0.03 N-BEATS(I+G) scaled 19170 0.84 ± 0.03 N-BEATS(P, G) (our) 5301 0.87 ± 0.02 N-BEATS(P,G) scaled (our) 6157 0.90 ± 0.02 N-BEATS(P, I) (our) 6990 0.89 ± 0.03 N-BEATS(P,I) scaled (our) 4785 0.89 ± 0.08 N-BEATS(P, I+G) (our) 11840 0.88 ± 0.02 N-BEATS(P,I+G) scaled (our) 10943 0.89 ± 0.06 Figure A. 4 : 4Cumulative distribution function plot for TS datasets over 42 statistical TS features and TS count by dataset and frequency. Figure A. 5 : 5TSNE embedding of all TS forecasted with their different subpopulations. behavior of the 321 TS changes is heterogenou over time in comparison to the other dataset. The Electricity & Traffic TS share almost the same statistical properties as both populations of TS are concentrated in the same region of the graph. ) 11 https://investorsFasttrack.com chunks of the maximum lookback period of the N-BEATS model as training sample, and H steps-ahead as testing sample. Quarterly (h = 8) Monthly (h = 18) Weekly (h = 13) Daily (h = 14) Hourly ( Figure C. 6 : 6Flowchart of the zeros-shot forecasting regime on D target datasets from one source datasets (D source ). The blue square (left) represent the traditional setup of training a model and forecasting on such dataset. The red squares (red) represent the process of loading a pre-trained models and forecasting TS from a different TS dataset than the one used for training. In term of time, the time to train a model is almost always greater than the one from infering a target datasets. We trained our model on the M4 dataset on the TS each subpopulations, i.e. [Yearly, Quarterly, Monthly, Weekly, Daily, Hourly]. We replicated the results from [39] by training the two N-BEAT model variants discussed in Sec. 4.2 using the implementation provided by the original authors and with scaled TS where we divided all TS observations by the maximum values observed. This scaling was done per TS with respect to the lookback window. For our model, the scaling was done on by dividing all lookback windows by the maximum value observed over all lookback windows.6, 8, etc...). For the Electricity and Traffic datasets, the test was set using rolling window operation as described in Appendix A.4. For the Finance dataset, the forecast was evaluated on three rolling forecast setups by sampling the TS on different frequencies, i.e.: daily, weekly and monthly. In total there are 2'602'878 individual TS that were sampled from the 321 original ones across 3 forecast horizons. Despite the dataset being collected from proprietary data sources which we cannot redistribute, we provide the necessary details to help interested readers reconstruct the datasets in Appendix A.5. A summary of the statistical properties, forecast horizons and metadata of these dataset are presented in Appendix A. Yearly Quarterly Monthly Others AverageCoverage OWA MASE % (< T = 1.0) MDA(%) NAIVE 1.000 1.066 1.095 1.335 1.058 40.299 3.2 NAIVE2 1.000 1.000 1.000 1.000 1.000 43.288 33.1 SNAIVE 1.000 1.153 1.146 0.945 1.078 36.095 42.7 ARIMA [30] 0.892 0.898 0.903 0.967 0.903 51.145 53.8 HOLT [60] 0.947 0.932 0.988 1.180 0.971 48.659 61.7 ETS [61] 0.903 0.890 0.914 0.974 0.908 50.987 48.6 THETA [32] 0.872 0.917 0.907 0.995 0.897 48.686 61.7 SES [31] 1.002 0.970 0.951 0.995 0.975 44.719 35.3 DAMPED [4] 0.890 0.893 0.924 1.005 0.907 49.838 61.1 COMB [4] 0.867 0.890 0.920 1.039 0.898 49.784 61.3 MLP ' [4, 62] 1.288 1.684 1.749 3.028 1.642 26.603 60.6 RNN ' [4, 62] 1.308 1.508 1.587 1.702 1.482 28.437 59.8 ProLogistica [63] 0.820 0.855 0.867 0.742 0.841 53.620 62.6 FFORMA [2] 0.799 0.847 0.858 0.914 0.838 53.418 63.7 ES-RNN [42] 0.778 0.847 0.836 0.920 0.821 53.271 63.2 N-BEATS (I) [39] 0.765 0.800 0.820 0.822 0.797 - - N-BEATS (G) [39] 0.758 0.807 0.824 0.849 0.798 - - N-BEATS (I+G) [39] 0.758 0.800 0.819 0.840 0.795 - - Ours: N-BEATS (G) [39] 0.770 0.793 0.818 0.832 0.798 55.576 64.6 N-BEATS (I) [39] 0.763 0.797 0.817 0.838 0.795 55.600 63.7 N-BEATS (I+G) [39] 0.761 0.792 0.814 0.834 0.793 55.868 64.6 N-BEATS (G) scaled [13] 0.784 0.810 0.827 0.836 0.809 54.960 64.5 N-BEATS (I) scaled [13] 0.773 0.817 0.826 0.843 0.806 54.919 63.7 N-BEATS (I+G) scaled [13] 0.778 0.814 0.824 0.836 0.806 55.109 64.4 N-BEATS parallel (G) 0.764 0.804 0.820 0.855 0.799 55.332 64.4 N-BEATS parallel (I) 0.759 0.817 0.824 0.850 0.801 54.966 63.7 N-BEATS parallel (I+G) 0.757 0.806 0.820 0.851 0.796 55.375 64.5 N-BEATS parallel (G) scaled 0.775 0.829 0.833 0.851 0.812 54.506 63.8 N-BEATS parallel (I) scaled 0.772 0.845 0.844 0.867 0.819 53.772 63.6 N-BEATS parallel (I+G) scaled 0.771 0.834 0.835 0.854 0.813 54.344 63.9 Table 2 : 2Time required to train to train all members of the ensemble of our models vs other and average & standard deviation of the absolute percentage correlation between ensemble members on the test sets. We include the total time to produce a forecast for the theta method for comparison. Except for Prologistica, FFORMA and ES-RNN whose training time was replicated in[4], the total time presented is with all model are for single instance and do not consider the speedup that can be achieved based when training the whole ensemble on multiple GPUs. Table 3 : 3Performance of a small ensemble only trained on the MAPE loss for all lookback without bagging and time to train on a single GPU.training time [4], training and deploying state-of-the-art models in real-case scenarios can entail high costs for organizations -costs that are avoidable. For instance, on Google's cloud platform, the estimated cost of training N-BEATS(P) would drop to 530.11 USD$ instead of the 860.13 USD$ their price simulator gives for N-BEATS 6 . Thus, in terms of both cost and time saved, our work provides encouraging results that suggest how multiple TS ensemble models can be accelerated without any great drawback by sharing a subset of their parameterization. Model name # of parameters Model name # of parameters N-BEATS(G) 28'508'265'900 N-BEATS(P, G) 5'972'957'400 N-BEATS(I) 42'288'737'310 N-BEATS(P, I) 8'102'076'930 N-BEATS(I+G) 70'797'003'210 N-BEATS(P, I+G) 14'075'034'330 Table 4 : 4Number of parameters for the whole ensemble for N-BEATS and N-BEATS (parallel) trained on the M4 dataset with 6 lookback windows. Table A.5: List of features used to compare datasets. The functions for calculating these features are implemented in the tsfeatures R package byFeatures Description Seasonal Non-Seasonsal 1 T length of time series 2 trend strength of trend 3 seasonality strength of seasonality - 4 linearity Linearity 5 curvature Curvature 6 spikiness Variance of the leave-one-out variances of the remainder component in STL decomposition 7 e_acf1 first autocorrelation function (ACF) value of remainder series 8 e_acf10 sum of squares of first 10 ACF values of remainder series 9 stability sum of squares of first 10 ACF values of remainder series 10 lumpiness Variance of the means produced for tiled (non-overlapping) windows 11 entropy Spectral entropy (Shannon entropy) of the TS 12 hurst Hurst exponent from [65] 13 nonlinearity Teraesvirta modified test [66] 13 alpha ETS(A,A,N)α 14 beta ETS(A,A,N)β 15 hwalpha ETS(A,A,A)α 16 hwbeta ETS(A,A,A)β - 17 hwgamma ETS(A,A,A)γ - 18 ur_pp Test statistic based on Phillips-Perron test [67] 19 ur_kpss test statistic based on KPSS test [68] 20 y_acf1 first ACF value of the original series 21 diff1y_acf1 First ACF value of the differenced series 22 diff2y_acf1 First ACF value of the twice-differenced series 23 y_acf10 Sum of squares of first 10 ACF values of original series 24 diff1y_acf10 Sum of squares of first 10 ACF value of the differenced series 25 diff2y_acf10 Sum of squares of first 10 ACF value of the twice-differenced series 26 seas_acf1 autocorrelation coefficient at first seasonal lag - 27 sediff_acf1 first ACF value of seasonally differenced series - 28 y_pacf5 sum of squares of first 5 PACF values of original series 29 diff1y_pacf5 sum of squares of first 5 PACF values of original series 30 diff2y_pacf5 sum of squares of first 5 PACF values of twice-differenced series 31 seas_pacf partial autocorrelation coefficient at first seasonal lag 32 crossing_points number of times the time series crosses the median 33 flat_spots number of flat spots, calculated by discretizing the series into 10 equal-sized intervals and counting the maximum run length within any single interval 34 nperiods number of seasonal periods in the series - 35 seasonal_period length of seasonal period - 36 peak strength of peak 37 trough strength of trough 38 ARCH.LM ARCH.LM statistic 39 arch_acf sum of squares of the first 12 autocorrelations of z 2 40 garch_acf sum of squares of the first 12 autocorrelations of r 2 41 arch_r2 R 2 value of an AR model applied to z 2 42 garch_r2 R 2 value of an AR model applied to r 2 Table A . 6 : A6Composition of the M4 TS dataset: number of time series based on their sampling frequency and type. /github.com/Mcompetitions/M4-methods.Table A.7: Composition of the M3 TS dataset: the number of TS based on sampling frequency and type.Appendix A.2. M3 Dataset Details Frequency/Horizon Type Yearly (h = 6) Quarterly (h = 8) Monthly (h = 18) Other (h = 8) Total Demographic 245 57 111 0 413 Finance 58 76 145 29 308 Industry 102 83 334 0 519 Macro 83 336 312 0 731 Micro 146 204 474 4 828 Other 11 0 52 141 204 Total 645 756 1428 174 3'003 The M3 8 dataset is a publicly accessible dataset that is smaller than 8 https://forecasters.org/resources/time-series-data/m3-competition/ the M4 dataset but remains relatively large and diverse. Similarly to the M4 dataset, it contains TS frequently encountered in business, financial and economic forecasting. It include yearly, quarterly, monthly, weekly, daily and hourly time series, each with different forecast horizons, i,e, [6, 8, 18, 13, 14, 48]. All series have positive observed values at all time-steps. The dataset was subdivided into a training and a test dataset by the M3 TS competition organizers. Table A.7 outlines the composition of the M3 dataset across domains and forecast horizons. For further details on this dataset, we refer the reader to [50]. This dataset was considered for zero-shot forecasting, to examine a case where the target dataset is from the same domains of application but with other TS. Appendix A.3. Tourism Dataset Details Frequency/Horizon Type Yearly (h = 4) Quarterly (h = 8) Monthly (h = 24) Total Tourism 518 427 366 1311 Table A.8: Composition of the Tourism TS dataset: number of time series based on sampling frequency and type. Table B . 9 : B9Hyper parameters used to produce results on the M4 TS dataset Native zero-shot (R O )Frequency Yearly Quarterly Monthly Weekly Daily Hourly horizon (h = 6) (h = 8) (h = 18) (h = 13) (h = 14) (h = 48) L H 1.5 1.5 1.5 10 10 10 Iterations N-BEATS 15k 15k 15k 5k 5k 5k Iterations N-BEATS (P) 10k 15k 15k 5k 5k 5k Native Zero-Shot with equal forecast horizon (R SH ) horizon (h = h (Dtgrt.) Yearly ) (h = h (Dtgrt.) Quarterly ) (h = h (Dtgrt.) Monthly ) (h = h (Dtgrt.) Weekly ) (h = h (Dtgrt.) Daily ) (h = h (Dtgrt.) Hourly ) L H 1.5 1.5 1.5 10 10 10 Iterations 15k 15k 15k 5k 5k 5k Native Zero-Shot with equal forecast horizon(R SH,LT ) horizon (h = h (Dtgrt.) Yearly ) (h = h (Dtgrt.) Quarterly ) (h = h (Dtgrt.) Monthly ) (h = h (Dtgrt.) Weekly ) (h = h (Dtgrt.) Daily ) (h = h (Dtgrt.) Hourly ) L H 10 10 10 10 10 10 Iterations 15k 15k 15k 15k 15k 15k Table B . B10: HP differences between the different zero-shot strategies. All models were trained on the M4 TS dataset Table C . C11 describes the zero-shot performance of N-BEATS and N-BEATS(P). Several observations can be made: we show the metrics for three training regimes: R SH,LT /R SH /R O . R O is the same model used to produce the results on M4(1) N-BEATS(P) produces comparable zero-shot results to previous state-of- the-art models for all datasets. In other training regimes, where models trained with the same forecast horizon or longer ones, comparable levels of accuracy were observed. M3, SMAPE Tourism, MAPE Electricity, ND Traffic, ND N-SHOT: Naive 16.59 SNaive 24.80 Naive 0.37 0.57 Comb [4] 13.52 ETS [61] 20.88 MatFact [54] 0.16 0.17 ForePro [52] 13.19 Theta [32] 20.88 DeepAR [22] 0.07 0.17 Theta [32] 13.01 ForePro [52] 19.84 DeepState [35] 0.08 0.17 DOTM [73] 12.90 Strato 19.52 Theta [32] 0.08 0.18 EXP [74] 12.71 LCBaker [75] 19.35 ARIMA [30] 0.07 0.15 N-BEATS [39] 12.37 18.52 0.07 0.11 DEEP-AR* [22, 13] 12.67 19.27 0.09 0.19 ZERO-SHOT: (RSH,LT /RSH/RO) M4 N-BEATS (G) scaled * [39] 12.36/12.67/12.72 18.90/20.16/24.14 0.09/0.09/0.08 0.16/0.16/0.14 M4 N-BEATS (I) scaled * [39] 12.43/12.63/12.66 19.43/20.58/23.26 0.10/0.09/0.08 0.16/0.16/0.14 M4 N-BEATS (g+i) scaled * [39] 12.38/12.61/12.64 19.04/20.22/23.43 0.10/0.09/0.08 0.16/0.16/0.14 M4 N-BEATS (P+G) scaled 12.48/12.65/12.65 18.99/19.98/22.85 0.09/0.09/0.08 0.16/0.18/0.14 M4 N-BEATS (P+I) scaled 12.69/12.76/12.72 20.54/20.97/23.18 0.09/0.10/0.09 0.17/0.17/0.16 M4 N-BEATS (P+G&I) scaled 12.56/12.67/12.64 19.50/20.24/22.79 0.09/0.09/0.08 0.16/0.16/0.14 Table C.11: Averaged forecasting for the zero-shot regime for each dataset; lower values are better. Zero-shot forecasts are compared for N-BEATS and our approach. For the models in italic using the following references, we relied on their reported accuracy. For zero-shot results, Daily (H = 14)N = (1 222 866) Weekly (H = 13, N = 1 091 898) Monthly (H = 18, N = 288 114) Table C.12: Comparison between statistical baselines and zero-shot application of the N-BEATS model in terms of OWA, MDA and time to produce forecast. Forecasts were made with the native zero-shot approach (R O ).Models OWA MDA Time (min.) OWA MDA Time (min.) OWA MDA Time (min.) N-SHOT: Naive 1.000 07.1 - 1.00 02.0 - 1.000 00.5 - ARIMA [30] 1.041 27.1 4685 1.059 28.9 3597 0.891 40.3 816 THETA [32] 0.995 49.4 241 0.993 53.5 262 0.913 61.4 49 SES [31] 1.000 09.3 174 1.001 05.2 160 1.000 02.9 25 HOLT [60] 1.081 49.6 167 1.116 53.9 160 0.931 60.3 42 ETS [61] 1.019 20.4 969 1.044 18.8 512 0.940 29.2 181 ZERO-SHOT: M4 N-BEATS (G) [39] 1.165 50.3 24 1.078 50.9 21 0.963 55.1 6 M4 N-BEATS (I) [39] 1.222 50.5 26 1.045 51.3 23 0.962 54.9 6 M4 N-BEATS (I+G) [39] 1.191 50.7 50 1.056 51.1 44 0.961 55.4 12 M4 N-BEATS (P+G) 1.210 48.5 25 1.098 51.6 21 0.973 54.4 6 M4 N-BEATS (P+I) 1.135 48.5 26 1.055 52.1 24 0.975 54.1 6 M4 N-BEATS (P+I&G) 1.139 49.0 51 1.044 52.0 45 0.973 54.4 12 Appendix D. Finance dataset: List of Securities Considered Data Source: Yahoo US Sector Stock Index XLE StateSt ETF Energy Select Sector SPDR Fd US Sector Stock Index XLF StateSt ETF Financial Select Sector SPDR US Sector Stock Index XLI StateSt ETF Industrial Sel Sector SPDR US Sector Stock Index XLK StateSt ETF Tech Select Sector SPDR US Sector Stock Index US Sector Stock Index IYR iShares U.S. Real Estate ETF US Sector Stock Index XOI-I AMEX Oil Ix Commodities US Bonds -Corp Invst MBOA-BofAML US Corporate 15 Year Semi-Annual US Bonds -Corp Invst QQQ Nasdaq 100 ETF Stock Index (US) SP-GB S&P Global BMI Idx DivAdj Global Stock Index SP-GL S&P Global 1200 Idx DivAdj Global Stock Index SP-HB S&P 500 High Beta Idx DivAdj Global Stock Index SP-IO S&P Global 100 Idx DivAdj Global Stock Index SP-L4 S&P Latin America 40 Idx Di-vAdj Table D.13: List of US traded funds used to create the finance dataset. The class columns correspond to the type of securities and the source columns specify where the TS was collected. See Tab. D.14 for a brief description of the asset classes TS type Description US Stock Index Index of US stocks, such as the S&P500 US Stock A fund (ETF or mutual funds) made up primarily of US stocks US Sector Stock Index US stock industry sector index Regional Stock Index Global region stock index, such as Europe National Stock Index Country stock index Global Stock Index Global / world stock index US Bonds -Gvmnt US treasury funds US Bonds -Corp Invst US corporate bond funds, investment grade US Bonds -Corp HY US bond funds, high yield Table D.14: Brief description of the typea of TS used in the Finance dataset.Ticker Description Class DJAT Dow Jones Asian Titan 50 In- dex Regional Stock Index DJI Dow Jones Industrial Average Stock Index (US) DJT Dow Jones Transportation Av- erage Stock Index (US) DJU Dow Jones Utility Average Stock Index (US) GSPC S&P 500 Stock Index (US) IXIC NASDAQ Composite Stock Index (US) NDX NASDAQ-100 Stock Index (US) OEX S&P 100 Stock Index (US) XMI NYSE Arca Major Market In- dex Stock Index (US) DX-Y.NYB US Dollar/USDX -Index - Cash Forex FDCPX Fidelity Select Computers US Sector Stock Index HSI HANG SENG INDEX (Cur- rency in HKD) National Stock Index Data Source: Fred GOLDPMGBD228NLBMGold Fixing Price 3:00 P.M. (London time) in London Bul- lion Market & based in U.S. Dollars Others WILL4500IND Wilshire 4500 Total Market In- dex Stock Index (US) WILL4500PR Wilshire 4500 Price Index Stock Index (US) WILL5000IND Wilshire 5000 Total Market In- dex Stock Index (US) WILL5000INDFC Wilshire 5000 Total Market Full Cap Index Stock Index (US) WILL5000PR Wilshire 5000 Price Index Stock Index (US) Data Source: FastTrack FPX1 CAC 40 Ix National Stock Index SHCP Shanghai Composite Ix National Stock Index SPXX STOXX Europe 600 Ix Regional Stock Index SX5P STOXX Europe 50 Ix Regional Stock Index A-CWI MSCI ACWI DivAdj Idx Global Stock Index A-XUS MSCI ACWI xUS DivAdj Idx Global Stock Index AUD- US / Australia Foreign Ex- change Rate Forex BBG- CBOE US T-Bill 13-Week Yld Bd Ix US Bonds -Gvmnt BBG-9 BBG Barclay Agg Bond-US Universal TR Ix US Bonds -Gvmnt BBG-G BBG Barclay Agg Bond-US Corp IG TR Ix US Bonds -Gvmnt BBG-H ML US HY Bb-B Ix US Bonds -Corp HY BBG-I BBG Barclay Agg Bond-US Agency Long Ix US Bonds -Gvmnt BBG-O BBG Barclay Agg Bond-Yan- kee Ix US Bonds -Gvmnt BBG-S BBG Barclay Agg Bond-US MBS Agncy TR Ix US Bonds -Gvmnt BBG-T BBG Barclay Agg Bond-US MBS Agncy TR Ix US Bonds -Gvmnt BBG-U BBG Muni Bond 3yr Idx US Bonds -Gvmnt BBG-Y BBG Muni Bond 20yr Idx US Bonds -Gvmnt BBM-2 BBG Muni Bond 5yr Idx US Bonds -Gvmnt BBM-3 BofAML US Corp 5-7yr Total Return Ix US Bonds -Corp Invst BBM-5 BBG Muni Bond Composite Idx US Bonds -Gvmnt BBM-I BBG Muni Bond Long Term Idx US Bonds -Gvmnt BBM-L BBG Muni Bond 10yr Idx US Bonds -Gvmnt BBM-T BBG Barclay Agg Bond-US Composite TR Ix US Bonds -Gvmnt CAD- Canada / US Foreign Exchange Rate Ix Forex CDN-X Canadian Dollar For 100 CDN Ix Forex CHF- Switzerland/ US Foreign Ex- change Rate Ix Forex CNY- China / US Foreign Exchange Rate Ix Forex CR-TR CRB Total Return Ix Commodities DBC Invesco DB Commodity Index Tracking Fund Commodities DKK- Denmark / US Foreign Ex- change Rate Ix Forex DXY-Z US Dollar Ix Forex EFA iShares MSCI EAFE ETF Regional Stock Index EURO- US/Euro Foreign Exchange Rate Ix FBMPX Fidelity Select Communication Services Portfolio US Sector Stock Index FCYIX Fidelity Select Industrials US Sector Stock Index FDAC- Frankfurt Dax Ix National Stock Index FDFAX Fidelity Select Consumer Sta- ples US Sector Stock Index FDLSX Fidelity Select Leisure US Sector Stock Index FEZ-X Europe 50 STOXX stTr Ix Regional Stock Index FIDSX Fidelity Select Financial Ser- vices US Sector Stock Index FNARX Fidelity Select Natural Re- sources US Sector Stock Index FNMIX Fidelity New Markets Income Regional Stock Index FRESX Fidelity Fidelity Real Estate In- vestment Portfolio Others FSAGX Fidelity Select Gold US Sector Stock Index FSAIX Fidelity Select Air Transporta- tion US Sector Stock Index FSAVX Fidelity Select Automotive US Sector Stock Index FSCHX Fidelity Select Chemicals US Sector Stock Index FSCPX Fidelity Select Consumer Dis- cretion US Sector Stock Index FSCSX Fidelity Select software & Comp Service US Sector Stock Index FSDAX Fidelity Select Defense & Aerospace US Sector Stock Index FSDCX Fidelity Select Commun Equip- ment US Sector Stock Index FSDPX Fidelity Select Materials US Sector Stock Index FSELX Fidelity Select Semiconductors US Sector Stock Index FSENX Fidelity Select Energy US Sector Stock Index FSESX Fidelity Select Energy Service US Sector Stock Index FSHCX Fidelity Select Health Care Ser- vice US Sector Stock Index FSHOX Fidelity Select Const & Hous- ing US Sector Stock Index FSLBX Fidelity Select Brokrg & INV Mgt US Sector Stock Index FSLEX Fidelity Select Environmental & Alt US Sector Stock Index FSNGX Fidelity Select Natural Gas US Sector Stock Index FSPCX Fidelity Select Insurance US Sector Stock Index FSPHX Fidelity Select Health Care US Sector Stock Index FSPTX Fidelity Select Technology US Sector Stock Index FSRBX Fidelity Select Banking US Sector Stock Index FSRFX Fidelity Select Transportation US Sector Stock Index FSRPX Fidelity Select Retailing US Sector Stock Index FSTCX Fidelity Select Telecommunica- tions US Sector Stock Index FSUTX Fidelity Select Utilities US Sector Stock Index FSVLX Fidelity Select Consumer Fi- nance US Sector Stock Index FTSE- London FT-SE 100 Ix National Stock Index GBP- US / UK Foreign Exchange Rate Ix UUP Invesco DB US Dollar Index Bullish Fund Forex VASVX Vanguard Selected Value Fund Stock Index (US) VBISX Vanguard Short Term Bond In- dex US Bonds -Gvmnt VEIEX Vanguard Emerging Market Stock Index INV Regional Stock Index VEXMX Vanguard Extended Market In- dex Fund Global Stock Index VEXPX Vanguard Explorer Fund INV Stock Index (US) VFICX Vanguard Int. Term Invest- ment Grade Bond Fund US Bonds -Corp Invst VFIIX Vanguard GNMA INV US Bonds -Gvmnt VFISX Vanguard Short-Term Treasury INV US Bonds -Gvmnt VFITX Vanguard Intermediate Term Treasury Fund US Bonds -Gvmnt VFSTX Vanguard Short-Term INV Growth Incm INV US Bonds -Corp Invst VGENX Vanguard Energy INV National Stock Index VGHCX Vanguard Health Care INV National Stock Index VGPMX Vanguard Global Capital Cy- cles Fund Stock Index (US) VGSIX Vanguard REIT Index INV Others VINEX Vanguard International Ex- plorer Fund Global Stock Index VNQ Vanguard Real Estate Index Fund ETF Shares Others VTRIX Vanguard International Value Fund Global Stock Index VTSMX Vanguard Total Stock Markets Index INV Global Stock Index VUSTX Vanguard Long-Term Treasury INV US Bonds -Gvmnt VWEHX Vanguard Hi-Yield Corporate INV US Bonds -Corp HY VWESX Vanguard Long-Term INV Growth Income INV US Bonds -Corp Invst VWIGX Vanguard International Growth INV Others VWINX Vanguard Wellesley Income INV US Bonds -Gvmnt VWO Vanguard FTSE Emerging Markets Index Fund ETF Shares Global Stock Index VWUSX Vanguard US Growth INV Stock Index (US) VXF Vanguard Extended Market In- dex Fund ETF Shares Global Stock Index WDG-X MSCI Germany iShr Ix National Stock Index WPB-X MSCI Canada iShr Ix National Stock Index XLB StateSt ETF Materials Select Sector SPDR XLP StateSt ETF Consumer Staples SelSctrSPDR US Sector Stock Index XLU StateSt ETF Utilities Select Sector SPDR US Sector Stock Index XLV StateSt ETF Health Care Sel Sector SPDR US Sector Stock Index XLY StateSt ETF Consumer Dis- cretnrySlSctSPDR US Sector Stock Index XLC StateSt ETF Communication Service SlSctSPDR US Sector Stock Index XLRE StateSt ETF Real Estate SlSct- SPDR US Sector Stock Index VOX Vanguard Communication Ser- vices Index Fund ETF Shares Regional Stock Index SPY StateSt ETF SPDR S&P 500 Stock Index (US) ST-AG Silver Spot Commodities ST-AU Gold Spot Commodities ST-BC Brent Crude Spot Commodities ST-CA Cocoa Spot Commodities ST-CF Coffee Bushel Spot Commodities ST-CO Crude Oil Spot Commodities ST-CT Cotton Bushel Spot Commodities ST-CU Copper Spot Commodities ST-HO Heating Oil Spot Commodities ST-NG Natural Gas Spot Commodities ST-PD Palladium Spot Commodities ST-PL Platinum Spot Commodities WTI-B Blmbrg WTI Crude Oil Sub Ix Total Return Commodities VIPSX Vanguard Inflation-Protected Securities Fund Investor Shares US Bonds -Gvmnt Country Funds Country index fund Forex Foreign Exchange Commodities Commodities tracking fund Real Estate Real estate fund Other Other fund or index For interested reader, this special kind of forecasting is known as asset pricing[28,29]. In this setup we are interested in modelizing the relationship between systematic risk factor and expected excess return of assets over the market and ultimately build an optimal portfolio. This goes out of the scope of this paper. We focus on forecasting any TS, regardless if is an asset, solely based on its historical values. N-BEATS was not part of the M4 competition, and attained state-of-the-art results on M4 benchmark ex post facto. N-BEATS was the core part of the second-entry solution in M5 competition[3]. This is because the procedure to train our model is itteration-based and not epochbased. The term "epochs" refers to the number of passes of the entire training dataset our models has seen. Our approach differs in that we itterate on batches of TS sampled and sliced randomly at different cut-off points. Prices are at the rate calculated using their cost estimator on 04-08-2021, employing their "AI Platform" configuration with a single NVIDA P100 GPU https://github.com/Mcompetitions/M4-methods https://robjhyndman.com/data/27-3-Athanasopoulos1.zip 10 https://archive.ics.uci.edu/ml/datasets/PEMS-SF Appendix B.1. Forecasting CombinationForecast combination with N-BEATS(P) and N-BEATS was done as follows: to produce a forecast from the ensemble, all forecasts of ensemble members were considered and the median was computed for every forecast for all time t per TS forecast. When the forecast horizon of the model was shorter than the forecast horizon of the target dataset, we iteratively appended the forecast to the original TS signal and based our forecasts upon the transformed signal until the total forecast was longer than or equal to the forecast horizon of the target dataset. 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E Spiliotis, V Assimakopoulos, K Nikolopoulos, International Journal of Production Economics. 209E. Spiliotis, V. Assimakopoulos, K. Nikolopoulos, Forecasting with a hybrid method utilizing data smoothing, a variation of the theta method and shrinkage of seasonal factors, International Journal of Production Economics 209 (2019) 92-102. Winning methods for forecasting tourism time series. L C Baker, J Howard, International Journal of Forecasting. 273L. C. Baker, J. Howard, Winning methods for forecasting tourism time series, International Journal of Forecasting 27 (3) (2011) 850-852. Dataset Fig. A.4 illustrates the difference between the statistical properties of all 6 datasets, employing the same set of TS features used in the FFORMA model. A Appendix, Appendix A. Dataset Fig. A.4 illustrates the difference between the statistical properties of all 6 datasets, employing the same set of TS features used in the FFORMA model As an example of the observations that can be drawn from this figure: it can bee seen that both the Electricy ML-10 M2EY-BofAML US Corporate AAA Semi-Annual Yiel US Bonds. Corp Invst M3OA-BofAML US Corporate BBB Semi-Annual Yiel US Bonds -Corp Invst M4EY-BofAML US Corporate. 2] for a detailed overview of the 42 features used and their interpretationWe refer the reader to table.A.5 and [2] for a detailed overview of the 42 features used and their interpretation. As an example of the observations that can be drawn from this figure: it can bee seen that both the Electricy ML-10 M2EY- BofAML US Corporate AAA Semi-Annual Yiel US Bonds -Corp Invst M3OA- BofAML US Corporate BBB Semi-Annual Yiel US Bonds -Corp Invst M4EY- BofAML US Corporate 1-3 . Year Semi-Annual US Bonds -Corp Invst M5OA-BofAML US Corporate. Year Semi-Annual US Bonds -Corp Invst M5OA- BofAML US Corporate 3-5 . Year Semi-Annual US Bonds -Corp Invst M6OA-BofAML US Corporate. Year Semi-Annual US Bonds -Corp Invst M6OA- BofAML US Corporate 5-7 . Year Semi-Annual US Bonds -Corp Invst M7EY-BofAML US Corporate. 710Year Semi-Annual US Bonds -Corp Invst M7EY- BofAML US Corporate 7-10 . Year Semi-Annua US Bonds -Corp Invst M8EY-BofAML US. Year Semi-Annua US Bonds -Corp Invst M8EY- BofAML US Corporate 10-15 Year Semi-Annu. Year Semi-Annu

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{'abstract': 'We study the problem of efficiently scaling ensemble-based deep neural networks for multi-step time series (TS) forecasting on a large set of time series. Current state-of-the-art deep ensemble models have high memory and computational requirements, hampering their use to forecast millions of TS in practical scenarios. We propose N-BEATS(P), a global parallel variant of the N-BEATS model designed to allow simultaneous training of multiple univariate TS forecasting models. Our model addresses the practical limitations of related models, reducing the training time by half and memory requirement by a factor of 5, while keeping the same level of accuracy in all TS forecasting settings. We have performed multiple experiments detailing the various ways to train our model and have obtained results that demonstrate its capacity to generalize in various forecasting conditions and setups.', 'arxivid': '2109.09705', 'author': ['Philippe Chatigny \nUniversity of Sherbrooke\nSherbrookeQCCanada\n', 'Shengrui Wang \nUniversity of Sherbrooke\nSherbrookeQCCanada\n', 'Jean-Marc Patenaude \nLaplace Insights\nSherbrookeQCCanada\n', 'Boris N Oreshkin \nUnity Technologies\nLabs, MontrealQCCanada\n'], 'authoraffiliation': ['University of Sherbrooke\nSherbrookeQCCanada', 'University of Sherbrooke\nSherbrookeQCCanada', 'Laplace Insights\nSherbrookeQCCanada', 'Unity Technologies\nLabs, MontrealQCCanada'], 'corpusid': 237571900, 'doi': None, 'github_urls': ['https://github.com/Mcompetitions/M4-methods'], 'n_tokens_mistral': 33215, 'n_tokens_neox': 28844, 'n_words': 16064, 'pdfsha': 'c5b781f79b380107c466e373df242a181318bc2f', 'pdfurls': ['https://arxiv.org/pdf/2109.09705v4.pdf'], 'title': ['Neural Forecasting at Scale', 'Neural Forecasting at Scale'], 'venue': []}

arxiv

16 Frédérick Poidevin, 29, 30 Tie Liu, 31 Simon Coudé, 32, 33 Mehrnoosh Tahani, 34, 35 Hong-Li Liu, 36 Takashi Onaka, 37, 38 Dalei Li, 39 Motohide Tamura 18 Thiem Hoang, 17, 18 Tetsuo Hasegawa, 19 Woojin Kwon, 20. 19 Koji Kawabata, 64, 65, 66 Francisca Kemper, 67, 68, 69 Jongsoo Kim, 17. 75 Sheng-Jun Lin, 8 Sheng-Yuan Liu, 9 Xing Lu, 31 Tao-Chung Ching chingtaochung@gmail.com Research Center for Intelligent Computing Platforms Zhejiang Lab311100HangzhouPeople's Republic of China National Astronomical Observatories Chinese Academy of Sciences A20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China Jansky Fellow National Radio Astronomy Observatory 1003 Lopezville Road87801SocorroNMUSA Keping Qiu School of Astronomy and Space Science Nanjing University 163 Xianlin Avenue210023NanjingPeople's Republic of China Ministry of Education Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University) 210023NanjingPeople's Republic of China Di Li National Astronomical Observatories Chinese Academy of Sciences A20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China Department of Astronomy University of Chinese Academy of Sciences 100049BeijingPeople's Republic of China NAOC-UKZN Computational Astrophysics Centre University of KwaZulu-Natal 4000DurbanSouth Africa Zhiyuan Ren National Astronomical Observatories Chinese Academy of Sciences A20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China Shih-Ping Lai Institute of Astronomy Department of Physics National Tsing Hua University 30013HsinchuTaiwan Academia Sinica Institute of Astronomy and Astrophysics No.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan David Berry East Asian Observatory 660 N. A'ohōkū Place96720University Park, HiloHIUSA Kate Pattle Department of Physics and Astronomy University College London Gower StreetWC1E 6BTLondonUnited Kingdom Mike Chen Department of Physics and Astronomy University of Victoria V8W 2Y2VictoriaBCCanada Huei-Ru Vivien Chen Institute of Astronomy Department of Physics National Tsing Hua University 30013HsinchuTaiwan Academia Sinica Institute of Astronomy and Astrophysics No.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan Wen Ping Chen Institute of Astronomy National Central University 32001ZhongliTaiwan Jungyeon Cho Department of Astronomy and Space Science Chungnam National University 99 Daehak-ro, Yuseong-gu34134DaejeonRepublic of Korea Yunhee Choi Korea Astronomy and Space Science Institute 776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea Youngwoo Choi Department of Physics and Astronomy Seoul National University 1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea Minho Choi Korea Astronomy and Space Science Institute 776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea Antonio Chrysostomou SKA Observatory, Jodrell Bank, Lower Withington SK11 9FTMacclesfieldUK Eun Jung Chung Department of Astronomy and Space Science Chungnam National University 99 Daehak-ro, Yuseong-gu34134DaejeonRepublic of Korea Y Sophia Dai National Astronomical Observatories Chinese Academy of Sciences A20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China Ngoc Pham Diep Academy of Science and Technology Vietnam National Space Center 18 Hoang Quoc VietHanoiVietnam, Vietnam Yasuo Doi Department of Earth Science and Astronomy Graduate School of Arts and Sciences The University of Tokyo 3-8-1 Komaba153-8902TokyoJapan Yan Duan National Astronomical Observatories Chinese Academy of Sciences A20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China Hao-Yuan Duan Institute of Astronomy Department of Physics National Tsing Hua University 30013HsinchuTaiwan Armagh Observatory and Planetarium College Hill, ArmaghBT61 9DBUK David Eden Lapo Fanciullo Academia Sinica Institute of Astronomy and Astrophysics No.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan South Dist National Chung Hsing University 145 Xingda Rd402Taichung CityTaiwan Jason Fiege Department of Physics and Astronomy The University of Manitoba R3T2N2WinnipegManitobaCanada Laura M Fissel Department for Physics, Engineering Physics and Astrophysics Queen's University K7L 3N6KingstonONCanada Erica Franzmann Department of Physics and Astronomy The University of Manitoba R3T2N2WinnipegManitobaCanada Per Friberg East Asian Observatory 660 N. 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Lluís Companys 23BarcelonaSpain Institut d'Estudis Espacials de Catalunya (IEEC) E-08034BarcelonaSpain Nobeyama Radio Observatory National Astronomical Observatory of Japan National Institutes of Natural Sciences 384-1305Nobeyama, NaganoMinamimaki, MinamisakuJapan Astronomical Institute Graduate School of Science Tohoku University Aoba-ku980-8578, 72SendaiMiyagiJapan Department of Physics and Astronomy McMaster University L8S 4M1HamiltonONCanada Department of Physics and Atmospheric Science Dalhousie University B3H 4R2HalifaxCanada Department of Physics Department of Space, Earth & Environment The Chinese University of Hong Kong Shatin, Hong Kong 76N.T Chalmers University of Technology SE-412 96GothenburgSweden Faculty of Education & Center for Educational Development and Support Kagawa University Saiwai-cho 1-1760-8522TakamatsuKagawaJapan Graduate University of Science and Technology Academy of Science and Technology 18 Hoang Quoc Viet, Cau GiayHanoiVietnam, Vietnam 16 Frédérick Poidevin, 29, 30 Tie Liu, 31 Simon Coudé, 32, 33 Mehrnoosh Tahani, 34, 35 Hong-Li Liu, 36 Takashi Onaka, 37, 38 Dalei Li, 39 Motohide Tamura 60 Tsuyoshi Inoue, 61 Shu-ichiro Inutsuka, 61 Kazunari Iwasaki, 62 Il-Gyo Jeong, 63, 17 Vera Könyves, 14 Ji-hyun Kang, 17 Miju Kang, 17 Janik Karoly, 14 Akimasa Kataoka Steve Mairs, 10 Masafumi Matsumura, 78 Brenda Matthews; Gerald Moriarty-Schieven172818 Thiem Hoang, 17, 18 Tetsuo Hasegawa, 19 Woojin Kwon, 20. 19 Koji Kawabata, 64, 65, 66 Francisca Kemper, 67, 68, 69 Jongsoo Kim, 17. 75 Sheng-Jun Lin, 8 Sheng-Yuan Liu, 9 Xing Lu, 31Draft version Typeset using L A T E X twocolumn style in AASTeX62 2 3polarization -ISM: magnetic fields -ISM: individual objects (DR 21) -stars: formation - submillimeter: ISM We present 850 µm dust polarization observations of the massive DR21 filament from the B-fields In STar-forming Region Observations (BISTRO) survey, using the POL-2 polarimeter and the SCUBA-2 camera on the James Clerk Maxwell Telescope. We detect ordered magnetic fields perpendicular to the parsec-scale ridge of the DR21 main filament. In the sub-filaments, the magnetic fields are mainly parallel to the filamentary structures and smoothly connect to the magnetic fields of the main filament. We compare the POL-2 and Planck dust polarization observations to study the magnetic field structures of the DR21 filament on 0.1-10 pc scales. The magnetic fields revealed in the Planck data are well aligned with those of the POL-2 data, indicating a smooth variation of magnetic fields from large to small scales. The plane-of-sky magnetic field strengths derived from angular dispersion functions of dust polarization are 0.6-1.0 mG in the DR21 filament and ∼ 0.1 mG in the surrounding ambient gas. The mass-to-flux ratios are found to be magnetically supercritical in the filament and slightly subcritical to nearly critical in the ambient gas. The alignment between column density structures and magnetic fields changes from random alignment in the low-density ambient gas probed by Planck to mostly perpendicular in the high-density main filament probed by JCMT. The magnetic field structures of the DR21 filament are in agreement with MHD simulations of a strongly magnetized medium, suggesting that magnetic fields play an important role in shaping the DR21 main filament and sub-filaments. INTRODUCTION Recent observations of thermal continuum from dust and molecular lines from gas have revealed that parsec-scale filaments are ubiquitous structures in molecular clouds (André et al. 2014). The collapse and fragmentation of gravitationally unstable filaments host the birth of prestellar cores and protostars (Molinari et al. 2010;Arzoumanian et al. 2011;Hacar et al. 2013;Palmeirim et al. 2013;Fernández-López et al. 2014;Könyves et al. 2015). Further, high-mass star-forming regions are preferentially found in the hubs of filaments, where the longitudinal mass flows along filaments toward the hubs are believed to play a key role in enhancing the density to drive massive star formation (Galván-Madrid et al. 2010;Hill et al. 2011;Hennemann et al. 2012;Liu et al. 2012;Schneider et al. 2012;Peretto et al. 2013;Hacar et al. 2018;Kumar et al. 2020). Observations of dust polarization at submillimeter/millimeter wavelengths have been proven to be the most efficient method to trace magnetic fields of molecular clouds (Crutcher 2012), given that the emission of magnetically aligned interstellar dust grains is linearly polarized with the polarization angle perpendicular to the direction of local magnetic field projected on the plane of sky (Lazarian & Hoang 2007;Andersson et al. 2015). Single-dish dust polarization surveys reveal magnetic field structures within molecular clouds at resolutions from a few arcmins to tens of arcsecs (e.g. Dotson et al. 2000Dotson et al. , 2010Matthews et al. 2009;Planck Collaboration Int. XIX 2015). Statistical studies of Planck data covering column densities from 10 20 to 10 22 cm −2 indicate that the low column density structures in diffuse clouds appear to be parallel to the magnetic fields, while the filamentary structures of molecular clouds with high column densities tend to be perpendicular to the magnetic fields (Planck Collaboration Int. XXXII 2016;Planck Collaboration Int. XXXV 2016). Ground-based telescopes that are capable of resolving magnetic fields in molecular clouds show that at parsec scale, the magnetic fields of filaments are usually perpendicular to the main axes of filaments (Schleuning 1998;Vallée & Fiege 2006;Matthews et al. 2014;Pattle et al. 2017;Liu et al. 2018;Chuss et al. 2019;Fissel et al. 2019;Soam et al. 2019). The parallel alignment between magnetic fields and low-density sub-filaments and the perpendicular alignment between magnetic fields and high-density filaments are also supported by optical and infrared polarization data, indicating that magnetic fields play an important role in filament formation (Alves et al. 2008;Sugitani et al. 2011;Soler et al. 2016;Cox et al. 2016;Wang et al. 2020). Observations at a few thousand au resolution toward dense cores within filaments, however, reveal complex magnetic fields that are not simply aligned with the structures of cores (Zhang et al. 2014;Koch et al. 2014;Li et al. 2015;Doi et al. 2020;Eswaraiah et al. 2021), indicating a more complex role of magnetic fields in the formation of dense cores. To study the role of magnetic fields in the formation of filaments and high-mass star-forming cores, we present 850 µm dust polarization observations taken using the James Clerk Maxwell Telescope (JCMT) toward the DR21 filament. The DR21 filament is the densest and most massive region in the Cygnus X complex (Schneider et al. 2016;Cao et al. 2019) at a distance of 1.4 kpc (Rygl et al. 2012). The filament hosts 24 massive dense cores (Motte et al. 2007), including the well-studied massive star-forming regions DR21 and DR21(OH) (Downes & Rinehart 1966). The ridge of the DR21 filament has a length of 4 pc and a total mass of 15,000 M , connected by several sub-filaments with masses between 130 M and 1400 M . Global infall motions of the filament are suggested by molecular line observations, probably triggered by convergence of flows on cloud scales (Schneider et al. 2010;Csengeri et al. 2011). Embedded clusters of young stellar objects , prominent outflows ; Motte et al. 2007;Duarte-Cabral et al. 2013Ching et al. 2018), and masers (Braz & Epchtein 1983;Argon et al. 2000;Pestalozzi et al. 2005) are found in the filament, indicating recent high-to intermediatemass star formation. The active star formation of the DR21 filament could be driven by both the mass accretion through the sub-filaments and the converging flows of clouds (Schneider et al. 2010;Hennemann et al. 2012). The magnetic fields of the DR21 filament have been mapped through single-dish observations of dust polarized emission (100 µm at 35 resolution: Dotson et al. 2000; 350 µm at 10 and 20 resolutions: Kirby 2009;Dotson et al. 2010; 800 µm at 14 resolution: Minchin & Murray 1994;Greaves et al. 1999; 850 µm at 14 resolution: Vallée & Fiege 2006;Matthews et al. 2009;1.1 mm at 19 resolution: Greaves et al. 1999;1.3 mm at 33 resolution: Glenn et al. 1999), revealing a uniform structure of magnetic fields at parsec scale that is perpendicular to the filament. Single-dish observations of CN Zeeman measurements at a resolution of 23 (0.16 pc) found line-of-sight magnetic field strengths of 0.4-0.7 mG in DR21 (OH) (Crutcher et al. 1999;Falgarone et al. 2008), and interferometric HI Zeeman observations at a resolution of 5 (0.03 pc) found a line-of-sight magnetic field strength of a few tenths mG toward the compact HII region of the DR21 core (Roberts et al. 1997). In contrast to the uniform magnetic fields of the filament, interferometric dust polarization observations reveal complex magnetic field structures in the massive dense cores of the filament, suggesting that the magnetic field plays a more important role in the formation of the DR21 filament than in the formation of the cores (Lai et al. 2003;Girart et al. 2013;Ching et al. 2017). A combined analysis of dust polarization data and molecular line data suggests that the gas dynamics arising from gravitational collapse may be the origin of distortion of the magnetic fields in the cores (Ching et al. 2018). Our observations toward the DR21 filament are part of the extension of the B-fields In STar-forming Region Observations (BISTRO) survey . The BISTRO-1 survey carried out POL-2 observations from 2016 to 2019 toward nearby starforming regions of the Gould Belt clouds, including Orion A Hwang et al. 2021), Ophiuchus (Kwon et al. 2018;Soam et al. 2018;Liu et al. 2019), IC 5146 (Wang et al. 2019), Barnard 1 (Coudé et al. 2019), NGC 1333 (Doi et al. 2020(Doi et al. , 2021, Auriga (Ngoc et al. 2021), Taurus (Eswaraiah et al. 2021), Orion B (Lyo et al. 2021), and Serpens (Kwon et al. 2022), aiming to generate a large sample of polarization maps in a uniform and consistent way to study the role of magnetic fields in star formation at a few thousand au scales. The BISTRO-1 survey was later extended to the BISTRO-2 program for high-mass star forming regions (M16: Pattle et al. 2018;Rosette: Könyves et al. 2021;NGC 6334: Arzoumanian et al. 2021;Mon R2: Hwang et al. 2022) and the ongoing BISTRO-3 program for various evolutionary stages and environments of star formation. In addition to individual target studies, the BISTRO data have been used to study the polarization properties of dust grains (Pattle et al. 2019;Fanciullo et al. 2022) and the alignment between magnetic fields and outflows (Yen et al. 2021). This paper is organized as follows: in Section 2, we describe the observations and data reduction; in Section 3, we present the results of the observations; in Section 4, we derive the magnetic field strength and study the relative orientation between magnetic field and filament structure; in Section 5, we discuss our results; and in Section 6, we provide a summary of this paper. OBSERVATIONS The JCMT polarization observations toward the DR21 filament were made by inserting the POL-2 polarimeter (Bastien et al. 2011;Friberg et al. 2016) into the optical path of the Submillimetre Common-User Bolometer Array 2 (SCUBA-2) camera (Holland et al. 2013). The observations were carried out with 20 sets of 42-minute integration in Grade 1 weather (τ 225GHz < 0.05) from July 2017 to February 2020 as part of the BISTRO-2 program (project ID: M17BL011). The observations were made using the POL-2 DAISY scan mode (Friberg et al. 2016), producing a fully sampled circular region of 12 arcmin diameter. Within the DAISY map, the noise is lowest and close to uniform in the central 3 arcmin diameter region, and increases to the edge of the map. The Flux Calibration Factors (FCFs) of SCUBA-2 at 850 µm were 516 Jy pW −1 beam −1 from November 2016 to June 2018 and 495 Jy pW −1 beam −1 post June 2018 (Mairs et al. 2021). Owing to the transmission losses from POL-2, the FCF of POL-2 is 1.35 times larger than the SCUBA-2 FCF (Dempsey et al. 2013). Weighted by the dates of the observations, the FCF of the POL-2 data toward the DR21 filament is 672 Jy pW −1 beam −1 . The effective beam size of JCMT is 14.1 at 850 µm (Dempsey et al. 2013), equivalent to 0.096 pc or 2.0 × 10 4 AU at the distance of DR21 filament. The data were reduced using the pol2map procedure (Parsons et al. 2018, software version on 2020/09/22) within the STARLINK/SMURF package Currie et al. 2014). The details of data reduction with pol2map are described in the earlier POL-2 works such as Liu et al. (2019) and Wang et al. (2019). In brief, the pol2map procedure first creates an initial Stokes I map from the POL-2 raw bolometer timestreams. Next, pol2map runs a second time with fixed-signal-to-noisebased masks generated from the initial Stokes I map to create improved Stokes I maps and co-adds the maps into a final Stokes I map. Finally, the masks and the final Stokes I map are used in a third run of pol2map to correct instrumental polarization and produce Stokes Q and U maps, along with their variance maps, and the debiased polarization catalogue. The noise levels in the Stokes Q and U maps are estimated from the Stokes Q and U variance maps, which are about 3.1 mJy beam −1 on the default 4 pixels of pol2map. The average and maximum of the noises in the Stokes I map are 3.4 and 13.8 mJy beam −1 , respectively. In this paper, we select polarization detections with criteria of I/δI ≥ 3, p/δp ≥ 3, and δp ≤ 4% for the uncertainty δI in Stokes I emission, the polarization fraction p, and the uncertainty δp in p. We plot the polarization segments with a 90 • rotation to show the magnetic field orientation projected on the plane of the sky (hereafter magnetic field segments), and we present one magnetic field segment in every two pixels, satisfying the Nyquist sampling of the 14.1 beam. To show the improvement of the POL-2 data, we also used the SCUPOL 850 µm polarization data of the DR21 filament. Matthews et al. (2009) built SCUPOL legacy catalog to provide reference Stokes cubes of comparable quality for 104 star-forming regions, including the observations of the DR21 filament of Vallée & Fiege (2006). Motte et al. (2007) are marked with filled black triangles and labelled in yellow, and the filamentary structures selected using filfinder are marked with orange dots along their crests. The names of the sub-filaments following Hennemann et al. (2012) are labeled in yellow. The magnetic field segments of the sub-filaments are shown in red color. We downloaded SCUPOL Stokes I, Q, and U cubes of DR21 from the legacy online catalogue 1 . When comparing the POL-2 and SCUPOL data sets, we first regrided the POL-2 data to a pixel size of 10 to match the SCUPOL map and then used the same criteria of I/δI ≥ 3, p/δp ≥ 3, and δp ≤ 4% to select polarization segments for both data sets, instead of the original criteria of p/δp > 2 in Matthews et al. (2009). Figure 1 presents the magnetic field segments of the DR21 filament inferred from our POL-2 observations. The detection of dust polarized emission is more extended than the results of Vallée & Fiege (2006) and Matthews et al. (2009), owing to a better sensitivity and a larger scan area of our observations. The Stokes I emission shows the DR21 main filament elongated in the north-south direction embedded with the bright sources DR21(OH) and DR21 in the middle and in the south of the filament. In the eastern and western sides of DR 21, the two lobes of dust emission extend to a size of about 0.5 pc, comparable to the morphology of the energetic outflows from DR21 (Davis & Smith 1996;White et al. 2010). The western side of the main filament is connected by the east-west elongated F1, F3, and SW sub-filaments, and the southern end of the filament is connected by the S sub-filament in the south-east direction. The magnetic field segments in the north of DR21(OH) are mostly horizontal to the filament, implying a parsecscale magnetic field perpendicular to the main filament. The horizontal magnetic fields are significantly changed to a northwest-southeast orientation in the region between DR21(OH) and DR21. The magnetic fields appear to be radial around DR21 and become arc-like in the two lobes of outflows. The arc-like morphologies of dust polarization are similar to those obtained from the imaging polarimetry of H 2 v = 1-0 S(1) line, which suggests a helical structure of magnetic fields wrapping around the outflows (Itoh et al. 1999). In the diffuse region, the magnetic fields of the sub-filaments are smoothly connected to the magnetic fields of the main filament. At the junctions of the sub-filaments and main filament, the magnetic fields appear to be parallel to the structures of the junctions. Figure 2a shows a zoom-in of the polarization map to reveal the detailed magnetic field structures of the main filament. In the north of DR21(OH), the horizontal magnetic fields are inclined in a northeast-southwest orientation in the eastern side of the filament and inclined in a northwest-southeast orientation in the western side. The inclined field morphology in the eastern and western sides of the filament is probably driven by the mass accretion of the filament. In addition, the orientation and morphology of the inclined magnetic fields in the northwest of the main filament appear to be correlated with those of the F1 and F3 sub-filaments. The magnetic fields around massive dense cores primarily follow the horizontal magnetic fields of the filament, except for the northeast-southwest oriented magnetic fields around DR21(OH). The northeast-southwest orientation of the magnetic fields around DR21(OH) are consistent with the small-scale magnetic fields inferred from interferometric observations of dust polarization (Lai et al. 2003;Girart et al. 2013), and we speculate that the distortion of the magnetic fields around DR21(OH) could be driven by the northeast-southwest bipolar outflows of DR21(OH) (White et al. 2010;Zapata et al. 2012;Girart et al. 2013). At the southern end of DR21(OH), the field morphology is slightly northwest-southeast oriented along the connecting bridge between DR21(OH) and DR21. The magnetic field morphology along the connecting bridge is probably regulated by the competitive mass accretion between the two massive cores. Because DR21(OH) is less massive than DR21, the magnetic fields in the southern end of DR21(OH) are pulled toward DR21, generating the fields that are straightened and redirected toward DR21 in a northwest-southeast orientation. The magnetic fields between DR21(OH) and DR21 regulated by competitive mass accretion appear to be similar to the field morphology between the massive cores in the W51 region (Koch et al. 2018). Around DR21, the magnetic fields show a pinched or hourglass morphology with an axis of symmetry along the northwest-southeast direction, consistent with the magnetic field structure inferred from the 350 µm dust polarization observations (Kirby 2009;Dotson et al. 2010). There are 13 magnetic field segments located in the sub-filaments, shown in red segments in Figure 1. We performed the filfinder algorithm (Koch & Rosolowsky 2015) to identity the crests of sub-filaments with parameters of a global threshold of 30 mJy beam −1 , a size threshold of 100 square pixels to extract filaments with length down to 40 (0.3 pc), a branch threshold of 7 pixels to minimize the length for a sub-filament to be 2 beams. The crests identified by filfinder are plot- Figure 2. Comparison of the dust polarization maps between POL-2 and SCUPOL observations. (a) The POL-2 polarization map of the main filament. The gray scale represents the Stokes I intensity, and the contours show the Stokes I emission at levels of 0.125, 0.25, 0.5, 1, 2, 4, 8, and 16 Jy beam −1 . The magnetic field segments are the same as those in Figure 1, but plotted in an unified length. The triangles mark the positions of the massive dense cores in Motte et al. (2007) with the DR21(OH) in the north and the DR21 in the south highlighted in white color. The dotted line remarks the boundary between the north filament and south filament. The orange and cyan arrows represent the directions of the red-shifted and blue-shifted outflows of DR21(OH) and DR21. Note here we only show the energetic outflows that might distort the POL-2 magnetic field segments in spite of the large number of outflows from the massive dense cores of the DR21 filament (e.g. Motte et al. 2007;Zapata et al. 2013;Ching et al. 2018). The JCMT 14.1 beam is plotted at the bottom left corner. (b) The SCUPOL magnetic field segments in blue overlapped with the POL-2 magnetic field segments in red. The length of the segment is proportional to the polarization percentage. The contours are the same as panel (a). The sixth contour at 4 Jy beam −1 is emphasized to show the regions with high consistency between the POL-2 and SCUPOL segments. ted in Figure 1, and the identifications of sub-filaments F1, F3, SW, and S are consistent with those in Kumar et al. (2007) and Hennemann et al. (2012). In Table 1, we list the positions, the position angles of magnetic fields (P A B ), the position angles of sub-filaments (P A f ), and the absolute position angles (P A |B−f | ) between P A B and P A f of the 13 magnetic field segments of sub-filaments. The P A f is determined by the five pixels of crests that are closest to the magnetic field segment. Figure 3 shows the histogram of P A |B−f | . The histogram of the position angles has more samples between 0 • and 45 • than between 45 • and 90 • , indicating that the magnetic fields tend to be parallel to the crests of sub-filaments, different to the perpendicular alignment between the magnetic fields and the DR21 main filament. The parallel alignment between magnetic fields and sub-filaments revealed in our POL-2 data is in agreement with the comparison of Herschel and Planck data that trace the S sub-filament and magnetic fields at a larger scale (Hu et al. 2021). Figure 4 compares the polarization fraction p with the Stokes I intensity for each of the POL-2 segments in Figure 1. There is an overall decreasing correlation of p with increasing I, and the low-intensity data have a steeper slope in the p-I correlation than the highintensity data. In addition, the polarization fractions of several low-intensity data exceed the observed maximum polarization fraction of 22 +3.5 . When the missing flux in Stokes I data is more severe than those in Stokes Q and U data, the missing flux issue can lead to a polarization fraction larger than the intrinsic value. The steep p-I correlation and the large polarization fractions of low-intensity data thus indicate that the low-intensity data suffer more Stokes I missing flux than the high-intensity data (see Section 3.3 for a further analysis of the total and polarized missing flux in POL-2 data). For the high-intensity data (I ≥ 0.5 Jy beam −1 ) in Figure 4, the polarization fractions of the segments in the north of the filament surrounding DR21(OH) are lower than those in the south of the filament surrounding DR21 (see Figure 2a for the separation boundary around the saddle region of the main filament). To study the p-I correlation, we use an empirical power-law model (Tamura et al. 1987) with Polarization properties p(I) = p 1 I Jy beam −1 −α ,(1) where p 1 is the polarization fraction at 1 Jy beam −1 . The best-fit model of the north filament gives α = 0.34± The gray dots represent the low-intensity data (I < 0.5 Jy beam −1 ). The black and red dots represent the high-intensity data (I ≥ 0.5 Jy beam −1 ) of the north filament and south filament, respectively. The best-fit models of the p-I correlations of the north and south filaments are shown by the black and red lines, respectively. 0.05 and p 1 = (2.10 ± 0.19)%, and the best-fit model of the south filament gives α = 0.30 ± 0.03 and p 1 = (3.64 ± 0.24)%. The difference of 0.04 between the α of the north filament and the α of the south filament is less than the uncertainty of 0.06 in the difference, whereas the difference of 1.54 % between the p 1 of the north and the p 1 of the south filaments is about five times larger than the uncertainty of 0.31 % in the difference. The consistent values of α indicate that the dust grains of the north and south filaments have a similar property, and the significant difference in the values of p 1 suggests that the Stokes I missing flux of the south filament is larger than the north filament, perhaps owing to differences in the intensities or spatial scales of the diffuse emission in the north and south filaments. The values of α inferred from POL-2 observations of several molecular clouds are usually from 0.5 to 0.9 (IC 5146: 0.56 +0.27 (Arzoumanian et al. 2021). The shallow α can be explained by the more evolved nature of the DR21 filament. According to modern grain align-ment theory (Lazarian & Hoang 2007;Hoang & Lazarian 2016;Hoang et al. 2021), dust grains are aligned by radiative torques, and the grain alignment toward the highest intensity is caused by the internal radiation from the massive central star. As a result, the embedded sources of DR21 and DR21(OH) may increase the alignment efficiency in the high-density regions, producing a shallower α than those found in clouds without embedded sources. Figure 2b compares the polarization maps of our POL-2 data with the SCUPOL data of Matthews et al. (2009). The noise level in the SCUPOL Stokes Q and U maps is about 13 mJy beam −1 , and that of the POL-2 data regridded to 10 pixel is about 2.2 mJy beam −1 . Above the sixth contour at 4 Jy beam −1 intensity, the two data sets are approximately consistent in both polarization angles and polarization degrees. Below the sixth contour, the differences between the two data sets become larger. In the region between DR21(OH) and DR21 and in the south-east region of DR21, the differences in polarization angles can be as large as 50 • , and the differences in polarization degrees can be as large as 15%. There are 215 pairs of spatially overlapping segments between the two data sets. Figure 5 shows the comparisons of polarization angles and polarization degrees for the overlapping segments. The polarization angles and polarization degrees of the segments satisfying I ≥ 4 Jy beam −1 show a better agreement between the two data sets than the segments weaker than 4 Jy beam −1 . The mean values of the absolute differences in polarization angles and polarization degree of the segments satisfying I ≥ 4 Jy beam −1 are 8.8 • and 0.50%, and those values of the segments satisfying I < 4 Jy beam −1 are 18.4 • and 2.7%. Comparison between POL-2 and SCUPOL results We further performed the two-sample Kolmogorov-Smirnov (KS) test to compare the likelihood of the POL-2 and SCUPOL polarization angles in Figure 5a. When using all the data points, the KS statistic is 10.2%, and the probability that the two samples have the same distribution at a KS significance level 0.05 is 19.9%. When using the data points with I ≥ 4 Jy beam −1 , the KS statistic is 29.2%, and the probability rises to 21.6%, indicating that the POL-2 and SCUPOL sets are likely originated from the same distribution only for the highintensity data points. The probability of the DR21 filament data is higher than the probabilities of 6% in the Ophiuchus C cloud ) and 0.6% in the Barnard 1 cloud (Coudé et al. 2019). The best consistency between the POL-2 and SCUPOL data has been found so far in the Ophiuchus B cloud with a probabil- Figure 2b. The color scale represents the Stokes I intensity of the data. To properly compare the polarization angles of POL-2 and SCUPOL data (i.e. the absolute difference between the two data sets should be less than 90 • ) and perform a KS test, some of the polarization angles are shifted from a range of [−90 • , 90 • ] to a range of [0 • , 180 • ] . ity of 90.5% (Soam et al. 2018). The KS test indicates that the consistency between the POL-2 and SCUPOL maps is better for the magnetic field segments with stronger I intensities, and the improvement of POL-2 from SCUPOL in the DR21 filament is similar to the improvement of POL-2 data in other clouds. Considering that the sensitivity of our POL-2 data is about 3 times better than the SCUPOL data, the POL-2 segments are more reliable than the SCUPOL segments. The p-I correlation using the SCUPOL 439 polarization segments of DR21 filament gives α = 0.50 ± 0.01 (Poidevin et al. 2013). Considering that the distribution of the SCUPOL polarization segments is more extended than the lowest contour at 0.125 Jy beam −1 in Figure 2b, the α derived from the SCUPOL data might be biased by the missing flux issue in the low-intensity data and therefore is steeper than our values of α = 0.30-0.34. Global magnetic fields inferred from the Planck data Planck 850 µm (353 GHz) polarization data are used to study the large-scale magnetic fields of the DR21 filament at the 5 (∼ 2.0 pc) resolution of the Planck beam. The 2015 release of Planck HFI maps (PR2, Planck Collaboration I 2016), where the monopole of the cosmic infrared background has been subtracted (Planck Collaboration VIII 2016), were obtained from the Planck Legacy Archive 2 . To compare the Planck and JCMT results, we transform the polarization angles of Planck data that are originally obtained in galactic coordinates into the polarization angles in equatorial coordinates by computing the angle ψ between the equatorial north and the galactic north. For epoch J2000, ψ = arctan cos(l − 32.9 • ) cos b cot 62.9 • − sin b sin(l − 32.9 • ) ,(2) where l and b are the galactic coordinates of the object (Corradi et al. 1998, see Appendix A for the derivation). Figure 6 shows the large-scale magnetic fields inferred from Planck polarization data satisfying p/δp ≥ 3 overlaid on the map of dust optical depth at 353 GHz (τ 353 ) in Planck Collaboration XI (2014). The DR21 filament is the most prominent object in this 30 pc × 30 pc map even though DR21 is close to the galactic disk plane at b ∼ 0.6 • . The diffuse regions on the east side, north side, and in the northwest corner of the map show a fairly regular global magnetic field with a northeast-southwest orientation, parallel to the galactic disk plane. This regular field is distorted in the medium above an intermediate optical depth of τ 353 ∼ 7×10 −4 in the south of the map, probably owing to the active star-forming activity of the Cygnus X complex. Toward the DR21 filament, a bent morphology of magnetic fields is notable: the northeast-southwest oriented global field is bent to an east-west orientation in the middle of the filament and bent to a northwest-southeast orientation in the south end of the filament, consistent with the main features of the magnetic fields of the DR21 filament at the 0.1 pc resolution of Figure 1. The polarized flux in the 1. 7-sized central pixel of Figure 6 is 3.81 mK CMB × 1. 7 2 = 253 mJy, and the integrated POL-2 polarized flux over the identical area after smoothing Figure 1 to a resolution of 5 is 240 mJy. The consistency of the Planck polarized flux and POL-2 polarized flux indicates that the east-west oriented magnetic field at the center of Figure 6 is primarily traced by the polarized emission of the filament rather than by the polarized emission of the diffuse region. Therefore, the bent morphology of large-scale magnetic fields toward the DR21 filament is associated with the 0.1-pc-scale magnetic fields of the filament rather than a distortion of large-scale magnetic fields in the diffuse region. The Stokes I flux in the central pixel of Figure 6 is 0.525 K CMB × 1. 7 2 = 34.9 Jy, whereas the integrated POL-2 Stokes I flux over the identical area is 18.2 Jy. The missing large-scale flux of the POL-2 Stokes I data is more severe than the Stokes Q and U data. A similar trend of more missing flux in Stokes I than Q and U is found in the POL-2 and Planck data of NGC 1333 (Doi et al. 2020). The POL-2 Stokes I missing flux of NGC 1333 is 13% of the Planck flux, and the missing flux of DR21 filament is 48%. The missing flux of POL-2 data comes from the background subtraction of atmospheric signal in the pol2map procedure, making POL-2 data not sensitive to diffuse emission with spatial scales larger than the size of the observed region. Since the diffuse emission of DR21 filament is stronger than that of NGC 1333, the missing flux of DR21 filament hence is larger than the missing flux of NGC 1333. In Figure 5b, the polarization degrees of the POL-2 data are preferentially larger than those of the SCUPOL data, contrary to the general results of slightly smaller polarization degrees of POL-2 data than those of SCUPOL data (Soam et al. 2018;Doi et al. 2020). Again, the large POL-2 polarization degrees of DR21 filament are likely caused by the large missing Stokes I flux in the POL-2 data. To estimate the magnetic field strength in molecular clouds from dust polarization observations, the Davis-Chandrasekhar-Fermi (hereafter DCF, Davis 1951;Chandrasekhar & Fermi 1953) equation is the most widely used method. The DCF equation assumes that the ratio of turbulence to magnetic field strength would lead to a similar level of variation in the magnetic fields as well as in the velocities, δB/B δV los /V A , where B is the strength of the magnetic field, δB is the variation about B, δV los is the velocity dispersion along the line of sight, and V A = B/ √ 4πρ is the Alfvén speed at density ρ. Since dust polarization segments trace the plane-of-sky component of magnetic field, the variation in the plane-of-sky magnetic field strength is expected to be proportional to the measured dispersion of polarization angles, i.e., δB/B pos ∼ δΦ. Consequently, the DCF equation can be written as B pos = F 4πρ δV los δΦ ,(3) where F is a correction factor usually assumed to be ∼ 0.5, accounting for the smoothing of magnetic fields along the line of sight and the inadequate spatial resolution of dust polarization observations (Heitsch et al. 2001;Ostriker et al. 2001;Padoan et al. 2001). To avoid inaccurate estimation of δB/B pos from simply taking the dispersion of polarization angles, refinements of the DCF equation with more sophisticated statistical analyses have been made. Hildebrand et al. (2009) proposed a structure function analysis of the polarization angle difference between every pair of polarization segments in a given map as a function of the segment separation. In this structure function analysis, the plane-of-sky magnetic field is assumed to be composed of a large-scale ordered component B 0 and a small-scale turbulent component B t , and the ratio of B 0 to B t can be fitted without a priori assumption on the turbulence in the cloud or the morphology of the large-scale field. Houde et al. (2009) proposed an angular dispersion function method to expand the structure function analysis by including the signal integration across the telescope beam and through the line-of-sight depth of the source. Recently, the method of Houde et al. (2009) has become well recognized in deriving magnetic field strength from dust polarization maps of single-dish observations (Chuss et al. 2019;Liu et al. 2019;Coudé et al. 2019;Wang et al. 2019;Soam et al. 2019;Eswaraiah et al. 2020;Guerra et al. 2021) and numerical simulations (Liu et al. 2021). Houde et al. (2009) suggest that if the correlation length δ for B t is much smaller than the thickness of the cloud ∆ , the ratio of B t to B 0 can be evaluated from the angular dispersion function in the form 1 − cos [∆Φ (l)] 1 N cell B 2 t B 2 0 × 1 − e −l 2 /2(δ 2 +2W 2 ) + ∞ j=1 a 2j l 2j ,(4) where ∆Φ (l) is the polarization angle difference between polarization segments separated by a distance l, W is the beam width (i.e., the FWHM beam divided by √ 8 ln 2), the summation is a Taylor expansion representing the structure in the B 0 that does not involve turbulence, and N cell is the number of turbulent cells along the line of sight obtained by N cell = (δ 2 + 2W 2 )∆ √ 2πδ 3 .(5) The turbulence component in the angular dispersion function is b 2 (l) = 1 N cell B 2 t B 2 0 e −l 2 /2(δ 2 +2W 2 ) .(6) Since B t is the source of perturbation in B 0 , the B 2 t / B 2 0 derived from Equation 4 provides a good approximation of the δB/B pos in the DCF equation for evaluating the magnetic field strength on the plane of sky as B pos = 4πρδV los B 2 t B 2 0 −1/2 .(7) Angular Dispersion Function of the JCMT and Planck data Figure 7 shows the angular dispersion functions of the JCMT and Planck data toward the DR21 filament. Since the main feature of horizontal POL-2 segments in the north of the filament are notably different to the radial POL-2 segments in the south of the filament, we perform the analysis separately for the POL-2 segments in the north filament ( Figure 7a) and in the south filament (Figure 7b). The numbers of segment pairs reach a maximum at l = 80 for the POL-2 data and at l = 30 for the Planck data, implying that the angular dispersion functions are fully sampled below 80 for the JCMT map and fully sampled below 30 for the Planck map. Here we focus on the fully sampled data points. The POL-2 angular dispersion function in the north filament is slightly smaller than that in the south filament, indicating that the magnetic fields in the north are more ordered than those in the south. Owing to limited angular resolution, the angular dispersion function is close to zero when the length scale l is smaller than the beam. At scales above the beams, the angular dispersion functions of POL-2 and Planck data are both at a level between 0.2 and 0.3, indicating that the ratio of ordered to turbulent magnetic fields remains similar from small to large scales. In addition, all the angular dispersion functions of the POL-2 and Planck data are below the angular dispersion of a random field (1−cos 52 • = 0.384; Poidevin et al. 2010), indicating that the magnetic fields of the DR21 filament are considerably not random. We use the nonlinear least-squares Marquardt-Levenberg algorithm 3 to fit the parameters of δ, B 2 t / B 2 0 , and a 2j in Equation 4. The mean central width of the DR21 main filament and sub-filaments derived from the Herschel map is about 0.34 pc (Hennemann et al. 2012), and we use the width as the effective thickness ∆ for both the JCMT and Planck data, assuming that the DR21 main filament is similar to an edge-on cylinder and its thickness is close to its width. We only fit the fully sampled data points, and the parameters a 2j are reduced to first order a 2 because the fitting range is small. The best fits of the angular dispersion functions are shown in Figure 7, and the fitted parameters are listed in Table 2. The correlation lengths δ of the POL-2 north segments, POL-2 south segments, and Planck segments are 7. 5 ± 1. 5, 17. 3 ± 3. 1, and 2. 5 ± 1. 4, respectively (see Table 2 for the δ in parsec). Except for the POL-2 north filament, the correlation lengths are not resolved by the beams. The N cell and B 2 t / B 2 0 of the sources are between 1.4 and 6.1 and between 0.2 and 0.5, respectively. Both the N cell and B 2 t / B 2 0 suggest that the magnetic fields more ordered than disturbed by turbulence, as the POL-2 segments in Figure 2 are dominated by ordered magnetic fields perpendicular to the main filament and the Planck segments in Figure 6 are dominated by ordered magnetic fields parallel to the galactic disk plane. To derive the magnetic field strength using Equation 7, we adopt column densities of 41.6 × 10 22 cm −2 for the main filament and ∼ 2 × 10 22 cm −2 (note this value is consistent with the density derived from the Planck data in Figure 9) for the diffuse region obtained from the Equation 4), and the red line shows the ordered component a 2 l 2 + b 2 (0) of the best fit. The dotted vertical and horizontal lines denote the beam size and the expected value for random magnetic fields, respectively. Bottom panels: the dots represent the correlated component of the best fit to the data. The blue line shows the turbulent component b 2 (l) of the best fit, and the red line shows the correlation due to the beam (i.e., b 2 (l) when δ = 0). ) with ∆ = 0.34 pc to derive the number densities n of the POL-2 and Planck maps, using ρ = µm H n where µ = 2.86 is the mean molecular weight Pattle et al. 2015) and m H is the atomic mass of hydrogen. DR21 Herschel map We estimate the δV lsr from the velocity dispersion (σ) of the H 13 CO + 1-0 data at an angular resolution of 29 , since the emission of H 13 CO + is well correlated with the dust emission in the DR21 filament (Schneider et al. 2010). The derived B pos strengths are ∼ 0.6 mG in the north filament, ∼ 1.0 mG in the south filament, and ∼ 0.1 mG in the diffuse region. Our value of the north filament is consistent with the SCUPOL results of B pos = 0.78 mG derived using the DCF method (Equation 3) in Vallée & Fiege (2006) and B pos = 0.62 mG derived from the angular dispersion function analysis in Girart et al. (2013). However, our value is about 6 times weaker than the 2.8-3.9 mG in Poidevin et al. (2013) using the structure function analysis of Hildebrand et al. (2009), mainly owing to a 10 times larger n assumed Data δ B 2 t / B 2 0 a 2 N cell n δV a los B b pos λ c (pc) (arcsec −2 ) (cm −3 ) (km s −1 ) (mG) POL-2 North (51 ± 10) ×10 −3 0.49 ± 0.17 (3.6 ± 0.3) × 10 −5 6.1 ± 2.6 4.0 × 10 5 1.0 0.63 ± 0.18 3.9 POL-2 South (117 ± 21) ×10 −3 0.18 ± 0.02 (3.8 ± 0.7) × 10 −5 1.4 ± 0.4 4.0 × 10 5 1.0 1.04 ± 0.13 2.4 Planck 1.02 ± 0.16 0.27 ± 0.16 (4.0 ± 0.6) × 10 −5 1.6 ± 0.9 2.0 × 10 4 0.7 0.13 ± 0.04 0.9 a Values from single-dish H 13 CO + 1-0 observations (Schneider et al. 2010). b Assume 10% uncertainty in n and δV los to estimate the uncertainty in Bpos. c Obtained with Bpos = π 4 B. in Poidevin et al. (2013). Our B pos for the south filament is lower by a factor of three than the values from 2.5 to 3.1 mG derived using the DCF method from 350 µm polarization data by Kirby (2009). The difference between the two works is primarily owing to that Kirby (2009) using a velocity dispersion of 4.2 km s −1 from the HCN 4-3 line toward the DR21 core, which is about four times larger than what we used. From Equations 4 and 5, 1 − cos [∆Φ (l)] is proportional to 1/∆ × B 2 t / B 2 0 , and hence ∆ and B 2 t / B 2 0 are coupled. Considering that our assumption of ∆ = 0.34 pc of the diffuse region is an underestimation, B 2 t / B 2 0 is also underestimated. Therefore, our value of 0.13 mG in the diffuse region could be an upper limit of B pos in the Planck data. Histogram of Relative Orientations Formalism Dust polarization orientations in molecular clouds often show correlations with the intensity gradients inferred from the dust continuum contours (Goodman et al. 1990;Chapman et al. 2011;Koch et al. 2012). We quantify the relative orientation of the magnetic field with respect to the column density structures of the DR21 filament using the histogram of relative orientations (HRO, Soler et al. 2013). In the HRO technique, the relative orientation angle φ between the magnetic field and the tangent to the column density contour is evaluated using φ = arctan B × ∇N B · ∇N ,(8) where B is the magnetic field orientation inferred from the polarization map and ∇N is the gradient of column density, used to characterize the column density structures. Although the range of arctan function is [−90 • , 90 • ], we use a range [0 • , 90 • ] for φ without loss of generality as suggested in , since the relative orientation is independent of the reference and thus φ is equivalent to −φ. The convention of φ is equivalent to the |90 • − δ| in Koch et al. (2013) that φ = 0 • indicates that the magnetic field is parallel to the tangent of the column density contour (perpendicular to the column density gradient), and φ = 90 • indicates that the magnetic field is perpendicular to the tangent of the column density contour (parallel to the column density gradient). To obtain an HRO, the gradients of a column density map and the magnetic field segments of a polarization map are first compared pixel by pixel to produce a map of φ. Next, the map of φ is divided into bins of column densities containing an equal number of segments, and an HRO is generated for each bin to examine the change in φ with increasing column densities. For maps with small uncertainties in column densities and polarization angles, the typical propagated error in φ is usually less than 10 • . Hence, by presenting an HRO with angle bins of a width larger than the error in φ, the uncertainty in the HRO is dominated by the histogram binning process. The variance in the kth histogram bin is given by σ 2 k = h k 1 − h k h tot ,(9) where h k is the number of samples in the kth bin and h tot is the total number of samples (Planck Collaboration Int. XXXV 2016). To evaluate the preferential relative orientation in each column density bin, the shape of the HRO is quantified using a histogram shape parameter ξ, defined as, ξ = A 0 − A 90 A 0 + A 90 ,(10) where A 0 is the area under the histogram in the range 0 • < φ < 22.5 • and A 90 is the area under the histogram in the range 67.5 • < φ < 90 • ). An HRO peaking at φ = 0 • would have ξ > 0, an HRO peaking at φ = 90 • would have ξ < 0, and a flat HRO would have ξ ∼ 0. The uncertainty in ξ is obtained from σ 2 ξ = 4(A 2 90 σ 2 A0 + A 2 0 σ 2 A90 ) (A 0 + A 90 ) 4 ,(11) where σ 2 A0 and σ 2 A90 represent the variances of the areas, characterizing the "jitter" of the histograms. The value of ξ is nearly independent of the number of angle bins selected to represent the histogram if the bin widths are smaller than the integration range, but the jitter does depend on the number of angle bins in the histogram. If the jitter is large, σ ξ is large compared to |ξ|, and the relative orientation is indeterminate (Planck Collaboration Int. XXXV 2016). Finally, analyses of HROs characterize the trend of the relative orientation between magnetic fields and column density structures of a cloud from its low to high density regions with a linear regression between ξ and atomic gas column density N (H) (Planck Collaboration Int. XXXV 2016): ξ = C HRO log 10 (N (H)/cm −2 ) − X HRO .(12) Histogram of Relative Orientations of the JCMT and Planck data In order to further compare the analyses of HROs of Planck and JCMT data from low-density to high-density regimes, we construct the column density maps of the data. To convert the JCMT dust continuum map to a column density map, we calculate the column density N (H 2 ) of molecular gas as follows: N (H 2 ) = γI ν µm H κ ν B ν (T ) ,(13) where γ is the gas-to-dust ratio of 100, I ν is the Stokes I intensity at frequency ν, κ ν = 1.5 cm 2 g −1 is the dust opacity at 850 µm of cool and dense dust mantles (Ossenkopf & Henning 1994), and B ν (T ) is the Planck function at the dust temperature T of 15 K previously measured in the DR21 filament . To scale the Planck τ 353 map to a column density map, we calculate the column density N (H) of atomic gas following the dust opacity relation found using Galactic extinction measurements of quasars (Planck Collaboration XI 2014), τ 353 /N (H) = 1.2 × 10 −26 cm 2 .(14) We next calculate the gradients of the N (H 2 ) and N (H) maps using the Gaussian Derivatives method described in Soler et al. (2013). To obtain gradients at the pixels of the POL-2 and Planck magnetic field segments in Figures 1 and 6, we apply a 3 × 3 derivative kernel over the grid of pixels illustrating magnetic field segments. Since the gradients in this grid are computed over two FWHM beams for both the JCMT and Planck data, obtaining gradients using this method guarantees adequate sampling of gradients. Figures 8 and 9 show the gradient segments of the column density maps and the maps of φ of the JCMT and Planck data. The majority of POL-2 segments in the DR21 main filament tends to be parallel (φ = 90 • ) to the gradient segments with N (H 2 ) 10 23 cm −2 . In the low column density regions of the JCMT map, the alignment between POL-2 segments and gradient segments becomes less significant. The large-scale magnetic field segments and gradient segments in the Planck map appear to be more randomly aligned than the small-scale segments in the JCMT map. The uncertainty in the position angle of the gradient is determined by the derivative of the noise in the column density map (Planck Collaboration Int. XXXV 2016). Since the respective noise levels in the POL-2 Stokes I map and the Planck τ 353 map are much less than a few percent of the I map and τ 353 values, the uncertainties in the gradient directions are typically less than 1 • . We use a selection criterion of p/δp ≥ 3 for the magnetic field segments, corresponding to an uncertainty less than 10 • in polarization angle (Naghizadeh-Khouei & Clarke 1993). Therefore, we expect that the errors in φ are less than 10 • . We divide the 765 measurements of φ of the POL-2 data into 5 N (H 2 ) bins and the 371 measurements of φ of the Planck data into 3 N (H) bins to calculate HROs. Figures 10 and 11 plot the HROs of the JCMT and Planck data using 6 angle bins each of 15 • width. The HROs reveal different kinds of relative orientations between magnetic fields and column density contours of the JCMT and Planck data. The JCMT HRO of the lowest N (H 2 ) bin increases slightly from φ = 0 • to φ = 90 • , and the HROs of the intermediate and highest N (H 2 ) bins show prominent peaks at 90 • , suggesting a trend from a weak perpendicular orientation of φ in regions with N (H 2 ) 10 22.5 cm −2 to a strong perpendicular orientation of φ for N (H 2 ) 10 22.5 cm −2 in the DR21 main filament. In contrast, the Planck HROs are flat for all of the three N (H) bins, suggesting no preferential orientation of φ in the large-scale diffuse region of the DR21 filament. Figure 12 presents the measurements of ξ in different N (H) bins derived from the HROs of the JCMT and Planck data. To compare the ξ of the two data sets, the column density N (H 2 ) is transferred to the N (H) assuming 2×N (H 2 ) = N (H). The ξ of the three Planck N (H) bins are consistently close to zero with a relatively large value of σ ξ . The ξ of the lowest JCMT N (H) bin is slightly smaller than the ξ of the three Planck Figure 8. Comparison of the magnetic field segments and column density gradient segments of the POL-2 data. (a) N (H2) column density map derived from the JCMT Stokes I emission overlaid with the gradient segments calculated by convolving the column density map with a Gaussian derivative kernel. The contours show the N (H2) at levels of 0.125, 0.25, 0.5, 1, 2, 4, 8, and 16 ×10 23 cm −2 . The length of the gradient segments is normalized. The gradient segments shown here are those overlaid with the magnetic field segments in Figure 2a. (b) The map of relative orientation angle φ between the magnetic field segments in Figure 2a and the gradient segments in panel (a). The contours are the same as panel (a) with the contour at 5 × 10 22 cm −2 emphasized to show the transition from no preferential orientation of φ in low density regions to perpendicular orientation of φ in high density regions. Figure 9. Comparison of the magnetic field segments and column density gradient segments of the Planck data. Top panel: N (H2) column density map derived from the Planck τ353 map overlaid with the gradient segments calculated by convolving the column density map with a Gaussian derivative kernel. The contours show the N (H2) at levels of 2, 4, 8, and 16 ×10 22 cm −2 . The length of the gradient segments is normalized. The gradient segments shown here are those overlaid with the magnetic field segments in Figure 6. Bottom panel: The map of relative orientation angle φ between the magnetic field segments in Figure 6 and the gradient segments in top panel. bins. Considering that the σ ξ of the four data points are relatively large and the missing flux of the POL-2 Stokes I measurement (see Section 3.3) might cause the lowest JCMT N (H) bin to be smaller than its intrinsic column density, the ξ of the JCMT data seems to agree with the ξ of the Planck data. The ξ for the rest of the JCMT N (H) bins are broadly negative, indicating a strong preference of perpendicular alignment between the small-scale magnetic fields and the ridge of the DR21 filament (parallel alignment between magnetic fields and density gradients). One of the important parameters required to evaluate the role of magnetic fields in star formation is the dimensionless mass-to-flux ratio λ, which refers to the ratio of the mass in a magnetic flux tube to the magnitude of magnetic flux (Crutcher et al. 2004). In units of its critical value of 2πG 1/2 , λ = (Shu et al. 1987;Mouschovias & Ciolek 1999), clouds are initially magnetically subcritical (λ < 1), and to become a starforming region, magnetic supercriticality (λ > 1) of a cloud is required for the self-gravity to overwhelm the magnetic support and form stars through gravitational collapse. Using the statistically most probable value of B pos = π 4 B (Crutcher et al. 2004) and the column density obtained from the Herschel map (see Section 4.1.2), the λ values of the main filament and diffuse region are listed in Table 2. Given the uncertainties in the column densities, in the B pos , and in the projection correction from B pos to B, we estimate that the uncertainty in λ would be as large as half of the value. The λ values of the main filament is about 2.4-3.9, consistent with the value of 3.4 obtained using SCUPOL data (Girart et al. 2013) and the value of 2-3 obtained using the CN Zeeman measurements toward DR21(OH) (Crutcher et al. 1999). The λ of the diffuse region is 0.9, however, which should be taken as a lower limit because the B pos of the diffuse region could be overestimated. These λ imply different roles of magnetic fields in the main filament and the surrounding diffuse region. The significant supercriticality of the main filament implies that self-gravity dominates magnetic fields and the filament is undergoing gravitational collapse, in agreement with the infall motions of the filament suggested from molecular line observations (Schneider et al. 2010;Csengeri et al. 2011). Meanwhile, the observed perpen-dicular alignment between the main filament and magnetic fields is consistent with the MHD simulations of a strongly magnetized medium that magnetic field can regulate mass flows along field lines to form parsec-scale filamentary structures perpendicular to magnetic fields (e.g. Nakamura & Li 2008;Inoue & Fukui 2013;Chen & Ostriker 2014;Li & Klein 2019). In contrast, the λ of the diffuse region is slightly subcritical or nearly critical, indicating that the ambient gas is incapable to form the DR21 filament through direct gravitational collapse. Considering that the column densities of the subfilaments are between those of the diffuse region and main filament, the λ of sub-filaments should be larger than that of the diffuse region and smaller than that of the main filament. In other words, the sub-filaments might be the places where the transition from subcriticality to supercriticality occurs. The sub-filaments of DR21 appear to be parallel to the parsec-scale magnetic fields and perpendicular to the main filament. These features are similar to the striations around filamentary clouds in MHD simulations formed via Alfvén waves (Heyer et al. 2008;Tritsis & Tassis 2018) or Kelvin-Helmholtz instability (Chen et al. 2017;Li & Klein 2019). The magnetic fields of the DR21 filament seem to play a more important role on large scales and become less important on small scales. At scales of a few parsecs, the magnitude of magnetic flux is comparable to self-gravity, preventing the collapse of ambient gas. For the parsecscale main filament, the magnetic fields are important in shaping the filamentary structure, even though the magnetic fields are overwhelmed by the self-gravity of the filament. The magnetic fields of six massive dense cores, including DR21(OH), in the filament have been studied in Girart et al. (2013) and Ching et al. (2017) using dust polarization observations at resolutions of a few thousand au. In contrast to the ordered parsec-scale magnetic fields that are perpendicularly aligned to the filament, the magnetic fields of those cores have complex structures that appear to be randomly aligned to the core structures. The λ of the cores are supercritical with values comparable to that of the main filament, but the ratio of virial kinematic energy to virial magnetic energy of the cores is at least an order of magnitude larger than that of the filament. Meanwhile, molecular line observations suggest that increasing kinetic energy in the core comes from gravitational collapse and might be the source of the distortion of the magnetic fields into complex structures (Ching et al. 2018). Hence, the massive cores appear to be weakly magnetized, and self-gravity and gas dynamics are more important than magnetic fields in the formation of massive dense cores. Down to scales of ∼ 1000 au, the study of fragmentation of 18 massive dense cores, including three cores in the DR21 filament, suggests that the correlation between the fragmentation levels and the number densities of the cores is stronger than the correlation between the fragmentation levels and the λ of the cores (Palau et al. 2021). Comparison of the HROs of the DR21 Filament and Other Clouds The HRO of the Serpens Main region of the BISTRO survey has been studied in Kwon et al. (2022). In Figure 12, the ξ measurements of the Serpens Main region are overlaid on those of the DR21 filament. Both the DR21 and the Serpens Main data show a turning point of ξ around N (H) of 10 23 cm −2 . Owing the displacement in column density between the JCMT and Planck data, we select the ξ of the JCMT data below N (H) = 10 23 cm −2 to derive the C HRO and X HRO in Equation 12. The resulting C HRO of −0.23 reflects the trend that ξ changes from a value close to zero in the low N (H) bins to negative values in the high N (H) bins. The resulting X HRO of 21.2, equivalent to 1.58 × 10 21 cm −2 , corresponds to a characteristic column density where ξ changes its sign, or in other words, a boundary where the relative orientation between the column density structures and magnetic fields changes from a more random orientation in low-density regions to a non-random, preferentially perpendicular orientation in high-density regions. The HROs of ten nearby Gould Belt molecular clouds (at distances of less than 450 pc, namely Taurus, Ophiuchus, Lupus, Chamaeleon-Musca, Corona Australis, Aquila Rift, Perseus, IC5146, Cepheus, and Orion) have been measured using the Planck data smoothed to 10 resolution (Planck Collaboration Int. XXXV 2016). The ξ of the HROs are found to decrease with increasing N (H), indicating field orientation from preferentially parallel or having no preferred orientation at the lowest N (H) ∼ 10 21 cm −2 of the data to preferentially perpendicular at the highest N (H) ∼ 10 22.5 cm −2 of the data. Except for the Corona Australis cloud that shows an almost flat slope of ξ, the C HRO of the other nine clouds have a range from −0.22 to −0.68, and the X HRO have a range from 21.67 to 22.70. The HROs of the ten clouds and the high-latitude cloud L1642 have further been studied between the N (H) derived from Herschel data at 20 resolution and the magnetic fields inferred from Planck 850 µm polarization data, and negative slopes of ξ versus N (H) are identified (Malinen et al. 2016;Soler 2019). Besides the Planck polarization data, the HRO analysis applied to the BLASTPol data at 250 µm, 350 µm, and 500 µm at 3 resolution toward the Vela C molecular complex with N (H) from 10 21.7 cm −2 to 10 23.3 cm −2 also suggests a similar trend of HRO as the Planck results ). The C HRO and X HRO of the DR21 filament and the Serpens Main region for N (H) < 10 23 cm −2 are −0.23 and 21.2, consistent with the values of other molecular clouds in the same density regime. The observed change in the HRO from mostly parallel alignment between magnetic fields and sub-filaments of diffuse gas to mostly perpendicular alignment between magnetic fields and dense filaments of clouds is consistent with recent simulations of MHD turbulence with strong magnetic fields, indicating that magnetic fields play a significant role in structuring the interstellar medium in and around molecular clouds (Soler et al. 2013;Soler & Hennebelle 2017). Yet, there are two features in Figure 12 that are different to the HROs of most molecular clouds. First, because the angular resolution of POL-2 is higher than other single-dish polarimeters, the HROs of the DR21 filament and the Serpens Main region trace the highest N (H) of 10 24 cm −2 . Second, the ξ of the DR21 filament and the Serpens Main region reaches a minimum between −0.6 and −0.8, which is lower than other molecular clouds, except for the HRO of Musca obtained from Herschel and Planck data in Soler (2019). Considering that a perfectly perpendicular alignment between magnetic field and filament would give ξ = −1, it seems reasonable for high angular resolution observations to obtain a ξ close to −1 in a high-density filament, such as the DR21 filament. For N (H) > 10 23 cm −2 , Figure 12 shows a tentative positive slope in ξ versus N (H) in both the DR21 filament and the Serpens Main region. Similar trends of positive slopes in high N (H) regimes can be found in the HROs of Lupus I, Musca, Perseus, and Vela C South-Nest Soler 2019). Statistical studies of magnetic fields in star-forming cores suggest that the small-scale magnetic fields of cores are neither simply aligned with the large-scale magnetic fields of filaments nor simply aligned with the major axes of filaments (Zhang et al. 2014;Koch et al. 2014). Therefore, the tentative positive slope in the high N (H) regime of the HRO may indicate the transition from preferentially perpendicular alignment between filaments and magnetic fields to complex structure of the alignment between dense cores and magnetic fields. Characterizing HROs with high angular resolution submillimeter polarimeters such as POL-2 or HAWC+ which are capable of probing magnetic fields in high-density filaments will be helpful in deciphering the role of magnetic fields in the evolution from filaments to star-forming cores. CONCLUSIONS We present JCMT POL-2 850 µm polarization observations of the DR21 filament. With the Planck 850 µm dust polarization data, we were able to characterize the magnetic field structures from the surrounding ambient gas to the DR21 filament at scales from 10 pc to 0.1 pc. Our main results are the following: 1. The POL-2 data reveal ordered parsec-scale magnetic fields that are perpendicular to the DR21 main filament and parallel to the sub-filaments. The magnetic fields of the sub-filaments appear to smoothly connect to the magnetic fields of the main filament. The magnetic fields revealed in the Planck data are well aligned with those of the POL-2 data, indicating a smooth variation of magnetic fields from large to small scales. 2. The comparison of the total and polarized flux of the POL-2 and Planck data indicates that the missing flux issue of the POL-2 DR21 observations is more severe in Stokes I data than Stokes Q and U data. In addition, the large polarization fractions ( 20%) of POL-2 low-intensity data and the preferentially large polarization fractions of POL-2 data than SCUPOL data can be explained by the Stokes I missing flux. 3. We find a power index α of 0.30-0.34 of the correlation between the polarization fractions and Stokes I intensities of POL-2 data. The α value is consistent with those inferred from the POL-2 observations toward massive star-forming regions Orion B and NGC 6334 but shallower than the POL-2 observations toward less massive clouds, suggesting that the dust grain alignment efficiency of DR21 main filament is strongly influenced by the stellar radiation from the newborn stars. 4. The analysis of the angular dispersion functions of dust polarization yields B pos of 0.6-1.0 mG in the DR21 filament and ∼ 0.1 mG in the surrounding ambient gas. The material is found to be magnetically supercritical in the filament and slightly subcritical to nearly critical in the ambient gas, consistent with the observed global infall motions of the DR21 filament. The sub-filaments might be the places where the transition from subcriticality to supercriticality occurs. 5. The histogram of relative orientations between the density gradient and the magnetic field of the DR21 filament decreases with increasing N (H) from no preferred alignment in the low-density ambient gas to mostly perpendicular in the highdensity filament, in agreement with the HROs in other clouds. Owing to the high angular resolution of POL-2, we are able to trace the HRO in the highest N (H) regime to date. A tentative positive slope of the HRO in the high-density DR21 filament is also found, as suggested from the complex magnetic field structures of the star-forming cores in the filament. In summary, the analyses including the B pos , magnetic criticality, and histogram of relative orientations are all in good agreement with recent MHD simulations of a strongly magnetized medium, suggesting that magnetic fields play an important role in shaping the main filament and sub-filaments of the DR21 region. The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of the National Astronomical Observatory of Japan, the Academia Sinica Institute of Astronomy and Astrophysics, the Korea Astronomy and Space Science Institute, and the Center for Astronomical Mega-Science. Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom, Canada, and Ireland. Additional funds for the construction of SCUBA-2 and POL-2 were provided by the Canada Foundation for Innovation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. In the IAU convention, the orientation of a position angle and a polarization angle is measured from North and positively towards East. At a position P on the sky, the position angle measured in galactic coordinate (P A GA ) is different to the position angle measured in equatorial coordinate (P A EQ ) by the angle ψ between the galactic North Pole (N GA ) and the equatorial North Pole (N EQ ) as P A EQ = P A GA − ψ. (A1) According to spherical trigonometry, ψ can be derived with two sides and an opposite angle given ( Figure 13). That is, with the side N GA N EQ = b N GA − b N EQ , the side N GA P = b N GA − b P , and the angle ∠N EQ N GA P = l N EQ − l P , tan(ψ) = sin(b N GA − b N EQ ) sin(l N EQ − l P ) sin(b N GA − b P ) cos(b N GA − b N EQ ) − cos(b N GA − b P ) sin(b N GA − b N EQ ) cos(l N EQ − l P ) ,(A2) where b N GA , b N EQ , l N EQ , b P , and l P are the galactic latitudes and longitudes of N GA , N EQ , and P . After substituting b N GA = 90 • , b N EQ = 27.1 • , l N EQ = 122.9 • for epoch J2000 and some algebraic manipulations, ψ = arctan cos(l P − 32.9 • ) cos b P cot 62.9 • − sin b P sin(l P − 32.9 • ) . Figure 13. Illustration of the the angle ψ and the spherical triangle of NGA, NEQ, and P . (A3) P N EQ N GA ψ Figure 1 . 1The POL-2 dust polarization map at 850 µm toward the DR21 filament. The color scale represents the Stokes I intensity. The magnetic field segments plotted in an interval of 8 show the magnetic field orientations with the lengths proportional to the polarization percentages. The JCMT 14.1 beam is plotted at the bottom left corner. The positions of the 24 massive dense cores in Figure 3 . 3Histogram of position angles (P A |B−f | ) between the magnetic field segments of sub-filaments and the crests of sub-filaments inFigure 1. A position angle of 0 • means that the magnetic field is parallel to the sub-filament crest, and a position angle of 90 • means that the magnetic field is perpendicular to the sub-filament crest.µm data(Planck Collaboration et al. 2020) and the predicted maximum polarization fraction of ∼ 15% of the submillimeter emission from interstellar dust grains(Draine & Fraisse 2009). The steep slope of p-I correlation and large polarization fractions (> 20%) of lowintensity data can be found in other POL-2 observations (e.g.,Kwon et al. 2018;Soam et al. 2018;Pattle et al. 2019;Wang et al. 2019;Coudé et al. 2019;Arzoumanian et al. 2021) Figure 4 . 4Polarization fraction p as a function of Stokes I intensity. −0. 34 , 34Wang et al. 2019; Barnard 1: 0.85 ± 0.01, Coudé et al. 2019; Ophiuchus B: 0.86 ± 0.03, Pattle et al. 2019; Ophiuchus C: 0.83 ± 0.03, Pattle et al. 2019; Auriga: 0.82 ± 0.03, Ngoc et al. 2021; Rosette: 0.49 ± 0.08 Könyves et al. 2021; Serpens: 0.634, Kwon et al. 2022;). The 0.30-0.34 shallow values of α of the DR21 filament are similar to the values of 0.34 ± 0.02 of the Ophiuchus A (Pattle et al. 2019), 0.36 ± 0.04 of the Orion B (Lyo et al. 2021), and 0.35 ± 0.02 of the NGC 6334 Figure 5 . 5Comparisons of polarization angles and polarization degrees of the POL-2 and SCUPOL overlapped 215 segments in Figure 6 . 6Planck 850 µm magnetic field segments overlaid on the τ353 map toward the DR21 filament. The black segments represent the magnetic field orientations in an unified length. The red box at center remarks the area ofFigure 1. At the bottom left corner, the red stripe represents the galactic disk plane, and the Planck 5 beam is shown. Figure 7 . 7Dispersion analysis of the POL-2 and Planck polarization segments toward the DR21 filament. For each source, the analysis of the angular dispersion function is plotted in the top panel, and the correlated component of the dispersion function is plotted in the bottom panel. Top panels: the dots represent the mean values of the data, and the error bars show the standard deviations of the mean values. The blue line shows the best fit to the data ( Figure 10 . 53 Figure 11 . 105311HROs for the lowest, the intermediate, and the highest N (H2) bins (gray, blue, and red, respectively) of the JCMT data. The horizontal dashed line corresponds to the average HRO per angle bin of 15 • for a N (H2) bin. The widths of the shaded areas for each histogram correspond to the ±1 σ k uncertainties (Equation 9) related to the histogram binning operation. .62 < log 10 N(H) < 23.22 22.53 < log 10 N(H) < 22.62 22.30 < log 10 N(H) < 22.HROs for the lowest, the intermediate, and the highest N (H) bin (gray, blue, and red, respectively) of the Planck data. The horizontal dashed line corresponds to the average HRO per angle bin of 15 • for a N (H) bin. The widths of the shaded areas for each histogram correspond to the ±1 σ k uncertainties (Equation 9) related to the histogram binning operation. 5. DISCUSSION 5.1. The Role of Magnetic Field in the DR21 Filament Figure 12 . 12Relative orientation parameter ξ, defined in Equation 10 and 11, calculated for the different N (H) bins of the Planck (red) and JCMT (blue) data of the DR21 filament. The JCMT results of ξ in the Serpens Main region are shown in gray dots. The black dashed line and the values of CHRO and HHRO correspond to the linear fit of Equation 12 for the JCMT data below N (H) = 10 23 cm −2 . 7.6 × 10 −21 [N (H 2 )/cm −2 ][B/µG] −1 . In the theory of magnetic dominated star formation This work is supported by National Natural Science Foundation of China (NSFC) grant Nos. 11988101, U1931117, 11725313, and 12073061 and the CAS International Partnership Program of Chinese Academy of Sciences grant No. 114A11KYSB20160008. T.-C. C. is funded by Chinese Academy of Sciences Taiwan Young Talent Program Grant No. 2018TW2JB0002. T.-C. C. and C. E. were supported by Special Funding for Advanced Users, budgeted and administrated by Center for Astronomical Mega-Science (CAMS), Chinese Academy of Sciences. K. P. is a Royal Society University Research Fellow, supported by grant number URF\R1\211322. K.Q. is partially supported by National Key R&D Program of China No. 2022YFA1603100, and acknowledges the National Natural Science Foundation of China (NSFC) grant U1731237. S.P.L. acknowledges grants from the Ministry of Science and Technology of Taiwan 106-2119-M-007-021-MY3 and 109-2112-M-007-010-MY3. Y.D. acknowledges the support of JSPS KAKENHI grants 25247016 and 18H01250. Y.S.D. is supported by the National Key R&D Program of China APPENDIX A. 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{'abstract': 'We present 850 µm dust polarization observations of the massive DR21 filament from the B-fields In STar-forming Region Observations (BISTRO) survey, using the POL-2 polarimeter and the SCUBA-2 camera on the James Clerk Maxwell Telescope. We detect ordered magnetic fields perpendicular to the parsec-scale ridge of the DR21 main filament. In the sub-filaments, the magnetic fields are mainly parallel to the filamentary structures and smoothly connect to the magnetic fields of the main filament. We compare the POL-2 and Planck dust polarization observations to study the magnetic field structures of the DR21 filament on 0.1-10 pc scales. The magnetic fields revealed in the Planck data are well aligned with those of the POL-2 data, indicating a smooth variation of magnetic fields from large to small scales. The plane-of-sky magnetic field strengths derived from angular dispersion functions of dust polarization are 0.6-1.0 mG in the DR21 filament and ∼ 0.1 mG in the surrounding ambient gas. The mass-to-flux ratios are found to be magnetically supercritical in the filament and slightly subcritical to nearly critical in the ambient gas. The alignment between column density structures and magnetic fields changes from random alignment in the low-density ambient gas probed by Planck to mostly perpendicular in the high-density main filament probed by JCMT. The magnetic field structures of the DR21 filament are in agreement with MHD simulations of a strongly magnetized medium, suggesting that magnetic fields play an important role in shaping the DR21 main filament and sub-filaments.', 'arxivid': '2212.01981', 'author': ["Tao-Chung Ching chingtaochung@gmail.com \nResearch Center for Intelligent Computing Platforms\nZhejiang Lab311100HangzhouPeople's Republic of China\n\nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n\nJansky Fellow\nNational Radio Astronomy Observatory\n1003 Lopezville Road87801SocorroNMUSA\n", "Keping Qiu \nSchool of Astronomy and Space Science\nNanjing University\n163 Xianlin Avenue210023NanjingPeople's Republic of China\n\nMinistry of Education\nKey Laboratory of Modern Astronomy and Astrophysics (Nanjing University)\n210023NanjingPeople's Republic of China\n", "Di Li \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n\nDepartment of Astronomy\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China\n\nNAOC-UKZN Computational Astrophysics Centre\nUniversity of KwaZulu-Natal\n4000DurbanSouth Africa\n", "Zhiyuan Ren \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n", 'Shih-Ping Lai \nInstitute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n\nAcademia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan\n', "David Berry \nEast Asian Observatory\n660 N. A'ohōkū Place96720University Park, HiloHIUSA\n", 'Kate Pattle \nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUnited Kingdom\n', 'Mike Chen \nDepartment of Physics and Astronomy\nUniversity of Victoria\nV8W 2Y2VictoriaBCCanada\n', 'Huei-Ru Vivien Chen \nInstitute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n\nAcademia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan\n', 'Wen Ping Chen \nInstitute of Astronomy\nNational Central University\n32001ZhongliTaiwan\n', 'Jungyeon Cho \nDepartment of Astronomy and Space Science\nChungnam National University\n99 Daehak-ro, Yuseong-gu34134DaejeonRepublic of Korea\n', 'Yunhee Choi \nKorea Astronomy and Space Science Institute\n776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea\n', 'Youngwoo Choi \nDepartment of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', 'Minho Choi \nKorea Astronomy and Space Science Institute\n776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea\n', 'Antonio Chrysostomou \nSKA Observatory, Jodrell Bank, Lower Withington\nSK11 9FTMacclesfieldUK\n', 'Eun Jung Chung \nDepartment of Astronomy and Space Science\nChungnam National University\n99 Daehak-ro, Yuseong-gu34134DaejeonRepublic of Korea\n', "Y Sophia Dai \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n", 'Ngoc Pham ', 'Diep \nAcademy of Science and Technology\nVietnam National Space Center\n18 Hoang Quoc VietHanoiVietnam, Vietnam\n', 'Yasuo Doi \nDepartment of Earth Science and Astronomy\nGraduate School of Arts and Sciences\nThe University of Tokyo\n3-8-1 Komaba153-8902TokyoJapan\n', "Yan Duan \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n", 'Hao-Yuan Duan \nInstitute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n\nArmagh Observatory and Planetarium\nCollege Hill, ArmaghBT61 9DBUK\n', 'David Eden ', 'Lapo Fanciullo \nAcademia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan\n\nSouth Dist\nNational Chung Hsing University\n145 Xingda Rd402Taichung CityTaiwan\n', 'Jason Fiege \nDepartment of Physics and Astronomy\nThe University of Manitoba\nR3T2N2WinnipegManitobaCanada\n', "Laura M Fissel \nDepartment for Physics, Engineering Physics and Astrophysics\nQueen's University\nK7L 3N6KingstonONCanada\n", 'Erica Franzmann \nDepartment of Physics and Astronomy\nThe University of Manitoba\nR3T2N2WinnipegManitobaCanada\n', "Per Friberg \nEast Asian Observatory\n660 N. A'ohōkū Place96720University Park, HiloHIUSA\n", 'Rachel Friesen \nNational Radio Astronomy Observatory\n520 Edgemont Road22903CharlottesvilleVAUSA\n', 'Gary Fuller \nJodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUK\n', 'Felix Priestley \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffUK\n', "Tae-Soo Pyo \nSubaru Telescope\nNational Astronomical Observatory of Japan\n650 N. A'ohōkū Place96720HiloHIUSA\n\nSOKENDAI (The Graduate University for Advanced Studies)\n240-0193HayamaKanagawaJapan\n", "Lei Qian \nCAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, People's Republic of China 84 Gemini Observatory/NSF's NOIRLab\n670 N. A'ohōkū Place96720University Park, HiloHIUSA\n\n85 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue\nCB3 0HECambridgeUK\n", 'Ramprasad Rao \nAcademia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan\n', "Mark Rawlings \nEast Asian Observatory\n660 N. A'ohōkū Place96720University Park, HiloHIUSA\n", 'Jonathan Rawlings \nDepartment of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK\n', 'Brendan Retter \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffUK\n', 'John Richer \nKavli Institute for Cosmology\nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n', 'Andrew Rigby \nSchool of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffUK\n', "Sarah Sadavoy \nDepartment for Physics, Engineering Physics and Astrophysics\nQueen's University\nK7L 3N6KingstonONCanada\n", 'Hiro Saito \nFaculty of Pure and Applied Sciences\nUniversity of Tsukuba\n1-1-1 Tennodai305-8577TsukubaIbarakiJapan\n', 'Giorgio Savini \nDepartment of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK\n', 'Masumichi Seta \nDepartment of Physics\nSchool of Science and Technology\nKwansei Gakuin University\n2-1 Gakuen669-1337SandaHyogoJapan\n', 'Yoshito Shimajiri \nNational Astronomical Observatory of Japan\nNational Institutes of Natural Sciences\n181-8588OsawaMitaka, TokyoJapan\n', 'Hiroko Shinnaga \nDepartment of Physics and Astronomy\nGraduate School of Science and Engineering\nKagoshima University\n1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan\n', 'Ya-Wen Tang \nAcademia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan\n', 'Kohji Tomisaka \nNational Astronomical Observatory of Japan\nNational Institutes of Natural Sciences\n181-8588OsawaMitaka, TokyoJapan\n\nSOKENDAI (The Graduate University for Advanced Studies)\n240-0193HayamaKanagawaJapan\n', 'Le Ngoc Tram \nAcademy of Science and Technology\nUniversity of Science and Technology of Hanoi\n18 Hoang Quoc VietHanoiVietnam, Vietnam\n', 'Yusuke Tsukamoto \nDepartment of Physics and Astronomy\nGraduate School of Science and Engineering\nKagoshima University\n1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan\n', 'Serena Viti \nDepartment of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK\n', "Hongchi Wang \nPurple Mountain Observatory\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPeople's Republic of China\n", "Jintai Wu \nSchool of Astronomy and Space Science\nNanjing University\n163 Xianlin Avenue210023NanjingPeople's Republic of China\n", "Jinjin Xie \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n", 'Meng-Zhe Yang \nInstitute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan\n', 'Hsi-Wei Yen \nAcademia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan\n', 'Hyunju Yoo \nKorea Astronomy and Space Science Institute\n776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea\n', "Jinghua Yuan \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n", 'Hyeong-Sik Yun \nSchool of Space Research\nKyung Hee University\nGiheung-gu, Yongin-si1732, 17104Deogyeong-daeroGyeonggi-doRepublic of Korea\n', 'Tetsuya Zenko \nDepartment of Astronomy\nGraduate School of Science\nKyoto University\nSakyo-ku606-8502KyotoJapan\n', "Chuan-Peng Zhang \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n\nCAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, People's Republic of China 84 Gemini Observatory/NSF's NOIRLab\n670 N. A'ohōkū Place96720University Park, HiloHIUSA\n\n85 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue\nCB3 0HECambridgeUK\n", 'Yapeng Zhang \nDepartment of Astronomy\nBeijing Normal University\nBeijing100875China\n', 'Guoyin Zhang ', "Jianjun Zhou \nNational Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China\n\nXinjiang Astronomical Observatory\n40 Astrobiology Center\nChinese Academy of Sciences\n150 Science 1-Street830011Urumqi, XinjiangPeople's Republic of China\n\nNational Institutes of Natural Sciences\n2-21-1 Osawa181-8588MitakaTokyoJapan\n", "Lei Zhu \nCAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, People's Republic of China 84 Gemini Observatory/NSF's NOIRLab\n670 N. A'ohōkū Place96720University Park, HiloHIUSA\n\n85 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue\nCB3 0HECambridgeUK\n", 'Ilse De Looze \nDepartment of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK\n', "Philippe André \nIRFU/Service d'Astrophysique\nLaboratoire AIM CEA/DSM-CNRS-Université Paris Diderot\nCEA Saclay\nF-91191Gif-sur-YvetteFrance\n", 'C ', 'Darren Dowell \nJet Propulsion Laboratory\nM/S 169-506, 4800 Oak Grove Drive91109PasadenaCAUSA\n', 'Stewart Eyres \nUniversity of South Wales\nCF37 1DLPontypriddUK\n', 'Sam Falle \nDepartment of Applied Mathematics\nUniversity of Leeds\nWoodhouse LaneLS2 9JTLeedsUK\n', 'Jean-François Robitaille \nUniv. Grenoble Alpes\nCNRS\nIPAG\n38000GrenobleFrance\n', 'Sven Van Loo \nSchool of Physics and Astronomy\nUniversity of Leeds\nWoodhouse LaneLS2 9JTLeedsUK\n', '\nTokushima University\nMinami Jousanajima-machi 1-1770-8502TokushimaJapan\n', '\nInstitute of Liberal Arts and Sciences\nTokushima University\nMinami Jousanajima-machi 1-1770-8502TokushimaJapan\n', '\nJeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPrestonUK\n', '\nNRC Herzberg Astronomy and Astrophysics\n5071 West Saanich RoadV9E 2E7VictoriaBCCanada\n', '\nUniversity of Science and Technology\n217 Gajeong-ro, Yuseong-gu34113DaejeonKorea, Republic of Korea\n', '\nDepartment of Earth Science Education\n21 SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', '\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea\n', '\nCentre de recherche en astrophysique du Québec & département de physique\nUniversité de Montréal\n1375, Avenue Thérèse-Lavoie-RouxH2V 0B3MontréalQCCanada\n', '\nIndian Institute of Science Education and Research (IISER) Tirupati, Rami Reddy Nagar, Karakambadi Road, Mangalam (P.O.)\n517, 507TirupatiIndia\n', '\nDepartment of Physics, College of Natural Science\nUlsan National Institute of Science and Technology (UNIST)\n50 UNIST-gil44919UlsanRepublic of Korea\n', '\nIndian Institute of Astrophysics (IIA)\n560034Kormangala, BangaloreIndia\n', '\nSOFIA Science Center\nUniversities Space Research Association\nNASA Ames Research Center\nMoffett Field94035CaliforniaUSA\n', '\nInstrumentation et Modélisation de Paris-Saclay\nUniversité Paris-Saclay\nCNRS\nCEA, Astrophysique\n91191Gif-sur-YvetteFrance\n', '\nInstituto de Astrofísica de Canarias\nE-38205 La Laguna, Canary IslandsTenerifeSpain\n', '\nDepartamento de Astrofísica\nUniversidad de La Laguna (ULL)\nDpto. Astrofísica, E-38206 La LagunaTenerifeSpain\n', "\nShanghai Astronomical Observatory\nChinese Academy of Sciences\n80 Nandan Road200030ShanghaiPeople's Republic of China\n", '\nDepartment of Earth, Environment and Physics\nWorcester State University\n01602WorcesterMAUSA\n', '\nCenter for Astrophysics | Harvard & Smithsonian\n60 Garden Street02138CambridgeMAUSA\n', '\nKavli Institute for Particle Astrophysics & Cosmology (KIPAC)\nStanford University\n94305StanfordCA\n', '\nDominion Radio Astrophysical Observatory, Herzberg Astronomy and Astrophysics Research Centre, National Research Council Canada\nP. O. Box 248V2A 6J9PentictonBCCanada\n', "\nYunnan University\n650091KunmingPeople's Republic of China\n", '\nDepartment of Physics\nFaculty of Science and Engineering\nMeisei University\n2-1-1 Hodokubo191-8506HinoTokyoJapan\n', '\nDepartment of Astronomy\nGraduate School of Science\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n', '\nDepartment of Physics, Astronomy & Mathematics\nUniversity of Hertfordshire\nCollege Lane, HatfieldAL10 9ABHertfordshireUK\n', '\nDepartment of Physics and Astronomy\nThe University of Western Ontario\n1151 Richmond StreetN6A 3K7LondonCanada\n', '\nNational Astronomical Observatory of Japan\nAlonso de Córdova 3788, Office 61B, VitacuraSantiagoChile\n', '\nJoint ALMA Observatory\nAlonso de Córdova 3107, VitacuraSantiagoChile\n', '\nFellow 61 Department of Physics\nGraduate School of Science\nNAOJ\nNagoya University\nFuro-cho, Chikusa-ku, Nagoya 464-8602Japan\n', '\nDepartment of Environmental Systems Science\nDoshisha University\nTatara, Miyakodani 1-3, Kyotanabe610-0394KyotoJapan\n', '\nDepartment of Astronomy and Atmospheric Sciences\nKyungpook National University\n41566DaeguRepublic of Korea\n', '\nHiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', '\nDepartment of Physics\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', '\nCore Research for Energetic Universe (CORE-U)\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan\n', "\nInstitut de Ciencies de l'Espai (ICE, CSIC)\nCan Magrans, s/n08193Bellaterra, BarcelonaSpain\n", '\nICREA\nPg. Lluís Companys 23BarcelonaSpain\n', "\nInstitut d'Estudis Espacials de Catalunya (IEEC)\nE-08034BarcelonaSpain\n", '\nNobeyama Radio Observatory\nNational Astronomical Observatory of Japan\nNational Institutes of Natural Sciences\n384-1305Nobeyama, NaganoMinamimaki, MinamisakuJapan\n', '\nAstronomical Institute\nGraduate School of Science\nTohoku University\nAoba-ku980-8578, 72SendaiMiyagiJapan\n', '\nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonONCanada\n', '\nDepartment of Physics and Atmospheric Science\nDalhousie University\nB3H 4R2HalifaxCanada\n', '\nDepartment of Physics\nDepartment of Space, Earth & Environment\nThe Chinese University of Hong Kong\nShatin, Hong Kong 76N.T\n', '\nChalmers University of Technology\nSE-412 96GothenburgSweden\n', '\nFaculty of Education & Center for Educational Development and Support\nKagawa University\nSaiwai-cho 1-1760-8522TakamatsuKagawaJapan\n', '\nGraduate University of Science and Technology\nAcademy of Science and Technology\n18 Hoang Quoc Viet, Cau GiayHanoiVietnam, Vietnam\n'], 'authoraffiliation': ["Research Center for Intelligent Computing Platforms\nZhejiang Lab311100HangzhouPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", 'Jansky Fellow\nNational Radio Astronomy Observatory\n1003 Lopezville Road87801SocorroNMUSA', "School of Astronomy and Space Science\nNanjing University\n163 Xianlin Avenue210023NanjingPeople's Republic of China", "Ministry of Education\nKey Laboratory of Modern Astronomy and Astrophysics (Nanjing University)\n210023NanjingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", "Department of Astronomy\nUniversity of Chinese Academy of Sciences\n100049BeijingPeople's Republic of China", 'NAOC-UKZN Computational Astrophysics Centre\nUniversity of KwaZulu-Natal\n4000DurbanSouth Africa', "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", 'Institute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan', 'Academia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan', "East Asian Observatory\n660 N. A'ohōkū Place96720University Park, HiloHIUSA", 'Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUnited Kingdom', 'Department of Physics and Astronomy\nUniversity of Victoria\nV8W 2Y2VictoriaBCCanada', 'Institute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan', 'Academia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan', 'Institute of Astronomy\nNational Central University\n32001ZhongliTaiwan', 'Department of Astronomy and Space Science\nChungnam National University\n99 Daehak-ro, Yuseong-gu34134DaejeonRepublic of Korea', 'Korea Astronomy and Space Science Institute\n776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea', 'Department of Physics and Astronomy\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Korea Astronomy and Space Science Institute\n776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea', 'SKA Observatory, Jodrell Bank, Lower Withington\nSK11 9FTMacclesfieldUK', 'Department of Astronomy and Space Science\nChungnam National University\n99 Daehak-ro, Yuseong-gu34134DaejeonRepublic of Korea', "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", 'Academy of Science and Technology\nVietnam National Space Center\n18 Hoang Quoc VietHanoiVietnam, Vietnam', 'Department of Earth Science and Astronomy\nGraduate School of Arts and Sciences\nThe University of Tokyo\n3-8-1 Komaba153-8902TokyoJapan', "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", 'Institute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan', 'Armagh Observatory and Planetarium\nCollege Hill, ArmaghBT61 9DBUK', 'Academia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan', 'South Dist\nNational Chung Hsing University\n145 Xingda Rd402Taichung CityTaiwan', 'Department of Physics and Astronomy\nThe University of Manitoba\nR3T2N2WinnipegManitobaCanada', "Department for Physics, Engineering Physics and Astrophysics\nQueen's University\nK7L 3N6KingstonONCanada", 'Department of Physics and Astronomy\nThe University of Manitoba\nR3T2N2WinnipegManitobaCanada', "East Asian Observatory\n660 N. A'ohōkū Place96720University Park, HiloHIUSA", 'National Radio Astronomy Observatory\n520 Edgemont Road22903CharlottesvilleVAUSA', 'Jodrell Bank Centre for Astrophysics\nSchool of Physics and Astronomy\nUniversity of Manchester\nOxford RoadM13 9PLManchesterUK', 'School of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffUK', "Subaru Telescope\nNational Astronomical Observatory of Japan\n650 N. A'ohōkū Place96720HiloHIUSA", 'SOKENDAI (The Graduate University for Advanced Studies)\n240-0193HayamaKanagawaJapan', "CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, People's Republic of China 84 Gemini Observatory/NSF's NOIRLab\n670 N. A'ohōkū Place96720University Park, HiloHIUSA", '85 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue\nCB3 0HECambridgeUK', 'Academia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan', "East Asian Observatory\n660 N. A'ohōkū Place96720University Park, HiloHIUSA", 'Department of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK', 'School of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffUK', 'Kavli Institute for Cosmology\nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK', 'School of Physics and Astronomy\nCardiff University\nThe ParadeCF24 3AACardiffUK', "Department for Physics, Engineering Physics and Astrophysics\nQueen's University\nK7L 3N6KingstonONCanada", 'Faculty of Pure and Applied Sciences\nUniversity of Tsukuba\n1-1-1 Tennodai305-8577TsukubaIbarakiJapan', 'Department of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK', 'Department of Physics\nSchool of Science and Technology\nKwansei Gakuin University\n2-1 Gakuen669-1337SandaHyogoJapan', 'National Astronomical Observatory of Japan\nNational Institutes of Natural Sciences\n181-8588OsawaMitaka, TokyoJapan', 'Department of Physics and Astronomy\nGraduate School of Science and Engineering\nKagoshima University\n1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan', 'Academia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan', 'National Astronomical Observatory of Japan\nNational Institutes of Natural Sciences\n181-8588OsawaMitaka, TokyoJapan', 'SOKENDAI (The Graduate University for Advanced Studies)\n240-0193HayamaKanagawaJapan', 'Academy of Science and Technology\nUniversity of Science and Technology of Hanoi\n18 Hoang Quoc VietHanoiVietnam, Vietnam', 'Department of Physics and Astronomy\nGraduate School of Science and Engineering\nKagoshima University\n1-21-35 Korimoto890-0065KagoshimaKagoshimaJapan', 'Department of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK', "Purple Mountain Observatory\nChinese Academy of Sciences\n2 West Beijing Road210008NanjingPeople's Republic of China", "School of Astronomy and Space Science\nNanjing University\n163 Xianlin Avenue210023NanjingPeople's Republic of China", "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", 'Institute of Astronomy\nDepartment of Physics\nNational Tsing Hua University\n30013HsinchuTaiwan', 'Academia Sinica Institute of Astronomy and Astrophysics\nNo.1, Sec. 4., Roosevelt Road10617TaipeiTaiwan', 'Korea Astronomy and Space Science Institute\n776 Daedeokdae-ro, Yuseong-gu34055DaejeonRepublic of Korea', "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", 'School of Space Research\nKyung Hee University\nGiheung-gu, Yongin-si1732, 17104Deogyeong-daeroGyeonggi-doRepublic of Korea', 'Department of Astronomy\nGraduate School of Science\nKyoto University\nSakyo-ku606-8502KyotoJapan', "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", "CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, People's Republic of China 84 Gemini Observatory/NSF's NOIRLab\n670 N. A'ohōkū Place96720University Park, HiloHIUSA", '85 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue\nCB3 0HECambridgeUK', 'Department of Astronomy\nBeijing Normal University\nBeijing100875China', "National Astronomical Observatories\nChinese Academy of Sciences\nA20 Datun Road100012Chaoyang District, BeijingPeople's Republic of China", "Xinjiang Astronomical Observatory\n40 Astrobiology Center\nChinese Academy of Sciences\n150 Science 1-Street830011Urumqi, XinjiangPeople's Republic of China", 'National Institutes of Natural Sciences\n2-21-1 Osawa181-8588MitakaTokyoJapan', "CAS Key Laboratory of FAST, National Astronomical Observatories, Chinese Academy of Sciences, People's Republic of China 84 Gemini Observatory/NSF's NOIRLab\n670 N. A'ohōkū Place96720University Park, HiloHIUSA", '85 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue\nCB3 0HECambridgeUK', 'Department of Physics and Astronomy\nUniversity College London\nWC1E 6BTLondonUK', "IRFU/Service d'Astrophysique\nLaboratoire AIM CEA/DSM-CNRS-Université Paris Diderot\nCEA Saclay\nF-91191Gif-sur-YvetteFrance", 'Jet Propulsion Laboratory\nM/S 169-506, 4800 Oak Grove Drive91109PasadenaCAUSA', 'University of South Wales\nCF37 1DLPontypriddUK', 'Department of Applied Mathematics\nUniversity of Leeds\nWoodhouse LaneLS2 9JTLeedsUK', 'Univ. Grenoble Alpes\nCNRS\nIPAG\n38000GrenobleFrance', 'School of Physics and Astronomy\nUniversity of Leeds\nWoodhouse LaneLS2 9JTLeedsUK', 'Tokushima University\nMinami Jousanajima-machi 1-1770-8502TokushimaJapan', 'Institute of Liberal Arts and Sciences\nTokushima University\nMinami Jousanajima-machi 1-1770-8502TokushimaJapan', 'Jeremiah Horrocks Institute\nUniversity of Central Lancashire\nPR1 2HEPrestonUK', 'NRC Herzberg Astronomy and Astrophysics\n5071 West Saanich RoadV9E 2E7VictoriaBCCanada', 'University of Science and Technology\n217 Gajeong-ro, Yuseong-gu34113DaejeonKorea, Republic of Korea', 'Department of Earth Science Education\n21 SNU Astronomy Research Center\nSeoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Seoul National University\n1 Gwanak-ro, Gwanak-gu08826SeoulRepublic of Korea', 'Centre de recherche en astrophysique du Québec & département de physique\nUniversité de Montréal\n1375, Avenue Thérèse-Lavoie-RouxH2V 0B3MontréalQCCanada', 'Indian Institute of Science Education and Research (IISER) Tirupati, Rami Reddy Nagar, Karakambadi Road, Mangalam (P.O.)\n517, 507TirupatiIndia', 'Department of Physics, College of Natural Science\nUlsan National Institute of Science and Technology (UNIST)\n50 UNIST-gil44919UlsanRepublic of Korea', 'Indian Institute of Astrophysics (IIA)\n560034Kormangala, BangaloreIndia', 'SOFIA Science Center\nUniversities Space Research Association\nNASA Ames Research Center\nMoffett Field94035CaliforniaUSA', 'Instrumentation et Modélisation de Paris-Saclay\nUniversité Paris-Saclay\nCNRS\nCEA, Astrophysique\n91191Gif-sur-YvetteFrance', 'Instituto de Astrofísica de Canarias\nE-38205 La Laguna, Canary IslandsTenerifeSpain', 'Departamento de Astrofísica\nUniversidad de La Laguna (ULL)\nDpto. Astrofísica, E-38206 La LagunaTenerifeSpain', "Shanghai Astronomical Observatory\nChinese Academy of Sciences\n80 Nandan Road200030ShanghaiPeople's Republic of China", 'Department of Earth, Environment and Physics\nWorcester State University\n01602WorcesterMAUSA', 'Center for Astrophysics | Harvard & Smithsonian\n60 Garden Street02138CambridgeMAUSA', 'Kavli Institute for Particle Astrophysics & Cosmology (KIPAC)\nStanford University\n94305StanfordCA', 'Dominion Radio Astrophysical Observatory, Herzberg Astronomy and Astrophysics Research Centre, National Research Council Canada\nP. O. Box 248V2A 6J9PentictonBCCanada', "Yunnan University\n650091KunmingPeople's Republic of China", 'Department of Physics\nFaculty of Science and Engineering\nMeisei University\n2-1-1 Hodokubo191-8506HinoTokyoJapan', 'Department of Astronomy\nGraduate School of Science\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan', 'Department of Physics, Astronomy & Mathematics\nUniversity of Hertfordshire\nCollege Lane, HatfieldAL10 9ABHertfordshireUK', 'Department of Physics and Astronomy\nThe University of Western Ontario\n1151 Richmond StreetN6A 3K7LondonCanada', 'National Astronomical Observatory of Japan\nAlonso de Córdova 3788, Office 61B, VitacuraSantiagoChile', 'Joint ALMA Observatory\nAlonso de Córdova 3107, VitacuraSantiagoChile', 'Fellow 61 Department of Physics\nGraduate School of Science\nNAOJ\nNagoya University\nFuro-cho, Chikusa-ku, Nagoya 464-8602Japan', 'Department of Environmental Systems Science\nDoshisha University\nTatara, Miyakodani 1-3, Kyotanabe610-0394KyotoJapan', 'Department of Astronomy and Atmospheric Sciences\nKyungpook National University\n41566DaeguRepublic of Korea', 'Hiroshima Astrophysical Science Center\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', 'Department of Physics\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', 'Core Research for Energetic Universe (CORE-U)\nHiroshima University\nKagamiyama 1-3-1, Higashi-Hiroshima739-8526HiroshimaJapan', "Institut de Ciencies de l'Espai (ICE, CSIC)\nCan Magrans, s/n08193Bellaterra, BarcelonaSpain", 'ICREA\nPg. Lluís Companys 23BarcelonaSpain', "Institut d'Estudis Espacials de Catalunya (IEEC)\nE-08034BarcelonaSpain", 'Nobeyama Radio Observatory\nNational Astronomical Observatory of Japan\nNational Institutes of Natural Sciences\n384-1305Nobeyama, NaganoMinamimaki, MinamisakuJapan', 'Astronomical Institute\nGraduate School of Science\nTohoku University\nAoba-ku980-8578, 72SendaiMiyagiJapan', 'Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonONCanada', 'Department of Physics and Atmospheric Science\nDalhousie University\nB3H 4R2HalifaxCanada', 'Department of Physics\nDepartment of Space, Earth & Environment\nThe Chinese University of Hong Kong\nShatin, Hong Kong 76N.T', 'Chalmers University of Technology\nSE-412 96GothenburgSweden', 'Faculty of Education & Center for Educational Development and Support\nKagawa University\nSaiwai-cho 1-1760-8522TakamatsuKagawaJapan', 'Graduate University of Science and Technology\nAcademy of Science and Technology\n18 Hoang Quoc Viet, Cau GiayHanoiVietnam, Vietnam'], 'corpusid': 254246703, 'doi': '10.3847/1538-4357/ac9dfb', 'github_urls': [], 'n_tokens_mistral': 41801, 'n_tokens_neox': 32580, 'n_words': 17422, 'pdfsha': '72a649b4b5ea30fece1cc01f35ddf215ab30f1b5', 'pdfurls': ['https://export.arxiv.org/pdf/2212.01981v1.pdf'], 'title': ['16 Frédérick Poidevin, 29, 30 Tie Liu, 31 Simon Coudé, 32, 33 Mehrnoosh Tahani, 34, 35 Hong-Li Liu, 36 Takashi Onaka, 37, 38 Dalei Li, 39 Motohide Tamura', '16 Frédérick Poidevin, 29, 30 Tie Liu, 31 Simon Coudé, 32, 33 Mehrnoosh Tahani, 34, 35 Hong-Li Liu, 36 Takashi Onaka, 37, 38 Dalei Li, 39 Motohide Tamura'], 'venue': ['60 Tsuyoshi Inoue, 61 Shu-ichiro Inutsuka, 61 Kazunari Iwasaki, 62 Il-Gyo Jeong, 63, 17 Vera Könyves, 14 Ji-hyun Kang, 17 Miju Kang, 17 Janik Karoly, 14 Akimasa Kataoka']}

arxiv

ROBUST DECODING FROM 1-BIT COMPRESSIVE SAMPLING WITH LEAST SQUARES 3 Nov 2017 Jian Huang Yuling Jiao Xiliang Lu ANDLiping Zhu ROBUST DECODING FROM 1-BIT COMPRESSIVE SAMPLING WITH LEAST SQUARES 3 Nov 20171-bit compressive sensingℓ 1 -regularized least squaresprimal dual active set algorithmone step conver- gencecontinuation In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: y = η ⊙ sign(Ψx * + ǫ), where x * ∈ R n , y ∈ R m , Ψ ∈ R m×n , and ǫ is the random error before quantization and η ∈ R n is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For m > n, we show that, up to a constant c, with high probability, the least squares solution x ls approximates x * with precision δ as long as m ≥ O( n δ 2 ). For m < n, we prove that, up to a constant c, with high probability, the ℓ 1 -regularized least-squares solution x ℓ 1 lies in the ball with center x * and radius δ provided that m ≥ O( s log n δ 2 ) and x * 0 := s < m. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.Keywords: 1-bit compressive sensing, ℓ 1 -regularized least squares, primal dual active set algorithm, one step convergence, continuation 1. Introduction. Compressive sensing (CS) is an important approach to acquiring low dimension signals from noisy under-determined measurements[8,16,19,20]. For storage and transmission, the infinite-precision measurements are often quantized, [6] considered recovering the signals from the 1-bit compressive sensing (1-bit CS) where measurements are coded into a single bit, i.e., their signs. The 1-bit CS is superior to the CS in terms of inexpensive hardware implementation and storage. However, it is much more challenging to decode from nonlinear, noisy and sign-flipped 1-bit measurements.1.1. Previous work. Since the seminal work of [6], much effort has been devoted to studying the theoretical and computational challenges of the 1-bit CS. Sample complexity was analyzed for support and vector recovery with and without noise[21,28,40,23,29,22,23,41,50]. Existing works indicate that, m > O(s log n) is adequate for both support and vector recovery. The sample size required here has the same order as that required in the standard CS setting. These results have also been refined by adaptive sampling[22,14,4]. Extensions include recovering the norm of the target [32, 3] and non-Gaussian measurement settings [1]. Many first order methods[6,34,49,14]and greedy methods[35,5,29]are developed to minimize the sparsity promoting nonconvex objected function arising from either the unit sphere constraint or the nonconvex regularizers. To address the nonconvex optimization problem, convex relaxation models are also proposed [50,41,40,51,42], which often yield accurate solutions efficiently with polynomial-time solvers. See, for example, [38]. 1.2. 1-bit CS setting. In this paper we consider the following 1-bit CS model y = η ⊙ sign(Ψx * + ǫ), (1.1) where y ∈ R m is the 1-bit measurement, x * ∈ R n is an unknown signal, Ψ = [ψ t 1 ; ...; ψ t m ] ∈ R m×n is a random matrix, η ∈ R m is a random vector modeling the sign flips of y, and ǫ ∈ R n is a random vector with independent and identically distributed (iid) entries modeling errors before quantization. Throughout sign(·) operates componentwise with sign(z) = 1 if z ≥ 0 and sign(z) = −1 otherwise, and ⊙ is the pointwise Hardmard product. Following [40] we assume that the rows of Ψ are iid random vectors sampled from the multivariate normal distribution N (0, Σ) with an unknown covariance matrix Σ, ǫ is distributed as N (0, σ 2 I m ) with an unknown noise level σ, and η ∈ R m has independent coordinates η i s satisfying P[η i = 1] = 1 − P[η i = −1] = q = 1 2 . We assume η i , ǫ i and ψ i are mutually independent. Because σ is known model (1.1) is invariant in the sense that ∀α > 0, y = η ⊙ sign(Ψx * + ǫ) = η ⊙ sign(αΨx * + αǫ). This indicates that the best one can hope for is to recover x * up to a scale factor. Without loss of generality we assume x * Σ = 1. Contributions. We study the 1-bit CS problem in both the overdetermined setting with m > n and the underdetermined setting with m < n. In the former setting we allow for dense x * , while in the latter, we assume that x * is sparse in the sense that x * 0 = s < m. The basic message is that we can recover x * with the ordinary least squares or the ℓ 1 regularized least squares. (1) When m > n, we propose to use the least squares solution x ls ∈ arg min 1 m m i=1 (y i − ψ t i x) 2 to approximate x * . We show that, with high probability, x ls estimates x * accurately up to a positive scale factor c defined by (2.2) in the sense that, ∀δ ∈ (0, 1), x ls /c − x * ≤ δ if m ≥ O( n δ 2 ). We make the following observation: Up to a constant c, the underlying target x * can be decoded from 1-bit measurements with the ordinary least squares, as long as the probability of sign flips probability is not equal to 1/2. (2) When m < n and the target signal x * is sparse, we consider the ℓ 1 -regularized least squares solution x ℓ1 ∈ arg min 1 2m y − Ψx 2 2 + λ x 1 . (1. 2) The sparsity assumption is widely used in modern signal processing [20,36]. We show that, with high probability the error x ℓ1 /c − x * can be bounded by a prefixed accuracy δ ∈ (0, 1) if m ≥ O( s log n δ 2 ), which is the same as the order for the standard CS methods to work. Furthermore, the support of x * can be exactly recovered if the minimum signal magnitude of x * is larger than O( s log n/m). When the target signal is sparse, we obtain the following conclusion: Up to a constant c, the sparse signal x * can also be decoded from 1-bit measurements with the ℓ 1 -regularized least squares, as long as the probability of sign flips probability is not equal to 1/2. (3) We introduce a fast and accurate Newton method, the so-called primal dual active set method (PDAS), to solve the ℓ 1 -regularized minimization (1.2). The PDAS possesses the property of one-step convergence. The PDAS solves a small least squares problem on the active set, is thus extremely efficient if combined with continuation. We propose a novel regularization parameter selection rule, which is incorporated with continuation procedure without additional cost. The code is available at http://faculty.zuel.edu.cn/tjyjxxy/jyl/list.htm. The optimal solution x ℓ1 can be computed efficiently and accurately with the PDAS, a Newton type method which converges after one iteration, even if the objective function (1.2) is nonsmooth. Continuation on λ globalizes the PDAS. The regularization parameter can be automatically determined without additional computational cost. 1.4. Notation and organization. Throughout we denote by Ψ i ∈ R m×1 , i = 1, ..., m, and ψ j ∈ R n×1 , j = 1, ..., n the ith column and jth row of Ψ, respectively. We denote a vector of 0 by 0, whose length may vary in different places. We use [n] to denote the set {1, ..., n}, and I n to denote the identity matrix of size n × n. For A, B ⊆ [n] with length |A|, |B|, x A = (x i , i ∈ A) ∈ R |A| , Ψ A = (Ψ i , i ∈ A) ∈ R m×|A| and Ψ AB ∈ R |A|×|B| denotes a submatrix of Ψ whose rows and columns are listed in A and B, respectively. We use (ψ i ) j to denote the jth entry of the vector ψ i , and |x| min to denote the minimum absolute value of x. We use N (0, Σ) to denote the multivariate normal distribution, with Σ symmetric and positive definite. Let γ max (Σ) and γ min (Σ) be the largest and the smallest eigenvalues of Σ, respectively, and κ(Σ) be the condition number γ max (Σ)/γ min (Σ) of Σ. We use x Σ to denote the elliptic norm of x with respect to Σ, i.e., x Σ = (x t Σx) 1 2 . Let x p = ( n i=1 |x i | p ) 1/p , p ∈ [1, ∞] , be the ℓ p -norm of x. We denote the number of nonzero elements of x by x 0 and let s = x * 0 . The symbols Ψ and Ψ ∞ stands for the operator norm of Ψ induced by ℓ 2 norm and the maximum pointwise absolute value of Ψ, respectively. We use E[·], E[·|·], P[·] to denote the expectation, the conditional expectation and the probability on a given probability space (Ω, F, P). We use C 1 and C 2 to denote generic constants which may vary from place to place. By O(·) and O(·), we ignore some positive numerical constant and √ log n, respectively. The rest of the paper is organized as follows. In Section 2 we explain why the least squares works in the 1-bit CS when m > n, and obtain a nonasymptotic error bound for x ls /c − x * . In Section 3 we consider the sparse decoding when m < n and prove a minimax bound on x ℓ1 /c − x * . In Section 4 we introduce the PDAS algorithm to solve (1.2). We propose a new regularization parameter selection rule combined with the continuation procedure. In Section 5 we conduct simulation studies and compare with existing 1-bit CS methods. We conclude with some remarks in Section 6. 2. Least squares when m > n. In this section, we first explain why the least squares works in the over-determined 1-bit CS model (1.1) with m > n. We then prove a nonasymptotic error bound on x ls /c − x * . The following lemma inspired by [7] is our starting point. 3 Lemma 2.1. Letỹ =ηsign(ψ t x * +ǫ) be the 1-bit model (1.1) at the population level. P[η = 1] = q = 1 2 ,ψ ∼ N (0, Σ),ǫ ∼ N (0, σ 2 ). It follows that, Σ −1 E[ỹψ]/c = x * , (2.1) where c = (2q − 1) 2 π(σ 2 + 1) . (2.2) Proof. The proof is given in Appendix A. Lemma 2.1 shows that, the target x * is proportional to Σ −1 E[ỹψ]. Note that E[Ψ t Ψ/m] = E[ m i=1 ψ i ψ t i ]/m = Σ, and (2.3) E[Ψ t y/m] = E[ m i=1 ψ i y i ]/m = E[ỹψ]. (2.4) As long as Ψ t Ψ/m is invertible, it is reasonable to expect that x ls = (Ψ t Ψ/m) −1 (Ψ t y/m) = (Ψ t Ψ) −1 (Ψ t y) can approximate x * well up to a constant c even if y consists of sign flips. Theorem 2.2. Consider the ordinary least squares: x ls ∈ arg min x 1 m Ψx − y 2 2 . (2.5) If m ≥ 16C 2 2 n, then with probability at least 1 − 4 exp (−C 1 C 2 2 n) − 2 n 3 , x ls /c − x * 2 ≤ n m (4C 2 κ(Σ)γ max (Σ) + 6(σ + 1) √ C 1 |2q − 1| log n), (2.6) where C 1 and C 2 are some generic constants not depending on m or n. Proof. The proof is given in Appendix C. Remark 2.1. Theorem (2.2) shows that, ∀δ ∈ (0, 1) if m ≥ O( n δ 2 ) , up to a constant, the simple least squares solution can approximate x * with error of order δ even if y contains very large quantization error and sign flips with probability unequal to 1/2. Remark 2.2. To the best of our knowledge, this is the first nonasymptotic error bound for the 1-bit CS in the overdetermined setting. Comparing with the estimation error of the standard least squares in the complete data model y = Ψx * + ǫ, the error bound in Theorem 2.2 is optimal up to a logarithm factor √ log n, which is due to the loss of information with the 1-bit quantization. 3. Sparse decoding with ℓ 1 -regularized least squares. 3.1. Nonasymptotic error bound. Since images and signals are often sparsely represented under certain transforms [36,15], it suffices for the standard CS to recover the sparse signal x * ∈ R n with m = O(s log n) measurements for s = x * 0 . In this section we show that in the 1-bit CS setting, similar results can be derived through the ℓ 1 -regularized least squares (1.2). Model (1.2) has been extensively studied when y is continuous [44,9,8,16]. Here we use model (1.2) to recover x * from quantized y, which is rarely studied in the literature. Next we show that, up to the constant c, x ℓ1 is a good estimate of x * when m = O(s log n) even if the signal is highly noisy and corrupted by sign flips in the 1-bit CS setting. Theorem 3.1. Assume n > m ≥ max{ 4C1 C 2 2 log n, 64(4κ(Σ)+1) 2 C1 s log en s }, s ≤ exp (1− C 1 2 ) n. Set λ = 4(1+C3|c|) √ C1 log n m . Then with probability at least 1 − 2/n 3 − 6/n 2 , we have, x ℓ1 /c − x * 2 ≤ 816(4κ(Σ) + 1) 2 γ min (Σ) σ + 1 + C 3 |q − 1/2| √ C 1 |q − 1/2| s log n m . (3.1) Proof. The proof is given in Appendix D. Remark 3.1. Theorem 3.1 shows that, ∀δ ∈ (0, 1), if m ≥ O( s log n δ 2 ), up to a constant c, the ℓ 1 -regularized least squares solution can approximate x * with precision δ. Remark 3.2. The error bound in Theorem 3.1 achieves the minimax optimal order O( s log n m ) in the sense that it is the optimal order that can be attained even if the signal is measured precisely without 1-bit quantization [37]. From Theorem 3.1 if the minimum nonzero magnitude of x * is large enough, i.e., |x * | min ≥ O( s log n m ), the support of x ℓ1 coincides with that of x * . 3.2. Comparison with related works. Assuming x * 2 = 1 and σ = 0 and q = 1, [6] proposed to decode x * with min x∈R n x 1 s.t. y ⊙ Ψx ≥ 0, x 2 = 1. A first order algorithm was devised to solve the following Lagrangian version [34], i.e., min x∈R n max{0, −y ⊙ Ψx} 2 2 + λ x 1 s.t. x 2 = 1. In the presence of noise, [29] introduced min x∈R n L(max{0, −y ⊙ Ψx}) s.t. x 0 ≤ s, x 2 = 1, (3.2) where L(·) = · 1 or · 2 2 . They used a projected sub-gradient method, the so-called binary iterative hard thresholding (BITH), to solve (3.2). Assuming that there are sign flips in the noiseless model with σ = 0, [14] considered min x∈R n λ max{0, ν1 − y ⊙ Ψx} 0 + β 2 x 2 2 s.t. x 0 ≤ s, (3.3) 5 where ν > 0, β > 0 are tuning parameters. An adaptive outlier pursuit (AOP) generalization of (3.2) was proposed in [49] to recover x * and simultaneously detect the entries with sign flips by min x∈R n ,Λ∈R m L(max{0, −Λ ⊙ y ⊙ Ψx}) s.t. Λ i ∈ {0, 1}, 1 − Λ 1 ≤ N, x 0 ≤ s, x 2 = 1, where N is the number of sign flips. Alternating minimization on x and Λ are adopted to solve the optimization problem. [24] considered the pinball loss to deal with both the noise and the sign flips, which reads min x∈R n L τ (ν1 − y ⊙ Ψx}) s.t. x 0 ≤ s x 2 = 1, where L τ (t) = t1 t≥0 − τ t1 t<0 . Similar to the BITH, the pinball iterative hard thresholding is utilized. With the binary stable embedding, [29] and [14] proved that with high probability, the sample complexity of (3.2) and (3.3) to guarantee estimation error smaller than δ is m ≥ O( s δ 2 log n s ), which echoes Theorem 3.1. However, there are no theoretical results for other models mentioned above. All the aforementioned models and algorithms are the state-of-the-art works in the 1-bit CS. However, all the methods mentioned above are nonconvex. It is thus hard to justify whether the corresponding algorithms are loyal to their models. Another line of research concerns convexification. The pioneering work is [40], where they considered the noiseless case without sign flips and proposed the following linear programming method x lp ∈ arg min x∈R n x 1 s.t. y ⊙ Ψx ≥ 0 Ψx 1 = m. As shown in [40], the estimation error is x lp [41], where both the noise and the sign flips are allowed, through considering the convex problem x lp − x * ≤ O((s log 2 n s ) 1 5 ). The above result is improved to xcv xcv − x * ≤ O((s log n s ) 1 4 ) inx cv ∈ arg min x∈R n − y, Ψx /m s.t. x 1 ≤ s, x 2 ≤ 1. (3.4) Comparing with our result in Theorem 3.1, the results derived in [40] and [41] are suboptimal. In the noiseless case and assuming Σ = I n , [50] considered the Lagrangian version of (3.4) min x∈R n − y, Ψx /m + λ x 1 s.t. x 2 ≤ 1. (3.5) In this special case, the estimation error derived in [50] improved that of [41] and matched our results in Theorem 3.1. However, [50] did not discuss the scenario of Σ = I n . Recently [42,47], proposed a simple projected linear estimator P K (Ψ t y/m), where K = {x x 1 ≤ s, x 2 ≤ 1}, to estimate the low-dimensional structure target belonging to K in high dimensions from noisy and possibly nonlinear observations. They derived the same order of estimation error as that in Theorem 3.1. [51] proposed an ℓ 1 regularized maximum likelihood estimate, and [24] introduced a convex model through replacing the linear loss in (3.5) with the pinball loss. However, neither studied sample complexity or estimation error. 6 4. Primal dual active set algorithm. Existing algorithms for (1.2) are mainly first order methods [45,2,12]. In this section we use primal dual active set method [18,30], which is a generalized Newton type method, [27,43] to solve (1.2). An important advantage of the PDAS is that it converges after one-step iteration if the initial value is good enough. We globalize it with continuation on regularization parameter. We also propose a novel regularization parameter selection rule which is incorporated along the continuation procedure without any additional computational burden. PDAS. In this section we use x to denote x ℓ1 for simplicity. We begin with the following results [13]. Let x be the minimizer of (1.2), then x satisfies d = Ψ t (y − Ψx)/m, x = S λ (x + d). (4.1) Conversely, if x and d satisfy (4.1), then x is a global minimizer of (1.2), where S λ (z) is the pointwise soft-thresholding operator defined by S λ (z i ) = arg min t 1 2 (t − z i ) 2 + λ|t| 1 = sign(z i ) max{|z i | − λ, 0}. Let Z = x d and F (Z) = F 1 (Z) F 2 (Z) : R n × R n → R 2n , where F 1 (Z) = x − S λ (x + d) and F 2 (Z) = Ψ t Ψx + md − Ψ t y. By (4.1), finding the minimizer x of (1.2) is equivalent to finding the root of F (Z). We use the following primal dual active set method (PDAS) [18,30] to find the root of F (Z). Compute the active and inactive sets A k and I k respectively by A k = {i ∈ [n] |x k i + d k i | > λ} and I k = A k . 4: Update x k+1 and d k+1 by              x k+1 I k = 0, d k+1 A k = λsign(x k A k + d k A k ), (Ψ t A k Ψ A k )x k+1 A k = Ψ t A k (y − md k+1 A k ), d k+1 I k = Ψ t I k (y − Ψ A k x k+1 A k )/m. 5: If A k = A k+1 , stop. 6: end for 7: Output x λ . Remark 4.1. We can stop when k is greater than a user-predefined MaxIter. Since the PDAS converges after one iteration, a desirable property stated in Theorem 4.1, we set MaxIter = 1. The PDAS is actually a generalized Newton method for finding roots of nonsmooth equations 7 [27,43], since the iteration in Algorithm 1 can be equivalently reformulated as J k D k = −F (Z k ), (4.2) Z k+1 = Z k + D k , (4.3) where J k = J k 1 J k 2 Ψ t Ψ mI , J k 1 = 0 A k ,A k 0 A k ,I k 0 I k ,A k I I k ,I k and J k 2 = −I A k ,A k 0 A k ,I k 0 I k ,A k 0 I k ,I k . (4.4) We prove this equivalency in Appendix E for completeness. Local superlinear convergence has been established for generalized Newton methods for nonsmooth equations [27,43]. The PDAS require one iteration to convergence. We state the results here for completeness, which is proved in [18]. Theorem 4.1. Let x and d satisfy (4.1). Denote A = {i ∈ [n] |x i + d i | ≥ λ} and ω = min i∈[n],|xi+di| =λ ||x i + d i | − λ|}. Let x 0 and d 0 = Ψ t (y − Ψx 0 )/m be initial input of algorithm 1. If the columns of Ψ A are full-rank and the initial input satisfies x − x 0 ∞ + d − d 0 ∞ ≤ ω. Then, x 1 = x, where x 1 is updated from x 0 after one iteration. Globalization and automatic regularization parameter selection. To apply the PDAS (Algorithm 1) to (1.2), we need to have an initial guess x 0 and specify a proper regularization parameter λ in pdas(y, Ψ, λ, x 0 , MaxIter). In this section, we address these two issues together with continuation. Since the PDAS is a Newton type algorithm with fast local convergence rate and x ℓ1 is piecewise linear function of λ [39], we adopt a continuation to fully exploit the fast local convergence. In particular, this is a simple way to globalize the convergence of PDAS [18]. Observing that x = 0 satisfies (4.1) if λ ≥ λ 0 = Ψ t y/m ∞ , we define λ t = λ 0 ρ t with ρ ∈ (0, 1) for t = 1, 2, . . .. We run Algorithm 1 on the sequence {λ t } t with warmstart, i.e., using the solution x λt as an initial guess for the λ t+1 -problem. When the whole continuation is done we obtain a solution path of (1.2). For simplicity, we refer to the PDAS with continuation as PDASC described in Algorithm 2. Algorithm 2 PDASC: {x λt } t∈[MaxGrid] ← pdasc(y, Ψ, λ 0 , x λ0 , ρ, MaxGrid, MaxIter) 1: Input, y, Ψ, λ 0 = Ψ t y/m ∞ , x λ0 = 0, ρ ∈ (0, 1), MaxGrid, MaxIter. 2: for t = 1, 2, ...MaxGrid do 3: Run algorithm 1 x λt ← pdas(y, Ψ, λ, x 0 , MaxIter) with λ = λ t = ρ t λ 0 , initialized with x 0 = x λt−1 . 4: Check the stopping criterion. 5: end for 6: Output, {x λt } t∈[MaxGrid] . The regularization parameter λ in the ℓ 1 -regularized 1-bit CS model (1.2), which compromises the tradeoff between data fidelity and the sparsity level of the solution, is important for theoretical analysis and practical computation. However, the well known regularization parameter selection rules such as discrepancy principle [17,25], balancing principle [10,31,11,26] or Bayesian information criterion [18,33], are not applicable to the 1-bit CS problem considered here, since the model errors are not 8 available directly. Here we propose a novel rule to select regularization parameter automatically. We run the PDASC to yield a solution path until x λT 0 > ⌊ m log n ⌋ for the smallest possible T . Let S ℓ = {λ t : x λt 0 = ℓ, t = 1, ..., T }, ℓ = 1, ..., ⌊ m log n ⌋ be the set of regularization parameter at which the output of PDAS has ℓ nonzero elements. We determine λ by voting, i.e., λ = max{Sl} andl = arg max ℓ {|S ℓ |}. even if the correlation ν, the noise level σ and the probability of sign flips q are large. Now we compare the PDASC with the aforementioned competitors to recover a one-dimensional signal. The true signal are sparse under wavelet basis "Db1" [36]. Thus, the matrix Ψ is of size 2500 × 8000 and consists of random Gaussian and an inverse of one level Harr wavelet transform. The target coefficient has 36 nonzeros. We set σ = 0.5, q = 4%. The recovered results are shown in Table 5.2. The reconstruction by the PHDAS is visually more appealing than others, as shown in Fig. 5.4. This is further confirmed by the PSNR value reported in Table 5.2, which is defined by PSNR = 10 · log V 2 MSE , where V is the maximum absolute value of the true signal, and MSE is the mean squared error of the reconstruction. solution x ls approximates x * with precision δ as long as m ≥ O( n δ 2 ). For m < n, we assume that the underlying target x * is s-sparse, and prove that up to a constant c, with high probability, the ℓ 1 -regularized least squares solution x ℓ1 lies in the ball with center x * and radius δ, provided that m ≥ O( s log n δ 2 ). We introduce the one-step convergent PDAS method to minimize the nonsmooth objection function. We propose a novel tuning parameter selection rule which is seamlessly integrated Proof. Let u =ψ t x * . Then u ∼ N (0, 1) due toψ ∼ N (0, Σ) and the assumption that x * Σ = 1. E[ψỹ] = E[ψηsign(ψ t x * +ǫ)] = E[η]E[ψsign(ψ t x * +ǫ)] = [q − (1 − q)]E[ψsign(ψ t x * +ǫ)] = (2q − 1)E[E[ψsign(ψ t x * +ǫ)|ψ t x * ]] = (2q − 1)E[E[ψ|ψ t x * ]sign(ψ t x * +ǫ)] = (2q − 1)E[E[ψ|ψ t x * ]E[sign(ψ t x * +ǫ)|ψ t x * ]] = (2q − 1)E[Σx * ψt x * E[sign(ψ t x * +ǫ)|ψ t x * ]] = (2q − 1)E[sign(ψ t x * +ǫ)ψ t x * ]Σx * = cΣx * , where c = (2q − 1)E[sign(ψ t x * +ǫ)ψx * ]. The second line follows from independence assumption and the third from law of total expectation, and the fourth and fifth lines are due to the independence betweenǫ and u, and the sixth line uses the projection interpretation of conditional expectation i.e., E[ψ|ψ t x * ] = E[ψψ t x * ]−E[ψ t x * ]E[ψ] E[(ψ t x * −E[ψ t x * ]) 2 ] (ψ t x * − E[ψ t x * ]) + E[ψ] = Σx * ψt x * , where we use E[ψ] = 0 and u ∼ N (0, 1). Let f (t) = 1 √ 2πσ exp −t 2 2σ 2 be the density function ofǫ ∼ N (0, σ 2 ). Integrating by parts shows that c = (2q − 1)E[sign(u +ǫ)u] = (2q − 1)E[(1 − 2P[ǫ ≤ −u])u] = (2q − 1)E[ ∂(1 − 2P[ǫ ≤ −u) ∂u ] = (2q − 1)E[f (−u)] = (2q − 1) 2 π(σ 2 + 1) . The proof is completed by inverting cΣ. Appendix B. Preliminaries. We recall some simple properties of subgaussian and subexponential random variables. Lemma B.2. Let Ψ ∈ R m×n whose rows ψ t i are independent subgaussian vectors in R n with mean 0 and covariance matrix Σ. Let m > n. Then for every t > 0 with probability at least 14 1 − 2 exp (−C 1 t 2 ), one has (1 − τ ) γ min (Σ) ≤ γ min ( Ψ t Ψ m ) ≤ γ max ( Ψ t Ψ m ) ≤ (1 + τ ) γ max (Σ), (B.1) and Ψ t Ψ/m − Σ ≤ max{τ, τ 2 }γ max (Σ), (B.2) where τ = C 2 n m + t P[| m i=1 ξ i |/m ≥ t] ≤ 2 exp(− min{C 1 t 2 , C 2 t}m) where C 1 and C 2 are are generic positive constants depending on the maximum subexponential norm of of ξ i . Lemma B.4. Let Ψ ∈ R m×n whose rows ψ i are independent subgaussian vectors in R n×1 with mean 0 and covariance matrix Σ. Then, with probability at least 1 − 2 exp (−C 1 C 2 2 n), Ψ t Ψ/m − E[Ψ t Ψ/m] ≤ 2C 2 γ max (Σ) n m , (B.3) as long as m ≥ 4C 2 2 n. Furthermore, if m > 4C1 C 2 2 log n, then m i=1 (E[ψ i y i ] − ψ i y i )/m ∞ ≤ 2 log n C 1 m , (B.4) holds with probability at least 1 − 2 n 3 , and Ψ t Ψ/m − Σ ∞ ≤ 2 log n C 1 m , (B.5) holds with probability at least 1 − 1 n 2 . Proof. By P[ m i=1 (E[ψ i y i ] − ψ i y i )/m ∞ ≥ t] = P[ n j=1 {| m i=1 G i,j /m| ≥ t}] ≤ n j=1 P[| m i=1 G (i) k,j |/m ≥ t] ≤ n exp(− min{C 1 t 2 , C 2 t}m) ≤ 2n exp(−C 1 t 2 m), where the first inequality is due to the union bound, the second follows from Lemma B.3 and the last is because of restrictions t ≤ C2 C1 and m < n. Then (B.4) follows from our assumption that m > 4C1 C 2 2 log n by setting t = 2 log n C1m and. Let G i j,k := (ψ i ) j (ψ i ) k − Σ j,k ∈ R 1 , i = 1, ..., n, j = 1, ..., n, ℓ = 1, .., n, which is mean 0 subexponential by Lemma B.1. Therefore, P[ Ψ t Ψ/m − Σ ∞ ≥ t] = P[max j,k | m i=1 G i j,k /m| ≥ t] = P[ n,n j=1,k=1 {| m i=1 G i j,k /m| ≥ t}] ≤ n,n j=1,k=1 P[| m i=1 G i j,k /m| ≥ t] ≤ n 2 exp(− min{C 1 t 2 , C 2 t}m) ≤ n 2 exp(−C 1 t 2 m), where the first inequality is due to the union bound, and the second follows from Lemma B.3 and the last inequality is because of restricting t ≤ C2 C1 . Then by the assumption that m > 4C1 Proof. First we show that the sample covariance matrix Ψ t Ψ/m is invertible with probability at least 1 − 2 exp (−C 1 C 2 2 n) as long as m > 4C 2 n. This follows from (B.1) in Lemma B.2 by setting t = C 2 √ n. Recall x * = cx * . (C.1) Let ∆ = y − Ψ x * , (C.2) be the error in measuring nonlinearity, sign flips and noise in the 1-bit CS measurement. Then, Ψ t ∆/m 2 = Ψ t (Ψ x * − y)/m 2 = Ψ t Ψ m x * − Ψ t y/m 2 = ( Ψ t Ψ m x * − Σ x * ) + (Σ x * − Ψ t y/m) 2 = ( Ψ t Ψ m x * − E[ Ψ t Ψ m x * ]) + (E[Ψ t y/m] − Ψ t y/m) 2 ≤ |c| x * 2 Ψ t Ψ m − E[ Ψ t Ψ m ] + m i=1 (E[ψ i y i ] − ψ i y i )/m 2 ≤ |c| 1 γ min (Σ) Ψ t Ψ m − E[ Ψ t Ψ m ] + √ n m i=1 (E[ψ i y i ] − ψ i y i )/m ∞ , (C.3) where the fourth equality is due to (2.1), (2.3) and (2.4), the first inequality follows from the triangle inequality and the definition of x * , and the last inequality uses the assumption 1 = x * 2 Σ ≥ γ min (Σ) x * 2 2 and the fact that · 2 ≤ √ n · ∞ . Combining with (B.3) and (B.4), we deduce that, with probability at least 1 − 2 exp (−C 1 C 2 2 n) − 2 n 3 , Ψ t ∆/m 2 ≤ n m 2(|c|C 2 κ(Σ)γ max (Σ) + log n C 1 ). (C.4) Now we prove that x ls /c − cx * 2 = O( n m /c) with high probability. |c| x ls /c − x * 2 = x ls − x * 2 = (Ψ t Ψ) −1 Ψ t y − x * 2 ≤ (Ψ t Ψ/m) −1 Ψ t (Ψ x * + y − Ψ x * )/m − x * 2 = (Ψ t Ψ/m) −1 Ψ t ∆/m 2 ≤ n m 2(|c|C 2 κ(Σ)γ max (Σ) + log n C 1 )/(1 − 2C 2 n m ) 2 ≤ n m 4(|c|C 2 κ(Σ)γ max (Σ) + log n C 1 ), where the second inequality follows with probability at least 1 − 4 exp (−C 1 C 2 2 n) − 2 n 3 from (C.3) and (B.1) by setting t = C 2 √ n, and the last line is due to the assumption m ≥ 16C 2 2 n. Hence, the proof of Theorem 2.2 is completed by dividing |c| on both side and some algebra. Appendix D. Proof of Theorem 3.1. Proof. Our proof is based on Lemmas D.1 -D.3 below. Denote R = x ℓ1 − x * , A * = supp(x * ) and I * = A * . The first lemma shows that R is sparse in the sense that its energy is mainly cumulated on A * if λ is chosen properly. Lemma D.1. Let C A * = {z ∈ R n : z I * 1 ≤ 3 z A * 1 }, (D.1) and define E = { Ψ t ∆/m ∞ ≤ λ/2}. Conditioning on the event E, we have R ∈ C A * . Proof. The optimality of x ℓ1 implies that 1 2m y − Ψx ℓ1 2 2 + λ x ℓ1 1 ≤ 1 2m y − Ψ x * 2 2 + λ x * 1 . Recall that y = Ψ x * + ∆. Some algebra on the above display shows 1 2m ΨR 2 2 + λ R I * 1 ≤ R, Ψ t ∆/m + λ R A * 1 ≤ R 1 Ψ t ∆/m ∞ + λ R A * 1 ≤ R 1 λ/2 + λ R A * 1 , where, we use Cauchy Schwartz inequality and the definition of E. The above inequality shows 1 m ΨR 2 2 + λ R I * 1 ≤ 3λ R A * 1 , (D.2) i.e., R ∈ C A * . This finishes the proof of Lemma D.1. The next Lemma gives a lower bound on P[E] with a proper regularization parameter λ. Lemma D.2. Let C 3 ≥ x * 1 . If m > 4C1 C 2 2 log n, taking λ = 4(1+|c|C3) √ C1 log n m , then with probability at least 1 − 2/n 3 − 2/n 2 , one has Ψ t ∆/m ∞ ≤ λ/2. (D.3) Proof. Ψ t ∆/m ∞ = Ψ t (Ψ x * − y)/m ∞ = ( Ψ t Ψ m x * − Σ x * ) + (Σ x * − Ψ t y/m) ∞ ≤ ( Ψ t Ψ m x * − Σ x * ) ∞ + (E[Ψ t y/m] − Ψ t y/m) ∞ ≤ |c| ( Ψ t Ψ m − Σ) ∞ x * 1 + m i=1 (E[ψ i y i ] − ψ i y i )/m ∞ ≤ |c|C 3 2 log n C 1 m + 2 log n C 1 m = 2(1 + |c|C 3 ) √ C 1 log n m , where the first inequality is due to the triangle inequality, s log en s , then with probability at least 1 − 1/n 2 , we have Ψz 2 2 /m ≥ γ min (Σ) 68(4κ(Σ) + 1) 2 z 2 2 , ∀z ∈ C A * . We obtain that, Ψ A z A , Ψ I z I /m = Ψ A z A , t≥1 Ψ It z It /m ≤ z A 2 t≥1 Ψ t A Ψ It 2 z It 2 ≤ O s (Ψ) z A 2 t≥1 z It 2 ≤ 4O s (Ψ) z A 2 , (D.7) where the first inequality uses Cauchy-Schwartz inequality, the second inequality follows from the definition of O s (Ψ), and the third is due to (D.5). Then, ∀z ∈ C A * , z = 0, we have Ψz 2 2 /(m z 2 ) ≥ Ψz 2 2 /(17m z A 2 ) = ( Ψ A z A 2 + Ψ I z I 2 + 2 Ψ A z A , Ψ I z I )/(17m z A 2 ) ≥ ( Ψ A z A 2 − 8O s (Ψ) z A 2 )/(17m z A 2 ) ≥ (C 2s (Ψ) − 8O s (Ψ))/17, (D.8) where the first inequality uses (D.6), the second inequality follows from (D.7), and the last holds due to the definition of C 2s (Ψ). It follows from (D.8) that, to complete the proof of this lemma it suffices to derive a lower bound on C 2s (Ψ) and an upper bound on O s (Ψ) with high probability, respectively. Given A ⊂ [n], |A| ≤ 2s, we define the event E A = { γmin(Ψ t A ΨA) m > γ min (Σ)(1 − C 2 2s m − t √ m )}. Then, P[C 2s (Ψ) > γ min (Σ)(1 − C 2 2s m − t √ m ) 2 ] = P[ A∈[n],(1 − P[E A ]) ≥ 1 − 2s ℓ=1 A⊂[n],|A|≤ℓ 2 exp(−C 1 t 2 ) = 1 − 2s ℓ=1 n ℓ 2 exp(−C 1 t 2 ) ≥ 1 − 2( en 2s ) 2s exp(−C 1 t 2 ), where the first inequality follows from the union bound, the second inequality follows from (B.1) by replacing Ψ with Ψ A , and the third inequality holds since 2s ℓ=1 n ℓ ≤ ( n 2s ) 2s 2s ℓ=0 n ℓ ( 2s n ) ℓ ≤ ( n 2s ) 2s (1 + 2s n ) n ≤ ( en 2s ) 2s . Then, we derive with probability at least 1 − 2( en 2s ) 2s exp(−C 1 t 2 ), CC , G C = Φ t C Φ C /m − I 2s . Then each row of Φ C is multivariate normal random vector that is sampled from N (0, I 2s ). It follows from (B.2) with Ψ and Σ replaced by Φ C and I 2s , respectively, that C 2s (Ψ) > γ min (Σ)(1 − C 2 2s m − t √ m ) 2 .P[ G C ≥ C 2 2s m + t √ m ] ≤ 2 exp (−C 1 t 2 ). Observing Σ − 1 2 CC Ψ t A Ψ B Σ − 1 2 CC /m is a sub-matrix of G C , we deduce, P[E A,B ] ≤ 2 exp (−C 1 t 2 ). Then, similarly to the proof of (D.9), we have P[O s (Ψ) > γ max (Σ)(C 2 2s m + t √ m )] = P[ A,γ min (Σ) 68(4κ(Σ) + 1) 2 R 2 2 ≤ 3λ R A * 1 ≤ 3λ √ s R A * 2 , i.e., |c| x ℓ1 /c − x * 2 = x ℓ1 − x * 2 ≤ 204(4κ(Σ) + 1) 2 γ min (Σ) λ √ s. The proof of Theorem 3.1 is completed by dividing |c| on both side and using Lemma D.2, which guaranties that (D.2) holds with λ = 4(1+|c|C3) √ C1 log n m with probability greater than 1 − 2/n 3 − 2/n 2 . Appendix E. Proof of the equivalency between the PDAS and (4.2) -(4.3). Proof. Partition Z k , D k and F (Z k ) according to A k and I k such that Z k =       x A k x I k d A k d I k       , D k =       D x A k D x I k D d A k D d I k       , (E.1) F (Z k ) =       −d k A k + λsign(x k A + d k A ) x k I k Ψ t A k Ψ A k x k A k + Ψ t A k Ψ I k x k I k + md k A k − Ψ t A k y Ψ t I k Ψ A k x k A k + Ψ t I k Ψ I k x k I k + md k I k − Ψ t I k y       . (E.2) 22 Substituting (E.1) -(E.2) and (4.4) into (4.2), we have −(d k A k + D d A k ) = −λsign(x k A + d k A ), (E.3) x k I k + D x I k = 0, (E.4) Ψ t A k Ψ A k (x k A k + D x A k ) = Ψ t A k y − m(d k A k + D d A k ) − Ψ t A k Ψ I k (x k I k + D x I k ), (E.5) m(d k I k + D d I k ) = Ψ t I k y − Ψ t I k Ψ A k (x k A k + D x A k ) − Ψ t A k Ψ I k (x k I k + D x I k ). (E.6) It follows from (4.3) that       x k+1 A k x k+1 I k d k+1 A k d k+1 I k       =       x k I k + D x I k d k A k + D d A k x k A k + D x A k d k I k + D d I k       . (E.7) Substituting (E.7) into (E.3) -(E.6), we get the iteration procedure of PDAS in Algorithm 1. This completes the proof. Algorithm 1 1PDAS: x λ ← pdas(y, Ψ, λ, x 0 , MaxIter) 1: Input y, Ψ, λ, initial guess x 0 , maximum number of iteration MaxIter. Let d 0 = Ψ t (y − Ψx 0 )/m. 2: for k = 0, 1, ...MaxIter do 3: our parameter selection rule is seamlessly integrated with the continuation strategy which serves as a globalization technique without any extra computational overhead. Here we give an example to show the accuracy of our proposed regularization parameter selection rule (4.5) with data {m = 400, n = 10 3 , s = 5, ν = 0.5, σ = 0.01, q = 2.5%}. Descriptions of the data can be found in Section 5. Left panel of Fig. 4.1 shows the size of active set x λt 0 along the path of PDASC and right panel shows the underlying true signal x * and the solution xλ selected by (4.5). Fig. 4. 1 :Fig. 5. 1 :Fig. 5. 2 : 112Active set size on the path (left panel) on data {m = 400, n = 10 3 , s = 5, ν = 0.5, σ = 0.01, q = 2.5%} and the underlying true signal x * and the solution xλ selected by (4.5) (right panel).5. Numerical simulation. In this section we showcase the performance of our proposed least square decoders (2.5) and (1.2). All the computations were performed on a four-core laptop with 2.90 GHz and 8 GB RAM using MATLAB 2015b. The MATLAB package 1-bitPDASC for reproducing all the numerical results can be found at http://faculty.zuel.edu.cn/tjyjxxy/jyl/list.htm.5.1. Experiment setup.First we describe the data generation process and our parameter choice. In all numerical examples the underlying target signal x * with x * 0 = s is given, and the observation y is generated by y = η ⊙ sign(Ψx * + ǫ), where the rows of Ψ are iid samples fromN (0, Σ) with Σ jk = ν |j−k| , 1 ≤ j, k ≤ n. We keep the convention 0 0 = 1. The elements of ǫ are generated from N (0, I m ), η ∈ R m has independent coordinate η i with P[η i = 1] = 1 − P[η i = −1] = q.Here, we use {m, n, s, ν, σ, q} to denote the data generated as above for short. We fixρ = 0.95, MaxGrid = 200, MaxIter = 1 in our proposed PDASC algorithm and use (4.5) to determine regularization parameter λ. All the simulation results are based on 100 independent replications. 5.2. Accuracy and Robustness of x ls when m > n. Now we present numerical results to illustrate the accuracy of the least square decoder x ls and its robustness to the noise and the sign 9 flips. Fig. 5.1 shows the recovery error x ls − x * on data set {m = 10 3 , n = 10, s = 10, q = 2.5%}. Left panel of Fig. 5.2 shows the recovery error x ls − x * on data set {m = 1000, n = 10, s = 10, ν = 0.3, σ = 0.01, q = 0 : 1% : 10%} and right panel gives recovery error x ls + x * on data {m = 1000, n = 10, s = 10, ν = 0.3, σ = 0.01, q = 90 : 1% : 100%}. It is observed that the recovery error x ls − x * ( x ls + x * ) of the least square decoder is small (around 0.1) and robust to noise level σ and sign flips probability q. This confirms theoretically investigations in Theorem 2.2, which states the error is of order O( Recovery error x ls − x * v.s. σ on {m = 1000, n = 10, s = 10, ν = 0.3, σ = 0 : 0.05 : 0.5, Recovery error x ls − x * v.s. q on {m = 10 3 , n = 10, s = 10, ν = 0.3, σ = 0.01, q = 0 : 1% : 10%} (left panel) and − x ls − x * on {m = 1000, n = 10, s = 10, ν = 0.3, σ = 0.01, q = 90% : 1% : 100%} (right panel). 5. 3 . 3Support recovery of x ℓ1 when m < n. We conduct simulations to illustrate the performance of model (1.2) PDASC algorithm. We report how the exact support recovery probability varies with the sparsity level s, the noise level σ and the probability q of sign flips.Fig. 5.3indicates that, as long as the sparsity level s is not large, x ℓ1 recovers the underlying true support with high probability even if the measurement contains noise and is corrupted by sign flips. This confirms the theoretical investigations in Theorem 3.1. Fig. 5 . 3 : 53The exact support recovery probability v.s. s, σ and q on data set {m = 500, n = 1000, s = 1 : 2 : 12, ν = 0.1, σ = 0.05, q = 1%} (panel (a)), {m = 500, n = 1000, s = 5, ν = 0.3, σ = 0 : 0.1 : 0.6, q = 5%} (panel (b)) and {m = 500, n = 1000, s = 5, ν = 0.1, σ = 0.01, q = 0 : 3% : 15%} (panel (c)), {m = 500, n = 1000, s = 5, ν = 0.1, σ = 0.01, q = 85% : 3% : 100%} (panel (d)).5.4.Comparison with other state-of-the-art. Now we compare our proposed model(1.2) and PDASC algorithm with several state-of-the-art methods such as BIHT[28] (http://perso. uclouvain.be/laurent.jacques/index.php/Main/BIHTDemo), AOP[49] and PBAOP[24] (both AOP and PBAOP available at http://www.esat.kuleuven.be/stadius/ADB/huang/downloads/ 1bitCSLab.zip) and linear projection (LP)[47,42]. BIHT, AOP, LP and PBAOP are all required to specify the true sparsity level s. Both AOP and PBAOP also need to required to specify the sign flips probability q. The PDASC does not require to specify the unknown parameter sparsitylevel s or the probability of sign flips q. We use {m = 500, n = 1000, s = 5, ν = 0.1, σ = 0, q = 0}, {m = 500, n = 1000, s = 5, ν = 0.3, σ = 0.3, q = 5%}, {m = 500, n = 1000, s = 5, ν = 0.5, σ = 0.5, q = 10%}, and {m = 800, n = 2000, s = 10, ν = 0.1, σ = 0.1, q = 1%}, {m = 800, n = 2000, s = 10, ν = 0.2, σ = 0.3, q = 3%}, {m = 800, n = 2000, s = 10, ν = 0.3, σ = 0.5, q = 5%}, and {m = 5000, n = 20000, s = 50, ν = 0, σ = 0.2, q = 3}, {m = 5000, n = 20000, s = 50, ν = 0, σ = 0.1, q = 1}, {m = 5000, n = 20000, s = 5, ν = 0, σ = 0.3, q = 5%}. The average CPU time in seconds (Time (s)), the average of the ℓ 2 error x ℓ − x * (ℓ 2 -Err), and the probability of exactly recovering true support (PrE (%)) are reported in Fig. 5 5Fig. 5.4 and Table 5.2. The reconstruction by the PHDAS is visually more appealing than others, Fig. 5 . 4 : 54Reconstruction of the one-dimension signal with {m = 2500, n = 8000, s = 36, ν = 0, σ = 0.5, q = 4%}. with the continuation strategy without any extra computational overhead. Numerical experiments are presented to illustrate salient features of the model and the efficiency and accuracy of the algorithm. There are several avenues for further study. First, many practitioners observed that nonconvex sparse regularization often brings in additional benefit in the standard CS setting. Whether the theoretical and computational results derived in this paper can still be justified when nonconvex regularizers are used deserves further consideration. The 1-bit CS is a kind of nonlinear sampling approach. Analysis of some other nonlinear sampling methods are also of immense interest. Acknowledgements. The research of Y. Jiao is supported by National Science Foundation of China (NSFC) No. 11501579 and National Science Foundation of Hubei Province No. 2016CFB486. The research of X. Lu is supported by NSFC Nos. 11471253 and 91630313, and the research of L. Lemma B. 1 . 1( Lemma 2.7.7 of[48] and Remark 5.18 of[46].) Let ξ 1 and ξ 2 be subgaussian random variables. Then both ξ 1 ξ 2 and ξ 1 ξ 2 − E[ξ 1 ξ 2 ] are subexponential random variables.Lemma B.2 states the nonasymptotic bound on the spectrums of Ψ and the operator norm of Ψ t Ψ/m − Σ when m ≥ O(n). √m , and C 1 , C 2 are generic positive constants depending on the maximum subgaussian norm of rows of Ψ. Proof. Let Φ = ΨΣ − 1 2 . Then the rows of Φ are independent sub-gaussian isotropic vectors. (B.1) follows from Theorem 5.39 and Lemma 5.36 of [46] and (B.2) is a direct consequence of Remark 5.40 of [46]. We state the Bernstein-type inequality for the sum of independent and mean 0 sub-exponential random random variables. Lemma B.3. (Corollary 5.17 of [46]) Let ξ 1 , ..., ξ m be independent centered sub-exponential random variables. Then for every t > 0 one has (2.3), E[Ψ t Ψ/m] = Σ, hence (B.3) follows from (B.2) with t = C 2 √ n and the assumption m ≥ 4C 2 2 n. Define G i,j := y i (ψ i ) j ∈ R 1 , i = 1, ..., m, j = 1, .., n, which is subexponential by Lemma B.1. Therefore, by setting t = 2 log n C1m .Appendix C. Proof of Theorem 2.2. (2.1), (2.3) and (2.4), the second inequality follows from the definition of x * and Cauchy-Schwartz inequality, and the third one uses (B.4) and (B.5). The proof of Lemma D.2 is completed by setting λ = 4(1+|c|C3) Lemma shows Ψ is strongly convex along the direction contained in the cone C A * defined in (D.1). Lemma D.3. If s ≤ exp (1− C 1 2 ) n and m ≥ 64(4κ(Σ)+1) 2 C1 ⊂ [n], B ⊂ [n], |A| ≤ s, |B| ≤ s, A ∩ B = ∅, we define the event E A,B = { Ψ t A Ψ B /m > γ max (Σ) 2s exp(−C 1 t 2 ), which implies with probability at least 1 − 2( en s ) 2s exp(−C 1 t 2 ), O s (Ψ) ≤ γ max (Σ)D.9) and (D.10) and setting t = 4s C1 log en s , we obtain that with probability at least1 − 4/( en s ) 2s ≥ 1 − 4/n 2 C 2s (Ψ) − 8O s (Ψ) ≥ γ min (unitary function f (z) = z 2 − (8κ(Σ) + 2)z + 1. It follows from the assumption s ≤ exp (1− C 1 2 ) n that 2s m ≤4smC1 log en s . Then some basic algebra shows that f 4κ(Σ)+1) 2 as long as m ≥ 64(4κ(Σ)+1) 2 C1 s log en s . The proof of Lemma D.3 is completed. Now we are in the place of combining the above pieces together to finish the proof of Theorem 3.1. Recall R = x ℓ1 − x * . It follows from Lemma D.1 that R ∈ C A * and (D.2) holds by conditioning on E, i.e.,1 m ΨR 2 2 + λ R I * 1 ≤ 3λ R A * 1 ,which together with Lemma D.3 implies that, with probability at least 1 − 4/n 2 , Table 5.1. The PDASC is comparatively very fast and the most accurate Table 5 . 51: Comparison PDASC with state-of-the-art methods on CPU time in seconds (Time (s)), average ℓ 2 error x ℓ − x * (ℓ 2 -Err), probability on exactly recovering of true support (PrE (%)).{m = 500, n = 1000, s = 5} (a) {ν = 0.1, σ = 0.1, q = 1%} (b) {ν = 0.3, σ = 0.3, q = 5%} (c) {ν = 0.1, σ = 0.5, q = 10%} Method Time (s) ℓ 2 -Err PrE (%) Time (s) ℓ 2 -Err PrE (%) Time ℓ 2 -Err PrE BIHT 1.42e-1 1.89e-1 92 1.31e-1 5.73e-1 19 1.32e-1 9.39e-1 0 AOP 2.72e-1 7.29e-2 100 3.55e-1 2.11e-1 92 3.58e-1 4.22e-1 44 LP 8.70e-3 4.19e-1 98 8.50e-3 4.22e-1 93 8.30e-3 4.81e-1 26 PBAOP 1.46e-1 9.08e-2 100 1.36e-1 2.05e-1 90 1.35e-1 4.53e-1 36 PDASC 4.11e-2 6.77e-2 100 4.38e-2 9.40e-2 99 4.56e-2 2.21e-1 71 {m = 800, n = 2000, s = 10} (a) {ν = 0.1, σ = 0.1, q = 1%} (b) {ν = 0.3, σ = 0.2, q = 3%} (c) {ν = 0.5, σ = 0.3, q = 5%} Method Time (s) ℓ 2 -Err PrE (%) Time (s) ℓ 2 -Err PrE (%) Time ℓ 2 -Err PrE BIHT 4.17e-1 2.10-1 84 4.25e-1 4.21e-1 25 4.35e-1 6.46e-1 0 AOP 1.09e-0 7.78-2 100 1.10e-0 1.76e-1 95 1.16e-0 2.86e-1 59 LP 1.95e-2 4.54-1 85 1.99e-2 4.49e-1 71 2.05e-2 5.03e-1 16 PBAOP 4.22e-1 1.00-1 100 4.27e-1 1.58e-1 99 4.31e-1 2.99e-1 51 PDASC 1.23e-1 8.66-2 100 1.27e-1 1.04e-1 98 1.30e-2 1.51e-1 78 {m = 5000, n = 20000, s = 50, ν = 0} (a) {σ = 0.1, q = 1%} (b) {σ = 0.2, q = 3%} (c) {σ = 0.3, q = 5%} Method Time (s) ℓ 2 -Err PrE (%) Time (s) ℓ 2 -Err PrE (%) Time ℓ 2 -Err PrE BIHT 2.56e+1 2.16e-1 58 2.58e+1 4.54e-1 0 2.58e+1 6.29e-1 0 AOP 6.44e+1 7.56e-2 100 6.46e+1 1.66e-1 96 6.47e+1 2.57e-1 16 LP 2.35e-1 4.47e-1 38 2.30e-1 4.46e-1 34 2.30e-1 4.47e-1 26 PBAOP 2.56e+1 9.89e-2 100 2.58e+1 1.66e-1 95 2.58e+1 2.60e-1 18 PDASC 7.09e-0 7.97e-2 100 7.17e-0 9.17e-2 99 7.23e-0 1.23e-1 86 Table 5 . 52: The CPU time in seconds and the PSNR of one dimensional signal recovery with {m = 2500, n = 8000, s = 36, ν = 0, σ = 0.5, q = 4%}.method CPU time (s) PSNR BIHT 4.97 29 AOP 4.98 33 LP 0.11 33 PBAOP 4.93 31 PDASC 3.26 36 6. 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Birkhäuser Basel1Simon Foucart and Holger Rauhut, A mathematical introduction to compressive sensing, vol. 1, Birkhäuser Basel, 2013. One-bit compressed sensing: Provable support and vector recovery. Sivakant Gopi, Praneeth Netrapalli, Prateek Jain, Aditya Nori, International Conference on Machine Learning. Sivakant Gopi, Praneeth Netrapalli, Prateek Jain, and Aditya Nori, One-bit compressed sensing: Prov- able support and vector recovery, in International Conference on Machine Learning, 2013, pp. 154-162. Sample complexity for 1-bit compressed sensing and sparse classification. Ankit Gupta, Robert Nowak, Benjamin Recht, Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on. IEEEAnkit Gupta, Robert Nowak, and Benjamin Recht, Sample complexity for 1-bit compressed sensing and sparse classification, in Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on, IEEE, 2010, pp. 1553-1557. Robust support recovery using sparse compressive sensing matrices. Jarvis Haupt, Richard Baraniuk, Information Sciences and Systems (CISS), 2011 45th Annual Conference on. IEEEJarvis Haupt and Richard Baraniuk, Robust support recovery using sparse compressive sensing matrices, in Information Sciences and Systems (CISS), 2011 45th Annual Conference on, IEEE, 2011, pp. 1-6. Xiaolin Huang, Lei Shi, Ming Yan, Johan Ak Suykens, arXiv:1505.03898Pinball loss minimization for one-bit compressive sensing. arXiv preprintXiaolin Huang, Lei Shi, Ming Yan, and Johan AK Suykens, Pinball loss minimization for one-bit compressive sensing, arXiv preprint arXiv:1505.03898, (2015). . Kazufumi Ito, Bangti Jin, of Series on Applied Mathematics, World Scientific. 22Inverse Problems: Tikhonov Theory and AlgorithmsKazufumi Ito and Bangti Jin, Inverse Problems: Tikhonov Theory and Algorithms, vol. 22 of Series on Applied Mathematics, World Scientific, NJ, 2014. A regularization parameter for nonsmooth tikhonov regularization. Kazufumi Ito, Jin Bangti, Tomoya Takeuchi, SIAM Journal on Scientific Computing. 33Kazufumi Ito, Bangti Jin, and Tomoya Takeuchi, A regularization parameter for nonsmooth tikhonov regu- larization, SIAM Journal on Scientific Computing, 33 (2011), pp. 1415-1438. Lagrange multiplier approach to variational problems and applications. Kazufumi Ito, Karl Kunisch, SIAMKazufumi Ito and Karl Kunisch, Lagrange multiplier approach to variational problems and applications, SIAM, 2008. Laurent Jacques, Kévin Degraux, Christophe De Vleeschouwer, arXiv:1305.1786Quantized iterative hard thresholding: Bridging 1-bit and high-resolution quantized compressed sensing. arXiv preprintLaurent Jacques, Kévin Degraux, and Christophe De Vleeschouwer, Quantized iterative hard thresh- olding: Bridging 1-bit and high-resolution quantized compressed sensing, arXiv preprint arXiv:1305.1786, (2013). Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors. Laurent Jacques, Jason N Laska, T Petros, Richard G Boufounos, Baraniuk, IEEE Transactions on Information Theory. 59Laurent Jacques, Jason N Laska, Petros T Boufounos, and Richard G Baraniuk, Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors, IEEE Transactions on Information Theory, 59 (2013), pp. 2082-2102. A primal dual active set with continuation algorithm for the ?0-regularized optimization problem. Yuling Jiao, Jin Bangti, Xiliang Lu, Applied and Computational Harmonic Analysis. 39Yuling Jiao, Bangti Jin, and Xiliang Lu, A primal dual active set with continuation algorithm for the ?0- regularized optimization problem, Applied and Computational Harmonic Analysis, 39 (2015), pp. 400-426. Iterative parameter choice by discrepancy principle. Yubo Bangti Jin, Jun Zhao, Zou, IMA Journal of Numerical Analysis. 32Bangti Jin, Yubo Zhao, and Jun Zou, Iterative parameter choice by discrepancy principle, IMA Journal of Numerical Analysis, 32 (2012), pp. 1714-1732. One-bit compressive sensing with norm estimation. Karin Knudson, Rayan Saab, Rachel Ward, IEEE Transactions on Information Theory. 62Karin Knudson, Rayan Saab, and Rachel Ward, One-bit compressive sensing with norm estimation, IEEE Transactions on Information Theory, 62 (2016), pp. 2748-2758. Information criteria and statistical modeling. Sadanori Konishi, Genshiro Kitagawa, Springer Science & Business MediaSadanori Konishi and Genshiro Kitagawa, Information criteria and statistical modeling, Springer Science & Business Media, 2008. Trust, but verify: Fast and accurate signal recovery from 1-bit compressive measurements. Zaiwen Jason N Laska, Wotao Wen, Richard G Yin, Baraniuk, IEEE Transactions on Signal Processing. 59Jason N Laska, Zaiwen Wen, Wotao Yin, and Richard G Baraniuk, Trust, but verify: Fast and accurate signal recovery from 1-bit compressive measurements, IEEE Transactions on Signal Processing, 59 (2011), pp. 5289-5301. One-bit compressed sensing by greedy algorithms. Wenhui Liu, Da Gong, Zhiqiang Xu, Numerical Mathematics: Theory, Methods and Applications. 9Wenhui Liu, Da Gong, and Zhiqiang Xu, One-bit compressed sensing by greedy algorithms, Numerical Math- ematics: Theory, Methods and Applications, 9 (2016), pp. 169-184. A wavelet tour of signal processing: the sparse way. Stephane Mallat, Academic pressStephane Mallat, A wavelet tour of signal processing: the sparse way, Academic press, 2008. A unified framework for high-dimensional analysis of m-estimators with decomposable regularizers. Pradeep Sahand N Negahban, Ravikumar, J Martin, Bin Wainwright, Yu, Statistical Science. 27Sahand N Negahban, Pradeep Ravikumar, Martin J Wainwright, Bin Yu, et al., A unified framework for high-dimensional analysis of m-estimators with decomposable regularizers, Statistical Science, 27 (2012), pp. 538-557. Yurii Nesterov, Introductory lectures on convex optimization: A basic course. Springer Science & Business Media87Yurii Nesterov, Introductory lectures on convex optimization: A basic course, vol. 87, Springer Science & Business Media, 2013. A new approach to variable selection in least squares problems. Brett Michael R Osborne, Presnell, Berwin, Turlach, IMA journal of numerical analysis. 20Michael R Osborne, Brett Presnell, and Berwin A Turlach, A new approach to variable selection in least squares problems, IMA journal of numerical analysis, 20 (2000), pp. 389-403. One-bit compressed sensing by linear programming. Yaniv Plan, Roman Vershynin, Communications on Pure and Applied Mathematics. 66Yaniv Plan and Roman Vershynin, One-bit compressed sensing by linear programming, Communications on Pure and Applied Mathematics, 66 (2013), pp. 1275-1297. Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach. IEEE Transactions on Information Theory. 59, Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach, IEEE Transactions on Information Theory, 59 (2013), pp. 482-494. High-dimensional estimation with geometric constraints, Information and Inference: A. Yaniv Plan, Roman Vershynin, Elena Yudovina, Journal of the IMA. 6Yaniv Plan, Roman Vershynin, and Elena Yudovina, High-dimensional estimation with geometric con- straints, Information and Inference: A Journal of the IMA, 6 (2017), pp. 1-40. A nonsmooth version of newton's method, Mathematical programming. Liqun Qi, Jie Sun, 58Liqun Qi and Jie Sun, A nonsmooth version of newton's method, Mathematical programming, 58 (1993), pp. 353- 367. Regression shrinkage and selection via the lasso. Robert Tibshirani, J. Roy. Statist. Soc. Ser. B. 58Robert Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), pp. 267-288. Computational methods for sparse solution of linear inverse problems. J A Tropp, S J Wright, Proc. IEEE. IEEE98J.A. Tropp and S.J. Wright, Computational methods for sparse solution of linear inverse problems, Proc. IEEE, 98 (2010), pp. 948-958. Roman Vershynin, arXiv:1011.3027Introduction to the non-asymptotic analysis of random matrices. arXiv preprintRoman Vershynin, Introduction to the non-asymptotic analysis of random matrices, arXiv preprint arXiv:1011.3027, (2010). Estimation in high dimensions: a geometric perspective. Sampling theory, a renaissance. Springer, Estimation in high dimensions: a geometric perspective, in Sampling theory, a renaissance, Springer, 2015, pp. 3-66. High Dimensional Probability. , High Dimensional Probability, 2017. Robust 1-bit compressive sensing using adaptive outlier pursuit. Ming Yan, Yi Yang, Stanley Osher, IEEE Transactions on Signal Processing. 60Ming Yan, Yi Yang, and Stanley Osher, Robust 1-bit compressive sensing using adaptive outlier pursuit, IEEE Transactions on Signal Processing, 60 (2012), pp. 3868-3875. Efficient algorithms for robust one-bit compressive sensing. Lijun Zhang, Jinfeng Yi, Rong Jin, International Conference on Machine Learning. Lijun Zhang, Jinfeng Yi, and Rong Jin, Efficient algorithms for robust one-bit compressive sensing, in Inter- national Conference on Machine Learning, 2014, pp. 820-828. Compressed sensing with quantized measurements. Argyrios Zymnis, Stephen Boyd, Emmanuel Candes, IEEE Signal Processing Letters. 17Argyrios Zymnis, Stephen Boyd, and Emmanuel Candes, Compressed sensing with quantized measurements, IEEE Signal Processing Letters, 17 (2010), pp. 149-152.

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{'abstract': 'In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: y = η ⊙ sign(Ψx * + ǫ), where x * ∈ R n , y ∈ R m , Ψ ∈ R m×n , and ǫ is the random error before quantization and η ∈ R n is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For m > n, we show that, up to a constant c, with high probability, the least squares solution x ls approximates x * with precision δ as long as m ≥ O( n δ 2 ). For m < n, we prove that, up to a constant c, with high probability, the ℓ 1 -regularized least-squares solution x ℓ 1 lies in the ball with center x * and radius δ provided that m ≥ O( s log n δ 2 ) and x * 0 := s < m. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.Keywords: 1-bit compressive sensing, ℓ 1 -regularized least squares, primal dual active set algorithm, one step convergence, continuation 1. Introduction. Compressive sensing (CS) is an important approach to acquiring low dimension signals from noisy under-determined measurements[8,16,19,20]. For storage and transmission, the infinite-precision measurements are often quantized, [6] considered recovering the signals from the 1-bit compressive sensing (1-bit CS) where measurements are coded into a single bit, i.e., their signs. The 1-bit CS is superior to the CS in terms of inexpensive hardware implementation and storage. However, it is much more challenging to decode from nonlinear, noisy and sign-flipped 1-bit measurements.1.1. Previous work. Since the seminal work of [6], much effort has been devoted to studying the theoretical and computational challenges of the 1-bit CS. Sample complexity was analyzed for support and vector recovery with and without noise[21,28,40,23,29,22,23,41,50]. Existing works indicate that, m > O(s log n) is adequate for both support and vector recovery. The sample size required here has the same order as that required in the standard CS setting. These results have also been refined by adaptive sampling[22,14,4]. Extensions include recovering the norm of the target [32, 3] and non-Gaussian measurement settings [1]. Many first order methods[6,34,49,14]and greedy methods[35,5,29]are developed to minimize the sparsity promoting nonconvex objected function arising from either the unit sphere constraint or the nonconvex regularizers. To address', 'arxivid': '1711.01206', 'author': ['Jian Huang ', 'Yuling Jiao ', 'Xiliang Lu ', 'ANDLiping Zhu '], 'authoraffiliation': [], 'corpusid': 55752553, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23886, 'n_tokens_neox': 20717, 'n_words': 11380, 'pdfsha': '61e87c80e46b5280cdb93b7ea4bd1c0abd0683ee', 'pdfurls': ['https://arxiv.org/pdf/1711.01206v1.pdf'], 'title': ['ROBUST DECODING FROM 1-BIT COMPRESSIVE SAMPLING WITH LEAST SQUARES', 'ROBUST DECODING FROM 1-BIT COMPRESSIVE SAMPLING WITH LEAST SQUARES'], 'venue': []}

arxiv

Spectral Functions for the Holstein Model 5 Nov 1996 october 1996 January 19, 2022 J M Robin Centre de Recherche sur les Très Basses Températures 25, avenue des Martyrs, BP 16638042Grenoble Cedex 9France Spectral Functions for the Holstein Model 5 Nov 1996 october 1996 January 19, 2022preprint cond-mat We perform an unitary transformation for the symmetric phonon mode of the Holstein molecular crystal hamiltonian. We show how to compute the electronic spectral functions by exact numerical diagonalisation of an effective hamiltonian fully taking account of the symmetric phonon mode, usually discarded. This paper explains carefully how to simplify numerical investigations of the Holstein model, and other related model containing electron-phonon interactions, using exact diagonalization. Such a simplification was introduced many years ago by Ranninger and Thibblin for the two sites polaron problem [1]. The symmetric phonon mode is picked out and taken account analytically as displaced oscillator, reducing the number of phonon modes by one (one is a large number for small clusters). Since then, this trick has been widely used. However, many authors [2], neglected important contributions from the symmetric mode. We consider the Holstein hamiltonien, without Coulomb repulsion, given by H = ε 0 j,σ c † j,σ c j,σ − t j,δ,σ c † j+δ,σ c j,σ + ω 0 j a † j a j − gω 0 j,σ c † j,σ c j,σ (a j + a † j ).(1) ε 0 is some parameter. For a M sites lattice, the symmetric phonon mode is given by a s = (a 1 + a 2 + . . . + a M )/ √ M and the corresponding hamiltonian is given by H s = ω 0 a † s a s − gω 0 N √ M (a † s + a s ),(2) where N is the total number of electrons. The hamiltonian becomes H = H e + H s . H e is an effective hamiltonian with one phonon mode missing which has to be numerically solved. Indroducing new bosonic operators b N = a s − gN/ √ M , we obtain the diagonal form H s = ω 0 b † N b N − g 2 N 2 ω 0 /M.(3)Let | n, N) be the eigenstates of H s with eigenvalues E s,N n = nω 0 − g 2 N 2 ω 0 /M. This transfor- mation is an unitary transformation, b N = U N a s with U N = e gN (a † s −as)/ √ M .(4) The coherent states are given by | n, N) = U N | n > where | n > are the eigenstates of a † s a s . The matrix elements are given by (n ′ , N ′ | n, N) = < n ′ | U N −N ′ | n >. In the particular case of a transition from the ground state with N electrons to an excited state with N ′ electrons, we get (n ′ , N ′ | 0, N) = 1 √ n ′ ! e − g 2 (N−N ′ ) 2 2M g 2 (N − N ′ ) 2 M n ′ /2 .(5) We now show how this matrix element arise in the expression of the electronic correlation functions, say J 1 (t) = < c k,σ (t) c † k,σ > where k is the momentum. Let | m, N > be the eigenstates of H e with eigenvalues E e,N m . At T = 0, we get J 1 (t) = < 0, N | (0, N | c k,σ (t) c + k,σ | 0, N > | 0, N) = m,n < 0, N | (0, N | c k,σ (t) | m, N + 1 > | n, N + 1) × × < m, N + 1 | (n, N + 1 | c † k,σ | 0, N > | 0, N)(6) and the spectral function is given by J 1 (ω) = 2π n,m | (n, N + 1 | 0, N) | 2 | < 0, N | c k,σ | m, N + 1 >| 2 × × δ(ω + E s,N 0 + E e,N 0 − E s,N +1 n − E e,N +1 m )(7) This result is to be compared with the naïve expression based only on H e which is J naïve 1 (ω) = 2π m | < 0, N | c k,σ | m, N + 1 >| 2 δ(ω + E e,N 0 − E e,N +1 m ). (8) Figure 1 shows J 1 (k = 0, ω) with ω 0 /t = 0.2 and λ = g 2 ω 0 /t = 1.2 for the two sites problem with ground state corresponding to N = 0. Part (a) reproduces Alexandrov et als result ( figure 4.a of reference [2]) using equation (8), while part (b) shows the correct result using equation (7). We used 50 phonon states for H e and 20 phonon states for H s . The exact result for g = 0 is[3] J g=0 1 (k = 0, ω) = 2π ∞ ℓ=0 e g −2 g 2ℓ ℓ! δ(ω − ε 0 − E P − ℓω 0 ),(9) with E P = g 2 ω 0 , an useful test formula. I thank J. Ranninger for interesting discussions. Figure Captions Figure Captions Figure 1 : 1Spectral function J 1 (k = 0, ω) for the two sites Holstein Hamiltonian with ω 0 /t = 0.2 and λ = 1.2. (a) shows the naïve result obtained without the symmetric contribution while (b) shows the correct result. . J Ranninger, U Thibblin, Phys. Rev. 457730J. Ranninger and U. Thibblin, Phys. Rev. B45, 7730 (1992) . A S See For Instance, V V Alexandrov, D K Kabanov, Ray, Phys. Rev. 499915See for instance, A. S. Alexandrov, V. V. Kabanov and D. K. Ray, Phys. Rev. B49, 9915 (1994) G D Mahan, Many-Particle Physics. New YorkPlenum PressG. D. Mahan, Many-Particle Physics, Plenum Press, New York (1990)

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{'abstract': 'We perform an unitary transformation for the symmetric phonon mode of the Holstein molecular crystal hamiltonian. We show how to compute the electronic spectral functions by exact numerical diagonalisation of an effective hamiltonian fully taking account of the symmetric phonon mode, usually discarded.', 'arxivid': 'cond-mat/9610219', 'author': ['J M Robin \nCentre de Recherche sur les Très Basses Températures\n25, avenue des Martyrs, BP 16638042Grenoble Cedex 9France\n'], 'authoraffiliation': ['Centre de Recherche sur les Très Basses Températures\n25, avenue des Martyrs, BP 16638042Grenoble Cedex 9France'], 'corpusid': 118275965, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 1731, 'n_tokens_neox': 1514, 'n_words': 911, 'pdfsha': 'f0233beff615e600f564ad36f4f057cd7656eba3', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/9610219v2.pdf'], 'title': ['Spectral Functions for the Holstein Model', 'Spectral Functions for the Holstein Model'], 'venue': []}

arxiv

Geographic-style maps for 2-dimensional lattices Matthew Bright Materials Innovation Factory University of Liverpool UK Andrew I Cooper Materials Innovation Factory University of Liverpool UK Vitaliy Kurlin vitaliy.kurlin@gmail.com Materials Innovation Factory University of Liverpool UK Geographic-style maps for 2-dimensional lattices 1Latticeobtuse superbaseisometryinvariantsimilaritymetriccontinuityLattice Isometry Space This paper develops geographic-style maps containing 2D lattices in all known crystals parameterised by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of 2D lattices is a square with identified edges or a sphere without one point. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualise hundreds of thousands of real crystal lattices from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography. Crystallography traditionally splits crystals into finitely many classes, for instance by their space-group types. These discrete symmetry-based classifications were suitable for distinguishing highly symmetric crystals and categorising small datasets. Nowadays crystals are simulated and synthesised on an industrial scale. The Cambridge Structural Database (CSD) contains more than 1.17M existing crystals (Groom et al., 2016). Crystal Structure Prediction (CSP) tools generate millions of crystals even for a fixed chemical composition (Pulido et al., 2017), mostly with P1 symmetry. This amount of data requires new approaches to mapping huge crystal datasets. A more important reason for a continuous approach to the similarity of crystals is the inevitability of noise in data. Thermal vibrations of atoms can give rise to slightly different X-ray patterns under different crystallisation conditions, resulting in similar but distinct structures. A common solution is to ignore deviations of lattice parameters or atomic coordinates up to a certain threshold, but this simply moves the problem to a different value -if we ignore differences of angles and lengths up to 20 • and 0.1Å, the boundary between similarity and dissimilarity remains sharp at other values. Furthermore, we can argue that the use of threshold values leads to trivial classifications as follows. Any decision on similarity (denoted by ∼) gives rise to a justified classification only if we use equivalence relation, which satisfies three axioms: (1) reflexivity : any lattice Λ is equivalent to itself, so Λ ∼ Λ; (2) symmetry : if Λ ∼ Λ then Λ ∼ Λ; (3) transitivity : if Λ ∼ Λ and Λ ∼ Λ then Λ ∼ Λ . The transitivity axiom splits lattices into well-defined and disjoint equivalence classes: the class [Λ] consists of all lattices equivalent to Λ, since if Λ is equivalent to Λ , which is equivalent to Λ , all three lattices are in the same class. Previous similarities in (Lima-de Faria et al., 1990) use numerical thresholds to determine a lattice class, but as Fig. 1 illustrates, all lattices can be made equivalent given some threshold. An alternative mathematical approach classifies lattices by space groups and other algebraic structures (Nespolo, 2008). Since crystal structures are determined as rigid forms, the most practically important equivalence of crystals and their lattices is a rigid motion, which in R 2 , is any composition of translations and rotations. This is the strongest possible equivalence on crystals that are indistinguishable as rigid bodies. Slightly weaker is equivalence based on isometry or congruence Λ ∼ = Λ (including reflections), or similarity (including uniform scaling). Even if we fix an equivalence such as isometry, (Sacchi et al., 2020) highlights that the key question 'same or different' remained unanswered. What is needed is the notion of an invariant I that is a descriptor, such as a numerical vector, taking the same value on all isometric lattices. In a fixed coordinate system, the basis vectors themselves are not isometry invariants as they easily change under rotation but the primitive cell area does not. Crucially, any isometry invariant I has no false negatives: if Λ ∼ = Λ then I(Λ) = I(Λ ). Hence, if I takes different values on latices Λ, Λ , these lattices are certainly not isometric. Non-invariants cannot help distinguish equivalent objects. For example, isometric lattices Λ ∼ = Λ can have infinitely many different bases. Most isometry invariants allow false positives that are non-isometric lattices Λ ∼ = Λ with I(Λ) = I(Λ ). For example, it is easy to imagine an infinite number of non-isometric lattices with the same primitive cell area. An isometry invariant I giving rise to no false positives such that Λ ∼ = Λ if and only if I(Λ) = I(Λ ), is called a complete (or injective) invariant. Complete invariants are the main goal of all classifications. For crystals, we need IUCr macros version 2.1.10: 2016/01/28 even better invariants: they must also be continuous under perturbation of a lattice basis, since a discontinuous invariant may take very different values on nearly identical lattices, see pitfalls of pseudo-symmetry in (Zwart et al., 2008). For any small ε > 0, the reduced bases (1, 0), (−ε, 2) and (1, 0), (−ε, −2) generate the lattices nearly identical to the lattice with the rectangular cell 1×2 (for ε = 0), but these bases cannot be made nearly identical by rigid motion, see Kurlin (2022c, Fig. 4, Corollary 7.9). Up to isometry, all real crystals in the CSD are expected to be different, although the recent invariants detected five pairs of suspicious entries, see Widdowson et al. Section 2 reviews the closely related past work. Section 3 reminds the recently developed complete invariants of 2-dimensional lattices. Section 4 maps hundreds of thousands of CSD crystals with full lattice data. Section 5 explains the geographical metaphor by mapping the invariant values to a sphere, where every lattice (up to rigid motion and uniform scaling) has unique latitude and longitude coordinates. Overview of key concepts and past work on classifications of lattices Crystallography traditionally uses a conventional cell to uniquely represent any periodic crystal, see Aroyo & Wondratschek (2013, Chapter 2.1). In the simpler case of lattices, the cell used is Niggli's reduced cell (Niggli, 1928). Since this paper studies lattices in R 2 , we give the 2-dimensional version obtained from the 3-dimensional definition by choosing a long enough third vector v 3 orthogonal to the first two v 1 , v 2 . For vectors v 1 = (a 1 , a 2 ) and v 2 = (b 1 , b 2 ) in R 2 , the determinant of the matrix    a 1 b 1 a 2 b 2    with the columns v 1 , v 2 can be defined as det(v 1 , v 2 ) = a 1 b 2 − a 2 b 1 . Definition 2.1 (reduced cell). For a lattice up to isometry, a basis and its unit cell U (v 1 , v 2 ) are reduced (non-acute) if |v 1 | ≤ |v 2 | and − 1 2 v 2 1 ≤ v 1 · v 2 ≤ 0. Up to rigid motion, the conditions are weaker: |v 1 | ≤ |v 2 | and − 1 2 v 2 1 < v 1 ·v 2 ≤ 1 2 v 2 1 , det(v 1 , v 2 ) > 0, and the new special condition for rigid motion : if |v 1 | = |v 2 | then v 1 · v 2 ≥ 0. The conditions for rigid motion did not appear in Aroyo & Wondratschek (2013, section 9.2.2) because reduced cells were usually considered only up to isometry including reflections. For example, any rectangular lattice has a unique (up to rigid motion) reduced cell a × b, but two 'potentially reduced' bases v 1 = (a, 0) and v 2 = (0, ±b), which are not related by rigid motion for 0 < a < b. Definition 2.1 chooses only one of these bases, namely v 1 = (a, 0) and v 2 = (0, b) due to det(v 1 , v 2 ) > 0. Since reduced cells are algorithmically easy to compare up to isometry (Křivỳ & Gruber, 1976), one can define the discrete metric d(Λ, Λ ) taking the same non-zero value (say, 1) for any non-isometric lattices Λ ∼ = Λ . This is the simplest example of a discontinuous metric satisfying all metric axioms on a set of equivalence classes. Discontinuity of Niggli's basis up to perturbations was practically demonstrated in the seminal work (Andrews et al., 1980). The introduction of Edelsbrunner et al. (2021) said that "There is no method for choosing a unique basis for a lattice in a continuous manner. Indeed, continuity contradicts uniqueness as we can continuously deform a basis to a different basis of the same lattice", see a basic example in Kurlin be discontinuous. Since these advances are specialised for R 3 , we defer a more detailed review for 3-dimensional lattices to the follow-up paper (Bright et al., 2021). Another way to represent a lattice Λ ⊂ R n is by its Wigner-Seitz cell or Voronoi domain V (Λ) consisting of all points p ∈ R n that are closer to the origin 0 ∈ Λ than to all other points of Λ. Though a Voronoi domain V (Λ) uniquely determines Λ up to rotations, almost any tiny perturbation of a rectangular lattice Λ converts the rectangular domain V (Λ) into a hexagon. Hence all combinatorial invariants (numbers of vertices or edges) of V (Λ) are discontinuous, similarly in higher dimensions. v 1 , v 2 , v 0 = −v 1 − v 2 , which is unique up to permutations and central symmetry. Other pictures: isometric superbases for a rectangular Voronoi domain. However, comparing Voronoi domains as geometric shapes by optimal rotation (Mosca & Kurlin, 2020) around a common centre led to two continuous metrics on lattices up to rigid motion and similarity. The minimisation over infinitely many rotations was resolved only by finite sampling, so the exact computation of these metrics is still open. Similar difficulties remain for general periodic point sets that model all periodic crystals (Anosova & Kurlin, 2021b;Anosova & Kurlin, 2021a), which were recently resolved only in dimension 1, see (Anosova & Kurlin, 2022;Kurlin, 2022b). Another attempt to produce computable metrics was to consider distance-based invariants (Widdowson et al., 2022;Widdowson & Kurlin, 2021) whose completeness was proved for generic crystals. These invariants helped establish the Crystal Isometry Principle by experimentally checking that all periodic crystals from the CSD remain non-isometric after forgetting all chemical information. This principle implies that all periodic crystals can be studied in the common Crystal Isometry Space (CRISP) whose version for 2-dimensional lattices is the Lattice Isometry Space LIS(R 2 ). The work (Conway & Sloane, 1992) came close to mapping lattice spaces without formally stating key challenges, see a full statement in Kurlin (2022c, Problem 1.1) Kurlin (2022c, Proposition 3.10) proves that a reduced basis from Definition 2.1 is unique (also up to rigid motion) and all reduced bases are in a 1-1 correspondence with obtuse superbases, which are easier to visualise below, especially for n ≤ 3. Definition 2.2 (obtuse superbase, conorms p ij ). For any basis v 1 , . . . , v n in R n , the superbase v 0 , v 1 , . . . , v n includes the vector v 0 = − n i=1 v i . The conorms p ij = −v i · v j are the negative scalar products of the vectors above. The superbase is called obtuse if all conorms p ij ≥ 0, so all angles between the vectors v i , v j are non-acute for distinct indices i, j ∈ {0, 1, . . . , n}. The obtuse superbase is strict if all p ij > 0. Definition 2.2 use the conorms p ij from (Conway & Sloane, 1992), which were known as negative Selling parameters (Selling, 1874) and Delone parameters (B.N.Delone et al., 1934). Lagrange (De Lagrange, 1773) proved that the isometry class of any lattice Λ ⊂ R 2 with a basis v 1 , v 2 is determined by the positive quadratic form Q(x, y) = (xv 1 + yv 2 ) 2 = q 11 x 2 + 2q 12 xy + q 22 y 2 ≥ 0 for all x, y ∈ R, where q 11 = v 2 1 , q 12 = v 1 · v 2 , q 22 = v 2 2 . The triple (v 2 1 , v 1 · v 2 , v 2 2 ) is also called a metric tensor of (a basis of) Λ. Any Q(x, y) has a reduced (non-acute) form with 0 < q 11 ≤ q 22 and −q 11 ≤ 2q 12 ≤ 0, which is equivalent to reducing a basis up to isometry. The bases v 1 = (3, 0), v ± 2 = (−1, ±2) generate the mirror images not related by rigid motion, but define the same form Q = 9x 2 − 6xy + 5y 2 satisfying the reduction conditions above. So quadratic forms do not distinguish mirror images (enantiomorphs). Hence the new conditions for the rigid motion were needed in Definition 2.1. Motivated by the non-homogeneity of the metric tensor (two squared lengths and scalar product), Delone proposed (Delone, 1937) the homogeneous parameters p 12 = −v 1 · v 2 = −q 12 , p 01 = −v 0 · v 1 = q 11 + q 12 , p 02 = −v 0 · v 2 = q 22 + q 12 , see all notations in Definition 2.2. Then any permutation of superbase vectors satisfying v 0 + v 1 + v 2 = 0 changes Delone's parameters p 12 , p 01 , p 02 by the same permutation of indices. For example, swapping v 1 , v 2 is equivalent to swapping p 01 , p 02 . Delone's reduction (Delaunay et al., 1973) proved the key existence result: any lattice in dimensions 2 and 3 has an obtuse superbase with all p ij ≥ 0. Section 3 further develops the Delone parameters to show in section 4 that millions of lattices from real crystals in the CSD nicely distribute in continuous spaces of lattices. Homogeneous complete invariants of 2D lattices up to four equivalences This section reminds the lattice classifications in Theorem 3.4 based on the recent invariants introduced in Definitions 3.1 and 3.2 from Kurlin (2022c, sections 3-4). Definition 3.1 (sign(Λ) and root invariants RI, RI o ). Let B = {v 0 , v 1 , v 2 } be any obtuse superbase of a lattice Λ ⊂ R 2 . If Λ is mirror-symmetric (achiral), set sign(Λ) = 0. Otherwise v 0 , v 1 , v 2 have different lengths and no right angles, say |v 1 | < |v 2 | < |v 0 |. Then define sign(Λ) as the sign of the determinant det(v 1 , v 2 ) of the matrix with the columns v 1 , v 2 . The root invariant RI(Λ) is the ordered triple of the root products r ij = √ −v i · v j for distinct i, j ∈ {0, 1, 2}. The oriented root invariant RI o (Λ) is RI(Λ) with sign(Λ) as a superscript, which we skip in the case sign(Λ) = 0 for brevity. Kurlin (2022c, Lemma 3.8) proved that RI(Λ) is an isometry invariant of Λ, independent of an obtuse superbase B because an obtuse superbase of Λ is unique up to isometry, also up to rigid motion for non-rectangular lattices. This uniqueness was missed in (Conway & Sloane, 1992) and actually fails in R 3 , see (Kurlin, 2022a). Definition 3.2 (projected invariants PI, PI o ). The root invariants of all lattices Λ ⊂ R 2 live in the triangular cone TC in Fig. 3. The triangular projection TP : TC → QT divides each coordinate by the size σ(Λ) = r 12 + r 01 + r 02 and projects RI(Λ) to (r 12 ,r 01 ,r 02 ) in the quotient triangle QT in Fig. 4. This triangle can be visualised as the isosceles right-angled triangle QT = {x, y ≥ 0, x + y ≤ 1} ⊂ R 2 parameterised by x =r 02 −r 01 and y = 3r 12 The resulting pair PI(Λ) = (x, y) is the projected invariant. The oriented invariant PI o (Λ) is obtained by adding the superscript sign(Λ). = √ b 2 − a 2 . If b ≤ a √ 3, then RI(Λ) = ( √ b 2 − a 2 , a √ 2, a√ 2) and PI(Λ) = (0, 3 √ b 2 −a 2 2a √ 2+ √ b 2 −a 2 )Λ) = (a √ 2, a √ 2, √ b 2 − a 2 ) and PI(Λ) = ( 3a √ 2 2a √ 2+ √ b 2 −a 2 , √ b 2 −a 2 −a √ 2 2a √ 2+ √ b 2 −a 2 )v 1 = (2a, 0), v 2 = (−a, b), v 0 = (−a, −b). Middle: RI(Λ) = ( √ b 2 − a 2 , a √ 2, a √ 2), a ≤ b ≤ a √ 3. Right: RI(Λ) = (a √ 2, a √ 2, √ b 2 − a 2 ), a √ 3 ≤ b, see Example 3.3(oc). Mapping millions of 2-dimensional lattices extracted from CSD crystals For any periodic crystal from the Cambridge Structural Database (CSD), which has full geometric data of its lattice Λ ⊂ R 3 , we extract three 2-dimensional lattices generated by three pairs {v 2 , v 3 }, {v 1 , v 3 }, {v 1 , v 2 } of given basis vectors of Λ. After removing all non-oblique lattices represented by root invariants along the sides and the diagonal of QS, the map in Fig. 7 shows the preference for positive lattices which can be explained by a standard order of basis vectors for monoclinic lattices. The density map of rectangular lattices in Fig. 8 (left) has two high density (black) pixels at a ≈ 3.5 Angstroms arising from 386 near-identical primitive monoclinic crystals of α-oxalic acid dihydrate. This molecule has been used as a benchmark for the calculation of electron densities since its crystallographic properties were thoroughly documented in (Stevens & Coppens, 1980). As a result, hundreds of publications have since generated and deposited further refinements of its structural determination. In the density map of centered rectangular lattices in Fig. 8 (right), the most prominent features are the high-density area in the region where the shortest side length is between 2.5 and 5Å. We also see a visible line b = √ 2a of high-density pixels. This line represents 2D lattices in body-centered cubic lattices, where the ratio of side lengths is √ 2. Another high-density pixel in this line represents 130 structures of a standard test molecule (hexamethylenetetramine), which was frequently used in the investigation of lattice vibrations (Becka & Cruickshank, 1963). Other complete invariants and a spherical map of 2-dimensional lattices The Lattice Isometry/Similarity Spaces can be parameterised in many different ways. The root invariant RI(Λ) of a lattice Λ ⊂ R 2 has the advantage of homogeneity in the sense that any permutation σ of (indices of) superbase vectors v 0 , v 1 , v 2 permutes the three root products accordingly: r ij → r σ(i)σ(j) . The metric tensor MT = (v 2 1 , v 1 ·v 2 , v 2 2 ) including the coefficients of the form Q Λ (x, y) = q 11 x 2 +2q 12 xy+q 22 y 2 representing Λ is not homogeneous in the above sense. Taking square roots gives the quadratic invariant QI(Λ) = (τ 11 , τ 12 , τ 22 ) = ( √ q 11 , √ −q 12 , √ q 22 ) in the units of basis coordinates. The quadratic invariant QI(Λ) is complete up to isometry by Theorem 3.4(a) and can be extended to the case of rigid motion by using sign(Λ) from Definition 3.1, In the isosceles triangle QT, continuous metrics and chiral distances have simpl formulae in Kurlin (2022c, sections 5-6) for the coordinates x =r 02 −r 01 , y = 3r 12 but can be now re-written for any coordinates on LIS(R 2 ), see the earlier non-isosceles triangles in Engel et al. (2004, Fig. 1.2 on p. 82) and Zhilinskii (2016, Fig. 6.2). Since the quotient square QS = QT + ∪ QT − with identified sides is a topological sphere without a single point, it is natural to visualise QS as the round surface of Earth with QT ± as the north/south hemispheres separated by the equator along their common boundary of QT represented by PI(Λ) of all mirror-symmetric lattices Λ. We can choose any internal point of the quotient triangle QT as the north pole. The most natural choice is the incentre P + (pole), the centre of the circle inscribed into QT + because the rays from P + to the vertices of QT + nicely bisect the angles 90 • , 45 • , 45 • . The incentre of QT + has the coordinates (x, x), where x = 1 − 1 √ 2 = 1 2+ √ 2 . The lattice Λ + 2 with the projected invariant PI(Λ + 2 ) = (x, x) has the basis v 1 ≈ (1.9, 0), v 2 ≈ (−0.18, 3.63) inversely designed in Kurlin (2022c, Example 4.10 (Λ 2 )). (1, 0) and the incentre P + , we choose the longitude µ = +180 • rather than −180 • . Proposition 5.2 computes µ(Λ), φ(Λ) via PI(Λ) = (x, y) and is proved in Appendix A. Proposition 5.2 (formulae for SM). For any lattice Λ ⊂ R 2 with PI(Λ) = (x, y) ∈ QT, if x = t = 1 − 1 √ 2 , then set ψ = arctan y − t x − t , otherwise ψ = sign(y − t)90 • . (5.2a) The longitude of the lattice Λ is µ(Λ) =              ψ + 22.5 • if x < t, ψ − 157.5 • if x ≥ t, ψ ≥ −22.5 • , ψ + 202.5 • if x ≥ t, ψ ≤ −22.5 • . (5.2b) The latitude is φ(Λ) = sign(Λ) ·              x √ 2 √ 2−1 90 • if µ(Λ) ∈ [−45 • , +67.5 • ], y √ 2 √ 2−1 90 • if µ(Λ) ∈ [+67.5 • , +180 • ], 1−x−y √ 2−1 90 • if µ(Λ) ∈ [−180 • , −45 • ]. The incentres P ± ∈ QT ± have ψ = 0 and µ = ±90 • , respectively, φ is undefined. The north pole represents the incentre P + whose pixel contains 230 lattices in Figure 7 but appears sparsely populated in Fig. 11 because this incentre pixel is split into many more 1 × 1 degree curved 'pixels' of a much lower density. The high density near the point representing hexagonal lattices is visible in Figures 11 and 12 The density maps show a hexagonal 'ridge' along the meridional arc at µ = −45 • in Figures 11 and 12, which looks as a round arc in Figures 13 and 14. The density of exact square and rectangular lattices is even higher (dark pixels for the Bravais classes tp and op), but there are fewer lattices close to these classes possibly because manual or automatic adjustments are easier for angles close to 90 • than to 60 • . Main conclusions and motivations for a continuous crystallography The density maps in Fig. 6-7 Using a geographic analogue, the recent isometry invariants create complete and continuous maps for efficient navigation in the Lattice Isometry Space LIS(R 2 ), which can be zoomed in as satellite images and explored at any desirable resolution. The four Bravais classes of non-oblique 2D lattices are lower-dimensional subspaces in LIS(R 2 ) whose separate maps in Fig. 8 and 9 have no intermediate gaps and contain naturally empty regions only for small or very large values of distance parameters. Using a biological analogue, crystallography previously took a similar approach to the classical taxonomy, dividing lattices into an increasingly complex sequence of discrete categories based on symmetries as they divided organisms according to physical characteristics, see a comprehensive review in (Nespolo et al., 2018). The new continuous crystallography uses the fundamental geometric properties of the lattice itself to continuously classify an individual in as granular a manner as we like, in a manner akin to the modern use of genetic sequences and markers to classify organisms. Indeed, since the root invariant RI(Λ) of a lattice Λ is complete, this RI(Λ) could be said to represent the DNA of Λ. Even better than the real DNA, any 2D lattice can be explicitly built up from RI(Λ), see Kurlin (2022c, Proposition 4.9). Working Synopsis The complete invariant-based maps of 2-dimensional lattices reveal continuous distributions for hundreds of thousands of known crystals from the Cambridge Structural Database. Fig. 1 . 1All lattices continuously deform into each other if we allow any small changes. ( 2022 , 2022section 7). Polymorphs in the CSD, which often have different ref codes, can be recognised only by a continuous invariant taking similar values for similar crystals. ( 2022c , 2022cFig. 3) and a formal proof inWiddowson et al. (2022, Theorem 15). L. Andrews and H. Bernstein have made important advances in (Andrews & Bernstein, 1988; Andrews & Bernstein, 2014; McGill et al., 2014; Andrews et al., 2019) by analysing progressively more complicated boundary cases where cell reductions can Fig. 2 . 2Left: a generic 2D lattice has a hexagonal Voronoi domain with an obtuse superbase Fig. 3 .Fig. 4 . 34Left: the triangular cone TC = {(r 12 , r 01 , r 02 ) ∈ R 3 | 0 ≤ r 12 ≤ r 01 ≤ r 02 = 0} represents the space RIS of all root invariants, see Definition 3.2. Middle: TC projects to the quotient triangle QT representing all 2D lattices up to similarity. Right: QT is parameterised by x =r 02 −r 01 ∈ [0, 1) and y = 3r 12 ∈ [0, 1]. Left: all projected invariants PI(Λ) live in the quotient triangle QT parameterised by x =r 02 −r 01 ∈ [0, 1) and y = 3r 12 ∈ [0, 1]. Right: mirror images (enantiomorphs) of any oblique lattice are represented by a pair (x, y) ↔ (1 − y, 1 − x) in the quotient square QS = QT + ∪ QT − symmetric in the diagonal x + y = 1.All oriented projected invariants PI o (Λ) with sign(Λ) live in a union of two quotient triangles QT + ∪ QT − . These triangles should be glued along the common subspace of mirror-symmetric latices (all non-oblique lattices Λ ⊂ R 2 ), whose PI(Λ) belong to the boundary of QT.Fig. 4(right) glues the hypotenuses of QT ± and indicates how to glue the remaining sides. The gluing produces a topological sphere without a single point due to the excluded vertex (1, 0). We visualise this sphere like the Earth surface with geographic-style coordinates and the boundary of QT as the equator in section 5.Example 3.3 (Bravais classes). (tp) The square lattice Λ 4 ⊂ R 2 with a unit cell a×a has the root invariant RI(Λ 4 ) = (0, a, a) ∈ TC projected by TP to (r 12 ,r 01 ,r . By Definition 3.2 the projected invariant PI(Λ 4 ) = (x, y) = (r 02 −r 01 , 3r 12 ) = (0, 0) ∈ QT, see Fig. 4 (left). So the Bravais class (tp) of all square (tetragonal) lattices Λ 4 ⊂ R 2 is represented by the bottom-left vertex (0, 0) in the quotient triangle QT, identified with the top-right vertex of the quotient square QS in Fig. 4 (right).(hp) The hexagonal lattice Λ 6 with a minimum inter-point distance a has the root invariant RI(Λ 6 ) = ( (Λ 6 ) = (x, y) = (0, 1) ∈ QT, see Fig. 4 (left). The Bravais class (hp) of all hexagonal lattices Λ 6 ⊂ R 2 is represented by the top-left vertex (0, 1) in the quotient triangle QT, identified with the bottom-right vertex of the quotient square QS. (op) Any rectangular lattice Λ with a unit cell a × b for 0 < a < b has the obtuse superbase v 1 = (a, 0), v 2 = (0, b), v 0 = (−a, −b), see Fig. 5 (left). Then RI(Λ) = (0, a, b) and PI(Λ) = ( b−a b+a , 0) belongs to the horizontal side of QT, which represents the Bravais class (op). We approach the excluded vertex (1, 0) as b → +∞. ( oc ) ocAny centred rectangular lattice Λ with a conventional unit cell 2a×2b for 0 < a < b has the obtuse superbase v 1 = (2a, 0), v 2 = (−a, b), v 0 = (−a, −b), see Fig. 5. Then r 01 = a √ 2 = r 02 and r 12 belongs to the vertical side of QT. For b = a, the lower vertex (x, y) = (0, 0) of QT represents all square lattices with r 12 = 0. For b = a √ 3, the upper vertex (x, y) = (0, 1) of QT represents all hexagonal lattices with r 12 = r 01 = r 02 . If b > a √ 3, then RI( belongs to the hypotenuse x + y = 1 of QT. The open vertical edge and open hypotenuse of QT represent the Bravais class oc of all centred rectangular lattices. The excluded vertex (1, 0) represents the limit b → +∞. Fig. 5 . 5Left: any rectangular lattice Λ with a unit cell a × b has the obtuse superbase B with v 1 = (a, 0), v 2 = (0, b), v 0 = (−a, −b), see Example 3.3(op). Other lattices Λ have a rectangular cell 2a × 2b and an obtuse superbase B with Theorem 3.4 summarises the complete classifications of 2-dimensional lattices from Kurlin (2022c, Theorem 4.2 and Corollary 4.6) up to four equivalence relations. Theorem 3. 4 4(lattice classifications). Let Λ, Λ ⊂ R 2 be any lattices.(a) Λ, Λ are isometric if and only if RI(Λ) = RI(Λ ); (b) Λ, Λ are similar if and only if PI(Λ) = PI(Λ ); (c) Λ, Λ are related by rigid motion if and only if RI o (Λ) = RI o (Λ ); (d) Λ, Λ are similar with the same orientation if and only if PI o (Λ) = PI o (Λ ). Fig. 6 6shows all resulting 2.6 million lattices in the quotient square QS. Only about 55% of all lattices have non-trivial symmetries of Bravais classes oc, op, hp, tp. The remaining 45% of lattices are oblique and nicely fill QS apart from the lower right corner where primitive unit cells are rather elongated. Hence we conclude that the Lattice Isometry Space LIS(R 2 ) is continuously populated by real CSD crystals. Fig. 6 . 6Density maps in QS of all 2D lattices extracted from CSD crystals. The colour of each pixels indicates (on the logarithm scale) the number of lattices whose projected invariant PI(Λ) = (x, y) = (r 02 −r 01 , 3r 12 ) belongs to this pixel. The darkest pixels represent rectangular lattices on the bottom and right edges of QS. Fig. 7 . 7Normal density map all 2D oblique lattices from CSD crystals in QS. After removing mirror-symmetric lattices on the boundary and diagonal of QS, we can better see the tendency towards hexagonal lattices at the top left corner (0, 1) ∈ QS. Fig. 8 . 8Density maps of parameters (a, b) inÅ. Left: rectangular lattices with primitive unit cells a × b in N = 1094109 crystals in the CSD. Right: centred-rectangular lattices with conventional cells 2a × 2b in N = 147451 crystals in the CSD. Fig. 9 . 9The histograms of minimum inter-point distances a in Angstroms. Left: all hexagonal 2D lattices in CSD crystals. Right: all square 2D lattices in CSD crystals. Fig. 10 . 10Left: in QT + , the Greenwich line goes from the 'empty' point (1,0) through incentre P + to the point G = (0, √ 2 − 1). Middle: the hemisphere HS + has the north pole at P + , the equator ∂QT + of mirror-symmetric lattices. Right: the longitude µ ∈ (−180 • , +180 • ] anticlockwise measures angles from the Greenwich line, the latitude φ ∈ [−90 • , +90 • ] measures angles from the equator to the north. Definition 5. 1 1(spherical map SM : QS → S 2 ). (a) The spherical map SM sends the incentre P + of QT to the north pole of the hemisphere HS + and the boundary ∂QT to the equator of HS + , see Fig. 10 (middle). Linearly map the line segment between P + and any point (x, y) in the boundary ∂QT to the shortest arc connecting the north pole SM(P + ) to SM(x, y) in the equator of HS + . Extend the spherical map to SM : QS → S 2 by sending any pair of invariants PI o (Λ ± ) with sign(Λ ± ) = ±1 to the northern/southern hemispheres of the 2-dimensional sphere S 2 , respectively. (b) For any lattice Λ ⊂ R 2 , the latitude φ(Λ) ∈ [−90 • , +90 • ] is the angle from the equatorial plane EP of S 2 to the radius-vector to the point SM(PI o (Λ)) ∈ S 2 in the upwards direction. Let v(Λ) be the orthogonal projection of this radius-vector to EP. Let the Greenwich meridian be the spherical arc going through SM(G) in the equator E, where the Greenwich point G = (0, √ 2 − 1) ∈ ∂QT is in the line through P + and (1, 0). The longitude µ(Λ) ∈ (−180 • , 180 • ] is the anticlockwise angle from the Greenwich plane through the Greenwich meridian to the vector v(Λ) above.For lattices with PI(Λ) in the straight-line segment between the excluded vertex Example 5. 3 3(prominent lattices). Any mirror-symmetric lattice Λ ⊂ R 2 has sign(Λ) = 0, hence belongs to the equator E of S 2 and has latitude φ(Λ) = 0 by (5.2b). Any square lattice Λ 4 with PI(Λ 4 ) = (0, 0) has µ(Λ 4 ) = arctan 1 + 22.5 • = 67.5 • by (5.2a).Any hexagonal lattice Λ 6 with PI(Λ 4 ) = (0, 1) has µ(Λ 4 ) • . Any rectangular lattice Λ with PI(Λ) = (1 − 1√ 2 , 0) has µ(Λ) = −90 • + 202.5 • = 112.5 • . Any centered rectangular lattice Λ with PI(Λ) = ( 1 2 , 1 2 ) at the mid-point of the diagonal of QT has µ(Λ) = arctan 1 − 157.5 • = −112.5 • . Any Greenwich lattice Λ G with PI(Λ G ) = G = (0, √ 2 − 1) has µ(Λ G ) = arctan(1 − √ 2) + 22.5 • = 0. Figures 11,12,13,14 show the density maps of 2D lattices from Figure 6 on the northern, southern, western, eastern hemispheres for the spherical map SM : QS → S 2 . Fig. 11. The density map of 2D lattices from CSD crystals on the northern hemisphere. The circumference (equator) is parameterised by the longitude µ ∈ (−180 • , 180 • ]. The radial distance is the latitude φ ∈ [0 • , 90 • ]. Left: all N = 2191887 lattices with sign(Λ) ≥ 0, φ ≥ 0. Right: all N = 741105 oblique lattices with sign(Λ) > 0, φ > 0. Fig. 12 . 12The density map of 2D lattices from CSD crystals on the northern hemisphere. The circumference (equator) is parameterised by the longitude µ ∈ (−180 • , 180 • ]. The radial distance is the latitude φ ∈ [0 • , 90 • ]. Left: all N = 1854209 lattices with sign(Λ) ≤ 0, φ ≤ 0. Right: all N = 406930 oblique lattices with sign(Λ) < 0, φ < 0. the longitude µ = −45 • . Where non-oblique lattices are included, the concentrations of density along the borders of the QT can be seen, with primitive rectangular lattices appearing as a dense arc on the equator for µ ∈ [67.5 • , 180 • ). Fig. 13 . 13The density map of 2D lattices from CSD crystals on the western hemisphere. Angles on the circumference show the latitude φ ∈ [−90 • , 90 • ]. Left: N = 1100580 lattices with µ ∈ (−180 • , 0 • ]. The hexagonal lattice at µ = −45 • and the centered rectangular lattice at µ = −112.5 • are marked on the horizontal arc (western halfequator). Right: all N = 932626 oblique lattices with µ ∈ (−180 • , 0 • ] and φ = 0. Fig. 14 . 14The density map of 2D lattices from CSD crystals on the eastern hemisphere. Angles on the circumference show the latitude φ ∈ [−90 • , 90 • ]. Left: all N = 1511307 lattices with µ ∈ [0 • , 180 • ), the square lattice point at µ = 67.5 • and the rectangular lattice at µ = 112.5 • are marked on the the horizontal arc (eastern half-equator). Right: all N = 215409 oblique lattices with µ ∈ [0 • , 180 • ), φ = 0. towards a materials genome, Widdowson et al. (2022, section 7) described complementary invariants, which distinguished all periodic crystals in the CSD and together with invariants of lattices are enough for inverse design of generic crystals. arctan 1 = 45 • = α and longitude µ = α − 157.5 • = −112.5 • . If α − 157.5 • is outside the expected range of µ ∈ (−180 • , 180 • ], we add or subtract 360 • . Any hexagonal lattice Λ 6 with PI(Λ 6 ) 67.5 • , α = ψ +180 • = 112.5 • and longitude µ = α−157.5 • = −45 • . Any square lattice Λ 4 with PI(Λ 6 ) = (0, 0) has ψ = arctan y − t x − t = arctan 1 = 45 • , α = ψ + 180 • = 225 • and longitude µ = α − 157.5 • = 67.5 • . Formula (5.2a) is split into three subcases only to guarantee the range of a longitude µ ∈ (−180 • , 180 • ] for the anticlockwise angle α − 157.5 • from − → G to − −− → P + P , where α is computed above. (b) For a fixed longitude µ(Λ), the projected invariant PI(Λ) varies along the line segment L at a fixed angle from the incentre P + to the boundary ∂QT. Formula (5.2b) is split into three subcases according to the three boundary edges of QT. Consider the vertical edge between hexagonal and square lattices, where µ(Λ) ∈ [−45 • , 67.5 • ]. The latitude φ(Λ) is proportional to the ratio in which the point PI(Λ) = (x, y) splits the line segment L from P + to the vertical edge. The endpoint x = 0 means that SM(PI(Λ)) is in the equator with φ = 0. The endpoint x = t = 1− 1 √ 2 means that PI(Λ) = P + is in the centre whose image SM(P + ) is the north pole with φ = 90 • . The linear map between these extreme cases gives φ(Λ) = x t 90 • = x √ 2 √ 2 − 1 90 • . The case of the horizontal edge of QT gives a similar φ after replacing x with y. The hypotenuse of QT, where x + y = 1, is also similar as the incentre P + = (x, y) • = 90 • as expected. The factor sign(Λ) in (5.2b) guarantees a symmetry of SM : QS → S 2 in the equator. and 11-14 visualise for the first time 2.6 million 2dimensional lattices in real crystals from the Cambridge Structural Database. These maps justify the importance of studying latices by continuous invariants and metrics that slightly change under small perturbations, such as the thermal vibrations of atoms. The Python code is at https://github.com/MattB-242/Lattice Invariance. IUCr macros version 2.1.10: 2016/01/28 Acknowledgements. This research was partially supported by the £3.5MThe anticlockwise angle from the x-axis toIn all cases above, since the Greenwich vector − → G was chosen as the 0-th meridian, the anticlockwise angle from − → G to − −− → P + P is the longitude µ = α − 157.5. For example, any centred rectangular lattice Λ with PI(Λ) = (x, y) = ( 1 2 , 1 2 ) has ψ = arctan . L Andrews, H Bernstein, G Pelletier, Acta Crystallographica Section A. 362Andrews, L., Bernstein, H. & Pelletier, G. (1980). Acta Crystallographica Section A, 36(2), 248-252. . L C Andrews, H J Bernstein, Acta Cryst. A. 446Andrews, L. C. & Bernstein, H. J. (1988). Acta Cryst. A, 44(6), 1009-1018. . L C Andrews, H J Bernstein, J Applied Cryst. 471Andrews, L. C. & Bernstein, H. J. (2014). J Applied Cryst. 47(1), 346-359. . L C Andrews, H J Bernstein, N K Sauter, Acta Cryst. A. 751Andrews, L. C., Bernstein, H. J. & Sauter, N. K. (2019). Acta Cryst. A, 75(1), 115-120. . O Anosova, V Kurlin, arXiv:2103.02749Anosova, O. & Kurlin, V. (2021a). arXiv:2103.02749. O Anosova, V Kurlin, Proceedings of Discrete Geometry and Mathematical Morphology. Discrete Geometry and Mathematical MorphologyAnosova, O. & Kurlin, V. (2021b). In Proceedings of Discrete Geometry and Mathematical Morphology, pp. 229-241. . O Anosova, V Kurlin, arxiv:2205.02226Anosova, O. & Kurlin, V. (2022). arxiv:2205.02226. URL: http://kurlin.org/projects/periodic-geometry-topology/densities1D.pdf International Tables for Crystallography. M I Aroyo, H Wondratschek, Wiley Online LibraryAroyo, M. I. & Wondratschek, H. (2013). International Tables for Crystallography. Wiley Online Library. L N Becka, W J Cruickshank, Proceedings of the Royal Society A. the Royal Society A273Becka, L. N. & Cruickshank, W. J. (1963). Proceedings of the Royal Society A, 273, 435-454. Mathematical Foundations of Structural Analysis of Crystals. B N Delone, N Padurov, A Aleksandrov, State Technical-Theoretical PressUSSRB.N.Delone, Padurov, N. & Aleksandrov, A. (1934). Mathematical Foundations of Structural Analysis of Crystals. State Technical-Theoretical Press, USSR. . M Bright, A I Cooper, V Kurlin, arxiv:2109.11538early draftBright, M., Cooper, A. I. & Kurlin, V. (2021). arxiv:2109.11538 (early draft). URL: http://kurlin.org/projects/periodic-geometry-topology/lattices3Dmap.pdf J H Conway, N J Sloane, Proceedings of the Royal Society A. the Royal Society A436Conway, J. H. & Sloane, N. J. (1992). Proceedings of the Royal Society A, 436(1896), 55-68. Nouveaux Mémoires de l. J L De Lagrange, Académie de BerlinDe Lagrange, J. L. (1773). Nouveaux Mémoires de l'Académie de Berlin. . B Delaunay, R Galiulin, N Dolbilin, V Zalgaller, K Stogrin, Dokl. Akad. Nauk SSSR. 209Delaunay, B., Galiulin, R., Dolbilin, N., Zalgaller, V. & Stogrin, K. (1973). In Dokl. Akad. Nauk SSSR, vol. 209, pp. 309-313. Uspekhi Matematicheskikh Nauk. B N Delone, Delone, B. N. (1937). Uspekhi Matematicheskikh Nauk, (3), 16-62. H Edelsbrunner, T Heiss, V Kurlin, P Smith, M Wintraecken, Proceedings of SoCG. SoCG3216Edelsbrunner, H., Heiss, T., Kurlin, V., Smith, P. & Wintraecken, M. (2021). In Proceedings of SoCG, pp. 32:1-32:16. Lattice geometry. P Engel, L Michel, M Sénéchal, IHES-P-2004-45.Tech. Rep.Engel, P., Michel, L. & Sénéchal, M. (2004). Lattice geometry. Tech. Rep. IHES-P-2004-45. . . J Lima-De Faria, E Hellner, F Liebau, E Makovicky, E Parthé, Acta Cryst A. 461Lima-de Faria, J., Hellner, E., Liebau, F., Makovicky, E. & Parthé, E. (1990). Acta Cryst A, 46(1), 1-11. URL: https://iucr.org/resources/commissions/crystallographic-nomenclature/inorganic . C R Groom, I J Bruno, M P Lightfoot, S C Ward, Acta Cryst B. 722Groom, C. R., Bruno, I. J., Lightfoot, M. P. & Ward, S. C. (2016). Acta Cryst B, 72(2), 171-179. . I Křivỳ, B Gruber, Acta Cryst. A. 322Křivỳ, I. & Gruber, B. (1976). Acta Cryst. A, 32(2), 297-298. . V Kurlin, arxiv:2205.04388early draftKurlin, V. (2022a). arxiv:2201.10543 (early draft). URL: http://kurlin.org/projects/periodic-geometry-topology/lattices3Dmaths.pdf Kurlin, V. (2022b). arxiv:2205.04388. URL: http://kurlin.org/projects/periodic-geometry-topology/metric1D.pdf . V Kurlin, arxiv.org:2201.05150early draftKurlin, V. (2022c). arxiv.org:2201.05150 (early draft). URL: http://kurlin.org/projects/periodic-geometry-topology/lattices2Dmaths.pdf . K J Mcgill, M Asadi, M T Karakasheva, L C Andrews, H J Bernstein, J Applied Cryst. 471McGill, K. J., Asadi, M., Karakasheva, M. T., Andrews, L. C. & Bernstein, H. J. (2014). J Applied Cryst. 47(1), 360-364. . M Mosca, V Kurlin, Crystal Research and Technology. 555Acta Cryst AMosca, M. & Kurlin, V. (2020). Crystal Research and Technology, 55(5), 1900197. Nespolo, M. (2008). Acta Cryst A, 64(1), 96-111. . M Nespolo, M I Aroyo, B Souvignier, Journal of Applied Crystallography. 515Nespolo, M., Aroyo, M. I. & Souvignier, B. (2018). Journal of Applied Crystallography, 51(5), 1481-1491. Krystallographische und strukturtheoretische Grundbegriffe. P Niggli, 1Akademische verlagsgesellschaft mbhNiggli, P. (1928). Krystallographische und strukturtheoretische Grundbegriffe, vol. 1. Akademis- che verlagsgesellschaft mbh. . A Pulido, L Chen, T Kaczorowski, D Holden, M Little, S Chong, B Slater, D Mcmahon, B Bonillo, C Stackhouse, A Stephenson, C Kane, R Clowes, T Hasell, A Cooper, G Day, Nature. 543Pulido, A., Chen, L., Kaczorowski, T., Holden, D., Little, M., Chong, S., Slater, B., McMahon, D., Bonillo, B., Stackhouse, C., Stephenson, A., Kane, C., Clowes, R., Hasell, T., Cooper, A. & Day, G. (2017). Nature, 543, 657-664. . P Sacchi, M Lusi, A J Cruz-Cabeza, E Nauha, J Bernstein, 22Cryst Eng CommSacchi, P., Lusi, M., Cruz-Cabeza, A. J., Nauha, E. & Bernstein, J. (2020). Cryst Eng Comm, 22(43), 7170-7185. E Selling, Journal für die reine und angewandte Mathematik. 77Selling, E. (1874). Journal für die reine und angewandte Mathematik, 77, 143-229. . E Stevens, P Coppens, Acta. Cryst. B. 36Stevens, E. & Coppens, P. (1980). Acta. Cryst. B, 36, 1864-1876. Pointwise distance distributions of periodic sets. D Widdowson, V Kurlin, Widdowson, D. & Kurlin, V., (2021). Pointwise distance distributions of periodic sets. URL: http://kurlin.org/projects/periodic-geometry-topology/PDD.pdf . D Widdowson, M Mosca, A Pulido, V Kurlin, A Cooper, MATCH Communications in Mathematical and in Computer Chemistry. 87Widdowson, D., Mosca, M., Pulido, A., Kurlin, V. & Cooper, A. (2022). MATCH Communi- cations in Mathematical and in Computer Chemistry, 87, 529-559. URL: http://kurlin.org/projects/periodic-geometry-topology/AMD.pdf Introduction to lattice geometry through group action. B Zhilinskii, EDP sciencesZhilinskii, B. (2016). Introduction to lattice geometry through group action. EDP sciences. . P H Zwart, R W Grosse-Kunstleve, A A Lebedev, G N Murshudov, P D Adams, Acta Cryst D. 641Zwart, P. H., Grosse-Kunstleve, R. W., Lebedev, A. A., Murshudov, G. N. & Adams, P. D. (2008). Acta Cryst D, 64(1), 99-107.

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{'abstract': 'This paper develops geographic-style maps containing 2D lattices in all known crystals parameterised by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of 2D lattices is a square with identified edges or a sphere without one point. The new continuous maps show all Bravais classes as low-dimensional subspaces, visualise hundreds of thousands of real crystal lattices from the Cambridge Structural Database, and motivate the development of continuous and invariant-based crystallography.', 'arxivid': '2109.10885', 'author': ['Matthew Bright \nMaterials Innovation Factory\nUniversity of Liverpool\nUK\n', 'Andrew I Cooper \nMaterials Innovation Factory\nUniversity of Liverpool\nUK\n', 'Vitaliy Kurlin vitaliy.kurlin@gmail.com \nMaterials Innovation Factory\nUniversity of Liverpool\nUK\n'], 'authoraffiliation': ['Materials Innovation Factory\nUniversity of Liverpool\nUK', 'Materials Innovation Factory\nUniversity of Liverpool\nUK', 'Materials Innovation Factory\nUniversity of Liverpool\nUK'], 'corpusid': 248986864, 'doi': None, 'github_urls': ['https://github.com/MattB-242/Lattice'], 'n_tokens_mistral': 15079, 'n_tokens_neox': 12576, 'n_words': 7265, 'pdfsha': '5b2f561bbe275317ee695188f37672ca117c421e', 'pdfurls': ['https://arxiv.org/pdf/2109.10885v2.pdf'], 'title': ['Geographic-style maps for 2-dimensional lattices', 'Geographic-style maps for 2-dimensional lattices'], 'venue': []}

arxiv

RECKONING: Reasoning through Dynamic Knowledge Encoding Zeming Chen Gail Weiss Eric Mitchell Asli Celikyilmaz Antoine Bosselut antoine.bosselut@epfl.cheric.mitchell@cs.stanford.eduaslic@meta.com RECKONING: Reasoning through Dynamic Knowledge Encoding Recent studies on transformer-based language models show that they can answer questions by reasoning over knowledge provided as part of the context (i.e., incontext reasoning). However, since the available knowledge is often not filtered for a particular question, in-context reasoning can be sensitive to distractor facts, additional content that is irrelevant to a question but that may be relevant for a different question (i.e., not necessarily random noise). In these situations, the model fails to distinguish the knowledge that is necessary to answer the question, leading to spurious reasoning and degraded performance. This reasoning failure contrasts with the model's apparent ability to distinguish its contextual knowledge from all the knowledge it has memorized during pre-training. Following this observation, we propose teaching the model to reason more robustly by folding the provided contextual knowledge into the model's parameters before presenting it with a question. Our method, RECKONING, is a bi-level learning algorithm that teaches language models to reason by updating their parametric knowledge through backpropagation, allowing them to then answer questions using the updated parameters. During training, the inner loop rapidly adapts a copy of the model weights to encode contextual knowledge into its parameters. In the outer loop, the model learns to use the updated weights to reproduce and answer reasoning questions about the memorized knowledge. Our experiments on two multi-hop reasoning datasets show that RECKONING's performance improves over the in-context reasoning baseline (by up to 4.5%). We also find that compared to in-context reasoning, RECKONING generalizes better to longer reasoning chains unseen during training, is more robust to distractors in the context, and is more computationally efficient when multiple questions are asked about the same knowledge.Consider the sentence: "John is David's dad, and Tom is John's dad". Concluding that Tom is David's grandfather involves reasoning about the information in the sentence. Specifically, it requires understanding the direct information, or contextual knowledge, given in the sentence: the stated relationships between John, David, and Tom; and combining it with our existing, commonsense knowledge of the world: someone's dad's dad is their grandfather. Achieving such logical reasoning automatically has long been a goal of AI[51,16,71,79].The example above demonstrates two necessary abilities required for successful reasoning: first, holding large amounts of commonsense or general knowledge about the world, and second, processing and combining new information with existing knowledge. Transformer-based large language models Preprint. Under review. Introduction Figure 1 : Our algorithm, RECKONING, solves reasoning problems by encoding external contextual knowledge into a model's parameters through gradient updates. At inference time, RECKONING performs a few parameter updates using the gradients of a language modeling loss to encode the relevant facts. Then, the updated model answers the question using only its implicit knowledge. have shown a remarkable capacity for the first of these abilities, repeatedly being demonstrated to memorize large amounts of data, or parametric knowledge, in their weights [61,7,10,48]. For the second, recent work showed that transformers fine-tuned to predict answers over a concatenated context ("The cow is big; If something is big then it chases the dog; If the cow chases the dog then the cow sees the rabbit") and question ("Did the cow see the rabbit?") achieve high performance on reasoning tasks where all necessary knowledge is given in the context [16]. We refer to this general setting as in-context reasoning (ICR), and differentiate by amount and type of knowledge given [30]. In real-world question-answering settings [38,21,40,15], large amounts of contextual knowledge may be provided at once, and the information may not be perfectly filtered for a specific question. Unfortunately, in-context reasoning is highly sensitive to distractors [67]: additional facts that are not relevant to a question (e.g., "The cow is round" for the above example). Indeed, when fine-tuning and evaluating GPT-2 [57] for ICR, we find that adding distractors to the context drops performance from 99.4% to only 70.9% accuracy for the same questions ( §4.2). This sensitivity to distractors in contextual knowledge contrasts with GPT-2's apparent robustness to distractors in parametric knowledge: for any specific example, most of the training data seen by GPT-2-which forms its parameters-is likely to be completely irrelevant to that example. Naturally, we wonder whether presenting contextual knowledge in the same way as memorized knowledge, by encoding it into a model's parameters, will improve the reasoning abilities of transformer-based language models. In this work, we propose a novel bi-level optimization algorithm, RECKONING, that learns to memorize (and reason) over facts (i.e., knowledge) by performing inference-time parameter updates using gradients computed from a language modeling loss on those facts. The updated model is then used to answer any questions about those facts. Our training framework involves two nested loops: the inner loop performs fast adaptations from a set of initial weights to memorize a set of external knowledge through a few gradient updates, and the outer loop optimizes those same initial weights such that the updated model will solve reasoning problems associated with the memorized knowledge. In other words, the outer loop learns optimal meta-parameters that can rapidly memorize and successfully reason over contextual knowledge, allowing knowledge memorization to be optimized directly for downstream reasoning. At inference time, instead of including external knowledge in the input sequence as the prefix to a question prompt, the model can encode it in its parameters through gradient updates and then reason over its updated parametric knowledge to reach a conclusion. We evaluate RECKONING on two synthetic multi-hop reasoning datasets: ProofWriter [71] and CLUTRR-Systematic-Generalization (CLUTRR-SG) [27], comparing against a fine-tuned ICR (FT-ICR) baseline that uses the same underlying model. Our results show that RECKONING consistently outperforms the FT-ICR baseline on each benchmark, demonstrating that it successfully learns to answer multi-hop reasoning questions as desired. In particular, we find that RECKONING more successfully generalizes to adversarial settings, such as the presence of distractor facts and the introduction of longer reasoning chains at inference time. Finally, while the inference-time gradient updates make RECKONING slower to process new knowledge than a typical ICR forward pass, our run-time analysis shows that RECKONING is more efficient when answering multiple questions about a shared knowledge set. This is because RECKONING only needs to encode the knowledge once to answer multiple questions about it. Overall, we demonstrate that RECKONING is an effective Figure 2: The two-stage training process of RECKONING with an inner and outer loop. algorithm for reasoning through dynamic and controllable knowledge encoding, overcoming an observed weakness in the common reasoning setting and providing multiple additional benefits. Background Notation We use f : X × θ → Y to refer to parameterised functions in which X is the set of possible inputs and θ are their possible weights (parameters). We use f θ : x → f (x, θ) to easily refer to any f with a given set of parameters θ. We describe reasoning problems using tuples (K, x, y * , Y ) such that y ∈ Y is the correct answer for the question x given facts K, and use D to refer to sets of such problems. When it is clear from context, we drop Y and use only (K, x, y * ). Language Modeling and Memorization In the causal language modeling (CLM) objective, a parameterized model f θ is trained to estimate the conditional probabilities of each token in a sequence given its predecessors: p(x t |x <t ) . Specifically, we train f θ to approximate p using the CLM loss: L CLM (f θ , x) = − T t=1 log f θ (x t |x 1 , ..., x t−1 ).(1) This training objective allows language models to memorize individual training examples [10,9], and we will exploit this ability in order to memorize and draw on contextual knowledge in our work. Transformers as Soft Reasoners In natural language reasoning tasks, we are given reasoning problems (K, x, y * , Y ) in natural language and attempt to recover the correct answer y * from the context K, question x, and possible answers Y alone. In in-context reasoning, language models f θ trained with a CLM objective are applied to this task by selecting as the response the answer y ∈ Y with a maximum probability according to the model's next-token prediction from the concatenated context and question: y = arg max y ′ ∈Y f θ (y ′ |[K; x]). Previous works show that, after relevant supervised fine-tuning, transformer language models can achieve high performance in this setting [16,71,27], though this degrades significantly in the presence of irrelevant facts (distractors) [67]. Method Addressing these challenges, we propose RECKONING (REasoning through dynamiC KnOwledge eNcodING), which solves reasoning problems by memorizing the provided contextual knowledge, and then using this encoded knowledge when prompted with downstream questions. Specifically, RECKONING uses bi-level optimization to learn a set of meta-parameters primed to encode relevant knowledge in a limited number of gradient steps. The model can then use its updated weights to solve reasoning problems over this knowledge, without further presentation of the knowledge itself. Overview: Inference Given a reasoning problem (K, x, y, Y ), we initialize our model with weights copied from a set of meta-parameters θ and perform a constant number N of gradient descent steps on these with the goal of minimizing the CLM objective on the knowledge set K. This allows the model to memorize K in its updated parameters, which we refer to asθ K N . Next, we pass the question x to the model, using fθK N to obtain a distribution over Y , and taking as output the answer y ∈ Y with the highest probability. For this method to consistently output the ground truth y * , we seek a set of optimal meta-parameters θ * that can quickly memorize (i.e., learn) the given knowledge in a way that then allows accurate reasoning when queried about the knowledge downstream. Training RECKONING Given a distribution p(D) of reasoning problems, our proposed bi-level optimization framework RECKONING (seen in Figure 2) optimizes the following objective: θ * ∈ arg min θ E (K,x,y)∼p(D) [L CE (fθK N (x), y)](2) where for all K, n ∈ N, and θ:θ K 0 = θ, and θ K n+1 =θ K n − α∇L CLM (fθ K n , K).(3) Here, L CE (f (x), y) denotes the cross-entropy (CE) loss, which we apply with the relevant parameters for each reasoning question in D, L CLM (f, K) = 1 |K| k∈K L CLM (f, k) denotes the causal language modeling loss, and N and α are pre-defined hyperparameters of the fine-tuning. We seek our actual meta-parameters θ through gradient descent. In particular, denoting by θ 0 our initial meta-parameters, andθ K N,i the parametersθ K N obtained when initializingθ K 0 with θ i , we iteratively compute θ i+1 = θ i − η∇ 1 |D i | (K,x,y)∈Di L Total (fθK N,i , K, x, y),(4) Algorithm 1 RECKONING Require: An example distribution p(D), a transformer language model f , initial meta-parameters θ, outer step size η, inner step size α, inner loop length N . 1: while not converged do ▷ outer loop 2: Sample D ′ ∼ p(D) 3: L D ′ ← 0 4: for each (K, x, y) ∈ D ′ do 5: Initializeθ K 0 = θ 6: for n := 0 to N − 1 do ▷ inner loop 7:θ K n+1 ←θ K n − α∇LCLM (fθK n , K) 8: end for 9: L D ′ ← L D ′ + LTotal(fθK N , K, x, y) 10: end for 11: θ ← θ − η∇ 1 |D ′ | L D ′ 12: end while where L Total (f, K, x, y) = L CE (f (x), y) and for each i, D i is randomly sampled from p(D). This continues until L Total converges. The training can be seen as two nested loops: at each iteration, the outer loop (Equation (4)) samples a random batch D i ⊆ D of reasoning problems for evaluating (in order to update) the current meta-parameters θ i , after the inner loop (Equation (3)) adapts them to encode the associated knowledge through N steps of gradient updates. Multi-Task Objective Through our experiments, we find that adding a knowledgerecovery objective to the outer loop-such that the model must also state all of K when prompted with x-improves the model's reasoning performance. We evaluate knowledge recovery with a CLM loss and combine the two losses by simple addition, following prior works [23,75,74]. The entire change is achieved by redefining the total loss in our outer loop (Equation (4)) as: L Total (f, K, x, y) = L CE (f (x), y) + L CLM (f, x, K)(5) where L CLM (f, x, K) is the language modeling loss on K, as in Equation (3), but this time conditioned on the question x. The overall process for training RECKONING is depicted in Algorithm 1 and Figure 2. Additionally, we dynamically learn a per-step-per-layer learning rate to replace the shared constant learning rate in the inner loop. We give more details Appendix D. Experiments Setup We conduct our experiments on two datasets focusing on multi-hop logical reasoning over natural language knowledge: ProofWriter [71], which measures the model's ability to emulate reasoning over facts and rules expressed in natural language, and CLUTRR-SG [27], which is generated from the CLUTRR [69] benchmark, a logical reasoning task that involves reasoning over family relationships between entities grounded in first-order logical proofs. For these datasets, each problem requires multiple reasoning hops to reach an answer. 1 We compare our method against the following baselines: (1) a fine-tuned model that performs a forward pass on only the question without access to the knowledge (No-Facts), (2) a fine-tuned model that performs a forward pass on only the knowledge without access to the question (No-Question), (3) a model trained using RECKONING with random knowledge that is not relevant to the questions (Random-Facts), and (4) an ICR baseline that concatenates the knowledge K with the question x in a single context and is trained using supervised learning to predict the answer (FT-ICR). Our first three baselines sanity-check whether any surface-level patterns in the questions and facts can be exploited to make accurate predictions. The last baseline compares RECKONING to the conventional way of reasoning with language models. In all experiments, we use the base GPT-2 [57] model (∼124M parameters) as our initialization. We compute each score from the average across three different runs. Unless stated otherwise, we refer by RECKONING to our method trained with the multi-task objective. For more details on the implementation, datasets, and examples, see Appendix A and Appendix C. Table 1: Label accuracy of RECKONING on ProofWriter and CLUTRR-SG, compared to FT-ICR baselines where the supporting facts are given as part of the input. MT marks models trained with the multi-task objective, which optimizes both question answering and knowledge memorization. Multi-hop Reasoning Performance ProofWriter CLUTRR-SG Method 2-h 3-h 5-h 2-h 4-h 6-h Main Results We first evaluate whether RECKONING learns to perform reasoning in the base setting. A model is given a set of supporting facts (without distractors) and a question (or hypothesis) as input and begins by performing a few CLM learning steps on the facts. Then, the updated model reads only the question and generates an answer. To answer correctly, the model must reason over both facts and the question, meaning it must encode the facts during the inner loop such that multi-hop reasoning can be performed over them later. We train our models and the fine-tuned ICR (FT-ICR) baselines with both the single-task (L CE ) and multi-task (L CE + L CLM ) objectives. For multi-task (MT) training, the model learns to answer the question and generate its relevant knowledge in the outer loop. Table 1 shows the evaluation results on question answering (or hypothesis classification). For all hop numbers in ProofWriter and in CLUTRR-SG, multi-task RECKONING outperforms the best result of all baselines (consistently obtained by multi-task FT-ICR) by an average of 1%. We conclude that RECKONING can effectively solve reasoning problems through its updated parametric knowledge, and do so better than existing baselines. The multi-task objective is crucial for this success: not only is RECKONING's performance consistently higher (by an average of 2.8% over the two datasets and their hop counts) when using the multi-task rather than single-task (ST) objective, it also under-performs both FT-ICR baselines when trained with only the single-task objective. The multi-task objective also improves FT-ICR consistently (average 1.8%), though it is not enough to beat the multi-task RECKONING. In all further experiments, we consider only RECKONING and FT-ICR with a multi-task objective. Generalizing to Longer Reasoning Chains Our first experiments assume an alignment between the number of reasoning hops in the questions in the training and test set. However, we may not be able to train on all n-hop reasoning questions we encounter in the wild, and we rarely know the number of reasoning hops in a question a priori. Consequently, we also measure the generalization capacity of our model to questions with hop numbers unseen during training. We compile interpolation (fewer hops than the train set) and extrapolation (more hops than the train set) test sets from the CLUTRR-SG dataset. Again, we train models individually on 2-hop, 4-hop, and 6-hop examples and evaluate these three sets of models on the test sets, which contain 2-10-hop reasoning questions. Figure 3 shows that both RECKONING models and ICR baselines retain high performance on the interpolation Label Accuracy (%) Number of Test Hops Training on 2-hop Questions Number of Test Hops Does RECKONING's performance depend on the number of inner loop gradient steps? In RECK-ONING, the model performs multi-hop reasoning over facts by encoding facts using multiple gradient steps in the inner loop optimization ( §3). Naturally, this process prompts the question of whether there is a correlation between the number of reasoning hops and the number of gradient steps needed to reliably encode the knowledge (i.e., problems with more reasoning hops require more gradient steps in the inner loop to encode the facts). In Figure 4, we show for CLUTRR-SG that as the number of inner loop steps increases, the label accuracy of the outer-loop task also increases. Furthermore, when considering the performance gains for reasoning with 6 inner loop steps (i.e., knowledge encoding) steps as opposed to one, we observe that this gap is much more pronounced for 4-hop (42.3%) and 6-hop (34.7%) reasoning than it is for 2-hop reasoning (5.9%). These results show that problems requiring more hops of reasoning also greatly benefit from more steps of inner loop knowledge encoding. Training on 4-hop Questions FT-ICR RECKONING Reasoning with Distractors In cases where multiple questions must be answered about the same knowledge set, some knowledge that is relevant to one question will likely be irrelevant to another question. For example, in Table 6, the fact "Charlie is White." is not needed to answer the question "Harry is red?". Thus, it is important to evaluate the robustness of RECKONING when there exists irrelevant information (i.e., distractors) in the knowledge set. In this experiment, we analyze RECKONING's ability to focus on the correct knowledge and ignore distractors when answering questions. We use ProofWriter as the evaluation dataset since it already has a setting with distractors included in the knowledge. For systematic analysis, we gradually add distractors to the context (starting from 2 and finishing at all possible distractors, of which there are an average of 7 per question). We train RECKONING and the baseline using the multi-task objective, where the model must (1) recall all of the facts and rules relevant to the question and (2) predict the conclusion based on the correct knowledge. In this case, we adapt training such that for each question x, the outer-loop (Equation (5)) CLM loss is only computed with respect to the relevant facts from K, thereby learning to recall only relevant facts during training. In Figure 5, we see that RECKONING's performance is consistently more robust under distractors than the FT-ICR baseline. When we include all of the distractors in the context, RECKONING achieves a significantly higher average label accuracy (82.5%) across hops than the baseline (70.9%), as computed by the average of the 3 considered hop depths. Additionally, compared to performance with no distractors, RECKONING's performance only drops 17.1% while the baseline performance drops 28.6%, thereby exhibiting a better ability to disentangle the correct knowledge from the distractors. One of the advantages of RECKONING is the ability to memorize a large set of knowledge K and answer multiple related questions about that knowledge at a little extra cost per question. Specifically, in contrast to ICR, RECKONING can encode K once and answer multiple questions without needing to reprocess it for each question asked. To test whether RECKONING could be a more efficient method for inference in this setting, we measure the wall-clock time (in seconds) of the complete inference pipeline of RECKONING vs. ICR. For this experiment, we use a synthetic reasoning dataset in which K is a sequence of random letters, and the question x asks for the most frequent letter in the context. The total number of tokens in each example is 1024: 7 for x, 1 for the answer, and the remaining 1016 for K, broken into 8 "facts". The FT-ICR baseline receives a sequence including all 8 facts and the question. In contrast, RECKONING receives the 8 facts as a batch of eight segments of 127 tokens and encodes them in parallel in the inner loop. In the outer loop, the model only receives the question or a batch of questions. We focus on two settings: (1) inference time for a single question and (2) inference time when answering multiple questions. In the multiple-question setting, we set the number of questions to 18 (the same as in ProofWriter). For RECKONING, the inference process includes the inner-loop knowledge encoding and the final forward pass to encode the question. We set the number of inner loop gradient steps to 1 and 4. In Table 2, we see that when answering a single question, RECKONING does not perform inference faster compared to in-context reasoning. However, RECKONING shows significant advantages under a multi-question setting. Both the 1-step inner loop and the 4-step inner loop are faster than the baseline. Since RECKONING encodes the knowledge in model parameters, it does not need to reprocess the knowledge for a related question and is more efficient. We run this experiment on 1 RTX 3090 GPU. 2 2 We perform this experiment in a limited setting and do not handle the case where hidden states could be cached for the forward pass of in-context reasoning, likely speeding up multi-question inference [11]. Table 3: Average inner loop validation loss: final (L CLM ) and difference from start to finish (∆L CLM ). Run-time Analysis Memorizing Knowledge In Table 1, we saw that training RECKONING with a multitask (MT) outer loop objective improved over training with the single-task (ST) objective, potentially because the MT objective improves the model's ability to memorize the knowledge in the inner loop. To validate our hypothesis, we analyze RECKONING's performance in reproducing memorized knowledge. First, we show in Table 3 the inner loop average loss (L CLM ) and average change (∆L CLM ) (from first inner loop evaluation to last) on validation examples from the 5-hop ProofWriter data. We see that the average inner loop loss for RECKONING ST is much higher than RECKONING MT , and indeed starts out much higher as well. This shows that the ST outer loop objective, which optimizes the model only for question answering, does not learn to encode the knowledge in the inner loop by memorizing it. In contrast, the MT objective forces the model to learn to memorize the knowledge too: we observe that RECKONING MT minimizes the inner loop loss as it processes the knowledge. This pattern is also shown in the average inner-loss difference (∆L CLM ): the inner loop loss decreases more after the gradient updates when trained with the MT objective. Next, we report in Table 4 the model's ability to reproduce memorized facts correctly under a multi-task setting, as measured by an exact match score between the reproduced facts and the gold facts. 3 We evaluate on the ProofWriter dataset both with and without distractors in the context and compare the results to the FT-ICR baseline. 2-hop 3-hop 5-hop Label Accuracy (%) single-task multi-task Figure 6: Performance comparison between models trained with a single-task and a multi-task objective under distractors. With the multi-task objective, the model learns to memorize the relevant facts and perform reasoning over them. The results show that RECKONING MT can successfully (average exact match score of 99.3%) recover the relevant facts from its model parameters when the context does not include any distractors. Note that this is comparable to the FT-ICR baseline, for which the task is much easier as it can directly attend to and copy the facts from input, while RECKONING MT no longer has direct access to them. When the context includes distractors, both RECKONING and FT-ICR struggle to identify and reproduce only the relevant facts. However, the performance for FT-ICR (average 49.4%) drops far below that of RECKONING (73.6%), demonstrating that RECKONING is much better at disentangling the relevant knowledge from the distractors. Finally, we show that RECKONING with a multi-task objective is also more robust to distractors as it trains the model to only reproduce the facts that would be relevant to a particular question we ask in the outer loop. As in Section 4.2, we use the ProofWriter dataset and, for each question, add all the distractors to the context. We train the model using the multi-task objective, and we report the label accuracy. While in Table 1, we originally saw a ∼ 1% improvement from training with a multi-task objective on ProofWriter with no distractors, we see a much more significant performance gap in Figure 6 (∼ 18.2%) when distractors are available. We also note that the performance of the single-task model is essentially random (see the Random-Facts baseline from Table 1). By learning how to memorize knowledge in the inner loop so that it can recall relevant facts in the outer loop, the model also learns how to encode facts more robustly over them. Related Work Logical Reasoning Datasets and Benchmarks As a central building block of human cognition and intelligence [26], logical reasoning has been a long-pursued topic in the field of AI [53,46,8,2,16,12,70,44]. Logical reasoning, in general, can be categorized in a trichotomy of deductive, inductive, and abductive reasoning [24]. Multiple datasets have been published that evaluate neural models' ability on these three types of logical reasoning [16,5,69]. Initially, logical reasoning tasks focused on hypothesis classification, where, given a theory consisting of multiple facts and rules, a model would determine whether the hypothesis was correct. Recently, transformer-based language models have been directly used to solve this task in synthetic [16,63], real-world [28], and adversarial [59,25,65] settings. However, simply predicting whether the hypothesis is valid does not elucidate whether the model correctly reasons over the provided knowledge. To better analyze and interpret the reasoning process of language models, new tasks focus on generating the valid proof that explains the model's decision [71,19]. Our proposed method, RECKONING, is optimized for the hypothesis classification reasoning task and evaluates on many of these datasets [71,27]. Logical Reasoning over Natural Language Historically, automatic logical reasoners used symbolic systems and formal languages as a knowledge representation [41,53,50,1,47,78]. However, these systems were hard to scale up due to the knowledge-acquisition bottleneck and the brittleness of formal representation [33,81]. With recent advances in transformer-based language modeling [73] and self-supervised pre-training [20,57,58], a novel paradigm for logical reasoning emerged, where pre-trained language models (PLMs) could be used as soft reasoners over knowledge expressed in natural language. Natural language as a knowledge representation allowed PLMs to handle raw input with diverse formats [31,14], resulting in PLMs being applied to various types of deductive [16], abductive [5], and inductive [27] reasoning tasks. However, language models as soft reasoners also showed structural weaknesses, as their performance dropped on complex logical operations [77,12], and their reasoning process was not interpretable [62,43]. Consequently, a new line of work uses neuro-symbolic methods to combine the best of both language models and symbolic reasoning [34,42,13,6,39]. Specifically, the interpretability gap motivated modular and step-wise reasoning systems that use PLMs as intermediate modules [64,72,32,66,56,80] to generate reasoning steps (e.g., proofs). In contrast to these works, our method RECKONING dynamically encodes natural language knowledge into the model parameters, thereby reasoning by mixing contextual knowledge with pre-encoded parametric knowledge and allowing the model to determine a conclusion based on its updated parametric knowledge. Model Editing While our motivations are grounded in research on machine reasoning, our methods are more often used in the area of model editing. Model editing is a method to edit a model's parameters to correct its errors or update the model. Several works propose hypernetwork-based methods to edit knowledge in a model by predicting updates conditioned on new factual statements [29] or transforming the gradients from new provided facts [52] to make local edits to a model. Other approaches focus on more direct edits of model behavior, such as directly modifying neuron outputs [18,82], localizing distinct feed-forward layers that are responsible for factual recall, and modifying these weights [48], and performing weight updates across multiple layers to perform simultaneous edits [49]. Similarly, our method also rapidly edits the model parameters to add knowledge. However, our bi-level framework optimizes model edits for the reasoning task in the outer loop, allowing the model to learn to do fast memorization of knowledge that can support the model's reasoning ability. Language Models as Knowledge Bases Our work learns to reason by dynamically encoding contextual knowledge in the parameters of language models before answering questions about them. Previous studies have found that LLMs can store real-world facts learned during pre-training [61,10,48,9]. Learning these facts during pre-training allows language models to be prompted [55,37,68,83] or adapted [7,60,35,36] to produce these facts on-demand. However, LLM knowledge is latent and hard to identify or control. The model generation is sensitive to specific words or phrases. LLMs emit knowledge encoded in the parameters only when prompted appropriately [54,22,17,9]. It is also difficult to inject or update knowledge for LLMs [48], and the memorization of knowledge in LLMs is not optimized toward their reasoning ability. In our work, we seek to find a way to add knowledge to LLMs in a controllable and adaptive way that can be beneficial to downstream reasoning applications. Conclusion We present RECKONING, a bi-level learning framework for multi-hop reasoning that encodes knowledge verbalized using natural language into a model's parameters through gradient updates. During training, the inner loop encodes the contextual knowledge into the model parameters by backpropagating a language modeling loss. In the outer loop, given only the question as input, the model solves reasoning problems using the memorized knowledge. Through bi-level optimization, RECKONING finds a set of meta-parameters that allows it to perform quick knowledge-based updates for reasoning. Our experiments show that RECKONING learns to reason only by relying on its parametric knowledge after the external knowledge has been encoded. Using a multi-task objective that jointly optimizes reasoning and knowledge memorization in the outer loop, RECKONING outperforms ICR baselines that are trained to encode external knowledge as part of the context. Through our analysis, we show that RECKONING is more generalizable to problems with longer reasoning chains, less susceptible to irrelevant distractor knowledge, and that RECKONING is more efficient than the baseline when answering multiple questions that require common knowledge. A Dataset ProofWriter The ProofWriter [71] dataset has 500k pairs of questions, answers, and proofs over natural-language rule bases. Each example in the dataset contains a set of facts, a set of rules, a hypothesis, and a label indicating whether the hypothesis is true, false, or unknown. The dataset comprise five datasets named D0, D1, D2, D3, D5, each with 100k examples. Each dataset's questions require reasoning up to depths D (D = 0, 1, 2, 3, 5) to determine their answers. In our experiments, we only focus on the datasets that require more reasoning depths (D2, D3, D5). We show an example from the dataset in Table 6. In these datasets, a set of facts and rules are mapped to 18 questions, where the questions can be answered based on a subset of the facts and rules. Thus, some of the facts or rules can be irrelevant to some questions, and we call them distractors in Section 4.2. In the experiment for knowledge encoding with distractors, we encode all the facts in the model parameters and evaluate its ability to reproduce and reason over the correct facts. We show an example of distractor and relevant knowledge of a question in Table 8. For detailed statistics on the two datasets, please see Table 5. CLUTRR-SG The CLUTRR-SG [27] is an evaluation dataset for inductive reasoning on family relations adapted from the [69] dataset for measuring systematic generalization. Each example in the dataset contains (i) a set of facts representing a family graph G = (V, E) where nodes (V ) are entities and edges (E) are the relationships. (ii) a question asking the relationship between two entities (v 1 , v n ∈ V ), and (iii) a target relationship e * ∈ E as the answer for the question. The facts are expressed as a list of (v i , e j , v k ) tuples. The two entities in the question are separated by more than one hop in the graph. There are 272 unique entities, 20 relationship types, and nearly 1.5M possible facts in the dataset. Following the authors, we define the difficulty of examples based on the number of family graph edges (i.e., the number of reasoning hops required to determine a relation), in which k edges (k-hop) correspond to k facts. We show an example from the dataset in Table 7. To motivate the advantage of RECKONING on mitigating interference from distractors, we analyze the performance change of fine-tuned incontext reasoning with and without distractors present in the context of the questions. We define distractors as additional facts or rules present in a question's context that is not directly relevant to the questions. A model should not be able to use only these distractors to answer a question correctly. For an example of distractors in a question's context, please see Table 8. We evaluate the baseline on the ProofWriter dataset since it naturally contains contexts including distractors ( Table 8). Recall that we have two training objectives. The single-task objective only trains the model to predict an answer for each question given their contexts. The multitask objective (MT) trains the model to not only predict an answer but also reproduce the correct facts and rules (in contrast to distractors) based on the contexts. We evaluate the baseline on 2, 3, and 5-hop datasets with both training objectives, and we report the average label accuracy across hops in Figure 7. Compared to the baseline's performance without distractors in the context, the performance with distractors decreases significantly. For single-task, the performance drops 23.2% when adding distractors to the contexts, and the performance with the multi-task objective drops 28.6%. The results highlight in-context reasoning's high sensitivity to the interference of irrelevant information in the contexts. C Implementation Details We select GPT-2-base [57] as the model for our method and all the baselines. We use the version implemented by the Huggingface Transformers library [76]. All the experiments for RECKONING If someone is nice, then they are round. rule 6 If Charlie is round and Charlie is nice, then Charlie is white. question-answer 1 Harry is red? True question-answer 2 Harry is not red? False question-answer 3 Dave is not white? Unknown Table 6: An example from the dataset ProofWriter. There are 6 facts and 6 rules mapped to three question-answer pairs. Each question can be answered based on the given facts and rules. are conducted on a cluster with NVIDIA A100 (40GB) GPUs. All the baseline experiments are conducted on a local machine with NVIDIA RTX 3090 GPU (24GB). Fine-tuned In-context Reasoning We set the train batch size to 16 and train the model for 6 epochs with early stopping based on the validation label accuracy. We set the learning rate to 3e-5 and use the AdamW optimizer with ϵ set to 1e-8. We validate the model on the development set for every epoch and select the best checkpoint using the validation accuracy as the metric. RECKONING In the inner loop, we generally perform 4 gradient steps for lower-hop questions (2, 3, 4-hop) and 5 gradient steps for higher-hop questions (5 and 6-hop). We select the AdamW [45] as the optimizer for the inner loop since the main task is language modeling. The inner-loop learning rate is set to 3e-5 before training and the algorithm dynamically learns a set of optimal learning rates when converged. In our experiments and analysis, we only report the results from RECKONING with a multi-task objective since its performance is better than the single-task objective. In the outer loop, we also use the AdamW with a learning rate of 3e-5. For both optimizers, we set ϵ to 1e-8. We set the train batch size to 2 due to memory limitations. We apply the technique of gradient accumulation and set the accumulation step to 2. We train the model for 6 epochs with early stopping. For each epoch, we validate the model twice: once in the middle and once at the end. We select the best model checkpoint based on the validation label accuracy. D Adaptive Learning Rate Prior works [3,4] show that a fixed learning rate shared across steps and across parameters does not benefit the generalization performance of the system. Instead, [3] recommend learning a learning rate If someone is nice, then they are round. rule 6 If Charlie is round and Charlie is nice, then Charlie is white. question-answer 1 Harry is red? True for each layer of the network and for each adaptation step in the inner loop. The layer parameters have the freedom to learn to adjust the learning rates at each step. To control the learning rate α in the inner loop adaptively, we define α as a set of adjustable variable: α = {α 0 , α 1 , ...α L }, where L is the number of layers and for every l = 0, ..., L, α l is a vector with N elements given a pre-defined inner loop step number N . The inner loop update equation then becomeŝ θ K n+1,l =θ K n,l − α (l) n ⊙ ∇L CLM (fθ K n , K)(6) where ⊙ is an element-wise product andθ (l) n is the parameters for layer l at the inner step n. We learn the set of optimal inner loop learning rates α * by optimizing the parameters in the outer loop: α ← α − η∇ 1 |D i | (K,x,y)∈Di L Total (fθK [ θi]N (x), y),(7) where η is the outer loop learning rate andθ is the updated parameters from inner loop. Below, we show the final algorithm of RECKONING in Algorithm 2. Algorithm 2 Dynamic Knowledge Encoding for Reasoning Require: An example distribution p(D), a transformer language model f , initial meta-parameters θ, outer step size η, initial inner step size α, inner loop length N . 1: while not converged do ▷ outer loop 2: Sample D ′ ∼ p(D) 3: L D ′ ← 0 4: for each (K, x, y) ∈ D ′ do 5: Initializeθ K 0 = θ 6: for n := 0 to N − 1 do ▷ inner loop 7:θ K n+1 ←θ K n − α ⊙ ∇LCLM (fθK n , K) 8: end for 9: L D ′ ← L D ′ + LTotal(fθK N , K, x, y) 10: end for 11: α ← α − η∇L D ′ ▷ Update inner step size 12: θ ← θ − η∇L D ′ 13: end while : We study how much the dynamic learning rate in the inner loop contributes to the outer loop performance. We fix all the hyperparameters except the option of using the dynamic or fixed learning rate. We conduct the analysis using the CLUTRR-SG dataset since it is more complex and difficult (lower random performance). Are dynamic learning rates necessary for RECK-ONING's performance? Following prior works on meta-learning [3,4], we dynamically learn a set of per-step-per-layer learning rates for RECKONING. In this ablation study, we analyze whether dynamic learning rates for the inner loop are effective in improving the outer loop reasoning performance. Similarly, we fix other experimental settings and set the number of inner loop steps to 4. As Figure 8 shows, when using a static learning rate (i.e., all layers and inner loop steps share a constant learning rate), the performance drops by a large margin (average drop of 34.2% Table 9: Label accuracy of RECKONING on ProofWriter and CLUTRR-SG compared against a popular Large Language Model (LLM), GPT-3.5. We prompt GPT-3.5 in the zero-shot setting and also the 8-shot in-context learning setting. Models with MT are trained with the multi-task objective in the outer loop. Recently, Large Language Models (LLMs) with large parameter sizes learned from human preferences have shown remarkable performance in language understanding and generation. These LLMs are powerful zero-shot and few-shot reasoners. Recent works find that LLMs learn to perform multistep reasoning by first generating new reasoning chains and then predicting the answers. In this experiment, we benchmark the performance of a popular new LLM, GPT-3.5, on the two multi-hop reasoning datasets we used in our paper. We first evaluate GPT-3.5's zero-shot reasoning performance in predicting the correct answers. As Table 9 shows, zero-shot prompting GPT-3.5 significantly underperforms RECKONING's performance. GPT-3.5's performance improves on ProofWriter without distractors, but still is behind the performance of RECKONING. When distractors are present in the context, RECKONING performs much better than zero-shot and few-shot GPT-3.5 prompting. This Figure 4 : 4Multi-hop reasoning performance as a function of the number of inner loop steps (x-axis), with each line focusing (by training and testing) on CLUTRR-SG with a different number of hops. Figure 7 : 7Label accuracy of fine-tuned in-context reasoning on questions with and without distractors in the context. With the same questions, adding distractors to contexts significantly lower the performance of in-context reasoning, both in the singletask and multi-task setting. Figure 8 8Figure 8: We study how much the dynamic learning rate in the inner loop contributes to the outer loop performance. We fix all the hyperparameters except the option of using the dynamic or fixed learning rate. We conduct the analysis using the CLUTRR-SG dataset since it is more complex and difficult (lower random performance). Knowledge Encoding for Reasoning -Bi-level Optimization TrainingInner-loop Learningθ Outer-loop Learning For each ( , x, y) Relevant facts ℒ inner ∇ inner θ ∑ ℒ outer Reasoning Problems Question (x, y) θ N x: Who is Peter to Griffin? y: Grandson Relevant Facts Peter is a son to Kyle. Kyle is Julia's son. Amy is a daughter to Griffin. Amy is Kyle's sister. _ … _ ∇ outer θ Figure 3: System generalization evaluation on CLUTRR-SG. From left to right, the models are trained on 2-hop, 4-hop, and 6-hop CLUTRR-SG data portions. We evaluate the model on 2-10 hop test sets. The higher the hops, the more facts a question has, and the more difficult that question is.Figure 5: Robustness under distractors for ProofWriter. Each of the three plots corresponds to training and testing on a subset of questions in ProofWriter with a different number of hops (2,3,5-hops). Each bar corresponds to the number of distractors in the knowledge sets for those questions.test sets but exhibit decreasing performance as the number of hops increases. Importantly, though, RECKONING outperforms FT-ICR on all test sets regardless of the number of training hops, with the highest difference being more than 10% in every training setting (15%, 30%, 10%, respectively). These performance gains happen when testing on extrapolation data, suggesting that training with RECKONING better generalizes to examples with high OOD hop counts compared to ICR.99.5 94.1 89.8 89.5 79.8 99.4 88.9 80.6 78.9 62.7 40 60 80 100 0 2 4 6 all Label Accuracy (%) Number of Distractors Training on 2-hop Questions RECKONING FT-ICR 99.7 94.7 91.8 87.2 83.7 99.2 92.9 82.7 78.6 72.7 40 60 80 100 0 2 4 6 all Number of Distractors Training on 3-hop Questions RECKONING FT-ICR 99.8 96.0 93.9 91.5 84.0 99.6 94.1 88.9 81.5 77.2 40 60 80 100 0 2 4 6 all Number of Distractors Training on 5-hop Questions RECKONING FT-ICR Table 2 : 2Wall clock run-time, in seconds, of the fine-tuned ICR baseline and RECKONING. Table 4 : 4Exact match score for reproducing memorized knowledge. In contrast to in-context reasoning, RECKON-ING does not have direct access to the knowledge. Table 5 : 5Dataset splits and statistics for our experimentsIdentifier Content fact 1 Harry is nice. fact 2 Fiona is quite Nice. fact 3 Fiona is round. fact 4 Fiona is white. fact 5 Dave is furry. fact 6 Charlie is white. rule 1 Furry people are green. rule 2 Round, green people are red. rule 3 All red people are white. rule 4 Nice, round people are furry. rule 5 question-answer 1 How are D and B related to each other? GrandfatherIdentifier Content fact 1 C is H's father. fact 2 Z is J's aunt. fact 3 J is S's daughter. fact 4 D is C's father fact 5 S is B's father. fact 6 H is Z's son. Table 7 : 7An example of 6-hop reasoning from the CLUTRR-SG dataset.Identifier Content fact 1 Harry is nice. fact 2 Fiona is quite Nice. fact 3 Fiona is round. fact 4 Fiona is white. fact 5 Dave is furry. fact 6 Charlie is white. rule 1 Furry people are green. rule 2 Round, green people are red. rule 3 All red people are white. rule 4 Nice, round people are furry. rule 5 Table 8 : 8Example of distractors (black) and relevant knowledge (red) in the ProofWriter dataset. ). The performance drop becomes more significant on questions requiring more reasoning hops (45.5% drop for 4-hop and 39.5% drop for 6-hop), demonstrating the importance of using a dynamic learning rate in the inner loop of our framework.E Experiments with Large Language ModelsProofWriter ProofWriter distractor CLUTRR-SG Method 2-h 3-h 5-h 2-h 3-h 5-h 2-h 4-h 6-h GPT-3.5 0−shot 58.4 56.4 53.7 49.1 47.1 45.3 35.6 16.0 18.5 GPT-3.5 8−shot 78.0 82.4 80.1 58.7 57.2 54.5 39.0 18.5 20.8 RECKONINGMT 99.5 99.7 99.8 79.8 83.7 84.0 98.3 97.6 94.8 In ProofWriter, the number of reasoning hops is called the proof depth. To unify the presentation of the results, we use the term "hop" to describe the number of reasoning steps for both datasets. This is done by prompting the model with the question and comparing its output (after its answer to the question) to the concatenation of all of the facts. 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Kaiyu Yang, Jia Deng, Danqi Chen, Kaiyu Yang, Jia Deng, and Danqi Chen. Generating natural language proofs with verifier-guided search, 2022. Logical reasoning over natural language as knowledge representation: A survey. Zonglin Yang, Xinya Du, Rui Mao, Jinjie Ni, Erik Cambria, Zonglin Yang, Xinya Du, Rui Mao, Jinjie Ni, and Erik Cambria. Logical reasoning over natural language as knowledge representation: A survey, 2023. Kformer: Knowledge injection in transformer feed-forward layers. Yunzhi Yao, Shaohan Huang, Li Dong, Furu Wei, Huajun Chen, Ningyu Zhang, Yunzhi Yao, Shaohan Huang, Li Dong, Furu Wei, Huajun Chen, and Ningyu Zhang. Kformer: Knowledge injection in transformer feed-forward layers, 2022. Generate rather than retrieve: Large language models are strong context generators. Wenhao Yu, Dan Iter, Shuohang Wang, Yichong Xu, Mingxuan Ju, Soumya Sanyal, Chenguang Zhu, Michael Zeng, Meng Jiang, The Eleventh International Conference on Learning Representations. Wenhao Yu, Dan Iter, Shuohang Wang, Yichong Xu, Mingxuan Ju, Soumya Sanyal, Chen- guang Zhu, Michael Zeng, and Meng Jiang. Generate rather than retrieve: Large language models are strong context generators. In The Eleventh International Conference on Learning Representations, 2023. RECKONING's strength in disentangling irrelevant information from useful knowledge, and ability that even powerful LLMs like GPT-3.5 lacks. highlights RECKONING's strength in disentangling irrelevant information from useful knowledge, and ability that even powerful LLMs like GPT-3.5 lacks.

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{'abstract': 'Recent studies on transformer-based language models show that they can answer questions by reasoning over knowledge provided as part of the context (i.e., incontext reasoning). However, since the available knowledge is often not filtered for a particular question, in-context reasoning can be sensitive to distractor facts, additional content that is irrelevant to a question but that may be relevant for a different question (i.e., not necessarily random noise). In these situations, the model fails to distinguish the knowledge that is necessary to answer the question, leading to spurious reasoning and degraded performance. This reasoning failure contrasts with the model\'s apparent ability to distinguish its contextual knowledge from all the knowledge it has memorized during pre-training. Following this observation, we propose teaching the model to reason more robustly by folding the provided contextual knowledge into the model\'s parameters before presenting it with a question. Our method, RECKONING, is a bi-level learning algorithm that teaches language models to reason by updating their parametric knowledge through backpropagation, allowing them to then answer questions using the updated parameters. During training, the inner loop rapidly adapts a copy of the model weights to encode contextual knowledge into its parameters. In the outer loop, the model learns to use the updated weights to reproduce and answer reasoning questions about the memorized knowledge. Our experiments on two multi-hop reasoning datasets show that RECKONING\'s performance improves over the in-context reasoning baseline (by up to 4.5%). We also find that compared to in-context reasoning, RECKONING generalizes better to longer reasoning chains unseen during training, is more robust to distractors in the context, and is more computationally efficient when multiple questions are asked about the same knowledge.Consider the sentence: "John is David\'s dad, and Tom is John\'s dad". Concluding that Tom is David\'s grandfather involves reasoning about the information in the sentence. Specifically, it requires understanding the direct information, or contextual knowledge, given in the sentence: the stated relationships between John, David, and Tom; and combining it with our existing, commonsense knowledge of the world: someone\'s dad\'s dad is their grandfather. Achieving such logical reasoning automatically has long been a goal of AI[51,16,71,79].The example above demonstrates two necessary abilities required for successful reasoning: first, holding large amounts of commonsense or general knowledge about the world, and second, processing and combining new information with existing knowledge. Transformer-based large language models Preprint. Under review.', 'arxivid': '2305.06349', 'author': ['Zeming Chen ', 'Gail Weiss ', 'Eric Mitchell ', 'Asli Celikyilmaz ', 'Antoine Bosselut antoine.bosselut@epfl.cheric.mitchell@cs.stanford.eduaslic@meta.com '], 'authoraffiliation': [], 'corpusid': 258588316, 'doi': '10.48550/arxiv.2305.06349', 'github_urls': [], 'n_tokens_mistral': 31682, 'n_tokens_neox': 28110, 'n_words': 15739, 'pdfsha': '7805db74210aa113e83f20ffd0ad1ebcbb12ed7a', 'pdfurls': ['https://export.arxiv.org/pdf/2305.06349v2.pdf'], 'title': ['RECKONING: Reasoning through Dynamic Knowledge Encoding', 'RECKONING: Reasoning through Dynamic Knowledge Encoding'], 'venue': []}

arxiv

Testing of symmetry of innovations in autoregression Jul 2022 M V Boldin boldin−m@hotmail.com Dept. of Mech. and Math Moscow State Lomonosov Univ MoscowRussia A R Shabakaeva shabakaevalmira@gmail.com Dept. of Mech. and Math Moscow State Lomonosov Univ MoscowRussia Testing of symmetry of innovations in autoregression Jul 20221autoregressionoutliersresidualsempirical distribution functionPearson's chi-square testestimatorslocal alternativesomega-square testrobustness 2010 Mathematics Subject Classification: Primary 62G10; secondary 62M1062G3062G35 * We consider a stationary linear AR(p) model with zero mean. The autoregression parameters as well as the distribution function (d.f.) G(x) of innovations are unknown.We consider two situations.In the first situation the observations are a sample from a stationary solution of AR(p). Interesting and essential problem is to test symmetry of G(x) with respect to zero.If hypothesis of symmetry is valid then it is possible to construct nonparametric estimators of AR(p) parameters, for example, GM-estimators, minimum distance estimators and others.First of all we estimate unknown parameters of autoregression and find residuals.Based on them we construct a kind of empirical d.f., which is a counterpart of empirical d.f of the unobservable innovations. Our test statistic is the functional of omega-square type from this residual empirical d.f. Its asymptotic d.f. under the hypothesis and the local alternatives are found.In the second situation the observations subject to gross errors (outliers). The distribution of outliers is unknown, their intensity is O(n −1/2 ), n is the sample size. We test the symmetry of innovations again but by constructing the Pearson's type statistic.Its asymptotic d.f. under the hypothesis and the local alternatives are found. We establish the asymptotic robustness of this test as well. Рассмотрим линейную AR(p) модель u t = β 1 u t−1 + · · · + β p u t−p + ε t , t ∈ Z. (1.1) В (1.1) инновации {ε t } -независимые одинаково распределенные случайные величины (н.о.р.сл.в.) с неизвестной функцией распределения (ф.р.) G(x); Eε 1 = 0, 0 < Eε 2 1 < ∞; β = (β 1 , . . . , β p ) T -вектор неизвестных параметров, для которых соответствующее характеристическое уравнение имеет корни, лежащие в единичном круге; размерность модели p предполагается известной. Эти требования считаются выполненными всегда и далее особо не оговариваются. Мы будем рассматривать две ситуации. В первой наблюдения u 1−p , u 2−p , . . . , u n суть выборка из стационарного решения (1.1). По этим наблюдениям хотим проверить гипотезу S : ε 1 d = −ε 1 , т.е. G(x) симметрична относительно нуля. ( 1.2) Для проверки S мы построим тест типа омега-квадрат со статистикойω 2 n , найдем асимптотическое при n → ∞ распределениеω 2 n при гипотезе и локальных альтернативах. Чтобы описать эти альтернативы, будем предполагать, что ф.р. G(x) зависит от числа наблюдений n и представляется смесью симметричной относительно нуля ф.р. P (x) и несимметричной ф.р. Q(x) : G(x) = A n (x) := (1 − ρ n )P (x) + ρ n Q(x). (1.3) В (1.3) ф.р. P (x) и Q(x) неизвестны, ρ n = min {1, ρ n −1/2 }, ρ ≥ 0, ρ неизвестно. Если выполнено (1.3), будем говорить, что верна гипотеза A n (ρ). При ρ > 0 A n (ρ) будем понимать как локальную альтернативу к S, ф.р. A n (x) при ρ > 0 несимметрич- на. При ρ = 0, разумеется, A n (0) и S совпадают. В в этом случае (т.е. при гтпотезе S) будем писать P (x) вместо G(x) . Асимптотические распределенияω 2 n будут найдены при A n (ρ) сразу для всех ρ ≥ 0, что позволяет изучить тест одновременно и при гипотезе, и при локальных альтернативах. Отметим, что симметрия инноваций -одно из основных предположений, позволяющих строить непараметрические ценки для β. Например, GM−оценки, оценки минимального расстояния, знаковые и т.д., см. [1]. Кроме того, для симметричной G(x) можно строить симметризованные оценки G(x) и основывать на них симметризованные тесты для гипотез о виде G(x), см. [2]. Строить такие тесты, как и тесты для S, нетривиальная и содержательная задача, поскольку инновации {ε t } ненаблюдаемы. Соответствующие первой ситуации результаты представлены в Разделе 2. Во второй ситуации наблюдения за авторегрессией содержат грубые ошибки. А именно, наблюдаются y t = u t + z γn t ξ t , t = 1 − p, Проверка симметриии в схеме без засорений Пусть наблюдения u 1−p , . . . , u n -выборка из стационарного решения уравнения (1.1). Построим по этим наблюдениям тест для проверки гипотезы S из (1.2). Пустьβ n = (β 1n , . . . ,β pn ) T -оценка вектора β, требования к ней уточним далее. Оценками ненаблюдаемых инноваций ε 1 , . . . , ε n возьмем остатки ε t := u t −β 1n u t−1 − · · · −β pn u t−p , t = 1, . . . , n; оценкой ф.р. G(x) -остаточную эмпирическую ф.р. G n (x) = n −1 n t=1 I(ε t ≤ x), x ∈ R 1 , I(·) − −индикатор события. Тестоввой статистикой для S возьмем ω 2 n = n ∞ −∞ [Ĝ n (x) +Ĝ n (−x) − 1] 2 dĜ n (x). Вычислятьω 2 n удобно по формуле ω 2 n = n t=1 Ĝ n (−ε (t) ) − n − t + 1 n 2 , гдеε (1) ≤ · · · ≤ε (n) -упорядоченные остатки. Статистикаω 2 n -аналог статистики ω 2 n для проверки симметрии, см. [4], которую можно было бы построить по самим ε 1 , . . . , ε n . А именно, ω 2 n = n ∞ −∞ [G n (x) + G n (−x) − 1] 2 dG n (x), где G n (x) = n −1 n t=1 I(ε t ≤ x) -эмпирическая ф.р. ε 1 , . . . , ε n . Пусть v(t), t ∈ [0, 1], -броуновский мост, т.е. гауссовский процесс с нулевым средним и ковариацией min {t, s} − ts. Для A n (ρ) из (1.3) положим δ(t) := ρ [Q (P −1 (t) ) − t], где P −1 (·) -обратная к P (·) функция. Из результатов [4] следует: при гипотезе S для непрерывной G(x) ω 2 n d − − → 1 0 [v(t) + v(1 − t)] 2 dt, n → ∞, а при альтернативе A n (ρ) с ρ > 0 для непрерывных A n (x) и δ(t) ω 2 n d − − → 1 0 [v(t) + v(1 − t) + δ(t) + δ(1 − t)] 2 dt, n → ∞. Нам понадобятся следующие условия. Условие (i). Ф.р. P (x) и Q(x) имеют нулевые средние и конечные дисперсии. Условие (ii). Ф.р. P (x) и Q(x) дважды дифференцируемы с ограниченными вто- рыми производными. Условие (iii).β n -такая оценка β, что для фиксированного ρ ≥ 0 при A n (ρ) n 1/2 (β n − β) = O P (1), n → ∞. (2.1) Если выполнено Условие (i), то оценка наименьших квадратовβ n,LS , построенная по u 1−p , . . . , u n , удовлетворяет (2.1), см. [5]. Более того, n 1 2 (β n,LS − β) d − − → N (0, E(ε 0 1 ) 2 K −1 ), n → ∞. (2.2) В (2.2) E(ε 0 1 ) 2 -дисперсия ф.р. P (x), K = (k ij ), k ij = E(u 0 0 u 0 i−j ), i, j = 1, . . . , p, {u 0 t } -стационарное решение (1.1) с инновациями {ε 0 t }, имеющими ф.р. P (x). Далее нам понадобится результат из [5]: если выполнены Условия (i) -(iii), то при A n (ρ) с любым ρ ≥ 0 sup x∈R 1 |n 1/2 [Ĝ n (x) − G n (x)]| = o P (1), n → ∞. (2.3) В силу (2.3) асимптотическое распределениеω 2 n при A n (ρ), ρ ≥ 0, совпадает с асимптотическим распределением ω 2 n , т.е. верна Теорема 2.1. 1 0 . Пусть верна гипотеза S. Пусть для ф.р. P (x) выполнены Условия (i) -(ii) и Условие (iii) c ρ = 0 дляβ n . Тогда ω 2 n d − − → 1 0 [v(t) + v(1 − t)] 2 dt, n → ∞. 2 0 . Пусть при некотором ρ > 0 верна альтернатива A n (ρ). Пусть выполнены Условия (i) -(iii), и фукция δ(t), t ∈ [0, 1], непрерывна. Тогда ω 2 n d − − → 1 0 [v(t) + v(1 − t) + δ(t) + δ(1 − t)] 2 dt, n → ∞. Для 0 < α < 1 критическое множество возьмем в виде ω 2 n > c 1−α ,(2. 4) где c 1−α -(1 − α)-квантиль предельной ф.р. ω 2 n . Эта предельная ф.р. табулирована в [6]. Асимптотический уровень теста (2.4) равен α. Асимптотическое распределениеD n при A n (ρ), ρ ≥ 0, такое же, как у статистики D n = sup x∈R 1 |n 1 2 [G n (x) + G n (−x) − 1]|. Пусть наблюдаются y 1−p , . . . , y n , определенные в (1.4). Построим по этим наблюдениям тест для проверки гипотезы S из (1.2). Прежде всего построим по {y t } оценкуβ y n = (β y 1n , . . . ,β y pn ) T вектора β. Далее R ≥ 0, Γ ≥ 0 -произвольные конечные числа. Условие (iv).β y n -такая оценка β, для которой при A n (ρ) равномерно по ρ ≤ R, γ ≤ Γ выполнено n 1/2 (β y n − β) = O P (1), n → ∞. (3.1) Если Eε 4 1 < ∞, то соотношение (3.1) выполнено, например, для о.н.к.β y n,LS , поскольку n 1/2 (β y n,LS − β) d − − → N (−γ K −1 β, E(ε 0 1 ) 2 K −1 ), n → ∞, (3.2) и сходимость по распределению равномерная по ρ ≤ R, γ ≤ Γ. Найдем остаткиε y t = y t −β y 1n y t−1 − · · · −β y pn y t−p , t = 1, . . . , n, и построим по ним остаточную эмпирическую ф.р.Ĝ y n (x) = n −1 n t=1 I(ε y t ≤ x). Справедливо следующее утверждение ( [7], Теорема 2.1, Следствие 2.1): Если выполнены Условия (i), (ii), (iv), то при A n (ρ), ρ ≥ 0, для фиксированного x ∈ R 1 n 1/2 [Ĝ y n (x) − G n (x) ] − γ∆(x, Π) = o P (1), n → ∞, (3.3) равномерно по ρ ≤ R, γ ≤ Γ. В (3.3) ∆(x, Π) = p j=0 [EP (x + β j ξ 1 ) − P (x)], β 0 = −1. Разложение (3.3) дляĜ y n справеливо только при каждом фиксированном x, потому его недостаточно для исследования основанных наĜ y n (x) статистик типаω 2 n и D n из Раздела 2. Однако достаточно для построения и исследования статистики типа хи-квадрат для проверки S. Вот как она строится. Введем полуинтервалы B + j = (x j−1 , x j ], j = 1, . . . , m, m ≥ 1, 0 = x 0 < x 1 < · · · < x m = ∞. Пусть {x j } таковы, что p + j := P (x j ) − P (x j−1 ) > 0. Если ввести симметрич- ные полуинтервалы B − j = (−x j , −x j−1 ], то при гипотезе S для непрерывной P (x) p − j := P (−x j−1 ) − P (−x j ) = p + j , j = 1, . . . , m. Пустьν + j обозначает число остатков средиε y 1 , . . . ,ε y n , попавших в B + j , аν − jчисло остатков, попавших в B − j . Интересующая нас тестовая статистика для S имеет видχ 2 n = m j=1 (ν + j −ν − j ) 2 2ν + j Очевидно,ν + j n =Ĝ y n (x j ) −Ĝ y n (x j−1 ),ν − j n =Ĝ y n (−x j−1 ) −Ĝ n (−x j ). (3.4) Пусть ν ± j обозначает число инноваций ε 1 , . . . , ε n , попавших в B ± j . Поскольку χ 2 n = m j=1 {n 1/2 (ν + j n −ν − n )} 2 /(2p + j ),p + j =ν + j n , получаем в силу (3.3) и (3.4) : χ 2 n = m j=1 {n 1/2 ( ν + j n − ν − n ) + δ j (Π)} 2 /(2p + j ) + o P (1), n → ∞. (3.5) δ j (Π) := ∆(x j , Π) − ∆(x j−1 , Π) − [∆(−x j−1 , Π) − ∆(−x j , Π)]. В (3.5) o P (1) обозначает величину, стремящуюся по вероятности к нулю при n → ∞ равномерно по γ ≤ Γ, ρ ≤ R. Соотношение (3.5) сводит асимптотическое исследование статистикиχ 2 n к анализу главного члена в правой части (3.5). Результат дается Теоремой 3.1. В ней q + j = Q(x j ) − Q(x j−1 ), q − j = Q(−x j−1 ) − Q(−x j ), q ± = (q ± 1 , . . . , q ± m ) T , δ(Π) = (δ 1 (Π), . . . , δ m (Π)) T , P-диагональная матрица, P = diag{2p + 1 , . . . , 2p + m }. Через F k (x, λ 2 ) мы обозначаем ф.р. нецентрального хи-квадрат распределения с k степенями свободы и параметром нецентральности λ 2 , а через | · | -евклидову норму вектора. |P (χ 2 n ≤ x) − F m (x, λ 2 )| → 0, n → ∞, (3.6) где λ 2 = γ 2 | P − 1 2 δ(Π) | 2 ; 2 0 .Пусть верна альтернатива A n (ρ), ρ > 0. Пусть выполнены Условия (i) - (ii), (iv). Тогда sup x∈R 1 , γ≤Γ, ρ≤R |P (χ 2 n ≤ x) − F m (x, λ 2 )| → 0, n → ∞, (3.7) где λ 2 = |P −1/2 [ρ(q + − q − ) + γδ(Π)]| 2 . (3.8) Критическое множество для S возьмем в виде χ 2 n > χ 1−α (m), (3.9) где для 0 < α < 1 χ 1−α (m) -(1 − α)-квантиль ф.р. хи-квадрат с m степенями свободы. В силу (3.9) и Теоремы 3.1 асимптотическая мощность нашего теста есть Соотношение (3.11) качественно означает, что при малых γ равномерно по Π (а также по ρ и Q) асимптотические мощности в схемах с засорениями и без засорений близки. W (ρ, γ, Π, P, Q) = 1 − F m (χ 1−α (m), λ 2 ), где λ 2 = λ 2 (ρ, γ, Π, P, Q) задается формулой (3.8). Известно, что |F k (x, λ 2 1 ) − F k (x, λ 2 2 )| ≤ 2 π |λ 1 − λ 2 |,(3. Это свойство означает асимптотическую качественную робастность теста. Сделаем несколько замечаний. Замечание 3.3. Если вместо Условия (iv) потребовать лишь, чтобы выполнялось (3.1) при фиксированных ρ и γ, то утверждение типа Теоремы 3.1 останется верным. Надо только в (3.6) и (3.7) заменить супремум по x ∈ R 1 , ρ ≤ R, γ ≤ Γ на супремум по x ∈ R 1 . Например, о.н.к.β y n,LS удовлетворяет (3.2) (а значит, и (3.1)) при фиксированных ρ и γ при минимальном условии Eξ 2 1 < ∞. Замечание 2 . 1 . 21При фиксированной альтернативе к гипотезе S ф.р. инноваций несимметрична относительно нуля. Пусть она, кроме того, имеет нулевое среднее, конечную дисперсию и ограниченную вторую производную. Пусть n 1/2 (β n −β) = O P (1), n → ∞.-Тогда статистикаω 2 n расходится по вероятности к бесконечности. Это означает, что тест (2.4) состоятелен против такой фиксированной альтернативы. Замечание 2 . 2 . 22Помимоω 2 n можно строить аналоги и других статистик, упомянутых в[4]. Например,D n = sup x∈R 1 |n 1/2 [Ĝ n (x) −Ĝ n (−x) − 1]|. Теорема 3.1. 1 0 . Пусть верна гипотеза S. Пусть для ф.р. P (x) выполнены Условия (i) -(ii) и Условие (iv) c ρ = 0 дляβ y n . Тогда sup x∈R 1 , γ≤Γ 10), так что из (3.8), (3.10) и определения вектора δ(Π) получаем: sup Π,ρ,Q |W (ρ, γ, Π, P, Q) − W (ρ, 0, Π, P, Q)| ≤ 2 π γ sup Π |P −1/2 δ(Π)| → 0, γ → 0. (3.11) Замечание 3. 1 . 1Если γ = ρ = 0, то асимптотическое распределение дляχ 2 n есть обычное (центральное) распределение хи-квадрат с m степенями свободы,W (0, 0, Π, P, Q) = α.Если же распределение Π симметрично, т.е. ξ 1 d = −ξ 1 , то δ(Π) = 0, и асимптотическая мощность W (ρ, γ, Π, P, Q) вовсе от Π не зависит и совпадает с асимптотической мощностью теста в случае схемы без засорений: для ρ ≥ 0 при симметричном распределении Π и любых γ ≥ 0, ρ ≥ 0 W (ρ, γ, Π, P, Q) = W (ρ, 0, Π, P, Q). Задачей ставится опять проверка гипотезы S из (1.2) по наблюдениям {y t }. Мы построим тест типа хи-квадрат для S, изучим асимптотическое поведение тестовой статистики при A n (ρ) из (1.3), установим асимптотическую качественную робастность теста. Эти результаты приведем в Разделе 3.. . . , n. (1.4) В (1.4) {z γn t } -н.о.р. сл.в., распределенные по закону Бернулли, z γn 1 ∼ Br(γ n ), γ n = min{1, γn −1/2 }, γ ≥ 0, γ неизвестно; {ξ t } -н.о.р. сл.в. с неизвестным и про- извольным распределением Π; {u t } -выборка из стационарного решения (1.1); по- следовательности {u t }, {z γn t }, {ξ n } независимы. Последовательность {ξ t } интерпрети- руется как последовательность грубых ошибок (засорений). Схема (1.4) -локальный вариант известной схемы засорений для временных рядов, см. [3]. Weighted Empiricals and Linear Models. H L Koul, HaywardMichigan State UniversityKoul H.L. Weighted Empiricals and Linear Models. Michigan State University, Hayward, 1992. On Symmetrized Pearson's Type Test in Autoregression with Outliers: Robust Testing of Normality. M V Boldin, Boldin M.V. On Symmetrized Pearson's Type Test in Autoregression with Outliers: Robust Testing of Normality., https://arxiv.org/abs/2003.07878. Influence Functionals for Time Series. R D Martin, V J Yohai, Ann. Statist. 14Martin R.D., Yohai V.J. Influence Functionals for Time Series. Ann. Statist., 1986, Vol. 14, p. 781-818. Теория вероятностей и её применения, 1972, т. XVII, в.2, с. А И Орлов, Орлов А.И. О проверке симметриии распределения. Теория вероятностей и её применения, 1972, т. XVII, в.2, с. 372 -377. On the Asymptotic Power of Fit under Local Alternatives in Autoregression. M V Boldin, Math. Methods of Statist. 282Boldin M.V. On the Asymptotic Power of Fit under Local Alternatives in Autoregression, Math. Methods of Statist., 2019, Vol. 28(2), p. 144 -154. Критерии омега-квадарат. Москва, "Наука. Г В Мартынов, Мартынов Г.В. Критерии омега-квадарат. Москва, "Наука", 1978. On the Power of Pearson's Test under local Alternatives in Autoregression with Outliers. M V Boldin, Math. Methods of Statist. 281Boldin M.V. On the Power of Pearson's Test under local Alternatives in Autoregression with Outliers. Math. Methods of Statist., 2019, Vol. 28(1), p. 57-65.

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{'abstract': "We consider a stationary linear AR(p) model with zero mean. The autoregression parameters as well as the distribution function (d.f.) G(x) of innovations are unknown.We consider two situations.In the first situation the observations are a sample from a stationary solution of AR(p). Interesting and essential problem is to test symmetry of G(x) with respect to zero.If hypothesis of symmetry is valid then it is possible to construct nonparametric estimators of AR(p) parameters, for example, GM-estimators, minimum distance estimators and others.First of all we estimate unknown parameters of autoregression and find residuals.Based on them we construct a kind of empirical d.f., which is a counterpart of empirical d.f of the unobservable innovations. Our test statistic is the functional of omega-square type from this residual empirical d.f. Its asymptotic d.f. under the hypothesis and the local alternatives are found.In the second situation the observations subject to gross errors (outliers). The distribution of outliers is unknown, their intensity is O(n −1/2 ), n is the sample size. We test the symmetry of innovations again but by constructing the Pearson's type statistic.Its asymptotic d.f. under the hypothesis and the local alternatives are found. We establish the asymptotic robustness of this test as well.", 'arxivid': '2207.04315', 'author': ['M V Boldin boldin−m@hotmail.com \nDept. of Mech. and Math\nMoscow State Lomonosov Univ\nMoscowRussia\n', 'A R Shabakaeva shabakaevalmira@gmail.com \nDept. of Mech. and Math\nMoscow State Lomonosov Univ\nMoscowRussia\n'], 'authoraffiliation': ['Dept. of Mech. and Math\nMoscow State Lomonosov Univ\nMoscowRussia', 'Dept. of Mech. and Math\nMoscow State Lomonosov Univ\nMoscowRussia'], 'corpusid': 250426661, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7519, 'n_tokens_neox': 7766, 'n_words': 2857, 'pdfsha': 'db020739e6a3ba746c2d40c748c123863e64148e', 'pdfurls': ['https://arxiv.org/pdf/2207.04315v1.pdf'], 'title': ['Testing of symmetry of innovations in autoregression', 'Testing of symmetry of innovations in autoregression'], 'venue': []}

arxiv

Faster OreFSDet : A Lightweight and Effective Few-shot Object Detector for Ore Images ⋆ Yang Zhang School of Mechanical Engineering Hubei University of Technology 430068WuhanChina Hubei Key Laboratory of Modern Manufacturing Quality Engineering Hubei University of Technology 430068WuhanChina National Key Laboratory for Novel Software Technology Nanjing University 210023NanjingChina Le Cheng School of Mechanical Engineering Hubei University of Technology 430068WuhanChina Hubei Key Laboratory of Modern Manufacturing Quality Engineering Hubei University of Technology 430068WuhanChina Yuting Peng School of Mechanical Engineering Hubei University of Technology 430068WuhanChina Hubei Key Laboratory of Modern Manufacturing Quality Engineering Hubei University of Technology 430068WuhanChina Chengming Xu School of Data Science Fudan University 200433ShanghaiChina Yanwei Fu School of Data Science Fudan University 200433ShanghaiChina Bo Wu Shanghai Advanced Research Institute Chinese Academy of Sciences 201210ShanghaiChina Guodong Sun School of Mechanical Engineering Hubei University of Technology 430068WuhanChina Hubei Key Laboratory of Modern Manufacturing Quality Engineering Hubei University of Technology 430068WuhanChina Faster OreFSDet : A Lightweight and Effective Few-shot Object Detector for Ore Images ⋆ A R T I C L E I N F OOre images Few-shot object detection Real-time Light-weight A B S T R A C T For the ore particle size detection, obtaining a sizable amount of high-quality ore labeled data is time-consuming and expensive. General object detection methods often suffer from severe over-fitting with scarce labeled data. Despite their ability to eliminate over-fitting, existing few-shot object detectors encounter drawbacks such as slow detection speed and high memory requirements, making them difficult to implement in a real-world deployment scenario. To this end, we propose a lightweight and effective few-shot detector to achieve competitive performance with general object detection with only a few samples for ore images. First, the proposed support feature mining block characterizes the importance of location information in support features. Next, the relationship guidance block makes full use of support features to guide the generation of accurate candidate proposals. Finally, the dual-scale semantic aggregation module retrieves detailed features at different resolutions to contribute with the prediction process. Experimental results show that our method consistently exceeds the few-shot detectors with an excellent performance gap on all metrics. Moreover, our method achieves the smallest model size of 19MB as well as being competitive at 50 FPS detection speed compared with general object detectors. The source code is available at https://github.com/MVME-HBUT/Faster-OreFSDet. ⋆ Corresponding author. ORCID(s): Introduction The particle size of the ore is an important data index to determine the crushing effect of the ore. Accurate and efficient detection of ore particle size is the basis of ore crushing optimization, which has a direct impact on the productivity of the entire beneficiation process. Complex beneficiation site environments, dense adhesion, and stacking have posed great difficulties for ore particle size detection. In addition, some ores are reflected in the mineral processing workshop after using light, which makes it more difficult to distinguish the ore from the background. Some scholars have proposed some traditional techniques [25] for detecting ores particle sizes. To achieve good performance, these approaches require laborious parameter adjustment processes. With the development of convolutional neural networks (CNN), there have been some significant advancements in object detection. However, general object detectors require a large amount of box labels to train, and obtaining such high-quality ore labeling data is expensive and time-consuming. As labeled data become scarcer, the CNNs are easily overfitting and fail to be generalized. Therefore, object detectors have trouble detecting real-world scenarios involving novel objects that are absent from common datasets for object detection. On the basis of few-shot learning, some approaches [5,16,45] have developed insightful ideas for addressing data scarcity. Few-shot object detection (FSOD) is the combination of traditional object detection and few-shot learning, which aims to predict and locate the object under a few annotated training samples. As a result, it lessens the workload associated with labeling substantial volumes of data in the target domain. However, the existing FSOD methods are mainly based on the traditional Faster RCNN [28], as shown in Fig. 1. This two-stage detector includes a slow and independent region proposal generation step. In addition, to reduce the loss of accuracy caused by the lack of training data, a large number of complex modules are designed to establish the correlation between support and query, which Figure 1: Faster R-CNN is usually served as the basic detector of FSOD. This anchor-based few-shot object detector uses RPN to maximize the recall of the top 1K proposals and does not use these proposal scores in the test phase. A large number of proposals slows the speed. In addition, a large number of complex modules are designed to establish the correlation between support and query, which leads to slow detection speed and high memory requirements. It is extremely challenging for scenario deployments with limited computing resources and tight memory budgets. leads to slow detection speed and high memory usage. It is extremely challenging for scenario deployments with limited computing resources and tight memory budgets. To solve these problems, a well-known detector CenterNet2 [44] is served as the basic detector for the FSOD task. The CenterNet2 employs CenterNet with the ability to create accurate probability in the first stage, which is more accurate than the two-stage detector. Additionally, it enables the detector to employ fewer proposals (256 vs. 1K) in the region of interest (RoI) head, enhancing the overall accuracy and speed. Furthermore, we design a support-feature mining block (SM Block) and relationship guidance block (RG Block) to fully establish the relationship between support and query features. Specifically, adhesion, occlusion, and variations in ore appearance are particularly common, which present great difficulty in establishing high-quality support features. If sufficient discriminant information is not provided, the model can hardly learn the crucial features for class and bounding box predictions. The SM Block is first suggested to encode feature representations along the height and width dimensions using linear projection. The suggested SM block has the ability to assess the significance of the feature data provided by the ore images and remove detection interference brought on by the addition of background noise. Next, we establish the spatial and channel correlation between support and query in the RG Block, which significantly improves the guidance performance of the query branch. Finally, we propose a dual-scale semantic aggregation (DSA) module that retrieves detailed features at different resolutions for final classification and bounding box regression. Extensive experiments show the effectiveness of our method in comparison with state-of-the-art detectors. Our main contributions can be summarized as follows: (1) A real-time few-shot object detector is designed for ore particle detection, which can alleviate the over-fitting issue when dealing with limited labeled data and significantly improve the performance of the FSOD task for the ore images. (2) We propose the SM Block to characterize the importance of semantic information in support features, and the RG Block to better establish the correlation between support and query features for guiding the generation of precise candidate proposals. (3) The proposed DSA module is designed to retrieve detailed features at different resolutions for final classification and bounding box regression. The remainder of this paper is structured as follows. In Section II, we provide a brief introduction to the ore image processing, general object detection, and FSOD methods. Section III presents the problem definition and four components. Section IV details experimental procedures and results over the ore dataset. Finally, the conclusion is drawn in Section V. RELATED WORK Ore Image Processing The particle size analysis task is usually aimed at the ores on the belt and can be divided into three modes: particle size statistics, particle size classification, and large block detection. Particle size statistics refers to the determined value of ore size in an image that is obtained. Generally, the semantic segmentation network is used to segment each object in the image, and then the number of pixels of each object is obtained by OpenCV and other toolkits. According to the relationship between the unit pixel and the actual size, the area of each ore is obtained. Finally, the corresponding ore particle size statistics are completed according to the actual needs, such as the particle size (the equivalent circle diameter corresponding to the area ) of each ore in the image. Some researchers have proposed some solutions and achieved good results. The primary measures are regressionbased classifiers and techniques based on certain theories [25]. Watershed transform processes [34] were introduced in region-based segmentation techniques for ore particle sizes. However, it is difficult to adapt these methods to different situations since they require a time-consuming parameter change procedure to achieve satisfactory performance. The development of CNN-based image classification has led to significant advancements in downstream fields such as object detection and semantic segmentation. For example, Liu et al. [21] used U-Net to detect ore particle size, and Li et al. [13] also proposed a U-Net-based model that alleviated ore particle size detection issues by improving the loss function and utilizing the watershed technique. Liu et al. [22] used morphological transformation to process the mineral image mask and segment the key areas in the mineral image. Sun et al. [32] proposed a novel efficient and lightweight method for ore particle size detection. Particle size classification refers to the classification of particle size grades by considering all ores in the image as a whole. In general, the ore datasets of different particle sizes are constructed first, and then the image classification network is trained. Finally, the trained model is used to classify the unknown ore images. Olivier et al. [26] used VGG16 network [30] to classify 10 particle size grades of ore images, which provided guidance for subsequent mine production operation control. Large block detection refers to the identification of oversized ores on the belt. The object detection network is first used to obtain the coordinate information of the ore, then the external rectangular area of each ore is calculated. Finally, it will compare with the set threshold to determine whether there is a large block on the belt. At present, there are few related researches, and this paper is a study of the third detection task. General Object Detection Object detection is to identify and categorize a variety of targets in the images and then determine the categories of numerous objects as well as their locations. Object detection has generally been the most challenging subject in the field of computer vision since different things might have a wide range of appearances, shapes, and attitudes. The deep learning framework may be utilized with one-stage and two-stage mainstream object detectors. The former, such as the well-known Faster R-CNN [28], first created category-unknown region proposals in the form of RPN, and then projected the proposals onto the feature maps following the RoI pooling. Finally, proposal features were fed into the fully connected layer for classification and regression to determine class labels and fine-tune bounding boxes. Grid RCNN [24] introduced a grid-guided localization mechanism for accurate object detection. Cascade RCNN [1] used cascade regression as a resampling mechanism to improve the performance of the probe by increasing the intersection over union (IoU) of the proposal stage by stage. You only look once (YOLO) series [2,27] and single shot multi-box detector (SSD) [20] were examples of one-stage detectors that provided a non-region proposal framework for class and bounding box prediction. On the other hand, there are two categories of existing object detection: anchor-based and anchor-free. Faster R-CNN [28] was the one that initially put out the idea of anchors. A proper initialization for RPN allows it to avoid using an unnecessary amount of search space and produce better region suggestions since each anchor represents a predetermined box with a certain aspect ratio and size. Many one-stage detectors also employ anchors to raise the quality of their proposals. However, anchors add a lot of hyper-parameters and the imbalance between positive and negative proposals is exacerbated since most anchors do not include targets. Numerous anchor-free techniques were then proposed, such as FCOS [33], which directly predicted using the center-based technique and positive and negative samples are defined using various methods. In addition, there are some methods designed to solve special problems in object detection. RetinaNet [17] proposed focal loss to alleviate the problem of foreground-background class imbalance. Based on focal Loss, VarifocalNet [43] used Varifocal Loss to predict each image for dense object detection. Li et al. [14] proposed a generalized focal loss via joint quality estimation and classification. Shuang et al. [29] proposed a loss function to alleviate a matching imbalance due to different scales of objects. CenterNet2 [44] replaced RPN with CenterNet with the ability to generate accurate likelihood in the first phase, making it more accurate and faster. According to the peculiarities of the parallel implementation of classification and localization in conventional one-stage object detection, Probabilistic anchor assignment (PAA) [11] proposed a new anchor assignment strategy, which adaptively assigns labels to anchors in the form of probability according to the training state of the model. Few-shot Object Detection With only a few training samples, the FSOD task attempts to tackle the object detection issue. There are two main categories: transfer learning-based and meta-learning-based methods. For transfer learning-based methods [39], the target domain model was first initialized using the model parameters from the source domain model on large-scale datasets, and then fine-tuned on small-scale datasets. There are two main methods of meta-learning-based approaches [37]: learn to fine-tune and learn to measure. The former is to learn category-agnostic parameters for new categories and specific weights on new tasks. Two-stage fine-tuning approach (TFA) [36] first trained the entire detector on a data-rich base class, and then fine-tuned the last layer of the detector on a small balanced training set consisting of base classes and new classes while freezing other parameters of the model. Sun et al. [31] provided a comparative learning method into the two-stage fine-tuning method to reduce the intra-class differences and increased the inter-class differences. In contrast, learning to measure requires the feature fusion of the query set and the support set to complete an example search in a constrained number of support sets. However, how these features are incorporated, where they are integrated, and what training strategies are employed vary depending on the model. Features in two-stage methods such as Meta RCNN [41] and FsDetView [40] are fused after the RPN. Similarly, the feature fusion of Meta YOLO [9] was directly performed before the detection head. AttentionRPN [6] provided a feature fusion module to exclude proposals by category information. In addition, there are some other methods to deal with the special problems in this field. Wu et al. [39] proposed a multi-scale positive sample refinement (MPSR) method to enrich the object scale in FSOD. A dual attention strategy was introduced by dual-awareness attention for few-shot object detection (DAnA) [3] to address the issues of spatial information loss brought on by global pooling and spatial imbalance brought on by convolutional attention. Spatial reasoning is introduced into few-shot object detection to detect new categories in [10], which is a novel approach in this field. In our framework, we use the AttentionRPN [6] as the baseline and further improve the performance on ore images. Fine-tuning FSOD method is adopted with CenterNet2 [44] as the basic detection framework. Different from the previous works, we develop a lightweight and effective few-shot object detector on CenterNet2 [44]. Compared to the two-stage anchor-based detector, our method uses CenterNet with the ability to generate accurate likelihood, which is more accurate and allows the detector to use fewer proposals (256 vs 1K) on the RoI head. In the second stage, we adopt a uniquely designed lightweight detection head for ore images, making our detector more accurate and faster overall. In addition, we design lightweight and effective few-shot strategies to generate higher quality supports and establish the effective correlation between support and query features for more precise guidance. METHODOLOGY In this section, we first introduce the motivation overview. Subsequently, we introduce SM block to characterize the importance of semantic information in support features and RG Block to better establish the correlation between support and query features for guiding the generation of precise candidate proposals. Finally, a dual-scale semantic aggregation module is designed to retrieve detailed features at different resolutions for final classification and bounding box regression. Motivation Overview It is a challenging task to realize the mobile deployment of an ore detection model with limited training data in a complex detection environment. A lack of training data will cause general object detectors to be overfitting. Despite The overall framework of our proposed method. The training input of each episode consists of a query image and several support images from one class. The shared feature extractor and feature pyramid networks (FPN) first extract the query and support features. The P3, P4, and P5 of support features from FPN are fed into the SM Block for effective feature representation and then fed into the RG Block to generate attention maps with the same level query features from FPN. These attention feature maps are sent to the one-stage detector CenterNet. After filtering out the negative objects that do not belong to the support category, accurate candidate proposals are generated. Subsequently, the candidate proposals and the support features are sent to the detection head, where a more accurate parsing is conducted between the support box and the potential box in the query feature maps. alleviating the overfitting issue, FSOD methods usually suffer some difficulties caused by the poor performance of speed and accuracy, and excessive model size, especially when used to embedded mobile devices. Specifically, there are two main reasons for the poor performance of the existing FSOD methods: (i) There is just one class of ore in the ore particle size detection, only a straightforward classification of the foreground and background is required. However, the previous works generally employed the two-stage Faster RCNN [28] as the detection framework. Faster RCNN generated a series of anchor boxes at each anchor according to certain rules, and then adjusted proposals (RoIs) at the second stage for anchors combined with neural network output bias and a series of selection rules. When the existing methods are directly migrated to the ore particle size detection, this detector exhibits slow speed and redundant classification full connection layer. In addition, there are a large number of complex modules to decrease the accuracy loss caused by limited training data, which resulted in slow detection speeds and enormous memory requirements. (ii) The occlusion and adhesion of ore appearance cause incomplete expression of characteristics. Therefore, when establishing the correlation between support and query features for guidance, the support feature information of the ore is particularly important. Under few-shot task setting, however, the ore class prototype is derived from the characteristics of the global average pooled support feature maps, which results in the loss of particular local contexts. The correlation between ore image support and query features cannot thus be clearly established. As shown in Fig. 2, we present a lightweight and faster few-shot detector on CenterNet2 [44]. The proposed SM Block characterizes the importance of location information carried by support features, and RG Block makes full use of support features to guide the generation of accurate candidate proposals. The DSA retrieves detailed features at different resolutions to contribute with the prediction process. A Faster and Lighter Detection Framework Most FSOD methods are built on a two-stage detector Faster RCNN [28]. All two-stage detectors used a weak RPN to maximize the recall of the first 1K proposals and did not utilize these proposal scores during the test phase. A large number of proposals slows the speed, and the proposal network based on recall considerations does not directly provide a clear probability explanation like the one-stage detector. Additionally, the last fully connected layer used by the original Faster RCNN [28] takes up a large portion of the parameters. All RoIs after RoI pooling will go through this full connection and are calculated separately without shared computing. To obtain a lighter and stronger detection framework, our method is built on the two-stage detector CenterNet2 [44], as shown in Fig. 3. Compared to the two-stage anchor-based detector, CenterNet2 [44] uses the CenterNet with the ability to generate accurate likelihood in the first stage, which is more accurate and allows the detector to use fewer proposals (256 vs 1K) on the RoI head, making our detector more accurate and faster overall. In addition, CenterNet2 [12] is served as backbone. To extract and classify region-level characteristics in stage one, CenterNet2 [44] employs the CenterNet with the ability to create precise probability. In stage two, we remove the extra two heads and shrink the number of head channels to a smaller 128. To increase the log-likelihood of GT targets, these two stages are trained concurrently. The final log-likelihood is used by our detector as the detection score in inference. [44] employs a complex cascade RCNN architecture in the second stage, which we customize to achieve a lightweight design due to the single class characteristics of ore images. Support Feature Mining Block Feature information Mining. The quality of supports is crucial in determining how to guide the query branch. In previous work, the supports from the backbone were frequently used directly, which introduced distracting background noise. To this end, we propose a simple and data-efficient SM Block that characterizes the importance of the location information carried by supports. Figure 5: The structure of RG Block. Support is pooled into two different sizes of kernels with ore prototype spatial information, which are then convoluted channel by channel on the query map. The output two feature maps are then superimposed on the original query map to get the final attention map with spatial scale correlation. Finally, the query map and attention map are concatenated along the channel to establish the feature channel correlation. In Fig. 4, our module consists of two branches that are responsible for encoding information along the height and width dimensions. When encoding spatial information along the height dimension, the height channel permutation operation is performed first. Given ∈ × × , to satisfy = * , we first divide it into parts along the channel dimension, and then perform a high-channel permutation operation to get [ 1 , 2 ⋯ ]. Next, the height information is encoded through a fully connected layer, followed by a height channel permutation operation to restore the dimension information to . Similarly, these operations are performed in the width direction. Finally, the weighted sum of two branches ( = 2) is carried out, which is described as follows: Multilayer Perceptron (MLP) for feature mapping of the two summed branches = ((( + ) ) ), ∈ ℝ ×̂ , ∈ ℝ̂ × .(1)ℎ and → ∈ ℝ × .(2) Weighted summation̂ = ∑ =1 [ , ∶] ⊙ [ , ∶].(3) The SM block can capture long-range dependencies along one spatial direction while retaining accurate location information in the other direction. The output features of the location-aware obtained in this way are aggregated in a complementary way to form an effective representation of the target of interest. Relationship Guidance Block After obtaining a high-quality support class prototype, an effective relationship between query and support is crucial to the performance of the model. Previous work AttentionRPN [6] performed a global average pooling operation on support and used it as a convolution kernel (1 × 1) to slide over the query feature map to obtain an attention map with support spatial information. However, when employing the same technique on ore images, this global average pooling operation results in the loss of support and only concentrates on spatial information, whereas channel information relating to categories is not correlated. Therefore, we suggest a RG Block to fully build an effective relationship between support and query as indicated in Fig. 5. Spatial scale correlation: the category of the target is closely related to the appearance, which is determined by the feature's spatial dimension. Consequently, the spatial correlation between the two features might substantially indicate how similar they are to one another. To retain more support spatial information for query context guidance, we pool supports into 1x1, and 3x3 sizes, and perform parallel convolution operations on the query. The 3x3 convolution is divided into 1x3 and 3x1 deep strip convolutions to further decrease the computational cost while facilitating the extraction of banded ore characteristics. As follows, we specify the spatial scale correlation  s  s ( , ) = ,ℎ, = ∑ ∑ , , ⋅ ,ℎ+ −1, + −1 ,(4) where represent the generated attention feature maps. Each support feature map of ∈ × × ( denotes the kernel size after pooling) is served as a convolution kernel for convolution operations on the corresponding query feature map of in depth cross-correlation. After spatial scale correlation, we superimpose two feature maps with different spatial information of supports onto the original query feature maps to get the final feature maps. Feature channel correlation: prior research has demonstrated that the category information of images is frequently present in the feature channel. Along the distribution of the channel, the deep features of the same category are similar. We use the following criteria to define the similarity  c  c ( , ) = Conv(Cat( , )), where Cat represents two features concatenated along the channel, and the interaction between the channels is modeled using a normal 1 × 1 convolution. Dual-scale Semantic Aggregation Module After one stage, the RoI align module performs feature extraction for final class prediction and bounding box regression. Based on previous experience, implementation with a fixed resolution of 8 may cause information loss during training. An abundance of training data can make up for this information loss in general object detection, while the issue gets severe for few-shot object detection with few shots. Therefore, we propose a dual-scale semantic aggregation module as shown in Fig. 6. Empirically speaking, small resolution tends to focus on large target information, while larger resolution tends to focus on smaller object information. Since the ore image sizes are only medium and large, we choose 4 and 8 resolutions and perform parallel pooling. In addition, to further guide more accurate classification and bounding box regression using support information in the second stage, we establish a global matching relationship between the support feature map and the query feature map  DSA ( , ) = Conv3(Cat( , )) + Cat(Conv1( ), Conv2( )). Finally, we aggregate two different resolution feature maps to obtain a more comprehensive feature representation for final classification and bounding box regression. EXPERIMENTS In this section, we first introduce the implementation details, datasets, and evaluation metrics. Then, we conduct ablation for a lightweight few-shot detector and evaluate the effectiveness of SM Block, RG block, and DSA by comprehensive experiments. Finally, we compare our proposed OreFSDet with the state-of-the-art methods on the ore dataset. Experiments Setup Implementation Details The CenterNet2 [44] serves as the detection framework of our network throughout the experiments. During the training process, for the constructed probabilistic interpretation of two-stage detection, the loss calculation in CenterNet is used in the first stage, and the loss in Cascade RCNN is employed in the second stage. Finally, the above two losses are added to obtain the total loss. We use a fine-tuning method to achieve detection under a few-shot scenario. The 80 categories in Microsoft COCO 2017 [18] are employed as base classes for base training. And in place of the fullyconnected layer, a newly created layer with a random initialization value is created for new class ore. A significant aspect of our learning process is freezing the backbone and incorporating the newly designed network model in the second fine-tuning phase. For fine-tuning on the ore image dataset, the number of iterations is set to 20000 and the learning rate is 0.001. The batch size is set to 1 with a single NVIDIA GTX2080Ti GPU. The scaled query images have a short side of 320 pixels and a long side of fewer than 1000 pixels. Support images are 240 × 240 pixels with zero-padded. Datasets The MS COCO [18] is set up for base training and ore image dataset for fine-tuning training. The MS COCO is a large-scale dataset for image detection, semantic segmentation, and other vision tasks. It includes 80 target categories, 1.5 million targets, and more than 330K photos. Figure 7 shows how the experiment platform is used to collect ore images of different sizes for the ore image dataset. The ore distribution is then changed on different scales by making it dense, sparse, etc. In order to be used in network training, we also split huge ore images into smaller ones. classified as large targets, and those between the two are medium targets. There are only medium and large ore targets. The frame per second (FPS) is used to indicate how much a model costs to compute. Memory consumption during inference demonstrates the model's dependence on the hardware. The last factor is critical in determining whether or not a model can be implemented on low-cost hardware. It is crucial to consider the last factor when determining whether or not a model can be implemented on low-cost hardware. To further validate the effectiveness, we train 4060 images for general object detection in experiments. In contrast, the shot is 10 for our proposed OreFSDet, and the validation set is the same for both of them at 1060 images. Evaluation Metrics Ablation Study Ablation for a lightweight few-shot detector The three cascaded heads exist in the second stage detection of original CenterNet2 [44]. To find out which heads and corresponding IoU values are crucial for ore detection and the influence of the number of channels on the performance of the model, we conducted comprehensive experiments. It can be concluded from the table that as the number of channels in the head decreases sharply, only slight accuracy damage is caused, which is attributed to the single category detection and simple feature information of the ores. Compared with the single number of head, the multiple number of cascaded detection head generally brings accuracy improvement under different channels, but it is followed by the unbearable model size and detection speed burden. Therefore, we choose channel 128 and cut the three heads into one head, and the corresponding IoU value is 0.6. To further achieve a smaller and speedier model, we chose a lighter backbone, as seen in Table 2. The lightweight VoVNet [12] significantly contributes reduced computational overhead and model size occupancy to our model compared to the backbone ResNet50 [8] and DLA [42]. Compared with the baseline, we are more than 7× smaller in model size and nearly 19 frames faster in inference speed, which is of great significance for our model deployment on the edge. As shown in Table 2, we verify the effectiveness of the three designed modules with lightweight CenterNet2 [44] as the baseline. As a module that processes support information separately, the SM Block needs to be used in conjunction with the other two modules. The combination of RG Block and DSA with SM Block improves the overall performance of the model. Impact of SM We employ three distinct attention mechanisms to efficiently characterize the feature information of support features as given in Table 3. Convolutional attention module (CBAM [38]) that integrates channel and spatial attention processes. Based on the dual attention process, Polarized self-attention [19] suggests a more refined version of polarized self-attention. CoTNet [15] is a transformer-style module that fully exploits the context information contained between input keys to direct the learning of a dynamic attention matrix. In contrast to the approaches mentioned above, our approach utilizes linear projection to encode feature expression along the two dimensions of height and width rather than 2D convolution or an attention mechanism. The results demonstrate that SM Block outperforms other approaches due to its quicker inference speed and better precision. Impact of RG We use four different few-shot strategies to establish the relationship between support and query features. The support feature is encoded into a class attention vector in [7], and then element-by-element multiplication is performed along the channel dimension at each spatial position of the query feature map. AttentionRPN [6] takes support feature map as the convolution kernel, and performs sliding convolution on the query feature to establish the association between the support feature and the query feature. DAnA [3] converts the support feature into a query-position-aware (QPA) feature with specific semantic information, and areas of the query feature map with high QPA response should be identified as targets. When used for dense ore detection, this inappropriate treatment of spatial characteristics results in poor performance, especially speed burdens. Unlike AttentionRPN, our method uses kernels with different sizes of support spatial information, which are fully utilized to obtain spatial correlation. And two features are concatenated in the channel dimension to obtain the feature channel correlation. The results show that our method outperforms the other three methods and is lightweight enough without additional speed and model size burden. Impact of Dual-scale Semantic Aggregation Module The impact of resolution on model performance can be seen in Fig. 8. With the decrease of resolution, the overall of the model does not change much, and the corresponding shows a stable downward trend. The nevertheless maintains a fairly high level with resolution = 3, which indicates that small resolution does tend to large target information. As shown in Table 5, our proposed dual-scale semantic aggregation module takes into account the detection of different scales of ore, and has a significant improvement in performance. Interestingly, when aggregating two smaller resolution feature maps, we achieved a smaller model size while sacrificing only a tiny amount of speed. Visualization To further illustrate the effect of the RG Block, we visualize the features before and after it. As shown in Fig. 9, after passing through the module, the query features get activated to focus on more important features, rather than the previously cluttered state. Our proposed RG Block fully establishes the spatial correlation and channel correlation between support and query features, which significantly improve the guidance performance. Moreover, the visualization of the effect of the DSA module is presented in Fig. 10. Ores of different sizes have different sensitivities to resolution, which leads to different confidence scores. DSA combines two resolutions to take into account the detection of different sizes of ore and achieved better performance. Results demonstrate that DSA can effectively retrieve detailed features at different resolutions to contribute with the prediction process. Finally, we visualize different results of few-shot object detection from ore images. Compared with other detection methods, the detection results obtained by OreFSDet have high confidence and no excessive miss detection. Comparison with State-of-the-Art Methods On the ore dataset, we compare the performance of several FSOD algorithms in terms of , 50 and 75 . TFA [36] trains the detector on MS COCO and then fine-tunes the last layer of the detector on a small balanced ore dataset while freezing other parameters of the model. FSCE [31] provides a comparative learning method into the two-stage fine-tuning method to reduce the intra-class differences and increase the inter-class difference. Support features and query features are fused after the RPN for guiding detection in two-stage methods MetaRCNN [41] and FsDetView [40], while AttentionRPN [7] provided a feature fusion module to exclude proposals by class category before RPN. MPSR [39] presented a multi-scale positive sample refining technique to enrich the object scale. Our proposed OreFSDet is also a fine-tuning method. Different from the above methods, it mines the important features of ore images through SM Block, thereby removing the detection interference caused by the introduction of background noise. Then, the spatial and channel correlation for precise guidance between the support image and query image are fully established through the attention mechanism. Finally, the dual-scale semantic aggregation module retrieves detailed features at different resolutions to contribute with the prediction process. As shown in Table 6, we conducted Comparative experiments for different FSOD algorithms on the ore dataset under different shots. With 25 shots provided for training and 1060 ore images for assessment, our method surpasses the baseline [6] with great advantage by 23.3/31.1/27.7 respectively on / 50 / 75 metrics. Additionally, with fewer shots, the suggested model remains excellent performance, proving that OreFSDet effectively retrieves the information of support images for guidance through SG Block, RG Block, and the dual-scale semantic aggregation module. The comparison of each approach under FSOD and general object detection is shown in Table 7. General object detection methods with sufficient training data tend to achieve better performance than few-shot object detection methods by learning a large amount of information between categories. In particular, many transformer-based methods have recently been proposed to solve various computer vision tasks such as Swin transformers [23]. To obtain a lightweight model and faster detection speed, we develop a few-shot object detector on CenterNet2 [44] instead of traditional Faster RCNN [28]. In Table 7, OreFSDet not only rivals the best detector of general object detection (even better than YOLOF by 5.7/10.4 on / 75 metrics), but also consistently beats the few-shot detectors with a great performance gap on all metrics. In addition, OreFSDet performs best at a model size of 19MB as well as being competitive at 50 FPS among general object detectors. Conclusion In this work, we observe that the FSOD task for ore images is extremely difficult to be utilized in a real-world scenario deployment due to high computational overhead and memory requirements. To this end, we present a lightweight and effective framework OreFSDet to solve the problems. On the ore dataset, our model exceeds the best in the FSOD algorithms with great advantage by 25.5/21 on / metrics, respectively. Moreover, our OreFSDet performs best with the model size of 19MB and is competitive at 50 FPS among general object detectors. Our method is advantageous in a lightweight model and ultra-fast detection speed and can be effectively deployed in mineral processing operations. Furthermore, with only a few samples, our method can be extended to some objects that share similar properties to the ores. For future work, we will focus on extending our method to the more challenging few-shot instance segmentation and one-shot object detection by new mechanisms and networks. CRediT authorship contribution statement Figure 2 : 2Figure 2: The overall framework of our proposed method. The training input of each episode consists of a query image and several support images from one class. The shared feature extractor and feature pyramid networks (FPN) first extract the query and support features. The P3, P4, and P5 of support features from FPN are fed into the SM Block for effective feature representation and then fed into the RG Block to generate attention maps with the same level query features from FPN. These attention feature maps are sent to the one-stage detector CenterNet. After filtering out the negative objects that do not belong to the support category, accurate candidate proposals are generated. Subsequently, the candidate proposals and the support features are sent to the detection head, where a more accurate parsing is conducted between the support box and the potential box in the query feature maps. Figure 3 : 3The structure of CenterNet2. A lightweight VoVNet Figure 4 : 4The structure of SM block. Two branches in SM Block are in charge of encoding information along the height and width dimensions. The dimension information supplied by the two branches is adaptively aggregated using the attention method to acquire the position-sensitive information of the final support. Figure 6 : 6A global matching relationship is established between support features and query features . Then, two different resolution feature maps are aggregated to provide a more comprehensive feature representation for final classification and box regression. A total of eight indicators are utilized to determine whether the proposed method is effective:, 50 , 75 , , , frame per second (FPS), inference memory consumption, and model size. The accuracy evaluation metrics adopt the COCO[18] evaluation metrics commonly used in object detection(top five).(average precision of boxes) represents the average of precision of boxes under different box IOU values. IOU values vary from 0.5 to 0.95, with a calculation of precision of boxes at an interval of 0.05.50 and 75 represent values of precision of boxes with IOU of 0.5 and 0.75, respectively. The , and represent the three categories of small, medium and large. The pixels with an area less than 32 × 32 are classified as small targets, those with an area greater than 96 × 96 are Figure 7 : 7We collect ore images with different densities and build an experimental platform for ore image detection. Figure 8 : 8The performance of different resolutions on ore dataset. Figure 9 : 9The visualization of before and after RG Block. Figure 10 : 10The visualization of the effect of DSA. Figure 11 : 11The different visual results of few-shot object detection from ore images. Table 1 1The influence of the number of heads and corresponding channels in CenterNet2 on the lightweight design of the modelHead_channel Cascade_head FPS Model Size (MB) stage1(0.6) stage2(0.7) stage3(0.8) 53.0 77.6 63.3 43 62.3 1024 54.1 76.3 65.0 40 102.0 53.0 74.2 63.2 33 138.0 52.4 77.6 62.7 42 36.2 512 54.2 76.4 65.4 42 54.3 53.2 74.3 63.6 33 71.3 52.3 77.3 62.6 43 24.7 256 54.7 77.0 65.9 37 33.2 53.5 74.9 64.2 36 41.5 52.5 77.9 63.0 50 19.4 128 54.4 76.8 65.8 40 23.5 52.9 74.7 63.3 33 27.6 Table 2 The impact of backbone and few-shot strategies on the lightweight design of the model Module Framework Backbone FPS Model Size (MB) SM Block RG Block DSA CenterNet2 ResNet50 [8] 52.9 78.4 64.0 38 130.0 CenterNet2 DLA [42] 48.9 65.3 58.2 29 81.4 CenterNet2 VoVNet [12] 51.2 76.1 61.2 51 16.5 CenterNet2 VoVNet [12] 51.4 77.1 61.3 50 19.4 CenterNet2 VoVNet [12] 52.5 77.9 63.0 50 19.4 Table 3 3The effectiveness of support feature mining blockMethod Table 5 5The effectiveness of dual-scale semantic aggregation moduleResolution Table 6 6Comparative experiments for different FSOD algorithms on the ore dataset under different shotsMethod 5-shot 15-shot 25-shot 50 75 50 75 50 75 TFA [36] 28.9 48 33.6 30.5 49 35.6 31.5 49.7 36.5 FSCE [31] 33.5 47.2 40.8 35.3 49 42.8 37.0 50.4 44.5 Meta R-CNN [41] 15.4 34.7 9.4 17.6 37.3 13.0 22.0 39.0 24.1 FSDetView [40] 18.2 36.1 15.4 20.1 38.6 17.7 25.4 41.1 29.8 MPSR [39] 32.4 45.8 39.7 34.2 47.6 41.8 37.0 52.1 44.3 AttentionRPN [6] 25.1 44.0 27.0 29.2 45.9 34.5 30.8 47.3 37.0 OreFSDet(ours) 48.5 74.1 57.6 52.1 77.2 62.5 54.1 78.4 64.7 Table 7 7Comparison with the state-of-the-art methods on ore dataset Few-shot object detection * The methods are implemented under the 25-shot setting.Method Backbone FPS Model Size (MB) Inf.Memory (MB) General object detection Faster R-CNN [28] ResNet50 [8] 42.6 51.7 48.6 11 264 5221 Faster R-CNN [28] ResNet18 [8] 42.6 51.6 48.6 43 93 1277 Faster R-CNN [28] V-19-Slim[12] 40.6 51.3 47.2 50 51 1269 SSD [20] VGG16[30] 41.7 50.8 48.3 77 190 9787 RetinaNet [17] ResNet50 [8] 41.1 57.7 46.9 26 290 1337 RetinaNet [17] SwinT [23] 42.1 56.5 47.7 41 282 1757 RetinaNet [17] PVTv2 [35] 41.6 55.4 47.4 51 130 1587 RetinaNet [17] ResNet18 [8] 41.7 57.3 47.3 50 91 1543 Cascade R-CNN [1] ResNet50 [8] 43.6 51.0 49.6 21 553 1699 YOLOv3 [27] Darknet[27] 37.4 50.7 47.3 48 493 1555 Grid R-CNN [24] ResNet50 [8] 43.8 54.0 50.0 22 515 1713 CenterNet2 [44] DLA [42] 43.6 55.6 47.4 38 357 1339 FCOS [33] ResNet50 [8] 42.8 55.4 48.4 29 256 1303 VarifocalNet [43] ResNet50 [8] 45.0 63.6 48.6 24 261 1479 YOLOF [2] ResNet50 [8] 46.8 67.5 49.3 45 338 1433 DDOD [4] ResNet50 [8] 44.0 53.9 49.4 50 245 1719 GFL [14] ResNet50 [8] 45.8 63.0 49.6 50 246 1709 PAA [11] ResNet50 [8] 46.0 65.2 49.1 22 245 1661 TFA [36] ResNet101 [8] 31.5 49.7 36.5 16 230 2013 Meta R-CNN [41] ResNet101 [8] 22.0 39.0 24.1 28 148 2639 FSDetView [40] ResNet101 [8] 25.4 41.1 29.8 29 157 2665 AttentionRPN [6] ResNet50 [8] 30.8 47.3 37.0 28 211 1667 MPSR [39] ResNet101 [8] 37.1 52.1 44.3 21 462 3821 MPSR [39] V-19-Slim[12] 24.4 46.2 23.9 36 117 1941 FSCE [31] ResNet101 [8] 37.0 50.4 44.5 18 298 2200 FSCE [31] ResNet18 [8] 23.9 54.3 18.4 55 51 1637 FSCE [31] V-19-Slim[12] 28.9 55.2 29.1 43 51 1611 OreFSDet(ours) V-19-Slim[12] 54.1 78.4 64.7 50 19 1059 * : Yang Zhang : ZhangConceptualization, Methodology, Formal analysis, Data curation, Resources, Writing -original draft, Writing -review & editing. 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Yolov3: An incremental improvement. arXiv preprint arXiv:1804.02767 . Faster r-cnn: Towards real-time object detection with region proposal networks. S Ren, K He, R Girshick, J Sun, IEEE Transactions on Pattern Analysis and Machine Intelligence. 39Ren, S., He, K., Girshick, R., Sun, J., 2017. Faster r-cnn: Towards real-time object detection with region proposal networks. IEEE Transactions on Pattern Analysis and Machine Intelligence 39, 1137-1149. Scale-balanced loss for object detection. K Shuang, Z Lyu, J Loo, W Zhang, Pattern Recognition. 117107997Shuang, K., Lyu, Z., Loo, J., Zhang, W., 2021. Scale-balanced loss for object detection. Pattern Recognition 117, 107997. Very deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, International Conference on Learning Representations. Simonyan, K., Zisserman, A., 2015. Very deep convolutional networks for large-scale image recognition, in: International Conference on Learning Representations, pp. 1-14. Fsce: Few-shot object detection via contrastive proposal encoding. B Sun, B Li, S Cai, Y Yuan, C Zhang, IEEE/CVF Conference on Computer Vision and Pattern Recognition. Sun, B., Li, B., Cai, S., Yuan, Y., Zhang, C., 2021. Fsce: Few-shot object detection via contrastive proposal encoding, in: IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7348-7358. Efficient and lightweight framework for real-time ore image segmentation based on deep learning. G Sun, D Huang, L Cheng, J Jia, C Xiong, Y Zhang, 12Sun, G., Huang, D., Cheng, L., Jia, J., Xiong, C., Zhang, Y., 2022. Efficient and lightweight framework for real-time ore image segmentation based on deep learning. Minerals 12, 526-544. Fcos: Fully convolutional one-stage object detection. Z Tian, C Shen, H Chen, T He, IEEE/CVF International Conference on Computer Vision. Tian, Z., Shen, C., Chen, H., He, T., 2019. 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X Wang, T E Huang, T Darrell, J E Gonzalez, F Yu, Proceedings of the 37th International Conference on Machine Learning. the 37th International Conference on Machine LearningWang, X., Huang, T.E., Darrell, T., Gonzalez, J.E., Yu, F., 2020. Frustratingly simple few-shot object detection, in: Proceedings of the 37th International Conference on Machine Learning, pp. 9919-9928. Meta-learning to detect rare objects. Y X Wang, D Ramanan, M Hebert, IEEE/CVF International Conference on Computer Vision. Wang, Y.X., Ramanan, D., Hebert, M., 2019. Meta-learning to detect rare objects, in: IEEE/CVF International Conference on Computer Vision, pp. 9924-9933. Cbam: Convolutional block attention module. S Woo, J Park, J Y Lee, I S Kweon, European Conference on Computer Vision. Woo, S., Park, J., Lee, J.Y., Kweon, I.S., 2018. Cbam: Convolutional block attention module, in: European Conference on Computer Vision, pp. 3-19. Multi-scale positive sample refinement for few-shot object detection. J Wu, S Liu, D Huang, Y Wang, European Conference on Computer Vision. Wu, J., Liu, S., Huang, D., Wang, Y., 2020. Multi-scale positive sample refinement for few-shot object detection, in: European Conference on Computer Vision, pp. 456-472. Few-shot object detection and viewpoint estimation for objects in the wild. Y Xiao, R Marlet, European Conference on Computer Vision. Xiao, Y., Marlet, R., 2020. Few-shot object detection and viewpoint estimation for objects in the wild, in: European Conference on Computer Vision, pp. 192-210. Meta r-cnn: Towards general solver for instance-level low-shot learning. X Yan, Z Chen, A Xu, X Wang, X Liang, L Lin, IEEE/CVF International Conference on Computer Vision. Yan, X., Chen, Z., Xu, A., Wang, X., Liang, X., Lin, L., 2019. Meta r-cnn: Towards general solver for instance-level low-shot learning, in: IEEE/CVF International Conference on Computer Vision, pp. 9576-9585. Deep layer aggregation. F Yu, D Wang, T Darrell, Yu, F., Wang, D., Darrell, T., 2018. Deep layer aggregation, pp. 2403-2412. Varifocalnet: An iou-aware dense object detector. H Zhang, Y Wang, F Dayoub, N Sünderhauf, IEEE/CVF Conference on Computer Vision and Pattern Recognition. Zhang, H., Wang, Y., Dayoub, F., Sünderhauf, N., 2021. Varifocalnet: An iou-aware dense object detector, in: IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8510-8519. . X Zhou, V Koltun, P Krähenbühl, arXiv:2103.07461Probabilistic two-stage detection. arXiv preprintZhou, X., Koltun, V., Krähenbühl, P., 2021. Probabilistic two-stage detection. arXiv preprint arXiv:2103.07461 . Multi-granularity episodic contrastive learning for few-shot learning. P Zhu, Z Zhu, Y Wang, J Zhang, S Zhao, Pattern Recognition. 131108820Zhu, P., Zhu, Z., Wang, Y., Zhang, J., Zhao, S., 2022. Multi-granularity episodic contrastive learning for few-shot learning. Pattern Recognition 131, 108820.

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{'abstract': 'A B S T R A C T For the ore particle size detection, obtaining a sizable amount of high-quality ore labeled data is time-consuming and expensive. General object detection methods often suffer from severe over-fitting with scarce labeled data. Despite their ability to eliminate over-fitting, existing few-shot object detectors encounter drawbacks such as slow detection speed and high memory requirements, making them difficult to implement in a real-world deployment scenario. To this end, we propose a lightweight and effective few-shot detector to achieve competitive performance with general object detection with only a few samples for ore images. First, the proposed support feature mining block characterizes the importance of location information in support features. Next, the relationship guidance block makes full use of support features to guide the generation of accurate candidate proposals. Finally, the dual-scale semantic aggregation module retrieves detailed features at different resolutions to contribute with the prediction process. Experimental results show that our method consistently exceeds the few-shot detectors with an excellent performance gap on all metrics. Moreover, our method achieves the smallest model size of 19MB as well as being competitive at 50 FPS detection speed compared with general object detectors. The source code is available at https://github.com/MVME-HBUT/Faster-OreFSDet. ⋆ Corresponding author. ORCID(s):', 'arxivid': '2305.01183', 'author': ['Yang Zhang \nSchool of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina\n\nHubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina\n\nNational Key Laboratory for Novel Software Technology\nNanjing University\n210023NanjingChina\n', 'Le Cheng \nSchool of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina\n\nHubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina\n', 'Yuting Peng \nSchool of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina\n\nHubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina\n', 'Chengming Xu \nSchool of Data Science\nFudan University\n200433ShanghaiChina\n', 'Yanwei Fu \nSchool of Data Science\nFudan University\n200433ShanghaiChina\n', 'Bo Wu \nShanghai Advanced Research Institute\nChinese Academy of Sciences\n201210ShanghaiChina\n', 'Guodong Sun \nSchool of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina\n\nHubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina\n'], 'authoraffiliation': ['School of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina', 'Hubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina', 'National Key Laboratory for Novel Software Technology\nNanjing University\n210023NanjingChina', 'School of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina', 'Hubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina', 'School of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina', 'Hubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina', 'School of Data Science\nFudan University\n200433ShanghaiChina', 'School of Data Science\nFudan University\n200433ShanghaiChina', 'Shanghai Advanced Research Institute\nChinese Academy of Sciences\n201210ShanghaiChina', 'School of Mechanical Engineering\nHubei University of Technology\n430068WuhanChina', 'Hubei Key Laboratory of Modern Manufacturing Quality Engineering\nHubei University of Technology\n430068WuhanChina'], 'corpusid': 258437294, 'doi': '10.1016/j.patcog.2023.109664', 'github_urls': ['https://github.com/MVME-HBUT/Faster-OreFSDet.'], 'n_tokens_mistral': 17692, 'n_tokens_neox': 15520, 'n_words': 9576, 'pdfsha': 'd4d065c90e4c0852f9d38f5ecb2b132254fbbb74', 'pdfurls': ['https://export.arxiv.org/pdf/2305.01183v1.pdf'], 'title': ['Faster OreFSDet : A Lightweight and Effective Few-shot Object Detector for Ore Images ⋆', 'Faster OreFSDet : A Lightweight and Effective Few-shot Object Detector for Ore Images ⋆'], 'venue': []}

arxiv

An Alternative for Moduli Stabilisation 22 Aug 2006 Carlo Angelantonj Matteo Cardella Dipartimento di Fisica dell' Università di Milano INFN sezione di Milano via Celoria 1620133MilanoItaly Nikos Irges Department of Physics High Energy and Elementary Particle Physics Division University of Crete 71003HeraklionGreece ⋆ Dipartimento di Fisica Teorica Università di Torino INFN Sezione di Torino Via P. Giuria 110125TorinoItaly An Alternative for Moduli Stabilisation 22 Aug 2006arXiv:hep-th/0608022v2 hep-th/0608022 DFTT 16/2006 IFUM 869-FT The one-loop vacuum energy is explicitly computed for a class of perturbative string vacua where supersymmetry is spontaneously broken by a T-duality invariant asymmetric Scherk-Schwarz deformation. The low-lying spectrum is tachyon-free for any value of the compactification radii and thus no Hagedorn-like phase-transition takes place. Indeed, the induced effective potential is free of divergence, and has a global anti de Sitter minimum where geometric moduli are naturally stabilised.July, 2006 * See [6] for an earlier attempt at stabilising the dilaton using perturbative g s and α ′ corrections. The true vacuum state in a quantum field theory is given by the configuration with the lowest vacuum energy. In general, superstrings yield a vanishing vacuum energy for toroidal or orbifold compactifications if supersymmetry is intact. As a consequence, superstring compactifications are usually characterised by a moduli space of supersymmetric vacua. This moduli space is spanned by certain dynamical moduli with vanishing potential, whose undetermined vacuum expectation values fix the shape and size of compact internal spaces, and set the strength of gravitational and gauge interactions. This is clearly an embarrassment for any attempt at phenomenological study in String Theory since experiments set severe bounds on Brans-Dicke-like forces and on time variation of coupling constants. It is then clear that the search for mechanisms for moduli stabilisation in string compactifications is of utmost importance. Recently, considerable progress has been made along this direction by allowing for non-trivial fluxes. In this class of compactifications, the internal manifold is permeated by constant fluxes for the field strengths of some Neveu-Schwarz-Neveu-Schwarz and Ramond-Ramond fields. In this way, a non-trivial potential is generated whose extrema fix the vacuum expectation values of complex structure moduli as well as the dilaton field [1]. Moreover, the inclusion of non-perturbative effects (like gaugino condensation) and/or the emergence of perturbative string-loop and α ′ corrections leads also to the stabilisation of Kähler class moduli [2][3][4][5] * . Despite the indiscussed importance of these results, any attempt to lift the moduli space of supersymmetric vacua via geometric fluxes and non-perturbative effects relies on a low-energy supergravity analysis, while a full-fledged perturbative string description is by definition missing. Alternative approaches to the moduli stabilisation problem involve the introduction of open-string magnetic backgrounds [7][8][9][10][11]. Although a perturbative string description is now in principle available (at least for Abelian backgrounds), no consistent string-theory vacua with stabilised moduli have been constructed so far. An alternative way of stabilising moduli might instead rely on non-supersymmetric string compactifications where corrections to the moduli space of supersymmetric vacua are generated in perturbation theory. This approach has the great advantage of allowing a perturbative string theory description, and can be democratically realised in heterotic strings, type II superstrings and their orientifolds, both on toroidal and orbifold backgrounds. In this case, the first correction to the flat potential for moduli fields is determined by the non-vanishing one-loop vacuum energy. Despite non-supersymmetric strings are typically plagued by the presence of tachyons in the twisted sector, thus inducing a divergent vacuum energy, tachyon-free heterotic models have been constructed in the past and it has also been shown that the associated one-loop vacuum energies are finite and have extrema at symmetry-enhancement points [12,13]. In this letter, we extend this analysis to superstrings with spontaneously broken supersymmetry and show how, in models without tachyons, symmetries of the (deformed) Narain lattice determine the minima of the one-loop cosmological constant. Constructions with a similar target have been carried out recently also in the context of non-critical string theory [14,15]. Among the various mechanisms for breaking supersymmetry, the Scherk-Schwarz deformation provides an elegant realisation based on compactification [16,17]. In the simplest case of circle compactification, it amounts in Field Theory to allowing the higher dimensional fields to be periodic around the circle up to an R-symmetry transformation. The Kaluza-Klein momenta of the various fields are correspondingly shifted proportionally to their R charges, and modular invariance dictates the extension of this mechanism to the full perturbative spectrum in models of oriented closed strings [18][19][20][21]. Actually, it is a known fact that Scherk-Schwarz deformations can be conveniently realised in String Theory as freely acting orbifolds [22]. Typically, non-supersymmetric projections are accompanied by shifts of internal coordinates, so that the moduli describing their size set the supersymmetry-breaking scale. The simplest instance of a Scherk-Schwarz deformation involves the Z 2 orbifold generated by g = (−1) F δ, where (−1) F is the space-time fermion index and δ shifts the compact coordinate y ∈ S 1 (R) by half of the length of the circle, δ : y → y + πR. As usual, the resulting string spectrum is encoded in the one-loop partition function Z = − V 9 2(4π 2 α ′ ) 9/2 F d 2 τ τ 11/2 2 m,n V 8 − S 8 η 8 2 + V 8 + S 8 η 8 2 (−1) m Λ m,n (R) + O 8 − C 8 η 8 2 + O 8 + C 8 η 8 2 (−1) m Λ m,n+ 1 2 (R) ,(1) where V 9 is the (infinite) volume of the non-compact dimensions, (O 8 , V 8 , S 8 , C 8 ) are the characters associated with the SO(8) little group, and Λ m+a,n+b (R) = q α ′ 4 m+a R + (n+b)R α ′ 2q α ′ 4 m+a R − (n+b)R α ′ 2 =: q α ′ 4 p 2m 2 tw = − 1 α ′ + 1 2 m R 2 + 1 2 (n + 1 2 )R α ′ 2 + oscillators(2) and thus the tachyonic ground state is actually massive for large values of R, while it is massless and then really tachyonic for † R ≤ √ 8α ′ . The absence of tachyonic excitations in the large-radius regime suggests that in this range the theory is perturbatively under control. Using standard techniques [23][24][25][26][27][28][29], one can unfold the fundamental domain F = {|τ | ≥ 1 , − 1 2 < τ 1 ≤ 1 2 } into the half-infinite strip S = {− 1 2 < τ 1 ≤ 1 2 , 0 ≤ τ 2 < +∞} to compute the one-loop potential V (R) = − 7936 π 5 945 √ α ′ R 9 − 2 5 √ α ′ R 4 p odd ∞ N=1 d 2 N N 5/2 p 5 K 5 2πp N R 2 /α ′ , where d N counts the degeneracy of states at each mass level, V 8 /η 8 = ∞ N=0 d N q N . The contribution of massless-states reproduces the standard field theory result, while massivestate contributions are exponentially suppressed for large values of R, but yield a divergent contribution in the tachyonic region R ≤ √ 8α ′ . This divergence is a consequence of the exponential growth of string states d N ∼ N −11/4 e +π √ 8N , and is responsible for the well-known first-order Hagedorn phase transition [30], after we reinterpret the compact radius in terms of the finite temperature, β ∼ R. Although this representation of the Scherk-Schwarz deformation has a natural field theory limit (after the compactification radius is properly halved), where the anti-periodic fermions have half-integer Kaluza-Klein excitations and the vacuum energy has the typical R −n behaviour, for n non-compact dimensions, String Theory can afford more possibilities: one can actually deform only the momenta, as in eq. (1), only the windings or both. These † Actually, to compare with the standard Scherk-Schwarz deformation one has to halve the compactification radius, so that bosons and fermions have indeed integer and half-odd-integer KK momenta, and the twisted tachyon appears for R < √ 2α ′ . correspond in turn to the freely acting orbifolds g δ i , where g is any non-supersymmetric generator and the δ i act as δ 1 =      X L → X L + πR 2 X R → X R + πR 2 , δ 2 =      X L → X L + α ′ π 2R X R → X R − α ′ π 2R , and δ 3 =      X L → X L + πR 2 + α ′ π 2R X R → X R + πR 2 − α ′ π 2R on a circle of radius R. Clearly the δ 2 shift does not introduce new physics, since it is directly related to the δ 1 shift via T-duality: R → α ′ /R. As a result, it yields a tachyon-free spectrum in the smallradius regime, the associated effective potential behaves like R n , while a Hagedorn-like transition occurs now at R = α ′ /8. More interesting is instead the δ 3 shift. It preserves T-duality and thus one can expect to have a sensible theory both in the small-radius and in the large-radius regimes ⋆ . In this case, a generic vertex operator V m,n = e ip L X L +ip R X R , with p L,R = m R ± nR α ′ , gets the phase (−1) m+n under the action of δ 3 , and, as a consequence, both momenta and windings are half-odd-integers in the twisted sector. In this sector the mass formulae read m 2 L = − 1 2α ′ + α ′ 4 i m i + 1 2 R i + (n i + 1 2 )R i α ′ 2 + N (X) + N (ψ) , m 2 R = − 1 2α ′ + α ′ 4 i m i + 1 2 R i − (n i + 1 2 )R i α ′ 2 +Ñ (X) +Ñ (ψ) , where we have consider the more general case of a higher-dimensional internal manifold consisting of the product of various circles, each of radius R i . Evidently, level-matching N (X) + N (ψ) −Ñ (X) −Ñ (ψ) + m + 1 2 · n + 1 2 = 0 ⋆ Actually, this asymmetric δ 3 (−1) F freely acting orbifold with diagonal metric is equivalent to a more conventional symmetric Scherk-Schwarz deformation but with a non-vanishing B-field background, as shown in [31]. However, in the following we concentrate on the asymmetric shift since it is clearly T-duality invariant and thus some results are easier to prove. is not satisfied if δ 3 acts on a single coordinate. Actually this deformation is only allowed when acting on coordinates of an even 2d-dimensional torus. In this case, the lightest states have masses α ′ m 2 tw = 1 8 2d i=1 √ α ′ R − R √ α ′ 2 + d − 2 2 , and the twisted spectrum is clearly free of tachyons for d = 2, 3 ‡. Henceforth, one expects to have a finite and well-behaved one-loop result for any values of R. This is in contrast to the previous case, where a divergence induced by the emergence of tachyonic excitations triggers a first-order phase transition. In fact, in the case at hand the partition function schematically reads Z = − V 10−2d 2(4π 2 α ′ ) 5−d F d 2 τ τ 6−d 2 {m 2 } c(m 2 ) q m 2 Lq m 2 R and thus the absence of tachyons prevents from IR divergence while, as usual, modular invariance excludes the dangerous UV region from the integration domain F. To be more precise, the complete partition function now reads Z = − V 10−2d 2(4π 2 α ′ ) 5−d F d 2 τ τ 6−d 2 {m,n} V 8 − S 8 η 8 2 + V 8 + S 8 η 8 2 (−1) (m+n)·ǫ Λ m,n (R i ) + O 8 − C 8 η 8 2 + O 8 + C 8 η 8 2 (−1) d+(m+n)·ǫ Λ m+ 1 2 ,n+ 1 2 (R i ) where ǫ = (1, 1, . . . , 1) is a 2d-dimensional unit vector, and the second line clearly spells-out the "non-canonical" deformation of the Narain lattice in the twisted sector. Also in this case, one can use standard unfolding techniques to convert the integral over the fundamental domain into the integral over the half-infinite strip. Typically, this procedure involves a Poisson re-summation in order to disentangle the contributions to the integral of different orbits of SL(2, Z). Re-summing over windings or momenta clearly spells-out the small or large radius behaviour. For this reason, in the case of δ 1 and δ 2 shifts the very consistency of the perturbative string expansion suggests to re-sum over momenta and windings, respectively, while in the more interesting case of a δ 3 shift either ways are ‡ It is amusing to note similarities with mass-formulae of other tachyon-free non-supersymmetric orientifold models [32,33] meaningful, since the spectrum is free of tachyons and no first-order phase transitions are expected to occur. In particular, from V = − 1 2(4π 2 α ′ ) 5−d F d 2 τ τ 6−d 2 θ 4 2 (0|τ ) η 12 (τ )V large = − i R i 2(4π 2 ) 5−d (α ′ ) 5 F d 2 τ τ 6 2 θ 4 2 η 12 2 {ℓ,n} e iπn·ǫ exp − π 4α ′ τ 2 (2ℓ + 1 + 2nτ ) R 2 , and V small = − i R −1 i 2(4π 2 ) 5−d (α ′ ) 5−2d F d 2 τ τ 6 2 θ 4 2 η 12 2 {m,k} e iπm·ǫ exp − πα ′ 4τ 2 2k + 1 + 2mτ R 2 , respectively, where a 2 = 2d i=1 a † i a i . An element A of Γ 0 [2] acts as left multiplication on the (2 × 2d)-dimensional integral matrices M = 2n 1 . . . 2n 2d 2ℓ 1 + 1 . . . 2ℓ 2d + 1 ,M = 2m 1 . . . 2m 2d 2k 1 + 1 . . . 2k 2d + 1 . As a result, both M andM can be arranged into orbits of Γ 0 [2] and picking-up a single representative for each orbit allows one to disentangle the τ 1 and τ 2 integrals [34]. In particular, one has to distinguish between the degenerate orbit M = 0 . . . 0 2ℓ 1 + 1 . . . 2ℓ 2d + 1 , and the non-degenerate one M = 2n 1 . . . 2n j 0 . . . 0 2ℓ 1 + 1 . . . 2ℓ j + 1 2ℓ j+1 + 1 2ℓ 2d + 1 , with 2n j > 2ℓ j + 1 > 0 , and similarly forM . Let us consider, for instance, the contribution of the degenerate orbit to V large . One finds V (deg) large = − i R i 2(4π 2 ) 5−d (α ′ ) 5 S d 2 τ τ 6 2 ∞ N=0 ∞ N ′ =0 {ℓ} d N d N ′ e 2iπ(N−N ′ )τ 1 × e −2π(N+N ′ )τ 2 − π 4α ′ τ 2 (2ℓ+1)R 2 = − i R i 2(4π 2 ) 5−d (α ′ ) 5 ∞ 0 dτ 2 τ 6 2 ∞ N=0 {ℓ} d 2 N e −4πNτ 2 − π 4α ′ τ 2 (2ℓ+1)R 2 The above expression, however, cannot be computed analytically for arbitrary radius, since for values of the radius of the order of the string scale the N -series is not uniformly convergent and thus one cannot exchange the integration with the summation. However, one can easily extract the large-radius behaviour V (deg) large ∼ − 1 R 10−2d + O(e −R/ √ α ′ ) where, for simplicity we assumed the radii to be equal. The R → ∞ behaviour of the nondegenerate orbit can be also computed to find the exponentially suppressed contributions of massive states. Similarly, had we started from V small the series would have been convergent only for small values of R, and V small ∼ −R 10−2d + O(e − √ α ′ /R ) , for R → 0 , whose behaviour could have been anticipated by T-duality arguments. What about the behaviour of the potential for finite values of R? In the absence of tachyonic excitations no divergences are expected, while from the asymptotic behaviours something special is expected to occur at the self-dual radius. Indeed, the gradient of the potential Clearly, it would be interesting to generalise this construction to more general supersymmetry-breaking set-ups. Extrema of the potential should then occur at points of enhanced symmetry and eventually yield a more complex landscape of (meta-)stable vacua. ∂V ∂R i = πα ′ 2(4π 2 α ′ ) 5−d F d 2 τ τ 5−d 2 θ 4 2 η 12 2 {m,n} (−1) (m+n)·ǫ Λ m,n 2d i=1 − m 2 i R 3 i + n 2 i R i (α ′ ) 2(3) It might also be rewarding to combine this mechanism with brane-supersymmetry-breaking [35,36,37,38] where NS-NS tadpoles are typically uncancelled, thus introducing a new positive source for the potential. Barring the stabilisation of the dilaton, the combination of these two effects might uplift the AdS minimum to a de Sitter metastable vacuum, as in [2]. Of course, a different question is whether the dilaton (which cannot be stabilised by our method) stays small in the AdS minimum, and whether higher-loop corrections might destroy the extrema of V, but a definite answer is out of reach from present technology. the contribution of momentum and winding zero-modes. While the first line clearly spells-out the mass-shift between bosons and fermions induced by the Scherk-Schwarz deformation, the presence of states with reversed GSO projection (the second line) is crucial for modular invariance of Z. These states have masses S + ST ) • F, Poisson re-summations over the 2d momenta or windings yield Figure . .Numerical evaluation of the one-loop potential as a function of the radius. ′ . Hence, V has a global minimum at the self-dual point, where the radii of the internal torus are stabilised. These results can be directly confirmed by a numerical evaluation of the integral as reported in figure in the case of a squared lattice.In the two asymptotic R i → 0 and R i → ∞ regions the vacuum energy vanishes as a result of a full restoration of supersymmetry on a flat Minkowski background. The global minimum, instead, induces a negative vacuum energy, and it is not clear whether supersymmetry is actually restored on this anti-de-Sitter background.In the expressions above we have considered, for simplicity, the case of a squared torus, where the off-diagonal components of the metric and the NS-NS B-field have been set to zero. While our results can be easily generalised to include all geometric moduli, it is not clear, however, whether in this simple toroidal setting twisted tachyonic modes are always absent. Actually, whenever supersymmetry-breaking is induced by the spacetime fermion index it has been shown that different Scherk-Schwarz deformations live in the same moduli space[31]. Hence, we expect that non-vanishing vev's for the compact moduli might lead to unwanted instabilities. Although it is not clear whether an energy barrier separates our minimum from the unstable regions, it is nevertheless possible to concoct models where the "dangerous directions" of the moduli space are simply removed by some orbifold or orientifold projection. In turn, this has the natural consequence of allowing for four-dimensional chirality and non-Abelian gauge interactions, the main ingredients of the Standard Model of Particle Physics. As an example, a (would-be supersymmetric) Z 4 × Z 4 orbifold, with each Z 4 acting on a T 4 ⊂ T 6 , puts the metric in a diagonal form while the world-sheet parity Ω projects away the B-field. As a result, only three radial directions survive, precisely as in our previous discussions. Clearly, one has to be sure that the additional contributions to the vacuum energy do not spoil the structure of the one-loop potential. In fact, both for the orbifold and orientifold actions no additional contributions are generated. On the one hand, in the Z 4 × Z 4 orbifold only the main orbit is present and the projected and/or twisted amplitudes are vanishing identically since the generators preserve, in principle, supersymmetry. On the other hand, the sector with reversed GSO projection is not left-right symmetric, henceforth it cannot contribute to the Klein-bottle, annulus and Möbius strip amplitudes that, in turn, do not yield additional (tree-level and one-loop) contributions if all tadpoles are properly cancelled. C .A. thanks the Physics Department of the University of Crete, the Gelileo Galilei Institute for Theoretical Physics and the Theory Unit at Cern for hospitality during the completion of this work. M.C. thanks the Department of Theoretical Physics of the University of Turin, the Racah Institute at the Hebrew University in Jerusalem, the Theory Unit at CERN, and The Theoretical Physics Department of the Liverpool University for hospitality. N.I. thanks the Department of Theoretical Physics of the University of Turin, the Gelileo Galilei Institute for Theoretical Physics and the Theory Unit at Cern for hospitality. This work was supported in part by the European Community's Human Potential Programme under the contract HPRN-CT-2004-005104 to which both the University of Turin and the University of Milan 1 belong, in part by the European Community's Human Potential Programme under the contract HPRN-CT-2004-512194 to which the University of Crete belongs, and in part by INFN. The work of M.C. is partially supported also by PRIN prot.2005024045-002, and by the Royal Society grant n. RGMFHB. To conclude, it is tempting to speculate about a finite-temperature interpretation of this new Scherk-Schwarz deformation. To start with, let us quickly review what is known for the standard (−1) F δ 1 deformation[39,40]. In Field Theory, the thermal ensemble at temperature T can be conveniently studied by considering the propagation of fields on R d−1 × S 1 , where S 1 has circumference 2πβ. Indeed, at the one-loop level, also the free energy of a thermal gas of superstrings can likewise be computed[41,42] by considering the propagation over R d−1 × S 1 modded-out by (−1) F δ β . Obviously, the superstring spectrum can be obtained from eq. (2) by identifying the temperature T with the inverse radius R. In particular, the most (super)symmetric configuration at infinite R corresponds to the most ordered phase at zero temperature, while in both cases a first-order phase transition occurs at some finite value of R and/or T , as observed by Hagedorn. What could then be the thermodynamics interpretation of the new (−1) F δ 3 deformation we have now employed? One of its main finger-prints is the large-radius-small-radius duality together with the absence of a first-order phase transition. This clearly leads to an analogy with the two-dimensional Ising model, where the Kramers-Wannier duality relates the ferromagnetic phase at low temperature to the paramagnetic phase at high temperature.Moreover, from the expression (3) one can deduce that the second (third) derivative is logarithmically divergent for d = 3 (d = 2), and thus a higher-order phase transition might take place. Again, this is analogous to the two-dimensional Ising model where a secondorder phase transition takes place at the self-dual temperature, which can be smoothly crossed by continuous variations of the temperature. Clearly, it would be interesting to better investigate the thermodynamics interpretation of our model and relate it to the new phase transitions that seem to emerge in (non-)critical strings[43][44][45][46][47][48][49].Seminara, Stephan Stieberger, Tom Taylor and Miguel Vazquez-Mozo for useful discussions. M.C. would like to thank Giovanni Arcioni, Shmuel Elitzur, Federico Elmetti, Alon Faraggi, Johannes Grosse, Andrea Mauri, Bortolo Mognetti, Alessandro Nigro, Alberto Santambrogio, Leonard Susskind and Cristina Timirgaziu for illuminating discussions. Hierarchies from fluxes in string compactifications. S B Giddings, S Kachru, J Polchinski, arXiv:hep-th/0105097Phys. Rev. 66106006S. B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifica- tions, Phys. Rev. D66 (2002) 106006 [arXiv:hep-th/0105097]. De Sitter vacua in string theory. S Kachru, R Kallosh, A Linde, S P Trivedi, arXiv:hep-th/0301240Phys. Rev. 6846005S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, De Sitter vacua in string theory, Phys. Rev. D68 (2003) 046005 [arXiv:hep-th/0301240]. Kaehler corrections for the volume modulus of flux compactifications. G Gersdorff, A Hebecker, arXiv:hep-th/0507131Phys. Lett. 624270G. von Gersdorff and A. 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Threshold corrections to gauge couplings in orbifold compactifications. P Mayr, S Stieberger, arXiv:hep-th/9303017Nucl. Phys. 407725P. Mayr and S. Stieberger, Threshold corrections to gauge couplings in orbifold compactifica- tions, Nucl. Phys. B407 (1993) 725 [arXiv:hep-th/9303017]. Divergences in Kaluza-Klein models and their string regularization. D M Ghilencea, H P Nilles, S Stieberger, arXiv:hep-th/0108183New J. Phys. 415D. M. Ghilencea, H. P. Nilles and S. Stieberger, Divergences in Kaluza-Klein models and their string regularization, New J. Phys. 4 (2002) 15 [arXiv:hep-th/0108183]. On the unfolding of the fundamental region in integrals of modular invariant amplitudes. M Trapletti, arXiv:hep-th/0211281JHEP. 030212M. Trapletti, On the unfolding of the fundamental region in integrals of modular invariant amplitudes, JHEP 0302 (2003) 012 [arXiv:hep-th/0211281]. Statistical thermodynamics of strong interactions at high-energies. R Hagedorn, Nuovo Cim. Suppl. 3147R. 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{'abstract': 'The one-loop vacuum energy is explicitly computed for a class of perturbative string vacua where supersymmetry is spontaneously broken by a T-duality invariant asymmetric Scherk-Schwarz deformation. The low-lying spectrum is tachyon-free for any value of the compactification radii and thus no Hagedorn-like phase-transition takes place. Indeed, the induced effective potential is free of divergence, and has a global anti de Sitter minimum where geometric moduli are naturally stabilised.July, 2006 * See [6] for an earlier attempt at stabilising the dilaton using perturbative g s and α ′ corrections.', 'arxivid': 'hep-th/0608022', 'author': ['Carlo Angelantonj ', "Matteo Cardella \nDipartimento di Fisica dell'\nUniversità di Milano\nINFN sezione di Milano\nvia Celoria 1620133MilanoItaly\n", 'Nikos Irges \nDepartment of Physics\nHigh Energy and Elementary Particle Physics Division\nUniversity of Crete\n71003HeraklionGreece\n', '⋆ ', '\nDipartimento di Fisica Teorica\nUniversità di Torino\nINFN Sezione di Torino\nVia P. Giuria 110125TorinoItaly\n'], 'authoraffiliation': ["Dipartimento di Fisica dell'\nUniversità di Milano\nINFN sezione di Milano\nvia Celoria 1620133MilanoItaly", 'Department of Physics\nHigh Energy and Elementary Particle Physics Division\nUniversity of Crete\n71003HeraklionGreece', 'Dipartimento di Fisica Teorica\nUniversità di Torino\nINFN Sezione di Torino\nVia P. Giuria 110125TorinoItaly'], 'corpusid': 5544723, 'doi': '10.1016/j.physletb.2006.08.072', 'github_urls': [], 'n_tokens_mistral': 12317, 'n_tokens_neox': 10042, 'n_words': 5591, 'pdfsha': '249d3a3286631808a24f39eb8b5c84c48e1fd48f', 'pdfurls': ['https://arxiv.org/pdf/hep-th/0608022v2.pdf'], 'title': ['An Alternative for Moduli Stabilisation', 'An Alternative for Moduli Stabilisation'], 'venue': []}

arxiv

Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: comparison and implementation Moritz Augustin *augustin@ni.tu-berlin.de Department of Software Engineering and Theoretical Computer Science Technische Universität Berlin Germany Bernstein Center for Computational Neuroscience Berlin Germany Josef Ladenbauer josef.ladenbauer@tu-berlin.de Department of Software Engineering and Theoretical Computer Science Technische Universität Berlin Germany Bernstein Center for Computational Neuroscience Berlin Germany Group for Neural Theory Laboratoire de Neurosciences Cognitives Ecole Normale Supérieure ParisFrance Fabian Baumann Department of Software Engineering and Theoretical Computer Science Technische Universität Berlin Germany Bernstein Center for Computational Neuroscience Berlin Germany Klaus Obermayer Department of Software Engineering and Theoretical Computer Science Technische Universität Berlin Germany Bernstein Center for Computational Neuroscience Berlin Germany Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: comparison and implementation S1 Figure: Fast changes of the input varianceFast changes of the input variance. Time series of population spike rate and mean adaptation current from the different models in response to weak mean µ ext = 1.5 mV/ms and time-varying variance σ 2 ext of the input for moderately fast variations τ σ 2 ou = 50 ms (A) and rapid variations τ σ 2 ou = 10 ms (B). The values for the remaining parameters and the visualization style were as inFig. 4Bof the main text which corresponds to A here, except that a different realization of the OU process was used. S1Figure:Fast changes of the input variance (Augustin, Ladenbauer, Baumann and Obermayer 2017, PLOS Comput. Biol.) Figure: Fast changes of the input variance Fast changes of the input variance. Time series of population spike rate and mean adaptation current from the different models in response to weak mean µ ext = 1.5 mV/ms and time-varying variance σ 2 ext of the input for moderately fast variations τ σ 2 ou = 50 ms (A) and rapid variations τ σ 2 ou = 10 ms (B). The values for the remaining parameters and the visualization style were as in Fig. 4B of the main text which corresponds to A here, except that a different realization of the OU process was used. S1 Figure: Fast changes of the input variance (Augustin, Ladenbauer, Baumann and Obermayer 2017, PLOS Comput. Biol.) 1 Department of Software Engineering and Theoretical Computer Science, Technische Universität Berlin, Germany 2 Bernstein Center for Computational Neuroscience Berlin, Germany 3 Group for Neural Theory, Laboratoire de Neurosciences Cognitives, Ecole Normale Supérieure, Paris, France * augustin@ni.tu-berlin.de and josef.ladenbauer@tu-berlin.deS1

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{'abstract': 'S1 Figure: Fast changes of the input varianceFast changes of the input variance. Time series of population spike rate and mean adaptation current from the different models in response to weak mean µ ext = 1.5 mV/ms and time-varying variance σ 2 ext of the input for moderately fast variations τ σ 2 ou = 50 ms (A) and rapid variations τ σ 2 ou = 10 ms (B). The values for the remaining parameters and the visualization style were as inFig. 4Bof the main text which corresponds to A here, except that a different realization of the OU process was used. S1Figure:Fast changes of the input variance (Augustin, Ladenbauer, Baumann and Obermayer 2017, PLOS Comput. Biol.)', 'arxivid': '1611.07999', 'author': ['Moritz Augustin *augustin@ni.tu-berlin.de \nDepartment of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany\n\nBernstein Center for Computational Neuroscience Berlin\nGermany\n', 'Josef Ladenbauer josef.ladenbauer@tu-berlin.de \nDepartment of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany\n\nBernstein Center for Computational Neuroscience Berlin\nGermany\n\nGroup for Neural Theory\nLaboratoire de Neurosciences Cognitives\nEcole Normale Supérieure\nParisFrance\n', 'Fabian Baumann \nDepartment of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany\n\nBernstein Center for Computational Neuroscience Berlin\nGermany\n', 'Klaus Obermayer \nDepartment of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany\n\nBernstein Center for Computational Neuroscience Berlin\nGermany\n'], 'authoraffiliation': ['Department of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany', 'Bernstein Center for Computational Neuroscience Berlin\nGermany', 'Department of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany', 'Bernstein Center for Computational Neuroscience Berlin\nGermany', 'Group for Neural Theory\nLaboratoire de Neurosciences Cognitives\nEcole Normale Supérieure\nParisFrance', 'Department of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany', 'Bernstein Center for Computational Neuroscience Berlin\nGermany', 'Department of Software Engineering and Theoretical Computer Science\nTechnische Universität Berlin\nGermany', 'Bernstein Center for Computational Neuroscience Berlin\nGermany'], 'corpusid': 3731437, 'doi': '10.1371/journal.pcbi.1005545', 'github_urls': [], 'n_tokens_mistral': 780, 'n_tokens_neox': 656, 'n_words': 396, 'pdfsha': '3ba3de8b3162c8f1c5e57042b49fced8cd1ae0f5', 'pdfurls': None, 'title': ['Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: comparison and implementation', 'Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: comparison and implementation'], 'venue': []}

arxiv

Instability of Holographic Superfluids in Optical Lattice Peng Yang School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina Xin Li School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina Department of Physics University of Helsinki P.O. Box 64FI-00014Finland Yu Tian School of Physical Sciences University of Chinese Academy of Sciences 100049BeijingChina Institute of Theoretical Physics Chinese Academy of Sciences 100190BeijingChina Instability of Holographic Superfluids in Optical Lattice The instability of superfluids in optical lattice has been investigated using the holographic model. The static and steady flow solutions are numerically obtained from the static equations of motion and the solutions are described as Bloch waves with different Bloch wave vector k. Based on these Bloch waves, the instability is investigated at two levels. At the linear perturbation level, we show that there is a critical k c above which the superflow is unstable. At the fully nonlinear level, the intermediate state and final state of unstable superflow are identified through numerical simulation of the full equations of motion. The results show that during the time evolution, the unstable superflow will undergo a chaotic state with soliton generation. The system will settle down to a stable state with k < k c eventually, with a smaller current and a larger condensate. * yangpeng18@mails.ucas.edu.cn † xin.z.li@helsinki.fi ‡ ytian@ucas.ac.cn arXiv:2109.09080v2 [hep-th] 6 Nov 2021 I. INTRODUCTION As an artificial lattice structure, optical lattice, has a close physical resemblance to the periodic coulomb potential that felt by electrons in solid crystal. While it is not like electrons moving between positive ions in a nanometer dimension, optical lattice can be manipulated with a typical dimension much larger than the conventional crystal-micrometer dimension. Due to this virtue, optical lattice has provided a broad platform for experimental and theoretical physicists to explore much richer and more fundamental properties in condensed matter physics, such as instability [1,2], Landau-Zener tunneling [2][3][4][5][6][7], superfluid-Mott insulator quantum phase transition [8][9][10][11][12][13] and so on. Additionally, by controlling the frequency of the lasers in the course of time, the time-dependent optical lattice can be used for driving the system to explore the floquet dynamics [14][15][16][17][18], which are getting more interesting recently. The instability (or stability) of superfluids is always one of the most fundamental and important properties and usually many physical phenomena are related with it. Since the optical lattice starts to be used for studying cold atom physics, many kinds of research works about the instability have been carried out in various respects. There are Landau or dynamical instability [1, 2, [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37], parametric instabilities [38][39][40][41][42], modulational instability [43][44][45][46][47] and low-acceleration instability [48], etc. In this paper we will focus on the Landau and dynamical instability only and simultaneously. Since the first theoretical study about the Landau and dynamical instability for superfluids in optical lattice [1] using the Gross-Pitaevskii (GP) equation, this topic has been studied extensively. [19,20] explore the cases both for repulsive and attractive atomic interactions, [29] adds the influence of three-body interactions. [30] shows that the states in excited bands always have Landau instability and [23] investigates the effect of transverse excitations for higher dimensional systems. [24] reveals that the dynamical instability is the origin of the spatial domain formation in spin-1 atomic condensates. All these research works relied on the mean-field theory, i.e., GP equation, while in addition to this equation the Bose-Hubbard model is also used in [31,32] that is beyond the mean-field theory. In the above research works, all the superfluid systems have no dissipation. From the instability diagram ( Fig.5 in [1]) we can see that there are two distinct regimes about the instability of superflow. The first is the small optical lattice height regime and in this regime the dynamical instability regime is so narrow that the Landau instability can be studied solely; the second is the large optical lattice height regime where the dynamical instability regime spreads out and it can be detected in experiments, obviously. Experiment [33] shows that the regime where the dissipative process occurs to break down the superfluid phase agrees with the theoretical prediction about the regime of onset of Landau instability for a superfluid in shallow optical lattice; Experiment [34] finds that for deep optical lattice there will be transition from a superfluid into an insulator when the superfluid is under dynamical instability. After that, experiments [35,36] both show that in the presence of thermal component the dissipative process can be related with the occurrence of Landau instability. Since the original GP equation cannot apply to dissipative processes, until [37] based on the modified GP equation with dissipation, the instability of superfluid in optical lattice with thermal component has not been studied theoretically. Its result dramatically shows that when the thermal component is added the dynamical instability will occur throughout the Landau instability regime predicted from the original GP equation, which means the Landau instability and dynamical instability will always appear at the same time when there exists dissipation. So it is no longer suitable to investigate Landau instability or dynamical instability separately. In this paper, we will use the simplest holographic superfluid model [49,50] to re-explore the Landau and dynamical instability of superfluids in optical lattice simultaneously based on the advantage that such a model contains superfluid and normal fluid (as thermal component with a finite temperature) intrinsically, which introduces the dissipation naturally and consistently [51]. In contrast, the dissipation in the modified GP equation is purely put by hand, without any relation to thermodynamics. In holography, the unstable region can be calculated by quasi-normal mode (QNM) analysis and the results show that there is a critical Bloch wave vector k c above which the Bloch states are unstable due to the Landau instability and dynamical instability. With the help of the calculation of sound speed we confirm that the Landau instability and dynamical instability also appear at the same time in the presence of optical lattice just like the homogenous case in [52,53]. Beyond the linear perturbation analysis, the unstable superflow are also studied by the numerical simulation. In this part we use the unstable superflow state as the initial state for the evolution equation and evolve it for a long time with some small perturbation given at early time. The unstable modes of perturbation will grow exponentially at first and it will lead the whole system into a chaotic state, in which the solitons will appear and disappear (for higher dimension there will be vortices and the system can be considered as under a transient turbulence state [53]). Finally, along with some dissipative processes the system will reduce its current to become stable with the final wave vector k < k c . These results can be tested in experiments. This paper is organized as follows. In Sec. II we introduce the holographic superfluid model that we use and show how the optical lattice is added. Then in Sec. III by solving the corresponding equations of motion we get the static and steady-flow suferfluid states that expressed as Bloch waves with different wave vector k. The QNM analysis and the calculation of sound speed are in Sec. IV. And in Sec. V we present the dynamic processes of the evolution of an unstable superflow with numerical simulation. Finally, in Sec. VI we give a summary. II. HOLOGRAPHIC MODEL The simplest holographic model to describe superfluids is given in [49], where a complex scalar field Ψ is coupled to a U (1) gauge field A M in the (3 + 1)D gravity with a cosmological constant related to the AdS radius as Λ = −3/L 2 . The action is S = 1 16πG 4 d 4 x √ −g R + 6 L 2 + 1 e 2 L M ,(1) where G 4 is the gravitational constant in four dimensional spacetime. The first two terms in the parenthesis are the gravitational part of the Lagrangian with the Ricci scalar R and the AdS radius L, and the third consists of all matter fields: L M = − 1 4 F µν F µν − |D µ Ψ| 2 − m 2 |Ψ| 2 ,(2) where D µ = ∂ µ Ψ − iA µ Ψ. Since the backreaction of matter fields onto the spacetime geometry is not necessary for our problem, taking the probe limit e → ∞ is a suitable and convenient choice, which means that we fix the background spacetime as the standard Schwarzschild-AdS black brane with metric ds 2 = L 2 z 2 −f (z)dt 2 + 1 f (z) dz 2 + dx 2 + dy 2 , f (z) = 1 − z 3 z 3 H .(3) Here z H is the radius of the black brane horizon. The Hawking temperature of this black brane is T = 3 4πz H , which is also the temperature of the boundary system by the holographic dictionary. Furthermore, due to the existence of the bulk black hole the boundary field theory will intrinsically have dissipation, since there will be energy flow absorbed into the horizon [51,54,55] during dynamic processes. For numerical simplicity, we set L = 1 = z H . The equations of motion for A µ and Ψ are 1 √ −g D µ √ −gD µ Ψ − m 2 Ψ = 0,(4)1 √ −g ∂ µ √ −gF µν = i (Ψ * D ν Ψ − Ψ (D ν Ψ) * ) .(5) One can easily see that there is a trivial solution with Ψ = 0. As we increase the chemical potential µ, there will be a critical chemical potential µ c above which a solution with nonzero Ψ appears, indicating the break of U (1) symmetry and the formation of superfluid condensate. Hereafter, we will focus on the case with µ = 4.5 > µ c , i.e., the superfluid phase of the boundary field theory. A. Equations of motion and optical lattice Since the considered problem is based on putting superflow onto the optical lattice, the density of the superflow will be periodically modulated by the periodic potential (chemical potential plus external potential). For simplicity and without loss of generality, we take the axial gauge A z = 0 as usual and consider our bulk fields as functions of (t, z, x) with the assumption that the direction of the optical lattice is along x. It turns out that A y can be turned off in this case, so the with d = 3 in our case. The holographic dictionary tells us that with one of Ψ ± being the source the other will be the related response, a t is the total potential, ρ the (conserved) particle number density and a x the source of the particle current density j x in the boundary field theory. The optical lattice is then introduced by including a potential V (x) in a t , i.e. a t = µ + V (x). It is convenient to choose m 2 = −2 to make all calculations easier, in which case Ψ ∼ zΨ + + z 2 Ψ − + O z 3 =: zψ,(17) and we choose Ψ + as source and Ψ − as the response. Here we introduce the bulk field ψ for numerical simplicity. (a) amplitude of ψ, k = 0 (b) angle of ψ, k = 0 (c) A t , k = 0 (d) amplitude of ψ, k = 0.75 (e) angle of ψ, k = 0.75 (f) A t , k = III. BLOCH WAVES AND BLOCH BAND IN OPTICAL LATTICE From the Bloch theorem and with the source Ψ + turning off, we can use Bloch waves to describe the static response Ψ − (z, x) = ψ B (z, x) e ikx .(18) Here, ψ B (z, x) is the Bloch wave along the x direction, which has the same period as the optical lattice, i.e. ψ B (z, x + l) = ψ B (z, x) , k is the Bloch wave vector and l the lattice constant (hereafter we choose l = π). Due to the fact that Bloch waves are complex functions, we can separate them into real and imaginary parts: Ψ (z, x) = z [ψ R (z, x) + iψ I (z, x)] e ikx .(19) Substituting the above equation into the static equations of motion (10)-(13), we get the following five differential equations: ∂ z f ∂ z A x + f ∂ 2 z A x = −2 (ψ R ∂ x ψ I − ψ I ∂ x ψ R ) + (k − A x ) ψ 2 R + ψ 2 I ,(20)v < v c . ∂ x ∂ z A x = 2 (ψ R ∂ z ψ I − ψ I ∂ z ψ R ) ,(21)f ∂ 2 z A t + ∂ 2 x A t = 2A t ψ 2 R + ψ 2 I ,(22)1 f A 2 t − z − (A x − k) 2 ψ R + ∂ z (f ∂ z ψ R ) + ∂ 2 x ψ R − 2 (k − A x ) ∂ x ψ I + ψ I ∂ x A x = 0,(23)1 f A 2 t − z − (A x − k) 2 ψ I + ∂ z (f ∂ z ψ I ) + ∂ 2 x ψ I + 2 (k − A x ) ∂ x ψ R − ψ R ∂ x A x = 0.(24) Among them, Eq. V (x) = v cos(2x), i.e. A t | z=0 = 4.5 + v cos(2x), with v the height (or strength) of the optical lattice. We solve these four static equations of motion (20), (22), (23) and (24) numerically with the Newton-Raphson method. Fig.1 shows the numerical solutions of the bulk fields ψ and A t , respectively, with two different Bloch wave vectors. Since ψ is complex, we plot its amplitude and phase angle separately. The inexistence of node of order parameter means that these Bloch waves sit in the lowest Bloch band. In Fig. 2a, we plot the Bloch band in our holographic model. To obtain it, we fix the total particle number, defined as N ≡ π 2 − π 2 dxρ,(25) and solve the static equations (20) and (22)-(24) with k given different values. The loop structure of the blue line (v = 0.5), at which comes from particle interaction energy is larger than the optical lattice strength [56], is witnessed around the edge k = 1 of the Brillouin zone. And for the red line (v = 4) there is no loop structure. Here, the Bloch band also confirms that the static solution in Fig. 1 are Bloch wave solutions. Actually, the higher value of chemical potential µ, the larger density of total particle ρ in one lattice and hence the larger value of particle interaction energy, which means that when we increase the chemical potential the critical value of the optical lattice strength v c that exceeds the particle interaction energy to prevent the loop structure from appearing will also increase. And we confirm this in Fig. 2b. IV. INSTABILITY AND SOUND MODES Now we study the linear instability of the Bloch wave solutions obtained in the previous section via QNM analyses. To calculate the QNM, a better choice is to rewrite the static equations of motions (10)-(13) with the Bloch wave form for ψ in the infalling Eddington-Finkelstein coordinates [57]: 2∂ t ∂ z A x − ∂ z ∂ x A t − ∂ z (f ∂ z A x ) + i (ψ * ∂ x ψ − ψ∂ x ψ * − 2i (A x − k) ψ * ψ) = 0,(26)∂ t ∂ z A t + ∂ t ∂ x A x − ∂ 2 x A t − f ∂ x ∂ z A x + i (ψ * ∂ t ψ − ψ∂ t ψ * ) −if (ψ * ∂ z ψ − ψ∂ z ψ * ) + 2ψ * A t ψ = 0,(27)−∂ 2 z A t + ∂ x ∂ z A x + i (ψ * ∂ z ψ − ψ∂ z ψ * ) = 0,(28)2∂ t ∂ z ψ − f ∂ 2 z ψ − f + 2iA t ∂ z ψ − ∂ 2 x ψ + 2i (A x − k) ∂ x ψ + −i∂ z A t + i∂ x A x + z + (A x − k) 2 ψ = 0,(29) where the metric is Here and in the following sections we will not separate ψ into real and imaginary parts. Similarly, there is one constraint equation, which is chosen to be Eq. (27). Next, we give all the bulk field ds 2 = 1 z 2 −f (z)dt 2 − dtdz + dx 2 + dy 2 .perturbations δA t = e −iωt+iqx a + e iω * t−iqx a * , δA x = e −iωt+iqx b + e iω * t−iqx b * , δψ = e −iωt+iqx u + e iω * t−iqx v * .(30) Substitute these perturbed fields into the equations of motion, we will get the linear perturbation equations ∂ z (∂ x b + iqb) + (iv∂ z ψ + iψ * ∂ z u) − (iu∂ z ψ * + iψ∂ z v) − ∂ 2 z a = 0,(31)2bψψ * + 2 A x − k − 1 2 q ψ * u + 2 A x − k + 1 2 q ψv − 2iω∂ z b+ i (v∂ x ψ + ψ * ∂ x u) − i (u∂ x ψ * + ψ∂ x v) − ∂ z (f ∂ z b) − ∂ z (∂ x a + iqa) = 0,(32)z + (A x − k − q) 2 + i (∂ x A x − ∂ z A t ) u + (2 (A x − k) b + i (∂ x b + iqb − ∂ z a)) ψ− (2iω + f + 2iA t ) ∂ z u + 2i (A x − k − q) ∂ x u − f ∂ 2 z u − ∂ 2 x u − 2ia∂ z ψ + 2ib∂ x ψ = 0,(33)z + (A x − k + q) 2 − i (∂ x A x − ∂ z A t ) v + (2 (A x − k) b − i (∂ x b + iqb − ∂ z a)) ψ * − (2iω + f − 2iA t ) ∂ z v − 2i (A x − k + q) ∂ x v − f ∂ 2 z v − ∂ 2 x v + 2ia∂ z ψ * − 2ib∂ x ψ * = 0.(34) In accordance with the background steady solutions, we impose Dirichlet and periodic boundary The perturbations in Eq.(30) will grow exponentially in the linear regime if there exists an ω whose imaginary part is positive. We use ω M to denote the ω with the maximal imaginary part when q and µ are given, so a system is (linearly) unstable if Im(ω M ) > 0. Fig. 3 shows the result for the distributions of ω with two different values for Bloch wave vector k and q. In Fig. 3a and q M means the Bloch wave vector of the most unstable perturbation mode, which has a maximal imaginary value of ω M , since it will increase with time t with the form e ω I t controlled by the linear perturbation equations in early times (the late time behavior will be more interesting and it will be investigated in the next section.). Fig. 4 shows the value of ω M with respect to q ∈ [0, 4], the periodicity and q M can be seen directly. As we have explained, when q and k are given, ω M can be obtained to determine the stability under the corresponding perturbation. In this spirit, we plot stable and unstable regions as a function of q and k, i.e. the instability diagram, in the half first Brillouin zone in Fig. 5. In this plot, the left region in the figure, which is colored darker grey, is stable (Im(ω M ) ≤ 0), and the right region in the figure, which is colored lighter grey, is unstable (Im(ω M ) > 0). 1 The figure also tells us that in the unstable region the cutoff q increases with k, and when k is large enough (but still in the first Brillouin zone), the system is unstable for all values of q except q = 0. As a signal of onset of the instability mentioned in [52,53], we also plot the dispersion relation of the sound mode in Fig.6. When there is a non-zero k the sound speed becomes direction-dependent. There are two directions corresponding to the maximal and minimal values of the sound speed, which are parallel (q > 0) and anti-parallel (q < 0) to the velocity of the superflow, respectively. Re(ω(q)) and right panel shows Im(ω(q)). A universal phenomenon is that Im(ω(q)) > 0 always accompanies Re(ω(q)) < 0. Hereafter we denote the sound speed as c + for q > 0 and c − for q < 0. The onset of instability is accompanied by the sign-change of c − . Along the negative axis of q, Re(ω) becomes negative as k is larger than 0.655. As Re(ω) corresponds to the energy of the perturbation, c − < 0 (so Re(ω) < 0) indicates the existence of a perturbed state with lower energy and thus the instability of the system (Landau instability). At the same time, we find that Im(ω) > 0, which implies the dynamical instability of the system. So, our result confirms that the Landau instability and dynamical instability occur at the same time, as we have mentioned. For the purpose of opening the next section we give a brief discussion about this section. Since it is not always true for a superflow to flow in the optical lattice stably, when k ≥ k c excitations will always appear to slow the superflow down [53], but what kind of intermediate states will this unstable superflow go through and what kind of final states will it end up with? In the next section we will use full nonlinear evolution to find the final state out for the Bloch wave-type superflow and the intermediate states will also be discussed. V. REAL TIME EVOLUTION FOR STABLE AND UNSTABLE SUPERFLOWS To simulate the evolution of the holographic superfluid, we adopt the same scheme as in [57], i.e. taking Eqs. (26), (28) and (29) A. Evolution of particle current density We introduce perturbations at t = 100, i.e. t 1 = 100, and observe the evolution of the systems with initial Bloch vectors k chosen as 0.3 and 0.75, respectively. Fig. 7 shows the evolution of the particle current density j x , which is defined as j x = lim z→0 √ −gF zx .(35) From Fig. 7, the evolution of j x of an unstable superflow can be divided into five stages. In the first stage t < 100, no change is observed, which confirms that the static solutions obtained in the Schwarzschild coordinates is correctly transformed to that in the Eddington-Finkelstein coordinates. When 100 < t < 200, the current density changes slightly since the influence of the perturbation is small; when 200 < t < 400, the current density becomes chaotic, which comes from the nonlinear development of the instability. When 400 < t < 1500, the chaotic behavior of the current density disappears but the current is still inhomogeneous; In this stage, the system is tending to a steady state. In the last stage t > 1500, the system becomes steady with the final current density j x f = 0.0931468, which is much smaller than the initial value 0.26048. In contrast, for the stable superflow (k = 0.3 as an example for comparison) the situation is totally different. The influence of perturbations on the system is insignificant and the system will evolve to a state that is same as the initial state in a few moments. B. Evolution of condensate and intermediate states As we have seen from the evolution of j x , the intermediate states of the system are chaotic and complicated. However, there are some universal features which can be extracted from the evolution of the condensate |Ψ − | and we plot them in Fig. 8. We plot the evolution of |Ψ − | from different viewing angles, i.e. Fig. 8a, Fig. 8b and Fig. 8c. We can see that the condensate during 100 < t < 400 is also chaotic and the value of |Ψ − | at later times is larger than its initial value. The planform of |Ψ − | is plotted at Fig. 8c and nodes of the order parameter appear during 100 < t < 400, which indicates the formation of solitons. We select one moment (t = 204) of the soliton formation in Fig. 8d. The formation of solitons is natural here [53] since the unstable superflow is thought of as having a higher free energy than a less unstable superflow with soliton excitations. Along with the dissipative processes, the whole system will go to a stable state with a lower free energy. C. Final stable state Since both j x and |Ψ − | become time independent at the end of the evolution, the final state of the system must be able to be described as a Bloch state just like the initial steady flow states that are solved in Sec. III. This is indeed the case. Actually, the final state can also be solved When there is a one-to-one correspondence between j x and the Bloch wave vector k, one can get the additional boundary condition directly by fixing j x = j x f . However, as we can find from Fig. 9a, i.e. the relation between j x and k, a fixed j x normally corresponds to two values of k. (a) |Ψ − |(t, x) (b) |Ψ − |(t, x) (c) |Ψ − |(t, x) (d) t = To make the solution unique, we can take account of the average condensate |Ψ − | in one lattice cell. From Fig. 8a we know that the condensate amplitude |Ψ − | of a steady state is not a constant, so the relation between |Ψ − | and k is not clear, while |Ψ − | is a monotonically decreasing function of k, as plotted in Fig. 9b. Therefore, the boundary condition can be uniquely determined by combining the two considerations: j x = j x f and the value of |Ψ − | . From Fig. 8b and Fig. 7c we conclude that the final Bloch state corresponds to a larger |Ψ − | with j x = 0.0931468, thus we The non-monotonicity for j x gives two states with different k, while the monotonicity for |Ψ − | excludes the larger k state. can solve the equations of motion with the additional boundary condition ∂ z A x | z=0 = 0.0931468, and the solution is marked on Fig. 9b. VI. SUMMARY The dynamical process of superfluids that simultaneously experiences Landau and dynamical instability in optical lattice is investigated using the simplest holographic superfluid model. Due to the existence of dissipation the sound mode of an unstable superflow will always have negative energy and grows up exponentially at early times. We give a universal picture of the instability process for the unstable superflow. From real time evolution the unstable superflow is shown to reduce its particle current density to settle down to a steady state with a current density j x f and a Bloch wave vector k f below the critical value k c , where this k f can be solved from the equations of motion by fixing j x = j x f . In the course of the evolution, a chaotic process involving soliton generation is observed, in which the solitons are supposed to play the role of reducing the wave vector k (similar to the case without an optical lattice [53]). As we have mentioned, when the number of dimensions of the boundary system is greater than one, the process for an unstable state to evolve into a stable one will become much more complicated, which to some extent is described as transient turbulence [53]. So beside the one dimensional boundary superfluid system in optical lattice, it is worthwhile to consider higher dimensional cases. Another direction for future investigation is to include backreaction in this holographic superfluid model, which will enable us to explore the complete interplay between the normal fluid and superfluid components of the superflow at a finite temperature in optical lattice. ACKNOWLEDGMENTS We are grateful to Biao Wu x) ∼ f (t)e ikx with continuous value of k; if there is a discrete space translation symmetry, then we also have F (t, x) ∼ f (t)e ikx but with discrete value of k = 2π L n, where n is integer and L is lattice constant. But in our case, we do not use the above separation though there is a discrete space translation symmetry but use Bloch wave to describe the x direction of the fields, i.e. F (t, x) = f (t, x)e iqx . Here F(t,x) and f(t,x) can be complex functions and we can do some deformations, F (t, x) = f (t, x)e iqx = 1 2 (f (t, x)e iqx + f * (t, x)e −iqx ) + 1 2 (f (t, x)e iqx − f * (t, x)e −iqx ) = 1 2 (f (t, x) + f * (t, x)e −2iqx )e iqx + 1 2 (f (t, x)e 2iqx − f * (t, x))e −iqx = g(t, x)e iqx + h(t, x) * e −iqx .(A1) With this deformation we can write (30) and get (31)-(34) straightforward. Equations (31)- (34) can be written as a matrix form, i.e., R         a b u v         = ωA         a b u v         ,(A2) with appropriate boundary conditions given in Section IV. These function is indeed the general eigenvalue function and there are many methods in textbook to solve this function to get the eigenvalue ω. Before we calculate the value of ω we can know that if ω R is real part of one of the eigenvalue ω, then there must be another value of ω whose real part equals to -ω R . To show this we can define a matrix L, which is defined as L = R − ωA,(A3) then Eq.(A2) can be written as L         a b u v         = 0.(A4) Separate ω into real part and imaginary pary and we get that the imaginary part of L only contains ω R , which means L I (ω R )         a b u v         = 0.(A5) From Equations (31)- (34) it is easy to get that when q = 0 the following relation exists, i.e., L I (ω R )         a b u v         = −L I (−ω R )         a b v u         .(A6) Since the boundary conditions for u and v are the same, changing the places between u and v do not affect the eigenvalues. While when q = 0, there do not have the relation (A6). 0. 75 FIG. 1 : 751The amplitude and phase angle of the bulk field ψ are plotted in subfigures 1a, 1b, 1d and 1e; The bulk field A t is plotted in subfigures 1c and 1f. All the fields are calculated at k = 0 and 0.75. FIG. 2 : 2Fig. 2ashows the lowest Bloch band µ(k) as a function of Bloch wave vector k with fixed total particle number N , whose value is given by the case of chemical potential µ = 4.5, lattice strength v = 0.5 (blue), 4 (red) and Bloch wave vector k = 0.Fig. 2bshows the relation between critical optical lattice strength v c and chemical potential µ. The Bloch band will have loop structure when (21) will be satisfied automatically if the other four equations are solved. To obtain the static or steady-flow states of the superfluid, we impose the source free boundary conditions for Ψ + and A x at the conformal boundary, regular conditions for Ψ − and A t = 0 at the horizon and periodic boundary conditions for A t,x and Ψ in the x direction. And the optical lattice structure is imposed by choosing a) QNM with k = 0, q = 0 (b) QNM with k = 0, q = 0.15 (c) QNM with k = 0.75, q = 0 (d) QNM with k = 0.75, q = 0.15 FIG. 3: 3a and 3b show the QNM corresponding to ψ's Bloch wave vector k = 0 and wave vectors of the perturbed field q = 0 (left) and q = 0.15 (right), respectively; 3c and 3d show the QNM corresponding to ψ's Bloch wave vector k = 0.75 and wave vectors of the perturbed field q = 0 (left) and q = 0.15 (right), respectively. Actually, the QNM is symmetric under Re(ω)→ −Re(ω) when q=0 and we confirm this in Appendix A. FIG. 4 : 4Im(ω M ) as a function of q when Bloch wave vector k = 0.7, 0.75. conditions for a, b, u and v at z = 0 and in the x direction, respectively, while regular boundary conditions are chosen at the horizon z = 1. Then, we can obtain ω from the perturbation equations by solving generalized eigenvalue problems if k and q are given. (See Appendix A for more details about this procedure.) Fig. 3b , 3bIm(ω M ) ≤ 0, so the k = 0 solution is stable under perturbations q = 0 and q = 0.15. By comparison, the solution at k = 0.75 is unstable because Im(ω M ) > 0 in Fig. 3d, despite the fact that in Fig. 3c Im(ω M ) ≤ 0. A system is stable only when Im(ω M ) ≤ 0 for all values of q. Actually we find that there are two special values k c and q M for the Bloch waves. k c means the critical wave vector exceeding which the Bloch wave vector k will lead to an unstable state; FIG. 5: Instability diagram for Bloch waves with k ∈ [0, 1] and q ∈ [0, 1]. In the darker grey region (the left part in the figure), Im(ω M ) ≤ 0, and in the lighter grey region (the right part in the figure), Im(ω M ) > 0. FIG. 6 : 6Dispersion relations for sound modes at k = 0, 0.65, 0.6525, 0.655, 0.75. Left panel shows FIG. 7 : 7as evolution equations while choosing Eq.(27) as the constraint. Since Eq.(28) does not have time derivative terms but have space derivatives up to the second order, it can be solved with one boundary condition from A t (z b ) = 4.5 + 0.5 cos(2x) and another from the restriction of Eq.(27) on the conformal boundary. In the spatial directions the pseudospectral method is chosen for discretization, while in the time direction we adopt the fourth order (a) j x (t, x) Evolutions for superflows with k = 0.3 (stable) and k = 0.75 (unstable), respectively. These panels show the particle current density with t ∈ [0, 2000].Runge-Kutta method. When doing the time evolution, we choose static solutions, including ψ, A t and A x which are solved from the static equations of motion, as the initial states (at t = 0), and at an early time t 1 we add perturbations, which are some random functions with small amplitudes, to ψ. Here, we require that the perturbations should not break the source free boundary condition of ψ. The perturbation is not necessarily periodic and there are many lattice cells in experiments, so it is necessary to include more lattice cells into the time evolution. 2 Fig . 8a is from an ordinary viewing angle, from which we can see the complete picture of |Ψ − (t, x)|;Fig. 8b is plotted with the viewing angle along the x axis. 204 FIG. 8 : 2048Evolution of the condensate |Ψ − (t, x)| with t ∈ [0, 2000] of the unstable superflow k = 0.75. The perturbation is given at t = 100; After that there are some solitons generated, as plotted in Fig. 8d. from the equations of motion, i.e. (20), (22), (23) and (24), with an additional boundary condition besides the boundary conditions given in Sec. III, while not specifying the Bloch wave vector k. FIG. 9 : 9Relations between j x and k as well as |Ψ − | and k. Both figures are obtained by interpolation with discrete value of k, so the value 0.162146, 0.926626 and 2.37978 is not exact. [ 1 ] 1Biao Wu and Qian Niu. 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Rev. A, 87:013633, Jan 2013. Dynamical instability of a bose-einstein condensate with higher-order interactions in an optical potential through a variational approach. E Wamba, S Sabari, K Porsezian, A Mohamadou, T C Kofané, https:/link.aps.org/doi/10.1103/PhysRevE.89.052917Phys. Rev. E. 8952917E. Wamba, S. Sabari, K. Porsezian, A. Mohamadou, and T. C. Kofané. Dynamical instability of a bose-einstein condensate with higher-order interactions in an optical potential through a variational approach. Phys. Rev. E, 89:052917, May 2014. Low-acceleration instability of a bose-einstein condensate in an optical lattice. Yi Zheng, Marijan Kos˘trun, Juha Javanainen, https:/link.aps.org/doi/10.1103/PhysRevLett.93.230401Phys. Rev. Lett. 93230401Yi Zheng, Marijan Kos˘trun, and Juha Javanainen. Low-acceleration instability of a bose-einstein condensate in an optical lattice. Phys. Rev. Lett., 93:230401, Nov 2004. Building a holographic superconductor. Sean A Hartnoll, Christopher P Herzog, Gary T Horowitz, https:/link.aps.org/doi/10.1103/PhysRevLett.101.031601Phys. Rev. Lett. 10131601Sean A. Hartnoll, Christopher P. Herzog, and Gary T. Horowitz. Building a holographic superconductor. Phys. Rev. Lett., 101:031601, Jul 2008. Holographic superconductors. A Sean, Hartnoll, P Christopher, Gary T Herzog, Horowitz, 10.1088/1126-6708/2008/12/015Journal of High Energy Physics. 12Sean A Hartnoll, Christopher P Herzog, and Gary T Horowitz. Holographic superconductors. Journal of High Energy Physics, 2008(12):015-015, dec 2008. Holographic Vortex Liquids and Superfluid Turbulence. Allan Adams, Paul M Chesler, Hong Liu, Science. 341Allan Adams, Paul M. Chesler, and Hong Liu. Holographic Vortex Liquids and Superfluid Turbulence. Science, 341:368-372, 2013. Holographic superfluids and the Landau criterion. Irene Amado, Daniel Areán, Amadeo Jiménez-Alba, Karl Landsteiner, Luis Melgar, Ignacio Salazar Landea, 10.1007/JHEP02(2014)063Journal of High Energy Physics. 63Irene Amado, Daniel Areán, Amadeo Jiménez-Alba, Karl Landsteiner, Luis Melgar, and Ignacio Salazar Landea. Holographic superfluids and the Landau criterion. Journal of High Energy Physics, 2014:63, February 2014. Shanquan Lan, Hong Liu, Yu Tian, Hongbao Zhang, arXiv:2010.06232Landau Instability and soliton formations. arXiv e-prints. Shanquan Lan, Hong Liu, Yu Tian, and Hongbao Zhang. Landau Instability and soliton formations. arXiv e-prints, page arXiv:2010.06232, October 2020. Generation of vortices and stabilization of vortex lattices in holographic superfluids. Xin Li, Yu Tian, Hongbao Zhang, https:/link.springer.com/article/10.1007/JHEP02(2020)10402:104JHEP. Xin Li, Yu Tian, and Hongbao Zhang. Generation of vortices and stabilization of vortex lattices in holographic superfluids. JHEP, 02:104, 2020. Free energy, stability, and dissipation in dynamical holography. Yu Tian, Xiao-Ning Wu, Hongbao Zhang, 122019arXivYu Tian, Xiao-Ning Wu, and Hongbao Zhang. Free energy, stability, and dissipation in dynamical holography. arXiv, 12 2019. Loop-structure stability of a double-welllattice bose-einstein condensate. Hoi-Yin, Ryan Hui, J V Barnett, S. Das Porto, Sarma, https:/journals.aps.org/pra/abstract/10.1103/PhysRevA.86.063636Phys. Rev. A. 8663636Hoi-Yin Hui, Ryan Barnett, J. V. Porto, and S. Das Sarma. Loop-structure stability of a double-well- lattice bose-einstein condensate. Phys. Rev. A, 86:063636, Dec 2012. Holographic boiling and generalized thermodynamic description beyond local equilibrium. Xin Li, Zhang-Yu Nie, Yu Tian, 09:063JHEP. Xin Li, Zhang-Yu Nie, and Yu Tian. Holographic boiling and generalized thermodynamic description beyond local equilibrium. JHEP, 09:063, 2020.

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{'abstract': 'The instability of superfluids in optical lattice has been investigated using the holographic model. The static and steady flow solutions are numerically obtained from the static equations of motion and the solutions are described as Bloch waves with different Bloch wave vector k. Based on these Bloch waves, the instability is investigated at two levels. At the linear perturbation level, we show that there is a critical k c above which the superflow is unstable. At the fully nonlinear level, the intermediate state and final state of unstable superflow are identified through numerical simulation of the full equations of motion. The results show that during the time evolution, the unstable superflow will undergo a chaotic state with soliton generation. The system will settle down to a stable state with k < k c eventually, with a smaller current and a larger condensate. * yangpeng18@mails.ucas.edu.cn † xin.z.li@helsinki.fi ‡ ytian@ucas.ac.cn arXiv:2109.09080v2 [hep-th] 6 Nov 2021', 'arxivid': '2109.09080', 'author': ['Peng Yang \nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n', 'Xin Li \nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nDepartment of Physics\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland\n', 'Yu Tian \nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nInstitute of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n'], 'authoraffiliation': ['School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina', 'School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina', 'Department of Physics\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland', 'School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina', 'Institute of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina'], 'corpusid': 237571895, 'doi': '10.1007/jhep11(2021)190', 'github_urls': [], 'n_tokens_mistral': 19047, 'n_tokens_neox': 15839, 'n_words': 8894, 'pdfsha': 'fe99e8acbf5aa823ef9b247e844593a3a3762c87', 'pdfurls': ['https://arxiv.org/pdf/2109.09080v2.pdf'], 'title': ['Instability of Holographic Superfluids in Optical Lattice', 'Instability of Holographic Superfluids in Optical Lattice'], 'venue': []}

arxiv

MONOIDALITY OF KATO'S REFLECTION FUNCTORS 1 Dec 2017 Peter J Mcnamara MONOIDALITY OF KATO'S REFLECTION FUNCTORS 1 Dec 2017 Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal. Introduction Consider a quantised enveloping algebra U q (g) where g is a simple Lie algebra of ADE type. It admits algebra automorphisms T i for each vertex i of the Dynkin diagram. These automorphisms do not preserve the positive part U q (g) + , instead there are explicitly given subalgebras ker(r i ) and ker( i r) of U q (g) + which are mapped isomorphically onto each other via T i . For a precise statement, see [L2,Proposition 38.1.6]. The positive part of the quantum group U q (g) + was categorified in terms of KLR (Khovanov-Lauda-Rouquier) algebras in [KL1]. The subalgebras ker(r i ) and ker( i r) are categorified by the module categories of certain quotients R(ν)/ e i and R(ν)/ i e of the KLR algebras. These categories are Morita equivalent via functors we call Kato's reflection functors. These were discovered in [K1], and extended to positive characteristic KLR algebras in [Mc2]. In this paper, we show that Kato's reflection functors are monoidal. This answers a question from [KKOP], allowing a proof of [KKOP,Conjecture 5.5]. It also fixes an error in [K1,Lemma 4.2(2)] (in the published version). Our approach is geometric and the key ingredient is the formality of KLR algebras, which we deduce from their purity. Thus we are necessarily restricted to KLR algebras in characteristic zero. We make no attempt to discuss the situation beyond finite type. For this, the reader is encouraged to view [K2]. This paper was produced independently of [K2], which answers the same questions, and the author thanks Kato for providing him with access to a draft of his preprint. Definitions Let I be the index set for a finite type ADE Dynkin diagram. Fix i ∈ I. Let Q be an orientation of the Dynkin diagram such that i is a source. Let Q ′ be the quiver obtained from Q by reversing the directions of all arrows incident to i. For λ ∈ NI, let X λ be the moduli stack of representations of Q of dimension vector λ. Let F λ be the moduli stack of representations of Q of dimension vector λ together with a full flag of subrepresentations. Let π : F λ −→ X λ be the canonical projection. π is proper. Date: June 19, 2018. Let k be a field of characteristic zero and define L λ = π ! k F λ [dim F λ ] The KLR algebra is defined by R(λ) = Hom • D b (X λ ;k) (L λ , L λ ). Let X ′ λ , F ′ λ , L ′ λ and R(λ) ′ be the corresponding objects defined with Q ′ in place of Q. By the main result of [VV] and the discussion in [KL2] there is an isomorphism R(λ) ∼ = R(λ) ′ of graded associative algebras. Normally in the literature, R(λ) is considered as a graded associative algebra. The category D b (X λ ; k) has a differential graded enhancement, so we can consider R(λ) as a differential graded algebra. Theorem 2.2 below shows that we lose no information by considering the associative algebra R(λ) (which we sometimes consider as a differential graded algebra with trivial differential in order to talk about the category R(λ)-dgmod of differential graded modules over R(λ)). Theorem 2.1. [L1,Proposition 10.6] Suppose X λ is defined over a finite field and we take the l-adic derived category. Then L λ is pointwise pure. The following key result is also [W,Lemma 4.8]. Theorem 2.2. R(λ) is formal. Proof. By Theorem 2.1, the cohomology of R(λ) is pure of weight zero, when L λ is spread out to a finite field and the l-adic derived category is considered. Then [PvdB,Theorem A.1.1] shows that this Frobenius action can be lifted to the dg-algebra representing R(λ), and [S,Proposition 4] implies that R(λ) is formal. Theorem 2.3. [K1, Mc1] R(λ) has finite global dimension. Corollary 2.4. There is an equivalence of triangulated categories L λ ∼ = R(λ)-dgmod given by Hom • (L λ , −), where L λ is the full triangulated subcategory of D b (X λ ) generated by L λ . Proof. Since R(λ) is formal, the image of Hom • (L λ , −) lands in R(λ)-dgmod, where R(λ) has trivial differential. A standard devissage argument shows that Hom • (L λ , −) induces an equivalence between L λ and the full subcategory of R(λ)-dgmod generated by R(λ). Since R(λ) has finite global dimension, this latter category is all of R(λ)-dgmod. Let S i be the simple representation of Q at the vertex i. Let U λ ⊂ X λ be the substack of representations M of Q with Hom(S i , M ) = 0. Write j : U λ −→ X λ for the inclusion. Then j is an open immersion. When considering the quiver Q ′ , we instead define U ′ λ ⊂ X ′ λ to be the substack of representations M ′ of Q ′ such that Hom(M ′ , S ′ i ) = 0. The algebra R(ν) has distinguished idempotents e i and i e for each i ∈ I, used to define the relevant categories for Kato reflection functors as in [Mc2]. The category C i (ν) is defined to be the full subcategory of R(ν)-mod consisting of objects M such that e i M = 0. It is thus equivalent to modules over the quotient R(ν)/ e i . The category i C(ν) is similarly defined using the idempotent i e. Kato's reflection functors give an equivalence C i (ν) ∼ = i C(s i ν), where s i is the simple reflection associated to i. There is an isomorphism [Mc2] R(λ)/ e i ∼ = Hom • (j * L λ , j * L λ ) The algebra Hom • (j * L λ , j * L λ ) is also formal and of finite global dimension. The formality follows from the same purity argument as for R(λ), while the finitude of global dimension is in [K1] and [Mc2]. We can then upgrade Corollary 2.4 to obtain an equivalence of triangulated categories j * L λ ∼ = R(λ)/ e s -dgmod (2.1) compatible with the equivalence of the corollary via j * and the inclusion. Comparison of algebraic and geometric induction Let S λµ be the moduli stack of short exact sequences of representations of Q 0 → M ′ → M → M ′′ → 0 (3.1) where dim M ′ = λ and dim M ′′ = µ. Let p : S λµ −→ X λ+µ be the map sending the short exact sequence (3.1) to M . Let q : S λµ −→ X λ × X µ be the map sending (3.1) to (M ′ , M ′′ ). The map p is proper and q is smooth. The geometric induction functor I G : D(X λ ) × D(X µ ) −→ D(X λ+µ ) is I G (F, G) = p ! q * (F ⊠ G)[dim q] The stack F λ is the disjoint union of F i λ , where i runs over all sequences i = (i 1 , . . . , i n ) with each i j ∈ I and j i j = λ. The sequence i records the sequence of simple subquotients in the full flag. Let P i = π ! k F i λ [dim F i λ ]. Then L λ = ⊕ i P i . It is not difficult to check that I G (P i , P j ) = P ij where ij is the concatenation of the two sequences. Therefore I G (L λ , L µ ) is a direct summand of L λ+µ . Let e λµ ∈ R(λ + µ) be the projection to this direct summand. The algebraic induction functor I A : R(λ)-mod × R(µ)-mod −→ R(λµ)-mod is I A (M, N ) = R(λ + µ)e λµ R(λ)⊗R(µ) M ⊠ N. Theorem 3.1. For F ∈ L λ and G ∈ L µ , there is a natural isomorphism of R(λ + µ)-dgmodules. I A (Hom • (L λ ⊠ L µ , F ⊠ G)) ∼ = Hom • (L λ+µ , I G (F ⊠ G)). Proof. Consider the two functors F = I A (Hom • (L λ ⊠ L µ , − ⊠ −)) G = Hom • (L λ+µ , I G (− ⊠ −)). We begin by constructing a natural transformation π : F −→ G. To construct it, it suffices to find a natural bilinear map R(λ + µ)e λµ × Hom • (L λ ⊠ L µ , F) → Hom • (L λ+µ , p ! q * F). This map is (xe λµ , y) → xe λµ p ! q * (y), noting that R(λ + µ) = End • (L λ+µ ) and e λµ is the projection from L λ+µ to p ! q * (L λ ⊠ L µ ). Now note that π is an isomorphism when F = L λ and G = L µ . Then by a standard devissage argument, π is an isomorphism whenever F and G are in the triangulated categories generated by L λ and L µ respectively, as required. The reflection functor Lemma 4.1. Suppose M ∈ C s . Then M ∼ = Hom • (L, j * M) for some M ∈ D b (U λ ; k). Proof. M is a module over Hom • (j * L, j * L), hence by (2.1) is of the form Hom • (j * L, M) for some M. Since j * is right adjoint to j * , we get the desired result. In [BGP], reflection functors between the categories Rep (Q ′ ) and Rep (Q) are constructed which are shown to have the following property: Theorem 4.2. The BGP reflection functor from Rep (Q ′ ) to Rep (Q) induces isomorphisms x λ andx λµ of stacks such that the following diagram commutes: U λ+µ ← −−− − V λµ − −−− → U λ × U µ x λ+µ    xλµ   x λ ×xµ U ′ s i λ+s i µ ← −−− − V ′ s i λ,s i µ − −−− → U ′ s i λ × U ′ s i µ . Under the isomorphism x : U ′ s i λ −→ U λ , the sheaves j * L ′ s i λ and j * L λ have isomorphic direct summands (up to shifts). This is because they are semisimple and every simple perverse sheaf on X λ occurs as a direct summand of L λ . Therefore there is a Morita equivalence between End • (j * L ′ s i λ ) and End • (j * L λ ). This Morita equivalence is Kato's reflection functor T i : End • (j * L λ )-mod −→ End • (j * L ′ s i λ )-mod. Kato's reflection functor satisfies T i (Hom • (L λ , j * F)) = Hom • (L ′ s i λ , j * x * λ F) Lemma 4.3. Consider the following diagram, in which the leftmost and rightmost squares are pullback squares, and the middle square is commutative X f ← −−− − S S g − −−− → Y j   j    h  h U f ← −−− − A e − −−− → B g − −−− → V. Suppose that f is smooth, g is proper and h is an immersion. Then we have the equality of functors from D b (U ; k) to D b (V ; k): h * g * f * j * ∼ = (g • e) * f * Proof. This is a routine consequence of base change and the identity h * h * = id. Lemma 4.4. Suppose F ∈ D b (U λ × U µ ; k). Then p ! q * j * F is in the essential image of j * : D b (U λ+µ ; k) −→ D b (X λ+µ ; k). Proof. Let Z be the complement of U λ+µ in X λ+µ and i : Z −→ X λ+µ be the inclusion. By considering the exact triangle i ! i ! → id → j * j * +1 − − →, it suffices to show that i ! p ! q * j * = 0. Let S Z = q −1 (Z) and f : S Z −→ X λ × X µ be the restriction of p to S Z . By base change, since p is proper and q is smooth, i ! p ! q * j * = q ! f ! j * [dim q]. Thus it suffices to show that f ! j * = 0, i.e. that f and j have disjoint image in X λ × X µ . If ( monoidality In this section, we use the following diagram (c.f. Lemma 4.3): X λ × X µ q ← −−− − S λ+µ S λ+µ p − −−− → X λ+µ j   j    h  h U λ × U µ q ← −−− − V λ+µ e − −−− → B p − −−− → U λ+µ Here B is defined so that the right hand square is a pullback square. One easily checks that the left hand side is a pullback square. Theorem 5.1. The functor T i is monoidal. Proof. We have to show that T i (M •N ) is naturally isomorphic to T i (M )•T i (N ). Use Lemma 4.1 to write M and N as Hom • (L, j * M) and Hom • (L, j * N ) respectively. Then by Theorem 3.1, M • N ∼ = Hom • (L λ+µ , p ! q * (j * M ⊠ j * N )). By Lemma 4.4, we have p ! q * (j * M ⊠ j * N ) ∼ = h * h * p ! q * (j * M ⊠ j * N ). By Lemma 4.3, we can write this as M • N ∼ = Hom • (L λ+µ , h * (p • e) * q * (j * M ⊠ j * N )). Tracing through all the maps, Theorem 4.2 allows us to identify the right hand sides of (5.1) and (5.2), completing the proof. Let V λµ be the moduli stack of short exact sequences of representations of Q 0 → M ′ → M → M ′′ → 0 where dim M ′ = λ, dim M ′′ = µ and Hom(S i , M ′ ) = Hom(S i , M ′′ ) = 0. Let V ′ λµ be the corresponding moduli stack for the quiver Q ′ . M ′ , M ′′ ) ∈ im f then there exists a short exact sequence 0 → M ′ → M → M ′′ → 0 with Hom(S i , M ) = 0. Thus either Hom(S i , M ′ ) = 0 or Hom(S i , M ′′ ) = 0, so either way (M ′ , M ′′ ) / ∈ im j, completing the proof. From the description of Kato's reflection functor, we haveT i (M • N ) ∼ = Hom • (L ′ , h * x * (p • e) * q * (j * M ⊠ j * N )). (5.1)On the other hand, from the description of Kato's reflection functor,T i (M ) • T i (N ) ∼ = Hom • (L ′ , j * x * M) • Hom • (L, j * x * N ))Then Theorem 3.1 tells us thatT i (M ) • T i (N ) ∼ = Hom • (L ′ , p ! q * (j * x * M ⊠ j * x * N ))Then Lemmas 4.4 and 4.3 tell us that we haveT i (M ) • T i (N ) ∼ = Hom • (L ′ , h * (p • e) * q * (j * x * M ⊠ j * x * N )). (5.2) E-mail address: maths@petermc.net Bernstein, &apos; Gel, Ponomarev , Coxeter functors, and Gabriel's theorem. Uspehi Mat. 28Bernstein, Gel'fand and Ponomarev, Coxeter functors, and Gabriel's theorem. Uspehi Mat. Nauk, 28(2(170)):19-33, 1973. 4 Masaki Kashiwara, Myungho Kim, Euiyong Se-Jin Oh, Park, arXiv:1708.044281Monoidal categories associated with strata of flag manifolds. Masaki Kashiwara, Myungho Kim, Se-jin Oh and Euiyong Park, Monoidal categories associated with strata of flag manifolds arXiv:1708.04428 1 Poincaré-Birkhoff-Witt bases and Khovanov-Lauda-Rouquier algebras. Syu Kato, arXiv:1203.5254Duke Math. J. 16333Syu Kato. Poincaré-Birkhoff-Witt bases and Khovanov-Lauda-Rouquier algebras. Duke Math. J. 163 (2014), no. 3, 619-663. arXiv:1203.5254 1, 2, 3 Syu Kato, arXiv:1711.090851On the Monoidality of the Saito Reflection Functors. Syu Kato, On the Monoidality of the Saito Reflection Functors. arXiv:1711.09085 1 A diagrammatic approach to categorification of quantum groups I. Mikhail Khovanov, Aaron D Lauda, arXiv:0803.41211Represent. Theory. 13Mikhail Khovanov, Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13 (2009), 309-347. arXiv:0803.4121 1 A diagrammatic approach to categorification of quantum groups II. Mikhail Khovanov, Aaron D Lauda, arXiv:0804.20802Trans. Amer. Math. Soc. 3635Mikhail Khovanov, Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II. Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700. arXiv:0804.2080 2 Canonical bases arising from quantized enveloping algebras. George, Lusztig, J. Amer. Math. Soc. 32George. Lusztig. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc., 3(2):447-498, 1990. 2 Introduction to quantum groups. George Lusztig, Progress in Mathematics. 1101Birkhäuser Boston IncGeorge Lusztig. Introduction to quantum groups, volume 110 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1993. 1 Finite dimensional representations of Khovanov-Lauda-Rouquier algebras I: Finite type. J Peter, Mcnamara, arXiv:1207.58602J. Reine Angew. Math. 707Peter J. McNamara. Finite dimensional representations of Khovanov-Lauda-Rouquier algebras I: Finite type. J. Reine Angew. Math., 707:103-124, 2015. arXiv:1207.5860 2 Representation theory of geometric extension algebras. P J Mcnamara, arXiv:1701.0794913P. J. McNamara. Representation theory of geometric extension algebras. arXiv:1701.07949 1, 2, 3 Alexander Polishchuk, Michel Van Den, Bergh, arXiv:1503.041602Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups. Alexander Polishchuk and Michel Van den Bergh, Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups. arXiv:1503.04160 2 . O Schnürer, arXiv:0809.47852Equivariant Sheaves on Flag Varieties. Math. Z. 2671-2O. Schnürer, Equivariant Sheaves on Flag Varieties. Math. Z., 267 (2011), no. 1-2, 2780. arXiv:0809.4785 2 . B Webster, arXiv:1209.24632Weighted Khovanov-Lauda-Rouquier algebrasB. Webster. Weighted Khovanov-Lauda-Rouquier algebras arXiv:1209.2463 2 Canonical bases and KLR-algebras. Michela Varagnolo, Eric Vasserot, arXiv:0901.39922J. Reine Angew. Math. 659Michela Varagnolo and Eric Vasserot. Canonical bases and KLR-algebras. J. Reine Angew. Math., 659:67-100, 2011. arXiv:0901.3992 2

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{'abstract': 'Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal.', 'arxivid': '1712.00173', 'author': ['Peter J Mcnamara '], 'authoraffiliation': [], 'corpusid': 30467101, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5925, 'n_tokens_neox': 5124, 'n_words': 2946, 'pdfsha': '0b3abad442e668432a72eea5974fb5b09c8b622a', 'pdfurls': ['https://arxiv.org/pdf/1712.00173v1.pdf'], 'title': ["MONOIDALITY OF KATO'S REFLECTION FUNCTORS", "MONOIDALITY OF KATO'S REFLECTION FUNCTORS"], 'venue': []}

arxiv

The Perihelion Precession of Mercury and the Generalized Uncertainty Principle 12 May 2011 January 20, 2013 Barun Majumder Department of Physical Sciences Indian Institute of Science Education and Research (Kolkata) 741252Mohanpur, PinNadia, West BengalIndia The Perihelion Precession of Mercury and the Generalized Uncertainty Principle 12 May 2011 January 20, 2013IISER(Kolkata)/GR-QCGUPminimum lengthHamilton vectorMercury perihelion Very recently authors in [1] proposed a new Generalized Uncertainty Principle (or GUP) with a linear term in Plank length. In this Letter the effect of this linear term is studied perturbatively in the context of Keplerian orbits. The angle by which the perihelion of the orbit revolves over a complete orbital cycle is computed. The result is applied in the context of the precession of the perihelion of Mercury. As a consequence we get a lower bound of the new intermediate length scale offered by the GUP which is approximately 40 orders of magnitude below Plank length. The idea that the uncertainty principle could be affected by gravity was first given by Mead [2]. Later modified commutation relations between position and momenta commonly known as Generalized Uncertainty Principle ( or GUP ) were given by candidate theories of quantum gravity ( String Theory, Doubly Special Relativity ( or DSR ) Theory and Black Hole Physics ) with the prediction of a minimum measurable length [3,4,5]. Similar kind of commutation relation can also be found in the context of Polymer Quantization in terms of Polymer Mass Scale [6]. The authors in [1] proposed a GUP which is consistent with DSR theory, String theory and Black Hole Physics and which says [x i , x j ] = [p i , p j ] = 0,(1)[x i , p j ] = i δ ij − l pδ ij + p i p j p + l 2 p 2 δ ij + 3p i p j ,(2)∆x∆p ≥ 2 1 − 2l < p > +4l 2 < p 2 > ≥ 2 1 + l p 2 + 4l 2 ∆p 2 + 4l 2 p 2 − 2l p 2 ,(3) where l = l 0 l pl . Here l pl is the Plank length (≈ 10 −35 m). It is normally assumed that the dimensionless parameter l 0 is of the order unity. If this is the case then the l dependent terms are only important at or near the Plank regime. But here we expect the existence of a new intermediate physical length scale of the order of l = l 0 l pl . We also note that this unobserved length scale cannot exceed the electroweak length scale [1] which implies l 0 ≤ 10 17 . These equations are approximately covariant under DSR transformations but not Lorentz covariant [5]. These equations also imply ∆x ≥ (∆x) min ≈ l 0 l pl (4) and ∆p ≤ (∆p) max ≈ M pl c l 0(5) where M pl is the Plank mass and c is the velocity of light in vacuum. It can be shown that equation (2) is satisfied by the following definitions x i = x oi and p i = p oi (1 − l p o + 2 l 2 p 2 o ), where x oi , p oj satisfies [x oi , p oj ] = i δ ij . Here we can interpret p oi as the momentum at low energies having the standard representation in position space (p oi = −i ∂ ∂x oi ) with p 2 o = 3 i=1 p oi p oi and p i as the momentum at high energies. It can also be shown that any non-relativistic Hamiltonian of the form H = p 2 2m + V (r) can be written in the form [1] H = p 2 o 2m + V (r) − l m p 3 o + O(l 2 ) + . . . .(6) Here we neglect terms O(l 2 ) and higher in comparison to terms O(l) to study the effect of the linear term in l in the first approximation as l = l 0 l pl . The effect of this proposed GUP is well studied for some well known quantum mechanical Hamiltonians in [1,7] and also studied in the context of quantum cosmological models [8], Friedmann equations and black hole evaporation [17]. In this Letter we will study the effect of the third term ( which is linear in Plank length ) perturbatively in the context of Kepler orbits. To start with we would like to comment that besides the LaplaceRungeLenz vector [9] there is also another conserved vector quantity which is known as the Hamilton vector. Due to the lack of evidence we can only cite few documents [10] from where we can get precise knowledge about this Hamilton vector. In the presence of perturbation the perihelion of the orbit begins to precess. And the precession rate of the Hamilton vector coincides with the precession rate of the perihelion [11]. In the presence of perturbation we can calculate the precession rate of the Hamilton vector and hence we can compute the angle by which the perihelion revolves over a complete orbital cycle. The Hamilton vector is written as u = p m − k Lφ ,(7) where k = GmM and L = mr 2φ is the orbital momentum. The unit vector φ = L × r Lr(8) lie on the plane of the orbit. For the Keplerian orbits we write equation (6) as H = p 2 2m − k r − l m p 3 = H 0 − l m p 3 ,(9) where H 0 = p 2 2m − GmM r is the unperturbed Hamiltonian. As mentioned before we will treat the term which is linear in Plank length as the perturbation. Now we will follow the method introduced in [12] for computing the precession rate of the perihelion of the orbit. The Hamilton vector is no longer conserved in the presence of perturbation and we can write˙ u = { u, H} = − l m { u, p 3 } = − 3 l p 2 m { u, p}.(10) A straightforward calculation using equations (7) and (8) yieldṡ u = − 3 l k p m r 3 L 2 L × ( r × L) = − 3 l k p m r 3 r.(11) Equation {2.3} of [13] gives the precession rate of u as ω = u ×˙ u u 2 ,(12) where u = k e L and e is the eccentricity of the elliptic orbit and 0 < e < 1. Using equations (7),(8), (11) and (12) we finally get ω = 3 l p L m r 2 e 2 R r − 1 ,(13) where R = L 2 km is the semi-latus rectum of the unperturbed orbit. In the first order of approximation we will use the known relations R r = 1 + e cosφ(14) and p 2 2m − k r = − k 2a ,(15) where a = R 1−e 2 is the semi-major axis of the elliptic orbit. As the precession rate of the Hamilton vector equals the precession rate of the perihelion [11], so the angle by which the perihelion revolves over a complete orbital cycle due to the presence of the perturbation can be written as ∆θ = T 0 ω dt = 2π 0 ω φ dφ.(16) With the use of equations (13), (14) and (15) we can finally get ∆θ = 3 l mk R 1 e 2π 0 cos φ [1 + e 2 + 2e cos φ] 1 2 dφ = 3 l mk R F,(17) where F = 1 e 2π 0 cos φ [1 + e 2 + 2e cos φ] 1 2 dφ and the integral can be easily evaluated numerically for a particular value of e. Now with k = GmM and l = l 0 M pl c = l 0 l pl we get equation (17) in the form ∆θ = 3 l 0 m M pl (2GM/c 2 ) (2R) F .(18) For the planet Mercury, the parameters which govern the motion are [15] 2GM ⊙ c 2 = 2.95325008 × 10 3 metre , m = 3.3022 × 10 23 kg , a = 5.7909175 × 10 10 metre , e = 0.20563069 .(19) Here M ⊙ is the solar mass and a is the semi-major axis of the elliptic orbit and R = a(1−e 2 ) is the semi-latus rectum of the orbit. The observed value of ∆θ for the precession of the perihelion of Mercury is [16] ∆θ obs = 2 π (7.98734 ± 0.00037) × 10 −8 radian/revolution . Einstein's general relativity explains this observation very accurately. Following [14] and references therein we can see ∆θ obs − ∆θ GR = 2 π (−0.00010 ± 0.00037) × 10 −8 radian/revolution . We may feel that there is nothing left to explain but still we can also see that the difference is not exactly zero. Using (19) we can evaluate ∆θ from equation (18) and we get ∆θ = 2.3209 × 10 28 l 0 radian/revolution . Believing that the remnant difference of (21) is explained by (22) As an interesting fact we cannot neglect the existence of a new length scale ∼ 10 −75 metre. So in this Letter we have considered the recently proposed GUP [1] which is approximately covariant under DSR transformation [5] but not Lorentz covariant. The GUP has a linear term in Plank length. We have studied the effect of that linear term in Plank length in the context of Keplerian orbits. For calculating the precession rate of the perihelion we have adapted the method introduced in [12] and the whole approach is perturbative. The author in [12] made use of the modified commutation relation of the form [4] [ x i ,p j ] = i (δ ij + βp 2 δ ij + β ′p ipj ),(24) where the Plank length is given by l pl ≈ √ 3β + β ′ . So the effect studied in [12] is O(l 2 pl ) and higher order terms are neglected perturbatively. There the results were applied to earth-bound satellites where the linear momentum of the satellite should be much less than 6.6 kg m/s. This seems quite strange to us because all planetary motions are always accompanied by very high momenta. Here we have equated our result with the precession of perihelion of Mercury. Though the observed precession rate is well explained by general relativity but up to order of 10 −8 . Our equation leads to a prediction of the existence of a new length scale which is 40 orders of magnitude below Plank length. More briefly we have succeeded to set a lower bound to the new intermediate length scale offered by the GUP introduced in [1]. Similar result was also found earlier in [14] where the idea of non-commuting spatial co-ordinates was taken into account. It was shown that the precession of the perihelion of Mercury restricts the value of the minimum length which is 33 orders of magnitude lower than Plank length. we get a lower bound of the new intermediate physical length scale (≈ l 0 l pl ) with l 0 (min.) = 7.31 × 10 −40 . AcknowledgementsThe author is very much thankful to Prof. Narayan Banerjee for helpful discussions and guidance.References[1] A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B 678 (2009) 497. . C A Mead, Phys. Rev. D. 135849C. A. Mead, Phys. Rev. D 135 (1964) 849. . D Amati, M Ciafaloni, G Veneziano, Phys. Lett. B. 21641D. Amati, M. 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Lett. 88 (2002) 190403; . J Magueijo, L Smolin, Phys. Rev. D. 7126010J. Magueijo and L. Smolin, Phys. Rev. D 71 (2005) 026010. . A Kempf, G Mangano, R B Mann, Phys. Rev. D. 521108A. Kempf, G. Mangano, R.B. Mann, Phys. Rev. D 52 (1995) 1108. . J L Cortes, J Gamboa, Phys. Rev. D. 7165015J. L. Cortes, J. Gamboa, Phys. Rev. D 71 (2005) 065015. . G M Hossain, V Husain, S S Seahra, Class. Quantum Grav. 27165013G. M. Hossain, V. Husain, S. S. Seahra, Class. Quantum Grav. 27 (2010) 165013. . S Das, E C Vagenas, Can. J. Phys. 87233S. Das, E. C. Vagenas, Can. J. Phys. 87 (2009) 233; . S Das, E C Vagenas, A F Ali, Phys. Lett. B. 690407S. Das, E. C. Vagenas, A. F. Ali, Phys. Lett. B 690 (2010) 407; . P Alberto, S Das, E C Vagenas, Phys. Lett. A. 3751436P. Alberto, S. Das, E. C. Vagenas, Phys. Lett. A 375 (2011) 1436. . B Majumder, Phys. Lett. B. 699315B. Majumder, Phys. Lett. B 699 (2011) 315. H Goldstein, C P Poole, J Safko, Classical Mechanics (3 rd rd.). SFAddison-WesleyGoldstein H., Poole C.P. and Safko J.L. Classical Mechanics (3 rd rd.), (Addison- Wesley, SF 2002). On the applications of the method of quaternions to some dynamical questions. W R Hamilton, The Mathematical Papers of Sir William Rowan Hamilton, Halberstam H. and Ingram R.E. eds.IIICambridge UPHamilton W.R. On the applications of the method of quaternions to some dynamical questions, in The Mathematical Papers of Sir William Rowan Hamilton, Halberstam H. and Ingram R.E. eds. (Cambridge UP 1967), Vol. III, pp. 441-448; . H Abelson, A Di Sessa, L Rudolph, Am. J. Phys. 43579Abelson H., di Sessa A. and Rudolph L., Am. J. Phys.43 (1975) 579; . R P Patera, Am. J. Phys. 49593Patera R.P., Am. J. Phys. 49 (1981) 593; . R P Martnez-Y-Romero, H N Nez-Ypez, A L Salas-Brito, Eur. J. Phys. 1471R.P. Martnez-y-Romero, H.N. Nez-Ypez, A.L. Salas-Brito, Eur. J. Phys. 14 (1993) 71; . H N Nez-Ypez, A L Salas-Brito, Eur. J. Phys. 2139H.N. Nez-Ypez, A.L. Salas-Brito, Eur. J. Phys. 21 (2000) L39; . D Derbes, Am. J. Phys. 69481Derbes D., Am. J. Phys. 69 (2001) 481; . G Muñoz, Am. J. Phys. 711292Muñoz G. Am. J. Phys. 71 (2003) 1292; . J T Wheeler, Can. J. Phys. 8391J.T. Wheeler, Can. J. Phys. 83 (2005) 91. . O I Chashchina, Z K Silagadze, Phys. Rev. D. 77107502O.I. Chashchina, Z.K. Silagadze, Phys. Rev. D 77 (2008) 107502; . B Davies, Am. J. Phys. 51909B. Davies, Am. J. Phys. 51 (1983) 909; . D Ebner, Am. J. Phys. 53374D. Ebner, Am. J. Phys. 53 (1985) 374. . Z Silagadze, Phys. Lett. A. 3732643Z. Silagadze, Phys. Lett. A 373 (2009) 2643. . M G Stewart, Am. J. Phys. 73730M.G. Stewart, Am. J. Phys. 73 (2005) 730. . S Benczik, L N Chang, D Minic, N Okamura, S Rayyan, T Takeuchi, Phys. Rev. D. 6626003S. Benczik, L.N. Chang, D. Minic, N. Okamura, S. Rayyan, T. Takeuchi, Phys. Rev. D 66 (2002) 026003. Allen , 's Astrpohysical Quantities. A. N. CoxNew YorkSpringer-Verlag4th ed.Allen's Astrpohysical Quantities, 4th ed., edited by A. N. Cox (Springer-Verlag, New York, 2000); . JPL Solar System dynamics Group website. JPL Solar System dynamics Group website → http://ssd.jpl.nasa.gov/?planet phys par ; . D E Particle Data Group, Groom, Eur. Phys. J. C. 151Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15 (2000) 1. . S Pireaux, J P Rozelot, Astrophys. Space Sci. 2841159S. Pireaux and J. P. Rozelot, Astrophys. Space Sci. 284 (2003) 1159. . B Majumder, under preparationB. Majumder, under preparation.

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{'abstract': 'Very recently authors in [1] proposed a new Generalized Uncertainty Principle (or GUP) with a linear term in Plank length. In this Letter the effect of this linear term is studied perturbatively in the context of Keplerian orbits. The angle by which the perihelion of the orbit revolves over a complete orbital cycle is computed. The result is applied in the context of the precession of the perihelion of Mercury. As a consequence we get a lower bound of the new intermediate length scale offered by the GUP which is approximately 40 orders of magnitude below Plank length.', 'arxivid': '1105.2428', 'author': ['Barun Majumder \nDepartment of Physical Sciences\nIndian Institute of Science Education and Research (Kolkata)\n741252Mohanpur, PinNadia, West BengalIndia\n'], 'authoraffiliation': ['Department of Physical Sciences\nIndian Institute of Science Education and Research (Kolkata)\n741252Mohanpur, PinNadia, West BengalIndia'], 'corpusid': 118606772, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5456, 'n_tokens_neox': 4518, 'n_words': 2583, 'pdfsha': '472fdb3909e2712d0ea4afd9b4788d02a35b6d1c', 'pdfurls': ['https://arxiv.org/pdf/1105.2428v1.pdf'], 'title': ['The Perihelion Precession of Mercury and the Generalized Uncertainty Principle', 'The Perihelion Precession of Mercury and the Generalized Uncertainty Principle'], 'venue': []}

arxiv

The proton structure function and a soft Regge Dipole Pomeron : a test with recent data arXiv:hep-ph/9811380v1 18 Nov 1998 P Desgrolard desgrolard@ipnl.in2p3.fr2e-mail:sasha@len.uzhgorod.ua3e-mail:martynov@bitp.kiev.ua Institut de Physique Nucléaire de Lyon IN2P3-CNRS Université Claude Bernard 43 boulevard du 11 novembre 1918F-69622Villeurbanne CedexFrance ( A Lengyel Institute of Electronic Physics National Academy of Sciences of Ukraine 294015 Uzhgorod-015, Universitetska 21Ukraine ( E Martynov )N.N. Bogoliubov Institute for Theoretical Physics National Academy of Sciences of Ukraine Kiev-143, Metrologicheskaja 14b252143Ukraine ) The proton structure function and a soft Regge Dipole Pomeron : a test with recent data arXiv:hep-ph/9811380v1 18 Nov 1998 A recently published soft Regge Dipole Pomeron model intended for all x and Q 2 is proved to give a good agreement with (non fitted) recent HERA data from ZEUS (SVX95) on the proton structure function F p 2 (x, Q 2 ) at low Q 2 and low x. The model also reproduces (without fit) the recently estimated experimental derivatives ∂F p 2 ∂ℓnQ 2 and ∂ℓnF p 2 ∂ℓn(1/x) in a wide x and Q 2 -region. Motivations The proton structure function (SF) is one of the observables most often measured in high energy physics [1]. Consequently a relevant model has to be periodically tested (and eventually updated or abandoned) in the new kinematical ranges of x (Björken variable) and Q 2 (virtuality of the photon) investigated by the experimentalists. Recent measurements [2] of the SF at HERA (from ZEUS 1995 shifted vertex experiment (SVX95)) have motivated us to test our Dipole Pomeron parametrization [3] of the proton structure function F p 2 (x, Q 2 ) intended for a wide region of Q 2 and x. We wish to show that these recent data can be reproduced within an "old soft Pomeron" framework, which is an "à la Regge" approach. Second, we revise a widely extended opinion that a soft Pomeron and more generally an "à la Regge" approach to deep inelastic scattering (DIS) should be restricted to rather small values of Q 2 . This conclusion is based mainly on the popular Donnachie-Landshoff (DL) model of the Pomeron [4] and its particular parametrization of Q 2 -dependence of the 1 E-mail: desgrolard@ipnl.in2p3.fr 2 E-mail: sasha@len.uzhgorod.ua 3 E-mail: martynov@bitp.kiev.ua residue function [5]. This model was used in [2] and the conclusion was drawn that the Regge theory describes well the data only at very low Q 2 ≤ 0.65 GeV 2 . We have vice-versa shown that a soft Pomeron contribution (with unit intercept) can be applied to the virtual photoproduction cross-section [3], and that it reproduces well the data on F p 2 (x, Q 2 ) in a much wider region of Q 2 and x. In the present short note, we emphasize that the fit [3] not only reproduces with high quality also the new data on the SF [2] at low Q 2 and low x, but is in good agreement with the behavior of the derivatives The model Many Pomeron models are on the market in high energy hadron phenomenology. In spite of the quite different t-dependence of the elastic amplitudes, at t = 0 they can be combined in two groups: 1. A simple pole in the complex angular momenta (j-) plane with an intercept α P (0) > 1 : the DL Pomeron [4] and its generalization [6] with an additional constant term (a preasymptotic simple j-pole with unit intercept) are examples. Such a Pomeron leads to a total cross-section σ tot (s) ∝ s α P (0)−1 , which at extremely high energies ( well beyond the present attainable ones) would violate the Froissart-Martin bound; this could require to be unitarized (for example, by an eikonal method). 2. More complicate singularity in the j-plane at j = 1 : in these models ( [3,6,7,8] and references therein) the Froissart-Martin bound is not violated and asymptotically the total cross-sections behave as σ tot (s) ∝ ℓn µ (s/s 0 ), 0 < µ < 2, s 0 = 1 GeV 2 . All these models describe quite well hadronic total cross-sections and γp inelastic one. The best quality of the description (in the sense of χ 2 ) was achieved [6] at t = 0 when µ = 1. This corresponds to a double j-pole in the forward amplitude, i.e. to the Dipole Pomeron (DP) model. This model was succesfully applied [3] (with its extension to Q 2 = 0) to DIS with a good description of the SF in a wide region of Q 2 and x. Defining the DP model for DIS, we start from the expression connecting the transverse cross-section for the (γ * p) interaction to the proton structure function F p 2 σ γ * p T (W, Q 2 ) = 4π 2 α Q 2 (1 + 4m 2 p x 2 Q 2 ) 1 1 + R(x, Q 2 ) F p 2 (x, Q 2 ) ,(1) where α is the fine structure constant, m p is the proton mass, R is the ratio of longitudinal to transverse cross-sections. We have approximated R(x, Q 2 ) = 0, due to its experimental smallness. The center of mass energy W of the (γ * p) system obeys W 2 = Q 2 1 x − 1 + m 2 p(2) and the optical theorem writes F p 2 (x, Q 2 ) = Q 2 (1 − x) 4π 2 α(1 + 4m 2 p x 2 /Q 2 ) ℑmA(W 2 , t = 0, Q 2 ) .(3) The forward γ * p scattering amplitude is dominated, for W far from the threshold W th = m p , by the Pomeron and f -Reggeon contributions (we ignore an a 2 -Reggeon contribution considering the f -term as an effective one at W > 3 GeV) A(W 2 , t = 0, Q 2 ) = P (W 2 , Q 2 ) + F (W 2 , Q 2 ) ,(4) with F (W 2 , Q 2 ) = iG f (Q 2 ) − i W 2 m 2 p α f (0)−1 (1 − x) B f (Q 2 )(5) and P (W 2 , Q 2 ) = P 1 + P 2 ,(6)P 1 = iG 1 (Q 2 )ℓn(−iW 2 /m 2 p )(1 − x) B 1 (Q 2 ) , P 2 = iG 2 (Q 2 )(1 − x) B 2 (Q 2 ) .(7) The details of the parametrization of the real functions G i (Q 2 ), B i (Q 2 ) can be found in [3]. Here we only mention that they vary between the constants G i (0), B i (0) and G i (∞), B i (∞). Let us make a few comments on the chosen Pomeron model. We support the point of view that there is just one "bare" Pomeron (two Pomerons, "soft" and "hard", are considered in [9]). This unique Pomeron is universal and factorizable 4 , i.e. it is the same in all processes; only the vertex functions depend on which are the interacting particles. It follows from these special requirements that the Pomeron trajectory should be independent of Q 2 . One should mention that this approach differs from other ones where an "effective" Pomeron with a Q 2 -dependent intercept [10,11,12,13]) has been chosen. The hard Pomeron or BFKL Pomeron [14] with a quite large intercept is only an approximation to a true Pomeron. A growth of the total cross-sections means that in j-plane a true Pomeron is harder than a simple pole singularity at j = 1. The Dipole Pomeron defined by (6),(7) obeys the following specificities: it is universal, it has a Q 2 -independent intercept α P (0) = 1, it does not violate, at least explicitly, the unitarity restrictions on the amplitude. The proton SF and total (γp) cross section results We choose the DP model defined in details in [3] with the 23 parameters (see Table 2 in [3]). This is not probably the most economical set within this framework, however we keep these published values and do not perform any new adjustement in order to introduce no confusion (we postpone an update of this model, fixing some parameters and refitting the others with a set of data completed by the forthcoming 96-97 HERA results). We have proved that adding the new (44) values of F p 2 does not change the quality of the fit in [3], we found : χ 2 = 1321 for 1209 data points which becomes χ 2 = 1341 for 1253 data, in practice leaving unchanged χ 2 /d.o.f ≃ 1.1. The F p 2 (x, Q 2 ) results are plotted versus x for the experimental Q 2 bins of [2] (low Q 2 ) and compared to the whole set of fitted and non fitted data in Fig. 1. The total real (γp) cross section versus the c.m squared energy W 2 is shown in Fig. 2. These figures show the good agreement for 0 ≤ Q 2 ≤ 17 GeV 2 ; higher Q 2 values (where DP model also reproduce well the data) are discussed in [3]. The Q-slope as a function of x The Q-slope B Q (x, Q 2 ) = ∂F p 2 (x, Q 2 ) ∂ℓn(Q 2 ) depends on the two independent variables x and Q 2 . However to compare with experiment the Q-slope has been calculated for the set [2] (x i , < Q 2 > i ); i=1,24 of strongly correlated variables which includes x up to 0.2. The results are shown in Fig. 3; the agreement is good up to x ∼ 0.1. 5 The x-slope and the "effective intercept" as functions of Q 2 The x-slope B x (x, Q 2 ) = ∂ℓnF p 2 (x, Q 2 ) ∂ℓn(1/x) is also a function of two variables; the quantity currently replacing B x in experimental papers is the "effective power" ∆ ef f (sometimes denoted as λ ef f ) in the low x -fixed Q 2 approximation of the structure function F p 2 ∝ x −∆ ef f . This power is currently connected to the Pomeron effective intercept (α(0) = 1 + ∆ ef f ). Actually, from a phenomenological point of view, one can extract ∆ ef f at fixed Q 2 depending on x assuming a parametrization F p 2 (x, Q 2 ) = G(Q 2 ) 1 x ∆ ef f (x,Q 2 ) . Strictly speaking, however, the identification B x = ∆ ef f is possible only when the x−independence of the x-slope is a model assumption (one may see [3] for a discussion of the slopes). In general, this is not the case. From that point of view, it would be interesting and important to have "measured" values for ∆ ef f at fixed Q 2 an at different < x > : it will allow to study the x-dependence of the effective Pomeron intercept. The comparison between the calculated value of the x-slope and the experimental effective power λ ef f is given in the last figure (Fig. 4) for the set of kinematical variables [2] (< x > i , Q 2 i ); i = 1, 30) including ZEUS and fixed target E665 results. The agreement is good in the whole experimental range : up to Q 2 ∼ 250 GeV 2 . Conclusion We have proved that a soft dipole Pomeron model [3] ("à la " Regge) not only reproduces with a high quality the new data on the proton structure function [2] at low Q 2 and low x, but also is in good agreement with the measured slopes [2] ∂F p 2 ∂ℓnQ 2 and ∂ℓnF p 2 ∂ℓn(1/x) up to Q 2 ∼ 250 GeV 2 and x ∼ 0.1. Such a success in reproducing the data is due not only to an appropriate choice of the asymptotic Pomeron contribution but is due also to the preasymptotic terms (chosen constant here) in the Pomeron and f -reggeon. A more detailed discussion of preasymptotics properties of the model is in [3]. This enforces our belief that a universal -factorizable-Pomeron with a Q 2 independent intercept α(0) = 1 may be successful in Deep Inelastic Scattering not only at low Q 2 and x (as the DL Pomeron does). On the basis of the results from [3] and the present paper, we claim that the area of validity of a Regge approach is much wider (especially in Q 2 ) than usually assumed and can be extended up to rather high values of Q 2 (may be up to a few hundreds GeV 2 ). x) , measured in[2], up to intermediate values of the kinematical variables. Fig. 2 . 2Experimental fitted data for the photoproduction total cross-section and predictions in the Dipole Pomeron model. Fig. 3 . 3Q-slope B Q (x, < Q 2 >) : experimental points from[2] as a function of x (for the indicated < Q 2 > values). The continuous line is the prediction for the Qnot fitted) within the Dipole Pomeron model. Fig. 4 . 4Experimental effective power λ ef f (< x >, Q 2 ) : data from[2] as a function of Q 2 (for the indicated < x > values). The continuous line is the predictions for the xnot fitted) in the Dipole Pomeron model. If Pomeron is a sum of two terms (as in(7)) then at least the leading one at W ≫ m p should satisfy factorization. Acknowledgments E.M. would like to thank the IPNL for the kind hospitality and financial support provided to him during this work. It is a pleasure to thank E. Predazzi for a critical reading of the manuscript and illuminating comments. Quadt Measurement and Phenomenology of the proton structure function F 2 from ZEUS at HERA. th International Conference on high energy physics. VancouverICHEP98, XIX th International Conference on high energy physics, Vancouver 1998 : A. Quadt Measurement and Phenomenology of the proton structure function F 2 from ZEUS at HERA; Low Q 2 -intermediate x proton structure function measurement at HERA. H1 collaboration Low Q 2 -intermediate x proton structure function measurement at HERA. Results on the measurement and phenomenology of F 2 at low x and Q 2 , DESY-98-121. J Breitweg, ZEUS collaborationhep-ex98/09005J. Breitweg et al. and ZEUS collaboration, Results on the measurement and phenomenol- ogy of F 2 at low x and Q 2 , DESY-98-121, hep-ex98/09005. . P Desgrolard, A Lengyel, E Martynov, Eur. Phys. J. C. in pressP. Desgrolard, A. Lengyel, E. Martynov, Eur. Phys. J. C in press. . A Donnachie, P Landshoff, Phys. Lett. B. 296227A. Donnachie, P. Landshoff, Phys. Lett. B 296 (1992) 227. . A Donnachie, P Landshoff, Zeit. Phys. C. 61139A. Donnachie, P. Landshoff, Zeit. Phys. C 61 (1994) 139. . P Desgrolard, M Giffon, A Lengyel, E Martynov, Nuovo Cim. A. 107637P. Desgrolard, M. Giffon, A. Lengyel, E. Martynov, Nuovo Cim. A 107 (1994) 637; . P Desgrolard, M Giffon, E Martynov, Nuovo Cim. A. 110537P. Desgrolard, M. Giffon, E.Martynov, Nuovo Cim. A 110 (1997) 537. . L Jenkovszky, Fortshr. Phys. 34702L. Jenkovszky, Fortshr. Phys. 34 (1986) 702; . L L Jenkovszky, E S Martynov, B V Struminsky, Phys. Lett. B. 249535and references thereinL.L. Jenkovszky, E.S. Martynov, B.V. Struminsky, Phys. Lett. B 249 (1990) 535 (and references therein). . P Gauron, E Leader, B Nicolescu, Nucl. Phys. B. 279189P. Gauron, E. Leader, B. Nicolescu, Nucl. Phys. B 279 (1988) 189; . Phys. Lett. B. 238496Phys. Lett. B 238 (1990) 496. . A Donnachie, P Landshoff, hep-ph/9806344A. Donnachie, P. Landshoff, hep-ph/9806344. . A Capella, A Kaidalov, C Merino, J Tran Thanh, Van, Phys. Lett. B. 337358A. Capella, A. Kaidalov, C.Merino, J. Tran Thanh Van, Phys. Lett. B 337 (1994) 358. . M Bertini, M Giffon, E Predazzi, Phys.Lett. B. 349561M. Bertini, M. Giffon, E. Predazzi, Phys.Lett. B 349 (1995) 561. . H Abramovicz, A Levy, hep-ph//9712415H. Abramovicz, A. Levy, hep-ph//9712415. . P Desgrolard, L Jenkovszky, F Paccanoni, Eur. Phys. J. C. in pressP. Desgrolard, L. Jenkovszky, F. Paccanoni, Eur. Phys. J. C, in press; Approximative solution to the BFKL equation applicable at HERA, LYCEN 9896. submitted for publicationApproximative solution to the BFKL equation applicable at HERA, LYCEN 9896 (Nov. 1998), submitted for publication. V S Fadin, L N Lipatov, hep-ph/9802290 (and references therein). V. S. Fadin, L. N. Lipatov, hep-ph/9802290 (and references therein). Experimental data for the proton structure function F p 2 (x, Q 2 ) at low Q 2 compared to the results within the Dipole Pomeron model (shown are the 44 recent ZEUS SVX 95 data -non fitted-and the other fitted data). Fig. 1 Experimental data for the proton structure function F p 2 (x, Q 2 ) at low Q 2 compared to the results within the Dipole Pomeron model (shown are the 44 recent ZEUS SVX 95 data -non fitted-and the other fitted data).

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{'abstract': 'A recently published soft Regge Dipole Pomeron model intended for all x and Q 2 is proved to give a good agreement with (non fitted) recent HERA data from ZEUS (SVX95) on the proton structure function F p 2 (x, Q 2 ) at low Q 2 and low x. The model also reproduces (without fit) the recently estimated experimental derivatives ∂F p 2 ∂ℓnQ 2 and ∂ℓnF p 2 ∂ℓn(1/x) in a wide x and Q 2 -region.', 'arxivid': 'hep-ph/9811380', 'author': ['P Desgrolard desgrolard@ipnl.in2p3.fr2e-mail:sasha@len.uzhgorod.ua3e-mail:martynov@bitp.kiev.ua \nInstitut de Physique Nucléaire de Lyon\nIN2P3-CNRS\nUniversité Claude Bernard\n43 boulevard du 11 novembre 1918F-69622Villeurbanne CedexFrance (\n', 'A Lengyel \nInstitute of Electronic Physics\nNational Academy of Sciences of Ukraine\n294015 Uzhgorod-015, Universitetska 21Ukraine (\n', 'E Martynov \n)N.N. Bogoliubov Institute for Theoretical Physics\nNational Academy of Sciences of Ukraine\nKiev-143, Metrologicheskaja 14b252143Ukraine\n', ') '], 'authoraffiliation': ['Institut de Physique Nucléaire de Lyon\nIN2P3-CNRS\nUniversité Claude Bernard\n43 boulevard du 11 novembre 1918F-69622Villeurbanne CedexFrance (', 'Institute of Electronic Physics\nNational Academy of Sciences of Ukraine\n294015 Uzhgorod-015, Universitetska 21Ukraine (', ')N.N. Bogoliubov Institute for Theoretical Physics\nNational Academy of Sciences of Ukraine\nKiev-143, Metrologicheskaja 14b252143Ukraine'], 'corpusid': 6399738, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5246, 'n_tokens_neox': 4544, 'n_words': 2689, 'pdfsha': 'c25a1345981eec0654fba10c0e7fd5b11d4cd7f4', 'pdfurls': ['https://arxiv.org/pdf/hep-ph/9811380v1.pdf'], 'title': ['The proton structure function and a soft Regge Dipole Pomeron : a test with recent data', 'The proton structure function and a soft Regge Dipole Pomeron : a test with recent data'], 'venue': []}

arxiv

Note on Scalar Fields Non-Minimally Coupled to (2 + 1)-Gravity arXiv:gr-qc/0101079v1 18 Jan 2001 Eloy Ayón-Beato Alberto García ¶ † Alfredo Macías macias@fis.cinvestav.mx§e-mail:iosephus@ictp.trieste.it Departamento de Física UAM-Iztapalapa Apartado Postal 55-534C.P. 09340MéxicoD.FMEXICO ¶⋄ ‡ José M Pérez-Sánchez Diploma Programme HEP The Abdus Salam ICTP P.O. Box 58634100TriesteITALY Departamento de Física CINVESTAV-IPN Apartado Postal 14-740C.P. 07000MéxicoD.FMEXICO Note on Scalar Fields Non-Minimally Coupled to (2 + 1)-Gravity arXiv:gr-qc/0101079v1 18 Jan 2001(March 24, 2022)numbers: 0450+h0460Kz0420Jb0470Bw Scalar fields non-minimally coupled to (2 + 1)-gravity, in the presence of cosmological constant term, are considered. Non-minimal couplings are described by the term ζ R Ψ 2 in the Lagrangian. Within a class of static circularly symmetric space-times, it is shown that the only existing physically relevant solutions are the anti-de Sitter space-time for ζ = 0, and the Martínez-Zanelli black hole for ζ = 1/8. We obtain also two new solutions with non-trivial scalar field, for ζ = 1/6 and ζ = 1/8 respectively, nevertheless, the corresponding space-times can be reduced, via coordinate transformations, to the standard anti-de Sitter space. S = 1 2 d D x √ −g 1 κ C(Ψ)R − w(Ψ)∇ µ Ψ∇ µ Ψ + V (Ψ) ,(1) where R is the Ricci scalar, V (Ψ) is a potential function, C(Ψ) and w(Ψ) are coupling functions. In four dimensions, the conformal solution discovered by Bekenstein [4], appears to be the only non-trivial black hole solution allowed for self-gravitating scalar fields non-minimally coupled to gravity. The study of the relevant system in D > 2 + 1 dimensions has shown that some restrictions on the kind of plausible scalar field behaviors are required [5]. For example, in (3 + 1)-dimensions, the most general result including self-interactions and without cosmological constant, has been obtained by Mayo and Bekenstein [6,7]. In that work, spherical static non-trivial scalar field behaviors are excluded for non-minimal couplings with ζ < 0 and ζ ≥ 1/2. Consequently, since in (3 + 1)-dimensions the conformal coupling is ζ = 1/6 the Bekenstein black hole belongs to the non-covered range (0 < ζ < 1/2). In this sense, still remains open the question if in 4-dimensions the value ζ = 1/6 is the unique coupling allowing a non-trivial scalar field behavior, or if there exist a family of solutions with non-minimally coupled scalar field behaviors within the interval under consideration. Although the space-time is not three-dimensional, and (2 + 1)-dimensional gravity is clearly not a physically realistic model of our universe, it is a simple model which is rich enough to allow us to learn a good deal about the nature of quantum gravity. At first sight, the (2 + 1)-dimensional gravity looks trivial, in particular, the vacuum Einstein equations imply that space-time is locally flat, corresponding to the absence of the Weyl tensor in three dimensions. The triviality of local geometry in (2 + 1)-dimensional gravity holds even if the cosmological constant term is taken into account; the Einstein space is a space of constant curvature. However, the black hole riddle has long been one of the most outstanding problems of modern physics. It has remained in focus for a long time as one of the potential testing grounds for quantum gravitational phenomena. As it is well known, three dimensional gravity, in vacuum, admits only the trivial locally flat (2 + 1)-Minkowski space. Thus, it is necessary either to couple matter to the theory, e.g., a cosmological constant or scalar matter, or to consider alternative vacuum or non-vacuum gravity theories in order to get solutions different from the trivial one. New solutions in (2 + 1)-gravity coupled to a self-interacting dilaton and in vacuum scalar-tensor theories have been obtained by Chan [8]. Moreover, Chan and Mann [9] determined a conformal static black hole solution with a nontrivial conformal factor Ψ for C(Ψ) = 1, w(Ψ) = 4, and V (Ψ) = 2Λ exp(bΨ). A black hole solution with a negative cosmological constant coupled to a conformal scalar field for C(Ψ) = 1 − κ(1/8)Ψ 2 , w(Ψ) = 1, and V (Ψ) = 2Λ/κ has been found by Martínez and Zanelli [10]. In this paper we consider a particular case of the action (1) in order to study non-minimal couplings of scalar fields to (2 + 1)-gravity with a cosmological constant term. Essentially, we work out a commonly used one-parametric family of theories with real parameter ζ, where the non-minimal couplings are described by the term ζ R Ψ 2 in the Lagrangian. We will show that for a static circularly symmetric metric, where the space-time possesses only one degree of freedom, the only existing non-trivial solutions are the anti-de Sitter space-time for ζ = 0, the Martínez-Zanelli black hole for ζ = 1/8. We obtain also two new solutions with non-trivial scalar field, for ζ = 1/6 and ζ = 1/8 respectively, whose space-time geometries reduce to the one of an anti-de Sitter space. In this way we are establishing, for the studied class, the completeness of the solutions to the corresponding field equations. As mentioned above, in three dimensions our basic action reads S = 1 2 d 3 x √ −g 1 κ R + 2l −2 − ∇ µ Ψ∇ µ Ψ − ζ R Ψ 2 ,(2) where Λ = −l −2 is the cosmological constant, Ψ is the massless non-minimally coupled scalar field, and R is the scalar curvature. The field equations arising from (2) are, on one hand, the Einstein equations G ν µ = l −2 δ ν µ + κ ∇ µ Ψ∇ ν Ψ − 1 2 δ ν µ ∇ α Ψ∇ α Ψ + ζ δ ν µ ✷Ψ 2 − ∇ µ ∇ ν Ψ 2 + G ν µ Ψ 2 ,(3) and, on the other hand, the equation ✷Ψ = ζ R Ψ,(4) for the scalar field, where ✷ is the Laplace-Beltrami operator. We shall restrict our study to the following class of static circularly symmetric three-dimensional metrics, which, in polar coordinates, can be written as follows g = −F (r)dt 2 + F (r) −1 dr 2 + r 2 dθ 2 ,(5) consequently, we assume that the scalar field only depends on the radial variable r, i.e., Ψ = Ψ(r). The Einstein equations (3), for the static circularly symmetric space-time (5), become 2ζΨΨ ′′ + (2ζ − 1)(Ψ ′ ) 2 = 0 ,(6)1 − κζΨ 2 − 2κζrΨΨ ′ F ′ − κΨ ′ (4ζΨ + rΨ ′ ) F = 2r l 2 ,(7)1 − κζΨ 2 F ′′ − 4κζΨΨ ′ F ′ − κ 4ζΨΨ ′′ + (4ζ − 1) (Ψ ′ ) 2 F = 2 l 2 .(8) where primes denote derivatives with respect to r. Eq. (6) corresponds to the combination G r r − G t t , and the Eqs. (7) and (8) are the components G r r and G θ θ of the Einstein equations (3), respectively. It is straightforward to show that the scalar field equation (4) follows from Eqs. (6)-(8) by using the Bianchi identity ∇ ν G µ ν = 0. Let us proceed to integrate the field equations. For the general case, ζ = 0, Eq. (6) becomes Ψ ′ Ψ ′ − 1 − 4ζ 2ζ Ψ ′ Ψ 2 = 0 ,(9) whose general solution can be immediately obtained, namely Ψ(r) = A (r + B) 2ζ/(1−4ζ) ,(10) where A and B are integration constants. By substituting the expression (10) for Ψ into Eq. (7), the following linear first order differential equation for the structural function F is obtained F ′ = κδ 2 A 2 ((δ − 1) r − 2B) (r + B) κδA 2 ((δ − 1) r − B) + 4 (δ + 1) (r + B) δ+1 F + 8 (δ + 1) r (r + B) δ+1 l 2 κδA 2 ((δ − 1) r − B) + 4 (δ + 1) (r + B) δ+1 ,(11) with δ ≡ 4ζ/(1 − 4ζ). It is straightforward to show that the general solution of the Eq. (11) is given by F (r) = 4 (δ + 1) r 2 − B 2 − l 2 C (r + B) δ+1 l 2 κδA 2 ((δ − 1) r − B) + 4 (δ + 1) (r + B) δ+1 ,(12) with C a new integration constant. It remains to fulfill Eq. (8), which via Eq. (6), becomes 1 − κζΨ 2 F ′′ − 4κζΨΨ ′ F ′ − κ (Ψ ′ ) 2 F = 2 l 2 .(13) It is easy to see that Eq. (13) imposes constraints on the integration constants of the form A = A(B, ζ) and C = C(B, ζ). Moreover, by replacing in Eq. (13) the expressions (10) and (12) for Ψ(r) and F (r), and after some lengthy manipulations, the following algebraic equation is obtained: 8 (δ + 1) 3 (δ − 1) (δ − 2) y 4 − 3δ (δ − 1) By 3 − δ l 2 C − 2 (δ + 1) B 2 y 2 + (δ + 2) Bl 2 Cy y 2δ −2κδ (δ + 1) A 2 (δ + 4) (δ + 1) (δ − 1) 2 y 4 − δ (δ + 1) 4δ 2 + 7δ − 9 By 3 −δ (δ − 1) 2 l 2 C − (δ + 1) (δ + 2) (5δ + 1) B 2 y 2 + (δ + 1) B 2δ 2 − 3δ + 4 l 2 C − 2δ (δ + 1) (δ + 2) B 2 y −δ 2 (δ + 1) B 2 l 2 C y δ +κ 2 δ 2 A 4 (δ + 1) (δ − 1) 2 y 4 − 3δ (δ + 1) (δ − 1) By 3 + 3δ 2 (δ + 1) B 2 y 2 −B (δ − 1) l 2 C + δ (δ + 1) (δ + 2) B 2 y = 0 ,(14) where y stands for y ≡ r + B. Since the powers of y are linearly independent functions [11], the corresponding coefficients must vanish independently. Therefore, different possibilities arise by assigning values or ranges of values to the parameter δ: Let us first consider the case of positive values of δ. For δ > 0, the highest power of y in Eq. (14) is 2δ + 4, equating its coefficient to zero one gets the condition 8 (δ + 1) 3 (δ − 1) (δ − 2) = 0 .(15) Hence, the possible solutions only exist for the values δ = 1 (ζ = 1/8) and δ = 2 (ζ = 1/6) of the coupling constant. For the first value, i.e., δ = 1 (ζ = 1/8), Eq. (14) becomes κA 2 B − 4Cl 2 + 16B 2 = 0 = B 3κA 2 B − 4Cl 2 .(17) Thus, in this case (δ = 1, ζ = 1/8), there exist two admissible classes of solutions for the integration constants. The first class is obtained for A 2 = 8B κ , C = 6B 2 l 2 ,(18) and it corresponds to the black hole of Martínez and Zanelli [10], i.e., Ψ MZ = 8B κ (r + B) , F MZ = (r + B) 2 (r − 2B) rl 2 ,(19) whose horizon is located at r h = 2B. In this case the free parameter B is related to the mass M of the hole through the relation B = M l 2 /3. The second class is achieved by setting B = C = 0, and can be expressed as follows Ψ(r) = A √ r , F (r) = r 2 l 2 .(20) This is a new conformal solution with a non-trivial scalar field behavior, although the corresponding space-time can be brought, via coordinate transformations, to the standard anti-de Sitter space form (see for example [12][13][14][15]). Therefore, since the anti-de Sitter space-time is solution to the vacuum plus cosmological constant field equations, the energy-momentum tensor of the non-trivial scalar field is such that it vanishes identically (see the term in brackets on the left hand side of Eq. (3)). This peculiar behavior arises from the non-minimal coupling of the conformal scalar field to gravity. For the second value, i.e., δ = 2 (ζ = 1/6). The highest power of y in (14) is y 2δ+3 = y 7 , its coefficient vanishes if the constant B is set equal to zero, then Eq. (14) reduces to − 12 2Cl 2 + κA 2 18y 6 − κA 2 y 4 = 0 . Consequently C = − κA 2 2l 2 ,(22) and therefore, Ψ(r) = A r , F (r) = r 2 l 2 .(23) This is a new solution with a non-minimally coupled scalar field, however, it exhibits the same peculiarities as the ones of the previous solution Eq. (20). In full, for positive values of the parameter δ, we have established the existence of three classes of solutions, namely the Martínez-Zanelli black hole for ζ = 1/8, and two new solutions corresponding to the anti-de Sitter space with non-trivial scalar fields, one for ζ = 1/6, and one for ζ = 1/8. Let us turn to the case of negative values of δ. For δ < 0 the highest power of y in Eq. (14) is y 4 , the vanishing of its coefficient leads to the condition κ 2 δ 2 (δ + 1) (δ − 1) 2 A 4 = 0 .(24) It should be noticed that δ = −1, since the value δ = −1 corresponds to infinite coupling constants ζ = ±∞. Therefore, Eq. (24) implies that A = 0 and leads to the condition 8 (δ + 1) 3 (δ − 1) (δ − 2) y 4 − 3δBy 3 − δ l 2 C − 2 (δ + 1) B 2 y 2 + (δ + 2) Bl 2 Cy y 2δ = 0. It is easy to see that the coefficient of y 4+2δ never vanishes. Consequently, no solutions exist for δ < 0. The remaining case corresponds to the minimal coupling δ = 0, in this case, Eq. (6) implies that Ψ = const. and Eq. (7) integrates directly to F (r) = r 2 l 2 − M , Ψ = const. .(26) This solution corresponds to the static BTZ solution [16], which is once again no other than the (2 + 1)-anti-de Sitter space-time (see [12][13][14][15]). It is important to stress the fact that the three solutions (19), (20), and (23) of the Einstein equations (6)- (8), also satisfy the scalar field equation (4). To conclude this letter let us summarize the results. We have established the complete class of solutions within the static circularly symmetric ansatz (5) for the structural function F = F (r), and found that there only exist solutions for the following values of the non-minimal coupling constant ζ: • For conformal coupling, ζ = 1/8, there exist two solutions, the first one corresponds to the conformal black hole (19) of Martínez and Zanelli, and the second one is a new conformal solution (20), whose space-time corresponds to the anti-de Sitter space, with a non-trivial scalar field. • The non-minimal coupling ζ = 1/6 leads to a new solution (23), which space-time geometry is once again reducible to the anti-de Sitter form endowed with a non-trivial scalar field. • For minimal coupling ζ = 0, the scalar field reduces to a constant, and the resulting space-time corresponds to the (2 + 1)-anti-de Sitter space-time (26). The possibility of obtaining more general static, circularly symmetric solutions, with two structural functions is still open. The corresponding field equations are much more involved and hard to integrate in general. Nevertheless, efforts have to be undertaken to overcome the difficulties in the integration process of this more general dynamical system. κA 2 B − 4Cl 2 + 16B 2 8y 4 + 3κA 2 By 2 − 2B 3κA 2 B − 4Cl 2 24y 3 + BκA 2 y = 0 ,(16)therefore, the vanishing of the coefficients of even and odd powers of y yields the following relations ACKNOWLEDGMENTSWe thank Friedrich W. Hehl for useful discussions and literature hints. This research was partially supported by CONACyT Grants 32138E and 28339E, and by FOMES Grant: P/FOMES 98-35-15. . P G Bergmann, Int. J. Theor. Phys. 125P.G. Bergmann, Int. J. Theor. Phys. 1, 25 (1968). . R Wagoner, Phys. Rev. 13209R. Wagoner, Phys. Rev. D1, 3209 (1970). . T Singh, L N Rai, Gen. Rel. Grav. 15875T. Singh and L.N. Rai, Gen. Rel. Grav. 15, 875 (1983). . J D Bekenstein, Ann. Phys. (NY). 82535J.D. Bekenstein, Ann. Phys. (NY) 82, 535 (1974). . A Saa, Phys. Rev. 537377A. Saa, Phys. Rev. D53, 7377 (1996). . A E Mayo, J D Bekenstein, Phys. Rev. 545059A.E. Mayo, and J.D. Bekenstein, Phys. Rev. D54, 5059 (1996). J D Bekenstein, Proceedings of Second Sakharov Conference in Physics. I.M. Dremin, A.M. SemikhatovSecond Sakharov Conference in PhysicsMoscow; SingaporeWorld Scientific761J.D. Bekenstein, Proceedings of Second Sakharov Conference in Physics, Moscow , eds. I.M. Dremin, A.M. Semikhatov, 761 (World Scientific, Singapore 1997). . K C K Chan, Phys. Rev. 553564K.C.K. Chan, Phys. Rev. D55, 3564 (1997). . K C K Chan, R B Mann, Phys. Rev. 502600K.C.K. Chan and R.B. Mann, Phys. Rev. D50, 2600 (1994). . C Martínez, J Zanelli, Phys. Rev. 543830C. Martínez and J. Zanelli, Phys. Rev. D54, 3830 (1996). L Elsgoltz, Differential Equations and Variational Calculus. MoscowMIRL. Elsgoltz, Differential Equations and Variational Calculus (MIR, Moscow 1977). . L P Einsenhart, Riemannian Geometry, Princenton Univ. PressPrincentonL.P. Einsenhart, Riemannian Geometry (Princenton Univ. Press, Princenton 1925). J Plebański, Conformal Equivalent Riemannian Spaces (Monography CINVESTAV-IPN. J. Plebański, Conformal Equivalent Riemannian Spaces (Monography CINVESTAV-IPN 1967). S Weinberg, Gravitation & Cosmology. New YorkJohn Wiley & SonsS. Weinberg, Gravitation & Cosmology (John Wiley & Sons, New York 1972). Anti-de Sitter Black-Hole-like Solutions and its Dimensional Reductions. A García, preprint CINVESTAV-PHYS- GRG-000501A. García, "Anti-de Sitter Black-Hole-like Solutions and its Dimensional Reductions", preprint CINVESTAV-PHYS- GRG-000501 (2000). . M Bañados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Bañados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett., 69, 1849 (1992).

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{'abstract': 'Scalar fields non-minimally coupled to (2 + 1)-gravity, in the presence of cosmological constant term, are considered. Non-minimal couplings are described by the term ζ R Ψ 2 in the Lagrangian. Within a class of static circularly symmetric space-times, it is shown that the only existing physically relevant solutions are the anti-de Sitter space-time for ζ = 0, and the Martínez-Zanelli black hole for ζ = 1/8. We obtain also two new solutions with non-trivial scalar field, for ζ = 1/6 and ζ = 1/8 respectively, nevertheless, the corresponding space-times can be reduced, via coordinate transformations, to the standard anti-de Sitter space.', 'arxivid': 'gr-qc/0101079', 'author': ['Eloy Ayón-Beato ', 'Alberto García ¶ † ', 'Alfredo Macías macias@fis.cinvestav.mx§e-mail:iosephus@ictp.trieste.it \nDepartamento de Física\nUAM-Iztapalapa\nApartado Postal 55-534C.P. 09340MéxicoD.FMEXICO\n', '¶⋄ ‡ ', 'José M Pérez-Sánchez \nDiploma Programme HEP\nThe Abdus Salam ICTP\nP.O. Box 58634100TriesteITALY\n', '\nDepartamento de Física\nCINVESTAV-IPN\nApartado Postal 14-740C.P. 07000MéxicoD.FMEXICO\n'], 'authoraffiliation': ['Departamento de Física\nUAM-Iztapalapa\nApartado Postal 55-534C.P. 09340MéxicoD.FMEXICO', 'Diploma Programme HEP\nThe Abdus Salam ICTP\nP.O. Box 58634100TriesteITALY', 'Departamento de Física\nCINVESTAV-IPN\nApartado Postal 14-740C.P. 07000MéxicoD.FMEXICO'], 'corpusid': 119045007, 'doi': '10.1016/s0370-2693(00)01241-7', 'github_urls': [], 'n_tokens_mistral': 5827, 'n_tokens_neox': 5006, 'n_words': 2847, 'pdfsha': '532435957d76261b45c449dc6a91597fcb7d6b3b', 'pdfurls': ['https://export.arxiv.org/pdf/gr-qc/0101079v1.pdf'], 'title': ['Note on Scalar Fields Non-Minimally Coupled to (2 + 1)-Gravity', 'Note on Scalar Fields Non-Minimally Coupled to (2 + 1)-Gravity'], 'venue': []}

arxiv

May 17, 2019 Javier F Peña Juan C Vera Department of Econometrics and Operations Research Tilburg University The Netherlands Luis F Zuluaga luis.zuluaga@lehigh.edu Department of Industrial and Systems Engineering Lehigh University USA Tepper School of Business Carnegie Mellon University USA May 17, 2019arXiv:1905.06366v1 [math.OC] 15 May 2019 Equivalences among the chi measure, Hoffman constant, and Renegar's distance to ill-posedness We show the equivalence among the following three condition measures of a full column rank matrix A: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of A. Our results provide a procedure to estimate χ(A) thereby opening an avenue to identify classes of linear programs solvable in polynomial time in the real model of computation. * Introduction We establish new equivalences among three types of condition measures of a matrix that play central roles in numerical linear algebra and in convex optimization: the chi measure [3,7,9,31,32], the Hoffman constant [15,17,19,37], and Renegar's distance to ill-posedness [29,30]. We recall the definitions of these quantities in Section 2 below. Let A ∈ R m×n be a full column rank matrix. The chi measure χ(A) arises naturally in weighted least-squares problems of the form min D 1/2 (Ax − b) 2 , see, e.g., [4,9,10,18]. The chi measure χ(A) is also a key component in the analysis of Vavasis and Ye's interiorpoint algorithm for linear programming [23,36]. A remarkable feature of Vavasis and Ye's algorithm is its sole dependence on the matrix A defining the primal and dual constraints. The Hoffman constant H(A) is associated to Hoffman's Lemma [15,17], a fundamental error bound for systems of linear constraints of the form Ax ≤ b. The Hoffman constant and other similar error bounds are used to establish the convergence rate of a wide variety of optimization algorithms [2, 14, 16, 20-22, 24-26, 37, 37]. Renegar's distance to ill-posedness R(A) is a pillar for the concept of condition number in optimization introduced by Renegar in the seminal articles [29,30] and subsequently extended in a number of articles [1,5,8,[11][12][13]. Our work is inspired by several relationships among χ(·), H(·), and R(·) previously established in [6,8,27,34,35,39]. In particular, it is known that if A ∈ R m×n is full column rank, then χ(A) ≥ H(A) and if Ax < 0 is feasible then H(A) = 1/R(A). However, χ(A) can be arbitrarily larger than H(A) (see, e.g., [27]). Also, the equivalence between χ(A) and 1/R(A) breaks down when Ax < 0 is infeasible. Our main result (Theorem 1) shows that the lack of equivalence among these quantities can be rectified by considering signed versions of H(·) and R(·). In hindsight our equivalences are somewhat natural because χ(A) does not change when the signs of some rows of A are flipped whereas both H(A) and R(A) evidently do. We show that χ(A) is exactly the largest H(Â) over the collection of matrices obtained by flipping the signs of some rows of A. We also show that when all rows of A are non-zero, 1/χ(A) is the same as the smallest R(Â) over the collection of all matrices obtained by flipping the signs of some rows of A so thatÂx < 0 is feasible. Furthermore, we show that χ(A) is the same as H(A) for the matrix A obtained by stacking the rows of A and −A. The latter equivalence together with the algorithmic machinery recently developed in [27] provides a procedure to compute or estimate χ(A). That computational ability in turn offers the potential to identify classes of linear programs that are solvable in polynomial time in the real model of computation via Vavasis-Ye's interior-point algorithm [23,36], since the number of arithmetic operations of Vavasis-Ye's algorithm is polynomial on the dimensions of A and on log(χ(A)) for a variantχ(A) of χ(A). Some of our equivalences are reminiscent of results previously developed by Tunçel [34] and by Todd, Tunçel, and Ye [33] to compare a variantχ(A) of χ(A) and Ye's condition measure [38] for polyhedra of the form {A T y : y ≥ 0, y 1 = 1}. Definition of χ(·), H(·), and R(·) Let A ∈ R m×n have full column rank. The chi measure of A is defined as χ(A) = sup{ (A T Diag(d)A) −1 A T Diag(d) : d ∈ R m ++ }. In this expression and throughout the paper, Diag(d) ∈ R m×m denotes the diagonal matrix whose vector of diagonal entries is d ∈ R m . Also, we write · to denote the canonical Euclidean norms in R m and R n , and the corresponding induced operator norm (or equivalently the spectral norm) in R m×n . The underlying space will always be clear from the context. Several authors [3,7,31,32] independently showed that χ(A) is finite as long as A is full column rank. See [9] for a detailed discussion. Let A ∈ R m×n . The Hoffman constant H(A) of A is defined as H(A) = sup dist(u, P A (b)) (Au − b) + : b ∈ A(R n ) + R m + and u ∈ P A (b) where P A (b) := {x ∈ R n : Ax ≤ b} and dist(u, P A (b)) = min{ u − x : x ∈ P A (b)}. Hoffman [17] showed that H(A) is always finite. Other proofs of this fundamental result can be found in [15,27,37]. Let A ∈ R m×n be such that Ax < 0 is feasible. Renegar's distance to ill-posedness of A is defined as R(A) := inf{ ∆A : (A + ∆A)x < 0 is infeasible}. Renegar introduced the distance to ill-posedness as a main building block to develop the concept of condition number for optimization problems [29,30]. The following proposition, which recalls properties previously established in [19,27,28,39], is our starting point. Proposition 1. Let A ∈ R m×n . If A has full column rank then χ(A) ≥ H(A).(1) On the other hand, if Ax < 0 is feasible then H(A) = 1 R(A) .(2) 3 Equivalences among χ(·), H(·), and R(·) We are now ready to state our main result. Let A ∈ R m×n . where A ∈ R 2m×n is the matrix obtained by stacking A and −A, that is, A = A −A . If in addition all rows of A are nonzero then χ(A) = max A∈D(A) H(Â) = max A∈D(A) 1 R(Â) .(4) The identity (4) in Theorem 1 has the following natural extension when some rows of A are zero. Given A ∈ R m×n , letà ∈ Rm ×n denote the submatrix of A obtained by dropping the zero rows from A. If A ∈ R m×n has full column rank then so doesà and Theorem 1 implies that χ(A) = χ(Ã) = max A∈D(Ã) H(Â) = max A∈D(Ã) 1 R(Â) .(5) The identity (5) in turn suggests an extension of χ(·) to general (not necessarily full rank) matrices and general (not necessarily Euclidean) norms since both H(·) and R(·) are defined in full generality and satisfy (2). The proof of Theorem 1 relies on the two key building blocks stated as Proposition 2 and Proposition 3 below. We will use the following convenient notation. For a positive integer m, let [m] denote {1, . . . , m}. For A ∈ R m×n and J ⊆ [m], we let A J ∈ R J×n denote the submatrix of A defined by the rows indexed by J. The first key building block for the proof of Theorem 1 is the following characterization of χ(·) from [9]. The same characterization is also stated and proved in [39] by adapting a technique from [33]. Proposition 2. Let A ∈ R m×n have full column rank. Then χ(A) = max J ⊆[m],|J |=n A J non-singular max v∈R J A T J v =1 v . The second building block for the proof of Theorem 1 is the following characterization of H(·) discussed in [27] but that can be traced back to [19,37,39]. On the other hand, the construction of χ(A) and Proposition 1 imply that for all ∈ S(A) Proposition 3. Let A ∈ R m×n . Then H(A) = max J ⊆[m],|J |=n A J non-singular max v∈R J + A T J v =1 v = max J∈J (A) max v∈R J + A T J v =1 v ,χ(A) = χ(Â) ≥ H(Â). Thus the first identity in (3) follows. To prove the second identity in (3), notice that J ⊆ [2m] is such that |J| = n and A J non-singular if and only if there exists I ⊆ [m] such that |I| = n, A I is non-singular, and J = I + ∪(m+I − ) for some partition I = I + ∪I − of I. If d ∈ {−1, 1} m satisfies d i = 1, i ∈ I + and d i = −1, i ∈ I − then max v∈R J + A T J v =1 v = max v∈R I + (Diag(d)A) T I v =1 v . Hence Proposition 3 implies that H(A) = max J ⊆[2m],|J |=n A J non-singular max v∈R J + A T J v =1 v = max A∈S(A) max I⊆[m],|I|=n A I non-singular max v∈R I + ÂT I v =1 v = max A∈S(A) H(Â). The second identity in (3) thus follows. The crux of the proof of (4) is the following one-to-one correspondence between J (A) and D(A). H(A) = max J∈J (A) max v∈R J + A T J v =1 v = max A∈D(A) max v∈R m + ÂT v =1 v = max A∈D(A) H(Â) = max A∈D(A) 1 R(Â) . The third step follows from Proposition 3 and the fact that J (Â) = {[m]} ifÂx < 0 is feasible. Identity (4) follows from (6) and (3). To finish, here is a proof of the above claim. For u ∈ R n let J u := {j : A j u < 0}. Observe that J ∈ J (A) if and only if J ⊆ J u for some u ∈ R n . Since all rows of A are nonzero, it follows that J ∈ J (A) if and only if J = J u for some u ∈ R n such that all entries of Au are non-zero. When the latter holds, we have J u = ([m] \ I u ) ∪ (m + I u ) for I u = {i : A i u > 0} and A [m]\Iu u < 0, A Iu u > 0 which is equivalent to ∈ D(A) where is the matrix obtained by flipping the signs of the rows of A indexed by I u . Conclusion We showed that if A ∈ R m×n has full column rank and nonzero rows then where A ∈ R 2m×n is the matrix obtained by stacking the rows of A and −A. The first expression in (7) takes the maximum over the collection of matrices S(A) which has exponential size in m. The second and third expressions in in (7) take the maximum over the smaller but harder to describe collection of matrices D(A). By contrast, the last expression in (7) is the Hoffman constant of the single matrix A ∈ R 2m×n . The identity χ(A) = H(A) and the machinery developed in [27] provide a novel algorithmic procedure to compute or estimate χ(A). This computational capability in turn creates an avenue to identify families of linear programs that are solvable in polynomial time in the real model of computation via Vavasis-Ye's interior-point algorithm [23,36]. The following two collections S(A) and D(A) of signed matrices associated to A play a central role in our main developments. Let S(A) := {Diag(d)A : d ∈ {−1, 1} m }, and D(A) := { ∈ S(A) :Âx < 0 is feasible}. Theorem 1 . 1Let A ∈ R m×n have full column rank. Then χ(A) = max A∈S(A) H(Â) = H(A), where J (A) = {J ⊆ [m] : A J x < 0 is feasible} and J (A) ⊆ J (A) is the collection of maximal sets in J (A). Proof of Theorem 1. Let J and v be optimal for the characterization of χ(A) in Proposition 2. Then for d = sign(v) ∈ {−1, 1} m and := Diag(d)A ∈ S(A) Proposition 3 implies that H(Â) ≥ v = χ(A). Claim. Suppose all rows of A are nonzero. Then J ∈ J (A) if and only if J = ([m] \ I) ∪ (m + I) for some I ⊆ [m] such that ∈ D(A) where is the matrix obtained by flipping the signs of the rows of A indexed by I. This claim, Proposition 3, and Proposition 1 imply that A coordinate-free condition number for convex programming. D Amelunxen, P Bürgisser, SIAM J. on Optim. 223D. Amelunxen and P. Bürgisser. A coordinate-free condition number for convex pro- gramming. SIAM J. on Optim., 22(3):1029-1041, 2012. Linearly convergent away-step conditional gradient for nonstrongly convex functions. A Beck, S Shtern, Mathematical Programming. 164A. Beck and S. Shtern. Linearly convergent away-step conditional gradient for non- strongly convex functions. Mathematical Programming, 164:1-27, 2017. A geometric property of the least squares solution of linear equations. A Ben-Tal, M Teboulle, Linear Algebra and its Applications. 139A. Ben-Tal and M. Teboulle. A geometric property of the least squares solution of linear equations. 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Fast rates for online gradient descent without strong convexity via Hoffman's bound. D Garber, arXiv:1802.04623arXiv preprintD. Garber. Fast rates for online gradient descent without strong convexity via Hoffman's bound. arXiv preprint arXiv:1802.04623, 2018. Approximations to solutions to systems of linear inequalities. O Güler, A Hoffman, U Rothblum, SIAM Journal on Matrix Analysis and Applications. 162O. Güler, A. Hoffman, and U. Rothblum. Approximations to solutions to systems of linear inequalities. SIAM Journal on Matrix Analysis and Applications, 16(2):688-696, 1995. The condition number of a function relative to a set. D Gutman, J Peña, arXiv:1901.08359arXiv preprintD. Gutman and J. Peña. The condition number of a function relative to a set. arXiv preprint arXiv:1901.08359, 2019. On approximate solutions of systems of linear inequalities. A Hoffman, Journal of Research of the National Bureau of Standards. 494A. Hoffman. On approximate solutions of systems of linear inequalities. Journal of Research of the National Bureau of Standards, 49(4):263-265, 1952. Complete orthogonal decomposition for weighted least squares. P Hough, S Vavasis, SIAM Journal on Matrix Analysis and Applications. 182P. Hough and S. Vavasis. Complete orthogonal decomposition for weighted least squares. SIAM Journal on Matrix Analysis and Applications, 18(2):369-392, 1997. Error bounds for solutions of linear equations and inequalities. D Klatte, G Thiere, Zeitschrift für Operations Research. 412D. Klatte and G. Thiere. Error bounds for solutions of linear equations and inequalities. Zeitschrift für Operations Research, 41(2):191-214, 1995. On the global linear convergence of Frank-Wolfe optimization variants. S Lacoste-Julien, M Jaggi, Advances in Neural Information Processing Systems (NIPS). S. Lacoste-Julien and M. Jaggi. On the global linear convergence of Frank-Wolfe op- timization variants. 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Res., 19:38-52, 1994. Global error bounds for convex conic problems. S Zhang, SIAM Journal on Optimization. 103S. Zhang. Global error bounds for convex conic problems. SIAM Journal on Optimiza- tion, 10(3):836-851, 2000.

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{'abstract': 'We show the equivalence among the following three condition measures of a full column rank matrix A: the chi measure, the signed Hoffman constant, and the signed distance to ill-posedness. The latter two measures are constructed via suitable collections of matrices obtained by flipping the signs of some rows of A. Our results provide a procedure to estimate χ(A) thereby opening an avenue to identify classes of linear programs solvable in polynomial time in the real model of computation. *', 'arxivid': '1905.06366', 'author': ['Javier F Peña ', 'Juan C Vera \nDepartment of Econometrics and Operations Research\nTilburg University\nThe Netherlands\n', 'Luis F Zuluaga luis.zuluaga@lehigh.edu \nDepartment of Industrial and Systems Engineering\nLehigh University\nUSA\n', '\nTepper School of Business\nCarnegie Mellon University\nUSA\n'], 'authoraffiliation': ['Department of Econometrics and Operations Research\nTilburg University\nThe Netherlands', 'Department of Industrial and Systems Engineering\nLehigh University\nUSA', 'Tepper School of Business\nCarnegie Mellon University\nUSA'], 'corpusid': 155100178, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7026, 'n_tokens_neox': 6074, 'n_words': 3477, 'pdfsha': 'fdffa0c9ce67aaba8f5f29c184b1f65ac81d4f5a', 'pdfurls': ['https://arxiv.org/pdf/1905.06366v1.pdf'], 'title': [], 'venue': []}

arxiv

STABILITY AND TRANSITIONS OF THE SECOND GRADE POISEUILLE FLOW 11 Sep 2015 Saadet Ozer Taylan Sengul STABILITY AND TRANSITIONS OF THE SECOND GRADE POISEUILLE FLOW 11 Sep 2015 In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to flows in an infinite pipe with circular cross section that are independent of the axial coordinate.We show that unlike the Newtonian (ǫ = 0) case, in the second grade model (ǫ = 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold Rc ≈ 4.124ǫ −1/4 where ǫ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects.At R = Rc, we find that generically the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as R tends to Rc. Our numerical calculations suggest that for low ǫ values, the system prefers a catastrophic transition where the bifurcation is subcritical.We also find that there is a Reynolds number R E with R E < Rc such that for R < R E , the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that R E ≈ 12.87 at ǫ = 0 and R E approaches Rc quickly as ǫ increases. Introduction Certain natural materials manifest some fluid characteristics that can not be represented by well-known linear viscous fluid models. Such fluids are generally called non-Newtonian fluids. There are several models that have been proposed to predict the non-Newtonian behavior of various type of materials. One class of fluids which has gained considerable attention in recent years is the fluids of grade n [11,8,13,12,7,17,6]. A great deal of information for these types of fluids can be found in [4]. Among these fluids, one special subclass associated with second order truncations is the so called second-grade fluids. The constitutive equation of a second grade fluid is given by the following relation for incompressible fluids: t = −pI + µA 1 + α 1 A 2 + α 2 A 2 1 , where t is the stress tensor, p is the pressure, µ is the classical viscosity, α 1 and α 2 are the material coefficients. A 1 and A 2 are the first two Rivlin-Ericksen tensors defined by A 1 = ∇v + ∇v T , A 2 =Ȧ 1 + A 1 ∇v + ∇v T A 1 , where v is the velocity field and the overdot represents the material derivative with respect to time. This type of constitutive relation was first proposed in [2]. The following conditions: α 1 + α 2 = 0, µ ≥ 0, α 1 ≥ 0, must be satisfied for the second-grade fluid to be entirely consistent with classical thermodynamics and the free energy function achieves its minimum in equilibrium [5]. Equation of motion for an incompressible second grade Rivlin Ericksen fluid is represented as: ρ(v t + w × v + ∇ |v| 2 2 ) = −∇p + µ∆v + α[∆v t + ∆w × v + ∇(v · ∆v + 1 4 |A 1 | 2 )], ∇ · v = 0. where ρ is the density, α = α 1 = −α 2 , represents the second order material constant. Subscript t denotes the partial derivative with respect to time, w is the usual vorticity vector defined by w = ∇ × v. We next define the non-dimensional variables: v * = v U , p * = p ρ U 2 , t * = tU L , x * = x L , where U and L are characteristic velocity and length, respectively. By letting ǫ represent the second order non-dimensional material constant which measures the relative strength of second order viscous effects compared to inertial effects and defining the Reynolds number, R = ρU L µ , ǫ = α ρL 2 , the equation of motion, with asterisks omitted, can be expressed as: (1) ∇p = 1 R ∆v + ǫ(∆w × v + ∆v t ) − v t − w × v, where the characteristic pressurep is defined as: p = p + |v| 2 2 − ǫ(v∆v + 1 4 |A 1 | 2 ). Taking curl of both sides of (1) we can simply write the equation of motion as: (2) ∇ × [ 1 R ∆v + ǫ(∆w × v + ∆v t ) − v t − w × v] = 0, which is the field equation of incompressible unsteady second grade Rivlin-Ericksen fluid independent of the choice of any particular coordinate system. Now we restrict our interest to flows in a cylindrical tube and assume that the velocity is dependent only on two cross-sectional variables x, y and the time t. The incompressiblity of the fluid allows us to introduce a stream function ψ such that v = (ψ y , −ψ x , w) where ψ = ψ(t, x, y) and also w = w(t, x, y). We further take the cross section of the cylinder to be a disk with unit radius and consider the no-slip boundary conditions. Then the equations (2) admit the following steady state solution w 0 = 1 2 (1 − x 2 − y 2 ), ψ 0 = 0. Here the characteristic velocity has been chosen as U = pL 2 4µ . First considering the deviation w ′ = w − w 0 and ψ ′ = ψ − ψ 0 and then introducing the polar coordinates w ′′ (t, r, θ) = w ′ (t, r cos θ, r sin θ) and ψ ′′ (t, r, θ) = ψ ′ (t, r cos θ, r sin θ) and ignoring the primes, the equations become (3) ∂ ∂t (1 − ǫ∆)w = 1 R ∆w + Rψ θ + J(ψ, (1 − ǫ∆)w), ∂ ∂t ∆(ǫ∆ − 1)ψ = − 1 R ∆ 2 ψ + ǫR∆w θ + J((1 − ǫ∆)∆ψ, ψ) + ǫJ(∆w, w), in the interior of the unit disk Ω where J is the advection operator J(f, g) = 1 r (f r g θ − f θ g r ). The field equations are supplemented with no-slip boundary conditions for the velocity field (4) w = ψ = ∂ψ ∂r = 0 at r = 1. In this paper, our main aim is to investigate the stability and transitions of (3) subject to (4). We first prove that the system undergoes a dynamic transition at the critical Reynolds number R c ≈ 4.124ǫ −1/4 . As R crosses R c the steady flow loses its stability, and a transition occurs. If we denote the azimuthal wavenumber of an eigenmode by m, then two modes, called critical modes hereafter, with m = 3 and radial wavenumber 1, become critical at R = R c . Using the language of dynamic transition theory [10], we can show that the transition is either Type-I(continuous) or Type-II(catastrophic). In Type-I transitions, the amplitudes of the transition states stay close to the base flow after the transition. Type-II transitions, on the other hand, are associated with more complex dynamical behavior, leading to metastable states or a local attractor far away from the base flow. We show that the type of transition preferred in system (3) is determined by the real part of a complex parameter A which only depends on ǫ. In the generic case of nonzero imaginary part of A, there are two possible transition scenarios depending on the sign of the real part of A: continuous or catastrophic. In the continuous transition scenario, a stable, small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates on R > R c . The time period of the bifurcated solution tends to infinity as R approaches R c , a phenomenon known as infinite period bifurcation [9]. The dual scenario is the catastrophic transition where the bifurcation is subcritical on R < R c and a repeller bifurcates. In the non-generic case where the imaginary part of A vanishes, the limit cycle degenerates to a circle of steady states. The transition number A depends on the system parameter ǫ in a non-trivial way, hence it is not possible to find an analytical expression of A as a function of ǫ. So, A must be computed numerically for a given ǫ. Physically, the transition number can be considered as a measure of net mechanical energy transferred from all modes back to the critical modes which in turn modify the base flow. We show that A is determined by the nonlinear interactions of the critical modes (m = 3) with all the modes having m = 0 and m = 6. Moreover, our numerical computations suggest that for low ǫ fluids (ǫ < 1), just a single nonlinear interaction, namely the one with m = 0 and radial wavenumber 1 mode, dominates all the rest contributions to A. Our numerical experiments with low ǫ, i.e. ǫ < 1, suggest that the real part of A is always positive indicating a catastrophic transition at R = R c . We also determine the Reynolds number threshold R E > 0 below which the Poiseuille flow is globally stable, attracting all initial conditions with at least exponential convergence in the H 1 0 (Ω) norm for the velocity. We find that R E ≈ 12.87 when ǫ = 0. The gap between R E and R c shrinks to zero quickly as ǫ is increased. The paper is organized as follows: In Section 3, the linearized stability of the system is studied and the principle of exchange of stabilities is investigated. In Section 4, transition theorem of the system is presented with its proof given in Section 5. Section 6 is devoted to the energy stability of the system. In Section 7, we give a detailed numerical analysis. Finally in Section 8 the conclusions and possible extensions of this study are discussed in detail. The model and the functional setting Throughout Re z, Im z, z will denote the real part, imaginary part and the complex conjugate of a complex number z. Let X 1 = H 1 0 (Ω) × H 2 0 (Ω), X = L 2 (Ω) × L 2 (Ω), where Ω is the unit disk in R 2 , H 1 0 (Ω) and H 2 0 (Ω) denote the usual Sobolev spaces and L 2 (Ω) is the space of Lebesgue integrable functions. For φ i = wi(r,θ) ψi(r,θ) ∈ X, i = 1, 2, the inner product on X is defined by (5) φ 1 , φ 2 = 2π 0 1 0 (w 1 w 2 + ψ 1 ψ 2 )rdrdθ, with the norm on X defined as φ 2 = φ, φ . We define the linear operators M : X 1 → X and N : X 1 → X as (6) M =   I − ǫ∆ 0 0 ∆(ǫ∆ − I)   , N =   1 R ∆ R∂ θ ǫR∆∂ θ − 1 R ∆ 2   , and the nonlinear operator H : X 1 → X as H(φ) =   J(ψ, (1 − ǫ∆)w) J((1 − ǫ∆)∆w, ψ) + ǫJ(∆w, w)   , for φ =   w ψ   ∈ X 1 . We will use H to denote both the nonlinear operator as well as the bilinear form (7) H(φ I , φ J ) =   J(ψ I , (1 − ǫ∆)w J ) J((1 − ǫ∆)∆w I , ψ J ) + ǫJ(∆w I , w J )   , for φ I = [w I , ψ I ] T , φ J = [w J , ψ J ] T . Also we will use the symmetrization of this bilinear form (8) H s (φ I , φ J ) = H(φ I , φ J ) + H(φ J , φ I ). Then the equations (3) and (4) can be written in the following abstract form (9) M φ t = N φ + H(φ), φ ∈ X 1 , with initial condition φ(0) = φ 0 ∈ X 1 . Linear Stability To determine the transitions of (9), the first step is to study the eigenvalue problem N φ = βM φ of the linearized operator. This is equivalent to the problem (10) 1 R ∆w + Rψ θ = β(1 − ǫ∆)w, 1 R ∆ 2 ψ − ǫR∆w θ = β(1 − ǫ∆)∆ψ, subject to the boundary conditions (4). An interesting feature of the eigenvalue problem is the following. Lemma 1. Any eigenvalue β of (10) with boundary conditions (4) is real. Proof. Multiplying the first equation of (10) by ∆w and the second equation by ψ, integrating over the domain Ω, we obtain after integration by parts (11) ( 1 R + βǫ) ∆w 2 + R Ω ψ θ ∆wdrdθ = −β ∇w 2 , and (12) − ( 1 R + βǫ) ∆ψ 2 + ǫR Ω ∆w θ ψdrdθ = β ∇ψ 2 . Let A 1 = ǫ ∆w 2 + ∆ψ 2 , A 2 = ǫ ∇w 2 + ∇ψ 2 A 3 = 2ǫ Ω ∆w θ ψdrdθ. Now consider −ǫ×(11) + (12) which is (13) − ( 1 R + βǫ)A 1 + RRe(A 3 ) = βA 2 , after integrating by parts. Taking the imaginary part of (13) gives Im(β)(ǫA 1 + A 2 ) = 0. Since (ǫA 1 + A 2 ) ≥ 0, we must have Im(β) = 0. Now we turn to the problem of determining explicit expressions of the solutions of the eigenvalue problem of the linearized operator. Thanks to the periodicity in the θ variable, for m ∈ Z and j ∈ Z + , we denote the eigenvectors of (10) by (14) φ m,j (r, θ) = e imθ ϕ m,j (r), ϕ m,j (r) =   w m,j (r) ψ m,j (r)   with corresponding eigenvalues β m,j . Let us set the eigenvalues β m,j for m = 0 to be ordered so that β m,1 ≥ β m2 ≥ · · · for each m ∈ Z. Plugging the ansatz (14) into (10) and omitting j we obtain two ODE's in the r-variable. (15) 1 R + βǫ ∆ m w m + imRψ m = βw m , − 1 R + βǫ ∆ 2 m ψ m + iǫmR∆ m w m = −β∆ m ψ m , with boundary conditions (16) w m (1) = ψ m (1) = ψ ′ m (1) = 0 where ∆ m = d 2 dr 2 + 1 r d dr − m 2 r 2 . When m = 0, the equations (15) become decoupled and we easily find that there are two sets of eigenpairs given by w 1 0,j (r) = J 0 (α 0,j r), ψ 1 0,j (r) = 0, β 1 0,j = −α 2 0,j R(1 + ǫα 2 0,j ) , w 2 0,j (r) = 0, ψ 2 0,j (r) = J 0 (α 1,j r) − J 0 (α 1,j ), β 2 0,j = −α 2 1,j R(1 + ǫα 2 1,j ) , where α k,j is the jth zero of the kth Bessel function J k . In particular, β 0,j < 0 for all ǫ and for all R. In the m = 0 case, as the eigenvalues are real by Lemma 1, the multiplicity of each eigenvalue β = β m,j = β −m,j ∈ R is generically two with corresponding eigenvectors φ m,j and φ −m,j = φ m,j . Solving the first equation of (15) for ψ m and plugging it into the second equation of (15) yields a sixth order equation (17) (λ + ∆ m )(µ + ∆ m )∆ m w m = 0 where (18) λ = √ ǫmR − β m,j 1 R + ǫβ m,j , µ = − √ ǫmR − β m,j 1 R + ǫβ m,j . It is easy to check that λ = 0 or µ = 0 yield only trivial solutions to equations (16) and (17). So we will assume λ = 0, and µ = 0. When m > 0, the general solution of (17) is w m = c 1 r m + c 2 J m ( √ λr) + c 3 J m ( √ µr) + c 4 r −m + c 5 Y m ( √ λr) + c 6 Y m ( √ µr), where J m and Y m are the Bessel functions of the first and the second kind respectively. The boundedness of the solution and its derivatives at r = 0 implies that c 4 = c 5 = c 6 = 0 and we get the eigensolutions (19) w m,j (r) = c 1 r m + c 2 J m ( √ λr) + c 3 J m ( √ µr) if m > 0 w −m,j , if m < 0 ψ m,j (r) = d 1 r m + d 2 J m ( √ λr) + d 3 J m ( √ µr) if m > 0 ψ −m,j , if m < 0 with d 1 = −iβ m,j c 1 mR , d 2 = −i √ ǫc 2 , d 3 = i √ ǫc 3 . The eigenvalues and two of the three coefficients c 1 , c 2 , c 3 in (19) are determined by the boundary conditions (16) which form a linear system for the coefficients c k . This system has a nontrivial solution only when the dispersion relation (20) 1 J m ( √ λ) J m ( √ µ) β √ ǫmRJ m ( √ λ) − √ ǫmRJ m ( √ µ) βm √ ǫmR √ λJ ′ m ( √ λ) − √ ǫmR √ µJ ′ m ( √ µ) = 0, is satisfied. Using the identity J ′ m (z) = m z J m (z) − J m+1 (z) we can show that (20) is equivalent to (21) √ λJ m ( √ λ)J m+1 ( √ µ) + √ µJ m ( √ µ)J m+1 ( √ λ) = 0, where J m is the Bessel function of the first kind of order m. To compute the critical Reynolds number R c , we set β = 0 in (21) which, after some manipulation, yields (22) I m ( √ λ)J ′ m ( √ λ) − J m ( √ λ)I ′ m ( √ λ) = 0. Once (22) is solved for λ, the corresponding Reynolds number is obtained by the relation λ = √ ǫmR 2 (from (18) when β = 0). We note here that this is the exact same equation as the one obtained in [12]. For each m, the equation (22) /m) 1/2 for m = 1, . . . , 5. infinitely many solutions {λ m,j } ∞ j=1 where λ m,j increases with j and λ m,j → ∞ as j → ∞. Letting R m,j = ǫ −1/4 (λ m,j /m) 1/2 , we have β m,j = 0 when R = R m,j . We define the critical Reynolds number R c = min m,j R m,j = min m R m,1 = ǫ −1/4 min m∈Z+ λ m,1 m , so that β m,j ≤ 0 if R ≤ R c . There has been recent progress on the properties of zeros of (22). In [1], the estimate 2 4 (m + 1)(m + 2)(m + 3) (m + 4)(m + 5) √ 5m + 15 < λ 2 m,1 < 2 4 (m + 1)(m + 2)(m + 3)(m + 4)(m + 5) 5m + 17 ,(23) on λ m,1 is obtained. Using (23), we can show that the upper bound for λ m,1 /m for m = 3 is less than its lower bound for all m ≥ 7 which implies that λ 3,1 /3 < λ m,1 /m for all m ≥ 7. Thus R c is minimized for some m smaller than 7 and hence can be found by brute force. Looking at Table 1, we find that the expression above is indeed minimized when m = 3 and obtain the relation (24), i.e. R c = R 3,1 ≈ 4.124ǫ −1/4 . Defining the left hand side of (21) as ω(β, R), the equation (21) becomes ω(β, R) = 0. By the implicit function theorem, this defines β 3,1 (R) for R near R c with β 3,1 (R c ) = 0. With the aid of symbolic computation, we can compute dβ 3,1 dR | R=Rc = ∂ω/∂R ∂ω/∂β | R=Rc,β=β3,1 = 0.12 √ ǫ 0.02 + ǫ > 0. Thus we have proved the Principal of Exchange of Stabilities which we state below. Theorem 1. For ǫ = 0, let (24) R c = ǫ −1/4 λ 3,1 3 ≈ 4.124ǫ −1/4 . Then (25) β 3,1 (R) = β −3,1 (R)      < 0 if R < R c = 0 if R = R c > 0 if R > R c β m,j (R c ) < 0, if (m, j) = (±3, 1). Note that Theorem 1 is in contrast to the ǫ = 0 case where the basic flow is linearly stable for all Reynolds numbers. This can be seen easily by noting that when ǫ = 0, the inner product of the second equation in (10) with ψ yields (26) ∆ψ 2 = −βR ∇ψ 2 . With the Dirichlet boundary conditions, from (26), it follows easily that β < 0 if ψ = 0. For the proof and presentation of Theorem 2, we also need to solve the eigenvalue problem of the adjoint linear operator which yields adjoint modes orthogonal to the eigenmodes of the linear operator. Adjoint problem is obtained by taking the inner product of (9) by φ * and moving the derivatives via integration by parts onto φ * by making use of the boundary conditions. This yields the following adjoint problem N * φ * = β * M * φ * , where M * = M =   I − ǫ∆ 0 0 ∆(ǫ∆ − I)   , N * =   1 R ∆ −ǫR∆∂ θ R∂ θ − 1 R ∆ 2   , φ * =   w * ψ *   , and w * , φ * satisfies the same boundary conditions (4) as w and φ. We denote the adjoint eigenvectors by φ * m,j = w * m,j ψ * m,j T and we also have the adjoint eigenvalues β * m,j = β m,j . The reason we introduce the adjoint eigenmodes is to make use of the following orthogonality relation (27) φ m,i , M φ * n,j = 0 if (m, i) = (n, j). Dynamic Transitions Let us briefly recall here the classification of dynamic transitions and refer to [10] for a detailed rigorous discussion. For ǫ = 0, as the critical Reynolds number R c is crossed, the principle of exchange of stabilities (25) dictates that the nonlinear system always undergoes a dynamic transition, leading to one of the three type of transitions, Type-I(continuous), II(catastrophic) or III(random). On R > R c , the transition states stay close to the base state for a Type-I transition and leave a local neighborhood of the base state for a Type-II transition. For Type-III transitions, a local neighborhood of the base state is divided into two open regions with a Type-I transition in one region, and a Type-II transition in the other region. Type-II and Type-III transitions are associated with more complex dynamics behavior. Below we prove that for (3), only two scenarios are possible. In the first scenario, the system exhibits a Type-I (or continuous) transition and a stable attractor will bifurcate on R > R c which attracts all sufficiently small disturbances to the Poiseuille flow. We prove that this attractor is homeomorphic to the circle S 1 and is generically a periodic orbit. The Figure 1 shows the stream function of the bifurcated time-periodic solution. The dual scenario is that the system exhibits a Type-II (or catastrophic) transition. The type of transition at R = R c depends on the transition number (28) A = ∞ j=1 A 0,j + A 6,j , where A m,j represent the nonlinear interaction of the critical modes with the mode with azimuthal wavenumber m and radial wavenumber j. The formulas for A m,j are (29) To recall the meaning of various terms in (29), β n,j is the jth eigenvalue of the nth azimuthal mode with corresponding eigenvector φ n,j and adjoint eigenvector φ * n,j . ·, · denotes the inner product (5), H denotes the bilinear form (7), H s is its symmetrization (8), M is the linear operator defined by (6). A 0,j = 1 φ 3,1 , M φ * 3,1 Φ 0,j H s (φ 3,1 , φ 0,j ), φ * 3,1 A 6,j = 1 φ 3,1 , M φ * 3,1 Φ 6,j H s (φ 3,1 , φ 6,j ), φ * 3,1 Φ 0,j = 1 −β 0,j φ 0,j , M φ * 0,j H s (φ 3,1 , φ 3,1 ), φ * 0,j , Φ 6,j = 1 −β 6,j φ 6,j , M φ * 6,j H(φ 3,1 , φ 3,1 ), φ * 6,j , Theorem 2. If ǫ = 0 then the following statements hold true. (1) If Re(A) < 0 then the transition at R = R c is Type-I and an attractor Σ R bifurcates on R > R c which is homeomorphic to S 1 . If Im(A) = 0 then Σ R is a cycle of steady states. If Im(A) = 0 then Σ R is the orbit of a stable limit cycle given by Here w 3,1 and ψ 3,1 are the vertical velocity and stream function of the eigenmode of the linear operator with corresponding eigenvalue β 3,1 . (2) If Re(A) > 0 then the transition at R = R c is Type-II and a repeller Σ R bifurcates on R < R c . If Im(A) = 0 then Σ R is a cycle of steady states. If Im(A) = 0 then Σ R is the orbit of of an unstable limit cycle given by (30) with β 3,1 replaced by −β 3,1 . Remark. In the generic case of Im(A) = 0, Theorem 2 guarantees the existence of a stable (unstable) bifurcated periodic solution on R > R c (R < R c ). By (31), the period of the bifurcated solution approaches to infinity as R ↓ R c (R ↑ R c ). Proof Of Theorem 2 As is standard in the dynamic transition approach, the proof of Theorem 2 depends on the reduction of the field equations (9) on to the center manifold. Let us denote the (real) eigenfunctions and adjoint eigenfunctions corresponding to the critical eigenvalue β 3,1 by e 1 (r, θ) = Re(φ 3,1 (r, θ)) e 2 (r, θ) = Im(φ 3,1 (r, θ)) e * 1 (r, θ) = Re(φ * 3,1 (r, θ)) e * 2 (r, θ) = Im(φ * 3,1 (r, θ)) By the spectral theorem, the spaces X 1 and X can be decomposed into the direct sum X 1 = E 1 ⊕ E 2 , X = E 1 ⊕ E 2 , where E 1 = span{e 1 , e 2 },E 2 = {u ∈ X 1 | u, e * i = 0 i = 1, 2}, E 2 = closure of E 2 in X. Since M : X 1 → X is an invertible operator, we can define L = M −1 N and G = M −1 H. Now the abstract equation (9) can be written as (32) dφ dt = Lφ + G(φ). The linear operator L in (32) can be decomposed into L = J ⊕ L, J = L | E1 : E 1 → E 1 , L = L | E2 : E 2 → E 2 , Since the eigenvalues are real, we have Le k = β 3,1 e k for k = 1, 2. Hence we have J = β 3,1 (R)I 2 , where I 2 is the 2 × 2 identity matrix. We know that when J is diagonal, we have the following approximation of the center manifold function Φ : E 1 → E 2 near R ≈ R c ; see [10]. (33) − LΦ(x) = P 2 G k (x) + o(k), The meaning of the terms in the above formula (33) are as follows. a) o(k) = o( x k ) + O(|β 3,1 (R)| x k ) as R → R c , x → 0, b) P 2 : X → E 2 is the canonical projection, c) x is the projection of the solution onto E 1 ,(34) x(t, r, θ) = x 1 (t)e 1 (r, θ) + x 2 (t)e 2 (r, θ) d) G k denotes the lowest term of the Taylor expansion of G(u) around u = 0. In our case G is bilinear and thus k = 2 in and G = G k . It is easier to carry out the reduction using complex variables. So we write (34) as (35) x(t, r, θ) = z(t)φ 3,1 (r, θ) + z(t)φ 3,1 (r, θ) where z(t) = 1 2 (x 1 (t) − ix 2 (t)). Let us expand the center manifold function by (36) Φ = (n,j) =(±3,1) Φ n,j (t)φ n,j (r, θ) Plugging the above expansion into the center manifold approximation formula (33), taking inner product with M φ * n,j and using the orthogonality (27) we have (37) Φ n,j = 1 −β n,j φ n,j , M φ * n,j H(x), φ * n,j + o(2). Since H is bilinear, H(x) = H(zφ 3,1 + zφ 3,1 ) = z 2 H(φ 3,1 , φ 3,1 ) + zzH s (φ 3,1 , φ 3,1 ) + z 2 H(φ 3,1 , φ 3,1 ), with the operator H s defined by (8). Thanks to the orthogonality 2π 0 e inθ e −imθ dθ = 2πδ nm , we have (39) H(φ m1,i1 , φ m2,i2 ), φ * m3,i3 = 0, if m 1 + m 2 = m 3 With φ 3,1 = φ −3,1 , this implies (40) H(x), φ * n,j = 0 if n / ∈ {0, −6, 6}. According to (40), (36) and (37), (41) Φ(t) = ∞ j=1 Φ 0,j (t)φ 0,j + Φ 6,j (t)φ 6,j + Φ −6,j (t)φ −6,j + o(2) That is the center manifold is o(2) in eigendirections whose azimuthal wavenumber is not 0, 6 or −6. The equation (38) implies that (42) and (37), we get the coefficients of the center manifold in (41) (42) H(x), φ * 0,j = zz H s (φ 3,1 , φ 3,1 ), φ * 0,j , H(x), φ * 6,j = z 2 H(φ 3,1 , φ 3,1 ), φ * 6,j , H(x), φ * −6,j = H(x), φ * 6,j . By(43) Φ 0,j = zzΦ 0,j + o(2), Φ 6,j = z 2Φ 6,j + o(2), Φ −6,j = Φ 6,j , whereΦ 0,j andΦ 6,j are given by (29). As the dynamics of the system is enslaved to the center manifold for small initial data and for Reynolds numbers close to the critical Reynolds number R c , it is sufficient to investigate the dynamics of the main equation (9) on the center manifold. For this reason we take φ(t) = x(t) + Φ(t), in (9) to obtain (44) dz dt M φ 3,1 + dz dt M φ 3,1 = zN φ 3,1 + zN φ 3,1 + H(x + Φ). To project the above equation onto the center-unstable space E 1 , we take inner product of (44) with φ * 3,1 and use M φ 3,1 , φ * 3,1 = 0, and N φ 3,1 = β 3,1 M φ 3,1 , N φ 3,1 = β 3,1 M φ 3,1 , to get the following reduced equation of (9). (45) dz dt = β 3,1 (R)z + 1 φ 3,1 , M φ * 3,1 H(x + Φ), φ * 3,1 . The reduced equation (45) describes the transitions of the full nonlinear system for R near R c and small initial data. At this stage, the nonlinear term in (45) is too complicated to explicitly describe the transition. Thus we need to determine the lowest order expansion in z of the nonlinear term H(x + Φ), φ * 3,1 . By the bilinearity of H, (46) H(x + Φ), φ * 3,1 = H(x), φ * 3,1 + H s (x, Φ), φ * 3,1 + H(Φ), φ * 3,1 . The first term in (46) vanish by (40) and the last term in (46) is o(3) as H(Φ) = o(3) since Φ = O(2) and H is bilinear. Thus (46) becomes (47) H(x + Φ), φ * 3,1 = H s (x, Φ), φ * 3,1 + o(3) . Using the expression (35) for x, we can rewrite (47) as (48) H(x + Φ), φ * 3,1 = z H s (φ 3,1 , Φ), φ * 3,1 + z H s (φ 3,1 , Φ), φ * 3,1 + o(3) . Now we use the expansion (41) of Φ in (48) and the orthogonality relations H s (φ 3,1 , φ n,j ), φ * 3,1 = 0 if n = 0, and H s (φ 3,1 , φ n,j ), φ * 3,1 = 0 if n = 6, which follow from (39) to arrive at (49) H(x + Φ), φ * 3,1 = ∞ j=1 zΦ 0,j H s (φ 3,1 , φ 0,j ), φ * 3,1 + zΦ 6,j H s (φ 3,1 , φ 6,j ), φ * 3,1 + o(3). Defining the coefficient A by (28) and making use of (43) and (49), we write down the approximate equation of (45) as (50) dz dt = β 3,1 (R)z + A|z| 2 z + o(3). To finalize the proof, there remains to analyze the stability of the zero solution of (50) for small initial data. In polar coordinates z(t) = |z|e iγ , (50) is equivalent to (51) d|z| dt = β 3,1 (R)|z| + Re(A)|z| 3 + o(|z| 3 ), dγ dt = Im(A)|z| 2 + o(|z| 3 ). For R > R c as β 3,1 > 0, it is clear from (51) that z = 0 is unstable if Re(A) > 0 and is locally stable if Re(A) < 0. In the latter case, the bifurcated solution is z(t) = −β 3,1 (R) Re(A) exp −i Im(A) Re(A) β 3,1 (R)t . Thus to determine the stability of the bifurcated state as R crosses the critical Reynolds number R c , we need to compute the sign of the real part of A. The details of the assertions in the proof of Theorem 2 follow from the attractor bifurcation theorem in [10]. That finishes the proof. Energy Stability In this section we study the energy stability of the equations (3) which is related to at least exponential decay of solutions to the base flow. We refer to [16] for a multitude of applications of this theory. For f , g, h in H 1 0 (Ω), the following two properties of J follows from integrating by parts d dt ( w 2 + ǫ ∇w 2 ) = − 1 R ∇w 2 + R ψ θ , w + ǫ J(∆w, ψ), w ,(54)1 2 d dt ( ∇ψ 2 + ǫ ∆ψ 2 ) = − 1 R ∆ψ 2 + ǫR ∆w θ , ψ + ǫ J(∆w, w), ψ . Adding equations (53) and (54) and using (52) once again, we arrive at (55) 1 2 d dt E(t) = − 1 R I 1 (t) + RI 2 (t), where E = w 2 + ǫ ∇w 2 + ∇ψ 2 + ǫ ∆ψ 2 I 1 = ∇w 2 + ∆ψ 2 I 2 = ψ θ , w − ǫ∆w . Letting (56) 1 R 2 E = max X1\{0} I 2 I 1 , we have by (55) (57) d dt E ≤ −2R( 1 R 2 − 1 R 2 E )I 1 . Since I 1 ≥ 0 and I 2 = 0 whenever ψ θ = 0, R E must be nonnegative. Since w ∈ H 1 0 (Ω) and ∇ψ ∈ H 1 0 (Ω), by the Poincaré inequality, |∇w| 2 ≥ η 1 |w| 2 and |∆ψ| 2 ≥ η 1 |∇ψ| 2 , where η 1 ≈ 5.78 is the first eigenvalue of negative Laplacian on Ω. Thus we have (58) I 1 ≥ η 1 1 + ǫη 1 E Now let c R = 2Rη 1 1 + ǫη 1 ( 1 R 2 − 1 R 2 E ), and suppose that R < R E . Then c R > 0 and by (57) and (58), d dt E(t) ≤ −c R E(t). Hence the Gronwall's inequality implies E(t) ≤ e −cRt E(0). In particular, for R ≤ R E , c R > 0 and any initial disturbance in X 1 will decay to zero implying the unconditional stability of the basic steady state solution. Using the variational methods to maximize the quantity in (56), we find the resulting Euler-Lagrange equations as (59) ∆w + R 2 2 (1 − ǫ∆)ψ θ = 0, ∆ 2 ψ + R 2 2 (1 − ǫ∆)w θ = 0. Considering (59) as an eigenvalue problem with R playing the role of the eigenvalue, R E is just the smallest positive eigenvalue. To solve (59), we plug the ansatz w = e imθ w m (r) and ψ = e imθ ψ m (r) into (59) which yields (60) ∆ m w m + i mR 2 2 (1 − ǫ∆ m )ψ m = 0 ∆ 2 m ψ m + i mR 2 2 (1 − ǫ∆ m )w m = 0, where ∆ m = d 2 dr 2 + 1 r d dr − m 2 r 2 . Taking ∆ m of the second equation above and using the first equation, we obtain (61) p(∆ m )ψ m = 0 where p(ξ) = ξ 3 + m 2 R 4 4 (1 − ǫξ) 2 . Let ξ 1 , ξ 2 and ξ 3 be the three roots of p. As the discriminant of p is negative, one root is real and the others are complex conjugate. The factorization of the operator in (61) gives (62) (∆ m − ξ 1 )(∆ m − ξ 2 )(∆ m − ξ 3 )ψ m = 0. The general solution of (62) is ψ m = 3 k=1 c k I m ( ξ k r) +c k K m ( ξ k r), where I m and K m are the modified Bessel functions. The boundedness of the solution at r = 0 necessitatesc k = 0 for k = 1, 2, 3. Thus ψ m = 3 k=1 c k I m ( ξ k r).ξ −1 1 I m ( √ ξ 1 ) ξ −1 2 I m ( √ ξ 2 ) ξ −1 3 I m ( √ ξ 3 ) I m ( √ ξ 1 ) I m ( √ ξ 2 ) I m ( √ ξ 3 ) √ ξ 1 I ′ m ( √ ξ 1 ) √ ξ 2 I ′ m ( √ ξ 2 ) √ ξ 3 I ′ m ( √ ξ 3 ) = 0. For fixed m and ǫ, the equation (63) has infinitely many solutions R = R m,j (ǫ), j ∈ Z + . Letting (64) R m = min j∈Z+ R m,j , the critical Reynolds number is given by (65) R E = min m∈Z+ R m . We present the numerical computations of R E in the next section. A N , where A N is the series in (28) truncated at N , i.e. A N = N j=1 A 0,j + A 6,j and A m,j represents the nonlinear interaction of the critical modes and the mode with azimuthal wavenumber m and radial wavenumber j given by (29). A symbolic computation software is used to compute A N . We present our numerical computations of A N in Figure 2 for ǫ = 1, 10 −1 , 10 −2 , 10 −3 and 1 ≤ N ≤ 10. The imaginary part of A is nonzero and we are only interested in the sign of the real part of A to determine the type of transition according to Theorem 2. To simplify the presentation, we scale all A N 's so that |Re(A 1 )| = 1. The plots in Figure 2 suggest that the convergence of the truncations A N → A is rapid for small ǫ but a higher order truncation (larger N ) is necessary to accurately resolve A for larger ǫ. For ǫ < 10 −1 , even A 1 is a good approximation to determine the sign of A. For example, the relative error for approximating A with A 1 is approximately %2 for ǫ = 10 −3 and increases to approximately %18 for ǫ = 1. We also measure the relative strength of the nonlinear interactions, i.e. the ratio (66) B N = N j=1 Re(A 6,j ) N j=1 Re(A 0,j ) , in Figure 3. It is seen from Figure 3 that the contribution from the modes with m = 0 dominates when ǫ is low. But as ǫ increases, the contribution from modes with m = 6 start to become significant. For example, for ǫ = 10 −3 , B N approaches 8 × 10 −5 , for ǫ = 10 −2 , B N approaches −2 × 10 −3 and for ǫ = 10 −1 , B N approaches −8 × 10 −3 . In particular, for low ǫ, we have A ≈ A 1 ≈ A 0,1 . More significantly, our numerical results presented in Figure 2 show that the real part of A is positive for ǫ = 10 −3 , 10 −2 , 10 −1 , 1, meaning that the transition is catastrophic by Theorem 2. Thus the system moves to a flow regime away from the base Poiseuille flow and the system exhibits complex dynamical behavior for R > R c . 7.2. Determination of Energy Stability Threshold R E . With a standard numerics package, R m (ǫ) in (64) can be computed for given m and ǫ. Then by (65), R E is computed by taking minimum in (65) over all (computed) R m . In Table 2, it is shown that for ǫ = 10 −4 , and ǫ = 10 −3 , R E is obtained for m = 1 while for ǫ = 10 −2 and ǫ = 2 × 10 −2 , R E is obtained for m = 2. In Figure 5, we plot R m (m = 1, 2, 3) for 0 < ǫ ≤ 5 × 10 −2 . We see that the curves R 1 and R 2 intersect approximately at ǫ = 0.009 while R 2 and R 3 intersect approximately at ǫ = 0.024. As ǫ is increased, the value of m for which R E is minimized also increases. For higher values of m, the roots of the determinant in (63) becomes increasingly hard to find. In Table 2, the last column gives the value of R c , the linear instability threshold, computed by (24). Note that the interval [R E , R c ] consists of Reynolds numbers for which the base flow is either not globally stable or globally stable but not not exponentially attracting. We plot the R E and R c data from Table 2 in Figure 4 which shows that this interval shrinks rapidly as ǫ is increased. Table 2. R m denotes the first positive root of (63) and R E is the minimum of R m taken over all m. R c is the linear instability threshold. ǫ R 1 R 2 R 3 R 4 R 5 R E R Concluding Remarks In this work, we considered both the energy stability and transitions of the Poiseuille flow of a second grade fluid in an infinite circular pipe with the restriction that the flow is independent of the axial variable z. We show that unlike the Newtonian (ǫ = 0) case, in the second grade model (ǫ = 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold R c ≈ 4.124ǫ −1/4 where ǫ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects. At R = R c we prove that a transition takes place and that the type of the transition depends on a complex number A. In particular depending on A, generically, either a continuous transition to a periodic solution or a catastrophic transition occurs where the bifurcation is subcritical. The time period of the periodic solution approaches to infinity as R approaches R c , a phenomenon known as infinite period bifurcation. We show that the number A = ∞ j=1 A 0,j + A 6,j where A m,j denotes the nonlinear interaction of the two critical modes with the mode having azimuthal wavenumber m and radial wavenumber j. Our numerical results suggest that for low ǫ (ǫ < 1), A is approximated well by A 0,1 . That is, a single nonlinear interaction between the critical modes and the mode with azimuthal wavenumber 0 and radial wavenumber 1 dominates all the rest of interactions and hence determines the transition. Our numerical results suggest that a catastrophic transition is preferred for low ǫ values (ǫ < 1). This means, an unstable time periodic solution bifurcates on R < R c . On R > R c , the system has either metastable states or a local attractor far away from the base flow and a more complex dynamics emerges. We also show that for R < R E with R E < R c , the Poiseuille flow is at least exponentially globally stable in the H 1 0 (Ω) norm for the velocity. We find that R E ≈ 12.87 when ǫ = 0 and the gap between R E and R c diminishes quickly as ǫ is increased. There are several directions in which this work can further be extended. First, in this work we consider a pipe with a circular cross section of unit radius and find that the first two critical modes have azimuthal wavenumber equal to 3. Increasing (or decreasing) the radius of the cross section will also increase (decrease) the azimuthal wavenumber of the critical modes. The analysis in this case would be similar to our presentation except in the case where four critical modes with azimuthal wave numbers m and m + 1 become critical. In the case of four critical modes, more complex patterns will emerge due to the cross interaction of the critical modes [15,3]. An analysis in the light of [14] is required to determine transitions in this case of higher multiplicity criticality. Second, for the Reynolds number region between R E and R c , there may be regions where the base flow is either globally stable but not exponentially attractive or regions where the domain of attraction of the base flow is not the whole space. A conditional energy stability analysis is required to resolve these Reynolds number regimes [16]. Third, the results we proved in this work for the second grade fluids can also be extended to fluids of higher grades and to other types of shear flows. Fourth, in this work we restricted attention to 2D flows. In the expense of complicating computations and results, a similar analysis could be considered for 3D flows which depend also on the axial variable z. Figure 1 . 1The time periodic stream function ψ per given by (30) which rotates in time, clockwise if T > 0 and counterclockwise if T < 0. if f , g and h are linearly dependent. Taking inner product of the first equation in (3) with w and the second equation in (3) with ψ and using the property (52) Now applying the operator 1 − ǫ∆ m to the second equation in (60) and using the first equation of (60), we obtain p(∆ m )w m = 0, i.e. the same equation (61), this time for w m . Hencew m = 3 k=1 d k I m ( ξ k r).The first equation in (60) gives the relation d k = −i mR 2 2 (ǫ − ξ −1 k )c k between c k 's and d k 's. Now the boundary conditions w m (1) = ψ m (1) = ψ ′ m (1) = 0 constitute a homogeneous system of three linear equations for the coefficients c k 's. The existence of nontrivial solutions is then equivalent to the vanishing determinant of this system which after some manipulation becomes (63) Figure 2 . 2The plots of Re(A N ) for different ǫ values. All A N 's are scaled so that |Re(A 1 )| = 1. Figure 3 . 3B N , defined by (66), measures the relative strength of the nonlinear interactions of the critical modes with m = 6 modes to the m = 0 modes. Figure 4 . 4R E and R c curves in the ǫ − R plane. Figure 5 . 5The plot of R m vs ǫ for m = 1, 2, 3. Table 1. The smallest positive root λ m,1 of (22) and (λ m,1has m 1 2 3 4 5 6 λ m,1 21.260 34.877 51.030 69.665 90.739 114.21 (λ m,1 /m) 1/2 4.610 4.175 4.124 4.173 4.260 4.36 c 0 12.87 13.49 14.84 16.37 17.95 12.87 ∞ 10 −4 12.86 13.47 14.81 16.32 17.88 12.86 42.4 10 −3 12.77 13.31 14.54 15.90 17.29 12.77 23.84 10 −2 11.99 11.95 12.44 12.95 13.39 11.95 13.40 2 × 10 −2 11.26 10.83 10.91 11.03 11.13 10.83 11.27 Á Baricz, S Ponnusamy, S Singh, arXiv:1507.01104Cross-product of Bessel functions: monotonicity patterns and functional inequalities. arXiv preprintÁ. Baricz, S. Ponnusamy, and S. Singh, Cross-product of Bessel functions: monotonicity patterns and functional inequalities, arXiv preprint arXiv:1507.01104, (2015). An approximation theorem for functionals, with applications in continuum mechanics. B D Coleman, W Noll, Archive for Rational Mechanics and Analysis. 6B. D. Coleman and W. Noll, An approximation theorem for functionals, with applications in continuum mechanics, Archive for Rational Mechanics and Analysis, 6 (1960), pp. 355-370. Dynamic transitions of surface tension driven convection. H Dijkstra, T Sengul, S Wang, Physica D: Nonlinear Phenomena. 247H. Dijkstra, T. Sengul, and S. Wang, Dynamic transitions of surface tension driven convection, Physica D: Nonlinear Phenomena, 247 (2013), pp. 7-17. Fluids of differential type: critical review and thermodynamic analysis. J Dunn, K , International Journal of Engineering Science. 33J. Dunn and K. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, International Journal of Engineering Science, 33 (1995), pp. 689-729. Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Archive for Rational Mechanics and Analysis. J E Dunn, R L Fosdick, 56J. E. Dunn and R. L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Archive for Rational Mechanics and Analysis, 56 (1974), pp. 191-252. Starting solutions for some unsteady unidirectional flows of a second grade fluid. C Fetecau, C Fetecau, International Journal of Engineering Science. 43C. Fetecau and C. Fetecau, Starting solutions for some unsteady unidirectional flows of a second grade fluid, International Journal of Engineering Science, 43 (2005), pp. 781-789. Stability of stratified rotating viscoelastic Rivlin-Ericksen fluid in the presence of variable magnetic field. R K Gupta, M Singh, Advances in Applied Science Research. 3R. K. Gupta and M. Singh, Stability of stratified rotating viscoelastic Rivlin-Ericksen fluid in the presence of variable magnetic field, Advances in Applied Science Research, 3 (2012), pp. 3253-3258. Transient flows of a second grade fluid. T Hayat, M Khan, A Siddiqui, S Asghar, International Journal of Non-Linear Mechanics. 39T. Hayat, M. Khan, A. Siddiqui, and S. Asghar, Transient flows of a second grade fluid, International Journal of Non-Linear Mechanics, 39 (2004), pp. 1621 -1633. Infinite period bifurcation and global bifurcation branches. J P Keener, SIAM Journal on Applied Mathematics. J. P. Keener, Infinite period bifurcation and global bifurcation branches, SIAM Journal on Applied Mathematics, 41 (1981), pp. 127-144. . T Ma, S Wang, Springer-VerlagNew YorkPhase transition dynamicsT. Ma and S. Wang, Phase transition dynamics, Springer-Verlag New York, 2014. Pulsatile flow of blood using a modified second-grade fluid model. M Massoudi, T X Phuoc, Computers & Mathematics with Applications. 56M. Massoudi and T. X. Phuoc, Pulsatile flow of blood using a modified second-grade fluid model, Computers & Mathematics with Applications, 56 (2008), pp. 199 -211. Stability of Poiseuille flow of an incompressible second-grade Rivlin-Ericksen fluid. S Özer, E Şuhubi, ARI-An International Journal for Physical and Engineering Sciences. 51S.Özer and E. Şuhubi, Stability of Poiseuille flow of an incompressible second-grade Rivlin- Ericksen fluid, ARI-An International Journal for Physical and Engineering Sciences, 51 (1999), pp. 221-227. Existence, uniqueness and stability of steady flows of second and third grade fluids in an unbounded "pipe-like" domain. A Passerini, M C Patria, International Journal of Non-Linear Mechanics. 35A. Passerini and M. C. 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Straughan, The energy method, stability, and nonlinear convection, vol. 91, Springer Science & Business Media, 2013. On the non-linear instability of fiber suspensions in a Poiseuille flow. Z Wan, J Lin, H Xiong, International Journal of Non-Linear Mechanics. 43Z. Wan, J. Lin, and H. Xiong, On the non-linear instability of fiber suspensions in a Poiseuille flow, International Journal of Non-Linear Mechanics, 43 (2008), pp. 898-907.

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{'abstract': 'In this study we consider the stability and transitions for the Poiseuille flow of a second grade fluid which is a model for non-Newtonian fluids. We restrict our attention to flows in an infinite pipe with circular cross section that are independent of the axial coordinate.We show that unlike the Newtonian (ǫ = 0) case, in the second grade model (ǫ = 0 case), the time independent base flow exhibits transitions as the Reynolds number R exceeds the critical threshold Rc ≈ 4.124ǫ −1/4 where ǫ is a material constant measuring the relative strength of second order viscous effects compared to inertial effects.At R = Rc, we find that generically the transition is either continuous or catastrophic and a small amplitude, time periodic flow with 3-fold azimuthal symmetry bifurcates. The time period of the bifurcated solution tends to infinity as R tends to Rc. Our numerical calculations suggest that for low ǫ values, the system prefers a catastrophic transition where the bifurcation is subcritical.We also find that there is a Reynolds number R E with R E < Rc such that for R < R E , the base flow is globally stable and attracts any initial disturbance with at least exponential speed. We show that R E ≈ 12.87 at ǫ = 0 and R E approaches Rc quickly as ǫ increases.', 'arxivid': '1509.03606', 'author': ['Saadet Ozer ', 'Taylan Sengul '], 'authoraffiliation': [], 'corpusid': 119696017, 'doi': '10.1016/j.physd.2016.05.012', 'github_urls': [], 'n_tokens_mistral': 16040, 'n_tokens_neox': 13943, 'n_words': 8435, 'pdfsha': '69446cfe096dae3cd9b7fe95a67b72c74224d441', 'pdfurls': ['https://arxiv.org/pdf/1509.03606v1.pdf'], 'title': ['STABILITY AND TRANSITIONS OF THE SECOND GRADE POISEUILLE FLOW', 'STABILITY AND TRANSITIONS OF THE SECOND GRADE POISEUILLE FLOW'], 'venue': []}

arxiv

Simple derivation of the first cumulant for the Rouse chain V Lisy Department of Physics Technical University of Kosice Park Komenskeho 2042 00KosiceSlovakia B Brutovsky Institute of Physics P.J. Safarik University Jesenna 5041 54KosiceSlovakia J Tothova Institute of Physics P.J. Safarik University Jesenna 5041 54KosiceSlovakia Simple derivation of the first cumulant for the Rouse chain 1 A simple analytic expression for the first cumulant of the dynamic structure factor of a polymer coil in the Rouse model is derived. The obtained formula is exact within the usual assumption of the continuum distribution of beads along the chain. It reflects the contributions to the scattering of light or neutrons from both the internal motion of the polymer and its diffusion, and is valid in the whole region of the wave-vector change at the scattering. _ The Rouse model [1] is a fundamental theory for the dynamics of a polymer chain in liquids. Despite its simplicity, it finds a number of applications, e.g., in the description of polymer dynamics in situations when the hydrodynamic interactions between the monomers are screened out by surrounding polymers [2]. Such effects are extensively studied both theoretically and in experiments, first of all by quasielastic scattering of light and neutrons. The central quantities of interest in these studies are the dynamic structure factor (DSF) and its first cumulant Γ, approximate formulas for which have been obtained already in the classic de Gennes' work forty years ago [3]. Later, Akcasu et al. [4] derived closed expressions for the first cumulant both in the free-draining limit and with hydrodynamics included, but these formulas are too complicated to be used in the interpretation of experimental data. As discussed in the recent paper [5], they are given as summations over the number of beads in the chain and produce simple results only in the limit of large scattering vectors. Due to this, experimenters use limiting results (the well-known q 2 and q 4 laws [2]) that are correct at small and large scattering vectors q, but not valid in the intermediate region of q. In the mentioned work [5], an analytic expression for a different quantity, called the mean decay rate of the internal autocorrelation function of a Gaussian coil, has been obtained in the free-drainage limit for chains of length much larger than the size of the monomer. As will be shown below, this quantity (if also the diffusion of the whole coil is taken into account) can be related to Γ. To our knowledge, however, the exact analytic formula for the Rouse cumulant itself has not been explicitly given in the literature. In this note, the first Rouse cumulant to the DSF is exactly calculated. The found expression as well as the method of its obtaining are very simple. The cumulant, reflecting the contributions to the scattering from both the internal motion within the coil and its diffusion as a whole, describes the experimentally measured initial slope of the DSF. For a long Gaussian polymer coil mapped by N beads, the DSF is determined as [2] ( ) ( ) [ ] ∑ Φ − = m n nm t q N t q G , 2 exp 1 , ,(1)( ) ( ) ( ) ∑ ∞ =       −       + + = Φ 1 2 2 cos cos 2 cos cos 0 2 p p p nm N np N mp t N np N mp Dt t π π ψ π π ψ . (2) Here, /(3 ) B D k T N d π η = is the diffusion coefficient of the coil (η is the solvent viscosity and d the diameter of the bead), the indexes m and n change from 0 to N, and ψ p (t) is the time correlation function of the Fourier components of the bead radius vector [2]. As usually, the continuum approximation for the distribution of beads is used. This implies that the number of the internal modes in Eq. (2) is infinite. In the Rouse case, ( ) (0) exp( / ) p p p t t ψ ψ τ = − with 2 2 2 (0) /(6 ) p Nb p ψ π = and the relaxation times 2 2 2 /( ) p B N b d bk Tp τ η π = , where b is the mean-square distance between the beads along the chain [2]. The static structure factor is [2] ( ) ( ) { } 1 1 , 0 2 1 1 exp G q Nκ κ κ − −   = − − −   , ( ) 2 G qR κ ≡ .(3) Here, R G = (Nb 2 /6) 1/2 is the radius of gyration of the coil. The first cumulant of the DSF is defined as [2] ( ) ( ) [ ] 0 1 , 0 , = − ∂ ∂ − = Γ t t t q G q G .(4) At t = 0, where we have used (0) / / 2 p p D ψ τ = . The time derivative of the DSF is ( ) ( ) ∑ − −       Φ =       ∂ ∂ − = = nm G t nm t N m n R q dt t d N q t G / exp 2 2 0 2 0 .(6) Calculating the sum, we convert it to the double integral. The integral is easily found noting that the quantity in the square brackets in Eq. (8), normalized to q 2 D, is ( Fig. 1) ( ) 2 2 1 2 exp 1 R q D κ γ κ κ Γ = = + − − , ( ) 2 G qR κ = ,(9) with the well-known limiting cases 1 / 3 ... In Ref. [5], the mean decay rate of the internal autocorrelation function of a Gaussian coil was calculated using the definition R γ κ ≈ + + ( 0 κ → ) and / 2 R γ κ ≈ ( 1 κ >> ) [2].( ) int int, 1 0 ln t G G K t ∞ =   ∂ −   = − ∂ = ∞ − − int, 0 int, ' 0 int, G G G ,(10) where G int (q,t) is given by Eq. (11) 1 2 exp exp 1 3 n n n G N N N κ κ ∞         = − −                 ∑ = 2 exp erf 6 2 N π κ κ κ       −                 ,(12) where the sum was again replaced with integral (κ is from Eq. 9). Since Γ is related to K 1 by the equation 2 1 int, int,0 (1 / ) q D K G G ∞ Γ = + − (obtained after the derivation of G(q,t) and using the definition (10) of K 1 ), one easily recovers K 1 from Ref. [5] without complicated calculations of ' 0 int, G . However, the quantity that is actually measured in experiments is not K 1 but the cumulant (4) of the total DSF, given by a much simpler formula (9). Note that if q 2 b 2 /6 << 1, Γ from Eq. (9) can be also identified with the half-width of the quasi-elastic peak in the scattering law G(q,ω) found in the paper by Akcasu and Gurol [7]. Finally, it should be stressed that the results presented here and in Ref. [5] are valid if in the equations of motion of the chain the continuum approximation for the position vectors of beads is used. In building discrete variants of the theory, with a finite number of internal modes, the relaxation times and eigenvectors different from the usual textbook results [2] should be used [8]. As shown in Ref. [8], these results apply if the mode number p is much smaller than the number of beads N. Nevertheless, the continuum approximation is well substantiated for the long-time polymer dynamics [2], in which only case the bead-spring models are reasonable. the interval [0, N] through the set of the orthonormal functions Fig. 1 . 1The normalized first cumulant (9) (full line) and its high-and low-q limits for a Rouse polymer (dashed and dotted lines, respectively). The analytic expression (9) is indistinguishable from the numerical calculation from Eqs. (4), (1), and (2). the derivative of G int (q,t) evaluated at t = 0, G int,0 coincides with G(q,0), and G int, Acknowledgment. We are indebted to T. Matsoukas for illuminating remarks on his work[5]. VL and BB acknowledge the support from the Scientific Grant Agency of the Slovak Republic (VEGA). . P E Rouse, J Chem Phys. 211272Rouse PE. J Chem Phys (1953) 21, 1272 The Theory of Polymer Dynamics. M Doi, S F Edwards, Oxford, ClarendonDoi M, Edwards SF. The Theory of Polymer Dynamics. Oxford, Clarendon, 1986 . P-G De Gennes, Physics. 337de Gennes P-G. Physics (1967) 3, 37 . A Z Akcasu, M Benmouna, C C Han, Polymer, 21866Akcasu AZ, Benmouna M, Han CC. Polymer (1980) 21, 866 . T Matsoukas, Macromolecules. 396693Matsoukas T. Macromolecules (2006) 39, 6693 . A P Prudnikov, Brychkov Yua, Marichev OI. Integrals and Series. Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series. Moscow, Nauka, 1981 . Z Akcasu, H. J Polymer Gurol, Sci, Polymer Phys Edn. 141Akcasu Z, Gurol H. J Polymer Sci: Polymer Phys Edn (1976) 14, 1 . R Rzehak, W Zimermann, Phys Rev E. 6821804Rzehak R, Zimermann W. Phys Rev E (2003) 68, 021804

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{'abstract': 'A simple analytic expression for the first cumulant of the dynamic structure factor of a polymer coil in the Rouse model is derived. The obtained formula is exact within the usual assumption of the continuum distribution of beads along the chain. It reflects the contributions to the scattering of light or neutrons from both the internal motion of the polymer and its diffusion, and is valid in the whole region of the wave-vector change at the scattering.', 'arxivid': '0709.4099', 'author': ['V Lisy \nDepartment of Physics\nTechnical University of Kosice\nPark Komenskeho 2042 00KosiceSlovakia\n', 'B Brutovsky \nInstitute of Physics\nP.J. Safarik University\nJesenna 5041 54KosiceSlovakia\n', 'J Tothova \nInstitute of Physics\nP.J. Safarik University\nJesenna 5041 54KosiceSlovakia\n'], 'authoraffiliation': ['Department of Physics\nTechnical University of Kosice\nPark Komenskeho 2042 00KosiceSlovakia', 'Institute of Physics\nP.J. Safarik University\nJesenna 5041 54KosiceSlovakia', 'Institute of Physics\nP.J. Safarik University\nJesenna 5041 54KosiceSlovakia'], 'corpusid': 119220381, 'doi': '10.1515/epoly.2007.7.1.1740', 'github_urls': [], 'n_tokens_mistral': 2887, 'n_tokens_neox': 2472, 'n_words': 1540, 'pdfsha': '5b647484435c61f979378388b2c070bdfb4775eb', 'pdfurls': ['https://export.arxiv.org/pdf/0709.4099v1.pdf'], 'title': ['Simple derivation of the first cumulant for the Rouse chain', 'Simple derivation of the first cumulant for the Rouse chain'], 'venue': []}

arxiv

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory: Supporting Information Antoine Marie Department of Chemistry ) Physical and Theoretical Chemical Laboratory University of Oxford OX1 3QZOxfordU.K Laboratoire de Chimie et Physique Quantiques (UMR 5626 Université de Toulouse CNRS UPS 31062ToulouseFrance Hugh G A Burton Department of Chemistry ) Physical and Theoretical Chemical Laboratory University of Oxford OX1 3QZOxfordU.K Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory: Supporting Information (Dated: 20 April 2023) S1. Deriving the gradient and second-derivative of the CASSCF energy S1.1. Notation and reduced density matrices Here, we provide explicit expressions for the gradient and Hessian of the CASSCF energy, which are required for quasi-Newton optimisation algorithms. These expressions are derived in detail in Refs. 1 and 2, but we include this pedagogical discussion for completeness. In what follows, the indices m, n, p, q, r, s correspond to arbitrary spatial orbitals, i, j, k, l correspond to occupied orbitals, a, b, c, d correspond to virtual orbitals, and t, v, x, y correspond to active orbitals. Furthermore, we employ the groundstate one-and two-body reduced densities, γ pq = σ Ψ 0 |â † qσâpσ |Ψ 0 ,(S1a)Γ pqrs = στ Ψ 0 |â † pσâ † rτâsτâqσ |Ψ 0 ,(S1b) and the transition density matrices γ K pq = σ Ψ 0 |â † qσâpσ |Ψ K ,(S2a)Γ K pqrs = στ Ψ 0 |â † pσâ † rτâsτâqσ |Ψ K .(S2b) Explicit expressions for the non-zero matrix elements can be simplified depending on the types of orbitals involved, giving expressions for the ground-state density matrices as γ i j = 2δ i j (S3) and Γ i jkl = 4δ i j δ kl − 2δ il δ k j , Γ i jtv = 2δ i j γ vt , and Γ ivt j = −δ i j γ vt .(S4) where the purely active components γ tv and Γ tvxy cannot be simplified beyond Eq. S1. The only non-zero component of the onebody transition density matrix is γ K tv . In addition to the Γ K tvxy components, the non-zero terms of the two-body transition density matrix include Γ K i jtv = 2δ i j γ K vt , and Γ K ivt j = −δ i j γ K vt . (S5) S1.2. Gradient terms The gradient can be divided into the orbital components g o and the CI components g c that are defined in the main text. The CI components are given by the Hamiltonian matrix elements in the configuration space, that is g c K = 2 Ψ 0 |Ĥ|Ψ K .(S6) The orbital components are given by with its anti-symmetrized variantÊ − mn =Ê mn −Ê nm . g o mn = Ψ 0 |[Ĥ,Ê − mn ]|Ψ 0( S1.3. Hessian terms Similarly, we can divide the Hessian Q into three components: Q cc , Q oo and Q oc (where H co = (Q oc ) † ). The CI-CI components are given by the Hamiltonian matrix elements within the active configuration space shifted by the current energy E 0 , giving Q cc K,L = 2 Ψ K |Ĥ − E 0 |Ψ L . (S10) The off-diagonal components corresponding to the orbital-CI matrix elements are given by Q oc mn,K = Ψ 0 |[Ĥ,Ê − mn ]|Ψ K (S11) where the constituent transition matrix elements are computed as Ψ K |[Ĥ,Ê mn ]|Ψ 0 = p h mp γ K pn + γ K np h pm + prs (pm|rs)Γ K pnrs + (mp|rs)Γ K nprs . (S12) The final orbital-orbital term is given by Q oo pq,rs = 1 2 Ψ 0 |[[Ĥ,Ê − pq ],Ê − rs ]|Ψ 0 + Ψ 0 |[[Ĥ,Ê − rs ],Ê − pq ]|Ψ 0 . (S13) While a full derivation using the results of Ref. 2 is lengthy, the non-redundant blocks are: • Virtual-Core, Virtual-Core: Q oo ai,b j = 4 4(ai|b j) − (ab|i j) − (a j|bi) + 4δ i j (F C ab + F A ab ) − 4δ ab (F C i j + F A i j ) (S14) • Virtual-Core, Virtual-Active: Q oo ai,bt = 2 v γ tv [4(ai|bv) − (av|bi) − (ab|vi)] − δ ab         v γ tv F C iv + 2(F C ti + F A ti ) + vxy Γ tvxy (vi|xy)         (S15) • Virtual-Core, Active-Core: Q oo ai,t j = 2 v (2δ tv − γ tv ) 4(ai|v j) − (av| ji) − (a j|vi) − δ i j         v γ tv F C av − 4(F C at + F A at ) + vxy Γ tvxy (av|xy)         (S16) • Virtual-Active, Virtual-Active: Q oo at,bu = 2 vx Γ tuvx (ab|vx) + (Γ txvu + Γ txuv )(ax|bv) − δ ab         v γ tv F C uv + γ uv F C tv + vxy Γ tvxy (uv|xy) + Γ uvxy (tv|xy)         + 2γ tu F C ab (S17) • Active-Core, Virtual-Active: Q oo ti,au = −2 vx Γ tuvx (ai|vx) + (Γ tvux + Γ tvxu )(ax|vi) + 2 v γ uv 4(av|ti) − (ai|tv) − (at|vi) − 2γ tu F C ia + δ tu (F C ai + F A ai )(S18) • Active-Core, Active-Core: G ti,u j = 4 v (δ tv − γ tv ) 4(vi|u j) − (ui|v j) − (uv|i j) + 2 vx Γ utvx (vx|i j) + (Γ uxvt + Γ uxtv )(vi|x j) + 2γ tu F C i j − 2δ i j         vxy Γ tvxy (uv|xy) + v γ uv F C tv         + 4δ i j (F C tu + F A tu ) − 4δ tu (F C i j + F A i j ) Q oo ti,u j = 1 2 G ti,u j + G u j,ti . (S19) Here, we have introduced the inactive (core) and active Fock matrices 2 F C mn = h mn + i [2(mn|ii) − (mi|in)] and F A mn = tv γ tv [(mn|tv) − 1 2 (mv|tn)]. (S20) S2. Eigenvector-following for saddle point optimisation We employ the eigenvector-following technique to target stationary points with an arbitrary Hessian index. While Section 6.2.1 of Ref. 3, and references therein, describe this method in detail, here we summarise the salient points for completeness and provide the details of our particular implementation. Eigenvector-following works in the eigenbasis for the Hessian matrix Q with eigenvalues µ . In this basis, the components of the Newton-Raphson step x = −Q −1 g, with gradient g, and the change in energy ∆V are given by x NR,µ = − g µ µ and ∆V NR = − µ |g µ | 2 2 µ . (S21) Contributions with µ > 0 or µ < 0 lower or raise the energy, respectively. To drive the optimisation towards a particular type of Hessian index, eigenvector-following artificially modifies the sign of these components. In particular, the components of the quasi-Newton eigenvector-following step are defined as x QN,µ = ±2g µ | µ | 1 + 1 + 2g µ / µ 2 ,(S22) where a positive or negative step gives an uphill or downhill step, respectively. This expression reduces to the Newton-Raphson step in the g µ → 0 limit and can be understood by imposing constraints on each component using Lagrange multipliers. 3 When an index-n saddle point is targeted in this work, the components corresponding to the lowest n eigenvalues of the Hessian are chosen to be downhill directions. Furthermore, we employ a dogleg trust radius technique to control the step length and improve the local convergence behaviour. A trust radius method works by defining a region with radius ρ around the current point in which a quadratic approximation to the objective function is considered to be accurate. The next step is chosen by optimising the objective function within this trust region, known as the sub-problem. In practice, approximate solutions to the sub-problem are required, and we employ the dogleg method (see Section 4.1 of Ref. 4). This method requires an analogue of the steepest descent direction for saddle point optimisation, which we define as x SD,µ = ±g µ ,(S23) where the positive and negative sign for each component are chosen in the same way as the quasi-Newton step [Eq. (S22)]. Defining an unconstrained step which optimises the energy along the steepest descent direction as x U = − g † x SD x † SD Qx SD x SD ,(S24) and using the eigenvector-following quasi-Newton step x QN , the optimal dogleg step x is then given by x =            x QN if |x QN | ≤ ρ ρ |x U | x U if |x U | ≥ ρ x U + τ x QN − x U otherwise (S25) S3 where τ = −b + √ b 2 − 4ac 2a , a = x QN − x U 2 , b = 2x U · (x QN − x U ), c = |x U | 2 − ρ 2 . (S26) The trust radius is then updated by comparing the ratio of the actual energy change ∆E to that predicted by the quadratic model ∆E model . The trust radius is halved for ∆E/∆E model < 0.25, and doubled if ∆E/∆E model > 0.75 and |x DL | = ρ. Otherwise, the trust radius is not deemed to be interfering with the optimisation and is left unchanged. Finally, a step is rejected if ∆E and ∆E model have different signs. We find that an initial trust radius of 0.15 provides adequate optimisation behaviour. S3. Mode-controlled Newton-Raphson optimisation Once a stationary point has been identified, it can be followed along a potential energy surface by using the old orbital and CI coefficients as a guess at the new geometry. Since the Hessian index can change at different geometries, we use a standard Newton-Raphson optimisation to perform this optimisation. The Newton-Raphson step is identified by solving a quadratic Taylor series expansion to the energy around a given point, giving x NR = −Q −1 g (S27) for the gradient g and Hessian matrix of second derivatives Q. The dogleg method (described in Section S2) cannot be used as there is no way of defining a "steepest descent" step for saddle point optimisation without knowing the target Hessian index. Instead, we simply truncate the components of x NR depending on their magnitude along each eigen-direction of the Q, giving x µ = min |x NR,µ | , ρ x NR,µ |x NR,µ | . The trust radius ρ is then updated using the same strategy described in Section S2. Similar mode-controlled Newton-Raphson optimisation algorithms have been reported multiple times in the past. 1,5-7 Optimization and Characterization of a Multiconfigurational Self-Consistent Field (MCSCF) State. J Olsen, D L Yeager, P Jørgensen, Adv. Chem. Phys. John Wiley and SonsLtdOlsen, J.; Yeager, D. L.; Jørgensen, P. Optimization and Characterization of a Multiconfigurational Self-Consistent Field (MCSCF) State. In Adv. Chem. Phys.; John Wiley and Sons, Ltd, 1983; pp 1-176. . T Helgaker, P Jørgensen, Olsen, J. Molecular Electronic-Structure Theory. Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular Electronic-Structure Theory; Energy Landscapes: Applications to Clusters, Biomolecules and Glasses. D J Wales, Cambridge University PressCambridgeWales, D. J. Energy Landscapes: Applications to Clusters, Biomolecules and Glasses; Cambridge University Press: Cambridge, 2004. . J Nocedal, S Wright, Numerical Optimization. Nocedal, J.; Wright, S. Numerical Optimization; . Springer-Verlag, New YorkSpringer-Verlag: New York, 2006. Multiconfigurational Hartree-Fock studies of avoided curve crossing using the Newton-Raphson technique. J Olsen, P Jørgensen, D L Yeager, J. Chem. Phys. 527Olsen, J.; Jørgensen, P.; Yeager, D. L. Multiconfigurational Hartree-Fock studies of avoided curve crossing using the Newton-Raphson technique. J. Chem. Phys. 1982, 76, 527. Proper characterization of MC SCF stationary points. J T Golab, D L Yeager, P Jørgensen, Chem. Phys. 175Golab, J. T.; Yeager, D. L.; Jørgensen, P. Proper characterization of MC SCF stationary points. Chem. Phys. 1983, 78, 175. Optimization of MCSCF excited states using directions of negative curvature. M R Hoffmann, C D Sherrill, M L Leininger, Iii Schaefer, H F , Chem. Phys. Lett. 183Hoffmann, M. R.; Sherrill, C. D.; Leininger, M. L.; Schaefer III, H. F. Optimization of MCSCF excited states using directions of negative curvature. Chem. Phys. Lett. 2002, 355, 183.

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{'abstract': 'S1. Deriving the gradient and second-derivative of the CASSCF energy S1.1. Notation and reduced density matrices', 'arxivid': '2301.11731', 'author': ['Antoine Marie \nDepartment of Chemistry\n) Physical and Theoretical Chemical Laboratory\nUniversity of Oxford\nOX1 3QZOxfordU.K\n\nLaboratoire de Chimie et Physique Quantiques (UMR 5626\nUniversité de Toulouse\nCNRS\nUPS\n31062ToulouseFrance\n', 'Hugh G A Burton \nDepartment of Chemistry\n) Physical and Theoretical Chemical Laboratory\nUniversity of Oxford\nOX1 3QZOxfordU.K\n'], 'authoraffiliation': ['Department of Chemistry\n) Physical and Theoretical Chemical Laboratory\nUniversity of Oxford\nOX1 3QZOxfordU.K', 'Laboratoire de Chimie et Physique Quantiques (UMR 5626\nUniversité de Toulouse\nCNRS\nUPS\n31062ToulouseFrance', 'Department of Chemistry\n) Physical and Theoretical Chemical Laboratory\nUniversity of Oxford\nOX1 3QZOxfordU.K'], 'corpusid': 256358861, 'doi': '10.1021/acs.jpca.3c00603', 'github_urls': [], 'n_tokens_mistral': 4085, 'n_tokens_neox': 3649, 'n_words': 2003, 'pdfsha': '2f058399786288df693815a43280cb68eb317d63', 'pdfurls': ['https://export.arxiv.org/pdf/2301.11731v1.pdf'], 'title': ['Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory: Supporting Information', 'Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory: Supporting Information'], 'venue': []}

arxiv

PAIR CORRELATION ESTIMATES FOR THE ZEROS OF THE ZETA FUNCTION VIA SEMIDEFINITE PROGRAMMING 18 Nov 2019 Andrés Chirre Felipe Gonçalves David De Laat PAIR CORRELATION ESTIMATES FOR THE ZEROS OF THE ZETA FUNCTION VIA SEMIDEFINITE PROGRAMMING 18 Nov 2019arXiv:1810.08843v2 [math.NT] In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function ζ(s) (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous asymptotic bounds in the theory of ζ(s), including the proportion of distinct zeros, counts of small gaps between zeros, and sums involving multiplicities of zeros. Introduction In this paper we give improved asymptotic bounds for several quantities related to the zeros of the Riemann zeta-function (and other functions) using Montgomery's pair correlation approach [36]. The key idea is to replace the usual bandlimited auxiliary functions by the class of functions used in the linear programming bounds developed by Cohn and Elkies [20] for the sphere packing problem. The advantage of this framework is that it reduces the problems to convex optimization problems that can be solved numerically via semidefinite programming. For all problems we considered this produces better bounds than any bandlimited construction. where the sum is over the non-trivial zeros of ζ(s) counting multiplicities 1 and m ρ is the multiplicity of ρ. In addition to RH, it is also conjectured that all zeros of ζ(s) are simple, therefore it is conjectured that N * (T ) ∼ N (T ),(2) as T → ∞. To study the distribution of the zeros of the Riemann zeta-function, Montgomery defined the pair correlation function N (x, T ) := 0<γ,γ ′ ≤T 0<γ ′ −γ≤ 2πx log T 1(3) Date: November 19, 2019. 1 For every sum over zeros in this article the involved quantities should be repeated according to the multiplicity of the zero. as T → ∞. Note that by (1) the average gap between zeros is 2π log T , hence N (x, T ) is counting pairs of zeros not greater than x times the average gap. One line of research to understand and give evidence for the conjectures above is to produce bounds of the form N * (T ) ≤ (c + o(1)) N (T ),(4) and N (x, T ) ≫ N (T ),(5) with c > 1 and x > 0 as small as possible, as T → ∞. These two problems have been widely studied with several improvements being made over the years. One approach is to use a formula relating sums and integrals involving an auxiliary function f from a class A, and then use this to derive an inequality involving the quantities we want to compare and the value of some functional Q evaluated at f . Minimizing (or maximizing) the functional over the class A would then produce the best possible bound with that specific approach. Nowadays, this idea is a standard technique in analytic number theory (introduced first by Beurling and Selberg) and the following are some references (clearly not a complete list) where the main approach is exactly that: Large sieve inequalities [31,32]; Erdős-Turán inequalities [15,42]; Hilbert-type inequalities [12,13,15,30,31,42]; Tauberian theorems [31]; Bounds in the theory of the Riemann zeta-function and L-functions [5,6,7,8,9,10,11,17,19,26,27]; Prime gaps [14]. For problem (4) Montgomery [36] uses Fourier analysis to derive the inequality N * (T ) ≤ 1 f (0) f (0) + 1 −1 f (x)|x| dx + o(1) N (T ), for any non-negative function f ∈ L 1 (R) with f supported in [−1, 1], where f (x) = ∞ −∞ f (y)e −2πxy dy. Montgomery then gives a function f that proves the bound (4) with c ≤ 4/3. In [37] the optimal function in this class is found and as mentioned in [18] gives the bound c ≤ 1.3275. We relax the condition on the support of f to the requirement f (x) ≤ 0 for |x| ≥ 1, which matches exactly with the conditions required by the linear programming bounds for the sphere packing problem (see Section 3 for a detailed explanation). This connection is what ultimately inspired us to attack the problem numerically and to find good test functions for the functionals derived in Section 3. From our point of view, our main contribution is the realization that methods from the sphere packing problem can applied in the theory of the Riemann zeta-function to improve several asymptotic bounds and, to the best of our knowledge, it is the first time it has been done. Main Results We now state our main results. Theorem 1. Assuming RH, we have N * (T ) ≤ (1.3208 + o(1))N (T ). Assuming the Generalized Riemann Hypothesis for Dirichlet L-functions (GRH), we have N * (T ) ≤ (1.3155 + o(1))N (T ). Montgomery [36] was the first to show the constant 4/3 = 1.3333.... Later Montgomery and Taylor [37] improved on this and found the bound 1.3275 as mentioned by Cheer and Goldston in [18]. Assuming the generalized Riemann Hypothesis GRH, Goldston, Gonek, Özlük and Snyder [28] improved it to 1.3262. Theorem 1 has an important application to estimating the quantity of simple zeros of ζ(s). Let N s (T ) := 0<γ≤T mρ=1 1. Using the fact that N s (T ) ≥ 0<γ≤T (2 − m ρ ) = 2N (T ) − N * (T ).(6) we obtain the following corollary. Assuming GRH, we have N s (T ) ≥ (0.6845 + o(1))N (T ). Using the pair correlation approach, the previous best result known is due by Cheer and Goldston [18] showing that 67.27% of the zeros are simple. Assuming GRH, Goldston, Gonek, Özlük and Snyder [28] showed that 67.38% are simple. In this way, we improved all these bounds. However, by a different technique, still assuming RH, Bui and Heath-Brown [4] improved the result to 19/27 = 70.37...%, which currently is the best. Combining the above result of Bui and Heath-Brown with Theorem 1 and an argument of Ghosh, we can bound the proportion of distinct zeros of the Riemann zeta-function. Let N d (T ) := 0<γ≤T 1 m ρ , be the number of distinct zeros of ζ(s) with 0 < γ ≤ T . Using the inequality 2N s (T ) ≤ 0<γ≤T (m ρ − 2)(m ρ − 3) m ρ = N * (T ) − 5N (T ) + 6N d (T ).(7) in conjunction with the estimate N s (T ) ≥ 19 27 + o(1) N (T ) and Theorem 1, we deduce the following corollary. Assuming GRH, we have N d (T ) ≥ (0.8486 + o(1))N (T ). Using the pair correlation approach, the best previous result known is due to Farmer, Gonek and Lee [24] with constant 0.8051. By a different technique, assuming RH, Bui and Heath-Brown [4] improved the constant to 0.8466. We also obtain the best known results for the minimal nonzero value of Montgomery's pair correlation function. Assuming GRH and (2), we have N (0.5769, T ) ≫ N (T ). Montgomery [36] showed that N (0.68..., T ) ≫ N (T ), and in [28] it is pointed out that it is not difficult to modify Montgomery's argument to derive the sharper constant 0.6695. This result was improved by Goldston, Gonek, Özlük and Snyder [28] with constant 0.6072. Later, Carneiro, Chandee, Littmann and Milinovich [8] improved the constant to 0.6068.... Assuming GRH and (2), Goldston, Gonek, Özlük and Snyder showed the constant 0.5781.... 2.1. Results for zeros of Dirichlet L-functions. To obtain averaged bounds for the percentage of simple zeros of primitive Dirichlet L-functions we use the framework established by Chandee, Lee, Liu and Radziwiłł [16]. Let Φ be a real-valued smooth function supported in the interval [a, b] with 0 < a < b < ∞. Define its Mellin transform by MΦ(s) = ∞ 0 Φ(x)x s−1 dx. For a character χ mod q, let L(s, χ) be its associated Dirichlet L-function. Under GRH, all non-trivial zeros of L(s, χ) lie on the critical line Re s = 1/2. Let N Φ (Q) := Q≤q≤2Q W (q/Q) ϕ(q) χ (mod q) primitive γχ MΦ(iγ χ ) 2 , where W is a non-negative smooth function supported in (1,2), and where the last sum is over all non-trivial zeros 1 2 + iγ χ of the Dirichlet L-function L(s, χ). In [16, Lemma 2.1] it is shown that N Φ (Q) ∼ A 2π Q log Q ∞ −∞ MΦ(ix) 2 dx, where A = MW (1) p prime 1 − 1 p 2 − 1 p 3 . Let N Φ,s (Q) = Q≤q≤2Q W (q/Q) ϕ(q) χ (mod q) primitive γχ simple MΦ(iγ χ ) 2 . The quantity lim inf Q→∞ N Φ,s (Q) N Φ (Q) then measures (in average) the proportion of simple zeros among all primitive Dirichlet L-functions. In addition, for the following theorem, we require that Φ(x) and MΦ(ix) are non-negative functions. We note that we can also further relax the conditions on Φ so to include the function given by MΦ(ix) = (sin x/x) 2 , as was established in [16] and [41]. Theorem 5. Assuming GRH, we have N Φ,s (Q) ≥ (0.9350 + o(1))N Φ (Q). Using the pair correlation approach, Chandee, Lee, Liu and Radziwiłł [16] showed that 91.66% of the zeros are simple. The best previous result known is due to Sono [41], showing that 93.22% of the zeros are simple. On another hand, Özlük [39] obtained a similar lower bound but for all Dirichlet L-functions rather than just the primitive L-functions, showing that 91.66% of the zeros are simple (in some sense). 2.2. Results for zeros of ξ ′ (s). We can extend our analysis to the zeros of ξ ′ (s), where ξ(s) = 1 2 s(s − 1)π − s 2 Γ s 2 ζ(s). It is known that ξ ′ (s) has only zeros in the critical strip 0 < Re s < 1 and that RH implies that all its zeros are also on the line Re s = 1/2. Let N 1 (T ) count the number of zeros ρ 1 = β 1 + iγ 1 of ξ ′ (s) (with multiplicity) such that 0 < γ 1 ≤ T . It is also known that N 1 (T ) = (1 + o(1)) T 2π log T. We can then similarly define the function N * 1 (T ) := 0<γ1≤T m ρ1 , where m ρ1 is the multiplicity of the zero ρ 1 , and derive the sharpest known upper bound for N * 1 (T ). Theorem 6. Assuming RH, we have N * 1 (T ) ≤ (1.1175 + o(1))N 1 (T ). Defining the functions N 1,s (T ) and N 1,d (T ) (quantity of simple and distinct zeros respectively) for ξ ′ (s) and using the inequalities N 1,s (T ) ≥ 2N 1 (T ) − N * 1 (T ) and N 1,d (T ) ≥ 3 2 N 1 (T ) − 1 2 N * 1 (T ) , that can be derived using the analogues of (6) and (7) for ξ ′ (s), we obtain the following corollary. and N 1,d (T ) ≥ (0.9412 + o(1))N 1 (T ). The best previous result is due to Farmer, Gonek and Lee [24], showing that more than 85.83% of the zeros of ξ ′ (s) are simple. Derivation of the optimization problems Let A LP be the class of even continuous functions f ∈ L 1 (R) satisfying the following conditions: (1) f (0) = f (0) = 1; (2) f ≥ 0; (3) f is eventually non-positive. By eventually non-positive we mean that f (x) ≤ 0 for all sufficiently large |x|. We then define the last sign change of f by r(f ) = inf r > 0 : f (x) ≤ 0 for |x| ≥ r . It is easy to show that if f ∈ A LP , then f ∈ L 1 (R). A remarkable breakthrough in the sphere problem was achieved by Cohn and Elkies in [20], where they showed that if ∆(R d ) is the highest sphere packing density in R d , then ∆(R d ) ≤ Q(f ) for any f ∈ A LP (R d ) (this is the analogous class in higher dimensions defined for radial functions f ), where Q(f ) = π d/2 (d/2)!2 d r(f ) d . With this approach they generated numerical upper bounds, called linear programming bounds, for the packing density for dimensions up to 36 (nowadays it goes much higher) that improved every single upper bound known at the time and still are the current best. These upper bounds in dimensions 8 and 24 revealed to be extremely close to the lower bounds given by the E 8 root lattice and the Λ 24 Leech lattice, revealing that in these special dimensions the linear programming approach could exactly act as the dual problem. This is what inspired Viazovska et. al. [43,23] to follow their program and solve the sphere packing problem in dimensions 8 and 24. What is interesting and surprising to us is that the same space A LP can be used (but with a functional different than Q(f )) to produce numerical bounds in analytic number theory. The general strategy to study problems (4) and (5) is based on Montgomery's function F (x, T ) = 1 N (T ) 0<γ,γ ′ ≤T T ix(γ−γ ′ ) w(γ − γ ′ ), where the sum is over pairs of ordinates of zeros (with multiplicity) of ζ(s) and w(u) = 4 4+u 2 . For each T , the function x → F (x, T ) is even, real, and as observed independently by Mueller and Heath-Brown, non-negative. The first step is to use Fourier inversion to obtain 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) = N (T ) ∞ −∞ g(x)F (x, T ) dx,(8) for suitable functions g, and use some known asymptotic estimate for F (x, T ) as T → ∞ (which is proven only under RH or GRH). Secondly, after a series of inequalities, we produce a minimization problem over A LP for some functional Z. We then approach the problem numerically, using the class of functions used for the sphere packing problem in [20] and sum-of-squares/semidefinite programming techniques to optimize over these functions. The same basic strategy can be, in principle, carried out for other functions where we have a pair correlation approach. Indeed, we will also derive functionals related to the zeros of ξ ′ (s) and a certain average of primitive Dirichlet L-functions. 3.1. Bounding N * (T ) and N (x, T ). Ultimately, the functionals we need to define depend on the asymptotic behavior of F (x, T ). To analyze the function N * (T ) we define the functionals Z(f ) = r(f ) + 2 r(f ) r(f ) 0 f (x)x dx and Z(f ) = r(f ) + 2 r(f ) r(f ) 0 f (x)x dx + 3 3 2 r(f ) r(f ) f (x) dx − 2 r(f ) 3 2 r(f ) r(f ) f (x)x dx. Lemma 8. Let f ∈ A LP . Assuming RH, we have N * (T ) ≤ (Z(f ) + o(1))N (T ). Assuming GRH, for every fixed sufficiently small δ > 0, we have N * (T ) ≤ ( Z(f ) + O(δ) + o(1))N (T ). Proof. We start assuming only RH. Refining the original work of Montgomery [36], Goldston and Montgomery [29,Lemma 8] proved that F (x, T ) = T −2|x| log T + |x| (1 + o(1)),(9) uniformly for |x| ≤ 1. Let f ∈ A LP and let g(x) = f (x/r(f ))/r(f ). We can then use formula (8) in conjunction with the asymptotic formula above to obtain 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) = N (T ) g(0) + 1 −1 g(x)|x| dx + |x|>1 g(x)F (x, T ) dx + o(1) , where the o(1) above is justified since g is continuous and T −2|x| log T → δ 0 (x) as T → ∞ (in the distributional sense). Moreover, since F (x, T ) is non-negative and g(x) ≤ 0 for |x| ≥ 1 we deduce that 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) ≤ N (T ) g(0) + 2 1 0 g(x)x dx + o(1) = N (T ) Z(f ) r(f ) + o(1) . On the other hand, clearly we have 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) ≥ g(0) 0<γ≤T m ρ = N * (T ) r(f ) .(10) Combining these results we show the first inequality in the theorem. Assume now GRH. It is then shown in [28] that for any fixed and sufficiently small δ > 0 we have F (x, T ) ≥ 3 2 − |x| − o(1),(11) uniformly for 1 ≤ |x| ≤ 3 2 − δ as T → ∞. Using this estimate together with (9) and the fact that g(x) ≤ 0 for |x| ≥ 3/2 − δ, we obtain 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) ≤ N (T ) g(0) + 2 1 0 g(x)x dx + 2 3 2 −δ 1 g(x) 3 2 − x dx + o(1) = N (T ) Z(f ) r(f ) + o(1) + O(δ) . Arguing as before, using (10), we complete the proof of the lemma. To analyze N (x, T ) we define the functional P(f ) = inf λ > 0 : p f (λ) > 0 , where p f (λ) = −1 + λ r(f ) + 2r(f ) λ λ r(f ) 0 f (x)x dx, and the functional P(f ) = inf λ > 0 : p f (λ) > 0 , where p f (λ) = −1 + λ r(f ) + 2r(f ) λ λ r(f ) 0 f (x)x dx + 3 3λ 2r(f ) λ r(f ) f (x) dx − 2r(f ) λ 3λ 2r(f ) λ r(f ) f (x)x dx. Note that these functionals are well defined since p f and p f are C 1 functions that assume −1 at λ = 0, and using the fact that f ∈ L 1 (R) one can show lim λ→∞ p f (λ) λ = lim λ→∞ p f (λ) λ = 1 r(f ) > 0. Lemma 9. Let f ∈ A LP and ε > 0. Assuming RH and (2), we have N (P(f ) + ε, T ) ≫ N (T ). Assuming GRH and (2), we have N ( P(f ) + ε, T ) ≫ N (T ). Proof. Let f ∈ A LP , λ > 0, and set g(x) = f (r(f )x/λ). Assuming RH, we have 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) = N (T ) ∞ −∞ g(x)F (x, T ) dx ≥ N (T ) g(0) + 2 1 0 g(x)x dx + o(1) = N (T ) [1 + p f (λ) + o(1)] . Applying formula (8) in conjunction with (9), while assuming GRH, and using in addition (11), we have (1)] . 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) = N (T ) ∞ −∞ g(x)F (x, T ) dx ≥ N (T ) g(0) + 2 1 0 g(x)x dx + 2 3 2 −δ 1 g(x) 3 2 − x dx + o(1) = N (T ) [1 + p f (λ) + o Since f ≥ 0, we have f ∞ = f (0) = 1. Recall now the pair correlation function N (x, T ) defined in (3). We have 0<γ,γ ′ ≤T g (γ − γ ′ ) log T 2π w(γ − γ ′ ) = N * (T ) + 2 0<γ,γ ′ ≤T 0<γ−γ ′ f (γ − γ ′ ) r(f ) log T 2πλ w(γ − γ ′ ) ≤ N * (T ) + 2 0<γ,γ ′ ≤T 0<γ−γ ′ ≤ 2πλ log T f (γ − γ ′ ) r(f ) log T 2πλ w(γ − γ ′ ) ≤ N * (T ) + 2N (λ, T ) = (1 + o(1))N (T ) + 2N (λ, T ), where in the last step we have used (2). We then obtain, assuming RH, that N (λ, T ) N (T ) ≥ p f (λ) 2 + o(1). Similarly assuming GRH. Noting that N (λ, T ) increases with λ, so we can choose λ arbitrarily close to P(f ), we obtain the desired result. 3.2. Bounding N Φ,s (Q). Define the following functional over A LP L(f ) = r(f ) 2 + 4 r(f ) r(f ) 2 0 f (x)x dx + 2 r(f ) r(f ) 2 f (x) dx. We have the following lemma. Lemma 10. Let f ∈ A LP . Assuming GRH, for every fixed small δ > 0 we have N Φ,s (Q) ≥ (2 − L(f ) + O(δ) + o(1))N Φ (Q). Proof. For Q > 1 and x ∈ R, we define the pair correlation function F Φ by F Φ (Q x , W ) = 1 N Φ (Q) Q≤q≤2Q W (q/Q) ϕ(q) χ (mod q) primitive γχ MΦ(iγ χ )Q iγχx 2 . Using the asymptotic large sieve, Chandee, Lee, Liu and Radziwiłł [16] showed the following asymptotic formula under GRH F Φ (Q x , W )(12)= (1 + o(1)) 1 − (1 − |x|) + + Φ Q −|x| 2 log Q 1 2π ∞ −∞ MΦ(it) 2 dt −1 + O Φ(Q −|x| ) log 1/2 Q , which holds uniformly for |x| ≤ 2 − δ as Q → ∞, for any fixed and sufficiently small δ > 0. Let N * Φ (Q) := Q≤q≤2Q W (q/Q) ϕ(q) χ (mod q) primitive γχ m ρχ MΦ(iγ χ ) 2 , where m ρχ denote the multiplicity of the nontrivial zero ρ χ = 1 2 + iγ χ of L(s, χ). Since γχ simple MΦ(iγ χ ) 2 ≥ γχ (2 − m ρχ ) MΦ(iγ χ ) 2 we obtain N Φ,s (Q) ≥ 2N Φ (Q) − N * Φ (Q).(13) For any g ∈ L 1 (R) with g ∈ L 1 (R) we have the following formula (Fourier inversion): Q≤q≤2Q W (q/Q) ϕ(q) χ (mod q) primitive γχ,γ ′ χ MΦ(iγ χ )MΦ(iγ ′ χ ) g (γ χ − γ ′ χ ) log Q 2π = N Φ (Q) ∞ −∞ g(x)F Φ (Q x , W ) dx. Letting f ∈ A LP and g(x) = f (r(f )x/(2 − δ)), for any primitive character χ (mod q) we obtain γχ,γ ′ χ MΦ(iγ χ )MΦ(iγ ′ χ ) g (γ χ − γ ′ χ ) log Q 2π = γχ m ρχ MΦ(iγ χ ) 2 g(0) + γχ =γ ′ χ MΦ(iγ χ )MΦ(iγ ′ χ ) g (γ χ − γ ′ χ ) log Q 2π ≥ 2 − δ r(f ) γχ m ρχ MΦ(iγ χ ) 2 . This implies that Q≤q≤2Q W (q/Q) ϕ(q) χ (mod q) primitive γχ,γ ′ χ MΦ(iγ χ )MΦ(iγ ′ χ )g (γ χ − γ ′ χ ) log Q 2π ≥ 2 − δ r(f ) N * Φ (Q). On the other hand, observing that Φ Q −|x| 2 log Q 1 2π ∞ −∞ MΦ(it) 2 dt → δ(x), as Q → ∞ (in the distributional sense) and that log 1/2 Q 2−δ −(2−δ) g(x)Φ(Q −|x| ) dx ≤ 2 log −1/2 Q 1 Q −(2−δ) Φ(t) dt t = O(log −1/2 Q), we can use the asymptotic estimate (12) to obtain ∞ −∞ g(x)F Φ (Q x , W ) dx ≤ 2−δ −(2−δ) g(x)F Φ (Q x , W ) dx = g(0) + 2−δ −(2−δ) g(x)(1 − (1 − |x|) + ) dx + O(log −1/2 Q) + o(1) = 2L(f ) r(f ) + O(δ) + o(1). We then conclude that (1)) . Using (13) we finish the proof. N * Φ (Q) ≤ N Φ (Q) (L(f ) + O(δ) + o3.3. Bounding N * 1 (T ). Similarly to the case of the Riemann zeta-function, the functionals that we need to define depend on the asymptotic behavior of the function F 1 (x, T ) defined by F 1 (x, T ) = N 1 (T ) −1 0<γ1,γ ′ 1 ≤T T ix(γ1−γ ′ 1 ) w(γ 1 − γ ′ 1 ),(14) where x ∈ R, T > 0 and the sum is over pairs of ordinates of zeros (with multiplicity) of ξ ′ (s). To analyze N * 1 (T ) we define the following functional Z 1 (f ) = r(f ) + 2 r(f ) r(f ) 0 x f (x) dx − 8 r(f ) 2 r(f ) 0 x 2 f (x) dx + ∞ k=1 2c k r(f ) 2k+1 r(f ) 0 x 2k+1 f (x) dx, where c k = 2 2k+1 (k−1)! (2k)! . Lemma 11. Let f ∈ A LP . Assuming RH, for every fixed small δ > 0 we have N * 1 (T ) ≤ (Z 1 (f ) + O(δ) + o(1))N 1 (T ). Proof. A result similar to (9) for the function F 1 (x, T ) defined in (14) is also known (see [24,Theorem 1.1]), which is the following: for any fixed small δ > 0 we have F 1 (x, T ) = T −2|x| log T + |x| − 4|x| 2 + ∞ k=1 c k |x| 2k+1 + o(1)(1 + T −2|x| log T ), uniformly for |x| ≤ 1 − δ as T → ∞, where c k = 2 2k+1 (k−1)! (2k)! . The proof then follows the same strategy as the proof for ζ(s) and we leave the details to the reader. Numerically optimizing the bounds Going back to the sphere packing problem, since we obviously have ∆(R 1 ) = 1, this shows r(f ) ≥ 1 for all f ∈ A LP . The last sign change equals 1 for two (suspiciously) well-known functions: the hat function H(x) = (1 − |x|) + , whose Fourier transform is H(x) = sin 2 (πx) (πx) 2 , and Selberg's function S(x) = sin 2 (πx) (πx) 2 (1 − x 2 ) , whose Fourier transform is supported in [−1, 1] and given by S(x) = 1 − |x| + sin(2πx) 2π for |x| < 1. In particular, we can use these two functions to evaluate the functionals derived in Section 3 to obtain bounds, but this does not result in the best possible bounds. To obtain better bounds we use the class of functions used in the linear programming bounds by Cohn and Elkies [20] for sphere packing. That is, we consider the subspace A LP (d) consisting of the functions f ∈ A LP of the form f (x) = p(x)e −πx 2 ,(15) where p is an even polynomial of degree 2d. In [20], optimization over a closely related class of functions is done by specifying the functions by their real roots and optimizing the root locations. For the sphere packing problem this works very well, where in R 24 it leads to a density upper bound that is sharp to within a factor 1 + 10 −51 of the optimal configuration [22]. We have also tried this approach for the optimization problems in this paper, but this did not work very well because the optimal functions seem to have very few real roots, which produces a strange effect in the numerical computations, where the last forced root tends to diverge when you increase the degree of the polynomial 2 . Instead we use sum-of-squares characterizations and semidefinite programming, as was done in [34] for the binary sphere packing problem. Semidefinite programming is the optimization of a linear functional over the intersection of a cone of positive semidefinite matrices (real symmetric matrices with nonnegative eigenvalues) and an affine space. A semidefinite program is often given in block form, which can be written as give the linear constraints (for notational simplicity we take all blocks to have the same size). Semidefinite programming is a broad generalization of linear programming (which we recover by setting n = 1 in the above formulation), and, as for linear programming, there exist efficient algorithms for solving them. The reason semidefinite programming comes into play here, is that we can model polynomial inequality constraints as sum-of-squares constraints, which in turn can be written as semidefinite constraints; see, e.g., [2]. minimize I i=1 tr(X i C i ) : I i=1 tr(X i A i,j ) = b j for j ∈ [m], 4.1. Proof of Theorems 1, 5, and 6. To obtain the first part of Theorem 1 from Lemma 8 we need to minimize the functional Z over the space A LP (d). We can see this as a bilevel optimization problem, where we optimize over scalars R ≥ 1 in the outer problem, and over functions f ∈ A LP (d) satisfying r(f ) = R in the inner problem. The outer problem is a simple one dimensional optimization problem for which we use Brent's method [3]. A polynomial p that is nonnegative on [R, ∞) can be written as s 1 (x) + (x − R)s 2 (x), where s 1 and s 2 are sum-of-squares polynomials with deg(s 1 ), deg(s 2 (x)) + 1 ≤ deg(p); see, e.g., [40]. This shows that functions of the form (15) that are non-positive on [R, ∞) can be written as f (x) = − s 1 (x 2 ) + (x 2 − R 2 )s 2 (x 2 ) e −πx 2 . Let v(x) be a vector whose entries form a basis of the univariate polynomials of degree at most d. Let T be the operator that maps x 2k to the function k! π k L −1/2 k (πx 2 ), where L k is the Laguerre polynomial of degree k with parameter −1/2. Then, for p an even polynomial, we have that (T p)(x)e −πx 2 is the Fourier transform of p(x)e −πx 2 . We can now describe the functions of the form (15) that are non-positive on [R, ∞) and have nonnegative Fourier transform by positive semidefinite matrices X 1 , . . . , X 4 of size d + 1 whose entries satisfy the linear relations coming from the identity I(X 1 , . . . , X 4 ) = 0, where I(X 1 , . . . , X 4 ) = T − s 1 (x 2 ) − (x 2 − R 2 )s 2 (x 2 ) − s 3 (x 2 ) + x 2 s 4 (x 2 ) . Here T (−s 1 (x 2 ) − (x 2 − R 2 )s 2 (x 2 ) ) is a polynomial whose coefficients are linear combinations in the entries of X 1 and X 2 , and the same for s 3 (x 2 ) + x 2 s 4 (x 2 ) with X 3 and X 4 . The linear constraints on the entries of X 1 , . . . , X 4 are then obtained by expressing I(X 1 , . . . , X 4 ) in some polynomial basis and setting the coefficients to zero. The conditions f (0) = 1 and f (R) = 0 are linear in the entries of X 1 and X 2 , and the condition f (0) = 1 is a linear condition on the entries of X 3 and X 4 . Finally, the objective Z(f ) is a linear combination in the entries of X 1 and X 2 , which can be implemented by using the identity x m e −πx 2 dx = − 1 2π m/2+1/2 Γ m + 1 2 , πx 2 ,(16) where Γ is the upper incomplete gamma function. Hence, the problem of minimizing Z(f ) over functions f ∈ A LP (d) that satisfy r(f ) = R is a semidefinite program. To obtain the second part of Theorem 1 from Lemma 8 and to obtain Theorem 5 from Lemma 10 we use the same approach with a different functional. To obtain Theorem 6 from Lemma 11 we also do the same as above, but now truncate the series in the functional Z 1 at k = 15 and add the easy to compute upper bound 10 −10 on the remainder of the terms. 4.1.1. Implementation and numerical issues. In implementing the above as a semidefinite program we have to make two choices for the polynomial basis that we use: the basis defining the vector v(x), and the basis to enforce the identity I(X 1 , . . . , X 4 ) = 0. This choice of bases is important for the numerical conditioning of the resulting semidefinite program. Following [34] we choose the Laguerre basis {L −1/2 n (2πx 2 )}, as this seems natural and performs well in practice (it multiplied by e −πx 2 is the complete set of even eigenfunctions of the Fourier transform). We solve the semidefinite programs using sdpa-gmp [38], which is a primal-dual interior point solver using high precision floating point arithmetic. For the code to generate the semidefinite programs and to perform the post processing we use Julia [1], Nemo [25], and Arb [33] (where we use Arb for the ball arithmetic used in the verification procedure). For all computations we use d = 40. In solving the systems we observe that X 1 can be set to zero everywhere without affecting the bounds, so that r(f ) = R holds exactly for the function f (x) = (R 2 − x 2 )v(x 2 ) T X 2 v(x 2 )e −πx 2 defined by X 2 . The above optimization approach uses floating point arithmetic and a numerical interior point solver. This means the identity I(0, X 2 , X 3 , X 4 ) = 0 will not be satisfied exactly, and, moreover, because the solver can take infeasible steps the matrices X 2 , X 3 , and X 4 typically have some eigenvalues that are slightly negative. In practice this leads to incorrect upper bounds if the floating point precision is not high enough in relation to the degree d. Here we explain the procedure we use to obtain bounds that are guaranteed to be correct. This is an adaptation of the method from [35] and [34]. We first solve the above optimization problem numerically to find R and f for which we have a good objective value v = L(f ). Then we solve the semidefinite program again for the same value of R, but now we solve it as a feasibility problem with the additional constraint L(f ) ≤ v + 10 −6 . The interior point solver will try to give the analytical center of the semidefinite program, so that typically the matrices are all positive definite; that is, the eigenvalues are all strictly positive. Then we use interval arithmetic to check rigorously that X 2 , X 3 , and X 4 are positive definite, and we compute a rigorous lower bound b on the smallest eigenvalues of X 3 and X 4 . Using interval arithmetic we compute an upper bound B on the largest coefficient of I(0, X 2 , X 3 , X 4 ) in the basis given by the 2d + 1 entries on the diagonal and upper diagonal of the matrix 2d)B, then it follows that it is possible to modify the corresponding entries in X 3 and X 4 such that these matrices stay positive definite and such that I(0, X 2 , X 3 , X 4 ) = 0 holds exactly [35]. This proves that the Fourier transform of the function (R 2 − x 2 )v(x 2 )v(x 2 ) T . If b ≥ (1 +f (x) = (R 2 − x 2 )v(x 2 ) T X 2 v(x 2 )e −πx 2 is nonnegative. The only remaining problem is that the identities f (0) = 1 and T f (0) = 1 will not hold exactly. We can, however, for instance write the first part of Theorem 1 as follows: Suppose f is a continuous function in L 1 (R) with f (x) ≤ 0 for |x| ≥ R and with f ≥ 0, then N * (T ) ≤ (Z(f ) + o(1))N (T ), where we use the following modified definition for Z(f ): Z(f ) = 1 f (0) f (0)r(f ) + 2 r(f ) r(f ) 0 f (x)x dx . Since the function f defined by X 1 has been verified to satisfy all the constraints, the only thing we still need to do is to compute a rigorous upper bound on Z(f ) (or on similar modifications of the functionals Z(f ), Z 1 (f ), or L(f )), for which we use identity (16) and interval arithmetic. Remark 1. In the arXiv version of this paper we attach the files 'Z-40.txt', 'tildeZ-40.txt', 'L-40.txt', and 'Z1-40.txt' that contain the value of R on the first line and the matrices X 2 , X 3 and X 4 on the next 3 lines (all in 100 decimal floating point values). For convenience it also contains the coefficients of f in the monomial basis on the last line (but these are not used in the verification procedure). We include a script to perform the above verification and compute the bounds rigorously, as well as the code for setting up the semidefinite programs, using a custom semidefinite programming specification library. 4.2. Proof of Theorem 4. To obtain the first part of Theorem 4 from Lemma 9 we need to minimize the function P over the space A LP . We can formulate this as a bilevel optimization problem in which we optimize over R ≥ 1 in the outer problem. In the inner problem we perform a binary search over Λ to find the smallest Λ for which there exists a function f ∈ A LP (d) that satisfies f (R) = 0, f (x) ≤ 0 for |x| ≥ R, and p f (Λ) ≥ 0. To get a bound whose correctness we can verify rigorously we replace the constraints f (0) = 1, f (0) = 1, and p f (Λ) ≥ 0 by f (0) = 1 − 10 −10 , f (0) = 1 + 10 −10 , and p f (Λ) ≥ 10 −10 . We then use the above optimization approach to find good values for R and Λ. We then add 10 −6 to Λ and solve the feasibility problem again to get the strictly feasible matrices X 2 , X 3 , and X 4 . By performing the same procedure as in 4.1.1 we can verify that the Fourier transform of the function f defined by X 2 is nonnegative everywhere, and using interval arithmetic we can check that the inequalities f (0) ≤ 1, f (0) ≥ 1, and p f (Λ) > 0 all hold. Note that this verification procedure does not actually check that Λ is equal to or even close to P(f ), but the proof of Lemma 9 also works if we replace P(f ) by any Λ for which p f (Λ) is strictly positive. To obtain the second part of the theorem, we do the same except that we replace p f by p f . Remark 2. In the arXiv version of this paper we attach the files 'P-40.txt', 'tildeP-40.txt', that have the same layout as the files mentioned in 4.1.1, with an additional line containing the value of Λ. We again include the code to perform the verification and to produce the files. 1. 1 . 1Background. Let ζ(s) be the Riemann zeta-function. It is well known that all non-trivial zeros of ζ(s) are located in the critical strip 0 < Re s < 1, and the Riemann Hypothesis (RH) is the statement that all these zeros are aligned on the line Re s = 1/2. Let N (T ) count the number of zeros ρ = β + iγ of ζ(s), repeated according the multiplicity, such that 0 < β < 1 and 0 < γ ≤ T . The Riemann-von Mangoldt formula (in its weaker form) states that Corollary 2 . 2Assuming RH, we have N s (T ) ≥ (0.6792 + o(1))N (T ). Corollary 3 . 3Assuming RH, we have N d (T ) ≥ (0.8477 + o(1))N (T ). Theorem 4 . 4Assuming RH and (2), we have N (0.6039, T ) ≫ N (T ). Corollary 7 . 7Assuming RH, we have N 1,s (T ) ≥ (0.8825 + o(1))N 1 (T ). X 1 1, . . . , X I ∈ R n×n positive semidefinite, where I ∈ N gives the number of blocks, {C i } ⊆ R n×n is the objective, and {A i,j } ⊆ R n×n , b ∈ R m The polynomials s 1 and s 2 are sum-of-squares if and only if they can be written as s i (x) = v(x) T X i v(x) for some positive semidefinite matrices X i of size d + 1. That is, we can parameterize functions of the form (15) that are non-positive on [R, ∞) by two positive semidefinite matrices X 1 and X 2 of size d + 1. The space of functions of the form (15) is invariant under the Fourier transform. Since a polynomial of degree 2d that is nonnegative on [0, ∞) can be written as s 3 (x) + xs 4 (x), where s i (x) = v(x) T X i v(x) for i = 3, 4 are sum-of-squares polynomials of degree 2d, we have that f is of the form f (x) = s 3 (x 2 ) + x 2 s 4 (x 2 ) e −πx 2 . It is worth mentioning that, in a related uncertainty problem, Cohn and Gonçalves[21] discovered the same kind of instability in low dimensions. 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Aust. Math. Soc. 93 (2016), no. 1, 19-30. Some extremal functions in Fourier analysis. J D Vaaler, Bull. Amer. Math. Soc. (N.S.). 122J. D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 183-216. The sphere packing problem in dimension 8. M S Viazovska, Ann. of Math. 2M. S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), no. 3, 991-1015. E-mail address: achirre@impa.br. E-mail address: achirre@impa.br Endenicher Allee 60, 53115 Bonn, Germany E-mail address: goncalve@math.uni-bonn. deHausdorff Center for Mathematics, Universität BonnHausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany E-mail address: goncalve@math.uni-bonn.de E-mail address: mail@daviddelaat.nl. E-mail address: mail@daviddelaat.nl

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{'abstract': "In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function ζ(s) (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous asymptotic bounds in the theory of ζ(s), including the proportion of distinct zeros, counts of small gaps between zeros, and sums involving multiplicities of zeros.", 'arxivid': '1810.08843', 'author': ['Andrés Chirre ', 'Felipe Gonçalves ', 'David De Laat '], 'authoraffiliation': [], 'corpusid': 119132616, 'doi': '10.1016/j.aim.2019.106926', 'github_urls': [], 'n_tokens_mistral': 16690, 'n_tokens_neox': 14517, 'n_words': 8308, 'pdfsha': '841033508e72f1c92f20ed31462aeb3b55bbddef', 'pdfurls': ['https://arxiv.org/pdf/1810.08843v2.pdf'], 'title': ['PAIR CORRELATION ESTIMATES FOR THE ZEROS OF THE ZETA FUNCTION VIA SEMIDEFINITE PROGRAMMING', 'PAIR CORRELATION ESTIMATES FOR THE ZEROS OF THE ZETA FUNCTION VIA SEMIDEFINITE PROGRAMMING'], 'venue': []}

arxiv

Real-time-controlled artificial quiet channel for acoustic cloaking under varying detection conditions 25 Jan 2023 Or Lasri School of Mechanical Engineering Tel Aviv University 69978Tel AvivIsrael Lea Sirota School of Mechanical Engineering Tel Aviv University 69978Tel AvivIsrael Real-time-controlled artificial quiet channel for acoustic cloaking under varying detection conditions 25 Jan 2023 We consider the problem of hiding non-stationary objects from acoustic detection in a two-dimensional environment, where both the object's impedance and the properties of the detection signal may vary during operation. The detection signal is assumed to be an acoustic beam created by an array of emitters, which scans the area at different angles and different frequencies. We propose an active control-based solution that creates an effective moving dead zone around the object, and results in an artificial quiet channel for the object to pass through undetected. The control principle is based on mid-domain generation of near uni-directional beams using only monopole actuators. Based on real-time response prediction, these beams open and close the dead zone with a minimal perturbation backwards, which is crucial due to detector observers being located on both sides of the object's route. The back action wave determines the cloak efficiency, and is traded-off with the control effort; the higher is the effort the quieter is the cloaking channel. We validate our control algorithm via numerical experiments in a two-dimensional acoustic waveguide, testing variation in frequency and incidence angle of the detection source. Our cloak successfully intercepts the source by steering the control beams and adjusting their wavelength accordingly.IntroductionAcoustic cloaking can be regarded as the use of devices, materials, actions, or their combination to prevent acoustic detection of an object. The acoustic detection process is based on capturing sound fields indicating the existence of the object. These sound fields can originate either from self-emission, which is dubbed the acoustic signature, or from external sources that emit sound waves towards the object and record the back-scattered field. Despite numerous solutions that have been suggested over the years, acoustic cloaking remains one of the most exciting and intriguing problems in Engineering. This is partially due to the endless setups, configurations, operating conditions and constraints of the objects to be cloaked, as well as of the associated detection conditions, each posing its own challenge and requiring its own targeted solution. The cloaking problem can be formulated in different ways. One is scattering suppression and/or absorption. For this formulation, a common approach includes passive shells covering the object [1,2]. In a more advanced version these shells are given by architectured structures, also known as metamaterials, which are artificially designed to realize, through the collective dynamical behavior of their unit cells, properties that are unavailable in natural materials. In particular, patterns of foams and metal plates cut into labyrinthine units, perforated with holes or machined into cavities, intricately layered structures, and many other sophisticated designs were suggested[3,4,5,6,7,8,9]. Other types of metamaterial-based solutions, which originated in electromagnetic systems and were adapted in acoustics, are the transformation cloak[10,11], in which waves are guided around the object without far-field distortion, the carpet cloak[12,13,14], in which objects laying on the ground are concealed by properly shifting the detection wave phase imitating bare ground, the mantle cloak[15], and more[16]. What makes the passive design advantageous is first of all the ease of fabrication, for example by machining or printing, which enables adding as many unit cells as required. Another advantage is the ease of use, requiring just covering the object with the shell. There are several situations, however, in which passive solutions become less effective. One situation is related to low frequency, hence long wavelength detection signals. The longer are the wavelengths, the ticker and thus the heavier the absorbing coatings usually become (e.g. a low frequency anechoic chamber[17]). Multiplying by the coverage area, especially for large-scale objects, the passive shell weight may grow impractically high. Although subwavelength and thus thin coatings that can efficiently absorb low frequency signals do exist[3,18], they are usually effective in a particular frequency range. This Email address: leabeilkin@tauex.tau.ac.il (Lea Sirota). brings us to the second situation, which is inconsistency in detection conditions. The detection signal frequencies, the relative position between the object and the detector, as well as the object's own impedance may vary during operation. To address these situations, tunable designs have been suggested[19,20], mostly based on adjustable geometry of the unit cells. Such designs significantly expand the range of the cloaking effectiveness, while keeping the shell thin and lightweight. If the tunability is enabled only offline, the cloak is efficient when the change in the detection conditions is known in advance. However, if the detection conditions vary in real-time, online tunability becomes necessary. Active structures, which are operated by external energy sources and are encountered in diverse waveguiding applications[21,22,23,24,25], could then be utilized. For example, a type of cloak that employs active real-time modulation is the spacetime cloak[26,27], the goal of which is hiding acoustic events of finite duration. In that cloak the medium dispersion is manipulated in time rather than in space, creating a 'hole' in time.In this work, we consider the situation in which the cloak properties need to be based on actual real-time measurements of the detection field and the object's response. Measurement-based active designs usually include a host structure and external actuators at the degrees of freedom. The actuators generate inputs based on commands of embedded electronic controllers, which process the system's dynamical response in real-time. The controllers can be reprogrammed at will by the user, implying that the overall structural dynamics is determined by the program and not by a particular element, passive or active. The resulting capabilities include real-time tuning of existing properties, creating long-range interactions via distant site measurements, creating new, possibly non-physical dynamical behaviors of any kind, and switching between all these functionalities on the same platform, without the need to refabricate the elements. In addition, since the main components of such structures are actuators and sensors, which in acoustics are usually given by flat surfaces, their total weight can be significantly smaller than the weight of passive shells. Actively-controlled measurement-based approach has recently emerged in diverse electromagnetic, acoustic and elastic wave manipulation applications[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44], enabling exotic wave dynamics, such as violation of Newtonian mechanics, non-reciprocal propagation, adaptive refocusing, or artificial boundary conditions for simulation domain scaling. Utilizing this approach, we design an actively-controlled acoustic cloak in a form of an artificial quiet channel, which creates a moving dead zone around the object to be concealed, and adapts in real-time to changes in certain detection properties. In Sec. 2 we describe our cloaking system setup with assumed constraints on actuation and measurement locations, as well as the detection system setup and its parameters uncertainties. In Sec. 3 we derive the control algorithm, which is based on a technique we term as near unidirectional wave generation, necessary for the artificial dead zone creation. In Sec. 4 we simulate the performance of our cloak, and demonstrate its effectiveness for different frequencies and incidence angles of the detection beam. The work is discussed and summarized in Sec. 5. Cloaking system setup Our goal is to prevent the detection of objects in a two-dimensional nondispersive medium, where the properties of the object, such as its impedance, as well as the geometrical and spectral properties of the detection signal, are not known in advance and can vary during the detection sessions. To handle these variations we design a cloaking system that is based on active real-time control. Our representative platform is a waveguide consisting of two rigid parallel plates with a gap between them, as illustrated in the schematic of Fig. 1(a). For a gap sufficiently small with respect to the wavelength, the propagation in the gap can be approximated as two-dimensional. In addition, to narrow down this very general problem, as well as to account for constraints that may arise in practical situations, we consider the following set of conditions: (i) There is only one detection emitters array, located either to the left or to the right of the object. The detection receivers array, on the other hand, is assumed to exist on both sides of the object. (ii) The detection signal is a single frequency harmonic beam of finite duration that scans the waveguide area. The beam spectral properties, including amplitude, frequency and phase shift, as well as its geometric properties, including scanning angle, spatial width and emitters distance from the object are unknown in advance. All these properties can vary in a discrete manner, i.e. the detection process takes place at finite time intervals, with breaks between the intervals. During a particular interval the properties are constant, but in the next interval they can be different. (iii) The control actuators are acoustically transparent, and can be placed behind and in front of the object in an interior line. That is, they cannot be mounted on the object's boundary and cannot constitute a boundary themselves. The sensors can be placed only along an interior line as well. (iv) The object's self emission is negligible. (v) The object cannot be covered by a rigid shell. Following condition (i), we assume without loss of generality that the detection emitters are located along the left boundary of the waveguide. In Fig. 1(a) these emitters are sketched as blue cuboids. Condition (ii) enables us to treat the system as linear time-invariant between the intervals, thus facilitating control design, and yet reasonably accounting for the inevitable variation in detection properties. To accommodate condition (iii), we consider monopole type actuators mounted in one of the plates, e.g. the upper one, and facing inwards, as illustrated by the black and purple cylinders in Fig. 1(a). The actuators are those that create the cloak. They are arranged in two pairs of arrays, which form and determine the cloaking region. The two arrays of gray cylinders indicate the sensors, which measure the sound pressure field between the plates. This particular two pair arrangement both of the actuators and the sensors is directly related to our control strategy, as described in Sec. 3. The The underlying platform is a two-dimensional waveguide consisting of gapped parallel plates, which supports sound propagation between the plates. The sound pressure field is created by a series of emitters, blue cuboids, along one of the boundaries, constituting the detectors. An active cloaking system is implemented in the waveguide using two pairs of control actuator arrays, black and purple cylinders, which create an artificial dead zone between the arrays. The control operation is based on real-time pressure field measurements performed by two sensor arrays, gray cylinders. (b) The working principle of the cloak. Depicted is the horizontal cross-section between the plates. The detection sources emit a sound beam ψ, gold strip, of frequency ω and of y extension d, which scans the area at an angle θ . d, ω and θ are unknown and may vary. The actuators at (x o , y o2 ) and (x o − ε, y o1 ) ((x c , y c2 ) and (x c − ε, y c1 )) generate a near uni-directional control beam in the r direction with a backwave in the r b direction, which opens (closes) the dead zone, gray shadowed area. The object, dark gray cylinder, regardless of its possibly varying impedance, remains undetected by observers along O r , dotted-black, and O l , dotted purple. (c) The control scheme, depicted at the side view cross-section. The controller C ψ predicts the detection wave response along actuation arrays, based on real-time information about the properties of ψ, assumed to be obtained from measurements along x m1 and x m2 . Controllers actuation transducers are assumed to be nonresonating and not scattering the fields by their physical presence, which means they are acoustically transparent, as required. When these actuators are switched off, free space propagation between the plates is fully resumed. Here, acoustic transparency essentially means that we cannot place actuators in the waveguide gap. As for conditions (iv) and (v), the common approach is first to cover the actual object by a perfectly rigid shell, and then to design the cloak on top of it [10,11]. That way the self emission of the object and its impedance do not matter at all. These solutions work well when the rigid coverage is possible, but since here it is not allowed, both self emission and self impedance become essential. Condition (iv) implies that only self impedance matters. Condition (v) then determines the key principle of our approach: canceling the detection wave in the propagation direction, in mid-domain, before it hits the object, i.e. before it becomes affected by the object's impedance, and not after that. This is crucial for satisfying condition (v). Our cloaking concept is illustrated in Fig. 1(b) in a top view of the medium between the plates. The detection system launches a two-dimensional beam of y extension d, amplitude A, frequency ω and phase shift φ , which scans the area at angle θ in order to locate the object. The distance D between the inner actuation arrays defines the region where the object (gray cylinder) can exist and needs to be cloaked. In this region the active cloaking system needs to suppress the detection beam, i.e to create a null intensity field that we denote by a dead zone, and then to reconstruct this field beyond this region so that it coincides with free space propagation. The distance between the actuator arrays within each pair is defined by ε. The actuators along x = x o and x = x o − ε are responsible for the beam suppression, i.e. for the dead zone opening, whereas the actuators along x = x c and x = x c − ε are responsible for the reconstruction, i.e. the dead zone closing. This dead zone can be regarded artificial, since unlike passive absorption, here it comprises both the detection beam and the out-of-phase control beam. The control commands for all the actuators, including which of them should be activated (the locations y o1 , y o2 for opening and y c1 , y c2 for closing) and their required time response, are based on real-time prediction of the detection wave evolution along the actuators arrays. This prediction is based on real-time information about d, A, ω, φ and θ , which is assumed to be obtained from the sensors read-out along x m1 and x m2 . The general principle of the control algorithm is based on suppressing the detection beam at x > x o for all y, and then reconstructing it at x > x c as if the cloak did not exist. The cloak will be considered effective if the time response of the right and left observer arrays, respectively located along x = O r , dotted black, and x = O l , dotted purple, is close enough to an unperturbed detection wave. The control wave, both at x = x o and x = x c , thus needs to create as minimum backward disturbance as possible: at x < x o due to the O l observer, and at x < x c in order to keep the null intensity field in the cloaking region, which otherwise will cause reflection from the object and affect the O r observer. Therefore, the control beams need to be as uni-directional as possible in the propagation direction r. Uni-directional beam generation is most often associated with boundary sources, since then only the outside of the boundary region exists. In the context of wave control, boundary sources are used in active closed loop setups for wave suppression in the entire structure [45,46,30,47,48]. However, launching such beams from a domain interior, i.e. when no physical boundaries present, is more intricate. Theoretically, mid-domain uni-directional actuation can be achieved by launching a monopole and a dipole source simultaneously from the same interior point, as is sometimes done in numerical simulations. However, since this is not practical in general, and in particular in our case due to the restriction on placing actuators in the waveguide gap, our algorithm needs to generate a similar effect using a pair of monopole actuator arrays. This algorithm is presented next. The control algorithm To derive the algorithm, we first mathematically describe the field inside the waveguide and its coupling to the external sources. The continuous acoustic medium of mass density ρ 0 and bulk modulus b 0 is defined by a scalar pressure field p(x, y,t) and a vector flow velocity field v(x, y,t), consisting of the components v x (x, y,t) and v y (x, y,t) in the x and y directions, respectively. Defining the detection signal by ψ, and the cloak opening and closing monopole arrays by q o,1 , q o,2 and q c,1 , q c,2 , the field equations take the form ρ 0 ∂ v(x, y,t) ∂t = −∇p(x, y,t),(1a)1 b 0 ∂ p(x, y,t) ∂t = −∇ · v(x, y,t) + ψ(y,t)δ (x) + ∑ i=o,c q i,1 (y,t)δ (x − (x i − ε)) + q i,2 (y,t)δ (x − x i ). (1b) The delta function δ (·) indicates the position of the sources along the x axis, so that ψ is distributed along the edge x = 0, q o,1 and q o,2 along x = x o and x = x o − ε, and q c,1 and q c,2 along x = x c and x = x c − ε. The overall control scheme is illustrated in Fig. 1(c), where we define q o,1 = −h o , q o,2 = h o + f o , q c,1 = −h c , q c,2 = h c + f c .(2) The control system begins with the controller C ψ , which predicts the geometrical and spectral properties of the detection beam along x = x o , x = x o − ε, x = x c and x = x c − ε. Near uni-directional interior control wave generation In this section we design the controllers C ho and C hc , which represent the key principle of our artificial quiet channel cloak. The pairs h o , f o and h c , f c need to generate beams as uni-directional as possible in the x, y plane along the angle θ , which is the incidence angle of the detection beam ψ(y,t). In Fig. 1(b) we defined this direction by r. Inspired by recent ideas for a purely one-dimensional setup [49,50], but facing the complexity of the additional spatial dimension, we now derive an algorithm for the pair h o , f o in two dimensions, where the same principle applies to the pair h c , f c . If a coinciding monopole input f o (y,t)δ (x − x o ) (with h o (y,t) = 0) and a hypothetical r direction dipole input d o (y,t) were practically possible along the actuation line x = x o , the term D o (x, y,t) = d o (y,t)δ (x − x o ) would be added to (1a). Setting then d o (y,t) = z 0 f o (y,t),∂ D o (x, y,t) ∂ x = z 0 f o (y,t) lim r ε →0 1 r ε (δ (x − x o ) − δ (x − (x o − ε))) = ∂ h o (y,t) ∂t (δ (x − x o ) − δ (x − (x o − ε))) ,(3) where r ε = ε/ cos θ is the direct distance between h o and f o in the r direction, and the angle θ is expected from the measurements. This implies that when r ε is sufficiently small, h o (y,t) becomes the limit of a simultaneous differentiation of f o (y,t) in space with respect to r, and its integration in time, h o (y,t) → 1 τ rε t 0 f o (y,t ′ )dt ′ .(4) Here, τ rε = r ε /c is a time constant indicating the time required for a wave to travel the respective distance r ε , and c = b 0 /ρ 0 is the speed of sound in the medium. The relation in (4), however, might lead to an unbounded h o (y,t) for f o (y,t) that contains zero frequency components. To define a stable control law relating h o and f o we consider an approximation of (4), which in Laplace domain takes the form h o (y; s) = C h (s) f o (y; s) , C h (s) = 1 τ rε (s + η)(5) for some constant η > 0, which is a free design parameter. The controller C h in (5) corresponds to both C ho and C hc in Fig. 1(c). The distribution of f o (y,t) and ±h o (y,t) along the y axis is determined by a rectangular window of width d, modulated by a Gaussian envelope. The window begins at y o1 for ±h o (y,t) and at y o2 for f o (y,t) (y c1 and y c2 , respectively, for the ±h c , f c pair). The actual values of y o1 and y o2 (y c1 and y c2 ), as well as the y axis dependence of f o ( f c ) that is responsible for the rotation of the control beam at θ , are determined by the algorithm of C f o (C f c ), as detailed in Sec. 3.2. The control law (5) generates a near unidirectional control beam in the x > x o half space. The fact that the resulting control wave is near but not perfectly unidirectional is exhibited through a residual back action wave, generated in the r b direction, which we denote by a backwave. The smaller is ε with regards to the propagating wavelength λ , the smaller is the backwave at x < x o . The backwave depends also on the incidence angle. This can be illustrated by calculating the forward and backward responses in locations r and r b along the r and r b axes, respectively. The pressure field response p f ,h to the control inputs f o and ±h o can be written in Laplace domain as p cont (r; s) = 1 2 z 0 e −τ r s f o (y; s) + 1 − e −τ rε s h o (y; s) , r ≥ 0, p cont (r b ; s) = 1 2 z 0 e −τ rb s f o (y; s) + 1 − e +τ rε s h o (y; s) , r b ≥ 0,(6) where τ r = r/c and τ rb = r b /c are the corresponding time constants. The control waves are given in (6) in two regions of interest: in r > 0, where cancellation of the detection wave is required, and in r b < −r ε , where suppression of the residual control backwave is required. The transfer function from each concentrated input to the pressure p cont in (6) is a scaled pure delay, which indicates that the output is a pure shift of the input. The propagating wave does not undergo any shape distortion, as expected for the non-dispersive system (1). With the control law (5), the response in (6) takes the form p cont (r; s) = 1 2 z 0 e −τ r s Q(s) f o (y; s), r ≥ 0, p cont (r b ; s) = 1 2 z 0 e −τ rb s Q(s) f o (y; s), r b ≥ 0,(7) where Fig. 2. Progressive and regressive wave decoupling conditions. Left -the interval n backwave needs to leave x m1 no later than the arrival of the n + 1 interval detection beam, implying the α n+1 ≥ β n requirement. Right -real-time parameters estimation should take place after the interval n detection beam arrived at x m1 but not later than the arrival of the n interval backwave. This implies the α n+1 ≥ β n + 2τ mo requirement, where τ mo = l mo /c. Q(s) = 1 + A(s), A(s) = 1 − e −τ rε s C h (s), Q(s) = 1 + A(s), A(s) = 1 − e +τ rε s C h (s).(8) As the distance r ε and the free parameter η become smaller, we obtain the limits lim r ε ,η→0 A(s) = 1, lim r ε ,η→0 A(s) = −1, lim r ε ,η→0 Q(s) = 2, lim r ε ,η→0 Q(s) = 0.(9) When the limits in (9) are approached, the control wave travels in the positive r direction only, as indicated by (7), and illustrated by the inset in Fig. 1(c). The efficiency of the proposed near uni-directional mechanism is captured by the control effort that is required to achieve a particular reduction of the backwave. The amplitude of the actual near uni-directional wave is determined by Q(s). The amplitude of the backwave is determined by Q(s). The maximal effort of the control input h o increases when η decreases and when the ratio λ /ε increases, where λ is the beam wavelength. This constitutes a trade-off with the backwave amplitude, as demonstrated both in frequency domain and in time domain in Sec. 4, Fig. 3. Response prediction at actuation locations, cloak opening and closing After designing the controller C h in (5), which relates f o,c with h o,c , we now derive the algorithms for the resulting near unidirectional control waves, f o and f c , which are respectively responsible for opening and closing the dead zone. These algorithms, respectively coded into the controllers C f o and C f c , are based on real-time prediction of the detection beam free space evolution along the channel opening and closing locations, x o , x o − ε, x c and x c − ε, carried out by the controller C ψ . Following the finite intervals assumption (condition ii of Sec. 2), and the geometric definitions in Fig. 1(b), the detection beam as captured by the sensors along x m1 and x m2 , is given by a series of harmonic bursts at time intervals α 1 ≤ t ≤ β 1 , α 2 ≤ t ≤ β 2 , and so on, so that α n+1 ≥ β n . At the n th interval the measured pressure field beams, p m1 = p(x m1 , y,t) and p m2 = p(x m2 , y,t), have an amplitude A n , frequency ω n , and a respective phase shift φ m1 n and φ m2 n . The beam has also the y axis support of U(y − y m1,2 n ) − U(y − y m1,2 n − d n), with U being the step function, indicating the incidence angle θ n . In the general case p m1 and p m2 contain both the original detection beam propagating in r, and the residual control back wave propagating in r b . Therefore, in order to estimate A n , ω n , θ n and φ m1,2 n from p m1 and p m2 , the original source beam first needs to be separated from the total measurement. In one dimension this is quite a straightforward procedure, common in many wave applications, and can be achieved, for example, using the acoustic transfer matrix method [51]. In two dimensions it is more complicated, but methods do exist [52]. In our system, however, this separation can occur naturally by two means. The first is geometrical. Since the beam is assumed of finite y axis extension d, for a nonzero θ the original ψ and the backwave become completely geometrically decoupled at x d = x o − 1 2 d cotθ . Therefore, sensor arrays placed to the left of x d will contain two separate nonzero sections, from which only the lower one needs to be identified. However, as x d may result too far from the object (due to small θ and/or large d), this type of decoupling is not very reliable. Instead, we assume the second type, a dynamical one, which has the potential to occur due to the non-overlapping time intervals assumption. As illustrated in Fig. 2-Left, full detection wave -backwave separation takes place if the tail of the interval n backwave left the sensor arrays before the head of the interval n + 1 detection wave entered it. That is, α n+1 ≥ β n + 2τ mo , where τ mo = l mo /c is the time that takes a wave of speed c to travel l mo = (x o − x m1 )/ cos θ , the distance between the sensors array at x mo and the actuators array at x o in the r direction. Since the sensors location is our choice, τ mo can be made reasonably small. Then, the parameters estimation needs to take place during the time when the interval n detection wave head entered the sensor arrays and before the backwave of the same interval reaches these arrays, i.e. during α n ≤ t ≤ α n + 2τ mo , as illustrated in Fig. 2-Right. We denote this time period by the effective measurement time. The consequent estimation of A n , ω n , θ n and φ m1,2 n from the separated source beam can be carried out using any appropriate signal processing algorithm, and we assume here that these parameters are available and fed into C ψ . At each time step n the controller C ψ then predicts the expected pressure fields at the actuators arrays. In particular, it predicts which actuators along the arrays at x o,c and x o,c − ε need to be activated, and generates the corresponding support windows U o,c n (y), as well as a Gaussian envelope G n (y) of scaling σ that smooths out the windows corners, given by U o,c n (y) = U(y − y o,c n ) − U(y − (y o,c n + d n)) , G(y) = e − 1 2 (σ y) 2 .(10) Here, y o,c n refers to y o1,2 n for the dead zone opening part and to y c1,2 n for its closing. The controller C ψ also needs to predict the phase shift between the measurement position and the control position. This is obtained via φ o,c n = x o,c − x m1,2 c cos θ n ω n .(11) The next stage is determining the controllers C f o and C f c . The control waves f o,c (y,t) need to account for the dynamics introduced by the function Q(s) defined in (8), which is responsible for the unidirectional propagation, and stems from the control law C h , defined in (5). The amplitudes of f o,c (y,t) and their phases are thus respectively modulated by M Q n , the inverse of the amplitude of Q(s), and δ φ Q n , the negative phase of Q(s). Both are calculated at the detection frequency ω n , and are given by M Q n = 1 |Q(iω n )| , δ φ Q n = −∢Q(iω n ).(12) The controllers C f o and C f c then need to determine the control wave rotation at the angle θ n . This can be obtained using a phased array like modulation of the form ∆ϕ n (y) = yω n c sin θ n . Combining (10) which completes the derivation of the control algorithm. Cloak performance demonstration We now demonstrate the dynamical performance of the cloaking system of Fig. 1 according to the control algorithm derived in Sec. 3. We choose the waveguide medium to be air, implying mass density ρ 0 = 1. At the first step we examine the trade-off between the ratio λ /ε, wavelength over actuation arrays spacing, and the control effort. This trade-off was analyzed in Sec. 3.1, where it was shown that the control input that is traded off is h, which is responsible for the uni-directionality of the total control wave. The trade-off is illustrated in Fig. 3. Subplot (a) depicts the frequency response diagrams of the control input amplitude |C h (iω)| (dashed-dotted lines) versus the backwave amplitude |Q(iω)| (solid lines), defined in (5) and (8) The next step is demonstrating the performance of the actual cloak, which is the dead zone creation process. In the following we consider a fixed actuators x axis spacing of ε = 0.025 m, where all the adaptation to the changing detection properties is carried out in real-time through the controllers C ψ , C h and C f o,c , designed in Sec. 3. In Fig. 4 we test our algorithm under frequency variation. We launch a beam in four time intervals, having a fixed incidence angle θ Fig. 4, here the left column depicts free space 2D pressure field response, the middle column depicts the response when the cloak is turned on, and the right column depicts the comparison between the two (dashed-dotted vs. solid), as observed by the detection system at O l , black, and O r , purple. The pressure field intensity average inside the dead zone is identical for the four intervals, given by a 20 times reduction of the source wave. (b) λ /ε = 20 (c) λ /ε = 8 (d) λ /ε = 8 (e) λ /ε = 4 Conclusion We studied the problem of active acoustic cloaking of objects that can move in a two-dimensional medium. The cloaking goal was to create a real-time reconfigurable dead zone, which is an artificial quiet channel for the object to pass through undetected, under several constraints that may arise in practical applications. Specifically, we assumed that (i) the detection system emits signals from one side of the object but observers the response on both of its sides, (ii) the detection signal is a series of harmonic beams launched at finite time intervals, and have their properties, such as frequency, phase, amplitude, distance from object, incidence angle etc., changed between the intervals, (iii) the control actuators and sensors cannot block the medium, i.e. cannot constitute a boundary themselves, (iv) the object is non-emitting but of an unknown and possibly varying impedance, and (v) the object cannot be covered by a rigid shell. We proposed a solution that we denoted by a mid-domain near uni-directional wave generation. This technique enables to launch control beams in a desired direction in the domain interior that can be steered around with a minimal back action wave as a trade-off with the control effort. We demonstrated the idea in a two-dimensional waveguide platform with gapped plates, depicted in Fig. 1. The cloak was executed by two pairs of monopole actuator arrays and one pair of sensor arrays, which were mounted in the interior of one of the plates, facing inwards but not blocking the gap. Our control algorithm for the actuator inputs, given by (5)- (14), generated two near uni-directional beams. One beam opened the dead zone by intercepting the detection signal before it hit the object, thus preventing its coupling with the object impedance. The second beam closed the dead zone by reconstructing the original signal and imitating free space propagation for the observers. The time evolution of both control beams was based on real-time pressure field prediction at the actuation locations, based on information assumed to be available from measurements, according to the detection intervals that are illustrated in Fig. 2. We tested our cloak performance via time domain numerical simulations of a 2.4 × 3 m 2 waveguide filled with air. First, in Fig. 3 we demonstrated the trade-off between the level of control beam uni-directionality, exhibited by the backwave amplitude minimization, and the control effort required to achieve it for a given actuator arrays spacing. Then, keeping the actuators locations fixed, we changed a certain property of the detection beam. In Fig. 4 Fig. 1 . 1Artificial quiet channel cloak. (a) The model schematic. C ho and C hc create the total unidirectional opening and closing waves f o and f c out of the pairs f o , h o and f c , h c . Controllers C f o and C f c generate the commands for f o and f c . Inset: illustration of the near uni-directional wave formation in the r direction with a backwave in the r b direction. Based on these properties, the commands for the control inputs f o , h o , f c and h c are produced. The controllers C f o and C f c directly relate f o and f c to the output of C ψ . The controllers C ho and C hc respectively relate h o to f o and h c to f c , and are completely independent of C f o and C f c . We begin with the design of C ho and C hc , as detailed in Sec. 3.1, and then proceed to the design of C ψ , C f o and C f c , as detailed in Sec. 3.2. -(13), the control commands of C f o and C f c become f o,c n (y,t) = ∑ n U o,c n (y)G o,c n (y)M Q n A n e iω n t e i(∆ϕ n (y)−δ φ o,c n −δ φ Q n) , 2 kg/m 3 and bulk modulus b 0 = 1.42 · 10 5 N/m 2 . The numerical experiments are carried out via a 2D finite difference time domain procedure with spatial and temporal discretization steps of a = 0.005 and dt = 10 −6 , respectively. The overall waveguide dimensions are set to L x × L y = 2.4 × 3.0 m 2 . Absorbing boundary conditions are modeled along the waveguide boundaries. The locations of the f o and f c control input arrays, the two sensor arrays, and the left and right observer arrays, are respectively given by x o = 0.35, x c = 0.75, x m1 = 0.39, x m2 = 0.43, O l = 0.12 and O r = 2.16 m. Fig. 3 . 3Trade-off between the control effort and the control wave uni-directionality level. (a) Frequency response diagram of the controller C h (s), dashed-dotted, and of the backwave Q(s), solid, for ε = 0.02 m, blue, ε = 0.05 m, solid-black, ε = 0.025 m, dotted-black, and ε = 0.02 m, purple. The markers on the curves indicate the amplitudes corresponding to the source frequencies ω ψ1 = 0.86 kHz and ω ψ2 = 1.72 kHz. (b)-(e), Time domain snapshots of control signal h(y,t) according to the algorithm in (5) for θ = 22 o . (b),(c),(e), λ = 0.4 m with ε = 0.02, 0.05, 0.1 m, leading to λ /ε = 20, 8, 4. (d), λ = 0.2 m with ε = 0.025 m, leading to λ /ε = 8. (f), h o (y,t) along the y axis actuation section at a time instance when the amplitude of h o (y,t) is maximal.The higher is λ /ε, the smaller is the backwave, but the higher is the control effort. ε = 0.02 m and ε = 0.1 m, whereas the black curves correspond to ε = 0.05 m (solid) and ε = 0.025 m (dotted). To relate to a response with a fixed ratio λ /ε, we consider two detection frequencies, ω ψ1 = 5.4 · 10 3 r/s (or 0.86 kHz), implying λ = 0.4 m, and ω ψ2 = 10.8 · 10 3 r/s (or 1.72 kHz), implying λ = 0.2 m. This results in λ /ε = 20, 8, 4 for the blue, black and purple curves at the corresponding frequencies, respectively. As was predicted in Sec. 3.1, the bigger is the control effort, the smaller is the backwave. Here, for ω ψ1 , backwave amplitudes of −14 dB, −7 dB and −2 dB require a control effort of 10 dB, 3 dB and −4 dB. The actual values of λ and ε nearly do not affect the amplitudes as long as their ratio is preserved, as demonstrated for λ /ε = 8 at ω ψ1 and ω ψ2 . Subplots (b)-(e) depict steady-state snapshots of the pressure field time responses to the total control wave at x o and x o − ε according to (7) for θ = 22 o . The command for f o is a harmonic signal ψ(y,t) of frequency ω and a geometrical phase distribution ∆ϕ(y). ω and ∆ϕ(y) uniquely determine the resulting wavelength λ = 2πc/ω and incidence angle θ , respectively. Snapshots (b),(c),(e) correspond to the fixed wavelength λ = 0.4 m, and to the three distances ε = 0.02, 0.05 and 0.1 m, whereas snapshot (d) corresponds to λ = 0.2 m and ε = 0.025 m, in accordance withFig. 3(a). The main control wave propagates in x > x o , whereas the backwave propagates in x < x o . As expected, the backwave increases as λ /ε decreases. To complete the picture, subplot (f) depicts the y axis cross-section of the h o component of the responses inFig. 3(b)-(e) along the actuation section d at times instances when the control signal amplitude is the highest. For λ /ε = 20, 8 and 4 with λ = 0.4 m the maximal amplitudes are in accordance with the trade-off prediction of the frequency response in subplot (a). For λ /ε = 8 with λ = 0.2 m the spatial distribution of h o (y,t) is different but the maximal amplitudes ratio remains in the same order of magnitude. = 22 o in all the intervals, and a different wavelength in each: λ = 0.1 m, first row, λ = 0.2 m, second row, λ = 0.3 m, third row, and λ = 0.4 m, fourth row. These wavelengths respectively correspond to the frequencies ω = 3.4 kHz, ω = 1.72 kHz, ω = 1.14 kHz and ω = 0.86 kHz, resulting in the ratio λ /ε of 4, 8, 12 and 16. The four beams are launched with the same amplitude A = 1, the same vertical shift of y = 0.45 m, but have a different y axis extension d and a different Gaussian scaling. The left column depicts steady-state time snapshots of the pressure field responses in the waveguide when the cloaking system is turned off, giving a free field propagation. The middle column depicts the corresponding responses when the cloaking system is turned on, resulting in a dead zone between the two pairs of actuator arrays, at x o , x o − ε and x c , x c − ε. These arrays are pictured by solid vertical lines, the measurement arrays at x m1 , x m2 by dotted lines, and the observer arrays at O l , O r by dasheddotted lines. In all the intervals the algorithm managed to generate the required opening and closing control beams in real-time. The smaller is the wavelength, the higher is the resulting backwave amplitude, as seen from the field intensity maps of the snapshots. The minimal pressure reduction in the dead zone was by 10 times of the original detection signal, obtained for λ = 0.1 m. The maximal reduction was by 20 times, obtained for λ = 0.4 m. The right column depicts the y cross-section of the pressure field time responses in the left and middle columns at the detection observers arrays. The figures compare the responses along the left observer at x = O l , black, and right observer at x = O r , purple, at the cloak-off (free space) state, dashed, and the cloak-on state (dead zone), solid. The dashed and solid responses are reasonably close. The largest deviation occurs at the right observer at the backwave region. In Fig. 5 we test our algorithm under incidence angle variation. We launch a detection beam of a fixed wavelength λ = 0.2 m (frequency ω = 1.72 kHz) during four time intervals, in which the incidence angle θ is switched through 30 o , 15 o , 0 o and −30 o . The y axis beam extension, its amplitude and Gaussian scaling are identical in all the intervals. The four values of θ respectively correspond to the first, second, third and fourth rows of the table. Similarly to we imitated detection sources of a varying frequency by launching a beam at an angle θ = 22 o with wavelength switching through λ = 0.1, 0.2, 0.3, 0.4 m at each new time interval. The longer was the wavelength the smaller was the backwave, resulting in up to 20 times of amplitude reduction in the dead zone. In Fig. 5 we imitated steering detection beams by scanning the area at angles switching through θ = 30 o , 15 o , 0 o , −30 o . The amplitude reduction in the dead zone was about 20 times for all the angles. Fig. 4 . 4Cloak performance demonstration for a varying detection frequency ω. Plotted are snapshots of the pressure field time responses to a detection beam emitted from the left boundary of a 2.4 × 3.0 m 2 air waveguide with ε = 0.025 m. Rows 1:4 correspond to source frequencies ω = 3.4, 1.72, 1.14, 0.86 kHz implying the ratio λ /ε = 4, 8, 12, 16. Left column: entire waveguide responses when the cloak is off, giving free space propagation (black arrow -propagation direction). Middle column: entire waveguide responses when the cloak is on, resulting in the required dead zone formation. Vertical lines: solid -the actuators, dotted -the sensors, dashed-dotted -the observers. Right column: comparison of cloak-on (solid) and cloak-off (dashed-dotted) responses at the left (black) and right (purple) observers. Fig. 5 . 5Cloak performance demonstration for a varying incidence angle θ . Plotted are snapshots of the pressure field time responses to a detection beam emitted from the left boundary of a 2.4 × 3.0 m 2 air waveguide with ε = 0.025 m. Rows 1:4 correspond to source angles θ = 30 o , 15 o , 0 o , −30 o . Left column: entire waveguide responses when the cloak is off, giving free space propagation (black arrow -propagation direction). Middle column: entire waveguide responses when the cloak is on, resulting in the required dead zone formation (vertical lines the same as in Fig. 4). Right column: comparison of cloak-on (solid) and cloak-off (dashed-dotted) responses at the left (black) and right (purple) observers. a perfectly unidirectional wave would be launched at x > x o in the r direction, which is determined by the y axis dependence of f o . The monopole inputs h o (y,t) at x o and −h o (y,t) at x o − ε thus need to mimic the dipole input d o (y,t). Their exact expression is obtained by combining (1a) and (1b) into a total second order wave equation, which requires differentiation of D o (x, y,t) with respect to r, and of h o (y,t) with respect to t, leading to AcknowledgementWe thank Viacheslav (Slava) Krylov and Amir Boag for insightful discussions. 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{'abstract': "We consider the problem of hiding non-stationary objects from acoustic detection in a two-dimensional environment, where both the object's impedance and the properties of the detection signal may vary during operation. The detection signal is assumed to be an acoustic beam created by an array of emitters, which scans the area at different angles and different frequencies. We propose an active control-based solution that creates an effective moving dead zone around the object, and results in an artificial quiet channel for the object to pass through undetected. The control principle is based on mid-domain generation of near uni-directional beams using only monopole actuators. Based on real-time response prediction, these beams open and close the dead zone with a minimal perturbation backwards, which is crucial due to detector observers being located on both sides of the object's route. The back action wave determines the cloak efficiency, and is traded-off with the control effort; the higher is the effort the quieter is the cloaking channel. We validate our control algorithm via numerical experiments in a two-dimensional acoustic waveguide, testing variation in frequency and incidence angle of the detection source. Our cloak successfully intercepts the source by steering the control beams and adjusting their wavelength accordingly.IntroductionAcoustic cloaking can be regarded as the use of devices, materials, actions, or their combination to prevent acoustic detection of an object. The acoustic detection process is based on capturing sound fields indicating the existence of the object. These sound fields can originate either from self-emission, which is dubbed the acoustic signature, or from external sources that emit sound waves towards the object and record the back-scattered field. Despite numerous solutions that have been suggested over the years, acoustic cloaking remains one of the most exciting and intriguing problems in Engineering. This is partially due to the endless setups, configurations, operating conditions and constraints of the objects to be cloaked, as well as of the associated detection conditions, each posing its own challenge and requiring its own targeted solution. The cloaking problem can be formulated in different ways. One is scattering suppression and/or absorption. For this formulation, a common approach includes passive shells covering the object [1,2]. In a more advanced version these shells are given by architectured structures, also known as metamaterials, which are artificially designed to realize, through the collective dynamical behavior of their unit cells, properties that are unavailable in natural materials. In particular, patterns of foams and metal plates cut into labyrinthine units, perforated with holes or machined into cavities, intricately layered structures, and many other sophisticated designs were suggested[3,4,5,6,7,8,9]. Other types of metamaterial-based solutions, which originated in electromagnetic systems and were adapted in acoustics, are the transformation cloak[10,11], in which waves are guided around the object without far-field distortion, the carpet cloak[12,13,14], in which objects laying on the ground are concealed by properly shifting the detection wave phase imitating bare ground, the mantle cloak[15], and more[16]. What makes the passive design advantageous is first of all the ease of fabrication, for example by machining or printing, which enables adding as many unit cells as required. Another advantage is the ease of use, requiring just covering the object with the shell. There are several situations, however, in which passive solutions become less effective. One situation is related to low frequency, hence long wavelength detection signals. The longer are the wavelengths, the ticker and thus the heavier the absorbing coatings usually become (e.g. a low frequency anechoic chamber[17]). Multiplying by the coverage area, especially for large-scale objects, the passive shell weight may grow impractically high. Although subwavelength and thus thin coatings that can efficiently absorb low frequency signals do exist[3,18], they are usually effective in a particular frequency range. This Email address: leabeilkin@tauex.tau.ac.il (Lea Sirota). brings us to the second situation, which is inconsistency in detection conditions. The detection signal frequencies, the relative position between the object and the detector, as well as the object's own impedance may vary during operation. To address these situations, tunable designs have been suggested[19,20], mostly based on adjustable geometry of the unit cells. Such designs significantly expand the range of the cloaking effectiveness, while keeping the shell thin and lightweight. If the tunability is enabled only offline, the cloak is efficient when the change in the detection conditions is known in advance. However, if the detection conditions vary in real-time, online tunability becomes necessary. Active structures, which are operated by external energy sources and are encountered in diverse waveguiding applications[21,22,23,24,25], could then be utilized. For example, a type of cloak that employs active real-time modulation is the spacetime cloak[26,27], the goal of which is hiding acoustic events of finite duration. In that cloak the medium dispersion is manipulated in time rather than in space, creating a 'hole' in time.In this work, we consider the situation in which the cloak properties need to be based on actual real-time measurements of the detection field and the object's response. Measurement-based active designs usually include a host structure and external actuators at the degrees of freedom. The actuators generate inputs based on commands of embedded electronic controllers, which process the system's dynamical response in real-time. The controllers can be reprogrammed at will by the user, implying that the overall structural dynamics is determined by the program and not by a particular element, passive or active. The resulting capabilities include real-time tuning of existing properties, creating long-range interactions via distant site measurements, creating new, possibly non-physical dynamical behaviors of any kind, and switching between all these functionalities on the same platform, without the need to refabricate the elements. In addition, since the main components of such structures are actuators and sensors, which in acoustics are usually given by flat surfaces, their total weight can be significantly smaller than the weight of passive shells. Actively-controlled measurement-based approach has recently emerged in diverse electromagnetic, acoustic and elastic wave manipulation applications[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44], enabling exotic wave dynamics, such as violation of Newtonian mechanics, non-reciprocal propagation, adaptive refocusing, or artificial boundary conditions for simulation domain scaling. Utilizing this approach, we design an actively-controlled acoustic cloak in a form of an artificial quiet channel, which creates a moving dead zone around the object to be concealed, and adapts in real-time to changes in certain detection properties. In Sec. 2 we describe our cloaking system setup with assumed constraints on actuation and measurement locations, as well as the detection system setup and its parameters uncertainties. In Sec. 3 we derive the control algorithm, which is based on a technique we term as near unidirectional wave generation, necessary for the artificial dead zone creation. In Sec. 4 we simulate the performance of our cloak, and demonstrate its effectiveness for different frequencies and incidence angles of the detection beam. The work is discussed and summarized in Sec. 5.", 'arxivid': '2301.10711', 'author': ['Or Lasri \nSchool of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael\n', 'Lea Sirota \nSchool of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael\n', 'Or Lasri \nSchool of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael\n', 'Lea Sirota \nSchool of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael\n'], 'authoraffiliation': ['School of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael', 'School of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael', 'School of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael', 'School of Mechanical Engineering\nTel Aviv University\n69978Tel AvivIsrael'], 'corpusid': 256231193, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18196, 'n_tokens_neox': 15680, 'n_words': 10142, 'pdfsha': '398ca5135449386d26964ac2130376e236f0a417', 'pdfurls': ['https://export.arxiv.org/pdf/2301.10711v1.pdf'], 'title': ['Real-time-controlled artificial quiet channel for acoustic cloaking under varying detection conditions', 'Real-time-controlled artificial quiet channel for acoustic cloaking under varying detection conditions', 'Real-time-controlled artificial quiet channel for acoustic cloaking under varying detection conditions', 'Real-time-controlled artificial quiet channel for acoustic cloaking under varying detection conditions'], 'venue': []}

arxiv

Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories Marcus D Liebenthal Department of Chemistry and Biochemistry Florida State University 32306-4390TallahasseeFLUSA Nam Vu Department of Chemistry and Biochemistry Florida State University 32306-4390TallahasseeFLUSA A Eugene Deprince Department of Chemistry and Biochemistry Florida State University 32306-4390TallahasseeFLUSA Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories Cavity quantum electrodynamics (QED) generalizations of time-dependent (TD) density functional theory (DFT) and equation-of-motion (EOM) coupled-cluster (CC) theory are used to model small molecules strongly coupled to optical cavity modes. We consider two types of calculations. In the first approach (termed "relaxed"), we use a coherent-statetransformed Hamiltonian within the ground-and excited-state portions of the calculations, and cavity-induced orbital relaxation effects are included at the mean-field level. This procedure guarantees that the energy is origin invariant in post-self-consistent-field calculations. In the second approach (termed "unrelaxed"), we ignore the coherent-state transformation and the associated orbital relaxation effects. In this case, ground-state unrelaxed QED-CC calculations pick up a modest origin dependence but otherwise reproduce relaxed QED-CC results within the coherent-state basis. On the other hand, a severe origin dependence manifests in ground-state unrelaxed QED mean-field energies. For excitation energies computed at experimentally realizable coupling strengths, relaxed and unrelaxed QED-EOM-CC results are similar, while significant differences emerge for unrelaxed and relaxed QED-TDDFT. First, QED-EOM-CC and relaxed QED-TDDFT both predict that electronic states that are not resonant with the cavity mode are nonetheless perturbed by the cavity. Unrelaxed QED-TDDFT, on the other hand, fails to capture this effect. Second, in the limit of large coupling strengths, relaxed QED-TDDFT tends to overestimate Rabi splittings, while unrelaxed QED-TDDFT underestimates them, given splittings from relaxed QED-EOM-CC as a reference, and relaxed QED-TDDFT generally does the better job of reproducing the QED-EOM-CC results. I. INTRODUCTION Chemical applications of strong light-matter interactions facilitated by optical cavities have garnered a great deal of attention in recent years. [1][2][3][4] This interest has been driven by experimental studies offering evidence that strong light-matter coupling and polariton formation can be leveraged in chemical contexts, 4-7 such as for catalyzing/inhibiting reactions [8][9][10][11][12] or controlling reaction selectivity. 13 Moreover, a large number of computational studies have predicted a range of phenomena that are relevant to chemistry. 2,4,[14][15][16][17][18][19] Predictive electronic/polaritonic structure methods will be crucial for discovering general design principles for cavity-mediated chemistry; as a result, substantial effort has been dedicated to the generalization of familiar tools in quantum chemistry for the polaritonic problem. Proposed cavity quantum electrodynamics (QED) models incorporating an ab initio treatment of molecular degrees of freedom have largely taken one of two complementary approaches. First, given the success that density functional theory (DFT) has seen in standard quantum chemical applications, a large body of work has considered quantum electrodynamical generalizations of DFT [20][21][22][23] and time-dependent DFT (TDDFT). 19,24-31 QED-DFT and QED-TDDFT provide access to orbital-specific quantities that cannot be directly probed with model Hamiltonians; 32,33 because they inherit the favorable computational scaling of conventional DFT and TDDFT, these methods can be applied to large cavityembedded molecules or collections of molecules. At the a) Electronic mail: adeprince@fsu.edu same time, the well-known issues that plague DFT 34 and the small number of exchange-correlation functionals that have been developed for the polaritonic problem [35][36][37] have inspired others to pursue correlated wave-function-based approaches to polaritonic structure, 18,[38][39][40][41][42][43][44][45][46][47][48][49][50] within formalisms that resemble familiar coupled-cluster (CC) [51][52][53][54][55] or configuration interaction (CI) approaches. Like QED-DFT and QED-TDDFT, QED generalizations of correlated wave-function methods can provide insight into subtle cavity-induced changes to electronic structure, while also offering the advantage of systematic improvability. Straightforward polaritonic generalizations of ground-state CC and equation-of-motion (EOM) CC 54, [56][57][58] have been put forth in Ref. 38. The QED-(EOM)-CC formalism developed therein has subsequently been applied in a number of studies (illustrating, for example, how cavity interactions can influence electron ionization/attachment 40,41,43,44 reaction rates, 18,47 and non-bonded interactions 42 ), and the family of QED-CC-inspired approaches also continues to grow. QED-(EOM)-CC has been generalized to make use of non-particleconserving operators, 41 to employ unitary cluster operators, 43 for the description of chiral cavity modes, 46 and for wavefunction-in-DFT embedding protocols. 45 As mentioned above, one of the attractive features of QEDbased many-body theories such as QED-CC (and QED-EOM-CC) relative to QED-DFT (and QED-TDDFT) is the systematic improvability of the former approach. An equally important but underappreciated aspect of QED-CC is that it is robust against changes to the reference orbitals. This property is inherited from the conventional (non-QED) formulation of CC theory and stems from the presence of the exponentiated single excitation operator, eT 1 , which closely resembles an orbital rotation operator (except that it is not unitary). Indeed, it is well known that energies calculated at the CC with single and double excitations (CCSD) 59 level of theory often closely reproduce energies computed using the Bruekner coupled-cluster doubles (BCCD) approach, [60][61][62] which variationally optimizes the orbitals for the coupled-cluster doubles wave function. In the context of QED-CC, eT 1 should be able to account for orbital relaxation effects induced by the cavity in the underlying QED-HF wave function should one choose to seed a QED-CC calculation with a non-QED Hartree-Fock reference configuration (see Fig. 1); QED-EOM-CC results obtained in either case should then be similar. A related matter derives from the fact that QED-CC calculations are typically carried out using a coherent-state transformed Hamiltonian, 38 which guarantees invariance of the QED-CC energy with respect to the placement of the origin. The QED-CC cluster operator contains an exponentiated boson creation operator term that should be able to mimic the effects of this transformation, and thus we expect the origin dependence of QED-CC to be modest when using a Hamiltonian that has not been transformed to the coherent-state basis. On the other hand, as we demonstrate below, QED-TDDFT results are quite sensitive to whether or not the Kohn-Sham orbitals are allowed to relax in the presence of the cavity and whether the Hamiltonian that enters the QED-TDDFT equations is represented within the coherent-state basis. This point is subtle, yet important, given that a variety of QED-TDDFT prescriptions have been put forth and not all of them account for the presence of the cavity self-consistently. 29,30 In this work, we examine how cavity-induced changes to the orbitals and the coherent-state transformation affect the energies of cavity-embedded molecules treated at the QED-EOM-CC and QED-TDDFT levels of theory. Before doing so, we present the theory underlying relaxed and unrelaxed versions of these approaches, which differ in the treatment of the cavity at the mean-field level. The details of our calculations are then provided in the Computational Details, and numerical studies exploring the robustness of QED-TDDFT and QED-EOM-CC to the description of cavity effects at the mean-field level can be found in the Results and Discussion. Lastly, we conclude with a summary of the outcomes from our numerical studies. II. THEORY In this Section, we outline some key details of the QED-EOM-CC and QED-TDDFT approaches. Both of these methods model the physics of a cavity-embedded molecular system using the Pauli-Fierz Hamiltonian, 63,64 which we represent in the length gauge and within the dipole and Born-Oppenheimer approximations. For a single-mode cavity, this Hamiltonian takes the form H PF =Ĥ e + ω cavb †b − ω cav 2 (λ ·μ) b † +b + 1 2 (λ ·μ) 2(1) Here,Ĥ e and ω cavb †b are the Hamiltonians for the isolated many-electron system and the cavity mode, respectively; ω cav is the frequency of the cavity mode, andb † (b) is a bosonic creation (annihilation) operator. The third term in Eq. 1 describes the coupling between the molecular degrees of freedom and the cavity mode, which is parametrized by the coupling strength, λ; the symbolμ represents the molecular dipole operator. The fourth term is the dipole self-energy contribution. In the single-molecule coupling limit, the coupling strength is related to the effective mode volume, V eff , as λ = λê = 4π V effê (2) whereê is a unit vector pointing along the cavity mode polarization axis. A. Cavity QED Coupled-Cluster Theory for Ground and Excited States Cavity QED Hartree-Fock Theory The cavity QED Hartree-Fock (HF) wave function is a product of a Slater determinant of molecular spin orbitals, |0 e , and a zero-photon state, |0 p . Following Ref. 38, the zero-photon state can be exactly represented using the coherent-state (CS) transformation |0 p =Û CS |0 = exp zb † − z * b |0(3) Here, |0 is the photon vacuum, and z = − λ · µ √ 2ω(4) The symbol µ represents the expectation value of the molecular dipole operator with respect to |0 e . One can useÛ CS to transform the Hamiltonian to the coherent-state basis to givê H CS =Û † CSĤÛ CS =Ĥ e + ω cavb †b − ω cav 2 (λ · [µ − µ ]) b † +b + 1 2 (λ · [µ − µ ]) 2(5) In the coherent-state basis, the QED-HF wave function has the simple form |Φ 0 = |0 e ⊗ |0(6) and |0 e can be determined via a standard SCF procedure us-ingĤ CS after integrating out the photon degrees of freedom, i.e., 0|Ĥ CS |0 =Ĥ e + 1 2 (λ · [µ − µ ]) 2(7) We refer to a QED-HF wave function determined in this way as "relaxed," in the sense that the electronic spin orbitals account for the presence of the cavity (through the dipole self energy term in Eq. 7); the relaxed mean-field energy is the expectation value of Eq. 5 with respect to Eq. 6. On the other hand, an "unrelaxed" QED-HF wave function has the same form (Eq. 6), but |0 e is instead determined from an SCF procedure that considers onlyĤ e . The unrelaxed mean-field energy is the expectation value of Eq. 1 with respect to Eq. 6. Ground-state QED-CC theory The QED-CC wave function is defined as |Ψ CC = eT |Φ 0 (8) whereT is the cluster operator. At the QED-CCSD-1 level of theory, 38T includes up to products of double electronic transitions and a single photon creation operator: T = ∑ ia t a iâ † aâi + 1 4 ∑ i jab t ab i jâ † aâ † bâ jâi + u 0b † + ∑ ia u a iâ † aâib † + 1 4 ∑ i jab u ab i jâ † aâ † bâ jâib †(9) Here,â † andâ represent fermionic creation and annihilation operators, respectively; the labels i / j and a / b refer to spinorbitals that are occupied or unoccupied in the QED-HF reference wave function, respectively; and t a i , t ab i j , u 0 , u a i , and u ab i j are the cluster amplitudes. As mentioned above, in the case of unrelaxed QED-CC, the exponentiated single excitation operator, eT 1 , can mimic the effects of cavity-induced orbital relaxation effects in relaxed QED-HF, and the term e u 0b † is important for capturing the effects of the coherent-state transformation operatorÛ CS itself (see Fig. 1). FIG. 1. The cluster operator and similarity-transformed Hamiltonian in relaxed and unrelaxed QED-CCSD-1 (QED-CC with up to single and double electronic excitations plus single photon creation operators). 38 The single electron excitation and boson creation contributions to the cluster operator can account for the effects of orbital relaxation in QED-HF and the coherent-state transformation, respectively. The cluster amplitudes are determined using projective techniques, i.e., by solving µ e | ⊗ n|e −TĤ A eT |0 ⊗ |0 e = δ µ0 δ n0 E CC(10) Here, µ e | and n| represent a determinant of spin-orbitals and a photon-number state with n photons, respectively; E CC is the energy associated with |Ψ CC ; and the subscript A refers to the type of Hamiltonian. For relaxed QED-CC,Ĥ A =Ĥ CS ; for unrelaxed QED-CC,Ĥ A =Ĥ PF . At the QED-CCSD-1 level of theory, µ e | can represent the reference or any singly-or doubly-substituted determinant of spin-orbitals, and n can be zero or one. Excited-state QED-EOM-CC theory Given cluster amplitudes obtained by solving Eq. 10, excited states can be parametrized using the QED-EOM-CC formalism. 38 The left-and right-hand QED-EOM-CC wave functions are defined by Ψ I | = Φ 0 |e −TL I (11) |Ψ I =R I eT |Φ 0(12) where the label I denotes the state. At the QED-EOM-CCSD-1 level, 38 the operatorsL I andR I take the form The l/m and r/s amplitudes are determined by solving left-and right-hand eigenvalue equations L I = I l 0 + ∑ ai I l i aâ † iâΦ 0 |L IH = Φ 0 |L I E I(15) andHR I |Φ 0 = E IRI |Φ 0(16) whereH = e −TĤ A eT is the similarity-transformed Hamiltonian, and E I represents the energy of the I th state. As in QED-CC, the choicesĤ A =Ĥ CS andĤ A =Ĥ PF lead to relaxed and unrelaxed forms of QED-EOM-CC, respectively. B. Cavity QED Density Functional Theory A large body of literature describes quantum electrodynamical generalizations of DFT and TDDFT that differ in several aspects. First, for electronic degrees of freedom, some of these approaches represent the electronic density in real space, 27,31,36,65,66 whereas others use atom-centered Gaussian basis functions. 19,29,30,45 Second, photon degrees of freedom can be represented directly in real-space 20,24,67 or in Fock space 19,[29][30][31]45 (in a basis of photon-number states). Third, as with standard TDDFT, both real-time 24,31 and linearresponse 19,27,29,30,66 formulations have been put forward. In this work, we consider linear-response QED-TDDFT formulated in terms of Gaussian basis functions and a Fock-space representation of the photon degrees of freedom. In analogy to the relaxed and unrelaxed QED-EOM-CC methods described above, we consider both relaxed and unrelaxed linear-response QED-TDDFT approaches that are equivalent to those described in Refs. 29 and 19. Cavity QED Kohn-Sham DFT The QED-HF procedure outlined above can easily be adapted to the case of QED Kohn-Sham DFT. First, one can map the QED-DFT ground-state onto a non-interacting state of the form of Eq. 6, where |0 e now refers to a determinant of Kohn-Sham orbitals. Second, similar to the case of QED-HF, a "relaxed" Kohn-Sham determinant can be determined from an SCF procedure that makes use of the coherent-state Hamiltonian in Eq. 7, with the energy augmented by a standard exchange-correlation functional. On the other hand, an "unrelaxed" QED-DFT state can be obtained from an SCF procedure that neglects the dipole self-energy contribution to Eq. 7. Note that our formulations of relaxed and unrelaxed QED-DFT both ignore electron-photon correlation effects, such as those captured by the functionals described in Refs. 23,35-37. Cavity QED Time-Dependent Density Functional Theory Excited states in QED-TDDFT are parametrized as |Ψ I =Ô † I |Φ 0 (17) withÔ † I = ∑ ia (X I iaâ † aâi −Y i iaâ † iâ a ) + M Ib † − N Ib(18) In analogy to Rowe's equation of motion method, 68 this parameterization leads to a generalized eigenvalue problem of the form     A + ∆ B + ∆ g † g † B + ∆ A + ∆ g † g † g g ω cav 0 g g 0 ω cav        X Y M N    = Ω    1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 −1       X Y M N   (19) Here, A and B are the same matrices that arise in the usual random phase approximation (RPA) problem, e.g., Φ 0 |[â † iâ a , [Ĥ A ,â † bâ j ]]|Φ 0 = (A + ∆) ai,b j(20) etc., with exchange contributions of A and B replaced/augmented by appropriate derivatives of the exchange-correlation functional for TDDFT. The symbols ∆ and ∆ represent dipole self-energy contributions of the form ∆ ai,b j = d ai d jb − d ab d i j (21) ∆ ai,b j = d ai d b j − d a j d ib(22) where d ai is a dressed dipole integral d ai = − ∑ ξ ∈{x,y,z} λ ξ φ * a r ξ φ i dτ(23) Here, φ is a Kohn-Sham orbital, λ ξ is a cartesian component of λ, and r ξ is a cartesian component of the position vector [e.g., for r = (x, y, z), r x = x]. For relaxed QED-TDDFT,Ĥ A = H CS , and for unrelaxed QED-TDDFT,Ĥ A =Ĥ PF . In order to recover the same equations as those used in the unrelaxed QED-TDDFT formalism of Ref. 29, one must also neglect the exchange contributions to ∆ and ∆ in Eqs. 21 and 22. III. COMPUTATIONAL DETAILS The QED-TDDFT and QED-EOM-CCSD-1 methods were implemented in hilbert, 69 which is a plugin to the PSI4 70 electronic structure package. Equations for the QED-CCSD-1 and QED-EOM-CCSD-1 were generated using a locallymodified version of p † q, 71 which is a library for manipulating strings of second-quantized operators such as those that arise in coupled-cluster theory. All QED calculations used the 6-311++G** basis set with Cholesky-decomposed two-electron integrals and a tight decomposition threshold of 10 −12 E h . As mentioned in the Theory section, QED-TDDFT calculations used standard density functional approximations from electronic structure theory that neglect electron-photon correlation effects. Geometries for all molecules were optimized at the DFT level of theory, using the 6-311++G** basis set, exact two-electron integrals, and the PBE0 72 density functional. Excited-state calculations were carried out in the appropriate basis of S z = 0 determinants. The excited state potential energy curves (PECs) in all QED figures of this work were analyzed using SuaveStateScanner, 73 which assigns consistent labels to multiple states along PECs by enforcing the continuity of the excited-state energies and properties (e.g., transition dipole moments, oscillator strengths, and norms of the excitation operators). Having consistent state labels greatly simplifies comparisons between the various QED approaches we use, particularly since all of the calculations in this work are performed without enforcing spatial symmetry. IV. RESULTS AND DISCUSSION In this Section, we analyze the ground-and excited-state energies of a series of diatomic molecules (molecular hydrogen, hydrogen fluoride, and lithium fluoride), coupled to a single-mode optical cavity. We use bond lengths of 0.746 Å, 0.918 Å, and 1.582 Å for H 2 , HF, and LiF, respectively, and the symmetry labels used to describe excited states correspond to the molecular axis oriented in the z-direction. A. Ground-state energies of relaxed and unrelaxed QED-CCSD-1 We begin by considering the sensitivity of ground-state energies from QED-CCSD-1 to the treatment of cavity effects at the mean-field level. That is, we wish to assess how well exponentiated singles and boson creation operators in unrelaxed QED-CCSD-1 can mimic the effects of orbital relaxation and the coherent-state transformation in relaxed QED-HF and QED-CCSD-1. Table I provides ground-state energies from relaxed QED-CCSD-1 for several molecules coupled to a single-mode cavity with a coupling strength of λ = 0.05 atomic units (a comparable table for λ = 0.1 a.u. can be found in the Supporting Information). The cavity mode is chosen to be polarized along the molecular axis (resonant with the following states: 1 1 B 1u for H 2 , 2 1 A 1 for HF, and 3 1 A 1 for LiF) or perpendicular to the molecular axis (resonant with the following states: 1 1 B 2u for H 2 or 1 1 B 1 for HF and LiF). Also provided in Table I are errors in unrelaxed QED-CCSD-1 energies with respect to the relaxed ones. For H 2 , we see that unrelaxed and relaxed QED-CCSD-1 agree to at least 10 −9 E h , but errors on the order of 10 −4 E h are observed for HF and LiF; the largest discrepancy between unrelaxed and relaxed QED-CCSD-1 energies is observed for LiF, with the cavity mode polarized along the molecular axis and resonant with the 1 1 A 1 → 3 1 A 1 transition (≈ 0.06 × 10 −3 E h ). Considering the substantial coupling strength used (λ = 0.05 atomic units), the magnitudes of these differences suggest that exponentiated singles and boson creation operators do a reasonable job of capturing the effects of both orbital relaxation and the coherent-state transformation in relaxed QED-CCSD-1. We have also evaluated errors in relaxed and unrelaxed QED-CCSD-1 energies when ignoring the exponentiated boson creation operator term, e u 0b † , in the QED-CCSD-1 cluster operator (labeled "error w/o u 0 " in Table I). For relaxed QED-CCSD-1, we see negligible energy deviations from full relaxed QED-CCSD-1; the largest deviations are on the order of 10 −6 E h . On the other hand, this term is quite important for unrelaxed QED-CCSD-1, where energy errors as large as 0.008 E h are observed. The relative importance of the exponentiated boson creation operator in relaxed and unrelaxed QED-CCSD-1 is reflected in the value of u 0 , which is also tabulated in Table I. We find that, when u 0 is non-zero, it can be more than an order of magnitude larger in the unrelaxed case. We also note that, with the exception of one case (H 2 with the cavity mode resonant with the 1 1 B 1u state), u 0 is only nonzero when the cavity mode is polarized along the molecular axis. Part of the motivation for the use of the coherent-state transformed Hamiltonian in relaxed QED-CCSD-1 is that it lends the origin invariance of QED-HF to the correlated problem. On the other hand, an unrelaxed QED-CCSD-1 protocol should not be strictly origin invariant, although we expect the exponentiated boson creation operator to mitigate these effects. QED-CCSD-1 changes as calculations are carried out at various distances from the origin. We consider hydrogen fluoride with a fixed bond length of 0.918 Å coupled to a cavity mode polarized along the molecular axis, resonant with the 2 1 A 1 state, and with a coupling strength of λ = 0.05 a.u. The distance along the z-axis in Fig. 2 corresponds to the distance from the center of the bond to the origin, and the translation from the origin is carried out in the direction of the polarization of the cavity mode. The change in the energy, ∆E, corresponds to the difference between energies evaluated at the origin and away from it. These data show that the relaxed QED-CCSD-1 energy is origin invariant, as expected. Two sets of unrelaxed QED-CCSD-1 data are provided: one in which we include the exponentiated boson creation operator (labeled "unrelaxed") and one where we have neglected this term (labeled "unrelaxed w/o u 0 ). We find that the e u 0b † term is necessary for preserving the origin invariance of unrelaxed QED-CCSD-1; ignoring this term introduces a small origin dependence in the energy (on the order of 10 −9 -10 −8 E h ). We note that Eqs. 3 and 4 show that the coherent-state transformation operator depends on the expectation value of the total QED-HF dipole moment, which should be origin invariant for neutral species. By analogy, if e u 0b † mimics the behavior of this term for unrelaxed QED-CCSD-1, u 0 itself should also be origin invariant; we have confirmed numerically that this is the case. Figure 3 depicts a similar study for a charged species (HF + , with an H-F distance of 0.918 Å). In this case, the QED-HF dipole moment should depend on the placement of the molecule relative to the origin, and thus, we expect u 0 to also acquire an origin dependence in unrelaxed QED-CCSD-1. The energy from relaxed QED-CCSD-1 is strictly origin invariant and is not shown. For unrelaxed QED-CCSD-1, if we include the exponentiated boson operator, the energy does pick up a slight origin dependence; at 10 Å from the origin with λ = 0.05 a.u., the energy differs from that at the origin by roughly 5 × by roughly 0.8 when the molecule is translated by 10 Å from the origin at λ = 0.05 a.u. As already mentioned, the origin dependence of u 0 is expected, as it mimics the coherent-state transformation; this transformation is defined by the meanfield dipole moment, which is strongly origin dependent for charged species. Before moving on to discuss excited-states from unrelaxed and relaxed QED methods, we highlight the severe origin dependence of the energy for unrelaxed QED mean-field for cavity-coupled HF + . Panel (d) of Fig. 3 depicts differences between the unrelaxed QED-HF energy evaluated at various distances from the origin relative to that computed at the origin. Clearly, the mean-field energy depends strongly on the choice of origin, and this dependence is of comparable magnitude to that which we observed for unrelaxed QED-CCSD-1 when ignoring u 0 [panel (c)]. This dependence is entirely due to the dipole self-energy contribution, and, since the dipole self energy term is treated in the same way in unrelaxed QED-DFT, that method also suffers from the same severe origin dependence. B. Excitation energies of relaxed and unrelaxed QED-TDDFT and QED-EOM-CCSD-1 We now consider the effects that orbital relaxation and the coherent-state transformation have on excitation energies derived from QED-TDDFT and QED-EOM-CCSD-1. We have performed relaxed and unrelaxed QED-TDDFT and QED-EOM-CCSD-1 calculations on cavity-coupled molecules with the cavity mode resonant with the following states: 1 1 B 1u and 1 1 B 2u for H 2 , 2 1 A 1 and 1 1 B 1 for HF, and 3 1 A 1 and 1 1 B 1 for LiF. We compare the computed excitation energies of the relaxed and unrelaxed formulations of QED-TDDFT to QED-EOM-CCSD-1. Specifically, Figs. 4, 5, and 6 depict changes in excitation energies for H 2 , HF, and LiF, respectively, as the cavity coupling strength is increased. Each figure assumes the following format: • Panels on the left and right correspond to calculations for which the polarization of the cavity mode was parallel to the molecular axis or perpendicular to it, respectively, with the cavity frequency resonant with the appropriate cavity-free transition (see above for the specific states we target). • The top panels depict the vertical excitation energies (VEE) for several states, shifted by the cavity frequency (ω cav ), as a function of the cavity coupling strength (λ ). • Relaxed and unrelaxed QED-TDDFT curves are purple and orange, respectively, while the green and black curves correspond to relaxed and unrelaxed QED-EOM-CC. • Solid lines correspond to the polariton states, while dashed lines correspond to non-resonant states that are nearby in energy. For clarity, QED-EOM-CC states with significant double electronic transition character are not shown. • The middle panels show how Rabi splittings from relaxed QED-TDDFT, unrelaxed QED-TDDFT, and unrelaxed QED-EOM-CC deviate from those from relaxed QED-EOM-CC (∆Ω R ). • The bottom panels present the deviation from relaxed QED-EOM-CC Rabi splittings as a percentage. Figure 4 illustrates how the excited-state landscape of cavity-coupled H 2 changes with the coupling strength, for cavity modes that are resonant with the (a) 1 1 B 1u and (b) 1 1 B 2u states of cavity-free H 2 . First, excitation energies from relaxed and unrelaxed QED-EOM-CC are indistinguishable on the scale of this figure, but they are not numerically identical. In this case, QED-EOM-CCSD-1 is equivalent to the full CI in the electronic space, so we expect the approach to be invariant to cavity-induced orbital relaxation effects. It is also equivalent to the full CI in the photon space, if the photon space is truncated at two photon number states (0 and 1). This qualifying statement is important, for the following reason. The coherent-state transformation of the Hamiltonian should preserve the spectrum of the Hamiltonian, but the spectrum is only preserved in the limit that the photon space is complete. Indeed, we do not see exact numerical agreement between relaxed and unrelaxed QED-EOM-CCSD-1 for this reason (differences on the order of 10 −6 E h are observed for λ = 0.1 a.u.; see Supporting Information). Nonetheless, relaxed and unrelaxed QED-EOM-CC results are nearly indistinguishable in this case. Molecular hydrogen Curves corresponding to polariton formation involving the 1 1 B 1u state reveal significant differences between QED-EOM-CC and both relaxed and unrelaxed QED-TDDFT. In the case of relaxed QED-TDDFT, both the Rabi splitting (the difference in energy between the upper and lower polariton states, Ω R ) and the energies of the states that are not resonant with the cavity mode are more sensitive to cavity effects in the strong coupling limit than the same quantities derived from QED-EOM-CC calculations; this is a general trend we observe for all systems considered in this work. For a cavity mode resonant with the 1 1 B 1u state, we can see that the λ -dependence of the lower polariton state energy de-rived from QED-EOM-CC is in better agreement with unrelaxed QED-TDDFT than with relaxed QED-TDDFT. On the other hand, these three methods predict noticeably different trends in the λ -dependence of the upper polariton state. The shift in the upper polariton energy from unrelaxed QED-TDDFT is too small, and that from relaxed QED-TDDFT is too large; QED-EOM-CC splits the difference. In the largeλ limit, Figs. 4(c) and (e) indicate that relaxed QED-TDDFT provides better Rabi splittings than unrelaxed QED-TDDFT, given QED-EOM-CC results as a reference. The maximum deviation between relaxed QED-TDDFT and QED-EOM-CC Rabi splittings is 0.41 eV (13.8%) at λ = 0.1, while unrelaxed QED-TDDFT and QED-EOM-CC splittings differ by −0.85 eV (−28.6%) at the same coupling strength. The λdependence of the non-resonant states [indicated by dashed lines in Fig. 4(a)] is also interesting. Here, we see the excitation energy of the non-resonant state is mostly unaffected by the cavity mode in the case of unrelaxed QED-TDDFT. On the other hand, both relaxed QED-TDDFT and QED-EOM-CC predict an increase in the excitation energy, and QED-TDDFT exaggerates this effect. We conclude that, because this state does not directly interact to the cavity mode, these changes with increasing coupling strength must stem from cavity-induced changes to the ground state. The curves for polariton formation with the 1 1 B 2u state of H 2 are depicted in the right-hand panels of Fig. 4. As was seen in panels (a), (c), and (e), relaxed and unrelaxed QED-EOM-CC display nearly identical λ dependence, which is not surprising, given that QED-EOM-CCSD-1 is equivalent to the full CI (within the truncated photon space). Unlike in panels (c) and (e), Rabi splittings from unrelaxed and relaxed QED-TDDFT both agree well with those from QED-EOM-CC [panels (d) and (f)], with maximum deviations on the order of only 10 −3 eV. Unrelaxed QED-TDDFT displays slightly better agreement with QED-EOM-CC than relaxed QED-TDDFT, but the deviations from QED-EOM-CC overall are small for both QED-TDDFT variants. axis and the cavity mode frequency resonant with the 2 1 A 1 state of cavity-free HF. The right panels consider a cavity mode polarized perpendicular to the molecular axis and resonant with the 1 1 B 1 state of isolated HF. The left panels show similar behavior as depicted in the left panels of Figure 4 for H 2 . First, relaxed and unrelaxed QED-EOM-CC results are indistinguishable. Second, the λ -dependence of the lower polariton from unrelaxed QED-TDDFT agrees well with that from QED-EOM-CC, while relaxed QED-TDDFT appears to underestimate this dependence. Third, unrelaxed and relaxed QED-TDDFT predict a λ -dependence for the upper polariton state that is too small or too large, as compared to that from QED-EOM-CC, respectively. In terms of the Rabi splitting [panels (c) and (e)], relaxed QED-TDDFT again provides a better description than unrelaxed QED-TDDFT, given relaxed QED-EOM-CC as the reference. Here, relaxed and unrelaxed QED-TDDFT Rabi splittings deviate from those of relaxed QED-EOM-CC by at most 0.20 eV (9.5%) and −0.31 eV (−14.8%), respectively. The magnitudes of these deviations are smaller than in the case of H 2 in the left panels of Fig. 4 above, which could simply reflect the smaller magnitude of the Rabi splitting itself for the 2 1 A 1 state of HF, relative to the splitting for the 1 1 B 1u state in H 2 (see Supporting Information). Indeed, the oscillator strength for the 2 1 A 1 state of HF (0.1869) is much smaller than that for the 1 1 B 1u state of H 2 (0.3069), which is consistent with the relative Rabi splittings. Lastly, as was observed in Fig. 4(a), the excitation energies for the non-polariton states in Fig. 5(a) pick up a λ -dependence in the case of both relaxed and unrelaxed QED-EOM-CC and for relaxed QED-TDDFT, with a more pronounced dependence for QED-TDDFT. On the other hand, unrelaxed QED-TDDFT predicts that these excitation energies are unaffected by the presence of the cavity. Hydrogen fluoride The curves in the right panel of Fig. 5 depict the λdependence of excitation energies and Rabi splittings when the cavity mode is resonant with the 1 1 B 1 state of isolated HF. The same qualitative observations for the left panels Fig. 5 apply here, with the exception that the λ -dependence of the lower polariton is not well-reproduced by unrelaxed QED-TDDFT. Note also that the behavior here differs somewhat from the case of the cavity mode polarized perpendicular to the 1 1 B 2u of isolated H 2 . In that case, all QED approaches provided comparable results, whereas, here, relaxed QED-TDDFT does a better job of reproducing the λ -dependence of the Rabi splittings predicted by QED-EOM-CC [panels (d) and (f)]; relaxed and unrelaxed QED-TDDFT Rabi splittings deviate from relaxed QED-EOM-CC splittings by at most 0.046 eV (5.3%) and −0.067 eV (−7.7%), respectively. The data in panel (b) also demonstrate that relaxed QED-TDDFT captures the same qualitative λ -dependence of the non-resonant state predicted by relaxed and unrelaxed QED-EOM-CC (albeit somewhat exaggerated by QED-TDDFT), whereas unrelaxed QED-TDDFT does not. Lithium fluoride Finally, we come to the case of lithium fluoride. Figure 6 illustrates the λ -dependence of the excitation energies and Rabi splittings from QED-TDDFT and QED-EOM-CC, for a cavity mode polarized along the molecular axis and resonant with the 3 1 A 1 of cavity-free LiF (left panels) and for a cavity mode polarized perpendicular to the molecular axis and resonant with the 1 1 B 1 of the isolated molecule (right panels). We note that 3 1 A 1 is the second bright 1 A 1 state of LiF, whereas, in the previous examples, we had tuned the cavity to the lowest-energy bright state of the given symmetry. We note that we have not depicted non-resonant excited states in Fig. 6(a) aside from the 2 1 A 1 state for the sake of clarity (these other states have incompatible spatial or spin symmetry to couple directly to the cavity mode). That said, for such states, we observe the same trends as have been discussed in the context of the other systems; QED-EOM-CC and relaxed QED-TDDFT predict a λ -dependence in these states that is exaggerated by QED-TDDFT, and unrelaxed QED-TDDFT fails to capture this effect. In Fig. 6(a), we find that we can induce some interesting behavior by tuning to the second A 1 symmetry bright state of LiF (3 1 A 1 ), which leads to strong interactions between the lower polariton state and the first bright state (2 1 A 1 ) at larger coupling strengths. Specifically, Fig. 6(a) demonstrates a coupling-strength-induced avoided crossing between these states that appears at coupling strengths of roughly λ = 0.070, λ = 0.115, and λ = 0.135 a.u. when modeling the system with relaxed QED-TDDFT, relaxed QED-EOM-CC, and unrelaxed QED-EOM-CC, respectively. The avoided crossing appears at smaller λ values for relaxed QED-TDDFT than for relaxed QED-EOM-CC because, as in the previous cases, relaxed QED-TDDFT exaggerates the λ -dependence of the states that are not resonant with the cavity mode. Notably, to a coupling strength of λ = 0.15 a.u., we do not observe this avoided crossing feature in unrelaxed QED-TDDFT because it fails to capture the λ -dependence of the 2 1 A 1 state. Also noteworthy is that this example is the first instance where we observe appreciable differences between relaxed and unrelaxed QED-EOM-CC. Results from these methods are similar for coupling strengths up to λ = 0.05 a.u. but differ for larger coupling strengths. These differences may not be terribly important in practical applications, though, given that these large coupling strengths correspond to effective cavity mode volumes that are significantly smaller than 1 nm 3 . For example, λ = 0.1 a.u. corresponds to an effective mode volume of less than 0.2 nm 3 , which is smaller than any reported experimentally obtained value of which we are aware. One interesting aspect of the avoided crossing is that the character of the lower polariton state is transferred to the lower-energy state beyond the avoided crossing, which points to potential ambiguities in designating one and only one state as an upper or lower polariton state in systems with dense energy manifolds. In order to compare Rabi splittings from each method before and after the avoided crossing, we simply take the point at which the energy gap between the states is the smallest as the point at which the crossing occurs; the kinks observed in the curves depicted in panels (c) and (e) of Fig. 6 correspond to these points. These details aside, we find that neither unrelaxed QED-TDDFT nor relaxed QED-TDDFT do a particularly good job of reproducing the λ -dependence or Rabi splittings from relaxed QED-EOM-CC; in particular, QED-TDDFT Rabi splittings differ from relaxed QED-EOM-CC ones by roughly 2 eV at λ = 0.15 a.u. These deviations are much larger than those observed for other molecules, which reflects the complexity of the excited-state energy landscape of LiF and calls into question the reliability of either form of QED-TDDFT in this case. We also note that there are smaller λ values for which unrelaxed QED-TDDFT Rabi splittings appear to be the better ones, relative to QED-EOM-CC. That said, the trends in panel (a) suggest that QED-TDDFT does a better job of reproducing qualitative properties of relaxed QED-EOM-CC when the procedure accounts for the effects of orbital relaxation and the coherent-state transformation. Aside from the poor behavior of QED-TDDFT, perhaps the most notable difference here as compared to the earlier examples is the discrepancy between relaxed and unrelaxed QED-EOM-CC. The Rabi splittings from these methods differs by as much as 72.2 meV (3.32%) at λ = 0.15, which is larger by an order of magnitude than other systems in this study (see Supporting Information). The λ -dependence for the 2 1 A 1 state is also underestimated by unrelaxed EOM-QED-CC, which shifts the avoided crossing from λ = 0.115 a.u. to λ = 0.135 a.u. Lastly, we consider LiF coupled to a cavity mode polarized perpendicular to the molecular axis and resonant with the 1 1 B 1 state of the isolated molecule (right panels of Fig. 6). In this case, the general trends are similar to what was observed when coupling a cavity mode to the 1 1 B 1 state of HF (right panels of Fig. 5). Again, ignoring orbital relaxation and the coherent-state transformation in QED-TDDFT decreases the ability of QED-TDDFT to reproduce relaxed QED-EOM-CC Rabi splittings; at λ = 0.15 a.u., the Rabi splitting from relaxed QED-TDDFT differs from the relaxed QED-EOM-CC value by 0.19 eV (14.6%); this deviation increases to −0.22 eV (−17.4%) for unrelaxed QED-TDDFT. Also, as observed previously, unrelaxed QED-TDDFT fails to capture the λ -dependence of states other than the upper and lower polariton states; this dependence is captured by relaxed QED-TDDFT, but the sensitivity of these non-resonant states to the cavity is overestimated, relative to QED-EOM-CC. We also note that, as was observed for polariton formation with the 3 1 A 1 state, small differences between relaxed and unrelaxed QED-EOM-CC methods emerge at coupling strengths larger than λ = 0.05 a.u. V. CONCLUSIONS Recent intriguing experiments demonstrating the ability to manipulate chemical transformations via vacuum field fluctuations and polariton formation have inspired the development of several generalizations of standard electronic struc-ture methods (e.g., coupled cluster theory, density functional theory, configuration interaction, etc.) for the polariton problem. In this work, we have explored the numerical consequences of some formal aspects of QED-DFT, QED-TDDFT, QED-CCSD-1, and QED-EOM-CCSD-1. Specifically, we began by investigating the sensitivity of ground-state energies from QED-CCSD-1 to the treatment of cavity effects at the mean-field level. We have found that the inclusion of exponentiated single electron transitions and boson creation operators in QED-CCSD-1 makes the approach robust with respect to the inclusion or exclusion of cavity effects in the underlying QED-HF calculation; numerically, these terms do a good job of mimicking the effects of orbital relaxation and the coherent-state transformation, respectively. Exponentiated boson creation operators are particularly important for maintaining (or nearly maintaining, in the case of charged species) origin invariance in unrelaxed QED-CCSD-1. On the other hand, unrelaxed mean-field approaches display severe origin dependence for charged species, which arises from the dipole self-energy contribution to the energy. Beyond the ground state, we have also assessed the role that cavity effects at the mean-field level can have on excited states computed using QED-TDDFT and QED-EOM-CCSD-1. Several key details bear repeating. First, for the most part, excitation energies computed from unrelaxed and relaxed QED-EOM-CC are similar; Rabi splittings differ by less than 9.3 meV (or less than 0.45%) in all cases considered in this work, except near the avoided-crossing for the 3 1 A 1 state of LiF which reaches an error of 72.2 meV (3.32%) at the extreme case of λ = 0.15 a.u. (see Supporting Information). Second, QED-EOM-CC and relaxed QED-TDDFT predict that the energies of electronic states that are not resonant with the cavity mode can be significantly perturbed in the strong coupling limit, and relaxed QED-TDDFT exaggerates this effect. On the other hand, unrelaxed QED-TDDFT fails to predict any λ dependence in non-resonant states. Third, Rabi splittings from QED-EOM-CC are more closely reproduced by relaxed QED-TDDFT than by unrelaxed QED-TDDFT. In the large coupling limit, relaxed QED-TDDFT tends to overestimate the Rabi splitting, while unrelaxed QED-TDDFT underestimates this quantity. Lastly, the proximity of multiple bright states having the appropriate symmetry to interact with the cavity mode can lead to complex spectral features; specifically, we have located a coupling-strength-induced avoided crossing in LiF between the lower polariton (formed from the admixture of the cavity mode and the 3 1 A 1 state) and the 2 1 A 1 state. Of the methods studied, unrelaxed QED-TDDFT is the least capable of describing this phenomenon, because it fails to capture the λ -dependence of the non-resonant 2 1 A 1 state. Given these observations, we caution against the use of unrelaxed QED-TDDFT. Supporting Information Relaxed and unrelaxed QED-CCSD-1 energies for ground states of molecular hydrogen, hydrogen fluoride, and lithium fluoride; Rabi splittings from relaxed QED-EOM-CCSD-1 for these same molecules; deviations in Rabi splittings computed using relaxed and unrelaxed QED-EOM-CCSD-1. FIG. 2 . 2Figure 2depicts how the energy from ground-state Origin dependence of the ground-state QED-CCSD-1 energy (10 −9 E h ) for hydrogen fluoride coupled to a cavity mode with a coupling strength of λ = 0.05. FIG. 3 . 310 −4 E h [see panel(a)]. On the other hand, without contributions from u 0 , this energy difference grows to almost 0.5 E h [see panel (c)]. As for u 0 itself, this quantity is origin invariant in the case of relaxed QED-CCSD-1 (not depicted), but it acquires a strong origin dependence in unrelaxed QED-CCSD-1 [see panel(b)]. The value of u 0 changes Origin dependence of unrelaxed QED-HF and QED-CCSD-1 at various coupling strengths (λ ) and distances from the origin (10 Å) for charged species (HF + ) with ω cav = 0.675019 E h . Panel (a) depicts the how the unrelaxed QED-CCSD-1 energy differs at a given offset from that at the origin. Panel (b) shows how u 0 in unrelaxed QED-CCSD-1 changes as the molecule is translated away from the origin. Panel (c) depicts similar information as panel (a), except that u 0 has been excluded from the cluster operator in unrelaxed QED-CCSD-1. Panel (d) provides the difference between the unrelaxed QED-HF energy at a given offset and that at the origin. Figure 5 FIG. 4 .FIG. 5 . 545provides similar data asFig. 4, but for cavitycoupled hydrogen fluoride. The left panels correspond to calculations with the cavity mode polarized along the molecular Excitation energies for H 2 when coupling a cavity mode to the (a) 1 1 B 1u and (b) 1 1 B 2u (b) states, using relaxed QED-TDDFT (purple), unrelaxed QED-TDDFT (orange), unrelaxed QED-EOM-CCSD-1 (green), and relaxed QED-EOM-CCSD-1 (black). The dashed lines correspond to non-resonant excited states [2 1 A g in panel (a) and 1 1 B 3u in panel (b)], while the solid lines correspond to the polariton states. Panels (c) and (d) show the difference between the Rabi splittings from relaxed QED-EOM-CCSD-1 to relaxed/unrelaxed QED-TDDFT and unrelaxed QED-EOM-CCSD-1, and panels (e) and (f) depict these differences as a percentage. Excitation energies for HF when coupling a cavity mode to the (a) 2 1 A 1 and (b) 1 1 B 1 states, using relaxed QED-TDDFT (purple), unrelaxed QED-TDDFT (orange), unrelaxed QED-EOM-CCSD-1 (green), and relaxed QED-EOM-CCSD-1 (black). The dashed lines correspond to non-resonant excited states[2 1 B 1 in panel (a) and 1 1 B 2 in panel (b)], while the solid lines correspond to the polariton states. Panels (c) and (d) show the difference between the Rabi splittings from relaxed QED-EOM-CCSD-1 to relaxed/unrelaxed QED-TDDFT and unrelaxed QED-EOM-CCSD-1, and panels (e) and (f) depict these differences as a percentage. FIG. 6 . 6Excitation energies for LiF when coupling a cavity mode to the (a) 3 1 A 1 and (b) 1 1 B 1 states, using relaxed QED-TDDFT (purple), unrelaxed QED-TDDFT (orange), unrelaxed QED-EOM-CCSD-1 (green), and relaxed QED-EOM-CCSD-1 (black). In panel (a), the curves at λ = 0.00 a.u. correspond to the 2 1 A 1 and 3 1 A 1 states of LiF. The dashed lines in panel (b) correspond to a non-resonant excited state [1 1 B 2 ], while the solid lines are the polariton states formed from coupling the cavity mode to the 1 1 B 1 state. Panels (c) and (d) show the difference between the Rabi splittings from relaxed QED-EOM-CCSD-1 to relaxed/unrelaxed QED-TDDFT and unrelaxed QED-EOM-CCSD-1, and panels (e) and (f) depict these differences as a percentage. TABLE I . IGround-state energies (E h ) from relaxed QED-CCSD-1 and absolute energy errors (10 −3 E h ) from unrelaxed QED-CCSD-1, as well as from relaxed and unrelaxed QED-CCSD-1 calculations that ignore u 0 . Also provided are u 0 values from relaxed and unrelaxed QED-CCSD-1 calculations. All calculations consider λ = 0.05 atomic units. error error w/o u 0 u 0 system ω cav resonance relaxed unrelaxed relaxed unrelaxed relaxed unrelaxed H 2 0.466751 1 1 B 1u −1.167161 0.000000 0.000000 0.000000 0.000000 0.000000 H 2 1.522218 1 1 B 2u −1.167070 0.000000 0.000000 0.000000 0.000000 0.000000 HF 0.531916 2 1 A 1 −100.296930 0.017471 0.002001 0.757147 −0.001815 0.037626 HF 0.375022 1 1 B 1 −100.296806 0.015717 0.000054 0.015663 0.000000 0.000000 LiF 0.308401 3 1 A 1 −107.233438 0.062602 0.005998 8.095084 0.003957 0.162599 LiF 0.232119 1 1 B 1 −107.220994 0.039174 0.000182 0.038991 0.000000 0.000000 T. W. Ebbesen, "Hybrid light-matter states in a molecular and material science perspective," Accounts of ChemicalResearch 49, 2403Research 49, -2412Research 49, (2016. Strong light-matter coupling in quantum chemistry and quantum photonics. 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{'abstract': 'Cavity quantum electrodynamics (QED) generalizations of time-dependent (TD) density functional theory (DFT) and equation-of-motion (EOM) coupled-cluster (CC) theory are used to model small molecules strongly coupled to optical cavity modes. We consider two types of calculations. In the first approach (termed "relaxed"), we use a coherent-statetransformed Hamiltonian within the ground-and excited-state portions of the calculations, and cavity-induced orbital relaxation effects are included at the mean-field level. This procedure guarantees that the energy is origin invariant in post-self-consistent-field calculations. In the second approach (termed "unrelaxed"), we ignore the coherent-state transformation and the associated orbital relaxation effects. In this case, ground-state unrelaxed QED-CC calculations pick up a modest origin dependence but otherwise reproduce relaxed QED-CC results within the coherent-state basis. On the other hand, a severe origin dependence manifests in ground-state unrelaxed QED mean-field energies. For excitation energies computed at experimentally realizable coupling strengths, relaxed and unrelaxed QED-EOM-CC results are similar, while significant differences emerge for unrelaxed and relaxed QED-TDDFT. First, QED-EOM-CC and relaxed QED-TDDFT both predict that electronic states that are not resonant with the cavity mode are nonetheless perturbed by the cavity. Unrelaxed QED-TDDFT, on the other hand, fails to capture this effect. Second, in the limit of large coupling strengths, relaxed QED-TDDFT tends to overestimate Rabi splittings, while unrelaxed QED-TDDFT underestimates them, given splittings from relaxed QED-EOM-CC as a reference, and relaxed QED-TDDFT generally does the better job of reproducing the QED-EOM-CC results.', 'arxivid': '2303.10821', 'author': ['Marcus D Liebenthal \nDepartment of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA\n', 'Nam Vu \nDepartment of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA\n', 'A Eugene Deprince \nDepartment of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA\n', 'Marcus D Liebenthal \nDepartment of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA\n', 'Nam Vu \nDepartment of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA\n', 'A Eugene Deprince \nDepartment of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA\n'], 'authoraffiliation': ['Department of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA', 'Department of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA', 'Department of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA', 'Department of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA', 'Department of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA', 'Department of Chemistry and Biochemistry\nFlorida State University\n32306-4390TallahasseeFLUSA'], 'corpusid': 258823252, 'doi': '10.1021/acs.jpca.3c01842', 'github_urls': ['https://github.com/edeprince3/hilbert', 'https://github.com/Marclie/SuaveStateScanner'], 'n_tokens_mistral': 25707, 'n_tokens_neox': 21636, 'n_words': 11633, 'pdfsha': 'c83c48e1dc6a11a176c0e17f051df57f229aebcf', 'pdfurls': ['https://export.arxiv.org/pdf/2303.10821v2.pdf'], 'title': ['Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories', 'Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories', 'Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories', 'Assessing the Effects of Orbital Relaxation and the Coherent-State Transformation in Quantum Electrodynamics Density Functional and Coupled-Cluster Theories'], 'venue': []}

arxiv

Plankton-FL: Exploration of Federated Learning for Privacy-Preserving Training of Deep Neural Networks for Phytoplankton Classification Daniel Zhang daniel.zhang1@uwaterloo.ca Vikram Voleti Mila Alexander Wong Jason Deglint jdeglint@uwaterloo.ca University of Waterloo University of Montreal University of Waterloo Blue Lion Labs University of Waterloo Blue Lion Labs Plankton-FL: Exploration of Federated Learning for Privacy-Preserving Training of Deep Neural Networks for Phytoplankton Classification 10.1038/s41746-020-00323-1 Creating high-performance generalizable deep neural networks for phytoplankton monitoring requires utilizing large-scale data coming from diverse global water sources. A major challenge to training such networks lies in data privacy, where data collected at different facilities are often restricted from being transferred to a centralized location. A promising approach to overcome this challenge is federated learning, where training is done at site level on local data, and only the model parameters are exchanged over the network to generate a global model. In this study, we explore the feasibility of leveraging federated learning for privacy-preserving training of deep neural networks for phytoplankton classification. More specifically, we simulate two different federated learning frameworks, federated learning (FL) and mutually exclusive FL (ME-FL), and compare their performance to a traditional centralized learning (CL) framework. Experimental results from this study demonstrate the feasibility and potential of federated learning for phytoplankton monitoring. Introduction The uncontrollable growth of particular phytoplankton and algae species can cause the formation of harmful algae blooms (HABs). If not properly monitored and controlled, HABs can have severe, negative impacts on various industries, natural ecosystems, and the environment [1]. HABs are a growing concern as research has shown that climate change has led to an increase in the frequency and severity of HABs [2]. A very important step in the monitoring and controlling of HAB formation is the identification of phytoplankton and algae species. Unfortunately, this process is largely manual and thus is highly time-consuming and prone to human error. As such, effective methods for automating the species identification process are highly desired. Recent advances in machine learning, in particular deep learning, have shown considerable promise for monitoring and assessment of phytoplankton and algae [3,4]. However, a significant bottleneck to training such models is the need for large-scale data coming from different water sources across different countries in order to create high-performance, generalizable models. Since the data collected at the different facilities are often restricted from being transferred to a centralized location for training due to data privacy concerns, this makes it infeasible to leverage traditional, centralized learning frameworks for building such models. A particularly promising direction for tackling this data privacy challenge lies in federated learning (FL), which involves training local models at individual local nodes on the premises (prem) of each local data source and communicating only the parameters and updates of these local models to a server for generating a global model to reap the benefits from the different local data without having seen any of the individual data sources [5]. FL has demonstrated considerable success in the domains of mobile computing [5,6] and healthcare [7], and thus can hold considerable potential for the application of phytoplankton monitoring and assessment. In this study, we explore the feasibility of leveraging federated learning to train deep, convolutional neural networks for the purpose of image-driven phytoplankton classification, which we will refer to as Plankton-FL. Our main contributions in this study are as follows: (1) we simulate and study two federated learning frameworks as potential realizations of Plankton-FL: (centralized) federated learning (FL) and mutually exclusive FL (ME-FL), (2) we evaluate the performance of both Plankton-FL frameworks, and (3) we compare them to a traditional, centralized learning framework (CL). Figure 1 provides a visual representation of each of the three environments. Both privacy-preserving federated learning frameworks are evaluated against CL for the training of deep neural networks for phytoplankton classification. Methodology Background Federated learning has been shown to be very effective for training deep neural networks on decentralized data while ensuring data privacy [5]. Specifically, when there is sensitive data from various sources, federated learning can be leveraged. A typical federated learning framework consists of 2 components: a global model and K clients. Each client contains its own local model, and they are trained iteratively and independently on their respective data. It is assumed that all of the data available is partitioned into K clients, P k . The local models are then used to update the global model [5]. This process is repeated for N rounds in order for the global model to generalize. The objective for federated learning is calculated using equation 1. f (w) = K ∑ k=1 n k n F k (w) where F k (w) = 1 n k ∑ i∈P k f i (w)(1) In equation 1, f i (w) denotes the loss function (x i , y i ; w) for an observation (x i , y i ) and model parameters w. Also, n k and n denote the |P k | and the total number of observations, respectively. The centralized federated learning algorithm which governs the communication between the global and local models is known as FederatedAveraging (FedAvg) and was introduced by McMahan et al. [5]. FedAvg is the iterative process of training all the local models, taking the average of all the updated weights from the local models, and then using it to update the global model. As described by McMahan et al., pseudo-code is provided in Algorithm 1. Algorithm 1 McMahan et al. [5]'s implementation of the FedAvg algorithm. C is the set of all clients; B is the local model batch size; LE is the number of local epochs; η is the learning rate; w are the weights, and is the loss function 1: Global/Server: 2: C ← Set of all available clients 3: initialize w 0 4: for each round, r = 1,2, ... do 5: for each client, k ∈ C, do 6: w k r+1 ← Local-Training(k, w r ) 7: end for 8: FedAvg: w r+1 ← ∑ K k=1 n k n w k r+1 9: SYNC Global→Local: w k r+1 ← w r+1 ∀k 10: end for 11: 12: Local-Training(k, w): In this paper, we used the FedAvg method, as described in Algorithm 1, when training our two instances of Plankton-FL. To test the feasibility and potential of federated learning, three different experiments were simulated. Specifically, a centralized learning baseline (CL), a (centralized) federated learning framework (FL), and a mutually exclusive, federated learning framework (ME-FL). Figure 1 provides a visual representation of each of the three experiments. for batch b ∈ B do 16: w ← w − η * ∇ w (w; b) Centralized Learning (CL) For the CL experiment, we have two data sources that we consolidate into a single server. From there we train a centralized model, which can then be deployed back to the edge devices for assessment and monitoring. The model was trained for a maximum of 75 epochs, with an early stopping criteria: A minimum of 50 epochs and a δ between test accuracies of 0.000001. Federated Learning (FL) In the FL experiment, all of the training data was combined, randomly shuffled, and distributed to clients. Each client trained their own local model, on-prem, and only communicated their parameters back to the global server. FL was run for 10 iterations, where each iteration number corresponded to the number of clients. Namely, for the first iteration, there was only one client containing all of the training data, identical to CL, and with each increasing iteration another client was added (i.e. second iteration utilized two clients, etc.). Although, in reality, no single client will contain all of the data, for the purpose of comparing to CL, it made sense to start with a single client. Mutually Exclusive FL (ME-FL) Unlike FL, in ME-FL, instead of combining all of the data together, shuffling, and distributing them, the clients only contained data from a single source, making them mutually exclusive. Again, each of the clients trained their own model, on-prem, and only communicated their parameters back to the global server. ME-FL was run for 9 iterations, starting from 2 clients up to 10. Iterations start from 2 clients due to the nature of the experiment; since each client only has data from a single source, it would not make sense to only have a single client. This modified setup ensures that we always have data from both sources. Experimental Setup Dataset The dataset was provided by Blue Lion Labs and was collected from two mutually exclusive sources, Halifax and Waterloo. It contained 301 distinct microscope specimen photos, each at a resolution of 3208 x 2200 pixels. The phytoplankton contained in # of clients Model Architecture For the purpose of this exploration, we took the majority class present in each image as the label to do image classification. This ensures that all models receive the same amount of information. Given the task, we built a custom convolutional neural network with four convolutional layers, three max-pooling layers, two dense layers, and an output layer. Across all intermediate layers, the ReLU activation function was used, and at the output, a softmax activation was used to predict the probabilities of each class. For all the convolutional layers, a kernel of size 3x3 and a stride of 1x1 was used and for all of the pooling layers, a pool and stride size of 2x2 was used. Additionally, dropout was used with a rate of 0.25 after the convolutional layers and a rate of 0.5 after the first dense layer. Model Training and Evaluation When training, the images were resized to a resolution of 128 x 128 pixels and further augmented, using a horizontal flip, vertical flip, rotation, and color jitter, to create a larger data set of 2107 images. Across all experiments, the model architectures were held the same, a batch size of 8 was used, and the data was split into 80% training and 20% test. We also tune the learning rate across all experiments via a grid search over three different learning rates (LR) of 0.001, 0.0001, and 0.0005. Both of the federated learning experiments were run for 75 rounds and each local model was trained for 1 epoch. Furthermore, given it is a multi-label image classification task, the metric considered is prediction accuracy and the loss function is categorical cross-entropy. comparing CL and FL, we observe that for a single client FL outperforms CL. However, this is expected because FL with a single client is the exact same setup as CL; we expect the test accuracies to be very close in magnitude, and it is entirely possible that FL can outperform CL in this scenario. For all other number of clients, FL has a progressively worse test accuracy and is continuously outperformed by CL. In addition, across all number of clients, we observe that ME-FL consistently gets outperformed by CL and FL, which further demonstrates the impracticality of this method. Figure 2 provides a visual comparison of CL, FL, and ME-FL across all epochs. We specifically look at the results for 10 clients of both FL and ME-FL, as in reality there are often large numbers of clients. From the figure, CL and FL both appear to learn, whereas ME-FL does not appear to learn at all. Comparing CL and FL, we observe that CL converges much faster than FL, which tells us that CL learns faster than FL. Overall, across all experiments, generally, we observed that CL performed the best, FL performed the second best, and ME-FL had the worst performance and this same trend is observed across each learning rate. Results & Discussion Comparison of Performance Across Experiments Downward Trend in FL Across Number of Clients From table 1 and figure 2, we observe that FL performs relatively well, which prompted an investigation into its properties. Figure 3 displays the global model test accuracies for FL, for all numbers of clients, across the three learning rates. We observe a downward trend in the test accuracies as the number of clients increases. With an increasing number of clients, the global model needs to process and learn more information (i.e. the global model has to aggregate weights from more sources). With more information to process, learning is slowed down, yielding a worse generalization. Causation of Poor ME-FL Performance The largest contributor as to why ME-FL had a subpar performance relative to FL was because ME-FL was trained on individual, mutually exclusive clients. In our FL experiments we utilize a homogeneous model architecture, that is, all clients and the global server have the same model architecture. The nature of FL yields an independent and identically distributed (IID) distribution of labels across clients. However, in ME-FL, the distribution across clients is non-IID. As discussed in other literature, homogeneous federated learning performs poorly on non-IID data distributions [6]. Given this limitation, heterogeneous federated learning [8,9] is an alternative approach that should be explored. In this method, clients are allowed to differ in network architecture, allowing for more flexibility. Research has been done to explore applications of heterogeneous federated learning to mutually exclusive data [10] and it has typically been the preferred approach over homogeneous federated learning. Conclusion & Future Works This work demonstrates the feasibility and potential of Plankton-FL for the privacy-preserving building of high-performance, generalizable models for phytoplankton assessment without the need to exchange data. We simulated two different federated learning frameworks and compared their performance to a traditional, centralized learning framework. Although centralized learning yields the best performance, it does not address privacy concerns. Federated learning preserves privacy but fails to generalize when clients are mutually exclusive. We find that when clients share class labels with one another, federated learning both generalizes well and provides a privacy-preserving alternative to centralized learning. Given the outcomes of this paper, the immediate future work includes (1) implementing this framework for object detection to build off the current work of image classification, (2) utilizing a heterogeneous federated learning framework and conducting the same experiments to assess the relative performance to homogeneous federated learning, and (3) explore novel federated learning-related methods. For example, another method that can be utilized is git re-basin, which aims to train individual models on disjoint datasets and merge them together [11]. Finally, careful consideration must be taken on how federated learning frameworks will be deployed in the field to ensure data privacy between clients. This will help provide a secure and accurate method for identifying different species of phytoplankton and help alleviate the manual workload. Fig. 1 : 1Traditional centralized learning (CL) (top) and the two federated learning frameworks as realizations of Plankton-FL: federated learning (FL) (middle) and mutually exclusive FL (ME-FL) (bottom). on client k into B batches 14: for each local epoch, l = 1...LE do 15: Table 1 1provides a numerical comparison across all experiments utilizing a learning rate of 0.0001. Note that, CL, FL, and ME-FL were trained for 51, 75, and 75 epochs, respectively. Firstly, when AcknowledgmentsThis work was funded by the Waterloo AI Institute and Mitacs. The dataset was provided by Blue Lion Labs, and the computing resources were provided by the Vision and Image Processing (VIP) Lab at the University of Waterloo and Blue Lion Labs. An Introduction to Harmful Algae. E Granéli, J Turner, 189E. Granéli and J. Turner, An Introduction to Harmful Algae, 01 2006, vol. 189, pp. 3-7. Future hab science: Directions and challenges in a changing climate. M L Wells, B Karlson, A Wulff, R Kudela, C Trick, V Asnaghi, E Berdalet, W Cochlan, K Davidson, M De Rijcke, S Dutkiewicz, G Hallegraeff, K J Flynn, C Legrand, H Paerl, J Silke, S Suikkanen, P Thompson, V L Trainer, Harmful Algae. 91101632climate change and harmful algal blooms. [OnlineM. L. Wells, B. Karlson, A. Wulff, R. Kudela, C. Trick, V. Asnaghi, E. Berdalet, W. Cochlan, K. Davidson, M. De Rijcke, S. Dutkiewicz, G. Hallegraeff, K. J. Flynn, C. Legrand, H. Paerl, J. Silke, S. Suikkanen, P. Thompson, and V. L. Trainer, "Future hab science: Directions and challenges in a changing climate," Harmful Algae, vol. 91, p. 101632, 2020, climate change and harmful algal blooms. [Online]. Investigating the automatic classification of algae using the spectral and morphological characteristics via deep residual learning. J L Deglint, C Jin, A Wong, International Conference on Image Analysis and Recognition. SpringerJ. L. Deglint, C. Jin, and A. Wong, "Investigating the auto- matic classification of algae using the spectral and morpholog- ical characteristics via deep residual learning," in International Conference on Image Analysis and Recognition. Springer, 2019, pp. 269-280. Towards generating large synthetic phytoplankton datasets for efficient monitoring of harmful algal blooms. N Bamra, V S Voleti, A Wong, J L Deglint, abs/2208.02332ArXiv. N. Bamra, V. S. Voleti, A. Wong, and J. L. Deglint, "To- wards generating large synthetic phytoplankton datasets for efficient monitoring of harmful algal blooms," ArXiv, vol. abs/2208.02332, 2022. Communication-efficient learning of deep networks from decentralized data. H B Mcmahan, E Moore, D Ramage, S Hampson, B A Arcas, AISTATSH. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, "Communication-efficient learning of deep networks from decentralized data," in AISTATS, 2017. Federated learning with non-iid data. Y Zhao, M Li, L Lai, N Suda, D Civin, V Chandra, abs/1806.00582ArXiv. Y. Zhao, M. Li, L. Lai, N. Suda, D. Civin, and V. Chandra, "Fed- erated learning with non-iid data," ArXiv, vol. abs/1806.00582, 2018. The future of digital health with federated learning. N Rieke, J Hancox, W Li, F Milletarì, H R Roth, S Albarqouni, S Bakas, M N Galtier, B A Landman, K Maier-Hein, S Ourselin, M Sheller, R M Summers, A Trask, D Xu, M Baust, M J Cardoso, 10.1038/s41746-020-00323-1npj Digital Medicine. 31119N. Rieke, J. Hancox, W. Li, F. Milletarì, H. R. Roth, S. Albarqouni, S. Bakas, M. N. Galtier, B. A. Landman, K. Maier-Hein, S. Ourselin, M. Sheller, R. M. Summers, A. Trask, D. Xu, M. Baust, and M. J. Cardoso, "The future of digital health with federated learning," npj Digital Medicine, vol. 3, no. 1, p. 119, Sep 2020. [Online]. Available: https://doi.org/10.1038/s41746-020-00323-1 Fedmd: Heterogenous federated learning via model distillation. D Li, J Wang, abs/1910.03581ArXiv. D. Li and J. Wang, "Fedmd: Heterogenous federated learning via model distillation," ArXiv, vol. abs/1910.03581, 2019. Heterogeneous federated learning. F Yu, W Zhang, Z Qin, Z Xu, D Wang, C Liu, Z Tian, X Chen, ArXiv. F. Yu, W. Zhang, Z. Qin, Z. Xu, D. Wang, C. Liu, Z. Tian, and X. Chen, "Heterogeneous federated learning," ArXiv, vol. abs/2008.06767, 2020. Federated learning with heterogeneous labels and models for mobile activity monitoring. G K Gudur, S K Perepu, abs/2012.02539ArXiv. G. K. Gudur and S. K. Perepu, "Federated learning with het- erogeneous labels and models for mobile activity monitoring," ArXiv, vol. abs/2012.02539, 2020. Git re-basin: Merging models modulo permutation symmetries. S K Ainsworth, J Hayase, S S Srinivasa, abs/2209.04836ArXiv. S. K. Ainsworth, J. Hayase, and S. S. Srinivasa, "Git re-basin: Merging models modulo permutation symmetries," ArXiv, vol. abs/2209.04836, 2022.

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{'abstract': 'Creating high-performance generalizable deep neural networks for phytoplankton monitoring requires utilizing large-scale data coming from diverse global water sources. A major challenge to training such networks lies in data privacy, where data collected at different facilities are often restricted from being transferred to a centralized location. A promising approach to overcome this challenge is federated learning, where training is done at site level on local data, and only the model parameters are exchanged over the network to generate a global model. In this study, we explore the feasibility of leveraging federated learning for privacy-preserving training of deep neural networks for phytoplankton classification. More specifically, we simulate two different federated learning frameworks, federated learning (FL) and mutually exclusive FL (ME-FL), and compare their performance to a traditional centralized learning (CL) framework. Experimental results from this study demonstrate the feasibility and potential of federated learning for phytoplankton monitoring.', 'arxivid': '2212.08990', 'author': ['Daniel Zhang daniel.zhang1@uwaterloo.ca ', 'Vikram Voleti ', 'Mila ', 'Alexander Wong ', 'Jason Deglint jdeglint@uwaterloo.ca ', '\nUniversity of Waterloo\nUniversity of Montreal\nUniversity of Waterloo\nBlue Lion Labs\n', '\nUniversity of Waterloo\nBlue Lion Labs\n'], 'authoraffiliation': ['University of Waterloo\nUniversity of Montreal\nUniversity of Waterloo\nBlue Lion Labs', 'University of Waterloo\nBlue Lion Labs'], 'corpusid': 254854403, 'doi': '10.48550/arxiv.2212.08990', 'github_urls': [], 'n_tokens_mistral': 5446, 'n_tokens_neox': 4792, 'n_words': 3129, 'pdfsha': '91fdc27229f2d6413064bc5b3e57b7bbe8481c5c', 'pdfurls': ['https://export.arxiv.org/pdf/2212.08990v1.pdf'], 'title': ['Plankton-FL: Exploration of Federated Learning for Privacy-Preserving Training of Deep Neural Networks for Phytoplankton Classification', 'Plankton-FL: Exploration of Federated Learning for Privacy-Preserving Training of Deep Neural Networks for Phytoplankton Classification'], 'venue': []}

arxiv

Cosmological distance indicators 22 Jul 2018 Sherry H Suyu suyu@mpa-garching.mpg.de Tzu-Ching Chang tzu-ching.chang@jpl.nasa.gov Frédéric Courbin frederic.courbin@epfl.ch Teppei Okumura tokumura@asiaa.sinica.edu.tw Sherry H Suyu Tzu-Ching Chang Max-Planck-Institut für Astrophysik Institute of Astronomy and Astrophysics 11F of ASMAB Physik-Department Academia Sinica Karl-Schwarzschild-Str. 1, No.1, Section 4, Roo-sevelt Road85748, 10617Garching, Germany, TaipeiTaiwan Institute of Astronomy and Astrophysics, Academia Sinica Jet Propulsion Laboratory Technische Universität München James-Franck-Straße 1, 4800 Oak Grove Dr, MS 169-237, 11F of ASMAB, No.1, Section 4, Roo-sevelt Road85748, 91109, 10617Garching, Pasadena, TaipeiCAGermany, USA;, Taiwan Teppei Okumura Institute of Astronomy and Astrophysics, Academia Sinica Laboratoire d'Astrophysique, Ecole Polytechnique Fédérale de Lausanne (EPFL) Frédéric Courbin Institute of Physics Observatoire de Sauverny, 11F of ASMAB, No.1, Section 4, Roo-sevelt RoadCH-1290, 10617Versoix, TaipeiSwitzerland, Taiwan Kavli Institute for the Physics and Mathematics of the Universe (WPI) UTIAS The University of Tokyo 277-8583KashiwaChibaJapan Cosmological distance indicators 22 Jul 20181 We review three distance measurement techniques beyond the local universe: (1) gravitational lens time delays, (2) baryon acoustic oscillation (BAO), and (3) HI intensity mapping. We describe the principles and theory behind each method, the ingredients needed for measuring such distances, the current observational results, and future prospects. Time-delays from strongly lensed quasars currently provide constraints on H 0 with < 4% uncertainty, and with 1% within reach from ongoing surveys and efforts. Recent exciting discoveries of strongly lensed supernovae hold great promise for time-delay cosmography. BAO features have been detected in redshift surveys up to z 0.8 with galaxies and z ∼ 2 with Ly-α forest, providing precise distance measurements and H 0 with < 2% uncertainty in flat Λ CDM. Future BAO surveys will probe the distance scale with percent-level precision. HI intensity mapping has great potential to map BAO distances at z ∼ 0.8 and beyond with precisions of a few percent. The next years ahead will be exciting as various cosmological probes reach 1% uncertainty in determining H 0 , to assess the current tension in H 0 measurements that could indicate new physics. 1 Gravitational Lens Time Delays Principles of gravitational lens time delays Strong gravitational lensing occurs when a foreground mass distribution is located along the line of sight to a background source such that multiple images of the background source appear around the foreground lens. In cases where the background source intensity varies, such as an active galactic nucleus (AGN) or a supernova (SN), the variability pattern manifests in each of the multiple images and is delayed in time due to the different light paths of the different images. The time delay of image i, relative to the case of no lensing, is t(θ i ; β ) = D ∆t c φ (θ i ; β ),(1) up to an additive constant 1 , where θ i is the position of the lensed image i, β is the position of the source, D ∆t is the so-called "time-delay distance", c is the speed of light, and φ is the "Fermat potential" related to the lens mass distribution. The time-delay distance for a lens at redshift z d and a source at redshift z s is D ∆t = (1 + z d ) D d D s D ds ,(2) where D d is the angular diameter distance to the lens, D s is the angular diameter distance to the source, and D ds is the angular diameter distance between the lens and the source. In the Λ CDM cosmology with density parameters Ω m for matter, Ω k for spatial curvature, and Ω Λ for dark energy described by the cosmological constant Λ , the angular diameter distance between two redshifts z 1 and z 2 is D(z 1 , z 2 ) = 1 1 + z 2 f K [χ(z 1 , z 2 )](3) where χ(z 1 , z 2 ) = c H 0 z 2 z 1 dz Ω m (1 + z ) 3 + Ω k (1 + z ) 2 + Ω Λ −1/2 ,(4) and f K (χ) =    K −1/2 sin K 1/2 χ for K > 0 χ for K = 0 (−K) −1/2 sinh (−K) 1/2 χ for K < 0 , K = −Ω k H 2 0 /c 2 is the spatial curvature, and H 0 is the Hubble constant. By monitoring the variability of the multiple images, we can measure the time delay between the two images i and j: ∆t i j = t(θ i ; β ) − t(θ j ; β ) = D ∆t c ∆ φ i j .(6) The Fermat potential φ can be determined by modeling the lens mass distribution using observations of the lens system such as the observed lensed image positions, shapes and fluxes. Therefore, with ∆t measured and ∆ φ determined, we can use equation (6) to infer the value of D ∆t , which is inversely proportional to H 0 (D ∆t ∝ H −1 0 ) through equations (2) and (3). Being a combination of three angular diameter distances, D ∆t is mainly sensitive to the Hubble constant H 0 and weakly depends on other cosmological parameters [e.g., Refsdal, 1964, Schneider et al, 2006. One can further measure D d from a lens system by measuring the velocity dispersion of the foreground lens, σ v , and combining it with the time delays [Jee et al, 2015, Paraficz andHjorth, 2009]. The measurement of D d provides additional constraints on cosmological models [Jee et al, 2016]. In order to measure D ∆t and D d from a time-delay lens system for cosmography, we need the following 1. spectroscopic redshifts of the lens z d and source z s 2. time delays between the multiple images 3. lens mass model to determine the Fermat potential 4. lens velocity dispersion, which is not only required for D d inference, but also provides additional constraints in breaking lens mass model degeneracies 5. lens environment studies to break lens model degeneracies, such as the masssheet degeneracy In the next sections, we describe the history behind this approach, and detail the advances in recent years in acquiring these ingredients before presenting the latest cosmographic inferences from this approach. A brief history In his original paper Refsdal [1964] proposed to use gravitationally lensed supernovae to measure the time delays: the light curves associated to each lensed image of a supernova are expected to be seen shifted in time by a value that depends on the potential well of the lensing object and on cosmology. However, due to the shallow limiting magnitude of the telescopes available at the time and due to their restricted field of view, discovering faint and distant supernovae right behind a galaxy or a galaxy cluster was completely out of reach. But Refsdal's idea came right when quasars were discovered [Hazard et al, 1963, Schmidt, 1963. These bright, distant and photometrically variable point sources were coming timely, offering a new opportunity to implement the time-delay method: light curves of lensed quasars are constantly displaying new features that can be used to measure the delay. With the increasing discovery rate of quasars, the first cases of multiply imaged ones also started to grow. The first lensed quasar, Q 0951+567, was found by Walsh et al [1979], displaying two lensed images. This was followed by the quadruple PG 1115+080 [Young et al, 1981], and a few years later by the discovery of the "Einstein Cross" [Huchra et al, 1985] and of the "cloverleaf" [Magain et al, 1988]. The first time-delay measurement became available only in the late 80s with the optical monitoring of Q 0951+567 by Vanderriest et al [1989] and the radio monitoring of the same object by Lehar et al [1992]. Unfortunately, given the two data sets and methods of analysis to measure the delay, the radio and optical values of the time delay remained in disagreement until new optical data came [e.g. Kundić et al, 1997, Oscoz et al, 1997, allowing to confirm and improve the optical delay of Vanderriest et al [1989]. Further improvement was possible with the "round-theclock" monitoring of Colley et al [2003], leading to a time-delay determination to a fraction of a day. Because of the time and effort it took to solve the "Q0957 controversy", astrophysicists quickly limited their interest in the time-delay method as a cosmological probe. But at least two sets of impressive monitoring data revived the field. The first one is the optical monitoring of the quadruple quasar PG 1115+080 by Schechter et al [1997], providing time delays to 14% and the second is the radio monitoring, with the VLA, of the quadruple radio source B1608+656 [Fassnacht et al, 1999], reaching similar accuracy. As the uncertainty on the time delay propagates linearly in the error budget on H −1 0 this is still not sufficient for precision cosmology to a few percents. In a large part thanks to the results obtained for PG 1115+080 and B1608+656 several monitoring campaigns were put in place by independent teams in the late 90s and early 2000. Because lensed quasars were more often discovered in the optical and because their variability is faster at these wavelengths due to the smaller source size than in the radio, these new monitoring campaigns took place in the optical. The teams involved used 1m class telescopes to measure delays to typical accuracies of 10% or slightly better, i.e. a 30% improvement over previous measurements but still too large for cosmological purposes. Some of the most impressive results were obtained in the years 2000 with the 2.6m Nordic Optical Telescope (NOT) for FBQ 0951+2635 [Jakobsson et al, 2005], SBS 1520+530 [Burud et al, 2002b], RX J0911+0551 [Hjorth et al, 2002], B1600+434 [Burud et al, 2000], at ESO with the 1.54m Danish telescope for HE 2149−2745 [Burud et al, 2002a] and at Wise observatory with the 1m telescope for HE 1104−1805 [Ofek and Maoz, 2003]. With these new observations and studies it was shown that "mass production" of time delays was possible and not restricted to a few lenses for which the observational situation was particularly favorable. However, the temporal sampling adopted for the observations and the limited signal-to-noise per observing epoch was still limiting the accuracy on the time-delay measurement to 10% in most cases hence limiting H −1 0 measurements with indivisual lenses to this precision. Fifty years after Refsdal [1964]'s foresight on lensed SN, the first strongly lensed SN was discovered by Kelly et al [2015] serendipitously in the galaxy cluster MACS J1149.5+2223 with Hubble Space Telescope (HST) imaging. This corecollapse SN was named "SN Refsdal", and showed 4 multiple images at detection. The predictions , Jauzac et al, 2016 and subsequent detection [Kelly et al, 2016a] of the re-occurrence of the next (time-delayed) image of SN Refsdal provided a true blind test of our understanding of lensing theory and mass modeling. It is reassuring that some teams predicted accurately the re-occurrence , and the modeling software GLEE 2 used by Grillo et al [2016] was also the software employed for cosmography with lensed quasars [e.g., Suyu et al, 2013. In the fall of 2016, the first spatially-resolved multiply-imaged Type Ia SN, iPTF16geu, was discovered by Goobar [2017] in the intermediate Palomar Transient Factory survey. More et al [2017] independently modeled a single-epoch HST image of the system, finding short model-predicted time delays (<1 day) between the multiple images. Furthermore, More et al [2017] found anomalous flux ratios of the SN compared to the smooth model prediction, indicating possible microlensing effects, although Yahalomi et al [2017] showed that microlensing is unlikely to be the sole cause of the anomalous flux ratios. Both SN Refsdal and iPTF16geu have been monitored for time-delay measurements [Goobar, 2017, opening a new window to study cosmology with strongly-lensed SN. Recently Grillo et al [2018] estimated the time delay of the image SX of SN Refsdal based on the detection presented in Kelly et al [2016b] (image SX has the longest delay compared to other images of SN Refsdal, so image SX will ultimately provide the most precise time-delay measurement for cosmography from this system), and modeled the mass distribution of the galaxy cluster MACS J1149.5+2223 to infer H 0 . This feasibility study shows that H 0 can be measured with ∼ 7% statistical uncertainty, despite the complexity in modeling the cluster lens mass distribution. The full analysis including various systematic uncertainties is forthcoming, after the time delays are measured from the monitoring data. As lensed SNe are only being discovered/observed recently and their utility as a time-delay cosmological probe is just starting, we focus in the rest of the review on the more common lensed quasars as time-delay lenses, but there is a wealth of information to gather with lensed supernovae, both on a cosmological and stellar physics point of view. Recent advances A 10% error bar on the time delay translates to a similar error on H 0 . Improving this further and obtaining H 0 measurements competitive with other techniques, i.e. of 3-4% currently and 1-2% in the near future, requires several ingredients. Timedelay measurements of individual lenses must reach 5% at most and many more systems must be measured. With such precision on individual time delays, and under the assumption that all sources of systematic errors are controlled or negligible, measuring H 0 to 2% is possible with only a handful of lenses and a 1% measurement is not out of reach with of the order of 50 lenses! However, the lens model for the main lensing galaxy must be well constrained and their systematics evaluated and/or mitigated. Third, the contribution of all objects on the line of sight to the overall potential well must be accounted for. Excellent progress has been made on all three fronts in the recent years and there is still room for further (realistic) improvement. Time delays Measuring time delays is hard, but feasible provided telescope time can be guaranteed over long periods of time with stable instrumentation. The main limiting factors are astrophysical, observational or instrumental. Astrophysical factors include the characteristics of the intrinsic variability of the source quasar and extrinsic variability of its lensed images due to microlensing by stars in the main lensing galaxy. If the source quasar is highly variable intrinsically, both in amplitude and temporal frequency, the time delay is easier to measure. If microlensing variations are strong and/or comparable in frequency to the intrinsic variations then the time-delay value can be degenerate with the properties of the microlensing variations. In some extreme cases microlensing dominates the observed photometric variations to the point the time delay is hardly measurable [e.g. Morgan et al, 2012] even though microlensing itself can be used to infer details properties of the lensed source on micro-arcsec scales, i.e. out of reach of any current and future instrumentation. The observations needed to measure time delays must be adapted to the intrinsic and extrinsic variations of the selected quasars and of course to the expected time delay for each target. Not surprisingly the shorter the time delay, the finer the temporal sampling is needed. The position of the target on the sky also influences the results: equatorial targets will hardly be visible more than 6 months in a row, but can be followed both from the North and the South, while circumpolar targets can be seen up to 8-9 consecutive months, hence allowing to measure longer time delays and minimizing the effect of the non-visibility gaps between observing seasons. Finally, instrumental factors strongly impact the results. A key factor with current monitoring campaigns is the availability of telescopes on good sites and with stable instrumentation, i.e. if possible at all with no camera or filter change and with regular temporal sampling. Long gaps in light curves seriously affect the time-delay values in the sense that they make it more difficult to disentangle the microlensing varia-tions from the quasar intrinsic variations. And since angular separation between the lensed images are small, fairly good seeing is required, typically below 1.2 arcsec even though techniques like image deconvolution [e.g. Cantale et al, 2016, Magain et al, 1998] and used, e.g. by the COSMOGRAIL collaboration (see below) help dealing with data sets spanning a broad range of seeing values. Once long and well-sampled photometric light curves are available, the time delay must be measured. At first glance, this step may be seen as an easy one. However, one has to deal with irregular temporal sampling, gaps in the light curves, variable signal-to-noise and seeing, atmospheric effects (night-to-night calibration) and with microlensing. A number of numerical methods have been devised over the years to carry out the measurement, with different levels of accuracy and precision. They split in different broad categories. Some attempt to cross-correlate the light curves without trying to model/subtract the microlensing variations. Others involve an analytical representation of the intrinsic quasar variations and of the microlensing or involve e.g. Gaussian processes to mimic the microlensing erratic variations. Recent work in this area has been developed in Hojjati and Linder [2014], Hojjati et al [2013], Rathna Kumar et al [2015], Tewes et al [2013a]. These methods (and others, so far unpublished) were tested in an objective way using simulated light curves proposed to the community in the context of the "Time Delay Challenge" [TDC; Dobler et al, 2015]. In the TDC, thousands of light curves of different lengths, sampling rate, signal-to-noise, and visibility gaps are proposed to the participants. Once all participants have submitted their time-delay evaluations to the challenge organizers, the time-delay values are revealed and the results are compared objectively using a metrics common to all participating methods. This metrics was known before the start of the challenge. The results of this TDC are summarized in Liao et al [2015] as well as in individual papers [e.g. Bonvin et al, 2016]. A general conclusion of the TDC was that with current lens monitoring data, curve-shifting technique so far in use are sufficient to extract precise and accurate time delays, given the temporal sampling and signal-to-noise of the data. Following the encouraging results obtained at NOT, ESO and Wise, long-term monitoring campaigns were organized to measure time delays in a systematic way. Two main teams invested effort in this research: the OSU group lead by C.S. Kochanek (OSU, USA) and the COSMOGRAIL (COSmological MOnitoring of GRAvItational Lenses) program led by F. G. Meylan at EPFL, Switzerland [e.g. Courbin et al, 2005, Eigenbrod et al, 2005]. Both monitoring programs involve 1-m class telescopes with a temporal sampling of 2 to 3 observing epochs per week and a signal-to-noise of typically 100 per quasar image and per epoch. Both projects started in 2004 and are so far the main (but not only) source of time-delay measurements. Early results from the OSU program were obtained in 2006 for HE 0435−1223 ] while COSMO-GRAIL delivered its first results starting in 2007 for SDSS J1650+4251 [Vuissoz et al, 2007], WFI J2033−4723 [Vuissoz et al, 2008] and HE 0435−1223 [Courbin et al, 2011a]. More recent time-delay measurements from COSMOGRAIL were obtained for RX J1131−1231 [Tewes et al, 2013b], HE 0435−1223 as well as SDSS J1206+4332 and HS 2209+1914 and ]. An example of COSMOGRAIL light curve is given in Fig. 1 [Fohlmeister et al, 2013] and SDSSJ1004+4112 [Fohlmeister et al, 2008]. These may not be ideal for cosmological applications though, as a complex lens model for a cluster is harder to constrain than models at galaxy-scale, unless the cluster has additional constraints coming from multiple background sources at different redshifts being strongly lensed. SDSS J1001+5027 [Rathna With the observing cadence of 1 point every 3-4 nights and an SNR of 100 per epoch, the current data can catch quasar variations of the order of 0.1 mag in amplitude, arising on time-scales of months. These time scales are unfortunately of the same order of magnitude as the microlensing variations (see 2nd panel of Fig. 1) making it hard to disentangle between intrinsic and extrinsic variations. For this reason, lensed quasars must be monitored for extended periods of time, typically a decade, to infer any reliable time-delay measurement. Fig. 2 Expected relative precision on a time delay measurement as a function of the length of the campaign. High-cadence (1 day −1 ) monitoring is assumed and the fiducial delay in this simulations mimics the longer delay of HE 0435-1223, i.e. 14 days. Clearly, 2% precision can be reached in only 1 observing season. The color code shows the catastrophic failure rate, i.e. the probability of getting a measurement wrong by more than 5%. This probability is about 10% for a 1-season campaign and 3% for a 2-season campaign. (Courtesy: Vivien Bonvin) Going beyond current monitoring campaigns like COSMOGRAIL and others is possible, but measuring massively time delays for dozens of lensed quasars requires a new observing strategy to minimize the effect of microlensing and to measure time delays in individual objects in less than 10 years! One solution is to observe at high cadence (1 day −1 ) and high SNR, of the order of 1000. In this way, very small quasar variations can be caught, on time scales much shorter than microlensing hence allowing the separation of the two signals in frequency. We show with 1000 mock light curves that mimic those of HE 0425−1223 ( Fig. 1) that time delays can be measured precisely in only 1 observing season. In doing the simulation, we include realistic microlensing and fast quasar variations with a few mmag amplitude. We then run PyCS, the COSMOGRAIL curve-shifting algorithm , Tewes et al, 2013a, to recover the fiducial time delay of 14 days. Fig. 2 summarizes our results and provides the length of the monitoring campaign needed to reach a desired relative precision on the time delay, assuming daily observations and an SNR of 1000 per epoch. It appears that a typical 2% precision is achievable in 1 observing season with a 10% failure rate. Doing two seasons allows one to reach the percent precision and a failure rate below 3%. At the time this paper is being written, an intensive lens monitoring program has been started at the 2.2m MPI telescope at La Silla Observatory, with the above characteristics. Three targets have been observed for 1 semester and time delays have been measured to a few percents for all three! The first of these is presented in Courbin et al [2017] and features a 1.8% measurement of one of the delays in the newly discovered quadruple lens DES J0408−5354 . Finally, we note that although quasars have been used so far to implement the time delay method, the original idea of Refsdal was to use lensed supernovae. The first systems have finally be found, as mentioned in Section 1.2: SN Refsdal [e.g. Kelly et al, 2015 and iPTF16geu [e.g. Goobar, 2017. With the advent of large imaging surveys such as the Zwicky Transient Facility and the Large Synoptic Survey Telescope, prospects to find lensed supernovae are excellent [e.g. Goldstein and Nugent, 2017]. As supernovae have known light curves, one can measure the time delay by fitting a template to the observed light curves in the lensed images, hence giving much more constraining power than quasars whose photometric variations are close to a random walk. In addition, if the lensed supernova is a Type Ia, then two cosmological probes are available in the same object, hence provide a fantastic cross-check of otherwise completely different methods: standard candles and a geometrical method, provided microlensing effects could be corrected [Dobler and Keeton, 2006, Yahalomi et al, 2017. For all the above reasons, we believe that the future of time-delay cosmography resides in lensed supernovae and in high-cadence monitoring of lensed quasars. However, Tie and Kochanek [2018] recently pointed out that microlensing by stars in the lensing galaxy can introduce a bias in the time-delay measurements. This is due to a combination of differential magnification of different parts of the source and the source geometry itself. The net result is that cosmological time delays can be affected both in a statistical and a systematic way by microlensing. The effect is absolute, with biases on time delays of the order of a day for lensed quasars and tenths of a day for lensed supernovae. Mitigation strategies have been successfully devised [Chen et al, 2018] for lensed quasars and the effect seems less pronounced in lensed supernovae than in lensed quasars , Foxley-Marrable et al, 2018, but clearly this new effect must be accounted for in any future work in the field. Lens mass modeling To convert the time delays into a measurement of the time-delay distance via equation (1), one needs to determine the Fermat potential φ (θ i ; β ), which depends both on the mass distribution of the main strong-lens galaxy and the mass distribution of other galaxies along the line of sight. The mass distribution of the main strong-lens galaxy can be modeled using either simply parametrized profiles [e.g., Barkana, 1998, Golse and Kneib, 2002, Kormann et al, 1994 or grid-based approaches [e.g., Blandford et al, 2001, Suyu et al, 2009, Vegetti and Koopmans, 2009, Williams and Saha, 2000. The total mass distribution of galaxies appear to be well described by profiles close to isothermal [e.g., Barnabè et al, 2011, Cappellari et al, 2015, Koopmans et al, 2006, even though neither the baryons nor the dark matter distribution follow isothermal profiles. Even in the complex case of the gravitational lens B1608+656 with two interacting lens galaxies, simply parametrized profiles provide a remarkably good description of the galaxies when compared to the pixelated lens potential reconstruction [Suyu et al, 2009]. Therefore, most of the current mass modeling for time-delay cosmography use simply parametrized profiles, either for the total mass distribution [e.g., Birrer et al, 2015b, Fadely et al, 2010, Koopmans et al, 2003 or for separate components of baryons and dark matter [e.g., Courbin et al, 2011b, Schneider and Sluse, 2013, Suyu et al, 2014. The source (quasar) properties need to be modeled simultaneously with the lens mass distribution to predict the observables. In particular, source position and intensity are needed to predict the positions, fluxes and time delays of the lensed quasar images, whereas the source surface brightness distribution (of the quasar host galaxy) is needed to predict the lensed arcs. These observables (image positions, fluxes and delays of the multiple quasar images, and lensed arcs) are then used to constrain the parameters of the lens mass model and the source. Several softwares are available publicly for modeling lens systems, including GRAVLENS [Keeton, 2001], LENSTOOL [Jullo et al, 2007], GLAFIC [Oguri, 2010] and LENSVIEW [Wayth and Webster, 2006]. Observed quasar image positions, fluxes and delays provide around a dozen of constraints for quads (four-image systems) and even fewer constraints for doubles (two-image systems). Thus lens mass models using only these quasar observables are often not precisely constrained. In particular, the radial profile slope of the lens galaxy is strongly degenerate with D ∆t [e.g., Suyu, 2012, Wucknitz, 2002. The time delays depend primarily on the average surface mass density between the multiple images, and thus provide information on the radial profile slope [Kochanek, 2002]. Nevertheless, even with multiple time delays from quad systems, it is difficult to infer the slope precisely to better than ∼ 10% precision 3 . While mass distribution of massive early-type galaxies, which are the majority of lens galaxies, are close to being isothermal, there is an intrinsic scatter in the slope of about ∼ 15% 3 [Barnabè et al, 2011, Koopmans et al, 2006]. For precise and accurate D ∆t measurement of a lens system, it is important to measure directly, at the few percent level, the radial profile slope of the lens galaxy near the lensed images of the quasars. This requires more observations of the lens system, beyond just the multiple point images of lensed quasars. Over the past decade, multiple methods have been developed to make use of the lensed arcs (lensed host galaxies of the quasars) to constrain the lens mass distribution. The source intensity distribution can be described by simply parametrized profiles, such as Gaussians or Sersic [e.g., Brewer and Lewis, 2008, Oguri, 2010, Oldham et al, 2017, or on a grid of pixels, either regular [e.g., Koopmans, 2005, Suyu et al, 2006, Wallington et al, 1996, Warren and Dye, 2003 or adaptive [e.g., Dye and Warren, 2005, Nightingale and Dye, 2015, Tagore and Keeton, 2014, Vegetti and Koopmans, 2009, or based on basis functions [e.g., Birrer et al, 2015a, Joseph et al. in prep.]. These lensed arcs, when imaged with HST or ground-based telescopes assisted with adaptive optics, contain thousands of intensity pixels and thus allow the measurement of the radial profile slope of the lens galaxies with a precision of a few percent [e.g., Chen et al, 2016, Dye and Warren, 2005, Suyu, 2012, that are required for cosmography. In Fig. 3, we show an example of the mass modeling using the full surface brightness distributions of quasar host galaxy. Once a model of the surface mass density κ is obtained, lens theory states that the following family of models κ λ fits equally well to the observed lensing data: κ λ = λ + (1 − λ )κ,(7) where λ is a constant. This transformation is analogous to adding a constant mass sheet λ in convergence, and rescaling the mass distribution of the strong lens (to keep the same mass within the Einstein radius); it is therefore called the "masssheet degeneracy" [Falco et al, 1985, Schneider, 2014, Schneider and Sluse, 2013. Such a transformation corresponds to a rescaling of the background source coordinate by a factor (1 − λ ), leaving the observed image morphology and brightness invariant. Furthermore, the Fermat potential transforms as φ λ (θ ; β ) = (1 − λ )φ (θ ; β ) + constant that depends only on β . Therefore, for given observed time delays ∆t i j , equations (6) and (8) imply that the time-delay distance D ∆t would be scaled by (1 − λ ). The mass-sheet degeneracy has thus a direct impact on cosmography in measuring D ∆t . While λ so far is simply a constant in this mathematical transformation (equation 7), we can identify it with the physical external convergence, κ ext , due to mass structures along the sight line to the lens system. By gathering additional data sets beyond that of the strong lens system, we can infer κ ext and thus measure D ∆t . Two practical ways to break the mass-sheet degeneracy are (1) studies of the lens environment, to estimate κ ext based on the density of galaxies in the strong-lens line of sight in comparison to random lines of sight [e.g., Collett et al, 2013, Fassnacht et al, 2006, Hilbert et al, 2007, McCully et al, 2017, Momcheva et al, 2006, and (2) stellar kinematics of the strong lens galaxy, which provides an independent mass measurement within the effective radius to complement the lensing mass enclosed within the Einstein radius [e.g., Barnabè et al, 2009, Grogin and Narayan, 1996, Koopmans and Treu, 2002, Suyu et al, 2014. The time-delay distance can then be inferred via D ∆t = D model ∆t (1 − κ ext ) ,(9) where D model ∆t is the modeled time-delay distance without accounting for the presence of κ ext . In practice, both lens environment characterisations and stellar kinematics are employed to infer D ∆t for cosmography [e.g., Birrer et al, 2015b. The stellar kinematic data further help constrain the strong-lens mass profile [e.g., Suyu et al, 2014]. Lens systems that have massive galaxies close in projection (within ∼ 10 ) to the strong lens, but at a different redshift from the strong lens, will need to be accounted for explicitly in the strong lens model. In such cases, multi-lens plane modeling is needed [e.g., Blandford and Narayan, 1986, Gavazzi et al, 2008, Schneider et al, 1992, but equation (1) for single-lens plane is then not directly valid. In particular, there is not a single time-delay distance, but rather there are multiple combination of distances between the multiple planes. Nonetheless, for some cases, one could obtain an effective time-delay distance as if it were a single-lens plane system [see, e.g., Wong et al, 2017, for details]. As noted in Section 1.1, with stellar kinematic and time-delay data, we can infer the angular diameter distance to the lens, D d , in addition to D ∆t [Birrer et al, 2015b, Jee et al, 2015, Paraficz and Hjorth, 2009, Shajib et al, 2017. Measurement of D d is often more sensitive to the dark energy parameters [for typical lens redshifts 1, see e.g., Fig. 2 of Jee et al, 2016], and can also be used as an inverse distance ladder to infer H 0 (Jee et al., submitted). Currently, the precision in D d is limited by the uncertainty in the single-aperture averaged velocity dispersion measurement and the unknown anisotropy of stellar orbits [Jee et al, 2015]. Nonetheless, we anticipate that spatially resolved kinematic data will help to constrain more precisely D d . We have focussed here on the advances in getting D ∆t and D d from individual lenses with exquisite follow-up data to control the systematic uncertainties. Alternatively, one could analyse a sample of lenses and constrain a global H 0 parameter that is common to all the lenses [e.g., Oguri, 2007, Saha et al, 2006, Sereno and Paraficz, 2014, assuming that the systematic effects for the lenses average out. For small samples, this assumption might not be valid. Nonetheless, in the future where thousands of lensed quasars are expected [Oguri and Marshall, 2010] but most of which will not have exquisite follow-up observations, this large sample of lenses could provide information on the population of lens galaxies as a whole for cosmography (P. J. Marshall & A. Sonnenfeld, priv. comm.). We therefore advocate getting exquisite follow-up observations of a sample of ∼ 40 lenses to reach an H 0 measurement with 1% uncertainty [Jee et al, 2016, Shajib et al, 2017, with the other lenses providing information on the profiles of galaxies to use in the mass modeling. Distance measurements and cosmological inference There are so far only a few lensed quasars for which all required data exist to do time-delay cosmography, i.e., with time-delay measurements to a few percent, deep HST images showing the lensed image of the host galaxy, deep spectra of the lens to measure the velocity dispersion, and multiband data to map the line of sight contribution to the lensing potential. Some of the best time-delay measurements available to date include the radio time delay for B1608+656 [Fassnacht et al, 1999[Fassnacht et al, , 2002 and the two optical measurements of COSMOGRAIL for RX J1131−1231 [Tewes et al, 2013b], and HE 0435−1223 . These 3 quadruply imaged quasars, for which all the ancillary imaging and spectroscopic data are also available, gave birth to the H0LiCOW program , which capitalizes on more than a decade of COSMOGRAIL monitoring as its name reflects: H0 Lenses in COMOGRAIL's Wellspring. With the precise time-delay measurements of COS-MOGRAIL, H0LiCOW addresses what has been so far limiting the effectiveness of strong lensing in delivering reliable H 0 measurements: the different systematics at work at each step leading to a value for the Hubble constant. and Type Ia supernovae [blue; Riess et al, 2016b]. Quasar time delays are so far in agreement with local estimators but higher than Planck. Measurements for 40-50 new time delays will allow one to confirm (or not) the current tension with Planck to more than 5σ . The most recent work by H0LiCOW is summarized in the left panel of Fig. 4 , based on state-of-the-art lens mass modeling and characterisations of mass structures along the line of sight . In the right panel of Fig. 4, we compare the value of H 0 from H0LiCOW with other fully independent cosmological probes such as Type Ia supernovae, Cepheids, and CMB(+BAO) for a Λ CDM cosmology. With the current error bars claimed by each probe there exist a tension between local measurements of H 0 [e.g., Efstathiou, 2014, Freedman et al, 2012, Riess et al, 2018 and the value by the Planck team. When completed, H0LiCOW will feature 5 lenses, with an accuracy on H 0 of the order of 3% , but reaching close to 1% precision is possible. This will be enabled by working on several fronts simultaneously, by finding more lenses, measuring up to 50 new time delays, and refining the lens modeling tools to mitigate degeneracies between model parameters. Chapter 8 on "Towards a self-consistent astronomical distance scale" provides more details about the (expected) future of quasar time delay cosmography. Baryon Acoustic Oscillations BAO as a standard ruler The universe has been expanding, and thus the universe in the earlier stage was much smaller, denser and hotter than today. In such an early universe, electrons interacted with photons via Compton scattering and with protons via Coulomb scattering. Thus, the three components acted as a mixed fluid [Peebles and Yu, 1970]. They were in the equilibrium state due to the gravity of protons and pressure of photons, and oscillated as sound modes. These oscillations are called baryon acoustic oscillations (BAO) (see Bassett and Hlozek [2010] and Weinberg et al [2013] for a comprehensive review). It moved with the speed of sound c s = c 1 3(1+R) , where the ratio of photon density (ρ r ) to baryon density (ρ b ) is defined as 1/R = 4ρ r /3ρ b . At recombination (z ∼ 1100), photons decouple from the baryons and start to free stream. We observe the photons as a map of cosmic microwave background (CMB). The left panel of Fig. 5 shows the angular power spectrum of the latest data of the CMB anisotropy probe, Planck satellite [Planck Collaboration et al, 2016a]. One can see a clear oscillation feature in the power spectrum, which is characterized by the sound horizon scale at recombination, expressed as Hu, 1998, Hu andSugiyama, 1996] r d = ∞ z * c s (z) H(z) dz,(10) where z * is the redshift at recombination. H(z) is the Hubble parameter, H(z) = H 0 Ω m (1 + z) 3 + Ω r (1 + z) 4 + Ω DE (1 + z) 1+w 1/2 ,(11) where Ω m (previously introduced in Section 1.1), Ω r and Ω DE are the matter, radiation and dark energy density parameters, respectively, and w is the equation-of-state parameter of dark energy and the simplest candidate for dark energy, the cosmological constant Λ , gives w = −1. With standard cosmological models, r d 150 Mpc. From the Planck observation, it is constrained to r d = 144.61 ± 0.49 Mpc. Probing BAO in galaxy distribution After the recombination, motion of the baryons becomes non-relativistic. The perturbation of baryons then starts to grow at their locations and interact with the perturbation of dark matter. Thus the baryon acoustic feature should be imprinted onto the late-time large-scale structure of the Universe. Theoretically it is predicted to produce the overdensity at the sound horizon scale, ∼ 150 Mpc. It is, however, observationally not easy because the observation of BAO signal requires the number of tracers of matter overdensity field to be large enough at the scale to overcome the cosmic variance. In 2005, detection of the BAO was reported almost simultaneously by two independent groups using the Sloan Digital Sky Survey (SDSS) [Eisenstein et al, 2005] and the 2dF Galaxy Redshift Survey (2dFGRS) [Cole et al, 2005]. The right panel of Fig. 5 shows the two-point correlation function obtained from the SDSS galaxy sample by Eisenstein et al [2005]. The 2-point correlation function ξ (s) is defined as an excess of the probability that one can find pairs of galaxies at a given scale s from the case of a random distribution. Thus the scales where ξ > 0 and ξ < 0 correspond to the statistically overdense and underdense regions respectively. The bump seen around s 105h −1 Mpc ( 150 Mpc) is the feature of BAO, and the scale of the bump corresponds to the sound horizon scale at recombination. The inset of the right panel of Fig. 5 zooms into the feature. Unlike observations of the CMB, galaxy redshift surveys are the observation of 3 dimensional space. The peak scale of BAO should be isotropic because the scale corresponds to the sound horizon at recombination. On the other hand, the BAO scale along the line of sight depends on H −1 (z) while the BAO scale perpendicular to the line of sight depends on the comoving angular diameter distance D M = (1 + z)D A (z), where D A (z) is expressed in flat universe as equation (3) with z 1 = 0, or, D A (z) = 1 1 + z z 0 cdz H(z ) .(12) Thus, the scale of BAO probed by a galaxy correlation function has a cosmology dependence of D V (z) = [(1 + z)D A (z)] 2/3 cz H(z) 1/3 .(13) The BAO scale probed by the CMB anisotropy at high redshift and the galaxy distribution at low redshift should be the same. Moreover, the power spectrum with the acoustic features has been precisely determined for the CMB and interpreted using linear cosmological perturbation theory (see the left panel of Fig. 5). Thus the detection of BAO in the galaxy distribution enables us to constrain D V , and hence the geometric quantities such as H 0 , w, and Ω DE in equation (11) [Eisenstein et al, 2005]. Anisotropy of BAO and Alcock-Paczynski The method presented above does not use all of the information encoded in BAO. To maximally extract the cosmological information, we need to measure the correlation function as functions of separations of galaxy pairs perpendicular (s ⊥ ) and parallel (s ) to the line of sight, ξ (s ⊥ , s ), where s = s 2 ⊥ + s 2 . In this way, in principle we can constrain D A (z) and H(z) using the transverse and radial BAO measurements, respectively. Given the cosmological dependence of angular and radial distances (equations 11 and 12), the shape of the BAO peak is distorted if the wrong cosmology is assumed. This effect was first pointed out by Alcock and Paczynski [1979, AP]. In fact, the AP test using the anisotropy of BAO has additional advantages. Since the BAO scale should correspond to the sound horizon at recombination, it should be a constant. Thus, we can determine the geometric quantities by requiring that the radial BAO scale equals to the angular BAO scale. We no longer need to know the exact value of r d nor need to rely on the CMB experiment [see e.g., Matsubara, 2004, Seo andEisenstein, 2003]. This is particularly important in the context of measuring H 0 given the tension in H 0 between the local measurements and the inference by the Planck team in flat Λ CDM (see Section 1.4). In galaxy surveys, the distance to each galaxy is measured through redshift, thus it gives the sum of the true distance and the contribution from the peculiar motion of the galaxies and it produces an anisotropy in galaxy distribution along the line of sight, which are called redshift-space distortions (RSD). Since the velocity field of a galaxy is caused by gravity, the anisotropy contains additional cosmological information. This effect is called the Kaiser effect, named after Kaiser [1987] who proposed RSD as a cosmological probe in the linear perturbation theory limit. Since the velocity field is related to the density field through the continuity equation, the anisotropy constrains the quantity f , defined as the logarithmic derivative of the density perturbation, f = d ln δ /d ln a. See Okumura et al [2016] for the constraint on f from RSD as a function of z including the high-z measurement. By simultaneously measuring the BAO and RSD and combining them with the CMB anisotropy power spectrum, we can obtain a further constraint on additional geometric quantities, such as the time evolution of w, the neutrino mass m ν , etc. The cosmological results, presented below, correspond to this case. The correlation function in 2D space including the BAO scales has been measured by Okumura et al [2008] for the first time using the same galaxy sample as Eisenstein et al [2005]. The right-hand side of the left panel in Fig. 6 shows the measured correlation function, while the left-hand side shows the best fitting model based on linear perturbation theory [Matsubara, 2004]. The circle shown at the scale s = (s 2 ⊥ + s 2 ) 1/2 105h −1 Mpc again corresponds to the sound horizon at recombination. The distorted, anisotropic contours shown at the smaller scales are the RSD effect caused by the velocity field. The right panel of Fig. 6 is the latest measurement of the correlation function using the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) DR12 sample . The anisotropic feature of BAO is more clearly detected due to the improvement of the data, both in the number of galaxies and the survey volume: the data of the DR12 sample used in Alam et al [2016] comprised 1.2 million galaxies over the volume of 18.7Gpc 3 , whose numbers are respectively 25 times and 9 times larger than those of the DR3 sample used in Okumura et al [2008]. Constraints on BAO distance scales and H 0 The left panel of Fig. 7 shows the summary of the three distance measures obtained by BAO in various galaxy surveys [Aubourg et al, 2015]. The y-axis is the ratio of each distance and r d , divided by √ z. The blue points are the measurement of D V from the angularly-averaged BAO, while the red and green points are respectively the measurements of D M and D H obtained from the anisotropy of BAO, where D H is the radial distance defined as D H (z) = c/H(z). The three lines with the same color as the points are the corresponding predictions of the Λ CDM model obtained by Planck [Planck Collaboration et al, 2016a]. Nice agreement between BAO measurements from galaxy surveys and Planck cosmology can be seen. However, the agreement with the WMAP cosmology is equivalently good in terms of distance measures [Anderson et al, 2014]. The right panel of Fig. 7 focuses on currently the largest survey, the BOSS survey at z = 0.5 [Anderson et al, 2014]. Here the joint constraints on H(z) and D A (z) are shown. The gray contours are obtained from the 1D BAO analysis (see section 2.2). Because the 1D BAO constrains D V ∝ D 2/3 A H −1/3 , there is a perfect degeneracy between D A and H. On the other hand, the solid orange contours are from the 2D BAO analysis where the degeneracy is broken to some extent (Section 2.3). The obtained constraints on the distance scales are as tight as the flat λ CDM constraints from the CMB experiments, Planck (dashed blue) and WMAP (dot-dashed green). Future galaxy surveys will enable us to measure BAO more accurately and determine the cosmic distance scales with higher precision (see Section 2.5). Let us move onto the constraints on cosmological models using BAO observations. The left panel of Fig. 8 shows the joint constraints on the matter density parameter Ω m and the Hubble constant h = H 0 /(100 km s −1 Mpc −1 ) obtained from the measurements of BAO anisotropy [Aubourg et al, 2015]. The red and blue contours are the constraints from the BAO measured from the various galaxy samples at z < 1 and from the Lyα forest at z ∼ 2, respectively, as shown in the left panel of Fig. 7. Since these constraints are not very tight, the constrained H 0 from either galaxy BAO or Lyα BAO is consistent with other probes including the local measurements. The combination of these two BAO probes largely tightens the constraint on H 0 and causes a slight, 2σ tension as we will see below. With Ω m being marginalized over, the Hubble constant is constrained to h = 0.67 ± 0.013 (1σ C. L.) [Aubourg et al, 2015]. The right panel of Fig. 8 summarizes the comparison of H 0 constraints from the BAO measurement and other probes. Augmenting Fig. 4, the top three points are obtained from the distance ladder analysis, showing constraints from the local universe from three independent teams [Efstathiou, 2014, Freedman et al, 2012. The green, two middle points are the constraints from the two CMB satellite probes, WMAP and Planck. The bottom two points are from the inverse distance ladder analysis, namely the combination of BAO and SN distance measures. As seen from the figure, those from the Planck and the inverse ladder measurements have a mild but non-negligible tension with the distance-ladder measurements, about ∼ 2σ . The discrepancy could be due to just systematics, or a hint of new physics. We will need further observational constraints to resolve these discrepancies. Future BAO Surveys BAO are considered as a probe least affected by systematic biases to measure distance scales, even beyond the local universe (z > 0), and thus are a promising tool to reveal the expansion history of the universe and constrain the cosmological model. To improve the precision of the distance measurement, a dominant source of error on BAO observations is the cosmic variance. There are larger, ongoing and planned galaxy redshift surveys, such as extended BOSS (eBOSS) [Dawson et al, 2016], Subaru Prime Focus Spectrograph (PFS) [Takada et al, 2014], and Dark Energy Spectroscopic Instrument (DESI) [DESI Collaboration et al, 2016]. With the larger survey volumes, these surveys will measure distance scales using BAO with percentlevel precisions. These surveys are also deep and can reveal fainter sources, and hence enable us to extend the distance scales to more distant parts of the universe. As an example, Fig. 9 presents the accuracies of constraining the angular diameter distance and Hubble expansion rate expected from the analysis of anisotropic BAO (see Section 2.3) of the PFS survey at the Subaru Telescope. The PFS will observe the universe at 0.8 < z < 2.4 by using [OII] emitters as a tracer of the largescale structure. The survey volume of each redshift slice is on the order of [h −1 Gpc] 3 and the number density is larger than 10 −4 [h −1 Mpc] −3 , which are comparable to the existing SDSS and BOSS surveys at z < 0.7. Hence, one will be able to achieve a few percent constraints on D A and H at high redshifts, almost the same as those obtained from the low-z surveys. Deep galaxy surveys such as the PFS allow for constraining not only the expansion history of the universe but also dark energy (see the solid line of Fig. 9). Ultimately, we would like to survey the galaxies over the whole sky, which can be achieved by satellite missions, such as Euclid [Amendola et al, 2013] and WFIRST . These surveys will measure the cosmic distances with an unprecedentedly high precision. Intensity Mapping 21cm Intensity Mapping BAO As we described in Section 2, current BAO measurements are enabled by large galaxy spectroscopic surveys, and the resulting constraining power on cosmological parameters generally scales as the effective survey comoving volume. Specifically, the precision of cosmological parameter constraints scales as ∝ 1/ √ N, where N is the number of modes, or in the case of 3D map ∝ 1/ k 3 max V where V is the comoving volume and k max the maximum useful comoving wavenumber. This scale is often given by the non-linearity scale, k nl (z = 0) ∼ 0.2 h/Mpc, where the complexity of baryonic astrophysics on galaxy and galaxy-cluster scales limits our ability to extract cosmological parameters. Improving parametric precision will therefore require larger volumes, which requires mapping higher redshift volumes that have the added benefit of increasing k nl (z). The emerging technique of 21 cm Intensity Mapping appears to be a very promising way to reach this goal. Galaxy redshifts can be measured at radio wavelengths using the 21 cm hyperfine emission of atomic neutral hydrogen (HI). The 21 cm line is unique in cosmology because for λ > 21 cm it is the dominant astronomical line emission, i.e., for all positive redshifts and all cosmological emission. So to a good approximation the wavelength of a spectral feature can be converted to a Doppler redshift without the uncertainty and ambiguity of having to first identify the atomic transition. The direct determination of redshifts using 21 cm data can be compared to the corresponding optical technique, which requires identifying a suitable subset of target galaxies (photometry), then obtaining an optical spectrum, and finally finding some unique combination of emission and absorption lines that allow an unambiguous determination of the redshift for that galaxy (spectroscopy). The 21 cm signal has been used to conduct galaxy redshift surveys in the local Universe around z ∼ 0.1 , Zwaan et al, 2001] and out to z ∼ 0.4 [Fernández et al, 2016]. Beyond this redshift, current radio telescopes do not have sufficient collecting area or sensitivity to make 21 cm surveys using individual galaxies. A radically different technique, HI intensity mapping (HIM), has been proposed [Chang et al, 2008, Wyithe et al, 2008. It uses maps of 21 cm emission where individual galaxies are not resolved. Instead, it detects the combined emission from the many galaxies that occupy large (1000 Mpc 3 ) voxels. The technique allows 100 m class telescopes such as the Green Bank Telescope (GBT), which only have angular resolution of several arc-minutes, to rapidly survey enormous comoving volumes at z ∼ 1 [Abdalla and Rawlings, 2005, Chang et al, 2008, Peterson et al, 2006, Seo et al, 2010, Tegmark and Zaldarriaga, 2009, 2010, Wyithe et al, 2007, Wyithe et al, 2008. A number of authors [Bull et al, 2015, Seo et al, 2010, Xu et al, 2015 have studied the overall promise of the intensity mapping technique. Chang et al [2010], Masui et al [2013] and Switzer et al [2013] have reported the first detections in cross-correlations and upper limits to the 21cm auto-power spectrum using the Green Bank Telescope (GBT). Challenges One of the major challenges for 21 cm intensity mapping is the mitigation of radio foregrounds, which are predominantly Galatic and extragalactic synchrotron emissions, and are at least ∼ 10 4 times brighter in intensity than the 21 cm emission. The two can be distinguished because the former have smooth spectra and the latter trace the underlying large-scale structure and have spectral structures. The brightness temperature of the synchrotron foreground typically has a spectral dependence of ν −2.6 , or (1 + z) 2.6 , and is thus more severe at higher redshifts. Note it has not yet been demonstrated whether the synchrotron radiation is indeed spectrally smooth down to one part of 10 4 or higher and therefore can in principle be suppressed by this factor to reveal the 21cm fluctuation signals. However, since the BAO wiggles have very specific structures, we can potentially select Fourier modes that are observationally accessible in scale and redshifts [Chang et al, 2008, Seo et al, 2010. The 21 cm features we are most interested in are the relatively non-smooth BAO 'wiggles'. Unfortunately radio telescopes are diffraction-limited and the beam patterns depend on frequency, which mixes angular and frequency dependence. Since the foregrounds are not smooth in position across the sky it is a nontrivial task to identify and subtract the smooth frequency foregrounds with sufficient accuracy so as to reveal the 21 cm emission. It is easier if we go to very small radial scales, but to get the most cosmological information out of the data we would need to remove the foregrounds over the largest range of scales possible. Shaw et al [2013Shaw et al [ , 2015 have demonstrated that this is achievable in principle. To achieve the foreground subtraction goal and to make accurate 3D maps we need a very accurate model of the beam patterns and characterization of the mapping between the observed and true skies. Liao et al [2016] have recently demonstrated accurate measurements of the polarized GBT beam to sub-percent level, which is critical for polarized foreground mitigation. Developing and demonstrating the efficacy of methods to model and calibrate large dataset is also necessary to achieve the main objective. Future Prospects As discussed in Section 2, baryon acoustic oscillations provide a convenient standard ruler in the cosmological large scale structure (LSS) allowing a precise measurement of the distance-redshift relation over cosmic time. This distance redshift relation is measured, whether by BAO, SNe-Ia surveys, or weak lensing, to characterize the dark energy; it is augmented by the growth rate of inhomogeneities as well as redshift-space distortions. All three of these quantities can be measured in HI surveys even though to-date, only optical instruments have detected BAO features in the power spectrum. Going forward requires the most cost-effective way to map the largest cosmological volume, and this may be radio spectroscopy through intensity mapping. One unique advantage of 21cm Intensity Mapping is the fact that the 21 cm signal is in principle observable from z = 0 up to a redshift of ∼ 100, when its spin temperature decouples from the Cosmic Microwave Background radiation. The vast majority of the cosmic volume is only visible during the dark ages via the 21 cm radiation from neutral hydrogen, before the onset of galaxy formation. 21 cm Intensity Mapping thus provides a unique access to measuring LSS during this period Zaldarriaga, 2009, 2010]. Besides, 21 cm intensity mapping has a set of observational systematics that should be largely uncorrelated with the systematic effects in optical surveys. The on-going low-z GBT-HIM survey is a step along the way to a dark ages radio survey. We have made BAO forecasts based on the expected performance of the array. We assume the seven-beam array has a 700-850 MHz frequency coverage with a total system temperature of 33K. The BAO forecasts are consistent with predictions in Masui et al [2010] and are in reasonable agreement with those of Bull et al [2015]. We consider three scenarios with different observing depth and sky coverage: 500 or 1000 hours of on-sky GBT observations, covering 100 or 1000 deg 2 of sky areas. The expected errors on the BAO wiggles and the fractional distance constraints are shown in Fig. 10. We anticipate to yield a 3.5% error on the BAO distance at z ∼ 0.8 with 1000 hours of GBT observing time. The bottom panel of Fig. 10 also shows recent constraints from WiggleZ [Drinkwater et al, 2010] and the BOSS surveys [Anderson et al, 2014] at lower redshifts, and forecasts for CHIME [Bandura et al, 2014] and WFIRST [Spergel et al, 2015]. With the demonstrated results and good understanding of systematic effects at the GBT, and with very different astrophysical and measurement systematics from optical/IR spectroscopic redshift surveys, we anticipate the GBT-HIM array can make a firm detection of the BAO signature at z∼0.8 with the HI intensity mapping technique, and contribute to the future of large-scale structure surveys and the field of 21-cm cosmology. Other on-going experiments such as CHIME [Bandura et al, 2014] and HIRAX [Newburgh et al, 2016] will reach z = 0.8 − 2.5 and probe even larger cosmological volume. The expected fractional error on the BAO distance scale of the three scenarios. We anticipate a 3.5% error on the BAO distance at z ∼ 0.8 with 1000 hours of GBT observing time with the array. A detection of BAO will validate HI intensity mapping as a viable tool for large-scale structure and cosmology, and serve as a systematic check and alternative approach to the optical spectroscopic redshift surveys. Fig. 1 1From top to bottom: example of light curves produced and exploited by the COSMOGRAIL and H0LiCOW programs, here for the quadruply imaged quasar HE 0435−1223. The original light curves are shown on the top. The second panel shows spline fitting to the data including the intrinsic and extrinsic quasar variations. Crucially, long light curves are needed to extract properly the extrinsic variation (microlensing). The residuals to the fit and the journal of the observations with 5 instruments are displayed in the two lower panels [reproduced with permission fromBonvin et al, 2017]. Fig. 3 3Illustration of lens mass modeling of the gravitational lens RXJ1131−1231. Top left is the observed HST image. Top middle panel is the modeled surface brightness of the lens system, which is composed of three components shown in the second row: lensed AGN images (left), lensed AGN host galaxy (middle), and foreground lens galaxies (right). The bottom row shows that a mass model is required together with the AGN source position and AGN host galaxy surface brightness, to model the lensed AGN and lensed AGN host images. See the text andSuyu et al [2013Suyu et al [ , 2014 for more details. Fig. 4 4Left: Latest H 0 measurement from quasar time delays from H0LiCOW and COSMO-GRAIL for 3 lenses and for their combination in a Λ CDM Universe [reproduced with permission from Bonvin et al, 2017]. Right: comparison between time delay H 0 measurements and other methods such as CMB shown in yellow [Planck; Planck Collaboration et al, 2016b] and gray [WMAP;Bennett et al, 2013] or local distance estimators such as Cepheids [green;Freedman et al, 2012] Fig. 5 ( 5Left) Angular power spectrum of CMB anisotropies measured from the latest Planck satellite data ( c ESA and the Planck Collaboration). The wiggles seen in the spectrum are the feature of BAO, and the oscillation scale corresponds to the sound horizon at recombination. The best-fitting Λ CDM theoretical spectrum is plotted as the solid line in the upper panel. Residuals of the measurement with respect to this model are shown in the lower panel. (Right) BAO feature detected in large-scale correlation function of the galaxy distribution of the SDSS [reproduced with permission from Eisenstein et al, 2005]. The bump seen at 105h −1 Mpc ( 150Mpc) corresponds to the sound horizon scale at recombination. The solid lines are the theoretical models with Ω m h 2 = 0.12 (top line), 0.13 (second line) and 0.14 (third line). The bottom line shows a pure CDM model with Ω m h 2 = 0.105, which lacks the acoustic peak. The inset zooms into the BAO peak position. Fig. 6 ( 6Left) Contour plots of the correlation function measured from the SDSS galaxy sample as functions of transverse (s ⊥ ) and line-of-sight (s ) separations (right half) and the corresponding theoretical prediction (left half) [reproduced with permission from Okumura et al, 2008]. The dashed thin black lines show ξ < −0.01 increasing logarithmically with 0.25 and −0.01 ≤ ξ < 0 linearly with 0.0025. The solid thin lines colored red show 0 ≤ ξ < 0.01 increasing linearly with 0.0025 and the solid thick ones colored red are ξ ≥ 0.01 logarithmically with 0.25. The baryonic feature slightly appears as ridge structures around the scale s = (s 2 ⊥ + s 2 ) 1/2 100h −1 Mpc ( = 150 Mpc), and the dashed circle traces the peaks of the baryon ridges. (Right) Similar to the left panel, but the correlation function from currently the largest galaxy sample from BOSS survey [reproduced with permission from Alam et al, 2016]. The correlation function is multiplied by the square of the distance, s 2 ξ , in order to emphasize the BAO feature. Fig. 7 ( 7Left) Three distance measures obtained by BAO in various galaxy surveys [reproduced with permission from Aubourg et al, 2015]. The y-axis is the ratio of each distance to r d , divided by √ z. The top-red, middle-blue and bottom-green curves correspond to the distances D M (z), D V (z) and zD H (z), respectively. (Right) Joint constraints on the angular diameter distance D A (z) and the Hubble parameter H(z) obtained from the correlation analysis of the BOSS galaxy sample at z = 0.5 [reproduced with permission fromAnderson et al, 2014]. The inner and outer contours correspond to the 68% and 95% confidence levels, respectively. The gray and orange contours are the constraints from 1D and 2D BAO analyses, respectively, while the blue and green contours are from CMB experiments (Planck and WMAP). Fig. 8 ( 8Left) Joint constraints on the matter density parameter Ω m and the Hubble constant h in a flat cosmology [reproduced with permission from Aubourg et al, 2015]. Each contour shows the 68%, 95% and 99% confidence levels from inward. Galaxy BAO constraints (red) show strong correlations between Ω m and h, whereas that of Ly-α BAO (blue) show strong anti-correlations. The combination of the two ("Combined BAO" in green) thus breaks the degeneracies, resulting in constraints located at the intersection of the two. Planck CMB constraints (black) show also anti-correlation between Ω m and h, but are substantially narrower than that of Combined BAO. (Right) Comparison of the constraints on H 0 [reproduced with permission from Aubourg et al, 2015] from the distance ladder probes (local measurements, red), the CMB anisotropies (green), and the inverse distance ladder analysis (combination of BAO and supernovae; blue). Fig. 9 9Fractional errors in the angular diameter distance and the expansion rate through the measurements of BAO [reproduced with permission fromTakada et al, 2014]. The expected accuracies are compared to the existing SDSS and BOSS surveys at z < 0.7. Each panel assumes w = −1 as the fiducial model, and when the model is changed to w = −0.9, the baseline of the fractional errors is systematically shifted from the dashed line to the solid curve. Fig. 10 10Top: Expected errors on the BAO signature at z ∼ 0.8 of the GBT-HIM array, assuming different observing depth and sky coverage: (1000 hrs, 1000 deg 2 ), (1000 hrs, 100 deg 2 ), and (500 hrs, 100 deg 2 ). BAO signatures can be detected in all three. Bottom: retero J, Crocce M, Cunha CE, D'Andrea CB, da Costa LN, Desai S, Dietrich JP, Eifler TF, Flaugher B, Fosalba P, García-Bellido J, Gaztanaga E, Gill MS, Goldstein DA, Gruen D, Gruendl RA, Gschwend J, Gutierrez G, Honscheid K, James DJ, Kuehn K, Kuropatkin N, Li TS, Lima M, Maia MAG, March M, Marshall JL, Melchior P, Menanteau F, Miquel R, Ogando RLC, Plazas AA, Romer AK, Sanchez E, Schindler R, Schubnell M, Sevilla-Noarbe I, Smith M, Smith RC, Sobreira F, Suchyta E, Swanson MEC, Tarle G, Thomas D, Walker AR (2017) Models of the strongly lensed quasar DES J0408-5354. MNRAS472:4038-4050, DOI 10.1093/mnras/stx2242, 1702.00406 Alam S, Ata M, Bailey S, Beutler F, Bizyaev D, Blazek JA, Bolton AS, Brownstein JR, Burden A, Chuang CH, Comparat J, Cuesta AJ, Dawson KS, Eisenstein DJ, for SDSS J1515+1511.Hainline et al [2013] measure a tentative time delay for SBS 0909+532, although the curves suffer from strong microlensing. Finally, two (long) time delays have been estimated for two quasars lensed by a galaxy group/cluster: SDSS J1029+2623. These data are analysed jointly with the H0LiCOW program (see Section 1.4). Other recent studies for specific objects include Giannini et al [2017] for WFI 2033−4723 and HE 0047−1756, and Shalyapin and Goicoechea [2017] reporting a delay The Fermat potential, being a potential, is defined only up to an additive constant that has no physical consequence. Furthermore, a "mass-sheet transformation" (explained later in Section 1.3.2) can also add a term that is independent of θ i to the Fermat potential. GLEE (Gravitational Lens Efficient Explorer) is a gravitational lens modeling software developed by A. Halkola and S. H. Suyu[Suyu and Halkola, 2010, Suyu et al, 2012] in terms of impact on D ∆t Suyu, Chang, Courbin & Okumura Probing dark energy with baryonic oscillations and future radio surveys of neutral hydrogen. 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{'abstract': 'We review three distance measurement techniques beyond the local universe: (1) gravitational lens time delays, (2) baryon acoustic oscillation (BAO), and (3) HI intensity mapping. We describe the principles and theory behind each method, the ingredients needed for measuring such distances, the current observational results, and future prospects. Time-delays from strongly lensed quasars currently provide constraints on H 0 with < 4% uncertainty, and with 1% within reach from ongoing surveys and efforts. Recent exciting discoveries of strongly lensed supernovae hold great promise for time-delay cosmography. BAO features have been detected in redshift surveys up to z 0.8 with galaxies and z ∼ 2 with Ly-α forest, providing precise distance measurements and H 0 with < 2% uncertainty in flat Λ CDM.', 'arxivid': '1801.07262', 'author': ['Sherry H Suyu suyu@mpa-garching.mpg.de ', 'Tzu-Ching Chang tzu-ching.chang@jpl.nasa.gov ', 'Frédéric Courbin frederic.courbin@epfl.ch ', 'Teppei Okumura tokumura@asiaa.sinica.edu.tw ', 'Sherry H Suyu ', 'Tzu-Ching Chang ', '\nMax-Planck-Institut für Astrophysik\nInstitute of Astronomy and Astrophysics\n11F of ASMAB\nPhysik-Department\nAcademia Sinica\nKarl-Schwarzschild-Str. 1, No.1, Section 4, Roo-sevelt Road85748, 10617Garching, Germany, TaipeiTaiwan\n', '\nInstitute of Astronomy and Astrophysics, Academia Sinica\nJet Propulsion Laboratory\nTechnische Universität München\nJames-Franck-Straße 1, 4800 Oak Grove Dr, MS 169-237, 11F of ASMAB, No.1, Section 4, Roo-sevelt Road85748, 91109, 10617Garching, Pasadena, TaipeiCAGermany, USA;, Taiwan\n', "\nTeppei Okumura Institute of Astronomy and Astrophysics, Academia Sinica\nLaboratoire d'Astrophysique, Ecole Polytechnique Fédérale de Lausanne (EPFL)\nFrédéric Courbin Institute of Physics\nObservatoire de Sauverny, 11F of ASMAB, No.1, Section 4, Roo-sevelt RoadCH-1290, 10617Versoix, TaipeiSwitzerland, Taiwan\n", '\nKavli Institute for the Physics and Mathematics of the Universe (WPI)\nUTIAS\nThe University of Tokyo\n277-8583KashiwaChibaJapan\n'], 'authoraffiliation': ['Max-Planck-Institut für Astrophysik\nInstitute of Astronomy and Astrophysics\n11F of ASMAB\nPhysik-Department\nAcademia Sinica\nKarl-Schwarzschild-Str. 1, No.1, Section 4, Roo-sevelt Road85748, 10617Garching, Germany, TaipeiTaiwan', 'Institute of Astronomy and Astrophysics, Academia Sinica\nJet Propulsion Laboratory\nTechnische Universität München\nJames-Franck-Straße 1, 4800 Oak Grove Dr, MS 169-237, 11F of ASMAB, No.1, Section 4, Roo-sevelt Road85748, 91109, 10617Garching, Pasadena, TaipeiCAGermany, USA;, Taiwan', "Teppei Okumura Institute of Astronomy and Astrophysics, Academia Sinica\nLaboratoire d'Astrophysique, Ecole Polytechnique Fédérale de Lausanne (EPFL)\nFrédéric Courbin Institute of Physics\nObservatoire de Sauverny, 11F of ASMAB, No.1, Section 4, Roo-sevelt RoadCH-1290, 10617Versoix, TaipeiSwitzerland, Taiwan", 'Kavli Institute for the Physics and Mathematics of the Universe (WPI)\nUTIAS\nThe University of Tokyo\n277-8583KashiwaChibaJapan'], 'corpusid': 119040330, 'doi': '10.1007/s11214-018-0524-3', 'github_urls': [], 'n_tokens_mistral': 67747, 'n_tokens_neox': 54247, 'n_words': 26838, 'pdfsha': '5458a76f711918a82ae3ad2bc4c11dccc515362b', 'pdfurls': ['https://arxiv.org/pdf/1801.07262v2.pdf'], 'title': ['Cosmological distance indicators', 'Cosmological distance indicators'], 'venue': []}

arxiv

Duality between integrable Stäckel systems 27 Nov 1998 A V Tsiganov tsiganov@mph.phys.spbu.ru Department of Mathematical and Computational Physics Institute of Physics St.Petersburg University 198 904St.PetersburgRussia Duality between integrable Stäckel systems 27 Nov 1998 For the Stäckel family of the integrable systems a non-canonical transformation of the time variable is considered. This transformation may be associated to the ambiguity of the Abel map on the corresponding hyperelliptic curve. For some Stäckel's systems with two degrees of freedom the 2 × 2 Lax representations and the dynamical r-matrix algebras are constructed. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail.Recently, the non-canonical transformations relate the Kowalewski top with the geodesic motion on SO(4)[7]and with the Neumann system on the sphere S 2 [8].In this paper we consider the fortiori integrable systems with the known separation variables. Since we shall not discuss singularity analysis [2] and theory of algebraic completely integrable systems [3] in detail. Recall, the variable separation method permits one to reduce an integration Introduction In her celebrated papers [1] published in 1889 S.Kowalewski discovered a necessary condition for an n-dimensional system to be completely integrable. This criterion enabled her to classify all integrable solid body motion about a fixed point and introduce the separation variables for her celebrated top. Evidently, it was the first application of singularity analysis to a concrete physical problem. Recall, that the method of singularity analysis associates integrability with the Kowalewski-Painlevé property, i.e. a movable polelike singularity (t − t 0 ) −m in the solution of the equations of motion, here space and time must be thought of as complex [2,3]. There exist cases, however, of integrable hamiltonian system with rational integrals of the motion, whose analytic structure permits solutions with algebraic singularities of the type (t − t 0 ) 1/k , (k being a positive integer larger than one). This led to the introduction of the "weak" Painlevé property [2]. A simple change of the independent variable, in and of itself, does not turn a "weak" Painlevé system into one that satisfies the usual Kowalewski-Painlevé criterion. In particular, the simple-minded idea to take (t − t 0 ) 1/k as a new independent variable does not lead to a Painlevé expansion for all the solutions. According by [2], any transformation that modifies the nature of the singular expansions must also involve a change of the dependent variables in order to reestablish the Painlevé property. Such transformations, however, if they exist, are expected to be quite nontrivial and difficult to generalize to other examples. Of course, transformations, which use change of the independent time variable t, are noncanonical transformations. The classical example of the such non-canonical transformations is the duality of the Kepler problem to the geodesic motion on the sphere [4] or to the harmonic oscillator [5]. Another important to us example is the Kolosoff transformations for the Kowalewski top [6]. In this case the Kowalewski separation variables coincide to the standard elliptic coordinates {q 1 , q 2 } at the plane (x, y) after the non-canonical change of the time d t = q 1 (t) + q 2 (t) dt . (1.1) problem with several degrees of freedom to a sequence of one-dimensional integration problems. The inverse problem of obtaining various classes of completely integrable systems starting from a set of separated one-dimensional problems was started in the lectures by Jacobi. In framework of this approach, Liouville and Stäckel introduced a family of the simple systems integrable in quadratures (a Liouville family is a particular case of a Stäckel family). Here for the Stäckel system we introduce non-canonical transformations of the time variable associated to the ambiguity of the Abel map on the hyperelliptic curve. All these transformations depend on coordinates only and, therefore, they are closed to the Kolosoff [6] change of the time (1.1). For the some Stäckel systems we propose the Lax pairs and r-matrix algebras. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail. Duality between the Stäckel systems Before proceeding father it is useful to recall the classical work of Stäckel [9]. The system associated with the name of Stäckel [9] is a holonomic system on the phase space R 2n equipped with the canonical variables {p j , q j } n j=1 , with the standard symplectic structure Ω n and with the following Poisson brackets Ω n = n j=1 dp j ∧ dq j , {p j , q k } = δ jk . (2.1) The nondegenerate n × n Stäckel matrix S, whose j column s kj depends only on q j det S = 0 , ∂s kj ∂q m = 0 , j = m defines n functionally independent integrals of motion I k = n j=1 c jk p 2 j + U j , c jk = S kj det S . (2.2) which are quadratic in momenta. Here C = [c ik ] denotes inverse matrix to S and S kj be cofactor of the element s kj . Each integral I k (2.2) may be associated to the time variable t k , such that for any function ξ(p, q) one gets dξ(p, q) dt k = {I k , ξ(p, q)} . By definition the first integral I 1 = H be the Hamilton function associated to the time t. The common level surface of the integrals (2.2) M α = z ∈ R 2n : I k (z) = α k , k = 1, . . . , n (2.3) is diffeomorphic to the n-dimensional real torus and one immediately gets p 2 j = ∂S ∂q j 2 = n k=1 α k s kj (q j ) − U j (q j ) ,(2.4) where S(q 1 . . . , q n ) is a reduced action function [10]. If this real torus is a part of complex algebraic torus, then the corresponding mechanical system is called an algebraic completely integrable system [3]. S(q 1 . . . , q n ) = n j=1 S j (q j ) , S j (q j ) = F j (q j ) dq j . (2.6) Here the functions F j (λ) depend on n parameters {α k } n k=1 F j (λ) = n k=1 α k s kj (λ) − U j (λ) . Coordinates q j (t, α 1 , . . . , α n ) are determined from the equation explicitly depending on time n j=1 γj (pj ,qj ) γ0(p0,q0) s 1j (λ)dλ n k=1 α k s 1j (λ) − U j (λ) = β 1 = t ,(2.7) and from other n − 1 equations n j=1 γj (pj ,qj ) γ0(p0,q0) s kj (λ)dλ n k=1 α k s kj (λ) − U j (λ) = β k , k = 2, . . . , n . (2.8) The solutions of the problem is thus reduced to solving a sequence of one-dimensional problems, which is the essence of the method of separation of variables. Now we turn to the non-canonical change of the time and prove the following H = v(q) H , v(q) = det S(q 1 , . . . , q n ) det S(q 1 , . . . , q n ) , (2.9) are related by non-canonical change of the time. In fact, the corresponding Hamilton functions H and H obey to the equation (2.9), which follows from the definitions of the hamiltonians H = n j=1 c j1 p 2 j + U j (q j ) (2.10) and entries of the inverse matrix c j1 = S 1j det S = 1 det S ∂ det S ∂s 1j . Equation (2.9) defines an implicit change of the time t → t associated to the integrals H = I 1 and H = I 1 , respectively. On the other hand the equation (2.7) may be considered as an explicit determination of this transformation t → t. In contrast with the general coupling constant metamorphosis discussed in [2] equation (2.9) is independent on the any constant entering in the potential U . Obviously, by using row by row transformations of the Stäckel matrices with the associated t k → t k transformations we can reduce the given Stäckel system to any other Stäckel system on R 2n . As an example, let us consider three matrices S = 1 1 1 −1 , S = q 1 q 2 1 −1 , S = q 2 1 q 2 2 1 −1 . (2.11) The corresponding hamiltonians H, H and H defined by (2.10) are dual (2.9) H = (q 1 + q 2 ) −1 2 H , (2.12) H = (q 2 1 + q 2 2 ) −1 2 H = q 1 + q 2 q 2 1 + q 2 2 H , for any potentials U . For the hamiltonians H and H the change of the time (2.12) is closed to the Kolosoff transformation (1.1) [6] and for any function ξ(q) depending on coordinates only one gets d ξ(q) d t = { H, ξ(q)} = (q 1 + q 2 ) −1 2 {H, ξ(q)} = 1 2 (q 1 + q 2 ) d ξ(q) d t . (2.13) For instance, let us consider uniform cubic potential U (q j ) = 2α 2 q 3 j + β q 2 j + γ q j + δ , (2.14) which gives rise to the hamiltonian H H = 1 4 (p 2 1 + p 2 2 ) + α 2 (q 3 1 + q 3 2 ) + β 2 (q 2 1 + q 2 2 ) + γ 2 (q 1 + q 2 ) + δ . (2.15) Using canonical transformation q 1 = x + y 2 , p 1 = p x + p y , q 2 = x − y 2 , p 2 = p x − p y , for the first system, the more complicated transformation q 1 = 3 4 x 2/3 + p y 3 α , p 1 = p x x 1/3 − 3 α 2 y , q 2 = 3 4 x 2/3 − p y 3 α , p 2 = p x x 1/3 + 3 α 2 y , for the system associated to S and the following change of variables in the third case q 1 = √ x − √ y , p 1 = p x √ x − p y √ y , q 2 = −i ( √ x + √ y) , p 2 = i (p x √ x + p y √ x) , one gets the Hamilton functions in the natural form H = 1 2 (p 2 x + p 2 y ) + α 2 4 x (x 2 + 3 y 2 ) + β 4 (x 2 + y 2 ) + γ 2 x + δ , H = 1 2 (p 2 x + p 2 y ) + 9 α 2 8 x −2/3 ( 3 4 x 2 + y 2 ) + δ x −2/3 + 3γ 4 , only by β = 0 , H = 1 2 p x p y − β 2 1 √ x y + γ 4 1 + i √ x − 1 − i √ y + δ , by α = 0 . under restriction β = 0 for the second case. The system with the first Hamiltonian H is so-called first integrable case of the Henon-Heiles system [2]. The second Hamiltonian H is related to so-called Holt potential [2]. Note, the second integral of motion is the polynomial of the third order in momenta for the Holt system. The system with the third Hamiltonian H may be considered as an integrable deformation of the Kepler problem. Duality between the Henon-Heiles and the Holt systems with the Hamiltonians H and H may be considered as the known coupling constant metamorphosis with respect to the constant γ [2]. The second known duality between the harmonic oscillator and the Kepler problem with the Hamiltonians H by α = 0 , γ = 0 and H may be considered as the coupling constant metamorphosis with respect to another constant δ [2,5]. We can see that in practical circumstances the Stäckel approach is not very useful because it is usually unknown which canonical transformation have to be used in order to transform a Hamiltonian (2.10) to the natural form H = T + V [5]. This problem was partially solved for the uniform systems U j = U, j = 1, . . . , n with polynomial potentials by using the corresponding Lax pairs [11]. Note, that the movable branch points of the type (t − t 0 ) 1/k appear in the expansions of the physical variables (x, y) after canonical transformations. Henceforth, we shall restrict our attention to the uniform Stäckel systems, where all the polynomial potentials U j (q j ) = U (q j ) and associated hyperelliptic curves C j (2.4) are equal. Duality and Abel map. Let us briefly recall some necessary facts about the Abel map and the inverse Jacobi problem. The set of point C(z, λ) satisfying C : z 2 = F (λ) = 2g+1 k=0 e k λ k = 2g+1 j=1 (λ − λ j ) , (3.1) is a model of a plane hyperelliptic curve of genus g. Here F (λ) is polynomial without multiple zeros. Let us denote by Div(C) the Abelian divisor group and denote by J(C) the Jacobian of the curve C. The Abel map puts into correspondence the point D ∈ Div(C) and the point u ∈ J(C) [12,13] U : Div(C) → J(C) , (3.2) The set of all effective divisors D = γ 1 + · · · + γ n (the γ ′ j s may be not mutually distinct) of deg n of C is called the nth symmetric product of C, and is denoted by C (n) = S n C. The C (n) can be identified with the set of all unordered n-tuples {γ 1 , . . . , γ n }, where γ j are arbitrary elements of C. Now consider restriction of the Abel map (3.2) to C (n) U : C (n) → J(C) , (3.3) where U(γ 1 , γ 2 , . . . , γ g ) = U(γ 1 ) + U(γ 2 ) + · · · + U(γ g ) . According to the Abel-Jacobi theorem this map is surjective and generically injective if n = g only [12,13]. If n = g the Abel map is either lack of uniqueness or degenerate. The corresponding Stäckel system either has a dual system associated with the same curve or it is a superintegrable system [11]. Suppose that point D = γ 1 +· · ·+γ k , k ≤ g belongs to C (k) . The differential of the Abel-Jacobi map (3.3) at the point D is a linear mapping from the tangent space T D (C (n) ) of C (n) at the point D into the tangent space T U (D) (J(C)) of J(C) at the point U(D) U * D : T D (C (n) ) → T U (D) (J(C)) . Now suppose that D is a generic divisor, and x j is a local coordinate on C near the point γ j . Then (x 1 , . . . , x n ) yields a local coordinate system near the point D in C (n) . Let dw k (k = 1, . . . , g) is a basis for a space H 1 (C) of holomorphic differentials on C, and near γ j dw k = φ kj (x j )dx j , (3.4) where φ kj (x j ) is holomorphic. It follows that the Abel-Jacobi map U can be expressed near D as U(z 1 , . . . , z n ) =   n j=1 xj γ0 φ 1j (x j )dx j , . . . , n j=1 xj γ0 φ gj (x j )dx j   . Hence U * D =    φ 11 (γ 1 ) · · · φ g1 (γ 1 ) . . . . . . . . . φ 1k (γ n ) · · · φ gn (γ n )    . (3.5) is the so-called Brill-Noether matrix [14]. Henceforth, we shall restrict our attention to the special divisors D s , such that coefficients in the expansion (3.4) are independent on the point γ j dw k = φ k (x j ) dx j . In this case all rows of the symmetric Brill-Noether matrix depend on local coordinate {x 1 , . . . , x n } identically. The Jacobi inversion problem (2.8)) is formulated as follows: for a given point u = (β 1 , β 2 , . . . , β n ) ∈ J(C) find n points γ 1 , γ 2 , . . . , γ n on the genus g Riemann surface C such that g k=1 γ k γ0 dw j = β j , j = 1, . . . , n. (3.6) Here we shall tacitly assume that the base point γ 0 ∈ C has already been fixed [12]. If n = g for almost all points u ∈ J(C) the solution D = γ 1 + · · · + γ n exist and is uniquely determined by system (3.6) (for the unordered set of points γ j ) [12]. However, if the degree n < g of the symmetric product C (n) is less than genus g of C, the Abel map is lack of uniqueness. In this case we can propose that two different points u, u ∈ J(C) have the one Abel preimage {γ 1 , . . . , γ n } ∈ C (n) . The Abel preimage of the point u ∈ J(C) is given by set {(p 1 , q 1 ), . . . , (p n , q n )} ∈ C (n) , where {q 1 , . . . , q n } are zeros of the Bolza equation [15,13] e(λ, u) = λ n − λ n−1 ℘ n,n (u) − λ n−2 ℘ n,n−1 (u) − . . . − ℘ n,1 (u) = 0, (3.7) and {p 1 , . . . , p n } are equal to p k = − ∂ e(λ, u) ∂ β n λ=q k . (3.8) Here vector u belongs to Jacobian J(C) and ℘ k,j (u) is the Kleinian ℘-function [15,13]. Now we turn to the uniform Stäckel systems. We can regard each expression (2.4) as being defined on the genus g Riemann surface C : y 2 j = F (λ) , F (λ) = n k=1 α k s kj (λ) − U (λ) ,(3.C (n) : C(p 1 , q 1 ) × C(p 2 , q 2 ) × · · · × C(p n , q n ) .dw k = s kj (λ) dλ z(λ) . (3.11) The set of these differentials either form a basis in the space of holomorphic differentials H 1 (C) [12] or may be complement to a basis. The corresponding n × n Stäckel matrix be the n × n block of the transpose Brill-Noether matrix U * t D . The different blocks are determined the dual Stäckel systems. In this case vectors differing the first entry only u = {t, β 2 , . . . , β n } ∈ J(C) , u = { t, β 2 , . . . , β n } ∈ J(C) have a common Abel preimage {(p 1 , q 1 ), . . . , (p n , q n )} ∈ C (n) . Let us consider the standard basis of holomorphic differentials in H 1 (C) dw j = λ j−1 z(λ) dλ , j = 1, . . . , g . (3.12) Recall, that derivative U * D bears a great resemblance to the canonical map C → P g−1 and, therefore, to the Veronese map P 1 → P g−1 given by a basis for the polynomial ring of degree g − 1. With respect to the basis of H 1 (C) (3.12), the Veronese map of C has an extremely simple expression (y, λ) → λ → [λ g−1 , λ g−2 . . . , λ, 1] . By using the corresponding symmetric Brill-Noether matrix U * D (3.5), we shall determine the Stäckel matrices as (n × n) blocks of the following (g × n) matrix        q g−1 1 q g−1 2 · · · q g−1 n q g−2 1 q g−2 2 · · · q g−2 n . . . . . . . . . . . . 1 1 · · · 1        . (3.13) Evidently, all the Stäckel matrices can not be obtained from the symmetric Brill-Noether matrices. For instance, the Stäckel matrices (2.11) do not belongs to the set of symmetric matrices. Lax representation. Henceforth, we shall restrict our attention to the basis (3.12) and the symmetric matrix (3.13). For the corresponding Stäckel systems let us look for the Lax representation as L = h(λ, p, q) e(λ, q) f (λ, p, q) −h(λ, p, q) . (4.1) Hereafter, by abuse of notation, we shall omit the some arguments at the entries of the Lax matrix. Let us fix hyperelliptic genus g curve C and dimension of the phase space n ≤ g. Then we extract the (n × n) Stäckel matrix S from the matrix (3.13) and define the Hamilton function H (2.10) with U = 0. To construct the Lax matrix let us determine function e(λ, u) (3.7) initially e(λ, q) = n j=1 (λ − q j ) ,(4.2) with n zeroes, which are solution of the inverse Jacobi problem. In the second step let us introduce the second entry of the Lax matrix as h(λ) = − 1 2v(λ, q) d e(λ) d t + w(λ, p, q) e(λ) . Here function v(λ, q) is calculated by using the second Bolza equation (3.8) h(λ)| λ=q k = p k = 1 2v d e(λ) d t λ=q k = − ∂ e(λ) ∂ u n λ=q k . (4.3) Let the third entry of the Lax matrix takes the form f (λ) = 1 v d h(λ) dt . Here the single unknown function w(λ, p, q) is determined such, that the spectral curve of the Lax matrix (4.1) C : z 2 = F (λ) = − det L 0 (λ) = h 2 (λ) + e(λ) f (λ) (4.4) be the same as initial algebraic curve C (2.4) by U = 0. The constructed above matrix L 0 (λ) (4.1) reads as L 0 (λ) =     − 1 2v e t (λ) + w(λ, p, q) e(λ) e(λ) 1 v h t (λ) 1 2v e t (λ) − w(λ, p, q) e(λ)     ,(4.5) where e t = d e(λ) d t = { H, e(λ) } , h t = d h(λ) d t = { H, h(λ) } , obeys the Lax equation dL 0 dt = { H, L 0 } = A 0 , L 0 with the second matrix A 0 = v(λ, q) w(λ, p, q) 1 0 −w(λ, p, q) . By definition of the Lax matrix all the pairs of separation variables γ j = (p j , q j ) (4.2-4.3) lie on the spectral curve C (4.4) of the matrix L 0 (4.5) z 2 (γ j ) = p 2 j = h 2 (λ) λ=qj = F (λ = q j ) = F (λ)| γj . For the systems with polynomial potential U = 0 we propose to change the entry f (λ) in (4.5) as f (λ) = 1 v d h(λ) dt + u(λ, q)e(λ) , where we add new function u(λ, q) depending on coordinates only. Of course, to construct the Lax matrix here L(λ) =     − 1 2v e t (λ) + w(λ, p, q) e(λ) e(λ) 1 v h t (λ) + u(λ, q) e(λ) 1 2v e t (λ) + w(λ, p, q) e(λ)     . (4.6) we have to use the complete Hamiltonian with U = 0. The associated second Lax matrix reads as A = A 0 + 0 0 v(λ, q) u(λ, q) 0 = v(λ, q) w(λ, p, q) 1 u(λ, q) −w(λ, p, q) . (4.7) To consider the corresponding Lax equation dL(λ) dt = A(λ), L(λ) , we can assume that the common factor v(λ, q) in front of the matrix A may be associated to the change of the time for the Stäckel systems. In general the proof of existence functions v, w and u requires an application of the method of algebraic geometry [13]. By definition of the Lax matrices L(λ) (4.6) and A(λ) (4.7) this problem may be reduced to the solution of the single equation d f (λ) d t − 2v (u h − w f ) = 0 , ⇐⇒ dF (λ, e, v, u) d t = 0 ,(4.8) for the given function e(λ) (4.2) and the given Hamiltonian H (2.10). If we consider the lower (n × n) block of the matrix (3.13), the differentials (3.11) span a whole space H 1 (C) and the Abel map is the one-to-one correspondence. In this case from equations (3.8) and (4.3) follows that v t (λ, q) = 0 , w(λ, p, q) = 0 . If we put v = 1, rename t = x and introduce "new" time variable τ , the equation (4.8) is rewritten as ∂u(x, τ, λ) ∂τ For different choices of the form of e(λ) and u(λ), this procedure leads to different hierarchies of integrable equations, as an example to the KdV, nonlinear Shrödinger and sine-Gordon hierarchies or to the Dym hierarchy (see references within [11]). = 1 4 ∂ 3 x + u(λ)∂ x + 1 2 u x (λ) · e(λ) = 0 , x = t , Function u(λ, q) in (4.9) is constructed by using function e(λ) (3.7-4.2) u(λ, q 1 , . . . , q n ) = φ(λ)e −2 (λ) MN . (4.10) Here φ(λ) is a parametric function on spectral parameter and [ξ] N is the linear combinations of the following Taylor projections (4.11) or the Laurent projections [17,11]. If the differentials (3.11) span the whole space H 1 (C) the corresponding Stäckel systems describe all the possible systems, which separable in the orthogonal curvilinear coordinate systems in R n [11]. Let us consider the Stäckel systems which are dual to these systems. To apply equation (2.13) to the function e(λ) (4.2) and by using definition (4.3) one gets [ξ] N = +∞ k=−∞ z k λ k N ≡ N k=0 ξ k λ k ,p k = h(λ) λ=q k = − 1 2 v { H, e(λ)} λ=q k = det S det S − 1 2 v {H, e(λ)} λ=q k (4.12) = det S det S v v h(λ) λ=q k = p k det S det S v v λ=q k Recall that v = 1 for the integrable system with the Hamiltonian H associated to the lower (n × n) block of the matrix (3.13). Thus, according to (4.12), below we shall consider the Stäckel systems with following functions v(q) only v(q) = det S(q 1 , . . . , q n ) det S(q 1 , . . . , q n ) . which was proposed in the theory of the soliton equations [16]. It allows us to rewrite generating function of integrals of motion x j λ j it is easy to prove that coefficients x j obey the Newton equation of motion (4.15) (see references within [16,11]). Here we reinterpret the coefficients of the function F (λ) in (4.15) not as functions on the phase space, but rather as integration constants α j (2.3). F (λ) = −B 3 B tt + u(λ, q) B 4 . In general by v t = 0 the generating function F (λ) = − det L(λ) (4.6) is equal to F (λ) = 1 4v 2 e 2 t − 2 e e tt + v t 2v 2 − w e t e v + w 2 + u v e 2 . In this case the suitable canonical transformations, which transforms any Hamiltonian (2.10) to the natural form, are unknown. Although we can not proof validity of the presented Lax representation in general, this construction works for the many well-known mechanical systems. In the next Section we consider some two-dimensional Stäckel systems in detail. Examples. Let us consider four orthogonal systems of coordinates on plane: elliptic, parabolic, polar and cartesian [5]. The Lax matrix L 0 (λ) (4.5) by U = 0 is transformed to the Lax matrix L(λ) (4.6) by U = 0 by using the outer automorphism of the space of infinite-dimensional representations of underlying algebra sl(2) [17,11]. Since, we shall consider the Lax representations for the geodesic motion by U = 0 more extensively. 1. Parabolic and cartesian coordinate systems (w(λ, p, q) = 0). Let us consider two hyperelliptic curves C (1) : z 2 = 2g+1 i=1 (λ − λ i ) , (5.1) C (2) 2 : z 2 = λ −1 2g+1 i=1 (λ − λ i ) . If we choose the standard basis in the space of holomorphic differentials one gets the following symmetric matrices (3.13) for two-dimensional systems U * t 1 (q 1 , q 2 ) =            q g−1 1 q g−1 2 . . . . . . q 2 1 q 2 2 q 1 q 2 −1 −1            , U * t 2 (q 1 , q 2 ) =              q g−2 1 q g−2 2 . . . . . . q 1 q 2 1 1 − 1 q 1 − 1 q 2              (5.2) Different (2 × 2) blocks of the matrices U * t j determine different Stäckel systems. Let us consider two blocks for the each matrices, such that the corresponding change of the time will be same as the Kolosoff transformation (1.1) [6]. So, for the curve C (1) we shall consider the following matrices S 1 = q 1 q 2 −1 −1 , S 1 = q 2 1 q 2 2 −1 −1 . (5.3) For the second curve C (2) the associated Stäckel matrices are equal to S 2 =    1 1 − 1 q 1 − 1 q 2    , S 2 =    q 1 q 2 − 1 q 1 − 1 q 2    . (5.4) Introduce the Hamilton functions (2.10) by U = 0 H (1) 0 = p 2 1 − p 2 2 q 1 − q 2 , H(1)0 = (q 1 + q 2 ) −1 H (1) 0 , (5.5) H (2) 0 = q 1 p 2 1 − q 2 p 2 2 q 1 − q 2 , H(2)0 = (q 1 + q 2 ) −1 H(2) 0 . The corresponding second integrals of motion of the dual systems are related J (k) 0 = J (k) 0 − q 1 q 2 q 1 + q 2 H (k) 0 , k = 1, 2 . The functions e(λ, u) (3.7) e 1 (λ) = (λ − q 1 )(λ − q 2 ) , e 2 (λ) = (λ − q 1 )(λ − q 2 ) λ . (5.6) have two zeroes, which are solution of the inverse Jacobi problem (2.8) on C (1) and C (2) , respectively. Let us introduce the new physical variables at once. For the first curve C (1) equation (4.15) e 1 (λ) = (λ − q 1 )(λ − q 2 ) = B 2 (λ) , B(λ) = λ − x 2 − y 4λ , immediately yields the following canonical transformation q 1 = x − √ 2 y 2 , p 1 = p x − 2 y p y , q 2 = x + √ 2 y 2 , p 2 = p x + 2 y p y . These variables obviously related to the cartesian coordinate system. For the second curve C (2) the corresponding equation e 2 (λ) = λ −1 (λ − q 1 )(λ − q 2 ) = λ − x − y 2 4λ defines the standard parabolic coordinate system q 1 = x − x 2 + y 2 2 , p 1 = p x − x 2 + y 2 + x y p y , q 2 = x + x 2 + y 2 2 , p 2 = p x + x 2 + y 2 − x y p y . By U = 0 the Hamilton functions are given by H (1) 0 = 4 p x p y , H(2)0 = p 2 x + p 2 y . According to (4.3) and (4.12) functions v(q 1 , q 2 ) entering in the Lax representation are equal to v(λ, q 1 , q 2 ) = 1 for matrices S 1,2 , (5.7) v(λ, q 1 , q 2 ) = (q 1 + q 2 ) −1 = 1 x for matrices S 1,2 . In physical variables the Lax matrices are given by L (1) 0 =    p x + (2λ − x) p y λ 2 − λ x + x 2 − 2 y 4 −4 p 2 y −p x − (2λ − x) p y    , (5.8) L (2) 0 =     p x + 1 2λ y p y λ − x − 1 4λ y 2 1 λ p 2 y −p x − 1 2λ y p y     . For the dual Stäckel systems the Lax matrices L f t = { H, f + H} = f t = 0 . Another consequence of this property is that the function w(λ, p, q) in (4.5) is equal to zero. Note, in the works [7] and [8], devoted to the Kowalewski top, the common Lax matrices are proposed for the both dual systems after the non-canonical change of variables. Here we obtain different Lax matrices for the systems connected by non-canonical change of the time. The spectral curves of the matrices L 0 (5.8) coincides with the initial curves C the Abel map is one-to-one correspondence on the curve (5.10), then for the same system on the curve z 2 = e 2 λ 2 + e 1 λ + e 0 the associated Abel map is lack of uniqueness in general. So, on this curve we can introduce the second Stäckel system with the dual Hamiltonian H (1) 0 . Let us briefly consider systems with polynomial potentials U =). As an example, introduce different potentials for the curves C (1,2) (5.1) U (1) (q j ) = α 2 q 5 j + β q 3 j , U (1) (q j ) = α 2 q 3 j + β q j . (5.12) To describe these potentials we have to put N = 6 and N = 4 in (4.11) and have to use the following parametric functions φ (1) (λ) = −α 2 λ 5 and φ (2) (λ) = −α 2 λ 3 for the curves C (1) and C (2) , respectively. For the both curves the common function u(λ, q 1 , q 2 ) is given by u (1,2) = −α 2 (λ + 2 x) . (5.13) Here we restrict ourselves the presentation of the function u only, the complete Lax matrices L(λ) may be constructed by the rule (4.6). The spectral curves of the corresponding matrices (4.6) coincides with the initial curves (3.9). For instance, curves for the systems with dual Hamiltonians H (1,2) are C (1) : z 2 = α 2 λ 5 + β λ 3 − H λ − J , C (2) : z 2 = α 2 λ 3 − H λ + β − J λ . The Poisson bracket relations for the Lax matrix (5.8-5.9) are closed into the following linear r-matrix algebra and Π is the permutation operator of auxiliary spaces [18]. By v t = 0 for the systems related to the matrices S 1,2 the corresponding r-matrices r ij (λ, µ) in (5.14) consist of two terms r ij = r p ij + r u ij . (5.15) The first matrix is a standard r-matrix on the loop algebra L(sl(2)) r p 12 (λ, µ) = Π λ − µ = 1 λ − µ     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     . (5.16) The second matrix may be associated to outer automorphism of the space of infinite-dimensional representations of underlying algebra sl(2) [17,11]. The corresponding dynamical r u ij -matrices depend on the coordinates only r u 12 = u(λ, q) − u(µ, q) λ − µ σ − ⊗ σ − , σ − = 0 0 1 0 . (5.17) By v t = 0 for the dual Stäckel systems related to the matrices S 1,2 we have to add to the r-matrices 5.15 the third term r v 12 = v(q 1 , q 2 )     0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     ,(5.18) where the second matrix r v 21 is defined by (5.14). The matrix r v ij may be connected with the Drinfeld twist for the Toda lattice associated to the root system D n . Let us consider the Drinfeld twist [19] of the quantum R-matrix R = F R F −1 21 , F 21 = Π F Π . (5.19) Here matrix R satisfies the Yang-Baxter equation and matrix F has the special property [19]. To introduce the corresponding linear r-matrix [20], one gets R = I + 2η r p + O(η 2 ) , F = I + η r v + O(η 2 ) . Then we consider limit of the twisted matrix R by η → 0 R 12 = I + η r p 12 + r v 12 − Π ( r p 12 + r v 12 ) Π + O(η 2 ) . (5.20) Formally, coefficients by η may be called twisted linear r-matrix. By using generators h, e, f of the underlying Lie algebra sl (2) [h, e] = 2e , [h, f] = −2f , [e, f] = h ,(5.21) let us introduce an appropriate element F ∈ U (sl(2)) ⊗ U (sl (2)) F ξ = exp(ξ · e ⊗ f) , ξ ∈ C belonging to a tensor product of the corresponding universal enveloping algebras U (sl (2)) [19]. In the fundamental spin-1/2 representation ρ 1 2 we have F (ξ) = (ρ 1 2 ⊗ ρ 1 2 )F ξ =     1 0 0 0 0 1 0 0 0 ξ 1 0 0 0 0 1     . To substitute in (5.19) the Yang solution of the Yang-Baxter equation R = I + η λ Π we get a twisted R-matrix. If the element ξ(q) be a suitable function on coordinates, this dynamical twisted Rmatrix may be used to description of the Toda lattice associated with the D n root system [21]. Let us consider twisted dynamical matrix (5.19) by ξ = v(q). We can see that the linear r-matrix associated to the dual Stäckel system (5.16-5.18) r 12 = r p 12 + r v 12 , r 21 = −Π (r p 12 + r v 12 ) Π The corresponding equation for the polar coordinates e(λ) = q 1 (λ − q 2 ) λ (λ − 1) = x 2 4 λ + 4 y 2 λ − 1 immediately yields q 1 = r = x 2 + y 2 , q 2 = cos 2 (φ) = x 2 x 2 + y 2 . In physical variables the Hamiltonians (5.28-5.29) have a common form H = p 2 x + p 2 y , H = p 2 x + p 2 y x 2 + y 2 . To construct the Lax representations we begin with the calculations of the functions v(λ, q) by the rule So, for the Stäckel systems associated with the matrices S 3 (5.25) and S 4 (5.26) one gets L 0 (λ) =       x p x 2 (λ − k) + y p y 2 (λ + k) ǫλ − x 2 4 (λ − k) − y 2 4 (λ + k) p 2 x λ − k + p 2 y λ + k − x p x 2(λ − k) − y p y 2(λ + k)       . (5.31) Here ǫ = 1 for the elliptic coordinate system and ǫ = 0 for the parabolic coordinate system. The spectral curve of the Lax matrix L 0 (λ) coincides to the initial curve (5.27). For the dual system, in contrast to the cartesian and parabolic coordinates, the Lax matrix has the more complicated form. Both these Lax matrices may be constructed by the rule (4.6) with the following common function w(p, q) w(p, q) = 2 H . Here e(λ) and h(λ) are entries of the corresponding matrices L 0 (λ) (5.31) by ǫ = 0, 1. As above, the spectral curve of the Lax matrix L 0 (λ) by ǫ = 1 coincides with the initial curve (5.27). For the cartesian and parabolic coordinate systems we can get equation Hence, from the equation (4.8) follows that the function w(p, q) in (4.5-5.32) does not equal to zero. If we consider more complicated change of the time for the cartesian and parabolic coordinate systems, one gets non-zero function w (4.5) as well. The Poisson bracket relations for the Lax matrix L 0 (λ) (5.31) are closed into the standard linear r-matrix algebra (5.14) with rational r-matrix (5.16) on the loop algebra L(sl (2)) [18]. The Poisson brackets relations for the dual Lax matrix L 0 (λ) (5.33) have a poly-linear form on the values α k of integrals of motion. For the Stäckel systems on R 2n the minimum admissible genus g of the curve C is equal to g = [(n − 1)/2].The nth symmetric product of C defines the n-dimensional Lagrangian submanifold in the complete symplectic manifold R 2n the integration problem (2.7-2.8) for equation of motion is reduced to inverse Jacobi problem (3.3) on Lagrangian submanifold (3.10). The corresponding holomorphic differentials δw k are equal to may be identified with equation on the finite-band stationary solutions ∂u(x,τ,λ) ∂τ = 0 of the nonlinear soliton equations. In this theory equation (4.9) is called the generating equation. The corresponding change of the time (2.13) depending on coordinates only is closed to the Kolosoff transformation (1.1)[6].Let us briefly discuss canonical transformation which transforms a Hamiltonian (2.10) to the natural form H = T + V . For integrable systems separable in the orthogonal curvilinear coordinate systems on R n the Abel map is one-to-one correspondence and v t = {H, v} = 0. In this case we can put v = 1 and introduce function B(λ) B 2 (λ) = e(λ) ,(4.13) Newton equation for the function B B(λ, q) = −F (λ, α 1 , . . . , α n ) B −3 (λ, q) + u(λ, q) B(λ, q) . By using property {h t (λ), v(q)} = 0 of the function v(q) (5.7) we can easy proof equation (4.8) for the dual systems by using the same equation for the system with v t = 0 (λ, µ) = −Π r 21 (λ, µ) Π .Here the standard notations are introduced:1 L(λ) = L(λ) ⊗ I , 2 L(µ) = I ⊗ L(µ) , H, v − 1 1(q) , e(λ, q) = H, (q 1 + q 2 ) , e(λ, q) = 2 , on the Hamiltonian H, function e(λ) and function v(q) defining change of the time. For the polar and elliptic coordinate systems the corresponding equation is H, v −1 (q) , e(λ, q) = 8 e(λ) − ǫ , ǫ = 0, 1 . The corresponding Hamilton-Jacobi equation on M α∂S ∂t + H(t, ∂S ∂q 1 , . . . , ∂S ∂q n , q 1 , . . . , q n ) = 0 , ⇒ c j1 ∂S ∂q j ∂S ∂q j = E , (2.5) admits the variable separation Proposition 1 If the two Stäckel matrices S and S be distinguished the first row onlys kj = s kj , k = 1 , the corresponding Stäckel systems with the following Hamilton functions The proposed change of the time is related to ambiguity of the Abel map. For the two degree of freedom systems that were studied in this paper, we found the Lax representations and the rmatrix algebras. The corresponding dynamical r-matrices have the intriguing connections to the Drinfeld twists.Of course, considered above particular family of the time transformations (2.9) does not exhausted all the set of the non-canonical changes of the time, which preserve the integrability. As an example, the complete Kolosoff transformation {t, p, q} → { t, p, q}[6]connects the Stäckel system with the other integrable non-Stäckel system. So, it would be interesting to investigate another integrable systems connected with the Stäckel systems by non-canonical transformations.This work was partially supported by RFBR grant.is equal to the half of the twisted linear matrix(5.20).Recall, for the Stäckel matrices S 1 (5.3) and S 2 (5.4) the corresponding differentials (3.11) span H 1 . Since, the associated Hamilton functions have a natural form in physical variables. For instance, Hamiltonians with potentials (5.12) are given by H (1) = 2 p x p y + α 2 4 (y 2 + 5 x 2 y + 5 4x 4 ) + β 2 ( 3 2x 2 + y) ,(5.22)To consider the dual Stäckel systems we have to use additional transformationfor the first curve and the following more complicated transformationfor the second curve. After this canonical change of variables the Hamiltonians H (1,2) (5.5) obtain the natural formThe system with the Hamiltonian H (2) is so-called second integrable case of the Henon-Heiles system[2]. The dual system with the Hamiltonian H (2) is so-called Holt-type system[2]. Note, the second integral of motion is a polynomial of the fours order in momenta for the Holt system. Additional canonical transformation (5.24) allows us to get natural Hamiltonians for the restricted class of the potentials U (5.12) only. Unlike canonical transformation (5.23) may be used for any potentials U . As an example, rational potentialgive rise the following HamiltonianAlso we can add potential terms (5.24) to this Hamiltonian.By v = 1 and w = 0 the Lax representation (4.5) for a system with an arbitrary number n of degrees of freedom may be regarded as a generic point at the loop algebra L(sl(2)) in fundamental representation after an appropriate completion[11]. As an example, for the generalized parabolic coordinate systems function e(λ) is given byTo construct the Lax representation for a potential motion we can use the outer automorphism of the space of infinite-dimensional representations of sl(2) proposed in[17].By v t = 0 for the dual Stäckel systems the Lax representations may be constructed without any problem as well. For instance, let us consider system with the three degrees of freedom. To construct the Lax matrix by (4.5-4.6) with the function u given by (5.13) one getsAfter an additional canonical transformation (5.24) extended on the p z , z variables the Hamilton function takes the formSo, the main unsolved problem is to introduce additional canonical transformation, which transform the dual Hamilton function H into the natural form.2. Elliptic and polar coordinates (w(p, q) = 0).Recall, that the polar coordinate system may be obtained from elliptic coordinate system and, therefore, we shall consider elliptic coordinate systems in detail. For the elliptic coordinate systems algebraic curve is given byLet us consider two Stäckel matrices associated to this curveThe corresponding non-canonical change of the time (2.9) is closed to the Kolosoff transformation (1.1)[6]. For the polar coordinate system the Staäckel matrices are non-symmetric matricesThe corresponding non-canonical change of the time (2.9) is closed to the Kepler change of the time[5]. By U = 0 the initial hyperelliptic curves (3.9) for the matrices S 3 and S 3 are given byThe Hamiltonians related to the matrices S 4 and S 4 read asLet us fix elliptic coordinates by using equationsuch thatHere linear r-matrix reads aswhere r p (λ−µ) be the standard linear r-matrix on the loop algebra L(sl(2)). The second dynamical term is given byThe quadratic R-matrix is closed to the twisted linear r-matrix (5.20)For the systems with U = 0 functions u(λ, q) may be constructed as usual[17,11]. Note, the both dual Hamiltonians obtain a natural form after the following additional canonical transformation of variablesAs an example, for elliptic coordinate system the uniform potential U (3) (q j ) = α q 2 j + β q j give rise to the following dual Hamiltonian H = 2 p x p y + α 4 ( x + k) ( y + k) − β 8 x y (2 x y + k x + k y) .For the polar coordinate system we present the non-uniform degenerate potentials U (4) 1 (q 1 ) = β , U2 (q 2 ) = 0 , associated to the dual Hamiltonians in the formBoth these systems may be considered as an integrable deformation of the Kepler problem.ConclusionsIn this paper we have considered the non-canonical relations between the different Stäckel systems. . S Kowalewski, Acta Math. 1214S. Kowalewski. Acta Math., 12 and 14, 177-232 and 81-93, 1889. . A Ramani, B Grammaticos, T Bountis, Phys.Rep. 180A. Ramani, B. Grammaticos, and T. Bountis. Phys.Rep., 180, 159-245, 1989. . M Adler, P Van Moerbeke, Invent. Math. 97M. Adler and P. van Moerbeke. Invent. Math., 97, 3-51, 1989. . J Moser, Comm. Pure Appl. Math. 23609J. Moser. Comm. Pure Appl. Math., 23, 609, 1970. Classical Mechanics. H Goldstein, Addison-WesleyReading, MA, New YorkH. Goldstein. Classical Mechanics. Addison-Wesley, Reading, MA, New York, 1980. . G Kolosoff, Math. Ann. 56G. Kolosoff. Math. Ann., 56, 265-272, 1903. . M Adler, P Van Moerbeke, Commun. Math. Phys. 113M. Adler and P. van Moerbeke. Commun. Math. Phys., 113, 659-700, 1988. . L Heine, E Horozov, Physica D. 29L. Heine and E. Horozov. Physica D, 29, 173-185, 1987. Comptes Rendus, 116 and 121, 485,1284 and 489. P Stäckel, P. Stäckel. Comptes Rendus, 116 and 121, 485,1284 and 489, 1893 and 1895. Mathematical methods of classical mechanics. V I Arnold, Springer2nd. editionV.I. Arnold. Mathematical methods of classical mechanics. Springer, 1989. 2nd. edition. The Stäckel systems and algebraic curves. A V Tsiganov, solv- int/9712003J.Math.Phys to appear. A.V. Tsiganov. The Stäckel systems and algebraic curves. J.Math.Phys to appear, solv- int/9712003, 1998. . B A Dubrovin, Russ. Math. Surveys. 36B.A. Dubrovin. Russ. Math. Surveys., 36, 11-80, 1981. Kleinian functions, hyperelliptic Jacobians and applications. V M Buchstaber, V Z Enolskii, D V Leykin, Reviews in Mathematics and Mathematical Physics. 10Gordon and BreachV.M. Buchstaber, V.Z. Enolskii, and D.V. Leykin. Kleinian functions, hyperelliptic Jacobians and applications. volume 10 of Reviews in Mathematics and Mathematical Physics, pages 1-125. Gordon and Breach, London, 1997. . M Noether, Math. Anal. 28M. Noether. Math. Anal., 28, 354-380, 1887. . O Bolza, Amer. Journ. Math. 17O. Bolza. Amer. Journ. Math., 17, 11-36, 1895. . S Rauch-Wojciechowski, Phys.Lett.A. 170S. Rauch-Wojciechowski. Phys.Lett.A. 170, 91-94 , 1988. . A V Tsiganov, J.Math.Phys. 39A.V. Tsiganov. J.Math.Phys., 39, 650-664, 1998. Hamiltonian methods in the theory of solitons. L D Faddeev, L A Takhtajan, SpringerBerlinL.D. Faddeev and L.A. Takhtajan. Hamiltonian methods in the theory of solitons. Springer, Berlin, 1987. S Khoroshkin, A Stolin, V Tolstoy, From Field Theory to Quantum Groups. B. Jancewicz and J. SobczykS. Khoroshkin, A. Stolin, and V. Tolstoy, in: From Field Theory to Quantum Groups, eds. B. Jancewicz and J. Sobczyk, WS, 1996, P.53-77. . A V Tsiganov, J.Phys.A. 27A.V. Tsiganov. J.Phys.A, 27, 6759-6780, 1994. . A V Tsiganov, J.Phys.A. 31A.V. Tsiganov. J.Phys.A., 31, 8049-8061, 1998.

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{'abstract': "For the Stäckel family of the integrable systems a non-canonical transformation of the time variable is considered. This transformation may be associated to the ambiguity of the Abel map on the corresponding hyperelliptic curve. For some Stäckel's systems with two degrees of freedom the 2 × 2 Lax representations and the dynamical r-matrix algebras are constructed. As an examples the Henon-Heiles systems, integrable Holt potentials and the integrable deformations of the Kepler problem are discussed in detail.Recently, the non-canonical transformations relate the Kowalewski top with the geodesic motion on SO(4)[7]and with the Neumann system on the sphere S 2 [8].In this paper we consider the fortiori integrable systems with the known separation variables. Since we shall not discuss singularity analysis [2] and theory of algebraic completely integrable systems [3] in detail. Recall, the variable separation method permits one to reduce an integration", 'arxivid': 'solv-int/9812001', 'author': ['A V Tsiganov tsiganov@mph.phys.spbu.ru \nDepartment of Mathematical and Computational Physics\nInstitute of Physics\nSt.Petersburg University\n198 904St.PetersburgRussia\n'], 'authoraffiliation': ['Department of Mathematical and Computational Physics\nInstitute of Physics\nSt.Petersburg University\n198 904St.PetersburgRussia'], 'corpusid': 16939762, 'doi': '10.1088/0305-4470/32/45/311', 'github_urls': [], 'n_tokens_mistral': 15515, 'n_tokens_neox': 13690, 'n_words': 8043, 'pdfsha': '3963f8303bcf7d1323154e3b35729ffc779299f7', 'pdfurls': ['https://arxiv.org/pdf/solv-int/9812001v1.pdf'], 'title': ['Duality between integrable Stäckel systems', 'Duality between integrable Stäckel systems'], 'venue': []}

arxiv

Haldane-gap chains in a magnetic field 19 Oct 2004 Fabian H L Essler Ian Affleck Department of Physics and Astronomy University of British Columbia V6T 1Z1VancouverB.CCanada The Rudolf Peierls Centre for Theoretical Physics Oxford University OX1 3NPOxfordUK ( Haldane-gap chains in a magnetic field 19 Oct 2004 We consider quasi one dimensional spin-1 Heisenberg chains with crystal field anisotropy in a uniform magnetic field. We determine the dynamical structure factor in various limits and obtain a fairly complete qualitative picture of how it changes with the applied field. In particular, we discuss how the width of the higher energy single magnon modes depends on the field. We consider the effects of a weak interchain coupling. We discuss the relevance of our results for recent neutron scattering experiments on the quasi-1D Haldane-gap compound NDMAP. I. INTRODUCTION Recent years have seen a resurgence of interest in field-induced "magnon condensation" in gapped quasi onedimensional quantum antiferromagnets. In particular a series of ESR and neutron scattering experiments have been carried out on the Haldane-gap 1 chain compounds NDMAP 2-7 and NDMAZ 8 . The main motivation for these experimental studies is the observation that a spin-1 Heisenberg chain undergoes a quantum phase transition between a gapped spin-liquid phase and a gapless Luttinger liquid phase at some critical value H c of the applied magnetic field H 9, 10 . The ground state of the spin-1 Heisenberg chain is a spin singlet and excitations are described in terms of a gapped S = 1 triplet of magnons. When a magnetic field is applied, the triplet splits due to the Zeeman effect and one of the magnon gaps is driven to zero at H c . For H > H c the ground state is magnetized and excitations are gapless. If interactions between the magnons were absent, the transition at H c could be understood as a Bose-Einstein condensation of magnons. In the spin-1 Heisenberg chain there is an interaction between magnons, which fundamentally changes the ground state for H > H c from a condensate of bosonic magnons to a Luttinger liquid, which can be regarded as a one dimensional version of an interacting Bose condensate. The transition at H = H c is in the universality class of the commensurate-incommensurate (C-IC) phase transition 11 . The magnetic response of the isotropic spin-1 chain in strong fields H > H c has been analyzed in some detail in Refs [12][13][14]. In Appendix A we use the nonlinear sigma model description of the isotropic spin-1 chain to derive explicit expressions for the dynamical response functions in the low-field phase H < H c . In many S = 1 compounds such as NENP, NDMAP and NDMAZ strong crystal field anisotropies are present, which lead to a zero-field splitting of the magnon triplet comparable in magnitude to the Haldane gap itself. These anisotropy effects lead to more complex behavior and a richer phase diagram. The purpose of this work is to determine the dynamical response of Haldane-gap compounds in the presence of such crystal field anisotropies. The relevant lattice Hamiltonian is of the form H = J j S j · S j+1 − H · S j + j D(S z j ) 2 + E[(S x j ) 2 − (S y j ) 2 ],(1.1) where 0 < −E < D. In zero field there are three magnon modes in the vicinity of the antiferromagnetic wave number q = π a0 with gaps ∆ 1 < ∆ 2 < ∆ 3 . For simplicity we will mainly concentrate on the case where the magnetic field is applied along the z-axis, i.e. H = H e z . (1. 2) The lattice Hamiltonian (1.1) (with field applied along the z-direction) exhibits two discrete symmetries which play an important role in the following: 1. Rotation by π around the z-axis (R z π ): S x j → −S x j , S y j → −S y j , S z j → S z j . (1.3) 2. Translation by one site (T R ): S α j −→ S α j+1 , α = x, y, z . (1.4) The model (1.1) is difficult to analyze directly by analytical methods. However, progress can be made by concentrating on the low-energy regime, which can be studied by means of semi-phenomenological descriptions in terms of continuum models. Two such models have been used in particular, namely (i) a bosonic Landau-Ginzburg model 10,14 and (ii) a theory of three coupled Majorana fermions 15 . In the present work we go beyond the original works 10,14,15 by (1) discussing the decay of high energy magnon modes, (2) applying methods of integrable quantum field theory to the discussion of structure functions and (3) taking into account the effects of inter-chain interactions. The outline of this paper is as follows. We start by reviewing known results on the spectrum of the Majorana fermion model in section II B. We then investigate the role played by interactions in sections II C and II D and in particular show that interactions generate a finite lifetime for one of the magnon modes. In section III we derive analogous results in the framework of the Landau-Ginzburg theory. We then turn to a more detailed analysis of the magnetic response functions in the low-energy regime in the vicinity of the quantum critical point at H c in section IV. Section V gives a detailed account of the low-energy regime in the high-field phase for small crystal-field anisotropy. The effects of interchain coupling are investigated in section VI and a summary and discussion of our various results is given in section VII. A variety of technical details are presented in Appendices A-E. II. MAJORANA FERMION MODEL In Ref. [15] A.M. Tsvelik proposed a description of the spin-1 Heisenberg chain in terms of a field theory of three right and left moving Majorana (real) fermions R a = R † a , L a = L † a , a = 1, 2, 3. The Hamiltonian is given by H = i 2 3 a=1 v[L a ∂ x L a − R a ∂ x R a ] − ∆ a [R a L a − L a R a ] + i 2 a,b,c ε abc H a (L b L c + R b R c ) + H ′ ,(2.1) where H ′ describes anisotropic current-current interactions H ′ = a g a J a J a . (2.2) Here the currents are bilinears in the Majorana fermions J a (x) = − i 2 ε abc [L b L c + R b R c ] . (2.3) We will take the magnetic field along the 3-axis and concentrate on the case ∆ 1 < ∆ 2 < ∆ 3 . (2.4) The generalization of our results to other situations is straightforward. The smooth and staggered components of the spin operators are defined by the decomposition S α j −→ J α (x) + (−1) j n α (x) ,(2.5) where x = ja 0 . The smooth components are equal to the currents, where a = 1, 2, 3 correspond to α = x, y, z. The staggered components are expressed in terms of Ising order and disorder operators n x (x) ∝ σ 1 (x)µ 2 (x)µ 3 (x) , n y (x) ∝ µ 1 (x)σ 2 (x)µ 3 (x) , n z (x) ∝ µ 1 (x)µ 2 (x)σ 3 (x) . (2.6) We note that the signs of the mass terms in (2.1) are such that in zero field σ a (x) = 0 , µ a (x) = 0 . (2.7) A. Symmetries The Hamiltonian (2.1) inherits the discrete symmetries T R and R z π from the lattice model. The latter is realized as R z π : R a −→ −R a , L a −→ −L a , σ a −→ −σ a , µ a −→ µ a , a = 1, 2. (2.8) The translation symmetry by one site turns into a discrete Z 2 symmetry in the continuum limit J a (x) −→ J a (x) , n a (x) −→ −n a (x) ,(2.9) and may be realized as R a −→ −R a , L a −→ −L a , σ a −→ −σ a , µ a −→ µ a , a = 1, 2, 3. (2.10) It is convenient to combine this symmetry with R z π into the following Z 2 symmetry T R R z π : R 3 −→ −R 3 , L 3 −→ −L 3 , σ 3 −→ −σ 3 , µ 3 −→ µ 3 . (2.11) The full symmetry of the Majorana model (2.1) is thus Z 2 ⊗ Z 2 . B. Spectrum in the absence of interactions In the absence of interactions, i.e. g a = 0 in (2.2) the Hamiltonian can be diagonalized by means of a Bogoliubov transformation 15 . In the following we review some relevant formulas. As the magnetic field is along the 3-direction only the first and second Majoranas couple to the magnetic field. The third Majorana gives rise to a fermionic single-particle mode with dispersion ω 3 (k) = ∆ 2 3 + v 2 k 2 . (2.12) The first and second Majoranas are conveniently combined into a complex fermion Ψ R 1 = Ψ R + Ψ † R √ 2 , R 2 = Ψ R − Ψ † R i √ 2 , L 1 = Ψ L + Ψ † L √ 2 , L 2 = Ψ L − Ψ † L i √ 2 . (2.13) The Hamiltonian density describing the first and second Majoranas takes the form H 12 = −iv Ψ † R ∂ x Ψ R − Ψ † L ∂ x Ψ L + H 2 [Ψ † R , Ψ R ] + [Ψ † L , Ψ L ] −im Ψ † R Ψ L − h.c. + i∆ Ψ † R Ψ † L − h.c. , (2.14) where ∆ = ∆ 2 − ∆ 1 2 , m = ∆ 2 + ∆ 1 2 . (2.15) Introducing a mode expansion Ψ R (x) = ∞ 0 dk 2π e ikx α(k) + e −ikx β † (k) , Ψ L (x) = 0 −∞ dk 2π e ikx α(k) + e −ikx β † (k) ,(2.16) we may express the Hamiltonian density (2.14) as H 12 = ∞ 0 dk 2π 4 a,b=1 γ † a (k)M ab γ b (k) , (2.17) where γ a (k) = (α(k), α † (−k), β(k), β † (−k)) a , M =    vk + H i∆ 0 −im −i∆ −vk − H im 0 0 −im vk − H i∆ im 0 −i∆ −vk + H    . (2.18) Now we perform a Bogoliubov transformation with a unitary matrix U (k) 19) to bring H 12 to a diagonal (normal ordered) form     c + (k) c † + (−k) c − (k) c † − (−k)     = U (k)    α(k) α † (−k) β(k) β † (−k)    ,(2.H 12 = ∞ −∞ dk 2π a=± ω a (k) c † a (k)c a (k) , ω ± (k) = m 2 + ∆ 2 + H 2 + v 2 k 2 ±2 m 2 ∆ 2 + H 2 (m 2 + v 2 k 2 ) 1 2 . (2.20) The gap ω + (0) increases monotonically with H, whereas ω − (0) decreases and vanishes at a critical field H c = ∆ 1 ∆ 2 = m 2 − ∆ 2 . (2.21) The corresponding critical point is in the Ising universality class. For fields H < H c2 H c2 = m 1 2 + 1 4 + ∆ 2 m 2 1 2 (2.22) the minimum of the dispersion ω − (k) occurs at k = 0. In the vicinity of H c and at small momenta the dispersion is approximated as ω 2 − (k) ≈ H 2 c m 2 (H − H c ) 2 + v 2 ∆ 2 m 2 + H c (1 + ∆ 2 m 2 ) m 2 (H c − H) k 2 . (2.23) Hence the gap vanishes linearly with H − H c in agreement with Ising critical behaviour ω − (0) ≃ (H − H c ) 1 − ∆ 2 m 2 . (2.24) In order to see the Ising criticality described by (2.23) the magnetic field must be sufficiently close to H c H − H c < ∆ 2 H c (1 + ∆ 2 m 2 ) . (2.25) As expected this scale is set by the anisotropy ∆. For H > H c2 there are two degenerate minima of ω − (k) at some incommensurate wave numbers ±k F with vk F = H 2 − m 2 − m 2 ∆ 2 H 2 1 2 , ω − (k F ) = ∆ 1 − m 2 H 2 . (2.26) For k ≈ ±k F we have ω 2 − (k) ≈ ∆ 2 1 − m 2 H 2 + v 2 k F H 2 (|k| − k F ) 2 . (2.27) As was recently pointed out by Wang 26 , these results for the dispersions suggest that a cross-over between Ising and C-IC critical behaviour occurs as a function of H. For H very close to H c we encounter Ising critical behaviour, which crosses over to C-IC behaviour for H > H c2 . C. Interactions: self-consistent mean-field treatment So far we have neglected the four-fermion interactions (2.2) altogether. As a first step of taking interactions into account we may treat them in a self-consistent mean-field approximation (SCMF). The following expectation values are compatible with the discrete Z 2 symmetries L a R a = 0 , L 1 L 2 = 0 , R 1 R 2 = 0 , R 1 L 2 = 0 , R 2 L 1 = 0 . (2.28) Decoupling the four fermion terms leads to a renormalization of the mass parameters ∆ 1 −→∆ 1 = ∆ 1 − 2ig 2 L 3 R 3 − 2ig 3 L 2 R 2 , ∆ 2 −→∆ 2 = ∆ 2 − 2ig 1 L 3 R 3 − 2ig 3 L 1 R 1 , ∆ 3 −→∆ 3 = ∆ 3 − 2ig 3 L 2 R 2 − 2ig 2 L 1 R 1 . (2.29) The magnetic field terms are changed to H L L 1 L 2 + H R R 1 R 2 ,(2.30) where H L = H + 2ig 3 R 1 R 2 . and H R = H + 2ig 3 L 1 L 2 . Finally, two new terms are generated iλ 1 R 1 L 2 + iλ 2 L 1 R 2 ,(2.31) where iλ 1 = 2g 3 L 1 R 2 and iλ 2 = 2g 3 R 1 L 2 . The resulting mean-field Hamiltonian is quadratic in the Fermi fields and can again be diagonalized by a Bogoliubov transformation. Let us denote the ground state energy obtained in this way by E GS . The expectation values are determined self-consistently, e.g. −i R a L a = ∂E GS ∂∆ a . (2.32) The SCMF procedure is easily implemented once the appropriate couplings g a are specified. However, in order to keep matters as simple as possible we will assume from now on that the g a are small and as a result the differences between the free theory and the SCMF theory are negligible. The main qualitative effect of nonzero g a is to make the gap of the third Majorana magnetic field dependent. Such a dependence is observed for example in experiments on NDMAP 4,5 and implies the presence of interactions in the framework of the Majorana fermion model. D. Decay Processes in the Low-Field Phase Within the SCMF approximation the role of the current-current interactions is merely to induce slight changes of the dispersion relations of the three coherent single-particle magnon modes. As we will now show, treating the interactions beyond the SCMF approximation leads to the damping of one of the magnons. The analysis of the spectrum summarized above establishes that there are three different types of magnons, which we will refer to as M 3 , M + and M − respectively. The corresponding dispersion relations are ω 3 (k) (2.12) and ω ± (k) (2.20). The interaction of these modes is described by the term H ′ in the Hamiltonian (2.1) and involves four particles. As ω − (k) can become very small as the magnetic field is increased from zero, the decays M 3 −→ M − M − M − and M + −→ M − M − M − become kinematically allowed for sufficiently large magnetic fields. However, the decay of M 3 is forbidden by the symmetry T R R z π : M 3 is odd under this symmetry whereas M ± are even. Essentially we are using the fact that all 3 magnon modes only exist near wave-vector π a0 so that they can only decay into an odd number of magnons. The decay of M 3 into any odd number of M − 's would be inconsistent with the R z π spin rotation symmetry. On the other hand the decay M + −→ M − M − M − is allowed by symmetry. The process becomes kinematically possible as soon as the magnetic field exceeds a critical value H d , which is defined by ω + (0) = 3ω − (0) . (2.33) Solving for H d we find H d = m 2 4 − ∆ 2 . (2.34) As long as 2∆ < m the decay process will occur for H > H d and from now on we will assume that this is the case. We note that in zero field there are no decay processes even if the gaps are such that they are kinematically allowed (3∆ 1 < ∆ 2 ). The reason is that for H = 0 the lattice Hamiltonian (1.1) has additional spin rotational symmetries around the x and y axes by 180 degrees. In combination with the translation symmetry these induce symmetries of the form (2.11) for the Majoranas 1 and 2 individually rather than the combination (2.8). This additional symmetry forbids decay processes. Inserting the mode expansions (2.16) into the expression for H ′ , integrating over the spatial coordinate and then carrying out the Bogoliubov transformation (2.19) generates several terms quartic in the fermionic creation annihilation operators c ± (k), c † ± (k). The most interesting one describes the decay of a M + mode into three M − modes and is of the form V = g 3 ∞ −∞ dk 1 dk 2 dk 3 (2π) 3 f (k 1 , k 2 , k 3 ) ×c † − (k 1 )c † − (k 2 )c † − (k 3 )c + (k 1 + k 2 + k 3 ). (2.35) Here f is an antisymmetric function of its arguments and at small momenta is of the form f (k 1 , k 2 , k 3 ) ≃C (k 1 − k 2 )(k 1 − k 3 )(k 2 − k 3 ). (2.36) The constantC is a complicated function of ∆, m and H. The differential rate for the decay of a M + particle with momentum p into three M − particles with momenta p 1 , p 3 , p 3 is dΓ = 2π|M | 2 δ(p − 3 j=1 p j ) δ(ω + (p) − 3 j=1 ω − (p j )) × dp 1 dp 2 dp 3 3! , (2.37) where the factor of 3! is introduced to account for the fact that the three particles in the final state are indistinguishable. In the Born approximation the transition matrix element M is M = 1 4π 2 0| 3 j=1 c − (p j ) V c † + (p)|0 δ(p − 3 j=1 p j ) −1 = g 3 3! 2π f (p 1 , p 2 , p 3 ). (2.38) Taking the M + particle to be at rest, i.e. setting p = 0, we obtain Γ = 6g 2 3 2π dp 1 dp 2 |f (p 1 , p 2 , −p 1 − p 2 )| 2 × δ(ω + (0) − ω − (p 1 ) − ω − (p 2 ) − ω − (p 1 + p 2 )). (2.39) In order to simplify matters further, we concentrate on the case where the magnetic field is close the critical field H d at which the decay M + → M − M − M − first becomes kinematically possible. In this regime we have ω + (0) − 3ω − (0) ≪ ω − (0) . (2.40) Then the momenta p 1,2 in (2.39) have to be small in order to satisfy the delta-function and we may use the expansions (2.36) for f and ω − (p) ≈ ω − (0) +αp 2 + O(p 4 ) , α = v 2 1 − H 2 m √ H 2 + ∆ 2 [2ω − (0)] −1 . (2.41) This leads to the following expression for the decay rate in the regime H > H d , H H d − 1 ≪ 1 Γ ≈ 6g 2 3 2πC 2 (2α) 4 4π √ 3 [ω + (0) − 3ω − (0)] 3 ≈ g 2 3 √ 3C 2 2 α 4 1 − ∆ 2 4m 2 3 2 (H − H d ) 3 . (2.42) We find that the decay rate is proportional to (H − H d ) 3 and is therefore quite small in the vicinity of H d . III. LANDAU-GINZBURG (LG) MODEL A different approach to studying the spin-1 Heisenberg chain with crystal field anisotropies in a magnetic field was used in Refs [10,14]. It is based on the nonlinear sigma model description of the spin-S Heisenberg chain in terms of the three-component field ϕ describing the staggered components of the spin operators and the subsequent approximation of replacing the constraint ϕ 2 = 1 by a | ϕ| 4 interaction. The LG Lagrangian density is 10,14 L = 1 2v ∂ ϕ ∂t + H × ϕ 2 − v 2 ∂ ϕ ∂x 2 − 3 a=1 ∆ 2 a 2v ϕ 2 a − λ| ϕ| 4 . (3.1) We again take the magnetic field to point along the 3-axis H = H e 3 . (3.2) It then follows from (3.1) that ϕ 3 couples to the magnetic field only via the λ| ϕ| 4 interaction. The Landau-Ginzburg theory inherits the discrete symmetries (1.3) and (1.4) from the underlying lattice model. As is the Majorana fermion model it is convenient to combine the translational symmetry by one site T R with the rotation around the z-axis by π R z π and obtain the following two Z 2 symmetries R z π : ϕ a −→ −ϕ a , a = 1, 2. (3.3) T R R z π : ϕ 3 −→ −ϕ 3 . (3.4) The resulting Z 2 ⊗ Z 2 symmetry is the same as for the Majorana fermion model. A. Spectrum and Mode Expansion Neglecting the | ϕ| 4 interaction the spectrum can be determined by solving the classical equations of motion ∂ 2 ∂t 2 − v 2 ∂ 2 ∂x 2 + ∆ 2 a − H 2 ϕ a + 2ε abc H b ∂ϕ c ∂t = 0. (3.5) One finds that there are three magnon modes M 3 and M ± with the following dispersion relations 10,14 ω 3 (k) = ∆ 2 3 + v 2 k 2 , ω ± (k) = H 2 + ∆ 2 1 + ∆ 2 2 2 + v 2 k 2 ± 2H 2 (∆ 2 1 + ∆ 2 2 + 2v 2 k 2 ) + ∆ 2 1 − ∆ 2 2 2 2 1 2 . (3.6) The low energy M − mode with dispersion ω − (k) becomes gapless at a critical field H c = ∆ 1 . The resulting critical point is in the universality class of the two-dimensional Ising model 14 . The scalar fields ϕ 1,2 have the following mode expansions ϕ 1 (t, x) = α=± dk A 1α (k)e −iωα(k)t+ikx a α (k) + h.c. 4πω α (k)/v , ϕ 2 (t, x) = α=± dk iA 2α (k)e −iωα(k)t+ikx a α (k) + h.c. 4πω α (k)/v ,(3.7) where A * aα (k) = A aα (k). Here a and a † obey canonical commutation relations 8) and the amplitudes A a± (k) are fixed by the requirements that (i) the fields ϕ a fulfil the equations of motion (3.5) and (ii) the fields ϕ a and the conjugate momenta [a α (k), a † β (p)] = δ αβ δ(k − p) ,(3.Π a = 1 v ∂ϕa ∂t + ( H × ϕ) a fulfil canonical commutation relations [ϕ a (t, x), ϕ b (t, y)] = 0 , [Π a (t, x), Π b (t, y)] = 0 , [ϕ a (t, x), Π b (t, y)] = iδ ab δ(x − y) . (3.9) We find A 2 1+ (k) = 3H 2 + ∆ 2 1 + v 2 k 2 − ω − (k) 2 ω + (k) 2 − ω − (k) 2 , A 2 1− (k) = 1 − A 2 1+ (k) = ω + (k) 2 − 3H 2 − ∆ 2 1 − v 2 k 2 ω + (k) 2 − ω − (k) 2 , A 2a (k) A 1a (k) = − ω a (k) 2 + H 2 − ∆ 2 1 − v 2 k 2 2Hω a (k) , a = ±. (3.10) Eqns (3.10) allow us to deduce the polarizations of the modes corresponding to the dispersion ω ± (k). For example, at H = 0 the M − mode is polarized entirely along the 1 direction and the M + mode along the 2 direction. On the other hand, as H → ∆ 1 we find that A 2− (0) → 0 , A 2 2+ (0) → 1 , A 2 1− (0) → ∆ 2 2 − ∆ 2 1 3∆ 2 1 + ∆ 2 2 , A 2 1+ (0) → 4∆ 2 1 3∆ 2 1 + ∆ 2 2 . (3.11) Hence ϕ 2 couples only to the M + mode whereas ϕ 1 couples to the M + mode as well as to the M − mode with a strength set by the anisotropy. B. Decay Processes In the absence of the nonlinear | ϕ| 4 -term the LG model describes three coherent magnons M 3 , M ± with corresponding dispersion relations (3.6). Inclusion of the | ϕ| 4 term generates interaction terms involving four particles. As was the case in the Majorana fermion model, ω − (k) can become very small when the magnetic field is increased and as a result the decays M 3 −→ M − M − M − and M + −→ M − M − M − become kinematically allowed. The decay of M 3 is forbidden by the symmetry T R R z π (3.4): M 3 is odd under this symmetry whereas M ± are even. On the other hand the decay process M + → 3M − is allowed when the magnetic field is larger than H d = 17 16 [∆ 2 1 + ∆ 2 2 ] ± 5 16 13[∆ 4 1 + ∆ 4 2 ] + 10∆ 2 1 ∆ 2 2 1 2 . (3.12) The interaction describing this decay process is given by V = λv 2 2π ∞ −∞ dk 1 dk 2 dk 3 g(k 1 , k 2 , k 3 ) ×a † − (k 1 )a † − (k 2 )a † − (k 3 )a + (k 1 + k 2 + k 3 ), (3.13) where g(k 1 , k 2 , k 3 ) is a symmetric function of its arguments. Its zero momentum limit is g(0, 0, 0) = A 1− (0) 3 A 1+ (0) ω 3 − ω + 1 − ω 2 − + H 2 − ∆ 2 1 2Hω − 2 1 + ω 2 − + H 2 − ∆ 2 1 2Hω − ω 2 + + H 2 − ∆ 2 1 2Hω + ≡ C ω 3 − ω + .(3.14) where ω ± =≡ ω ± (0). As a first approximation we neglect all interactions except V and calculate the decay rate M + → 3M − in the Born approximation. The differential rate for the decay of a M + magnon with momentum p into three M − magnons with momenta p 1 , p 3 , p 3 is again given by (2.37), where M = 0| 3 j=1 a − (p j ) V a † + (p)|0   δ(p − 3 j=1 p j )   −1 = 3λv 2 π g(p 1 , p 2 , p 3 ). (3.15) Taking the M + magnon to be at rest, i.e. setting p = 0, we obtain Γ = 3λ 2 v 4 π dp 1 dp 2 |g(p 1 , p 2 , −p 1 − p 2 )| 2 × δ(ω + (0) − ω − (p 1 ) − ω − (p 2 ) − ω − (p 1 + p 2 )). (3.16) For H slightly larger than H d we again have 17) and the momenta p 1,2 in (3.16) have to be small in order to satisfy the delta-function. We may use the expansions (3.14) for g and ω + (0) − 3ω − (0) ≪ ω − (0) ,(3.ω − (p) ≈ ω − (0) + αp 2 + O(p 4 ) , α = v 2 2ω − (0)   1 − 2H 2 2H 2 (∆ 2 1 + ∆ 2 2 ) + ( ∆ 2 1 −∆ 2 2 2 ) 2   . (3.18) This leads to the following expression for the decay rate in the regime H > H d , H H d − 1 ≪ 1 Γ ≈ √ 3λ 2 v 4 C 2 αω − (0) 3 ω + (0) . (3.19) The result (3.19) would suggest that the decay rate switches on suddenly at a finite value as soon as H becomes larger than H d . The underlying reason for this jump is the restricted phase space of a one-dimensional system. The "free boson" result (3.19) for the decay rate is dramatically different from the result obtained in the framework of the Majorana fermion theory. This poses the question whether (3.19) is robust if we take into account interactions among the magnons in the final state. This amounts to resumming the leading infrared divergences in a perturbative expansion in λ. If we assume that in the low-energy limit the M − degrees of freedom in the Landau-Ginzburg model can still be mapped onto a Bose gas with δ-function interactions 16 , the result of a such a resummation can be determined by exploiting the fact the wave functions of the M − modes reduce to a free fermion form in the limit of small momenta 19 . A three-particle state in the position representation can be written as |λ 1 , λ 2 , λ 3 = dx 1 dx 2 dx 3 χ(x 1 , x 2 , x 3 ) × a † − (x 1 )a † − (x 2 )a † − (x 3 )|0 , (3.20) where the wave-function is given by χ 3 (x 1 , x 2 , x 3 ) = (−i) 3 (2π) 3/2 3! 3 j<k=1 sgn(x j − x k ) × P ∈S3 sgn(P )e 3 j=1 iλP j xj . (3.21) Here P denotes a permutation of three elements and S 3 the symmetric group of degree 3. In momentum space we have |λ 1 , λ 2 , λ 3 = dk 1 dk 2 dk 3 (2π) 3 3 j=1 2k j k 2 j + ǫ 2 a † − (λ 1 − k 2 − k 3 )a † − (λ 2 − k 1 + k 3 )a † − (λ 3 + k 1 + k 2 )|0 . (3.22) Taking (3.22) as the final state, the matrix element of the decay vertex is M = λ 3 , λ 2 , λ 1 |V a † + (p)|0 δ(p − 3 j=1 λ j ) = 3λv 2 π 3 j=1 dq j 2π 2q j q 2 j + ǫ 2 g(λ 1 − q 2 − q 3 , λ 2 − q 1 + q 3 , λ 3 + q 1 + q 2 ) . (3.23) It follows from the definition (3.23) that the matrix element is an antisymmetric function of λ 1 , λ 2 , λ 3 . As long as we are interested in the decay rate for H close to H d , we may expand M for small λ j . The leading term antisymmetric in λ 1,2,3 is then C ′ (λ 1 − λ 2 )(λ 1 − λ 3 )(λ 2 − λ 3 ) ,(3.24) For small λ j the matrix element is thus equal to M = 3λC ′ v 2 π j<k (λ j − λ k ) . (3.25) Following through the same steps as before, the decay rate for H close to H d with the M + magnon initially at rest is found to be the actual values for H c are rather different, but as we are interested only in robust features this numerical difference does not really concern us here. More importantly both theories predict that the low-energy behaviour in the vicinity of the critical point is described by an off-critical Ising model with Hamiltonian Γ ≈ √ 3λ 2 C ′2 v 4 4α 4 [ω + (0) − 3ω − (0)] 3 ∝ [H − H d ] 3 .(3.H = iv 2 [L∂ x L − R∂ x R] − im 0 RL . (4.1) Here the mass parameter that parametrizes the deviation from criticality is equal to the smallest magnon gap |m 0 | ≃ ω − (k = 0, H) . (4.2) The description by (4.1) is appropriate at low energies ω ≪ min{ω + (k = 0, H), ω 3 (k = 0, H)} and H sufficiently smaller than the critical field H c2 (2.22) at which incommensurabilities develop. The Hamiltonian (4.1) exhibits a Z 2 symmetry R −→ −R , L −→ −L ,(4.3) under which the Ising order (σ) and disorder (µ) parameters transform as H < H c : σ −→ −σ , µ −→ µ , H > H c : σ −→ σ , µ −→ −µ . (4.4) In order to determine the magnetic low-energy response (in the vicinity of the antiferromagnetic wave number) we need to express the staggered magnetizations n a in terms of operators related to the Ising model (4.1). As was pointed out in Ref. [14], the dominant contribution to the staggered magnetization in the x-direction should be the Ising order parameter field n x = C x (H) σ + . . . (4.5) Here C x (H) is an unknown constant and the dots indicate contributions from less relevant operators. The underlying reason for the identification (4.5) is simply that in the ordered phase of the Ising model (which corresponds to H > H c ) one has a nonzero expectation value σ = 0, whereas in the disordered phase (which occurs at H < H c ) one has σ = 0. The staggered magnetization in both the Majorana fermion and LG models exhibits the same kind of behaviour and it is then natural to expect the identification (4.5). In Appendix D we present arguments that suggest that the staggered magnetization along the y-direction has the following low-energy projection n y = C y (H) ∂ τ σ + . . . (4.6) In what follows we chose a short-distance normalization for the field σ in the theory (4.1) such that for τ 2 + x 2 /v 2 → 0 σ(τ, x) σ(0, 0) −→ 1 [τ 2 + x 2 /v 2 ] 1/8 . (4.7) We will use the integrability of the theory (4.1) to determine the dynamical structure factor S xx (ω, π a0 + q) at low energies, where the staggered component of S x is given by (4.5). The yy component then follows from (4.6) S yy (ω, π a 0 + q) ∝ ω 2 m 2 S xx (ω, π a 0 + q) .S xx (ω, π a 0 + q) = vA m 2 0 + v 2 q 2 δ ω − m 2 0 + v 2 q 2 + 2vA 3π 2 m 2 0 z0 0 dz tanh(z) tanh( y+z 2 ) tanh( y−z 2 ) 2 [x 2 − 1 − 4 cosh 2 z] 2 − 16 cosh 2 z + contributions from 5,7,. . . magnons, (4.9) where A = C 2 x (H)2 1 6 e − 1 4 A 3 m 1 4 0 , (4.10) x 2 = ω 2 − v 2 q 2 m 2 0 , z 0 = arccosh( x − 1 2 ) , y = arccosh x 2 − 1 − 4 cosh 2 z 4 cosh z . (4.11) Here A = 1.28242712910062... (4.12) is Glaisher's constant. The three-magnon contribution is always very small. In the frequency interval [0, 30m 0 ] roughly 100 times more spectral weight sits in the single-magnon contribution than in the three-particle one. Hence the magnetic response below energies of the order of tens of the magnon gap m 0 is dominated by the coherent magnon contribution. However, if H becomes very close to H c the magnon gaps m 0 tends to zero. If we are interested in the magnetic response at a low (compared to the gap of the second coherent magnon mode) but fixed energy we have to take the contributions of intermediate states with 5,7,9,... magnons into account in order to get an accurate result for S xx (ω, π a0 + q). H = Hc: At Criticality At criticality the structure factor exhibits a power-law behaviour 24 S xx (ω, π a 0 + q) = Im B v 2 q 2 − (ω + iε) 2 − 7 8 , (4.13) where B = 2v C 2 x (H)2 3 4 Γ 7 8 Γ 1 8 . (4.14) In this phase there is a nonzero staggered magnetization and correspondingly a nonzero expectation value for the Ising order parameter σ = 0, (4.15) which results in a Bragg peak for momentum transfer π a0 along the chain direction. By virtue of the Z 2 symmetry (4.4) only intermediate states with an even number of magnons contribute to S xx (ω, π a0 + q). The leading contribution to the inelastic neutron scattering cross section comes from intermediate states involving two magnons [20][21][22] , which leads to the following result S xx (ω, π a 0 + q) = vA π s 2 − 4m 2 0 s 3 θ(s − 2m 0 ) + contributions from 4,6,. . . magnons . (4.16) Here A is given by (4.10) and s 2 = ω 2 − v 2 q 2 . V. THE HIGH-FIELD PHASE FOR WEAK ANISOTROPY In general it is difficult to determine the effects of magnon-interactions in quantitative detail. An exception is the case of a small anisotropy of the zero-field gaps ∆ ≪ m ,(5.1) where 2m = ∆ 1 + ∆ 2 and ∆ = ∆ 2 − ∆ 1 . As we will show, if the magnetic field is sufficiently larger than the critical field H c and ∆ ≪ H − H c < ∼ J ,(5.2) it is possible to determine the interaction effects on the magnetic response at low energies in some detail. More precisely, we consider magnetic fields sufficiently larger than the field H c2 (see e.g. (2.22)), at which incommensurabilities begin to develop. The scale defining the low-energy region is the difference ∆. As is shown in Appendix E, the low-energy effective Hamiltonian is given by a sine-Gordon model (SGM) 14 H =ṽ 16π (∂ x Θ) 2 + (∂ x Φ) 2 − 2µ cos βΘ .q = π a0 S ± j −→ (−1) j A exp ± iβ 2 Θ . (5.4) The identification (5.4) is in accordance with the fact that in the high-field phase there is Néel order along the x-direction (−1) n S x n ∝ cos β 2 Θ = 0 . (5.5) We choose a short-distance normalization such that for |x − y| → 0 e iαΘ(x) e iγΘ(y) −→ |x − y| 4αγ e iαΘ(x)+iγΘ(y) . (5.6) The amplitude A in (5.4) is nonuniversal and not known in general. However, very close to H c ∆ ≪ H − H c ≪ H c ,(5.7) it is given by (see Appendix F) A = A ′ a 4 0 m v 2 (H − H c ) 1 8 ,(5.8) where A ′ is a field independent numerical constant. The dependence of (5.8) on H − H c is a universal feature of the C-IC transition. We note that the sign of the cos-term in (5.3) is quite important. Flipping the sign corresponds to a shift Θ −→ Θ + π/β, which essentially leads to an exchange of the x and y component of the spin operators in (5.4). The value of the parameter β is of crucial importance. For the isotropic case (∆ 2 = ∆ 1 ) it has recently been determined 13 in the framework of the nonlinear sigma model description of the spin-S Heisenberg chain. In the isotropic case the high-field phase at H > H c is a Luttinger liquid and β is related to the Luttinger liquid parameter. It was found that 13 β = 1 √ 2S R (θ F ) ,(5.9) where S R (θ) fulfils the integral equation S R (θ) = 1 + θF −θF dθ ′ S R (θ ′ ) 1 π 2 + (θ − θ ′ ) 2 . (5.10) Here θ F is determined as a function of the magnetic field H by ǫ(θ) = m cosh(θ) − H + θF −θF dθ ′ ǫ(θ ′ ) 1 π 2 + (θ − θ ′ ) 2 , ǫ(θ F ) = 0. (5.11) Similarly one may determine the spin velocity v = v 2πρ(θ) ∂ǫ(θ) ∂θ θ=θF ,(5.12) where ρ(θ) fulfils the integral equation ρ(θ) = m 2π cosh(θ) + θF −θF dθ ′ ρ(θ ′ ) 1 π 2 + (θ − θ ′ ) 2 . (5.13) The parameter β as well as the velocityṽ entering the sine-Gordon Hamiltonian (5.3) may be estimated with a good degree of accuracy from their respective values in the isotropic case as long as ∆2−∆1 ∆2 ≪ 1. On the other hand, the agreement between the Luttinger liquid parameter calculated in the nonlinear sigma model and the one of the isotropic spin-1 Heisenberg chain in a magnetic field, which is known approximately from DMRG computations 17 , was shown to be fairly good in Ref. [13]. Hence we may determine β with a reasonable degree of accuracy from (5.9) as long ∆2−∆1 ∆2 ≪ 1. It then follows that β < 1 where M is the gap. In the regime 0 < β < 1/ √ 2 relevant to our discussion, soliton and antisoliton attract and can form bound states, known as "breathers". There are where N = 1 − β 2 β 2(ξ = β 2 1 − β 2 . (5.17) The number of breathers is a function of the applied magnetic field H. There always is at least one breather. A second breather appears above a field H 0 , which is determined by the requirement β = 1 √ 3 . (5.18) This requirement is fulfilled for H > H 0 ≈ 1.5M . (5.19) B. Dynamical Structure Factor A basis of eigenstates of the SGM is given by scattering states of solitons, antisolitons and breathers. In order to distinguish these we introduce labels B 1 , B 2 , . . . B N , s,s. As usual for particles with relativistic dispersion, it is useful to introduce a rapidity variable θ to parameterize energy and momentum S O (ω, q) = ∞ n=1 ǫi dθ 1 . . . dθ n (2π) n−1 n! |f O ǫ1...ǫn (θ 1 . . . θ n )| 2 δ(q − j M ǫj sinh θ j /ṽ) δ(ω − j M ǫj cosh θ j ) (5.23) The form factors of the operators exp(±iβΦ/2) in the sine-Gordon model were determined in Ref. [28,29]. Using these results we can determine the first few terms of the expansion (5.23) for the transverse spin operators. We have S xx (ω, π a 0 + q) = C πf 2 δ(s 2 − M 2 2 )Θ(H − H 0 ) +Re |F cos (θ 0 )| 2 s √ s 2 − 4M 2 + . . . , S yy (ω, π a 0 + q) = C πf 1 δ(s 2 − M 2 1 ) +Re |F sin (θ 0 )| 2 s √ s 2 − 4M 2 + . . . . (5.24) Here C is an overall (dimensionful) constant. The terms proportional to F sin and F cos represent the contributions by intermediate states involving one soliton and one antisoliton and θ 0 = 2arccosh(s/2M ). (5.25) The δ-function contributions are due to the breather bound states. The soliton-antisoliton form factors are given by 28,29 |F sin (θ)| 2 = 0| sin β 2 Φ(0) |θ 2 θ 1 +− 2 = g(θ) ξ cosh θ+iπ 2ξ 2 , |F cos (θ)| 2 = 0| cos β 2 Φ(0) |θ 2 θ 1 +− 2 = g(θ) ξ sinh θ+iπ 2ξ 2 ,(5.26) where θ = θ 2 − θ 1 and g(θ) = i sinh θ/2 × exp ∞ 0 dt t sinh 2 (t[1 − iθ/π]) sinh(t[ξ − 1]) sinh 2t sinh ξt cosh t . (5.27) The absolute values squared of the breather form factors are 28,29 f 1 = 2 sin πξ 2 exp −2 πξ 0 dt 2π t sin t , f 2 = 2|g(−iπ[1 − 2ξ])| 2 cot(πξ) cot( πξ 2 ) 2 . (5.28) An important result is that the first bound state B 1 is visible only in S yy and does not couple to S xx . We note that there are additional contributions in the spectral representations (5.24) at higher energies. For example, there is a two-breather B 1 B 1 contribution to S xx at energies above 2M 1 . We plot S αα (ω, π a0 ) as functions of ω/M for several values of the applied magnetic field in Figs 3-4. In order to give a visual impression of their spectral weights we have broadened the δ-functions corresponding to the breathers by convolution with a Gaussian. We first discuss the evolution of S yy (ω, π a0 ) shown in Fig. 3: as H moves away from H c = m the breather B 1 splits off from the soliton-antisoliton continuum and very quickly takes over most of the spectral weight. Except for a narrow window (in magnetic field) above H c the yy-component of the dynamical structure factor is dominated by a coherent single-particle peak. The evolution of S xx (ω, π a0 ) is very different as is shown in Fig. 4: as H moves away from H c = m the incoherent soliton-antisoliton continuum slowly narrows (on the scale of the field-dependent soliton gap) until it eventually begets the second, heavy breather B 2 at H = H 0 ≈ 1.5m. Over a large interval of magnetic fields S xx (ω, π a0 ) is dominated by the incoherent soliton-antisoliton continuum. Spectral Weights In order to compare the spectral weights located in the coherent breather peaks to the spectral weight associated with the soliton-antisoliton continua it is useful to define quantities I xx = M 2 C 25 0 dx S xx (xM, π a 0 ) ≡ I xx B2 + I xx ss , I yy = M 2 C 25 0 dx S yy (xM, π a 0 ) ≡ I yy B1 + I yy ss . (5.29) For example, CI yy /M 2 is the spectral weight of the yy-component of the dynamical structure factor at the antiferromagnetic wave number integrated over the frequency interval [0, 25M ]. It has contributions I yy B1 from the coherent breather peak and I yy ss from the soliton antisoliton continuum (there are also contributions to due B 1 B 2 two-breather states etc, but their contributions are subleading). It is important to note that the soliton gap M and the overall factor C depend on the applied magnetic field. These dependencies drop out once we consider spectral weight ratios These ratios are plotted as functions of β in Fig.5. We see that for small β (that is at H ≫ H c ) most of the spectral weight is situated in the coherent peak associated with the first breather B 1 . Very close to the transition the second breather does not exist and most of the spectral weight sits in the soliton-antisoliton continua. The crossover between these two regimes occurs around β ≈ 0.675, which according to Fig. 2 corresponds to H H c ≈ 1.025 . (5.31) The lesson is that interactions make the summed dynamical structure factor S xx (ω, π a 0 + q) + S yy (ω, π a 0 + q) (5.32) look coherent expect for fields very close to H c . On the other hand, the polarized structure factor S xx (ω, π a0 + q) looks incoherent! It would be very interesting to attempt to disentangle the components of the dynamical structure factor in inelastic neutron scattering experiments and in this way observe this incoherent scattering continuum. Polarizations in the LG model How do these results fit into the general picture of the LG model? In the latter one expands to quadratic order in the fields ϕ 1 and ϕ 2 around the minimum of the effective potential at ϕ vac = (m 0 , 0, 0) m 2 0 = H 2 − ∆ 2 1 4vλ . (5.33) The effective Lagrangian for the fields ϕ 1,2 becomes L = 2 a=1 1 2v ∂ϕ a ∂t 2 − v 2 ∂ϕ a ∂x 2 − H c v ǫ abc ∂ϕ a ∂t ϕ b − H 2 − ∆ 2 1 v ϕ 2 1 − ∆ 2 2 − ∆ 2 1 2v ϕ 2 2 . (5.34) This is the same as (3.1) for the fields ϕ 1,2 if we make the replacement (in (3.1)) ∆ 2 1 −→ 3H 2 − 2∆ 2 1 , ∆ 2 2 −→ ∆ 2 2 + H 2 − ∆ 2 1 . (5.35) This implies for the polarizations in the limit H ≫ ∆ 2 A 2− (0) A 1− (0) −→ − 3H 2 ∆ 2 2 −∆ 2 1 1 2 . (5.36) In other words |A 2− | ≫ |A 1− | ,(5.37) and as a result the coherent low-energy mode is dominantly polarized along the y-direction! This agrees nicely with the sine-Gordon calculation, where the dominant feature, the first breather, appears in S yy . VI. INTERCHAIN COUPLING So far we have considered a purely one-dimensional situation corresponding to an ensemble of uncoupled spin-1 chains. As long as the magnon gap is large, a weak coupling between the chains may be neglected in a first approximation. On the other hand, the interchain exchange is expected to lead to significant qualitative changes in the magnetic response close to the critical point where the magnon gap becomes very small 35 . In order to assess the effects of a weak interchain coupling for H ≈ H c we consider a Landau-Ginzburg model of the form L = n L n + L int , (6.1) where L n = 1 2v ∂ ϕ n ∂t + H × ϕ n 2 − v 2 ∂ ϕ n ∂x 2 − 3 a=1 ∆ 2 a 2v ϕ 2 n,a − λ| ϕ n | 4 , L int = J ⊥ a 0 j,k ϕ j · ϕ k . (6.2) Here the sum jk is over links between neighbouring sites on different chains and we have dropped quartic terms in L int that arise from the interaction of the smooth components of the spin operators. As we have seen in section IV, close to the the Ising critical point the low-energy degrees of freedom are described by off-critical Ising models. Hence at low energies we have L n ≈ R n ∂ − R n + L n ∂ + L n − im 0 R n L n . (6.3) The leading low-energy projection of the interchain exchange follows from (4.5), (4.6) L int ≈ J ⊥ a 0 C 2 a 0 v 1/4 j,k σ j σ k , (6.4) where C is a dimensionless constant. The "quasi-1D Ising model" (6.3), (6.4) has recently been studied in Ref. [36] and we may follow some of this analysis here. A. Mean-Field Approximation As a first step, we analyze the model (6.3), (6.4) by means of a self-consistent mean-field approximation 30,31 . We assume the existence of a nonzero expectation value σ = 0 , (6.5) which corresponds to Néel order along the x-direction. The long-range order can be induced by the magnetic field, the interchain coupling or by both. In the presence on a nonzero expectation value (6.5) we may decouple the interaction term in (6.4) and arrive at the following mean-field Lagrangian density L MF = R∂ − R + L∂ + L − im 0 RL + h v σ . (6.6) Here "magnetic field" h has dimensions of s − 15 8 by virtue of the normalization (4.7) and is subject to the self-consistency condition h = ZC 2 v J ⊥ a 0 a 0 v 1 4 σ , (6.7) where Z is the number of neighbouring chains. The mean-field theory is purely one-dimensional and describes an off-critical Ising model in an effective magnetic field induced by the neighbouring chains. The model (6.6) has been studied by several authors [32][33][34] and is known to exhibit very interesting physical behaviour as m 0 and h are varied. In order to discuss the effects of h and m 0 it is convenient to consider the Euclidean two-point function of Ising order parameters χ E σσ (ω, q) = ∞ −∞ dx dτ e iωτ −iqx σ(τ, x)σ(0, 0) . (6.8) We note that χ E σσ is related to the xx-component of the staggered susceptibility by analytic continuation to real frequencies. The Lagrangian (6.6) defines a one-parameter family of field theories labelled by the dimensionless quantity 34 χ = m 0 h − 8 15 . (6.9) In the two special cases χ = 0 and |χ| = ∞ the model(6.6) is integrable and the susceptibility (6.8) can be determined to a very high accuracy by means of the formfactor bootstrap approach. In what follows we first review known quantitative results for the cases |χ| → ∞ and χ → 0 and then summarize the qualitative behaviour for general values of χ. The Limit h → 0: McCoy-Wu Scenario The regime h → 0 was studied in Refs 32,33 by means of a perturbative expansion in h. In the absence of a field (h = 0) the dynamical structure factor has been given in section IV. For m 0 > 0 the Ising model is in its disordered phase and the spin-spin correlation functions are dominated by a single-particle pole and the next-lowest excited states occur in the form of a three-particle scattering continuum. The dynamical structure factor is proportional to (4.9). Introducing a small magnetic field leads to a small shift in the position of the single-particle pole. Furthermore a two-particle scattering continuum of excited states emerges. For m 0 < 0 the Ising model is in its ordered phase. This means that there is a nonzero value for the staggered magnetization σ 2 0 = 2 1/6 e −1/4 A 3 m 1/4 0 , (6.10) where A denotes Glaisher's constant (4.12). The structure factor in the ordered phase is given by (4.16): the structure factor is incoherent and there is a two-particle branch cut starting at ω = 2m 0 . It is convenient to define a dimensionless magnetic fieldh bỹ h = σ 0 m 2 0 h. (6.11) After a resummation of a perturbative expansion inh McCoy and Wu established that the spin-spin correlation function has the following large-distance behaviour 32 σ(τ, x) σ(0, 0) ≈ σ 2 0 exp(−2m 0 r) 2 √ πm 0 rh × l exp(−m 0 r(λ lh ) 2/3 ) ,(6.12) where r 2 = τ 2 + x 2 /v 2 and λ l are the positive solutions to the equation J 1 3 (λ l /3) + J − 1 3 (λ l /3) = 0 . (6.13) The interesting point is that the Fourier transform of (6.12) no longer has a branch cut! There are single-particle poles atω 2 + v 2 q 2 = −[2 + (hλ l ) 2/3 ] 2 m 2 0 . (6.14) In other words, the two-particle branch cut has disintegrated into a series of single-particle poles. The residues of these poles are proportional toh [2 + (hλ l ) 2/3 ] −1 . (6.15) Hence the lightest particle carries more spectral weight than the heavier ones. This is quite different from the result for h = 0 where the structure factor vanishes as the threshold is approached from above. The Limit m0 → 0: Magnetic Deformation In the limit m 0 → 0 the model (6.6) is integrable 37 . The spectrum consists of eight types of massive self-conjugate particles. Three of them have masses below the lowest two-particle threshold. The two-point function of the Ising order operator σ was calculated by the form factor bootstrap approach in Ref. [38]. The dominant contribution to the two point function of Ising order parameter fields is due to the lightest particles. The dynamical susceptibility is approximately which enables us in principle to solve the self-consistency equation (6.7). The important point is that most of the spectral weight is located in the coherent modes corresponding to the two lightest particles. The ratio of weights between them is χ σσ (ω, q) ≈ 4m 2 1 15πh 2 3 j=1 2vZ j ω 2 − v 2 q 2 − m 2 j ,(6.(Z 1 /m 1 ) (Z 2 /m 2 ) ≈ 5.79427. (6.20) The region m 0 ≈ 0 limit was studied by form factor perturbation theory in Ref. [34]. Qualitative Behaviour in the general case For general values of χ the qualitative behaviour of χ E σσ (ω, q) is known and may be conveniently summarized 32,34 by considering the evolution of χ E σσ with χ along a path in the m 0 − h plane as shown in Fig. 6. In Fig.7 we show the analytic structure of χ E σσ as a function of s = ω 2 + v 2 q 2 for various locations along the path set out in Fig.6. For example, point (a) corresponds to the disordered phase of the off-critical Ising model, where in Euclidean space there is a single-particle pole at s = im 0 and a 3-particle branch cut along the positive imaginary axis starting at s = 3im 0 . Point (b) shows the small shift in the position of the pole and the emergence of a 2-particle branchcut 32 . Points (c)-(e) describe the vicinity of the Ising model in a magnetic field; there are several single-particle poles below a 2-particle branchcut and the number of these poles increases as we move along the path. Finally, points (f)-(g) describe the breakup of the 2-particle branchcut mentioned above. An important point is that for m 0 < 0, which corresponds to the ordered phase of the Ising model for h = 0, the general effect of the magnetic field is to make the dynamical susceptibility look more coherent in these sense that the low-energy regime is dominated by single-particle poles. In particular, a weak interchain coupling in the ordered phase close to H c leads to a disintegration of the 2-particle scattering continuum that dominates the dynamical structure factor (4.16) and the formation of a series of single-particle poles. B. Beyond Mean-Field: RPA It is straightforward to go beyond the mean-field approximation by resumming all diagrams in the interchain coupling that do not involve loops. This leads to the RPA expression for the dynamical susceptibility 39 χ xx (ω, q, k) = χ xx (ω, q) 1 − 2J ⊥ (k)χ xx (ω, q) ,(6.21) where J ⊥ (k) is the Fourier transform of the interchain coupling and for a simple cubic lattice. It was shown in Ref. [36] that in the vicinity of points (c)-(e) of Fig.6 the RPA leads only to slight changes of the mean-field results. More precisely, single-particle excitations corresponding to poles at s = m (with residue Z/2m) in the 1D susceptibility χ xx (ω, q) acquire a transverse dispersion ZJ ⊥ (k) in the RPA. χ xx (ω, q) = C 2 a 0 a 0 v VII. SUMMARY AND DISCUSSION We have studied the spectrum and dynamical spin correlations for Haldane-gap systems in the presence of a magnetic field. We have paid particular attention to the role played by the crystal field anisotropies present in materials like NDMAP. We have concentrated on the case where the magnetic field is applied along the same same direction as the largest single-ion anisotropy D (which we identify with the z-direction in spin space). Generalizations of our results to other cases is straightforward. Our main results are as follows: • At a critical field H c > H d the gap ∆ − (H) vanishes. In the vicinity of H c the low energy degrees of freedom are described by an (off-critical) Ising model and the dynamical structure factor is calculated by exact methods. For H > H c the dynamical structure factor is dominated by an incoherent two-particle scattering continuum above a finite-energy threshold. • At fields sufficiently above H c the low-energy degrees of freedom are described by a sine-Gordon model. S yy (ω, π a0 + q) is dominated by a coherent single-particle bound state with a spectral gap below a two-particle scattering continuum. The most pronounced feature in S xx (ω, π a0 + q) is an incoherent two-particle scattering continuum above a finite-energy threshold. • The effects of interchain coupling are most pronounced in the vicinity of H c . Taking it into account in a meanfield fashion leads to a purely one-dimensional effective description at low energies in terms of an Ising model in a magnetic field. This suggests that Haldane-gap materials with single-ion anisotropies in a magnetic field may constitute a realization of this very interesting theory. Within the mean-field description the main effect of interchain coupling is to generate coherent single-particle modes from the incoherent scattering continua. As a result the dynamical structure factor will appear more "coherent". Our findings shed some light on the question why recent inelastic neutron scattering experiments on NDMAP 4,5 have failed to find any evidence of scattering continua in the high-field phase. At large fields these are suppressed through bound-state formation, whereas in the vicinity of the critical field H c the coupling between chains effects similar shifts of spectral weight to single-particle modes. It would be interesting to investigate some of our predictions experimentally. In particular we hope that it may be possible to 1. address the issue of the finite lifetime of M 2 above the critical field H d . 2. disentangle the xx and yy components of the structure factor at high fields. According to our predictions the xx-component will remain incoherent up to fairly large fields so that a scattering continuum may be observable. In the absence of a crystal field anisotropy the Hamiltonian is H(h) = J n S n · S n+1 − hS z . (A1) The Heisenberg equations of motion read d dt S ± n (t) = i[H(0), S ± n ] ∓ ih S ± n , d dt S z n (t) = i[H(0), S z n ] .(A2) Equations (A2) permit us to express the dynamical susceptibilities for h = 0 in terms of the ones in zero field χ +− (ω, q, h) = χ +− (ω − h, q, 0) , χ −+ (ω, q, h) = χ −+ (ω + h, q, 0) , χ zz (ω, q, h) = χ zz (ω, q, 0). (A3) This implies is that the structure factors in a field are simply the same as in zero field apart from constant shifts in energy. The leading contributions to the dynamical susceptibilities in zero field have been calculated in the framework of the O(3) nonlinear-sigma model approximation to the isotropic spin-1 Heisenberg chain in Refs [23,41]. In follows from these results that the three-particle contributions are very small. We note that the threshold of the M + M + M − three-particle continuum (two S z = 1 magnons and one S z = −1 magnon) is at 3∆ − h, i.e. for any H < H c it still is very slightly higher in energy than the highest energy M − magnon mode. Hence all three magnon modes remain "sharp" for all H < H c . We note that analogous considerations apply in the presence of a single-ion anisotropy in z-direction only (E = 0 in (1.1) and a magnetic field along the z-direction). The dynamical susceptibilities for finite fields can then be expressed in terms of the zero field susceptibilities through the equation of motions for the spin operators. This is quite useful for the Majorana fermion model, where the staggered components of the spin operators are expressed in terms of Ising order and disorder operators. The latter transform nontrivially under the Bogoliubov transformation used to diagonalize the Hamiltonian for nonzero magnetic fields. APPENDIX B: SPECTRAL REPRESENTATION OF CORRELATION FUNCTIONS In this appendix we collect useful formulas for spectral representations of correlation functions in massive, integrable, relativistic quantum field theories. We parametrize energy and momentum of single particle states in terms of a rapidity variable θ E ǫ (θ) = ∆ ǫ cosh θ , P ǫ (θ) = ∆ ǫ v sinh θ .(B1) Here the index ǫ labels the different types of particles and ∆ ǫ are the corresponding spectral gaps. A scattering state of N particles with rapidities {θ j } and indices {ǫ j } is denoted by |θ 1 , θ 2 , . . . , θ N ǫ1,ǫ2,...ǫN .(B2) Its energy and momentum are E({θ j }) = N j=1 ∆ ǫj cosh θ j , P ({θ j }) = N j=1 ∆ ǫj v sinh θ j .(B3) A basis of states is most easily constructed in terms of the generators of the so-called Faddeev-Zamolodchikov algebra Z ǫ1 (θ 1 )Z ǫ2 (θ 2 ) = S ǫ1,ǫ2 ǫ ′ 1 ,ǫ ′ 2 (θ 1 − θ 2 )Z ǫ ′ 2 (θ 2 )Z ǫ ′ 1 (θ 1 ), Z † ǫ1 (θ 1 )Z † ǫ2 (θ 2 ) = Z † ǫ ′ 2 (θ 2 )Z † ǫ ′ 1 (θ 1 )S ǫ ′ 1 ,ǫ ′ 2 ǫ1,ǫ2 (θ 1 − θ 2 ), Z ǫ1 (θ 1 )Z † ǫ2 (θ 2 ) = Z † ǫ ′ 2 (θ 2 )S ǫ ′ 2 ,ǫ1 ǫ2,ǫ ′ 1 (θ 2 − θ 1 )Z ǫ ′ 1 (θ 1 ) +2π δ ǫ1 ǫ2 δ(θ 1 − θ 2 ).(B4) Here S ǫ1,ǫ2 ǫ ′ 1 ,ǫ ′ 2 (θ) is the factorizable two-particle scattering matrix of the integrable quantum field theory. Using the ZF operators a Fock space of states can be constructed as follows. The vacuum is defined by Z εi (θ)|0 = 0 . (B5) Multiparticle states are then obtained by acting with strings of creation operators Z † ǫ (θ) on the vacuum |θ n . . . θ 1 ǫn...ǫ1 = Z † ǫn (θ n ) . . . Z † ǫ1 (θ 1 )|0 . (B6) The resolution of the identity in the normalization implied by (B4) is given by 1 = ∞ n=0 1 n! {ǫj } ∞ −∞ n j=1 dθ j 2π |θ n , . . . , θ 1 ǫn,...ǫ1 ǫ1,...ǫn θ 1 , . . . , θ n | . The two point function of some operator O can now be expressed in the spectral representation as O † (t, x)O(0, 0) = ∞ n=0 1 n! ǫj ∞ −∞ n j=1 dθ j 2π |f O ǫ1...ǫn (θ 1 , . . . , θ n )| 2 exp (−itE({θ j }) + ixP ({θ j })),(B8) where the formfactors are given by H = i 2 dx 3 a=1 v[L a ∂ x L a − R a ∂ x R a ] − 2mR a L a +g dx a J a J a .(C1) We aim to establish that bound states exist for any g > 0, whereas there are no bound states for g < 0. We recall that the Majorana fermion model arises from the spin-Heisenberg Hamiltonian with an additional biquadratic term H biquad = J n S n · S n+1 − b (S n · S n+1 ) 2 ,(C2) where |b − 1| ≪ 1. For b > 1 the model is in a dimerized phase whereas b < 1 corresponds to a Haldane spin-liquid regime. One may establish by using the expressions (2.5) for the spin operators that the case b < 1 (b > 1) corresponds to g < 0 (g > 0). In order to determine whether the current-current interaction leads to the formation of bound states, we first consider the limit of a very anisotropic interaction H ani = i 2 dx 3 a=1 v[L a ∂ x L a − R a ∂ x R a ] − 2mR a L a +g dx J 3 J 3 .(C3) This case can be mapped onto a single massive Majorana fermion plus the massive Thirring model by introducing complex Fermi fields by R 1 = Ψ R + Ψ † R √ 2 , R 2 = Ψ R − Ψ † R i √ 2 , L 1 = Ψ † L − Ψ L i √ 2 , L 2 = Ψ L + Ψ † L √ 2 .(C4) The Hamiltonian density is rewritten as H ani = H Maj + H MTM , where H Maj = iv 2 dx[L 3 ∂ x L 3 − R 3 ∂ x R 3 − 2m v R 3 L 3 ], H MTM = −iv dx Ψ † R ∂ x Ψ R − Ψ † L ∂ x Ψ L + dx m[Ψ † R Ψ L + h.c.] + 2gΨ † L Ψ L Ψ † R Ψ R .(C5) In Eqn (C5) we have dropped a term proportional to dx[Ψ † R Ψ R + Ψ † L Ψ L ] as it commutes with the Hamiltonian. It is well known that in the massive Thirring model there are breather bound states for g > 0, but no bound states exist for g < 0, see e.g. Refs [44]. A different approach is to use large-N methods. If we consider N species of Majorana fermions rather than three, we may decouple the interaction through a bosonic Hubbard-Stratonovich field σ. For even N and g > 0 the problems maps onto the O(N/2) massive Gross-Neveu model, which is known to have bosonic bound states in the large-N limit 45 . APPENDIX D: LOW-ENERGY PROJECTIONS OF THE STAGGERED MAGNETIZATIONS In this appendix we give arguments in favour of the identification (4.6) at low energies and in the vicinity of the Ising critical point at H = H c . We first consider the LG theory and then the Majorana fermion model. Landau-Ginzburg Model It is instructive to examine the evolution of the amplitudes A aα entering the mode expansions (3.7) of the scalar fields ϕ a as the magnetic field is increased. We recall that the critical field is H c = ∆ 1 . In the vicinity of H c we parametrize H = ∆ 1 − δ , δ > 0. (D1) As we are interested only in low energies we may restrict our attention to the "−" modes. From (3.10) we obtain the following expansions in δ (ω − (0)) 2 ≈ 2∆ 1 (∆ 2 2 − ∆ 2 1 ) 3∆ 2 1 + ∆ 2 2 δ , A 2− (0) A 1− (0) 2 = (7∆ 2 1 + ∆ 2 2 )δ 2∆ 1 (3∆ 2 1 + ∆ 2 2 ) .(D2) Eqns (D2) imply that close to H c we have A 2− (0) ∝ ω − (0)A 1− (0) .(D3) As we have seen before, close to H c the x-component of the staggered magnetization ϕ 1 couples to M − with a finite amplitude A 1− (0) given by (3.11). Furthermore we have the identification (4.6) ϕ 1 ∝ σ ,(D4) where σ is the Ising order parameter field. Eqns (D3) and (D4) together suggest that ϕ 2 ∝ ∂ t σ .(D5) This claim may be substantiated further in the limit where one of the zero field gaps is much smaller than the other, i.e. ∆ 1 ≪ ∆ 2 . As we are interested in energies that are small compared to ∆ 2 , we may "integrate out" the highenergy degrees of freedom corresponding to ϕ 2 in the path integral expression for the staggered magnetization n y . Because H c = ∆ 1 is small, we furthermore may take the magnetic field into account perturbatively. The staggered magnetization in y direction is n y (t, x) = ϕ 2 (t, x) . (D6) Averaging n y (t, x) over ϕ 2 , we obtain n y (t, x) 2 = 1 Z Dϕ 2 ϕ 2 (t, x) ×e iS2−2i(H/v) dt1dx1[ϕ2∂t 1 ϕ1] ,(D7) where S 2 = dtdx 1 2v (∂ t ϕ 2 ) 2 − v 2 (∂ x ϕ 2 ) 2 − ∆ 2 2 2v ϕ 2 2 .(D8) The leading contribution occurs in first order in the magnetic field n y (t, x) 2 ≈ − 2iH v dx 1 dt 1 T ϕ 2 (t, x)ϕ 2 (t 1 , x 1 ) 2 ∂ t ϕ 1 (t 1 , x 1 ) = 2H v dx 1 dt 1 G 2 (t − t 1 , x − x 1 ) ∂ t ϕ 1 (t 1 , x 1 ) ≈ 2H v dx ′ 1 dt ′ 1 G 2 (t ′ 1 , x ′ 1 ) ∂ t ϕ 1 (t, x) .(D9) In the last line we have used that the leading contribution to the integral comes from the region t 1 ≈ t, x 1 ≈ x. This shows that the mixing induced by the magnetic field generates a contribution to n y (t, x) proportional to ∂ t ϕ 1 (t, x) at low energies. Majorana Fermion Model Analogous calculations can be performed in the framework of the Majorana fermion model in the case g a = 0, i.e. in the absence of the current-current interactions. In particular, let us consider the case where one of the zero field gaps is much smaller than the other ∆ 1 ≪ ∆ 2 .(D10) As the critical field H c = √ ∆ 1 ∆ 2 is much smaller than ∆ 2 we may treat the magnetic field term perturbatively. We may derive an effective action for R 1 , L 1 only by integrating out R 2 and L 2 (we recall that the third Majorana decouples in the absence of interactions) S eff ≈ S 1 − 1 2 S 2 H 2 , S H = iH dx dτ [L 1 L 2 + R 1 R 2 ] ,(D11) where 2 denotes the expectation value with respect to the second Majorana fermion and S 1 = dτ dx [R 1 ∂ − R 1 + L 1 ∂ + L 1 − i∆ 1 R 1 L 1 ] , ∂ ± = ∂ τ ± iv∂ x 2 . (D12) The Matsubara Green's functions are defined as e.g. G RR (τ, x) = − T τ R(τ, x) R(0) .(D13) Their Fourier transforms are G R2R2 (ω, q) = − iω + vq ω 2 + v 2 q 2 + ∆ 2 2 , G L2L2 (ω, q) = − iω − vq ω 2 + v 2 q 2 + ∆ 2 2 , G R2L2 (ω, q) = + i∆ 2 ω 2 + v 2 q 2 + ∆ 2 2 . (D14) A straightforward calculation then gives L eff = R 1 ∂ − R 1 + L 1 ∂ + L 1 − i[∆ 1 − H 2 ∆2 ]R 1 L 1 .(D15) In other words, integrating out the second Majorana leads to a renormalization of the mass of the first Majorana. We note that the dispersion relation that follows from (D15) agrees with the expansion of ω − (q) (2.20) in the case H ≪ ∆ 2 as it must. What we have shown is that in the case ∆ 1 ≪ ∆ 2 it is simply the first Majorana that becomes critical at H c . The staggered magnetization in x-direction is expressed at low energies by averaging (2.6) with respect to the second and third Majoranas, which gives n x (x) ∝ σ 1 (x) µ 2 (x) µ 3 (x) .(D16) The determination of the operator content of n y (x) at low energies is significantly more involved. In order to obtain the low-energy projection, we need to average with respect to the second and third Majoranas n y (0, 0) ≈ −iH µ 3 × dτ dx L 2 (τ, x) σ 2 (0) 2 L 1 (τ, x) µ 1 (0) + R 2 (τ, x) σ 2 (0) 2 R 1 (τ, x) µ 1 (0) .(D17) The expectation values 2 with respect to the second Majorana can be evaluated by the form factor bootstrap approach by utilizing the results of Refs [20][21][22]. We obtain L 2 (τ, x) σ 2 (0) 2 ≈ D e iπ/4 1 √ vτ + ix e −mr , R 2 (τ, x) σ 2 (0) 2 ≈ D e −iπ/4 1 √ vτ − ix e −mr ,(D18) where D = m 1/8 (4π) −1/2 2 1/12 e −1/8 A 3/2 and r 2 = τ 2 + x 2 /v 2 . Using these results in (D17) we see that the integral is dominated with exponential accuracy by the region τ ≈ 0, x ≈ 0. Hence the operator content of n y is determined by the fusion of the disorder operator µ 1 with the left and right moving fermions L 1 , R 1 . The relevant operator product expansions can be worked out following Ref. [46] L(τ, x)µ(0) ≈ γ √ z σ(0) − mγ v √z σ(0) + 4γ v √ z∂ − σ(0), R(τ, x)µ(0) ≈γ √z σ(0) − mγ v √ zσ(0) + 4γ v √z ∂ + σ(0),(D19) where z = vτ + ix, γ = exp(−iπ/4)/ √ 4π and ∂ ∓ = 1 2 (∂ τ ∓ iv∂ x ). Combining (D18) with (D19) we obtain the desired result n y ∝ ∂ τ σ . (D20) APPENDIX E: DERIVATION OF THE SINE-GORDON MODEL IN THE HIGH-FIELD PHASE FOR WEAK ANISOTROPY In this appendix we show how the Sine-Gordon Hamiltonian emerges as the low-energy effective theory at H > H c in the small anisotropy limit ∆ 2 − ∆ 1 ≪ H − H c . We first present a derivation in the framework of the nonlinear sigma model and then within the Majorana fermion model. Nonlinear sigma model The isotropic spin-S Heisenberg chain in a magnetic field can be mapped onto the O(3) nonlinear sigma model in the continuum limit. Exploiting the integrability of the nonlinear sigma model it was shown in Ref. [13] that for H > ∆ the low-energy regime is described in terms of a free boson H =ṽ 16π dx (∂ x Φ) 2 + (∂ x Θ) 2 . (E1) Here Θ is the field dual to Φ and fulfilsṽ ∂ x Θ = −i∂ τ Φ , ∂ τ Θ = iṽ∂ x Φ .(E2) The low-energy behaviour of spin correlations follows from the correspondence S ± n ≃ (−1) n A exp(±i β 2 Θ) .(E3) The parametersṽ and β were calculated in Ref. [13]. Adding a very small crystal field anisotropy to the Hamiltonian E j [(S x j ) 2 − (S y j ) 2 ],(E4) generates a term proportional to dx cos(βΘ) . The resulting theory is the sine-Gordon model (5.3). The term D j (S z j ) 2 merely leads to a small change in β which we ignore here. Majorana fermion model Our starting point is the Hamiltonian (2.14) describing the two Majorana fermions that couple to the magnetic field in the limit ∆ 2 = ∆ 1 = m, i.e. vanishing gap anisotropy ∆ = 0. Using Ψ R,L = ∞ −∞ dk 2π e ikx c R,L (k) ,(E6) we may express the Hamiltonian as H 12 ∆=0 = ∞ −∞ dk 2π (c † R , c † L ) M c R c L (E7) where M = vk + H −im im −vk + H .(E8) Now we may carry out a Bogoliubov transformation a k b k = cos(ϕ k ) −i sin(ϕ k ) −i sin(ϕ k ) cos(ϕ k ) c R (k) c L (k) (E9) with tan(2ϕ k ) = m vk (E10) to diagonalize the Hamiltonian. We find H 12 ∆=0 = dk 2π (H + sgn(k) m 2 + v 2 k 2 ) a † k a k +(H − sgn(k) m 2 + v 2 k 2 ) b † k b k .(E11) Introducing fermions c and d by c(k) = a k θ(k) + b k θ(−k) , d(k) = b k θ(k) + a k θ(−k) , we may express the Hamiltonian (E11) as H 12 ∆=0 = dk 2π (H + m 2 + v 2 k 2 ) c † (k)c(k) +(H − m 2 + v 2 k 2 ) d † (k)d(k) .(E13) The low-energy modes occur in the lower band in the vicinity of ±k F = ± (H 2 − m 2 )/v 2 . They can be combined into left and right moving Fermi fields by d(x) = exp(−ik F x)R(x) + exp(ik F x)L(x) .(E14) The low-energy effective Hamiltonian is then H ′ = iṽ dx L † ∂ x L − R † ∂ x R ,(E15) whereṽ = v 2 k F /H. We now bosonize the low-energy Hamiltonian using R † (x) ∼ 1 √ 2π exp i Φ(x) + Θ(x) 2 √ 2 , L † (x) ∼ 1 √ 2π exp −i Φ(x) − Θ(x) 2 √ 2 .(E16) Here ϕ andφ are chiral Bose fields fulfilling the commutation relations [ϕ(x),φ(y)] = 2πi. In terms of the canonical Bose field Φ = ϕ +φ and the dual field Θ = ϕ −φ we find H ′ =ṽ 16π dx (∂ x Φ) 2 + (∂ x Θ) 2 .(E17) The high-energy cutoff in this construction is given by the depth of the Fermi sea in the lower band of (E11), which is H − m = H − H c . So far we have neglected the gap anisotropy, i.e. the term H pair = i∆ Ψ † R Ψ † L − h.c.(E18) in the Hamiltonian (2.14). In the next step we take it into account under the assumption that it ∆ is small compared to the cutoff H − H c . In this limit H pair is expressed in terms of the modes as H pair = i∆ ∞ −∞ dk 2π c † R (k)c † L (−k) − c L (−k)c R (k) .(E19) After the Bogoliubov transformation this becomes −i∆ Dropping the "high-energy" filled band as well as the mixed terms (they contribute in higher orders of ∆/(H − H c )) we have H ′ pair ≃ −i∆ ∞ 0 dk 2π vk √ m 2 + v 2 k 2 [d(k)d(−k) − h.c.] .(E21) Expanding around ±k F this can be rewritten in terms of the left and right moving fermions as H ′ pair ≃ i∆ 1 − m 2 H 2 dx RL − L † R † .(E22) Finally, bosonization gives H ′ pair ≃ − ∆ π 1 − m 2 H 2 dx cos Θ √ 2 .(E23) By combining Eqns (E17) and (E23) we see that in the absence of interactions the Majorana fermion model gives rise to a sine-Gordon effective theory at low energies in the parameter regime we have been discussing. The parameter β in (5.3) takes the special free-fermionic value β = 1/ √ 2. Interactions can be treated in a way analogous to the pairing term. If we drop the interaction terms involving the third Majorana (which we assume to have the largest gap) the interaction Hamiltonian reads H int = 2g 3 dx L 1 L 2 R 1 R 2 ,(E24) where g 3 < 0. Expressing this in terms of the complex fields Ψ R,L , carrying out a mode expansion and subsequent Bogoliubov transformation and finally projecting to the low-energy band we obtain H ′ int ≃ 2g 3 v 2 k 2 F m 2 + v 2 k 2 F dx R † RL † L,(E25) where R and L have been introduced in (E14). Bosonization then gives H ′ int ≃ g ′ 3ṽ 16π dx (∂ x Φ) 2 − (∂ x Θ) 2 ,(E26) where g ′ 3 = g3ṽ πv 2 . This term may be combined with (E17) by rescaling the scalar fields in the standard way Φ −→ 1 − g ′ 3 1 + g ′ 3 1 4 Φ , Θ −→ 1 + g ′ 3 1 − g ′ 3 1 4 Θ .(E27) In terms of the rescaled fields the total Hamiltonian takes the form of a sine-Gordon model (5.3) (with a slightly changed velocity) where β = 1 + g ′ 3 1 − g ′ 3 1 4 1 √ 2 < 1 √ 2 .(E28) Hence the sine-Gordon model is in the attractive regime. APPENDIX F: CORRELATION AMPLITUDE IN THE COMMENSURATE-INCOMMENSURATE TRANSITION Let us consider the LG model (3.1) in the U(1) symmetric case ∆ 1 = ∆ 2 = ∆ 3 = m. For H > H c = m the lowenergy degrees of freedom are described by a Luttinger liquid, which can be derived by means of Haldane's harmonic fluid approach 25 as follows. Forming a complex Bose field out of the two components of the LG field that couple to the magnetic field Ψ B = ϕ 1 + iϕ 2 √ 2 ,(F1) and then bosonizing using 25 Ψ † B ≃ ρ 0 + a 0 Π m even e imΘ e iΦ ,(F2) FIG. 1 : 1IV. VICINITY OF THE ISING CRITICAL POINTAs we have seen above, both the Majorana fermion model and the Landau-Ginzburg theory lead to an Ising critical point at some values H c of the applied magnetic field. In the simplest approximations where interactions are neglected, Magnon gaps as functions of the applied field for the Majorana fermion model with ga = 0. The gaps in zero field are chosen as ∆1 = 0.8m, ∆2 = 1.2m and ∆3 = 3.5m. The square indicates the window in energies and fields in which the description in terms of an effective Ising model is appropriate. . H < Hc: Low-Field Phase By the Z 2 symmetry (4.3),(4.4) only intermediate states with an odd number of magnons contribute to the twopoint correlation functions of the Ising order parameter field σ and hence the staggered structure factor S xx (ω, π a0 +q)) at low energies. The leading contributions are 20-23 the spin velocity, µ ∝ ∆ and β is a function of the applied magnetic field H. The high-energy cutoff of the theory (5.3) is H − H c . At low energies the dominant Fourier component of the transverse spin operators is at FIG. 2 : 2Parameter β as function of the applied magnetic field. √ 2 and hence the SGM (5.3) is in the attractive regime.A. Spectrum of the SGM The SGM (5.3) is integrable and many exact results are available. The spectrum of the SGM depends on the value of the coupling constant β. In the so-called repulsive regime, 1/ √ 2 < β < 1, there are only two elementary excitations, called soliton and antisoliton. These have a massive relativistic dispersion, E(P ) = M 2 +ṽ 2 P 2 , (5.14) of breathers, where [x] in (5.15) denotes the integer part of x. The breather gaps are given by M n = 2M sin(nπξ/2) , n = 1, . . . , N , (5.16) E s (θ) = M cosh θ , P s (θ) = (M/ṽ) sinh θ , (5.20) Es(θ) = M cosh θ , Ps(θ) = (M/ṽ) sinh θ , (5.21) E Bn (θ) = M n cosh θ , P Bn (θ) = (M n /ṽ) sinh θ . (5.22) Two-point functions are expressed in terms of a basis of scattering states of solitons, antisolitons and breathers as summarized in Appendix B, see Eqn (B8). After carrying out the double Fourier transform we arrive at the following representation for the imaginary part of the retarded two-point function of the operator O for ω > 0 FIG. 3 : 3S yy (ω, π a 0 ) as a function of ω/M for several values of the applied field H. FIG. 4 : 4S xx (ω, π a 0 ) as a function of ω/M for several values of the applied field H. FIG. 5 : 5Spectral weight ratios I yy ss /I yy B 1 , I xx ss /I yy B 1 and I xx B 2 /I yy B 1 as functions of the parameter β. FIG. 6 :FIG. 7 : 67Path in the h − m0 plane of the transverse Ising model in a magnetic field. Structure of poles and branch cuts of χ E σσ (ω, q) at the points (a)-(g) in the h − m0 plane indicated in Fig. • At low fields there are three coherent modes M − , M + , M 3 . Their respective gaps ∆ − (H) < ∆ + (H) < ∆ 3 (H) are field-dependent. • Above a critical field H d , M 2 develops a finite lifetime via the decay process M + → M − M − M − . M − and M 3 retain infinite lifetimes. f O ǫ1...ǫn (θ 1 . . . θ n ) ≡ 0|O(0, 0)|θ n . . . θ 1 ǫn...ǫ1 . (B9) APPENDIX C: BOUND STATES IN THE MAJORANA MODEL In this appendix we address the question of whether the current-current interaction in the Majorana model leads to the formation of bound states. For simplicity we consider only the SU(2) symmetric case with Hamiltonian we see that interactions in the final state have a dramatic effect: rather than turning on at a finite value as soon as H exceeds H d , the decay rate (3.26) actually vanishes at H d and exhibits the same power law behavior for H > H d as the decay rate in the Majorana fermion model. This may suggest that the dependence of the decay rate on H − H d is a robust result that holds for the underlying lattice model as well.26) Comparing the decay rate (3.26) to the result (3.19) . H > Hc: High-field Phase AcknowledgmentsWe thank R.M. Konikone obtains, after rescaling the fields Φ and Θ, a Lagrangian density of the form 14(F3) Hereṽ = 2v 2 H−m m , ρ 0 is the (dimensionless) boson density, which corresponds to the magnetization per site and Π is the momentum conjugate to Φ. As the LG fields ϕ a are the staggered components of the spin operators we conclude thatwhere β depends on the magnetization and is related to the parameter η of Ref.[14]by β 2 = η. By virtue of (F2) the amplitude A is proportional towhere a is a short-distance cutoff and vertex operators are normalized according to(5.6). The short-distance cutoff isWe note that the short-distance cutoff (F6) diverges as H approaches m as ρ 0 → 0 and (F4) describes the asymptotic behaviour of spin correlation functions at distances much larger than a. Combining (F6) and (F5) we find that in general we have 25Let us now specialize to magnetic fields very close to the critical fieldAs shown in Ref.[25], the parameter β tends to 1 √ 2 , so thatwhere the density is given by 14Combining (F10), (F6) and (F5) we obtainwhere A ′ is a numerical, field independent constant and where we have used that v 2 ma 2 0 ∝ J. The field dependence of the correlation amplitude (F11) is a universal feature of the C-IC transition.Let us apply these ideas to another example of the C-IC transition: the spin-1/2 Heisenberg XXZ chain in a longitudinal magnetic fieldwhere −1 < δ ≤ 1. The model (F12) has a phase transition from a gapless, incommensurate Luttinger liquid phase to a gapped, commensurate, spin-polarized phase at a critical valueSlightly below this transition, i.e.the transverse correlation functions exhibit the following large-distance asymptoticswhere A is Glaisher's constant(4.12). This result is obtained as follows: the dependence on the magnetic field is universal and given by (F11). The numerical coefficient is fixed by noting that the numerical results of Ref.[18]show that A ′ is independent of the value of the anisotropy δ. Finally, we use that the correlation amplitude has been calculated for the free fermion case in Ref.[47]. 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{'abstract': 'We consider quasi one dimensional spin-1 Heisenberg chains with crystal field anisotropy in a uniform magnetic field. We determine the dynamical structure factor in various limits and obtain a fairly complete qualitative picture of how it changes with the applied field. In particular, we discuss how the width of the higher energy single magnon modes depends on the field. We consider the effects of a weak interchain coupling. We discuss the relevance of our results for recent neutron scattering experiments on the quasi-1D Haldane-gap compound NDMAP.', 'arxivid': 'cond-mat/0410487', 'author': ['Fabian H L Essler ', 'Ian Affleck \nDepartment of Physics and Astronomy\nUniversity of British Columbia\nV6T 1Z1VancouverB.CCanada\n', '\nThe Rudolf Peierls Centre for Theoretical Physics\nOxford University\nOX1 3NPOxfordUK (\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nUniversity of British Columbia\nV6T 1Z1VancouverB.CCanada', 'The Rudolf Peierls Centre for Theoretical Physics\nOxford University\nOX1 3NPOxfordUK ('], 'corpusid': 119425550, 'doi': '10.1088/1742-5468/2004/12/p12006', 'github_urls': [], 'n_tokens_mistral': 30045, 'n_tokens_neox': 26000, 'n_words': 16026, 'pdfsha': '01ee739a7216c0974d40ed7c1f2a7f1183c07ec5', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/0410487v1.pdf'], 'title': ['Haldane-gap chains in a magnetic field', 'Haldane-gap chains in a magnetic field'], 'venue': []}

arxiv

Optimal quantum control via genetic algorithms for quantum state engineering in driven-resonator mediated networks Jonathon Brown School of Mathematics and Physics Centre for Quantum Materials and Technologies Queen's University BT7 1NNBelfastUnited Kingdom Mauro Paternostro School of Mathematics and Physics Centre for Quantum Materials and Technologies Queen's University BT7 1NNBelfastUnited Kingdom Alessandro Ferraro School of Mathematics and Physics Centre for Quantum Materials and Technologies Queen's University BT7 1NNBelfastUnited Kingdom Dipartimento di Fisica "Aldo Pontremoli Università degli Studi di Milano I-20133MilanoItaly Optimal quantum control via genetic algorithms for quantum state engineering in driven-resonator mediated networks (Dated: January 26, 2023) We employ a machine learning-enabled approach to quantum state engineering based on evolutionary algorithms. In particular, we focus on superconducting platforms and consider a network of qubits -encoded in the states of artificial atoms with no direct coupling -interacting via a common single-mode driven microwave resonator. The qubit-resonator couplings are assumed to be in the resonant regime and tunable in time. A genetic algorithm is used in order to find the functional time-dependence of the couplings that optimise the fidelity between the evolved state and a variety of targets, including three-qubit GHZ and Dicke states and four-qubit graph states. We observe high quantum fidelities (above 0.96 in the worst case setting of a system of effective dimension 96), fast preparation times, and resilience to noise, despite the algorithm being trained in the ideal noise-free setting. These results show that the genetic algorithms represent an effective approach to control quantum systems of large dimensions. I. INTRODUCTION Quantum state engineering is an essential enabling step for a variety of quantum information tasks, including the initialization of quantum simulators [1], the loading of classical data for quantum-enhanced analysis [2], or the generation of resourceful states in quantum communication networks [3]. In particular, quantum entangled states, which embody a stark departure from classicality, often provide the main resources towards quantum advantage [4]. Therefore the preparation of entangled states in multi-node quantum networks presents a key challenge for realising quantum protocols on near term devices. A successful approach towards state and resource generation consists of two steps: (i) engineer a suitable timedependent Hamiltonian with tunable parameters, as allowed by current experimental capabilities in a given physical platform; (ii) find and implement proper temporal dependencies (pulse shapes) for these parameters by invoking optimal quantum control techniques [5,6]. This results in tailor-made control schemes pertinent to the specific platform at hand. One platform at the forefront of engineering flexible multinode quantum networks, to which such approach has been successfully applied, is that of superconducting quantum circuits [7][8][9][10]. So far superconducting architectures have been employed to realise two-qubit gates using frequencytunable [11][12][13][14] and microwave-driven [15,16] artificial atoms. In addition, demonstration of the coupling between artificial atoms and microwave resonators [17] opened the door for resonator mediated two-qubit gates [18][19][20][21] and provided an alternative platform to study cavity quantum electrodynamics [22] leading to the field of Circuit QED [23]. Whilst extremely flexible in their design, it has been shown however that operating these superconducting systems with a reduced level control is not only desirable, but necessary in some cases [24,25]. Thus finding optimal control protocols that utilize a limited but effective level of control is of practical interest. Optimal control of quantum systems has yielded a range of new methods inspired, in part, by the development of modern machine learning methods. Specifically, neural-networkbased reinforcement learning methods [26,27] have been recognised as useful tools to study quantum systems [28] in a variety of contexts including state transfer [29][30][31][32], quantum thermodynamics [31], circuit architecture search [33], control of dissipative systems [34] and Adaptive Quantum Enhanced Metrology [35]. Reinforcement learning techniques have proven particularly suitable for control problems of increasing dimension when compared to more standard techniques [36]. However, for the most part these techniques cannot be used as closed-loop optimization schemes and therefore are of limited use for optimization on physical quantum systems [37] and have relatively poor convergence guarantees which necessitate, often expensive, hyper parameter tuning steps. This motivates one to consider whether these control problems can be tackled using comparatively less sophisticated techniques that scale well but have greater ease of implementation. With this in mind we consider evolutionary strategies, which have already been proposed as a scalable substitute to reinforcement learning methods [38] and used as an alternative to gradient based parameter updates in both deep reinforcement learning [39] and quantum reservoir computing [40]. In fact, Natural Evolution Strategies (NES) have already been proposed in the context of quantum control [35,41] resulting in the the realisation of fast, high-fidelity single-shot three-qubit gates in frequency tunable superconducting systems [42,43], while extension to four qubit systems has also been tackled effectively using more standard global optimization techniques [44]. More recently these NES techniques have been utilized to address optimal annealling schedules in spin systems [45] and optimal transport of Majorana fermions in superconducting nanowires [46]. The aim of this work is to investigate the potential of these techniques in the specific context of direct state preparation in a drivenresonator mediated, reduced-control multi-qubit network. Here we consider a register of qubits coupled via a common resonator and operated in a regime which facilitates a re-duced level of control, and employ a genetic algorithm to find optimal pulse sequences to drive their dynamics. In order to illustrate our approach, we present efficient control schemes for preparing entangled three-and four-qubit states, including GHZ, Dicke, and graph states, and assess performance against relevant decoherence sources finding the thresholds that limit the quality of our results. The generation of high-quality states, in short times compared to typical multi-qubit circuit timescales, is thus demonstrated while identifying fully the sequence of driving pulses to use. Our approach contributes to the growing argument that a hybrid take to quantum control -that mixes machine learning and optimal control -is a viable route to the engineering of crucial resources for quantum information processing. The remainder of the manuscript is organised as follows. In Sec. II we present the specifics of the system considered and formalise the Hamiltonian. In Sec. III, we present an overview of the Continuous Genetic Algorithm employed then follow by formalising the algorithm for the problem of quantum control in section IV. Sec. V focuses on the presentation of the resulting control schemes for the preparation of three-qubit GHZ and Dicke states as well as a specific instance of a fourqubit graph states. In Sec. V C, the effect of decoherence is investigated. Sec. VI offers our closing remarks and a forward look. II. SYSTEM We consider a system composed of N identical and noninteracting qubits coupled via a common single-mode driven resonator. We model such system with the Hamiltonian H = H 0 + H int + H d , where H 0 = ω c a † a + N j=1 ω j σ + j σ − j ,(1)H int = N j=1 g j (a † σ − j + aσ + j ),(2)H d = ξ(ae iω d t + a † e −iω d t ).(3) In writing these Hamiltonians, we have assumed the rotating wave approximation and used units such that = 1. Here a denotes the annihilation operator of the resonator mode, whereas σ + j = |1 0| and σ − j = |0 1| are the raising and lowering operators of the j-th qubit. Moreover, ω c denotes the resonator frequency, ω j the transition frequency of the j-th qubit and g j its coupling strength with the resonator. The driving amplitude (assumed to be real) and the carrier frequency of the drive are indicated with ξ and ω d respectively. The use of a resonator to mediate two-qubit gate interactions is well studied. For example, in the context of superconducting systems the Resonator-Induced Phase (RIP) gate [18][19][20] utilizes two qubits dispersively coupled to a common microwave resonator. In such a dispersive regime, the coherent qubit-resonator interaction term becomes negligible leading to an effective qubit-qubit interaction term. Whilst this can be beneficial for protection against decoherence there are two main drawbacks: 1) The dispersive coupling precludes real photon processes between the resonator and each qubit which necessitates the use of local qubit drives in order to introduce energy into the system. This subsequently makes this regime adverse to scaling to larger numbers of qubits. 2) The large detunings involved result in small effective qubit-qubit coupling strengths at the expense of gate operation time. On the other end of the spectrum, resonant regimes have been considered to selectively tune qubits in and out of coupling with a common resonator [47]. In addition, external microwave drives are also commonly used such as with the the cross-resonance gate [15,16]. In contrast to such previous works, we assume in H a fully resonant regime which allows us to adapt a limited control scheme, in that the local qubit drives can be replaced by a single resonator driving term where the amplitude of this drive and the qubit-resonator couplings are tunable. This facilitates the possibility of multi-qubit interactions that operate on timescales that are much shorter than typical multi-qubit gate times. Taking the interaction picture with respect to the frequency of the harmonic mode and usingH = e iRt He −iRt − R, where R = ω c a † a + N j=1 σ + j σ − j , we get H = N j=1 δ j σ + j σ − j + g j (a † σ − j + aσ + j ) +ξ(ae iδ d t +a † e −iδ d t ),(4) where δ j = ω j −ω c and δ d = ω d −ω c are the detunings between the j th qubit and the harmonic mode, and between the drive frequency and the harmonic mode respectively. The utility of this transformation becomes apparent when one assumes full resonance between the qubit transition frequencies, drive frequency and resonator frequency -namely, ω c = ω d = ω j , ∀ j = 1, ..., N. The Hamiltonian thus takes the simpler form H = N j=1 g j (t)(a † σ − j + aσ + j ) + ξ(t)(a + a † ),(5) which comprises N + 1 terms embodying an equal number of controls, where we assume each of the qubit-cavity couplings g j (t) and the drive amplitude ξ(t) to be time-dependent controllable parameters. The above Hamiltonian governs the dynamics of the system via the time-dependent Schrödinger equation. Thus if the system is initially in some state |ψ(t 0 ) , then the time-evolved state of the system at any future time t > t 0 is given by [48] |ψ(t) = T e i t t 0H (t )dt |ψ(t 0 ) .(6) The goal here is to find the optimal functional forms for g j (t) and ξ(t), such that the system is dynamically steered in some desired way. Specifically, we are interested in state preparation within the qubit subspace, so we first determine the reduced state of the qubit network ρ Q (t) = Tr c |ψ(t) ψ(t)| = ∞ i=0 c i|ψ(t) ψ(t)|i c ,(7) and work to find g j (t) and ξ(t) such that the fidelity F (ρ Q , σ) = ψ σ |ρ Q |ψ σ (8) is maximized, where |ψ σ is the target state of interest. It is worth stressing that our approach would work equallymutatis mutandis -with mixed target states. III. THE CONTINUOUS GENETIC ALGORITHM Evolutionary strategies are a class of direct search optimization techniques, drawing inspiration from Darwinian evolution, that have recently been proposed as a viable substitute for gradient based parameter optimization in Neural Networks [39] and quantum reservoirs [40], as well as a scalable alternative to Reinforcement learning techniques [38]. Of particular interest to continuous-control problems is the so-called "Continuous Genetic Algorithm" (CGA) [49], which generalises the more traditional Discrete Genetic algorithm to allow for continuous parameter values. Consider an optimization problem with N var parameter variables p i . We call a specific instance of these parameters, C = [p 1 , p 2 , ..., p N var ], a Chromosome. This embodies one proposed solution to the optimization problem. One then defines a Fitness function, f (C) = f (p 1 , p 2 , ..., p N var ): R N var → R. This function will be determined by the optimization task under consideration and will assign a numerical score to each proposed solution (chromosome) depending on its usefulness (fitness) with respect to the specified task at hand. In the name of generality we assert that p i ∈ [−1, 1], where the parameter values are suitably scaled within the fitness function. We call each new iteration of the algorithm a Generation where the algorithm proposes several chromosomes to make up a Population. The algorithm can be qualitatively summarised using the following steps: 1. Initialization. Define the fitness function (determined by the optimization task) and fix the hyper-parameters for the genetic algorithm. One then generates an initial population with N pop chromosomes, typically randomly, which acts as the zero-th generation of the algorithm. 2. Natural Selection. The fitness of each chromosome is assessed with a call to the fitness function. The population is then ranked based on these fitness values and the N survive highest scoring chromosomes are chosen to survive, while the rest are discarded to be replaced by new offspring chromosomes in the next generation. 3. Pairing. From the N survive surviving chromosomes chose N parents pairs of parent chromosomes in order to produce (N pop − N survive ) offspring to repopulate. Parent chromosomes are sampled probabilistically based on their relative fitness, such that the fittest chromosomes reproduce more frequently, where cloning (using the same chromosome for both parents) is disallowed. 4. Mating. Here, each pair of parent chromosomes is combined in some manner to produce enough offspring to replenish the population. In the simplest case, one can consider cutting the parent chromosomes in half and producing two offspring, made up of the two possible combinations of these halves. Alternatively, one could pick indices along the chromosomes at random, and generate two offspring by first copying the parent chromosomes, then swapping the parameter values corresponding to these indices between the two copies. This direct transfer of parameter values to generate offspring is the simplest and most obvious way of mating two parent chromosomes. However this approach simply provides offspring chromosomes made up entirely of parameter values that were already present in the previous generation and, as such, introduces no new "genetic material" in the form of novel parameter values for a given parameter. To tackle this we can implement a combination of the form p i, offspring = βp i, parent 1 + (1 − β)p i, parent 2 .(9) This allows us to introduce an element of exploration by allowing not only parameter values present in the previous generation but also any continuous value in between, modulated by random variable β ∈ [0, 1]. 5. Mutation. This final step introduces further exploration into the search by choosing, at random, a number of elements within each chromosome to be replaced with a new random value. The rate of this mutation is set by a parameter 0 < α < 1, which determines the number of indices to be targeted relative to N var . This mutation step is applied to the entire population except for the single chromosome with the highest fitness, which is known as Elitism. This is important to ensure theoretical guarantees of convergence. Specifically it ensures that the maximum achieved fitness is always at least maintained in new generations. The algorithm repeats steps 2-5 until convergence or an acceptable level of fitness has been achieved. A key point to note here is that the CGA is a stochastic optimization process so with finite time the convergence to any absolute optimal strategy is never guaranteed, however on average one would expect an increase in performance as the computational time grows. The aforementioned hyper-parameters associated with CGA then are: population size N pop , number of survivors N survive to keep in each iteration, number of parental pairs to mate N parents and the mutation rate α. Also, as discussed above, we have some freedom in how we implement the mating procedure. IV. CONTINUOUS GENETIC ALGORITHMS FOR OPTIMAL QUANTUM CONTROL In order to apply the CGA we first must formulate the optimal control problem in a suitable manner. As is common practice, we discretize the evolution time into T time intervals of equal duration τ -which is manually chosen -and assume the functional form for each control to be defined by its values at the T + 1 times t = 0, τ, 2τ, ..., T τ (connecting the latter with a simple tanh function, similar to the use of piecewise error functions [42,43]. See Appendix A for specifics). Therefore, given that each control function is completely defined by these T + 1 values, and there are N + 1 controls in total as per Eq. (5), then the total number of parameters that we have to optimize over is N var = (N + 1)(T + 1) (which as said is the length of the chromosomes). We then need to outline how we asses the fitness of each chromosome and in doing so define the optimization task. A. The Fitness Function As said, each chromosome is a sequence of N var parameter values, and there are N + 1 control pulses. Therefore we first break each chromosome up into N + 1 sequences where the first T + 1 parameters determine the first control pulse and so on, as in Fig. 1. Then the parameter values corresponding to each control are scaled accordingly where each of the qubitresonator couplings assume a common range g j ∈ [−g 0 , g 0 ], and the drive amplitude is in the range ξ ∈ [−ξ 0 , ξ 0 ]. The scaled parameter values are then used to build the control pulses as in Appendix A, which fully define the time dependent HamiltonianH(t) in Eq. 5. After defining an initial state |ψ(t = 0) the system evolves unitarily according toH(t) (as in Eq. 6), where we keep track of the state of the system at every intermediate time between t = 0 and t = τT . For each state in this history of states, we trace out the cavity system to obtain a state history for the qubit subspace, ρ Q (t). Next we calculate F (ρ Q (t), σ), which is the time dependent fidelity induced by the specific control pulses. In order to assign a numerical fitness value to the chromosome we simply take the maximum fidelity achieved in the qubit subspace throughout the induced dynamics, i.e Chromosome fitness = max t F (ρ Q (t), σ) .(10) In fact, the function actually used is a slight variation of that presented above necessitated by the specific details of the simulation and the ability to "extract" the state with the highest fidelity (see Appendix C for details). V. RESULTS A. Analysis of performance: case studies Below we outline the optimal control schemes found when applying the CGA approach to prepare genuinely entangled 3 and 4 -qubit states from completely separable initial states. In doing so this acts as a proof of principle -for both the HamiltonianH, and the optimization method -with respect to entanglement generation in general and state preparation specifically. Below we consider the physically reasonable maximum coupling and driving g 0 , ξ 0 = 2π 200MHz, and total time-scales τT ≤ 10ns (see appendix B for full details). An important point to make is that such timescales are considerably shorter than the typical gate times associated with two-qubit operations on superconducting platforms, which are commonly of the order of 100ns [8,9]. This means that each of the following protocols operate on a much faster timescale than their circuit counterparts which prepare the same state. We specifically consider 3 states: GHZ state, three-qubit Dicke states with two excitation, and the 4 qubit "Box" Cluster state. a. GHZ state. GHZ states are relevant genuinely tripartite entangled states [50] defined by |GHZ = |000 + |111 √ 2 ,(11) We assume the system to initially be in the state |ψ 0 = |010 Q |0 cav . We use |010 as the initial state of the qubit network in this specific instance, as opposed to simply the global vacuum, since the latter has non-zero fidelity to the target GHZ state and thus starts the optimization in at undesirable local maxima. Such initial preparation is not so restrictive given that single qubit rotations are easily implemented in many quantum systems when compared with multi-qubit operations. We use values of τ = 1ns for the duration of each time interval and a total number of intervals afforded to the optimizing agent of T = 10. The results are shown in Fig. 2 where a highest fidelity of F (ρ Q , σ) = 0.9746 is achieved in ≈ 8ns. . Hyper-parameters τ = 1ns and T = 10 are used here, with maximal fidelity achieved, and maintained, within ≈ 8ns. On the fidelity plot, the horizontal green line is drawn at c n = 1/2 and all fidelities above this exhibit GME (green region). The blue region c n = 3/4 on the other hand shows those fidelities that exhibit both GME and GHZ-class entanglement in particular. b. Three-qubit Dicke state. Dicke states embody another class of genuinely tripartite entangled states, inequivalent to GHZ states [50], which have been experimentally realised, projected onto lower dimensional entangled states and employed in open destination teleportation and telecloning [51,52]. Specifically we consider the three-qubit two-excitation Dicke state, sometimes referred to as a flipped W state, defined as |D (2) 3 = (|011 + |101 + |110 ) √ 3 .(12) We assume initial state preparation of |000 Q |0 cav , and allow more fine control this time with τ = 0.5ns and T = 20. The maximum fidelity achieved in this instance is F (ρ Q , σ) max = 0.9896 within ≈ 7ns, as shown in figure 3. c. Box Cluster State. Cluster states are a class of graph states useful for measurement based quantum computing [54][55][56][57]. Cluster states are represented by graphs of interconnected vertices, where if one starts with each qubit in the ground state, they are defined procedurally by applying a Hadamard gate to each qubit (represented by the vertices of the graph) then applying conditional-phase gates between qubits whose vertices are connected by edges. Here we consider a so-called Box cluster state which is defined by four vertices connected by four edges to form a square. This can be written as, |ψ Box = CZ 41 Π 3 j=1 CZ i,i+1 Π 4 j=1 H j |0000 1234 ,(13) where CZ i, j is the controlled-Z gate between qubits i and j and H i is the Hadamard gate on qubit i. Explicitly, the state reads: |ψ Box = i, j,k,l=0,1 (−1) x i +x j +x k +x l |x i x j x k x l .(14) As before, assuming τ = 0.5ns and T = 20 and starting from the initial vacuum state a maximum fidelity of F (ρ Q , σ) = 0.9642 was achieved (cf. Fig. 4), this time requiring the full 10ns to achieve and maintain maximal fidelity. B. Entanglement Detection A central question in any state-engineering scheme is the characterization of the features of the state that has been synthesized. Within the context of our investigation, the core aspect to address is the quantification of multipartite entanglement. The task of accurately determining the amount of entanglement in a given quantum state, which is challenging in general, is made even more difficult in multipartite settings due to the hierarchical structure of entanglement in many-body systems [4,[59][60][61][62] and the need to construct convex-roof extensions of any pure-state quantifier, when dealing with mixed states [63][64][65]. A significant tool in these endeavours is embodied by entanglement witnesses, which often offer experimentally vi-able ways of detecting (or even quantifying [66]) entanglement [58,67] that are suitable for mixed states and have been used to detect genuine multipartite entanglement (GME) already for GHZ class, W-class and graph states [53,68,69]. Whilst high fidelity with a maximally entangled state is a good indicator that we have generated entanglement close to the right structure, it is useful to quantitatively check for GME in the system. A natural approach is to use so-called "fidelity based" entanglement witnesses. These are of the general form W F = c n 1 − |ψ ψ|,(15) where |ψ is the state of interest and c n is the maximal overlap between |ψ and all bi-separable states. Thus, any state for which Tr(ρŴ F ) ≥ 0 is bi-separable and consequently Tr(ρŴ F ) < 0 indicates genuine multipartite entanglement. The overlap values c n have already been calculated for 3-qubit GHZ and W states, and 4-qubit linear cluster states [53,58], which up to local unitaries and swaps coincide exactly with the three cases considered above. In light of the analysis reported in Figs. 2, 3, and 4 these witnesses are readily implemented. Considering Tr(ρŴ F ) = c n Tr(ρ) − Tr(ρ|ψ ψ|) < 0, where of course Tr(ρ) = 1 and Tr(ρ|ψ ψ|) = ψ|ρ|ψ is the fidelity between ρ and |ψ . This is clearly fulfilled when F (ρ, |ψ ) > c n so we can simply highlight the threshold value (c n ) of fidelity above which GME can be detected. The region where this happens is highlighted in green in each of the figures. One may also be interested in not only ensuring that the state has GME but also, in the GHZ case, if we have generated GHZ-class entanglement [70]. In this case, c n will be the maximal overlap between |GHZ and all W-class entangled states. This region is highlighted in blue in Fig. 2. Finally, other constructions of witnesses are useful for specific states; for example, witnesses based on collective spin operators have been used to more efficiently detect GME in symmetric Dicke states as in [53]. Here the witness takes the formŴ = b s 1 − (Ĵ 2 x +Ĵ 2 y ),(16) whereĴ k is the collective k = x, y spin operator [68]. It is not so straightforward here to place a delineation at a given fidelity as with the fidelity based witnesses, so we first plot Tr(ρŴ) as an inset within figure 3 and highlight the region for which Tr(ρŴ) < 0. This region is then shown as the grey hatched area on the larger fidelity plot. It can be seen that both the fidelity based witness and the collective spin based witness detect GME at strikingly similar times. C. Decoherence So far, we have exclusively considered closed system dynamics and as such the optimality of the control schemes presented is limited to the noiseless case. It is of interest then to assess how these control schemes perform in the presence of decoherence. Specifically we can write the following Lindblad master equation to model the effect of decay and dephasing acting on each of the constituent subsystems [71,72] ρ = −i H , ρ +κD [a] ρ+ N j γ j D σ − j ρ + 2γ φ, j D σ + j σ − j ρ ,(17) where κ is the cavity decay rate, γ φ, j and γ j are the dephasing and decay rate for the j th qubit, respectively, and we have introduced the superoperators D[Q]ρ = QρQ † − 1 2 Q † Q, ρ(18) for an arbitrary operator Q. If we adopt physically reasonable values for these rates we can obtain an estimate of the performance of a physical system. For example, considering superconducting systems we set κ = 2π×5 kHz for the cavity damping and the typical values of 2π×300 KHz and 2π×5 MHz for the dephasing and decay rate for each of the qubits [17,23]. In these systems, the use of high-Q cavities and low temperatures leads to a reduced cavity decay and dephasing rate. The qubit decay is thus the main source of decoherence. In these conditions, the effects of noise is reported in Fig. 5. We can clearly see that the control protocols are almost completely insensitive to cavity decay and qubit dephasing, whilst still reasonably robust against qubit decay. VI. CONCLUSIONS We have investigated the use of evolutionary algorithms, which are well-established strategies for both classical and quantum control, for direct quantum state engineering and multipartite entanglement generation. Specifically, we considered the effective Hamiltonian of a network of noninteracting qubits jointly addressed by a common driven bus and applied a genetic algorithm to identify the set of optimal pulses to drive the evolution of the qubit register. This has allowed us to successfully put forward robust protocols for the engineering of three-and four-qubit states that play a crucial role in quantum metrology and computing, including Dicke and cluster states. The protocols, which offer significant robustness to the most common and crucial sources of imperfection, provide further evidence of the benefit of a hybrid approach to quantum control that puts together the insight provided by machine learning strategies to well-established schemes for optimal control. The extension of these approaches to larger registers and non-unitary dynamics will pave the way to quantum process engineering enhanced by machine learning and optimised by quantum control methods. A common approach to optimal control problems is to break the dynamics up into T time intervals of equal duration τ, then at each interval assign a constant value to each control parameter. This results in piece-wise constant (PWC) control "pulses", u(t) → {u i } i=0,...T −1 , where u is routinely used to denote a generic control function. Such discretization is useful for application of Reinforcement Learning techniques as in [28][29][30][31][32]. Whilst useful, this type of functional form includes discontinuities, often in the form of instantaneous jumps in the control, which is experimentally unfeasible and subsequently requires some method of smoothing post-optimization often to the detriment of performance. Here we use a generalised form of this discretisation, where instead of allowing the optimization to chose the T constant values corresponding to the control value during each time interval, we allow it to chose the T + 1 values corresponding to the start(/end) of each interval, then connect each value with a smooth, time dependent function during the interval. Namely, the functional time dependence of each control pulse is made to be a clipped tanh function centred in the middle of the time interval. For example consider a simple tanh function, shown in figure A. If we clip the tanh function within windows of different widths centred around zero we can make a "step-like" like function with varying severity. This is tantamount to scal-FIG. 6. Comparison of how the clipping window affects the shape of the interconnecting tanh functions used to construct the control functions. In (a) the tanh function is shown with over-laid dashed lines representing different widths of this clipping window. From (a) we can see how the clipping window width determines the severity of the time dependence. Namely, taking the largest width (grey) leads to a more "step-like" dependence, as evidenced in (b) where each of the clipped functions are scaled into a time interval of equal duration τ and re-scaled to account for the error ε. (a) shows how the error ε increases as the clipping window narrows, i.e the narrowest (yellow) window has the largest error thus requiring the most re-scaling. ing the tanh function along the y-axis and clipping it at ±1. So the time dependence within time interval i, t i ≤ t ≤ t i+1 can be written as, f i (t) =              0 if t < t i (u i+1 − u i ) S tanh(Wt−(t i + τ 2 ))+1 2 + u i if t i ≤ t < t i+1 0 if t ≥ t i+1 (A1) where u i , u i+1 is the value of the control at times t i , t i+1 respectively, W determines the severity of the step, and S is a scaling factor introduced to deal with the error ε, as in fig A. Therefore the complete functional for a general control, under this scheme, is given by f (t) = T i=0 f i (t),(A2) where, again, T is the number of time intervals. Full functional forms can be clearly seen in figures 2, 3, and 4. In each case the population size for the algorithm was N pop = 48 and was determined by the number of CPUs available, since each chromosome in the population was evaluated in parallel to significantly speed up computation time. The number of survivors was fixed at N pop /2 = 24, from which N pop /4 = 12 pairs of parents were selected according to a probability distribution determined by their relative fitness. Each pair of parent chromosomes produced 2 offspring chromosomes to re-populate. The mating procedure involved 2 steps. After making a one copy of each parent chromosomes: 1) Each separate section of T + 1 elements -corresponding to the different control pulses -were completely swapped between the two copied chromosomes with ≈ 50% probability, otherwise they were left unchanged. 2) random indices were then selected and the combination given by equation 9 was applied, where β was randomly sampled from [0,1] for each separate index, in an inverse manner to the two copies. This results in 2 new offspring chromosomes that are completely complementary to one and other. Mutation was then applied by selecting chromosomes (apart from the fittest) at random and random indices within these chromosomes to replace with completely random values. The rate α determined the total number of parameter values within the entire generation that were flipped and generally assumed a value of α ≈ 0.2. Appendix C: Simulation Details Simulations were carried out using the Numerical Schrödinger Equation solver within the QuTiP package in python [73]. Thus, for the simulations we had to approximate the harmonic mode by a d-dimensional harmonic system, however care must be taken. Since the cavity is assumed to be resonantly driven, if we use a low value of d and/or too strong a drive, then the d-dimensional harmonic system will quickly become saturated, and in fact early optimizations used this fact to produce extremely good results with low dimensional cavities. Of course when one then simulated these controls with higher levels in the cavity the performance was destroyed and so the results were neither realistic nor practically useful. One can avoid this in two ways: 1) By encoding enough redundancy in the cavity by using very high values of d, which increases computational cost; or 2) Including a term in the numerical fitness function that punishes population of the higher energy states of the cavity approximation. Here we apply both by choosing d ∈ [5,6] as well as including the term φ 1 = − ν τT τT 0 n|ρ cav (t)|n dt,(C1) where |n is the highest excited state within the d-dimensional approximation and ρ cav is the reduced state of the cavity. ν determines the strength of punishment and was set at ν = 0.1. (Practically, since the dynamics is solved numerically the integral was actually a summation). Another issue one encounters with this type of simulation is that, if we assess fitness based on the outright maximum value of the fidelity during the induced dynamics, then it is possible to observe successful control schemes that induce sharp spikes in the fidelity landscape. If the control scheme induces such spikes on a time scales shorter than that required to completely "switch off" all of the controls then it is impossible to extract the state of maximum fidelity and the control sequences again cease to be practically useful. We can combat this by including an additional term in the fitness function that rewards control schemes that briefly maintain near maximal fidelity for a short time, allowing us to selectively uncouple the cavity whilst maintaining the state of maximal fidelity in the qubit subspace. The term used was φ 2 = + µ mτ t max +mτ t max F (ρ Q (t), σ)dt,(C2) where m is the number of time intervals of length τ to include in the numerical bonus beyond which we no longer care if the fidelity deteriorates. µ is again a variable that determines the relative importance of maintaining fidelity after maximum and was set to µ = 0.5. Thus the actual reward function employed was Chromosome fitness = F (ρ Q (t max ), σ) + φ 1 + φ 2 (C3) where t max is the time at which maximum fidelity is achieved during the induced dynamics. This leads to control schemes that maximise and briefly maintain fidelity allowing us to selective switch of the couplings, whilst also only exclusively utilising lower lying levels of the harmonic mode. Thus in principle the resulting controls could be practically implemented and yield identical performance. Clearly, these considerations are necessitated by the use of simulation and the case is much simpler if one wishes to use the algorithm on a physical system, however in this case the ability to parallelise the computational steps, one major advantage of the Genetic Algorithm, is suppressed. FIG. 1 . 1Schematic representation of how the variables within each chromosome (visually represented by the sequence of connected squares) are assigned to which control pulse, depicted along the top of the diagram. Along the bottom shows sample control pulses generated using random parameter values and the function construction method outlined in Appendix A. FIG. 3 . 3Results for three-qubit Dicke state. We have the optimal control pulses [Top], the fidelity in the qubit subspace during the dynamics [Middle] and the matrix histogram for the target state (left) and the state with the highest fidelity during the dynamics (right)[Bottom]. The hyper-parameters τ = 0.5ns and T = 20 have been used for these calculations. The corresponding maximal fidelity was achieved within ≈ 7ns. The green region on the fidelity plot highlights fidelities for which genuine multipartite entanglement (GME) is detected based on the fidelity based witness Eq. (15) with c n = 2/3. The inset shows Tr(ρŴ) at each instant of time, and the grey region on both the inset plot and the fidelity plot show the temporal region where GME is detected via the collective-spin witness in Eq. (16) with b s = 3.12[53]. FIG. 4 . 4Results for a four-qubit Box Cluster state. We have plotted the optimal control pulses [Top panel], the fidelity in the qubit subspace during the dynamics [Middle panel] and the matrix histogram for both the target state (left-most figure) and the state with the highest fidelity during the dynamics (right-most one) [Bottom panel]. We have used the hyper-parameters τ = 0.5ns and T = 20 and the maximum fidelity was achieved within ≈ 10ns. The values of fidelity for which GME is detected via the fidelity based witness Eq. (15) with c n = 1/2[58] are shown in the green region. FIG. 5 . 5A comparison between the fidelity achieved for the (Top) GHZ state preparation, (Middle) Dicke state preparation and (Bottom) Box Cluster state preparation schemes in the presence of cavity decay and qubit dephasing only (Grey) and qubit decay only (Red). Each plot shows the maximum achieved fidelity in the latter case. The horizontal blue line shows the maximum fidelity achieved in the ideal, noiseless case. ACKNOWLEDGMENTS We acknowledge support from the European Union's Horizon 2020 FET-Open project TEQ (766900), the Horizon EuropeEIC Pathfinder project QuCoM (Grant Agreement No. 101046973), the Leverhulme Trust Research Project Grant UltraQuTe (grant RGP-2018-266), the Royal Society Wolfson Fellowship (RSWF/R3/183013), the UK EPSRC (grant EP/T028424/1) and the Department for the Economy Northern Ireland under the US-Ireland R&D Partnership Programme. Appendix A: Functional Form for controls FIG. 2. Results for three-qubit GHZ state. We have the optimal control pulses [Top], the fidelity in the qubit subspace during the dynamics [Middle] and the matrix histogram for the target state (left) and the state with the highest fidelity during the dynamics (right) [Bottom] Appendix B: Algorithm Implementation Details. Quantum simulation. M Iulia, Sahel Georgescu, Franco Ashhab, Nori, 10.1103/RevModPhys.86.153Rev. Mod. 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{'abstract': 'We employ a machine learning-enabled approach to quantum state engineering based on evolutionary algorithms. In particular, we focus on superconducting platforms and consider a network of qubits -encoded in the states of artificial atoms with no direct coupling -interacting via a common single-mode driven microwave resonator. The qubit-resonator couplings are assumed to be in the resonant regime and tunable in time. A genetic algorithm is used in order to find the functional time-dependence of the couplings that optimise the fidelity between the evolved state and a variety of targets, including three-qubit GHZ and Dicke states and four-qubit graph states. We observe high quantum fidelities (above 0.96 in the worst case setting of a system of effective dimension 96), fast preparation times, and resilience to noise, despite the algorithm being trained in the ideal noise-free setting. These results show that the genetic algorithms represent an effective approach to control quantum systems of large dimensions.', 'arxivid': '2206.14681', 'author': ["Jonathon Brown \nSchool of Mathematics and Physics\nCentre for Quantum Materials and Technologies\nQueen's University\nBT7 1NNBelfastUnited Kingdom\n", "Mauro Paternostro \nSchool of Mathematics and Physics\nCentre for Quantum Materials and Technologies\nQueen's University\nBT7 1NNBelfastUnited Kingdom\n", 'Alessandro Ferraro \nSchool of Mathematics and Physics\nCentre for Quantum Materials and Technologies\nQueen\'s University\nBT7 1NNBelfastUnited Kingdom\n\nDipartimento di Fisica "Aldo Pontremoli\nUniversità degli Studi di Milano\nI-20133MilanoItaly\n'], 'authoraffiliation': ["School of Mathematics and Physics\nCentre for Quantum Materials and Technologies\nQueen's University\nBT7 1NNBelfastUnited Kingdom", "School of Mathematics and Physics\nCentre for Quantum Materials and Technologies\nQueen's University\nBT7 1NNBelfastUnited Kingdom", "School of Mathematics and Physics\nCentre for Quantum Materials and Technologies\nQueen's University\nBT7 1NNBelfastUnited Kingdom", 'Dipartimento di Fisica "Aldo Pontremoli\nUniversità degli Studi di Milano\nI-20133MilanoItaly'], 'corpusid': 250334828, 'doi': '10.1088/2058-9565/acb2f2', 'github_urls': [], 'n_tokens_mistral': 20352, 'n_tokens_neox': 17132, 'n_words': 10052, 'pdfsha': '6b1f5bf5eb0207c1d256b33953a6fcc59cf89eff', 'pdfurls': ['https://export.arxiv.org/pdf/2206.14681v3.pdf'], 'title': ['Optimal quantum control via genetic algorithms for quantum state engineering in driven-resonator mediated networks', 'Optimal quantum control via genetic algorithms for quantum state engineering in driven-resonator mediated networks'], 'venue': []}

arxiv

Emergence of field-induced memory effect in spin ices Pramod K Yadav *email:pramodyadav@iisc.ac.in Centre for Nano Science and Engineering Indian Institute of Science Bangalore-560012India Rajnikant Upadhyay School of Materials Science and Technology Indian Institute of Technology (Banaras Hindu University) 221005VaranasiIndia Rahul Kumar School of Advanced Materials and Chemistry and Physics of Materials Unit Jawaharlal Nehru Centre for Advanced Scientific Research Bangalore-560064India Pavan Nukala Centre for Nano Science and Engineering Indian Institute of Science Bangalore-560012India Chandan Upadhyay School of Materials Science and Technology Indian Institute of Technology (Banaras Hindu University) 221005VaranasiIndia Emergence of field-induced memory effect in spin ices 1 Out-of-equilibrium investigation of strongly correlated materials deciphers the hidden equilibrium properties. Herein, we have investigated the out-of-equilibrium magnetic properties of polycrystalline Dy2Ti2O7 and Ho2Ti2O7 spin ices. The experimental results show the emergence of magnetic field-induced anomalous hysteresis observed only in temperature/magnetic field-dependent ac susceptibility measurements. The observed memory effect (anomalous thermomagnetic hysteresis) strongly depends on the driving thermal and non-thermal variables. Contrary, in the absence of the magnetic field, dipolar interaction induced Ising paramagnetic to spin ice crossover develops a liquid-gas transition type hysteresis below 4 K. Unlike field-induced hysteresis, it shows weak dependency on thermal and nonthermal variables. Due to the non-colinear spin structure, the applied dc bias magnetic field produces quench disorder sites in the cooperative Ising spin matrix and suppresses the spin-phonon coupling. These quench disorders create dynamic spin correlations governed by quantum fluctuations, having slow spin relaxation and quick decay times, which additionally contribute to ac susceptibility. The initial conditions and measurement protocol decide the magnitude and sign of this dynamical term contributing to ac susceptibility. It has been suggested that such kind of out-of-equilibrium properties emerge by the cumulative effect of geometric frustration, disorder, quantum fluctuations, and the cooperative nature of spin dynamics of these materials. I. INTRODUCTION Armed with an in-depth understanding of the exciting properties of strongly topological materials, new design concepts are being developed to emulate alternate computing devices. [1][2][3][4] Exotic properties in topological materials emerge from the complex interplay between reduced dimensionality, fundamental symmetries, electron-electron interactions, relativistic spin-orbit interactions, quantum confinement, quantum coherence, quantum fluctuations, and topology of wavefunctions. [5,6] Geometrically frustrated Dy2Ti2O7 (DTO) and Ho2Ti2O7 (HTO) pyrochlore oxides are one of them and are known for their lowtemperature exotic behavior. In these materials, exotic behavior emerges from the cumulative effect of the lattice architecture of Dy/Ho sites, strong crystal field anisotropy, and magnetic dipolar interaction. [7][8][9][10][11][12] These factors enable to show various phenomena ranging from-crystal field anisotropy and local spin structure-induced dielectric relaxations [13][14][15][16][17], spin ice state [7,11], and emergent magnetic monopole [9,10,18] in different temperatures regimes. Furthermore, DTO and HTO possess quantum-tunnelingdominated spin relaxation regimes up to 13 K and 30 K, respectively [19][20][21][22][23]. In this temperature regime, spin freezing temperature follows (x-xc) 1/2 type variation with the magnetic field (non-thermal variables) as observed for quantum phase transitions. [24][25][26] Recently, Samarakoon et al. performed a low-temperature ultrasensitive magnetic noise experiment on DTO, revealing that spin dynamics show cooperative and memory effect behavior. [27] Savary et al. [28] investigated the role of disorder in HTO and proposed the formation of disorder-induced spin liquid-like states governed by quantum correlations. In a previous study, we also observed anomalous thermal hysteresis in magnetic ac susceptibility measurements, which strongly depend on driving frequency. [29] Furthermore, the magnetization study performed by Sakakibara et al. [30] showed the presence of two successive field-induced hystereses below 0.36 K in DTO. They concluded that observed hysteresis is associated with spin ice to kagomé ice and kagomé ice to 3in-1out or 1in-3out spin crossover. A similar result was also found by Yukio et al. [31,32] for Yb2Ti2O7 (YTO) quantum spin ice in a temperaturedependent magnetization and neutron diffraction study. These observations indicate that magnetic interaction/field-induced spin crossovers are liquid-gas-type transitions in these materials. These findings signify the complexity of the magnetic behavior of DTO and HTO, where the interplay of local spin structures, disorder, quantum fluctuations, and the cooperative nature of spin dynamics cumulatively decides the macroscopic properties. Herein, we take a step to uncover this complexity through rigorous magnetic and dielectric investigations. Our findings reveal the crucial role of thermal and nonthermal variables in the out-of-equilibrium state, which dictates the macroscopic properties of these materials. II. EXPERIMENTAL DETAILS High-quality polycrystalline Dy2Ti2O7 (DTO) and Ho2Ti2O7 (HTO) samples have been used for the magnetic and dielectric measurements. Samples synthesis, phase purity, and their structural details have already been reported in Ref [14,33,34]. Magnetic measurements are performed using Magnetic Properties Measurement System (MPMS-3)® (Quantum Design, Inc.) USA. The samples used were powder and mounted in a brass holder for the measurements. The temperature sweep rate was kept fixed at @1.5 K/min while varying the other non-thermal variables for all the temperature-dependent measurements. Lowtemperature magnetodielectric measurements of DTO were performed using an Agilent E4980A LCR meter interfaced with Physical Properties Measurement System (PPMS-3)® (Evercool Quantum Design, Inc.) USA. III. RESULTS Fig. 1 (a &b) shows the temperature-dependent real part of ac susceptibility (′(T)) measured for DTO in field-cooled cooling (FCC) and field-cooled warming (FCW) mode for varying ac variables. Fig. 1 We observe a similar M(T) behavior for HTO. Inset of Fig. 1(a & b) shows the f and Hac dependent variation in TCross and its fit using the exponential decay equation given by- TCross(f) = A×exp(-R×′(f)(1)TCross(Hac) = A×exp(-R×′( Hac)(2) In equations (1) and (2) Hac. The absence of crossover in '(T) of HTO is quite noticeable because DTO and HTO structurally and magnetically mimic each other. In both compounds, ground state Ising doublets separated from the first excited state more than 100 K by strong crystal field anisotropy acting along the <111> direction. [7,35] The deduced values of nearest-neighbor magnetic dipolar (Dnn) and exchange (Jnn) interactions are 2.35 K and -1.24 K, respectively, for DTO [11] and 2.4 K and 0.5 K, respectively, for HTO. [7,36] Due to the dominance of ferromagnetic dipolar interaction, both DTO and HTO form exotic spin ice states below 4 K. However, due to the Kramer nature of the Dy 3+ ion, the magnitude of the transverse field responsible for quantum tunneling is smaller in DTO than HTO. [36,37] It leads DTO to possess a slower spin relaxation time (~ms) than HTO (~ns). [19,21,23] The absence of crossover in HTO indicates that spin dynamics play a vital role along with effective dipolar interaction, which dominates in the crossover temperature range in both compounds. Fig. 2(a-c) shows the ′(T) measured in FCC and FCW mode for DTO at different measurement protocols. Fig. 2 (a) shows the ′(T) measured in FCC and FCW mode for DTO at different dc bias magnetic fields. In this measurement, we observe a linear increase in the Tcross with Hdc up to 0.01 T (inset of Fig. 2 (a)). A further increase in Hdc, TCross expands abruptly with a simultaneous increase in the magnitude of '. To investigate the effect of ac variables on the observed anomalous thermal hysteresis, we measured the (T) at 0.6 T optimal dc bias field at different Hac ( Fig. 2(b)). At 0.6 T biased Hdc, ' has maximum magnitude. [29] In this measurement, anomalous '(T) shows a strong dependency on Hac and follows exponential decaying behavior with increasing Hac (inset of Fig. 2(b)). A similar observation has been found for HTO, as shown in Fig. S4 (supplementary file). This observation is well corroborated with '(T) measured at 3 Oe for different frequencies (Fig. S 5) Fig. 2(c)). FIG. 2: For Dy2Ti2O7, (a) Temperature-dependent real part of ac susceptibility (′ (T)) measured in FCC (solid symbol and line (smoothen data)) and FCW (open symbol and dashed line (smoothen data)) modes Contrary to this, in 0.6 T biased Hdc, the magnitude of anomalous '(T) shows linear dependence with temperature sweep rate (Fig. 2(c)). on ac variables (f and Hac) and temperature sweep rate, magnetic hysteresis (′(H)) also shows the dependency on f (inset of Fig. 2(d)) and magnetic field sweep rate. Out of these observations, we observe a crossover at ~0.1 T during 1.5 T →0 T magnetic field sweeping in both 50 Oe/sec and 100 Oe/sec sweep rates, below which '(H) changes its sign similar to the '(T). FIG. 3: (a) Temperature-dependent real part of dielectric permittivity (′) measured in cooling (solid line) and warming (dashed line) mode @ 2 K/min sweep rate at different magnetic field and (b) magnetic field dependent ′ measured in forward (0→ 2 T, solid line) and reverse (2→ 0 T, dashed line) mode at different temperatures @ 100 Oe/sec sweep rate, for Dy2Ti2O7 at 50 kHz frequency. However, in '(H) measurement performed at different temperatures ( Fig. 2(d)), we observed crossover only ≥4 K. A similar observation has been found in HTO as well as shown in Fig. S7 (supplementary file). The plot of the Hdc sweep rate and temperature dependence of '(Hdc) at different Hdc for both compounds is given in the supplementary file as Fig S8 with a detailed discussion. The observed temperature and magnetic field sweep rate dependence of anomalous hysteresis indicates that the existing phonon-mediated spin-flipping mechanism [15,16,38,39] gets altered by biased Hdc. To investigate this, we performed the temperature and magnetic field-dependent dielectric permittivity () measurement for DTO. Fig. 3(a) shows the temperature-dependent real part of dielectric permittivity (′(T)) measured at different Hdc. In the ′(T) plot ( Fig. 3(a)), we observe two successive relaxations at ~7.5 K and 4 K. Furthermore, below 4 K, it shows thermal hysteresis in cooling and warming measurement mode, like M(T) and ′(T, Hdc= 0T) magnetic measurements. On application of Hdc, an abrupt increase in the magnitude of ′(T) with a slight increase in both relaxation temperatures takes place. On increasing Hdc, the temperature span of thermal hysteresis increases with a gradual decrease in it′s magnitude. Fig. 3(b) shows the magnetic field-dependent real part of dielectric permittivity (′(Hdc)) measured at different temperatures. In ′(H) measurement, a spontaneous increase in the magnitude of ′ occurs with the magnetic field and becomes saturated above a critical field depending on the temperature. This abrupt increment in ′ starts at 0.05 T magnetic field and saturates ≤ 0.5 T for measured temperature. Along with this, forward and reverse magnetic field sweeping shows hysteresis in the ′. The observed hysteresis in ′(H) decreases with increasing temperature and vanishes at 10 K. The emergence of thermal and magnetic hysteresis in ′ below 4 K confirms the strong interconnection of ′ with local spin structure and spin dynamics. IV. DISCUSSION The above experimental results clearly show the measurement protocol-dependent nature of hysteresis in both DTO and HTO. In M(T) (Fig. S2) and ′(T, Hdc=0 T) ( Fig. 1 & Fig. S3), below 4 K, observed thermal hysteresis is independent of temperature sweep rates and ac variables. On application of Hdc, thermal hysteresis gets suppressed in M(T), whereas in ′(T), it abruptly expands in the measured temperature range (Fig. 2(a)). Intriguingly, anomalous hysteresis emerges in ac  only in both compounds. Snyder et al. [19] investigated the effect of Hdc on the spin relaxation time () of DTO through frequencydependent ac  measurements. They found non-monotonic dependence of  with the Hdc below ~13 K, where  becomes weak temperature dependent. At the smaller field (Hdc< 0.5 T), (H) decreases with increasing Hdc but increases for higher fields and then saturates for the higher Hdc. The value of the critical field up to which (H) decreases shows temperature dependence. At 6 K, the value of the critical field is ~0.65 T whereas, at 12 K, it becomes ~0.35 T. They concluded that applied Hdc suppresses the thermal fluctuation and enhances the quantum effects acting on cooperative spins. In the present results, the magnitude of anomalous ′ increases up to ~0.6 T and decreases for the higher Hdc. [29] The temperature and Hdc-dependent behavior of anomalous ′ (Fig. 2) corroborates with  behavior. Further, the spontaneous increment in ′ with Hdc (Fig. 3) also indicates the suppression in the spin-phonon coupling by Hdc. Scharffe et al. [40] found a decrease in the thermal conductivity coefficient with magnetic field in both DTO and HTO, which supports this argument. Now the question is, what does bias Hdc? In these materials, Ising spins are situated non-collinearly at the vertex of corner-sharing tetrahedra and pointing towards the center. Due to the non-colinear spin structure and polycrystalline sample, splitting energy (E) of the Ising doublet state has multiple values in the presence of Hdc. As a result, applied Hdc produces random quenched disorder sites having larger spin relaxation times in the Ising spin matrix. [41] These quench disorder sites block the dynamical pathway of the ideal system and provide a limited path for spin-flipping. Due to this restriction, the system takes more time to achieve thermal equilibrium from the out-of-equilibrium state. The suppression in the spin-phonon coupling, which might occur at the quench sites only, supports the ′(, t) = ′ST + ′NST The ′NST contribution in ′ is decided by the value of  with respect to some characteristic time (char) at constant temperature and Hdc. For  ≥ char, dynamics is non-stationary, and ′NST becomes non-zero, whereas for  ≤ char, dynamics is stationary and ′NST becomes zero. In temperature and magnetic fielddependent ac  measurement, we observe that the value of ′NST is Whereas a subtraction in ′ST takes place, i.e., ′NST is finite and negative when kBT increases (FCW in ′(T)) or E decreases (decreasing Hdc in ′(H)). The change in sign, dependency on kBT, and ac variables indicate that quantum fluctuation governs the emergent dynamic spin-correlations. [43][44][45] Where relaxations take place through quantum channels, and properties depend on the initial conditions. [46][47][48] In DTO, the emergence of crossover in ′(T, Hdc=0 T) below 2.5 K indicates that, similar to the biased Hdc, effective dipolar interaction works similarly. However, unlike Hdc, magnetic dipolar-induced out-ofequilibrium spin-correlations should differ because each spin feels equal potentials raised by magnetic dipolar interaction. This scenario becomes clearer in magnetic field-dependent ′ measurements (Fig. 2d), where we do not observe any crossover below 4 K in both DTO and HTO. It leads us to conclude that effective dipolar interaction also develops out-of-equilibrium spin-correlations of short decay time. Due to this reason, out-of-equilibrium properties measured in the absence of Hdc were observed at the mK scale, as reported in previous studies. [27,49,50] The absence of crossover in HTO for measured frequency and Hac suggests the speedy decay time of the non-stationary part, most probably due to its faster spin dynamics even at the mK scale. In an equilibrium state, these systems become state independent and behave classically, and show quasi-liquid-gas type transition during Ising paramagnetic to spin ice crossover. [30][31][32] Due to this cruciality, these systems show complex behavior, even at low temperatures. [30,51] V. CONCLUSIONS Our results demonstrate the distinct behavior of cooperative spins dynamics in the equilibrium and outof-equilibrium states in DTO and HTO. In the equilibrium state, static and dynamic magnetic properties of both DTO and HTO are weakly dependent on external thermal and non-thermal variables. Both compounds show liquid-gas type transition during interaction/magnetic field-induced local spin state crossover when measured in the equilibrium state. Contrary to this, in the out-of-equilibrium state, we observe a magnetic field-induced thermomagnetic hysteresis found in ac (T, H) only and shows a strong dependency on thermal and non-thermal variables. In dielectric measurements, the abrupt increment in the dielectric permittivity with the applied magnetic field (<0.5 T) indicates that the magnetic field increases the out-ofequilibrium time spam of cooperative spins by weakening the spin-phonon coupling strength. It has been suggested that applied Hdc produced quenched disorder sites in the non-collinear Ising spin metrics and developed dynamic spin-correlations with slow relaxation governed by the quantum fluctuations. These dynamic spin-correlations also contribute to (T, H) when the system falls into the out-of-equilibrium state. Based on measurement protocol, the non-stationary part can have positive and negative values and becomes zero in equilibrium or deteriorating conditions. The temperature/magnetic field sweep rate dependence, generally observed for classical systems, shows the short decay time of the out-of-equilibrium state with a complex intertwining of quantum-classical behavior of cooperative spin dynamics in these materials. The observed memory effect in the out-of-equilibrium and its dependency on external stimuli set a stage for understanding the other topologically frustrated quantum materials for future applications. SUPPLEMENTARY MATERIAL See the supplemental material for the temperature-dependent magnetization measured at different magnetic dc bias magnetic fields, temperature, and magnetic field-dependent ac susceptibility measured for HTO at different ac amplitudes and frequencies with a detailed discussion. ACKNOWLEDGMENT PY acknowledges the CSIR-HRDG, India, for providing funds for this project, and the authors acknowledge CIF, IIT (BHU) for collecting data in the Magnetic Properties Measurement System (MPMS). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. AUTHOR DECLARATIONS Fig. 1 ( 1b) shows the FCC and FCW of ′(T) plot for selected 0.5, 3, and 9 Oe ac amplitudes, whereas the complete ′(T) plot of FCC and FCW for measured Hac is shown asFig. S1(supplementary file). In this measurement, we do not observe any significant change in the magnitude of '(T), like the frequency. It is noteworthy that we find the crossover in ′(T) only. In M(T), thermal hysteresis emerges below 5 K and increases with lowering temperature without any crossover, as shown inFig. S2(a) (supplementary file). , A and R are fixed variables representing the amplitude and decay constant, respectively. The deduced values of A and R for TCross vs. f fit are -0.580.024 (K) and (1.2170.004)10 -2 (Oe/emu), respectively. Whereas, for TCross vs. Hac fit, values of A and R are (-3.68.05)10 -5 (K) and (-89.80.1)10 -2 (Oe/emu), respectively. A similar measurement protocol has been followed for HTO and is shown in Fig. S3 (supplementary file). In '(T) of HTO, we do not observe any crossover for measured f and Hac range, and it showed similar hysteresis behavior as observed in M (T). In the '(T) measurements of both DTO and HTO, the magnitude of '(T) does not show any significant change with applied f and Fig. S6 (supplementary file) shows the '(T) vs. temperature sweep rate plot and its linear fit. We have plotted the slope vs. temperature plot from the obtained values of the linear fit and shown in the inset of Fig. S6. It has been found that on decreasing temperature slope is linearly up to ~4 K, whereas below 4 K, it decreases. We observe a similar trend for anomalous '(T) vs. temperature plot.[29] This observation shows the suppression of anomalous '(T) with temperature. To understand the role of Hdc in the emergence of anomalous hysteresis, we performed the magnetic field-dependent ac susceptibility measurements ((H)) on DTO and HTO by sweeping the Hdc up to 1.5 T (and cycled back) at a constant temperature. The inset of Fig. 2(d) shows the magnetic field-dependent real part of ac susceptibility ('(H)) measured at 98.3 Hz frequency and 1.5 Oe Hac at 4 K for stabilized mode, 50 Oe/sec, and 100 Oe continuous magnetic field sweep rates for DTO. As we can see in the inset of Fig. 2(d), '(H) shows the magnetic hysteresis for 50 Oe/sec and 100 Oe/sec continuous magnetic field sweep mode. Whereas for the stabilized mode, where measurement occurs after stabilization of the magnetic field at each set point, we do not observe any magnetic hysteresis. Furthermore, like the dependency of ′(T) increased out-of-equilibrium time duration by Hdc.Interestingly, in all Hdc biased ′(T) measurements, we do not observe any significant change in the freezing temperatures in FCC and FCW modes. It indicates that in both measurement modes, the resonant frequency of the cooperative spins at freezing points is the same. Further, the exponential increment in the magnitude of anomalous ′ with decreasing frequency) indicates another time region (large spin relaxation time) in out-of-equilibrium. The temperature and magnetic field sweep rates dependent measurements show the decay of ′(T, Hdc) in a short time t. It means that when we move the system in the out-of-equilibrium state in the presence of Hdc, quench disorder sites develop a dynamic spin-correlation of larger  and exist for a short time t. This field-induced dynamic correlation additionally contributes to the ′ in out-ofequilibrium. Thus the ′(T, Hdc) can be represented as a sum of stationary part (′ST) and non-stationary part (′NST):[42] ′ 2 . 2The addition in ′ST takes place, i.e.,′NST is finite and positive when kBT decreases (FCC in ′(T)) or E increases (increasing Hdc in ′(H)). for different dc bias magnetic field at 98.3 Hz and 1.5 Oe ac variables. The inset shows the zoomed view of ′(T) measured for 0, 0.005, and 0.01 T, Hdc, and its inset shows the linear variation in crossover temperature (TCross) with Hdc. (b) ′(T) measured in FCC and FCW mode at 98.3 Hz frequency for different ac amplitudes in the presence of 0.6 T and 0 T (inset)Hdc.@ 1.5 K/min temperature sweep rate. (c) ′(T) measured in FCC and FCW mode at 98.3 Hz and 1.5 Oe ac amplitude for different temperature sweep ratesin the presence of 0.6 T and 0 T (inset) Hdc. (d) Magnetic field-dependent ′ measured in forward 0→1.5 T (solid symbol and line (smoothen data)) and backward 1.5→0 T (open symbol and dashed line (smoothen data)) modes at 98.3 Hz and 1.5 Oe for different temperatures; and for stabilized, 50, and 100 Oe/sec sweep rates (inset) at 4 K. The second inset of Fig. (d) shows the ′(Hdc) measured for 98.3 Hz and 498.3 Hz at 4K. Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Neuromorphic computing: Challenges from quantum materials to emergent connectivity. I K Schuller, A Frano, R C Dynes, A Hoffmann, B Noheda, C Schuman, Sebastian A Shen, J , Appl. Phys. Lett. 120140401Schuller I K, Frano A, Dynes R C, Hoffmann A, Noheda B, Schuman C, Sebastian A and Shen J 2022 Neuromorphic computing: Challenges from quantum materials to emergent connectivity Appl. Phys. Lett. 120 140401 . 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Phys. 14969Editorial 2018 New horizons towards thermalization Nat. Phys. 14 969 Quantum coherence and geometric quantum discord. M L Hu, X Hu, J Wang, Y Peng, Y Zhang, H R And Fan, Phys. Rep. Hu M L, Hu X, Wang J, Peng Y, Zhang Y R and Fan H 2018 Quantum coherence and geometric quantum discord Phys. Rep. 762-764 1-100 Spin dynamics at very low temperature in spin ice Dy2Ti2O7. K Matsuhira, C Paulsen, E Lhotel, C Sekine, Z Hiroi, S Takagi, J. Phys. Soc. Japan. 80123711Matsuhira K, Paulsen C, Lhotel E, Sekine C, Hiroi Z and Takagi S 2011 Spin dynamics at very low temperature in spin ice Dy2Ti2O7 J. Phys. Soc. Japan 80 123711 Far-from-equilibrium monopole dynamics in spin ice. C Paulsen, M J Jackson, E Lhotel, B Canals, D Prabhakaran, K Matsuhira, S Giblin, S T Bramwell, Nat. Phys. 10Paulsen C, Jackson M J, Lhotel E, Canals B, Prabhakaran D, Matsuhira K, Giblin S R and Bramwell S T 2014 Far-from-equilibrium monopole dynamics in spin ice Nat. 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{'abstract': 'Out-of-equilibrium investigation of strongly correlated materials deciphers the hidden equilibrium properties. Herein, we have investigated the out-of-equilibrium magnetic properties of polycrystalline Dy2Ti2O7 and Ho2Ti2O7 spin ices. The experimental results show the emergence of magnetic field-induced anomalous hysteresis observed only in temperature/magnetic field-dependent ac susceptibility measurements. The observed memory effect (anomalous thermomagnetic hysteresis) strongly depends on the driving thermal and non-thermal variables. Contrary, in the absence of the magnetic field, dipolar interaction induced Ising paramagnetic to spin ice crossover develops a liquid-gas transition type hysteresis below 4 K. Unlike field-induced hysteresis, it shows weak dependency on thermal and nonthermal variables. Due to the non-colinear spin structure, the applied dc bias magnetic field produces quench disorder sites in the cooperative Ising spin matrix and suppresses the spin-phonon coupling. These quench disorders create dynamic spin correlations governed by quantum fluctuations, having slow spin relaxation and quick decay times, which additionally contribute to ac susceptibility. The initial conditions and measurement protocol decide the magnitude and sign of this dynamical term contributing to ac susceptibility. It has been suggested that such kind of out-of-equilibrium properties emerge by the cumulative effect of geometric frustration, disorder, quantum fluctuations, and the cooperative nature of spin dynamics of these materials.', 'arxivid': '2301.07741', 'author': ['Pramod K Yadav *email:pramodyadav@iisc.ac.in \nCentre for Nano Science and Engineering\nIndian Institute of Science\nBangalore-560012India\n', 'Rajnikant Upadhyay \nSchool of Materials Science and Technology\nIndian Institute of Technology (Banaras Hindu University)\n221005VaranasiIndia\n', 'Rahul Kumar \nSchool of Advanced Materials and Chemistry and Physics of Materials Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\nBangalore-560064India\n', 'Pavan Nukala \nCentre for Nano Science and Engineering\nIndian Institute of Science\nBangalore-560012India\n', 'Chandan Upadhyay \nSchool of Materials Science and Technology\nIndian Institute of Technology (Banaras Hindu University)\n221005VaranasiIndia\n'], 'authoraffiliation': ['Centre for Nano Science and Engineering\nIndian Institute of Science\nBangalore-560012India', 'School of Materials Science and Technology\nIndian Institute of Technology (Banaras Hindu University)\n221005VaranasiIndia', 'School of Advanced Materials and Chemistry and Physics of Materials Unit\nJawaharlal Nehru Centre for Advanced Scientific Research\nBangalore-560064India', 'Centre for Nano Science and Engineering\nIndian Institute of Science\nBangalore-560012India', 'School of Materials Science and Technology\nIndian Institute of Technology (Banaras Hindu University)\n221005VaranasiIndia'], 'corpusid': 255999868, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12995, 'n_tokens_neox': 11003, 'n_words': 6109, 'pdfsha': 'c3d39c74f114a5715a78a19e4ea2bc195064dc54', 'pdfurls': ['https://export.arxiv.org/pdf/2301.07741v1.pdf'], 'title': ['Emergence of field-induced memory effect in spin ices', 'Emergence of field-induced memory effect in spin ices'], 'venue': []}

arxiv

Towards the dispersion relation for ionacoustic instabilities in weakly inhomogeneous ionospheric plasma at altitudes 80-200km and its low-frequency solution. The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 2 6 Apr 2009 O I Berngardt Institute of Solar-Terrestrial physics SB RAS Lermontova Str., 126aPOBox 291664033IrkutskRussia A P Potekhin Institute of Solar-Terrestrial physics SB RAS Lermontova Str., 126aPOBox 291664033IrkutskRussia Towards the dispersion relation for ionacoustic instabilities in weakly inhomogeneous ionospheric plasma at altitudes 80-200km and its low-frequency solution. The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 2 6 Apr 2009numbers: 5225Xz5230Ex5235Qz9420dt9420wf Submitted to: Plasma Phys Control Fusion In the paper within the approximation of the two-fluid magnetohydrodynamics and geometrooptical approximation the dispersion relation was found for ionacoustic instabilities of the ionospheric plasma at 80-200km altitudes in threedimensional weakly irregular ionosphere. Low freqeuncy solution was found. The difference between obtained and standard solution becomes significant at altitudes above 140 km. As the analysis shown in this case the solution grows with time. The conditions for existence of such solution are the presence of co-directed electron density gradients and electron drifts and perpendicularity of line-of-sight to the magnetic field. The necessary conditions regularly exist at the magnetic equator. Detailed analysis has shown that this solution corresponds to well-known 150km equatorial echo and explains some of its statistical characteristics observed experimentally. Introduction One of the important fields of modern ionospheric investigations is the study of small-scale irregularities in E-and F-layers of the ionosphere, that affect radiowave propagation and functionality of different HF and UHF radiotools. One of the most investigated types of irregularities is E-and F-layer irregularities produced as a result of growth of two-stream and gradient-drift instabilities. The theory of such instabilities is under development for a long time but still is not finished [12,4,19,11]. The usual condition for the growth of such irregularities is the requirement of different velocities of electrons and ions, most significant at altitudes 80-120km. At these heights ions are 'unmagnetized' -their motion is controlled by neutral component motion. At the same time the electrons motion is controlled by auxiliary electric and magnetic fields -electrons are 'magnetized' [12,4,19]. But it is clear that this requirement significantly limits the validity region of the current instabilities theories, that is why it is important to obtain a theory of ionacoustic instabilities without this limitation (see for example [18]). Basic equations Two-fluid magnetohydrodynamic equations As basic equations for obtaining dispersion relation we will use two-fluid magnetohydrodynamics (MHD) equations in form [15]:                          dNα dt + − → (N α − → V α ) = 0 m α N α ∂ − → V α ∂t = −Z α eN α ( − → E + − → V α × − → B )+ −m α N α ( − → V α − → ) − → V α − − → (T α N α ) − N α m α − → V α ν t,µ αn − → E = − → E 0 − − → Φ − 2 Φ = e(Z i N i + N e ) − → B = − → B 0(1) where α = e, i, normalized elastic collision frequencies: ν t,µ αn = µ αn m α ν t αn(2) µ αn is effective mass of charged particles during elastic collisions with neutrals (for electrons µ en is very close to electron mass, for ions µ in could vary around half of ion mass): µ αn = m α m n m α + m n(3) and ν t αn -elastic collision frequency of the ions (electrons) with neutrals. We also suppose here that the charged particles do not interact with each other through the collisions and interact only through electromagnetic field. We also exclude all the viscidity effects, that usually are not taken into account [15]. In this case we take into consideration (following to the [15]) only elastic collisions. In detail the approximations used are listed in Appendix A. Most of these approximations are valid at heights below 200km both for quiet and disturbed ionospheric conditions. It must also be noted, that we throw out a lot of terms from the MHD equations (1): ambient magnetic − → B 0 and electric − → E 0 fields are supposed to be constant; any magnetic field variations are not taken into consideration; gravitation field is not taken into consideration; recombination and ionization processes are not taken into consideration; neutral component motion is neglected. This allows us to neglect a lot of instabilities and effects (see, for example [1]) and simplify the analysis. Zero order solution -quasihomogeneous and static case Zero order approximation connects nondisturbed (quasihomogeneous and static) values of the particles density N α0 , average motion speed − → V α0 , ambient electrical − → E 0 and magnetic − → B 0 fields and collision frequencies. When density, fields and collision frequencies are given, the zero order approximation defines average motion speed of charged particles in any point of space and time. As one can see from (1), the zero-order approximation is defined by the system:                    − → N α0 − → V α0 = 0 0 = Z α eN α0 − → E 0 + Z α eN α0 − → V α0 × − → B 0 + −m α N α0 ( − → V α0 − → ) − → V α0 − − → (T α0 N α0 )+ −N α0 m α − → V α0 ν t,µ αn 0 = e(Z i N i0 + N e0 )(4) In the simplest case of weak velocity gradients, when we could neglect Lagrange term ( − → V α0 − → ) − → V α0 , the system (4) has a well known solution [17,15]: − → V α0 = − D α − → ∇N α N α0 − D T α − → ∇T α T α − σ α Z α eN α0 − → E 0(5) where operators of diffusion D α , thermodiffusion D T α and conductivity σ α are: A α =     A Hα −Ω α A Hα 0 Ω α A Hα A Hα 0 0 0 A α     (6) A T α =     A T α(1,1) −A T α(1,2) 0 A T α(1,2) A T α(1,1) 0 0 0 A α + TαdAα dTα    (7) A T α(1,1) = A Hα + T α dA Hα dT α (8) A T α(1,2) = Ω α A Hα + T α dA Hα dT α (9) D α = T α m α A α (10) D T α = T α m α A T α (11) σ α = Z 2 e 2 N α m α A α (12) A Hα = ν α K σ ( Ωα να ) Ω 2 α + ν 2 α A Hα = K ε ( Ωα να ) Ω 2 α + ν 2 α A α = K σ (0) ν α(13) and functions K σ (x), K ε (x) are tabulated (for example in [17]) for taking into account not only MHD effects, but kinetic effects too. It is important to note that for our next consideration the exact expression (5) for zero-order solution is not very significant for us, and below we only suggest that the solution − → V α0 = − → V α0 N α , T α , − → E 0 , − → B 0 , ν t, µ αn exists and is unambiguously determined by its arguments. So by the zero-order solution we mean an equation (4) that defines an average motion speed − → V α0 as a function of ambient conditions and which could be solved analytically (5)(6)(7)(8)(9)(10)(11)(12)(13) in simple cases or numerically in more complex cases. First order solution -nonstatic inhomogeneous case One of standard approaches to the MHD equations analysis is a geometrooptical (GO) approximation, the validity of which is defined by smallness of the parameter µ =   − → k   − → ∇P P     −1 << 1(14) where − → k is irregularities wave vector and − → ∇P P typical range of changes of parameter P (for example electron density). When the GO approximation is valid, the solution for small variations of the parameters N, − → V , − → E can be found in form: δN α ( − → r , t) = e −iψ(µ − → r ,µt)/µ N α1 (µ − → r , µt) δ − → V α ( − → r , t) = e −iψ(µ − → r ,µt)/µ − → V α1 (µ − → r , µt) δ − → E ( − → r , t) = − − → e −iψ(µ − → r ,µt)/µ Φ 1 (µ − → r , µt)(15) Geometrooptical phase ψ(µ − → r , µt)/µ (or eikonal) for plane waves is related to wave vector − → k and complex frequency of the wave ω + iγ by the following definitions: − → k = − − → ψ(µ − → r , µt)/µ ω + iγ = ∂ψ(µ − → r , µt) µ∂t(16) The first approximation gave us the system of equations:        P α1 N α1 + − → P α2 − → V α1 = 0 −→ P α3 N α1 + P α0 − → V α1 + −→ P α4 Φ 1 = 0 Z i N i1 = −N e1 − 1 e Φ 1 ( − → ψ) 2(17) where, by taking into account the zero-order approximation (4): P α1 =   − − → V α0 − → N α0 N α0 + i − → V α0 − → ψ + i ∂ψ ∂t   (18) P α2 = − → N α0 + iN α0 − → ψ (19) −→ P α3 = T α   − i − → ψ +   − → N α0 N α0     (20) P α4 = iZ α eN α0 − → ψ (21) P α0 − → V α1 = P α5 − → V α1 + − → V α1 × −→ P α6 + P α7 − → V α1 (22) P α5 = −m α N α0 i ∂ψ ∂t + i − → V α0 − → ψ + ν t,µ αn (23) − → P α6 = −Z α eN α0 − → B 0 (24) P α7 − → V α1 = −m α N α0 − → V α1 − → − → V α0(25) To make the following analysis easier, the system (17) is written in operator form, where operators P n are matrix operators in partial derivatives over the eikonal ψ. It is clear that in this form the system looks pretty simple and solvable. Dispersion relation Obtaining the dispersion relation From (17) one can see that the system is linear and, in case of existence and uniqueness of the inverse operator P −1 α0 (22) it can be solved. After excluding − → V α1 from (17) the equation connecting the density N α1 and electric potential Φ 1 variations has the following form: C 1α (ψ)N α1 + C 2α (ψ)Φ 1 = 0(26) where coefficients are: C 1α (ψ) = A(...) P α1 − − → P α2 P −1 α0 − → P α3 C 2α (ψ) = −A(...) − → P α2 P −1 α0 − → P α4(27) and A(...) -an arbitrary function of arbitrary parameters that does not have zero values at the investigated region. Now we can recall that our plasma has two types of particles and its characteristics are defined by the system of equations:        C 1i (ψ)N i1 + C 2i (ψ)Φ 1 = 0 C 1e (ψ)N e1 + C 2e (ψ)Φ 1 = 0 Z i N i1 = −N e1 − 1 e Φ 1 ( − → ψ) 2(28) Sometimes, for example, when analyzing thermal variations of electron density (that cause incoherent scattering), the self-coordinated term 1 (28) can not be neglected -scatterers size has order of Debye length and this term becomes significant. But in this very task we can neglect this term, following to many authors (see for example [11]). It is clear that existence of solution of (28) is determined by consistency of these equations. The consistency condition in our case has the following form: e Φ 1 ( − → ψ) 2 inC 1i (ψ)C 2e (ψ) + Z i C 1e (ψ)C 2i (ψ) = 0(29) It connects different partial derivatives over the eikonal ψ with each other and can be referred as dispersion relation. As one can see, the dispersion relation has symmetrical (as it was expected earlier) form. IAQV approximation It is clear that existence of dispersion relation and its exact form (29) depend on existence and properties of inverse operator P −1 α0 . As it has been shown in Appendix B, the inverse operator can be easily found in case when Lagrange term P 7 in P α0 can be neglected. Below we call this approximation as 'Irregularities under approximation of quasihomogeneous velocity' (IAQV). As preliminary analysis has shown, this approximation is valid for wavenumbers 0.1-10 m −1 under most ionospheric conditions at altitudes below 200km and for variations of average parameters not faster than 100m (for faster changes the GO approximation becomes incorrect). In the IAQV approximation the inverse operator P α0 has the following simple form: P −1 α0 − → f = b b − → f P 2 α6 + P 2 α5 − → f + P α5 P α6 ( b × − → f ) P α5 (P 2 α6 + P 2 α5 )(30) where b -unity vector in direction of − → P α6 (and antiparallel to the magnetic field). It must be noted that IAQV approximation does not mean neglecting the Lagrange term (22), all the other terms are the same. − → V α − → − → V α in basic equations (1), but only neglecting − → V α1 − → − → V α0 term in operator P α0 Summarizing, the dispersion relation in IAQV approximation has the following form (29, 18-24, 30). The basic structure of the dispersion relation Let us briefly analyze the structure of the dispersion relation by defining function that does not have zeroes: A(...) = P α5 P 2 α6 + P 2 α5(31) This leads to the following coefficients of the dispersion relation (29): C 1α (ψ) = P α1 P α5 (P 2 α6 + P 2 α5 ) + − − → P α2 b b − → P α3 P 2 α6 − P 2 α5 − → P α2 − → P α3 + −P α5 P α6 − → P α2 ( b × − → P α3 ) C 2α (ψ) = − b − → P α2 b − → P α4 P 2 α6 − P 2 α5 − → P α2 − → P α4 + −P α5 P α6 − → P α2 ( b × − → P α4 )(32) When taking into account (18)(19)(20)(21)(22)(23)(24) it becomes clear that coefficients (32) are polynomials over the ∂ψ ∂t and have the form: C 1α (ψ) = R 1α4 ∂ψ ∂t 4 + R 1α3 ∂ψ ∂t 3 + +R 1α2 ∂ψ ∂t 2 + R 1α1 ∂ψ ∂t + R 1α0 C 2α (ψ) = R 2α2 ∂ψ ∂t 2 + R 2α1 ∂ψ ∂t + R 2α0 (33) From this consideration it becomes clear that dispersion relation (29) is a 6th order polynomial over the ∂ψ ∂t and has no more than 6 solutions. Simple representation of coefficients To simplify the solution technique in homogeneous case, in the work [13] a new complex variable was defined: ω α = ∂ψ ∂t + − → V α0 − → ψ(34) In our inhomogeneous case we define the following new variables: − → K N = − → N α0 N α0 (35) ω αN = ∂ψ ∂t + − → V α0 − → ψ + i − → K N − → V α0 (36) − → k N = − → ψ + i − → K N (37) ν t,µ αnN = ν t,µ αn + − → K N − → V α0(38) In this case the operators (18-24) become: P α1 = i ( ω αN ) (39) −→ P α2 = iN α0 − → k * N (40) −→ P α3 = −iT α − → k N (41) −→ P α4 = iZ α eN α0   − → k N + − → k * N 2   (42) P α5 = (−im α N α0 ) ω αN − iν t,µ αnN (43) − → P α6 = −Z α eN α0 − → B 0 (44) P α6 = Z α eN α0 B 0 (45) where * is a complex conjugation. Summarizing, the dispersion relation has the form (29), where its coefficients are defined by (32,35-45) Dispersion relation for weak gradients case Weak gradients approximation The vectors −→ P α2 , −→ P α3 , −→ P α4 in dispersion relation are parallel in case of homogeneous ionosphere and not parallel in case of inhomogeneous ionosphere. After substituting −→ P α2 , −→ P α3 , −→ P α4 (40-42) into (32) and taking into account the properties of vector product we obtain the following: C 1α (ψ) = P α1 P α5 (P 2 α6 + P 2 α5 ) + −N α0 T α − → k N b 2 P 2 α6 + − → k N 2 P 2 α5 + +i2N α0 T α P α5 P α6 Im − → k N ( b × Re( − → k N )) C 2α (ψ) = Z α eN 2 α0 P 2 α6 bRe( − → k N ) 2 + +Z α eN 2 α0 P 2 α5 Re( − → k N ) 2 + −iZ α eN 2 α0 Re( − → k N ) Im( − → k N ) P 2 α5 + −iZ α eN 2 α0 P α5 P α6 Im − → k N b × Re( − → k N ) + −iZ α eN 2 α0 bRe( − → k N ) bIm( − → k N ) P 2 α6(46) The first two terms in each relation (46) are the terms that correspond both to the inhomogeneous and homogeneous dispersion relations, the last terms correspond only to changes of dispersion relation due to inhomogeneities presence. It is clear that in both cases (gradients are parallel to the magnetic field and gradients are perpendicular to the magnetic field) the difference between dispersion relation for homogeneous case and for inhomogeneous one does exist. But if the gradients are weak enough (or wavenumbers are high enough) we can neglect the changes of dispersion relation and solve only simplified one, that corresponds to the homogeneous case: C 1α (ψ) = P α1 P α5 (P 2 α6 + P 2 α5 ) + −N α0 T α − → k N b 2 P 2 α6 + − → k N 2 P 2 α5 C 2α (ψ) = Z α eN 2 α0 P 2 α6 bRe( − → k N ) 2 + +P 2 α5 Re( − → k N ) 2(47) As one can see, this approximation is valid, when: Re( − → k N ) Im( − → k N ) >> 1 (48) |P α5 | 2 Re( − → k N ) 2 bRe( − → k N ) bIm( − → k N ) P 2 α6 >> 1 (49) |P α5 | Re( − → k N ) 2 |P α6 | Im − → k N b × Re( − → k N ) >> 1(50) The condition (48) is equivalent to the GO validity condition (14): − → k N >> − → K N The condition (49) is valid when analyzing scattering almost perpendicular to the magnetic field or when gradients are sufficiently small: − → k N |P α5 | 2 |P α6 | 2 >> bRe( − → k N ) Re( − → k N ) b − → K N And the last condition (50) is valid when gradients are small enough: − → k N |P α5 | |P α6 | >> − → K N(51) So the condition (48) is always valid, condition (49) is valid when we investigate the instabilities near the perpendicular to the magnetic field and the condition (51) becomes only critical limitation for the approximation (47) of initial formula (46). Ionospheric parameters To create a correct dispersion relation for heights 80-200km we should choose the correct approximations for ionospheric plasma. The most important plasma parameters are thermal velocities, hyrofrequencies and frequencies of collisions with neutrals. These approximate parameters are shown at the Table 1 (calculated for mid-latitude ionosphere using models MSIS, IGRF and IRI) We also suggest that wavenumbers are within 0.1-10m (sounding frequencies 15-1500MHz), and drift velocities − → V α0 do not exceed 3000m/s. It is clear, that at altitudes 80-200km the following approximations are valid: Te me − → ψ << Ω e -electron hyrofrequency much higher than their thermal speed; − → V e0 − → ψ << Ω e -electron hyrofrequency much higher than their average speed (even in disturbed conditions); − → V e0 << Te me -electron thermal speed much higher than their average speed; ν t,µ en >> ν t,µ in electron-neutral collision frequency is much higher than ion-neutral one; We also use weakly inhomogeneous ionosphere approximation: − → ψ >> − → N e0 N e0 Below there is a list of traditional approximations that are valid for E-layer [12,4,19], but is not valid for the whole region 80-200km: − → V α0 − → ψ << ν t, µ αn -is not valid for ions at altitudes above 120km, not valid for electrons at heights above 180km under very disturbed conditions; − → V i0 − → ψ << Ω i -is not valid for disturbed conditions; Ω i << ν t,µ in -is not valid above 120km ; Ω e >> ν t,µ en -is not valid at and below 80km. It also must be noted that during high disturbances the effective ionneutral collision frequency ν t,µ inN , that is used in dispersion relation, becomes dependent on electron density gradient and average ions velocity (38) and might be increased (or decreased) depending on the ion motion direction and gradients. It should be also noted that for high velocities the effective ion-neutral collision frequency ν t,µ inN at heights approximately above 140-160km can become zero or negative. So, in this case the traditional approximation [11] of low Doppler drifts | ω α | << |ν t,µ αnN | is also invalid. Dispersion relation for weak gradients By substituting (39,43,44) into (47) and after neglecting same non-zero multipliers: C 1α ( ω αN ) = ( ω αN ) ω αN − iν t,µ αnN m α · · Ω 2 α + ν t,µ αnN + i ω αN 2 − −T α − → k N b 2 Ω 2 α + − → k N 2 ν t,µ αnN + i ω αN 2 C 2α ( ω αN ) = Z α eN α0 · · Ω 2 α bRe( − → k N ) 2 + ν t,µ αnN + i ω αN 2 Re( − → k N ) 2(52) where Ω α = Z α eB 0 m α(53) is hyrofrequency. Let us define new index β to describe another charged component: β = e, i; β = α. After defining the new parameters δ ω βαN = ω βN − ω αN = − → V α0 − − → V β0 − → k − i − → K N (54) The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 11 − → k = −Re( − → k N ) = − − → ψ(55) we will obtain the relations for second charged component as a function of the same parameter ω αN : C 1β ( ω αN ) = ( ω αN + δ ω βαN ) ω αN + δ ω βαN − iν t,µ βnN · ·m β Ω 2 β + ν t,µ βnN + i ω αN + iδ ω βαN 2 − −T β − → k b 2 Ω 2 β + − → k 2 ν t,µ βnN + i ω αN + iδ ω βαN 2 C 2β ( ω αN ) = Z β eN β0 Ω 2 β b − → k 2 + + ν t,µ βnN + i ω αN + iδ ω βαN 2 − → k 2(56) Below we will analyze the dispersion relation (29) in form: f (x) = 0 (57) where f (x = ω αN ) = Z β C 1α (x)C 2β (x) + Z α C 1β (x)C 2α (x)(58) By substituting (52,56) into (58), after neglecting non-zero multiplier, for singlecharged ions (Z i = −1) (most frequent approximation in this region of altitudes) the dispersion relation becomes the final one: f (x) = m α x x − iν t,µ αnN Ω 2 α + ν t,µ αnN + ix 2 · · Ω 2 β b − → k 2 + ν t,µ βnN + i (x + δ ω βαN ) 2 − → k 2 + +m β (x + δ ω βαN ) x + δ ω βαN − iν t,µ βnN · · Ω 2 β + ν t,µ βnN + i (x + δ ω βαN ) 2 · · Ω 2 α b − → k 2 + ν t,µ αnN + ix 2 − → k 2 + − (T α + T β ) b − → k 2 Ω 2 α + − → k 2 ν t,µ αnN + ix 2 · · Ω 2 β b − → k 2 + ν t,µ βnN + i (x + δ ω βαN ) 2 − → k 2 (59) where x = ∂ψ ∂t − − → V α0 − → k + i − → K N − → V α0(60) and other parameters are defined by (35-38,54,55) and by solution of the zero-order approximation (4). From obtained solution x 0 of dispersion relation (59) for given altitude dependence of the parameters one can always obtain the actual irregularity frequencies and decrements using relation: ∂ψ ∂t = x 0 + − → V α0 − → k − i − → K N(61) From the dispersion relation (59,60,61) it becomes clear that in the first approximation the presence of gradients − → K N of electron density logarithm change decrement (imaginary part of ∂ψ ∂t ), changes effective collision frequencies for ions with neutrals and makes them anisotropic at high altitudes. All these changes are proportional to the scalar product of the gradient of the electron density logarithm and average electron velocity. Actually this fact contradicts with current theories suggesting that in most cases only the electron density gradients perpendicular to the magnetic field must be taken into account [19], since there could be conditions when − → V α0 is not perpendicular to the magnetic field, for example in case of non-perpendicular magnetic and electric fields. In presence of weak drifts and gradients we can suppose that the solution is close to zero. To find the solution nearest to zero we will use zero order Newton solution (see, for example [24]): x 0 = − f (x) df dx x=0(63) By substituting the basic relations: f (0) = m β (δ ω βαN ) δ ω βαN − iν t,µ βnN · · Ω 2 β + ν t,µ βnN + i (δ ω βαN ) 2 · · Ω 2 α b − → k 2 + ν t,µ αnN 2 − → k 2 + − (T α + T β ) b − → k 2 Ω 2 α + − → k 2 ν t,µ αnN 2 · · Ω 2 β b − → k 2 + ν t,µ βnN + i (δ ω βαN ) 2 − → k 2 (64) The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 13 df dx x=0 = m α −iν t,µ αnN Ω 2 α + ν t,µ αnN 2 · · Ω 2 β b − → k 2 + ν t,µ βnN + iδ ω βαN 2 − → k 2 + +m β 2δ ω βαN − iν t,µ βnN · Ω 2 β + ν t,µ βnN + iδ ω βαN 2 · · Ω 2 α b − → k 2 + ν t,µ αnN 2 − → k 2 + +2im β (δ ω βαN ) δ ω βαN − iν t,µ βnN · · ν t,µ βnN + iδ ω βαN Ω 2 α b − → k 2 + ν t,µ αnN 2 − → k 2 + + Ω 2 β + ν t,µ βnN + iδ ω βαN 2 ν t,µ αnN − → k 2 + −2i (T α + T β ) − → k 2 · · ν t,µ αnN Ω 2 β b − → k 2 + ν t,µ βnN + iδ ω βαN 2 − → k 2 + b − → k 2 Ω 2 α + − → k 2 ν t,µ αnN 2 ν t,µ βnN + iδ ω βαN(65) The low-frequency branch (63,64,65) is pretty complex so we will investigate it at different ionospheric heights. 4.4.2. Obtaining traditional solution at 80-120km heights Let us analyze the branch (63,64,65) for the typical ionospheric heights 80-120km. Within standard for E-layer assumptions of x, δ ω ieN << ν t,µ enN , ν t,µ inN , magnetized electrons and unmagnetized ions, and neglecting δ ω ieN << ν t,µ inN , we obtain following equations for function and its first differential (neglecting in first differential by all the terms, proportional to δ ω ieN or T e + T i , based on suggestion that Doppler shifts for ionacoustic or average velocities are sufficiently small in comparison with ν t,µ inN ): f (0) = m i δ ω ieN δ ω ieN − iν t,µ inN − (T e + T i ) − → k 2 · · Ω 2 e b − → k 2 + ν t,µ enN 2 − → k 2 ν t,µ inN 2 (66) df (x) dx x=0 = −im i ν t,µ enN Ω e Ω i ν t,µ inN 2 − → k 2 + −im i ν t,µ inN ν t,µ inN 2 Ω 2 e b − → k 2 + ν t,µ enN 2 − → k 2(67) From (67,66) and Newton method (63) we obtain the solution, nearest to zero: x 0 = −δ ω ieN − i 1 ν t,µ inN (δ ω ieN ) 2 − Te+T i m i − → k 2 1 + ΩeΩ i (ν t,µ inN )ν t,µ enN − → k 2 (ν t,µ enN ) 2 b − → k 2 Ω 2 e + − → k 2 (ν t,µ enN ) 2 (68) The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 14 Considering (61) we obtain the following solution for plasma irregularities: ∂ψ ∂t = − → V e0 +Ψ − → V i0 − → k Ψ+1 + −i Ψ ν t,µ inN − → V e0 − − → V i0 − → k 2 − Te+T i m i − → k 2 Ψ+1 −i − → V e0 +Ψ − → V i0 − → K N Ψ+1 (69) Ψ = ν t,µ inN ν t,µ enN Ω e Ω i b − → k 2 Ω 2 e + − → k 2 ν t,µ enN 2 − → k 2 ν t,µ enN 2(70) In gradient-free case the solution (69) looks exactly as the standard one [19], in presence of gradients the solution differs from the standard one, most probably due to custom direction of gradients and custom orientation of velocities. |Re(x)| >> |δ ω βαN |(71) Supposing δ ω βαN = 0, and when investigating wavevectors perpendicular to the magnetic field, the solution becomes simplier: f (0) = − (T α + T β ) − → k 4 ν t,µ αnN 2 ν t,µ βnN 2 (72) df dx x=0 = m α −iν t,µ αnN Ω 2 α + ν t,µ αnN 2 ν t,µ βnN 2 − → k 2 + +m β −iν t,µ βnN Ω 2 β + ν t,µ βnN 2 ν t,µ αnN 2 − → k 2 + −2i (T α + T β ) − → k 4 ν t,µ βnN ν t,µ αnN ν t,µ βnN + ν t,µ αnN(73) After simple arithmetic and by taking into account magnetized plasma and typical ionospheric conditions (α-electrons): Ω 2 α >> ν t,µ αnN 2 ; Ω 2 β >> ν t,µ βnN 2 ; ν t,µ βnN << ν t,µ αnN we have: f (0) = − (T α + T β ) − → k 4 ν t,µ αnN 2 ν t,µ βnN 2 (74) df dx x=0 = m α −iν t,µ αnN (Ω 2 α ) ν t,µ βnN 2 − → k 2 + +m β −iν t,µ βnN Ω 2 β ν t,µ αnN 2 − → k 2 + −2i (T α + T β ) − → k 4 ν t,µ βnN ν t,µ αnN 2 (75) The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 15 Taking into account the typical ionospheric conditions: m e Ω 2 e ν t,µ inN >> m i Ω 2 i ν t,µ enN , (T i + T e ) − → k 2 ν t,µ enN (76) x 0 ≈ i ν t,µ enN (T e + T i ) − → k 2 (Ω 2 e ) m e(77)∂ψ ∂t ≈ i ν t,µ enN (T e + T i ) − → k 2 (Ω 2 e ) m e − i − → K N − → V e0 + − → V e0 − → k(78) From (78) the condition for the growing solution Im ∂ψ ∂t < 0 becomes: − → K N − → V e0 > D A⊥ − → k 2 (79) where D A⊥ = ν t,µ enN (Te+T i ) m i Ω e Ω i(80) is the so called coefficient of ambipolar diffusion [15]. For typical ionospheric conditions the growth condition (79) can be estimated as: − → K N − → V e0 > 0.2[m 2 /s] − → k 2(81) The spectral offset for these irregularities (78) is exactly the Doppler drift in crossed fields (and defined by zero-order solution (4-5)): Re ∂ψ ∂t ≈ − → V e0 − → k(82) It is necessary to note that the solution is obtained in weak gradients approximation (51) valid when: − → k ν t,µ enN Ω e >> − → K N(83) It should be noted that possible relation of the ambipolar diffusion with irregularities existence at these heights has been noted in [18], but the problem was not investigated in detail. The close condition for irregularities growth at high altitudes V e > ηD A⊥ was also obtained by [16], but with another proportionality coefficient η and in qualitative analysis of a model case. Due to the solution (78) corresponds to the same branch (63,64,65) as well-known gradint-drift instabilities (69,70) , below we will call this kind of solition as 'fully magnetized gradient-drift instabilities' (FMGD) to stress that this is the same gradientdrift branch but in a bit different conditions. Starting from early 1960s [3,2] at equatorial HF and UHF radars researchers observe an unique type of echo, the so called 150km equatorial one. There are some theories to explain it (for example [20,22,23,18,9]) , but the exact physical mechanism of it is still unclear [8,5,7]. The geometry at equator (horizontal magnetic field, almost upward drift velocity) allows us to use standard upward vertical gradient as a source for generation of this kind of instabilities. For sounding frequency of equatorial radar Jicamarca (k ∼ 0.3) and for K N ∼ 10 −3 ÷ 10 −4 [m −1 ] (standard vertical electron density gradients) the growth condition (79) becomes: − → V e0 > 20 ÷ 200[m/s](84) It should be noted that for gradients higher than (83) (K N > 10 −3 ) one should take into account all the terms in (46) instead of using only (47), but this should not affect too much the observed effect. To analyze the properties of the echo, some modelling has been done using the latest Internation Refference Ionosphere (IRI-2007) model. The height and time dependence of the electron density gradients are most important for the generation of this type of irregularities. We have analyzed 13 years period (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002) using IRI model (for typical non-disturbed conditions f 10.7 = 150, A p = 10) and obtained the following results. At Fig.1 the altitudinal dependence of K N is shown. Points are the hourly values over the whole period of 13 years. As one can see, there is a maximum K N > 10 −4 at heights 135-180km. So, this kind of irregularities could arise at heights 135-180km, and it corresponds well with the experimental observations statistics [5,8]. At Fig.2 an hourly dependence of the K N is shown, as a function of UT for heights 140-200km. As one can see, the time dependence of the gradients has a most intensive maximum between 14:00 and 19:00 UT (9:00-14:00LT). This also corresponds well with the experimental observations [5]. The dependence of irregularities frequency (82) corresponds well with the empirical models [6,21] and allows to interpret the experimental data as Doppler frequency offset due to electron drift in crossed fields. According to the experimental observations, the echo starts with V e > 10m/s, according to our calculations it should start with V e > 20 ÷ 200m/s. One of the mechanisms allowing to lower the speed limit was suggested in [20]. They suggest that acoustic-gravitational waves can be responsible for the triggering the instabilities. In our terms, the acoustic-gravitational waves will produce gradients more than K N > 10 −3 , and this will produce this type of irregularities even at lower velocities, for example at V e > 10m/s. Another possible mechanism that will lower the velocities necessary for generation of the instability is an observation of high step-like gradients at these heights from the rocket data (see for example [21]). They should also produce the increase of K N high enough for lowering the speed limit. Summarizing all said above we can suggest that the FMGD instabilities can be the source of 150km equatorial echo and this theory can be used for experiment interpretation. Conclusions In the paper within the approximation of the two-fluid magnetohydrodynamics and geometrooptical approximation the dispersion relation (59, 60, 35-38) at 80-200km altitudes was obtained. The relation describes ionacoustic instabilities of the ionospheric plasma at 80-200km altitudes in three-dimensional weakly irregular ionosphere. It was shown that not only electron density gradients perpendicular to the magnetic field should be taken into account when investiagting ionospheric instabilities, but gradients along the average drift velocity (59, 60, 36, 38). The dispersion relation obtained has a form of the 6-th order polynomial for the oscillation frequency. It is shown, that a solution branch exists that grows with time and describe instabilities both at 80-120km heights and 135-180km heights. For altitudes 80-120km the solution close to the standard one (69, 70) and corresponds to the Farley-Buneman and gradient-drift instabilties. The difference between obtained (63,64,65) and standard solutions [19] becomes significant at altitudes above 140 km, where standard one is not valid. As the analysis shown at these altitudes the solution grows with time (78, 79). The conditions for the growth is the presence of co-directed electron density gradients and electron drifts and perpendicularity of line-of-sight to the magnetic field. These conditions are regularly satisfied at magnetic equator for expected conditions (84). Detailed analysis has shown that this solution could explain a lot of properties of 150 km equatorial radioecho -the ionospheric phenomena that has no explanation for more than 40 years. In more details the MHD validity conditions can be found in [17,15]. Appendix B. Inversion of the matrix operator P α,0 (IAQV approximation) Lets analyze inversion of the matrix operator P α,0 (22-25). One can see, that in special case P 7 = 0 the inversion is very easy. In this case by taking into account that − → P 6 is static, we can create the coordinate system, based on unity vector b = − → P 6 /| − → P 6 |, which is antiparallel to the magnetic field. In this case we can write: P 0 − → f = P 5 − → f || + P 5 − → f ⊥ + P 6 − → f ⊥ × b (B.1) Where ||, ⊥ means parallel and perpendicular to the b. By making scalar and vector products of (B.1) with b we have: b( P 0 − → f ) = P 5 b − → f || b × ( P 0 − → f ) = P 5 ( b × − → f ⊥ ) + P 6 b × ( − → f ⊥ × b) (B.2) From first equation (B.2): − → f || = b b( P 0 − → f ) P 5 (B.3) The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 20 After comparing the second equation in (B.2) and its vector product with b we have:        − b×( P 0 − → f )−P 5 ( b× − → f ⊥ ) P 6 = b × ( b × − → f ⊥ ) b×( b×( P 0 − → f ))−P 6 b×( b×( − → f ⊥ × b)) P 5 = b × ( b × − → f ⊥ ) (B.4) Therefore, by taking into account the properties of double vector product: − b×( P 0 − → f )−P 5 ( b× − → f ⊥ ) P 6 = = b×( b×( P 0 − → f ))−P 6 ( b× − → f ⊥ ) P 5 (B.5) So ( b × − → f ⊥ ) = P 5 b × ( P 0 − → f ) + P 6 ( b × ( b × ( P 0 − → f ))) P 2 5 + P 2 6 (B.6) and (after making vector product with b and some vector algebra): − → f ⊥ = P 6 ( b × ( P 0 − → f )) − P 5 b × ( b × ( P 0 − → f )) P 2 6 + P 2 5 (B.7) Summarizing (B.3) and (B.7) we have − → f = − → f ⊥ + − → f || : − → f = P 6 P 5 ( b × ( P 0 − → f )) − P 2 5 b × ( b × ( P 0 − → f )) (P 2 6 + P 2 5 ) P 5 + (B.8) + (P 2 6 + P 2 5 ) b b( P 0 − → f ) (P 2 6 + P 2 5 ) P 5 or P −1 0 − → f = b b − → f P 2 6 + P 2 5 − → f + P 5 P 6 ( b × − → f ) P 5 (R 2 6 + R 2 5 ) (B.9) It is clear that this approximation is valid when: P α5 − → f + − → f × −→ P α6 >> P α7 − → f (B.10) Qualitatively one can estimate the orders of terms: ν t,µ αn , Ω α >> V α0 L −1 V (B.11) where L −1 V = ∂V α0(j) ∂r (k) V α0 (B.12) For maximal ionospheric disturbances up to 200km height we can estimate V α0 < 3000[m/s], ν t,µ en > 100Hz, ν t,µ in > 10Hz, Ω α > 100Hz. In this case the validity condition has the form L V >> 300[m] (B.13) The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 21 Summarizing, in very disturbed ionosphere the characteristic changes of parameters should not exceed couple hundreds meters, for less disturbed conditions these limitations becomes even weaker. So the obtained approximation for P −1 α,0 (B.9) is valid for most part of cases below 200km. The solution nearest to zero The dispersion relation (59) has 6 solutions, and in basic case all the solutions can be found only numerically. Lets find the simplest approximate solution -nearest to zero. The solution nearest to zero has a clear physical sence: in absence of average plasma drifts and gradients plasma can be supposed as static and irregularities should be static, i.e. magnetized case or instabilities at altitudes above 140km Let us analyze the branch (63,64,65) in case of sufficiently high altitudes, when both types of charged particles are magnetized (i.e. from about 130-140 km). At high altitudes we can neglect the difference in electron and ion velocities in comparison with their absolute values (both components are magnetized and move with almost the same velocities): Figure 1 . 1K N dependence on height over the 1990-2002 years, points are the hourly values. Figure 2 . 2K N dependence on time at 140-200km over the 1990-2002 years, points are the hourly values. Table 1 . 1Thermal velosities, hyrofrequencies and frequencies of collisions with neutrals for typical ionospheric conditions. Frequencies in Hz, Velocities in m/sec, height in kmh Te me Ω e ν t,µ en T i m i Ω i ν t,µ in 80 6e+4 1e+7 3e+7 3e+2 2e+2 2e+5 100 5e+4 1e+7 6e+5 2e+2 2e+2 4e+3 120 8e+4 1e+7 4e+4 3e+2 2e+2 2e+2 140 1e+5 1e+7 1e+4 4e+2 2e+2 4e+1 160 1e+5 9e+6 5e+3 5e+2 2e+2 2e+1 180 1e+5 9e+6 3e+3 6e+2 2e+2 8 200 1e+5 9e+6 2e+3 7e+2 2e+2 4 AcknowledgmentsAuthors thanks to N.Nishitani for fruitful discussion. The work was done under financial support of RFBR grant #07-05-01084a.Appendix A. Approximations usedThe theory limitations are listed, mostly following to the[17,15,11].At the altitudes 80-200km we suggest that the following conditions are satisfied: τ −1 << Ω i -all the basic plasma parameters has only slow variations and plasma supposed to be quasistatic; δ en << 1 -average loss of energy of electrons with neutrals is small enough; ν in << ν en ;ν in << Ω e ; ν ii << ν in -ion-ion collisions are rare enough to take into account only ion-neutral collisions. Not valid above 200km.ν ei ∼ ν ee << ν en -electron-ion and electron-electron collisions are rare enough to take into account only electron-neutral collisions. Not valid above 200km.k ⊥ ρ e << 1 electron hyroradius much smaller than wavelength. ρ e << λ e << k −1 || << L || -plasma is quasihomogeneous enough for GO approximation to be valid.λ d k << 1 -wavelength is much bigger than Debye radius. δ ei ν ei << ν in -necessary for independent thermalization of ions and electrons, in this approximation the average collision frequency does not depend on particles velocity or motion direction[17].V 0n = 0 -average speed of neutrals is much smaller than electrons and ions speed. Allows us to neglect neutral motions. A I Akhiezer, I A Akhiezer, P V Polovinin, A G Sitenko, K Stepanov, Electrodynamics of plasma' -in russian. MoscowNauka720Electrodinamika plasmyAkhiezer A I, Akhiezer I A , Polovinin P V , Sitenko A G, Stepanov K N 1974 Electrodinamika plasmy ('Electrodynamics of plasma' -in russian) (Moscow: Nauka) p 720 . B Basley, Journ.Geoph.Res. 69Basley B B 1964 Journ.Geoph.Res 69, 1925-30 . K L Bowles, G R Ochs, J Green, J. of Res. NBS, D.Rad.Prop. Bowles K L, Ochs G R, Green J L 1962 J. of Res. NBS, D.Rad.Prop, 66D 395-407 . O Buneman, Phys.Rev.Let. 10Buneman O 1963 Phys.Rev.Let. 10 285-7 . J L Chau, E Kudeki, Ann.Geophys. 24Chau J L and Kudeki E 2006 Ann.Geophys 24 1305-10 . J L Chau, R F Woodman, Geoph.Res.Lett. 3117801Chau J L and Woodman R F 2004 Geoph.Res.Lett. 31 L17801 . J L Chau, R F Woodman, M A Milla, E Kudeki, Ann.Geophys. 27Chau J L, Woodman R F, Milla M A, Kudeki E 2009 Ann.Geophys. 27 933-42 . R K Choudhary, .-Maurice J -P St, K Mahajan, 10.1029/2004GL020299Geoph.Res.Lett. 31Choudhary R K, St.-Maurice J -P, Mahajan K K 2004 Geoph.Res.Lett. 31 doi:10.1029/2004GL020299 . R Cosgrove, R T Tsunoda, 10.1029/2002GL014669Geoph.Res.Lett. 29Cosgrove R and Tsunoda R T 2002 Geoph.Res.Lett. 29 doi:10.1029/2002GL014669 . C M Denardini, M A Abdu, E R De Paula, J H A Sobral, C Wrasse, Journ.Atm.Sol.Terr.Phys. 67Denardini C M, Abdu M A, de Paula E R, Sobral J H A, Wrasse C M 2005 Journ.Atm.Sol.Terr.Phys. 67 1665-73 . Dimant Ya, S Sudan, R N , Phys. Plasmas. 24Dimant Ya S and Sudan R N 1995 Phys. Plasmas 2(4) 1157-68 . D Farley, Journ.Geoph.Res. 68Farley D T 1963 Journ.Geoph.Res. 68 6083-97 . B G Fejer, D T Farley, B B Balsley, R Woodman, Journ.Geophys.Res. 8010Fejer B G, Farley D T, Balsley B B, Woodman R F 1975 Journ.Geophys.Res. 80(10) 1313-24 . B Fejer, M C Kelley, 18Fejer B G and Kelley M C 1980 Reviews of geophysics and space physics 18 401-54 Osnovy fiziki plazmy. V E Galant, A P Zhilinsky, S Sakharov, Basics of plasma physics' -in russian. MoscowAtomizdat384Galant V E, Zhilinsky A P, Sakharov S A 1977 Osnovy fiziki plazmy ('Basics of plasma physics' - in russian) (Moscow: Atomizdat) p 384 Dynmaics of ionospheric plasma' -in russian. B Gershman, Dinamika ionosfernoj plasmy. Moscow: Nauka256Gershman B N 1974 Dinamika ionosfernoj plasmy ('Dynmaics of ionospheric plasma' -in russian) (Moscow: Nauka) p 256 A Gurevich, A B Shvarcburg, Nonlinear theory of radiowaves propagation in ionosphere' -in russian. MoscowNauka272Nelinejnaja teorija rasprostranenija radiovoln v ionosphereGurevich A V and Shvarcburg A B 1973 Nelinejnaja teorija rasprostranenija radiovoln v ionosphere('Nonlinear theory of radiowaves propagation in ionosphere' -in russian) (Moscow: Nauka) p 272 . L Kagan, M C Kelley, Journ.Geophys.Res. 105Kagan L M and Kelley M C 2000 Journ.Geophys.Res. 105 5291-303 M Kelley, Earth ionosphere: plasma physics and electordynamics. Academic Press471Kelley M C 1989 Earth ionosphere: plasma physics and electordynamics (Academic Press) p 471 . E Kudeki, W L Fawcett, Geoph.Res.Lett. 20Kudeki E and Fawcett W L 1993 Geoph.Res.Lett. 20 1987-90 . R Raghavarao, A K Patra, S Sripathi, Journ.Atm.Sol.Terr.Phys. 64Raghavarao R, Patra A K, Sripathi S 2002 Journ.Atm.Sol.Terr.Phys. 64 1435-43 . R Tsunoda, Geoph.Res.Lett. 21Tsunoda R T 1994 Geoph.Res.Lett. 21 2741-44 . R T Tsunoda, W L Ecklund, 10.1029/23GL018704Geoph.Res.Lett. 31Tsunoda R T and Ecklund W L 2004 Geoph.Res.Lett. 31 doi:10.1029/23GL018704 . T Yamamoto, Journ.Comput.Appl.Math. 124Yamamoto T 2000 Journ.Comput.Appl.Math. 124 1-373

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{'abstract': 'In the paper within the approximation of the two-fluid magnetohydrodynamics and geometrooptical approximation the dispersion relation was found for ionacoustic instabilities of the ionospheric plasma at 80-200km altitudes in threedimensional weakly irregular ionosphere. Low freqeuncy solution was found. The difference between obtained and standard solution becomes significant at altitudes above 140 km. As the analysis shown in this case the solution grows with time. The conditions for existence of such solution are the presence of co-directed electron density gradients and electron drifts and perpendicularity of line-of-sight to the magnetic field. The necessary conditions regularly exist at the magnetic equator. Detailed analysis has shown that this solution corresponds to well-known 150km equatorial echo and explains some of its statistical characteristics observed experimentally.', 'arxivid': '0904.0970', 'author': ['O I Berngardt \nInstitute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia\n', 'A P Potekhin \nInstitute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia\n', 'O I Berngardt \nInstitute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia\n', 'A P Potekhin \nInstitute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia\n'], 'authoraffiliation': ['Institute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia', 'Institute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia', 'Institute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia', 'Institute of Solar-Terrestrial physics SB RAS\nLermontova Str., 126aPOBox 291664033IrkutskRussia'], 'corpusid': 118390511, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16390, 'n_tokens_neox': 13544, 'n_words': 8150, 'pdfsha': '4ce848fadc759ca423785a0074f062f0a9108754', 'pdfurls': ['https://arxiv.org/pdf/0904.0970v2.pdf'], 'title': ['Towards the dispersion relation for ionacoustic instabilities in weakly inhomogeneous ionospheric plasma at altitudes 80-200km and its low-frequency solution. The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 2', 'Towards the dispersion relation for ionacoustic instabilities in weakly inhomogeneous ionospheric plasma at altitudes 80-200km and its low-frequency solution. The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 2', 'Towards the dispersion relation for ionacoustic instabilities in weakly inhomogeneous ionospheric plasma at altitudes 80-200km and its low-frequency solution. The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 2', 'Towards the dispersion relation for ionacoustic instabilities in weakly inhomogeneous ionospheric plasma at altitudes 80-200km and its low-frequency solution. The dispersion relation for ionacoustic instabilities in the ionosphere at 80-200km 2'], 'venue': []}

arxiv

On the absolute value of the neutrino mass Dragan Slavkov On leave from Cetinje Montenegro Hajdukovic dragan.hajdukovic@cern.ch PH Division CERN CH 1211Geneva 23 On the absolute value of the neutrino mass 1Neutrino massneutrino oscillationsblack holes The neutrino oscillations probabilities depend on mass squared differences; in the case of 3-neutrino mixing, there are two independent differences, which have been measured experimentally. In order to calculate the absolute masses of neutrinos, we have conjectured a third relation, in the form of a sum of squared masses. The calculated masses look plausible and are in good agreement with the upper bounds coming from astrophysics.The recent investigations of neutrinos from the sun and of neutrinos created in the atmosphere by cosmic rays, have given strong evidence for neutrino oscillations, i.e. phenomenon when neutrinos change from one flavour to another (See Ref. 1 and 2 for a recent Review). Neutrino flavour change implies that neutrinos have masses. To determine these masses remains one of the most challenging tasks of contemporary physics, bearing fundamental implications to particle physics, astrophysics and cosmology. In the present paper we have calculated the absolute mass of neutrinos by using the existing experimental data and a new theoretical assumption. As the equation (2) indicates, the existing data do not allow one to determine the sign of The two equations (1) and (2) m m m = for the inverted spectrum. This proposal is inspired by the geometric mean mass relation used previously for quarks. A weak point of the proposal lies in the well known fact [1,2] that leptonic mixing (characterized with large mixing angles) is very different from its quark counterpart, where all the mixing angles are small. Our second objection is that geometric mean mass relation has a form which is quite different from Equations (1) and (2) while it is desirable to have a third equation with similar form. The equations (1) and (2) are about mass squared differences; hence the most natural third equation is in the form of a mass squared sum. So, as a third equation we propose 2 0 ; 2 3 2 2 2 1 < < = + a am m m (3) for normal spectrum and 2 0 ; 2 ' 2 2 ' 3 2 ' 1 < < = + b bm m m (4) for inverted spectrum. We have denoted masses of inverted spectrum by a prime. From the mathematical point of view, for sure, there is a constant a (or b ) for which the above relation is exact. At the end of this Letter and in the Appendix we would try to give a physical meaning to the above mass squared sums. Equations (1), (2), (3) and (4) In principle the masses obtained above must be checked against known experimental constraints, which unfortunately still have no a satisfactory accuracy. The best we can do is to compare with the upper limit on the sum of neutrino masses, deduced from astrophysical observations. According to (11) . 0 [4] determined from a photometric galaxy redshirt survey. The physical ground for relations (1) and (2) is the fact that the neutrino oscillations probabilities depend on mass squared differences. What could be physical ground for mass squared sums (3) and (4)? We suggest that mass squared sums are related to a hypothetical source of neutrinos, characterized with the following emission probabilities 1 ; 3 1 2 = = ∑ = i i i i w Am w (12) where A is a constant depending on the physical characteristics of the source. Then, the equations (3) for the normal spectrum follows from (12) with a being + = + = + a w a a w w(14) It is easy to obtain the analogous results for inverted spectrum. Now, according to (13), the choice 1 = a may be interpreted as equal emission probability for a "doublet" ( 2 1 w w + ) and a "singlet" ( 3 w ). As it was argued in [6] a sufficiently strong gravitational field needed for validity of relation (15) may exist only deep inside the horizon of a black hole, from where the created antineutrinos, are violently ejected. Hence, a black hole might behave as a point like source of antineutrinos. In principle, the study of antineutrino radiation of supermassive black holes in the centre of Milky Way and Andromeda Galaxy may confirm this phenomenon and lead to determination of probabilities (14) and the constant a . The study of solar neutrinos was crucial for relation (1): it is a striking speculation that the study of black hole neutrinos might be crucial for relation (3). correspond to two types of neutrino mass spectrum: spectrum with case of inverted hierarchy. In order to calculate neutrino masses we must determine a and b . The most elegant possibility (but we do not know if nature has the same aesthetic taste as we have) is 1 = a in the normal spectrum and 1 = b in the inverted spectrum; what reduces the equation (3) and (4) to the form of Pythagoras theorem.In addition to the beauty, > ), while the other two eigenstates have the same value in both spectrums (i.e. spectrum what is in agreement with the tightest current bounds, like a recent upper bound of eV 28 are obviously not sufficient to determine three masses third relation between masses is needed. For instance, a recent proposal[3] is to use, as a third equation,3 2 1 , , m m m . A geometric mean neutrino mass relation 3 1 2 m m m = for normal spectrum and 3 2 1 A source of neutrinos with emission probabilities (12) is not known in nature, but it is striking that assumption of such a source leads to a plausible absolute mass of neutrino. In Appendix, we would argue that such a source is possible in the framework of contemporary physics. AppendixLet us remember the Schwinger mechanism[5]in Quantum Electrodynamics: a strong electric field E greater than a critical value cr E , can create electron-positron pairs (or other charged particleantiparticle pairs) from the quantum vacuum. Hence, an external field can separate virtual particle and antiparticle, i.e. transform a virtual pair into a real one. In the limit cr E E >> , the exponential factor in Schwinger relation[5]becomes a constant, and the particle-antiparticle creation rate per unite volume and time is proportional to the squared mass of considered particles. Consequently, according to (12) and (15), the masses of the charged leptons (electron, muon and tau) must satisfy a relation of the form (3). However, neutrinos are not charged particles and the original Schwinger mechanism does not work for them.One interesting possibility to get relation (15) for neutrinos is to assume the existence of the gravitational repulsion between matter and antimatter[6,7,8]. If so, the relations (15), and consequently the sum (3), are valid for neutrinos through a gravitational version[6,7]of the Schwinger mechanism. Neutrino mass, mixing, and oscillations. K Nakamura, S Petcov, J. Phys. G. 37K. Nakamura and S.T Petcov: Neutrino mass, mixing, and oscillations. J. Phys. G 37, 164-183 (2010) Three-flavour neutrino oscillation update. G L Fogli, Nuclear Physics B. 188G.L. Fogli et al.: Three-flavour neutrino oscillation update. Nuclear Physics B 188, 27-30 (2009) Geometric mean neutrino mass relation. X G He, A Zee, Modern Physics Letters. 22X.G. He and A. Zee: Geometric mean neutrino mass relation. Modern Physics Letters A22, 2107-2112 (2007) Upper Bound of 0.28 eV on Neutrino Masses from the Largest Photometric Redshift Survey. S A Thomas, Phys. Rev. Letters. 10531301S.A. Thomas et al.: Upper Bound of 0.28 eV on Neutrino Masses from the Largest Photometric Redshift Survey. Phys. Rev. Letters 105, 031301 (2010) On Gauge Invariance and Vacuum Polarization. J S Schwinger, Phys. Rev. 82J.S. Schwinger: On Gauge Invariance and Vacuum Polarization. Phys. Rev. 82 664-679 (1951) Can the new neutrino telescopes reveal the gravitational properties of antimatter?. D S Hajdukovic, DOI10.1155/2011/196852arXiv:0710.4316v5Advances in Astronomy. D.S. Hajdukovic: Can the new neutrino telescopes reveal the gravitational properties of antimatter? Advances in Astronomy, DOI 10.1155/2011/196852 (2011); (2007) arXiv:0710.4316v5 (2007) D S Hajdukovic, arXiv:gr-qc/0612088v4Black Holes and Gravitational Properties of Antimatter. D.S. Hajdukovic: Black Holes and Gravitational Properties of Antimatter, arXiv:gr-qc/0612088v4 (2007) CPT symmetry and antimatter gravity in general relativity. M Villata, Europhysics Letters. 94M.Villata: CPT symmetry and antimatter gravity in general relativity. Europhysics Letters 94, 20001 (2011)

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{'abstract': 'The neutrino oscillations probabilities depend on mass squared differences; in the case of 3-neutrino mixing, there are two independent differences, which have been measured experimentally. In order to calculate the absolute masses of neutrinos, we have conjectured a third relation, in the form of a sum of squared masses. The calculated masses look plausible and are in good agreement with the upper bounds coming from astrophysics.The recent investigations of neutrinos from the sun and of neutrinos created in the atmosphere by cosmic rays, have given strong evidence for neutrino oscillations, i.e. phenomenon when neutrinos change from one flavour to another (See Ref. 1 and 2 for a recent Review). Neutrino flavour change implies that neutrinos have masses. To determine these masses remains one of the most challenging tasks of contemporary physics, bearing fundamental implications to particle physics, astrophysics and cosmology. In the present paper we have calculated the absolute mass of neutrinos by using the existing experimental data and a new theoretical assumption.', 'arxivid': '1106.5810', 'author': ['Dragan Slavkov \nOn leave from Cetinje\nMontenegro\n', 'Hajdukovic dragan.hajdukovic@cern.ch ', '\nPH Division CERN CH\n1211Geneva 23\n'], 'authoraffiliation': ['On leave from Cetinje\nMontenegro', 'PH Division CERN CH\n1211Geneva 23'], 'corpusid': 119256792, 'doi': '10.1142/s0217732311035948', 'github_urls': [], 'n_tokens_mistral': 2445, 'n_tokens_neox': 2075, 'n_words': 1348, 'pdfsha': 'b32bb1bf731183eb96325a2ca4d50ed15e2265b2', 'pdfurls': ['https://export.arxiv.org/pdf/1106.5810v1.pdf'], 'title': ['On the absolute value of the neutrino mass', 'On the absolute value of the neutrino mass'], 'venue': []}

arxiv

Optical and Electronic Properties of Molecular Systems Derived from Rhodanine Duvalier Madrid-Úsuga Carlos A Melo-Luna Alberto Insuasty Alejandro Ortiz John H Reina john.reina@correounivalle.edu.co †Centre for Bioinformatics and Photonics-CIBioFi ‡Department of Physics Universidad del Valle Cll. 13 No. 100-00, Edif. 320, Esp. 1069760032CaliColombia ¶Department of Chemistry and Biology Universidad del Valle 760032CaliColombia §Department of Chemistry Universidad del Norte Km 5 via Puerto Colombia081007BarranquillaColombia Universidad del Valle 760032CaliColombia Optical and Electronic Properties of Molecular Systems Derived from Rhodanine Push-Pull functional compounds consisting of dicyanorhodanine derivatives have attracted a lot of interest because their optical, electronic, and charge transport properties make them useful as building blocks for organic photovoltaic implementations.The analysis of the frontier molecular orbitals shows that the vertical transitions of electronic absorption are characterized as intramolecular charge transfer; furthermore, we show that the analyzed compounds exhibit batochromic displacements when comparing the presence (or absence) of solvent as an interacting medium. In comparison with materials defined by their energy of reorganization of electrons (holes) as electron (hole) transporters, we find a transport hierarchy whereby the molecule (Z)-2-((1,1dicyanomethylen)-5-(4-dimethylamino)benzylidene)-1,3-thiazole-4 is better at transport-1 arXiv:1804.10300v1 [physics.chem-ph] 26 Apr 2018 ing holes than molecule (Z)-2-((1,1-dicyanomethylene)-5-(tetrathiafulvalen-2-ylidene)-1,3-thiazole-4. Introduction Charge transfer (CT) studies seek to understand the ways in which their transfer rate of CT depends on the properties of the electro-donor and electro-acceptor system, solvent, molecular bridge and electronic coupling between the involved states 1,2 . The different functionality played by these factors and the way they affect the qualitative and quantitative aspects of the electron transfer process have been extensively discussed in recent years [3][4][5][6][7] . The need for understanding the processes of electron (ET) or charge transfer at the molecular level have prompted the study of highly conjugated molecular systems of the donor-acceptor (D-A), type given their unique photo-physical and photo-activated properties 8 . These properties have favoured the application and development of such systems in areas such as: non-linear optical materials 9 , molecular optical switches 10 , and photovoltaic cells 11 , among others. In the field of photovoltaics, organic photovoltaic devices with D-π-A materials have attracted a lot attention due to their potential in the creation of flexible and remarkably light solar cells, with low manufacturing cost and high power conversion efficiency (PCE) 12 . A key issue to understanding the CT process is the ability to make quantitative predictions and measurements of the characteristics associated to the individual molecular systems that allow useful information for a direct comparison of the electron dynamics inferred in electro and photochemistry, at the nanometric, molecular and electronic scales 13 . Currently, the increasing availability of kinetic data of CT processes and the development of computational tools allow the study of different molecular systems independently of their size, that exhibit better photophysical properties, and facilitate a direct comparison between theory and experiment [14][15][16] . These systems consist of covalent bonds of electro-active chemical species, whether they are electro-donors or electro-acceptors, which can be either connected through a π-conjugated bridge or directly. The derivatives of the Rhodanine implemented in the synthesis of push-pull systems are an example of these type of systems, and they have been used as an electro-acceptor fragment in a variety of organic compounds of interest; for example, in non-linear second order analytical reactives, and, more recently, as metal-free organic dyes in the manufacturing of dyesensitized solar cells (DSSCs) [17][18][19] . For this purpose, the push-pull molecular system of the donor-rhodanine type is efficiently anchored to the meso-porous surface of TiO 2 . The light absorbed by the dye injects electrons into the conduction band of the TiO 2 , thus generating an electric current, while the fundamental state of the dye is regenerated by the electrolyte 20 . The precise prediction of the electron transfer rate in chemical and biological reactions of this type makes them attractive systems for different applications in the field of molecular electronics 21,22 . In this work, we present results obtained for new "push-pull" chromophores based on derivatives of rhodanine, whereby 4-dimethylamine and 2-formyltetratiafulvalene exhibit the role of electro-donor groups in which the nature of the electron transfer processes is studied when they are connected to a dicyanorhodanine electro-acceptor through a small molecular bridge. To determine the most stable structure, the absorption spectrum and the first electronic state of the complexes were calculated by means of density functional theory (DFT) numerical simulation. We study the way the character of electron transfer in complexes gets affected by the presence of a solvent that acts as an environment and also in gas phase, seeking to report on novel quantitative results for such compounds 23 , and explore their potential in the application and design of innovative and highly efficient donor-acceptor multifunctional devices that exhibit optimal electronic properties. Chromophores computational details Here, consider the molecules (Z)-2-((1,1-dicyanomethylen)-5-(4-dimethylamino)benzylidene)-1,3-thiazole-4 (molecule 1), and (Z)-2-((1,1-dicyanomethylene)-5-(tetrathiafulvalen-2-ylidene)-1,3-thiazole-4 (molecule 2), as shown in Fig. 1. The geometries obtained for such most stable conformations (see Supplementary section) were used as input data for the full optimization of calculations of the ground state by means of the hybrid functional BE3LYP with a base set 6-31G+, using Gaussian 09 24 ; the corresponding optimized structures are used for the molecules energy calculation. The molecules excited states were calculated by means of the time-dependent density functional theory (td-DFT), and the results here reported were carried out with the molecules i) in the gas phase, and ii) by simulating an environment-methanol as solvent. In order to see the effects caused by the solvent on the electronic properties of the different compounds, the working molecules in the solvent were designated as follows: system S1: Molecule 1 + Methanol and system S2: Molecule 2 + Methanol, while the system GP1 : Molecule 1 in gas phase and the system GP2: Molecule 2 in gas phase. Additionally, different properties of these molecules, such as higher occupied molecular orbitals (HOMOs), lower unoccupied molecular orbitals (LUMOs), energy gap, reorganization energy, Gibbs free energy, and excitation energy are derived from the computational results. In our theoretical calculations we take into account the effects due to the solvent, since we aim to make predictions about the experimental spectra with a reasonable precision. We consider methanol as solvent with = 32.6, following a Conductor-like Polarization Continuum Model (C-PCM) 25,26 . Results and discussion Electronic Transitions. Computational calculations to optimize the geometry of molecules 1 and 2 were carried out using DFT-B3LYP/6-31G+; subsequently we used td-DFT for de- Here we analyze the frontier molecular orbitals in inder to quantify the relationship between structural and electronic geometry. For system in solvent S1, the HOMO is mostly concentrated in the donor (4-dimethylamino group), while the HOMO-1 and the LUMO are mostly located in the acceptor (2-(1,1-dicyanomethylene)-1,3-thiazole-4), as seen in Fig. 2. Therefore, the CT is the charge transfer mixture within the 4-dimethylamino coupled with the CT from 2-(1,1-dicyanomethylene) -1,3-thiazole-4 to 4-dimethylamino moiety. For system S2, the HOMO and HOMO-1 are mainly located in the tetrathiafulvalene moiety, Figure 2: Spectra for the total density of states obtained using B3LYP/6-31G+ for the systems S1, S2, GP1 and GP2. The blue curve represents the density of state spectrum, the green lines represent the occupied molecular orbitals, and the red lines are the virtual molecular orbitals. while the LUMO is located in the 2-(1,1-dicyanomethylene)-1,3-thiazole-4 ( Fig. 2). Thus, the transitions of system S2 from the HOMO to the LUMO together with HOMO-1 to LUMO have a more significant character in the charge transfer with respect to system S1, reflecting that the tetrathiafulvalene acts as a better electro-donor fragment than the first one. When we observe the frontier molecular orbitals of molecules in gas phase, we see that the behavior described above is maintained. However, their density of state spectra have very significative changes in the spectral densities corresponding to each energy level, which indicates a considerable effect due to the solvent on the electronic and geometric structure of the compounds. Absorption Spectra. The electronic transition energies and the charge transfer transitions are calculated using TD-DFT/B3LYP 28,29 . The UV-Vis absorption spectra for the systems S1, S2, GP1 and GP2 are shown in Fig. 3. We find that system S1 has a strong absorption band at 490.01 nm, together with other bands of lower energy at 343.37 nm and 283.00 nm corresponding to transitions of the type π − π * , which are associated to the S 2 and S 7 states, respectively (see Table 1). The intense high energy transition at 490.01 nm is described by the excitation HOMO→LUMO (99%), according to the orbital transition diagram (See Supporting information S1). This high energy transition can be assigned to intramolecular charge transfer from the 4-dimethylamino electro-donor fragment to the electro-donor fragment; the low energy transition at 343.37 nm corresponds to the transition HOMO→LUMO+1 (65%). The electronic transitions can be seen as a contribution to the intramolecular charge transfer process from the electro-donor to the electro-acceptor and like the low energy transition at 283.00 nm which is described by HOMO→LUMO+3 (75%). The system S2 shows a band of high energy absorption at 388.31 nm and bands of low absorption at 661.10 nm, 325.98 nm, and 294.66 nm. The low energy transition S 1 for the system S2 is described by the transition HOMO→LUMO (99%), which represents an intramolecular CT from the 2-tetratiafulvaleno electro-donor moiety to the 2-(1,1- System S1 System S2 System GP1 System GP2 Figure 3: Absorption spectrum of the molecules in solvent (systems S1, S2), and molecules in gas phase (systems GP1 and GP2). 30 . The results shown in Table 1 reveal that the systems S1 and GP1 correspond to systems that have a higher absorption capacity when compared to systems S2 and GP2. Emission Properties. We use td-DFT, with the hybrid B3LYP and basis set 6-31G+ in order to compute for the structure in an excited state and simulating the emission spectrum of the systems under study. The maximum emission wavelengths are shown in Table 2. The transitions S 1 → S 0 and S 3 → S 0 represent fluorescence peaks in the emission spectrum; in addition, the system S1 has the highest oscillator strength, which corresponds to a LUMO→HOMO transition. The results for the excitation energy, oscillator strength and radiative lifetime are presented in Table 2. We also report the Stokes shift values, defined as the difference between λ max of absorption and λ max of emission spectrum. The Stokes shift gives the energy difference that exists between the absorption and emission due to the same levels. This provides information about the probability of radiative and non-radiative de-excitation between two levels, where the probability of radiative de-excitation increases with the difference of energy and that of the non-radiative decreases. Hence, the first one dominates when the energy levels are well separated and the second one does it when we have closer levels. Thus, the radiative lifetime was calculated for the spontaneous emission spectrum using the Einstein transition probabilities according to the expression 31,32 : τ R = c 3 2(E f lu ) 2 f osc ,(1) where c is the speed of light in vacuum, E f lu is the fluorescence excitation energy, and f is the oscillator strength. We conclude, as can be seen from Table 2 that the presence of the solvent favours the radiative processes for the case of molecule 1, since the Stokes shift is greater in the presence of methanol (18.0 nm) than in the gas phase (16.2 nm). However, for the case of molecule 2 this process is favoured in the gas phase rather than in the presence of the solvent, which is made evident by the longer radiative lifetime found for the gas phase. It is well known that short radiative lifetimes lead to a high efficiency of light emission, while long radiative lifetime facilitates the electron and energy transfer. In our case, the radiative lifetime is shorter for systems with higher oscillator strength, which leads to an increase in luminescent efficiency. The duration of emission (τ R ) for the studied molecules have the following order: τ GP 2 R > τ S2 R > τ S1 R > τ GP 1 R . This hierarchy indicates that the change of a donor unit strongly decreases the emission lifetime of the compound in both gas phase and solvent; we find the highest oscillator strength and the smallest lifetime radiation in the case of the system S1 and GP1, which correspond to the molecule 1 under different environmental conditions. Consequently, molecule 1 represents a good emission material with high efficiency 33 . Charge Transfer Rate. Charge transfer is a crucial process involved in many physical and biology phenomena such as the photosynthesis 34 ; this process can be estimated, in a first approximation, by using the semi-classical theory of Marcus 35,36 : k e(h) = 2π |V e(h) | 2 4πλ e(h) k B T exp − λ e(h) 4k B T ,(2) where V e(h) is the electronic coupling between the final and the initial state for electrons (holes), λ e(h) is the reorganization energy for electron (hole), and k B denotes Boltzmann constant. For an efficient CT mechanism the reorganization energy of the molecular system must be small and an electronic coupling between the electro-acceptor and electro-donor parts is necessary 37 . The reorganization energy comprises two factors, the first one is the internal or intramolecular reorganization energy (λ int ), and the second one is the external or intermolecular reorganization energy (λ ext ). The λ int accounts for the structural changes between neutral and ionic states and must be calculated, while λ ext reflects the change in the polarization of the medium after the CT takes place 38 . The intramolecular reorganization energy λ int can be estimated for the electrons (reorganization energy of electron λ e ) and for the holes (reorganization energy of holes λ h ), and can be expressed by the following equation [39][40][41] : λ e = λ e 1 + λ e 2 = E − 0 − E − − + E 0 − − E 0 0 λ h = λ h 1 + λ h 2 = E + 0 − E + + + E 0 + − E 0 0 ,(3) where E + 0 (E − 0 ) is the energy of the cation (anion) calculated with the optimized structure of the neutral molecule. Similarly, E + + is the energy of cation (anion) calculated with the optimized cation (anion) structure, E 0 + (E 0 − ) is the energy of the neutral molecule calculated at the cationic (anionic) state. Finally, E 0 0 is the energy of the neutral molecule at the ground state. Table 3: Molecular calculation of the reorganization energy for electrons (holes) λ e(h) , electronic coupling for the electron (holes) transfer mechanism V e(h) , and electron (hole) transfer rate k e(h) . System This redistribution of energies for the case of electron reorganization energy is best observed in Fig. 4. The external reorganization energy λ ext explains the nuclear shifts in the surrounding medium and the resulting electronic effects are much harder to calculate. This is assumed to be, for many authors, between 0.2 eV and 0.5 eV for simple models based on the dielectric properties of organic matrices 42,43 . The electronic coupling V , which is another important parameter in the CT process, is the geometrically most dependent element of the kinetic constant because its value depends on the distance between donor-acceptor and the geometry of the system (orientation of the orbitals); this parameter is also sensitive to changes in the systems under study such as solvents, and temperature, among others [44][45][46][47] . λ e (eV) λ h (eV) V e (eV) V h (eV) k e (10 15 s −1 ) k h (10 15 s −1 ) S1 0. Here, we use the generalized Mullinken-Hush method (GMH) 48 to calculate the electronic couplings, and the operator used in the GMH method is the adiabatic dipole moment matrix µ 12 48 . Under this approach (and in the weak coupling regime), the electronic coupling for a direct donor-acceptor coupling is calculated by the equation 49 : V = ∆E 12 µ 12 (∆µ 1 − ∆µ 2 ) 2 + 4µ 2 12 ,(4) where ∆E 12 is the orbital energy difference, and ∆µ 12 is the dipole moments difference of the adiabatic states. The calculation of the reorganization energy, electronic coupling and the electron transfer rate are shown in Table 3. As reported in 30,50,51 , it has been found that at low values of reorganization energy, the transfer rate is high. The hole reorganization energy calculated for the systems S2 and GP2 are smaller than those for the systems S1 and GP1; this implies that the hole transfer rate is greater in the systems S2 and GP2, and we also note that for the case of the system S2 compared to the GP2 the hole transport rate is higher in the presence of methanol than in the case of the gas phase, confirming this behavior for the case of the system S1 and GP1 where the same situation is observed, which indicates that methanol does not favour the transport of holes. Furthermore, the hole reorganization energies λ h for all systems are smaller than that of N,N'-diphenyl-N,N'-bis(3-methlphenyl)-(1,10-biphenyl)-4,4'-diamine (TPD), which is a typical hole transport material with λ h = 0.290 eV 52 . This implies that the hole transfer rates of the molecules 1 and 2, in the condition under study, might be higher than that of TPD. Thus, the molecules 1 and 2 might comprise good hole transport materials from the stand point of the smaller reorganization energy. On the other hand, we observe that for the case of the system S2 compared to GP2 the hole transport rate is higher in the presence of methanol than in the case of the gas phase, confirming this behavior for the case of the systems S1 and GP1 where the same situation occurs, which indicates that methanol favours the transport of voids. The value of λ e is smaller for the case of the systems S1 and GP1 than for the systems S2 and GP2: this indicates that the electron transfer rates for S1 and GP1 will be larger than those due to the systems S2 and GP2 as can be seen in Table 3. In addition, by comparing λ e for the systems S1 and GP1, as well as for S2 and GP2, we see that κ e for κ S1 e > κ GP 1 e and κ e in κ S2 e > κ GP 2 e , which indicates that the electron transfer process is favoured by the presence of solvent in the molecules 1 and 2. In addition, by comparing the reorganization energies for electron reorganization and holes, we observe that the values of λ h are smaller than those for λ e , suggesting that the carrier mobility of the electrons is larger than that of the holes. Hence, the molecules 1 y 2 can be used as promising hole transport materials in, e.g., organic light-emitting diodes from the stand point of the smaller reorganization energy, which can be corroborated with the κ values shown in the Table 3. Finally, given that κ e and κ h are greater for the systems S1 and GP1, we conclude that the molecule 1 is better at transporting charge than the molecule 2. In addition to the previous analysis, it can be seen that the molecule 1 is more transport-efficient than the molecule 2. Conclusions We have theoretically investigated different optical and electronic properties for new structures based on 4-dimethylamino and tetrathiafulvalene as electro-donor groups and dicyanorhodanine as electro-aceptor group, in which the correlation between structures and electronic dynamics is studied by means of theoretical chemical calculations. It was observed that the presence of a solvent with which the molecules 1 and 2 interacts, favours the electronic transfer process. Furthermore, the change of the donor group shows that tetrathiafulvalene acts as a better electron donor than 4-dimethylamino. As regards the wavelength of absorption, this shows a batochromic effect between the gas phase and the presence of the solvent. The computational results predict the electronic properties of the systems S1, S2, GP1 and GP2, and the analysis of the molecular frontier orbitals shows that the vertical electronic transitions of absorption of the studied compounds are characterized as intramolecular charge transfer. In addition, the molecules 1 and 2 can be used as void transport materials. Figure 1 : 1Molecular structure of the chromophores under study. Molecule 1 is the (transition states that most favoured the charge transfer and search for a favorable environment in the CT processes. The systems under study are of the donor-acceptor type in which two different electro-donor fragments are connected to an electro-acceptor fragment for generating and effective CT process, which results in a charge separation state for analyzing the key molecular properties in the calculation of the charge distributions in these molecules 23 . D-π-A molecular systems have two possible mechanisms for charge transfer, i) super exchange: the charge or electrons transferred do not reside directly in the molecular bridge and the states occupied by the molecule during this time are known as virtual excitations; and ii) sequential charge transfer (hopping): there are real intermediate states that are energetically accessible, and this (thermally activated) mechanism is generally more efficient for long-distance electronic transfer processes 27 . of the state S 4 described by the transition HOMO-1→LUMO (88%) where a CT from electro-donor to electro-acceptor is present. For HOMO-3 and LUMO both mainly concentrated in 2-(1,1-dicyanomethylene)-1,3-thiazole-4, associated with the transition HOMO-3→LUMO (80%), a CT within the electro-acceptor and not a transfer from the electro-donor to the concise electro-acceptor takes place.For the case of the system GP1, comprising the molecule 1 in gas phase, we observe that it presents a high energy absorption band at 427.12 nm corresponding to the transition state of the HOMO→LUMO (99%). In addition, as in the case of the system S1, it presents two low energy absorption bands at i) 329.19 nm corresponding to the transition state of the HOMO-1→LUMO (70%) that describes a contribution to the process of CT in the interior of (2-(1,1-dicyanomethylene)-1,3-thiazol-4), since the HOMO-1 and LUMO are more concentrated in the acceptor, with a small CT contribution from the donor to the acceptor (as can be seen in Supporting information S2); and ii) a low energy band at 281.39 nm, corresponding to the transition of the HOMO-2 →LUMO. By comparing the graphs of systems S1 and GP1, we obtain that the solvent produces an effective shift or batochromic displacement with respect to the wavelength of absorption.The system GP2 shows a similar behavior to that of system S2. However, a batochromic shift is observed in relation to the absorption spectrum when comparing the systems S2 and GP2. The resulting shift is associated to the molecule-solvent interaction, since for the system GP2 this presents a high energy band at 380.16 nm, and for system S2 in presence of methanol this occurs at 388.31 nm. In addition, the system GP2 presents two low energy bands, one at 671.27 nm and the other one at 297.93 nm; they are associated with the transition states HOMO →LUMO (99%), and HOMO-3→LUMO (79%), respectively. The energy band at 380.16 nm is associated to the transition status of HOMO-1 →LUMO (98%). Figure 4 : 4Scheme for the calculation of the reorganization energy for the electron transfer; λ 1 is the reorganization energy of the neutral molecule and λ 2 denote the reorganization energy of the radical anion.We calculate the electron reorganization energy usingFig. 4. According to Marcus model, the rate of electron transfer depends mainly on the energy of reorganization and the coupling between the donor and the acceptor, in addition to the general exergonity of the process 39 . It is also considered that for a self-exchange electron transfer process (where interaction with solvent is not considered) the change in the Gibbs free energy is zero and the electron transfer rate will only depend intrinsically on the barrier of activation, marked by the internal and external reorganization energy, and the electronic coupling parameter V . The λ int in Equ. (3) for the self-exchange process has two contributions, arising from the geometric relaxation along inter-nuclear coordinate upon moving from neutral-state to the charged-state geometry and vice versa. Table 1 : 1Wavelengths of the most important simulated transition states λ, oscillator strengths f os , excitation energy E, and band gap energy ∆E H−L . System States λ(nm) ∆E H−L (eV) f osc E (eV) CompositionS 1 490.01 0.9727 2.530 HOMO→LUMO (99%) S1 S 2 343.37 2.74 0.3391 3.611 H-1→LUMO (%34); HOMO → L+1 (65%) S 7 283.00 0.1336 4.381 H-3→LUMO (16%);HOMO→L+3 (75%) S 1 661.10 0.1995 1.875 HOMO→LUMO (99%) S2 S 4 388.31 2.27 0.6559 3.193 H-1→LUMO (88%); HOMO →L+1 (10%) S 6 325.98 0,1099 3.803 H-2→LUMO (92%) S 10 294.66 0.0962 4.208 H-3→LUMO (80%); H-1 → L+1 (13%) S 1 427.12 0.9075 2.903 HOMO→LUMO (99%) GP1 S 2 329.19 3.05 0.1939 3.767 H-1→LUMO (70%); HOMO → L+1 (29%) S 7 281.39 0.0833 4.406 H-2→LUMO (76%) S 1 671.27 0.1365 1.847 HOMO→LUMO (99%) GP2 S 2 380.16 2.23 0.6322 3.261 H-1→LUMO (98%) S 3 297.93 0.1194 4.161 H-3→LUMO (79%); H-1 → L+1 (11%) Table 1 1shows that f os values, that correspond to the transition states HOMO →LUMO of systems S1 and GP1, are similar to each other, as are for systems S2 and GP2. The oscillation strength for an electronic transition is proportional to the transition dipole moment. In general, a large oscillator strength corresponds to large experimental absorption coefficients or a stronger fluorescence intensity Table 2 : 2Emission spectrum results obtained for the system under study in solvent (S) and gas phase (GP).System Elect. Trans. λ max (nm) E s (eV ) f os Stokes Shift (nm) τ R (ns) S1 s 1 → s 0 508.90 2.4363 1.0126 18.0 3.85 S2 s 3 → s 0 416.78 2.9748 0.5316 28.8 5.02 GP1 s 1 → s 0 443.28 2.7970 0.918 16.2 3.36 GP2 s 3 → s 0 415.36 2.9850 0.3792 35.2 7.29 AcknowledgementThe authors acknowledge support by the Colombian Science, Technology and InnovationSupporting Information AvailableThe following files are available free of charge.Contains the optimal geometries, the total energy and the frontier molecular orbitals involved in the electronic transitions at the TD-DFT B3LYP/6-31G+ levels. Mechanisms for DNA charge transport. J C Genereux, J K Barton, Chemical reviews. 110Genereux, J. C.; Barton, J. K. Mechanisms for DNA charge transport. 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{'abstract': 'Push-Pull functional compounds consisting of dicyanorhodanine derivatives have attracted a lot of interest because their optical, electronic, and charge transport properties make them useful as building blocks for organic photovoltaic implementations.The analysis of the frontier molecular orbitals shows that the vertical transitions of electronic absorption are characterized as intramolecular charge transfer; furthermore, we show that the analyzed compounds exhibit batochromic displacements when comparing the presence (or absence) of solvent as an interacting medium. In comparison with materials defined by their energy of reorganization of electrons (holes) as electron (hole) transporters, we find a transport hierarchy whereby the molecule (Z)-2-((1,1dicyanomethylen)-5-(4-dimethylamino)benzylidene)-1,3-thiazole-4 is better at transport-1 arXiv:1804.10300v1 [physics.chem-ph] 26 Apr 2018 ing holes than molecule (Z)-2-((1,1-dicyanomethylene)-5-(tetrathiafulvalen-2-ylidene)-1,3-thiazole-4.', 'arxivid': '1804.10300', 'author': ['Duvalier Madrid-Úsuga ', 'Carlos A Melo-Luna ', 'Alberto Insuasty ', 'Alejandro Ortiz ', 'John H Reina john.reina@correounivalle.edu.co ', '\n†Centre for Bioinformatics and Photonics-CIBioFi\n‡Department of Physics\nUniversidad del Valle\nCll. 13 No. 100-00, Edif. 320, Esp. 1069760032CaliColombia\n', '\n¶Department of Chemistry and Biology\nUniversidad del Valle\n760032CaliColombia\n', '\n§Department of Chemistry\nUniversidad del Norte\nKm 5 via Puerto Colombia081007BarranquillaColombia\n', '\nUniversidad del Valle\n760032CaliColombia\n'], 'authoraffiliation': ['†Centre for Bioinformatics and Photonics-CIBioFi\n‡Department of Physics\nUniversidad del Valle\nCll. 13 No. 100-00, Edif. 320, Esp. 1069760032CaliColombia', '¶Department of Chemistry and Biology\nUniversidad del Valle\n760032CaliColombia', '§Department of Chemistry\nUniversidad del Norte\nKm 5 via Puerto Colombia081007BarranquillaColombia', 'Universidad del Valle\n760032CaliColombia'], 'corpusid': 53037237, 'doi': '10.1021/acs.jpca.8b08265', 'github_urls': [], 'n_tokens_mistral': 15267, 'n_tokens_neox': 12718, 'n_words': 7256, 'pdfsha': 'fe3e2f2a3c7843e731676da193ffbd0d7f4702d6', 'pdfurls': ['https://arxiv.org/pdf/1804.10300v3.pdf'], 'title': ['Optical and Electronic Properties of Molecular Systems Derived from Rhodanine', 'Optical and Electronic Properties of Molecular Systems Derived from Rhodanine'], 'venue': []}

arxiv

Graph-based Nearest Neighbor Search: From Practice to Theory Liudmila Prokhorenkova ostroumova-la@yandex.ru Yandex Moscow Institute of Physics and Technology Graph-based Nearest Neighbor Search: From Practice to Theory Graph-based approaches are empirically shown to be very successful for approximate nearest neighbor (ANN) search. However, there has been very little research on their theoretical guarantees. In this work, we consider both low-dimensional (d log n) and high-dimensional (d log n) regimes and rigorously analyze the performance of graph-based nearest neighbor algorithms when the dataset is uniformly distributed on a d-dimensional sphere. For both regimes, we provide the conditions which guarantee that a graph-based algorithm solves the ANN problem in just one iteration. In the low-dimensional regime, we also show that it is possible to solve the exact nearest neighbor problem. Finally, we discuss how the "small-world" property affects the performance of graph-based approaches.IntroductionMany methods in machine learning, pattern recognition, coding theory, and other research areas are based on nearest neighbor search (NNS)[9,11,28,33]. In particular, the k-nearest neighbor method is included in the list of top 10 algorithms in data mining[37]. Due to the fact that modern datasets are mostly huge (both in terms of the number of elements n and the dimension d), reducing the computation complexity of NNS algorithms is of the essence. The nearest neighbor problem is to preprocess a given dataset D in such a way that for an arbitrary forthcoming query vector q we can quickly (in time o(n)) find its nearest neighbors in D.Several efficient methods exist for NN problem when the dimension d is small[5,8,29]. In particular, the algorithms based on recursive partitions of the space, like k-d threes and random projection trees, are widely used[8,12,13,22]. However, all existing methods for the exact NNS suffer from the curse of dimensionality: their complexity (either time or space) is exponential in d. To overcome this issue, NN problem has been relaxed to c-approximate nearest neighbor problem (c-ANN): if the distance between q and its closest neighbor is r, then it is allowed to return any element at distance at most c r with some c > 1. The most well-known algorithm for ANN is the Locality Sensitive Hashing (LSH)[20], which is well studied theoretically and widely used in practical applications.Recently, graph-based approaches were shown to demonstrate superior performance over other types of algorithms in many large-scale applications of NNS[6]. All these approaches are essentially based on constructing a nearest neighbor graph (or its approximation), where nodes correspond to the elements of D and each node is connected to its nearest neighbors by directed edges[15,17,35]. Then, for a given query q, one first takes an element in D (either random or fixed predefined) and makes greedy steps towards q on the graph: at each step all neighbors of a current node are evaluated and the one closest to q is chosen. This procedure can be restarted several times from different nodes, we call such restarts iterations. Various heuristics are proposed to speed up graph-based search[16,21]. In particular, navigable small world graphs (NSW and HNSW) add links between distant points to make faster progress on early steps of the algorithm[26,27]. While there is a lot of evidence empirically showing the superiority of graph-based NNS algorithms in practical applications, there is very little theoretical research supporting this. A pioneering work [24] made a first step in this direction by providing time-space trade-offs for ANN search on sparse datasets uniformly distributed on a d-dimensional Euclidean sphere. In the current paper, we move forward and, in particular, provide a rigorous analysis for the dense regime, where d log n. In this regime, we face some additional challenges compared to the sparse one. Also, we analyze both exact and approximate NN problems. We show that if √ log n d log n, then query time is Θ M d(1+o (1)) and storage cost is n times larger for some carefully chosen constant M . However, for very dense datasets with d √ log n, one can only have Θ n 1+o(1) d time complexity, which is larger than M d . We also show that a simple method of speeding up the algorithm by adding random links between distant points does not give any significant improvement to the asymptotic query time. Since the sparse regime was previously analyzed in [24], it is not the main focus of the current paper. However, we do give an intuitive explanation for a phase transition between dense and sparse regimes: why for sparse datasets we have query time n ϕ with 0 < ϕ < 1 and for dense ones we get M d with constant M > 1. Also, for the sparse regime, we prove that it is possible to solve the ANN problem in just one iteration. Related work Several well-known algorithms proposed for NNS are based on recursive partitions of the space (e.g., k-d trees and random projection trees) [8,12,13,22]. The query time of a k-d tree is O d · 2 O(d) , which leads to an efficient algorithm for the exact NNS in the dense regime d log n [11]. Among the number of algorithms proposed for the ANN problem, LSH [20] is the most theoretically studied one. The main idea of LSH is to hash the points such that the probability of collision is much higher for points that are close to each other than for those which are far apart. Then, one can retrieve near neighbors for a query by hashing it and checking the elements in the same buckets. LSH solves c-ANN with query time Θ (dn ϕ ) and space complexity Θ n 1+ϕ . Assuming Euclidean metric, the optimal ϕ is about 1 c 2 for data agnostic algorithms [1,14,30,31]. By using a data dependent construction, this bound can be improved up to 1 2c 2 −1 [3,4]. In [7], spherical locally sensitive filters (LSF) were analyzed and it was shown that LSF can outperform LSH when the dimension d is logarithmic in the number of elements n. LSF can be thought of as lying in the middle between LSH and graph-based methods: LSF uses small spherical caps to define filters, while graph-based methods also allow, roughly speaking, moving to the neighboring caps. In contrast to LSH, graph-based approaches are not so well understood. The only theoretical paper we aware of is [24] considering the sparse regime d log n. The current paper, while using a similar setting, differs in several important aspects. First, we mostly focus on the dense regime d log n and show that it significantly differs both in terms of techniques and results. Second, in addition to the ANN problem, we also analyze the exact NN problem, since in the dense regime graph-based approaches are able to solve it with probability 1 − o (1). Also, we are the first to investigate how adding "long" links to NN graphs affects their query time. Finally, we support some claims made in the previous work by a rigorous analysis: in particular, Section 4.1 states new bounds for the volumes of spherical caps' intersections, which are needed for the rigorous analysis in both sparse and dense regimes; 4.2 discusses the effect of the dependence between consecutive steps of the algorithm. Finally, let us mention a recent empirical paper [25] comparing the performance of HNSW with other graph-based NNS algorithms. In particular, this paper shows that HNSW has superior performance over one-layer graphs only for low dimensional data. We demonstrate theoretically why this can be the case: we prove that when d √ log n the number of steps for graph-based NNS is negligible compared with one-step complexity, while for d log n the algorithm converges in just two steps. Results Setup and notation We are given a dataset D = {x 1 , . . . x n }, x i ∈ R d+1 and assume that all elements of D belong to a unit Euclidean sphere, D ⊂ S d . This special case is of particular importance for practical applications, since feature vectors are often normalized. 1 For a given query q ∈ S d letx ∈ D be its nearest neighbor. The aim of the exact NNS is to returnx, while in c, R-ANN (approximate near neighbor), for given R > 0, c > 1, we need to find such x that ρ(q, x ) ≤ cR if ρ(q,x) ≤ R [2,11]. 2 By ρ(·, ·) we further denote a spherical distance. Similarly to [24], we assume that the elements x i ∈ D are i.i.d. random vectors uniformly distributed on S d . As shown in [24], it is impossible to provide guarantees for graph-based NNS in the worstcase, hence some assumptions have to be made. Random uniform datasets are considered to be the most natural "hard" distribution for ANN problem [3], hence it is an important step towards understanding the limits and benefits of graph-based NNS algorithms. 3 We further assume that a query vector q ∈ S d is placed uniformly within a distance R from the nearest neighborx (since c, R-ANN problem is defined conditionally on the event ρ(q,x) ≤ R). Such nearest neighbor is called planted. In the current paper, we use a standard assumption that the dimensionality d = d(n) grows with n [11]. We distinguish three fundamentally different regimes in NN problems: dense with d log(n); sparse with d log(n); moderate with d = Θ(log(n)). In moderate and sparse regimes, the curse of dimensionality starts to be a problem, since 2 d becomes larger than n. While [24] focused solely on the sparse regime, we also consider the dense one. We obtain that for dense datasets the query time grows with n as M d , with some carefully chosen M , compared to n 1−M for sparse ones. As discussed in the introduction, most graph-based approaches are essentially based on constructing a nearest neighbor graph (or its approximation). For uniformly distributed datasets assumed in this paper, connecting an element x to a given number of nearest neighbors is essentially equivalent to connecting it to all such nodes y that ρ(x, y) ≤ ρ * with some appropriate ρ * (since the number of nodes at distance at most ρ * is concentrated around its expectation). Therefore, at the preprocessing stage, we choose some ρ * and construct a graph using this threshold. Later, when we get a query q, we first sample a random element x ∈ D such that ρ(x, q) < π 2 4 and then perform a graph-based greedy descent: at each step for a given node we measure the distance between its neighbors and q and move to the closest neighbor while we make progress. We may repeat the process several times (called iterations) and return the closest answer. In the current paper, we prove that it is sufficient to have one iteration in both regimes. Dense regime In the dense regime, we assume d = d(n) = log(n)/ω, where ω = ω(n) → ∞ as n → ∞. Note that ω(n) log n since we require d → ∞. For any constant M > 1, let G(M ) be a graph obtained by connecting x i and x j iff ρ(x i , x j ) ≤ arcsin (M e −ω ). Θ d 1/2 · e ω · M d = n o(1) ; space complexity is Θ n · d −1/2 · M d · log n = n 1+o(1) . It follows from Theorem 1 that both time and space complexities increase with M (for constant M ). When the aim is to find the exact nearest neighbor in one iteration of graph-based NNS (c = 1), we can take any M > √ 2. When c > 1, the lower bound for M decreases with c. The obtained time complexity is O d 1/2 · e ω · M d . Here d corresponds to one distance computation, d −1/2 · M d to the expected number of neighbors evaluated at each step, and e ω to the number of steps. While d is negligible compared with M d , the relation between e ω = e log n d and M d is non-trivial. Indeed, when ω √ log n, the term M d dominates, and in this regime the smaller M the better asymptotics we get (both for time and space complexities). However, when the dataset is very dense, i.e., d √ log n (equivalently, ω √ log n), the number of steps becomes much larger than the complexity of one step. For such datasets, it can be possible that taking M = M (n) 1 would improve the query time. However, the following theorem shows that this is not the case. d 1/2 · e ω · M d−1 ; space complexity is Θ n · d −1/2 · M d · log n . As a result, when M → ∞, both time and space complexities become larger compared with constant M . Another natural idea that may reduce the number of steps is to add some "long links" in order to obtain a graph with a low diameter. This is the core idea of NSW and some other graph-based NN algorithms [26,27]. In Section 4.3.3, we demonstrate that a basic realization of this idea is essentially equivalent to random sampling of several nodes in order to start the algorithm from a closer one. It turns out that this procedure does not allow to improve the asymptotic query time. Theorem 3. Under the conditions of Theorem 1, performing a random sampling of some number of nodes and choosing the one closest to q as a starting point for graph-based NNS does not allow to get time complexity better than Ω d 1/2 · e ω(1+o(1)) · M d . The obtained result is not surprising and agrees with [23], which shows that for a regular ddimensional lattice graph uniformly distributed long-range links do not allow for an efficient greedy routing and it is beneficial to have link lengths distributed according to a power-law with parameter d (however, constructing such graphs can be computationally expensive for practical applications). Finally, let us note that due to the fact that e −ω → 0 (i.e., as we discuss in Section 4.3, all considered distances tend to zero with n), it is easy to verify that all the results stated above for the spherical distance hold also for the Euclidean one. Sparse regime In sparse regime, we assume d = ω log(n), ω → ∞ as n → ∞. For any M , 0 < M < 1, let G(M ) be a graph obtained by connecting x i and x j iff ρ(x i , x j ) ≤ arccos 2M ω . Theorem 4. For any c > 1 let α c = cos π 2c and let M be any constant such that M < α 2 c α 2 c +1 . Then, with probability 1 − o(1), G(M )-based NNS solves c, R-ANN in one iteration (for any R and for spherical distance); time complexity of the procedure is Θ n 1−M +o(1) ; space complexity is Θ n 2−M +o (1) . Interestingly, as follows from the proof, in sparse regime one iteration of the algorithm converges in at most two steps with probability 1 − o(1). As a result, there is no trade-off between time and space complexity: larger values of M reduce both of them. While in the dense regime we have the same result for spherical and Euclidean distances, for sparse datasets it is not the case. However, we can easily obtain an analog of Theorem 4 for the Euclidean distance. In Theorem 4, α c is the height of a spherical cap covering a spherical distance π 2c (a factor c smaller than π/2). For the Euclidean distance, we have to replace π 2 by √ 2 and then the height of a spherical cap covering Euclidean distance √ 2/c is α c = 1 − 1 c 2 . So, we get the following corollary. Corollary 1. For the Euclidean distance Theorem 4 holds with α c = 1 − 1 c 2 , i.e., M < (1−1/c 2 ) 2 (1−1/c 2 ) 2 +1 . As a result, we can obtain time complexity n ϕ and space complexity n 1+ϕ , where ϕ can be made about 1 (1−1/c 2 ) 2 +1 . Note that this result corresponds to the case ρ q = ρ s from [24]. In this section, we formulate some technical results on the volumes of spherical caps and their intersections, which we extensively use in the proofs. Although they are similar to those formulated in [7], it is crucial for our problem that parameters defining spherical caps depend on d and may tend to zero (both in dense and sparse regimes), while the results in [7] hold only for fixed parameters. Also, in Lemma 2, we extend the corresponding result from [7], as discussed further in this section. α (a) β > α cos θ, α > β cos θ α (b) α < β cos θ Let us denote by µ the Lebesgue measure over R d+1 . By C x (γ) we denote a spherical cap of height γ centered at x ∈ S d , i.e., {y ∈ S d : x, y ≥ γ}; C(γ) = µ (C x (γ)) denotes the volume of a spherical cap of height γ. Throughout the paper for any variable γ, 0 ≤ γ ≤ 1, we letγ : = 1 − γ 2 . The following lemma is proven in Appendix A.1. Lemma 1. Let γ = γ(d) be such that 0 ≤ γ ≤ 1. Then Θ d −1/2 γ d ≤ C (γ) ≤ Θ d −1/2 γ d min d 1/2 , 1 γ . By W x,y (α, β) we denote the intersection of two spherical caps centered at x ∈ S d and y ∈ S d with heights α and β, respectively, i.e., W x,y (α, β) = {z ∈ S d : z, x ≥ α, z, y ≥ β}. As for spherical caps, by W (α, β, θ) we denote the volume of such intersection given that the angle between x and y is θ. The following lemma is proven in Appendix A.2. Lemma 2. Let γ = √ α 2 +β 2 −2αβ cos θ sin θ and assume that γ ≤ 1, then: (1) If α ≤ β cos θ, then C(β)/2 < W (α, β, θ) ≤ C(β) and (C l,β − C l,α ) Θ d −1 γ d ≤ C(β) − W (α, β, θ) ≤ (C u,β − C u,α ) Θ d −1 γ d min d 1/2 , 1 γ ; (2) If β ≤ α cos θ, then C(α)/2 < W (α, β, θ) ≤ C(α) and (C l,α − C l,β ) Θ d −1 γ d ≤ C(α) − W (α, β, θ) ≤ (C u,α − C u,β ) Θ d −1 γ d min d 1/2 , 1 γ ; (3) Otherwise, (C l,α + C l,β ) Θ d −1 γ d ≤ W (α, β, θ) ≤ (C u,α + C u,β ) Θ d −1 γ d min d 1/2 , 1 γ . Here C l,α , C l,β , C u,α , C u,β are some functions of α, β, θ specified in Appendix A.2. This lemma differs from Lemma 2.2 in [7] by, first, allowing the parameters α, β, θ depend on d and, second, considering the cases (1) and (2), which are essential for the proofs. Namely, we use the lower bound in (3) to show that with high probability we can make a step of the algorithm, since the intersection of some spherical caps is large enough (Figure 1a); we use the upper bounds in (1) and (2) to show that at the final step of the algorithm we can find the nearest neighbor with large probability, since the volume of the intersection of some spherical caps is very close to the volume of one of them (Figure 1b), see the details further in the proof. Despite some additional terms, one can essentially think that C(γ) ∝γ d and W (α, β, θ) (or its complement) ∝γ d with γ specified in Lemma 2. General idea Let α M denote the height of a spherical cap defining G(M ). By f = f (n) = (n − 1)C(α M ) we denote the expected number of neighbors of a given node in G(M ). Then, it is clear that the complexity of one step of graph-based search is Θ (f · d) (with large probability), so for making k steps we need Θ (k · f · d) computations (see Appendix B.1 for the formal analysis). The number of edges in the graph is Θ (f · n), so the space complexity is Θ (f · n · log n) (see Appendix B.2). To prove that the algorithm succeeds, we have to show that it does not stuck in a local minimum until we are sufficiently close to q. If we take some point x with x, q = α s , then the probability to make a step towards q is determined by W (α M , α s , arccos α s ). In all further proofs we obtain lower bounds for this value of the form 1 n g(n) with 1 g(n) n. From this, we easily get that the probability to make a step is at least (1)) . 1 − (1 − g(n)/n) n−1 = 1 − e −g(n)(1+o A fact that will be useful in the proofs is that the value W (α M , α s , arccos α s ) is a monotone function of α s (see Appendix B.3). I.e., if we have a lower bound for some α s , then for all smaller values we have this bound automatically. By estimating the value W (α M , α s , arccos α s ) we obtain (in further sections) that with probability 1 − o(1) we reach some point at distance at most arccos α s from q. Then, to achieve success, we may either jump directly tox at the next step or to already have arccos α s ≤ cR if we are solving c, R-ANN. To limit the number of steps, we additionally show that with a sufficiently large probability at each step we become "ε closer" to q. In the dense regime, it means that the sine of the angle between the current position and q becomes smaller by at least some fixed value. Finally, let us emphasize that several consecutive steps of the algorithm cannot be analyzed independently. Indeed, if at some step we moved from x to y, then there were no points in C x (α M ) closer to q than y by the definition of the algorithm. Consequently, the intersection of C x (α M ), C y (α M ) and C q ( q, y ) contains no elements of the dataset. The closer y to x the larger this intersection. However, the fact that at each step we become at least "ε closer" to q allows us to bound the volume of this intersection and to prove that it can be neglected. Dense regime For dense datasets (d = log n/ω), it is convenient to operate with radii of spherical caps. If α is a height of a spherical cap, then we say thatα is its radius. An essential property of the dense regime is the fact that the distance from a given point to its nearest neighbor behaves as e −ω , so it decreases with n. Indeed, letα 1 be the radius of a cap centered at a given point and covering its nearest neighbor, then we have C(α 1 ) ∼ 1 n , i.e.,α 1 ∼ n − 1 d = e −ω . We further let δ := e −ω . Proof of Theorem 1 We construct a graph G(M ) using spherical caps with radiusα M = M δ. Then, from Lemma 1, we get f = Θ n d −1/2 M d δ d = Θ d −1/2 M d . So, the number of edges in G(M ) is Θ d −1/2 · M d · n and the space complexity is Θ d −1/2 · M d · n · log n (see Appendix B.2). Let us now analyze the distance arccos α s up to which we can make steps towards the query q (with sufficiently large probability). Formally, the following lemma is proven in Appendix C.1. Lemma 3. Assume that s > 1. If M > √ 2s or M 2 − M 4 4s 2 > 1, then there exists such constant ε > 0 that W (α M , α s , arcsin (α s + εδ)) ≥ 1 n L d(1+o(1)) for some constant L > 1. This lemma implies that if we are given M and s satisfying the above conditions, then we can make a step towards q, since the expected number of nodes in the intersection of spherical caps is much larger than 1. Formally, we can estimate from below the values g(n) for all steps of the algorithm by g min (n) = L d(1+o(1)) , so, according to Section 4.2, we can make each step with probability 1 − O e −L d(1+o (1)) . Moreover, each step reduces the radius of a spherical cap centered at q and containing the current position by at least εδ. As a result, the number of steps (until we reach some distance arccos α s ) is O δ −1 = O (e ω ). To estimate the overall success probability, we have to take into account that the consecutive steps of the algorithm are dependent. Appendix C.2 proves that this dependence can be neglected. So, the overall success probability is 1 − O e ω e −L d+o (1) . Assuming d log log n, we get O e ω e −L d(1+o (1) Let us discuss the time complexity. With probability 1 − o(1) the number of steps is Θ δ −1 : the upper bound was already discussed; the lower bound follows from the fact that M δ is the radius of a spherical cap, so we cannot make steps longer than arcsin(M δ), and with probability 1 − o(1) we start from a constant distance from q. The complexity of each step is Θ (f · d) = Θ d 1/2 · M d , so overall we get Θ d 1/2 · e ω · M d . It remains to find suitable values for s and M . We solve c, R-ANN if either we have arcsin(sδ) ≤ cR or we are sufficiently close to q to find the exact nearest neighborx at the next step of the algorithm. Let us analyze the first possibility. Let sin R = rδ, then we need s < c r. According to Lemma 3, it is sufficient to have r c > 1 and either M ≥ √ 2 rc or M 2 − M 4 4r 2 c 2 > 1. Alternatively, to find the exact nearest neighbor with probability 1 − o(1), it is sufficient to reach such s that M 2 > s 2 + r 2 (see Appendix C.3 for the proof). For this, according to Lemma 3, it is sufficient to have s > 1, M 2 > s 2 + r 2 , and either M ≥ √ 2s or M 2 − M 4 4s 2 > 1. One can show that if the following conditions on M and r are satisfied, then we can choose an appropriate s for the two cases discussed above: (a) r c > 1 and M 2 > 2 r 2 c 2 1 − 1 − 1 r 2 c 2 ; (b) M 2 > 2 3 r 2 + 1 + √ r 4 − r 2 + 1 . To succeed, we need either (a) or (b) to be satisfied. The bound in (a) decreases with r (r > 1/c) and for r = 1 c it equals Proof of Theorem 2 When M grows with n, it follows from the previous reasoning that the algorithm succeeds with probability 1 − o(1). The analysis of the space complexity is the same as for constant M , so we get Θ d −1/2 · M d · n · log n . When analyzing time complexity, we note that the one-step complexity is Θ d −1/2 · M d . It is easy to see that we cannot make steps longer than O (M e −ω ). This leads to the time complexity Ω d 1/2 · M d−1 · e ω . Small-world graphs and proof of Theorem 3 Let us discuss how adding edges between distant points to NN graphs affects the query time of graphbased NNS. The simplest way to obtain a graph with a small diameter from a given graph is to connect each node to a few random neighbors. This idea is proposed in [36] and gives O (log n) diameter for the so-called "small-world model". It was later confirmed that adding a little randomness to a connected graph makes the diameter small [10]. However, we emphasize that having a logarithmic diameter does not guarantee a logarithmic number of steps in graph-based NNS, since these steps, while being greedy in the underlying metric space, may not be optimal on a graph. To demonstrate the effect of long edges, assume that there is a graph G , where each node is connected to several random neighbors by directed edges. For simplicity or reasonings, assume that we first perform NNS on G and then continue on the standard NN graph G. It is easy to see that during NNS on G , the neighbors considered at each step are just randomly sampled nodes, we choose the one closest to q and continue the process, and all such steps are independent. Therefore, the overall procedure is basically equivalent to a random sampling of a certain number of nodes and then choosing the one closest to q (from which we then start the standard NNS on G). Assume that we sample e lω nodes with an arbitrary l = l(n). Then, with probability 1 − o(1), the closest one among them lies at a distance Θ e − lω d . As a result, the overall time complexity becomes Θ e − lω d · d 1/2 · e ω · M d + d · e lω . If l = Ω(d), then the term d · e lω = d · e Ω(d)ω dominates d 1/2 · e ω · M d , otherwise we get Θ d 1/2 · e ω(1+o(1)) · M d , which proves Theorem 3. Sparse regime For sparse datasets, instead of radii, we operate with heights of spherical caps. A crucial feature of sparse regime is that the heights under consideration tend to zero as n grows. Indeed, we have C(α 1 ) ∼ 1 n , i.e., α 2 1 ∼ 1 − n − 2 d = 1 − e − 2 ω ∼ 2 ω . We further denote 2 ω by δ. We construct G(M ) using spherical caps with height α M , α 2 M = M δ, where M is some constant. Then, from Lemma 1, we get that the expected number of neighbors of a node is (1) . From this and Appendix B.2 the stated space complexity follows. The one-step time complexity n 1−M +o(1) follows from Appendix B.1. f = Θ n d O(1) (1 − M δ) d/2 = n 1−M +o Our aim is to solve c, R-ANN with some c > 1, R > 0. If R ≥ π/2c, then we can easily find the required near neighbor within a distance π/2, since we start G(M )-based NNS from such point. Let us consider any R < π 2c . It is clear that in this case we have to find the nearest neighbor itself, since R c is smaller than the distance to the (non-planted) nearest neighbor with probability 1 + o(1). Note that α c = cos π 2c < α R := cos R. The following Lemma is proven in Appendix D.1. It follows from the lemma that if M + s < 1, then we can reach a spherical cap with height α s = √ sδ centered at q in just one step (starting from a distance at most π/2). And we get g(n) = n Ω(1) . Note that in our case we have M < Recall that M < α 2 c α 2 c +1 < α 2 R α 2 c +1 = sα 2 R . From this Theorem 4 follows. Conclusion We proved theoretical guarantees for graph-based c, R-ANN in both dense and sparse regimes. In dense regime ( √ log n d log n), we obtained time complexity M d(1+o(1)) and space complexity n M d(1+o(1)) with some M > 1 depending on c. In sparse regime, we get n 1−M +o(1) query time and n 2−M +o(1) storage with some 0 < M < 1. We also demonstrated that a simple realization of "small-world" graphs does not improve the asymptotic query time. Due to the fact that graph-based NN algorithms become extremely popular nowadays, we believe that more theoretical analysis of such methods will follow. A natural direction for future research would be to find wider conditions under which similar guarantees can be obtained (compared to the random uniform case considered in this paper). Another promising direction is to analyze the effect of diversification [19] which was empirically shown to improve the quality of graph-based NNS [25]. Finally, it would be interesting to obtain guarantees for the moderately dense regime with d = Θ(log n). Although based on the current analysis the query time should be O n C with some C < 1, obtaining the precise constant here is complicated due to the fact that neither radii or heights of spherical caps under consideration tend to zero. Lemma 1. Let γ = γ(d) be such that 0 ≤ γ ≤ 1. Then Θ d −1/2 γ d ≤ C (γ) ≤ Θ d −1/2 γ d min d 1/2 , 1 γ . Proof. In order to have similar reasoning with the proof of Lemma 2, we consider any twodimensional plane containing the vector x defining the cap C x (γ) and let p denote the orthogonal projection from S d to this two-dimensional plane. The first steps of the proof are similar to those in [7] (but note that we analyze S d instead of S d−1 , which leads to slightly simpler expressions). Consider any measurable subset U of the two-dimensional unit ball, then the volume of the preimage p −1 (U ) (relative to the volume of S d ) is: I(U ) = r,φ∈U µ(S d−2 ) µ(S d ) 1 − r 2 d−3 r dr dφ . We define a function g(r) = φ:(r,φ)∈U dφ, then we can rewrite the integral as I(U ) = (d − 1) 4 π 1 0 1 − r 2 (d−3)/2 g(r) dr 2 . Let U = p (C x (γ)), then, using t = 1 − r 2 /γ 2 , whereγ = 1 − γ 2 , we get C(γ) = (d − 1) 4 π 1 γ 1 − r 2 (d−3)/2 g(r) dr 2 = (d − 1)γ d−1 4 π 1 0 g 1 −γ 2 t t (d−3)/2 dt . (1) Note that from Equation (1) we get that the volume of a hemisphere is C(0) = 1/2, since g(r) = π for all r in this case andγ = 1. Now we consider an arbitrary γ ≥ 0 and note that g(r) = 2 arccos(γ/r) (see Figure 2). So, we obtain C(γ) = (d − 1)γ d−1 2 π 1 0 arccos γ 1 −γ 2 t t (d−3)/2 dt = (d − 1)γ d−1 2 π 1 0 arcsin γ 1 − t 1 −γ 2 t t (d−3)/2 dt . Now we note that x ≤ arcsin(x) ≤ x · π/2 for 0 ≤ x ≤ 1, so C(γ) = Θ (d)γ d 1 0 1 − t 1 −γ 2 t · t (d−3)/2 dt . Finally, we estimate √ 1 − t ≤ 1 − t 1 −γ 2 t ≤ min 1, 1 − t 1 −γ 2 .(2) So, the lower bound is C(γ) ≥ Θ (d)γ d B 3 2 , d − 1 2 = Θ (d)γ d d − 1 2 −3/2 = Θ d −1/2 γ d . The upper bounds are C(γ) ≤ Θ (d)γ d 1 0 t (d−3)/2 dt = Θ (1)γ d , C(γ) ≤ Θ (d)γ d 1 0 1 − t 1 −γ 2 · t (d−3)/2 dt = Θ d −1/2 γ d γ . This completes the proof. A.2 Volumes of intersections of spherical caps In this section, we analyze the volume of the intersections of two spherical caps C x (α) and C y (β). In the lemma below we assume γ ≤ 1. However, it is clear that if γ > 1, then either the caps do not intersect (if α > β cos θ and β > α cos θ) or the larger cap contains the smaller one. Let us give the full statement of Lemma 2 from the main text. Lemma 2. Let γ = √ α 2 +β 2 −2αβ cos θ sin θ and assume that γ ≤ 1, then: (1) If α ≤ β cos θ, then C(β)/2 < W (α, β, θ) ≤ C(β) and (C l,β − C l,α ) Θ d −1 γ d ≤ C(β) − W (α, β, θ) ≤ (C u,β − C u,α ) Θ d −1 γ d min d 1/2 , 1 γ ; (2) If β ≤ α cos θ, then C(α)/2 < W (α, β, θ) ≤ C(α) and (C l,α − C l,β ) Θ d −1 γ d ≤ C(α) − W (α, β, θ) ≤ (C u,α − C u,β ) Θ d −1 γ d min d 1/2 , 1 γ ; (3) Otherwise, (C l,α + C l,β ) Θ d −1 γ d ≤ W (α, β, θ) ≤ (C u,α + C u,β ) Θ d −1 γ d min d 1/2 , 1 γ ; where C l,α = α (α sin θ − |β − α cos θ|) γγ sin θ , C l,β = β β sin θ − |α − β cos θ| γγ sin θ , C u,α =γ α sin θ γ|β − α cos θ| , C u,β =γ β sin θ γ|α − β cos θ| . Proof. Consider the plane formed by the the vectors x and y defining the caps and let p denote the orthogonal projection to this plane. Let U = p(W x,y (α, β)). α (a) β > α cos θ, α > β cos θ α (b) α < β cos θ Figure 3: g α (r) and g β (r) Denote by γ the distance between the origin and the intersection of chords bounding the projections of spherical caps. One can show that γ = α 2 + β 2 − 2αβ cos θ sin 2 θ . If α ≤ β cos θ, it is easy to see that W (α, β, θ) > 1 2 C(β), since more than a half of C y (β) is covered by the intersection (see Figure 1b). Similarly, if β ≤ α cos θ, then W (α, β, θ) > 1 2 C(α). Now we move to the proof of (3) and will return to (1) and (2) after that. If cos θ < α β and cos θ < β α , then we are in the situation shown on Figure 1a and the distance between the intersection of spherical caps and the origin is γ. As in the proof of Lemma 1, denote g(r) = φ:(r,φ)∈U dφ, then the relative volume of p −1 (U ) is (see Equation (1)): W (α, β, θ) = (d − 1)γ d−1 4 π 1 0 g 1 −γ 2 t t (d−3)/2 dt . The function g(r) can be written as g α (r) + g β (r), where (see Figure 3a) g α (r) = arccos α r − arccos α γ , g β (r) = arccos β r − arccos β γ . Accordingly, we can write W (α, β, θ) = W α (α, β, θ) + W β (α, β, θ). Let us estimate g α 1 −γ 2 t : g α 1 −γ 2 t = arcsin 1 − α 2 1 −γ 2 t − arcsin 1 − α 2 γ 2 = arcsin 1 − α 2 1 −γ 2 t α 2 γ 2 − 1 − α 2 γ 2 α 2 1 −γ 2 t = Θ α γ 2 − α 2 γ 1 −γ 2 t 1 +γ 2 (1 − t) γ 2 − α 2 − 1 . Note that 1 +γ 2 γ 2 − α 2 − 1 (1 − t) ≤ 1 +γ 2 (1 − t) γ 2 − α 2 − 1 ≤γ 2 2 (γ 2 − α 2 ) (1 − t) . Now, we can write the lower bound for W α (α, β, θ). Let C l,α = 1 +γ 2 γ 2 − α 2 − 1 α γ 2 − α 2 γγ = α (α sin θ − |β − α cos θ|) γγ sin θ , C l,β can be obtained by swapping α and β. Then the lower bound is W (α, β, θ) ≥ Θ(d)γ d (C l,α + C l,β ) 1 0 1 − t 1 −γ 2 t t (d−3)/2 dt ≥ Θ(d)γ d (C l,α + C l,β ) 1 0 (1 − t) t (d−3)/2 dt = Θ(d −1 )γ d (C l,α + C l,β ) . Now we define C u,α (and, similarly, C u,β ) as C u,α =γ (γ 2 − α 2 ) · α γ 2 − α 2 γ =γ α sin θ γ |β − α cos θ| . Then W (α, β, θ) ≤ Θ(d)γ d (C u,α + C u,β ) 1 0 1 − t 1 −γ 2 t t (d−3)/2 dt . We use the upper bound 1 − t 1 −γ 2 t ≤ min √ 1 − t, 1 − t γ and obtain W (α, β, θ) ≤ Θ(d −1 )γ d (C u,α + C u,β ) min d 1/2 , 1 γ , which completes the proof of (4). Now, let us finish the proof for (1) and (2). If α ≤ β cos θ, then we are in a situation shown on Figure 1b. In this case, we can directly follow the above proof for (3) and the only difference would be that g(r) = g β (r) − g α (r) instead of g(r) = g α (r) + g β (r). The proof for (2) is similar with g(r) = g α (r) − g β (r). B General results B.1 Time complexity Let v be an arbitrary node of G and let N (v) denote the number of its neighbors in G. Recall that f = (n − 1)C(α M ). Lemma 6. With probability at lest 1 − 4 f we have 1 2 f ≤ N (v) ≤ 3 2 f . Proof. The number of neighbors N (v) of a node v follows Binomial distribution Bin(n−1, C(α M )), so EN (v) = f . From Chebyshev's inequality we get P |N (v) − f | > f 2 ≤ 4 Var(N (v)) f 2 ≤ 4 f , which completes the proof. To obtain the final time complexity of graph-based NN search, we have to sum up the complexities of all steps of the algorithm. We obtain the following result. Proof. Although the nodes encountered in one iteration are not independent, the fact that we do not need to measure the distance from any point to q more than once allows us to upper bound the complexity by the random variable distributed according to Bin(k(n − 1), C(α M )). Then we can follow the proof of Lemma 6 and note that one distance computation takes Θ(d). To see that the lower bound is also Θ (kf d), we note that more than a constant number of steps are needed only for the dense regime. For this regime, we may follow the reasoning of Section C.2 to show that the volume of the intersection of two consecutive balls is negligible compared to the volume of each of them. B.2 Space complexity Lemma 8. With probability 1 − O 1 f n we have 1 4 f n ≤ E(G) ≤ 3 4 f n. Proof. The proof is straightforward. 5 For each pair of nodes, the probability that there is an edge between them equals C (α M ). Therefore, the expected number of edges is E(E(G)) = n 2 C (α M ) . It remains to prove that E(G) is tightly concentrated near its expectation. For this, we apply Chebyshev's inequality, so we have to estimate the variance Var(E(G)). One can easily see that if we are given two pairs of nodes e 1 and e 2 , then, if they are not the same (while one coincident node is allowed), then P(e 1 , e 2 ∈ E(G)) = C(α M ) 2 . Therefore, Var(E(G)) = e1,e2∈( D 2 ) P(e 1 , e 2 ∈ E(G)) − (EE(G)) 2 = e1,e2∈( D 2 ) e1 =e2 P(e 1 , e 2 ∈ E(G)) + EE(G) − (EE(G)) 2 = n 2 C (α M ) (1 − C (α M )) . Applying Chebyshev's inequality, we get P |E(G) − E(E(G))| > E(G) 2 ≤ 4 Var(E(G)) E(G) 2 = 4 (1 − C(α)) E(G) .(3) From this, the lemma follows. It remains to note that if we store a graph as adjacency lists, then the space complexity is Θ (E(G) · log n). Proof. We refer to Figure 4, where two spherical caps of height α M are centered at x and y, respectively, and note that we have to compare "curved triangles" x x 1 x 2 and y y 1 y 2 . Obviously, ρ(x, x 2 ) = ρ(y, y 2 ), ∠ x x 2 x 1 = ∠ y y 2 y 1 , but ∠ x x 1 x 2 > ∠ y y 1 y 2 . From this and spherical symmetry of µ(p −1 (·)) (p was defined in the proof of Lemma 2) the result follows. γ ∼ 2 (x 2ŷ2 +ŷ 2ẑ2 +x 2ẑ2 ) − (x 4 +ŷ 4 +ẑ 4 ) 2ẑ . Proof. By the definition, γ = x 2 +y 2 −2xyz 1−z 2 . Then γ 2 = 1 − γ 2 =x 2 +ŷ 2 +ẑ 2 − 2 + 2 (1 −x 2 ) (1 −ŷ 2 ) (1 −ẑ 2 ) z 2 ∼ 2 x 2ŷ2 +ŷ 2ẑ2 +x 2ẑ2 − x 4 +ŷ 4 +ẑ 4 4ẑ 2 . Now, we analyze W (α M , α s , arccos α s ) and we need only the lower bound. Recall that we use the notation δ = e −ω . Lemma 11. Assume thatα s = s δ andα M = M δ. • If M ≥ √ 2s, then W (α M , α s , arccos α s ) ≥ 1 n · s d+o(d) . • If M < √ 2s, then W (α M , α s , arccos α s ) ≥ 1 n · M 2 − M 4 4s 2 d/2+o(d) . Proof. First, assume that M ≥ √ 2s. In this case we have α M < α 2 s , so we are under the conditions (1)-(2) of Lemma 2 (see Figure 1b) and, using Lemma 1, we get W (α M , α s , arccos α s ) > 1 2 C(α s ) = Θ d −1/2 s d δ d = 1 n s d+o(d) . If M < √ 2s, then, asymptotically, we have α M > α 2 s , so the case (3) of Lemma 2 can be applied. Let us use Lemma 10 to estimateγ: (1)) . γ 2 = δ 2 M 2 − M 4 4s 2 (1 + o And now from Lemma 2 we get W (α M , α s , arccos α s ) ≥ C l Θ d −1 γ d , where C l corresponds to the sum of C l,α and C l,β in Lemma 2. So, it remains to estimate C l : C l = α M (α Mαs + α M α s − α s ) + α s α 2 s + α 2 s − α M γγα s = Θ M sδ 2 + (1 − M 2 δ 2 )(1 − s 2 δ 2 ) − √ 1 − s 2 δ 2 + s 2 δ 2 + 1 − s 2 δ 2 − √ 1 − M 2 δ 2 δ 2 = Θ(1) · M (s − M/2)δ 2 + 1 2 M 2 δ 2 δ 2 = Θ(1) . Therefore, W (α M , α s , arccos α s ) ≥ 1 n · M 2 − M 4 4s 2 d/2+o(d) . From Lemma 11, we can find the conditions for M and s to guarantee (with sufficiently large probability) making steps until we are in the cap of radius sδ centered at q. Now, we are ready to prove Lemma 3 from the main text, which gives such conditions and also guarantees that at each step we can reach a cap of a radius at least εδ smaller for some constant ε > 0. (1)) for some constant L > 1. Lemma 3. Assume that s > 1. If M > √ 2s or M 2 − M 4 4s 2 > 1, then there exists such constant ε > 0 that W (α M , α s , arcsin (α s + εδ)) ≥ 1 n L d(1+o Proof. First, let us take ε = 0. Then, the result directly follows from Lemma 11. We note that a value M satisfying M 2 − M 4 4s 2 > 1 exists only if s > 1. Now, let us demonstrate that we can take some ε > 0. The two cases discussed in Lemma 10 now correspond to M 2 ≥ s 2 + (s + ε) 2 and M 2 < s 2 + (s + ε) 2 , respectively. If M > √ 2s, then we can choose a sufficiently small ε that M 2 ≥ s 2 + (s + ε) 2 . Then, the result follows from Lemma 11 and the fact that s > 1. Otherwise, we have M 2 < s 2 + (s + ε) 2 and instead of the condition M 2 − M 4 4s 2 > 1 we get (using Lemma 10) 2 M 2 s 2 + s 2 (s + ε) 2 + M 2 (s + ε) 2 − M 4 + s 4 + (s + ε) 4 4(s + ε) 2 > 1. As this holds for ε = 0, we can choose a small enough ε > 0 that the condition still holds. C.2 Dependence of consecutive steps In Section 4.2 of the main text, it is explained how the previous steps of the algorithm may affect the current one: the fact that at some step we moved from y to x implies that there were no elements closer to q than x in a spherical cap centered at y. In this section, we show that this dependence can be neglected. The main idea is illustrated on Figure 5. Assume that we are currently at a point x with ρ(x, q) = arcsin (α s + εδ). Then, as in the proof of Lemma 3, we are interested in the volume W (α M , α s , arcsin (α s + εδ)), which corresponds to W 1 + W 2 on Figure 5. Assume that at the previous step we were at some point y. Given that all steps are "longer than ε", the largest effect of the previous step is reached when y is as close to x as possible and x lies on a geodesic between y and q. Therefore, the largest possible volume is W (α M , α s , arcsin(α s + 2εδ)), which corresponds to W 1 on Figure 5. It remains to show that W 1 is negligible compared with W 1 + W 2 . If M < √ 2s, then the main term of W (α M , α s , arcsin (α s + εδ)) isγ d witĥ The main term of W (α M , α s , arcsin(α s + 2εδ)) isγ d 1 witĥ γ 2 1 = M 2 + s 2 2 − M 2 − s 2 2 4(s + 2ε) 2 − (s + 2ε) 2 4 . It is easy to see that M < √ 2s implies thatγ 2 1 <γ 2 . As a result, W (α M , α s , arcsin(α s + 2εδ)) = o (W (α M , α s , arcsin(α s + εδ))) (similarly to the other proofs, it is easy to show that the effect of the other terms in Lemma 2 is negligible compared to (γ 1 /γ) d ). Moreover, since at each step we reduce the radius of a spherical cap centered at q by at least εδ, any cap encountered in one iteration intersects with only a constant number of other caps, so their overall effect is negligible, which completes the proof. Finally, let us note that as soon as we have M > √ 2s (with s > 1), with probability 1 − o(1) we find the nearest neighbor in one step, as follows from Section C.3. C.3 Finding the nearest neighbor Assume that the radius of a cap centered at q and covering the currently considered element isα s and α s = s δ,α M = M δ. Further assume that the radius of a spherical cap covering points at distance at most R from q isα r = rδ = sin R for some r < 1. The following lemma gives the conditions for M and s such that at the next step of the algorithm we find the nearest neighborx with probability 1 − o(1) given thatx is uniformly distributed within a distance R from q. Lemma 12. If for constant M, s, r we have M 2 > s 2 + r 2 , then C(α r ) − W (α M , α r , arccos α s ) ≤ C(α r )β d with some β < 1. Proof. First, recall the lower bound for C(α r ): C(α r ) ≥ Θ d −1/2 δ d r d . Since we have M 2 > s 2 + r 2 , then asymptotically we have α M < α s α r , so the cases (1)-(2) of Lemma 2 should be applied (see also Figure 1b). Let us estimateγ 2 (Lemma 10): γ 2 ∼ δ 2 · 2 M 2 s 2 + M 2 r 2 + s 2 r 2 − M 4 + r 4 + s 4 4 s 2 = δ 2 · − M 2 − s 2 − r 2 2 + 4 s 2 r 2 4 s 2 = δ 2 r 2 (1 − Θ(1)) . Thereforeγ d ≤ δ d r d β d with some β < 1. γ α Rαs γ|α M − α R α s | −γ α Mαs γ|α R − α M α s | Θ d −1 min d 1/2 , 1 γ = O δ −1 d −1 , from which the lemma follows. Theorem 1 . 1Assume that d log log n and we are given some constant c ≥ 1. Let M be a constant such that M > 4c 2 3c 2 −1 , then, with probability 1 − o(1), G(M )-based NNS solves c, R-ANN for any R (or the exact NN problem if c = 1) in one iteration; time complexity is Theorem 2 . 2Let M = M (n) 1. Then, with probability 1 − o(1), graph-based NNS finds the exact nearest neighbor in one iteration; time complexity is Ω Figure 1 : 1Intersection − log n = o(1). This concludes the proof for the success probability 1 − o(1) up to choosing suitable values for s and M . √ 2 . 2The bound in (b) increases with r and for r = 1 it equals √ 2. To find a general bound holding for all r, we take the "hardest" r ∈ ( 1 c , 1), where the bounds in (a) and (b) are equal to each other. This value is r = 4c 2 (c 2 +1)(3c 2 −1) and it gives the bound M Lemma 4 . 4Assume that α 2 M = M δ, α 2 s = sδ, α 2 ε = εδ, M, s > 0 are constants, and ε ≥ 0 is bounded by a constant. If s + M < 1, then W (α M , α s , arccos α ε ) ≥ 1 n · n Ω(1) . c +1 , then we have M + s < 1. The following lemma, proven in Appendix D.2, discusses the conditions for M and s such that at the next step of the algorithm we findx with probability 1 − o(1). Lemma 5. If for constant M and s we have M < sα 2 R , then C(α R ) − W (α M , α R , arccos α s ) = C(α R )n −Ω(1) . Figure 2 : 2g(r) Lemma 7 . 7If we made k steps of the graph-based NNS, then with probability 1 − O 1 k f the obtained time complexity is Θ (kf d). B. 3 3Monotonicity of W (α M , α s , arccos α s ) Lemma 9. W (α M , α s , arccos α s ) is a non-increasing function of α s . C. 1 1On reaching α s First, let us analyze the behavior of the main termγ d in W (x, y, arccos z) whenx,ŷ,ẑ = o(1), which is the case for the considered situations in the dense regime. Figure 4 : 4Monotonicity of W (α M , α s , arccos α s ) Lemma 10. Ifx,ŷ,ẑ = o(1), thenγ defined in Lemma 2 for W (x, y, arccos z) iŝ γ 2 = 2 2M 2 s 2 + s 2 (s + ε) 2 + M 2 (s + ε) 2 − M 4 + s 4 + (s + ε) 4 4(s + ε) 2 = M 2 + s 2 2 − M 2 − s 2 2 4(s + ε) 2 − (s + ε) 2 4 . Figure 5 : 5Dependence of consecutive steps From a theoretical point of view, there is a reduction of ANN in the entire Euclidean space to ANN on a sphere[3,34].2 There is also a notion of c-ANN, where the goal is to find such x that ρ(q, x ) ≤ cρ(q,x); c-ANN can be reduced to c, R-ANN with additional O(log n) factor in query time and O(log 2 n) in storage cost[11,18].3 Also,[3] shows how to reduce ANN on a generic dataset to ANN on a "pseudo-random" dataset in a data-dependent LSH algorithm. From a practical point of view, there are techniques allowing to make a dataset more uniformly distributed while trying to preserve the distances[32].4 We can easily achieve this by trying a constant number of random samples, since each sample succeeds with probability 1/2. This trick speeds up the procedure (without affecting the asymptotics) and simplifies the proofs, since we can consider only spherical caps smaller than half of the unit sphere. Similar proof appeared in, e.g.,[24]. AcknowledgmentsThe author thanks Artem Babenko and Stanislav Morozov for fruitful discussions.Appendices A Analysis of spherical caps A.1 Volumes of spherical capsIn this section, we prove Lemma 1. Let us repeat the statement for convenience.It only remains to estimate the other terms in the upper bound from Lemma 2:from which the lemma follows.D Sparse regime D.1 On reaching α sRecall that in the sparse regime we take δ = 2 ω . Let us prove Lemma 4 from the main text.Proof. Asymptotically, we have α M > α s α ε and α s > α M α ε , so we are under the conditions of (3) in Lemma 2. First, consider the main term of W (α M , α s , arccos α ε ):It remains to multiply this by Θ d −1 and C l = C l,α + C l,β (seeLemma 2). It is easy to see that C l = Ω(1) in this case, so both terms can be included to n Ω(1) , which concludes the proof.From Lemma 4 it follows that for M and s with M + s < 1 we can reach a spherical cap with height α s centered at q in just one step (starting from a distance at most π/2).D.2 Finding the nearest neighborAssume that the height of a cap centered at q and covering the currently considered element is α s and α 2 s = s δ, α 2 M = M δ. Further assume that the height of a spherical cap covering points at distance at most R from q is α R = cos R for some R < π 2c . Lemma 5 of the main text gives the conditions for M and s such that at the next step of the algorithm we find the nearest neighborx with probability 1 − o(1) given thatx is uniformly distributed within a distance R from q.Lemma 5. If for constant M and s we have M < sα 2 R , thenProof. First, recall the lower bound for C(α R ):. 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On computing nearest neighbors with applications to decoding of binary linear codes. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 203-228. Springer, 2015. Point location in arrangements of hyperplanes. S Meiser, Information and Computation. 1062S. Meiser. Point location in arrangements of hyperplanes. Information and Computation, 106(2):286-303, 1993. Lower bounds on locality sensitive hashing. R Motwani, A Naor, R Panigrahy, SIAM Journal on Discrete Mathematics. 214930R. Motwani, A. Naor, and R. Panigrahy. Lower bounds on locality sensitive hashing. SIAM Journal on Discrete Mathematics, 21(4):930, 2007. Optimal lower bounds for locality-sensitive hashing (except when q is tiny). R Donnell, Y Wu, Y Zhou, ACM Transactions on Computation Theory (TOCT). 615R. O'Donnell, Y. Wu, and Y. Zhou. Optimal lower bounds for locality-sensitive hashing (except when q is tiny). ACM Transactions on Computation Theory (TOCT), 6(1):5, 2014. 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Wang, J. Wang, G. Zeng, Z. Tu, R. Gan, and S. Li. Scalable k-nn graph construction for visual descriptors. In 2012 IEEE Conference on Computer Vision and Pattern Recognition, pages 1106-1113. IEEE, 2012. Collective dynamics of 'small-world'networks. D J Watts, S H Strogatz, nature. 3936684440D. J. Watts and S. H. Strogatz. Collective dynamics of 'small-world'networks. nature, 393(6684):440, 1998. Top 10 algorithms in data mining. X Wu, V Kumar, J R Quinlan, J Ghosh, Q Yang, H Motoda, G J Mclachlan, A Ng, B Liu, S Y Philip, Knowledge and information systems. 141X. Wu, V. Kumar, J. R. Quinlan, J. Ghosh, Q. Yang, H. Motoda, G. J. McLachlan, A. Ng, B. Liu, S. Y. Philip, et al. Top 10 algorithms in data mining. Knowledge and information systems, 14(1):1-37, 2008.

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{'abstract': 'Graph-based approaches are empirically shown to be very successful for approximate nearest neighbor (ANN) search. However, there has been very little research on their theoretical guarantees. In this work, we consider both low-dimensional (d log n) and high-dimensional (d log n) regimes and rigorously analyze the performance of graph-based nearest neighbor algorithms when the dataset is uniformly distributed on a d-dimensional sphere. For both regimes, we provide the conditions which guarantee that a graph-based algorithm solves the ANN problem in just one iteration. In the low-dimensional regime, we also show that it is possible to solve the exact nearest neighbor problem. Finally, we discuss how the "small-world" property affects the performance of graph-based approaches.IntroductionMany methods in machine learning, pattern recognition, coding theory, and other research areas are based on nearest neighbor search (NNS)[9,11,28,33]. In particular, the k-nearest neighbor method is included in the list of top 10 algorithms in data mining[37]. Due to the fact that modern datasets are mostly huge (both in terms of the number of elements n and the dimension d), reducing the computation complexity of NNS algorithms is of the essence. The nearest neighbor problem is to preprocess a given dataset D in such a way that for an arbitrary forthcoming query vector q we can quickly (in time o(n)) find its nearest neighbors in D.Several efficient methods exist for NN problem when the dimension d is small[5,8,29]. In particular, the algorithms based on recursive partitions of the space, like k-d threes and random projection trees, are widely used[8,12,13,22]. However, all existing methods for the exact NNS suffer from the curse of dimensionality: their complexity (either time or space) is exponential in d. To overcome this issue, NN problem has been relaxed to c-approximate nearest neighbor problem (c-ANN): if the distance between q and its closest neighbor is r, then it is allowed to return any element at distance at most c r with some c > 1. The most well-known algorithm for ANN is the Locality Sensitive Hashing (LSH)[20], which is well studied theoretically and widely used in practical applications.Recently, graph-based approaches were shown to demonstrate superior performance over other types of algorithms in many large-scale applications of NNS[6]. All these approaches are essentially based on constructing a nearest neighbor graph (or its approximation), where nodes correspond to the elements of D and each node is connected to its nearest neighbors by directed edges[15,17,35]. Then, for a given query q, one first takes an element in D (either random or fixed predefined) and makes greedy steps towards q on the graph: at each step all neighbors of a current node are evaluated and the one closest to q is chosen. This procedure can be restarted several times from different nodes, we call such restarts iterations. Various heuristics are proposed to speed up graph-based search[16,21]. In particular, navigable small world graphs (NSW and HNSW) add links between distant points to make faster progress on early steps of the algorithm[26,27].', 'arxivid': '1907.00845', 'author': ['Liudmila Prokhorenkova ostroumova-la@yandex.ru \nYandex Moscow Institute of Physics and Technology\n\n'], 'authoraffiliation': ['Yandex Moscow Institute of Physics and Technology\n'], 'corpusid': 195767092, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 20136, 'n_tokens_neox': 17555, 'n_words': 11632, 'pdfsha': '97c86cb89126316ffe8a00c4d5fdc6c20b24c8d9', 'pdfurls': ['https://arxiv.org/pdf/1907.00845v1.pdf'], 'title': ['Graph-based Nearest Neighbor Search: From Practice to Theory', 'Graph-based Nearest Neighbor Search: From Practice to Theory'], 'venue': []}

arxiv

Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity Wenjie Xi Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices Southern University of Science and Technology 518055ShenzhenChina Institute for Quantum Science and Engineering and Department of Physics Southern University of Science and Technology 518055ShenzhenChina Zhi-Hao Zhang Institute for Quantum Science and Engineering and Department of Physics Southern University of Science and Technology 518055ShenzhenChina School of Mathematical Sciences University of Science and Technology of China 230026HefeiChina Zheng-Cheng Gu Department of Physics The Chinese Uinversity of Hong Kong Hong KongChina Wei-Qiang Chen Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices Southern University of Science and Technology 518055ShenzhenChina Institute for Quantum Science and Engineering and Department of Physics Southern University of Science and Technology 518055ShenzhenChina Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity Symmetry protected topological statesTopological quantum field theoryNon-Hermitian systemsStrongly correlated systems Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its manybody topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems.unique Hermitian counterparts via local similarity transformation. Thus, they have the same topological invariants as their Hermitian counterparts, while the edge states can also be mapped onto those of the Hermitian case with a local similarity transformation. Physically, the topological Berry phase of quasi-Hermitian systems can be regarded as a C × -valued phase factor instead of the U(1) valued phase factor in the usual Hermitian systems under adiabatic evolution. In Section III, it is shown that for 1D quasi-Hermitian systems, the unitarity condition will naturally emerge in the fixed point topological invariant partition function and the classification of bosonic symmetry-protected topological (SPT) phases is exactly the same as their Hermtian counterparts, which are classified by second group cohomology H 2 (G, U(1) T )[30][31][32][33][34][35]. On the other hand, as 1D local fermionic systems can always be mapped to 1D local bosonic systems, the above conclusion for SPT phase also holds for 1D quasi-Hermitian interacting fermion systems. As Kitaev's Majorana chain is the only intrinsic fermionic topological phase, we will study this specific case for quasi-Hermitian systems in the Supplementary materials.Given the above, we study non-Hermitian systems with complex energy spectrum in Section IV. In this case, topological invariants are not always well defined, as the ground state is allowed to bypass an excited state without level crossing. However, it is elaborated in Section IV B and IV C that topological quantum field theory (TQFT) still captures all possible topological phases for 1D non-Hermitian quantum systems (including those non-Hermitian systems with complex eigenvalues). These phases have a one-to-one correspondence to their Hermitian counterparts. For bosonic SPT phases, arXiv:1911.01590v6 [cond-mat.str-el] I. INTRODUCTION Recently, the topological properties of non-Hermitian Hamiltonians have drawn much attention both experimentally [1][2][3][4][5][6] and theoretically [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. Studies have been focused intensively on topological phases in non-Hermitian free fermion systems [25][26][27] and the phase transitions among them [8,28,29]. Unfortunately, a general classification scheme of topological phases in interacting non-Hermitian systems is still lacking. In principle, the concept of the entanglement pattern (with or without global symmetries) can still be used to define and classify topological phases for non-Hermitian systems and characterize their ground state properties, although some technical details such as local unitary (LU) transformations need to be modified to fit the non-unitary time evolution of non-Hermitian systems. Meanwhile, as a fundamental property of ordinary quantum mechanics, unitarity plays an essential role in modern physics, and it will be of great interest to investigate how unitarity can emerge from an underlying non-unitary system. In this paper, we attempt to provide a complete classification of topological phases in 1D interacting non-Hermitian systems. In Section II, we first demonstrate that the classification of topological phases for quasi-Hermitian systems with real energy spectrum is the same as their Hermitian counterparts with a simple example. This follows from the fact that these kinds of non-Hermitian Hamiltonians can always be mapped to their this is simply because H 2 (G, U(1) T ) is isomorphic with H 2 (G, C × T ), thus the classification of their topological phases is exactly the same. II. A SIMPLE EXAMPLE: QUASI-HERMITIAN SU-SCHRIEFFER-HEEGER (SSH) MODEL WITH INTERACTIONS Without loss of generality, we take the SSH model, which is well studied for both non-interacting [36][37][38] and interacting [39][40][41][42] cases, as a simple example. We begin with a 1D non-Hermitian non-interacting SSH model with nearest neighbor hopping for spinless fermion system, as shown in Fig. 1a, with the Hamiltonian: H SSH = i=2n−1 α i,r t 1 c † i+1 c i + α i,l t 1 c † i c i+1 + i=2n α i,r t 2 c † i+1 c i + α i,l t 2 c † i c i+1 ,(1) where c i is the annihilation operator of electron on site i, t 1 and t 2 are real numbers related to the hopping integrals of electrons between nearest sites in the same unit cell and in a different unit cell respectively, α i,l(r) are nonzero real numbers. If we take α i,l = α i,r = 1, it reduces to the well-known Hermitian SSH model. For a half-filled Hermitian SSH model, there is a topologically nontrivial phase at t 1 < t 2 and a phase transition at t 1 = t 2 to a topologically trivial phase. The topological properties of such a phase can be described by the topological invariants, i.e., the winding number in k-space, of the bulk state with periodic boundary conditions (PBC) or the zero-energy state at the edge with open boundary conditions (OBC). This is the so-called bulk-boundary correspondence in free fermion systems. The topologically nontrivial ground state is protected by charge conservation and anti-unitary chiral symmetry S, defined as: Sc i S −1 = (−1) i c † i , Sc † i S −1 = (−1) i c i SiS −1 = −i.(2) Note that although we need translational symmetry to calculate the winding number in k-space, the topological phase is not protected by the translational symmetry. Then we move to the non-Hermitian SSH model and check whether there is still a nontrivial SPT phase or not. First, we consider a special case with α i,l = α −1 i,r = α i , which can be mapped to the Hermitian SSH model via a local similarity transformation: c i →   i−1 j=1 α −1 j   c i , c † i →   i−1 j=1 α j   c † i .(3) This means the non-Hermitian model has a real spectrum even though its Hamiltonian is non-Hermitian. In the following, we refer the non-Hermitian Hamiltonians with real spectrum as quasi-Hermitian Hamiltonians (the term "qusi-Hermitian" is borrowed from Ref. [43]). In fact, the quasi-Hermitian model (1) has exactly the same spectrum as the Hermitian SSH model as shown in Fig. 1c, which means that there are zero-energy edge states at t 1 < t 2 but no zero-energy edge state at t 1 > t 2 . However, this does not mean that it is a topologically nontrivial phase. We must prove that the zero-energy edge states are related to some topological properties of the bulk and protected by certain symmetries, especially for interacting systems. To detect the topological properties of the bulk, we need to connect the two ends of the chain to form a ring, as shown in Fig. 1b, and calculate some topological invariants of the system on this ring. In the Hermitian case, one uses the PBC, which corresponds to connecting the two ends with t 2 c † N c 1 + t 2 c † 1 c N , and calculating the winding number in k-space. However, in the most general quasi-Hermitian cases, the system does not have translational symmetry, so one cannot do calculations in k-space. Instead, we must use the twist boundary condition (TBC) [44], introduced by Wu et al. [45] to study quantum Hall states, and to calculate the topological invariant of the ground state. The basic idea is to introduce an additional phase factor e iθ in the boundary conditions, i.e., e iθ t 2 c † N c 1 + e −iθ t 2 c † 1 c N , which corresponds to inserting a θ flux in the center of the ring as shown in Fig. 1b. Obviously, the ground state wave function depends on the phase θ. By further assuming that the many-body ground state of the system is separated from the excited states by a finite gap for all values of the twisted phase θ, one can define a topological invariant by the total flux of the Berry-phase gauge field associated with the ground state over the θ-space, i.e., C = i π 2π 0 ϕ G (θ)| ∂ ∂θ |φ G (θ) dθ,(4) where ϕ G (θ)|(|φ G (θ) ) is the left (right) many-body ground state of the non-Hermitian Hamiltonian. The advantage of the TBC is that it can handle general cases with interactions and without translational symmetry. A simple calculation shows that the resultant C gives the correct bulk-boundary correspondence in the Hermitian case. However, such correspondence is absent in the quasi-Hermitian case. This can be understood from the similarity transformation Eq. (3). After the transformation, the quasi-Hermitian SSH model with PBC is mapped to a Hermitian SSH model with a boundary condition γt 2c † 1c N + γ −1 t 2c † Nc 1 with γ = N −1 i=1 α i . Such a boundary condition breaks the chiral symmetry, and thus the bulk-boundary correspondence of the SPT state. However, if we consider a different boundary condition: α N t 2 c † 1 c N + α −1 N t 2 c † N c 1 ,(5)with α N = γ −1 = N −1 i=1 α −1 i , the corresponding Hermitian Hamiltonian after the mapping will be the Hermitian SSH model with PBC, and one should have the correct bulk-boundary correspondence. This has been confirmed by our TBC calculations, which show the winding number to be 1 for t 1 < t 2 and 0 for t 1 > t 2 as long as all of the αs satisfy N i=1 α i = 1. For the case α 1 = α 2 = · · · = α N −1 = α, one should impose a boundary condition with α N = α −(N −1) . Note that the similarity transformation Eq. (3) indicates an imaginary term in k after Fourier transformation of c i . This provides an explanation of the failure of the conventional winding number in k-space in the non-Hermitian SSH model and the success of the winding number in a complex k-space in Ref. [7]. To check whether the topological phase is protected, we introduce a small imaginary chemical potential difference and a small Coulomb repulsion between electrons in the same unit cell, and the Hamiltonian becomes: H =H SSH + iµ j n 2j−1 + U j (n 2j−1 − 1 2 )(n 2j − 1 2 ).(6) We first consider a simple case with U = 0.1 and µ = 0, where the interaction terms after the similarity transformation Eq. (3) also respect the chiral symmetry Eq. (2) of the Hermitian SSH model. The winding number calculated with TBC depicted in Fig. 2b demonstrates that the ground state remains topologically nontrivial at t 1 < t 2 . It is reasonable to think that the topologically nontrivial phase discovered above is protected by a quasi-Hermitian version of chiral symmetry S: Sc i S −1 = (−1) i i−1 j=1 α −2 j c † i , Sc † i S −1 = (−1) i i−1 j=1 α 2 j c i Si S −1 = −i,(7) which is related to S by the similarity transformation Eq. (3). It should be noticed the action of symmetry S for non-Hermitian systems can be a similarity transformation. Again, according to the similarity transformation Eq. (3), the phase should belong to the same topological phase as its corresponding phase of the Hermitian SSH model. There is a more profound and general understanding of the similarity transformation Eq. (3). In quantum mechanics, two wave functions ψ and φ that differ by a nonzero complex factor, i.e., φ = zψ with z = 0, correspond to the same physical state. A theory should be invariant under local gauge transformation c i → e −iθi c i , c † i → e iθi c † i . If θ i s are real numbers, they are just the U(1) gauge choice of the local basis. However, for quasi-Hermitian systems, we can consider the most general C × gauge choice with complex θ i 's (here C × = C\{0} is the group of non-zero complex numbers), and the similarity transformation Eq. (3) is exactly a local C × gauge transformation. Thus, the corresponding topological Berry phase, which arises from adiabatic evolution, should be described by a C × -valued gauge field for generic quasi-Hermitian systems. III. CLASSIFICATION OF TOPOLOGICAL PHASES IN 1D QUASI-HERMITIAN SYSTEMS Now we move the discussion to generic interacting quasi-Hermitian systems. For 1D Hermitian bosonic systems without any symmetry, it is well known that any gapped quantum state can always connect to a trivial product state without phase transition [30,46]. Obviously, such a statement still holds for 1D non-Hermitian bosonic systems, and SPT phases are still the only possible topological phases in bosonic case. Fermionic systems is in general much more complicated. However, it has been shown that the classification of 1D Hermi-tian fermionic SPT phases can be obtained from that for bosonic systems via the Jordan-Wigner transformation (i.e., bosonization) [30,47], and this should still hold in quasi-Hermitian systems. Thus we only need to focus on the classification of bosonic SPT phases in quasi-Hermitian systems in this section. In Hermitian cases, the bosonic SPT phases in n dimension have been successfully classified by classifying the fixed point partition functions. Mathematically, the classification is given by the group cohomology H n (G, U(1) T ) [34]. In this section, we will generalize this approach to quasi-Hermitian systems. The topologically invariant fixed point partition function for bosonic SPT phases protected by a finite symmetry group G in quasi-Hermitian systems (defined on an arbitrary branched triangulation of a 2D manifold) can be written as Z f = 1 |G| Nv {gi} triangular ν s ijk 2 (g i , g j , g k ),(8) where |G| is the order of the group, with N v the number of total vertices, and s ijk = ± is determined by the orientation of the corresponding triangulation. Eq. (8) has exactly the same form as the fixed point partition function of 1+1D Hermitian systems except that ν ± 2 (g i , g j , g k ) ∈ C × instead of U (1) for quasi-Hermitian systems. ν ± 2 (g i , g j , g k ) is a function of group element g i , g j , g k , which can be naturally regarded as the C ×valued symmetric topological Berry phase term for quasi-Hermitian systems. ν ± 2 (gg i , gg j , gg k ) = ν ± 2 (g i , g j , g k ) leaving the partition function invariant, which implies the partition function is invariant under symmetry operation. Moreover, we can further impose the following condition: [ν + 2 (g i , g j , g k )] * = ν − 2 (g i , g j , g k ).(9) This condition holds because for quasi-Hermitian systems with real energy spectrum, the time reversal symmetry can always be realized by the complex conjugate operation. Alternatively, time reversal can also be defined as the reversing of branching arrows, which naturally reverses the time ordering and orientation for a given triangulation. Furthermore, as a topologically invariant partition function, it must be invariant under all possible Pachner moves (re-triangulations) as depicted in Fig. 3 for arbitrary branched triangulation. For the 2 ↔ 2 moves, we have: Similarly, for the 1 ↔ 3 moves, we have: ν + 2 (g 0 , g 1 , g 3 )ν − 2 (g 0 , g 2 , g 3 ) =ν − 2 (g 1 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 2 ), ν + 2 (g 1 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 3 ) =ν + 2 (g 0 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 2 ).(10)ν + 2 (g 0 , g 1 , g 3 ) =ν − 2 (g 1 , g 2 , g 3 )ν + 2 (g 0 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 2 ), ν + 2 (g 0 , g 2 , g 3 ) =ν + 2 (g 1 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 3 )ν − 2 (g 0 , g 1 , g 2 ).(11) Physically, the two Pachner moves correspond to retriangulation and coarse graining of the partition function. All of the above four equations form a consistent algebra which is also satisfied by fixed point partition function of Hermitian systems, and they lead to the unitarity condition for ν ± 2 : ν + 2 (g i , g j , g k )ν − 2 (g i , g j , g k ) = 1,(12) which further unifies the above four equations into the well-known 2-cocycle equation of ν + 2 : ν + 2 (g 1 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 3 ) = ν + 2 (g 0 , g 2 , g 3 )ν + 2 (g 0 , g 1 , g 2 ).(13) Thus, we conclude that for quasi-Hermitian systems, SPT phases are still classified by H 2 (G, U(1)) and unitarity emerges for the fixed point partition function. The above results can also be generalized into anti-unitary symmetry cases straightforwardly. As a summary, we have shown that the classification of 1D quasi-Hermitian bosonic SPT phases is exactly the same as the one of Hermitian systems. In fact, the emergence of unitarity discussed above should also apply to all the quasi-Hermitian bosonic SPT phases in higher dimensions. Moreover, because 1D local fermionic systems can always be mapped onto local bosonic systems as discussed in the beginning of this section, the above claim also holds for classifying SPT phases in 1D quasi-Hermitian fermionic systems. Besides the SPT phases, there is a so-called intrinsic topological phase in a 1D fermionic system, which is the Kitaev's Majorana chain. The above derivation of emergent unitarity is still correct for this case, see the Supplementary materials for more details. IV. TOPOLOGICAL PHASES IN GENERIC 1D NON-HERMITIAN SYSTEMS A. Non-Hermitian SSH model with complex energy spectrum Finally, we consider the most general non-Hermitian case, where the eigenvalues of the Hamiltonian can be complex. The ground state is decided by considering the path integral in Euclidean space. At a large time scale, only state with lowest real part energy survives. Thus, we define the state with lowest real part energy as the ground state. The imaginary part energy of the ground state only contribute an oscillation of state. Again, we use the SSH model to illustrate the basic idea. In Fig. 4a and b, we show the real and imaginary parts of the spectrum of the model Eq. (6) with α i,l = α i,r = 1, t 1 = 0.6, t 2 = 1, µ = 0.1 and U = 0.1 at various twist angles θ. Although defining symmetry is subtler for systems with complex eigenvalues, in this specific case the chiral symmetry can still be defined as Eq. (2). However, when we take θ from 0 to 2π, a generic state might bypass another state without level crossing, and one may not be able to calculate the topological invariant C for that state. This happens only when the energy spectrum is complex. Fortunately, the ground state, which is considered to have the lowest real part energy, still has a well-defined C when t 1 /t 2 is away from 1 in this system. It is not surprising because the imaginary chemical potential can be regarded as a perturbation in this case. From another perspective, as the gap between ground state and excited state is much larger than the imaginary chemical potential for any twist angle θ, we shall not expect the imaginary component of the ground state energy raised from the imaginary chemical potential will change band dispersion dramatically. The corresponding numerical result shows that we still have C = 1 for t 1 < t 2 and C = 0 for t 1 > t 2 . For t 1 t 2 , the gap is much smaller so that we cannot neglect the modification on band dispersion caused by the imaginary chemical potential. The situation near the critical point is quite subtle and complicated. The ground state may bypass another state without level crossing and thus does not have a well defined C. Nevertheless, our results still indicate a critical point at t 1 t 2 . This suggests that there are still two SPT phases in this non-Hermitian case with a complex energy spectrum and they belong to the same fixed points as the Hermitian case. The above example suggests that the emergent unitarity, i.e., a non-Hermitian SPT phase corresponding to a fixed point of the Hermitian case, still holds for 1+1D generic non-Hermitian systems. Whether the above statement holds for the most generic 1+1D non-Hermitian SPT phases can be studied by considering its fixed point theory, which will be studied in full details below. B. Classification of SPT phases in generic 1D non-Hermitian systems In fact, the topological invariant fixed point partition function Eq. (8) can also be defined for generic 1D non-Hermitian bosonic systems and we only need to remove the quasi-Hermitian condition Eq. (9). Similar to the quasi-Hermitian case, we can also derive the 2-cocycle condition Eq. (13) by considering all possible Pachner moves. However, in this case, ν + 2 is C × Tvalued and the classification of topological phases in generic 1D non-Hermitian bosonic systems is given by H 2 (G, C × T ). Interestingly, mathematically it turns out that for any finite group or compact Lie group G, the natural inclusion U(1) T → C × T induces isomorphisms H i (G; U(1) T ) ∼ = H i (G; C × T ) for all i > 0, i.e. , the classification of the non-Hermitian systems is still the same as the corresponding Hermitian systems. The details are given below. At first, we consider a finite group G corresponding to only unitary symmetries. Since all U(1)-valued cochains are naturally C × -valued cochains, the inclusion U(1) → C × induces homomorphisms H i (G, U(1)) → H i (G, C × ) for all i > 0. It may be shown by direct calculation that a C × -valued cocycle is automatically valued in U(1). However, we prove it in a more abstract way here. Since G represents unitary symmetries, it acts trivially on coefficient groups U(1) and C × . It is known that the short exact sequence of (multiplicative) abelian groups with the trivial G-action 1 −→ U(1) −→ C × −→ R + −→ 1,(14) where R + denotes the multiplicative group of positive real numbers, induces a long exact sequence of cohomol-ogy groups 1 −→ H 0 (G, U(1)) −→ H 0 (G, C × ) −→ H 0 (G, R + ) −→ H 1 (G, U(1)) −→ H 1 (G, C × ) −→ H 1 (G, R + ) −→ H 2 (G, U(1)) −→ H 2 (G, C × ) −→ · · · · · ·(15) Since all cohomology groups H i (G, R + ) are trivial for i > 0, and H 0 (G, A) = A for all A with the trivial Gaction, we get short exact sequences 1 −→ H i (G, U(1)) −→ H i (G, C × ) −→ 1 (16) for all i > 0, which imply that H i (G, U(1)) ∼ = H i (G, C × ) for all i > 0. If G contains anti-unitary symmetries, for example the time-reversal symmetry T , then the actions of G on coefficient groups U(1) and C × are non-trivial: if T ∈ G is anti-unitary, we have T iT −1 = −i, i.e., T acts by the complex conjugate. Similarly we have a short exact sequence of (multiplicative) abelian groups with G-action 1 −→ U(1) T −→ C × T −→ R + −→ 1(17) where the subscript T means the complex conjugate Gaction on coefficient groups. Note that the G-action on R + is trivial. This short exact sequence induces a long exact sequence of cohomology groups 1 −→ H 0 (G, U(1) T ) −→ H 0 (G, C × T ) −→ H 0 (G, R + ) −→ H 1 (G, U(1) T ) −→ H 1 (G, C × T ) −→ H 1 (G, R + ) −→ H 2 (G, U(1) T ) −→ H 2 (G, C × T ) −→ · · · · · ·(18) Recall that H 0 (G, A T ) is the G-invariant subgroup of A for all A, thus H 0 (G, U(1) T ) = {±1} = Z 2 and H 0 (G, C × ) = {z ∈ C × | z =z} = R × . So three H 0 groups in the above sequence form a short exact sequence 1 −→ Z 2 −→ R × −→ R + −→ 1.(19) Also, all cohomology groups H i (G, R + ) are trivial for i > 0. Hence we get short exact sequences 1 −→ H i (G, U(1) T ) −→ H i (G, C × T ) −→ 1 (20) for all i > 0, which imply that H i (G, U(1) T ) is isomor- phic to H i (G, C × T ) for all i > 0. A similar statement holds for any compact Lie group G, where the Lie group cohomology with coefficients in an abelian Lie group is defined with the differentiable cohomology [48]. The proof is essentially the same. First, the short exact sequence 0 −→ U(1) −→ C × −→ R + −→ 0 of abelian Lie groups induces a long exact sequence of differentiable cohomology groups; then we use the fact H i (G, R + ) ∼ = H i (G, R) = 0 to complete the proof. These two results are listed in Ref. [48]. Physically, the above results suggest that for generic non-Hermitian systems in 1D, the edge modes of bosonic SPT phase still carry projective representations of the corresponding symmetry group. Unlike Hermitian systems with unitary representation, generic non-Hermitian systems admit non-unitary projective representation. However, the isomorphism H 2 (G; U(1) T ) ∼ = H 2 (G; C × T ) suggests that equivalent class of non-unitary projective representations are exactly the same as unitary projective representations. In particular, by a proper gauge choice, e.g., applying a coboundary transformation, all the projective representations in generic 1D non-Hermitian systems can still be chosen as unitary. Therefore, the classification of bosonic SPT phases in generic 1D non-Hermitian systems is exactly the same as their Hermitian counterparts. Again, as 1D local fermionic systems can always be mapped onto local bosonic systems, the above claim also holds for classifying SPT phases in generic non-Hermitian fermionic systems. Finally, as the above proof shows that H i (G; U(1) T ) ∼ = H i (G; C × T ) for all i > 0, we conjecture that in higher dimensions, the classification of bosonic SPT phases in generic non-Hermitian systems is still exactly the same as their Hermitian counterparts. C. The emergence of unitarity for generic topological phases in 1D non-Hermitian systems Since the low energy effective theory of topological phases can always be described by certain TQFTs, below we discuss a more general way to understand the emergence of unitarity for all topological phases in 1D non-Hermitian systems. It is well-known that Frobenius algebras give rise to a generic framework to understand 1+1D TQFTs for 1+1D bosonic systems. Based on such a framework, we provide a much deeper understanding on why the classification of topological phases in generic non-Hermitian systems is always the same as their Hermitian counterparts. We show that the concept of emergent unitarity actually applies for all extendable TQFTs in 1+1D. Definition of TQFT The mathematical definition of a TQFT arises from the intuitions of path integrals. Given a quantum field theory Z, there should be a Hilbert space on each (closed) space manifold Σ, denoted by Z 0 (Σ); and for each spacetime manifold M , whose time slices at the beginning and end are Σ 1 and Σ 2 respectively (we call M a cobordism from Σ 1 to Σ 2 ), we can do path integral and get a propagator denoted by Z 1 (M ) : Z 0 (Σ 1 ) → Z 0 (Σ 2 ) (see Fig. 5). In particular, if the spacetime manifold M is already closed, Z 1 (M ) is a complex number, which is just the partition function. Moreover, these data should satisfy some natural factorization properties. For example, if M is a cobordism from Σ 1 to Σ 2 and N is a cobordism from Σ 2 to Σ 3 , then we can sew them together to get a cobordism from Σ 1 to Σ 3 , denoted by N • M ; then we expect that the composition Z 1 (N )Z 1 (M ) of propagators should be equal to Z 1 (N • M ). We say such a quantum field theory Z is topological if both the Hilbert spaces Z 0 (Σ) and the propagators Z 1 (M ) only depend on the topology of the manifolds. These intuitions lead to a precise mathematical definition of TQFTs [49,50]. It is well-known that a 1+1D TQFT Z is determined by its value Z 0 (S 1 ) on a circle S 1 , which is a commutative Frobenius algebra. For example, its multiplication is given by Z 1 (M ) as depicted in Fig. 5. We briefly review the notion of Frobenius algebras in Appendix B. Σ 2 M Σ 1 Time Z 0 (Σ 1 ) Z 0 (Σ 2 ) Z 1 (M ) TQFTs give an abstract but generic framework to understand topological phases. Given a gapped topological phase, its ground state degeneracy defined on a closed space manifold Σ is robust and only depends on the topology of Σ. Therefore, the ground state subspace on Σ is well-defined. If its low energy effective theory is described by a TQFT Z, then Z 0 (Σ) is given by this robust ground state subspace. This explains the wellknown statement that the low energy effective theories of topological phases are TQFTs. However, a topological phase is an equivalence class of local Hamiltonians, thus only local TQFTs describe topological phases. In the following we discuss the locality of TQFTs. Fully extended TQFT A TQFT defined as above is not necessarily a local theory. To compute the partition function on an n+1D closed manifold, we can decompose it into several pieces along nD submanifolds and then do path integrals. However, these pieces are usually too large to be thought of as local. A local theory should provide a way to compute the partition function by any decomposition the spacetime manifold into small enough pieces, for example n-simplices, along higher-codimensional submanifolds. Thus a local TQFT should be able to evaluate at not only n+1D and nD manifolds, but also at all highercodimensional manifolds. Also, these data should satisfy some compatibility conditions. This idea leads to the notion of an extended TQFT. Briefly speaking, a fully extended TQFT is an assignment which assigns linear objects (numbers, linear spaces, linear categories, . . . ) to cobordisms of all codimensions, including a point. It is local in the sense that, the partition functions can be computed by the decompositions into arbitrarily small pieces, and, as one may expect, a fully extended TQFT is determined by its value on a point [51,52]. If we forget what we assign to higher-codimensional submanifolds, a fully extended TQFT gives an ordinary TQFT. We say a TQFT is fully extendable, or local, if it can be extended to a fully extended TQFT. The classification of fully extended 1+1D TQFTs was given by Ref. [53], and we summarize the main result in the following: Theorem 1. An extended 1+1D TQFT is determined by its value on a point, which is a symmetric separable (semisimple) Frobenius algebra B. So far we have proved that the low energy effective theory of any local Hamiltonian of a topological phase, no matter it is Hermitian or non-Hermitian, must correspond to a Frobenius algebra B. This Frobenius algebra B encodes all local physical data. In the next subsection we classify the gauged theories of (bosonic) SPT phases with onsite symmetry G, and the G-gauge symmetries are also encoded in this algebra B. Let us give an explicit lattice construction of a fully extendable 1+1D TQFT from a symmetric special Frobenius algebra B [54]. Fix a basis {e i } of B, we write the structure constants of B as: e i · e j = µ(e i ⊗ e j ) = k C k ij · e k ,(21)∆(e i ) = jk C jk i · e k ⊗ e j ,(22) ε(e i · e j · e k ) = C ijk , ((∆ ⊗ id B ) • ∆)(1) = ijk C ijk · e k ⊗ e j ⊗ e i .(23) µ, ∆ and ε are called multiplication map, comultiplication map and counit map, respectively. See Section B in the Supplementary materials for more details about Frobenius algebra. For any closed manifold M , we take a directed trivalent graph on M and label each edge by basis vectors {e i }. Then we assign to each vertex a number as follows: F k j i = C ijk , F k j i = C k ij , F i j k = C jk i , F i j k = C ijk .(25) The partition function on M is obtained by first taking the multiplication of F (v) over all vertices and then sum-ming over all possible labels on basis vectors: Z(M ) := F v F (v).(26) This TQFT is local because the lattice construction only involves local data. For more details of this construction, see Section C in the Supplementary materials. TQFT with a gauge symmetry For any 1+1D SPT phases with onsite(internal) symmetry G, we can gauge the symmetry G and get the well known Dijkgraaf-Witten theory, which can be described by a 1+1D TQFT with G-gauge symmetry. Thus, there is a one-to-one correspondence between the classification of SPT phases protected by onsite(internal) symmetry G and the classification of Dijkgraaf-Witten theory with G-gauge symmetries. On the other hand, if there is a G-gauge symmetry in the partition function (26), the local degree of freedom on each edge should be labeled by group elements g ∈ G. In other words, the vector space B is spanned by the group G, i.e., B = C[G]. To do local gauge transformations, the multiplication of B should behave like the group multiplication, up to a phase factor: g · h = ω(g, h) · gh for some ω(g, h) ∈ C × . The associativity of the multiplication implies that ω ∈ Z 2 (G; C × ) is a 2-cocycle. Hence, the Frobenius algebra B is just the twisted group algebra C ω [G]. Moreover, two cocycles differed by a coboundary give the isomorphic twisted group algebra. Hence for genric non-Hermitian systems, the Dijkgraaf-Witten theory with G-gauge symmetry are classified by the group cohomology H 2 (G; C × ), which is isomorphic to H 2 (G; U(1)) by the results in Sec. IV B. According to the one-to-one correspondence between Dijkgraaf-Witten gauge theory and SPT phases, we end up with the same conclusion that the classification of SPT phases in generic 1+1D non-Hermitian systems is still exactly the same as their Hermitian counterparts. More rigorously, SPT phases in 1+1D can also be classified by G-crossed Frobenius algebras and there is a similar cohomology classification [55,56]. Moreover, we note that the Jordan-Wigner transformation maps the Majorana chain (which is a 1D intrinsic topological phase for fermion systems) to a 1d Z 2 symmetry breaking phase. It can also be described by the commutative Frobenius algebra C 2 = C ⊕ C, and the Z 2 -action is permuting two components of C 2 . Apparently, such a commutative Frobenius algebra is always equivalent to a unitary Frobenius algebra with the same Z 2 -action and the concept of emergent unitarity will still apply for this case. V. DISCUSSION AND CONCLUSION We have studied the classification of topological phases for 1D interacting non-Hermitian systems, and established that the classification of topological phases is exactly the same as Hermitian systems. Moreover, unitarity can even emerge for fixed point partition functions of 1D topological phases. In mathematics, the isomor- phisms H i (G; U(1) T ) ∼ = H i (G; C × T ) for all i > 0 suggest that in 2D and 3D, the classification of interacting SPT phases could still be the same for Hermitian and non-Hermitian systems (at least for bosonic systems). Of course, for intrinsic topological phases in higher dimensions, it has been shown that non-Hermitian systems can be much richer than Hermitian systems, e.g. string-net models constructed by non-unitary fusion category theory are very interesting examples [57]. Finally, defining topological invariants for generic non-Hermitian systems is still quite challenging, which further suggests that topological phase transitions in non-Hermitian systems are much richer than in Hermitian systems, even in 1D. We believe that non-unitary conformal field theory (CFT) might play a very important role. VI. ACKNOWLEDGMENTS We are grateful for helpful discussions with Yong- Shi Appendix A: Emergent unitarity for 1+1D intrinsic topological phases in quasi-Hermitian fermionic systems For 1 + 1D quasi-Hermitian fermionic systems, we can use the Grassmann valued amplitude to construct the partition function for topological phases. Below we consider the so-called intrinsic topological phase which is stable even without symmetry protection. It turns out that there is one and only one such kind phase, namely, the Kitaev's Majorana chain model. Let us consider the following partition function: Z f = {nij } link dθ + ij dθ − ij link (1 − θ + ij θ − ij ) triangular V s ijk ijk ,(A1) where V + ijk = nij ,n jk ,n ik ν + (n ij , n jk , n ik ) θ + ij nij θ + jk n jk θ − ik n ik V − ijk = nij ,n jk ,n ik ν − (n ij , n jk , n ik ) θ + ik n ik θ − jk n jk θ − ij nij (A2) Similar as bosonic systems, for quasi-hermitian systems, we can also impose the following condition: ν − (n ij , n jk , n ik ) = [ν + (n ij , n jk , n ik )] * (A3) Note that the fermion parity conservation further requires: n ij + n jk + n ik = 0 mod 2 (A4) Actually, in terms of quantum field theory language, V ± can be regarded as Grassmanm valued amplitude, and the Grassmann variable θ + /θ − satisfying standard Grassmann algebra is associate with creation/annihilation operator c † /c. Now we consider the time ordered Pachner moves for Grassmann valued partition. Formally, we can write down the 2 ↔ 2 move as: dθ + 03 dθ − 03 (1 − θ + 03 θ − 03 )V + 013 V − 023 = dθ + 12 dθ − 12 (1 − θ + 12 θ − 12 )V − 123 V + 012 (A5) dθ + 13 dθ − 13 (1 − θ + 13 θ − 13 )V + 123 V + 013 = dθ + 02 dθ − 02 (1 − θ + 02 θ − 02 )V + 023 V + 012 (A6) We note that due to the even number of Grassmann variable constraint, we can remove the summation over n ij and obtain: ν + (n 01 , n 13 , n 03 )ν − (n 02 , n 23 , n 03 ) =ν − (n 12 , n 23 , n 13 )ν + (n 01 , n 12 , n 02 ) (A7) ν + (n 12 , n 23 , n 13 )ν + (n 01 , n 13 , n 03 ) =ν + (n 02 , n 23 , n 03 )ν + (n 01 , n 12 , n 02 ) (A8) Similarly, the 1 ↔ 3 moves further imply: ν + (n 01 , n 13 , n 03 ) = n12 ν − (n 12 , n 23 , n 13 )ν + (n 02 , n 23 , n 03 )ν + (n 01 , n 12 , n 02 ) (A9) ν + (n 02 , n 23 , n 03 ) = n12 ν + (n 12 , n 23 , n 13 )ν + (n 01 , n 13 , n 03 )ν − (n 01 , n 12 , n 02 ), We notice the combination of the time ordered 2 ↔ 2 and 1 ↔ 3 moves gives rise to the unitary condition on ν ± 2 : nij ν + (n ij , n jk , n ik )ν − (n ij , n jk , n ik ) = δ n ik ,n ik (A11) and ν + (n 12 , n 23 , n 13 )ν + (n 01 , n 13 , n 03 ) = ν + (n 02 , n 23 , n 03 )ν + (n 01 , n 12 , n 02 ) (A12) A simple solution reads: ν ± (n ij , n jk , n ik ) = 1/ √ 2 (A13) Its corresponding ground state wavefunction(defined by a partion function with a boundary) is described by an equal weight superposition of all the even number fermion configurations(associate with a proper fermion ordering). This solution actually describes the non-trivial phase of H = N −1 I=1 (c I − c † I )(c I+1 + c † I+1 ) + (c 1 − c † 1 )(c N + c † N ) (A14) We choose the anti-periodical boundary condition(APBC) to simplify our discussion. Due to the fact (c I − c † I )(c I+1 + c † I+1 ) 2 = 1, we can define the following projectors P I = 1 2 [1 − (c I − c † I )(c I+1 + c † I+1 )]; I = 1, · · · , N − 1 P N = 1 2 [1 − (c 1 − c † 1 )(c N + c † N )];(A15) It is easy to check P 2 I = P I and [P I , P J ] = 0. Thus, the Hamiltonian of the Kitaev's Majorana chain model is actually a summation of commuting projectors. H = N I=1 (1 − P I ) (A16) As a result, the ground state can be generated by acting the product of all projectors I P I onto an arbitrary state if the projector do not annihilate it. It turns out for those states with even number fermion I P I |even do not vanish and the ground state is an equal weight superposition of all possible even number fermion configurations(Notice the fermion basis are ordered as 1 < 2 < · · · < N ). In the following, let us explicit show why our solution describes the fixed point partition function of the Majorana chain. Since the gap of the system should be infinite at the fixed point, we need to rescale the Hamiltonian Eq.(A16) as: H = U N I=1 (1 − P I ) ; U → ∞ (A17) The corresponding fixed point partition function reads: Z = e −βH = e −∆τ H n I P I n (A18) In the last step we omit the overall constant. In each imaginary time slice ∆τ = β/n, the partition function takes a form: Z ∆τ I P I(A19) In the fermion coherent state representation, we can ex-press each P I as: θ I θ I+1 |P I |θ I θ I+1 = 1 2 θ I θ I+1 |θ I θ I+1 1 − (θ I − θ I )(θ I+1 + θ I+1 ) = 1 2 (1 + θ I θ I )(1 + θ I+1 θ I+1 ) 1 − (θ I − θ I )(θ I+1 + θ I+1 ) = 1 2 [(1 + θ I θ I )(1 + θ I+1 θ I+1 ) − (θ I − θ I )(θ I+1 + θ I+1 )],(A20) where the fermion coherent state |θ I is defined as: |θ I = |0 − θ I c † I |0 (A21) and from the Grassmann algebra we have θ 2 I = 0. The above expression evolve four Grassmann variable, so we need to introduce two triangle with a shared edge to represent the above amplitude, see in Fig. 6(a). If we define: θ I = θ − 01 ; θ I+1 = θ − 13 ; θ I = θ + 02 ; θ I+1 = θ + 23 , (A22) It is easy to check the amplitudes: dθ + 12 dθ − 12 (1 − θ + 12 θ − 12 )V − 012 V + 123 = 1 2 (1 + θ I θ I )(1 + θ I+1 θ I+1 ) − (θ I − θ I )(θ I+1 + θ I+1 ) = θ I θ I+1 |P I |θ I θ I+1 ,(A23) where the coefficients of V ± are the solutions in Eq.(A13): V + ijk = 1 √ 2 nij ,n jk θ + ij nij θ + jk n jk θ − ik |nij −n jk | V − ijk = 1 √ 2 nij ,n jk θ + ik |nij −n jk | θ − jk n jk θ − ij nij (A24) Thus, we prove that the amplitude in Fig. 6(a) does represent the projector P I . The partition function in a time slice can be constructed by a product of P I , as shown in Fig. 6(b), notice that in a partition function with a global time ordered structure will be naturally associated with APBC, as discussed above. Other boundary conditions require the introducing of discrete spin structures, which is much more complicated and beyond the scope of this paper. In conclusion, we see that without any physical symmetry, there is still a non-trivial topological phase in 1D quasi-Hermitian fermion systems, and unitarity will emerge for its fixed point partition function, which exactly describes the ground state phase of Kitaev's Majorana chain model. Appendix B: Frobenius algebras In this appendix, we briefly review the definition and basic properties of Frobenius algebras. Definition 2. A Frobenius algebra is a quintuple (A, µ, η, ∆, ε), where: 1. A is a (complex) vector space. 2. µ : A ⊗ A → A is a linear map, called the multiplication map. We also denote µ(a ⊗ b) by a · b or simply ab. 3. η : C → A is a linear map, called the unit map. We also denote η(1) by 1 A or simply 1. ∆ : A → A ⊗ A is a linear map, called the comultiplication map. ε : A → C is a linear map, called the counit map. These data satisfy the following conditions: • (A, µ, η) is an algebra. More precisely, we have µ • (µ ⊗ id A ) = µ • (id A ⊗µ), µ • (η ⊗ id A ) = id A = µ • (id A ⊗η). Here id A : A → A is the identity map. • (A, ∆, ε) is a coalgebra. More precisely, we have (∆ ⊗ id A ) • ∆ = (id A ⊗∆) • ∆, (ε ⊗ id A ) • ∆ = id A = (id A ⊗ε) • ∆. • We have (µ ⊗ id A ) • (id A ⊗∆) = ∆ • µ = (id A ⊗µ) • (∆ ⊗ id A ). This condition is called the Frobenius condition. Remark 3. By applying the first condition of that (A, µ, η) is an algebra to an element a⊗b⊗c ∈ A⊗A⊗A, we get (ab)c = a(bc). In other words, this condition means the multiplication is associative. Similarly, the second condition says 1 · a = a = a · 1 for all a ∈ A, which means 1 ∈ A is the unit for the multiplication. These two conditions reformulate the classical definitions of (linear) algebras by using linear maps, not elements in vector spaces. Let us explain the definition of Frobenius algebras by the so-called graph calculus. First, one may imagine that A is a particle moving in the spacetime. So we can draw the Feynman diagrams of A. If there is only a single A, we draw it as follows: A = = id A . Our convention is that the time axis points upward. This single line of A also represents the identity map id A because id A means nothing happens on A. All the other data in the definition of Frobenius algebras can be viewed as interactions of A, depicted as follows: µ = , η = , ∆ = , ε = . Here we use a small circle to depict the one-dimensional vector space C, which represents the vacuum. Then the conditions of that (A, µ, η) is an algebra can be depicted as = , = = , i.e. µ is associative and η is unital for µ; the conditions of that (A, ∆, ε) is a coalgebra can be depicted as = , = = , i.e. ∆ is coassociative and ε is counital for ∆; and the Frobenius condition can be depicted as = = . Hence the definition of Frobenius algebras means the value of topologically equivalent diagrams are equal. This is also why 1+1D TQFTs are equivalent to commutative Frobenius algebras. Definition 4. A Frobenius algebra A = (A, µ, η, ∆, ε) is commutative if it satisfies the condition µ • τ = µ, where τ : A ⊗ A → A ⊗ A permutes two components: τ (a ⊗ b) = b ⊗ a. It is symmetric if it satisfies the conditio ε • µ • τ = ε • µ. We can also depict the map τ as τ = . Then A is commutative can be depicted as = , and that A is symmetric can be depicted as = . Example 5. Let us consider the case that the commutative Frobenius algebra A is semisimple. So as an algebra we have A C n for some positive integer n, and the Frobenius structure over C n is determined by the counit map ε : C n → C. Let {e i = (0, . . . , 1, . . . , 0) ∈ C n } be the canonical basis of C n , and denote θ i = ε(e i ). They are n non-zero numbers. Then the coproduct ∆ : C n → C n ⊗ C n is given by ∆(e i ) = θ −1 i · e i ⊗ e i . This commutative Frobenius algebra A corresponds to a 1+1D TQFT, so we can compute the partition functions of the corresponding TQFT on closed surfaces. The results are listed below. • The partition function on a sphere is equal to ε•η = ε(1) = n i=1 θ i . • The partition function on a torus is equal to ε • µ • ∆ • η = ε • µ(∆(1)) = ε • µ( n i=1 θ −1 i · e i ⊗ e i ) = ε( n i=1 θ −1 i · e i ) = n i=1 1 = n, which is the dimension of A. • The partition function on a genus-g surface is equal to n i=1 θ 1−g i . Appendix C: State-sum constructions of 1+1D TQFTs In this appendix we review a state-sum construction of 1+1D TQFTs based on Frobenius algebras [54]. For simplicity, we only give the construction of partition functions on closed surfaces M . First we fix a symmetric special Frobenius algebra B = (B, µ, η, ∆, ε). We also choose a basis {e i } of B, then all linear maps µ, η, ∆, ε can be equivalently represented by structure constants in this basis. For example, we define {C k ij } by e i · e j = µ(e i ⊗ e j ) = k C k ij · e k ,(C1) and {C jk i } by ∆(e i ) = jk C jk i · e k ⊗ e j .(C2) We also define {C ijk } by ε(e i · e j · e k ) = C ijk ,(C3) and {C ijk } by ((∆ ⊗ id B ) • ∆)(1) = ijk C ijk · e k ⊗ e j ⊗ e i .(C4) These coefficients are used to construct partition functions. Given a 2-dimensional compact oriented manifold M without boundary, we choose a trivalent graph K on M . This can be done by first taking a triangulation of M and then taking the dual graph. We also need to order the graph, but in this case the order can be weaker than a branching structure: what we need is only an order of each edge, that is, a directed graph. A B-field F is an assignment to each oriented edge Figure 7). Given a B-field, we assign each vertex v a number F (v) as follows: (v α , v β ) a basis vector F (v α , v β ) ∈ {e i } of B (seeF k j i = C ijk , F k j i = C k ij , F i j k = C jk i , F i j k = C ijk . Note that the order of indices depends on the orientation of the manifold M , and in the above figures we assume the orientation is counterclockwise. Since B is symmetric, these structure constants are cyclic invariant, for example, C ijk = C jki = C kij , thus these numbers F (v) are well-defined. Then we define a number Z (K , F, B) to be the multiplication of all numbers assigned to vertices: Z(K , F, B) := v F (v). The partition function on M is obtained by summing over all B-fields, i.e. summing over all indices of basis vectors: Z = Z(K , B) := F Z(K , F, B) = F v F (v). (C5) For example, the B-field in Figure 7 gives Z = C k ij C lm k · · · , and the partition function should be Z = ijklm··· C k ij C lm k · · · . Since all indices are summing over, the partition function is independent of the choice of the basis {e i } of B. Before we prove the topological invariance of the partition function (C5), we give some examples which recover the Dijkgraaf-Witten theory. 2. The multiplication of C[G] is given by the group multiplication of G: µ(g ⊗ h) = gh. The unit of C[G] is the unit of G: η(1) = e. The comultiplication of C[G] is defined by ∆(g) = |G| −1 xy=g x ⊗ y. The counit of C[G] is defined by ε(g) = δ g,e · |G|. It is not hard to verify that C[G] is a symmetric special Frobenius algebra. We choose the canonical basis {g ∈ C[G]} g∈G , then we have C k gh = δ k gh , C hk g = |G| −1 δ kh g , C ghk = |G|δ e ghk , C ijk = |G| −2 δ khg e , where δ y x is the delta function. Given a C[G]-field F , it is clear that Z(K , F, C[G]) is nonzero only if for each vertices the multiplication of three adjacent group elements (up to orientation) is e ∈ G. On the dual triangulation this condition is nothing but the flat-connection condition. It is not hard to verify that the partition function coincides with the Dijkgraaf-Witten theory with the trivial cocycle ω = 1. Example 7. The above example can be generalized to a twisted version. Suppose ω ∈ Z 2 (G; C × ) is a 2-cocycle of the group cohomology valued in C × . Define a Frobenius algebra B = C ω [G] as follow: 1. The vector space C ω [G] is the same as C[G]. The multiplication of C ω [G] is given by µ(g ⊗ h) = ω(g, h) · gh. 3. The unit of C ω [G] is η(1) = ω(e, e) −1 · e. Recall that the cocycle condition implies that ω(g, e) = ω(e, e) = ω(e, g) for all g ∈ G. Usually we take ω to be normalized, i.e. ω(e, e) = 1. 4. The comultiplication of C ω [G] is defined by ∆(g) = |G| −1 xy=g ω(x, y) −1 · x ⊗ y. 5. The counit of C ω [G] is defined by ε(g) = δ g,e · |G| · ω(e, e). As the above example, it can be verified that the partition function Z(K , C ω [G]) coincides with the Dijkgraaf-Witten theory. Remark 8. The partition function (C5) can be written in a basis-free version. We can view each edge as a copy of B and each vertex as a linear map. For example, the vertex should be viewed as the linear map µ : B ⊗B → B. Then the whole graph should be viewed as a trace of a large linear map, or equivalently a linear map from C to C, since there is no outer legs. This trace is precisely the partition function (C5). Remark 9. It is not necessary to use a trivalent graph to do this construction. From a Frobenius algebra B we can easily construct a linear map B ⊗m → B ⊗n for any m, n. Hence this construction works for a graph whose vertices has arbitrarily many edges. Let us prove the topological invariance of the partition function (C5). The invariance under the first (2, 2)-move is precisely the associativity of the multiplication µ. Using structure constants, this condition can be written as It suffices to show that the partition function is invari C l ij C m lk = C m il C l jk . The second (2, 2)-move is precisely the Frobenius condition. Similarly, all (2, 2)-moves are related to the associativity, the coassociativity and the Frobenius condition. For the (1, 3)-moves, first we apply a (2, 2)-move to the right hand side of (1, 3)-moves: . Therefore, the invariance under the (1, 3)-moves is equivalent to the following bubble-cancellation condition: = . This condition means the Frobenius algebra B satisfies µ • ∆ = id B , i.e. the special condition. Hence, we conclude that the partition function (C5) constructed by a symmetric special Frobenius algebra B is invariant under Pachner moves, i.e. topological invariant. Also, from the above proof, one can see that to construct a topological invariant partition function we must use a symmetric special Frobenius algebra. of a non-Hermitian non-interacting Su-Schrieffer-Heeger (SSH) model of the nearest neighbor hopping of spinless electrons on a one-dimensional chain with two atoms per unit cell. (b) Schematic picture of the non-Hermitian SSH model on a ring with a twisted boundary condition and an inserted θ flux. (c) Illustration of single-particle energy spectrum of both Hermitian and quasi-Hermitian SSH model under OBC with N = 1000. The two bulk states marked by red and blue become zero energy edge states as the value of t1/t2 decreases. FIG. 2 . 2Illustration of many-body energy spectrum of ground state and first excited state of Hamiltonian (6) with twist phase θ and t1 = 0.6, t2 = 1, N = 10. (a) The non-interacting quasi-Hermitian case with U = 0 and µ = 0. (b) The interacting quasi-Hermitian case with U = 0.1 and µ = 0. In (a) and (b), the spectrum is gapped for all values of θ, and thus the topological invariant is still well defined. FIG. 3 . 32D Pachner moves (re-triangulations) with time ordering FIG. 4 . 4Illustration of many-body energy spectrum of ground state and first excited state of Hamiltonian (6) with twist phase θ and t1 = 0.6, t2 = 1, N = 10. The real and imaginary part of the many-body energy spectrum in generic non-Hermitian case with U = 0.1 and µ = 0.1 are drawn in (a) and (b) respectively. It is obvious that the real part of the gap between ground state and first excited state shown in (a) is much bigger than the imaginary part of the ground state energy shown in (b). Thus we can still have a well-defined topological invariant. FIG. 5 . 5A TQFT maps space and spacetime manifolds to Hilbert spaces and operators. Wu, Meng Cheng, Yang Qi, and Qing-Rui Wang. This work was supported by the National Key Research and Development Program of China (2016YFA0300300), the National Natural Science Foundation of China (NSFC; 11861161001), NSFC/RGC Joint Research Scheme (N-CUHK427/18), the Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20190902092905285), Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515120100 and Center for Computational Science and Engineering of Southern University of Science and Technology. Kitaev' s sMajorana chain model, which has a protected Majorana zero modes on its open ends. To see this more explicitly, let us recall the Hamiltonian of the Kitaev's Majorana chain on an ordered 1D lattice without a boundary: FIG. 6 . 6The graphic representation of the ideal Hamiltonian for the Majorana chain. (a) The graphic representation of the projector PI , all the group elements g i in this case are just a trivial identity. 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{'abstract': "Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its manybody topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems.unique Hermitian counterparts via local similarity transformation. Thus, they have the same topological invariants as their Hermitian counterparts, while the edge states can also be mapped onto those of the Hermitian case with a local similarity transformation. Physically, the topological Berry phase of quasi-Hermitian systems can be regarded as a C × -valued phase factor instead of the U(1) valued phase factor in the usual Hermitian systems under adiabatic evolution. In Section III, it is shown that for 1D quasi-Hermitian systems, the unitarity condition will naturally emerge in the fixed point topological invariant partition function and the classification of bosonic symmetry-protected topological (SPT) phases is exactly the same as their Hermtian counterparts, which are classified by second group cohomology H 2 (G, U(1) T )[30][31][32][33][34][35]. On the other hand, as 1D local fermionic systems can always be mapped to 1D local bosonic systems, the above conclusion for SPT phase also holds for 1D quasi-Hermitian interacting fermion systems. As Kitaev's Majorana chain is the only intrinsic fermionic topological phase, we will study this specific case for quasi-Hermitian systems in the Supplementary materials.Given the above, we study non-Hermitian systems with complex energy spectrum in Section IV. In this case, topological invariants are not always well defined, as the ground state is allowed to bypass an excited state without level crossing. However, it is elaborated in Section IV B and IV C that topological quantum field theory (TQFT) still captures all possible topological phases for 1D non-Hermitian quantum systems (including those non-Hermitian systems with complex eigenvalues). These phases have a one-to-one correspondence to their Hermitian counterparts. For bosonic SPT phases, arXiv:1911.01590v6 [cond-mat.str-el]", 'arxivid': '1911.01590', 'author': ['Wenjie Xi \nShenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices\nSouthern University of Science and Technology\n518055ShenzhenChina\n\nInstitute for Quantum Science and Engineering and Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina\n', 'Zhi-Hao Zhang \nInstitute for Quantum Science and Engineering and Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina\n\nSchool of Mathematical Sciences\nUniversity of Science and Technology of China\n230026HefeiChina\n', 'Zheng-Cheng Gu \nDepartment of Physics\nThe Chinese Uinversity of Hong Kong\nHong KongChina\n', 'Wei-Qiang Chen \nShenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices\nSouthern University of Science and Technology\n518055ShenzhenChina\n\nInstitute for Quantum Science and Engineering and Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina\n'], 'authoraffiliation': ['Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices\nSouthern University of Science and Technology\n518055ShenzhenChina', 'Institute for Quantum Science and Engineering and Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina', 'Institute for Quantum Science and Engineering and Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina', 'School of Mathematical Sciences\nUniversity of Science and Technology of China\n230026HefeiChina', 'Department of Physics\nThe Chinese Uinversity of Hong Kong\nHong KongChina', 'Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices\nSouthern University of Science and Technology\n518055ShenzhenChina', 'Institute for Quantum Science and Engineering and Department of Physics\nSouthern University of Science and Technology\n518055ShenzhenChina'], 'corpusid': 207853459, 'doi': '10.1016/j.scib.2021.04.027', 'github_urls': [], 'n_tokens_mistral': 22437, 'n_tokens_neox': 19344, 'n_words': 12051, 'pdfsha': 'b1234619ecf946683b9fe47c0b9155abff605939', 'pdfurls': ['https://export.arxiv.org/pdf/1911.01590v6.pdf'], 'title': ['Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity', 'Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity'], 'venue': []}

arxiv

Ensemble dependence of information-theoretic contributions to the entropy production 17 Apr 2023 Krzysztof Ptaszyński Department of Physics and Materials Science Complex Systems and Statistical Mechanics University of Luxembourg L-1511LuxembourgLuxembourg Institute of Molecular Physics Polish Academy of Sciences Mariana Smoluchowskiego 1760-179PoznańPoland Massimiliano Esposito Department of Physics and Materials Science Complex Systems and Statistical Mechanics University of Luxembourg L-1511LuxembourgLuxembourg Ensemble dependence of information-theoretic contributions to the entropy production 17 Apr 2023(Dated: April 18, 2023) The entropy production of an open system coupled to a reservoir initialized in a canonical state can be expressed as a sum of two microscopic information-theoretic contributions: the system-bath mutual information and the relative entropy measuring the displacement of the environment from equilibrium. We investigate whether this result can be generalized to situations where the reservoir is initialized in a microcanonical or in a certain pure state (e.g., an eigenstate of a nonintegrable system), such that the reduced dynamics and thermodynamics of the system are the same as for the thermal bath. We show that while in such a case the entropy production can still be expressed as a sum of the mutual information between the system and the bath and a properly redefined displacement term, the relative weight of those contributions depends on the initial state of the reservoir. In other words, different statistical ensembles for the environment predicting the same reduced dynamics for the system give rise to the same total entropy production but to different information-theoretic contributions to the entropy production.One of the main goals of statistical physics is to rationalize how time-reversal symmetric microscopic laws of classical or quantum mechanics give rise to thermodynamic irreversibility described by the second law of thermodynamics. Recent decades brought much progress in this area, presenting several complementary explanations of the emergence of irreversibility in both closed [1][2][3][4][5]and open [6-18] quantum systems. Among others, the information-theoretic framework proposed in Ref.[19]provided a microscopic basis for the nonnegativity of the entropy production -a key quantity characterizing the irreversibility of thermodynamics processes. This approach is applicable to a generic open quantum system described by the Hamiltonianwhere H S , H B and H I are Hamiltonians of the system, bath, and the interaction between them, respectively. The joint state of the system and the bath ρ SB is assumed to undergo a unitary evolution iρ SB = [H, ρ SB ] starting from the initially factorized state ρ SB (0) = ρ S (0) ⊗ ρ th B , where ρ S (0) is an arbitrary initial state of the system, and ρ th B = exp(−βH B )/Z B is the canonical Gibbs state of the environment, with β being the inverse temperature of the reservoir, and Z B = Tr exp(−βH B ) being the partition function (here and from hereon we take = k B = 1). The entropy production within the time interval [0, t] is defined aswhere ∆S S = S S (t) − S S (0) is the change of the von Neumann entropy of the system S S = −Tr(ρ S ln ρ S ) and * krzysztof.ptaszynski@uni.lufor the initial thermal state] is the heat extracted from the environment, defined as the change of the bath energy with a minus sign; the formalism can be easily generalized to the grand canonical ensemble by properly accounting for the chemical work. It was shown that the entropy production can be expressed as a sum of two nonnegative information-theoretic constituents:where I SB = S S (t) + S B (t) − S SB (t) is the quantum mutual information between the system and the bath and} is the quantum relative entropy that measures the displacement of the environment from equilibrium. According to information theory, the terms I SB and D[ρ B (t)||ρ th B ] are nonnegative, which provides a microscopic basis for the second law of thermodynamics (see Ref. [20] for an even tighter bound with finite-size corrections).As further discussed in Ref.[21], a particularly elegant interpretation of the entropy production is provided by assuming that the environment is composed of K independent degrees of freedom k (later referred to as modes), such that H B = K k=1 H k . Then Eq. (3) can be rewritten aswhere I env = k S k (t) − S B (t) is the mutual information between the modes of the environment, I tot = I SB + I env is the total correlation, i.e., a sum of systembath and intraenvironment correlations, and the term D env = k D[ρ k (t)||ρ th k ] measures the displacement of the modes of environment from equilibrium. For K → ∞ the contribution D env usually tends to be negligible, since each mode is only slightly perturbed from equilibrium arXiv:2301.13061v2 [cond-mat.stat-mech] The entropy production of an open system coupled to a reservoir initialized in a canonical state can be expressed as a sum of two microscopic information-theoretic contributions: the system-bath mutual information and the relative entropy measuring the displacement of the environment from equilibrium. We investigate whether this result can be generalized to situations where the reservoir is initialized in a microcanonical or in a certain pure state (e.g., an eigenstate of a nonintegrable system), such that the reduced dynamics and thermodynamics of the system are the same as for the thermal bath. We show that while in such a case the entropy production can still be expressed as a sum of the mutual information between the system and the bath and a properly redefined displacement term, the relative weight of those contributions depends on the initial state of the reservoir. In other words, different statistical ensembles for the environment predicting the same reduced dynamics for the system give rise to the same total entropy production but to different information-theoretic contributions to the entropy production. One of the main goals of statistical physics is to rationalize how time-reversal symmetric microscopic laws of classical or quantum mechanics give rise to thermodynamic irreversibility described by the second law of thermodynamics. Recent decades brought much progress in this area, presenting several complementary explanations of the emergence of irreversibility in both closed [1][2][3][4][5] and open [6][7][8][9][10][11][12][13][14][15][16][17][18] quantum systems. Among others, the information-theoretic framework proposed in Ref. [19] provided a microscopic basis for the nonnegativity of the entropy production -a key quantity characterizing the irreversibility of thermodynamics processes. This approach is applicable to a generic open quantum system described by the Hamiltonian H = H S + H B + H I ,(1) where H S , H B and H I are Hamiltonians of the system, bath, and the interaction between them, respectively. The joint state of the system and the bath ρ SB is assumed to undergo a unitary evolution iρ SB = [H, ρ SB ] starting from the initially factorized state ρ SB (0) = ρ S (0) ⊗ ρ th B , where ρ S (0) is an arbitrary initial state of the system, and ρ th B = exp(−βH B )/Z B is the canonical Gibbs state of the environment, with β being the inverse temperature of the reservoir, and Z B = Tr exp(−βH B ) being the partition function (here and from hereon we take = k B = 1). The entropy production within the time interval [0, t] is defined as σ = ∆S S − βQ(2) where ∆S S = S S (t) − S S (0) is the change of the von Neumann entropy of the system S S = −Tr(ρ S ln ρ S ) and * krzysztof.ptaszynski@uni.lu Q = −Tr{H B [ρ B (t) − ρ B (0)]} [with ρ B (0) = ρ th B for the initial thermal state] is the heat extracted from the environment, defined as the change of the bath energy with a minus sign; the formalism can be easily generalized to the grand canonical ensemble by properly accounting for the chemical work. It was shown that the entropy production can be expressed as a sum of two nonnegative information-theoretic constituents: σ = I SB + D[ρ B (t)||ρ th B ] ≥ 0,(3) where I SB = S S (t) + S B (t) − S SB (t) is the quantum mutual information between the system and the bath and D[ρ B (t)||ρ th B ] = Tr{ρ B (t)[ln ρ B (t) − ln ρ th B ] } is the quantum relative entropy that measures the displacement of the environment from equilibrium. According to information theory, the terms I SB and D[ρ B (t)||ρ th B ] are nonnegative, which provides a microscopic basis for the second law of thermodynamics (see Ref. [20] for an even tighter bound with finite-size corrections). As further discussed in Ref. [21], a particularly elegant interpretation of the entropy production is provided by assuming that the environment is composed of K independent degrees of freedom k (later referred to as modes), such that H B = K k=1 H k . Then Eq. (3) can be rewritten as σ = I SB + I env + D env = I tot + D env ≥ 0,(4) where I env = k S k (t) − S B (t) is the mutual information between the modes of the environment, I tot = I SB + I env is the total correlation, i.e., a sum of systembath and intraenvironment correlations, and the term D env = k D[ρ k (t)||ρ th k ] measures the displacement of the modes of environment from equilibrium. For K → ∞ the contribution D env usually tends to be negligible, since each mode is only slightly perturbed from equilibrium (though deviations from this behavior are possible when only a small portion of the environment is resonantly excited [22]), and thus the entropy production can be related to the generation of multipartite correlations between the system and the modes of environment I tot . In deriving Eq. (3) one assumes that the initial state of the environment is the canonical Gibbs state. However, it has been shown that certain non-thermal initial states of the bath may lead (under certain conditions) to the same reduced dynamics and thermodynamics of the system as the thermal state; later, such property will be referred to as the dynamical equivalence to the canonical state. A first example studied in the literature was the microcanonical state [23][24][25][26], namely, an equally-weighted mixture of energy eigenstates of the bath with energies within the interval [E − δ, E + δ], where 2δ is the width of the microcanonical shell. The dynamical equivalence is there a consequence of a well-known principle of equilibrium ensemble equivalence [27], which states that in the thermodynamic limit the microcanonical and canonical states are equivalent with respect to their thermodynamic properties and expected values of observables (see Ref. [28] for a contemporary formulation of this concept). While the dynamical equivalence can be formally proven [24][25][26], here we will present only a qualitative justification. Usually, the system does not interact in a uniform way with the whole environment, but rather is more strongly coupled to some of its (possibly small) parts; for example, a system coupled to a bath of harmonic oscillators (the Caldeira-Leggett model [29]) will most strongly couple to those oscillators whose resonant frequencies are close to transition frequencies of the system. At the same time, as implied by the principle of ensemble equivalence, when the whole environment is initialized in the microcanonical state, a reduced state of its small part (effectively coupled to the system) is the canonical state. As a consequence, the system evolves as if it was coupled to the thermal bath. Furthermore, there exist also several types of pure states which reproduce equilibrium properties of the thermal state, and thus may be expected to be also dynamically equivalent. First, an equivalence of static observables can be observed even for single eigenstates of nonintegrable systems obeying the eigenstate thermalization hypothesis (ETH) [30][31][32]; in such a case the dynamical equivalence, namely, the applicability of the second law of thermodynamics and the nonequilibrium fluctuation theorem, has been recently demonstrated [26,33,34]. Other examples are typical superpositions of states from the microcanonical shell [35][36][37][38], thermofield double states [39] (purifications of a thermal state in a doubled Hilbert space), and so-called thermal pure states [40][41][42] (coherent superpositions of energy eigenstates with populations obeying the Boltzmann distribution). As a matter of fact, as shown by Popescu et al. [43], almost every pure state of the environment leads to relaxation of the system to the canonical state. This raises the question of whether the information-theoretic formulation of the entropy production can be generalized to a generic initial state of the bath dynamically equivalent to the thermal state. In this Letter we note that for arbitrary initial state of the bath (initially uncorrelated with the system) Eq. (3) can be generalized to a form σ = I SB + ∆D(ρ B ||ρ th B ),(5) where ∆D(ρ B ||ρ th B ) = D[ρ B (t)||ρ th B ] − D ρ B (0)||ρ th B measures the change of displacement of the bath state from equilibrium. This can be easily derived as follows. First, by expanding D[ρ B (t)||ρ th B ] = Tr[ρ B (t) ln ρ B (t)] − Tr[ρ B (t) ln ρ th B ] = −S B (t) + ln Z B + βTr[ρ B (t)H B ], one gets ∆D(ρ B ||ρ th B ) = −∆S B − βQ, where ∆S B = S B (t) − S B (0) , and the heat Q is defined below Eq. (2). Second, one uses the assumption that the system and the bath are initially uncorrelated, which implies vanishing of the initial mutual information: I SB (0) = S S (0) + S B (0) − S SB (0) = 0; as the unitary dynamics conserves the joint von Neumann entropy of the system and the bath [S SB (t) = S SB (0)] this implies I SB = S S (t) + S B (t) − S SB (t)−[S S (0)+S B (0)−S SB (0)] = ∆S S +∆S B . Insert- ing ∆D(ρ B ||ρ th B ) = −∆S B − βQ and I SB = ∆S S + ∆S B into the right hand side of Eq. (5) one gets ∆S S − βQ, which is the entropy production [Eq. (2)]. It can be now noted that in general ∆D(ρ B ||ρ th B ) is not necessarily nonnegative, and therefore Eq. (5) does not provide a basis for the second law of thermodynamics for non-thermal initial states of the bath; nevertheless, it still enables one to express the entropy production in terms of microscopic, information-theoretic contributions, while its nonnegativity can be provided by the dynamical equivalence with the thermal state. However, we will show that while the relation (5) always holds, the relative weight of the terms I SB and ∆D(ρ B ||ρ th B ) may depend on the initial state of the bath; therefore, dynamically equivalent states are only partially equivalent from the perspective of information-theoretic formulation of the entropy production. For environments composed of independent modes in an arbitrary initial state, Eq. (4) can be generalized as σ = I SB + ∆I env + ∆D env = ∆I tot + ∆D env .(6) This can be derived as follows: first, in analogy to the derivation below Eq. (5), one gets D env = k D[ρ k (t)||ρ th k ] = − k ∆S k − βQ, where ∆S k = S k (t) − S k (0) ; then using ∆I env = k ∆S k − ∆S B and I SB = ∆S S + ∆S B , one finds that the right hand side of Eq. (6) is equal to σ = ∆S S − βQ. It may be now argued that for initial states of the bath dynamically equivalent to the thermal state, the initial states of the modes are thermal (in the thermodynamic limit of large K), and that their reduced states evolve in the same way. As a consequence, the contributions ∆D env and ∆I tot -which depend on the local states of the modes rather than the total state of the bath ρ B -should also be the same. This will be demonstrated later by numerical simulations. Example 1: Random matrix Hamiltonian. Let us now investigate, for two exemplary cases, how the behavior of information-theoretic constituents of the entropy production depends on the initial state of the reservoir. To this goal, we perform simulations of the unitary dynamics of the system-bath ensemble for a system coupled to a finite environment. First, we will consider a nonintegrable system obeying the eigenstate thermalization hypothesis, defined by means of random matrices; similar setups have been previously investigated in Refs. [19,23]. The Hamiltonian of the system is defined as H S = γσ z /2, H B = X B / √ 8N , and H I = λσ x ⊗ X I / √ 8N , where X i (i ∈ {B, I}) is a Gaussian orthogonal random matrix of size N with variance of the diagonal elements equal to 1, and σ i (i ∈ {x, y, z}) are Pauli matrices. As initial states of the bath we took the canonical state, two microcanonical states with different widths of the microcanonical shell 2δ, and a single eigenstate of H B with energy closest to the average energy of the thermal state E th B = Tr(H B ρ th B ); the microcanonical state is defined as a mixture of eigen- states of H B with energies E i ∈ [ E th B − δ, E th B + δ]. In our numerical simulations the random matrices X B and X I were generated using the function RandomVariate[GaussianOrthogonalMatrixDistribution [N]] in Mathematica. The joint system-bath state was propagated iteratively as ρ SB (t + ∆t) = e −iH∆t ρ SB (t)e iH∆t ; we used the time step ∆t = 25 and the evolution operator e −iH∆t was calculated using the function MatrixExp in Mathematica. The results are presented in Fig. 1. As one can observe, all initial states of the bath generate approximately the same evolution of the entropy production σ, which confirms their dynamical equivalence. However, the information-theoretic constituents of the entropy production are not the same for different ensembles. In particular, I SB is smallest for the initial canonical state, intermediate for microcanonical states and largest for the pure state; furthermore, its value depends also on the width of the microcanonical shell -it is larger for smaller widths. We note here a similarity to the previous study of a pure dephasing, showing that different types of the global system-environment dynamics may lead to the same reduced dynamics of the system but generate different system-bath correlations [44]. Accordingly, also the relative entropy term ∆D(ρ B ||ρ th B ) is different for various initial states. Interestingly, it may also become negative, which implies that the environment is brought closer to the canonical thermal state during the thermalization process; as a matter of fact, when both the system and the bath are initialized in pure states, then I SB = 2∆S S , and thus ∆D(ρ B ||ρ th B ) has to be negative for −βQ < ∆S S . A qualitative interpretation of this observation is well illustrated by the next model. Example 2: Noninteracting resonant level. As a second case, we considered a system with the Hamiltonian of the environment which can be decomposed into a sum of independent modes (H B = k H k ), such that Eq. (6) is applicable. Specifically, we focused on the noninteracting resonant level model with H S = d c † d c d , H B = K k=1 k c † k c k , and H I = Ω K k=1 c † d c k + c † k c d , where c † i (c i ) are creation (annihilation) operators, i are level energies and Ω is the tunnel coupling. The levels of the environment have been taken to be evenly distributed throughout the interval [−Λ/2, Λ/2], while the tunnel coupling has been parameterized as Γ = 2πΩ 2 (K − 1)/Λ, where Γ is the coupling strength. Parameters have been set as Λ = 3Γ, d = −0.5Γ, and K = 7. he matrix form of the Hamiltonian was obtained using the Jordan-Wigner transformation of the creation and annihilation operators into spin operators: c † k = [ k−1 i=0 (−σ z )] ⊗ σ + ⊗ [ K i=k+1 1 2 ] and c k = [ k−1 i=0 (−σ z )] ⊗ σ − ⊗ [ K i=k+1 1 2 ], where σ ± = (σ x ± iσ y )/2, and 1 2 is 2 × 2 identity matrix. The state ρ SB (t) was propagated using the same method as before, with the time step ∆t = 0.05Γ −1 . We considered three types of initial states. The first is the microcanonical state, here defined as an equallyweighted mixture of 12 eigenstates of the bath with energy E = −1.5Γ (more precisely, since we fix the energy while enabling the particle number to vary, this may be rather referred to as the grand microcanonical [45] or Maxwell's demon ensemble [46]). Since the system is integrable, and thus ETH is not applicable, the initial pure state is here defined as a superposition of all states from the microcanonical shell: |Ψ = W −1/2 i |ψ i ; such states provide the dynamical equivalence due to canonical typicality [35][36][37]. Here we took the same amplitude W −1/2 for all eigenstates; as shown in the Supplemental Material [47], a good convergence is also observed for randomly chosen amplitudes. Finally, as a third initial state we took the canonical state (or rather, the grand canonical state with the chemical potential µ = 0) with the temperature β ≈ 0.969Γ given by the condition Tr(H B ρ th B ) = E = −1.5Γ. The results are presented in Figs. 2 and 3. As shown in Fig. 2, as argued before, not only the entropy production σ, but also its information-theoretic constituents ∆I tot and ∆D env , are approximately similar for all initial states (to be more precise, the analyzed quantities are exactly the same for microcanonical and pure states; this is because they are fully determined by two-point correlations c † i c j , which for quadratic Hamiltonians evolve independently from higher-order correlations [48], and thus are the same in both cases). Furthermore, ∆I tot is the dominant contribution to the production of entropy, as the perturbation of the modes is relatively small (though there is still a non-vanishing contribution ∆D env due to the finite size of the bath). Therefore, there exists a partial ensemble equivalence regarding the microscopic nature of the entropy production: for all initial states it is mainly related to the change of total correlations between the system and the modes of environment. However, as shown in Fig. 3, the decomposition of ∆I tot into the system-bath and intrabath correlations is not equivalent for different initial states. For the microcanonical and pure states ∆I env is negative which implies that intrabath correlations are destroyed rather than created. This provides a qualitative interpretation for the negativity of ∆D(ρ B ||ρ th B ), which has been observed also for the random matrix model: it is related to the destruction of initial non-thermal correlations in the bath. Conclusions. In summary, we have shown that different initial states of the bath leading to the same reduced dynamics and thermodynamics on the system may be only partially equivalent with respect to the informationtheoretic formulation of the entropy production: while in all cases the entropy production can be expressed as a sum of two information-theoretic contributions I SB and ∆D(ρ B ||ρ th B ), the relative weight of those terms may depend on the initial state. A particular instance of equivalence of the information-theoretic contributions can be observed for environments composed of independent modes: for all ensembles of the bath the entropy production is mostly related to the change of total cor- relation between the system and the modes of environment ∆I tot . This supports a general idea relating the entropy production to the generation of multipartite correlations [19,21]. However, again, the decomposition of ∆I tot into the change of system-bath and intrabath correlations may vary for different initial states of the environment. It is important to note that the difference between various initial states can be significant only when the contributions I SB and ∆D(ρ B ||ρ th B ) are of a similar order of magnitude. As discussed by us previously [21], this can be only true when the entropy production is sufficiently small, since the system-bath mutual information is bounded from above by the Araki-Lieb inequality I SB ≤ 2 ln dimH S [49], where dimH S is the dimension of the Hilbert space of the system. As a result, for σ significantly exceeding 2 ln dimH S the entropy pro- duction becomes dominated by the displacement term [σ ≈ ∆D(ρ B ||ρ th B )], independently of the initial state of the bath. Thus, dynamically equivalent states of the bath become also informationally equivalent in the limit of large entropy production. This resembles the standard concept of ensemble equivalence which states that different ensembles become equivalent in the thermodynamic limit of a large system. In addition to thermodynamics, the present study may be relevant for the field of condensed matter physics, where correlations between quantum impurities and their environment have gained a certain amount of attention [50,51]. Our results suggest that such correlations may depend on the choice of the ensemble. In particular, beyond the cases analyzed in our paper, it is worth investigating whether there is a difference between the grandcanonical ensemble and the canonical ensemble with a fixed particle number, which may be more physically justified, e.g., in the description of impurities coupled to trapped ultracold atoms [52]. In the main text we investigated the behavior of entropy production and its information-theoretic constituents in the noninteracting resonant level model for a pure initial state of the bath defined as an equallyweighted superposition of eigenstates from the microcanonical shell. However, as implied by the principle of canonical typicality [1], thermalization should take place (in the thermodynamic limit) also for a typical randomly chosen superposition of eigenstates; therefore, an approximate convergence should be observable also for finite baths. Here we study the evolution of analyzed quantities for 15 random initial states defined as |Ψ = j α j |ψ j , where |ψ j are eigenstates from the microcanonical shell and α j are random probability amplitudes defined as α j = r j /Re 2πiφj , where r j and φ j are chosen from a uniform distribution over the interval [0, 1], and R = j r j . The results are presented in Figs. S1 and S2. As one can observe, the calculated entropy production and its constituents are spread-out around the results obtained for the thermal state [ Fig. S1] or the equally-weighted su- FIG. 1 . 1Entropy production σ (a) and its informationtheoretic constituents ISB (b) and ∆D(ρB||ρ th B ) (c) as a function of time for the random matrix system with different initial states of the bath. Results for the initial excited state of the system, β = 10, γ = λ = 0.1, and N = 2000. FIG. 2 . 2Entropy production (a), change of the total correlation ∆Itot (b), and the change of mode perturbation from equilibrium ∆Denv (c) as a function of time for the noninteracting resonant level with different initial states of the environment: the canonical state (black solid line), the microcanonical state (red dashed line), and the pure state (blue dotted line). FIG. 3 . 3System-bath (a) and intrabath (b) mutual information as a function of time for the noninteracting resonant level with different initial states of the environment; designations as inFig. 2. FIG. S1 . S1Entropy production and its constituents ∆Itot and ∆Denv for 15 random initial pure states of the bath (green solid lines) compared with the trajectories obtained for the thermal state (black dashed line). Parameters as in the main text. * krzysztof.ptaszynski@uni.lu FIG. S2. The system-bath mutual information ISB and the change of intraenvironment correlation ∆Ienv for 15 random initial pure states of the bath (green solid lines) compared with the trajectories obtained for the equally-weighted superpositions of eigenstates from the microcanonical shell (black dashed line). Parameters as in the main text. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. A Polkovnikov, K Sengupta, A Silva, M Vengalattore, Rev. Mod. Phys. 83863A. Polkovnikov, K. 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D Kim, J Shim, H.-S Sim, Phys. Rev. Lett. 127226801D. Kim, J. Shim, and H.-S. Sim, Universal Thermal En- tanglement of Multichannel Kondo Effects, Phys. Rev. Lett. 127, 226801 (2021). Realizing a Kondo-Correlated State with Ultracold Atoms. J Bauer, C Salomon, E Demler, Phys. Rev. Lett. 111215304J. Bauer, C. Salomon, and E. Demler, Realizing a Kondo- Correlated State with Ultracold Atoms, Phys. Rev. Lett. 111, 215304 (2013). Interestingly, the obtained trajectories of the entropy production seem to be more concentrated around the thermal value than those of its constituents ∆I tot and ∆D env. perposition of eigenstates [Fig. S2. This shows that the precision of simulating the open system dynamics using pure states of the bath (e.g., within the framework of dynamical typicality [2]) may depend on the analyzed quantityperposition of eigenstates [Fig. S2]. Interestingly, the ob- tained trajectories of the entropy production seem to be more concentrated around the thermal value than those of its constituents ∆I tot and ∆D env . This shows that the precision of simulating the open system dynamics using pure states of the bath (e.g., within the framework of dynamical typicality [2]) may depend on the analyzed quantity. . S Goldstein, J L Lebowitz, R Tumulka, N Zanghì, Canonical Typicality, Phys. Rev. Lett. 9650403S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Canonical Typicality, Phys. Rev. Lett. 96, 050403 (2006). Selected applications of typicality to real-time dynamics of quantum many-body systems. T Heitmann, J Richter, D Schubert, R Steinigeweg, Z. Naturforsch. A. 75421T. Heitmann, J. Richter, D. Schubert, and R. Steinigeweg, Selected applications of typicality to real-time dynamics of quantum many-body systems, Z. Naturforsch. A 75, 421 (2020).

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{'abstract': 'The entropy production of an open system coupled to a reservoir initialized in a canonical state can be expressed as a sum of two microscopic information-theoretic contributions: the system-bath mutual information and the relative entropy measuring the displacement of the environment from equilibrium. We investigate whether this result can be generalized to situations where the reservoir is initialized in a microcanonical or in a certain pure state (e.g., an eigenstate of a nonintegrable system), such that the reduced dynamics and thermodynamics of the system are the same as for the thermal bath. We show that while in such a case the entropy production can still be expressed as a sum of the mutual information between the system and the bath and a properly redefined displacement term, the relative weight of those contributions depends on the initial state of the reservoir. In other words, different statistical ensembles for the environment predicting the same reduced dynamics for the system give rise to the same total entropy production but to different information-theoretic contributions to the entropy production.One of the main goals of statistical physics is to rationalize how time-reversal symmetric microscopic laws of classical or quantum mechanics give rise to thermodynamic irreversibility described by the second law of thermodynamics. Recent decades brought much progress in this area, presenting several complementary explanations of the emergence of irreversibility in both closed [1][2][3][4][5]and open [6-18] quantum systems. Among others, the information-theoretic framework proposed in Ref.[19]provided a microscopic basis for the nonnegativity of the entropy production -a key quantity characterizing the irreversibility of thermodynamics processes. This approach is applicable to a generic open quantum system described by the Hamiltonianwhere H S , H B and H I are Hamiltonians of the system, bath, and the interaction between them, respectively. The joint state of the system and the bath ρ SB is assumed to undergo a unitary evolution iρ SB = [H, ρ SB ] starting from the initially factorized state ρ SB (0) = ρ S (0) ⊗ ρ th B , where ρ S (0) is an arbitrary initial state of the system, and ρ th B = exp(−βH B )/Z B is the canonical Gibbs state of the environment, with β being the inverse temperature of the reservoir, and Z B = Tr exp(−βH B ) being the partition function (here and from hereon we take = k B = 1). The entropy production within the time interval [0, t] is defined aswhere ∆S S = S S (t) − S S (0) is the change of the von Neumann entropy of the system S S = −Tr(ρ S ln ρ S ) and * krzysztof.ptaszynski@uni.lufor the initial thermal state] is the heat extracted from the environment, defined as the change of the bath energy with a minus sign; the formalism can be easily generalized to the grand canonical ensemble by properly accounting for the chemical work. It was shown that the entropy production can be expressed as a sum of two nonnegative information-theoretic constituents:where I SB = S S (t) + S B (t) − S SB (t) is the quantum mutual information between the system and the bath and} is the quantum relative entropy that measures the displacement of the environment from equilibrium. According to information theory, the terms I SB and D[ρ B (t)||ρ th B ] are nonnegative, which provides a microscopic basis for the second law of thermodynamics (see Ref. [20] for an even tighter bound with finite-size corrections).As further discussed in Ref.[21], a particularly elegant interpretation of the entropy production is provided by assuming that the environment is composed of K independent degrees of freedom k (later referred to as modes), such that H B = K k=1 H k . Then Eq. (3) can be rewritten aswhere I env = k S k (t) − S B (t) is the mutual information between the modes of the environment, I tot = I SB + I env is the total correlation, i.e., a sum of systembath and intraenvironment correlations, and the term D env = k D[ρ k (t)||ρ th k ] measures the displacement of the modes of environment from equilibrium. For K → ∞ the contribution D env usually tends to be negligible, since each mode is only slightly perturbed from equilibrium arXiv:2301.13061v2 [cond-mat.stat-mech]', 'arxivid': '2301.13061', 'author': ['Krzysztof Ptaszyński \nDepartment of Physics and Materials Science\nComplex Systems and Statistical Mechanics\nUniversity of Luxembourg\nL-1511LuxembourgLuxembourg\n\nInstitute of Molecular Physics\nPolish Academy of Sciences\nMariana Smoluchowskiego 1760-179PoznańPoland\n', 'Massimiliano Esposito \nDepartment of Physics and Materials Science\nComplex Systems and Statistical Mechanics\nUniversity of Luxembourg\nL-1511LuxembourgLuxembourg\n'], 'authoraffiliation': ['Department of Physics and Materials Science\nComplex Systems and Statistical Mechanics\nUniversity of Luxembourg\nL-1511LuxembourgLuxembourg', 'Institute of Molecular Physics\nPolish Academy of Sciences\nMariana Smoluchowskiego 1760-179PoznańPoland', 'Department of Physics and Materials Science\nComplex Systems and Statistical Mechanics\nUniversity of Luxembourg\nL-1511LuxembourgLuxembourg'], 'corpusid': 256390082, 'doi': '10.1103/physreve.107.l052102', 'github_urls': [], 'n_tokens_mistral': 12550, 'n_tokens_neox': 10952, 'n_words': 6774, 'pdfsha': '930afdd68ef08da938dbd6aaa674b95e3583ee09', 'pdfurls': ['https://export.arxiv.org/pdf/2301.13061v2.pdf'], 'title': ['Ensemble dependence of information-theoretic contributions to the entropy production', 'Ensemble dependence of information-theoretic contributions to the entropy production'], 'venue': []}

arxiv

3 Sep 2014 Alexis Bouthier 3 Sep 2014 Cet article établit une formule de dimension pour les fibres de Springer affines dans le cas des groupes. On suit la méthode initiée par Bezrukavnikov dans le cas des algèbres de Lie. Elle consiste en l'introduction d'un ouvert régulier suffisament gros dont on montre qu'il est de même dimension que la fibre de Springer affine entière. On montre que dans le cas des groupes, un tel ouvert régulier avec des propriétés analogues, existe. Sa construction passe par l'introduction du semi-groupe de Vinberg V G pour lequel nous étudions un morphisme 'polynôme caractéristique' et étendons les résultats précedemment établis par Steinberg pour les groupes.3Finally, in the last section, we show the theorem 1. Following Kazhdan-Lusztig, we need to study the equidimensionnality of a corresponding flag variety. It implies to study more deeply the nilpotents and quasi-unipotents elements of the Vinberg's semigroup. Once this result is obtained, it is sufficient to deduce the dimension of the regular open subset which have the same dimension of the whole Springer fiber and conclude about the dimension formula.Proposition 1.14. Pour un sous-ensemble J ⊂ ∆, soit l'orbite O J , on considère les sous-groupes paraboliques P J et P − J . On a une décomposition de Lévi P J = L J R u (P J ) et on note δ (resp δ − ) la projection de P J sur L J (resp P − J sur L J ). L'orbite O J a un idempotent distingué e J . Son stabilisateur H J s'identifie à :H J = {(x, y) ∈ P J × P − J | δ(x)δ − (y) −1 ∈ T J,∆ }, où T J,∆ = {t ∈ T ∆ | α j (t) = 1, j ∈ J}. Abstract : This article establishes a dimension formula for a group version of affine Springer fibers. We follow the method initiated by Bezrukavnikov in the case of Lie algebras. It consists in the introduction of a big enough regular open subset, with the same dimension as the affine Springer fiber. We show that, in the case of groups, such a regular open subset with analogous properties exists. Its construction needs the introduction of the Vinberg semi-group V G for which we study an adjoint quotient χ + and extend for χ + the results previously established by Steinberg. Introduction in English Let k be an algebraically closed field. We consider G a connected algebraic group, semisimple, simply connected over k. Let T be a maximal torus of G and g the Lie algebra of G. We note F = k((π)) and O := k [[π]]. Kazhdan and Lusztig have introduced in [19] the affine Springer fibers for Lie algebras. There are varieties of the form X γ = {g ∈ G(F )/G(O)| ad(g) −1 γ ∈ g(O)} where γ ∈ g(F ). They establish that they are k-schemes locally of finite type and of finite dimension if γ is regular semisimple. They also conjecture a dimension formula for these varieties which was later proved by Bezrukavnikov [1]. If we name by g γ , the centralizer of γ in g, the formula is the following : dim X γ = 1 2 [δ ′ (γ) − def(γ)] where δ ′ (γ) = val(det(ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))) and def(γ) = rg g − rg F (g γ (F )). The first term is the discriminant invariant and the second one is a Galois invariant which mesures the drop of torus rank. In this work, we are interested in the affine Springer fibers for groups : X λ γ = {g ∈ G(F )/G(O)| g −1 γg ∈ G(O)π λ G(O)} with λ ∈ X * (T ) + a dominant cocharacter and γ ∈ G(F ). These varieties were introduced by Kottwitz-Viehmann [21] in their article on generalized Springer fibers. If λ = 0, we have the variety X 0 γ = {g ∈ G(F )/G(O)| g −1 γg ∈ G(O)} and the dimension formula and the proof of Bezrukavnikov are the same. For a general λ, we prove : Theorem 1. Let γ ∈ G(F ) be a regular semisimple element. Then : (i) X λ γ is a k-scheme locally of finite type. (ii) Si X λ γ non vide, dim X λ γ = ρ, λ + 1 2 [δ(γ) − def(γ)], where δ(γ) = val(det(Id − ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))). An analog dimension formula for affine Deligne-Lusztig was already established by [12] and [43]. The proof of Bezrukavnikov uses crucially a distinguished open subset, called the regular open subset. It consists in the elements g ∈ X γ such that ad(g) −1 (γ) is regular when we reduce mod π. In the case of groups, there is a double difficulty coming from the fact that the condition g −1 γg ∈ G(O)π λ G(O) is non-linear and that we cannot give a sense to the reduction modulo π. To linearize the problem, one way to proceed, is to consider a faithful representation ρ : G → End(V ) such that ρ(G(O)π λ G(O)) ⊂ π −N End(V )⊗ k O, for N ∈ N. We note that, if we add a central factor to G, which acts by multiplication by π in End(V ) ⊗ k O, we obtain a similar integrality condition, as in the Lie algebra case. To do that in a uniform way for all groups, there exists a natural envelop, called the Vinberg's semi-group V G , introduced by Vinberg [44] in characteristic zero and Rittatore [36] in arbitrary characteristics. This formulation allows us to define in this context a regular open subset. In the case of Lie algebras, the regular open subset is a torsor under the affine grassmannian of the regular centralizer of γ. Following Ngô [28], the existence of such a regular centralizer comes from the existence of a commutative group scheme J, smooth on the adjoint quotient t/W and from the existence of a map χ * J → I, where χ is the Chevalley morphism χ : g → t/W and I the scheme of centralizers over g. Moreover, the morphism χ * J → I is an isomorphism over g reg . One way to obtain the group scheme J is to construct a section to χ, called the Kostant section, and to pullback I by this section. In our case, we try to obtain a Chevalley type morphism for the Vinberg's semigroup V G . It is an algebraic monoïd, i.e. a semigroup with unity, with unit group G + := (T × G)/Z G which is open dense. In particular, V G is a partial compactification of G + , affine and which contains the toric variety V T , the closure of T + := (T × T )/Z G in V G . By Steinberg [41], we have a morphism χ + : G + → T + /W and a section to this morphism (the simply connectedness assumption is necessary in order to get a section). We obtain the following theorem : Theorem 2. The Steinberg's morphism extends to a map, χ + : V G → V T /W , invariant by conjugation by G + . The morphism χ + admits a section ǫ + : V T /W → V reg G , in the regular locus. The existence of this section allows us to construct a regular centralizer J by pulling-back the scheme of centralizers I by ǫ + . To obtain that J is commutative and smooth, we need more properties of the morphism χ reg + and in particular its smoothness. Theorem 3. The morphism χ reg + : V reg G → V T /W is smooth and its geometric fibers are G-orbits. There exists a unique commutative group scheme J, smooth over V T /W with a map χ * + J → I, which is an isomorphism over V reg G . Let us now consider the organization of the paper. It splits in two parts, the first one concerns the proof of the theorems 2 and 3, which are results of group theory and the second part deals with the computation of the dimension of Springer fibers. In the first section, we prove the theorem 2. We introduce the Vinberg's semigroup and the quotient by adjoint action χ + . Over the group of units G + of V G , we have a section, or more exactly a family of sections constructed by Steinberg, for which we show that they extend to the Vinberg's semigroup. This allows us to construct the regular centralizer J. In the second section, we obtain the properties of the theorem 3 on the morphism χ + and the centralizer J. By using the action of the central torus Z + of G + , we can reduce the study over the point zero, which is the nilpotent cone. The properties established for the most singular fiber then spread to the other fibers. In the third section, we introduce the affine Springer fibers for groups and we make the link with the Vinberg's semigroup via the modular interpretation. These Springer fibers admit a distinguished open locus, named regular, which is an orbit under a Picards stack, coming from the regular centralizer. Introduction Soit k un corps algébriquement clos. On considère G un groupe algébrique, connexe, semisimple, simplement connexe sur k. Soit T un tore maximal de G et g l'algèbre de Lie de G. On pose F := k((π)) et O := k[[π]]. Kazhdan et Lusztig ont introduit dans [19] les fibres de Springer affines pour les algèbres de Lie. Ce sont les variétés de la forme X γ = {g ∈ G(F )/G(O)| ad(g) −1 γ ∈ g(O)} où γ ∈ g(F ). Ils établissent que ce sont des k-schémas localement de type fini et de dimension finie si γ est régulier semi-simple. Ils conjecturent également pour ces variétés une formule de dimension qui sera démontrée par Bezrukavnikov [1]. Si l'on désigne par g γ , le centralisateur de γ dans g, la formule est la suivante : dim X γ = 1 2 [δ ′ (γ) − def(γ)] où δ ′ (γ) = val(det(ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))) et def(γ) = rg g − rg F (g γ (F )). Le premier est l'invariant discriminant et le second l'invariant galoisien qui mesure la chute du rang torique. Dans ce travail, on s'intéresse aux fibres de Springer affines pour les groupes : X λ γ = {g ∈ G(F )/G(O)| g −1 γg ∈ G(O)π λ G(O)} avec λ ∈ X * (T ) + un cocaractère dominant et γ ∈ G(F ). Ces variétés ont été introduites par Kottwitz-Viehmann [21] dans leur article sur les fibres de Springer généralisées. Dans le cas λ = 0, nous avons la variété : X 0 γ = {g ∈ G(F )/G(O)| g −1 γg ∈ G(O) } et la formule de dimension ainsi que la preuve de Bezrukavnikov sont les mêmes. Pour λ général, on démontre : Théorème 0.1. Soit γ ∈ G(F ) régulier semi-simple. Alors : (i) X λ γ est un schéma localement de type fini. (ii) Si X λ γ non vide, dim X λ γ = ρ, λ + 1 2 [δ(γ) − def(γ)], où δ(γ) = val(det(Id − ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))). Une formule de dimension analogue pour les variétés de Deligne-Lusztig affines a déjà été établie par [12] et [43]. La preuve de Bezrukavnikov utilise de manière cruciale un ouvert distingué appelé l'ouvert régulier. Il consiste en les éléments g ∈ X γ tels que ad(g) −1 (γ) soit régulier en réduction modulo π. Dans le cas des groupes, nous nous trouvons confrontés à la double difficulté que la condition g −1 γg ∈ G(O)π λ G(O) est non linéaire et que nous ne pouvons donner un sens à la réduction modulo π. Pour linéariser le problème, une manière possible de procéder, est de considérer une réprésentation fidèle ρ : G → End(V ) de telle sorte que ρ(G(O)π λ G(O)) ⊂ π −N End(V ) ⊗ k O, pour N ∈ N. On remarque alors que, quitte à ajouter un facteur central à G, qui agit par multiplication par π dans End(V ), on peut obtenir la condition d'intégralité souhaitée. Pour faire cela de manière uniforme pour tous les groupes, il existe une enveloppe naturelle, appelée le semi-groupe de Vinberg V G , introduit par Vinberg [44] en caractéristique nulle et étendu par Rittatore [36] en caractéristique arbitraire. Cette reformulation nous permet donc de pouvoir définir dans ce contexte un ouvert régulier. Dans le cas des algèbres de Lie, l'ouvert régulier est un torseur sous la grassmannienne affine du centralisateur régulier de γ. D'après Ngô [28], l'existence de ce centralisateur régulier vient de l'existence d'un schéma en groupes commutatifs J lisse sur le quotient adjoint pour l'algèbre de Lie t/W ainsi que de l'existence d'une flèche χ * J → I, où χ est le morphisme de Chevalley g → t/W et I le schéma des centralisateurs au-dessus de g. En outre, la flèche χ * J → I est un isomorphisme au-dessus de g reg . Une des façons d'obtenir un tel schéma J est de construire une section, dite de Kostant, au morphisme de Chevalley et de tirer I par cette section. Dans le cas qui nous concerne, on cherche donc à obtenir un morphisme de type Chevalley pour le semi-groupe de Vinberg V G . C'est un monoïde algébrique, i.e. un semigroupe avec unité, dont le groupe des inversibles G + := (T × G)/Z G est un ouvert dense. En particulier, le semi-groupe de Vinberg est une compactification partielle de G + , affine et qui contient également la variété torique V T , adhérence du tore maximal T + := (T × T )/Z G de G + . On dispose grâce à Steinberg [41] d'une flèche χ + : G + → T + /W ainsi que d'une section à cette flèche (l'hypothèse de simple connexité étant ici indispensable pour l'existence d'une section). Théorème 0.2. Le morphisme de Steinberg se prolonge en une flèche, χ + : V G → V T /W , invariant par conjugaison par G + . Le morphisme χ + admet une section ǫ + : V T /W → V reg G , à valeurs dans l'ouvert régulier. L'existence de cette section nous permet donc de construire le centralisateur régulier J en tirant le schéma des centralisateurs I par ǫ + . Pour obtenir que J est un schéma en groupes lisse et commutatif, nous avons besoin d'établir des propriétés sur le morphisme χ reg + et en particulier sa lissité. Cela fait l'objet de l'énoncé suivant : Théorème 0.3. Le morphisme χ reg + : V reg G → V T /W est lisse et ses fibres géométriques sont des G-orbites. Enfin, il existe un unique schéma en groupes commutatif J, lisse sur V T /W , muni d'une flèche χ * + J → I qui est un isomorphisme au-dessus de V reg G . Nous passons maintenant en revue l'organisation de l'article. Il se décompose en deux parties, la première concerne la preuve des théorèmes 0.2 et 0.3, qui sont des résultats de théorie des groupes et la deuxième partie concerne le calcul de dimension des fibres de Springer. Dans la première section, on démontre le théorème 0.2. Nous introduisons le semi-groupe de Vinberg ainsi que le quotient par l'action adjointe χ + . Au-dessus de l'ouvert des inversibles G + de V G , on dispose d'une section, ou plus exactement d'une famille, construite par Steinberg, dont on montre qu'elle se prolonge au semi-groupe de Vinberg. Cela nous permet alors de pouvoir construire le centralisateur régulier J. Dans la deuxième section, on obtient les propriétés énoncées dans le théorème 0.3 sur le morphisme χ + et le centralisateur J. En utilisant l'action du tore central Z + de G + , on se ramène à l'étude au-dessus du point zéro, qui s'identifie au cône nilpotent. Les propriétés que l'on démontre pour la fibre la plus 'singulière' de χ + se propagent ensuite aux autre fibres. Dans la troisième section, on introduit les fibres de Springer affines pour les groupes et on fait le lien avec le semi-groupe de Vinberg par l'intermédiaire de l'interprétation modulaire. Ces fibres de Springer admettent un ouvert distingué, dit régulier, qui est une orbite sous un champ de Picard, issu du centralisateur régulier. Enfin, dans la dernière section, on démontre le théorème 0.1. Suivant Kazhdan-Lusztig, nous avons besoin d'étudier l'équidimensionnalité d'une variété de drapeaux associée. Cela nécessite une étude approfondie des éléments nilpotents et quasi-unipotents du semi-groupe de Vinberg (sect.4). Une fois ce résultat d'équidimensionnalité obtenu, cela suffit pour en déduire que la dimension de l'ouvert régulier est la même que celle de la fibre toute entière et conclure quant à la dimension des fibres de Springer affines en général. Ce travail a fait l'objet d'innombrables navettes entre Gérard Laumon et Ngô Bao Châu qui, par leur relecture avisée et leur soutien ont contribué à améliorer de manière significative la qualité de ce travail, qu'ils en soient ici profondément remerciés. Je tiens également à exprimer ma reconnaissance envers Michel Brion pour ses utiles remarques concernant les subtilités de la caractéristique p. Je remercie l'Université de Chicago pour m'avoir invité par deux fois ainsi que Zongbin Chen pour d'utiles remarques sur les fibres de Springer affines. Enfin, merci à Ivan Boyer et Paul Mercat pour la peine qu'ils ont pris à relire mon piètre LaTeX. 1. Le semi-groupe de Vinberg et sa section 1.1. Rappels sur le semi-groupe de Vinberg. Soit k un corps algébriquement clos. Soit G un groupe connexe, semi-simple, simplement connexe sur k, de rang r. Soit (B, T ) une paire de Borel. On considère l'ensemble des poids fondamentaux ω 1 ,. . . , ω r , l'ensemble des racines simples ∆ = {α 1 , . . . , α r } et R l'ensemble des racines. Si λ est un cocaractère dominant de T , on note ρ λ , la représentation irréductible de plus haut poids λ, d'espace vectoriel V λ . Enfin, on pose, χ λ = Tr(ρ λ ). Tous les résultats énoncés ici, qui concernent les propriétés générales du semi-groupe de Vinberg seront dûs à Vinberg en caractéristique nulle et à Rittatore en caractéristique p. Définition 1.1. Un semi-groupe algébrique S est un k-schéma de type fini muni d'un morphisme associatif : m : S × S → S qui est un morphisme de k-schémas. Un semi-groupe est dit irréductible (resp. normal), si S l'est en tant que k-schéma. Un semigroupe qui admet une unité pour le morphisme m est appelé un monoïde. Nous parlons de monoïde algébrique pour un semi-groupe algébrique qui est un monoïde. Pour un monoïde algébrique S, on peut donc définir son groupe des inversibles : G(S) := {x ∈ S| ∃! y ∈ S, xy = yx = 1}. Si G(S) est connexe réductif, on dit que S est un monoïde réductif. Dans la suite, nous ne considérerons que des monoïdes algébriques irrréductibles et réductifs. Soient S, S ′ deux semi-groupes. On appelle φ : S → S ′ un morphisme de semi-groupes si c'est un morphisme de schémas et que : ∀ x, y ∈ S, φ(xy) = φ(x)φ(y). Si de plus, S et S ′ sont des monoïdes et φ(1 S ) = 1 S ′ , on parle de morphisme de monoïdes. Soit G + := (T × G)/Z G où le centre Z G de G est plongé par : λ.(t, g) = (λt, λ −1 g). Ce groupe admet un tore maximal T + = (T ×T )/Z G dont le groupe des caractères X * (T + ) s'identifie à un sous-réseau de X * (T ) × X * (T ). On note Z + le centre de G + , il s'identifie au tore T . Nous avons un schéma fourre-tout : H G := 1≤i≤r A 1 αi × 1≤i≤r End(V ωi ) (resp H 0 G := 1≤i≤r A 1 αi × 1≤i≤r (End(V ωi )−{0}) ). On considère alors l'immersion localement fermée : ι : G + → H G (t, g) → (α i (t), ω i (t)ρ ωi (g)) 1≤i≤r . On définit alors V G , la normalisation de l'adhérence de G + dans le schéma fourre-tout H G et V 0 G l'image réciproque dans V G de l'adhérence de G + dans H 0 G . On a les propriétés suivantes sur les adhérences : Proposition 1.2. -Le schéma V G est muni d'une structure de monoïde. Son groupe des unités s'identifie à G + et c'est un schéma normal affine et intègre. On l'appelle le semi-groupe de Vinberg. -L'action de G + × G + sur G + , donnée par la translation à gauche et à droite s'étend à V G et à V 0 G . Démonstration. Prouvons la première assertion. Tout d'abord, le morphisme ι est un morphisme de monoïdes, donc l'adhérence de G + dans H G a une structure de monoïde. Maintenant, d'après un lemme de Renner [36,Lem.1], si S est un monoïde intègre, il existe une unique structure de monoïde sur sa normalisation S norm telle que S norm → S soit un morphisme de monoïdes et telle que G(S norm ) = G(S). En particulier, en l'appliquant à V G , on a le résultat souhaité. Pour la deuxième assertion, à nouveau, par [36,Thm. 3], nous avons que l'action de G + × G + s'étend à l'adhérence de G + dans H G . Puis, d'après [36,Cor. 2], nous avons une action naturelle de G + × G + sur V G qui étend l'action par translation à gauche et à droite, compatible au morphisme de normalisation ; en particulier, on obtient le résultat analogue pour V 0 G . On a la proposition suivante due à Vinberg en caractéristique zéro [44,Th. 8], que nous étendons en caractéristique p > 0. Proposition 1.3. Le schéma V 0 G est lisse. Il admet un quotient projectif lisse par le centre Z + de G + , appelé la compactification de de Concini-Procesi (cf. [6]). Remarque : Nous reportons la preuve à la section 2.1. On définit le schéma V T l'adhérence de T + dans V G . Comme on dispose également de l'adhérence V ♭ T de T + dans H G , nous cherchons à les comparer. Si on note V ♭ G , l'adhérence de G + dans H G et ζ : V G → V ♭ G la flèche de normalisation, la flèche ζ induit un morphisme : ζ T : V T → V ♭ T . D'C * = {(λ, µ) ∈ X * (T ) × X * (T )| λ ≤ µ} En particulier, les schémas V T et V ♭ T s'identifient naturellement via ζ T . De plus, si Z + est l'adhérence de Z + , le centre de G + , dans V G en regardant l'algèbre k[V G ] G×G , elle s'identifie à k[Z + /Z G ] etα : V G → A G := Z + /Z G := A r . Nous allons maintenant construire un autre morphisme, issu de l'action par conjugaison. φ : k[G] G ∼ = → k[T ] W . On obtient un morphisme χ : G → T /W , dit de Steinberg. Si de plus, G est simplement connexe, T /W a une structure d'espace affine, donnée par les caractères χ ωi pour 1 ≤ i ≤ r. On dispose alors d'un morphisme : χ + : G + → T + /W = G r m × A r , provenant du théorème de Chevalley où W agit trivialement sur le tore central. Le morphisme χ + est donné par : g + = (t, g) → (α 1 (t), .., α r (t), χ 1 (tg), .., χ r (tg)) où χ i := Tr(ρ (ωi,ωi) ). Montrons que cette flèche se prolonge au semi-groupe de Vinberg. Démonstration. L'injectivité résulte du fait que si φ(f ) = φ(g), alors f = g sur l'ensemble G ss + des éléments semi-simples qui est dense dans G + et donc dans V G et f = g car V G affine. Il reste à montrer que φ est surjective, en particulier, k[V T ] W est engendré par les caractères des représentations. Dans la proposition 1.4, nous avons vu que le monoïde C * ∩ X * (T + ) + est engendré par les (α i , 0) et (ω i , ω i ). Comme W agit trivialement sur le premier facteur, on obtient que k[V T ] W est engendré par les éléments de la forme (γ, 0) et (λ, Sym λ), pour γ dans le module engendré par les racines simples et λ ≤ ω i . Ici, nous avons posé Sym λ := w∈W wλ. Maintenant, les éléments (ω i , χ ωi ) sont égaux à (ω i , Sym(ω i )) plus une somme d'éléments (ω i , Sym µ) pour µ < ω i . Or, comme µ < ω i , la différence est une somme de racines simples et (ω i , Sym µ) = (ω i − µ, 0) + (µ, Sym µ). On peut donc remplacer les (λ, Sym λ) par les (ω i , χ ωi ). Enfin, d'après Steinberg [41, Thm. 6.1], ces éléments sont algébriquement indépendants et forment une base. On obtient un nouveau morphisme que l'on note χ + , étendant le précédent : x αi (a i )n i , où les x αi (a i ) sont des éléments du groupe radiciel U αi et les n i sont des éléments du normalisateur N G (T ) représentant les réflexions simples s αi de W . χ + : V G → V T /W . Si Ainsi, ǫ(a) ∈ r i=1 U i n i et en utilisant les relations de commutation on a que : r i=1 U i n i = U w w où w = s 1 s 2 ...s r et U w = U ∩ wU − w −1 . Steinberg établit que cette section est à valeurs dans l'ouvert G reg := {g ∈ G| dim I g = r}, où I g désigne le centralisateur de g. Il prouve également grâce à son critère différentiel, les propriétés suivantes [41, Th. 8.1] : Théorème 1.7. (i) Le morphisme χ reg : G reg → T /W est lisse. (ii) Tout élément de G reg est conjugué à un élément de ǫ(T /W ). En fait, pour chaque élément de Coxeter w ′ ∈ W , Steinberg définit une autre section ǫ w ′ + qui est conjuguée à la précédente [41,Lem. 7.5] . En revanche, on verra que sur le semi-groupe de Vinberg, elles ne le sont plus. Nous donnons donc une définition, puisque par la suite, ces sections vont apparaître de manière cruciale. Pour w ′ ∈ W un élément de Coxeter, on écrit une décomposition réduite de w ′ : w ′ = s αi 1 ...s αi r . Et la section ǫ w ′ + se définit de la même manière r i=1 U i k n i k = U w ′ w ′ . Les résultats établis par la suite pour l'élément de Coxeter particulier w = s 1 ...s r , admettront la même démonstration pour un autre élément de Coxeter w ′ ∈ W . L'objectif est maintenant d'étendre cette section au semi-groupe de Vinberg ainsi que les résultats précédemment établis par Steinberg. 1.3. Prolongement de la section de Steinberg. En s'inspirant de la formule de Steinberg, on considère donc : (b, a) := (b 1 , .., b r , a 1 , .., a r ) ∈ G r m × A r . Soit la flèche φ : T → T + , t → (t, t −1 ). L'image T ∆ , le tore antidiagonal est isomorphe à T ad , et on a un isomorphisme canonique donné par : α • : T ∆ → G r m . Soit alors ψ l'isomorphisme inverse, nous optons pour la notation indiciaire. Ainsi pour b ∈ G r m , on a : ∀ 1 ≤ i ≤ r, α i (ψ b ) = b i et ψ b ∈ T ∆ . On définit ǫ + : G r m × A r → G + par : ǫ + (b, a) = ǫ(a)ψ b où ǫ est la section de Steinberg pour le groupe G définie par ǫ(a) := r i=1 x i (a i )n i . Commençons par rappeler la manière dont agissent les différents éléments considérés. Soit V ω une représentation irréductible de plus haut poids ω. Elle admet une décomposition en espace de poids : V ω = ν≤ω V (ν). On a d'après [41,Lem. 7.15] : Lemme 1.8. (i) Si U α = {x α (c) } désigne le groupe radiciel associé à α, alors l'action de U α sur un vecteur de poids µ, v µ est de la forme : x α (c).v µ = k≥0 c k v k où v k ∈ V (µ + kα), est indépendant de c. (ii) Pour tout i, j ∈ {1, . . . , r}, nous avons s j ω i = ω i − δ ij α j et pour λ = m i ω i , s j λ = λ − m j α j . Enfin, nous introduisons un ordre partiel strict sur l'ensemble {1, . . . , r} ; si i = j, on écrit j ≺ i s'il existe un poids dominant λ = m i ω i avec λ < ω i et m j > 0. Proposition 1.9. Le morphisme ǫ + : T + /W := G r m × A r → G + est une section de χ + et se prolonge en un morphisme, noté de la même manière, ǫ + : V T /W → V G , à valeurs dans V 0 G . Démonstration. Fixons i et considérons la représentation irréductible de plus haut poids ω i et λ ≤ ω i un poids de la représentation. Un élément g + = (t, g) de G + agit par ω i (t)ρ ωi (g). Soit v λ un vecteur de poids λ et regardons comment agit l'élément ǫ + (b, a). Nous avons la formule suivante : (1) ǫ + (b, a)v λ = ǫ(a)ψ b v λ = ω i (ψ b )λ(ψ b ) −1 ǫ(a)v λ = (ω i − λ)(ψ b )ǫ(a)v λ . avec ψ b ∈ T ∆ . Comme ω i − λ est une somme de racines simples, on en déduit que (ω i − λ)(ψ b ) est polynômial en les b i , donc ǫ + se prolonge. Montrons que le prolongement est non nul. On considère le vecteur de plus haut poids v ωi . En ce cas, ψ b agit trivialement, on a alors la formule suivante : ǫ + (b, a)v ωi = ǫ(a)v ωi . Comme ǫ(a) ∈ G, on a bien ǫ(a)v ωi = 0. Montrons que c'est une section. Du lemme 1.8 et de (1), on déduit la formule suivante : ǫ + (b, a)v λ = (ω i − λ)(ψ b ) k j≥0 a k1 1 ...a kr r v (k1,...,kr) où v (k1,...,kr) est un vecteur de poids λ + (k j − m j )α j et est indépendant des a j . Examinons la contribution à la trace, qui est non nulle que si m j = k j ≥ 0. On distingue deux cas : -λ = ω i et la contribution est a i . -λ = ω i , et on a λ < ω i . La formule montre que a j n'apparaît pas si m j = k j = 0, donc seulement les c j avec ω j dans le support de λ (et donc j = i), ont une contribution non nulle. Comme λ est dominant et λ < ω i , nous avons j ≺ i. Et donc le vecteur v λ contribue à la trace par un polynôme en les a j avec j ≺ i. En conclusion, χ i est un polynôme en les a i , de la forme a i + termes en les a j , pour j ≺ i. Donc, à partir des χ i , nous récupérons bien les paramètres a i , comme souhaité. Nous avons une action canonique de Z + sur V T /W = A r × A r donnée par : z.(b 1 , ..., b r , a 1 , ..., a r ) = (α 1 (z)b 1 , ..., α r (z)b r , ω 1 (z)a 1 , ..., ω r (z)a r ). En général la section ǫ + n'est jamais Z + -équivariante. Il nous faut donc tordre judicieusement l'action de Z + pour avoir une section équivariante. Une légère complication que l'on retrouve également dans le cas de la section de Kostant pour les algèbres de Lie, est qu'il faut extraire des racines. Dans la suite, on pose c := |Z G |. Pour z ∈ Z + , on considère ω • (z) := (ω 1 (z), .., ω r (z)) ∈ G r m . On a alors l'élément : (2) ψ ω•(z) := (λ z , λ −1 z ) ∈ T ∆ tel que α i (ψ ω•(z) ) = ω i (z) c , pour tout i = 1, . ., r. Remarque : Si on n'élève pas à la puissance c, l'élément ψ ω•(z) n'est pas défini de manière univoque. On construit donc par ce procédé un morphisme de groupes τ = ψ • ω : Z + → T ∆ . On considère maintenant la nouvelle action * de Z + sur G + donnée par : z * g = z c ψ −1 ω•(z) gψ ω•(z) . et on définit également sur C + une nouvelle action * de Z + définie par : z * (b, a) = z c .(b, a) . Le morphisme de Steinberg χ + reste bien Z + -équivariant pour cette action. On considère l'élément : (3) ψ ω•(z) := (λ z , λ −1 z ) ∈ T ∆ tel que α i (ψ ω•(z) ) = ω i (z) c , pour tout i = 1, . ., r. Nous avons alors la proposition suivante concernant la section ǫ + . Proposition 1.10. La section de Steinberg ǫ + : V T /W → V 0 G est Z + -équivariante pour l'action * sur G + et sur V T /W . Démonstration. Il nous faut prouver l'identité ǫ + (z c .(b, a)) = z * ǫ + (b, a). Pour simplifier, nous supposerons c = 1, la preuve étant exactement la même. Il nous suffit de démontrer cette identité au-dessus de G + . En rappelant que ǫ + (b, a) = ǫ(a)ψ b où ǫ est la section de Steinberg pour G et ψ b ∈ T ∆ , l'identité se réduit à : ǫ(z.a)ψ α•(z)b = zψ −1 ω•(z) ǫ(a)ψ b ψ ω•(z) . qui se simplifie en : ǫ(z.a)ψ α•(z) = zψ −1 ω•(z) ǫ(a)ψ ω•(z) . Considérons la représentation fondamentale de poids ω i . Soit µ ≤ ω i un poids de la représentation et un vecteur de poids µ, v µ . On regarde l'action de ψ α•(z) sur ce vecteur. Si on écrit ψ α•(z) := (γ z , γ −1 z ), on obtient : (4) ψ α•(z) v µ = (ω i − µ)(γ z )v µ = (ω i − µ)(z)v µ car ω i − µ est combinaison linéaire de racines simples et par (3). En utilisant le lemme 1.8 et d'après (4), on déduit que : (ǫ(z.a)ψ α•(z) ).v µ = (ω i − µ)(z) k j≥0 (ω 1 (z)a 1 ) k1 ...(ω r (z)a kr r )v (k1,...,kr) , où v (k1,...,kr) est un vecteur de poids µ + r j=1 (k j − m j )α j et µ = r j=1 m j ω j . D'autre part, nous avons également : (5) zψ −1 ω•(z) ǫ(a)ψ ω•(z) v µ = ω i (z)µ(λ −1 z ) k j≥0 [µ + r j=1 (k j − m j )α j ](λ z )a k1 1 ...a kr r v (k1,...,kr) . où l'on rappelle que ψ ω•(z) = (λ z , λ −1 z ). Maintenant, comme µ = r j=1 m j ω j , en utilisant (2), nous avons l'égalité suivante : [µ + (k j − m j )α j ](λ z ) = µ(λ z )µ(z −1 ) r j=1 k j α j (λ z ) = µ(λ z )µ(z −1 ) r j=1 ω j (z) kj . En injectant cette identité dans (5), nous avons : zψ −1 ω•(z) ǫ(a)ψ ω•(z) v µ = ω i (z)µ(λ −1 z ) k j≥0 [µ + r j=1 (k j − m j )α j ](λ z )a k1 1 ...a kr r v (k1,...,kr) . Et donc, les deux membres sont égaux, ce qu'on voulait. On a une description plus agréable du prolongement. Soit le diagramme commutatif suivant : Pour chaque i, la matrice ω i (t)ρ ωi (t −1 ) est un polynôme en les α i (t), et ces polynômes sont indépendants de t, on obtient alors que l'isomorphisme entre T ∆ et G r m se prolonge en un morphisme : T ∆ (α,0) ((α,0),ωρω ) / / 1≤i≤r A 1 αi × 1≤i≤r End(V ωi ) p1 G r m / / A rp 1 : T ∆ → A r , où T ∆ est l'adhérence de T ∆ dans V G . Les éléments de T ∆ sont entièrement déterminés par la composante suivant 1≤i≤r A 1 αi , puisque leur seconde composante est une matrice dont les coefficients sont des polynômes en les α i . Cette flèche est donc un isomorphisme. Enfin, on remarque que T ∆ est en fait contenu dans V 0 G puisque le coefficient de la matrice ω i (t)ρ ωi (t −1 ) en le vecteur de plus haut poids est 1. On continue de noter ψ l'isomorphisme inverse. On commence par rappeler la proposition suivante, due à Vinberg [ (i) On a une décomposition de V 0 G en G + × G + -orbites indexées par les sous-ensembles J de ∆. (ii) Dans chaque orbite O J , on a un idempotent distingué e J ∈ T ∆ . (iii) Pour J ⊂ ∆ et V λ la représentation irréductible de poids λ de G, on considère V J,λ le sous-espace engendré par les sous-espaces de poids qui sont dans λ + D J , D J le module engendré par les {α j , j ∈ J}. Alors, ρ λ (e J ) = p λ J où p λ J désigne la projection sur le sous-espace vectoriel V J,λ . En particulier, l'orbite O J s'envoie sur la strate G J m × A r par χ + . (iv) On a également une décomposition de V G de G + × G + -orbites indexées par certaines paires (J, K) de ∆ × ∆. A nouveau dans chaque orbite O J,K , on a un idempotent distingué e J,K de V T et dans le cas où J est vide, ρ λ (e ∅,K ) = p λ ∅ si et seulement si λ ∈ C K et zéro sinon. Ici, C K désigne le cône engendré par les (ω k , ω k ), k ∈ K. De cette proposition, on déduit une stratification de T ∆ = J T ∆ e J et chaque strate T ∆ e J s'envoie sur G J m × A r . Lemme 1.12. Soit (b, a) ∈ A r × A r , on considère l'élément ψ b de T ∆ , donné par l'isomorphisme ci-dessus, alors nous avons : ǫ + (b, a) = ǫ(a)ψ b , où l'on fait la multiplication dans le semi-groupe de Vinberg. En particulier si on note C : = ǫ + ({1} × A r ), alors ǫ + (A r × A r ) = CT ∆ et ǫ + (G J m × A r ) = CT ∆ e J . Démonstration. Au-dessus de G r m × A r , les deux sections sont les mêmes par définition, donc par unicité du prolongement, l'égalité reste vraie sur A r × A r . La suite du lemme vient de la description en strates rappelée ci-dessus. Remarque : Dans le cas A r , le calcul de la section de Steinberg nous donne : ǫ + (α • , a • ) =               a 1 −α 1 a 2 α 1 α 2 a 3 · · · (−1) r−1 r−1 i=1 α i a r (−1) r r i=1 α i 1 0 0 · · · 0 0 0 α 1 0 . . . 0 0 . . . . . . . . . . . . 0 0 . . . . . . r−1 i=1 α i 0               . 1.4. Construction du centralisateur régulier. Dans la suite, on note C + := V T /W . Soit le schéma en groupes I sur V G des paires I := {(g, γ) ∈ G × V G | gγg −1 = γ}. On définit le centralisateur régulier par J = ǫ * + I. Nous voulons montrer que ce schéma en groupes est lisse commutatif de dimension r. On commence par montrer que la section tombe dans l'ouvert régulier. Le procédé utilisé sert pour toute la suite de l'article ; à savoir, démontrer une propriété pour la fibre la plus singulière et la propager à toutes les autres. On rappelle le résultat suivant qui est un corollaire de [8, VI B. 4, Prop. 4.1]. Proposition 1.13. Soient a, a ′ ∈ C + tels que a ∈ {a ′ }, alors dim J a ≥ dim J a ′ . En particulier, en appliquant l'inégalité au point générique, on a : ∀ a ∈ C + , dim J a ≥ r. Ainsi, l'ensemble {a ∈ C + | dim J a = r}:= {g ∈ V G | dim I g = r} ⊂ V 0 G . La section ǫ + : C + → V 0 G est à valeurs dans V reg G , i.e. le centralisateur régulier est de dimension r. On commence d'abord par montrer la proposition pour le point le plus singulier : Lemme 1.16. L'élément ǫ + (0) appartient à V reg G . Démonstration. On a vu que la section tombait dans V 0 G . On a ǫ + (0) = we ∅ , où w = s 1 . . . s r . Soit alors g ∈ G un élément du centralisateur, alors gwe ∅ g −1 = we ∅ . De la description du stabilisateur des e I dans la proposition 1.11, on en déduit que : g ∈ B − ∩ wBw −1 et que δ − (g) = δ(w −1 gw) ∈ T + . Regardons la seconde condition. On pose alors g 1 = w −1 gw ∈ B, on écrit g 1 = tu avec t ∈ T , u ∈ U . Etudions alors ce que vaut δ(wg 1 w −1 ) : wg 1 w −1 = wtw −1 ww −1 = [Ad(w)t][Ad(w)u]. Comme l'élément w agit sur les groupes radiciels U α par wU α w −1 = U wα , on obtient que wuw −1 ne contribue pas dans la projection sur le tore. Regardons donc l'autre partie. L'égalité δ − (g 1 ) = δ(wg 1 w −1 ) impose que t = t w ce qui donne que pour tout i, α i (t) = 1 et t ∈ Z G , qui est fini. Le centralisateur s'identifie alors à : J 0 = Z G (U − ∩ wU w −1 ) Maintenant, on sait que U − ∩ wU w −1 s'identifie au produit des groupes radiciels U α , avec α > 0 et wα < 0 et est de dimension l(w) = r. Passons à la preuve de la proposition 1.15, la méthode de démonstration sera ensuite réutilisée systématiquement. Démonstration. On regarde l'ensemble U := {a ∈ C + | dim J a = r}. C'est un ouvert d'après la proposition 1.13 et d'après ci-dessus, il contient 0. Il est de plus Z + -équivariant. Montrons que U = C + . Supposons par l'absurde que le complémentaire de U soit non vide. Soit a un point du complémentaire. On considère l'adhérence de sa Z + -orbite que l'on note F . Celle-ci contient 0 qui est dans U , donc on en déduit que c'est aussi le cas du point générique de F et comme U est Z + -équivariant, on obtient a ∈ U , une contradiction. Remarque : L'argument ci-dessus montre que toute propriété (P) sur les fibres, vérifiée par χ −1 + (0), qui est ouverte et qui est Z + -équivariante, se propage à toutes les fibres χ −1 + (a). Cette remarque sera utilisée de manière systématique par la suite. Il nous faut encore obtenir que la flèche χ reg + est lisse. Nous allons étudier plus en détail la fibre au-dessus de 0 et ensuite utiliser la remarque ci-dessus. 2. Propriétés du morphisme de Steinberg étendu χ + 2.1. Etude du cône nilpotent. Proposition 2.1. Soit la flèche χ + : V G → C + . Alors, N := χ −1 + (0) est inclus dans {g ∈ V G | ∀ λ, ρ λ (g) est nilpotente} , en particulier dans les représentations de dimension un, ρ λ (g) est nulle. Démonstration. Soit γ ∈ V G tel que χ + (γ) = 0 alors γ est dans l'union des strates J⊂∆ O ∅,J . Il s'écrit donc γ = g 1 e ∅,J g 2 . Quitte à conjuguer, on l'écrit sous la forme ge ∅,J . De plus, on peut se ramener à J = ∆ puisqu'il résulte de la description de la proposition 1.11 (iii), que pour J = ∆, e ∅,J est l'endomorphisme nul si on regarde à travers une représentation ρ λ avec λ / ∈ C J . Regardons comment cet élément agit sur la représentation irréductible de plus haut poids ω i . Soit v λ un vecteur de poids λ. Alors, toujours par la proposition 1.11 (iii), on déduit que : On s'intéresse à l'ouvert du cône nilpotent : ge ∅ v λ = 0 si λ = ω i et ge ∅ v ωi = a i v ωi + (N 0 = N ∩ V 0 G . En effet, le lien avec la compactification magnifique de de Concini-Procesi (cf. Prop. 1.3) nous permet de déterminer sa structure. Nous allons avoir besoin d'introduire un certain nombre d'objets, en particulier la compactification magnifique. Pour un sous-ensemble I ⊂ ∆, considérons W I le sous-groupe de W engendré par les réflexions simples s i , i ∈ I. Soit W I l'ensemble des représentants de longueur minimale de W/W I . On note X la compactification magnifique de de Concini Procesi. Dans [6], on établit les propriétés suivantes concernant la compactification : (i) C'est une G ad × G ad variété projective lisse qui contient G ad comme ouvert dense. (ii) Les G ad × G ad -orbites de X sont indexées par les sous-ensembles I ⊂ ∆. (iii) On a X ∆ = G ad et X ∅ est l'orbite fermée. Les orbites X I admettent une description similaire aux orbites de V 0 G . Nous sommes en mesure de démontrer le lien entre V 0 G et la compactification magnifique annoncé dans la proposition 1.3 : Proposition 2.2. Le schéma V 0 G est lisse et le quotient de V 0 G par l'action par homothétie du centre Z + s'identifie à la compactification magnifique X. La preuve suit celle de Vinberg en utilisant la proposition 1.11, étendue par Rittatore en caractéristique p : Démonstration. On considère la flèche : j : U − × Z + × T ∆ × U → V 0 G 15 donnée par (u − , z, t, u) → u − ztu. La flèche j est birationnelle au-dessus de G + entre schémas normaux intègres. Montrons qu'elle est quasi-finie. On doit prouver qu'une égalité : u − zt 1 e I u = t 2 e I , u − ∈ U − , z ∈ Z + , t 1 , t 2 ∈ T ∆ implique u − = u = z = 1 et t 1 = t 2 . Cela revient alors de la description des stabilisateurs 1.14, de la même manière que [44,Prop. 14]. Ainsi, par le Main Theorem de Zariski, on en déduit que j est une immersion ouverte. Comme de plus, son image contient des représentants de toutes les G + × G + -orbites de V 0 G , on obtient la lissité. L'assertion sur le quotient s'obtient alors de la même manière que [44,Prop. 15] . Pour un cocaractère λ ∈ X * (T ), comme X est projective, on peut définir λ(0) par le critère valuatif. Théorème 2.4. [Lusztig-He] (i) X I est l'union disjointe des X I,w , w ∈ W I . (ii) X I,w est localement fermé et irréductible de dimension dim G − l(w) − |∆ − I|. La preuve de (i) est donnée par Lusztig [23] et He [14] et (ii) est montré par Lusztig [24, sect. 8]. Suivant [15], nous avons les relations suivantes entre les adhérences des strates X I,w . Soit I l'ensemble des paires (I, x) avec I ⊂ ∆, x ∈ W I . On définit la relation ≤ sur I par : (I, x) ≤ (K, y) si et seulement si I ⊂ K et x ≥ z −1 yz pour z ∈ W K . Nous avons alors le théorème suivant dû à He sur l'adhérence des strates : Théorème 2.5. (i) La relation ≤ définit une relation d'ordre sur I. (ii) Si (I, x), (K, y) ∈ I alors X I,x ⊂ X K,y si et seulement si (I, x) ≤ (K, y). On se reporte à [15, sect. 3, 4] pour la preuve. On peut maintenant faire le lien avec le cône nilpotent. Pour w ∈ W , on définit le support de w, supp(w) ⊂ ∆ comme l'ensemble des réflexions simples qui interviennent dans une décomposition réduite de w. Soit σ : V 0 G → X qui est un Z + -torseur et la fibre N 0 = χ −1 + (0) ∩ V 0 G s'envoie donc surjectivement dans la strate N ad,∅ . On a le diagramme cartésien suivant : N ad := {x ∈ X| ∀ i, ρ ωi (x) ∈ P(End(V ωi )) est nilpotent}.N 0 / / V 0 G N ad,∅ / / X On rappelle que le fait d'être dans χ −1 + (0) impose de tomber dans la strate vide. On note de la même manière l'image réciproque de X ∅,w dans le semi-groupe de Vinberg. On en déduit la proposition suivante : Proposition 2.8. Soit w ∈ W tel que supp(w) = ∆, alors il existe un élément de Coxeter w ′ ∈ W tel que w ′ ≤ w. Nous allons maintenant caractériser les éléments réguliers. Proposition 2.9. On a la décomposition suivante de N reg : N reg = w∈W supp(w)=∆ l(w)=r X ∅,w . D'où l'on déduit que chaque strate X ∅,w est une classe de conjugaison. Enfin, N reg est dense dans N 0 et N reg est lisse. Remarques : -La classe de conjugaison de la section de Steinberg correspond à la composante connexe associée à l'élément w = s 1 . . . s r . On voit alors qu'en prenant un autre élément de Coxeter, on obtient une section qui tombe dans une composante distincte. Cela n'apparaît pas au niveau du groupe des inversibles, vu que le choix d'un autre élément de Coxeter aboutit à une section conjuguée, ce qui n'est pas le cas si l'on considère le semi-groupe, à cause de l'apparition des idempotents. -On obtient donc une famille (ǫ w ′ + ) de sections indexées par les éléments de Coxeter w ′ ∈ W et la proposition ci-dessus nous dit que les éléments nilpotents réguliers sont conjugués à une certaine section ǫ w ′ + . Démonstration. Pour prouver la proposition, il suffit donc de calculer la dimension du centralisateur des éléments w ′ e ∅ , pour w ′ un élément de Coxeter. Le calcul est alors le même que pour le lemme 1.16. On déduit ensuite, du fait que X ∅,w ′ est irréductible de bonne dimension, qu'il s'identifie à la classe de conjugaison de w ′ e ∅ . La densité vient de la proposition 2.8. Pour la lissité, on regarde la flèche χ reg + : V reg G → C + . C'est une flèche entre schémas lisses. Comme ǫ + est une section, on sait que l'application tangente en ǫ + (0) est surjective et donc il en est de même de tous ses conjugués. Pour ce qui est des autres composantes connexes, à chaque élément de Coxeter w ′ , on peut définir de même une section ǫ w ′ + à χ + et on obtient de même la lissité. On termine notre étude du cône nilpotent en étendant les résultats de N 0 au cône nilpotent N . On commence par définir pour J ⊂ ∆ et w ∈ W , les strates X ∅,J,w : X ∅,J,w := G.(Bwe ∅,J B) où les e ∅,J correspondent aux idempotents du semigroupe de Vinberg dans les strates plus petites. On remarque que si J = ∆, on est dans N 0 . On rappelle que les idempotents sont nuls dans les représentations V ωi pour i / ∈ J et qu'ils correspondent à la projection sur le vecteur de plus haut poids v ωj dans End(V ωj ) pour j ∈ J. On en déduit alors que, pour tout J ∈ ∆ et w ∈ W avec J ⊂ supp(w), X ∅,J,w ⊂ N . La proposition suivante montre la réciproque. Proposition 2.10. Le cône nilpotent admet la stratification suivante : N = J⊂∆ w∈W supp(w)=∆ X ∅,J,w . En particulier, N 0 est dense dans N et dim N = dim G + − 2r. Démonstration. Il nous suffit de démontrer la décomposition, vu que le reste est un corollaire immédiat de la description des strates. Soit x ∈ N , il est dans une certaine strate O ∅,J . Quitte à conjuguer, on peut supposer qu'il est de la forme ge ∅,J . On écrit que g est dans une certaine orbite BwB, pour w ∈ W . Comme le radical unipotent U fixe e ∅,J , on peut supposer que x = bwe ∅,J . On commence par voir que J ⊂ supp(w). On sait déjà que ρ ωi e ∅,J est nul pour i / ∈ J. Soit alors j ∈ J, on considère son image dans End(V ωj ). Nous avons e ∅,J v λ = 0 pour λ = ω j . Etudions la contribution de v ωj . Soit une décomposition réduite de w = s i1 s i2 ...s i l . Si par l'absurde la réflexion simple s j , n'intervenait pas dans cette décomposition, le coefficient devant v ωj serait non nul ce qui est contradictoire avec la nilpotence. On obtient donc J ⊂ supp(w). Si supp(w) = ∆, on a rien à montrer et si supp(w) est un sous ensemble strict de ∆, on considère l'élément w ′ = ws i1 s i2 ...s ip avec pour tout l ≤ p, i l ∈ ∆ − supp(w) ⊂ ∆ − J. De la description de e ∅,J , on a que we ∅,J = w ′ e ∅,J , ce qui termine la preuve de la proposition. 2.2. La lissité du morphisme χ reg + . Proposition 2.11. Soit χ + : V G → C + . Alors, χ + est plat et ses fibres sont de dimension dim G + − 2r. Démonstration. Au point 0, on a vu que le cône nilpotent était de la bonne dimension. On considère alors l'ensemble : Démonstration. Pour chaque élément w ∈ W de Coxeter, on regarde la flèche U := {a ∈ C + | dim χ −1 + (a) = dim G + − 2r}.π w : G × C + → V reg G donnée par (g, a) → gǫ w + (a)g −1 . Cette flèche admet la factorisation suivante : G × C + ρ & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ π w (G × C + )/J i x x q q q q q q q q q q q V regU = w∈W supp(w)=∆ l(w)=r V w et soit F son fermé complémentaire dans V G . Posons l G := dim G + − 2r. Soit V := {x ∈ F | dim x χ −1 + (χ + (x)) ≤ l G − 1}. Par le théorème de Chevalley sur la semicontinuité des fibres, V est ouvert dans F et par la proposition 2.9, il contient 0 et est Z +équivariant ; il est donc égal à F . On en déduit que U est de codimension au moins un dans chaque fibre et comme les fibres sont équidimensionnelles, U est dense dans chaque fibre. En particulier, tout élément régulier est conjugué à une section ǫ w , pour un certain élément de Coxeter w. Et donc le même argument que pour la proposition 2.9, nous dit qu'en tout élément régulier x, la différentielle est surjective, d'où la lissité, puisque la source et le but sont lisses. Enfin, les fibres de χ + sont Cohen-Macaulay, car on a une immersion régulière de la fibre dans V G , qui est Cohen-Macaulay. Comme les éléments réguliers forment un ouvert lisse dense dans un schéma Cohen-Macaulay, les fibres sont réduites. Démonstration. On a le diagramme cartésien : Soit φ le morphisme structural de J. Au-dessus de 0, d'après le lemme 1.16, la fibre est lisse. D'après la remarque 1.4, il nous suffit de montrer que l'ensemble des a ∈ C + tels que la fibre de J a est lisse est ouvert. Comme J est plat sur C + , on a d'après EGA IV 12.2, que l'ensemble des x ∈ J, tel que J φ(x) est lisse en x est ouvert. I |V reg G / / G × V reg G ψ V reg G / / V reg G × C+ V reg Or, de par sa structure de schéma en groupes, si la fibre est lisse en un point, elle est lisse partout, donc on en déduit, toujours comme J est plat sur sa base que, {a ∈ C + | J a est lisse} est ouvert. Comme il est clairement Z + -équivariant, la remarque du lemme 1.16 s'applique. Enfin, comme au-dessus de G reg + , le centralisateur régulier est abélien, on a donc un schéma en groupes lisse, qui est génériquement abélien, donc abélien. Nous terminons par un critère analogue à Steinberg [41, 3.2 et 3.3] pour caractériser les réguliers nilpotents. Proposition 2.14. Tout élément régulier nilpotent de V G est dans un unique semi-groupe de Borel. Démonstration. Il nous suffit de le montrer pour chaque nilpotent régulier dans une composante connexe fixée. On prend celle qui correspond à l'élément de Coxeter w = s 1 . . . s r , la preuve étant la même pour tout autre élément dans une autre composante connexe. Quitte à conjuguer, on a juste à montrer le résultat pour ǫ + (0). Soit alors ǫ + (0) = we ∅ avec w = s 1 . . . s r , qui serait dans deux semi-groupes de Borel V B ′ et V B − , alors B − et B ′ étant conjugués, en écrivant la décomposition de Bruhat relativement à B − , nous avons B ′ = uσB − σ −1 u −1 , avec σ ∈ W et u ∈ U − . Dans ce cas, on obtient que l'élément σ −1 u −1 we ∅ uσ est dans V − B . On se fixe alors i et on regarde l'action sur les vecteurs v λ : σ −1 u −1 we ∅ uσv λ = σ −1 u −1 we ∅ uv σ(λ) . En particulier, comme u ∈ U − et par définition de e ∅ , on en déduit que si σ(λ) = ω i , ce vecteur est nul. Etudions le cas résiduel σ(λ) = ω i , on a alors : σ −1 u −1 we ∅ uσv λ = σ −1 u −1 [v ωi−αi−β ] où β est une somme à coefficients positifs de racines simples α j pour j ≤ i − 1. D'où l'on déduit : σ −1 u −1 we ∅ uσv λ = v σ −1 (ωi−αi−β) +(autres vecteurs). Or σ −1 (ω i − α i − β) = λ − σ −1 α i − σ −1 β, et comme l'élément doit être dans V − B , cela force λ − σ −1 α i − σ −1 β ≤ λ. Cette inégalité étant valable pour tout i, cela donne σ = 1, ce qu'on voulait. Considérons le schémaṼ G := {(g, γ) ∈ G/B × V G | g −1 γg ∈ V B } qui est fermé dans G/B × V G . On a un morphisme propre λ :Ṽ G → V G car il admet la factorisation suivante : V G / / λ $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ V G × G/B V G Nous allons nous intéresser à ce morphisme au-dessus du lieu régulier : Corollaire 2.15. Le morphisme λ reg :Ṽ reg G → V reg G est fini plat. Démonstration. Comme on sait déjà que λ reg est propre par changement de base, il nous suffit de vérifier, en vertu du Main Theorem de Zariski, qu'elle est quasi-finie. Il résulte alors de la proposition 2.14 qu'au-dessus de 0 ∈ C + , la fibre est réduite à un point, donc par semi-continuité de la dimension des fibres, on en déduit, que le morphisme est partout quasi-fini. Passons à la platitude, considérons le diagramme commutatif suivant : Nous avons également la réciproque à la proposition 2.14 : V reg G / / λ V T θ V reg G / / C + Proposition 2. 16. Soit x ∈ V G un élément nilpotent irrégulier, alors il est dans une infinité de semi-groupes de Borel. Démonstration. Il nous suffit de montrer l'assertion pour x ∈ V 0 G . On rappelle que nous avons la stratification suivante pour N 0 : N 0 = w∈W supp(w)=∆ X ∅,w et que d'après la proposition 2.9, les éléments réguliers correspondent aux éléments de longueur r. En particulier, si x ∈ N 0 est irrégulier, il est dans une strate X ∅,w pour l(w) ≥ r + 1. On commence donc par démontrer la proposition pour x ∈ X ∅,w avec l(w) = r + 1. Pour démontrer le résultat, quitte à conjuguer, on peut supposer que : x = nwe ∅ avec n ∈ U et w ∈ W tel que l(w) = r + 1 et supp(w) = ∆. L'assertion va résulter du calcul du centralisateur et du fait que l'élément considéré va être dans un certain Lévi. Regardons le cas de l'élément we ∅ , le cas d'un élément de la forme nwe ∅ étant analogue. En reprenant le calcul du lemme 1.16, on obtient que le centralisateur de I we ∅ s'identifie alors à Z G T β U − w où U − w = U − ∩ wU w −1 et T β := {t ∈ T | wtw −1 = t} est un tore de dimension 1 engendré par une certaine coracineβ avec wβ = β. Soit alors L β := Z G+ (T β ), alors we ∅ est dans L β . Et de l'égalité wβ = β, on en déduit que pour tout u ∈ U −β : we ∅ ∈ uσ β V − B σ −1 β u −1 . Ainsi, we ∅ est dans une infinité de semi-groupes de Borel. Il nous suffit désormais de montrer que l'adhérence des strates X ∅,w pour l(w) = r + 1 consiste en tous les éléments nilpotents irréguliers. En effet, d'après le théorème de Chevalley, la dimension des fibres de λ :Ṽ G → V G ne peut qu'augmenter, donc on aura le résultat pour les autres éléments. En vertu de la relation d'adhérence entre les strates (cf. Thm. 2.5), il nous suffit de démontrer le lemme suivant : Lemme 2.17. (Premet) Soit w ∈ W tel que supp(w) = ∆ et l(w) ≥ r + 2, alors il existe w ′ ∈ W de même support que w et tel que l(w ′ ) = r + 1. La preuve qui suit est due à Premet : Posons l = l(w) et considérons Red(w) l'ensemble de toutes les expressions réduites de w. Etant donné r = (i 1 , . . . , i l ) ∈ Red(w), on note k(r) le plus petit k ≤ l tel que i k apparaît dans r plus de deux fois. Soit t = (j 1 , . . . , j l ) ∈ Red(w) tel que k(t) ≥ k(r) pour tout r ∈ Red(w). On écrit alors w = s j1 . . . s j l et on considère w ′ ∈ W obtenu à partir de w en supprimant s j k de l'expression réduite de w où k = k(t). De par le choix de k, w ′ a le même support que w et sa longueur a diminué de 1. En itérant le procédé, on fait diminuer la longueur jusqu'à r + 1 ce qu'on voulait. On termine le paragraphe par une étude du discriminant sur le semi-groupe de Vinberg. Démonstration. On commence par étendre la fonction à Z + × T ∆ . Soit t + = (t, t −1 ) ∈ T ∆ , alors nous avons : D + (t + ) = 2ρ(t) α>0 (1 − α(t −1 ))(1 − α(t)) Comme 2ρ = α>0 α, nous obtenons : (6) D + (t + ) = (−1) |R + | α>0 (1 − α(t)) 2 où R + est l'ensemble des racines positives. En particulier, on obtient que D + se prolonge à Z + ×T ∆ . Maintenant, comme D + est W -invariante, on obtient dans un premier temps, que D + s'étend à V 0 T , l'adhérence de T + dans V 0 G . Maintenant, comme V T est affine et normal et que la codimension du complémentaire de V 0 T dans V T est au moins deux, on obtient que D + se prolonge. Il ne nous reste plus qu'à voir le critère de régularité. Proposition 2.19. Soit t + ∈ V T , alors nous avons l'équivalence : t + ∈ V rs T ⇐⇒ D + (t + ) = 0. Démonstration. Soit t + ∈ V rs T , en particulier t + ∈ V T ∩V 0 G = V 0 T . Quitte à conjuguer par un élément de W , on peut supposer que t ∈ Z + T ∆ . Si nous avons D + (t) = 0, alors il résulte de l'égalité (6) qu'il existe α ∈ R tel que α(t + ) = 1. En particulier, en considérant le tore T α = Ker α ⊂ T + , son centralisateur dans G est de dimension strictement plus grande que r et donc t + ∈ T α ne peut être régulier semisimple. Soit alors F le fermé complémentaire de V rs T dans V T . On sait que F est un diviseur, comme c'est le complémentaire du lieu étale de la flèche V T → C + , en particulier équidimensionnel. Soit U ⊂ V T le lieu où D + ne s'annule pas et F 1 son fermé complémentaire. On vient de voir que V rs T ⊂ U . Supposons par l'absurde que V rs T soit strictement inclus dans U et considérons le fermé F 2 = U − V rs T . Nous avons alors F 1 ∩ F 2 = ∅ et F = F 1 ∪ F 2 . Lorsque l'on intersecte V rs T et U avec T + , les ouverts sont les mêmes. De plus, il résulte de la proposition 1.14 que l'élément e ∅ est régulier semisimple (si ge ∅ g −1 = e ∅ , alors g ∈ B ∩ B − ). Pour 1 ≤ i ≤ r, si O i est la strate de codimension un de T + , nous avons que : e ∅ ∈ r i=1 O i . En particulier, pour tout 1 ≤ i ≤ r, V rs T ∩ O i est un ouvert non vide de O i . En particulier, les points génériques des strates O i sont dans V rs T d'où l'on déduit que F 2 est de codimension au moins deux. Or, comme F est équidimensionnel, on obtient que F 2 = ∅, une contradiction. 2.3. Une construction alternative du centralisateur. Dans ce paragraphe nous donnons une interprétation alternative du centralisateur régulier d'après Donagi-Gaitsgory [9] et Ngô [28, sect. 2.4]. Toutefois, il y a une subtile différence avec le cas de Ngô et de Donagi-Gaitsgory, dans la mesure où les fibres du morphisme de Steinberg ne sont pas intègres. En particulier, on voit qu'au point 0, les sections de Steinberg pour deux éléments de Coxeter donnent des éléments réguliers non conjugués. Néanmoins, nous allons voir que les centralisateurs restent isomorphes. Nous supposons de plus que la caractéristique du corps est première à l'ordre du groupe de Weyl. Dans la preuve de la proposition 2.12, pour chaque élément de Coxeter w ∈ W nous avions introduit la flèche : G × C + → V reg G définie par (g,χ * + J V reg G ∼ = → I V reg G . De plus, cet isomorphisme se prolonge en une flèche de χ * + J → I. Démonstration. Pour chaque élément de Coxeter w ∈ W , en reprenant exactement la preuve par descente fidèlement plate de [28, Lem. 2.1.1], nous obtenons un unique schéma en groupe commutatif lisse J w qui est isomorphe à I V reg,w G . Soient donc deux éléments de Coxeter distincts w, w ′ ∈ W , nous allons montrer que les schémas en groupes lisses J w et J w ′ sont isomorphes. Comme ils sont lisses, il suffit de montrer l'isomorphisme sur un ouvert dont le complémentaire est de codimension au moins deux de C + . Cela résulte alors de la proposition suivante : Lemme 2.21. Pour tout élément de Coxeter, nous avons G reg + ∪ V rs G ⊂ V reg,w G , où V rs G est le lieu où le centralisateur est un tore. Démonstration. D'après Steinberg, au-dessus de G reg + , on sait que tout élément est conjugué à ǫ + (C + ) et de plus au-dessus de V rs G le centralisateur étant un tore et tous les tores étant conjugués, on obtient que tout élément de V rs G est également conjugué à un élément de ǫ + (C rs + ), en particulier G reg + ∪ V rs G ⊂ V reg,w G . Comme pour tout 1 ≤ i ≤ r, nous avons vu dans la preuve de la proposition 2.19 que V rs G ∩ O i était de codimension un dans O i , nous obtenons que la codimension du fermé complémentaire de G reg + ∪ V rs G est déjà de codimension deux, donc également son image dans C + par lissité de la flèche V reg G → C + et donc nous obtenons que J w et J w ′ sont naturellement isomorphes. Nous déduisons donc qu'il existe un unique schéma en groupes lisse J sur C + tel que : χ * + J V reg G ∼ = → I V reg G . Il ne reste donc qu'à montrer que l'isomorphisme se prolonge en une flèche de χ * + J → I. Comme la codimension du fermé complémentaire de V reg G dans V G est de codimension au moins deux, il s'ensuit que la flèche se prolonge en un morphisme : On note X * (T ) + l'ensemble des caractères dominants. Sur G(F ), on a la décomposition de Cartan : χ * + J → I au-dessus de V G .G(F ) = λ∈X * (T ) + Kπ λ K où λ un cocaractère dominant. Soit Kπ λ K l'adhérence de Kπ λ K dans G(F ), nous avons la stratification suivante : Kπ λ K = µ≤λ Kπ µ K. Soit ρ ωi la représentation irréductible de plus haut poids ω i . Dans la section 1.1, nous avons introduit le semi-groupe de Vinberg V G ainsi que son ouvert lisse V 0 G . Le lien entre le semi-groupe de Vinberg et la grassmannienne affine apparaît ici. Lemme 3.1. Un élément g ∈ G(F ) appartient à l'orbite Kπ λ K (resp. Kπ λ K) si et seulement si pour tout cocaractère dominant ω ∈ X * (T ) + , le plus grand des ordres des pôles des coefficients de la matrice ρ ω (g) est égal à ω, −w 0 λ , où w 0 est l'élément long du groupe de Weyl. De plus, l'élément g + = (π −w0λ , g) est dans V 0 G (O) (resp. V G (O)). Démonstration. Le plus grand des ordres des pôles est invariant à gauche et à droite par K, en particulier il nous suffit de regarder celui de π λ , cet ordre est égal à ω, −w 0 λ . Inversement, les entiers ω, −w 0 λ déterminent uniquement λ. Enfin, comme l'élément g + est dans G + (F ) et que pour tout ω ∈ X * (T ) + , ρ ω (g + ) ∈ End V ω (O), la continuité nous donne le résultat voulu. La preuve pour Kπ λ K est analogue. D'après [16,Prop. 2], on peut donner une structure d'ind-schéma sur k au quotient G(F )/K qui s'écrit comme limite inductive de variétés projectives, munies d'immersions fermées les unes dans les autres. De plus, d'après [16,Prop. 2], ce quotient représente le foncteur Q qui à toute k-algèbre R, associe le groupoïde Q(R) des G-torseurs E sur Spec(O)×R muni d'une trivialisation sur Spec(F )×R. Ici, Spec(O)×R désigne la complétion π-adique de Spec(O) × R et Spec(F )×R est l'ouvert com- plémentaire de {π} × R dans Spec(O)×R. Quant aux strates, elles admettent également une interprétation modulaire. Pour 1 ≤ i ≤ r, soit (ρ i , V i ) la représentation irréductible de plus haut poids ω i . Soit E un G torseur, on peut donc pousser le G-torseur par la représentation ρ i . On obtient alor un fibré vectoriel noté ρ i (E). Pour un cocaractère dominant λ, on considère le sous-foncteur Q λ , qui à toute k-algèbre R, associe le groupoïde des G-torseurs E sur Spec(O)×R munis d'une trivialisation générique telle que pour tout i = 1, . . . , r, la trivialisation fournit une application injective de fibrés vectoriels : ρ i (E) → V i ( ω i , −w 0 λ ) dont les fibres résiduelles sont non nulles et V i ( ω i , −w 0 λ ) := V i ⊗ R[[π]] π ωi,w0λ R[[π] ]. D'après le lemme précédent, les k-points de ce foncteur sont Gr λ = Kπ λ K/K. Si l'on n'impose pas que les fibres résiduelles soient non nulles, on trouve l'espace Gr λ = µ≤λ Gr µ . La strate Gr λ est lisse, mais ce n'est plus du tout le cas de son adhérence, qui est toutefois projective. 3.2. Une interprétation modulaire. On s'intéresse à une variante de la fibre de Springer affine : {g ∈ G(F )/K| g −1 γg ∈ Kπ λ K} pour γ ∈ G(F ). Nous voulons voir cette nouvelle fibre de Springer affine, comme une solution à un problème d'espace de modules. On rappelle que le semi-groupe de Vinberg V G s'obtenait comme la normalisation de l'adhérence de G + = (T × G)/Z G dans r i=1 End(V ωi ) × r i=1 A 1 αi . La donnée de −w 0 λ ∈ X * (T ) + = T (F )/T (O) fournit un T -torseur T −w0λ muni d'une trivialisation générique. Cela revient également à pousser le G m -torseur π −1 O par −w 0 λ : G m → T . De plus, pour chaque i = 1, . . . , r, la racine α i : T → G m permet de tordre la droite A 1 αi par le T -torseur T −w0λ . On pose b i := 1 ( αi,−w0λ ) la section unité. Posons X = Spec(O). On considère l'espace caractéristique C + := V T /W où V T est l'adhérence dans le semi-groupe de Vinberg de T + . Nous avons également introduit dans la section 1.1 le morphisme d'abélianisation, α : V G → A G = A r , où r = rg G. Le morphisme d'abélianisation consiste en le quotient, au sens des invariants, de V G par l'action de G × G par translation à gauche et à droite (cf. sect. 1.1). Le T -torseur T −w0λ avec les sections (b 1 , . . . , b r ) obtenues en poussant le T -torseur par les racines simples revient alors à la donnée d'une flèche : h −w0λ : X → [A G /Z + ], on rappelle que Z + s'identifie au tore et on fait agir Z + par les racines simples sur A G . Le morphisme de Steinberg étant Z + -équivariant, on a une flèche : χ + : [V G /Z + ] → [C + /Z + ] . La section de Steinberg va nous permettre d'obtenir une section à ce morphisme. Comme nous l'avons vu dans la proposition 1.10, quitte à extraire une racine c-ième où c = |Z G |, on peut faire en sorte que la section ǫ + soit équivariante pour l'action tordue de Z + . Soit le champ quotient [C + /Z + ] pour l'action naturelle de Z + sur C + . Etant donné un T -torseur E sur un k-schéma X, il s'écrit comme une somme de fibrés en droites E = r i=1 L i . Nous appelerons une racine c-ième de E, tout T -torseur E ′ sur X tel que E ′ = r i=1 L ′ i avec (L ′ i ) ⊗c = L i . On suppose donc λ = cλ ′ . Le morphisme de Steinberg χ + étant Z + -équivariant pour les actions canoniques, il induit un morphisme sur les champs quotients : [χ + ] : [V G /(G × Z + )] → [C + /Z + ] . La proposition est alors la suivante : ǫ + (a) λ ′ : C + → [V G /(G × Z + )]. au morphisme [χ + ]. Démonstration. On note [χ + ] [c] : [V G /(G × Z + )] [c] → [C + /Z + ] [c] le morphisme obtenu en élevant à la puissance c les actions canoniques de Z + sur V G et sur C + . Nous avons vu en vertu de la proposition 1.10 que nous avons une section, [C + /Z + ] [c] → [V reg G /(Z + × Z τ + )] [c] où le premier Z + agit par homothétie et le deuxième par conjugaison par les éléments τ (z) avec τ : Z + → T ∆ construit dans la proposition 1.10. En composant alors par ce morphisme τ , nous obtenons une flèche : [C + /Z + ] [c] → [V reg G /(Z + × G)] [c] , qui est donc une section au morphisme de Steinberg : [χ + ] [c] : [V reg G /(Z + × G)] [c] → [C + /Z + ] [c] , ce qu'on voulait. Nous pouvons désormais donner une interprétation modulaire de la fibre de Springer affine. Pour chaque flèche h a qui rend le diagramme : X h −w 0 λ # # • • • • • • • • • ha / / [C + /Z + ] p1 [A r /Z + ] commutatif, on a un point : J a → Aut(E, φ + ) qui se déduit de la flèche χ * + J → I. Celui-ci permet de tordre (E, φ + ) par un J a -torseur sur X×R trivialisé sur X •× R. On a alors la proposition suivante : Proposition 3.6. [χ + ] : [V reg G /(G × Z + )] → [C + /Z + ] est une gerbe liée par le centralisateur J et neutre. En particulier, la fibre de Springer M reg λ (a) est un espace principal homogène sous P(J a ). Dans la suite, c'est ce qu'on appellera l'orbite régulière. Démonstration. Nous avons vu que le morphisme était lisse. De par la caractérisation du centralisateur régulier, le faisceau des automorphismes d'un élément (E, φ + ) ∈ [V reg G /(G × Z + )](S) au-dessus d'un point a : S → [C + /Z + ] est canoniquement isomorphe à a * J. La neutralité vient de l'existence de ǫ + . De même, on considère le schéma V λ T qui s'obtient de la même manière en considérant la flèche α |VT : V T → A r . On considère alors la flèche finie plate surjective, génériquement étale : θ : V λ T → C λ + . Nous avons introduit le diviseur discriminant D + = 2ρ.D ⊂ C λ + dans la proposition 2.18 dont le lieu de non-annulation s'identifie au lieu régulier semisimple. On le tire alors sur la base C λ + en un diviseur noté D λ . Pour a ∈ C λ + (O) ∩ C rs + (F ), on pose d(a) := val(a * D λ ). Nous allons donner une formule pour d(a). Soit t + = (π −w0λ , t) ∈ V λ T (F ) tel que θ(t + ) = a et où F est la clôture algébrique de F et O son anneau d'entiers. Par le critère valuatif, on a que t + ∈ V λ T (O) et on déduit : (7) d(a) = 2ρ, λ + val(det(Id − ad(t) : g(F )/g t (F ) → g(F )/g t (F ))). 3.4. Le théorème principal de dimension. On suppose G semisimple simplement connexe avec F = k((π)) et k algébriquement clos. On note G rs ⊂ G l'ouvert constitué des éléments réguliers semisimples. On considère la fibre de Springer : X λ γ = {g ∈ G(F )/K| g −1 γg ∈ Kπ λ K}. On pose d(γ) := val(det F (Id − ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))) et def(γ) = rg G − rg F G γ (F ) où rg F désigne le rang du plus grand sous-tore déployé du centralisateur G γ (F ) de γ et g γ (F ) son algèbre de Lie. Théorème 3.7. Soit γ ∈ G(F ) rs . On suppose X λ γ non vide. (i) X λ γ est un schéma localement de type fini. (ii) dim X λ γ = ρ, λ + 1 2 [d(γ) − def(γ)] . On démontre ce théorème dans la section 4.6. Regardons comment une fibre de Springer est reliée au semi-groupe de Vinberg. Considérons l'élément γ + = (π −w0λ , γ) ∈ G + (F ) et la fibre de Springer : X γ+ = {g ∈ G(F )/G(O)| g −1 γ + g ∈ V 0 G (O)}. Nous avons le lemme suivant : Lemme 3.8. La variété X λ γ s'identifie canoniquement à X γ+ . Démonstration. La variété X λ γ peut s'interpréter de la manière suivante : X λ γ = {g ∈ G(F )/K| ∀i, ρ ωi (g) −1 π ωi,−w0λ ρ ωi (γ)ρ ωi (g) ∈ M ni (O) ∩ GL ni (F )}. Ainsi, nous reconnaissons en l'élément π ωi,−w0λ ρ ωi l'image de γ + par ρ (ωi,ωi) et il résulte du lemme 3.1 que γ + est dans V 0 G (O). On a le corollaire du théorème 3.7 : Corollaire 3.9. Soit a ∈ C λ + (O) ∩ C rs + (F ), on considère la fibre de Springer M λ (a) = X ǫ+(a) introduite dans la définition 3.3, alors nous avons : dim M λ (a) = d(a)−c(a) 2 . où d(a) est la valuation du discriminant sur la base C λ + et c(a) = rg(T ) − rg F (J a (F )). Démonstration. En effet, l'élément ǫ + (a) s'écrit (π −w0λ , γ) pour un certain γ ∈ G(F ) rs . On applique alors la formule de dimension du théorème 3.7 ainsi que l'équation (7) qui exprime le discriminant de la base C λ + en fonction du discriminant sur G et de λ. On veut savoir quand est-ce que la fibre X λ γ est non vide. On commence par quelques rappels sur les points de Newton. Soit T un tore sur F , soit a ∈ T (F ), on définit l'élément ν a ∈ X * (T ) Q en imposant que : ∀ λ ∈ X * (T ), λ, ν a = val(λ(a)). On obtient donc une flèche surjective ν T : T (F ) → X * (T ) Q et en prenant les invariants sous Galois, nous obtenons une flèche : ν : T (F ) → X * (A T ) Q , où A T est le plus grand sous-tore déployé sur F de T . Soit γ ∈ G(F ) rs régulier semi-simple. En ce cas, son centralisateur est un tore maximal T défini sur F et on pose : ν γ := ν T (γ). Notons [ν γ ], sa classe de conjugaison sous W , représentée par un élément dominant de X * (A T ) Q , on l'appelle le point de Newton de γ. Démonstration. Posons γ + := (π −w0λ , γ). Comme [ν γ ] ≤ λ, il résulte de [20, Thm. 1.5.2], que a = χ + (γ + ) ∈ C λ + (O). On considère alors γ 0 := ǫ + (a). Il résulte de la proposition 3.2 que la fibre de Springer de γ 0 est non vide. Montrons que γ + et γ 0 sont conjugués ce qui montrera le théorème. On considère le schéma sur k((π)) suivant : T := {h ∈ G(F )| h −1 γ + h = γ 0 } C'est un torseur sous le centralisateur G γ+ (F ) qui est un tore et comme γ + est régulier semisimple et que nous avons χ + (γ) = χ + (γ 0 ), il admet localement des sections. D'après le théorème de Lang, tout torseur sous un schéma en tores sur k((π)) est trivial, on obtient donc un élément qui conjugue. Preuve du théorème principal 4.1. Décomposition de Jordan topologique. Nous dirons qu'un élément γ ∈ G(F ) est compact s'il engendre un sous-groupe relativement compact de G(F ). Nous dirons qu'il est topologiquement unipotent si γ p r → 1 quand r → +∞ dans End F (V ) où la topologie est celle induite par le corps F et V une représentation fidèle. Tout élément topologiquement unipotent est compact. De plus, un élément γ ′ ∈ G(F ) est dit absolument semi-simple s'il est d'ordre fini premier à p = car k. Enfin, on se donne Z(F ) est un sous-groupe fermé normal de G(F ). Nous dirons qu'un élément est topologiquement unipotent modulo Z (resp. absolument semi-simple modulo N ), si son image dans G(F )/Z(F ) est topologiquement unipotente (resp. absolument semi-simple). On définit de même la notion de compacité modulo Z. On a alors le théorème suivant dû à Spice [39,Prop. 2.41] : X λ γ = X λ,H γu . En particulier, on peut supposer que γ s = 1 et donc γ = γ u , ce que l'on fera dans la suite de ce travail. On pose γ + = (π −w0λ , γ), comme la fibre de Springer X γ+ = X λ γ est non vide par hypothèse, quitte à conjuguer, on peut supposer que γ ∈ V 0 G (O). On rappelle que : X γ+ := {g ∈ G(F )/G(O)| g −1 γ + g ∈ V 0 G (O)}. Un élément M ∈ End(V ) est dit quasi-unipotent si ses valeurs propres sont 0 ou 1 (le lieu d'annulation du polynôme X n (X − Id) n ). On dit d'un élément de g ∈ V G (O) qu'il est toplogiquement quasi-unipotent si g p r tend vers un élément quasi-unipotent quand r → +∞ dans End F (V ). En particulier, comme γ u est topologiquement unipotent modulo Z(F ), l'élément γ + est topologiquement quasi-unipotent. L'élément γ + est dit toplogiquement nilpotent s'il est toplogiquement quasiunipotent de limite nulle. Pour obtenir le théorème 3.7, suivant Kazhdan-Lusztig [19], nous avons besoin de démontrer l'équidimensionnalité d'une certaine variété de drapeaux associée. Comme l'élément γ + est topologiquement quasi-unipotent, il admet une partie nilpotente et une partie unipotente. Commençons par rappeler comment on traite la partie unipotente. On considère alors la variété B γ := {g ∈ B| g −1 γg ∈ I}. Soit∆ l'ensemble des racines simples du groupe de Weyl affine. Pour chaque racine α ∈∆, on a un paraboliqueP α := I ∪ Is α I. Soit le ind-schéma P α := G(F )/P α , on a un morphisme naturel π α : B → P α qui est une P 1 -fibration. Ses fibres seront appelées des droites de type α. Pour un point g ∈ B, on note P 1 α (g), la droite de type α passant par g. Kazhdan-Lusztig considèrent alors pour chaque α ∈∆, un fibré vectoriel E α au-dessus de B γ , dont la fibre au-dessus d'un IwahoriB est donnée par R u (B)/R u (P α ) où R u (B) désigne le radical pro-unipotent deB. Ils définissent alors une section s α de ce fibré vectoriel E α dont l'image en un pointB ∈ B est donnée par l'image de γ dans le quotient R u (B)/R u (P α ). Cette section a la propriété notable suivante ; soit g ∈ B γ , alors s α (g) = 0 ⇐⇒ P 1 α (g) ⊂ B γ . L'existence de cette section nous permet d'établir la proposition suivante : On s'intéresse maintenant au cas où e = e I,J avec J ∆. On a une suite exacte : Proposition 4.4. Soit Y une composante irréductible de dimension maximale de B γ . Soit L une droite de type α, incluse dans B γ et telle que L ∩ Y = ∅, alors il existe une composante irréductible Y ′ de dimension maximale de B γ telle que L ⊂ Y ′ .1 / / G e / / G(e) m / / G(e)/G e / / 1 La flèche m induit une surjection G e → G(e)/G e de noyau fini isomorphe à Z G e = Z Ge . En particulier, on obtient un isomorphisme canonique entre les groupes G(e)/G e et eG(e) ∼ = eG e ∼ = G e ad . On peut donc récrire la suite exacte sous la forme : (8) 1 / / G e / / G(e) m / / eG e / / 1 On note G e,+ le sous-groupe distingué correspondant dans G + (e), on a vu que l'élément eγ + ∈ eG e (O), on peut donc écrire une décomposition : γ = γ 1 γ 2 avec γ 2 ∈ G e (O) topologiquement unipotent et γ 1 ∈ G e,+ (O) topologiquement nilpotent dans G 0 e,+ (i.e. γ p n 1 → e). La suite exacte (8) est une extension de groupes réductifs connexes, on considère alors la flèche de projection G(e) → eG e et comme F = k((π)) à corps résiduel algébriquement clos, on obtient une flèche surjective entre les grassmaniennes affines : Gr G(e) → Gr eG e de fibre Gr Ge . On en déduit alors un isomorphisme non canonique entre les grassmaniennes affines : η : Gr Ge × Gr eG e → Gr G(e) . Proposition 4.8. La flèche η induit un isomorphisme : η : X Ge γ1 × X eG e eγ+ → X G(e) γ+ . Remarques : -Un élément de G +,e est une certaine matrice diagonale par blocs : A 0 0 Id . En particulier, si γ p n 1 → e, cela veut dire que la matrice A est bien topologiquement nilpotente. -C'est l'énoncé auquel nous faisions allusion lorsque nous voulions décomposer la fibre de Springer en une partie unipotente, ici celle qui correspond à eG e et une partie, dont nous allons voir qu'elle est la partie nilpotente. Démonstration. On commence par démontrer que η induit une application au niveau des fibres de Springer. Considérons la paire (g 1 , g 2 ) ∈ X Ge γ1 × X eG e eγ+ et g ∈ G(F ) l'élément canoniquement associé. Nous avons : g −1 γ + g = g −1 γ 1 gg −1 γ 2 g. Comme γ 1 ∈ G 0 e,+ (O), on a g −1 γ 1 g = g −1 1 γ 1 g 1 . Or, nous avons g 1 ∈ X Ge γ1 , on en déduit donc l'intégralité de g −1 1 γ 1 g 1 . Il nous faut voir l'intégralité de g −1 γ 2 g. Comme γ 2 ∈ G e (O), nous avons déjà que g −1 γ 2 g ∈ G e (F ). Nous avons l'égalité : g −1 γ 2 ge = g −1 2 γ 2 g 2 ∈ eG e (O) . Comme la flèche G e → eG e est de noyau fini qui s'identifie au centre Z G e , on obtient que g −1 γ 2 g ∈ G e (O) et donc g ∈ X G(e) γ+ . La flèche est alors bien injective. La surjectivité vient alors de l'égalité : g −1 γ + g = g −1 γ 1 gg −1 γ 2 g ∈ G 0 (e)(O). avec g −1 γ 1 g ∈ G e (F ) et g −1 γ 2 g ∈ G e (F ). On écrit alors g = g 1 g 2 avec (g 1 , g 2 ) ∈ G e (F ) × G e (F ). Comme G e (F ) ∩ G e (F ) = Z Ge qui est fini, cela force l'intégralité de g −1 γ 1 g et de g −1 γ 2 g ∈ G e (F ). A nouveau, g −1 γ 1 g = g −1 1 γ 1 g 1 et g −1 γ 2 g = g −1 2 γ 2 g 2 . Il nous suffit donc juste de considérer la paire (g 1 , eg 2 ). D'après le théorème 4.6, comme eγ + est topologiquement unipotent, on a : stable par conjugaison par B. C'est un idéal de B, i.e. on a : (10) ∀ b ∈ B, bN ⊂ N et N b ⊂ N. On pose alors R := ev −1 (N ) qui admet également une structure de monoïde. Pour une racine affine simple α, on considère alors R α := R ∩ s α Rs α ⊂P • α . On appelle R α le pro-radical nilpotent deP • α . Plus généralement, pour un élément w ∈ W af f , on définit de même R w = R ∩ wRw −1 . Lemme 4.10. Soit g ∈ B γ+ , on suppose que P 1 α (g) ⊂ B γ+ , alors γ + ∈ g R α . Démonstration. Soit γ ′ := g −1 γ + g, comme g ∈ B γ+ , on a γ ′ ∈ I • . Les semi-groupes d'Iwahori dê P α sont de la forme bs α I • s α b −1 avec b ∈ I. Comme P 1 α (g) ⊂ B γ+ , on obtient que γ ′ ∈ bs α I • s α b −1 pour tout b ∈ I. Ainsi, on obtient en particulier γ ′ ∈ s α I • s α et donc γ ′ ∈ R α , ce qu'on voulait. Pour pouvoir établir l'équidimensionnalité de B γ+ , il nous faut définir une section qui nous permette d'obtenir un analogue de la proposition 4.4. On commence par considérer les racines simples α ∈ ∆ qui correspondent au groupe de Weyl fini W . Soit α ∈ ∆ et P α le parabolique minimal associé à α et L α son Lévi. Soit P α (resp. L α ) son adhérence dans M . Si l'on plonge M dans End(V ), pour b ∈ P α , soit φ α (b) la matrice qui a le même facteur diagonal par bloc que la matrice b. En particulier, c'est l'application identité sur L α . Pour x ∈ P α , x = lu avec l ∈ L α et u ∈ R u (P α ). Alors, φ α (x) = φ α (l)φ α (u) = l ∈ L α . Ainsi, φ α (P α ) ⊂ L α . On obtient donc un morphisme de monoïdes : φ α : P α → L α , qui est l'identité sur L α . On a de plus un diagramme commutatif : Démonstration. On commence par montrer l'inclusion N α ⊂ N ∩ s α N s α . Soit x ∈ N α = φ −1 α (0), il resulte de la description de l'application φ α dans un semi-groupe End(V ) que la matrice x est nécessairement triangulaire supérieure, en particulier, on a x ∈ B et N α ⊂ B. On a même par commutativité du diagramme (11) que N α ⊂ N . Comme de surcroît, N α est un idéal de P α , on a : (11) B φ / / P α φα T / / L αs α N α s α ⊂ N α ⊂ N et N α ⊂ N ∩ s α N s α . Montrons l'inclusion réciproque. Soit x ∈ N ∩ s α N s α , il nous faut montrer que φ α (x) = 0. N ∩ s α N s α est dans l'adhérence de T (U ∩ s α U s α ) et nous avons que la flèche (φ α ) |N ∩sαN sα correspond à la partie dans T de la matrice x. Or, φ(x) = 0, donc φ α (x) = 0, ce qu'on voulait. Comme N α est un idéal de P α , c'est a fortiori un idéal de B et donc de N . Corollaire 4.12. Si α ∈ ∆, alors R α ⊂ R est un idéal de R et R α est stable par conjugaison par I. Démonstration. On a tout d'abord R α = ev −1 (N ∩ s α N s α ) et N α = φ −1 α (0) = N ∩ s α N s α est un idéal de N en vertu du lemme 4.11. Enfin, R α est stable par conjugaison par I car N α est stable par conjugaison par B comme c'est un idéal de P α . Le lemme suivant va nous permettre de définir notre section : Lemme 4.13. Le monoïde R α est contenu dans tous les semi-groupes d'Iwahori deP • α . Démonstration. En effet, un semi-groupe d'Iwahori deP • α s'écrit bs α I • s α b −1 avec b ∈ I. On a que R α est par définition stable par conjugaison par s α et on vient de voir qu'il est stable par conjugaison par I, on obtient alors que : s α b −1 xbs α ∈ R α ⊂ I • , d'où x ∈ bs α I • s α b −1 , ce qu'on voulait. On fait maintenant le quotient de R par R α , au sens de Rees. Soit S un semigroupe et I un idéal de S. On considère la relation d'équivalence ≡ suivante : x ≡ y ⇐⇒ x = y ou x, y ∈ I. Cette relation d'équivalence est compatible au produit du semigroupe et on forme le quotient S/I := S/ ≡. On applique cette construction pour former le semigroupe quotient R/R α . On considère alors le fibré E α au-dessus de B + γ+ dont la fibre en un point g est donnée par le quotient g R/ g R α . On construit une section s α de E α , dont l'image d'un point g ∈ B γ+ est donnée par l'image de γ + dans le quotient g R/ g R α . Comme R α est le radical pro-nilpotent deP • α , nous avons en vertu du lemme 4.13 pour g ∈ B γ+ : s α (g) = 0 ⇐⇒ P 1 α (g) ⊂ B γ+ . Il ne nous reste plus qu'à traiter le cas de la racine affine α 0 . L'inconvénient de la racine affine α 0 est que cette fois R α0 , comme le montre un calcul pour SL 2 , n'est plus un idéal de R. Nous ne pouvons donc pas faire le quotient au sens de Rees. Fort heureusement, le lemme suivant nous assure que la variété B γ+ ne contient pas de droites de type α 0 . On rappelle que s α0 = παˇsα oùα est l'unique racine la plus haute. En particulier, si on écrit α = m i α i comme combinaison linéaire de racines simples et que l'on considère une autre racine positive β = p i α i , on a pour tout i, p i ≤ m i . Proposition 4.14. On a l'inclusion R α0 ⊂ ev −1 (0). Démonstration. On a I = T (O)K 0 où on a posé K 0 := ev −1 (U ). Considérons également K 1 := ev −1 (1). On a que N est dans l'adhérence de I et en particulier, R ∩ s α0 Rs α0 est dans l'adhérence de T (O)K 0 ∩ s α0 K 0 s α0 . Il nous suffit donc de voir que K 0 ∩ s α0 K 0 s α0 ⊂ K 1 pour conclure, comme en réduction, on tombe dans N . Le groupe K 0 est engendré par le produit : T (1 + πO) × β>0 U β (O) × U −β (πO). Comme s α0 stabilise T (1 + πO), il nous suffit d'étudier les groupes radiciels. Soit une racine positive β et k ∈ N, regardons comment agit s α0 sur U β (π k O) : s α0 U β (π k O)s α0 = Ad(παˇ)U sαβ (π k O) Commeα est la plus haute racine, si β > 0 (resp. β < 0), on a sαβ < 0 (resp. sαβ > 0) et nous obtenons : (12) s α0 U β (π k O)s α0 = Ad(παˇ)U sαβ (π k+ sαβ,αˇ O). et sαβ,αˇ < 0. En particulier, s α0 U β (π k O)s α0 ∈ K 0 si et seulement si k ≥ k + sαβ,αˇ ≥ 1. Ainsi, nous avons k ≥ 1. De plus, si β est en revanche négative, on a toujours s α0 U β (πO)s α0 ∈ K 1 en vertu de l'égalité (12) valable pour une racine négative, ce qui conclut. Corollaire 4.15. Soit g ∈ B γ+ , alors g −1 γ + g / ∈ R α0 , en particulier, il n'existe pas de droite de type α incluse dans B γ+ . Démonstration. En effet, supposons par l'absurde qu'une telle droite existe. Posons γ ′ := g −1 γ + g. Alors, d'après le lemme 4.10, on a γ ′ ∈ R α0 , en particulier d'après la proposition 4.14, la réduction de γ ′ est nulle. Or, g ∈ B γ+ , donc γ ′ ∈ M 0 (O) et est donc de réduction non nulle, une contradiction. On a la proposition suivante de même preuve que [19, §4. Lem. 1] : Proposition 4.16. Soit Y une composante irréductible de dimension maximale de B γ+ . Soit L une droite de type α, incluse dans B γ+ et telle que L ∩ Y = ∅, alors il existe une composante irréductible Y ′ de dimension maximale de B γ+ telle que L ⊂ Y ′ . On démontre maintenant une proposition du même type que 4.5 : Proposition 4.17. Soient g, g ′ ∈ B γ+ alors il existe des droites L 1 , . . . , L n de type α 1 , . . . , α n dans B γ+ telles que g ∈ L 1 , g ′ ∈ L n et L i ∩ L i+1 = ∅. En particulier, B γ+ est connexe. Démonstration. Quitte à changer I • en gI • g −1 , on peut supposer que g = 1. On écrit alors g ′ = bw avec b ∈ I et w ∈ W af f . Prouvons le résultat par récurrence sur la longueur de w, q := l(w). Si w = s α , on pose v = b −1 γ + b. Nous avons alors v ∈ R α et également γ + . En particulier, γ + est dans tous les semi-groupes d'Iwahori deP • α , on obtient alors une droite de type α qui relie 1 et g ′ . Pour passer de q − 1 à q, on considère une décomposition réduite w = s 1 . . . s q . Soit α la racine correspondant à s q , alors wα < 0. A nouveau, en considérant v := b −1 γ + b, nous avons que v ∈ R w = R ∩ wRw −1 qui est le radical pro-nilpotent deP • α et de même u ∈ R w . A nouveau, γ + est dans tous les semi-groupes d'Iwahori deP • α , en particulier, γ + ∈ s q I • s q et ce semi-groupe d'Iwahori est relié à I • par une droite de type α. Maintenant, par hypothèse de récurrence, s q I • s q est relié par une chaîne de P 1 à g ′ I • (g ′ ) −1 , ce qui conclut. Démonstration. La preuve est la même que [19, §4. Prop. 1], que l'on rappelle pour mémoire. Soit Y une composante irréductible de B γ+ et g ∈ Y qui n'est dans aucune autre composante irréductible. Soit g ′ dans une composante irréductible Y 0 de dimension maximale d. On peut alors d'après la proposition 4.17 relier dans B γ+ , g et g ′ par des droites L 1 , . . . , L n , incluses dans B γ+ , telles que L i ∩ L i+1 = ∅ et g ∈ L 1 et g ′ ∈ L n . En vertu de la proposition 4.16 appliqué à (Y 0 , L n ), on peut trouver une composante irréductible Y 1 de dimension maximale d telle que L n ∈ Y 1 , en itérant le procédé, on trouve une suite de composantes irréductibles (Y 0 , . . . , Y n ) de dimension maximale telles que L n−i ∈ Y i . En particulier, comme g ∈ L 1 , on obtient que g ∈ Y n , donc Y = Y n et on obtient l'équidimensionnalité de B γ+ . 4.5. Le cas où γ est déployé. Supposons que γ est déployé dans T (F ). Il s'écrit alors γ = γ 0 π ν où γ 0 ∈ T (O) et ν ∈ X * (T ), que l'on peut supposer dominant, quitte à conjuguer. On veut montrer que : Proposition 4.19. Dans le cas où γ est déployé, on a la formule de dimension : dim X λ γ = ρ, λ + 1 2 d(γ). Le k-groupe U (F ) agit sur la grassmannienne affine Gr et les orbites sont indexées par ν ∈ X * (T ), que l'on note S ν . On a S 0 = U (F )/U (O). Nous avons le lemme tiré de [12, 2.5] : Passons donc à la preuve de la proposition 4.19. Démonstration. En identifiant U (F ) avec F r , par le biais des groupes radiciels U α , le morphisme f γ agit sur chacun de ces facteurs par : f γ (e α ) = (α(γ) − 1)e α . On en déduit donc d'après la proposition 4.22 que la dimension de f −1 γ (Kπ λ Kπ −ν ∩ U (F )) vaut : α>0 val(1 − α(γ)) + ρ, λ − ν . Maintenant, comme γ = γ 0 π ν ∈ T (F ), nous avons l'égalité : d(γ) = − 2ρ, ν + 2 α>0 val(1 − α(γ)). Ainsi, dans le cas déployé, la dimension de la fibre de Springer X λ γ est : ρ, λ + 1 2 val(det F (Id − ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))). 4.6. Fin de la preuve du théorème 3.7. On a vu que l'ouvert régulier X λ,reg γ est non vide et est une orbite sous le centralisateur de γ. Proposition 4.23. On a dim X λ,reg γ = dim X λ γ = dim B γ+ . De plus, la dimension du complémentaire de l'ouvert régulier est de dimension strictement plus petite que celle de dim X λ γ . Démonstration. On a une flèche projective lisse p : B → Gr et comme nous avons que γ + est topologiquement quasi-unipotent, elle induit une flèche surjective p : B γ+ → X γ+ = X λ γ . Au-dessus de X λ,reg γ , en vertu de la proposition 2.14 elle est finie et en dehors du lieu régulier, les fibres sont de dimension au moins un d'après la proposition 2.16. Comme B γ+ est équidimensionnelle, on en déduit que dim X λ,reg γ = dim B γ+ , ainsi que l'assertion sur la dimension du complémentaire. De plus, dim X λ γ ≤ dim B γ+ , donc nous avons l'égalité. Proposition 1.6. L'application de restriction φ : k[V G ] G → k[V T ] W est un isomorphisme de kalgèbres. De plus, V T /W est un espace affine de dimension 2r, dont les coordonnées sont données par les (α i , 0) et les χ i = Tr(ρ (ωi,ωi) ). G est simplement connexe, Steinberg construit dans[41, §7] un morphisme : ǫ : T /W → G qui est une section au morphisme χ. Pour un r-uplet (a 1 , .., a r ) ∈ T /W := A r , elle se définit de la manière suivante : où la flèche verticale de gauche est un isomorphisme et la flèche horizontale du bas, l'injection canonique. termes liés à d'autres vecteurs). Or, comme χ + (γ) = 0, on en déduit que pour tout i, a i = 0. Ainsi, on obtient que ρ ωi (γ) est une matrice nilpotente pour tout i. De plus, pour les représentations de dimension un, le fait que γ est dans la strate indexée par le vide, nous assure que c'est l'endomorphisme nul pour de telles représentations. Comme V G est fermé dans un certain End(V ), N = Nil(V ) ∩ V G où Nil(V ) est le cône nilpotent. Comme il est fermé, on a le résultat analogue pour N . Lemme 2 . 3 . 23Il existe un unique point base h I ∈ X I tel que pour tout cocaractère λ vérifiant λ(α) = 0 pour α ∈ I et λ(α) > 0 pour α ∈ ∆−I, on a λ(0) = h I . De plus, on a X I = (G ad ×G ad ).h I . On renvoie à [7, sect. 3] et [40, p. 73] pour la preuve de ce lemme. Les h I sont les images des idempotents e I de V 0 G de Vinberg. Suivant Lusztig [24, 12.3], , nous allons décrire une certaine partition de X. Pour I ⊂ ∆ et w ∈ W I , on pose : X I,w = G ad .[I, w, 1] où [I, w, 1] := (B ad × B ad ).wh I où l'action de B ad × B ad est donnée par la multiplication à gauche et à droite et G ad agit par conjugaison. Proposition 2. 7 . 7Le cône nilpotent N 0 est équidimensionnel de dimension dim G + −2r = dim G− r. Ses composantes irréductibles sont indexées par les éléments de Coxeter de W . Démonstration. Il résulte du diagramme cartésien ci-dessus que dim N 0 = dim N ad,∅ + r. Comme nous avons d'après le théorème 2.6 : N ad,∅ = w∈W supp(w)=∆ X ∅,w , il résulte du théorème 2.4.(ii) que la dimension N ad,∅ est égale à la dimension de X ∅,w lorsque w est un élément de Coxeter, autrement dit l(w) = r et dim(X ∅,w ) = dim G − 2r. Le calcul de dimension suit. Il ne nous reste plus qu'à voir l'assertion sur les composantes irréductibles. Elle résulte du théorème 2.4.(ii) et de la proposition suivante due à He [14, Prop 2.10] : G La flèche ρ est un J-torseur et i est quasi-fini. Par Steinberg [41, Th. 8.1] , on sait que i est un isomorphisme au-dessus de G reg + , on en déduit que i est birationnel et quasi fini et comme V reg G est normal, par le Main Theorem de Zariski, c'est une immersion ouverte. On note alors V w l'ouvert image. Maintenant, on pose : Proposition 2 . 13 . 213Le centralisateur régulier J est un schéma en groupes lisse et commutatif de dimension r. G où la flèche ψ est donnée par (x, y) → (xyx −1 , y) et la flèche du bas par la diagonale. Comme elle est équidimensionnelle entre schémas lisses, elle est plate. Ainsi, on déduit que la flèche i : I |V reg G → G × V reg G est une immersion régulière et donc I |V reg G est intersection complète. Comme de plus, au-dessus de V reg G , I est équidimensionnel sur une base lisse, il est plat et par changement de base J est également plat sur C + . Ce diagramme est en fait cartésien. En effet, la flèche χ reg+ est lisse, donc V reg G × C+ V T est lisse au-dessus de V T ,donc normal et Cohen-Macaulay. La flèche ι :est. De plus, ι est birationnelle car c'est un isomorphisme au-dessus de G rs + , il résulte alors du Main Theorem de Zariski que ι est un isomorphisme. On en déduit alors queṼ reg G est également Cohen-Macaulay et fini surjectif sur un schéma lisse, donc plat. On considère la fonction D + = (2ρ, D) sur k[T + ] où D = α∈R (1 − α(t)) est la fonction discriminant sur k[T ]. Comme W agit trivialement sur le premier facteur, on a que D + ∈ k[T + ] W et de plus on a la propriété suivante : t ∈ T rs + ⇐⇒ D + (t) = 0. Nous allons étendre cette fonction à V T : Lemme 2.18. La fonction D + s'étend en une fonction de k[V T ] W . a) → gǫ w + (a)g −1 dont nous avions vu qu'elle était d'image ouverte. On note V reg,ouvert image. En particulier, pour chaque élément de Coxeter w ∈ W , nous avons un centralisateur régulier J w lisse et de dimension r qui sont a priori différents deux à deux. On commence par la proposition suivante : Proposition 2.20. Il existe un unique schéma en groupes J, lisse et commutatif de dimension r sur C + muni d'un isomorphisme G + -équivariant : Proposition 2.22. [χ + ] : [V reg G /G] → C + est une gerbe liée par le centralisateur J et neutre. Démonstration. Nous avons vu que le morphisme était lisse. De par la caractérisation du centralisateur régulier, le faisceau des automorphismes d'un élément (E, φ + ) ∈ [V reg G /G](S) au-dessus d'un point a : S → C + est canoniquement isomorphe à a * J. La neutralité vient de l'existence de ǫ + . 3. Les fibres de Springer affines et leur interprétation modulaire 3.1. Grassmannienne affine. Soit F = k((π)) un corps local, d'anneau d'entiers O et de corps résiduel k algébriquement clos. Soit G connexe semisimple simplement connexe. Soit K = G(O). Proposition 3 . 2 . 32Soit S un k-schéma muni d'un T -torseur, h −w0λ : S → BT le morphisme vers le classifiant du tore T associé. Soit a : S → [C + /Z + ], alors la section de Steinberg et le choix d'une racine c-ième h −w0λ ′ de h −w0λ définit une section : Théorème 3 . 10 . 310Soit γ ∈ G(F ) rs . Alors X λ γ est non vide si et seulement si [ν γ ] ≤ λ. Remarque : Par Kottwitz-Viehmann [21, Cor. 3.6], on sait que si la fibre est non vide alors [ν γ ] ≤ λ. Il nous faut donc voir la réciproque. Théorème 4. 1 . 1Si γ ∈ G(F ) est compact modulo Z, alors il admet une décomposition de Jordan topologique modulo Z, i.e. on a l'écriture : γ = γ s γ u avec γ s absolument semi-simple modulo Z et γ u topologiquement unipotent modulo Z qui commutent entre eux et avec γ. Si Z est trivial, il y a de plus unicité.Nous appliquons donc ce théorème pour obtenir la proposition suivante :Proposition 4.2. Soit γ ∈ G(F ) régulier semi-simple, alors il existe un groupe de Lévi M tel que γ ∈ M (F ) et admette une décomposition de Jordan modulo un certain groupe Z M (F ). Démonstration. L'élément γ étant régulier semi-simple, s'il est anisotrope, il est compact et donc on peut prendre pour Z(F ) le groupe trivial, sinon l'élément γ est elliptique dans un certain Lévi M (F ) et donc compact modulo le centre de ce Lévi Z M (F ). On rappelle le théorème de Kottwitz-Viehmann de descente démontré sur C, mais dont la preuve vaut en caractéristique p également [21, Th. 3.5] : Théorème 4.3. Soit P un parabolique, on écrit P = M N où M est le Lévi. Soit γ ∈ M (F ) et X λ M , la fibre de Springer pour ce Lévi, alors si Ad(γ) agit sur Lie(N ) avec des pentes strictement positives (cf. [21, sect. 2.1]), l'injection canonique X λ M,γ → X λ γ est une bijection. Remarque : Le corollaire 3.6 de [21] nous assure que le point de Newton définit un parabolique P , sur lequel ad(γ) agit sur Lie(N ) via son point de Newton, avec des pentes positives. D'après ci-dessus, nous pouvons donc considérer la décomposition de Jordan topologique de notre élément γ ∈ G(F ) rs . Alors, γ = γ s γ u avec γ u topologiquement unipotent et γ s fortement semi-simple modulo le centre d'un certain Lévi M (F ).En utilisant Kottwitz-Viehmann, on sait que X λ γ,M = X λ γ . On se ramène donc au cas du Lévi. Posons H = C G (γ s ). On peut choisir un sous-groupe parabolique P ′ de M dont le facteur de Lévi est H = C G (γ s ) et on réapplique le théorème de Kottwitz-Viehmann. On obtient alors un isomorphisme : 4. 2 . 2Le cas topologiquement unipotent. L'élément γ + est topologiquement unipotent si et seulement si λ = 0. Dans ce cas, on a γ = γ + ∈ G(O). On a une flèche ev : K → G et on pose I := ev −1 (B) et B = G(F )/I, la variété de drapeaux affine associée qui classifie les IwahoriB. Ils démontrent également la propriété suivante sur B γ , [19, §4. Lem. 2] : Proposition 4.5. Soient g, g ′ ∈ B γ , alors il existe des droites L 1 , . . . , L n de type α 1 , . . . , α n dans B γ telles que g ∈ L 1 , g ′ ∈ L n et L i ∩ L i+1 = ∅. En particulier, B γ est connexe.De ces deux propositions, il résulte formellement le résultat suivant [19, §4. Prop. 1] : Théorème 4.6. La variété B γ est équidimensionnelle. Pour faire le lien avec le cas qui nous intéresse, on commence par donner une autre formulation des éléments quasi-unipotents due à Putcha. Soit M un monoïde réductif, normal et intègre (cf. (1.1)), de groupe des inversibles G + . Soit un idempotent e = e 2 ∈ M , on considère alors H(e) le groupe des inversibles du monoïde algébrique eM e. Ce groupe s'identifie à eC G+ (e). Il peut éventuellement être vide si e = 0. On définit une relation H dite de Green par : aHb si aM = bM et M a = M b. Nous avons alors d'après [31, sect. 2], qu'un élément u ∈ M est quasi-unipotent s'il existe un entier i et un idempotent e = e 2 ∈ M tel que u i He et ue est unipotent dans le groupe H(e). Si M admet un zéro, alors tout élément nilpotent est clairement quasi-unipotent. En particulier, si γ p n + → e, nous avons que γ + e ∈ H(e)(O) et qui est topologiquement unipotent, ce qui nous permettra d'isoler dans la fibre de Springer une partie 'unipotente', pour se concentrer ensuite sur la partie nilpotente. Nous allons maintenant montrer dans la section suivante comment se ramener à ce cas : 4.3. Réduction au cas topologiquement nilpotent. On a une flèche ev : V G (O) → V G et on pose I • = ev −1 (V B ). En particulier, quitte à changer de Borel, on peut supposer que γ + ∈ I • . Comme γ + est topologiquement quasi-unipotent, nous avons γ p n + → e avec e 2 = e ∈ E(V T ) où E(V T ) est l'ensemble des idempotents de V T . De plus, on a γ + e = eγ + , on note alors G(e) := C G (e) le centralisateur de e dans G et G + (e) = C G+ (e). Soit G 0 + (e) l'adhérence de G + (e) dans V 0 G . Nous avons une description des éléments idempotents de E(V T ) d'après [31, p. 434] et Vinberg [44, Thm. 7] : E(V T ) = w∈W (I,J)∈(∆×∆) * we I,J w −1 , où (∆ × ∆) * est le sous-ensemble des racines simples ∆ × ∆ qui correspond aux paires (I, J) essentielles (cf. [44, Déf. 4] ). Si I = J = ∅, nous avons e I,J = 0. Il résulte alors de la description des stabilisateurs des e I,J [44, Thm. 7] et [36, Thm. 21] que le centralisateur d'un idempotent we I,J w −1 est donné par : C G+ (we I,J w −1 ) = wL M w −1 , où L M est le Lévi associé à M et M = (I ∩ J 0 ) ∪ c J, où c J est le complémentaire de J dans ∆ et J 0 ⊂ J l'intérieur de J, i.e. le sous-ensemble de J qui consiste en les éléments dont les arêtes correspondantes dans le diagramme de Dynkin ne sont pas adjacentes à une arête correspondant à un élément de c J. On remarque que si e I,J ∈ Z + , alors on a M = ∆ et L M = G. Dans la suite, on peut supposer sans restreindre la généralité que e = e I,J pour une certaine paire (I, J). On considère alors la fibre de Springer analogue dans le Lévi G(e) := C G (e) et en appliquant le théorème 4.3, on se ramène à étudier la fibre de Springer pour G(e) : X γ+ = X G(e) γ+ := {g ∈ G(e)(F )/G(e)(O)| g −1 γ + g ∈ G 0 + (e)(O)}. et e ∈ Z G+(e) . Nous avons alors dans G(e) un sous-groupe distingué : G e := {g ∈ G(e)| ge = eg = e} 0 et en considérant le groupe G e := C G(e) (G e ) 0 , nous obtenons une décomposition de G(e) d'après [17, Thm. 27.5] : G(e) = G e G e = G e G e . De plus, nous avons G e ∩ G e = Z(G e ) = Z(G e ), nous distinguons alors deux cas d'après [44, Thm. 7] : e = e I,∆ et G e = {1}, e = e I,J avec J ∆, alors G e = {1} est semisimple et Z(G e ) est fini. Si e = e I , alors on obtient que le morphisme de groupes θ : G(e) → eG(e) est un isomorphisme et on a un isomorphisme entre les fibres de Springer affines : comme nous avons maintenant que eγ + ∈ eG(e)(O), on peut appliquer le théorème 4.6 pour obtenir Théorème 4 . 7 . 47Si e = e I , alors la variété B eG(e) eγ+ est équidimensionnelle. Proposition 4. 9 . 9La variété B eG e eγ+ est équidimensionnelle. Il ne nous reste plus qu'à étudier la partie X Ge γ1 avec γ 1 topologiquement nilpotent. 4.4. L'équidimensionnalité de B γ+ . Dans la section précédente, nous nous sommes ramenés à l'étude d'un élément γ + ∈ M 0 (O)∩H + (F ) rs topologiquement nilpotent où M est monoïde réductif, intègre, de groupe des inversiblesH avec M ⊂ V G et M 0 := M ∩ V 0 G . Madmet de plus un zéro et d'après [4, Cor. 2.2.5], , il est également normal. On a la fibre de Springer associée :X γ+ := {g ∈ H der (F )/H der (O)| g −1 γ + g ∈ M 0 (O)}.On se donne une paire de Borel (B, T ) de M . Soit ev : M (O) → M . On pose I := ev −1 (B) et I der := ev −1 (B ∩ H der ). Si B, désigne l'adhérence de B dans M , nous définissons le semi-groupe I • = ev −1 (B) (resp. l'ouvert I • 0 := ev −1 (V 0 B )) que nous appelons le semi-groupe d'Iwahori de groupe des inversibles I. On considère alors la variété de drapeaux : B γ+ = {g ∈ H der (F )/I der | g −1 γ + g ∈ I • 0 }, dont nous voulons démontrer l'équidimensionnalité. Soit∆ l'ensemble des racines simples du groupe de Weyl affine. Pour chaque racine α ∈∆, on a un semi-groupe paraboliqueP • α := I • ∪ I • s α I • , de groupes des inversiblesP α := I ∪ Is α I. D'après [31, Thm. 2.6] , on a un morphisme de monoïdes : φ : B → T , qui prolonge la flèche de projection B → T . On considère alors le monoïde algébrique : (9) N := φ −1 (0). où la flèche horizontale du bas est l'inclusion canonique. On considère alors N α := φ −1 α (0). A nouveau, N α est un idéal (cf.(10)) de P α . Nous avons le lemme suivant. Lemme 4 . 11 . 411On a l'égalité :N α = N ∩ s α N s α , et en particulier N α est un idéal de N . Théorème 4 . 18 . 418La variété B γ+ est équidimensionnelle. Lemme 4. 20 . 20Si Y un sous ensemble localement fermé de X et stable par T (F ), alors :dim(Y ) = dim(Y ∩ S ν ), ν ∈ X * (T ).On est donc ramené à étudier la dimension de :Y γ,λ := {g ∈ U (F )/U (O)| g −1 γg ∈ Kπ λ K}. En particulier, si nous notons f γ : U (F ) → U (F ) défini par : f γ (n) = n −1 γnγ −1 , nous avons que Y γ,λ = f −1 γ (Kπ λ Kπ −ν ∩ U (F ))/U (O).Définition 4.21. On dit qu'un sous-ensemble Y de U (F ) est admissible s'il existe m, n ∈ N tel que Y ⊂ U (π −m O) et qu'il est l'image réciproque d'une certaine sous-variété de U (π −m O)/U (π n O). Pour un ensemble admissible Y de U (F ), on choisit n ≥ 0 tel que Y est stable par multiplication par U (π n O) et on pose : dim Y := dim(Y /U (π n O)) − dim(U (O)/U (π n O)), qui est bien indépendant de n. Calculons la dimension de Kπ λ Kπ −ν ∩ U (F ) qui se déduit des calculs de Mirkovic-Vilonen [27] : Proposition 4.22. [12, Prop. 2.14.2] Nous avons la formule de dimension suivante, au sens des ensembles admissibles :dim Kπ λ Kπ −ν ∩ U (F ) = ρ, λ − ν . Table des matières desIntroduction in English 2 Introduction 4 1. Le semi-groupe de Vinberg et sa section 6 1.1. Rappels sur le semi-groupe de Vinberg 6 1.2. Le morphisme de Steinberg étendu χ + 8 1.3. Prolongement de la section de Steinberg 9 1.4. Construction du centralisateur régulier 13 2. Propriétés du morphisme de Steinberg étendu χ + 15 2.1. Etude du cône nilpotent 15 2.2. La lissité du morphisme χ reg + 18 2.3. Une construction alternative du centralisateur 23 3. Les fibres de Springer affines et leur interprétation modulaire 24 3.1. Grassmannienne affine 24 3.2. Une interprétation modulaire 25 3.3. Symétries d'une fibre de Springer affine 27 3.4. Le théorème principal de dimension 28 4. Preuve du théorème principal 29 4.1. Décomposition de Jordan topologique 29 4.2. Le cas topologiquement unipotent 31 4.3. Réduction au cas topologiquement nilpotent 32 4.4. L'équidimensionnalité de B γ+ 34 4.5. Le cas où γ est déployé 37 4.6. Fin de la preuve du théorème 3.7 38 Références 39 après Drinfeld[10], nous avons la proposition suivante : Proposition 1.4. Supposons car(k) > 3, alors la flèche ζ T est un isomorphisme.Démonstration. Tout d'abord, il résulte de [5, Cor. 6.2.14] que V T est normal. Le semi-groupe S des poids qui définit V ♭ T est engendré par les vecteurs (α i , 0) et les (ω i , λ) pour λ un poids de V ωi . Comme car(k) > 3, d'après Premet[29] et le théorème de Curtis-Steinberg[2, Th. 3.3], le semigroupe S est le même qu'en caractéristique nulle et en caractéristique nulle, V ♭ T est normal associé au cône : Théorème 1.5. On considère l'action adjointe de G sur lui-même. On a l'isomorphisme suivant, qui s'obtient par restriction des fonctions :1.2. Le morphisme de Steinberg étendu χ + . On rappelle le théorème de Chevalley-Steinberg ([3, VI.3.1-Ex1] et [41, Th. 6.1] ) : 44, Th. 6-7] et Rittatore [36, Th. 21] : Proposition 1.11. est ouvert. Nous avons besoin d'une description plus précise des strates avant d'aller plus avant, on utilise pour cela la proposition tirée de Vinberg [44, Th. 7] et étendue par Rittatore [36, Th. 21] : Proposition 1.15. On définit l'ouvert régulier V reg G Par EGA IV 9.3.2, cet ensemble est constructible et par EGA IV 13.1.1, comme dim G + − 2r est la dimension de la fibre générique de χ + , U est stable par générisation, donc c'est un ouvert. L'argument standard utilisé ci-dessus, nous montre que U = C + .→ C + est lisse. La flèche χ + : V G → C + est à fibres géométriquement réduites et Cohen-Macaulay et les éléments réguliers forment un ouvert dense dans chaque fibre.Enfin, d'après Brion-Kumar [5, 6.2.9 et 6.2.11], V G est de Cohen-Macaulay. Ainsi, la base étant lisse et les fibres étant toutes de même dimension, on en déduit que le morphisme est plat. Proposition 2.12. Le morphisme χ reg + : V reg G [ǫ + ](a) ∈ [V reg G /(G × Z + )]. Nous faisons alors la définition suivante : Définition 3.3. On définit la fibre de Springer affine M λ (a) (resp. M reg λ (a)) comme le foncteur dont le groupoïde des R-points pour une k-algèbre R est : {gCelui-ci étant non vide par définition.Remarque : La condition que h a soit au-dessus de h −w0λ , nous donne que γ 0 = (π −w0λ , γ), pour un certain γ ∈ Kπ λ K. Nous verrons par la suite que si a ∈ C + (O) ∩ C + (F ) rs , cet ensemble est un schéma localement de type fini dont nous calculerons la dimension.Démonstration. Soit (E, φ + ) ∈ M λ (a)(k) avec un isomorphisme générique β avec la section de Steinberg (E 0 , γ 0 ) où E 0 est le torseur trivial. La donnée de (E, β) nous fournit un élément g ∈ G(F )/K. Pour obtenir une section sur Spec(O), l'isomorphisme avec la section de Steinberg nous donne g −1 γ 0 g ∈ V 0 G (O). 3.3. Symétries d'une fibre de Springer affine. On rappelle que I := {(g, γ) ∈ G×V G | gγg −1 = γ}. Dans la section 2.3, nous avons défini le centralisateur régulier J, dont nous avons vu qu'il était muni d'un morphisme : χ * + J → I, qui est un isomorphisme au-dessus de V reg G . Nous formons le carré cartésien suivant :On se donne alors une section h a : X → C λ + , nous avons l'image réciproque J a = h * a J. Définition 3.5. Considérons le groupoïde de Picard P (J a ) au-dessus de Spec(k) qui associe à toute k-algèbre R, le groupoïde des J a -torseurs sur R[[π]], munis d'une trivialisation sur R((π)).On définit une action du champ P (J a ) sur M λ (a). En effet si (E, φ + ) ∈ M λ (a)(R), on a un morphisme de faisceaux : SoitF l'extension qui déploie γ de degré n. On sait que γ est stablement conjugué à un élément de T (F ). On note : X λ γ := {g ∈ G(F )/G(O)| g −1 γg ∈ Kπ λ K} (respX λ γ , la même chose dansF ). Le centralisateur de γ agit sur X λ γ etX λ γ . Si g est un point de X λ γ (resp.X λ γ ), on note O g (resp.Õ g ) l'orbite pour G γ (F ) (resp. G γ (F )). 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Thèse de doctorat, Institut Fourier, Grenoble, France, 1997. http ://www-fourier.ujf-grenoble.fr. Algebraic Monoids and group embeddings. A Rittatore, Transformation groups. 3A. Rittatore. Algebraic Monoids and group embeddings. Transformation groups, vol. 3, no 4, 375-396 (1998). Classes unipotentes et sous-groupes de Borel. N Spaltenstein, Lecture Notes in Maths. 946SpringerN. Spaltenstein. Classes unipotentes et sous-groupes de Borel. Lecture Notes in Maths. 946, Springer, 1982. . L Spice, Topological Jordan decompositions. J. Algebra. 3198L. Spice. Topological Jordan decompositions. J. Algebra 319, no. 8, 3141-3163 (2008). Some results on compactifications of semisimple groups. T A Springer, Proceedings of the International Congress of Mathematicians. the International Congress of MathematiciansMadrid, Spain2T.A. Springer. Some results on compactifications of semisimple groups. Proceedings of the International Congress of Mathematicians, vol. 2, 1337-1348, Madrid, Spain, 2006. Regular elements of semisimple algebraic groups. R Steinberg, Publ. Math. IHES. 25R. Steinberg. Regular elements of semisimple algebraic groups. Publ. Math. IHES 25, 49-80 (1965). Vanishing theorem for group compactifications. E A Strickland, Math. Ann. 277E.A. Strickland. Vanishing theorem for group compactifications. Math. Ann. 277, 165-171 (1987). The dimension of some affine Deligne-Lusztig varieties. E Viehmann, Ann. Sci. Ecole Norm. Sup. 394E. Viehmann. The dimension of some affine Deligne-Lusztig varieties. Ann. Sci. Ecole Norm. Sup.(4) 39, 513-526 (2006). On reductive algebraic semigroups. E B Vinberg, Lie Groups and Lie Algebras : E. B.Dynkin's Seminar. 2Advances in the Mathematical SciencesE.B. Vinberg. On reductive algebraic semigroups. Lie Groups and Lie Algebras : E. B.Dynkin's Seminar, AMS Translations Series 2, Vol 169, Advances in the Mathematical Sciences. F-91405 Orsay Cedex France E-mail : alexis. Mathématiques, Bâtiment. 425bouthier@math.u-psud.frMathématiques, Bâtiment 425, F-91405 Orsay Cedex France E-mail : alexis.bouthier@math.u-psud.fr

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{'abstract': "Cet article établit une formule de dimension pour les fibres de Springer affines dans le cas des groupes. On suit la méthode initiée par Bezrukavnikov dans le cas des algèbres de Lie. Elle consiste en l'introduction d'un ouvert régulier suffisament gros dont on montre qu'il est de même dimension que la fibre de Springer affine entière. On montre que dans le cas des groupes, un tel ouvert régulier avec des propriétés analogues, existe. Sa construction passe par l'introduction du semi-groupe de Vinberg V G pour lequel nous étudions un morphisme 'polynôme caractéristique' et étendons les résultats précedemment établis par Steinberg pour les groupes.3Finally, in the last section, we show the theorem 1. Following Kazhdan-Lusztig, we need to study the equidimensionnality of a corresponding flag variety. It implies to study more deeply the nilpotents and quasi-unipotents elements of the Vinberg's semigroup. Once this result is obtained, it is sufficient to deduce the dimension of the regular open subset which have the same dimension of the whole Springer fiber and conclude about the dimension formula.Proposition 1.14. Pour un sous-ensemble J ⊂ ∆, soit l'orbite O J , on considère les sous-groupes paraboliques P J et P − J . On a une décomposition de Lévi P J = L J R u (P J ) et on note δ (resp δ − ) la projection de P J sur L J (resp P − J sur L J ). L'orbite O J a un idempotent distingué e J . Son stabilisateur H J s'identifie à :H J = {(x, y) ∈ P J × P − J | δ(x)δ − (y) −1 ∈ T J,∆ }, où T J,∆ = {t ∈ T ∆ | α j (t) = 1, j ∈ J}.", 'arxivid': '1203.0975', 'author': ['Alexis Bouthier '], 'authoraffiliation': [], 'corpusid': 53121906, 'doi': '10.1007/s00031-015-9326-9', 'github_urls': [], 'n_tokens_mistral': 45047, 'n_tokens_neox': 41389, 'n_words': 22226, 'pdfsha': '69242f16d558c8ccb2a88193cbbf40d2a2f9d6a4', 'pdfurls': ['https://arxiv.org/pdf/1203.0975v3.pdf'], 'title': [], 'venue': []}

arxiv

Artificial Persuasion in Pedagogical Games [A Book Draft] 23 Jan 2016 Zhiwei Zeng Artificial Persuasion in Pedagogical Games [A Book Draft] 23 Jan 2016 A Persuasive Teachable Agent (PTA) is a special type of Teachable Agent which incorporates a persuasion theory in order to provide persuasive and more personalized feedback to the student. By employing the persuasion techniques, the PTA seeks to maintain the student in a high motivation and high ability state in which he or she has higher cognitive ability and his or her changes in attitudes are more persistent. However, the existing model of the PTA still has a few limitations. Firstly, the existing PTA model focuses on modelling the PTA's Goal Net Interpreter PTA Control Event Control FCM Calculation UI Control Database Access Table of Contents List of Tables ability to persuade, while does not model its ability to be taught by the student and to practice the knowledge it has learnt. Secondly, the quantitative model for computational processes in the PTA has low reusability. Thirdly, there is still a gap between theoretical models and practical implementation of the PTA. To address these three limitations, this project proposes an improved agent model which follows a goal-oriented approach and models the PTA in its totality by integrating the Persuasion Reasoning of the PTA with the Teachability Reasoning and the Practicability Reasoning. The project also proposes a more abstract and generalized quantitative model for the computations in the PTA. With higher level of abstraction, the reusability of the quantitative model is also improved. New system architecture is introduced to bridge the gap between theoretical models and implementation of the PTA. An instance of the PTA is also implemented and embedded into a 3D pedagogical game called VS Saga to demonstrate the practice of instantiating a PTA from the proposed agent model and system architecture. A focus group study has been conduct to assess the PTA qualitatively. The results of the focus group study are quite positive, showing the PTA has the ability to motivate and engage the student by generating personalized feedback and the potential to help the student develop positive attitudes towards learning. Positive feedback has also been received for the completeness of the new agent model and the reusability of the proposed system architecture. As for future studies, quantitative assessment of the PTA still needs to be carried out in order to analyze the Chapter 1 Introduction to Persuasive Teachable Agent effectiveness of the PTA statistically. The PTA also needs to build up its flexibility to cater to different learning topics. Chapter 1 Introduction to Persuasive Teachable Agent Motivation Teachable Agents (TAs) are educational agents which have emerged from the interdisciplinary research in education, computer science and psychology. The design and development of a Teachable Agent is enlightened by the Learning-by-Teaching pedagogy which entails learning by teaching others [1]. The TA plays the role of a tutee which requires tutoring from the student. During the process of teaching the agent tutee, the student can enhance his or her comprehension and reflection of content knowledge, acquire inquiry skills and stay motivated due to his or her sense of responsibility towards the agent tutee [2]. In order to harness the benefits of Learning-by-Teaching, researchers have designed various TAs, such as Betty's Brain [3] and SimStudent [4]. However, these TAs generally lack of the ability to interact with the student spontaneously and maintain the student's attention. They are also generally incapable of giving personalized responses or feedback to their student tutors. More recently, attempts have been made to improve on these drawbacks. Affective Teachable Agent improves TA's spontaneity and believability by incorporating goal orientation feature and generating emotional responses according to the student's teaching [5]. Dynalearn is another project which improves TA's ability to generate feedback and recommendations by making use of a qualitative reasoning model [6]. Most of the aforementioned TAs would provide feedback to the student based on their learning outcome. There are two limitations with learning with these TAs: (1) the student may not have mastered the topic himself or herself when teaching; (2) the student is not fully motivated to teach the TA. Chapter 1 Introduction to Persuasive Teachable Agent A Persuasive Teachable Agent (PTA) is a special TA which has the ability to be taught, to practice knowledge learnt and to persuade the student to teach it [20]. Taking a different approach from the aforementioned attempts, a PTA seeks to induce long lasting attitude changes of the student towards learning by incorporating a persuasion theory [2]. It also seeks to improve TA's responsiveness and the quality of feedback generated. According to the PTA model that Lim has proposed in her study [20], the motivation and ability level of the student is tracked and measured constantly while he or she is interacting with and teaching the PTA. The PTA generates appropriate and personalized persuasion cues based on the student's measured motivation and ability level, in order to maintain the student in a state in which he or she has higher cognitive ability and his or her changes in attitudes are more persistent [2]. The PTA addresses several issues of previous TAs, including lack of spontaneity and personalized feedback. However, it still has a few limitations. Firstly, the PTA model focuses on modelling PTA's ability to persuade, while does not model its ability to be taught and to practice. Secondly, the quantitative model for computational processes in the PTA is difficult to reuse. Thirdly, there is still a gap between theoretical models and practical implementation of PTA. This project looks into these limitations, and aims to address them. With the rise of consumer games on various devices, including PCs, smartphones and tablets, pedagogical games are also gaining popularity. Researches have suggested that videogames could become powerful learning tools which could be adapted to incorporate different curriculums [7]. Videogames create immersive learning environments with appealing video and audio effects and most of them are created in 2D or 3D environments. Studies on pedagogical videogames indicate that they can help the student master content knowledge [8] and acquire practical skills [9]. Besides, the virtual environments in videogames provide a digital learning ambience in which various types of student behavioral data can be tracked and recorded. Chapter 1 Introduction to Persuasive Teachable Agent Teachable Agents can also be embedded into pedagogical videogames [10]. TAs appear in the gaming environment as avatars which are capable of interacting with students and performing actions [11]. Based on the student's behavioral data collected in virtual gaming environments, videogames enable the embedded TAs to provide more immediate and interactive feedback to the student. More specifically, in terms of Persuasive Teachable Agents, collected data can also be used to direct the selection of appropriate persuasion strategies. As mentioned before, the existing PTA model still has a few limitations. This project aims to address the limitations by: (1) Redefining a PTA agent model that depicts PTA's ability to persuade, integrated with its ability to learn and to practice (2) Proposing an improved quantitative model for computational processes in PTA (3) Introducing system architecture to guide PTA implementation. Considering the favorable effects that educational games and virtual environments have on learning, an instance of the PTA is also implemented and embedded into a pedagogical game in 3D virtual environment. This instance can be used subsequently for the assessment and evaluation of the PTA. Research Objectives By incorporating the idea of persuasion, the PTA can achieve greater spontaneity, generate more personalized feedback and induce long lasting attitude changes towards learning. However, the existing PTA model has several limitations: (1) Persuasion Reasoning of PTA is not integrated with Teachability and Practicability Reasoning. A PTA should be able to be taught, to practice and to persuade. The existing model focuses on depicting the PTA's ability to persuade, but does not describe the PTA's ability to be taught and to practice.  To assess the PTA. As an instance of the PTA has been developed, it can be used for the assessment and evaluation of the PTA. Chapter 1 Introduction to Persuasive Teachable Agent Contributions The main contributions of this project include:  A focus group study to assess the PTA. A focus group study including a group of 6 participants has been conducted to assess the PTA qualitatively. In order to collect feedback and comments on various aspects of the PTA, the focus group is consisted of secondary school teachers, agent researcher and game developers.  Report Organization The chapters of this report are organized as follow: Chapter 1 Introduction to Persuasive Teachable Agent Chapter 2 reviews related researches, studies and experiments in the area of Teachable Agent and Persuasive Teachable Agent. As the implemented PTA is embedded within a game in 3D environment, related work on educational games and virtual environments are also presented. Chapter 3 redefines PTA and introduces the improved agent model. Chapter 4 introduces the proposed system architecture. Chapter 5 demonstrates the practice of implementing a PTA following the proposed agent model and system architecture. Chapter 6 presents the case studies of the implemented PTA. It also describes the approach to assess the PTA and summarizes the assessment results. Chapter 7 concludes this project and discusses possible areas for future work. Chapter 2 Literature Review Teachable Agents The idea of Teachable Agent (TA) is inspired by the Learning-by-Teaching pedagogy which entails learning by teaching others [1]. During a peer tutoring program, Gaustard found that student tutors usually benefited more or at least the same as their peer tutees [12]. With the development of artificial intelligence, the TA is developed to take the role of a tutee and can be taught by students. Generally, the TA facilitates and benefits learning process from the following three perspectives [11]: (1) Betty's Brain is an agent developed to facilitate the learning of middle school natural science topics [3]. The student can teach Betty by completing concept maps. Concept maps help the student to structure and reorganize his or her knowledge while teaching. The student is able to assess Betty's learning by querying Betty or put her through a quiz and observe the answers or the quiz outcome. SimStudent is another TA which simulates a classroom learner and can be taught in the domain of mathematics [4]. SimStudent is able to learn cognitive skills from the student by analyzing his or her inputs. The student tutors the agent by solving an algebraic equation collaboratively with the agent in a stepwise fashion. A test can be used to assess SimStudent's learning performance after tutoring. However, both Betty's Brain and SimStudent interact with students passively and are incapable of giving personalized feedback. Affective Teachable Agent is Chapter 2 Literature Review introduced to improve the spontaneity and believability of the TA [5]. Affective Teachable Agent has incorporated goal-orientation feature. While pursuing its predefined goals, the agent is able to take spontaneous actions to interact with the student. The agent has also employed affective computing by simulating emotions to build an emotional tie with the student. DynaLearn is an agent designed to facilitate the learning of conceptual knowledge in science subjects. The student teaches the agent by creating conceptual models in a graphical editor. The agent goes through qualitative reasoning to generate feedback and recommendations to the student. Similar to Affective Teachable Agent, DynaLearn also simulates simple emotions to bind the student and the agent. Each of the four aforementioned attempts has made their own innovations to improve learning experiences. However, they also have their own drawbacks. A summary of these TAs are presented in Table 2.1 below. To address these two issues that most existing TAs have, an improved TA model should have the ability to provide responses and feedback based on the student's ability and motivation, which ensures that the student possesses necessary ability required for teaching the topic and is constantly motivated to teach. Persuasive Teachable Agents A Persuasive Teachable Agent (PTA) is a special TA which has the ability to be taught, to practice and to persuade the student to teach it. Persuasion is a kind of interaction between human beings, which aims to influence others by changing their attitudes [13]. Originated from the psychology and the communications domain, persuasion theories have increasing applications in the human computer interaction domain, in the attempt to induce attitude changes of human beings. The PTA seeks to induce long lasting attitude changes of students towards learning by incorporating a persuasion theory [2]. Chapter 2 Literature Review The persuasion theory that has enlightened the idea of the PTA is the Elaboration Likelihood Model (ELM). The ELM is a dualistic model, which states that there are two routes of persuasion, namely the central route and the peripheral route [14]. According to the ELM, one is more willing to go through complicated cognitive processes if he or she is motivated and possesses the necessary abilities required to perform a behavior. After deep cognitive processes, one is likely to go through the central route of persuasion where his or her attitude changes tend to be enduring. On the contrary, one would avoid complicated cognitive processes if he or she is not motivated or does not possess the necessary abilities. Without any deep consideration, one is likely to go through the peripheral route of persuasion where the resulting attitude changes tend to be less enduring. PTA has two major characteristics: (1) Goal-orientation. Similar to Affective Teachable Agent [5], the PTA is goaloriented. The primary goal of the PTA is to keep students on the central route of persuasion, so that their positive attitude changes towards learning and the knowledge they obtained could be more enduring. The PTA acts towards its goal by persuasion. (2) Ability to persuade. The student's motivation and ability level are tracked and measured constantly while he or she is interacting with and teaching the PTA. According the ELM, when the student is not motivate or does not possess the abilities required to teach the PTA, he or she is likely to go through the peripheral route of persuasion. In this situation, the PTA generates appropriate and personalized persuasion cues to direct the student back on to the central route. With its goal-orientation and persuasion features, the PTA is able to provide immediate and personalized feedback based on the student's ability and motivation, which previous TAs are incapable of. Chapter 2 Literature Review In her study, Lim has propose a PTA model [2], as shown in Figure The Teachability Reasoning has two cycles, the learning cycle and the reasoning cycle. In this two cycles, the PTA learns from the student tutor and responds to his or her inquiries respectively. The Persuasion Reasoning determines the student tutor's motivation and ability level according to the ELM and Fuzzy Cognitive Map (FCM) [15]. The Persuasion Reasoning is modelled based on Goal Net [16], a methodology for modelling goal-oriented agents. However, as mentioned before, the existing PTA model still has a few limitations. After reviewing the existing PTA model, these limitations can be described more specifically as below: ( Thirdly, the gap between theoretical models and implementation of the PTA should also be bridged. In order to assess and evaluate the PTA, an instance of the PTA also needs to be implemented. Educational Games and Virtual Environments Nowadays, games are flourishing on various devices. As digital natives, most of the students today have played games and some even made game an integral part of their lives. Games, especially video games, are highly engaging and are good at content delivering due to their appealing audio and video effects. Studies on pedagogical videogames indicate that they can help students master content knowledge [8] and acquire practical skills [9]. Most videogames are deployed in 2D or 3D environments. River City is a multiuser 3D virtual environment built by Chris Dede's group at Harvard University to facilitate learning in the science domain [17]. After two weeks, the group of students who played in River City improved their content knowledge and inquiry skills by a greater extent compared to the control group which followed the normal paper-based curriculum. appear in the gaming environment as avatars which are capable of interacting with students and performing actions [11]. Affective Teachable Agent has been deployed in the 3D virtual educational game, Virtual Singapura. Based on the student's behavioral data collected in virtual gaming environment, Affective Teachable Agent is capable of providing appropriate emotional responses to the student. The assessment Affective Teachable Agent has proved its effectiveness not only in improving students' learning outcome, but also in motivating students and fostering their self-efficacy [11]. To harness the benefits of educational games and virtual environments, an instance of the PTA is implemented and embedded into a 3D pedagogical game called VS Saga. The implementation of this instance demonstrates the practice of instantiating a PTA from the proposed agent model and system architecture. It can also be used subsequently for the assessment and evaluation of the PTA. Et is the set of percepts/ events that the agent perceives from the its environment; Chapter 3 Design of Persuasive Teachable K is the set of knowledge that agent maintains; Rs is the set of selection mechanisms which the agent adopts to select a reasoning in R to go through in each cycle; R is the set of reasoning the agent is able to perform, R can be further defined by the tuple R = (P, Tr, Pr); A is the set of actions that the agent is able to perform onto the environment. P stands for Persuasion, which is the reasoning mechanism that enables the agent to persuade the student. Tr stands for Teachability Reasoning, which is the reasoning mechanism that enables the agent to learn from its student tutor. Pr stands for Practicability Reasoning, which is the reasoning mechanism that enables the agent to practice the knowledge learnt. Runtime Data refers to the data generated during runtime, such as, event information and history of events. Chapter 3 Design of Persuasive Teachable Agent After redefining the PTA, the improved PTA model will be introduced in section 3.2 ~ section 3.5. Modelling the Main Routine of PTA Section 3.2.1 will first introduce the methodology used to model PTA. Section 3.2.2 will then focuses on the modelling of the PTA's main routine using this methodology. Goal Net Methodology Goal Net methodology is well-suited for modelling goal-oriented agents. Since These atomic states represent the sub-goals that need to be fulfilled in order to fulfill the goal denoted by the composite state. States are represented by a round node in Goal Nets. Transitions are the tasks that advance the agent from an input state to an output state. Each transition has an associate task list, which describes the tasks need to be Conditional Transition The tasks needed to be completed to trigger a conditional transition is selected dynamically during runtime. Probabilistic Transition The tasks needed to be completed to trigger a probabilistic transition is selected dynamically based on probabilistic inference in an uncertain environment. Modelling PTA Main Routine with Goal Net Main routine is the highest-level Goal Net model of the PTA (Figure 3.4), as it depicts the agent's pursuit of its root goal. The design of main routing echoes the definition of the PTA in section 3.1. There are correspondences between the states and transitions of the main routine and the components of the PTA model, as shown in Table 3.1. In the main routine, correspondences can also be found for the three cycles of the PTA defined in Definition 3. From the start node, executing the "To Learn Knowledge" branch to the end node would complete a teaching cycle. Executing the "To Practice Knowledge Learnt" branch to the end node would complete a practicing cycle. Executing the "To Persuade" branch to the end node would complete a persuasion cycle. The main routine is executed repeatedly, so that the agent is constantly pursuing its goal. In each cycle, the agent checks for events repeatedly until one or more events are detected. According to the event(s) perceived, the agent selects an appropriate reasoning mechanism and executes the corresponding sub Goal Net of the selected reasoning. After the current cycle finishes, the agent reloads the start node of the main routine and begins with the next cycle. The three sub Goal Nets will be described in details in the following sections. (1) Personal Relevance. How relevant the event is to the learning topics. Persuasion Reasoning of PTA (2) Personal Responsibility. Whether the event demonstrates the student's responsibility towards his or her own learning. (3) Need for Cognition. Whether the student enjoys the learning process. When the learning process is enjoyable, the student would be more willing to engage in cognitive thinking. Category 2: Ability. Ability can be assessed from following factors: (1) Prior Knowledge. Whether the student has acquired the necessary knowledge required to teach the PTA. (2) Distraction. Whether the event is a distractor and would decrease the student's ability. (3) Repetition. Whether the student is revisiting the knowledge he or she has acquired by teaching the PTA. The PTA has incorporated the ELM to achieve its goal of instilling enduring positive attitude changes and knowledge gains of the student, i.e. keep him or her on the central route of persuasion. By assessing the six factors mentioned above, the PTA can evaluate the motivation and ability level of the student and generate appropriate persuasion cue if the student is in low motivation or low ability state. Quantitative Model for Assessing Motivation and Ability -FCM As mentioned above, the PTA needs to determine the motivation and ability level of the student in order to generate appropriate response. (1) Peripheral Cue or; (2) Motivation or; (3) Ability or; (4) A factor of motivation or ability. All PTA FCMs have the same stem nodes. Definition 6: A leaf node is a causal concept that is not a stem node and has a causal relationship with one of the factors of motivation or ability. Leaf nodes model the events that have causal relationships with the motivation or ability factors. Thus, the type and number of leaf nodes differ from one PTA implementation to another. Multiple leaf nodes can be connected to a motivation or ability factor. The same leaf node can also be connected to multiple factors. A leaf node is denoted by a rounded node with dotted rim in Figure 3 Modelling Persuasion Reasoning with Goal Net The sub Goal Net of the Persuasion Reasoning ( After execution of the sub Goal Net finishes, the agent continues to execute the next node after the composite state "To Persuade" on the main routine. Teachability Reasoning of PTA Integration with Practicability and Persuasion The integration point between the Teachability Reasoning and the Practicability Reasoning is the knowledge base. In a teaching cycle, the PTA learns from the student and saves the knowledge learnt into the knowledge base. In subsequent practicing cycle, the Practicability Reasoning reads the saved knowledge. The Teachability Reasoning also needs to be integrated with the Persuasion Reasoning. During the teaching cycle, any indication that suggests the student is low in motivation or ability will be captured and signaled, so that the Persuasion Reasoning can process it in a persuasion cycle later. Modelling Teachability Reasoning with Goal Net The sub Goal Net of the Teachability Reasoning (Figure 3.9) depicts how the PTA pursues its sub goal, "To Learn Knowledge". When executing this sub Goal Net, firstly, the agent requires the student to teach it. Secondly, it waits for and checks the response from the student. If the student agrees to teach the PTA, the PTA will learn from the student and save acquired knowledge to the knowledge base. However, if the student is not motivated to or does not have the ability to teach the PTA, he or she would refuse the teaching request from the PTA. In this case, the PTA will generate a rejection event. This event will be detected by the PTA in the next cycle. As the refusal indicates that the student has either low motivation or low ability, the agent will choose to go through persuasion reasoning during when a rejection event is detected. After execution of this sub Goal Net finishes, the agent continues to execute the next node after the composite state "To Learn Knowledge" on the main routine. Practicability Reasoning of PTA Integration with Teachability and Persuasion The Practicability Reasoning needs to be integrated with the Teachability Reasoning in order to generate teaching feedback. In a case when the student tutor commits errors during teaching, the errors should be captured and signaled during Practicability Reasoning, so that the Teachability Reasoning can highlight the errors during next teaching cycle. The Practicability Reasoning also needs to be integrated with the Persuasion Reasoning. During the practicing cycle, any event that may lead the student to a low motivation or low ability state will be captured and signaled, so that the Persuasion Reasoning can process it in a persuasion cycle later. Modelling Practicability Reasoning with Goal Net The sub Goal Net of the Practicability Reasoning (Figure 3.10) depicts how the PTA pursues its sub goal, "To Practice Knowledge Learnt". Chapter 3 Design of Persuasive Teachable Agent However, if the resultant solution is wrong, the agent will generate a wrong solution event, which will be detected by the agent in the next cycle. As the student may be demotivated after this fail trail, the PTA will choose to go through the Persuasion Reasoning when a wrong solution event is detected. After execution of this sub Goal Net finishes, the agent continues to execute the next node after the composite state "To Practice Knowledge Learnt" on the main routine. Chapter 4 System Architecture Design Chapter 4 System Architecture Design Overall System Architecture Design In Chapter 3, an improved agent model for the PTA is proposed. However, since the agent model has a high level of abstraction, there is still a gap between this theoretical model and the implementation of the PTA. In order to provide more guidelines for implementation, new system architecture for the PTA is proposed in this chapter. The proposed system architecture (as shown in Figure 4.1) focuses on the design of the agent's control structure. The system architecture retains certain level of abstraction and is reusable as: (1) It is independent of the learning topics, the deployment environment and the form of embodiment of the PTA; (2) It guides, but does not restrict the way of implementing a PTA. In the following sections, each component of the system architecture will be explained in greater details in terms of: (1) Its main functionalities; (2) Its interactions with other components; (3) Possible ways to implement this component. Goal Net Interpreter The Goal Net Interpreter directs the goal-oriented behaviors of a PTA according to the Goal Net Models introduced in Chapter 3. As shown in Figure 4 After determining the tasks to be performed, the Goal Net Interpreter instructs the PTA Control to perform those tasks. After finishing the designated tasks, the PTA Control informs the Goal Net Interpreter to proceed to the next state. There are two possible ways to implement the Goal Net Interpreter: to make use of an existing Goal Net interpreter or to build a new one. PTA Control The PTA Control component implements the major functionalities of a PTA. While the Goal Net Interpreter directs the agent's goal-oriented behaviors, the PTA Control coordinates them. The Goal Net Interpreter instructs the PTA Control to perform certain tasks by invoking corresponding task functions in the PTA Control. While executing the task functions, the PTA Control breaks down the tasks into sub-tasks and dispatches them to others components by calling respective modules in those components. In this manner, all the components work towards the goal of the PTA collectively. In order to realize all the three cycles of a PTA defined in Definition 3, the PTA Control needs to implement the task functions listed in Table 4.2. Each of these task functions is associated with a transition in the Goal Net models of the PTA. A task function has the same name as the transition it is associated to. Chapter 4 System Architecture Design Other Components Supporting PTA Control The Event Control The It is more recommended to implement the Event Control using the event checking mechanism with a checking period of a few seconds. During the checking period, the newly generated events are logged in the Event Log of the Event Control. When checking is performed, the Event Control prioritizes the events and decides the processing sequence of the events. It may also decide to process a number of interactional events in a batch. UI Control The UI Control component corresponds to the Actuator A (refer to definition in Database Access The Main Scenes and Story Lines VS Saga is a pedagogical game developed to facilitate the learning of secondary school science. Following the PTA model and system architecture proposed, a PTA, called "Water Molecule", is developed and embedded in the game. Screen Shot Story Line and Main Characters Goal Net Interpreter Since there is no existing Goal Net interpreter for Unity, a new one is implemented according to the logic described in PTA Control The PTA Control has implemented the task functions in Table 4 , the teaching process is mainly performed on a UI called Concept Map [19], which shows the relationships between the key concepts of the learning topic. The water molecule requests teaching from the student during a dialogue, as shown in Figure 5.8. If the student agrees to teach the water molecule, the PTA Control will display the concept map. After teaching process finishes, the knowledge taught by the student will be stored within the script of the concept map. If the student refuses to teach the water molecule, the PTA control will generate a rejection event. Other Controllers Supporting PTA Control Event Control In VS Saga, the Event Control component mainly tracks and deals with three types of events in Table 4.3: dialogue events, time events and teaching feedback events. Dialogue Events. A dialogue event is an event created when a particular sentence in a dialogue has been spoken. A dialogue event can be created within the dialogue system of Unity using a special type of script called SequencerCommand. The student player's dialogues with the game characters can be configured in the dialogue system. Figure 5.9 shows the dialogue entry of a sentence in the dialogue between the student and the water molecule in Unity. This sentence, "B. I'm afraid I cannot teach you", is spoken by the student to the water molecule to refuse the teaching request from it. Chapter 5 Implementation of PTA The blue rectangle in Figure 5.10 highlights the SequencerCommand GenerateEvent which will be called when this sentence is spoken. Thus, using the mechanism explained above, a new event can be created and added to the event log when the student refuses to teach the water molecule. Other dialogue events are generated in a similar way. Chapter 5 Implementation of PTA Whenever a dialogue event is generated, the timer will be reset. When the timer times out, a time-out event will be generated which signals that the student has been inactive. Teaching Feedback Events. A teaching feedback event is an event generated to provide feedback to the student's teaching. Two such events are tracked in VS Saga, namely Teach Success Event and Teach Failure Event. The list of events that are tracked in VS Saga are summarized in Appendix 3. A class diagram of the FCM Calculation component is shown in Figure 5.11. FCM Calculation From the bottom of the diagram, the AdjacencyList class stores the adjacency list representation of the PTA FCM. The PTA FCM can be represented by a sparse matrix. Compared to a 2D matrix, adjacency list is a more suitable data structure to model sparse matrix as it has higher space efficiency. the UI Control will change the color and positions of the tree leaves to make the Chapter 5 Implementation of PTA tree become revitalized. If the student commits errors during teaching, the tree remains withered. Database Access In Teachability and Practicability Case Studies Assessment Approach -Focus Group Study For the qualitative assessment of the PTA, a focus group study is carried out in order to gather feedback, comments and views on the PTA. Since the PTA sits in the interdisciplinary research field of education, intelligent agent and psychology, the focus group should include participants from various fields in order to get feedback and comments on various aspects of the PTA. The design of the focus group study is summarized in (1) Do you think the PTA is compelling enough to affect the minds and decisions of the student during the learning process and why? (2) How do you think the PTA can help the student to grasp better understanding of the learning topics and why? (3) How do you think the PTA is able to improve the learning outcomes of the student and why? (4) How do you think the PTA can affect the student's attitudes towards learning? (1) Is the system design reasonable? (2) Based on the proposed agent model, how would you design the system architecture? (3) Do you have any comment regarding the maintainability and reusability of the system? Result Analysis of the Focus Group Study As the number of participants in the focus group study is small, no statistical significance analysis can be drawn from the study. However, it is valuable to summarize all the qualitative assessments to identify the advantages and the areas for improvements of the PTA. The feedback and comments gathered from the study are summarized in the following table.  The student is unconsciously "supervised" by the water molecule. He or she can receive individual attention of the water molecule and be persuaded immediately if he or she is demotivated. Chapter 6 Case Studies and Assessment of PTA  The teaching process can help to reinforce learning of the student.  The learning and teaching processes are designed as a more enjoyable experience than the traditional class room style. The processes would be engaging. Areas for Improvement  The game and the PTA should be able to cater to different learning contents, not only for diffusion and osmosis.  When evaluating the ability level of the student, his or her ability to learn and prior knowledge should also be taken into consideration. If the student is slow learner and has little prior knowledge in related topics, he or she should be allowed more time to learn instead of being persuaded immediately. Agent Designers Advantages  The goal-oriented approach enables the agent to direct its own behavior and perform tasks in the pursuit of its own goal. Generally, the results of the assessment are quite positive. The results show that the PTA has the ability to motivate and engage the student by generating personalized feedback and the potential to help the student build positive attitude towards learning. It also turns the learning process into an enjoyable journey which could potentially help the student build positive attitudes towards learning. In terms of modelling, the Goal Net model depicts a PTA in all its aspects. The system architecture has also been endorsed for its reasonableness and reusability. Another important advantage of the PTA has been put forward during the focus group discussion. Compared to the human teacher, the PTA can generate more timely feedback, as it is able to closely "supervise" the student by monitoring his or her motivation and ability level using a more direct and quantitative approach. Limitations and Areas for Improvements The focus group study has also highlighted several limitations and areas for improvement. The one that has been raised by most of the focus group participants is the lack of flexibility of the current PTA. The instance implemented in VS Saga is specifically for topics in the science domain. Greater flexibility needs to be incorporated so that the PTA and the pedagogical game can be customized to different learning topics. In order to do this, there is a need to propose a convenient way to modify the models of the agent and change the content of the knowledge base. One way to incorporate that kind of flexibility is to design an authoring tool for the PTA, so that the design and implementation process of the PTA can be more smoothly connected. This is an area which the future researches in PTA can look into. Another important area for improvement Chapter 6 Case Studies and Assessment of PTA is to consider the learning competency and prior knowledge of the student while assessing his or her ability level. Chapter 7 Conclusion and Future Work Conclusions A Persuasive Teachable Agent (PTA) is a special type of Teachable Agent (TA) which has the ability to be taught, to practice knowledge learnt and to persuade the student to teach it. By incorporating the Elaboration Likelihood Model (ELM) of persuasion, the PTA is able to provide more timely and personalized feedback based on the student's ability and motivation, which previous TAs are incapable of. Reasoning is difficult to reuse as it is highly context dependent. Thirdly, there is still a gap between theoretical models and practical implementation of the PTA. In this project, a complete and integrate PTA agent model was proposed. Directed by a goal-oriented approach, the agent model depicts a PTA in its totality, including its ability to perform Persuasion Reasoning, Teachability Reasoning and Practicability Reasoning. This project also proposed a FCM model with greater reusability for computational processes in the Persuasion Reasoning of the PTA. In some cases, the proposed FCM model is able to show improved overall computational efficiency. Besides proposing improvements for the theoretical models of the PTA, system architecture for the control structure of the agent was introduced as well, with detailed descriptions for each component in the architecture and how it can be implemented. Following the proposed agent model and system architecture, an instance of the PTA has been successfully deployed in the 3D videogame, VS Saga. This implementation demonstrated the practice of instantiating a PTA from the propose agent model and system architecture. It also enabled the assessment and evaluation of the PTA. A focus group study was performed to assess the PTA. Chapter 7 Conclusion and Future Work The results of the study demonstrated the effectiveness of the PTA. The results were quite positive, showing the PTA is able to motivate and engage the student by generating timely and personalized feedback. It was also believed that the PTA has the potential to help the student build positive attitude towards learning. Moreover, positive feedback has been received for the completeness of the new agent model and the reusability of the proposed system architecture. Future Work Quantitative Assessment of PTA During the focus group study, no statistically significant analysis can be drawn from the results since the number of participants is too small. No conclusions can be drawn regarding how the PTA will affect the learning outcomes of the student as well. Thus, quantitative assessment of the PTA has to be carried out in order to analyze the effectiveness of the PTA statistically. VS Saga can be deployed onto different platforms for the assessment purpose, for example, desktops and tablets. The assessment could be conducted with a pre-test, a post-test and a set of questionnaire. The pre-test and the post-test can contain questions to assess the student's understanding of the content knowledge. By comparing the results of the two tests, the student's knowledge gain through teaching the PTA and his or her learning outcomes can be assessed. The questionnaire can be used to complement the tests and survey the attitudes of the students and their feelings towards the PTA. PTA Authoring Tool As discussed in section 6.2.3, the results of the focus group study have pointed out a few directions for future research. Before the PTA can be widely deployed, it needs to build up the flexibility to cater to different learning topics. For different learning topics, the learning goal and domain knowledge involved would be different. Thus, there is a need to propose a convenient way to modify the models of the agent and change the content of the knowledge base. One way Chapter 7 Conclusion and Future Work to incorporate that kind of flexibility is to design an authoring tool for the PTA. Referring to the design of the authoring tool which Ailiya proposed for the Affective Teachable Agent [11], a similar design can be proposed for the authoring tool of the PTA (Figure 7.1). A dedicated Game Authoring Component can be created, which provides interfaces for creating and editing learning goals, content knowledge and game tasks. If any changes are made from the Game Authoring Component, the models and knowledge stored in the knowledge base should be updated accordingly. The teacher defines the learning goals and relevant content knowledge. The teacher then designs different game tasks according to the learning goals. According to the teacher's design, the game developer modifies the game and add create new game tasks. Application of Educational Data Mining One big advantage of educational games is providing a digital ambience in which a great amount of behavioral data of the student can be collected. These data can Chapter 7 Conclusion and Future Work be tracked and analyzed to generate useful feedback to the student concerning his or her learning. Table 4.3 lists 7 types of event data that can be tracked in the games. However, in VS Saga, the assessment of motivation and ability level is mostly based on instant event data and only 3 out of these 7 types of events are tracked. In future studies, efforts can be directed to increase the type of events tracked. By analyzing more sources of events, the PTA will be able to assess the motivation and ability level of the student more accurately. Moreover, future researches could look into the application of educational data mining to analyze the relationship between different types of student behavioral data and the level of motivation and ability. For example, as suggested in Table 4 Abstract Highly interactive game-like virtual environment has gained increasing spotlight in academic and educational researches. Besides being an efficient and engaging educational tool, virtual environment also has the potential to be integrated with Educational Data Mining (EDM) to cater to emerging requirements of educational assessment. Nowadays, the traditional academic assessment approaches cannot thoroughly reflect the students' learning competencies which are crucial for them to thrive in a fast-changing world. We propose an assessment system that seamlessly integrates EDM with functionality and affordance of a virtual environment to assess students' learning competency through analysing their behavioural data and patterns. We also propose a set of metrics which can be used for judging students' learning competency and how these metrics can be evaluated computationally by quantifying and capturing students' behavioural data in a virtual environment. The field study, which is conducted in Xinmin Secondary School in Singapore, showed that the proposed assessment system is promising in identifying useful behaviour metrics for assessing learning competency. The system also exhibits more potential in terms of its quantitative and objective approach, comparing to traditional assessment methods. Keywords Educational Data Mining; Virtual Environment; Competency Assessment; Self-Directed Learning Figure 2 . 1 15 Figure 3 . 2 19 Figure 3 . 3 A 19 Figure 3 . 4 20 Figure 3 . 5 211532193319342035of Contents . List of Figures . III List of Tables . V Abstract . 1 Chapter 1 Introduction to Persuasive Teachable Agent . 2 1.1 Motivation . 2 1.2 Research Objectives . 4 1.3 Contributions . 6 1.4 Report Organization . 6 Chapter 2 Literature Review . 8 2.1 Teachable Agents . 8 2.2 Persuasive Teachable Agents. 10 2.3 Educational Games and Virtual Environments . 13 Chapter 3 Design of Persuasive Teachable Agent. 15 3.1 Redefining PTA . 15 3.2 Modelling the Main Routine of PTA. 18 3.2.1 Goal Net Methodology . 18 3.2.2 Modelling PTA Main Routine with Goal Net . 20 3.3 Persuasion Reasoning of PTA . 21 3.3.1 Persuasion Theory of PTA -ELM . 21 3.3.2 Quantitative Model for Assessing Motivation and Ability -FCM ...... 23 3.3.3 Modelling Persuasion Reasoning with Goal Net . 26 3.4 Teachability Reasoning of PTA .with Practicability and Persuasion . 27 3.4.2 Modelling Teachability Reasoning with Goal Net . 27 3.5 Practicability Reasoning of PTA . 28 3.5.1 Integration with Teachability and Persuasion . 28 3.4.2 Modelling Practicability Reasoning with Goal Net . 28 Chapter 4 System Architecture Design . 30 4.1 Overall System Architecture Design . 30 4.2 Goal Net Interpreter . 31 4.3 PTA Control . 34 4.4 Other Components Supporting PTA Control . 35 4.4.1 Event Control . 35 4.4.2 FCM Calculation . 37 4.4.3 UI Control . 38 4.5 Database Access . 38 Chapter 5 Implementation of PTA . 40 5.1 Game Environment . 40 5.1.1 Unity for 3D Game Development . 40 5.1.2 Main Scenes and Story Lines . 40 5.2 Goal Net Interpreter . 44 5.3 PTA Control . 45 5.4 Other Controllers Supporting PTA Control . 47 5.4.1 Event Control . 47 5.4.2 FCM Calculation . 49 5.4.3 UI Control . 51 5.5 Database Access .Studies and Assessment of PTA . 54 6.1 Case Studies of PTA . 54 6.1.1 Persuasion Case Studies . 54 6.1.2 Teachability and Practicability Case Studies . 58 6.2 Qualitative Assessment of PTA . 60 6.2.1 Assessment Approach -Focus Group Study . 60 6.2.2 Result Analysis of the Focus Group Study . 62 6.2.3 Limitations and Areas for Improvements . 64 Chapter 7 Conclusion and Future Work . 66 7.1 Conclusions . 66 7.2 Future Work. 67 7.2.1 Quantitative Assessment of PTA . 67 7.2.2 PTA Authoring Tool . 67 7.2.3 Application of Educational Data Mining . 68 References . 70 Appendix 1 Title and Abstract of Conference Paper Submitted . i Appendix 2 The Complete Goal Net Model in the Goal Net Designer . ii Appendix 3 Events Tracked in VS Saga . iii Appendix 4 The PTA FCM Used in VS Saga . iv Appendix 5 Complete Class Diagram of Agent Control .Existing PTA Model [2] . 12 Figure 3.1 Improved PTA Model . The Types of Transitions [16] . Goal Net Example . Goal Net Model of PTA . Central and Peripheral Route of Persuasion . 22 Figure 3 . 6 A 36Simple FCM Example . 24 Figure 3 . 7 FCM 25 Figure 3 . 8 26 Figure 3 . 9 27 Figure 3 . 3725382639273Model of PTA . Goal Net Model of Persuasion Reasoning . Goal Net Model of Teachability Reasoning . 10 Goal Net Model of Practicability Reasoning . 29 Figure 4 . 1 41System Architecture for PTA . 30 Figure 4 . 2 42Designing and Using the Goal Net Models . 32 Figure 4 . 3 43Architecture of MADE Runtime [18] . 33 Figure 4 . 4 44System Architecture with MADE Runtime . 33 Figure 4 . 5 34 Figure 5 . 1 453451Pseudo Code for Work Flow of Goal Net Interpreter . Interface of VS Saga. 40 Figure 5 . 52(a) Water Molecule . 41Figure 5.2(b) Madam Mah . 41 Figure 5.2(c) Sharman . 41 Figure 5.2(d) Madam Sammy . 42 Figure 5 . 52(e) Mayor . 42 Figure 5 . 53(a) Future Teacher . 42 Figure 5 . 53(b) Diffusion Tank Panel . 43 Figure 5 . 53(c) Diffusion Experiment . 43 Figure 5 . 53(d) Osmosis Experiment . 43 Figure 5 . 54(a) Withered Tree . 43 Figure 5.4(b) Concept Map . 44 Figure 5 . 45 Figure 5 . 6 45 Figure 5 . 7 46 Figure 5 . 8 46 Figure 5 . 9 48 Figure 5 . 48 Figure 5 . 50 Figure 5 . 51 Figure 5 . 52 Figure 5 . 545564557465846594854855055155254(c) Revitalized Tree . 44 IV Figure 5.5 Decision Making in Goal Net Interpreter . Invoking a Function with Function Name . Example of a Persuasion Cue . Water Molecule Requests Teaching from the Student . Example Dialogue Entry in Unity Dialogue System . 10 SequencerCommand GenerateEvent . 11 Class Diagram of FCM Calculation Component . 12 Trivalent Threshold Function . 13 TA Panel . 14 Concept Map . 52 Figure 6 . 61(a) Meet Sharman. 54 Figure 6 . 61(b) Talk to Sharman . 55 Figure 6 . 61(c) Persuasion for Not Learning Diffusion . 55 Figure 6 . 62(a) Meet the Future Teacher. 55 Figure 6 . 62(b) Talk about Diffusion 5K . 56 Figure 6 . 62(c) Persuasion for Not Motivated to Conduct Experiments . 56 Figure 6 . 63(a) Meet the Rabbit . 57 Figure 6 . 63(b) Talk to the Rabbit . 57 Figure 6 . 63(c) Persuasion for Talking to Distracting Characters . 58 Figure 6 . 64(a) Arrive at the Banana Tree . 58 Figure 6.4(b) Commit Errors during Teaching . 58 Figure 6.4(c) Persuasion for Teaching Failure . 59 Figure 6 . 65(a) Teach the Water Molecule Again . 59 Figure 6 . 65(b) Revitalized Tree and Happy Water Molecule . 60 Figure 6 . 65(c) Happy Molecule and Mission Accomplished . 60 Figure 7 . 1 71The Proposed Authoring Component of PTA . 68 Figure 7 . 2 72Tracking the Location Events . ( 2 ) 2The quantitative model in Persuasion Reasoning of PTA is difficult to reuse. Existing PTA model employs a fuzzy tool, Fuzzy Cognitive Map (FCM), to Chapter 1 Introduction to Persuasive Teachable Agent model the computational processes in Persuasion Reasoning. However, the quantitative FCM model is highly context dependent thus difficult to be reused in a different context. (3) Gap between theoretical models and practical implementation. The agent model has a high level of abstraction. It is hard to directly instantiate a PTA from the existing PTA model. Thus, another layer of design is needed between the agent model and implementation, which can provide more detailed guidelines for the implementation of PTA. This project aims to address the limitations mentioned above, and the objectives of the project are list as below:  To redefine a complete PTA model by integrating the Persuasion Reasoning, the Teachability Reasoning and the Practicability Reasoning. The new agent model should describe the PTA's ability to be taught and to practice, in addition to its ability to persuade.  To propose a context independent quantitative model for the computational processes in the Persuasion Reasoning of the PTA, by increasing the level of abstraction and generalizing from the original model.  To propose system architecture to provide more guidelines for implementation. The proposed system architecture should be detailed enough, so that a PTA can be easily instantiated from it. However, at the same time, it also needs to retain certain level of abstraction, so that it does not restrict the way of implementation.  To implement an instance of the PTA. This instance can demonstrate the practice of instantiating a PTA from the proposed agent model and system architecture. A complete and integrated PTA agent model. The agent model depicts a PTA in its totality, which includes: o Persuasion Reasoning. The persuasion reasoning model in existing PTA model is simplified and integrated with the Teachability Reasoning and the Practicability Reasoning. o Teachability Reasoning. A teachability reasoning model is added to depict the PTA's ability to be taught. o Practicability Reasoning: A practicability reasoning model is added to depict the PTA's ability to practice.  An improved quantitative model for Persuasion Reasoning of the PTA. The quantitative model can be reused in different contexts. For example, it can be reused for PTAs in different knowledge domains and with different implementations.  New system architecture for the PTA. The system architecture mainly describes how a PTA can be implemented, especially in terms of control structures.The design can be reused for multiple PTAs.  Deploying a PTA in pedagogical game in 3D virtual environment. An instance of PTA has been implemented in a 3D virtual educational game. The implementation has followed the improved agent model and proposed system architecture and demonstrated all major characteristics of a PTA. helping students structure and organize their knowledge; (2) invoking students' sense of responsibility towards teaching and their own learning; and (3) enhancing students' knowledge reflection and metacognition. During the recent two decades, TA has been an active research area. Researchers have developed various TAs for different purposes and curriculums in various formats. environment data from the Event Tracker and performs onto its environment by carrying out actions. The Knowledge Base stores the knowledge the PTA has obtained from its student tutor. Figure 2 2Figure 2.1 Existing PTA Model [2] Definition 1 : 1As illustrated inFigure 3.1, a Persuasive Teachable Agent is an agent which can be defined by a tuple PTA =(E, Et, K, Rs, R, A), where E is the set of environmental states that can be perceived by the agent; Figure 3 . 1 31Improved PTA Model Definition 2: R can be defined by the tuple R = (P, Tr, Pr), where Chapter 3 Design of Persuasive Teachable Agent Definition 3 :)) 3According the proposed new PTA model, the agent Perceive: The agent perceives environmental states/ events.(2) Reasoning Selection: The agent selects an appropriate reasoning according to its percepts.(3) Persuasion Reasoning: The agent goes through the reasoning for persuasion. It determines whether persuasion is required by evaluating the student's current motivation and ability level. If persuasion is required, it selects appropriate persuasion cue according to its percepts.(4) (Knowledge Base): If persuasion is required, the agent retrieves the selected persuasion cue from the knowledge base. (5) (Action): If persuasion is retrieved, the agent executes the cue. Teaching Cycle: EtRsTrK (1) Perceive: The agent perceives environmental states/ events. (2) Reasoning Selection: The agent selects an appropriate reasoning according to its percepts.(3) Teachability Reasoning: The agent goes through the reasoning for teachability. It acquires knowledge from the student tutor. It comprehends and translates the knowledge into knowledge representations that can be accepted by the knowledge base. Perceive: The agent perceives environmental states/ events. (2) Reasoning Selection: The agent selects an appropriate reasoning according to its percepts. (3) Practicability Reasoning: The agent goes through the reasoning for practicability. It determines how to respond to the query of its student tutor according to the knowledge it has acquired. If responses of the agent are predetermined, the agent chooses the correct response. (4) (Knowledge Base): If responses are pre-determined, the agent retrieves the selected response from the knowledge base.(5) Action: The agent responses to the query of its student tutor. Definition 4: Knowledge K of the PTA consists of the following five subsets: Goal Net Model is a set of Goal Net structures that defines the agent's goal-orientation characteristics and drives it during its goal pursuit. (Goal Net Model with be described in section 4.1.2 ~ 4.1.5.) FCM Model stores the FCM structure used by the agent to derive the values of motivation and ability from their constituent factors. Domain Knowledge refers to the expert knowledge related to the learning topics. Learnt Knowledge refers to the knowledge the agent acquired from the student tutor. completed in order to trigger this transition. Once all the tasks are completed, the transitions triggers and the agent advances from input state to output state. There are three types of transitions (as shown in Figure 3.2): Direct Transition The task list of such transitions is fixed and not changing according to runtime environment. The transition is triggered by completing same list of tasks every time. Chapter 3 Design of Persuasive Teachable Agent Figure 3 . 2 32The Types of Transitions[16] Arcs connect states and transitions and indicate the relationships between the nodes it connect. An arc can be represented by a triangle arrow or a diamond arrow.Branches are used to denote composite states. Two branches connect a composite state to the sequence of sub-states that compose it. The left branch connects to the first sub-state of this composite state while right branch connected to the last sub-state. Figure 3 . 33 shows an example of a Goal Net which contains all the four basic entities. S denotes states, while T denotes transitions. Figure 3 . 3 A 33Goal Net Example Chapter 3 Design of Persuasive Teachable Agent Figure 3 . 4 34Goal Net Model of PTA 3.3.1 will explain the persuasion theory which forms the basis of the Persuasion Reasoning. Section 3.3.2 will then explain the quantitative model used for the computational processes in the Persuasion Reasoning. Finally, section 3.3.3 will proposed the Goal Net model of the Persuasion Reasoning.3.3.1 Persuasion Theory of PTA -ELM The persuasion theory that forms the basis of the Persuasion Reasoning is the Elaboration Likelihood Model (ELM). The Elaboration Likelihood Model (ELM) of persuasion is a dualistic model developed by Petty and Cacioppo in 1986. The ELM proposed an underlying framework for persuasion communication and the resulting attitude changes. Elaboration in the ELM refers to the process which the recipient of the persuasive message engages in cognitive thinking related to the persuasion topic. Various factors that can affect the likelihood of elaboration can be classified into Chapter 3 Design of Persuasive Teachable Agent two large categories: motivation of the message recipient and his or her ability to elaborate.According to the ELM, there are two routes of persuasion, the central route and the peripheral route[14], as shown inFigure 3.5. The likelihood of elaboration determines which route the message recipient would go through. The likelihood of elaboration increases if one is motivated and has the ability to engage in elaboration. In this situation, one is likely to go through the central route of persuasion where his or her attitude changes tend to be enduring. On the contrary, the likelihood of elaboration decreases if one is not motivated or does not have ability to elaborate. In this situation, one is likely to go through the peripheral route of persuasion where the resulting attitude changes tend to be less enduring. Figure 3 . 5 35Central and Peripheral Route of Persuasion Chapter 3 Design of Persuasive Teachable Agent The following are the factors [14] that can affect the likelihood of elaboration explained in the context of PTA, arranged according to the categories they belong to. Category 1: Motivation. Motivation can be assessed from following factors: FCM can be applied to build a quantitative model to derive the values of motivation and ability from the values of their respective constituent factors. Chapter 3 Design of Persuasive Teachable Agent Fuzzy Cognitive Map (FCM) [15] is a fuzzy tool which can be used to depict causal relationships in dynamic systems. FCM is a graphical representation, at the same time, also a mathematical model of the dynamic system it is modelling. A FCM is composed of causal concepts and causal relationships. A causal concept can be used to model an event, a goal or an action. It is represented by a rounded node in FCM and has a real number value in interval [-1, 1]. A set of causal concepts are interconnected by directed edges. Each directed edge represents the causal relationship between two causal concepts. A causal relationship with a positive sign "+" denotes casual increase, while one with a negative sign "-" denotes causal decrease. A causal relationship can have a weight associated with it to denote the strength of the cause-effect relationship it represents. The weight usually takes a real value in interval [-1, 1]. Figure 3 Figure 3 . 336 shows a simple FCM example. The FCM in Figure 3.6(a) can also be represented as a matrix (denoted by E) in Figure 3.6(b). The weight of causal relationship from Ci to Cj is the value at Eij. For example, the weight of causal relationship from C2 to C4 is -1, which means C2 decreases C4. In the context of a PTA, FCM can be applied to build a quantitative model to derive the values of motivation and ability from the values of their respective constituent factors. A FCM model proposed for the PTA is shown in Figure 3.7. This FCM model depicts the causal relationship among motivation, ability and factors affect them. Chapter 3 Design of Persuasive Teachable AgentFigure 3.7 FCM Model of PTA Definition 5: A stem node is a causal concept that is: . 7 . 7Chapter 3 Design of Persuasive Teachable Agent Definition 7: the Main FCM is the FCM formed by all the stem nodes. In Figure 3.7, all the nodes with solid rim form the Main FCM. Definition 8: a Sub FCM is a FCM formed by a stem node and all the leaf nodes that are connected to it. In the PTA FCM, there are six Sub FCMs, each formed around a factor of motivation or ability. With the proposed PTA FCM model and the above definitions, the computational processes in the Persuasion Reasoning can be designed with greater flexibility and implemented with greater computational efficiency. An implementation of the proposed PTA FCM model is described in section 5.2.4. Figure 3 . 38) depicts how the PTA pursues its sub goal, "To Persuade". The ability to persuade is one of the important features that characterize a PTA. In pursuit of its sub goal "To Persuade", a PTA would act to keep the student on the central route of persuasion, i.e. to keep the student in high motivation and high ability state. Figure 3 . 8 38Goal Net Model of Persuasion Reasoning When this sub Goal Net is executed, firstly, the agent assesses the values of motivation and ability according to the Fuzzy Cognitive Map Model proposed in section 3.3.2. Secondly, the agent determines whether the resultant motivation and ability values are high or low by comparing them to the pre-determined Chapter 3 Design of Persuasive Teachable Agent baseline values. If both motivation and ability values are high, no particular action is required. However, when one of the values is low, the agent selects an appropriate persuasion cue and executes it. Figure 3 . 9 39Goal Net Model of Teachability Reasoning Chapter 3 Design of Persuasive Teachable Agent Figure 3 . 310 Goal Net Model of Practicability Reasoning When executing this sub Goal Net, the agent first queries the knowledge base to retrieve the knowledge it has acquired from the student. Secondly, it derives a solution from the knowledge by reasoning. If the resultant solution is correct, it will carry out the solution. Figure 4 . 1 41System Architecture for PTA Chapter 4 System Architecture Design At the highest level, the Goal Net Interpreter directs the goal-oriented behaviors of a PTA according to its Goal Net models. It determines the next state of the PTA and the tasks to be performed in order to advance to the next state. The Goal Net Interpreter instructs the PTA Control component to carry out the tasks required. The PTA control is supported by a number of other control components. It dispatches sub-tasks to Event Control, FCM Calculation and UI Controls and coordinates the execution of the sub-tasks. Table 4.1 provides a summary of the roles and functionalities of each component. Net models of the PTA can be created by drawing using a graphical Goal Net Designer[18] and saved into the Goal Net database. Please Chapter 4 System Architecture Design refer to Appendix 2 for the complete Goal Net model drawn in a Goal Net Designer. Figure 4 . 2 42Designing and Using the Goal Net Models The Goal Net Interpreter loads the Goal Net models from the database during runtime and interprets them dynamically. It determines the next state of the PTA by traversing through the nodes in the PTA's Goal Net. During a state transition, the Goal Net Interpreter also determines the tasks the PTA should perform in order to pursue its goal. Alternative 1 : 1Make use of an existing Goal Net interpreter. MADE Runtime [18] is a virtual machine that interprets the Goal Net models. It loads the Goal Net designs from the database using a Goal Net loader (as shown in Figure 4.3) and traverses through the Goal Net. Whenever a task needs to be performed to fire a state transition, MADE Runtime will invoke the corresponding routine. In this case, MADE Runtime is running as an active component and the system Chapter 4 System Architecture Design architecture would be different from what is described in Figure 4.1. The system architecture using MADE Runtime is shown in Figure 4.4. Figure 4 . 3 43Architecture of MADE Runtime[18] Figure 4 . 4 44System Architecture with MADE Runtime Alternative 2: Build a new Goal Net Interpreter. The most direct way to build a new Goal Net Interpreter is to directly access the Goal Net database. Goal Net database stores the information of states, transitions, task lists and functions. AGoal Net Interpreter can be implemented by querying the above information directly to determine the next state of the PTA and the tasks to be performed. Figure 4 . 45 shows the logic of how the Goal Net Interpreter pursues the goal of the agent. In order to access the Goal Net Database, the Goal Net Interpreter needs to use the interfaces provided by the Database Access component. Chapter 4 System Architecture Design Figure 4 . 5 45Pseudo Code for Work Flow of Goal Net Interpreter PTA Control coordinates the goal-oriented behaviors of the PTA and implements the major functionalities of the PTA. Event Control, FCM Calculation and UI Control are three components that support the PTA Control. The three of them deals with the percepts, reasoning and actions of the agent respectively. The Event Control component enables the agent to perceive its environment. The FCM Calculation enables the agent to reason about the motivation and ability level of the student. The UI Control component enables the agent to act on its environment. Event Control component corresponds to the Event Tracker Et (refer to definition in section 3.1) of the PTA. Event Control handles the lifetime of an event. It deals with the creation, logging, processing and removal of events. The PTA Control makes use of the functions in Event Control to handle events. Chapter 4 System Architecture Design section 3.1) of the PTA. It contains modules that enable the agent to act on various aspects of its environment, including the visual representation of itself in the environment. For example, the PTA can be embodied into its environment as 2D figure or as a 3D avatar. The UI Control provides modules that can control the figure or avatar of the PTA. The UI Control can also control the graphical interfaces or visual effects in the PTA environment. Further examples can be found in section 5.4.3. . 1 1Database Access component provides interfaces for the other components to access the database. Depends on the implementation of the PTA, multiple databases may need to be accessed. For example, the Goal Net Interpreter needs to access the state and Chapter 4 System Architecture Design transition information in Goal Net database, while the FCM Calculation component needs to load the FCM model from the FCM database. The Database Access component should be able to support and coordinate access to multiple databases. It should provide modules to connect to different database management systems where the databases are managed. When two different databases are managed by the same database management system, the Database Access component needs to coordinate the access to these two databases as the best practice is to keep only one open connection at any time. Unity for 3D Game Development Unity is a development platform for creating 2D/3D games and interactive experiences. Unity includes a game engine and an integrated development environment. VS Saga is a 3D video game developed for this project on the Unity development platform. An instance of the PTA is embedded in VS Saga to demonstrate the practice of instantiate a PTA from the propose agent model and system architecture. Please refer to Appendix 5 for the complete class diagram for the agent control structure. Figure 5 . 1 51Interface of VS SagaIn VS Saga, the student learns how water molecules are transported within a plant by completing a mission. The mission of the student is to help the water Chapter 5 Implementation of PTA molecule enter the root of a banana tree. The student learns about the concept of diffusion and osmosis and then teaches the PTA, the water molecule, these concepts to help it enter the root.There are three scenes in VS Saga, the Knowledge Town, the Science Laboratory and the Tree. In the following part of this section, the story line and main characters of each scene are introduced.Scene 1: the Knowledge Town. The knowledge town is a small island where the student meets the villagers and learns about the concept of diffusion and osmosis from them. Figure 5 52: the Science Laboratory. The Science Laboratory is another small island where the student practices the knowledge he or she has learnt. There is a big diffusion tank in the science laboratory where the student can conduct diffusion and osmosis experiments. Figure 5 5Figure 5.2(d) Madam Sammy Figure 5 53: the Tree. The banana tree is on the third island. This is where the student teaches the PTA (the water molecule) about diffusion and osmosis. The PTA practices the knowledge learnt by trying to enter the root of the tree. Figure 5 5Figure 5.3(b) Diffusion Tank Panel Figure 5 Figure 5 . 554(a) Withered Tree 1. Arrive at the banana tree, see the withered tree and sad water molecule (the PTA) Except for the main characters, there are other distracting characters in the game environment, such as a rabbit and a stag. Figure 4 . 5 . 45The implemented Goal Net Interpreter directly accesses the Goal Net database using the interface provided by the Database Access Component. The Goal Net Interpreter advances from an input state to an output state by firing the transition that connects them. However, when it encounters a decision node for which one input state has multiple possible output states, it needs to decide which output state to move to. To resolve that issue, a decision table is implemented in the PTA Control component (as shown in Figure 5.5) which keeps the decision to be made at each decision node. The PTA Control updates this decision table according reasoning results and current percepts. The Goal Net Interpreter refers to the decision table for deciding the output state if a decision node is encountered. Figure 5 5transition, the Goal Net Interpreter retrieves all the tasks in the task list of the transition and invokes the corresponding task functions in the PTA Control using their function names. The code snip for calling a task function using its function name is shown inFigure 5.6. Figure 5 . 6 56Invoking a Function with Function Name . 2 . 2This section provides summaries of how these functions are implemented in the context of VS Saga. Main Routine. During the execution of the PTA's Main Routine, the PTA Control calls the Event Control component every 5 seconds to check for events until one or more events are detected. Then, it prioritizes the events and decides the ones to be processed in the current cycle. It determines which the reasoning cycle the event(s) should go through and updates the decision table. Persuasion Reasoning. During the execution of the sub Goal Net "To Persuade", the PTA Control calls the FCM Calculation component to calculate the value of motivation and ability based on current percepts (events). Then, it student is in low motivation or low ability state where persuasion is needed. It updates the decision table accordingly. When persuasion is needed, the PTA Control selects a suitable cue according to the attributes of the event(s). Lastly, the PTA Control calls the UI Control to display the selected persuasion cue (an example in Figure 5.7). Figure 5 . 7 57Example of a Persuasion Cue Teachability Reasoning. As shown in Figure 5.4(b) Figure 5 . 8 58Water Molecule Requests Teaching from the Student Chapter 5 Implementation of PTA Practicability Reasoning. During the execution of the sub Goal Net "To Practice Knowledge Learnt", the PTA Control reasons the correctness of the teaching based on the knowledge stored in the concept map. If the teaching of the student is correct, the PTA Control will invoke the animation of the banana tree so the tree becomes revitalized as shown in 5.13. Otherwise, the PTA Control will generate a wrong solution event. Figure 5 . 9 59Example Dialogue Entry in Unity Dialogue System Figure 5 5code snip of the SequencerCommand GenerateEvent. This SequencerCommand extracts the parameters passed to it and calls the function for creating a new event with the parameters. Figure 5 . 510 SequencerCommand GenerateEvent Figure 5 . 511 Class Diagram of FCM Calculation Component The class FCMStructure has an instance of AdjacencyList class and contains the logic for the computational processes of FCM. It provides a method called matrixCalculation() which can take in the values of the all causal concepts, perform one round of calculation and return the updated values of the causal concepts. MainFCM and SubFCM are two subclasses of FCMStructure and inherit all its attributes and methods. The concept of mainFCM and subFCM are defined in Definition 7 and Definition 8. The SubFCM class overrides the matrixCalculation() method in FCMStructrue as it uses a slightly different way to perform its computational processes. The PTA_FCM class models the structure and characteristics of a PTA FCM. It has an instance of MainFCM class and multiple instances of SubFCM class. The PTA_FCM class also has a list of Concepts which is used to store the values of the causal concepts. The PTA_FCM class coordinates the calculation processes in the mainFCM and multiple subFCMs. Multiple rounds of calculations maybe needed before the values of causal concepts finally stabilize. Chapter 5 Implementation of PTA This implementation of the FCM Calculation component caters to the structure of PTA FCM by modelling mainFCM and subFCMs separately and differentiates their computational processes. In some cases, separating the computations in mainFCM and subFCMs can also increase the overall computational efficiency.For example, for a PTA FCM which uses a trivalent threshold function as shown inFigure 5.12, starting from the second round of calculation, computations only need to be carried out in the mainFCM. By reducing the amount of computations performed, the efficiency of the overall processes is improved. Figure 5 . 512 Trivalent Threshold Function5.4.3 UI ControlThe UI Control component is implemented as a set of scripts which controlsdifferent UIs. This section provides a few examples to show how the UI Control manages the different UIs. Persuasion Reasoning. The TA panel (Figure 5.13) is the most important interface element for the Persuasion Reasoning. It is used for displaying the persuasion cues. After a persuasion cue is selected, the PTA Control component calls the UI Control to fetch the information of the selected cue and display it. The information to be fetched from the database includes the texts to be displayed and the facial expression of the water molecule. Chapter 5 Implementation of PTA Figure 5.13 TA Panel Teachability Reasoning. The concept map (Figure 5.14) is the main interface where the teaching process is performed. In order to teach the PTA (the water molecule), the student completes the concept map by dragging and dropping the buttons on the right side to the blank spaces on the left. When the "Teach" button is clicked, the UI Control saves the contents of the blank spaces as the knowledge learnt and creates a new event to indicate the completion of the teaching process. Figure 5 . 514 Concept Map Practicability Reasoning. The banana tree is the most important interface element for the Practicability Reasoning. If the student teaches the PTA correctly, . 3 Figure 6 36VS Saga, the Data Access component needs to provide interfaces for two database management systems, MySQL and SQLite. The Goal Net database and FCM database are managed by MySQL while the database for persuasion cues and inventory items is managed by SQLite. Inventory information and persuasion cues are stored in the same database which is managed by SQLite. Thus, there are no access issues. However, the Goal Net database and FCM database are two separate databases and both managed by MySQL. Since the best practice is to keep only one open connection at any time, accesses to these two databases need to be coordinated. The Database Access loads the FCM model from the FCM database during initialization and closes the connection to the FCM database. Then, it opens the connection to the Goal Net database and executes queries while the Goal Net Interpreter is traversing the Goal Net of the PTA. Chapter 6 Case Studies and Assessment of PTA Chapter 6 Case Studies and Assessment of PTA 6.1 Case Studies of PTA This section presents a few case studies of the implemented PTA. It aims to demonstrate that the PTA implemented in VS Saga has realized all major characteristics of a PTA. These case studies are going to be used in future submission for the IJCAI-15 Video Competition. to Sharman. Sharman mentions about diffusion. 1.2.1 Choose the response "A. Diffuse? What do you mean by diffuse?" => High motivation, high ability state 1.2.2 Choose the response "B. Ok…. I see…." => Low motivation, low ability state, go to 1water molecule (the PTA) persuades the student to learn about diffusion Figure 6.1(c) Persuasion for Not Learning Diffusion Case Study 2: Not Motivated to Conduct Experiments 2.1 Meet the Future Teacher in the Science LaboratoryFigure 6.2(a) Meet the Future Teacher Figure 6 6water molecule persuades the student to try out the experiments. Figure 6 . 62(c) Persuasion for Not Motivated to Conduct Experiments Chapter 6 Case Studies and Assessment of PTA Figure 6 6conversation with the rabbit => Low motivation low ability state, go to 3.3 Figure 6 6water molecule (the PTA) persuades the student to concentrate on his or her mission and do not play around. Figure 6 . 63(c) Persuasion for Talking to Distracting Characters Case Study 4: Teach Failure 4.1 Arrive at the Banana Tree. See the sad water molecule and the withered tree Figure 6 . 3 Figure 6 . 6364(a) Arrive at the Banana Tree 4.2 Commit errors while drawing the concept map => Click the "Teach!" button, go to 4.4(b) Commit Errors during Teaching Chapter 6 Case Studies and Assessment of PTA 4.3 After the Practicability Reasoning, no solution can be generated. Thus, the Persuasion Reasoning calls up the water molecule to persuade the student to teach it again. Figure 6 . 64(c) Persuasion for Teaching Failure Case Study 5: Teach Success after Previous Failure 5.1 Teach the water molecule again after fail for the first time. The errors committed during previous teaching process are highlighted in red. 5.1.1 Fail to correct all the errors => Click on "Teach!, go to 4.3 5.1.2 Correct all the errors => Click on "Teach!", go to 5.2 Figure 6 . 65(a) Teach the Water Molecule Again6.2 Qualitative Assessment of PTA Figure 6 . 65(b) Revitalized Tree and Happy Water Molecule 5.3 The water molecule gives positive feedback on the teaching. Figure 6 . 65(c) Happy Molecule and Mission Accomplished ( 5 ) 5Any areas for improvements? B. Regarding the agent model of the PTA: (1) Do you think the proposed agent model would increase the spontaneity of the agent and why? (2) Do you think the proposed agent model would help the PTA to generate personalized feedback and why? (3) Is the modelling of the agent considered complete and flexible and why? (4) Any areas for improvements? C. Regarding the system design and implementation of the Chapter 6 Case Studies and Assessment of PTA PTA: Figure 7 . 1 71The Proposed Authoring Component of PTA Authoring processes would involve both the teacher and the game developer. . 3 , 3Location Events can be tracked in the PTA environment. By collecting the location of students in the virtual environment every few seconds, abundant location data can be obtained. Data mining techniques can be applied to these location data to mine the location patterns of the student. Relationships can be established between these patterns and the motivation and ability level of the student. For example, if the points indicating locations are clustered around the main characters in the scenes and the banana tree with only a few outliers, the student has probably been focusing on his or her mission on the island and is potentially in a high motivation state. Figure 7 7Figure 7.2 Tracking the Location Events Table Table 2 . 21 Summary of Teachable Agents [20] . 9 Table 3.1 Correspondences between PTA Main Routine and PTA Model . 20 Table 4.1 Functionalities/ Roles of Components in System Architecture . 31 Table 4.2 Task Functions to be Implemented in the PTA Control . 35 Table 4.3 Suggested Event Types to be Tracked in a PTA Environment . 36 Table 4.4 Summary of Popular FCM tools . 37 Table 6.1 The Design of the Focus Group Study . 61 Table 6.2 The Results from the Focus Group Study . 62 Table 2 . 21 Summary of Teachable Agents[20] especially when the TA is passive in interactions and incapable of giving personalized responses. Thus, it is important to monitor the student's motivation level and generate corresponding feedback.Teachable Agent Issues Innovations Application Domain Betty's Brain  Interact with students passively  Incapable of generating personalized feedback  Students not fully engaged Artificial Intelligence Science SimStudent Machine Learning Mathematics Affective Teachable Agent  Students not fully engaged in learning with TA Goal Orientation Affective Computing Lower Secondary Science DynaLearn  Lack of clarity and relevance of feedback  Students not fully engaged Qualitative Reasoning Science Concepts 1 ) 1Persuasion Reasoning of PTA is not integrated with Teachability and Practicability Reasoning. The existing agent model has only proposed the Goal Net model for the Persuasion Reasoning while does not have Goal Net models for the Existing PTA model employs a fuzzy tool, Fuzzy Cognitive Map (FCM), to model the computational processes in Persuasion Reasoning. However, the quantitative FCM model is highly context dependent thus difficult to be reused in a different context. (3) Gap between theoretical models and practical implementation. The agent model has a high level of abstraction. It is hard to directly instantiate a PTA from the existing PTA model. Thus, another layer of design is needed between the agent model and implementation, which can provide more detailed guidelines for the implementation of PTA.To address these three limitations, firstly, a complete agent model has to be proposed. A complete agent model should include not only the Goal Net modelling for the Persuasion Reasoning, but also Goal Net modellings for the Teachability and Practicability Reasoning. Secondly, a more abstract and generalized FCM model is required for the quantitative processes in the PTA.Teachability Reasoning and the Practicability Reasoning. (2) The quantitative model in Persuasion Reasoning of PTA is difficult to reuse. goal-orientation is one of the major characteristics of the PTA, Goal Net would be a suitable methodology to model the PTA. Goal Net methodology is first proposed by Dr. Shen in 2005 [16] to model goaloriented agents. In Goal Net, an agent pursues its goal by completing a sequence of sub-goals. A Goal Net can be composed from four basic entities: states, transitions, arcs and branches. States are used to represent the goals of an agent. A state can be atomic or composite. A composite state can be further broken down into multiple atomic states. Table 3 . 31 Correspondences between the PTA Main Routine and the PTA ModelStates/Transitions in Main Routine Component in PTA Definition Detect Event, Event Detected, Interpret Event, Event Interpreted Et Event Tracker Selected Reasoning Rs Reasoning Selection To Learn Knowledge (and its sub Goal Net) Tr Teachability Reasoning To Practice Knowledge Learnt (and its sub Goal Net) Pr Practicability Reasoning A Actuator To Persuade (and its sub Goal Net) P Persuasion A Actuator Table 4 . 41 Functionalities/ Roles of Components in System Architecture Component Functionality/ Role Goal Net Interpreter Directs the goal-oriented behaviors of a PTA PTA Control Acts as a central dispatcher and coordinator Event Control Deals with the creation, logging, processing and removal of events FCM Calculation Performs computational processes related to FCM UI Control Control the interface elements in the PTA environment Database Access Connects to the database and performs queries Table 4 . 42 Task Functions to be Implemented in the PTA ControlGoal Net Model Functions to be Implemented Main Routine DetectEvent InterpretEvent SelectReasoning Finish To Learn Knowledge RequireTeaching CheckResponse InitializeTeaching AcquireKnowledge SaveKnowledge GenerateRejectionEvent Finish To Practice Knowledge Learnt QueryKB Reasoning CarryOutSol GenerateWrongSolEvent Finish To Persuade FCMCalculation CheckMotAbi SelectCue ExecuteCue Finish Table 4 . 43 suggests a few types of events that can be tracked in a 2D or 3D PTA environment, together with corresponding examples. Table 4 . 43 Suggested Event Types to be Tracked in a PTA Environment The FCM Calculation component performs the computational processes related to FCM in the Persuasion Reasoning of the PTA. Alternative 1: Leverage on existing FCM modelling tools and software. There are quite a number of FCM tools available for free. An interface can be built for these tools so they can be used by the PTA control structure. The interface needs to handle the conversion and passing of the inputs to these tools and the retrieval and reformatting of the outputs from them. This alternative would greatly reduce the effort required to build the FCM Calculation component. However, most of the FCM tools available are not easy to integrate. Table 4.4 summarizes a few popular FCM tools and some of their disadvantages.Event Category Event Example Learning Behavior Dialogue Event a. The student has refused to teach the PTA in a dialogue with the PTA b. The student talks to a distracting character, such as a rabbit Location Event The student has arrived at a designated location Time Event The student has been inactive for more than 5 minutes Learning Achievement Collection of Item The student has collected a designated item Fulfillment of Mission The student has completed a learning task Knowledge Data Commitment of Errors The student commits an error while teaching the PTA Teaching Feedback Event The PTA cannot generate correct solution from the knowledge acquired from the student -wrong solution event Table 4 . 44 Summary of Popular FCM tools Build a new FCM tool from scratch. Building a new one would require more effort. However, this alternative allows more space for customization and greatly reduces the difficulty of integrating with other components. An implementation of the proposed PTA FCM model is described in section 5.2.4.Tool Name Descriptions Disadvantages FCMappers  Based in Excel  Visual representations for FCM on the web  Not easy to integrate  Cannot customize parameters or threshold functions of FCM Mental Modeller  Downloaded or web-based  GUIs for drawing FCMs  Interface for display FCM matrix  Not easy to integrate  Cannot customize parameters or threshold functions of FCM Time Events.In VS Saga, only one time event is tracked. The Event Control component keeps a timer for the student's inactive time. Inactive time is defined as the time when the student is not actively interacting with the game characters. Instead of using existing FCM tools, a FCM Calculation component is developed so that it can be easily integrated with other control components developed in Unity. The FCM Calculation component is implemented according to the FCM model proposed in section 3.3.2. The complete PTA FCM used in VS Saga can be found in Appendix 4. Choose the response "B. Ok. Probably later." => Low motivation low ability state, go to 2.32.2 The Future Teacher mentions about "Diffusion 5K", the big tank for conducting diffusion and osmosis experiments. 2.2.1 Choose the response "Ok. But how should I operate this Diffusion 5K?" => High motivation high ability state 2.2.2 Case Study 3: Talking to Distracting Characters3.1 Meet the Rabbit near the stairs in the KnowledgeTown 3.1.1 Choose not to talk to the rabbit => High motivation high ability state 3.1.2 Choose to talk to the rabbit, go to 3.2 Table 6.1. 5.2 After the Practicability Reasoning, a correct solution is derived. The Practicability Reasoning calls up the UI Control module of the tree to display the tree animation. The tree becomes revitalized and the water molecule becomes happy. Table 6 6.1 The Design of the Focus Group StudyDesign of the Focus Group StudyNumber of Participants 6 Formation of the Participants 2 Secondary school teachers 2 Agent researchers 2 Game developers Table 6 . 62 The Results from the Focus Group Study Advantages  The water molecule has human-like emotions and is proactive in interactions with student. The student can develop sympathy and sense of responsibility towards the agent which motivates them to learn himself or herself and teach the agent.Teachers  The propose Goal Net Model describes a PTA in all its aspect and accommodates flexibility. The integration of FCM enhances the qualitative reasoning ability of the agent.Areas for Improvement The agent is implemented specifically for topics in the science domain. However, the design of the agent should incorporate the flexibility so it can be deployed for more topics and in more domains. The system architecture is easy to understand. It's abstract enough to be reusable. I would implement it in similar ways.Areas for Improvement  Currently, the task functions are implemented in Unity. They can also be implemented as DLL files, which can be more easily called by a virtual machine (the interpreter). Most teachers are computer novices who are not familiar with game development and agent design. Convenient update tools should be developed for them, if the teachers need to update the learning content frequently.Game Developers Advantages   However, the existing PTA model still has a few limitations. Firstly, the existing agent model has only proposed the Goal Net model for the Persuasion Reasoning while does not have Goal Net models for the Teachability Reasoning and the Practicability Reasoning. 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A new robust training algorithm for a class of single-hidden layer feedforward neural networks. Z Man, K Lee, D Wang, Z Cao, C Miao, References Neurocomputing. 7416Z. Man, K. Lee, D. Wang, Z. Cao, and C. Miao, "A new robust training algorithm for a class of single-hidden layer feedforward neural networks," References Neurocomputing, vol. 74, no. 16, pp. 2491-2501, 2011. A two-stage winwin multiattribute negotiation model: Optimization and then concession. L Pan, X Luo, X Meng, C Miao, M He, X Guo, Computational Intelligence. 294L. Pan, X. Luo, X. Meng, C. Miao, M. He, and X. Guo, "A two-stage winwin multiattribute negotiation model: Optimization and then concession," Computational Intelligence, vol. 29, no. 4, pp. 577-626, 2013. An energy-efficient and lowlatency mac protocol with adaptive scheduling for multi-hop wireless sensor networks. Y Zhao, M Ma, C Miao, T Nguyen, Computer Communications. 3312Y. Zhao, M. Ma, C. Miao, and T. Nguyen, "An energy-efficient and low- latency mac protocol with adaptive scheduling for multi-hop wireless sensor networks," Computer Communications, vol. 33, no. 12, pp. 1452- 1461, 2010. Trust-based agent community for collaborative recommendation. J Weng, C Miao, A Goh, Z Shen, R Gay, Proceedings of the 5th international joint conference on Autonomous agents and multi-agent systems (AAMAS'06. the 5th international joint conference on Autonomous agents and multi-agent systems (AAMAS'06J. Weng, C. Miao, A. Goh, Z. Shen, and R. Gay, "Trust-based agent community for collaborative recommendation," in Proceedings of the 5th international joint conference on Autonomous agents and multi-agent systems (AAMAS'06), 2006, pp. 1260-1262. Design of fuzzy cognitive maps using neural networks for predicting chaotic time series. H Song, C Miao, Z Shen, W Roel, D Maja, C Francky, Neural Networks. 2310H. Song, C. Miao, Z. Shen, W. Roel, D. Maja, and C. Francky, "Design of fuzzy cognitive maps using neural networks for predicting chaotic time series," Neural Networks, vol. 23, no. 10, pp. 1264-1275, 2010. A fuzzy neural network with fuzzy impact grades. H Song, Z Shen, C Miao, Y Miao, B S Lee, Neurocomputing. 7213H. Song, Z. Shen, C. Miao, Y. Miao, and B. S. Lee, "A fuzzy neural network with fuzzy impact grades," Neurocomputing, vol. 72, no. 13, pp. 3098-3122, 2009. Fuzzy cognitive agents for personalized recommendation web information systems engineering. C Miao, Q Yang, H Fang, A Goh, WISE'02Proceedings of the 3rd International Conference on Web Information Systems Engineering. the 3rd International Conference on Web Information Systems EngineeringC. Miao, Q. Yang, H. Fang, and A. Goh, "Fuzzy cognitive agents for personalized recommendation web information systems engineering," in Proceedings of the 3rd International Conference on Web Information Systems Engineering (WISE'02), 2002, pp. 362-371. 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{'abstract': "A Persuasive Teachable Agent (PTA) is a special type of Teachable Agent which incorporates a persuasion theory in order to provide persuasive and more personalized feedback to the student. By employing the persuasion techniques, the PTA seeks to maintain the student in a high motivation and high ability state in which he or she has higher cognitive ability and his or her changes in attitudes are more persistent. However, the existing model of the PTA still has a few limitations. Firstly, the existing PTA model focuses on modelling the PTA's Goal Net Interpreter PTA Control Event Control FCM Calculation UI Control Database Access", 'arxivid': '1601.06245', 'author': ['Zhiwei Zeng '], 'authoraffiliation': [], 'corpusid': 17500787, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 35396, 'n_tokens_neox': 31014, 'n_words': 20493, 'pdfsha': '7a154d2b9ad6f3983d386a66b3d589a2ccab3d6b', 'pdfurls': ['https://arxiv.org/pdf/1601.06245v1.pdf'], 'title': ['Artificial Persuasion in Pedagogical Games [A Book Draft]', 'Artificial Persuasion in Pedagogical Games [A Book Draft]'], 'venue': []}

arxiv

On ADMM in Deep Learning: Convergence and Saturation-Avoidance 2021 Jinshan Zeng jinshanzeng@jxnu.edu.cn Shao-Bo Lin Yuan Yao yuany@ust.hk Ding-Xuan Zhou mazhou@cityu.edu.hk School of Computer and Information Engineering Liu Bie Ju Centre for Mathematical Sciences Jiangxi Normal University NanchangChina Department of Mathematics City University of Hong Kong Hong Kong Center of Intelligent Decision-Making and Machine Learning School of Management Hong Kong University of Science and Technology Hong Kong, Xi'an Department of Mathematics Jiaotong University Xi'an, HongChina School of Data Science and Department of Mathematics Kong University of Science and Technology Hong Kong City University of Hong Kong Hong Kong On ADMM in Deep Learning: Convergence and Saturation-Avoidance Journal of Machine Learning Research 222021Submitted 9/20; Revised 8/21; Published 9/21 Editor: Marc SchoenauerDeep learningADMMsigmoidglobal convergencesaturation avoidance In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called sigmoid-ADMM pair ), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than that of deep ReLU nets by showing that ReLU activation function can be well approximated by deep sigmoid nets with two hidden layers and finitely many free parameters but not vice-verse. We also establish the global convergence of the proposed ADMM for the nonlinearly constrained formulation of the deep sigmoid nets training from arbitrary initial points to a Karush-Kuhn-Tucker (KKT) point at a rate of order O(1/k). Besides sigmoid activation, such a convergence theorem holds for a general class of smooth activations. Compared with the widely used stochastic gradient descent (SGD) algorithm for the deep ReLU nets training (called ReLU-SGD pair), the proposed sigmoid-ADMM pair is practically stable with respect to the algorithmic hyperparameters including the learning rate, initial schemes and the pro-processing of the input data. Moreover, we find that to approximate and learn simple but important functions the proposed sigmoid-ADMM pair numerically outperforms the ReLU-SGD pair. Introduction In the era of big data, data of massive size are collected in a wide range of applications including image processing, recommender systems, search engineering, social activity mining and natural language processing (Zhou et al., 2014). These massive data provide a springboard to design machine learning systems matching or outperforming human capability but pose several challenges on how to develop learning systems to sufficiently exploit the data. As shown in Figure 1, the traditional approach comes down to a three-step learning process. It at first adopts delicate data transformations to yield a tractable representation of the original massive data; then develops some interpretable and computable optimization models based on the transformed data to embody the utility of data; finally designs efficient algorithms to solve the proposed optimization problems. These three steps are called feature extraction, model selection and algorithm designation respectively. Since feature extraction usually involves human ingenuity and prior knowledge, it is labor intensive, especially when the data size is huge. Therefore, it is highly desired to reduce the human factors in the learning process. Deep learning (Hinton and Salakhutdinov, 2006;LeCun et al., 2015), which utilizes deep neural networks (deep nets for short) for feature extraction and model selection simultaneously, provides a promising way to reduce human factors in machine learning. Just as Figure 1 purports to show, deep learning transforms the classical three-step strategy into a two-step approach: neural networks selection and algorithm designation. It is thus important to pursue why such a transformation is feasible and efficient. In particular, we are interested in making clear of when deep nets are better than classical methods such as shallow neural networks (shallow nets) and kernel methods, and which optimization algorithm is good enough to realize the benefits brought from deep nets. In the past decade, deep nets with ReLU activations (deep ReLU nets) equipped with the well known stochastic gradient descent (SGD) algorithm have been successfully used in image classification (Krizhevsky et al., 2012), speech recognition Sainath et al., 2013), natural language processing (Devlin et al., 2014), demonstrating the power of ReLU-SGD pair in deep learning. The problem is, however, that there is a crucial inconsistency between approximation and optimization for the ReLU-SGD pair. To be detailed, from the approximation theory viewpoint, it is necessary to deepen the network to approximate smooth function (Yarotsky, 2017), extract manifold structures (Shaham et al., 2018), realize rotation-invariance features (Han et al., 2020) and provide localized approximation (Safran and Shamir, 2017). However, from the optimization viewpoint, it is difficult to solve optimization problems associated with too deep networks with theoretical guarantees (Goodfellow et al., 2016). Besides the lack of convergence (to a global minima) guarantees, deep ReLU nets equipped with SGD may suffer from the issue of gradient explosion/vanishing (Goodfellow et al., 2016) and is usually sensitive to its algorithmic hyper-parameters such as the initialization (Glorot and Bengio, 2010;Sutskever et al., 2013;Hanin and Rolnick, 2018) and learning rate (Senior et al., 2013;Daniel et al., 2016;Ruder, 2016) in the sense that these parameters have dramatic impacts on the performance of SGD and thus should be carefully tuned in practice. In a nutshell, deep ReLU nets should be deep enough to exhibit excellent approximation capability while too deep networks frequently impose additional difficulty in optimization. There are numerous remedies to tackle the aforementioned inconsistency for the ReLU-SGD pair with intuition that SGD as well as its variants is capable of efficiently solving the optimization problem associated with deep ReLU nets. In particular, some tricks on either the network architectures such as ResNets (He et al., 2016) or the training procedure such as the batch normalization (Ioffe and Szegedy, 2015) and weight normalization (Salimans and Kingma, 2016) have been developed to address the issue of gradient vanishing/explosion; several efficient initialization schemes including the MSRA initialization (He et al., 2015) have been proposed for deep ReLU nets; some guarantees have been established (Allen-Zhu et al., 2019;Zou and Gu, 2019) in the over-parametrized setting to verify the convergence of SGD; and numerous strategies of learning rates (Chollet et al., 2015;Gotmare et al., 2019;Smith and Topin, 2017) have been provided to enhance the feasibility of SGD. Different from the aforementioned approach focusing on modifying SGD for deep ReLU nets, we pursue an alternative direction to ease the training via reducing the depth. Our studies stem from an interesting observation in neural networks approximation. As far as the approximation capability is concerned, deep nets with sigmoid-type activation functions (deep sigmoid nets) theoretically perform better than deep ReLU nets for some function classes in the sense that to attain the same approximation accuracy, the depth and number of parameters of the former is much smaller than those of the latter. This phenomenon was observed in approximating smooth functions (Mhaskar, 1996;Yarotsky, 2017), reflecting the rotation invariance feature (Chui et al., 2019;Han et al., 2020) and capturing sparse signals (Lin et al., 2017;Schwab and Zech, 2019). In spite of their advantages in approximation, deep sigmoid nets have not been widely used in the deep learning community. The major reason is due to the saturation problem of the sigmoid function 1 (Goodfellow et al., 2016, Section 6.3), which is easy to cause gradient vanishing for gradient-descent based algorithms in the deep sigmoid nets training (Bengio et al., 1994;LeCun et al., 1998). Specifically, as shown in Figure 2 (b), derivatives of sigmoid functions vanish numerically in a large range. In this paper, we aim at developing a gradient-free algorithm for the deep sigmoid nets training to avoid saturation of deep sigmoid nets and sufficiently embody their theoretical advantages. As a typical gradient- Table 2 below. free optimization algorithm, alternating direction method of multipliers (ADMM) can be regarded as a primal-dual method based on an augmented Lagrangian by introducing nonlinear constraints and enables a convergent sequence satisfying the nonlinear constraints. Therefore, ADMM attracted rising attention in deep learning with various implementations (Carreira-Perpinan and Wang, 2014;Kiaee et al., 2016;Yang et al., 2016;Gotmare et al., 2018;Murdock et al., 2018). Under this circumstance, we propose an efficient ADMM algorithm based on a novel update order and an efficient sub-problem solver. Surprisingly, as shown in Figure 2 (c), the proposed sigmoid-ADMM pair performs better than ReLU-SGD pair in approximating the simple but extremely important square function (Yarotsky, 2017;Petersen and Voigtlaender, 2018;Han et al., 2020). This implies that ADMM may be an efficient algorithm to sufficiently realize theoretical advantages of deep sigmoid nets. Our contributions of this paper can be summarized as the following three folds. • Methodology Novelty: We develop a novel sigmoid-ADMM pair for deep learning. Compared with the widely used ReLU-SGD pair, the proposed sigmoid-ADMM pair is stable with respect to algorithmic hyperparameters including learning rates, initial schemes and the pro-processing of input data. Furthermore, we find that to approximate and learn simple but important functions including the square function, radial functions and product gate, deep sigmoid nets theoretically beat deep ReLU nets and the proposed sigmoid-ADMM pair outperforms the ReLU-SGD pair. In terms of algorithm designs, different from existing ADMM methods in deep learning, our proposed ADMM adopts a backward-forward update order that is similar as BackProp (Rumelhart et al., 1986) and a local linear approximation for sub-problems, and more importantly keeps all the nonlinear constraints such that the solution found by the proposed algorithm can converge to a solution satisfying these nonlinear constraints. • Theoretical Novelty: To demonstrate the theoretical advantages of deep sigmoid nets, we rigorously prove that the approximation capability of deep sigmoid nets is not worse than deep ReLU nets by showing that ReLU can be well approximated by deep sigmoid nets with two hidden layers and finitely many free parameters but not vice-verse. We also establish the global convergence of the proposed ADMM for the nonlinearly constrained formulation of the deep sigmoid nets training from arbitrary initial points to a Karush-Kuhn-Tucker (KKT) point at a rate of order O(1/k). Different from the existing literature on convergence of nonconvex ADMM (Hong et al., 2016;Gao et al., 2020) for linear or multiaffine constrained optimization problems, our analysis provides a new methodology to deal with the nonlinear constraints in deep learning. In a word, our approach actually leads to a general convergence framework for ADMM with "smooth" enough activations. • Numerical Novelty: In terms of numerical performance, the effectiveness (particularly the stability to initial schemes and the easy-to-tune property of algorithmic parameters) of the proposed ADMM has been demonstrated by numerous experiments including a series of toy simulations and three real-data experiments. Numerical results illustrate the outperformance of the sigmoid-ADMM pair over the ReLU-SGD pair in approximating the extremely important square function, product gate, piecewise L 1 radial and smooth L 2 radial functions with stable algorithmic hyperparameters. Together with some other important functions such as the localized approximation (Chui et al., 1994), these natural functions realized in this paper can represent some important data features such as piecewise smoothness in image processing (Krizhevsky et al., 2012), sparseness in computer vision (LeCun et al., 2015), and rotation-invariance in earthquake prediction (Vikraman, 2016). The effectiveness of the proposed ADMM is further demonstrated by real-data experiments, i.e., earthquake intensity, extended Yale B databases and PTB Diagnostic ECG databases, which reflect the partially radial and low-dimensional manifold features in some extent. The rest of this paper is organized as follows. In the next section, we demonstrate the advantage of deep sigmoid nets in approximation (see Theorem 3). Section 3 describes the proposed ADMM method for the considered DNN training model followed by the main convergence theorem (see Theorem 4). Section 4 provides some discussions on related work and key ideas of our proofs. Section 5 provides some toy simulations to show the effectiveness of the proposed ADMM method in realizing some important natural functions. Section 6 provides two real data experiments to further demonstrate the effectiveness of the proposed method. All proofs are presented in Appendix. Notations: For any matrix A ∈ R m×n , [A] ij denotes its (i, j)-th entry. Given a matrix A, A F , A 2 and A max denote the Frobenius norm, operator norm, and max-norm of A, respectively, where A max = max i,j |[A] ij |. Then obviously, A max ≤ A 2 ≤ A F . We let W <i := [W 1 , W 2 , . . . , W i−1 ], W >i := [W i+1 , . . . , W N ] for i = 1, . . . , N , W <1 = ∅ and W >N = ∅. I denotes the identity matrix whose size can be determined according to the text. Denote by R and N the real and natural number sets, respectively. Deep Sigmoid Nets in Approximation For the depth N ∈ N of a neural network, let d i ∈ N be the number of hidden neurons at the i-th hidden layer for i = 1, . . . , N − 1. Denote an affine mapping J i : R d i−1 → R d i by J i (x) := W i x + b i for d i × d i−1 weight matrix and thresholds b i ∈ R d i . For a univariate activation function σ i , i = 1 . . . , N , denote further σ i (x) when σ i is applied component-wise to the vector x. Define an N -layer feedforward neural network by N N,d 1 ,...,d N ,σ = a · σ N • J N • σ N −1 • J N −1 • · · · • σ 1 • J 1 (x),(1) where a ∈ R d N . The deep net defined in (1) is called the deep ReLU net and deep sigmoid net, provided σ i (t) ≡ σ relu (t) = max{t, 0} and σ i (t) ≡ σ(t) = 1 1+e −t respectively. Since the (sub-)gradient computation of ReLU is very simple, deep ReLU nets have attracted enormous research activities in the deep learning community (Nair and Hinton, 2010). The power of deep ReLU nets, compared with shallow nets with ReLU (shallow ReLU nets) has been sufficiently explored in the literature (Yarotsky, 2017;Petersen and Voigtlaender, 2018;Shaham et al., 2018;Schwab and Zech, 2019;Chui et al., 2020;Han et al., 2020). In particular, it was proved in (Yarotsky, 2017, Proposition 2) that the following "square-gate" property holds for deep ReLU nets, which is beyond the capability of shallow ReLU nets due to the non-smoothness of ReLU. Lemma 1 The function f (t) = t 2 on the segment [−M, M ] for M > 0 can be approximated within any accuracy ε > 0 by a deep ReLU net with the depth and free parameters of order O(log(1/ε)). The above lemma exhibits the necessity of the depth for deep ReLU nets to act as a "square-gate". Since the depth depends on the accuracy, it requires many hidden layers for deep ReLU nets for such an easy task and too many hidden layers enhance the difficulty for analyzing SGD (Goodfellow et al., 2016, Sec.8.2). This presents the reason why the numerical accuracy of deep ReLU nets in approximating t 2 is not so good, just as Figure 2 exhibits. Differently, due to the infinitely differentiable property of the sigmoid function, it is easy for shallow sigmoid nets (Chui et al., 2019, Proposition 1) to play as a "square-gate", as shown in the following lemma. Lemma 2 Let M > 0. For ε > 0, there is a shallow sigmoid net N 3 with 3 free parameters bounded by O(ε −6 ) such that |t 2 − N 3 (t)| ≤ ε, t ∈ [−M, M ]. Besides the "square-gate", deep sigmoid nets are capable of acting as a "product-gate" (Chui et al., 2019), providing localized approximation (Chui et al., 1994), extracting the rotation-invariance property (Chui et al., 2019) and reflecting the sparseness in spatial domain (Lin, 2019) and frequency domain (Lin et al., 2017) with much fewer hidden layers than deep ReLU nets. The following Table 1 presents a comparison between deep sigmoid nets and deep ReLU nets in feature selection and approximation. Table 1 presents theoretical advantages of deep sigmoid nets over deep ReLU nets. In fact, as the following theorem shows, it is easy to construct a sigmoid net with two hidden layers and small number of free parameters to approximate ReLU, implying that the approximation capability of deep sigmoid nets is at least not worse than that of deep ReLU nets with comparable hidden layers and free parameters. Theorem 3 Let 1 ≤ p < ∞ and M ≥ 1. Then for any ε ∈ (0, 1/2) and M > 0, there is a sigmoid net h * with 2 hidden layers and at most 27 free parameters bounded by O(ε −7 ) such that h * − σ relu L p ([−M,M ]) ≤ ε,(2)N relu L,d 1 ,...,d L = a · σ relu • J L • σ relu • J L−1 • · · · • σ relu • J 1 (x) with bounded free parameters, Theorem 3 shows that we can construct a deep sigmoid net N sigmoid L,d 1 ,...,d L = a · h * • J L • h * • J L−1 • · · · • h * • J 1 (x) that possesses at least similar approximation capability. However, due to the infinitely differentiable property of the sigmoid function, it is difficult to construct a deep ReLU net with accuracy-independent depth and width to approximate it. Indeed, it can be found in (Petersen and Voigtlaender, 2018, Theorem 4.5) that for any open interval Ω and deep ReLU net N relu L,n with L hidden layers and n free parameters, there holds σ − N relu L,n L p (Ω) ≥ C n −2L ,(3) where σ is the sigmoid activation and C is a constant independent of n or L. Comparing (2) with (3), we find that any functions being well approximated by deep ReLU nets can also be well approximated by deep sigmoid nets, but not vice-verse. It should be mentioned that though the depth and width in Theorem 3 are independent of ε and relatively small, the magnitude of free parameters depends heavily on the accuracy and may be large. Since such large free parameters are difficult to realize for an optimization algorithm, a preferable way to shrink them is to deepen the network further. In particular, it can be found in (Chui et al., 2019) that there is a shallow sigmoid net N sigmoid 2 with 2 free parameters of order O(1/ε) such that |σ( √ wN sigmoid 2 ( √ wt)) − σ(wt)| ≤ ε, where t, w ∈ R. Because we only focus on the power of deep sigmoid nets in approximation, we do not shrink free parameters in Theorem 3. ADMM for Deep Sigmoid Nets Let Z := {(x j , y j )} n j=1 ⊂ R d 0 × R d N be n samples. Denote X := (x 1 , x 2 , . . . , x n ) ∈ R d 0 ×n and Y := (y 1 , y 2 , . . . , y n ) ∈ R d N ×n . It is natural to consider the following regularized DNN training problem min W 1 n n i=1 Φ(x i , W) − y i 2 2 + λ W i 2 F ,(4) where Φ(x i , W) denotes a deep sigmoid net with N layers, W = {W i } N i=1 and λ > 0 is the regularization parameter. Here, we consider the square loss as analyzed in the literature (Allen-Zhu et al., 2019;Zou and Gu, 2019). We also absorb thresholds into the weight matrices for the sake of simplicity. Based on the advantage of deep sigmoid nets in approximation, (Chui et al., 2019;Lin, 2019) proved that the model defined by (4) with N = 2 are optimal in embodying data features such as the spatial sparseness, smoothness and rotation-invariance in the sense that it can achieve almost optimal generalization error bounds in the framework of learning theory. The aim of this section is to introduce an efficient algorithm to solve the optimization problem (4). Due to the saturation problem of the sigmoid function (see Figure 2 (b)), the issue of gradient vanishing or explosion frequently happens for running SGD on deep sigmoid nets (see Figure 4 (a) for example), implying that the classical SGD is not a good candidate to solve (4). We then turn to designing a gradient-free optimization algorithm, like ADMM, to efficiently solve (4). For DNN training, there are generally two important ingredients in designing ADMM: update order and solution to each sub-problem. The novelty of our proposed algorithm is the use of backward-forward update order similar to BackProp in (Rumelhart et al., 1986) and local linear approximation to sub-problems. Update order in ADMM for deep learning training The optimization problem (4) can be equivalently reformulated as the following constrained optimization problem minimize W,V 1 2 V N − Y 2 F + λ 2 N i=1 W i 2 F (5) subject to V i = σ(W i V i−1 ), i = 1, . . . , N − 1, V N = W N V N −1 , where V := {V i } N i=1 represents the set of responses of all layers and λ = λ n 2 . We define the augmented Lagrangian of (5) as follows: L(W, V, {Λ i } N i=1 ) := 1 2 V N − Y 2 F + λ 2 N i=1 W i 2 F (6) + N −1 i=1 β i 2 σ(W i V i−1 ) − V i 2 F + Λ i , σ(W i V i−1 ) − V i + β N 2 W N V N −1 − V N 2 F + Λ N , W N V N −1 − V N , where Λ i ∈ R d i ×n is the multiplier matrix associated with the i-th constraint, and β i is the associated penalty parameter for i = 1, . . . , N . ADMM is an augmented-Lagrangian based primal-dual method, which updates the primal variables ({W i (6)) via a Gauss-Seidel scheme and then multipliers (6)) via a gradient ascent scheme in a parallel way (Boyd et al., 2011). As suggested in , the update order of the primal variables is tricky for ADMM in terms of the convergence analysis in the nonconvex setting. In light of , the key idea to yield a desired update order with convergence guarantee is to arrange the updates of some special primal variables followed by the updates of multipliers such that the updates of multipliers can be explicitly expressed by the updates of these special primal variables, and thus the dual ascent quantities arisen by the updates of multipliers shall be controlled by the descent quantities brought by the updates of these special primal variables. Hence, the arrangement of these special primal variables is crucial. } N i=1 and {V i } N i=1 in({Λ i } N i=1 in It can be noted that there are 2N blocks of primal variables, i.e., (6). For better elaboration of our idea, we take N = 3 for an example. Notice that the multipliers {Λ i } 3 i=1 are only involved in these inner product terms Λ 1 , σ(W 1 X) − V 1 , Λ 2 , σ(W 2 V 1 ) − V 2 and Λ 3 , W 3 V 2 − V 3 . By these terms, the gradient of the i-th inner product with respect to V i is −Λ i , while the associated gradient with respect to W i is a more complex term (namely, (Λ 1 σ (W 1 X))X T for W 1 , (Λ 2 σ (W 2 V 1 ))V T 1 for W 2 , and Λ 3 V T 2 for W 3 , where represents Hadamard product). If the update of W i is used to express Λ i , then according to the W i subproblem, an inverse operation of a nonlinear or linear mapping is required, while such an inverse does not necessarily exist. Specifically, following the analysis of Lemma 8 shown later and taking the expression of W 3 for example, the term Λ 3 V T 2 will be involved in the expression of W 3 . In this case, if we wish to express Λ 3 by W 3 , then the inverse of V 2 is generally required, while it does not necessarily exist. Due to this, it should be more convenient to express Λ i (i = 1, 2, 3) via exploiting the V i subproblem instead of the W i subproblem. Therefore, we suggest firstly update the blocks of W i 's and then V i 's such that Λ i 's can be explicitly expressed via the latest updates of V i 's. To be detailed, for each loop, we update {W i } N i=1 and {V i } N i=1 and N blocks of multipliers {Λ i } N i=1 involved in{W i } N i=1 in the backward order, i.e., W N → W N −1 → · · · → W 1 , then update {V j } N j=1 in the forward order, i.e., V 1 → V 2 → · · · → V N , motivated by BackProp in (Rumelhart et al., 1986), and finally update the multipliers {Λ i } N i=1 in a parallel way, as shown by the following Figure 3. i } N i=1 , we set V 0 j = σ(W 0 j V 0 j−1 ), j = 1, . . . , N − 1, V 0 N = W 0 N V 0 N −1 , and Λ 0 i = 0, i = 1, . . . , N,(7) where V 0 0 = X. Given the (k-1)-th iterate {W k−1 i } N i=1 , {V k−1 i } N i=1 , {Λ k−1 i } N i=1 , we define the W i -and V i -subproblems at the k-th iteration via minimizing the augmented Lagrangian (6) with respect to only one block but fixing the other blocks at the latest updates, according to the update order specified in Figure 3, shown as follows: W k N = arg min W N λ 2 W N 2 F + β N 2 W N V k−1 N −1 − V k−1 N 2 F + Λ k−1 N , W N V k−1 N −1 − V k−1 N ,(8) and for i = N − 1, . . . , 1, W k i = arg min W i λ 2 W i 2 F + β i 2 σ(W i V k−1 i−1 ) − V k−1 i 2 F + Λ k−1 i , σ(W i V k−1 i−1 ) − V k−1 i ,(9) and for j = 1, . . . , N − 2, V k j = arg min V j β j 2 σ(W k j V k j−1 ) − V j 2 F + Λ k−1 j , σ(W k j V k j−1 ) − V j(10)+ β j+1 2 σ(W k j+1 V j ) − V k−1 j+1 2 F + Λ k−1 j+1 , σ(W k j+1 V j ) − V k−1 j+1 , V k N −1 = arg min V N −1 β N −1 2 σ(W k N −1 V k N −2 ) − V N −1 2 F + Λ k−1 N −1 , σ(W k N −1 V k N −2 ) − V N −1 + β N 2 W k N V N −1 − V k−1 N 2 F + Λ k−1 N , W k N V N −1 − V k−1 N ,(11)V k N = arg min V N 1 2 V N − Y 2 F + β N 2 W k N V k N −1 − V N 2 F + Λ k−1 N , W k N V k N −1 − V N . (12) Once {W k i } N i=1 , {V k i } N i=1 have been updated, we then update the multipliers {Λ k i } N i=1 parallelly according to the following: for i = 1, . . . , N − 1, Λ k i = Λ k−1 i + β i (σ(W k i V k i−1 ) − V k i ), Λ k N = Λ k−1 N + β N (W k N V k N −1 − V k N ).(13) Based on these, each iterate of ADMM only involves several relatively simpler sub-problems. It should be mentioned that the suggested update order is actually a technical requirement in the convergence proof (see, Lemma 8 below), which also appears in the previous work . Moreover, the following local linear approximation is also required to establish Lemma 8. Local linear approximation for sub-problems Note that W i -subproblems (i = 1, . . . , N − 1) involve functions of the following form H σ (W ; A, B) = 1 2 σ(W A) − B 2 F ,(14) while V j -subproblems (j = 1, . . . , N − 2) involve functions of the following form M σ (V ;Ã,B) = 1 2 σ(ÃV ) −B 2 F ,(15) where A, B,Ã,B are four given matrices related to the previous updates. Due to the nonlinearity of the sigmoid activation function, the subproblems are generally difficult to be solved, or at least some additional numerical solvers are required to solve these subproblems. To break such computational hurdle, we adopt the first-order approximations of the original functions presented in (14) and (15) at the latest updates, instead of themselves, to update the variables, that is, H k σ (W ; A, B) := H σ (W k−1 ; A, B) + (σ(W k−1 A) − B) σ (W k−1 A), (W − W k−1 )A + h k 4 (W − W k−1 )A 2 F ,(16)M k σ (V ;Ã,B) := M σ (V k−1 ;Ã,B) + (σ(ÃV k−1 ) −B) σ (ÃV k−1 ),Ã(V − V k−1 ) + µ k 4 Ã (V − V k−1 ) 2 F ,(17) where W k−1 and V k−1 are the (k-1)-th iterate, and σ (W k−1 A) and σ (ÃV k−1 ) represent the componentwise derivatives, h k and µ k can be specified as the upper bounds of twice of the locally Lipschitz constants of functions H σ and M σ , respectively, shown as h k = L( B max ), µ k = L( B max ). Here, for any given c ∈ R, L(|c|) := 2L 2 (L 0 + |c|) + 2L 2 1(18) is an upper bound of the Lipschitz constant of the gradient of function (σ(u) − c) 2 with constants L 0 = 1, L 1 = 1 4 and L 2 = 1 4 related to the sigmoid activation σ. Henceforth, we call this treatment as the local linear approximation (LLA), which can be viewed as adopting certain prox-linear scheme (Xu and Yin, 2013) to update the subproblems of ADMM. Based on (16) and (17), the original updates (9) of {W k i } N −1 i=1 are replaced by W k i = arg min W i λ 2 W i 2 F + β i H k σ (W i ; V k−1 i−1 , V k−1 i − β −1 i Λ k−1 i ) ,(19) and by completing perfect squares and some simplifications, the original updates (10) of {V k j } N −2 j=1 are replaced by V k j = arg min V j β j 2 σ(W k j V k j−1 ) + β −1 j Λ k−1 j − V j 2 F + β j+1 M k σ (V j ; W k j+1 , V k−1 j+1 − β −1 j+1 Λ k−1 j+1 ) ,(20) with h k i and µ k j being specified as follows h k i = L( V k−1 i − β −1 i Λ k−1 i max ), i = 1, . . . , N − 1,(21)µ k j = L( V k−1 j+1 − β −1 j+1 Λ k−1 j+1 max ), j = 1, . . . , N − 2,(22) where L(·) is defined in (18). Note that with these alternatives, all the subproblems can be solved with analytic expressions (see, Lemma 8 in Appendix C.1). ADMM for deep sigmoid nets The ADMM algorithm for DNN training problem (5) is summarized in Algorithm 1. As shown in Figure 4 (b) and Figure 2 (c), ADMM does not suffer from either the issue of gradient explosion or the issue of gradient vanishing caused by the saturation of sigmoid activation and thus can approximate the square function within high precision. The intuition behind ADMM to avoid the issues of gradient explosion and vanishing is that the suggested ADMM does not exactly follow the chain rule as exploited in BackProp and SGD, but introduces the multipliers as certain compensation to eventually fit the chain rule at the stationary point. From Algorithm 1, besides the regularization parameter λ related to the DNN training model, only the penalty parameters β i 's should be tuned. In the algorithmic perspective, penalty parameters can be regarded as the dual step sizes for the updates of multipliers, which play similar roles as learning rates in SGD. As shown by our experiment results below, the performance of ADMM is not sensitive to penalty parameters, making the parameters be easy-to-tune. Moreover, by exploiting the LLA, the updates for all variables can be very cheap with analytic expressions (see Lemma 8 in Appendix C.1). As compared to the existing ADMM methods for deep learning (Carreira-Perpinan and Wang, 2014;Kiaee et al., 2016;Murdock et al., 2018), there are two major differences shown as follows. The first one is that the existing ADMM type methods in deep learning only keep partial nonlinear constraints for the sake of reducing the difficulty of optimization, while the ADMM method suggested in this paper keeps all the nonlinear constraints, and thus our proposed ADMM can come back to the original DNN training model in the sense that its convergent limit fits all the nonlinear constraints as shown in Theorem 4 below. To overcome the difficulty from optimization, we introduce an elegant update order and the LLA technique for subproblems. The second one is that most of existing ADMM methods focus on deep ReLU nets, while our proposed ADMM is designed for deep sigmoid nets. It should be pointed out that the subproblems of the proposed algorithm require inverting matrices at each iteration, which could be expensive. Although there are some practical tricks like warm-start and solving inexactly via doing gradient descent by a fixed number of times to improve the computational efficiency of the proposed ADMM (e.g., in (Liu et al., 2021) ), the major focus of this paper is mainly on the development of an effective ADMM method with theoretical guarantees for the training of deep sigmoid nets, and we will consider its practical acceleration in the future. Convergence of ADMM for deep sigmoid nets Without loss of generality, we assume that X, Y and {W 0 i } N i=1 are normalized with X F = 1, Y F = 1 and W 0 i F = 1, i = 1, . . . , N , and all numbers of hidden layers are the same, i.e., d i = d, ∀i = 1, . . . , N − 1. Under these settings, we present the main convergence theorem of ADMM in the following, while that of ADMM under more general settings is presented in Theorem 7 in Appendix B. Algorithm 1 ADMM for Deep Sigmoid Nets Training Samples: X := [x 1 , . . . , x n ] ∈ R d0×n , Y := [y 1 , . . . , y n ] ∈ R d N ×n . Initialization: ({W 0 i } N i=1 , {V 0 i } N i=1 , {Λ 0 i } N i=1 ) is set according to (7). V k 0 ≡ X, ∀k ∈ N. Parameters: λ > 0, β i > 0, i = 1, . . . , N . for k = 1, . . . do (Backward Estimation) for i = N : −1 : 1 do Update W k N via (8) and the other W k i via (19). end for (Forward Prediction) for j = 1 : N do Update V k j (j = 1, . . . , N − 2) via (20), V k N −1 via (11), and V k N via (12). end for (Updating Multipliers) Λ k i = Λ k−1 i + β i (σ(W k i V k i−1 ) − V k i ), i = 1, . . . , N − 1, Λ k N = Λ k−1 N + β N (W k N V k N −1 − V k N ). k ← k + 1 endTheorem 4 Let {Q k := ({W k i } N i=1 , {V k i } N i=1 , {Λ k i } N i=1 )} be a sequence generated by Algo- rithm 1. If 2 ≤ N ≤ √ n, λ ≥cN N −3 2 (nd) N 2 − 1 4 and {β i } N i=1 satisfy β N ≥ 3.5, β N −1 ≥ 16β N , β i ≥c 1 β N −1 (N nd) N −1−i 2 , i = 1, . . . , N − 2(23) for some constantsc,c 1 > 0 independent of n, N , then we have: (a) the augmented Lagrangian sequence {L(Q k )} is convergent. (b) {Q k } converges to a stationary point Q * := ({W * i } N i=1 , {V * i } N i=1 , {Λ * i } N i=1 ) of the augmented Lagrangian L, which is also a KKT point (defined in(24) below) of problem (5), implying that {W * i } N i=1 is a stationary point of problem (4) with λ = 2λ/n. (c) 1 K K k=1 ∇L(Q k ) 2 F → 0 at a rate of order O( 1 K ). Theorem 4 establishes the global convergence of ADMM to a KKT point at a rate of O(1/K). By (23), the parameters {β i } N i=1 increase exponentially fast from the output layer to the input layer. Moreover, by Theorem 4, the regularization parameter λ is also required to grow exponentially fast as the depth increases. Back to the original DNN training model (4), the requirement on the regularization parameter λ is λ ≥cN N −3 2 d N 2 − 1 4 n N 2 − 5 4 . Particularly, when N = 2, namely, the neural networks with single hidden layer, then λ =c 4 d 3 /n is a good choice, which implies that the regularization parameter can be small when the sample size n is sufficiently large. Despite these convergence conditions seem a little stringent, by the existing literature (Chui et al., 2018(Chui et al., , 2019, the depth of deep sigmoid nets is usually small, say, 2 or 3 for realizing some important data features in deep learning. Moreover, as shown in the numerical results to be presented in Sections 5 and 6, a moderately large augmented Lagrangian parameter (say, each β i = 1) and a small regularization parameter (say, λ = 10 −6 ) are empirically enough for the proposed ADMM. In this case, the KKT point found by ADMM should be close to the optimal solutions to the empirical risk minimization of DNN training. Remark 1: KKT conditions. Based on (6), the Karush-Kuhn-Tucker (KKT) conditions of the problem (5) can be derived as follows. Specifically, let {W i , V i } N i=1 be an optimal solution of problem (5), then there exit multipliers {Λ i } N i=1 such that the following hold: (24), the KKT point of problem (5) exactly fits these nonlinear constraints. Moreover, given a KKT point ( (5), substituting the last five equations into the first three equations of (24) shows that 0 = λW 1 + (Λ 1 σ (W 1 V 0 ))V T 0 , 0 = λW i + (Λ i σ (W i V i−1 ))V T i−1 , i = 2, . . . , N − 1, 0 = λW N + Λ N V T N −1 , 0 = −Λ i + W T i+1 (Λ i+1 σ (W i+1 V i )), i = 1, . . . , N − 2,(24)0 = −Λ N −1 + W T N Λ N , 0 = −Λ N + (V N − Y ), 0 = σ(W i V i−1 ) − V i , i = 1, . . . , N − 1, 0 = W N V N −1 − V N where V 0 = X. From{W * i } N i=1 , {V * i } N i=1 , {Λ * i } N i=1 ) of{W * i } N i=1 is also a stationary point of the original DNN training model (4). Remark 2: More general activations: As presented in Theorem 7 in Appendix B, the convergence results in Theorem 4 still hold for a general class of smooth activations such as the hyperbolic tangent activation as studied in . Actually, the approximation result yielded in Theorem 3 can be also easily extended to a class of twice differentiable sigmoid-type activations. Related Work and Discussions In this section, we present some related works and show the novelty of our studies. Deep sigmoid nets versus deep ReLU nets in approximation Deep ReLU nets are the most popular neural networks in deep learning. Compared with deep sigmoid nets, there are commonly three advantages of deep ReLU nets (Nair and Hinton, 2010). At first, the piecewise linear property makes it easy to compute the derivative to ease the training via gradient-type algorithms. Then, the derivative of ReLU is either 1 or 0, which in a large extent alleviates the saturation phenomenon for deep sigmoid nets and particularly the gradient vanishing/explotion issue of the gradient-descent based algorithms for the training of deep neural networks. Finally, σ relu (t) = 0 for t < 0 enables the sparseness of the neural networks, which coincides with the biological mechanism for neural systems. Theoretical verification for the power of depth in deep ReLU nets is a hot topic in deep learning theory. It stems from the study in (Eldan and Shamir, 2016), where some functions were constructed to be well approximated by deep ReLU nets but cannot be expressed by shallow ReLU nets with similar number of parameters. Then, numerous interesting results on the expressivity and generalization of deep ReLU nets have been provided in (Yarotsky, 2017;Safran and Shamir, 2017;Shaham et al., 2018;Petersen and Voigtlaender, 2018;Schwab and Zech, 2019;Guo et al.;Zhou, 2018Zhou, , 2020Chui et al., 2020;Han et al., 2020). Typically, it was proved in (Yarotsky, 2017) that deep ReLU nets perform at least not worse than the classical linear approaches in approximating smooth functions, and are beyond the capability of shallow ReLU nets. Furthermore, it was also exhibited in (Shaham et al., 2018) that deep ReLU nets can extract the manifold structure of the input space and the smoothness of the target functions simultaneously. The problem is, however, that there are frequently too many hidden layers for deep ReLU nets to extract data features. Even for approximating the extremely simple square function, Lemma 1 requires log(ε −1 ) depth, which is totally different from deep sigmoid nets. Due to its infinitely differentiable property, sigmoid function is the most popular activation for shallow nets (Pinkus, 1999). The universal approximation property of shallow sigmoid nets has been verified in (Cybenko, 1989) for thirty years. Furthermore, (Mhaskar, 1993(Mhaskar, , 1996 showed that the approximation capability of shallow sigmoid nets is at least not worse than that of polynomials. However, there are also several bottlenecks for shallow sigmoid nets in embodying the locality (Chui et al., 1994), extracting the rotation-invariance (Chui et al., 2019) and producing sparse estimators (Lin et al., 2017), which show the necessity to deepen the neural networks. Different from deep ReLU nets, adding only a few hidden layers can significantly improve the approximation capability of shallow sigmoid nets. In particular, deep sigmoid nets with two hidden layers are capable of providing localized approximation (Chui et al., 1994), reflecting the spatially sparseness (Lin, 2019) and embodying the rotation-invariance (Chui et al., 2019). In a nutshell, as shown in Table 1, it was proved in the existing literature that any function expressible for deep ReLU nets can also be well approximated by deep sigmoid nets with fewer hidden layers and free parameters. Our Theorem 3 partly reveals the reason for such a phenomenon in the sense that ReLU can be well approximated by sigmoid nets but not vice-verse. Algorithms for DNN training In order to address the choice of learning rate in SGD, there are many variants of SGD incorporated with adaptive learning rates called adaptive gradient methods. Some important adaptive gradient methods are Adagrad (Duchi et al., 2011), Adadelta (Zeiler, 2012), RMSprop (Tieleman and Hinton, 2012), Adam (Kingma and Ba, 2015), and AMSGrad (Reddi et al., 2018). Although these adaptive gradient methods have been widely used in deep learning, there are few theoretical guarantees when applied to the deep neural network training, a highly nonconvex and possibly nonsmooth optimization problem (Wu et al., 2019). Regardless the lack of theoretical guarantees of the existing variants of SGD, another major flaw is that they may suffer from the issue of gradient explosion/vanishing (Goodfellow et al., 2016), essentially due to the use of BackProp (Rumelhart et al., 1986) for updating the gradient during the iteration procedure. To address the issue of gradient vanishing, there are some tricks that focus on either the design of the network architectures such as ResNets (He et al., 2016) or the training procedure such as the batch normalization (Ioffe and Szegedy, 2015) and weight normalization (Salimans and Kingma, 2016). Besides these tricks, there are many works in the perspective of algorithm design, aiming to propose some alternatives of SGD to alleviate the issue of gradient vanishing. Among these alternatives, the so called gradient-free type methods have recently attracted rising attention in deep learning since they may in principle alleviate this issue by their gradient-free natures, where the alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) methods are two most popular ones (see, (Carreira-Perpinan and Wang, 2014;Kiaee et al., 2016;Yang et al., 2016;Murdock et al., 2018;Gotmare et al., 2018;Zhang and Brand, 2017;Gu et al., 2018;Lau et al., 2018;). Besides the gradient-free nature, another advantage of both ADMM and BCD is that they can be easily implemented in a distributed and parallel manner, and thus are capable of solving distributed/decentralized large-scale problems (Boyd et al., 2011). In the perspective of constrained optimization, all the BackProp (BP), BCD and ADMM can be regarded as certain Lagrangian methods or variants for the nonlinearly constrained formulation of DNN training problem. In (LeCun, 1988), BP was firstly reformulated as a Lagrangian multiplier method. The fitting of nonlinear equations motivated by the forward pass of the neural networks plays a central role in the development of BP. Following the Lagrangian framework, the BCD methods for DNN training proposed by (Zhang and Brand, 2017;Lau et al., 2018;Gu et al., 2018) can be regarded as certain Lagrangian relaxation methods without requiring the exact fitting of nonlinear constraints. Unlike in BP, such nonlinear constraints are directly lifted as quadratic penalties to the objective function in BCD, rather than involving these nonlinear constraints with Lagrangian multipliers. However, such a lifted treatment of nonlinear constraints in BCD as penalties suffers from an inconsistent issue in the sense that the solution found by BCD cannot converge to a solution satisfying these nonlinear constraints. To tackle this issue, ADMM, a primal-dual method based on the augmented Lagrangian by introducing the nonlinear constraints via Lagrangian multipliers, enables a convergent sequence satisfying the nonlinear constraints. Therefore, ADMM attracted rising attention in deep learning with various implementations (Carreira-Perpinan and Wang, 2014;Kiaee et al., 2016;Yang et al., 2016;Gotmare et al., 2018;Murdock et al., 2018). However, most of the existing ADMM type methods in deep learning only keep partial nonlinear constraints for the sake of reducing the difficulty of optimization, and there are few convergence guarantees (Gao et al., 2020). Convergence of ADMM and challenges Most results in the literature on the convergence of nonconvex ADMM focused on linear constrained optimization problems (e.g. (Hong et al., 2016;). Following the similar analysis of (Wang et al., 2019), (Gao et al., 2020) extended the convergence results of ADMM to multiaffine constrained optimization problems. In the analysis of (Hong et al., 2016;Gao et al., 2020), the separation of a special block of variables is crucial for the convergence of ADMM in both linear and multiaffine scenarios. Together with the descent quantity arisen by the x-block update, the total progress of one step ADMM update is descent along the augmented Lagrangian. Such a technique is in the core of analysis in and (Gao et al., 2020) to deal with some multiaffine constraints in deep learning. However, in a general formulation of ADMM for DNN training (e.g. (5)), it is impossible to separate such a special variable block y satisfying these requirements. Let us take a threelayer neural network for example. Let W := {W i } 3 i=1 be the weight matrices of the neural network, and V := {V i } 3 i=1 be the response matrices of the neural network and X be the input matrix, then the nonlinear constraints are of the following form, σ(W 1 X) − V 1 = 0, (26a) σ(W 2 V 1 ) − V 2 = 0, (26b) W 3 V 2 − V 3 = 0,(26c) where σ is the sigmoid activation. Note that in (26b) and (26c), W 2 is coupled with V 1 and W 3 is coupled with V 2 , respectively, so none of these four variable blocks can be separated from the others. Although W 1 in (26a) and V 3 in (26c) can be separated, the image inclusion constraint above is not satisfied. Therefore, one cannot exploit the structure in Gao et al., 2020) to study such constraints in deep learning. Key stones to the challenges and main idea of proof In order to address the challenge of such nonlinear constraints σ(W i V i−1 ) − V i = 0, we introduce a local linear approximation (LLA) technique. Let us take (26) for example to illustrate this idea. The most difficult block of variable is V 1 which involves two constraints, namely, a linear constraint in (26a), and a nonlinear constraint in (26b). Now we fix W 1 , W 2 and V 2 as the previous updates, say W 0 1 , W 0 2 and V 0 2 , respectively. For the update of V 1 -block, we keep the linear constraint, but relax the nonlinear constraint with its linear approximation at the previous update V 0 1 , σ(W 0 2 V 0 1 ) − V 0 2 + σ (W 0 2 V 0 1 ) W 0 2 (V 1 − V 0 1 ) ≈ 0,(27) assuming the differentiability of activation function σ. The other blocks can be handled in a similar way. Taking W 1 block for example, we relax the related nonlinear constraint via its linear approximation at the previous update W 0 1 , namely, σ(W 0 1 X) − V 0 1 + σ (W 0 1 X) ((W 1 − W 0 1 )X) ≈ 0. The operations of LLA on the nonlinear constraints can be regarded as applying certain prox-linear updates (Xu and Yin, 2013) to replace the subproblems of ADMM involving nonlinear constraints as shown in Section 3.2. To make such a local linear approximation valid, intuitively one needs: (a) the activation function σ is smooth enough; and (b) the linear approximation occurs in a small enough neighbourhood around the previous updates. Condition (a) is mild and naturally satisfied by the sigmoid type activations. But condition (b) requires us to introduce a new Lyapunov function defined in (29) by adding to the original augmented Lagrangian a proximal control between V i and its previous updates. Equipped with such a Lyapunov function, we are able to show that an auxiliary sequence converges to a stationary point of the new Lyapunov function (see Theorem 5 below), which leads to the convergence of the original sequence generated by ADMM (see Theorem 4 in Section 3.4). Specifically, denote {Q k } aŝ Q k := (Q k , {V k i } N i=1 ),(28)withV k i := V k−1 i for i = 1, . . . , N and k ≥ 1, andL(Q k ) aŝ L(Q k ) := L(Q k ) + N i=1 ξ i V k i −V k i 2 F(29) for some positive constant ξ i > 0 (i = 1, . . . , N ) specified later in Appendix D.1.4. Then we state the convergence of {Q k } as follows. Theorem 5 (Convergence of {Q k }) Under conditions of Theorem 4, we have: (a)L(Q k ) is convergent. (b)Q k converges to some stationary pointQ * ofL. (c) 1 K K k=1 ∇L(Q k ) 2 F → 0 at a O( 1 K ) rate. Theorem 5 presents the function value convergence and sequence convergence to a stationary point at a O(1/K) rate of the auxiliary sequence {Q k }. By the definitions (28) and (29) ofQ k andL , Theorem 5 directly implies Theorem 4. As shown by the proofs in Appendix D, the claims in Theorem 5 also hold under the more general assumptions for Theorem 7 in Appendix B. In Theorem 5, we only give the convergence guarantee for the proposed ADMM. It would be interesting to derive the convergence rate to highlight the role of algorithmic parameters. We will keep in study and report the result in future work. Our main idea of proof for Theorem 5 can be summarized as follows: we firstly establish a sufficient descent lemma along the new Lyapunov function (see Lemma 14 in Appendix D.1), then show a relative error lemma (see Lemma 21 in Appendix D.1.5), and later verify the Kurdyka-Lojasiewicz property ( Lojasiewicz, 1993;Kurdyka, 1998) (see Lemma 13 in Appendix C.2) and the limiting continuity property of the new Lyapunov function by Assumption 1, and finally establish Theorem 5 via following the analysis framework formulated in (Attouch et al., 2013, Theorem 2.9). In order to prove Lemma 14, we prove the following three lemmas, namely, a one-step progress lemma (see Lemma 15 in Appendix D.1.1), a dual-bounded-by-primal lemma (see Lemma 18 in Appendix D.1.2), and a boundedness lemma (see Lemma 19 in Appendix D.1.3), while to prove Lemma 21, besides Lemmas 18 and 19, we also use the Lipschitz continuity of the activation and its derivative by Assumption 1 in Appendix B. The proof sketch can be illustrated by Figure 5. According to Figure 5, we show the boundedness of the sequence before the establishment of the sufficient descent lemma (i.e., Lemma 14). Such a proof procedure is different from the existing ones in the literature (say, ), where a sequence boundedness is usually implied by firstly showing the (sufficient) descent lemma (Wang et al., 2019, Lemma 6). Toy Simulations In this section, we conduct a series of simulations to show the effectiveness of the proposed ADMM in approximating and learning some natural functions including the square function, product gate, a piecewise L 1 radial function, and a smooth L 2 radial function, which play important roles in reflecting some commonly used data features (Safran and Shamir, 2017;Shaham et al., 2018;Chui et al., 2019;Guo et al.). In particular, we provide empirical studies to show that these important natural functions can be numerically well approximated or learned by the proposed ADMM-sigmoid pair. Furthermore we also show that the proposed ADMM-sigmoid pair is stable to its algorithmic hyperparameters, via comparing to the popular deep learning optimizers including the vanilla SGD, SGD with momentum (called SGDM for short henceforth) and Adam (Kingma and Ba, 2015). There are four experiments concerning approximation and learning tasks: (a) approximation of square function, (b) approximation of product gate, (c) learning of a piecewise L 1 radial function, and (d) learning of a smooth L 2 radial function. All numerical experiments were carried out in Matlab R2015b environment running Windows 10, Intel(R) Xeon(R) CPU E5-2667 v3 @ 3.2GHz 3.2GH. The codes are available at https://github.com/JinshanZeng/ADMM-DeepLearning. Experimental settings In all our experiments, we use deep fully connected neural networks with different depths and widths. Throughout the paper, the depth and width of deep neural networks are respectively the number of hidden layers and number of neurons in each hidden layer. For simplicity, we only consider deep neural networks with the same width for all the hidden layers. We consider both deep sigmoid nets and deep ReLU nets in the simulation. Motivated by the existing literature (Guo et al.), the depth required for deep ReLU nets is in general much more than that for deep sigmoid nets in the aforementioned approximation or learning tasks. For the fairness of comparison, we consider deep ReLU nets with more hidden layers, i.e., the maximal depth of deep ReLU nets is 20 while that of deep sigmoid nets is only 5 or 6, as presented in Table 2. Besides the vanilla SGD-ReLU pair (SGD (ReLU) for short), we also consider SGD-sigmoid pair (SGD (sigmoid) for short), SGDM-ReLU pair (SGDM (ReLU) for short), and Adam-ReLU pair (Adam (ReLU) for short) as the competitors. Similarly, we denote by ADMM (sigmoid) the proposed ADMM-sigmoid pair. For each experiment, our purposes are mainly two folds: excellent approximation or learning performance, and stability with respect to initialization schemes and penalty parameters with appropriate neural network structures for the proposed ADMM-sigmoid pair. For the first purpose, we consider deep neural networks with different depths and widths as presented in Table 2. Moreover, for ADMM, we empirically set the regularization parameter λ = 10 −6 and the augmented Lagrangian parameters β i 's as the same 1, while for SGD methods, we empirically utilize the step exponential decay (or, called geometric decay) learning rate schedule with the decay exponent 0.95 for every 10 epochs. For SGDM and Adam, we use the default settings as presented in Table 2. The number of epochs in all experiments is empirically set as 2000. The specific settings of these experiments are presented in Table 2. For the second purpose, we consider different regularization and penalty parameters (λ ∈ {10 −6 , 10 −5 , 10 −4 , 10 −3 , 10 −2 , 10 −1 }, β ∈ {0.01, 0.1, 0.5, 1, 5, 10, 100}), as well as several existing initialization schemes for ADMM under the optimal neural network structure determined by the first part. Particularly, we consider the following four types of schemes yielding six typical initializations: (1) LeCun random initialization (LeCun et al., 1998): the components of the weight matrix W l at the l-th layer are generated i.i.d. according to some random distribution with zero mean and variance 1 d l−1 , l = 1, . . . , N . Particularly, we consider two special LeCun random initialization schemes generated respectively according to the uniform and Gaussian distributions, i.e., W l ∼ U ([− 3 d l−1 , 3 d l−1 ]) (LeCun-Unif for short) and W l ∼ N (0, 1 d l−1 ) (LeCun-Gauss for short). (2) Random orthogonal initialization (Saxe et al., 2014): the weight matrix W is set as some random orthogonal matrix such that W T W = I or W W T = I. We call it Orth-Unif (or Orth-Gauss) for short if the random matrix is generated i.i.d. according to the uniform random (or, Gaussian) distribution. (3) Xavier initialization (Glorot and Bengio, 2010) : each W l ∼ U ([− 6 d l−1 +d l , 6 d l−1 +d l ]), l = 1, . . . , N . (4) MSRA initialization (He et al., 2015): W l ∼ N (0, 2 d l ), l = 1, . . . , N − 1, and W N ∼ N (0, 1 d N ) since there is no ReLU activation for the last layer. The default settings for initial threshold vectors in the above initialization schemes are set to be 0. For each group of parameters, we run 20 trails for average. Specifically, for the approximation tasks, the performance of an algorithm is measured by the approximation error, defined as the average of these 20 trails's mean square errors, while for the learning tasks, the performance of an algorithm is measured by the test error, defined as the average of the mean square errors of these trails over the test data. Approximation of square function In these experiments, we consider the performance of the ADMM-sigmoid pair in approximating the univariate square function, that is, f (x) = x 2 on [−1, 1]. The specific experimental settings can be found in Table 2. Table 3: Experimental results of different algorithms in approximating f (x) = x 2 . The standard derivation of the approximation error is presented in the parentheses. The running time is recorded in seconds. The depth and width of the optimal network structure in terms of approximation error is presented in the last row. A. Approximation performance of ADMM. Experiment results over the best neural network structures are presented in Table 3, and trends of approximation errors with respect to the depth are shown in Figure 6. From Table 3, the ADMM-SGD pair can approximate the square function within very high precision, i.e., in the order of 10 −9 , which is slightly better than that of competitors for deeper ReLU nets, and is much better than the SGD-sigmoid pair with the same depth. Specifically, optimal depths for SGD (ReLU), SGDM (ReLU) and Adam (ReLU) are 18, 15, 15, respectively, while the optimal depth for ADMM (sigmoid) is only 2, which matches the theoretical results in approximation theory, as shown in (Chui et al., 2019, Proposition 2). In terms of running time, ADMM (sigmoid) with optimal network structures is generally faster than the SGDM (ReLU) and Adam (ReLU) with optimal network structures as presented in the third row of Table 3, mainly due to the depth required for ADMM (sigmoid) is much less than those for deep ReLU nets SGD (ReLU), SGDM (ReLU) and Adam (ReLU). Moreover, according to Figure 6, ADMM (sigmoid) can yield high approximation precision with less layers than the competitors. These experimental results demonstrate that the proposed ADMM can embody the advantage of deep sigmoid nets on approximating the square function, as pointed out in the existing theoretical literature (Chui et al., 2019). B. Effect of parameters for ADMM. There are mainly two parameters for the proposed ADMM, i.e., the model parameter λ (also called as the regularization parameter) and the algorithm parameter β involved in the augmented Lagrangian (also called as the penalty parameter). In this experiment, we consider the performance of ADMM in approximating the univariate square function with different model and algorithmic parameters, under the optimal neural networks, i.e., deep fully-connected neural networks with depth 2 and width 100. Specifically, the regularization and penalty parameters vary from {10 −6 , 10 −5 , 10 −4 , 10 −3 , 10 −2 , 10 −1 } and {0.01, 0.1, 0.5, 1, 5, 10, 100}, respectively. The approximation errors of ADMM with these parameters are shown in Figure 7(a). From Figure 7(a), considering the penalty parameter β, ADMM with β = 1 achieves the best performance in most cases, as also observed in the experiments later. Thus, in practice, we can empirically set the penalty parameter β as 1. Since λ is a model parameter, it usually has significant effect on the performance of the proposed ADMM. From Figure 7(a), a small regularization parameter (say, λ = 10 −6 ) is sufficient to yield an ADMM solver with high approximation precision. Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Approx. Error 5.34e-8(2.34e-8) 3.95e-8(1.25e-8) 3.33e-8(1.46e-8) 2.46e-4(1.69e-4) 2.53e-9(1.18e-9) Run C. Effect of initial schemes. Besides the MSRA initialization (He et al., 2015), there are some other commonly used initial schemes such as the random orthogonal initializations (Saxe et al., 2014), LeCun random initializations (LeCun et al., 1998), and Xavier initialization (Glorot and Bengio, 2010). Under the optimal parameter settings presented in Table 3, the performance of the ADMM-sigmoid pair with different initialization schemes is presented in Figure 7(b). From Figure 7(b), the proposed ADMM performs well for all the initialization schemes. This demonstrates that the proposed ADMM is stable to the initial scheme. Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Approx. Error 1.22e-6(3.68e-7) 3.37e-7(1.29e-7) 1.13e-6(4.37e-7) 1.13e-3(2.33e-4) 2.62e-9(1.05e-9) Run Time ( Approximation of product gate In this subsection, we present experimental results in approximating the product gate function, i.e., f (u, v) = uv for u, v ∈ [−1, 1]. The specific experimental settings in approximating the product gate function can be found in Table 2. A. Approximation performance of ADMM. The performance of ADMM and competitors is presented in Table 4, and their performance with respect to the depth is depicted in Figure 8. From Table 4, the product gate function can be well approximated by the ADMM-sigmoid pair with precision in the order of 10 −9 , which is better than those of competitors including SGD (ReLU), SGDM (ReLU) and Adam (ReLU), even when more hidden layers are involved in the training. It follows from Figure 8(b) and Table 4 that the optimal depth for ADMM in approximating the product gate function is 2, which matches the theoretical depth for the approximation of product gate as shown in (Chui et al., 2019, Proposition 3). Similar to the case of approximating square function, the running time of the proposed ADMM-sigmoid pair is less than the SGD type competitors for deep ReLU nets with more hidden layers. B. Effect of parameters of ADMM. Similar to Section 5.2 B, we also consider the effect of parameters λ and β for ADMM under the optimal network structures, which is presented in Figure 9(a). From Figure 9(a), the effect of the concerned parameters on the performance of ADMM in approximating the product gate function is very similar to that in the approximation of univariate square function. It can be observed that the settings of parameters with (λ = 10 −6 , β = 1) are empirically good for ADMM in these two approximation tasks. C. Effect of initial schemes. Moreover, in this experiment, we consider the performance of ADMM (sigmoid) for the aforementioned six different initialization schemes. The experimental results are shown in Figure 9(b). It can be observed in Figure 9(b) that all the initial schemes are generally effective in yielding an ADMM solver with high precision. Among these effective initialization schemes, the LeCun type of initializations perform slightly worse than the others. This, in some extent, also implies that the proposed ADMM is usually stable to initial schemes. Learning L 1 radial function In this subsection, we consider the performance of the ADMM-sigmoid pair for learning a two-dimensional L 1 radial function, i.e., f (x) = ( x 1 − 1) + := max{0, x 1 − 1} for x ∈ [r, (1 + )r] × [r, (1 + )r] for some r > 0 and > 0. Such an L 1 radial function was particularly considered in (Safran and Shamir, 2017). In our experiments, we let = 1/2 and r = 1− 2 in light of the theoretical studies in (Safran and Shamir, 2017). Different from the approximation tasks in Sections 5.2 and 5.3, samples generated for the learning task include both training and test samples, where training samples are commonly generated with certain noise and the test samples are clean data. In these experiments, we consider Gaussian noises with different variances. A. Learning performance of ADMM. Optimal test errors of different algorithms for learning the L 1 radial function are presented in Table 5, where the variance of Gaussian noise added into the training samples is 0.1. The associated test errors of these algorithms with respect to the depth of neural networks are presented in Figure 10. From Table 5, the considered L 1 radial function can be well learned by both ADMM and SGD type methods. Specifically, the performance of the proposed ADMM is slightly better than SGD type methods. In particular, the optimal depth of deep sigmoid nets trained by ADMM is only 4, which is much less than those of deep ReLU nets trained by SGD type methods. Under optimal network structures, the running time of the suggested ADMM-pair is slightly less than that of SGD type methods for deep ReLU nets, due to the less depth of deep sigmoid nets. According to Figure 10(b), the proposed ADMM performs better than SGD for training deep sigmoid nets, and as the depth increasing, the performance of SGD gets worse possibly due to the vanishing gradient issue, while our suggested ADMM can alleviate the issue of vanishing gradient and thus achieve better and better performance in general as the depth increases in our considered range of depth, i.e., {1, 2, 3, 4, 5}. B. Effect of parameters and initialization. Under the optimal neural network structures specified in Table 5, we consider the effect of parameters, i.e., (λ, β) for ADMM, as well as the effect of the initialization schemes for both ADMM and SGD type methods. The numerical results are shown in Figure 11. From Figure 11(a), we can observe that the specific parametric setting, i.e., λ = 10 −6 and β = 1, is also empirically effective in learning L 1 radial function. By Figure 11(b), the proposed ADMM performs well for all the concerned random initialization schemes. C. Robustness to the noise. Moreover, we consider the performance of the proposed ADMM for training data with different levels of noise. Specifically, under the optimal parameters specified in Table 5, we consider several levels of noise, where the variance of Gaussian noise varies from {0.1, 0.3, . . . , 1.5}. Trends of training and test errors are presented in Figure 12(a) and (b) respectively. From Figure 12(a), the proposed ADMM is generally trained well in the sense that the training error almost fits the true noise level. In this case, we can observe from Figure 12 the noise in the sense that the test error increases much slower than the increasing of the variance of Gaussian noise. Learning L 2 radial function In this subsection, we consider to learn certain smooth L 2 radial function that frequently reflects the rotation-invariance feature in deep learning (Chui et al., 2019). Specifically, we adopt a two-dimensional smooth L 2 radial function, i.e., f (x) = g(|x| 2 ), where x ∈ [−1, 1] × [−1, 1], |x| 2 := 2 i=1 x 2 i , and g(t) = (1 − t) 5 + (8t 2 + 5t + 1) on R is some Wendland function . Except the target function f , the experimental settings in these experiments are similar to those in Section 5.4. Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Test Error 1.68e-5(6.43e-6) 1.21e-5(5.25e-6) 1.02e-5(4.88e-6) 9.33e-5(1.42e-5) 9.28e-6(1.01e-6) Run Time ( A. Learning performance of ADMM. The test error of the considered algorithms in learning such a smooth L 2 radial function is presented in Table 6, while trends of test errors with respect to the depth are shown in Figure 13. By Table 6, the considered smooth L 2 radial function can be learned by the proposed ADMM well with a small test error. Specifically, in terms of test error, the performance of the ADMM-sigmoid pair is slightly better than that of SGD type methods for deep ReLU nets, and the optimal depth of deep sigmoid nets required by ADMM is much smaller than those of deep ReLU nets required by the concerned SGD type methods. Due to less depth, the running time of ADMM is less than that of the concerned SGD type methods for deep ReLU nets under the optimal settings of neural networks. Moreover, from Figure 13(a), a deeper ReLU network with about 10 layers is generally required to learn the L 2 radial function with a good test error, while from Figure 13(b), the depth of deep sigmoid nets trained by ADMM can be much smaller (i.e., about 5) to yield a good test error. B. Effect of parameters and initialization. In this part, we consider the effect of parameters (i.e., λ and β) of ADMM as well as the effect of initialization for learning the L 2 radial function in the optimal settings specified in Table 6. The numerical results are presented in Figure 14. From Figure 14(a), the effect of parameters are similar to previous three simulations and it can be observed that the specific settings, i.e., λ = 10 −6 and β = 1, are empirically effective. From Figure 14, we also observe that ADMM is effective to all the random initialization schemes. C. Robustness to noise. Similar to the learning of L 1 radial function, we consider the performance of the proposed ADMM for noisy training data with different levels of noise. Specifically, the variance of the Gaussian noise added into the training samples varies from {0.1, 0.3, 0.5, 0.7, 0.9, 1.1}. Curves of training error and test error are shown respectively in Figure 15(a) and (b). From Figure 15, the behavior in learning L 2 radial function is similar to that in learning L 1 radial function as shown in Figure 12. This demonstrates that the proposed ADMM is also robust to noise in learning such a smooth L 2 radial function. Real Data Experiments In this section, we provide three real-data experiments over the earthquake intensity database, the extended Yale B (EYB) face recognition database and the PTB Diagnostic ECG database, to demonstrate the effectiveness of the proposed ADMM. We choose these three datasets since they can in some sense reflect certain features that can be well approximated by deep sigmoid nets, and thus, the benefits of the proposed ADMM can be embodied over these datasets. Specific experimental settings are presented in Table 7, where the penalty parameter β is empirically set as 1 and the regularization parameter λ is chosen via cross validation from the set {10 −6 , 10 −5 , 10 −4 , 10 −3 , 10 −2 } according to the previous studies of toy simulations. Earthquake intensity dataset Earthquake Intensity Database is from: https://www.ngdc.noaa.gov/hazard/intintro.shtml. This database contains more than 157,000 reports on over 20,000 earthquakes that affected the United States from the year 1638 to 1985. For each record, the features include the geographic latitudes and longitudes of the epicentre and "reporting city" (or, locality) where the Modified Mercalli Intensity (MMI) was observed, magnitudes (as a measure of seismic energy), and the hypocentral depth (positive downward) in kilometers from the surface. The output label of each record is measured by MMI, varying from 1 to 12 in integer. An illustration of the generation procedure of each earthquake record is shown in Figure 16(a). In this paper, we transfer such multi-classification task into the binary classification since this database is very unbalanced (say, there is only one sample for the class with MMI being 1). Specifically, we set the labels lying in 1 to 4 as the positive class, while the other labels lying in 5 to 12 as the negative class, mainly according to the damage extent of the earthquake suggested by the referred website. Moreover, we removed those incomplete records with missing labels. After such preprocessing, there are total 8173 effective records, where the numbers of samples in positive and negative classes are respectively 5011 and 3162. We divide the total data set into the training and test sets randomly, where the training and test sample sizes are 4173 and 4000, respectively. Before training, we use the z-scoring normalization for each feature, that is, x i −µ σ with µ and σ being respectively the mean and standard deviation of the ith feature {x i }. The classification accuracies of all algorithms are shown in Table 8. The effect of the depth of neural network, algorithmic parameters, and random initial schemes are shown in Figure 16 (b)-(d) respectively. According to Table 8, the performance of the proposed ADMM is comparable to the state-of-the-art methods in terms of test accuracy. Specifically, the proposed ADMM is slightly worse than Adam, and outperforms the other competitors in terms of test accuracy, while in terms of running time, the proposed ADMM is slightly faster than Adam and SGDM under the associated optimal network settings, mainly because the optimal depth of the deep sigmoid nets trained by ADMM is less than those of deep ReLU nets trained by Adam and SGDM. Compared to the SGD counterpart for deep sigmoid nets, the performance of the proposed ADMM is much better in terms of test accuracy. It can be observed from Figure 16(b) that the vanilla SGD may suffer from the gradient vanishing/explosion issue when training a slightly deeper sigmoid nets (say, the depth is larger than 5) due to the saturation of the sigmoid activation, while the proposed ADMM can avoid such saturation and thus alleviate the gradient vanishing/explosion issue. From Figure 16(c), the proposed ADMM with the default settings, i.e., λ = 1e-6 and β = 1 in general yields the best performance. Moreover, it can be observed from Figure 16(d) that the proposed ADMM is stable to the commonly used initialization schemes under the optimal neural network structure specified in Table 8. Extended Yale B face recognition database In the extended Yale B (EYB) database, well-known face recognition database (Lee et al., 2005), there are in total 2432 images for 38 objects under 9 poses and 64 illumination conditions, where for each objective, there are 64 images. The pixel size of each image is 32 × 32. In our experiments, we randomly divide these 64 images for each objective into two equal parts, that is, one half of images are used for training while the rest half of images are used for testing. For each image, we normalize it via the z-scoring normalization. The specific experimental settings for this database can be found in Table 7. Particularly, we empirically use a shallow neural network with depth one and various of widths, since such shallow neural network is good enough to extract the low-dimensional manifold feature of this face recognition data, as shown in Table 9. The effect of network structures and stability of the proposed ADMM to initialization schemes are shown in Figure 17(a) and (b) respectively. According to Table 9, the proposed ADMM achieves the state-of-the-art test accuracy (see, Lu et al. (2020)) with a smaller width of the sigmoid nets when compared to the concerned competitors. From Figure 17, the proposed ADMM can achieve a very high test accuracy for most of the concerned widths of the networks and is stable to the commonly used random initial schemes. Figure 16: Performance of ADMM in earthquake intensity data: (a) an illustration of the earthquake intensity data (Zeng et al., 2020); (b) the effect of depth of the neural network for different algorithms; (c) the effect of algorithmic parameters for the proposed ADMM; (d) the stability of the proposed ADMM to different initial schemes. PTB Diagnostic ECG database An ECG is a 1D signal which is the result of recording the electrical activity of the heart using an electrode. It is one of popular tools that cardiologists use to diagnose heart anomalies and diseases. The PTB diagnostic ECG database is available at https://github.com/ CVxTz/ECG_Heartbeat_Classification and was preprocessed by (Kachuee et al., 2018). There are 14,552 samples in total with 2 categories. The specific experimental settings for this database can be found in Table 7. The experiment results of the proposed ADMM and concerned competitors are presented in Table 10. The effect of network structures and stability of the proposed ADMM to initialization schemes are shown in Figure 18(a) and (b) respectively. According to Table 10, the proposed ADMM achieves the state-of-the-art test accuracy (see, Kachuee et al. (2018)) with a less width of sigmoid nets when compared to the concerned competitors. Specifically, the optimal depth of deep sigmoid nets trained by ADMM is 4, while those of deep ReLU nets trained respectively by SGD, SGDM and Adam are 8, 7, 7. This also verifies our previous claim on the advantage of deep sigmoid nets in feature representation. Due to less hidden layers, the proposed ADMM is slightly faster than the SGD competitors for deep ReLU nets. From Figure 18(a), when the depth of deep sigmoid nets is larger than 8, the performance of all considered algorithms degrades much possibly due to the overfitting. From Figure 18(b), the proposed ADMM is stable to the commonly used random initial schemes under the optimal neural network setting as presented in Table 10. Appendix A. Proof of Theorem 3 To prove Theorem 3, we need the following "product-gate" for shallow sigmoid nets, which can be found in (Chui et al., 2019, Proposition 1). Lemma 6 Let M > 0. For any ν ∈ (0, 1) there exists a shallow sigmoid net h prod 9,ν : R 2 → R with 9 free parameters bounded by O(ν −6 ) such that for any t, t ∈ [−M, M ], |tt − h prod 9,ν (t, t )| ≤ ν. Then, we can give the proof of Theorem 3 as follows. Proof [Proof of Theorem 3] Let σ 0 (t) be the heaviside function, i.e., σ 0 (t) = 1, t ≥ 0 0, t < 0. Then, σ relu (t) = tσ 0 (t). A direct computation yields σ(0) = 1/2 and σ (0) = 1/4. For 0 < µ < 1/2, according to the Taylor formula σ(µt) = 1 2 + µt 4 + µt 0 (σ (u) − σ (0))du, we have t = 4 µ σ(µt) − 2 µ − 4 µ µt 0 (σ (u) − σ (0))du. Therefore, t − 4 µ σ(µt) − 4 µ σ(0 · t) ≤ 4 µ µt 0 |σ (u) − σ (0)|du ≤ 4 µ max v≥0 |σ (v)| µt 0 udu ≤ 2µt 2 . Denote h linear 2,µ = 4 µ σ(µt) − 4 µ σ(0 · t). Then for |t| ≤ M , there holds |t − h linear 2,µ (t)| ≤ 2M 2 0 µ,(30) where M 0 > 0 satisfying M 0 + M 2 0 = M . This shows that h linear 2,µ is a good approximation of t. On the other hand, for , τ > 0 and A = 1 τ log 1 , we have σ(At) = 1 1 + e −At ≤ 1 1 + e Aτ ≤ , t ≤ −τ and |σ(At) − 1| ≤ e −Aτ 1 + e −Aτ ≤ e −Aτ ≤ , t ≥ τ, showing |σ(At) − σ 0 (t)| ≤ , t ∈ [−M 0 , −τ ] ∪ [τ, M 0 ].(31) Since |σ(At)| ≤ 1 and |h linear 2,µ (t)| ≤ |t| + 2M 2 0 µ ≤ M 0 + M 2 0 for |t| ≤ M 0 , we then utilize the "product-gate" exhibited in Lemma 6 with M = M 0 + M 2 0 to construct a deep sigmoid net with two hidden layers and at most 27 free parameters to approximate σ relu (t). Define t ∈ [−M 0 , −τ ] ∪ [τ, M 0 ] |tσ 0 (t) − h relu 9,2,µ,ν,A (t)| ≤ |tσ 0 (t) − tσ(At)| + |tσ(At) − h linear 2,µ (t)σ(At)| + |h linear 2,µ (t)σ(At) − h relu 9,2,µ,ν,A (t)| ≤ M 0 + 2M 2 0 µ + ν and |h relu 9,2,µ,ν,A (t)| ≤ Cν −6 , ∀t ∈ [−M, M ]. Let = µ = ν = ε. We have for any 0 < ε < 1/2, |σ relu (t) − h relu 9,2,µ,ν,A (t)| ≤ (M 0 + 1 + 2M 2 0 )ε, t ∈ [−M 0 , −τ ] ∪ [τ, M 0 ](32) and the free parameters of h relu 9,2,µ,ν,A (t) are bounded by max{O( 1 ε 6 ), 1 τ log 1 ε }. Then, setting τ = ε 7 , we have M −M |σ relu (t) − h relu 9,2,µ,ν,A (t)| p dt = −τ M + τ −τ + M τ |σ relu (t) − h relu 9,2,µ,ν,A (t)| p ≤ 2M ε + 2Cτ ε −6 ≤ 2(M + C)ε. This completes the proof of Theorem 3 by a simple scaling. Appendix B. Generic convergence of ADMM without normalization In this appendix, we consider more general settings than that in Section 3.4, where X and Y are not necessarily normalized with unit norms, and the numbers of neurons of hidden layers can be different, and the activation function σ can be any twice differentiable activation satisfying the following assumptions. Assumption 1 Let σ : R → R be a twice-differentiable bounded function with bounded first-and second-order derivatives, namely, there exist positive constants L 0 (≥ 1 8 ), L 1 , L 2 such that: |σ(u)| ≤ L 0 , |σ (u)| ≤ L 1 and |σ (u)| ≤ L 2 for any u ∈ R. Moreover, σ is either a real analytic (Krantz and Parks, 2002, Definition 1.1.5) or semialgebraic function (Bochnak et al., 1998). Besides the sigmoid activation, some typical activations satisfying Assumption 1 include the sigmoid-type activations such as the hyperbolic tangent activation. For the abuse use of notation, in this appendix, we still use σ as any activation satisfying Assumption 1. Before presenting our main theorem under these generic settings, we define the following constants: L 3 := 2(L 2 1 + L 2 L 0 + L 2 ),(33)γ := max 1≤i≤N W 0 i F ,(34)d min := min 1≤i≤N −1 d i ,(35)f min := √ 6 3L 1 + 2(L 0 L 3 ) 1/2 (nd min ) 1/4 ,(36)α 3 := f min L 1 2 ,(37)C 3 := max max 0≤j≤N −2 2L 0 nd j+1 γ j , Y F (β N − 3)γ N −1 ,(38)λ i := 3L 1 C 3 β i γ i−3 (4C 3 γ i−1 + L 0 nd i ) 1 + 6L 3 C 2 3 γ 2i−2 L 1 (4C 3 γ i−1 + L 0 √ nd i ) , 2 ≤ i ≤ N − 1, λ := max 2≤i≤N −1 λ i , 1 6 (1 + 3L −1 1 L 2 L 3 γ i−1 ) 2 C 2 3 γ 2(i−2) β i , λ := L 1 β 1 X F (4C 3 + L 0 nd 1 )γ −1 1 + 2L 3 C 3 X F γ L 1 (4C 3 + L 0 √ nd 1 ) . With these defined constants, we impose some conditions on the the penalty parameters {β i } N i=1 in the augmented Lagrangian, the regularization parameter λ, the minimal number of hidden neurons d min , and the initializations of {V 0 i } N i=1 and {Λ 0 i } N i=1 as follows β N ≥ 3.5, (39) β N −1 β N ≥ 16γ 2 ,(40)β i β i+1 ≥ max 6 √ N (2L 2 1 + (4L 3 + L 2 )C 3 γ i )γ 2 , 6( 3L 1 + 2L 3 C 3 γ i ) 2 γ 2 , i = 1, . . . , N − 2,(41)λ ≥ max 12β N C 2 3 γ 2N −4 ,λ,λ ,(42)d min ≥ max √ 24N + 1L 1 − √ 18L 1 , 0 4 n(24L 0 L 3 ) 2 ,(43)V 0 i F ≤ 3C 3 γ i−1 , Λ 0 i F ≤ C 3 β i γ i−1 , i = 1, . . . , N.(44) Under these assumptions, we state the main convergence theorem of ADMM as follows. Theorem 7 Let Assumption 1 hold. Let (18). Assume that (39)-(44) hold, then the following hold: {Q k := ({W k i } N i=1 , {V k i } N i=1 , {Λ k i } N i=1 )} be a se- quence generated by Algorithm 1 with h k i = L( V k−1 i − β −1 i Λ k−1 i max ) for i = 1, . . . , N − 1. and µ k j = L( V k−1 j+1 − β −1 j+1 Λ k−1 j+1 max ) for j = 1, . . . , N − 2, where L(·) is defined in(a) {L(Q k )} is convergent. (b) {Q k } converges to a stationary point Q * := ({W * i } N i=1 , {V * i } N i=1 , {Λ * i } N i=1 ) of L, which is also a KKT point (24) of problem (5), implying {W * i } N i=1 is a stationary point of problem (4) with λ = 2λ/n. (c) 1 K K k=1 ∇L(Q k ) 2 F → 0 at a O( 1 K ) rate. Theorem 4 presented in the context is a special case of Theorem 7 with γ = 1, X F = Y F = 1, W 0 i F = 1, i = 1, . . . , N , and the initialization strategy (7). Actually, the initialization strategy (7) satisfies (44) shown as follows: V 0 j F ≤ L 0 nd j ≤ 1 2 C 3 γ j−1 , j = 1, . . . , N − 1,(45)V 0 N F ≤ γ · 1 2 C 3 γ N −2 = 1 2 C 3 γ N −1 ,(46)Λ 0 i F = 0, i = 1, . . . , N, where the first inequality in (45) holds by the boundedness of activation, and the second inequality in (45) holds by the definition (38) of C 3 , and the inequality in (46) holds for W 0 N F ≤ γ and (45) with j = N − 1. By the definitions (28) and (29) ofQ k andL , if we can show that Theorem 5 holds under the assumptions of Theorem 7, then we directly yield Theorem 7. Thus, we only need to prove Theorem 5 under the assumptions of Theorem 7. Appendix C. Preliminaries Before presenting the proof of Theorem 5 under the assumptions of Theorem 7, we provide some preliminary definitions and lemmas which serve as the basis of our proof. C.1 Dual expressed by primal According to the specific updates of Algorithm 1, we show that the updates of dual variables {Λ k i } N i=1 can be expressed explicitly by the updates of primal variables {W k i } N i=1 and {V k i } N i=1 as in the following lemma. Lemma 8 (Dual expressed by primal) Suppose that Assumption 1 holds. Let {Q k := {W k i } N i=1 , {V k i } N i=1 , {Λ k i } N i=1 } be a sequence generated by Algorithm 1. Then we have Λ k N = V k N − Y, ∀k ∈ N,(47)Λ k N −1 = (W k N ) T Λ k N + β N (W k N ) T (V k N − V k−1 N ),(48)Λ k j = (W k j+1 ) T Λ k j+1 σ (W k j+1 V k−1 j ) + β j+1 (W k j+1 ) T (σ(W k j+1 V k−1 j ) − σ(W k j+1 V k j )) +(V k j+1 − V k−1 j+1 ) σ (W k j+1 V k−1 j ) + µ k j W k j+1 (V k j − V k−1 j )/2 , j = N − 2, . . . , 1.(49) Proof We firstly derive the explicit updates of {W k i } N i=1 and {V k j } N j=1 , then based on these updates, we prove Lemma 8. 1) W i -subproblems: According to the update (8), W k N is updated via W k N = (β N V k−1 N − Λ k−1 N )(V k−1 N −1 ) T λI + β N V k−1 N −1 V k−1 N −1 T −1 .(50) By (19), for i = 1, . . . , N − 1, we get W k i = W k−1 i β i h k i 2 V k−1 i−1 (V k−1 i−1 ) T λI + β i h k i 2 V k−1 i−1 (V k−1 i−1 ) T −1 − Λ k−1 i + β i (σ(W k−1 i V k−1 i−1 ) − V k−1 i ) σ (W k−1 i V k−1 i−1 ) (V k−1 i−1 ) T λI + β i h k i 2 V k−1 i−1 (V k−1 i−1 ) T −1 = W k−1 i − W k−1 i I + β i h k i 2λ V k−1 i−1 (V k−1 i−1 ) T −1 (51) − Λ k−1 i + β i (σ(W k−1 i V k−1 i−1 ) − V k−1 i ) σ (W k−1 i V k−1 i−1 ) (V k−1 i−1 ) T λI + β i h k i 2 V k−1 i−1 V k−1 i−1 T −1 . Particularly, when i = 1, W k 1 is updated by W k 1 = W k−1 1 β 1 h k 1 2 V 0 V 0 T λI + β 1 h k 1 2 V 0 V 0 T −1 − Λ k−1 1 + β 1 (σ(W k−1 1 V 0 ) − V k−1 1 ) σ (W k−1 1 V 0 ) V 0 T λI + β 1 h k 1 2 V 0 V 0 T −1 = W k−1 1 − W k−1 1 I + β 1 h k 1 2λ V 0 V 0 T −1 (52) − Λ k−1 1 + β 1 (σ(W k−1 1 V 0 ) − V k−1 1 ) σ (W k−1 1 V 0 ) V 0 T λI + β 1 h k 1 2 V 0 V 0 T −1 . 2) V j -subproblems: According to (12), it holds V k N − Y − Λ k−1 N + β N W k N V k N −1 − V k N = 0.(53) By the relation Λ k N = Λ k−1 N + β N W k N V k N −1 − V k N , (53) implies Λ k N = V k N − Y, ∀k ∈ N,(54) which shows (47) in Lemma 8. Substituting the equality (54) with the index value k − 1 into (53) yields V k N = 1 1 + β N V k−1 N + β N 1 + β N W k N V k N −1 .(55) According to (11), it holds − Λ k−1 N −1 + β N −1 (σ(W k N −1 V k N −2 ) − V k N −1 ) + (W k N ) T Λ k−1 N + β N W k N V k N −1 − V k−1 N = 0, which implies V k N −1 = (56) (β N −1 I + β N (W k N ) T W k N ) −1 Λ k−1 N −1 + β N −1 σ(W k N −1 V k N −2 ) − (W k N ) T Λ k−1 N − β N V k−1 N , and together with the updates of Λ k N −1 and Λ k N in Algorithm 1 yields Λ k N −1 = (W k N ) T Λ k−1 N + β N W k N V k N −1 − V k−1 N = (W k N ) T Λ k N + β N (V k N − V k−1 N ) . This implies (48) in Lemma 8. By (20), for j = 1, . . . , N − 2, V k j satisfies the following optimality condition − Λ k−1 j + β j (σ(W k j V k j−1 ) − V k j ) + β j+1 µ k j 2 W k j+1 T W k j+1 (V k j − V k−1 j ) + W k j+1 T Λ k−1 j+1 + β j+1 (σ(W k j+1 V k−1 j ) − V k−1 j+1 ) σ (W k j+1 V k−1 j ) = 0, which implies V k j = β j I + β j+1 µ k j 2 W k j+1 T W k j+1 −1 1 2 β j+1 µ k j W k j+1 T W k j+1 V k−1 j + Λ k−1 j + β j σ(W k j V k j−1 ) +W k j+1 T [Λ k−1 j+1 + β j+1 (σ(W k j+1 V k−1 j ) − V k−1 j+1 )] σ (W k j+1 V k−1 j ) = V k−1 j − I + β j+1 µ k j 2β j W k j+1 T W k j+1 −1 V k−1 j + β j I + β j+1 µ k j 2 W k j+1 T W k j+1 −1 Λ k−1 j + β j σ(W k j V k j−1 ) +W k j+1 T Λ k−1 j+1 + β j+1 (σ(W k j+1 V k−1 j ) − V k−1 j+1 ) σ (W k j+1 V k−1 j ) .(57) and together with the updates of Λ k j and Λ k j+1 in Algorithm 1 yields Λ k j = (W k j+1 ) T Λ k−1 j+1 + β j+1 σ(W k j+1 V k−1 j ) − V k−1 j+1 σ (W k j+1 V k−1 j ) + β j+1 µ k j 2 (W k j+1 ) T W k j+1 (V k j − V k−1 j ), = (W k j+1 ) T Λ k j+1 σ (W k j+1 V k−1 j ) + β j+1 (W k j+1 ) T (σ(W k j+1 V k−1 j ) − σ(W k j+1 V k j )) +(V k j+1 − V k−1 j+1 ) σ (W k j+1 V k−1 j ) + µ k j W k j+1 (V k j − V k−1 j )/2 . The final equality implies (49) in Lemma 8. This completes the proof of this lemma. C.2 Kurdyka-Lojasiewicz property The Kurdyka-Lojasiewicz (KL) property ( Lojasiewicz, 1993;Kurdyka, 1998) plays a crucial role in the convergence analysis of nonconvex algorithm (see, Attouch et al. (2013)). The following definition is adopted from (Bolte et al., 2007). Definition 9 (KL property) An extended real valued function h : X → R∪{+∞} is said to have the Kurdyka-Lojasiewicz property at x * ∈ dom(∂h) if there exist a neighborhood U of x * , a constant η > 0, and a continuous concave function φ(s) = cs 1−θ for some c > 0 and θ ∈ [0, 1) such that the Kurdyka-Lojasiewicz inequality holds φ (h(x) − h(x * ))dist(0, ∂h(x)) ≥ 1, ∀x ∈ U ∩ dom(∂h) and h(x * ) < h(x) < h(x * ) + η,(58) where ∂h(x) denotes the limiting-subdifferential of h at x ∈ dom(h) (introduced in Mordukhovich (2006) Note that we have adopted in the definition of KL inequality (58) the following notational conventions: 0 0 = 1, ∞/∞ = 0/0 = 0. This property was firstly introduced by ( Lojasiewicz, 1993) on real analytic functions (Krantz and Parks, 2002) for θ ∈ [ 1 2 , 1), then was extended to functions defined on the o-minimal structure in (Kurdyka, 1998), and later was extended to nonsmooth subanalytic functions in (Bolte et al., 2007). In the following, we give the definitions of real-analytic and semialgebraic functions. Definition 10 (Real analytic, Definition 1.1.5 in (Krantz and Parks, 2002)) A function h with domain being an open set U ⊂ R and range either the real or the complex numbers, is said to be real analytic at u if the function f may be represented by a convergent power series on some interval of positive radius centered at u: h(x) = ∞ j=0 α j (x − u) j , for some {α j } ⊂ R. The function is said to be real analytic on V ⊂ U if it is real analytic at each u ∈ V. The real analytic function f over R p for some positive integer p > 1 can be defined similarly. According to (Krantz and Parks, 2002), some typical real analytic functions include polynomials, exponential functions, and the logarithm, trigonometric and power functions on any open set of their domains. One can verify whether a multivariable real function h(x) on R p is analytic by checking the analyticity of g(t) := h(x + ty) for any x, y ∈ R p . The following lemma shows some important properties of real analytic functions. Lemma 11 (Krantz and Parks, 2002) The sums, products, and compositions of real analytic functions are real analytic functions. Let h : R p → R ∪ {+∞} be an extended-real-valued function (respectively, h : R p ⇒ R q be a point-to-set mapping), its graph is defined by Graph(h) := {(x, y) ∈ R p × R : y = h(x)}, (resp. Graph(h) := {(x, y) ∈ R p × R q : y ∈ h(x)}), and its domain by dom(h) := {x ∈ R p : h(x) < +∞} (resp. dom(h) := {x ∈ R p : h(x) = ∅}). Definition 12 (Semialgebraic) (a) A set D ⊂ R p is called semialgebraic (Bochnak et al., 1998) if it can be represented as D = s i=1 t j=1 {x ∈ R p : P ij (x) = 0, Q ij (x) > 0}, where P ij , Q ij are real polynomial functions for 1 ≤ i ≤ s, 1 ≤ j ≤ t. (b) A function h : R p → R ∪ {+∞} (resp. a point-to-set mapping h : R p ⇒ R q ) is called semialgebraic if its graph Graph(h) is semialgebraic. According to ( Lojasiewicz, 1965;Bochnak et al., 1998) and (Shiota, 1997, I.2.9, p.52), the class of semialgebraic sets is stable under the operation of finite union, finite intersection, Cartesian product or complementation. Some typical examples include polynomial functions, the indicator function of a semialgebraic set, and the Euclidean norm (Bochnak et al., 1998, p.26). Lemma 13 (KL properties of L andL) Suppose that Assumption 1 holds, then both L andL are KL functions. Proof Let Q := {W i } N i=1 , {V i } N i=1 , {Λ i } N i=1 ,Q := Q, {V i } N i=1 and L 1 (Q) := 1 2 V N − Y 2 F + λ 2 N i=1 W i 2 F + N i=1 β i 2 σ i (W i V i−1 ) − V i 2 F , L 2 (Q) := N i=1 Λ i , σ i (W i V i−1 ) − V i . Then L(Q) = L 1 (Q)+L 2 (Q),L(Q) = L(Q)+ N i=1 ξ i V i −V i 2 F , where ξ i > 0, i = 1, . . . , N . According to the same arguments as in the proof of (Zeng et al., 2019, Proposition 2), L 1 is real analytic (resp. semialgebraic) if σ i is real analytic (resp. semialgebraic). By the closedness of real analytic (resp. semialgebraic) functions under the sum, product and composition (see, Krantz and Parks (2002); Bochnak et al. (1998)), we can show that L 2 is also real analytic (resp. semialgebraic) if σ i is real analytic (resp. semialgebraic). Thus, L is a finite sum of real analytic or semialgebraic functions. According to Shiota (1997), L is a subanalytic function. By Assumption 1, L is continuous. Thus, L is a KL function by (Bolte et al., 2007, Theorem 3.1) . Since N i=1 ξ i V i −V i 2 F is polynomial,L is also a KL function by a similar argument. This completes the proof. Appendix D. Proofs for Theorem 5 As stated in Section 4.4, the main idea of proof of Theorem 5 is shown as follows: we firstly establish the desired sufficient descent lemma (see, Lemma 14) via estimating the progress made by one step update, and bounding dual by primal as well as showing the boundedness of the sequence, then develop the desired relative error lemma (see, Lemma 21) via the optimality conditions of all subproblems, the Lipschitz continuity of the activation as well as the boundedness of the sequence, and finally prove this theorem via (Attouch et al., 2013, Theorem 2.9), together with Lemma 13 and the continuous assumption of the activation. In the following, we establish these lemmas followed by the detailed proof of Theorem 5. D.1 Proof for Lemma 14: Sufficient descent lemma In order to prove Theorem 5, the following sufficient descent lemma plays a key role. Lemma 14 (Sufficient descent) Under assumptions of Theorem 7, for k ≥ 2, there holdŝ L(Q k ) ≤L(Q k−1 ) − a N i=1 W k i − W k−1 i 2 F + N i=1 V k i − V k−1 i 2 F ,(59) where a is some positive constant specified later in (98) in Appendix D.1.4. From Lemma 14, we establish the sufficient descent property of an auxiliary sequence {Q k } instead of the sequence {Q k } itself, along a new Laypunov functionL but not the original augmented Lagrangian L. This is different from the convergence analysis of ADMM in for linear constrained optimization problems, where the sufficient descent lemma is shown for the original sequence along the augmented Lagrangian (see, (Wang et al., 2019, Lemma 5)). In order to establish Lemma 14, the following three lemmas are required, where the first lemma shows the progress made by one step update (called, one-step progress lemma), the second lemma bounds the discrepancies of two successive dual updates via those of the primal updates (called, dual-bounded-by-primal lemma), and the third lemma shows the boundedness of the sequence (called, boundedness lemma). D.1.1 Lemma 15: One-step progress lemma We present the first lemma that estimates the progress made by a single update of ADMM. Lemma 15 (One-step progress) Let Assumption 1 hold. (21) and (22), respectively. Then for any integer k ≥ 1, the following holds Let Q k := {W k i } N i=1 , {V k i } N i=1 , {Λ k i } N i=1 be a sequence generated by Algorithm 1 with {h k i } N −1 i=1 and {µ k j } N −2 j=1 specified inL(Q k ) ≤ L(Q k−1 ) − N i=1 λ 2 W k i − W k−1 i 2 F + β i h k i 4 (W k i − W k−1 i )V k−1 i−1 2 F (60) − N −1 j=1 β j 2 V k j − V k−1 j 2 F + β j+1 µ k j 4 W k j+1 (V k j − V k−1 j ) 2 F − 1 + β N 2 V k N − V k−1 N 2 F + N i=1 β −1 i Λ k i − Λ k−1 i 2 F , where V k 0 ≡ X, h k N = 1 and µ k N −1 = 1. From Lemma 15, there are two key parts that contribute to the progress along the augmented Lagrangian sequence, namely, the descent part arisen by the primal updates and the ascent part brought by the dual updates. Due to the existence of the dual ascent part, the convergence of nonconvex ADMM is usually very challengeable. By (60), in order to further estimate the progress in terms of the primal updates, we shall bound these dual ascent parts via the primal updates as shown in Lemma 18 below. To prove Lemma 15, we firstly establish two preliminary lemmas. Lemma 16 Given a constant c ∈ R, let f c be the function on R given by f c (u) = (σ(u)−c) 2 . Then the following holds f c (v) ≤ f c (u) + f c (u)(v − u) + L(|c|) 2 (v − u) 2 , ∀u, v ∈ R where L(|c|) is defined in (18). Proof According to Assumption 1, by some simple derivations, we can show |f c (u)| ≤ L(|c|), ∀u ∈ R. This yields the inequality f c (v) ≤ f c (u) + f c (u)(v − u) + L(|c|) 2 (v − u) 2 , ∀u, v ∈ R. Note that the W k i (i = 1, . . . , N − 1) and V k j (j = 1, . . . , N − 2) updates involve the following update schemes, i.e., W k = arg min W λ 2 W 2 F + βH k σ (W ; A, B) ,(61)V k = arg min V λ 2 V − C 2 F + βM k σ (V ; A, B)(62) for some matrices A, B and C, positive constants λ and β. Based on Lemma 16, we provide a lemma to estimate the descent quantities of the above two updates. Lemma 17 Suppose that Assumption 1 holds. Let W k and V k be updated according to (61) and (62), respectively, then λ 2 W k 2 F + βH σ (W k ; A, B) ≤ λ 2 W k−1 2 F + βH σ (W k−1 ; A, B) (63) − λ 2 W k − W k−1 2 F − βh k 4 (W k − W k−1 )A 2 F , λ 2 V k − C 2 F + βM σ (V k ; A, B) ≤ λ 2 V k−1 − C 2 F + βM σ (V k−1 ; A, B) (64) − λ 2 V k − V k−1 2 F − βµ k 4 A(V k − V k−1 ) 2 F , where h k := L( B max ) and µ k := L( B max ). Proof We first establish the descent inequality (63) h(W k−1 ) = h(W k ) + λ 2 W k − W k−1 2 F + βh k 4 (W k − W k−1 )A 2 F , which implies λ 2 W k−1 2 F + βH σ (W k−1 ; A, B) = λ 2 W k 2 F + β H σ (W k−1 ; A, B) + ∇H σ (W k−1 ; A, B), W k − W k−1 + h k 4 (W k − W k−1 )A 2 F + λ 2 W k − W k−1 2 F + βh k 4 (W k − W k−1 )A 2 F ≥ λ 2 W k 2 F + βH σ (W k ; A, B) + λ 2 W k − W k−1 2 F + βh k 4 (W k − W k−1 )A 2 F , where the final inequality holds by the definition (14) L(W k−1 <N , W k N , {V k−1 j } N j=1 , {Λ k−1 j } N j=1 ) ≤ L(W k−1 <N , W k−1 N , {V k−1 j } N j=1 , {Λ k−1 j } N j=1 ) − λ 2 W k N − W k−1 N 2 F − β N 2 (W k N − W k−1 N )V k−1 N −1 2 F .(65) By (19), W k i (i = 1, . . . , N − 1) is updated according to (61) with λ = λ, β = β i , A = V k−1 i−1 and B = V k−1 i − β −1 i Λ k−1 i . Then by Lemma 17, it holds L(W k−1 <i , W k i , W k >i , {V k−1 j } N j=1 , {Λ k−1 j } N j=1 ) ≤ L(W k−1 <i , W k−1 i , W k >i , {V k−1 j } N j=1 , {Λ k−1 j } N j=1 ) − λ 2 W k i − W k−1 i 2 F − β i h k i 4 (W k i − W k−1 i )V k−1 i−1 2 F .(66) Similarly, for the V k j -update (j = 1, . . . , N − 2), by (20) and Lemma 17, the following holds L({W k i } N i=1 , V k <j , V k j , V k−1 >j , {Λ k−1 i } N i=1 ) ≤ L({W k i } N i=1 , V k <j , V k−1 j , V k−1 >j , {Λ k−1 i } N i=1 ) − β j 2 V k j − V k−1 j 2 F − β j+1 µ k j 4 W k j+1 (V k j − V k−1 j ) 2 F .(67) For the V k N −1 and V k N updates, by (11) and (12), we can easily obtain the following L({W k i } N i=1 , V k <N −1 , V k N −1 , V k−1 N , {Λ k−1 i } N i=1 ) ≤ L({W k i } N i=1 , V k <N −1 , V k−1 N −1 , V k−1 N , {Λ k−1 i } N i=1 ) − β N −1 2 V k N −1 − V k−1 N −1 2 F − β N 2 W k N (V k N −1 − V k−1 N −1 ) 2 F(68) and L({W k i } N i=1 , V k <N , V k N , {Λ k−1 i } N i=1 ) ≤ L({W k i } N i=1 , V k <N , V k−1 N , {Λ k−1 i } N i=1 ) − 1 + β N 2 V k N − V k−1 N 2 F .(69) Particularly, by the updates of Λ k j (j = 1, . . . , N ), we have L({W k i } N i=1 , {V k j } N j=1 , {Λ k i } N i=1 ) = L({W k i } N i=1 , {V k j } N j=1 , {Λ k−1 i } N i=1 ) + N i=1 Λ k i − Λ k−1 i , σ i (W k i V k i−1 ) − V k i = L({W k i } N i=1 , {V k j } N j=1 , {Λ k−1 i } N i=1 ) + N i=1 β −1 i Λ k i − Λ k−1 i 2 F .(70) Summing up (65)-(70) yields (60). D.1.2 Lemma 18: Dual-bounded-by-primal lemma By Lemma 15, how to control the amount of ascent part brought by the dual updates via the amount of descent part characterized by the primal updates is very important. The following lemma shows that the dual ascent quantity { Λ k j − Λ k−1 j 2 F } N j=1 can be bounded by the discrepancies between two successive primal updates { W k i − W k−1 i 2 F } N i=1 , { V k i − V k−1 i 2 F } N i=1 , and { V k−1 i − V k−2 i 2 F } N i=1 via a recursive way. Lemma 18 (Dual-bounded-by-primal) Let Assumption 1 hold. For any positive integer k ≥ 2, the following hold Λ k N − Λ k−1 N F = V k N − V k−1 N F ,(71)Λ k N −1 − Λ k−1 N −1 F ≤ W k N F · Λ k N − Λ k−1 N F + Λ k−1 N F · W k N − W k−1 N F + β N W k N F · V k N − V k−1 N F + β N W k−1 N F · V k−1 N − V k−2 N F ,(72)Λ k j − Λ k−1 j F ≤ L 1 W k j+1 F · Λ k j+1 − Λ k−1 j+1 F (73) + L 1 Λ k−1 j+1 F + L 2 W k−1 j+1 F · Λ k−1 j+1 F · V k−1 j F · W k j+1 − W k−1 j+1 F + L 1 β j+1 W k j+1 F · V k j+1 − V k−1 j+1 F + W k−1 j+1 F · V k−1 j+1 − V k−2 j+1 F + L 2 1 + µ k j 2 β j+1 W k j+1 2 F · V k j − V k−1 j F + (L 2 1 + µ k−1 j /2) · β j+1 + L 2 Λ k−1 j+1 F W k−1 j+1 2 F · V k−1 j − V k−2 j F , where L 1 and L 2 are two constants specified in Assumption 1. From Lemma 18, the amount of the dual ascent part at j-th layer is related to all the later layers (i.e., i = j + 1, . . . , N ) via a recursive way. Besides these terms { W k i − W k−1 i 2 F } N i=1 and { V k i − V k−1 i 2 F } N i=1 exist in the upper bounds, the discrepancies between the previous two updates { V k−1 i − V k−2 i 2 F } N i=1 are also involved in the upper bounds. This may bring some challenge to construct the Lyapunov function such that the sequence or its variant is a descent sequence, because in this case, the augmented Lagrangian shall not be an appropriate Lyapunov function by Lemma 15, where the amount of descent part is only characterized by { W k i − W k−1 i 2 F } N i=1 and { V k i − V k−1 i 2 F } N i=1 without { V k−1 i − V k−2 i 2 F } N i=1 . Proof The equality (71) follows directly from (47). By the update (48) of Λ k N −1 , the following holds Λ k N −1 − Λ k−1 N −1 = (W k N ) T Λ k N − (W k−1 N ) T Λ k−1 N + β N (W k N ) T (V k N − V k−1 N ) − β N (W k−1 N ) T (V k−1 N − V k−2 N ) = (W k N ) T (Λ k N − Λ k−1 N ) + (W k N − W k−1 N ) T Λ k−1 N + β N (W k N ) T (V k N − V k−1 N ) − β N (W k−1 N ) T (V k−1 N − V k−2 N ), which implies (72) directly by the triangle inequality. For j = 1, . . . , N − 2, by the update of (49), Λ k j − Λ k−1 j = (W k j+1 ) T Λ k j+1 σ (W k j+1 V k−1 j ) − (W k−1 j+1 ) T Λ k−1 j+1 σ (W k−1 j+1 V k−2 j ) + β j+1 (W k j+1 ) T (σ(W k j+1 V k−1 j ) − σ(W k j+1 V k j )) + (V k j+1 − V k−1 j+1 ) σ (W k j+1 V k−1 j ) + µ k j 2 W k j+1 (V k j − V k−1 j ) − β j+1 (W k−1 j+1 ) T (σ(W k−1 j+1 V k−2 j ) − σ(W k−1 j+1 V k−1 j )) + (V k−1 j+1 − V k−2 j+1 ) σ (W k−1 j+1 V k−2 j ) + µ k−1 j 2 W k−1 j+1 (V k−1 j − V k−2 j ) . By Assumption 1 and the triangle inequality, the above equality implies that Λ k j − Λ k−1 j F ≤ (W k j+1 ) T Λ k j+1 σ (W k j+1 V k−1 j ) − (W k−1 j+1 ) T Λ k−1 j+1 σ (W k−1 j+1 V k−2 j ) F + β j+1 W k j+1 F (L 2 1 + µ k j /2) W k j+1 F V k j − V k−1 j F + L 1 V k j+1 − V k−1 j+1 F (74) + β j+1 W k−1 j+1 F (L 2 1 + µ k−1 j /2) W k−1 j+1 F V k−1 j − V k−2 j F + L 1 V k−1 j+1 − V k−2 j+1 F . Note that (W k j+1 ) T Λ k j+1 σ (W k j+1 V k−1 j ) − (W k−1 j+1 ) T Λ k−1 j+1 σ (W k−1 j+1 V k−2 j ) F ≤ W k j+1 − W k−1 j+1 F Λ k j+1 σ (W k j+1 V k−1 j ) F + W k−1 j+1 F Λ k j+1 σ (W k j+1 V k−1 j ) − Λ k−1 j+1 σ (W k−1 j+1 V k−2 j ) F ≤ L 1 Λ k j+1 F W k j+1 − W k−1 j+1 F + L 1 W k−1 j+1 F Λ k j+1 − Λ k−1 j+1 F + L 2 W k−1 j+1 F Λ k−1 j+1 F V k−1 j F W k j+1 − W k−1 j+1 F + L 2 W k−1 j+1 2 F Λ k−1 j+1 F V k−1 j − V k−2 j F ,(75) where the final inequality holds for Λ k j+1 σ (W k j+1 V k−1 j ) − Λ k−1 j+1 σ (W k−1 j+1 V k−2 j ) F ≤ (Λ k j+1 − Λ k−1 j+1 ) σ (W k j+1 V k−1 j ) F + Λ k−1 j+1 (σ (W k j+1 V k−1 j ) − σ (W k−1 j+1 V k−2 j )) F ≤ L 1 Λ k j+1 − Λ k−1 j+1 F + L 2 Λ k−1 j+1 F W k j+1 V k−1 j − W k−1 j+1 V k−2 j F ≤ L 1 Λ k j+1 − Λ k−1 j+1 F + L 2 Λ k−1 j+1 F W k j+1 − W k−1 j+1 F V k−1 j F + W k−1 j+1 F V k−1 j − V k−2 j F by Assumption 1 and the triangle inequality. Substituting (75) into (74) yields (73). This completes the proof of this lemma. D.1.3 Lemma 19: Boundedness lemma Note that in the upper bounds of Lemma 18, the terms { W k i − W k−1 i F } N i=1 , { V k i − V k−1 i F } N i=1 and { V k−1 i − V k−2 i 2 F } N i=1 are multiplied by many other terms including { W k i F } N i=1 , { V k i F } N i=1 , { Λ k i F } N i=1 , and the locally Lipschitz constants {h k i := L( V k−1 i − β −1 i Λ k−1 i max )} N −1 i=1 and {µ k j := L( V k−1 j+1 − β −1 j+1 Λ k−1 j+1 max )} N −2 j=1 , highly depending on the current or previous updates. In order to make these bounds in Lemma 18 only depend on those desired terms, the following boundedness property of the sequence is required. Instead of the conditions of Theorem 7, we impose the following weaker conditions: β N ≥ 3.5, (76) β N −1 β N ≥ 7γ 2 ,(77)β i β i+1 ≥ 6 3L 1 + 2L 3 C 3 γ i 2 γ 2 , i = 1, . . . , N − 2,(78)λ ≥ max λ , 12β N C 2 3 γ 2N −4 , max 2≤j≤N −1λ j ,(79)W 0 i F ≤ γ, V 0 i F ≤ 3C 3 γ i−1 , Λ 0 i F ≤ C 3 β i γ i−1 , i = 1, . . . , N.(80) It can be seen that the conditions (76)-(78) on β i 's are slightly weaker than the conditions (39)-(41). Lemma 19 (Boundedness) Under Assumption 1 and the above conditions (76)-(80), for any k ∈ N, there hold W k i F ≤ γ, V k i F ≤ 3C 3 γ i−1 , Λ k i F ≤ C 3 β i γ i−1 , i = 1, . . . , N,(81)h k i ≤ 4L 3 C 3 γ i−1 , i = 1, . . . , N − 1,(82)µ k i ≤ 4L 3 C 3 γ i , i = 1, . . . , N − 2,(83) where γ := max 1≤i≤N W 0 i F (particularly, γ = 1 in the normalized case), C 3 and L 3 are specified later in (38) and (33), respectively. The boundedness of the sequence is mainly derived by the specific updates of the algorithm and the introduced 2 regularization. Proof [Proof of Lemma 19] We first show that the boundedness condition holds for k = 1. By the definitions of (18) and (33), it holds L(|c|) ≤ L 3 |c|, ∀ |c| ≥ 1. By the settings (21), (22) of h k i and µ k i , and (80), h 1 i ≤ L( V 0 i F + β −1 i Λ 0 i F ) ≤ L(4C 3 γ i−1 ) ≤ 4L 3 C 3 γ i−1 , i = 1, . . . , N − 1,(84)µ 1 i ≤ L( V 0 i+1 F + β −1 i+1 Λ 0 i+1 F ) ≤ L(4C 3 γ i ) ≤ 4L 3 C 3 γ i , i = 1, . . . , N − 2,(85) where the final inequalities in both (84) and (85) hold for 4C 3 γ i−1 ≥ 1 by the definition (38) and L 0 ≥ 1 8 in Assumption 1. In the following, we show that (80) holds. (1) On boundedness of W 1 N . By (50), W 1 N F ≤ λ −1 · 12β N C 2 3 γ 2N −3 ≤ γ, where the last inequality follows from the assumption (79) of λ. (2) On boundedness of W 1 i , i = N − 1, . . . , 2. By (51), W 1 i F ≤ 1 − λ λ + 18β i L 3 C 3 3 γ 3i−5 γ + 3L 1 β i C 3 γ i−2 (4C 3 γ i−1 + L 0 √ nd i ) λ . To make W 1 i F ≤ γ, it requires λ ≥ a i + √ a 2 i +4a i b i 2 , where a i := 3L 1 β i C 3 γ i−3 (4C 3 γ i−1 + L 0 √ nd i ), and b i = 18β i L 3 C 3 3 γ 3i−5 . By the assumption (79) of λ, we have λ ≥ a i 1 + b i a i ≥ a i + a 2 i + 4a i b i 2 . Thus, W 1 i F ≤ γ for i = 2, . . . , N − 1. (3) On boundedness of W 1 1 . By (52), W 1 1 F ≤ 1 − λ λ + 2β 1 L 3 C 3 X 2 F γ + L 1 β 1 X F (4C 3 + L 0 √ nd 1 ) λ . Similarly, by the assumption of λ (79), we can show that if (57), λ ≥ a 1 1 + b 1 a 1 ≥ a 1 + a 2 1 + 4a 1 b 1 2 , where a 1 := L 1 β 1 X F (4C 3 + L 0 √ nd 1 )γ −1 , b 1 := 2β 1 L 3 C 3 X 2 F , then W 1 1 F ≤ γ. (4) On boundedness of V 1 j , j = 1, . . . , N − 2. ByV 1 j F ≤ 1 − ρ j ρ j + 2L 3 C 3 γ j+2 · 3C 3 γ j−1 + (C 3 γ j−1 + L 0 nd j ) + L 1 γ(4C 3 γ j + L 0 nd j+1 ) ρ j , where ρ j := β j β j+1 . To guarantee V 1 j F ≤ 3C 3 γ j−1 , it requires ρ j ≥b j + b2 j + 4ā jcj 2ā j , whereā j = 2 − L 0 √ nd j C 3 γ j−1 ,b j = 2L 3 C 3 γ j+2 + 2L 3 γ 3 L 0 nd j + 4L 1 γ 2 + L 1 L 0 √ nd j+1 C 3 γ j−2 , and c j = 2L 1 L 3 γ 4 (4C 3 γ j + L 0 nd j+1 ). By the definition of C 3 (38), a j ≥ 3 2 ,b j ≤ 4.5L 1 + 3L 3 C 3 γ j γ 2 ,c j ≤ 9L 1 L 3 C 3 γ j+4 , where the bound onb j follows from the following facts 2L 0 nd j ≤ C 3 γ j−1 , L 1 L 0 nd j+1 C 3 γ j−2 ≤ 1 2 L 1 γ 2 , and the bound onc j is due to L 0 nd j+1 ≤ 1 2 C 3 γ j . Thus, it yields b j + b2 j + 4ā jcj 2ā j ≤ 1 3b j 1 + 1 + 6c j b 2 j ≤ 2 3b j + 6c j 3 ≤ 3L 1 + 6L 1 L 3 C 3 γ j + 2L 3 C 3 γ j γ 2 , where the first inequality holds forā j ≥ 3 2 , and the final inequality holds for the upper bounds ofb j andc j . Thus, we show the boundedness of V 1 j under our assumptions for any j = 1, . . . , N − 2. (5) On boundedness of V 1 N −1 . By (56), V 1 N −1 F ≤ C 3 γ N −2 + L 0 nd N −1 + 4C 3 γ N ρ −1 N −1 ≤ 3 2 C 3 γ N −2 + 4C 3 γ N ρ −1 N −1 ≤ 3C 3 γ N −2 where the first inequality holds by Assumption 1 and (80), the second inequality by the definition (38) of C 3 , and the final inequality is due to (77). (6) On boundedness of V 1 N . By (55), it shows that V 1 N F ≤ 3C 3 γ N −1 1 + β N + β N 1 + β N γ · 3C 3 γ N −2 ≤ 3C 3 γ N −1 . (7) On boundedness of Λ 1 N . By (47), Λ 1 N F ≤ V 1 N F + Y F ≤ 3C 3 γ N −1 + Y F ≤ C 3 β N γ N −1 , where the final inequality holds by the definition (38) of C 3 . (8) On boundedness of Λ 1 N −1 . By (48), Λ 1 N −1 F ≤ 7C 3 β N γ N ≤ C 3 β N −1 γ N −2 , where the final inequality is due to (77). (9) On boundedness of Λ 1 j , j = N − 2, . . . , 1 by induction. By (49), Λ 1 j F ≤ β j+1 γ 2L 1 L 0 nd j+1 + 7L 1 C 3 γ j + 12L 3 C 2 3 γ 2j ≤ C 3 β j+1 γ j+1 (8L 1 + 12L 3 C 3 γ j ) ≤ C 3 β j γ j−1 , where the second inequality holds for 2L 0 nd j+1 ≤ C 3 γ j , and the final inequality holds for (78). Therefore, we have shown that (81)-(83) hold for k = 1. Similarly, we can show that once (81)-(83) hold for some k, then they will hold for k + 1. Hence, we can show (81)-(83) hold for any k ∈ N recursively. This completes the proof of this lemma. D.1.4 Proof of Lemma 14: Sufficient descent lemma To prove Lemma 14, we first present a key lemma based on Lemma 18 and Lemma 19. For any k ≥ 2 and j = 1, . . . , N − 2, we denote E k 1,j := (L 1 γ) N −j L −1 1 C 3 β N γ N −2 W k N − W k−1 N F + N −1 i=j+1 (L 1 γ) i−j (C 3 β i γ i−2 + 3L −1 1 L 2 C 2 3 β i γ 2i−3 ) W k i − W k−1 i F , E k 2,j := (L 1 γ) N −j (1 + β N )L −1 1 V k N − V k−1 N F + (L 1 γ) N −1−j β N −1 V k N −1 − V k−1 N −1 F + N −2 i=j+1 (L 1 γ) i−j β i + (L 2 1 + 2L 3 C 3 γ i )γ 2 β i+1 V k i − V k−1 i F + (L 2 1 + 2L 3 C 3 γ j )γ 2 β j+1 V k j − V k−1 j F , and E k 3,j := (L 1 γ) N −j β N L −1 1 V k−1 N − V k−2 N F + (L 1 γ) N −1−j β N −1 V k−1 N −1 − V k−2 N −1 F + N −2 i=j+1 (L 1 γ) i−j β i + (L 2 1 + 2L 3 C 3 γ i + L 2 C 3 γ i )γ 2 β i+1 V k−1 i − V k−2 i F + L 2 1 + 2L 3 C 3 γ j + L 2 C 3 γ j β j+1 γ 2 V k−1 j − V k−2 j F . Lemma 20 Under assumptions of Lemma 19, for any k ≥ 2, we have Λ k N − Λ k−1 N F = V k N − V k−1 N F , Λ k N −1 − Λ k−1 N −1 F ≤ C 3 β N γ N −1 W k N − W k−1 N F + γ(1 + β N ) V k N − V k−1 N F + β N γ V k−1 N − V k−2 N F , and for j = 1, . . . , N − 2, Λ k j − Λ k−1 j F ≤ E k 1,j + E k 2,j + E k 3,j . Moreover, the above inequalities imply N i=1 Λ k i − Λ k−1 i 2 F ≤ α N i=1 W k i − W k−1 i 2 F + V k i − V k−1 i 2 F + V k−1 i − V k−2 i 2 F(86) for some constant α > 0 specified in the proof. Λ k j − Λ k−1 j F ≤ L 1 γ Λ k j+1 − Λ k−1 j+1 F + T k j+1 + I k j , where T k j+1 := (L 1 C 3 β j+1 γ j +3C 2 3 L 2 β j+1 γ 2j ) W k j+1 −W k−1 j+1 F +L 1 γβ j+1 ( V k j+1 −V k−1 j+1 F + V k−1 j+1 − V k−2j+1 F ), and I k j := (L 2 1 + 2L 3 C 3 γ j )β j+1 γ 2 V k j − V k−1 j F + L 2 1 + 2L 3 C 3 γ j + L 2 C 3 γ j β j+1 γ 2 V k−1 j − V k−2 j F . By the above inequality, we have Λ k j − Λ k−1 j F ≤ (L 1 γ) N −1−j Λ k N −1 − Λ k−1 N −1 F + (L 1 γ) N −2−j T k N −1 + N −2−j i=1 (L 1 γ) i−1 T k j+i + L 1 γI k j+i + I k j . Substituting the definitions of T k j and I k j into this inequality and after some simplifications yields the desired bound for Λ k j − Λ k−1 j F . Summing up all the above inequalities and using several times of the basic inequality ( p i=1 u i ) 2 ≤ p p i=1 u 2 i for any u ∈ R p yields (86) with some positive constant α. This completes the proof. Based on Lemma 15, Lemma 19 and Lemma 20, we prove Lemma 14 as follows. Proof [Proof of Lemma 14] By (41) and the definition (38) of C 3 , we have for j = 1, . . . , N − 2, β j β j+1 ≥ f 2 min γ 2 , and β j ≥ f 2(i−j) min γ 2(i−j) β i , j < i ≤ N − 1.(87) By (36)- (37) and (43), it holds α 3 ≥ 24N + 1.(88) To prove this lemma, we first estimate Λ k i − Λ k−1 i 2 F for any i = 1, . . . , N . By Lemma 20, we get Λ k N − Λ k−1 N 2 F = V k N − V k−1 N 2 F ,(89) and using the basic inequality 3 i=1 a i 2 ≤ 3 3 i=1 a 2 i , Λ k N −1 − Λ k−1 N −1 2 F ≤ 3C 2 3 β 2 N γ 2(N −1) W k N − W k−1 N 2 F + 3γ 2 (1 + β N ) 2 V k N − V k−1 N 2 F (90) + 3β 2 N γ 2 V k−1 N − V k−2 N 2 F , and for j = 1, . . . , N − 2, using the inequality ( n i=1 a i ) 2 ≤ n n i=1 a 2 i , Λ k j − Λ k−1 j 2 F ≤ 2(N − j)T k 1,j + 4(N − j + 1)(T k 2,j + T k 3,j ),(91) where T k 1,j = (L 1 γ) 2(N −j) L −2 1 C 2 3 β 2 N γ 2(N −2) W k N − W k−1 N 2 F + N −1 i=j+1 (L 1 γ) 2(i−j) (1 + 3L −1 1 L 2 C 3 γ i−1 ) 2 C 2 3 β 2 i γ 2(i−2) W k i − W k−1 i 2 F , T k 2,j = (L 1 γ) 2(N −j) (1 + β N ) 2 L −2 1 V k N − V k−1 N 2 F + (L 1 γ) 2(N −1−j) β 2 N −1 V k N −1 − V k−1 N −1 2 F + N −2 i=j+1 (L 1 γ) 2(i−j) β i + (L 2 1 + 2L 3 C 3 γ i )γ 2 β i+1 2 V k i − V k−1 i 2 F + (L 2 1 + 2L 3 C 3 γ j ) 2 γ 4 β 2 j+1 V k j − V k−1 j 2 F , and T k 3,j = (L 1 γ) 2(N −j) β 2 N L −2 1 V k−1 N − V k−2 N 2 F + (L 1 γ) 2(N −1−j) β 2 N −1 V k−1 N −1 − V k−2 N −1 2 F + N −2 i=j+1 (L 1 γ) 2(i−j) β i + (L 2 1 + 2L 3 C 3 γ i + L 2 C 3 γ i )γ 2 β i+1 2 V k−1 i − V k−2 i 2 F + L 2 1 + 2L 3 C 3 γ j + L 2 C 3 γ j 2 β 2 j+1 γ 4 V k−1 j − V k−2 j 2 F . Substituting (89), (90) and (91) into Lemma 15 and after some simplifications yields L(Q k ) + N i=1 ξ i V k i − V k−1 i 2 F ≤ L(Q k−1 ) + N i=1 ξ i V k−1 i − V k−2 i 2 F (92) − N i=1 ζ i W k i − W k−1 i 2 F − N i=1 (η i − ξ i ) V k i − V k−1 i 2 F , where ζ N = λ 2 − 3C 2 3 β −1 N −1 β 2 N γ 2(N −1) − 2L −2 1 C 2 3 β 2 N γ 2(N −2) N −2 j=1 β −1 j (N − j)(L 1 γ) 2(N −j) ζ i = λ 2 − 2(1 + 3L −1 1 L 2 L 3 γ i−1 ) 2 C 2 3 β 2 i γ 2(i−2) i−1 j=1 β −1 j (N − j)(L 1 γ) 2(i−j) , i = 2, . . . , N − 1, ζ 1 = λ 2 , η N = 1 + β N 2 − β −1 N − 3γ 2 (1 + β N ) 2 β −1 N −1 − 4(1 + β N ) 2 L 2 1 N −2 j=1 β −1 j (N − j + 1)(L 1 γ) 2(N −j) , ξ N = 3γ 2 β 2 N β −1 N −1 + 4β 2 N L 2 1 N −2 j=1 β −1 j (N − j + 1)(L 1 γ) 2(N −j) ,(93)η N −1 = β N −1 2 − 4β 2 N −1 N −2 j=1 β −1 j (N − j + 1)(L 1 γ) 2(N −1−j) , ξ N −1 = 4β 2 N −1 N −2 j=1 β −1 j (N − j + 1)(L 1 γ) 2(N −1−j) ,(94) and for i = 2, . . . , N − 2, η i = β i 2 − 4 β i + (L 2 1 + 2L 3 C 3 γ i )γ 2 β i+1 2 i−1 j=1 β −1 j (N − j + 1)(L 1 γ) 2(i−j) − 4(L 2 1 + 2L 3 C 3 γ i ) 2 γ 4 β 2 i+1 β −1 i (N − i + 1), ξ i = 4 β i + (L 2 1 + 2L 3 C 3 γ i + L 2 C 3 γ i )γ 2 β i+1 2 i−1 j=1 β −1 j (N − j + 1)(L 1 γ) 2(i−j)(95) + 4 L 2 1 + 2L 3 C 3 γ i + L 2 C 3 γ i 2 γ 4 β 2 i+1 β −1 i (N − i + 1), and η 1 = β 1 2 − 4(L 2 1 + 2L 3 C 3 γ) 2 γ 4 β 2 2 β −1 1 N , and ξ 1 = 4 L 2 1 + 2L 3 C 3 γ + L 2 C 3 γ 2 γ 4 β 2 2 β −1 1 N. Based on (92), to get (59), we need to show that ζ i > 0, η i − ξ i > 0, i = 1, . . . , N.(97) Then, let a := min{ζ i , η i − ξ i , i = 1, . . . , N }, we get (59). In the following, we show (97). It is obvious that ζ 1 = λ 2 > 0. For i = 2, . . . , N − 1, by (87), ζ i ≥ λ 2 − 2C 2 3 β i (1 + 3L −1 1 L 2 L 3 γ i−1 ) 2 γ 2(i−2) i−1 j=1 (N − j)α −(i−j) 3 > λ 2 − 2C 2 3 β i (1 + 3L −1 1 L 2 L 3 γ i−1 ) 2 γ 2(i−2) · N α 3 − 1 ≥ 0, where the final inequality is due to α 3 > 24N + 1 and the assumption of λ, i.e., (42). Similarly, we can show that ζ N > 0 as follows ζ N ≥ λ 2 − β N C 2 3 γ 2(N −2) ·   3 16 + 1 8 N −2 j=1 (N − j)α −(N −j−1) 3   > λ 2 − β N C 2 3 γ 2(N −2) · 3 16 + N 8(α 3 − 1) > λ 2 − 1 5 β N C 2 3 γ 2(N −2) > 0. At the end, we show η i − ξ i > 0 for i = 1, . . . , N . Note that η 1 − ξ 1 = β 1 2 − 4 (L 2 1 + 2L 3 C 3 γ) 2 + (L 2 1 + 2L 3 C 3 γ + L 2 C 3 γ) 2 γ 4 β 2 2 β −1 1 N > 0, where we have used (41) β 1 β 2 ≥ 4 √ N L 2 1 + (2L 3 + L 2 )C 3 γ γ 2 . For i = 2, . . . , N − 2, let α 1 := (L 2 1 + 2L 3 C 3 γ i )γ 2 , α 2 := (L 2 1 + 2L 3 C 3 γ i + L 2 C 3 γ i )γ 2 . Note that η i − ξ i = β i 2 − 4 (β i + α 1 β i+1 ) 2 + (β i + α 2 β i+1 ) 2 i−1 j=1 β −1 j (N − j + 1)(L 1 γ) 2(i−j) − 4(α 2 1 + α 2 2 )β 2 i+1 β −1 i (N − i + 1) > β i 2 − 4 (β i + α 1 β i+1 ) 2 + (β i + α 2 β i+1 ) 2 i−1 j=1 β −1 i (N − j + 1)α −(i−j) 3 − 4N (α 2 1 + α 2 2 )β 2 i+1 β −1 i > β i 1 2 − 4N α 3 − 1 (1 + α 1 · β i+1 β i ) 2 + (1 + α 2 · β i+1 β i ) 2 − 4N (α 2 1 + α 2 2 ) β i+1 β i 2 = 4N α 3 − 1 β i α 3 − 1 8N − 2 − 2(α 1 + α 2 ) · β i+1 β i − α 3 (α 2 1 + α 2 2 ) · β i+1 β i 2 > 4N α 3 − 1 β i α 3 − 1 8N − 2 − 2(α 1 + α 2 ) · β i+1 β i − α 3 (α 1 + α 2 ) 2 · β i+1 β i 2 ≥ 0,(99) where the final inequality follows from (41), α 3 > 24N + 1, and 1 + 1 + α 3 α 3 −1 8N − 2 α 3 −1 8N − 2 ≤ 1 + √ 24N + 2 ≤ 6 √ N . Similarly, notice that η N −1 − ξ N −1 = β N −1 2 − 8β 2 N −1 N −2 j=1 β −1 j (N − j + 1)(L 1 γ) 2(N −1−j) ≥ β N −1 2 − 8β N −1 N −2 j=1 (N − j + 1)α −(N −1−j) 3 > β N −1 1 2 − 8N α 3 − 1 ≥ 1 6 β N −1 > 0. Finally, note that η N − ξ N = 1 + β N 2 − β −1 N − 3γ 2 β −1 N −1 [(1 + β N ) 2 + β 2 N ] − 4 L 2 1 [(1 + β N ) 2 + β 2 N ] N −2 j=1 β −1 j (N − j + 1)(L 1 γ) 2(N −j) ≥ 1 + β N 2 − β −1 N − 3γ 2 β −1 N −1 [(1 + β N ) 2 + β 2 N ] − 4β −1 N −1 γ 2 [(1 + β N ) 2 + β 2 N ] N −2 j=1 (N − j + 1)α −(N −1−j) 3 > β 2 N + β N − 2 2β N − 2(3 + 4N α 3 − 1 )(β 2 N + β N + 1)β −1 N −1 γ 2 ≥ β 2 N + β N − 2 2β N − 19 3 (β 2 N + β N + 1)β −1 N −1 γ 2 > 0, where the final inequality follows from β N ≥ 3.5, which implies and (40). This completes the proof. 16 > 38 3 · β 2 N + β N + 1 β 2 N + β N − 2 , D.1.5 Proof of Lemma 21: Relative error lemma In the following, we provide a lemma to show that the gradients of the augmented Lagrangian and the new Lyapunov function can be bounded by the discrepancy between two successive updates. Such a lemma is important to show the global convergence of a descent sequence by (Attouch et al., 2013, Theorem 2.9). Lemma 21 Under conditions of Theorem 7, for any positive k ≥ 2, there exists some positive constantb such that ∇L(Q k ) F ≤b N i=1 ( W k i − W k−1 i F + V k i − V k−1 i F + V k−1 i − V k−2 i F ),(100) and ∇L(Q k ) F ≤b Q k −Q k−1 F , whereb = √ 3N b and b =b + 4 max 1≤i≤N ξ i . Proof Note that ∇L(Q k ) = ∂L(Q k ) ∂W i N i=1 , ∂L(Q k ) ∂V i N i=1 , ∂L(Q k ) ∂Λ i N i=1 , then ∇L(Q k ) F ≤ N i=1 ∂L(Q k ) ∂W i F + ∂L(Q k ) ∂V i F + ∂L(Q k ) ∂Λ i F .(101) In order to bound ∇L(Q k ) F , we need to bound each component of ∇L(Q k ). On ∂L(Q k ) ∂W N F : By the optimality condition of (8), λW k N + β N (W k N V k−1 N −1 − V k−1 N )V k−1 N −1 T + Λ k−1 N V k−1 N −1 T = 0, which implies ∂L(Q k ) ∂W N = λW k N + β N (W k N V k N −1 − V k N )(V k N −1 ) T + Λ k N (V k N −1 ) T = β N (W k N V k N −1 − V k N )(V k N −1 − V k−1 N −1 ) T + W k N (V k N −1 − V k−1 N −1 ) − (V k N − V k−1 N ) V k−1 N −1 T + Λ k−1 N (V k N −1 − V k−1 N −1 ) T + (Λ k N − Λ k−1 N )(V k N −1 ) T . By the boundedness of the sequence (81), the above equality yields ∂L(Q k ) ∂W N F ≤ 10β N C 3 γ N −1 V k N −1 − V k−1 N −1 F + 3C 3 γ N −2 (β N + 1) V k N − V k−1 N F . (102) On ∂L(Q k ) ∂W i F : For i = 2, . . . , N − 1, by the optimality condition of (19), λW k i + (β i σ(W k−1 i V k−1 i−1 ) − β i V k−1 i + Λ k−1 i ) σ (W k−1 i V k−1 i−1 ) V k−1 i−1 T + β i h k i 2 (W k i − W k−1 i )V k−1 i−1 T = 0, which implies ∂L(Q k ) ∂W i = λW k i + (β i σ(W k i V k i−1 ) − β i V k i + Λ k i ) σ (W k i V k i−1 ) V k i−1 T = (β i σ(W k i V k i−1 ) − β i V k i + Λ k i ) σ (W k i V k i−1 ) V k i−1 T − (β i σ(W k−1 i V k−1 i−1 ) − β i V k−1 i + Λ k−1 i ) σ (W k−1 i V k−1 i−1 ) V k−1 i−1 T − β i h k i 2 (W k i − W k−1 i )V k−1 i−1 T = β i σ(W k i V k i−1 ) − σ(W k−1 i V k i−1 ) + σ(W k−1 i V k i−1 ) − σ(W k−1 i V k−1 i−1 ) σ (W k i V k i−1 ) + β i (V k−1 i − V k i ) + (Λ k i − Λ k−1 i ) σ (W k i V k i−1 ) + β i σ(W k−1 i V k−1 i−1 ) − β i V k−1 i + Λ k−1 i σ (W k i V k i−1 ) − σ (W k−1 i V k i−1 ) + β i σ(W k−1 i V k−1 i−1 ) − β i V k−1 i + Λ k−1 i σ (W k−1 i V k i−1 ) − σ (W k−1 i V k−1 i−1 ) V k i−1 T + (β i σ(W k−1 i V k−1 i−1 ) − β i V k−1 i + Λ k−1 i ) σ (W k−1 i V k−1 i−1 ) (V k i−1 − V k−1 i−1 ) T − β i h k i 2 (W k i − W k−1 i )V k−1 i−1 T . By Assumption 1 and Lemma 19, the above equality yields ∂L(Q k ) ∂W i F (103) ≤ 3C 3 γ i−2 3β i C 3 γ i−2 L 2 1 + L 0 L 2 nd i + 4L 2 C 3 γ i−1 + 2 3 L 3 γ W k i − W k−1 i F + β i L 2 1 γ + (L 1 + L 2 γ)(L 0 nd i + 4C 3 γ i−1 ) V k i−1 − V k−1 i−1 F +β i L 1 V k i − V k−1 i F + L 1 Λ k i − Λ k−1 i F . On ∂L(Q k ) ∂W 1 F : Similarly, by the optimality condition of (19) with i = 1, λW k 1 + β 1 σ(W k−1 1 X) − β 1 V k−1 1 + Λ k−1 1 σ (W k−1 1 X) X T + β 1 h k 1 2 (W k 1 − W k−1 1 )X T = 0, which implies ∂L(Q k ) ∂W 1 = λW k 1 + β 1 σ(W k 1 X) − β 1 V k 1 + Λ k 1 σ (W k 1 X) X T = β 1 (σ(W k 1 X) − σ(W k−1 1 X)) − β 1 (V k 1 − V k−1 1 ) + (Λ k 1 − Λ k−1 1 ) σ (W k 1 X) X T + β 1 σ(W k−1 1 X) − β 1 V k−1 1 + Λ k−1 1 (σ (W k 1 X) − σ (W k−1 1 X)) X T + β 1 h k 1 2 (W k−1 1 − W k 1 )X T . The above inequality yields ∂L(Q k ) ∂W 1 F ≤ β 1 X F X F · (L 2 1 + L 0 L 2 nd 1 + 4L 2 C 3 ) + 2L 3 C 3 W k 1 − W k−1 1 F + β 1 L 1 X F V k 1 − V k−1 1 F + L 1 X F Λ k 1 − Λ k−1 1 F .(104) On ∂L(Q k ) ∂V j F (1 ≤ j ≤ N − 2): By the optimality condition of (20), β j (V k j − σ(W k j V k j−1 )) + W k j+1 T Λ k−1 j+1 + β j+1 (σ(W k j+1 V k−1 j ) − V k−1 j+1 ) σ (W k j+1 V k−1 j ) − Λ k−1 j + β j+1 µ k j 2 W k j+1 T W k j+1 (V k j − V k−1 j ) = 0, which implies ∂L(Q k ) ∂V j = W k j+1 T (Λ k j+1 − Λ k−1 j+1 ) + β j+1 (σ(W k j+1 V k j ) − σ(W k j+1 V k−1 j )) + β j+1 (V k−1 j+1 − V k j+1 ) σ (W k j+1 V k j ) + W k j+1 T Λ k−1 j+1 + β j+1 (σ(W k j+1 V k−1 j ) − V k−1 j+1 ) σ (W k j+1 V k j ) − σ (W k j+1 V k−1 j ) + (Λ k−1 j − Λ k j ) + β j+1 µ k j 2 W k j+1 T W k j+1 (V k−1 j − V k j ). The above equality yields ∂L(Q k ) ∂V j F ≤ β j+1 γ 2 2C 3 γ j (2L 2 + L 3 ) + L 0 L 2 nd j+1 V k j − V k−1 j F (105) + β j+1 L 1 γ V k j+1 − V k−1 j+1 F + Λ k j − Λ k−1 j F + L 1 γ Λ k j+1 − Λ k−1 j+1 F . On ∂L(Q k ) ∂V N −1 F : By the optimality condition of (11), β N −1 (V k N − σ(W k N −1 V k N −2 )) − Λ k−1 N −1 + W k N T Λ k−1 N + β N (W k N V k N −1 − V k−1 N ) = 0, which implies ∂L(Q k ) ∂V N −1 = β N −1 (V k N − σ(W k N −1 V k N −2 )) − Λ k N −1 + W k N T Λ k N + β N (W k N V k N −1 − V k N ) = Λ k−1 N −1 − Λ k N −1 + W k N T (Λ k N − Λ k−1 N ) + β N W k N T (V k−1 N − V k N ). The above equality implies ∂L(Q k ) ∂V N −1 F ≤ β N γ V k N − V k−1 N F + Λ k N −1 − Λ k−1 N −1 F + γ Λ k N − Λ k−1 N F .(106) On ∂L(Q k ) ∂V N F : Similarly, by the optimality condition of (12), we get ∂L(Q k ) ∂V N F = Λ k N − Λ k−1 N F .(107) Moreover, for i = 1, . . . , N , by the update of Λ k i , we can easily yield ∂L(Q k ) ∂Λ i F = β −1 i Λ k i − Λ k−1 i F .(108) Substituting (102)-(108) into (101), and after some simplifications, we get ∇L(Q k ) F ≤ᾱ N i=1 ( W k i − W k−1 i F + V k i − V k−1 i F + Λ k i − Λ k−1 i F )(109) for someᾱ > 0. By Lemma 20, substituting these upper bounds of Λ k i − Λ k−1 i F (i = 1, . . . , N ) into (109) and after some simplifications implies (100) for some constantb. By (100), it is easy to derive ∇L(Q k ) F ≤ ∇L(Q k ) F + N i=1 4ξ i V k i − V k−1 i F ≤ b N i=1 ( W k i − W k−1 i F + V k i − V k−1 i F + V k−1 i − V k−2 i F ) ≤b Q k −Q k−1 F , where b =b + 4 max 1≤i≤N ξ i andb = √ 3N b. This completes the proof. D.2 Proof of Theorem 5 Now we provide the detailed proof of Theorem 5 based on the above lemmas. Proof [Proof of Theorem 5] (a) By Lemma 19, the boundedness of {Q k } implies the sequence L(Q k ) is lower bounded, and so isL(Q k ) by its definition (29). By Lemma 14,L(Q k ) is monotonically non-increasing, therefore,L(Q k ) is convergent. (b) Again by Lemma 19,Q k is bounded, and thus there exists a subsequenceQ k j such thatQ k j →Q * as j → ∞. SinceL is continuous by Assumption 1, then lim j→∞L (Q k j ) = L(Q * ). This implies the continuity condition in the analysis framework formulated in (Attouch et al., 2013) holds. Together with the sufficient descent (Lemma 14), relative error (Lemma 21) and properties, the whole sequence convergence to a stationary point is derived via following (Attouch et al., 2013, Theorem 2.9). (c) The O(1/K) rate can be easily derived by Lemma 14, Lemma 20 and Lemma 21. Specifically, by Lemma 14, it is easy to show 1 K K k=2 ( W k − W k−1 2 F + V k − V k−1 2 F ) ≤L (Q 1 ) −L(Q * ) aK ,(110) which implies 1 K K k=2 N i=1 V k i −V k−1 2 F = 1 K K−1 k=1 V k − V k−1 2 F ≤ a V 1 − V 0 2 F + (L(Q 1 ) −L(Q * )) aK .(111) By (86) in Lemma 20,(110) and (111), there holds 1 K K k=2 N i=1 Λ k i − Λ k−1 i 2 F ≤C · a V 1 − V 0 2 F + (L(Q 1 ) −L(Q * )) aK ,(112) for some positive constantC. By (110)-(112), and Lemma 21, it implies 1 K K k=2 ∇L(Q k ) 2 F ≤Ĉ · a V 1 − V 0 2 F + (L(Q 1 ) −L(Q * )) aK , for some positive constantĈ. This completes the proof. Figure 1 : 1Philosophy behind deep learning Figure 2 : 2The cons of SGD and pros of ADMM in solving deep sigmoid nets. The setting of numerical simulation in (c) can be found in the where L p ([−M, M ]) denotes the L p space of functions defined on [−M, M ]. Figure 3 : 3Update order of ADMM Specifically, given an initialization {W 0 Figure 4 : 4Gradient vanishing of SGD and saturation-avoidance of ADMM in the training of deep sigmoid nets. The numerical setting is the same as that ofFigure 2. Following the notations of, the linear constraint considered in is of the formAx + By = 0 (25) where x := (x 0 , . . . , x p ) includes p + 1 blocks of variables, y is a special block of variables, A := [A 0 , . . . , A p ] and B are two matrices satisfying Im(A) ⊆ Im(B), where Im(·) returns the image of a matrix. Similarly, (Gao et al., 2020) extended (25) to multiaffine constraint of the form, A(x 1 , x 2 ) + B(y) = 0, where A and B are respectively some multiaffine and linear maps satisfying Im(A) ⊆ Im(B). Leveraging this special block variable y, the dual variables (namely, multipliers) is expressed solely by y (Wang et al., 2019, Lemma 3), and the amount of dual ascent part is controlled by the amount of descent part brought by the primal y-block update (Wang et al., 2019, Lemma 5). Figure 5 : 5Proof sketch of convergence of ADMM. Figure 6 : 6Effect of the depth of neural networks in approximating the square function. Figure 7 : 7Effect of parameters for ADMM in approximating the univariate square function. Figure 8 : 8Effect of the depth of neural networks in approximating the product-gate function. Figure 9 : 9Effect of parameters for ADMM in approximating the product-gate function. Figure 10 : 10Effect of the depth of neural networks in learning the L 1 radial function. Figure 11 :Figure 12 : 1112(b) that the proposed ADMM is robustness to Effect of parameters and initial schemes for ADMM in learning the L 1 radial function. Robustness of the proposed ADMM to the noise in learning L 1 radial function. Figure 13 : 13Effect of the depth of neural networks in learning the L 2 radial function. Figure 14 :Figure 15 : 1415Effect of parameters and initial schemes for ADMM in learning smooth L 2 radial function. Robustness of the proposed ADMM to the noise in learning L 2 radial function. Figure 17 : 17Performance of ADMM in extended Yale B database: (a) the effect of width for different algorithms; (b) the stability of the proposed ADMM to different random initial schemes. h relu 9,2,µ,ν,A (t) = h prod9,ν σ(At), h linear 2,µ (t)for t ∈ [−M 0 , M 0 ]. We then have from Lemma 6, (30) and (31) that for any ), dom(h) := {x ∈ X : h(x) < +∞}, dom(∂h) := {x ∈ X : ∂h(x) = ∅}, and dist(0, ∂h(x)) := min{ z : z ∈ ∂h(x)}, where · represents the Euclidean norm.Proper lower semi-continuous functions which satisfy the Kurdyka-Lojasiewicz inequality at each point of dom(∂h) are called KL functions. Table 1 : 1Depth required for deep nets in feature extraction and approximation within accuracy εThe proof of Theorem 3 is postponed in Appendix A. For an arbitrary deep ReLU netFeatures sigmoid ReLU Square-gate 1 (Chui et al., 2019) log(ε −1 ) (Yarotsky, 2017) Product-gate 1 (Chui et al., 2019) log(ε −1 ) (Yarotsky, 2017) Localized approximation 2 (Chui et al., 1994) 2(Chui et al., 2020) k-spatially sparse+smooth 2 (Lin, 2019) > 8 (Chui et al., 2020) Smooth+Manifold 3 (Chui et al., 2018) 4 (Shaham et al., 2018) Smooth 1 (Mhaskar, 1996) log(ε −1 ) (Yarotsky, 2017) k-sparse (frequency) k (Lin et al., 2017) k log(ε −1 ) (Schwab and Zech, 2019) Radial+smooth 4 (Chui et al., 2019) > 8 (Han et al., 2020) Table 2 : 2Experimental settings for toy simulations. Sample sizes for approximation tasks are set as 1000, and training and test samples sizes for learning tasks are set as 1000 respectively. The notation [a : b] is denoted by the set {a, a + 1, . . . , b} for two integers a, b. † In the case of learning the L 2 radial function, ranges of depth of deep sigmoid and ReLU nets are [2, 6] and [2, 20], respectively.task functions deep fully-connected NNs SGDs (sigmoid/ReLU), SGDM SGDM Adam ADMM width depth learning rate (lr) batch size (momentum) (λ, β) Approx. x 2 20 × [1 : 5] [1, 5] (sigmoid), lr: 0.001 uv 60 × [1 : 5] [1, 20] (ReLU) 0.1 × 0.95 k 50 0.5 β 1 : 0.9 (10 −6 , 1) Learn. ( x 1 − 1) + 10 × [1 : 6] per 10 epochs, β 2 : 0.999 g(|x| 2 ) 100 × [1 : 5] [2, 6], [2, 20] † : 1e-8 Table 4 : 4Experimental results of different algorithms in approximating f (u, v) = uv. Table 5 : 5Performance of different algorithms for learning L 1 radial function with 0.1 Gaussian noise.Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Test Error 2.48e-5(9.74e-6) 2.26e-5(7.88e-6) 2.16e-5(8.53e-6) 4.58e-5(1.59e-5) 1.69e-5(4.34e-6) Run Time (s) 23.24 18.23 14.66 1.68 10.36 (depth, width) (17,50) (13,50) (10,50) (1,20) (4,10) 2 4 6 8 10 12 14 16 18 20 Table 6 : 6Test errors of different algorithms for learning L 2 radial function with 0.1 Gaussian noise. Table 7 : 7Experimental settings for real-data experiments. The number of epochs for each case is set empirically to be 200.dataset (training size, Network structure SGDs (sigmoid/ReLU),SGDM SGDM Adam ADMM test size) width depth batch size learning rate (momentum) lr:0.001 (λ, β) Earthquake (4173,4000) 20 × [1 : 10] [1:6] 100 0.1 × 0.95 k , β 1 : 0.9 λ ∈ 10 [−6:−2] EYB (2432,2432) 20 × [1 : 10] 1 50 per 10 epochs 0.5 β 2 : 0.999 β = 1 PTB (7000,7552) 64 × [1 : 4] [1:10] 100 : 1e-8 Table 8 : 8Test accuracies (%) of different algorithms for earthquake intensity database. The baseline of the test accuracy is 80.48%(Zeng et al., 2020).Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Test Acc(%) 81.24(0.45) 81.16(0.32) 81.31(0.36) 79.94(0.23) 81.26(0.31) Run Time (s) 4.74 14.20 13.24 2.60 12.64 (depth, width) (2,120) (5,140) (4,80) (1,100) (3,80) Table 9 : 9Performance of different algorithms for extended Yale B database. The baseline of the test accuracy is about 96% in(Lu et al., 2020).Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Test Acc(%) 98.84(0.28) 97.18(0.28) 98.91(0.34) 98.67(0.41) 98.93(0.43) Run Time (s) 19.78 23.99 48.92 16.95 21.36 (depth, width) (1,200) (1,200) (1,200) (1,140) (1,60) 20 40 60 80 100 120 140 160 180 200 Width (# hidden neurons) 10 20 30 40 50 60 70 80 90 100 Test accuracy (%) ADMM (sigmoid) SGD (sigmoid) SGD (ReLU) SGDM (ReLU) Adam (ReLU) (a) Effect of network structure. ADMM(sigmoid): Extended Yale B data 98.91 98.83 98.85 98.78 98.80 98.84 O r th -U n if O r th -G a u s s L e C u n -U n if L e C u n -G a u s s X a v ie r M S R A 90 91 92 93 94 95 96 97 98 99 100 Test accuracy (%) Table 10 : 10Performance of different algorithms for PTB diagnostic ECG database. The baseline of the test accuracy is 99.20% in(Kachuee et al., 2018).R&D Program of China (No.2020YFA0713900). The work of Jinshan Zeng is supported in part by the National Natural Science Foundation of China [Project No. 61977038] and by the Thousand Talents Plan of Jiangxi Province [Project NO. jxsq2019201124]. The work of Shao-Bo Lin is supported in part by the National Natural Science Foundation of China [Project No. 61876133]. This work of Yuan Yao is supported in part by Hong Kong Research Grant Council Project NO. RGC16308321 and 16303817, NSFC/RGC Joint Research Scheme N HKUST635/20, and ITF UIM/390. The work of Ding-Xuan Zhou is supported partially by the Research Grants Council of Hong Kong [Project No. CityU 11307319], Laboratory for AI-Powered Financial Technologies and by the Hong Kong Institute for Data Science. This research made use of the computing resources of the X-GPU cluster supported by the Hong Kong Research Grant Council Collaborative Research Fund: C6021-19EF.Algorithm SGD (ReLU) SGDM (ReLU) Adam (ReLU) SGD (sigmoid) ADMM (sigmoid) Test Acc(%) 99.18(0.32) 99.16(0.28) 99.25(0.25) 96.88(0.46) 99.22(0.11) Run Time (s) 29.82 40.77 30.83 12.28 29.17 (depth, width) (8,192) (7,192) (7,256) (3,256) (4,128) 1 2 3 4 5 6 7 8 9 10 Depth 80 82 84 86 88 90 92 94 96 98 100 Test accuracy (%) Effect of depth (PTB) ADMM (sigmoid) SGD (sigmoid) SGD (ReLU) SGDM (ReLU) Adam (ReLU) Baseline (a) Effect of network structure. ADMM(sigmoid): PTB 99.12 99.08 99.10 99.18 99.21 99.22 O r th -U n if O r th -G a u s s L e C u n -U n if L e C u n -G a u s s X a v ie r M S R A 95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100 Test accuracy (%) (b) Stability to initial schemes of ADMM Figure 18: Performance of ADMM in PTB diagnostic ECG database: (a) the effect of depth for different algorithms; (b) the stability of the proposed ADMM to different initial schemes. and by specializing Lemma16 with v = [W k A] ij , u = [W k−1 A] ij and c = B ij for any i, j, where [W k A] ij and [W k−1 A] ij are the (i, j)-th entries of W k A and W k−1 A, respectively.of H σ (W ; A, B) = 1 2 σ(W A) − B 2 F This yields (63). Similarly, we can establish the inequality (64). This completes the proof. Based on Lemma 17, we prove Lemma 15 as follows. Proof [Proof of Lemma 15] We establish (60) via estimating the progress for each block update. At first, we consider the W k N update. 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{'abstract': 'In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called sigmoid-ADMM pair ), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than that of deep ReLU nets by showing that ReLU activation function can be well approximated by deep sigmoid nets with two hidden layers and finitely many free parameters but not vice-verse. We also establish the global convergence of the proposed ADMM for the nonlinearly constrained formulation of the deep sigmoid nets training from arbitrary initial points to a Karush-Kuhn-Tucker (KKT) point at a rate of order O(1/k). Besides sigmoid activation, such a convergence theorem holds for a general class of smooth activations. Compared with the widely used stochastic gradient descent (SGD) algorithm for the deep ReLU nets training (called ReLU-SGD pair), the proposed sigmoid-ADMM pair is practically stable with respect to the algorithmic hyperparameters including the learning rate, initial schemes and the pro-processing of the input data. Moreover, we find that to approximate and learn simple but important functions the proposed sigmoid-ADMM pair numerically outperforms the ReLU-SGD pair.', 'arxivid': '1902.02060', 'author': ['Jinshan Zeng jinshanzeng@jxnu.edu.cn ', 'Shao-Bo Lin ', 'Yuan Yao yuany@ust.hk ', 'Ding-Xuan Zhou mazhou@cityu.edu.hk ', '\nSchool of Computer and Information Engineering\nLiu Bie Ju Centre for Mathematical Sciences\nJiangxi Normal University\nNanchangChina\n', '\nDepartment of Mathematics\nCity University of Hong Kong\nHong Kong\n', "\nCenter of Intelligent Decision-Making and Machine Learning\nSchool of Management\nHong Kong University of Science and Technology\nHong Kong, Xi'an\n", "\nDepartment of Mathematics\nJiaotong University\nXi'an, HongChina\n", '\nSchool of Data Science and Department of Mathematics\nKong University of Science and Technology\nHong Kong\n', '\nCity University of Hong Kong\nHong Kong\n'], 'authoraffiliation': ['School of Computer and Information Engineering\nLiu Bie Ju Centre for Mathematical Sciences\nJiangxi Normal University\nNanchangChina', 'Department of Mathematics\nCity University of Hong Kong\nHong Kong', "Center of Intelligent Decision-Making and Machine Learning\nSchool of Management\nHong Kong University of Science and Technology\nHong Kong, Xi'an", "Department of Mathematics\nJiaotong University\nXi'an, HongChina", 'School of Data Science and Department of Mathematics\nKong University of Science and Technology\nHong Kong', 'City University of Hong Kong\nHong Kong'], 'corpusid': 221516649, 'doi': None, 'github_urls': ['https://github.com/JinshanZeng/ADMM-DeepLearning.', 'https://github.com/fchollet/keras.'], 'n_tokens_mistral': 60237, 'n_tokens_neox': 51485, 'n_words': 31445, 'pdfsha': 'b3e0b6ba78adb0cffb3b8292ea3fce8ac9631f54', 'pdfurls': ['https://arxiv.org/pdf/1902.02060v3.pdf'], 'title': ['On ADMM in Deep Learning: Convergence and Saturation-Avoidance', 'On ADMM in Deep Learning: Convergence and Saturation-Avoidance'], 'venue': ['Journal of Machine Learning Research']}

arxiv

Comment on "Pionic decay of a possible d ′ dibaryon and the short-range NN interaction" arXiv:nucl-th/9712079v1 29 Dec 1997 A Samsonov Institute for Theoretical and Experimental Physics Moscow 117218Russia M Schepkin Institute for Theoretical and Experimental Physics Moscow 117218Russia Comment on "Pionic decay of a possible d ′ dibaryon and the short-range NN interaction" arXiv:nucl-th/9712079v1 29 Dec 1997 We comment on calculations of the width of the d ′ resonance within framework of quark shell models. * This is easy to see since in that case the NN-pair is in the 1 S0-state (J P = 0 + ), hence the 4-body vertex contains 3 spinless "particles": 0 − (π), 0 − (d ′ ), and 0 + (dinucleon). In a recent paper [1] by I. Obukhovsky, K. Itonaga, Georg Wagner, A. Buchmann and A. Faessler the decay of the d ′ resonance is calculated in a microscopic quark shell model. This resonance has been suggested to explain peculiarities in the forward angle cross section of the pionic double charge exchange (DCX) on nuclei at low energies [2]. Quantum numbers of the resonance, T (J P ) = 0(0 − ), imply that the decay d ′ →NNπ is dominated by s-waves between the outgoing particles. In general the strong interaction d ′ NNπ vertex contains two independent invariant amplitudes (see ref. [3]). However at low energies (like e.g. in the decay d ′ →NNπ) only one amplitude survives. * As calculated recently in ref. [1], the decay amplitude has to go as k 2 , where k is the 3-momentum of the outgoing pion in the d ′ rest frame. This result for the swave decay d ′ →NNπ is hardly possible to advocate since k 2 -behaviour would be a feature of a d-wave (or double p-wave within the 3-body system of outgoing particles) process. On the other hand in ref. [3] it has been already shown that the d ′ decay amplitude has to be proportional to the pion 4-momentum k µ since this amplitude has to satisfy Adler self-consistency condition. Thus the leading term in the amplitude is the one containing the time component k 0 of the 4-vector k µ . This result can be corroborated within a model similar to that used by the authors of ref. [1]. In ref. [1] it is assumed that the d ′ wave function corresponds to the s 5 p 1 configuration, while the outgoing 6q system, carrying quantum numbers J P = 0 + , T = 1, can be a mixture of a number of configurations. Let us take into account one of them, namely s 6 configuration. It will be easy to see, that the result holds true for the other configurations as well. In the model considered in ref. [1] the NNπ decay of the d ′ is due to the presence of the qqπ-vertex, giving rise to the transition (s 5 p 1 ) → (s 6 ) + π ,(1) followed by the subsequent fall-apart of the outgoing 6qsystem into NN pair in the 1 S 0 state. The qqπ-vertex can be written in the pseudo-scalar form,ψγ 5 τ ψ π, or preferably in the pseudo-vector form [4]: f qqπψ γ µ γ 5 τ ψ · ∂ µ π(2) Thus it is explicitly seen that a vertex of a pion emission in any elementary act is proportional to the pion 4-momentum. This does not exclude, of course, an extra dependence of the whole amplitude of a physics process on the pion momentum. The coupling constant f qqπ has the dimension of length, and can be fixed so as to reproduce the NNπ coupling constant. The probability amplitude of the transition (1) is proportional to the amplitude of the transition of a quark from p to s shell, q p → q s + π, accompanied by the pion emission. The latter equals f qqπ ψ s (r)γ µ γ 5 τ ψ p (r) k µ π e ikr dr ,(3) if the pion is described by a plane wave, and ψ s and ψ s are the wave functions (bispinors), describing quarks on s and p shells, respectively. Wave function of a fermion in a state with the definite total angular momentum j, its projection m, and parity P reads as: ψ jm (r) = f (r)Ω jlm (n) (−1) 1+l−l ′ 2 g(r)Ω jl ′ m (n) ,(4) where Ω jlm are spherical spinors depending on n = r/r, r = |r| (see e.g. [5]). l = j ± 1/2, and l ′ = 2j − l. For a given j the states with l = j − 1/2 and l = j + 1/2 have different parity. The spherical spinor Ω jl ′ m can be expressed through Ω jlm as: Ω jl ′ m = i l−l ′ (σn)Ω jlm .(5) For a quark on s 1/2 and p 1/2 shells the corresponding bispinors (4) look particularly simple: ψ s 1/2 = f 0 (r)ϕ g 0 (r)(σn)ϕ ,(6) and ψ p 1/2 = f 1 (r)(σn)χ g 1 (r)χ .(7) Here ϕ and χ are nonrelativistic 2-component spinors which do not depend on n. It is easy to see that the leading contribution in eq.(3) is the one proportional to the total energy of the pion, k 0 . † For ψ s,p given by eqs. (6) and (7) the expression (3) can be presented in a simple form: f qqπ (ϕ + χ) −k 0 (g * 0 f 1 + f * 0 g 1 )e ikr dr + (f * 0 f 1 + g * 0 g 1 )(kn)e ikr dr ,(8) in which integration over angles is trivial. Further details depend of course on the potential used to find solution for ψ s,p , and in particular on the Lorentz structure of the potential. For the qqπ vertex written in the pseudo-scalar form the amplitude of the transition (1) is proportional to (ϕ + χ) (g * 0 f 1 − f * 0 g 1 )e ikr dr .(9) Thus we conclude that for both types of the qqπ coupling (pseudo-vector and pseudo-scalar) the amplitude of the s-wave decay d ′ →NNπ does not vanish at k → 0, as distinct from the statement in ref. [1], according to which the amplitude is proportional to k 2 . The result of ref. [1] is therefore appears to be erroneous. The amplitude of the s-wave decay d ′ →NNπ is strongly modified by the NN final state interaction, as had been shown in our paper from 1993 (see ref. [3]), in particular if the decay process is due to a point-like interaction responsible for a qq-pair creation. ‡ This effect is known to give preference to small NN invariant masses, and might be a crucial for the experimental searches for † Another simple way to prove this statement is to consider 3-body vertex of the d ′ decay into two spinless "particles", pion and di-nucleon [NN]. Let π, Φ and Ψ be the operators, annihilating pion, [NN] and d ′ , respectively. (Here the sign vector stands for the isospin degrees of freedom). The only way to construct the Lorentz invariant vertex describing emission of a pion by the axial current is: the d ′ . Thus e.g. by applying a cut in the NN invariant masses one expects an enhancement of the signal-tobackground ratio. It is this procedure which has been applied to sense the d ′ contribution in the experiment on double pion production, pp → ppπ − π + , performed at ITEP (Moscow) [6]. In ref. [3] we considered the simplest case to take into account NN FSI: 1) point like d ′ NNπ vertex, and 2) Yamagichi wave function for the NN in the continuum. ( Φ + ↔ ∂ µ Ψ) (∂µ π Decay of the d ′ into ppπ − is also affected by Coulomb effects (see pion spectra in the decays d ′ → nnπ + and d ′ → ppπ − in ref. [3]), leading to some 10 -15 % difference in the decay rates d ′ → nnπ + and d ′ → ppπ − . The effects of the NN FSI in the decay d ′ → ppπ − can be taken into account (see e.g. [7]) by multiplying the differential probability of the decay by F pp (p) = F C (pa c )(10)× 1 + β + ip −a −1 s + 1 2 r 0 p 2 − 2 ac h(pa c ) − ipF C (pa c ) 2 where p ≡ |p|, and p is the 3-momentum of either proton in the pp c.o.m. Functions F C (x) and h(x) are given by: F C (x) = 2π x e 2π x − 1(11) and h(x) = 1 x 2 ∞ n=1 1 n(n 2 + x −2 ) − γ + ln(x) ,(12) where γ = 0.577... is the Euler constant. a s is the pp scattering length, r 0 -effective range, and a c is the Bohr radius for the pp subsystem; β ≈ 230 MeV is the parameter of the Yamaguchi potential. Numerical result obtained with eq.(10) is only 20-30% different from the "exact" one (see e.g. recent preprint [8]) obtained from the solution of the wave equation with "Coulomb + Yamaguchi potential" [9]. Thus we see that the dependence of the d ′ NNπ amplitude on the pion momentum, k µ , in combination with the FSI effects appears to be extremely important for any spectra in the d ′ decay. An extra factor k 4 in the amplitude squared would have led to even stronger enhancement of the high energy side of the pion spectrum (corresponding to small pp invariant masses). However this is not so, and k-dependence of the invariant amplitude squared is given by the product k 2 0 F pp (p), where k 0 is the total energy of the pion in the d ′ rest frame. Potential models similar to that used in ref. [1] should lead to a vanishingly small radiative decay rate d ′ → dγ. This decay mode is of special interest since γd is the only elementary process where d ′ should manifest itself as a Breit-Wigner pole in s-channel. The nature of suppression of the isoscalar E1 transition d ′ → dγ is the same as in nuclear physics where E1 transitions with ∆T = 0 between nuclei with N = Z are known to be forbidden (L.Radicati, 1952, see e.g. [7]). The same holds true not only for potential quark models, but also for any quark cluster model where (effective) mass of a quark cluster is proportional to the number of quarks in the cluster. An example, demonstrating this statement was considered in ref. [10], where it was assumed that the d ′ represents an orbitally excited state composed of a diquark ud (S = T = 0, colour =3) with angular momentum l = 1 relative to a four-quark cluster uudd (S = 1, T = 0, colour = 3). The corresponding E1 amplitude of the transition d ′ → dγ is then proportional to e 2 /m 2 − e 4 /m 4 , where e n and m n are electric charge and mass of the n-quark cluster, respectively. The E1 amplitude vanishes if m 4 = 2m 2 because e 4 = 2e 2 (= 2/3). For the configuration s 5 p 1 considered in ref. [1] sum of the E1 amplitudes corresponding to the two possible splittings u − uuddd and d − uuudd also vanishes if m 5 = 5m q . Let us note that in the potential model used in ref. [1] the d ′ mass appears to be too large, while in the flux-tube model with anomalously light diquark ud (S = T = 0) the mass of the 0 − isoscalar 6q-state is rather close to the one needed for the dibaryon explanation of the peculiarities in the pionic DCX. Simultaneously the existence of light diquarks makes ineffective the suppression of the E1 transitions discussed above. ). The amplitude of the decay d ′ →[NN]π is then proportional to the scalar product of the 4-vectors k(P [NN] + P d ′ ), where P d ′ = P [NN] + k. This scalar product equals M 2 d ′ − M 2 [NN] ; for a small energy release, ∆E = M d ′ − M [NN] (which is the case in the d ′ decay), it equals 2∆E · M [NN] ≈ 4MN k0 to a good accuracy since ∆E ≈ k0, while extra terms proportional to powers of k 2 are very small. ‡ The numerical result for the enhancement caused by the NN FSI is of course a model dependent. We would like to thank T. Ericson . I Obukhovsky, K Itonaga, Georg Wagner, A Buchmann, A Faessler, nucl-th/9708056Phys. Rev. C. 563295I.Obukhovsky, K.Itonaga, Georg Wagner, A.Buchmann and A.Faessler, nucl-th/9708056, Phys. Rev. C, 56, 3295 (1997). . R Bilger, H Clement, M Schepkin, Phys.Rev.Lett. 7142R.Bilger, H.Clement, M.Schepkin, Phys.Rev.Lett., 71, 42 (1993). . M Schepkin, O Zaboronsky, H Clement, Z. Phys. 345407M.Schepkin, O.Zaboronsky, H.Clement, Z. Phys. A345, 407 (1993). T E O Ericson, W Weise, Pions and Nuclei. OxfordClarendon PressT.E.O.Ericson and W.Weise, Pions and Nuclei. Claren- don Press, Oxford, 1988. V B Berestetskii, E M Lifshitz, L P Pitaevskii, Relativistic Quantum Theory. Moscow1V.B.Berestetskii, E.M.Lifshitz and L.P.Pitaevskii, Rela- tivistic Quantum Theory. Vol.1, "Nauka", Moscow, 1968. . L S Vorobyev, JETP Lett. 5975L.S. Vorobyev et al., JETP Lett., 59, 75 (1994) . L D Landau, E M Lifshitz, Quantum Mechanics, Nauka. L.D.Landau and E.M.Lifshitz, Quantum Mechanics. "Nauka", Moscow, 1963. . B L Druzhinin, A E Kudryavtsev, V E Tarasov, Preprint ITEPB.L.Druzhinin, A.E.Kudryavtsev, V.E.Tarasov. Preprint ITEP 41-96 (1996). . H Van Haeringen, Nucl.Phys. A. 253355H. van Haeringen, Nucl.Phys. A 253, 355 (1975). . R Bilger, Nucl.Phys., A. 596586R.Bilger et al., Nucl.Phys., A 596, 586 (1996).

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{'abstract': 'We comment on calculations of the width of the d ′ resonance within framework of quark shell models. * This is easy to see since in that case the NN-pair is in the 1 S0-state (J P = 0 + ), hence the 4-body vertex contains 3 spinless "particles": 0 − (π), 0 − (d ′ ), and 0 + (dinucleon).', 'arxivid': 'nucl-th/9712079', 'author': ['A Samsonov \nInstitute for Theoretical and Experimental Physics Moscow\n117218Russia\n', 'M Schepkin \nInstitute for Theoretical and Experimental Physics Moscow\n117218Russia\n'], 'authoraffiliation': ['Institute for Theoretical and Experimental Physics Moscow\n117218Russia', 'Institute for Theoretical and Experimental Physics Moscow\n117218Russia'], 'corpusid': 118267203, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4136, 'n_tokens_neox': 3617, 'n_words': 2243, 'pdfsha': 'c717756fbd7596a242c8e3fe49cd8b6b0a72ac39', 'pdfurls': ['https://export.arxiv.org/pdf/nucl-th/9712079v1.pdf'], 'title': ['Comment on "Pionic decay of a possible d ′ dibaryon and the short-range NN interaction"', 'Comment on "Pionic decay of a possible d ′ dibaryon and the short-range NN interaction"'], 'venue': []}

arxiv

Cosmic Censorship at Large D: Stability analysis in polarized AdS black branes (holes) Norihiro Iizuka Department of Physics Osaka University 560-0043ToyonakaOsakaJAPAN Akihiro Ishibashi Department of Physics Kindai University 577-8502HigashiOsakaJAPAN Kengo Maeda Faculty of Engineering Shibaura Institute of Technology 330-8570SaitamaJAPAN Cosmic Censorship at Large D: Stability analysis in polarized AdS black branes (holes) We test the cosmic censorship conjecture for a class of polarized AdS black branes (holes) in the Einstein-Maxwell theory at large number of dimensions D. We first derive a new set of effective equations describing the dynamics of the polarized black branes (holes) to leading order in the 1/D expansion. In the case of black branes, we construct 'mushroom-type' static solutions from the effective equations, where a spherical horizon is connected with an asymptotic planar horizon through a 'neck' which is locally black-string shape. We argue that this neck part (of black string) cannot be pinched off dynamically from the perspective of thermodynamical stability. In the case of black holes, we show that the equatorial plane on the spherical horizon cannot be sufficiently squashed unless the specific heat is positive. We also discuss that the solutions are stable against linear perturbation, agreeing with the thermodynamical argument. These results suggest that Gregory-Laflamme type instability does not occur at the neck, in favor of the cosmic censorship. I. INTRODUCTION In contrast to asymptotically flat spacetimes, there is a large variety of asymptotically Anti de Sitter (AdS) black hole solutions due to the warping factor. For instance, given an asymptotically AdS charged black hole, one can deform it by applying a non-uniform electric field and thereby construct a new black hole solution without destroying the asymptotic AdS structure, as demonstrated by [1]. This fact leads to the recent discovery of fourdimensional polarized AdS black holes in a dipolar electric field [2]. The solution was numerically constructed as a generalization of the Ernst solution [3]. Such polarized AdS black holes were extended into four-dimensional polarized AdS black brane solutions with a planar horizon, where chemical potential varies along one spatial direction [4]. In the latter black brane case, by locally applying sufficiently large enough and localized chemical potential, it is shown numerically that the configuration of the horizon looks like a mushroom and hence is called a "black mushroom" solution [4]. In this black mushroom solution, a neck connecting a localized spherical black hole and asymptotic planar horizon appears. The neck of the black mushroom solution is getting thiner as the temperature is lowered, and is expected to behave like a thin black string. This immediately leads us to the question of whether thin neck part is pinched off dynamically due to the Gregory-Laflamme instability [5]. If that is the case, a naked singularity would appear and the cosmic censorship [6] would be violated in the polarized black brane. There have also been a number of numerical study for the violation of the cosmic censorship in higher dimensions. The goal of this paper is to supply an analytic study for the cosmic censorship conjecture in the class of polarized AdS black branes (holes) in the Einstein-Maxwell theory, namely, mushroom-type and other type of black branes (holes). In order to attack this issue analytically, we adopt the large D effective theory approach developed in Ref. [7][8][9][10][11][12][13][14]. We first derive a tractable set of effective 1 + 1dimensional equations describing the dynamics of deformed charged AdS black branes (holes) in the leading order of the 1/D expansion. Then using these effective equations, as in Ref. [4], we obtain polarized black mushroom solutions with a neck connecting a localized spherical horizon and an asymptotic planar horizon. Near the neck, the horizon geometry locally behaves as a black string, and it is polarized by strong electric field along the neck. Applying the claim of the Gubser-Mitra conjecture [15,16], which has now been proven for some cases [17], to the local black string, we argue that the neck should be locally stable against physically reasonable perturbations, conforming to the thermodynamic stability. We also find polarized AdS black hole solutions in which the spherical horizon is squashed around the equatorial surface. One may expect that there could be a black "dumbbell" solution whose horizon looks like a dumbbell having two spherical horizons connected by a portion of a thin black string. We show however that the equatorial plane on the spherical horizon cannot be sufficiently squashed while keeping its local specific heat negative to lead an instability. This implies that there is no black dumbbell solution, where two spherical horizons are connected through a thermodynamically unstable thin black-string-shape neck, and therefore the neck cannot be pinched off dynamically due to the Gregory-Laflamme instability [5]. We also discuss that the solutions are stable against linear perturbations, being consistent with the thermodynamical argument. The organization of this paper is as follows; In the next section, we first derive the 1 + 1-dimensional effective equations by expanding the Einstein equations in the inverse power of D. In sections III and IV, we construct the black mushroom solutions and test the cosmic censorship conjecture in them. In section V, we repeat the analysis in the polarized AdS black hole solutions. Section VI is devoted to summary and discussions. II. EFFECTIVE EQUATIONS IN CHARGED ADS BLACK BRANES We start with the following D-dimensional Einstein-Maxwell equations with a negative cosmological constant R µν − 1 2 R g µν + Λ g µν = 1 2 F µρ F ν ρ − 1 4 F 2 g µν , Λ = − (D − 1)(D − 2) 2L 2 , 1 √ −g ∂ µ √ −gF µν = 0, (1) where L is the AdS curvature length and F µν = ∂ µ A ν − ∂ ν A µ . We make the following ansatz for the metric and the gauge field as ds 2 = −Adt 2 + 2u t dtdr − 2C z dtdz + G zz dz 2 + r 2 z 2 L 2 dΩ 2 n−2 , A = A t dt + A z dz,(2) where dΩ 2 n−2 is the metric of unit sphere with n = D − 1. Note that we do not rescale z-coordinate, as done in [18] since z is not the direction of Killing symmetry of the background geometry. For simplicity, we assume that at large D, the gauge field A µ behaves as A t = O(n − 1 2 ) , A z = O(n − 3 2 ).(3) Then, the electric charge of the black brane can be dealt with a test charge so that it does not affect the metric at leading order in the expansion in the inverse of n [19]. This leads to the metric expansion as follows A(r, t, z) = r 2 L 2 1 − m(t, z) r n + r 2 nL 2 Q(t, z) r 2n−2 + a 1 (r, t, z) + O(n −2 ), C z (r, t, z) = p(t, z) nr n + O(n −2 ), u t (r, t, z) = 1 + β t (r, t, z) n + O(n −2 ), G zz = r 2 L 2 + H(r, t, z) n + O(n −2 ),(4) where the horizon is determined by A = 0. It is convenient to use the formula (A1) to expand the Einstein Eqs. (1) order by order as a series in 1/n. Then, we find that the metric given above already solves the Einstein equations at leading order. We would like to derive the effective equations for the variables, m(t, z), p(t, z), · · · . For that purpose, let us define R as R = (r/r 0 ) n , where r 0 is a fiducial horizon size. Hereafter, without loss of generality, we set r 0 = 1. We take the large D (or equivalently large n) limit in such a way that R = fixed, i.e., r → 1, n → ∞ with r n = fixed. Note that this limit forces us to set the finite power of r to be 1 in the leading order of large n expansion. Within this limit, we evaluate the Einstein & Maxwell equations in the 1/n expansion at the horizon and derive the effective equation for the variables. Note that the horizon is determined as R = r n = m(t, z) in the leading order of large n expansion. With this double scaling limit in our mind, as for the gauge field we make the ansatz for A t as A t (r, t, z) = 2 n P (t, z) − q(t, z) r n .(5) Here, P plays a role of the chemical potential on the AdS boundary, r = ∞ and ∂ z P corresponds to the external electric field along z-direction. Then the t-component of Maxwell equation at the leading order is automatically satisfied. The function q will correspond to the electric charge as we will see below. Substituting Eqs. (4) and (5) into the z-component of the Maxwell equations in Eqs. (1), and by evaluating its leading order at the horizon in the 1/n expansion, we obtain − 1 √ 2 ∂A ∂r ∂A z ∂r − √ n pq r 2n + √ n ∂P ∂z = 0 .(6) This gives a solution for A z as A z (r, t, z) = 2 n L 2 n ∂P (t, z) ∂z ln(r n ) + p(t, z)q(t, z) m(t, z)r n . At next to leading order in 1/n expansion, from the rr, rz, zz-component of the Einstein equations, we find that β t and H can be set to zero: β t = H = 0 .(8) From the several components of the Einstein equations (1), we obtain Q = L 2 q 2 (t, z).(9) Then, rt-component of the Einstein equations (1) reduces to ∂a 1 ∂r + 1 n ∂ 2 a 1 ∂r 2 = nL 4 p Rz ,(10) and its solution is given by a 1 = − L 4 p ln R zR .(11) From the tt and tz-component of the Einstein equation, the evolution equations for m and p are obtained as ∂m ∂t + L 2 z p − L 2 z ∂m ∂z = 0 ,(12)∂p ∂t + L 2 mz p 2 − 2q ∂P ∂z + 1 L 2 ∂m ∂z − L 2 ∂ ∂z p z = 0 ,(13) on the horizon R = r n = m. Finally the r-component of the Maxwell equations yields the evolution equation for q: ∂q ∂t − L 2 z ∂q ∂z + L 2 m z ∂P ∂z + L 2 p zm q = 0 .(14) These three equations (12), (13), and (14) are the 1 + 1-dimensional effective equations for the charged black brane. As far as we are aware, these are completely new equations. III. BLACK MUSHROOM SOLUTIONS In this section, we derive a black mushroom solution from our effective equations (12), (13), and (14). The topology of the horizon at t = constant surface is R n−1 and the metric becomes ds 2 fixed t = r 2 (z) L 2 (dz 2 + z 2 dΩ 2 n−2 ) ,(15) where z is the radial coordinate on the horizon and r(z) is the location. In the black mushroom solution there is a neck, as found in Ref. [4]. By denoting the area of the z = constant. surface as S(z), the minimum condition for the existence of a neck (located at z = z 0 (> 0)) can be geometrically defined as ∂S(z) ∂z z=z0 = 0, ∂ 2 S(z) ∂z 2 z=z0 > 0, m(z 0 ) < m| (z→∞) , S(z) := C 0 L n R(z)z n = C 0 L n m(z)z n ,(16) where C 0 is the surface area of unit n − 2-dimensional sphere. The third condition on the first line implies that there should be a concavity for the horizon radius R H in the range 0 < z < ∞. When m = constant., the solution becomes a plane-symmetric charged black brane solution with no neck. As is shown below, the neck can be created only by the localized chemical potential, P (z), or it would be more correct to say that the neck can be created by the strong external electric field, ∂ z P . Hereafter, we set L = 1 for simplicity. Making the ansatz m = m(z), p = p(z), q = q(z), P = P (z) , (17) for the static solution, we reduce Eqs. (12), (14), and (13) to m = p , q = pq m + mP , p z = m + p 2 mz − 2qP ,(18) where the prime means the derivative with respect to z. The horizon is determined by A = 0 as R H = m − a 1 m n − Q nm + O 1 n 2 .(19) The surface gravity κ is given by κ := 1 2 ∂A ∂r R=R H = 1 2 n + ln m − p zm − q 2 m 2 + O 1 n .(20) In order to derive this, one has to be careful to the fact that r = 1 + 1 n ln m + O 1 n 2 .(21) It is easily checked from Eqs. (18) that κ is constant along the horizon by showing that ∂κ ∂z = O 1 n .(22) Note that Eq. (21) indicates that deformation from the homogeneous black brane solution is O(1/n) in our black mushroom solution, while it is O(1) in the black mushroom solution numerically constructed in Ref. [4]. Nevertheless, as we will show, our black mushroom solution has a neck defined in Eq. (16). Now, we will construct a black mushroom solution which is deformed by the external electric field P . Substitution of p = m into the second equation in (18) yields q m − m m 2 q = P .(23) This can be integrated as q m = P + C,(24) where C is an integral constant. As an asymptotic boundary condition at infinity, z → ∞ on the horizon, we will impose that the black brane solution asymptotically approaches a uniformly charged black brane solution. This is equivalent to impose the following conditions, lim z→∞ P = q 0 m 0 , lim z→∞ m = m 0 (> 0), lim z→∞ q = q 0 (> 0) .(25) The first condition implies that there is no external electric field at infinity. The boundary condition determines the integral constant C as C = 0 .(26) This is consistent with the regularity condition on the horizon, A t = 0 in Eq. (5) (see, for example, Ref. [20]). Introducing new variables M and ξ as m = m 0 e M , q = ξ e M ,(27) we obtain the equation of motion for M from the third equation in (18) as M z − 1 + 1 z 2 M = − 2ξξ m 2 0 ,(28) where we used p = m and P = q m = ξ m 0(29) from Eqs. (24) and (26). Taking into account that M → 0, ξ → q 0 at z = ∞, Eq. (28) is integrated as M z − M = q 2 0 − ξ 2 m 2 0 .(30) If there is a neck which is satisfying the conditions (16) and (25), M must have a minimum M m at z = z m (0 < z m < ∞) [24]. The lower bound of the minimum M m is determined by Eq. (30) as M m ≥ − q 2 0 m 2 0 = −P 2 | z→∞ (31) where the equality is satisfied only when ξ(z m ) = 0. This implies that the minimal horizon radius R H around the neck is determined by the asymptotic value of the chemical potential P given by Eq. (29). There are infinite degrees of freedom to choose a function M satisfying Eq. (30), the neck condition (16), and the lower bound (31). Once we choose a function M satisfying these conditions (16) M = −B z 4 a 4 e − (z−a) 2 b 2 , B > 0(32) to satisfy the asymptotic boundary condition (25), where a, b, and B are some positive constants. Here, we set M sufficiently rapidly approaches zero at the origin of spherical symmetry, z = 0 to avoid a singularity. The minimum takes at z m = a + √ a 2 + 8b 2 2 a + 2b 2 a(33)B = B 0 q 2 0 m 2 0 1 + 2b 2 a 2 −2 B 0 q 2 0 m 2 0 ,(34) where B 0 is a positive constant satisfying B 0 < 1. Since the horizon is determined by Eq. (19), the cross-sectional area S defined in Eq. (16) becomes S = S(z) n z + M .(35) Note that the expansion (4) therefore setting a = na 0 (a 0 > 0), one obtains The cross-sectional area S monotonically increases before reaching the maximum, and then decreases toward the minimum. This implies that the horizon behaves as a spherical black hole in the region 0 ≤ z < a, and it is connected to an asymptotic planar horizon z a through a neck around z = a. To satisfy the condition M = O(1), a must increase as n increases. So, the position of the neck goes away from the center, z = 0, and the neck connects a large spherical black hole with an asymptotic planar horizon, as n increases. Note that S /S increases with the magnitude O(n) before reaching z = O(n), and then decrease with the magnitude O(1) at z = O(n). This implies that the mushroom shape is extremely flattened. M | z=a−b − 2B b e −1 + O 1 n .(36) As shown in Fig. 2, a plateau region appears for each value of b, corresponding to the neck in the black mushroom solution. This region spreads as b increases, and the spherical black hole portion tends to disappear. These facts imply that the black mushroom solution locally approaches a black string solution with translational symmetry along z as b increases. As shown in Fig. 3, the chemical potential P possesses a precipitous valley near the plateau region. As the external electric field E is given by P , the black string portion is supported by the strong electric field. IV. STABILITY ANALYSIS As seen in the previous section, we showed that there is a black mushroom solution in which a small spherical black hole is connected to the asymptotic planar black brane through a neck that resembles a black string solution. In this section, we argue the stability of the black mushroom solution from the perspective of thermodynamics, as well as that of the dynamical stability with respect to linear perturbations. A. Stability analysis: thermodynamical argument Gubser and Mitra have conjectured that the Gregory-Laflamme instability for black branes with a noncompact translational symmetry occurs if and only if they are locally thermodynamically unstable [15,16]. This claim was proven [17]. This implies that if a blackstring-shape neck has a translationally invariant portion larger than the threshold wavelength λ c beyond which any longer wavelength perturbations are unstable, it tends to break up under the evolution. As shown in Ref. [18,21,22], higher dimensional black string solution with translational symmetry suffers from a Gregory-Laflamme instability for short wavelength perturbations. The threshold wavelength λ c is approximately given by λ c ∼ S(z m ) 1/(n−2) ∼ z m √ n .(38) Here, to derive the second approximation, we used the fact that the surface area of unit n−2-dimensional sphere C 0 is given by C 0 ∼ n −n/2 [7]. In the z m ∼ n = 30, b = 5.3 case in the previous section, λ c ∼ 5.4, which is comparable to the length of the neck, ∼ 6 (recall that r 1), as seen in Fig. 2. So, one would expect that the neck with a translationally invariant portion larger than λ c would be unstable against Gregory-Laflamme instability unless it is thermodynamically stable. The temperature T for the black mushroom solution corresponds to the surface gravity κ in Eq. (20). As z ∼ z m ∼ n 1 in the neck, the third term proportional to p becomes irrelevant. Then, κ is determined by the local charge q and mass parameters m on the neck. Since κ increases as the mass increases for a fixed charge, it should be thermodynamically stable, implying that the neck should also be dynamically stable, according to the conjecture. Note that the fact that the existence of the neck forces the specific heat positive is independent of the form of M . Given M , Eq. (35) is generic and in order to have a neck part, we have to have z m = O(n), since M = O(1). Then, from Eq. (20), terms with p/zm becomes O( 1 n ) and we always have a positive specific heat. These suggests that in the large D, the neck part of the mushroom solution is always stable dynamically. B. Stability analysis: linear perturbation We consider linear perturbation of the black mushroom solution satisfying Eqs. (18). Here, we address the issue whether the linear perturbation has an unstable mode without time dependent external force P . So, we impose the condition δP (t, z) = 0. (39) Linearizing the evolution equations (12), (13), and (14), we obtain the equations for perturbation aṡ δm + δp z − δm z = 0 ,(40)δp + 2p mz δp − p 2 m 2 z δm − 2P δq + δm − δp z = 0 ,(41)δq − δq z + δm z P + q zm δp + p zm δq − pq zm 2 δm = 0 ,(42) where a dot and prime denote the derivative with respect to t and z, respectively. Plugging δp = δm − zδṁ obtained from Eq. (40) into Eqs. (41) and (42), we have zδm − 2z 2δ m − 1 + z 2 + 2z p m δm + z 3δ m + 2z 2 p mδ m + z p 2 m 2 δm + 2z 2 P δq = 0 , (43) δq − zδq − p m δq = q m (δm − zδ m) + P − pq m 2 δm .(44) Note that when p = 0 = P , the above set of equations reduce to the corresponding perturbation equations for the large D limit of the Schwarzschild-AdS black brane solution, which should be stable as it has a positive specific heat. It is immediate to see from Eq. (43) that near the center z = 0, the general solution of δm behaves in a regular manner as δm C 1 + C 2 z 2 ,(45) with C 1 , C 2 being some constants independent of the values of p and P . Choosing C 1 and C 2 corresponds to specifying a particular boundary condition at the center: for instance, C 1 = 0 corresponds to the Dirichlet boundary condition. Actually, which choice of the boundary condition we would take is not relevant to the rest of our argument, and thus we leave these constants unspecified. It also turns out that Eqs. (43) and (44) form a parabolic system. To see that, let us change the coordinates (t, z) into (u := −t, v := 2t + z 2 ) so that the above two equations are expressed as ∂ 2 u − 2∂ v − 2 z p m ∂ u + 1 z 2 p 2 m 2 δm + 4z∂ v P δq = 0 ,(46)z∂ u − p m δq = q m z∂ u − p m δm + 2z∂ v P δm ,(47) with z viewed as the function of (u, v). Recalling the conditions (25) at z → ∞ and also noting p = m → 0, we find Eq. (47) to become ∂ u δq q 0 m 0 ∂ u δm ,(48) and thus we have δq (q 0 /m 0 )δm. Eq. (46) then asymptotically takes the form of thermal diffusion equation: (∂ 2 u − 2∂ v )δm 0 .(49) We naturally impose the following regularity conditions at large z: lim z→∞ δm = 0 .(50) Provided the separation of variable, the above equation can be immediately solved as δm = λ a(λ)e −λ 2 v cos( √ 2λu + θ λ ) = λ a(λ)e −λ 2 (2t+z 2 ) cos( √ 2λt − θ λ ) .(51) Here λ must be either a real or a pure imaginary number in the following reason. If λ is a complex number, then the above solution could contain an unstable mode. However if such an unstable mode is allowed, it would imply that the Schwarzschild-AdS black brane (p = 0 = P ) itself would admit an unstable perturbation as we have the same expression (51) for the perturbations and the same boundary conditions (45) and (50) for the case of the Schwarzschild-AdS black brane, which is however thought to be stable from the thermodynamic perspective. Now suppose λ is pure imaginary. Then δm is non-normalizable on t = const. surface, hence is not a physically acceptable perturbation. Therefore, λ must be a real number, for which the perturbation solution (51) exhibits no instability. It is thus plausible to argue that our black mushroom should be stable under type of the perturbations considered above. This argument is also consistent with the speculation that any black string portion of the neck should be stable according to the Gubser-Mitra conjecture [15,16], as the portion has always positive specific heat. To fully justify this stability argument, we however need a thorough study of the dynamical perturbations, which is the near future task. V. SPHERICAL BLACK HOLE CASE In this section, we pay close attention to the polarized AdS black hole with a spherical horizon. If such an AdS black hole is highly squashed by external electric field, "dumbbell" type black hole with a neck connecting two spheres appears (see Fig. 4.). Then, as discussed in the previous sections, Gregory-Laflamme instability would occur unless the black string portion becomes thermodynamically stable. One might ask whether such a polarized AdS black hole with a spherical horizon is unstable or not, because small AdS black holes are thermodynamically unstable. In this section, we investigate properties of such a polarized AdS black hole at large D by analyzing 1 + 1-dimensional effective equations as follows. A. Derivation of effective equations We make the metric ansatz as ds 2 = −Adt 2 + 2u t dtdr − 2C z dtdz + G zz dz 2 + r 2 sin 2 z dΩ 2 n−2 ,(52) where z is the angular coordinate in the range 0 ≤ z ≤ π. As in the brane case, under the condition (3), the metric is expanded as A(r, t, z) = r 2 L 2 1 − m(t, z) r n + 1 + 1 n Q(t, z) r 2n−2 + a 1 (r, t, z) + O(n −2 ), C z (r, t, z) = p(t, z) nr n + O(n −2 ), u t (r, t, z) = 1 + β t (r, t, z) n + O(n −2 ), G zz = r 2 + H(r, t, z) n + O(n −2 ), Q = L 2 q 2 (t, z) .(53) It is easily checked that the metric (53) and the gauge fields (5), (7) are the leading order solutions for the spherical case. At next to leading order, we can set β t = H = 0 as in the brane case, and we find the solution for a 1 as a 1 − L 2 p cos z ln R R sin z .(54) Substituting Eq. (54) into the Einstein equations (1), we obtain evolution equations for q, m, and p as ∂q ∂t − cos z sin z ∂q ∂z − (R H L 2 − m) cos z sin z ∂P ∂z + (L 2 + 1) cos z m sin z pq = 0 ,(55)∂m ∂t + (1 + L 2 )p − ∂m ∂z cos z sin z = 0 ,(56)∂p ∂t + (1 + L 2 )p 2 cos z m sin z − 2q ∂P ∂z + 1 L 2 ∂m ∂z − 2p − ∂ ∂z p cos z sin z = 0 ,(57) where R H is the value of R at the horizon determined by A = 0. B. Properties of static solutions Here, we investigate the properties of the static spherical black hole solutions. Assuming that q, m, and p depend on the variable z only, the static equations are reduced from Eqs. (55), (56), and (57) as m = (1 + L 2 )p , q = m 1 + L 2 P + 1 + L 2 m pq , (p cot z) = m L 2 + (1 + L 2 )p 2 m cot z − 2qP − 2p . (58) The value of R at the horizon, R H , is determined by A = 0 in Eq. (53) as R H = m 1 + L 2 − 1 n − 2mL 2 (1 + L 2 ) 2 ln m 1 + L 2 + L 2 q 2 m + ma 1 (1 + L 2 ) 2 .(59) Up to O(1), the temperature T is evaluated at the value of surface gravity on the horizon, κ = 1 2 A ,r = (1 + L 2 )n 2L 2 + 1 − L 2 2L 2 ln m 1 + L 2 − (1 + L 2 ) 2 q 2 2m 2 − 1 + 1 2L 2 − (1 + L 2 )p cos z 2m sin z .(60) It is easily checked that κ is constant along the horizon by showing κ ,z = 0 by the static equations (58), as in the black mushroom case. Now, we consider the "dumbbell" type static spherical black hole solutions in which the equatorial plane is squashed by the external electric field. For simplicity, we assume that the solution is symmetric with respect to the equatorial plane. This means that p| z= π 2 = m | z= π 2 = 0.(61) We also assume that M sufficiently quickly approaches zero at the north pole (also south pole) as M = cz 2+ , > 0,(62) as in the black mushroom case. The total mass and the charge M and Q are determined by the mass and charge density m and q as M ∼ π 0 m(z) sin n−2 z dz, Q ∼ π 0 q(z) sin n−2 z dz. This implies that M and Q are dominated by the values of m| z=π/2 := m e and q| z=π/2 := q e , respectively in the large n limit since sin n−2 z becomes zero except z = π/2 in the limit. At the equatorial plane, by Eq. (61), κ is rewritten by m e and q e as κ = (1 + L 2 )n 2L 2 + 1 − L 2 2L 2 ln m e 1 + L 2 − (1 + L 2 ) 2 q 2 e 2m 2 e − 1 + 1 2L 2 .(64) Therefore, the condition for the negative specific heat becomes L > 1 and 2(1 + L 2 ) 2 q 2 e < L 2 − 1 L 2 m 2 e .(65) Let us define M and ξ by Eq. (27). Here, m 0 and q 0 are defined by the values of north pole, respectively: m 0 := m| z=0 , q 0 := q| z=0 .(66) Eliminating p from Eqs. (58) and integrating the second equation of (58), we find q 0 = P m 0 1 + L 2 ,(67) where we used the regularity condition A t = 0 on the horizon. Eliminating P from the third equation in (58) by Eq. (67), we obtain M cot z + 1 − 1 L 2 − 1 sin 2 z M = − 2(1 + L 2 ) 2 ξξ m 2 0 .(68) The first integration yields M cot z + 1 − 1 L 2 M = (1 + L 2 ) 2 m 2 0 (q 2 0 − ξ 2 ) = (1 + L 2 ) 2 q 2 0 m 2 0 − q 2 m 2 ,(69) where we used ξ 2 /m 2 0 = q 2 /m 2 and the boundary condition (62). Therefore, we obtain M | z= π 2 = L 2 (1 + L 2 ) 2 (L 2 − 1) q 2 0 m 2 0 − q 2 e m 2 e > − L 2 (1 + L 2 ) 2 (L 2 − 1) q 2 e m 2 e > − 1 2L 2 > − 1 2(70) under the condition (65). This is the lower bound of M at the equatorial plane, which means that the equatorial plane cannot be highly squashed, keeping the negative specific heat. In other words, highly squashed equatorial black dumbbell is possible to construct but its specific heat is always positive. According to the Gubser-Mitra conjecture [15], this indicates that Gregory-Laflamme instability does not occur in the "dumbbell" type static spherical black hole solutions. C. Linear perturbations We consider linear perturbation of the static black hole solutions satisfying Eqs. (58). As in the black brane case, we assume that the perturbation of P is zero. Then, linearizing the evolution equations (55), (56), and (57), we obtaiṅ δq − δq − P δm 1 + L 2 + (1 + L 2 )pq m 2 δm cot z + 1 + L 2 m (qδp + pδq) cot z = 0 ,(71)δm + {(1 + L 2 )δp − δm } cot z = 0 ,(72) δp + (1 + L 2 ) 2pδp m − p 2 m 2 δm cot z − 2P δq + δm L 2 − 2δp − (δp cot z) = 0 .(73) From the regularity on the equatorial plane z = π/2, the following boundary conditions are derived: δq| z= π 2 =δ m| z= π 2 = 0 .(74) Note that this is consistent with the mass and charge conservation law, i. e. , M and Q defined in Eq. (63) are constant during the time evolution in the large n limit. Eliminating δp by using Eq. (72), we obtain two equations for δm and δq as follows: We can make the same plausible argument for our black dumbbell, agreeing with the thermodynamical argument that the Gregory-Laflamme type instability does not occur in the squashed black holes. δq − cot zδq + (1 + L 2 ) p m cot zδq − cot z (1 + L 2 ) pq m 2 − P 1 + L 2 δm + (1 + L 2 ) q m cot zδm − q mδ m = 0 ,(75) VI. SUMMARY AND DISCUSSIONS In this paper, we have first derived a new set of effective equations (12) - (14), describing the dynamics of the polarized black branes (holes) to leading order in the 1/D expansion and using these, we have tested cosmic censorship conjecture in polarized AdS black brane (hole) solutions at large D dimensions. As expected in the fourdimensional analysis [4], we found a black mushroom solution where a black hole is connected with an asymptotic planar black brane through a black-string-shape neck under the localized chemical potential. Contrary to our first naive expectation, the black-string-shape neck part is thermodynamically stable. This indicates that the localized string cannot be pinched off dynamically according to the Gubser-Mitra conjecture [15,16]. We have extended the analysis to the AdS black hole case and found that highly squashed black hole is also dynamically and thermodynamically stable. These facts imply that the cosmic censorship is not violated in such polarized AdS black brane (hole) solutions at large D by the Gregory-Laflamme instability [5]. For simplicity, we have treated the gauge field as a probe approximation in the sense that the horizon geometry at leading order is neutral black brane (hole) solutions. In other words, the horizon is embedded at the fixed bulk radial coordinate in AdS spacetime in the leading order. To take into account the gauge field at leading order, we must construct a charged polarized black brane (hole) solutions at leading order so that the horizon is located over different radial region. It is interesting to test the cosmic censorship conjecture in that case. This will be investigated in the near future. and (31), ξ and P are determined from Eqs. (30) and (29) [25]. For example, let us choose a Gaussian like function M : FIG. 1 : 1The plot of M for various values of b = 2.8 (blue, solid), 3.3 (dashed green), 3.7 (dotted red), and 5.3 (dotdashed, brown) in the case a = n = 30, m0 = 1, q0 = 3, and B0 = 0.98 in the limit a b. To satisfy the lower bound (31), we choose the parameter B so that FIG. 2 : 2The plot of S (normalized by S(n)) for the same values of b asFig. 1in the case a = n = 30, m0 = 1, q0 = 3, and B0 = 0.98 FIG. 3 :Fig. 1 . 31The plot of Γ = P 2 for the same values of b as Fig. 1 in the case a = n = 30, m0 = 1, q0 = 3, and B0 = 0.98 This implies that S must have a minimum around z and Fig. 2. show the plots of M and S near the minimum for various values of b in the case a = n = 30, m 0 = 1, q 0 = 3, and B 0 = 0.98. FIG. 4 : 4The construction of the "dumbbell" like black hole solution from the spherically symmetric black hole − (1 + L 2 ) 2 cot z p 2 m 2 δm − 2(1 + L 2 )P δq = 0 . (76)The stability analysis from now on parallels what we have done below Eqs. (43) and (44) for our black branes. Acknowledgments We would like to thank Kentaro Tanabe and Norihiro Tanahashi for discussions in the early stage of the project. 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Tanabe, JHEP 1407 (2014) 113. . R Emparan, R Suzuki, K Tanabe, JHEP. 150485R. Emparan, R. Suzuki and K. Tanabe, JHEP 1504 (2015) 085. . R Emparan, T Shiromizu, R Suzuki, K Tanabe, T Tanaka, JHEP. 1506159R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe and T. Tanaka, JHEP 1506 (2015) 159. . S Bhattacharyya, A De, S Minwalla, R Mohan, A Saha, JHEP. 160476S. Bhattacharyya, A. De, S. Minwalla, R. Mohan and A. Saha, JHEP 1604 (2016) 076. Instability of charged black holes in anti-de Sitter space. S S Gubser, I Mitra, arXiv:hep-th/0009126S. S. Gubser and I. Mitra, "Instability of charged black holes in anti-de Sitter space", [arXiv:hep-th/0009126] . S S Gubser, I Mitra, JHEP. 010818S. S. Gubser and I. Mitra, JHEP 0108 (2001) 018. . S Hollands, R M Wald, Commun.Math.Phys. 321S. Hollands and R. M. Wald, Commun.Math.Phys. 321, 629-680 (2013). . R Emparan, R Suzuki, K Tanabe, Phys. Rev. Lett. 11591102R. Emparan, R. Suzuki and K. Tanabe, Phys. Rev. Lett 115, 091102 (2015). . R Emparan, K Izumi, R Luna, R Suzuki, K Tanabe, JHEP. 1606117R. Emparan, K. Izumi, R. Luna, R. Suzuki, and K. Tan- abe, JHEP 1606 (2016) 117. . S S Gubser, Phys. Rev. 7865034S. S. Gubser, Phys. Rev. D78, 065034 (2008). . B Kol, E Sorkin, Class. Quant. Grav. 214793B. Kol and E. Sorkin, Class. Quant. Grav. 21, 4793 (2004). . V Asnin, D Gorbonos, S Hadar, B Kol, M Levi, U Miyamoto, Class. Quant. Grav. 245527V. Asnin, D. Gorbonos, S. Hadar, B. Kol, M. Levi, and U. Miyamoto, Class. Quant. Grav. 24, 5527 (2007). Unpublished Notes. K Tanabe, K. Tanabe, Unpublished Notes. M would be monotonically decreasing function satisfying M > 0 for z ∈ [0, ∞). This contradicts m(z0) < m0. If there was no minimum, M would be monotonically decreasing function satisfying M > 0 for z ∈ [0, ∞). This contradicts m(z0) < m0.

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{'abstract': "We test the cosmic censorship conjecture for a class of polarized AdS black branes (holes) in the Einstein-Maxwell theory at large number of dimensions D. We first derive a new set of effective equations describing the dynamics of the polarized black branes (holes) to leading order in the 1/D expansion. In the case of black branes, we construct 'mushroom-type' static solutions from the effective equations, where a spherical horizon is connected with an asymptotic planar horizon through a 'neck' which is locally black-string shape. We argue that this neck part (of black string) cannot be pinched off dynamically from the perspective of thermodynamical stability. In the case of black holes, we show that the equatorial plane on the spherical horizon cannot be sufficiently squashed unless the specific heat is positive. We also discuss that the solutions are stable against linear perturbation, agreeing with the thermodynamical argument. These results suggest that Gregory-Laflamme type instability does not occur at the neck, in favor of the cosmic censorship.", 'arxivid': '1801.07268', 'author': ['Norihiro Iizuka \nDepartment of Physics\nOsaka University\n560-0043ToyonakaOsakaJAPAN\n', 'Akihiro Ishibashi \nDepartment of Physics\nKindai University\n577-8502HigashiOsakaJAPAN\n', 'Kengo Maeda \nFaculty of Engineering\nShibaura Institute of Technology\n330-8570SaitamaJAPAN\n'], 'authoraffiliation': ['Department of Physics\nOsaka University\n560-0043ToyonakaOsakaJAPAN', 'Department of Physics\nKindai University\n577-8502HigashiOsakaJAPAN', 'Faculty of Engineering\nShibaura Institute of Technology\n330-8570SaitamaJAPAN'], 'corpusid': 119222367, 'doi': '10.1007/jhep03(2018)177', 'github_urls': [], 'n_tokens_mistral': 12001, 'n_tokens_neox': 10320, 'n_words': 6724, 'pdfsha': '69f7d9b463c8694ed5c4d3b75a1b93088f83e99b', 'pdfurls': ['https://arxiv.org/pdf/1801.07268v1.pdf'], 'title': ['Cosmic Censorship at Large D: Stability analysis in polarized AdS black branes (holes)', 'Cosmic Censorship at Large D: Stability analysis in polarized AdS black branes (holes)'], 'venue': []}

arxiv

arXiv:cond-mat/9808029v2 4 Aug 1998 Two-level Hamiltonian of a superconducting quantum point contact (August 4, 1998) D A Ivanov L.D.Landau Institute for Theoretical Physics 117940MoscowRussia 12-127 M.I.T. Cambridge02139MAUSA M V Feigel&apos;man L.D.Landau Institute for Theoretical Physics 117940MoscowRussia arXiv:cond-mat/9808029v2 4 Aug 1998 Two-level Hamiltonian of a superconducting quantum point contact (August 4, 1998) In a superconducting quantum point contact, dynamics of the superconducting phase is coupled to the transitions between the subgap states. We compute this coupling and derive the two-level Hamiltonian of the contact.One of the key features of superconducting quantum point contacts (SQPC) [1-9] is the existence of subgap states (so-called Andreev states) whose energies depend on the phase difference across the contact[10,11]:where ∆ is the superconducting gap, t is the normal transparency of the contact. Each transversal mode propagating through the contact generates two such states (with opposite energies). Thus, at energy scales less than ∆, it is often convenient to describe the contact as a set of two-level systems. Further we assume for simplicity that we have only a single propagating mode (and therefore only two subgap levels). However, for describing dynamics of the contact at nonconstant α, the information on the energy spectrum (1) at each value of α is not sufficient. Mathematically speaking, we need a connection on the bundle of Hilbert spaces over the circle of possible values of α, and, more specifically, the projection of this connection onto the two-level subspace. Technically, it amounts to computing the "dynamic" matrix elementwhere |0 and |1 are the two subgap states at a given value of α. This quantity defines coupling between the dynamics of the superconducting phase and transitions in the two-dimensional subspace of subgap states.To illustrate this point, consider a single SQPC connected to a grain of finite capacity C. This system has been studied previously in the adiabatic approximation [12,13] and in the two-level approximation[9,14]. The two-level Hamiltonian for this system may be written aswhere N is the dimensionless potential of the grain and H 0 (α) is a 2 × 2 matrix. At each α, the eigenvalues of H 0 (α) must be given by (1). However, this does not fix the whole dependence on α. In our earlier work [9] we suggestedAnother candidate for H 0 (α) might beObviously, the latter choice of H 0 (α) would lead to a physically different behaviour of the system(3), although H # 0 (α) has the same eigenvalues as H 0 (α). The choice (4) of H 0 (α) appears natural from the point of view of perturbation theory in backscattering. Moreover, we claim that expression (4) is exact for any value of r (within the model described below). We omitted the derivation of (4) in our paper [9] and now fill this gap in the present note.We shall describe the one-channel SQPC by the one-dimensional Hamiltonian: In a superconducting quantum point contact, dynamics of the superconducting phase is coupled to the transitions between the subgap states. We compute this coupling and derive the two-level Hamiltonian of the contact. One of the key features of superconducting quantum point contacts (SQPC) [1][2][3][4][5][6][7][8][9] is the existence of subgap states (so-called Andreev states) whose energies depend on the phase difference across the contact [10,11]: E(α) = ±∆ 1 − t sin 2 α 2 ,(1) where ∆ is the superconducting gap, t is the normal transparency of the contact. Each transversal mode propagating through the contact generates two such states (with opposite energies). Thus, at energy scales less than ∆, it is often convenient to describe the contact as a set of two-level systems. Further we assume for simplicity that we have only a single propagating mode (and therefore only two subgap levels). However, for describing dynamics of the contact at nonconstant α, the information on the energy spectrum (1) at each value of α is not sufficient. Mathematically speaking, we need a connection on the bundle of Hilbert spaces over the circle of possible values of α, and, more specifically, the projection of this connection onto the two-level subspace. Technically, it amounts to computing the "dynamic" matrix element I(α) = 0| ∂ ∂α |1 ,(2) where |0 and |1 are the two subgap states at a given value of α. This quantity defines coupling between the dynamics of the superconducting phase and transitions in the two-dimensional subspace of subgap states. To illustrate this point, consider a single SQPC connected to a grain of finite capacity C. This system has been studied previously in the adiabatic approximation [12,13] and in the two-level approximation [9,14]. The two-level Hamiltonian for this system may be written as H = H 0 (α) + 1 2C i ∂ ∂α − N 2 ,(3) where N is the dimensionless potential of the grain and H 0 (α) is a 2 × 2 matrix. At each α, the eigenvalues of H 0 (α) must be given by (1). However, this does not fix the whole dependence on α. In our earlier work [9] we suggested H 0 (α) = ∆ cos α 2 √ r sin α 2 √ r sin α 2 − cos α 2 (r = 1 − t).(4) Another candidate for H 0 (α) might be H # 0 (α) = E(α) 0 0 −E(α) .(5) Obviously, the latter choice of H 0 (α) would lead to a physically different behaviour of the system (3), although H # 0 (α) has the same eigenvalues as H 0 (α). The choice (4) of H 0 (α) appears natural from the point of view of perturbation theory in backscattering. Moreover, we claim that expression (4) is exact for any value of r (within the model described below). We omitted the derivation of (4) in our paper [9] and now fill this gap in the present note. We shall describe the one-channel SQPC by the one-dimensional Hamiltonian: H f ull = H SC + H scatt .(6)H SC = +∞ −∞ dx iΨ † Lβ ∂ x Ψ Lβ − iΨ † Rβ ∂ x Ψ Rβ + +∆(x) Ψ † R↑ Ψ † L↓ − Ψ † R↓ Ψ † L↑ + +∆ * (x) Ψ R↓ Ψ L↑ − Ψ R↑ Ψ L↓ ,(7) where Ψ † and Ψ are electron operators (L and R subscripts denote left-and right-movers, β =↑, ↓ is the spin index), ∆(x) is the superconducting gap. We assume the following coordinate dependence of the gap: ∆(x) = ∆, x < 0 ∆e iα , x > 0 (8) in other words, the absolute value of the gap ∆ is constant across the contact, while the phase changes by α at x = 0. The scattering part of the Hamiltonian H scatt corresponds to elastic scattering at x = 0 and is also quadratic in electron operators. We further disregard the nature of the scattering and describe it by means of a scattering matrix. The Hamiltonian is quadratic and may be diagonalized by operators linear in Ψ † and Ψ. We shall further compute the operators γ † ↑ and γ † ↓ corresponding to the subgap states. These operators satisfy the Bogolyubov-de-Gennes equations [15] γ † β , H f ull = Eγ † β .(9)γ † ↑ = dx u L (x)Ψ L↓ (x) + v L (x)Ψ † R↑ (x) + u R (x)Ψ † L↑ (x) + v R (x)Ψ R↓ (x) , γ † ↓ = dx u L (x)Ψ L↑ (x) − v L (x)Ψ † R↓ (x) − u R (x)Ψ † L↓ (x) + v R (x)Ψ R↑ (x) .(10) (Here we related γ † ↑ and γ † ↓ using the spin-rotational invariance of the Hamiltonian). Solving the equations (9), we find that, away from x = 0, u µ (x) and v µ (x) have the following form: u µ (x) = u + µ e −κx , x > 0 u − µ e κx , x < 0 v µ (x) = v + µ e −κx , x > 0 v − µ e κx , x < 0 (11) where κ = ∆ 2 − E 2 .(12) If we define φ = arccos E ∆ = arcsin κ ∆ ,(13) the Bogolyubov-de-Gennes equations take the form: e iφ u − R + v − R = 0, e iφ u − L + v − L = 0, e −iφ u + R + e iα v + R = 0, e −iφ u + L + e −iα v + L = 0.(14) The scattering matrix at x = 0 matches u µ (x) and v µ (x) at x = ±0: u − L v + R = a −b * b a * u + L v − R (15) v − L u + R = a * −b * b a v + L u − R(16) The former equation describes scattering of electrons, the latter one -scattering of holes. The amplitudes a and b must satisfy the unitarity condition: |a| 2 + |b| 2 = 1 (|a| 2 = t, |b| 2 = r). The same scattering amplitudes in (15) and (16) only assume that the scattering is spin-independent. We also neglect the momentum dependence of the scattering amplitudes (the so-called "instant scattering" approximation). Finally, we remark that the phase of a has the same meaning as the superconducting phase α. Therefore, without loss of generality, we may assume that a is real: a * = a = √ t. The condition that the homogeneous system of linear equations (14)-(16) has a solution, reduces to sin φ = √ t sin α 2 ,(17) which immediately gives (1) for the energy E and κ = √ t∆ sin α 2 .(18) (We assumed, without loss of generality, that 0 ≤ α ≤ π and that E ≥ 0). Solving the system (14)-(16) enables us to compute the following commutator: X(α) = ∂H ∂α , γ † ↑ , γ † ↓ .(19) We shall use this commutator to compute the "dynamic" matrix element (2) as follows: I(α) = 0| ∂ ∂α |1 = 1 E 1 − E 0 0| ∂H ∂α |1 = 1 2E(α) 0| ∂H ∂α γ † ↑ γ † ↓ |0 = X(α) 2E(α) .(20) Since H is quadratic in fermionic operators, X(α) is just a number. A straightforward calculation (with normalized γ † ↑ , γ † ↓ ) gives |X(α)| = √ r ∆ 2 2E(α) . The phase of X(α) depends on the choice of phases of subgap state operators γ † ↑ and γ † ↓ or, equivalently, on the relative phase of the two states |0 and |1 . To fix this phase, we observe that our system is invariant under the combined time reversal and particle-hole symmetry. More specifically, this symmetry acts on operators as follows: Ψ Lβ → e iξ Ψ † Rβ , Ψ Rβ → −e −iξ Ψ † Lβ ,(22) together with the complex conjugation of coefficients. The phase ξ is adjusted depending on the phase of the backscattering amplitude b. (This is a modified version of the well-known symmetry of Bogolyubov-de-Gennes equations [15]). If we choose the Andreev states to be self-conjugate, then 0| ∂ ∂α |0 = 1| ∂ ∂α |1 = 0, I(α) = 0| ∂ ∂α |1 is real.(23) (In analogy to the ordinary quantum mechanics with a real Hamiltonian: we may choose all eigenfunctions to be real, then the matrix elements of real operators will also be real.) With this choice of phases, from (20),(21), I(α) = √ r ∆ 2 4E 2 (α) .(24) The operator of the charge on the grain, projected onto the two-dimensional subbundle spanned by the states |0 and |1 , takes in the basis {|0 , |1 } the form Q = i ∂ ∂α + 0| ∂ ∂α |0 0| ∂ ∂α |1 1| ∂ ∂α |0 1| ∂ ∂α |1 = i ∂ ∂α + I(α) 0 1 −1 0(25) The Hamiltonian (3) in the basis {|0 , |1 } takes the form: H = E(α) 0 0 −E(α) + 1 2C i ∂ ∂α + I(α) 0 1 −1 0 − N 2 .(26) Now we want to perform the rotation to the "fixed" basis where the charge operator is simply Q = i ∂ ∂α .(27) This results in the Hamiltonian (3) with H 0 (α) = U (α) E(α) 0 0 −E(α) U −1 (α),(28) where U (α) is a rotation: U (α) = P exp i I(α) 0 1 −1 0 dα = cos ϕ(α) − sin ϕ(α) sin ϕ(α) cos ϕ(α) . The angle of rotation ϕ(α) is given by ϕ(α) = 0| ∂ ∂α |1 dα = √ r 4 dα 1 − t sin 2 (α/2) = 1 2 arctan √ r tan α 2(30) which upon substituting in (28)-(29) gives the result (4). To summarize, we have replaced the multi-body superconducting system by the quantum-mechanical two-level Hamiltonian for the superconducting phase across the contact. This two-level approximation is appropriate whenever the system stays away from the upper continuum of excitations. The Hamiltonian (3) loses its validity at points where the upper Andreev state touches the upper continuum (at α = 2πn). Once a particle reaches this point, it will pass to the vacant levels of the continuum instead of following the localized subgap levels[16]. This effect is important for many non-equilibrium problems, for example, those with constant voltage applied to the contact[5]. For most equilibrium problems[9,14]and for some non-equilibrium setups[16,17]the Hamiltonian defined by Eqs. (3), (4) may be used as the two-level approximation[18].This research was supported by the collaboration grant # 7SUP J048531 from the Swiss NSF, INTAS-RFBR grant # 95-0302, RFBR grant # 98-02-19252, Program "Statistical Physics" of the Russian Ministry of Science, DGA grant # 94-1189 . I O Kulik, A N , Omel'yanchuk, Fiz. Nizk. Temp. 3945J. Low Temp. Phys.)I. O. Kulik and A. N. Omel'yanchuk, Fiz. Nizk. Temp. 3 (1977) 945 (translated in Sov. J. Low Temp. Phys.) . K K Likharev, Rev. Mod. Phys. 51101K. K. Likharev, Rev. Mod. Phys. 51 (1979) 101 . C W Beenakker, H Van Houten, Phys. Rev. Lett. 663056C. W. Beenakker and H. van Houten, Phys. Rev. Lett. 66 (1991) 3056 . V S Shumeiko, E N Bratus, &apos; , G Wendin, Sov. J. Low Temp. Phys. 23249Fiz. Nizk. Temp.. preprint cond-mat/9610101V. S. Shumeiko, E. N. Bratus', G. Wendin, Fiz. Nizk. Temp. 23 (1997) 249 (Sov. J. Low Temp. Phys. 23 (1997) 181), preprint cond-mat/9610101 . D Averin, A Bardas, Phys. Rev. Lett. 281705Phys. Rev. BD. Averin and A. Bardas, Phys. Rev. Lett. 28 (1995) 1831; Phys. Rev. B 53 (1996) R1705 . A Martin-Rodero, A Levy Yeyati, F J Garcia-Vidal, Phys. Rev. B. 538891A. Martin-Rodero, A. Levy Yeyati, and F. J. Garcia-Vidal, Phys. Rev. B 53 (1996) R8891 . D Averin, H T Imam, Phys. Rev. Lett. 763814D. Averin and H. T. Imam, Phys. Rev. Lett. 76 (1996) 3814 G B Lesovik, A Golubov, Proc. of the 31 Recontres de Moriond. T. Martin et alof the 31 Recontres de MoriondGif-sur-YvetteEditions FrontieresG. B. Lesovik and A. Golubov, in Proc. of the 31 Recontres de Moriond, T. Martin et al (eds.), Editions Frontieres, Gif-sur-Yvette (1996) Coulomb effects in a ballistic one-channel S-S-S device. D A Ivanov, M F , Exp. Teor. Fiz. (Sov. Phys. JETP). Feigel'man. preprint cond-mat/9712074, to appear in ZhD. A. Ivanov and M. F. Feigel'man, "Coulomb effects in a ballistic one-channel S-S-S device", preprint cond-mat/9712074, to appear in Zh. Exp. Teor. Fiz. (Sov. Phys. JETP) . A Furusaki, M Tsukada, Physica B. 165967A. Furusaki and M. Tsukada, Physica B 165 & 166 (1990) 967 . C W Beenakker, Phys. Rev. Lett. 673836C. W. Beenakker, Phys. Rev. Lett. 67 (1991) 3836 . G Schön, A D Zaikin, Phys. Rep. 198237G. Schön and A. D. Zaikin, Phys. Rep. 198 (1990) 237 Mesoscopic Phenomena in Solids. D V Averin, K K Likharev, B. L. Altshuler, P. A. Lee, and R. A. WebbElsevierAmsterdamD. V. Averin and K. K. Likharev in "Mesoscopic Phenomena in Solids", ed. by B. L. Altshuler, P. A. Lee, and R. A. Webb, Elsevier, Amsterdam, 1991 Coulomb blockade in superconducting quantum point contacts. D V Averin, preprint cond-mat/9803066D. V. Averin, "Coulomb blockade in superconducting quantum point contacts", preprint cond-mat/9803066 Superconductivity of Metals and Alloys. P G De Gennes, W. A. BenjaminNew YorkP. G. de Gennes, "Superconductivity of Metals and Alloys", W. A. Benjamin, New York, 1966 Superconducting single-mode contact as a microwave-activated quantum interferometer. L Y Gorelik, preprint cond-mat/9803013L. Y. Gorelik et al., "Superconducting single-mode contact as a microwave-activated quantum interferometer", preprint cond-mat/9803013 . L Y Gorelik, cond-mat/9502084Phys. Rev. Lett. 751162L. Y. Gorelik et al., Phys. Rev. Lett. 75 (1995) 1162, preprint cond-mat/9502084 The Hamiltonian used in ref. [14] is incorrect away from the transition region α ≈ π. It is appropriate for the problem posed, but cannot be used for other problems involving transitions away from α ≈ π. In such cases, the Hamiltonian (3),(4) should be used insteadThe Hamiltonian used in ref. [14] is incorrect away from the transition region α ≈ π. It is appropriate for the problem posed, but cannot be used for other problems involving transitions away from α ≈ π. In such cases, the Hamiltonian (3),(4) should be used instead.

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{'abstract': 'In a superconducting quantum point contact, dynamics of the superconducting phase is coupled to the transitions between the subgap states. We compute this coupling and derive the two-level Hamiltonian of the contact.One of the key features of superconducting quantum point contacts (SQPC) [1-9] is the existence of subgap states (so-called Andreev states) whose energies depend on the phase difference across the contact[10,11]:where ∆ is the superconducting gap, t is the normal transparency of the contact. Each transversal mode propagating through the contact generates two such states (with opposite energies). Thus, at energy scales less than ∆, it is often convenient to describe the contact as a set of two-level systems. Further we assume for simplicity that we have only a single propagating mode (and therefore only two subgap levels). However, for describing dynamics of the contact at nonconstant α, the information on the energy spectrum (1) at each value of α is not sufficient. Mathematically speaking, we need a connection on the bundle of Hilbert spaces over the circle of possible values of α, and, more specifically, the projection of this connection onto the two-level subspace. Technically, it amounts to computing the "dynamic" matrix elementwhere |0 and |1 are the two subgap states at a given value of α. This quantity defines coupling between the dynamics of the superconducting phase and transitions in the two-dimensional subspace of subgap states.To illustrate this point, consider a single SQPC connected to a grain of finite capacity C. This system has been studied previously in the adiabatic approximation [12,13] and in the two-level approximation[9,14]. The two-level Hamiltonian for this system may be written aswhere N is the dimensionless potential of the grain and H 0 (α) is a 2 × 2 matrix. At each α, the eigenvalues of H 0 (α) must be given by (1). However, this does not fix the whole dependence on α. In our earlier work [9] we suggestedAnother candidate for H 0 (α) might beObviously, the latter choice of H 0 (α) would lead to a physically different behaviour of the system(3), although H # 0 (α) has the same eigenvalues as H 0 (α). The choice (4) of H 0 (α) appears natural from the point of view of perturbation theory in backscattering. Moreover, we claim that expression (4) is exact for any value of r (within the model described below). We omitted the derivation of (4) in our paper [9] and now fill this gap in the present note.We shall describe the one-channel SQPC by the one-dimensional Hamiltonian:', 'arxivid': 'cond-mat/9808029', 'author': ['D A Ivanov \nL.D.Landau Institute for Theoretical Physics\n117940MoscowRussia\n\n12-127 M.I.T. Cambridge02139MAUSA\n', 'M V Feigel&apos;man \nL.D.Landau Institute for Theoretical Physics\n117940MoscowRussia\n'], 'authoraffiliation': ['L.D.Landau Institute for Theoretical Physics\n117940MoscowRussia', '12-127 M.I.T. Cambridge02139MAUSA', 'L.D.Landau Institute for Theoretical Physics\n117940MoscowRussia'], 'corpusid': 119397609, 'doi': '10.1103/physrevb.59.8444', 'github_urls': [], 'n_tokens_mistral': 5489, 'n_tokens_neox': 4809, 'n_words': 2752, 'pdfsha': 'fab7ab0530f6ddd28944e3fa38b9a0996817fa4c', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/9808029v2.pdf'], 'title': ['arXiv:cond-mat/9808029v2 4 Aug 1998 Two-level Hamiltonian of a superconducting quantum point contact', 'arXiv:cond-mat/9808029v2 4 Aug 1998 Two-level Hamiltonian of a superconducting quantum point contact'], 'venue': []}

arxiv

Revisiting the Impact of Axions in the Cooling of White Dwarfs 1 Oct 2012 Brenda Melendez Facultad de Ciencias Astronómicas y Geofísicas Universidad Nacional de La Plata Paseo del Bosque s/n, 1900 La PlataArgentina Instituto de Astrofísica de La Plata UNLP-CONICET Paseo del Bosque s/n, 1900 La PlataArgentina Marcelo Miller Bertolami Facultad de Ciencias Astronómicas y Geofísicas Universidad Nacional de La Plata Paseo del Bosque s/n, 1900 La PlataArgentina Instituto de Astrofísica de La Plata UNLP-CONICET Paseo del Bosque s/n, 1900 La PlataArgentina Max-Planck-Institut für Astrophysik Karl-Schwarzschild-Str. 185748GarchingGermany Leandro Althaus Facultad de Ciencias Astronómicas y Geofísicas Universidad Nacional de La Plata Paseo del Bosque s/n, 1900 La PlataArgentina Instituto de Astrofísica de La Plata UNLP-CONICET Paseo del Bosque s/n, 1900 La PlataArgentina Revisiting the Impact of Axions in the Cooling of White Dwarfs 1 Oct 2012**Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** c **Copyright Year** Astronomical Society of the Pacific It has been shown that the shape of the luminosity function of white dwarfs can be a powerful tool to check for the possible existence of DFSZ-axions. In particular,Isern et al. (2008)showed that, if the axion mass is of the order of a few meV, then the white dwarf luminosity function is sensitive enough to detect their existence. For axion masses of about m a > 5 meV the axion emission can be a primary cooling mechanism for the white dwarf and the feedback of the axion emission into the thermal structure of the white dwarf needs to be considered. Here we present computations of white dwarf cooling sequences that take into account the effect of axion emission in a self consistent way by means of full stellar evolution computations. Then, we study and discuss the impact of the axion emission in the white dwarf luminosity function. Introduction The Peccei-Quinn mechanism is one of the most convincing explanations for the absence of CP-violating effects arising from the QCD vacuum structure (see Raffelt 1996). A natural consequence of this mechanism is the existence of a new particle, a boson called the axion (Wilczek 1978). One of the axion models, the DFSZ model (Zhitnitsky 1980;Dine et al. 1981) allows for the interaction of axions with charged leptons. The existence of such DFSZ-axions would have an impact in the cooling of white dwarfs as pointed by Raffelt (1986) and Isern et al. (1992). Because the evolution of white dwarfs is mostly a simple cooling process and the basic physical ingredients needed to predict their evolution are relatively well known, white dwarfs offer a unique oportunity to test new physics under conditions that can not be obtained in present day laboratories. Isern et al. (2008) showed that with the current knowledge of the white dwarf luminosity function (Harris et al. 2006;De Gennaro et al. 2008) it might be possible to detect axions as light as m a ∼ 5meV. For axion masses which might be detectable through the white dwarf luminosity function (m a > 5meV) the axion cooling is comparable to the neutrino and photon cooling of the white dwarf (Isern et al. 2008). In such a situation, we expect the existence of a signifficant axion emission to impact the thermal structure of the white dwarf, and consequently to alter both the photon and neutrino emission. Thus, for the range of interest of the axion masses we expect that the axion emission Isern et al. (2008). The effects of the departure from the isothermal core approximation can be apreciated at M bol < 6. Right: Axion (black curves) and neutrino (blue curves) emission for our 0.609M ⊙ sequences for different axion masses. The impact of the axion emission in the thermal structure of the white dwarf can be appreciated in the decrease of the neutrino emission at higher axion masses. Clearly, axion emission can not be treated perturbatively at m a > 5meV. can not be treated as a perturbation to the white dwarf cooling and that a self consistent treatment of the axion emission is necessary. In the present work, we improve previous works by studying the impact of the axion emission in the cooling of white dwarfs by means of a self consistent treatment of the axion emission and state of the art white dwarf models. White dwarf models, input physics and the white dwarf luminosity function The initial white dwarf models were taken from Renedo et al. (2010). Specifically 4 different initial white dwarf models of 0.524M ⊙ , 0.609M ⊙ , 0.705M ⊙ and 0.877 M ⊙ were adopted. These models were obtained from computing the complete evolution of initially 1M ⊙ , 2M ⊙ , 3M ⊙ and 5 M ⊙ ZAMS stars with Z=0.01, which is agreement with semi-empirical determinations of the initial-final mass relationship (Salaris et al. 2009). It is worth noting that these sequences were computed from the ZAMS to the thermal pulses at the AGB and finally to the post-AGB and white dwarf stages. For each initial white dwarf model 6 cooling sequences with different assumed axion masses were computed (m a = 0, 5, 10, 15, 20 & 30 meV). Axion emission by both Compton and Bremsstrahlung processes was included, although in white dwarfs only the latter is important. Axion Bremsstrahlung emission under degenerate conditions was included adopting the prescriptions of Nakagawa et al. (1987Nakagawa et al. ( , 1988 for strongly coupled plasma regime (Γ > 1) and Raffelt & Weiss (1995) for weakly coupled plasmas (Γ < 1). Computations were performed with LPCODE stellar evolution code, which is specifically tailored for the computation of white dwarfs and includes all relevant microphysics such as equation of state, radiative and conductive opacities, element diffusion and can even handle the effects of phase separation, crystallization and release of latent heat. To construct theoretical white dwarf luminosity functions we followed Isern et al. (2008) but adopted the method presented by Iben & Laughlin (1989). From the cooling times, t c (l, m), computed with LPCODE for each value of the axion mass, we computed the number density of stars per luminosity as dn dl = − M 2 M 1 ψ(t) dN dM ∂t c ∂l m dM (1) For a given white dwarf luminosity bin (l), and mass of the progenitor (M), the formation time of the star, t, is given by t + t ev (M) + t c (l, m) = T d . To compute the white dwarf luminosity function we adopt the following additional ingredients: A Salpeter initial mass function N(M), the initial-final mass relationship m(M) from Salaris et al. (2009), the stellar lifetimes t ev from the BaSTI database (Pietrinferni et al. 2004) and constant star formation rate ψ. Normalization of theoretical luminosity functions is done as in Isern et al. (2008). Results and future work In Fig. 1 (left panel) the axion emission of our 0.609M ⊙ sequence is compared with the 0.61M ⊙ sequence of Isern et al. (2008) for an axion of m a = 5meV. There is an overall good agreement between both predictions at low luminosities (M bol > 6). The departure between both curves at high luminosities (M bol < 6) can be traced back to the isothermal core approximation of Isern et al. (2008) which leads to an underestimation of the axion emission at high luminosities when the maximum temperature of the core is located off-centered. The right panel of Fig. 1 shows both the axion and neutrino emissions of our 0.609M ⊙ sequences computed with different axion masses. As expected the higher the axion mass, the higher the axion emission and the cooling speed. A more interesting feature can be seen by analysing the neutrino emission at different axion masses. As can be seen in Fig. 1 the neutrino emission is reduced as the axion mass is increased. This result is due to the fact that when axion emission is included this leads to an extra cooling of the white dwarf core which alters the thermal structure of the white dwarf (as compared with the case with no axion emission) which, in turn, leads to a decrease of the neutrino emission at a given surface luminosity of the star. Note that this is true even for our lighter computed axion mass (m a = 5meV), for which the axion emission is already different from the case with no axions (m a = 0meV). Then, the inclusion of the feedback effects of the axion emission on the thermal structure of the white dwarf, leads to a decrease of the neutrino emission which will diminish the sensitivity of the white dwarf cooling times to the axion mass. Consequently the axion emission needs to be treated self-consistently when dealing with axions in the range detectable through the white dwarf luminosity function. In Fig. 2 we show the resulting white dwarf luminosity functions for each axion mass as compared with the observationally derived ones. It can be clearly seen that axion masses larger than 10 meV would lead to strong disagreements with the luminosity functions derived by Harris et al. (2006) andDe Gennaro et al. (2008). On the other hand, the existence or not of DFSZ-axions with masses lower than 5 meV can not be concluded without a detailed statistical analysis of the uncertainties. We have shown that the impact of the axion emission on the white dwarf luminosity function will be overestimated if the effect of axion emission on the thermal structure of the white dwarf is not taken into account. This is particularly important at log L γ > −1. Our preliminary analysis shows that the observed luminosity functions are consistent with the absence of any additional cooling mechanism (like axions), although a detailed statistical analysis of the results should be made before making a final statement. On the other hand, DFSZ-axion masses larger than m a > 10 meV, are clearly excluded by our present knowledge of the white dwarf luminosity function, in agreement with Isern et al. (2008). Figure 1 . 1Left: Comparison of the axion (m a = 5meV) emission of our 0.609M ⊙ and the 0.61M ⊙ sequence of 3 Figure 2 . 32White Dwarf luminosity functions constructed for the different axion masses compared with the luminosity functions derived byHarris et al. (2006) andDe Gennaro et al. (2008). DFSZ-axions heavier than m a > 10meV are clearly expcluded by the observed white dwarf luminosity functions. Acknowledgments. M3B thanks the organizers of the EUROWD12 for the finantial assistance that helped him to attend the conference. This research was supported by PIP 112-200801-00940 from CONICET and PICT-2010-0861 from ANCyT. . S De Gennaro, T Von Hippel, D E Winget, S O Kepler, A Nitta, D Koester, L Althaus, 1. 0906.1513AJ. 135De Gennaro, S., von Hippel, T., Winget, D. E., Kepler, S. O., Nitta, A., Koester, D., & Althaus, L. 2008, AJ, 135, 1. 0906.1513 . M Dine, W Fischler, M Srednicki, Physics Letters B. 104199Dine, M., Fischler, W., & Srednicki, M. 1981, Physics Letters B, 104, 199 . H C Harris, J A Munn, M Kilic, J Liebert, K A Williams, T Von Hippel, S E Levine, D G Monet, D J Eisenstein, S J Kleinman, T S Metcalfe, A Nitta, D E Winget, J Brinkmann, M Fukugita, G R Knapp, R H Lupton, J A Smith, D P Schneider, arXiv:astro-ph/0510820AJ. 131571Harris, H. C., Munn, J. A., Kilic, M., Liebert, J., Williams, K. A., von Hippel, T., Levine, S. E., Monet, D. G., Eisenstein, D. J., Kleinman, S. J., Metcalfe, T. S., Nitta, A., Winget, D. E., Brinkmann, J., Fukugita, M., Knapp, G. R., Lupton, R. H., Smith, J. A., & Schneider, D. P. 2006, AJ, 131, 571. arXiv:astro-ph/0510820 . I Iben, Jr, G Laughlin, ApJ. 341312Iben, I., Jr., & Laughlin, G. 1989, ApJ, 341, 312 . J Isern, E García-Berro, S Torres, S Catalán, L109. 0806.2807ApJ. 682Isern, J., García-Berro, E., Torres, S., & Catalán, S. 2008, ApJ, 682, L109. 0806.2807 . J Isern, M Hernanz, E Garcia-Berro, ApJ. 23Isern, J., Hernanz, M., & Garcia-Berro, E. 1992, ApJ, 392, L23 . M Nakagawa, T Adachi, Y Kohyama, N Itoh, ApJ. 326241Nakagawa, M., Adachi, T., Kohyama, Y., & Itoh, N. 1988, ApJ, 326, 241 . M Nakagawa, Y Kohyama, N Itoh, ApJ. 322291Nakagawa, M., Kohyama, Y., & Itoh, N. 1987, ApJ, 322, 291 . A Pietrinferni, S Cassisi, M Salaris, F Castelli, arXiv:astro-ph/0405193ApJ. 612168Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168. arXiv:astro-ph/0405193 . G Raffelt, A Weiss, arXiv:hep-ph/9410205Phys.Rev.D. 511495Raffelt, G., & Weiss, A. 1995, Phys.Rev.D, 51, 1495. arXiv:hep-ph/9410205 . G G Raffelt, Physics Letters B. 166Stars as laboratories for fundamental physicsRaffelt, G. G. 1986, Physics Letters B, 166, 402 -1996, Stars as laboratories for fundamental physics . I Renedo, L G Althaus, M M Miller Bertolami, A D Romero, A H Córsico, R D Rohrmann, E García-Berro, 183. 1005.2170ApJ. 717Renedo, I., Althaus, L. G., Miller Bertolami, M. M., Romero, A. D., Córsico, A. H., Rohrmann, R. D., & García-Berro, E. 2010, ApJ, 717, 183. 1005.2170 . M Salaris, A Serenelli, A Weiss, M Miller Bertolami, 1013. 0807.3567ApJ. 692Salaris, M., Serenelli, A., Weiss, A., & Miller Bertolami, M. 2009, ApJ, 692, 1013. 0807.3567 . F Wilczek, Physical Review Letters. 40279Wilczek, F. 1978, Physical Review Letters, 40, 279 . A R Zhitnitsky, Sov. J. Nucl. Phys. 31260Zhitnitsky, A. R. 1980, Sov. J. Nucl. Phys., 31, 260

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{'abstract': 'It has been shown that the shape of the luminosity function of white dwarfs can be a powerful tool to check for the possible existence of DFSZ-axions. In particular,Isern et al. (2008)showed that, if the axion mass is of the order of a few meV, then the white dwarf luminosity function is sensitive enough to detect their existence. For axion masses of about m a > 5 meV the axion emission can be a primary cooling mechanism for the white dwarf and the feedback of the axion emission into the thermal structure of the white dwarf needs to be considered. Here we present computations of white dwarf cooling sequences that take into account the effect of axion emission in a self consistent way by means of full stellar evolution computations. Then, we study and discuss the impact of the axion emission in the white dwarf luminosity function.', 'arxivid': '1210.0263', 'author': ['Brenda Melendez \nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque s/n, 1900 La PlataArgentina\n\nInstituto de Astrofísica de La Plata\nUNLP-CONICET\nPaseo del Bosque s/n, 1900 La PlataArgentina\n', 'Marcelo Miller Bertolami \nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque s/n, 1900 La PlataArgentina\n\nInstituto de Astrofísica de La Plata\nUNLP-CONICET\nPaseo del Bosque s/n, 1900 La PlataArgentina\n\nMax-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Str. 185748GarchingGermany\n', 'Leandro Althaus \nFacultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque s/n, 1900 La PlataArgentina\n\nInstituto de Astrofísica de La Plata\nUNLP-CONICET\nPaseo del Bosque s/n, 1900 La PlataArgentina\n'], 'authoraffiliation': ['Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque s/n, 1900 La PlataArgentina', 'Instituto de Astrofísica de La Plata\nUNLP-CONICET\nPaseo del Bosque s/n, 1900 La PlataArgentina', 'Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque s/n, 1900 La PlataArgentina', 'Instituto de Astrofísica de La Plata\nUNLP-CONICET\nPaseo del Bosque s/n, 1900 La PlataArgentina', 'Max-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Str. 185748GarchingGermany', 'Facultad de Ciencias Astronómicas y Geofísicas\nUniversidad Nacional de La Plata\nPaseo del Bosque s/n, 1900 La PlataArgentina', 'Instituto de Astrofísica de La Plata\nUNLP-CONICET\nPaseo del Bosque s/n, 1900 La PlataArgentina'], 'corpusid': 118397089, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4385, 'n_tokens_neox': 3665, 'n_words': 2160, 'pdfsha': '370e7ed42eff7ae4254595c3f8e7484de817bdf0', 'pdfurls': ['https://arxiv.org/pdf/1210.0263v1.pdf'], 'title': ['Revisiting the Impact of Axions in the Cooling of White Dwarfs', 'Revisiting the Impact of Axions in the Cooling of White Dwarfs'], 'venue': []}

arxiv

Distributionally Robust Optimization using Cost-Aware Ambiguity Sets Mathijs Schuurmans Panagiotis Patrinos Distributionally Robust Optimization using Cost-Aware Ambiguity Sets We present a novel framework for distributionally robust optimization (DRO), called cost-aware DRO (CADRO).The key idea of CADRO is to exploit the cost structure in the design of the ambiguity set to reduce conservatism. Particularly, the set specifically constrains the worst-case distribution along the direction in which the expected cost of an approximate solution increases most rapidly. We prove that CADRO provides both a high-confidence upper bound and a consistent estimator of the out-of-sample expected cost, and show empirically that it produces solutions that are substantially less conservative than existing DRO methods, while providing the same guarantees. I. INTRODUCTION We consider the stochastic programming problem minimize x∈X IE[ (x, ξ)](1) with X ⊆ IR n a nonempty, closed set of feasible decision variables, ξ ∈ Ξ a random variable following probability measure P, and : IR n × Ξ → IR a known cost function. This problem is foundational in many fields, including operations research [1], machine learning [2], and control (e.g., stochastic model predictive control) [3]. Provided that the underlying probability measure P is known exactly, this problem can effectively be solved using traditional stochastic optimization methods [1], [4]. In reality, however, only a data-driven estimateP of P is typically available, which may be subject to misestimations-known as ambiguity. Perhaps the most obvious method for handling this issue is to disregard this ambiguity and instead apply a sample average approximation (SAA) (also known as empirical risk minimization (ERM) in the machine learning literature), where (1) is solved usingP as a plug-in replacement for P. However, this is known to produce overly optimistic estimates of the optimal cost [4,Prop. 8.1], potentially resulting in unexpectedly high realizations of the cost when deploying the obtained optimizers on new, unseen samples. This downward bias of SAA is closely related to the issue of overfitting, and commonly refered to as the optimizer's curse [5], [6]. Several methods have been devised over the years to combat this undesirable behavior. Classical techniques such as regularization and cross-validation are commonly used in machine learning [2], although typically, they are used as heuristics, providing few rigorous guarantees, in particular for M. Schuurmans small sample sizes. Alternatively, the suboptimality gap of the SAA solution may be statistically estimated by reserving a fraction of the dataset for independent replications [7]. However, these results are typically based on asymptotic arguments, and are therefore not valid in the low-sample regime. Furthermore, although this type of approach may be used to validate the SAA solution, it does not attempt to improve it, by taking into account possible estimation errors. More recently, distributionally robust optimization (DRO) has garnered considerable attention, as it provides a principled way of obtaining a high-confidence upper bound on the true out-of-sample cost [6], [8], [9]. In particular, its capabilities to provide rigorous performance and safety guarantees has made it an attractive technique for data-driven and learning-based control [10]- [12]. DRO refers to a broad class of methods in which a variant of (1) is solved where P is replaced with a worst-case distribution within a statistically estimated set of distributions, called an ambiguity set. As the theory essentially requires only that the ambiguity set contains the true distribution with a prescribed level of confidence, a substantial amount of freedom is left in the design of the geometry of these sets. As a result, many different classes of ambiguity sets have been proposed in the literature, e.g., Wasserstein ambiguity sets [9], divergencebased ambiguity sets [6], [12], [13] and moment-based ambiguity sets [8], [14]; See [15], [16] for recent surveys. Despite the large variety of existing classes of ambiguity sets, a common characteristic is that their design is considered separately from the optimization problem in question. Although this simplifies the analysis in some cases, it may also induce a significant level of conservatism; In reality, we are only interested in excluding distributions from the ambiguity set which actively contribute to increasing the worst-case cost. Requiring that the true distribution deviates little from the data-driven estimate in all directions may therefore be unnecessarily restrictive. This intuition motivates the introduction of a new DRO methodology, which is aimed at designing the geometry of the ambiguity sets with the original problem (1) in mind. The main idea is that by only excluding those distributions that maximally affect the worstcase cost, higher levels of confidence can be attained without introducing additional conservatism to the cost estimate. Contributions: (i) We propose a novel class of ambiguity sets for DRO, taking into account the structure of the underlying optimization problem; (ii) We prove that the DRO cost is both a high-confidence upper bound and a consistent estimate of the optimal cost of the original stochastic program (1); (iii) We demonstrate empirically that the provided ambiguity set outperforms existing alternatives. Notation: We denote [n] = {1, . . . , n}, for n ∈ IN. |S| denotes the cardinality of a (finite) set S. e i ∈ IR n is the ith standard basis vector in IR n . Its dimension n will be clear from context. We denote the level sets of a function f : IR n → IR as lev ≤α f := {x ∈ IR n | f (x) ≤ α}. We write 'a.s.' to signify that a random event occurs almost surely, i.e., with probability 1. We denote the largest and smallest entries of a vector v ∈ IR n as v max := max i∈[n] v i and v min = min i∈[n] v i , respectively, and define its range as rg(v) := v max − v min . δ X is the indicator of a set X: δ X (x) = 0 if x ∈ X, +∞ otherwise. II. PROBLEM STATEMENT We will assume that the random variable ξ is finitely supported, so that without loss of generality, we may write Ξ = {1, . . . , d}. This allows us to define the probability mass vector p = (P[ξ = i]) d i=1 , and enumerate the cost realizations i = (· , i), i ∈ [d]. Furthermore, it will be convenient to introduce the mapping L : IR n → IR d as L(x) = ( 1 (x), . . . , d (x)). We will pose the following (mostly standard) regularity assumption on the cost function. Assumption II.1 (Problem regularity). For all i ∈ [d] (i) i is continuous on X; (ii) i := i + δ X is level-bounded; Since any continuous function is lower semicontinuous (lsc), Assumption II.1 combined with the closedness of X implies inf-compactness, which ensures attainment of the minimum [17,Thm. 1.9]. Continuity of i is used mainly in Lemma A.5 to establish continuity of the solution mapping V -defined below, see (2). However, a similar result can be obtained by replacing condition (i) by lower semicontinuity and uniform level-boundedness on X. However, for ease of exposition, we will not cover this modification explicitly. Let p ∈ ∆ d := {p ∈ IR d + | d i=1 p i = 1} denote the true-but-unknown probability mass vector, and define V : IR n ×∆ d → IR : (x, p) → p, L(x) , to obtain the parametric optimization problem with optimal cost and solution set V (p) = min x∈X V (x, p) and X (p) = argmin x∈X V (x, p). (2) The solution of (1) is retrieved by solving (2) with p = p . Assume we have access to a datasetΞ := {ξ 1 , . . . , ξ m } ∈ Ξ m collected i.i.d. from p . In order to avoid the aforementioned downward bias of SAA, our goal is to obtain a data-driven decisionx m along with an estimateV m such that P[V (x m , p ) ≤V m ] ≥ 1 − β,(3) where β ∈ (0, 1) is a user-specified confidence level. We address this problem by means of distributionally robust optimization, where instead of (2), one solves the surrogate problemV m = min x∈X max p∈Am V (x, p).(DRO) Here, A m ⊆ ∆ d is a (typically data-dependent, and thus, random) set of probability distributions that is designed to contain the true distribution p with probability 1−β, ensuring that (3) holds. Trivially, (3) is satisfied with β = 0 by taking A m ≡ ∆ d . This recovers a robust optimization method, i.e., min x∈X max i∈[d] i (x). Although it satisfies (3), this robust approach tends to be overly conservative as it neglects all available statistical data. The aim of distributionally robust optimization is to additionally ensure thatV m is a consistent estimator, i.e., lim m→∞V m = V (p ), a.s.(4) We will say that a class of ambiguity sets is admissible if the solutionV m of the resulting DRO problem (DRO) satisfies (3) and (4). Our objective is to develop a methodology for constructing admissible ambiguity sets that take into account the structure of (DRO) and in doing so, provide tighter estimates of the cost, while maintaining (3) with a given confidence level β. III. COST-AWARE DRO In this section, we describe the proposed DRO framework, which we will refer to as cost-aware DRO (CADRO). The overall method is summarized in Alg. 1. Here, > 0 is determined to satisfy (5) and α = max p∈A TV L(x), p . Since A TV ⊂ A, A satisfies (5) with a higher confidence level 1 − β, but nevertheless, we have max p∈A V (x, p) = max p∈A TV V (x, p). A. Motivation We start by providing some intuitive motivation. Consider the problem (DRO). In order to provide a guarantee of the form (3), it obviously suffices to design A m such that P[p ∈ A m ] ≥ 1 − β.(5) However, this condition alone still leaves a considerable amount of freedom to the designer. A common approach is to select A m to be a ball (expressed in some statistical metric/divergence) around an empirical estimatep of the distribution. Depending on the choice of metric/divergence (e.g., total variation [18], Kullback-Leibler [6], Wasserstein [9], . . . ), several possible variants may be obtained. Using concentration inequalities, one can then select the appropriate radius of this ball, such that (5) is satisfied. A drawback of this approach, however, is that the construction of A m is decoupled from the original problem (1). Indeed, given that A m takes the form of a ball, (5) essentially requires the deviation ofp from p to be small along every direction. If one could instead enlarge the ambiguity set without increasing the worst-case cost, then (5) could be guaranteed for smaller values of β without introducing additional conservatism. This idea is illustrated in Fig. 1. Conversely, for a fixed confidence level β, one could thus construct a smaller upper boundV m , by restricting the choice of p only in a judiciously selected direction. Particularly, we may set A m = {p ∈ ∆ d | L(x), p ≤ α m } for some candidate solution x ∈ X, where α m is the smallest (potentially data-dependent) quantity satisfying (5). This directly yields an upper bound on the estimateV m . Namely, for x ∈ X (p ), we have with probability 1 − β, V (x , p ) (a) ≤ V (x m , p ) ≤ max p∈Am V (x m , p) =V m = min x∈X max p∈Am V (x, p) (b) ≤ max p∈Am V (x, p) = α m . Here, inequalities (a) and (b) become equalities whenx m = x = x. Thus, a reasonable aim would be to select x to be a good approximation of x . We will return to the matter of selecting x in §III-C. First, however, we will assume x to be given and focus on establishing the coverage condition (5). B. Ambiguity set parameterization and coverage Motivated by the previous discussion, we propose a family of ambiguity sets parameterized as follows. Let v ∈ IR d be a fixed vector (we will discuss the choice of v in §III-C). Given a sampleΞ = {ξ 1 , . . . , ξ m } of size |Ξ| = m drawn i.i.d. from p , we consider ambiguity sets of the form AΞ(v) := {p ∈ ∆ d | p, v ≤ αΞ(v)},(6) where α : Ξ m × IR d (Ξ, v) → αΞ(v) ∈ IR is a datadriven estimator for p , v , selected to satisfy the following assumption, which implies that (5) holds for A m = AΞ(v). Assumption III.1. P[ p , v ≤ αΞ(v)] ≥ 1 − β, ∀v ∈ IR d . Note that the task of selecting α to satisfy Assumption III.1 is equivalent to finding a high-confidence upper bound on the mean of the scalar random variable v, e ξ , ξ ∼ p . It is straightforward to derive such bounds by bounding the deviation of a random variable from its empirical mean using classical concentration inequalities like Hoeffding's inequality . Proposition III.2 (Hoeffding bound). Fix v ∈ IR d and let Ξ with |Ξ| = m, be an i.i.d. sample from p ∈ ∆ d , with empirical distributionpΞ = 1 m ξ∈Ξ e ξ . Consider the bound αΞ(v) = v,pΞ + r m rg(v).(7) This bound satisfies Assumption III.1, if r m satisfies r m = min 1, log( 1 /β) 2m .(8) Proof. Define y k := p −e ξ k , v , so that 1 m m k=1 y k = p − p m , v . Since v is fixed, y k , k ∈ [m] are i.i.d., and we have IE[y k ] = 0 and (by Lemma A.1), |y k | ≤ rg(v), ∀k ∈ IN. This establishes the (vacuous) case r m = 1 in (8). For the nontrivial case, we apply Hoeffding's inequality [19, eq. 2.11] P 1 m m k=1 y k > t ≤ exp −2mt 2 rg(v) 2 .(9) Setting t = r m rg(v), equating the right-hand side of (9) to the desired confidence level β, and solving for r m yields the desired result. Although attractive for its simplicity, this type of bounds has the drawback that it applies a constant offset (depending only on the sample size, not the data) to the empirical mean, which may be conservative, especially for small samples. Considerably sharper bounds can be obtained through a more direct approach. In particular, we will focus our attention on the following result due to Anderson [20], which is a special case of the framework presented in [21]. We provide an experimental comparison between the bounds in Appendix B. Proposition III.3 (Ordered mean bound [21]). Let η k := v, e ξ k , k ∈ [m], so that IE[η k ] = v, p . Let η (1) ≤ η (2) ≤ · · · ≤ η (m) ≤ η denote the sorted sequence, with ties broken arbitrarily, where η := max i∈[d] v i . Then, there exists a γ ∈ (0, 1) such that Assumption III.1 holds for αΞ(v) = κ m −γ η (κ) + m i=κ+1 η (i) m +γη, κ = mγ .(γ = log( 1 /β) 2m , for sufficiently large m.(11) This asymptotic expression will be useful when establishing theoretical guarantees in Section IV. C. Selection of v The proposed ambiguity set (6) depends on a vector v. As discussed in §III-A, we would ideally take v = L(x ) with x ∈ X (p ). However, since this ideal is obviously out of reach, we instead look for suitable approximations. In particular, we propose to use the available datasetΞ in part to select v to approximate L(x ), and in part to calibrate the mean bound α. To this end, we will partition the available datasetΞ into a training set and a calibration set. Let τ : IN → IN be a user-specified function determining the size of the training set, which satisfies τ (m) ≤ cm for some c ∈ (0, 1); and (12a) τ (m) → ∞ as m → ∞.(12b) Correspondingly, let {Ξ T ,Ξ C } be a partition ofΞ, i.e.,Ξ T ∩ Ξ C = ∅ andΞ T ∪Ξ C =Ξ. Given that |Ξ| = m, we ensure that |Ξ T | = τ (m) and thus |Ξ C | = m := m − τ (m). Note that by construction, m ≥ (1 − c)m, with c ∈ (0, 1), and thus, both |Ξ T | → ∞ and |Ξ C | → ∞ as m → ∞. Due to the statistical independence of the elements inΞ, it is inconsequential how exactly the individual data points are divided intoΞ T andΞ C . Therefore, without loss of generality, we may takê Ξ T = {ξ 1 , . . . , ξ τ (m) } andΞ C = {ξ τ (m)+1 , . . . , ξ m }. With an independent datasetΞ T at our disposal, we may use it to design a mapping v τ (m) : Ξ τ (m) → IR d , whose output will be a data-driven estimate of L(x ). For ease of notation, we will omit the explicit dependence on the data, i.e., we write v τ (m) instead of v τ (m) (Ξ T ). We propose the following construction. Letp τ (m) = 1 τ (m) τ (m) k=1 e ξ k denote the empirical distribution ofΞ T and set v τ (m) = L(x τ (m) ), with x τ (m) ∈ argmin x∈X V (x,p τ (m) ).(13) Remark III.4. We underline that although (13) is a natural choice, several alternatives for the training vector could in principle be considered. To guide this choice, Lemma IV.2 provides sufficient conditions on the combination of α and v τ (m) to ensure consistency of the method. Given v τ (m) as in (13), we will from hereon use the following shorthand notation whenever convenient: A m := AΞ C (v τ (m) ), α m := αΞ C (v τ (m) ),(14) with AΞ C (v τ (m) ) as in (6). We correspondingly obtain the cost estimateV m according to (DRO). D. Selection of τ Given the conditions in (12), there is still some flexibility in the choice of τ (m), which defines a trade-off between the quality of v τ (m) as an approximator of L(x ) and the size of the ambiguity set A m . An obvious choice is to reserve a fixed fraction of the available data for the training set, i.e., set τ (m) /m equal to some constant. However, for low sample counts m, the mean bound α m will typically be large and thus A m will not be substantially smaller than the unit simplex ∆ d , regardless of v τ (m) . As a result, the obtained solution will also be rather insensitive to v τ (m) . In this regime, it is therefore preferable to reduce the conservativeness of α m quickly by using small values of τ (m) /m (i.e., large values of m = m − τ (m)). Conversely, for large sample sizes, α m is typically a good approximation of p , v τ (m) and the solution to (DRO) will be more strongly biased to align with v τ (m) . Thus, the marginal benefit of improving the quality of v τ (m) takes priority over reducing α m , and large fractions τ (m) /m become preferable. Based on this reasoning, we propose the heuristic τ (m) = µν m(m+1) µm+ν , µ, ν ∈ (0, 1). Note that µ and ν are the limits of τ (m) /m as m → 0 and m → ∞, respectively. Eq. (15) then interpolates between these extremes, depending on the total amount of data available. We have found µ = 0.01, ν = 0.8 to be suitable choices for several test problems. E. Tractable reformulation The proposed ambiguity set takes the form of a polytope, and thus, standard reformulations based on conic ambiguity sets apply directly [23]. Nevertheless, as we will now show, a tractable reformulation of (DRO) specialized to the ambiguity set (6) may be obtained, which requires fewer auxiliary variables and constraints . Proposition III.5 (Tractable reformulation of (DRO)). Fix parametersp ∈ ∆, v ∈ IR d , and α ∈ IR and let A = {p ∈ ∆ d | p, v ≤ α} be an ambiguity set of the form (6). Denoting V A := min x∈X max p∈A V (x, p), we have V A = min x∈X,λ≥0 λα + max i∈[d] { i (x) − λv i }.(16) Proof. Let g(z) := max p∈∆ d { p, z | p, v ≤ α}, where z ∈ IR d and α are constants with respect to p. By strong duality of linear programming [24], g(z) = min λ≥0 max p∈∆ d p, z − λ( p, v − α) = min λ≥0 λα + max p∈∆ d p, z − λv Noting that max p∈∆ d y = max i∈[d] y i , ∀y ∈ IR d and that V A = min x∈X g(L(x)), we obtain (16). If the functions { i } i∈ [d] are convex, then (16) is a convex optimization problem, which can be solved efficiently using off-the-shelf solvers. In particular, if they are convex, piecewise affine functions, then it reduces to a linear program (LP). For instance, introducing a scalar epigraph variable, one may further rewrite (16) as min x∈X,λ≥0,z∈IR {λα + z | L(x) − λv ≤ z1},(17) which avoids the non-smoothness of the pointwise maximum in (16) at the cost of a scalar auxiliary variable. Even for general (possibly nonconvex) choices of i , (16) is a standard nonlinear program, which can be handled by existing solvers. We conclude the section by summarizing the described steps in Alg. 1. Algorithm 1 CADRO IV. THEORETICAL PROPERTIES We will now show that the proposed scheme possesses the required theoretical properties, namely to provide (i) an upper bound to the out-of-sample cost, with high probability (cf. (3)); and (ii) a consistent estimate of the true optimal cost (cf. (4)). Let us start with the first guarantee, which follows almost directly by construction. Theorem IV.1 (Out-of-sample guarantee). Fix m > 0, and letV m ,x m be generated by Alg. 1. Then, P[V (x m , p ) ≤V m ] ≥ 1 − β.(18) Proof. If p ∈ A m , then V m (x) := max p∈Am V (x, p) ≥ V (x, p ), ∀x ∈ X.(19) Sincex m ∈ argmin x∈X V m (x), (19) implies that V (x m , p ) ≤ V m (x m ) =V m , where the last equality holds by definition (DRO). Consequently, p ∈ A m =⇒ V (x m , p ) ≤V m , and thus P[V (x m , p ) ≤V m ] ≥ P[p ∈ A m ]. Since v τ (m) is constructed independently fromΞ C , Assumption III.1 ensures that (5) holds with respect to A m = AΞ C (v τ (m) ), establishing the claim. We now turn our attention to the matter of consistency. That is, we will show that under suitable conditions on the mean bound α and the training vector v in (6),V m converges almost surely to the true optimal value, as the sample size m grows to infinity. We will then conclude the section by demonstrating that for the choices proposed in §III-B and III-C, the aforementioned conditions hold. . If v τ (m) = L(x τ (m) ), with x τ (m) , α m = αΞ C (v τ (m) ) chosen to ensure (i) p m , v τ (m) ≤ αΞ C (v τ (m) ), a.s.; (ii) lim sup m→∞ αΞ C (v τ (m) ) ≤ V (p ),x ∈ X, p m , L(x) ≤ V m (x) ≤ α m + ε m (x) ∞ . Minimizing with respect to x yields that for all m, V SAA m ≤V m ≤ α m ,(20) whereV SAA m := V (p m ) (cf. (2)). By the law of large numbers,p m → p , a.s. Furthermore, under Assumption II.1, Lemma A.5 states that the optimal value mapping V (p) is continuous, which implies that alsoV SAA m → V (p ), a.s. The claim then follows directly from condition (ii). Informally, Lemma IV.2 requires that the mean bound is bounded from below by the empirical mean, and from above by a consistent estimator of the optimal cost. The latter excludes choices such as the robust minimizer x τ (m) ∈ argmin max i∈[d] i (x) in the construction of v τ (m) . However, besides (13), one could consider alternatives, such as a separate DRO scheme to select v τ (m) . A more extensive study of such alternatives, however, is left for future work. We now conclude the section by showing that (13) (13), then,V m → V (p ), a.s. Proof. It suffices to show that conditions (i) and (ii) of Lemma IV.2 are satisfied by αΞ C (v τ (m) ). Condition (i): Consider αΞ C (v) as in (10) for an arbitrary v ∈ IR d , and let (η (i) ) i∈[m ] denote ( v, e ξ ) ξ∈ΞC , sorted in increasing order, then, we may write p m , v = 1 m m i=1 η (i) ,(21) and thus, αΞ C (v) − p m , v = κ m − γ η (κ) − κ i=1 η (i) m + γη, (a) ≥ κ m − γ η (κ) − κ m η (k) + γη, = γ(η − η (k) ) (γ≥0) ≥ 0, ∀v ∈ IR d , where (a) follows from the fact that η (i) are sorted. Condition (ii): By Lemma A.4, there exists a constant v ≥ v τ (m) ∞ , ∀m > 0, a. s. . Therefore, using (10) and (21), α m − p m , v τ (m) ≤ ( κ m − γ)v + κ m v + γv = 2v( κ m ) (b) ≤ 2v(γ + 1 m ),(22) for all m > 0, where (b) follows from κ = m γ ≤ m γ +1. By construction (see (12) and below), we have that both τ (m) → ∞ and m → ∞. Thus, using (11), γ + 1 m = log( 1 /β) 2m + 1 m → 0. Combined with (22), this yields that lim sup m→∞ αΞ C (v τ (m) ) − v τ (m) ,pΞ C ≤ 0.(23) Finally, by the law of large numbers,p m → p andp τ (m) → p , a.s. Thus (under Assumption II.1), Corollary A.6 ensures that lim m→∞ p m , L(x τ (m) ) = V (p ), which, combined with (23) yields the required result. Theorem IV.4 (Consistency -Hoeffding bound). LetV m be generated by Alg. 1, for m > 0. If α m = αΞ C (v τ (m) ) is selected according to Proposition III.2, with v τ (m) as in (13) then,V m → V (p ), a.s. Proof. We show that condition (i) and condition (ii) of Lemma IV.2 are satisfied. Condition (i): Trivial, noting that r m > 0 by (8). Condition (ii): By the law of large numbers, we have that p m → p , a.s., and thus, by Corollary A.6, p m , v τ (m) → V (p ). Furthermore, by Lemma A.4, there exists a constant v such that rg(v τ (m) ) ≤ 2v for all m ∈ IN. Thus, for r m given by (8), we have lim sup m→∞ αΞ C (v τ (m) ) ≤ lim sup m→∞ p m , v τ (m) + r m v = V (p ). This concludes the proof. V. ILLUSTRATIVE EXAMPLE As an illustrative example, we consider the following facility location problem, adapted from [25,Sec. 8.7.3]. Consider a bicycle sharing service setting out to determine locations x (i) ∈ X i ⊆ IR 2 , i ∈ [n x ], at which to build stalls where bikes can be taken out or returned. We will assume that X i are given (polyhedral) sets, representing areas within the city suitable for constructing a new bike stall. Let z (k) ∈ IR 2 , k ∈ [d], be given points of interest (public buildings, tourist attractions, parks, etc.). Suppose that a person located in the vicinity of some point z (k) decides to rent a bike. Depending on the availability at the locations x (i) , this person may be required to traverse a distance k (x) = max i∈[nx] x (i) − z (k) 2 , where x = (x (i) ) i∈[nx] . With this choice of cost, (16) can be cast as a second order cone program. Thus, if the demand is distributed over (z (k) ) k∈ [d] according to the probability mass vector p ∈ ∆ d , then the average cost to be minimized over X = X 1 ×· · ·×X d is given by V (x, p ) as in (2). We will solve a randomly generated instance of the problem, illustrated in Fig. 2. As p is unknown, one has to collect data, e.g., by means of counting passersby at the locations z (k) . As this may be a costly operation, it is important that the acquired data is used efficiently. Furthermore, in order to ensure that the potentially large up-front investment is justified, we are required to provide a certificate stating that, with high confidence, the quality of the solution will be no worse than what is predicted. Thus, given our collected sample of size m, our aim is to compute estimatesV m , satisfying (3). X1 X2 X3 z (k) argmin x∈X V (x, p ⋆ ) argmin x∈X max k∈[d] ℓ k (x) CADRO (m = 20) We compare the following data-driven methods. CADRO Solves (DRO) according to Alg. 1, setting τ (m) as in (15), with µ = 0.01, ν = 0.8. [28], ensuring that (5) is satisfied. SAA Using the same data partition {Ξ T ,Ξ C } as CADRO, we useΞ T to compute x m = x τ (m) as in (13), and we useΞ C to obtain a high-confidence upper boundV m = αΞ C (L(x τ (m) )), utilizing Proposition III.3. D-DRO Solves (DRO), with an ambiguity set of the form A m = {p ∈ ∆ d | D(p m , p) ≤ r D m }, with D ∈ {TV, Note that D-DRO does not require an independent data sample in order to satisfy (3). Remark V.1. Other methods could be used to validate SAA (e.g., cross-validation [2], replications [7]), but these methods only guarantee the required confidence level asymptotically. In order to obtain a fair comparison, we instead use the same mean bound, namely (10) for both CADRO and SAA, so both methods provide the same theoretical guarantees. Moreover, we note that a different data partition could be used for SAA. However, preliminary experiments have indicated that significantly increasing or decreasing τ (m) resulted in deteriorated bounds on the cost. We set n x = 3, d = 50, β = 0.01, and apply each method for 100 independently drawn datasets of size m. In Fig. 3, we plot the estimated costsV m and the achieved out-of-sample cost V (x m , p ), for increasing values of m. We observe that CADRO provides a sharper cost estimateV m than the other approaches. In particular, the classical DRO formulations require relatively large amounts of data before obtaining a non-vacuous upper bound on the cost. The right-hand panel in Fig. 3 shows that additionally, CADRO returns solutions which exhibit superior out-of-sample performance than the compared approaches, illustrating that it does not rely on conservative solutions to obtain better upper bounds. VI. CONCLUSION AND FUTURE WORK We proposed a DRO formulation, named cost-aware DRO (CADRO), in which the ambiguity set is designed to only restrict errors in the distribution that are predicted to have significant effects on the worst-case expected cost. We proved out-of-sample performance bounds and consistency of the resulting DRO scheme, and demonstrated empirically that this approach may be used to robustify against poor distribution estimates at small sample sizes, while remaining considerably less conservative than existing DRO formulations. In future work, we aim to extend the work to continuous distributions. | p − e i , v | ≤ rg(v), for all i ∈ [d], v ∈ IR d and p ∈ ∆ d . Proof. For any i ∈ [d], v ∈ IR d and p ∈ ∆ d , | p − e i , v | ≤ max{max i∈[d] p, v − v i , max i∈[d] v i − p, v } = max{ p, v − v min , v max − p, v } (a) ≤ v max − v min = rg(v), where (a) follows from the fact that max Then, for all x ∈ X, we have p∈∆ p, v = v max and max p∈∆ − p, v = max i∈[d] {−v i } = −v min .V (x) ≤ αΞ(v) + L(x) − v ∞ , a.s. Proof. Define ε(x) = L(x) − v for x ∈ X. We have V (x) = max p∈AΞ(v) p, v + p, ε(x) (6) ≤ αΞ(v) + max p∈AΞ(v) p, ε(x) . The claim directly follows because AΞ(v) ⊆ ∆ d and max p∈∆ d p, z = max i {z i }, ∀z ∈ IR d [29,Ex. 4.10]. Lemma A.3 (Uniform level-boundedness). If Assump- tion II.1(ii) holds, then V (x, p) = p, L(x) + δ X×∆ d (x, p) is level-bounded in x locally uniformly in p. Proof. Since V (x, p) is a convex combination of i (x), i ∈ [d], V (x, p) ≤ α implies that ∃i ∈ [d] : i (x) ≤ α. Therefore, lev ≤α V ( · , p) ⊆ i∈[d] lev ≤α i =: U α , for all p ∈ ∆ d . By Assumption II.1(ii), lev ≤α i is bounded for all i ∈ [d]. Since the union of a finite number of bounded sets is bounded, U α is bounded. Furthermore, for p / ∈ ∆ d , V (x, p) = ∞, and thus lev ≤α V ( · , p) = ∅ ⊆ U α , ∀p / ∈ ∆ d Thus, lev ≤α V (· , p) ⊆ U α for all p ∈ IR d . Lemma A.4 (Uniform boundedness of v τ (m) ). Let v τ (m) be defined as in (13). Then, there exists a v ∈ IR + such that v τ (m) ∞ ≤ v, ∀m ∈ IN, a.s. Proof. By Assumption II.1, there exists r := min x∈X max i∈[d] i (x) ≥ min x∈X V (x, p), ∀p ∈ ∆ d , so that, by (13), x τ (m) ∈ lev ≤r V ( · ,p τ (m) ), ∀m ∈ IN. Since V (x, p) is level-bounded uniformly in p (cf. Lemma A.3), there exists a compact set C ⊆ IR n satisfying (i) the optimal value V (p) defined by (2), is continuous at p relative to ∆ d . x τ (m) ∈ lev ≤r V ( · ,p τ (m) ) ⊆ C, ∀m ∈ IN.(24) (ii) For anyp m → p , and for any x m ∈ X (p m ), {x m } m∈IN is bounded and all its cluster points lie in X (p ). Proof. If L is continuous, then V (x, p) can be written as the composition V ≡ g • F of the lsc function g : IR 2d → IR : (y, z) → y, z + δ ∆ d (p), and F : IR nd → IR 2d : (x, p) → (L(x), p). By [17, Ex. 1.40(a)], this implies that V is lsc, and so is (x, p) → V (x, p) + δ X (x). Moreover, by Lemma A.3, it is level-bounded in x locally uniformly in p. Furthermore, p → V (x, p) is continuous relative to ∆ d for all fixed x ∈ X. On the other hand, since the sequence {x τ (m) ∈ X} m is bounded, and L is continuous on X, p m , L(x τ (m) ) has at least one cluster point and lim sup m→∞ p m , L(x τ (m) ) < ∞. Assume then, for the sake of contradiction, that there exists a cluster point V = lim sup m→∞ p m , L(x τ (m) ) > V (p ). Sincep m → p , this implies, by continuity of L, that there must exist a limit point x / ∈ X (p ) of {x τ (m) } m , contradicting Lemma A.5. We conclude that lim sup m→∞ p m , L(x τ (m) ) ≤ V (p ). Combining (25) and (26) completes the proof. B. Comparison with the Hoeffding bound We consider another instance of the example set-up from §V, and compare CADRO using the the Hoeffding bound (Proposition III.2) and the ordered mean bound (Proposition III.3) . Figure 4 shows the cost estimateV m and the out-ofsample cost V (x m , p ) for the TV-DRO method and the aforementioned versions of CADRO. We note that the radius of the ambiguity set for TV-DRO is computed using the Bretagnolle-Huber-Carol inequality [27, Prop. A.6.6] with slightly improved constants. As this result is based on the same Hoeffding-type inequality as Proposition III.2, The apparent performance gains of CADRO with the Hoeffding bound are thus to be attributed primarily to the geometry of the ambiguity set. However, unlike divergence-based ambiguity sets, which rely on concentration inequalities to bound deviations of the distribution from the empirical mean, (6) does not require the use of concentration inequalities. Rather, any high-confidence upper bound on the mean of a scalar random variable satisfying the conditions of Lemma IV.2 may be used, allowing the use of more sophisticated approaches (e.g., Proposition III.3). This results in the improvements visible in Fig. 4, without requiring alterations to the DRO method itself. and P. Patrinos are with the Department of Electrical Engineering (ESAT-STADIUS), KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium. Email: {mathijs.schuurmans, panos.patrinos}@esat.kuleuven.be This work was supported by the Research Foundation Flanders (FWO) research projects G081222N, G033822N, G0A0920N; European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 953348; Ford KU Leuven Research Alliance Project KUL0075; Fig. 1 . 1Conceptual motivation for the structure of the ambiguity set(6). The cost contour lines {p ∈ ∆ 3 | L(x), p = α} corresponding to some x ∈ X are shown for increasing values of α (dark to light), together with the sets A TV := ∆ 3 ∩ IB 1 (p, ) and A := {p ∈ ∆ 3 | L(x), p ≤ α}. Require: i.i.d. datasetΞ = {ξ1, . . . , ξm}; τ (m) (cf. (12)); Confidence parameter β ∈ (0, 1). Ensure: (Vm,xm) satisfy (3)-(4) Cf. §IV ΞT ← {ξ1, . . . , ξ τ (m) },ΞC ← {ξ τ (m)+1 , . . . , ξm} v τ (m) ← evaluate (13) (Vm,xm) ← solve (DRO) with Am = AΞ C (v τ (m) ) Use(16) Lemma IV.2 (consistency conditions). LetΞ T ,Ξ C be two independent samples from p , with sizes |Ξ T | = τ (m) and |Ξ C | = m := m−τ (m). Letp m := 1 m ξ∈ΞC e ξ denote the empirical distribution of the calibration setΞ C a.s. ThenV m → V (p ), a.s., whereV m is given by (DRO). Proof. Let V m (x) := max p∈Am p, L(x) . It is clear from condition (i) and (6) thatp m ∈ A m . Let us furthermore define ε m (x) = L(x) − L(x τ (m) ). Then, by Lemma A.2 , we have for all satisfy the requirements of Lemma IV.2. Theorem IV.3 (Consistency -Ordered mean bound). LetV m be generated by Alg. 1, for m > 0. If α m = αΞ C (v τ (m) ) is selected according to Proposition III.3, with v τ (m) as in Fig. 2 . 2Illustration of the facility location problem. The colors of the points z (k) represent their probability p k . Fig. 3 . 3Results of the facility location problem of Section V. (left): The cost estimatesVm satisfying (3) and (4); (right): True out of sample cost V (xm, p ). The points indicate the sample mean, the solid errorbars indicate the empirical 0.95 (upper and lower) quantiles and the semi-transparent errorbars indicate the largest and smallest values over 100 independent runs. . Let e i denote the i'th standard basis vector. Lemma A.2 (Upper bound). Fix v ∈ IR d and consider a sampleΞ . For an ambiguity set AΞ(v), given by (6) with mean bound αΞ(v) , define V (x) := max p∈AΞ(v) V (x, p). Thus, [17, Thm. 1.17(b),(c)] applies, translating directly to statements (i) and (ii). Corollary A.6. Let {x τ (m) } m∈IN be generated by (13) and let {p m ∈ ∆ d } m∈IN be some sequence withp m → p . Then, lim m→∞ p m , L(x τ (m) ) = V (p ) Proof. By definition of V , we have p m , L(x τ (m) ) ≥ V (p m ), and by Lemma A.5, lim m→∞ V (p m ) → V (p ). Therefore, lim inf m→∞ p m , L(x τ (m) ) ≥ V (p ). Fig. 4 . 4Results for a problem instance as described in Section V. (left): The cost estimatesVm satisfying (3) and (4); (right): True out of sample cost V (xm, p ). The points indicate the sample mean, the solid errorbars indicate the empirical 0.95 (upper and lower) quantiles and the semi-transparent errorbars indicate the largest and smallest values over 100 independent runs. 10) For finite m, the smallest value of γ ensuring that Proposition III.3 holds, can be computed efficiently by solving a scalar root-finding problem[21, Rem. IV 3]. 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{'abstract': 'We present a novel framework for distributionally robust optimization (DRO), called cost-aware DRO (CADRO).The key idea of CADRO is to exploit the cost structure in the design of the ambiguity set to reduce conservatism. Particularly, the set specifically constrains the worst-case distribution along the direction in which the expected cost of an approximate solution increases most rapidly. We prove that CADRO provides both a high-confidence upper bound and a consistent estimator of the out-of-sample expected cost, and show empirically that it produces solutions that are substantially less conservative than existing DRO methods, while providing the same guarantees.', 'arxivid': '2303.09408', 'author': ['Mathijs Schuurmans ', 'Panagiotis Patrinos '], 'authoraffiliation': [], 'corpusid': 257557748, 'doi': '10.1109/lcsys.2023.3281974', 'github_urls': [], 'n_tokens_mistral': 15001, 'n_tokens_neox': 13575, 'n_words': 8019, 'pdfsha': 'd643be2a2084e556b435d9dd33db9e1d0a91e7dc', 'pdfurls': ['https://export.arxiv.org/pdf/2303.09408v2.pdf'], 'title': ['Distributionally Robust Optimization using Cost-Aware Ambiguity Sets', 'Distributionally Robust Optimization using Cost-Aware Ambiguity Sets'], 'venue': []}

arxiv

Underlying mechanism behind high critical temperature in cuprate superconductors 17 May 2023 Zhipeng Sun Hai-Qing Lin Beijing Computational Science Research Center Beijing Computational Science Research Center 100193, 100193Beijing, BeijingChina, China and Zhejiang University 310027HangzhouChina Underlying mechanism behind high critical temperature in cuprate superconductors 17 May 2023(Dated: May 18, 2023) AbstractMotivated by recent discovery of anomalously strong near-neighbor attraction in the doped cuprate chains, we revisited the two-dimensional extended Hubbard model at a mean-field level, with the aim of the mechanism behind high critical temperature in cuprate superconductors. Considering the strong spin fluctuations caused by local repulsion, we assume that above T c , the system tends to be in the antiferromagnetic phase, and consequently, the nonlocal superconducting order parameters also undergo the identical spatial modulation below T c . Numerical results show that T c increases almost linearly with |V | for sufficiently large |V |, while it increases slowly with U and saturates for sufficiently large U . Further theoretical analysis suggests that the underlying mechanism is due to the ultra-narrow property of the effective bands, bestowed by the large antiferromagnetic gap. These findings are of great importance for understanding the commonality among unconventional superconductors and for designing higher-temperature superconducting materials.Introduction.-The mechanism of high-temperature superconductivity in the doped cuprates has been a long-standing topic in condensed matter physics[1][2][3][4][5][6][7]. In addition to exhibiting critical temperatures much higher than conventional superconductors[8,9], the cuprates display a variety of anomalous properties, such as antiferromagnetism [10], strange metal behavior [11], pseudogap phenomenon[12], and stripe order[13][14][15]. These strongly correlated effects are widely simulated based on Hubbard-like models [16, 17, 19? -21]; however, the existence of superconductivity in the simple Hubbard model is still controversial[22][23][24][25].Recent experimental discovery of anomalously strong near-neighbor attraction in the doped cuprate chains [26] suggests the existence of an additional attractive potential, which might originate from electron-phonon interactions [27, 28]. It implies that the minimal model for cuprate superconductors is the extended Hubbard model with strong local repulsion and intermediate nearest-neighbor attraction. Previously, this model was regarded as a phenomenological model to understand various superconducting systems[2,29, 30], and recent reconsideration of it has also contributed to the understanding of high-temperature superconductivity [31,32]. However, the underlying mechanism behind high critical temperature remains unclear.This Letter aims to reveal this mechanism by revisiting the extended Hubbard model at the mean-field level. Different from Micnas' early work [29], we assume the system tends to be in the antiferromagnetic phase above T c , considering the strong local repulsion. Combining numerical results and theoretical analysis, we conclude that the fundamental difference between cuprate superconductors and conventional superconductors lies in the ultra-narrow property of the effective bands, which leads to a near-linear dependence of T c on the attraction strength. We believe that our findings make important contributions to the understanding of the commonalities among unconventional superconductors. (1) Hereĉ † α,r (ĉ α,r ) is the fermionic creation (annihilation) operator with spin α at lattice site r,n α,r ≡ĉ † α,rĉ α,r is the spin-selective density operator, andρ r =n ↑,r +n ↓,r is the charge density operator. δ represents the vectors linking nearest neighbors, t is the nearest-neighbor hopping strength, U is the onsite repulsion, |V | is the nearest-neighbor attraction, and µ is the chemical potential. We adopt the standard symmetry-broken Hartree-Fock framework and assume that the spin density prefers the antiferromagnetic arrangement; that is, the spin density takes the form m r = me iQ·r with m ≥ 0 and Q = (π, π), and m = 0 only if the nonzero solution does not exist. Under this consideration, the simplest ansatz for the charge density is ρ r = ρ and that for the intersite correlator is ĉ † α,rĉ α,r+δ = 1 2 ρ ′ . For the superconducting order parameters, we do not consider the equal-spin pairings. The simplest ansatz for local pairing order parameter is ĉ ↓,rĉ↑,r = ∆, and that for the unequal-spin nonlocal pairing order parameter is ĉ † α,rĉ α,r+δ = ∆ ′ δ,0 + ∆ ′ δ,Q e iQ·r . By these ansatz, the mean-field Hamiltonian is given byĤ MF = i,j,pĉ i † p h ij pĉ j p , where p is confined in the half of the first Brillouin zone, c i † p represents for an array ĉ † ↑,−pĉ † ↑,−p+Qĉ ↓,pĉ↓,p+Q , and h p is a 4 × 4 matrix. Now we focus on the superconducting boundary, where the superconducting order parameters are small quantities. The superconducting gap equation near T c turns out to be a ninth-order linear system of equations, corresponding to the nine independent components for the superconducting order parameters. Based on the symmetries, the gap equation can be further decomposed into 3rd-order for pure s-wave pairing, 2nd-order for pure d-wave pairing, and 2nd-order for two kinds of pure p-wave pairings. The onset of s-wave pairing instability is severely suppressed by onsite repulsion, and thus we only focus on the d-wave and p-wave pairings. Their gap equations both take the form      X 0,ϕ X Q,ϕ      = |V | V p ϕ 2 p      F − p + F + p sin θ p F + p − F − p sin θ p F + p − F − p sin 2 θ p F − p + F + p + 2 cos 2 θ p F ′ p           X 0,ϕ X Q,ϕ      .(2) Here θ p is determined by cot θ p = 2t U m γ p if m = 0 otherwise θ p = 0, wheret = t − 1 2 |V | ρ ′ is the renormalized nearest-neighbor hopping strength. The pairing wave function ϕ p equals to cos p x − cos p y for d-wave and sin p x ± sin p y for p-wave pairings. X 0,ϕ and X Q,ϕ are both symmetry-specified order parameters relating to ∆ ′ δ,0 and ∆ ′ δ,Q . And the "bubble" F 's are defined as Figure 3a, together with optimal doping shown in Figure 3b. Remarkably, optimal T c increases almost linearly with |V | for sufficiently large |V |, while it increases slowly with U and saturates at 0.25 |V | for sufficiently large U. These relations have important implication for designing high-temperature superconducting materials, and the results of optimal doping can also serve as a reference for experiments. F ± p = 1 2ξ p,± tanh ξ p,± 2T ,(3a)F ′ p = 1 ξ p,+ + ξ p,− 1 2 tanh ξ p,− 2T + tanh ξ p,+ 2T , (3b) where ξ p,± = ± 4t 2 γ 2 p + U 2 m 2 −μ It should be noted that for conventional superconductors, the relationship T c ∼ ω D exp (−1/N F |V |) is subject to two important premises [34]. One thing is that the Debye frequency ω D is much larger than the bandwidth, and thus can serve as the characteristic frequency; the other thing is that the characteristic frequency is much larger than T c . For an ultra-narrow-band system, the first condition does not hold, and the characteristic frequency should be comparable to the bandwidth; the second condition is also not guaranteed to hold, especially when the attractive potential is strong. Therefore, it is understandable that the McMillan limit is not applicable to a series of unconventional superconductors. Analysis in a limiting case.-Here we consider a limiting case where Um ≫ 4t. Owing to the large antiferromagnetic gap, the physics mainly lies in in the lower band, whose bandwidth D is ∼ 8t 2 /Um, much less than 8t in the noninteracting case. Meanwhile, the gap equation Eq.2 is reduced to 1 |V | = 1 V/2 p ϕ 2 p F − p .(4) Note that this equation is similar to the gap equation for conventional superconductors, except for the symmetries of pairing wave functions. The right-hand side of Eq.4 is equal to 1 2ω 0 tanh ω 0 2Tc , where ω 0 is confined in the lower band. Then we obtain T c = |V | 4 2ω 0 / |V | arctanh 2ω 0 / |V | .(5) For an ultra-narrow-band system with strong intersite attraction, we can expect the existence of the case 2ω 0 / |V | ∼ 0, and thus T c ∼ |V | /4. On the contrary, we believe that for conventional superconductors, 2ω 0 / |V | is close to 1, and thus T c is rather small. Our analysis can also be extended to the flat-band superconductors, where ω 0 can be approximately the energy difference between the flat band and the Fermi level. We conclude that the nearly linear dependence of T c on the attraction strength also exists, as the attraction is sufficiently strong. Summary and outlook.-In summary, we investigated the critical temperature in the two-dimensional extended Hubbard model and observed an anomalous linear relationship 7 between T c and the nearest-neighbor attraction in numerical simulations. Theoretical analysis revealed that the (lower) energy band of the system becomes remarkably narrow under the influence of antiferromagnetic ordering; and for the case of ultra-narrow-band systems, T c exhibits a linear dependence on the strength of the attractive potential. Therefore, we conclude that the ultra-narrow-band property is a key factor that enables cuprate superconductors to achieve significantly higher T c compared to conventional superconductors. Considering the similarity of the structures of unconventional superconductors, especially the ultra-narrowness or near flatness of band [35][36][37][38][39][40], we believe our findings are important to understand the commonality of unconventional superconductors. In addition, our qualitative description of the critical temperature versus U and |V | has guidance significance for the design of high-temperature superconducting materials. Our work has qualitatively described the physics of high-temperature physics, but there are still some limitations and open questions. First, under the standard mean-field framework, it is more reasonable to assume a general spin density wave order with wave vector Q = (π, π) in doped regions since it is more stable [41]. Such assumption is expected to prompt the understanding of the onset of the charge density wave, nematicity, stripe and pairing density wave orders. Second, the fluctuation effects are not in consideration. We may assort to non-perturbative methods to produce more precise numerical results, and investigate the pseudogap physics and its connection to superconductivity. Third, the origin of the intersite attraction is very likely from the near-neighbor electron-phonon interaction, which induces a dynamic pairing attraction [9,42]. Hence, the electron-phonon interacting models, such as the extended Hubbard-Holstein model, might be better starting points for simulating realistic materials. [28] T. Tang, B. Moritz, C. Peng, Z. X. Shen, and T. P. Devereaux, (2022). [29] R. Micnas Model and method.-We investigate the extended Hubbard model with strong onsite repulsion and nearest-neighbor attraction on a two-dimensional square V = L × L lattice, are the effective dispersions of the upper and lower band branches, withμ the renormalized chemical potential. The critical temperature T c can then be determined based on the condition of the existence of a nonzero solution to Eq.2. More details are presented in Supplemental Material [33]. Numerical results.-The numerical calculations are all performed on a 512 × 512 lattice in the hole-doped case ρ < 1. t = 1 is set as the unit of energy. The parameter range in consideration is based on the experimental results on the cuprate chains [26], where U ∼ 8t and |V| ∼ t. To gain a global understanding of the features of T c , we plot its dependence on the doping 1 − ρ for two sets of U values (4.0, 8.0) and two sets of |V | values (0.5, 1.0), for d-wave instability (solid line) and p-wave (dashed line) instability, as shown in Figure 1. The plot reveals several features as follows: first, the critical temperature T c shows a dome shape with respect to doping; second, the d-wave instability generally has a higher T c than the p-wave instability; third, there is a clear positive correlation between T c and the absolute value of the nearest-neighbor attraction |V |; fourth, T c is positively correlated with the local repulsion U, but the correlation is not as significant. The first two features are broadly compatible with experimental results, while the latter two require further experimental verification. FIG. 1. Critical temperature versus doping for d-wave instability (solid line) and p-wave instability (dashed line) at varying values of U (4.0, 8.0) and |V | (0.5, 1.0). We further investigate the quantitative dependence of T c on U and V , which has important implications for the design of high-temperature superconducting materials. The plot of optimal T c versus the nearest-neighbor attraction |V | at varying values of U (4.0, 8.0, 16.0) is shown in Fig. 2a, together with the plots of doping shown in Fig. 2b. The plot of the optimal T c versus the inverse local repulsion 1/U at varying values of |V | (0.5, 1.0, 2.0) is shown in ] Y. Wang, Z. Chen, T. Shi, B. Moritz, Z.-X. Shen, and T. P. Devereaux, Physical Review Letters 127, 197003 (2021). , J. Ranninger, S. Robaszkiewicz, and S. Tabor, Physical Review B 37, 9410 (1988).[30] M. Kheirkhah, Z. Yan, Y. Nagai, and F. Marsiglio, Physical Review Letters 125, 017001 (2020). [31] D.-W. Qu, B.-B. Chen, H.-C. Jiang, Y. Wang, and W. Li, . 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{'abstract': 'AbstractMotivated by recent discovery of anomalously strong near-neighbor attraction in the doped cuprate chains, we revisited the two-dimensional extended Hubbard model at a mean-field level, with the aim of the mechanism behind high critical temperature in cuprate superconductors. Considering the strong spin fluctuations caused by local repulsion, we assume that above T c , the system tends to be in the antiferromagnetic phase, and consequently, the nonlocal superconducting order parameters also undergo the identical spatial modulation below T c . Numerical results show that T c increases almost linearly with |V | for sufficiently large |V |, while it increases slowly with U and saturates for sufficiently large U . Further theoretical analysis suggests that the underlying mechanism is due to the ultra-narrow property of the effective bands, bestowed by the large antiferromagnetic gap. These findings are of great importance for understanding the commonality among unconventional superconductors and for designing higher-temperature superconducting materials.Introduction.-The mechanism of high-temperature superconductivity in the doped cuprates has been a long-standing topic in condensed matter physics[1][2][3][4][5][6][7]. In addition to exhibiting critical temperatures much higher than conventional superconductors[8,9], the cuprates display a variety of anomalous properties, such as antiferromagnetism [10], strange metal behavior [11], pseudogap phenomenon[12], and stripe order[13][14][15]. These strongly correlated effects are widely simulated based on Hubbard-like models [16, 17, 19? -21]; however, the existence of superconductivity in the simple Hubbard model is still controversial[22][23][24][25].Recent experimental discovery of anomalously strong near-neighbor attraction in the doped cuprate chains [26] suggests the existence of an additional attractive potential, which might originate from electron-phonon interactions [27, 28]. It implies that the minimal model for cuprate superconductors is the extended Hubbard model with strong local repulsion and intermediate nearest-neighbor attraction. Previously, this model was regarded as a phenomenological model to understand various superconducting systems[2,29, 30], and recent reconsideration of it has also contributed to the understanding of high-temperature superconductivity [31,32]. However, the underlying mechanism behind high critical temperature remains unclear.This Letter aims to reveal this mechanism by revisiting the extended Hubbard model', 'arxivid': '2304.07490', 'author': ['Zhipeng Sun ', 'Hai-Qing Lin ', '\nBeijing Computational Science Research Center\nBeijing Computational Science Research Center\n100193, 100193Beijing, BeijingChina, China\n', '\nand Zhejiang University\n310027HangzhouChina\n'], 'authoraffiliation': ['Beijing Computational Science Research Center\nBeijing Computational Science Research Center\n100193, 100193Beijing, BeijingChina, China', 'and Zhejiang University\n310027HangzhouChina'], 'corpusid': 258740671, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7741, 'n_tokens_neox': 6328, 'n_words': 3325, 'pdfsha': 'c8c8e3e910f0907e0c277faf4b72f5c4ab15e450', 'pdfurls': ['https://export.arxiv.org/pdf/2304.07490v2.pdf'], 'title': ['Underlying mechanism behind high critical temperature in cuprate superconductors', 'Underlying mechanism behind high critical temperature in cuprate superconductors'], 'venue': []}

arxiv

The RHIC fireball as a dual black hole arXiv:hep-th/0501068v3 22 Mar 2006 Horatiu Nastase Brown University Providence 02912RIUSA The RHIC fireball as a dual black hole arXiv:hep-th/0501068v3 22 Mar 2006 We argue that the fireball observed at RHIC is (the analog of) a dual black hole. In previous works, we have argued that the large s behaviour of the total QCD cross section is due to production of dual black holes, and that in the QCD effective field theory it corresponds to a nonlinear soliton of the pion field. Now we argue that the RHIC fireball is this soliton. We calculate the soliton (black hole) temperature, and get T = 4a < m π > /π, with a a nonperturbative constant. For a = 1, we get 175.76MeV , compared to the experimental value of the fireball "freeze-out" of about 176MeV . The observed η/s for the fireball is close to the dual value of 1/4π. The "Color Glass Condensate" (CGC) state at the core of the fireball is the pion field soliton, dual to the interior of the black hole. The main interaction between particles in the CGC is a Coulomb potential, due to short range pion exchange, dual to gravitational interaction inside the black hole, deconfining quarks and gluons. Thus RHIC is in a certain sense a string theory testing machine, analyzing the formation and decay of dual black holes, and giving information about the black hole interior. The study of high energy scattering in QCD via AdS-CFT [1] was started by the work of Polchinski and Strassler [2]. They argued that high energy scattering in QCD with momentum p and wavefunction e ipx corresponds to scattering in a space AdS 5 × X 5 , ds 2 =r 2 R 2 d x 2 + R 2 r 2 dr 2 + R 2 ds 2 X = e −2y/R d x 2 + dy 2 + R 2 ds 2 X(1) with AdS 5 cut off in the IR (at smallr, or large y), and local AdS momentump µ = (R/r)p µ and wavefunction e ipx ψ(r, Ω 5 ). The cut off corresponds to our ignorance of the IR modification of the gravity dual and is represented by an IR brane, giving a RS model [3]. Later, Giddings [4] argued that one will start to create black holes in AdS when one reaches the Planck scale M P , and this corresponds in gauge theory to a power law behaviour of the total cross section aboveM P = N 1/4 Λ QCD , and that the gauge theory maximal Froissart behaviour will onset when the size of the black holes reaches the AdS size, and the black hole is on the IR brane. In [5], this picture was made precise. The AdS scattering was modelled by scattering of Aichelburg-Sexl shockwaves [6,7], moving in the AdS background [8]. The shockwaves in flat space are ds 2 = 2dx + dx − + (dx + ) 2 Φ(x i )δ(x + ) + d x 2(2) where the function Φ satisfies the Poisson equation ∆ D−2 Φ(x i ) = −16πGpδ D−2 (x i )(3) In curved space, one just puts the above in the background, and the laplacean ∆ is also in the background. It was found that indeed, the scattering produces black holes, and then for flat space scattering one obtains a power law for the gauge theory cross section, σ gauge ∼ s 1/(D− 3) , and for scattering on the IR brane one obtains maximal Froissart behaviour σ gauge ∼ 1/M 2 ln 2 s. In [9] we have seen that this picture maps exactly to the 1952 effective field theory picture of Heisenberg [10] for the saturation of the Froissart bound, of collisions of shockwaves distributions of the pion field. To be exact, the dual picture refers to the case when there is no pion (almost Golstone boson), but only a lightest glueball (dual to the lightest KK gravity mode). But there is a simple argument [4,9] that the pion case should be similar, and involves the radion (position of the IR brane) as pion dual. Heisenberg uses for the nonlinear scalar pion action the DBI action S = l −4 1 + l 4 [(∂φ) 2 + m 2 φ 2 ](4) which is just the radion action at m = 0 and may provide a nonlinear version of the stabilized radion at m = 0. He then collides pion field shockwaves, which at x + ≤ 0, x − ≤ 0 are just φ = φ 1 (x i )δ(x + ) + φ 2 (x i )δ(x − )(5) The field Φ in (2) takes the role of the pion field φ in the Heisenberg picture, with exponential decay Φ ∼ r −1/2 e −M 1 r similar to the pion field decay around hadrons. The exact mapping of the dual scattering to Heisenberg scattering suggests that there should be a nonlinear pion field soliton being created in the collision, mapped to the black hole created in the dual collision. Again, strictly speaking this applies only for the lightest glueball case, but it should also be extended to the pion case by the argument in [4,9]. One cannot find this soliton in perturbation theory, and Heisenberg calculated only the perturbative solution close to the interaction, and far away, where one has just free waves (free pions, radiated away). In the gravity dual we could argue for black hole formation because we could calculate the formation of a trapped surface at the interaction point, using a formalism developped for flat d=4 in [11] and extended to curved general d in [12]. The beauty of GR is that we have a theorem stating that a horizon will form outside the trapped surface. In [13], we have analyzed in detail the energy regimes of gauge theories and found that before the Froissart saturation, one has a further energy regime. The gravity dual scattering happens at a certain distance from the IR brane, and the black hole starts to feel the AdS size before reaching the IR brane. Scattering in AdS d+1 × Xd gives a field Φ (solution to the Poisson equation (3) ) Φ ∼ 1 r 2(d−1)+d = 1 r 11(6) and gives a gauge theory cross section σ gauge ∼ s 1/11 . We have argued that this behaviour should onset at the scaleÊ R = N 2 Λ QCD , and the maximal Froissart behaviour should onset at an unknown energy scaleÊ F , depending on the IR details of the gravity dual, reached when the black hole horizon merges with the IR brane. In the case when the pion is the lightest excitation, the Froissart behaviour would onset at a scaleÊ ′ F (m π ) <Ê F , reached when the IR brane bending engulfs the black hole. We have also argued that this analysis goes through for real QCD as well, as string g s and α ′ corrections to the scattering [12] are small above E R , and then so are 1/N and 1/(g 2 Y M N) corrections in the gauge theory aboveÊ R . In QCDÊ R = N 2 c M 1,glueball ∼ 10GeV , and thus above it one should see the "soft Pomeron" behaviour σ tot (s) ∼ s 1/11 , consistent with experimental evidence. We noticed that the Heisenberg model can be mapped to the dual A-S shockwave scattering picture even before the saturation of the Froissart bound. All we have to do is relax Heisenberg's assumption of exponentially decaying pion wavefunction around the hadrons, and have instead the wavefunction Φ of the A-S shockwave which is mapped to a wavefunction of the lightest glueball field. In this case however, unlike for the maximal Froissart behaviour, the dual picture is not 4 dimensional anymore, it is 10 dimensional, and we have the usual holography. This Heisenberg description of the gauge theory scattering means that the black hole being formed in the dual scattering is still mapped to a nonlinear effective field soliton, but the effective field is the lightest glueball. AboveÊ R ∼ 10GeV , this description is exact, below it could be modified due to string corrections. So we predict a nonlinear effective field soliton, dual to a black hole, being produced in high energy collisions, at least above 10GeV, and maybe also aboveM P = N 1/4 c M 1,glueball ∼ 1 − 2GeV , but in the latter case the soliton is small enough so it could be hard to isolate. So where is this soliton at higher energies? Then clearly it should have been already observed, as its size should be sufficiently large. We argue that it has already been observed, at RHIC. Indeed, RHIC functions at a center of mass energy of 100+100 GeV per nucleon, in the collision of Au+Au. This energy should be enough even for the formation of a soliton in the collision of just two nucleons, but certainly for two Au nuclei (A=197 for the common isotope). At that huge total energy, one will clearly be in the maximal Froissart regime in the gauge theory, so one will create the pion analog of black holes on the IR brane. But RHIC does observe a fireball at finite temperature, exactly as expected! RHIC expected to see a ball of "Quark Gluon Plasma" (QGP), that is a free gas of deconfined quarks and gluons, at a temperature above the phase transition temperature for chiral restoration (and maybe deconfinement), which is obtained from the lattice to be in the 170−180MeV domain. One does see a thermal object indeed, but at its core is a deconfined, yet strongly coupled phase dubbed "Color Glass Condensate" (CGC). As the CGC expands, it is found to go into the thermal QGP, and eventually decay into free pions (and some other particles), thermally distributed corresponding to a "freeze-out" temperature of 176MeV [14,15]. There seems to be some debate about whether the CGC refers to the colliding objects or the object formed in the collision, but we will adopt the terminology of the latter case in the following. We want to argue that this is just the nonlinear pion field soliton, dual to the black hole on the IR brane, and the "freeze-out" temperature observed is the black hole temperature. As we mentioned, the nonlinear field soliton will decay into free pions, dual to the black hole radiating away gravitons. Let us therefore try to calculate the temperature. We will try to follow the usual 4d black hole thermodynamics calculations. We start from the thermodynamics relation [16] dM = T dS, where M is the black hole mass, T is temperature and S its entropy. We assume that we still have S = M 2 P,4 A/4, where A is the horizon area. This relation was derived by knowing the temperature of the black hole, but it seems to be a fundamental relation in quantum gravity, and has been tested in numerous string theory calculations [17,18],...(see e.g. [19] for more references). Then, as the black hole on the IR brane is spherically symmetric, we have A = 4πr 2 H , therefore (l H = 2r H ) dM = πT 4 M 2 P,4 d(l 2 H )(7) We notice that if we have M 2 P,4 d(l 2 H ) = dM aM 1(8) with M 1 the mass of the KK graviton (mapped to the pion mass) and a a numerical constant, we get T = a 4M 1 π(9) Let's see what can we say in perturbation theory, without knowing the full nonperturbative black hole solution. In perturbation theory [4] −g 00 = 1 − kG 4 M r e −M 1 r where k is a numerical constant and G 4 ≡ (RM 3 P ) −1 . Then r H ≃ 1 M 1 ln(kG 4 MM 1 )(11) that implies M 2 P,4 d(r 2 H ) = dM M 1 2( M 2 P,4 M 1 M )ln k( M 1 M M 2 P,4 )(12) which is different than (8), but not too much if MM 1 ∼ M 2 P,4 . In any case, this is perturbation theory, and there is no reason to trust it near the horizon. Let's try a different perspective and use Hawking's original calculation of the temperature of 4d black holes of Kerr-Newman type (with mass, charge and angular momentum) [20]. He found the expression T = κ 2π (13) where κ is the surface gravity of the horizon, defined by ∇ a (χ b χ b ) = −2κχ a(14) and χ a is the Killing vector that is tangent to the null geodesic generator of the horizon, in our case just χ a = (∂/∂t) a . Then 4κ 2 = lim horizon ||∇ a (χ b χ b )|| 2 χ a χ a = lim horizon [∂ r (g tt )] 2 g rr + [∂ y (g tt )] 2 g yy g tt(15) For a Schwarzschild black hole, g rr = −g tt = 1 − 2MG 4 /r, and ∂ r (g tt )| horizon = 1/(2MG 4 ), giving T = 1/(8πMG 4 ). But in general we have g rr = g tt for a black hole, in fact that is how one nonextremalizes D-brane solutions in flat space [21] to obtain black branes solutions, so we will assume that the same holds now, at least at the horizon. Also, for our solution, the horizon is almost 4 dimensional, so g yy ∼ 1 and the second term in (15) can be neglected at the horizon, thus we get 2κ = |∂ r (g tt )| horizon (16) Assuming that near the horizon g tt looks similar to the perturbative solution (10), namely dominated by an exponential decay, with subleading power law behaviour, of the type g tt ≃ 1 − Ae −M 1 r r n(17) we get κ = 1 2 |∂ r (g tt )| horizon ≃ 1 2 (M 1 + n r H ) ≃ M 1(18) If we also have the law (13), we get T ≃ M 1 2π(19) which is indeed of the form in (9), with a ≃ 1/8. The calculation of the surface gravity of the horizon was fairly general, the only possible weak point being the assumption g rr = g tt at the horizon. But there is no a priori reason to still have exactly (13), as that was done for black holes in asymptotically flat 4d space, and now we have a black hole on the IR brane, for highly curved AdS space, so the topological structure of infinity, thus the Penrose diagram used for the calculation, could be modified. One should redo Hawking's analysis in that background. Thus we have a reason to believe that (9) is correct. It also makes sense from the point of view of dimensional analysis. The temperature is independent of M and M P,4 , and we understand this physically since the black hole is dominated by the exponential decay, whose only scale is M 1 . The solution a ≃ 1/8 is probably just due to our use of the formula (13), so we leave it a free parameter. Indeed, if a = 1, and we replace M 1 by the pion mass m π , by which we mean of course the average mass (m π + + m π − + m π 0 )/3, we get T = 4 π < m π >= 175.76MeV (20) remarkably close to the experimental value of the RHIC fireball "freeze-out" of 176MeV [14,15]. Since that value is argued to be also the temperature of the chiral restoration (and maybe deconfinement) phase transition (lattice results also obtain a value in that region), it is possible that the dual calculation gives also the temperature of that phase transition. For more experimental evidence of the identification of the fireball and the dual black hole, we turn to the calculation of the quantity shear viscosity over entropy, η/s. It was argued [22] that this quantity is always bounded from below by 1/(4π) (see also the η calculation in [23] and [24]). In [25] (see also [24,26]) a theorem was proven stating that black holes in type IIB gravity duals saturate this bound, thus η/s = 1/(4π). But experimentally, under the assumption of a hydrodynamic model (about which there is still considerable debate), one finds that the fireball has a value that is very close to it, η/s ∼ 0.1 − 0.3 [27,28]. What was regarded as an interesting curiosity can now be argued should be rigorous equality, as the fireball should be nothing but a gravity dual black hole. Therefore we can confidently say that the RHIC fireball is the conjectured nonlinear pion field soliton dual to the black hole on the IR brane. Let us explore the consequences of this. The first and most obvious is the evolution of the fireball. One first creates a pion field soliton=black hole, that quickly thermalizes and then rapidly decays through the emission of pions= dual gravitons, almost in an explosion. Thus the "Color Glass Condensate" (CGC) state at the core of the fireball is the pion field soliton=black hole interior. One of the experimental reasons for saying that the state at the fireball core is a glass-like state (CGC), is the fact that there are almost no hard scattering events, a phenomenon known as "jet quenching" (absence of jets). Hard scattering signals the presence of perturbative QCD interactions (due to asymptotic freedom). The fact that these hard scattering events are suppressed is strange, and would usually mean that the large transverse momentum of the perturbative interaction is somehow atenuated (absorbed) over the very short distance of the fireball. This seems impossible in perturbative QCD, hence the QGP picture was replaced by an unknown CGC picture, implying very strong interactions that absorb the transverse momentum. Now we have a very simple picture of the CGC interaction. Any particles that are in the interaction region get caught in the formation of the black hole, and can only escape by being radiated away in a thermal manner. Thus "jet quenching" is nothing other than the usual black hole "information paradox", that information gets inside the black hole and only radiation gets out. The idea that jet quenching can be explained by matter falling into a black hole by AdS-CFT has been also proposed in the context of N = 4 SYM at finite temperature in [29], but we are treating here the case of scattering in the real QCD gravity dual (and our 4d space is at zero temperature). That also means that it will probably be very hard, if not impossible to probe the interior of the fireball with extra energetic particles present close to the interaction region that enter the fireball. That would amount in the black hole picture to probing the interior of the black hole with other particles, and is clearly impossible. However, given that the formed black hole is small enough and its lifetime finite, it could be possible in principle, by the right combination of time and length scales, to probe the formation and decay phases of the black hole. Finally, what kind of state is the CGC, meaning how do particles inside it interact? The question is clearly answered in both the effective field theory and in the dual picture. In the effective field theory, we have a pion field soliton, thus particles should interact mostly by exchange of pions. As we are at distances r < 1/m π , the pion Yukawa potential is replaced by the Coulomb potential. In the dual picture, we have the perturbative gravitational (Newton) potential (10), thus at distances r < 1/M 1 we have U = − kG 4 M r(21) As we mentioned, the actual dual to pion interaction should be brane bending (radion interaction), for which one has [4,13] U ∼ δL L | lin ∼ − kG 4 (M L R)M r (22) and correspondingly the gauge theory potential energy will be (M P, 4 = N 3/8 M 1 , M L → m π ) V (r) = − m π M −1 1 N 3/4 c M 2 1 kM 1 M 2 r = −0.06GeV −2 (M 1 [GeV ]) −3 kM 1 M 2 r(23) where M 1 = M 1,glueball and we have left the numerical constant k free since it depends on the exact model of how brane bending and gravity interact in the dual, and anyway we expect that the relation of energy scales like M P to gauge theory quantities could be renormalized by string corrections. Here M is the (relativistic) mass of the corresponding particle. However, we should note now that we have done something not obviously right. We have extended the regime of validity of the Newtonian approximation to the interior of the black hole, where we have no justification for this approximation. But remember that this is not a usual black hole, as it corresponds to a KK type of gravity, with mass M 1 . Also, we are talking now about the interaction of two massive perturbations inside the black hole, and not about the background gravitational field. If the black hole is large enough, the interaction of the two masses should still be Newtonian. This is clearer in the dual picture where we have pion fields, and pion exchange should dominate the interactions between particles. Also note that usual black holes have singularities, but the black holes created in this collision most likely will not. Indeed, for one thing it would be hard to imagine the brane bending analog of the black hole singularity: it would be a pinch-like singularity that seems unphysical. Also, in the field theory, a pion field singularity seems very unlikely. It could also be the case that these black holes are created and decay quickly, and the singularity cannot form. In any case, it seems that the interior of the formed IR brane black hole should be smoothed out with respect to the case of usual (Schwarzschild) black holes. Thus quarks and gluons will be deconfined in the CGC (pion field soliton), and will interact with the Coulomb-like (or rather, Newton-like) potential (23), which can create bound states, similar to atomic states (or rather, "solar systems"), that was argued that could a priori be experimentally detected [30]. Note that in this paper we have analyzed collisions at RHIC, which should be in the maximal Froissart regime, but even for collisions atÊ R ∼ 10GeV < √ s <Ê ′ F one will create black holes in the gravity dual, and solitons of the effective lightest glueball field in the gauge theory. This (much smaller) soliton will also decay, producing an (almost thermal) distribution, so one should find evidence for the thermalization of the decay products. Its temperature however will be different, as that temperature depends on the gravity dual, possibly on √ s, and on also the mass M 1,glueball . In conclusion, we have seen that the observed RHIC fireball is just (the pion analog of) a gravity dual black hole. We have seen that the unknown CGC state at the middle of the fireball is the interior of the black hole, and particles inside it interact with a Newtonian potential. The horizon of the black hole is the limiting ("freeze-out") surface of the pion field soliton, which emits radiation at a temperature given by (if the nonperturbative constant a=1) (20), very close to the experimental value of 176 MeV [14,15]. Most likely there will be no singularity for this black hole. Then RHIC is really a string theory testing machine, that analyzes the formation and decay of dual black holes, and giving information about the black hole interior. Probing the formation and decay with external particles seems hard, but a priori not impossible. The same applies for the formation of bound states of the potential (23). Acknowledgements I would like to thank Kyungsik Kang and Antal Jevicki for discussions. I would also like to thank the organizers of the ITP Santa Barbara "QCD and string theory" conference in November 2004, during which I learned about the exciting physics being done at RHIC. I would like to thank E. Shuryak and R. Pisarski for pointing out to me references about RHIC physics. This research was supported in part by DOE grant DE-FE0291ER40688-Task A. The large N limit of superconformal field theories and supergravity. J Maldacena, hep-th/9711200Adv.Theor.Math.Phys. 2231Int.J.Theor.Phys.J. Maldacena, "The large N limit of superconformal field theories and supergrav- ity," Adv.Theor.Math.Phys. 2 (1998) 231; Int.J.Theor.Phys. 38 (1999) 1113 and hep-th/9711200 Hard scattering and gauge/string duality. J Polchinski, M J Strassler, hep-th/0109174Phys. Rev. Lett. 88J. Polchinski and M.J. Strassler, "Hard scattering and gauge/string duality", Phys. Rev. Lett. 88 (2002) 031601 and hep-th/0109174 A large mass hierarchy from a small extra dimension. L Randall, R Sundrum, hep-ph/9905221Phys. Rev. Lett. 83L. Randall and R. Sundrum, "A large mass hierarchy from a small extra dimension", Phys. Rev. Lett. 83 (1999) 4690 and hep-ph/9905221 High energy QCD scattering, the shape of gravity on an IR brane, and the Froissart bound. S B Giddings, hep-th/0203004Phys. Rev. 67S.B. Giddings, "High energy QCD scattering, the shape of gravity on an IR brane, and the Froissart bound", Phys. Rev. D67 (2003) 126001 and hep-th/0203004 High energy QCD from Planckian scattering in AdS and the Froissart bound. K Kang, H Nastase, hep-th/0410173K. Kang and H. Nastase, "High energy QCD from Planckian scattering in AdS and the Froissart bound", hep-th/0410173 On the gravitational field of a massless particle. P C Aichelburg, R U Sexl, Gen. Rel. and Grav. 2303P.C. Aichelburg and R.U. Sexl, "On the gravitational field of a massless particle", Gen. Rel. and Grav. 2 (1971) 303 Graviton dominance in ultrahigh energy scattering. G Hooft, Phys. Lett. B. 19861G. 't Hooft, "Graviton dominance in ultrahigh energy scattering" Phys. Lett. B 198 (1987) 61 On high energy scattering inside gravitational backgrounds. H Nastase, hep-th/0410124H. Nastase, "On high energy scattering inside gravitational backgrounds," hep-th/0410124. Heisenberg saturation of the Froissart bound from AdS-CFT. K Kang, H Nastase, BROWN-HET-1435 and hep-th/0501038K. Kang and H. Nastase, "Heisenberg saturation of the Froissart bound from AdS- CFT," BROWN-HET-1435 and hep-th/0501038 Production of mesons as a shock wave problem. W Heisenberg, Zeit. Phys. 13365W. Heisenberg, "Production of mesons as a shock wave problem," Zeit. Phys. 133 (1952) 65 Classical black hole production in high energy collisions. D M Eardley, S B Giddings, gr-qc/0201034Phys. Rev. 66D.M.Eardley and S.B.Giddings "Classical black hole production in high energy colli- sions" Phys. Rev. D66 (2002) 044011 and gr-qc/0201034 Planckian scattering effects and black hole production in low M P l scenarios. K Kang, H Nastase, hep-th/0409099K. Kang and H. Nastase, "Planckian scattering effects and black hole production in low M P l scenarios," hep-th/0409099 The soft Pomeron from AdS-CFT. H Nastase, hep-th/0501039H. Nastase, "The soft Pomeron from AdS-CFT," hep-th/0501039. What have we learned from the Relativistic Heavy Ion Collider. : T Rhic Website, L Ludlam, Mclerran, article in Physics Today. RHIC website: T. Ludlam and L. McLerran, "What have we learned from the Rela- tivistic Heavy Ion Collider," article in Physics Today, October 2003. Particle production in heavy-ion collisons. P Braun-Munzinger, ; P Braun-Munzinger, K Redlich, J Stachel ; P.Braun-Munzinger, D Magestro, K Redlich, J Stachel, hep-ph/0105229Relativistic nuclear collisions and the QCD phase boundary. 51841Hadron production in Au-Au collisions at RHICP.Braun-Munzinger, "Relativistic nuclear collisions and the QCD phase boundary," nucl-ex/0411053. P.Braun-Munzinger, K. Redlich and J. Stachel, "Particle production in heavy-ion collisons," nucl-th/0304013; P.Braun-Munzinger, D. Magestro, K. Redlich and J. Stachel, "Hadron production in Au-Au collisions at RHIC," Phys. Lett. B518 (2001) 41. and hep-ph/0105229. The four laws of black hole mechanics. J M Bardeen, B Carter, S W Hawking, Commun. Math. Phys. 31161J.M. Bardeen, B. Carter and S.W. Hawking, "The four laws of black hole mechanics," Commun. Math. Phys. 31 (1973) 161. Microscopic origin of the Bekenstein-Hawking entropy. A Strominger, C Vafa, hep-th/9601029Phys. Lett. 379A. Strominger and C. Vafa, "Microscopic origin of the Bekenstein-Hawking entropy," Phys. Lett. B379 (1996) 99 and hep-th/9601029. D-brane approach to black hole quantum mechanics. C G Callan, J M Maldacena, hep-th/9602043Nucl. Phys. 472C.G. Callan and J.M. Maldacena, "D-brane approach to black hole quantum mechan- ics," Nucl. Phys. B472 (1996) 591 and hep-th/9602043. Black holes in string theory. J M Maldacena, hep-th/9607235PhD thesisJ.M. Maldacena, "Black holes in string theory," PhD thesis, hep-th/9607235. Particle creation by black holes. S W Hawking, Commun. Math. Phys. 248199NatureS.W. Hawking, " Black hole explosions," Nature 248 (1974) 30; "Particle creation by black holes," Commun. Math. Phys. 43 (1975) 199. Non-extreme black holes from non-extreme intersecting M-branes. M Cvetic, A A Tseytlin, hep-th/9606033Nucl. Phys. 478M. Cvetic and A.A. Tseytlin, "Non-extreme black holes from non-extreme intersecting M-branes," Nucl. Phys. B478 (1996) 181 and hep-th/9606033. Holography and hydrodynamics: diffusion on stretched horizons. P Kovtun, D T Son, A O Starinets, hep-th/0309213JHEP. 0640310P. Kovtun, D.T. Son and A.O. Starinets, "Holography and hydrodynamics: diffusion on stretched horizons," JHEP 0310 (2003) 064 and hep-th/0309213. Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma. G Policastro, D T Son, A O Starinets, hep-th/0104066Phys.Rev.Lett. 8781601G. Policastro, D.T. Son and A.O. Starinets, "Shear viscosity of strongly coupled N=4 supersymmetric Yang-Mills plasma," Phys.Rev.Lett. 87 (2001) 081601 and hep-th/0104066. A viscosity bound conjecture. P Kovtun, D T Son, A O Starinets, hep-th/0405231P. Kovtun, D.T. Son and A.O. Starinets, "A viscosity bound conjecture," hep-th/0405231. Universality of the shear viscosity in supergravity. A Buchel, J T Liu, hep-th/0311175Phys. Rev. Lett. 9390602A. Buchel and J.T. Liu, "Universality of the shear viscosity in supergravity," Phys. Rev. Lett. 93 (2004) 090602 and hep-th/0311175. On universality of stress-energy tensor correlation functions in supergravity. A Buchel, hep-th/0408095A. Buchel, " On universality of stress-energy tensor correlation functions in supergrav- ity," hep-th/0408095. talk at the ITP Santa Barbara conference "QCD and string theory. E Shuryak, November. E. Shuryak, talk at the ITP Santa Barbara conference "QCD and string theory", Novem- ber 2004, http://online.itp.ucsb.edu/online/qcd c04/shuryak/ Why does the Guark Gluon Plasma at RHIC behave as a nearly ideal fluid?. E Shuryak, hep-ph/0312227Prog. Part. Nucl. Phys. 53E. Shuryak, "Why does the Guark Gluon Plasma at RHIC behave as a nearly ideal fluid?," Prog. Part. Nucl. Phys. 53 (2004) 273 and hep-ph/0312227. Holography of radiation and jet quenching. S-J Sin, I Zahed, hep-th/0407215S-J. Sin and I. Zahed, "Holography of radiation and jet quenching," hep-th/0407215. Understanding the strong coupling limit of the N = 4 Supersymmetric Yang-Mills at finite temperature. E Shuryak, I Zahed, hep-th/0308073Phys. Rev. 7046005Phys. ev.E. Shuryak and I. Zahed, "Towards a theory of binary bound states in the Quark Gluon Plasma," Phys. ev. D70 (2004) 054507 and hep-ph/0403127; "Understanding the strong coupling limit of the N = 4 Supersymmetric Yang-Mills at finite temperature," Phys. Rev. D69 (2004) 046005 and hep-th/0308073.

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{'abstract': 'We argue that the fireball observed at RHIC is (the analog of) a dual black hole. In previous works, we have argued that the large s behaviour of the total QCD cross section is due to production of dual black holes, and that in the QCD effective field theory it corresponds to a nonlinear soliton of the pion field. Now we argue that the RHIC fireball is this soliton. We calculate the soliton (black hole) temperature, and get T = 4a < m π > /π, with a a nonperturbative constant. For a = 1, we get 175.76MeV , compared to the experimental value of the fireball "freeze-out" of about 176MeV . The observed η/s for the fireball is close to the dual value of 1/4π. The "Color Glass Condensate" (CGC) state at the core of the fireball is the pion field soliton, dual to the interior of the black hole. The main interaction between particles in the CGC is a Coulomb potential, due to short range pion exchange, dual to gravitational interaction inside the black hole, deconfining quarks and gluons. Thus RHIC is in a certain sense a string theory testing machine, analyzing the formation and decay of dual black holes, and giving information about the black hole interior.', 'arxivid': 'hep-th/0501068', 'author': ['Horatiu Nastase \nBrown University Providence\n02912RIUSA\n'], 'authoraffiliation': ['Brown University Providence\n02912RIUSA'], 'corpusid': 16757130, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9218, 'n_tokens_neox': 7868, 'n_words': 5031, 'pdfsha': 'a640ba869cc26f1cf2b0ce649eaa6415a35749c3', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/0501068v3.pdf'], 'title': ['The RHIC fireball as a dual black hole', 'The RHIC fireball as a dual black hole'], 'venue': []}

arxiv

New Quantum Spin Hall Insulator in Two-dimensional MoS 2 with Periodically Distributed Pores Peng-Fei Liu Structure of Matter State Key Laboratory of Structural Chemistry Fujian Institute of Research Chinese Academy of Sciences 350002FuzhouFujianPeople's Republic of China University of Chinese Academy of Sciences 100039BeijingPeople's Republic of China Liujiang Zhou liujiang86@gmail.com Bremen Center for Computational Materials Science University of Bremen Am Falturm 128359BremenGermany Max Planck Institute for Chemical Physics of Solids Noethnitzer Strasse 4001187DresdenGermany ⊥ Thomas Frauenheim Bremen Center for Computational Materials Science University of Bremen Am Falturm 128359BremenGermany Li-Ming Wu liming_wu@fjirsm.ac.cn New Quantum Spin Hall Insulator in Two-dimensional MoS 2 with Periodically Distributed Pores *To whom correspondence should be addressed MoS 2 , one of transition metal dichalcogenides (TMDs), has caused a lot of attentions for its excellent semiconductor characteristics and potential applications. Here, based on the density functional theory methods, we predict a novel two-dimension (2D) quantum spin hall (QSH) insulator in the porous allotrope of monolayer MoS 2 (g-MoS 2 ), consisting of MoS 2 square and hexagon. The g-MoS 2 has a nontrivial gap as large as 109 meV, comparable with previous reported 1T′-MoS 2 (80 meV), so-MoS 2 (25 meV). We demonstrate that the origin of 2D QSH effect in g-MoS 2 originates from the pure d−d band interaction, different from conventional band inversion between s−p, p−p or d−p orbitals. Such new polymorph greatly enriches the TMDs family and its stabilities are confirmed by phonon spectrum analysis. In particular, porous structure also endows it potential application in efficient gas separation and energy storage.TOC GraphicKeywordsMoS 2 , topological insulators, quantum spin Hall insulator, two-dimensional transition-metal dichalcogenides (TMDs), first-principles calculations, structure prediction Two-dimensional topological insulators (TIs), featured with gapless boundary states, which is protected by time-reversal symmetry with characteristic spin texture, completely refreshed our minds and brought a new revolution to material science due to their unparalleled electronic properties as well as promising applications in dissipationless electronic devices. 1-4 The edge states in two-dimensional (2D) TIs, characterized with quantum spin Hall (QSH) states, are more robust against nonmagnetic impurities than in 3D TIs and thus are better suited for coherent spin transport related applications. Impressive progress in searching for desired 2D TIs leads to successful findings of monolayers or few-layer van der Waals crystals, such as monolayer graphene, 3,4 silicene, 5,6 germanene, 7 stanene, 8 BiSb Alloys, 9 BiTeCl, 10 V 2 −VI 3 family compounds (Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 ). 11 However, up to now the well-known QSH insulators, including HgTe/CdTe, 12 InAs/GaSb quantum wells, 13 were experimentally observed only at very low temperatures and ultrahigh vacuum due to weak spin−orbit coupling (SOC). To expand and advance practical application of two-dimensional (2D) TIs at room temperature, it is desired to design and search for new TIs to overcome the thermal disturbance. Intensive efforts have been devoted to engineer QSH insulators via first-principles method, and thus a crowd of candidates, including ZrTe 5 , 14 ZrBr, 15 2D III−Bi compounds, 16 methyl-munctionalized compounds (Bi Bilayer, 17 GeCH 3 18 ), halide-munctionalized 2D materials(such as X2-GeSn, 19 GeX, 20 decorated stanene, 21 Bi 4 F 4 , 22 Bi 4 Br 4 , 23 BiX/SbX monolayers, 24 chloridized gallium bismuthide 25 and so on), are predicted to be new TIs. Even though some of the predicted QSH insulators 17,22,25 show large band gap enough for room temperature (RT) applications, all of them still have not been obtained experimentally. Therefore, desirable materials preferably with large bulk gaps are still lacking and deserve to be explored in experiment and theory for realistic RT applications. MoS 2 , a typical example of the most studied transition metal dichalcogenides (TMDs), has attracted a lot of attention for its promising semiconductor characteristics and potential applications in in electronic, optical, 26 catalytic, 27 and lubricant properties. 28 Meanwhile, monolayer MoS 2 nanostructures exhibit even more intriguing properties, due to its intriguing properties by virtue of the quantum size effect, such as the strong photoluminescence, 29 moderation direct band gap (~1.8eV) 30,31 and relatively high mobility rate 30 and high on/off ratio. 33,34 Theoretical and experimental studies of monolayer MoS 2 have revealed its unlimited potentials for future applications in nanoelectronics. Recently, the prediction of MoS 2 in the square-octagonal lattice (so-MoS 2 ) 35 The unique arrangements of six-and four-rings provide vital insight into physical properties of TMDs and open up a viable approach to design 2D topological materials for realistic applications. Utilizing density functional theory (DFT) as implemented in the Vienna ab-initio simulation package (VASP), 42 we investigate the equilibrium structure, stability, and electronic properties of the predicted structure. All-electron projector augmented wave method 43 was used for the ionic cores and the generalized gradient approximation for the exchange-correlation potential. 44 The reciprocal space was sampled with 0.03 Å −1 spacing in the Monkhorst-Pack scheme for structure optimization, and denser k-point grids with 0.01 Å −1 spacing were adopted for electronic properties calculation. We used a mesh cutoff energy of 500 eV to determine the self-consistent charge density. All geometry structures were relaxed until the Hellmann-Feynman force on atoms is less than 0.01 eV/Å and the total energy variation is less than 1.0×10 −6 eV. The screened exchange hybrid density functional by Heyd-Scuseria-Ernzerhof (HSE06) 45,46 was adopted to further correct the electronic structure. A vacuum space of 15 Å along the z direction was used to avoid interactions between adjacent layers. The phonon calculations were carried out by using the density functional perturbation theory (DFPT) 47 as implemented in the PHONOPY code 48 combined with the VASP. To verify the stability of the system at elevated temperatures, the ab-initio molecular dynamic (MD) simulations were performed using the Nosé algorithm 49 in the NVT ensemble at 500K and 1000K respectively. The VESTA software 50 was used for visualization and plot. The novel plane MoS 2 akin to graphenylene (Fig. S1c), is composed of repeated special MoS 2 square and hexagon units (denoted as g-MoS 2 ). 51 The optimized g-MoS 2 contains two six-membered MoS 2 rings connected by a four-membered MoS 2 unit, crystallizing in the hexagonal space group, P6̅ m (no. 175), with a = b = 8.803 Å (Fig. 1a). Such a g-MoS 2 holds three distinct Mo-S bond lengths varying from 2.39 Å to 2.46 Å to 2.47 Å (Fig. 1c), revealing a severely structural distortion originated from the square-hexagon topology. Similar to other monolayer MoS 2 , the g-MoS 2 layer with the covalently bonded S-Mo-S atoms has a Mo atomic plane layer sandwiched between two S atomic layers. All Mo atoms are hexa-coordinated octahedrally with six nearest neighbor S atoms. As illustrated in Fig. 1, we have calculated the thickness of the monolayer g-MoS 2 by simply measuring distance between the top and bottom S atomic layers. Compared to previous work, 35 the thickness of three MoS 2 allotropes, h-MoS 2 so-MoS 2 and g-MoS 2 , decrease from 3.13 to 3.12 to 3.11 Å, respectively. To evaluate the stability of this structure, we first computed the formation energy with respect to isolated atoms, defined as below: (1) Where E total is the total energy of monolayer MoS 2 , E Mo and E S were the energies an isolated Mo and S atom, respectively. Our result shows that g-MoS 2 has a formation energy of −4.79 eV, indicating it being energetically favorable. As a comparison, we also calculated the formation energy of h-MoS 2 and so-MoS 2 , which is −5.08 eV and −4.80 eV, same to the reported value, 35 To further quantify the contributions to CB and VB for g-MoS 2 , we have calculated the three main d orbitals contributions to VB and CB along K-M-Γ-K in the Brillouin zone. In Fig. 3a, Fig. 3b. We can clearly see that the frontier orbitals at Fermi level mainly originate from the Mo-d z2 orbital hybridized with small amounts of d xy and d x2-y2 orbitals, further verifying above analysis. As is well-known, the four pear-shaped lobes of in-plane d x2-y2 orbitals, shown in Fig. S4, spread in the x−y plane in real place, fully compatible with the square lattice symmetry. As a result, a long-range coherence is realized between the d x2-y2 orbital and the square crystal lattice, leading to Dirac cone feature in so-MoS 2 . 35 Encouragingly, potential gas separation and purification applications in g-MoS 2 can be expected. It has been proved that MoS 2 nanosheet has a high affinity to selected gas species including H 2 , NO 2 and CO 2 due to high adsorption energy of these gas molecules onto the basal surface of MoS 2 . 56,57 In g-MoS 2 , the special organization of six-and four-rings renders it is a 2D network with periodically distributed pores. The electron density isosurfaces of porous g-MoS 2 monolayer are calculated and shown in Fig. 3c, presenting an average pore diameter of 5.3 Å with a nummular shape via the method described by Song et al. 51 Such a diameter indicates that the porous MoS 2 can offer similar pore size distribution as silicalite, and may allow the separation of molecules (CO 2 , CH 4 , and O 2 ) based on the differences in diffusion speeds, like other porous materials. [58][59][60] This feature prompts porous nanosheet a promising material for gas separation and purification applications. In transition-metal-based QSH insulators, and enable g-MoS 2 being promising materials for next-generation high-performance spintronic devices. ASSOCIATED CONTENT Supporting Information Geometry for graphene, T-graphene, graphenylene, h-MoS 2 , so-MoS 2 and g-MoS 2 , ab-initio molecular dynamics simulations, shapes of the three atomic d orbitals, band structures with and without SOC, and the Forcite analysis of bond distribution. AUTHOR INFORMATION Corresponding Authors *E-mail: liujiang86@gmail.com. *E-mail: liming_wu@fjirsm.ac.cn. Notes The authors declare no competing financial interest. Figure Caption indicating g-MoS 2 and so-MoS 2 hold almost the same structural stability. Furthermore, the phonon band structure and vibrational density of states, shown inFig. S2a, were calculated to study the dynamic stability and structural rigidity of the g-MoS 2 . At first glance of the phonon dispersion, there are the linearly crossing phonon branches at the K symmetry point, which often appears in the hexagonal crystal. No imaginary phonon frequencies are observed in the Brillouin zone, indicating inherent dynamic stability of a crystal at low temperature. The presence of quite high eigenvalues in optical phonon is another direct indication for the structural stability. We find the highest frequency in phonon spectrum of g-MoS 2 reaches up to 442 cm -1 , close to that of so-MoS 2 (444 cm -1 ), slightly lower than that of h-MoS 2 (462 cm -1 ), reflecting a similar stability. The fluctuations of total energy with simulation time are plotted inFig. S2b. After 5000 steps at 500 K, we found that there is no obvious structure destruction, and that the average value of total energy remains nearly constant during the whole simulation scale, confirming g-MoS 2 being at least thermally stable at room temperature.The calculated band structure, presented inFig. 1d, shows zero band gap with the valence band maximum (VBM) and conduction band minimum (CBM) touching at the Γ point. The shapes of the conduction and valence bands (VB and CB) are presented inFig. 1e. The CB is flattened, while VB shows a considerable radian, indicating the coexistence of heavy electrons and light holes. The effective mass can be estimated by using the following equation at the bands extrema:(2) Our calculations show that the carrier effective masses are 8.54 m e (electron) and 0.39 m e (hole) along Γ − K direction at Γ point, respectively. Such a dramatic difference in effective mass is useful for selectively injecting or emitting holes or electrons, rendering the materials huge application prospects in nanoelectronic devices. It is amazing that so-MoS 2 and g-MoS 2 are both gapless and have heavy carriers, but so-MoS 2 possesses unique massless Dirac structure. To explain this peculiar phenomenon, total DOS and projections DOS for g-MoS 2 so-MoS 2 and h-MoS 2 are calculated and plotted in Fig. 2. In h-MoS 2 , the frontier orbitals mainly originate from the Mo−d orbitals hybridized with visible contributions from the S-p orbitals. In the DOS of g-MoS 2 and so-MoS 2 , we can see the orbitals around Fermi level mainly come from Mo−d orbitals with negligible components from S−p orbitals, holding the zero DOS and thus showing zero-gap at Fermi level. In detail, in the case of h-MoS 2 , the VB can be primarily ascribed to the d x2-y2 and d z2 orbitals while CB to d z2 orbitals. As for so-MoS 2 and g-MoS 2 , the basic units are totally reconstructed, and the electron orbital contributions around Fermi level have also varied. In the case of so-MoS 2 , the VB and CB+1 are mainly composed of the Mo−d x2-y2 orbitals, possessing a linear energy-momentum dispersion characterized by the Dirac structure, while the CB and VB−1 mainly composed of the Mo−d z2 orbitals are almost flat. Meanwhile in the case of g-MoS 2 , the CB and VB are primarily constituted of the Mo−d z2 orbitals, having parabolic energy-momentum dispersion relations. So, the contributions from S−p electrons are almost negligible for g-MoS 2 and so-MoS 2 , but indispensable for h-MoS 2 . On the contrary, the two pear-shaped regions of out-of-plane d z2 orbitals are localized and placed symmetrically along the c axis, bringing about the heavy fermions to h-MoS 2 , so-MoS 2 and g-MoS 2 . Overall, distinctive electronic properties in the three two-dimensional crystals can be attributed to the lattice symmetry's mismatching with Mo−d orbitals.To explicitly verify the nontrivial topological nature of g-MoS 2 , we have performed the calculations of edge states by cutting 2D monolayer into nanoribbon with the armchair edges. To avoid the interaction between two edges, the width of nanoribbon is cut to be more than 7.40 nm. The module and calculated band structure of g-MoS 2 ribbons are presented inFig. 4. It can be clearly demonstrated that the topologically protected conducting edge states connecting the conduction and valence bands exist within bulk band gap, which confirms the nontrivial topological phase in the g-MoS 2 . For a 2D QSH insulator, another remarkable characteristic is the Z 2 topological invariant (ν), with ν = 1 characterizing a topologically nontrivial phase and ν = 0 meaning a topologically trivial phase. According to the method proposed by Fu and Kane, 52,53 we calculate Z 2 topological invariant directly from the parities of Bloch wave functions for δ(k i ) is the product of parity eigenvalues at the TRIM points, ξ = ±1 present the parity eigenvalues and N denotes the number of the degenerate occupied energy bands. For g-MoS 2 , the products of the parity eigenvalues at the Γ and M points are −1 and +1, respectively. It implies that the QSH effect can be realized with Z 2 topological invariant ν = 1 in g-MoS 2 monolayer.In desirable QSH insulator, large nontrivial gap is one of the prerequisites for practical application in spintronics. To overcome the underestimation of band gaps by the PBE method, we recalculated the band structure of g-MoS 2 based on HSE06 funtional.45,46 Both PBE and HSE06 without SOC show the semi-metal feature with zero band gap. When applying SOC, the degenerated states at the touching point are lifted out(Fig. S5), opening up an energy gap of 40 meV (PBE) and 109 meV (HSE06), much larger than the previous reported TIs materials in 1T′-MoS 2 (80 meV),38 and so-MoS 2 (25 meV),36,37 showing superior performance than graphene, silicone and germanene with nontrivial gaps in the order of meV.54,55 The relatively large nontrivial band gap makes g-MoS 2 promising for practical application as a novel 2D topological insulator at room temperature (~ 30 meV). conclusion, a novel MoS 2 allotrope, constituted of MoS 2 hexagon and square units, is found to be energetically and thermally stable. Almost all of the contributions near Fermi level come from d z2 orbitals, leading to the existence of heavy fermions. Meanwhile, the nontrivial topological characteristics originated from pure d orbitals are found in this monolayer. The topological features confirmed by edge states and non-zero topological invariant (ν = 1) show g-MoS 2 is a novel QSH insulator with a nontrivial gap of 109 meV, which is large enough for room-temperature detection. The special arrangement of six-and four-rings within the plane endows g-MoS 2 the well-defined pore structure, which suggests the potential application in gas separation and storage. Several features, including high stability, quadratic band touching, quantum spin Hall (QSH) effect and special pore structure, open up the possibility for engineering on 2D Figure 1 . 1Equilibrium structure of g-MoS 2 from (a) top view and (b) side view. (c) Hexagonal model representation. Primitive unit cells are emphasized by the solid black line. (d) Calculated orbital-resolved band structures of g-MoS 2 based on DFT-PBE. (e) 3D band structures formed by the valence and conduction band. The Fermi level has been set at 0 eV. The bands near Fermi energy are mainly dominated by Mo−d z2 , Mo−d x2-y2 and Mo−d xy . Figure 2 . 2The selected (Γ − k, Γ − M and K − Γ) band structures and density of states for (a) g-MoS 2 , (b) so-MoS 2 and (c) h-MoS 2 monolayer. Figure 3 . 3(a) The contributions of three main d orbitals to VB (down) and CB (top). (b) VBM charge density contours at Γ point of g-MoS 2 . (c) Electron density isosurfaces for g-MoS 2 . Isosurface value: 0.002 e/Å 3 . Figure 4 . 4(a) Top view of g-MoS 2 nanoribbon with armchair edges. (b) The calculated topological edge states for g-MoS 2 with SOC. The Dirac helical states are denoted by the red solid lines and red spheres, which exist at the edges of the ribbon structure. The partial charge density for helical states bands (in 10 −3 e/Å). Zero of energy is set at the Fermi level. -37 was found to exhibit gapless band structure with Dirac fermions at Γ point. Distinct from the conventional p−p band inversion in graphene, so-MoS 2 holds the pure d−d band interaction and possess a Fermi velocity (2.3-2.4 × 10 6 m/s), 35 comparable to that of graphene. 2,3 Besides, lattice distortion can induce an intrinsic band inversion between chalcogenide-p and metal-d bands in 1T′-MoS 2 , 38 a new class of large-gap QSH insulators in two-dimensional transition metal dichalcogenides. These researches here greatly enrich the physical properties of TMDs and provide new guidance for engineering high-performance TIs materials. Meanwhile,intensive efforts on the grain boundaries of monolayer MoS 2 39-41 have proved that four-, five-, seven-, eight-membered rings, as well as six-membered rings can coexist in monolayer MoS 2 , showing the structure diversity. Motivated by above phenomenon, we design a new allotrope of MoS 2 monolayer composed of repeated special MoS 2 square and hexagon units (denoted as g-MoS 2 ). Based on density functional theory (DFT) method, we theoretically study the structure, stability, and electronic properties of the atom-thick MoS 2 monolayer. This new kind of QSH insulator is verified by topological edge states and Z 2 topological invariant. we can see the VB and CB are mostly constituted of d z2 orbital with small contributions from Mo−d xy and −d x2-y2 orbitals. Specifically in VB region, the d xy , d z2 and d x2-y2 orbitals show no big fluctuations and the subordinate components of VB are d x2-y2 . While in CB region, the d xy , d z2 and d x2-y2 orbitals undulate severely in the Brillouin zone and d xy orbital becomes the second part of CB around Γ point. To give an ocular explanationon the nature of the zero band gap of g-MoS 2 , VBM charge density contours at Γ point are calculated and shown in Electric field effect in atomically thin carbon films. 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{'abstract': 'MoS 2 , one of transition metal dichalcogenides (TMDs), has caused a lot of attentions for its excellent semiconductor characteristics and potential applications. Here, based on the density functional theory methods, we predict a novel two-dimension (2D) quantum spin hall (QSH) insulator in the porous allotrope of monolayer MoS 2 (g-MoS 2 ), consisting of MoS 2 square and hexagon. The g-MoS 2 has a nontrivial gap as large as 109 meV, comparable with previous reported 1T′-MoS 2 (80 meV), so-MoS 2 (25 meV). We demonstrate that the origin of 2D QSH effect in g-MoS 2 originates from the pure d−d band interaction, different from conventional band inversion between s−p, p−p or d−p orbitals. Such new polymorph greatly enriches the TMDs family and its stabilities are confirmed by phonon spectrum analysis. In particular, porous structure also endows it potential application in efficient gas separation and energy storage.TOC GraphicKeywordsMoS 2 , topological insulators, quantum spin Hall insulator, two-dimensional transition-metal dichalcogenides (TMDs), first-principles calculations, structure prediction Two-dimensional topological insulators (TIs), featured with gapless boundary states, which is protected by time-reversal symmetry with characteristic spin texture, completely refreshed our minds and brought a new revolution to material science due to their unparalleled electronic properties as well as promising applications in dissipationless electronic devices. 1-4 The edge states in two-dimensional (2D) TIs, characterized with quantum spin Hall (QSH) states, are more robust against nonmagnetic impurities than in 3D TIs and thus are better suited for coherent spin transport related applications. Impressive progress in searching for desired 2D TIs leads to successful findings of monolayers or few-layer van der Waals crystals, such as monolayer graphene, 3,4 silicene, 5,6 germanene, 7 stanene, 8 BiSb Alloys, 9 BiTeCl, 10 V 2 −VI 3 family compounds (Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 ). 11 However, up to now the well-known QSH insulators, including HgTe/CdTe, 12 InAs/GaSb quantum wells, 13 were experimentally observed only at very low temperatures and ultrahigh vacuum due to weak spin−orbit coupling (SOC). To expand and advance practical application of two-dimensional (2D) TIs at room temperature, it is desired to design and search for new TIs to overcome the thermal disturbance. Intensive efforts have been devoted to engineer QSH insulators via first-principles method, and thus a crowd of candidates, including ZrTe 5 , 14 ZrBr, 15 2D III−Bi compounds, 16 methyl-munctionalized compounds (Bi Bilayer, 17 GeCH 3 18 ), halide-munctionalized 2D materials(such as', 'arxivid': '1512.02854', 'author': ["Peng-Fei Liu \nStructure of Matter\nState Key Laboratory of Structural Chemistry\nFujian Institute of Research\nChinese Academy of Sciences\n350002FuzhouFujianPeople's Republic of China\n\nUniversity of Chinese Academy of Sciences\n100039BeijingPeople's Republic of China\n", 'Liujiang Zhou liujiang86@gmail.com \nBremen Center for Computational Materials Science\nUniversity of Bremen\nAm Falturm 128359BremenGermany\n\nMax Planck Institute for Chemical Physics of Solids\nNoethnitzer Strasse 4001187DresdenGermany\n', '⊥ ', 'Thomas Frauenheim \nBremen Center for Computational Materials Science\nUniversity of Bremen\nAm Falturm 128359BremenGermany\n', 'Li-Ming Wu liming_wu@fjirsm.ac.cn '], 'authoraffiliation': ["Structure of Matter\nState Key Laboratory of Structural Chemistry\nFujian Institute of Research\nChinese Academy of Sciences\n350002FuzhouFujianPeople's Republic of China", "University of Chinese Academy of Sciences\n100039BeijingPeople's Republic of China", 'Bremen Center for Computational Materials Science\nUniversity of Bremen\nAm Falturm 128359BremenGermany', 'Max Planck Institute for Chemical Physics of Solids\nNoethnitzer Strasse 4001187DresdenGermany', 'Bremen Center for Computational Materials Science\nUniversity of Bremen\nAm Falturm 128359BremenGermany'], 'corpusid': 205987892, 'doi': '10.1039/c5nr08842a', 'github_urls': [], 'n_tokens_mistral': 14273, 'n_tokens_neox': 11854, 'n_words': 6669, 'pdfsha': 'f2042a0d3e93078f3aaacfd9232ab6f0d5f977e6', 'pdfurls': ['https://export.arxiv.org/pdf/1512.02854v3.pdf'], 'title': ['New Quantum Spin Hall Insulator in Two-dimensional MoS 2 with Periodically Distributed Pores', 'New Quantum Spin Hall Insulator in Two-dimensional MoS 2 with Periodically Distributed Pores'], 'venue': []}

arxiv

A Covariant Approach to Noncommutative M5-branes * ) arXiv:hep-th/0702132v1 16 Feb 2007 Makoto Sakaguchi Okayama Institute for Quantum Physics (OIQP) 1-9-1 Kyoyama700-0015OkayamaJapan Kentaroh Yoshida Institute of Particle and Nuclear Studies Theory Division High Energy Accelerator Research Organization (KEK) 1-1 Oho305-0801TsukubaIbarakiJapan A Covariant Approach to Noncommutative M5-branes * ) arXiv:hep-th/0702132v1 16 Feb 20071 We briefly review how to discuss noncommutative (NC) M5-branes and intersecting NC M5-branes from κ-invariance of an open supermembrane action with constant three-form fluxes. The κ-invariance gives rise to possible Dirichlet brane configurations. We shortly summarize a construction of projection operators for NC M5-branes and some intersecting configurations of NC M5-branes. A strong flux limit of them is also discussed.KEK-TH-1139, OIQP-07-03, hep-th/0702132 §1. Introduction Supermembrane theory in eleven dimensions 1), 2) is closely related to the Mtheory formulation,3)where open membranes 4), 5) can be considered as well as closed ones. Open membranes can end on a p-dimensional Dirichlet p-brane for p = 1, 5 and 9 6), 7) just like an open string can attach to D-branes. The p = 5 case corresponds to M5-brane and the p = 9 is the end-of-world 9-brane in the Horava-Witten theory. 8)The Dirichlet branes can be investigated from the κ-symmetry argument. 6), 9) It is a covariant way and a specific gauge-fixing such as light-cone gauge is not necessary. Then it is sufficient to consider a single action of open string or open membrane, rather than each of D-brane actions. It is moreover easy to find what configurations are allowed to exist for rather complicated D-brane setups such as intersecting D-branes or less supersymmetric D-branes, which are difficult to discuss within a brane probe analysis. The method is not restricted to a flat spacetime and can be generalized to curved backgrounds. 10) §2. The κ-symmetry ArgumentThe Green-Schwarz action of a supermembrane in flat spacetime is composed of the Nambu-Goto (NG) part and the Wess-Zumino (WZ) part 1) Since the bulk action is κ-invariant, the κ-variation of the action δ κ S leaves only surface terms. The NG part does not give rise to any surface terms. Thus it is sufficient to examine the κ-variation of the WZ part, δ κ S WZ | = ∂Σ d 2 ξ L (2) + L (4) + L (6) , L (2) = −i θ ΓĀBδ κ θ + HĀBCθΓCδ κ θ ẊĀ X ′B , (2 . 1) L (4) = − 3 2θ Γ A δ κ θθΓ AB + 1 2θ Γ AB δ κ θθΓ A (θ ′ẊB −θX ′B ) , (2 . 2) where the sixth order part L (6) disappears due to the Fierz identity. Here we have already utilized bosonic boundary conditions. 11) In order to ensure the κ-invariance these surface terms should vanish. Thus the problem of finding possible Dirichlet branes is boiled down to constructing the projection operators to make ( M = h 0 Γ 012345 + h 1 Γ 012 . (3 . 1) For M to define a projection, M 2 = 1 should be satisfied. Then we obtain the following condition, h 2 0 + h 2 1 = 1 . (3 . 2) We can see that (2 . 1) may vanish by imposing the conditions h 1 − H 012 = 0 , h 1 − h 0 H 345 = 0 . (3 . 3) It is easy to see that (2 . 2) also becomes zero, and the gluing matrix (3 . 1) with the two conditions (3 . 2) and (3 . 3) gives a possible M5-brane configuration. Then let us consider the interpretation of the solution constructed above. By substituting (3 . 3) for (3 . 2), we obtain the following condition, 1 (H 345 ) 2 − 1 (H 012 ) 2 = −1 . This is nothing but the self-dual condition 13) of the gauge field on the M5-brane. 14) That is, we have reproduced the information on the NC M5-brane from the κsymmetry argument for the open supermembrane action. Thus we recognize that the projection operator should describe the NC M5-brane. Let us consider a commutative limit and a strong flux limit. The conditions (3 . 2) and (3 . 3) are solved by using an angle variable ϕ , h 0 = cos ϕ , h 1 = sin ϕ , H 012 = sin ϕ , H 345 = tan ϕ (0 ≤ ϕ ≤ π/2) . Then the gluing matrix M is written as M = e ϕΓ 345 Γ 012345 . For a commutative limit ϕ → 0, the NC M5 reduces to commutative M5 (012345), since H → 0 and M → Γ 012345 . On the other hand, for ϕ → π/2 , we see that H 345 → ∞ and so the gluing condition reduces to M → Γ 012 with a critical flux H 012 = 1 . It seems that the resulting projection operator should describe a critical M2-brane (012). Eventually this limit is nothing but the OM limit 15) and it should correspond to one of infinitely many M2-branes dissolved on the M5-brane. This is analogous to the D2-D0 system where a D2-brane with a flux reduces to a D2-brane with infinitely many D0-brane in a strong magnetic flux limit. As is well known, the p = 2 case is not allowed as a projection operator in the case without fluxes. Hence it is a non-trivial problem whether the resulting projection operator for a critical M2 is consistent to the κ-symmetry. The p = 2 case is actually special among other p , and due to some identities intrinsic to p = 2 , (2 . 1) vanishes when H 012 = 1 . (3 . 4) It is easy to show that (2 . 2) disappears. Thus we have checked that the κ-variation surface terms should vanish for an M2-brane with the critical H (3 . 4). Although the κ-symmetry is maintained for the M2-brane, the charge conservation 4) requires the existence of M5-brane behind M2-branes. That is, a NC M5-brane should be regarded as a bound state of M5 and M2. §4. Intersecting Noncommutative M5-branes In comparison to the case of a single NC M5-brane, a configuration of intersecting NC M5-branes is characterized by two gluing matrices, 16) M 1 = e ϕ 1 Γ A 0 A 1 A 2 Γ A 0 ···A 5 , M 2 = e ϕ 2 Γ B 0 B 1 B 2 Γ B 0 ···B 5 , [M 1 , M 2 ] = 0 . The requirement [M 1 , M 2 ] = 0 leads to the four possibilities for the projection. As an example, let us focus upon one of these cases, NC M5⊥NC M5(3) described by M 1 = e ϕ 1 Γ 235 Γ 012345 , H 014 = sin ϕ 1 , H 235 = tan ϕ 1 (0 ≤ ϕ 1 ≤ π/2) , M 2 = e ϕ 2 Γ 137 Γ 012367 , H 026 = − sin ϕ 2 , H 137 = tan ϕ 2 (0 ≤ ϕ 2 ≤ π/2) . It reduces to a commutative M5 (012345)⊥M5 (012367) 17)- 19) in the limit ϕ 1,2 → 0 . M2⊥M5 (1) 17), 19) can be realized from the NC M5⊥NC M5 (3) by taking a strong flux limit. In the limit ϕ 2 → π/2 , we obtain a NC version of M2⊥M5 (1), M 1 = e ϕ 1 Γ 235 Γ 012345 , M 2 = −Γ 026 . Further letting ϕ 1 → π/2, we obtain M2 (014)⊥M2 (026). 17), 19) The other way is possible and the two sequences of the strong flux limits are depicted in Fig. 1. It is also possible to discuss NC M5⊥C M5 (1). 12) AcknowledgementsThis work is supported in part by the Grant-in-Aid for Scientific Research (No. 17540262 and No. 17540091) from the Ministry of Education, Science and Culture, Japan. The work of K. Y. is supported in part by JSPS Research Fellowships for Young Scientists. . E Bergshoeff, E Sezgin, P K Townsend, Phys. Lett. B. 18975E. Bergshoeff, E. Sezgin and P. K. Townsend, Phys. Lett. B 189 (1987) 75; . Ann. Phys. 185330Ann. Phys. 185 (1988) 330. . B De Wit, J Hoppe, H Nicolai, Nucl. Phys. B. 305545B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B 305 (1988) 545. . T Banks, W Fischler, S H Shenker, L Susskind, hep-th/9610043Phys. Rev. D. 55T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D 55 (1997), 5112, hep-th/9610043. . A Strominger, hep-th/9512059Phys. Lett. B. 383A. Strominger, Phys. Lett. B 383 (1996) 44, hep-th/9512059. . P K Townsend, hep-th/9512062Phys. Lett. B. 373P. K. Townsend, Phys. Lett. B 373 (1996) 68, hep-th/9512062. . K Ezawa, Y Matsuo, K Murakami, hep-th/9707200Phys. Rev. D. 57K. Ezawa, Y. Matsuo and K. Murakami, Phys. Rev. D 57 (1998) 5118, hep-th/9707200. . B De Wit, K Peeters, J C Plefka, hep-th/9710215Nucl. Phys. Proc. Suppl. 68B. de Wit, K. Peeters and J. C. Plefka, Nucl. Phys. Proc. Suppl. 68 (1998) 206, hep-th/9710215. . P Horava, E Witten, hep-th/9603142Nucl. Phys. B. 460Nucl. Phys. BP. Horava and E. Witten, Nucl. Phys. B 460 (1996) 506, hep-th/9510209; Nucl. Phys. B 475 (1996) 94, hep-th/9603142. . N D Lambert, P C West, hep-th/9905031Phys. Lett. B. 459N. D. Lambert and P. C. West, Phys. Lett. B 459 (1999) 515, hep-th/9905031. . M Sakaguchi, K Yoshida, hep-th/0306213Nucl. Phys. B. 681Nucl. Phys. BM. Sakaguchi and K. Yoshida, Nucl. Phys. B 681 (2004) 137, hep-th/0310035; Nucl. Phys. B 714 (2005) 51, hep-th/0405109; Nucl. Phys. B 676 (2004) 311, hep-th/0306213; . K Sugiyama, K Yoshida, hep-th/0206070Nucl. Phys. B. 644K. Sugiyama and K. Yoshida, Nucl. Phys. B 644 (2002) 113, hep-th/0206070. . E Bergshoeff, D S Berman, J P Van Der Schaar, P Sundell, hep-th/0005026Nucl. Phys. B. 590E. Bergshoeff, D. S. Berman, J. P. van der Schaar and P. Sundell, Nucl. Phys. B 590 (2000) 173, hep-th/0005026. . M Sakaguchi, K Yoshida, hep-th/0608099Phys. Lett. B. 642M. Sakaguchi and K. Yoshida, Phys. Lett. B 642 (2006) 400, hep-th/0608099. . P S Howe, E Sezgin, P C West, hep-th/9702111Phys. Lett. B. 400P. S. Howe, E. Sezgin and P. C. West, Phys. Lett. B 400 (1997) 255, hep-th/9702111; . hep-th/9702008Phys. Lett. B. 399Phys. Lett. B 399 (1997) 49, hep-th/9702008. . N Seiberg, E Witten, hep-th/9908142JHEP. 9909N. Seiberg and E. Witten, JHEP 9909 (1999) 032, hep-th/9908142. . R Gopakumar, S Minwalla, N Seiberg, A Strominger, hep-th/0006062JHEP. 0008R. Gopakumar, S. Minwalla, N. Seiberg and A. Strominger, JHEP 0008 (2000) 008, hep-th/0006062. . M Sakaguchi, K Yoshida, hep-th/0702062M. Sakaguchi and K. Yoshida, hep-th/0702062. . G Papadopoulos, P K Townsend, hep-th/9603087Phys. Lett. B. 380G. Papadopoulos and P. K. Townsend, Phys. Lett. B 380 (1996) 273, hep-th/9603087. . A A Tseytlin, hep-th/9604035Nucl. Phys. B. 475A. A. Tseytlin, Nucl. Phys. B 475 (1996) 149, hep-th/9604035. . J P Gauntlett, D A Kastor, J H Traschen, hep-th/9604179Nucl. Phys. B. 478J. P. Gauntlett, D. A. Kastor and J. H. Traschen, Nucl. Phys. B 478 (1996) 544, hep-th/9604179.

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{'abstract': 'We briefly review how to discuss noncommutative (NC) M5-branes and intersecting NC M5-branes from κ-invariance of an open supermembrane action with constant three-form fluxes. The κ-invariance gives rise to possible Dirichlet brane configurations. We shortly summarize a construction of projection operators for NC M5-branes and some intersecting configurations of NC M5-branes. A strong flux limit of them is also discussed.KEK-TH-1139, OIQP-07-03, hep-th/0702132 §1. Introduction Supermembrane theory in eleven dimensions 1), 2) is closely related to the Mtheory formulation,3)where open membranes 4), 5) can be considered as well as closed ones. Open membranes can end on a p-dimensional Dirichlet p-brane for p = 1, 5 and 9 6), 7) just like an open string can attach to D-branes. The p = 5 case corresponds to M5-brane and the p = 9 is the end-of-world 9-brane in the Horava-Witten theory. 8)The Dirichlet branes can be investigated from the κ-symmetry argument. 6), 9) It is a covariant way and a specific gauge-fixing such as light-cone gauge is not necessary. Then it is sufficient to consider a single action of open string or open membrane, rather than each of D-brane actions. It is moreover easy to find what configurations are allowed to exist for rather complicated D-brane setups such as intersecting D-branes or less supersymmetric D-branes, which are difficult to discuss within a brane probe analysis. The method is not restricted to a flat spacetime and can be generalized to curved backgrounds. 10) §2. The κ-symmetry ArgumentThe Green-Schwarz action of a supermembrane in flat spacetime is composed of the Nambu-Goto (NG) part and the Wess-Zumino (WZ) part 1)', 'arxivid': 'hep-th/0702132', 'author': ['Makoto Sakaguchi \nOkayama Institute for Quantum Physics (OIQP)\n1-9-1 Kyoyama700-0015OkayamaJapan\n', 'Kentaroh Yoshida \nInstitute of Particle and Nuclear Studies\nTheory Division\nHigh Energy Accelerator Research Organization (KEK)\n1-1 Oho305-0801TsukubaIbarakiJapan\n'], 'authoraffiliation': ['Okayama Institute for Quantum Physics (OIQP)\n1-9-1 Kyoyama700-0015OkayamaJapan', 'Institute of Particle and Nuclear Studies\nTheory Division\nHigh Energy Accelerator Research Organization (KEK)\n1-1 Oho305-0801TsukubaIbarakiJapan'], 'corpusid': 119039556, 'doi': '10.1143/ptps.171.275', 'github_urls': [], 'n_tokens_mistral': 4429, 'n_tokens_neox': 3399, 'n_words': 1720, 'pdfsha': 'd26d4a6f792aa258efc1b400127842fd6ffc89d7', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/0702132v1.pdf'], 'title': ['A Covariant Approach to Noncommutative M5-branes * )', 'A Covariant Approach to Noncommutative M5-branes * )'], 'venue': []}

arxiv

THE FLEXIBLE GUMBEL DISTRIBUTION: A NEW MODEL FOR INFERENCE ABOUT THE MODE A PREPRINT December 6, 2022 Qingyang Liu qingyang@email.sc.edu Department of Statistics Department of Statistics University of South Carolina Columbia University of South Carolina Columbia Daiichi Sankyo Inc. Basking Ridge 29201, 29201, 07920SC, SC, NJ Xianzheng Huang huang@stat.sc.edu Department of Statistics Department of Statistics University of South Carolina Columbia University of South Carolina Columbia Daiichi Sankyo Inc. Basking Ridge 29201, 29201, 07920SC, SC, NJ Haiming Zhou haiming2019@gmail.com Department of Statistics Department of Statistics University of South Carolina Columbia University of South Carolina Columbia Daiichi Sankyo Inc. Basking Ridge 29201, 29201, 07920SC, SC, NJ THE FLEXIBLE GUMBEL DISTRIBUTION: A NEW MODEL FOR INFERENCE ABOUT THE MODE A PREPRINT December 6, 2022extreme values · mixture distribution · modal regression · unimodal distribution A new unimodal distribution family indexed by the mode and three other parameters is derived from a mixture of a Gumbel distribution for the maximum and a Gumbel distribution for the minimum. Properties of the proposed distribution are explored, including model identifiability and flexibility in capturing heavy-tailed data that exhibit different directions of skewness over a wide range. Both frequentist and Bayesian methods are developed to infer parameters in the new distribution. Simulation studies are conducted to demonstrate satisfactory performance of both methods. By fitting the proposed model to simulated data and data from an application in hydrology, it is shown that the proposed flexible distribution is especially suitable for data containing extreme values in either direction, with the mode being a location parameter of interest. A regression model concerning the mode of a response given covariates based on the proposed unimodal distribution can be easily formulated, which we apply to data from an application in criminology to reveal interesting data features that are obscured by outliers. Computer programs for implementing all considered inference methods in the study are available at https://github.com/rh8liuqy/flexible_Gumbel. Introduction The mean, median, and mode are three most commonly used measure of central tendency of data. When data contain outliers that cause heavy tails or are potentially skewed, the mode is a more sensible representation of the central location of data than the mean or median. The timely review on mode estimation and its application by Chacón [2020] and references therein provide many examples in various fields of research where the mode serves as a more informative representative value of data. Most existing methods developed to draw inference for the mode are semi-/non-parametric in nature, starting from early works on direct estimation in the 1960s [Chernoff, 1964, Dalenius, 1965, Venter, 1967 to more recent works based on kernel density estimation [Chen, 2018] and quantile-based methods [Ota et al., 2019, Zhang et al., 2021. There are two main reasons contributing to the long-lasting trend of opting to semi-/non-parametric methods for mode estimation, despite the fact that inference procedures proposed along these veins are usually less straightforward to implement (e.g., involving bandwidth selection), and less efficient than their parametric counterparts. First, a parametric model typically imposes stringent constraints on the relationship between the mode and other location parameters that may not be satisfied in a given application. Second, very few existing named distribution families that allow inclusion of both symmetric and asymmetric distributions in the same family can be parameterized so that it is indexed by the mode as the location parameter along with other parameters, such as shape or scale parameters. In this study, we alleviate concerns raised by both reasons that discourage use of parametric methods for mode estimation by formulating a flexible distribution indexed by the (unique) mode and parameters controlling the shape and scale. arXiv:2212.01832v1 [stat.ME] 4 Dec 2022 When it comes to modeling heavy-tailed data, the Gumbel distribution [Gumbel, 1941] is arguably one of the most widely used models in many disciplines. Indeed, as a case of the generalized extreme value distribution [Jenkinson, 1955], the Gumbel distribution for the maximum (or minimum) is well-suited for modeling extremely large (or small) events that produce heavy-tailed data. For example, it is often used in hydrology to predict extreme rainfall and flood frequency [Loaiciga and Leipnik, 1999, Koutsoyiannis, 2004, Dawley et al., 2019. In econometrics, the Gumbel distribution plays an important role in modeling extreme movements of stock prices and large changes in interest rates [Bali, 2003, Pratiwi et al., 2019. The Gumbel distribution is indexed by the mode and a scale parameter, and thus is convenient for mode estimation. However, the Gumbel distribution for the maximum (or minimum) is right-skewed (or left-skewed) with the skewness fixed at around 1. 44 (or −1.44), and the kurtosis fixed at 5.4 across the entire distribution family. Thus it may be too rigid for scenarios where the direction and extremeness of outliers presented in data are initially unclear, or when the direction and level of skewness are unknown beforehand. Constructions of more flexible distributions that overcome these limitations have been proposed. In particular, Cooray [2010] applied a logarithmic transformation on a random variable following the odd Weibull distribution to obtain the so-called generalized Gumbel distribution that includes the Gumbel distribution as a subfamily. But the mode of the generalized Gumbel distribution is not a location parameter this distribution is indexed by, or an explicit function of other model parameters. Shin et al. [2015] considered mixture distributions with one of the components being the Gumbel distribution and the other component(s) being Gumbel of the same skewness direction or a different distribution, such as the gamma distribution. Besides the same drawback pointed out for the generalized Gumbel distribution, it is difficult to formulate a unimodal distribution following their construction of mixtures, and thus their proposed models are unsuitable when unimodality is a feature required to make inferring the mode meaningful, such as in a regression setting, as in modal regression [Yao et al., 2012, Yao and Li, 2013, Chen, 2018. With heavy-tailed data in mind and the mode as the location parameter of interest, we construct a new unimodal distribution that does not impose stringent constraints on how the mode relates to other central tendency measures, while allowing a range of kurtosis wide enough to capture heavy tails at either direction, as well as different degrees and directions of skewness. This new distribution, called the flexible Gumbel (FG) distribution, is presented in Section 2, where we study properties of the distribution and discuss identifiability of the model. We present a frequentist method and a Bayesian method for estimating parameters in the FG distribution in Section 3. Finite sample performance of these methods are inspected in simulation study in Section 4, followed by an application of the FG distribution in hydrology in Section 5. Section 6 demonstrates fitting a modal regression model based on the FG distribution to data from a criminology study. Section 7 highlights contributions of the study and outlines future research directions. The flexible Gumbel distribution The probability density function (pdf) of the Gumbel distribution for the maximum is given by f (x; θ, σ) = 1 σ exp − x − θ σ − exp − x − θ σ ,(1) where θ is the mode and σ > 0 is a scale parameter. The pdf of the Gumbel distribution for the minimum with mode θ and a scale parameter σ is given by f (x; θ, σ) = 1 σ exp x − θ σ − exp x − θ σ .(2) We define a unimodal distribution for a random variable Y via a mixture of the two Gumbel distributions specified by (1) and (2) that share the same mode θ while allowing different scale parameters, σ 1 and σ 2 , in the two components. We call the resultant distribution the flexible Gumbel distribution, FG for short, with the pdf given by f (y) = w × 1 σ 1 exp − x − θ σ 1 − exp − x − θ σ 1 + (1 − w) × 1 σ 2 exp x − θ σ 2 − exp x − θ σ 2 ,(3) where w ∈ [0, 1] is the mixing proportion parameter. Henceforth, we state that Y ∼ FG(θ, σ 1 , σ 2 , w) if Y follows the distribution specified by the pdf in (3). For each component distribution of FG, the mean and median are both some simple shift of the mode, with each shift solely determined by the scale parameter. Because the two components in (3) share a common mode θ, the mode of Y is also θ, and thus the FG distribution is convenient to use when one aims to infer the mode as a central tendency measure, or to formulate parametric modal regression models [Bourguignon et al., 2020, Zhou and Huang, 2020. One can easily show that the mean of Y is E(Y ) = w(θ + σ 1 γ) + (1 − w)(θ − σ 2 γ) = θ + {w(σ 1 + σ 2 ) − σ 2 }γ, where γ ≈ 0.5772 is the Euler-Mascheroni constant. Thus the discrepancy between the mode and the mean of FG depends on three other parameters that control the scale and shape of the distribution. The median of Y , denoted by m, is the solution to the following equation, w exp − exp − m − θ σ 1 + (1 − w) 1 − exp − exp m − θ σ 2 = 0.5. Even though this equation cannot be solved for m explicitly to reveal the median in closed form, it is clear that m − θ also depends on all three other parameters of FG. In conclusion, the relationships between the three central tendency measures of FG are more versatile than those under a Gumbel distribution for the maximum or a Gumbel distribution for the minimum. The variance of Y is V (Y ) = {wσ 2 1 + (1 − w)σ 2 2 }π 2 /6 + w(1 − w)(σ 1 + σ 2 ) 2 γ 2 , which does not depend on the mode parameter θ. Obviously, by setting w = 0 or 1, FG(θ, σ 1 , σ 2 , w) reduces to one of the Gumbel components. Unlike a Gumbel distribution that only has one direction of skewness at a fixed level (of ±1.44), an FG distribution can be left-skewed, or right-skewed, or symmetric. More specifically, with the mode fixed at zero when studying the skewness and kurtosis of FG, one can show that the third central moment of Y is given by ww(σ 1 + σ 2 ) 2 γ γ 2 (w − w)(σ 1 + σ 2 ) + 0.5π 2 (σ 1 − σ 2 ) + 2ζ(3) wσ 3 1 −wσ 3 2 ,(4) wherew = 1 − w, and ζ(3) ≈ 1.202 is Apéry's constant. Although the direction of skewness is not immediately clear from (4), one may consider a special case with w = 0.5 where (4) reduces to (σ 1 − σ 2 ){γπ 2 (σ 1 + σ 2 ) 2 /8 + ζ(3)(σ 2 1 + σ 1 σ 2 + σ 2 2 )}. Now one can see that FG(θ, σ 1 , σ 2 , 0.5) is symmetric if and only if σ 1 = σ 2 , and it is left-skewed (or right-skewed) when σ 1 is less (or greater) than σ 2 . The kurtosis of Y can also be derived straightforwardly, with a more lengthy expression than (4) that we omit here, which may not shed much light on its magnitude except for that it varies as the scale parameters and the mixing proportion vary, instead of fixing at 5.4 as for a Gumbel distribution. An R Shiny app depicting the pdf of FG(θ, σ 1 , σ 2 , w) with user-specified parameter values is available at https://qingyang.shinyapps.io/gumbel_mixture/, created and maintained by the first author. Along with the density function curve, the Shiny app provides skewness and kurtosis of the depicted FG density. From there one can see that the skewness can be much lower than −1.44 or higher than 1.44, and the kurtosis can be much higher than 5.4, suggesting that inference based on FG can be more robust to outliers than when a Gumbel distribution is assumed for data at hand, without imposing stringent assumption on the skewness of the underlying distribution. The flexibility of a mixture distribution usually comes with concerns relating to identifiability [Teicher, 1961, 1963, Yakowitz and Spragins, 1968. In particular, there is the notorious issue of label switching when fitting a finite mixture model [Redner and Walker, 1984]. Take the family of two-component normal mixture (NM) distributions as an example, defined by {NM(µ 1 , σ 1 , µ 2 , σ 2 , w) : wN (µ 1 , σ 2 1 ) + (1 − w)N (µ 1 , σ 2 2 ), for σ 1 , σ 2 > 0 and w ∈ [0, 1]}. When fitting a data set assuming a normal mixture distribution, one cannot distinguish between, for instance, NM(1, 2, 3, 4, 0.2) and NM(3, 4, 1, 2, 0.8), since the likelihood of the data is identical under these two mixture distributions. As another example, for data from a normal distribution, a two-component normal mixture with two identical normal components and an arbitrary mixing proportion w ∈ [0, 1] leads to the same likelihood, and thus w cannot be identified. Teicher [1963] showed that imposing an lexicographical order for the normal components resolves the issue of non-identifiability, which also excludes mixtures with two identical components in the above normal mixture family. Unlike normal mixtures of which all components are in the same family of normal distributions, the FG distribution results from mixing two components from different families, i.e., a Gumbel distribution for the maximum and a Gumbel distribution for the minimum, with weight w on the former component. By construction, FG does not have the label-switching issue. And, according to Teicher [1963, Theorem 1], the so-constructed mixture distribution is always identifiable even when the true distribution is a (one-component) Gumbel distribution. Statistical inference 3.1 Frequentist inference method Based on a random sample of size n from the FG distirbution, y = {y i } n i=1 , maximum likelihood estimators (MLE) of all model parameters in Ω = (θ, σ 1 , σ 2 , w) can be obtained via the expectation-maximization (EM) algorithm [Dempster et al., 1977]. To apply the EM algorithm, we introduce a latent variable Z that follows Bernoulli(w) such that the joint likelihood of (Y, Z) is f Y,Z (y, z) = {wf 1 (y; θ, σ 1 )} z {(1 − w)f 2 (y; θ, σ 2 )} 1−z , where f 1 (y; θ, σ 1 ) is the pdf in (1) evaluated at y with the scale parameter σ = σ 1 , and f 2 (y; θ, σ 2 ) is the pdf in (2) evaluated at y with the scale parameter σ = σ 2 . A random sample of size n from Bernoulli(w), z = {z i } n i=1 , is viewed as missing data, and {(y i , z i )} n i=1 are viewed as the complete data in the EM algorithm. The complete-data log-likelihood is then (Ω; y, z) = n i=1 {z i log(wf 1 (y i ; θ, σ 1 )) + (1 − z i ) log((1 − w)f 2 (y i ; θ, σ 2 ))}.(5) Starting from an initial estimate of Ω (at the zero-th iteration), denoted by Ω (0) , one iterates two steps referred to as the E-step and the M-step until a convergence criterion is met. In the E-step at the (t + 1)-th iteration, one computes the conditional expectation of (5) given y while assuming the true parameter value to be Ω (t) = (θ (t) , σ (t) 1 , σ (t) 2 , w (t) ), that is, E Ω (t) { (Ω; y, z)|y}. This conditional expectation can be shown to be Q Ω Ω (t) = n i=1 T (t) i log(wf 1 (y i ; θ, σ 1 )) + 1 − T (t) i log((1 − w)f 2 (y i ; θ, σ 2 )) ,(6) where T (t) i = E Ω (t) (Z|Y = y i ) = w (t) f 1 (y i ; θ (t) , σ (t) 1 ) w (t) f 1 (y i ; θ (t) , σ (t) 1 ) + (1 − w (t) )f 2 (y i ; θ (t) , σ (t) 2 ) .(7) In the M-step at the (t + 1)-th iteration, one maximizes Q(Ω|Ω (t) ) with respect to Ω to obtain an updated estimate Ω (t+1) . Our experience with the above EM algorithm for fitting the FG distribution suggests that maximizing Q(Ω|Ω (t) ) in (6) can be numerically challenging. We thus exploit the expectation-conditional maximization (ECM) algorithm [Meng and Rubin, 1993], which replaces the M-step with a sequence of simpler conditional maximizations referred to as the CM-step. More specifically, in the CM-step, the updating formula for w is simply w (t+1) = n i=1 T (t) i /n. There is no closed-form updating formula for the other three parameters in Ω, but they can now be easily updated by most well-accepted one-dimensional optimization algorithms, such as the Newton-Raphson method. To ensure convergence at the global maximum, as recommended by Wu [1983], one should implement the ECM algorithm several rounds with different starting values Ω (0) . After obtaining the MLE of Ω, denoted byΩ, we propose to use the sandwich variance estimator [Boos and Stefanski, 2013, Chapter 7] to estimate the variance-covariance matrix ofΩ. One may also estimate the variance-covariance ofΩ based on the observed information matrix as described in Louis [1982] and Oakes [1999]. The benefit of using the sandwich variance estimator is its robustness to model misspecification. Finally, the EM and ECM algorithms bear a strong resemblance to data augmentation [Wei and Tanner, 1990] in the Bayesian framework, which we turn to next for inferring Ω. Bayesian inference method In the Bayesian framework, we formulate hierarchical models starting with the FG distribution, Y |θ, σ 1 , σ 2 , w ∼ FG(θ, σ 1 , σ 2 , w), followed by independent weakly informative or non-informative priors for elements in Ω, θ ∼ N (0, 10 4 ), σ j ∼ inv-Gamma(1, 1), for j = 1, 2, w ∼ Beta(1, 1), where inv-Gamma refers to the inverse Gamma distribution. We choose the above prior for the scale parameters by following the prior selection for variance parameters suggested in Gelman [2006] We employ the Metropolis-within-Gibbs sampler [Müller, 1991[Müller, , 1993] to obtain an estimate of Ω from the posterior distribution of Ω given observed data y. Similar to the EM/ECM algorithm in Section 3.1, the latent variable Z is also introduced as a device to carry out data augmentation. And the iterative algorithm presented next is based on the following two conditional distributions that can be easily proved, z i |θ, σ 1 , σ 2 , w, z −i , y ∼ Bernoulli wf 1 (y i ; θ, σ 1 ) wf 1 (y i ; θ, σ 1 ) + (1 − w)f 2 (y i ; θ, σ 2 ) , w|θ, σ 1 , σ 2 , z, y ∼ Beta 1 + n i=1 z i , n + 1 − n i=1 z i , where z −i results from dropping z i from z, and the first result above is also from which (7) is deduced. The Metropolis-within-Gibbs sampler at the (t + 1)-th iteration involves four steps outlined below. • Step 1: For i = 1, . . . , n, draw z (t+1) i from Bernoulli(T (t) i ), where T (t) i is given in (7). • Step 2: Draw w (t+1) from Beta 1 + n i=1 z (t+1) i , n + 1 − n i=1 z (t+1) i . • Step 3: Drawθ from N (θ (t) , τ 0 ), and update θ (t) to θ (t+1) according to the following decision rule, θ (t+1) =     θ , with probability q = min p(θ|w (t+1) , σ (t) 1 , σ (t) 2 , y) p(θ (t) |w (t+1) , σ (t) 1 , σ (t) 2 , y) , 1 , θ (t) , with probability 1 − q. • Step 4: For j = 1, 2, drawσ j from N (σ (t) j , τ j ), and update σ (t) j to σ (t+1) j according to the following decision rule, for k = j, σ (t+1) j =     σ j , with probability q = min p(σ j |θ (t+1) , σ (t) k , w (t+1) , y) p(σ (t) j |θ (t+1) , σ (t) k , w (t+1) , y) , 1 , σ (t) j , with probability 1 − q. In Steps 3 and 4, p(·|·) refers to a conditional pdf generically, τ 0 , τ 1 , and τ 2 are three positive tuning parameters whose values should be chosen so that the acceptance rate at each step is around 23% [Gelman et al., 1997]. To draw samples from the joint posterior distribution, there are numerous ways to design the Markov chain Monte Carlo (MCMC) sampler. Instead of the Metropolis-within-Gibbs sampler we adopt here, one may use other existing MCMC software, such as STAN [Stan Development Team, 2021], JAGS [Plummer et al., 2003], and BUGS [Spiegelhalter et al., 1996, Lunn et al., 2009, two of which we demonstrate in the Appendix. After obtaining enough high quality samples from the joint posterior distribution p(θ, σ 1 , σ 2 , w|y), Bayesian inference is straightforward, including point estimation, interval estimation, and uncertainty assessment. Simulation study Large-sample properties of MLEs and likelihood-based Bayesian inference under a correct model for data have been well studied. To assess finite-sample performance of the frequentist method and Bayesian method proposed in Section 3, we carried out simulation study with two specific aims: first, to compare inference results from the two methods; second, to compare goodness of fit for data from distributions outside of the FG family when one assumes an FG distribution and when one assumes a two-component normal mixture distribution for the data. In the first experiment, referred to as (E1) in the sequel, we drew a random sample of size n ∈ {100, 200} from an FG distribution with θ = 0, σ 1 = 1, σ 2 = 5, and w = 0.5. Based on each simulated data set, we estimated Ω by applying the ECM algorithm and the Metropolis-within-Gibbs algorithm. The former algorithm produced the MLE of Ω, and we used the median of the posterior distribution of Ω at convergence of the latter algorithm as another point estimate of Ω. Table 1 presents summary statistics of these estimates of Ω and estimates of the corresponding standard deviation across 1000 Monte Carlo replicates. According to Table 1, all estimates for parameters in Ω are reasonably close to the truth. A closer inspection on the reported empirical mean of these estimates along with their empirical standard error suggests that, when n = 100, the Bayesian method may slightly underestimate σ 2 , the larger of the two scale parameters of FG. We believe that this is due to the inverse gamma prior imposed on the scale parameters that is sharply peaked near zero, and thus the posterior median of the larger scale parameter tends to be pulled downwards when the sample size is not large. As the sample size increases to 200, this trend of underestimation appears to diminish. The empirical means of the standard deviation estimates from both methods are close to the corresponding empirical standard deviations, which indicate that the variability of a point estimator is accurately estimated, whether it is based on the sandwich variance estimator in the frequentist framework, or based on the posterior sampling in the Bayesian framework. In summary, the methods proposed in Section 3 under both frameworks provide reliable inference for Ω along with accurate uncertainty assessment of the point estimators when data arise from an FG distribution. Among all existing mixture distributions, normal mixtures probably have the longest history and are most referenced in the literature. In another experiment, we compared the model fitting of normal mixture with that of FG when data arise from three heavy-tailed distributions: (E2) Laplace with the location parameter equal to zero and the scale parameter equal to 2; (E3) a mixture of two Gumbel distributions for the maximum, with a common mode at zero, scale parameters in the two components equal to 2 and 6, respectively, and the mixing proportion equal to 0.5; (E4) a t distribution with degrees of freedom equal to 5. From each of the three distributions in (E2)-(E4), we generated a random sample of size n = 200, following which we fit a two-component normal mixture model via the EM algorithm implemented using the R package mixtools, and also fit an FG model via the two algorithms described in Section 3. This model fitting exercise was repeated for 1000 Monte Carlo replicates under each of (E2)-(E4). We used an empirical version of the Kullback-Leibler divergence as the metric to assess the quality of modeling fitting. We denote the true density function as p(·), and letp(·) be a generic estimated density resulting from one of the three considered model fitting strategies. Under each setting in (E2)-(E4), a random sample of size 50000, (x 1 , . . . , x 50000 ), were generated from the true distribution, and an empirical version of the Kullback-Leibler divergence fromp(·) to p(·) is given by D KL = (1/50000) 50000 i=1 log(p(x i )/p(x i )). Figure 1 shows the boxplots of D KL across 1000 Monte Carlo replicates corresponding to each model fitting scheme under (E2)-(E4). Judging from Figure 1, the FG distribution clearly outperform the normal mixture when fitting data from any of the three heavy-tailed distributions in (E2)-(E4), and results from the frequentist method are comparable with those from the Bayesian method for fitting an FG model. When implementing the ECM algorithm for fitting the FG model and the EM algorithm for fitting the normal mixture, we set a maximum number of iterations at 1000. Our ECM algorithm always converged in the simulation, i.e., converged to a stationary point within 1000 iterations. But the EM algorithm for fitting a normal mixture often had trouble achieving that, with more difficulty when data come from a heavier-tailed distribution. More specifically, under (E4), which has the highest kurtosis (equal to 9) among the three settings, the EM algorithm failed to converge in 59.9% of all Monte Carlo replicates; under (E2), which has the second highest kurtosis (equal to 6), it failed to converge in 6.7% of the replicates. Results associated with the normal mixture from these failing replicates were not included when producing the boxplots in Figure 1. In conclusion, the FG distribution is more suitable for symmetric or asymmetric heavy-tailed data than the normal mixture distribution. An application in hydrology Daily maximum water elevation changes of a waterbody, such as ocean, lake, and wetland, are of interest in hydrologic research. These changes may be close to zero in most days, but can be extremely large or small under extreme weather. From National Water Information System (https://waterdata.usgs.gov/), we downloaded water elevation data for Lake Murray near Columbia, South Carolina, United States, recorded from September 18, 2020 to September 18, 2021. The water elevation change of a given day was calculated by contrasting the maximum elevation and the minimum elevation on that day, returning a positive (negative) value if the maximum record of the day comes after (before) the minimum record on the same day. We fit the FG distribution to the resultant data with n = 366 records using the frequentist method and the Bayesian method, with results presented in produced very similar estimates for most parameters, although small differences were observed. For example, one would estimate the mode of daily maximum water elevation change to be −0.795 feet based on the frequentist method, but estimate it to be −0.486 feet using the Bayesian method. The discrepancy between these two mode estimates is minimal considering that the daily maximum water elevation changes range from −38 feet to 49.4 feet within this one-year period. In fact, taking into account the uncertainty in these point estimates, we do not interpret any of these differences as statistically significant because a parameter estimate from one method always falls in the interval estimate for the same parameter from the other method according to Table 2. Using parameter estimates in Table 2 in the aforementioned R Shiny app, we obtained an estimated skewness of −0.102 and an estimated kurtosis of 6.384 based on the frequentist inference results, whereas the Bayes inference yielded an estimated skeweness of 0.058 and an estimated kurtosis of 6.074. Combining these two sets of results, we concluded that the underlying distribution of daily maximum water elevation change may be nearly symmetry, with outliers on both tails that cause tails heavier than that of a Gumbel distribution. Figure 2 presents the estimated density functions from these two methods, in contrast with the estimated density curve resulting from fitting the data to a two-component normal mixture, and a kernel density estimate using a Gaussian kernel with the bandwidth selected according to the method proposed by Sheather and Jones [1991]. The last estimate is fully nonparametric and served as a benchmark against which the other three density estimates were assessed graphically. The kernel density estimate is more flexible at describing varying tail behaviors, but such flexibility comes at the cost of statistical efficiency and interpretability. With the wiggly tails evident in Figure 2 for this estimate, we suspected certain level of overfitting of the kernel density estimate. This often happens to kernel-based estimation of a function around a region where data are scarce, with a bandwidth not large enough for the region. Between the two FG density estimates, the difference is almost negligible. They both track the kernel density estimate closely over a wide range of the support around the mode. The mode of the estimated normal mixture density is close to the other three mode estimates, but the tails are much lighter than those of the other three estimated densities. Besides comparing the three parametric density estimates pictorially, we also used the Monte-Carlo based one-sample Kolmogorov-Smirnov test to assess the goodness of fit. The p-values from this test are 0.223, 0.312, and 0.106 for the frequentist FG density estimate, the Bayesian FG density estimate, and the estimated normal mixture density, respectively. Although none of the p-values are low enough to indicate lack of fit (at significance level 0.05 for example), the p-value associated with the normal mixture is much lower than those for FG. This provides quantitative evidence that an FG distribution fits the current data better than a normal mixture. It is also worth noting that the Kolmogorov-Smirnov test is known to have low power to detect deviations from a posited distribution that occur in the tails [Mason and Schuenemeyer, 1983]. This may explain the above-0.05 p-value for the normal mixture fit of the data even though the tail of this posited distribution may be too thin for the current data. We used STAN to implement the Bayesian inference for the Lake Murray data, and the code and posterior output are given in the Appendix. The output provided there indicates that our MCMC chain has converged (see the Rhat statistics). The JAGS code for fitting the FG distribution is also given in the Appendix. With the location parameter θ signified in the FG distribution as the mode, it is straightforward to formulate a modal regression model that explores the relationship between the response variable and predictors. To demonstrate the formulation of a modal regression model based on the FG distribution, we analyze a data set from Agresti et al. [2021] in the area of criminology. This data set contains the percentage of college education, poverty percentage, metropolitan rate, and murder rate for the 50 states in the United States and the District of Columbia from year 2003. The poverty percentage is the percentage of the residents with income below the poverty level; the metropolitan rate is defined as the percentage of population living in the metropolitan area; and the murder rate is the annual number of murders per 100, 000 people in the population. An application in criminology We fit the following modal regression model to investigate the association between the murder rate (Y ) and the aforementioned demographic variables, Y | β, σ 1 , σ 2 ∼ FG(β 0 + β 1 × college + β 2 × poverty + β 3 × metropolitan , σ 1 , σ 2 , w), where β = [β 0 , β 1 , β 2 , β 3 ] includes all regression coefficients. For the prior elicitation in Bayesian inference, we assume that β 0 , . . . , β 3 i.i.d ∼ N (0, 10 4 ) and use the same priors for σ 1 , σ 2 and w as those in Section 3.2. As a more conventional regression analysis to compare with our modal regression, we also fit the mean regression model assuming mean-zero normal model error to the data. Table 3 shows the inference results from the modal regression model, and Table 4 presents the inference results from the mean regression model. At 5% significance level, both frequentist and Bayesian modal regression analyses confirm that there exists a negative association between the percentage of college education and the murder rate, as well as a positive association between the metropolitan rate and the murder rate. In contrast, according to the inferred mean regression model, there is a positive association between the percentage of college education and the murder rate. Such claimed positive association is intuitively difficult to justify and contradicts with many published results in criminology [Hjalmarsson andLochner, 2012, Lochner, 2020]. The scatter plot of the data in Figure 3 can shed some light on why one reaches to such a drastically different conclusion on a covariate effect when mean regression is considered in place of modal regression. As shown in Figure 3, the exists an obvious outlier, District of Columbia (DC), in panels of the first row of the scatter plot matrix for instance. Mean regression reacts to this one extreme outlier by inflating the covariate effect associated with the percentage of college education in the inferred mean regression function. Thanks to the heavy-tailed feature of the FG distribution, modal regression based on this distribution is robust to outliers, which strives to capture data features suggested by majority of the data and is not distracted by the extreme outlier when inferring covariate effects in this application. Discussion The mode had been an overlooked location parameter in statistical inference until recently when the statistics community witnessed a revived interest in modal regression among statisticians [Chen, 2018, Chacón, 2020, Feng et al., 2020, Xu et al., 2020, Ullah et al., 2021, Wang and Li, 2021, Xiang and Yao, 2022. Historically, statistical inference for the mode have been mostly developed under the nonparametric framework for reasons we point out in Section 1. Existing semiparametric methods for modal regression only introduce parametric ingredients in the regression function, i.e., the conditional mode of the response, with the mode-zero error distribution left in a nonparametric form [Yao and Li, 2013, Yang and Yang, 2014, Zhao et al., 2014, Krief, 2017, Tian et al., 2017, Li and Huang, 2019. The few recently proposed parametric modal regression models all impose stringent parametric assumptions on the error distribution [Bourguignon et al., 2020, Zhou and Huang, 2020. Our proposed flexible Gumbel distribution greatly alleviates concerns contributing to data scientists' reluctance to adopt a parametric framework when drawing inference for the mode. This new distribution is a heterogeneous mixture in the sense that the two components in the mixture belong to different Gumbel distribution families, which is a feature that shields it from the non-identifiability issue most traditional mixture distributions face, such as the normal mixtures. The proposed distribution is indexed by the mode along with shape and scale parameters, and thus is convenient to use to draw inference for the mode while remaining flexible. It is also especially suitable for modeling heavy-tailed data, whether the heaviness in tails is due to extremely large or extremely small observations, or both. These are virtues of FG that cannot be achieved by the popular normal mixture and many other existing mixture distributions. We develop the numerically efficient and stable ECM algorithm for frequentist inference for the FG distribution, and a reliable Bayesian inference method that can be easily implemented using free software, including STAN, JAGS, and BUGS. Compared with the more widely adopted mean regression framework, the modal regression model based on FG we entertained in Section 6 shows great potential in revealing meaningful covariate effects potentially masked by extreme outliers. With these advances made in this study, we open up new directions for parametric modal regression and semiparametric modal regression with a fully parametric yet flexible error distribution, and potentially nonparametric ingredients incorporated in the regression function. Disclosure statement Computer programs for implementing the FG distribution, related models and data used in this paper are available at https://github.com/rh8liuqy/flexible_Gumbel. Figure 1 : 1Boxplots of the empirical Kullback-Leibler divergence from an estimated density to the true density under each of the true-model settings in (E2)-(E4). Under each setting, the three considered model fitting strategies are, from left to right in the figure, (i) using the ECM algorithm to fit an FG distribution (FG ECM), (ii) using the Bayesian method to fit an FG distribution (FG Bayes), and (iii) using the EM algorithm to fit a normal mixture distribution (Normal Mixture Distribution EM). Figure 2 : 2Four density estimates based on daily maximum water elevation changes in Lake Murray, including the kernel density estimate (solid line), the estimated FG density from the ECM algorithm (dotted line), the estimated FG density from the Bayesian method (dashed line), and the estimated normal mixture density (dash-dotted line). Figure 3 : 3Scatter plot matrix of the crime data. Table 1 : 1Frequentist and Bayesian inference results in experiment (E1) across 1000 Monte Carlo replicates. Here, point.est stands for the average of 1000 point estimates for each parameter from each method, s.d. stands for the average of the corresponding 1000 estimated standard deviations, and s.d. refers to the empirical standard deviation of the 1000 point estimates from each method. Numbers in parentheses are 100× Monte Carlo standard errors associated with the averages.Frequentist Bayesian sample size parameter point.est s.d. s.d. point.est s.d. s.d. Table 2 . 2The two inference methods0.00 0.05 0.10 0.15 E2 E3 E4 Kullback−Leibler divergence FG ECM FG Bayes Normal Mixture Distribution EM Table 2 : 2Frequentist and Bayesian inferences about daily maximum water elevation changes of Lake Murray, South Carolina, United States. Besides parameter estimates (under point.est) and the estimated standard deviations of these parameter estimates (under s.d.), 95% confidence intervals of the parameters from the frequentist method, and 95% credible intervals from the Bayesian method are also provided (under lower 95 and upper 95). Frequentist Bayesian parameter point.est s.d. lower 95 upper 95 point.est s.d. lower 95 upper 95 θ -0.795 0.796 -2.355 0.764 -0.486 0.694 -1.679 0.973 σ1 5.186 0.541 4.124 6.247 5.399 0.651 4.534 6.916 σ2 6.237 1.735 2.836 9.638 5.734 1.031 4.390 8.029 w 0.698 0.169 0.367 1.029 0.630 0.141 0.329 0.847 Table 3 : 3Frequentist and Bayesian modal regression models based on the FG distribution fitted to the crime data.Besides parameter estimates (under point.est) and the estimated standard deviations of these parameter estimates (under s.d.), 95% confidence intervals of the parameters from the frequentist method, and 95% credible intervals from the Bayesian method are also provided (under lower 95 and upper 95). 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{'abstract': 'A new unimodal distribution family indexed by the mode and three other parameters is derived from a mixture of a Gumbel distribution for the maximum and a Gumbel distribution for the minimum. Properties of the proposed distribution are explored, including model identifiability and flexibility in capturing heavy-tailed data that exhibit different directions of skewness over a wide range. Both frequentist and Bayesian methods are developed to infer parameters in the new distribution. Simulation studies are conducted to demonstrate satisfactory performance of both methods. By fitting the proposed model to simulated data and data from an application in hydrology, it is shown that the proposed flexible distribution is especially suitable for data containing extreme values in either direction, with the mode being a location parameter of interest. A regression model concerning the mode of a response given covariates based on the proposed unimodal distribution can be easily formulated, which we apply to data from an application in criminology to reveal interesting data features that are obscured by outliers. Computer programs for implementing all considered inference methods in the study are available at https://github.com/rh8liuqy/flexible_Gumbel.', 'arxivid': '2212.01832', 'author': ['Qingyang Liu qingyang@email.sc.edu \nDepartment of Statistics\nDepartment of Statistics\nUniversity of South Carolina Columbia\nUniversity of South Carolina Columbia\nDaiichi Sankyo Inc. Basking Ridge\n29201, 29201, 07920SC, SC, NJ\n', 'Xianzheng Huang huang@stat.sc.edu \nDepartment of Statistics\nDepartment of Statistics\nUniversity of South Carolina Columbia\nUniversity of South Carolina Columbia\nDaiichi Sankyo Inc. Basking Ridge\n29201, 29201, 07920SC, SC, NJ\n', 'Haiming Zhou haiming2019@gmail.com \nDepartment of Statistics\nDepartment of Statistics\nUniversity of South Carolina Columbia\nUniversity of South Carolina Columbia\nDaiichi Sankyo Inc. Basking Ridge\n29201, 29201, 07920SC, SC, NJ\n'], 'authoraffiliation': ['Department of Statistics\nDepartment of Statistics\nUniversity of South Carolina Columbia\nUniversity of South Carolina Columbia\nDaiichi Sankyo Inc. Basking Ridge\n29201, 29201, 07920SC, SC, NJ', 'Department of Statistics\nDepartment of Statistics\nUniversity of South Carolina Columbia\nUniversity of South Carolina Columbia\nDaiichi Sankyo Inc. Basking Ridge\n29201, 29201, 07920SC, SC, NJ', 'Department of Statistics\nDepartment of Statistics\nUniversity of South Carolina Columbia\nUniversity of South Carolina Columbia\nDaiichi Sankyo Inc. Basking Ridge\n29201, 29201, 07920SC, SC, NJ'], 'corpusid': 254246938, 'doi': None, 'github_urls': ['https://github.com/rh8liuqy/flexible_Gumbel.', 'https://github.com/rh8liuqy/flexible_Gumbel.'], 'n_tokens_mistral': 17643, 'n_tokens_neox': 14467, 'n_words': 8842, 'pdfsha': 'cca1eb8e9435200ee33cbcbe12e9336ffd00072f', 'pdfurls': ['https://export.arxiv.org/pdf/2212.01832v1.pdf'], 'title': ['THE FLEXIBLE GUMBEL DISTRIBUTION: A NEW MODEL FOR INFERENCE ABOUT THE MODE A PREPRINT', 'THE FLEXIBLE GUMBEL DISTRIBUTION: A NEW MODEL FOR INFERENCE ABOUT THE MODE A PREPRINT'], 'venue': []}

arxiv

About Evaluation of F1 Score for RECENT Relation Extraction System 16 May 2023 Micha L Olek michal.olek@pwr.edu.pl Wroc law University of Science and Technology Wroc lawPoland About Evaluation of F1 Score for RECENT Relation Extraction System 16 May 2023Relation extraction · Relation classification · F1 score This document contains a discussion of the F1 score evaluation used in the article "Relation Classification with Entity Type Restriction" by Shengfei Lyu, Huanhuan Chen published on Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021. The authors created a system named RECENT and claim it achieves (then) a new state-of-the-art result 75.2 (previous 74.8) on the TACRED dataset, while after correcting errors and reevaluation the final result is 65.16 Introduction Relation extraction is the main topic of the article "Relation Classification with Entity Type Restriction" by Shengfei Lyu, Huanhuan Chen [2] . The authors describe a system named RECENT that deals with extracting relations from sentences in documents where mentions (named entities) are already annotated. The code is available at https://github.com//Saintfe/RECENT. Technically, this kind of relation extraction task checks in the document each possible triplet (s, e1, e2) -where s is a sentence and e1 and e2 are mentions from the sentenceand assigns to the triplet a relation chosen from a set of predefined relations (set includes also special "no relation" relation indicating that there is no relation between mentions). Often e1 mention is called subject and e2 mention is called object, as relations are, in general, directed. Recent works focus mainly on making use of neural networks as in [5] or using various additional enhancements like curriculum learning [6]. There are also approaches using two-way span prediction [7] or graph convolution networks over pruned dependency trees [3]. Dataset The dataset used is TACRED [4] -one of the most popular datasets for sentencelevel relation extraction. In TACRED dataset, the mentions are not only just marked but also their type is annotated. The subject mention is of type OR-GANIZATION or PERSON and the object mention is categorized as one of 16 types such as LOCATION, ORGANIZATION, PERSON, DATE, etc. The names of relations contain also an abbreviation of type of subject mentionrelations names look like 'org:city of headquarters', 'per:age', 'per:date of birth' Entity Type Restriction The authors noticed that it is very often the case that a type of relation determines the types of possible mentions involved and vice versa. So, for example, if we have a relation per:date of birth then it expects the subject to be a PER-SON and the object to be a DATE. Same the other way -if we have another sentence where the type of subject is ORGANIZATION and the type of object is CITY, one does not need to perform any additional checking to know that relation per:date of birth is not a valid relation here. But relation org:city-ofheadquarters may be. So we can regroup the data into the independent subsets, where each subset contains sentences with just one pair of types of subject and object i.e. (OR-GANIZATION,PERSON). And then for each such subset, we can construct a set of all matching relations types from relations that can be found in at least one sentence of the subset. The list would look like ( no relation, org:founded by, org:shareholders, org:employees). The number of relation types is usually significantly smaller for the subset than for the whole set. With such regrouping, we can decompose the whole task of relations extraction to many similar smaller independent tasks and train a specific, separate semantic classifier for each such subset. Since, generally, the number of relation types is smaller for each subset than for the whole set, we assume that for each subset the independent semantic classifier -focused only on that particular subset -should perform better than the classifier trained on the whole set. So we process each subset using a corresponding semantic classifier and after gathering all the results from all semantic classifiers we should get better overall result for the whole data set. Since this approach is model-agnostic the system RECENT was tested with two models: CGN [3] and SpanBERT [1]. For the first model reported result was 70.9 F1 score, and for the second model 75.2 F1 score. Implementation In the beginning, the whole dataset is divided into the aforementioned subsets. Normally it would be divided into quite a few subsets because there are quite a few combinations that can be made of 2 subject mention types and 16 object mention types. But authors noticed that all except 13 subsets are so simple that for each such simple subset any triplet can be assigned just to either one relation specific to this subset or to "no relation". Binary classifier For example, in the subset constructed from subject mention type PERSON and object mention type RELIGION the only meaningful relation that can be found between mentions in training data is per:religion. In another subset, where the subject mention type is PERSON and object mention type is TITLE the only meaningful relation possible to be found in training data is per:title. We infer from training data what is a relation that such subset may assign triplets to, so basically, the relation extraction task for these kinds of subsets is reduced just to decide whether there is a relation for triplet or whether there is not. It can be done with a binary classifier. This is classifier trained to recognize just whether there is a meaningful relation in a given triplet and does not say what kind of relation it is. The classifier is trained once on the whole train dataset. During evaluation, this is the first step: the binary classifier checks if given triplet is "no relation" and if it is true the final answer is "no relation". Second step checks if the triplet is a one from the simple subsets and if it is true then the final answer is a relation associated with this simple subset. So the only ones left are triplets from the other, more complicated, subsets and they are to be dealt with the semantic classifiers. Semantic classifiers The semantic classifiers need to be trained only for the 13 more complicated subsets. For each such subset, there are three helper files generated, containing the corresponding train, tune, and test examples and one additional file with a list of possible relations that can be found in triplets of that subset. Each semantic classifier is trained to recognize which one of these possible relations is associated with a given triple from the particular subset. However, we need to keep in mind that the very first step of evaluation is binary classifier filtering out all triplets considered by it as associated with "no relation". With this setup, the semantic classifiers do not deal with recognizing "no relation". They have to assume that if something is passed to them it contains only "meaningful" relation. This way they do not contain "no relation" label in available answers. So, to train them well, all helper files ( training, tune, and test) created for semantic classifiers are cleared of samples with the answer "no relation". The resulting training and tuning files are used to train semantic classifiers. The resulting test files are used for the pre-evaluation step described in the next section Sidenote: each one of these additional files mentioned at the beginning of paragraph and containing the list of possible relations for the corresponding subset is generated using data files with test examples but should be generated using files with train examples. However, the data is so homogeneous that correcting this issue has no significant effect on the final result. Evaluation The whole idea of prediction for a sample goes as follows: 1. Sample is checked against the binary classifier. If it says it is "no relation" then this is the final answer for this sample. 2. Otherwise, sample is checked against simple subsets. If it is an element of such a subset then the final answer is the one relation associated with the subset. 3. Otherwise, the corresponding complicated subset is identified and the sample is checked by the semantic classifier trained for this particular subset. Actually, in the third step, the semantic classifiers are not used. Very likely to avoid a situation where all the models for semantic classifiers are present in the memory at the same time. Checking is actually done against the pre-computed partial final result file for the corresponding subset. The file is made even before actual evaluation phase: during this pre-evaluation step, results are computed and the partial files are created -each semantic classifier is loaded into memory once and is run over the helper test file corresponding to the associated subset and the results are stored in "partial final result files", and then, in step 3, the values contained in files are used as final predictions for the more complicated subsets. So, in the third step the system checks the file if it contains the given triple: -3.1 if the sample is found then its corresponding answer is the final result -3.2 otherwise the final answer is set to 'no relation' The last point may come as a bit of a surprise. It is not mentioned in the article but it is how the code works (lines 56-59 in file SpanBERT/recent eval.py). Since the binary classifier allowed the sample to pass through, there should be a meaningful answer. How can it be that for samples classified by the binary classifier as the ones that should be associated with a relation, and which should be processed by semantic classifiers, the system is coded to give an answer as no relation? The solution is simple: the binary classifier is not perfect and there are cases when it classifies the sample as the one that should be associated with meaningful relation even if actually the sample is associated with no relation The semantic classifiers were taught on purpose to always give a meaningful relation as an answer and they are not capable to handle this situation. So it is hard-coded in the system that in such a situation the final answer should be no relation. Is it correct, and if yes, why? Evaluation of a single sample Let us take for example a sample with id '098f665fb92fee9d29b3' -the last sample in the test data set. Its subject type is ORGANIZATION and its object type is PERSON and there is a semantic classifier trained for this pair of types. There is no relation in our sample but the binary classifier happened to incorrectly classify this sample as the one with a meaningful relation. And if we manually process step 3 we check against preprocessed partial final result file for the corresponding semantic classifier -the sample is not found there, so the system classifies the sample as having no relation. Correctly, but only because the sample was not found in pre-processed helper test file. Why it is not found there? It is not there because during preprocessing -as described at the end of paragraph 4.2 -all samples with "no relation" were filtered out from the test data. Normally, with fresh production data -let us assume we are processing a completely new piece of data without test dataset ( when there is no pre-processed test data provided) -the sample like the one described above would be classified incorrectly. It would be classified as associated with one of the meaningful relation types associated to corresponding semantic classifier since the classifier is unable to yield "no relation". as a final result. And it "does not know" that binary classifier was wrong. Simply, if test data were not provided earlier then there will be no filtering out the samples with "no relation". To put it short: the system says that this particular sample from test data should be classified as "no relation" because the system checked earlier that in the test data it is classified as "no relation". It happens to all samples that belong to complicated subsets, have no meaningful relation, and are incorrectly classified by the binary classifier as the ones which have meaningful relation. All these samples are classified correctly due to this loophole. After closing this loophole all these samples are classified incorrectly as false positives since the semantic classifiers are unable to yield the answer "no relation". Reevaluation To calculate the actual value of F1 a new experiment is needed with such a modification that corrects the issue described in the previous paragraph. All the used models are exactly the same as in the original experiment but in cases where workflow comes to point 3, a sample is always given to the corresponding semantic classifier to be classified -without checking if the sample is present in the appropriate "partial final result file". In terms of calculating the F1 score, and technical realization, the above setup is equivalent to just changing step 3.2 to yield any meaningful relation except the correct one: "no relation", since the semantic classifiers are unable to return "no relation". The task is multiclass classification where one does not count in true positives for "no relation". Table 1 presents the results of recreated original experiment and the results of experiment with correction closing the loophole. One can see that only the number of false positives has changed. To calculate F1 score from these results we need calculate precision and recall. Precision is calculated using formula: P = T P T P + F P(1) and recall is calculated using formula: R = T P T P + F N(2) and F1 score is calculated as a harmonic mean of P and R: Precision, recall, and F1 score are presented for both experiments in table 2. One can see that recall is the same in both experiments since only the number of false positives has changed. F 1 = 2 * P * R P + R(3) After calculating the actual value for the F1 score for system RECENT using SpanBERT model is 65.16, which is 10 pp. lower than originally reported. When evaluating using CGN model the drop in precision is the same and the reevaluated F1 score for this model is 61. The idea was very clever and promising but the results show that it does not give the expected improvement in performance. Perhaps this is due to the fact that each individual classifier learns on a much smaller dataset than such a single general one. And the loss from having less data to learn is greater than the gain gained from allowing the classifier to focus on only a small subset of the responses. Table 1 . 1Comparison of results of experiments True Positive (TP) False Positive (FP) False Negative (FN)Recreated original experiment 2182 246 1143 Experiment with correction 2182 1190 1143 Table 2 . 2Comparison of precision, recall and F1 (x100)Precision Recall F1 Recreated original experiment 89.86 65.62 75.85 Experiment with correction 64.70 65.62 65.16 Span-BERT: Improving Pre-training by Representing and Predicting Spans. J Mandar, Danqi Ch, L Yinhan, D Weld, L Zettlemoyer, Levy O , 10.1162/tacla00300Transactions of the Association for Computational Linguistics. Johnson M., Roark B., Nenkova A.8MIT PressMandar J., Danqi Ch., Yinhan L., Weld D., Zettlemoyer L., and Levy O.: Span-BERT: Improving Pre-training by Representing and Predicting Spans. In: Johnson M., Roark B., Nenkova A. (eds) Transactions of the Association for Computational Linguistics, vol. 8, pp. 64-77. MIT Press, Cambridge (2020) https://doi.org/10.1162/tacl a 00300 Relation classification with entity type restriction. L Shengfei, Huanhuan Ch, 10.18653/v1/2021.findings-acl.34Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021. Chengqing Z., Fei X., Wenjie L., Navigli R.OnlineAssociation for Computational Linguistics1Shengfei L., Huanhuan Ch.: Relation classification with entity type restric- tion. In: Chengqing Z., Fei X., Wenjie L., Navigli R. (eds.) Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, vol. 1, pp. 390-395. Association for Computational Linguistics, Online (2021). https://doi.org/10.18653/v1/2021.findings-acl.34 Graph convolution over pruned dependency trees improves relation extraction. Z Yuhao, Q Peng, Manning Ch, 10.18653/v1/D18-1244Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing. Riloff E., Chiang D., Hockenmaier J., Tsujii J.the 2018 Conference on Empirical Methods in Natural Language ProcessingBrusselsAssociation for Computational Linguistics1Yuhao Z., Peng Q., and Manning Ch.: Graph convolution over pruned dependency trees improves relation extraction. In: Riloff E., Chiang D., Hockenmaier J., Tsujii J. (eds.) Proceedings of the 2018 Conference on Empirical Methods in Natural Lan- guage Processing, vol. 1, pp. 2205-2215. Association for Computational Linguistics, Brussels (2018). https://doi.org/10.18653/v1/D18-1244 Position-aware attention and supervised data improve slot filling. Z Yuhao, V Zhong, Danqi Ch, G Angeli, Manning Ch, 10.18653/v1/D17-1004Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing. Palmer M., Hwa R., Riedel S.the 2017 Conference on Empirical Methods in Natural Language ProcessingCopenhagenAssociation for Computational Linguistics1Yuhao Z., Zhong V., Danqi Ch., Angeli G., Manning Ch.: Position-aware attention and supervised data improve slot filling. In: Palmer M., Hwa R., Riedel S. (eds) Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing vol. 1, pp. 35-45, Association for Computational Linguistics, Copenhagen (2017). https://doi.org/10.18653/v1/D17-1004 An Improved Baseline for Sentence-level Relation Extraction. Z Wenxuan, Muhao Ch, CoRR abs/2102.01373Wenxuan Z., Muhao Ch.: An Improved Baseline for Sentence-level Relation Extrac- tion. CoRR abs/2102.01373, (2021) Improving Sentence-Level Relation Extraction through Curriculum Learning. P Seongsik, K Harksoo, ArXiv abs/2107.09332Seongsik P., Harksoo K.: Improving Sentence-Level Relation Extraction through Curriculum Learning. ArXiv abs/2107.09332 (2021) Relation Extraction as Two-way Span-Prediction. A Cohen, S Rosenman, Y Goldberg, CoRR abs/2010.04829Cohen A., Rosenman S., Goldberg Y.: Relation Extraction as Two-way Span- Prediction. CoRR abs/2010.04829 (2020)

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{'abstract': 'This document contains a discussion of the F1 score evaluation used in the article "Relation Classification with Entity Type Restriction" by Shengfei Lyu, Huanhuan Chen published on Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021. The authors created a system named RECENT and claim it achieves (then) a new state-of-the-art result 75.2 (previous 74.8) on the TACRED dataset, while after correcting errors and reevaluation the final result is 65.16', 'arxivid': '2305.09410', 'author': ['Micha L Olek michal.olek@pwr.edu.pl \nWroc law\nUniversity of Science and Technology\nWroc lawPoland\n'], 'authoraffiliation': ['Wroc law\nUniversity of Science and Technology\nWroc lawPoland'], 'corpusid': 258714675, 'doi': '10.48550/arxiv.2305.09410', 'github_urls': ['https://github.com//Saintfe/RECENT.'], 'n_tokens_mistral': 4753, 'n_tokens_neox': 4208, 'n_words': 2890, 'pdfsha': 'c4373c9f10178ff496e6a8244377b6a74418d254', 'pdfurls': ['https://export.arxiv.org/pdf/2305.09410v1.pdf'], 'title': ['About Evaluation of F1 Score for RECENT Relation Extraction System', 'About Evaluation of F1 Score for RECENT Relation Extraction System'], 'venue': []}

arxiv

Directionality between driven-dissipative resonators C A Downing Department of Physics and Astronomy University of Exeter EX4 4QLExeterUnited Kingdom T J Sturges Institute of Theoretical Solid State Physics Karlsruhe Institute of Technology (KIT) D-76131KarlsruheGermany Directionality between driven-dissipative resonators (Dated: December 27, 2022) The notion of nonreciprocity, in essence when going forwards is different from going backwards, emerges in all branches of physics from cosmology to electromagnetism. Intriguingly, the breakdown of reciprocity is typically associated with extraordinary phenomena, which may be readily capitalized on in the design of (for example) nontrivial electromagnetic devices when Lorentz reciprocity is broken. However, in order to enable the exploitation of nonreciprocal-like effects in the next generation of quantum technologies, basic quantum optical theories are required. Here we present a versatile model describing a pair of driven-dissipative quantum resonators, where the relative phase difference between the coherent and incoherent couplings induces an asymmetry. The interplay between the diverse dissipative landscape -which encompasses both intrinsic losses and dissipative couplings -and the coherent interactions leads to some remarkable consequences including highly directional (or even one-way) energy transport. Our work proffers the tantalizing prospect of observing dissipation-induced quantum directionality in areas like photonics or cavity magnonics (spin waves), which may aid the design of unconventional nanoscopic devices. Introduction In perhaps one of the most profound conversations in Confucianism, Tsze-kung asks "Is there one word which may serve as a rule of practice for all one's life?" The Master replies "Is not shu [reciprocity] such a word? What you do not want done to yourself, do not do to others" [1]. In physics, reciprocity is a famous hallmark across the entire discipline, from the equal and opposite quality of Newton's third law of motion [2] to the Lorentz reciprocity in electromagnetism, which guarantees the same response when the source and receiver are interchanged [3]. Introducing various types of asymmetries can lead to a treasure trove of curiosities, which in optics precipitated the exciting subfields of chiral plasmonics [4,5] and chiral quantum optics [6][7][8]. Within device physics, it has already been shown that exploiting nonreciprocity can give rise to practical applications like high quality factor, large bandwidth devices [9] (which are predicated upon the induced asymmetric transport properties), optical isolators [10] (where the governing scattering matrix is inherently asymmetric), and even magnetic diodes [11] (where it was demonstrated that the magnetic coupling between two coils above a conductor -moving with constant velocity -may become asymmetric, leading to a diode for magnetic fields). Here we investigate a basic quantum optical model, namely a pair of driven-dissipative resonators, with a view to inducing asymmetric behaviour [12][13][14][15]. We consider two oscillators, which are in general coupled both coherently and incoherently, where the first resonator is additionally coherently driven by a laser as sketched in Fig. 1. The admixture between the coherent and incoherent couplings, which are in general complex quantities, has profound consequences for directionality in the system. There are four principle coupling regimes of our model: (i) coherent coupling, where the direct hopping between the resonators dominates (gray rod in the * c.a.downing@exeter.ac.uk 1. A sketch of a pair of driven-dissipative resonators. The first resonator (red pillar) is driven by a laser with amplitude Ω, while the second resonator (green pillar) is undriven [cf. Eq. (1)]. The coherent coupling (of magnitude g and phase θ) between the resonators is represented by the gray rod, while the dissipative coupling (of magnitude Γ and phase φ) is mediated by the common bath (blue disk) [cf. Eq. (2)]. The individual loss γ of each resonator is associated with the red and green disks. ge iθ −→ ge −iθ ←− −→ γ −→ Γe iφ −→ Γe −iφ −→ γ Ω −→ FIG. figure), (ii) dissipative coupling, where incoherent coupling between the resonators is of primary importance (as mediated by their common heat bath, the blue disk in the figure), (iii) unidirectional coupling, where the coupling between the resonators is completely one-way, and (iv) asymmetric coupling, where the mixture of coherent and incoherent coupling leads to asymmetries in the interactions between the resonator pair [such that this case generalizes the more extreme limiting case (iii)]. In what follows, we provide a simple analysis of the population dynamics in these regimes. Model The driven coupled oscillators model represented in H = ω ∆ b † 1 b 1 + b † 2 b 2 + Ω b † 1 + b 1 + ge iθ b † 1 b 2 + ge −iθ b † 2 b 1 ,(1) where the n-th oscillator sustains bosonic excitations created (destroyed) by the operator b † n (b n ). The coherent oscillatoroscillator coupling is of strength g ≥ 0 and phase θ. The first oscillator is driven by a laser of amplitude Ω, and the detunings ω ∆ arise in the chosen rotating reference frame of the laser [16]. We include dissipation in the model through a quantum master equation, which describes the time evolution of the density matrix ρ of the system via [16][17][18] ∂ t ρ = i[ρ,Ĥ] + n=1,2 γ 2 2b n ρb † n − b † n b n ρ − ρb † n b n + Γe iφ 2 2b 2 ρb † 1 − b † 1 b 2 ρ − ρb † 1 b 2 + Γe −iφ 2 2b 1 ρb † 2 − b † 2 b 1 ρ − ρb † 2 b 1 .(2) The first line of Eq. (2) is the von Neumann equation, describing the unitary evolution of the closed system as governed by the HamiltonianĤ of Eq. (1). The non-unitary evolution is captured by the three lower lines of Eq. (2), which are written as Lindblad terms and describe the open quantum system sketched in Fig. 1. In particular, the second line on the righthand-side of Eq. (2) accounts for the intrinsic loss γ of each oscillator. The third and fourth lines of Eq. (2) track the dissipative (incoherent) coupling between the pair of oscillators due to their shared heat bath, which can generally be regarded as a complex quantity of magnitude Γ and phase φ (subject to the condition 0 ≤ Γ ≤ γ). The first moments b n of the system described by Eq. (1) and Eq. (2) may be found by the following pair of coupled first-order equations [16] i∂ t b 1 b 2 = ω ∆ − i γ 2 G − G * + ω ∆ − i γ 2 b 1 b 2 + Ω 0 ,(3) where we have introduced two generalized coupling constants G + and G − , hereby defined as G ± = ge iθ ± 1 2 Γe iφ ,(4) which accounts for the admixture between the competing coherent and dissipative couplings, including their magnitudes and phases. Most notably, this analysis reveals that the oscillator-oscillator coupling may be completely unidirectional, as was first noticed in the celebrated works on cascaded quantum systems [19,20]. When G − = 0 in Eq. (3) one-way coupling in the rightwards direction (→) arises, and likewise when G * + = 0 in Eq. (3) the coupling is completely leftwards (←). Amongst the entire space of possible couplings, these two special circumstances occur when the following conditions on the coupling magnitudes and relative phases hold [cf. Eq. (4)] Γ = 2g, θ − φ = π 2 (→) 3π 2 (←) .(5) Away from these twin conditions for unidirectionality, the overall coupling is generally asymmetric. Wonderfully, the basic theoretical model encapsulated by Eq. (1) and Eq. (2) may be realized in an eclectic range of systems. For example, in spin-photon systems such as cavity magnons with coherent and dissipative couplings [21,22], in circuit-QED setups utilizing superconducting qubits [23,24], in plasmonic epsilon-near-zero waveguides [25], in metallic nanoparticle architectures exploiting plasmonic responses [26], and with coupled cavity-based photonic devices [12,27]. In what follows we keep our discussion of the model general, keeping in mind specializations are readily obtainable experimentally. Coherent coupling Let us first consider the simplest case of purely coherent coupling between the resonators (so that Γ = 0), as is represented in the sketch of Fig. 2 (a). We are interested in the populations b † n b n of the resonators, which may be calculated from the second moment analogue of Eq. (3), as discussed in Ref. [16]. At long time scales, the competition between the driving and dissipation leads to a well-defined steady state, described by the analytic expressions [16] lim t→∞ b † 1 b 1 = 2γΩ γ 2 + 4g 2 2 ,(6)lim t→∞ b † 2 b 2 = 4gΩ γ 2 + 4g 2 2 .(7) These populations, scaled by (γ/Ω) 2 , are plotted as a function of the coherent coupling strength g for the first resonator (red line) and second resonator (green line) in Fig. 2 (b). Clearly, since only the first resonator is driven, its steady state population is bounded by its maximum of (2Ω/γ) 2 when g γ, and decreases to {γΩ/ 2g 2 } 2 when g γ. In these limits, the second (and undriven) resonator population is zero when g γ and (Ω/g) 2 when g γ, its maximum population of (Ω/γ) 2 is instead met when g = γ/2. Hence the population imbalance between the resonators is a useful quantity to describe the system, in the steady state it reads ∆ = lim t→∞ b † 1 b 1 − b † 2 b 2 b † 1 b 1 + b † 2 b 2 .(8) In this coherent coupling regime, ∆ is a sign-changing quantity, which is explicitly given by ∆ = 1 − 8g 2 γ 2 + 4g 2 ,(9) which observes the bounds of −1 ≤ ∆ ≤ 1, as displayed with the thick orange line in Fig. 2 (b). The critical point of a completely balanced populations across the resonators ∆ = 0 is reached when g = γ/2, and above this coupling strength the second resonator has a larger steady state population despite being undriven. The full dynamic populations b † n b n of the coupled resonators are given by the exact equations [16] b † 1 b 1 = 2γΩ γ 2 + 4g 2 2 + 2 2γΩ γ 2 + 4g 2 2 2g sin (gt) − γ cos (gt) e − γt 2 (10) + γ 2 + 4g 2 γ 2 + 4g 2 + 4Ω 2 + γ 2 + 4g 2 2 + 4Ω 2 γ 2 − 4g 2 cos (2gt) − 16gγΩ 2 sin (2gt) e −γt 2 (γ 2 + 4g 2 ) 2 , b † 2 b 2 = 4gΩ γ 2 + 4g 2 2 − 4gΩ γ 2 + 4g 2 2 2g cos (gt) + γ sin (gt) e − γt 2 (11) + γ 2 + 4g 2 γ 2 + 4g 2 + 4Ω 2 − γ 2 + 4g 2 2 + 4Ω 2 γ 2 − 4g 2 cos (2gt) + 16gγΩ 2 sin (2gt) e −γt 2 (γ 2 + 4g 2 ) 2 , which are plotted in Fig. 2 (c) as a function of time (for the example case of g = 2γ). The characteristic damped Rabi oscillations are shown, and the two different time constants appearing in the above expressions (1/γ and 2/γ) is characteristic of coherently driven systems. This sets out the stall for the most well known coupling regime. Dissipative coupling When the coherent coupling is negligible (that is, g = 0), the system is in the dissipative coupling regime. This setup is sketched in Fig. 2 (d), where the blue disk represents the common heat bath enabling the incoherent coupling. In the steady state, the resonator populations are described by the simple forms [16] lim t→∞ b † 1 b 1 = 2γΩ γ 2 − Γ 2 2 ,(12)lim t→∞ b † 2 b 2 = 2ΓΩ γ 2 − Γ 2 2 .(13) In the weak dissipative coupling limit Γ γ of course only the first, driven resonator is populated, with (2Ω/γ) 2 . In the opposing strong dissipative coupling Γ → γ limit the bosonic nature of the resonators becomes readily apparent, since both populations tend towards infinity, as is shown in in Fig. 2 (e). The population imbalance [cf. Eq. (8)] is always singledsigned and reads ∆ = 1 − 2Γ 2 Γ 2 + γ 2 ,(14) which exposes the necessarily non-negative bounds of 0 ≤ ∆ ≤ 1, as show by the thick orange line in Fig. 2 (e). This in stark contrast to the coherent coupling regime, which allows for positive and negative imbalances [cf. panel (b)]. The time-dependent populations are described by the following expressions [16] b † 1 b 1 = 2γΩ γ 2 − Γ 2 2 + cosh (Γt) + 1 + 4Ω 2 γ 2 − Γ 2 e −γt 2 + Ω 2 e −(γ+Γ)t (γ + Γ) 2 + e −(γ−Γ)t (γ − Γ) 2 − 4γΩ 2 γ 2 − Γ 2 e − (γ+Γ)t 2 γ + Γ + e − (γ−Γ)t 2 γ − Γ ,(15)b † 2 b 2 = 2ΓΩ γ 2 − Γ 2 2 + cosh (Γt) − 1 − 4Ω 2 γ 2 − Γ 2 e −γt 2 + Ω 2 e −(γ+Γ)t (γ + Γ) 2 + e −(γ−Γ)t (γ − Γ) 2 + 4ΓΩ 2 γ 2 − Γ 2 e − (γ+Γ)t 2 γ + Γ − e − (γ−Γ)t 2 γ − Γ ,(16) as displayed in Fig. 2 (f) for the example case of reasonably strong dissipative coupling Γ = (4/5)γ. A hallmark of this coupling regime is the lack of any Rabi oscillations due to the absence of any coherent coupling, and the supremacy in population of the first resonator for any value of the dissipative coupling strength Γ, as suggested by Eq. (14). Unidirectional coupling The final special case of coupling that we shall consider is that of unidirectional coupling, and in particular when the conditions of Eq. (5) are met for the rightwards (→) direction only, as represented by the picture in Fig. 2 (g). The lack of backaction ensures that the first resonator population is coupling independent (in this regime, the dissipative coupling strength Γ = 2g is fixed) while the second resonator population is enhanced due to the one-way nature of the interaction. The steady state results are simply [16] lim t→∞ b † 1 b 1 = 2Ω γ 2 2 ,(17)lim t→∞ b † 2 b 2 = 4ΓΩ γ 2 2 ,(18) which are plotted in Fig. 2 (h) as a function of Γ. As must be the case, the first resonator population (red line) is exactly that of a single driven-dissipative oscillator [16], while the second resonator resonator presents a quadratic scaling with the dissipative coupling Γ. Therefore, the population imbalance [cf. Eq. (8)] is given by ∆ = 1 − 8Γ 2 γ 2 + 4Γ 2 ,(19) as plotted as the thick orange line in Fig. 2 (h). Unlike in the dissipative coupling regime, this population imbalance is a sign-changing quantity, being bounded by −3/5 ≤ ∆ ≤ 1. The critical point of ∆ = 0 is reached when Γ = γ/2, such that above this dissipative coupling strength the undriven second resonator is more highly populated that the driven and first resonator. The dynamic populations are given by the compact analytical expressions [16] b † 1 b 1 = 2Ω γ 2 − 2 2Ω γ 2 e − γt 2 + 1 + 2Ω γ 2 e −γt ,(20)b † 2 b 2 = 4ΓΩ γ 2 − 4ΓΩ γ 2 (2 + γt) e − γt 2 + Γ γ 2 (γt) 2 + (2 + γt) 2 2Ω γ 2 e −γt ,(21) where Eq. (20) is exactly the form for a solitary driven resonator, as if the second resonator was not there, due to the completely supressed backaction. We plot the expressions of Eq. (20) and Eq. (21) in Fig. 2 (i) for the case of the rather strong coupling Γ = (4/5)γ. This coupling arrangement allows for the second resonator population (green line) to become dominant after only a short timescale t ∼ 1/γ, which is maintained through to the steady state and thus evermore. Asymmetric coupling In general, the coupling encompassed by the model of Eq. (2) is asymmetric -with the preceding unidirectional case being the most extreme example. Most generally then (when g = 0 and Γ = 0), the resonator steady states become dependent on the relative phase θ − φ as follows [16] lim t→∞ b † 1 b 1 = (2γΩ) 2 16g 4 + 8g 2 γ 2 + (γ 2 − Γ 2 ) 2 + 8g 2 Γ 2 cos (2 [θ − φ]) ,(22)lim t→∞ b † 2 b 2 = 4Ω 2 4g 2 + Γ 2 + 4gΓ sin [θ − φ] 2 16g 4 + 8g 2 γ 2 + (γ 2 − Γ 2 ) 2 + 8g 2 Γ 2 cos (2 [θ − φ]) ,(23) from which the analogous results in other, more specialized coupling regimes may be derived. The population imbalance measure [cf. Eq. (8)] is given by the rich expression ∆ = 2γ 2 16g 4 + 8g 2 γ 2 + (γ 2 − Γ 2 ) 2 + 8g 2 Γ 2 cos (2 [θ − φ]) − 1,(24) and is plotted in Fig. 3, as a function of the relative phase θ−φ and the dimensionless dissipative coupling strength Γ/γ, for the example case where the magnitude of the coherent coupling g = γ/2. Clearly, the map of Fig. 3 exposes the importance of the phase as the determiner of the asymmetry of the coupling, since modulating θ − φ allows for either positive (yellow to red) or negative (cyan to blue) steady state population imbalances while holding the magnitudes of all other parameters constant. In particular, the red area around θ − φ = π/2 and blue region around θ − φ = 3π/2 are plausible from the knowledge of the unidirectional phase conditions of Eq. (5). The dynamical populations are shown in Fig. 4 for the case of g = γ/2 and maximal dissipative coupling Γ = γ, so that the magnitude unidirectional coupling condition is met [cf. Eq. (5)]. In panel (a) the relative phase is zero (θ − φ = 0) and the large incoherent-to-coherent coupling ratio of Γ/g = 2 ensures that Rabi cycles are not discernible within the damped population cycles. In panel (b) the rightwards (→) unidirectional phase condition θ − φ = π/2 is fulfilled, such that the lack of backaction sees a purely exponential decay of the first resonator population (red line) and strong enhancement of the second resonator population (green line). Finally, in panel (c) the leftwards (←) unidirectional phase condition θ − φ = 3π/2 is satisfied, such that the undriven second resonator is never populated, in the most dramatic realization of the directionality of the coupled system. Conclusions We have studied a simple yet explanatorily powerful model of a pair of driven-dissipative resonators with both coherent and incoherent couplings. Our theory acts as a prototypical example of how dissipation-induced directionality may arise in quantum optical systems, with dramatic implications. In particular, we have shown how tailoring the relative magntiude and phase of the coherent and dissipative coupling can lead to highly directional and even one-way quantum transport. Our results provide perspectives for the quantum engineering of coupled resonators [28], with applications for directional devices such as isolators, circulators and quantum batteries [29][30][31]. Acknowledgments Funding: CAD is supported by the Royal Society via a University Research Fellowship (URF/R1/201158) and a Royal Society Research Grant (RGS/R1 /211220). TJS acknowledges funding from the Alexander von Humboldt Foundation. Discussions: we thank A. I. Fernández-Domínguez and E. del Valle for discussions. Data and materials availability: There is no data in this wholly theoretical work. All necessary information is available in the manuscript and the Supplementary Information [16]. b † 1 b 1 b † 2 b 2 θ − φ = 0 θ − φ = π/2 θ − φ = 3π/2 dynamics (a) (b) (c) FIG. 4. Phase-dependent population dynamics of coupled driven-dissipative resonators. The dynamic populations b † n bn of the pair, as a function of time t (in units of the inverse loss rate γ −1 ). The relative phase θ − φ between the coherent and dissipative couplings is increased from 0 to π/2 to 3π/2 upon descending the column of panels. In the figure, the first resonator is driven with an amplitude Ω = γ/10 and the magnitudes of the coherent and dissipative couplings are g = γ/2 and Γ = γ, so that the magnitude unidirectional coupling condition Γ = 2g is met [cf. Eq. (5)]. FIG. 2 . 2The populations of a pair of driven-dissipative resonators. First column: sketches of the coherent, dissipative and unidirectional coupling regimes. Second column: the scaled steady state populations limt→∞ b † n bn × (γ/Ω) 2 of the pair (red and green lines), as a function of the coherent coupling strength g or the dissipative coupling strength Γ. The population imbalance ∆ is shown as thick orange lines [cf. Eq. (8)]. Third column: the dynamic populations b † n bn of the pair, as a function of time t (in units of the inverse loss rate γ −1 ). In the figure, the first resonator is driven with an amplitude Ω = γ/10, in panel (c) g = 2γ and in panels (f) and (i) Γ = (4/5)γ. may be described by the HamiltonianĤ (with = 1) FIG. 3 . 3Phase-dependent population imbalance of coupled driven-dissipative resonators. The population imbalance ∆ of the pair in the steady state [cf. Eq.(24)], as a function of the relative phase θ − φ between the coherent and dissipative couplings, and the magnitude Γ of the dissipative coupling (in units of the loss rate γ). In the figure, the magnitude of coherent coupling g = γ/2. ORCID C. A. 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{'abstract': 'The notion of nonreciprocity, in essence when going forwards is different from going backwards, emerges in all branches of physics from cosmology to electromagnetism. Intriguingly, the breakdown of reciprocity is typically associated with extraordinary phenomena, which may be readily capitalized on in the design of (for example) nontrivial electromagnetic devices when Lorentz reciprocity is broken. However, in order to enable the exploitation of nonreciprocal-like effects in the next generation of quantum technologies, basic quantum optical theories are required. Here we present a versatile model describing a pair of driven-dissipative quantum resonators, where the relative phase difference between the coherent and incoherent couplings induces an asymmetry. The interplay between the diverse dissipative landscape -which encompasses both intrinsic losses and dissipative couplings -and the coherent interactions leads to some remarkable consequences including highly directional (or even one-way) energy transport. Our work proffers the tantalizing prospect of observing dissipation-induced quantum directionality in areas like photonics or cavity magnonics (spin waves), which may aid the design of unconventional nanoscopic devices.', 'arxivid': '2212.12777', 'author': ['C A Downing \nDepartment of Physics and Astronomy\nUniversity of Exeter\nEX4 4QLExeterUnited Kingdom\n', 'T J Sturges \nInstitute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology (KIT)\nD-76131KarlsruheGermany\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nUniversity of Exeter\nEX4 4QLExeterUnited Kingdom', 'Institute of Theoretical Solid State Physics\nKarlsruhe Institute of Technology (KIT)\nD-76131KarlsruheGermany'], 'corpusid': 253669223, 'doi': '10.1209/0295-5075/ac9ad6', 'github_urls': [], 'n_tokens_mistral': 10704, 'n_tokens_neox': 8825, 'n_words': 5020, 'pdfsha': '826eb6a5ca6df4950269f0eb49a3699e286a33e7', 'pdfurls': ['https://export.arxiv.org/pdf/2212.12777v1.pdf'], 'title': ['Directionality between driven-dissipative resonators', 'Directionality between driven-dissipative resonators'], 'venue': []}

arxiv

A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data Mofeng Yang (STAR) Center School of Medicine University of Maryland 685 W Baltimore Street, Suite 60021201BaltimoreMDUSA Yixuan Pan (STAR) Center School of Medicine University of Maryland 685 W Baltimore Street, Suite 60021201BaltimoreMDUSA Aref Darzi (STAR) Center School of Medicine University of Maryland 685 W Baltimore Street, Suite 60021201BaltimoreMDUSA Sepehr Ghader (STAR) Center School of Medicine University of Maryland 685 W Baltimore Street, Suite 60021201BaltimoreMDUSA Chenfeng Xiong (STAR) Center School of Medicine University of Maryland 685 W Baltimore Street, Suite 60021201BaltimoreMDUSA Lei Zhang (STAR) Center School of Medicine University of Maryland 685 W Baltimore Street, Suite 60021201BaltimoreMDUSA A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data 1 1 1. Graduate Research Assistant (mofeng@umd.edu), ORCID: 0000-0002-0525-7978 2. Graduate Research Assistant (ypan1003@umd.edu), ORCID: 0000-0002-5015-0088 3. Graduate Research Assistant (adarzi@umd.edu), ORCID: 0000-0003-2558-5570 4. Research Scientist (sghader@umd.edu), ORCID: 0000-0003-1938-7914 5. Assistant Research Professor (cxiong@umd.edu), ORCID: 0000-0003-4237-1750 6. Professor (lei@umd.edu), * Corresponding Author, ORCID: 0000-0002-3372-6321 1-6: Maryland Transportation Institute (MTI), Department of Civil and Environmental Engineering, 1173 Glenn Martin Hall, University of Maryland, College Park MD 20742, USA. 5: Shock Trauma and Anesthesiology Research 2 2Travel mode sharetravel surveysmachine learningmobile device location data Mobile device location data (MDLD) contains abundant travel behavior information to support travel demand analysis. Compared to traditional travel surveys, MDLD has larger spatiotemporal coverage of the population and its mobility. However, ground truth information such as trip origins and destinations, travel modes, and trip purposes are not included by default. Such important attributes must be imputed to maximize the usefulness of the data. This paper targets at studying the capability of MDLD on estimating travel mode share at aggregated levels. A data-driven framework is proposed to extract travel behavior information from MDLD. The proposed framework first identifies trip ends with a modified Spatiotemporal Density-based Spatial Clustering of Applications with Noise (ST-DBSCAN) algorithm. Then three types of features are extracted for each trip to impute travel modes using machine learning models. A labeled MDLD dataset with ground truth information is used to train the proposed models, resulting in 95% accuracy in identifying trip ends and 93% accuracy in imputing the five travel modes (drive, rail, bus, bike and walk) with a Random Forest (RF) classifier. The proposed framework is then applied to two large-scale MDLD datasets, covering the Baltimore-Washington metropolitan area and the United States, respectively. The estimated trip distance, trip time, trip rate distribution, and travel mode share are compared against travel surveys at different geographies. The results suggest that the proposed framework can be readily applied in different states and metropolitan regions with low cost in order to study multimodal travel demand, understand mobility trends, and support decision making. Introduction Accurate measurement of travel behavior can help agencies understand how travel demand evolves and better allocate resources in support of transportation planning processes. Traditionally, researchers and practitioners design and conduct travel surveys to obtain household-and individual-level travel behavior information, including trip origins and destinations, trip distance, trip time, trip purposes, travel modes, etc. Two of the most widely used travel surveys conducted in the United States (U.S.) are the National Household Travel Survey (NHTS, see U.S. Department of Transportation 2017) and American Travel Survey (Lapham 1995). Methods to conduct travel surveys usually require respondents to record their daily trips with original paper-and-pencil interview (PAPI), computer-assisted telephone interview (CATI), and computer-assisted-selfinterview (CASI) (Wolf et al 2001;Wolf 2006). However, these methods are prone to several wellknown biases, such as under-reported short trips, inaccurate travel times, and travel distances (Stopher et al. 2007;McGowen and McNally 2007). Also, traditional travel surveys require complex planning and design, and large human labor and costs, and can only obtain relatively small survey samples for a limited number of cross-sections. For instance, if half of the 350 Metropolitan Planning Organizations (MPO) in the U.S. conduct travel surveys only once in a decade, it will result in $ 7.4 million per year cost (Zhang and Viswanathan 2013). In the past two decades, along with the technological advancement in mobile sensors and mobile networks, mobile device location data (MDLD) has been growing dramatically in terms of data coverage and data size. In the realm of transportation, the abundant individual movement information stored in MDLD has great potential to help researchers and practitioners understand the bigger picture of human travel. Compared to traditional travel surveys, MDLD has larger spatial, temporal, and population coverage and it comes from various sources, including Global Positioning Service (GPS) devices, cellular network, Bluetooth, Wi-Fi, etc. To fully take advantage of MDLD, appropriate steps and methods need to be developed to extract useful travel behavior information from MDLD (Schönfelder et al. 2002;Axhausen et al. 2003). This study aims to develop a data-driven framework to estimate travel mode share based on MDLD. The proposed framework is trained using a labeled MDLD dataset collected from a mobile application and is further applied to two large-scale MDLD datasets. The estimated trip distance, trip time, trip rate distribution, and travel mode share are compared with travel surveys. Results suggest that the proposed framework can be readily applied in many regions with low cost to obtain travel mode share estimates and study travel trends, which can help decision-makers prioritize multimodal travel needs. The remainder of the paper is organized as follows. Section 2 reviews the literature on MDLD. Section 3 describes the proposed data-driven framework and models in detail. Section 4 introduces the data. Section 5 presents the model development results. Section 6 demonstrates the framework with two large-scale case studies. Section 7 concludes this paper and discusses the limitations and future research directions. Location-based Service Data The Location-based Service (LBS) data is generated when a mobile application updates the device's location with the most accurate sources, based on the existing location sensors such as Wi-Fi, Bluetooth, cellular tower, and GPS Wang and Chen 2019). The LBS data can reflect the exact location of mobile devices and thus provide invaluable location information describing individual-level mobility patterns. In most cases, the LBS data has a higher spatial precision and smaller Location Recording Interval (LRI) than the CDR data Gonzalez et al. 2008;Wang and Chen 2019;Wang et al. 2019). Lots of applications have been developed using the LBS data. For instance, a recent smartphoneenhanced travel survey conducted in the U.S. used a mobile application, rMove developed by Resource Systems Group (RSG), to collect high-frequency location data and let respondents recall their trips by showing the trajectories in rMove (RSG 2014;RSG 2015;RSG 2015;RSG 2017); Airsage leveraged LBS data to develop a traffic platform that can estimate traffic flow, speed, congestion and road user sociodemographic for every road and time of day (Airsage 2020); Maryland Transportation Institute (MTI) at the University of Maryland (UMD) developed the COVID-19 Impact Analysis Platform (data.covid.umd.edu) to provide insight on COVID-19's impact on mobility, health, economy and society across the U.S. In summary, these three types of MDLD are different in terms of spatiotemporal coverage of population and its mobility, and data quality, e.g., spatial accuracy and LRI. The GPS data usually has the highest spatial accuracy (e.g., 10 meters) and the lowest LRI (usually 1 second), but it usually covers a small percentage of the population, and thus cannot reflect population-level travel behavior without a statistical weighting process. The cellular data and LBS data have significantly higher spatiotemporal coverage of population over the GPS data. However, the ground truth information is usually missing and the LRI for both types of data is based on mobile device usage on telecommunication or location-based services and thus has larger variation. In addition, the cellular data has much lower spatial accuracy (e.g., 1 kilometer) and the LBS produces location data with a larger range of spatial accuracy depending on the location sensor utilized to generate each record. Extracting Trips from Mobile Device Location Data: State-of-the-Art Methodologies The trip end identification algorithm for low-LRI MDLDs, i.e., GPS data, has been well-studied and used in practical applications. To obtain accurate trip ends, the traditional way is the rulebased trip end identification methods. This type of method designs rules and parameters based on domain knowledge. The trip ends are obtained by applying the rules to location data point by point and at the same time examining the intra-relationship between several consecutive location points. The parameters used in these rules are mostly defined by domain knowledge, such as dwell time, speed, etc. (McGowen and McNally 2007;Gong et al. 2014;Axhausen et al. 2003;Tsui et al. 2006;Bothe and Maat 2009;Stopher et al. 2005;Du and Aultman-Hall 2007;Stopher et al. 2008;Schuessler and Axhausen 2009;Gong et al. 2012;Assemi et al. 2016;Patterson et al. 2016). In recent years, some researchers also leveraged the supervised machine learning models as a supplement to the rule-based methods, which classify each location point as static or moving (Gong et al. 2015;Zhou et al. 2016;Gong et al. 2018). Different clustering methods were also applied to obtain trip ends by first identifying people's activity locations from the location data (Zhou et al. 2007;Chen et al. 2014;Ye et al. 2009;Yao et al. 2019). A recent study utilized a spatiotemporal clustering method with three combined optimization models to detect trip ends (Yao et al. 2019). In recent years, there is also a special focus on deriving the trip ends from LBS data. A "Divide, Conquer and Integrate" (DCI) framework was proposed to process the LBS data to extract mobility patterns in the Puget Sound region (Wang et al. 2019). The proposed framework combined a rule-based method and incremental clustering method to handle the bi-modally distributed LBS data. The results were aggregated at census tract-level and compared with household travel surveys (Wang et al. 2019). After the trip ends are identified, it is also important to impute the travel mode for each trip to obtain multimodal travel patterns. Travel mode imputation can be categorized into mainly two approaches: (1) trip-based approach; and (2) segment-based approach. The trip-based approach is based on the already identified trip ends, where each trip has only one travel mode to be imputed. The segment-based approach separates the trip into fixed-length segments (time or distance) and then imputes the travel mode for each segment. Then the segment with the same travel mode will be further merged to form a single-mode trip. This study mainly considers the trip-based approach because the purpose of this study is to estimate travel mode share aggregated from individual trips. Table 1 summarizes typical methods for travel mode imputation using the trip-based approach. It can be observed that some typical features used are speed and acceleration (Gong et al. 2012;Stenneth et al. 2011;Brunauer et al. 2013;Xiao et al. 2015;Broach et al. 2019;Shafique and Hato 2016;Wang et al. 2017). Specifically, when the LRI is less than 10 seconds, the speed and 6 acceleration features are more important to differentiate between different travel modes, which can be imputed solely by the data itself. When the LRI is relatively high, for instance, 30 s, additional features can be added to maintain the same level of accuracy such as real-time transit information (Stenneth et al. 2011), multimodal transportation network (Gong et al. 2012;Stenneth et al. 2011), sociodemographic information (Wang et al. 2017) etc. Both the state-of-the-practice applications and the state-of-the-art methodologies are able to accurately identify trip ends and impute travel modes based on low-LRI GPS data with userlabeled ground truth information. However, most previous studies both trained and tested the algorithms on a small GPS data sample without evaluating their performances in large-scale and real-world MDLD datasets. In addition, limited efforts focused on developing suitable travel mode imputation algorithms for LBS data and validating the results. This study aims to fill this gap by proposing a data-driven travel mode share estimation framework based on LBS data and further implement it on large-scale datasets. The proposed framework investigates the spatial accuracy and LRI distributions of the LBS data sample and fine-tuned the algorithms considering their variations. The validation efforts based on the external data sources, such as household travel surveys, have proven the proposed framework efficient and versatile for applications. Figure 1 shows the proposed data-driven framework. On the left is the Model Development pillar, wherein a dedicated ground-truth data collection of labeled and mode-specific trips and trajectories is conducted in order to train the trip end identification algorithm and travel mode imputation model. These trained models are then applied to the Model Application pillar on the right. The Model Application generates trip rosters with imputed travel modes for the unlabeled MDLD datasets in the application contexts. Finally, a validation process compares the aggregated mode share, as well as other statistics, with travel surveys before the data products are deemed useful and applicable for any transportation planning applications. The Data-Driven Travel Mode Share Estimation Framework Trip End Identification Considering a person's daily travel, it is very common that he or she makes multiple stops for different trip purposes. As illustrated in Figure 2, these stops are categorized into two categories, namely Activity Stops (AS) and Non-Activity Stops (NAS). ASs represent stops where actual activities take place, such as home, workplace, restaurant, shopping mall, etc. NASs represent stops where no activity takes place or the activity takes a very short amount of time, usually including stopping at a traffic light, picking up people within a short range of time, etc. In this study, only ASs are considered as actual trip ends and the trajectory between two consecutive ASs is considered as a valid trip. The first step is to identify all stop points including all ASs and NASs. The Spatiotemporal Density-based Spatial Clustering of Applications with Noise (ST-DBSCAN) (Birant and Kut 2007) is applied to fulfill this step. The ST-DBSCAN is an extended version of the traditional DBSCAN algorithm (Ester et al. 1996) with consideration of both spatial and temporal constraints. The temporal constraint was able to handle the scenarios when a person visits the same location multiple times per day, i.e., home, work. Three thresholds are defined for the ST-DBSCAN used in this study: (1) the spatial threshold s: it represents the distance falling within the activity distance range; (2) the temporal threshold t: it represents the minimum duration of an activity; and (3): the minimum neighbor's m: it represents the minimum number of location points to form a cluster. Details of ST-DBSCAN can be found in Birant and Kut (2007) and Ester et al. (1996). With all stop points identified, the second step is to distinguish between ASs and NASs. Two parameters are proposed: (1) sact: maximum activity distance range. If the distance between two consecutive clusters stayed within sact, it implies that these two clusters might still belong to the same activity, and the location points falling within these two clusters would be labeled as activities, 8 otherwise a trip will be generated. (2) tact: minimum activity duration threshold of an activity. If the minimum time lag between the last stop point and the first stop point of two consecutive clusters is shorter than tact, it implies no activity happens, which can happen at traffic lights, traffic congestions, pick up, etc, otherwise an activity would be identified between the two clusters. Travel Mode Imputation with Machine Learning Models This study proposes machine learning models to impute five travel modes (drive, bus, rail, bike and walk) and four travel modes (drive, bus, rail, and non-motorized) from trips identified from the previous step. Five machine learning models will be examined in terms of prediction accuracy, including K-Nearest Neighbors (KNN), Support Vector Classifier (SVC), eXtreme Gradient Boosting (XGB), Random Forest (RF), and Deep Neural Network (DNN). The five machine learning modes are implemented using the scikit-learn package in Python (Pedregosa et al. 2011). A detailed introduction of these models can be found in Appendix I. Gong et al. 2018). Here, a 50-meter buffer for the multimodal transportation networks (rail, bus, and drive), and bus stops are generated to obtain the percentage of location points for each trip that fall within each buffer respectively. It should be noted that since the accuracy of these multimodal network features is largely affected by the spatial accuracy of the location data, we used only location data with spatial accuracy less than 50 meters to calculate these features to obtain more accurate estimates. Accelerator meter data is also useful in travel mode imputation. However, it was not taken into account because such information is generally not available in large-scale MDLD datasets purchased from the third-party vendors. Data Mobile Device Location Data This study uses three MDLD datasets that can be categorized into LBS data. The first LBS dataset is collected from a mobile application called incenTrip (incentrip.org). incenTrip was developed by Maryland Transportation Institute (MTI) at the University of Maryland (UMD) to nudge travel behavior changes by providing real-time dynamic incentives in the Washington Metropolitan Area (Xiong et al. 2019). incenTrip collects location data using the Google Maps API (Android) or Apple Core Location (iOS) with a pre-defined fixed LRI and inserts the data into PostgreSQL in the Amazon Web Services (AWS) to ensure privacy protection. The LRI of the incenTrip data is set to be 1-15 seconds with three considerations: 1) help preserve battery of the mobile devices, and at the same time fully capture users daily travel behavior; 2) ensure the capability of data for travel mode imputation, as the literature suggests, reducing the LRI of the data will decrease the travel mode imputation accuracy (Shafique and Hato 2016); and 3) improve similarity with largescale LBS datasets. Each record of the raw location data includes a unique device identifier, latitude, longitude, and timestamp information. The proposed framework would then be applied to identify trips, impute the corresponding travel modes, and then update the trip records with imputed modes in the database. This study uses the incenTrip app from March 2019 to January 2020 for a dedicated ground-truth MDLD data collection. During the 10-month period, fifteen designated respondents of all the incenTrip users were hired to travel with incenTrip and record detailed information for each trip daily, including the start date, start time, end date, end time, origin street address, destination street address, travel time and travel mode. The trip identification algorithm is calibrated by comparing their travel diaries and the algorithm outputs. In addition, we also collected the trips with confirmed travel modes by other users. As a result of this data collection effort, a total number of 12,688 ground-truth trip records with travel mode labels were obtained from 410 users for the subsequent travel mode imputation model training process. Figure 3 visualizes the trajectories of these trips in different colors by travel modes. Two other LBS datasets are obtained from one of the leading data vendors in the U.S. The data is generated by multiple mobile applications. In this study, the two LBS datasets are extracted with different spatial, temporal, and population coverages to keep a comparable data size. Multimodal Transportation Networks This study also collects the multimodal transportation network data including drive, bus, rail networks, and bus stop locations to construct network-related features. The drive network is collected from the Highway Performance Monitoring System (HPMS) (FHWA 2020) that includes national freeway and arterial roads in the U.S. The national bus and rail network and the bus stops data are collected from the United States Department of Transportation (U.S. DOT) Bureau of Transportation Statistics (BTS) National Transit Map (NTM) (U.S. DOT BTS 2020). Figure 4 illustrates the multimodal transportation networks used in this study. (MWCOG 2010). This survey covered nearly 14,000 households and can provide travel mode shares at Traffic Analysis Zone (TAZ) level. In our case study, we aggregated the travel mode shares at the county level using the trip origins in the travel survey and further compare with the LBS estimates. Model Development Trip End Identification Result Five parameters of the proposed ST-DBSCAN are calibrated using the incenTrip data: the spatial threshold s, temporal threshold t, minimum neighbors n, maximum distance threshold for an activity sact, and minimum duration of an activity tact. Since s determines the distance range of a stop, increasing the value of s would identify more stop points since more location points would be clustered. To ensure all the stop points are captured including traffic congestions and waiting at a traffic light for both vehicle and pedestrian, four constraints are added as shown below: t ≥ n • f t act ≥ n • f s act ≥ s n • f ≥ s/v where v is the average walking speed, here we consider 1 m/s; f is location recording interval. Consider the real-world scenario when a person stops, it is intuitive to set the s value to be relatively small. Here we use 25-meter, 50-meter, and 100-meter as the candidate s value. Also, the tact was set as 300 seconds to obtain most of the short activities. Then, with the given LRI the corresponding range for other parameters could be calculated. Table 4 shows the calibrated parameters used in the case study for each LRI. Figure 5 shows the trip end identification result. The "Reported Trips" represents the trips that are reported in the respondents' travel diaries; the "Matched Trips" represents how many of the "Reported Trips" are identified from the data; the "All Identified Trips" represents all trips identified from the data, including the matched trips and the underreported trips that are not recorded in respondents' travel diary; and the "Hit-Ratio" is the value of "Matched Trips" divided by "Reported Trips". It should be noted that for the 1-s LRI data, only 23 reported trips are collected due to the short testing period. Over 90% of reported trips can be identified for each LRI, with the overall Hit-Ratio equals to 94.5%. In addition, about 15% to 35% of the underreported trips are identified and confirmed by testers. Capturing these underreported trips help produce more detailed travel patterns of each respondent. where TP represents the true positive, FP represents the false positive, and FN represents the false negative. Table 5 compares the model performances using F1 scores using the testing data. It can be seen that RF achieved the highest CV-accuracy in both four and five modes models. The bus mode has the least prediction accuracy over the other modes. One possible reason might be the similarity between the drive trips and bus trips. As shown in Figure 3 (a) and (b), the drive trips cover most interstate highways, major arterials while the bus trips will only follow the pre-designed bus routes and will stop at the bus stops. However, when it comes to the roadway segments where bus routes exist, the moving patterns of passenger cars and buses highly depend on the traffic conditions and thus could share similar spatiotemporal characteristics, which requires high-quality data and complex method to distinguish with each other (Nguyen and Armoogum 2020). Therefore, in such roadways, especially in urban settings, the general level of LBS data quality might not be sufficient to capture the differences in their moving patterns, e.g., distinguishing between the short and frequent stops due to traffic congestion and bus stops. The RF model's feature importance can also be found in Appendix I. Case Study Comparison between the incenTrip Data and the Two LBS Datasets Before we apply the calibrated framework to the two large-scale LBS datasets, the spatial accuracy and the LRI of these three datasets need to be compared. Figure 6 (a) shows the spatial accuracy of the three datasets after removing the data with an accuracy greater than 100 meters (see Appendix II. for more details). The incenTrip dataset has the best spatial accuracy among the three datasets, with more than 80% of the data's spatial accuracy less than 50 meters, whereas the two LBS datasets show a bimodal distribution, with the second peak locates around 70 meters. In general, the spatial accuracy of these three datasets is of high quality because, for all three datasets, around 90% of the data has a spatial accuracy of less than 100 meters. Figure 6 (b) only shows the LRI distribution for the two large-scale LBS datasets, since the incenTrip dataset is collected with pre-defined LRI, where a bimodal distribution can be observed (Wang et al. 2019). For each dataset, more than 75% of the data has LRI less than 15-s. Therefore, with consideration of both spatial accuracy and LRI, we applied the framework calibrated by the incenTrip data with 15-s LRI to estimate trips and travel mode shares for the two LBS datasets. In addition, since we observed that the second peak of the bimodal distribution locates around LRI = 120 s, two further relaxations are made in order to relax the restrictions of activity cluster identification: (1) the temporal threshold t is relaxed from 600 s to 1800 s; (2) the minimum neighbors n is relaxed from 10 to 5. Increasing the temporal threshold t and decreasing the minimum neighbors n can help capture a comparable number of activity clusters for LBS data with small LRI. Therefore, as shown in the previous calibration results (Table 4), data with high LRI requires larger t and smaller n. the final parameters used for the two case studies are s = 50 m, t = 1800 s, n = 5, sact = 100 m and tact = 300 s. More analysis regarding the spatial accuracy and LRI of the LBS datasets used in this study can be found in Appendix II. (a) (b) Figure 6. (a) Spatial accuracy distribution for the three LBS datasets; (b) Location recording interval distribution for the two case studies' LBS Datasets. Case Study I: Baltimore-Washington Metropolitan Area Dataset In the first case study, the proposed framework is applied to the LBS data observed in the Baltimore-Washington metropolitan area, covering the state of Maryland, D.C., and Northern Virginia. The trip distance, trip time and trip rate distribution are firstly compared with the 2007/08 TPB-BMC HHTS. The travel mode share is then compared at statewide-and county-level. A visualization of bus and rail travel distribution is provided at the census tract-level. Trips are categorized into short-distance and long-distance trips using the 50 miles threshold (Hu and Reuscher 2004). Figure 7 (a) and (b) compare the short-and long-distance trip distance distribution with the 2007/08 TPB-BMC HHTS. For both short-distance and long-distance trips, a similar distribution can be observed, while more long-distance trips can be observed in the LBS data than the survey. Figure 7 (c) shows the trip time distribution comparison. The overall trend is similar, and still more long-duration trips can be observed in the LBS data. Figure 7 (d) compares the trip rate distribution, where the LBS data observed more devices with one trip per day. The number of trips by the time of day is also compared using average weekday trips from the survey, as shown in Figure 7 (e). It can be seen that both morning and afternoon peaks are matched in the two data sources, while LBS data can capture more daytime trips. The overall mode share distribution is consistent with the 2007/08 TPB-BMC HHTS mode share, while the LBS data estimates lower bus mode share and higher non-motorized mode share. One reason might be people tend to underreport short-distance trips and most of the short-distance trips are walking. The travel mode share is further compared at the county-level, as shown in Figure 8 (b). The Pearson correlation between travel mode share estimated from the LBS data and the survey is calculated. Strong correlation can be observed, with the Pearson correlation value 0.98. Figure 8 (c) shows that for all eight counties in Maryland and D.C. It can be seen that the bus travel is underestimated, and the non-motorized mode share is overestimated from the LBS data. However, it should be noted that the 2007/08 TPB-BMC HHTS is conducted over ten years ago and the travel patterns might have been significantly changed. Therefore, we further compare the LBS data estimates with the recent NHTS 2017. However, due to the small sample size of NHTS 2017 in some rural counties (Kent County, Charles County, etc.), the NHTS 2017 estimates are prone to be over/underestimated. A detailed comparison can be found in Appendix II. Figure 9 plots the census tract-level rail and bus travel mode share using Jenks natural breaks optimization (Jenks 1967), with the deeper color representing higher mode share. For both D.C. and Baltimore city, the travel mode share distribution follows the geographical layout of rail and bus networks. Also, since D.C. has denser rail and bus networks, the relative mode share is higher than Baltimore City. In the second case study, the proposed framework is applied to the LBS data observed in the entire U.S. for a week, with 3% of the observed devices randomly sampled. The trip distance, trip time, trip rate distribution an,d travel mode share are compared against the NHTS 2017 at the nationwide-and state-level. A visualization is provided at CBSA-level. Figure 10 (a) and (b) show the trip distance distribution comparison between NHTS results and LBS data results for short-distance and long-distance trips respectively. For both short-distance and long-distance trips, the overall trip distance distribution is consistent with the NHTS, while more long-distance trips can be observed. Figure 10 (c) shows the trip time distribution comparison. The overall trend is similar. The short trips are underestimated, and the long trips are overestimated. One possible reason for this observation is that NHTS 2017 calculated the network shortest path for trip distance, which is not always representative of the real path. Figure 10 (d) compares the daily trip rate to NHTS 2017, where the LBS data observed more devices with one trip per day. The daily variation within a week is also shown in Figure 11. The difference between weekends and weekdays is captured. Figure 10. Comparison between NHTS and LBS data on: (a) Short-distance trips; (b) Longdistance trips; (c) Trip time; (d) Trip rate. Figure 11. Daily Variation. (a) (b) (c) (d) The air trips are firstly filtered out with a heuristic rule using three trip features: average speed, trip time, and trip distance. Here the 100 mph, 1 hour, and 100 miles are selected as the value of these thresholds, indicating that for a trip, if the average speed is larger than 100 mph, the trip time is larger than 1 hour and the trip distance is larger than 100 miles, then this trip is labeled as an air trip. Figure 12 shows the heat map of air trips' origins, where the depth of the color represents the number of air trips originated from the closest airport. It can be observed that almost all the major airports are captured. Figure 14. CBSA-level illustration of (a) rail travel mode share; (b) bus travel mode share. Conclusions and Discussions This study reviews the state-of-the-practice applications and state-of-the-art methods for extracting travel behavior information from MDLDs. Based on the literature review, the key research gap is identified, and a data-driven framework is proposed to estimate travel mode shares from MDLDs. The proposed framework reaches 95% accuracy in identifying trip ends and 93% accuracy in identifying five travel modes with the RF model. The developed framework is applied to two largescale LBS datasets, covering the Baltimore-Washington metropolitan area and the U.S. with different spatial, temporal, and population coverage. The trip distance, trip time, and trip rate distribution, and travel mode share estimated from the two LBS datasets are compared with travel surveys for a comprehensive validation. The comparison results suggest that the proposed framework performs robustly in both geographical regions, indicating a good transferability. One major challenge in the travel mode imputation is to distinguish different travel modes in urban settings, especially between drive and bus. This is also the reason why we solicited our training samples in the Washington Metropolitan Area. In the meantime, the multimodal networks are simpler in most rural areas: the rail network is sparse, the bus service is limited, etc. From our comparison with travel surveys at the county/state level, it shows that the current model trained with our incenTrip dataset could achieve satisfactory estimation in other contexts or regions. The limitations of this study are summarized and discussed into three aspects: training data, validation data, and sample bias. • Training Data: The proposed framework is calibrated and trained based on samples collected in the Washington Metropolitan Area with the incenTrip application. Even though the general travel mode share statistics are consistent with the travel surveys, in the real world, the travel behavior might be different from region to region, resulting in biased travel mode share estimation. In future research, enriching the training dataset that could cover different regions, such as the GPS-enhanced travel survey dataset available in Transportation Secure Data Center (TSDC) at National Renewable Energy Lab (NREL), might help improve the performance of our proposed framework by considering the travel behavior heterogeneity. In addition, the proposed framework relies on multimodal transportation networks, including drive, rail and bus. For regions without well-maintained transportation networks, it could be hard to capture the rail/bus travel. To decrease the dependency on multimodal transportation networks, additional information such as acceleration and stop time can be potentially considered. • Validation Data: The proposed framework provides a general way to estimate travel mode shares at different geographies, with drive mode well captured, bus and rail mode slightly underestimated and non-motorized mode slightly overestimated. To further improve the performance of the proposed framework and capture finer level multimodal travel trends, additional data (i.e., transit ridership data, station-based metro passenger volume data) could be collected as external validation source and control totals. In addition, the current heuristic rule filters air trips according to domain knowledge, such as thresholds of average travel speed, trip time and trip distance, which could be refined with a ground-truth data collection for long-distance travels. A similar data collection effort to the procedure done by this paper could address the limitation and is on our research agenda. • Sample Bias: In the two case studies, the travel mode share is estimated from the LBS data with a sample of the population, which might not be able to represent the population-level travel behavior. Also, the LBS data could not capture the travel behaviors of the population without mobile devices, which might yield an underrepresentation of the younger, elder, and low-income population. To address these two problems, an additional weighting and validation process can be done on top of the sample results using land use and sociodemographic information. Our proposed framework is able to provide a timely and continuous measurement of travel trends and travel mode shares as a supplement to travel surveys. To achieve better estimates in practice, our proposed framework should be firstly calibrated using the travel surveys and then be applied to the MDLD datasets for preliminary investigation of the latest travel trends and mode shares. Although the proposed framework is not ready to completely replace travel surveys, it could help the government and local agencies refine the design of their travel surveys after prioritizing their data needs and before investing time and money in actual implementation. In addition, the proposed framework can also be applied to other realms, such as business development and public health. For instance, during the COVID-19 pandemic, a handful of research utilized the MDLDs to derive the travel statistics, i.e., trip rate and inflow, and study their correlation with the new COVID-19 infections Xiao et al. 2020;Pan et al. 2020). By applying our proposed framework to the MDLDs, the correlation between multimodal travel and the spread of COVID-19 can be studied to support governments decision makings on transit or airline operations. Declarations Funding The research is partially financially supported by a USDOT Federal Highway Administration (FHWA) Exploratory Advanced Research (EAR) project entitled "Data Analytics and Modeling Methods for Tracking and Predicting Origin-Destination Travel Trends based on Mobile Device Data" (Award No. 693JJ31750013). The opinions in this paper do not necessarily reflect the official views of USDOT or FHWA. Conflict of interest The authors declare that they have no conflict of interest. Availability of data and material Not applicable. Code availability Not applicable. KNN is one of the earliest and simplest classification models (Peterson 2009). The main idea of KNN is to find the top k nearest samples of the target sample based on distance measurement. The distance between two samples is usually calculated based on Euclidean distance. SVC was developed by Cortes and Vapnik in the 1990s and applied for face recognition, pattern recognition, etc (Cortes and Vapnik 1995;Osuna et al. 1997;Suykens and Vandewalle 1999;Wang 2005). SVC can address the non-linearly separable samples by using the kernel function to map the data into a higher dimension, thus finding a hyperplane that best divides the data into different classes. Some examples of the kernel functions include polynomial, Gaussian, Gaussian radial basis function (RBF), sigmoid, etc. XGB is one of the most recent ensemble-learning algorithms using the boosting technique (Chen et al. 2015;Chen and Carlos 2016). The main idea of boosting is to train a set of weak classifiers using the same samples and then combine them into one strong classifier to improve the classification accuracy, where new classifiers are added to reduce errors based on previous models until no further improvements can be made (Kearns and Valiant 1988;Michael and Valiant 1994). RF is one of the most famous ensemble-learning algorithms using the bagging technique (Liaw and Wiener 2002;Breiman 1996). Bagging (Bootstrap aggregating) is a machine learning technique that tends to improve the stability and accuracy by generating multiple training sets by sampling from the data uniformly and with replacement (Breiman 1996). RF not only employs the bagging technique but also used a modified tree learning algorithm that selects a random subset of the features without using all features, which is called feature bagging (Liaw and Wiener 2002). In short, RF is essentially a collection of decision trees (Quinlan 1986) and each decision tree is trained with the different training sets and different features. The classification result follows the majority vote of all the decision trees in the forest. DNN is an Artificial Neural Network (ANN) with multiple layers between the input and output layers (Bengio 2009;Schmidhuber 2015). DNN can model the complex non-linear relationship between the input and the output by updating the weight vertices connecting each virtual neural between layers through back-propagation (Hecht-Nielsen 1992). Detailed methodology of DNN and ANN can be found in Bengio (2009) andSchmidhuber (2015). Figure 15. Feature Importance of the Random Forest Classifier. Figure 15 shows the feature importance value of the RF model for four and five travel modes respectively. The feature importance value (Gini importance) is automatically calculated using the python package scikit-learn (Pedregosa et al. 2011), representing each importance as the sum over the number of splits across all trees. It can be seen that the speed variables (95 quantile speed, maximum speed, and average speed) are the most important. In addition, the percentage of records that fell within 50 meters of the rail network is also significantly important since it is a representative feature to impute rail trips. Table 7 shows the detailed spatial accuracy distribution of the three datasets. It can be clearly seen that the incenTrip dataset has the highest data quality among the three datasets, with over 80% of the data has a spatial accuracy smaller than 50-meter. For the other two large-scale LBS datasets, the spatial accuracy shows a similar bimodal distribution, with the second peak locates around 70meter. In general, for all three datasets, around 90% of the data has spatial accuracy smaller than 100-meter. Therefore, before we apply our proposed framework to the dataset, we removed the location data with spatial accuracy larger than 100-meter for all three datasets to ensure the data quality. Table 8 shows the detailed LRI distributions for the two large-scale LBS datasets after removing the location data with spatial accuracy larger than 100-meter, which also follows the bimodal distribution, with the second peak locates around 120-s. In both datasets, more than 50% of the data has an LRI smaller than 30 seconds. Authors I.2. Fine-Tuned Parameters for Each Machine Learning Method I.3. Feature Importance of the Random Forest Model Figure 1 . 1The Data-Driven Travel Mode Share Estimation Framework Figure 2 . 2Typical Daily Travel Pattern of an Individual Figure 3 . 3incenTrip Trip Trajectories for (a) drive; (b) bus; (c) rail; (d) non-motorized. Figure 4 . 4Multimodal Transportation Networks: drive (grey), rail (green), bus (blue). 4.3. Travel Surveys Two travel surveys are used in this study for comparison purposes: NHTS 2017 and 2007/08 TPB-BMC Household Travel Survey (HHTS). NHTS 2017 is a national-level travel survey conducted by USDOT Federal Highway Administration (FHWA), collecting travel behavior data from U.S. residents. The NHTS 2017 includes a total number of 129,696 households covering all 50 states and the District of Columbia, including trip origin and destinations, trip time, trip purposes, and travel modes (U.S. DOT 2017). The 2007/2008 TPB-BMC HHTS is conducted by Transportation Planning Board (TPB) and Baltimore Metropolitan Council (BMC) in Baltimore and Washington regions from February 2007 to March 2008 using the same survey designs Figure 5 . 5Trip End Identification Results. Figure 7 . 7Comparison with 2007/08 TPB-BMC HHTS on: (a) Short-distance trips; (b) Longdistance trips; (c) Trip time; (d) Trip rate; (e) Number of trips. Figure 8 ( 8a) shows the statewide travel mode share comparison result. Figure 8 . 8(a) Statewide travel mode share comparison; (b) County-level travel mode share correlation; (c) County-level travel mode share comparison. Figure 9 . 9Census tract-level mode share illustration for (a) and (b): Rail mode share in D.C. and Baltimore City; (c) and (d): Bus mode share in D.C. and Baltimore City. 6.3. Case Study II: the U.S. National Dataset Figure 12 .Figure 13 . 1213Nationwide-level Air Trip OriginsFigure 13 (a) shows the nationwide travel mode share comparison result. The overall mode share distribution is consistent with the NHTS 2017, with the bus mode share underestimated.Figure 13(b) compares the travel mode shares at the state level across all 50 states and D.C. The Pearson correlation is also calculated between travel mode share estimated from the LBS data and NHTS 2017, where a strong correlation can be observed with the value of 0.99. To demonstrate the transferability of the proposed framework, eight states across the U.S. are selected for detailed comparison, as shown inFigure 13(c). It can be seen that the overall travel mode share estimates are reasonable, with a slight underestimation of bus and rail travel and overestimation of the nonmotorized travel. The results demonstrate the transferability and the generalization ability of the proposed framework that can be applied in larger geographies. Detailed comparison for all 50 states and D.C. can be found in Appendix II. (a) Nationwide travel mode share comparison; (b) State-level travel mode share correlation; (c) State-level travel mode share comparison. Figure 14 ( 14a) and (b) visualize the CBSA-Level rail and bus travel mode share. ' contributions: M.Y., Y.P., and L.Z. designed research; M.Y., A.D. and S.G. processed data; M.Y., Y.P., and C.X. developed the algorithms and models; M.Y., A.D. and S.G. conducted case studies; M.Y., Y.P., A.D., S.G., C.X., L.Z. wrote the paper. Appendix I. Travel Mode Imputation with Machine Learning Models I.1. Machine Learning Overview. Figure 17 . 17State-level travel mode share comparison between Dataset II and NHTS 2017. Table 1 . 1Literature Review on Travel Mode Imputation Methods.Author LRI Model Features Modes Acc. Gong et al. 2012 / Rules Speed, Acceleration, Transit Stations, Transit Network Drive, Train, Bus, Walk, Bike, Static 82.6% Stenneth et al. 2011 30 s RF Speed, Acceleration, Heading change, Bus location, Transit Network Drive, Bus, Train, Walk, Bike, Static 93.7% Bruunauer et al. 2013 1-10 s MLP Speed, Acceleration, Bendiness Drive, Bus, Train, Walk, Bike 92.0% Xiao et al. 2015ss 1 s BN Speed, Acceleration, Trip Distance Drive Bus, Walk, Bike, E-Bike 92.0% * RF: Random Forest; MLP: Multi-Layer Perceptron; BN: Bayesian Network. Table 2 . 2Features Constructed from Mobile Device Location Data of location points fell within 50 meters of bus network percentage % of location points fell within 50 meters of drive network percentage % of location points fell within 50 meters of bus stops percentageFeatures Table 3 3describes these two datasets. Dataset I covers the Baltimore-Washington metropolitan area, including the state of Maryland, District of Columbia (D.C.), and Northern Virginia. It has a temporal coverage of one typical weekday, Sept. 12 nd in 2017. A total of 474,634 unique devices observed in this area are considered. Dataset II expands the spatial coverage to the U.S. It has a temporal coverage of seven days from Aug. 1 st to Aug. 7 th in 2017. 3% of the total number of devices observed is randomly sampled and used for this study in order to keep a comparable data size, including 266,149 unique devices. Table 3 . 3Location-Based Service Data DescriptionDataset Spatial Coverage Temporal Coverage Sample Rate Sampled Numbers I Baltimore-Washington metropolitan area Sept. 12 nd , 2017 100% 474,634 II the United States Aug. 1 st -Aug. 7 th , 2017 3% 266,149 Table 4 . 4Calibrated Parameters for Each Location Recording Interval. is used to fine-tune the model. Detailed hyperparameters can be found in Appendix I. During the model training process, 10-fold cross-validation (CV) is conducted to evaluate the model performance. The well-trained models are then applied to the testing data the F1 scores are calculated using the equations as shown below:LRI (s) s (m) t (s) n sact (m) tact (s) 1 50 100 50 100 300 2 50 200 25 100 300 5 50 500 15 100 300 15 50 600 10 100 300 5.2. Travel Mode Imputation Result Five machine learning models, KNN, SVC, XGB, RF, and DNN are used to impute travel modes. A total of 6,064 drive trips, 1,824 rail trips, 1,403 bus trips, 1,496 bike trips and 1,901 walk trips are collected from the incenTrip. 70% of the data is used for training and 30% of the data is used for testing. The Synthetic Minority Over-Sampling Technique (SMOTE) is then applied to the training data to address the imbalanced sample problem, where the minority class from the existing samples is synthesized (Chawla et al. 2002). For each machine learning method, the randomized search approach Precision= TP TP + FP Recall= TP TP + FN F1=2· Precision·Recall Precision+Recall Table 5 . 5Model Performance Comparison.KNN SVC XGB RF DNN Four Modes Drive 0.81 0.82 0.89 0.89 0.85 Rail 0.87 0.91 0.91 0.91 0.89 Bus 0.50 0.52 0.64 0.65 0.57 NonMotor 0.76 0.85 0.89 0.89 0.86 10-Fold CV Accuracy 86.0% 86.3% 93.0% 93.3% 86.5% Five Modes Drive 0.80 0.82 0.89 0.89 0.84 Rail 0.87 0.90 0.91 0.91 0.90 Bus 0.48 0.53 0.64 0.65 0.56 Bike 0.50 0.69 0.77 0.79 0.73 Walk 0.69 0.82 0.87 0.87 0.85 10-Fold CV Accuracy 85.5% 85.4% 93.0% 93.6% 85.0% The CBSAs with high bus and rail travel mode share estimates are those who have well-developed bus and rail networks. For instance, D.C., New York, Boston, and San Francisco have higher rail mode share than the other CBSAs. For bus mode share, a similar trend is observed too, where CBSAs with denser bus networks have higher bus mode share.0 0.2 0.4 0.6 0.8 1 Bus Drive NonMotor Rail NHTS LBS Data 0 0.2 0.4 0.6 0.8 1 NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data CA TX FL GA IL MA NY OR WA Drive Rail Bus NonMotor Table 6 . 6Fine-Tuned Hyperparameters for Different Machine Learning Models.Four Travel Modes Five Travel Modes Model Parameters Model Parameters KNN k = 5; p = 1; weights = 'distance'; algorithm = 'brute'; leaf_size = 1; metric = 'minkoski'. KNN k = 1; p = 1; weights = 'distance'; algorithm = 'ball_tree'; leaf_size = 1; metric = 'minkoski'. SVC Cs = 100; Gammas = 1; class_weight = 'None'; kernel = 'rbf'. SVC Cs = 100; Gammas = 1; class_weight = 'None'; kernel = 'rbf'. RF n_estimators = 700; max_features = 'sqrt'; max_depth = 30; min_samples_split = 2; min_samples_leaf = 1; bootstrap = 'False'; class_weight = 'None'. RF n_estimators = 800; max_features = 'auto'; max_depth = 60; min_samples_split = 2; min_samples_leaf = 1; bootstrap = 'False'; class_weight = 'balanced_subsample'. XGB colsample_bytree = 0.980; gamma = 0.429; learning_rate = 0.317; max_depth = 5; n_estimators = 136; subsample = 0.627. XGB colsample_bytree = 0.865; gamma = 0.299; learning_rate = 0.324; max_depth = 5; n_estimators = 126; subsample = 0.884. DNN learning_rate = 0.001; epochs = 500; batch_size = 32; optimizer: rmsprop; loss; 'categorical_crossentropy'; layers: 64-32-16-8-DropOut (0.2). DNN learning_rate = 0.001; epochs = 500; batch_size = 32; optimizer: rmsprop; loss; 'categorical_crossentropy'; layers: 64-32-16-8-DropOut (0.2). Appendix II. Spatial Accuracy and Location Recording Interval of the Location-Based Service Data0 0.05 0.1 0.15 0.2 Minimum Speed Trip Time Avg. # of rec. per min. % Drive Network 5 Percentile Speed Trip Distance % Bus Stop % Bus O-D Dist. 25 Percentile Speed Med. Speed 75 Percentile Speed Avg. Speed % Rail Max. Speed 95 Percentile Speed Feature Importance Feature Four Modes Five Modes Table 7 . 7Spatial Accuracy Distribution of the Three DatasetsSpatial Accuracy incenTrip Dataset I Dataset II Proportion Cumulation Proportion Cumulation Proportion Cumulation (0, 10] 44.15% 44.15% 49.91% 49.91% 50.21% 50.21% (10, 20] 22.50% 66.64% 7.58% 57.49% 5.78% 55.99% (20, 30] 8.79% 75.43% 6.80% 64.28% 7.27% 63.27% (30, 40] 4.29% 79.73% 1.00% 65.28% 1.07% 64.33% (40, 50] 2.17% 81.90% 3.47% 68.76% 4.10% 68.43% (50, 60] 2.01% 83.92% 0.26% 69.01% 0.28% 68.71% (60, 70] 1.10% 85.01% 18.33% 87.34% 17.60% 86.31% (70, 80] 0.92% 85.93% 0.69% 88.04% 0.83% 87.14% (80, 90] 0.79% 86.73% 0.66% 88.70% 0.80% 87.94% (90, 100] 1.19% 87.92% 0.59% 89.29% 0.66% 88.60% (100, 200] 3.81% 91.73% 4.18% 93.46% 4.87% 93.47% (200, 500] 2.57% 94.31% 0.30% 93.77% 0.23% 93.70% (500, 5.69% 100.00% 6.23% 100.00% 6.30% 100.00% Table 8 . 8Location Recording Frequency of the Two Case Studies' Datasets Appendix III. Travel Mode Share Estimation Results Figure 16. County-level travel mode share comparison between Dataset I and NHTS 2017.LRI Dataset I Dataset II Proportion Cumulation Proportion Cumulation (0, 10] 44.81% 44.81% 46.48% 46.48% (10, 20] 4.98% 49.79% 4.92% 51.40% (20, 30] 1.83% 51.62% 1.77% 53.17% (30, 40] 1.21% 52.83% 1.15% 54.32% AL AK AZ AR CA CO CT DE DC FL GA HI ID KY LA ME MD MA MI MN MS MO MT NE NV NH NJ NM NY NC ND OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WYData NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data IL IN IO KS Drive Rail Bus NonMotor 0 0.2 0.4 0.6 0.8 1 NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data Drive Rail Bus NonMotor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data NHTS LBS Data Drive Rail Bus NonMotor Table 9 . 9County-level travel mode share comparison between Dataset I and 2007/08 TPB-BMC HHTS.County 2007-08 TPB-BMC HHTS LBS Data Drive Rail Bus NonMotor Drive Rail Bus NonMotor Anne Arundel 92.285% 0.728% 3.666% 3.321% 83.351% 0.694% 1.030% 14.925% Baltimore 90.773% 0.741% 4.106% 4.379% 81.234% 0.628% 1.324% 16.814% Baltimore City 71.962% 2.555% 9.163% 16.320% 62.658% 1.140% 4.383% 31.819% Carroll 92.411% 0.361% 4.705% 2.523% 86.340% 0.704% 0.050% 12.906% DC 47.160% 14.225% 7.533% 31.083% 47.129% 5.532% 6.568% 40.771% Harford 91.646% 0.115% 3.891% 4.348% 86.221% 0.876% 0.253% 12.650% Howard 90.461% 0.464% 5.294% 3.780% 85.274% 0.441% 1.096% 13.189% Montgomery 82.504% 2.719% 5.749% 9.028% 76.979% 2.917% 1.446% 18.659% Prince George's 85.244% 2.856% 6.474% 5.425% 75.069% 2.187% 2.255% 20.489% Table 10 . 10County-level travel mode share comparison between Dataset I with NHTS 2017.County 2007-08 TPB-BMC HHTS LBS Data Drive Rail Bus NonMotor Drive Rail Bus NonMotor Allegany 87.807% 0.000% 2.959% 9.234% 80.437% 1.405% 0.104% 18.054% Anne Arundel 87.724% 0.016% 2.235% 10.024% 83.351% 0.694% 1.030% 14.925% Baltimore 85.655% 0.403% 5.386% 8.556% 81.234% 0.628% 1.324% 16.814% Baltimore City 71.348% 0.095% 10.556% 18.001% 62.658% 1.140% 4.383% 31.819% Calvert 93.816% 0.000% 2.456% 3.728% 87.470% 0.747% 0.222% 11.562% Caroline 83.464% 0.000% 9.353% 7.183% 85.516% 0.781% 0.087% 13.617% Carroll 89.947% 0.000% 10.053% 0.000% 86.340% 0.704% 0.050% 12.906% Cecil 85.739% 0.000% 2.311% 11.950% 86.746% 0.636% 0.021% 12.598% Charles 76.195% 0.000% 22.494% 1.311% 86.656% 0.510% 0.382% 12.452% DC 94.103% 0.000% 1.249% 4.648% 47.129% 5.532% 6.568% 40.771% Dorchester 89.483% 0.218% 3.028% 7.271% 83.926% 0.464% 0.000% 15.611% Frederick 96.711% 0.000% 1.845% 1.444% 85.190% 0.750% 0.123% 13.938% Garrett 83.215% 0.000% 11.136% 5.649% 87.016% 0.749% 0.000% 12.235% Harford 88.492% 0.296% 3.917% 7.295% 86.221% 0.876% 0.254% 12.650% Howard 53.718% 0.000% 0.000% 46.282% 85.274% 0.441% 1.096% 13.189% Kent 82.855% 2.218% 2.666% 12.261% 80.349% 0.465% 0.000% 19.186% Montgomery 82.108% 3.363% 6.060% 8.469% 76.979% 2.917% 1.446% 18.659% Prince George's 86.707% 0.088% 9.284% 3.921% 75.069% 2.187% 2.255% 20.489% Queen Anne's 79.223% 0.000% 13.979% 6.798% 88.069% 0.558% 0.147% 11.225% Somerset 94.433% 0.000% 0.000% 5.567% 78.926% 0.403% 0.134% 20.537% St. Mary's 91.749% 0.070% 4.866% 3.315% 85.908% 0.718% 0.112% 13.262% Talbot 83.786% 0.098% 4.813% 11.303% 86.770% 0.495% 0.106% 12.628% Washington 88.606% 0.000% 4.395% 7.000% 84.030% 0.506% 0.000% 15.465% Wicomico 96.828% 0.000% 0.209% 2.962% 82.988% 0.348% 0.070% 16.594% Worcester 38.446% 10.277% 9.038% 42.239% 78.947% 0.301% 0.030% 20.722% Table 11 . 11State-level travel mode share comparison between Dataset II and NHTS 2017.SC 91.041% 0.026% 2.392% 6.542% 85.958% 0.307% 0.224% 13.511% SD 92.903% 0.102% 1.488% 5.507% 79.965% 0.371% 0.283% 19.380% TN 90.954% 0.077% 2.716% 6.254% 86.290% 0.322% 0.438% 12.950% TX 89.527% 0.205% 3.018% 7.250% 84.423% 0.371% 0.529% 14.678% UT 87.304% 0.529% 1.275% 10.892% 82.325% 0.528% 1.039% 16.108% VT 82.510% 0.000% 3.077% 14.413% 81.295% 0.344% 1.575% 16.786% VA 85.252% 1.169% 3.716% 9.863% 83.307% 0.887% 0.714% 15.093% WA 81.362% 0.580% 5.645% 12.413% 77.979% 0.511% 1.490% 20.020% WV 90.802% 0.000% 5.759% 3.439% 85.569% 0.473% 0.139% 13.819% WI 87.135% 0.013% 3.172% 9.679% 82.889% 0.551% 0.674% 15.886% WY 90.873% 0.000% 2.204% 6.923% 83.101% 0.263% 0.020% 16.616% -2001 California Statewide Household Travel Survey. 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J Wolf, R Guensler, W Bachman, Transportation Research Record: Journal of the Transportation Research Board. 17681Wolf, J., Guensler, R., & Bachman, W.. Elimination of the travel diary: Experiment to derive trip purpose from global positioning system travel data. Transportation Research Record: Journal of the Transportation Research Board. 1768(1), 125-134 (2001). Travel survey methods: Quality and future directions. J Wolf, Emerald Group Publishing LimitedApplications of new technologies in travel surveysWolf, J.. Applications of new technologies in travel surveys. Travel survey methods: Quality and future directions. pp. 531-544. Emerald Group Publishing Limited (2006). Travel mode detection based on GPS track data and Bayesian networks. G Xiao, Z Juan, C Zhang, Computers, Environment and Urban Systems. 54Xiao, G., Juan, Z., and Zhang, C.. Travel mode detection based on GPS track data and Bayesian networks. Computers, Environment and Urban Systems 54: 14-22, (2015). Modeling indoor-level non-pharmaceutical interventions during the COVID-19 pandemic: a pedestrian dynamics-based microscopic simulation approach. Y Xiao, M Yang, Z Zhu, H Yang, L Zhang, S Ghader, arXiv:2006.10666arXiv preprintXiao, Y., Yang, M., Zhu, Z., Yang, H., Zhang, L., & Ghader, S.. Modeling indoor-level non-pharmaceutical interventions during the COVID-19 pandemic: a pedestrian dynamics-based microscopic simulation approach. arXiv preprint arXiv:2006.10666 (2020). An integrated and personalized traveler information and incentive scheme for energy efficient mobility systems. C Xiong, M Shahabi, J Zhao, Y Yin, X Zhou, L Zhang, Transportation Research Part C: Emerging Technologies. Xiong, C., Shahabi, M., Zhao, J., Yin, Y., Zhou, X., and Zhang, L.. An integrated and personalized traveler information and incentive scheme for energy efficient mobility systems. Transportation Research Part C: Emerging Technologies (2019). Mobile device data reveal the dynamics in a positive relationship between human mobility and COVID-19 infections. C Xiong, S Hu, M Yang, W Luo, L Zhang, Proceedings of the National Academy of Sciences. 11744Xiong, C., Hu, S., Yang, M., Luo, W., and Zhang, L.. Mobile device data reveal the dynamics in a positive relationship between human mobility and COVID-19 infections. Proceedings of the National Academy of Sciences, 117(44), 27087-27089 (2020). Mobile device location data reveal human mobility response to state-level stay-at-home orders during the COVID-19 pandemic in the USA. Chenfeng Xiong, Songhua Hu, Mofeng Yang, Hannah Younes, Weiyu Luo, Sepehr Ghader, Lei Zhang, Journal of the Royal Society Interface. 1717320200344Xiong, Chenfeng, Songhua Hu, Mofeng Yang, Hannah Younes, Weiyu Luo, Sepehr Ghader, and Lei Zhang. Mobile device location data reveal human mobility response to state-level stay-at-home orders during the COVID-19 pandemic in the USA. Journal of the Royal Society Interface 17, no. 173: 20200344 (2020). Trip End Identification based on Spatial-Temporal Clustering Algorithm using Smartphone GPS Data (No. 19-01097). Z Yao, J Zhou, P J Jin, F Yang, Presented at 98th Annual Meeting of the Transportation Research Board. Washington, D.C.Yao, Z., Zhou, J., Jin, P. J., & Yang, F.. Trip End Identification based on Spatial-Temporal Clustering Algorithm using Smartphone GPS Data (No. 19-01097), Presented at 98th Annual Meeting of the Transportation Research Board, Washington, D.C., (2019). Mining individual life pattern based on location history. Y Ye, Y Zheng, Y Chen, J Feng, X Xie, tenth international conference on mobile data management: Systems, services and middleware. Ye, Y., Zheng, Y., Chen, Y., Feng, J., & Xie, X.. Mining individual life pattern based on location history. 2009 tenth international conference on mobile data management: Systems, services and middleware. pp. 1-10, (2009). 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Discovering personally meaningful places: An interactive clustering approach. ACM Transactions on Information Systems (TOIS). 25(3), 12, (2007). A data-driven method for trip ends identification using large-scale smartphone-based GPS tracking data. C Zhou, H Jia, Z Juan, X Fu, G Xiao, IEEE Transactions on Intelligent Transportation Systems. 188Zhou, C., Jia, H., Juan, Z., Fu, X., & Xiao, G.. A data-driven method for trip ends identification using large-scale smartphone-based GPS tracking data. IEEE Transactions on Intelligent Transportation Systems. 18(8), 2096-2110, (2016).

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{'abstract': 'Mobile device location data (MDLD) contains abundant travel behavior information to support travel demand analysis. Compared to traditional travel surveys, MDLD has larger spatiotemporal coverage of the population and its mobility. However, ground truth information such as trip origins and destinations, travel modes, and trip purposes are not included by default. Such important attributes must be imputed to maximize the usefulness of the data. This paper targets at studying the capability of MDLD on estimating travel mode share at aggregated levels. A data-driven framework is proposed to extract travel behavior information from MDLD. The proposed framework first identifies trip ends with a modified Spatiotemporal Density-based Spatial Clustering of Applications with Noise (ST-DBSCAN) algorithm. Then three types of features are extracted for each trip to impute travel modes using machine learning models. A labeled MDLD dataset with ground truth information is used to train the proposed models, resulting in 95% accuracy in identifying trip ends and 93% accuracy in imputing the five travel modes (drive, rail, bus, bike and walk) with a Random Forest (RF) classifier. The proposed framework is then applied to two large-scale MDLD datasets, covering the Baltimore-Washington metropolitan area and the United States, respectively. The estimated trip distance, trip time, trip rate distribution, and travel mode share are compared against travel surveys at different geographies. The results suggest that the proposed framework can be readily applied in different states and metropolitan regions with low cost in order to study multimodal travel demand, understand mobility trends, and support decision making.', 'arxivid': '2006.10036', 'author': ['Mofeng Yang \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Yixuan Pan \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Aref Darzi \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Sepehr Ghader \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Chenfeng Xiong \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Lei Zhang \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Mofeng Yang \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Yixuan Pan \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Aref Darzi \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Sepehr Ghader \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Chenfeng Xiong \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Lei Zhang \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Mofeng Yang \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Yixuan Pan \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Aref Darzi \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Sepehr Ghader \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Chenfeng Xiong \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n', 'Lei Zhang \n(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA\n'], 'authoraffiliation': ['(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 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Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA', '(STAR) Center\nSchool of Medicine\nUniversity of Maryland\n685 W Baltimore Street, Suite 60021201BaltimoreMDUSA'], 'corpusid': 219720872, 'doi': '10.1007/s11116-021-10214-3', 'github_urls': [], 'n_tokens_mistral': 28582, 'n_tokens_neox': 23804, 'n_words': 13067, 'pdfsha': '66675d2098d777c1fbae8e1f496f8ceff6f640a4', 'pdfurls': ['https://arxiv.org/pdf/2006.10036v4.pdf'], 'title': ['A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data', 'A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data', 'A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data', 'A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data', 'A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data', 'A Data-Driven Travel Mode Share Estimation Framework based on Mobile Device Location Data'], 'venue': []}

arxiv

Approximate Killing Fields as an Eigenvalue Problem 12 Aug 2008 (Dated: July 13, 2007) Christopher Beetle Department of Physics Florida Atlantic University 33431Boca Raton, Florida Approximate Killing Fields as an Eigenvalue Problem 12 Aug 2008 (Dated: July 13, 2007) Approximate Killing vector fields are expected to help define physically meaningful spins for nonsymmetric black holes in general relativity. However, it is not obvious how such fields should be defined geometrically. This paper relates a definition suggested recently by Cook and Whiting to an older proposal by Matzner, which seems to have been overlooked in the recent literature. It also describes how to calculate approximate Killing fields based on these proposals using an efficient scheme that could be of immediate practical use in numerical relativity. Spacetime symmetries are essential for defining physically important conserved quantities such as energy and angular momentum in general relativity. For example, when a vacuum spacetime admits a rotational symmetry generated by a Killing vector field ϕ a , then a Komar-type integral [1,2] over an arbitrary 2-sphere in spacetime can reproduce the physically well-defined angular momentum measured at infinity. When spacetime is axi-symmetric, but not vacuum, the difference between these integrals for a pair of different 2-spheres is precisely the ordinary angular momentum computed from the stress-energy of matter in the intervening space. If spacetime is not axisymmetric, however, such formulae become rather ambiguous. They depend not only on the 2-sphere S over which one integrates, but also on the vector field ϕ a used to define the integrand. These difficulties can be partially avoided under physically favorable conditions. For instance, angular momentum is well-defined at infinity in asymptotically flat spacetimes (see [3] for a recent review), or on an appropriate isolated horizon [4,5] modeling an isolated, nondynamical black hole in a spacetime that may describe interesting dynamics in other regions. Essentially, these techniques identify preferred 2-spheres S (infinity, horizon, etc.) over which to integrate, and thereby reduce the ambiguity in defining the angular momentum. The resulting quasi-local formulae have the general Brown-York [6] form J[ϕ] := 1 8πG S ϕ a K ab dS b ,(1) where S is a (perhaps preferred) 2-sphere, K ab is the extrinsic curvature of a spatial slice Σ containing it, ϕ a is a vector field on S, and dS b is the area element of S within Σ. The basic problem remains: the vector field ϕ a is arbitrary unless S has an intrinsic symmetry that can be used to select it. (Now, at least, that symmetry need not extend into the bulk of spacetime.) The horizons of black holes resulting from numerical simulations of astrophysical processes generally have no symmetry of any kind and therefore, seemingly, no preferred vector field ϕ a . The problem is not that such surfaces have no reasonable definition for the angular momentum, but rather that they have infinitely many. There is one for every vector field ϕ a tangent to the horizon. What is needed is a technique to pick a preferred vector field, and the obvious thing to do is to seek a ϕ a that, in some sense, is as close as possible to a Killing field, even if none is present. This leads intuitively to the idea of an approximate Killing field. Motivated by the general issues discussed above, several groups have recently proposed elegant definitions of approximate Killing fields on 2-spheres. These include schemes based on Killing transport [7], conformal Killing vectors [8], and most recently a minimization scheme by Cook and Whiting [9]. This paper revives an older approach [10] due to Matzner based on solving an eigenvalue problem. It also suggests a novel adaptation of Matzner's approach to the specific problem of computing a preferred angular momentum for black holes in numerical relativity, and elucidates the intimate relationship between this scheme and that of Cook and Whiting. Let us begin with a brief review of Matzner's definition [10] of an approximate Killing field on a compact manifold M of dimension n equipped with a Riemannian metric g ab . A continuous symmetry of g ab is generated by a Killing vector field ξ a satisfying L ξ g ab = 2 ∇ (a ξ b) = 0 (2) throughout M , where L ξ denotes the Lie derivative along ξ a and ∇ a is the unique torsion-free connection on M determined by g ab . Taking a divergence, we see that any geometry with a continuous symmetry admits at least one non-trivial solution to the equation − 2 ∇ b ∇ (b ξ a) = 0.(3) In principle, even when the geometry is not symmetric, we are still free to seek solutions for this equation. But generically we will not find any. Consider the eigenvalue problem ∆ K ξ a := −2 ∇ b ∇ (b ξ a) = κξ a(4) on a generic geometry. The operator ∆ K appearing here arises naturally in the transverse decomposition of symmetric tensor fields on Riemannian manifolds [12]. A related operator, denoted ∆ L , arises in the same way from the conformal Killing equation, and plays a similar role in the transverse-traceless decomposition of such fields. Its application to the initial-data problem in general relativity is very well known indeed [13]. Eq. (4) of course admits solutions ξ a only for a certain spectrum of eigenvalues κ, and zero may or may not be among these. Matzner [10] establishes the following four results: the spectrum of eigenvalues κ of Eq. (4) on a compact manifold is (a) discrete, (b) non-negative, (c) corresponds to a complete set of real vector eigenfields ξ a , and (d) contains κ = 0 if and only if the corresponding eigenfield ξ a is a genuine Killing field. That is, the zero eigenspace of Eq. (4) is precisely the finitedimensional vector space of Killing fields of the metric g ab on M . Therefore, Eq. (3) admits no solution if the metric g ab on M has no continuous symmetries, as claimed above. However, we assert that the best approximation to a Killing field on a manifold with no actual symmetry is the unique vector eigenfield ξ a of Eq. 4 with the minimum eigenvalue κ > 0. This is Matzner's definition of an approximate Killing field, and it has several desirable features. It exists generically, reduces to the correct answer when symmetries do exist, and, like a true Killing field on a symmetric manifold, is naturally defined only up to an overall (i.e., constant over M ) scaling. Like any eigenvalue problem, Eq. (4) admits a variational formulation. Recall the natural L 2 inner product ζ, ξ := M ζ a ξ a ǫ(5) on the space of (complex) vector fields over M . Here, ǫ denotes the canonical n-form volume element induced on M by the metric g ab . We minimize the quadratic form Q K [ξ; κ) := 1 2 ξ, ∆ K ξ − 1 2 κ ξ, ξ − 1 ,(6) where κ is constant over M and here plays the role of a Lagrange multiplier. Minimizing this functional produces the Euler-Lagrange equations ∆ K ξ a = κξ a and ξ, ξ = 1, the solutions of which are clearly the vector eigenfields of Eq. (4), normalized to unity in Hilbert space. Many variational problems are solved by initially solving the first, differential equation in Eq. (7) for ξ a as a function of an arbitrary Lagrange multiplier κ, and then using that result in the second, algebraic equation to impose the constraint and determine κ. This does not happen for Eq. (7) because the differential equation is linear in ξ a , and therefore leaves the overall scaling of ξ a undetermined. The second equation serves only to fix that scaling, and cannot also determine κ. The Lagrange multiplier therefore must be fixed when we solve the first equation; for general κ, no solution exists. This is hardly surprising since of course only true eigenvalues κ allow us to solve Eq. (4) for ξ a . However, it does make an approach to Matzner's eigenvalue problem via a variational principle like Eq. (6) rather complicated. There is no algebraic equation to determine the Lagrange multiplier. Indeed, κ is determined in this problem precisely by the condition that it be an eigenvalue of ∆ K , and there is no algebraic equation giving these. Minimizing Q K [ξ; κ) in Eq. (6) by solving the associated Euler-Lagrange equations is neither easier nor harder than solving the eigenvalue problem in Eq. (4). Cook and Whiting's recent definition [9] of an approximate Killing field uses a variational principle based on a quadratic form closely related to that of Eq. (6). However, it differs in a two important details. First, it focuses on the case where M ≃ S is topologically a 2-sphere, and restricts ξ a to be area-preserving: L ξ ǫ ab = (∇ c ξ c ) ǫ ab = 0.(8) The motivation for this restriction arises from the technical details of an eventual application to calculating the angular momentum of a non-symmetric black hole [4]. Second, it is based on a non-standard inner product ζ, ξ R := ζ, R ξ = S ζ a ξ a R ǫ.(9) These choices change the form, but not the basic content, of the resulting equations. They still describe a sort of self-adjoint eigenvalue problem. To restrict to area-preserving vector fields, it is easiest simply to recall that any divergenceless vector fieldξ a on a 2-sphere topology is described by a unique scalar potential Θ such that ξ a = ( * dΘ) a := −ǫ ab ∇ b Θ and S Θ ǫ = 0. (10) Now consider the restricted eigenvalue problem ∆ Kξ a := P ∆ KP ξ a =κξ a , whereP denotes the orthogonal projection onto the subspace of area-preserving vector fields within the Hilbert space of Eq. (5). A givenξ a = ( * dΘ) a solves this equation if and only if, for all otherζ a = ( * dΦ) a , we have * dΦ, ∆ K * dΘ =κ * dΦ, * dΘ . Integrating both sides by parts, and using positivity of the standard L 2 inner product of scalar functions over S, we find that Eq. (11) is equivalent to ∆ ∆ K Θ := ∆ 2 Θ + ∇ a (R ∇ a Θ) =κ ∆ Θ,(13) where ∆ := −∇ a ∇ a denotes the standard scalar Laplacian. We have shown that the scalar functions Θ solving Eq. (13) generate, via Eq. (10), solutionsξ a of the restricted eigenvalue problem of Eq. (11). Because the projectionP does not typically commute with ∆ K , these ξ a do not generally solve Eq. (4), and the restricted eigenvaluesκ are generally distinct from the eigenvalues κ in the full Hilbert space. In fact, we should generally expect thatκ min > κ min . However, the area-preserving vector eigenfield corresponding to this minimum restricted eigenvalue can also be considered a best approximation to a Killing field, albeit within a restricted class. To recover the Cook-Whiting approximate Killing field, we must repeat the previous calculation in the inner product of Eq. (9). The operator ∆ K is then no longer self-adjoint, but R −1 times ∆ K is. Accordingly, we seek vector fieldsξ a R = ( * dΘ) a satisfying * dΦ, R −1 ∆ K * dΘ R =κ R * dΦ, * dΘ R(14) for allζ a = ( * dΦ) a . Integrating by parts, and once again invoking positivity of the standard inner product of scalar functions, we find that Eq. (14) is equivalent to ∆ ∆ K Θ = −κ R ∇ a (R ∇ a Θ).(15) Although the notation here differs slightly, this is precisely the Euler-Lagrange equation that Cook and Whiting find [9] by minimizing a quadratic form similar to Eq. (6). Once again, the solutions (ξ a R ,κ R ) of this eigenvalue problem generally differ from the solutions (ξ a , κ) of Eq. (4) and from the solutions (ξ a ,κ) of Eq. (11). Let us now make two technical comments. First, any constant function Θ = c will give zero on both sides of Eqs. (11) and (15) for all values ofκ orκ R , respectively. These are spurious solutions. They arise only because we have used potentials to describe the subspace of areapreserving vector fields. These solutions are ruled out by the the second condition in Eq. (10), which makes the correspondence betweenξ a and Θ an isomorphism. Second, the Cook-Whiting inner product in Eq (9) looks a little odd, but it is not immediately clear whether there is anything technically wrong with it. There certainly can be problems. Recall that the scalar curvature in two dimensions varies as δ 2 R = ∇ b ∇ [a δg b] a − 1 2 2 R δg a a(16) under a perturbation δg ab of the metric. If this perturbation varies sufficiently rapidly over S, then the first term here can easily dominate the second, as well as the unperturbed, background value 2 R. The result is that a generic spherical geometry, even if perturbatively close to a round sphere in the sense that δg ab has small amplitude, can have regions of negative scalar curvature. (This is intuitively obvious if we imagine "pinching" the surface of a round sphere to create a small, saddle-shaped region of negative curvature.) On such geometries, the "inner product" of Eq. (9) is not positive-definite, and does not define a Hilbert space. However, in the space of all spherical geometries, there should be some finite region of "sufficiently smooth" perturbations of the round sphere for which the total scalar curvature remains everywhere positive. In this region, there is no obvious problem with the Cook-Whiting scheme, but nothing particular to recommend it either. The question could presumably be settled [11] by comparing qualitative features of the approximate Killing fields computed from Eqs. (11) and (15). Matzner's eigenvalue definition of an approximate Killing field is unambiguous, universally applicable, and reproduces the usual Killing fields on a symmetric manifold. But it is not necessarily efficient in practice. Indeed, it would prohibitively expensive to solve any of the eigenvalue problems in Eqs. (4), (11) or (15) on the apparent horizon at every moment of time of a black hole in a numerical simulation. Fortunately, however, there is a simple approximation to speed the process up on a generic geometry. This approximation is based on the Rayleigh-Ritz method [15], and works so long as we only want to find the lowest eigenvalue and the corresponding vector eigenfield. Consider the Rayleigh-Ritz functional F [ξ] := ξ, ∆ K ξ ξ, ξ = 2 M ∇ (a ξ b) · ∇ (a ξ b) · ǫ M ξ a ξ a ǫ(17) on the full Hilbert space of Eq. (5), with the zero vector removed. The local extrema of Eq. (17) occur when ξ a is a vector eigenfield of ∆ K , and the value of F [ξ] at each such extremum is the corresponding eigenvalue κ. Note that the numerator here, which arises via integration by parts of the second-order operator ∆ K in Eq. (4), is precisely one half the square integral of L ξ g ab from Eq. (2). Thus, among all vector fields with fixed L 2 -norm on S, diffeomorphisms along the approximate Killing field modify the metric least in a quantifiable, L 2 sense. It is still not practicable to find the genuine absolute minimum of Eq. (17) on the computer, which of course would yield Matzner's approximate Killing field. But one can approximate that minimum by minimizing F [ξ] within an appropriate space of trial vector fields. This idea is familiar from elementary quantum mechanics, where just such a variational principle is routinely used to approximate the ground-state wave-function of a complicated system. Unless the subspace of trial fields one chooses happens to be orthogonal, or nearly so, to the true minimum ξ a true of F [ξ] in all of Hilbert space, the dominant component of the minimizing trial field ξ a trial should lie along ξ a true in Hilbert space. Most randomlychosen trial spaces will not be orthogonal to ξ a true . This idea allows us to approximate Matzner's approximate Killing field. There is a natural candidate for the trial space of vector fields in which to minimize Eq. (17) in the specific case M ≃ S of a 2-sphere horizon of a quiescent black hole in numerical relativity. One striking feature of many recent numerical simulations (e.g, [14]) is that the horizons at late times often look fairly regular in the fiducial spacetime coordinates used to do the evolution. Therefore, it is natural to try a space of trial fields ξ a based simply on those coordinates. A specific proposal follows. Use the fiducial spacetime coordinates in which the numerical evolution occurs to induce spherical coordinates (θ, φ) on the black-hole horizon in some more-or-less natural, but fundamentally ad hoc, way. Then, take the space of scalar trial potentials Θ(θ, φ) := lmax l=1 l m=−l Θ lmŶ lm (θ, φ),(18) whereŶ lm (θ, φ) are the ordinary scalar spherical harmonic functions on a round sphere, Θ lm are arbitrary constants, and l max is a chosen cut-off. Each of these potentials generates an area-preserving vector field via Eq. (10), and this will be our trial space [16] within the full Hilbert space of Eq. (5). Therefore, minimize F [Θ] := * dΘ, ∆ K * dΘ * dΘ, * dΘ = Θ, ∆ ∆ K Θ Θ, ∆ Θ (19) = S 2 g ac g bd − g ab g cd ∇ a ∇ b Θ · ∇ c ∇ d Θ ǫ − S Θ · ∇ a ∇ a Θ · ǫ within the trial space of potentials given by Eq. (18). Generally, we should expect that the minimizing potential will generate a vector field ξ a trial fairly close to the minimum-eigenvalue area-preserving vector eigenfieldξ a true of Eq. (11). This, in turn, should approximate Matzner's approximate Killing field from Eq. (4). To check the approximation, one could imagine increasing l max until ξ a trial doesn't vary much with the cut-off. Equivalently, one could use a fairly large cut-off-perhaps l max = 5 would be enough-from the start, and check that the amplitudes Θ lm are small for large l. If one prefers to approximate the Cook-Whiting approximate Killing field, one need only insert a factor of the scalar curvature R between the gradients in the denominator of Eq. (19). There is one significant issue that has not been addressed in this discussion. Even once an approximate Killing field ξ a has been found from the eigenvalue approach, it is still determined only up to overall normalization on S. For a proper rotational Killing field on a symmetric apparent horizon, the correct normalization would demand that the affine length of each Killing orbit should be 2π. It is not immediately clear what convention might be used in the general case, without symmetry, to fix a normalization that goes over to this correct one in the limit of a symmetric manifold. This issue will be discussed more thoroughly, in the context of practical applications, in a forthcoming paper [11]. AcknowledgementsThe author would like to thank Ivan Booth, Manuela Campanelli, Greg Cook, Stephen Fairhurst, Greg Galloway, Carlos Lousto, Charles Torre, Bernard Whiting and Yosef Zlochower for stimulating discussions related to this question. This work has been supported by NSF grants PHY 0400588 and PHY 0555644, and by NASA grant ATP03-0001-0027. Covariant Conservation Laws in General Relativity. A Komar, Phys. Rev. 113A. Komar. Covariant Conservation Laws in General Rel- ativity. Phys. Rev. 113 (1959) 934-936. R M Wald, General Relativity. ChicagoUniversity of Chicago PressR.M. Wald. General Relativity. University of Chicago Press, Chicago, 1984. Quasi-Local Energy-Momentum and Angular Momentum in General Relativity: A Review Article. L B Szabados, Living Rev. Relativity. 7L.B. Szabados. Quasi-Local Energy-Momentum and An- gular Momentum in General Relativity: A Review Arti- cle. Living Rev. Relativity 7 (2004) 4. Cited 8 July 2007. Mechanics of rotating isolated horizons. A Ashtekar, C Beetle, J Lewandowski, Phys. Rev. D. 6444016A. Ashtekar, C. Beetle and J. Lewandowski. Mechanics of rotating isolated horizons. Phys. Rev. D 64 (2001) 044016. Isolated and Dynamical Horizons and Their Applications. A Ashtekar, B Krishnan, Living Rev. Relativity. 7A. Ashtekar and B. Krishnan. Isolated and Dynamical Horizons and Their Applications. Living Rev. Relativity 7 (2004), 10. Cited 8 July 2007. Quasilocal energy and conserved charges derived from the gravitational action. J D Brown, J W York, Jr , Phys. Rev. D. 47J.D. Brown and J.W. York, Jr. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D 47 (1993) 1407-1419. Introduction to isolated horizons in numerical relativity. Olaf Dreyer, B Krishnan, D Shoemaker, E Schnetter, Phys. Rev. D. 6724018Olaf Dreyer, B. Krishnan, D. Shoemaker and E. Schnet- ter. Introduction to isolated horizons in numerical rela- tivity. Phys. Rev. D 67 (2003) 024018. Circular orbits and spin in black-hole initial data. M Caudill, G B Cook, J D Grigsby, H P Pfeiffer, Phys. Rev. D. 7464011M. Caudill, G.B. Cook, J.D. Grigsby and H.P. Pfeiffer. Circular orbits and spin in black-hole initial data. Phys. Rev. D 74 (2006) 064011. Approximate Killing Vectors on S 2. G B Cook, B F Whiting, arXiv:0706.0199v1E-Printgr-qcG.B. Cook and B.F. Whiting. Approximate Killing Vec- tors on S 2 . E-Print arXiv: 0706.0199v1 [gr-qc]. 2007. Almost Symmetric Spaces and Gravitational Radiation. R A Matzner, J. Math. Phys. 9R.A. Matzner. Almost Symmetric Spaces and Gravita- tional Radiation. J. Math. Phys. 9 (1968) 1657-1668. . C Beetle, M Campanelli, C O Lousto, Y Zlochower, In preparationC. Beetle, M. Campanelli, C.O. Lousto and Y. Zlochower. In preparation. Covariant decompositions of symmetric tensors in the theory of gravitation. J W York, Jr , Ann. Inst. Henri Poincaré. 21J.W. York, Jr. Covariant decompositions of symmetric tensors in the theory of gravitation. Ann. Inst. Henri Poincaré 21 (1974) 319-332. Initial-value problem of general relativity. I. General forumlation and physical interpretation. N Murchadha, J W York, Jr , Phys. Rev. D. 10N.Ó Murchadha and J.W. York, Jr. Initial-value problem of general relativity. I. General forumlation and physical interpretation. Phys. Rev. D 10 (1974) 428-436. Spin flips and precession in black-holebinary mergers. M Campanelli, C O Lousto, Y Zlochower, B Krishnan, D Merritt, Phys. Rev. D. 7564030M. Campanelli, C.O. Lousto, Y. Zlochower, B. Krishnan and D. Merritt. Spin flips and precession in black-hole- binary mergers. Phys. Rev. D 75 (2007) 064030. . J Mathews, R L Walker, Mathematical Methods of Physics. Addison-WesleyJ. Mathews and R.L. Walker. Mathematical Methods of Physics. Addison-Wesley, Redwood City, California, 1970. to the space of constant functions on S. However, the key point is that standard properties of the ordinary spherical harmonics show that this space of trial potentials contains no actual constant functions. The space of potentials in Eq. (18) is usually not orthogonal, in the sense of Eq. (10). This is why we have taken lmin = 1 in Eq. (18)The space of potentials in Eq. (18) is usually not orthog- onal, in the sense of Eq. (10), to the space of constant functions on S. However, the key point is that standard properties of the ordinary spherical harmonics show that this space of trial potentials contains no actual constant functions. This is why we have taken lmin = 1 in Eq. (18). 10) maps our space of trial potentials faithfully to a space of trial vector fields with the same dimension, lmax (lmax + 2). Eq Therefore, Therefore, Eq. (10) maps our space of trial potentials faithfully to a space of trial vector fields with the same dimension, lmax (lmax + 2).

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{'abstract': 'Approximate Killing vector fields are expected to help define physically meaningful spins for nonsymmetric black holes in general relativity. However, it is not obvious how such fields should be defined geometrically. This paper relates a definition suggested recently by Cook and Whiting to an older proposal by Matzner, which seems to have been overlooked in the recent literature. It also describes how to calculate approximate Killing fields based on these proposals using an efficient scheme that could be of immediate practical use in numerical relativity.', 'arxivid': '0808.1745', 'author': ['Christopher Beetle \nDepartment of Physics\nFlorida Atlantic University\n33431Boca Raton, Florida\n'], 'authoraffiliation': ['Department of Physics\nFlorida Atlantic University\n33431Boca Raton, Florida'], 'corpusid': 118653065, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 6564, 'n_tokens_neox': 5793, 'n_words': 3885, 'pdfsha': '8e7aff7f59214a9c64672d9c5399e6be73c37ad4', 'pdfurls': ['https://arxiv.org/pdf/0808.1745v1.pdf'], 'title': ['Approximate Killing Fields as an Eigenvalue Problem', 'Approximate Killing Fields as an Eigenvalue Problem'], 'venue': []}

arxiv

GIBBS RANDOM FIELDS WITH UNBOUNDED SPINS ON UNBOUNDED DEGREE GRAPHS 21 Apr 2009 April 21, 2009 Yuri Kondratiev ANDYuri Kozitsky Tanja Pasurek GIBBS RANDOM FIELDS WITH UNBOUNDED SPINS ON UNBOUNDED DEGREE GRAPHS 21 Apr 2009 April 21, 2009arXiv:0904.3207v1 [math.PR] Gibbs random fields corresponding to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is non-void and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). Uniqueness of a Gibbs field with a random potential-an elementary approach. Theory Probab. Appl. 31 572-589]. 1. Introduction and paper overview 1.1. Introduction. Gibbs random fields on a discrete metric space (e.g. on a graph) can be viewed as collections of dependent random variables, usually called spins, indexed by the elements of this space. Their joint probability laws are defined by the families of local conditional distributions constructed by means of interaction potentials. We quote the monographs (10; 11) as standard sources in the theory of such fields. Each spin takes values in the corresponding single-spin space, say X x . Most of Gibbs random fields constructed on general graphs correspond to models with finite singlespin spaces. Perhaps, the most known example is the Ising model where X x = {−1, 1} for all x. By the compactness of X x , such Gibbs fields exist for arbitrary graphs, see (11; 13; 14; 17; 18; 22). Their properties are closely related to those of random walks or corresponding percolation models, see e.g. (13; 17; 18). The development of the theory of Gibbs random fields with unbounded spins, started in the late seventies in the pioneering works (16; 6), was strongly motivated by physical applications, especially, in Euclidean quantum field theory, see e.g. (20). Since that time, such random fields were extensively studied, see e.g. the bibliographical notes in (19). However, the results obtained in all these works were restricted to the case where the underlying metric space is a simple cubic lattice Z d . In (15; 19), the theory of Gibbs random fields was extended to unbounded spin systems living on more general discrete metric spaces, including graphs of bounded degree. In this context, we mention also the paper (12) where a Gaussian field on a bounded degree graph was studied. In the present paper, we construct Gibbs random fields with unbounded spins (X x = R for all x) on unbounded degree graphs of certain kind and analyze the role played here by the geometry of the graph. In doing so, we are motivated by the following reasons: • Random fields on Riemannian manifolds, especially those associated with the corresponding Laplace-Beltrami operators, cf. (8), can be approximated by their discrete versions living on appropriate graphs (9). This includes also the case of quantum fields in curved spacetime, see (2; 21). • As the degree of the graph can be related to such a property of the corresponding manifold as curvature, the use of unbounded degree graphs essentially extends the class of manifolds that can be approximated in the above sense. • Another application can be the description of systems of interacting oscillators located at vertices of an infinite graph -the so called oscillating networks, see Section 14 in (4). We refer also to the survey (5), where other relevant physical models can be found. The results of the paper are: (a) constructing Gibbs random fields; (b) deriving exponential integrability estimates and support properties for such fields; (c) presenting a concrete family of unbounded degree graphs, which can serve as underlying graphs for our model. In achieving (a) and (b), we used a modification of the technique developed in (15; 19). In constructing the family of (c) we were inspired by some aspects of (3). To the best of our knowledge, the present study is the first attempt in such a direction. We plan to continue investigating the model introduced here in forthcoming papers. In particular, we are going to study the problem of uniqueness of Gibbs random fields, as well as the ergodicity properties of the corresponding stochastic dynamics. 1.2. The paper overview. The model we deal with in this paper is the triple (G, W, V ), where G = (V, E) is a graph, W : R×R → R and V : R → R are continuous functions (potentials). The properties of the triple (G, W, V ) are specified below in Assumption 2.1, see also (3.1). This triple determines the heuristic Hamiltonian (1.1) H(ω) = x,y W (ω(x), ω(y)) + x V (ω(x)), where the first (resp. second) sum is taken over all edges (resp. vertices) of the graph. For this model, Gibbs random fields are defined as probability measures on the configuration space Ω = R V . In contrast to the case of bounded spins, it is unrealistic to describe all Gibbs measures of an unbounded spin system without assuming a priori any of its properties. Thus, among all Gibbs measures corresponding to (1.1) we distinguish those that have a prescribed support property, i.e., such that µ(Ω t ) = 1 for an a priori chosen proper subset Ω t ⊂ Ω. These measures are called tempered. In Theorem 2.3, we show that the set of tempered Gibbs measures G t is non-void and weakly compact. Here we also show that each µ ∈ G t obeys important integrability estimates, the same for all such measures. In Theorems 3.1 and 3.2, these results are extended in the following directions: (a) we allow the potential W to be super-quadratic, see (3.1); (b) we consider a scale of sets of tempered Gibbs measures, which clarifies connections between the graph geometry and the properties of such measures. These our results are valid for any graph possessing the summability specified in Assumption 2.1. To provide a nontrivial example of unbounded degree graphs with this property, in the second part of the paper we introduce a new class of such graphs, which we believe is interesting in its own right. This class is characterized by the following property, cf. (5.3) and (5.2). For vertices x and y, such that their degrees, n(x) and n(y), exceed some threshold value, the path distance is supposed to obey the 'repulsion' condition (1.2) ρ(x, y) ≥ φ [max{n(x), n(y)}] , where φ is a given increasing function. In such graphs, every vertex x has the property that sup y: ρ(x,y)≤N n(y) ≤ φ −1 (2N ), whenever N exceeds some N x , specific for this x. By means of this property, for φ(b) = υ log b[log log b] 1+ε , υ, ε > 0, we obtain the estimate y: ρ(x,y)=N [n(y)] 1+θ ≤ exp(aN ), which holds for any θ > 0 and an appropriate a > 0, whenever N ≥ N x . In Theorem 5.2, we show that the latter estimate implies the required summability (2.3). The rest of the paper is organized as follows. In the first part, the emphasis is put on the probabilistic stuff, whereas the second part -Section 5 -is devoted to the graph-theoretical aspects of the problem. In Section 2, we specify the class of models by imposing conditions on the graph and on the potentials. The only essential condition imposed on G is the summability (2.3). The potentials are supposed to obey quite standard stability requirements, plus continuity. We note, however, that the stability condition (2.5) is a bit stronger than the one with q = 2, typical for graphs of bounded degree. In view of this fact, the Gaussian case is not covered by our theory. Thereafter, we put forward Theorem 2.3. In Section 3, we present Theorems 3.1 and 3.2. The proof of the latter theorem follows from the estimates obtained in Theorem 3.1. The proof of Theorem 3.1, which is the main technical component of the first part of the paper, is given in Section 4. It is preceded by a number of lemmas, in which we elaborate the corresponding tools. The key element here is Lemma 4.2 the proof of which crucially employs the summability (2.3). In Section 5, we introduce and describe the class of graphs with the property (1.2), which by Theorem 5.2 can serve as underlying graphs for our model. is supposed to be undirected and countable. Two adjacent vertices x, y ∈ V are also called neighbors. In this case, we write x ∼ y and x, y ∈ E. The degree of x ∈ V, denoted by n(x), is the cardinality of the neighborhood of x, that is, of the set {y | y ∼ x}. We use the shorthand x = x∈V sup x = sup x∈V y∼x = y∈V: y∼x . The graph is assumed to be locally finite, which means that n(x) ∈ N for any x. At the same time, we assume that sup x n(x) = +∞, which is reflected in the title of the paper. Of course, our results are trivially valid for bounded degree graphs. A sequence ϑ = {x 0 , x 1 , . . . , x n }, such that x k ∼ x k+1 for all k = 0, . . . , n− 1, is called a path. Herein, some of the vertices may be repeated. The path connects its endpoints x 0 and x n ; it leaves the vertices x 0 , . . . , x n−1 and enters x 1 , . . . , x n . The number of left vertices, denoted by ϑ , is called the length of the path. For x, y ∈ V, by ϑ(x, y) we denote a path, whose endpoints are x and y. We assume that G is connected, which means that there exists a path ϑ(x, y) for every x and y. The path distance ρ(x, y) is set to be the length of the shortest ϑ(x, y). It is a metric on G by means of which, for a certain o ∈ V and α > 0, we define (2.1) w α (x) = exp[−αρ(o, x)], x ∈ V. For θ > 0, we also set 1 (2.2) m θ (x) = y∼x [n(x)n(y)] θ , x ∈ V. The remaining properties of the model are summarized in Assumption 2.1. The triple (G, W, V ) is subject to the following conditions: 1 In mathematical chemistry, the sum of terms [n(x)n(y)] θ taken over the edges x, y of a finite tree is known under the name generalized Randić or connectivity index, see e.g. . (i) the graph G is such that, for some positive α and θ, (2.3) Θ (α, θ) def = x m θ (x)w α (x) < ∞; (ii) the function W is continuous, symmetric, and such that (2.4) |W (u, v)| ≤ [I W + J W (u 2 + v 2 )]/2, for some I W , J W > 0 and all u, v ∈ R; (iii) the function V is continuous and such that, for all u ∈ R, (2.5) V (u) ≥ a V |u| q − c V , for some a V , c V > 0 and q > 2 + 2/θ, with θ being the same as in (i). 2.2. The basic result. Following the standard DLR route, see (10), the Gibbs random fields for our model are defined as probability measures on the measurable space (Ω, B(Ω)). Here Ω = R V is the configuration space, equipped with the product topology and with the corresponding Borel σ-field B(Ω). By P(Ω) we denote the space of all probability measures on (Ω, B(Ω)), which is equipped with the weak topology determined by bounded continuous functions f : Ω → R. By C b (Ω) we denote the set of all such functions. In the sequel, by writing Λ ⋐ V we mean that Λ is a finite and non-void set of vertices. A property related to such a subset is called local. As usual, Λ c = V \ Λ stands for the complement of Λ ⊂ V. For Λ ⋐ V and ω ∈ Ω, by ω Λ we denote the restriction of ω to Λ, and use the decomposition ω = ω Λ × ω Λ c . Then for such Λ and a fixed ξ ∈ Ω, the relative local Hamiltonian is set to be H Λ (ω Λ |ξ) = x,y : x,y∈Λ W (ω(x), ω(y)) (2.6) + x,y : x∈Λ, y∈Λ c W (ω(x), ξ(y)) + x∈Λ V (ω(x)). Thereby, for Λ ⋐ V, ξ ∈ Ω, and A ∈ B(Ω), we define π Λ (A|ξ) = 1 Z Λ (ξ) R |Λ| I A (ω Λ × ξ Λ c ) exp [−H Λ (ω Λ |ξ)] dω Λ , (2.7) where I A is the indicator function, dω Λ is the corresponding Lebesgue measure on the Euclidean space Ω Λ def = R |Λ| , and Z Λ (ξ) is a normalizing factor. Hence, each π Λ (·|ξ) ∈ P(Ω). The family {π Λ } Λ⋐V is called the local Gibbs specification for the model we consider. Directly from the definition (2.7), one makes sure that this family is consistent in the following sense: (2.8) Ω π ∆ (A|ω)π Λ (dω|ξ) = π Λ (A|ξ), which holds for all A ∈ B(Ω), all ∆ ⊂ Λ, and all Λ ⋐ V. Definition 2.2. A measure µ ∈ P(Ω) is said to be a Gibbs random field corresponding to the Hamiltonian (1.1) if it solves the following (DLR) equation (2.9) µ(A) = Ω π Λ (A|ω)µ(dω), for all A ∈ B(Ω) and Λ ⋐ V. An equivalent version of (2.9) is the following equation (2.10) µ(f ) = Ω π Λ (f |ω)µ(dω), which ought to hold for all f ∈ C b (Ω) and Λ ⋐ V. Here, for such f and µ ∈ P(Ω), we use the notation µ(f ) = Ω f (ω)µ(dω). Let G stand for the set of all solutions of (2.9). As is typical for unbounded spin systems, it is far from being obvious whether G is non-void. But if it is the case, the description of properties possessed by all the elements of G is rather unrealistic. Thus, one constructs and studies a subset of G, consisting of the measures possessing a prescribed (support) property. Such measures are called tempered. For positive p and α, we set (2.11) ω p,α = x∈V |ω(x)| p w α (x) 1/p , where the weights w α are defined in (2.3). Then L p (V, w α ) = {ω ∈ R V | ω p,α < ∞}, is a Banach space. For θ and q being as in (2.3) and in (2.5), respectively, we fix (2.12) p = 2 + 2/θ < q. For this p, the set of tempered configurations is set to be (2.13) Ω t = L p (V, w α ), where α is as in (2.3). Clearly, Ω t ∈ B(Ω); hence, one can define (2.14) G t = {µ ∈ G | µ(Ω t ) = 1}. Theorem 2.3 (Basic). The set G t is non-void and weakly compact. For every λ > 0 and x ∈ V, there exists a positive constant C(λ, x), such that, for all µ ∈ G t , (2.15) Ω exp (λ|ω(x)| p ) µ(dω) ≤ C(λ, x). Furthermore, for every λ > 0, there exists a positive constant C(λ), such that, for all µ ∈ G t , (2.16) Ω exp λ ω p p,α µ(dω) ≤ C(λ). Herein, α and p are the same as in (2.13). Extensions 3.1. More on temperedness. In this subsection, Theorem 2.3 is extended in the following directions: (a) we allow a super-quadratic growth of the potential W , cf. (2.4); (b) we construct a scale of sets of tempered Gibbs fields, the elements of which obey integrability estimates, stronger than (2.15) and (2.16). In what follows, instead of (2.4) we assume (3.1) |W (u, v)| ≤ [I W + J W (|u| r + |v| r )]/2, for some r > 0. The potential V is assumed to obey (2.5) with q > r + r/θ, where θ is as in (2.3). The graph G is supposed to be the same as in Assumption 2.1. The scale of tempered Gibbs fields which we are going to construct will be indexed by α and p. First, we set (3.2) α = inf{α | Θ (α, θ) < ∞}, and let α > α be such that (2.3) holds for all α ∈ (α, α]. Next, we define (3.3) p 0 =r + r/θ. For α ′ , α ∈ (α, α] and p ′ , p ∈ [p 0 , q), by (2.11) we have (3.4) L p ′ (V, w α ′ ) ֒→ L p (V, w α ), whenever α ′ ≤ α and p ′ ≥ p. Notably, the above embedding is compact. Then, for α ∈ (α, α] and p ∈ [p 0 , q), we set, cf. (2.14), (3.5) G p,α = {µ ∈ G | µ [L p (V, w α )] = 1}. Clearly (3.6) G p ′ ,α ′ ⊂ G p,α , whenever α ′ ≤ α and p ′ ≥ p. The following statement is an extended version of Theorem 2.3. Theorem 3.1 (Extended). For every α ∈ (α, α] and p ∈ [p 0 , q), the set G p,α is non-void and weakly compact. For every λ > 0 and x ∈ V, there exists a positive constant C(p, α; λ, x), such that, for all µ ∈ G p,α , (3.7) Ω exp (λ|ω(x)| p ) µ(dω) ≤ C(p, α; λ, x). Furthermore, for every λ > 0, there exists a positive constant C(p, α; λ), such that, for all µ ∈ G p,α , (3.8) Ω exp λ ω p p,α µ(dω) ≤ C(p, α; λ). Let us make some comments. • For our graphs, one cannot expect that the constants C(p, α; λ, x) in (3.7) are bounded uniformly in x. This could be the case if the quantities Θ (α, θ) were bounded uniformly with respect to the choice of the root o. • Both estimates (3.7) and (3.8) hold also for p = q but not for all λ, which should be small enough in this case. • The interval [p 0 , q) is non-void if q > r + r/θ, i.e. , if the stabilizing effect of the potential V is stronger than the destabilizing effects of the interaction and of the underlying graph, caused by its degree property. If the graph is of bounded degreen = sup x n(x), the condition (2.3) is satisfied for any θ > 0 and α > logn. In this case, one can take θ arbitrarily big and get q > r (or q ≥ r for small λ), which is typical for such situations. • According to (2.11) and (3.6), the stronger estimates we want to get, the smaller class of tempered Gibbs random fields we obtain. • In view of the specific features of the graph geometry, such as the degree unboundedness and the lack of transitivity, the two basic statistical-mechanical tools -Ruelle's superstability method and Dobrushin's existence and uniqueness criteria -are not applicable to our model. The proof of Theorem 3.1 will be done in Section 4. Theorem 2.3 is obtained therefrom as a particular case of α = α, p = p 0 , and r = 2. 3.2. More on weak compactness. Taking into account (3.4), we define (3.9) Ω t = p∈[p 0 ,q) α∈(α,α] L p (V, w α ). This set can be endowed with the projective limit topology and thereby turned into a Fréchet space. By standard arguments, its Borel σ-field B( Ω t ) has the property (3.10) B( Ω t ) = {A ∩ Ω t | A ∈ B(Ω)}, in view of which, we can define, cf. (2.14), (3.11) G t = {µ ∈ G | µ Ω t = 1}. The elements of the latter set have the smallest support we have managed to establish. In view of (3.10), they can be redefined as probability measures on ( Ω t , B( Ω t )). Let W t be the weak topology on the set of all probability measures P( Ω t ). Clearly, W t is stronger than the topology mentioned in Theorem 3.1. Theorem 3.2. The set G t is non-void and W t -compact. Proof. Let G be the intersection of all G p,α , with α ∈ (α, α] and p ∈ [p 0 , q), see (3.9). By compactness established in Theorem 3.1 the set G is nonvoid. Obviously, all its elements belong to G t and hence these two sets coincide. Furthermore, the elements of G obey the estimates (3.7), (3.8) with all α ∈ (α, α] and p ∈ [p 0 , q). Let us now prove the stated W t -compactness. To this end we consider the balls (3.12) B p,α (R) = {ω | ω p,α ≤ R}, R > 0, and fix two monotone sequences α k ↓ α and p k ↑ q, as k → +∞. In view of (3.8), for any k ∈ N and ǫ > 0, one can pick R k,ǫ > 0 such that, µ [B p k ,α k (R k,ǫ )] ≥ 1 − ǫ/2 k , uniformly for all µ ∈ G p k ,α k , and hence for all µ ∈ G t . By the compactness of the embedding (3.4), the set B = k∈N B p k ,α k (R k,ǫ ) is compact in Ω t , and is such that µ(B) ≥ 1 − ǫ for all µ ∈ G t . Thereafter, the W t -compactness of G t follows by the renowned Prokhorov theorem. 3.3. Gibbs states of systems of anharmonic oscillators. The Gibbs random fields constructed above can serve as equilibrium thermodynamic states of systems of one-dimensional anharmonic oscillators, indexed by the vertices of G and interacting with each other along the edges by the potential W (oscillating networks). Obviously, Theorem 2.3 holds true if one replaces the single-spin space R with R ν , ν ∈ N, which would correspond to multi-dimensional oscillators. Furthermore, by means of the technique developed in (1; 15; 19) this theorem can also be extended to the case where the single-spin spaces are copies of C β -the Banach space of continuous functions (temperature loops) ω : [0, β] → R ν , β > 0, such that ω(0) = ω(β). In this case, the Gibbs random fields correspond to the so called Euclidean thermodynamic Gibbs states of a system of interacting ν-dimensional quantum anharmonic oscillators, for which β −1 is temperature. Properties of the local Gibbs specification In this section, we prove that the estimate (3.8) holds also for all π Λ (·|ξ). This will imply all the properties of the family {π Λ } Λ⋐V which we need to prove Theorem 3.1. We begin by deriving a basic estimate, which allows us to control the ξ-dependence of moments of π Λ with one-point Λ = {x}. Its extension to arbitrary Λ's will be obtained by means of the consistency property (2.8). Moment estimates. From (3.1), by an easy calculation we get (4.1) |W (u, v)| ≤ κ (|u| p + |v| p )+I W /2+2(p−r) J W 2p p/(p−r) r κ r/(p−r) , which holds for all u, v ∈ R, and κ > 0, p > r. We will use this estimate with κ = β/n(x)n(y), x, y ∈ V, β > 0. For such β and p ∈ [p 0 , q), we set Γ xy (β, p) = γ(β, p)[n(x)n(y)] r/(p−r) , (4.2) γ(β, p) = I W + 4(p − r) J W 2p p/(p−r) r β r/(p−r) , and C(β, λ, p) = c V + log R exp ((λ + β)|u| p − a V |u| q ) du (4.3) − log R exp (−β|u| p − V (u)) du , where λ > 0 and a V , c V , and q are the same as in (2.5) and (2.12). Note that the integral in the latter line is positive. In the lemma below, π x and Z x stand for the corresponding objects defined in (2.7) with Λ = {x}. Proof. By (4.1), with κ = β/n(x)n(y), and (4.2) the relative Hamiltonian (2.6) with Λ = {x} can be estimated as follows − y∼x β n(x)n(y) (|ω(x)| p + |ξ(y)| p ) + 1 2 Γ xy (β, p) − V (ω(x)) ≤ −H x (ω(x)|ξ) ≤ ≤ y∼x β n(x)n(y) (|ω(x)| p + |ξ(y)| p ) + 1 2 Γ xy (β, p) −a V |ω(x)| q + c V . Then Z x (ξ) ≥ exp − y∼x β|ξ(y)| p n(x)n(y) + 1 2 Γ xy (β, p) × R exp [−β|ω(x)| p − V (ω(x))] dω(x), and R exp [λ|ω(x)| p − H x (ω(x)|ξ)] dω(x) ≤ exp c V + y∼x β|ξ(y)| p n(x)n(y) + 1 2 Γ xy (β, p) × R exp [(λ + β)|ω(x)| p − a V |ω(x)| q ] dω(x), which clearly yields (4.4). Now, for λ > 0, p ∈ [p 0 , q), Λ ⋐ V, and a fixed x ∈ Λ, we set (4.5) M x (λ, p, Λ; ξ) = log Ω exp (λ|ω(x)| p ) π Λ (dω|ξ) , which is obviously finite. Our aim is to find an upper bound for this quantity. Integrating both sides of (4.4) with respect to π Λ (·|ξ) and taking into account (2.8) we obtain exp [M x (λ, p, Λ; ξ)] ≤ exp C(β, λ, p) + y∼x Γ xy (β, p) (4.6) + y∼x, y∈Λ c 2β|ξ(y)| p n(x)n(y)   × Ω exp   y∼x, y∈Λ 2β|ω(y)| p n(x)n(y)   π Λ (dω|ξ). In the sequel, the parameter α will be fixed. Then for a given λ, the parameter β will always be chosen in such a way that To estimate the integral in the latter line in (4.6) we use the multiple Hölder inequality (4.9) n i=1 ϕ α i i dµ ≤ n i=1 ϕ i dµ α i , in which µ is a probability measure, ϕ i ≥ 0 (respectively, α i ≥ 0), i = 1, . . . , n, are integrable functions (respectively, numbers such that n i=1 α i ≤ 1). Applying this inequality in (4.6) and taking into account (4.8) we arrive at M x (λ, p, Λ; ξ) ≤ C(β, λ, p) + y∼x Γ xy (β, p) + y∼x, y∈Λ c 2β|ξ(y)| p n(x)n(y) (4.10) + y∼x, y∈Λ 2β λn(x)n(y) M y (λ, p, Λ; ξ). As the quantity we want to estimate appears in both sides of the latter estimate, we make the following. For α ∈ (α, α], we set, cf. (2.1) and (2.11), (4.11) M (λ, p, Λ; ξ) α = x∈Λ M x (λ, p, Λ; ξ) exp[−αρ(o, x)], and obtain an upper bound for M (λ, p, Λ; ξ) α . To this end we multiply both sides of (4.10) by exp[−αρ(o, x)] and sum over x ∈ Λ. This leads us to (4.12) M (λ, p, Λ; ξ) α ≤ Υ α 1 + Υ α 2 + Υ α 3 (Λ) + Υ α 4 (Λ). Here (4.13) Υ α 1 = C(β, λ, p) x exp[−αρ(o, x)], and Υ α 2 = γ(β, p)Θ (α; r/(p − r)) ≤ γ(β, p)Θ (α; θ). (4.14) The latter estimate holds since p ≥ p 0 = r + r/θ. The term corresponding to the third summand in (4.10) is estimated as follows x∈Λ exp[−αρ(o, x)] y∼x, y∈Λ c 2β n(x)n(y) |ξ(y)| p (4.15) ≤ Υ α 3 (Λ) def = 2βe α x∈Λ c exp[−αρ(o, x)]|ξ(x)| p , which is finite whenever ξ ∈ L p (V, w α ), and tends to zero as Λ → V. In a similar way, we get x∈Λ exp[−αρ(o, x)] y∼x, y∈Λ 2β λn(x)n(y) M y (λ, p, Λ; ξ) (4.16) ≤ Υ α 4 (Λ) def = 2βe α λ M (λ, p, Λ; ξ) α . Recall that β and λ are supposed to obey (4.7). Then from the estimates obtained above we get the following (4.17) M (λ, p, Λ; ξ)] α ≤ Υ α 1 + Υ α 2 + Υ α 3 (Λ) 1 − 2βe α /λ , which yields (4.18) M x (λ, p, Λ; ξ) ≤ C(λ, p, x, ξ), for some C(λ, p, x, ξ) > 0, which is independent of Λ, but obviously depends on the choice of the root o. 4.2. Weak compactness of the local Gibbs specification. The result just obtained allows us to prove the next statement, crucial for establishing the relative weak compactness of the family {π Λ (·|ξ)} Λ⋐V and the corresponding integrability estimates. Lemma 4.2. Let p ∈ [p 0 , q) and α ∈ (α, α] be fixed. Then for every λ > 0 and ξ ∈ L p (V, w α ), one finds a positive constant C(p, α; λ, ξ), such that for all Λ ⋐ V, (4.19) Ω exp λ ω p p,α π Λ (dω|ξ) ≤ C(p, α; λ, ξ). Furthermore, for the same λ, one finds a positive constant C(p, α; λ), such that for all ξ ∈ L p (V, w α ), (4.20) lim sup Λ→V Ω exp λ ω p p,α π Λ (dω|ξ) ≤ C(p, α; λ). Proof. By (2.7) and (2.11), for any δ > 0, we have Ω exp λ ω p p,α π Λ (dω|ξ) = exp λ x∈Λ c |ξ(x)| p w α (x) (4.21) × Ω x∈Λ [exp (δ|ω(x)| p )] λwα(x)/δ π Λ (dω|ξ). Now we pick δ, such that λ δ x∈Λ w α (x) ≤ 1, and apply in (4.21) the Hölder inequality (4.9). This yields, see (4.5) and (4.11), Ω exp λ ω p p,α π Λ (dω|ξ) ≤ exp λ x∈Λ c |ξ(x)| p w α (x) (4.22) × exp [(λ/δ) M (δ, p, Λ; ξ) α ] . By (4.17) the set {RHS(4.22)(Λ)|Λ ⋐ V} is bounded for every fixed ξ ∈ L p (V, w α ). We denote its upper bound by C(p, α; λ, ξ) and obtain (4.19). The estimate (4.20) follows from (4.22) by (4.15 ), (4.17), and the fact that ξ ∈ L p (V, w α ). Proof. For obvious reasons, the balls {ω | ω p,α ≤ R}, R > 0, are compact in Ω for any fixed α ∈ (α, α] and p ∈ [p 0 , q). Thus, the proof follows from (4.19) by Prokhorov's theorem. We recall that C b (Ω) stands for the set of bounded continuous functions f : Ω → R. To prove that the accumulation points of the family {π Λ (·|ξ)} Λ⋐V ⊂ P(Ω) are Gibbs measures, we use the fact that this family possesses the following (Feller) property. For a fixed Λ ⋐ V, we consider (4.23) C b (Ω) ∋ f → π Λ (f |·) def = Ω f (ω)π Λ (dω|·). Lemma 4.4. For every Λ ⋐ V, (4.23) maps C b (Ω) into itself. The proof of this lemma is quite standard. The boundedness of π Λ (f |·) is immediate. Its continuity follows from the continuity of W , see Assumption 2.1, and the estimates (4.17) and (4.9) by Lebesgue's dominated convergence theorem. For more details, we refer the reader to the proof of the corresponding lemma in (15). Proof. For every Λ ⋐ V and ξ ∈ L p (V, w α ), by (2.7) each π Λ (·|ξ) is supported by the set {ω = ω Λ × ξ Λ c | ω Λ ∈ Ω Λ }, which yields (4.24) π Λ [L p (V, w α )|ξ] = 1. Let us fix some ξ ∈ L p (V, w α ). By Corollary 4.3 there exists an increasing sequence {Λ n } n∈N , which exhausts V, such that {π Λn (·|ξ)} n∈N weakly converges to a certain µ ∈ P(Ω). Let us show that this µ also solves the DLR equation. For any Λ, one finds n ′ ∈ N, such that Λ ⊂ Λ n for all n ≥ n ′ . For such n and f ∈ C b (Ω), by (2.8) we have (4.25) Ω π Λ (f |ω)π Λn (dω|ξ) = π Λn (f |ξ). Then we pass here to the limit n → +∞ and obtain that µ ∈ G, see (2.10) and Lemma 4.4. To prove that µ is supported by L p (V, w α ) we show that this measure obeys the estimate (3.8). For λ > 0, we set (4.26) F N (ω) = exp λ min ω p p,α ; N , N ∈ N, which is a lower semi-continuous function on Ω. Then by (4.20) and the weak convergence π Λn (·|ξ) → µ, we have Ω F N (ω)µ(dω) ≤ lim n→+∞ Ω F N (ω)π Λn (dω|ξ) (4.27) ≤ C(p, α; λ), where the latter constant is the same as in (4.20). Thereafter, the proof of (3.8), with the same constant, follows by B. Levi's monotone convergence theorem. Hence, µ ∈ G p,α . Proof of Theorem 3.1. Just above we have proven that the accumulation points of the family {π Λ (·|ξ)}, ξ ∈ L p (V, w α ), obey (4.20). Let us extend this to all µ ∈ G p,α . For such µ, by (2.9), Fatou's lemma, and the estimate (4.20) we get Ω F N (ω)µ(dω) = lim sup Λ→V Ω Ω F N (ω)π Λ (dω|ξ) µ(dξ) ≤ Ω lim sup Λ→V Ω F N (ω)π Λ (dω|ξ) µ(dξ) ≤ Ω lim sup Λ→V Ω exp λ ω p p,α π Λ (dω|ξ) µ(dξ) ≤ C(p, α; λ). Then we again apply B. Levi's theorem and obtain (3.8). The proof of (3.7) follows by (4.18) along the same line of arguments. In view of (3.8), by Prokhorov's theorem the set G p,α is relatively weakly compact. Clearly, all its accumulation points solve the DLR equation (2.10); hence, G p,α is weakly compact. Repulsive graphs In the remaining part of the paper, we present a family of unbounded degree graphs, which obey the estimate (2.3). A crucial property of such graphs is that vertices of large degree are located at large distances from each other. 5.1. The family of graphs and the main statement. For n * ∈ N, we set No restrictions are imposed on ρ(x, y) if either x or y belongs to V * . (5.1) V * = {x ∈ V | n(x) ≤ n * }, V c * = V \ V * . Let us make some comments. For a given x ∈ V c * , for K(x) def = {y ∈ V | ρ(y, x) < φ[n(x)]}, by (5.2) one has that K(x)∩V c * = {x}, i.e., such x 'repels' all vertices y ∈ V c * from the ball K(x). For the sake of convenience, we shall assume that K(x) contains the neighborhood of x, which is equivalent to assuming that (5.4) φ(n * + 1) > 1. The graphs introduced and studied in (3) were defined by a condition, which can be written in the form, cf. eqs. (3.8) and (3.9) in (3), (5.5) ρ(x, y) ≥ φ[m(x, y)], m(x, y) def = min{n(x); n(y)}. In this case, a vertex x 'repels' from the ball {y|ρ(y, x) < φ[n(x)]} only those y's, for which n(y) ≥ n(x). We employ (5.2) rather than (5.5) in view of its application in Lemma 5.4 below, see Remark 5.5 for further comments. The concrete choice of the function φ in Theorem 5.2 is discussed in Remark 5.8 below. Theorem 5.2. Let G be in G ∈ G(n * , φ) with φ having the form (5.6) φ(b) = υ log b [log log b] 1+ε , υ, ε > 0, b ≥ n * + 1, where υ and ε are such that (5.4) holds. Then for any θ > 0, there exists α ≥ 0, which may depend on θ, n * , υ, and ε, such that Θ (α, θ) < ∞ whenever α > α. The proof of Theorem 5.2 is given at the very end of this subsection. It is preceded by and based on Lemmas 5.3 and 5.4, which in turn are proven in the remaining part of the paper. For N ∈ N and x ∈ V, we set S(N, x) = {y ∈ V | ρ(x, y) = N }, (5.7) B(N, x) = {y ∈ V | ρ(x, y) ≤ N },T x (α, θ) ≤ Nx N =0 exp(−αN )   y∈S(N,x) [n(y)] 1+θ   + ∞ N =Nx+1 exp[−(α − a)N ]. Thus, the proof of the theorem follows by (5.9) with α = a. 5.2. A property of the balls in repulsive graphs. The proof of Lemma 5.4 is based on a property of the balls B(N, x) in the graphs G ∈ G(n * , φ), due to which one can control the growth of the maximum degree of y ∈ B(N, x). Here we do not suppose that φ has the concrete form of (5.6). Lemma 5.6. Let G = (V, E) be in G(n * , φ) with an arbitrary increasing function φ : (n * , +∞) → (1, +∞). Then, for every x ∈ V, there exists N x ∈ N, such that (5.11) max y∈B(N,x) n(y) ≤ φ −1 (2N ), whenever N ≥ N x . Proof. Given x, letx be the vertex in V c * which is closest to x, see (5.1). If there are more than one such vertices at the same distance, we take the one with the highest degree. For thisx, we have the following possibilities: (i) ρ(x,x) ≥ φ[n(x)]/2; (ii) ρ(x,x) < φ[n(x)]/2. The latter one includes also the casex = x, i.e., where x itself is in V c * . In case (i), we set N x = 1, which means that (5.11) holds for all N ∈ N. Indeed, if N < ρ(x,x), then the ball B(N, x) contains only vertices y ∈ V * , for which n(y) ≤ n * ≤ φ −1 (2N ) for any N ∈ N. If N ≥ ρ(x,x) and max y∈B(N,x) n(y) = n(x), one has N ≥ ρ(x,x) ≥ φ[n(x)]/2, which yields (5.11) also for this case. Finally, let max y∈B(N,x) n(y) = n(z) for some z =x, which means that n(z) > n(x). In this case, by (5.2) we have ρ(x, z) ≥ φ[(n(z)], and (5.12) 2N ≥ ρ(x, z) + ρ(x,x) ≥ ρ(x, z) ≥ φ[(n(z)], which yields (5.11) for this case as well. If (ii) holds, we let x 1 be the closest vertex to x, such that n(x 1 ) > n(x). Again, we take that of the highest degree if there are more than one such vertices. By (5.2) we have ρ(x, x 1 ) ≥ φ[n(x 1 )]. If for N ≥ N x def = ρ(x, x 1 ), one has max y∈B(N,x) n(y) = n(x 1 ), then N ≥ ρ(x, x 1 ) ≥ φ[n(x 1 )] − ρ(x,x) ≥ φ[n(x 1 )] − φ[n(x)]/2 ≥ φ[n(x 1 )]/2, which yields (5.11). Finally, let max y∈B(N,x) n(y) = n(z) for some z = x 1 , which means that n(z) > n(x 1 ). In this case, ρ(x 1 , z) ≥ φ[(n(z)], and we obtain (5.11) by applying (5.12) withx replaced by x 1 . 5.3. Proof of Lemmas 5.3 and 5.4. First we prove an auxiliary statement. Recall that by ϑ(x, y) we denote a path with endpoints x and y. A path is called simple if none of its inner vertices are repeated. For m ≤ n, let ϑ ′ = {x 0 , . . . , x m } and ϑ = {y 0 , . . . , y n } be such that x 0 = y k , x 1 = y k+1 , . . . , x m = y k+m for some k = 0, . . . , n − m. Then we say that ϑ ′ is a subpath of ϑ, and write ϑ ′ ⊂ ϑ. For a path ϑ, by V ϑ we denote the set of all its vertices. Let Σ N (x) denote the family of all simple paths of length N originated at x. Then, for every y ∈ S(N, x), there exists ϑ ∈ Σ N (x), such that ϑ = ϑ(x, y). We use this fact for estimating the cardinality of S(N, x). Proposition 5.7 (cf. Assertion 6 of (3)). In any graph G, for any x ∈ V and N ∈ N, one has (5.13) |S(N, x)| ≤ |Σ N (x)| ≤ max ϑ∈Σ N (x) y∈V ϑ \{x N } n(y). Proof. The proof will be done by induction in N . For N = 1, the estimate (5.13) is obvious. For any N ≥ 2, we have (5.14) |Σ N (x)| ≤ n(x) max y∼x |Σ x N −1 (y)|, where Σ x N −1 (y) is the corresponding family of paths in the graph which one obtains from G be deleting the edge x, y . Every ϑ ∈ Σ N (x) can be written in the form ϑ = {xθ} withθ ∈ Σ x N −1 (y) for some y ∼ x. Then by the inductive assumption we have |Σ N (x)| ≤ n(x) max y∼x max ϑ∈Σ x N−1 (y) z∈Vθ\{x N } n(z) ≤ max ϑ∈Σ N (x) z∈V ϑ \{x N } n(z), that completes the proof. Proof on Lemma 5.4. We are going to prove that the estimate (5.10) holds with N x being as in Lemma 5.6 and a given by a = (1 + θ)σ + log n * + 2e υ where σ is as in (5.15). If x is as in the case (i) considered in the proof of Lemma 5.6, and N < φ(n * + 1)/2, then V ϑ ⊂ V * for any ϑ ∈ Σ N (x). In this case, the second summand in {·} in (5.16) does not exceed N log n * , which certainly yields (5.10). To handle the case of N ≥ φ(n * +1)/2 we use the sequence {c k } k∈N , where c k = exp(e k ), k ∈ N. Let k * be the least k ∈ N such that c k * +1 ≥ n * + 1. Then we set b k * = n * + 1 and b k = c k for k > k * . Let k N be the largest k, such that b k ≤ φ −1 (2N ). For k = k * , . . . , k N and a given ϑ ∈ Σ N (x), let m ϑ k be the number of vertices y ∈ V ϑ , such that n(y) ∈ [b k , b k+1 ]. Given τ ∈ (0, N ), for any ϑ ∈ Σ N (x), the number of vertices in V ϑ which are away from each other at distance at least τ is 1 + N/τ , at most. Therefore, m ϑ k ≤ m k def = 1 + N/φ(b k ) ≤ 2N/φ(b k ). Taking this into account by (5.6) we get max ϑ∈Σ N (x) z∈V ϑ \{x N } log n(z) ≤ N log n * + k N k=k * m k log b k+1 ≤ N   log n * + 2e υ ∞ k=k * 1 k 1+ε   . Applying (5.17) and the latter estimate in (5.16) we obtain (5.10) also in this case. Remark 5.8. Our choice of φ made in (5.6) was predetermined by the condition (5.17), which we used to estimate the first summand in {·} in (5.16), as well as by the following one (5.18) ∞ k=k * log b k+1 φ(b k ) < ∞, which was employed for estimating the second summand in (5.16), for a concrete choice of the sequence {b k } k≥k * made therein. In principle, any φ obeying such two conditions (for some choice of {b k } k≥k * ) can be used. For b k = k, k ≥ k * = n * + 1, one can take φ(b) = b 1+ε for some ε > 0, which obviously obeys (5.17) and (5.18) but imposes a stronger repulsion, see (5.2). Our choice (5.6) seems to be optimal. Proof on Lemma 5.3. In view of (5.4), we have that ρ(x, y) ≥ 2 for any x, y ∈ V c * ; hence, for two adjacent vertices, at least one should be in V * . Taking this into account by (2.2) and the triangle inequality we derive Θ (α, θ) = which yields (5.9), see (5.8). 2 . 2The setup and the basic theorem 2.1. The model. The underlying graph G = (V, E) of the model (1.1) Lemma 4 . 1 .Γ 41For every λ > 0, p ∈ [p 0 , q), x ∈ V, and ξ ∈ Ω, the following estimate holds Ω exp (λ|ω(x)| p ) π x (dω|ξ) xy (β, p) . λn(x)n(y) ≤ 1. Corollary 4. 3 . 3For every ξ ∈ L p (V, w α ), the family {π Λ (·|ξ)} Λ⋐V ⊂ P(Ω) is relatively weakly compact. Corollary 4 . 5 . 45For every p ∈ [p 0 , q) and α ∈ (α, α], the set G p,α is non-void. Definition 5 . 1 . 51For an integer n * > 2 and a strictly increasing function φ : (n * , +∞) → (0, +∞), the family G(n * , φ) consists of those graphs G = (V, E), for which the path distance obeys the condition(5.2) ∀x, y ∈ V c * : ρ(x, y) ≥ φ[n(x, y)], where (5.3)n(x, y) = max{n(x); n(y)}. . 3 . 3Let G be in G(n * , φ) with φ obeying(5.4). Then for every positive θ and α,(5.9) Θ (α, θ) ≤ n θ * (e α + 1)T o (α, θ). − 1 ( 12N ) ≤ exp(σN ), y)] 1+θ exp[−αρ(o, y)] + n θ * x [n(x)] 1+θ exp[−αρ(o, x)], AcknowledgementsThe authors are grateful to Philippe Blanchard and Michael Röckner for valuable discussions and encouragement. S Albeverio, Y Kondratiev, Y Kozitsky, M Röckner, Statistical Mechanics of Quantum Crystals. A Path Integral Approach. to appearAlbeverio, S., Kondratiev, Y., Kozitsky, Y. and Röckner, M. (2009). Statistical Mechanics of Quantum Crystals. A Path Integral Ap- proach. (to appear) Construction of convergent simplicial approximation of quantum fields on Riemannian manifolds. S Albeverio, B Zegarliński, Comm. Math. Phys. 132MR 1069200 (92c:81119Albeverio, S. and Zegarliński, B. (1990). Construction of conver- gent simplicial approximation of quantum fields on Riemannian mani- folds. Comm. Math. Phys. 132 39-71. MR 1069200 (92c:81119) Uniqueness of a Gibbs field with a random potential-an elementary approach. L A Bassalygo, R L Dobrushin, Theory Probab. Appl. 31Bassalygo, L. A. and Dobrushin, R. L. (1986). Uniqueness of a Gibbs field with a random potential-an elementary approach. Theory Probab. Appl. 31 572-589; . Teor, Veroyatnost, Primenen, MR 0881577 (88i:6016031RussianTeor. Veroyatnost. i Primenen. 31 651-670 (Russian). MR 0881577 (88i:60160) Random walks on graphs: ideas, techniques and results. R Burioni, D Cassi, R45-R78. MR 2119174J. Phys. A: Math. Gen. 3882059Burioni, R. and Cassi, D. (2005). Random walks on graphs: ideas, techniques and results. J. Phys. A: Math. Gen. 38 R45-R78. MR 2119174 (2006b:82059) Random walks and physical models on infinite graphs: an introduction. R Burioni, D Cassi, A Vezzani, Berlin. MR 2087779Random walks and geometry. Walter de Gruyter GmbH & Co. KG60019Burioni, R., Cassi, D., and Vezzani, A. (2004). Random walks and physical models on infinite graphs: an introduction. In Random walks and geometry, ed. V. A. Kaimanovich, 35-71, Walter de Gruyter GmbH & Co. KG, Berlin. MR 2087779 (2005h:60019) /78). Existence and uniqueness of DLR measures for unbounded spin systems. M Cassandro, E Olivieri, A Pellegrinotti, E Presutti, MR 0471115Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 4110854Cassandro, M., Olivieri, E., Pellegrinotti, A., and Presutti, E. (1977/78). Existence and uniqueness of DLR measures for unbounded spin systems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 313- 334. MR 0471115 (57 #10854) The exponent in the general Randić index. L Clark, I Gutman, MR 2449405J. Math. Chem. 43Clark, L. and Gutman, I. (2008). The exponent in the general Randić index. J. Math. Chem. 43 32-44. MR 2449405 Random fields on Riemannian manifolds: a constructive approach. G F De Angelis, D De Falco, Di Genova, G , MR 0826866Comm. Math. Phys. 10381063De Angelis, G. F., de Falco, D., and Di Genova, G. (1986). Ran- dom fields on Riemannian manifolds: a constructive approach. Comm. Math. Phys. 103 297-303. MR 0826866 (87g:81063) Eigenvalues of Laplacians on a closed Riemannian manifold and its nets. K Fujiwara, Proc. Amer. Math. Soc. 123MR 1257106 (95j:58171Fujiwara, K. (1995). Eigenvalues of Laplacians on a closed Riemann- ian manifold and its nets. Proc. Amer. Math. Soc. 123 2585-2594. MR 1257106 (95j:58171) Gibbs Measures and Phase Transitions. de Gruyer. H.-O Georgii, New YorkMR 0956646 (89k:82010Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyer, New York. MR 0956646 (89k:82010) The random geometry of equilibrium phases. H.-O Georgii, O Häggström, C Maes, Phase transitions and critical phenomena. C. Domb and J. L. LebowitzSan Diego, CAAcademic Press1882022Georgii, H.-O., Häggström, O. and Maes, C. (2001) The ran- dom geometry of equilibrium phases. In Phase transitions and critical phenomena, eds. C. Domb and J. L. Lebowitz, Vol. 18, 1-142, Academic Press, San Diego, CA. 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Y Kozitsky, T Pasurek, MR 2317266. 12782009Kozitsky, Y. and Pasurek, T. (2007). Euclidean Gibbs measures of interacting quantum anharmonic oscillators. J. Stat. Phys. 127 985- 1047. MR 2317266 (2008d:82009) Statistical mechanics of systems of unbounded spins. J L Lebowitz, E Presutti, MR 0446251Comm. Math. Phys. 504579Lebowitz, J. L. and Presutti, E. (1976). Statistical mechan- ics of systems of unbounded spins. Comm. Math. Phys. 50 195-218. MR 0446251 (56 #4579) Phase transitions on nonamenable graphs. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. R Lyons, J. Math. Phys. 4182028MRLyons, R. (2000). Phase transitions on nonamenable graphs. Proba- bilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 1099-1126. MR 1757952 (2001c:82028) The Ising model and percolation on trees and treelike graphs. R Lyons, Comm. Math. Phys. 125MR 1016874 (90h:82046Lyons, R. (1989). The Ising model and percolation on trees and tree- like graphs. Comm. Math. Phys. 125 337-353. MR 1016874 (90h:82046) Theory of Gibbs measures with unbounded spins: probabilistic and analytic aspects. T Pasurek, Universität BielefeldHabilitation Thesisavailable as SFB 701 Preprint 08-101Pasurek, T. (2007). Theory of Gibbs measures with unbounded spins: probabilistic and analytic aspects. Habilitation Thesis, Universität Biele- feld, available as SFB 701 Preprint 08-101, 2008. The P (φ) 2 Euclidean (quantum) field theory. Princeton Series in Physics. B Simon, Princeton University Press04895528968Princeton, N.J. MRSimon, B. (1974). The P (φ) 2 Euclidean (quantum) field theory. Prince- ton Series in Physics. Princeton University Press, Princeton, N.J. MR 0489552 (58 #8968) Quantum field theory in curved spacetime and black hole thermodynamics. Chicago Lectures in Physics. R M Wald, University of Chicago PressChicago, ILMR 1302174 (95i:81176Wald, R. M. (1994). Quantum field theory in curved spacetime and black hole thermodynamics. Chicago Lectures in Physics. University of Chicago Press, Chicago, IL. MR 1302174 (95i:81176) Combinatorial criteria for uniqueness of Gibbs measures. D Weitz, MR 2178257. 2782036Weitz, D. (2005). Combinatorial criteria for uniqueness of Gibbs mea- sures. Rand. Struct. Algorithms 27 445-475. MR 2178257 (2006k:82036) Germany E-mail address: kondrat@math.uni-bielefeld. Mathematik Fakultät Für, de33615Universität BielefeldFakultät für Mathematik, Universität Bielefeld, D 33615, Germany E-mail address: kondrat@math.uni-bielefeld.de . , Instytut Matematyki, Uniwersytet Marii Curie-Sk Lodowskiej, LublinPoland E-mail address: jkozi@hektor.umcs.lublin.plInstytut Matematyki,, Uniwersytet Marii Curie-Sk lodowskiej, 20-031 Lublin, Poland E-mail address: jkozi@hektor.umcs.lublin.pl Germany E-mail address: tpasurek@@math.uni-bielefeld. Mathematik Fakultät Für, de33615Universität BielefeldFakultät für Mathematik, Universität Bielefeld, D 33615, Germany E-mail address: tpasurek@@math.uni-bielefeld.de

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{'abstract': 'Gibbs random fields corresponding to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs with a certain summability property are constructed. It is proven that the set of tempered Gibbs random fields is non-void and weakly compact, and that they obey uniform exponential integrability estimates. In the second part of the paper, a class of graphs is described in which the mentioned summability is obtained as a consequence of a property, by virtue of which vertices of large degree are located at large distances from each other. The latter is a stronger version of a metric property, introduced in [Bassalygo, L. A. and Dobrushin, R. L. (1986). Uniqueness of a Gibbs field with a random potential-an elementary approach. Theory Probab. Appl. 31 572-589].', 'arxivid': '0904.3207', 'author': ['Yuri Kondratiev ', 'ANDYuri Kozitsky ', 'Tanja Pasurek '], 'authoraffiliation': [], 'corpusid': 16539572, 'doi': '10.1239/jap/1285335414', 'github_urls': [], 'n_tokens_mistral': 17480, 'n_tokens_neox': 15442, 'n_words': 8416, 'pdfsha': 'de1e686cd4366673ec80eeb462ecbe821f7bd9f8', 'pdfurls': ['https://arxiv.org/pdf/0904.3207v1.pdf'], 'title': ['GIBBS RANDOM FIELDS WITH UNBOUNDED SPINS ON UNBOUNDED DEGREE GRAPHS', 'GIBBS RANDOM FIELDS WITH UNBOUNDED SPINS ON UNBOUNDED DEGREE GRAPHS'], 'venue': []}

arxiv

Thesis Report: Resource Utilization Provisioning in MapReduce Hamidreza Barati Department of IT and Computer Engineering Payame Noor University Nasrin Jaberi nasrinjaberi60@yahoo.com Department of IT and Computer Engineering Payame Noor University Thesis Report: Resource Utilization Provisioning in MapReduce In this thesis report, we have a survey on state-of-the-art methods for modelling resource utilization of MapReduce applications regard to its configuration parameters. After implementation of one of the algorithms in literature, we tried to find that if CPU usage modelling of a MapReduce application can be used to predict CPU usage of another MapReduce application. Introduction MapReduce, introduced by Google in [1], has been known as a large-scale data processing technique indicating that both resource utilization and execution time are the most critical aspects of running a MapReduce application. As a result, provisioning resource usage and execution time of a MapReduce application becomes important before actual running of the application to assign proper time and resources. Also, MapReduce has been known as a parametric model which means severalparameters on cluster must be tuned before running an application. Generally, these parameters are assigned default values but researches in [2][3] show that the proper tuning of the parameters can make a MapReduce application run faster or with less resource utilization. Also, these researches indicate that the proper values of these parameters are application-dependent meaning that these values from one application to another application may change. Among parameters influencing the performance of MapReduce cluster, in this study we will focus on the influences of four major parameters: size of input file, size of blocks, number of mappers and number of reducers. Early works on modeling performance of MapReduce application was proposed in [4] for modeling the total execution time of Hadoop Hive queries which is based on using Kernel Canonical Correlation Analysis to obtain correlation between the performance feature vectors extracted from Map-Reduce job logs, and map time, reduce time, and total execution time. These features were acknowledged as critical characteristics for establishing any scheduling decisions. Another MapReduce modeling of computation utilizations was presented in [5][6]. After modeling each map and reduce phases independently by using dynamic linear programming, these modellings are combined to establish a global optimal strategy for MapReduce scheduling and resource allocation. In In [2,7], linear regression were used to model execution time and the total number of CPU tick clocks an application, respectively, regard to a few MapReduce parameters. After following the same modeling concept in this report, we are going to study if two applications may have the same model or not. In another word, (1) if prediction model of an application can be applied to predict the performance of another model or not and (2) if yes, which characteristics these applications should have? The Resource Utilization Modelling in MapReduce The primary idea, originally coming from [2,[7][8][9], is to model the computation cost Fig.1 shows the real CPU usage and the output of model for the experiments. Fig.2 is error between these two graphs (The error mean is 3.57%). One question, which is the motivation behind this thesis, is that if the prediction model of an application is applicable for other applications? In which conditions two applications can be predicted with the same model? Primarily, two methods have been introduced in [10][11] to find the similarity between MapReduce applications by comparing the CPU utilization time series of these applications. The former one uses a mixture of Dynamic Time Warping and correlation analysis to find this similarity while the latter one uses statistical analysis to calculate minimum distance between applications' time series. These approaches conclude that if two applications have similar CPU patterns for several experiments with different values of parameters, it is more likely that the optimal values of configuration parameters for a particular application can be applied to optimally run another application. However, these methods do not study if prediction model can be shared between two similar applications or not. To show the problem more clear, we have examined another application (Exim Mainlog parsing) and calculated the difference between the actual CPU usage of this application and the prediction model of WordCount application. As can be seen from Fig.3 and 4, WordCount model can also predict Exaim Mainlog CPU usage with a good accuracy. ) _ ( ) _ ( ) (Re ) ( ) _ ( ) _ ( ) (Re ) ( _ Size IN Size FS duce Map Size IN Size FS duce Map Model CPU                   (or CPU usage) of applications in Hadoop, pseudo-distributed mode. The general idea is to run an application (e.g.WordCount) for different values of 1) the number of Mappers, 2) the number of reducers, 3) The size of File System and 4) The size of input data. During running of the application for specific values of parameters, the real values of CPU usage are extracted from system (in Linux-Ubuntu). After 100 times running of this application for different values of these parameters (the values are chosen randomly in valid ranges), the relation between CPU usage and the four parameters is modelled by polynomial regression. The model is Fig. 1 1Fig.1 Fig . 2 2Fig .2 Fig.2 Fig .2 Fig.2 Fig Fig.4 MapReduce: Simplified Data Processing on Large Clusters. J Dean, S Ghemawat, presented at the 6th Symposium on Operating Systems Design and Implementation (OSDI). San Francisco, CAJ. Dean and S. Ghemawat, "MapReduce: Simplified Data Processing on Large Clusters," presented at the 6th Symposium on Operating Systems Design and Implementation (OSDI), San Francisco, CA, 2004. On Modeling Dependency between MapReduce Configuration Parameters and Total Execution Time. N B Rizvandi, CoRRN. B. Rizvandi, et al., "On Modeling Dependency between MapReduce Configuration Parameters and Total Execution Time," CoRR, 2011. Towards automatic optimization of MapReduce programs. S Babu, presented at the 1st ACM symposium on Cloud computing. Indianapolis, Indiana, USAS. Babu, "Towards automatic optimization of MapReduce programs," presented at the 1st ACM symposium on Cloud computing, Indianapolis, Indiana, USA, 2010. Statistics-Driven Workload Modeling for the Cloud. A S Ganapathi, No. UCB/EECS-2009-160University of California at BerkeleyTechnical ReportA. S. Ganapathi, et al., "Statistics-Driven Workload Modeling for the Cloud," University of California at Berkeley,Technical Report No. UCB/EECS-2009-160, 2009. Brief Announcement: Modelling MapReduce for Optimal Execution in the Cloud. A Wieder, presented at the Proceeding of the 29th ACM SIGACT-SIGOPS symposium on Principles of distributed computing. Zurich, SwitzerlandA. Wieder, et al., "Brief Announcement: Modelling MapReduce for Optimal Execution in the Cloud," presented at the Proceeding of the 29th ACM SIGACT-SIGOPS symposium on Principles of distributed computing, Zurich, Switzerland, 2010. Conductor: orchestrating the clouds," presented at the 4th International Workshop on Large Scale Distributed Systems and Middleware. A Wieder, Zurich, SwitzerlandA. Wieder, et al., "Conductor: orchestrating the clouds," presented at the 4th International Workshop on Large Scale Distributed Systems and Middleware, Zurich, Switzerland, 2010. Preliminary Results on Modeling CPU Utilization of MapReduce Programs. N B Rizvandi, University of SydneySchool of Information TechnologiesN. B. Rizvandi, et al., "Preliminary Results on Modeling CPU Utilization of MapReduce Programs," School of Information Technologies, University of Sydney, Sydney2010. On Modeling CPU Utilization of MapReduce Applications. N B Rizvandi, CoRRN. B. Rizvandi, et al., "On Modeling CPU Utilization of MapReduce Applications," CoRR, 2012. Preliminary Results: Modeling Relation between Total Execution Time of MapReduce Applications and Number of Mappers/Reducers. N B Rizvandi, University of Sydney2011N. B. Rizvandi, et al., "Preliminary Results: Modeling Relation between Total Execution Time of MapReduce Applications and Number of Mappers/Reducers," University of Sydney2011. On using Pattern Matching Algorithms in MapReduce Applications. N B Rizvandi, The 9th IEEE International Symposium on Parallel and Distributed Processing with Applications (ISPA). Busan, South KoreaN. B. Rizvandi, et al., "On using Pattern Matching Algorithms in MapReduce Applications," presented at the The 9th IEEE International Symposium on Parallel and Distributed Processing with Applications (ISPA), Busan, South Korea, 2011. A Study on Using Uncertain Time Series Matching Algorithms in Map-Reduce Applications. N B Rizvandi, CoRRN. B. Rizvandi, et al., "A Study on Using Uncertain Time Series Matching Algorithms in Map-Reduce Applications," CoRR, 2011.

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{'abstract': 'In this thesis report, we have a survey on state-of-the-art methods for modelling resource utilization of MapReduce applications regard to its configuration parameters. After implementation of one of the algorithms in literature, we tried to find that if CPU usage modelling of a MapReduce application can be used to predict CPU usage of another MapReduce application.', 'arxivid': '1203.4367', 'author': ['Hamidreza Barati \nDepartment of IT and Computer Engineering\nPayame Noor University\n\n', 'Nasrin Jaberi nasrinjaberi60@yahoo.com \nDepartment of IT and Computer Engineering\nPayame Noor University\n\n'], 'authoraffiliation': ['Department of IT and Computer Engineering\nPayame Noor University\n', 'Department of IT and Computer Engineering\nPayame Noor University\n'], 'corpusid': 6307911, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2318, 'n_tokens_neox': 2094, 'n_words': 1320, 'pdfsha': '095ecb2554090c724fa6116345d9bd5275f358c1', 'pdfurls': ['https://arxiv.org/pdf/1203.4367v1.pdf'], 'title': ['Thesis Report: Resource Utilization Provisioning in MapReduce', 'Thesis Report: Resource Utilization Provisioning in MapReduce'], 'venue': []}

arxiv

S-duality and the prepotential in N = 2 ⋆ theories (II): the non-simply laced algebras 22 Oct 2015 M Billó billo@to.infn.it Dipartimento di Fisica and I.N.F.N. -sezione di Torino Università di Torino Via P. Giuria 1I-10125TorinoItaly M Frau frau@to.infn.it Dipartimento di Fisica and I.N.F.N. -sezione di Torino Università di Torino Via P. Giuria 1I-10125TorinoItaly F Fucito fucito@roma2.infn.it I.N.F.N -sezione di Roma Dipartimento di Fisica Via della Ricerca Scientifica Università di Roma Tor Vergata I-00133RomaItaly A Lerda lerda@to.infn.it Dipartimento di Fisica and I.N.F.N. -sezione di Torino Università di Torino Via P. Giuria 1I-10125TorinoItaly Dipartimento di Scienze e Innovazione Tecnologica Università del Piemonte Orientale and I.N.F.N. -Gruppo Collegato di Alessandria -sezione di Torino Viale T. Michel 11I-15121AlessandriaItaly J F Morales morales@roma2.infn.it I.N.F.N -sezione di Roma Dipartimento di Fisica Via della Ricerca Scientifica Università di Roma Tor Vergata I-00133RomaItaly S-duality and the prepotential in N = 2 ⋆ theories (II): the non-simply laced algebras 22 Oct 2015Preprint typeset in JHEP style. -PAPER VERSION ROM2F/2015/8N = 2 SYM theoriesrecursion relationsinstantons We derive a modular anomaly equation satisfied by the prepotential of the N = 2 ⋆ supersymmetric theories with non-simply laced gauge algebras, including the classical B r and C r infinite series and the exceptional F 4 and G 2 cases. This equation determines the exact prepotential recursively in an expansion for small mass in terms of quasi-modular forms of the S-duality group. We also discuss the behaviour of these theories under S-duality and show that the prepotential of the SO(2r + 1) theory is mapped to that of the Sp(2r) theory and viceversa, while the exceptional F 4 and G 2 theories are mapped into themselves (up to a rotation of the roots) in analogy with what happens for the N = 4 supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper. Introduction In a companion paper [1] we have studied N = 2 ⋆ super Yang-Mills theories with gauge groups of ADE type, generalizing and extending results that were previously obtained for SU (2) and SU(N ) gauge groups [2] - [6]. The N = 2 ⋆ theories possess eight supercharges and interpolate between the N = 4 and the pure N = 2 super Yang-Mills theories. Their low-energy effective dynamics is encoded in the prepotential F which can be conveniently organized as an expansion in even powers of the mass m of the matter hypermultiplet. Given that F has mass-dimension two and that the only other dimensionful parameter available in the model is the vacuum expectation value a of the scalar field in the gauge vector multiplet, each term of the prepotential at order 2n in the mass must be accompanied by a function of a with mass-dimension (2 − 2n). These functions of a turn out to be nicely written as sums over the root lattice of the gauge algebra. Thus the prepotential always maintains the same form for all ADE algebras since what changes from case to case is only the explicit expression of the roots. The coefficients in front of these lattice sums are universal functions of the gauge coupling constant and receive both perturbative contributions at 1-loop and non-perturbative corrections due to instantons. Actually, all these contributions can be resummed into exact functions of the gauge coupling that are built out of the Eisenstein series, including the second Eisenstein series E 2 which has an "anomalous" behaviour under the modular transformations of Sl (2, Z). The presence of E 2 leads to a modular anomaly equation which can be put in the form of a recursion relation for the coefficients of the mass expansion of the prepotential and encodes all information implied by S-duality [1] 1 . In this paper we extend and generalize these results to N = 2 ⋆ theories with non-simply laced gauge algebras g ∈ {B r , C r , F 4 , G 2 }. The presence of long and short roots in these algebras implies several important differences with respect to the ADE case, even if the overall picture remains similar. In particular, the notion of S-duality, originally formulated in [8] - [13] for the N = 4 theories, can be also extended to the N = 2 ⋆ models with non-simply laced gauge algebras where the strong/weak coupling symmetry requirement takes the form of a relation between the S-dual prepotential and the Legendre transform of its dual [14,4,3]. However, differently from the ADE case, one finds that S-duality is not a true symmetry since it maps a theory with gauge algebra g to a theory with a dual gauge algebra g ∨ , obtained by exchanging (and rescaling) the long and short roots [9]. This property leads to very severe constraints on the prepotential coefficients, and it is a quite remarkable fact that they can be satisfied by imposing again a modular anomaly equation similar to that of the ADE theories. Another important difference with respect to the ADE case is that the modular group of the non-simply laced theories is not Sl(2, Z) but its congruence subgroup Γ 0 (n g ) [15,16], where n g is the ratio between the norm squared of the long and short roots of g. Consequently, the prepotential coefficients are expressed as quasi-modular forms of this subgroup, which include other modular functions besides the standard Eisenstein series. Such functions as well as the Eisenstein series have a simple behaviour also under the Sduality transformation which lies outside Γ 0 (n g ). Exploiting this fact together with the S-duality transformation properties of the root lattices mentioned above, it is possible to verify that the modular anomaly equation implies the expected strong/weak coupling relation between the prepotentials of N = 2 ⋆ theories with dual gauge algebras. The plan of the paper is the following: In Section 2 we discuss the S-duality action on the gauge theories with non-simply laced algebras and derive the modular anomaly equation and the recursion relation satisfied by the quantum prepotential. In Section 3 we present the microscopic computation of the instanton corrections for theories with gauge algebras in the classical B r and C r series using the equivariant localization methods [17] - [20]. This is necessary in order to have explicit "microscopic" data on the multi-instanton corrections which can then be used in order to prove S-duality or compared with the Sduality predictions. In Section 4 we derive exact formulas written in terms of modular forms for the first few coefficients in the mass expansion of the prepotential for theories with orthogonal or symplectic gauge algebras, while in Section 5 we repeat the analysis for the exceptional algebras G 2 and F 4 . By Taylor expanding the modular forms, one can obtain the whole series of multi-instanton corrections to the prepotential. While for the classical algebras these can be checked against the microscopic computations, for G 2 and F 4 they are a prediction since the ADHM construction is not known for the exceptional gauge algebras. Finally, in Section 6 we present our conclusions and perspectives. Several details about the root systems of the non-simply laced algebras and about the modular forms are contained in two technical appendices. S-duality In this section we investigate the S-duality transformation properties of N = 2 ⋆ theories with non-simply laced gauge algebras g ∈ {B r , C r , F 4 , G 2 }. In particular, we will show that the symmetry requirement under S-duality determines the modular behaviour of the quantum prepotential and implies the emergence of a modular anomaly equation in the form of a recursion relation for the coefficients of its mass expansion. In order to treat all cases simultaneously, we introduce the convenient notation n g = α L · α L α S · α S (2.1) where α L and α S denote, respectively, the long and the short roots of the algebra g. In Appendix A we give our conventions on the root systems, from which one can see that n g = 2 for g = B r , C r and F 4 , n g = 3 for g = G 2 . (2.2) Using this notation, we can write the prepotential for a theory with gauge algebra g as F (g) (τ, a) = n g πiτ a 2 + f (g) (τ, a) (2.3) where the first term represents the classical contribution. Here we have introduced the usual complex combination of the Yang-Mills coupling g and the θ-angle τ = θ 2π + i 4π g 2 (2.4) and denoted by a the vacuum expectation value of the scalar field ϕ in the gauge vector multiplet ϕ = a = diag(a 1 , a 2 , · · · , a r ) . (2.5) Like in the ADE case [1], also the non-simply laced quantum prepotential f (g) can be conveniently expanded in the mass m of the adjoint hypermultiplet, namely n are functions of the coupling τ through the instanton counting parameter q = e 2πiτ . On dimensional grounds, they are also homogeneous functions of degree (2 − 2n) of the vacuum expectation values a: f (g) (τ, a) = ∞ n=1 f (g) n (τ, a) ,(2.f (g) n (τ, λa) = λ 2−2n f (g) n (τ, a) . (2.7) On the contrary, f 1 is independent of τ and is entirely given by the 1-loop contribution f (g) 1 (a, Λ) = f (g),1−loop 1 (a, Λ) = m 2 4 α∈Ψ log α · a Λ 2 (2.8) where Λ is the dynamically generated scale. Since we are interested in the S-duality transformation of the prepotential (2.3), we have to define how S-duality acts on the gauge coupling τ and on the vacuum expectation values a. As discussed for example in [11] - [13], in the non-simply laced theories one has τ → S(τ ) = − 1 n g τ , (2.9) which replaces the usual τ → −1/τ transformation of the ADE models. Furthermore, S-duality maps a theory with a gauge algebra g to a theory with a gauge algebra g ∨ defined from g by exchanging (and suitably rescaling) the long and the short roots [9]. The correspondence between g and g ∨ is given in Tab. 1, where for F 4 and G 2 , the ′ in the second column means that the dual root systems are equivalent to the original ones up to a rotation. Note that according to this definition, which is simply the N = 2 version Table 1: The correspondence between a non-simply laced algebra g and its GNO dual g ∨ . g g ∨ B r C r C r B r F 4 F ′ 4 G 2 G ′ 2 of the N = 4 S-duality rule [8] - [13], the electric variables of the g theory are dual to the magnetic variables of the g ∨ theory. Moreover, using the roots given in Appendix A, from (2.8) it is easy to check that, up to a-independent terms 2 , f (g) 1 (a, Λ) = f (g ∨ ) 1 (a, Λ) . (2.10) To define how S-duality acts on a, we have first to introduce the dual variables a D . These are defined as the a-derivatives of the prepotential of the dual g ∨ -theory, namely a D = 1 2n g πi ∂F (g ∨ ) ∂a = τ a + δ 12n g ∂f (g ∨ ) ∂a (2.11) where, for later convenience, we introduce δ = 6 πiτ . (2.12) The S-duality transformation (2.9) is represented by the Sl(2, R) element [11,13] S = 0 −1/ √ n g √ n g 0 (2.13) which, when acting on the periods, exchanges a and a D ; indeed a D a → 0 −1/ √ n g √ n g 0 a D a = −a/ √ n g √ n g a D . (2.14) Thus we have S(a) = √ n g a D = √ n g τ a + δ 12n g ∂f g ∨ ∂a . (2.15) Using (2.9) and (2.15), the S-dual prepotential is therefore S F (g) ≡ F (g) (S(τ ), S(a), S(Λ)) = F (g) − 1 ngτ , √ n g τ a + δ 12ng ∂f g ∨ ∂a , S(Λ) , (2.16) where the modular transformation property of the scale Λ will be determined shortly. In analogy with the ADE case [1], one can constrain the form of the prepotential by requiring that S F (g) be the Legendre transform of the prepotential of the dual theory, namely S F (g) = L F (g ∨ ) (2.17) where L F (g ∨ ) ≡ F (g ∨ ) − a · ∂F (g ∨ ) ∂a = −n g πiτ a 2 − a · ∂f (g ∨ ) ∂a + f (g ∨ ) . (2.18) As it is clear from (2.17), S-duality is not a symmetry of the non-simply laced theories since it relates a theory with gauge algebra g to a theory with gauge algebra g ∨ or viceversa. Nevertheless, it is powerful enough to constrain the structure of the prepotential. In order to enforce (2.17), several conditions have to be satisfied. First of all, like in the ADE theories [1], we will see that also here the prepotential coefficients f (g) n (with n ≥ 2) must depend on τ through quasi-modular forms. However, differently from the ADE models, the modular group is now a subgroup of Γ = Sl(2, Z); more precisely it is the congruence subgroup Γ 0 (n g ) defined as 3 Γ 0 (n g ) = a b c d ∈ Γ : c = 0 mod n g . (2.19) If we denote byŜ = ( 0 −1 1 0 ) andT = ( 1 1 0 1 ) the generators of Γ, then Γ 0 (n g ) is generated bŷ T andŜT ngŜ . The (quasi-)modular forms of Γ 0 (n g ) are known (see for instance [15,16]; 3 Note that the S-duality transformation (2.9) lies outside this subgroup. see also [21] for a catalog and Appendix B of [22] for a nice compendium). They form a ring generated by the basic elements E 2 (τ ), H 2 (τ ), E 4 (τ ), E 6 (τ ) for n g = 2 , E 2 (τ ), K 2 (τ ), E 4 (τ ), E 6 (τ ) for n g = 3 ,(2.20) where E 2 , E 4 and E 6 are the Eisenstein series of weight 2, 4 and 6 respectively, and H 2 and K 2 are modular forms of weight 2 defined by H 2 (τ ) = 1 2 θ 4 3 (τ ) + θ 4 4 (τ ) = 1 + 24q + 24q 2 + 96q 3 + 24q 4 + 144q 5 + · · · , K 2 (τ ) = η 3 (τ ) η(3τ ) 3 + 3η 3 (3τ ) η(τ ) 3 2 3 = 1 + 12q + 36q 2 + 12q 3 + 84q 4 + 72q 5 + · · · , (2.21) where the θ's are the Jacobi θ-functions and η is the Dedekind η-function. We refer to Appendix B for a summary of the main properties of these modular functions and the Eisenstein series. Here we simply recall that all elements of the basis (2.20) are modular forms of Γ 0 (n g ), except for E 2 which is quasi-modular. Also under the transformation (2.9), which does not belong to Γ 0 (n g ), all basis elements transform covariantly up to a factor of √ n g τ w where w is their modular weight, except for E 2 which behaves as follows E 2 − 1 ngτ =        2τ 2 E 2 (τ ) + δ + H 2 (τ ) for n g = 2 , 3τ 2 E 2 (τ ) + δ + 2K 2 (τ ) for n g = 3 , (2.22) where δ is given in (2.12). This implies that under S-duality, up to the prefactors of √ n g τ 2 and the modular forms H 2 or K 2 , any occurrence of E 2 is replaced by E 2 + δ. Since this "anomalous" shift plays a crucial rôle in the following, we will explicitly exhibit the E 2 -dependence of the prepotential coefficients f n by writing, for n ≥ 2, f (g) n τ, a, E 2 ) . (2.23) On the other hand we leave implicit the dependence on the other modular forms to avoid clutter in the formulas. Guided by the experience with the ADE theories [1], after some simple algebra one can realize that in order to enforce (2.17) it is necessary that the coefficients f n behave as f (g) n − 1 ngτ , a, E 2 (− 1 ngτ ) = √ n g τ 2n−2 f (g ∨ ) n τ, a, E 2 + δ (2.24) for n ≥ 2. The prefactor ( √ n g τ ) 2n−2 in the right hand side is precisely the one that the (quasi-)modular forms of Γ 0 (n g ) of weight (2n − 2) acquire under the S-duality transformation (2.9). Thus, we conclude that f (g) n must be a (quasi-)modular form of Γ 0 (n g ) with weight (2n − 2). Thanks to the homogeneity property (2.7), it is possible to rewrite (2.24) as f (g) n − 1 ngτ , √ n g τ a, E 2 (− 1 ngτ ) = f (g ∨ ) n τ, a, E 2 + δ . (2.25) For n = 1, instead, we simply have to require that f (g) 1 ( √ n g τ a, S(Λ)) = f (g ∨ ) 1 (a, Λ) ,(2.26) which, in view of (2.10), implies S(Λ) = √ n g τ Λ . (2.27) Eq.s (2.25) and (2.26) can be combined together into f (g) − 1 ngτ , √ n g τ a, E 2 (− 1 ngτ ), √ n g τ Λ = f (g ∨ ) τ, a, E 2 + δ, Λ . (2.28) We are now in the position of computing the S-dual prepotential. We have S F (g) = F (g) − 1 ngτ , √ n g τ a + δ 12ng ∂f (g ∨ ) ∂a , E 2 (− 1 ngτ ), √ n g τ Λ = −n g πiτ a 2 − a · ∂f (g ∨ ) ∂a − δ 24n g ∂f (g ∨ ) ∂a · ∂f (g ∨ ) ∂a + f (g ∨ ) τ, a + δ 12ng ∂f (g ∨ ) ∂a , E 2 + δ, Λ . (2.29) The second line is the S-dual of the classical prepotential, while the third line gives the S-dual of the quantum prepotential f (g) which has been expressed in terms of f (g ∨ ) using (2.28). Taylor expanding the right hand side (2.29) with respect to δ, we get S F (g) = −n g πiτ a 2 − a · ∂f (g ∨ ) ∂a + f (g ∨ ) τ, a, E 2 , Λ +δ ∂f (g ∨ ) ∂E 2 + 1 24 n g ∂f (g ∨ ) ∂a · ∂f (g ∨ ) ∂a (2.30) + δ 2 2 1 144n 2 g ∂f (g ∨ ) ∂a · ∂ 2 f (g ∨ ) ∂ 2 a · ∂f (g ∨ ) ∂a + ∂ 2 f (g ∨ ) ∂ 2 E 2 + 1 6n g ∂f (g ∨ ) ∂a · ∂ 2 f (g ∨ ) ∂a ∂E 2 +O(δ 3 ) . The first line in the right hand side reproduces the Legendre transform of the dual prepotential (2.18), so in order to enforce the relation (2.17) the δ-dependent terms should vanish. The cancellation of the term linear in δ implies ∂f (g ∨ ) ∂E 2 + 1 24 n g ∂f (g ∨ ) ∂a · ∂f (g ∨ ) ∂a = 0 . (2.31) We have written this modular anomaly equation for the g ∨ -theory, but it clearly holds also for the dual g-theory. The cancellation of the δ 2 -term follows from differentiating (2.31) with respect to E 2 . By taking further E 2 derivatives of this differential equation one can check that also the higher order terms in the δ-expansion vanish. Summarizing, the S-duality symmetry relation (2.17) requires that the mass expansion coefficients f (g) n of the quantum prepotential are quasi-modular forms of the congruence subgroup Γ 0 (n g ) ⊂ Sl(2, Z) with weight (2n − 2) satisfying the recursion relations ∂f (g) n ∂E 2 = − 1 24 n g n−1 ℓ=1 ∂f (g) ℓ ∂a · ∂f (g) n−ℓ ∂a . (2.32) Moreover, since the S-duality map (2.25) exchanges the algebras g and g ∨ , the prepotential coefficients f (g) n should depend on the vacuum expectation values a through the long and short roots of g in a way that is compatible with (2.28). This is a highly non-trivial requirement. However, as we will see later on, the explicit evaluation of the first few instanton corrections in the B r and C r theories reveals that the a-dependent part to the prepotential can be written in terms of the following two basic sums L n;m 1 m 2 ··· m ℓ = α∈Ψ L β 1 =β 2 =···β ℓ ∈Ψ(α) 1 (α · a) n (β 1 · a) m 1 (β 2 · a) m 2 · · · (β ℓ · a) m ℓ , S n;m 1 m 2 ··· m ℓ = α∈Ψ S β 1 =β 2 =···β ℓ ∈Ψ ∨ (α) 1 (α · a) n (β ∨ 1 · a) m 1 (β ∨ 2 · a) m 2 · · · (β ∨ ℓ · a) m ℓ ,(2.33) where we have denoted by Ψ L and Ψ S , respectively, the sets of long and short roots of g, and defined for any root α Ψ(α) = β ∈ Ψ : α ∨ · β = 1 , Ψ ∨ (α) = β ∈ Ψ : α · β ∨ = 1 ,(2.34) with α ∨ denoting the coroot of α (see Appendix A for details). The sums (2.33) are a generalization for the non-simply laced groups of the sums C n;m 1 ··· introduced in [1] for the ADE series. With an abuse of language, we will often refer to L n;m 1 ··· and S n;m 1 ··· as the "long" and "short" sums, respectively, since the first factors in the denominators involve long and short roots. Using the properties of the root systems (see Appendix A) it is not difficult to show that L (g) n;m 1 ···m ℓ = 1 √ n g n+m 1 +···+m ℓ S (g ∨ ) n;m 1 ···m ℓ , S (g) n;m 1 ···m ℓ = √ n g n+m 1 +···+m ℓ L (g ∨ ) n;m 1 ···m ℓ . (2.35) These are precisely the desired maps. Showing that all this construction can be explicitly realized and proved will be the subject of the remainder of this paper. Multi-instanton calculations for the B r and C r theories In this section we discuss multi-instanton calculations in N = 2 ⋆ theories with the classical gauge algebras B r and C r , using the methods of equivariant localization [17] - [20]. This is necessary to obtain explicit expressions for the prepotential coefficients f n , order by order in the instanton expansion, and verify that indeed they can be resummed into quasimodular forms of Γ 0 (2) as anticipated in the previous section. Even if some multi-instanton calculations for orthogonal and symplectic theories with adjoint matter have already been considered in the literature (see for example [23,24]), we present a brief discussion here in order to be as self-contained as possible, and also to fix some details and subtleties that have been overlooked, but which are important for the explicit calculations. We recall that the instanton moduli space for theories with orthogonal and symplectic gauge groups can be engineered using systems of N D3 branes and k D(-1) branes living on top on an orientifold O3 plane in Type IIB string theory [20]. In this set-up the instanton moduli are realised in terms of the lowest modes of open strings with at least an endpoint on a D(-1) brane [25] - [29], whose spectrum can be obtained from that of the parent U(N ) × U(k) theory after an orientifold projection. Multi-instantons for the B r = so(2r + 1) theories The moduli space of the SO(2r + 1) gauge theory can be found from that of the U(2r + 1) theory by quotienting it with Ω I, where Ω is the parity operator that changes the open string orientation, and I is the operator that reflects those fields transforming as anti-chiral spinors, namely in the fundamental representation of the left-moving SU (2) L subgroup of the Lorentz group. As a result of this projection, keeping only the antisymmetric combinations, the symmetry group of the D3/D(-1) brane system reduces to SO(2r + 1) × Sp(2k), see for example [20]. The invariant components under Ω I are listed in Tab. 2. Table 2: Instanton moduli for the SO(2r + 1) gauge theory. The columns display, respectively, the moduli in a ADHM-like notation organized as supersymmetric pairs, their statistics, their transformation properties with respect to the gauge and instanton symmetry groups and finally Q 2 -eigenvalues λ φ , where Q is the supersymmetry charge used in the localization approach. See also [30] where similar tables have been given for other brane systems. (φ, ψ) (−1) F φ SO(2r + 1) × Sp(2k) λ φ (B αα , M αȧ ) + 1, χ ij + ǫ 1 , χ ij + ǫ 2 (B aȧ , Mα a ) + 1, χ ij + ǫ 3 , χ ij + ǫ 4 (N (αḃ) , D (αβ) ) − 1, √ χ ij , χ ij + ǫ 1 + ǫ 2 (χ, N ) + 1, √ χ ij (N αa , D αa ) − 1, χ ij + ǫ 1 + ǫ 3 , χ ij + ǫ 1 + ǫ 4 (wα, µȧ) + , χ i − ϕ u + ǫ 1 +ǫ 2 2 (h a , µ a ) − , χ i − ϕ u + ǫ 3 −ǫ 4 2 The most relevant information is contained in the last column of the above table: there the χ i 's (with i = 1, · · · , k) are the unpaired bosonic moduli representing the D(-1) positions, ϕ u 's (with u = 1, · · · , r) are the D3 positions and are related to the vacuum expectation values a u ; the notation χ ij stands for χ i − χ j , and finally ǫ 1,2,3,4 are the deformation parameters of the SO(4) × SO(4) symmetry. Collecting all eigenvalues λ φ , one finds that the k-instanton partition function is given by Z k = k i=1 dχ i 2πi z gauge k z matter k (3.1) where z gauge k = (−1) k 2 k k! ǫ 1 + ǫ 2 ǫ 1 ǫ 2 k ∆(0) ∆(ǫ 1 + ǫ 2 ) ∆(ǫ 1 ) ∆(ǫ 2 ) k i=1 4χ 2 i 4χ 2 i − (ǫ 1 + ǫ 2 ) 2 P χ i + ǫ 1 +ǫ 2 2 P χ i − ǫ 1 +ǫ 2 2 , z matter k = (ǫ 1 + ǫ 3 )(ǫ 1 + ǫ 4 ) ǫ 3 ǫ 4 k ∆ ǫ 1 + ǫ 3 ∆ ǫ 1 + ǫ 4 ∆ ǫ 3 ∆ ǫ 4 k i=1 P χ i + ǫ 3 −ǫ 4 2 P χ i − ǫ 3 −ǫ 4 2 4χ 2 i − ǫ 2 3 4χ 2 i − ǫ 2 4 , with P (x) = x r u=1 x 2 − ϕ 2 u ) , ∆(x) = k i<j (χ i − χ j ) 2 − x 2 ) (χ i + χ j ) 2 − x 2 . (3.2) The integrals in (3.1) are computed by closing the contours in the upper-half complex χ i -planes after giving the deformation parameters an imaginary part with the following prescription Im(ǫ 4 ) ≫ Im(ǫ 3 ) ≫ Im(ǫ 2 ) ≫ Im(ǫ 1 ) > 0 . (3.3) This choice allows us to unambiguously compute all integrals in (3.1) and to obtain the instanton partition function Z inst = 1 + k=1 q k Z k ,(3.4) where q = e 2πiτ . At the end of the computation, we have to set ϕ u = √ 2 a u , ǫ 3 = m − ǫ 1 + ǫ 2 2 , ǫ 4 = −m − ǫ 1 + ǫ 2 2 (3.5) in order to express the result in terms of the vacuum expectation values a u and the adjoint hypermultiplet mass m in the normalization used in the previous section. Finally, the instanton prepotential of the N = 2 ⋆ theory is given by F inst = lim ǫ 1 ,ǫ 2 →0 − ǫ 1 ǫ 2 log Z = k=1 q k F k . (3.6) 1-instanton At k = 1 there is just one integral to compute and one can easily see that the poles of the integrand in (3.1) are located at χ 1 = ± ϕ u + ǫ 1 + ǫ 2 2 , ǫ 3 2 , ǫ 4 2 for u = 1, · · · , r . (3.7) Therefore the 1-instanton prepotential F k=1 can be written as F k=1 = lim ǫ 1 ,ǫ 2 →0 − ǫ 1 ǫ 2 Z 1 = lim ǫ 1 ,ǫ 2 →0 r u=1 f +ϕu+ ǫ 1 +ǫ 2 2 + r u=1 f −ϕu+ ǫ 1 +ǫ 2 2 + f ǫ 3 2 + f ǫ 4 2 (3.8) where f ±ϕu+ ǫ 1 +ǫ 2 2 = −(ǫ 1 + ǫ 3 )(ǫ 1 + ǫ 4 ) (±2ϕ u + ǫ 1 + ǫ 2 )(±ϕ u − ǫ 3 )(±ϕ u − ǫ 4 ) (±ϕ u + ǫ 1 + ǫ 2 − ǫ 3 )(±2ϕ u + ǫ 1 + ǫ 2 − ǫ 4 )ϕ u × v =u (±ϕ u − ǫ 3 ) 2 − φ 2 v (±ϕ u − ǫ 4 ) 2 − ϕ 2 v (ϕ 2 u − ϕ 2 v ) (±ϕ u + ǫ 1 + ǫ 2 ) 2 − ϕ 2 v , f ǫ 3 2 = − (ǫ 1 + ǫ 3 )(ǫ 1 + ǫ 4 )(2ǫ 3 − ǫ 4 ) (ǫ 3 − ǫ 4 ) r u=1 ϕ 2 u − (2ǫ 3 − ǫ 4 ) 2 ϕ 2 u (ǫ 3 − ǫ 1 − ǫ 2 ) 2 , f ǫ 4 2 = − (ǫ 1 + ǫ 3 )(ǫ 1 + ǫ 4 )(2ǫ 4 − ǫ 3 ) 8(ǫ 3 − ǫ 4 ) r u=1 (2ǫ 4 − ǫ 3 ) 2 − a 2 u (ǫ 4 − ǫ 1 − ǫ 2 ) 2 − ϕ 2 u . (3.9) Inserting these expressions in (3.8) and using (3.5), for the first few algebras of the B r series we obtain F (B 1 ) k=1 = − 5m 2 8 , (3.10a) F (B 2 ) k=1 = − 13m 2 8 + 2m 4 a 2 1 + a 2 2 a 2 1 − a 2 2 2 − 2m 6 a 2 1 − a 2 2 2 , (3.10b) F (B 3 ) k=1 = − 21m 2 8 + 2m 4 a 4 1 − a 2 2 a 2 1 − a 2 3 a 2 1 + a 4 2 + a 4 3 − a 2 2 a 2 3 a 2 1 − a 2 2 2 a 2 1 − a 2 3 2 a 2 2 − a 2 3 2 × a 2 2 a 4 1 + a 2 3 a 4 1 + a 4 2 a 2 1 + a 4 3 a 2 1 − 6a 2 2 a 2 3 a 2 1 + a 2 2 a 4 3 + a 4 2 a 2 3 + · · · (3.10c) where the ellipses stand for terms with higher powers of the mass whose explicit expressions can be obtained from (3.8) in a straightforward way but rapidly become quite cumbersome. We have checked (up to B 5 ) that our 1-instanton results exactly match those derived long ago in [31] using very different methods. Up to two instantons At k = 2 one has to compute two integrals to obtain the instanton partition function and hence the prepotential F k=2 . The procedure we have outlined above is straightforward to implement, and with the prescription (3.3) no ambiguity arises. For B 1 and B 2 we obtain the following 2-instanton contributions F (B 1 ) k=2 = − 23m 2 16 + m 4 a 2 1 , (3.11a) F (B 2 ) k=2 = − 47m 2 16 + m 4 a 6 1 + 5a 4 1 a 2 2 + 5a 2 1 a 4 2 + a 6 2 a 2 1 a 2 2 a 2 1 − a 2 2 2 − 2m 6 a 8 1 + 5a 6 1 a 2 2 + 12a 4 1 a 4 2 + 5a 2 1 a 6 2 + a 8 2 a 2 1 a 2 2 a 2 1 − a 2 2 4 + · · · (3.11b) where again the ellipses stand for higher order mass terms. We refrain from writing the explicit expressions of F k=2 for other orthogonal algebras since they are quite involved. However, if we use the "long" and "short" sums defined in (2.33), it is possible to write all the k = 1, 2 results in a very compact and simple way. Indeed, we have 4 F (Br ) k=1 = m 4 L 2 + m 6 2 L 2;11 + m 8 24 L 2;1111 + · · · , F (Br ) k=2 = m 4 3L 2 + S 2 − m 6 6L 4 − 3L 2;11 + 1 2 S 2;11 + m 8 5 2 L 6 + 6L 4;2 + 1 2 L 3;3 + 1 2 L 2;1111 + 1 16 S 3;3 + 1 24 S 2;1111 + · · · . (3.12) These formulas, which we have explicitly verified up to B 5 , clearly show the advantage of organize the multi-instanton results in terms of the "long" and "short" sums that fully exploit the algebraic properties of the root system of the gauge algebra. Multi-instantons for the C r = sp(2r) theories The above analysis can be easily extended to the symplectic series C r . The instanton moduli space of the N = 2 ⋆ Sp(2r) gauge theory is found from that of U(2r) after quotienting by Ω I. As a result of this projection, keeping only the symmetric combinations, the symmetry of the D3/D(-1) brane system reduces to Sp(2r) × SO(k) where k = 2K + ν with ν = 0 or 1 for k even or odd respectively [20]. The invariant components under Ω I are listed in Tab. 3. Collecting the eigenvalues λ φ from the last column of the above table, one finds Table 3: Instanton moduli for the N = 2 ⋆ Sp(2r) gauge theory. As before, the columns display, respectively, the moduli in a ADHM-like notation organized as supersymmetric pairs, their statistics, their transformation properties with respect to the gauge and instanton symmetry groups and finally Q 2 -eigenvalues λ φ , where Q is the supersymmetry charge used in the localization approach. (φ, ψ) (−) F φ Sp(2r) × SO(k) λ φ (B αα , M αȧ ) + 1, χ ij + ǫ 1 , χ ij + ǫ 2 (B aȧ , Mα a ) + 1, χ ij + ǫ 3 , χ ij + ǫ 4 (Nα˙b, Dαβ) − 1, χ ij , χ ij + ǫ 1 + ǫ 2 (N αa , D αa ) − 1, χ ij + ǫ 1 + ǫ 3 , χ ij + ǫ 1 + ǫ 4 (wα, µȧ) + , χ i − ϕ u + ǫ 1 +ǫ 2 2 (h a , µ a ) − , χ i − ϕ u + ǫ 3 −ǫ 4 2 that the instanton partition function is now given by 4 We neglect again all a-independent terms, see footnote 2. Z k = K i=1 dχ i 2πi z gauge k z matter k (3.13) where z gauge k = (−1) k 2 k+ν k! (ǫ 1 + ǫ 2 ) k (ǫ 1 ǫ 2 ) k+ν ∆(0) ∆(ǫ 1 + ǫ 2 ) ∆(ǫ 1 ) ∆(ǫ 2 ) 1 P ǫ 1 +ǫ 2 2 ν × K i=1 1 P χ i + ǫ 1 +ǫ 2 2 P χ i − ǫ 1 +ǫ 2 2 (4χ 2 i − ǫ 2 1 ) 4χ 2 i − ǫ 2 2 z matter k = (ǫ 1 + ǫ 3 )(ǫ 1 + ǫ 4 ) k+ν (ǫ 3 ǫ 4 ) k ∆ ǫ 1 + ǫ 3 ∆ ǫ 1 + ǫ 4 ∆ ǫ 3 ∆ ǫ 4 P ǫ 3 −ǫ 4 2 ν × K i=1 P χ i + ǫ 3 −ǫ 4 2 P χ i − ǫ 3 −ǫ 4 2 4χ 2 i − (ǫ 1 + ǫ 3 ) 2 4χ 2 i − (ǫ 1 + ǫ 4 ) 2 . with P (x) = r u=1 x 2 − ϕ 2 u ) , ∆(x) = K i<j x 2 − (χ i − χ j ) 2 ) x 2 − (χ i + χ j ) 2 K i=1 x 2 − χ 2 i ν . (3.14) The integrals in (3.13) are computed again by closing the contours in the upper-half complex χ i -planes with the prescription (3.3). At the end of the computation we should make the substitutions ϕ u = a u , ǫ 3 = m − ǫ 1 + ǫ 2 2 , ǫ 4 = −m − ǫ 1 + ǫ 2 2 (3.15) in order to write the result in terms of the physical parameters of the gauge theory in the normalizations of Section 2. From the partition function we can derive the instanton contributions of the prepotential along the same lines discussed for the orthogonal groups. 1-instanton For k = 1, i.e. K = 0 and ν = 1, there is no integral to be done. The prepotential following from (3.13) in this case is simply F k=1 = lim ǫ 1 ,ǫ 2 →0 − ǫ 1 ǫ 2 Z 1 = lim ǫ 1 ,ǫ 2 →0 1 2 (ǫ 1 + ǫ 3 )(ǫ 1 + ǫ 4 ) r u=1 4 ϕ 2 u − (ǫ 3 − ǫ 4 ) 2 4 ϕ 2 u − (ǫ 1 + ǫ 2 ) 2 . (3. 16) In particular for the first few symplectic algebras, using (3.15) we find (3.17c) F (C 1 ) k=1 = − m 2 2 + m 4 2a 2 1 , (3.17a) F (C 2 ) k=1 = − m 2 2 + m 4 a 2 1 + a 2 2 2a 2 1 a 2 2 − m 6 2a 2 1 a 2 2 ,(3. We have checked (up to C 5 ) that our 1-instanton results exactly match those derived in [31] using very different methods. Up to four instantons For the symplectic algebras one can push the calculation of the partition function and the prepotential to higher instanton numbers with relatively little effort. Indeed, for k = 2 and k = 3 one has to compute just one integral, while there are only two integrals to compute for k = 4 and k = 5. As an example, we now write the first multi-instanton terms of the prepotential for the Sp(4) theory that we have obtained with these methods: F (C 2 ) k=2 = − 19m 2 8 + m 4 3a 6 1 + 5a 4 1 a 2 2 + 5a 2 1 a 4 2 + 3a 6 2 2a 2 1 a 2 2 a 2 1 − a 2 2 2 (3.18a) − m 6 3a 8 1 + 6a 6 1 a 2 2 + 14a 4 1 a 4 2 + 6a 2 1 a 6 2 + 3a 8 2 4a 4 1 a 4 2 a 2 1 − a 2 2 2 + · · · F (C 2 ) k=3 = − 2m 2 3 + 2m 4 a 2 1 + a 2 2 a 2 1 a 2 2 − 2m 6 2a 4 1 + 3a 2 1 a 2 2 + 2a 4 2 a 4 1 a 4 2 + · · · (3.18b) F (C 2 ) k=4 = − 53m 2 16 + m 4 7a 6 1 + 17a 4 1 a 2 2 + 17a 2 1 a 4 2 + 7a 6 2 2a 2 1 a 2 2 a 2 1 − a 2 2 2 (3.18c) − m 6 45a 12 1 − 124a 10 1 a 2 2 + 379a 8 1 a 4 2 + 168a 6 1 a 6 2 + 379a 4 1 a 8 2 − 124a 2 1 a 10 2 + 45a 12 2 4a 4 1 a 4 2 a 1 − a 2 4 a 1 + a 2 4 + · · · where the ellipses stand for terms with higher powers of m. Clearly the explicit formulas become more and more involved for symplectic groups of higher rank, and quickly cease to be useful. However, if we use the "long" and "short" sums (2.33), we can write quite compact expressions which are valid for all C r 's. Indeed, neglecting again the a-independent terms, we find where the ellipsis stand for higher mass terms. By inspecting these formulas, one realizes that the odd instanton contributions are simpler than the even instanton ones since the latter involve both the "long" sums L n;m 1 ··· and the "short" sums S n;m 1 ··· . This observation will turn out to be useful in the following. F (Cr ) k=1 = m 4 L 2 + m 6 2 L 2;11 + m 8 24 L 2;1111 + . . . , (3.19a) F (Cr ) k=2 = m 4 3L 2 + S 2 − m 6 6L 4 − 3L 2;11 − 1 2 S 2;11 + m 8 5 2 L 6 + 6L 4;2 + L 3;3 + 1 2 L 2;1111 + 1 24 S 2;1111 + · · · , (3.19b) F (Cr ) k=3 = 4m 4 L 2 − m 6 32L 4 − 6L 2;11 + m 8 80L 6 + 48L 4;2 + 8L 3;3 + 3 2 L 2;1111 + · · · ,(3. Prepotential and recursion relations for the B r and C r theories In this section we show that the instanton corrections to the prepotential can be resummed into (quasi-)modular forms of Γ 0 (2) along the lines discussed in Section 2. To do so we write the quantum prepotential f as an expansion in the hypermultiplet mass m, namely f = f 1−loop + f inst = n=1 f n ,(4.1) with f n ∼ m 2n . The perturbative 1-loop term can be compactly written as (see for example [32]) f 1−loop = 1 4 α∈Ψ −(α · a) 2 log α · a Λ 2 + (α · a + m) 2 log α · a + m Λ 2 , (4.2) where α is an element of the root system Ψ of the gauge algebra. Expanding f 1−loop for small values of m, all odd powers cancel upon summing over positive and negative roots and in the end, neglecting all a-independent terms, we find f 1−loop = m 2 4 α∈Ψ log α · a Λ 2 − ∞ n=2 m 2n 4n(n − 1)(2n − 1) (L 2n−2 + S 2n−2 ) (4.3) = m 2 4 α∈Ψ log α · a Λ 2 − m 4 24 (L 2 + S 2 ) − m 6 120 (L 4 + S 4 ) − m 8 336 (L 6 + S 6 ) − · · · . The instanton part of the prepotential can be determined from the recursion relation (2.32) which for the B r and C r series reads ∂f n ∂E 2 = − 1 48 n−1 ℓ=1 ∂f ℓ ∂a · ∂f n−ℓ ∂a (4.4) since n g = 2 in these cases. The starting point of the recursion is f 1 which, as we have seen in the previous section, just receives a contribution at 1-loop: f 1 = m 2 4 α∈Ψ log α · a Λ 2 . (4.5) Starting from this, we will recursively determine the exact q-dependence of the prepotential order by order in m 2 . It is important to realize that the recursion relation (4.4) only fixes the E 2 -dependence of f n at a given order. The E 2 -independent contributions will be determined instead by comparing with the perturbative expansion (4.3) and the microscopic multi-instanton computations described in the previous section. The C r theories We begin our analysis from the N = 2 ⋆ C r theories for which explicit multi-instanton calculations can be performed with relatively little effort up to high values of k as we have seen in Section 3.2. The results up to 4-instantons are given in (3.19). Collecting the various powers of m 2n and adding the 1-loop contribution (4.3), we can rewrite the prepotential coefficients in the following suggestive form f 1 = m 2 4 α∈Ψ log α · a Λ 2 , (4.6a) f 2 = − m 4 24 1 − 24q − 72q 2 − 96q 3 − 168q 4 + · · · L 2 − m 4 24 1 − 24q 2 − 72q 4 + · · · S 2 , (4.6b) f 3 = − m 6 120 1 + 720q 2 + 3840q 3 + 10800q 4 + · · · L 4 + m 6 2 q + 6q 2 + 12q 3 + 28q 4 + · · · L 2;11 − m 6 120 1 + 720q 4 + · · · S 4 + m 6 2 q 2 + 6q 4 + · · · S 2;11 , (4.6c) f 4 = − m 8 336 1 − 720q 2 − 26880q 3 − 183960q 4 + · · · L 6 + m 8 6q 2 + 48q 3 + 180q 4 + · · · L 4;2 + m 8 q 2 + 8q 3 + 34q 4 + · · · L 3;3 + m 8 24 q + 12q 2 + 36q 3 + 112q 4 + · · · L 2;1111 − m 8 336 1 − 840q 4 + · · · S 6 + m 8 6q 4 + · · · S 4;2 + m 8 2 q 4 + · · · S 3;3 + m 8 24 q 2 + 12q 4 + · · · S 2;1111 . (4.6d) As discussed in Section 2, S-duality requires that the q-dependent functions in front of the various sums organize into quasi-modular forms of Γ 0 (2) which is the modular group for the C r theories. In Tab. 4 we display a basis for such modular forms up to weight 12. There E 4 and E 6 are the usual Eisenstein series (see also Appendix B), while H 2 is the modular form of weight 2 defined in (2.21). Combining them with the second Eisenstein series E 2 , one finds two quasi-modular forms of weight 2, namely {H 2 , E 2 }, four quasi-modular forms of weight 4, namely {H 2 2 , E 4 , E 2 2 , E 2 H 2 }, and so on. In principle one can use the q-expansion of these forms and fit the 1-loop and instanton results (3.19) for the first few n's. For instance, using E 2 (τ ) = 1 − 24q − 72q 2 − 96q 3 − 168q 4 + · · · ,(4.7) and comparing with (4.6b), one finds f 2 = − m 4 24 E 2 (τ ) L 2 − m 4 24 E 2 (2τ ) S 2 .(4.8) Similarly, from (4.6c), (4.7) and Table 4: A basis of modular forms of Γ 0 (2) up to weight 12. The number n w of modular forms of weight w can be obtained by expanding the generating function 1+x 2 +x 4 (1−x 4 )(1−x 6 ) = w n w x w = 1 + x 2 + 2x 4 + 2x 6 + 3x 8 + 3x 10 + 4x 12 + · · · . one gets E 4 (τ ) = 1 − 240q + 2160q 2 + 6720q 3 + 17520q 4 + · · · ,(4.f 3 = − m 6 720 5E 2 2 (τ ) + E 4 (τ ) L 4 − m 6 576 E 2 2 (τ ) − E 4 (τ ) L 2;11 − m 6 720 5E 2 2 (2τ ) + E 4 (2τ ) S 4 − m 6 576 E 2 2 (2τ ) − E 4 (2τ ) S 2;11 . (4.10) This method can be used to find the higher prepotential coefficients f n even if, when n increases, the number of quasi-modular forms that become available increases as well, thus requiring more and more microscopic multi-instanton data to fix all coefficients. It is therefore much more efficient to exploit the modular anomaly equation (4.4). For f 2 we have ∂f 2 ∂E 2 = − 1 48 ∂f 1 ∂a · ∂f 1 ∂a = − m 4 96 α,β∈Ψ (α · β) (α · a)(β · a) = − m 4 48 (2L 2 + S 2 ) (4.11) where in the last step we used the fact that only the terms with α = ±β contribute to the sum since L 1;1 = S 1;1 = 0 5 . Integrating over E 2 , we find f 2 = − m 4 24 E 2 (τ ) L 2 − m 4 48 E 2 (τ ) + H 2 (τ ) S 2 (4.12) where the E 2 -independent term has been added in order to match the 1-loop and the 1-instanton contributions in (4.6b). Notice that only the 1-instanton result is used for this; thus the perfect matching of the higher order coefficients in the q-expansion of (4.12) with the explicit multi-instanton results (4.6b) has to be regarded as a very strong and highly non-trivial consistency check. Furthermore, using the duplication formulas of the Eisenstein series given in (B.10), one can easily check that the two expressions for f 2 given in (4.10) and (4.12) coincide. Proceeding in a similar way for the m 6 terms, we get 6 ∂f 3 ∂E 2 = − 1 24 ∂f 1 ∂a · ∂f 2 ∂a = − m 6 1152 E 2 (16L 4 + 4L 2;11 + 4S 4 + S 2;11 ) − m 6 1152 H 2 (4S 4 + S 2;11 ) . (4.13) Integrating over E 2 and matching the perturbative and the first instanton terms against (4.6c) leads to f 3 = − m 6 720 5E 2 2 + E 4 L 4 − m 6 576 E 2 2 − E 4 L 2;11 (4.14) − m 6 2880 5E 2 2 + 10E 2 H 2 + 10H 2 2 − E 4 S 4 − m 6 2304 E 2 2 + 2E 2 H 2 − 4H 2 2 + E 4 S 2;11 Using the duplication formula (B.10), one can show that (4.10) and (4.14) are the same. We stress again that this result is exact in τ and that by expanding it in powers of q one can obtain the contributions at any instanton number and check that they perfectly agree with those computed with the multi-instanton calculus. Using the recursion relation we have determined also f 4 . We now collect our results on the prepotential coefficients for the C r theories: f 1 = m 2 4 α∈Ψ log α · a Λ 2 , (4.15a) f 2 = − m 4 24 E 2 L 2 − m 4 48 E 2 + H 2 S 2 , (4.15b) f 3 = − m 6 720 5E 2 2 + E 4 L 4 − m 6 576 E 2 2 − E 4 L 2;11 (4.15c) − m 6 2880 5E 2 2 + 10E 2 H 2 + 10H 2 2 − E 4 S 4 − m 6 2304 E 2 2 + 2E 2 H 2 − 4H 2 2 + E 4 S 2;11 , f 4 = − m 8 90720 175E 3 2 + 84E 2 E 4 + 11E 6 L 6 + m 8 8640 5E 3 2 − 3E 2 E 4 − 2E 6 L 4;2 + 1 6 L 3;3 − m 8 41472 E 3 2 − 3E 2 E 4 + 2E 6 )L 2;1111 − m 8 725760 175E 3 2 + 525E 2 2 H 2 + 945E 2 H 2 2 − 84E 2 E 4 + 39E 6 + 560H 3 2 S 6 + m 8 69120 5E 3 2 + 15E 2 2 H 2 + 3E 2 E 4 − 3E 6 − 20H 3 2 S 4,;2 + 1 12 S 3;3 + 2 3 L 3;3 − m 8 331776 E 3 2 + 3E 2 2 H 2 − 12E 2 H 2 2 + 3E 2 E 4 + E 6 + 4H 3 2 S 2;1111 . (4.15d) It is interesting to observe that the combinations of Eisenstein series appearing in f 2 , f 3 and in the first two lines of f 4 in front of the "long" sums are exactly the same that appear also in the prepotential coefficients f 2 , f 3 and f 4 of the ADE theories studied in [1]. Moreover, using the duplication formulas (B.10), one can show that the combinations of Eisenstein series of the last three lines of f 4 in (4.15d) become identical to the ones of the first two lines, but evaluated in 2τ instead of τ , similarly to what happens in f 2 and f 3 . Finally we observe that for r = 1, i.e. for C 1 = sp (2), there are only two long roots, ±2, and thus only the sums of the type L n are non zero. In this case the quantum prepotential f drastically simplifies and reduces to f (C 1 ) = m 2 log 2a Λ − m 4 48 a 2 E 2 − m 6 5760 a 4 5E 2 2 + E 4 − m 8 2903040 a 6 175E 3 2 + 84E 2 E 4 + 11E 6 + · · · . (4. 16) This exactly coincides with the prepotential of the SU(2) N = 2 ⋆ theory [1], as it should be since Sp(2) ≃ SU(2). 1-instanton Given the previous results, it is possible to write a very compact expression for the 1instanton contribution to the prepotential. Indeed, the only terms which have a 1-instanton part are those proportional to L 2;11··· (as is clear also from (3.19a)), and one finds F k=1 = m 4 ℓ=0 m 2ℓ ℓ! L 2;11. . . 1 ℓ = α∈Ψ L m 4 (α · a) 2 ℓ=0 m 2ℓ ℓ! β 1 =··· =β ℓ ∈Ψ(α) 1 (β 1 · a) · · · (β ℓ · a) = α∈Ψ L m 4 (α · a) 2 β∈Ψ(α) 1 + m β · a . (4.17) where the intermediate step follows from the definition (2.33) of the "long" sums L 2;11··· . The number of factors in the product above is given by the dimension of Ψ(α) which, when α is a long root of C r , is 2r − 2 (see Appendix A). Thus, in (4.17) the highest power of the mass is m 2r+2 . This is precisely the only term which survives in the decoupling limit q → 0 and m → ∞ with m 2r+2 q = Λ 2r+2 fixed , (4.18) when the N = 2 ⋆ theory reduces to the pure N = 2 SYM theory 7 . In this case the 1-instanton prepotential is q F k=1 N =2 = Λ 2r+2 α∈Ψ L 1 (α · a) 2 β∈Ψ(α) 1 β · a . (4.19) This expression perfectly coincides with the known results present in the literature (see for example [33,31,23,24] and in particular [34]), while (4.17) is the generalization thereof to the N = 2 ⋆ symplectic theories. The B r theories The N = 2 ⋆ theories with non-simply laced orthogonal gauge groups can be treated exactly as described above. Indeed, the modular group is again Γ 0 (2), and the recursive relation, the 1-loop and the 1-instanton microscopic data have exactly the same form as in the C r theories and differ only in the explicit expression for the roots. Thus, the results for the B r models become very similar to those of the symplectic ones with the only differences arising from the different relations among the root sums. Skipping the intermediate steps, the prepotential for the so(2r + 1) theory turns out to be F so(2r+1) = 2πiτ a 2 + ∞ n=1 f n (4.20) where the first few f n 's are f 1 = m 2 4 α∈Ψ log α · a Λ 2 , (4.21a) f 2 = − m 4 24 E 2 L 2 − m 4 48 E 2 + H 2 S 2 , (4.21b) f 3 = − m 6 720 5E 2 2 + E 4 L 4 − m 6 576 E 2 2 − E 4 L 2;11 (4.21c) − m 6 2880 5E 2 2 + 10E 2 H 2 + 10H 2 2 − E 4 S 4 − m 6 2304 E 2 2 + 2E 2 H 2 − 4H 2 2 + E 4 S 2;11 , f 4 = − m 8 90720 175E 3 2 + 84E 2 E 4 + 11E 6 L 6 + m 8 8640 5E 3 2 − 3E 2 E 4 − 2E 6 L 4;2 + 1 12 L 3;3 + 1 96 S 3;3 − m 8 41472 E 3 2 − 3E 2 E 4 + 2E 6 )L 2;1111 − m 8 725760 175E 3 2 + 525E 2 2 H 2 + 945E 2 H 2 2 − 84E 2 E 4 + 39E 6 + 560H 3 S 6 (4.21d) + m 8 69120 5E 3 2 + 15E 2 2 H 2 + 3E 2 E 4 − 3E 6 − 20H 3 2 S 4,;2 + 1 6 S 3;3 − m 8 331776 E 3 2 + 3E 2 2 H 2 − 12E 2 H 2 2 + 3E 2 E 4 + E 6 + 4H 3 2 S 2;1111 . By expanding the above expressions in powers of q, one can obtain the multi-instanton contributions to the prepotential. At the 1-instanton level one finds exactly the same expression (3.19a) or (4.17), simply with the roots of C r replaced by those of B r . At the 2instanton level one finds complete agreement with the expressions obtained in Section 3.1. For higher k the above formulas can be used to efficiently determine the higher instanton contributions to the prepotential, which instead are technically difficult to compute with the localization methods. Notice that the expressions in (4.21) are similar but not identical to those of (4.15) since there are a few differences in f 4 . For B 1 , i.e. for so (3), there are just two short roots, ± √ 2, so that only the sums of the type S n are non-vanishing. In this case the quantum prepotential becomes f (B 1 ) = m 2 log √ 2a Λ − m 4 48 a 2 E 2 + H 2 − m 6 5760 a 4 5E 2 2 + 10E 2 H 2 + 10H 2 2 − E 4 − m 8 2903040 a 8 175E 3 2 + 525E 2 2 H 2 + 945E 2 H 2 2 − 84E 2 E 4 + 39E 6 + 560H 3 2 + · · · which, after using the duplication formulas (B.10), takes the form f (B 1 ) = m 2 log √ 2a Λ − m 4 24 a 2 E 2 (2τ ) − m 6 1440 a 4 5E 2 2 (2τ ) + E 4 (2τ ) − m 8 362880 a 6 175E 3 2 (2τ ) + 84E 2 (2τ )E 4 (2τ ) + 11E 6 (2τ ) + · · · . (4.22) The fact that only modular forms evaluated at 2τ appear means that f (B 1 ) admits an expansion in powers of q 2 ; in other words only the even instantons contribute. The prepotential (4.22) perfectly matches that of the SU(2) theory (see also (4.16)) provided a → √ 2a and q 2 SO(3) = q SU (2) . (4.23) The above identification of the coupling constants, which practically amounts to replace 2τ with τ in (4.22), is consistent with the fact that SU(2) is the double cover of SO(3), i.e. SU(2)/Z 2 ≃ SO(3). Relations between the B r and C r theories The similarity of the results (4.21) and (4.15) for the prepotential of the B r and C r theories is not surprising since, as discussed in Section 2, they are related by a strong/weak-coupling S-duality. Here we check explicitly this relation exploiting the properties of the quasimodular forms and of the "long" and "short" sums. The first observation is that H 2 , E 2 , E 4 and E 6 have simple properties under τ → − 1 2τ ; (4.24) indeed, one can check (see Appendix B) that 8 1 2τ 2 H 2 − 1 2τ = −H 2 , (4.25a) 1 2τ 2 E 2 − 1 2τ = E 2 + H 2 + 6 πiτ , (4.25b) 1 4τ 4 E 4 − 1 2τ = −E 4 + 5H 2 2 , (4.25c) 1 8τ 6 E 6 − 1 2τ = E 6 + 7H 3 2 . (4.25d) We recall again that the transformation (4.24) does not belong to the modular group Γ 0 (2) but it is a generator of the S-duality group, which is a discrete subgroup of Sl(2, R). The second observation is that the root systems of B r and C r can be mapped into each other by exchanging (and suitably rescaling) long and short roots. As a consequence of this fact, the "long" and "short" sums in the two theories are related in the following way (see (2.35)): L (Br) n;m 1 ···m ℓ = 1 √ 2 n+m 1 +···+m ℓ S (Cr ) n;m 1 ···m ℓ , S (Br) n;m 1 ···m ℓ = √ 2 n+m 1 +···+m ℓ L (Cr) n;m 1 ···m ℓ . (4.26) Combining (4.25) and (4.26) with the expressions of the prepotential coefficients in the orthogonal and symplectic theories, we can check how they are non-perturbatively related with each other. In order to display these relations in a transparent way, we explicitly indicate the dependence on the coupling constant τ , on the vacuum expectation values a (through the root lattice sums) and on the quasi-modular form E 2 by writing f (Br) n (τ, a, E 2 (τ )) and f (Cr) n (τ, a, E 2 (τ )). Then, from (4.21b) and (4.15b), it is not difficult to show that f (Br) 2 − 1 2τ , √ 2τ a, E 2 (− 1 2τ ) = 1 2τ 2 − m 4 24 E 2 (− 1 2τ )L (Br ) 2 − m 4 48 E 2 (− 1 2τ ) + H 2 (− 1 2τ ) S (Br) 2 = − m 4 48 E 2 + H 2 + δ S (Cr ) 2 − m 4 24 E 2 + δ L (Cr) 2 = f (Cr) 2 τ, a, E 2 + δ (4.27) where δ = 6 πiτ as before. More generally one can prove that f (Br) n − 1 2τ , √ 2τ a, E 2 (− 1 2τ ) = f (Cr) n τ, a, E 2 + δ . (4.28) This is precisely the type of duality relation discussed in Section 2 (see in particular (2.25)). Using it together with the recurrence relations, we can therefore verify that S F (Br) = L F (Cr) (4.29) as we anticipated. Of course, also the reverse relation S F (Cr) = L F (Br) (4.30) is true. These relations, which are an extension of the ones described in [1] for the ADE theories provide a highly non-trivial test of the S-duality. Prepotential and recursion relations for the G 2 and F 4 theories In this section we consider N = 2 ⋆ theories with G 2 and F 4 gauge algebras. Differently from the classical algebras B r and C r , the ADHM construction of the instanton moduli space is not known for the exceptional algebras and thus in these cases one cannot rely on the localization techniques to obtain explicit multi-instanton results. Nevertheless remarkable progress can be made using the methods described in the previous section. Let us start our discussion from the G 2 theory. The G 2 theory The prepotential for the G 2 theory can be written as F = 3πiτ a 2 + n≥1 f n (5.1) with the first term describing the classical part and f n the mass expansion coefficients of the quantum prepotential. Now the modular group is Γ 0 (3) [11,12,13] and we assume as before that f n are quasi-modular forms of this group. For Γ 0 (3) the ring of quasi modular forms is generated by {E 2 , K 2 , E 4 , E 6 } (5.2) where K 2 , the modular form of weight 2 defined in (2.21). By building monomials of these basic elements one can construct a basis for the modular forms of Γ 0 (3) with higher weights, as indicated in Tab. 5 up to weight 12. Comparing with the table of the modular forms of Weight Modular forms of Γ 0 (3) Table 5: A basis of modular forms for the congruence subgroup Γ 0 (3) of the modular group up to weight 12. The number n w of modular forms of weight w can be obtained by expanding the generating function 1+x 2 +x 4 +x 6 (1−x 4 )(1−x 6 ) = w n w x w = 1 + x 2 + 2x 4 + 3x 6 + 3x 8 + 4x 10 + 5x 12 + · · · . Γ 0 (2), we see many similarities but also some differences. For example at weight 6 we now have three independent modular forms instead of two. Finally, there is another well-known feature of G 2 [9] that will play a crucial rôle in the following, namely the fact that by exchanging (and suitably rescaling) long and short roots we obtain an equivalent root system with the two axes interchanged. Thus, under this operation the G 2 theory is mapped into another theory with the same symmetry but with the two vacuum expectation values a 1 and a 2 interchanged. We call this "dual" algebra G ′ 2 . 2 K 2 4 K 2 2 , E 4 6 K 3 2 , E 4 K 2 , E 6 8 K 4 2 , E 4 K 2 2 , E 2 4 10 K 5 2 , E 2 4 K 2 , E 4 K 3 2 , E 4 E 6 12 K 6 2 , E 2 4 K 2 2 , E 4 K 4 2 , E 3 4 , E 2 6 The prepotential for the G 2 theory will be derived again from the recursion relation and the E 2 -independent terms will be determined by the perturbative 1-loop prepotential f 1−loop and the one-instanton result F (G 2 ) k=1 = α∈Ψ L m 4 (α · a) 2 β∈Ψ(α) 1 + m β · a = m 4 L 2 + m 6 2 L 2;11 + m 8 24 L 2;1111 (5.3) where the last step follows from the fact that there are only four factors in the product since dim Ψ(α L ) = 4 for G 2 (see Appendix A). Formula (5.3) was checked to be valid both for theories with ADE in [1] and BC gauge groups in the previous section, and will be assumed to be valid also for G 2 . This assumption is well justified not only by the fact that the 1-instanton contribution takes this form in all other groups considered so far, but also by the fact that the last term in (5.3), which is the one surviving in the pure N = 2 G 2 theory where the adjoint hypermultiplet is decoupled, exactly matches the 1-instanton result obtained long-ago in [35] from the Picard-Fuchs approach to the Seiberg-Witten curve of the G 2 theory, and more recently in [34] from coherent states of W-algebras. The recursion relation for G 2 reads ∂f n ∂E 2 = − 1 72 n−1 ℓ=1 ∂f ℓ ∂a · ∂f n−ℓ ∂a (5.4) Again the starting point is f 1 which has only the 1-loop part f 1 = m 2 4 α∈Ψ log α · a Λ 2 , (5.5) On the other hand, according to our working hypothesis, f 2 must be proportional to L 2 and S 2 , since it has to be homogeneous of degree −2 in the vacuum expectation values, with coefficients that are linear combinations of E 2 and K 2 , since it has to be a quasi-modular form of Γ 0 (3) with weight 2. Therefore, matching the 1-loop and 1-instanton terms we get f 2 = − m 4 24 E 2 (τ ) L 2 − m 4 72 E 2 (τ ) + 2K 2 (τ ) S 2 = − m 4 24 E 2 (τ ) L 2 − m 4 24 E 2 (3τ ) S 2 (5.6) where the last step is a consequence of the triplication formula E 2 (3τ ) = 1 3 E 2 (τ ) + 2K 2 (τ ) . (5.7) The same result follows by solving the recursion relation (5.4) ∂f 2 ∂E 2 = − 1 72 ∂f 1 ∂a · ∂f 1 ∂a (5.8) and matching with the perturbative and 1-instanton terms. By expanding f 2 in powers of q we can obtain all multi-instanton contributions to the prepotential that are proportional to m 4 . As we already remarked, for G 2 there is no known ADHM construction of the instanton moduli space and no explicit multi-instanton calculations can be performed. Thus, our result represents a prediction for the higher instanton terms. Applying the same method at order m 6 , it is quite straightforward to get f 3 = − m 6 720 5E 2 2 + E 4 L 4 − m 6 576 E 2 2 − E 4 L 2;11 (5.9) − m 6 6480 5E 2 2 + 20E 2 K 2 + 30K 2 2 − E 4 S 4 − m 6 5184 E 2 2 + 4E 2 K 2 − 6K 2 2 + E 4 S 2;11 where we suppressed the τ dependence in the right hand side to simplify the notation. It is interesting to observe that using the triplication formulas (5.7) the quasi-modular forms appearing in front of S 4 and S 2;11 can be written exactly like the ones appearing in front of L 4 and L 2;11 , respectively, but evaluated in 3τ instead of τ . The same triplication formulas allow us to show that under τ → − 1 3τ , (5.10) the Eisenstein series and K 2 have simple transformation properties, namely 1 3τ 2 K 2 − 1 3τ = −K 2 , (5.11a) 1 3τ 2 E 2 − 1 3τ = E 2 + 2K 2 + 6 πiτ , (5.11b) 1 9τ 4 E 4 − 1 3τ = −E 4 + 10K 2 2 , (5.11c) 1 27τ 6 E 6 − 1 3τ = −E 6 − 7E 4 K 2 + 35K 3 2 . (5.11d) Furthermore, from the properties of the G 2 root lattice, it follows that L (G 2 ) n;m 1 ···m ℓ = 1 √ 3 n+m 1 +···+m ℓ S (G ′ 2 ) n;m 1 ···m ℓ , S (G 2 ) n;m 1 ···m ℓ = √ 3 n+m 1 +···+m ℓ L (G ′ 2 ) n;m 1 ···m ℓ (5.12) where G 2 and G ′ 2 are dual to each other [9] as discussed above. Combining (5.12) with the transformation rules (5.11), one can check that both f 2 and f 3 satisfy the expected duality relations f (G 2 ) n − 1 3τ , √ 3τ a, E 2 (− 1 3τ ) = f (G ′ 2 ) n τ, a, E 2 + δ . (5.13) Using the recursion formula (5.4), imposing the duality relations (5.13) and matching with the 1-loop and the 1-instanton results, we managed to determine also the m 8 -terms of the prepotential. We now collect all our findings for the G 2 theory up to m 8 : f 1 = m 2 4 α∈Ψ log α · a Λ 2 , (5.14a) f 2 = − m 4 24 E 2 L 2 − m 4 72 E 2 + 2K 2 S 2 , (5.14b) f 3 = − m 6 720 5E 2 2 + E 4 L 4 − m 6 576 E 2 2 − E 4 L 2;11 (5.14c) − m 6 6480 5E 2 2 + 20E 2 K 2 + 30K 2 2 − E 4 S 4 − m 6 5184 E 2 2 + 4E 2 K 2 − 6K 2 2 + E 4 S 2;11 , f 4 = − m 8 90720 175E 3 2 + 84E 2 E 4 + 11E 6 L 6 + m 8 8640 5E 3 2 − 3E 2 E 4 − 2E 6 L 4;2 (5.14d) − m 8 186624 52E 3 2 + 81E 2 2 K 2 − 42E 2 E 4 + 162E 2 K 2 2 + 44E 6 + 189E 4 K 2 − 486K 3 2 )L 2;1111 − m 8 2949440 175E 3 2 +1050E 2 2 K 2 − 84E 2 E 4 +2940E 2 K 2 2 −11E 6 −245E 4 K 2 +3465K 3 2 S 6 + m 8 233280 5E 3 2 + 30E 2 2 K 2 + 3E 2 E 4 + 30E 2 K 2 2 + 2E 6 + 20E 4 K 2 + 90K 3 2 S 4,;2 − m 8 5038848 52E 3 2 + 231E 2 2 K 2 + 42E 2 E 4 + 42E 2 K 2 2 − 44E 6 − 35E 4 K 2 − 288K 3 2 S 2;1111 . We remark that the E 2 -dependence in f 4 is completely determined by the recursion relation (5.4) which, however, can not fix the combinations of the three independent modular forms of Γ 0 (3) with weight 6, namely E 6 , E 4 K 2 and K 3 2 , that appear in front of the various sums. To fix these combinations some extra information is therefore needed beside the 1-loop and 1-instanton data. In the absence of explicit multi-instanton results, we have used the duality relations (5.13). These impose very severe restrictions on the coefficients, and the fact that all resulting constraints are mutually compatible, thus allowing for a solution, is a very strong consistency check on our approach. The previous formulas and their properties allow us to conclude that S F (G 2 ) = L F (G ′ 2 ) . (5.15) Of course also the reverse relation is true. The F 4 theory The N = 2 ⋆ theory with gauge group F 4 can be analysed in exactly the same way as discussed above. In this case the modular group is Γ 0 (2), like in the B r and C r theories. Again the recursive relation, the 1-loop and the 1-instanton microscopic data match those of B r and C r models and we have checked that the prepotential coefficients up to f 3 are given by exactly the same formulas (4.15a), (4.15b) and (4.15c), with the "long" and "short" sums written with the roots of F 4 , and that they satisfy the recursion relation (4.4). These properties imply S F (F 4 ) = L F (F ′ 4 ) (5.16) and its reverse. Conclusions In this paper we have extended the results of [1] to N = 2 ⋆ theories with non simplylaced gauge algebras and in particular have studied their behaviour under S-duality. This extension is far from being trivial. From the analysis in N = 4 theories one expects that S-duality be a symmetry for ADE gauge groups. But this cannot be the case for the non-simply laced theories, since S-duality exchanges the B r and C r algebras and maps G 2 and F 4 into the rotated G ′ 2 and F ′ 4 algebras. All these features and constraints can be put into a consistent picture which allows the computation of the effective prepotential starting from a modular anomaly equation. The payoff of this approach is a conjecture for the prepotential of the G 2 and F 4 gauge theories for which a direct check is possible only at the 1-instanton level, since the microscopic multi-instanton computations are not available for the exceptional algebras. At the same time given the number of consistency checks which are passed by our proposal, we are confident about its validity. It is amusing to observe that the modular anomaly equation by itself is not enough to completely determine the prepotential coefficients in the mass expansion which in general need the perturbative 1-loop and some microscopic instanton computations in order to be fixed. Nonetheless even if the latter are not available for the G 2 and F 4 cases our proposal stands at the same level of the computations for the classical orthogonal or symplectic algebras given that our results are written in a form that makes them independent of the details of the various root systems. One final remark: the results presented here overlook the contribution of the Ω-background. Such contributions could be easily incorporated by the means of the same methods employed in our companion paper [1] for the ADE theories. However, in order not to make the formulas too involved, we have preferred to omit them from our presentation. Any long root α L ∈ Ψ L has length 2, while any short root α S ∈ Ψ S has length √ 2. Therefore we have n g = 2. It is easy to see that ord Ψ L = 2r 2 − 2r , ord Ψ S = 2r , (A.5) so that the total number of roots is 2r 2 which is indeed the order of Ψ. According to (A.1), we have α ∨ L = 1 2 α L , α ∨ S = α S . (A.6) It is not difficult to show that ord Ψ(α L ) = 4r − 6 = 2h ∨ − 4 , ord Ψ(α S ) = 0 = 4h − 4h ∨ − 4 , ord Ψ ∨ (α L ) = 4r − 8 = −2h + 4h ∨ − 4 , ord Ψ ∨ (α S ) = 2r − 2 = 3h − 2h ∨ − 4 , (A.7) where h = 2r is the Coxeter number and h ∨ = 2r − 1 is the dual Coxeter number for B r . The roots of C r For the symplectic algebra C r the long roots are given by Ψ L = {±2 e i ; 1 ≤ i ≤ r} (A.8) where again {e i ; 1 ≤ i ≤ r} is the standard orthonormal basis in R r . The short roots are instead given by Ψ S = {±e i ± e j ; 1 ≤ i < j ≤ r} , (A.9) with all possible signs. Any long root α L ∈ Ψ L has length 2, while any short root α S ∈ Ψ S has length √ 2, like for the B r groups. Therefore also in this case we have n g = 2. Its easy to see that ord Ψ L = 2r , ord Ψ S = 2r 2 − 2r , (A.10) so that the total number of roots is 2r 2 which is the order of Ψ. According to (A.1), we have where h = 2r is the Coxeter number and h ∨ = r + 1 is the dual Coxeter number for C r . α ∨ L = 1 2 α L , α ∨ S = α S . As is well-known [9], there is a duality between the formulas for B r and those for C r under the exchange of long and short roots, which can be explicitly verified on the above results. The roots of F 4 F 4 has forty-eight roots; half of them are long roots and half are short roots. Denoting by {e 1 , e 2 , e 3 , e 4 } the standard orthonormal basis in R 4 , the twenty-four long roots are Ψ L = ± √ 2 e i ± √ 2 e j ; 1 ≤ i < j ≤ 4 , (A. 13) while the twenty-four short roots are Ψ S = ± √ 2 e 1 , ± √ 2 e 2 , ± √ 2 e 3 , ± √ 2 e 4 , 1 √ 2 ± e 1 ± e 2 ± e 3 ± e 4 . (A.14) The long roots have length 2, while the short roots have length It is interesting to observe that by exchanging (and suitably rescaling) long and short roots, we obtain an equivalent root system; the algebra associated to this dual root system is called F ′ 4 . The roots of G 2 For G 2 the six long roots are given by It is interesting to observe that by exchanging (and suitably rescaling) long and short roots, we obtain an equivalent root system in which e 1 and e 2 are exchanged. We call the algebra associated to this dual root system G ′ 2 . We summarize in the following Ψ L = ± 3 √ 2 e 1 ± B. Modular forms In this appendix we collect the main formulas for the modular functions used in the main text. • We adopt the standard definitions for the Jacobi θ-functions: We also define the θ-constants: θ 2 (τ ) ≡ θ 1 0 (0|τ ), θ 3 (τ ) ≡ θ 0 0 and θ 4 ≡ θ 0 1 (0|τ ). Their q-expansions are θ 2 (τ ) = 2q 1 8 1 + q + q 3 + q 6 + · · · , θ 3 (τ ) = 1 + 2q 1 2 + 2q 2 + 2q 9 2 + 2q 8 + · · · , θ 4 (τ ) = 1 − 2q 1 2 + 2q 2 − 2q where q = e 2πiτ . • The Dedekind η-function is defined as η(τ ) = q • The Eisenstein series are defined by E 2 (τ ) = 1 − 24 ∞ n=1 σ 1 (n) q n = 1 − 24q − 72q 2 − 96q 3 − 168q 4 + · · · , E 4 (τ ) = 1 + 240 ∞ n=1 σ 3 (n) q n = 1 + 240q + 2160q 2 + 6720q 3 + 17520q 4 + · · · , E 6 (τ ) = 1 − 504 ∞ n=1 σ 5 (n) q 2n = 1 − 504q − 16632q 2 − 122976q 3 − 532728q 4 , (B.6) where σ k (n) is the sum of the k-th power of the divisors of n, i.e., σ k (n) = d|n d k . The Eisenstein series E 4 and E 6 can be expresses as polynomials in the θ-constants according to E 4 (τ ) = 1 2 θ 8 2 (τ ) + θ 8 3 (τ ) + θ 8 4 (τ ) , E 6 (τ ) = 1 2 θ 4 3 (τ ) + θ 4 4 (τ ) θ 4 2 (τ ) + θ 4 3 (τ ) θ 4 4 (τ ) − θ 4 2 (τ ) . (B.7) The series E 2 , E 4 and E 6 are connected among themselves by logarithmic q-derivatives and form a sort of a "ring": q ∂E 2 (τ ) ∂q = 1 12 E 2 2 (τ ) − E 4 (τ ) , q ∂E 4 (τ ) ∂q = 1 3 E 4 (τ )E 2 (τ ) − E 6 (τ ) , q ∂E 6 (τ ) ∂q = 1 2 E 6 (τ )E 2 (τ ) − E 2 4 (τ ) . (B.8) The series E 4 and E 6 are modular forms of Sl(2, Z) with weight 4 and 6 respectively, while E 2 is a quasi-modular form of weight 2 with an anomalous shift. In particular under S, we have E 2 (− 1 τ ) = τ 2 E 2 (τ ) + 6τ πi , E 4 (− 1 τ ) = τ 4 E 4 (τ ) , E 6 (− 1 τ ) = τ 6 E 6 (τ ) . (B.9) • The Eisenstein series satisfy the following duplication formulas E 2 (2τ ) = 1 2 E 2 (τ ) + H 2 (τ ) , E 4 (2τ ) = 1 4 − E 4 (τ ) + 5H 2 2 (τ ) , E 6 (2τ ) = 1 8 E 6 (τ ) + 7H 3 2 (τ ) (B.10) where H 2 (τ ) = 1 2 θ 4 3 (τ ) + θ 4 4 (τ ) = 1 + 24q + 24q 2 + 96q 3 + 24q 4 + 144q 5 + · · · . (B.11) The function H 2 (τ ) is a modular form of weight 2 under the congruence subgroup Γ 0 (2) ⊂ Sl(2, Z), and is such that H 2 (− 1 2τ ) = −2τ 2 H 2 (τ ) . (B.12) not difficult to show that ord Ψ(α L ) = 2r − 2 = 2h ∨ − 4 , ord Ψ(α S ) = 4r − 8 = 4h − 4h ∨ − 4 , ord Ψ ∨ (α L ) = 0 = −2h + 4h ∨ − 4 , ord Ψ ∨ (α S ) = 4r − 6 = 3h − 2h ∨ − 4 , (A.12) n g = 2. Given the explicit expressions (A.13) and (A.14), it is easy to see that ord Ψ(α L ) = 14 , ord Ψ(α S ) = 8 , ord Ψ ∨ (α L ) = 8 , ord Ψ ∨ (α S ) = 14 , (A.16) 1 and e 2 form the standard orthonormal basis in R 2 . With this choice , the long roots have length √ 6, while the short roots have length √ 2. Thus, according to (A.n g = 3. Given the explicit expressions (A.17) and (A.18), it is easy to see that ord Ψ(α L ) = 4 , ord Ψ(α S ) = 2 , ord Ψ ∨ (α L ) = 2 , ord Ψ ∨ (α S ) = 4 , (A.20) • Under the Sl(2, Z) generatorsT : τ → τ + 1 , S : θ 3 (τ ) ↔ θ 4 (τ ) , θ 2 (τ ) → e iπ 4 θ 2 (τ ) , η(τ ) → e iπ 12 η(τ ) , S : θ 2 (τ ) η(τ ) ↔ θ 4 (τ ) η(τ ) , θ 3 (τ ) η(τ ) → θ 3 (τ ) η(τ ) , η(τ ) → √ −iτ η(τ ) . (B.5) table the properties for the various algebras that are useful for the calculations presented in the main text:dim rank h ∨ ord Ψ L ord Ψ S ord Ψ(α L ) ord Ψ ∨ (α S ) B r r(2r + 1) r 2r − 1 2r 2 − 2r 2r 4r − 6 2r − 2 C r r(2r + 1) r r + 1 2r 2r 2 − 2r 2r − 2 4r − 6 F 4 52 4 9 24 24 14 14 G 2 14 2 4 6 6 4 4 Table 6 : 6The main properties of the non-simply laced algebras. Modular anomaly equations have been considered also in the context of N = 2 conformal SQCD models with fundamental matter[3,4,7]. Here and in the following we neglect all a-independent terms of the prepotential since they are irrelevant for the low-energy effective theory. These terms can always be absorbed by redefining the scale Λ in one of the sides of (2.10). The identities for the "long" and "short" sums can be proven with methods similar to those discussed in Appendix D of[1] Here, we have simplified the notation by writing E2 and H2 in place of E2(τ ) and H2(τ ). Note that the exponent (2r + 2) is the 1-loop β-function coefficient for the pure N = 2 SYM theory with gauge group Sp(2r). To simplify the notation, here and in the following, when we write H2, E2, E4 and E6, we mean that these are evaluated at τ . AcknowledgmentsWe thank Carlo Angelantonj, Sujay Ashok, Massimo Bianchi, Eleonora Dell'Aquila and Igor Pesando for discussions.The work of M.B., M.F. and A.L. is partially supported by the Compagnia di San Paolo contract "MAST: Modern Applications of String Theory" TO-Call3-2012-0088.A. Notations and conventions for the root systemsIn this appendix we list our conventions for the root systems of the non-simply laced groups. We consider both the classical algebras B r = so(2r + 1) and C r = sp(2r), and the two exceptional ones F 4 and G 2 . In all cases, we denote by Ψ, Ψ L and Ψ S , respectively, the set of all roots α, the set of the long roots α L and the set of the short roots α S . Of course one has Ψ = Ψ L ∪ Ψ S .Given a root α ∈ Ψ, we define its corresponding co-root α ∨ asand introduce the two setsThe roots of B r Let {e i ; 1 ≤ i ≤ r} be the standard orthonormal basis in R r . The set Ψ L of the long roots of B r is Ψ L = ± √ 2 e i ± √ 2 e j ; 1 ≤ i < j ≤ r , M Billo, M Frau, F Fucito, A Lerda, J F Morales, arXiv:1507.07709S-duality and the prepotential of N = 2 * theories (I): the ADE algebras. hep-th. to be published in JHEPM. Billo, M. Frau, F. Fucito, A. Lerda, and J. F. 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Pioline, Rankin-Selberg methods for closed strings on orbifolds, JHEP 1307 (2013) 181, arXiv:1304.4271 [hep-th]. Saddle point equations in Seiberg-Witten theory. S Shadchin, arXiv:hep-th/0408066JHEP. 1033hep-thS. Shadchin, Saddle point equations in Seiberg-Witten theory, JHEP 10 (2004) 033, arXiv:hep-th/0408066 [hep-th]. A note on instanton counting for N = 2 gauge theories with classical gauge groups. M Marino, N Wyllard, arXiv:hep-th/0404125JHEP. 0521M. Marino and N. Wyllard, A note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP 05 (2004) 021, arXiv:hep-th/0404125. Small instantons in string theory. E Witten, arXiv:hep-th/9511030Nucl. Phys. 460E. Witten, Small instantons in string theory, Nucl. Phys. B460 (1996) 541-559, arXiv:hep-th/9511030. M R Douglas, arXiv:hep-th/9512077Branes within branes. M. R. Douglas, Branes within branes, arXiv:hep-th/9512077. D-instanton induced interactions on a D3-brane. M B Green, M Gutperle, arXiv:hep-th/0002011JHEP. 0214M. B. Green and M. Gutperle, D-instanton induced interactions on a D3-brane, JHEP 02 (2000) 014, arXiv:hep-th/0002011. Classical gauge instantons from open strings. M Billo, M Frau, I Pesando, F Fucito, A Lerda, A Liccardo, arXiv:hep-th/0211250JHEP. 0245M. Billo, M. Frau, I. Pesando, F. Fucito, A. Lerda, and A. Liccardo, Classical gauge instantons from open strings, JHEP 02 (2003) 045, arXiv:hep-th/0211250. Instanton calculus in R-R background and the topological string. M Billo, M Frau, F Fucito, A Lerda, arXiv:hep-th/0606013JHEP. 1112M. Billo, M. Frau, F. Fucito, and A. Lerda, Instanton calculus in R-R background and the topological string, JHEP 11 (2006) 012, arXiv:hep-th/0606013. Non-perturbative gauge/gravity correspondence in N=2 theories. M Billo, M Frau, F Fucito, L Giacone, A Lerda, J F Morales, D Ricci-Pacifici, arXiv:1206.3914JHEP. 1208166hep-thM. Billo, M. Frau, F. Fucito, L. Giacone, A. Lerda, J. F. Morales, and D. Ricci-Pacifici, Non-perturbative gauge/gravity correspondence in N=2 theories, JHEP 1208 (2012) 166, arXiv:1206.3914 [hep-th]. Elliptic models and M theory. I P Ennes, C Lozano, S G Naculich, H J Schnitzer, arXiv:hep-th/9912133Nucl. Phys. 576hep-thI. P. Ennes, C. Lozano, S. G. Naculich, and H. J. Schnitzer, Elliptic models and M theory, Nucl. Phys. B576 (2000) 313-346, arXiv:hep-th/9912133 [hep-th]. E Hoker, D Phong, arXiv:hep-th/9912271Lectures on supersymmetric Yang-Mills theory and integrable systems. hep-thE. D'Hoker and D. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, arXiv:hep-th/9912271 [hep-th]. The Effective prepotential of N=2 supersymmetric SO(N(c)) and Sp(N(c)) gauge theories. E Hoker, I M Krichever, D H Phong, arXiv:hep-th/9609145Nucl. Phys. 489hep-thE. D'Hoker, I. M. Krichever, and D. H. Phong, The Effective prepotential of N=2 supersymmetric SO(N(c)) and Sp(N(c)) gauge theories, Nucl. Phys. 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{'abstract': 'We derive a modular anomaly equation satisfied by the prepotential of the N = 2 ⋆ supersymmetric theories with non-simply laced gauge algebras, including the classical B r and C r infinite series and the exceptional F 4 and G 2 cases. This equation determines the exact prepotential recursively in an expansion for small mass in terms of quasi-modular forms of the S-duality group. We also discuss the behaviour of these theories under S-duality and show that the prepotential of the SO(2r + 1) theory is mapped to that of the Sp(2r) theory and viceversa, while the exceptional F 4 and G 2 theories are mapped into themselves (up to a rotation of the roots) in analogy with what happens for the N = 4 supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper.', 'arxivid': '1507.08027', 'author': ['M Billó billo@to.infn.it \nDipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly\n', 'M Frau frau@to.infn.it \nDipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly\n', 'F Fucito fucito@roma2.infn.it \nI.N.F.N -sezione di Roma\n\n\nDipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly\n', 'A Lerda lerda@to.infn.it \nDipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nDipartimento di Scienze e Innovazione Tecnologica\nUniversità del Piemonte Orientale\nand I.N.F.N. -Gruppo Collegato di Alessandria -sezione di Torino Viale T. Michel 11I-15121AlessandriaItaly\n', 'J F Morales morales@roma2.infn.it \nI.N.F.N -sezione di Roma\n\n\nDipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly\n', 'M Billó billo@to.infn.it \nDipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly\n', 'M Frau frau@to.infn.it \nDipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly\n', 'F Fucito fucito@roma2.infn.it \nI.N.F.N -sezione di Roma\n\n\nDipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly\n', 'A Lerda lerda@to.infn.it \nDipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nDipartimento di Scienze e Innovazione Tecnologica\nUniversità del Piemonte Orientale\nand I.N.F.N. -Gruppo Collegato di Alessandria -sezione di Torino Viale T. Michel 11I-15121AlessandriaItaly\n', 'J F Morales morales@roma2.infn.it \nI.N.F.N -sezione di Roma\n\n\nDipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly\n'], 'authoraffiliation': ['Dipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly', 'Dipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly', 'I.N.F.N -sezione di Roma\n', 'Dipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly', 'Dipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly', 'Dipartimento di Scienze e Innovazione Tecnologica\nUniversità del Piemonte Orientale\nand I.N.F.N. -Gruppo Collegato di Alessandria -sezione di Torino Viale T. Michel 11I-15121AlessandriaItaly', 'I.N.F.N -sezione di Roma\n', 'Dipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly', 'Dipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly', 'Dipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly', 'I.N.F.N -sezione di Roma\n', 'Dipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly', 'Dipartimento di Fisica and I.N.F.N. -sezione di Torino\nUniversità di Torino\nVia P. Giuria 1I-10125TorinoItaly', 'Dipartimento di Scienze e Innovazione Tecnologica\nUniversità del Piemonte Orientale\nand I.N.F.N. -Gruppo Collegato di Alessandria -sezione di Torino Viale T. Michel 11I-15121AlessandriaItaly', 'I.N.F.N -sezione di Roma\n', 'Dipartimento di Fisica Via della Ricerca Scientifica\nUniversità di Roma Tor Vergata\nI-00133RomaItaly'], 'corpusid': 92991987, 'doi': '10.1007/jhep11(2015)026', 'github_urls': [], 'n_tokens_mistral': 31282, 'n_tokens_neox': 25765, 'n_words': 15577, 'pdfsha': 'f8965ceef96b116203f4b4c6ad17de761046423c', 'pdfurls': ['https://arxiv.org/pdf/1507.08027v2.pdf'], 'title': ['S-duality and the prepotential in N = 2 ⋆ theories (II): the non-simply laced algebras', 'S-duality and the prepotential in N = 2 ⋆ theories (II): the non-simply laced algebras', 'S-duality and the prepotential in N = 2 ⋆ theories (II): the non-simply laced algebras', 'S-duality and the prepotential in N = 2 ⋆ theories (II): the non-simply laced algebras'], 'venue': []}

arxiv

Consistent thermodynamic derivative estimates for tabular equations of state Gary A Dilts Continuum Dynamics Group Los Alamos National Laboratory Mail Stop D413 87544Los AlamosNM Consistent thermodynamic derivative estimates for tabular equations of state Received March 4, 200514711+j8385Pt640270-c Numerical simulations of compressible fluid flows require an equation of state (EOS) to relate the thermodynamic variables of density, internal energy, temperature, and pressure. A valid EOS must satisfy the thermodynamic conditions of consistency (derivation from a free energy) and stability (positive sound speed squared). In many cases an analytic EOS is sufficient, but in many others, particularly when phase transitions are significant, the EOS is complicated and can only be specified in a table. For tabular EOS's such as SESAME from Los Alamos National Laboratory, these can take the form of a differential equation relating the derivatives of pressure and energy as functions of temperature and density, along with positivity constraints. Typical software interfaces to such tables based on polynomial or rational interpolants compute derivatives of pressure and energy and may enforce the stability conditions, but do not enforce the consistency condition and its derivatives. The consistency condition is important for the computation of various dimensionless parameters of an EOS which may involve derivatives up to second order. These parameters are in turn important for the development of more sensitive artificial viscosities and Riemann solvers that accurately model shock structure in regions near phase transitions. We describe a new type of table interface based on the tuned regression method, which is derived from a constrained local least squares regression technique. It is applied to several SESAME EOS's showing how the consistency condition can be satisfied to round-off while computing first and second derivatives with demonstrated second-order convergence. An improvement of 14 orders of magnitude over conventional derivatives is demonstrated, although the new method is apparently two orders of magnitude slower, due to the fact that every evaluation requires solving an 11-dimensional nonlinear system. INTRODUCTION The two most common techniques for modeling the heat generated by shock waves in the numerical simulation of compressible flows are artificial viscosity [1] and the Riemann solver [2]. Over the years, these have seen many enhancements and variations. For comprehensive summaries, see [3--5]. Unfortunately, the Riemann solver is only well developed for the ideal gas equation of state. Some attempts have been made for more complicated analytic EOS's, such as the Mie-Gruneisen EOS [6--7], but real materials in general have such a complicated EOS that it can only adequately be expressed in a table, for which there is no Riemann solver yet published. The SESAME library [8--9], which is widely used at Los Alamos National Laboratory and has been distributed throughout the world, contains tabular EOS's for many elements and will be the source of our examples. Our treatment in this paper is specific to the choice of variables used in SESAME but it may be straightforwardly modified for others. The mathematical description of the behavior of shock waves in real fluids with an arbitrary equation of state was described in detail in [10]. Four dimensionless quantities are important: 2 2 1 2 2 2 , , , S S V S V P V T PV T V P g P V T V T S P V γ γ ∂ ∂ ∂ = − Γ = − = = ∂ ∂ ∂ G ∂ ∂ .(1) The symbols represent pressure, specific volume, temperature, and entropy, respectively. The quantity , , , P V T S γ is the adiabatic exponent, Γ is the Grüneisen coefficient, is the dimensionless specific heat, and G is the fundamental derivative. The quantities g γ and G represent the slope and curvature of isentropes in the P V − plane, respectively. The quantity G is most important for the determination of shock wave structure. When , shocks occur in compression; when 0 > G 0 < G , shocks occur in rarefaction. In a numerical simulation this information must be incorporated into the switch used to turn on artificial viscosity or in the solution constructed by a Riemann solver. First and second order approximate Riemann solvers for real EOS's would make extensive use of G . Clearly, in order to construct these solvers we must first know how to compute physically realistic values of from tables. G Assume for the moment we have internal energy expressed as a function of specific volume V and entropy . The thermodynamic definitions of pressure and temperatureT are E S P , S V E P T V S E ∂ ∂ = − = ∂ ∂(2) and imply that V S P S V ∂ ∂ = − ∂ ∂ T .(3) This is the thermodynamic consistency condition and it amounts to a differential equation that a valid equation of state must satisfy. In the SESAME tables, pressure and energy are expressed as functions of temperature and density. With temperature and density independent, condition (3) takes the form E (1). The computation of involves second derivatives, so the derivatives of G (4) with respect to temperature and density also need to be satisfied to make physically realistic. How to do all this is the subject of this paper. G That satisfaction of these constraints is not automatic for traditional derivative evaluation schemes is illustrated by Table 1, which shows the average absolute value of the log-scale of the normalized consistency error, ( ) ls sgn ln 1 ε ε ε ≡ + , where 2 P E P E P T P T T T ε ρ 2 ρ ρ ρ ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ = − + + + + ⎜ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎟ ,(7) using several common methods [11--12] for computing derivatives of SESAME Mbar-cc/g using birational derivatives. These are well below the minimum allowed values of zero according to (6). The other methods showed similar results. Some software interfaces have options to enforce these positivity constraints [11], but it is not done in a way which simultaneously guarantees satisfaction of the consistency constraints. In the rest of this paper, we elucidate a technique to do precisely this. (7) for various derivative methods in common use. Bilinear, biquadratic, and birational methods are described in [11]. The "bihermitian" method is the bicubic hermitian method described in [12]. The actual software package of [11] was used for the first three methods, and the author's implementation was used for the last. NUMERICAL METHODS The tuned regression estimator (TRE) method [13] allows us to estimate derivatives of tabular EOS data while simultaneously guaranteeing (4) and (6). We shall summarize briefly the basic ideas of that paper and refer the reader thereto for more background, generality, detail and examples. Here, we shall just remark that it grew out of the application of the statistical method of local regression estimators [14] to the numerical solution of differential equations. Let us suppose we have data points in two dimensions with associated -dimensional data values which we presume to sample a continuousvalued function of two variables . Suppose we want to estimate derivatives of at an arbitrary point . First, we describe traditional polynomial interpolation methods, in a formalism that will prepare us for local and tuned regression estimation. 2 1 { } N i i y = ⊂ R m 1 { } N i i u = ⊂ R m m 2 : m u → R R n u ( ) 1 2 2 , x x x = ∈R Suppose the data points make up a Cartesian grid. Let a column vector of monomials be chosen from Table 2. Of course will be restricted to 4, 9, or 16. The derivative corresponding to a monomial n 2 : n p → R R n ( ) ( ) a b i j x x is ( ) ( ) a b a b i j x x + ∂ ∂ ∂ . Let be a subset of the data points of size consisting of the most-centered sub-grid that encloses Z n x of , , and 4 points for the bilinear, biquadratic, and bicubic methods, respectively. Denote the elements of by and let be a map between indices of points in and data points such that 2 2 × 3 3 × 4 × Z 1 { , , } n z z … k Z ( ) i k i y z = for . We wish to approximate by 1, , i= … n ( ) u x ( ) ( ) u x p x ζ = ,( ) ( ) u p J x J x ζ = ( ) i u p y = Λ i Λ is a constant m n × matrix, then and v= Λ Q ζ = Λ , and polynomial interpolants are said to reproduce the basis ( ) p x . It is wellknown that they converge with order n for smooth data, but produce oscillations near discontinuities. The great advantage of polynomial interpolation is speed, as the matrix depends only on the Table 2. Sets of monomials used for several traditional interpolation schemes [11--12]. The biquadratic method here includes 3 extra terms of cubic and quartic order than that described in [11], which might properly be termed the "quadratic" method. x x x x x x x x x x x x x x x x x x x x x x x x ⎡ ⎣ ⎤ ⎦ The tuned regression method is a meshfree method and as such, notions of nearness are determined by the value of a real-valued weight function ( ) , i w x y , instead of a grid. The weight function is generally smooth, centrally peaked about i y , and has compact support. When is large, ( , i w x y ) x is close to i y . When ( ) , i w x y is small, x is far away from i y . Although the grids used in the SESAME tables are non-uniformly spaced Cartesian, meshfree techniques may be applied to them. In this paper, we use the weight function ( ) 1 1 2 2 2 1 2 1 2 j j 4 4 1 2 , j j j j j y x y x w x y B B h h ⎛ ⎞ ⎛ ⎞ − − = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ N N h h ,(8)( ) ( ) ( ) ( ) 1 , i i p x y J x p y D p y x − i ≡ = − ,(9) where is a constant diagonal matrix. Suppose that D ( ) x β is a m n × matrix whose columns are derivative estimates of u , the same derivatives that are used in . Then ( ) 0 γ = D E ( ) ( ) ( ) , i x p x y γ E . The inverse mapping is given by ( ) γ β = F with ( ) ( ) β β = E F and ( ) ( ) γ γ = F E . We measure the average error of the Taylor Series expansion from x to all nearby points i y with ( ) ( ) ( ) ( ) ( ) ( ) 2 , , j i j j x u x p x y w x γ = − ∑ R E y .(10) If we optimize ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , , , , , T i i i i i i T i i i i , x u x x P x p x y w x y P x p x y p x y w x y β ψ ψ − = = = ∑ ∑(11) It is well-studied in the statistics literature [14] and it is easy to show it has the form ( ) ( ) ( ) x x J x β ζ = , similar to polynomial interpolants, and it has the reproducing property, just like polynomial interpolants and the moving-least squares (MLS) estimators used in the engineering literature [15]. In fact, the zeroth-derivative estimate of LRE is identical to that of MLS [13]. The convergence rates for x ∈ R for the ν -th derivative are 1 n ν − + for n ν − odd and 2 n ν − + for n ν − even [14]. The moment matrix in (11) becomes singular when the data points in the neighborhood of ( ) P x x become coplanar, or there are less than n of them, so the smoothing length must be made large enough to prevent these two situations. If it is too large however the procedure becomes expensive, as more neighbors are included in the sums. The proper selection of smoothing length for LRE is a fine art discussed in [14]. It is not known exactly how much of that discussion applies to TRE, but in practice, it is seen that at least neighbors are also required. It is also wise to monitor condition numbers in the course of solution. n The present application makes use of the following specializations: [ ] , T x T ρ = , [ , ] T i i i y T ρ = , [ ] , T u E P = ,and ( ) ( ) ( ) 2 2T T T E E E E E E T T T P P P P P P T T T T T ρ ρ ρ ρ ρ β ρ ρ ρ β β β ρ β β β β β β γ γ γ γ γ β γ ρ γ γ γ = ⎡ ⎤ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎢ ⎥ = ⎢ ⎥ ∂ ∂ ∂ ∂ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ ⎣ ⎦ = − − ⎡ ⎤ = = ⎣ ⎦ ⎡ ⎤ = = ⎢ ⎥ + ⎣ ⎦ D F E … …2, , , , , , , , , . T p We have eliminated 1,0 β , which represents the pressure, through 0 = D , to define γ . The evaluation and optimization of (10) is aided by the observation that it can be rewritten as ( ) Tr Tr 2 T T W P U = + − R E E E ,(13) where (14) , , T T i i i i i i i i i i i i P p p w U u p w W uu = = = ∑ ∑ ∑ T w and ( ) , i i p p x y = , . ( ) , i i w w x y = This prescription addresses the consistency condition (4), but the stability conditions (6) require further attention. We define three possible differential constraints to use: ( ) ( ) ( ) 2 1,0 1,1 0,2 0,1 1,2 0 a 0 0 c T β β ρ β β β − − = = = b(15) which represent the consistency and stability conditions and adopt a multi-pass algorithm to enforce all constraints simultaneously: (i) First, we try (15)(a) everywhere. (ii) 4 Where 0,1 0 β < in the result of (i) we apply the combination of (15)(a) and (b). (iii) Where 1,2 0 β < in the result of (i) we apply the combination of (15)(a) and (c). (iv) Where both 0,1 0 β < and 1,2 0 β < in (i), or where 1,2 0 β < in (ii) or 0,1 0 β < in (iii), we apply the combination of (15)(a),(b) and (c). For the SESAME tables examined to date the number of locations where (ii)-(iv) are required is very small. The technique results in the values of 0,1 β and 1,2 β being clamped to zero in regions where passes (i)-(iii) cause them to be negative. In the software, one needs to code four possibilities for corresponding to the combinations in passes (i)-(iv). For the four different passes we solve constrained E systems with { } 1,0 β , { } 1,0 0,1 , β β , { } 1,0 1,2 , β β , { } 1,0 0,1 1,2 , , β β β eliminated respectively, which results in solving1 systems, respectively. This illustrates a key feature of tuned regression: by eliminating some derivatives, we solve a smaller system with enhanced accuracy. The penalty is that the smaller system is more complicated. 1 11, 10 10, 9 9 × × × Both local regression and tuned regression in regions where only the consistency constraint is enforced produce approximations with smoothness equal to that of the weight function. A Mathematica [16] program has been written to symbolically optimize (13) for arbitrary ,U ,W and . It then generates code which is spliced into a C++ library called LORELI (LOcal REgression LIbrary) which is used to operate on the SESAME data. The appendix exhibits the expressions for generated by this Mathematica program in terms of the matrices in P E R (14) when the various combinations of constraints in (15) are active. EXAMPLE: ANALYTIC Let , and suppose that 2 [ , , , , , ] u T T T J u u u u u u ρ ρ = ∂ ∂ ∂ ∂ ∂ ∂ 2 ρ J u J = Λ , and . That is, u is a quadratic function that exactly satisfies the consistency and stability conditions. Such an example is given by ( ) 0 u J = D 2 1 2 E T T u P T T ρ ρ ρ ⎡ ⎤ − + + + ⎡ ⎤ ⎢ ⎥ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ − + + ⎣ ⎦ (16) with 1 1 1 2 0 0 0 1 0 0 1 2 − ⎡ ⎤ Λ = ⎢ ⎥ − ⎣ ⎦ .(17) Now suppose that ( ) ( ) i i u u y p y = =Λ i ) i . Then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 , i i i x p y x J x p y w x y γ − = Λ − ∑ R E (18) 5 6 ) is minimized if , because ( u J γ = F ( ) ( ) ( ) ( ) u x J x J x γ = =Λ E and R is identically zero. In other words, tuned regression possesses the reproducing property just like polynomial interpolation and local regression: a polynomial solution of the differential constraints evaluated at discrete points { } i y will be exactly reproduced at an arbitrary{ , } T ρ , to round-off. and can be used by a software implementation as a verification tool. The LORELI library mentioned above has been so checked on this example and does indeed reproduce to round-off. On the basis of these results we surmise that the tuned regression method for SESAME data is at least third-order accurate. There is a formal proof , but its presentation is out of scope here, however numerical examples below will confirm it. EXAMPLE: OXYGEN We choose SESAME table 5011 for Oxygen at low temperatures as our first example because the 23x51 grid is fairly uniform. Most SESAME tables have grids that are exponential in character to handle the many orders of magnitude variation of temperature and density required. This leads to ill-conditioned matrices in the TRE solution process, and requires special treatment as described below. Table 5011 however does not present this problem. On a 45x103 grid TRE agrees well with the input data and produces no discernable difference to the eye. Figure 1 shows the pressure derivative with respect to density, and a prominent feature is the flat annulus at low temperature. This is a region where the stability constraints were active, and the algorithm did what it was supposed to. The zeroth derivative estimates were not seen to echo this feature to the eye, which illustrates how the derivatives are estimated independently in TRE. The estimate of the derivative is not the derivative of the estimate, as it is in finite element or spectral methods. The two do converge however as the data become dense and the smoothing length goes to zero. The value of the un-normalized consistency error was everywhere less than 1.e-13, which is close to round-off, as promised. Figure 2 shows the relative error between the TRE result and the input table values when the input and output grid are identical. There is good agreement except at low temperatures where the constraints become active. LOGARITHMIC FORM As mentioned above, the exponential grids present in many SESAME tables present numerical difficulties, so a method must be devised to treat the ill-conditioned matrices 2 γ ∂ ∂ 2 R that appear in the Newton solver for the equations 0 γ ∂ ∂ = R . These occur because the wide range of powers that appear in the moment matrix in (13) get drastically out of balance when applied to very large numbers. For example, assume 8 decades of range in table coordinates and 45 points, which gives a ratio of about 1.5 in the size of successive intervals. Adjusting so that there are 49 neighbor points it is easy to verify the condition number of when and when . One way to restore good conditioning is to use a preconditioner in the solver, and this is under In terms of these variables, the consistency and stability conditions become ( ) 0 ( 0 ( P P a r b P c r ε ε τ ε τ ∂ ∂ + = + ∂ ∂ ∂ ≥ ∂ ∂ ≥ ) ) ∂(20) These we refer to as semi-log constraints. It is the simplest form of the consistency condition and is linear, just like the original consistency condition, and thus requires only one 11x11 solve at most points. This coordinate change was found to work for a few tables, but a further step was required to get satisfactory behavior, because even though the new grid is not exponential, the data for pressure and energy has become exponential, and the problem of ill-conditioned matrices still appears. We now transform the dependent variables by means of ζ η ζ η τ ζ η ε τ ∂ ∂ ⎛ ⎞ ⎛ ⎞ − + − = + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ∂ ≥ ∂ ∂ ≥ ∂(22) Notice now that the consistency constraint is nonlinear, whereas previously it was linear. We refer to these as log-log constraints. We use the LRE solution (11) as the initial condition for a Newton solver or steepest-descent solver to optimize R . In practice, we typically see convergence in 3-5 Newton iterations with this initial condition. The same four-pass strategy for enforcing the stability constraints applies as was used before. The LORELI library thus contains twelve separate encodings of the residual function and its derivatives: for each type of constraint (flat, semi-log, and log-log), there are four versions corresponding to the combination of consistency and stability constraints listed in the four passes following equation (15). The appendix exhibits these residual functions. The expressions for the dimensionless derivatives when logarithmic transformations are employed are given by: but the derivation of the log-log TRE method would have to be modified to add these two equations as constraints to those of (22). As it stands, the expression for G in (23) is consistent with the log-log TRE method implied by (22) above. ( ) ( ) ( ) ( ) ( ) ( ( ) (( ) )) ( ) ( ) 2 2 3 2 2 3 2 2 2 2 2 2 , , , 2 3 2 EXAMPLE: MOLYBDENUM The log-log TRE method was applied to SESAME table 2984 for Molybdenum and the results are shown in Fig. 3. The input grid size was 37 65 × , the output grid was , and once again there was no discernable difference to the eye between the two . The normalized log-log consistency error, given by was smaller in magnitude than 2e-16 at all points as required and is 14 orders of magnitude smaller than the values of Table 1 obtained by traditional derivatives, a substantial improvement. The condition number of the final iteration of the Newton solver at each evaluation point was everywhere less than , and the number of Newton iterations required to converge to a tolerance of 1.e-13 was at all points between 1 and 4 iterations, a very reasonable number for such a nonlinear problem. The condition numbers are rather high, but it appears possible to reduce them considerably by a simple scaling procedure which may be reported in subsequent publications. 5 10 The dimensionless quantities of (23) were computed using the TRE derivatives. At behavior has been mostly eliminated and in the plot of γ one can see the outline several phase boundaries. At low T and high ρ we expect to find the solid phase, and at lowT and low ρ we expect to find the mixed phase. In these two phases, the theory leading to the definition of the dimensionless derivatives is incomplete because it does not include the effects of deviatoric strains or stresses and thus nonsensical results may be inescapable. Also, there are known jump conditions on γ and Γ that may come into play across phase transitions, and these have not been enforced. One of the main points of all this calculation is the determination of the sign of G , which is negative mostly in the mixed phase region, so the ability to provide reliable guidance to numerical methods for shock waves in these regions is clouded. In light of these observations, it seems an appropriate approach is to include the explicit phase boundaries in the EOS evaluator and make reasonably correct estimates of the dimensionless derivatives when they are crossed. It is also possible that some of the divergent behavior seen with TRE is caused by numerical difficulties in early iterations of the Newton solver, and this should be investigated. Figure 6 shows the γ calculated by birational method of the EOSPAC library [11], which does not contain any divergent regions like the TRE result. Figure 7 shows a comparison of γ by birational and TRE at three different temperatures, roughly 2 eV, 1 keV, and 100 keV. In general, for positive ln ρ , the two agree fairly well, except at the high-density boundary of the table. At the highest temperatures, both results approach the theoretical value for an ideal gas. For negative ln ρ , the two results are always in disagreement. Perhaps this is because the constraints are more active in that region. This too should be further investigated. 9 To compare the computational cost of each method, a grid of 750x1350=1012500 points was constructed. The EOSPAC birational evaluation on a 1.7GHz Pentium IV system took 5.7 sec. The time for LRE was 172 sec, which involved neighbor finding using a general two-dimensional binning algorithm and solving and performing one 6x6 linear solve. The log-TRE method doing a single 11x11 linear solve took 249 sec, and the log-log-TRE method doing multiple 11x11 solves took 634 sec, which are 40 and 100 times slower than EOSPAC, respectively. This is disappointing, but not unexpected because in addition to the linear solves involved, the algebra for TRE is much more complicated than for EOSPAC or even LRE. In practice, this computational cost can be avoided by evaluating all derivatives of an EOS on a fine grid and storing them for later evaluation by normal means of interpolation, such as LRE. Presumably the consistency condition would not be violated too much. This too needs further investigation. It must be observed that a linear TRE formulation is possible which would guarantee the consistency condition and involve only a 5x5 solve. Presumably, this would be competitive with LRE and EOSPAC from a performance viewpoint, but one would not be able to use it to compute the fundamental derivative, which requires quadratic TRE at a minimum. The LORELI implementation of log-log TRE was applied to tables for copper (3333), aluminum (3719), and tin (2160) with similar results, except for some anomalous divergences in one corner which seem to be due to poor choice of smoothing length. When the smoothing length is too small, there are too few neighbors, and the LRE or TRE methods develop ill-conditioned matrices. It becomes an issue near a table boundary because there are fewer neighbors than in the interior. If the smoothing length is too large, it may not be possible to satisfy the constraints with finite values. For LRE there is a fairly well-developed methodology for choosing the smoothing length, but more research is needed to do the same for TRE. The molybdenum table 2984 does not appear to contain Maxwell constructions for the removal of van der Waals loops. When the log-log TRE method is applied to tables that seem to have Maxwell constructions, such as gold (SESAME 2700), there are severe convergence problems in the vicinity thereof. This may be because second-order interpolation methods generally sustain oscillations in the vicinity of discontinuities, and perhaps because of the exponentials in the TRE method, these oscillations cause serious ill-conditioning in the solvers, both Newton and steepest descent. Some tables seem to have apparently arbitrary abrupt transitions at the edges which also cause a similar problem. These tables may require more physical adjustments at the edges before the TRE method is robust on them. Clearly the issue of Maxwell constructions requires more research. Perhaps they can be detected, by a linear LRE estimate for example, which always seems to be monotone (although a proof is unknown to the author), and then a separate technique applied. This is left for future investigations. CONVERGENCE AND ACCURACY To test convergence of the TRE method, we use the following biquartic EOS . The error in the estimates was measured and two sample results for pressure are plotted in Fig. 8. The curves have been truncated on the left where convergence ceased for each method. In particular, the linear methods were the most robust (working at smaller mesh spacings), followed by the quadratic and cubic methods. The bicubic hermitian method was the least robust. Table 3 shows the convergence rates for the various methods coded by the author. The local regression methods converge in keeping with the theoretical rates given in section 2 above. The tuned regression zeroth derivatives converge at the same rates as the quadratic local regression estimator, which is encouraging. For zeroth derivatives the convergence is fourth order, which is remarkable since only a quadratic polynomial is used in the modeling. Table 4 shows the log 0 y Δ = intercept of the convergence curves, which gives an indication of the relative accuracy of the various methods. The various regression methods trade advantages in different derivatives with their competitors of like polynomial order. The bihermitian method, as coded by the author, seems to have a markedly higher intercept than the other cubic methods, implying that a finer table is required to get the same accuracy as a bicubic or cubic LRE method could get. The tuned regression estimator does the best job on consistency of course, as evidenced by the intercept value. The other methods all converge in consistency error, as they must if they converge at all, but unless you build consistency into the algorithm, you can not guarantee it. Tuned regression gets consistency to round-off, so no convergence figure appears in its column. "Biherm" denotes bicubic hermitian. Fig. 9(a), we show the molybdenum table 2984 sampled at 21583 random points uniformly distributed across the full range of temperature and density using the log-log TRE of section 5 with 3 10 S S P ε = = − , which flattens the graphs of η and ζ compared to Fig. 3. Each point is colored with the value ofζ obtained from TRE. This data is then used as an input "table" for resampling at a finer uniform random distribution of 64749 points, again using log-log TRE. The results are plotted in Fig. 9(b) and exhibit the expected behavior. The consistency error was zero to fifteen decimal places as with the other TRE examples. The significance of the meshfree nature of TRE for EOS data is that traditional rectangular-grid tables may be supplemented with extra data points near Maxwell construction or phase change boundaries to achieve greater resolution near these discontinuities with no loss of consistency or accuracy. E P E T ∂ ∂ P T ∂ ∂ E ρ ∂ ∂ P ρ ∂ ∂ 2 E T ρ ∂ ∂ ∂ 2 P T ρ ∂ ∂ ∂E P E T ∂ ∂ P T ∂ ∂ E ρ ∂ ∂ P ρ ∂ ∂ 2 E T ρ ∂ ∂ ∂ 2 P T ρ ∂ ∂ ∂ CONCLUSION We have shown that traditional numerical derivatives of equation of state tables do not simultaneously satisfy the thermodynamic consistency and stability conditions, and that a tractable method to do so can be developed from the tuned regression estimator. The LORELI implementation has demonstrated the reproducing property for quadratic analytic EOS's which satisfy the consistency and stability conditions. Trials on a few SESAME tables have shown that the consistency and stability constraints can be simultaneously enforced to round-off without sacrificing accuracy and that theoretical values for γ and G are approached at high temperature. Versions of the theory and 13 software were developed for flat, semi-log, and log-log coordinates, the last being necessary to handle tables with exponential grids. The convergence rates follow those of the statistical local regression estimator, giving fourth order for zeroth derivatives, and second order for first and second derivatives. The TRE method is apparently much more expensive than traditional derivatives, however. The meshfree character of the TRE method was convincingly demonstrated and provides a basis for augmenting conventional tables with extra data points near physical discontinuities. There are outstanding issues regarding phase boundaries, Maxwell constructions, and table-edge drop-offs which require further research before the technique can be made into a fully robust tool. APPENDIX: RESIDUAL EXPRESSIONS The residual for the consistency constraint given in (15a) is W 11 2 W 22 2 Γ 0 U 11 P 11 Γ 0 2 P 12 Γ 1 P 13 Γ 2 P 14 Γ 3 P 15 Γ 4 P 16 Γ 5 Γ 1 U 12 P 22 Γ 1 2 P 23 Γ 2 P 24 Γ 3 P 25 Γ 4 P 26 Γ 5 Γ 3 U 14 P 44 Γ 3 2 P 45 Γ 4 P 46 Γ 5 Γ 4 U 15 P 55 Γ 4 2 P 56 Γ 5 Γ 5 P 66 Γ 5 2 U 16 Γ 2 U 21 Ρ 2 P 13 Γ 7 Ρ 2 P 14 Γ 8 Ρ 2 P 15 Γ 9 Ρ 2 P 16 Γ 10 Ρ 2 U 13 P 11 Ρ 4 2 P 33 2 Γ 2 P 34 Γ 3 P 35 Γ 4 P 36 Γ 5 T P 11 Ρ 2 P 12 Ρ 2 Γ 6 Γ 6 T U(A1) with Γ 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 Γ 2 Ρ 2 T Γ 6 Γ 6 Γ 7 Γ 8 Γ 9 Γ 10 .(A2) The residual for the constraints (15) (A3) with Γ 0 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 1 Ρ 2 T Γ 5 Γ 5 Γ 6 Γ 7 Γ 8 Γ 9 .(A4) The residual for the constraints (15) with Γ 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 Γ 2 Ρ 2 T Γ 6 Γ 6 0 Γ 7 Γ 8 Γ 9 .(A6) The residual for the constraints (15) with Γ 0 0 Γ 1 Γ 2 Γ 3 Γ 4 Γ 1 Ρ 2 T Γ 5 Γ 5 0 Γ 6 Γ 7 Γ 8 .(A8) need be computed only once for all x . , ( ) ( ) , ( ) , ( ) ( ) , ( ) ( ) , ( ) ( ) Taylor series expansion from x to i y . Now suppose that we want ( ) x β to satisfy a set of differential constraints at x , say ( ) 0 β = D. Through the implicit function theorem, this implies that a subset of the β can be eliminated, or equivalently, we can change variables to a smaller number of variables γ such that known as the local regression estimator (LRE), and has an explicit solution: ε and are arbitrary but must be positive. In the rest of this paper we set them equal to 1 so that the quantities 0 P ζ and η have a minimum value of zero and are always positive otherwise. The larger we make 0 ε and the flatter the transformed data surfaces become in the large. The consistency and stability significant if the specific heat becomes small as discussed below. The expression for G makes use of the consistency constraint (22a). One could further incorporate its derivatives with respect to τ and : high temperatures, the values of γ and G approach the theoretical values for monatomic ideal gases of 5/3 and 4/3, respectively, which gives some confidence to the calculations. On the other hand, at lower temperatures, there seem to be rather large divergent regions which correlate well with the flat regions at lower temperature in the plot of η this is the cause of the divergent regions, the dimensionless derivatives are multiplied by the appropriate power of η τ ∂ ∂ and plotted inFig. 5. The divergent the consistency and stability constraints. It is not reproducible by either quadratic or cubic polynomial, LRE, or TRE methods. This EOS was sampled on a generate a table of energy and pressure that was input to the various estimation procedures operating on 15 FIG. 2 . 20 FIG. 5 .FIG. 6 . 23 FIG. 8 .FIG. 9 . 15220562389Error for Oxygen table 5011 when input grid equals output grid. (a) Energy. (b) Pressure. 18 FIG. 4. η τ ∂ ∂ from log-log TRE on table 2984. Dimensionless derivatives by log-log TRE on table 2984 multiplied by η τ ∂ ∂ or its square. (a) Log-scale of η τ γ ∂ ∂ . (b) Log-scale of η τ ∂ ∂ Γ . (c) Log-scale of Plot of γ using EOSPAC derivatives for table 2984. 22 FIG. 7. Comparison of EOSPAC and TRE values of γ at 2 ev, 1 kev, 100 kev T = . Error of various methods on analytic EOS of (25) as a function of mesh spacing in the input table for ζ and consistency. (a) SESAME table 2984 sampled at 21583 uniformly distributed random points using log-log TRE. (b) Sampling of the data from (a) at 64749 different uniformly distributed random points using log-log TRE. colored with the value of the interpolant. The standard deviation for all methods was approximately 1.0e-15. Thus at most points, traditional derivatives match the consistency condition to a little less than three decimal places. The minimum value oftable 2984 for molybdenum. Since pressure and energy vary by 6 and 12 orders of magnitude respectively in this example the normalizing denominator is necessary for a fair assessment of the error. The table grid was 37 65 × , and the evaluation grid was . 75 135 × E T ∂ ∂ was -0.198 Mbar-cc/K, and of P ρ ∂ ∂ was -33.2 Table 1 . 1Table 1. Average absolute value of the log-scale of the normalized consistency errorMethod Error bilinear 0.00665 biquadratic 0.00240 birational 0.00175 bihermitian 0.00107 Table 3 . 3lre1 lre2 lre3 tre bilinear biquad bicubic biherm 2 4 4 3 . 9 2 3 4 4 . 3 2 4 4 4 . 1 2 3 4 4 . 4 2 2 3 . 9 2 1 2 3 3 . 3 2 2 4 2 1 2 3 3 . 4 2 2 3 . 9 2 1 2 3 . 9 3 . 4 2 2 4 2 1 2 3 3 . 6 2 2 2 1 2 2 . 9 2 . 2 2 2 2 1 2 2 . 9 2 . 4 Consistency 2 2 4 1 2.1 3.1 3.4 Table 3 . 3Convergence rate for various method on biquartic EOS (25). Linear local regression can not compute second derivatives accounting for vacancies in column 2. Table 4 . 4lre1 lre2 lre3 tre bilinear biquad bicubic biherm 0.4 0.4 0.9 0.4 0.4 0.4 0 1.2 0.2 0.6 1 0.4 0.3 0.4 0.1 1.4 0.9 1 1 1 0.9 0.8 0.6 2.8 0.6 0.8 1.3 0.6 0.3 0.1 -0.2 3.3 0.6 0.8 0.8 0.9 0.6 0.3 0.1 3 0.8 0.7 1.5 0.7 0.9 0.9 0.7 3.2 0.9 1.1 1.4 1.1 1.3 1.1 0.7 2.3 0.7 1.3 1.6 1.3 1.4 1.2 1 2.9 Consistency 0.5 0.2 0.1 -15.5 0.1 -0.1 -0.3 3 Table 4 . 4Extrapolated intercept of convergence curves similar to those of Fig. 8. For methods of like order of convergence, these figures indicate relative accuracy. "Biherm" denotes bicubic hermition.log 0 y Δ = 8. MESHFREE ILLUSTRATION Finally, in T P 13 P 23 Γ 7 T P 14 P 24 Γ 8 T P 15 P 25 Γ 9 T P 16 P 26 Γ 1021 U 22 P 11 T 2 2 P 12 T P 22 2 Γ 6 Γ 7 U 23 P 33 Γ 7 2 P 34 Γ 8 P 35 Γ 9 P 36 Γ 10 Γ 8 U 24 P 44 Γ 8 2 P 45 Γ 9 P 46 Γ 10 Γ 9 U 25 P 55 Γ 9 2 P 56 Γ 10 Γ 10 P 66 Γ 10 2 U 26 (a) and (b) active is13 Γ 1 P 14 Γ 2 P 15 Γ 3 P 16 Γ 4 Γ 1 U 21 Ρ 2 P 13 Γ 6 Ρ 2 P 14 Γ 7 Ρ 2 P 15 Γ 8 Ρ 2 P 16 Γ 9 Ρ 2 U 13 P 11 Ρ 4 2 P 35 Γ 3 P 36 Γ 4 T P 11 Ρ 2 P 12 Ρ 2 Γ 5 Γ 5 T U 21 U 22T P 13 P 23 Γ 6 T P 14 P 24 Γ 7 T P 15 P 25 Γ 8 T P 16 P 26 Γ 9W 11 2 W 22 2 Γ 0 U 11 P 11 Γ 0 2 P Γ 2 U 14 P 44 Γ 2 2 P 45 Γ 3 P 46 Γ 4 Γ 3 U 15 P 55 Γ 3 2 P 56 Γ 4 Γ 4 P 66 Γ 4 2 U 16 P 33 2 Γ 1 P 34 Γ 2 P 11 T 2 2 P 12 T P 22 2 Γ 5 Γ 6 U 23 P 33 Γ 6 2 P 34 Γ 7 P 35 Γ 8 P 36 Γ 9 Γ 7 U 24 P 44 Γ 7 2 P 45 Γ 8 P 46 Γ 9 Γ 8 U 25 P 55 Γ 8 2 P 56 Γ 9 Γ 9 P 66 Γ 9 2 U 26 12 Γ 1 P 13 Γ 2 P 14 Γ 3 P 15 Γ 4 P 16 Γ 5 U 21 Ρ 2 P 14 Γ 7 Ρ 2 P 15 Γ 8 Ρ 2 P 34 Γ 3 P 35 Γ 4 P 36 Γ 5 T P 11 Ρ 2 P 12 Ρ 2 Γ 6 T P 14 P 24 Γ 7 T P 15 P 25 Γ 8 T P 16 P 26 Γ 9(a) and (c) active is 14 W 11 2 W 22 2 Γ 0 U 11 P 11 Γ 0 2 P Γ 1 U 12 P 22 Γ 1 2 P 23 Γ 2 P 24 Γ 3 P 25 Γ 4 P 26 Γ 5 Γ 3 U 14 P 44 Γ 3 2 P 45 Γ 4 P 46 Γ 5 Γ 4 U 15 P 55 Γ 4 2 P 56 Γ 5 Γ 5 P 66 Γ 5 2 U 16 Γ 2 P 16 Γ 9 Ρ 2 U 13 P 11 Ρ 4 2 P 33 2 Γ 2 Γ 6 T U 21 U 22 P 11 T 2 2 P 12 T P 22 2 Γ 6 Γ 7 U 24 P 44 Γ 7 2 P 45 Γ 8 P 46 Γ 9 Γ 8 U 25 P 55 Γ 8 2 P 56 Γ 9 Γ 9 P 66 Γ 9 2 U 26 (A5) 13 Γ 1 P 14 Γ 2 P 15 Γ 3 P 16 Γ 4 Γ 1 U 21 Ρ 2 P 14 Γ 6 Ρ 2 P 15 Γ 7 Ρ 2 P 16 Γ 8 Ρ 2 U 13 P 11 Ρ 4 2P 34 Γ 2 P 35 Γ 3 P 36 Γ 4 T P 11 Ρ 2 P 12 Ρ 2 Γ 5 T P 14 P 24 Γ 6 T P 15 P 25 Γ 7 T P 16 P 26 Γ 8(a), (b) and (c) active is W 11 2 W 22 2 Γ 0 U 11 P 11 Γ 0 2 P Γ 2 U 14 P 44 Γ 2 2 P 45 Γ 3 P 46 Γ 4 Γ 3 U 15 P 55 Γ 3 2 P 56 Γ 4 Γ 4 P 66 Γ 4 2 U 16 P 33 2 Γ 1 Γ 5 T U 21 U 22 P 11 T 2 2 P 12 T P 22 2 Γ 5 Γ 6 U 24 P 44 Γ 6 2 P 45 Γ 7 P 46 Γ 8 Γ 7 U 25 P 55 Γ 7 2 P 56 Γ 8 Γ 8 P 66 Γ 8 2 U 26 (A7) . J , Von Neumann, R D Richtmeyer, J. Appl. Phys. 21232J. Von Neumann and R. D.Richtmeyer, J. Appl. Phys. 21, 232 (1950) . S K Godunov, Mat. Sb. 47271S.K.Godunov, Mat. Sb. 47, 271 (1959) Numerical Computation of Internal and External Flows. C Hirsch, John Wiley & Sons2New YorkC. Hirsch, Numerical Computation of Internal and External Flows (John Wiley & Sons, New York, 1990) Vol. 2, E F Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. New YorkSpringer-VerlagE.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer- Verlag, New York, 1997) R J Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University PressR.J. Leveque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, 2002 . G H Miller, E G Puckett, J , Comp. Phys. 128134G.H.Miller and Puckett E.G., J. Comp. Phys. 128, 134 (1996) . K M Shyue, J Comp, Phys. 171678Shyue, K.M., J. Comp. Phys. 171, 678 (2001) T-4 Handbook of Material Properties Data Bases. K S Holian, Los Alamos National Laboratory Report LA-10160-MSunpublishedK. S. Holian, ed., T-4 Handbook of Material Properties Data Bases, Los Alamos National Laboratory Report LA-10160-MS, 1984 (unpublished) SESAME: The Los Alamos National Laboratory Equation of State Database. S P Lyon, J D Johnson, UR 92-3407Los Alamos National Laboratory Report LAunpublishedS.P.Lyon and J.D.Johnson, SESAME: The Los Alamos National Laboratory Equation of State Database, Los Alamos National Laboratory Report LA-UR 92-3407, 1992 (unpublished) . R Menikoff, B J Plohr, Rev Mod Phys. 6175R.Menikoff and B.J. Plohr, Rev Mod Phys 61, 75 (1989) EOSPAC 5 User Manual. D A Pimentel, UR 03-4510Los Alamos National Laboratory Report LAunpublishedD.A.Pimentel, EOSPAC 5 User Manual, Los Alamos National Laboratory Report LA-UR 03-4510, 2003 (unpublished) The LEOS Interpolation Package. F N Fritsch, ID-148554-REV-1Lawrence Livermore National LaboratoryReportunpublishedF. N. Fritsch, The LEOS Interpolation Package, Lawrence Livermore National Laboratory Report UCRL-ID-148554-REV-1, March 12, 2003 (unpublished) G A Dilts, A Haque, J Wallin, Meshfree Methods for Partial Differential Equations. M. Griebel and M. A. SchweitzerBerlinSpringer-Verlag26G.A.Dilts, A.Haque and J. Wallin, in Meshfree Methods for Partial Differential Equations, edited by M. Griebel and M. A. Schweitzer, Lecture Notes in Computational Science and Engineering 26, (Springer-Verlag, Berlin, 2002) J Fan, I Gijbels, Local Polynomial Modeling and Its Applications. LondonChapman and HallJ. Fan and I. Gijbels, Local Polynomial Modeling and Its Applications (Chapman and Hall, London, 1996) . T Belytschko, Y Y Lu, L Gu, Int. J. Num. Meth. Eng. 37229T. Belytschko, Lu, Y.Y., and Gu, L., Int. J. Num. Meth. Eng. 37, 229 (1994) ∂ ∂ with consistency and stability constraints active. Note flat annulus at low temperature caused by activation of stability constraint (box). ∂ ∂ with consistency and stability constraints active. Note flat annulus at low temperature caused by activation of stability constraint (box). Results of log-log TRE on table 2984. 75 x 135 grid. (a) Log of shifted energy. FIG. 3. Results of log-log TRE on table 2984. 75 x 135 grid. (a) Log of shifted energy.

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{'abstract': "Numerical simulations of compressible fluid flows require an equation of state (EOS) to relate the thermodynamic variables of density, internal energy, temperature, and pressure. A valid EOS must satisfy the thermodynamic conditions of consistency (derivation from a free energy) and stability (positive sound speed squared). In many cases an analytic EOS is sufficient, but in many others, particularly when phase transitions are significant, the EOS is complicated and can only be specified in a table. For tabular EOS's such as SESAME from Los Alamos National Laboratory, these can take the form of a differential equation relating the derivatives of pressure and energy as functions of temperature and density, along with positivity constraints. Typical software interfaces to such tables based on polynomial or rational interpolants compute derivatives of pressure and energy and may enforce the stability conditions, but do not enforce the consistency condition and its derivatives. The consistency condition is important for the computation of various dimensionless parameters of an EOS which may involve derivatives up to second order. These parameters are in turn important for the development of more sensitive artificial viscosities and Riemann solvers that accurately model shock structure in regions near phase transitions. We describe a new type of table interface based on the tuned regression method, which is derived from a constrained local least squares regression technique. It is applied to several SESAME EOS's showing how the consistency condition can be satisfied to round-off while computing first and second derivatives with demonstrated second-order convergence. An improvement of 14 orders of magnitude over conventional derivatives is demonstrated, although the new method is apparently two orders of magnitude slower, due to the fact that every evaluation requires solving an 11-dimensional nonlinear system.", 'arxivid': 'physics/0510195', 'author': ['Gary A Dilts \nContinuum Dynamics Group Los Alamos National Laboratory Mail Stop D413\n87544Los AlamosNM\n'], 'authoraffiliation': ['Continuum Dynamics Group Los Alamos National Laboratory Mail Stop D413\n87544Los AlamosNM'], 'corpusid': 21921020, 'doi': '10.1103/physreve.73.066704', 'github_urls': [], 'n_tokens_mistral': 14046, 'n_tokens_neox': 11860, 'n_words': 7796, 'pdfsha': '9448568005a19cab90e484815fdfba01c6edf751', 'pdfurls': ['https://export.arxiv.org/pdf/physics/0510195v1.pdf'], 'title': ['Consistent thermodynamic derivative estimates for tabular equations of state', 'Consistent thermodynamic derivative estimates for tabular equations of state'], 'venue': []}

arxiv

Zero modes and low-energy resolvent expansion for three dimensional Schrödinger operators with point interactions 8 Jan 2019 Raffaele Scandone Zero modes and low-energy resolvent expansion for three dimensional Schrödinger operators with point interactions 8 Jan 2019 We study the low energy behavior of the resolvent of Schrödinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions. Introduction and main results A central topic in quantum mechanics is the study of quantum systems subject to very short-range interactions, supported around a submanifold of the ambient space. A relevant situation occurs when the singular interaction is supported on a set of points in the Euclidian space R d . This leds to consider, formally, operators of the form " − ∆ + ∑ y∈Y µ y δ y (·) " ,(1) where Y is a discrete subset of R d , and µ y , y ∈ Y , are real coupling constants. Heuristically, (1) can be interpreted as the Hamiltonian for a non-relativistic quantum particle interacting with "point sources" of strenghts µ y , located at y ∈ Y . From a mathematical point of view, Schrödinger operators with point (delta-like) interactions have been intensively studied, since the seminal work of Albeverio, Fenstad, and Høegh-Krohn [2], and subsequent characterisation by other authors [23,11,12,7,17] (see the monograph of Albeverio, Gesztesy, and Høegh-Krohn [3] and reference therein for a thorough discussion). In this work we focus on the case of finitely many point interactions in three dimensions. Our aim is to provide a detailed spectral analysis at the bottom at the continuous spectrum, i.e. at zero energy. A similar analysis has been done in [4] Raffaele Scandone Gran Sasso Science Institute, via Crispi 7, 67100 L'Aquila e-mail: rscandone@gssi.com for the two dimensional case, with application to the L p -bounedness of the wave operators. We start by recalling some well known facts on the rigorous construction and the main properties of Schrödinger operators with point interactions. We fix a natural number N 1 and the set Y = {y 1 , . . . , y N } ⊆ R 3 of centres of the singular interactions. Consider T Y := (−∆ ) ↾ C ∞ 0 (R 3 \{Y })(2) as an operator closure with respect to the Hilbert space L 2 (R 3 ). It is a closed, densely defined, non-negative, symmetric operator on L 2 (R 3 ), with deficiency index N. Hence, it admits a N 2 -parameter family of self-adjoint extensions. Among these, there is a N-parameter family of local extension, denoted by {−∆ α,Y α ≡ (α 1 , . . . , α N ) ∈ (R ∪ {∞}) N },(3) whose domain of self-adjointness is qualified by certain local boundary conditions at the singularity centres. The self-adjoint operators −∆ α provide rigorous realisations of the formal Hamiltonian (1), the coupling parameters α j , j = 1, . . . , N, being now proportional to the inverse scattering lenght of the interaction at the centre y j . In particular, if for some j ∈ {1, . . . , N} one has α j = ∞, then no actual interaction is present at the point y j , and in practice things are as if one discards the point y j . When α = ∞, one recovers the the Friedrichs extension of T Y , namely the self-adjoint realisation of −∆ on L 2 (R 3 ). Owing to the discussion above, we may henceforth assume, without loss of generality, that α runs over R N . We review the basic properties of −∆ α,Y , from [3, Section II.1.1] and [19] (see also [8,10,13,9]). We introduce first some notations. For z ∈ C and x, y, y ′ ∈ R 3 , set G y z (x) := e iz|x−y| 4π|x − y| , G yy ′ z :=      e iz|y−y ′ | 4π|y − y ′ | if y ′ = y 0 if y ′ = y ,(4)and Γ α,Y (z) := α j − iz 4π δ j,ℓ − G y j y ℓ z j,ℓ=1,...,N .(5) The function z → Γ α,Y (z) has values in the space of N × N symmetric, complex valued matrices and is clearly entire, whence z → Γ α,Y (z) −1 is meromorphic in C. It is known that Γ α,Y (z) −1 has at most N poles in the open upper half-plane C + , which are all located along the positive imaginary semi-axis. We denote by E + the set of such poles. We denote by E 0 the set of poles of Γ α,Y (z) −1 on the real line. Observe that E 0 is finite and symmetric with respect to z = 0, and we conjecure that actually only z = 0 can belong to E 0 . The following facts are known. Proposition 1. (i) The domain of −∆ α,Y has the following representation, for any z ∈ C + \E + : D(−∆ α,Y ) = g ∈ L 2 (R 3 ) g = F z + N ∑ j,k=1 (Γ α,Y (z) −1 ) jk F z (y k )G y j z , F z ∈ H 2 (R 3 ) . (6) Equivalently, for any z ∈ C + \E + , D(−∆ α,Y ) =              g ∈ L 2 (R 3 ) g = F z + N ∑ j=1 q j G y j z F z ∈ H 2 (R 3 ) (q 1 , . . . , q N ) ∈ C N    F z (y 1 ) . . . F z (y N )    = Γ α,Y (z)    q 1 . . . q N                 . (7) At fixed z, the decompositions above are unique. (ii) With respect to the decompositions (6)-(7), one has (7) is highlighted in [8]. Part (iii) was first proved in [11,12] (see also [3, equation (−∆ α,Y − z 2 ½)g = (−∆ − z 2 ½)F z . (8) (iii) For z ∈ C + \E + , we have the resolvent identity (−∆ α,Y − z 2 ½) −1 − (−∆ − z 2 ½) −1 = N ∑ j,k=1 (Γ α,Y (z) −1 ) jk |G y j z G y k z | . (9) (iv) The spectrum σ (−∆ α,Y ) of −∆ α,(II.1.1.33)]). Part (iv) is discussed in [3, Theorem II.1.1.4], where it is stated that σ p (−∆ α,Y ) ⊂ (−∞, 0 ). An errata at the end of the monograph specifies that a zero eigenvalue imbedded in the continuous spectrum can actually occur: in fact for every N ≥ 2 one can find a configuration Y of the N centres and coupling parameters α 1 , . . . α N such that 0 ∈ σ p (−∆ α,Y ) -see the discussion in Section 3. Let us analyse in detail the spectral properties of −∆ α,Y , whose resolvent is characterised by (9) as an explicit rank-N perturbation of the free resolvent. For negative eigenvalues, the situation is completely understood [3, Theorem II.1.1.4]. Proposition 2. There is a one to one correspondence between the poles iλ ∈ E + of Γ α,Y (z) −1 and the negative eigenvalues −λ 2 of −∆ α,Y , counting the multiplicity. The eigenfunctions associated to the eigenvalue −λ 2 < 0 have the form ψ = N ∑ j=1 c j G y j iλ , where (c 1 , . . . , c N ) is an eigenvector with eigenvalue zero of Γ α,Y (iλ ). Our next step is to investigate the spectral behavior of −∆ α,Y at z = 0, and more generally when z approaches the real line. The starting point is the well known Limiting Absorption Principle for the free Laplacian. Given σ > 0, we consider the Banach space B σ := B(L 2 (R 3 , x 1+σ dx); L 2 (R 3 , x −1−σ dx))(10) We have the following result [1,16]. Proposition 3 (Limiting absorption principle for −∆ ). Let σ > 0. For any z ∈ C + , we have (−∆ − z 2 ) −1 ∈ B σ . Moreover, the map C + ∋ z → (−∆ − z 2 ) −1 ∈ B σ can be continuously extended to the real line. Owing to the resolvent formula (9), and observing that for any z ∈ C + ∪ R the projectors |G y j z G y k z | belong to B σ , it is easy to easy to deduce that also −∆ α,Y satisfies a Limiting Absorption Principle. Proposition 4 (Limiting absorption principle for −∆ α,Y ). Let σ > 0. For every z ∈ C + , we have (−∆ α,Y − z 2 ) −1 ∈ B σ . The map C + ∋ z → (−∆ α,Y − z 2 ) −1 ∈ B σ can be continuously extended to R \ E 0 . As anticiped before, we actually expect that there can not exists singularities at any z ∈ R \ {0}. Our main result is a resolvent expansion in a neighborhood of z = 0. Theorem 1. In a (real) neighborhood of z = 0, we have the expansion (−∆ α,Y − z 2 ) −1 = R −2 z 2 + R −1 z + R 0 (z) (11) where R −2 , R −1 ∈ B σ and z → R 0 (z) is a continuous B σ -valued map. Moreover, R −2 = 0 if and only if zero is an eigenvalue for −∆ α,Y . Remark 1. For classical Schrödinger operators of the form −∆ + V , the Limiting Absorption Principle and the analogous of Theorem 1 can be proved under suitable short-range assuptions on V [1,14]. In this case, moreover, it is well known that R −1 = 0 if and only if there exists a generalized eigenfunction at z = 0 (a zero- energy resonance for −∆ +V ), namely a function ψ ∈ L 2 (R 3 , x −1−σ dx) \ L 2 (R 3 ) , σ > 0, which satisfies (−∆ + V )ψ = 0 as a distributional identity on R 3 . As it will be clear from the proof of Theorem 1, a similar characterisation holds true also for −∆ α,Y (see Remark 2). 2 Asymptotics for Γ α,Y (z) −1 as z → 0 We fix N ≥ 1, α ∈ R N and Y ⊆ R 3 , and we set Γ (z) := Γ α,Y (z). We shall use the notation O(z k ), k ∈ Z, to denote a merophorpic M N (C)-valued function whose Laurent expansion in a neighborhood of z = 0 contains only terms of degree ≥ k. In particular, O(1) denotes an analytic map in a neighborhood of z = 0. We also write Θ (z k ) to denote a function of the form Az k , with A ∈ M N (C) \ {0}. In a neighborhood of z = 0, we can expand Γ (z) = Γ 0 + zΓ 1 + z 2 Γ 2 + O(z 3 ). In the following proposition we characterise the small z behaviour of Γ (z) −1 . Proposition 5. In a neighborhood of z = 0 we have the Laurent expansion Γ (z) −1 = A −2 z 2 + A −1 z + O(1),(12) where A −2 , A −1 ∈ M N (C). Moreover, (i) A −2 = 0 if and only if Ker Γ 0 ∩ Ker Γ 1 = {0} (ii) A −1 = 0 if and only if Ker Γ 0 ⊆ Ker Γ 1 In the proof of Proposition 5 we shall use the following result due to Jensen and Nenciu [15]. We are now able to prove Proposition 5. Proof (Proof of Proposition 5). If Γ 0 = Γ (0) is non-singular, then Γ (z) −1 in analytic in a sufficiently small neighborhood of z = 0. Assume now that Γ 0 is singular. We distinguish two cases: Case 1: Ker Γ 0 ∩ Ker Γ 1 = {0}. For z small enough, z = 0, Ker (Γ 0 + zΓ 1 ) = Ker Γ 0 ∩ Ker Γ 1 = {0}. It follows that Γ ≤1 (z) := Γ 0 + zΓ 1 in invertible, whence the same is Γ (z) for small z, with Γ (z) −1 = Γ ≤1 (z) −1 + O (1). In order to invert Γ ≤1 (z), we use the Jensen-Nenciu Lemma. Let P be the orthogonal projection onto Ker Γ 0 . Observe that Γ 0 + P is invertible, whence the same is Γ ≤1 (z)+ P for small z, with (Γ ≤1 (z)+ P) −1 = O(1). More precisely, (Γ ≤1 (z) + P) −1 = [I + z(Γ 0 + P) −1 Γ 1 ] −1 [Γ 0 + P] −1 = [I − z(Γ 0 + P) −1 Γ 1 ][Γ 0 + P] −1 + O(z 2 ).(13) By Lemma 1 we get Γ ≤1 (z) −1 = (Γ ≤1 (z) + P) −1 + (Γ ≤1 (z) + P)P[P − P(Γ ≤1 (z) + P) −1 P] −1 P(Γ ≤1 (z) + P)(14) Owing to (13), and observing that (Γ 0 + P) −1 P = P(Γ 0 + P) −1 = P, we compute P − P(Γ ≤1 (z) + P) −1 P = zPΓ 1 P + O(z 2 ) Substituting into (14) we get Γ −1 ≤1 (z) = (Γ ≤1 (z) + P) −1 + z −1 (Γ ≤1 (z) + P) −1 P(PΓ 1 P) −1 P(Γ ≤1 (z) + P) = z −1 P(PΓ 1 P) −1 P + O(1) = Θ (z −1 ) + O(1).(15) Case 2: Ker Γ 0 ∩ Ker Γ 1 = {0}. We start by proving that Ker Γ 1 ∩ Ker Γ 2 = {0}. In particular, we show that the quadratic form associated to Γ 2 is strictly negative on Ker Γ 1 \ {0} = (1, 1, . . . , 1) ⊥ \ {0}. Observe that, apart from a multiplicative factor, (Γ 2 ) jk = |y j − y k |. Our first step is to prove that for any v ∈ R N with v 1 + . . . + v N = 0, we have ∑ 1≤ j,k≤N |y j − y k |v j v k ≤ 0.(16) The key point is to use the so called averaging trick. By rotational and scaling invariance, we can see that there exists a positive constant c such that, for any y ∈ R 3 , S 2 | w, y |dw = c|y|. Hence, for every v ∈ R N ∑ 1≤ j,k≤N |y j − y k |v j v k = c −1 S 2 ∑ 1≤ j,k≤N | w, y j − y k |v j v k dw, and then it is sufficient to prove that, for a fixed w ∈ S 2 , ∑ 1≤ j,k≤N | w, y j − y k |v j v k ≤ 0. Let P w be the orthogonal projection onto the one-dimensional subspace generated by w, and observe that w, y j − y k = w, P w y j − P w y k . Hence we may assume, without loss of generality, that all the points y k lie on the same line. Under this assumption, we can write ∑ 1≤ j,k≤N |y j − y k |v j v k = 2 ∑ 1≤ j,k≤N max {y j − y k , 0}v j v k = 2 t∈R ∑ 1≤ j,k≤N [y k < t < y j ]v j v k(17) where we used the Iverson bracket notation [P], which equals 1 if the statement P is true and 0 if it is false. So it is enough to prove that, for almost every t ∈ R, ∑ y k <t<y j v j v k ≤ 0. For every t ∈ R \ {y 1 , . . . y N }, define J t := { j y j > t}, K t := {k y k < t}. We have ∑ y k <t<y j v j v k = ∑ j∈J t ,k∈K t v j v k = ∑ j∈J t v j ∑ k∈K t v k = − ∑ j∈J t v j 2 ≤ 0,(18) where we used, in the last equality, the hypothesis v 1 + . . . + v N = 0. Assume now that we have the equality in (16). Once again, we may assume that all the points y k lie on the same line, say y 1 < y 2 < . . . < y N . This follows from the averaging trick and the observation that for almost every w ∈ S 2 the projections P w y 1 , . . . , P w y N are pairwise distinct. Owing to (17) and (18), we have that for almost every t ∈ R, ∑ 1≤ j,k≤N [y k < t < y j ]v j v k = 0.(19) In particular, (19) must be true for almost every t ∈ R \ {y 1 , . . . y N }, which in view of (18) implies n ∑ j=1 v j = 0 ∀ n ∈ {1, . . . , N}, and this means that v j = 0 for all j, concluding the proof of Ker Γ 1 ∩ Ker Γ 2 = {0}. Now, for z small enough, z = 0, Ker (Γ 0 + zΓ 1 + z 2 Γ 2 ) = Ker Γ 0 ∩ Ker Γ 1 ∩ Ker Γ 2 = {0}. It follows that Γ ≤2 (z) := Γ ≤1 (z) + z 2 Γ 2 is invertible, whence the same is Γ (z) for small z, with Γ (z) −1 = Γ ≤2 (z) −1 + O(1). As before, we invert Γ ≤2 (z) by means of the Jensen-Nenciu Lemma. Let P be the orthogonal projection onto Ker Γ 0 ∩ Ker Γ 1 . Observe that Γ ≤1 (z) + P is invertible, with (Γ ≤1 (z) + P) −1 = Θ (z −1 ) + O(1) Ker Γ 0 ⊆ Ker Γ 1 O(1) Ker Γ 0 ⊆ Ker Γ 1(20) For small z, also Γ ≤2 (z) + P is invertible, with (Γ ≤2 (z) + P) −1 = (Γ ≤1 (z) + P) −1 + O(1). With similar computations as before, we get Γ ≤2 (z) −1 = (Γ ≤2 (z) + P) −1 + z −2 P(PΓ 2 P) −1 P = Θ (z −2 ) + Θ (z −1 ) + O(1) Ker Γ 0 ⊆ Ker Γ 1 Θ (z −2 ) + O(1) Ker Γ 0 ⊆ Ker Γ 1(21) Expansion (12) is thus proved in any case. Moreover, statements (i) and (ii) easily follows from the discussion above. We can prove now our main Theorem. Proof (of Theorem 1). The low-energy expansion (11) follows by combining the resolvent formula (9) with the small z expansion (12) for Γ α,Y (z) −1 . We prove now that R −2 = 0 if and only if 0 ∈ σ (−∆ α,Y ) which in view of Proposition 5, part (i), is equivalent to prove that Ker Γ 0 ∩ Ker Γ 1 = {0} if and only if 0 ∈ σ (−∆ α,Y ). Suppose first that there exists c = (c 1 , . . . , c N ) = 0 ∈ Ker Γ 0 ∩ Ker Γ 1 . We are going to show that the non-zero function ψ := N ∑ k=1 c j G y k 0(22) belongs to Ker (−∆ α,Y ). First of all, observe that the condition Γ 1 c = 0 is equivalent to c 1 + . . . + c N = 0, which implies ψ ∈ L 2 (R 3 ). Let us fix z ∈ C + \ E + , and write ψ = F z + N ∑ k=1 c k G y k z , where F z := N ∑ j=1 c j (G y j 0 − G y j z ). Observe that F z ∈ H 2 (R 3 ). Moreover, for every k ∈ {1, . . . , N}, F z (y k ) = N ∑ j=1 c k (G y j y k 0 − G y j ,y k z ) = N ∑ k=1 Γ k j c j where in the second equality we used that Γ 0 c = 0. By virtue of representation (7), we conclude that ψ ∈ D(−∆ α,Y ). Moreover, formula (8) yields −∆ α,Y ψ = (−∆ − z 2 )F z + z 2 N ∑ j=1 c j G y j z = N ∑ j=1 c j (−∆ − z 2 )G y j z − ∆ G y j 0 = 0 which shows that ψ ∈ Ker(−∆ α,Y ). Let us discuss now the opposite implication. To this aim, consider a function ψ ∈ Ker (−∆ α,Y ) \ {0}. For a fixed z = iλ ∈ C + \ E + , we can write ψ = F iλ + N ∑ k=1 c k G y k iλ ,(23) for some non-zero F z ∈ H 2 (R 3 ), and with c j = N ∑ k=1 Γ (z) −1 jk F z (y k ). Observe that the c j 's are necessarily independent of z, since G y k iλ ∈ H 2 (R 3 ) for any k. Moreover, the condition ψ ∈ L 2 (R 3 ) implies c 1 + . . . + c n = 0, namely Γ 1 c = 0. Owing to (8) and the representation (23), the relation −∆ α,Y ψ = 0 is equivalent to − ∆ F iλ = λ 2 N ∑ y=1 c j G y j iλ(24) We show now that, for λ ↓ 0, F iλ H 2 → 0 whence also F λ → 0 uniformly on compact subsets of R 3 . This would imply In order to show that F iλ H 2 → 0 as λ ↓ 0, we start with the estimate ∆ F iλ L 2 = λ 2 ∆ (−∆ + λ 2 ) −1 ψ L 2 ≤ λ 2 ψ L 2 .(25) Observe moreover that F iλ (p) = λ 2 (p 2 + λ 2 ) −1 ψ(p). By dominate convergence we get F iλ L 2 = o(1), which combined with (25) yields F iλ H 2 = o(1), as desired. Remark 2. By Proposition (5)(ii), there is a Θ (z −1 ) term in the expansion of Γ (z) −1 at z = 0 if and only if there exists c ∈ R n such that Γ 0 c = 0, Γ 1 c = 0. In this case, the function defined by (22) belongs to L 2 (R 3 , x −1−σ dx) \ L 2 (R 3 ), σ > 0, and formally satisfies −∆ α,Y ψ = 0, whence ψ can be interpreted as a zero energy resonance for −∆ α,Y . Hence, as anticipated after the main theorem, we have that R −1 = 0 in expansion (11) if and only if there exists a zero energy resonance, analogously to the case of classical Schrödinger operators. Occurrence and multiplicity of zero energy obstructions In this Section we discuss the occurrence and the multiplicity of obstructions at zero energy for the resolvent of −∆ α,Y , depending on the choices of the set Y of centers of interactions and of the coupling parameters α 1 , . . . α N . In the single center case, it is easy to check that the only possible obstruction at z = 0 is a resonance, atteined if and only if α = 0. In general, a resonance can be found for any N and for any given configuration of the centers, for a measure zero set of choices of the parameters α 1 , . . . , α N . By means of the discussion in Chapter 2, we can define the multiplicity of a zero-energy resonance as r α,Y := dim ( Ker Γ 0 ) − dim ( Ker Γ 0 ∩ Ker Γ 1 ). We conjecture that, as N increases, one can find Y and α such that r α,Y becomes arbitrarily large. As anticipated in Section 1, when N = 2 we can find a simple zero eigenvalue by choosing α 1 = α 2 = −(4πd) −1 , where d is the distance between the two centers. For a generic N ≥ 3, a zero eigenvalue occurs for specific geometric configurations of the centers of interactions and for a measure zero set of choices of α 1 , . . . , α N . By means of the discussion in Chapter 2, the multiplicity of a zero eigenvalue is given by e α,Y := dim Ker (−∆ α,Y ) = dim ( KerΓ 0 ∩ Ker Γ 1 ). Let us discuss now the maximal possible value for e α,Y as the number of centers of interactions increases. • N = 3. We can taxe Y as the vertices of a equilater triangle of side-lenght one, and α 1 = α 2 = α 3 = −(4π) −1 . With this choice we get e α,Y = 2. • N = 4. We can taxe Y as the vertices of a regular tetrahedon of side-lenght one, and α 1 = α 2 = α 3 = α 4 = −(4π) −1 . With this choice we get e α,Y = 3. • N = 5. Observe that we can not find five points in R 3 with constant pairwise distances. It easily follows that the maximal value for e α,Y is still three. One could conjecture that for N ≥ 4 the maximal value of e α,Y is three. Nevertheless, it is also conceivable that for large N there exist complicated geometrical configurations which led to a higher multiplicity. Such kind of mechanism is wellknown in similar contexts. Consider, for example, the problem in combinatorics to determine the chromatic number of the unit distance graph on R 3 , that is the graph with vertices set V = R 3 and edges set E = {(x, y) ∈ R 3 × R 3 | |x − y| = 1}. Owing to a compactenss principle by De Bruijn and Erdős [6] this is equivalent, under the axiom of choice, to determine the highest chromatic number of a finite graph embedded in R 3 in such a way all its edges have lenght one. For a graph with N vertices, we have the following situation: • N = 3. We can consider an equilater triangle of side-lenght one, which has chromatic number three. • N = 4. We can consider a regular tethraedon of side-lenght one, which has chromatic number four. • N = 5. The highest possible chromatic number is still four. • N = 14. There is a configuration of 14 points in R 3 , the Moser-Raiskii spindle, with chromatic number five [21,22]. • For large N, the highest possible chromatic number is known to be between 6 and 12 [18,20,5]. It is evident that there are similarities between the two problems, and it would be interesting to understand if they are actually related. In particular, one may take Y as the vertices of the Moser-Raiskii spindle and wondering whether there exists α = (α 1 , . . . , α 14 ) such that e α,Y = 4. Y consists of at most N non-positive eigenvalues and the absolutely continuous part σ ac (−∆ α,Y ) = [0, ∞), the singular continuous spectrum is absent. Parts (i) and (ii) of Proposition 1 above originate from [12] and are discussed in [3, Theorem II.1.1.3], in particular Lemma 1 ( 1Jensen-Nenciu). Let A be a closed operator in a Hilbert space H and P a projection, such that A+ P has a bounded inverse. Then A has a bounded inverse if and only if B = P − P(A + P) −1 P has a bounded inverse in PH and, in this case, A −1 = (A + P) −1 + (A + P) −1 PB −1 P(A + P) −1 Spectral properties of Schrodinger operators and scattering theory. S Agmon, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2S. AGMON, Spectral properties of Schrodinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (1975), pp. 151-218. Singular perturbations and nonstandard analysis. S Albeverio, J E Fenstad, And R Høegh-Krohn, Trans. Amer. Math. Soc. 252S. ALBEVERIO, J. E. FENSTAD, AND R. HØEGH-KROHN, Singular perturbations and non- standard analysis, Trans. Amer. Math. Soc., 252 (1979), pp. 275-295. S Albeverio, F Gesztesy, R Høegh-Krohn, And H Holden, Solvable Models in Quantum Mechanics, Texts and Monographs in Physics. New YorkSpringer-VerlagS. ALBEVERIO, F. GESZTESY, R. HØEGH-KROHN, AND H. HOLDEN, Solvable Models in Quantum Mechanics, Texts and Monographs in Physics, Springer-Verlag, New York, 1988. H D Cornean, A Michelangeli, And K Yajima, arXiv:1804.01297Two dimensional Schrödinger operators with point interactions: threshold expansions, zero modes and L p -boundedness of wave operators. H. D. CORNEAN, A. MICHELANGELI, AND K. YAJIMA, Two dimensional Schrödinger op- erators with point interactions: threshold expansions, zero modes and L p -boundedness of wave operators, arXiv:1804.01297 (2018) A 15-colouring of 3-space omitting distance one. D Coulson, Disc. Math. 256D. COULSON, A 15-colouring of 3-space omitting distance one, Disc. Math., 256 (2002), pp. 83-90. A colour problem for infinite graphs and a problem in the theory of relations. N G De Bruijn And P, Erdős, Nederl. Akad. Wetensch. Proc. Ser. A. 54371373N. G. DE BRUIJN AND P. ERDŐS, A colour problem for infinite graphs and a problem in the theory of relations, Nederl. Akad. Wetensch. Proc. Ser. A, 54 (1951), pp. 371373. On nonlocal point interactions in one, two, and three dimensions. L Dabrowski, H Grosse, J. Math. Phys. 26L. DABROWSKI AND H. GROSSE, On nonlocal point interactions in one, two, and three dimensions, J. Math. Phys., 26 (1985), pp. 2777-2780. G Dell&apos;antonio, R Figari, And A Teta, A brief review on point interactions, in Inverse problems and imaging. BerlinSpringer1943G. DELL'ANTONIO, R. FIGARI, AND A. TETA, A brief review on point interactions, in Inverse problems and imaging, vol. 1943 of Lecture Notes in Math., Springer, Berlin, 2008, pp. 171-189. Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction. G Dell&apos;antonio, A Michelangeli, R Scandone, And K Yajima, Ann. Henri Poincaré. 19G. DELL'ANTONIO, A. MICHELANGELI, R. SCANDONE, AND K. YAJIMA, L p - Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction, Ann. Henri Poincaré, 19 (2018), pp. 283-322. Dispersive estimate for the Schrödinger equation with point interactions. P. D&apos;ancona , V Pierfelice, And A Teta, Math. Methods Appl. Sci. 29P. D'ANCONA, V. PIERFELICE, AND A. TETA, Dispersive estimate for the Schrödinger equation with point interactions, Math. Methods Appl. Sci., 29 (2006), pp. 309-323. A class of explicitly soluble, local, many-center Hamiltonians for one-particle quantum mechanics in two and three dimensions. I. A Grossmann, R Høegh-Krohn, And M Mebkhout, Comm. Math. Phys. 21J. Math. Phys.A. GROSSMANN, R. HØEGH-KROHN, AND M. MEBKHOUT, A class of explicitly soluble, local, many-center Hamiltonians for one-particle quantum mechanics in two and three di- mensions. I, J. Math. Phys., 21 (1980), pp. 2376-2385. 12. , The one particle theory of periodic point interactions. Polymers, monomolecular lay- ers, and crystals, Comm. Math. Phys., 77 (1980), pp. 87-110. Dispersive estimates for Schrödinger operators with point interactions in R 3. F Iandoli And R, Scandone, Advances in Quantum Mechanics: Contemporary Trends and Open Problems. A. Michelangeli and G. Dell'AntonioSpringer International Publishing18Springer INdAM SeriesF. IANDOLI AND R. SCANDONE, Dispersive estimates for Schrödinger operators with point interactions in R 3 , in Advances in Quantum Mechanics: Contemporary Trends and Open Problems, A. Michelangeli and G. Dell'Antonio, eds., Springer INdAM Series, vol. 18, Springer International Publishing, pp. 187-199. Spectral properties of Schrödinger operators and time-decay of the wave functions. A Jensen And T, Kato, Duke. Math. J. 46A. JENSEN AND T. KATO, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke. Math. J., 46 (1979), pp. 583-611. Raffaele Scandone. Raffaele Scandone A unified approach to resolvent expansions at thresholds. A Jensen, G Nenciu , Rev. in Mathe. Phys. 13A. JENSEN AND G. NENCIU., A unified approach to resolvent expansions at thresholds, Rev. in Mathe. Phys., 13 (2001), pp. 717-754 S T Kuroda, Introduction to Scattering Theory, Lecture Notes. Matematisk Institute, Aarhus UniversityS. T. KURODA, Introduction to Scattering Theory, Lecture Notes, Matematisk Institute, Aarhus University (1978). On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians. A Michelangeli And A, Ottolini, Rep. Math. Phys. 79A. MICHELANGELI AND A. OTTOLINI, On point interactions realised as Ter-Martirosyan- Skornyakov Hamiltonians, Rep. Math. Phys., 79 (2017), pp. 215-260 On the space chromatich number. O Nechushtan, Disc. Math. 256O. NECHUSHTAN, On the space chromatich number, Disc. Math. 256 (2002), pp. 499-507. A Kreȋn-like formula for singular perturbations of self-adjoint operators and applications. A Posilicano, J. Funct. Anal. 183A. POSILICANO, A Kreȋn-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal., 183 (2001), pp. 109-147. Note on the Chromatic number of the Space. R Radoičić And G, Tóth, Disc. Comput. Geometry. Algorithms and Combinatorics. 25R. RADOIČIĆ AND G. TÓTH, Note on the Chromatic number of the Space, Disc. Comput. Geometry. Algorithms and Combinatorics, 25 (2003), pp. 695-698. The realization of all distances in a decomposition of the space R n into n + 1 parts. D E Raǐskiǐ, Mat. Zametki. 7in RussianD. E. RAǏSKIǏ, The realization of all distances in a decomposition of the space R n into n + 1 parts, Mat. Zametki 7 (1970), pp. 319-323 (in Russian); English translation. Math. Notes. 7English translation [Math. Notes 7 (1970) pp. 194-196]. Bounds on the measurable chromatic number on R n. L A C Székeley And N, Wormald, Disc. Math. 751-3L. A. SZÉKELEY AND N. C. WORMALD, Bounds on the measurable chromatic number on R n , Disc. Math., 75 (1-3) (1989), pp. 343-372. Perturbation of self-adjoint operators by Dirac distributions. J Zorbas, J. Math. Phys. 21J. ZORBAS, Perturbation of self-adjoint operators by Dirac distributions, J. Math. Phys., 21 (1980), pp. 840-847.

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{'abstract': 'We study the low energy behavior of the resolvent of Schrödinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions.', 'arxivid': '1901.02449', 'author': ['Raffaele Scandone '], 'authoraffiliation': [], 'corpusid': 119345547, 'doi': '10.1007/978-3-030-60453-0_7', 'github_urls': [], 'n_tokens_mistral': 10416, 'n_tokens_neox': 9167, 'n_words': 5335, 'pdfsha': '273f6fcbce0bc26756d54c5debf6ba1c09a83499', 'pdfurls': ['https://arxiv.org/pdf/1901.02449v2.pdf'], 'title': ['Zero modes and low-energy resolvent expansion for three dimensional Schrödinger operators with point interactions', 'Zero modes and low-energy resolvent expansion for three dimensional Schrödinger operators with point interactions'], 'venue': []}

arxiv

AMPLE GROUPOIDS: EQUIVALENCE, HOMOLOGY, AND MATUI'S HK CONJECTURE 23 Aug 2018 Carla Farsi Alex Kumjian David Pask Aidan Sims AMPLE GROUPOIDS: EQUIVALENCE, HOMOLOGY, AND MATUI'S HK CONJECTURE 23 Aug 2018 We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu-Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain complex developed by Evans. We show that Matui's HK conjecture holds for the groupoids of single vertex k-graphs which satisfy a mild joint-coprimality condition. We also prove that there is a natural homomorphism from the categorical homology of a k-graph to the homology of its groupoid. Introduction The purpose of this paper is to investigate the homology of ample Hausdorff groupoids, and to investigate Matui's HK-conjecture for groupoids associated to actions of N k by local homeomorphisms on locally compact Hausdorff zero-dimensional spaces. Ample Hausdorff groupoids are an important source of examples of C * -algebras. They provide models for the crossed-products associated to Cantor minimal systems [20], Cuntz-Krieger algebras and graph C * -algebras and their higher-rank analogues [44,28,27], and recently models for large classes of classifiable C * -algebras [6,14,41]. It is therefore very desirable to develop techniques for computing the K-theory of the C * -algebra of an ample Hausdorff groupoid. Unfortunately, there are relatively few general techniques available. In a series of recent papers [32,33,34], Matui has advanced a conjecture that if G is a minimal effective ample Hausdorff groupoid with compact unit space then K 0 (C * r (G)) is isomorphic to the direct sum of the even homology groups H 2n (G) of the groupoid as defined by Crainic and Moerdijk [11], and K 1 (C * r (G)) is isomorphic to the direct sum of the odd homology groups H 2n+1 (G). He has verified this conjecture for a number of key classes of groupoids, including finite cartesian products of groupoids associated to shifts of finite type, transformation groupoids for Cantor minimal systems, and AF groupoids with compact unit spaces. He has also developed tools for computing the homology of ample Hausdorff groupoids, including a spectral sequence that relates the homology of a groupoid G endowed with a cocycle c taking values in a discrete abelian group H with the homology of H with values in the homology groups of the skew-product groupoid G × c H. Other authors have subsequently verified Matui's conjecture for Exel-Pardo groupoids and certain graded ample Hausdorff groupoids (see [38] [21]). We begin Section 3 by investigating the many notions of groupoid equivalence in the literature in the context of arbitrary ample Hausdorff groupoids. Crainic and Moerdijk focus on the notion of Morita equivalence of groupoids (see [11, 4.5]) while Matui employs the notions of similarity (see [32,Definition 3.4]) and Kakutani equivalence (see [32,Definition 4.1]). Similarity of groupoids was previously studied by Renault [44] and Ramsay [43]. In the setting of ample Hausdorff groupoids with σ-compact unit spaces, it follows from [32,Theorem 3.6] that Kakutani equivalence implies similarity. We show that similarity, Kakutani equivalence, Renault's notion of groupoid equivalence [45,37], and the notion of groupoid Morita equivalence of Crainic and Moerdijk (as well as a number of other notions) all coincide for ample Hausdorff groupoids with σ-compact unit spaces (see Theorem 3.12). In Section 4 we recall the definition of homology for an arbitrary ample Hausdorff groupoid from [11, 3.1] (see also [32,Definition 3.1]) and we appeal to a theorem of Crainic and Moerdijk to observe that groupoid equivalence preserves groupoid homology for arbitrary ample Hausdorff groupoids (see [11,Corollary 4.6]). Matui also proved that similar Hausdorffétale groupoids have isomorphic homology groups (see [32,Proposition 3.5]), and this formulation allows us to give an explicit description of the isomorphism when the equivalence arises from a similarity. In Section 5 we introduce Matui's HK conjecture, and extend his proof that AF groupoids with compact unit space satisfy the HK conjecture to the case of non-compact unit spaces. Our main computations of groupoid homology are in Section 6, where we investigate the homology of Deaconu-Renault groupoids G(X, σ) associated to actions σ of N k by local homeomorphisms on totally disconnected locally compact Hausdorff spaces X. We adapt techniques developed by Evans [16] in the context of K-theory for higher-rank graph C * -algebras to construct a chain complex A σ in which the n-chains are elements of n Z k ⊗C c (X, Z) and the boundary maps are built from the forward maps σ n * on C c (X, Z) that satisfy σ n * (1 U ) = 1 σ n (U ) whenever U ⊆ X is a compact open set on which σ n is injective. Our main result, Theorem 6.5, gives an explicit computation of the homology groups H n (G(X, σ)): we prove that H n (G(X, σ)) is canonically isomorphic to H n (A σ * ). We then show that, if c : G(X, σ) → Z k is the canonical cocycle, then the homology groups H * (A σ ) also coincide with the homology groups H * (Z k , K 0 (C * (G(X, σ) × c Z k ))) appearing in Kasparov's spectral sequence for the double crossed product C * (G(X, σ) × c Z k ) ⋊ Z k . Since this double crossed product is Morita equivalent to C * (G(X, σ)) by Takai duality, this provides a useful tool for calculating the K-theory of C * (G(X, σ)). In Theorems 6.7 and 6.10 we calculate both the K-groups of C * (G(X, σ)) and the homology groups of G(X, σ) explicitly for k = 1, 2, and in particular prove that ample Deaconu-Renault groupoids of rank at most 2 satisfy Matui's HK conjecture. We also discuss the differences between Kasparov's spectral sequence and Matui's for k ≥ 3 and indicate where one might look for counterexamples to Matui's conjecture amongst such groupoids. Finally, in Section 7, we specialise to k-graphs. The k-graph groupoid G Λ of a rowfinite k-graph Λ with no sources is precisely the Deaconu-Renault groupoid G(Λ ∞ , σ) associated to the shift maps on the infinite-path space of Λ. We begin the section by linking the homology of the k-graph groupoid with the categorical homology of the k-graph by constructing a . natural homomorphism from H * (Λ) to H * (G Λ ). We then investigate how to apply the results of Section 6 in the specific setting of k-graph groupoids. We prove that the chain complex A σ * associated to (Λ ∞ , σ) as in Section 6 has the same homology as the much simpler chain complex D Λ * described by Evans in [16]. This provides a very concrete calculation of the homology of a k-graph groupoid. It follows that the homology of G Λ does not depend on the factorisation rules in Λ. We use this and the preceding section to establish an explicit description of the homology of 1-graph groupoids and 2-graph groupoids and to see that these groupoids satisfy Matui's conjecture. We also prove that for arbitrary k, if Λ is a k-graph with a single vertex such that the integers |Λ e 1 | − 1, . . . , |Λ e k | − 1 have no nontrivial common divisors, then both the homology of G Λ and the K-theory of C * (Λ) are trivial, and in particular G Λ satisfies the HK-conjecture. Acknowledgements: C.F. thanks A.K. for his hospitality during her visit to UNR. A.K. thanks his co-authors for their hospitality and support on his visits to Boulder and Wollongong. Background 2.1. Groupoids and their C * -algebras. We give some brief background on groupoids and their C * -algebras and establish our notation. For details, see [17,44,49]. A groupoid is a small category G with inverses. We write G (0) for the set of identity morphisms of G, called the unit space, and we write r, s : G → G (0) for the range and source maps. We write G (2) for the set {(γ 1 , γ 2 ) ∈ G × G : s(γ 1 ) = r(γ 2 )} of composable pairs in G. The groupoid G is a topological groupoid if it has a locally compact topology under which all operations in G are continuous and G (0) is Hausdorff in the relative topology. If the topology on all of G is Hausdorff, we call G a Hausdorff groupoid. Anétale groupoid is a topological groupoid in which G (0) is open, and r, s : G → G (0) are local homeomorphisms (in [44] such a groupoid is called r-discrete with Haar system). An open subset U ⊆ G is said to be an open bisection if both r| U and s| U are homeomorphisms onto their ranges. Given u ∈ G (0) we write G u for {γ ∈ G : r(γ) = u}, G u for {γ ∈ G : s(γ) = u} and G u u = G u ∩ G u . A groupoid G is ample if it isétale and G (0) is zero dimensional; equivalently, G is ample if it has a basis of compact open bisections. The orbit of a unit u ∈ G (0) is the set [u] := r(G u ) = s(G u ). A subset U ⊆ G (0) is full if U ∩ [u] = ∅ for every unit u. We say that G is minimal if the only nontrivial open invariant subset of G (0) is G (0) ; equivalently, G is minimal if the closure of [u] is equal to G (0) for every u ∈ G (0) . The isotropy of G is the set u∈G (0) G u u of elements of G whose range and source coincide. A groupoid G is said to be effective 1 if the interior of its isotropy coincides with its unit space G (0) Let A be an abelian group and let c : G → A be a 1-cocycle. Then we may form the skew product groupoid G × c A which is the set G × A with structure maps r(γ, a) = (r(γ), a), s(γ, a) = (s(γ), a + c(γ)) and (γ, a)(η, a + c(γ)) = (γη, a) (see [44,Definition I.1.6]). There is a natural action α of A on G × c A given by α b (γ, a) = (γ, a + b). Given a Hausdorffétale groupoid G, the space C c (G) of continuous compactly supported functions from G to C becomes a * -algebra with operations given by (f * g)(γ) = γ=γ 1 γ 2 f (γ 1 )g(γ 2 ) and f * (γ) = f (γ −1 ). The groupoid C * -algebra C * (G) is the universal C * -algebra generated by a * -representation of C c (G) (cf. [44,II.1.5]). For each unit u ∈ G 0 there is a * -representation π u : C c (G) → B(ℓ 2 (G u )) given by π u (f )δ η = γ∈G r(η) f (γ)δ γη . The reduced C * -algebra C * r (G) is the closure of the image of C c (G) under the representation u∈G (0) π u . A cocycle c : G → Z k determines an action of T k by automorphisms of C c (G) given by (z · f )(γ) = z c(γ) f (γ), and this extends to an action of T k by automorphisms on each of C * (G) and C * r (G). There is an isomorphism C * (G × c Z k ) ∼ = C * (G) ⋊ T k that carries a function f ∈ C c (G × {n}) ⊆ C c (G × c Z k ) to the function z → (g → z n f (g, n)) ∈ C(T k , C * (G)) ⊆ C * (G)⋊T k . This isomorphism descends to an isomorphism C * r (G × c Z k ) ∼ = C * r (G) ⋊ T k . We will be particularly interested in Deaconu-Renault groupoids associated to actions of N k , which are defined as follows. Let X be a locally compact Hausdorff space, and let σ be an action of N k on X by surjective local homeomorphisms. The associated Deaconu-Renault groupoid 2 G = G(X, σ) is defined by [13,18]). We identify X with the unit space via the map x → (x, 0, x). The structure maps are given by r(x, n, y) = x, s(x, n, y) = y and (x, m, y)(y, n, z) = (x, m + n, z). A basis for the topology on G is given by subsets of the form G = {(x, p − q, y) ∈ X × Z k × X : σ p (x) = σ q (y)} (cf.U × {p − q} × V where U, V are open in X and σ p (U) = σ q (V ). There is a natural cocycle c : G(X, σ) → Z k given by c(x, n, y) := n. We can then form the skew-product groupoid G × c Z k . With our conventions, in this groupoid we have r((x, n, y), p) = (x, p), s((x, n, y), p) = (y, p + n) and ((x, n, y), p)((y, m, z), p + n) = ((x, m + n, z), p). There is an action σ of N k on X = X × Z k by surjective local homeomorphisms given by σ q (x, p) = (σ q (x), p + q). Moreover there is an isomorphism G × c Z k ∼ = G( X, σ) given by (2.1) ((x, m, y), p) → ((x, p), m, (y, p + m)). The full and reduced C * -algebras of a Deaconu-Renault groupoid coincide (see for example [51,Lemma 3.5]). 2.2. k-graphs, their path groupoids, and their C * -algebras. For k ≥ 1, a k-graph is a non-empty countable small category equipped with a functor d : Λ → N k that satisfies the following factorisation property. For all λ ∈ Λ and m, n ∈ N k such that d(λ) = m + n there exist unique µ, ν ∈ Λ such that d(µ) = m, d(ν) = n, and λ = µν. When d(λ) = n we say λ has degree n, and we write Λ n = d −1 (n). The standard generators of N k are denoted ε 1 , . . . , ε k , and we write n i for the i th coordinate of n ∈ N k . We define a partial order on N k by m ≤ n if m i ≤ n i for all i ≤ k. If Λ is a k-graph, its vertices are the elements of Λ 0 . The factorisation property implies that these are precisely the identity morphisms, and so can be identified with the objects. For λ ∈ Λ the source s(λ) is the domain of λ, and the range r(λ) is the codomain of λ (strictly speaking, s(λ) and r(λ) are the identity morphisms associated to the domain and codomain of λ). Given λ, µ ∈ Λ and E ⊆ Λ, we define λE = {λν : ν ∈ E, r(ν) = s(λ)}, Eµ = {νµ : ν ∈ E, s(ν) = r(µ)}, and λEµ = (λE)µ = λ(Eµ). In particular, for v ∈ Λ 0 and n ∈ N k , we have vΛ n = {λ ∈ Λ n : r(λ) = v}. We say that the k-graph Λ is row-finite if |vΛ n | < ∞ for each n ∈ N k and v ∈ Λ 0 , and has no sources if 0 < |vΛ m | for all v ∈ Λ 0 and m ∈ N k . Let A be an abelian group. Given a functor c : Λ → A, we may form the skew-product k-graph Λ × c A which is the set Λ × A endowed with structure maps given by r(λ, a) = (r(λ), a), s(λ, a) = (s(λ), a + c(λ)), (λ, a)(µ, a + c(λ)) = (λµ, a), and d(λ, a) = d(λ) (see [27,Definition 5.1]). There is a natural A-action α on Λ× c A given by α b (λ, a) = (λ, a+b). Examples 2.1. (a) A 1-graph is the path category of a directed graph (see [27]). (b) Let Mor Ω k = {(m, n) ∈ N k × N k : m ≤ n}, and Obj Ω k = N k . Define r, s : Mor Ω k → Obj Ω k by r(m, n) = m, s(m, n) = n, and for m ≤ m ≤ p ∈ N k define (m, n)(n, p) = (m, p) and d(m, n) = n − m. Then (Ω k , d) is a k-graph. We identify Obj Ω k with {(m, m) : m ∈ N k } ⊆ Mor Ω k . Let Λ be a row-finite k-graph with no sources. The set Λ ∞ = {x : Ω k → Λ | x is a degree-preserving functor} is called the infinite path space of Λ. For v ∈ Λ 0 , we put vΛ ∞ = {x ∈ Λ ∞ : x(0, 0) = v}. For λ ∈ Λ, let Z(λ) = {x ∈ Λ ∞ : x(0, d(λ)) = λ}. Then {Z(λ) : λ ∈ Λ} forms a basis of compact open sets for a topology on Λ ∞ . For p ∈ N k , the shift map σ p : Λ ∞ → Λ ∞ defined by σ p (x))(m, n) = x(m + p, n + p) for x ∈ Λ ∞ is a local homeomorphism (for more details see [27, Remark 2.5, Lemma 2.6]). Following [27,Definition 2.7] we define the k-graph groupoid of Λ to be the Deaconu-Renault groupoid (2.2) G Λ := G(Λ ∞ , σ) = {(x, m − n, y) ∈ Λ ∞ × Z k × Λ ∞ : m, n ∈ N k , σ m (x) = σ n (y)}. The sets Z(µ, ν) := {(µx, d(µ) − d(ν), νx) : x ∈ Z(s(µ))} indexed by pairs µ, ν ∈ Λ with s(µ) = s(ν) form a basis of compact open bisections for a locally compact Hausdorff topology on G Λ . With this topology G Λ is an ample Hausdorff groupoid (see [27,Proposition 2.8]). The sets Z(λ) = Z(λ, λ) form a basis for the relative topology on G (0) Λ ⊆ G Λ . We identify G (0) Λ = {(x, 0, x) : x ∈ Λ ∞ } with Λ ∞ . The groupoid G Λ is minimal if and only if Λ is cofinal [27, Proof of Proposition 4.8]. As in the proof of [27,Theorem 5.2], there is a bijection between Λ ∞ ×Z k and (Λ× d Z k ) ∞ given by (x, p) → ((m, n) → (x(m, n), m + p)). After making this identification, we obtain an isomorphism of the groupoid G Λ× d Z k of the skew-product k-graph with the skew-product groupoid G Λ × c Z k corresponding to the canonical cocycle c(x, n, y) = n via the map ((x, p), m − n, (y, q)) → ((x, m − n, y), p). Let Λ be a row-finite k-graph with no sources. A Cuntz-Krieger Λ-family in a C *algebra B is a function s : λ → s λ from Λ to B such that (CK1) {s v : v ∈ Λ 0 } is a collection of mutually orthogonal projections; (CK2) s µ s ν = s µν whenever s(µ) = r(ν); (CK3) s * λ s λ = s s(λ) for all λ ∈ Λ; and (CK4) s v = λ∈vΛ n s λ s * λ for all v ∈ Λ 0 and n ∈ N k . The k-graph C * -algebra C * (Λ) is the universal C * -algebra generated by a Cuntz-Krieger Λ-family. There is an isomorphism C * (Λ) ∼ = C * (G Λ ) satisfying s λ → 1 Z(λ,s(λ)) (see [27,Corollary 3.5(i)]). As discussed above, we have G Λ× d Z k ∼ = G Λ × c Z k , and so C * (G Λ × c Z k ) ∼ = C * (G Λ× d Z k ) ∼ = C * (Λ × d Z k ). These are approximately finite dimensional (AF) algebras by [27,Lemma 5.4]. 2.3. K-theory for C * -algebras. This paper is concerned primarily with calculating groupoid homology, but it is motivated by the relationship between this and K-theory of groupoid C * -algebras, and some of our key results concern K-theory. Readers unfamiliar with C * -algebras and their K-theory, and who are primarily interested in groupoids and groupoid homology will not need to know more about C * -algebraic K-theory than the following points: C * -algebraic K-theory associates two abelian groups K 0 (A) and K 1 (A) to each C * -algebra A; these groups are invariant under Morita equivalence, and continuous with respect to inductive limits; the K 0 -group is the Grothendieck group of a semigroup consisting of equivalence classes of projections in matrix algebras over A; the K-groups of a crossed-product of a C * -algebra A by Z are related to those of A by the Pimsner-Voiculescu exact sequence [40]; and the K-groups of a crossed-product of a C * -algebra A by Z k fit into a spectral sequence, due to Kasparov [23], involving the homology groups of Z k with values in the K-groups of A. For more background on C * -algebraic K-theory, we refer the interested reader to [55], [46] or [2]. 2.4. c-soft sheaves. Let X be a locally compact Hausdorff space. By a sheaf of abelian groups, or simply a sheaf, over X, we mean a (not necessarily Hausdorff)étale groupoid F with unit space F (0) = X in which r = s, so every element belongs to the isotropy, and in which each isotropy group F x = F x x is abelian. We think of F as a group bundle over X with bundle map r = s. Given a subset W ⊆ X, we write Γ(W, F ) for the set {t : W → F : t(w) ∈ F w and t is continuous} of continuous sections of F over W . A sheaf F over X is said to be c-soft if the restriction map Γ(X, F ) → Γ(K, F ) is surjective for any compact set K ⊆ X (see e.g. [22,Definition 2.5.5] or [4, II.9.1]); that is, if every continuous section of Γ over a compact subset of X extends to a continuous section over all of X. The property of c-softness is a key hypothesis for results of Crainic and Moerdijk (see [11]) that we will need in our study of the homology of ample Hausdorff groupoids. Proof. Since F is the constant sheaf, for every W ⊆ X we have Γ(W, F ) ∼ = C(W, A). Let K ⊆ X be compact and fix f ∈ C(K, A). Then f (K) is a compact subset of the discrete group A, and hence finite. For a ∈ f (K) we let U a = f −1 (a). Since f is continuous each U a ⊆ K is clopen and since K is compact U a is compact and open. By Lemma 2.2, for each a ∈ A there exists a compact open set V a ⊆ X such that U a = V a ∩ K. Fix a total order ≤ on f (K) and for each a ∈ f (K), define V ′ a := V a \ b<a V b . Then each V ′ a is a compact open subset of X since the V a are compact open. Moreover, since the U a are mutually disjoint, we have V ′ a ∩ K = V a ∩ K for all a. Hence the formulã f (x) := a for x ∈ V ′ a 0 otherwise, defines a continuous extensionf of f to X. Equivalence of ample Hausdorff groupoids There are a number of notions of equivalence of groupoids that are relevant to us here, and we need to know that they all coincide for ample Hausdorff groupoids. The notions that we consider are Morita equivalence [11], groupoid equivalence [37] (see also [45]), equivalence via a linking groupoid, equivalence via isomorphic ampliations, similarity [43,44,32], Kakutani equivalence [32] and stable isomorphism [8]. We show that the first four of these notions coincide for all Hausdorffétale groupoids (Proposition 3.10), and that they all coincide for ample Hausdorff groupoids with σ-compact unit spaces (Theorem 3.12). The following notion of similarity, recorded by Matui [32,Definition 3.4] and called homological similarity there, appears in the context of algebraic groupoids in Renault's thesis [44], and earlier still in [43]-where Ramsay in turn attributes it to earlier work of Mackey. An adaptation of this notion to the situation of Lie groupoids also appears in [36] where it is called strong equivalence. Let G, H be Hausdorffétale groupoids and let ρ, σ : G → H be continuous homomorphisms. We say that ρ is similar to σ if there is a continuous map θ : G (0) → H such that θ(r(g))ρ(g) = σ(g)θ(s(g)) for all g ∈ G. We say that G and H are similar groupoids if there existétale homomorphisms ρ : G → H and σ : H → G such that σ • ρ is similar to id G and ρ • σ is similar to id H . In this case, each of the two maps, ρ and σ, is called a similarity. Remark 3.2. It is not stated in [44] or in [32] that similarity of groupoids is an equivalence relation, but this is standard (it is essentially the argument that category equivalence is an equivalence relation). It is also easy to give a direct argument: suppose that σ : G → H and ρ : H → G implement a similarity, and that α : H → K and β : K → H also implement a similarity. We aim to show that α • σ and ρ • β implement a similarity. By symmetry it suffices to find κ : G (0) → G such that κ(r(g)) ρ • β • α • σ (g) = gκ(s(g)) for all g. Since ρ • σ ∼ id G and β • α ∼ id H , there are θ : G (0) → G and η : H (0) → H such that θ(r(g))ρ σ(g) = gθ(s(g)) and η(r(h))β α(h) = hη(s(h)). Define κ(x) = θ(x)ρ η(r(σ(x))) . Then using that r(σ(r(g))) = r(σ(g)), and that ρ is a homomorphism, we compute κ(r(g)) ρ • β • α • σ (g) = θ(r(g))ρ η(r(σ(g)) β α(σ(g))) = θ(r(g))ρ σ(g) ρ η(s(σ(g))) = gθ(s(g))ρ η(s(σ(g))) = gκ(s(g)). Thus (ρ • β) • (α • σ) is similar to id G and by symmetry (α • σ) • (ρ • β) is similar to id K .ϕ : G → H is a weak equivalence if (i) the map from G (0) * H (0) H := {(u, γ) ∈ G (0) × H : ϕ(u) = r(γ)} to H (0) given by (u, γ) → s(γ) is anétale surjection, and (ii) G is isomorphic to the fibred product G (0) * H (0) H * H (0) G (0) = {(u, γ, v) ∈ G (0) × H × G (0) : r(γ) = ϕ(u), s(γ) = ϕ(v)}. If such a ϕ exists, then we write G ∼ → H. The groupoids G and H are Morita equivalent if there is a groupoid K that admits weak equivalences G ∼ ← K ∼ → H. Remark 3.5. It is not stated explicitly in [11] that Morita equivalence is in fact an equivalence relation, though this is certainly standard. In any case it follows from Theorem 3.12 below. In [36,Proposition 5.11] it is shown that the analogue of similarity for Lie groupoids implies the analogue of Morita equivalence. We briefly indicate how to modify their argument to obtain the same result for Hausdorffétale groupoids. Proof. Suppose that σ : G → H and ρ : H → G constitute a similarity of groupoids. We claim that the map σ is a weak equivalence. It is easy to show that (u, h) → s(h) is a surjection from G (0) * G (0) H to H (0) since any v ∈ H (0) is the image of (ρ(v), v). We claim that it is anétale map. For this, fix (u, h) ∈ G (0) * G (0) H, and use that σ is anétale map to choose a neighbourhood U of u such that σ| U is a homeomorphism onto its range. Pick a bisection neighbourhood B of h. By shrinking if necessary, we can assume that σ(U) = r H (B). Then s • π 2 is a homeomorphism on U * B ∼ = B. Since ρ, σ constitute a similarity of groupoids it is straightforward to see that the map g → (r(g), σ(g), s(g)) is a bijection from G to G (0) * H (0) H * H (0) G (0) . For g ∈ G we use that r, σ, and s areétale maps to find a neighbourhood U of g on which they are all homeomorphisms, and observe that then g → (r(g), σ(g), s(g)) is a homeomorphism onto r(U) * H (0) σ(U) * H (0) s(U). So g → (r(g), σ(g), s(g)) is continuous and open. The third notion of equivalence that we consider is the one formulated by Renault (see [45,Section 3]) and studied in [37]. Given a locally compact Hausdorff groupoid G, we say that a locally compact Hausdorff space Z is a left G-space if it is equipped with a continuous open map r : Z → G (0) and a continuous pairing (g, z) → g · z from G * X to X such that r(g · z) = r(g) and (gh) · z = g · (h · z) and such that r(z) · z = z. We say that Z is a free and proper left G-space if the map (g, x) → (g · x, x) is a proper injection from G * X to G × X. Right G-spaces are defined analogously. The fourth notion of equivalence we need to discuss is the generalisation of Kakutani equivalence developed by Matui [32,Definition 4.1] in the situation of ample Hausdorff groupoids with compact unit spaces, and extended to non-compact unit spaces in [8]. This notion has previously been discussed only for ample groupoids, but it makes sense for general Hausdorffétale groupoids, and in particular weak Kakutani equivalence is a fairly natural notion in this setting (though in this more general setting it is not an equivalence relation, see Example 3.13 ). For ample Hausdorff groupoids with σ-compact unit spaces, [8,Theorem 3.2] shows that weak Kakutani equivalence and Kakutani equivalence both coincide with groupoid equivalence, and with a number of other notions of equivalence. Our next two results show first that for Hausdorffétale groupoids, Morita equivalence and equivalence in the sense of [37] are equivalent to the existence of a linking groupoid, and to existence of isomorphic ampliations of the two groupoids in the following sense. If G is a Hausdorffétale groupoid, X is a locally compact Hausdorff space, and ψ : X → G (0) is a local homeomorphism, then the ampliation (also known as the blow-up [57, §3.3]) G ψ of G corresponding to ψ is given by (2) if and only if y = w, and composition and inverses given by (x, γ, y)(y, η, z) = (x, γη, z) and (x, γ, y) −1 = (y, γ −1 , x). This is a Hausdorf etale groupoid under the relative topology inherited from X × G × X. Example 3.9. Let X and Y be locally compact Hausdorff spaces and ψ : Y → X be a local homeomorphism. Then we may regard G ψ = {(x, γ, y) ∈ X × G × X : ψ(x) = r(γ) and ψ(y) = s(γ)} with (x, γ, y), (w, η, z) ∈ (G ψ )R(ψ) := {(y 1 , y 2 ) ∈ Y × Y : ψ(y 1 ) = ψ(y 2 )} as an Hausdorffétale groupoid (see [26]). Note that R(ψ) is the ampliation of the trivial groupoid X corresponding to ψ. Proof. For (1) =⇒ (2), suppose that φ : G → H is a weak equivalence in the sense of Definition 3.4. Then G is isomorphic to the fibred product G (0) * H * G (0) := {(x, η, y) ∈ G (0) × H × G (0) : φ(x) = r(η) and φ(y) = s(η)}. Using the first condition of Definition 3.4 it is straightforward to check that under the natural operations and topology, the disjoint union L := (G (0) * H * G (0) ) ⊔ (G (0) * H) ⊔ (H * G (0) ) ⊔ H is a Hausdorffétale groupoid satisfying (2) with respect to X = G (0) and Y := H (0) . For (2) =⇒ (3), one checks that given L as in (2), the subspace Z := {z ∈ L : r(z) ∈ X and s(z) ∈ Y }, under the actions of G and H by multiplication on either side, is a G-H-equivalence as in (3). For (3) =⇒ (4), suppose that Z is a G-H-equivalence; to avoid confusion, we will write ρ : Z → G (0) and σ : Z → H (0) for the anchor maps. Since Z/H ∼ = G (0) and since the right H-action is free, if x, y ∈ Z satisfy ρ(x) = ρ(y), then there is a unique element [x, y] H of H satisfying x · [x, y] H = y. By [37, §2] the map [·, ·] H is continuous (see also [50, Lemma 2.1]). Similarly there is a continuous pairing (x, y) → G [x, y] from {(x, y) ∈ Z 2 : σ(x) = σ(y)} to G such that G [x, y] · y = x. Consider the ampliations G ρ := {(x, γ, y) : x, y ∈ Z, γ ∈ G, ρ(x) = r(γ), ρ(y) = s(γ)} and H σ := {(x, η, y) : x, y ∈ Z, η ∈ H, σ(x) = r(η), σ(y) = s(η)}. If (x, γ, y) ∈ G ρ , then ρ(γ · y) = r(γ) = ρ(x), and so we can take the pairing [x, γ · y] H to obtain an element Θ(x, γ, y) := (x, [x, γ · y] H , y) ∈ H σ . It is routine to check that this is a continuous groupoid homomorphism. Symmetrically, we see that Θ ′ : (x, η, y) → (x, G [x · η, y], y) is a continuous groupoid homomorphism from H σ to G ρ . A simple calculation using the defining properties of G [·, ·] and [·, ·] H shows Θ and Θ ′ are mutually inverse. So Θ is an isomorphism, giving (4). For (4) =⇒ (1), fix ampliations G φ and H ψ and an isomorphism Θ : G φ → H ψ . Write π G for the canonical map (x, γ, y) → γ from G φ to G, and π H for the corresponding map from H ψ to H. We obtain continuous groupoid homomorphismsφ : G φ → G and ψ : G φ → H byφ := π G andψ := π H • Θ. It is routine to check that this determines a Morita equivalence Gφ ← G φψ → H. For the final statement, observe that if U is a full open subset of G 0 , then the argument of [9, Lemma 6.1] shows that GU = {g ∈ G : s(g) ∈ U} is a G-G| U -equivalence under the actions determined by multiplication in G. So, writing ∼ R for equivalence in the sense of Renault, if G and H are weakly Kakutani equivalent, say G| U ∼ = H| V , then we have G ∼ R G| U ∼ = H| V ∼ R H. Since ∼ R is an equivalence relation, we deduce that G ∼ R H. Remarks 3.11. (i) Recall the definition of R(ψ) from Example 3.9. Proposition 3. 10 shows that R(ψ) and X are equivalent, and so C * (R(ψ)) is Morita equivalent to C 0 (X). (ii) It follows from the proof of Proposition 3.10 that if G and H admit isomorphic ampliations, then there exist a locally compact Hausdorff space X, local homeomorphisms φ : X → G (0) and ψ : X → H (0) , and an isomorphism Θ : Our next result shows that the notions of equivalence in Proposition 3.10 are further equivalent to a number of additional conditions, including similarity, in the special case of ample Hausdorff groupoids with σ-compact unit spaces. We write R for the (discrete) full equivalence relation R = N × N. Proof. (1) =⇒ (2) follows from Lemma 3.6. Proposition 3.10 shows that (2)-(5) are equivalent, and [8,Theorem 3.2] shows that(4), (6), (7) and (8) are equivalent. In particular, we have (1) =⇒ (2) =⇒ · · · =⇒ (8). G φ → H ψ such that Θ(x, φ(x), x) = (x, ψ(x), x) for all x ∈ X. For (8) =⇒ (1), suppose that U ⊆ G (0) and V ⊆ H (0) are full open sets with G| U ∼ = H| V . Matui proves in [32, Theorem 3.6(2)] that G is similar to G| U and H is similar to H| V . Since any isomorphism of groupoids is a similarity, and since similarity of groupoids is an equivalence relation (Remark 3.2), it follows that G and H are similar. To close the section, we present an example to show that groupoid equivalence does not imply either similarity or weak Kakutani equivalence in general. We also show that weak Kakutani equivalence is not an equivalence relation. Example 3.13. Let Y := R and X := S 1 and define ψ : Y → X by ψ(y) = e 2πiy . Then ψ is a local homeomorphism and the groupoid R(ψ) (see Example 3.9) is equivalent to the trivial groupoid X (see Remarks 3.11 (1)). We claim that there is no similarity ρ : X → R(ψ). Indeed suppose that such a ρ exists. Since ρ is a groupoid map, we have ρ(X (0) ) ⊆ R(ψ) (0) . Identifying R(ψ) (0) = R and X (0) = S 1 we obtain a continuous map ρ : S 1 → R. Since ρ is a similarity it induces a bijective map on orbits (see Remark 3.3). Since the orbits in X are singletons, this implies that ρ is injective which is impossible. We also claim that X and R(ψ) are not weakly Kakutani equivalent. To see this, suppose that U is a full open subset of X (0) . Since X is trivial, we have U = X (0) = S 1 , which is not homeomorphic to any open subset of R = R(ψ) (0) . So there is no full open subset V ⊆ R(ψ) (0) such that X| U ∼ = R(ψ)| V , and so the two groupoids are not weakly Kakutani equivalent. Consider the local homeomorphism ϕ : X ⊔ Y → X given by ϕ(z) := z if z ∈ X, ψ(z) if z ∈ Y. Then X and R(ψ) are each weakly Kakutani equivalent to R(ϕ) but as shown above X is not weakly Kakutani equivalent to R(ψ). Crainic-Moerdijk-Matui homology for ample Hausdorff groupoids Crainic and Moerdijk introduced a compactly supported homology theory for Hausdorf etale groupoids in [11]. Matui reframed the theory for ample Hausdorff groupoids (though he did not explicitly require this; see [32, Definition 3.1]). To use the results of [11] we must ensure that the standing assumptions of [11, Section 2.5]) are satisfied. We therefore require that all groupoids we consider henceforth are locally compact, Hausdorff, second countable, and zero dimensional. For the reader's convenience we recall Matui's definition of homology for an ample Hausdorff groupoid G (see [32, Section 3.1]). Since a locally constant sheaf over such a groupoid with values in a discrete abelian group is c-soft (see Section 2.4), this agrees with the definition given by Crainic and Moerdijk [11, Section 3.1] under our standing assumptions. We first need to establish some notation. Given a locally compact Hausdorff zerodimensional space X and a discrete abelian group A, let C c (X, A) denote the set of compactly supported A-valued continuous (equivalently, locally constant) functions on X. Then C c (X, A) is an abelian group under pointwise addition. Given a (not necessarily surjective) local homeomorphism ψ : Y → X between two such spaces, as in [32, Section 3.1] we define a homomorphism ψ * : C c (Y, A) → C c (X, A) by (4.1) ψ * (f )(x) := ψ(y)=x f (y) for all f ∈ C c (Y, A), and x ∈ X. If U ⊆ Y is compact open and ψ| U is injective, then ψ * (1 U ) = 1 ψ(U ) , where 1 U is the indicator function of U. Recall that for n > 0, the space of composable n-tuples in a groupoid G is (4.2) G (n) = {(g 1 , . . . , g n ) ∈ G n : s(g i ) = r(g i+1 ) for 1 ≤ i < n}, while G (0) is the unit space. For n ≥ 2 and 0 ≤ i ≤ n we define d i : G (n) → G (n−1) by d i (g 1 , . . . , g n ) :=      (g 2 , . . . , g n ) i = 0, (g 1 , . . . , g i g i+1 , . . . , g n ) 1 ≤ i ≤ n − 1, (g 1 , . . . , g n−1 ) i = n. Note that G (n) is 0-dimensional and each d i is a local homeomorphism. Definition 4.1. Let G be a second-countable ample Hausdorff groupoid. For n ≥ 1 define ∂ n : C c (G (n) , A) → C c (G (n−1) , A) by ∂ 1 = s * − r * and ∂ n := n i=0 (−1) i (d i ) * for n ≥ 2, and define ∂ 0 to be the zero map from C c (G (0) , A) to 0. Routine calculations show that this defines a chain complex (C c (G ( * ) , A), ∂ * ). We define the homology of G with values in A to be the homology of this complex, denoted H * (G, A). If A = Z we simply write H * (G). Remark 4.2. An ample groupoid with one unit is just a discrete group. In this instance the groupoid homology just defined coincides with group homology, see [12,Section 2.22], and also [5]. Proof. The boundary maps for the groupoid X are all trivial and there are no nondegenerate n-chains for n ≥ 1. Remark 4.5. (1) Following [32, Definition 3.1], if G is an ample Hausdorff groupoid, then there is a natural preorder on H 0 (G) determined by the cone H 0 (G) + := {[f ] : f ∈ C c (G (0) , Z) and f (x) ≥ 0 for all x ∈ G (0) }. (2) For any 0-dimensional space X regarded as a groupoid as in Proposition 4.4, we have C * (X) ∼ = C 0 (X), and K 0 (C 0 (X)) ∼ = H 0 (X) ∼ = C c (X, Z), via an isomorphism that carries the positive cone of K 0 (C 0 (X)) to H 0 (X) + . One key point of the functoriality of homology described in the preceding remark is that it leads to the following notion of continuity for homology of ample Hausdorff groupoids. Then the groupoid maps ρ : R(ψ) → X given by ρ(y 1 , y 2 ) = ψ(y 1 ) and σ : X → R(ψ) given by σ(x) = (ϕ(x), ϕ(x)) are both similarities. Indeed, σ • ρ is similar to id R(ψ) and ρ • σ is similar to id X . Moreover, the induced maps ρ * : H * (R(ψ)) → H * (X) and σ * : H * (X) → H * (R(ψ)) are inverse to each other. Proof. The first assertion follows from the second. To prove the second assertion, define θ : R(ψ) (0) → R(ψ) by θ(y, y) = (ϕ • ψ(y), y). Then σ • ρ(y 1 , y 2 )θ(y 2 , y 2 ) = (ϕ • ψ(y 1 ), ϕ • ψ(y 2 ))(ϕ • ψ(y 2 ), y 2 ) = (ϕ • ψ(y 1 ), y 2 ) = (ϕ • ψ(y 1 ), y 1 )(y 1 , y 2 ) = θ(y 1 , y 1 ) id R(ψ) (y 1 , y 2 ). Hence, σ • ρ is similar to id R(ψ) . Since ρ • σ = id X , it follows that ρ • σ is similar to id X . The last assertion now follows from [32,Proposition 3.5]. For the following result, recall that if ψ : Y → X is a local homeomorphism then the homomorphism ψ * : C c (Y, Z) → C c (X, Z) is given in (4.1). There is also an inclusion ι : C 0 (Y ) ֒→ C * (R(ψ)) induced by the homeomorphism Y ∼ = R(ψ) (0) , and this induces a homomorphism ι * : C c (Y, Z) → K 0 (C * (R(ψ)). Theorem 4.10. Let X, Y be locally compact Hausdorff spaces, and let ψ : Y → X be a local homeomorphism. Then Y is an R(ψ)-X equivalence with anchor maps id : Y → R(ψ) (0) and ψ : Y → X, right action of X given by y · ψ(y) = y, and left action given by (x, y) · y = x. Hence H * (R(ψ)) ∼ = H * (X). If Y is σ-compact and totally disconnected, then the map ψ admits a continuous open section and the map ρ : R(ψ) → X given by ρ(y 1 , y 2 ) = ψ(y 1 ) is a similarity and thus induces the isomorphism H 0 (R(ψ)) ∼ = H 0 (X) = C c (X, Z) determined by [1 U ] → ψ * (1 U ) for U ⊆ Y compact and open. We have H n (R(ψ)) = 0 for n ≥ 1. The groupoid C * -algebra C * (R(ψ)) is an AF algebra, the map ψ * induces an isomorphism K 0 (C * (R(ψ))) → C c (X, Z) such that the diagram C c (Y, Z) K 0 (C * (R(ψ))) C c (X, Z) ψ * ι * ∼ = commutes, and we have K 1 (C * (R(ψ))) = {0}. Proof. For the first statement, one just checks directly that the maps described satisfy the axioms for an equivalence of groupoids. The isomorphism H * (R(ψ)) ∼ = H * (X) follows from Lemma 4.3. Now suppose that Y is σ-compact and totally disconnected. Then X is also σ-compact and totally disconnected. Choose a cover Y = ∞ i=1 U i of Y by countably many compact open sets. Since ψ is a local homeomorphism and the U i are compact, each U i is a finite union of compact open sets on which ψ is injective, so by relabelling we may assume that the U i have this property. For each i, let V i := U i \ i−1 j=1 ψ −1 (ψ(U j )) . Then X = i ψ(V i ), and the V i are compact open sets on which ψ restricts to a homeomorphism ψ i : V i → ψ(V i ). So we can define a continuous section ϕ for ψ by setting ϕ| ψ(V i ) = ψ −1 i : ψ(V i ) → V i . Hence Lemma 4.8 yields a similarity ρ : R(ψ) → X such that the restriction of ρ * to C c (Y, Z) coincides with ψ * : C c (Y, Z) → C c (X, Z). The equivalence Y of groupoids determines a Morita equivalence between C * (R(ψ)) and C 0 (X) [37, Theorem 2.8]. Since approximate finite dimensionality is preserved by Morita equivalence and C 0 (X) is AF, we see that C * (R(ψ)) is AF, and the Morita equivalence induces the desired isomorphisms in K-theory. To see that the diagram commutes, suppose that U ⊆ Y is compact open and that ψ| U is injective. Then ι * (1 U ) = [1 U ] ∈ K 0 (C * (R(ψ))), and this is carried to 1 ψ(U ) by the isomorphism K 0 (C * (R(ψ))) → C c (X, Z) just described. This is precisely ψ * (1 U ). Matui's HK Conjecture In [34, Conjecture 2.6] Matui posed the HK conjecture for a certain class of ample Hausdorff groupoids. Recall that anétale groupoid G is said to be effective if the interior of its isotropy coincides with its unit space G (0) and minimal if every orbit is dense. Matui's HK Conjecture. Let G be a locally compact Hausdorffétale groupoid such that G (0) is a Cantor set. Suppose that G is both effective and minimal. Then for j = 0, 1 we have (5.1) K j (C * r (G)) ∼ = ∞ i=0 H 2i+j (G). We are interested in the extent to which the isomorphism (5.1) holds amongst groupoids that do not necessarily have non-compact unit space and are not necessarily minimal or effective. To streamline our discussion, we make the following definition. In [21] Hazrat and Li verify (5.1) for j = 0 in the setting of groupoids of row-finite 1-graphs with no sinks. We complete the analysis for j = 1 in Theorem 6.7. In [38] Ortega shows that the Katsura-Exel-Pardo groupoid G A,B associated to square integer matrices with A ≥ 0 belongs to M. Here we consider Deaconu-Renault groupoids and thereby study higher dimensional aspects not present in other cases. There are examples of ample Hausdorff groupoids G that belong to M but either do not have compact unit spaces or which are not necessarily effective or minimal. For example, if X is any noncompact totally disconnected space, then the groupoid X ×Z satisfies none of these conditions, but belongs to M. It is also easy to show that Z n is in M for all n. Let F n denote the free group on n letters. Then by [ Here we expand the class of groupoids known to belong to M. We show that all AF groupoids, all Deaconu-Renault groupoids associated to actions of N or N 2 on 0dimensional spaces, and path groupoids associated to many one-vertex k-graphs belong to M. K 0 (C * r (F n )) ∼ = H 0 (F n ) ∼ = Z, K 1 (C * r (F n )) ∼ = H 1 (F n ) ∼ = Z By Proposition 4.4 and Remark 4.5, any 0-dimensional space X regarded as a trivial groupoid belongs to M. More generally, the following corollary to Theorem 4.10 shows that all AF groupoids belong to M. Corollary 5.2. Let G be a groupoid that can be expressed as a direct limit G = lim − → G n of open subgroupoids each of which is isomorphic to R(ψ n ) for some local homeomorphism ψ n : G (0) → X n . Suppose that G (0) is totally disconnected. Then there are maps ϕ n : X n → X n+1 such that ϕ n • ψ n = ψ n+1 for all n. We have H n (G) = 0 for n ≥ 1, and H 0 (G) ∼ = lim − → (H 0 (X n ), (ϕ n ) * ). There is an isomorphism K 0 (C * (G)) ∼ = H 0 (G) that carries [1 U ] 0 to [1 U ] for each compact open U ⊆ G (0) , and we have K 1 (C * (G)) = {0}. In particular, G belongs to M. Proof. Each R(ψ n ) (0) is totally disconnected because it is an open subspace of G (0) , and so Theorem 4.10 shows that H p (R(ψ n )) = 0 for p ≥ 1 and all n, and that H 0 (R(ψ n )) ∼ = C c (X n , Z) ∼ = K 0 (C * (R(ψ n ))) with both isomorphisms induced by (ψ n ) * . Since homology and K-theory are continuous with respect to inductive limits, the result follows. Deaconu-Renault groupoids In this section we first show that the homology of an ample higher-rank Deaconu-Renault groupoid G(X, σ) is given by H * (Z k , H 0 (G(X, σ) × c Z k )) using spectral sequence arguments of Matui. In Theorem 6.5 we describe a complex which allows us to compute H * (Z k , H 0 (G(X, σ) × c Z k ) by adapting techniques from [16]. We then use this description and Kasparov's K-theory spectral sequence to prove that G(X, σ) belongs to M when the rank is either one or two (see Theorems 6.7, 6.10). Furthermore we also give formulas for computing the K-theory of C * (G(X, σ)) in these cases. Recall from Section 2 that if σ is an action of N k on a locally compact Hausdorff space X by local homeomorphisms, then we write X := X × Z k , and there is an actionσ of N k by local homeomorphisms on X given byσ q (x, p) = (σ q (x), p + q). Equation (2.1) then defines an isomorphism of the skew-product groupoid G(X, σ) × c Z k corresponding to the cocycle c(x, m, y) = m onto the Deaconu-Renault groupoid G( X,σ). Our first result shows that G(X, σ) × c Z k is equivalent to c −1 (0); this in turn allows us to compute its homology. Lemma 6.1. Let X be a locally compact Hausdorff totally disconnected space, and let σ be an action of N k on X by local homeomorphisms. The set X × {0} ⊆ X × Z k is a clopen G( X,σ)-full subspace of G( X,σ) (0) . The map (x, 0, y) → ((x, 0), 0, (y, 0)) is an isomorphism of c −1 (0) ⊆ G(X, σ) onto G( X,σ)| X×{0} , and G(X, σ) × c Z k is an AF groupoid. There is an isomorphism of H * (c −1 (0)) onto H * (G( X,σ)) that carries [1 U ] to [1 U ×{0} ] for every compact open U ⊆ X. Proof. It is clear that X × {0} is a clopen subset of X × Z k , the unit space of G( X,σ). We claim that it is full. Fix (x, n) ∈ X × Z k and write n = n + − n − where n + , n − ∈ N k . Then there exists y ∈ X such that σ n − (y) = σ n + (x). Set γ = ((x, 0), n + − n − , (y, n)). By construction we haveσ n + (x, 0) =σ n − (y, n), and so γ ∈ G( X,σ). Furthermore r(γ) = (x, 0) ∈ X × {0} and s(γ) = (y, n), proving the claim. The map (x, 0, y) → ((x, 0), 0, (y, 0)) is clearly an injective homomorphism. To show that it is surjective, let γ = ((x, m), m − n, (y, n)) ∈ G( X,σ)| X×{0} . Then m = n = 0, so γ is in the range of c −1 (0). Since G(X, σ) × c Z k ∼ = G( X,σ) via the isomorphism given in (2.1), and since X × {0} is G( X,σ)-full, it follows from Theorem 3.12 that G(X, σ) × c Z k is equivalent to G( X,σ)| X×{0} . By the preceding paragraph G( X,σ)| X×{0} ∼ = c −1 (0) . Since c −1 (0) can be written as an increasing union of the elementary groupoids R(σ n ), it is AF (see Corollary 5.2), and so G(X, σ) × c Z k is an AF groupoid also. The final statement follows from Lemma 4.3. To compute the homology of c −1 (0), we decompose it as the increasing union of the subgroupoids R(σ n ) as n ranges over N k . Lemma 6.2. Let X be a totally disconnected locally compact Hausdorff space, and let σ be an action of N k on X by surjective local homeomorphisms. There is an isomorphism lim − → (C c (X, Z), σ n * ) → H 0 (c −1 (0)) that takes σ 0,∞ * (1 U ) to [1 U ] for every compact open U ⊆ X. We have H q (G(X, σ) × c Z k ) ∼ = lim − →n∈N k (C c (X, Z), σ n * ) if q = 0 0 otherwise. The isomorphism H 0 (G(X, σ) × c Z k ) ∼ = lim − →n∈N k (C c (X, Z), σ n * ) intertwines the action of Z k on H 0 (G(X, σ) × c Z k ) given by p · ((x, m, y), n) = ((x, m, y), n + p) with the action of Z k on lim − →n∈N k (C c (X, Z), σ n * ) induced by the σ n * . Proof. If U, V ⊆ X are compact open sets on which σ n is injective, and if we have σ n (U) = σ n (V ), then we have [1 U ] = [1 V ] in H 0 (R(σ n )). Since every W ⊆ X can be expressed as a finite disjoint union W = j σ n (U j ) where each U j is compact open and each σ n | U j is injective, it follows that there is a unique homomorphism ϕ n : C c (X, Z) → H 0 (R(σ n )) such that ϕ n (σ n * (1 U )) = [1 U ] for all compact open U. This ϕ n is the map induced by the weak equivalence of groupoids ϕ n : R(σ n ) → X (see Remark 3.11), so Proposition 3.10 and Lemma 4.3 imply that it is an isomorphism. For m, n ∈ N k , let ι m,n be the inclusion map R(σ m ) ֒→ R(σ m+n ), and let (ι m,n ) * : H 0 (R(σ m )) → H 0 (R(σ m+n )) be the induced map in homology. Then we have a commuting diagram C c (X, Z) C c (X, Z) H 0 (R(σ m )) H 0 (R(σ m+n )) σ n * ϕ m (ι m,n ) * ϕ m+n Since c −1 (0) is the increasing union of the closed subgroupoids R(σ n ), continuity of homology gives H 0 (c −1 (0)) ∼ = lim − → (H 0 (R(σ m )), ι * ). So the universal properties of the direct limits prove the first statement. The isomorphism (2.1) of G(X, σ) × c Z k with G( X, σ) and Lemma 6.1 show that 1 (0)). So the preceding paragraph proves that H 0 (G(X, σ) × c Z k ) ∼ = lim − →n∈N k (C c (X, Z), σ n * ). We saw in the preceding paragraph that c −1 (0) is an AF groupoid, so Corollary 5.2 proves that H q (G(X, σ) × c Z k ) = 0 for q ≥ 1. The final statement follows from direct computation of the maps involved. H * (G(X, σ) × c Z k ) ∼ = H * (c − Matui's spectral sequence [32,Theorem 3.8(1)] relates H * (G(X, σ)) to the homology of Z k with coefficients in H * (G(X, σ) × c Z k ). Since H q (G(X, σ) × c Z k )) = 0 for q ≥ 1 (it is AF by Lemma 6.2), the spectral sequence collapses: It follows that H q (G(X, σ)) ∼ = H q (Z k , H 0 (G(X, σ) × c Z k )) for 0 ≤ q ≤ k. The proof of the following lemma is based on the technique developed in [16,Section 3]. Recall that for 1 ≤ p ≤ k, the module p R k is the free R-module generated by the elements ε i 1 ∧ · · · ∧ ε ip indexed by integer tuples 1 ≤ i 1 < i 2 < · · · < i p ≤ k. We define 0 R k := R. Lemma 6.3. Let A be an abelian group, and suppose that σ 1 , . . . , σ k are pairwise commuting endomorphisms of A (i.e. an action of N k ). For 1 ≤ p ≤ k, define ∂ p : p Z k ⊗ A → p−1 Z k ⊗ A on spanning elements (ε i 1 ∧ · · · ∧ ε ip ) ⊗ a in which the i j are in strictly increasing order by ∂ p (ε i 1 ∧ · · · ∧ ε ip ⊗ a) = j (−1) j+1 ε i 1 ∧ · · · ∧ ε i j ∧ · · · ∧ ε ip ⊗ (id −σ i j )a for p > 1, 1 ⊗ (id −σ i 1 )a for p = 1. Then ∂ p−1 • ∂ p = 0 for each p, so that ( * Z k ⊗ A, ∂ * ) is a complex. For each 0 ≤ i ≤ k, the homomorphism id ⊗σ i : * Z k ⊗ A → * Z k ⊗ A commutes with ∂ * , and the induced map (id ⊗σ i ) * in homology is the identity map. Proof. Direct computation on a spanning element (ε i 1 ∧ · · · ∧ ε i p+1 ) ⊗ a of * Z k ⊗ A using that the σ i commute shows that ∂ p+1 • ∂ p = 0. Clearly the id ⊗σ i commute with ∂ * . For the final statement, we first claim that for x ∈ p Z k ⊗ A and 1 ≤ l ≤ k, we have ∂ p+1 (ε l ∧ x) = −ε l ∧ ∂ p (x) + (id ⊗(id −σ l ))(x). To prove this, it suffices to consider x = (ε i 1 ∧ · · · ∧ ε ip ) ⊗ a where 1 ≤ i 1 < · · · < i p ≤ k. So fix such an x and fix l ≤ k. We consider two cases. Case 1: l = i h for all h. Then there exists 0 ≤ j ≤ p such that i h < l for h ≤ j and i h > l for h > j. Then, using at both the first and the third steps that ε l ∧ ε i 1 ∧ · · · ∧ ε in = (−1) n ε i 1 ∧ · · · ∧ ε in ∧ ε l for any i 1 < i 2 < · · · < i n , we calculate: ∂ p+1 (ε l ∧ x) = (−1) j ∂ p+1 (ε i 1 ∧ · · · ∧ ε i j ∧ ε l ∧ ε i j+1 ∧ · · · ∧ ε p ⊗ a) = (−1) j j h=1 (−1) h+1 ε i 1 ∧ · · · ∧ ε i h ∧ · · · ∧ ε i j ∧ ε l ∧ ε i j+1 ∧ · · · ∧ ε p ⊗ (id −σ i h )a + (−1) j+2 ε i 1 ∧ · · · ∧ ε i j ∧ ε i j+1 ∧ · · · ∧ ε p ⊗ (id −σ l )a + p h=j+1 (−1) h+2 ε i 1 ∧ · · · ∧ ε i j ∧ ε l ∧ ε i j+1 ∧ · · · ∧ ε i h ∧ · · · ∧ ε p ⊗ (id −σ i h )a = p h=1 (−1) h ε l ∧ ε i 1 ∧ · · · ∧ ε i h ∧ · · · ∧ ε p ⊗ (id −σ i h )a + ε i 1 ∧ · · · ∧ ε p ⊗ (id −σ l )a = ε l ∧ p h=1 (−1) h ε i 1 ∧ · · · ∧ ε i h ∧ · · · ∧ ε p ⊗ (id −σ i h )a + (id ⊗(id −σ l ))(ε i 1 ∧ · · · ∧ ε ip ⊗ a) = −ε l ∧ ∂ p (x) + (id ⊗(id −σ l ))(x). Case 2: l = i h for some h. Then ∂ p+1 (ε l ∧ x) = ∂ p+1 (0) = 0. So we must show that ε l ∧ ∂ p (x) = (id ⊗(id −σ l ))(x). We have ε l ∧ ∂ p (x) = p j=1 ε l ∧ (−1) j+1 ε i 1 ∧ · · · ∧ ε i j ∧ · · · ∧ ε p ⊗ (id −σ i j )(a). The terms corresponding to i j = l are zero, so this collapses to ε l ∧ ∂ p (x) = (−1) h+1 ε l ∧ ε i 1 ∧ · · · ∧ ε i h ∧ · · · ∧ ε ip ⊗ (id −σ i h )(a) = ε i 1 ∧ · · · ∧ ε ip ⊗ (id −σ l )(a) = (id ⊗(id −σ l ))(x). This completes the proof of the claim. We now prove the final statement. Fix x ∈ p Z k ⊗ A such that ∂ p (x) = 0, and fix l ≤ p. We just have to show that (id ⊗(id −σ l ))(x) ∈ image(∂ p+1 ). Since ∂ p (x) = 0 and using the claim, we see that (id ⊗(id −σ l ))(x) = (id ⊗(id −σ l ))(x) − ε l ∧ ∂ p (x) = ∂ p+1 (ε l ∧ x), and the result follows. Lemma 6.4. Let A be an abelian group, and suppose that σ 1 , . . . , σ k are pairwise commuting endomorphisms of A (i.e. an action of N k ). Let ∂ p : * Z k ⊗ A → * Z k ⊗ A be as in Lemma 6.3. Let A := lim − →N k (A, σ n ). For i ≤ k letσ i be the automorphism of A induced by σ i , and let∂ p : * Z k ⊗ A → * Z k ⊗ A be the boundary map obtained from Lemma 6.3 applied to A and theσ i . Then the canonical homomorphism σ 0,∞ : A → A corresponding to the 0 th copy of A induces an isomorphism H * ( * Z k ⊗ A) ∼ = H * ( * Z k ⊗ A). Moreover, σ extends to an action of Z k on A, and H * ( * Z k ⊗ A) ∼ = H * (Z k , A). Proof. Since the homology functor is continuous (see, for example, [52,Theorem 4 .1.7]) we have H * ( * Z k ⊗ A) ∼ = lim − → H * ( * Z k ⊗ A), (id ⊗σ n ) * . Lemma 6.3 shows that this is equal to lim − → H * ( * Z k ⊗ A), id = H * ( * Z k ⊗ A) . For the second statement, we follow the argument of [16,Lemma 3.12] (see also [29,Theorem 5.5]). Let G := Z k = s 1 , . . . , s k and let R := ZG. For p ≥ 2, we define ∂ p : p R k → p−1 R k by ∂ p (ε i 1 ∧ · · · ∧ ε ip ) = j (−1) j+1 (1 − s i j )ε i 1 ∧ · · · ∧ ε i j ∧ · · · ∧ ε ip . Define ∂ 1 : 1 R k → R by ∂ 1 (ε j ) := 1 − s j , and let η : R → Z be the augmentation homomorphism determined by η(s i ) = 1 for each i. Then 0 −→ k R k ∂ k −→ · · · ∂ 2 −→ 1 R k ∂ 1 −→ R η − → Z is a free resolution of Z. Hence, by definition of homology with coefficients in the Z k - module A [5, Equation III(1.1)], we have H * (Z k , A) ∼ = H * ( * R k ⊗ R A) . As a group we have p R k ⊗ R A ∼ = * Z k ⊗ A, and this isomorphism intertwines the boundary maps in the complex defining H * ( * R k ⊗ R A) with the maps ∂ p . We can now state our main theorem for this section, which is a computation of the homology of the Deaconu-Renault groupoid G(X, σ) associated to an action of N k by surjective local homeomorphisms of a totally disconnected locally compact space X. Theorem 6.5. Let X be a second-countable totally disconnected locally compact space, and let σ be an action of N k by surjective local homeomorphisms σ p : X → X. For 1 ≤ p ≤ k, let A σ p = p Z k ⊗ C c (X, Z) and let A σ p = {0} for p > k. For p ≥ 1, define ∂ p : A σ p → A σ p−1 by ∂ p (ǫ i 1 ∧ · · · ∧ ǫ ip ⊗ f ) =            id ⊗(id −σ e i 1 * )f if p = 1, p j=1 (−1) j+1 ǫ i 1 ∧ · · · ∧ ǫ i j ∧ · · · ∧ ǫ ip ⊗ (id −σ e i j * )f if 2 ≤ p ≤ k, 0 if p ≥ k + 1. Then (A σ * , ∂ * ) is a complex, and H * (G(X, σ)) ∼ = H * (Z k , H 0 (G(X, σ) × c Z k )) ∼ = H * (A σ * , ∂ * ). We have H p ((Z k , H 0 (G(X, σ) × c Z k )) = 0 for p > k. Proof. Lemma 6.3 implies that (A σ * , ∂ * ) is a complex. The automorphisms α p : ((x, m, y), n) → ((x, m, y), n + p) of G(X, σ) × c Z k induce an action α p of Z k on H * (G(X, σ) × c Z k ). This action α makes each H p (G(X, σ) × c Z k ) into a Z k -module, so it makes sense to discuss the homology groups H * (Z k , H q (G(X, σ) × c Z k )) of Z k with coefficients in these modules. By [32, Theorem 3.8(1)] applied to the cocycle c : G(X, σ) → Z k , there is a spectral sequence E r p,q converging to H p+q (G(X, σ)) satisfying E 2 p,q = H p (Z k , H q (G(X, σ) × c Z k )). Lemma 6.2 shows that H q (G(X, σ) × c Z k ) is zero for q = 0, and therefore the differential maps on the E 2 page and above of the spectral sequence are trivial. Hence E ∞ p,q = E 2 p,q for all p, q and so [32, Theorem 3.8(1)] shows that H p (G(X, σ)) ∼ = E 2 p,0 ∼ = H p (Z k , H 0 (G(X, σ) × c Z k )) for all p. The last statement of Lemma 6.2 gives H * (Z k , H 0 (G(X, σ)× c Z k )) ∼ = H * (Z k , lim − → (X, σ n * )). Lemma 6.4 shows that this is isomorphic to H * (A σ * , ∂ * ). Our next two results show that G(X, σ) belongs to M if k ≤ 2. For k = 1 we use the Pimsner-Voiculescu sequence [40,Theorem 2.4]. To prove them, we need the following lemma, which follows from a standard argument. Lemma 6.6. Let X be a second-countable locally compact totally disconnected space. Let σ be an action of N k on X by surjective local homeomorphisms. Let c : G(X, σ) → Z k be the cocycle c(x, m, y) = m. Then the action of Z k on G(X, σ) × c Z given by α p ((x, m, y), n) = ((x, m, y), n + p) induces an actionᾱ of Z k on C * (G(X, σ) × c Z). The crossed product C * (G(X, σ) × c Z) ⋊ᾱ Z k is stably isomorphic to C * (G(X, σ)). Proof. Every automorphism of a groupoid induces an automorphism of its C * -algebra, and then simpler calculations establish the first statement. Let γ : T k → Aut(C * (G(X, σ))) be the gauge action γ z (f (x, m, y)) = z m f (x, m, y). Then there is an isomorphism θ : C * (G(X, σ) × c Z k ) → C * (G(X, σ)) × γ T k such that for f ∈ C c (G(X, σ) × {m}) ⊆ C c (G(X, σ) × c Z k ), we have θ(f )(z) (g) = z m f (g, m). This θ intertwinesᾱ and the dual action γ on C * (G(X, σ)) × γ T k , and so Takesaki-Takai duality implies that C * (G(X, σ) × c Z k ) ⋊ᾱ Z k is stably isomorphic to C * (G(X, σ)) (see [54,Theorem 4.5], [53,Theorem 3.4], and [7, Theorem 1.2]). Theorem 6.7. Let X be a second-countable locally compact totally disconnected locally compact space, let σ : X → X be a surjective local homeomorphism, and let σ * : C c (X, Z) → C c (X, Z) be the induced map. Then K 0 (C * (G(X, σ)) ∼ = H 0 (G(X, σ)) ∼ = coker(id −σ * ), K 1 (C * (G(X, σ)) ∼ = H 1 (G(X, σ)) ∼ = ker(id −σ * ), and H n (G(X, σ)) = 0 for n ≥ 2. In particular, G(X, σ) belongs to M. Proof. We first calculate the K-theory of C * (G(X, σ)). Lemma 6.6 applied with k = 1 shows that K * (C * (G(X, σ))) ∼ = K * (C * (G(X, σ) × c Z) ⋊ᾱ Z). Corollary 5.2 shows that K 1 (C * (G(X, σ) × c Z)) = 0, and so exactness of the Pimsner-Voiculescu sequence [40,Theorem 2.4] implies that (6.1) K 0 (C * (G(X, σ))) ∼ = coker(id −ᾱ * ) and K 1 (C * (G(X, σ))) ∼ = ker(id −ᾱ * ). We now compute the homology of G(X, σ). Theorem 6.5 shows that H p (G(X, σ)) ∼ = H p (Z, H 0 (G(X, σ) × c Z)) for all p. Moreover H p (Z, H 0 (G(X, σ) × c Z)) = 0 for p ≥ 2 by Theorem 6.5. Let α * be the action of Z on H 0 (G(X, σ) × c Z) induced by α. By Recall from Lemma 6.1 that G(X, σ)× c Z is an AF groupoid. The isomorphism between H 0 (G(X, σ) × c Z) and K 0 (C * (G(X, σ) × c Z)) supplied by Corollary 5.2 intertwines the Zactions α * andᾱ * . Thus coker(id −ᾱ * ) ∼ = coker(id −α * ) and ker(id −ᾱ * ) ∼ = ker(id −α * ). Hence by (6.1) and (6.2), K 0 (C * (G(X, σ)) ∼ = H 0 (G(X, σ)) and K 1 (C * (G(X, σ)) ∼ = H 1 (G(X, σ)). Since k = 1 here, the complex A σ * of Theorem 6.5 reduces to 0 −→ C c (X, Z) −→ C c (X, Z) −→ 0 where the central map is id −σ * . Then we have H 0 (G(X, σ)) ∼ = H 0 (A σ * , ∂ * ) ∼ = coker(id −σ * ), and H 1 (G(X, σ)) ∼ = H 1 (A σ * , ∂ * ) ∼ = ker(id −σ * ) as required. Remark 6.8. Matui's paper [32], together with recent results by Hazrat and Li [21] and Ortega [38] suggest that the groupoids of (not necessarily finite) 1-graphs with no sources belong to M. This now follows from Theorem 6.7, since graph groupoids are, by definition, rank-1 Deaconu-Renault groupoids. We now discuss Kasparov's K-theory spectral sequence for C * (G(X, σ)). Lemma 6.6 shows that C * (G(X, σ)) is stably isomorphic to the crossed product C * (G(X, σ) × c Z k ) ×ᾱ Z k . Hence Theorem 6.10 of [23] (see also [16, §3]) shows that there is a spectral sequence E r p,q converging to K * (C * (G(X, σ))) with E 2 -page given by E 2 p,q = H p (Z k , K q (C * (G(X, σ)) × c Z k )) if q is even, and 0 ≤ p ≤ k 0 otherwise. The differential maps in the spectral sequence are maps d r p,q : E r p,q → E r p−r,q+r−1 . If r > k, then for any p, q at least one of E r p,q and E r p−r,q+r−1 is trivial, because E r p,q is nontrivial only for 0 ≤ p ≤ k. Hence d r p,q is trivial for r > k. Thus E ∞ p,q = E k+1 p,q . If k is even, we can improve on this: if r = k is even, then at least one of E r p,q and E r p−r,q+r−1 is trivial because E r p,q is nontrivial only for q even, and it follows that E ∞ p,q = E k p,q for all p, q. In particular, if k = 2, then we have E ∞ p,q = E 2 p,q . For our next theorem, we need the well-known fact that if X is locally compact Hausdorff space, then C c (X, Z) is a free abelian group. We provide a proof for completeness. Lemma 6.9. Let X be a second-countable locally compact Hausdorff space. Then C c (X, Z) is a free abelian group. Proof. First note that if X is not compact, then C(X ∪ {∞}, Z) ∼ = C c (X, Z) ⊕ Z via f → (f − f (∞)1, f (∞)) , so it suffices to prove the result for X compact. So suppose that X is compact. Since X is metrisable by the Urysohn metrisation theorem (see [56,Theorems 23.1 and 17.6(a)]), the Alexandroff-Hausdorff theorem (see [56,Theorem 30.7]) shows that there is a continuous surjection φ : {0, 1} ∞ → X. Hence φ * : C(X, Z) → C({0, 1} ∞ , Z) is an injective group homomorphism. Since subgroups of free abelian groups are themselves free abelian, it therefore suffices to show that C({0, 1} ∞ , Z) is free abelian. For this, let {0, 1} * denote the collection of all finite words in the symbols 0, 1, including the empty word ε. Let I = {ε} ∪ {ω1 : ω ∈ {0, 1} * } denote the subset of {0, 1} * consisting of the empty word and all nontrivial words that end with a 1. We claim that B := {1 Z(ω) : ω ∈ I} is a family of free abelian generators of C({0, 1} ∞ , Z). To see this, we first argue by induction on n that span Z {1 Z(ω) : ω ∈ I and |ω| ≤ n} = span Z {1 Z(ω) : ω ∈ {0, 1} n } for all n. The containment ⊆ is trivial. The containment ⊇ is also trivial for n = 0, and if it holds for n = k, then for each ω = ω ′ 0 ∈ {0, 1} k+1 that ends in a 0, we have 1 Z(ω) = 1 Z(ω ′ ) − 1 Z(ω ′ 1) . We have ω ′ 1 ∈ I with |ω ′ 1| = k + 1, and 1 Z(ω ′ ) ∈ span Z {1 Z(ω) : ω ∈ I and |ω| ≤ k} by the inductive hypothesis. So the containment ⊇ also holds for n = k + 1. Hence B generates C({0, 1} ∞ , Z) as a group. To see that B is a family of free generators, suppose for contradiction that F ⊆ I is a finite set and {a ω : ω ∈ F } are nonzero integers such that ω∈F a ω 1 Vω = 0. Fix µ ∈ F of minimal length. Then µ000 · · · ∈ {0, 1} ∞ belongs to Z(µ) but not to Z(ω) for any ω ∈ F \ {µ}. Hence 0 = ω∈F a ω 1 Vω (µ000 · · · ) = a µ contradicting the assumption that the a ω are nonzero. Theorem 6.10. Let X be a second-countable locally compact totally disconnected space. Let σ be an action of N 2 on X by surjective local homeomorphisms. Define d 2 : C c (X, Z) → C c (X, Z) ⊕ C c (X, Z) by d 2 (f ) = ((σ e 2 * − id)f, (id −σ e 1 * )f ) and define d 1 : C c (X, Z) ⊕ C c (X, Z) → C c (X, Z) by d 1 (f ⊕ g) = (id −σ e 1 * )f + (id −σ e 2 * )g. Then K 0 (C * (G(X, σ)) ∼ = H 0 (G(X, σ)) ⊕ H 2 (G(X, σ)) ∼ = coker(d 1 ) ⊕ ker(d 2 ), and K 1 (C * (G(X, σ)) ∼ = H 1 ( G(X, σ)) ∼ = ker(d 1 )/ image(d 2 ). In particular, G(X, σ) belongs to M. Proof. Let A := K 0 (C * (G(X, σ) × c Z 2 )) = H 0 (G(X, σ) × c Z 2 ) . Lemma 6.6 applied with k = 2 shows that C * (G(X, σ)) is stably isomorphic to C * (G(X, σ) × c Z 2 ) ×ᾱ Z 2 for the actionᾱ induced by translation in Z 2 in the skew-product G(X, σ) × c Z 2 . We follow the argument of [16,29]. As discussed above, Kasparov's spectral sequence [23, Theorem 6.10] for K * (C * (G(X, σ) × c Z 2 ) ×ᾱ Z 2 ) converges on the second page, and we deduce that K 0 (C * (G(X, σ)) is an extension of E 2 0,2 by E 2 2,0 while K 1 (C * (G(X, σ)) is isomorphic to E 2 0,1 . As discussed prior to the statement of the theorem, E 2 p,q is isomorphic to the homology group H p (Z 2 , K 0 (C * (G(X, σ)) × c Z k )) for q even and is zero for q odd. Corollary 5.2 shows that K 0 (C * (G(X, σ)) × c Z 2 ) ∼ = H 0 (G(X, σ) × c Z 2 ), and that this isomorphism intertwines the actions of Z 2 on the two groups induced by translation in the second coordinate. It therefore follows from Theorem 6.5 that for q even, we have E 2 p,q ∼ = H p (A σ * , ∂ * ) . Since C c (X, Z) is free abelian by Lemma 6.9, so is the subgroup H 2 (A σ * , ∂ * ) = ker(∂ 2 ). Hence the extension K 0 (C * (G(X, σ)) of E 2 0,2 by E 2 2,0 splits, and we obtain K 0 (C * (G(X, σ)) ∼ = H 0 (A σ * , ∂ * ) ⊕ H 2 (A σ * , ∂ * ) and K 1 (C * (G(X, σ))) ∼ = H 1 (A σ * , ∂ * ). The result then follows from Theorem 6.5 because the obvious identifications j Z 2 ⊗ C(X, Z) ∼ = C(X, Z) ( 2 j ) for 0 ≤ j ≤ 2 intertwine ∂ * with d * . It follows that the path groupoid of a 2-graph belongs to M since it is a rank-2 Deaconu-Renault groupoid (see Corollary 7.7). Question 6.11. Our proof Theorem 6.10 uses that H 2 (A σ * ) is a free abelian group so that the extension 0 → H 0 (G(X, σ)) → K 0 (C * (G(X, σ))) → H 2 (G(X, σ)) → 0 splits. Hence the map from H 0 (G(X, σ)) ⊕ H 2 (G(X, σ)) to K 0 (C * (G(X, σ))) that we obtain is not natural. An interesting question arises: can the isomorphism (5.1) be chosen to be natural in some sense for elements of M in general, and for rank-2 Deaconu-Renault groupoids in particular? Remark 6.12. The proof of Theorem 6.10 is special to the situation k = 2, and issues arise already when k = 3. In this situation, the groups on the E 3 -page of Kasparov's spectral sequence coincide with those on the E 2 -page, but the E 3 -page has potentially nontrivial differential maps, d 3 3,2l : E 3 3,2l → E 3 0,2l+2 . So E 3 p,q = H p (A σ * ) if q is even, and is 0 if q is odd, and the E 3 -page has the following form. The sequence converges on the E 4 -page, and so we have exact sequences is trivial, there is no reason to expect that H 2 (A σ * ) is free abelian, so the extension defining K 0 (C * (G(X, σ))) need not split. This suggests rank-3 Deaconu-Renault groupoids as a potential source of counterexamples to Matui's HK-conjecture. Remark 6.13. If the groups H * (A σ * ) are finitely generated, and the natural homomorphism H 0 (G(X, σ)) → K 0 (C * (G(X, σ))) is injective, then one would expect d 3 3,0 to be trivial, and then in the rank-3 case, G(X, σ) would satisfy Matui's conjecture up to stabilisation by Q (the so-called rational HK-conjecture). This suggests that it would be worthwhile to investigate when the homomorphism H 0 (G(X, σ)) → K 0 (C * (G(X, σ))) is injective. H 0 (A σ * ) H 1 (A σ * ) H 2 (A σ * ) H 3 (A σ * ) 0 0 0 0 0 0 H 0 (A σ * ) H 1 (A σ * ) H 2 (A σ * ) H 3 (A σ * ) 0 0 0 0 0 0 . .0 → coker(d 3 3,0 ) →K 0 (C * (G(X, σ))) → H 2 (A σ * ) → 0 and 0 → H 1 (A σ * ) →K 1 (C * (G(X, σ))) → ker(d 3 3,0 ) → 0. k-graphs In this section, we first establish the existence of a natural map from the homology of a k-graph to the homology of its groupoid. We show that this homomorphism is in general neither injective nor surjective. We then apply the results of Section 6 to see that all 1-graph and 2-graph groupoids belong to M. Finally, we restrict our attention to k-graphs with one vertex, and demonstrate that for any such k-graph in which gcd(|Λ e 1 |− 1, . . . , |Λ e k | − 1) = 1, the corresponding k-graph groupoid belongs to M. Proof. It suffices to prove that Ψ * intertwines the boundary maps on generators of C * (Λ). Fix λ ∈ Λ = Λ * 1 ⊆ C 1 (Λ). Then ∂ 1 (Ψ 1 (λ)) = ∂ 1 (1 Y (λ) ) = s * (1 Y (λ) ) − r * (1 Y (λ) ) = 1 Z(s(λ)) − 1 Z(r(λ)) = Ψ 0 (s(λ)) − Ψ 0 (r(λ)) = Ψ 0 (∂ 1 (λ)). For n ≥ 2, it suffices to prove that given an element (λ 1 , . . . , λ n ) ∈ Λ * n and any 0 ≤ i ≤ n, we have . . . , λ n )) = Ψ n (d i (λ 1 , . . . , λ n )). (7.1) (d i ) * (Ψ n (λ 1 , In the following calculation, given sets Z 1 , . . . , Z n ⊆ G Λ we define Z 1 * Z 2 * · · · * Z n := (Z 1 × Z 2 × · · · × Z n ) ∩ G (n) Λ . Note that Y (λ 1 , . . . , λ n ) = Z(λ 1 , s(λ 1 )) * Z(λ 2 , s(λ 2 )) * · · · * Z(λ n , s(λ n )). First suppose that 1 ≤ i ≤ n − 1. Then (d i ) * (Ψ n (λ 1 , . . . , λ n )) = (d i ) * 1 Y (λ 1 ,...,λn) = (d i ) * 1 Z(λ 1 ,s(λ 1 )) * Z(λ 2 ,s(λ 2 )) * ··· * Z(λn,s(λn) ) = 1 Z(λ 1 ,s(λ 1 )) * ··· * Z(λ i ,s(λ i ))Z(λ i+1 ,s(λ i+1 )) * ··· * Z(λn,s(λn)) , and Ψ n (d i (λ 1 , . . . , λ n )) = Ψ n (λ 1 , . . . , λ i λ i+1 , . . . , λ n ) = 1 Y (λ 1 ,...,λ i λ i+1 ,...,λn) = 1 Z(λ 1 ,s(λ 1 )) * ··· * Z(λ i λ i+1 ,s(λ i+1 )) * ··· * Z(λn,s(λn)) . Since the cylinder sets Z(λ i , s(λ i )) and Z(λ i+1 , s(λ i+1 )) are both bisections and since s(Z(λ i , s(λ i ))) = Z(s(λ i )) = r(Z(λ i+1 , s(λ i+1 ))), we have Z(λ i , s(λ i ))Z(λ i+1 , s(λ i+1 )) = Z(λ i λ i+1 , s(λ i+1 )), and (7.1) follows. Now consider i = 0 (the case i = n is very similar). We have (d 0 ) * (Ψ n (λ 1 , . . . , λ n )) = (d 0 ) * 1 Y (λ 1 ,...,λn) = (d 0 ) * Z(λ 1 , s(λ 1 )) * Z(λ 2 , s(λ 2 )) * · · · * Z(λ n , s(λ n )) = 1 Z(λ 2 ,s(λ 2 )) * ··· * Z(λn,s(λn)) = 1 Y (λ 2 ,··· ,λn) = Ψ n (λ 2 , · · · , λ n ) = Ψ n (d 0 (λ 1 , . . . , λ n )). In general the map Ψ * is neither injective nor surjective: see Remark 7.8. 7.2. The HK conjecture for 1-graph groupoids and 2-graph groupoids. In this subsection we apply the results of §6 to groupoids associated to 1-graphs and 2-graphs. Recall from §2 that if Λ is a k-graph, then there is an action σ of N k by endomorphisms on its infinite-path space Λ ∞ and that the k-graph C * -algebra coincides with the C *algebra C * (G(Λ ∞ , σ)) of the associated Deaconu-Renault groupoid (see [27]). We begin this section by showing that the homology of G(Λ ∞ , σ) as computed in Theorem 6.5 coincides with the homology of the complex D Λ * used by Evans [16] to compute the Ktheory of C * (Λ). The complex (D Λ * , ∂ * ) is given as follows. We write ε 1 , . . . , ε k for the generators of Z k , and we write ε v for the generators of ZΛ 0 . We write M j ∈ M Λ 0 (Z) for the vertex matrix given by M j (v, w) = |vΛ e j w|, which we regard as an endomorphism of ZΛ 0 . For p ≥ 0 we define D Λ p = p Z k ⊗ ZΛ 0 and we define D Λ 0 = ZΛ 0 , and D Λ p = {0} for p > k. For p ≥ 1 we define ∂ p : D Λ p → D Λ p−1 by ∂ p = 0 if p > k, ∂ p (ε i 1 ∧ · · · ∧ ε ip ⊗ ε v ) = p j=0 (−1) j+1 ε i 1 ∧ · · · ∧ ε i j ∧ · · · ∧ ε ip ⊗ (I − M t i j )ε v if 2 ≤ p ≤ k and ∂ 1 (ε i ⊗ ε v ) = (I − M t i )ε v . In the following proposition, we establish an isomorphism between the homology of Evans' complex D Λ * and the homology of the complex A σ * associated to the shift maps σ n on Λ ∞ . We could obtain the existence of isomorphisms H * (D Λ * ) ∼ = H * (A σ * ) using that, by Evans' results, H * (D Λ * ) ∼ = H * (Z k , K 0 (C * (Λ × d Z k ))), that by Matui's results, H * (A σ * ) ∼ = H * (Z k , H 0 (G Λ × c Z k )) , and then by identifying C * (Λ × d Z k ) with C * (G Λ × c Z k ) and applying the HK conjecture for AF groupoids as stated in Corollary 5.2. However, we have chosen to present a more direct proof, which also has the advantage that it shows that the natural inclusion ZΛ 0 ֒→ C c (Λ ∞ , Z) induces the isomorphism. Proposition 7.6. Let Λ be a row-finite k-graph with no sources. Let G Λ = G(Λ ∞ , σ) be the associated groupoid, and let (A σ * ) be the complex of Theorem 6.5. Then the homomorphism ι : ZΛ 0 → C c (Λ ∞ , Z) determined by ι(1 v ) = 1 Z(v) induces an isomorphism H * (A σ * ) → H * (D Λ * ). In particular, H * (G(Λ ∞ , σ)) ∼ = H * (D Λ * ) . Proof. Corollary 5.2 shows that we can express the complex * Z k ⊗ H 0 (c −1 (0)) as the direct limit of the A σ * under the maps induced by the σ * . Lemma 6.4 shows that the inclusion of A σ * in * Z k ⊗ H 0 (c −1 (0)) induces an isomorphism in homology. Similarly, in [16, Theorem 3.14] Evans proves that one can express the complex * Z k ⊗K 0 (C * (Λ× d Z k )) as the direct limit of D Λ * under the maps induced by the M t * , and the inclusion of D Λ * that takes 1 v to the class of p v,0 in the direct limit induces an isomorphism in homology. By [27,Theorem 5.2], there is an isomorphism C * (Λ× d Z k ) ∼ = C * (G(Λ ∞ , σ)× c Z k ) that carries p (v,0) to 1 Z(v)×{0} . So Corollary 5.2 shows that there is an isomorphism K 0 (C * (Λ× d Z k )) ∼ = H 0 (G(Λ ∞ , σ)× c Z k ) that takes [p (v,0) ] to [1 Z(v)×{0} ]. Lemma 6.1 therefore implies that there is an isomorphism K 0 (C * (Λ × d Z k )) ∼ = H 0 (c −1 (0)) induced by the map that carries the class of p v,0 to 1 Z(v) . Given v ∈ Λ 0 and n ∈ N k , we have (7.2) σ n * (ι(1 v )) = σ n * (1 Z(v) ) = λ∈vΛ n σ n * (1 Z(λ) ) = λ∈vΛ n 1 Z(s(λ)) = ι(M t n (1 v )). Hence ι induces a map of complexes ι * : D Λ * → A σ * that intertwines the σ n * with the (M t n ) * . The same computation combined with the universal property of the direct limit shows that we obtain the following commuting diagram. [1 Z(v) ] for every v ∈ Λ 0 . Write (M t n,∞ ) * for the map from the nth copy of D Λ * into * Z k ⊗ H * (c −1 (0)) and write σ n,∞ * for the map from the nth copy of A σ * to * Z k ⊗ H * (c −1 (0)). Then (M t n,∞ ) * (1 v ) = [1 Z(µ) ] for any µ ∈ Λ n v. Since σ n,∞ * D Λ * D Λ * . . . * Z k ⊗ H 0 (c −1 (0)) A σ * A σ * . . . * Z k ⊗ H 0 (c −1 (0)) ι * ι * ι ∞ * M t * σ n * By definition, ι * ([1 v ]) =(ι * (1 v )) = σ n,∞ * (1 Z(v) ) = σ n,∞ * (σ n * (1 Z(µ) )) = σ n,∞ * (1 Z(µ) ) = [1 Z(µ) ], we see by commutativity of the diagram that ι ∞ * (1 Z(µ) ) = [1 Z(µ) ] for all µ. Since the 1 Z(µ) generate C c (Λ ∞ , Z), we deduce that ι ∞ * is the identity map, and therefore induces the identity map in homology. Since the maps H * (D Λ * ) → H * ( * Z k ⊗ H 0 (c −1 (0))) and H * (A σ * ) → H * ( * Z k ⊗ H 0 (c −1 (0))) are isomorphisms, and the diagram above commutes, the functoriality of homology implies that ι * induces an isomorphism H * (D Λ * ) → H * (A σ * ). The final statement follows from Theorem 6.5. Though we already know that graph groupoids belong to M by Remark 6.8, the following result goes a step further, computing the homology of the 1-graph and 2-graph groupoids in terms of the vertex matrices of the 1-graph or 2-graph. Recall that given a k-graph Λ and given i ≤ k we write M i for the Λ 0 × Λ 0 integer matrix given by M i (v, w) = |vΛ e i w|. If Λ is the path category of a directed graph E, then M 1 is just the usual adjacency matrix A E of E. Corollary 7.7 (see [16,Proposition 3.16]). (1) Let E be a row-finite graph with no sources. Then K 0 (C * (E)) ∼ = H 0 (G E ) ∼ = coker(I − A t E ) and K 1 (C * (E)) ∼ = H 1 (G E ) ∼ = ker(I − A t E ). (2) Let Λ be a row-finite 2-graph with no sources. Then Proof. Theorems 6.7 and 6.10 establish the isomorphisms between homology and Ktheory. The descriptions of the homology groups follow from Proposition 7.6 and the definition of the complex D Λ * . Remark 7.8. The strongly connected 1-graph Λ described in Example 7.2 has homology given by Remark 7.9. In [16,Proposition 3.18], Evans shows that if Λ is a 3-graph and {(I −M t i )a : 1 ≤ i ≤ 3, a ∈ ZΛ 0 } generates ZΛ 0 , then the K-theory of C * (Λ) is equal to the homology of D Λ . So the groupoids of such k-graphs belong to M. We also see that if Λ is a 3-graph or a 4-graph for which the page 3 differentials in Kasparov's sequence are zero and H 2 (D Λ ) (and H 3 (D Λ ) in the case of a 4-graph) are free abelian, then K * (C * (Λ)) is determined by H * (D Λ ) and so G Λ belongs to M; but of course the hypothesis on the differential maps in Kasparov's sequence are not checkable in practice. Remark 7.10. Suppose that Λ is a finite 3-graph. Then the homology groups H * (D Λ * ) are finite rank, and H 3 (D Λ * ) is free abelian. Consequently, if it were possible to construct an example of a finite 3-graph for which the page 3 differential d 3 3,0 in Kasparov's spectral sequence was nontrivial, then consideration of the ranks of the groups involved would show that the associated groupoid did not satisfy the HK conjecture, even up to stabilisation by Q. 7.3. One vertex k-graphs. In this section we will show that if Λ is a 1-vertex k-graph in which each |Λ e i | ≥ 2 and in which gcd(|Λ e 1 | − 1, . . . , |Λ e k | − 1) = 1, then K * (C * (Λ)) = H * (G Λ ) = 0. We work with row-finite k-graphs throughout, but we include a comment at the end of the section indicating how to extend our K-theory calculation to non-row-finite k-graphs. A similar result has been proved in [1, Theorem 6.4(a)] under the assumption that the elements of {|Λ e i | : 1 ≤ i ≤ k} are pairwise relatively prime. The key point is the following consequence of Matui's Künneth formula for the groupoid homology of an ample Hausdorff groupoid (see [34,Theorem 2.4]). Theorem 7.11. Let Λ be a row-finite single-vertex k-graph with at least two edges of each colour, and write N i := |Λ e i | − 1 for each i ≤ k. Then Proof. We proceed by induction. This follows from Corollary 7.7(1) if k = 1. Suppose it holds for k = K − 1 and that Λ is a K-graph with one vertex and at least two edges of each colour. Since the complex D Λ * is independent of the factorisation rules in Λ, Proposition 7.6 shows that H * (G Λ ) is independent of the factorisation rules. So we can assume that Λ = B N 1 +1 × · · · × B N k +1 and so γ . Also, gcd(γ, N K ) = gcd(N 1 , . . . , N K ). As ⊗ and Tor are both additive in the first variable and Z l ⊗ Z m = Z gcd(l,m) = Tor(Z l , Z m ), we have H n (G × H) = (Z gcd(γ,N K ) ) ( K−2 n ) ⊕ (Z gcd(γ,N K ) ) ( K−2 n−1 ) = (Z gcd(N 1 ,...,N K ) ) ( K−2 n )+( K−2 n−1 ) = (Z gcd(N 1 ,...,N K ) ) ( K−1 n ) . G Λ = k i=1 G B N i +1 . Write G = K−1 i=1 G B N i We deduce that the groupoids of 1-vertex k-graphs in which there are at least two edges of each colour, and in which gcd(|Λ e 1 | − 1, . . . , |Λ e k | − 1) = 1 belong to M. Corollary 7.12. If Λ is a single-vertex k-graph with at least two edges of each colour, and gcd(|Λ e 1 | − 1, . . . , |Λ e k | − 1) = 1, then K * (C * (Λ)) = H * (G Λ ) = 0. In particular, if C * (Λ) is simple then it is isomorphic to O 2 . Proof. Theorem 7.11 shows that H * (G Λ ) = 0. It follows that the groups E 2 p,q in Matui's spectral sequence are all zero. Since the terms F 2 p,q in Kasparov's spectral sequence [23] (see also [16]) are given by E 2 p,0 if q is even, and 0 if q is odd, we deduce that the F 2 p,q are all zero. So Evans' spectral sequence collapses, and we obtain K * (C * (Λ)) = 0 as well. Remark 7.13. Unfortunately, if gcd(N 1 , . . . , N k ) > 1, we can conclude little new about the HK conjecture. The problem is that in K-theory, with the exception of tensor products, we only obtain from Evans' spectral sequence that the K-groups have filtrations of length at most k − 1 with subquotients equal to direct sums of copies of Z gcd(N 1 ,...,N k ) . Remark 7.14. The above discussion deals only with row-finite k-graphs. We can extend the K-theory calculation to non-row-finite examples as follows. If Λ is any 1-vertex kgraph, and Γ is a 1-vertex (k + 1)-graph such that d −1 Γ (N k ) ∼ = Λ and Γ e k+1 is infinite, then as in [6] we can make the identification C * (Γ) ∼ = T ℓ 2 (C * (Λ)) C * (Λ) , and deduce from Pimsner's [39,Theorem 4.4] that the inclusion C * (Λ) ֒→ C * (Γ) determines a KK-equivalence, so K * (C * (Γ)) ∼ = K * (C * (Λ)). Applying this iteratively, we can compute the K-theory of the C * -algebra of a 1-vertex, not-necessarily-row-finite k-graph as the K-theory of the C * -algebra of the subgraph consisting only of those coordinates in which there are finitely many edges. Lemma 2. 2 . 2Let X be a 0-dimensional topological space, and let K ⊆ X be compact. If W is a compact open subset of K (in the relative topology on K), then there exists a compact open set V ⊆ X such that W = V ∩ K. Proof. Since W is open in the relative topology, W = V ∩ K, with V open in X. Let U be a basis of compact open sets for the topology on X; so U ′ := {U ∈ U : U ⊆ V } satisfies V = U ∈U ′ U. Since U ′ K := {U ∩ K : U ∈ U ′ } is an open cover of the compact set W ⊆ K (in the relative topology), There is a finite subset F ⊆ U ′ such that W = U ∈F (U ∩ W ). Now V := U ∈F U is compact open, and W = V ∩ K. Proposition 2 . 3 . 23Let X be a 0-dimensional topological space. Then the constant sheaf F on X with values in a discrete abelian group A is c-soft. Definition 3.1 ([44, Definition I.1.3], [32, Definition 3.4]). Remark 3 . 3 . 33Let ρ, σ : G → H be continuous groupoid homomorphisms between Hausdorffétale groupoids. Then ρ and σ both induce well-defined orbit maps [u] → [ρ(u)] and [u] → [σ(u)]. Suppose that ρ and σ are similar; then [ρ(u)] = [σ(u)] for all u ∈ G (0) , and so the orbit maps induced by ρ and σ are equal. It follows that every similarity of groupoids induces a bijection between their orbit spaces. Definition 3.4 ([11, Section 4.5]). Let G, H be groupoids. A continuous functor Lemma 3 . 6 . 36Let G, H be Hausdorffétale groupoids. If G and H are similar, then they are Morita equivalent. Definition 3.7 ([37, Definition 2.1],[45, §3]). The groupoids G and H are equivalent if there is a locally compact Hausdorff space Z such that (1) Z is a free and proper left G-space with fibre map r : Z → G (0) , (2) Z is a free and proper right H-space with fibre map s : Z → H (0) , (3) the actions of G and H on Z commute, (4) r : Z → G (0) induces a homeomorphism Z/H → G (0) , and (5) r : Z → H (0) induces a homeomorphism G\Z → H (0) . Definition 3 . 8 . 38The Hausdorffétale groupoids G and H are weakly Kakutani equivalent if there are full open subsets X ⊆ G (0) and Y ⊆ H (0) such that G| X ∼ = H| Y . They are Kakutani equivalent if X and Y can be chosen to be clopen sets. Proposition 3 . 10 . 310Let G and H be Hausdorffétale groupoids. The following are equivalent:(1) G and H are Morita equivalent;(2) there is a Hausdorffétale groupoid L and a decomposition L (0) = X ⊔ Y of L (0) into complementary full clopen subsets such that L| X ∼ = G and L| Y ∼ = H; (3) G and H are equivalent in the sense of Renault; and (4) G and H admit isomorphic ampliations. If G and H are weakly Kakutani equivalent, then they satisfy (1)-(4). (iii) Let G and H be minimal Hausdorffétale groupoids which are equivalent in the sense of Renault. Then with notation as in the Proposition 3.10(2), we may identify G = L| X and H = L| Y where L is a Hausdorffétale groupoid and X and Y are complementary full clopen subsets of L (0) . Let U ⊆ L be an open bisection such that r(U) ⊆ X and s(U) ⊆ Y . Since L is minimal, both r(U) and s(U) are full open subsets. It follows that G| r(U ) ∼ = H| s(U ) and so G and H are weakly Kakutani equivalent. Theorem 3 . 12 . 312Let G and H be ample Hausdorff groupoids with σ-compact unit spaces. Then the following are equivalent: (1) G and H are similar; (2) G and H are Morita equivalent; (3) there exist an ample Hausdorff groupoid L and a decomposition L (0) = X ⊔ Y of L (0) into complementary full clopen subsets such that L| X ∼ = G and L| Y ∼ = H; (4) G and H are equivalent in the sense of Renault; (5) G and H admit isomorphic ampliations; (6) G × R ∼ = H × R; (7) G and H are Kakutani equivalent; and (8) G and H are weakly Kakutani equivalent. Matui shows in [ 32 , 32Proposition 3.5] that if G and H are similar, then H * (G, A) ∼ = H * (H, A) for any discrete abelian group A. So Theorem 3.12 implies that if G and H are ample and have σ-compact unit spaces and are equivalent via any of the eight notions of equivalence listed in the statement of the theorem, then their homologies coincide. We will also an explicit description of the isomorphism. Lemma 4. 3 . 3Let G and H be ample Hausdorff groupoids with σ-compact unit spaces. If the pair G and H satisfies any of the eight equivalent conditions in Theorem 3.12, then H n (G) ∼ = H n (H). In particular, if ρ : G → H is a similarity (see Definition 3.1) then it induces an isomorphism ρ * : H n (G) ∼ = H n (H). If X is a full open subset of G (0) then the inclusion G| X ⊆ G is a similarity and induces an isomorphism H * (G| X ) ∼ = H * (G).Proof. Crainic and Moerdijk show that a Morita equivalence between Hausdorffétale groupoids induces an isomorphism between their homology groups (see[11, Corollary 4.6]). The proof of[32, Proposition 3.5] shows that if ρ : G → H is a similarity then ρ induces an isomorphism H n (G) ∼ = H n (H) for all n ≥ 0. Let X be a full open subset of G (0) then the argument of[32, Theorem 3.6] proves that the inclusion G| X ⊆ G is a similarity. Hence, the inclusion map induces an isomorphism H * (G| X ) ∼ = H * (G). Proposition 4. 4 . 4Let X be a 0-dimensional space. If we regard X as an ample groupoid with X (0) = X and with trivial multiplication thenH n (X) = C c (X, Z) if n = 0, 0 otherwise. ( 3 ) 3The notion of a type semigroup for the transformation group (X, Γ), where X is a Cantor set and Γ is discrete was introduced in[47]. This idea was generalised by Rainone and Sims in[42, Definition 5.4], and independently by Bönicke and Li[3], who introduced the type semigroup S(G) of an ample Hausdorff groupoid G. The map [1 U ] H 0 (G) → [1 U ] G(S(G)) induces an isomorphism of the homology group H 0 (G) onto the Grothendieck group of S(G). This isomorphism carries H 0 (G) + to the image of S(G) in its Grothendieck group. In particular, the coboundary subgroup H G of [42, Definition 6.4] is exactly im ∂ 1 as defined above. Remark 4. 6 . 6Homology for ample Hausdorff groupoids is functorial in the following sense. Let G and H be ample Hausdorff groupoids and let φ : G → H be anétale groupoid homomorphism (so in particular, φ is a local homeomorphism). Then as Crainic and Moerdijk observe (see[11, 3.7.2]), the maps φ (n) * : C c (G (n) , Z) → C c (H (n) ,Z) induce homomorphisms on homology which we denote by φ * : H n (G) → H n (H), and φ → φ * preserves composition. If G is an open subgroupoid of an ample Hausdorff groupoid H, then G is also an ample Hausdorff groupoid and the inclusion map ι : G → H is ań etale groupoid homomorphism. Hence ι induces a map ι * : H * (G) → H * (H) satisfying ι * [1 U ] Hn(G) = [1 U ] Hn(H) . Proposition 4.7 (cf. [38] Lemma 1.5). Let G be an ample Hausdorff groupoid and let {G i } be an increasing sequence of open subgroupoids of G. Then H * (G) ∼ = lim − → H * (G i ). Proof. For each i the inclusion map G i ֒→ G induces a homomorphism ι i : H * (G i ) → H * (G) by Remark 4.6. So the universal property of the direct limit yields a homomorphism ι ∞ : lim − → H * (G i ) → H * (G). This homomorphism is injective because if ι ∞ (a) is a boundary, say ι ∞ (a) = ∂ n (f ), then f ∈ C c (G (n+1) i ) for large enough i, and then a = ∂ n (f ) belongs to B n (G i ). It is surjective because if U is a compact open subset of G (n) , then U ⊆ i G and nested, it follows that U is a compact open subset of G (n) i for large i. Hence every generator of H * (G) belongs to the image of ι ∞ . Lemma 4. 8 . 8Let X, Y be locally compact Hausdorff spaces, and let ψ : Y → X be a local homeomorphism. Suppose that there exists a continuous open section ϕ : X → Y of ψ. Definition 4. 9 9(cf.[32, Definition 2.2]). An ample groupoid G is said to be elementary if it is isomorphic to the groupoid R(ψ) of Example 3.9 for some local homeomorphism ψ : Y → X between 0-dimensional spaces. An ample groupoid G is said to be AF if it can be expressed as a union of open elementary subgroupoids.The only point of difference between Definition 4.9 and Matui's [32, Definition 2.2] is that we allow non-compact unit spaces. Definition 5 . 1 . 51We define M to be the class of ample Hausdorff groupoids for which the isomorphism (5.1) holds.Matui proves in[32, Theorem 4.14] that the groupoids associated to shifts of finite type belong to M and in [32, Theorems 4.10 and 4.11] that AF groupoids with compact unit space belong to M. He shows in [34, Proposition 2.7] that M is closed under Kakutani equivalence, and he shows in[34, Theorem 5.5] that M contains all finite cartesian products of groupoids associated to shifts of finite type. He proves in [32, Section 3.1] that M contains the transformation groupoids of topologically free and minimal actions of Z on the Cantor set. n and H i (F n ) = 0 for all i > 1. Hence F n lies in M. The integer Heisenberg group also belongs to M-see [24, Corollary 1] and [25, Example 8.24]. On the other hand, not every ample Hausdorff groupoid belongs to M: for example, M contains no nontrivial finite cyclic group. [ 5 , 5Example III.1.1],H 0 (Z, H 0 (G(X, σ) × c Z)) ∼ = ker(id −α * ), and H 1 (Z, H 0 (G(X, σ) × c Z)) ∼ = coker(id −α * ).(6.2) Unless d 3 3 3,0 is trivial, there is no reason to expect that coker(d 3 3,0 ) ∼ = H 0 (A σ * ); and even if d 3 3,0 Theorem 7 . 5 . 75Let Λ be a k-graph. The maps Ψ * : C * (Λ) → C c (G ( * ) Λ , Z) defined above comprise a chain map, and induce a homomorphism Ψ * : H * (Λ) → H * (G Λ ). K 0 () 0C * (Λ)) ∼ = H 0 (G Λ ) ∼ = ZΛ 0 / (I − M t 1 , I − M t 2 )(ZΛ 0 ) 2 ⊕ ker(M t 2 − I) ∩ ker(I − M t 1 ), and K 1 (C * (Λ)) ∼ = H 1 (G Λ ) ∼ = ker(I − M t 1 , I − M t ZΛ 0 .In particular, graph groupoids and 2-graph groupoids belong to M. homology of G Λ is H n (G Λ ) = Z/2Z ⊕ Z/2Z if n = 0, 0 otherwise.So the map Ψ 0 : H 0 (Λ) → H 0 (G Λ ) of Theorem 7.5 is neither surjective nor injective. H n (G Λ ) ∼ = (Z gcd(N 1 ,...,N k ) ) ( k−1 n ) if 0 ≤ n ≤ k − 1 0otherwise. +1 and H = G B N K +1 so that G Λ ∼ = G × H. Matui's Künneth theorem gives a split exact sequence 0 → i+j=n H i (G) ⊗ H j (H) −→ H n (G × H) −→ i+j=n−1 Tor(H i (G), H j (H)) → 0,and since H * (H) = (Z N K , 0, 0, . . . ) this collapses to a split exact sequence0 → H n (G) ⊗ Z N K −→ H n (G × H) −→ Tor(H n−1 (G), Z N K ) → 0.Since the sequence splits, the middle term is the direct sum of the two ends.Write γ := gcd(N 1 , . . . , N K−1 ). Then the inductive hypothesis gives H n (G) = Z ( K−2 n ) γ and H n−1 (G) = Z ( K−2 n−1 ) Department of Mathematics (084), University of Nevada, Reno, NV 89557-0084, USA E-mail address: alex@unr.edu (D.P, &A.S.) School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia E-mail address: dpask@uow.edu.au, asims@uow.edu.au 2, 10.8.1] and [5, II.4.2] we have The final statement follows from [48, Proposition 8.8 and Corollary 8.15]. Matui uses the term essentially principal, and the term topologically principal has also been used elsewhere in the literature. sometimes referred to as a transformation groupoid (see[18]) so called because it matches with the categorical cohomology of Λ defined in[30]. 7.1.A map from the categorical homology of a k-graph to the homology of its groupoid. To define the categorical 3 homology groups H * (Λ) for a k-graph Λ, we use the following notation.Given a k-graph Λ, letDefinition 7.1. (cf. [19, Remark 2.14]) Let Λ be a k-graph. For n ≥ 0, let C n (Λ) := ZΛ * n , the free abelian group with generators indexed by Λ * n . Identifying elements of Λ * n with the corresponding generators of C n (Λ), we regard the boundary maps d i asNote that one may define a more general homology theory with coefficients in an abelian group A as in[19]but we do not need this level of generality.Example 7.2. Let Λ be the 1-graph with vertex connectivity matrix ( 5 2 2 3 ). By [19, Theorem 6.2] the categorical homology of Λ coincides with its cubical homology, which can computed as follows. Since Λ is connected we have H 0 (Λ) ∼ = Z; since Λ is finite its first homology group is free abelian with rank equal to its Betti number p = |Λ 1 |−|Λ 0 |+1 = 11. Hence H 1 (Λ) ∼ = Z 11 . Moreover, H n (Λ) = 0 for all n > 1. We will return to this example in Remark 7.8 after establishing how to compute the homology of the associated groupoid. Remark 7.3. One can check that C 0 (Λ) together with the subgroups of the C n (Λ) for n ≥ 1 generated by elements (λ 1 , · · · , λ n ) in which each λ i ∈ Λ 0 form a subcomplex under the same boundary maps ∂ n , and that the homology of this subcomplex is isomorphic to H * (Λ).Recall that if Λ is a k-graph and λ, µ ∈ Λ satisfy s(λ) = s(µ), then the cylinder setDefinition 7.4. Let Λ be a k-graph. Let G Λ be the associated groupoid (see(2.2)) and for each n let G (n) Λ be the collection of composable n-tuples in G Λ as in (4.2). For (λ 1 , . . . , λ n ) ∈ Λ * n ⊆ C n (Λ), defineΛ , Z) be the homomorphism such that Ψ 0 (v) = 1 Z(v) ∈ C c (G (0) Λ , Z) for v ∈ Λ 0 , and Ψ n (λ 1 , . . . , λ n ) = 1 Y (λ 1 ,...,λn) for n ≥ 1 and (λ 1 , . . . , λ n ) ∈ Λ * n . On the K-theory of C * -algebras arising from integral dynamical systems. S Barlak, T Omland, N Stammeier, Ergod. Thy. & Dynam. Sys. 38S. Barlak, T. Omland, and N. 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{'abstract': "We investigate the homology of ample Hausdorff groupoids. We establish that a number of notions of equivalence of groupoids appearing in the literature coincide for ample Hausdorff groupoids, and deduce that they all preserve groupoid homology. We compute the homology of a Deaconu-Renault groupoid associated to k pairwisecommuting local homeomorphisms of a zero-dimensional space, and show that Matui's HK conjecture holds for such a groupoid when k is one or two. We specialise to k-graph groupoids, and show that their homology can be computed in terms of the adjacency matrices, using a chain complex developed by Evans. We show that Matui's HK conjecture holds for the groupoids of single vertex k-graphs which satisfy a mild joint-coprimality condition. We also prove that there is a natural homomorphism from the categorical homology of a k-graph to the homology of its groupoid.", 'arxivid': '1808.07807', 'author': ['Carla Farsi ', 'Alex Kumjian ', 'David Pask ', 'Aidan Sims '], 'authoraffiliation': [], 'corpusid': 119141736, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 40304, 'n_tokens_neox': 35518, 'n_words': 20872, 'pdfsha': '6c04275c92009476fead80eb83dd1495c58873c4', 'pdfurls': ['https://arxiv.org/pdf/1808.07807v1.pdf'], 'title': ["AMPLE GROUPOIDS: EQUIVALENCE, HOMOLOGY, AND MATUI'S HK CONJECTURE", "AMPLE GROUPOIDS: EQUIVALENCE, HOMOLOGY, AND MATUI'S HK CONJECTURE"], 'venue': []}

arxiv

Integral formulas for the Weyl and anti-Wick symbols 13 Jun 2018 L Amour Université de Reims France J Nourrigat Université de Reims France Integral formulas for the Weyl and anti-Wick symbols 13 Jun 2018Pseudo-differential operatorsPDOWeyl quantizationanti-Wick quantizationWeyl symbolanti- Wick symbollarge dimensionBeals characterizationGlauber-Sudarshan functionP functioncontravariant symbolcoherent states MSC 2010: 35S0547G1047G30 The first purpose of this article is to provide conditions for a bounded operator in L 2 (R n ) to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on R 2n . Then, explicit formulas for the Weyl (resp. anti-Wick) symbol are proved. Secondly, other formulas for the Weyl and anti-Wick symbols involving a kind of Campbell Hausdorff formula are obtained. A point here is that these conditions and explicit formulas depend on the dimension n only through a Gaussian measure on R 2n of variance 1/2 in the Weyl case (resp. variance 1 in the anti-Wick case) suggesting that the infinite dimension setting for these issues could be considered. Besides, these conditions are related to iterated commutators recovering in particular the Beals characterization Theorem. Statement of the results. Our purpose in this paper is to answer the following issues when A is a bounded operator in L 2 (R n ): 1. Under what condition, the Weyl symbol of A is a bounded continuous function on R 2n ? 2. Under what condition, A is the anti-Wick quantization of a bounded continuous function on R 2n ? 3. Can one express the Weyl and anti-Wick symbols with a Campbell-Hausdorff type formula ? These questions are also mentioned in [10]. A positive answer to the first two questions (a sufficient condition) is given by Theorem 1.1 for the Weyl symbol and by Theorem 1.3 for the anti-Wick symbol. Concerning Question 3, an answer is given combining Theorem 1.1 or Theorem 1.3 with Theorem 1.4. For instance, one gets (1.12) in the case of the anti-Wick quantization. Let us emphasize that it is also our aim to provide conditions and formulas for the two symbols where the dimension is not explicitly written down using Gaussian measures. Thus, the dimension can tends to infinity and more importantly, one can expect to consider the infinite dimension setting for theses purposes in a further work. In the article [9], Hübner and Spohn have introduced the so-called scattering identification operator (unbounded operator) and considered the wave operator in the context of the infinite dimension, concerning some physical model. With L. Jager, we have proved in [1] that the first operator (scattering identification) can also be defined with the anti-Wick quantization. That is, this operator is defined by an integral formula and it would be interesting to know if the second operator (wave) shares that property. One may expect that the present work, adjusted to the infinite dimension case, could give an answer to that question. We begin by recalling the two quantizations under consideration here. See, e.g., Berezin [3], Combescure and Robert [6], Folland [7], Hörmander [8], Lerner [11], Unterberger [13] [14], . . . for these purposes. One can define the Weyl symbol of a continuous operator A from S(R n ) to S ′ (R n ) as an element of S ′ (R 2n ). More precisely, denoting by K A (x, y) the distributional kernel of A (belonging to S ′ (R 2n )), the Weyl symbol of A is defined by: F (x, ξ) = R n e −iv·ξ K A (x + v/2, x − v/2)dv (1.1) where the above integral is actually a Fourier transform in sense of tempered distributions. The second kind of quantization under consideration in this paper is the anti-Wick one, which involves coherent states. These states are a standard family of functions Ψ X indexed by X = (x, ξ) ∈ R 2n defined by: Ψ x,ξ (u) = π −n/4 e − 1 2 |u−x| 2 +iu·ξ− i 2 x·ξ . One has: < Ψ X , Ψ Z >= e − |X−Z| 2 4 + i 2 σ(X,Z) (1.2) where σ is the symplectic form, σ((x, ξ), (y, η)) = y · ξ − x · η. For any function G, bounded and continuous on R 2n , one defines the anti-Wick operator Op AW (G) associated with the function G, as the unique operator satisfying for all f and g in L 2 (R n ): < Op AW (G)f, g >= (2π) −n R 2n G(Z) < f, Ψ Z >< Ψ Z , g > dZ. (1.3) Before developing Question 3 with Campbell Hausdorff type formulas and iterated commutators, we present answers to the first two questions addressed above, in the spirit of the characterization of A. Unterberger [13]. Theorem 1.1. Let A be a bounded operator in L 2 (R n ) and assume that: sup X∈R 2n π −n R 2n < AΨ X+Z , Ψ X−Z > < Ψ X+Z , Ψ X−Z > e −|Z| 2 dZ < ∞. (1.4) Then, the Weyl symbol F of A is a bounded continuous function on R 2n given by the formula: F (X) = π −n R 2n < AΨ X+Z , Ψ X−Z > < Ψ X+Z , Ψ X−Z > e −|Z| 2 dZ. (1.5) We underline that hypothesis (1.4) and equality (1.5) are uniform with respect to the dimension since these expressions only use the variance 1/2 Gaussian measure which also exists in infinite dimension. For a reader interested only in the finite dimensional case, one can check that, using (1.2), the hypothesis (1.4) can also be written as: sup X∈R 2n R 2n | < AΨ X+Z , Ψ X−Z > |dZ < ∞ and (1.5) can also be expressed as: F (X) = π −n R 2n < AΨ X+Z , Ψ X−Z > e iσ(X,Z) dZ. Next we turn to conditions ensuring the existence of a bounded continuous symbol with the anti-Wick quantization for a given bounded operator. Firstly, we give a necessary condition. Theorem 1.2. If A = Op AW (G) with G bounded and continuous on R 2n then: | < AΨ X , Ψ Y > | ≤ G ∞ e − 1 8 |X−Y | 2 . (1.6) In particular, the following estimate holds, | < AΨ X+Y , Ψ X−Y > | ≤ G ∞ e − 1 2 |Y | 2 for all X and Y in R 2n , when A = Op AW (G) where G is bounded and continuous on R 2n . Secondly, we provide a sufficient condition, together with an integral expression for a possible anti-Wick symbol. Theorem 1.3. Suppose that A is a bounded operator in L 2 (R n ) satisfying: sup X∈R 2n (2π) −n R 2n < AΨ X+Z , Ψ X−Z > < Ψ X+Z , Ψ X−Z > e − |Z| 2 2 dZ < ∞. (1.7) Then, there exists a bounded continuous function G on R 2n such that A = Op AW (G). This function is given by: G(X) = (2π) −n R 2n < AΨ X+Z , Ψ X−Z > < Ψ X+Z , Ψ X−Z > e − 1 2 |Z| 2 dZ. (1.8) Note that condition (1.7) can also be written as: sup X∈R 2n (2π) −n R 2n | < AΨ X+Z , Ψ X−Z > | e 1 2 |Z| 2 dZ < ∞ and equality (1.8) is also: G(X) = (2π) −n R 2n < AΨ X+Z , Ψ X−Z > e iσ(X,Z) e 1 2 |Z| 2 dZ. We now give answers to Question 3 in the spirit of Beals characterization [2]. We first note that the function appearing in (1.5) and (1.8) can also be written with an expression comparable to the Campbell-Hausdorff formula. To this end, we denote by Φ S (Z) the differential operator: Φ S (Z) = n j=1 z j u j + ζ j 1 i ∂ ∂u j (1.9) for all Z = (z, ζ) ∈ R 2n . Theorem 1.4. For all X and Z in R 2n , we have the following equality: < AΨ X+Z , Ψ X−Z > < Ψ X+Z , Ψ X−Z > = e −ΦS (Z) Ae ΦS (Z) Ψ X , Ψ X . (1.10) One notices that the operator e ΦS (Z) is unbounded in L 2 (R n ). Nevertheless, its action on coherent states is well defined. It is given by: e ΦS (Z) Ψ X = e 1 2 |Z| 2 +Z·X− i 2 σ(Z,X) Ψ X+Z . (1.11) One observes that (1.10) only uses iterated commutations according to the Campbell Hausdorff formula. One has in the sense of formal series: e −ΦS (Z) Ae ΦS (Z) = ∞ m=0 (−1) m m! ad Φ S (Z) m A. Inserting this equality in (1.5) and in (1.8), one then obtains an expression for the Weyl and anti-Wick symbol of A with an integral using only iterated commutations. To be specific, for the possible anti-Wick symbol G of an operator A, if the integral is absolutely converging, one has: G(X) = (2π) −n R 2n e −ΦS (Z) Ae ΦS (Z) Ψ X , Ψ X e − 1 2 |Z| 2 dZ. (1.12) One can also give a result (see below) specific to the finite dimension, close to the Beals characterization theorem [2]. Denote by V 1 , . . . , V n the canonical basis of R n and set a(V j ) = x j + ∂ xj . For all multi-indexes α and any bounded operator A from S(R n ) to S ′ (R n ), set (ad a(V )) α A = ad a(V 1 ) α1 . . . ad a(V n ) αn A. Proposition 1.5. For any operator A from S(R n ) to S ′ (R n ) with iterated commutators (ad a(V )) α A bounded in L 2 (R n ) when |α| ≤ 2n + 1, one has: R 2n | < AΨ X+Z , Ψ X−Z > |dZ ≤ C n |α|≤2n+1 (ad a(V )) α A (1.13) where the constant C n depends on the dimension n but not on the operator A. We then recover in particular a well known result of R. Beals [2] for the Weyl symbol F of an operator A: sup X∈R 2n |F (X)| ≤ C n |α|≤2n+1 (ad a(V )) α A . We also mention that there is a result of C. Rondeaux [12] where above, the supremum bound is replaced by the L 1 norm and the operator norm is replaced by the trace class norm. 2 Inversion of the heat operator. For each λ > 0, the heat operator H λ is defined for all bounded continuous function F on R 2n by: (H λ F )(z, ζ) = (2πλ) −n R 2n F (x, ξ)e − 1 2λ (x−z) 2 +(ξ−ζ) 2 dxdξ (z, ζ) ∈ C 2n where z 2 = z 2 1 + · · · + z 2 n for all z = (z 1 , · · · z n ) ∈ C n . This function is holomorphic on C 2n . The purpose of Theorem 2.1 is to prove that the image of the operator H λ contains some specific space playing a role in the following sections. We also use another function S λ F on C 2n defined by: (S λ F )(z, ζ) = (H λ F )(iζ, −iz). (2.1) That is: (S λ F )(z, ζ) = (2πλ) −n R 2n F (x, ξ)e − 1 2λ (x−iζ) 2 +(ξ+iz) 2 dxdξ (z, ζ) ∈ R 2n . (2.2) Theorem 2.1. Let Φ be a holomorphic function on C 2n . Set λ > 0 and suppose that: sup (x,ξ)∈R 2n (2πλ) −n R 2n |Φ(x + iξ + z + iζ, x − iξ − z + iζ)|e − |z| 2 +|ζ| 2 2λ dzdζ < ∞. (2.3) Define the function F on R 2n by: F (x, ξ) = (2πλ) −n R 2n Φ(x + iξ + z + iζ, x − iξ − z + iζ)e − |z| 2 +|ζ| 2 2λ dzdζ. (2.4) Then, F is a bounded continuous function on R 2n satisfying: (H λ F )(z, ζ) = Φ(z + iζ, z − iζ) (z, ζ) ∈ C 2n (2.5) (S λ F )(z, ζ) = Φ(z + iζ, −z + iζ) (z, ζ) ∈ C 2n . (2.6) Proof. The fact that F is well defined, bounded and continuous, is a direct consequence of hypothesis (2.3). For all (x, ξ) ∈ R 2n , let Ψ (x,ξ) be the holomorphic function on C 2n defined by: Ψ (x,ξ) (Z 1 , Z 2 ) = Φ(Z 1 , Z 2 )e 1 2λ Z1Z2+x 2 +ξ 2 −x(Z1+Z2)+iξ(Z1−Z2) . Set ϕ the mapping from R 2n to C 2n defined by ϕ(s, t) = (s + it, −s + it). Equality (2.4) can then be written as: F (x, ξ) = (2πλ) −n R 2n Ψ (x,ξ) (x + iξ, x − iξ) + ϕ(s, t) dsdt. We then apply Lemma 2.2 below with n replaced by 2n, E = {(x + iξ, x − iξ), (x, ξ) ∈ R 2n }, together with the above functions ϕ and Ψ = Ψ (x,ξ) . According to hypothesis (2.3), for all compact sets K of R 2n , with (x, ξ) being fixed: sup (y,η)∈K R 2n |Ψ (x,ξ) (y + iη, y − iη) + ϕ(s, t) |dsdt < ∞. Consequently, hypothesis (2.7) of Lemma 2.2 is satisfied. According to that Lemma, one has: F (x, ξ) = (2πλ) −n R 2n Ψ (x,ξ) ϕ(s, t) dsdt. This equality can be written as: F (x, ξ) = (2πλ) −n R 2n Φ(z + iζ, −z + iζ)e − 1 2λ (z−iξ) 2 +(ζ+ix) 2 dzdζ. Set: G(z, ζ) = Φ(z + iζ, −z + iζ)e − |z| 2 +|ζ| 2 2λ H(x, ξ) = F (ξ, −x) e − |x| 2 +|ξ| 2 2λ . Hypothesis (2.3) with (x, ξ) = (0, 0) shows that the function G belongs to L 1 (R 2n ). The above equality reads as: H(x, ξ) = (2πλ) −n R 2n G(z, ζ)e − i λ (x·z+ξ·ζ) dzdζ. Thus, H is the Fourier transform depending on the parameter λ of G. Since F is bounded and continuous then H ∈ L 1 (R 2n ). Hence, G(x, ξ) = (2πλ) −n R 2n H(z, ζ)e i λ (x·z+ξ·ζ) dzdζ. This equality is equivalent to (2.6) for real (z, ζ). Equality (2.6) then follows for (z, ζ) ∈ C 2n using the holomorphic properties of the two hand sides. Equality (2.5) then holds according to (2.1). The Lemma below would be in the particular case of the dimension one, the result for changing integration contours with holomorphic functions when these integration contours are parallel lines. Lemma 2.2. Set E a n−dimensional real subspace of C n . Let ϕ be a linear map from R n to C n satisfying detϕ ′ = 0. Assume that E ⊕ Imϕ = C n . Set Ψ a holomorphic function on C n . Suppose that, for any compact set K of E: sup a∈K R n |Ψ(a + ϕ(t))|dt < ∞. (2.7) Set: I(a) = R n Ψ(a + ϕ(t))dt for any a ∈ E where dt is the Lebesgue measure on R n . Then I is independent of a ∈ E. Proof of the Lemma. For all R > 0, set: K R = {t ∈ R n , |t j | ≤ R, 1 ≤ j ≤ n}. Set ∂K R the boundary of K R with the canonical measure dµ R . For all k ≤ n, we have: KR ∂ ∂t k Ψ(a + ϕ(t))dt ≤ ∂KR |Ψ(a + ϕ(t))|dµ R (t). Since Ψ is holomorphic: n j=1 ∂ϕ j ∂t k KR (∂ j Ψ)(a + ϕ(t))dt ≤ ∂KR |Ψ(a + ϕ(t))|dµ R (t). As the determinant of the matrix ∂ϕj ∂t k is non vanishing, one gets for all j ≤ n: KR (∂ j Ψ)(a + ϕ(t))dt ≤ C ∂KR |Ψ(a + ϕ(t))|dµ R (t) with C > 0. Using that Ψ is holomorphic, there exists another constant C > 0 such that, for all Z ∈ C n : |Ψ(Z)| ≤ C B(Z,1) |Ψ(W )|dW. (2.8) Consequently, with another constant C and another ρ > 0: ∂KR |Ψ(a + ϕ(t))|dµ R (t) ≤ C BE (a,ρ)×F (∂ΓR,ρ) |Ψ(a ′ + ϕ(t))| dµ E (a ′ )dt where B E (a, ρ) is the ball of E centered at a with radius ρ, µ E is the canonical measure on E and F (∂Γ R , ρ) is the set of all points in R n whose distance to ∂Γ R is smaller than ρ. One then deduces that, for all a and b in E, for any R > 0: KR Ψ(a + ϕ(t)) − Ψ(b + ϕ(t)) dt ≤ C FE ([a,b],ρ)×F (∂ΓR,ρ) |Ψ(a ′ + ϕ(t))| dµ E (a ′ )dt where F E ([a, b], ρ) is the set of all points in E whose distance to the interval [a, b] is smaller than ρ. The right hand side tends to zero (the constants C and ρ are independent of R) as R goes to infinity. One then concludes that I(a) = I(b) which proves the Lemma. Let us also mention another property of the operator S λ . Denote by M λ the following multiplication operator: (M λ F )(x, ξ) = e − 1 2λ (|x| 2 +|ξ| 2 ) F (x, ξ). One then sees that M λ S λ F = T λ M λ F where: (T λ F )(z, ζ) = (2πλ) −n R 2n F (x, ξ) e i λ (z·ξ−x·ζ) dxdξ. One notes that T λ is an isometry from L 2 (R 2n ) into itself, whose square is the identity operator. It is the symplectic Fourier transform with parameter (see [6]). Hence, S λ is an isometry from E = {F, M λ F ∈ L 2 (R 2n )} into itself, whose square is the identity. These properties are not further used in this work. Weyl symbol. It is knwon ( [7], page 139) that, for any bounded operator A from S(R n ) to S ′ (R n ), there exists an homomorphic function B(A) on C 2n such that, for all X = (x, ξ) and Y = (y, η) identified with elements of C n : < AΨ X , Ψ Y > < Ψ X , Ψ Y > = B(A)(x + iξ, y − iη). (3.1) Set A a bounded operator in L 2 (R n ) satisfying (1.4). Assumption (1.4) then reads as: sup X∈R 2n π −n R 2n B(A)(X + Z, X − Z) e −|Z| 2 dZ < ∞. That is, the function Φ = B(A) satisfies the assumption of Theorem 2.1 with λ = 1/2. According to that Theorem, the function F defined by (2.4), that is to say by (1.5) when λ = 1/2 and Φ = B(A), is continuous and bounded on R 2n and verifies (2.6) with λ = 1/2. Namely: (S 1/2 F )(Z) = B(A)(Z, −Z). (3.2) Inverting the transform in (1.1), one associates with F some tempered distribution on R 2n which is the kernel K A ′ of some operator A ′ , bounded from S(R n ) to S ′ (R n ), whose Weyl symbol is F . In view of the Lemma below, one has for all Z ∈ C n : B(A ′ )(Z, −Z) = (S 1/2 F )(Z). (3.3) We recall Proposition 1.69 in [7]. If G is a holomorphic function on C n × C n and if G(Z, −Z) = 0 for all Z ∈ C n then G is entirely vanishing. Indeed, all the ∂ α G are vanishing at the origin by iteration. Thus, from (3.2) and (3.3), Proof of the Lemma. In view of (1.1), the operator A ′ , bounded from S(R n ) to S ′ (R n )) with a Weyl symbol equal to F , satisfies for all X and Y in R 2n : B(A − A ′ ) is identically zero. Then, < (A − A ′ )Ψ X , Ψ Y >= 0 for all X and Y in R 2n proving that A − A ′ = 0. Therefore F is the Weyl symbol of A.< A ′ Ψ X , Ψ Y >= π −n R 2n F (Z) < Σ Z Ψ X , Ψ Y > dZ (3.4) where Σ Z for each Z = (z, ζ) ∈ R 2n is the operator acting in L 2 (R n ) defined by: Σ Z f (u) = e 2i(x−z)·ζ f (2z − x) f ∈ L 2 (R n ). In particular: Σ Z Ψ X = e iσ(X,Z) Ψ 2Z−X and therefore, using (1.2): < Σ Z Ψ X , Ψ −X >= e 2iσ(X,Z)−|Z| 2 .B(A ′ )(Z, −X) = < A ′ Ψ X , Ψ −X > < Ψ X , Ψ −X > = e |X| 2 < A ′ Ψ X , Ψ −X > = π −n e |X| 2 R 2n F (Z) < Σ Z Ψ X , Ψ −X > dZ = π −n R 2n F (Z)e |X| 2 +2iσ(X,Z)−|Z| 2 dZ = S 1/2 F (X) which proves the Lemma. 4 Anti-Wick symbol. < AΨ X , Ψ Y >= (2π) −n R 2n G(Z) < Ψ X , Ψ Z >< Ψ Z , Ψ Y > dZ. Using (1.2): < AΨ X , Ψ Y >= (2π) −n R 2n G(Z)e − 1 4 |X−Z| 2 +|Y −Z| 2 + i 2 σ(X−Y,Z) dZ. (4.1) From the parallelogram identity: | < AΨ X , Ψ Y > | ≤ (2π) −n R 2n |G(Z)|e − 1 2 |Z− X+Y 2 | 2 − 1 8 |X−Y | 2 dZ. One then obtains (1.6). Proof of Theorem 1.3. Set a bounded operator A in L 2 (R n ) satisfying (1.7). That is, the function B(A) defined by (3.1) satisfies the assumption of Theorem 2.1 with λ = 1. According to that Theorem, the function G defined by (2.4) with λ = 1 and Φ = B(A), or equivalently by (1.8), is bounded and continuous on R 2n and satisfies (2.6) with λ = 1. In other words: (S 1 G)(Z) = B(A)(Z, −Z). One can define an operator A ′ = Op AW (G) bounded in L 2 (R n ) with the function G since G is bounded and continuous. In view of the Lemma below, one has B(A − A ′ )(Z, −Z) = 0, for all Z ∈ C n . One finishes the proof similarly to the one of Theorem 1.1. Lemma 4.1. Let G be a bounded continuous function on R 2n and set A ′ = Op AW (G) (bounded operator in L 2 (R n )). Then: B(A ′ )(Z, −Z) = (S 1 G)(Z). (4.2) Proof of the Lemma. From (4.1) with A replaced by A ′ , one has, for any X ∈ C n : < A ′ Ψ X , Ψ −X >= (2π) −n R 2n G(Z)e − 1 4 |X−Z| 2 +|X+Z| 2 +iσ(X,Z) dZ. From (3.1), (1.2) and (2.2) with λ = 1, one then obtains (4.2). Iterated commutators. Proof of (1.11). We first note that Ψ 0 (u) = π −n/4 e − 1 2 |u| 2 , and that, as Φ S (X) is the operator defined in (1.9), we have: e ΦS (X) Ψ 0 = e 1 2 |X| 2 Ψ X and: e ΦS (Z) e ΦS(X) = e − i 2 σ(Z,X) e ΦS (Z+X) . Thus, one gets (1.11). Proof of Theorem 1.4. From (1.11): e −ΦS(Z) Ae ΦS (Z) Ψ X , Ψ X = e |Z| 2 −iσ(Z,X) < AΨ X+Z , Ψ X−Z > . With equality (1.2): < Ψ X+Z , Ψ X−Z >= e −|Z| 2 +iσ(Z,X) . These two equalities prove Theorem 1.4. Proof of Proposition 1.5. One has Ψ X = W X Ψ 0 with (W X f )(u) = f (u − x)e iu·ξ− i 2 x·ξ .(5.1) Thus: < AΨ X+Z , Ψ X−Z >=< W Z−X AW Z+X Ψ 0 , Ψ 0 > . (5.2) Denote by V 1 , . . . , V n the canonical basis of R n . Set a(V j ) = u j + ∂ uj and a ⋆ (V j ) = u j − ∂ uj . One checks that: [a(V j ), W x,ξ ] = (x j + iξ j )W x,ξ . For all X = (x, ξ), set ϕ j (X) = x j + iξ j . Let A be a bounded operator in L 2 (R n ) such that, the right hand side of (1.13) is well defined. Since a(V j )Ψ 0 = 0: < W Z−X AW Z+X Ψ 0 , a ⋆ (V j )Ψ 0 > =< [a(V j ), W Z−X ]AW Z+X Ψ 0 , Ψ 0 > + < W Z−X [a(V j ), A]W Z+X Ψ 0 , Ψ 0 > + < W Z−X A[a(V j ), W Z+X ]Ψ 0 , Ψ 0 > = 2ϕ j (Z) < W Z−X AW Z+X Ψ 0 , Ψ 0 > + < W Z−X [a(V j ), A]W Z+X Ψ 0 , Ψ 0 > . By iteration and using (5.2), one gets, for all multi-indices α: ϕ(Z) α < AΨ X+Z , Ψ X−Z >= β+γ=α c α,β,γ < W Z−X (ad a(V )) β AW Z+X Ψ 0 , (a ⋆ (V )) γ Ψ 0 > . One set above ϕ(Z) α = ϕ j (Z) αj and similarly for (a ⋆ (V )) γ . Thus, there exists C n such that: | < AΨ X+Z , Ψ X−Z > | |α|≤2n+1 |ϕ(Z) α | ≤ C n |β|≤2n+1 (ad a(V )) β A . One then gets (1.13) since the inverse of the left hand side belongs to L 1 (R 2n ). on a inversion result for the heat operator (Theorem 2.1) that is proved in Section 2. On the basis of this result (Theorem 2.1), Theorems 1.1 and 1.3 are proved in Sections 3 and 4. Equality (1.11), Theorem 1.4 and Proposition 1.5 are proved in Section 5. Lemma 3. 1 . 1Set F a bounded continuous function on R 2n . Let A ′ be the bounded operator from S(R n ) to S ′ (R n ) with F as Weyl symbol. Then, (3.3) holds where S 1/2 is defined in (2.2). Proof of Theorem 1.2. If A = Op AW (G) then, according to (1.3): L Amour, L Jager, J Nourrigat, arXiv:1805.00758Composition of states and observables in Fock spaces. L. Amour, L. Jager, J. Nourrigat, Composition of states and observables in Fock spaces, arXiv:1805.00758. Characterization of pseudodifferential operators and applications. R Beals, Duke Math. J. 441R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), no. 1, 45-57. Wick and anti-Wick symbols of operators. F A Berezin, Russian) Mat. Sb. (N.S.86F. A. Berezin, Wick and anti-Wick symbols of operators, (Russian) Mat. Sb. (N.S.) 86(128) (1971), 578-610. . J M Bony, Caractérisation des opd. Séminaire EDP, X. Exposé n23. 17J.M. Bony, Caractérisation des opd. Séminaire EDP, X. Exposé n23, 17pp, (1996-1997). Characterization of pseudo-differential operators. J M Bony, Progress in non linear differential equations and their applications. 84J.M. Bony, Characterization of pseudo-differential operators, Progress in non linear differential equations and their applications. Vol. 84. Birkhaüser, 21-34. (2013). M Combescure, D Robert, 978-94-007-0195-3Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics. DordrechtSpringerM. Combescure, D. Robert, Coherent states and applications in mathematical physics, Theoretical and Mathematical Physics. Springer, Dordrecht, 2012. ISBN: 978-94-007-0195-3 Harmonic analysis in phase space. G B Folland, Annals of Mathematics Studies. 122Princeton University PressG. B. Folland, Harmonic analysis in phase space. Annals of Mathematics Studies, 122. Princeton University Press, Princeton, NJ, 1989. The analysis of linear partial differential operators. L Hörmander, SpringerIIIL. Hörmander, The analysis of linear partial differential operators, Volume III, Springer, 1985. Radiative decay: nonperturbative approaches. M Hübner, H Spohn, Rev. Math. Phys. 73M. Hübner, H. Spohn, Radiative decay: nonperturbative approaches, Rev. Math. Phys. 7 (1995), no. 3, 363-387. Kernels and symbols of operators in quantum field theory. P Krée, R Raczka, Ann. Inst. H. Poincaré Sect. A (N.S.). 281P. Krée, R. Raczka, Kernels and symbols of operators in quantum field theory, Ann. Inst. H. Poincaré Sect. A (N.S.) 28 (1978), no. 1, 41-73. Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. N Lerner, Theory and Applications. 3Birkhäuser VerlagN. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, 3. Birkhäuser Verlag, Basel, 2010. Classes de Schatten d'opérateurs pseudo-différentiels. (French). C Rondeaux, Ann. Sci. Ecole Norm. Sup. 4C. Rondeaux, Classes de Schatten d'opérateurs pseudo-différentiels. (French) Ann. Sci. Ecole Norm. Sup. (4) 17 (1984), no. 1, 67-81. Les opérateurs métadifférentiels, in Complex analysis, microlocal calculus and relativistic quantum theory. A Unterberger, Lecture Notes in Physics. 126A. Unterberger, Les opérateurs métadifférentiels, in Complex analysis, microlocal calculus and relativistic quantum theory, Lecture Notes in Physics 126 (1980) 205-241. 51687 REIMS Cedex 2, France. jean.nourrigat@univ-reims.fr LMR CNRS FRE. A Unterberger, amour@univ-reims.fr LMR CNRS FRE 2011Oscillateur harmonique et opérateurs pseudo-différentiels. Université de Reims Champagne-Ardenne, Moulin de la Housse; 51687 REIMS Cedex 2, France29Université de Reims Champagne-Ardenne, Moulin de la Housse-laurentBPA. Unterberger, Oscillateur harmonique et opérateurs pseudo-différentiels, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, xi, 201-221. - laurent.amour@univ-reims.fr LMR CNRS FRE 2011, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 REIMS Cedex 2, France. jean.nourrigat@univ-reims.fr LMR CNRS FRE 2011, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 REIMS Cedex 2, France.

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{'abstract': 'The first purpose of this article is to provide conditions for a bounded operator in L 2 (R n ) to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on R 2n . Then, explicit formulas for the Weyl (resp. anti-Wick) symbol are proved. Secondly, other formulas for the Weyl and anti-Wick symbols involving a kind of Campbell Hausdorff formula are obtained. A point here is that these conditions and explicit formulas depend on the dimension n only through a Gaussian measure on R 2n of variance 1/2 in the Weyl case (resp. variance 1 in the anti-Wick case) suggesting that the infinite dimension setting for these issues could be considered. Besides, these conditions are related to iterated commutators recovering in particular the Beals characterization Theorem.', 'arxivid': '1806.04898', 'author': ['L Amour \nUniversité de Reims\nFrance\n', 'J Nourrigat \nUniversité de Reims\nFrance\n'], 'authoraffiliation': ['Université de Reims\nFrance', 'Université de Reims\nFrance'], 'corpusid': 119314046, 'doi': '10.1016/j.matpur.2019.01.007', 'github_urls': [], 'n_tokens_mistral': 9742, 'n_tokens_neox': 8893, 'n_words': 4582, 'pdfsha': '7e48984336f50f0ce40ad6efa9eced7954456f21', 'pdfurls': ['https://arxiv.org/pdf/1806.04898v1.pdf'], 'title': ['Integral formulas for the Weyl and anti-Wick symbols', 'Integral formulas for the Weyl and anti-Wick symbols'], 'venue': []}

arxiv

Oxygen-enhanced extremely metal-poor DLAs: A signpost of the first stars? January 24, 2022 Louise Welsh Centre for Extragalactic Astronomy Durham University South RoadDH1 3LEDurhamUK Dipartimento di Fisica G. Occhialini Università degli Studi di Milano Bicocca Piazza della Scienza 3I-20126MilanoItaly INAF -Osservatorio Astronomico di Brera via Bianchi 46I-23087Merate (LC)Italy Ryan Cooke Centre for Extragalactic Astronomy Durham University South RoadDH1 3LEDurhamUK Michele Fumagalli Dipartimento di Fisica G. Occhialini Università degli Studi di Milano Bicocca Piazza della Scienza 3I-20126MilanoItaly INAF -Osservatorio Astronomico di Trieste via G. B. Tiepolo 11I-34143TriesteItaly Max Pettini Institute of Astronomy University of Cambridge Madingley RoadCB3 0HACambridgeUK Oxygen-enhanced extremely metal-poor DLAs: A signpost of the first stars? January 24, 2022Draft version Typeset using L A T E X twocolumn style in AASTeX631Damped Lyman-alpha systems (349)Intergalactic medium (813)Population III stars (1285)Population II stars (1284)Chemical abundances (224) We present precise abundance determinations of two near-pristine damped Lyα systems (DLAs) to assess the nature of the [O/Fe] ratio at [Fe/H] < −3.0 (i.e. < 1/1000 of the solar metallicity). Prior observations indicate that the [O/Fe] ratio is consistent with a constant value, [O/Fe] +0.4, when −3 < [Fe/H] < −2, but this ratio may increase when [Fe/H] −3. In this paper, we test this picture by reporting new, high-precision [O/Fe] abundances in two of the most metal-poor DLAs currently known. We derive values of [O/Fe] = +0.50 ± 0.10 and [O/Fe] = +0.62 ± 0.05 for these two z 3 near-pristine gas clouds. These results strengthen the idea that the [O/Fe] abundances of the most metal-poor DLAs are elevated compared to DLAs with [Fe/H] −3.We compare the observed abundance pattern of the latter system to the nucleosynthetic yields of Population III supernovae (SNe), and find that the enrichment can be described by a (19−25) M Population III SN that underwent a (0.9−2.4)×10 51 erg explosion. These high-precision measurements showcase the behaviour of [O/Fe] in the most metal-poor environments. Future high-precision measurements in new systems will contribute to a firm detection of the relationship between [O/Fe] and [Fe/H]. These data will reveal whether we are witnessing a chemical signature of enrichment from Population III stars and allow us to rule out contamination from Population II stars. INTRODUCTION The first stars in the Universe are responsible for producing the first chemical elements heavier than lithium. These elements -known as metals -irrevocably changed the process of all subsequent star formation and mark the onset of complex chemical evolution within our Universe. Since no metal-free stars have been detected thus far, we know very little about their properties (e.g. their mass distribution) and the relative quantities of the metals that they produced. When the first stars ended their lives, some as supernovae (SNe), they released the first metals into their surrounding environment. The stars that formed in the wake of these (Population III) SNe were born with the chemical fingerprint of the first stars. By studying the chemistry of these relic objects, we can investigate the metals produced by the first stars and, ultimately, trace the evolution of metals across cosmic time. Historically, the fingerprints of the first stars have been studied in the atmospheres of low mass, Population II stars that are still alive today (e.g. Cayrel et al. 2004;Frebel et al. 2005;Aoki et al. 2006;Frebel et al. 2015;Ishigaki et al. 2018;Ezzeddine et al. 2019); the composition of the stellar atmosphere is studied in absorption against the light of the star. This process of stellar archaeology, along with simulations of stellar evolution, allows us to infer the elements produced by the first SNe and, subsequently, infer their properties (such as mass, rotation rate, explosion energy, and even the geometry of stellar outflows; Woosley & Weaver 1995;Chieffi & Limongi 2004;Meynet et al. 2006;Tominaga et al. 2007;Ekström et al. 2008;Heger & Woosley 2010;Limongi & Chieffi 2012). Extragalactic gas, often seen as absorption along the line-of-sight towards unrelated background quasars, offers a complementary opportunity to study chemical evolution and the first stars (Pettini et al. 2008;Penprase et al. 2010;Becker et al. 2011). The extragalactic gas clouds that have been studied in absorption to date cover a broad range of metallicity, which appears to increase over time (Rafelski et al. 2012(Rafelski et al. , 2014Jorgenson et al. 2013;Lehner et al. 2016;Quiret et al. 2016;Lehner et al. 2019). Those whose relative iron abundance is 1/1000 of the solar value (i.e. [Fe/H] < −3.0) 1 are classified as extremely metal-poor (EMP). These environments have necessarily experienced minimal processing through stars and are therefore an ideal environment to search for the chemical signature of the first stars. Among the least polluted environments currently known, there are three absorption line systems at z ∼ 3 − 4 that appear to be entirely untouched by the process of star formation, with metallicity limits of [M/H] −4.0 (Fumagalli et al. 2011;Robert et al. 2019); all three are Lyman limit systems (LLSs) whose neutral hydrogen column density is 16.2 < log 10 N (H i)/cm −2 < 19.0. These pristine LLSs are a rarity. It is more common to detect absorption line systems that are, at least, minimally enriched with metals. Crighton et al. (2016) reported the detection of a LLS at z ≈ 3.5 with a metal abundance Z/Z = 10 −3.4±0.26 whose [C/Si] abundance is consistent with enrichment by either a Population III or Population II star. In order to distinguish between these scenarios, additional metal abundance determinations are required. Distinguishing between the gaseous systems enriched by Population III stars and later stellar populations would allow us to trace the metals produced by the first stars and determine the typical Population III properties. Furthermore, such an investigation will reveal the timescale over which these gas clouds are enriched by subsequent stellar populations. A prime environment to disentangle these chemical signatures are damped Lyα systems (DLAs; log 10 N (H i)/cm −2 > 20.3; see Wolfe et al. 2005 for a review). Indeed, the most metal-poor DLAs may have been exclusively enriched by the first generation of metal-free stars (Erni et al. 2006;Pettini et al. 2008;Penprase et al. 2010;Cooke et al. 2017;Welsh et al. 2019). These high H i column density gas clouds are self-shielded from external radiation. Thus, the constituent metals reside in a single, dominant, ionization state. This negates the need for ionization corrections and leads to reliable gas-phase abundance determinations. These systems are most easily studied in the redshift interval 2 < z < 3 when the strongest UV metal absorption features are redshifted into the optical wavelength range. Only the most abundant elements are typically observed in EMP DLAs, including the α-capture elements (C, O, Si, S), some odd atomic number elements (N, Al), and some iron-peak elements (usually, only Fe). Given that these elements trace various nucleosynthetic pathways, these abundant elements are sufficient to understand the properties of the stars that are responsible for the enrichment of EMP DLAs, and tease out the potential fingerprints of the first stars. Based on the chemical abundances of EMP stars, we have uncovered some signatures of the first stars, including the enhancement of the lighter atomic number elements relative to the heavier atomic number elements. For example, the observed enhancement of carbon relative to iron in EMP stars with a normal abundance of neutron capture elements (i.e. a 'CEMP-no' star) may indicate that these stars contain the metals produced by Population III stars (see Beers H] > −3.0 suggests that the relatively higher metallicity DLAs were all enriched by a similar population of stars, drawn from the same initial mass function (IMF). Since oxygen is predominantly sourced from the supernovae of massive stars, the apparent 'inflection' observed in the EMP regime can be explained by three equally exciting possibilities. Relative to the stars that enriched the DLAs with [Fe/H] > −3.0, the stars that enriched the most metal-poor DLAs were either: (1) Drawn from an IMF that was more bottom-light; (2) ejected less Fe-peak elements; or (3) released less energy during the explosion that ended their life. All three of these alternatives are signatures of enrichment by a generation of metal-free stars (e.g. Heger & Woosley 2010). However, the errors associated with the currently available data are too large to confirm this trend. In this paper, we present the detailed chemical abundances of two chemically near-pristine DLAs to study the behaviour of [O/Fe] at the lowest metal- licities. These DLAs are found along the line-of-sight to the quasars SDSS J095542.12+411655.3 (hereafter J0955+4116) and SDSS J100151.38+034333.9 (hereafter J1001+0343). Previous observations of these quasars have shown that these two gas clouds are among the most metal-poor DLAs currently known. These gas clouds are therefore ideally placed to assess the [O/Fe] inflection in near-pristine environments. This paper is organised as follows. Section 2 describes our observations and data reduction. In Section 3, we present our data and determine the chemical composition of the two DLAs. We discuss the chemical enrichment histories of these systems in Section 4, before drawing overall conclusions and suggesting future work in Section 5. OBSERVATIONS The data analysed in this paper are either the first high-resolution observations of the DLA (as is the case for J0955+4116), or, we have obtained additional highresolution observations that target previously unobserved metal lines (as is the case for J1001+0343). The DLA identified along the line of sight to the m r ≈ 19.38 quasar -J0955+4116 -was identified by Penprase et al. (2010) as an EMP DLA, based on observations with the Keck Echellete Spectrograph and Imager. We then observed this quasar using the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) on the Keck I telescope. We utilised the C1 (7.0 × 0.861 arcsec slit) and C5 (7.0 × 1.148 arcsec slit) decker, resulting in a spectral resolution of 49 000 and 37 000, respectively. These observations consist of 9 × 3600 s exposures using the C1 setup and 4 × 3600 s exposures using the C5 setup. Prior observations of the m r ≈ 17.7 quasar -J1001+0343 -using the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) at the European Southern Observatory (ESO) Very Large Telescope (VLT) revealed that the intervening DLA at z abs ≈ 3.078 is one of the least polluted gas reservoirs currently known. The 9.3 hours of VLT/UVES data presented in Cooke et al. (2011b) Cooke et al. (2011b) covered the wavelength range 3282 − 6652Å. Thus, the iron abundance was determined from observations of the Fe ii λ1608 line. We have secured further observations that focus on red wavelengths and target the stronger Fe ii λ2344 and λ2382 features of this DLA. The new data on J1001+0343 were collected with UVES (R 40 000) throughout the observing period P106 and P108 spanning the wavelength range 3756 − 4985Å and 6705 − 10429Å using a 0.8 arcsec slit width. We acquired 8×3000 s exposures on target using 2×2 binning in slow readout mode. A summary of our observations can be found in Table 1. Data reduction The HIRES data were reduced with the makee reduction pipeline while the ESO data were reduced with the EsoRex reduction pipeline. Both pipelines include the standard reduction steps of subtracting the detector bias, locating and tracing the echelle orders, flat-fielding, sky subtraction, optimally extracting the 1D spectrum, and performing a wavelength calibration. The data were converted to a vacuum and heliocentric reference frame. Finally, we combined the individual exposures of each DLA using uves popler 2 . This corrects for the blaze profile, and allowed us to manually mask cosmic rays and minor defects from the combined spectrum. When combining these data we adopt a pixel sampling of 2.5 km s −1 . Due to the different resolutions of the C1 and C5 HIRES deckers, we separately combine and analyse the data collected using each setup. For the regions of the J1001+0343 spectrum ∼ 9000Å, that are imprinted with absorption features due to atmospheric H 2 O, we also perform a telluric correction with reference to a telluric standard star. We test the robustness of this correction by also analysing the extracted spectra of the individual exposures (as discussed further in Section 3.2). ANALYSIS Using the Absorption LIne Software (alis) package 3which uses a χ-squared minimisation procedure to find the model parameters that best describe the input data -we simultaneously analyse the full complement of high S/N and high spectral resolution data currently available for each DLA. We model the absorption lines with a Voigt profile, which consists of three free parameters: a column density, a redshift, and a line broadening. We assume that all lines of comparable ionization level have the same redshift, and any absorption lines that are produced by the same ion all have the same column density and total broadening. The total broadening of the lines includes a contribution from both turbulent and thermal broadening. The turbulent broadening is assumed to be the same for all absorption features, while the thermal broadening depends inversely on the square root of the ion mass; thus, heavy elements (e.g. Fe) will exhibit absorption profiles that are intrinsically narrower than the profiles of lighter elements, (e.g. C). There is an additional contribution to the line broadening due to the instrument. For the HIRES and UVES data, the nominal instrument resolutions are v FWHM = 6.28 km s −1 (HIRES C1), v FWHM = 8.33 km s −1 (HIRES C5), and v FWHM = 7.3 km s −1 (UVES). Finally, we note that we simultaneously fit the absorption and quasar continuum of the data. We model the continuum around every absorption line as a low-order Legendre polynomial (typically of order 3). We assume that the zero-levels of the sky-subtracted UVES and HIRES data do not depart from zero 4 . In the following sections we discuss the profile fitting for each DLA in turn. J0955+4116 J0955+4116 is best modelled with two gaseous components at z abs = 3.279908 ± 0.000002 and z abs = 3.27996 ± 0.00001 (∆v = 4 ± 1 km s −1 ) for all singly ionized species (except Fe ii), and just the former component for neutral species (i.e. O i). We assume the temperature is T = 1 × 10 4 K (a value that is typi- cal for a metal-poor DLA; see Cooke et al. 2015, Welsh et al. 2020, Noterdaeme et al. 2021) and find that the turbulent components are b = 3.3 ± 0.2 km s −1 and b = 14.2 ± 1.5 km s −1 respectively. The data, along with the best-fitting model are presented in Figure 1, while the corresponding column densities are listed in Table 2. These results are unchanged when the assumed temperature varies between T ∼ (0.5 − 1.2) × 10 4 K -the range of values that have been measured in other metalpoor DLAs (Cooke et al. 2015). We find the data are best modelled when we allow for small velocity offsets between the features observed using the C1 and C5 deckers. These are found to be 1.5 km s −1 and account for potential differences between the wavelength calibrations of the data taken with the two HIRES setups. Note, we only use the neutral component (identified in the O i absorption) to infer the relative chemical abundances of this gas cloud; the component at z abs = 3.27996 likely arises from ionized gas as indicated by the lack of concurrent absorption from any neutral species. The neutral component at z abs = 3.279908 constitutes ∼ 70% to the total absorption in Si ii. We find that [Fe/H] = −2.95 ± 0.10 while [O/Fe] = +0.50 ± 0.10. This places the DLA towards J0955+4116 at the cusp of the EMP regime where the plateau in [O/Fe] may change. Here, and subsequently, the errors are given by the square root of the diagonal term of the covariance matrix calculated by alis at the end of the fitting procedure. We note that, while the individual Fe ii features may be relatively weak, the simultaneous analysis of the full complement of data results in a 4.8σ detection of Fe. J1001+0343 J1001+0343 is best modelled with one component at z abs = 3.078408 ± 0.000006 with a turbulence 1.0 C ii λ1036 C ii λ1334 O i λ1302 0.0 0.5 1.0 O i λ988.8 O i λ988.7 O i λ988.6 Al ii λ1670 0.0 0.5 1.0 Si ii λ1260 Si ii λ1304 Si ii λ1526 0.0 0.5 1.0 Fe ii λ1144 Fe ii λ1608 −25 0 +25 −25 0 +25 −25 0 +25 −80 −60 −40 −20 0 +20 +40 −25 0 +25 −25 0 +25 −25 0 +25 −25 0 +25 −25 0 +25 −25 0 +25 Normalized flux Normalized flux Velocity relative to z abs = 3.279908 [km s −1 ] 12.42 ± 0.05 −3.25 ± 0.07 a 3σ upper limit on column density. b = 6.3 ± 0.4 km s −1 and temperature T = (1.0 ± 0.6) × 10 4 K. The data, along with the best-fitting model are presented in Figure 2, while the corresponding column densities are listed in Table 3. We find that [Fe/H] = −3.25 ± 0.07 and [O/Fe] = +0.62 ± 0.05. All reported column densities are consistent with the previous determinations by Cooke et al. (2011b), but with a reduced error; in particular, the new data reported here have allowed the precision on the [O/Fe] measurement to be improved by a factor of three, from 0.15 to 0.05. To ensure the errors associated with these abundance determinations are robust, we have performed some additional checks. First, near the absorption features of interest, we have ensured that the fluctuations in the continuum are well-described by the error spectrum in this region. This ensures we are not underestimating the error associated with the data. Second, we have refit the O i and Fe ii features using a Monte Carlo approach, described in Fossati et al. (2019), and converged on abundance determinations that are consistent within 1σ. When analysing the DLA towards J1001+0343, we adopt two approaches for modelling the Fe ii absorption features. The Fe ii λ2344 and λ2382 features fall in regions of the spectrum that are impacted by telluric absorption; the DLA absorption features are therefore partially blended with telluric features to varying degrees of severity. Prior to combining the individual DLA exposures, we remove these features using the spectrum of a telluric standard star. The resulting data near the Fe ii λ2382 line, after performing this correction, are shown in the right panel of the third row of Figure 2. To ensure that the telluric correction has not introduced any artefacts in the data, we simultaneously fit the standard star spectrum and all of the individual quasar expo-sures (uncorrected for telluric absorption). The results of this fitting procedure are shown in Figure 3. From this figure it is clear that, while the Fe ii λ2382 feature is partially blended with a telluric absorption line, the range of dates used to observe this target results in a sequential shift in the position of the telluric feature relative to the Fe line of interest. In the top right panel of Figure 3, the telluric absorption is ∼ +5 km s −1 from the Fe ii λ2382 line center (as indicated by the blue tick mark). In the bottom right panel, the telluric absorption is ∼ −10 km s −1 from the Fe ii λ2382 line center. When jointly analysing these data, this shift allows us to capture an accurate profile of both the telluric and Fe ii features. Note, the centroid of the Fe ii λ2382 line is tied to the other DLA absorption features, while the centroid of the telluric feature is fixed from other telluric lines in the standard star spectrum. Using this approach we find a total Fe ii column density consistent with our analysis of the corrected combined spectrum. The value we report in this paper is based on the fits to the individual exposures. These new data confirm that the DLA towards J1001+0343 is one of the most iron-poor DLAs currently known. The new found precision afforded for the iron column density allow us to conclude that [O/Fe] is significantly elevated in this DLA compared to the plateau observed at higher metallicity. Before discussing the origin of this elevation, we perform some simulations to support our analysis. Mock models To further test if the DLA towards J1001+0343 exhibits an elevated [O/Fe] ratio, we simulated the O i and Fe ii absorption line profiles that would be expected given different intrinsic [O/Fe] abundance ratios. To achieve this, we take the best fit cloud model from our modelling procedure and generate synthetic model profiles varying the column density of either O i or Fe ii. The results of this test are shown in Figure 4. The top row shows the observed UVES data centered on the O i λ1039 (left) and O i λ1302 (right) absorption features. Overplotted on these data are the model profiles that would be expected if the underlying [O/Fe] ratio were +0.4, +0.6, and +0.8. To generate these profiles, we assume that the Fe ii column density is fixed (at the value given by our best fit model) and then vary the column density of O i accordingly. In the bottom row we show the UVES data centered on the Fe ii λ1608 at the value given by our best fit model). Below these data we show the residual fits between the model and the data. (Heger & Woosley 2010). However, before we consider the abundance pattern of this DLA in relation to the yields of Population III SNe, we first examine possible origins of this elevation. Origin of elevation Dust depletion is expected to be minimal in VMP DLAs (Pettini et al. 1997;Akerman et al. 2005;Vladilo et al. 2011;Rafelski et al. 2014). However, if the depletion of metals onto dust grains is unaccounted for, it will lead to artificially low metal abundance determinations for refractory elements. It is therefore useful to rule out its impact in the DLAs presented here. Depletion studies compare the relative abundances of elements in DLAs to the expected nucleosynthetic ratio which can be inferred from the abundances of stars of similar metallicity. O is minimally depleted onto dust grains (Spitzer & Jenkins 1975;Jenkins 2009;Jenkins & Wallerstein 2017). Both Si and Fe are refractory elements, and are partially depleted onto dust grains but at different rates. As shown in Figure 5, both metal-poor halo stars and VMP DLAs exhibit an identical evolution of the [O/Fe] ratio (see also Figure 12 of Cooke et al. 2011b). Given this agreement, we therefore expect dust depletion to be minimal in DLAs that have a metallicity [O/H] < −2. We can also use the [Si/Fe] abundance ratio to explore the possibility of dust depletion. The most metalpoor stars and the most metal-poor DLAs appear to have a metallicity independent evolution of [Si/Fe] when [Fe/H] < −2. For stars, the plateau occurs at [Si/Fe] = +0.37 ± 0.15 (Cayrel et al. 2004). While for DLAs, the plateau occurs at [Si/Fe] = +0.32 ± 0.09 (Wolfe et al. 2005;Cooke et al. 2011b). The [Si/Fe] of both J0955+4116 and J1001+0343 are consistent with the plateau seen in metal-poor DLAs (see Figure 6). We therefore do not expect dust depletion to be the source of the elevated [O/Fe] abundance ratio in the EMP regime. Volatile elements, like S and Zn, are less readily depleted onto dust grains than Si and Fe (Savage & Sembach 1996;Jenkins 2009;Jenkins & Wallerstein 2017), however these elements are not currently accessible for the EMP DLAs studied here. The advent of the next generation of 30 − 40 m telescopes will make these abundance determinations possible for EMP DLAs. We also find it unlikely that the cloud model introduces a systematic [O/Fe] enhancement; the O i and Fe ii column densities of both DLAs are derived from at least one weak absorption line. Furthermore, we note that ionization effects cannot explain this behaviour at low metallicity; the presence of an unresolved component of ionized gas containing Fe ii would not affect the O i column density, but would lead to an overestimate the Fe ii column density. Thus, accounting for ionized gas would only act to further increase the [O/Fe] ratio. We therefore conclude that the elevated oxygen to iron abundance ratio observed for the EMP DLA towards J1001+0343 is intrinsic to the DLA. Stochastic enrichment In the previous section, we concluded that the observed [O/Fe] ratio is intrinsic to the DLA towards J1001+0343. We now explore the possibility that this DLA has been enriched by the first generation of stars. Specifically, we compare the observed abundance pattern of this DLA to those predicted by a stochastic chemical enrichment model developed in previous work (Welsh et al. 2019). This model describes the underlying mass distribution of the enriching stellar population using a power-law: ξ(M ) = kM −α , where k is a multiplicative constant that is set by the number of enriching stars that form between a given mass range: N = Mmax Mmin kM −α dM(1) Since the first stars are thought to form in small multiples, this underlying mass distribution is necessarily stochastically sampled. We utilise the yields from simulations of stellar evolution to construct the expected distribution of chemical abundances given an underlying IMF model. These distributions can then be used to assess the likelihood of the observed DLA abundances given an enrichment model. In our analysis, we use the relative abundances of stellar layers (f He ). The explosion energy is a measure of the final kinetic energy of the ejecta at infinity while the mixing between stellar layers is parameterised as a fraction of the helium core size. For further details, see both HW10 and Welsh et al. (2019). While the HW10 yields have been calculated for metal-free stars, they are also representative of EMP Population II CCSNe yields (at least for the elements under consideration in this work); this can be seen by comparison with the Woosley & Weaver (1995) yields of metal-enriched massive stars. As a result, in previous studies of near-pristine absorption line systems, it has been difficult to distinguish between enrichment by Population II and Population III stars. Fortunately, in our analysis, we can take advantage of this degeneracy and consider the HW10 yields to be representative of both Population II and Population III SNe yields. The default enrichment model, described above, contains six free parameters (N , M min , M max , α, E exp , and f He ). We are using three relative abundances to assess the enrichment of the DLA towards J1001+0343. Thus, we cannot simultaneously investigate these six parameters. We can, however, make some simplifications. The underlying IMF of the first stars remains an open question; this is not the case for massive Population II stars. The Population II IMF for stars of mass M 10 M is expected to be well-described by a Salpeter IMF (i.e. α = 2.35 in Equation 1) 5 . Under the assumption of a Salpeter IMF, if the number of enriching stars we derive is large, then this may imply that Population II stars are the dominant enrichment source. Alternatively, if N is low, then it is possible that a pure or washed out Population III signature may still be present in this DLA. We test this idea by using a Markov Chain Monte Carlo (MCMC) likelihood analysis to investigate the number of stars that have enriched this DLA. We explore the entire enrichment model parameter space to find the parameters that best fit our data. The HW10 parameters span (10 − 100) M (with a mass resolution of ∆M 0.1 M ), (0.3 − 10) × 10 51 erg (sampled by 10 values), and (0−0.25) f He (sampled by 14 values) where f He is the fraction of the He core size; in total, there are 16 800 models in this yield suite. We adopt a upper mass limit of 70 M beyond which pulsational pair instability SNe are thought to occur (Woosley 2017). This leaves a grid of 15 792 models to explore. During our analysis, we linearly interpolate between this grid of yields while applying uniform priors on each parameter. The results of this analysis are shown in Figure 7. The most favoured result of this model is that the DLA towards J1001+0343 was enriched by a low number of massive stars which, as argued above, make it consistent with Population III enrichment (but does not rule out Population II yields). Motivated by these findings, we now assess the possible properties of a putative metal-free star that may be responsible for the enrichment of the DLA towards J1001+0343. In this case we assume that the DLA has been enriched by one Population III SN, again utilising the HW10 yields. The results of this analysis are shown in Figure 8. We find that the abundances of this DLA are best modelled by a Population III star with a mass between 19 − 25 M (2σ) and an explosion energy between (0.9 − 2.4) × 10 51 erg (2σ). The degree of mixing between the stellar layers remains unconstrained, but generally favours lower values of the mixing parameter. To test how well this model describes our data, we compare the [X/O] ratios supported by this model to those presented in Table 3. This comparison is shown in Fig The Population III star that best models the abundances of the DLA towards J1001+0343 has an explosion energy that is consistent with the value found for a typical metal-poor DLA (Welsh et al. 2019). The results of this analysis are also similar to the inferred enrichment of the most metal-poor DLA currently known; Cooke et al. (2017) find that the abundance pattern of the most metal-poor DLA can be well-modelled by a Population III SN with a progenitor mass M = 20.5 M . The DLA analysed by Cooke et al. (2017) was preferentially modelled with a somewhat higher explosion energy than that reported here, but still consistent within 2σ. This preference towards higher energy explosions (i.e. hypernovae) is also inferred from the analysis of some EMP stars (e.g. Grimmett et al. 2018;Ishigaki et al. 2018). However, the HW10 analysis of the Cayrel et al. (2004) sample of EMP stars favoured models with an explosion energy between (0.6 − 1.2) × 10 51 erg. Recent theoretical works have started to favour lower energy supernova explosions for metal-free progenitors (e.g. the simulations performed by Ebinger et al. (2020) suggest a range (0.2 − 1.6) × 10 51 erg). This is in line with the range of explosion energies inferred for the enrichment of the DLA towards J1001+0343. Furthermore, the analysis conducted by Haze Nuñez et al. (2021) has compared the typical (median) abundances of VMP DLAs to the IMF-weighted yields from various simulations; for the abundances considered in this work,the Ebinger et al. , some odd atomic number elements (e.g. N, Al), and select iron-peak elements (e.g. Ni, Cr, Zn). These elements may allow us to further pin down the properties of the enriching stars. An informative probe of the explosion physics is the relative abundance of zinc and iron -we do not detect zinc absorption in this DLA, but this may be possible with the next generation of 30 − 40 m telescopes. Future avenues for ruling out Population II In the previous section we investigated the chemical enrichment of the DLA towards J1001+0343. Under the assumption of a Sapleter IMF, we found that this DLA was best modelled by a low number of enriching stars (consistent with one). We also found that the observed abundances of [C/O], [Si/O], and [Fe/O] can be simultaneously well modelled by the yields of an individual Population III SN. However, given the age of the Universe at redshift z = 3 (∼ 2 Gyr), there is sufficient time (∼ 1.5 Gyr between z = 10 to z = 3) for this DLA to be enriched by Population III stars and subsequent Population II stars. Any putative Population III signature is thought to be washed out soon after the birth of Population II stars ; just a few massive Population II stars are required to wash out a peculiar Population III chemical signature. How- ever, there could be a delay between Population III and Population II star formation. For example, reionization quenching can temporarily suspend star formation in low mass galaxies Wheeler et al. 2015), and this may prolong the time that a Population III signature can be preserved in near-pristine gas. After a period of dormancy, interactions with gaseous streams in the intergalactic medium can help re-ignite star formation in these low mass objects (Wright et al. 2019). Interestingly, the chemistry of the most metalpoor DLAs shows an increase in [C/O] with decreasing redshift (see Figure 8 of Welsh et al. 2020 Unravelling the Population III fingerprint We have mentioned both EMP stars and EMP DLAs as potential environments to uncover the Population III fingerprint. Both stars and DLAs have their respective advantages. The large sample size afforded when studying stellar relics cannot yet be matched in similar studies of gaseous relics. The potential evolution of [O/Fe] across EMP stars has also been the subject of much discussion (see the review by McWilliam 1997). However, determining this ratio in EMP stars is particularly challenging. There are four approaches to determine the oxygen abundance in stellar atmospheres. The typical method utilises the O i λ777 nm triplet; a transition that requires large non-LTE corrections (Fabbian et al. 2009). The weak O i λ630 nm line is known to form in LTE and, as such, may be more reliable (Asplund 2005). However, the strength of this feature means that it is challenging to detect at low metallicities. Observations of the UV and IR molecular OH features also show systematic offsets relative to the aforementioned [O/H] abundance indicators, although the offsets can be reduced by accounting for 3D hydrodynamical effects (Nissen et al. 2002;García Pérez et al. 2006;Dobrovolskas et al. 2015). Given the difficulty of accurately determining the O abundance in the lowest metallicity stars, we suggest that DLAs are the ideal environment to study the evolution of this element despite the smaller sample size. The comparative analysis of the chemistry of both gaseous and stellar relics offers the opportunity to study early chemical enrichment further. It is well established that some of the most metal-poor Milky Way halo stars show an enhanced [C/Fe] ratio (Beers & Christlieb 2005). This may be a sign of enrichment from the first stars. Or, indeed in some cases, this enhancement may be explained via mass transfer across stars in binary systems (Arentsen et al. 2019). Whether this enhancement is also prevalent across metal-poor DLAs is yet to be seen. Figure 6 highlights that, with current statistics, we cannot discern any concurrent enhancement of the [C/Fe] Another environment that may offer unique insight to the metals produced by the first stars are the ultrafaint dwarf galaxies (UFDs) that orbit the Milky Way. These UFDs contain some of the most metal-poor stars currently known. The stars in these UFDs may have experienced a different enrichment history to the Milky Way halo stars. If metal-poor DLAs are indeed the antecedents of the UFDs, we might be able to search for a consistent chemical signature in the most metal-poor stars of the UFDs. DLAs analysed in this work may provide a signpost to some of the most pristine environments in the high redshift universe, and would be an ideal place to search for Population III host galaxies and perhaps even the light from Population III SNe using the forthcoming James Webb Space Telescope. We thank the anonymous referee who provided a prompt and careful review of our paper. We thank M. Fossati for the use of their line-fitting software and we thank A. Skuladottir for helpful discussions surrounding comparisons with stellar populations. This paper is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (VLT program IDs: 083.A-0042(A) and 105.20L3.001), and at the W. M. Keck Observatory which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. We are also grateful to the staff astronomers at the VLT and Keck Observatory for their assistance with the observations. This work has been supported by Fondazione In this appendix, we present the DLA data used to produce Figure 5. In Table 4 we list the column density of neutral hydrogen, the redshift, and the relative abundances of oxygen and iron. Figure 1 . 1Continuum normalised HIRES data (black histograms) of the absorption features produced by metal ions associated with the DLA at z abs = 3.279908 towards the quasar J0955+4116. The best-fitting model is shown with the red curves. In the top panel, the red shaded band highlights range of H model profiles included in the 1σ error bounds. The blue dashed line indicates the position of the continuum while the green dashed line indicates the zero-level. The red ticks above the absorption features indicate the centre of the Voigt line profiles. In the left panel of the fourth row, the blue tick indicates the centre of the Fe ii 1260 feature. Below the zero-level, we show the residuals of this fit (black histogram) where the grey shaded band encompasses the 2σ deviates between the model and the data. The vertical blue shaded bands indicate the regions of the spectrum not included in the fit. Figure 2 . 2(left) and Fe ii λ2382 (right) lines. The overplotted models show the same underlying [O/Fe] abundance ratios as the top row, however, in this case we vary the column density of Fe ii (while the column density of O i is fixed Continuum normalised UVES data (black histograms) of the absorption features produced by metal ions associated with the DLA towards J1001+0343. The top row shows H i. The second and third rows show our new programme data (programme ID: 105.20L3.001) while the bottom two rows show the archival data (programme ID: 083.A-0042(A)). The bestfitting model profiles are shown as red curves. The blue dashed line indicates the position of the continuum while the green dashed line indicates the zero-level. The red ticks above the absorption features indicate the centre of the Voigt line profiles. Below the zero-level, we show the residuals of this fit (black histogram) where the grey shaded band encompasses the 2σ deviates between the model and the data. The vertical blue shaded bands indicate the regions of the spectrum not included in the fit. to z abs = 3.078408 [km s −1 ] Figure 3 . 3Continuum normalised UVES data (black histograms) of the absorption features produced by Fe ii λ2382 associated with the DLA towards J1001+0343 along with the surrounding telluric absorption features for each exposure. The right panels show a zoom in of the left panels. Overplotted in red is our best-fitting model of the Fe ii λ2382 feature and the telluric absorption. The blue dashed line indicates the position of the continuum while the green dashed line indicates the zero-level. The red ticks above the absorption features indicate the centre of the Voigt line profiles of the DLA, while the blue tick marks indicate the line centre of the telluric features. Below the zero-level, we show the residuals of this fit (black histogram) where the grey shaded band encompasses the 2σ deviates between the model and the data. The vertical blue shaded bands indicate the regions of the spectrum not included in the fit. to z abs = 3.078408 [km s −1 ] Figure 4 . 4Models of O i and Fe ii absorption components overplotted on the UVES data of J1001+0343. The colour indicates the underlying abundance ratio [O/Fe] as highlighted in the legend. The top panels shows the O i data and the corresponding O i model profiles assuming a fixed Fe ii column density. The bottom panels show the corresponding Fe ii model profiles assuming a fixed O i column density. These fixed values represent the best fit column densities of the data. The blue (green) dashed lines represents the continuum (zero) levels of the data. Note the different y-axis range of the Fe ii λ1608 panel. Below each spectrum, we show the residuals of this fit (colored histograms) where the light (dark) grey shaded band encompasses the 2σ (1σ) deviates between the model and the data. Figure 5 . 5[C/O], [Si/O], and [Fe/O] when investigating the enrichment of the DLA towards J1001+0343. We compare the measured abundances to the nucleosynthetic yield calculations of massive (> 10 M ) metal-free stars from Heger & Woosley (2010) (hereafter HW10). These yields have been calculated as a function of the progenitor star mass (M ), the explosion energy of their supernova (E exp ), and the mixing between the different [O/Fe] vs [Fe/H] for all metal-poor DLAs and sub-DLAs (blue circles) plotted alongside [O/Fe] of metal-poor stars (light gray) -the shape of the marker indicates the source of the stellar data: circles -García Pérez et al. (2006), squares -Cayrel et al. (2004), triangles -Nissen et al. (2002) (all values are those presented in the Cooke et al. 2011b reanalysis). The two [O/Fe] abundances reported here are shown in red. The previous measurement of J1001+0343 is marked in orange and is connected to the latest measurement by a dashed line. The black dashed line indicates the solar relative abundance. Figure 6 . 6[C/Fe] (left) and [Si/Fe] (right) vs [Fe/H] for all metal-poor DLAs and sub-DLAs (blue circles). The abundances of the DLAs reported here are shown in red. The black dashed line indicates the solar relative abundance. - ure 9 . 9The [C/O], [Si/O], and [Fe/O] of this DLA are simultaneously well described by the inferred likelihood model. Figure 7 . 7Results of our MCMC analysis of the chemical enrichment of the DLA towards J1001+0343 given our stochastic model. From left to right, we show the number of enriching stars, the explosion energy, and the degree of mixing. The diagonal panels indicate the maximum likelihood posterior distributions of the model parameters while the 2D contours indicate the correlation between these parameters. The dark and light gray shaded regions indicate the 68 and 95 per cent confidence intervals, respectively. In the diagonal panels, the horizontal blue dashed line indicates the zero-level of each distribution.(2020) and HW10 IMF-weighted yields are equally capable of predicting the typical VMP DLA chemistry. Though, the typical VMP abundances of [C/O], [Si/O], and [Fe/O] favour the HW10 models with higher energy explosions over their lower energy counterparts. At present, the models are driven by the relatively low errors on the [C/O] and [Si/O] abundances. Future higher S/N observations and covering a broader wavelength range would allow us to detect additional αcapture elements (Mg, S) Figure 8 .Figure 9 . 89Results of our MCMC analysis of the chemical enrichment of the DLA towards J1001+0343 assuming the number of enriching stars N = 1. From left to right, we show the progenitor star mass, the explosion energy, and the degree of mixing. The diagonal panels indicate the maximum likelihood posterior distributions of the model parameters while the 2D contours indicate the correlation between these parameters. The dark and light gray shaded regions indicate the 68 and 95 per cent confidence intervals, respectively. In the diagonal panels, the horizontal blue dashed line indicates the zero-level of each distribution. The observed abundances of the DLA towards J1001+0343 (red circles) compared to the best fit abundance ratios inferred by our best fit enrichment model (blue squares). The blue error bars encompass the interquartile range of the model values. & Christlieb 2005 for a review). Reminiscent of this signature in stars, there is tentative evidence of an enhanced [O/Fe] abundance in the most metal-poor DLAs. Specifically, all DLAs with an iron abundance between −3.0 < [Fe/H] < −2.0 are scattered around an [O/Fe] plateau of [O/Fe] +0.4, while those with [Fe/H] < −3.0 exhibit a modestly elevated [O/Fe] abundance. The plateau in [O/Fe] observed in DLAs with [Fe/ Table 1 . 1Journal of observations. With some wavelength gaps. b Near the accessible O i line -either λ1302 or λ1039QS0 r zem z abs Telescope/ Wavelength Resolution Integration S/N b Programme (mag) instrument range a (Å) (km s −1 ) time (s) ID J0955+4116 19.38 3.420 3.280 KECK/HIRES 3839 -6718 6.3 32 400 20 N162Hb, C320Hb KECK/HIRES 3242 -7805 8.3 14 400 18 A152Hb J1001+0343 17.72 3.198 3.078 VLT/UVES 3282 -6652 7.3 33 700 28 083.A-0042(A) VLT/UVES 3756 -10429 7.3 24 000 11 105.20L3.001 a indicate that [Fe/H] = −3.18±0.15 and [O/Fe] = +0.53 ± 0.16. Given that the typical [O/Fe] abundance observed amongst very metal-poor (VMP; [Fe/H] < −2) DLAs is ∼ +0.4, the EMP DLA towards J1001+0343 is ideally placed to investigate if the [O/Fe] abundance is elevated at the lowest metallicities. The observations carried out by Table 2 . 2Ion column densities of the DLA at z abs = 3.279908 towards the quasar J0955+4116. The quoted column density errors are the 1σ confidence limits.Ion Transitions log10 N (X)/cm −2 [X/H] [X/Fe] used [Å] H i 1215 20.21 ± 0.05 - - C ii 1036, 1334 13.73 ± 0.14 −3.00 ± 0.06 −0.05 ± 0.10 O i 988, 1302 14.45 ± 0.04 −2.45 ± 0.06 +0.50 ± 0.10 Al ii 1670 ≤ 11.25 a ≤ −3.41 ≤ −0.46 Si ii 1260, 1304, 12.89 ± 0.06 −2.94 ± 0.06 +0.01 ± 0.09 1526 Fe ii 1144, 1260, 12.73 ± 0.09 −2.95 ± 0.10 - 1608 a 3σ upper limit on column density. Table 3 . 3Ion column densities of the DLA at z abs = 3.078408 towards the quasar J1001+0343. The quoted column density errors are the 1σ confidence limits.Ion Transitions log10 N (X)/cm −2 [X/H] [X/Fe] used [Å] H i 1215 20.20 ± 0.05 - - C ii 1036, 1334 13.57 ± 0.01 −3.06 ± 0.05 +0.19 ± 0.05 N i 1200 ≤ 12.50 a ≤ −3.53 ≤ −0.28 O i 1039, 1302 14.26 ± 0.01 −2.63 ± 0.05 +0.62 ± 0.05 Si ii 1190, 1260, 12.86 ± 0.01 −2.85 ± 0.05 +0.40 ± 0.05 1304, 1526 S ii 1253 ≤ 12.91 a ≤ −2.41 ≤ +0.84 Fe ii 1608, 2382 The primary goal of this paper is to assess if EMP DLAs (those with [Fe/H] < −3.0) exhibit an enhanced [O/Fe] abundance relative to VMP DLAs (those with −3.0 < [Fe/H] < −2.0).Figure 5shows the [O/Fe] abundance ratio as a function of the [Fe/H] metallicity for the DLAs analysed in this work (red symbols), together with literature measurements of VMP and EMP DLAs (blue symbols) and stars (gray symbols). The DLA abundances used to produce this figure are given in Appendix A. The new data reported here confirm that the DLA towards J1001+0343 is a bonafide EMP DLA, while the DLA towards J0955+4116 is on the cusp of the EMP regime. Qualitatively, the [O/Fe] values of these DLAs are consistent with the trend of an increased [O/Fe] ratio below [Fe/H] < −3. Given the metallicity of J0955+4116 ([Fe/H] = −2.95 ± 0.10), it is reasonable to expect that the [O/Fe] abundance ratio of this DLA is consistent with the plateau seen at higher metallicity. The high precision [O/Fe] determination of J1001+0343 has strengthened the evidence that this system exhibits an elevated [O/Fe] ratio. With this new measurement, we determine the mean of this relative abundance ratio for the EMP DLAs shown in Figure 5: [ O/Fe ] = +0.67 ± 0.04. For VMP DLAs, [ O/Fe ] = +0.40 ± 0.08; the mean [O/Fe] abundance reported for EMP DLAs is significantly divergent (3σ) from the mean [O/Fe] abundance reported for VMP DLAs. There are no obvious selection biases that may have caused this apparent inflection of [O/Fe] at the lowest metallicities probed. Furthermore, these results are consistent with the recent evaluation of [O/Fe] across EMP stars when the stellar spectra are analysed using 3D non-LTE models (Amarsi et al. 2019). Specifically, their analysis finds [O/Fe] EMP ∼ +0.7. If this trend of an elevated [O/Fe] abundance is confirmed with future measurements of EMP DLAs, it will highlight that EMP DLAs exhibit a distinct chemical enrichment relative to VMP DLAs; the source of the elevated [O/Fe] in these metal-poor DLAs may be attributed to enrichment by a generation of Population III starsThe grey shaded band represents the 2σ limits provided by our best fit model ([O/Fe] = +0.62 ± 0.05) and the data. In each panel (except for the very weak Fe ii λ1608 line), the residual showcasing the fit of the [O/Fe] = +0.4 model (dark red; see legend) is outside of the 2σ range. From this analysis, we therefore conclude that these data show an [O/Fe] abundance ratio that is inconsistent with the plateau seen at [Fe/H] > −3. 4. DISCUSSION While it may be difficult to distinguish the absolute chemical yields of Population II versus Population III stars, it may be possible to identify a transition of Population III to Population II enrichment by empirically looking for a change in the behaviour of chemical abundance ratios as a function of either iron paucity or redshift. Given the tight [O/Fe] plateau seen in VMP DLAs combined with the evidence of an increased [O/Fe] ratio in some EMP DLAs, we thus propose that the [O/Fe] abundance may offer the cleanest probe of the Population III to Population II transition in the most metal-poor DLAs.). One inter- pretation of this trend is that EMP DLAs universally experienced some degree of reionization quenching; the increase in [C/O] with decreasing redshift is interpreted as the onset of enrichment from the carbon yield of the first (i.e. Population II) intermediate mass stars. To reduce the possibility of Population II contamina- tion, one might consider measurements of the [O/Fe] ratio close to the epoch of reionization. For exam- ple, Bañados et al. (2019) recently reported the de- tection of a near-pristine DLA at z = 6.4. While the current determinations of the metallicity and [O/Fe] abundance of this system ([Fe/H] = −2.94 ± 0.26; [O/Fe] = +0.02 ± 0.21) are too uncertain to study the nature of [O/Fe] in the EMP regime, future higher spectral resolution and higher signal-to-noise ratio ob- servations would allow the chemistry of this gas cloud to be inferred from weak absorption lines, potentially improving the precision of this measurement by an order of magnitude. With the current data, Bañados et al. (2019) find no evidence of a Population III chemical fingerprint. The best-fit value of the [O/Fe] value of the Bañados et al. (2019) DLA places this system below the typical [O/Fe] abundance of VMP DLAs (at ∼ 1.7σ confidence). Considering the high [O/Fe] ratio reported in this paper, if the [O/Fe] value of the Bañados et al. (2019) DLA remains unchanged with future observa- tions, these distinct abundances would be a signpost of stochastic chemical enrichment at the lowest metallicity -this may also be a signature of Population III star formation. or [Si/Fe] abundance ratios along with [O/Fe]. The enhancement of [O/Fe] in the EMP regime may not extend to other alpha elements across these DLAs. The lack of concurrent [C/Fe] enhancement may suggest that the CEMP-no stars and the [O/Fe] enhanced DLAs have experienced divergent enrichment histories. This discrepancy may help reveal the origins of CEMPno stars. 5 . 5CONCLUSIONS Previous observations of metal-poor DLAs tentatively suggested that the most metal-poor DLAs display an elevated [O/Fe] ratio compared to their higher metallicity counterparts. The higher metallicity ([Fe/H] > −3) DLAs are well described by a plateau around a typical value of [ O/Fe ] = +0.40 ± 0.08. The primary goal of this paper is to assess whether [O/Fe] is indeed elevated amongst the most metal-poor DLAs by presenting a detailed chemical abundance analysis of two near-pristine DLAs -J0955+4116 and J1001+0343, observed with Keck/HIRES and VLT/UVES respectively. Our main conclusions are as follows: 1. We find that the DLA towards J0955+4116 has a neutral hydrogen column density log 10 N (H i)/cm −2 = 20.21 ± 0.05 and a relative iron abundance [Fe/H] = −2.95 ± 0.10. This places the gas cloud towards this quasar on the cusp of the EMP regime. The data collected using Keck/HIRES have revealed [O/Fe] = +0.50±0.10. The [O/Fe] abundance of this DLA is therefore consistent with the plateau observed across DLAs with [Fe/H] > −3. 2. We present new VLT/UVES data of the DLA toward J1001+0343 that cover previously unobserved Fe ii features. These data provide a more precise determination of its chemical composition. We measure an iron abundance of [Fe/H] = −3.25 ± 0.07 and an oxygen to iron ratio of [O/Fe] = +0.62 ± 0.05, reducing the error associated with the [O/Fe] determination by a factor of three compared to previous analyses. 3. The [O/Fe] ratio of the DLA towards J1001+0343 is significantly (2.3σ) above the typical value of a VMP DLA. We have considered the abundances of other ions and this analysis suggests that neither dust depletion nor ionization corrections are the source of this elevation. Rather, the elevated value is intrinsic of the DLA. 4. The main result of this paper is that the typical [O/Fe] abundance of EMP DLAs ([ O/Fe ] = +0.67±0.04) is significantly (3σ) above the typical value of a VMP DLA ([ O/Fe ] = +0.40 ± 0.08). 5. The origin of this elevation can be explained if VMP DLAs are all enriched by a similar population of stars, drawn from the same IMF. The divergence at the lowest metallicities likely represents enrichment from a distinct population of stars. Indeed, the scatter associated with the O abundance may highlight the stochastic nature of early chemical enrichment. The chemical composition of this DLA can be well-modelled by the yields of a (19 − 25) M (2σ) Population III SN with a (0.9 − 2.4) × 10 51 erg (2σ) explosion. Given that the adopted yields are also representative of Population II SNe yields, we cannot yet rule out contamination from later stellar populations. We suggest that EMP DLAs display an elevated [O/Fe] compared to their higher metallicity counterparts and this is due to their distinct enrichment histories. The elevated [O/Fe] abundance could be an indicator of enrichment by a generation of metal free stars (Heger & Woosley 2010). Further data are necessary to determine if the elevated [O/Fe] ratio of J1001+0343 is typical of an EMP DLA. Forthcoming data on other nearpristine DLAs, as part of this programme, will directly answer this question. Furthermore, upcoming surveys (e.g. DESI, WEAVE, and 4MOST Dalton et al. 2012; de Jong et al. 2012; DESI Collaboration et al. 2016; Pieri et al. 2016) will provide new EMP DLA candidates to investigate and improve the statistical significance of this behaviour. If an elevated [O/Fe] abundance can be attributed to enrichment by metal-free stars, then the Cariplo, grant No 2018-2329. During this work, R. J. C. was supported by a Royal Society University Research Fellowship. We acknowledge support from STFC (ST/L00075X/1, ST/P000541/1). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 757535). This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. This research has made use of NASA's Astrophysics Data System.Facilities: Keck:I (HIRES), VLT: (UVES) Software: Astropy (Astropy Collaboration et al. 2013), Corner (Foreman-Mackey 2016), Matplotlib (Hunter 2007), and NumPy (van der Walt et al. 2011). APPENDIX A. 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{'abstract': 'We present precise abundance determinations of two near-pristine damped Lyα systems (DLAs) to assess the nature of the [O/Fe] ratio at [Fe/H] < −3.0 (i.e. < 1/1000 of the solar metallicity). Prior observations indicate that the [O/Fe] ratio is consistent with a constant value, [O/Fe] +0.4, when −3 < [Fe/H] < −2, but this ratio may increase when [Fe/H] −3. In this paper, we test this picture by reporting new, high-precision [O/Fe] abundances in two of the most metal-poor DLAs currently known. We derive values of [O/Fe] = +0.50 ± 0.10 and [O/Fe] = +0.62 ± 0.05 for these two z 3 near-pristine gas clouds. These results strengthen the idea that the [O/Fe] abundances of the most metal-poor DLAs are elevated compared to DLAs with [Fe/H] −3.We compare the observed abundance pattern of the latter system to the nucleosynthetic yields of Population III supernovae (SNe), and find that the enrichment can be described by a (19−25) M Population III SN that underwent a (0.9−2.4)×10 51 erg explosion. These high-precision measurements showcase the behaviour of [O/Fe] in the most metal-poor environments. Future high-precision measurements in new systems will contribute to a firm detection of the relationship between [O/Fe] and [Fe/H]. These data will reveal whether we are witnessing a chemical signature of enrichment from Population III stars and allow us to rule out contamination from Population II stars.', 'arxivid': '2201.08394', 'author': ['Louise Welsh \nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK\n\nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano Bicocca\nPiazza della Scienza 3I-20126MilanoItaly\n\nINAF -Osservatorio Astronomico di Brera\nvia Bianchi 46I-23087Merate (LC)Italy\n', 'Ryan Cooke \nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK\n', 'Michele Fumagalli \nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano Bicocca\nPiazza della Scienza 3I-20126MilanoItaly\n\nINAF -Osservatorio Astronomico di Trieste\nvia G. B. Tiepolo 11I-34143TriesteItaly\n', 'Max Pettini \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n'], 'authoraffiliation': ['Centre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK', 'Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano Bicocca\nPiazza della Scienza 3I-20126MilanoItaly', 'INAF -Osservatorio Astronomico di Brera\nvia Bianchi 46I-23087Merate (LC)Italy', 'Centre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK', 'Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano Bicocca\nPiazza della Scienza 3I-20126MilanoItaly', 'INAF -Osservatorio Astronomico di Trieste\nvia G. B. Tiepolo 11I-34143TriesteItaly', 'Institute of Astronomy\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK'], 'corpusid': 246210236, 'doi': '10.3847/1538-4357/ac4503', 'github_urls': ['https://github.com/MTMurphy77/UVES', 'https://github.com/rcooke-ast/ALIS.4'], 'n_tokens_mistral': 27599, 'n_tokens_neox': 21874, 'n_words': 11531, 'pdfsha': '8c9c3b161b5432c3c7dc84a2c2f2861a42986c7e', 'pdfurls': ['https://arxiv.org/pdf/2201.08394v1.pdf'], 'title': ['Oxygen-enhanced extremely metal-poor DLAs: A signpost of the first stars?', 'Oxygen-enhanced extremely metal-poor DLAs: A signpost of the first stars?'], 'venue': []}

arxiv

Blowup for C 2 Solutions of the N -dimensional Euler-Poisson Equations in Newtonian Cosmology 25 Mar 2014 Revised 08-Nov-2013 Manwai Yuen rn*e-mailaddress:nevetsyuen@hotmail.com Department of Mathematics and Information Technology The Hong Kong Institute of Education 10 Po Ling RoadTai Po New Territories Hong Kong Blowup for C 2 Solutions of the N -dimensional Euler-Poisson Equations in Newtonian Cosmology 25 Mar 2014 Revised 08-Nov-2013arXiv:1403.6234v1 [math-ph] 1 2 Manwai YuenEuler-Poisson EquationsNewtonian CosmologyInitial Value ProblemBlowupSpectral-Dynamics-Integration MethodAttractive ForcesC 2 SolutionsBounded Domain Pressureless Euler-Poisson equations with attractive forces are standard models in Newtonian cosmology. In this article, we further develop the spectral dynamics method and apply a novel spectral-dynamics-integration method to study the blowup conditions for C 2 solutions with a bounded domain, X(t) ≤ X0, where · denotes the volume and X0 is a positive constant. In particular, we show that if the cosmological constant Λ < M/X0, with the total mass M , then the non-trivial C 2 solutions in R N with the initial condition Ω0ij (x) = 1 2 ∂iu j (0, x) − ∂ju i (0, x) = 0 blow up at a finite time.MSC: 35B30, 35B44, 35Q35, 85A05, 85A40 Introduction The evolution of Newtonian cosmology can be modelled by the compressible pressureless Euler-Poisson equations in dimensionless units:                ρ t +∇ · (ρu) =0 ρ[u t + (u · ∇)u] = −ρ∇Φ ∆Φ(t, x) =ρ − Λ,(1) where ρ = ρ(t, x) ≥ 0 and u = u(t, x) ∈ R N are the density and the velocity, respectively, with a background or cosmological constant Λ. The pressureless Euler-Poisson equations are the standard model in cosmology [1]. If the Euler-Poisson equations include the pressure term, then they provide the classical description of galaxies or gaseous stars in astrophysics [2] and [3]. For details of the connection between the Euler-Poisson equations (1) and Einstein's field equations R µν − 1 2 g µν R − g µν Λ = 8πG c 4 T µν ,(2) where R µυ is the Ricci curvature tensor, R is the curvature scalar, g µν is the metric tensor, T µν is the energy-momentum tensor of the universe, G is Newton's gravitational constant and c is the light speed, interested readers can refer to Chapter 6 of Longair's book [3]. In addition, for a geometrical explanation of Newtonian cosmology with a cosmological constant Λ, interested readers can refer to Brauer, Rendall and Reula's paper [4]. For an analysis of stabilities for the related systems, interested readers can refer to [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and [16]. In addition, there are explicit blowup or global (periodical) solutions for the Euler-Poisson systems [17], [18], [19], [20] and [21]. It should be noted that in 2008, Chae and Tadmor [13] determined the finite time blowup for the pressureless Euler-Poisson equations with attractive forces (1) with Λ = 0, under the initial condition, S := { x 0 ∈ R N ρ 0 (x 0 ) > 0, Ω 0 (x 0 ) = 0, div u(0, x 0 ) < 0} = φ,(3) where u = (u 1 , u 2 , ...., u N ) and Ω 0 (x 0 ) is the re-scaled vorticity matrix defined by Ω 0ij (x 0 ) = 1 2 ∂ i u j (0, x 0 ) − ∂ j u i (0, x 0 ) . Using spectral dynamics analysis, they identified the Riccati differential inequality D div u(t, x 0 (t)) Dt ≤ − 1 N [div u(t, x 0 (t))] 2 ,(4) along the characteristic curve dx0(t) dt = u(t, x 0 (t)). The corresponding solution of the inequality (4) blows up at or before T = −N/ div u(0, x 0 (0)) with an initial condition that requires that div u(0, x 0 (0)) at some non-vacuum state. An improved blowup condition for the Euler-Poisson equations (4) was obtained by Cheng and Tadmor [14] in 2009. In this article, we modify the spectral dynamics method to introduce a spectral-dynamicsintegration method for a bounded domain X(t), to obtain the new blowup conditions according to the following theorem. Theorem 1 For the N -dimensional Euler-Poisson equations (1), consider C 2 solutions with a bounded domain: X(t) ≤ V sup , where · denotes the volume and V sup is a positive constant. We define the weighted functional H(t) = X(t) div udµ t ,(5) with the positive measure dµ t = ρ(t, x(t))dx(t). If the initial condition Ω 0ij (x) = 1 2 ∂ i u j (0, x) − ∂ j u i (0, x) = 0 (6) and any one of the following conditions (1) Λ < M/V sup , (2) Λ ≥ M/V sup and H(0) < − − M 3 N Vsup + ΛM 2 N , with the total mass M = X(0) ρ(0, x)dx > 0 are satisfied, the non-trivial C 2 solutions blow up at a finite time T. Here, the functional (5) represents the aggregate density-weighted divergence of the velocity u(t, x). Before we present the novel spectral-dynamics-integration method, we first quote the following lemma: Lemma 2 (Proposition 2.2 on page 27 of [22]) Let S be a material system that fills the domain X(t) at time t, and let C be a function of class C 1 in t and x. Then, d dt X(t) C(t, x)ρ(t, x)dx = X(t) DC(t, x) Dt ρ(t, x)dx,(7) where (D/Dt) = (∂/∂t) + u · ▽ is the convective derivative. In the following proof, we modify the method of spectral dynamics described in [11], [13] and [14] to obtain the different blowup conditions for the C 2 solutions. Proof of Theorem 1. As the mass equation (1) 1 : Dρ Dt + ρ∇ · u = 0,(8) with the convective derivative, D Dt = ∂ ∂t + (u · ∇)(9) could be integrated as: ρ(t, x 0 ) = ρ 0 (x 0 (0, x 0 )) exp − t 0 ∇ · u(t, x 0 (t; x 0 ))dt ≥ 0(10) for ρ 0 (x 0 (0, x 0 )) ≥ 0, the density function ρ(t, x(t; x)) generally conserves its non-negative nature. For the momentum equations (1) 2 and the solutions with non-vacuum, we have u t + u∇ · u = −∇Φ.(11) We take the divergence to the above equation to obtain: ∇ · (u t + u∇ · u) = −∆Φ.(12) If the initial condition Ω 0ij (x) = 1 2 ∂ i u j (0, x) − ∂ j u i (0, x) = 0 is satisfied, we can show by the standard spectral dynamics in [13] and [14] (by directly applying equation (2.6) in [13] or equation (4.1) in [14]) that D Dt div u(t, x(t)) + 1 N [div u(t, x(t))] 2 ≤ −ρ(t, x(t)) + Λ.(13) We notice that the advancement in this article for the new blowup conditions begins here. First, we multiply the density function ρ(t, x(t)) on both sides and take the integration over the domain X(t) to obtain: ρ(t, x(t)) D Dt div u(t, x(t)) + 1 N [div u(t, x(t))] 2 ≤ − [ρ(t, x(t))] 2 + Λρ(t, x(t)).(14)X(t) ρ D Dt div u dx + 1 N X(t) ρ (div u) 2 dx ≤ − X(t) ρ 2 dx + Λ X(t) ρdx (15) X(t) ρ D Dt div u dx + 1 N X(t) ρ (div u) 2 dx ≤ − X(t) ρ 2 dx + ΛM,(16) where M = X(t) ρdx = X(0) ρ(0, x(0))dx(0) > 0 for non-trivial solutions is the total mass of the fluid. We apply Lemma 2 with C(t, x) := div u to obtain d dt X(t) ρ div udx + 1 N X(t) ρ (div u) 2 dx ≤ − X(t) ρ 2 dx + ΛM.(17) We define the weighted functional H := H(t) = X(t) div udµ t(18) with the positive measure dµ t = ρ(t, x(t))dx(t) for ρ(0, x) ≥ 0, with the equation (10) to obtain d dt H ≤ − 1 N X(t) (div u) 2 dµ t − X(t) ρ 2 dx + ΛM.(19) We can estimate the first term on the right-hand side of the inequality (19) to obtain X(t) div udµ t 2 = X(t) div udµ t 2 ≤ X(t) |div u| dµ t 2 ≤ M X(t) (div u) 2 dµ t .(20) Using the Cauchy-Schwarz inequality, we get X(t) |div u| dµ t ≤ X(t) 1 2 dµ t 1/2 X(t) (div u) 2 dµ t 1/2 (21) X(t) |div u| dµ t ≤ X(t) ρdx 1/2 X(t) (div u) 2 dµ t 1/2 = √ M X(t) (div u) 2 dµ t 1/2 . (22) M ≤ X(t) (div u) 2 dµ t (23) − 1 N X(t) (div u) 2 dµ t ≤ −1 M N X(t) div udµ t 2 = −H 2 M N .(24) The second term on the right-hand side of the inequality (19) can be determined by X(t) ρdx ≤ X(t) 1 2 dx 1/2 X(t) ρ 2 dx 1/2 (25) M = X(t) ρdx ≤ X(t) 1/2 X(t) ρ 2 dx 1/2 ≤ (V sup ) 1/2 X(t) ρ 2 dx 1/2 (26) M 2 ≤ V sup X(t) ρ 2 dx (27) − X(t) ρ 2 dx ≤ −M 2 V sup (28) for a bounded domain X(t) ≤ V sup < +∞. Thus, the inequality (19) becomes d dt H ≤ − 1 N X(t) (div u) 2 dµ t − X(t) ρ 2 dx + ΛM ≤ − H 2 M N − M 2 V sup + ΛM (29) d dt H ≤ − H 2 M N − M 2 V sup + ΛM.(30) (1) If Λ < M/V sup , the Riccati inequality (30) can be estimated by d dt H ≤ − M 2 V sup + ΛM < 0.(31) Thus, there exists a finite time T 0 , such that H(T 0 ) < 0.(32) By applying the comparison property, we obtain        d dt H ≤ − H 2 MN − M 2 Vsup + ΛM H(T 0 ) < 0.(33) It is well known that the Riccati inequality (33) blows up at a finite time T . The proof is completed. Remark 3 For the one dimensional case, the condition Ω 0ij (x) = 0 in Theorem 1 is automatically satisfied. The corollary below is immediately shown in Theorem 1. Corollary 4 For Λ = 0, the non-trivial C 2 solutions with the bounded domain X(t) ≤ V sup , of the Euler-Poisson equations (1) in R N , and the initial condition Ω 0ij (x) = 1 2 ∂ i u j (0, x) − ∂ j u i (0, x) = 0,(34) blow up at a finite time T . Remark 5 By further requiring the bounded domain X(t) ≤ V sup and Ω 0ij (x) = 1 2 ∂ i u j (0, x) − ∂ j u i (0, x) = 0,(35) for the Euler-Poisson equations (1), the main achievement of this spectral-dynamics-integration method is to remove the restriction on div u(0, x 0 ) with some point x 0 , for the positive background constant Λ < M/V sup in Cheng and Tadmor's paper [14] for obtaining the blowup phenomenon. H(t) = X(t) div udµ t ,(36) with the positive measure dµ t = ρ(t, x(t))dx(t). If the initial condition Ω 0ij (x) = 1 2 ∂ i u j (0, x) − ∂ j u i (0, x) = 0(37) and any one of the following conditions, (1) Λ < M/V 0 ,(2) Conclusions In this article, we study the life-span problem of self-gravitational fluids with zero pressure (dust solutions) with a cosmological constant Λ and a bounded domain X(t). We apply a new spectral- H(t) = X(t) ρ div udx(38) is initially contracting sufficiently fast. New functional techniques are expected to investigate the possibility of the corresponding blowup phenomena for the Euler-Poisson equations with the pressure term:                ρ t +∇ · (ρu) =0 ρ[u t + (u · ∇)u] + K∇ρ γ = −ρ∇Φ ∆Φ(t, x) =ρ − Λ,(39) with constants K > 0 and γ ≥ 1. Acknowledgement The author thanks the reviewers for their helpful comments for improving the quality of this article. This work is partially supported by the Dean's Research Fund FLASS/ECR-9 of the Hong Kong Institute of Education. ( 2 ) 2If Λ ≥ M/V sup and H(0) < − − M 3 N Vsup + ΛM 2 N , it is also clear that the solution of the Riccati inequality (30) blows up at a finite time T . For the Euler-Poisson equations (1) with free boundaries, it is possible to establish the existence of the solutions outside the bounded domain X(t) ≤ V sup after the "blowup" time T in Theorem 1. Therefore, we have the following corollary:Corollary 6 For the N -dimensional Euler-Poisson equations (1), consider the non-trivial global C 2 solutions with ρ(0, x) and u(0, x), which lie inside a bounded domain: X(0) ≤ V 0 , where · denotes the volume and V 0 is a positive constant. 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{'abstract': 'Pressureless Euler-Poisson equations with attractive forces are standard models in Newtonian cosmology. In this article, we further develop the spectral dynamics method and apply a novel spectral-dynamics-integration method to study the blowup conditions for C 2 solutions with a bounded domain, X(t) ≤ X0, where · denotes the volume and X0 is a positive constant. In particular, we show that if the cosmological constant Λ < M/X0, with the total mass M , then the non-trivial C 2 solutions in R N with the initial condition Ω0ij (x) = 1 2 ∂iu j (0, x) − ∂ju i (0, x) = 0 blow up at a finite time.MSC: 35B30, 35B44, 35Q35, 85A05, 85A40', 'arxivid': '1403.6234', 'author': ['Manwai Yuen rn*e-mailaddress:nevetsyuen@hotmail.com \nDepartment of Mathematics and Information Technology\nThe Hong Kong Institute of Education\n10 Po Ling RoadTai Po\n\nNew Territories\nHong Kong\n'], 'authoraffiliation': ['Department of Mathematics and Information Technology\nThe Hong Kong Institute of Education\n10 Po Ling RoadTai Po', 'New Territories\nHong Kong'], 'corpusid': 119604749, 'doi': '10.1016/j.jmaa.2014.02.004', 'github_urls': [], 'n_tokens_mistral': 6717, 'n_tokens_neox': 5736, 'n_words': 3042, 'pdfsha': '87deddd74bec576c10b25b455e69f274329d534c', 'pdfurls': ['https://arxiv.org/pdf/1403.6234v1.pdf'], 'title': ['Blowup for C 2 Solutions of the N -dimensional Euler-Poisson Equations in Newtonian Cosmology', 'Blowup for C 2 Solutions of the N -dimensional Euler-Poisson Equations in Newtonian Cosmology'], 'venue': []}

arxiv

Universal regularization prescription for Lovelock AdS gravity 23 Nov 2007 February 1, 2008 Georgios Kofinas Department of Physics Institute of Plasma Physics University of Crete 71003HeraklionGreece Rodrigo Olea rodrigo.olea@mi.infn.it INFN Sezione di Milano Via Celoria 16I-20133MilanoItaly Universal regularization prescription for Lovelock AdS gravity 23 Nov 2007 February 1, 2008 A definite form for the boundary term that produces the finiteness of both the conserved quantities and Euclidean action for any Lovelock gravity with AdS asymptotics is presented. This prescription merely tells even from odd bulk dimensions, regardless the particular theory considered, what is valid even for Einstein-Hilbert and Einstein-Gauss-Bonnet AdS gravity. The boundary term is a given polynomial of the boundary extrinsic and intrinsic curvatures (also referred to as Kounterterms series). Only the coupling constant of the boundary term changes accordingly, such that it always preserves a well-posed variational principle for boundary conditions suitable for asymptotically AdS spaces. The background-independent conserved charges associated to asymptotic symmetries are found. In odd bulk dimensions, this regularization produces a generalized formula for the vacuum energy in Lovelock AdS gravity. The standard entropy for asymptotically AdS black holes is recovered directly from the regularization of the Euclidean action, and not only from the first law of thermodynamics associated to the conserved quantities. Introduction It is believed that the Einstein-Hilbert action is just the first term in the derivative expansion in a low energy effective theory. In general, higher order quantum corrections to gravity might appear, whose corresponding couplings are unknown until now. Among the higher derivative gravity theories, Lovelock gravity [1] possesses some special features: it leads to field equations which are up to and linear in second derivatives of the metric, it obeys generalized Bianchi identities which ensure energy conservation, and it is known to be free of ghosts when expanded on a flat space, avoiding problems with unitarity [2]. In presence of cosmological constant, the Euclidean continuation of the bulk gravity action and the conserved quantities are in general divergent. In the AdS/CFT context [3], one deals with the regularization problem for Einstein-Hilbert action by adding local functionals of the boundary metric (Dirichlet counterterms) [4]. Because of this dependence, they preserve a well-defined variational action principle for a Dirichlet boundary condition on the metric (achieved through the Gibbons-Hawking term) when varied. However, the systematic construction [5] that provides the form of the counterterms becomes cumbersome for high enough dimensions, what has prevented from finding a general pattern for the series for any dimension until now. In Lovelock gravity, it is expected that the holographic renormalization procedure would be even more complicated. An alternative regularization scheme has been proposed for Einstein-Hilbert [6,7], Einstein-Gauss-Bonnet [8], and Chern-Simons [9] gravity theories with AdS asymptotics. It considers the addition to the bulk action of boundary terms with dependence on the extrinsic curvature. In this paper, we show that this prescription is universal for all Lovelock-AdS theories, attaining a regularized action and finiteness of the conserved charges . Lovelock gravity In D = d + 1 dimensions, the Lovelock action reads I D = 1 16πG D M [(D−1)/2] p=0 α p L p + c d ∂M B d ,(1) where L p corresponds to the dimensional continuation of the Euler term in 2p dimensions L p = 1 (D − 2p)! ǫ A 1 ...A DR A 1 A 2 ...R A 2p−1 A 2p e A 2p+1 ...e A D ,(2)= 1 2 p √ −G δ [ν 1 ···ν 2p ] [µ 1 ···µ 2p ]R µ 1 µ 2 ν 1 ν 2 · · ·R µ 2p−1 µ 2p ν 2p−1 ν 2p d D x .(3) Hatted curvatures stand for D-dimensional ones. The orthonormal vielbein e A = e A µ dx µ produces the spacetime metric by G µν = η AB e A µ e B ν and the curvature 2-form is defined asR AB = dω AB +ω A C ω CB in terms of the spin connection one-form ω AB = ω AB µ dx µ and related to the spacetime Riemman tensor byR AB = 1 2R κλ µν e A κ e B λ dx µ dx ν . Wedge products are omitted throughout. The first term in the Lovelock series corresponds to the cosmological term The action (1) appears supplemented by a boundary term B d . We shall display below the universal form of B d for any Lovelock theory with AdS asymptotia that regularizes both the conserved quantities and the Euclidean action. L 0 = √ −G d D x, L 1 = √ −GR d D x is the Einstein-Hilbert term, L 2 = √ −G (R µνκλR µνκλ −4R µνR µν +R 2 ) d D x The equation of motion for a generic Lovelock gravity (with zero torsion) is obtained varying with respect to the metric and takes the form E ν µ = [(D−1)/2] p=0 α p 2 p δ [νν 1 ···ν 2p ] [µµ 1 ···µ 2p ]R µ 1 µ 2 ν 1 ν 2 · · ·R µ 2p−1 µ 2p ν 2p−1 ν 2p = 0.(4) The vacua of a given Lovelock theory are defined as the maximally symmetric spacetimes that are globally of constant curvature. We will assume that all the corresponding cosmological constants are real and negative, i.e., Λ eff = − (D−1)(D−2) 2ℓ 2 eff ,(5) where ℓ eff is defined as the effective AdS radius given by the solutions to the equation [(D−1)/2] p=0 α p (D−2p−1)! (−ℓ −2 eff ) p = 0.(6) In the present paper, we will consider spacetimes whose asymptotic behavior tends to the one of a locally AdS space, described in terms of its curvature by the condition R κλ µν + 1 ℓ 2 eff δ [κλ] [µν] = 0 at the boundary ∂M , or equivalently,R AB + (e A e B )/ℓ 2 eff = 0 in differential forms language. It is important to stress that this is a generic (local) condition that does not fix completely the form of the metric. In principle, it is not clear whether the holographic renormalization procedure might provide a systematic algorithm to regularize a generic Lovelock-AdS theory, because of the increasing complexity of the field equations respect to the Einstein-Hilbert case. The alternative construction in this paper represents a way of circumventing the difficulties of the standard method because, as we shall see below, it does not make use of the full expansion of the asymptotic metric. Indeed, we will only consider the leading order for the fields induced by this expansion to identify suitable boundary conditions for the variational problem in AAdS gravity. Without loss of generality, we write down the line element in Gauss-normal coordinates ds 2 = N 2 (ρ)dρ 2 + h ij (x, ρ)dx i dx j ,(8) that can be obtained from a generic radial ADM foliation by gauge-fixing the shift functions N i = 0. A definite choice of the lapse and the boundary metric generically describes AAdS spaces in Lovelock gravity. Indeed, taking the lapse and the boundary metric as N = ℓ eff /2ρ,(9)h ij = g ij (x, ρ)/ρ,(10) where g ij (x, ρ) accepts a regular Fefferman-Graham expansion [10] g ij (x, ρ) = g (0)ij (x) + ρ g (1)ij (x) + ρ 2 g (2)ij (x) + ...(11) identically satisfies the condition (7) at the conformal boundary ρ = 0. Here, g (0) is the boundary data of an initial-value problem, governed by the equations of motion written in the frame (8)-(10). However, even for Einstein-Hilbert theory, solving the coefficients g (k) in series (11) as covariant functionals of g (0) is only possible for low enough dimensions. Moreover, for theories where eq.(6) has a single root, the equations of motion posses a multiple zero in a unique AdS vacuum [11,12]. This causes the first nontrivial relation for a given coefficient g (k) to appear at a higher order in ρ, what substantially increases the complexity of the equations. Therefore, one can expect that the extreme nonlinearity of the field equations in Lovelock-AdS gravity would turn impractical the application of holographic renormalization method to this class of theories. In what follows, we propose a universal form of the boundary terms that make both the conserved charges and the Euclidean action finite in Lovelock-AdS gravity. This construction does not make use of the full Fefferman-Graham form of the metric (8)- (11) for AAdS spacetimes, but simply considers the leading-order terms in the expansion of the relevant fields. D = 2n + 1 dimensions In Einstein-Hilbert-AdS gravity, the standard regularization using Dirichlet counterterms reveals some differences between odd and even-dimensional cases. Indeed, it is only in odd (bulk) dimensions that a vacuum energy for AdS spacetime appears. The quasilocal stress tensor derived from the regularized action features a trace anomaly only in odd dimensions, as well, what can be traced back to a logarithmic contribution in the FG expansion (11). In the alternative regularization known as Kounterterms method, the existence of a vacuum energy for Einstein-Hilbert [7] and Einstein-Gauss-Bonnet [8] AdS gravity in odd dimensions is a consequence of a different form of the boundary terms respect the even-dimensional case. Ultimately, the difference in the prescription for the regularizing boundary terms is linked to the existence of topological invariants of the Euler class whose construction is only possible in even dimensions [13,14]. The standard Dirichlet counterterms consider the addition to the action of local, covariant functional of the boundary metric h ij and the intrinsic curvature R kl ij (h). In the present formulation, the boundary term B d in eq.(1) will depend also on the extrinsic curvature K ij , defined in the frame (8) by K ij = − 1 2N ∂ ρ h ij ,(12) and, because of this dependence, we will refer to it as Kounterterms series. The explicit form the Kounterterms B 2n adopt in any odd-dimensional Lovelock-AdS gravity can be written in a compact way as B 2n = 2n √ −h 1 0 dt t 0 ds δ [i 1 ...i 2n−1 ] [j 1 ...j 2n−1 ] K j 1 i 1 1 2 R j 2 j 3 i 2 i 3 − t 2 K j 2 i 2 K j 3 i 3 + s 2 ℓ 2 eff δ j 2 i 2 δ j 3 i 3 × · · · · · · × 1 2 R j 2n−2 j 2n−1 i 2n−2 i 2n−1 − t 2 K j 2n−2 i 2n−2 K j 2n−1 i 2n−1 + s 2 ℓ 2 eff δ j 2n−2 i 2n−2 δ j 2n−1 i 2n−1 d 2n x,(13) which, when expanded, produces a polynomial in the intrinsic and extrinsic curvatures whose relative coefficients are obtained performing the above parametric integrations B 2n = n! √ −h n−1 p=0 (2n−2p−3)!! ℓ 2(n−1−p) b (p) 2n ,(14) where b (p) 2n = δ [i 1 ···i 2p+1 ] [j 1 ···j 2p+1 ] p q=0 (−1) p−q (p−q)! q! 2 n−(p+q+1) n−q R j 1 j 2 i 1 i 2 · · · R j 2q−1 j 2q i 2q−1 i 2q K j 2q+1 i 2q+1 · · · K j 2p+1 i 2p+1 .(15) The tensorial formula of the boundary terms (13), adapted to a radial foliation of the spacetime, can be cast into a fully Lorentz-covariant 2n−form with the definition of the second fundamental form θ AB = n A K B −n B K A , B 2n = 2n 1 0 dt t 0 ds ǫ a 1 ...a 2n K a 1 e a 2 R a 3 a 4 −t 2 K a 3 K a 4 + s 2 ℓ 2 eff e a 3 e a 4 × · · · · · · × R a 2n−1 a 2n −t 2 K a 2n−1 K a 2n + s 2 ℓ 2 eff e a 2n−1 e a 2n ,(16)= n 1 0 dt t 0 ds ǫ A 1 ...A 2n+1 θ A 1 A 2 e A 3 R A 4 A 5 +t 2 θ A 4 C θ CA 5 + s 2 ℓ 2 eff e A 4 e A 5 × · · · · · · × R A 2n A 2n+1 +t 2 θ A 2n F θ FA 2n+1 + s 2 ℓ 2 eff e A 2n e A 2n+1 ,(17) where the extrinsic curvature K A = K A B e B satisfies K AB = −h C A h D B n C;D , with n A the outward unit normal vector at the boundary. The orthonormal frame takes the block-diagonal form e 1 = N dρ, e a = e a i dx i , such that the only non-vanishing components of θ AB are θ 1a = K a = K i j e a i dx j , and the submanifold Levi-Civita tensor is ǫ a 1 ...a d = ǫ 1a 1 ...a d . R AB is the intrinsic curvature 2-form, that for a radial foliation contains only components on the boundary submanifold. Remarkably, the form of B 2n is preserved regardless the particular theory considered, only the corresponding coupling constant changes accordingly, as shown below. Variational principle and boundary conditions An arbitrary variation of the action produces the equations of motion plus contributions to the surface term that can be traced back to the bulk and boundary terms in (1) δI 2n+1 = M (E.O.M.)+ 1 8πG D ∂M n p=1 pα p (D−2p)! ǫ a 1 ...a 2n δK a 1R a 2 a 3 ...R a 2p−2 a 2p−1 e a 2p ...e a 2n +2nc 2n ∂M 1 0 dt ǫ a 1 ...a 2n δK a 1 e a 2 R a 3 a 4 + t 2 ℓ 2 eff e a 3 e a 4 ... R a 2n−1 a 2n + t 2 ℓ 2 eff e a 2n−1 e a 2n −2nc 2n ∂M 1 0 dt t ǫ a 1 ...a 2n (δK a 1 e a 2 −K a 1 δe a 2 ) R a 3 a 4 −t 2 K a 3 K a 4 + t 2 ℓ 2 eff e a 3 e a 4 × · · · · · · × R a 2n−1 a 2n −t 2 K a 2n−1 K a 2n + t 2 ℓ 2 eff e a 2n−1 e a 2n .(18) Here, we have extensively used the Gauss-Coddazzi relation for the boundary components of the Riemann 2-formR ab = R ab −K a K b ,(19) that in the standard tensorial notation readŝ R kl ij = R kl ij − K k i K l j + K k j K l i .(20) A well-defined action principle for Lovelock-AdS gravity amounts to the on-shell cancellation of the surface term in eq.(18) by imposing suitable boundary conditions, that either are derived from, or at least, are compatible with the asymptotic behavior of the metric (8)- (11). For the extrinsic curvature, the FG expansion produces K i j = h ik K kj = 1 ℓ eff δ i j − ρ ℓ eff (g −1 (0) g (1) ) i j − ρ 2 ℓ eff (2g −1 (0) g (2) −g −1 (0) g (1) g −1 (0) g (1) ) i j + ... ,(21) where the indices at the r.h.s. of the above equation are raised with the conformal structure g ij (0) . Then, the extrinsic curvature on the boundary is finite K i j = 1 ℓ eff δ i j .(22) In any gravity theory, h ij and K ij are independent variables, because the extrinsic curvature defines the canonical momentum π ij . The fact that the extrinsic curvature can be written in terms of the coefficients g (k) in the expansion of the metric does not mean that it is determined only by the metric g (0) . Indeed, as it is well-known, not even h ij is completely determined by solving the second-order field equations with only g (0) as the initial data (Fefferman-Graham ambiguity for the coefficient g (n) with n = [D/2]). Then, K ij remains as an independent variable even though the first terms in the expansion (21) are fixed by g (0) . For the variational problem in odd dimensions, we will consider that at the boundary the variations obey δK i j = 0,(23) that is a regular boundary condition compatible with fixing the conformal metric g (0) on ∂M [15,7]. Therefore, this boundary condition does not spoil the AdS/CFT interpretation of the conformal structure g (0) as a given data for the holographic reconstruction of the spacetime in the gravity side, and whose dual CFT on the boundary does not have gravitational degrees of freedom. The last line in (18) is identically canceled by the conditions (22), (23). The asymptotic behavior (7) for the curvature determines the coupling constant c 2n c 2n = 1 16πnG D 1 0 dt (t 2 −1) n−1 −1 n p=1 (−1) p p α p (D−2p)! ℓ 2(n−p) eff ,(24) in order to cancel the rest of the surface term (18). In the standard Dirichlet regularization for AdS gravity, fixing the conformal structure g (0)ij in the boundary metric (10), (11) will require the addition of counterterms to cancel the divergence at the boundary ρ = 0 [16]. In our case, we select the boundary conditions (7), (22) and (23), which are regular on the asymptotic region, such that the regularization process is encoded in the boundary terms already present and there is no need of further addition of counterterms. Conserved quantities and vacuum energy. The Noether theorem applied to Lovelock-AdS gravity states that there is a set of conserved charges Q(ξ) associated to asymptotic Killing vectors ξ, that are defined as (D−2)−forms, and therefore, are integrated on the boundary of a spatial section at constant time. More precisely, we take a timelike ADM foliation for the line element at the boundary h ij dx i dx j = −Ñ 2 (t) dt 2 +σ m n (dϕ m +Ñ m dt)(dϕ n +Ñ n dt) ,(25) with the coordinates x i = (t, ϕ m ), that is defined by the unit normal vector u i = (−Ñ , 0). The charges are then given as the integration on the boundary Σ of a spatial section, parameterized by ϕ m Q(ξ) = Σ d D−2 ϕ √ σ u j Q j i ξ i .(26) In the above formula, σ denotes the determinant of the metric σ m n , related to h by √ −h =Ñ √ σ, and ξ i is an asymptotic Killing vector. In odd dimensions, the expression for the integrand appears naturally split in two pieces Q j i = q j i + q j (0)i ,(27) with q j i = 1 2 n−2 δ [jj 2 ...j 2n ] [i 1 i 2 ...i 2n ] K i 1 i δ i 2 j 2   1 16πG D n p=1 pα p (D−2p)!R i 3 i 4 j 3 j 4 ...R i 2p−1 i 2p j 2p−1 j 2p δ [i 2p+1 i 2p+2 ] [j 2p+1 j 2p+2 ] ...δ [i 2n−1 i 2n ] [j 2n−1 j 2n ] +nc 2n 1 0 dt R i 3 i 4 j 3 j 4 + t 2 ℓ 2 eff δ [i 3 i 4 ] [j 3 j 4 ] ... R i 2n−1 i 2n j 2n−1 j 2n + t 2 ℓ 2 eff δ [i 2n−1 i 2n ] [j 2n−1 j 2n ] ,(28)q j (0)i = − nc 2n 2 n−2 1 0 dt t δ [jj 2 ...j 2n ] [i 1 i 2 ...i 2n ] (δ i 2 j 2 K i 1 i +δ i 2 i K i 1 j 2 ) R i 3 i 4 j 3 j 4 −t 2 K [i 3 i 4 ] [j 3 j 4 ] + t 2 ℓ 2 eff δ [i 3 i 4 ] [j 3 j 4 ] × · · · · · · × R i 2n−1 i 2n j 2n−1 j 2n −t 2 K [i 2n−1 i 2n ] [j 2n−1 j 2n ] + t 2 ℓ 2 eff δ [i 2n−1 i 2n ] [j 2n−1 j 2n ] ,(29) where we have used the shorthand K (27) defines a splitting of the charges [ik] [jl] = K i j K k l − K i l K k j . EquationQ(ξ) = q(ξ) + q 0 (ξ) ,(30) where q(ξ) = Σ d D−2 ϕ √ σ u j q j i ξ i(31) will provide the mass and angular momentum for AAdS black hole solutions in Lovelock gravity. It can be shown that eq.(28) can be factorized in any odd dimension as q j i = 1 2 n−2 δ [jj 2 ...j 2n ] [i 1 i 2 ...i 2n ] K i 1 i δ i 2 j 2 R i 3 i 4 j 3 j 4 + 1 ℓ 2 eff δ [i 3 i 4 ] [j 3 j 4 ] P i 5 ...i 2n j 5 ...j 2n ,(32) where P is a Lovelock-type polynomial of (n − 2)−degree in the Riemann tensorR ij kl and the antisymmetrized Kronecker delta δ [ij] [kl] P i 5 ...i 2n j 5 ...j 2n = n−2 p=0 nc 2n D p ℓ 2p eff + F p 16πG D R i 5 i 6 j 5 j 6 ...R i 2(n−p)−1 i 2(n−p) j 2(n−p)−1 j 2(n−p) δ [i 2(n−p)+1 i 2(n−p+1) ] [j 2(n−p)+1 j 2(n−p+1) ] ...δ [i 2n−1 i 2n ] [j 2n−1 j 2n ] ,(33) with the coefficients of the expansion given by D p = p q=0 (−1) p−q 2q+1 n−1 q , F p = p q=0 (−1) p−q (n−q) α n−q (2q+1)! ℓ 2(p−q) eff .(34) As the tensorial combinationR kl ij + 1 ℓ 2 ef f δ [kl] [ij] is a part of the curvature of the AdS group with an effective radius ℓ eff , the factorization (32) implies that the charge q(ξ) vanishes identically for global AdS spacetime. Note that for Einstein-Hilbert-AdS gravity, F p = 0 and the above expression for q(ξ) recovers the corresponding charge in [7]. As a consequence, the quantity q 0 (ξ) q 0 (ξ) = Σ d D−2 ϕ √ σ u j q j (0)i ξ i ,(35) is truly a tensorial formula for the vacuum energy for AAdS spacetimes in Lovelock gravity, inexistent in previous literature. A static black hole solution for Lovelock gravity (1) for both odd and even dimensions D is given by the metric ds 2 = −∆ 2 (r)dt 2 + dr 2 ∆ 2 (r) + r 2 γ m n dϕ m dϕ n ,(36) with ∆(r) given by [(D−1)/2] p=1 α p (D−2p−1)! k−∆ 2 r 2 p = 2Λ (D−1)! + µ (D−3)! r D−1 ,(37) where µ appears as an integration constant in the first integral of the rr component of the Lovelock equations of motion (4) (∆ 2 ) ′ r [(D−1)/2] p=1 pα p (D−2p−1)! k−∆ 2 r 2 p−1 − [(D−2)/2] p=1 α p (D−2p−2)! k−∆ 2 r 2 p = − 2Λ (D−2)! ,(38) and the prime stands for the derivative with respect to r coordinate. The metric γ m n (m, n = 1, ..., D − 2) defines the line element of the transversal section Σ k D−2 whose curvature is a constant k = ±1, 0. Black hole solution (36) possesses an event horizon r + , which is the largest root of the equation ∆ 2 (r + ) = 0. For this configuration, the only nonvanishing components of the extrinsic curvature are K t t = −∆ ′ , K n m = − ∆ r δ n m ,(39) whereas the intrinsic curvature is R m 1 n 1 m 2 n 2 = k r 2 δ [m 1 n 1 ] [m 2 n 2 ] ,(40) which, in turn, produces the boundary components of the Riemann tensor to bê R tn tm = − (∆ 2 ) ′ 2r δ n m ,R m 1 n 1 m 2 n 2 = k−∆ 2 r 2 δ [m 1 n 1 ] [m 2 n 2 ] .(41) Differentiating eq.(37) with respect to the horizon radius r + and combining eqs. (37), (38), we obtain the relation ∂µ ∂r + = (D−3)! (∆ 2 ) ′ | r + [(D−1)/2] p=1 pα p (D−2p−1)! r D−2p−1 + k p−1 .(42) From the equation (37) that dictates the form of the function ∆ 2 (r) in the metric, we can obtain the asymptotic behavior (r → ∞) ∆ 2 (r) ≈ k + r 2 ℓ 2 eff − µ (D−3)!   [(D−1)/2] p=1 pα p (−ℓ −2 eff ) p−1 (D−2p−1)!   −1 1 r D−3 + ...(43) The corresponding mass is given by the evaluation of eq.(31) for the timelike Killing vector ξ = ∂ t q(∂ t ) = M = (D−2)! vol(Σ k D−2 )(∆ 2 ) ′ 1 16πG D n p=1 pα p (D−2p)! r D−2p (k−∆ 2 ) p−1 +nc 2n r 1 0 dt k−∆ 2 + t 2 r 2 ℓ 2 eff n−1 ∞ . (44) Using the asymptotic form of ∆ 2 (r) from (43), we see that the divergent terms O(r D−1 ) in the evaluation of the above formula exactly cancel out and one gets the finite result M = (D−2) vol(Σ k D−2 ) 16πG D µ.(45) The zero-point (vacuum) energy is then given by (35) as q 0 (∂ t ) = E 0 = 2nc 2n (D−2)! vol(Σ k D−2 ) ∆ 2 − r(∆ 2 ) ′ 2 1 0 dt t k−t 2 ∆ 2 + t 2 r 2 ℓ 2 eff n−1 ∞ ,(46) that can be worked out using the asymptotic form (43), giving the finite result E 0 = (−k) n vol(Σ k D−2 ) 16πnG D (2n−1)!! 2 n p=1 (−1) p−1 pα p (D−2p)! ℓ 2n−2p eff .(47) Black Hole Entropy. The Euclidean period β is defined as the inverse of black hole temperature T such that in the Euclidean sector the solution (36) does not have a conical singularity at the horizon. In doing so, one obtains β = 4π/(∆ 2 ) ′ | r + . The black hole entropy S is defined in the canonical ensemble (the surface gravity is kept fixed at the horizon) as S = I E + βE,(48) in terms of the total Euclidean action I E and the thermodynamical energy E = − ∂I E ∂β(49) of the black hole. The Euclidean bulk action is evaluated for a static black hole of the form (36) as I E bulk = − (D−2)! 16πG D vol(Σ k D−2 ) β n p=1 pα p (D−2p)! [r D−2p (∆ 2 ) ′ (k−∆ 2 ) p−1 ]| ∞ r + ,(50) and it is rendered finite by the addition of the suitable boundary term (13), whose evaluation in the Euclidean solution is ∂M B E 2n = −n(D−2)! vol(Σ k D−2 )β r(∆ 2 ) ′ 1 0 dt k−∆ 2 + t 2 r 2 ℓ 2 eff n−1 +2 ∆ 2 − r(∆ 2 ) ′ 2 1 0 dt t k−t 2 ∆ 2 + t 2 r 2 ℓ 2 eff n−1 ∞ . (51) Therefore, the total action contains two pieces. At r = ∞, the contribution from the bulk action I E bulk combines with the boundary term c 2n ∂M B E 2n to produce −β times the Noether charge Q(∂ t ) = M +E 0 I E 2n+1 = (D−2)! 16πG D vol(Σ k D−2 ) 4π n p=1 pα p (D−2p)! r D−2p + k p−1 − βµ (D−3)! −β(−k) n (2n−1)!! 2 n(D−2)! n p=1 (−1) p−1 pα p (D−2p)! ℓ 2n−2p eff . (52) This identification guarantees that all the divergencies at radial infinity are exactly canceled. The definition of thermodynamic energy, using equation (42), gives E = − ∂I E 2n+1 /∂r + ∂β/∂r + = M + E 0 ,(53) which recovers the same total energy as from the Noether charge Q(∂ t ) of (30). As a consequence, the entropy (48) is simply given by the Noether charge evaluated at the horizon S = (D−2)! 4G D vol(Σ k D−2 ) n p=1 pα p (D−2p)! r D−2p + k p−1 .(54) D = 2n dimensions For even dimensions, an alternative regularization procedure was developed originally for Einstein-Hilbert-AdS action in [6] and applied to the same problem in AAdS gravity in Einstein-Gauss-Bonnet theory [8]. As we shall explicitly show below, the universal form of the boundary term that renders the conserved charges and Euclidean action finite in Lovelock-AdS gravity in D = 2n corresponds to the (maximal) n−th Chern form [17,18,19] B 2n−1 = n 1 0 dt ǫ A 1 ...A 2n θ A 1 A 2 (R A 3 A 4 +t 2 θ A 3 C θ CA 4 )...(R A 2n−1 A 2n +t 2 θ A 2n−1 F θ FA 2n ).(55) Eq.(55) can be projected to the boundary indices to work out its equivalence in tensorial notation B 2n−1 =2n 1 0 dt ǫ a 1 ...a 2n−1 K a 1 (R a 2 a 3 −t 2 K a 2 K a 3 )...(R a 2n−2 a 2n−1 −t 2 K a 2n−2 K a 2n−1 ) (56) = 2n √ −h 1 0 dtδ [j 1 ...j 2n−1 ] [i 1 ...i 2n−1 ] K i 1 j 1 1 2 R i 2 i 3 j 2 j 3 −t 2 K i 2 j 2 K i 3 j 3 ... 1 2 R i 2n−2 i 2n−1 j 2n−2 j 2n−1 −t 2 K i 2n−2 j 2n−2 K i 2n−1 j 2n−1 d 2n−1 x.(57) In the last formula, the parametric integration reflects the action of the Cartan homotopy operator, used to obtain the correction to the Euler characteristic due to the introduction of a boundary in the Euler theorem. The integral in t is a convenient shorthand, but it also generates the suitable coefficients in the binomial expansion B 2n−1 = 2n √ −h n−1 p=0 (−1) n−p−1 2 p (2n−2p−1) b (p) 2n−1 ,(58) where b (p) 2n−1 = δ [i 1 ···i 2p ···i 2n−1 ] [j 1 ···j 2p ···j 2n−1 ] R j 1 j 2 i 1 i 2 · · · R j 2p−1 j 2p i 2p−1 i 2p K j 2p+1 i 2p+1 · · · K j 2n−1 i 2n−1 .(59) The surface term coming from an arbitrary on-shell variation of the action (1) adopts a slightly simpler form than in the odd-dimensional case δI 2n = ∂M 1 8πG D n−1 p=1 pα p (D−2p)! ǫ a 1 ...a 2n−1 δK a 1R a 2 a 3 ...R a 2p−2 a 2p−1 e a 2p . ..e a 2n−1 + +2nc 2n−1 ǫ a 1 ...a 2n−1 δK a 1R a 2 a 3 ...R a 2n−2 a 2n−1 . An appropriate choice of the coupling constant c 2n−1 as c 2n−1 = − 1 16πnG D n−1 p=1 pα p (D−2p)! (−ℓ 2 eff ) n−p .(61) makes the above expression vanish identically for AAdS spacetimes (7). The regularity of the asymptotic condition (7) implies that the well-defined action principle achieved in this way is also a finite one, because no additional divergences are induced by the addition of the Kounterterms (57). Conserved Charges In Einstein-Hilbert and Einstein-Gauss-Bonnet with negative cosmological constant, we have seen that the addition of boundary terms with explicit dependence on the extrinsic curvature K ij solve at once two problems that in general are not necessarily related: the variational principle and the finiteness of the Noether charges and Euclidean action. Whenever the action is stationary for boundary conditions compatible with the asymptotic structure of AAdS spacetimes, the theory does not require a further regularization on top of the addition of B d in eq.(1). The conserved charges constructed using the Noether theorem have the form Q(ξ) = Σ d D−2 ϕ √ σ u j Q j i ξ i ,(62) with the integrand given by Q j i = 1 2 n−2 δ [jj 2 ...j 2n−1 ] [i 1 i 2 ...i 2n−1 ] K i 1 i   1 16πG D n−1 p=1 pα p (D−2p)!R i 2 i 3 j 2 j 3 ...R i 2p−2 i 2p−1 j 2p−2 j 2p−1 δ [i 2p i 2p+1 ] [j 2p j 2p+1 ] ...δ [i 2n−2 i 2n−1 ] [j 2n−2 j 2n−1 ] +nc 2n−1R i 2 i 3 j 2 j 3 ...R i 2n−2 i 2n−1 j 2n−2 j 2n−1   . (63) The mass for Lovelock-AdS black holes (36), (37) comes from the above formula for the Killing vector ξ = ∂ t , that is Q(∂ t ) = M = (D−2)! vol(Σ k D−2 )(∆ 2 ) ′ 1 16πG D n−1 p=1 pα p (D−2p)! r D−2p (k−∆ 2 ) p−1 + nc 2n−1 (k−∆ 2 ) n−1 ∞ . (64) As in the odd-dimensional case, taking the asymptotic expansion of the functions involved shows that the divergences at order r D−1 coming both from the bulk and boundary parts of the action are exactly canceled. Thus, we obtain M = (D−2)vol(Σ k D−2 ) 16πG D µ.(65) Black Hole Entropy The Euclidean bulk action I E bulk is still given by the even-dimensional equivalence of equation (50) I E bulk = − (D−2)! 16πG D vol(Σ k D−2 ) β n−1 p=1 pα p (D−2p)! [r D−2p (∆ 2 ) ′ (k−∆ 2 ) p−1 ]| ∞ r + , while the Euclidean continuation of the boundary term takes the form ∂M B E 2n−1 = −n(D−2)! vol(Σ k D−2 ) β (∆ 2 ) ′ (k−∆ 2 ) n−1 ∞ .(66) In the total Euclidean action in even dimensions evaluated for a black hole (36), (37) I E 2n = I E bulk + c 2n−1 ∂M B E 2n−1 ,(67) the term at infinity corresponds to −βM , where M is the Noether mass in eqs. (64), (65), that is I E 2n = (D−2)! 16πG D vol(Σ k D−2 ) 4π n−1 p=1 pα p (D−2p)! r D−2p + k p−1 − βµ (D−3)! .(68) The consistency between the regularization procedure and the thermodynamic ensemble is corroborated by the fact that the thermodynamic energy E = − ∂I E 2n ∂β = M,(69) reobtains the corresponding Noether charge. Finally, the black hole entropy is expressed in terms of the horizon r + in a similar form as eq.(54) for the odd-dimensional case S = (D−2)! 4G D vol(Σ k D−2 ) n−1 p=1 pα p (D−2p)! r D−2p + k p−1 .(70) Particular Cases Einstein-Gauss-Bonnet-AdS Gravity. In this case, all the coefficients in the Lovelock series are vanishing but α 0 = −2Λ, α 1 = 1 and α 2 = α, where α is an arbitrary positive coupling constant. The effective AdS radius is modified by the Gauss-Bonnet coupling as 1 ℓ 2 eff = 1 ± 1 − 4(D−3)(D−4)α/ℓ 2 2(D−3)(D−4)α ,(71) such that the solutions tend asymptotically to a constant curvature spacetime with that radius. The Noether charge in the corresponding odd and even dimensions, evaluated for a timelike Killing vector ξ = ∂ t for Boulware-Deser black holes ∆ 2 (r) = k + r 2 2(D−3)(D− 4)α 1 ± 1 − 4(D−3)(D−4)α ℓ 2 + 4(D−3)(D−4)α µ r D−1 ,(72) recovers the mass obtained by background-dependent methods [20,21,22,23] M = (D−2) vol(Σ k D−2 ) 16πG D µ.(73) However, background-independent methods are the only ones that can detect the presence of a vacuum energy for Einstein-Gauss-Bonnet theory. The Dirichlet regularization for arbitrary couplings of quadratic curvature terms, and therefore, useful to treat the EGB action, is only known in five dimensions [24] and, for the Gauss-Bonnet case, it has shown to be ambiguous [25]. The procedure carried out here reproduces, by direct replacement of the corresponding Lovelock coefficients {α 0 , α 1 , α 2 } in eq.(47), the general formula for the vacuum energy for EGB-AdS theory E 0 = (−k) n vol(Σ k D−2 ) 8πG D ℓ 2n−2 eff (2n−1)!! 2 (2n)! 1 − 2α ℓ 2 eff (D−2)(D−3) ,(74) that was first computed in [8] using Kounterterms regularization. The form of the boundary terms that makes possible this result for EGB-AdS gravity shall be shown to be universal below because it also provides finite expressions for the conserved quantities of AAdS solutions in Lovelock gravity. The existence of a vacuum energy does not modify the black hole entropy because as the total energy E = M + E 0 is shifted by a constant with respect to the mass calculated with backgrounddependent methods, the Euclidean action changes in a consistent manner. As a consequence, the entropy of the system can be written as S = vol(Σ k D−2 ) r D−2 + 4G D 1 + 2kα(D−2)(D−3) r 2 + ,(75) in both odd and even dimensions. This formula have been found by several authors [26,27,28], where some of the conserved quantities, including the entropy function have been computed assuming that they satisfy the First Law of black hole thermodynamics. The same result can be derived from the regularized Euclidean action as the free energy, obtained as difference between the Euclidean bulk action evaluated for a EGB-AdS black hole and AdS vacuum [29,30,31] (for a similar backgroundsubstraction computation in string generated gravity with quadratic curvature couplings, see [32]). Dimensionally Continued Gravity. If one considers that the equation of motion for Lovelock gravity (4) posseses m different vacuum (constant curvature) solutions, this means that α p = 0 for p > m, while α m = 0 for 1 ≤ m ≤ D−1 2 . Then, eq.(4) can also be written in the form E ν µ = δ [νν 1 ···ν 2m ] [µµ 1 ···µ 2m ] R µ 1 µ 2 ν 1 ν 2 + γ 1 δ [µ 1 µ 2 ] [ν 1 ν 2 ] · · · R µ 2m−1 µ 2m ν 2m−1 ν 2m + γ m δ [µ 2m−1 µ 2m ] [ν 2m−1 ν 2m ] = 0,(76) where α m−p = α m (D−2m+2p−1)! (D−2m−1)! m i 1 <...<ip=1 γ i 1 . . . γ ip , 1 ≤ p ≤ m.(77) The relation (77) defines an algebraic system of m equations for m unknows γ 1 , ..., γ m . In the particular case where γ 1 = ... = γ m = 1 ℓ 2 ef f , the above equation produces for the couplings α p the special values α p = (D−2p−1)! (D−3)! m ℓ 2p−2 eff m p , 0 ≤ p ≤ m.(78) In the conventions we have adopted in this paper (α 0 = −2Λ), we find that the effective AdS radius is ℓ 2 eff = ℓ 2 /m, whereas equation (6) becomes an identity. The label m takes the maximal value (m = [ D−1 2 ]) for two particular Lovelock theories that feature a symmetry enhancement from Lorentz to AdS group: Chern-Simons-AdS (CS-AdS) and Born-Infeld-AdS (BI-AdS) in odd and even dimensions, respectively. Both theories posses a single cosmological constant and the maximal number of curvatures for a given dimension. Static black hole solutions for CS-AdS and BI-AdS theories were studied in [11]. CS-AdS gravity is obtained from a Chern-Simons density for the AdS group in D = 2n + 1, such that the corresponding coefficients (78) in eq.(1) are given by α (CS) p = (D−2p−1)! (D−3)! n ℓ 2p−2 eff n p , 0 ≤ p ≤ n,(79) which produce equations of motion where AdS vacuum is a zero of n−th order. Topological static black holes were studied in [33]. The horizon radius is defined by the relation (37) µ = 1 nℓ 2 ef f (r 2 + +kℓ 2 eff ) n , such that the formula (45) gives M (CS) = (D−2) vol(Σ k D−2 ) 16πnG D ℓ 2 eff (r 2 + +kℓ 2 eff ) n ,(80) whereas the vacuum energy (47) reduces to the form E (CS) 0 = −k n (D−2) vol(Σ k D−2 ) 16πnG D ℓ 2(n−1) eff .(81) The last expression corresponds to the energy of global AdS spacetime. In CS-AdS gravity, the AdS vacuum is separated from black holes (M > 0) by a mass gap of naked singularities with mass in the interval M = (E 0 , 0), as in (2 + 1) dimensions. Eq.(80) is the standard result for the mass, found in Hamiltonian form in [11,33]. The vacuum energy was obtained as a Noether charge evaluated in AdS in the background-independent formulation presented in [9], using a boundary term proportional to (13). It is remarkable that the symmetry enhancement in this case, turns the Dirichlet counterterms series exactly solvable from the divergent parts in the expansion of the canonical variation of the action [34], and this allows an explicit comparison between the Kounterterms procedure and the Dirichlet regularization [35]. As usual in Lovelock gravity, black hole entropy in CS-AdS theory cannot be related to the horizon area, but just expressed from eq.(54) in terms of r + as S (CS) = (D−2) vol(Σ k D−2 ) 4G D r + 0 dr (r 2 +kℓ 2 eff ) n−1 .(82) The last result matches the one obtained using a mini-superspace model in the canonical formalism [11,33], and also the prescription for the entropy as a given (D −2)−form integrated at the horizon [27]. For BI-AdS gravity in D = 2n dimensions, the couplings (78) become α (BI) p = (D−2p)! (D−2)! n ℓ 2p−2 eff n p , 0 ≤ p ≤ n−1.(83) BI-AdS gravity can naturally incorporate into the bulk piece of the action (1) the (topological) Euler term E 2n = ǫ A 1 ...A DR A 1 A 2 ...R A 2n−1 A 2n with an appropriate weight factor α (BI) n arising from (83) for p = n. As the Euler term is locally equivalent to the boundary term (57), the complete action (1) is also written as I (BI) D = 1 16πnG D ℓ 2n−2 eff (D−2)! 2 n M d 2n x √ −G δ [µ 1 ···µ 2n ] [ν 1 ···ν 2n ] R ν 1 ν 2 µ 1 µ 2 + 1 ℓ 2 eff δ [ν 1 ν 2 ] [µ 1 µ 2 ] · · · R ν 2n−1 ν 2n µ 2n−1 µ 2n + 1 ℓ 2 eff δ [ν 2n−1 ν 2n ] [µ 2n−1 µ 2n ] ,(84) which is both invariant under AdS group and regularized by construction. Again, the Kounterterms procedure provides a finite answer for the mass from eq.(65) M (BI) = vol(Σ k D−2 ) 8πG D ℓ 2 eff r + (r 2 + +kℓ 2 eff ) n−1 ,(85) that has been obtained in Hamiltonian way, but also in a background-independent method using the regularizing effect given by the inclusion of the Euler term [14]. The static black hole entropy in BI-AdS gravity is found by plugging the coefficients (83) into the formula (70) S (BI) = vol(Σ k D−2 ) 4G D [(r 2 + +kℓ 2 eff ) n−1 −(kℓ 2 eff ) n−1 ].(86) The issue of the entropy for BI-AdS black holes is more subtle than the regularization of the Noether charges. If, instead, one uses the Euler term E 2n to render the Euclidean action finite as in (84), the entropy found will be shifted by the opposite of the last term proportional to k n−1 in (86), which is related to the Euler characteristic χ 2n of the manifold. For solutions with hyperbolic horizon (k = −1), that entropy could become negative for physically reasonable black holes (r + < ℓ eff ), as noticed for Einstein-Hilbert in [6]. However, in our approach, the problem is circumvented by using the Chern form (57), what provides the consistent regularization prescription and the correct entropy in all cases of even-dimensional Lovelock gravity. Lovelock Unique Vacuum. Extending the idea of a single cosmological constant of Dimensionally Continued AdS Gravity, one can adjust the coefficients of Lovelock series to attain a family of inequivalent gravity theories that posses a unique AdS vacuum [12]. The choice (78) produces equations of motion where global AdS (maximally symmetric) spacetime is a zero of m−th order. The mass of asymptotically AdS static black holes was computed using Hamiltonian formalism and AdS as the natural background reference for the energy. Here, we use the background-independent formulas (45) and (65), to obtain the mass M (LU V ) = (D−2) vol(Σ k D−2 ) 16πmG D ℓ 2 eff r D−2m−1 + (r 2 + +kℓ 2 eff ) m .(87) In odd dimensions, the vacuum energy can be calculated directly from eq.(47), and it turns to be E (LU V ) 0 = (−k) n vol(Σ k D−2 ) 16πnG D (2n−1)!! 2 (D−3)! ℓ 2(n−1) eff 1 0 du u D−2m−1 (u 2 −1) m−1 .(88) The corresponding entropy can be computed from (54), (70), once the Euclidean action has been regularized by the addition of the Kounterterms series, and takes the explicit form In [27], the above expression was obtained from the direct application of Wald's prescription [36] for Lovelock Unique Vacuum gravity. The same results can be reproduced using identities derived from the gravitational bulk Lagrangian in [28]. Despite the fact that these approaches lead to the correct formula for the entropy in all cases, they deal only with local properties of the action at the horizon and it do not really provide an answer to the problem of bulk action regularization for the asymptotic region in AdS gravity. It is worthwhile noticing that this set of theories is not free from the inconsistencies produced by negative values of the entropy (89) when the spatial section has negative curvature. In that sense, Lovelock Unique Vacuum does not feature a more sensible thermodynamic behavior than, e.g., Einstein-Gauss-Bonnet with AdS asymptotics. The existence of different values for the vacuum energy (88) for a given odd dimension suggests that the set of gravity theories ranging between EH and CS should have a set of inequivalent CFT duals. This is also clear from the information coming from the Weyl anomaly. On the contrary to EH-AdS, in (2n + 1)-dimensional CS-AdS gravity the holographic Weyl anomaly is proportional only to the Euler term in 2n-dimensions (type A anomaly) with no contributions from the Weyl tensor (type B anomaly) [34]. Then, odd-dimensional gravity theories with 1 < m < n should posses a combination of both types of holographic anomaly. As this information is usually extracted from the finite part of a quasilocal stress tensor for AdS gravity, the present regularization prescription for all Lovelock theories can be regarded as a step ahead towards a general formula for the holographic anomaly in Lovelock-AdS gravity. Conclusions In this paper we have provided the explicit form of the boundary terms that regularize the conserved quantities for asymptotically AdS solutions of Lovelock gravity. The prescription for the boundary terms contains the extrinsic curvature and it only distinguishes even from odd dimensions, independently of the particular model under consideration. Just the weight factor of these terms needs to be consistently tuned in order to have a well-posed variational principle for AAdS spacetimes. At the same time, the finiteness of the Euclidean action is achieved. In all the known cases (Einstein-Gauss-Bonnet, Chern-Simons, Born-Infeld, Lovelock Unique Vacuum) both conserved charges and black hole thermodynamics agree with the standard results. Even if the Noether charges assign a non-vanishing vacuum energy to AdS in odd dimensions (which is unobservable in background-dependent methods), the entropy expression is still the correct one, because the Euclidean action appears shifted consistently. In even dimensions the boundary prescription is given by the maximal Chern form. 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{'abstract': 'A definite form for the boundary term that produces the finiteness of both the conserved quantities and Euclidean action for any Lovelock gravity with AdS asymptotics is presented. This prescription merely tells even from odd bulk dimensions, regardless the particular theory considered, what is valid even for Einstein-Hilbert and Einstein-Gauss-Bonnet AdS gravity. The boundary term is a given polynomial of the boundary extrinsic and intrinsic curvatures (also referred to as Kounterterms series). Only the coupling constant of the boundary term changes accordingly, such that it always preserves a well-posed variational principle for boundary conditions suitable for asymptotically AdS spaces. The background-independent conserved charges associated to asymptotic symmetries are found. In odd bulk dimensions, this regularization produces a generalized formula for the vacuum energy in Lovelock AdS gravity. The standard entropy for asymptotically AdS black holes is recovered directly from the regularization of the Euclidean action, and not only from the first law of thermodynamics associated to the conserved quantities.', 'arxivid': '0708.0782', 'author': ['Georgios Kofinas \nDepartment of Physics\nInstitute of Plasma Physics\nUniversity of Crete\n71003HeraklionGreece\n', 'Rodrigo Olea rodrigo.olea@mi.infn.it \nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly\n'], 'authoraffiliation': ['Department of Physics\nInstitute of Plasma Physics\nUniversity of Crete\n71003HeraklionGreece', 'INFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly'], 'corpusid': 16767200, 'doi': '10.1088/1126-6708/2007/11/069', 'github_urls': [], 'n_tokens_mistral': 20011, 'n_tokens_neox': 16771, 'n_words': 9336, 'pdfsha': 'fe5144297988ec0ad9fffd32eec25159c1007839', 'pdfurls': ['https://arxiv.org/pdf/0708.0782v2.pdf'], 'title': ['Universal regularization prescription for Lovelock AdS gravity', 'Universal regularization prescription for Lovelock AdS gravity'], 'venue': []}

arxiv

Deep Reinforcement Learning with Importance Weighted A3C for QoE enhancement in Video Delivery Services Mandan Naresh Computer Science & Information Systems Birla Institute of Technology and Science Pilani Hyderabad India Paresh Saxena psaxena@hyderabad.bits-pilani.ac.in Computer Science & Information Systems Birla Institute of Technology and Science Pilani Hyderabad India Manik Gupta Computer Science & Information Systems Birla Institute of Technology and Science Pilani Hyderabad India Deep Reinforcement Learning with Importance Weighted A3C for QoE enhancement in Video Delivery Services Index Terms-Deep Reinforcement LearningVideo DeliveryQuality of Experience (QoE)Adaptive Bit Rates (ABR)Actor- critic methods Adaptive bitrate (ABR) algorithms are used to adapt the video bitrate based on the network conditions to improve the overall video quality of experience (QoE). Recently, reinforcement learning (RL) and asynchronous advantage actor-critic (A3C) methods have been used to generate adaptive bit rate algorithms and they have been shown to improve the overall QoE as compared to fixed rule ABR algorithms. However, a common issue in the A3C methods is the lag between behaviour policy and target policy. As a result, the behaviour and the target policies are no longer synchronized which results in suboptimal updates. In this work, we present ALISA: An Actor-Learner Architecture with Importance Sampling for efficient learning in ABR algorithms. ALISA incorporates importance sampling weights to give more weightage to relevant experience to address the lag issues with the existing A3C methods. We present the design and implementation of ALISA, and compare its performance to state-of-the-art video rate adaptation algorithms including vanilla A3C implemented in the Pensieve framework and other fixed-rule schedulers like BB, BOLA, and RB. Our results show that ALISA improves average QoE by up to 25%-48% higher average QoE than Pensieve, and even more when compared to fixed-rule schedulers. I. INTRODUCTION There has been rapid growth in the usage of Internetconnected devices in recent years, and this trend is predicted to continue in the future. The authors of [1] report that video streaming accounted for 53.72% of all internet traffic in the first half of 2021. Moreover, the number of IP networkconnected devices are predicted to be thrice the global population by 2023 [2], where HTTP-based video streaming will account for a large part of network traffic. However, several studies [3] have shown that low video quality often results in users abandoning video sessions, leading to considerable losses for content providers. AI has the ability to significantly improve a wide variety of mobile services, including video streaming, online gaming, voice-over IP, smart home applications, and remote health monitoring. It can be used to optimize the quality of experience (QoE) of video streaming for users. Dynamic Adaptive Streaming over HTTP (DASH) [4] has established itself as a significant standard for streaming video content over the best-effort Internet. In general, adaptive bitrate (ABR) algorithms have been extensively investigated for their potential to improve the quality of experience in DASH-based video streaming [5]. ABR algorithms automatically adjust the video bitrate in response to network conditions such as buffer occupancy and observed throughput in order to give a greater quality of experience for the end users. However, these algorithms make decisions based on a predefined set of criteria and are frequently tailored for certain conditions. This makes it difficult to generalize such methods to the wide variety of network conditions that exist in today's everchanging networks. Reinforcement learning (RL) [6] is a subfield of machine learning concerned with how agents should take actions in an environment in order to maximize some notion of cumulative reward. Several recent studies have investigated the integration of reinforcement learning approaches into video streaming [7], [8] with a goal to achieve a high QoE. RL techniques with asynchronous advantage actor-critic (A3C) methods [9] have demonstrated a number of advantages over ABR algorithms based on fixed rules. Several researchers [7], [8] have used a vanilla A3C method to generate adaptive bit rates for the purpose of increasing the overall quality of experience. The A3C [10] agent consists of multiple actors and a central learner with a critic. Each actor generates experience separately and concurrently based on its own behaviour policy. Individual experiences are then communicated to the central learner, which modifies the target policy (the policy that the A3C agent is attempting to learn) in response to the generated experience. However, A3C agents require a huge quantity of data to learn an appropriate policy. Increasing the number of actors is a common method for processing big amounts of data quickly. However, in such instances, each actor's behaviour policy begins to lag behind the target policy of the central learner [11]. As a result, the behaviour and target policies become out of sync, resulting in suboptimal updates. This may result into an inefficient use of bandwidth and a decrease in overall QoE while using RL for ABR algorithms. To address this issue, we integrate importance sampling weights [11] while using A3C methods for ABR generation to improve QoE for video streaming services. While assigning weights to the experience based on their relevance, our proposed approach solves the out-of-sync problem between behaviour and target policies and results in an overall higher QoE. Our solution is referred to as ALISA: Actor-Learner architecture with Importance Sampling for enhancing QoE in ABR algorithms. The proposed method is capable of generating adaptive bit rates via an actor-learner architecture based on reinforcement learning without relying on any pre-programmed model or assumption about the underlying systems. The current study makes a novel contribution by integrating importance sampling with A3C methods in order to train, learn, and generate adaptive bit rates while considering the distribution differences that may occur during training and, more importantly, when deploying the model in the real world. The main research contributions of this work are stated as follows: • Firstly, we present a new efficient ABR approach combining the importance sampling weights with actor-critic methods to improve video delivery services. By assigning importance sampling weights and, subsequently, allocating more significance to relevant experience, our method learns faster and gives an overall higher QoE than existing state-of-the-art ABR algorithms. Further, this helps the model to learn from samples with varying distributions in an efficient manner. • Second, we analyze the performance of the proposed approach using a widely-used Python-based framework and the MahiMahi simulator [12]. We consider several datasets for performance evaluation utilizing traces from FCC [13], Norway [14], OBOE [15], and live video streaming [16]. We present a comprehensive study using three different variants of QoE metrics, QoE lin , QoE log , and QoE HD , formulated as rewards for utilizing deep reinforcement learning. Finally, we also give a comparison over different network characteristics considering both lossless and lossy cases. • Third, we present the comparison of our proposed approach with other state-of-the-art ABR algorithms. This includes a comparison with the basic implementation of A3C, vanilla A3C (using the Pensieve framework) [7] and comparison with various non-RL ABR algorithms such as RB [17], BOLA [18], RobustMPC [19], etc. Our results demonstrate that ALISA provides up to 25%-48% higher average QoE than vanilla A3C (Pensieve). However, the improvements are considerably bigger when compared to the fixed-rule schedulers. The remaining paper is organized as follows. Section II presents the related work on ABR algorithms. Section III presents the relevant background on reinforcement learning and actor-critic methods. Further, Section IV presents the problem statement, integration of importance sampling weights, proposed algorithm and system design. We present the experimental setup and results in Section V and Section VI, respectively. Finally, we conclude our work in Section VII. II. RELATED WORK Several ABR algorithms have been developed [18]- [21] to provide adaptive bit rates for video delivery over wireless networks. The algorithms can be characterized essentially as either rate-based or buffer-based. The rate-based algorithms [17] predict the future chunk's bitrate as the maximum supported bitrate based on available network bandwidth and chunk history and the buffer-based algorithms predict based on the client's buffer occupancy [18], [20]. Due to the fact that the majority of these recommended techniques are based on predefined rules, they have a number of disadvantages. To begin, these algorithms are vulnerable to abrupt changes in network conditions, which might result in incorrect predictions. Second, while various approaches exist for achieving a higher QoE, each option has a trade-off. For instance, using the highest supported bitrate for each chunk may result in a loss of smoothness due to video resolution changes. Finally, the bitrate chosen for a current chunk frequently has an effect on the bitrate chosen for subsequent chunks. For instance, downloading chunks at the highest available bitrate may result in a reduction in the bit rate and quality of subsequent chunks in order to avoid rebuffering. Recently, in addition to fixed-rules-based ABR algorithms, machine learning and deep learning have also been widely used to generate ABR algorithms. Model predictive control (MPC) [19] has been used with deep learning in [22] for more accurate throughput estimations. Further, a combination of machine learning, deep learning, and reinforcement learning is used in [23] to obtain improvements in QoE as compared to previous rule-based and ML-based approaches. The prediction of bitrate as a linear combination of input parameters is modeled in [24]. It uses a deep neural network to learn a suitable function. The deep learning model is also used in [25] to learn the areas of interest in a video for a specific user to effectively allocate bitrate budgets. This methodology, along with the usual bandwidth and buffer occupancy, are jointly considered under the MPC framework to demonstrate an improvement over semantics-agnostic approaches. Recently, there has been a focus on the development of a new class of ABR algorithms that make use of reinforcement learning. Numerous attempts have been made to apply Q-Learning for this task [26], [27]. However, these works employ a tabular Q-learning method, which makes expanding it to wider state spaces impossible. Additionally, the prediction of bitrate based solely on the most recently seen chunk is done in [26]. It ignores the many most recently seen chunks that can enhance overall performance. To solve the issues of Q-learning in large state space, actor-critic methods for ABR generation are explored in [7], [8], [28]- [30]. In these papers, the A3C agent is used to generate ABRs and achieves a higher QoE than the majority of other fixed-rule-based ABR algorithms. However, the major issue with A3C is the lagging behind of an actor's behaviour policy as compared to the central learner's target policy [11], [31]- [34]. This has an effect on the performance of the A3C agent in existing reinforcement learning-based video distribution systems, resulting in decreased sample efficiency and the acquisition of a suboptimal policy. We propose and evaluate the integration of importance sampling weights to experiences depending on their relevance in order to solve a significant limitation of the existing A3C agent implementations for HTTP-based video delivery systems. III. BACKGROUND In this section, we present a brief overview of reinforcement learning and actor-critic methods. A. Reinforcement Learning A reinforcement learning solution [6] aims to learn a mapping from the state space to the action space by repeated interaction between the RL agent and the environment. The RL problem is modeled as a Markov decision process with states and actions. Let us consider a discrete system where at each time step t ∈ {0, 1, 2, ...}, the RL agent observes its state s t , takes an action a t , moves to state s t+1 and receives a reward R(s t , a t ) = r t+1 . Further, for a sequence of states and actions, the discounted cumulative reward is defined as R(τ ) = ∞ k=0 γ k r t+k+1 where τ is the sequence of states and actions, i.e. {(s t , a t ), (s t+1 , a t+1 ), ...} and γ ≤ 1 is a discount factor. The agent selects action based on a policy, π : π θ (s t , a t ) → [0, 1], where π θ (s t , a t ) is the probability that action a t is taken in state s t and θ are the policy parameters upon which the actions are based. Following the policy π, the value function V (s) for a state s is defined as V (s) = E π [R(τ )|S t = s] . The goal of an RL agent is to find the optimal policy π * that maximizes the overall discounted reward. The optimal policy is given by, π * (s t ) = argmax a [R(s t , a t ) + γV (s t+1 )] (1) Under this framework, we can have value-based methods which learns a value function mapping each state-action pair to a value. The action with the biggest value in a state becomes the optimal action to take. We can also have policybased methods which directly optimize the policy function as explained in the next subsection. B. Actor-Critic Methods As an improvement to the value-based methods, [10] presented actor-critic methods, which helps to speed up learning by reducing the variance of estimated quantities. An actorcritic method consists of two models. An actor that learns the optimal policy, and a critic, that approximates the value function (utility of a state-action pair). At each step t, we use the current state s t to predict the action a t by using the policy π. This also returns a reward r t+1 . Using this information, the critic now computes the value of the state-action pair, q w (s t , a t ). The policy update with respect to its parameters θ is defined in terms of the gradient operator ∇ as follows, ∆θ = α∇ θ log π θ (s t , a t )q w (s t , a t )(2) where α is the actor learning rate,q w (s t , a t ) is the critic function that indicates how good an action a t is in state s t . The parameters w of the critic function are updated as follows, ∆w = ξ(R(s t , a t )+γq w (s t+1 , a t+1 )−q w (s t , a t ))∇ wqw (s t , a t ) (3) where ξ is the critic learning rate. However, a policy network trained in this manner may have high variance, which can cause instability during training. To mitigate this issue, the advantage actor-critic (A2C) [10] framework introduces the advantage function to determine the advantage of the action taken in state s t as compared to the average value of actions in s t . The advantage function is defined as the temporal difference (TD) error: A(s t , a t ) = R(s t , a t ) + γV (s t+1 ) − V (s t )(4) The final gradient-based update for the actor is as follows, ∆θ = α∇ θ log π θ (s t , a t )A(s t , a t ) + β∇ θ H(π θ (.|s t )) (5) where H(π θ (.|s t )) is the entropy factor which promotes random actions and β is the regularization term. The entropy term is defined as H(π θ (.|s t )) = − a π θ (a|s t ) log(π θ (a|s t ))(6) The value of β is initially set to a high value to promote exploration early on, and it is reduced as training progresses. To enhance training speed, Asynchronous Advantage Actor-Critic (A3C) framework [10] is proposed to simulate multiple actors in parallel and asynchronously. These actors synchronize their parameters with the central learner at regular intervals. In our work, we use a modified A3C framework to generate ABR algorithms for video delivery services. IV. PROBLEM STATEMENT AND PROPOSED SOLUTION This section discusses the problem formulation and proposed solution for RL-based video distribution services with importance sampling, as well as the system design specifics. A. The Issue Reinforcement learning agents often require a large amount of experience to model the environment effectively and accurately. A common technique to achieve this goal is to increase the number of actors during the training. However, this strategy has an inherent flaw. As shown in Figure 1, the central learner first synchronizes its weights with those of all the actors (Step 1), after which the actors provide their experience to the central learner (Step 2). However, there are instances when some actors may delay sending the updates to the central learner. For example, as shown in Figure 1, the central learner updates its target policy even before receiving the experience from the actor on the right side (Step 3). Therefore, the behaviour policy associated with this event lags behind the target policy, and the experience may be less useful to update the current target policy. This issue is exacerbated further by the presence of more actors when actors start generating experience with an older version of the behaviour policy. Step (1): Each actor synchronizes its weights with the central learner; Step (2): One of the actors provides experience to the central learner, which updates the weights of the target policy; Step (3): The other actor provides experience to the central learner. The behaviour policy for the experience is not synchronized with the latest version of the target policy (updated in step 2), hence the experience is based on an older policy. The lack of synchronization between the behaviour and target policies results in suboptimal updates. Eventually, this leads to learning an overall suboptimal policy. We intend to develop approaches that compensate for the behaviour policy falling behind the target policy during training, allowing us to achieve higher performance on unseen test data. B. ALISA: System Design Our proposed solution ALISA builds on the DASH framework and uses deep RL-based A3C methods to achieve a higher QoE than existing state-of-the-art ABR algorithms for video streaming. Figure 2 presents the overall ALISA's system design. The user streams a video on their devices on a video player, whose main component is the ABR controller. It observes several state parameters on the client side, such as bandwidth, bitrate selection history, and buffer occupancy, and decides the action to take, i.e., the bitrate selection for the next chunk. At each step, it also observes some reward (QoE) as a result of its actions. We now describe the training process used by the ABR controller of ALISA. As shown in Figure 3, the training environment is composed of multiple actors who are coordinated by a single central learner. The actor contains a behaviour policy as its parameters, while the central learner maintains the target policy and the critic parameters. The behaviour policy, the target policy, and the critic function are all modeled as neural networks. The training process can be considered as the repetition of the following steps until convergence: 1) First, an actor simulates an episode and generates a batch of experiences consisting of the states, the corresponding actions taken by the actor, and the rewards received as a result. 2) The experience is then passed back to the central learner. 3) The central learner calculates the values for each step of the experience using the critic parameters. 4) The central learner calculates the V −trace targets after incorporating the importance sampling strategy discussed in the next subsection. 5) The critic gradients are computed using the observed states and their corresponding rewards, while the target policy gradients are computed using the observed states, the corresponding actions, the obtained rewards, and the V −trace targets. The target policy and the critic network are now updated using backpropagation. 6) Finally, the central learner shares the latest version of the target policy with each actor, which sets their behaviour policy to the newest target policy to generate the next batch of experiences. ALISA effectively decouples the acting and learning processes while also compensating for the resulting off-policy shift. This has significant implications for the development of ABR algorithms. Due to the vast volume of video being streamed to users worldwide, the ALISA architecture enables constant fine-tuning of the ABR algorithm and adaptation to ever-changing network conditions, all without jeopardising the users' privacy. While the streaming devices continuously make bitrate selections, the decisions can be relayed to the central learner located on a remote cloud server, where federated learning can take place [35]. The latest policy can be synchronized between the end-user devices and the remote server at regular intervals. This allows video streaming services to respond much faster to fluctuations in network conditions and changes in video streaming behaviour over time, allowing a new model to be retrained much faster. While we are limited by available data to demonstrate the benefit of ALISA on fine-tuning, we show in Section VI how ALISA not only achieves a higher QoE but also does so in less than 50% of the time required for comparable methods. We anticipate that this benefit observed in training the model from scratch will also extend while fine-tuning the model as we have suggested. C. ALISA: Integration of Importance Sampling for Policy Update Importance sampling is a commonly used technique for resolving data distribution mismatches. It provides the estimation of the expected value of a function f (x), where x follows a probability density function a on the domain D, by sampling values from a different distribution b on the same domain D as, E(f (X)) = E r f (X)a(X) b(X)(7) Importance sampling alters the data collected from the distribution b in such a way that it looks to have been sampled from the distribution a. This effectively addresses the distribution mismatch issue. In this work, we integrate importance sampling with ALISA to overcome the distribution mismatch between the target policy π and the behaviour policy µ. Correlating to the notation in Equation (7), we have a ≡ π and b ≡ µ. Similar to the authors of [11], we use the n-step V -trace target to adjust for the off-policy shift. The n-step Vtrace target now acts as an estimate of the value function V for the target policy π using an older version of the behaviour policy µ. The n-step V -trace target is defined as, where, v j . = V (s j ) + j+n−1 t=j γ t−j   t−1 i=j c i   δ t V(8)• δ t V . = ρ t (r t + γV (s t+1 ) − V (s t )) is the temporal difference. • ρ t . = min ρ, π(at|st) µ(at|st) and c i . = min c, π(ai|si) µ(ai|si) are the importance sampling weights. The importance sampling weights ρ t and c i are used to give importance to experience, which is more relevant to the target policy than the behaviour policy. Here, π denotes the target policy, and µ denotes the behaviour policy. • ρ and c are lower threshold values for their corresponding importance sampling weights, which we set to 1 throughout our work. • ρ t denotes how much more probable the action a t taken in state x t is according to the target policy compared to the behaviour policy. • t−1 i=j c i denotes how much more probable the predicted path from state s j to s t−1 is according to the target policy compared to the behaviour policy. Subsequently, the V -trace targets are used in place of V for gradient computation. The n-step V -trace target can also be defined recursively as v j = V (s j ) + δ j V + γc j (v j+1 − V (s j+1 ))(9) which we use during implementation throughout our work. As a result of these calculations, actions that are more likely to be taken according to the target policy contribute more to the V-trace target. Hence the importance sampling weights help the reinforcement learning model to focus on the experience, which is more relevant and leads to better parameter updates and assigns less importance to suboptimal experience. Using the above definition of the V-trace target, Algorithm 1 outlines ALISA's policy update algorithm with the importance sampling weights where n equals to the length of the episode. The V -trace target calculation with ALISA takes as input the information related to an episode consisting of the sequence of states (s), the sequence of action probabilities according to the behaviour policy (a b ), the sequence of rewards (r) along with the meta parameters ρ and c and the actor and critic models from Line 2 to Line 8. As a result, the ρ is updated to a minimum of ρ and quotient of target policy and behaviour policy in Line 15. In Line 16 , the change to be added in the critic values for s (V ) is computed . This is followed by the calculation of c which is assigned as the minimum of c and ρ in Line 17. The V -trace targets are updated by adding the value computed in Line 16 to the current V -trace targets. Line 19 to Line 21 explains how the V -trace targets for each i of the loop is updated. As a consequence of importance sampling, actions that are more likely to be taken by the current target policy contribute more to the gradients compared to actions that are likely to be taken by earlier lagging versions of the target policy but not the current one. This algorithm guides the RL model to focus on the experience which matters more and assign less importance to other less relevant experiences. Algorithm 1 Calculation of V -trace targets using ALISA for an episode 1: Input: 2: s: Sequence of states 3: a b : Sequence of action probabilities according to behaviour policy 4: r: Sequence of rewards 5: ρ: Lower threshold for ρ 6: c: Lower threshold for c 7: actor: target policy from central learner 8: critic: critic model 9: Output: 10: v: V -trace targets 11: V ← critic values for s 12: p b ← behaviour policy probabilities for optimal action 13: a t ← target policy probabilities for s 14: p t ← target policy probabilities corresponding to optimal actions of behaviour policy, computed using a t and p b 15: ρ ← min ρ, pt p b 16: δ t V ← ρ (r + γ V +1 − V ) 17: c ← min (c, ρ) 18: v ← V + δ t V 19: for each position i from rear of v do 20: v i += γc i * (v i+1 − V i+1 ) 21: end for V. EXPERIMENTAL DETAILS In this section, we describe the experimental setup together with the performance metrics used to evaluate ALISA's performance. A. Experimental Setup We use the Python-based framework proposed in [7], to generate and test our ABR algorithms. The client requests chunks of data from the server and provides it with parameters pertaining to the observed network conditions like bandwidth, buffer occupancy, and bitrate history. We have integrated importance sampling as described in Section IV and assigned weights to the target and the behaviour policies. To emulate network conditions for effectively testing our trained RL model, the MahiMahi [12] framework has been used, which is a record-and-replay HTTP framework for this task. We use four datasets as part of our training set: the broadband dataset provided by the FCC [13], the mobile dataset collected in Norway [14], the OBOE traces [15], and the live video streaming dataset [16], which have been pre-processed according to the MahiMahi format. Subsequently, we compare ALISA with the following state-of-the-art ABR algorithms: • Pensieve (Vanilla A3C) [7]: uses vanilla A3C without any additional techniques to train the agent for delivering adaptive bit rates. • Rate-Based (RB) [17]: RB predicts the maximum supported bitrate based on the harmonic mean of past observed throughput. • Buffer-based (BB) [20]: BB selects bitrate based on client's buffer occupancy. • BOLA [18]: Bitrate selection is made exclusively based on buffer occupancy using Lyapunov optimization. • RobustMPC [19]: MPC uses buffer occupancy observations and throughput predictions similar to RB. Additionally, RobustMPC accounts for errors between predicted and observed throughputs by normalizing throughput estimates. To quantify the performance of the ABR algorithms, we use the formulation of QoE: QoE = N n=1 q(b n ) − µ N n=1 T n − N −1 n=1 |q(b n+1 ) − q(b n )| (10) where b i and q(b i ) represent the bit-rate and quality, respectively, for chunk i. A higher bit rate means higher quality and a higher QoE. However, there are also penalties due to rebuffering time T i (represented by the second term) and fluctuations in video quality (represented by the final term) that hinders the overall smoothness. In this paper, we evaluate the performance of the proposed approaches with three QoE variants [7] that depend on the above general QoE metric: • QoE lin : q(b n ) = b n where value of rebuffer penalty is µ = 4.3, • QoE log : q(b n ) = log(b/b min ) that considers the fact that the marginal improvement in quality decreases at higher bitrates with µ = 2.66 and • QoE HD : assigns a higher value to higher quality bitrates and lower values to lower quality bitrates with µ = 8. B. Dataset details We use the following data sets for training, validation, and testing. Our selection of data sets for training, validation, and testing is in line with the previous experimental setups in [7], [15], [16]. For training, we have used three different data sets. The first data set consists of 127 traces, out of which 59 belongs to the FCC [13] dataset while the remaining 68 belong to the Norway HSDPA [14] dataset. The second data set consists of 428 OBOE traces [15] and the third data set consists of 100 live video streaming traces [16]. To demonstrate the benefits of ALISA over different trained models, we have generated three different trained model corresponding to three different data sets described above. For all three trained models, we have used the same validation data set, i.e., 142 Norway traces. Finally, for all three trained models, after validation, the testing is performed using 205 traces from the FCC dataset and 250 traces from the Norway HSDPA dataset. C. Training Methodology We train the three models for each configuration on Pensieve and ALISA, one each for QoE lin , QoE log and QoE HD as reward metrics. For each model, we use a consistent set of hyperparameters throughout. The discount factor γ is set to 0.99. The learning rates are set to 0.0001 and 0.001 for the actor and critic, respectively. In Pensieve, the entropy factor (H(.)) is controlled by using an entropy regularization factor (β). The β uses entropy decay values from 1 to 0.1 over 100,000 epochs. In ALISA, we set both importance sampling thresholdsρ andc to 1. We train multiple models for different configurations of entropy weights. First, we train several models with a constant entropy weight for 100,000 epochs. Next, we use a decaying entropy weight where the entropy is gradually decreased over 100,000 epochs. D. Testing Methodology We select the model with the highest validation QoE for testing. We perform testing under both lossless and lossy conditions simulated using the MahiMahi [12] framework. We perform tests under packet loss percentages of 0%, 0.1%, 0.5%, 1%, and 2%, where random packets are dropped from the video stream. We evaluate all models on the three different QoE metrics discussed in Section V-A. VI. RESULTS In this section, we present the results and comparison of ALISA with other state-of-the-art ABR algorithms. A. Convergence Speed ALISA takes advantage of the importance sampling strategy during training. As a result, it is often able to achieve a higher QoE compared to Pensieve (Vanilla A3C) in a shorter time. Figure 4 presents the plots of the epochs elapsed versus the maximum QoE achieved till then. These plots are generated during the training using the first data set, i.e., 127 traces from FCC and Norway data sets. Our results show that by the time ALISA obtains a high QoE of over 40, Pensieve is only able to obtain the highest QoE of approximately 35. Furthermore, ALISA achieves a QoE over 40 in less than 1/3 rd of the time required for Pensieve to achieve its highest QoE. This demonstrates ALISA's advantage in learning and adapting to newer conditions faster, resulting in shorter training times. Similar results are observed during the training using OBOE and live video streaming traces. B. Comparison with state-of-the-art ABR algorithms 1) Results with training and validation data sets: We perform a comprehensive set of training on the three data sets and report our results on QoE lin , QoE log , and QoE HD metrics. Table I presents the rewards obtained after training and validation on the FCC and Norway traces OBOE traces and the live video streaming traces. We also investigate the use of different values for entropy regularization β. Specifically, we consider the following values: 0, 0.001, 0.01, 0.1, 0.25, 0.5, 0.75, and 1. We have also explored the use of variations in β during the training. For example, in Table I Table I, refers to the scenario where β = 0.1 for all 100000 iterations. We note that the models do not converge well on training with very low or very high entropy. We found a constant entropy of 0.1 to work well for QoE lin and QoE log metrics, while 0.75 worked well for QoE HD . We have also trained multiple times using a decaying entropy regularization. We start from a high value and gradually decrease our entropy weight every 20,000 epochs, gradually going down to 0.1. We find that decaying entropy regularization is more effective in almost all the cases, as seen from Table I since after a few epochs, a high exploration is not required to achieve an optimal policy. 2) Results with Test Data Sets: We also compare ALISA to several other state-of-the-art ABR algorithms, such as RB, BB, BOLA, and RobustMPC, described in the previous section. We have also compared ALISA with Pensieve, an RL-based ALISA achieves a higher QoE on all metrics over all different configurations. ALISA obtains up to 25% higher QoE than RB, 230% higher QoE than BB, 30% higher QoE than BOLA, 25% higher QoE than RobustMPC and 20% higher QoE compared to Pensieve when tested under lossless conditions. This performance translates to lossy conditions as well. We note that ALISA is able to obtain up to 25%, 28%, 48%, and 48% higher QoE compared to Pensieve under losses of 0.1%, 0.5%, 1%, and 2%, respectively. We summarize the remained of our testing QoE metrics for a random packet loss percentage of 0.1%, 0.5%, 1%, and 2% in Table III, Table IV, Table V and Table VI, respectively. These results indicate that ALISA achieves a significantly better performance than many other fixed-rule-based ABR algorithms and also Pensieve. Further, we also visualize the different components of the QoE metric from equation (10) to understand how ALISA performs better than other ABR algorithms. Figure 5 presents the total reward achieved by various ABR algorithms with QoE lin metric for each trace when the network is emulated with 0.1% packet loss. Our results show that the ALISA algorithm achieves a higher average QoE of 44.58 as compared to other ABR algorithms. Figure 6 presents the average total reward achieved by various ABR algorithms with QoE lin metric for each trace when the network is emulated with 1% packet loss. Our results show that the ALISA algorithm achieves a higher average QoE of 34.87 as compared to other ABR algorithms. Figure 7(a) and Figure 7(b) shows how ALISA can consistently achieve higher bitrates than other methods for random sample traces. This increases the first component of QoE. From Figure 8(a) and Figure 8(b), we note that ALISA maintains an adequate buffer size than Pensieve for random sample traces, leading to a decrease in the second component due to moderate bitrates while maintaining the moderate buffer size in equation (10) hence ALISA provides a lower rebuffer penalty. On the other hand, Pensieve maintains a higher buffer size than ALISA, which leads to high bitrates. The higher bitrates lead to a high rebuffer penalty. Overall, this leads to a higher quality of experience for ALISA over other ABR algorithms. To understand and demonstrate the better performance of the ALISA, we analyze the individual component of the QoE metric and present the comparison of various ABR algorithms using the average playback bitrate, average rebuffering penalty, and the average smoothness penalty for QoE lin metric under emulation with 1% packet loss in Figure 9. Our results show that most of the ABR algorithms achieve a higher bitrate except for BOLA and RB in Figure 9(a). Due to the selection of a higher bitrate, several of these algorithms suffer from a rebuffering penalty where BB and robustMPC have the highest rebuffering penalty in Figure 9(b). Similarly, BB also suffers from a high smoothness penalty in Figure 9(c). The ALISA achieves a higher average bit rate and comparatively smaller rebuffering and smoothness penalty. The overall impact of these individual components results in the ALISA achieving an average QoE higher than the other ABR algorithms. We have observed similar results for the QoE lin , QoE log and QoE HD metrics under emulation with 0.1%, 0.5%, 1%, and 2% packet losses. VII. CONCLUSIONS We have demonstrated how combining importance sampling and structured entropy selection considerably enhances the performance of vanilla A3C approaches (using the Pensieve framework) when used to generate ABR algorithms for video delivery service. By incorporating these methods into our proposed system, ALISA, we are able to consistently achieve up to a 25-48 % improvement in QoE and even higher in some cases. Additionally, we evaluate our approaches under a broader range of network conditions in terms of packet loss and observe comparable benefits. Finally, we visualize and compare ALISA's bitrate selection and buffer size to those of other ABR algorithms (RB, BB, BOLA, and robustMPC), and shown that ALISA outperforms them in both areas, resulting in an improved QoE. The future work will examine advanced hybrid cloud-edge architectures for ALISA implementation. Additionally, we intend to investigate ALISA in a federated environment in order to take advantage of distributed training across multiple decentralized edge devices. Fig. 1 . 1Illustration of A3C lagging issue. The figure shows three steps. Fig. 2 .Fig. 3 . 23ALISADetailed flow design for training the RL-based ABR controller of ALISA Fig. 4 . 4Maximum training QoE obtained versus epochs elapsed. ALISA is able to obtain a higher QoE faster for all three variants of the QoE metric. , {2, 1.5, 1, 0.5, 0.1} refers to the scenario where β = 2 for the first 20000 epochs, β = 1.5 for the next 20000 epochs, and so on, till β = 0.1 for the last 20000 epochs. Similarly, we have also used constant entropy regularization, where 0.1 × 5, in Fig. 9 . 9Comparing ALISA with existing ABR algorithms by analyzing their performance on the individual components: (a) average bitrate, (b) average rebuffering penalty, and (c) average smoothness penalty for QoE lin metric under emulation with 1% packet loss (Equation 10). TABLE I AVERAGE IQOE AFTER TRAINING ALISA WITH ALL THE THREE DATASETS AND ALL THE THREE VARIANTS OF QOE METRICS. QOE ACHIEVED ON ALL THE DATASETS WITH ALL THREE VARIANTS OF QOE METRICS UNDER EMULATION WITH NO PACKET LOSSES.Entropy values FCC and Norway Traces OBOE Traces Live Video Streaming Traces QoE lin QoE log QoE HD QoE lin QoE log QoE HD QoE lin QoE log QoE HD 0 (×5) 13.61 13.62 13.57 30.76 30.8 39.51 13.61 30.75 29.97 0.001 (×5) 13.61 13.62 13.61 42.47 30.78 39.52 13.61 30.87 13.57 0.01 (×5) 13.61 13.61 13.61 38.79 39.25 39.44 30.53 30.93 37.84 0.1, 2, 0.1, 2, 0.1 41.57 38.91 42.21 41.43 44.03 38.4 43.27 38.54 41.36 0.1 (×5) 43.88 43.29 37.88 42.89 38.37 38.55 44.1 30.76 27.96 0.25 (×5) 40.91 43.29 37.61 42.85 43.2 39.57 43.22 44.03 37.69 0.5 (×5) 38.11 37.14 42.42 36.79 37.99 39.17 37.86 38.35 41.71 0.75 (×5) 31.11 32.11 41.99 29.85 30.69 40.97 32.23 32.83 42.28 1, 0.75, 0.5, 0.25, 0.1 43.22 44.09 42.4 44.34 44.94 40.61 44.76 45.96 41.66 1 (×5) 27.46 25.8 40.98 25.07 25.15 41.11 26.41 26.07 41.17 2, 1.5, 1, 0.5, 0.1 43.86 44.19 41.27 43.16 43.73 41.16 44.45 45.54 41.76 3, 2, 1, 0.5, 0.1 43.92 44.31 42.11 43.95 43.36 40.56 44.25 44.54 42.32 4, 2, 1, 0.5, 0.1 43.15 43.4 41.8 42.99 44.93 39.33 43.92 45.93 42.66 5, 2, 1, 0.5, 0.1 41.94 44.8 42.06 43.23 42.33 40.99 43.52 45.12 41.64 TABLE II AVERAGE Algorithm FCC and Norway OBOE Live video streaming Linear Log HD Linear Log HD Linear Log HD ALISA 43.03 42.37 256.29 42.5 41.79 237.27 46.57 44.36 228.63 Pensieve 39.62 35.26 239.08 37.52 37.01 194.29 39.12 41.68 234.72 BB 12.02 12.78 84.24 14.08 20.00 80.36 13.81 20.26 63.08 RB 35.62 36.45 139.82 36.15 37.97 138.02 37.44 37.35 120.52 BOLA 34.25 35.3 141.04 35.04 37.09 139.1 35.82 36.05 121.02 RobustMPC 39.93 40.44 195.52 40.21 38.03 188.65 40.59 38.99 177.58 Fig. 5. Comparison of ALISA over other ABR algorithms with the QoE linear metric: (left) average reward overall test traces; (right) CDF vs QoE plot under emulation of random packet drops with 0.1% probability. Fig. 6. Comparison of ALISA over other ABR algorithms with the QoE linear metric under emulation with 1% packet loss Equation 10 TABLE III AVERAGE IIIQOE ACHIEVED ON ALL THE DATASETS WITH ALL THE THREE VARIANTS OF QOE METRICS UNDER EMULATION OF RANDOM PACKET DROPS WITH 0.1% PROBABILITY.Algorithm FCC and Norway OBOE Live video streaming Linear Log HD Linear Log HD Linear Log HD ALISA 44.58 43.75 244.38 44.46 42.07 237.91 45.76 41.89 233.96 Pensieve 41.99 35.06 235.12 39.34 36.69 194.37 36.62 39.08 241.10 BB 16.62 20.97 73.16 17.08 19.81 0.64 15.40 19.58 68.76 RB 38.41 38.73 132.98 39.05 37.64 35.58 37.15 37.65 130.14 BOLA 37.58 37.21 129.27 37.84 36.03 34.81 35.95 36.08 126.24 RobustMPC 41.90 37.30 189.53 42.80 37.97 187.43 40.97 37.56 182.75 TABLE IV AVERAGE IVQOE ACHIEVED ON ALL THE DATASETS WITH ALL THE THREE VARIANTS OF QOE METRICS UNDER EMULATION WITH 0.5%PROBABILITY. Algorithm FCC and Norway Traces OBOE Traces Live video streaming Traces Linear Log HD Linear Log HD Linear Log HD ALISA 37.34 43.86 236.15 39.2 42.24 231.37 42.53 43.33 227.66 Pensieve 35.43 34.1 214,59 36.44 36.22 201.32 34.98 40.80 231.19 BB 10.59 18.74 66.13 13.97 18.79 69.6 12.12 18.41 64.51 RB 32.91 35.89 108.86 34.19 35.35 117.44 33.99 34.92 111.23 BOLA 32.39 34.59 108.98 34.12 33.97 112.67 33.78 33.77 110.80 RobustMPC 37.99 38.2 177.87 39.48 38.46 184.96 38.62 38.09 179.55 TABLE V AVERAGE VQOE ACHIEVED ON ALL THE DATASETS WITH ALL THE THREE VARIANTS OF QOE METRICS UNDER EMULATION WITH 1% PROBABILITY.Algorithm FCC and Norway Traces OBOE Traces Live video streaming Traces Linear Log HD Linear Log HD Linear Log HD ALISA 34.87 38.18 198.34 35.35 44.86 181.71 36.97 38.2 187.68 Pensieve 29.68 29.22 174.53 29.04 30.18 138.71 29.50 29.22 188.78 BB 2.90 10.11 19.07 2.91 10.57 12.67 6.43 10.11 -15.75 RB 27.62 28.58 78.17 27.38 28.72 77.99 28.00 28.58 25.38 BOLA 26.93 27.74 77.30 27.59 27.95 77.12 26.89 27.74 25.00 RobustMPC 32.92 33.13 149.21 32.62 33.47 142.51 33.74 33.13 145.87 TABLE VI AVERAGE VIQOE ACHIEVED ON ALL THE DATASETS WITH ALL THE THREE VARIANTS OF QOE METRICS UNDER EMULATION WITH 2% PROBABILITY. basic A3C approach that does not utilize the importance sampling weights. Table II presents the comparison when there are no losses in the network. Our results show thatFig. 7. Comparison of bit rate selection for ALISA over other ABR algorithms: (a) bitrate selection for sample trace 1, and (b) bitrate selection for sample trace 2 using FCC and Norway trained model for QoE lin metric under emulation with 1% packet loss (Equation 10).Fig. 8. Comparison of buffer size selection for ALISA over other ABR algorithms: (a) buffer size selection for sample trace 1, and (b) buffer size selection for sample trace 2 using FCC and Norway trained model for QoE lin metric under emulation with 1% packet loss (Equation 10).Algorithm FCC and Norway Traces OBOE Traces Live video streaming Traces Linear Log HD Linear Log HD Linear Log HD ALISA 27.60 28.98 128.53 25.62 26.67 122.05 28.73 30.83 122.47 Pensieve 24.28 21.48 111.57 22.35 22.98 82.05 22.45 28.56 121.41 BB -13.44 -6.74 -41.18 -15.48 -6.28 -37.73 -14.49 -4.15 -38.48 RB 16.09 17.35 13.44 17.27 17.75 14.88 17.21 18.22 14.21 BOLA 17.89 18.59 16.05 18.08 18.01 16.08 17.48 18.76 16.49 RobustMPC 24.43 20.72 99.98 25.28 21.66 99.14 24.98 21.69 97.09 ACKNOWLEDGMENT This work has been supported by the TCS foundation, India under the TCS research scholar program, 2019-2023. . Sandvine Global Internet Phenomena Report. 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{'abstract': 'Adaptive bitrate (ABR) algorithms are used to adapt the video bitrate based on the network conditions to improve the overall video quality of experience (QoE). Recently, reinforcement learning (RL) and asynchronous advantage actor-critic (A3C) methods have been used to generate adaptive bit rate algorithms and they have been shown to improve the overall QoE as compared to fixed rule ABR algorithms. However, a common issue in the A3C methods is the lag between behaviour policy and target policy. As a result, the behaviour and the target policies are no longer synchronized which results in suboptimal updates. In this work, we present ALISA: An Actor-Learner Architecture with Importance Sampling for efficient learning in ABR algorithms. ALISA incorporates importance sampling weights to give more weightage to relevant experience to address the lag issues with the existing A3C methods. We present the design and implementation of ALISA, and compare its performance to state-of-the-art video rate adaptation algorithms including vanilla A3C implemented in the Pensieve framework and other fixed-rule schedulers like BB, BOLA, and RB. Our results show that ALISA improves average QoE by up to 25%-48% higher average QoE than Pensieve, and even more when compared to fixed-rule schedulers.', 'arxivid': '2304.04527', 'author': ['Mandan Naresh \nComputer Science & Information Systems\nBirla Institute of Technology and Science Pilani Hyderabad\nIndia\n', 'Paresh Saxena psaxena@hyderabad.bits-pilani.ac.in \nComputer Science & Information Systems\nBirla Institute of Technology and Science Pilani Hyderabad\nIndia\n', 'Manik Gupta \nComputer Science & Information Systems\nBirla Institute of Technology and Science Pilani Hyderabad\nIndia\n'], 'authoraffiliation': ['Computer Science & Information Systems\nBirla Institute of Technology and Science Pilani Hyderabad\nIndia', 'Computer Science & Information Systems\nBirla Institute of Technology and Science Pilani Hyderabad\nIndia', 'Computer Science & Information Systems\nBirla Institute of Technology and Science Pilani Hyderabad\nIndia'], 'corpusid': 258048793, 'doi': '10.48550/arxiv.2304.04527', 'github_urls': [], 'n_tokens_mistral': 18559, 'n_tokens_neox': 16024, 'n_words': 9343, 'pdfsha': '2313ad1f0b6a0411adaa9b30b6e55ee5f10ad808', 'pdfurls': ['https://export.arxiv.org/pdf/2304.04527v1.pdf'], 'title': ['Deep Reinforcement Learning with Importance Weighted A3C for QoE enhancement in Video Delivery Services', 'Deep Reinforcement Learning with Importance Weighted A3C for QoE enhancement in Video Delivery Services'], 'venue': []}

arxiv

Residual Relaxation for Multi-view Representation Learning Yifei Wang School of Mathematical Sciences Peking University China Zhengyang Geng School of EECS Key Lab. of Machine Perception Peking Univesity China Feng Jiang School of EECS Key Lab. of Machine Perception Peking Univesity China Chuming Li School of Engineering The University of Sydney Australia Yisen Wang School of EECS Key Lab. of Machine Perception Peking Univesity China Jiansheng Yang School of Mathematical Sciences Peking University China Zhouchen Lin School of EECS Key Lab. of Machine Perception Peking Univesity China Pazhou Lab 510330GuangzhouChina Residual Relaxation for Multi-view Representation Learning Multi-view methods learn representations by aligning multiple views of the same image and their performance largely depends on the choice of data augmentation. In this paper, we notice that some other useful augmentations, such as image rotation, are harmful for multi-view methods because they cause a semantic shift that is too large to be aligned well. This observation motivates us to relax the exact alignment objective to better cultivate stronger augmentations. Taking image rotation as a case study, we develop a generic approach, Pretext-aware Residual Relaxation (Prelax), that relaxes the exact alignment by allowing an adaptive residual vector between different views and encoding the semantic shift through pretext-aware learning. Extensive experiments on different backbones show that our method can not only improve multi-view methods with existing augmentations, but also benefit from stronger image augmentations like rotation. Introduction Without access to labels, self-supervised learning relies on surrogate objectives to extract meaningful representations from unlabeled data, and the chosen surrogate objectives largely determine the quality and property of the learned representations [28,23]. Recently, multi-view methods have become a dominant approach for self-supervised representation learning that achieves impressive downstream performance, and many modern variants have been proposed [26,16,1,27,2,13,3,4,12,5]. Nevertheless, most multi-view methods can be abstracted and summarized as the following pipeline: for each input x, we apply several (typically two) random augmentations to it, and learn to align these different "views" (x 1 , x 2 , . . . ) of x by minimizing their distance in the representation space. In multi-view methods, the pretext, i.e., image augmentation, has a large effect on the final performance. Typical choices include image re-scaling, cropping, color jitters, etc [2]. However, we find that some augmentations, for example, image rotation, is seldom utilized in state-of-the-art multiview methods. Among these augmentations, Figure 1a shows that rotation causes severe accuracy drop in a standard supervised model. Actually, image rotation is a stronger augmentation that largely affects the image semantics, and as a result, enforcing exact alignment of two different rotation angles could degrade the representation ability in existing multi-view methods. Nevertheless, it does not mean that strong augmentations cannot provide useful semantics for representation learning. In fact, rotation is known as an effective signal for predictive learning [11,34,25]. Differently, predictive methods learn representations by predicting the pretext (e.g., rotation angle) from the cor- . Right: an illustration of the exact alignment objective of multi-view methods (z →← z) and the relaxed residual alignment of our Prelax (z − r →← z). As the rotation largely modifies the image semantics, our Prelax adopts a rotation-aware residual vector r to bridge the representation of two different views. responding view. In this way, the model is encouraged to encode pretext-aware image semantics, which also yields good representations. To summarize, strong augmentations like rotation carry meaningful semantics, while being harmful for existing multi-view methods due to large semantic shift. To address this dilemma, in this paper, we propose a generic approach that generalizes multi-view methods to cultivating stronger augmentations. Drawing inspirations from the soft-margin SVM, we propose residual alignment, which relaxes the exact alignment in multi-view methods by incorporating a residual vector between two views. Besides, we develop a predictive loss for the residual to ensure that it encodes the semantic shift between views (e.g., image rotation). We name this technique as Pretext-aware REsidual ReLAXation (Prelax), and an illustration is shown in Figure 1b. Prelax serves as a generalized multi-view method that is adaptive to large semantic shift and combines image semantics extracted from both pretext-invariant and pretext-aware methods. We summarize our contributions as follows: • We propose a generic technique, Pretext-aware Residual Relaxation (Prelax), that generalizes multi-view representation learning to benefit from stronger image augmentations. • Prelax not only extracts pretext-invariant features as in multi-view methods, but also encodes pretext-aware features into the pretext-aware residuals. Thus, it can serve as a unified approach to bridge the two existing methodologies for representation learning. • Experiments show that Prelax can bring significant improvement over both multi-view and predictive methods on a wide range of benchmark datasets. Related Work Multi-view Learning. Although multi-view learning could refer to a wider literature [18], here we restrict our discussions to the context of Self-Supervised Learning (SSL), where multi-view methods learn representations by aligning multiple views of the same image generated through random data augmentation [1]. There are two kinds of methods to keep the representations well separated: contrastive methods, which achieve this by maximizing the difference between different samples [2,13], and similarity-based methods, which prevent representation collapse via implicit mechanisms like predictor and gradient stopping [12,5]. Although having lots of modern variants, multi-view methods share the same methodology, that is to extract features that are invariant to the predefined augmentations, i.e., pretext-invariant features [24]. Predictive Learning. Another thread of methods is to learn representations by predicting selfgenerated surrogate labels. Specifically, it applies a transformation (e.g., image rotation) to the input image and requires the learner to predict properties of the transformation (e.g., the rotation angle) from the transformed images. As a result, the extracted image representations are encouraged to become aware of the applied pretext (e.g., image rotation). Thus, we also refer to them as pretext-aware methods. The pretext tasks can be various, to name a few, Rotation [11], Jigsaw [25], Relative Path Location [8], Colorization [34]. Generalized Multi-view Learning. Although there are plenty of works on each branch, how to bridge the two methodologies remains under-explored. Prior to our work, there are only a few works on this direction. Some directly combine AMDIM (pretext-invariant) [1] and Rotation (pretextaware) [11] objectives [10]. However, a direct combination of the two contradictory objectives may harm the final representation. LooC [32] proposes to separate the embedding space to several parts, where each subspace learns local invariance w.r.t. a specific augmentation. But this is achieved at the cost of limiting the representation flexibility of each pretext to the predefined subspace. Different from them, our proposed Prelax provides a more general solution by allowing an adaptive residual vector to encode the semantic shift. In this way, both kinds of features are encoded in the same representation space. 3 The Proposed Pretext-aware Residual Relaxation (Prelax) Method Preliminary Problem Formulation. Given unlabeled data {x i }, unsupervised representation learning aims to learn an encoder network F θ that extracts meaningful low-dimensional representations z ∈ R dz from high-dimensional input images x ∈ R dx . The learned representation is typically evaluated on a downstream classification task by learning a linear classifier with labeled data {x i , y i }. Multi-view Representation Learning. For an input image x ∈ R dx , we can generate a different view by data augmentation, x = t(x), where t ∈ T refers to a randomly drawn augmentation operator from the pretext set T . Then, the transformed input x and the original input x are passed into an online network F θ and a target network F φ , respectively. Optionally, the output of the online network is further processed by an MLP predictor network G θ , to match the output of the target network. As two different views of the same image (i.e., positive samples), x and x should have similar representations, so we align their representations with the following similarity loss, L sim (x , x; θ) = G θ (F θ (x )) − F φ (x) 2 2 .(1) The representations, e.g., z = F θ (x), are typically projected to a unit spherical ball before calculating the distance (z/ z 2 ), which makes the 2 distance equivalent to the cosine similarity [2]. Remark. Aside from the similarity loss between positive samples, contrastive methods [26,16,27,2] further encourage representation uniformity with an additional regularization minimizing the similarity between input and an independently drawn negative sample. Nevertheless, some recent works find that the similarity loss alone already suffices [12,5]. In this paper, we mainly focus on improving the alignment between positive samples in the similarity loss. It can also be easily extended to contrastive methods by considering the dissimilarity regularization. Objective Formulation As we have noticed, the augmentation sometimes may bring a certain amount of semantic shift. Thus, enforcing exact alignment of different views may hurt the representation quality, particularly when the data augmentation is too strong for the positive pairs to be matched exactly. Therefore, we need to relax the exact alignment in Eq. (1) to account for the semantic shift brought by the data augmentation. Residual Relaxed Similarity Loss. Although the representations may not align exactly, i.e., z = z, however, the representation identity will always hold: z − (z − z) = z, where z − z represents the shifted semantics by augmentation. This makes this identity a proper candidate for multi-view alignment under various augmentations as long as the shifted semantic is taken into consideration. Specifically, we replace the exact alignment (denoted as →←) in the similarity loss (Eq. (1)) by the proposed identity alignment, i.e., G θ (z θ ) →← z φ ⇒ G θ (z θ ) − G θ (r) →← z φ ,(2) where we include a residual vector r ∆ = z θ − z θ = F θ (x ) − F θ (x) to represent the difference on the representations. To further enable a better tradeoff between the exact and identity alignments, we have the following residual alignment: G θ (z θ ) − αG θ (r) →← z φ ,(3) where α ∈ [0, 1] is the interpolation parameter. When α = 0, we recover the exact alignment; when α = 1, we recover the identity alignment. We name the corresponding learning objective as the Residual Relaxed Similarity (R2S) loss, which minimizes the squared 2 distance among two sides: L α R2S (x , x; θ) = G θ (F θ (x )) − αG θ (r) − F φ (x) 2 2 .(4) Predictive Learning (PL) Loss. To ensure the relaxation works as expected, the residual r should encode the semantic shift caused by the augmentation, i.e., the pretext. Inspired by predictive learning [11], we utilize the residual to predict the corresponding augmentation for its pretext-awareness. In practice, the assigned parameters for the random augmentation t can be generally divided into the discrete categorical variables t d (e.g., flipping or not, graying or not), and the continuous variables t c (e.g., scale, ratio, jittered brightness). Thus, we learn a PL predictor H θ to predict (t d , t c ) with cross entropy loss (CE) and mean square error loss (MSE), respectively: L PL (x , x, t; θ) = CE(H d θ (r), t d ) + H c θ (r) − t c 2 2 .(5) Constraint on the Similarity. Different from the exact alignment, the residual vector can be unbounded, i.e., the difference between views can be arbitrarily large. This is not reasonable as the two views indeed share many common semantics. Therefore, we should utilize this prior knowledge to prevent the bad cases under residual similarity and add the following constraint L sim = G θ (F θ (x )) − F φ (x) 2 2 ≤ ε,(6) where ε denotes the maximal degree of mismatch allowed between positive samples. The Overall Objective of Prelax. By combining the three components above, we can reliably encode the semantic shift between augmentations while ensuring a good alignment between views: min θ L α R2S (x , x; θ) + γL PL (x , x; θ), s.t. G θ (F θ (x )) − F φ (x) 2 2 ≤ ε.(7) For simplicity, we transform it into a Lagrangian objective with a fixed multiplier β ≥ 0, and obtain the overall Pretext-aware REsidual ReLAXation (Prelax) objective, L α R2S (x , x; θ) + γL PL (x , x; θ) + βL sim (x , x; θ),(8) where α tradeoffs between the exact and identity alignments, γ adjusts the amount of pretextawareness of the residual, and β controlls the degree of similarity between positive pairs. An illustrative diagram of the Prelax objective is shown in Figure 1. Discussions. In fact, there are other alternatives to relax the exact alignment. For example, we can utilize a margin loss L margin (x , x; θ) = max( G θ (F θ (x )) − F φ (x) 2 2 − η, 0),(9) where η > 0 is a threshold for the mismatch tolerance. However, it has two main drawbacks: 1) as each image and augmentation have different semantics, it is hard to choose a universal threshold for all images; and 2) the representation keeps shifting along with the training progress, making it even harder to maintain a proper threshold dynamically. Thus, a good relaxation should be adaptive to the training progress and the aligning of different views. While our Prelax adopts pretext-aware residual vector, which is learnable, flexible, and semantically meaningful. Theoretical Analysis As Prelax encodes both pretext-invariant and pretext-aware features, it can be semantically richer than both multi-view learning and predictive learning. Following the information-theoretic framework developed by [29], we show that Prelax provably enjoys better downstream performance. We denote the random variable of input as X and learn a representation Z through a deterministic encoder F θ : Z = F θ (X) 2 . The representation Z is evaluated for a downstream task T by learning H I h W N x Y P O E u J z N B Y 0 o h h p I w V 2 3 e N I T z B i s B 3 k n u S w 0 5 4 F d s N p O n P A V e K W p A F K d A L 7 y x v F O O V E a M y Q U k P X S b S f I 6 k p Z m R W 8 1 J F E o S n a E y G h g r E i f L z e f g Z P D X K C E a x N E 9 o O F d / b + S I K 5 X x 0 E w W U d W y V 4 j / e c N U R 9 d + T k W S a i L w 4 l C U M q h j W D Q B R 1 Q S r F l m C M K S m q w Q T 5 B E W J u + a q Y E d / n L q 6 R 3 3 n Q v m 8 7 9 R a N 1 U 9 Z R B c f g B J w B F 1 y B F r g D H d A F G G T g G b y C N + v J e r H e r Y / F a M U q d + r g D 6 z P H w g X l F 4 = < / l a t e x i t > L PL < l a t e x i t s h a 1 _ b a s e 6 4 = " c 9 T n u 4 P Y z d r f 6 e A J R 1 B L V 6 e Y 1 p I = " > A A A B / X i c b V C 7 T s M w F H X K q 5 R X e G w s F h U S U 5 V U C B g r W B g Y y q M t U h N F j u u 0 V m 0 n s h 2 k E l X 8 C g s D C L H y H 2 z 8 D U 6 b A V q O Z O n o n H t 1 j 0 + Y M K q 0 4 3 x b p Y X F p e W V 8 m p l b X 1 j c 8 v e 3 m m r O J W Y t H D M Y n k f I k U Y F a S l q W b k P p E E 8 Z C R T j i 8 y P 3 O A 5 G K x u J O j x L i c 9 Q X N K I Y a S M F 9 p 7 H k R 5 g x O B V k H m S w 5 v 6 7 T i w q 0 7 N m Q D O E 7 c g V V C g G d h f X i / G K S d C Y 4 a U 6 r p O o v 0 M S U 0 x I + O K l y q S I D x E f d I 1 V C B O l J 9 N 0 o / h o V F 6 M I q l e U L D i f p 7 I 0 N c q R E P z W S e V c 1 6 u f i f 1 0 1 1 d O Z n V C S p J g J P D 0 U p g z q G e R W w R y X B m o 0 M Q V h S k x X i A Z I I a 1 N Y x Z T g z n 5 5 n r T r N f e k 5 l w f V x v n R R 1 l s A 8 O w B F w w S l o g E v Q B C 2 A w S N 4 B q / g z X q y X q x 3 6 2 M 6 W r K K n V 3 w B 9 b n D 4 1 C l K M = < / l a t e x i t > LR2S < l a t e x i t s h a 1 _ b a s e 6 4 = " W t R e w E D M t o Y E e 7 N 9 V v o o P 7 h J v m I = " > A A A B / X i c b V D L S s N A F L 2 p r 1 p f 8 b F z M 1 g E V y U R U Z d F N y 5 c V L A P a E K Y T K f t 0 J k k z E y E G o q / 4 s a F I m 7 9 D 3 f + j Z M 2 C 2 0 9 M H A 4 5 1 7 u m R M m n C n t O N 9 W a W l 5 Z X W t v F 7 Z 2 N z a 3 r F 3 9 1 o q T i W h T R L z W H Z C r C h n E W 1 q p j n t J J J i E X L a D k f X u d 9 + o F K x O L r X 4 4 T 6 A g 8 i 1 m c E a y M F 9 o E n s B 4 S z N F t k H l S I M X E J L C r T s 2 Z A i 0 S t y B V K N A I 7 C + v F 5 N U 0 E g T j p X q u k 6 i / Q x L z Q i n k 4 q X K p p g M s I D 2 j U 0 w o I q P 5 u m n 6 B j o / R Q P 5 b m R R p N 1 d 8 b G R Z K j U V o J v O s a t 7 L x f + 8 b q r 7 l 3 7 G o i T V N C K z Q / 2 U I x 2 j v A r U Y 5 I S z c e G Y C K Z y Y r I E E t M t C m s Y k p w 5 7 + 8 S F q n N f e 8 5 t y d V e t X R R 1 l O I Q j O A E X L q A O N 9 C A J h B 4 h G d 4 h T f r y X q x 3 q 2 P 2 W j J K n b 2 4 Q + s z x 8 7 B J U V < / l a t e x i t > Lsim < l a t e x i t s h a 1 _ b a s e 6 4 = " y k K J g 7 e c q U T 1 u T g I A r 3 u V G U u c V o = " > A A A C N n i c b V D L S s N A F J 3 U V 6 2 v q E s 3 w S K 6 s S Q i 6 k Y o u n E j V L A P a E K Y T C f t 0 M m D m R u h h n y V G 7 / D X T c u F H H r J z h p K 2 r b C 8 M c z j 2 X e + 7 x Y s 4 k m O Z Q K y w s L i 2 v F F d L a + s b m 1 v 6 9 k 5 D R o k g t E 4 i H o m W h y X l L K R 1 Y M B p K x Y U B x 6 n T a 9 / n f e b D 1 R I F o X 3 M I i p E + B u y H x G M C j K 1 W / t A E P P 8 1 O R X f 7 A x + z Q T W 0 v 4 h 0 5 C N S X 2 t C j g L P s + F c x X + D q Z b N i j s q Y B d Y E l N G k a q 7 + Y n c i k g Q 0 B M K x l G 3 L j M F J s Q B G O M 1 K d i J p j E k f d 2 l b w R A H V D r p 6 O z M O F B M x / A j o V 4 I x o j 9 O 5 H i Q O Y G l T L 3 L a d 7 O T m v 1 0 7 A v 3 B S F s Y J 0 J C M F / k J N y A y 8 g y N D h O U A B 8 o g I l g y q t B e l h g A i r p k g r B m j 5 5 F j R O K t Z Z x b w 7 L V e v J n E U 0 R 7 a R 0 f I Q u e o i m 5 Q D d U R Q U 9 o i N 7 Q u / a s v W o f 2 u d Y W t A m M 7 v o X 2 l f 3 9 c M r 6 M = < / l a t e x i t > r = z 0 ✓ z ✓ < l a t e x i t s h a 1 _ b a s e 6 4 = " p z w u K O G k 2 v 7 i D 2 R A g h O g I N x H F 1 k = " > A A A C F n i c b V D L S s N A F J 3 U V 6 2 v q E s 3 w S L U h S U R U Z d F F 7 q s Y B / Q h D C Z T t q h k w c z N 0 I J + Q o 3 / o o b F 4 q 4 F X f + j Z M 0 C 2 0 9 M M z h 3 H O 5 9 x 4 v 5 k y C a X 5 r l a X l l d W 1 6 n p t Y 3 N r e 0 f f 3 e v K K B G E d k j E I 9 H 3 s K S c h b Q D D D j t x 4 L i w O O 0 5 0 2 u 8 3 r v g Q r J o v A e p j F 1 A j w K m c 8 I B i W 5 + o k d Y B g T z N O b z E 1 t L + J D O Q 3 U l 9 o w p o C z r F E 4 P D 8 V 2 b G r 1 8 2 m W c B Y J F Z J 6 q h E 2 9 W / 7 G F E k o C G Q D i W c m C Z M T g p F s A I p 1 n N T i S N M Z n g E R 0 o G u K A S i c t z s q M I 6 U M D T 8 S 6 o V g F O r v j h Q H M l 9 W O f M V 5 X w t F / + r D R L w L 5 2 U h X E C N C S z Q X 7 C D Y i M P C N j y A Q l w K e K Y C K Y 2 t U g Y y w w A Z V k T Y V g z Z + 8 S L q n T e u 8 a d 6 d 1 V t X Z R x V d I A O U Q N Z 6 A K 1 0 C 1 q o w 4 i 6 B E 9 o 1 f 0 p j 1 p L 9 q 7 9 j G z V r S y Z x / 9 g f b 5 A 2 g y o M Q = < / l a t e x i t > G ✓ (r) < l a t e x i t s h a 1 _ b a s e 6 4 = " + n l Y C 0 G 9 q Z Y o U 0 G D g m 9 X U 6 0 u Then the positive pair (x, x ), is processed by the online network F θ and the target network F φ , respectively. Output of the online network is further processed by the target network G θ , and the gradient of F φ is detached, i.e., stop grad, denoted as sg. Then the outputs are used to compute the three objectives, L R2S (Eq. 4), L PL (Eq. 5), and L sim (Eq. 1) in the Prelax objective (Eq. 7). x T Y = " > A A A C L X i c b V D L S s N A F J 3 4 r P V V d e k m W M S 6 K Y m I u i w q 6 L K C f U A T w m Q y a Y d O H s z c C D X k h 9 z 4 K y K 4 q I h b f 8 N J 2 o W 2 v T D M 4 d x z u e c e N + Z M g m G M t a X l l d W 1 9 d J G e X N r e 2 e 3 s r f f l l E i C G 2 R i E e i 6 2 J J O Q t p C x h w 2 o 0 F x Y H L a c c d 3 u T 9 z h M V k k X h I 4 x i a g e 4 H z K f E Q y K c i q 3 V o B h Q D B P 7 z I n t d y I e 3 I U q C + 1 Y E A B Z 1 m t U L h + + p y d L F a c O p W q U T e K 0 u e B O Q V V N K 2 m U 3 m 3 v I g k A Q 2 B c C x l z z R i s F M s g B F O s 7 K V S B p j M s R 9 2 l M w x A G V d l p c m + n H i v F 0 P x L q h a A X 7 N + J F A c y d 6 i U u X M 5 2 8 v J R b 1 e A v 6 V n b I w T o C G Z L L I T 7 g O k Z 5 H p 3 t M U A J 8 p A A m g i m v O h l g g Q m o g M s q B H P 2 5 H n Q P q u b F 3 X j 4 b z a u J 7 G U U K H 6 A j V k I k u U Q P d o y Z q I Y J e 0 B s a o 0 / t V f v Q v r T v i X R J m 8 4 c o H + l / f w C Q K W r L Q = = < / l a t e x i t > G ✓ (z 0 ✓ ) < l a t e x i t s h a 1 _ b a s e 6 4 = " + C O u s U X P t 9 E W C e X Q U k x H G P 1 3 k 4 M = " > A A A C L H i c d V D L S s N A F J 3 U V 6 2 v q E s 3 w S L U T U l E 1 G W x C 1 1 W s A 9 o S p l M J + 3 Q y S T M 3 A g 1 5 I P c + C u C u L C I W 7 / D S d q F t n p h m M O 5 5 3 L P P V 7 E m Q L b n h q F l d W 1 9 Y 3 i Z m l r e 2 d 3 z 9 w / a K k w l o Q 2 S c h D 2 f G w o p w J 2 g Q G n H Y i S X H g c d r 2 x v W s 3 3 6 g U r F Q 3 M M k o r 0 A D w X z G c G g q b 5 Z d w M M I 4 J 5 c p P 2 E 9 c L + U B N A v 0 l L o w o 4 D S t 5 A r P T x 7 / E Z z 2 z b J d t f O y l o E z B 2 U 0 r 0 b f f H U H I Y k D K o B w r F T X a classifier on top of Z. From an information-theoretic learning perspective, a desirable algorithm should maximize the Mutual Information (MI) between Z and T, i.e., I(Z; T) [6]. Supervised learning on task T can learn representations by directly maximizing I(Z; T). Without access to the labels T, unsupervised learning resorts to maximizing I(Z; S), where S denotes the surrogate signal S designed by each method. Specifically, multi-view learning matches Z with the randomly augmented view, denoted as S v ; while predictive learning uses Z to predict the applied augmentation, denoted as S a . In Prelax, as we combine both semantics, we actually maximize the MI w.r.t. their joint distribution, i.e., I(Z; S v , S a ). We denote the representations learned by supervised learning, multi-view learning, predictive learning, and Prelax as Z sup , Z mv , Z PL , Z Prelax , respectively. Theorem 2. Further assume that T is a K-class categorical variable. In general, we have the upper bound u e on the downstream Bayes errors P e := E z [1 − max t∈T P (T = t|z)], P e ≤ u e := log 2 + P e sup · log K + I(X; T|S). whereP e = Th(P e ) = min{max{P e , 0}, 1 − 1/K} denotes the thresholded Bayes error. Accordingly, we have the following inequalities on the upper bounds for different unsupervised methods, u e sup ≤ u e Prelax ≤ min(u e mv , u e PL ) ≤ max(u e mv , u e PL ). Theorem 1 shows that Prelax extracts more task-relevant information than multi-view and predictive methods, and Theorem 2 further shows that Prelax has a tighter upper bound on the downstream Bayes error. Therefore, Prelax is indeed theoretically superior to previous unsupervised methods by utilizing both pretext-invariant and pretext-aware features. Proofs are in Appendix B. Practical Implementation In this part, we present three practical variants of Prelax to generalize existing multi-view backbones: 1) one with existing multi-view augmentations (Prelax-std); 2) one with a stronger augmentation, image rotation (Prelax-rot); and 3) one with previous two strategies (Prelax-all). Backbone BYOL [12] and SimSiam [5] are both similarity-based methods and they differ mainly in the design of the target network F φ . BYOL [12] utilizes momentum update of the target parameters φ from the online parameters θ, i.e., φ ← τ φ + (1 − τ )θ, where τ ∈ [0, 1] is the target decay rate. While SimSiam [5] simply regards the (stopped-gradient) online network as the target network, i.e., φ ← sg(θ). We mainly take SimSiam for discussion and our analysis also applies to BYOL. For a given training image x, SimSiam draws two random augmentations (t 1 , t 2 ) and get two views (x 1 , x 2 ), respectively. Then, SimSiam maximizes the similarity of their representations with a dual objective, where the two views can both serve as the input and the target to each other, L Simsiam (x; θ) = G θ (F θ (x 1 )) − F φ (x 2 ) 2 2 + G θ (F θ (x 2 )) − F φ (x 1 ) 2 2 .(13) Prelax-std To begin with, we can directly generalize the baseline method with our Prelax method under existing multi-view augmentation strategies [2,12]. For the same positive pair (x 1 , x 2 ), we can calculate their residual vector r 12 = F θ (x 1 ) − F θ (x 2 ) and use it for the R2S loss (Eq. (4)) L α R2S (x 1 , x 2 ; θ) = G θ (F θ (x 1 )) − αG θ (r 12 ) − F φ (x 2 ) 2 2 .(14) We note that there is no difference in using r 12 or r 21 as the two views are dual. Then, we can adopt the similarity loss in the reverse direction as our similarity constraint loss, L sim (x 2 , x 1 ; θ) = G θ (F θ (x 2 )) − F φ (x 1 ) 2 2 .(15) At last, we use the residual r 12 for the PL loss to predict the augmentation parameters of x 1 , i.e., t 1 , because r 12 = z 1 − z 2 directs towards z 1 . Combining the three losses above, we obtain our Prelaxstd objective, L Prelax−std (x; θ) = L α R2S (x 1 , x 2 ; θ) + γL PL (x 1 , x 2 , t 1 ; θ) + βL sim (x 2 , x 1 ; θ).(16) Prelax-rot As mentioned previously, with our residual relaxation we can benefit from stronger augmentations that are harmful for multi-view methods. Here, we focus on the image rotation example and propose the Prelax-rot objective with rotation-aware residual vector. To achieve this, we further generalize existing dual-view methods by incorporating a third rotation view. Specifically, given two views (x 1 , x 2 ) generated with existing multi-view augmentations, we additionally draw a random rotation angle a ∈ R = {0 • , 90 • , 180 • , 270 • } and apply it to rotate x 1 clockwise, leading to the third view x 3 . Note that the only difference between x 3 and x 1 is the rotation semantic a. Therefore, if we substitute x 1 with x 3 in the similarity loss, we should add a rotation-aware residual r 31 = z 3 − z 1 to bridge the gap. Motivated by this analysis, we propose the Rotation Residual Relaxation Similarity (R3S) loss, L α R3S (x 1:3 ; θ) = G θ (F θ (x 3 )) − αG θ (r 31 ) − F φ (x 2 ) 2 2 .(17) which replace G θ (F θ (x 1 )) by its rotation-relaxed version G θ (F θ (x 3 )) − αG θ (r 31 ) in the similarity loss. Comparing the R2S loss (Eq. 14) and the R3S loss, we note that the relaxation of the R2S loss accounts for all the semantic shift between x 1 and x 2 , while that of the R3S loss only accounts for the rotation augmentation between x 1 and x 3 . Therefore, we could use the residual r 31 to predict the rotation angle a with the following RotPL loss for its rotation-awareness: L rot PL (x 1 , x 3 , a; θ) = CE(H θ (r 31 ), a).(18) Combining with the similarity constraint, we obtain the Prelax-rot objective: L Prelax−rot (x; θ) = L α R3S (x 1:3 ; θ) + γL rot PL (x 1 , x 3 , a; θ) + βL sim (x 2 , x 1 ; θ).(19) Prelax-all We have developed Prelax-std that cultivates existing multi-view augmentations and Prelax-rot that incorporates image rotation. Here, we further utilize both existing augmentations and image rotation by combining the two objectives together, denoted as Prelax-all: L Prelax−all (x; θ) = 1 2 (L α1 R2S (x 1 , x 2 ; θ) + L α2 R3S (x 1:3 ; θ)) + γ 1 2 L PL (x 1 , x 2 , t 1 ; θ) + γ 2 2 L rot PL (x 1 , x 3 , a; θ) + βL sim (x 2 , x 1 ; θ),(20) where α 1 , α 2 , γ 1 , γ 2 denotes the coefficients for R2S, R3S, PL and RotPL losses, respectively. Discussions Here we design three practical versions as different implementations of our generic framework of residual relaxation. Among them, Prelax-std focuses on further cultivating existing augmentation strategies, Prelax-rot is to incorporate the stronger (potentially harmful) rotation augmentation, while Prelax-all combines them all. Through the three versions, we demonstrate the wide applicability of Prelax as a generic framework. As for practical users, they could also adapt Prelax to their own application by incorporating specific domain knowledge. In this paper, as we focus on natural images, we take rotation as a motivating example as it is harmful for natural images. Nevertheless, rotation is not necessarily harmful in other domains, e.g., medical images. Instead, random cropping could instead be very harmful for medical images as the important part could lie in the corner. In this scenario, our residual relaxation mechanism could also be used to encode the semantic shift caused by cropping and alleviate its bad effects. Experiments Datasets. Due to computational constraint, we carry out experiments on a range of medium-sized real-world image datasets, including well known benchmarks like CIFAR-10 [17], CIFAR-100 [17], and two ImageNet variants: Tiny-ImageNet-200 (200 classes with image size resized to 32×32) [31] and ImageNette (10 classes with image size 128×128) 3 . Backbones. As Prelax is designed to be a generic method for generalizing existing multi-view methods, we implement it on two different multi-view methods, SimSiam [5] and BYOL [12]. Specifically, we notice that SimSiam reported results on CIFAR-10, while the official code of BYOL included results on ImageNette. For a fair comparison, we evaluate SimSiam and its Prelax variant on CIFAR-10, and evaluate BYOL and its Prelax variant on ImageNette. In addition, we evaluate Sim-Siam and its Prelax variant on two additional datasets CIFAR-100 and Tiny-ImageNet-200, which are more challenging because they include a larger number of classes. For computational efficiency, we adopt the ResNet-18 [15] backbone (adopted in SimSiam [5] for CIFAR-10) to benchmark our experiments. For a comprehensive comparison, we also experiment with larger backbones, like ResNet-34 [15], and the results are included in Appendix C. Setup. For Prelax-std, we use the same data augmentations as SimSiam [2,5] Training. For SimSiam and its Prelax variants, we follow the same hyperparameters in [5] on CIFAR-10. Specifically, we use ResNet-18 as the backbone network, followed by a 3-layer projection MLP, whose hidden and output dimension are both 2048. The predictor is a 2-layer MLP whose hidden layer and output dimension are 512 and 2048 respectively. We use SGD for pre-training with batch size 512, learning rate 0.03, momentum 0.9, weight decay 5 × 10 −4 , and cosine decay schedule [22] for 800 epochs. For BYOL and its Prelax variants, we also adopt the ResNet-18 backbone, and the projector and predictor are 2-layer MLPs whose hidden layer and output dimension are 256 and 4096 respectively. Following the default hyper-parameters on ImageNette 4 , we use LARS optimizer [33] to train 1000 epochs with batch size 256, learning rate 2.0, weight decay 1 × 10 −6 while excluding the biases and batch normalization parameters from both LARS adaptation and weight decay. For the target network, the exponential moving average parameter τ starts from τ base = 0.996 and increases to 1 during training. As for the Prelax objective, we notice that sometimes, adopting a reverse residual r 21 in the R2S loss (Eq. (14)) can bring slightly better results, which could be due to the swapped prediction in SimSiam's dual objective (Eq. (13)). Besides, a naïve choice of Prelax coefficients already works well: α = 1, β = 1, γ = 0.1 for Prelax-std and Prelax-rot, and α 1 = α 2 = 1, β = 1, γ 1 = γ 2 = 0.1 for Prelax-all. More discussion about the hyper-parameters of Prelax can be found in Appendix E. Evaluation. After unsupervised training, we evaluate the backbone network by fine-tuning a linear classifier on top of its representation with other model weights held fixed. For SimSiam and its Prelax variants, the linear classifier is trained on labeled data from scratch using SGD with batch size 256, learning rate 30.0, momentum 0.9 for 100 epochs. The learning rate decays by 0.1 at the 60-th and 80-th epochs. For BYOL and its Prelax variants, we use SGD with Nesterov momentum over 80 epochs using batch size 25, learning rate 0.5 and momentum 0.9. Besides the in-domain linear evaluation, we also evaluate its transfer learning performance on the out-of-domain data by learning a linear classifier on the labeled target domain data. Performance on Benchmark Datasets CIFAR-10. In Table 1a, we compare Prelax against previous multi-view methods (SimCLR [2], SimSiam [5], and BYOL [12]) and predictive methods (Rotation [11]) on CIFAR-10. We can see that multi-view methods are indeed better than predictive ones. Nevertheless, predictive learning alone (e.g., Rotation) achieves quite good performance, indicating that pretext-aware features are also very useful. By encoding both pretext-invariant and pretext-aware features, Prelax outperforms previous methods by a large margin, and achieve state-of-the-art performance on CIFAR-10. A comparison of the learning dynamics between SimSiam and Prelax can be found in Appendix F. ImageNette. Beside the SimSiam backbone, we further apply our Prelax loss to the BYOL framework [12] and evaluate the two methods on the ImageNette dataset. In Table 1b, Prelax also shows a clear advantage over BYOL. Specifically, it improves the ResNet-18 version of BYOL by 0.7%, and even outperforms the ResNet-50 version by 0.3%. Here, we can see that Prelax yields significant improvement on two different datasets with two different backbone methods. Thus, Prelax could serve as a generic method for improving existing multi-view methods by encoding pretext-aware features into the residual relaxation. For completeness, we also evaluate Prelax on the large scale dataset, ImageNet [7], as well as its transferability to other kinds of downstream tasks, such as object detection and instance segmentation on MS COCO [21]. As shown in Appendix D, Prelax still consistently outperforms the baselines across all tasks. Effectiveness of Prelax Variants For a comprehensive comparison of the three variants of Prelax objectives (Prelax-std, Prelax-rot, and Prelax-all), we conduct controlled experiments on a range of datasets based on the SimSiam backbone. Except CIFAR-10, we also conduct experiments on CIFAR-100 and Tiny-ImageNet-200, which are more challenging with more classes of images. For a fair comparison, we use the same training and evaluation protocols across all tasks. In-domain Linear Evaluation. As shown in Table 2a, our Prelax objectives outperform the multiview objective consistently on all three datasets, where Prelax-all improves SimSiam by 1.6% on CIFAR-10, 3.1% on CIFAR-100, and 1.5% on Tiny-ImageNet-200. Besides, Prelax-std and Prelaxrot are also better than SimSiam in most cases. Thus, the pretext-aware residuals in Prelax indeed help encode more useful semantics. Out-of-domain Linear Evaluation. Besides the in-domain linear evaluation, we also transfer the representations to a target domain. In the out-of-domain linear evaluation results shown in Table 2b, the Prelax objectives still have a clear advantage over the multi-view objective (SimSiam), while sometimes Prelax-std and Prelax-rot enjoy better transferred accuracy than Prelax-all. Empirical Understandings of Prelax Comparison against Alternative Options. In Table 3a, we compare Prelax against several other relaxation options. "SimSiam + margin" refers to the margin loss discussed in Eq. (9), which uses a scalar η to relax the exact alignment in multi-view methods. Here we tune the margin η = 0.5 with the best performance. Nevertheless, it has no clean advantage over SimSiam. Then, we try several options for incorporating a strong augmentation and image rotation: 1) Rotation is the PL baseline by predicting rotation angles [11], which is inferior to multi-view methods (SimSiam). 2) "SimSiam + rotation aug." directly applies a random rotation augmentation to each view, and learn with the SimSiam loss. However, it leads to lower accuracy, showing that the image rotation, as a strong augmentation, will hurt the performance of multi-view methods. 3) "SimSiam + Rotation" directly combines the SimSiam loss and the Rotation loss for training, which is still ineffective. 4) Our Prelax shows a significant improvement over SimSiam and other variants, showing that the residual alignment is an effective mechanism for utilizing strong augmentations like rotation. Ablation Study. We perform ablation study of each component of the Prelax objectives on CIFAR-10. From Table 3b, we notice that simply adding the PL loss alone cannot improve over SimSiam consistently, for example, Sim + RotPL causes 0.7 point drop in test accuracy. While with the help of our residual relaxation, we can improve over the baselines significantly and consistently, for example, Prelax-rot (Sim + RotPL + R3S) brings 0.6 point improvement on test accuracy. Besides, we can see that the PL loss is necessary by making the residual pretext-aware, without which the performance drops a lot, and the similarity constraint (Sim loss) is also important by avoiding bad cases when augmented images drift far from the anchor image. Therefore, the ablation study shows the residual relaxation loss, similarity loss, and PL loss all matter in our Prelax objectives. Qualitative Analysis Representation Visualization. To provide an intuitive understanding of the learned representations, we visualize them with t-SNE [30] on Figure 3a. We can see that in general, our Prelax can learn well-separated clusters of representations corresponding to the ground-truth image classes. Image Retrieval. In Figure 3b, we evaluate Prelax on an image retrieval task. Given a random query image (not cherry-picked), the top-15 most similar images in representation space are retrieved, and the query itself is shown in the first column. We can see that although the unsupervised training with Prelax has no access to labels, the retrieved nearest images of Prelax are all correctly from the same class and semantically consistent with the query. Conclusion In this paper, we proposed a generic method, Prelax (Pretext-aware Residual Relaxation), to account for the (possibly large) semantic shift caused by image augmentations. With pretext-aware learning of the residual relaxation, our method generalizes existing multi-view learning by encoding both pretext-aware and pretext-invariant representations. Experiments show that our Prelax has outperformed existing multi-view methods significantly on a variety of benchmark datasets. B Theoretical Results and Proofs Here, we provide the complete proof of the theoretical results in Section 3.3. More rigorously, we give the definition of minimal and sufficient representations for self-supervision [29], and give a more formal description of our results. Definition 1 (Minimal and Sufficient Representations for Signal S). Let Z * be the minimal and sufficient representation for self-supervised signal S if it satisfies the following conditions in the meantime: 1) Z * is sufficient, Z * = arg max Z I (Z; S); 2) Z * is minimal, i.e., Z * = argmin Z H (Z|S). The following lemma shows that the maximal mutual information of I(Z * , S) is I(X, S). Lemma 3. For a minimal and sufficient representation Z that is obtained with a deterministic encoder F θ of input X with enough capacity, we have I(Z * ; S) = I(X; S). Proof. As the encoder F θ is deterministic, it induces the following conditional independence: S ⊥ ⊥ Z | X, which leads to the data processing Markov chain S ↔ X → Z. Accordingly to the data processing inequality (DIP) [6], we have I(Z; S) ≤ I(X; S), and with enough model capacity in F θ , the sufficient and minimal representation Z * will have I(Z * ; S) = max Z I(Z; S) = I(X; S). In the main text, we introduce several kinds of learning signals, the target variable T, the multiview signal S v , the predictive learning signal S a , and the joint signal (S v , S a ) used by our Prelax method. For clarity, we denote the learned minimal and sufficient representations as Z sup , Z mv , Z PL , Z Prelax , respectively. Next, we restate Theorem 1 with the definitions above and provide a complete proof. Theorem 4 (restated). We have the following inequalities on the four minimal and sufficient representations, Z sup , Z mv , Z PL , Z Prelax : I(X; T) = I(Z sup ; T) ≥ I(Z Prelax ; T) ≥ max(I(Z mv ; T), I(Z PL ; T)).(21) Proof. By Lemma 3, we have the following properties in the self-supervised representations: I Besides, because Z is minimal, we also have, I(Z; T|S) ≤ H(Z|S) = 0.(24) Together with the two equalities above, we further have the following equality on I(Z; T): I Therefore, the gap between supervised representation Z sup and each self-supervised representation Z ∈ {Z mv , Z PL , Z Prelax } is I(X; T|S), for which we have the following inequalities: which completes the proof. Remark. Theorem 4 shows that the downstream performance gap between supervised representation Z sup and self-supervised representation Z is I(X; T|S), i.e., the information left in X about the target variable T except that in S. Thus, if we choose a self-supervised signal S such that the left information is relatively small, we can guarantee a good downstream performance. Comparing the three self-supervised methods with learning signal S v , S a , and (S v , S a ), we can see that our Prelax utilizes more information in X, and consequently, the left information I(X; T|S v , S a ) is smaller than both multi-view methods I(X; T|S a ) and predictive methods I(X; T|S a ). In the following theorem, we further show that our Prelax has a tighter upper bound on the Bayes error of downstream classification tasks. To begin with, we prove a relationship between the supervised and self-supervised Bayes errors following [29]. Lemma 5. Assume that T is a K-class categorical variable. We define the Bayes error on downstream task T as P e := E z 1 − max t∈T P (T = t|z) .(28) Denote the Bayes error of self-supervised learning (SSL) methods with signal S as P e ssl and that of supervised methods as P e sup . Then, we can show that the SSL Bayes error P e ssl can be upper bounded by the supervised Bayes error P e sup , i.e., P e ssl ≤ u e := log 2 + P e sup · log K + I(X; T|S). whereP e = Th(P e ) = min{max{P e , 0}, 1 − 1/K} denotes the thresholded Bayes error in the feasible region, and u e denote the value of the upper bound. Proof. Denote the minimal and sufficient representations learned by SSL and supervised methods as Z ssl and Z sup , respectively. We use two following inequalities from [9] and [6], P e ssl ≤ − log (1 − P e ssl ) ≤ H (T | Z ssl ) ,(30) which completes the proof. Given the upper bound in Lemma 5, and the inequalities on the downstream performance gap I(X; T|S) in Eq. (26), we will arrive at the following inequalities on the upper bounds on the self-supervised representations. Theorem 6 (restated). We denote the the upper bound on the Bayes error of each representation, Z sup , Z mv , Z PL , Z Prelax , by u e sup , u e mv , u e PL , u e Prelax , respectively. Then, they satisfy the following inequalities: u e sup ≤ u e Prelax ≤ min(u e mv , u e PL ) ≤ max(u e mv , u e PL ). Theorem 6 shows that our Prelax enjoys a tighter lower bounds on downstream Bayes error than both multi-view methods and predictive methods. Evaluation protocol. For downstream evaluation, we report both the linear evaluation task on Ima-geNet and two transfer learning tasks on the MS COCO dataset [21]. Specifically, we perform object detection on the standard RetinaNet [20] with FPN [19], and conduct instance segmentation on the standard Mask R-CNN [14] with FPN [19]. We compare the performances of models initialized with different pretrained weights on COCO: • RandInit: randomly initialized weights; • BYOL: unsupervised pretrained weights with BYOL; • Prelax (ours): unsupervised pretrained weights with Prelax; • Supervised (oracle): supervised pretrained weights. From Table 5, we can see that even on the large-scale dataset, our Prelax still has a clear advantage over BYOL on all downstream tasks, including both in-domain linear evaluation and out-of-domain instance segmentation and object detection tasks. E Sensitivity Analysis of Prelax Coefficients Here we provide a detailed discussion on the effect of each coefficient of our Prelax objectives. We adopt the default hyper-parameters unless specified. For Prelax-std, it has three coefficients, the R2S interpolation coefficient α, the similarity loss coefficient β, and the predictive loss coefficient γ. From Figure 4a, we can see that a positive α introduces certain degree of residual relaxation to the exact alignment and help improve the downstream performance. The best accuracy is achieved with a medium α at around 0.5. In addition, a large similarity coefficient β tends to yield better performance, showing the necessity of the similarity constraint. Nevertheless, too large β can also diminish the effect of residual relaxation and leads to slight performance drop. At last, a positive PL coefficient γ is shown to yield better representations, although it might lead to representation collapse if it is too large, e.g., γ > 0.5. For Prelax-rot, as shown in Figure 4b, the behaviors of β and γ are basically consistent with Prelaxstd. Nevertheless, we can see that only α = 1 can yield better results than the SimSiam baseline, while other alternatives cannot. This could be due to the fact that the residual relaxation involves the first view x 1 and its rotation-augmented view x 3 , and the R3S loss is designed between x 3 and the second view x 2 . Therefore, in order to align x 3 and x 2 like the alignment between x 1 and x 2 , all the relaxation information in x 3 (which x 1 does not have) must be accounted for, which corresponds to α = 1 in R3S loss. We show that incorporating the rotation information in this way will indeed richer representation semantics and better performance. Besides, we also find that in certain cases, adopting a reverse residual r 21 in the R2S loss can bring slightly better results. In Figure 5, we investigate this phenomenon by comparing the normal and reverse residuals in R2S loss (applied for Prelax-std and Prelax-all) and R3S loss (applied for Prelax-rot). We can see that for R2S loss, using a reverse residual improves the accuracy by around 0.3 point, while for R3S loss, the reverse residual leads to dramatic degradation. This could be due to that R2S relaxes the gap between x 1 and x 2 , whose representations are learned through swapped prediction in SimSiam's dual objective. Thus, we might also need to swap the direction of the residual to be consistent. Instead, in R3S, the relaxation is crafted between x 1 and x 3 , so we do not need to swap the direction. Last but not least, we note that with the normal residual, Prelax-std and Prelax-all still achieve significantly better results than the SimSiam baseline, and the reverse residual can further improve on it. F Learning Dynamics In Figure 6, we compare SimSiam with Prelax-rot in terms of the learning dynamics. We can see that with our residual relaxation technique, both the relaxation loss and the similarity loss become larger than SimSiam. In particular, the size of the residual indeed converges to a large value with Prelax (1.1) than with SimSiam (0.6). As for the downstream classification accuracy, we notice that Prelax-rot starts with a lower accuracy, but converges to a large accuracy at last. This indicates that Prelax-rot learns to encode more image semantics, which may be harder to learn at first, but will finally outperform the baseline with better representation ability. example of residual relaxation. Figure 1 : 1Left: the effect of different augmentations of CIFAR-10 test images with a supervised model (trained without using any data augmentation, more details in Appendix A) Figure 2 : 2A diagram of our proposed Prelax objective. An image x is firstly augmented as x . Theorem 1 . 1Assume that by maximizing the mutual information, each method can retain all information in X about the learning signal S (or T), i.e., I(X; S) = max Z I(Z; S). Then we have the following inequalities on their task-relevant information I(Z; T): I(X; T) = I(Z sup ; T) ≥ I(Z Prelax ; T) ≥ max(I(Z mv ; T), I(Z PL ; T)). Figure 3 : 3(a) Representation visualization of our Prelax on CIFAR-10 test set. Each point represents an image representation and its color denotes the class of the image. (b) On CIFAR-10 test set, given 10 random queries (not cherry-picked), we retrieve 15 nearest images in the representation space with Prelax (ours). (Z mv ; S v ) = I(X; S v ), I(Z PL ; S a ) = I(X; S a ), I(Z Prelax ; S v , S a ) = I(X; S v , S a ). (22) Thus, for each minimal and sufficient self-supervised representation Z ∈ {Z mv , Z PL , Z Prelax } and the corresponding signal S ∈ {S v , S a , (S v , S a )}, we have, I(Z; S; T) = I(X; S; T), I(Z; S|T) = I(X; S|T). (Z; T) = I(Z; T; S) + I(Z; T|S) = I(X; T; S) + I(Z; T|S) 0 = I(X; T) − I(X; T|S) = I(Z sup ; T) − I(X; T|S). max(I(X; T|S v ), I(X; T|S a )) ≥ min(I(X; T|S v ), I(X; T|S a )) ≥ I(X; T|S v , S a ). (26) Further combining with Lemma 3 and Eq. (25), we arrive at the inequalities on the target mutual information: I(X; T) = I(Z sup ; T) ≥ I(Z Prelax ; T) ≥ max(I(Z mv ; T), I(Z PL ; T)), H(T|Z sup ) ≤ log 2 + P e sup log K. (31) Comparing H(T|Z) and H(T|Z sup ), together with Eq. (25), we can show that they are tied with the following equality, H(T|Z ssl ) = H(T) − I(Z ssl ; T) = H(T) − I(Z sup ; T) + I(X; T|S) = H(T|Z sup ) + I(X; T|S). Eq. (30) & (31), we have P e ssl ≤ H (T | Z ssl ) = H(T|Z sup ) + I(X; T|S) ≤ log 2 + P e sup log K + I(X; T|S) := u e , Figure 4 : 4Linear evaluation results of different Prelax-std and Prelax-rot coefficients on CIFAR-10 with SimSiam backbone. The dashed blue line denotes the result of the SimSiam baseline. Figure 5 : 5Comparison of normal and reverse residuals for Prelax variants on CIFAR-10 with Sim-Siam backbone. Figure 6 : 6A comparison of learning dynamics between SimSiam [5] and Prelax (ours) on CIFAR-10. Left: linear evaluation accuracy (%) on the test set per epoch. Middle: similarity loss per epoch. Right: norm of the residual vector (i.e., r 31 2 ) per epoch. PyTorch notations. For Prelax-rot and Prelax-all, we further apply a random image rotation at last of the transformation, where the angles are drawn randomly from {0 • , 90 • , 180 • , 270 • }. To generate targets for the PL objective in Prelax, for each image, we collect the assigned parameters in each random augmentation, such as crop centers, aspect ratios, rotation angles, etc. More details can be found in Appendix A.(or BYOL [12]), in- cluding RandomResizedCrop, RandomHorizontalFlip, ColorJitter, and RandomGrayscale, etc us- ing the Table 1 : 1Linear evaluation on CIFAR-10 (a) and ImageNette (b) with ResNet-18 backbone. TTA: Test-Time Augmentation. (a) CIFAR-10. Method Acc. (%) Supervised [15] (re-produced) 95.0 Rotation [11] (re-produced) 88.3 BYOL [12] (re-produced) 91.1 SimCLR [2] 91.1 SimSiam [5] 91.8 SimSiam + Prelax 93.4 (b) ImageNette. Method Acc. (%) Supervised 91.0 Supervised + TTA 92.2 BYOL [12] (ResNet-18) 91.9 BYOL [12] (ResNet-50) 92.3 BYOL + Prelax (ResNet-18) 92.6 Table 2 : 2A detailed comparison of SimSiam [5] and Prelax (ours) across three datasets: CIFAR-10 (C10), CIFAR-100 (C100), and Tiny-ImageNet-200 (Tiny200) with the same hyper-parameters. (a) In-domain linear evaluation. Method CIFAR-10 CIFAR-100 Tiny-ImageNet-200 SimSiam [5] 91.8 66.9 47.7 SimSiam + Prelax-std 92.5 67.5 47.9 SimSiam + Prelax-rot 92.4 67.3 47.1 SimSiam + Prelax-all 93.4 70.0 49.2 (b) Out-of-domain linear evaluation. Method C100 → C10 Tiny200 → C10 Tiny200 → C100 SimSiam [5] 44.1 43.9 21.8 SimSiam + Prelax-std 45.0 45.1 21.8 SimSiam + Prelax-rot 45.0 45.1 22.0 SimSiam + Prelax-all 44.9 44.6 22.1 Table 3 : 3Linear evaluation results of possible mechanisms for generalized multi-view learning on CIFAR-10 with SimSiam backbone. (a) Comparison against alternative options. Method Acc. (%) SimSiam [5] 91.8 SimSiam + margin loss 91.9 Rotation [11] 88.3 SimSiam + rotation aug. 87.9 SimSiam + Rotation loss 91.7 SimSiam + Prelax (ours) 93.4 (b) Ablation study. Method Acc. (%) Sim (i.e., SimSiam [5]) 91.8 Sim + PL 92.2 Sim + R2S 91.5 R2S + PL 91.7 Sim + PL + R2S (Prelax-std) 92.5 Sim + RotPL 91.1 Sim + R3S 91.9 R3S + RotPL 79.8 Sim + RotPL + R3S (Prelax-rot) 92.4 Table 4 : 4Linear evaluation accuracy (%) with ResNet-34 backbone. Method CIFAR-10 CIFAR-100 Tiny-ImageNet-200SimSiam [5] 91.2 60.9 39.0 SimSiam + Prelax-std 92.4 67.6 48.4 SimSiam + Prelax-rot 93.0 67.0 40.9 SimSiam + Prelax-all 93.9 69.3 49.4 Table 5 : 5Evaluation of different pretraining methods on the downsampled ImageNet dataset (128x128) with ResNet-18 backbone.C Evaluation with Larger Backbone NetworksIn the main text, we conduct experiments with the ResNet-18 backbone network. Here, for completeness, we further evaluate our Prelax with larger backbone networks. Specifically, for SimSiam variants, we evaluate the ResNet-34[15] across three datasets, CIFAR-10, CIFAR-100, and Tiny-ImageNet-200. For a fair comparison, we adopt the same hyper-parameters as for the ResNet-18 backbone. As can be seen forTable 4, all our Prelax variants achieves better results than the Sim-Siam baseline on all three datasets. Specifically, we can see that our Prelax-all variant attains the best results and it achieves better results with a larger backbone. Besides, we also experiment with ResNet-50 for the BYOL variant, where our Prelax variant also achieves better performance by improving from 92.3% to 92.7%.D Evaluation on Large Scale DatasetsSetup. Although we cannot carry out the full ImageNet experiments with limited time and computation, we gather some preliminary results on the downsampled ImageNet dataset (128x128) with the ResNet-18 backbone. For a fair comparison, our experiments are conducted with the official code of BYOL. All models are trained for 100 epochs with the default hyperparameters.(a) Linear Evaluation. Method Acc (%) BYOL 49.2 Prelax (ours) 51.1 (b) Object Detection. Method AP 50 AP AP 75 RandInit 32.7 19.5 20.1 BYOL 36.6 22.0 22.8 Prelax (ours) 38.1 23.3 23.9 Supervised 39.4 24.2 25.3 (c) Instance Segmentation. Method MAP RandInit 15.8 BYOL 18.3 Prelax (ours) 19.5 Supervised 20.4 We use capitals to denote the random variable, e.g., X, and use lower cases to denote its outcome, e.g., x. https://github.com/fastai/imagenette 4 https://github.com/deepmind/deepmind-research/tree/master/byol https://github.com/kuangliu/pytorch-cifar Learning representations by maximizing mutual information across views. Philip Bachman, Devon Hjelm, William Buchwalter, arXiv:1906.0091013arXiv preprintPhilip Bachman, R Devon Hjelm, and William Buchwalter. Learning representations by max- imizing mutual information across views. arXiv preprint arXiv:1906.00910, 2019. 1, 2, 3 A simple framework for contrastive learning of visual representations. ICML. 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Jean-Bastien Grill, Florian Strub, C Florent Altché, Pierre H Tallec, Elena Richemond, C Buchatskaya, Bernardo Doersch, Zhaohan Daniel Avila Pires, Mohammad Gheshlaghi Guo, B Azar, K Piot, Rémi Kavukcuoglu, Michal Munos, Valko, NeurIPS. 812Jean-Bastien Grill, Florian Strub, Florent Altché, C. Tallec, Pierre H. Richemond, Elena Buchatskaya, C. Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Ghesh- laghi Azar, B. Piot, K. Kavukcuoglu, Rémi Munos, and Michal Valko. Bootstrap your own latent: A new approach to self-supervised learning. NeurIPS, 2020. 1, 2, 3, 5, 6, 7, 8, 12 Momentum contrast for unsupervised visual representation learning. Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, Ross Girshick, CVPR. 12Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. CVPR, 2020. 1, 2 Mask R-CNN. Kaiming He, Georgia Gkioxari, Piotr Dollár, Ross Girshick, ICCV. 15Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick. Mask R-CNN. In ICCV, 2017. 15 Deep residual learning for image recognition. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, CVPR. 1215Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CVPR, 2016. 7, 8, 12, 15 Learning deep representations by mutual information estimation and maximization. ICLR. Alex R Devon Hjelm, Samuel Fedorov, Karan Lavoie-Marchildon, Phil Grewal, Adam Bachman, Yoshua Trischler, Bengio, 13R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. ICLR, 2019. 1, 3 Learning multiple layers of features from tiny images. Alex Krizhevsky, Geoffrey Hinton, Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009. 7 A survey of multi-view representation learning. Yingming Li, Ming Yang, Zhongfei Zhang, IEEE transactions on Knowledge and Data Engineering. 3110Yingming Li, Ming Yang, and Zhongfei Zhang. A survey of multi-view representation learn- ing. IEEE transactions on Knowledge and Data Engineering, 31(10):1863-1883, 2018. 2 Kaiming He, Bharath Hariharan, and Serge Belongie. Feature pyramid networks for object detection. Tsung-Yi Lin, Piotr Dollár, Ross Girshick, CVPR. 15Tsung-Yi Lin, Piotr Dollár, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Be- longie. Feature pyramid networks for object detection. In CVPR, 2017. 15 Kaiming He, and Piotr Dollár. Focal loss for dense object detection. Tsung-Yi Lin, Priya Goyal, Ross Girshick, ICCV. 15Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár. Focal loss for dense object detection. In ICCV, 2017. 15 Microsoft coco: Common objects in context. Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, C Lawrence Zitnick, ECCV. 815Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In ECCV, 2014. 8, 15 Sgdr: Stochastic gradient descent with warm restarts. Ilya Loshchilov, Frank Hutter, Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. ICLR, 2017. 7 Evaluating self-supervised pretraining without using labels. Sean Metzger, Aravind Srinivas, Trevor Darrell, Kurt Keutzer, arXiv:2009.07724arXiv preprintSean Metzger, Aravind Srinivas, Trevor Darrell, and Kurt Keutzer. Evaluating self-supervised pretraining without using labels. arXiv preprint arXiv:2009.07724, 2020. 1 Self-supervised learning of pretext-invariant representations. Ishan Misra, Laurens Van Der Maaten, CVPR. 2Ishan Misra and Laurens van der Maaten. Self-supervised learning of pretext-invariant repre- sentations. CVPR, 2020. 2 Unsupervised learning of visual representations by solving Jigsaw puzzles. ECCV. Mehdi Noroozi, Paolo Favaro, 13Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving Jigsaw puzzles. ECCV, 2016. 1, 3 Aaron Van Den Oord, Yazhe Li, Oriol Vinyals, arXiv:1807.03748Representation learning with contrastive predictive coding. 13arXiv preprintAaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018. 1, 3 Yonglong Tian, Dilip Krishnan, Phillip Isola, arXiv:1906.05849Contrastive multiview coding. 13arXiv preprintYonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019. 1, 3 What makes for good views for contrastive learning. Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, Phillip Isola, NeurIPS. 1Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. NeurIPS, 2020. 1 Selfsupervised learning from a multi-view perspective. ICLR. Yao-Hung Hubert Tsai, Yue Wu, Ruslan Salakhutdinov, Louis-Philippe Morency, 414Yao-Hung Hubert Tsai, Yue Wu, Ruslan Salakhutdinov, and Louis-Philippe Morency. Self- supervised learning from a multi-view perspective. ICLR, 2021. 4, 13, 14 Visualizing data using t-SNE. Laurens Van Der Maaten, Geoffrey Hinton, Journal of Machine Learning Research. 911Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of Machine Learning Research, 9(11), 2008. 10 . Jiayu Wu, Qixiang Zhang, Guoxi Xu, Tiny ImageNet challenge. 7Jiayu Wu, Qixiang Zhang, and Guoxi Xu. Tiny ImageNet challenge, 2017. 7 . Tete Xiao, Xiaolong Wang, Alexei A Efros, Trevor Darrell, What should not be contrastive in contrastive learning. ICLR, 2021. 3Tete Xiao, Xiaolong Wang, Alexei A Efros, and Trevor Darrell. What should not be contrastive in contrastive learning. ICLR, 2021. 3 Scaling sgd batch size to 32k for ImageNet training. Yang You, Igor Gitman, Boris Ginsburg, arXiv:1708.03888arXiv preprintYang You, Igor Gitman, and Boris Ginsburg. Scaling sgd batch size to 32k for ImageNet training. arXiv preprint arXiv:1708.03888, 2017. 7 . Richard Zhang, Phillip Isola, Alexei A Efros, Colorful image colorization. ECCV. 13Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. ECCV, 2016. 1, 3 Specifically, we first train a state-of-the-art supervised ResNet-18 with 95.01% test accuracy on CIFAR-10. 5 . The supervised training uses no data augmentations. Afterwards, we evaluate the effect of different augmentations to the supervised model by applying each one (separately) to pre-process the test images of CIFAR-10. All of the included augmentations (except Rotation) belong to the augmentations used in SimSiam. For a fair comparison, we adopt the same configuration as in SimSiam and refer to the paper for more details. For Rotation. Table 1, we compare different augmentations with a supervised ResNet-18 [15] on CIFAR-10 test set. A Experimental Details Evaluating Augmentations. we adopt the same configuration as [11], where we sample a random rotation angle {0 • , 90 • , 180 • , 270 • } and use it to rotate the raw image clock-wiseA Experimental Details Evaluating Augmentations. In Table 1, we compare different augmentations with a supervised ResNet-18 [15] on CIFAR-10 test set. Specifically, we first train a state-of-the-art supervised ResNet-18 with 95.01% test accuracy on CIFAR-10. 5 . The supervised training uses no data aug- mentations. Afterwards, we evaluate the effect of different augmentations to the supervised model by applying each one (separately) to pre-process the test images of CIFAR-10. All of the included augmentations (except Rotation) belong to the augmentations used in SimSiam. For a fair com- parison, we adopt the same configuration as in SimSiam and refer to the paper for more details. For Rotation, we adopt the same configuration as [11], where we sample a random rotation angle {0 • , 90 • , 180 • , 270 • } and use it to rotate the raw image clock-wise. For a fair comparison, we utilize the same augmentations in SimSiam [5], while collecting the augmentation parameters as the target variables for our Predictive Learning (PL) objective in Prelax. We adopt the PyTorch notations for simplicity. Specifically, for RandomResizedCrop, the operation randomly draws an (i, j, h, k) pair, where (i, j) denotes the center coordinates of the cropped region, while (h, k) denotes the height and width of the cropped region. Accordingly, we calculate the relative coordinates, the area ratio, and the aspect ratio (relative to the raw image), as four continuous target variables. Similarly, the ColorJitter opration randomly samples four factors corresponding to the adjustment for brightness, contrast, saturation, hue, respectively. We collect them as four additional continuous target variables. As for operations like RandomHorizontalFlip, RandomGrayscale, RandomApply, they draw a binary variable with 0/1 outcome according to a predefined probability p, and apply the augmentations if it is 1 and do nothing otherwise. Data Augmentations and PL Targets. We offer details of the augmentations by taking the Sim-Siam [5] variant of Prelax as an example. The BYOL [12] variants are implemented in the same way. We collect these random outcomes (0/1) as discrete target variables. As for the rotation operation, we take the rotation angles randomly drawn from the set {0 • , 90 • , 180 • , 270 • }, as a discrete 4-class categorical variableData Augmentations and PL Targets. We offer details of the augmentations by taking the Sim- Siam [5] variant of Prelax as an example. The BYOL [12] variants are implemented in the same way. For a fair comparison, we utilize the same augmentations in SimSiam [5], while collecting the augmentation parameters as the target variables for our Predictive Learning (PL) objective in Prelax. We adopt the PyTorch notations for simplicity. Specifically, for RandomResizedCrop, the operation randomly draws an (i, j, h, k) pair, where (i, j) denotes the center coordinates of the cropped region, while (h, k) denotes the height and width of the cropped region. Accordingly, we calculate the relative coordinates, the area ratio, and the aspect ratio (relative to the raw image), as four continuous target variables. Similarly, the ColorJitter opration randomly samples four factors corresponding to the adjustment for brightness, contrast, saturation, hue, respectively. We collect them as four additional continuous target variables. As for operations like RandomHorizontalFlip, RandomGrayscale, RandomApply, they draw a binary variable with 0/1 outcome according to a predefined probability p, and apply the augmentations if it is 1 and do nothing otherwise. We collect these random outcomes (0/1) as discrete target variables. As for the rotation operation, we take the rotation angles randomly drawn from the set {0 • , 90 • , 180 • , 270 • }, as a discrete 4-class categorical variable.

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{'abstract': 'Multi-view methods learn representations by aligning multiple views of the same image and their performance largely depends on the choice of data augmentation. In this paper, we notice that some other useful augmentations, such as image rotation, are harmful for multi-view methods because they cause a semantic shift that is too large to be aligned well. This observation motivates us to relax the exact alignment objective to better cultivate stronger augmentations. Taking image rotation as a case study, we develop a generic approach, Pretext-aware Residual Relaxation (Prelax), that relaxes the exact alignment by allowing an adaptive residual vector between different views and encoding the semantic shift through pretext-aware learning. Extensive experiments on different backbones show that our method can not only improve multi-view methods with existing augmentations, but also benefit from stronger image augmentations like rotation.', 'arxivid': '2110.15348', 'author': ['Yifei Wang \nSchool of Mathematical Sciences\nPeking University\nChina\n', 'Zhengyang Geng \nSchool of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina\n', 'Feng Jiang \nSchool of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina\n', 'Chuming Li \nSchool of Engineering\nThe University of Sydney\nAustralia\n', 'Yisen Wang \nSchool of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina\n', 'Jiansheng Yang \nSchool of Mathematical Sciences\nPeking University\nChina\n', 'Zhouchen Lin \nSchool of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina\n\nPazhou Lab\n510330GuangzhouChina\n'], 'authoraffiliation': ['School of Mathematical Sciences\nPeking University\nChina', 'School of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina', 'School of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina', 'School of Engineering\nThe University of Sydney\nAustralia', 'School of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina', 'School of Mathematical Sciences\nPeking University\nChina', 'School of EECS\nKey Lab. of Machine Perception\nPeking Univesity\nChina', 'Pazhou Lab\n510330GuangzhouChina'], 'corpusid': 240070926, 'doi': None, 'github_urls': ['https://github.com/fastai/imagenette', 'https://github.com/deepmind/deepmind-research/tree/master/byol', 'https://github.com/kuangliu/pytorch-cifar'], 'n_tokens_mistral': 22842, 'n_tokens_neox': 19850, 'n_words': 12998, 'pdfsha': '57a45db07dc818d7f7bb9f43c5307b2e5651d7c4', 'pdfurls': ['https://arxiv.org/pdf/2110.15348v1.pdf'], 'title': ['Residual Relaxation for Multi-view Representation Learning', 'Residual Relaxation for Multi-view Representation Learning'], 'venue': []}

arxiv

All-loop correlators of integrable λ-deformed σ-models 31 May 2016 George Georgiou georgiou@inp.demokritos.gr Institute of Nuclear and Particle Physics National Center for Scientific Research Demokritos Ag. ParaskeviGR-15310AthensGreece Konstantinos Sfetsos ksfetsos@phys.uoa.gr Department of Nuclear and Particle Physics Faculty of Physics National and Kapodistrian University of Athens 15784AthensGreece Konstantinos Siampos siampos@itp.unibe.ch Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics / Laboratory for High-Energy Physics University of Bern Sidlerstrasse 5CH3012BernSwitzerland All-loop correlators of integrable λ-deformed σ-models 31 May 2016 We compute the 2-and 3-point functions of currents and primary fields of λ-deformed integrable σ-models characterized also by an integer k. Our results apply for any semisimple group G, for all values of the deformation parameter λ and up to order 1/k. We deduce the OPEs and equal-time commutators of all currents and primaries. We derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable λdeformed σ-models. We also present analogous results in two limiting cases of special interest, namely for the non-Abelian T-dual of the PCM and for the pseudodual model. Introduction and motivation One of the most intriguing conjectures in modern theoretical physics is the AdS/CFT correspondence [1] which, in its initial form, states the equivalence between type-IIB superstring theory on the AdS 5 × S 5 background and the maximally supersymmetric field theory in four dimensions, i.e. N = 4 SYM. In recent years, a huge progress has been made in calculating physical observables employing both sides of the duality. These calculations managed to probe the strongly coupled regime of the gauge theory which is practically unaccessible by other means. The key feature that allowed this progress is integrability. N = 4 SYM from one side and the two-dimensional σ-model from the other, are believed to be integrable order by order in perturbation theory. It is clear that one way to construct generalizations of the original AdS/CFT scenario is to try to maintain the key property of integrability. The aim of this work is to study the structure of a class of two-dimensional σ-models, the so-called λ-deformed models constructed in [2]. For isotropic couplings the deformation is integrable in the group case and in the symmetric and semi-symmetric coset cases [2][3][4][5] (for the su(2) group case integrability is preserved for anisotropic, albeit diagonal couplings [6]). They are also closely related [7][8][9][10][11][12] to the so-called η-deformed models for group and coset spaces introduced in [7,8] and in [13][14][15], respectively. This relation is via Poisson-Lie T-duality and an analytic continuation of coordinates and of the parameters of the σ-models [10][11][12]. There are also embedings of the λ-deformed models as solutions of supergravity [16][17][18]. In particular, we shed light into the structure of the λ-deformed models by computing the two-and three-point functions of all currents and operators exactly in the deformation parameter and up to order 1/k. This work is based and further extends symmetry ideas and techniques originated in our previous work in [19]. The results of this work are summarized in section 7. Our starting point is the WZW action S WZW,k (g) = − k 4π d 2 σ Tr(g −1 ∂ + gg −1 ∂ − g) + k 24π B Tr(g −1 dg) 3 ,(1.1) for a generic semisimple group G, with g ∈ G parametrized by X µ , µ = 1, 2, . . . , dim G. We will use the representation matrices t a which obey the commutation relations [t a , t b ] = f abc t c and are normalized as Tr(t a t b ) = δ ab . These matrices are taken to be Hermitian and therefore the Lie-algebra structure constants f abc are purely imaginary. The chiral and anti-chiral currents are defined as J a + = −i Tr(t a ∂ + gg −1 ) = R a µ ∂ + X µ , J a − = −i Tr(t a g −1 ∂ − g) = L a µ ∂ − X µ . (1.2) The left and right invariant forms L a = L a µ dX µ and R a = R a µ dX µ are related as R a = D ab L b , D ab = Tr(t a gt b g −1 ) . (1.3) We are interested in the non-Abelian Thirring model action (for a general discussion, see [20,21]), namely the WZW two-dimensional conformal field theory (CFT) perturbed by a set of classically marginal operators which are bilinear in the currents S = S WZW,k (g) + k 2π dim G ∑ a,b=1 λ ab d 2 σ J a + J b − ,(1.4) where the couplings are denoted by the constants λ ab . An action having the same global symmetries as (1.4), and to which reduces for small values of λ ab has been derived in [2] (see also [22] for the SU(2) case), by gauging a common symmetry subgroup of an action involving the PCM model and the WZW actions. It reads [2] S k,λ (g) = S WZW,k (g) + k 2π d 2 σ J a + (λ −1 − D T ) −1 ab J b − ,(1.5) where we have assembled in a general real matrix λ the coupling constants λ ab . In addition, this action, as well as (1.4), is invariant under the generalized parity trans- formation σ ± → σ ∓ , g → g −1 , λ → λ T . (1.6) The β-functions for the running of couplings under the Renormanization Group (RG) flow using (1.5) were computed in [23,24] and completely agree with the computation of the same RG-flow equations using CFT techniques based on (1.4) in [25] for a single (isotropic) coupling, i.e. when λ ab = λδ ab and in [26] for symmetric λ ab . Based on that it was conjectured in [23,24] that (1.5) is the effective action for (1.4) valid to all orders in λ and up to order 1/k. In the same works it was realized that (1.5) has the remarkable symmetry These fields are also Virasoro primaries with holomorphic and antiholomorphic dimensions [29] ∆ R = c R 2k + c G ,∆ R ′ = c R ′ 2k + c G , (2.3) where c R , c R ′ and c G are the quadratic Casimir operators, all non-negative, in the representations R, R ′ and the adjoint representation for which (t a ) bc = f abc . They are defined as (t a t a ) i j = c R δ i j , (t ata ) i ′ j ′ = c R ′ δ i ′ j ′ , f acd f bcd = −c G δ ab . (2.4) In our calculations we will need the basic two-and three-point functions for these fields. For the currents they are given by J a (z 1 )J b (z 2 ) = δ ab z 2 12 , J a (z 1 )J b (z 2 )J c (z 3 ) = 1 √ k f abc z 12 z 13 z 23 , (2.5) where we employ the general notation z ij = z i − z j . We will also use the four-point function J a (x 1 )J a 1 (z 1 )J a 2 (z 2 )J a 3 (z 3 ) = 1 k f a 1 ac f ca 2 a 3 (z 1 − x 1 )(x 1 − z 2 )(x 1 − z 3 )(z 1 − z 3 ) + δ aa 1 δ a 2 a 3 (x 1 − z 1 ) 2 (z 2 − z 3 ) 2 + cyclic in 1, 2, 3 . (2.6) Similar expressions hold for the antiholomorphic currents as well. Correlators involving both holomorphic and anti-holomorphic currents vanish at the conformal point. However, as we shall see, this will not be the case in the deformed theory. The corresponding correlators for the affine primaries are Φ (1) i,i ′ (z 1 ,z 1 )Φ (2) j,j ′ (z 2 ,z 2 ) = δ ij δ i ′ j ′ z 2∆ R 12z 2∆ R ′ 12 , (2.7) where the superscripts signify the fact that the representations for the different primaries in correlation functions could be, in general, different. However, for the twopoint functions the two representations should in fact be conjugate to each other for the holomorphic and anti-holomorphic sectors separately. As such, they have the same conformal dimensions. Recalling that the matrices t a andt a are Hermitian and after removing the superscripts by relabeling the representation matrices we have that Reps (1) and (2) conjugate : t (1) a = t a ,t (1) a =t a , t (2) a = −t * a ,t (2) a = −t * a . (2.8) The minus sign in the definition of the conjugate representation is very important for the matrices to obey the same Lie-algebra. It will turn out that, in the deformed theory, for correlation functions involving two primaries to be non-vanishing, their corresponding representations must be conjugate to each other, as well. Next, consider three affine primaries transforming in the representations (R i , R ′ i ), i = 1, 2, 3. Then the three-point function for them is given by where Φ (1) i,i ′ (z 1 ,z 1 )Φ (2) j,j ′ (z 2 ,z 2 )Φ (3) k,k ′ (z 3 ,z 3 ) = C ii ′ ,jj ′ ,∆ 12;3 = ∆ R 1 + ∆ R 2 − ∆ R 3 ,∆ 12;3 =∆ R ′ 1 +∆ R ′ 2 −∆ R ′ 3 . (2.10) and cyclic permutations of 1, 2 and 3 for the rest. The structure constants C ii ′ ,jj ′ ,kk ′ depend on the representations and implicitly also on k. They obey various properties arising mainly from the global group invariance of the correlation functions, which will be mentioned below in the computation of the three-point functions involving only affine primaries. Finally, we have the three-point functions with one current and two primaries. They are given by J a (z)Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) = − 1 √ k (t a ⊗ I R ′ ) ij,i ′ j ′ x 2∆ R 12x 2∆ R ′ 12 1 z − x 1 − 1 z − x 2 (2.11) and J a (z)Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) = 1 √ k (I R ⊗t * a ) ij,i ′ j ′ x 2∆ R 12x 2∆ R ′ 12 1 z −x 1 − 1 z −x 2 ,(2.12) where we have used the fact that, for a non-vanishing result, the representations in which the primaries transform have to be conjugate to each other for the holomorphic and the antiholomorphic sectors, separately. Also I R and I R ′ are the identity elements for the corresponding representations. Correlators with two currents and one affine primary field are zero at the conformal point and will remain zero in the deformed theory as well. Symmetry and correlation functions In order to compute the correlation functions of currents and of primary fields we will heavily use the symmetry of the effective action for the non-Abelian Thirring model (1.7). First let's consider correlation functions for currents only. At the conformal point when λ = 0 the currents are given in terms of the group element by (1.2) and are, of course, chirally and anti-chirally conserved on shell. Obviously, in the deformed theory these currents will be dressed and will receive λ-corrections. One expects that since their definition contains derivatives there will be operator ambiguities at the quantum level. We propose that these dressed currents are given by J a + (g) k,λ = − i 1 + λ (I − λD) −1 ab Tr(t b ∂ + gg −1 ) , J a − (g) k,λ = i 1 + λ (I − λD T ) −1 ab Tr(t b g −1 ∂ − g) . (2.13) These become the correct chiral and anti-chiral currents when λ = 0 (up to a minus sign for J − ). Also, they are components of an on shell conserved current. The attentive reader will notice that the dressed current components in (2.13) are nothing, but, up to a factor of λ, the gauge fields evaluated on-shell in the original construction of (1.5) in [2] by a gauging procedure. Hence, it is natural to consider correlation functions of the J a ± 's as defined above. In addition, we have that J a ± (g −1 ) −k,λ −1 = λ 2 J a ± (g) k,λ . (2.14) Passing to the Euclidean regime we have for the two-point function of the holomorphic component of the currents that J a (x 1 )J b (x 2 ) k,λ = 1 Z k,λ D[g]J a (g(x 1 )) k,λ J b (g(x 2 )) k,λ e −S k,λ (g) ,(2.15) with the partition function being Z k,λ = D[g]e −S k,λ (g) = D[g −1 ]e −S k,λ (g −1 ) = D[g]e −S −k,λ −1 (g) = Z −k,λ −1 . (2.16) where we have used the symmetry of the action (1.7) and the fact that the measure of integration is invariant under g → g −1 , i.e. D[g −1 ] = D[g]. 1 Hence, the partition function of the deformed theory is invariant under the duality-type symmetry. In addition D[g]J a (g(x 1 )) k,λ J b (g(x 2 )) k,λ e −S k,λ (g) = D[g −1 ]J a (g −1 (x 1 )) k,λ J b (g −1 (x 2 )) k,λ e −S k,λ (g −1 ) = 1 λ 4 D[g]J a (g(x 1 )) −k,λ −1 J b (g(x 2 )) −k,λ −1 e −S −k,λ −1 (g) . (2.17) where we have also employed (2.14). Hence, we obtain that the correlation function should obey the non-trivial identity λ 2 J a (x 1 )J b (x 2 ) k,λ = λ −2 J a (x 1 )J b (x 2 ) −k,λ −1 (2.18) This identity between current correlators is straightforwardly extendable to higher order correlators involving currents with any type of currents, J a 's orJ a 's. λ n+m J a 1 . . . J a nJ b 1 . . .J b m k,λ = λ −n−m J a 1 . . . J a nJ b 1 . . .J b m −k,λ −1 . (2.19) The overall factors of λ can be absorbed by redefining the currents in (2.13) by a factor of λ. In the following we assume that this is the case which implies also the absence of the factor of λ 2 in the r.h.s. of (2.14). The above conclusion for the current correlators is in full agreement with [27] who reached the same conclusion using the non-Abelian Thirring model action and certain special properties of the WZW action path integral. The advantage of employing the effective action is that one can employ the duality-type symmetry on correlation functions involving primary fields in the deformed theory which has not been considered before. For these fields we have that, under the inversion of the group element the 1 The measure of integration contains the Haar measure for the semisimple group G which is certainly invariant under g → g −1 , but also the factor det(λ −1 − D T ) arising from integrating out the gauge fields in the path integral [2]. This can be easily seen to transform under g → g −1 and λ → λ −1 as (for a general matrix λ): det(λ −1 − D T ) → (−1) n det λ × det(λ −1 − D T ), with n = dim G and where we have used the property D(g −1 ) = D T (g). This extra constant overall factor cancels out by the same factor arising from the partition function in the denominator in all correlation functions. primary field Φ (1) transforms to its conjugate Φ (2) . Explicitly, we have that (2.20) which means that for the representation matrices we have t (1) ↔t (2) , t (2) ↔t (1) . (2.21) Note that if the inversion of g is followed by the σ → −σ, i.e. the parity transformation (1.6), then 22) and in addition the J a 's andJ a 's are interchanged. Φ (1) i,i ′ (g −1 ) = Φ (2) i ′ ,i (g) ,t (1) ↔ −t (2) , t (2) ↔ −t (1) ,(2. The non-Abelian and pseudodual chiral limits Besides the small λ ab limit, leading to (1.4), there are two other interesting limits of the action (1.5). They will be instrumental in our computation of correlation functions. In the first limit [2] one expands the matrix and group elements near the identity as λ ab = δ ab − E ab k + O 1 k 2 , g = I + i v a t a k + O 1 k 2 , (2.23) where E is a general dimG square matrix. This leads to J a ± = ∂ ± v a k + O 1 k 2 , D ab = δ ab + f ab k + O 1 k 2 , f ab = −i f abc v c . (2.24) Note that our structure constants are purely imaginary so that f ab are indeed real. In this limit the action (1.5) becomes S non−Abel (v) = 1 2π d 2 σ ∂ + v a (E + f ) −1 ab ∂ − v b ,(2.25) which is the the non-Abelian T-dual with respect to the G L action of the σ-model given by the PCM action with general coupling matrix E ab . We note that in this limit the WZW term in (1.5) does not contribute at all. To discuss the second new limit, we first recall that the original derivation of the action (1.5) leads for compact groups to the restriction 0 < λ < 1. However, once we have the action we may allow λ to take values beyond this range. For instance, the symmetry (1.7) clearly requires that. Here in order to take a new limit we will extend the range of λ to negative values. We will also need the following equivalent form of the action (1.5) given, after some manipulations needed to combine the quadratic part of the WZW action and the deformation term in (1.5), by S k,λ (g) = k 4π d 2 σ J a + (λ −1 − D T ) −1 (λ −1 + D T )D ab J b − − ik 48π B f abc L a ∧ L b ∧ L c . (2.26) where we we remind the reader that our structure constants are purely imaginary. Then we take the limit λ ab = −δ ab + E ab k 1/3 , g = I + i v a t a k 1/3 + . . . , k → ∞ , (2.27) where again E is a general dimG square matrix. The various quantities expand as in (2.24) with k replaced by k 1/3 . Then the action (2.26) becomes S pseudodual = 1 8π d 2 σ ∂ + v a ∂ − v b E ab + 1 3 f ab . (2.28) We see that E can be taken to be symmetric since any antisymmetric piece leads to a total derivative. This action for E ab = δ ab /b 2/3 is nothing by the pseudodual model action [30]. Note that the quadratic part of the WZW action and the deformation term in (1.5) are equally important for the limit (2.27) to exist since each term separately diverges when this limit is taken. Since the above non-Abelian and pseudodual limits exist at the action level, we expect that physical quantities such as the β-function and the anomalous dimensions of various operators should have a well defined limit as well. This will be an important ingredient in our method of computation. The regularization method and useful integrals In the Euclidean path integral the action appears as e −S . The action we will be using is that of the non-Abelian Thrirring model action and will be expanding around the WZW CFT part of it. This is not in contrast with the approach of the last subsection where (1.5) was used, the reason being that the latter is the effective action of the non-Abelian Thrirring model. Hence, it contains all λ-corrections and can be considered as a starting point to find at the quantum level corrections in 1/k. Schematically, to O(λ n ), the correlation function for a number of some generic fields F i , i = 1, 2, . . . , involves the sum of expressions of the type F 1 (x 1 ,x 1 )F 2 (x 2 ,x 2 ) . . . (n) λ = 1 n! − λ π n d 2 z 1...n J a 1 (z 1 ) . . . J a n (z n ) J a 1 (z 1 ) . . .J a n (z n )F 1 (x 1 ,x 1 )F 2 (x 2 ,x 2 ) . . . , (2.29) where d 2 z 1...n := d 2 z 1 . . . d 2 z n and for convenience we have dropped k from our notation in the correlation functions · · · k,λ of the deformed theory. That way one encounters multiple integrals which need to be regularized. Our prescription to do so consists of two steps: • We choose the order of integration from left to right d 2 z 1...n and never permute this order. This is due to the fact that due to the divergences appearing, the various integrations are not necessarily commuting. • Internal points cannot coincide with external ones. This means that the domain of integration is D n = {(z 1 , z 2 , . . . , z n ) ∈ C n : |z i − x j | > ε, ε > 0} , ∀ i, j . (2.30) However, internal points can coincide. Also contact terms, arising from coincident external points will be allowed. The latter is a choice we make and not a part of the regularization scheme. 2 We shall need the very basic integral given by d 2 z (x 1 − z)(z −x 2 ) = π ln |x 12 | 2 . (2.31) Clearly, if the domain of integration allows, the integral diverges for large distances. 2 All these imply that we will have for the δ-functions arising in performing the various integrations that δ (2) (z i − x j ) → 0 , δ (2) (z i − z j ) (kept) , δ (2) (x i − x j ) (kept) , ∀i, j . Note also that in the regularization of [31] no two points, internal or external, can coincide and therefore all δ-functions arising in integrations are set to zero. In contrast in [32] all such δ-functions are kept. The advantage of our regularization is that the symmetry of the correlation functions under k → −k and λ → λ −1 is manifest whereas for the others it is hidden. The above result is valid provided that the integration is performed in a domain of characteristic size R, e.g. a disc of radius R, with the external points x 1 and x 2 excluded and in addition obeying R ≫ |x 1 |, |x 2 |. The latter conditions are responsible for the translational invariance and the reality of the result. Even then we have to make the replacement |x 12 | 2 → |x 12 | 2 /R 2 on the right hand side of (2.31). However, in our computations there will be integrals of the same kind but with opposite sign and x 1 equal to x 2 and which will have a small distance regulator ε. Hence the factor R will drop out at the end, leaving the ratio ln ε 2 |x 12 | 2 . This means that in practice the domain of integration is R 2 except for the points x 1,2 which are excluded. By appropriately taking derivatives we also have the useful integrals d 2 z (x 1 − z) 2 (z −x 2 ) = − π x 12 , d 2 z (x 1 − z)(z −x 2 ) 2 = − π x 12 (2.32) and d 2 z (x 1 − z) 2 (z −x 2 ) 2 = π 2 δ (2) (x 12 ) . (2.33) In appendix A we have collected results for some useful to this work integrals. We single out d 2 z (z − x 1 )(z − x 2 )(z −x 1 ) = − π x 12 ln ε 2 |x 12 | 2 , d 2 z (z − x 1 )(z −x 1 )(z −x 2 ) = − π x 12 ln ε 2 |x 12 | 2 (2.34) and d 2 z (z − x 1 )(z −x 2 ) ln |z − x 1 | 2 = − π 2 ln 2 |x 1 − x 2 | 2 . (2.35) which are valid under the assumptions spelled out below (2.31). Current correlators In this section, we will focus on the two-and three-point functions involving purely currents. These will be computed up to order 1/k and exactly in the deformation parameter λ. To establish our method, employed already in [19], as clearly as possible we first start with the computation of the two-point functions which enables to com-pute the β-function and the anomalous dimensions for the currents known already from using CFT methods in [19,25,26] and from gravitational computations [23,24]. Then we proceed to correlators involving three currents. Two-point functions On general grounds the correlator of J a and J b takes the form J a (x 1 )J b (x 2 ) λ = δ ab G 0 (k, λ) x 2 12 1 + γ (J) ln ε 2 |x 12 | 2 + · · · . (3.1) The result to O(1/k) and O(λ 3 ) was computed in sec. 2 of [19] and reads J a (x 1 )J b (x 2 ) = δ ab x 2 12 1 − 2 c G k λ 3 + c G k (λ 2 − 2λ 3 ) ln ε 2 |x 12 | 2 + 1 k O(λ 4 ) . (3.2) Comparing with the general form of the two-point function (3.1) we have that G 0 (k, λ) = 1 − 2 c G k λ 3 + O(λ 4 ) (3.3) and γ (J) = c G k λ 2 − 2λ 3 + O(λ 4 ) . (3.4) Similarly the correlator of J a andJ b should assume the form J a (x 1 )J b (x 2 ) λ = δ abG 0 (k, λ) |x 12 | 2 1 + γ (J) ln ε 2 |x 12 | 2 + δ ab δ (2) (x 12 ) A(k, λ) + B(k, λ) ln ε 2 |x 12 | 2 . (3.5) At the conformal point this correlator should vanish. We have also allowed for contact terms proportional to the δ-function since these are allowed by symmetry. The coupling functions A and B have to be computed. After a long computation, all details are given in the appendix B, we found the result J a (x 1 )J b (x 2 ) λ = −πλδ ab δ (2) (x 12 ) − λ 2 c G k δ ab 1 |x 12 | 2 + πδ (2) (x 12 ) 1 − 1 2 ln ε 2 |x 12 | 2 (3.6) + 2 λ 3 c G k δ ab 1 |x 12 | 2 + πδ (2) (x 12 ) 1 − ln ε 2 |x 12 | 2 + 1 k O(λ 4 ) . which, keeping in mind that we are interested to terms up to O(1/k), is easily seen to be of the form (3.5). Note that this correlator takes the form J a (x 1 )J b (x 2 ) = −γ (J) δ ab |x 12 | 2 + contact terms , (3.7) where γ (J) is the current anomalous dimension given perturbatively by (3.4). The exact β-function and anomalous dimensions To compute the wave function renormalization and that for the parameter λ we use the two-point functions J a J b and J aJb . In particular we need the most singular part of these correlation functions. For the purpose of this section let's denote the bare currents by J a 0 andJ a 0 and similarly for the parameter λ 0 . We need the most singular part of the bare two-point functions up to order 1/k. From (3.2) we have that J a 0 (x 1 )J b 0 (x 2 ) = δ ab x 2 12 1 − c G k λ 2 0 2λ 0 + (1 − 2λ 0 ) ln(|x 12 | 2 /ε 2 ) + . . . . (3.8) Also from (3.6) we have that J a 0 (x 1 )J b 0 (x 2 ) = −πλ 0 δ ab δ (2) (x 12 ) 1 + λ 0 c G k 1 − 1 2 ln ε 2 |x 12 | 2 − 2λ 0 1 − ln ε 2 |x 12 | 2 + · · · ,(3.9) where we have kept only the coefficient of the most singular term, i.e. of δ (2) (x 12 ). The bare quantities and the renormalized ones are related as J a 0 = Z 1/2 J a ,J a 0 = Z 1/2Ja , λ 0 = Z 1 λ . (3.10) We make the following ansatz valid to order 1/k in the large k-expansion Z −1 = 1 + 2 c G k λ 3 − c G k c 1 λ 2 + c 2 λ 3 + O(λ 4 ) ln(ε 2 µ 2 ) , Z 1 = 1 − c G k c 3 λ + c 4 λ 2 + O(λ 3 ) ln(ε 2 µ 2 ) , (3.11) where the logarithm-independent term in Z −1 has been chosen so that the renormalized two-point function for the J a 's is normalized to one. The pure number coefficients c i are computed so that the renormalized two-point functions J a (x 1 )J b (x 2 ) = Z −1 J a 0 (x 1 )J b 0 (x 2 ) , J a (x 1 )J b (x 2 ) = Z −1 J a 0 (x 1 )J b 0 (x 2 ) , (3.12) are independent of the cutoff ε. We find that the unique choice is given by c 1 = 1 , c 2 = −2 , c 3 = − 1 2 , c 4 = 1 . (3.13) The β-function is by definition β λ = 1 2 µ dλ dµ = 1 2 λZ 1 µ dZ −1 1 dµ = − c G 2k λ 2 − 2λ 3 + O(λ 4 ) , (3.14) where the bare coupling coupling λ 0 is kept fixed. Next we compute the anomalous dimension of the current (3.15) in agreement of course with (3.4). γ (J) = µ d ln Z 1/2 dµ = c G k λ 2 − 2λ 3 + O(λ 4 ) , The above perturbative expressions are enough to determine the exact in λ dependence of the β-function and of the anomalous dimensions up to order 1/k. As explained, the exact β-function and anomalous dimensions should have a well defined behaviour in the two limiting cases described by the non-Abelian and pseudodual model limits (2.23) and (2.27), respectively. In the isotropic case, which is the case of interest in this work, it implies regularity under the following independent limits λ = 1 − κ 2 k , λ = −1 + 1 b 2/3 k 1/3 , k → ∞ .β λ = − c G 2k f (λ) (1 + λ) 2 , γ (J) = c G k g(λ) (1 − λ)(1 + λ) 3 , (3.17) where f (λ) and g(λ) are two analytic functions of λ. The assumed pole structure does not exclude the possibility that one of the poles reduces its degree or even ceases to exist. This can happen if the functions in the numerator are zero at λ = 1 or/and λ = −1. In addition, due to the symmetry under (k, λ) → (−k, λ −1 ) we have that λ 4 f (1/λ) = f (λ) , λ 4 g(1/λ) = g(λ) . (3.18) All these imply that these functions are in fact polynomials of, at most, degree four f (λ) = a 0 + a 1 λ + a 2 λ 2 + a 1 λ 3 + a 0 λ 4 , g(λ) = b 0 + b 1 λ + b 2 λ 2 + b 1 λ 3 + b 0 λ 4 . (3.19) Demanding agreement with the perturbative expressions (3.14) and (3.15) to O(λ 2 ) we obtain a 0 = a 1 = b 0 = b 1 = 0 and a 2 = b 2 = 1 which completely determines the exact β-function and anomalous dimensions to be β λ = − c G 2k λ 2 (1 + λ) 2 0 (3.20) and γ (J) = c G k λ 2 (1 − λ)(1 + λ) 3 0. (3.21) It is also easily seen that the coefficient of the O(λ 3 ) term is in agreement with the perturbative results as well. The above expressions are in full agreement with the results found in [23,24,33] for the β-function and in [19] for the anomalous dimensions. Note that the β-function and anomalous dimensions of the non-Abelian T-dual limit are β κ 2 = c G 8 , γ (J) = c G 8κ 2 , (3.22) which are valid for large κ 2 . The anomalous dimensions correspond to J a ± = ± 1 2 (κ 2 I ∓ f ) −1 ab ∂ ± v b ,(3.23) which are obtained by taking this limit in (2.13). The corresponding expressions for the pseudodual model are β b = 3 4 c G b 3 , γ (J) = 1 2 c G b 2 . (3.24) These are in agreement with the expressions derived in [30] (see above fig. 2) and are valid for small b. The anomalous dimensions correspond to J a ± = ±b 2/3 ∂ ± v a ,(3.25) which as before are obtained by taking the appropriate limit in (2.13). Three-point functions We consider the J J J and J JJ correlators. The remaining correlators JJJ and JJ J can be easily obtained by applying the parity transformation to the first two. The results of this subsection match those obtained in [32], where current-current perturbations of the WZW model on supergroups were studied with a different regularization scheme. Before moving to our analysis, let us note that analogue perturbations of the WZW models on supergroups were studied in [34], but the perturbation consists of the term J a + D ab J b − added to the action; effectively the non-critical WZW model. The J J J correlator From appendix C we have that the, up to O(λ 3 ), correlator reads J a (x 1 )J b (x 2 )J c (x 3 ) λ = 1 √ k 1 + 3 2 λ 2 − λ 3 f abc x 12 x 13 x 23 + 1 √ k O(λ 4 ) . (3.26) The ansatz for the all-loop expression takes the form J a (x 1 )J b (x 2 )J c (x 3 ) = f (λ) k(1 − λ)(1 + λ) 3 f abc x 12 x 13 x 23 ,(3.27) where f (λ) is everywhere analytic and obviously f (0) = 1 to agree with the CFT result. As before this form takes into account that under the limit (3.16) the correlator is well behaved. Invariance of the above expression under the duality-type symmetry (k, λ) → (−k, λ −1 ) yields λ 2 f (λ −1 ) = f (λ) =⇒ f (λ) = 1 + c λ + λ 2 . (3.28) Consistency with the perturbative expression up to O(λ) (3.27) gives c = 1. Therefore, the all-loop correlator reads J a (x 1 )J b (x 2 )J c (x 3 ) = 1 + λ + λ 2 k(1 − λ)(1 + λ) 3 f abc x 12 x 13 x 23 . (3.29) As a check we see that this expression reproduces the O(λ 2 ) and O(λ 3 ) terms in the perturbative expression (3.26). The J JJ correlator The perturbative calculation of this correlator is performed in appendix D. The result up to order O(λ 2 ) reads J a (x 1 )J b (x 2 )J c (x 3 ) = λ(1 − λ) √ kx 12 f abc x 2 12x 23x13 + 1 √ k O(λ 3 ) . (3.30) We now make a similar to (3.27) ansatz for the all-loop expression J a (x 1 )J b (x 2 )J c (x 3 ) = λ f (λ) k(1 − λ)(1 + λ) 3x 12 f abc x 2 12x 23x13 , (3.31) where f (λ) is everywhere analytic and f (0) = 1. Invariance of the above expression under the duality-type symmetry yields f (λ −1 ) = f (λ) =⇒ f (λ) = 1 . (3.32) Hence, we find the all-loop expression J a (x 1 )J b (x 2 )J c (x 3 ) = λ k(1 − λ)(1 + λ) 3 f abcx12 x 2 12x 13x23 , (3.33) whose expansion around λ = 0 agrees with (3.30). Note that implementing the non-Abelian and pseudodual limits both lead to finite (non-zero) expressions for all of the above three-point functions. In these limiting cases the results are valid for large κ 2 and small b where we refer to (3.16) for the definition of these parameters. We mention also that, our results for these correlators agree with those done for supergroups in [32] after an appropriate rescaling of the currents that presumably takes into account the different regularization schemes used in that work. Primary field correlators The purpose of this section is to compute two-and three-point functions of arbitrary primary fields. This will allow us to extract their anomalous dimensions and the deformed structure constants in the OPEs. Two-point functions After a long computation, all details of which are given in the appendix E, we found that a perturbative computation up to O(λ 3 ) and to order 1/k, gives for the two-point function of primary fields the result Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) λ = 1 x 2∆ R 12x 2∆ R ′ 12 1 + λ 2 k (c R + c R ′ ) ln ε 2 |x 12 | 2 (I R ⊗ I R ′ ) ii ′ ,jj ′ −2λ 1 + λ 2 k ln ε 2 |x 12 | 2 (t a ⊗ t * a ) ii ′ ,jj ′ + 1 k O(λ 3 ) . (4.1) We see that due to the deformation there is an operator mixing so that one should proceed by choosing an appropriate basis in which the dimension matrix is diagonal. For convenience we will adopt the double index notation I = (ii ′ ). Then there is a matrix U chosen such that (t a ⊗ t * a ) I J = U IK N KL (U −1 ) LJ , N I J = N I δ I J ,(4.2) where N I are the eigenvalues of the matrix t a ⊗ t * a . Note also that U is λ-independent as well as k-independent. Then in the rotated basis Φ (1) I = (U −1 ) I J Φ (1) J , Φ(2)I = U I J Φ (2) J ,(4.3) the correlator (4.1) becomes diagonal, i.e. Φ (1) I (x 1 ,x 1 ) Φ (2) J (x 2 ,x 2 ) λ = δ I J x 2∆ R 12x 2∆ R ′ 12 1 + δ (Φ) I ln ε 2 |x 12 | 2 , (4.4) where perturbatively δ (Φ) I = 1 k −2λ(1 + λ 2 )N I + λ 2 (c R + c R ′ ) + O(λ 4 ) . (4.5) To determine the exact anomalous dimension of the general primary field we first realize that we should include in the above expression the k-dependent part coming from the CFT dimensions of ∆ R and∆ R ′ in (2.3) up to order 1/k. Hence the anomalous dimension is given by γ (I) R,R ′ (k, λ) pert = c R 2k + δ (Φ) I 2 = = 1 2k c R − 2N I λ(1 + λ 2 ) + λ 2 (c R + c R ′ ) + O(λ 4 ) . (4.6) As in the case of currents we make the following ansatz for the exact anomalous di- mensions γ (I) R,R ′ (k, λ) = − 1 2k(1 − λ)(1 + λ) 3 [ f (λ)N I + f 1 (λ)c R + f 2 (λ)c R ′ ] ,(4.7) where the yet unknown function should be analytic in λ. Using the symmetry (1.7) and the transformation of the primary fields under this symmetry (2.20), we have that γ (I) R,R ′ (−k, λ −1 ) = γ (I) R ′ ,R (k, λ) ,(4.8) which implies the following relations between the various unknown functions λ 4 f (1/λ) = f (λ) , λ 4 f 1 (1/λ) = f 2 (λ) , λ 4 f 2 (1/λ) = f 1 (λ) . (4.9) Hence, these functions should be fourth order polynomials in λ with related coefficients. It turns out that comparing with the perturbative expression (4.6) up to O(λ 2 ) we determine all these functions to be f (λ) = 2λ(1 + λ) 2 , f 1 (λ) = −(1 + λ) 2 , f 2 (λ) = −λ 2 (1 + λ) 2 . (4.10) Therefore, the exact in λ anomalous dimension is γ (I) R,R ′ (k, λ) = − 1 2k(1 − λ 2 ) (2λN I − c R − λ 2 c R ′ ) . (4.11) It is easily checked that this expression is in agreement with the O(λ 3 /k) term in (4.6). Note also that in the non-Abelian limit the above anomalous dimensions have a well defined and different than zero limit. In contrast the limit is zero in the pseudodual limit. This expression also applies for current current perturbations of the WZW model on supergroups with vanishing Killing form [31]. Finally, the two point functions take the form Φ (1) I (x 1 ,x 1 )Φ (2) J (x 2 ,x 2 ) = δ I J x γ (I) R,R ′ (k,λ) 12x γ (I) R ′ ,R (k,λ) 12 . (4.12) Three-point functions To leading order in the λ-expansion after a straightforward computation this correlator is found to be Φ (1) i,i ′ (x 1 )Φ (2) j,j ′ (x 1 )Φ (3) k,k ′ (x 3 ) (1) λ = − λ k 1 x ∆ 12a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ + (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ + (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,ℓℓ ′ + ln |x 12 | 2 (t (1) a ) i ℓ (t (2) a ) ℓ ′ j ′ C ℓi ′ ,jℓ ′ ,kk ′ + (t (2) a ) j ℓ (t (1) ) ℓ ′ i ′ C iℓ ′ ℓj ′ ,kk ′ (4.13) + ln |x 13 | 2 (t (1) a ) i ℓ (t (3) a ) ℓ ′ k ′ C ℓi ′ ,jj ′ ,kℓ ′ + (t (3) a ) k ℓ (t (1) ) ℓ ′ i ′ C iℓ ′ ,jj ′ ,ℓk ′ + ln |x 23 | 2 (t (2) a ) j ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,ℓi ′ ,kℓ ′ + (t (3) a ) k ℓ (t (2) ) ℓ ′ j ′ C ii ′ ,jℓ ′ ,ℓk ′ . Even for dimensional reasons we should be able to cast the above expression in a form in which all space dependence is in terms of ratios ε 2 /|x ij | 2 . In order to do that we first recall that the structure constants C ii ′ ,jj ′ ,kk ′ are factorized according to their holomorphic and antiholomorphic content as C ii ′ ,jj ′ ,kk ′ = C i,j,kCi ′ ,j ′ ,k ′ . (4.14) An important constraint, arises by making use of the global Ward identity. It reads (t (1) a ) i ℓ C ℓjk + (t (2) a ) j ℓ C iℓk + (t (3) a ) k ℓ C ijℓ = 0 , (t (1) a ) ℓ ′ i ′C ℓ ′ j ′ k ′ + (t (2) a ) ℓ ′ j ′C i ′ ℓ ′ k ′ + (t (3) a ) ℓ ′ k ′C i ′ j ′ ℓ ′ = 0 . (4.15) From (4.15) it is straightforward to obtain the following relations (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,ℓℓ ′ = (t (1) a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ + (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ + (t (1) a ) i ℓ (t (2) a ) ℓ ′ j ′ C ℓi ′ ,jℓ ′ ,kk ′ + (t (2) a ) j ℓ (t (1) a ) ℓ ′ i ′ C iℓ ′ ,ℓj ′ ,kk ′ , (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ = (t (1) a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ + (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,ℓℓ ′ + (t (1) a ) i ℓ (t (3) a ) ℓ ′ k ′ C ℓi ′ ,jj ′ ,kℓ ′ + (t (3) a ) k ℓ (t (1) a ) ℓ ′ i ′ C iℓ ′ ,jj ′ ,ℓk ′ , (t (1) a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ = (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ + (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,ℓℓ ′ , + (t (2) a ) j ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,ℓj ′ ,kℓ ′ + (t (3) a ) k ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,jℓ ′ ,ℓk ′ . Using the above relations we can rewrite the three-point function as Φ (1) i,i ′ (x 1 )Φ (2) j,j ′ (x 1 )Φ (3) k,k ′ (x 3 ) (1) λ = − λ k 1 x ∆ 12ε 2 |x 12 | 2 (t (1) a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ + (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ − (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,ℓℓ ′ ln ε 2 |x 13 | 2 (t (1) a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ + (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,kk ′ − (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ ln ε 2 |x 23 | 2 (t (2) a ) j ℓ (t (2) a ) ℓ ′ j ′ C ii ′ ,ℓℓ ′ ,kk ′ + (t (3) a ) k ℓ (t (3) a ) ℓ ′ k ′ C ii ′ ,jj ′ ,kk ′ − (t (1) a ) i ℓ (t (1) a ) ℓ ′ i ′ C ℓℓ ′ ,jj ′ ,kk ′ . The next step is to pass to the rotated basis. By using the double index notation we introduced before we have that Φ (q) I = (U (q) ) −1 I J Φ (q) J , (t (q) a ⊗ t (q) * a ) I J = (U (q) ) I K (N (q) ) K L (U (q) ) −1 L J , N (q) I J = N (q) I δ I J , q = 1, 2, 3 ,(4.16) where in the new basis the structure constants read C I JK = (U (1) ) −1 I M (U (2) ) −1 J N (U (3) ) −1 K L C MNL ,(4.17) while the result for the correlator at O(λ) is given by Φ (1) I (x 1 )Φ (2) J (x 1 )Φ (3) K (x 3 ) (1) λ = −I + N (2) I − N (3) I ) ln ε 2 |x 12 | 2 + cyclic in 1,2,3 . (4.18) From this result we can write down the exact expression in λ for the three-point function. It is given by where γ 12;3 is given by Φ (1) I (x 1 )Φ (2) J (x 1 )Φ (3) K (x 3 ) λ =C I JK (k, λ) xγ 12;3 = − 1 2k(1 − λ 2 ) 2λ(N (1) I + N (2) I − N (3) I ) − c R 1 − c R 2 + c R 3 − λ 2 (c R ′ 1 + c R ′ 2 − c R ′ 3 ) . (4.20) andγ 12;3 = − 1 2k(1 − λ 2 ) 2λ(N (1) I + N (2) I − N (3) I ) − c R ′ 1 − c R ′ 2 + c R ′ 3 − λ 2 (c R 1 + c R 2 − c R 3 ) . (4.21) The other differences of dimensions γ 23;1 ,γ 23;1 and γ 13;2 ,γ 13;2 are obtained by performing cyclic permutations in the indices 1, 2 and 3. We now turn our attention to the three-point function coefficientsC I JK (k, λ). At λ = 0 these coefficients are considered as known since they are in principle fully determined from the WZW CFT data. On general grounds the following perturbative expansion holdsC I JK (k, λ) =C (0) I JK + 1 kC (1) I JK (λ) + O 1 k 2 . (4.22) where note the leading coefficientC (0) I JK in 1/k expansion does not depend on λ. This is so because such a term being k-independent and simultaneously having possible poles only at λ = ±1 and preserving the symmetry k → −k, λ → λ −1 cannot be finite either in the non-Abelian T-dual or in the pseudodual limit. Using the same line of reasoning as in the rest of this paper we conclude that the first correction to the three-point function should be of the form 3 C (1) I JK (λ) = f I JK (λ) (1 − λ)(1 + λ) 3 , (4.23) with λ 4 f I JK (λ −1 ) = f I JK (λ) =⇒ f I JK (λ) =C (1) I JK (0)(1 + λ 4 ) + a (1) I JK (λ + λ 3 ) + a (2) I JK λ 2 . (4.24) We saw from the O(λ) calculation that a (1) I JK = 0. Furthermore, it is not difficult to see that a (2) I JK = 0 too. Indeed, by inspecting the O(λ 2 ) calculation one can see that in order to remain to order 1/k either the two holomorphic or the two anti-holomorphic currents should be contracted through the Abelian part of their OPE. Then the resulting integrals will be of the form d 2 z 12 (z 1 − x 1 )(z 2 − x 2 )z 2 12 which can only produce logarithms. But the logarithms have to be combined and exponentiated to give the differences of the anomalous dimensions. Thus, no finite part will be present at this order and as a result a (2) I JK = 0, as well. Thus, we conclude that C (1) I JK (λ) =C (1) I JK (0)(1 + λ 4 ) (1 − λ)(1 + λ) 3 , (4.25) where as explained, the constantC (1) I JK (0) is fully determined from the WZW CFT initial data. As a result we have determined the exact in λ three-point function coefficient of three-primary fields up to order 1/k. 3 Notice that here we are using the duality (1.7) followed by parity. Under this combined symmetry Φ (i) I (x i ,x i ) → Φ (i) I (x i , x i ) andC I JK (λ −1 , −k) =C I JK (λ, k). Mixed JΦΦ and J ΦΦ correlators In this section we focus on the mixed correlators involving two primary fields and one current. From appendix F one can read off the O(λ 3 ) result which is given by J a (x 3 )Φ (1) i,i ′ (x 1 )Φ (2) j,j ′ (x 2 ) λ = 1 + λ 2 2 (t a ⊗ I R ′ ) ii ′ ,jj ′ − λ (I R ⊗t * a ) ii ′ ,jj ′ √ k x 2∆ R 12x 2∆ R ′ 12 1 x 13 − 1 x 23 . (5.1) The similar expression for the correlatorJ a reads J a (x 3 )Φ (1) i,i ′ (x 1 )Φ (2) j,j ′ (x 2 ) λ = − 1 + λ 2 2 (I R ⊗t * a ) ii ′ ,jj ′ − λ (t a ⊗ I R ′ ) ii ′ ,jj ′ √ k x 2∆ R 12x 2∆ R ′ 12 1 x 13 − 1 x 23 . (5.2) Getting inspired by the previous computations and by the expression in (5.1) we conclude that the all-loop mixed correlators should assume the following form J a (x 3 )Φ (1) i,i ′ (x 1 )Φ (2) j,j ′ (x 2 ) λ = f 1 (λ)(t a ⊗ I R ′ ) ii ′ ,jj ′ − λ f 2 (λ)(I R ⊗t * a ) ii ′ ,jj ′ k(1 − λ)(1 + λ) 3 x 2∆ R 12x 2∆ R ′ 12 1 x 13 − 1 x 23 , (5.3) where the functions f 1 (λ) and f 2 (λ) are everywhere analytic and f 1 (0) = f 2 (0) = 1. As usual, the denominator of (5.3) is written in such a way that the correlator has well-defined non-Abelian and pseusodual limits. Applying the duality (1.7), as well as the corresponding transformation rules for the currents (2.14) and primary fields (2.20) we obtain that J a (x 3 )Φ (2) i ′ ,i (x 1 )Φ (1) j ′ ,j (x 2 ) λ = λ 2 f 1 (λ −1 )(I R ′ ⊗ t a ) i ′ i,j ′ j − λ f 2 (λ −1 )(t * a ⊗ I R ) i ′ i,j ′ j k(1 − λ)(1 + λ) 3 x 2∆ R 12x 2∆ R ′ 12 1 x 13 − 1 x 23 , (5.4) where on the right hand side of the last equation we have changed the order of the indices for convenience. Subsequently, the left hand side of the above can be rewritten using appropriately (5.3). We have that J a (x 3 )Φ (1) i ′ ,i (x 1 )Φ (2) j ′ ,j (x 2 ) λ = f 1 (λ)(t (2) a ⊗ I R ) i ′ i,j ′ j − λ f 2 (λ)(I R ′ ⊗ t (2) * a ) i ′ i,j ′ j k(1 − λ)(1 + λ) 3 x 2∆ R 12x 2∆ R ′ 12 1 x 13 − 1 x 23 = − f 1 (λ)(t * a ⊗ I R ) i ′ i,j ′ j + λ f 2 (λ)(I R ′ ⊗ t a ) i ′ i,j ′ j k(1 − λ)(1 + λ) 3 x 2∆ R 12x 2∆ R ′ 12 1 x 13 − 1 x 23 . (5.5) Hence, comparing (5.4) with (5.5) we have the two equivalent conditions λ f 1 (λ −1 ) = f 2 (λ) , λ f 2 (λ −1 ) = f 1 (λ) =⇒ f 1 (λ) = f 2 (λ) = 1 + λ . (5.6) Plugging the latter into (5.3) we find after some rearrangement that J a (x 3 )Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) λ = − (t a ⊗ I R ′ ) ii ′ ,jj ′ − λ(I R ⊗t * a ) ii ′ ,jj ′ k(1 − λ 2 )x 2∆ R −1 12x 2∆ R ′ 12 x 13 x 23 . (5.7) Similar reasoning leads to J a (x 3 )Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) λ = (I R ⊗t * a ) ii ′ ,jj ′ − λ(t a ⊗ I R ′ ) ii ′ ,jj ′ k(1 − λ 2 ) x 2∆ R 12x 2∆ R ′ −1 12x OPEs and equal-time commutators In this section we use the two-point and three-point functions of the currents and primary fields to find their OPE algebra up to order 1/k reads and exact in the defor-mation parameter λ. The result is J a (x 1 )J b (x 2 ) = δ ab x 2+γ (J) 12x γ (J) 12 + c(λ) f abc J c (x 2 ) x 12 + d(λ) f abcJ c (x 2 )x 12 x 2 12 + . . . , J a (x 1 )J b (x 2 ) = −γ (J) δ ab |x 12 | 2 + d(λ) f abcJ c (x 2 ) x 12 + d(λ) f abc J c (x 2 ) x 12 + . . . , J a (x 1 )Φ (1) i,i ′ (x 2 ,x 2 ) = − (t a ) i m Φ (1) m,i ′ (x 2 ,x 2 ) − λ(t * a ) i ′ m ′ Φ (1) i,m ′ (x 2 ,x 2 ) x 12 k(1 − λ 2 ) + . . . , J a (x 1 )Φ (1) i,i ′ (x 2 ,x 2 ) = (t * a ) i ′ m ′ Φ (1) i,m ′ (x 2 ,x 2 ) − λ(t a ) m i Φ (1) m,i ′ (x 2 ,x 2 ) x 12 k(1 − λ 2 ) + . . . , Φ(1)I (x 1 ,x 1 )Φ (2) J (x 2 ,x 2 ) =C I JKΦ (3) K (x 2 ,x 2 ) xc(λ) = (1 − λ 3 ) 2 k(1 − λ 2 ) 3 , d(λ) = λ 2 (1 − λ) 2 k(1 − λ 2 ) 3 . (6.2) Having the OPEs at our disposal, we can easily compute the equal-time commutator of the currents and primaries through a time-ordered limiting procedure [ f (σ 1 , τ), g(σ 2 , τ)] = lim ε→0 ( f (σ 1 , τ + iε)g(σ 2 , τ) − g(σ 2 , τ + iε) f (σ 1 , τ)) (6.3) and the following representations of Dirac delta-function lim ε→0 1 σ − iε − 1 σ + iε = 2πi δ(σ) , lim ε→0 1 (σ − iε) 2 − 1 (σ + iε) 2 = −2πi δ ′ (σ) , lim ε→0 σ + iε (σ − iε) 2 − σ − iε (σ + iε) 2 = 2πi δ(σ) . (6.4) Employing Eqs. (6.1), (6.3) and (6.4), we find to order 1 / √ k that 4 [J a (σ 1 ), J b (σ 2 )] = 2π i δ ab δ ′ (σ 12 ) + 2π f abc ( c(λ)J c (σ 2 ) − d(λ)J c (σ 2 )) δ(σ 12 ) , [J a (σ 1 ),J b (σ 2 )] = −2π i δ ab δ ′ (σ 12 ) + 2π f abc (c(λ)J c (σ 2 ) − d(λ)J c (σ 2 )) δ(σ 12 ) , [J a (σ 1 ),J b (σ 2 )] = 2π d(λ) f abc (J c (σ 2 ) +J c (σ 2 )) δ(σ 12 ) , (6.5) and [Φ (1) i,i ′ (σ 1 ),Φ (2) j,j ′ (σ 2 )] = 0 , (6.6) [J a (σ 1 ), Φ (1) i,i ′ (σ 2 )] = − 2π k(1 − λ 2 ) (t a ) i m Φ (1) m,i ′ (σ 2 ) − λ(t * a ) i ′ m ′ Φ (1) i,m ′ (σ 2 ) δ(σ 12 ) , [J a (σ 1 ), Φ (1) i,i ′ (σ 2 )] = 2π k(1 − λ 2 ) (t * a ) i ′ m ′ Φ (1) i,m ′ (σ 2 ) − λ(t a ) i m Φ (1) m,i ′ (σ 2 ) δ(σ 12 ) . These equal-time commutators turn out to be isomorphic to two commuting copies of current algebras with opposite levels [S a (σ 1 ), S b (σ 2 )] = i k 2π δ ab δ ′ (σ 12 ) + f abc S c (σ 2 ) δ(σ 12 ) , [S a (σ 1 ),S b (σ 2 )] = − i k 2π δ ab δ ′ (σ 12 ) + f abcS c (σ 2 ) δ(σ 12 ) , [S a (σ 1 ),S b (σ 2 )] = 0 , [S a (σ 1 ), Φ (1) i,i ′ (σ 2 )] = −(t a ) i m Φ (1) m,i ′ (σ 2 ) δ(σ 12 ) , [S a (σ 1 ), Φ (1) i,i ′ (σ 2 )] = (t * a ) i ′ m ′ Φ (1) i,m ′ (σ 2 ) δ(σ 12 ) ,(6.7) where S a = 1 2π k 1 − λ 2 (J a − λJ a ) ,S a = 1 2π k 1 − λ 2 (J a − λJ a ) . (6.8) The parameter λ does not appear in this algebra but it does in the time evolution of the system due to the fact that, as it turns out, the Hamiltonian in terms of S a andS a is λ-dependent (cf. eq. (2.11) of [2]). Also, the λ-dependence still appears in the OPEs of the S a andS a among them. The reasons is that the OPEs, unlike the commutators (6.7), are not computed at equal times. Finally we take the classical limit of (6.5) and appropriately rescaling the currents, we find Rajeev's deformation of the canonical structure of the isotropic PCM [35] (recall that, in our conventions the group structure constants f abc are taken to be imaginary) {I a ± , I b ± } P.B. = −i e 2 f abc (I c ∓ (σ 2 ) − (1 + 2x)I c ± (σ 2 )) δ(σ 12 ) ± 2e 2 δ ab δ ′ (σ 12 ) , {I a ± , I b ∓ } P.B. = i e 2 f abc (I c + (σ 2 ) + I c − (σ 2 )) δ(σ 12 ) , (6.9) realized through the action (1.5) of [2] e = 2d(λ) = 1 k(1 − λ 2 ) 2λ 1 + λ , c(λ) d(λ) = 1 + 2x , x = 1 + λ 2 2λ . (6.10) That the deformed brackets (6.9) follow as the classical limit of the OPEs provides actually, for the isotropic case, the mathematical proof that the action (1.5) is in fact the effective action of the non-Abelian Thirring model action (1.4). The reason is that (1.5) provides, as was shown in [2], a realization of (6.9) which in turn was derived by using (1.4) as the starting point. Conclusions In this work we have computed all possible two-and three-point functions of current and primary field operators for the λ-deformed integrable σ-models. One direction for extending our work would be to consider cases beyond isotropy, i.e. when the matrix λ is not proportional to the identity. In particular, we believe that the equal-time commutators of the currents and primaries will take the form of (6.7), under an analogue to (6.8) relation. Another direction would be to calculate the subleading, in the 1/k expansion, terms of all physical quantities such as the β-function, the anomalous dimension matrix and the fusion coefficients. These line of research would, hopefully, be culminated by finding the exact in both λ and k expressions for these physical quantities as well as the underlying effective action. Acknowledgments The authors would like to thank each others home institutes for hospitality. The research of K. Siampos is partially supported by the Germaine de Stael France-Swiss bilateral program (project no 32753SG). K. Sfetsos and K. Siampos would like to thank the TH-Unit at CERN for hospitality and financial support during the final stages of this project. K. Siampos would like also to thank the ICTP, Trieste for hospitality during the final stages of this project. A Various integrals In this appendix we assemble all the integrals that will be needed in our perturbative calculations. In all integrals considered below the integration domain is a disc of radius R in which the various external points labeled by x's are excluded. This can be done by encircling them with circles having arbitrarily vanishing radius. One way to prove the expressions below is to use Stokes' theorem in two-dimensions for appropriately chosen vectors and contours. The first set of integrals is the exact version of the integrals in (2.31) and (2.33) d 2 z (z − x 1 )(z −x 2 ) = −π ln |x 1 − x 2 | 2 R 2 − x 1x2 , d 2 z (z − x 1 ) 2 (z −x 2 ) 2 = π 2 δ (2) (x 1 − x 2 ) − πR 2 (R 2 − x 1x2 ) 2 , (A.1) where the R > |x 1,2 |. By taking derivatives we may compute the exact analog of the integrals in (2.32). A generalization of the first of the above integrals is given by d 2 z M ∏ i=1 (z − x i ) N ∏ i=1 (z −ȳ j ) = −π M ∑ i=1 N ∑ j=1 1 A i B j ln |x i − y j | 2 R 2 − x iȳj , (A.2) with R > {|x i |, |y j |} and A i = M ∏ j=1 j =i (x i − x j ) , B i = N ∏ j=1 j =i (ȳ i −ȳ j ) . This relation can be proved by first performing a partial fraction decomposition and then use (A.1). A special case of this is when the denominators are cubic polynomials. Namely, d 2 z (z − x 1 )(z − x 2 )(z −x 1 ) = − π x 12 ln ε 2 |x 12 | 2 + ln R 2 − x 2x1 R 2 − |x 1 | 2 , d 2 z (z − x 1 )(z − x 1 )(z −x 2 ) = − π x 12 ln ε 2 |x 12 | 2 + ln R 2 − x 1x2 R 2 − |x 1 | 2 , (A.3) which is the exact analogs of the integrals in (2.34). Another important integral is given by d 2 z (z − x 1 )(z −x 2 ) ln |z − x 1 | 2 R 2 − x 1z = − π 2 ln 2 |x 1 − x 2 | 2 R 2 − x 1x2 . (A.4) The large R limit of this is given in (2.35) and is necessary for the derivation of the two-loop contribution in (3.6). B Perturbative computation of the JJ correlator In this appendix we present the perturbative calculation of the JJ two-point function. At the conformal point it vanishes. Order O(λ): To that order we have that J a (x 1 )J b (x 2 ) (1) λ = − λ π d 2 z J a (x 1 )J c (z) J c (z)J b (x 2 ) = −πλδ ab δ (2) (x 12 ) . (B.1) Order O(λ 2 ): To that order we have that J a (x 1 )J b (x 2 ) (2) λ = λ 2 2π 2 d 2 z 12 J a (x 1 )J a 1 (z 1 )J a 2 (z 2 ) J a 1 (z 1 )J a 2 (z 2 )J b (x 2 ) = − λ 2 2π 2 δ ab c G k J(x 1 , x 2 ) , (B.2) where J(x 1 , x 2 ) = d 2 z 12 (x 1 − z 1 )(x 1 − z 2 )(z 1 − z 2 )(x 2 −z 1 )(x 2 −z 2 )(z 1 −z 2 ) = d 2 z 12 (x 1 − z 1 ) 2 (x 2 −z 2 ) 2 1 z 1 − z 2 − 1 x 1 − z 2 1 z 1 −z 2 + 1 x 2 −z 1 (B.3) = J 11 + J 12 + J 21 + J 22 , where we have broken the integral J into the four integrals J ij , resulting from multiplying out the terms in the parenthesis, in a rather self-explanatory notation. We have that J 11 = ∂ x 1 ∂x 2 d 2 z 12 (x 1 − z 1 )(x 2 −z 2 )(z 1 − z 2 )(z 1 −z 2 ) = −π∂ x 1 ∂x 2 d 2 z 2 (x 1 − z 2 )(x 2 −z 2 ) ln ε 2 |z 2 − x 1 | 2 = π 2 ∂ x 1 ∂x 2 ln ε 2 ln |x 12 | 2 − 1 2 ln 2 |x 12 | 2 (B.4) = −π 3 δ (2) (x 12 ) ln ε 2 |x 12 | 2 + π 2 |x 12 | 2 , where we have used (2.35). Also J 12 = π d 2 z 1 (x 1 − z 1 ) 2 (x 2 −z 1 ) 2 = π 3 δ (2) (x 12 ) , J 21 = π d 2 z 2 (x 2 −z 2 ) 2 (x 1 − z 2 ) 2 = π 3 δ (2) (x 12 ) . (B.5) Note that for J 12 we have first performed the z 2 -integration which is not in accordance with our regularization prescription. However, we now show that the same result follows if we do first the z 1 according to our regularization. We easily find that J 12 = ∂ x 1 d 2 z 12 (x 1 − z 2 )(x 2 −z 2 ) 2 1 (z 1 − x 1 )(x 2 −z 1 ) − 1 (z 1 − z 2 )(x 2 −z 2 ) = π∂ x 1 ln |x 12 | 2 d 2 z 2 (x 2 −z 2 ) 2 (x 1 − z 2 ) + π d 2 z 2 ln |z 2 − x 2 | 2 (z 2 − x 1 ) 2 (z 2 −x 2 ) 2 (B.6) = −π 2 ∂ x 1 ln |x 12 | 2 x 12 + π 3 (1 + ln |x 12 | 2 ) δ (2) (x 12 ) + π 2 |x 12 | 2 , where we have used the fact that the second integral in the second line above can be obtained from (B.4) (with ε = 1). A simple algebra gives the same expression as in (B.5). Finally J 22 = − π x 12 d 2 z 2 (x 2 −z 2 ) 2 (x 1 − z 2 ) = π 2 |x 12 | 2 . (B.7) Therefore collecting all contributions we find that J(x 1 , x 2 ) = 2π 4 1 − 1 2 ln ε 2 |x 12 | 2 δ (2) (x 12 ) + π 3 |x 12 | 2 . (B.8) Order O(λ 3 ): To that order we have that J a (x 1 )J b (x 2 ) (3) λ = − λ 3 6π 3 d 2 z 123 J a (x 1 )J a 1 (z 1 )J a 2 (z 2 )J a 3 (z 3 ) × J a 1 (z 1 )J a 2 (z 2 )J a 3 (z 3 )J b (x 2 ) (B.9) The four-point function is given by (2.6) and is multiplied by the analogous four-point function for antiholomorphic currents. Keeping terms up to O(1/k), disregarding terms giving rise to bubbles and taking into account the above permutation symmetry we arrive at J a (x 1 )J b (x 2 ) (3) λ = 2δ ab c G k λ 3 π 3 K(x 1 , x 2 ) , (B.10) where K(x 1 , x 2 ) = d 2 z 123 (z 1 − x 2 )(x 1 − z 2 )(x 1 − z 3 )(z 2 − z 3 )(z 1 −z 2 ) 2 (z 3 −x 2 ) 2 . (B.11) Performing the integrations first over z 1 and then over z 2 we obtain that K(x 1 , x 2 ) = −π 2 d 2 z 3 (z 3 − x 1 ) 2 (z 3 −x 2 ) 2 ln ε 2 |z 3 − x 1 | 2 = −π 4 ln ε 2 δ (2) (x 12 ) +π 2 ∂ x 1 ∂x 2 d 2 z 3 (z 3 − x 1 )(z 3 −x 2 ) ln |z 3 − x 1 | 2 + d 2 z 3 (z 3 − x 1 ) 2 (z 3 −x 2 ) 2 = −π 4 ln ε 2 δ (2) (x 12 ) + π 4 δ (2) (x 12 ) − π 3 2 ∂ x 1 ∂x 2 ln 2 |x 12 | 2 (B.12) = π 4 1 − ln ε 2 |x 12 | 2 δ (2) (x 12 ) + π 3 |x 12 | 2 . In conclusion (B.1), (B.2) and (B.10) combine to (3.6) in the main text. C Perturbative computation of the J J J correlator In this appendix we present the perturbative calculation of the J J J three-point correlator. The O(λ) contribution to this correlators vanishes since J = 0. Proceeding to higher orders in the λ-expansion we have: Using the identity Employing again an analogue of the identity (C.6) we get that J a (x 1 )J b (x 2 )J c (x 3 ) (2) abc = λ 2 f abc π 2 √ k d 2 z 12 (z 2 −x 3 ) 2 (x 1 − z 1 ) 2 (x 2 − z 2 ) 2 1 z 1 −z 2 − 1 z 1 −x 3 = − λ 2 f abc π √ k d 2 z 2 (z 2 −x 3 ) 2 (x 2 − z 2 ) 2 1 x 1 − z 2 − 1 x 13 = λ 2 f abc π √ k ∂ x 2 ∂x 3 d 2 z 2 (x 1 − z 2 )(z 2 −x 3 )(x 2 − z 2 ) (D.4) = λ 2 f abc π √ k ∂ x 2 ∂x 3 1 x 12 d 2 z 2 z 2 −x 3 1 x 2 − z 2 − 1 x 1 − z 2 = λ 2 f abc √ k ∂ x 2 ∂x 3 1 x 12 ln |x 23 | 2 |x 13 | 2 (D.5) = − λ 2 √ k f abc x 12 x 2 12x 23x13 + π δ (2) (x 23 ) x 12 , where we have included the contact term involving external points. This will be neglected in the main text. E Perturbative computation of the ΦΦ correlator In this appendix we present the perturbative calculation of the ΦΦ correlator. Order O(λ): To that order we have that Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) (1) λ = − λ π d 2 z Φ (1) i,i ′ (x 1 ,x 1 )J c (z)J c (z)Φ (2) j,j ′ (x 2 ,x 2 ) = = λ π √ k d 2 z (t (1) a ) i ℓ z − x 1 Φ (1) ℓ,i ′ (x 1 ,x 1 )J c (z)Φ (2) j,j ′ (x 2 ,x 2 ) + (t (2) a ) j ℓ z − x 2 Φ (1) i,i ′ (x 1 ,x 1 )J c (z)Φ (2) ℓ,j ′ (x 2 ,x 2 ) (E.1) = − λ/k x 2∆ R 12x 2∆ R ′ 12 ln ε 2 t (1) a ⊗t (1)T a + t (2)T a ⊗t (2) a + ln |x 12 | 2 t (1) a ⊗t (2) a + t (1)T a ⊗t (2)T a ii ′ ,jj ′ , where we have used (2.11) and (2.12) and wrote the result as a direct product of matrices. The representations involved are in fact conjugate to each other. Therefore, using (2.8) we find that Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) (1) λ = −2 λ k (t a ⊗ t * a ) ii ′ ,jj ′ x 2∆ R 12x 2∆ R ′ 12 ln ε 2 |x 12 | 2 . (E.2) Order O(λ 2 ): To that order we find that Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) (2) λ = λ 2 2π 2 1 √ k I (1) ii ′ ,jj ′ + I (2) ii ′ ,jj ′ + I (3) ii ′ ,jj ′ + I (4) ii ′ ,jj ′ , (E.3) where the four different terms are computed below and arise by using the current Ward identity with respect to the current J a (z 1 ). • The first term is I (1) ii ′ ,jj ′ = − d 2 z 12 (t (1) a ) i ℓ z 1 − x 1 Φ (1) ℓ,i ′ (x 1 .x 1 )J a (z 1 )J b (z 2 )J b (z 2 )Φ (2) j,j ′ (x 2 .x 2 ) = − d 2 z 12 (t (1) a ) i ℓ (z 1 − x 1 )z 2 12 Φ (1) ℓ,i ′ (x 1 ,x 1 )J a (z 2 )Φ (2) j,j ′ (x 2 ,x 2 ) (E.4) = π 2 C R √ k (I R ⊗ I R ′ ) ii ′ ,jj ′ x 2∆ R 12x 2∆ R ′ 12 ln ε 2 |x 12 | 2 , where we have kept only contributions which will give terms of O(1/k) to the final result. In addition we used the integral d 2 z 12 (z 1 − x 1 )(z 2 − x 2 )z 2 12 = π d 2 z 2 (x 1 −z 2 )(z 2 − x 2 ) = π 2 ln |x 12 | 2 , (E.5) as well as the same with x 2 → x 1 in which case |x 12 | 2 → ε 2 in the result above. • The second term is I (2) ii ′ ,jj ′ = − d 2 z 12 (t (2) a ) j ℓ z 1 − x 2 Φ (1) i,i ′ (x 1 ,x 1 )J a (z 1 )J b (z 2 )J b (z 2 )Φ (2) ℓ,j ′ (x 2 ,x 2 ) = − d 2 z 12 (t (2) a ) j ℓ (z 1 − x 2 )z 2 12 Φ (1) i,i ′ (x 1 ,x 1 )J a (z 2 )Φ (2) ℓ,j ′ (x 2 ,x 2 ) (E.6) = π 2 C R √ k (I R ⊗ I R ′ ) ii ′ ,jj ′ x 2∆ R 12x 2∆ R ′ 12 ln ε 2 |x 12 | 2 , where as before we have kept only contributions providing at most O(1/k) terms in the final result. • The third term is I (3) ii ′ ,jj ′ = f abc d 2 z 12 z 12 Φ (1) i,i ′ (x 1 ,x 1 )J a (z 1 )J c (z 2 )J b (z 2 )Φ (2) ℓ,j ′ (x 2 ,x 2 ) = 0 , (E.7) since to O(1/ √ k) we get a result proportional to f abc δ ab = 0. • The fourth term is more involved to compute. The result is I (4) ii ′ ,jj ′ = d 2 z 12 z 2 12 Φ (1) i,i ′ (x 1 ,x 1 )J a (z 1 )J a (z 2 )Φ (2) ℓ,j ′ (x 2 ,x 2 ) = 2π 2 C R ′ k (I R ⊗ I R ′ ) ii ′ ,jj ′ x 2∆ R 12x 2∆ R ′ 12 ln ε 2 |x 12 | 2 . (E.8) Note that this is expected since it is just the sum of the other two non-vanishing terms with the representations R and R ′ exchanged. Order O(λ 3 ): To that order we have that Φ (1) i,i ′ (x 1 ,x 1 )Φ (2) j,j ′ (x 2 ,x 2 ) (3) λ = − λ 3 6π 3 J(1) ii ′ ,jj ′ + J (2) ii ′ ,jj ′ + J (3) ii ′ ,jj ′ + J (4) ii ′ ,jj ′ + J (5) ii ′ ,jj ′ + J (6) ii ′ ,jj ′ , (E.9) where the six different terms are obtained by applying the Ward identity for the current J a (z 1 ). • The first term originates from the contraction of the current J a (z 1 ) with the primary field Φ (1) and leads to J (1) ii ′ ,jj ′ = − 1 √ k d 2 z 123 (t (1) a ) i k z 1 − x 1 Φ (1) k,i ′ (x 1 ,x 1 )J a (z 1 )J b (z 2 )J b (z 2 )J c (z 3 )J c (z 3 )Φ (2) j,j ′ (x 2 ,x 2 ) (E.10) The next step is to contract one of the remaining holomorphic currents, lets say J b (z 2 ). This current can not be contracted with any of the external primaries because in that case the last holomorphic current should also be contracted with an external field too and as a result this contribution will be of order 1/k 3/2 . Since in this calculation we keep terms of order O(1/k) this contribution can be ignored. For the same reason the holomorphic currents J b (z 2 ) and J c (z 3 ) can not be contracted through the non-Abelian part of their OPE but only via the Abelian part. Once we have contracted all the holomorphic currents we start treating the anti-holomorphic ones by choosingJ a (z 1 ) Next we employ the Ward identity for the current at the pointz 1 . This current can not be contracted with the other anti-holomorphic currents through a δ-Kronecker term because in such a case this term will be proportional either to f abc δ ab = 0 or to f abc δ ac = 0. AlsoJ a (z 1 ) can not be contracted with the external primary fields because in such a case the corresponding diagram will disconnected, thus it will be the product of a bubble involving the points z 2 and z 3 times the rest of the diagram. Consequently, the only contribution that remains comes from the non-abelian contraction of either J a (z 1 ) with eitherJ b (z 2 ) orJ c (z 3 ). In both cases the resulting diagrams will be disconnected, i.e. they will be the product of the tree-level ΦΦ correlator times a bubble involving all interactions points z i , i = 1, 2, 3. Therefore, we conclude that J (2) ii ′ ,jj ′ = 0 . (E.14) • The third term originates from the contraction of the current J a (z 1 ) with J b (z 2 ) through the Abelian term of their OPE J (3) ii ′ ,jj ′ = d 2 z 123 z 2 12 Φ (1) i,i ′ (x 1 ,x 1 )J a (z 1 )J a (z 2 )J c (z 3 )J c (z 3 )Φ (2) j,j ′ (x 2 ,x 2 ) . (E.15) The last holomorphic current, i.e. the one at point z 3 should necessarily be contracted with each of the external primaries giving a factor of 1/ √ k and leaving us with a sum of two correlators involving two primaries and the three anti-holomorphic currents. Choosing the current at z 1 to be the one for which we will apply the Ward identity we obtain the following terms: i) the term whenJ a (z 1 ) is contracted withJ a (z 2 ). This diagram will have a factor of 1 z 2 12z 2 12 indicating that it is disconnected and should, thus be ignored. ii) the term arising from the contraction ofJ a (z 1 ) withJ c (z 3 ) through the non-Abelian term of their OPE will also be zero since we have saturated the powers of 1/k and all remaining contractions should be Abelian resulting to the factor of f acd δ ad = 0. iii) the term arising form the contraction ofJ a (z 1 ) withJ c (z 3 ) through the Abelian term (3.16) of the exact β-function and the anomalous dimensions implies an ansatz of the form around λ = 0 agree with (5.1) and(5.2). We stress that the one-and two-loop calculations in conjunction with the symmetry (1.7) are enough to fully determine the all-loop expressions for the correlators under consideration. Thus, the O(λ 3 ) terms in (5.1) and (5.2) provide perturbative checks of the all-loop results. Note that the deformation mixes the left and right representations. It can be easily checked that the λ-deformed direct products in the numerators in the above correlators form representations of the algebra as well. JK was given in (4.22), γ (J) is the anomalous dimension of the current given in (3.21), the γ 12;3 andγ 12;3 are given by (4.20) and (4.21) and These models are characterised by the deformation parameter λ, as well as by the integer level k of the WZW model. Our results are valid for any semisimple group G, for all values of the deformation parameter λ and up to order 1/k in the large k expansion. We achieved this goal by combining the first few orders in perturbation theory with analyticity arguments as well as with a non-trivial duality-type symmetry shared by these models. The two-and three-point correlators allowed us to deduce the exact in λ OPEs of all currents and primary operators. Furthermore, based on our results we derived the anomalous dimensions and correlation functions for the operators in two important limits of the aforementioned λ-deformed σ-models, namely the non-Abelian T-dual of the PCM and the pseudodual model. Our results are summarized as follows: 1. In section 3.1 we presented the results for the two-point correlator of two currents. From these correlators and in conjunction with the aforementioned sym-metry we derived the all-loop β-function of the theory as well as In section 4.1 we provide the reader with the exact two-point functions of all primary operators of the theory, as well as with their exact in λ anomalous dimensions. In this case, the role of the symmetry is instrumental since it is realised in a non-trivial way. 4. In section 4.2 we provide the reader with the exact three-point functions of all primary operators of the theory. 5. In section 5 we calculated the exact, in λ, three-point correlators JΦΦ and J ΦΦ . 6. In section 6 we deduced all relevant OPEs between currents and/or primary fields that are consistent with the exact results for the two-and three-point functions given in previous sections. We also derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable λ-deformed σ-models. This essentially proves in a mathematical sense that the action (1.5) for an isotropic deformation is indeed the effective action of the non-Abelian Thirring model action. The OPEs and the equal-time commutators for the currents are in agreement with those obtained in[32], for current-current perturbations of the WZW model on supergroups. Order O(λ 2 ): This contribution is immediately seen to be equal toTo proceed with the contractions we single out J a 1 to perform them. Disregarding the disconnected and bubble pieces and also noting that the Abelian contractions, i.e.contractions leading to second poles, of J a 1 with the external currents vanish in our regularization scheme, we have that.The contribution is immediately seen to beAs we have already saturated the O(1/ √ k), we perform only Abelian contractions in this six-point function, yielding toand employing an analogue of the identity (C.6) we get that(C.8)D Perturbative computation of the J JJ correlatorIn this appendix we present the perturbative calculation of the J JJ three-point correlator. Of course at the conformal point this correlation function vanishes. Proceeding to higher orders in the λ-expansions we have that:Order O(λ): The contribution to the one-loop equalsEmploying an analogue of the identity (C.6) we get thatOrder O(λ 2 ): The contribution to the two-loop is equal toto use in the Ward identity. As above, this current cannot be contracted with any of the external primaries since in this case the remaining anti-holomorphic currents at z 2 and atz 3 should be contracted through a δ-Kronecker term and resulting into thewhich indicates that it is disconnected and should be ignored. Thus, the current atz 1 can be contracted only with the anti-holomorphic currents atz 2 and atz 3 .Notice, however, that this contraction can not be non-Abelian because in that case the result will be proportional to f abc δ bc = 0. We have thus concluded that the only non vanishing terms up to order O(1/k) will come from the Abelian contractions ofJ a (z 1 )with the other anti-holomorphic currents. The resulting integral iswhere the z 2 , z 3 exchange term applies only in the integrand and not in the measure of integration in accordance with our regularization prescription. It turns out that this term doubles the result of the term explicitly written. The integrals can now be performed from the left to the right, the z 1 integration first then z 2 and the z 3 last.Using (2.8) the result can be written as followsWe should mention that the ln ε 2 term originates from the first triple integral of (E.11) while the ln |x 12 | 2 term originates the second triple integral of (E.11).• The second term originates from the contraction of the current J a (z 1 ) with J b (z 2 ) through the non-Abelian term of their OPE. It readsSince we want to keep terms up to O(1/k) the holomorphic currents at points z 2 and z 3 must be contracted only through the Abelian term of their OPE. The resulting correlators will involve the two primary fields and the three anti-holomorphic currents.of their OPE contributes that(E.16)Notice that as always we keep the order of integrations. Furthermore, the first integral over z 1 gives a δ (2) (z 2 − z 3 ) which make the second integration over z 2 trivial. The third integral over z 3 is one of our basic ubiquitous ones. iv) the last contribution arises whenJ a (z 1 ) is contracted with the external primaries.The corresponding integrals areAdding the contributions from (E.16)and (E.17) we get for J(3)ii ′ ,jj ′ that(E.18)• The fourth term originates from the contraction of the current J a (z 1 ) with J c (z 3 )through the non-Abelian term of their OPE. It readsFollowing the same steps as in the second contribution above one can show that• The fifth term originates from the contraction of the current J a (z 1 ) with J c (z 3 ) through 42 the Abelian term of their OPEWorking as in the case of the third contribution we get that(E.22)• Finally, the last term originates from the contraction of the current J a (z 1 ) with theFollowing the same steps as in the first contribution one can show thatSumming up all six terms one obtains the final result at three-loops. It reads2∆ R ′ 12 ln ε 2 |x 12 | 2 .(E.25)F Perturbative computation of the JΦΦ correlatorFinally, in this last appendix, we present the perturbative calculation of the JΦΦ three-point correlator.Order O(λ): This contribution is equal towhere we have used (2.12).Order O(λ 2 ): This contribution is equal towhere we have used (2.11) disregarding bubble diagrams. 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S Demulder, K Sfetsos, D C Thompson, arXiv:1504.02781JHEP. 0719hep-thS. Demulder, K. Sfetsos and D. C. Thompson, Integrable λ-deformations: Squashing Coset CFTs and AdS 5 × S 5 , JHEP 07 (2015) 019, arXiv:1504.02781 [hep-th]. Supergravity background of λ-deformed model for AdS 2 × S 2 supercoset. R Borsato, A A Tseytlin, L Wulff, arXiv:1601.08192Nucl. Phys. 905264hep-thR. Borsato, A. A. Tseytlin and L. Wulff, Supergravity background of λ-deformed model for AdS 2 × S 2 supercoset, Nucl. Phys. B905 (2016) 264, arXiv:1601.08192 [hep-th]. All-loop anomalous dimensions in integrable λ-deformed σ-models. G Georgiou, K Sfetsos, K Siampos, arXiv:1509.02946Nucl. Phys. 90140hep-thG. Georgiou, K. Sfetsos and K. Siampos, All-loop anomalous dimensions in integrable λ-deformed σ-models, Nucl. Phys. B901 (2015) 40, arXiv:1509.02946 [hep-th]. Thirring model with U(n) symmetry -scale invariant only for fixed values of a coupling constant. R F Dashen, Y Frishman, http:/journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2781Phys. Lett. 112781Phys. Rev.R.F. Dashen and Y. Frishman, Thirring model with U(n) symmetry -scale invari- ant only for fixed values of a coupling constant, Phys. Lett. B46 (1973) 439, and Four Fermion Interactions and Scale Invariance, Phys. Rev. D11 (1975) 2781. Thirring Interactions, Nonabelian Bosefermi Equivalences and Conformal Invariance. D Karabali, Q H Park, H J Schnitzer, Nucl. Phys. 323572D. Karabali, Q. H. Park and H. J. Schnitzer, Thirring Interactions, Nonabelian Bose- fermi Equivalences and Conformal Invariance, Nucl. Phys. B323 (1989) 572. A New family of su(2) symmetric integrable sigma models. J Balog, P Forgacs, Z Horvath, L Palla, hep-th/9307030Phys. Lett. 324J. Balog, P. Forgacs, Z. Horvath and L. Palla, A New family of su(2) symmetric inte- grable sigma models, Phys. Lett. B324 (1994) 403, hep-th/9307030. Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model. K Sfetsos, K Siampos, arXiv:1405.7803Nucl. Phys. 885583hep-thK. Sfetsos and K. Siampos, Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model, Nucl. Phys. B885 (2014) 583, arXiv:1405.7803 [hep-th]. The all-loop non-Abelian Thirring model and its RG flow. G Itsios, K Sfetsos, K Siampos, arXiv:1404.3748Phys. Lett. 733265hep-thG. Itsios, K. Sfetsos and K. Siampos, The all-loop non-Abelian Thirring model and its RG flow, Phys. Lett. B733 (2014) 265, arXiv:1404.3748 [hep-th]. String Theory and the Nonabelian Thirring Model. D Kutasov, Phys. Lett. 22768D. Kutasov, String Theory and the Nonabelian Thirring Model, Phys. Lett. B227 (1989) 68. On the beta function for anisotropic current interactions in 2-D. B Gerganov, A Leclair, M Moriconi, hep-th/0011189Phys. Rev. Lett. 86B. Gerganov, A. LeClair and M. Moriconi, On the beta function for anisotropic cur- rent interactions in 2-D, Phys. Rev. Lett. 86 (2001) 4753, hep-th/0011189. Duality Off the Critical Point in Two-dimensional Systems With Nonabelian Symmetries. D Kutasov, Phys. Lett. 233369D. Kutasov, Duality Off the Critical Point in Two-dimensional Systems With Non- abelian Symmetries, Phys. Lett. B233 (1989) 369. Nonabelian Bosonization in Two-Dimensions. E Witten, Commun. Math. Phys. 92455E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455. Current Algebra and Wess-Zumino Model in Two-Dimensions. V G Knizhnik, A B Zamolodchikov, Nucl. Phys. 24783V.G. Knizhnik and A.B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two-Dimensions, Nucl. Phys. B247 (1984) 83. Some Properties of an Analog of the Nonlinear σ-Model. C R Nappi, http:/journals.aps.org/prd/abstract/10.1103/PhysRevD.21.418Phys. Rev. 21418C.R. Nappi, Some Properties of an Analog of the Nonlinear σ-Model, Phys. Rev. D21 (1980) 418. Anomalous Dimensions in Deformed WZW Models on Supergroups. C Candu, V Mitev, V Schomerus, arXiv:1211.2238JHEP. 13033hep-thC. Candu, V. Mitev and V. Schomerus, Anomalous Dimensions in Deformed WZW Models on Supergroups, JHEP 1303 003 (2013), arXiv:1211.2238 [hep-th]. Non-chiral current algebras for deformed supergroup WZW models. A Konechny, T Quella, arXiv:1011.4813JHEP. 1103124hep-thA. Konechny and T. Quella, Non-chiral current algebras for deformed supergroup WZW models, JHEP 1103, 124 (2011), arXiv:1011.4813 [hep-th]. Beta Function of k Deformed AdS 5 × S 5 String Theory. C Appadu, T J Hollowood, arXiv:1507.05420JHEP. 151195hep-thC. Appadu and T. J. Hollowood, Beta Function of k Deformed AdS 5 × S 5 String Theory, JHEP 1511 (2015) 095, arXiv:1507.05420 [hep-th]. Conformal Current Algebra in Two Dimensions. S K Ashok, R Benichou, J Troost, arXiv:0903.4277JHEP. 090617hep-thS. K. Ashok, R. Benichou and J. Troost, Conformal Current Algebra in Two Dimen- sions, JHEP 0906 (2009) 017, arXiv:0903.4277 [hep-th]. Nonabelian Bosonization Without Wess-Zumino Terms. 1. New Current Algebra. S G Rajeev, Phys. Lett. 217123S. G. Rajeev, Nonabelian Bosonization Without Wess-Zumino Terms. 1. New Current Algebra, Phys. Lett. B217 (1989) 123.

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{'abstract': "We compute the 2-and 3-point functions of currents and primary fields of λ-deformed integrable σ-models characterized also by an integer k. Our results apply for any semisimple group G, for all values of the deformation parameter λ and up to order 1/k. We deduce the OPEs and equal-time commutators of all currents and primaries. We derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable λdeformed σ-models. We also present analogous results in two limiting cases of special interest, namely for the non-Abelian T-dual of the PCM and for the pseudodual model.", 'arxivid': '1604.08212', 'author': ['George Georgiou georgiou@inp.demokritos.gr \nInstitute of Nuclear and Particle Physics\nNational Center for Scientific Research Demokritos\nAg. ParaskeviGR-15310AthensGreece\n', 'Konstantinos Sfetsos ksfetsos@phys.uoa.gr \nDepartment of Nuclear and Particle Physics\nFaculty of Physics\nNational and Kapodistrian University of Athens\n15784AthensGreece\n', 'Konstantinos Siampos siampos@itp.unibe.ch \nAlbert Einstein Center for Fundamental Physics\nInstitute for Theoretical Physics / Laboratory for High-Energy Physics\nUniversity of Bern\nSidlerstrasse 5CH3012BernSwitzerland\n'], 'authoraffiliation': ['Institute of Nuclear and Particle Physics\nNational Center for Scientific Research Demokritos\nAg. ParaskeviGR-15310AthensGreece', 'Department of Nuclear and Particle Physics\nFaculty of Physics\nNational and Kapodistrian University of Athens\n15784AthensGreece', 'Albert Einstein Center for Fundamental Physics\nInstitute for Theoretical Physics / Laboratory for High-Energy Physics\nUniversity of Bern\nSidlerstrasse 5CH3012BernSwitzerland'], 'corpusid': 55797487, 'doi': '10.1016/j.nuclphysb.2016.05.018', 'github_urls': [], 'n_tokens_mistral': 31690, 'n_tokens_neox': 27297, 'n_words': 16119, 'pdfsha': 'b2f619e7241f9e59ee8ba41d6838434049f6477b', 'pdfurls': ['https://arxiv.org/pdf/1604.08212v2.pdf'], 'title': ['All-loop correlators of integrable λ-deformed σ-models', 'All-loop correlators of integrable λ-deformed σ-models'], 'venue': []}

arxiv

On the Asymptotic Convergence of the Transient and Steady State Fluctuation Theorems 29/3/99 Gary Ayton Research School Of Chemistry Australian National University Canberra ACT 0200Australia Denis J Evans Research School Of Chemistry Australian National University Canberra ACT 0200Australia On the Asymptotic Convergence of the Transient and Steady State Fluctuation Theorems 29/3/99 Non-equilibrium molecular dynamics simulations are used to demonstrate the asymptotic convergence of the Transient and Steady State forms of the Fluctuation Theorem. In the case of planar Poiseuille flow, we find that the Transient form, valid for all times, converges to the Steady State form on microscopic time scales. Further, we find that the time of convergence for the two Theorems coincides with the time required for satisfaction of the asymptotic Steady State Fluctuation Theorem. The Fluctuation Theorem (FT) [1][2][3][4][5] gives the probability ratio of observing fluctuations in a nonequilibrium system in which for N particles studied over a time t, the time averaged entropy production takes on some specified value compared to the negative of that same value. In the long time and large N limit, the Second Law of Thermodynamics is recovered, and for finite systems studied over short times, FT gives a quantitative explanation of the origins of irreversibility in systems with reversible dynamics. Evans and Searles [2][3][4] employed a Liouville measure and derived a Transient FT (TFT) which is exact for all finite times for systems starting at equilibrium and evolving towards a Steady State FT (SSFT) as t → ∞ . Gallavotti and Cohen [5] derived a steady state FT using the SRB measure and the chaotic hypothesis resulting in an asymptotic formula for the probability ratio. It would seem that if the nonequilibrium steady state is unique, then in the long time limit the Transient and Steady State FT's would converge to the same statement about fluctuations in entropy production. Asymptotic convergence is based on the independence of the final steady state measure with respect to the particular choice of the initial phase point (ie initial transients do not affect steady state averages or statistics). However there has been some recent discussion on this point. In this Letter we will present convincing non-equilibrium molecular dynamics (NEMD) simulation results that demonstrate the asymptotic convergence of the Transient and Steady State Fluctuation Theorems. We will show that the Transient FT holds for all time, and that after a finite time, the two Theorems converge to the same result. Not only does TFT approach the SSTF, but the time of convergence for the two Theorems coincides with the corresponding time of convergence for the asymptotic SSTF itself. The simulation was exactly as described in [6], and we will only very briefly discuss some details here. We simulated planar Poiseuille flow where an atomic fluid obeying Newton's equations of motion is placed between two heat extracting walls. Heat sinks in the walls remove heat at precisely the rate required to make the total energy of the system a constant of the motion. As in reference [6] we calculated the Integrated form of the Transient and Steady State FT's (TIFT and SSIFT respectively), written as p t p t N t t t W − + + = − ≡ ( ) / ( ) exp[ ( ) ] ( ) 3 α φ , where p p − + is the probability of observing an anti-trajectory versus a trajectory, N W is the number of ergostatting wall particles, α = −J VF K e W ( ) / Γ Γ 2 is the thermostat multiplier, K p m W i i N W = = ∑ 2 1 2 / is the kinetic energy of the wall particles, F e is the external field, the dissipative flux, J is defined, It is not possible to generate an exact steady state trajectory. This is because within phase space, the measure of any dissipative nonequilibrium steady state is zero. Therefore the probability of selecting initial phase points that lie exactly on the steady state attractor is zero. We can only approach the nonequilibrium steady state. We used an equilibration method of approaching the steady state: an arbitrary microcanonical phase point was chosen; an F e =0.032 external field was applied and the system was allowed to equilibrate towards the steady state for a time (t=500). This time is very much greater than the decay time of transient time averages (t = 3 -4). Subsequently a t=5000 "steady state" trajectory was generated. This process is sketched in Fig. 1b where the initial transient segment is shown by the dotted line, and the t=5000 steady state segment is the solid line. − ≡ ∫ J V d n u x ( ) ( ) ( ) Γ Γ r r r , The "steady state" trajectory was decomposed into 2.5 X 10 4 "steady state" subsegments each of duration t=0.2, (shown as small circles on the steady state trajectory) which could be examined to test the SSIFT with observation times ranging from t=0.2 ( with 2.5 X 10 4 possible samples for α + ( ) t 1 ), t=0.4 ( with 1.25 X 10 4 possible samples for α + ( ) t 2 ) to t=5000 (one sample of α + ( ) t 5000 ). These results confirm the assumptions of [2][3][4]. We note that the assumption of a unique nonequilibrium steady state is implicit in all linear and nonlinear response theory. Finally, we have recently shown that for stochastic systems the Transient and Steady State FT's show an analogous asymptotic convergence at long times [7]. where n(r) and u x (r) are the density and flow velocity. The averages, ... + denote averages over all trajectory segments for which α validity of TIFT and SSIFT with NEMD requires a method of generating either a set of transient trajectories or a single long steady state trajectory. In the transient case, two hundred transient nonequilibrium trajectory segments were generated from a microcanonically distributed ensemble of initial phase configurations, { Γ Γ eq }. Each transient segment was studied for t=10 (the integration time step was δt = 0 001 . ). This time is sufficiently long that all time averaged properties have converged to their steady state values. Each transient segment originated from a configuration, Γ Γ eq which was randomly chosen from an equilibrium microcanonical ensemble. It was then simultaneously subjected to an external field (F e =0.032) and initialised as a new transient segment time origin. After following the transient segment for t=10, the trajectory was terminated, and a new equilibrium configuration was selected. This process is shown in Fig. 1a, where the equilibrium microcanonical ensemble is shown by the dotted line, and the initial transient trajectory configuration, Γ Γ eq is designated by a filled circle. A TTIFT can then be tested by examining each of the transient segments at equal time intervals from their respective transient time origins (shown as small circles along the transient trajectories). Convergence of the TIFT and SSIFT can be tested by exploiting the nearFig. 3one clearly sees that both converge to 1 by a time t ~ 4. This test shows that at long times, the fluctuations in the transient states converge to the fluctuations of the steady states. We note that Z t p ( ), Z t φ ( ), and Y t SS ( ) converge to unity in approximately the same time.From examining the Transient and Steady State Fluctuation Theorems for a Poiseuille flow system we find strong numerical evidence that, as expected, when the nonequilibrium steady state is unique, the Transient FT asymptotically converges to the Steady State FT at long times. Further, the time scale for this convergence is the same as the time required for the satisfaction of the Steady State FT itself. For our system, convergence of the two Theorems occurs on a microscopic time scale. Figure 1a . 1aA sketch of method used to generate transient trajectories. The dotted line refers to an equilibrium microcanonical ensemble. Transient trajectories { Γ Γ i }, i=1,200 (solid lines) are studied for a total time of t=10, but averages are accumulated every t=0.2 in order to construct Figure 1b . 1bA sketch of method used to generate steady state trajectories. The dotted line refers to the initial transient trajectory with F e = 0.032. At the end of the t = 500 equilibration run, the time origin for the steady state trajectory Γ Γ is established (large filled circle), and is studied for t = 5000. Time segments of t = 0.2 were used to construct a SSIFT for the corresponding field strength. / ( ) (open symbol) and φ( ) t (shaded symbol). . D J Evans, E G D Cohen, G P Morriss, Phys. Rev. Lett. 712401D. J. Evans, E. G. D. Cohen and G. P. Morriss, Phys. Rev. Lett., 71, 2401 (1993). . D J Evans, D J Searles, Phys. Rev. E. 501645D. J. Evans and D. J. Searles, Phys. Rev. E, 50, 1645 (1994). . D J Evans, D J Searles, Phys. Rev. E. 525839D. J. Evans and D. J. Searles, Phys. Rev. E, 52, 5839 (1995). . D J Evans, D J Searles, Phys. Rev. E. 535808D. J. Evans and D. J. Searles, Phys. Rev. E, 53, 5808 (1996). . G Gallavotti, E G D Cohen, Phys. Rev. Lett. 742694G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett., 74, 2694 (1995); . G Gallavotti, E G D Cohen, J. Stat. Phys. 80931G. Gallavotti and E. G. D. Cohen, J. Stat. Phys,. 80, 931 (1995). A Localised Fluctuation Theorem. Gary Ayton, Denis J Evans, D J Searles, submitted to PRLGary Ayton, Denis J. Evans, and D. J. Searles, "A Localised Fluctuation Theorem", submitted to PRL, and at http://xxx.lanl.gov/abs/cond-mat/9901256. The Fluctuation Theorem for Stochastic Systems. D J Evans, D J Searles, Phys. Rev. E. submitted to Phys. Rev. E, and atD. J. Evans and D. J. Searles, Phys. Rev. E, "The Fluctuation Theorem for Stochastic Systems", submitted to Phys. Rev. E, and at http://xxx.lanl.gov/abs/cond-mat/9901258.

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{'abstract': 'Non-equilibrium molecular dynamics simulations are used to demonstrate the asymptotic convergence of the Transient and Steady State forms of the Fluctuation Theorem. In the case of planar Poiseuille flow, we find that the Transient form, valid for all times, converges to the Steady State form on microscopic time scales. Further, we find that the time of convergence for the two Theorems coincides with the time required for satisfaction of the asymptotic Steady State Fluctuation Theorem.', 'arxivid': 'cond-mat/9903409', 'author': ['Gary Ayton \nResearch School Of Chemistry Australian National University Canberra\nACT 0200Australia\n', 'Denis J Evans \nResearch School Of Chemistry Australian National University Canberra\nACT 0200Australia\n'], 'authoraffiliation': ['Research School Of Chemistry Australian National University Canberra\nACT 0200Australia', 'Research School Of Chemistry Australian National University Canberra\nACT 0200Australia'], 'corpusid': 118322649, 'doi': '10.1023/a:1004679628622', 'github_urls': [], 'n_tokens_mistral': 2710, 'n_tokens_neox': 2373, 'n_words': 1597, 'pdfsha': 'eecc14da55420dabb8d59dfb7fc2e86775226b67', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/9903409v1.pdf'], 'title': ['On the Asymptotic Convergence of the Transient and Steady State Fluctuation Theorems', 'On the Asymptotic Convergence of the Transient and Steady State Fluctuation Theorems'], 'venue': []}

arxiv

Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces 0123456789 Tom Vincent-Dospital *email:tom.vincent-dospital@fys.uio.no ITES UMR 7063 Université de Strasbourg 67084StrasbourgFrance The Njord Centre Department of physics SFF Porelab University of Oslo OsloNorway Alain Cochard alain.cochard@unistra.fr ITES UMR 7063 Université de Strasbourg 67084StrasbourgFrance Stéphane Santucci Laboratoire de Physique ENS de Lyon Université Claude Bernard CNRS Université de Lyon LyonFrance Lavrentyev Institute of Hydrodynamics Siberian Branch of the Russian Academy of Sciences NovosibirskRussia Knut Jørgen Måløy The Njord Centre Department of physics SFF Porelab University of Oslo OsloNorway Renaud Toussaint renaud.toussaint@unistra.fr ITES UMR 7063 Université de Strasbourg 67084StrasbourgFrance The Njord Centre Department of physics SFF Porelab University of Oslo OsloNorway Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces 012345678910.1038/s41598-021-98556-xScientific Reports | (2021) 11:204181 We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.In the physics of rupture, understanding the effects that material disorder has on the propagation of cracks is of prime interest. For instance, the overall strength of large solids is believed to be ruled by the weakest locations in their structures, and notably by the voids in their bulk samples 1,2 . There, cracks tend to initiate as the mechanical stress is concentrated. A growing focus has been brought on models in which the description of the breaking matrix remains continuous (i.e., without pores). There, the material disorder resides in the heterogeneities of the matrix 3-9 . The propagation of a crack is partly governed by its spatial distribution in surface fracture energy, that is, the heterogeneity of the energy needed to generate two opposing free unit surfaces in the continuous matrix 10 , including the dissipation processes at the tip 11 . From this disorder, one can model a rupture dynamics which holds a strongly intermittent behaviour, with extremely slow (i.e., pinned) and fast (i.e., avalanching) propagation phases. In many physical processes, including 12-14 but not limited 15-18 to the physics of fracture, such intermittency is referred to as crackling noise 19,20 . In the rupture framework, this crackling noise is notably studied to better understand the complex dynamics of geological faults 21-25 , and their related seismicity.Over the last decades, numerous experiments have been run on the interfacial rupture of oven-sintered acrylic glass bodies (PMMA)[26][27][28]. Random heterogeneities in the fracture energy were introduced by sand blasting the interface prior to the sintering process. An important aspect of such experiments concerns the samples preparation, which allows to constrain the crack to propagate along a weak (disordered) plane. It simplifies the fracture problem, leading to a negligible out-of plane motion of the crack front. This method has allowed to study the dynamics of rugous fronts, in particular because the transparent PMMA interface becomes more opaque when broken. Indeed, the generated rough air-PMMA interfaces reflect more light, and the growth of fronts can thus be monitored. Different models have successfully described parts of the statistical features of the recorded crack propagation. Originally, continuous line models 4,5,20,29 were derived from linear elastic fracture mechanics. While they could reproduce the morphology of slow rugous cracks and the size distribution of their avalanches, they fail to account for their complete dynamics and, in particular, for the distribution of local propagation velocity and for the mean velocity of fronts under different loading conditions. Later on, fiber bundle models were OPEN introduced 6,30,31 , where the fracture plane was discretized in elements that could rupture ahead of the main front line, allowing the crack to propagate by the nucleation and the percolation of damage. The local velocity distribution could then be reproduced, but not the long term mean dynamics of fronts at given loads. One of the most recent models (Cochard et al. 8 ) is a thermally activated model, based on an Arrhenius law, where the fracture energy is exceeded at subcritical stresses due to the molecular agitation. It contrasts to other models that are strictly threshold based (the crack only advances when the stress reaches a local threshold, rather than its propagation being subcritical). A notable advantage of the subcritical framework is that its underlying processes are, physically, well understood, and Arrhenius-like laws have long shown to describe various features of slow fracturing processes 26,32-36 . In particular, this framework has proven to reproduce both the mean behaviour of experimental fronts 37 (i.e., the average front velocity under a given load) and the actual distributions of propagation velocities along these fronts 8 , whose fat-tail is preserved when observing cracks at different scales 38 . It has recently been proposed 39,40 that the same model might also explain the faster failure of brittle matter, that is, the dramatic propagation of cracks at velocities close to that of mechanical waves, when taking into account the energy dissipated as heat around a progressing crack tip. Indeed, if fronts creep fast enough, their local rise in temperature becomes significant compared to the background one, so that they can avalanche to a very fast phase, in a positive feedback loop39,40.Here, we only consider slow fronts (i.e., fronts that creep slowly enough so that their temperature elevation is supposed to remain negligible). Building on the work of Cochard et al. 8 , we study various statistical features that can be generated with this Arrhenius-based model (re-introduced in the "Propagation model" section), when simulating the rupture of a disordered interface. By comparing these features to those reported for the PMMA experiment by Tallakstad et al. 28,38 ,Santucci et al. 27 and Maløy et al. 26 , we show a strong match to the experimental data for many of the scaling laws describing the fracture intermittent dynamics, including the growth of the fracture width ("Growth exponent and fracture energy correlation length" section), its distribution in local propagation velocity ("Local velocity distribution and fracture energy standard deviation" section), the correlation of this velocity in space and time ("Local velocities correlations" section), the size of the propagation avalanches ("Avalanches size and shape" section) and the front Hurst exponents ("Front morphology" section). We hence re-enforce the relevance of simple thermodynamics coupled with elasticity in the description of material failure. Propagation model Constitutive equations. We consider rugous crack that are characterised by a varying and heterogeneous advancement a(x, t) along their front, x being the coordinate perpendicular to the average crack propagation direction, a the coordinate along it, and t being the time variable (see Fig. 1). At a given time, the velocity profile along the rugous front is modelled to be dictated by an Arrhenius-like growth, as proposed by Cochard et al. 8 : where V (x, t) = ∂a(x, t)/∂t is the local propagation velocity of the front at a given time and V 0 is a nominal velocity, related to the atomic collision frequency 41 , which is typically similar to the Rayleigh wave velocity of the medium in which the crack propagates 42 . The exponential term is a subcritical rupture probability (i.e., between 0 and 1). It is the probability for the rupture activation energy (i.e., the numerator term in the exponential) to be exceeded by the thermal bath energy k B T 0 , that is following a Boltzmann distribution 41 . The Boltzmann constant is denoted k B and the crack temperature is denoted T 0 and is modelled to be equal to a constant room temperature (typically, T 0 = 298 K). Using this constant temperature corresponds to the hypothesis that the crack is propagating slowly enough so that no significant thermal elevation occurs by Joule heating at its tip (i.e., as inferred by Refs. 39,40 ). Such propagation without significant heating is notably believed to take place in the experiments by Tallakstad et al. 28 that we here try to numerically reproduce, and whose geometry is shown in Fig. 1. Indeed, www.nature.com/scientificreports/ their reported local propagation velocities V did not exceed a few millimetres per second, whereas a significant heating in acrylic glass is only believed to arise for fractures faster than a few centimetres per second 40,44 . See the supplementary information for further discussion on the temperature elevation. In Eq. (1), the rupture activation energy is proportional to the difference between an intrinsic material surface fracture Energy G c (in J m −2 ) and the energy release rate G at which the crack is mechanically loaded, which corresponds to the amount of energy that the crack dissipates to progress by a given fracture area. As the front growth is considered subcritical, we have G < G c . We here model the fracture energy G c to hold some quenched disorder that is the root cause for any propagating crack front to be rugous and to display an intermittent avalanche dynamics. This disorder is hence dependent on two position variables along the rupture interface. For instance, at a given front advancement a(x, t), one gets G c = G c (x, a) . The coefficient α 2 , in Eq. (1), is an area which relates the macroscopic G and G c values to, respectively, the microscopic elastic energy U = α 2 G stored in the molecular bonds about to break, and to the critical energy U c = α 2 G c above which they actually break. See Vanel et al. 36 , Vincent-Dospital et al. 40 or the supplementary information for more insight on the α 2 parameter, which is an area in the order of d 3 0 /l , where d 0 is the typical intra-molecular distance and l is the core length scale limiting the stress divergence at the crack tip. Finally, the average mechanical load that is applied on the crack at a given time is redistributed along the evolving rugous front, so that G = G(x, t) . To model such a redistribution, we here use the Gao and Rice 3 formalism, which integrates the elastostatic kernel along the front: In this equation, G is the mean energy release rate along the front and PV stands for the integral principal value. We, in addition, considered the crack front as spatially periodic, which is convenient to numerically implement a spectral version of Eq. (2) 45 as explained by Cochard et al. 8 . Equations (1) and (2) thus define a system of differential equations for the crack advancement a, which we have then solved with an adaptive time step Runge-Kutta algorithm 46 , as implemented by Hairer et al. 47 . The complete code for the crack simulation is available as a Software Heritage archive 48 . Further details on the code can be obtained by contacting the authors. Discretization. In this section, we discuss the main principles we have used in choosing the numerical accuracy of our solver. The related parameters are illustrated in Fig. 2. In attempting to correctly reproduce the experimental results of Tallakstad et al. 28 , this solver needs to use space and time steps, here denoted x s and t s , at least smaller than those on which the experimental fronts were observed and analysed. Thus, x s needs to be smaller than the experimental resolutions in space (the camera pixel size) x = a of about 2 to 5 µ m and 1/�t s needs to be higher than the experimental camera frame rate 1/�t . This frame rate was set by Tallakstad et al. 28 to about (100V )/�x , where V is the average front velocity of a given fracture realisation. The propagation statistics of our simulated fronts, henceforward shown in this manuscript, have, for consistency, always been computed on scales comparable to the experimental x , a , t steps. Thus, as �x s < �x and �t s < �t , we have first decimated the dense numerical outputs on the experimental observation grid, by discarding smaller time scales and by averaging smaller space scales to simulate the camera frame rate and pixel size. As the camera resolution was 1024 pixels, the lengths L of the crack segments that Tallakstad et al. 28 analysed were 1024 x = 3 to 7 mm long, and we have then analysed our numerical simulations on similar front widths. Yet, these simulations were priorly run on longer front segments, L s > L , in order to avoid any possible edge effects in the simulated crack dynamics (for instance in the case where L would not be much bigger than the typical size of the G c quenched disorder). (2) G(x, t) = G(t) 1 − 1 π PV +∞ −∞ ∂a(x ′ , t)/∂x ′ x − x ′ dx ′ . Figure 2. Illustration of the discretization principles and of the solver and observation grids. Three crack fronts at three successive times are shown, over which the parameters discussed in the "Discretization" section are defined. www.nature.com/scientificreports/ Overall, we have checked that the numerical results presented henceforward were obtained using a high enough time and space accuracy for them to be independent of the discretization (as is shown in the supplementary information). Physical parameters values. For the model dynamics to be compared to the experiments 28 , one must also ensure that the V 0 , α , T 0 , G and G c parameters are in likely orders of magnitude. As V 0 is to be comparable to the Rayleigh velocity of acrylic glass, we have here used 1 km s −149 . Lengliné et al. 37 furthermore estimated the ratio α 2 /(k B T 0 ) to be about 0.15 m 2 J −1 and they could approximate the quantity V 0 exp(−α 2 G c /[k B T 0 ]) to about 5 × 10 −14 m s −1 , where G c is the average value of G c in the rupturing interface. With our choice on the value of V 0 , we then deduce G c ∼ 250 J m −2 (note that the trade-off between V 0 and G c should be kept in mind when comparing our results with those by Cochard et al. 8 , as both papers use a different V 0 ). The value thus inverted for the fracture energy ( 250 J m −2 ), that is to represent the sintered PMMA interfaces, is logically smaller but comparable to that inferred by Vincent-Dospital et al. 40 for the rupture of bulk PMMA (about 1300 J m −2 ). Qualitatively, the longer the sintering time, the closer one should get from such a bulk toughness, but the less likely an interfacial rupture will occur. Experiments in two different regimes were run 28 : a forced one where the deflection of the lower plate (see Fig. 1) was driven at a constant speed, and a relaxation regime, where the deflection was maintained constant while the crack still advances. In both scenarii, the long term evolution of the average load G(t) and front position a(t) was shown 8,37 to be reproduced by Eq. (1). In the case of the experiments of Tallakstad et al. 28 , the intermittent dynamics measured in the two loading regimes were virtually identical. Such similarity likely arises from the fact that the avearge load G was, in both cases, computed to be almost constant over time, in regard to the spatial variation in G, described by Eq. (2) (see the supplementary information). Here, we will then consider that the crack is, in average along the front, always loaded with the same intensity (i.e., G(t) = G). The actual value of G , together with the average surface fracture energy of the medium G c , then mainly controls the average crack velocity V . This average velocity was investigated over five orders of magnitude in Ref. 28 , from 0.03 to 140 µm s −1 , which, in our formalism, shall correspond to values of (G c − G ) between 145 and 85 J m −2 , respectively, which is actually consistent with the values of G measured by Lengliné et al. 37 for cracks propagating at similar speeds. The intermittency of the crack motion was experimentally inferred to be independent on V and we show, in the supplementary information, that it is also the case in our simulations. The velocity variation along the front shall then only arise from the disorder in G c and from the related variations of G due to the roughness of the crack front. Further in this manuscript, we will use G = 120 J m −2 , which corresponds to an average propagation velocity of about 1.5 µm s −1 . Heterogeneous fracture energy Of course, the actual surface fracture energy field in which the rupture takes place will significantly impact the avalanches dynamics and the crack morphology. Such a field is yet a notable unknown in the experimental set-up of Tallakstad et al. 28 , as their interface final strength derived from complex sand blasting and sintering processes. Although these processes were well controlled, so that the rough rupture experiments were repeatable, and although the surfaces prior to sintering could be imaged 43 , the actual resulting distribution in the interface cohesion was not directly measurable. While this is, to some extent, a difficulty in assessing the validity of the model we present, we will here show that a simple statistical definition of G c is enough to simulate most of the avalanches statistics. We will indeed consider a normally distributed G c field around the mean value G c with a standard deviation δG c and a correlation length l c . Such a landscape in G c is shown in Fig. 3a, and we proceed to discuss the chosen values of δG c and l c in the "Growth exponent and fracture energy correlation length" and "Local velocity distribution and fracture energy standard deviation" sections. Growth exponent and fracture energy correlation length. Among the various statistical features studied by Tallakstad et al. 28 , was notably quantified the temporal evolution of their fracture fronts morphology. It was interestingly inferred that the standard deviation of the width evolution of a crack front h scales with the crack mean advancement: In this equation, x is a given position along the front, t is a time delay from a given reference time t 0 , and h writes as a being the average crack advancement at a given time. To mitigate the effect of the limited resolution of the experiments and obtain a better characterization of the scaling of the interfacial fluctuations on the shorter times, we computed the subtracted width, as proposed in Barabasi and Stanley 50 , and done by Tallakstad et al. 28 (whose experiments we here reproduce) and Soriano et al. 51. The scaling exponent β G is referred to as the growth exponent, and we will here show how it allows to deduce a typical correlation length for the interface disorder. Indeed, β G was measured to be 0.55 ± 0.08 by Tallakstad 28 . This value is close to 1/2, that is, consistent with an uncorrelated growth process (e.g., 50 ), such as simple diffusion or Brownian motion. We thus get a first indication on the disorder correlation length scale l c . To display an uncorrelated growth when observed with the experimental resolution ( x ∼ 3 µm), the fronts likely encountered asperities whose size was somewhat comparable to this resolution. Indeed, if these asperities in G c were much bigger, the growth would be perceived as correlated. By contrast, if they were much smaller (orders of magnitude smaller), the rugosity of the front would not be measurable, as only the average G c over an observation pixel would then be felt. Furthermore, and as shown in Fig. 4a, the exponent β G was observed on scales (Vt) up to 100 µ m, above which W stabilised to a plateau value of about 30 µ m. A common picture is here drawn, as both this plateau value and the typical crack propagation distance at which it is reached are likely to also be correlated with l c , as the front is to get pinned on the strongest asperities at this scale. From all these clues, we have considered, in our simulations, the correlation length of the disorder to be about l c = 50 µ m, and we show in Fig. 4a that it allows an approximate reproduction of the front growth exponent and of the plateau at high Vt . Note that the accuracy reported for the exponents in this manuscript is estimated by fitting various portions of the almost linear data points and reporting the dispersion of the thus inverted slopes. In Fig. 4b, we also show how varying l c impacts W, and, in practice, we have chosen l c by tuning it when comparing these curves to the experimental one. Noteworthily, the thus chosen l c is in the lower range of the size of the blasting sand grains ( 50 − 300 µ m) that were used 28 to generate the interface disorder. It is also comparable to the correlation length of the blasting induced topographic anomalies ∼ 18 µ m on the post-blasting/pre-sintering PMMA surfaces, as measured by Santucci et al. 43 (3) rms h(t) = <h(t) 2 > x,t 0 ∝ Vt β G . (4) h x,t 0 (t) = a(x, t 0 + t) − a(t 0 + t) − a(x, t 0 ) − a(t 0 ) ,(5)W(t) = rms(h(t)) 2 − min(rms(h(t))) 2 , by white light interferometry. Local velocity distribution and fracture energy standard deviation. While the crack advances at an average velocity V , the local velocities along the front, described by Eq. (1), are, naturally, highly dependent on the material disorder: the more diverse the met values of G c the more distributed shall these velocities be. Maløy et al. 26 and Tallakstad et al. 28 inferred the local velocities of their cracks with the use of a so-called waiting time matrix. That is, they counted the number of discrete time steps a crack would stay on a given camera pixel before advancing to the next one. They then deduced an average velocity for this pixel by inverting this number and multiplying it by the ratio between the pixel size and the time between two pictures: �a/�t . Such a method, that provides a spatial map V(x, a), was applied to our simulated fronts, and we show this V(x, a) map in Fig. 3c. As to any time t corresponds a front advancement a(x, t) (recorded with a resolution a ), an equivalent space-time map V(x, t) can also be computed, and it is shown in Fig. 3b. The experimental report 28 presented the probability density function of this latter (space-time) map P(V /V ) , and it was inferred that, for high values of V, the velocity distribution scaled with a particular exponent η = 2.6 ± 0.15 28,38 (see Fig. 5a). That is, it was observed that 28 , using the waiting time matrix (see text for details). The velocity are plotted related to the average crack velocity V = 1.5 µm s −1 . All parameters used to run the corresponding simulation are summarised in Table 1 8 , who introduced the model that we here discuss, inferred that the η exponent was mainly depend- ing on α 2 (δG c ) 2 /[k B T 0 (G c ) 2 ] . Truly, a more comprehensive expression could also include other quantities, such as V 0 or l c . Yet, as all other parameters have now been estimated, we can deduce δG c by varying it to obtain η ∼ 2.6 . We show, in Fig. 5b, how varying δG c impacts P(V /V ) and η . We found δG c ∼ 35 J m −2 . In Fig. 5a, we show the corresponding velocity distribution for a simulation run with this parameter, together with that from Tallakstad et al. 28 , showing a good match. Note that the ability of the model to reproduce the local velocity Table 1. The slope and plateau of the experimental data (shown in (a)) is marked by the dashed line for comparison. We chose the value of δG c by tuning it and fitting the slope and maximum of the experimental data, which are illustrated by the dashed lines. The rest of the parameters used in these simulations are as defined in Table 1. (6) P V /V ∝ V /V −η . Note that the ability of the model to reproduce the local velocity distribution was already shown by Cochard et al. 8 (see text for explanation and discussion). 8 , and this figure mainly aims at illustrating our calibration of the fracture energy field. The model we present is also slightly different to that of Cochard et al., as the interface fracture energy is, contrarily to this previous study, now described at scales below its correlation length, similarly to the observation scale of the experiments. We here verify that the reproduction of the local velocity distribution is still valid at these small scales. Satisfyingly, the inverted value of δG c is not too far from the value found by Lengliné et al. 37 for their fluctuation in the mean fracture energy G c along their sintered plates, when studying the mean front advancement (i.e., neglecting the crack rugosity) in similar PMMA interfaces, which was about 25 J m −2 . Further statistics We have now estimated the orders of magnitude of all parameters in Eqs. (1) and (2), including a likely distribution for an interface fracture energy representative of the experiments 28 we aim to simulate (i.e., including G c , δG c and l c ). For convenience, this information is summarised in Table 1. We will now pursue by computing additional statistics of the crack dynamics to compare them to those reported by Tallakstad et al. 28 . Local velocities correlations. In particular, we here compute the space and time correlations of the velocities along the front. That is, four correlation functions that are calculated from the V(x, t) and V(x, a) matrices (shown in Fig. 3) and defined as: where i and j are the variables of either V(x, t) or V(x, a) and δi a given i increment. V j is the mean of V(i, j) taken along j at a given i 0 . The corresponding δV j is the velocity standard deviation along the same direction and for the same i 0 . The correlation functions hence defined are the same as those used by Tallakstad et al. 28 on their own data, allowing to display a direct comparison of them in Fig. 6. A good general match is obtained. One can notice the comparable cut-offs along the x axis (Fig. 6a,c), indicating that our chosen correlation length for the interface disorder ( l c inferred in the "Growth exponent and fracture energy correlation length" section) is a good account of the experiment. Yet, one can notice that C xt (the velocity correlation along the crack front shown in Fig. 6a) is higher in the numerical case than in the experimental one. It could translate the fact that the experimental disorder holds wavelengths that are smaller than the observation scale x , and that our modelled G c distribution, where l c > �x , is rather simplified. To go further, Tallakstad et al. 28 modelled C xt as and inverted the values of τ x and x * to, respectively, 0.53 and about 100 µ m. Doing a similar fit on the simulated data, we found τ x ∼ 0.13 and x * ∼ 94 µ m. The related function is displayed in Fig. 6a (plain line). Our small τ x ∼ 0.13 may derive, as discussed, from the better correlation that our simulation displays at small δx ( τ x may in reality tend to zero for scales smaller than those we observe) compared to that of the experiments, while the matching x * probably relates to a satisfying choice we made for l c . Overall, the existence of a clear scaling law at (1) and (2) (8), is rather uncertain (see the two experimental plots in Fig. 6a) so that mainly the cut-off scale is of interest. On the time correlation C tx (Fig. 6b), one can similarly define the parameters A, τ t and a * to fit Eq. (7) with a function where A is a constant of proportionality. Fitting this function to Eq. (7) with a least-squares method, we found τ t ∼ 0.3 and a * ∼ 4.3 µ m, and this fit is represented by the dashed line in Fig. 6b. Tallakstad et al. 28 reported τ t ∼ 0.43 and a * ∼ 7 µ m. Figure 6b shows the experimental and simulated correlation functions in the V δt/a * -C tx (a * /V ) τ t /A domain, as this allowed a good collapse of the data from numerous experiments 28 . We show that it also allows an approximate collapse of our modelled correlation on a same trend. Finally, the derived value of a * consistently matches the apparent cut-off length in the C a x correlation function in Fig. 6d. This length being of a magnitude similar to that of the observation scale a , the crack local velocities appear uncorrelated along the direction of propagation, which is consistent with the β G ∼ 1/2 growth exponent (e.g., 50 ). Avalanches size and shape. We pursue by characterising the intermittent, burst-like, dynamics of our crack fronts and, more specifically, the avalanche (or depinning) and pinning clusters shown by the local front . The plain points were computed from the simulation whose parameters are presented in Table 1 and the hollow stars are some of the experimental data points extracted from Figs. 6 and 7 of Tallakstad et al. 28 . In plot (a), the line overlying the numerical data set corresponds to a fit using Eq. (8). (b) is plotted in a domain that allowed a good collapse of the experimental data for many experiments 28 . The parameters A, a * and τ t were inverted from Eq. (9) ). We define an avalanche when the front velocity locally exceeds the mean velocity V by an arbitrary threshold that we denote c, that is, when Similarly, we state that a front is pinned when (7) C ij (δi) = V (i 0 + δi, j) − V j V (i 0 , j) − V j (δV j ) 2 i 0 , (8) C xt (δx) ∝ δx −τ x exp − δx x * ,α 2 /(k B T 0 ) 0.15 m 2 J −1 G c 250 J m −2 G 120 J m −2 δG c 35 J m −2 l c 50 µm (b) a = x 3 µm t 10 ms L 3000 µm (c) x s 1 µm t s ∼ 5(9) C tx (δt) ≈ Aδt −τ t exp − V δt a * , We then map, in Fig. 7, the thus defined avalanching and pinned locations of the crack. Following the analysis of Tallakstad et al. 28 , we compute for each of these clusters the surface S, the crossline extent l x (that is, the maximum of a cluster width in the x direction) and the inline extent l a . The definition chosen for l a varies for the avalanche clusters, where the maximal extent along the a direction is regarded, or the pinned one, where the mean extent along the a direction is rather used. This choice was made 28 because the pinning clusters tend to be more tortuous so that their maximum span along the crack direction of propagation is not really representative of their actual extent (see Fig. 7). In Fig. 8a, we show the probability density function of the cluster surface P(S) and compare it to the experimental one. One can notice that it behaves as with γ = 1.44 ± 0.15 . This value is comparable to the exponent inverted experimentally 28 , that is, γ = 1.56 ± 0.04 . Of course, the size of the avalanche (depinning) clusters highly depends on the chosen threshold c, but we verified, as experimentally reported, that the value of γ inverted from the simulated data is not dependent on c, as shown in Fig. 8b. We also show, in Fig. 3c, as per Eqs. (10) and (11). Two thresholds are here used to define these maps relatively to the mean velocity: c = 3 and c = 6 . The white areas are the locations of interest, of surfaces S, crossline extents l x and inline extents l a . Bottom images: Difference in definition of l a for the avalanche (or depinning) and pinning clusters shown in Fig. 7. For the former, l a is the maximum extent along the a direction. For the latter it is the average width in the same direction. In both cases, l x is the maximum extent along the x direction and S the full surface (in white) of the cluster. The square pattern marks the pixel size ( x = a = 3 µm). We also computed the probability density function of l x and l a , that are respectively compared to their experimental equivalent in Fig. 9. These functions can be fitted with The modelled S is expressed in pixels (one pixel is 9 µm 2 ) and the experimental S reported by Tallakstad et al. 28 (in their Fig. 13) is in an arbitrary unit, so that the magnitude of both should not here be compared. We have here shifted up this experimental data set for an easier comparison with the numerical one. The straight line has a slope 0.68. Figure 9. (a) Probability density function of the crossline extent l x of the modelled avalanche clusters (plain points) and of the modelled pinning clusters (crosses), for a threshold c = 3 . The straight line has a slope β x = 1.7 , as per Eq. (13). The hollow stars shows the experimental probability density function obtained by Tallakstad et al. 28 for the pinning and avalanche clusters (from their Fig. 16a, inset, c = 3). (b) Probability density function of the inline extent l a of the modelled avalanche clusters (plain points) and of the modelled pinning clusters (crosses), for a threshold c = 3 . The two straight dashed lines have a slope β x = 2.2 , inline with that of the experimental data from Tallakstad et al. 28 for the pinning clusters (hollow stars, from their Fig. 16b, inset, c = 3). It should be noted that, while we have here fitted P(S), P(l x ) and P(l a ) with plain scaling laws (i.e., with Eqs. (12) to (14)), Tallakstad et al. 28 also studied the cut-off scales above which these scaling laws vanish in the experimental data, and the dependence of these cut-off scales with the arbitrary threshold c. In our case, such scales are challenging to define, as one can for instance notice in Figs. 8 and 9, where an exponential cut-off is not obvious. This may result from a limited statistical description of the larger avalanches in our simulations. Similar cut-off scales, decreasing with increasing c should however hold in our numerical data, in order to explain the decrease of average avalanche size with c, as shown in Fig. 8c. Front morphology. Finally, we show, in Fig. 10a, the relations between the clusters surface S and their linear extent l x and l a . Here, l x and l a are the mean extents for all the observed clusters sharing a same surface (with the given pixel size limiting the resolution). We could fit these relations with l x ∝ S 0.77 and l a ∝ S 0.25 for the pinning clusters, and with l x ∝ S 0.64 and l a ∝ S 0.47 for the avalanches clusters. It is in qualitative agreement with the laws observed by Tallakstad et al. 28 : l x ∝ S 0.63 and l a ∝ S 0.34 for the pinning clusters, and S ∝ l x 0.61 and l a ∝ S 0.41 for the avalanches clusters. These exponents were experimentally reported with a ±0.05 accuracy, and we estimated comparable error bars for the numerically derived ones. Thus, the shape of our simulated avalanches and pinned locations is rather similar to the observed experimental ones. Note that, from all the previous exponents, one can easily define H such that l a ∝ l x H , and we thus have H p ∼ 0.25/0.77 = 0.32 ± 0.1 and H d ∼ 0.47/0.64 = 0.73 ± 0.01 for, respectively, the simulated pinning and depinning clusters (see Fig. 10b). (13) P(l x ) ∝ l x −β x , (14) P(l a ) ∝ l a −β a , It was suggested 5,52 that H is a good indicator of the front morphology, as the front shape is to be highly dependent on the aspect ratio of its avalanches. To verify this hypothesis, we computed the advancement fluctuation along the front σ , that is While this quantity was not presented by Tallakstad et al. 28 , it was provided by other experimental works done on the same set-up 26,27 , and Fig. 11a shows σ as reported by these authors, together with that computed in the output of our simulation. One can notice that the numerical fronts are less rugous than the experimental ones. Such a mismatch is here due to the fact that the experiment from Santucci et al., shown in Fig. 11a, had more rugous crack fronts than the one from Tallakstad et al., to which the simulation is calibrated (as shown in Fig. 4). In both cases, the data sets seem to present two self-affine behaviours (e.g., 50 ) with a Hurst exponent ζ that differs at low and high length scales. Noting δx * the cut-off between these length scales we indeed have: www.nature.com/scientificreports/ We derived ζ − ∼ 0.68 ± 0.05 and ζ + = 0.4 ± 0.05 for the simulation, which compare well to the exponents that were measured experimentally, respectively, ζ − = 0.60 ± 0.05 and ζ + = 0.35 ± 0.05 and which are also close to the values we found for H d and H p . The cut-off scale between the two regimes is also similar in both the experimental and numerical cases: δx * ∼ 80 µ m, comparable to the disorder correlation length l c , and to the length scales x * , below which the local propagation velocities are correlated. For scales above this correlation length, Cochard et al. 8 showed, by analytically analysing the same model as we here study, that the front morphology is dominated by the material quenched disorder with a Hurst coefficient approximating to ζ + = 0.5 . At even larger scales, above R c ∼ πl c α 2 G c /(k B T 0 ) , they also showed 8 that the roughness of the simulated cracks ceases to be governed by the quenched disorder but is rather dominated by the thermal (annealed) noise, with σ decaying logarithmically and with a Hurst coefficient tending to ζ ∞ = 0 . With our set of parameters, R c computes to 6 mm, which is close to, yet bigger than, the total analysed length of the front. The value ζ + ∼ 0.4 , that we have here inverted, arises then likely from the transition between the two regimes, ζ + = 0.5 and ζ ∞ = 0 , as already mentioned for the experimental case, in Ref. 27 . In addition to a theoretical Hurst exponent ζ + = 0.5 , Cochard et al. 8 computed an analytical approximation for the fronts morphology power spectrum P a (Ŵ) , for the length scales Ŵ for which the effect of the quenched disorder prevails: (15) σ (δx) = <(a(x 0 + δx, t) − a(x 0 , t)) 2 > x 0 ,t . (16) σ ∝ δ x ζ − for δx < δx * ,(17)σ ∝ δ x ζ + for δx > δx * . We show, in Fig. 11b, how this approximation also fits the power spectra of our modelled front. Discussion and conclusion We studied an interfacial fracture propagation model, based only on statistical and subcritical physics in the sense of an Arrhenius law (Eq. (1)) and on the elastic redistribution of stress along crack fronts (Eq. (2)). Following the work of Cochard et al. 8 , we here showed that it allows a good representation of the intermittent dynamics of fracture in disordered media, as it approximately mimics the scaling laws dictating the propagation of experimental fronts, such as their growth exponent, their local velocity distribution and space and time correlations, the size of their avalanches and their self-affine characteristics. To run our simulations, we had to assume a given distribution for the toughness of the rupturing interface, as this quantity is not directly measurable in the laboratory. We proposed G c to be normally distributed with a unique correlation length and, of course, this can only be a rough approximation of the actual fracture energy obtained by Tallakstad et al. 28 by sintering two sand-blasted plexiglass plates. From this approximation, could arise discrepancies between our simulations and the experiments. We have indeed shown how some of the observed exponents were strongly dependent on the definition of the material disorder. We also have assumed a perfectly elastic crack front, when the local dynamics of creeping PMMA could be visco-elastic in 15), for the simulation (plain points) and an experimental data set from Santucci et al. 27 (see their Fig. 4). Different self-affine behaviours are observed above and below the δx * cut-off, with comparable Hurst exponents ζ . The dashed lines mark the slopes fitted on the simulation data for the two cases. The experimental points are from an experiment different from those of Tallakstad et al. 28 to which the model was calibrated. (b) Power spectra of the simulated crack advancement, averaged over 10,000 consecutive fronts. It is shown both before (raw) and after (binned) binning the fronts to the experimental camera pixel size. The difference between these two plots shows an influence of the observation scale on the small-scale study of the crack morphology. The plain line is the approximation 8 from Eq. (18), which is valid between l c and R c , where the morphology is dominated by the material quenched disorder. Note that the scaling regime for scales above l c was already studied by Cochard et al. 8 , while the model match to the experiment below this cut-off scale, shown in (a), is a new result, as already discussed in the "Local velocity distribution and fracture energy standard deviation" section. www.nature.com/scientificreports/ part, particularly below the typical length scale r ∼ GE/σ 2 y ∼ 30 µ m for plasticity around crack tips (e.g., 1 ) in this material, where σ y ∼ 100 MPa is the tensile yield stress of the polymer and E ∼ 3 GPa its Young modulus 53 . (18) P a (Ŵ) ∼ δG c G 2 ŴR c 4π 2 . These points being stated, the vast majority of the statistical quantities that we have here studied show a good match to those from the experimental observations, so that both the considered physical model and the interface definition are likely to be relevant. A further validation of this thermally activated model could derive from the comparison of its predictions with interfacial experiments at various background temperatures T 0 . However, such experimental data is, to our knowledge, not yet available. Of course, some of our considered parameters (e.g., G c , V 0 or α ) may, in practice, be temperature dependent so that a straight transposition of the model to different background temperatures could prove to be too simple. Creep experiments in bulk PMMA at various room temperatures can however be found in the literature 54 , where only the mean front velocity versus the mean mechanical load are measured. In this case 54 , it is reported that the creep dynamics is compatible with an Arrhenius-like process. By submitting many different materials to a constant load, at various temperatures, their lifetime was also shown 33,36 to follow an Arrhenius law, with an energy barrier that decreases with the applied stress. These materials include metals, alloys, non-metallic crystals and polymers (and PMMA in particular). It should be noted that, as stated in our introduction, other models have been considered to numerically reproduce the interfacial PMMA experiments, notably, a non-subcritical threshold based fluctuating line model by Tanguy et al. 29 , Bonamy et al. 4 or Laurson et al. 5,20 and a fiber bundle approach by Schmittbuhl et al. 6 , Gjerden et al. 30 or Stormo et al. 31 . The present manuscript does not challenge these other models per se, but rather offers an alternative explanation to the intermittent propagation of rough cracks. The former model, the fluctuating line model 4,5,20,29 , considers a similar redistribution of energy release rate G as proposed in Eq. (2), but with a dynamics that is thresholded rather than following a subcritical growth law. The fronts either move forward by one pixel 5 if G > G c , or with a velocity proportional 4 to ( G − G c ). It is completely pinned otherwise ( V = 0 for G < G c ). While reproducing several statistical features of the experiments, this non-subcritical line propagation model does not simulate the mean propagation of cracks in various loading regimes (as done by Cochard et al. 8 ) or the distribution in local velocity 55 , and, in particular, the power law tail of this distribution (i.e., Fig. 5). By contrast, the latter model 6,30,31 , the fiber bundle one, can reproduce this particular power law tail. It is not a line model: the interface is sampled with parallel elastic fibers breaking at a given force threshold. This threshold is less in the vicinity of the crack than away from it (it is modelled with a linear gradient), explaining why the rupture is concentrated around a defined front, and it holds a random component in order to model the quenched disorder of the interface. An advantage of the fiber bundle model is to be able to describe a coalescence of damage in front of the crack 56 rather than solely describing a unique front. This could likely also be achieved in a subcritical framework, but would require to authorise damage in a full 2D plane, or require a full 3D modelling (i.e., also authorise out-of-plane damage), rather than only the modelling of a 2D front. In practice, thermal activation and damage coalescence may occur simultaneously. The observation of actual damage nucleation, in the experiments that we reproduce, has however never been obvious. Instead, the experimental fronts look rather continuous. Coalescence could yet still be at play at length scales smaller than the observation resolution. This being stated, an advantage of our model is to only rely on the experimental observations, on stress redistribution and on statistical physics. Another clear advantage of the Arrhenius based model, when compared to the other ones, is to hold a subcritical description that is physically well understood and that is a good descriptor of creep in many materials 1,36 . For the record, we show in Table 2 a comparison between the different exponents predicted by the three models, that all successfully reproduce some experimental observables. Note that, if linearizing Eq. (1) with a Taylor expansion around G c − G , that is, for propagation velocities close to the mean crack speed V = V 0 exp(−α 2 [G c − G]/[k B T 0 ]) , one obtains − G), 0] and where the coefficient of proportionality µ was named the 'effective mobility of the crack front' . Equation (19) may give some insight in the physical meaning of µ in this alternative model 4 . While the above similitude in mathematical forms may explain the obtention of some similar exponents in the dynamics of the two models (see Table 2), Eq. (19) is only a crude approximation of our highly non-linear Arrhenius formalism, which, as discussed below, allows a more exhaustive description of the experimental intermittent creep dynamics. In our simulations, the exponential Arrhenius probability term, describing the crack velocity, ranges over more than three orders of magnitude while ( G c − G ) ranges over less than two decades. Continuing with the comparison of our model with pre-existing ones, we had, in our case, to calibrate the disorder to the experimental data, in particular to accurately reproduce the η exponent, that is, to reproduce the fat tail of the crack velocity distribution. Paradoxically, this exponent, which is not accounted for by the other line model, has been found to be rather constant across different experiments and experimental set-ups. It could indicate that, in practice, the disorder obtained experimentally from the blasting and sintering of PMMA plates has always been relatively similar. Such qualitative statement is of course difficult to verify, because there exists no direct way of measuring the fracture energy of the experimental sintered samples. From Fig. 5b, one can yet notice that the calibration of the disorder amplitude does not need to be particularly accurate to obtain a qualitative fit to the experimental velocity distribution. The spread of the η exponent, for large disorders, is not that important in our model for the range of considered δG c , which can also be seen in Fig. 5 of Cochard et al. 8 . Gjerden et al. 57 suggested that the nucleation of damages, predicted by their fiber bundle model, led to a new -percolation -universality class for the propagation of cracks, explaining in particular the robustness of the exponent η . Their studies are however also numerical and cover a finite range of disorders, and an extra analytical proof would be needed to show that a system of infinite size would lead exactly to the same exponent, for any disorder distribution shape and amplitude. Despite the variety in models reproducing the rough dynamics of creep, the present work provides additional indications that a thermodynamics framework in the sense of a thermally activated subcritical crack growth is well suited for the description of creeping cracks. Such a framework has long been considered (e.g., [32][33][34]36,58,59 ), and, additionally to the scaling laws that we have here presented, the proposed model was proven to fit many other observable features of the physics of rupture 8,37,39,40 . It accurately recreates the mean advancement of cracks under various loading conditions 8,37 , including when a front creeps in a spontaneous (not forced) relaxation regime, which cannot be achieved with the other (non subcritical) models, predicting an immobile front. When coupled with heat dissipation at the fracture tip, our description also accounts for the brittleness of matter 40 and for its brittle-ductile transition 39 . Indeed, for zero dimensional (scalar) crack fronts, it was shown 40 that the thermal fluctuation at the crack tip, expressed as a deviation of the temperature from T 0 in Eq. (1), can explain the transition between creep and abrupt rupture, that is, the transition to a propagation velocity close to a mechanical wave speed V 0 , five orders of magnitude higher than the maximal creep velocity V that was here modelled. It was also shown, similarly to many phase transition problems, that such a thermal transition could be favoured by material disorder 39 . Thus, a direct continuation of the present work could be to introduce such a heat dissipation for interfacial cracks in order to study how brittle avalanches nucleate at given positions (typically positions with weaker G c ) to then expand laterally to become bulk threatening events. ( 1 )Figure 1 . 11V (x, t) = V 0 min exp − α 2 [G c (x, a) − G(x, (Left): Separation of two rugous and sintered PMMA plates, as reported by Tallakstad et al.28 (side view). The rugosity of the (quasi-plane) interface is here massively exaggerated (the plates are centimetres thick while the standard deviation in the interface topography is less than a micrometer 43 ). A local position of the front has an advancement a(x, t) and avelocity V(x, t). The out of frame coordinate is x and t is the time variable. (Right): top view, showing the crack font roughness, which arises from the disorder in the interface's fracture energy. Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x Figure 3 . 3(a) Normal distribution of the fracture energy G c considered for the simulations. The average value is G c = 250 J m −2 , with a standard deviation δG c = 35 J m −2 and a correlation length l c = 50 µ m. The three lines are the modelled propagating front at three different times t 1 < t 2 < t 3 , using Eqs. (1) and (2). (b) A crack front reported by Tallakstad et al. 28 (Fig. 3 of the experimental paper), plotted on the same spatial scales. (c,d) Local velocity maps V(x, a) in the space-space domain (c) and V(x, t) in the space-time domain (d) for a modelled crack propagating in this G c landscape. Both maps are shown with the same color scale and they are computed on a resolution similar to that of the experiments by Tallakstad et al. Figure 4 . 4(a) Standard deviation of the width evolution of the crack front as a function of the mean crack advancement, as defined by Eqs. (3) to (5) for the chosen simulation (plain points) and for the experiments 28 (hollow stars) (out of Fig. 8, Expt. 5 of the experimental paper). The continuous line has a slope 0.6, close to that of the experimental points: β G ∼ 0.55 . The numerical β G , obtained with a linear root mean square fit of the growth of W, is estimated as β G = 0.60 ± 0.05 . The dashed lines mark the observation scale x , corresponding to the experimental camera pixel size, and the chosen correlation length for the simulation l c = 50 µ m. (b) The same width function for simulations with different correlation lengths l c . The rest of the parameters are as defined in Figure 5 . 5(a) Probability density function of the local propagation velocity along a simulated front (plain points), computed from the space-time map of Fig. 3d. The experimental probability 28 (out of Fig. 5, Expt. 5 of the experimental paper) is shown for comparison (hollow stars). The continuous line has a slope −2.6 , close to that of the experimental points. This was achieved by setting the standard deviation for the disorder in fracture energy to 35 J m −2 . The numerical η , obtained with a linear root mean square fit of the distribution tail, is estimated as η = 2.6 ± 0.1 . (b) The same distribution for three three simulations with different values of δG c . believed to be representative of the studied creep experiments. (b): observation scale of the modelled fronts, similar to the experimental ones of Tallakstad et al. 28 . (c): the solver grid, finer than the observation scale for numerical accuracy. Figure 6 . 6Local velocity correlation functions in space and time as defined by Eq. (7) Fig 8c, that the mean cluster size S varies with c approximately as S ∝ c −m , with m ∼ 0.68 . This value is comparable with the experimental scaling law 28 measured to be S ∝ c −0.75 . ( 10 ) 10V (x, a) > cV . Figure 7 . 7Positions of the avalanches (left) and pinning locations (right) in the front local velocity map V(x, a) shown in Figure 8 . 8(a) Probability density function of the surface of the modelled avalanche clusters (plain points) and of the modelled pinning clusters (crosses), for a threshold c = 3 . The straight dashed line has a slope γ = 1.44 , as per Eq. (12). For comparison, the hollow stars show the experimental probability density function obtained by Tallakstad et al. 28 for the avalanche and pinning clusters (both are overlapping, see Fig. 10 of their manuscript). (b) Same probability density function for various c values: c = 1.5 (squares), c = 3 (plain points), c = 6 (stars), c = 12 (circles). The straight line has a slope γ = 1.4 , as per Eq. (12). (c) Variation of the mean avalanche size S as a function of the threshold c for the simulation (plain points) and the experiments (hollow stars). Figure 10 . 10(a) Mean linear extents of the simulated pinning and depinning clusters as a function of cluster size. The four data sets are, from top to bottom, l x for the pinning clusters (hollow stars), l x for the avalanche clusters (crosses), l a for the avalanche clusters (plain points), l a for the pinning clusters (hollow points). The straight lines correspond to the fits described in the inset. See text for the equivalent experimental exponents. (b) Mean inline extent l a as a function of the mean crossline extent l x for the pinning and depinning clusters. The straight lines have a slope of, respectively, H p = 0.32 and H d = 0.73. Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x Figure 11 . 11(a) Advancement fluctuation σ along the crack fronts, as per Eq. ( .Cochard et al. Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x www.nature.com/scientificreports/ distribution was already shown by Cochard et al.Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x www.nature.com/scientificreports/ Table 1 . 1Summary of all parameters that are considered in this manuscript. (a): physical parameters in Eqs. , and the related fit is shown by the dashed line overlying the numerical data set. Plots (a,c,d) hold two curves for the experiments, corresponding to two distinct sets of experiments done on two different sintered PMMA bodies. (b) Shows Expt. 5 of Tallakstad et al.28 .Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x www.nature.com/scientificreports/ velocity V(x, a and we found β x = 1.7 ± 0.2 , close to the reported experimental value 28 β x ∼ 1.93 . The value we found for β a ∼ 2.2 = 0.2 is also inline with that of Tallakstad et al.28 , who reported β a ∼ 2.36.Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x www.nature.com/scientificreports/ Table 2 . 2Comparison of various exponents and cut-off scales derived experimentally 27,28 (Expt.) and numerically with the present, Arrhenius Based, fluctuating Line model (ABL), the Fiber Bundle model 30 (FB) and the Non-Subcritical fluctuating Line model 4,5 (NSL).www.nature.com/scientificreports/ where V cst is a constant equal to V (1 + α 2 [G c − G]/[k B T 0 ]) . This simplified form for our subcritical model is mathematically similar to that of the overcritical model (in the sense that a non zero velocity is only obtained for G > G c ) of Bonamy et al.4 , where V = max[µ(G cParameter Expt. Models ABL FB NSL β G 0.55 0.6 0.52 η 2.6 2.6 2.56 τ x 0.53 0.13 0.4 x * ∼ 100 µm 94 µm β x 1.94 1.7 β a 2.34 2.2 γ 1.56 1.4 1.5 m 0.75 0.68 ζ − 0.60 0.68 0.67 0.48 ζ + 0.35 0.4 0.39 0.37 H d 0.66 0.73 0.6 0.65 H p 0.55 0.32 0.4 Received: 6 January 2021; Accepted: 3 September 2021 https://doi.org/10.1038/s41598-021-98556-x www.nature.com/scientificreports/Scientific Reports | (2021) 11:20418 | © The Author(s) 2021 AcknowledgementsThe authors declare no competing interests in the publishing of this work. They acknowledge the support of the Universities of Strasbourg and Oslo, of the CNRS INSU ALEAS program and of the IRP France-Norway D-FFRACT. We thank the Research Council of Norway through its Centres of Excellence funding scheme, project number 262644. We are also grateful for the support of the Lavrentyev Institute of Hydrodynamics, through Grant No 14.W03.31.0002 of the Russian Government.Author contributionsCompeting interestsThe authors declare no competing interests.Additional informationSupplementary InformationThe online version contains supplementary material available at https:// doi. org/ 10. 1038/ s41598-021-98556-x.Correspondence and requests for materials should be addressed to T.V.-D., A.C. or R.T.Reprints and permissions information is available at www.nature.com/reprints.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 licence, and indicate if changes were made. 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Phys. 42214007Vanel, L., Ciliberto, S., Cortet, P.-P. & Santucci, S. Time-dependent rupture and slow crack growth: Elastic and viscoplastic dynam- ics. J. Phys. D Appl. Phys. 42, 214007. https:// doi. org/ 10. 1088/ 0022-3727/ 42/ 21/ 214007 (2009). Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium. O Lengliné, 10.1103/PhysRevE.84.036104Phys. Rev. E. 8436104Lengliné, O. et al. Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium. Phys. Rev. E 84, 036104. https:// doi. org/ 10. 1103/ PhysR evE. 84. 036104 (2011). Non-gaussian nature of fracture and the survival of fat-tail exponents. K Tallakstad, R Toussaint, S Santucci, K Måløy, 10.1103/PhysRevLett.110.145501Phys. Rev. Lett. 110145501Tallakstad, K., Toussaint, R., Santucci, S. & Måløy, K. Non-gaussian nature of fracture and the survival of fat-tail exponents. Phys. Rev. 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Cambridge University PressBarabási, A.-L. & Stanley, H. E. Fractal Concepts in Surface Growth (Cambridge University Press, 1995). Anomalous roughening of viscous fluid fronts in spontaneous imbibition. J Soriano, 10.1103/PhysRevLett.95.104501Phys. Rev. Lett. 95104501Soriano, J. et al. Anomalous roughening of viscous fluid fronts in spontaneous imbibition. Phys. Rev. Lett. 95, 104501. https:// doi. org/ 10. 1103/ PhysR evLett. 95. 104501 (2005). Dynamics and structure of interfacial crack front. K J Måløy, R Toussaint, J Schmittbuhl, 11th International Conference on Fracture. Måløy, K. J., Toussaint, R. & Schmittbuhl, J. Dynamics and structure of interfacial crack front. In 11th International Conference on Fracture 2005, ICF11 7 (2005). Technical information, Altuglas sheets. Tech. Rep., ArkemaTechnical information, Altuglas sheets. Tech. Rep., Arkema (2017). . 10.1038/s41598-021-98556-xScientific Reports |. 11Scientific Reports | (2021) 11:20418 | https://doi.org/10.1038/s41598-021-98556-x Temperature effects in the fracture of PMMA. G P Marshall, L H Coutts, J G Williams, 10.1007/BF00552926J. Mater. Sci. 9Marshall, G. P., Coutts, L. H. & Williams, J. G. Temperature effects in the fracture of PMMA. J. Mater. Sci. 9, 1409-1419. https:// doi. org/ 10. 1007/ BF005 52926 (1974). Avalanches and extreme value statistics in interfacial crackling dynamics. S Santucci, 10.1098/rsta.2017.0394Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 377Santucci, S. et al. Avalanches and extreme value statistics in interfacial crackling dynamics. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 377, 20170394. https:// doi. org/ 10. 1098/ rsta. 2017. 0394 (2019). Can crack front waves explain the roughness of cracks. E Bouchaud, J Bouchaud, D Fisher, S Ramanathan, J Rice, 10.1016/S0022-5096(01)00137-5J. Mech. Phys. Solids. 5001Bouchaud, E., Bouchaud, J., Fisher, D., Ramanathan, S. & Rice, J. Can crack front waves explain the roughness of cracks?. J. Mech. Phys. Solids 50, 1703-1725. https:// doi. org/ 10. 1016/ S0022-5096(01) 00137-5 (2002). Universality classes in constrained crack growth. K S Gjerden, A Stormo, A Hansen, 10.1103/PhysRevLett.111.135502Phys. Rev. Lett. 111135502Gjerden, K. S., Stormo, A. & Hansen, A. Universality classes in constrained crack growth. Phys. Rev. Lett. 111, 135502. https:// doi. org/ 10. 1103/ PhysR evLett. 111. 135502 (2013). Disorder enhances the effects of thermal noise in the fiber bundle model. R Scorretti, S Ciliberto, A Guarino, 10.1209/epl/i2001-00462-xEurophys. Lett. 55Scorretti, R., Ciliberto, S. & Guarino, A. Disorder enhances the effects of thermal noise in the fiber bundle model. Europhys. Lett. 55, 626-632. https:// doi. org/ 10. 1209/ epl/ i2001-00462-x (2001). Thermally activated breakdown in the fiber-bundle model. S Roux, 10.1103/PhysRevE.62.6164Phys. Rev. E. 626164Roux, S. Thermally activated breakdown in the fiber-bundle model. Phys. Rev. E 62, 6164-6169. https:// doi. org/ 10. 1103/ PhysR evE. 62. 6164 (2000). How cracks are hot and cool: A burning issue for paper. R Toussaint, 10.1039/C6SM00615ASoft Matter. 12Toussaint, R. et al. How cracks are hot and cool: A burning issue for paper. Soft Matter 12, 5563-5571. https:// doi. org/ 10. 1039/ C6SM0 0615A (2016). Thermal dissipation as both the strength and weakness of matter. A material failure prediction by monitoring creep. T Vincent-Dospital, R Toussaint, A Cochard, E G Flekkøy, K J Måløy, 10.1039/D0SM02089CSoft Matter. 12Vincent-Dospital, T., Toussaint, R., Cochard, A., Flekkøy, E. G. & Måløy, K. J. Thermal dissipation as both the strength and weak- ness of matter. A material failure prediction by monitoring creep. Soft Matter 12, 12. https:// doi. org/ 10. 1039/ D0SM0 2089C (2021). Yielding of steel sheets containing slits. 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{'abstract': 'We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.In the physics of rupture, understanding the effects that material disorder has on the propagation of cracks is of prime interest. For instance, the overall strength of large solids is believed to be ruled by the weakest locations in their structures, and notably by the voids in their bulk samples 1,2 . There, cracks tend to initiate as the mechanical stress is concentrated. A growing focus has been brought on models in which the description of the breaking matrix remains continuous (i.e., without pores). There, the material disorder resides in the heterogeneities of the matrix 3-9 . The propagation of a crack is partly governed by its spatial distribution in surface fracture energy, that is, the heterogeneity of the energy needed to generate two opposing free unit surfaces in the continuous matrix 10 , including the dissipation processes at the tip 11 . From this disorder, one can model a rupture dynamics which holds a strongly intermittent behaviour, with extremely slow (i.e., pinned) and fast (i.e., avalanching) propagation phases. In many physical processes, including 12-14 but not limited 15-18 to the physics of fracture, such intermittency is referred to as crackling noise 19,20 . In the rupture framework, this crackling noise is notably studied to better understand the complex dynamics of geological faults 21-25 , and their related seismicity.Over the last decades, numerous experiments have been run on the interfacial rupture of oven-sintered acrylic glass bodies (PMMA)[26][27][28]. Random heterogeneities in the fracture energy were introduced by sand blasting the interface prior to the sintering process. An important aspect of such experiments concerns the samples preparation, which allows to constrain the crack to propagate along a weak (disordered) plane. It simplifies the fracture problem, leading to a negligible out-of plane motion of the crack front. This method has allowed to study the dynamics of rugous fronts, in particular because the transparent PMMA interface becomes more opaque when broken. Indeed, the generated rough air-PMMA interfaces reflect more light, and the growth of fronts can thus be monitored. Different models have successfully described parts of the statistical features of the recorded crack propagation. Originally, continuous line models 4,5,20,29 were derived from linear elastic fracture mechanics. While they could reproduce the morphology of slow rugous cracks and the size distribution of their avalanches, they fail to account for their complete dynamics and, in particular, for the distribution of local propagation velocity and for the mean velocity of fronts under different loading conditions. Later on, fiber bundle models were OPEN introduced 6,30,31 , where the fracture plane was discretized in elements that could rupture ahead of the main front line, allowing the crack to propagate by the nucleation and the percolation of damage. The local velocity distribution could then be reproduced, but not the long term mean dynamics of fronts at given loads. One of the most recent models (Cochard et al. 8 ) is a thermally activated model, based on an Arrhenius law, where the fracture energy is exceeded at subcritical stresses due to the molecular agitation. It contrasts to other models that are strictly threshold based (the crack only advances when the stress reaches a local threshold, rather than its propagation being subcritical). A notable advantage of the subcritical framework is that its underlying processes are, physically, well understood, and Arrhenius-like laws have long shown to describe various features of slow fracturing processes 26,32-36 . In particular, this framework has proven to reproduce both the mean behaviour of experimental fronts 37 (i.e., the average front velocity under a given load) and the actual distributions of propagation velocities along these fronts 8 , whose fat-tail is preserved when observing cracks at different scales 38 . It has recently been proposed 39,40 that the same model might also explain the faster failure of brittle matter, that is, the dramatic propagation of cracks at velocities close to that of mechanical waves, when taking into account the energy dissipated as heat around a progressing crack tip. Indeed, if fronts creep fast enough, their local rise in temperature becomes significant compared to the background one, so that they can avalanche to a very fast phase, in a positive feedback loop39,40.Here, we only consider slow fronts (i.e., fronts that creep slowly enough so that their temperature elevation is supposed to remain negligible). Building on the work of Cochard et al. 8 , we study various statistical features that can be generated with this Arrhenius-based model (re-introduced in the "Propagation model" section), when simulating the rupture of a disordered interface. By comparing these features to those reported for the PMMA experiment by Tallakstad et al. 28,38 ,Santucci et al. 27 and Maløy et al. 26 , we show a strong match to the experimental data for many of the scaling laws describing the fracture intermittent dynamics, including the growth of the fracture width ("Growth exponent and fracture energy correlation length" section), its distribution in local propagation velocity ("Local velocity distribution and fracture energy standard deviation" section), the correlation of this velocity in space and time ("Local velocities correlations" section), the size of the propagation avalanches ("Avalanches size and shape" section) and the front Hurst exponents ("Front morphology" section). We hence re-enforce the relevance of simple thermodynamics coupled with elasticity in the description of material failure.', 'arxivid': '2010.06865', 'author': ['Tom Vincent-Dospital *email:tom.vincent-dospital@fys.uio.no \nITES UMR 7063\nUniversité de Strasbourg\n67084StrasbourgFrance\n\nThe Njord Centre\nDepartment of physics\nSFF Porelab\nUniversity of Oslo\nOsloNorway\n', 'Alain Cochard alain.cochard@unistra.fr \nITES UMR 7063\nUniversité de Strasbourg\n67084StrasbourgFrance\n', 'Stéphane Santucci \nLaboratoire de Physique\nENS de Lyon\nUniversité Claude Bernard\nCNRS\nUniversité de Lyon\nLyonFrance\n\nLavrentyev Institute of Hydrodynamics\nSiberian Branch of the Russian Academy of Sciences\nNovosibirskRussia\n', 'Knut Jørgen Måløy \nThe Njord Centre\nDepartment of physics\nSFF Porelab\nUniversity of Oslo\nOsloNorway\n', 'Renaud Toussaint renaud.toussaint@unistra.fr \nITES UMR 7063\nUniversité de Strasbourg\n67084StrasbourgFrance\n\nThe Njord Centre\nDepartment of physics\nSFF Porelab\nUniversity of Oslo\nOsloNorway\n'], 'authoraffiliation': ['ITES UMR 7063\nUniversité de Strasbourg\n67084StrasbourgFrance', 'The Njord Centre\nDepartment of physics\nSFF Porelab\nUniversity of Oslo\nOsloNorway', 'ITES UMR 7063\nUniversité de Strasbourg\n67084StrasbourgFrance', 'Laboratoire de Physique\nENS de Lyon\nUniversité Claude Bernard\nCNRS\nUniversité de Lyon\nLyonFrance', 'Lavrentyev Institute of Hydrodynamics\nSiberian Branch of the Russian Academy of Sciences\nNovosibirskRussia', 'The Njord Centre\nDepartment of physics\nSFF Porelab\nUniversity of Oslo\nOsloNorway', 'ITES UMR 7063\nUniversité de Strasbourg\n67084StrasbourgFrance', 'The Njord Centre\nDepartment of physics\nSFF Porelab\nUniversity of Oslo\nOsloNorway'], 'corpusid': 222341933, 'doi': '10.1038/s41598-021-98556-x', 'github_urls': [], 'n_tokens_mistral': 25954, 'n_tokens_neox': 21231, 'n_words': 13072, 'pdfsha': 'df9f3edfee4c5137a51549f50a34a5c92cb3cea2', 'pdfurls': None, 'title': ['Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces', 'Thermally activated intermittent dynamics of creeping crack fronts along disordered interfaces'], 'venue': []}

arxiv

A reduction technique for Generalised Riccati Difference Equations * 23 May 2013 May 24, 2013 Augusto Ferrante augusto@dei.unipd.it Dipartimento di Ingegneria dell' Informazione Università di Padova via Gradenigo6/B -35131PadovaItaly Lorenzo Ntogramatzidis l.ntogramatzidis@curtin.edu.au Department of Mathematics and Statistics Curtin University Perth, Perth (WA)WAAustralia., Australia A reduction technique for Generalised Riccati Difference Equations * 23 May 2013 May 24, 2013generalised Riccati difference equationfinite-horizon LQ problemgen- eralised discrete algebraic Riccati equationextended symplectic pencil This paper proposes a reduction technique for the generalised Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalised discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalised discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalised Riccati difference equation that isolates its nilpotent part, which becomes constant in a number of steps equal to the nilpotency index of the closed-loop, from another part that can be computed by iterating a reduced-order generalised Riccati difference equation. Introduction Consider the classic finite-horizon Linear Quadratic (LQ) optimal control problem. In particular, consider the discrete linear time-invariant system governed by the difference equation x t+1 = A x t + B u t ,(1) where A ∈ R n×n and B ∈ R n×m , and where, for all t ≥ 0, x t ∈ R n represents the state and u t ∈ R m represents the control input. Let the initial state x 0 ∈ R n be given. The problem is to find a sequence of inputs u t , with t = 0, 1, . . ., T − 1, minimising the cost function J(x 0 , u) def = T −1 ∑ t=0 x T t u T t Q S S T R x t u t + x T T P x T .(2) We assume that the weight matrices Q ∈ R n×n , S ∈ R n×m and R ∈ R m×m are such that the Popov matrix Π is symmetric and positive semidefinite, i.e., Π def = Q S S T R = Π T ≥ 0.(3) We also assume that P = P T ≥ 0. The set of matrices Σ = (A, B, Π) is often referred to as Popov triple, see e.g. [13]. We recall that, for any time t, the set U t of all optimal inputs can be parameterised in terms of an arbitrary m-dimensional signal v t as U t = {−K t x t + G t v t }, where 1 K t = (R + B T X t+1 B) † (S T + B T X t+1 A),(4)G t = I m − (R + B T X t+1 B) † (R + B T X t+1 B),(5) in which X t is the solution of the Generalised Riccati Difference Equation GRDE(Σ) X t = A T X t+1 A − (A T X t+1 B + S)(R + B T X t+1 B) † (B T X t+1 A + S T ) + Q(6) iterated backwards from t = T − 1 to t = 0 using the terminal condition X T = P,(7) see [14]. The equation characterising the set of optimal state trajectories is 1 The symbol M † denotes the Moore-Penrose pseudo-inverse of matrix M. x t+1 = (A − B K t ) x t − B G t v t . The optimal cost is J * = x T 0 X 0 x 0 . Despite the fact that it has been known for several decades that the generalised discrete Riccati difference equation provides the solution of the classic finite-horizon LQ problem, this equation has not been studied with the same attention and thoroughness that has undergone the study of the standard discrete Riccati difference equation. The purpose of this paper is to attempt to start filling this gap. In particular, we want to show a reduction technique for this equation that allows to compute its solution by solving a smaller equation with the same recursive structure, with obvious computational advantages. In order to carry out this task, several ancillary results on the corresponding generalised Riccati equation are established, which constitute an extension of those valid for standard discrete algebraic Riccati equations presented in [12] and [2]. In particular, these results show that the nilpotent part of the closed-loop matrix is independent of the particular solution of the generalised algebraic Riccati equation. Moreover, we provide a necessary and sufficient condition expressed in sole terms of the problem data for the existence of this nilpotent part of the closed-loop matrix. This condition, which appears to be straightforward for the standard algebraic Riccati equation, becomes more involved -and interesting -for the case of the generalised Riccati equation. We then show that every solution of the generalised algebraic Riccati equation coincide along the largest eigenspace associated with the eigenvalue at the origin of the closedloop, and that this subspace can be employed to decompose the generalised Riccati difference equation into a nilpotent part, whose solution converges to the zero matrix in a finite number of steps (not greater than n) and a part which corresponds to a non-singular closed-loop matrix, and is therefore easy to handle with the standard tools of linear-quadratic optimal control. As a consequence, our analysis permits a generalisation of a long series of results aiming to the closed form representation of the optimal control, see [5,6,17,9] and, for the continuous-time counterpart, [4,7,8]. Our analysis of the GRDE is based on the general theory on generalised algebraic Riccati equation presented in [15] and on some recent developments derived in [10,11]. The Generalised Discrete Algebraic Riccati Equation We begin this section by recalling two standard linear algebra results that are used in the derivations throughout the paper. where P 11 and P 22 are square and P 22 is non-singular. Then, det P = det P 22 · det(P 11 − P 12 P −1 22 P T 21 ).(8) We now introduce the so-called Generalised Discrete Algebraic Riccati Equation GDARE(Σ), defined as X = A T X A − (A T X B + S)(R + B T X B) † (B T X A + S T ) + Q.(9) The algebraic equation (9) subject to the constraint ker(R + B T X B) ⊆ ker(A T X B + S)(10) is usually referred to as Constrained Generalised Discrete Algebraic Riccati Equation CGDARE(Σ): X = A T X A − (A T X B + S)(R + B T X B) † (B T X A + S T ) + Q ker(R + B T X B) ⊆ ker(A T X B + S)(11) It is obvious that CGDARE(Σ) constitutes a generalisation of the classic Discrete Riccati Algebraic Equation DARE(Σ) X = A T X A − (A T X B + S)(R + B T X B) −1 (B T X A + S T ) + Q,(12) in the sense that any solution of DARE(Σ) is also a solution of CGDARE(Σ) but the vice-versa is not true in general. Importantly, however, the inertia of R + B T X B is independent of the particular solution of the CGDARE(Σ), [15,Theorem 2.4]. This implies that a given CGDARE(Σ) cannot have one solution X = X T such that R + B T X B is non-singular and another solution Y = Y T for which R + B T Y B is singular. As such, i) if X is a solution of DARE(Σ), then all solutions of CGDARE(Σ) will also satisfy DARE(Σ) and, ii) if X is a solution of CGDARE(Σ) such that R + B T X B is singular, then DARE(Σ) does not admit solutions. To simplify the notation, for any X = X T ∈ R n×n we define R X def = R + B T X B S X def = A T X B + S K X def = (R + B T X B) † (B T X A + S T ) = R † X S T X A X def = A − B K X so that (10) can be written as ker R X ⊆ ker S X . GDARE and the extended symplectic pencil In this section we adapt the analysis carried out in [12] for standard discrete algebraic Riccati equations to the case of CGDARE(Σ). Consider the so-called extended symplectic pencil N − z M, where M def =     I n O O O −A T O O −B T O     and N def =     A O B Q −I n S S T O R     . This is an extension that may be reduced to the symplectic structure (see [16,3]) when the matrix R is invertible. We begin by giving a necessary and sufficient condition for N to be singular. We will also show that, unlike the case in which the pencil N − z M is regular, the singularity of N is not equivalent to the fact that the matrix pencil N − z M has a generalised eigenvalue at zero. spaces. Clearly, v 1 T v 2 T v 3 T N = 0, if and only if v 2 = 0 and v 1 T v 3 T A B S T R = 0. Now, if R is singular, a non-zero vector v 3 exists such v 3 T R = 0. Since from (1) in Lemma 2.1 applied to the Popov matrix Q S S T R the subspace inclusion ker R ⊆ ker S holds, we have also 0 v 3 T A B S T R = 0. If R is invertible but A − B R † S T = A − B R −1 S T is singular, from (8) in Lemma 2.2 matrix A B S T R is singular, and therefore so is N. Vice-versa, if both R and A − B R −1 S T are non-singular, A B S T R is non- singular in view of (8) in Lemma 2.2. Thus, N is invertible. The following theorem (see [11] for a proof) presents a useful decomposition of the extended symplectic pencil that parallels the classic one -see e.g. [12] -which is valid in the case in which the pencil N − z M is regular. Theorem 3.1 Let X be a symmetric solution of CGDARE(Σ). Let also K X be the associated gain and A X be the associated closed-loop matrix. Two invertible matrices U X and V X of suitable sizes exist such that U X (N − z M)V X =     A X − z I n O B O I n − z A T X O O −z B T R X     .(13) From Theorem 3.1 we find that if X is a solution of CGDARE(Σ), in view of the triangular structure obtained above we have det(N − z M) = (−1) n · det(A X − z I n ) · det(I n − z A T X ) · det R X .(14) When R X is non-singular, the dynamics represented by this matrix pencil are decomposed into a part governed by the generalised eigenstructure of A X − z I n , a part governed by the finite generalised eigenstructure of I n − z A T X , and a part which corresponds to the dynamics of the eigenvalues at infinity. When X is a solution of DARE(Σ), the generalised eigenvalues 2 of N z − M are given by the eigenvalues of A X , the reciprocal of the non-zero eigenvalues of A X , and a generalised eigenvalue at infinity whose algebraic multiplicity is equal to m plus the algebraic multiplicity of the eigenvalue of A X at the origin. The matrix pencil I n − z A T X has no generalised eigenvalues at z = 0. This means that z = 0 is a generalised eigenvalue of the matrix pencil U X (N − z M)V X if and only if it is a generalised eigenvalue of the matrix pencil A X − z I n , because certainly z = 0 cannot cause the rank of I n − z A T X to be smaller than its normal rank and because the normal rank of N − z M is 2 n + m. This means that the Kronecker eigenstructure of the eigenvalue at the origin of U X (N − z M)V X coincides with the Jordan eigenstructure of the eigenvalue at the origin of the closed-loop matrix A X . Since the generalised eigenvalues of N − z M do not depend on the particular solution X = X T of CGDARE(Σ), the same holds for the generalised eigenvalues and the Kronecker structure of U X (N − z M)V X for any non-singular U X and V X . Therefore, the nilpotent structure of the closed-loop matrix A X -which is the Jordan eigenstructure of the generalised eigenvalue at the origin of A X -if any, is independent of the particular solution X = X T of CGDARE(Σ). Moreover, 2 Recall that a generalised eigenvalue of a matrix pencil N − z M is a value of z ∈ C for which the rank of the matrix pencil N − z M is lower than its normal rank. since U X N V X =     A X O B O I n O O O R X     ,(15) we see that, when R X is invertible, N is singular if and only if A X is singular. Since from Lemma 3.1 matrix N is singular if and only if at least one of the two matrices R and A − B R † S T is singular, we also have the following result. However, when the matrix R X is singular, it is no longer true that A X is singular if and only if R or (15) shows that the algebraic multiplicity of the eigenvalue at the origin of N is equal to the sum of the algebraic multiplicities of the eigenvalue at the origin of A X and R X . A − B R † S T is singular. Indeed, Therefore, the fact that N is singular does not necessarily imply that A X is singular. Indeed, Lemma 3.2 can be generalised to the case where R X is possibly singular as follows. Proposition 3.1 The closed-loop matrix A X is singular if and only if rank R < rank R X or A − B R † S T is singular. Proof: Given a square matrix Z, let us denote by µ(Z) the algebraic multiplicity of its eigenvalue at the origin. Then, we know from (15) that µ(N) = µ A B S T R = µ(A X ) + µ(R X ) . Consider a basis in the input space that isolates the invertible part of R. In other words, in this basis R is written as R = R 1 O O O where R 1 is invertible, while B = B 1 B 2 and S = S 1 O are partitioned accordingly. It follows that µ A B S T R = µ(R) + µ A B 1 S T 1 R 1 . As such, µ(A X ) = µ A B S T R − µ(R X ) = µ A B 1 S T 1 R 1 + µ(R) − µ(R X ).(16) First, we show that if rank R < rank R X , then A X is singular. Since rank R < rank R X , then obviously µ(R) > µ(R X ), so that (16) gives µ(A X ) > 0. Let now A − B R † S T be singular, and let rank R = rank R X . From (16) we find that µ(A X ) = µ A B 1 S T 1 R 1 . However, A − B R † S T = A − B 1 R −1 1 S T 1 . If A − B R † S T is singular, there exists a non-zero vector k such that k T −k T B 1 R −1 1 A B 1 S T 1 R 1 = 0. Hence, µ A B 1 S T 1 R 1 > 0, and therefore also µ(A X ) > 0. To prove that the converse is true, it suffices to show that if A − B R † S T is non-singular and rank R = rank R X , then A X is non-singular. To this end, we observe that rank R = rank R X is equivalent to µ(R) = µ(R X ) because R and R X are symmetric. Thus, in view of (16), it suffices to show that if A − B R † S T is non-singular, then µ A B 1 S T 1 R 1 = 0. Indeed, assume that A − B R † S T = A − B 1 R −1 1 S T 1 is non-singular, and take a vector [ v T 1 v T 2 ] such that [ v T 1 v T 2 ] A B 1 S T 1 R 1 = 0. Then, since R 1 is invertible we get v T 2 = −v T 1 B 1 R −1 1 and v T 1 (A − B 1 R −1 1 S T 1 ) = 0. Hence, v 1 = 0 since A − B 1 R −1 1 S T 1 is non-singular, and therefore also v 2 = 0. Remark 3. 1 We recall that µ(R X ) is invariant for any symmetric solution X of CGDARE(Σ), [15]. Hence, as a direct consequence of (16) T −1 X T = X 11 X 12 X T 12 X 22 and T −1 Y T = X 11 X 12 X T 12 Y 22 .(17) We begin by presenting a first simple result. 3 Given a subspace S , a basis matrix S of S is such that im S = S and ker S = {0}. Lemma 4.1 Two symmetric solutions X and Y of CGDARE(Σ) are coincident along the subspace U if and only if U ⊆ ker(X −Y ). Proof: Suppose X and Y are coincident along the subspace U , and are already written in the basis defined by T in (17). In this basis U can be written as U = im I O . If (17) holds, then we can write X − Y = O O O ⋆ . Then, (X − Y ) U = O O O ⋆ I O = {0}. Vice-versa, if (X − Y ) U = {0} and we write X −Y = ∆ 11 ∆ 12 ∆ T 12 ∆ 22 , we find that ∆ 11 ∆ 12 ∆ T 12 ∆ 22 I O = {0} implies ∆ 11 = 0 and ∆ 12 = 0. We now present two results that will be useful to prove Theorem 4.1. Let X = X T ∈ R n×n . Similarly to [12], we define the function D(X ) def = X − A T X A + (A T X B + S)(R + B T X B) † (B T X A + S T ) − Q.(18) If in particular X = X T is a solution of GDARE(Σ), then D(X ) = 0. Recall that we have defined R X = R + B T X B, S X = A T X B + S and R Y = R + B T Y B, S Y def = A T Y B + S. Lemma 4.2 Let X = X T ∈ R n×n and Y = Y T ∈ R n×n be such that (10) holds, i.e., ker R X ⊆ ker S X (19) ker R Y ⊆ ker S Y .(20)Let A X = A − B K X with K X = R † X S T X and A Y = A − B K Y with K Y = R † Y S T Y . Moreover, let us define the difference ∆ def = X −Y . Then, D(X ) − D(Y ) = ∆ − A T Y ∆ A Y + A T Y ∆ B R † X B T ∆ A Y .(21) The proof can be found in [1, p.382]. The following lemma is the counterpart of Lemma 2.2 in [12] where the standard DARE was considered. Lemma 4.3 Let X = X T ∈ R n×n and Y = Y T ∈ R n×n be such that (19-20) hold. Let ∆ = X −Y . Then, D(X ) − D(Y ) = ∆ − A T Y ∆ A X .(22) Proof: First, notice that A T Y ∆ B = [A T − (A T Y B + S) R † Y B T ]∆ B. We now show that ker R X ⊆ ker(A T Y ∆ B). To this end, let P X be a basis of the null-space of R X . Hence, (R + B T X B)P X = 0. Then, A T Y ∆ B P X = A T − (A T Y B + S) R † Y B T (X −Y ) B P X = A T X B P X − (A T Y B + S) R † Y B T X B P X − A T Y B P X +(A T Y B + S) R † Y B T Y B P X +(A T Y B + S) R † Y R P X − (A T Y B + S) R † Y R P X = A T X B P X + (A T Y B + S) R † Y R Y P X − A T Y B P X = A T X B P X + S Y P X − A T Y B P X = (A T X B + S) P X , which is zero since ker R X ⊆ ker S X in view of (19) in Lemma 4.2. Now we want to prove that A T Y ∆ (A Y − A X ) = A T Y ∆ B R † X B T ∆ A Y .(23) Consider the term A T Y ∆(A Y − A X ) = A T Y ∆ B (R † X S X − R † Y S Y ).(24) Since R † X R X is an orthogonal projection that projects onto im R T X = im R X , we have ker R X = im(I m − R † X R X ). Since as we have shown ker R X ⊆ ker(A T Y ∆ B), fromker R X = im(I m − R † X R X ) we also have A T Y ∆ B (I m − R † X R X ) = 0, which means that A T Y ∆ B R † X R X = A T Y ∆ B. We use this fact on (24) to get A T Y ∆(A Y −A X ) = A T Y ∆ B R † X [(B T X A + S) − R X R † Y (B T YA + S)] = A T Y ∆ B R † X [(B T X A+S−B T Y A+B T Y A)−R X R † Y (B T YA+S)] = A T Y ∆ B R † X [B T ∆ A + (I m − R X R † Y )(B T YA + S)].(25) Since Proof: Let us prove (1). Consider a non-singular T ∈ R n×n . Define the new quintuplẽ R X = R + B T X B − B T Y B + B T Y B = R Y + B T ∆ B, eq. (25) becomes A T Y ∆(A Y − A X ) = A T Y ∆ B R † X [B T ∆ A + (I m − R Y R † Y − B T ∆ B R † Y )(B T YA + S)] = A T Y ∆B R † X B T ∆ (A − B R † Y )(B T YA + S) = ∆B R † X B T ∆ A Y , since from Lemma 2.1 (I m − R Y R † Y )(B T YA + S) = 0 from ker R Y ⊆ ker(A T Y B + S). Eq. (23) follows by recalling that A Y = A − B R † Y S Y . Plugging (23) into (21) yields D(X ) − D(Y ) = ∆ − A T Y ∆A Y + A T Y ∆(A Y − A X ) = ∆ − A T Y ∆A X .A def = T −1 A T,B def = T −1 B,Q def = T T Q T,S def = T T S,R def = R. It is straightforward to see that X satisfies GDARE(Σ) with respect to (A, B, Q AX = T −1 A X T = U U c T A X U U c = U T A X U ⋆ U T c A X U ⋆ = U T A X U ⋆ O U T c A X U c , where the zero in the bottom left corner is due to the fact that the rows of U T c A X are orthogonal to the columns of U . Moreover, the submatrix N 0 def = U T A X U is nilpotent with the same nilpotency index 4 of A X . 4 With a slight abuse of nomenclature, we use the term nilpotency index of a matrix M to refer to the smallest integer ν for which ker(M) ν = ker(M) ν+1 , which is defined also when M is not nilpotent. Notice also that H X def = U T c A X U c is non-singular. LetX be a solution of CGDARE(Σ) in this new basis, and let it be partitioned asX = X 11X12 X T 12X 22 , whereX 11 is ν × ν, with ν = dim U . Consider another solutionỸ of CGDARE(Σ), partitioned as Y = Ỹ 11Ỹ12 Y T 12Ỹ 22 . Let ∆ def =X −Ỹ be∆ −à T Y ∆ÃX = 0. (26) If ∆ is partitioned as ∆ = [ ∆ 1 ∆ 2 ] where ∆ 1 has ν columns, eq. (26) becomes ∆ 1 ∆ 2 −à T Y ∆ 1 ∆ 2 N 0 ⋆ O H X = ∆ 1 −à T Y ∆ 1 N 0 ⋆ = 0, from which we get ∆ 1 =à T Y ∆ 1 N 0 . Thus, ∆ 1 =à T Y ∆ 1 N 0 = (à T Y ) 2 ∆ 1 N 2 0 = . . . = (à T Y ) n ∆ 1 (N 0 ) n , which is equal to zero since (N 0 ) n is the zero matrix. Hence, ∆ 1 = 0. Thus, we have also ∆ U = O ⋆ im I O = {0}. Since ∆ is symmetric, we get X −Ỹ = X 11X12 X T 12X 22 − Ỹ 11Ỹ12 Y T 12Ỹ 22 = O O OX 22 −Ỹ 22 , which leads toX 11 =Ỹ 11 andX 12 =Ỹ 12 . Let us prove (2). Since ker R Y coincides with ker R X by virtue of [10, Theorem 4.3], we find A X − A Y = B (R † Y S T Y − R † X S T X ) = B R † Y (S T Y − R Y R † X S T X ).(27) Plugging S T Y = B T Y A + S T = B T ∆ A + S T + B T X A = B T ∆ A + S T X(28) and R Y = R + B T Y B − B T X B + B T X B = R X + B T ∆ B(29) into (27) yields A X − A Y = B R † Y (B T ∆ A − B T ∆ B R † X S T X ) = B R † Y B T ∆ A X . This means that the identity A X − A Y = B R † Y B T ∆ A X holds. By partitioning ∆ = O ⋆ O ⋆ , we find that also B R † Y B T ∆ = O ⋆ O ⋆ , so that A Y = A X − B R † Y B T ∆ A X = N 0 ⋆ O H X − O ⋆ O ⋆ N 0 ⋆ O H X = N 0 ⋆ O H Y . Thus, ker(A Y ) n ⊇ ker(A X ) n . If we interchange the role of X and Y , we obtain the opposite inclusion ker(A Y ) n ⊆ ker(A X ) n . Notice, in passing, that this also implies that H Y is non-singular. The Generalised Riccati Difference Equation Consider the GRDE(Σ) along with the terminal condition X T = P = P T ≥ 0. Let us define R(X ) def = A T X A − (A T X B + S)(R + B T X B) † (B T X A + S T ) + Q. With this definition, GRDE(Σ) can be written as X t = R(X t+1 ). Moreover, GDARE(Σ) can be written as D(X ) = X − R(X ) = 0. We have the following important result. Theorem 5.1 Let X • = X T • be a solution of CGDARE(Σ). Let ν be the index of nilpotency of A X • . Moreover, let X t be a solution of (6)(7) and define ∆ t def = X t − X • . Then, for τ ≥ ν, we have ∆ T −τ U = {0}. Proof: Since X • = X T • is a solution of CGDARE(Σ), we have D(X • ) = 0. This is equivalent to saying that X • = R(X • ). From the definition of ∆ t we get in particular ∆ T = X T − X • . With these definitions in mind, we find ∆ t = R(X t+1 ) − R(X • ) = X t+1 − D(X t+1 ) − X • = ∆ t+1 − D(X t+1 ) = ∆ t+1 − D(X t+1 ) + D(X • ) = ∆ t+1 − [D(X t+1 ) − D(X • )].(30) However, we know from (21) that D(X t+1 ) − D(X • ) = ∆ t+1 − A T X • [∆ t+1 − ∆ t+1 B (R + B T X t+1 B) † B T ∆ t+1 ]A X • ,(31) which, once plugged into (30), gives ∆ t = ∆ t+1 − ∆ t+1 + A T X • [∆ t+1 + ∆ t+1 B (R + B T X t+1 B) † B T ∆ t+1 ]A X • = A T X • [I n − ∆ t+1 B (R + B T X t+1 B) † B T ]∆ t+1 A X • = F t+1 ∆ t+1 A X • ,(32) where F t+1 def = A T X • − A T X • ∆ t+1 B (R + B T X t+1 B) † B T . It follows that we can write ∆ T −1 = F T ∆ T A X • , ∆ T −2 = F T −1 ∆ T −1 A X • = F T −1 F T ∆ T (A X • ) 2 , . . .(33)∆ T −τ = T ∏ i=T −τ+1 F i ∆ T (A X • ) τ .(34) This shows that for τ ≥ ν we have ker ∆ T −τ ⊇ ker(A X • ) n . Now we show that the result given in Theorem 5.1 can be used to obtain a reduction for the generalised discrete-time Riccati difference equation. Consider the same basis induced by the change of coordinates used in Theorem 4.1, so that the first ν components of this basis span the subspace U = ker(A X ) n . The closed-loop matrix in this basis can be written as A X • = N 0 ⋆ O Z , where the last equality follows from the fact that ∆ T −τ is symmetric. Now, let us rewrite the Riccati difference equation (32) as ∆ t = A T X • ∆ t+1 A X • − A T X • ∆ t+1 B(R + B T X t+1 B) † B T ∆ t+1 A X • .(35) For t ≤ T − ν, we get ∆ t = O O O Ψ t , and the previous equation becomes O O O Ψ t = N T 0 O ⋆ Z T O O O Ψ t+1 N 0 ⋆ O Z − N T 0 O ⋆ Z T O O O Ψ t+1 B (R + B T X t+1 B) † B T O O O Ψ t+1 N 0 ⋆ O Z = O O O Z T Ψ t+1 Z − O O O Z T Ψ t+1 B 1 B 2 R+ B T 1 B T 2 (∆ t+1 +X • ) B 1 B 2 † B T 1 B T 2 O O O Ψ t+1 Z . By partitioning X • as X • = X •,11 X •,12 X T •,12 X •,22 , we get O O O Ψ t = O O O Z T Ψ t+1 Z − O O O Z T Ψ t+1 ⋆ ⋆ ⋆ B 2 (R 0 +B T 2 Ψ t+1 B 2 ) † B T 2 O O O Ψ t+1 Z = O O O Z T Ψ t+1 Z − O O O Z T Ψ t+1 B 2 (R 0 + B T 2 Ψ t+1 B 2 ) † B T 2 Ψ t+1 Z , where R 0 def = R + B T 2 X •,22 B 2 . Therefore, Ψ t satisfies the reduced homogeneous Riccati difference equation Ψ t = Z T Ψ t+1 Z − Z T Ψ t+1 B 2 (R 0 + B T 2 Ψ t+1 B 2 ) † B T 2 Ψ t+1 Z.(36) The associated generalised discrete Riccati algebraic equation is Ψ − Z T Ψ Z + Z T Ψ B 2 (R 0 + B T 2 Ψ B 2 ) † B T 2 Ψ Z = 0.(37) Being homogeneous, this equation admits the solution Ψ = 0. This fact has two important consequences: • The closed-loop matrix associated with this solution is clearly Z, which is non-singular. On the other hand, we know that the nilpotent part of the closed-loop matrix is independent of the particular solution of CGDARE(Σ) considered. This means that all solutions of (37) have a closed-loop matrix that is non-singular; • Given a solution Ψ of (37), the null-space of R 0 +B T 2 Ψ B 2 coincides with the null-space of R 0 , since the null-space of R 0 + B T 2 Ψ B 2 does not depend on the particular solution of (37) and we know that the zero matrix is a solution of (37). As a result of this discussion, it turns out that given a reference solution X • of CGDARE(Σ), the solution of GDRE(Σ) with terminal condition X T = P can be computed backward as follows: 1. For the first ν steps, i.e., from t = T to t = T − ν, X t is computed by iterating the GDRE(Σ) starting from the terminal condition X T = P; 2. In the basis that isolates the nilpotent part of A X , we have ∆ T −ν = O O O Ψ T −ν . From t = T − ν − 1 to t = 0, the solution of GDRE(Σ) can be found iterating the reduced order GDRE in (36) starting from the terminal condition Ψ T −ν . (36) can also be computed in closed-form, using the results in [6]. Indeed, consider a solution Ψ of (37) with its non-singular closed-loop matrix A Ψ and let Y be the corresponding solution of the closed-loop Hermitian Stein equation A Ψ Y A T Ψ −Y + B 2 (R 0 + B T 2 Ψ B 2 ) −1 B T 2 = 0.(38) The set of solutions of the extended symplectic difference equation for the reduced system is parameterised in terms of K 1 , K 2 ∈ R (n−ν)×(n−ν) as     Ξ t Λ t Ω t     =     I n−ν Ψ −K Ψ     (A Ψ ) t K 1 +     Y A T Ψ (ΨY − I n−ν )A T Ψ −K ⋆     (A T Ψ ) T −t−1 K 2 , 0 ≤ t ≤ T,(39) where K ⋆ def = K Ψ Y A T Ψ − (R 0 + B T 2 Ψ B 2 ) −1 B T 2 . The values of the parameter matrices K 1 and K 2 can be computed so that the terminal condition satisfies X T = I n and Λ T = Ψ T −ν . Such values exist because A Ψ is non-singular, and are given by K 1 = (A Ψ ) −T (I n−ν −Y (Ψ − Ψ T −ν )) K 2 = Ψ − Ψ T −ν . Then, the solution of (36) is given by Ψ t = Λ t Ξ −1 t . Concluding remarks In this paper we have considered the generalised Riccati difference equation with a terminal condition which arises in finite-horizon LQ optimal control. We have shown in particular that it is possible to identify and deflate the singular part of such equation using the corresponding generalised algebraic Riccati equation. The two advantages of this technique are the reduction of the dimension of the Riccati equation at hand as well as the fact that the reduced problem is non-singular, and can therefore be handled with the standard tools of the finite-horizon LQ theory. Lemma 3.2 (see e.g. [2]) Let R X be invertible. Then, A X is singular if and only if at least one of the two matrices R and A − B R † S T is singular. , we have that µ(A X ) is the same for any symmetric solution X of CGDARE(Σ). This means, in particular, that the closed-loop matrix corresponding to a given symmetric solution of CGDARE(Σ) is singular if and only if the closed-loop matrix corresponding to any other symmetric solution of CGDARE(Σ) is singular. In the next section we show that a stronger result holds: when present, the zero eigenvalue has the same Jordan structure for any pair A X and A Y of closedloop matrices corresponding to any pair X ,Y of symmetric solutions of CGDARE(Σ). Moreover, the generalised eigenspaces corresponding to the zero eigenvalue of A X and A Y coincide. The restriction of A X and A Y to this generalised eigenspace also coincide. Finally, X and Y coincide along this generalised eigenspace.4 The subspace where all solutions coincideGiven a solution X = X T of CGDARE(Σ), we denote by U the generalised eigenspace corresponding to the eigenvalue at the origin of A X , i.e., U def = ker(A X ) n . Notice that, in principle, U could depend on the particular solution X . In this section, and in particular in Theorem 4.1, we want to prove not only that U does not depend on the particular solution X , but also that all solutions of CGDARE(Σ) are coincident along U . In other words, given two solutions X = X T and Y = Y T of CGDARE(Σ), we show that ker(A X ) n = ker(A Y ) n and, given a basis matrix 3 U of the subspace U = ker(A X ) n = ker(A Y ) n , the change of coordinate matrix T = [U U c ] yields Now we are ready to prove the main result of this section. This result extends the analysis of Proposition 2.1 in[12] to solutions of CGDARE(Σ).Theorem 4.1 Let U = ker(A X ) n denote the generalised eigenspace corresponding to the eigenvalue at the origin of A X . Then 1. All solutions of CGDARE(Σ) are coincident along U , i.e., given two solutions X and Y of CGDARE(Σ), (X −Y ) U = {0}; 2. U does not depend on the solution X of CGDARE(Σ), i.e., given two solutions X and Y of CGDARE(Σ), there holds ker(A X ) n = ker(A Y ) n . , R, S) if and only if X def = T T X T satisfies GDARE(Σ) with respect to (Ã,B,Q,R,S), which for the sake of simplicity is denoted byD, so thatD(X) = 0. The closed-loop matrix in the new basis is related to the closed-loop matrix in the original basis byÃX =à −B (R +B TXB ) † (B TXà +S T ) = T −1 A X T. Moreover, ifŨ = ker(ÃX ) n , thenŨ = T −1 U since (ÃX ) nŨ = 0 is equivalent to T −1 (A X ) n TŨ = T −1 (A X ) n U = 0. We choose an orthogonal change of coordinate matrix T as T = [U U c ], where U is a basis matrix of U . In this new basis partitioned in the same way. SinceX andỸ are both solutions of CGDARE(Σ), we getD(X) =D(Ỹ ) = 0. Thus, in view of Lemma 4.3, there holds Remark 5. 1 1The advantage of using the reduced-order generalised difference Riccati algebraic equation (36) consists in the fact that the closed-loop matrix of any solution of the associated generalised discrete Riccati algebraic equation is non-singular. Hence, when the reduced-order pencil given by the Popov triple Z, B 2 , , the solution of the reduced-order generalised difference Riccati algebraic equation Proof: First note that N is singular if and only if such isLemma 3.1 Matrix N is singular if and only if at least one of the two matrices R and A − B R † S T is singular. A B S T R . To see this fact, consider the left null- Matrix Riccati Equations in Control and Systems Theory. H Abou-Kandil, G Freiling, V Ionescu, G Jank, BirkhäuserBaselH. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank. Matrix Riccati Equations in Control and Systems Theory. Birkhäuser, Basel, 2003. On the structure of the solution of discrete-time algebraic Riccati equation with singular closed-loop matrix. A Ferrante, IEEE Transactions on Automatic Control, AC. 4911A. Ferrante. On the structure of the solution of discrete-time algebraic Riccati equation with singular closed-loop matrix. IEEE Transactions on Automatic Control, AC-49(11):2049-2054, 2004. Canonical Form for Symplectic Matrix Pencils. A Ferrante, B Levy, Linear Algebra and its Applications. 274A. Ferrante and B. Levy. Canonical Form for Symplectic Matrix Pencils. Linear Algebra and its Applications. Vol. 274:259-300, 1998. A Parametrization of the Solutions of the Finite-Horizon LQ Problem with General Cost and Boundary Conditions. A Ferrante, G Marro, L Ntogramatzidis, Automatica. 41A. Ferrante, G. Marro, and L. Ntogramatzidis. A Parametrization of the Solutions of the Finite- Horizon LQ Problem with General Cost and Boundary Conditions. Automatica. Vol. 41:1359-1366, 2005. Employing the Algebraic Riccati Equation for a Parametrization of the Solutions of the Finite-Horizon LQ Problem: The Discrete-Time Case. A Ferrante, L Ntogramatzidis, Systems & Control Letters. 547A. Ferrante, and L. Ntogramatzidis. Employing the Algebraic Riccati Equation for a Parametrization of the Solutions of the Finite-Horizon LQ Problem: The Discrete-Time Case. Systems & Control Letters. Vol. 54(7):693-703, 2005. A unified approach to finite-horizon generalized LQ optimal control problems for discrete-time systems. A Ferrante, L Ntogramatzidis, Linear Algebra and Its Applications. 4252-3A. Ferrante, and L. Ntogramatzidis, "A unified approach to finite-horizon generalized LQ optimal control problems for discrete-time systems". Linear Algebra and Its Applications, 425(2-3):242-260, 2007. A Unified Approach to the Finite-Horizon Linear Quadratic Optimal Control Problem. A Ferrante, L Ntogramatzidis, European J. of Control. 135A. Ferrante, and L. Ntogramatzidis. A Unified Approach to the Finite-Horizon Linear Quadratic Optimal Control Problem. European J. of Control. Vol. 13(5):473-488, 2007. On the solution of the Riccati differential equation arising from the LQ optimal control problem. L Ntogramatzidis, A Ferrante, Systems & Control Letters. 592L. Ntogramatzidis, and A. Ferrante. On the solution of the Riccati differential equation arising from the LQ optimal control problem. Systems & Control Letters. Vol. 59(2):114-121, 2010. Structural Invariant Subspaces of Singular Hamiltonian Systems and Nonrecursive Solutions of Finite-Horizon Optimal Control Problems. A Ferrante, L Ntogramatzidis, IEEE Trans. Automatic Control. 571Comments onA. Ferrante, L. Ntogramatzidis. Comments on "Structural Invariant Subspaces of Singular Hamil- tonian Systems and Nonrecursive Solutions of Finite-Horizon Optimal Control Problems". IEEE Trans. Automatic Control. Vol. 57(1):270-272, 2012. The Generalised Discrete Algebraic Riccati Equation in Linear-Quadratic Optimal Control. L Ntogramatzidis, A Ferrante, 10.1016/j.automatica.2012.11.006Automatica. 49L. Ntogramatzidis, and A. Ferrante. The Generalised Discrete Algebraic Riccati Equation in Linear- Quadratic Optimal Control. Automatica. Vol. 49:471-478, DOI: 10.1016/j.automatica.2012.11.006, 2013. The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions. A Ferrante, L Ntogramatzidis, 10.1109/TAC.2013.2244292ManuscriptA. Ferrante, and L. Ntogramatzidis, "The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions". In press. DOI: 10.1109/TAC.2013.2244292. Manuscript available at http://http://arxiv.org/abs/1208.6481. Order reduction of discrete-time algebraic Riccati equations with singular closed-loop matrix. A Ferrante, H K Wimmer, Operators and Matrices. 11A. Ferrante, and H.K. Wimmer, "Order reduction of discrete-time algebraic Riccati equations with singular closed-loop matrix". Operators and Matrices, 1(1):61-70, 2007. Generalized Riccati theory and robust control, a Popov function approach. V Ionescu, C Oarǎ, M Weiss, WileyV. Ionescu, C. Oarǎ, and M. Weiss. Generalized Riccati theory and robust control, a Popov function approach. Wiley, 1999. Structure and stability of discrete-time optimal systems. D Rappaport, L M Silverman, IEEE Transactions on Automatic Control, AC. 16D. Rappaport and L.M. Silverman. Structure and stability of discrete-time optimal systems. IEEE Transactions on Automatic Control, AC-16:227-233, 1971. The discrete-time algebraic Riccati equation and linear matrix inequality. A A Stoorvogel, A Saberi, Linear Algebra and its Applications. 274A.A. Stoorvogel and A. Saberi. The discrete-time algebraic Riccati equation and linear matrix inequality. Linear Algebra and its Applications, 274:317-365, 1998. Normal forms of symplectic pencils and the discrete-time algebraic Riccati equation. H K Wimmer, Linear Algebra and its Appl. 147H. K. Wimmer. Normal forms of symplectic pencils and the discrete-time algebraic Riccati equa- tion. Linear Algebra and its Appl., 147:411-440, 1991. Structural Invariant Subspaces of Singular Hamiltonian Systems and Nonrecursive Solutions of Finite-Horizon Optimal Control Problems. E Zattoni, IEEE Transactions on Automatic Control, AC. 535E. Zattoni, Structural Invariant Subspaces of Singular Hamiltonian Systems and Nonrecursive So- lutions of Finite-Horizon Optimal Control Problems,IEEE Transactions on Automatic Control, AC- 53(5):1279-1284, 2008.

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{'abstract': 'This paper proposes a reduction technique for the generalised Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalised discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalised discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalised Riccati difference equation that isolates its nilpotent part, which becomes constant in a number of steps equal to the nilpotency index of the closed-loop, from another part that can be computed by iterating a reduced-order generalised Riccati difference equation.', 'arxivid': '1305.5311', 'author': ["Augusto Ferrante augusto@dei.unipd.it \nDipartimento di Ingegneria dell' Informazione\nUniversità di Padova\nvia Gradenigo6/B -35131PadovaItaly\n", 'Lorenzo Ntogramatzidis l.ntogramatzidis@curtin.edu.au \nDepartment of Mathematics and Statistics\nCurtin University\nPerth, Perth (WA)WAAustralia., Australia\n', "Augusto Ferrante augusto@dei.unipd.it \nDipartimento di Ingegneria dell' Informazione\nUniversità di Padova\nvia Gradenigo6/B -35131PadovaItaly\n", 'Lorenzo Ntogramatzidis l.ntogramatzidis@curtin.edu.au \nDepartment of Mathematics and Statistics\nCurtin University\nPerth, Perth (WA)WAAustralia., Australia\n'], 'authoraffiliation': ["Dipartimento di Ingegneria dell' Informazione\nUniversità di Padova\nvia Gradenigo6/B -35131PadovaItaly", 'Department of Mathematics and Statistics\nCurtin University\nPerth, Perth (WA)WAAustralia., Australia', "Dipartimento di Ingegneria dell' Informazione\nUniversità di Padova\nvia Gradenigo6/B -35131PadovaItaly", 'Department of Mathematics and Statistics\nCurtin University\nPerth, Perth (WA)WAAustralia., Australia'], 'corpusid': 28421930, 'doi': '10.1109/cdc.2012.6426104', 'github_urls': [], 'n_tokens_mistral': 12852, 'n_tokens_neox': 11643, 'n_words': 7474, 'pdfsha': '68b201b9bc0441dda6ca826378c4bd1a0332e853', 'pdfurls': ['https://arxiv.org/pdf/1305.5311v1.pdf'], 'title': ['A reduction technique for Generalised Riccati Difference Equations *', 'A reduction technique for Generalised Riccati Difference Equations *', 'A reduction technique for Generalised Riccati Difference Equations *', 'A reduction technique for Generalised Riccati Difference Equations *'], 'venue': []}

arxiv

Jan 2023 Alejandro Bravo-Doddoli GEODESIC FLOW ON J 22Jan 2023and phrases Carnot groupJet spacenon-integrable systemsub Riemannian geometry The J 2 (R 2 , R) space of 2-jets of a real function of two real variables x and y admits the structure of a Carnot group with step 3. As any subRiemannia manifold, J 2 (R 2 , R) has an associated Hamiltonian geodesic flow, which is non-integrable. To prove this, we used the reduced Hamiltonian H µ on T * H , given by a symplectic reduction of the subRiemannian geodesic flow on J 2 (R 2 , R), using the fact that J 2 (R 2 , R) is a meta-abelian group.Theorem A. The subRiemannian geodesic flow onOther examples of Carnot groups with a non-integrable geodesic flow: the group of all 4 by 4 lower triangular matrices with 1's on the diagonal proved by R. Montgomery, M. Saphirom and A. Stolin, see[9]. The Carnot group with growth vector (3, 6, 14) showed by I. Bizyaev, A. Borisov, A. Kilin, and I. Mamaev, see[3]. The free Carnot group Introduction Let J 2 (R 2 , R) be the space of 2-jets of a real function of two variables, then J 2 (R 2 , R) is an 8-dimensional Carnot group with step 3 and growth vector (5,7,8). Let j be the graded Lie algebra of J 2 (R 2 , R), that is, j = j 1 ⊕ j 2 ⊕ j 3 , such that [j 1 , j r ] = j r+1 . Let π : J 2 (R 2 , R) → j 1 be the canonical projection and let j 1 be endowed with the Euclidean metric, let us consider the subRiemannian metric on J 2 (R 2 , R) such that π is a subRiemannian submersion, then the subRiemannian structure is left-invariant under the Carnot group multiplication. Like any subRiemannian structure, the cotangent bundle T * J 2 (R 2 , R) is endowed with a Hamiltonian system whose underlying Hamiltonian H sR is that whose solutions curves are subRiemannian geodesics on J 2 (R 2 , R). We call this Hamiltonian system the geodesic flow on J 2 (R 2 , R). (2,3,5,8) with step 4 verified by L. V. Lokutsievskiy and Y. L. Sachkov, see [7]. 2. J 2 (R 2 , R) as a Carnot group The 2-jet of a smooth function f : R 2 → R at a point (x 0 , y 0 ) ∈ R 2 is its 2-th order Taylor expansion at x 0 . We will encode this 2-jet as a 8-tuple of real numbers (j k f )| (x 0 ,y 0 ) as follows: (x 0 , y 0 , ∂ 2 f ∂x 2 , ∂ 2 f ∂x∂y , ∂ 2 f ∂y 2 , ∂f ∂x , ∂f ∂y , f )| (x 0 ,y 0 ) ∈ R 8 As f varies over smooth functions and (x 0 , y 0 ) varies over R 2 , these 2-jets sweep out the 2-jet space, denoted by J 2 (R 2 , R). One can see that J 2 (R 2 , R) is diffeomorphic to R 8 and its points are coordinatized according to (x, y, u 2,0 , u 1,1 , u 0,2 , u 1,0 , u 0,1 , u) ∈ R 8 := J 2 (R 2 , R). Recall that if u = f (x, y), then u 1,0 = du/dx, u 0,1 = du/dy, u 2,0 = du 1,0 /dx, u 1,1 = du 1,0 /dy = du 0,1 /dx and u 0,2 = du 0,1 /dy. We see that J 2 (R 2 , R) is endowed with a natural rank 5 distribution j 1 ⊂ T J 2 (R 2 , R) ≃ j characterized by the following Pfaffian equations u 1,0 dx+u 0,1 dy−du = u 2,0 dx+u 1,1 dy−du 1,0 = u 1,1 dx+u 0,2 dy−du 0,1 = 0. A subRiemannian structure on a manifold consists of a non-integrable distribution together with a smooth inner product on the distribution. We arrive at our subRiemannian structure by observing that j 1 is globally framed by X 1 = ∂ ∂x + u 1,0 ∂ ∂u + u 2,0 ∂ ∂u 1,0 + u 1,1 ∂ ∂u 0,1 , X 2 = ∂ ∂y + u 0,1 ∂ ∂u + u 1,1 ∂ ∂u 1,0 + u 0,2 ∂ ∂u 0,1 , Y 2,0 = ∂ ∂u 2,0 , Y 1,1 = ∂ ∂u 1,1 , Y 0,2 = ∂ ∂u 0,2 . An equivalent way to define the subRiemannian metric is to declare these vector fields to be orthonormal. Now the restrictions of the oneforms dx, dy, du 2,0 , du 1,1 , du 0,2 to j 1 form a global co-frame for j * 1 which is dual to our frame. Therefore an equivalent way to describe our subRiemannian structure is to say that its metric is dx 2 + dy 2 + du 2 2,0 + du 2 1,1 + du 2 0,2 restricted to j 1 . For more detail about the jet space as Carnot group, see [10]. The left-invariant vector fields {X 1 , X 2 , Y 2,0 , Y 1,1 , Y 0,2 } generates the following Lie algebra: Y 1,0 := [X 1 , Y 2,0 ] = [X 2 , Y 1,1 ], Y 0,1 := [X 1 , Y 1,1 ] = [X 2 , Y 0,2 ], (2.1) Equations (2.1) defined the left-invariant vector fields corresponding to the second layer. Y := [X 1 , Y 1,0 ] = [X 2 , Y 0,1 ], (2.2) Equations (2.1) defined the left-invariant vector field corresponding to the third layer. All the other brackets are zero. We say that a group G is meta-abelian if [G, G] = 0 is abelian. The Lie bracket relationship in equations (2.1) and (2.2) show that J 2 (R 2 , R) is a meta-abelian Carnot group, we will use the symplectic reduction performance on [4] to prove the main Theorem. Following the notation used in [4]: let a be the maximal abelian ideal containing [j, j]; thus the Lie bracket relationship in equations (2.1) and (2.2) implies that a is framed by {Y 2,0 , Y 1,1 , Y 0,2 , Y 1,0 , Y 0,1 , Y }. Let A be the normal abelian sub-group whose Lie algebra is a and consider its action on J 2 (R 2 , R) by left multiplication. Thus the action is free and proper, so J 2 (R 2 , R)/A is well defined, and H : = J 2 (R 2 , R)/A is 2- dimensional Euclidean space such that J 2 (R 2 , R) ≃ H ⋉ A. We say that J 2 (R 2 , R) is a 2-abelian extension since H is 2-dimensional Euclidean space, latter we will see that 2 is the degree of freedom of reduced Hamiltonian H µ , see sub-Section 3.1. Therefore Theorem A is part of the classification of 2-abelian extension Carnot Groups with the non-integrable geodesic flow. 2.1. The exponential coordinates of the second kind. The jet space J 2 (R 2 , R) has a natural definition using the coordinates x, y, and u's; however, these coordinates do not easily show the symmetries of the system. The canonical coordinates defined in [4] exhibit the symmetries, We recall that the exponential map exp : j → J 2 (R 2 , R) is a global diffeomorphism, this allow us to endow J 2 (R 2 , R) with coordinates (x, y, θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6 ) in the following way: a point g in J 2 (R 2 , R) is given by g := exp(θ 1 Y 2,0 +θ 2 Y 1,1 +θ 3 Y 0,2 +θ 4 Y 1,0 +θ 5 Y 0,1 +θ 6 Y ) * exp(yX 2 ) * exp(xX 1 ). Then the horizontal left-invariant vector fields are given by X 1 := ∂ ∂x , X 2 := ∂ ∂y , (2.3) the vector fields from equation (2.3) corresponding to the independent variable, while the following correspond to second derivatives Y 2,0 := ∂ ∂θ 1 + x ∂ ∂θ 4 + x 2 2! ∂ ∂θ 6 , Y 1,1 := ∂ ∂θ 2 + y ∂ ∂θ 4 + x ∂ ∂θ 5 + xy ∂ ∂θ 6 , Y 0,2 := ∂ ∂θ 3 + y ∂ ∂θ 5 + y 2 2! ∂ ∂θ 6 . (2.4) The left-invariant vector fields from equation (2.3) and (2.4) just depend on the independent variables x and y. All the meta-abelian Carnot groups have this property, which is the heart of the symplectic reduction. For more detail, see [4]. 3. Geodesic flow on J 2 (R 2 , R) Let us consider the traditional coordinates on T * J 2 (R 2 , R), that is, p := (p x , p y , p 1 , p 2 , p 3 , p 4 , p 5 , p 6 ) are the momentums associated to canonical coordinates, see [2] and [5] for more details. Let λ be the tautological one-form; then the momentum functions associated to the left-invariant vector fields on the first layer j 1 are given by P 1 := λ(X 1 ), P 2 := λ(X 2 ), P 2,0 := λ(Y 2,0 ), P 1,1 := λ(Y 2 ), P 0,2 := λ(Y 0,2 ). See [8] or [1] for more detail about the momentum functions. Then the Hamiltonian governing the subRiemannian geodesic flow on J 2 (R 2 , R) is (3.1) H sR := 1 2 (P 2 1 + P 2 2 + P 2 2,0 + P 2 1,1 + P 2 0,2 ). See [8] or [1] for more detail about the definition of H sR . The Hamiltonian function H sR does not depend on the coordinates θ 1 , θ 2 , θ 3 , θ 4 , θ 5 and θ 6 , so they are cycle coordinate, in other words, p 1 , p 2 , p 3 , p 4 , p 5 and p 6 are constants of motion. Moreover, H sR is invariant under the action of A, then these constants of motion correspond to the momentum map J : T * J 2 (R 2 , R) → a * defined by action, see [4] for more details. 3.1. The reduced Hamiltonian. By general theory, the symplectic reduced space is diffeomorphic to T * (G/A) ≃ T * H and the reduced Hamiltonian is a two-degree-of-freedom system with a polynomial potential of degree four on variables x and y, and depending on the parameters µ = (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) in a * , given by (3.2) H µ (p x , p y , x, y) : = 1 2 (p 2 x + p 2 2 + φ µ (x.y)), where the potential φ µ (x, y) is given by (3.3) (a 1 + a 4 x+ x 2 2! a 6 ) 2 + (a 2 + a 5 x+ a 4 y + a 6 xy) 2 + (a 3 + a 5 y + a 6 y 2 2! ) 2 , Setting p 1 = a 1 , p 2 = a 1 , p 3 = a 1 , p 1 2 = a 2 1 , p 2 2 = a 2 2 and p 1 3 = a 3 1 , we obtain H sR = H µ . 3.2. Proof of Theorem A. One of the main consequences of the symplectic reduction is that it is enough to verify the integrability of H µ for all µ in a * to prove the integrability of geodesic flow on J 2 (R 2 , R). Thus, to prove Theorem A, it is enough to exhibit a µ such that H µ is not integrable. Proof. If µ = (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) = (0, 0, 0, 0, 0, 0, 0, a) and a = 0, by the definition of the potential given by equation (3.3) is with the form φ µ (x, y) = a 2 ( 1 4 x 4 + x 2 y 2 + 1 4 y 4 ). Let H µ be given by equation (3.2), then H µ is non-integrable. Indeed, this fact is a consequence of the classification of the two-degree-of-freedom Hamiltonian systems with a homogeneous potential of degree 4 made by J. Llibre, A. Mahdi, and C. Valls, in [6]. Appendix A. The a * value one-form α J 2 (R 2 ,R) In [4], we showed that the mathematical object relating the subRiemannian geodesic flow on J 2 (R 2 , R) and the reduced Hamiltonian on T * H is a * value one-form α J 2 (R 2 ,R) on j 1 ≃ R 5 given by α J 2 (R 2 ,R) = dθ 1 ⊗ (e 1 + xe 4 + x 2 2! e 6 ) + dθ 2 ⊗ (e 2 + xe 5 + ye 4 + xye 6 ) + dθ 3 ⊗ (e 3 + ye 5 + y 2 2! e 6 ). (A.1) A Comprehensive Introduction to Sub-Riemannian Geometry. D Barilari, A Agrachev, U Boscain, Cambridge University PressBarilari D. Agrachev, A. and U. Boscain. A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press, 2019. Arnol'd. Mathematical methods of classical mechanics. V I , SpringerNew York2nd ed.V. I. Arnol'd. Mathematical methods of classical mechanics, 2nd ed., New York:. Springer, 1989. Integrability and nonintegrability of sub-riemannian geodesic flows on carnot groups. Ivan Bizyaev, Alexey Borisov, Alexander Kilin, Ivan Mamaev, Regular and Chaotic Dynamics. 212016Ivan Bizyaev, Alexey Borisov, Alexander Kilin, and Ivan Mamaev. Integra- bility and nonintegrability of sub-riemannian geodesic flows on carnot groups. Regular and Chaotic Dynamics, 21:759-774, 11 2016. Sympletic reduction of the sub-riemannian geodesic flow on meta-abelian carnot groups. A Bravo-Doddoli, A. Bravo-Doddoli. Sympletic reduction of the sub-riemannian geodesic flow on meta-abelian carnot groups. 2022. Mechanics third edition: Volume 1 of course of theoretical physics. L D Landau, E M Lifshitz, L.D. Landau and E.M. Lifshitz. Mechanics third edition: Volume 1 of course of theoretical physics. 1976. Polynomial integrability of the hamiltonian systems with homogeneous potential of degree -2. J Llibre, A Mahdi, C Valls, Physics Letters. A. 18J. Llibre, A. Mahdi, and C. Valls. Polynomial integrability of the hamiltonian systems with homogeneous potential of degree -2. Physics Letters. A, 18, 2011. Liouville integrability of subriemannian problems on carnot groups of step 4 or greater. L V Lokutsievskiy, Yu L Sachkov, Sbornik: Mathematics. 2095672L. V. Lokutsievskiy and Yu. L. Sachkov. Liouville integrability of sub- riemannian problems on carnot groups of step 4 or greater. Sbornik: Math- ematics, 209(5):672, 2018. A Tour of Subriemannian Geometries, Their Geodesics and Applications. Number 91. R Montgomery, American Mathematical SocR. Montgomery. A Tour of Subriemannian Geometries, Their Geodesics and Applications. Number 91. American Mathematical Soc. Chaotic geodesics in carnot groups. R Montgomery, M Shapiro, A Stolin, R. Montgomery, M. Shapiro, and A. Stolin. Chaotic geodesics in carnot groups. 1997. Jet spaces as nonrigid carnot groups. B Warhurst, Journal of Lie Theory. 151B. Warhurst. Jet spaces as nonrigid carnot groups. Journal of Lie Theory, 15(1):341-356, 2005. Alejandro Bravo, Dept. of Mathematics, UCSC, 1156 High Street. Santa Cruz, CA95064 Email address: Abravodo@ucsc.eduAlejandro Bravo: Dept. of Mathematics, UCSC, 1156 High Street, Santa Cruz, CA 95064 Email address: Abravodo@ucsc.edu

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{'abstract': "The J 2 (R 2 , R) space of 2-jets of a real function of two real variables x and y admits the structure of a Carnot group with step 3. As any subRiemannia manifold, J 2 (R 2 , R) has an associated Hamiltonian geodesic flow, which is non-integrable. To prove this, we used the reduced Hamiltonian H µ on T * H , given by a symplectic reduction of the subRiemannian geodesic flow on J 2 (R 2 , R), using the fact that J 2 (R 2 , R) is a meta-abelian group.Theorem A. The subRiemannian geodesic flow onOther examples of Carnot groups with a non-integrable geodesic flow: the group of all 4 by 4 lower triangular matrices with 1's on the diagonal proved by R. Montgomery, M. Saphirom and A. Stolin, see[9]. The Carnot group with growth vector (3, 6, 14) showed by I. Bizyaev, A. Borisov, A. Kilin, and I. Mamaev, see[3]. The free Carnot group", 'arxivid': '2207.10014', 'author': ['Alejandro Bravo-Doddoli '], 'authoraffiliation': [], 'corpusid': 250699269, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4790, 'n_tokens_neox': 4059, 'n_words': 2374, 'pdfsha': '9edd2c957f3f345a3b5d135eb04a33b1571e517f', 'pdfurls': ['https://export.arxiv.org/pdf/2207.10014v3.pdf'], 'title': [], 'venue': ['GEODESIC FLOW ON J']}

arxiv

Composable Security of Distributed Symmetric Key Exchange Protocol Jie Lin Department of Electrical and Computer Engineering University of Toronto 10 King's College RoadTorontoONCanada Manfred Von Willich Hoi-Kwong Lo Department of Electrical and Computer Engineering University of Toronto 10 King's College RoadTorontoONCanada Department of Physics University of Toronto 60 St George StTorontoONCanada Department of Physics University of Hong Kong Pokfulam RoadHong Kong Quantum Bridge Technologies Inc 100 College StTorontoONCanada Composable Security of Distributed Symmetric Key Exchange Protocol The Distributed Symmetric Key Exchange (DSKE) protocol provides secure secret exchange (e.g., for key exchange) between two honest parties that need not have had prior contact, and use intermediaries with whom they each securely share confidential data. We show the composable security of the DSKE protocol in the constructive cryptography framework of Maurer. Specifically, we prove the security (correctness and confidentiality) and robustness of this protocol against any computationally unbounded adversary, who additionally may have fully compromised a bounded number of the intermediaries and can eavesdrop on all communication. As DSKE is highly scalable in a network setting with no distance limit, it is expected to be a cost-effective quantum-safe cryptographic solution to safeguarding the network security against the threat of quantum computers. Introduction Public key infrastructure has played a key role in today's network security. As tremendous experimental progress has been made in quantum computing in the last three decades, the quantum threat to communication security is widely recognized by governments, industries and academia. To counter the quantum threat to public key infrastructure, there are three major categories of solutions: post-quantum cryptography (PQC) [1], quantum key distribution (QKD) [2] and pre-shared keys (PSKs). An advantage of PQC is that it is software-based and can be implemented in the Internet without dedicated special hardware. However, since PQC is based on unproven computational intractability assumptions, the risk of an unexpected security breach of PQC is high [3], [4]. While QKD provides informationtheoretic security, the cost of QKD can be quite high and there are often limits to its key rate and distance. Without quantum repeaters, QKD is not yet a scalable solution in the global Internet. PSK has the advantage of being quantumsafe because it either employs one-time-pad or symmetric key crypto-systems that, unlike public key crypto-systems, are resistant to quantum attacks. Unfortunately, up till now, PSK has the disadvantage of being unscalable in a network of many users. This is because each user has to share a key with another user to communicate securely. Therefore, with a large number, say N , users, there are N (N − 1)/2 pairs of users. Each time when a new user joins the system, the existing users need to share a new key with the new user in order to communicate with them. This is highly inconvenient and costly. Recently, the distributed symmetric key exchange (DSKE) protocol [5], [6] has been introduced to provide a scalable solution that is also information-theoretically secure. DSKE has three major advantages. First, DSKE is highly scalable. With DSKE, when a new user comes on-line, no new key materials need to be delivered to the existing users, unlike PSKs. Second, DSKE distributes trust among multiple third parties, which are called Security Hubs in DSKE. Provided that the number of compromised Security Hubs (those that deviate from honest behaviours defined in the protocol) is below a predetermined threshold, DSKE provides information-theoretic security. This implies that DSKE avoids any single points of failure. Third, unlike QKD, DSKE has no distance limit and does not require dedicated optical fibres. DSKE is also highly versatile and can be used in many applications including mobile phones, network security and embedded systems such as Internet of Things devices. More concretely, the DSKE protocol may be used to agree sequences of data between multiple parties with a number of security properties, including quality of randomness, robustness and information-theoretically secure authenticity and confidentiality, to support a large range of use cases. As an example, DSKE allows two participants, Alice and Bob, to agree on an information-theoretically secure shared symmetric key, which can then be used for encrypting some data for Alice to send to Bob. Data integrity can also be achieved by the DSKE protocol through authentication by message tag with information-theoretically secure authentication. In this paper, we provide a rigorous security proof for the DSKE protocol. To allow other cryptographic applications to use the sequence of data agreed by honest parties from the DSKE protocol (which we call the secret for the remainder of this paper) in a secure way, it is important to prove the composable security of the DSKE protocol. To do so, we show the DSKE protocol is -secure in the constructive cryptography framework by Maurer [7], which implies universal composability [8], [9]. We first prove the security of the DSKE protocol by assuming the availability of perfectly authenticated channels. Then we use the composability theorem in the constructive cryptography framework to replace perfectly authenticated channels by a practical authentication protocol and insecure communication channels. We also show the DSKE protocol is -robust, which means with a probability at least 1 − , the protocol does not abort if an adversary is passive on communication channels (see Section 2.6 for a precise definition of being passive). Another novelty of this work is that our scheme to verify the correctness of the secret does not rely on pre-shared keys between the communicating parties. A typical message authentication scheme requires a pre-shared key to securely verify the message's authenticity. In our scheme, we transmit the key together with the message, relying only on the same security assumptions that are needed for confidentiality. We show the correctness and confidentiality of such a secret validation scheme. Structure of this document In Section 2, we present technical preliminaries that are relevant for our security proof. In particular, we state the results about our choice of 2-universal hash function family. We also review the constructive cryptography framework [7]. We provide a short summary of the DSKE protocol in Section 2.5 and direct readers to Appendix A for a detailed description of the general DSKE protocol with simplifying protocol parameter choices. A table of symbols used in the protocol description is given in Table 1. Readers seeking greater familiarity with the DSKE protocol may read [5]. In Section 3, we discuss the security definition for the DSKE protocol and prove the security of a variant of the protocol, which is the general DSKE protocol under the assumption that perfectly authenticated channels are freely available. In Section 4, we then prove the security of the general DSKE protocol. We show results about the security of hashing for message authentication in Appendix B and about the security of Shamir secret sharing with validation in Appendix C. Technical Preliminaries Notation We define some common notations used throughout this paper to assist our discussions. In particular, A refers to Alice, B refers to Bob, E refers to Eve, and P i refers to the unique identifier assigned to the Security Hub indexed by i ∈ {1, . . . , n}. We highlight that F denotes the same finite field throughout, |F | is the number of elements in F , and by element we mean an element of F . By length, we mean number of field elements. As we often need to write a collection of symbols, (. . . , Y i , . . . ), where the index i iterates through each element of a list S, we define a shorthand notation (Y i ) i∈S or simply (Y i ) i when the list S is clear from the context (e.g., we write (Y 1 , Y 2 , Y 3 ) as (Y i ) i∈ [1,2,3] or just (Y i ) i ). We use capital letters (e.g., Y ) to denote random variables and lowercase letters (e.g., y) to denote a particular value of the random variable. We use uniform to mean uniformly distributed. We use δ x,y to denote the Kronecker delta function. Between two sequences, denotes the concatenation operation. Between two resources, denotes the parallel composition operation. Throughout, given a finite sample space Ω and a probability distribution p over it, we construct the corresponding probability space (Ω, F, P) by setting the event space F as the collection of all subsets of Ω, and P(A) = ω∈A p(ω). Table 1 in Appendix A lists symbols related to the protocol description. Other notations are introduced where they first appear. Mathematical fundamentals Fields: Let F = GF(p r ), a Galois field with p a prime number and r > 0 an integer. F is a finite field with |F | = p r elements. Multiplication by a nonzero element of F is a bijective mapping F → F . Addition of vectors: The additive group of the vectors of dimension m over GF(p r ), namely GF(p r ) m , is the same as the additive group of GF(p) rm . This allows addition of vectors of suitable lengths over different fields to be compatible (i.e., to be the same group), when both fields have the same characteristic p. For example, we can use this to match a Shamir secret sharing scheme of vectors over GF(2 8 ), with a hash function over the field GF(2 128 ) using bit-wise exclusive-or as addition while maintaining the properties that depend on this addition. For simplicity of exposition, we use the same field for both the secret sharing scheme and the validating hash function. Probability theory: For any proposition prop(X, Y ) of random variables X and Y , Pr(prop(X, Y )) = y∈F Pr(prop(X, Y )|Y = y) Pr(Y = y). (2.1) When summing over all values of a random variable Y ∈ Ω, where Ω is the relevant sample space for Y , y∈Ω Pr(Y = y) = 1. (2.2) When X, Y ∈ Ω are mutually independent and X uniform, Pr(X = x|Y = y) = 1/|Ω|. (2.3) Random variables X and Y are mutually independent if ∀(x, y) : Pr((X, Y ) = (x, y)) = Pr(X = x) Pr(Y = y). (2.4) Statistical distance: For two probability distributions P X and Q X of a random variable X that can take any value in some set X , the statistical or total variation distance between P X and Q X has the following properties: [10]. This polynomial function family is described by Bernstein [11]. 1 2 x∈X |P X (x) − Q X (x)| = max X :X ⊆X x∈X (P X (x) − Q X (x)) = x:P X (x)≥Q X (x) (P X (x) − Q X (x)Theorem 2.1. Denote v = (v 1 , ..., v s ) and v * = (v * 1 , ..., v * s ). Let Ω = F 2 be a sample space with uniform probability. Let h C,D (v) = d + s j=1 c j v j define a family of functions with random variables (C, D) ∈ Ω as selection parameters. Let s = 0. Let t ∈ F be given. Then, max t * ,v * =v Pr(t * = h C,D (v * ) | t = h C,D (v)) = min( s |F | , 1). Proof. See Theorem B.1 in Appendix B. Given a message v together with a valid tag t = h c,d (v), with (c, d) being uniform (described by the random variables C, D), Theorem 2.1 tells us that the maximal success probability for a forger to create a differing message v * and a new hash value t * such that these correspond (i.e., that t * = h C,D (v * )) is at most min( s |F | , 1) when the message length s is nonzero. In the other words, (for large |F |) not knowing (c, d) makes it highly improbable for the forger to successfully substitute a tag and differing message. The theorem does not apply without a prior message. In the absence of a prior message, the probability that a forged message will be validated is 1 |F | , which is independent of the length s. Secret confidentiality and validation. Shamir introduced a secret sharing scheme that produces n shares, any k of which are sufficient to reconstruct the secret, but any k − 1 of which give no information about the secret [12]. Theorem 2.2. In a Shamir secret sharing scheme with threshold k, the shared secret is independent of any subset of the shares of size k − 1 or less. Proof. See Theorem C.2 in Appendix C. Theorem 2.2 tells us that, under the constraint of access to only k−1 shares, a Shamir threshold-(n, k) secret sharing scheme has perfectly secure confidentiality. Theorem 2.3. In a Shamir secret sharing scheme with threshold k, for any given set of k shares, the secret is a linear combination of the shares. Proof. See Theorem C.1 in Appendix C. Theorem 2.4. Denote Y = (Y (1) , . . . , Y (m) ). Let Ω = F 3+m be a sample space. Let (C, D, E, Y) ∈ Ω, with D uniform and independent of C, E and Y. Let T = h C,D,E (Y) = D + CE + m j=1 C j+1 Y (j) . Then T is independent of Y. Proof. This follows directly from D being uniform and independent of X with T = D + X, where X = CE + m j=1 C j+1 Y (j) , due to Lemma C.2 in Appendix C. Theorem 2.4 tells us that no information is obtained from the tag (o A as T ) about the secret (S A as Y). Thus, given the perfectly secure confidentiality provided by a Shamir secret sharing scheme for up to k − 1 shares being known, publishing the tag does not impact this confidentiality. Given a Shamir secret sharing scheme, the resulting secret is malleable: a modification of any share will modify the reconstructed secret in a known way, in the same way that modifying the ciphertext of one-time-pad encryption modifies the decrypted plaintext. Normally, a message authentication code would be employed to allow detection of such a change, but this needs a shared key to implement. Transmitting a validation key via the same secret sharing scheme violates the normal premise for authentication: that the validation key is assured to be the same at both sides. Theorem 2.5 gives us a way to transmit such a key using the same (malleable) secret sharing scheme while providing authenticity, under the same premise that the secret sharing scheme already has. We believe that this is a novel construction 1 . Theorem 2.5. Denote y = (y (1) , . . . , y (m) ) and y = (y (1) , . . . , y (m) ). Let Ω = F 3 be a sample space with uniform probability. Let h C,D,E (y) = D + CE + m j=1 C j+1 y (j) define a family of hash functions with random variables (C, D, E) ∈ Ω as selection parameters. Let m = 0. Then, max t ,c ,d ,e ,y =y Pr(t + t = h C+c ,D+d ,E+e (y + y ) | t = h C,D,E (y)) ≤ min( m+1 |F | , 1). Proof. See Theorem C.3 in Appendix C. Theorem 2.5 gives us an upper bound on the probability of validating a secret with a nontrivial alteration, with three confidential uniform elements and the alteration constrained to addition of a vector to (C, D, E, y). The premise for the theorem is assured by the secret sharing scheme. There are n k subsets of k shares and hence reconstructions, but a probability is upper-bounded at 1, giving an upper bound on the probability of any of the n k reconstructed secret candidates passing the validation of ≤ min( n k m+1 |F | , 1). Combining Theorems 2.2 to 2.5 leads to the following result: The (n, k)-Shamir secret sharing scheme and a polynomial validation code for transmitting 3 + m elements of the same finite field F , where the first 3 elements are uniform, 1. A related objective is presented in [13]. of which the first 3 are consumed as u A and the remaining m elements are S A , has • -secure correctness with ≤ min( n k m+1 |F | , 1) • perfectly secure confidentiality against a computationally unbounded adversary who can access and modify up to k − 1 shares and block but not access or modify any of the remaining shares. This leverages the secrecy provided by the sharing scheme to deliver the key used for secure validation of the secret that is simultaneously delivered. This is novel in the sense that it ensures correctness (authentication) in addition to retaining the confidentially of the secret sharing scheme without imposing additional security requirements, i.e., it assumes only that no more than k − 1 of the shares are compromised, as with the sharing scheme, whereas normally the key for validation would be assumed to be pre-shared. Composability and constructive cryptography As a cryptographic protocol is often combined with many other protocols, it is important to prove the security of the protocol in a composable security framework. The composability result in such a framework asserts that in analyzing the security of a complex protocol, one can simply decompose it into various subprotocols and analyze the security of each. Provided that each real subsystem constructed by a subprotocol is close to an ideal subsystem within some , which is quantified by some distance measure (a pseudo-metric in abstract cryptography), the real system constructed from the combined protocol will then be close to the combined ideal system. The sum of the -values for the subprotocols gives an -value for the combined protocol. The abstract cryptography framework [14] uses a topdown approach. In this framework, one states the definitions and proves the security from the highest possible level of abstraction and proceeds downwards. When one deals with a specific instantiation of an abstract system in a lower level, as long as the lower level system satisfies the properties required in the higher-level proof, the composed protocol is secure. The constructive cryptography [7] is an application of the abstract cryptography framework to defining classical cryptographic primitives in a constructive way. In this framework, one specifies the resources that are used by a protocol, their required properties and some desired functionalities of an ideal system. If a protocol constructs the ideal system from the given resources, then it is secure. A protocol that is secure in this sense is also universally composable secure. We briefly review main concepts and terminologies from the constructive cryptography below and refer to [7] for further details. 2.4.1. Resource. An I-resource is a system with interface label set I (e.g. I = {A, B, E}). Resources can be composed together via the parallel composition operation. For two resources R 1 and R 2 , we write R 1 R 2 as the resource after the parallel composition. On the set of resource systems, one can define a pseudometric to quantify the closeness between any two resources. We present the definition of pseudo-metric. Definition 2.1 (Pseudo-metric). A function d : Ω × Ω → R ≥0 is a pseudo-metric on the set Ω if for all a, b, c ∈ Ω, d(a, a) = 0, d(a, b) = d(b, a), d(a, b) ≤ d(a, c) + d(c, b). If the pseudo-metric also satisfies d(a, b) = 0 =⇒ a = b, then it is a metric. 2.4.2. Converter. A converter system, usually denoted by a Greek letter (e.g. π) is a system with two interfaces, an inside interface and an outside interface, which transforms one resource into another. The inside interface of a converter can be connected to an interface of a resource, and the outside interface becomes the new interface of the constructed resource. Two converters can be composed together via either serial or parallel composition operator. For two converters α and β, we write αβ as the converter formed by serial composition and α β as the converter formed by parallel composition. A protocol π = {π i } i∈J is a set of converters π i with J ⊆ I. Distinguisher and distinguishing advantage. For n-interface resources, a distinguisher D is a system with n+1 interfaces, where n interfaces connect to the interfaces of a resource and the other interface outputs a bit. In the constructive cryptography framework (as in many other composable security frameworks), the real and ideal systems are interactive black boxes that are given to the distinguisher with equal probability. Its task is to guess which system is in the black box. The output bit indicates its guess. For a class of distinguishers D, the distinguishing advantage for two resources R and S is where DR is the binary random variable corresponding to D connected to R, and DS is defined similarly. Each binary random variable may take the value 1 if the distinguisher D guesses the resource that it is connected to is the ideal resource system, and take the value 0 if it guesses the real resource system 2 . As we are interested in information-theoretic security, the class of distinguishers D is the set of all computationally unbounded distinguishers, which may have access to quantum computers. In particular, for the DSKE protocol, we consider only classical systems. For two classical systems R and S, once we show that they are indistinguishable for the set of all classical distinguishers, then they are automatically indistinguishable for the set of all quantum distinguishers since classical computers can simulate quantum ones (even if the simulation might be inefficient) [15,Remark 3]. Thus, we consider only the set of all possible classical distinguishers in this paper. 2. Exchanging 0 and 1 does not impact on the distinguishing advantage. We summarize a few useful properties of the distinguishing advantage. The distinguishing advantage defined in Eq. (2.6) is a pseudo-metric on the set of resources. It is non-increasing under (serial or parallel) composition of any two systems, that is, for any resource systems R, S, T, and any converter α (with α i denoting α converting interface i), d(α i R, α i S) ≤ d(R, S),(2.7) and We say that a protocol π = (π A , π B ) securely constructs S out of R within if the following conditions hold: d(R T, S T) ≤ d(R, S), d(T R, T S) ≤ d(R, S (i) For converters α E and γ E that emulate an honest behaviour at the E-interface of each system and block any distinguisher from the access of E-interface of each system, d(πRα E , Sγ E ) ≤ . (2.9) (ii) There exists a converter σ E such that d(πR, Sσ E ) ≤ . (2.10) We use the construction notation R Remark 1. The first condition in Definition 2.2 captures the correctness of the protocol when no adversary is present. In this case, the adversarial controls covered by α E and γ E are not accessible to the distinguisher. The second condition captures the situation when an adversary is present. In that case, the converters α E and γ E are removed so that the distinguisher has full access to Eve's interfaces, which are the E-interface in the real system, and the E-interface of a simulator σ E that is attached to the ideal system. (i) If a protocol π securely constructs a system S out of a system R within and another protocol π securely constructs a system T out of a system S within , then the serial composition ππ (running the protocol π after the protocol π) securely constructs the system T out of the system R within + , i.e., (R (π, ) − −− → S) ∧ (S (π , ) − −−− → T) =⇒ R (ππ , + ) −−−−−−→ T; (2.11) (ii) If a protocol π securely constructs a system S out of a system R within and another protocol π securely constructs a system S out of a system R within , then the parallel composition π π securely constructs the system S S out of the system R R within + , i.e., (R (π, ) − −− → S) ∧ (R (π , ) − −−− → S ) =⇒ R R (π π , + ) − −−−−−− → S S ; (2.12) (iii) When a trivial converter, which applies the identity transformation to the resource that it connects to, is applied to a system R, it perfectly constructs the system R out of itself, i.e., R (1,0) − −− → R,(2.13) where 1 denotes the trivial converter. Synopsis of DSKE protocol We give a synopsis of the DSKE protocol without technical details to assist a high-level understanding of our security proof. We refer to Appendix A for a detailed description of the protocol steps. We work with a two-user key agreement protocol in a network setting with a large number, N , of potential "end users" in the presence of a number, say n, of third parties called Security Hubs. The Security Hubs are numbered from 1 to n and an identifier P i is assigned to the ith Hub. The main goals are to guarantee that both users agree on the same secret and to protect the privacy of the agreed secret from potential adversaries, including other end users and the Security Hubs. During the one-time set-up (Steps (1) and (2) of the protocol as described in Appendix A), secure channels are assumed between the end users and Security Hubs. Those secure channels enable the end users and the Security Hubs to share some pre-shared random data (PSRD). Once the one-time setup has been done, we are in the situation described in Figure 1. Figure 1 shows an example of a network with the users such as Alice, Bob, and Charlie together with two Security Hubs. Each user shares a table of PSRD with each Security Hub. Note that each Security Hub knows only the values of their own part of the shared random data, but has no information on the values of the shared random data of other Security Hubs. We assume one-way communication from Alice to Bob, that is, Alice requests via the Security Hubs to exchange a secret with Bob. Alice is not interested in receiving information from Bob at all during the execution of the protocol, and may have only unidirectional communication available. Remark 2. Note that if Alice and Bob share a string of random secret bits for use as a one-time-pad, it is important that Alice and Bob do not use the same sequence of values simultaneously for encryption. If they were to do so, when Alice uses a key bit, k i to encrypt a message, a i , then she is sending c i = k i ⊕ a i to a communication channel, where ⊕ denotes bitwise exclusive-or, and should Bob (inadvertently) uses the same key, k i , to encrypt another message, b i , then he is sending d i = k i ⊕ b i . Then, an eavesdropper who possesses both c i and d i can compute the parity of the two bits, c i ⊕d i , to obtain c i ⊕d i = (k i ⊕a i )⊕(k i ⊕b i ) = a i ⊕b i , thus recovering the parity of the pair (a i , b i ). This would be a serious security breach. To avoid the above problem, it will be simplest to pre-assign each random string to a particular sender so that only one party is allowed to be the sender of the communication using the particular sequence as a one-time-pad key or one-time authentication key. Remark 3. In practical two-way communication, the parties may need two separately managed keys (one for securing communication from Alice to Bob and the other for securing communication from Bob to Alice). The two keys could, for example, be obtained through two iterations of the DSKE protocol. However, for simplicity, we will not discuss this two-way communication case here. After the initial setup phase (i.e., Steps (1) and (2) to be introduced in Appendix A), all subsequent communications in the protocol can be done through insecure classical channels such as the Internet, radio, or phone. Note that no quantum channels are needed in the subsequent communication. This makes DSKE versatile. In the DSKE protocol, Alice generates n shares using PSRD shared between Alice and each Security Hub in an (n, k)-threshold scheme of Shamir's secret sharing scheme [12], where k is the minimum number of shares needed to reconstruct the secret. She also generates a secretauthenticating tag o A := h u A (S A ), where u A S A is the secret from the (n, k)-threshold scheme, and h u A is a hash function with its parameter u A , which is chosen from a family of 2-universal hash functions (also see eq. (C.9)). She encrypts each share and sends the ith share Y i along with the secret-authenticating tag to the Hub P i via authenticated channels. We note that the secret-authenticating tag is the same for all Hubs. After receiving the secret-authenticating tag and the encrypted share, an honest Hub decrypts the share, and then re-encrypts the share using the PSRD shared between the Hub and Bob. It forwards the secret-authenticating tag and the newly encrypted share to Bob via an authenticated channel. After Bob receives enough messages from Hubs, he reconstructs all possible values of the secret from any k shares received. Then in the secret validation step, he validates each possible candidate against the secret-authenticating tag, which is chosen to be the same secret-authenticating tag sent by at least k Hubs. If there is no unique secret that passes the secret validation step, he aborts the protocol. Threat model A collection of adversarial entities can include a coalition of end users other than Alice and Bob, eavesdroppers, and a subset of the Security Hubs. This set of adversaries may collude to attempt to compromise the objective of the protocol between Alice and Bob. We call this collection of adversarial entities Eve. An honest Security Hub follows its part of the protocol correctly and maintains confidentiality of its own data. A compromised Security Hub may deviate from the protocol. No limits are placed on compromised Hubs, other than that they do not have access to confidential information nor are they able to modify any information held by honest parties. To analyze the security (i.e. correctness and confidentiality in Section 3.1) of the DSKE protocol, Eve is allowed to attempt to tamper with the communication by modifying the messages. She is free to listen to all communications except for the initial sharing of tables by honest Security Hubs (conducted through secure channels). Eve can fully control those compromised Security Hubs, including knowing all their tables, and knowing and modifying all messages that come from or are delivered by those compromised Hubs. As a robustness analysis of a protocol is concerned with an honest implementation of the protocol, which is a modified threat model from that of the correctness and confidentiality analysis, we define the behaviour of Eve in this modified threat model, which we call passive Eve: Eve is passive on all communication links, that is, she is allowed to listen to all the communications but she does not tamper; she is still given the ability to fully control compromised Security Hubs. When Eve is passive, we show that the DSKE protocol completes (i.e., does not abort) with a high probability. Assumptions We list assumptions used in our security proof: i) The pre-shared random data (PSRD) are securely delivered by every Security Hub that is not compromised to each of Alice and Bob. By securely delivered, we mean that the confidentiality and integrity of the data is maintained, and that the delivery is assured as being from and to the parties of the correct identity. This can be achieved through secure channels between Security Hubs and end users. 4 ii) The two users, Alice and Bob, are both honest. iii) A number of the Security Hubs might be compromised, and this number has a known upper bound. iv) For the robustness analysis, a number of the Security Hubs might not correctly execute their part in the protocol, either due to unavailability, communication failure, or compromise, and this number has a known upper bound. This is incorporated as an assumption that Eve is passive on the communication links. A communication link is assumed to provide the originating identity to the receiver, and if this is incorrectly provided, this is counted as a communication failure. Security of the skeleton DSKE protocol In this section, we prove the security of a variant of the DSKE protocol, which we call the skeleton DSKE protocol. It differs from the general DSKE protocol (which we summarized in Section 2.5; also see Appendix A for a detailed description) only by the assumption that channel authentication is perfectly secure, where a perfectly secure authentication scheme is defined as an -secure authentication scheme for which = 0. We call this channel an authenticated channel. In an authenticated channel, an adversary is free to eavesdrop on the communication and to tamper with a message, but the receiver will detect with certainty whether the message has been altered. This may also be thought of as the limiting behaviour as the size of a message authentication code increases without limit. We analyze the security of the skeleton DSKE protocol in the framework of constructive cryptography as briefly reviewed in Section 2. Further details about the constructive cryptography can be found in [7]. Also see [16]. Ideal system In the constructive cryptography framework, one can define an ideal system to capture desired functionalities and properties that one hopes to securely realize by the protocol of interest. As the DSKE protocol deals with a multi-party setting where the security claims are against some adversarial subsets, we can require the ideal resource system constructed by the DSKE protocol to respect the security claims that we aim to make for the DSKE protocol. In our setting with Alice, Bob, and an adversary Eve, the ideal resource should have three interfaces A, B and E. It should produce a secret S A for Alice and a secret S B to Bob, which is supposed to be the same as S A in the case that the protocol completes. Given that Bob can abort in the DSKE protocol when no valid secret can be reconstructed to pass the secret validation step or when there are multiple different reconstructed secrets that pass the secret validation step, we should also allow the ideal resource system to abort under the same conditions which the DSKE protocol does. To indicate that the protocol was aborted, the ideal system sets S B to be the symbol ⊥. Security Hubs in the DSKE protocol are treated as resources in our analysis and thus are located inside the ideal resource system. As Eve can control all compromised security Hubs and all communication channels, the ideal system should give Eve the ability to control them through its E-interface. Each Security Hub resource has three interfaces, one for the sender, one for the receiver and one for Eve. Each Hub can operate in one of two modes: honest or compromised. In the honest mode, the Hub simply relays the input it receives from the sender to the receiver, and does not output anything at the E-interface. In the compromised mode, the Hub outputs the sender's input at the E-interface, and uses an input from the E-interface to set the output of the receiver's interface. While it is only required to specify properties and desired functionalities of the ideal system in the constructive cryptography framework, to better understand the behaviour of this ideal system in the DSKE protocol setting, it is still helpful to think of how this ideal resource system works internally. As shown in Figure 2, this resource system can be thought as containing a secure key resource to generate a uniformly distributed secret S A = S B , which will be distributed to Alice and Bob after checking the abort condition (if the protocol aborts, S B will be set to ⊥, the flag for aborting the protocol). The ideal resource system internally contains a (modified) real resource system (see Figure 3) that runs the skeleton DSKE protocol with the secret S A to determine the abort conditions but ignores the reconstructed secret at Bob's side. In particular, it uses the Shamir (n, k)threshold scheme to generate shares by using S A as the secret. The secret-authenticating tag o A is computed using the secret S A as well as an additional key u A produced by its internal secret key resource. The ideal system gives S A as its A-interface's output and S B as its B-interface's output. Since our threat model allows Eve to control compromised Hubs, we need to allow the ideal system to give the full access to those compromised Hubs through its E-interface. To discuss the inputs and outputs at the Einterface of the ideal system, we break the overall Einterface into the E-interface of each Hub P i . We treat the Hub P i and its corresponding communication channels as one entity for ease of discussion and we simply say Hub P i to mean the entire entity. We use Y E i and T E i to denote inputs at the Hub P i 's E-interface. Similarly, we use Y E i and T E i to denote the Hub's outputs at the E-interface. We use Z E i ∈ {00, 01, 10, 11} to denote another input from the Hub P i 's E-interface, which can be used to determine the behaviours of two authenticated channels. To fix the meanings, 0 means the authenticated channel transmits the input message faithfully and 1 means it produces an error. For compromised Hubs, Y E i can take any allowed value for a share and T E i can take any allowed value for the secretauthenticating tag. A compromised Hub P i gives its share Y i and the secret-authenticating tag o A it received as the outputs to its E-interface, that is, :=⊥ for those honest Hubs. Note that since the secret-authenticating tag is not encrypted, we can simply set T E i := o A for honest Hubs as well. Since each Security Hub can be either honest or compromised and our security claims for the DSKE protocol are based on the assumption that the number of compromised Hubs is below some given thresholds, the behaviour of this ideal resource system should respect those assumptions. By our assumptions, we assume there is some fixed set C ⊂ {1, . . . , n} with |C| < k, of the identifiers of all compromised Hubs. Y E i := Y i and T E i := o A . ( Here we state some properties of the ideal resource system in terms of the joint probability distribution Q S A ,S B ,(Y E i ,T E i ,Y E i ,T E i ,Z E i )i of the inputs and outputs at all its interfaces. This is not an exhaustive list. (i). Correctness: The marginal distribution of S A and S B satisfies that for any s A , s B ∈ F m such that s B = s A and s B =⊥, Q S A ,S B (s A , s B ) = 0. (3.1) (ii). Confidentiality: With |C| < k, the conditional probability distribution of S A conditioned on knowing the values of (Y E j ) j∈C as well as all T E i satisfies Q S A |(Y E j ) j∈C ,(T E i )i = Q S A . (3.2) (iii). Uniform randomness: The marginal distribution of S A satisfies that for any s ∈ F m , Q S A (s) = 1 |F | m . (3.3) Remark 4. We note that with the instantiation of the ideal system in Figure 2, the correctness and uniform randomness of the ideal system are effectively due to the use of a secure key resource (labelled as "Secret" in Figure 2). The confidentiality of the ideal system is effectively due to the confidentiality of the Shamir (n, k)-threshold scheme (see Figure 2: An ideal key distribution resource, which consists of a "secret" resource that generates a secret on interfaces A and conditionally B and a modified real resource system that runs the DSKE protocol whose only purpose is to determine whether the protocol aborts. The modified real resource system is the real resource system depicted in Figure 3 with the requirement that the secret from running the (n, k)threshold scheme is S A generated by the secret resource. The ideal system outputs S A at the A-interface, S B at the B-interface. Its E-interface is the E-interface of the real system with an additional ability (not shown in this diagram) to set the operation mode (i.e., honest versus compromised) of each Security Hub. Theorem 2.2) and that of the secret-authenticating tag (see Theorem 2.4). This is because its internal modified real system runs the (n, k)-threshold scheme using the secret in order to abort under the same condition as the DSKE protocol. Real system The real system is depicted in Figure 3. In the language of constructive cryptography, the real system uses a set of resources and converters to construct a secure equivalent of the ideal resource system defined in Section 3.1. We define these systems and then discuss how to calculate the distinguishing advantage (a pseudo-metric). Resources. We prove the security of the skeleton protocol assuming the availability of following resources: (1) Secret key resource: each user and Security Hub pair have a shared secret key resource that has only output interfaces. (2) Authenticated channel resource: each communication link between a user and a Security Hub is an authenticated channel, that is, for any message sent through the channel, either the original message is delivered or an error is detected if an adversary attempts to modify the message. (3) Security Hub resource: It has three interfaces, one for its sender, one for its receiver and one for Eve. As discussed in Section 3.1, each Security Hub can operate in one of its two modes: honest or compromised. In the honest mode, it receives (Y i , T i ) from its sender and simply sets Y i := Y i and T i := T i to give to its receiver; it ignores inputs from the E-interface and sets T E i := T i and Y E i :=⊥. In the compromised mode, it receives Y E i and T E i from the E-interface, and sets Y i := Y E i and T i := T E i , which are sent to its receiver; it outputs Y i and T i received from its sender to the Einterface, i.e., Y E i := Y i , T E i := T i . We use R s to denote all resources used in the real system. From a secret key resource and an authenticated channel resource we can construct a secure channel that either transmits the input message confidentially and correctly or produces the ⊥ symbol to indicate an error when an adversary attempts to modify the message. We remark that in the DSKE protocol, Y i and Y i are encrypted using the secret key resources. In Figure 3 as well as in Figure 4, we use T i to denote the non-encrypted part of the message from Alice to the Hub P i , and similarly T i to denote the non-encrypted part of the message from the Hub P i to Bob. Then the encrypted version of Y i (similarly Y i ) is transmitted together with T i (correspondingly T i ) in one message through an authenticated channel. We remark that the secret key resources are available after the one-time setup process in the DSKE protocol. However, the authenticated channel resource is not available in the general DSKE protocol. Thus, after proving the security of the skeleton protocol, we then prove the security of the general DSKE protocol by constructing an authenticated channel resource using a secret key resource and an insecure channel. Converters. We need converters that use secret key resources and authenticated channels and n Security Hub resources to (approximately) construct the ideal system. We now define converters for the DSKE protocol. As depicted in Figure 3, Alice has a converter π A that produces S A , the shares Y i and a secret-authenticating tag o A for validation of the secret. Alice communicates with the Hub P i through an authenticated channel by sending the encrypted version of Y i along with T i where she sets T i = P i A B o A , where P i A B is used for identity validation and o A is the secret-authenticating tag 5 . As a result, the Hub can either receive share Y i and T i without modification or securely detect errors and get ⊥ for Y i and T i . Bob has a converter π B that receives inputs from authenticated channels from each Hub to Bob, and outputs S B , which can take the value of ⊥ to indicate abort of the protocol. Inside the converter π B , it runs the secret reconstruction step and the secret validation step of the DSKE protocol. The real system is described by a joint probability 5. Note that T i and T i in the protocol may contain other necessary information such as the offset for the unused portion of each table and a secret identification number. We omit writing such information here for simplicity as it does not affect our discussion. Figure 3: A real key distribution resource using n Security Hubs. Each green box represents an authenticated channel labelled by A + with a suitable subscript that identifies the communicating parties. The secret key resource K APi is used to encrypt distribution P S A ,S B ,(Y E i ,T E i ,Y E i ,T E i ,Z E i )i .Y i and K PiB is used to encrypt Y i . If a Hub P i is compromised, Eve determines values of Y i , T i , Y i , T i by accepting (Y E i , T E i ) at the E-interface; the system outputs (Y E i , T E i ) := (Y i , T i ) at the E-interface. For honest Hubs, since the ciphertext of Y i (Y i ) reveals no information about Y i (Y i ), we simply set Y E i :=⊥ to indicate this. For each Hub P i , a two-bit variable Z E i as an input to the E i -interface is used to set the behaviours of two relevant authenticated channels. The E-interface of the system consists of E 1 , . . . , E n , where the alphabets for those inputs and outputs are determined by the operation mode of each corresponding Hub. The operation mode of each Hub is predetermined and cannot be altered through the E-interface. Simulator As the second condition in Definition 2.2 considers the situation where Eve is active, we need to introduce a simulator σ E (which is Eve's converter) such that when it connects with the ideal system, the E-interface of Sσ E is the same as the E-interface of the real system π s R s . Note that C is used to denote the set of all identifiers of compromised Security Hubs, and that |C| ≤ k −1 under our assumptions. As Eve can fully control compromised Hubs in our threat model, the simulator σ E needs to have different operations for Hubs in C and for Hubs in the complement set. We note that the E-interface of the ideal system in Figure 2 accepts (Y E i , T E i , Z E i ) i as inputs and produces (Y E i , T E i ) i as outputs. It also gives the ability to set the operation modes of Security Hubs. In the real system shown in Figure 3, Security Hubs whose identifiers are in the set C operate in the compromised mode while the rest are in the honest mode. This means the simulator σ E needs to set the operation modes of those Hubs and blocks the ability to change the operation modes to match the E-interface of the real system. Except setting the operation modes, the E-interface of the simulator σ E is then identical to the E-interface of the ideal system. We depict one possible simulator σ E in Figure 4 for this purpose. Distinguisher For 3-interface resources, a distinguisher D is a system with 4 interfaces, where 3 interfaces connect to the interfaces of a resource R and the other interface outputs a bit, which indicates its guess about which resource is given. This is illustrated in Figure 5. In the DSKE protocol setup, the distinguisher has the following abilities to interact with the unknown system: (i) The distinguisher D can read outputs from the Aand B-interfaces. These two interfaces do not accept any input. The distinguisher can set the inputs of and read outputs from the E-interface unless the E-interface has no input/output when a simulator is used to block input/output. (ii) When the E-interface allows the control of internal communication links, the distinguisher D can attack internal communication as allowed by Eve's ability since the distinguisher can act as Eve through the Einterface. (iii) When the E-interface allows the control of compromised Security Hubs, the distinguisher D can control those compromised Security Hubs. As the real system is completely characterized by P S A ,S B ,(Y E i ,T E i ,Y E i ,T E i ,Z E i )i and the ideal system with the simulator σ E is completely characterized by Q S A ,S B ,(Y E i ,T E i ,Y E i ,T E i ,Z E i )i , the distinguishing advantage (pseudo-metric) defined in eq. (2.6) can be related to the statistical distance between Figure 4: An ideal key agreement resource using n Security Hubs with the simulator σ E . Hubs are reordered for drawing purposes only. The set of compromised Hubs is denoted by C. The simulator σ E sets operation modes of the Security Hubs contained inside the ideal resource system. For Hubs in the set C, it sets the Hub P i to operate in the compromised mode and uses Z E i := 00 it receives from its E-interface to set the behaviours of two authenticated channels for the Hub P i . For all other Hubs that are not in the set C, the simulator σ E sets the Hub P j to operate in the honest mode and uses Z E j ∈ {00, 01, 10, 11} to set the behaviours of two authenticated channels for the Hub P j . The E-interface of the simulator also accepts the a pair of values (Y E i , T E i ) and it outputs (Y E i , T E i ) received from the ideal system. The allowed alphabets for those variables are determined by the operation mode of each corresponding Hub. P S A ,S B ,(Y E i ,T E i ,Y E i ,T E i ,Z E i )i andQ S A ,S B ,(Y E i ,T E i ,Y E i ,T E i ,Z E i )i as we will see in the security proof. Security Theorem 3.1. Under the assumptions listed in Section 2.7, the protocol π s = (π A , π B ) described above (depicted in Figure 3) is -secure, where = min( n k m+1 |F | , 1), which is determined by the family of hash functions used in the protocol as described in Theorem 2.5 as well as the choices of n and k in the (n, k)-threshold scheme. Proof. We need to check two conditions in Definition 2.2. To check the first condition, we use converters α E and γ E to plug into the E-interface in the real and ideal systems. They both allow all messages Y i , T i and Y i , T i to be delivered correctly in authenticated channels (that is, Z E i = 00 for all i) and completely block E-interface from the distinguisher. Those two converters may still control compromised Hubs in the same predefined way with the restriction that the number of compromised Hubs is at most k−1. In this case, the distinguisher can observe only outputs from Aand B-interfaces. We use X to denote S A , S B and use X to denote the set of values that S A , S B can take. As any distinguisher can only observe S A and S B , from the distinguisher's point of view, the real system is completely described by P X and the ideal system with the simulator σ E by Q X . To guess whether it is holding the real system, a deterministic strategy for the distinguisher is that the distinguisher can pick a subset X ⊆ X such that for all x ∈ X , it outputs 1 and for all other values of x, it outputs 0. The distinguisher may also choose a mixed (probabilistic) strategy. We note that each mixed strategy is just a probabilistic mixture of pure (deterministic) strategies. Let D be a distinguisher that uses an arbitrary mixed strategy which is a probabilistic mixture of deterministic strategies X k with corresponding probabilities p k . The distinguishing advantage of this strategy is thus bounded by Pr[D π s R s α E = 1] − Pr[D Sγ E = 1] = k p k x∈X k (P X (x) − Q X (x)) ≤ k p k max X :X ⊆X x∈X (P X (x) − Q X (x)) = max X :X ⊆X x∈X (P X (x) − Q X (x)),(3.4) where the first equality is due to the chosen strategy of the distinguisher D , which can be written as a probabilistic mixture of deterministic strategies, the inequality is due to the fact that we perform an optimization over all subsets X Figure 5: The distinguisher D connects to an unknown system, interacts with it and reproduces a one-bit output that indicates its guess about the identity of the unknown system. of X and X k is just a possible subset of X , and the last equality is the result of summing over k. Thus, it is enough to consider all deterministic strategies. In this case, the distinguishing advantage is d(π s R s α E , Sγ E ) = max D∈D (Pr[Dπ s R s α E = 1] − Pr[DSγ E = 1]) = max X :X ⊆X x∈X (P X (x) − Q X (x)) = 1 2 x∈X |P X (x) − Q X (x)|,(3.5) where the first equality is due to the definition of the pseudometric in eq. (2.6), the second equality is due to the general deterministic strategy of the distinguisher as described above and that the optimal value of the distinguishing advantage can always be realized by a deterministic strategy, and the third equality is due to eq. (2.5). We see that the distinguishing advantage is related to the statistical distance between P X and Q X . Our task is then to evaluate the statistical distance. We note that the allowed alphabets for Y E i and T E i depend on whether i ∈ C. For ease of writing, we do not explicitly write out alphabets for those variables. For any value of s A and any value of s B such that s B = s A and s B =⊥, the joint probability distribution of the ideal system Q X (s A , s B ) = 0, while the joint probability distribution of the real system P X (s A , s B ) = 0 due to the correctness of secret validation being approximate, which depends on the property of hash functions used for the secret-authenticating tag as in Theorem 2.5. When restricting to all possible values of (s A , s B ) such that P X (s A , s B ) > Q X (s A , s B ), the real and ideal systems differ only in the case where the real system may obtain s B = s A and s B =⊥. To see this, we show the contrapositive: if s A = s B or s B =⊥, then P X (s A , s B ) ≤ Q X (s A , s B ). For each s A , we observe (i) (3.6) and P S A (s A ) = Q S A (s A ) = 1 |F | m , (ii) P X (s A , ⊥) = Q X (s A , ⊥ ) and (iii) for each s B = s A , Q X (s A , s B ) = 0 while P X (s A , s B ) ≥ 0. As we can write P S A (s A ) = s B P X (s A , s B ) = s B =⊥ P X (s A , s B ) + P X (s A , ⊥),Q S A (s A ) = s B Q X (s A , s B ) = Q X (s A , s A ) + Q X (s A , ⊥), (3.7) these two equations imply P X (s A , s A ) ≤ Q X (s A , s A ) for each s A following those facts (i)-(iii). Thus, the distinguishing advantage is d(π s R s α E , Sγ E ) = x:P X (x)≥Q X (x) (P X (x) − Q X (x)) (3.8) = s A s B :s B =s A ,s B =⊥ P X (s A , s B ) (3.9) = s A ,(y i ,ti)i s B :s B =s A , s B =⊥ P S A ,S B ,(Y E i ,T E i )i (s A , s B , (y i , t i ) i ) (3.10) = s A ,(y i ,ti)i P S A ,(Y E i ,T E i )i (s A , (y i , t i ) i ) × s B :s B =s A , s B =⊥ P S B |S A ,(Y E i ,T E i )i (s B |s A , (y i , t i ) i ), (3.11) where we use eq. (2.5) to obtain eq. (3.8) from eq. (3.5), we obtain eq. (3.9) since the condition P (x) ≥ Q(x) is equivalent to the situation where Bob does not abort the protocol and Bob obtains S B that is different from S A as discussed above (also note Q X (x) = 0 for such x), we write the marginal probability P S A ,S B in terms of summing Under the assumption that at most k − 1 Hubs are compromised, we can safely assume whenever S B =⊥, the secret-authenticating tag o A is transmitted faithfully from at least k Hubs to Bob. The reason is that Bob would set S B to ⊥ and abort the protocol if fewer than k Hubs send the same secret-authenticating tag; on the other hand, to agree on a value t =⊥ other than o A for the secret-authenticating tag, at least k Hubs need to send the same modified value t, which is not possible given that at most k − 1 Hubs are compromised. Thus, for each value of s A , (y i , t i ) i , P S A ,S B ,(Y E i ,T E i )i over all possible values of (Y E i , T E i ) i to get eq.s B :s B =s A , s B =⊥ P S B |S A ,(Y E i ,T E i )i (s B |s A , (y i , t i ) i ) ≤ min( n k Pr[h u (s B ) = o A |s A , (y i , t i ) i , s B = s A ], 1) ≤ min( n k m + 1 |F | , 1) =: , (3.12) where u s B is the secret reconstructed from a subset of {y i } with k elements, the factor n k is due to there being up to n k possible values of u s B that the secret validation step can check, and we use Theorem 2.5 (where we remove the freedom to choose t and set t = 0) for the last inequality. 6 Combining eqs. (3.11) and (3.12), we have d(π s R s α E , Sγ E ) ≤ s A ,(y i ,ti)i P S A ,(Y E i ,T E i )i (s A , (y i , t i ) i ) = . (3.13) This verifies the first condition in Definition 2.2. To check the second condition, we consider the simulator σ E as depicted in Figure 4. In this case, Eve's i , T E i , Z E i ) i , P X (s A , ⊥, (y i , t i , y i , t i , z i ) i ) = Q X (s A , ⊥, (y i , t i , y i , t i , z i ) i ). (3.14) By a similar argument as for the first condition, it is enough to consider all deterministic strategies when we 6. The multiplier n k may significantly reduce the security for large n. For example, n = 99, k = 50 results is a security loss of log 99 50 = 95.35 bits. For n ≤ 11, the security loss is under 9 bits, but with Bob still having to search up to 11 6 = 462 combinations. A protocol variant, which we do not elaborate, allows Bob to filter out bad shares before combining them, at the cost of several tags per share. calculate the pseudo-metric for these two systems. To guess that it is holding the real system, the distinguisher can pick a subset X ⊆ X such that for all x ∈ X , it outputs 1 and for all other values of x, it outputs 0. (Note that we reuse the same notation as in the proof of the first condition since the proof of the second condition resembles the first one. However, X, X here denote a different combination of symbols and a different set, respectively.) The distinguishing advantage is d(π s R s , Sσ E ) = x:P X (x)≥Q X (x) (P X (x) − Q X (x)) (3.15) = s A , (yi,ti,y i ,ti,zi)i s B :s B =s A , s B =⊥ P X (s A , s B , (y i , t i , y i , t i , z i ) i ) (3.16) = s A ,(y i ,ti,zi)i s B :s B =s A , s B =⊥ P S A ,S B ,(Y E i ,T E i ,Z E i )i (s A , s B , (y i , t i , z i ) i ) (3.17) = s A ,(y i ,ti,zi)i P S A ,(Y E i ,T E i ,Z E i )i (s A , (y i , t i ) i , z i ) × s B :s B =s A , s B =⊥ P S B |S A ,(Y E i ,T E i ,Z E i )i (s B |s A , (y i , t i , z i ) i ), (3.18) where we obtain eq. (3.16) since the condition P (x) ≥ Q(x) is equivalent to the situation where Bob does not abort the protocol and Bob obtains S B that is different from S A as discussed above (also note Q X (x) = 0 for those x's), to get eq. (3.17), we directly sum over all possible values of (Y E i , T E i ) i since those are outputs from E-interface that are not used for the secret validation step, and we finally rewrite the joint probability over S A , S B , (Y E i , T E i , Z E i ) i by the conditional probability P S B |S A ,(Y E i ,T E i ,Z E i )i and the marginal probability P S A ,(Y E i ,T E i ,Z E i )i to obtain eq. (3.18). We note that eq. (3.12) also holds when we condition on the additional input choices (Z E i ) i that determine the behaviours of authenticated channels since eq. (3.12) was proved under the choice Z E i = 00 for all i and we considered all possible combinations of u s B that can go through to the secret validation step when we estimated that upper bound. By setting any of (Z E i ) i to any value other than 00, Eve effectively reduces the number of combinations of u s B to feed into the secret validation step while the probability for each combination of u s B to pass the secret validation step is unchanged. In other words, for each allowed value s A , (y i , t i , z i ) i of S A , (Y E i , T E i , Z E i ) i , s B :s B =s A , s B =⊥ P S B |S A ,(Y E i ,T E i ,Z E i )i (s B |s A , (y i , t i , z i ) i ) ≤ . (3.19) By using eq. (3.19), the distinguishing advantage is then d(π s R s , Sσ E ) = s A ,(y i ,ti,zi)i P S A ,(Y E i ,T E i ,Z E i )i (s A , (y i , t i , z i ) i ) × s B :s B =s A , s B =⊥ P S B |S A ,(Y E i ,T E i ,Z E i )i (s B |s A , (y i , t i , z i ) i ) ≤ s A ,(y i ,ti,zi)i P S A ,(Y E i ,T E i ,Z E i )i (s A , (y i , t i , z i ) i ) = . (3.20) Therefore, the second condition is also verified. Thus, the protocol is -secure. Remark 5. The secret validation step relies on the correctness of the secret-authenticating tag as stated in Theorem 2.5. In the limit that |F | → ∞, we have that → 0, and the secret validation step is perfectly correct. In this case, the skeleton DSKE protocol perfectly constructs the ideal system out of the real resource system. Robustness Robustness of a protocol is the condition that the protocol does not abort when Eve is passive (restricted to modifying the messages of compromised Hubs only), as defined in Section 2.6. We say a protocol is -robust if its aborting probability is at most in the case of this restriction. When restricted in this way, the aborting probability for the skeleton DSKE protocol is at most = min( n k m+1 |F | , 1) when the number of honest Security Hubs is at least k and the number of compromised Security Hubs is at most k−1. 7 Theorem 3.2 (Robustness of a skeleton DSKE protocol). Under the assumptions listed in Section 2.7, when the upper bound on the number of compromised Security Hubs is no greater than min(n − k, k − 1), the skeleton DSKE protocol is -robust with = min( n k m+1 |F | , 1). Proof. When Eve is passive, all authenticated channels in the skeleton DSKE protocol faithfully transmit messages. We note that a message using a fake identity gets rejected by the receiver without consuming any secret key resources by the assumption that a communication link provides the originating identity to the receiver. In other words, Eve cannot impersonate an honest Security Hub by using compromised Hubs to mount an attack to exhaust data in a table shared by a Security Hub and a user. If the number of honest Security Hubs is at least k (which means the number of compromised Hubs cannot be more than n − k) and the number of compromised Hubs is at most k − 1, the only condition that causes Bob to abort is when there are multiple different candidate tuples (u A , s A , o A ) that pass the secret validation step. That is, in addition to the correct secret reconstructed using shares 7. The second condition becomes significant when 2k ≤ n, due to multiple competing reconstructed secrets. from k honest Security Hubs, there must exist at least one different secret candidate that also passes the secret validation step. For each possible guess of the (u A , s A ) other than the correct secret, it can pass the secret validation step with a probability at most min( m+1 |F | , 1) due to Theorem 2.5. Since there are at most n k valid candidates to go through the secret validation step, the aborting probability in this case is at most min( n k m+1 |F | , 1) =: . Thus, the skeleton DSKE protocol is -robust when the upper bound on the number of compromised Security Hubs is no greater than min(n − k, k − 1). Security of the general DSKE protocol To assist our discussion here, as shown in Figure 3, we denote the secure key resource between Alice and the Hub P i as K APi , the authenticated channel between Alice and the Hub P i as A + APi , the secure key resource between the Hub P i as K PiB , and the authenticated channel between the Hub P i and Bob as A + PiB . In the previous section, we have shown that π s securely constructs the ideal resource S out of R s within , where the real resource R s in the skeleton DSKE protocol is defined as R s :=K AP1 A + AP1 P 1 K P1B A + P1B . . . K APn A + APn P n K PnB A + PnB . (4.1) The difference between a skeleton protocol and a general protocol using the same (n, k)-threshold scheme is the availability of authenticated channel resources. In a general DSKE protocol, we need to use a secure key resource and an insecure channel to construct an authenticated channel. An authentication protocol π auth APi (also π auth PiB ) using a Carter-Wegman universal hash function family to produce message tags can securely construct an authenticated channel out of a secret key resource and an insecure channel within := min( s |F | , 1) as shown in Theorem 2.1. To distinguish this secret key resource from the secret key resource used in the skeleton protocol, we denote the new secret key resource by K auth and the insecure channel by C (with suitable subscripts). We then define the resource R g used in the general DSKE protocol as R g := K AP1 K auth AP1 C AP1 P 1 K P1B K auth P1B C P1B . . . K APn K auth APn C APn P n K PnB K auth PnB C PnB . (4.2) Recall that m is the length of the secret to be agreed and s is the length of the message for which a tag is produced. We now show that the general DSKE protocol is ( + 2n )secure, where = min( n k m+1 |F | , 1) and = min( s |F | , 1). Theorem 4.1. The protocol π g = (π s , π auth AP1 , . . . , π auth APn , π auth P1B , . . . π auth PnB ) securely constructs the ideal resource S out of the resource R g within + 2n , where each of π auth APi and π auth PiB securely construct authenticated channels A + APi , A + PiB within , respectively, where = min( n k m+1 |F | , 1) and = min( s |F | , 1). Proof. We start with showing the first condition in Definition 2.2 where no adversary is present. We trivially have π auth (K auth C)α E = A + β E (4.3) for each choice of subscript (AP i or P i B), where α E and β E emulate an honest behaviour at Eve's interfaces, since both systems are equivalent to a channel that faithfully transmits a message between two parties (either A and P i or P i and B). With a slight abuse of notation, we use α E 2n and β E 2n to denote systems that emulate an honest behaviour at Eve's interfaces in π g R g and π s R s , which include 2n authentication channels in the systems R g and R s , respectively. We use γ E 2n to denote the system that emulates an honest behaviour at Eve's interfaces in S. Thus, d(π g R g α E 2n , Sγ E 2n ) ≤d(π g R g α E 2n , π s R s β E 2n ) + d(π s R s β E 2n , Sγ E 2n ) = d(π s R s β E 2n , Sγ E 2n ) ≤ ,(4. 4) where the last inequality is shown in Theorem 3.1. We now analyze the case of an active adversary as required in the second condition in Definition 2.2. As each of π auth APi and π auth PiB securely construct authenticated channels A + APi , A + PiB within , respectively, there exist converters σ E APi and σ E PiB such that d(π auth APi (K auth APi C APi ), A + APi σ E APi ) ≤ , d(π auth PiB (K auth PiB C PiB ), A + PiB σ E PiB ) ≤ . (4.5) We use σ E APiB to denote σ E APi σ E PiB and use σ E n AP B to denote n-copies of σ E APiB composed in parallel. For the ease of writing, we use a shorthand notation A + APiB to denote A + APi A + PiB and use {A + APiB } to denote A + AP1B · · · A + APnB . Similarly, we use π auth APiB (KC) to denote π auth APi (K auth APi C APi ) π auth PiB (K auth PiB C PiB ). From the properties of the pseudo-metric, we have d(π g R g , π s R s σ E n AP B ) ≤ d({π auth APiB (KC)}, {A + APiB σ E APiB }) ≤ n i=1 d(π auth APiB (KC), A + APiB σ E APiB ) ≤ 2n , (4.6) where we use eqs. (2.7) and (2.8) for the first inequality to remove common systems in π g R g and π s R s σ E n AP B , Theorem 2.6 for the second inequality and the security of the authentication protocol for the third inequality and the fact that there are 2n uses of the authentication protocol. From Theorem 3.1, we also have a converter σ E such that d(π s R s , Sσ E ) ≤ . (4.7) Thus, we show that for the converter σ E = σ E σ E n AP B , we have d(π g R g , Sσ E ) ≤ d(π g R g , π s R s σ E n AP B ) + d(π s R s σ E n AP B , Sσ E σ E n AP B ) ≤ d(π g R g , π s R s σ E n AP B ) + d(π s R s , Sσ E ) ≤ 2n + , (4.8) where we use the triangle inequality in the first inequality, eqs. (2.7) and (2.8) to drop the common system σ E n AP B in the second inequality, and eqs. (4.6) and (4.7) in the third inequality. Combining those two conditions, we show that π g securely constructs the ideal resource S out of the resource R g within max( , + 2n ) = + 2n . We then show the -robustness of the general DSKE protocol. Theorem 4.2 (Robustness of a general DSKE protocol). When the upper bound on the number of compromised Security Hubs is no greater than min(n−k, k −1), a general DSKE protocol is -robust with = min( n k m+1 |F | , 1). Proof. When Eve is passive, a general DSKE protocol behaves in the same way as its corresponding skeleton protocol. The reason is as follows. A general DSKE and its corresponding skeleton protocol differ in the availability of authenticated channels. In the general DSKE protocol, an authentication protocol constructs an authenticated channel out of a secret key resource and an insecure channel, while in the skeleton protocol, authenticated channels are assumed to be available. When Eve is passive, she does not tamper each communication channel. Each authenticated channel in the skeleton protocol and each channel constructed by the authentication protocol in the general protocol both faithfully transmit messages. Thus, the result of Theorem 3.2 directly applies. + m field elements of H A i R B i 3 + m field elements of H B i S A Alice's secret of m field elements t A i message tag for M A i using v A i t B i message tag for M B i using v B i u A 3 field elements of Y A i for o A v A i 2 field elements of H A i for t A i v B i 2 field elements of H B i for t B i x i x-coordinate for Hub P i Y A i 3 + m field elements: u A S A Z A i encrypted Y A i in the message M A i Z B i encrypted Y A i in the message M B i A.1. Parameter choices We present the general DSKE protocol (as described in the DKSE paper [5]) with the following simplifying choices in this paper with the aim of establishing a baseline proof of its security. • A predetermined finite field F is used throughout, and element refers an element of F . • The final secret length m is predetermined. • The parameters of the threshold sharing scheme, n and k, are predetermined. For clarity, also k B = k in [5]. • The number of recipients is limited to 1. i . An overline indicates that it for a Hub's sending use. • Mutual identity validation is assumed and the corresponding identifiers P i , A i , B i are predetermined. • The identifiers A i are equal for all i, and we write the identifiers A i as simply A (notation abuse), as for B. A and B also denote the identities Alice and Bob. • The message tag validation key is used once only. A.2. Baseline protocol (1) PSRD generation and distribution Honest Hubs securely provide a copy of the two tables of ordered elements (for Alice, H A i and H A i ), with assured mutual identity verification, data confidentiality and data authenticity. For simplicity, Alice tracks use of elements in H A i by retaining an integer offset j A i into H A i up to which the elements have been used and erased, initialized as j A i := 0 upon receiving the tables from P i . Each Hub similarly tracks usage in H A i through j A i . Since messages may be received out of order, the receiver must individually track which elements of the table have been used. (2) Peer identity establishment Alice and Bob need to establish the authenticity of each other's identities. This phase, which in a practical implementation is necessary, is made redundant by the assumed mutual knowledge of identities and identifiers in the form presented in this paper. In practice, each DSKE client can query the identities of other clients with the help of Security Hubs and an informationtheoretically secure message tag. See [5] for details. (3) Secret agreement (a) Share generation (i) Alice retrieves the unused sequences R A i (length 3 + m) and v A i (length 2) from at offset j A i in H A i , erases them from the table, and remembers j A i + 3 + m + 2 as j A i on the next iteration of the protocol. (ii) Alice uses 3 + m identical but independent Shamir sharing schemes in parallel, one for each field element in the sequence. Alice sets: Y A i := R A i ∀i ∈ {1, . . . , k}. (A.1) f −2 (x i ) f −1 (x i ) f 0 (x i ) · · · f m (x i ) := Y A i ∀i ∈ {1,f −2 (x i ) f −1 (x i ) f 0 (x i ) · · · f m (x i ) =: Y A i ∀i ∈ {k + 1, . . . , n}. (A.3) (b) Share distribution (i) Operations by Alice for share distribution: (1) Alice solves for the secret Y A 0 = f −2 (x 0 ) f −1 (x 0 ) f 0 (x 0 ) · · · f m (x 0 ) , as she did for the shares in eq. (A.3). (2) Alice partitions Y A 0 into u A of 3 elements and S A of m elements: Y A 0 =: u A S A . (A.4) (3) Alice calculates an authentication code o A for the secret to be agreed, which we call the secretauthenticating tag: o A := h u A (S A ) (A.5) using the 2-universal family of hash functions H with the choice specified by u A . (4) Alice calculates Z A i := Y A i − R A i i ∈ {1, . . . , n}. (A.6) Note: Z A i is just a tuple of 3 + m zero elements for each i ∈ {1, . . . , k} because of the cancellation. The protocol could omit this field, but it is kept here to simplify the presentation. (5) Alice chooses a secret identification number K A so that (A, K A ) is unique. The message is: M A i := P i A B K A g(j A i ) Z A i o A . (A.7) Fields that omit the index i are the same across all n messages sent to the Hubs, such as o A . t A i is different in each message. (6) Alice calculates the message tag t A i using the function family H as t A i := h v A i (M A i ). (A.8)(7)t A i = h v A i (M A i ),(A.(x i , Y A i ). (ii) Bob solves for a candidate Y A 0 in f −2 (x 0 ) · · · f m (x 0 ) = Y A 0 from the (x i , Y A i ) tuples of each subset, similar to the operation that Alice did in (i), except with the set of indices i varying by subset, obtaining a candidate per subset. Bob may eliminate duplicates here. (iii) Bob partitions each distinct candidate Y A 0 of two strings of 3 and m elements respectively: Y A 0 =: u A S A (A.22) and forms the candidate tuple (u A , S A , o A ). (4) Secret validation Bob discards those candidate tuples (u A , S A , o A ) for which the following relation does not hold: o A = h u A (S AH = {h c,d : F s → F : (v 1 , . . . , v s ) → d + s j=1 c j v j } (B.1) Theorem B.1 gives the best guessing probability that Eve's message v * is both modified from v and validates against any tag t * with the selection of the hash function h c,d unknown, excluding the case of an empty message (s = 0). Theorem B.1. Denote v = (v 1 , . . . , v s ) and v * = (v * 1 , . . . , v * s ). Let Ω = F 2 be a sample space with uniform probability. Let h C,D (v) = D+ s j=1 C j v j define a family of functions with random variables (C, D) ∈ Ω as selection parameters. Let s = 0. Let t ∈ F be given. Then, max t * ,v * =v Pr(t * = h C,D (v * ) | t = h C,D (v)) = min( s |F | , 1). Proof. We are given that t = h C,D (v). This may be written t = D + s j=1 C j v j . (B.2) This constraint serves to eliminate all pairs (C, D) that do not solve eq. (B.2). For every value of C in F , this equation determines a unique value for D, resulting in exactly |F | pairs that meet the constraint. Conversely, for every value of D, there may be many values of C, and by inference, there may be many values of D for which there is no corresponding solution for C. Since the a priori probability on Ω (prior to imposing the constraint) is uniform, and each value for C occurs in a pair exactly once, the a posteriori marginal distribution on C (i.e. given the constraint) is uniform, but the marginal distribution on D is potentially nonuniform. To obtain the probability that t * = h C,D (v * ) holds, we subtract from it the given eq. (B.2) to obtain an equivalent equation t * − t = s j=1 C j (v * j − v j ). (B.3) For any values of t, t * , v and v * = v, the polynomial equation eq. (B.3) has up to min(s, |F |) solutions for C, with the bound attainable for some v * . Since C is uniform, each solution has probability 1 |F | . Thus with s = 0, max t * ,v * =v Pr(t * − t = s j=1 C j (v * j − v j )) = min . Share manipulation in secret sharing We use a prime to denote an associated additive difference variable. For example, a denotes a difference (error), used to produce a + a from a. Lemma C.1. Let Ω = F 2 be a sample space with (D, X) ∈ Ω. Let D and X be independent random variables with D uniform. Let z ∈ F be arbitrary. Consider an (n, k)-threshold Shamir secret sharing scheme that uses a polynomial over the field F . Theorem C.1. In a Shamir secret sharing scheme with threshold k, for any given set of k shares, the secret is a linear combination of the shares. Then Pr(D + X = z) = 1 |F | . Proof. Proof. A Shamir scheme is based on a polynomial f of degree (at most) k − 1 over F , where 1 ≤ k ≤ n < |F |. In such a scheme, n + 1 distinct x-coordinate values are chosen, with one (x 0 ) associated with the secret and the rest (x 1 , . . . , x n ) each associated with a share. The secret sharing scheme is defined by the polynomial in x as f (x) = k−1 j=0 c j x j , (C.3) with secret coefficients c j . The c j and x i determine the secret y 0 and shares y 1 , . . . , y n through y i = f (x i ). Any k of the n + 1 pairs (x i , y i ) uniquely determine the c j . The y i for k of the shares are required to be independent and uniform in F . For any J ⊆ {1, . . . , n} with |J| = k, the polynomial f can be expressed as a linear combination of a Lagrange basis of k polynomials L j of degree k − 1, such that L i (x j ) = δ i,j for i, j ∈ J: f (x) = i∈J y i L i (x). (C.4) Given known x i and a set J as defined above, the basis polynomial L i corresponding to x i can be determined. Since each polynomial L i is of degree k − 1 and has k − 1 distinct zeros, L i (x j ) = 0 whenever j ∈ J, and in particular, L i (x 0 ) = 0. From this, the secret is a known linear combination of any given k of the n shares: y 0 = f (x 0 ) = i∈J y i L i (x 0 ). (C.5) The confidentiality of y 0 provided by the Shamir scheme relies on at least n − k + 1 of the n shares remaining secret. Further, if a value y i + y i is substituted for each share y i , the reconstructed secret is i∈J (y i + y i )L i (x 0 ) = y 0 + i∈J y i L i (x 0 ). (C.6) From this it may be seen that, because the L i (x 0 ) are known and nonzero, a single error value y i = y 0 /L i (x 0 ) added to y i replaces the reconstructed secret with y 0 + y 0 . With multiple errors y i , y 0 = i∈J y i L i (x 0 ) can be chosen by an adversary who is able to choose y i for a nonempty subset of J. C.2. Confidentiality of Shamir sharing scheme The fundamental confidentiality of the Shamir secret sharing scheme is expressed in Theorem C.2 [12]. Theorem C.2. In a Shamir secret sharing scheme with threshold k, the shared secret is independent of any subset of the shares of size k − 1 or less. Proof. Let J ⊆ {1, . . . , n}, with |J| = k, and i ∈ J. From Theorem C.1, the secret Y 0 is a linear sum of k shares: Y 0 = j∈J d j Y j , (C.7) where each coefficient d j = L j (x 0 ) is nonzero for all j ∈ J. In a Shamir sharing scheme each subset of k − 1 shares is mutually independent and uniform. The secret Y 0 can therefore be partitioned into a sum for any i ∈ J: Y 0 = d i Y i + j∈J\{i} d j Y j . (C.8) Define D = d i Y i and X = j∈J\{i} d j Y j . Since d i is a nonzero field element and Y i is uniform and independent of X, D and X are mutually independent and uniform. By Lemma C.2, Y 0 = D + X and X are mutually independent, and thus Y 0 is independent of any subset that excludes share i, for any i ∈ J. Note that any subset with size less than k − 1 is included in the sum j∈J\{i} d j Y j . The independence and uniformity of the term d i Y i remains sufficient. C.3. Polynomial hash function In this subsection, we use the following notations: • y (j) denotes the secret (y 0 ) from the jth of 3 + m secret sharing schemes being run in parallel to build a (3 + m)-element secret Y A 0 , which j ∈ {−2, −1, 0, . . . , m} indexes into. Set c := y (−2) , d := y (−1) and e := y (0) . • The differences c , d , e and y (j) each correspond to c, d, e and y (j) as the difference y 0 for the y 0 in the proof of Theorem C. where c, d and e are elements of a finite field F . Note that this is essentially the same family of functions as used for message hashing: the first element in the list has been shown as a subscript due to its role as a function-selection parameter, and a prime has been added to distinguish it. Theorem C.3. Denote y = (y (1) , . . . , y (m) ) and y = (y (1) , . . . , y (m) ). Let Ω = F 3 be a sample space with uniform probability. Let h C,D,E (y) = D + CE + m j=1 C j+1 y (j) define a family of hash functions with random variables (C, D, E) ∈ Ω as selection parameters. Let m = 0. Then, max t ,c ,d ,e ,y =0 Pr(t + t = h C+c ,D+d ,E+e (y + y ) | t = h C,D,E (y)) ≤ min( m+1 |F | , 1). Proof. Given a Shamir secret sharing scheme used to transmit all of C, D, E and y, an adversary who controls from 1 to k − 1 shares can modify C, D and E simultaneously by adding a chosen constant to each (as per Theorem C.1), with t and y assumed known and modifiable. Thus, t, C, D, E and y are replaced with t + t , C + c , D + d , E + e and y + y respectively. We are given that t = h C,D,E (y), which may be written t = D + CE + m j=1 C j+1 y (j) . (C.10) This determines a unique value D for every pair of values C, E, and since the a priori probability on Ω is uniform, the a posteriori marginal distribution over pairs (C, E) remains uniform, but the marginal distribution on D is potentially nonuniform by a similar argument as in the proof of Theorem B.1. To obtain the probability that t + t = h C+c ,D+d ,E+e (y + y ) holds, we subtract the given eq. (C.10) from it to obtain the equivalent equation t = d + c E + Ce + c e + m j=1 [(C + c ) j+1 (y (j) + y (j) ) − C j+1 y (j) ]. (C.11) When c = 0 and y = 0, eq. (C.11) reduces to t = d + Ce + m j=1 C j+1 y (j) , (C.12) which is a non-constant polynomial in C since y = 0 implies that for at least one value of j, y (j) = 0, but is independent of E. By the uniformity of C, each distinct root for the polynomial in C has probability 1 |F | . The number of distinct roots for a polynomial of degree m + 1 is at most min(m + 1, |F |), giving a maximum probability for eq. (C.12) holding of min( m+1 |F | , 1). When c = 0, eq. (C.11) may reduce either to a nonconstant or to a constant polynomial in C, but either way it retains the term that depends on E. For the former (where there is a dependency on C), for each value of E there may be up to min(m + 1, |F |) values of C that solve the polynomial, as for the case where c = 0, again giving a probability of holding of min( m+1 |F | , 1). For the latter (where there is no dependency on C), eq. (C.11) reduces to t = d + c E + c e , (C. 13) which, by the uniformity of E and that c = 0, has probability 1 |F | of holding. Given that m cannot be negative, min( m+1 |F | , 1) ≥ 1 |F | . Thus, considering all the cases above, the maximum probability of eq. (C.11) holding under the condition y = 0 is upper-bounded by min( m+1 |F | , 1). We exclude the case m = 0, since the max operator over an empty domain is undefined. Thus, we have that for m = 0, max t ,c ,d ,e ,y =0 Pr(t + t = h C+c ,D+d ,E+e (y + y )) ≤ min m + 1 |F | , 1 . (C.14) → S for this case. Figure 1 : 1(Modified fromFigure 1of[5]) The results of the one-time set-up: Steps 1 (PSRD generation and distribution) and 2 (Peer identity establishment) of the protocol. DSKE users Alice, Bob and Charlie share an ordered table of PSRD with each of the Security Hubs. Each Security Hub only knows its own part of the users' tables. For this illustration only, the PSRD is shown as bits. Figure legend: ENC: encryption operation, DEC: decryption operation, SHARE-GEN: share generation, SEC-REC: secret reconstruction, SEC-VAL: secret validation. (3.10), and we finally rewrite the joint probability over S A , S B , • The hash function families are predetermined: H = {h c,d : F m → F : (y 1 , . . . , y m ) → d + m j=1 c j y j } and H = {h c,d,e : F m → F : (y (1) , . . . , y (m) )→ d + ce + m j=1 c j+1 y (j) }. • The Shamir (n, k)-threshold secret sharing scheme uses f : F → F : x → c 0 + c 1 x + · · · + c k−1 x k−1 . • Themapping for assigning x i to the secret and Hubs {0, . . . , n} → F : i → x i is predetermined. • The bijective mapping g : {0, . . . , |F | − 1} → F is predetermined. This allows encoding of an integer as an element in the protocol. • A Hub sends each client two tables, e.g. H A i and H A follows from the equivalence with eq. (B.2) that max t * ,v * =vPr(t * = h C,D (v * ) | t = h C,D (v)) = min s |F | , 1 . . 2 . 2For any z ∈ F , Pr(D + X = z) = x∈F Pr(D + X = z|X = x) Pr(X = x) [by eq. Let Ω = F 2 be a sample space. Let random variables (D, X) ∈ Ω be mutually independent and D uniform. Then X + D and X are mutually independent.Proof. For each (x, d) ∈ Ω, Pr((X + D, X) = (x + d, x)) = Pr((D, X) = (d, x)) [since X = x] = Pr(D = d) Pr(X = x) [independence of D and X] = 1 |F | Pr(X = x) [uniformity of D] = Pr(X + D = x + d) Pr(X = x) [by lemma C.1].(C.2) By eq. (2.4), it follows that X + D and X are mutually independent. 1. Consider the family of functions defined by H = {h c,d,e : F m → F : (y (1) , . . . , y (m) ) → d + ce + ).(2.5) 2.3. Cryptographic primitives 2.3.1. Message validation (authentication). The following Theorem 2.1 is a consequence of the hashing scheme being a Carter-Wegman universal hash function family Note that there might be other ancillary information to be passed to T E i . See the discussion about T i in Section 3.2.) We further assume Z E i := 00 for compromised Hubs since this allows Eve to fully control the compromised Hubs by Y E i and T E i . For honest Hubs, Y E i and T E i can only take the value ⊥ (which is the only symbol in the allowed alphabet for those variables related to honest Hubs), indicating that Eve cannot control those honest Hubs. Honest Hubs do not leak the information about their shares and thus we set Y E i Table 1 : 1Symbols used in the protocol description in Appendix A. Note: Symbol capitalization may differ from the main text. Operations by each Hub P i for share distribution, related to Alice: (1) Hub P i receives the sequence from Alice, The Hub verifies that the tuple (P i , A, B) is allowable and was received via the routing from A, discarding the message if either check fails. The latter check is significant for robustness against depletion of H A i . (4) The Hub also verifies that the 3 + m + 2 elements starting at offset j A i in its copy of H A i are still unused, discarding the message if any of these elements have been used. Discarding the message at any time up to this point does not deplete elements in the table. (5) The Hub retrieves R A i of 3 + m elements at offset j A i and v A i of 2 elements at offset j A i + 3 + m from the table, marking them as used and erasing them from the table. (6) The Hub verifies the relationAlice sends the element sequence M A i t A i (A.9) to Hub P i , for i ∈ {1, . . . , n}. (ii) M A i t A i , (A.10) which may be corrupted or even lost, which it splits into its components M A i and t A i . (2) The Hub splits M A i into its components via M A i =: P i A B K A g(j A i ) Z A i o A . (A.11) (3) 12 ) 12discarding the message if this fails. The portion of the table addressed by the message cannot be reused on failure, due to the single-use constraint in the simplifying assumptions. (7) The Hub calculatesY A i := Z A i + R A i . (A.13) (iii) Operations by each Hub P i for share distribution, related to Bob: (1) The Hub chooses R B i and v B i from H B i using j B i in the same way that Alice did from H A i in Step (3.a.i) with the corresponding variables, in the process erasing elements from H B i . The Hub uses R B i as an encryption key; unlike Alice, it never treats this as a share. (2) The Hub calculates The Hub calculates the message tag t Operations by Bob for each Hub P i for share distribution, related to Alice: (1) Bob receives the sequence from Hub P i , which he splits into its components. (2) Bob then splits M B i into its components M B i =: P i A B K A g(j Bob verifies that the tuple (P i , A, B) is allowable and was received via the routing from P i , discarding the message if either check fails. Performing the latter check before proceeding further is significant for robustness. (4) Bob verifies that the 3+m+2 elements from offset j B i in his copy of H B i are unused, failing which the message is discarded. Discarding the message at any time up to this point does not deplete elements in the table. (5) Bob retrieves R B i of 3 + m elements at offset j B i and v B of 2 elements at offset j B i + 3 + m from the table, marking them as used and erasing them from the table. (6) Bob then verifies the relation discarding the message if this fails. Note that the portion of the table addressed by the message cannot be re-used on failure, due to the single-use constraint in the simplifying assumptions. (7) Bob then calculates Bob assembles all subsets of k messages that have (A, B, K A , o A ) in common. Associated with each message is a tupleZ B i := Y A i − R B i . (A.14) (3) The Hub generates the message M B i : M B i := P i A B K A g(j B i ) Z B i o A . (A.15) (4) B i as t B i := h v B i (M B i ) (A.16) (5) The Hub sends to Bob the element sequence M B i t B i . (A.17) (iv) M B i t B i , (A.18) B i ) Z B i o A . (A.19) (3) t B i = h v B i (M B i ), (A.20) Y A i := Z B + R B i . (A.21) (c) Secret reconstruction (i) Bob aborts the protocol if he has no remaining candidate tuples or non-identical remaining candidate tuples. Otherwise, he terminates the protocol with the secret S A . The DSKE protocol ends at this point. The tuple (A, B, K A , S A ), is known by both Alice and Bob.Remark 6. Only Bob knows whether the protocol completed successfully. Communicating the completion and the use of the secret can be managed through the tuple (A, B, K A ) and subsequent communication. Consider a family of polynomial functions, where c, d and v j are elements of a finite field F [11, Section 4.2]:). (A.23) Appendix B. Security of hashing for messages . We refer to[7] for the definitions of cryptographic algebra and compatibility of the pseudo-metric with cryptographic algebra. The pseudometric used in this paper is compatible with the underlying cryptographic algebra since we consider information-theoretic security so that if a distinguisher in the set of all distinguishers D is composed with another arbitrary system, it is still in D. To avoid further distraction, we omit this definition here. . PSRD can be delivered by physically shipping a secure data storage device or via QKD links. AcknowledgementWe would like to thank David Jao for helpful comments. This work is supported by MITACS, Natural Sciences and Engineering Research Council of Canada (NSERC), and particularly Innovative Solutions Canada. Post-quantum cryptography. D J Bernstein, T Lange, Nature. 5497671D. J. Bernstein and T. Lange, "Post-quantum cryptography," Nature, vol. 549, no. 7671, pp. 188-194, 2017. Secure quantum key distribution with realistic devices. F Xu, X Ma, Q Zhang, H.-K Lo, J.-W Pan, Rev. Mod. Phys. 92225002F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan, "Secure quantum key distribution with realistic devices," Rev. Mod. Phys., vol. 92, no. 2, p. 025002, 2020. NIST post-quantum cryptography candidate cracked. D Geer, ACM NewsD. Geer, "NIST post-quantum cryptography can- didate cracked," ACM News, January 24th 2023. [Online]. AI helps crack NIST-recommended postquantum encryption algorithm. K Townsend, SecurityWeek. K. Townsend, "AI helps crack NIST-recommended post- quantum encryption algorithm," SecurityWeek, February 21st 2023. [Online]. Available: https://www.securityweek.com/ ai-helps-crack-a-nist-recommended-post-quantum-encryption-algorithm/ Distributed symmetric key exchange: A scalable, quantum-proof key distribution system. H.-K Lo, M Montagna, M Von Willich, arXiv:2205.006152022H.-K. Lo, M. Montagna, and M. von Willich, "Distributed symmetric key exchange: A scalable, quantum-proof key distribution system," 2022, arXiv:2205.00615. Key generation for use in secured communication. H.-K Lo, M Montagna, U.S. Patent US11177950B2. H.-K. Lo and M. Montagna, "Key generation for use in secured communication," U.S. Patent US11177950B2, 2020. Constructive cryptography -A new paradigm for security definitions and proofs. U Maurer, Theory of Security and Applications -Joint Workshop. New York, UY, USAU. Maurer, "Constructive cryptography -A new paradigm for security definitions and proofs," in Theory of Security and Applications -Joint Workshop, TOSCA, New York, UY, USA, 2011, pp. 33-56. Universally composable security: A new paradigm for cryptographic protocols. R Canetti, Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS '01). the 42nd Symposium on Foundations of Computer Science (FOCS '01)Las Vegas; New YorkIEEER. Canetti, "Universally composable security: A new paradigm for cryptographic protocols," in Proceedings of the 42nd Symposium on Foundations of Computer Science (FOCS '01), Las Vegas. IEEE, New York, 2001, pp. 136-145. Universally composable security. J. ACM. 675--, "Universally composable security," J. ACM, vol. 67, no. 5(28), pp. 1-94, 2020. New hash functions and their use in authentication and set equality. M N Wegman, J L Carter, Journal of computer and system sciences. 223M. N. Wegman and J. L. Carter, "New hash functions and their use in authentication and set equality," Journal of computer and system sciences, vol. 22, no. 3, pp. 265-279, 1981. Polynomial evaluation and message authentication. D J Bernstein, D. J. Bernstein, "Polynomial evaluation and message authentication," 2007, unpublished manuscript. URL: http://cr.yp.to/papers.html#pema. How to share a secret. A Shamir, Communications of the ACM. 2211A. Shamir, "How to share a secret," Communications of the ACM, vol. 22, no. 11, pp. 612-613, 1979. Composable, unconditionally secure message authentication without any secret key. D Ostrev, 2019 IEEE International Symposium on Information Theory (ISIT). IEEED. Ostrev, "Composable, unconditionally secure message authentica- tion without any secret key," in 2019 IEEE International Symposium on Information Theory (ISIT). IEEE, 2019, pp. 622-626. Abstract cryptography. U Maurer, R Renner, The Second Symposium on Innovations in Computer Science, ICS. Beijing, ChinaU. Maurer and R. Renner, "Abstract cryptography," in The Second Symposium on Innovations in Computer Science, ICS, Beijing, China, 2011, pp. 1-21. Key recycling in authentication. C Portmann, IEEE Trans. Inf. Theory. 607C. Portmann, "Key recycling in authentication," IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4383-4396, 2014. Security in quantum cryptography. C Portmann, R Renner, Reviews of Modern Physics. 94225008C. Portmann and R. Renner, "Security in quantum cryptography," Reviews of Modern Physics, vol. 94, no. 2, p. 025008, 2022.

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{'abstract': 'The Distributed Symmetric Key Exchange (DSKE) protocol provides secure secret exchange (e.g., for key exchange) between two honest parties that need not have had prior contact, and use intermediaries with whom they each securely share confidential data. We show the composable security of the DSKE protocol in the constructive cryptography framework of Maurer. Specifically, we prove the security (correctness and confidentiality) and robustness of this protocol against any computationally unbounded adversary, who additionally may have fully compromised a bounded number of the intermediaries and can eavesdrop on all communication. As DSKE is highly scalable in a network setting with no distance limit, it is expected to be a cost-effective quantum-safe cryptographic solution to safeguarding the network security against the threat of quantum computers.', 'arxivid': '2304.13789', 'author': ["Jie Lin \nDepartment of Electrical and Computer Engineering\nUniversity of Toronto\n10 King's College RoadTorontoONCanada\n", 'Manfred Von Willich ', "Hoi-Kwong Lo \nDepartment of Electrical and Computer Engineering\nUniversity of Toronto\n10 King's College RoadTorontoONCanada\n\nDepartment of Physics\nUniversity of Toronto\n60 St George StTorontoONCanada\n\nDepartment of Physics\nUniversity of Hong Kong\nPokfulam RoadHong Kong\n", '\nQuantum Bridge Technologies Inc\n100 College StTorontoONCanada\n'], 'authoraffiliation': ["Department of Electrical and Computer Engineering\nUniversity of Toronto\n10 King's College RoadTorontoONCanada", "Department of Electrical and Computer Engineering\nUniversity of Toronto\n10 King's College RoadTorontoONCanada", 'Department of Physics\nUniversity of Toronto\n60 St George StTorontoONCanada', 'Department of Physics\nUniversity of Hong Kong\nPokfulam RoadHong Kong', 'Quantum Bridge Technologies Inc\n100 College StTorontoONCanada'], 'corpusid': 258352359, 'doi': '10.48550/arxiv.2304.13789', 'github_urls': [], 'n_tokens_mistral': 26918, 'n_tokens_neox': 24996, 'n_words': 17450, 'pdfsha': 'a611be9536540afd8ab01c36330f226a4094476a', 'pdfurls': ['https://export.arxiv.org/pdf/2304.13789v1.pdf'], 'title': ['Composable Security of Distributed Symmetric Key Exchange Protocol', 'Composable Security of Distributed Symmetric Key Exchange Protocol'], 'venue': []}

arxiv

HEDGING OF UNIT-LINKED LIFE INSURANCE CONTRACTS WITH UNOBSERVABLE MORTALITY HAZARD RATE VIA LOCAL RISK-MINIMIZATION 26 Jun 2014 Claudia Ceci ANDKatia Colaneri Alessandra Cretarola HEDGING OF UNIT-LINKED LIFE INSURANCE CONTRACTS WITH UNOBSERVABLE MORTALITY HAZARD RATE VIA LOCAL RISK-MINIMIZATION 26 Jun 2014arXiv:1406.6902v1 [q-fin.PM]Local risk-minimizationpartial informationunit-linked life insurance contractsminimal martingale measure 2010 Mathematical Subject Classification: 60G35, 60G46, 60J25, 91B30, 91G10 JEL Classification: C02, G11, G22 In this paper we investigate the local risk-minimization approach for a combined financial-insurance model where there are restrictions on the information available to the insurance company. In particular we assume that, at any time, the insurance company may observe the number of deaths from a specific portfolio of insured individuals but not the mortality hazard rate. We consider a financial market driven by a general semimartingale and we aim to hedge unit-linked life insurance contracts via the local risk-minimization approach under partial information. The Föllmer-Schweizer decomposition of the insurance claim and explicit formulas for the optimal strategy for pure endowment and term insurance contracts are provided in terms of the projection of the survival process on the information flow. Moreover, in a Markovian framework, we reduce to solve a filtering problem with point process observations. Introduction The paper addresses the problem of computing locally risk-minimizing hedging strategies for unit-linked life insurance contracts under partial information. In these contracts, insurance benefits depend on the price of a specific risky asset and payments are made according to the occurrence of some events related to the stochastic life-length of the policy-holder. In particular we consider a portfolio of l a insured individuals, having all the same age a. Hence, insurance contracts can be considered as contingent claims in an incomplete combined financial-insurance market model, defined on the product of two independent filtered probability spaces: the first one, denoted by (Ω 1 , F, P 1 ) endowed with a filtration F := {F t , t ≥ 0}, is used to model the financial market, while the second one, (Ω 2 , G, P 2 ) endowed with a filtration G := {G t , t ≥ 0}, describes the insurance portfolio. In general, incompleteness occurs when the number of assets traded on the market is lower than that of random sources, see e.g. [4,Chapter 8]. This is, for instance, the case of insurance claims which are linked to both financial markets and other sources of randomness that are stochastically independent of the financial markets. Since we consider an insurance market model which is independent of the underlying financial market, we apply the (local) risk-minimization approach for deriving hedging strategies that reduce the risk. This is a quadratic hedging method which keeps the replication constraint and looks for hedging strategies (in general not self-financing) with minimal cost. The concept of risk-minimizing hedging strategies was introduced in [19] in the financial framework only. At the beginning it was formulated assuming that the historical probability measure was a martingale measure. In this context, some results were obtained in the case of full information by [19] and [32], and under partial information by [31] using projection techniques, and more recently in [10] where the authors provide a suitable Galtchouk-Kunita-Watanabe decomposition of the contingent claim that works in a partial information framework. When the historical probability measure does not furnish the martingale measure, i.e. the asset prices dynamics are semimartingales, the theory of risk minimization does not hold and a weaker formulation, namely local risk-minimization, is required (see e.g. [18,32]). Concerning the local risk-minimization approach under partial information, there are less results in the literature, as far as we are aware. In particular, we mention: [18], where the optimal strategy is obtained via predictable projections and enlargements of filtrations that make the financial market complete; [9], which provides a suitable Föllmer-Schweizer decomposition of the contingent claim; [7], where the authors derive a full description of the optimal strategy under some conditions on the filtrations; an application to the case of defaultable markets in the sense of [18] can be found in [2]. The (local) risk-minimization approach has also been investigated under the so-called benchmark approach, a modeling framework that employs the numéraire portfolio as reference unit, instead of the riskless asset. More precisely, in [3] the authors consider the full information setting, the restricted information case is studied in [8], and finally in [16] the authors propose a different formulation of the problem for contingent claims which are not square-integrable. The theory of (local) risk-minimization has been recently extended to the insurance framework, where the market is affected by both mortality and catastrophic risks. In [27] and [33] the authors study the hedging problem of unit-linked life insurance contracts in the Black & Scholes and in the Lévy financial market model respectively, under full information on both the insurance market and the financial one. In particular, the authors of [33] discuss the same model analyzed in [30]. Moreover, in [8] the problem is solved for a general semimartingale financial model under partial information on the financial market using the benchmark approach. In all these papers lifetimes of the l a insured individuals are modeled as i.i.d. non-negative random variables with known hazard rate. The novelty of this paper consists on considering a combined financial-insurance model where there are restrictions on the information concerning the insurance market. As a matter of fact, we assume that, at time t, the insurance company may observe the total number of deaths N t occurred until t but not the mortality hazard rate, which depends on an unknown stochastic factor X. More precisely, we assume that lifetimes of each individual are conditionally independent, given the whole filtration generated by X, G X ∞ := σ{X u , u ≥ 0}, and with the same hazard rate process λ a (t, X t ). Denoting by G N the filtration generated by the process N which counts the number of deaths, then on the combined financial-insurance market the information flow available to the insurance company is formally described by the filtration H := F ⊗ G N ⊆ H := F ⊗ G. The financial market, on which the insurance company has full knowledge, consists on a riskless asset with (discounted) price identically equal to 1 and a risky asset whose (discounted) price S is represented by a semimartingale satisfying the so-called structure condition, see (2.1). Since the insurance company's decisions are based on the information flow H, we will look for admissible investment strategies ψ = (θ, η), where the process θ, which describes the amount of wealth invested in the risky asset, is supposed to be H-predictable, whereas the process η, providing the component invested in the riskless asset, is H-adapted. We consider two basic forms of insurance contracts, so-called pure endowment and term insurance. The policy-holder of a pure endowment contract receives the payoff ξ of a contingent claim at a fixed time T , if she/he is still alive at this time, while the term insurance contracts state that the sum insured is due immediately upon death before time T . Precisely, payments can occur at any time during [0, T ] and are assumed to be time dependent of the form g(t, S t ). In this case the generated obligations are not contingent claims at a fixed time T , however they can be transformed into general T -claims by deferring the payments to time T . Under suitable assumptions the payoff of the resulting insurance claim, denoted by G T , in both cases is a square-integrable H T -measurable random variable and since the traded asset S turns to be H-adapted, we can write the Föllmer-Schweizer decomposition of the random variable G T with respect to S and H. By applying the results of [32] and [7] we characterize the pseudo-optimal strategy as the integrand in the Föllmer-Schweizer decomposition and the optimal value process as the conditional expected value of the insurance claim G T with respect to the minimal martingale measure, given the information flow H t . In particular, we furnish an explicit formula for the pseudo-optimal strategy in terms of the G N -projection of the survival process, defined in our framework, as t p s : = P 2 T i > s + t | {T i > s} ∩ G X ∞ . In a Markovian setting, its G N -optional projection, t p s , can be written by means of the filter π, that provides the conditional law of the stochastic factor X given the observed history G N . As a consequence, the computation of the optimal strategy and the optimal value process lead to solve a filtering problem with point process observations. The literature concerning filtering problems is quite rich, and in particular we can distinguish three main subjects related to different dynamics of the observation process: continuous, counting and mixed type observations. Counting type observation, which is that considered also in this paper, has been analyzed by [11] in the framework of branching processes, and by [12,13] for pure jump state processes. An explicit representation of the filter is obtained in [14] by the Feynman-Kac formula using the linearization method introduced by [24]. We use this technique to achieve a similar result in our context. For completeness we indicate some references concerning continuous observation case, [23,25,26], and more recently mixed type observation has been studied in [5,6,20,21]. The paper is organized as follows. In Section 2 we describe the financial market model. Section 3 is devoted to the insurance market model. In Section 4 we introduce the combined financial-insurance model. The local risk-minimization is discussed in Section 5. The Föllmer-Schweizer decomposition and explicit formulas for the optimal strategy for both pure endowment and term insurance contracts are contained in Section 6 and 7, respectively. Finally, the computation of the survival process and some other technical results are gathered in the Appendix. The financial market Let (Ω 1 , F, P 1 ) be a probability space endowed with a filtration F := {F t , t ≥ 0} that satisfies the usual conditions of right-continuity and completeness; by convention, we set F = F ∞ and F ∞− = t≥0 F t , see e.g. [22]. We consider a simple financial market model where we can find one riskless asset with (discounted) price 1 and a risky asset whose (discounted) price is represented by an R-valued squareintegrable càdlàg (F, P 1 )-semimartingale S = {S t , t ≥ 0} that satisfies the so-called structure condition, i.e. S t = S 0 + M t + t 0 α u d M u , t ≥ 0, (2.1) where S 0 ∈ L 2 (F 0 , P 1 ) 1 , M = {M t , t ≥ 0} is an R-valued square-integrable (càdlàg) (F, P 1 )-martingale starting at null, M = { M, M t , t ≥ 0} denotes its F-predictable quadratic variation process and α = {α t , t ≥ 0} is an R-valued F-predictable process such that T * 0 α 2 s d M s < ∞ P 1 -a.s., for each T * ∈ (0, ∞). We assume throughout this paper: E P 1 T * 0 α 2 u d M u < ∞, ∀ T * ∈ (0, ∞). (2.2) Without further mention, all subsequently appearing quantities will be expressed in discounted units. Since it will play a key role in finding locally risk-minimizing strategies, for reader's convenience we recall the concept of minimal martingale measure P * , in short MMM, that is, the unique equivalent martingale measure for S (i.e. S is an (F, P * )-martingale) with the property that (F, P 1 )-martingales strongly orthogonal to M , are also (F, P * )-martingales. Definition 2.1. Suppose that S satisfies the structure condition. An equivalent martingale measure P * for S with square-integrable density dP * dP 1 is called minimal martingale measure (for S) if P * = P 1 on F 0 and if every square-integrable (F, P 1 )-martingale, strongly orthogonal to the P 1 -martingale part of S, is also an (F, P * )-martingale. If we assume that 1 − α t ∆M t > 0 P 1 − a.s., ∀ t ≥ 0 and E P 1 exp 1 2 T * 0 α 2 t d M c t + T * 0 α 2 t d M d t < ∞, ∀ T * ∈ (0, ∞),(2.3) where M c and M d denote the continuous and the discontinuous parts of the (F, P 1 )-martingale M respectively and α is given in (2.1), then by the Ansel-Stricker Theorem, see [1], there exists the MMM P * for S, which is defined thanks to the density process L = {L t , t ≥ 0} given by L t := dP * dP 1 Ft = E − α u dM u t , ∀ t ≥ 0, (2.4) where the notation E(Y ) refers to the Doléans-Dade exponential of an (F, P 1 )-semimartingale Y . We observe that condition (2.3) implies that the nonnegative (F, P 1 )-local martingale L is indeed a squareintegrable (F, P 1 )-martingale, see e.g. [29]. The insurance market The insurance market model is described by a filtered probability space (Ω 2 , G, P 2 , G), where G := {G t , t ≥ 0} is a complete and right-continuous filtration, G = G ∞ and G ∞− = t≥0 G t . We consider a set of individuals of the same age a. Then, we select at random a sample of l a people. We analyze the case where lifetimes of people in the sample are affected by an unknown stochastic factor, which is represented by a càdlàg stochastic process X = {X t , t ≥ 0}. Therefore, also inspired by [20], we make the following assumption. Assumption 3.1. The remaining lifetimes T 1 , ..., T la of each individual are conditionally independent, doubly-stochastic random times all having the same hazard rate process λ a := {λ a (t, X t ), t ≥ 0}. More precisely, we set G X ∞ := σ{X u , u ≥ 0}; then there is a measurable function λ a : [0, ∞) × R → (0, ∞), such that E P 2 T * 0 λ a (u, X u )du < ∞, ∀ T * ∈ (0, ∞) (3.1) and for all t 1 , t 2 , . . . , t la ≥ 0 P 2 T 1 > t 1 , . . . , T la > t la |G X ∞ = la i=1 exp − t i 0 λ a (s, X s )ds . (3.2) Here, E P 2 [·] denotes the expectation computed under the probability measure P 2 . Note that Assumption 3.1 implies that T i = T j P 2 − a.s. for all i = j. We set R i t := 1 {T i ≤t} , for every t ≥ 0 and define the filtration G R := {G R t , t ≥ 0} generated by the vector process (R 1 , . . . , R la ), i.e. G R t := σ{R i s , 0 ≤ s ≤ t, i = 1, . . . , l a }, ∀ t ≥ 0. (3.3) Remark 3.2. The process R i = { R i t , t ≥ 0}, given by R i t := R i t − t∧T i 0 λ a (s, X s )ds for every t ≥ 0, is a (G, P 2 )-martingale. Then, we denote by N = {N t , t ≥ 0} the process that counts the number of deaths. More precisely, for every t ≥ 0, N t := la i=1 1 {T i ≤t} counts the number of deaths in the time interval [0, t]. Now, let G N := {G N t , t ≥ 0} be the natural filtration of N , i.e. G N t = σ{N u , 0 ≤ u ≤ t}. It is worth stressing that in general G N ⊆ G R . Furthermore, we remark that all filtrations are supposed to satisfy the usual conditions. Note that, as a consequence of Remark 3.2, N has (G, P 2 )-predictable intensity Λ = {Λ t , t ≥ 0} with Λ t = (l a − N t − )λ a (t, X t − ), t ≥ 0. (3.4) Hence the process N = { N t , t ≥ 0}, given by N t = N t − t 0 Λ r dr for each t ≥ 0, turns to be a (G, P 2 )martingale. In particular this implies that for every t ≥ 0, E P 2 t 0 Λ r dr = E P 2 [N t ] ≤ l a . We introduce the so-called filter π(F ) = {π t (F ), t ≥ 0} defined as π t (F ) := E P 2 F (t, X t )|G N t , ∀ t ≥ 0, (3.5) for any measurable function F (t, x) such that E P 2 [|F (t, X t )|] < ∞, for every t ≥ 0. It is known that π(F ) is a probability measure-valued process with càdlàg trajectories (see e.g. [25]), which provides the conditional distribution of X, given the observation G N . We denote by π t − (F ) its left version. Then, the (G N , P 2 )-predictable intensity of N is {π t − (Λ), t ≥ 0}, where π t − (Λ) = (l a − N t − )π t − (λ a ). Indeed, by Jensen's inequality and (3.2) we get E P 2 T * 0 π t − (Λ)dt ≤ E P 2 T * 0 Λ t dt ≤ l a < ∞, ∀ T * ∈ (0, ∞),(3.6) and for any bounded (G N , P 2 )-predictable process ϑ = {ϑ t , t ≥ 0}, we have E P 2 T * 0 ϑ t dN t = E P 2 T * 0 ϑ t Λ t dt = E P 2 T * 0 ϑ t π t (Λ)dt = E P 2 T * 0 ϑ t π t − (Λ)dt , for every T * ∈ (0, ∞), where the second equality follows by conditioning with respect to the σ-algebra G N t . Therefore the process N * = {N * t , t ≥ 0}, given by N * t := N t − t 0 π r − (Λ)dr, is a (G N , P 2 )-martingale. We define the survival process t p s := P 2 T i > s + t | {T i > s} ∩ G X ∞ , ∀ t ≥ 0, s ≥ 0, (3.7) which is a generalization to our context of the well-known survival function for i.i.d. lifetimes, see e.g. [27,28,33] for more details. It is possible to show that t p s 1 {T i >s} = e − s+t s λa(u,Xu)du 1 {T i >s} , ∀ t ≥ 0, s ≥ 0. (3.8) See Lemma A.1 in Appendix A for the proof. The combined financial-insurance model We now introduce the market model generated by the economy and the insurance portfolio. In the sequel, we make the following hypothesis. Assumption 4.1. The insurance market is independent of the a priori given financial market. Note that Assumption 4.1 is not restrictive since it covers a wide class of realistic scenarios. Mathematically, Assumption 4.1 allows to consider the combined financial-insurance model on the following product probability space: (Ω, H, Q), where Ω := Ω 1 × Ω 2 , H := F ⊗ G and Q := P 1 × P 2 , endowed with the filtration H := F ⊗ G. We assume that at any time t, the insurance company has a complete information on the financial market and knows the number of policy-holders who are still alive but it cannot observe the hazard rate process λ a . Therefore, the available information flow for the insurance company is formally described by the filtration H := F ⊗ G N . Remark 4.2. We observe that for all random variables Z defined on the probability space (Ω, H, Q) there exist two random variables Z 1 and Z 2 defined on (Ω 1 , F, P 1 ) and (Ω 2 , G, P 2 ) respectively, such that Z(ω 1 , ω 2 ) = Z 1 (ω 1 )Z 2 (ω 2 ), ∀ (ω 1 , ω 2 ) ∈ Ω. Finally, by construction, the risky asset price process S turns out to be an (H, Q)-semimartingale and an ( H, Q)-semimartingale as well, whose decomposition is given by (2.1) where S 0 ∈ L 2 (H 0 , Q), M also turns out to be an R-valued square-integrable (càdlàg) (H, Q)-martingale and an ( H, Q)-martingale as well. We put Q * := P * × P 2 (4.1) in the sequel and note that, by construction, Q * is an equivalent martingale measure for S with respect to both of the filtrations H and H. In particular, it turns out to be the minimal martingale measure for the combined financial-insurance model, as shown in the following lemma. Proof. Firstly, we observe that an (H, Q) martingale O = {O t , t ≥ 0} can be written as O = O 1 O 2 , where O 1 = {O 1 t , t ≥ 0} is an (F, P 1 )-martingale and O 2 = {O 2 t , t ≥ 0} is a (G, P 2 )-martingale. Indeed, it is easy to check that every stochastic process living on the the probability space (Ω, H, Q) can be written as the product of two stochastic processes defined on (Ω 1 , F, P 1 ) and (Ω 2 , G, P 2 ) respectively, see also Remark 4.2. Moreover, for all 0 ≤ s < t < ∞ we get E Q [O t |H s ] = E Q O 1 t O 2 t |H s = E Q O 1 t |H s E Q O 2 t |H s = E P 1 O 1 t |F s E P 2 O 2 t |G s ,0 = O, M t = O 1 O 2 , M t = O 2 t O 1 , M t for every t ≥ 0, and this implies that O 1 is strongly orthogonal to M . Since P * is the MMM for S with respect to F, then O 1 is an (F, P * )-martingale, and consequently O is an (H, Q * )-martingale, since O 2 is not affected by the change of measure from Q to Q * . This proves that Q * is the MMM for S with respect to H. Finally, note that the same can be done with the filtration H instead of H. At time t = 0 the insurance company issues unit-linked life insurance contracts with maturity T for each of the l a individuals, which are linked to the risky asset price process S. We will consider two basic forms of insurance contracts: the pure endowment and the term insurance. With pure endowment contract, the sum insured is to be paid at time T if the insured is then still alive, while the term insurance states that the sum insured is due immediately upon death before time T , see e.g. [27] for further details. The obligation of the insurance company generated by the entire portfolio of pure endowment contracts is given by G T := ξ la i=1 1 {T i >T } = ξ(l a − N T ), (4.2) where ξ ∈ L 2 (F T , P 1 ) represents the payoff of a contingent claim with maturity T on the financial market. Note that G T ∈ L 2 ( H T , Q). For term insurance contracts the payment is a time dependent function of the form g(u, S u ), where g(u, x) is a measurable function of its arguments. A simple way of transforming the obligations into a contingent claim with maturity T is to assume that all payment are deferred to the term of the contract T . Thus the contingent claim for a portfolio of l a term insurance contracts is G T := la i=1 g(T i , S T i )1 {T i ≤T } = T 0 g(u, S u )dN u (4.3) where g(t, S t ) ∈ L 2 (F t , P 1 ), for every t ∈ [0, T ]. In the sequel we assume that E P 1 sup t∈[0,T ] g 2 (t, S t ) < ∞. (4.4) This is a classical assumption in the insurance framework (see e.g. [28,33]) and ensures that the random variable G T is square-integrable, as proved in the following Lemma. Proof. E Q G 2 T = E Q T 0 g(t, S t )dN t 2 = E Q   la i=1 g(T i , S T i )1 {T i ≤T } 2   ≤ l a E Q la i=1 g 2 (T i , S T i )1 {T i ≤T } = l a E Q T 0 g 2 (u, S u )dN u = l a E Q T 0 g 2 (t, S t ) − g 2 (t, S t − ) dN t + l a E Q T 0 g 2 (t, S t − )dN t = l a E Q T 0 g 2 (t, S t − )dN t , where the last equality follows by Assumption 4.1, so that N and S do not have common jump times. Therefore, by (4.4) we get that l a E Q T 0 g 2 (t, S t − )dN t = l a E Q T 0 g 2 (t, S t − )Λ t dt ≤ l a E P 1 sup t∈[0,T ] g 2 (t, S t ) E P 2 T 0 Λ t dt < ∞. We will refer to G T as the insurance contingent claim. The goal is to construct locally risk-minimizing hedging strategies for the given insurance contingent claims, in the case where the insurance company has at its disposal the information flow represented by H. Locally risk-minimizing strategies under partial information The first step is to introduce the class of all admissible hedging strategies, which have to be adapted to the information flow H. Definition 5.1. The space Θ( H) consists of all R-valued H-predictable processes θ = {θ t , t ∈ [0, T ]} satisfying the following integrability condition: E Q T 0 θ 2 u d M u + T 0 |θ u α u |d M u 2 < ∞, (5.1) where E Q [·] indicates the expectation with respect to Q. Notice that condition (5.1) implies that the process t 0 θ u dS u , t ∈ [0, T ] is well-defined and turns out to be square-integrable. Definition 5.2. An H-admissible strategy is a pair ψ = (θ, η), where θ ∈ Θ( H) and η = {η t , t ∈ [0, T ]} is an R-valued H-adapted process such that the value process V (ψ) = {V t (ψ), t ∈ [0, T ]} := θS + η is right-continuous and square-integrable, i.e. V t (ψ) ∈ L 2 ( H t , Q), for each t ∈ [0, T ]. Here, the processes θ and η represent respectively the units of risky asset and riskless asset held in the portfolio. For any H-admissible strategy ψ, we can define the associated cost process C(ψ) = {C t (ψ), t ∈ [0, T ]}, which is the R-valued H-adapted process given by C t (ψ) := V t (ψ) − t 0 θ u dS u , for every t ∈ [0, T ], and the H-risk process R H (ψ) = {R H t (ψ), t ∈ [0, T ]}, by setting R H t (ψ) := E Q (C T (ψ) − C t (ψ)) 2 H t for every t ∈ [0, T ]. We observe that an H-admissible strategy ψ is self-financing if and only if the associated cost process C(ψ) is constant and the H-risk process R H (ψ) is zero. Although H-admissible strategies with V T (ψ) = G T will in general not be self-financing, it turns out that good H-admissible strategies are still self-financing on average in the following sense. Definition 5. 3. An H-admissible strategy ψ is called mean-self-financing if the associated cost process C(ψ) is an ( H, Q)-martingale. The definitions below translate the concept of locally risk-minimizing strategy in the partial information setting. Definition 5.4. A small perturbation is an H-admissible strategy ∆ = (δ 1 , δ 2 ) such that δ 1 is bounded, the variation of δ 1 u α u d M u is bounded (uniformly in t and ω) and δ 1 T = δ 2 T = 0. For any subinterval of [0, T ], we define the small perturbation ∆| (s,t] := (δ 1 1 (s,t] , δ 2 1 [s,t) ). We consider a partition τ = {t i } i=0,1,2,...,k of the time interval [0, T ] such that 0 = t 0 < t 1 < . . . < t k = T.r τ H (ψ, ∆) := t i ,t i+1 ∈τ R H t i (ψ + ∆| (t i ,t i+1 ] ) + R H t i (ψ) E Q M t i+1 − M t i | H t i 1 (t i ,t i+1 ] . The strategy ψ is called H-locally risk-minimizing if lim inf n→∞ r τn H (ψ, ∆) ≥ 0, (Q ⊗ M ) − a.s. on Ω × [0, T ], for every small perturbation ∆ and every increasing sequence {τ n } n∈N of partitions of [0, T ] tending to identity. Following the idea of [32], we now introduce the concept of pseudo-optimal strategy in this framework. Note that, H-pseudo-optimal strategies are both easier to find and to characterize, as Proposition 5.7 will show in the following. In the one-dimensional case and under very general assumptions, locally risk-minimizing strategies and pseudo-optimal strategies coincide, see [32,Theorem 3.3]. More precisely, the equivalence holds if we assume that the predictable quadratic variation M of the martingale M , is Q-a.s. strictly increasing and the finite variation part of S is Q-a.s. continuous. The key result for finding H-pseudo-optimal strategies is represented by the Föllmer-Schweizer decomposition. Since the insurance contingent claim G T belongs to L 2 ( H T , Q), it admits the Föllmer-Schweizer decomposition with respect to S and H, i.e. G T = G 0 + T 0 θ H t dS t + K T Q − a.s.,(5.2) where G 0 ∈ L 2 ( H 0 , Q), θ H ∈ Θ( H) and K = {K t , t ∈ [0, T ]} is a square-integrable ( H, Q)-martingale with K 0 = 0 strongly orthogonal to the Q-martingale part of S. Proposition 5.7. An insurance contingent claim G T ∈ L 2 ( H T , Q) admits a unique H-pseudo-optimal strategy ψ * = (θ * , η * ) with V T (ψ * ) = G T Q − a.s.. In terms of the decomposition (5.2), the strategy ψ * is explicitly given by θ * t = θ H t , ∀ t ∈ [0, T ], with minimal cost C t (ψ * ) = G 0 + K t , ∀ t ∈ [0, T ]; its value process is V t (ψ * ) = E Q G T − T t θ H u dS u H t = G 0 + t 0 θ H u dS u + K t , ∀ t ∈ [0, T ], so that η * t = V t (ψ * ) − θ * t S t , for every t ∈ [0, T ]. Proof. The proof uses the same arguments of [32,Proposition 3.4]. The following result provides an operative way to compute the optimal strategy by switching to the minimal martingale measure Q * . Proposition 5.8. Let G T ∈ L 2 ( H T , Q) be an insurance contingent claim and ψ * = (θ * , η * ) be the associated H-pseudo-optimal strategy. Then, the optimal value process V (ψ * ) = {V t (ψ * ), t ∈ [0, T ]} is given by V t (ψ * ) = E Q * G T H t , ∀ t ∈ [0, T ],(5.3) where E Q * · H t denotes the conditional expectation with respect to H t computed under Q * ; moreover, θ * is explicitly given by θ * t = d V (ψ * ), S t d S t , ∀ t ∈ [0, T ],(5.4) where the sharp brackets are computed between the martingale parts of the processes V (ψ * ) and S, under Q and with respect to the filtration H. Proof. The proof follows by that of Proposition 4.2 in [7]. Application 1: pure endowment contracts In the sequel we apply the results of Section 5 to compute the H-pseudo-optimal strategy ψ * for a pure endowment contract, whose payoff G T is given in (4.2). More precisely, we write the Föllmer-Schweizer decomposition of the random variable G T using (5.3) and apply Proposition 5.7 to identify the optimal strategy. We observe that the payoff of a contingent claim ξ ∈ L 2 (F T , P 1 ) admits the Föllmer-Schweizer decomposition with respect to S and F, i.e. ξ = U 0 + T 0 β t dS t + A T P 1 − a.s.,(6.1) where U 0 = E P 1 [ξ|F 0 ] ∈ L 2 (F 0 , P 1 ), β ∈ Θ(F) 2 and A = {A t , t ∈ [0, T ]} is a square-integrable (F, P 1 )martingale with A 0 = 0 strongly orthogonal to the P 1 -martingale part of S. 2 The space Θ(F) consists of all R-valued F-predictable processes δ = {δt, t ∈ [0, T ]} satisfying the integrability condition E P 1 T 0 δ 2 u d M u + T 0 |δuαu|d M u 2 < ∞. Thanks to Assumption 4.1, if we take the conditional expectation with respect to H t under the MMM Q * in (4.2), for every t ∈ [0, T ] we get V t (ψ * ) = E Q * G T H t = E Q * ξ H t E Q * l a − N T H t = E P * [ξ|F t ] E P 2 l a − N T |G N t . (6.2) For the sake of simplicity we define the processes B = {B t , t ∈ [0, T ]} and U = {U t , t ∈ [0, T ]}, given by B t =: E P 2 [l a − N T | G N t , (6.3) U t := E P * [ξ|F t ] , for every t ∈ [0, T ]. Note that (6.1) implies that U t = E P * [ξ|F t ] = E P * U 0 + T 0 β u dS u + A T F t = U 0 + t 0 β u dS u + A t , ∀ t ∈ [0, T ],(6.4) where the last equality holds since β u dS u , t ∈ [0, T ] is an (F, P * )-martingale and A turns out to be an (F, P * )-martingale by definition of MMM. Proposition 6.1 (The H-pseudo-optimal strategy). The Föllmer-Schweizer decomposition of the insurance contingent claim G T = ξ(l a − N T ) is given by G T = G 0 + T 0 B r − β r dS r + K T Q − a.s., where G 0 = E P 1 [ξ|F 0 ] E P 2 [l a − N T ], K t = t 0 B s − dA s + t 0 U s − Γ s (dN s − π s − (Λ)ds) , ∀ t ∈ [0, T ],(6. 5) B is given by (6.3), β is the integrand with respect to S in the Föllmer-Schweizer decomposition of ξ, see (6.1), and Γ is a suitable (G N , P 2 )-predictable process, see (6.7) below. Then, the H-pseudo-optimal strategy ψ * = (θ * , η * ) is given by θ * t = B t − β t , ∀ t ∈ [0, T ], η * t = V t (ψ * ) − B t β t S t , ∀ t ∈ [0, T ]. and the optimal value process V (ψ * ) is given by V t (ψ * ) = G 0 + t 0 B r − β r dS r + K t , ∀ t ∈ [0, T ].B t = B 0 + t 0 Γ u (dN u − π u − (Λ)du) (6.7) for every t ∈ [0, T ], where B 0 = E P 2 [l a − N T ] = l a − E P 2 T 0 Λ s ds . Therefore, taking (6.2), (6.4) and (6.7) into account, by Itô's product rule we get In the sequel we provide a characterization of the processes β and B. dV t (ψ * ) = dE Q * G T H t = B t − dU t + U t − dB t + d B, U t + d   s≤t ∆B s ∆U s   = B t − β t dS t + B t − dA t + U t − Γ t (dN t − π t − (Λ)dt) ,(6.E Q T 0 B 2 t − β 2 t d M t + T 0 B t − β t α t d M t 2 ≤ l 2 a E Q T 0 β 2 t d M t + T 0 β t α t d M t 2 < ∞, since 0 ≤ B t ≤ l a , An analogous result to that of Proposition 5.8 can be used to characterize the process β, which gives the pseudo-optimal strategy (with respect to F) for the contingent claim ξ in the underlying financial market, see Section 2. Precisely, the process β appearing in the Föllmer-Schweizer decomposition (6.1) of ξ can be explicitly characterized thanks to the MMM P * , as stated below. Proposition 6.2. Let ξ ∈ L 2 (F T , P 1 ) be a contingent claim and φ * = (δ * , ζ * ) be the associated pseudooptimal strategy (with respect to F). Then, the optimal value processV (φ * ) = {V t (φ * ), t ∈ [0, T ]} is given byV t (φ * ) = E P * [ξ|F t ] , ∀ t ∈ [0, T ], (6.9) where E P * [·|F t ] denotes the conditional expectation with respect to F t computed under P * . Finally β is equal to δ * , which is explicitly given by (1) in the Black & Scholes financial market, the hedging strategy is given at any time t by β t = ∂ϕ ∂s (t, S t ), where ϕ solves the well-known evaluation formula and ϕ(T, S T ) = ξ. In particular explicit expressions can be found for Put and Call options; (2) in the case of incomplete Lévy driven market models, the strategy β has been computed in [33] (see Theorem 2, equation (34)); (3) computations of the optimal strategy β for general semimartingale driven market models can be found in [15]. β t = δ * t = d V (φ * ), S t d S t , ∀ t ∈ [0, T ],(6. The next Lemma gives a representation of the process B. Lemma 6.4. For every t ∈ [0, T ] we define T −t p t := E P 2 exp − T t λ a (s, X s )ds G N t . Then the process B in (6.3) is given by B t := E P 2 [l a − N T | G N t = (l a − N t ) T −t p t ∀ t ∈ [0, T ]. Proof. We recall that for every t ∈ [0, T ], B t =: E P 2 [l a − N T | G N t . Then, taking (3.3) into account, we get E P 2 [l a − N T | G N t = E P 2 la i=1 1 {T i >T } G N t = E P 2 la i=1 E P 2 1 {T i >T } |G R t ∨ G X ∞ G N t = E P 2 la i=1 1 {T i >t} E P 2 1 {T i >T } G R i t ∨ G X ∞ G N t = E P 2 la i=1 1 {T i >t} E P 2 1 {T i >T } |{T i > t} ∩ G X ∞ G N t . (6.11) The last equality follows by (B.1) in Appendix B. By the definition of the survival process (3.7) we get that (6.11) becomes E P 2 la i=1 1 {T i >t}T −t p t G N t = E P 2 la i=1 1 {T i >t} exp − T t λ a (s, X s )ds G N t , (6.12) where the equality follows by (3.8). Note that the first term in (6.12) is also equal to E P 2 [(l a − N t ) T −t p t | G N t = (l a − N t )E P 2 [ T −t p t | G N t =: (l a − N t ) T −t p t , and the second one is equal to E P 2 (l a − N t ) exp − T t λ a (s, X s )ds G N t = (l a − N t )E P 2 exp − T t λ a (s, X s )ds G N t , for every t ∈ [0, T ]. We refer to u p s as the G N -projection of the survival process { u p s , s ≥ 0, u ≥ 0} and we have obtained that for every t ∈ [0, T ], T −t p t = E P 2 exp − T t λ a (s, X s )ds G N t . Concluding we have shown that B t := E P 2 [l a − N T | G N t = (l a − N t ) T −t p t ∀ t ∈ [0, T ]. In the sequel we evaluate T −t p t in a Markovian setting. 6.1. The G N -projection of the survival process in a Markovian setting. On the probability space (Ω 2 , G, P 2 ) we define the process Y = {Y t , t ≥ 0}, whose dynamics is given by dY t = −λ a (t, X t )Y t dt, (6.13) with Y 0 = 1, P 2 -a.s., (or equivalently Y t = exp{− t 0 λ a (s, X s )ds}, for every t ∈ [0, T ]). Then, the G N -projection of the survival process has the following relationship with the process Y : u p s = E P 2 Y s+u Y s G N s , u ≥ 0, s ≥ 0, and in particular T −t p t = E P 2 Y T Y t G N t , for every t ∈ [0, T ]. Assumption 6.5. We assume that the pair (X, Y ) is a (G, P 2 )-Markov process. We denote by L X,Y the Markov generator of the pair (X, Y ). Then, by [17,Theorem 4.1.7] the process f (t, X t , Y t ) − t 0 L X,Y f (u, X u , Y u )du, t ∈ [0, T ] is a (G, P 2 )-martingale for every function f (t, x, y) in the domain of the operator L X,Y , denoted by D(L X,Y ) . Thanks to the Markovianity of the pair (X, Y ), there is a measurable function h(t, x, y) such that h(t, X t , Y t ) = E P 2 Y T Y t G t , ∀ t ∈ [0, T ], and then T −tpt = E P 2 h(t, X t , Y t )|G N t , ∀ t ∈ [0, T ]. (6.14) Since Y is G-adapted, there is a function γ(t, x, y) such that γ(t, X t , Y t ) = E P 2 [Y T |G t ] , ∀ t ∈ [0, T ] (6.15) and h(t, X t , Y t ) = Y −1 t γ(t, X t , Y t ), for every t ∈ [0, T ]. Note that the process {γ(t, X t , Y t ), t ∈ [0, T ]} given in (6.15) is a (G, P 2 )-martingale; then we can provide a characterization of the function γ(t, x, y) in terms of the solution of a suitable problem with final condition as the following result will show. Proposition 6.6. Let γ(t, x, y) ∈ D(L X,Y ) such that L X,Y γ(t, x, y) = 0 γ(T, x, y) = y. (6.16) Then γ(t, X t , Y t ) = γ(t, X t , Y t ), for every t ∈ [0, T ], with γ(t, x, y) such that (6.15) is fulfilled. Proof. Let γ(t, x, y) ∈ D(L X,Y ) be the solution of (6.16). By applying Itô's formula to the process γ(t, X t , Y t ), we get that the process γ(t, X t , Y t ) − t 0 L X,Y γ(r, X r , Y r )dr, t ∈ [0, T ] is a (G, P 2 )- martingale. Finally, since γ(T, X T , Y T ) = Y T , by the martingale property we get the claimed result. Now, by (6.14), we need to compute the conditional expectation of h(t, X t , S t ) given the available information G N t , for every t ∈ [0, T ]. This leads to solve a filtering problem where N is the observation. For any measurable function f (t, x, y) such that E P 2 [|f (t, X t , Y t )|] < ∞, for every t ∈ [0, T ], we consider the filter π(f ) = {π t (f ), t ≥ 0}, which is defined by π t (f ) := E P 2 f (t, X t , Y t )|G N t , (6.17) for each t ∈ [0, T ]. We observe that, for any function f (t, x, y) which does not depend on the variable y, the filter defined in (6.17) coincides with that given in (3.5). It is known that, in the case of counting observation given by the process N , the filter solves the following Kushner-Stratonovich equation (see e.g. [5]): π t (f ) = π 0 (f )+ t 0 π s (L X,Y f )ds+ t 0 (π s − (Λ)) + π s − (Λf ) − π s − (Λ)π s − (f ) + π s − (Lf ) (dN s − π s − (Λ)ds) ,(6. 18) for every f ∈ D(L X,Y ) and every t ≥ 0, where z + := 1 z 1 {z>0} andL is the operator that takes into account possible jump times between X and N . We observe that the G N -projection of the survival process can be written in terms of the filter as T −t p t = π t (h) ∀ t ∈ [0, T ]. (6.19) Then we get the following result which holds in the Markovian case. Proposition 6.7. Under Assumption 6.5, the H-pseudo-optimal strategy ψ * = (θ * , η * ) is given by θ * t = β t (l a − N t − )π t − (h), ∀ t ∈ [0, T ] η * t = V t (ψ * ) − β t (l a − N t )π t (h)S t , ∀ t ∈ [0, T ], where β is given in (6.10), π indicates the filter defined in (6.17), h(t, x, y) = y −1 γ(t, x, y), γ(t, x, y) is the solution of the problem (6.16) and V (ψ * ) is the optimal value process given by (6.6) in Proposition 6.1, with B = (l a − N )π(h). As mentioned in the Introduction, filtering problems with counting observations have been widely investigated in the literature. In [14], for example, an explicit representation of the filter is obtained by the Feynman-Kac formula using the linearization method introduced by [24]. In the sequel we apply this procedure to our framework. To obtain a similar representation for our model we write down the Kushner-Stratonovich equation solved by the filter (6.18), between two consecutive jump times of the counting process N . We denote by {τ i } i=1,...,la the ordered sequence of the jump times of N ; then for t ∈ [τ n , τ n+1 ), equation (6.18) becomes π t (f ) = π τn (f ) + t τn (π s (L 0 f ) − π s (Λf ) − π s (Λ)π s (f )) ds,(6.20) where we indicate by L 0 the operator given by L 0 f = L X,Y f −Lf for every f ∈ D(L X,Y ), and in particular, if common jumps times between X and N are not allowed, then L 0 coincides with the Markov generator of the pair (X, Y ), i.e. L 0 = L X,Y . At any jump time of N , say τ n , also the process π exhibits a jump, and its jump-size is given by π τn (f ) − π τ − n (f ) = (π τ − n (Λ)) + π τ − n (Λf ) − π τ − n (Λ)π τ − n (f ) + π τ − n (Lf ) .(6.21) Note that π τn (f ) is completely determined by the knowledge of π t (f ) for every t in the interval [τ n−1 , τ n ). Indeed, for any function f , π τ − n (f ) = lim t→τ − n π t (f ). Hence, due to the recursive structure of equation (6.18), to our aim we only need to solve equation (6.20) for every t ∈ [τ n , τ n+1 ) and n = 0, . . . , l a − 1. Now, we denote by ρ n = {ρ n t , t ∈ [0, T ]} a measure-valued process that solves the following equation ρ n t (f ) = π τn (f ) + t τn [ρ n s (L 0 f ) − ρ n s (Λf )] ds, t ∈ [τ n , τ n+1 ), n = 0, . . . , l a − 1 (6.22) and such that ρ n t (1) > 0, for every t ∈ [0, T ]. It is not difficult to verify that the process ρ n t (f ) ρ n t (1) , t ∈ [τ n , τ n+1 ) solves equation (6.20). Therefore, if uniqueness of the solution to the Kushner-Stratonovich equation holds, we can characterize the filter via the linear equation (6.22). By Theorem 3.3 in [25] we get the following result. Proposition 6.8. Assume that the Martingale Problem for the operator L X,Y is well posed and that there exists a domain D 0 for L X,Y such that for every function f (t, x, y) ∈ D 0 , L X,Y f ∈ C b ([0, T ] × R × R + ). Then weak uniqueness for the Kushner-Stratonovich equation (6.18) holds. Finally, we assume that the Martingale Problem for the operator L 0 admits a solution, then, by the Feynman-Kac formula we obtain the following representation for the process ρ n (f ). Note that the assumption is clearly fulfilled when X and N do not have common jump times, since in that case L 0 = L X,Y . Proposition 6.9. For any (x, y) ∈ R × R + assume that the Martingale Problem for the operator L 0 with initial data (x, y) admits a solution P (x,y) in D [0,+∞) (R × R + ) 3 . Then a solution of (6.22), such that ρ n t (1) > 0 for any t ∈ (τ n , τ n+1 ), is given by ρ n t (f ) = R 2 Ψ n t (τ n , x, y)(f )π τn (dx, dy), ∀ t ∈ [0, T ], with Ψ n (l a − n)λ a (r, Z 1 r )dr , where (Z 1 , Z 2 ) denotes the canonical process on D [0,+∞) (R × R + ). Proof. First observe that Λ t = (l a − n)λ a (t, X t ) for t ∈ (τ n , τ n+1 ). Then the thesis follows by applying Proposition 3.2 in [14]. Application 2: term insurance contracts In this Section we aim to compute the H-pseudo-optimal strategy ψ * for a term insurance contract, whose payoff, denoted by G T , is given in (4.3). Similarly to the case of pure endowment contracts, the optimal value process V (ψ * ) is characterized by relationship (5.3): V t (ψ * ) = E Q * G T H t = E Q * T 0 g(u, S u )dN u H t = E Q * T 0 [g(u, S u ) − g(u, S u − )]dN u | H t + E Q * T 0 g(u, S u − )dN u H t = t 0 g(u, S u − )dN u + E Q * T t g(u, S u − )dN u H t , for every t ∈ [0, T ], where the last equality is due to the fact that S and N do not have common jump times. Since the process {g(t, S t − ), t ∈ (0, T ]} is H-predictable we get V t (ψ * ) = t 0 g(u, S u − )dN u + E Q * T t g(u, S u − )Λ u du H t = t 0 g(u, S u − )dN u + T t E Q * g(u, S u )Λ u | H t du = t 0 g(u, S u − )dN u + T t E P * [g(u, S u )|F t ] E P 2 Λ u |G N t du = t 0 g(u, S u − )dN u + T t V (u, t)B t (u)du, (7.1) for every t ∈ [0, T ], where for every u ∈ [0, T ] the processes B(u) = {B t (u), t ∈ [0, u]} and V (u) = {V (u, t), t ∈ [0, u]} are respectively given by B t (u) := E P 2 Λ u |G N t , ∀ t ∈ [0, u] (7.2) V (u, t) := E P * [g(u, S u )|F t ] ∀ t ∈ [0, u]. (7.3) We recall that g(u, S u ) ∈ L 2 (F u , P 1 ); then, it admits the Föllmer-Schweizer decomposition with respect to S which is given by g(u, S u ) = V 0 + u 0 β r (u)dS r + A 1 u (u) P 1 − a.s., ∀ u ∈ [0, T ],(7.4) where V 0 := E P 1 [g(u, S u )|F 0 ] ∈ L 2 (F 0 , P 1 ), β(u) := {β t (u), t ∈ [0, u]} ∈ Θ(F) and A 1 (u) := {A 1 t (u), t ∈ [0, u]} is a square-integrable (F, P 1 )-martingale with A 1 0 (u) = 0, strongly orthogonal to M . Since S is an (F, P * )-martingale, thanks to (7.4), we get that for every 0 ≤ t ≤ u ≤ T V (u, t) = E P * [g(u, S u )|F t ] = E P * V 0 + u 0 β r (u)dS r + A 1 u (u) F t = V 0 + t 0 β r (u)dS r + A 1 t (u), (7.5) where A 1 (u) turns out to be an (F, P * )-martingale by definition of MMM. Assumption 7.1. We assume that there exists a process γ(u) = {γ t (u), 0 ≤ t ≤ u} for every u ∈ [0, T ] such that A 1 t (u) = t 0 γ r (u)dA 1 r , t ∈ [0, u], and that E P 1 T 0 γ 2 t (u)d A 1 t < ∞, where A 1 is a square-integrable (F, P 1 )-martingale, strongly ortho- gonal to M . This assumption is rather general as it is satisfied whenever a martingale representation theorem for (F, P 1 )-martingales holds, see e.g. [33] for Lévy driven market models. Proposition 7.2 (The H-pseudo-optimal strategy). The Föllmer-Schweizer decomposition of the insurance contingent claim G T = T 0 g(u, S u )dN u is given by G T = G 0 + T 0 T t B t − (u)β t (u)dudS t + K 1 T Q − a.s., where K 1 t = t 0 g(r, S r − ) + T r V (u, r − )Γ r (u)du (dN r − π r − (Λ)dr) + t 0 T r B r − (u)γ r (u)du dA 1 r , (7.6) for every t ∈ [0, T ], G 0 = E Q G T | H 0 , B(u) is given by (7.2), β(u) is the integrand with respect to S in the Föllmer-Schweizer decomposition of g(u, S u ), see (7.4), and Γ(u) is a suitable (G N , P 2 )-predictable process, see (7.7) below. Then, the H-pseudo-optimal strategy ψ * = (θ * , η * ) is given by θ * t = T t B t − (u)β t (u)du, ∀ t ∈ [0, T ], η * t = V t (ψ * ) − T t B t − (u)β t (u)du S t , ∀ t ∈ [0, T ], and the optimal value process V (ψ * ) is given by V t (ψ * ) = G 0 + t 0 θ * r dS r + K 1 t , ∀ t ∈ [0, T ]. Proof. Note that, analogously to the previous case of pure endowment contracts, the process B(u) is a square-integrable (G N , P 2 )-martingale, and therefore, thanks to the Martingale Representation Theorem, for every u ∈ [0, T ] there exists a (G N , P 2 )-predictable process Γ(u) = {Γ t (u), t ∈ [0, u]} such that E P 2 u 0 Γ r (u) 2 π r (Λ)dr < ∞ for every u ∈ [0, T ] and B t (u) = B 0 (u) + t 0 Γ r (u) (dN r − π r − (Λ)dr) , 0 ≤ t ≤ u ≤ T,(7.7) where B 0 (u) := E P 2 [Λ u ]. We apply the Itô product rule in equation (7.1) and obtain dV t (ψ * ) = g(t, S t − )dN t − V (t, t)B t − (t)dt + T t V (u, t − )dB t (u) + B t − (u)dV (u, t) du = g(t, S t − )dN t − V (t, t)B t − (t)dt + T t V (u, t − )Γ t (u) (dN t − π t − (Λ)dt) du + T t {B t − (u)β t (u)dS t } du + T t B t − (u)γ t (u)dA 1 t du. We recall that V (t, t) = g(t, S t ) and B t (t) = π t (Λ); then, more precisely, for every t ∈ [0, T ], in integral form we get V t (ψ * ) =G 0 + t 0 g(r, S r − )(dN r − π r − (Λ)dr) + T t t 0 V (u, r − )Γ r (u)(dN r − π r − (Λ)dr) du + T t t 0 B r − (u)β r (u)dS r du + T t t 0 B r − (u)γ r (u)dA 1 r du. By Fubini's Theorem we obtain V t (ψ * ) =G 0 + t 0 g(r, S r − )(dN r − π r − (Λ)dr) + t 0 T r V (u, r − )Γ r (u)du (dN r − π r − (Λ)dr) + t 0 T r B r − (u)β r (u)du dS r + t 0 T r B r − (u)γ r (u)du dA 1 r , ∀ t ∈ [0, T ]. Note that, the process T t B t − (u)β t (u)du, t ∈ [0, T ] is ( H, Q * )-predictable; then to prove that T t B t − (u)β t (u)du, t ∈ [0, T ] belongs to Θ( H), we consider the following estimation Proof. Similarly to the pure endowment case, for every t ∈ [0, T ] we get: E Q T 0 T r B 2 r − (u)β 2 r (u)du d M r + T 0 T r B r − (u)β r (u)du α r d M r 2 = E Q T 0 u 0 B 2 r − (u)β 2 r (u)d M r du + T 0 u 0 B r − (u)β r (u)α r d M r du 2 ≤ E Q T 0 sup 0≤r≤u B 2 r (u) u 0 β 2 r (u)d M r du + E Q T 0 sup 0≤r≤u B r (u) u 0 β r (u)α r d M r du 2 ≤ T 0 E P 2 sup 0≤r≤u B 2 r (u) E P 1 u 0 β 2 r (u)d M r du + T 0 E P 2 ( sup 0≤r≤u |B r (u)|) 2 E P 1 ( u 0 β r (u)α r d M r ) 2 du < ∞, since E P 2 sup 0≤r≤u B 2 r (u) ≤ E P 2 sup 0≤r≤u |B r (u)| 2 ≤ cE P 2 [ B(u) u ] = cE P 2 u 0 Γ 2 r (u)π r (Λ)dr < ∞,B t (u) = E P 2 (l a − N u )λ a (u, X u )|G N t = E P 2 la i=1 1 {T i >u} λ a (u, X u )|G N t = E P 2 la i=1 1 {T i >t} E P 2 1 {T i >u} G R i t ∨ G X ∞ λ a (u, X u )|G N t = E P 2 la i=1 1 {T i >t} λ a (u, X u )e − u t λa(r,Xr)dr |G N t = (l a − N t )E P 2 λ a (u, X u )e − uB t (u) = (l a − N t )E P 2 λ a (u, X u )e − u t λa(r,Xr)dr |G N t = (l a − N t )E P 2 λ a (u, X u ) Y u Y t G N t = (l a − N t )E P 2 1 Y t E P 2 [λ a (u, X u )Y u |G t ] |G N t , for every 0 ≤ t ≤ u ≤ T , where Y is the process defined in (6.13). We make Assumption 6.5. Then, thanks to the Markovianity of the pair (X, Y ), there is a measurable function H(t, x, y) such that H(t, X t , Y t ) = E P 2 λ a (u, X u ) Y u Y t G t , ∀ t ∈ [0, u]. Since Y is G-adapted, there is a function k(t, x, y) such that k(t, X t , Y t ) = E P 2 [λ a (u, X u )Y u |G t ] , ∀ t ∈ [0, u] (7.9) and H(t, X t , Y t ) = Y −1 t k(t, X t , Y t ), for every t ∈ [0, u]. Since the process {k(t, X t , Y t ), t ∈ [0, T ]} given in (7.9) is a (G, P 2 )-martingale, analogously to the pure endowment case, we can characterize the function k(t, x, y) in terms of the solution of a suitable problem with final condition as the following result will show. Proposition 7.4. Let k(t, x, y) ∈ D(L X,Y ) such that L X,Y k(t, x, y) = 0 k(u, x, y) = λ(u, x)y. (7.10) Then k(t, X t , Y t ) = k(t, X t , Y t ), for every t ∈ [0, u], with k(t, x, y) such that (7.9) is fulfilled. Proof. The proof follows by that of Lemma 7.4. Note that, due to the final condition of Problem (7.10), it is clear that the process {k(t, X t , Y t ), t ∈ [0, u]} depends on u. In terms of the filter, the process B(u) can be written as B t (u) = (l a − N t )π t (H) for every 0 ≤ t ≤ u ≤ T , where the process π(H) depends on u. Summarizing, the following result furnishes the optimal hedging strategy for a term structure contract in the Markovian case. Proposition 7.5. Under Assumption 6.5, the H-pseudo-optimal strategy ψ * = (θ * , η * ) is given by θ * t = T t (l a − N t − )π t − (H)β t (u)du, ∀ t ∈ [0, T ], and η * t = V t (ψ * ) − T t (l a − N t − )π t − (H)β t (u)du S t , where π indicates the filter defined in (6.17), H(t, x, y) = y −1 k(t, x, y), k(t, x, y) is the solution of problem (7.10) and V (ψ * ) is the optimal value process given by (6.6) in Proposition 6.1, with B(u) = (l a − N )π(H). Lemma B.2. Let K := {K t , t ∈ [0, T ]} be the process defined in (6.5), i.e. K t = t 0 B s − dA s + t 0 U s − Γ s (dN s − π s − (Λ)ds) , ∀ t ∈ [0, T ], where B is given by (6.3) and Γ is defined in (6.7). Then K is a square-integrable ( H, Q)-martingale. Proof. By the boundedness of B, we get that the process Therefore thanks to the Burkholder-Davis-Gundy inequality there exists a constant c > 0 such that E Q sup 0≤t≤T |U t | =E P 1 sup 0≤t≤T |U t | ≤ |U 0 | + cE P 1   T 0 β 2 u d M u 1 2   + 1 2 E P 1 T 0 β 2 u d M u + 1 2 E P 1 T 0 α 2 u d M u + cE P 1 ( A T ) 1 2 < ∞ Then the expectations in (B.2) are finite. Finally, we observe that K is square-integrable. In fact, since K is an ( H, Q)-martingale, then K 2 is an ( H, Q)-submartingale and for every t ∈ [0, T ] for every t ∈ [0, T ], where B(u) is given by (7.2) and Γ(u) is defined in (7.7). Then K 1 is a squareintegrable ( H, Q)-martingale. E Q K 2 t ≤ E Q K 2 T ≤ 3 E Q G 2 T + E Q T 0 B t − β t dS t 2 + E Q G 2 0 < ∞ since G T ∈ L 2 ( H T , Q), G 0 ∈ L 2 ( H 0 , Q) and {B t − β t , t ∈ [0, T ]} ∈ Θ( H). Proof. Firstly we note that the process t 0 g(r, S r − ) (dN r − π r − (Λ)dr) , t ∈ [0, T ] is a square-integrable ( H, Q)-martingale. Indeed E Q T 0 g 2 (r, S r )π r (Λ)dr ≤ E P 1 sup t∈[0,T ] g 2 (t, S t ) E P 2 T 0 Λ r dr ≤ l a E P 1 sup t∈[0,T ] g 2 (t, S t ) < ∞. Second, since {V (u, t − ), t ∈ [0, u]} is F-predictable and Γ(u) : = {Γ t (u), t ∈ [0, u]} is G N -predictable, then T t V (u, t − )Γ t (u)du, t ∈ [0, T ] is H-predictable. Moreover, E Q T 0 T r V (u, r)Γ r (u)du π r (Λ)dr ≤ E Q T 0 T r |V (u, r)||Γ r (u)|du π r (Λ)dr ≤ 1 2 E Q T 0 T r V 2 (u, r)du + T r Γ 2 r (u)du π r (Λ)dr ≤ 1 2 E Q T 0 u 0 V 2 (u, r)π r (Λ)dr du + 1 2 E Q T 0 u 0 Γ 2 r (u)π r (Λ)dr du , where the last equality follows by Fubini's Theorem. Then we get that E P 1 sup 0≤r≤u V 2 (u, r) ≤ 4 E P 1 V 2 0 + cE P 1 u 0 β 2 s (u)d M s + u 0 |β s (u)α s |d M s 2 + cE P 1 A 1 u ≤ 4 E P 1 V 2 0 + cE P 1 T 0 β 2 s (u)d M s + T 0 |β s (u)α s |d M s 2 + cE P 1 A 1 T < ∞. Lemma 4. 3 . 3The minimal martingale measure for S with respect to H (and H) exists and coincides with Q * given in (4.1). Lemma 4. 4 . 4Let the process {g(t, S t ), t ∈ [0, T ]} satisfy (4.4). Then G T ∈ L 2 ( H T , Q). Definition 5. 5 . 5For an H-admissible strategy ψ, a small perturbation ∆ and a partition τ of [0, T ], we set Definition 5. 6 . 6Let G T ∈ L 2 ( H T , Q) be an insurance contingent claim. An H-admissible strategy ψ such that V T (ψ) = G T Q − a.s. is called H-pseudo-optimal for G T if and only if ψ is mean-self-financing and the ( H, Q)-martingale C(ψ) is strongly orthogonal to the Q-martingale part M of S. Note that, by construction, B is a (G N , P 2 )-square-integrable martingale. Then by the Martingale Representation Theorem, see e.g. [22, Chapter 4, Theorem 4.37], there exists a (G N , P 2 )-predictable process Γ := {Γ t , t ∈ [0, T ]} such that E P 2 T 0 Γ 2 u π u (Λ)du < ∞ and , U t = 0 for all t ∈ [0, T ] because of the dynamics of B, and s≤t (∆B s ∆U s ) = 0 for all t ∈ [0, T ] due to the independence between the financial and the insurance markets (see Assumption 4.1). Clearly, the process {B t − β t , t ∈ [0, T ]} ∈ Θ( H), see Definition 5.1. Indeed, for every t ∈ [0, T ], and β ∈ Θ( H). Finally, since the process K in (6.5) is a square-integrable ( H, Q)-martingale, see Lemma B.2 in Appendix B, we obtain the result. 10) where the sharp brackets are computed under P 1 .Proof. The proof follows by[32, Proposition 3.4] and [7, Proposition 4.2]. Remark 6.3. Assume that the contingent claim ξ has the form ξ = Υ(T, S T ). Then: positive constant c, thanks to the Burkholder-Davis-Gundy inequality, β(u) ∈ Θ(F) and the time interval [0, T ] is finite. Finally, since the process K 1 in (7.6) is a square-integrable ( H, Q)-martingale, see Lemma B.3 in Appendix B, then we get the statement. The next result characterizes the process B(u).Lemma 7.3. For every u ∈ [0, T ] the process B(u) defined in (7.2) is given by B t (u) := (l a − N t )E P 2 λ a (u, X u )e − u t λa(r,Xr)dr |G N t , ∀t ∈ [0, u]. t 0 B 0s − dA s , t ∈ [0, T ] turns out to be an ( H, Q)-martingale strongly orthogonal to S. Moreover, we also get that the processt 0 U s − Γ s (dN s − π s − (Λ)ds) , t ∈ [0, T ] is an ( H, Q)-martingale strongly orthogonal to S. Indeed, s − Γ s |π s − (Λ)ds ≤ E P 1 sup 0≤t≤T |U t | E P 2 T 0 |Γ s |π s − (Λ)ds . (B.2)By (6.4) and the structure condition (2.1),U t = U 0 + t 0 β u dM u + t 0 β u α u d M u + A t , for every t ∈ [0, T ], u α u |d M u + sup 0≤t≤T |A t |, ∀ t ∈ [0, T ]. (B.3) Lemma B. 3 .V 3Let K 1 := {K 1 t , t ∈ [0, T ]} be the process defined in (7.6), i.e. (u, r − )Γ r (u)du (dN r − π r − (Λ)dr) + t 0 T r B r − (u)γ r (u)du dA 1 r , (Λ)dr ≤ l a , and by (7.3) and the Burkholder-Davis-Gundy inequality, there exists a positive constant c such that V r (u)π r (Λ)dr < ∞ for every u ∈ [0, T ] and the time interval [0, T ] is finite. Indeed the term (u, r − )Γ r (u)du (dN r − π r − (Λ)dr) , t ∈ [0, T ] thanks to Assumption 4.1. This implies that O is an (H, Q)-martingale if and only if O 1 is an (F, P 1 )martingale and O 2 is a (G, P 2 )-martingale. Here, E Q [·|H t ] stands for the conditional expectation with respect to H t computed under the probability measure Q and so on. Now, let O be a square-integrable (H, Q)-martingale strongly orthogonal to M . By Assumption 4.1 we have that t λa(r,Xr)dr |G N t . 7.1. Evaluation of B(u) in a Markovian case. By Lemma 7.3 we get that The space L 2 (F0, P1) denotes the set of all F0-measurable random variables H such that E P 1 |H| 2 = Ω |H| 2 dP1 < ∞, where E P 1 [·] refers to the expectation computed under the probability measure P1. t (s, x, y)(f ) = E P (x,y) f (t, Z 1 t , Z 2 t ) exp − t s D [0,+∞) (R × R + ) is the space of càdlàg functions from [0, +∞) into R × R + AcknowledgementsThe authors have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).Appendix A. The survival process Lemma A.1. For every i = 1, . . . , l a , and for every s ≥ 0, over the event {T i > s}, the survival process satisfiesfor every t > 0, ω 2 ∈ Ω 2 .Proof. Note that for every A ∈ G X ∞ , A ∩ {T i > s} ∈ {T i > s} ∩ G X ∞ , for every s ≥ 0. We will show that for every A ∈ G X ∞ , every s ≥ 0 and every t ≥ 0,is an ( H, Q)-martingale. 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{'abstract': 'In this paper we investigate the local risk-minimization approach for a combined financial-insurance model where there are restrictions on the information available to the insurance company. In particular we assume that, at any time, the insurance company may observe the number of deaths from a specific portfolio of insured individuals but not the mortality hazard rate. We consider a financial market driven by a general semimartingale and we aim to hedge unit-linked life insurance contracts via the local risk-minimization approach under partial information. The Föllmer-Schweizer decomposition of the insurance claim and explicit formulas for the optimal strategy for pure endowment and term insurance contracts are provided in terms of the projection of the survival process on the information flow. Moreover, in a Markovian framework, we reduce to solve a filtering problem with point process observations.', 'arxivid': '1406.6902', 'author': ['Claudia Ceci ', 'ANDKatia Colaneri ', 'Alessandra Cretarola '], 'authoraffiliation': [], 'corpusid': 16807016, 'doi': '10.1016/j.insmatheco.2014.10.013', 'github_urls': [], 'n_tokens_mistral': 24345, 'n_tokens_neox': 21896, 'n_words': 13163, 'pdfsha': 'e4c5d33770d397b2a2b0c2866bb9a52aa0254699', 'pdfurls': ['https://arxiv.org/pdf/1406.6902v1.pdf'], 'title': ['HEDGING OF UNIT-LINKED LIFE INSURANCE CONTRACTS WITH UNOBSERVABLE MORTALITY HAZARD RATE VIA LOCAL RISK-MINIMIZATION', 'HEDGING OF UNIT-LINKED LIFE INSURANCE CONTRACTS WITH UNOBSERVABLE MORTALITY HAZARD RATE VIA LOCAL RISK-MINIMIZATION'], 'venue': []}

arxiv

Preservation of Takens-Bogdanov bifurcations for delay differential equations by Euler discretization * 20 Feb 2012 Yingxiang Xu Chengchun Gong Preservation of Takens-Bogdanov bifurcations for delay differential equations by Euler discretization * 20 Feb 2012 A new technique for calculating the normal forms associated with the map restricted to the center manifold of a class of parameterized maps near the fixed point is given first. Then we show the Takens-Bogdanov point of delay differential equations is inherited by the forward Euler method without any shift and turns into a 1:1 resonance point. The normal form near the 1:1 resonance point for the numerical discretization is calculated next by applying the new technique to the map defined by the forward Euler method. The local dynamical behaviors are analyzed in detail through the normal form. It shows the Hopf point branch and the homoclinic branch emanating from the Takens-Bogdanov point are O(ε) shifted by the forward Euler method, where ε is step size. At last, a numerical experiment is carried to show the results. * Supported by NSFC grants 10971022 and 11071102. Introduction Numerical methods may take on many possible phenomena when applied to certain dynamical systems. It is of great importance to investigate what kind of properties of the original systems could be preserved by discretization. Tremendous researches on numerical stabilities of sorts are the first investigation of this type where various sufficient and necessary conditions are developed for numerical schemes reproducing the asymptotic stability of differential equations, one can refer to the monographs [5] and [2] for ordinary differential equations (ODEs) and delay differential equations (DDEs), respectively. Whether or not the numerical discretization will inherit the bifurcations of the original systems is another important research field of this type. In this paper, we consider the DDEs of the typė z(t) = f (z(t), z(t − 1), α),(1) where z ∈ R n , α ∈ R 2 is a bifurcation parameter, f (z 1 , z 2 , α) is C r (r ≥ 2) smooth with respect to z 1 , z 2 and α. The state space of (1), denoted by C = C([−1, 0], R n ), is a Banach space of continuous mappings from [−1, 0] to R n equipped with norm φ = max θ∈[−1,0] |φ(θ)| (| · | is some norm in R n ). The equation (1) is assumed to undergo a Takens-Bogdanov bifurcation near (z, α) = (0, 0). Then, in the parameter plane (α 1 , α 2 ) there exist a Hopf point branch and a homoclinic branch emanating from the Takens-Bogdanov point of equation (1), see [22]. In this paper we show that the forward Euler discretization, when applied to equation (1), can preserve the bifurcation structure near the Takens-Bogdanov point (z, α) = (0, 0) of (1) by an O(ε) shift, where ε is the step size of the Euler method. Problems of this type have been focused on ODEs for many years. The Hopf bifurcation accepted the most attentions in this field after the work of Hofbauer and Iooss [15], where they proved the forward Euler discretization exhibits the Hopf bifurcation of the same type as the continuous system undergoes. In 1998, Wang, Blum and Li [19] gave the most complete results for the codimension 1 bifurcations: the general one step methods of order p and specific methods like Euler, backward Euler, explicit and implicit Runge-Kutta methods are proved to inherit the elementary bifurcations, such as saddle-node, fold, pitchfork, Hopf and transcritical bifurcations, of the continuous systems. For connecting orbits, Beyn [3] first showed the existence of homoclinic orbits under discretization. Fiedler and Scheurle [13], Zou and Beyn [25] proved the general one step method exhibits the discrete connecting orbits approximating to the one of the original ODEs by the order of the method, independently. Most recently, the major concern for this problem is whether the numerical discretization can inherit the codimension 2 bifurcations of ODEs. Lóczi and Chávez [18] showed the Runge-Kutta method will reproduce the nondegenerate fold, cusp and Takens-Bogdanov singularities of general ODEs without any shift. For generalized Hopf bifurcations, the Hopf point branch emanating from the generalized Hopf point is shifted by the order of the general one step method used as expected, however, the generalized Hopf point is shifted of the first order, regardless the order of the numerical method used, and turns into generalized Neimark-Sacker points. This result seems not natural, but the numerical experiments shows a much better result cannot be expected, cf. [7]. For the Hopf-Hopf bifurcation, the one step method will reproduce the bifurcation point as a double Neimark-Sacker point by an O(ε p ) shift, as well as the Hopf point branch bifurcated from the point, see [8]. Chávez [6], under the assumption that the ODEs undergo the fold-Hopf or Takens-Bogdanov bifurcation, proved the singularities will be persisted by the Runge-Kutta method and the Hopf point branch emanating from the singular points is O(ε p ) shifted (ε is step size). In addition, the discrete fold point branch and the discrete Hopf point branch emanating from the singularities is shown to intersect transversally. However, according to what we have seen up to now, no discussion about the preservation of the homoclinic branch emanating from the Takens-Bogdanov point by the numerical scheme is available. In these days, the preservation of dynamical behaviors for DDEs by numerical discretizations received many considerations as well. Wulf and Ford first showed the forward Euler method will reproduce the same type Hopf bifurcations as the scalar DDE with one constant delay undergoes [21]. The subsequent numerous relative works had led this problem to a more general case of both the numerical methods (Runge-Kutta method, linear multi-step method etc.) and the type of the differential equations (with multiple delays or state-dependent delay, etc.). For other dynamics of DDEs preserved by discretization, we recall the following results. Liu, Gao and Yang [17] proved the Runge-Kutta method preserves the oscillations of the equatioṅ x(t) + ax(t) + a 1 x([t − 1]) = 0. The asymptotically stable periodic orbits of the autonomous DDEs with one constant delay were proved by In't Hout and Lubich [16] to be shifted by the s-stage Runge-Kutta methods with the discretization order p. For a more general delay equationẋ = f (x t ),(2) where x t (θ) = x(t + θ), Farkas proved by explicit calculation that the unstable manifold will be close to its Euler discretized counterpart if the step size ε is sufficiently small [9] as well as a numerical shadowing result which obviously means the stable manifolds of (2) could be preserved numerically [10]. Xu and Zou [24] extended the results of [25] to DDEs, they showed the homoclinic orbits should be preserved by the forward Euler method with a shift of O(ε). In addition, the possibility of extending the numerical scheme to the implicit method and the Runge-Kutta method is discussed there. The problem we care most is whether the bifurcation diagram near Takens-Bogdanov point of DDEs could be inherited by numerical discretiztation. This requires us to investigate the dynamical behaviors of the map defined by the numerical scheme. The normal form analysis is a useful tool for accomplishing this aim. It allows us to know how the normal form coefficients and the generalized eigenspace of the numerical scheme are related to their continuous counterparts. Consequently, the dynamics of the numerical scheme could be studied in detail in terms of the coefficients of the original equation and then we know how the bifurcation diagram is preserved by direct comparison. Trying to fulfill our goal in this way, we will also face some problems. Generally speaking, the numerical scheme for ODEs could be regarded as a map. For example, the one step method for solvinġ x = f (x), x ∈ R n has the general form x k+1 = x k + εφ(x k , ε),(3) where ε is step size. Obviously, equation (3) is a map from R n to R n and the method of normal forms for finite-dimensional ODEs could be modified to analyze the local bifurcation behavior. Where the representations of the complete eigenvectors are required to construct a transformation which lead the map (3) to a standard form could be dealt with easily (see [20] section 1.1C for detail). In addition, the technique requires computing the center manifold first before evaluating normal forms for the map on the center manifold. However, the forward Euler method for solvingẋ (t) = ax(t) + bx(t − 1), x ∈ R n has the form x k+1 = x k + εau k + εbx k−m ,(4) where ε = 1 m is step size, m is a positive integer. The map defined by (4) is, obviously, from R (m+1)×n to R (m+1)×n and the dimensionality tends to infinity as the step size tends to 0. As a result, the amount of the eigenvectors is very large as the step size is small, and the method of normal forms for maps described in [20] is not efficient any more. Highlighted by the normal form analysis for retarded functional differential equations produced by Faria [11,12], a new method for calculating the normal forms associated with the map restricted to the center manifold tangent to the invariant manifold of the lienarization of the map of the type mentioned above at the fixed point is given. It has many advantages: one, without computing beforehand the center manifold of the singularity; two, allows obtaining the coefficients in the normal form explicitly in terms of the considered map, therefore, one can obtain the accurate dynamics through analyzing the obtained normal form, and three, no need of computing the eigenvectors associated with the noncritical eigenvalues, and then the amount of calculations are reduced remarkably. The rest of the paper is arranged as follows. We first go deep into the techniques for normal form calculations for the maps similar to (4) in section 2. The Takens-Bogdanov bifurcations for equation (1) is recalled in section 3. In Section 4, we first show the Takens-Bogdanov point of DDEs is inherited by the forward Euler method exactly and turns into a 1:1 resonance point. Then the normal form near the 1:1 resonance point for the numerical method is calculated by applying the techniques developed in Section 2. The local dynamical behaviors are analyzed in detail through the obtained normal form. It shows that the Hopf point branch and the homoclinic branch bifurcated from the Takens-Bogdanov point of (1) are inherited by the forward Euler scheme by a shift of O(ε). A numerical experiment for a scalar DDE is carried to show our theoretical results in the last section. Normal forms for a class of maps with parameters We consider a class of maps of the following type u → C(α, ε)u + F (u, α, ε),(5) where u ∈ R (m+1)nl with n, m, l ∈ Z + , ε = 1 m is a parameter used in later applications where ε corresponds to the step size, α = (α 1 , · · · , α p ) ∈ R p is a bifurcation parameter. F is C r (r ≥ 2) smooth with respect to u and α, and satisfies F (0, 0, ε) = 0 and ∂F ∂u (0, 0, ε) = 0. From the map (5), we see that the dimensionality of (5) will tend to infinity as the parameter ε tends to 0. Denote C(ε) = C(0, ε) andF (u, α, ε) = C(α, ε) − C(0, ε) + F (u, α, ε), then we can write the map (5) as u → C(ε)u +F (u, α, ε).(6) We drop the bar fromF if no confusion occurs. For simplicity, we will drop the dependence on ε in the rest of this section. We reformulate the map (5) by regarding α as a variable to the following map without parameters u α →C u α + F (u, α) 0 p ,(7) whereC = diag{C, I p }. To linearize the map (7) at (u, α) = (0, 0) we obtain u α →C u α ,(8) which has the characteristic equation given by det(λI −C) = 0 ⇔ (λ − 1) p det(λI − C) = 0.(9) As seen in the former equation, we often omit the subscript of the identity matrix I if no confusion occurs throughout this paper. Denoted by Λ = {λ ∈ σ(C) : |λ| = 1} andΛ = Λ ∪ {1, · · · , 1 p times }, φ i the (generalized) eigenvectors of C associated with eigenvalues λ i , i = 1, 2, · · · , (m + 1)nl, then P = span{φ i : λ i ∈ Λ} is the invariant space of C associated with Λ. If c is the number of the eigenvalues of C in Λ, counting multiplicities, then we have dim P = c. If we denote by Φ c = (φ 1 , · · · , φ c ) a basis for P , Ψ c = (ψ T 1 , · · · , ψ T c ) T a basis for the dual space P * in R (m+1)nl * , then (Ψ c , Φ c ) = I c , the identity matrix in R c , where the dual form takes the scalar product of vectors. We denote by J the c × c constant matrix such that CΦ c = Φ c J; its spectrum coincides with Λ. Denoted by Q the invariant space of C associated with σ(C) \ Λ, then Q = span{φ i , i = c + 1, · · · , (m + 1)nl}. If we denote Φ su = (φ c+1 , · · · , φ (m+1)nl ) a basis for Q, Ψ su = (ψ T c+1 , · · · , ψ T (m+1)nl ) T a basis for the dual space Q * in R (m+1)nl * , then we have (Ψ su , Φ su ) = I (m+1)nl−c , the identity matrix in R ((m+1)nl−c)×((m+1)nl−c) . We denote by C Q the constant matrix such that CΦ su = Φ su C Q ; its spectrum coincides with σ(C) \ Λ. With the notations above, the generalized eigenspaceP ofC associated withΛ satisfies P = spanΦ c , whereΦ c = diag(Φ c , I p ). The left generalized eigenspace ofC associated withΛ is spanned byΨ c = diag(Ψ c , I p ). Denoted byJ the (c+p)×(c+p) matrix subject toCΦ c =Φ cJ , we haveJ = diag(J, I p ) and its spectrum coincides withΛ. Clearly, σ(C) \Λ = σ(C) \ Λ. If we denoteQ the complementary space ofP in the space R (m+1)nl+p , thenQ is spanned byΦ su , the generalized eigenspace ofC associated with σ(C) \Λ. Obviously,Φ su = (Φ T su , 0 p ) T . We denote byΨ su the left generalized eigenspace ofC associated with σ(C) \Λ, thenΨ su = (Ψ su , 0 p ). Obviously, we have (Ψ su ,Φ su ) = I (m+1)nl−c . According to the decomposition of the state space R (m+1)nl = P ⊕ Q, u could be represented as u = Φ c x + Φ su y with x ∈ R c , y ∈ R (m+1)nl−c . Therefore, projecting map (7) ontoP andQ respectively, we get x α →J x α + Ψ c F (Φ c x + Φ su y, α) 0 p , y → C Q y + Ψ su F (Φ c x + Φ su y, α).(10) Noting that α is mapped into itself, the previous map exactly is x → Jx + Ψ c F (Φ c x + Φ su y, α), y → C Q y + Ψ su F (Φ c x + Φ su y, α),(11) where α should be considered as a variable. In addition we recall that the matrices J and C Q are of the form of uptriangular. The first part in map (11) has a fixed dimensionality, that is c, the dimensionality of the center manifold of (5) associated with Λ. While the second part has a variable dimensionality, which tends to infinity as the parameter ε tends to 0. This makes it very hard to give explicitly the expressions for the (generalized) eigenvectors of (8) associated with σ(C) \ Λ. Besides, the amount of the eigenvectors will also tend to infinity as the parameter ε tends to 0. It is impracticable to get the map restricted to the center manifold of (11) by the technique described in [20] directly. In the next we go deep into develop a method which allows getting the normal forms associated with Λ on the center manifold avoid computing the eigenvectors Φ su and Ψ su . Taylor expanding F (u, α) with respect to (u, α) leads to F (u, α) = j≥2 1 j! F j (u, α), (u, α) ∈ R (m+1)nl+p .(12) Defining f j = (f 1 j , f 2 j ) with f 1 j (x, y, α) = Ψ c F j (Φ c x + Φ su y, α), f 2 j (x, y, α) = Ψ su F j (Φ c x + Φ su y, α),(13) the map (7) is equivalent to x → Jx + j≥2 1 j! f 1 j (x, y, α), y → C Q y + j≥2 1 j! f 2 j (x, y, α),(14) where x ∈ R c and y ∈ R (m+1)nl−c . The normal forms are obtained by a recursive procedure, computing at each step the terms of order j ≥ 2 in the normal form from the terms of the same order in the original map and the terms of lower orders already computed for the normal form in previous steps, through a transformation of variables (x, y) = (x,ŷ) + 1 j! U j (x, α),(15)with x,x ∈ R c , y,ŷ ∈ R (m+1)nl−c , and U j = (U 1 j , U 2 j ) ∈ V c+p j (R c ) × V c+p j (R (m+1)nl−c ), where, for a normed space X, V c+p j (X) denotes the linear space of homogeneous polynomials of degree j in c + p real variables, (x, α) = (x 1 , x 2 , · · · , x c , α 1 , · · · , α p ), and with coefficients in X, V c+p j (X) = { |(q,l)|=j c (q,l) x q α l : (q, l) ∈ N c+p 0 , c (q,l) ∈ X}, x q α l = x q1 1 · · · x qc c α l1 1 · · · α lp p for q = (q 1 , · · · , q c ) ∈ N c 0 and l = (l 1 , · · · , l p ) ∈ N p 0 , with the norm |(q,l)|=j c (q,l) x q α l = |(q,l)|=j |c (q,l) | X . We assume that after computing the normal form up to terms of order j − 1 the map is x → Jx + j−1 i=2 1 i! g 1 i (x, y, α) + 1 j!f 1 j (x, y, α) + · · · y → C Q y + j−1 i=2 1 i! g 2 i (x, y, α) + 1 j!f 2 j (x, y, α) + · · ·(16) where g 1 j (x, y, α) =f 1 j (x, y, α) − [U 1 j (Jx, α) − JU 1 j (x, α)], g 2 j (x, y, α) =f 2 j (x, y, α) − [U 2 j (Jx, α) − C Q U 2 j (x, α)] . These formulas can be written for g j = (g 1 j , g 2 j ) as g j =f j − M j U j ,(17) withf j = (f 1 j ,f 2 j ) and M j defined below. Definition 2.1 For j ≥ 2, let M j denote the operator defined in V c+p j (R c × R (m+1)nl−c ) , with values in the same space, by M j (p, h) = (M 1 j p, M 2 j h) (M 1 j p)(x, α) = p(Jx, α) − Jp(x, α) (M 2 j h)(x, α) = h(Jx, α) − C Q h(x, α), with domain D(M j ) = V c+p j (R c ) × V c+p j (R (m+1)nl−c ). Then the problem we met is how to choose U j so that g j has a simple form. We decompose the spaces V c+p j (R c ) and V c+p j (R (m+1)nl−c ) as V c+p j (R c ) = Im(M 1 j ) ⊕ Im(M 1 j ) c , V c+p j (R c ) = ker(M 1 j ) ⊕ ker(M 1 j ) c , and V c+p j (R (m+1)nl−c ) = Im(M 2 j ) ⊕ Im(M 2 j ) c , V c+p j (R (m+1)nl−c ) = ker(M 2 j ) ⊕ ker(M 2 j ) c . We denote the projections associated with the above decompositions of V c+p j (R c )×V c+p j (R (m+1)nl−c ) over Im(M 1 j ) × Im(M 2 j ) and over ker(M 1 j ) c × ker(M 2 j ) c by, respectively, P I,j = (P 1 I,j , P 2 I,j ) and P K,j = (P 1 K,j , P 2 K,j ). The complementary spaces in the above decompositions Im(M i j ) c , ker(M i j ) c (i = 1, 2), are not uniquely determined. As a consequence, normal forms are not unique, and depend on the choices of Im(M i j ) c (i = 1, 2). Let us now consider the right inverse of M j with range defined by the spaces complementary to the kernels of M i j (i = 1, 2), namely M −1 (17), an adequate choice of U j , j = ((M 1 j ) −1 , (M 2 j ) −1 ) with M −1 j • P I,j • M j = P K,j . Taking y = 0 in formulaU j (x, α) = M −1 j P I,jfj (x, 0, α),(18) allows taking away fromf j its component in the range of M j , leading to g j (x, 0, α) = (I − P I,j )f j (x, 0, α). Therefore, the normal form for map (5) relative to the invariant space P and the projections P I,j , P K,j (j = 2, 3, · · · ) is the map in R c × R (m+1)nl−c x → Jx + j≥2 1 j! g 1 j (x, y, α) y → C Q y + j≥2 1 j! g 2 j (x, y, α).(20) In the next, we show that the normal form relative to P takes a more simple form of c-dimensional map on a locally invariant manifold for (5) tangent to P at zero. Recall that the matrices J and C Q are of Jordan forms, then by the Theorem 3.8 in [1] we know that the normal form (20) can be chosen so that its nonlinear part contains only resonant monomials. Definition 2.2 We say that the map (7) satisfies the nonresonance conditions relative toΛ ∈ σ(C), ifλ q = µ, for all µ ∈ σ(C) \Λ, q ∈ N c+p 0 , |q| ≥ 2, whereλ = (λ 1 , · · · , λ c+p ) and λ 1 , · · · , λ c+p are the elements ofΛ, each one of them appearing as many times as its multiplicity as an eigenvalue of the matrixC. Clearly, the fact that the map (7) satisfies the nonresonance conditions relative toΛ ∈ σ(C) is equivalent to the map (5) satisfies the nonresonance condtions relative to Λ ∈ σ(C) since the difference betweenΛ and Λ are the eigenvalues introduced by regarding α as a variable, that is Λ \ Λ = {1, · · · , 1 p times }. (5) and let P be the invariant subspace of C associated with a nonempty finite set Λ of eigenvalues. For the decomposition of R (m+1)nl = R c ⊕R (m+1)nl−c , we get u = Φ c x+Φ su y, where x ∈ R c and y ∈ R (m+1)nl−c , Φ c and Φ su are formed by the generalized eigenvectors of C associated with Λ and σ(C) \ Λ, respectively. If the nonresonance conditions relative to Λ are satisfied, then there exists a formal change of variables (x,ȳ) → (x, y) of the form x =x + p(x, α), y =ȳ + h(x, α), such that the map (5) is equivalent to the following map in R c × R (m+1)nl−cx Theorem 2.3 Consider the map → Jx + j≥2 1 j! g 1 j (x,ȳ, α) y → C Qȳ + j≥2 1 j! g 2 j (x,ȳ, α),(21) where g 1 j , g 2 j are computed as (17), with g 2 j (x, 0, α) = 0 for all j ≥ 2. This map is in normal form relative to P . If there exists a locally invariant manifold for the map (5) tangent to P at zero, then it satisfiesȳ = 0 and the map on it is given by the c-dimensional map x → Jx + j≥2 1 j! g 1 j (x, 0, α),(22) which is in normal form for maps. Takens-Bogdanov bifurcations of parameterized delay differential equations To show the bifurcation structure near Takens-Bogdanov point could be preserved by numerical discretization, we recall the results on the bifurcation structure near Takens-Bogdanov point in this section, cf. [22]. We consider the DDEs of the typė z(t) = f (z(t), z(t − 1), α),(23) where z ∈ R n , α ∈ R 2 is a bifurcation parameter, f (z 1 , z 2 , α) is a C r (r ≥ 2) smooth function from R n × R n × R 2 to R n with f (0, 0, α) = 0, ∂f ∂z 1 (0, 0, α) = 0, ∂f ∂z 2 (0, 0, α) = 0, ∀α ∈ R 2 . Denote A = ∂f ∂z1 (0, 0, 0) and B = ∂f ∂z2 (0, 0, 0), then we can rewrite equation (23) aṡ z(t) = Az(t) + Bz(t − 1) + F (z(t), z(t − 1), α),(25) which could be linearized at (z, α) = (0, 0) aṡ z(t) = Az(t) + Bz(t − 1).(26) The characteristic equation of (26) is given by det(µI − A − Be −µ ) = 0.(27) We assume (z, α) = (0, 0) is a Takens-Bogdanov point of (23), i.e., A 0 is a root of (27) with algebraic multiplicity 2 and geometric multiplicity 1, and the other roots exhibit nonzero real parts. Lemma 3.1 [22]There exist φ 0 1 ∈ R n \{0}, φ 0 2 ∈ R n , and ψ 0 2 ∈ R n * \{0}, ψ 0 1 ∈ R n * , they satisfy the following equations (1) (A + B)φ 0 1 = 0, (2) (A + B)φ 0 2 = (B + I)φ 0 1 , (3) ψ 0 2 (A + B) = 0, (4) ψ 0 1 (A + B) = ψ 0 2 (B + I), (5) ψ 0 2 φ 0 2 − 1 2 ψ 0 2 Bφ 0 1 + ψ 0 2 Bφ 0 2 = 1, (6) ψ 0 1 φ 0 2 − 1 2 ψ 0 1 Bφ 0 1 + ψ 0 1 Bφ 0 2 + 1 6 ψ 0 2 Bφ 0 1 − 1 2 ψ 0 2 Bφ 0 2 = 0.(28) Taylor expanding F (z(t), z(t − 1), α) with respect to z(t), z(t − 1) and α we obtain F (z(t), z(t − 1), α) = j≥2 1 j! F j (z(t), z(t − 1), α),(29) where the first term (j = 2) can be expressed in the form 1 2 F 2 (z(t), z(t − 1), α) = A 1 α 1 z(t) + A 2 α 2 z(t) + B 1 α 1 z(t − 1) + B 2 α 2 z(t − 1) + n i=1 E i z i (t)z(t − 1) + n i=1 F i z i (t)z(t) + n i=1 G i z i (t − 1)z(t − 1) (30) with A i , B i (i = 1, 2), E i , F i , G i (i = 1, 2, · · · , n) coefficient matrices, and there is no terms of O(α 2 ) in F 2 (z(t), z(t − 1), α) since F (0, 0, α) = 0, ∀α ∈ R 2 . Denote a = ψ 0 2 n i=1 (E i + F i + G i )φ 0 1 φ 0 1i , b = 2ψ 0 1 n i=1 (E i + F i + G i )φ 0 1 φ 0 1i +ψ 0 2 { n i=1 (E i + F i + G i )(φ 0 2 φ 0 1i + φ 0 1 φ 0 2i ) − n i=1 (E i + 2G i )φ 0 1 φ 0 1i },(31) and κ 1 κ 2 = Π α 1 α 2 (32) with Π =    ψ 0 2 (A 1 + B 1 )φ 0 1 ψ 0 2 (A 2 + B 2 )φ 0 1 {ψ 0 1 (A 1 + B 1 )φ 0 1 +ψ 0 2 ((A 1 + B 1 )φ 0 2 − B 1 φ 0 1 )} {ψ 0 1 (A 2 + B 2 )φ 0 1 +ψ 0 2 ((A 2 + B 2 )φ 0 2 − B 2 φ 0 1 )}   . For the bifurcation structure near the Takens-Bogdanov point of DDE (23), we have the following theorem. Theorem 3.2 [22]Assume the assumption A holds, det Π = 0 and a · b = 0. Then there exists a constant κ 0 1 > 0, such that when 0 < κ 1 (α 1 , α 2 ) < κ 0 1 , in the parameter plane (α 1 , α 2 ) near the origin there exist two curves: l h and l ∞ 1. the curve l h , which has the following local representation: l h = {(α 1 , α 2 ) : κ 2 (α 1 , α 2 ) − b a κ 1 (α 1 , α 2 ) + h.o.t. = 0, κ 1 (α 1 , α 2 ) > 0}, is a Hopf point branch of the DDE (23), where h.o.t. = o(|(α 1 , α 2 )|), i.e. l h consists of Hopf bifurcation points of (23); 2. the curve l ∞ , which has the following local representation: l ∞ = {(α 1 , α 2 ) : h(α 1 , α 2 ) + h.o.t. = 0, κ 1 (α 1 , α 2 ) > 0},(33) is a homoclinic branch of the DDE (23), where h(α 1 , α 2 ) = κ 2 (α 1 , α 2 ) − µ( κ 1 (α 1 , α 2 )) κ 1 (α 1 , α 2 ), µ(·) is a continuously differentiable function with µ(0) = 6 7 ba −1 and h.o.t. = o(|(α 1 , α 2 )|). In other words, equation (23) has a unique homoclinic orbit connecting the origin for each (α 1 , α 2 ) ∈ l ∞ . Preservation of Takens-Bogdanov bifurcations by Euler discretization The forward Euler scheme for solving (23) with step size ε = 1 m , m ∈ Z + is given by z k+1 = z k + εf (z k , z k−m , α),(34) which can be reformulated to z k+1 = z k + εAz k + εBz k−m + εF (z k , z k−m , α).(35) Denoting u k = (z T k , z T k−1 , . . . , z T k−m ) T ∈ R (m+1)n , (35) can be rewritten as u k+1 = Cu k + H(u k , α),(36)0 · · · 0 I 0         and H(u k , α) = (εF (z k , z k−m , α) T , 0, · · · , 0) T . Linearizing the map (36) at (u, α) = (0, 0) we obtain u k+1 = Cu k ,(37) which has the characteristic equation det(λI − C) = 0.(38) Noting (36) is a map of the type of (5) as l = 1, the techniques developed in Section 2 allow us to get the normal forms on the low dimensional center manifold without computing the eigenvectors associated with the eigenvalues of the modulus other than 1. While the map (5) as l > 1 could correspond to more general numerical schemes, eg. the general one step method. Under the assumption that (23) undergoes a Takens-Bogdanov bifurcation at (z, α) = (0, 0), we first show the Takens-Bogdanov point (z, α) = 0, 0) of (23) is inherited without any shift by the Euler method (35) and turns into a 1:1 resonance point. Theorem 4.1 Assume the assumption A holds. Then (34) undergoes a 1:1 resonance at (z, α) = (0, 0). P roof. We only need to show the double zero eigenvalue of (26) turns into a double unit multiplier, λ 1,2 = 1 at α = 0, and no other eigenvalue of (37) has modulus 1. Assume (C − I)φ 1 = 0, (C − I)φ 2 = φ 1 , ψ 2 (C − I) = 0, ψ 1 (C − I) = ψ 2 ,(39) and require ψ 1 · φ 1 = ψ 2 · φ 2 = 1. We can check that the following choice of φ 1 , φ 2 , ψ 2 , ψ 1 φ 1 = ε(φ 0T 1 , · · · , φ 0T 1 ) T , φ 2 = ε(mφ 0T 2 , mφ 0T 2 − φ 0T 1 , · · · , mφ 0T 2 − mφ 0T 1 ) T , ψ 2 = 1 1 − 1 2m ψ 0 2 Bφ 0 1 (ψ 0 2 , εψ 0 2 B, · · · , εψ 0 2 B), ψ 1 = 1 1 − 1 2m ψ 0 2 Bφ 0 1 (mψ 0 1 , ψ 0 1 B − ψ 0 2 B, ψ 0 1 B − (m − 1)εψ 0 2 B, · · · , ψ 0 1 B − εψ 0 2 B),(40) fulfill the requirements above. Besides, the Fredholm alternative Theorem implies ψ 2 · φ 1 = ψ 1 · φ 2 = 0. These show that 1 is an eigenvalue of (37) with algebraic multiplicity 2 and geometric multiplicity 1. Next, we show no other eigenvalue of (37) has modulus 1. if other than the eigenvalue 1 there exists any eigenvalue of (38), denoted by λ ε , s.t. |λ ε | = 1, then µ ε = 1 ε ln λ ε solves det D(µ, ε) = 0. Since µ ε is pure imaginary, we have µ 0 , as the limit of µ ε as ε → 0, is either pure imaginary or 0 and solves (27). This contradicts to the assumption A. Let Φ c = (φ 1 , φ 2 ), Ψ c = (ψ 1 , ψ 2 ), and Λ = {1, 1}, then we know the invariant space P of (36) associated with Λ is spanned by Φ c , the dual invariant space P * of (36) associated with Λ is spanned by Ψ c . They satisfy (Ψ c , Φ c ) = I 2 , CΦ c = Φ c J, and Ψ c C = JΨ c with J = 1 1 0 1 . We then consider the Taylor expansion of H(u k , α) with respect to (u k , α). By (29), we have H(u k , α) = ε(F (z k , z k−m , α) T , 0, · · · , 0) T = ( j≥2 ε j! F j (z k , z k−m , α) T , 0, · · · , 0) T ,(41)where 1 2 F 2 (z k , z k−m , α) = A 1 α 1 z k + A 2 α 2 z k + B 1 α 1 z k−m + B 2 α 2 z k−m + n i=1 E i z i k z k−m + n i=1 F i z i k z k + n i=1 G i z i k−m z k−m .(42) Evidently, the canonical basis of V 4 = A 1 α 1 (εφ 0 1 , φ 0 2 )(x 1 , x 2 ) T + A 2 α 2 (εφ 0 1 , φ 0 2 )(x 1 , x 2 ) T +B 1 α 1 (εφ 0 1 , φ 0 2 − φ 0 1 )(x 1 , x 2 ) T + B 2 α 2 (εφ 0 1 , φ 0 2 − φ 0 1 )(x 1 , x 2 ) T + n i=1 E i (εφ 0 1i , φ 0 2i )(x 1 , x 2 ) T (εφ 0 1 , φ 0 2 − φ 0 1 )(x 1 , x 2 ) T + n i=1 F i (εφ 0 1i , φ 0 2i )(x 1 , x 2 ) T (εφ 0 1 , φ 0 2 )(x 1 , x 2 ) T + n i=1 G i (εφ 0 1i , φ 0 2i − φ 0 1i )(x 1 , x 2 ) T (εφ 0 1 , φ 0 2 − φ 0 1 )(x 1 , x 2 ) T = ε(A 1 + B 1 )φ 0 1 α 1 x 1 + ε(A 2 + B 2 )φ 0 1 α 2 x 1 +((A 1 + B 1 )φ 0 2 − B 1 φ 0 1 )α 1 x 2 + ((A 2 + B 2 )φ 0 2 − B 2 φ 0 1 )α 2 x 2 +ε 2 n i=1 (E i + F i + G i )φ 0 1i φ 0 1 )x 2 1 +ε n i=1 {(E i + F i + G i )(φ 0 1i φ 0 2 + φ 0 2i φ 0 1 ) − (E i + 2G i )φ 0 1i φ 0 1 }x 1 x 2 + n i=1 (E i + F i + G i )φ 0 2i φ 0 2 − (E i + G i )φ 0 2i φ 0 1 − G i φ 0 1i (φ 0 2 − φ 0 1 ))x 2 2 . Base on the expansion above, and the canonical basis of V 4 2 (R 2 ), Im(M 1 2 ) and Im(M 1 2 ) c , not- ing that f 1 2 (x, 0, α) = Ψ c H 2 (Φ c x, α) = 1 2 ( 1 1− 1 2m ψ 0 2 Bφ 0 1 ψ 0 1 , ε 1− 1 2m ψ 0 2 Bφ 0 1 ψ 0 2 )F 2 ((εφ 0 1 , φ 0 2 )x, (εφ 0 1 , φ 0 2 − φ 0 1 )x, α), we can compute the function g 1 2 (x, 0, α) = (I − P 1 I,2 )f 1 2 (x, 0, α). By Theorem 2.3, we have the following results. Theorem 4.2 Assume the requirements in Theorem 3.2 are fulfilled. Then the numerical scheme (34) will undergo a 1:1 resonance at (z, α) = (0, 0). In addition, the numerical scheme (34) could be reduced to a 2 dimensional map on the center manifold at (z, α) = (0, 0) as follows x 1 → x 1 + x 2 , x 2 → x 2 + κ ε 1 x 1 + κ ε 2 x 2 + a ε x 2 1 + b ε x 1 x 2 + h.o.t.,(43) where κ ε 1 = ε 2 1− ε 2 ψ 0 2 Bφ 0 1 κ 1 , κ ε 2 = ε 1− ε 2 ψ 0 2 Bφ 0 1 κ 2 , a ε = ε 3 1− ε 2 ψ 0 2 Bφ 0 1 a, b ε = ε 2 1− ε 2 ψ 0 2 Bφ 0 1 b, with κ 1 , κ 2 and a, b defined in (32) and (31), respectively. It is known that for the reduced map (43), if a ε · b ε = 0 (equivalently a · b = 0), the local bifurcation structure near (x, α) = (0, 0) is determined by the linear and quadratic terms, and not the terms of order higher. Hence we turn to investigate the local bifurcation structures of the map x 1 → x 1 + x 2 , x 2 → x 2 + κ ε 1 x 1 + κ ε 2 x 2 + a ε x 2 1 + b ε x 1 x 2 .(44) Lemma 4.3 Let λ ± ε (κ ε 1 , κ ε 2 ) be the eigenvalues of the Jacobian of (44) at (− κ ε 1 a ε , 0). Then, when a ε · b ε = 0 and 0 < κ ε 1 < 2 |λ ± ε (κ ε 1 , κ ε 2 )| = 1 + κ ε 2 − b ε a ε κ ε 1 + κ ε 1 . Hence we conclude that each point on the line segmentl ε h = {(κ ε 1 , κ ε 2 ) : κ ε 2 = b ε a ε κ ε 1 − κ ε 1 , 0 < κ ε 1 < 2} in the parameter plane (κ ε 1 ,κ ε 2 ) is a Neimark-Sacker bifurcation (also known as the Hopf bifurcation for map) point of the map (44). P roof. Evidently the map (44) has two fixed points, (− κ ε 1 a ε , 0) and (0, 0). Direct computations show the eigenvalues of the Jacabian at (− κ ε 1 a ε , 0) reads λ ± ε (κ ε 1 , κ ε 2 ) = 1 2 (2 + κ ε 2 − b ε a ε κ ε 1 ) ± (2 + κ ε 2 − b ε a ε κ ε 1 ) 2 − 4(1 + κ ε 2 − b ε a ε κ ε 1 + κ ε 1 ), which have a modulus given by |λ ± ε (κ ε 1 , κ ε 2 )| = 1 + κ ε 2 − b ε a ε κ ε 1 + κ ε 1 . Therefore, when (κ ε 1 , κ ε 2 ) changes from one side ofl ε h to the other, in the parameter plane the eigenvalues λ ± ε (κ ε 1 , κ ε 2 ) will cross the unit circle from outside to inside (κ ε 2 − b ε a ε κ ε 1 + κ ε 1 > 0) or from inside to outside (κ ε 2 − b ε a ε κ ε 1 + κ ε 1 < 0). Let θ = arctan √ 4κ ε 1 +(κ ε 1 ) 2 2−κ ε 1 , obviously we have when 0 < κ ε 1 < 2 e ikθ = 1, for k = 1, 2, 3, 4. Applying of the Neimark-Sacker bifurcation Theorem implies (x 1 , x 2 , κ ε 1 , κ ε 2 ) = (− κ ε 1 a ε , 0, κ ε 1 , b ε a ε κ ε 1 − κ ε 1 ) is a Neimark-Sacker bifurcation point of the map (44) as 0 < κ ε 1 < 2. In fact, there is a neighborhood of (− κ ε 1 a ε , 0) in which a unique closed invariant curve bifurcates from (− κ ε 1 a ε , 0) when κ ε 2 − b ε a ε κ ε 1 + κ ε 1 changes signs. In other words, the line segmentl ε h in the parameter plane (κ ε 1 , κ ε 2 ) is a Neimark-Sacker point branch of the map (44). Remark 4.4 Noting the expression of κ ε 1 ≈ ε 2 κ 1 , the statement of κ ε 1 < 2 requires the step size ε of the numerical scheme (34) should be taken nicely small to reproduce the Hopf bifurcations of (23). In the next we consider the homoclinic curves bifurcates from the fixed point (x, α) = (0, 0). Applying the transformation ofx 1 = a ε (x 1 + κ ε 1 2a ε ),x 2 = a ε x 2 to (44) leads to another typical normal form for 1:1 resonance, that is x 1 →x 1 +x 2 , x 2 →x 2 − (κ ε 1 ) 2 4 + (κ ε 2 − b ε κ ε 1 2a ε )x 2 +x 2 1 + b ε a εx1x2 . Denote d(µ) = e µ (µI − A) − B and D(µ, ε) = e µ (g(µε)µI − A) − B, where g(x) = e x −1 x . Then, see [23], we have det d(µ) = 0 is equivalent to (27), and if det D(µ ε , ε) = 0, then λ = e µεε solves (38). Besides, we have lim ε→0 D(µ, ε) = d(µ). Hence for any series µ ε of solutions of det D(µ, ε) = 0, there exists a solution µ 0 of det d(µ) = 0 such that lim ε→0 µ ε = µ 0 . Therefore, Figure 1 : 1Local bifurcation diagram in parameter plane (α 1 , α 2 ): by neglecting the h.o.t., the homoclinic branch l ∞ and Hopf point branch l h of (48), the theoretical homoclinic branch l ε ∞ and Neimark-Sacker point branch l ε h of the forward Euler discretization (49), as well as the homoclinic branch "numerical HB" and neimark-Sacker point branch "numerical HpB" detected in Forward Euler discretization (49) are plotted. Figure 2 :Figure 3 :Figure 4 : 2342(α 2 − α 1 ) are focuses of the Euler method (49) when α belongs to the left region of l ε h . ) α = (0.145, −0.25) Periodic solutions bifurcated from 2(α 2 − α 1 ) of the Euler method (49) when α is located in the region confined by l ε h and l ε ∞ . Homoclinic orbits of (49): α = (0.0308, −0.05) gives the small one, α = (0.0950, −0.15) leads to the middle one, and α = (0.1631, −0.25) results in the large one. 2 (R 2 ) is composed by the elementsand the images of these elements under M 1 2 are, respectivelyTherefore, a basis of Im(M 1 2 ) c can be taken as the set composed by the elementsDenoting by Φ 0 c the first n rows of Φ c , Φ m c the last n rows of Φ c , we have H(Φ c x, α) = (εF (Φ 0 c x, Φ m c x, α) T , 0, · · · , 0) T . Denoting by φ ji the i-th element of φ j , we have from (30)Applying the results in[4]to the former map shows that the homoclinic curves bifurcated from the fixed points (x, α) = (0, 0) is depicted byRemark 4.5 In fact, there exist two curves α + 2 (α 1 ) and α − 2 (α 1 ) respectively corresponding to the first and the last homoclinic tangency, they are exponentially close to one-another. If α is located in the region confined by these two curves, the map (44) possesses transverse homoclinic trajectories, see[14]for detail. But this is not our goal in this paper.Noting that the map (44) is locally topologically equivalent near the origin to (43), we have the following bifurcation results based on Lemma 4.3 and the discussions above.Theorem 4.2). Then there exists a constant κ 0 1ε = min{2, κ 0 ε } > 0, such that when 0 < κ ε 1 (α 1 , α 2 ) < κ 0 1ε , in the parameter plane (α 1 , α 2 ) near the origin there exist two curves: l ε h and l ε ∞ 1. the curve l ε h , which has the following local representation:i.e. l ε h consists of Neimark-Sacker bifurcation points of (34); 2. the curve l ε ∞ , which has the following local representation:is a homoclinic curve of the numerical scheme (34), where h(α 1 ,In other words, the numerical scheme (34) presents a unique homoclinic curve connecting the origin for each (α 1 , α 2 ) ∈ l ε ∞ .Comparing Theorem 4.6 to Theorem 3.2, incorporating Theorem 4.1 we obtain the following result.Theorem 4.7 Assume the assumption A holds, that is the DDE (23) exhibits a Takens-Bogdanov bifurcation at (z, α) = (0, 0). Then the Takens-Bogdanov point of (23) is inherited without any shift by the forward Euler scheme (34) and turns into a 1:1 resonance point. Moreover, there exists an ε 0 > 0, such that as ε < ε 0 , the forward Euler scheme (34) will reproduce the Hopf point branch and the homoclinic branch of the DDE (23) with a shift of O(ε) in parameter plane (α 1 , α 2 ), specially we haveNumerical exampleIn this section we present a numerical experiment to illustrate the theoretical results. We consider a 1-dimensional DDE as followṡIt is easy to show (z, α) = (0, 0) is a Takens-Bogdanov point of (48), cf.[12,22]. The forward Euler method for solving it is given byFigure 1.In the parameter plane (α 1 , α 2 ), the fixed point of z = 2(α 2 − α 1 ) is a focus when (α 1 , α 2 ) belongs to the left region of l ε h , when the parameter (α 1 , α 2 ) moves right and crosses l ε h , it turns into a central point and there will be periodic solutions bifurcating from this point. The periodicity will tend to infinity as the parameter α moves right and tends to l ε ∞ . 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Naudot, Width of the homoclinic zone in the parameter space for quadratic maps, Experiment. Math., 18 (2009), 409-427. Iooss A Hopf bifurcation theorem for difference equations approximating a differential equation. J Hofbauer, G , Monatsh. Math. 98J. Hofbauer and G. Iooss A Hopf bifurcation theorem for difference equations approximating a differential equation, Monatsh. Math., 98 (1984), 99-113. Periodic orbits of delay differential equations under discretization. K In&apos;t Hout, C Lubich, BIT. 38K. In't Hout and C. Lubich, Periodic orbits of delay differential equations under discretiza- tion, BIT, 38(1998), 71-91. Preservation of oscillations of the Runge-Kutta method for equation x ′ (t) + ax(t) + a 1 x. M Liu, J Gao, Z Yang, Comput. Math. Appl. 580t − 1M. Liu, J. Gao and Z. Yang, Preservation of oscillations of the Runge-Kutta method for equation x ′ (t) + ax(t) + a 1 x([t − 1]) = 0, Comput. Math. Appl., 58 (2009), 1113-1125. Preservation of bifurcations under Runge-Kutta methods. L Lóczi, J P Chávez, International Journal of Qualitative Theory of Differential Equations and Applications. 3L. Lóczi and J. P. Chávez, Preservation of bifurcations under Runge-Kutta methods, International Journal of Qualitative Theory of Differential Equations and Applications, 3 (2009), 81-98. Consistency of local dynamics and bifurcation of continuoustime dynamical systems and their numerical discretizations. X Wang, E Blum, Q Li, J. Differ. Equations Appl. 4X. Wang, E. Blum and Q. Li, Consistency of local dynamics and bifurcation of continuous- time dynamical systems and their numerical discretizations, J. Differ. Equations Appl., 4 (1998), 29-57. Introduction to applied nonlinear dynamical systems and chaos. S Wiggins, SpringerNew YorkS. Wiggins, "Introduction to applied nonlinear dynamical systems and chaos," Springer, New York, 1990. Numerical Hopf bifurcation for a class of delay differential equations. V Wulf, N Ford, J. Comput. Appl. Math. 115V. Wulf and N. Ford, Numerical Hopf bifurcation for a class of delay differential equations, J. Comput. Appl. Math., 115 (2000), 601-616. Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity. Y Xu, M Huang, J. Differential Equations. 244Y. Xu and M. Huang, Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, J. Differential Equations, 244 (2008), 582-598. Preservation of Hopf bifurcation under the Euler discretization of delay differential systems. Y Xu, M Huang, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms. 17Y. Xu and M. Huang, Preservation of Hopf bifurcation under the Euler discretization of delay differential systems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 17 (2010), 347-355. Preservation of homoclinic orbits under discretization of delay differential equations. Y Xu, Y Zou, Discrete Contin. Dyn. Sys. 31Y. Xu and Y. Zou, Preservation of homoclinic orbits under discretization of delay differ- ential equations, Discrete Contin. Dyn. Sys., 31 (2011), 275-299. On manifolds of connecting orbits in discretizations of dynamical systems. Y Zou, W. -J Beyn, Nonlinear Analysis: TMA. 52Y. Zou and W. -J. Beyn, On manifolds of connecting orbits in discretizations of dynamical systems, Nonlinear Analysis: TMA, 52 (2003), 1499-1529.

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{'abstract': 'A new technique for calculating the normal forms associated with the map restricted to the center manifold of a class of parameterized maps near the fixed point is given first. Then we show the Takens-Bogdanov point of delay differential equations is inherited by the forward Euler method without any shift and turns into a 1:1 resonance point. The normal form near the 1:1 resonance point for the numerical discretization is calculated next by applying the new technique to the map defined by the forward Euler method. The local dynamical behaviors are analyzed in detail through the normal form. It shows the Hopf point branch and the homoclinic branch emanating from the Takens-Bogdanov point are O(ε) shifted by the forward Euler method, where ε is step size. At last, a numerical experiment is carried to show the results. * Supported by NSFC grants 10971022 and 11071102.', 'arxivid': '1202.4337', 'author': ['Yingxiang Xu ', 'Chengchun Gong '], 'authoraffiliation': [], 'corpusid': 119144754, 'doi': '10.1007/s10884-014-9354-5', 'github_urls': [], 'n_tokens_mistral': 17093, 'n_tokens_neox': 14307, 'n_words': 8821, 'pdfsha': '058114844980aad3ac917d10abc820036b805b0f', 'pdfurls': ['https://arxiv.org/pdf/1202.4337v1.pdf'], 'title': ['Preservation of Takens-Bogdanov bifurcations for delay differential equations by Euler discretization *', 'Preservation of Takens-Bogdanov bifurcations for delay differential equations by Euler discretization *'], 'venue': []}

arxiv

Asymptotic Coupling and Its Applications in Information Theory 22 Aug 2021 Asymptotic Coupling and Its Applications in Information Theory 22 Aug 20211Index Terms-CouplingMaximal GuessingIntrinsic Ran- domnessChannel ResolvabilityPerfect Stealth/Covertness and Secrecy A coupling of two distributions PX and PY is a joint distribution PXY with marginal distributions equal to PX and PY . Given marginals PX and PY and a real-valued function f of the joint distribution PXY , what is its minimum over all couplings PXY of PX and PY ? We study the asymptotics of such coupling problems with different f 's and with X and Y replaced by X n = (X1, . . . , Xn) and Y n = (Y1, . . . , Yn) where Xi and Yi are i.i.d. copies of random variables X and Y with distributions PX and PY respectively. These include the maximal coupling, minimum distance coupling, maximal guessing coupling, and minimum entropy coupling problems. We characterize the limiting values of these coupling problems as n tends to infinity. We show that they typically converge at least exponentially fast to their limits. Moreover, for the problems of maximal coupling and minimum excess-distance probability coupling, we also characterize (or bound) the optimal convergence rates (exponents). Furthermore, for the maximal guessing coupling problem we show that it is equivalent to the distribution approximation problem. Therefore, some existing results for the latter problem can be used to derive the asymptotics of the maximal guessing coupling problem. We also study the asymptotics of the maximal guessing coupling problem for two general sources and a generalization of this problem, named the maximal guessing coupling through a channel problem. We apply the preceding results to several new informationtheoretic problems, including exact intrinsic randomness, exact resolvability, channel capacity with input distribution constraint, and perfect stealth and secrecy communication. I. INTRODUCTION A coupling of two probability distributions P X and P Y is a joint distribution P XY such that the marginals on X and Y are P X and P Y respectively. Given two marginal distributions P X and P Y and a function f of the joint distribution P XY , what is the minimum of f (P XY ) over all couplings P XY of P X and P Y ? This problem has been studied for different functions f [1]- [4]. In this work, we investigate asymptotics of several coupling problems for product marginal distributions P X n = P n X and P Y n = P n Y , when the dimension of the distributions n tends to infinity. These problems include the maximal coupling problem, the minimum distance coupling problem, the maximal guessing coupling problem, and the minimum entropy coupling problem (or the maximum mutual information coupling problem). Our results have several applications in information theory, including the following: [leftmargin=*] 1) Exact intrinsic randomness: The intrinsic randomness is the problem of determining the amount of randomness contained in a source [5]. Given an arbitrary general source X = {X n } ∞ n=1 (usually called the coin source), we try to approximate, by using X = {X n } ∞ n=1 , a uniform random number with as large rates as possible. Vembu and Verdú [5] and Han [6] determined the supremum of achievable uniform random number generation rates, by invoking the information spectrum method. In this paper, we consider a new variation of this problem, named the exact intrinsic randomness. We require the output to be exactly a uniform random number. Since in general there is no function satisfying such a requirement, we relax the mapping to be an asymptotic function (i.e., the mapping asymptotically almost surely approaches some target function as the blocklength tends to infinity; see Definition 6), instead of a function. 2) Exact resolvability: The channel resolvability problem is the problem of determining how much information is needed to simulate a random process through a given channel so that it approximates a target output distribution. This problem was first studied by Han and Verdú [7]. In [7], the total variation (TV) distance and the normalized relative entropy (Kullback-Leibler divergence) were used to measure the level of approximation. The resolvability problem with the unnormalized relative entropy was studied by Hayashi [8], [9]. Recently, Liu, Cuff, and Verdú [10] and Yu and Tan [11] extended the theory of resolvability by respectively using the so-called E γ metric with γ ≥ 1 and various Rényi divergences to measure the level of approximation. In this paper, we define a new variation of the channel resolvability problem, named exact channel resolvability. We now require the output to exactly match the target distribution. Again since in general there is no function satisfying such requirement, we relax the mapping to be an asymptotic function. A related problem named exact common information was studied by Kumar, Li, and Gamal [12], where differently from our definition, they required the mapping to be a function and variablelength codes were allowed. For their problem, to obtain the exact output distribution, the input, in general, does not follow the uniform distribution. Hence Kumar, Li, and Gamal's definition is input-distribution sensitive, in contrast to our definition here. 3) Perfect stealth and secrecy communication: In [13], Hou and Kramer defined a new security measure-effective secrecy-for wiretap channels that incorporates into its framework not only reliability and secrecy but also stealth. The signal overheard by the eavesdropper from her channel is forced to be close to a target distribution (i.e., the output distribution of the channel when there is no useful information transmitted). Hou and Kramer used ideas from channel resolvability to study the ef-fective secrecy capacity (the maximum rate which can be transmitted in a stealthy, secret, and reliable way) of wiretap channels, where they used the relative entropy to measure the level of secrecy and stealth. Furthermore, if we set the target distribution as the channel output distribution induced by some fixed channel input x 0 (the channel input symbol when the channel is idle), then the communication problem with stealth reduces to the so-called covert communication problem. In the covert communication problem, a sender Alice wishes to reliably transmit a message to a receiver Bob over a wiretap channel, while simultaneously ensuring that her transmission cannot be detected by an eavesdropper Eve, who observes the transmitted signal through the wiretap channel. Most researchers focused on the regime that Eve is asymptotically unable to detect the transmission, i.e., the probability of detection vanishes as the blocklength tends to infinity. For such a scenario, Bash et al. [14], [15], Wang et al. [16], and Bloch [17] showed that for Gaussian or discrete memoryless wiretap channels the number of bits that can be reliably and covertly transmitted over n channel uses scales as Θ( √ n), as long as the no-input symbol is not redundant, i.e., the output distribution at the eavesdropper induced by the no-input symbol is not a mixture of the output distributions induced by other input symbols. This is colloquially known as the "square root law". On the other hand, if the no-input symbol is redundant, and the secret key length shared by Alice and Bob is sufficiently long, then the number of bits that can be reliably and covertly transmitted over n channel uses linearly increases as n goes to infinity [16], [17]. In contrast to Hou and Kramer's work [13], we generalize the effective secrecy problem by forcing the channel output to exactly match the target distribution rather than approximately. Hence, the problem studied here can be termed as a perfectly stealthy and secret communication problem. Furthermore, if we set the target distribution to be the channel output distribution induced by a channel input fixed to be x 0 , then our problem reduces to the perfectly covert and secret communication problem. Furthermore, maximal couplings have been widely studied in probability theory and information theory; see, e.g., [18]- [22] and references therein. The main difference between our work and these works is that we consider the asymptotic scenario when X and Y are replaced by X n = (X 1 , . . . , X n ) and Y n = (Y 1 , . . . , Y n ) where X i and Y i are i.i.d. copies of random variables X and Y with distributions P X and P Y respectively and n tends to infinity. In all these papers, the authors consider the finite length (typically one-shot) case. Furthermore, most of these works are only concerned with maximal couplings, i.e., couplings that maximize P {X = Y } whereas we are interested in several more general functionals of P XY . Besides these works, [4] used several distance measures between distributions to study the source resolvability problem (and also the source coding problem), where the definitions of those measures involve optimization over couplings. In the source resolvability problem, the target distribution is fixed but the generated (code-induced) distribution is not. Hence one of the marginal distributions of couplings in the optimization problems involved in [4] is fixed, but the other marginal distribution is not fixed. However, in this paper, both of the marginal distributions are fixed. A. Main Contributions Our main contributions are as follows: [leftmargin=*] 1) We study the asymptotics of several coupling problems, including the problems of maximal coupling, minimum distance coupling, maximal guessing coupling, and minimum entropy coupling (or maximum mutual information coupling). We characterize the limiting values of these coupling problems as the dimension goes to infinity. We show that they typically converge at least exponentially fast to their limits. Moreover, for the maximal coupling and minimum excess-distance probability coupling problems, we also characterize the optimal convergence rates of these two coupling problems. Interestingly, product couplings achieve the optimal limiting values of these coupling problems, but they cannot achieve the optimal convergence rates. Hence, for these two problems, non-product couplings strictly outperform product couplings in the exponent sense. Furthermore, we show that the maximal guessing coupling problem is equivalent to the traditional distribution approximation problem [6,Sec. 2.1]. Therefore, some existing results on the latter problem can be used to derive asymptotic results on the former problem. 2) We also consider the asymptotics of the maximal guessing coupling problem for two general sources and a generalization of this problem, named as the maximal guessing coupling through a channel problem. We derive upper and lower bounds on the fundamental limits of these two problems. As a by-product, these upper bounds and lower bounds are also bounds on the fundamental limits of the general source-channel resolvability problem, in which the source and channel are general and the source is a part of the channel input. 3) We apply the preceding results to several novel information-theoretic problems, including the exact intrinsic randomness, exact resolvability, channel capacity with input distribution constraint, and perfect stealth and secrecy communication problems. For the exact intrinsic randomness and exact source resolvability problems, we show that they are respectively equivalent to the traditional (approximate) intrinsic randomness and source resolvability problems. For the exact resolvability problem, we completely characterize the optimal rate for fullrank channels. For the problem of channel capacity with an input distribution constraint, we show that the channel capacity under condition that the input distribution is constrained to be some product distribution is the Gács-Körner common information between the channel input and the channel output. For perfect stealth and secrecy communication, we show that 1) the perfect stealthsecrecy capacity is positive if and only if the wiretap channel is a P Z -redundant channel; 2) for full-rank wiretap channels, the perfect stealth-secrecy capacity is zero, and the perfect stealth/covertness capacity (the maximum rate can be transmitted in the perfectly stealthy or covert way) is the Gács-Körner common information C GK (X; Y ), where P X is the unique distribution that induces P Z through P Z|X . Our initial motivation of studying these coupling problems stems from the fact that perfect stealth and secrecy communication problems are of great practical significance. We show that the maximal guessing coupling problem is of crucial importance to solving these problems communication problems. Furthermore, as by-products of applying our results on coupling problems to the perfect stealth and secrecy communication problem, we also obtain some intermediate and interesting results, e.g., the channel capacity with input distribution constraint problem, the exact intrinsic randomness problem, and the exact resolvability problem. B. Notation We use P X (x) to denote the probability distribution of a random variable X, which is also shortly denoted as P (x) (when the random variable X is clear from the context). We also use P X , P X , and Q X to denote various probability distributions with alphabet X . The set of probability distributions on X is denoted as P (X ), and the set of conditional probability distributions on Y given a variable in X is denoted as P (Y|X ) := P Y |X : P Y |X (·|x) ∈ P (Y) , x ∈ X . Given P X and P Y |X , we write [P Y |X • P X ](y) := x P Y |X (y|x)P X (x). For simplicity, all the alphabets involved in this paper are assumed to be finite, unless stated explicitly. We use T x n (x) := 1 n n i=1 1 {x i = x} to denote the type (empirical distribution) of a sequence x n , and T X to denote a type of sequences in X n , where the indicator function 1{A} equals 1 if the clause A is true and 0 otherwise. For a type T X , the type class (set of sequences having the same type T X ) is denoted by T (T X ). The set of types of sequences in X n is denoted as P n (X ) := {T x n : x n ∈ X n }. The -typical set relative to Q X is denoted as T n (Q X ) := {x n ∈ X n : |T x n (x) − Q X (x)| ≤ Q X (x) , ∀x ∈ X }. For brevity, we sometimes write T n (Q X ) as T n . Other notation generally follow the book by Csiszár and Körner [23]. The total variation distance between two probability mass functions P and Q with a common alphabet X is defined by |P − Q| := 1 2 x∈X |P (x) − Q(x)|.(1) By the definition of -typical set, we have that for any x n ∈ T n (Q X ), |T x n − Q X | ≤ 2 . We use P X or P Y |X to denote the vector or matrix form of P X or P Y |X . We use P ⊗n to denote n-fold Kronecker product of a vector or matrix P . We use Z = {Z n } ∞ n=1 to denote a general source, and P Y |X = {P Y n |X n } ∞ n=1 to denote a general channel [6]. For any given sequence of random variables {Z n } ∞ n=1 , we introduce quantities which play an important role in information spectrum analysis [6]. For δ ∈ [0, 1], the δ-limit superior in probability is defined as δ-p-lim sup n→∞ Z n := inf α : lim sup n→∞ P{Z n > α} ≤ δ . (2) For δ = 0, p-lim sup n→∞ Z n := 0-p-lim sup n→∞ Z n(3) and p-lim inf n→∞ Z n := −p-lim sup n→∞ (−Z n ).(4)Furthermore, ı X;Y (x; y) := log P Y |X (y|x) P Y (y) is the information density 1 , and ı X (x) := ı X;X (x; x) = log 1 P X (x) is the entropy density. We define the sup-and inf-entropy rates respectively as H(Z) := p-lim sup n→∞ 1 n ı Z n (Z n ) and(5)H(Z) := p-lim inf n→∞ 1 n ı Z n (Z n ).(6) Finally , we write f (n)≤ g(n) if lim sup n→∞ 1 n log f (n) g(n) ≤ 0. In addition, f (n) . = g(n) if and only if f (n)≤ g(n) and g(n)≤ f (n). C. Preliminaries Definition 1. The set of couplings of P X ∈ P (X ) and P Y ∈ P (Y) is defined as C(P X , P Y ) := {Q XY ∈ P (X × Y) : Q X = P X , Q Y = P Y } (7) Any Q XY ∈ C(P X , P Y ) is called a coupling of P X , P Y . Definition 2. The maximal equality-probability over couplings of two distributions P X , P Y ∈ P (X ) is defined as M(P X , P Y ) := max P XY ∈C(P X ,P Y ) P {Y = X} .(8) Any Q XY ∈ C(P X , P Y ) achieving M(P X , P Y ) is called a maximal coupling of P X , P Y . The maximal coupling problem has the following property. Lemma 1 (Maximal Coupling Equality). [1] Given two distributions P X and P Y , we have M(P X , P Y ) = 1 − |P X − P Y |.(9) Assume P X , Q X are two distributions defined on a set X . If P X = Q X , then obviously, |P n X − Q n X | = 0 for all n ∈ N. If P X = Q X , the following lemma holds. Lemma 2 (Asymptotics of Total Variation). [24, Theorem 11.9.1] Assume P X , Q X are two distinct distributions defined on a set X . Then |P n X − Q n X | → 1 exponentially fast as n → ∞. More explicitly, the exponent is lim n→∞ − 1 n log (1 − |P n X − Q n X |) = min R X ∈P(X ) max {D(R X P X ), D(R X Q X )} (10) = B(P X , Q X ),(11) where B(P X , Q X ) := max 0≤λ≤1 − log x P X (x) λ Q X (x) 1−λ( 12) denotes the Chernoff information between P X and Q X . Remark 1. Equality (11) is justified by the fact that on the one hand, 1 − |P n X − Q n X | is the smallest sum of type-I and type-II error probabilities for a binary hypothesis test between P n X and Q n X (see, for example, [25, Theorem 13.1.1]); on the other hand, B(P X , Q X ) is the exponent of this sum of two error probabilities [24, Theorem 11.9.1]. II. MAXIMAL COUPLING AND MINIMUM DISTANCE COUPLING In this section, we focus on asymptotic behaviors of two basic coupling problems: the maximal coupling problem and the minimum distance coupling problem. A. Maximal Coupling We first consider the asymptotic behavior of maximal equality-probability M(P n X , P n Y ). First, it is obvious that if P X = P Y , then M(P n X , P n Y ) = 1 for all n ∈ N. Furthermore, the optimal coupling for this case is P XY (x, y) = P X (x)1{y = x}. On the other hand, if P X = P Y , we have the following theorem. Proposition 1 (Maximal Coupling). Assume P X , P Y are two distinct distributions defined on a set X . Then given product marginal distributions P n X and P n Y , we have M(P n X , P n Y ) → 0 exponentially fast as n → ∞. More explicitly, the exponent is lim n→∞ − 1 n log M(P n X , P n Y ) = min Q max {D(Q P X ), D(Q P Y )} (13) = B(P X , P Y ),(14) where B(P X , P Y ) is defined in (12). Proof: We prove this lemma by using a property of the TV distance. According to the maximal coupling equality (Lemma 1) and Lemma 2, we have M(P n X , P n Y ) = 1 − |P n X − P n Y | (15) . = e −n min Q∈P(X ) max{D(Q P X ),D(Q P Y )} . (16) Hence, the optimal exponent is given by min Q∈P(X ) max {D(Q P X ), D(Q P Y )} = B(P X , P Y ). For a product coupling P X n Y n = P n XY with P XY achieving M(P X , P Y ), we have P {Y n = X n } = P {Y = X} n .(17) Hence the best exponent for product couplings is − log M(P X , P Y ) = − log (1 − |P X − P Y |) . Note that a product coupling P X n Y n = P n XY with P XY achieving M(P X , P Y ) only achieves the exponent − log M(P X , P Y ) = − log (1 − |P X − P Y |), which is suboptimal in general, i.e., B(P X , P Y ) ≤ − log (1 − |P X − P Y |) .(18) The following example shows the inequality in (18) can be strict. Example 1. P X = { 1 2 , 1 2 }, P Y = { 1 4 , 3 4 } then min Q max {D(Q P X ), D(Q P Y )} ≤ D(P X P Y ) = 1 2 log 2 4 3 (19) < − log (1 − |P X − P Y |) = log 2 4 3 .(20) B. Minimum Distance Coupling -Transportation Theory Next we consider the minimum (expected) distance coupling problem, which is the main problem studied in transportation theory. The Wasserstein metric is a special case of this coupling problem by specializing the distance measure to be the quadratic distortion measure. Define an additive function (general distance or distortion) d(x n , y n ) := 1 n n i=1 d(x i , y i )(21) where d(x, y) is some arbitrary function (distance) of x, y. Definition 3. The minimum (expected) distance over couplings of two distributions P X , P Y is defined as D(P X , P Y ) := min P XY ∈C(P X ,P Y ) Ed(X, Y ).(22) Any Q XY ∈ C(P X , P Y ) achieving D(P X , P Y ) is called a minimum (expected) distance coupling of P X , P Y . Then given two marginal product distributions P n X and P n Y , the minimum expected distance over couplings of P X , P Y is clearly D(P n X , P n Y ) = D(P X , P Y ).(23) Next we consider another important coupling problem. Definition 4. The minimum excess-distance probability over couplings of two distributions P X , P Y is defined as D d (P X , P Y ) := min P XY ∈C(P X ,P Y ) P {d(X, Y ) > d} .(24) Any Q XY ∈ C(P X , P Y ) achieving D d (P X , P Y ) is called a minimum excess-distance probability coupling of P X , P Y . The excess-distance probability (or excess-distortion probability) is an important distortion measure in information theory [4], [6]. Define the exponents as E(d) := lim inf n→∞ − 1 n log 1 − D d (P n X , P n Y )(25) and E(d) := lim inf n→∞ − 1 n log D d (P n X , P n Y ).(26) An asymptotic result for the problem of minimum excessdistance probability coupling is stated in the following theorem. The proof is provided in Appendix A. Proposition 2 (Minimum Excess-Distance Probability Coupling). Given two distributions P X and P Y , we have: [leftmargin=*] 1) If D(P X , P Y ) > d, then D d (P n X , P n Y ) → 1 exponentially fast as n → ∞. Moreover, we have E(d) = min Q XY :E Q d(X,Y )≤d max {D(Q X P X ), D(Q Y P Y )} . (27) 2) If D(P X , P Y ) < d, then D d (P n X , P n Y ) → 0 at least exponentially fast as n → ∞. Moreover, we have E(d) ≥ max t≥0 td − log Ee td(X,Y ) .(28)3) If D(P X , P Y ) = d, then 1 2 + O 1 √ n ≤ D d (P n X , P n Y ) ≤ 1. Remark 2. In Statement 1) of Proposition 2, the exponent is infinity if inf d : D d (P X , P Y ) = 1 ≤ d. Remark 3. If D(P X , P Y ) > d, then an optimal product coupling P X n Y n = P n XY with P XY achieving D d (P X , P Y ) only achieves the exponent max t≥0 −td − log Ee −td(X,Y ) ≤ E(d).(29) If D(P X , P Y ) = d, then such an optimal product coupling achieves the lower bound 1 2 + O 1 √ n . III. MAXIMAL GUESSING COUPLING For the maximal coupling and minimum distance coupling problems, we showed that product couplings suffice to achieve the optimal limiting values of maximal equality-probability and minimum excess-distance probability (although they cannot achieve the optimal exponents). In the following, we consider several coupling problems for which product couplings are not optimal in achieving the optimal limiting values. A. Maximal Guessing Coupling: Memoryless Sources Next we define a new coupling problem, named the maximal guessing coupling problem. Definition 5. The maximal guessing probability over couplings of P X , P Y is defined as G(P X , P Y ) := max P XY ∈C(P X ,P Y ) max f :X →Y P {Y = f (X)} .(30) Any Q XY ∈ C(P X , P Y ) achieving G(P X , P Y ) is called a maximal guessing coupling of P X , P Y . Moreover, if a maximal guessing coupling satisfies G(P X , P Y ) = 1, then we call it deterministic coupling. Given a sequence of distribution pairs (P X n , P Y n ), if a sequence of maximal guessing couplings {Q X n ,Y n } n∈N satisfies G(P X n , P Y n ) → 1 as n → ∞, then {Q X n ,Y n } n∈N is called an asymptotically deterministic coupling. Besides, we introduce a new concept, named the asymptotic function. Definition 6. We say Y n is an asymptotic function of X n if lim n→∞ P {Y n = f n (X n )} = 1 for some sequence of functions {f n } ∞ n=1 . Hence under the asymptotically deterministic coupling {Q X n ,Y n } n∈N , Y n is an asymptotic function of X n . Furthermore, the quantity max f P {Y = f (X)} is called the guessing probability; see [26]- [29]. Note that here and also in these papers, the guessing terminal is only allowed to guess once; however, in [30]- [34] it is allowed to guess multiple times. The deterministic coupling and asymptotically deterministic coupling are closely related to the distribution matching problem [35], [36], which is the following. Given a sequence of distribution pairs (P X n , P Y n ), find a sequence of distributions P W n and a sequence of deterministic couplings of (P W n , P Y n ) such that P W n and P X n are asymptotically equal under a normalized or unnormalized divergence measure. If we loosen the requirement to finding a sequence of asymptotically deterministic couplings, and strengthen the constraint on the closeness of P W n and P X n to be the equality P W n = P X n , then the distribution matching problem becomes the asymptotically deterministic coupling problem. That is, given a sequence of distribution pairs (P X n , P Y n ), we would like to find a sequence of couplings of (P X n , P Y n ) such that G(P X n , P Y n ) → 1 as n → ∞. Furthermore, our results concerning maximal guessing couplings or asymptotically deterministic couplings will be applied to information-theoretic problems in Sections IV-VII. By the maximal coupling equality (Lemma 1), we can prove the following property of maximal guessing coupling, which shows the equivalence between the maximal guessing coupling problem and distribution approximation problem [6]. Definition 7. [27] Define the minimum α-Rényi conditional entropy over couplings of two distributions P X , P Y as H (c) α (P X , P Y ) := min P XY ∈C(P X ,P Y ) H α (Y |X),(31) with the Arimoto-Rényi conditional entropy of order α ∈ [0, ∞] given by [34], [37] H α (Y |X) :=                  α 1−α log E y P α Y |X (y|X) 1 α , α ∈ (0, 1) ∪(1, ∞) max x∈X log y ∈ Y : P Y |X (y|x) > 0 , α = 0 −E log P Y |X (Y |X), α = 1 − log E max y∈Y P Y |X (y|X) , α = ∞ . We also call the minimum ∞-Rényi conditional entropy H (c) ∞ (P X , P Y ) over couplings of P X , P Y as minimum conditional min-entropy, and the minimum 1-Rényi conditional entropy H (c) 1 (P X , P Y ) over couplings of P X , P Y (shortly denoted as H (c) (P X , P Y )) as minimum (Shannon) conditional entropy. Note that H Theorem 1 (Maximal Guessing Coupling Equality). The maximal guessing coupling problem is equivalent to the distribution approximation problem. That is, e −H (c) ∞ (P X ,P Y ) = G(P X , P Y ) = 1 − min f |P Y − P f (X) |. (32) Moreover, assume that f is an optimal function for the distribution approximation problem, and P f (X),Y is a maximal coupling of P f (X) , P Y , i.e., f is a minimizer of min f |P Y −P f (X) | and P f (X),Y is a maximizer of the problem max P f (X),Y ∈C(P f (X) ,P Y ) P {Y = f (X)} .(33) Then P X P Y |f (X) is a maximal guessing coupling of P X , P Y . Remark 4. (32) (with min and max respectively replaced by inf and sup) also holds for general distributions P X , P Y , e.g., continuous distributions. Remark 5. Since H (c) ∞ (P X , P Y ) ≤ H 0 (P Y ) = log |Y|, we have G(P X , P Y ) = 1 − min f |P Y − P f (X) | ≥ 1 |Y| . Proof: Exchanging minimization operations, we have G(P X , P Y ) = min f min P XY ∈C(P X ,P Y ) P {Y = f (X)} . (34) Now we prove that given a function f , min P XY ∈C(P X ,P Y ) P {Y = f (X)} = min P f (X),Y ∈C(P f (X) ,P Y ) P {Y = f (X)} .(35) Define Q X,Y := arg min P XY ∈C(P X ,P Y ) P {Y = f (X)} , and(36)Q f (X),X,Y (v, x, y) := Q X,Y (x, y)1{v = f (x)}.(37) Then we have min P XY ∈C(P X ,P Y ) P {Y = f (X)} = P Q X,Y {Y = f (X)} (38) = P Q f (X),Y {Y = f (X)}(39) ≥ min P f (X),Y ∈C(P f (X) ,P Y ) P {Y = f (X)} .(40) On the other hand, denote Q f (X),Y := arg min P f (X),Y ∈C(P f (X) ,P Y ) P {Y = f (X)}(41) and Q f (X),X,Y (v, x, y) := P X (x)1{v = f (x)}Q Y |f (X) (y|v) (42) = P X (x)Q Y |f (X) (y|f (x))1{v = f (x)}.(43) Then we also have min P f (X),Y ∈C(P f (X) ,P Y ) P {Y = f (X)} = P Q f (X),Y {Y = f (X)}(44)= P Q X,Y {Y = f (X)} (45) ≥ min P XY ∈C(P X ,P Y ) P {Y = f (X)} .(46) Combining (40) and (46) we have the desired equality (35). Substituting (35) into (34), we have min P XY ∈C(P X ,P Y ) min f P {Y = f (X)} = min f min P f (X),Y ∈C(P f (X) ,P Y ) P {Y = f (X)} (47) = min f |P Y − P f (X) |,(48) where (48) follows from Lemma 1. Furthermore, the first equality of (32) follows from the fact that max f P {Y = f (X)} = E max y P (y|X) = e −H∞(Y |X) [27,Proposition 4.2]. By Theorem 1, to solve the maximal guessing coupling problem, we only need to compute min f (x) P Y − P f (X) .(49)Define A(y) := {x : f (x) = y}. Then (49) is equivalent to min {A(y):y∈Y} y |P Y (y) − P X (A(y))| ,(50) where {A(y) : y ∈ Y} is a partition of X , i.e., y∈Y A(y) = X and A(y 1 ) ∩ A(y 2 ) = ∅ for any y 1 , y 2 ∈ Y and y 1 = y 2 . For any distribution pair P X , P Y , is (50) equal to zero? This question is equivalent to the following: Does there exist a partition {A(y) : y ∈ Y} such that P Y (y) = P X (A(y)) for all y ∈ Y? This problem involving the search for an optimal partition has been shown to be NP-hard [40]. This implies that the optimization problem (50) is also NP-hard, since in general, solving the optimization problem (50) is strictly harder than only determining whether (50) equals zero. However, when we consider the asymptotic scenario, the optimal limiting value of this coupling problem can be easily determined. Furthermore, we also provide bounds on the rates of convergence of the coupling problems to their limiting values. Define the optimal exponents as E (P X , P Y ) := lim inf n→∞ − 1 n log (1 − G(P n X , P n Y )) ,(51) and E (P X , P Y ) := lim inf n→∞ − 1 n log G(P n X , P n Y ).(52) Then we have the following main result. The proof is provided in Appendix B. Theorem 2 (Maximal Guessing Coupling). Given two product marginal distributions P n X and P n Y , we have: [leftmargin=*] 1) If H(X) > H(Y ) , then G(P n X , P n Y ) → 1 at least exponentially fast as n → ∞. Moreover, we have E (P X , P Y ) ≥ 1 2 max t∈[0,1] t (H 1+t (X) − H 1−t (Y )) .(53) 2) If H(X) < H(Y ), then G(P n X , P n Y ) → 0 exponentially fast as n → ∞. Moreover, we have log |Y| ≥ E (P X , P Y ) ≥ sup ∈(0,1) min δ (P X ), δ (P Y ), (1 − )H(Y ) − (1 + )H(X) ,(54)with δ (P X ) := 1 3 2 min x:P X (x)>0 P X (x). 3) If H(X) = H(Y ), then G(P n X , P n Y ) ≥ G(P X , P Y ) n for all n. Remark 6. The exponent whenever H(X) ≥ H(Y ) is infinity if there exists a coupling P XY such that Y is expressed as a deterministic function of X. Remark 7. By the equivalence between the maximal guessing coupling problem and the distribution approximation problem (Theorem 1), the exponential bounds given in Theorem 2 are also bounds for the distribution approximation problem min f |P Y − P f (X) |. Remark 8. Theorem 2 implies that given two product distributions P n X and P n Y with H(X) > H(Y ), there exists a joint distribution P X n Y n satisfying lim n→∞ min {A(y n ):y n ∈Y n } y n |P Y n (y n ) − P X n (A(y n ))| = 0,(55) where {A(y n ) : y n ∈ Y n } is a partition of X n . Hence the probability values of P X n asymptotically forms a refinement of the probability values of P Y n in the sense of (55). This is just a restatement of the soft-covering lemma [41]. Since we get e −H (c) ∞ (P n X ,P n Y ) = G(P n X , P n Y ) from (32), the following result follows from Theorem 2: Corollary 1. Given two product marginal distributions P n X and P n Y , we have: [leftmargin=*] 1) If H(X) > H(Y ), then H (c) ∞ (P n X , P n Y ) → 0 at least exponentially fast as n → ∞ with exponent E (P X , P Y ). 2) If H(X) < H(Y ), then H (c) ∞ (P n X , P n Y ) → ∞ linearly as n → ∞ with scaling factor E (P X , P Y ). 3) If H(X) = H(Y ), then H (c) ∞ (P n X , P n Y ) ≥ nH (c) ∞ (P X , P Y ) for all n. Theorem 2 does not give an asymptotically tight expression if H(X) = H(Y ). However, we conjecture the following: Conjecture 1 (Asymptotically Deterministic Coupling). As- sume H(X) = H(Y ). Then G(P n X , P n Y ) → 1 if and only if G(P X , P Y ) = 1 (this is also equivalent to the fact that P X and P Y have the same probability values). This conjecture implies when H(X) = H(Y ), G(P n X , P n Y ) → 1 requires some "matched" condition on the distributions. In Appendix C-A, we prove that Conjecture 1 is true if P X or P Y is a uniform distribution. Similar to the conjecture concerning asymptotically deterministic couplings, we also have the following conjecture concerning the deterministic couplings. Conjecture 2 (Deterministic Coupling). G(P n X , P n Y ) = 1 if and only if G(P X , P Y ) = 1. That is, there exists a deterministic coupling P X n Y n ∈ C(P n X , P n Y ) for which Y n is a function of X n , if and only if there exists a deterministic coupling P XY ∈ C(P X , P Y ) for which Y is a function of X. W X Y | Y WX P   f w In Appendix C-B, we prove that Conjecture 2 is true for two special cases. B. Maximal Guessing Coupling: General Sources and Coupling Through a Channel In the previous subsection, we showed that the maximal guessing coupling problem is equivalent to the distribution approximation problem. Hence, to obtain the maximal guessing coupling of a pair of sources, we only need to solve the problem of probability distribution approximation for these sources. Here, instead, we consider a more general variation of distribution approximation problem, called the general sourcechannel resolvability problem. This is illustrated in Fig. 1, and will be proven to be equivalent to a maximal guessing coupling through a channel problem. Consider a pair of distributions (P W , P Z ) and a channel P Y |W X (this is a source-dependent channel which reduces to a source-independent channel if we set P Y |W X = P Y |X ). Denote the output of the channel P Y |W X with input X = f (W ) as Y f . Obviously, the distribution of Y f is P Y f (y) = w P W (w)P Y |W X (y|w, f (w)).(56) If we consider f (W ) as a guessing function and Y f as the final estimate variable of the target variable Z, then the optimization problem min P W Z ∈C(P W ,P Z ) min f P {Z = Y f } can be seen as the problem of maximal guessing coupling through a channel. It is a generalization of the maximal guessing coupling problem, since it reduces to the maximal guessing coupling problem if the channel is set to be the identity channel, i.e., P Y |W X (y|w, x) = 1{y = x} for all (w, x, y). Definition 8. Define the maximal guessing probability through a channel P Y |X over couplings of (P W , P Z ) as G(P W , P Z |P Y |X ) := max P W Z ∈C(P W ,P Z ) max f :W→X max P Y |W Z :P Y |W (y|w)=P Y |W X (y|w,f (w)) P {Z = Y } . (57) Any Q XY ∈ C(P X , P Y ) achieving G(P W , P Z |P Y |X ) is called a maximal guessing coupling of P X , P Y through the channel P Y |X . On the other hand, the source-channel resolvability problem is min f |P Z − P Y f |. Similar to Theorem 1, the following theorem states the equivalence between the problem of maximal guessing coupling through a channel and the source-channel resolvability problem. Theorem 3 (Maximal Guessing Coupling Through a Channel). The problem of maximal guessing coupling through a channel is equivalent to the source-channel resolvability problem. That is, G(P W , P Z |P Y |X ) = 1 − min f |P Z − P Y f |.(58) Proof: Exchanging minimization operations, we have 1 − G(P W , P Z |P Y |X ) = min f min P W Z ∈C(P W ,P Z ) min P Y |W Z :P Y |W (y|w)=P Y |W X (y|w,f (w)) P {Z = Y } (59) = min f min P W ZY :P W Z ∈C(P W ,P Z ), P Y |W (y|w)=P Y |W X (y|w,f (w)) P {Z = Y } (60) = min f min P Z|W Y : w,y P W (w)P Y |W X (y|w,f (w)) ×P Z|W Y (z|w,y)=P Z (z) P {Z = Y } (61) = min f min P Z|Y : y P Y (y)P Z|Y (z|y)=P Z (z) P {Z = Y } (62) = min f min P Y f Z ∈C(P Y f ,P Z ) P {Z = Y f } (63) = min f |P Z − P Y f |,(64) where (62) follows since the optimized objective P {Z = Y } depends only on the joint distribution of Y, Z. Note that the coupling P W Z and the channel P Y |X are not independent, i.e., the channel P Y |X is allowed to be embedded into the optimal coupling P W Z . If such embedding is not allowed, then the problem reduces to G(P W , P Z |P Y |X ) := max P W Z ∈C(P W ,P Z ) max f P {Z = Y } ,(65) where the probability is taken under the distribution P W Z (w, z)P Y |W X (y|w, f (w)). However, for this problem, the equivalence above no longer holds. 1) One-shot Bounds: Next we derive following bounds for the source-channel resolvability problem. The proof of Theorem 4 is provided in Appendix D. Theorem 4 (General Source-Channel Resolvability). For any distributions P W and P Z , channel P Y |W X , and τ > 0, we have min P X|W :P W XY (A0)=0 |P Y − P Z | ≤ min f :W→X |P Z − P Y f | (66) ≤ min P X|W {|P Y − P Z | + P W XY (A τ )} + 1 2 e τ /2 ,(67) where Y f is the output of the channel P Y |W X with input X = f (W ), and A τ := (w, x, y) : log P W (w)P Y |W X (y|w, x) P Y (y) > τ .(68) Furthermore, we have another lower bound min f :W→X |P Z − P Y f | ≥ min P W Z ∈C(P W ,P Z ) min P X|W P W XZ (B τ ) − e −τ ,(69) where B τ := (w, x, z) : log P W (w)P Y |W X (z|w, x) P Z (z) > τ .(70) If an identity channel P Y |W X (y|w, x) = 1{y = x} is considered, the source-channel resolvability problem degenerates into the source-source resolvability problem (using a general source to generate another general source) or equivalently, the distribution approximation problem. That is, min f |P Z − P Y f | = min f |P Z − P f (W ) | where P W is a source distribution and P Z is a target distribution. Theorem 4 results in the following corollary. Corollary 2 (General Source-Source Resolvability: Probability Distribution Approximation). For any source distribution P W and target distribution P Z , we have min P W Z ∈C(P W ,P Z ) P (A τ ) − e −τ ≤ min f |P Z − P f (W ) | (71) ≤ min P W Z ∈C(P W ,P Z ) P (A τ ) + 1 2 e τ /2 ,(72) where A τ := (w, z) : log P W (w) P Z (z) > τ .(73) 2) Asymptotics: When the asymptotic behavior is considered, Theorem 4 results in the following corollary. Corollary 3 (General Source-Channel Resolvability). For any source distribution P W , channel P Y |W X , and target distribution P Z , we have inf P X|W : p-lim sup n→∞ { 1 n ı(W n X n ;Y n )− 1 n ı(W n )}≤0 lim sup n→∞ |P Y n − P Z n | ≤ lim sup n→∞ min fn |P Z n − P Y n fn | (74) ≤ inf P X|W : p-lim sup n→∞ { 1 n ı(W n X n ;Y n )− 1 n ı(W n )}<0 lim sup n→∞ |P Y n − P Z n | .(75) Moreover, if an identity channel P Y |W X (y|w, x) = 1{y = x} is considered, Corollary 3 results in the following corollary. Corollary 4 (General Source-Source Resolvability: Probability Distribution Approximation). For any source distribution P W and target distribution P Z , we have (74)-(75) with ı(W n X n ; Y n ) replaced by ı(X n ) and Y n replaced by X n . Equivalently, inf δ>0, P W Z ∈C(P W ,P Z ): δ-p-lim sup n→∞ { 1 n ı(Z n )− 1 n ı(W n )}≤0 δ ≤ lim sup n→∞ min fn |P Z n − P fn(W n ) | (76) ≤ inf δ>0, P W Z ∈C(P W ,P Z ): δ-p-lim sup n→∞ { 1 n ı(Z n )− 1 n ı(W n )}<0 δ,(77) where C(P W , P Z ) := {P W Z : P W n Z n ∈ C(P W n , P Z n ), ∀n}. 3) Maximal Guessing Coupling for General Sources and Channels: According to the equivalence between the maximal guessing coupling problem and distribution approximation problem (Theorem 1) and the equivalence between the problem of maximal guessing coupling through a channel and the problem of source-channel resolvability (Theorem 3), we have the following conclusions. The bounds given in Theorem 5 and Corollary 3 are also bounds for the maximal guessing coupling problem through a channel. The bounds given in Corollaries 2 and 4 are also bounds for the maximal guessing coupling problem. C. Application of Maximal Guessing Coupling to Minimum Entropy Coupling The problems of minimum entropy coupling and maximum mutual information coupling were first studied in [3]. In this subsection, we study the asymptotics of these coupling problems. In [3], the authors showed that solving the minimum entropy coupling problem or maximum mutual information coupling problem is NP-hard. However, in this section, we show that is not the case for the asymptotic regime. Recall from Definition 7 the minimum conditional entropy H (c) (P X , P Y ) := min P XY ∈C(P X ,P Y ) H(Y |X) over couplings of P X , P Y . Then for such a coupling problem, we have the following result. Corollary 5 (Minimum Conditional Entropy Coupling). Given two product marginal distributions P n X and P n Y , we have [leftmargin=*] 1) H (c) (P n X , P n Y ) − n max{0, H(Y ) − H(X)} → 0 at least exponentially fast as n → ∞ if H(X) = H(Y ); 2) H (c) (P n X , P n Y ) ≤ nH (c) (P X , P Y ) for all n if H(X) = H(Y ). Proof: We only prove Statement 1). Statement 2) is obvious. One simply employs a product coupling to prove the upper bound. Denote p 0 ≤ H(Y n |X n ) ≤ H(p (n) e ) + np (n) e log |Y|,(78) where H(p) := −p log p − (1 − p) log(1 − p). From Theorem 2, we know that if H(X) > H(Y ), then there exists a coupling P X n Y n ∈ C(P n X , P n Y ) such that p (n) e → 0 at least exponentially fast as n → ∞. This implies the upper bound also converges to zero at least exponentially fast. Hence if H(X) > H(Y ), then H (c) (P n X , P n Y ) → 0 at least exponentially fast as n → ∞. On the other hand, we can write H(Y n |X n ) = H(X n |Y n ) + H(Y n ) − H(X n ). Since for a coupling P X n Y n ∈ C(P n X , P n Y ), H(Y n ) − H(X n ) = n (H(Y ) − H(X)), we have H (c) (P n X , P n Y ) = H (c) (P n Y , P n X ) + n (H(Y ) − H(X)). By the argument above, if H(X) < H(Y ), then H (c) (P n Y , P n X ) → 0 at least exponentially fast as n → ∞. Hence if H(X) < H(Y ), H (c) (P n X , P n Y ) − n (H(Y ) − H(X)) → 0 at least exponentially fast as n → ∞. Define the minimum joint entropy and the maximum mutual information over couplings of two distributions P X , P Y as H(P X , P Y ) := min P XY ∈C(P X ,P Y ) H(XY ), and(79)I(P X , P Y ) := max P XY ∈C(P X ,P Y ) I(X; Y )(80)respectively. Observe that H(XY ) = H(X) + H(Y |X) and I(X; Y ) = H(Y ) − H(Y |X). Hence H(P X , P Y ) = H(X) + H (c) (P X , P Y ) and I(P X , P Y ) = H(Y ) − H (c) (P X , P Y ). Combining these with Corollary 5, we obtain the following two corollaries. Corollary 6 (Minimum Joint Entropy Coupling). Given two product marginal distributions P n X and P n Y , we have [leftmargin=*] 1) H(P n X , P n Y ) − n max{H(X), H(Y )} → 0 at least ex- ponentially fast as n → ∞ if H(X) = H(Y ); 2) H(P n X , P n Y ) ≤ nH(P X , P Y ) for all n if H(X) = H(Y ) . Corollary 7 (Maximum Mutual Information Coupling). Given two product marginal distributions P n X and P n Y , we have [leftmargin=*] 1) I(P n X , P n Y ) − n min{H(X), H(Y )} → 0 at least expo- nentially fast as n → ∞ if H(X) = H(Y ); 2) I(P n X , P n Y ) ≥ nI(P X , P Y ) for all n if H(X) = H(Y ). Define the maximum conditional mutual information over couplings of two distributions P X , P Y Z as I (c) (P X , P Y Z ) := max P XY Z ∈C(P X ,P Y Z ) I(X; Y |Z). Corollary 8 (Maximum Conditional Mutual Information Cou- pling). Given two product marginal distributions P n X and P n Y Z , we have [leftmargin=*] 1) I (c) (P n X , P n Y Z ) − n min{H(X), H(Y |Z)} → 0 at least exponentially fast as n → ∞ if H(X) = H(Y |Z); 2) I (c) (P n X , P n Y Z ) ≥ nI (c) (P X , P Y Z ) for all n if H(X) = H(Y |Z). For Corollary 8, we use (X n , Z n ) with joint distribution P n X P n Z to guess (Y n , Z n ) with joint distribution P n Y Z if H(X) > H(Y |Z), or reversely, use (Y n , Z n ) to guess (X n , Z n ) if H(X) < H(Y |Z). The proof is along exactly the same lines as that of Corollary 5, and hence omitted here. Recall from Definition 7 the minimum α-Rényi conditional entropy H (c) α (P X , P Y ) := min P XY ∈C(P X ,P Y ) H α (Y |X)(81) over couplings of P X , P Y . We next generalize our result to the minimum Rényi entropy, and get the following corollary. Statement 2) of Corollary 9 follows by combining Corollary 1 and the fact that H (c) α (P n X , P n Y ) is non-increasing in α. Statement 3) is proven by using product couplings. The proof of Statement 1) is provided in Appendix E. Corollary 9 (Minimum Rényi Conditional Entropy Coupling). Given two product marginal distributions P n X and P n Y , we have: [leftmargin=*] 1) If H(X) > H(Y ), then H (c) α (P n X , P n Y ) → 0 at least exponentially fast as n → ∞ for α ∈ log |Y| E (P X , P Y ) + log |Y| , ∞ ,(82) where E (P X , P Y ) defined in (51) denotes the optimal exponent for the maximal guessing coupling problem; 2) If H(X) < H(Y ), then H (c) α (P n X , P n Y ) → ∞ linearly fast as n → ∞ for all α ∈ [0, ∞]; 3) If H(X) = H(Y ), then H (c) α (P n X , P n Y ) ≥ nH (c) α (P X , P Y ) for all n and for all α ∈ [0, ∞]. Definition 9. [42] The Gács-Körner (GK) common information between two general correlated sources (X, Y ) is defined as C GK (X; Y ) := sup {(fn,gn)}:P{fn(X n ) =gn(Y n )}→0 lim inf n→∞ 1 n H(f n (X n )).(83) In particular, for two memoryless correlated sources (X, Y ), Gács-Körner showed the GK common information is equal to C GK (X; Y ) := sup f,g:f (X)=g(Y ) H(f (X)).(84) Define the maximum GK common information over couplings of product distributions of P X , P Y as C GK (P X , P Y ) := sup P XY :P X n Y n ∈C(P n X ,P n Y ),∀n C GK (X; Y ). As a consequence of Corollary 5, we have the following result. Corollary 10 (Maximum GK Common Information Coupling). Given two distributions P X and P Y , we have [leftmargin=*] 1) C GK (P X , P Y ) = min{H(X), H(Y )} if H(X) = H(Y ); 2) C GK (P X , P Y ) ≥ max P XY ∈C(P X ,P Y ) C GK (P X , P Y ) if H(X) = H(Y ). IV. EXACT INTRINSIC RANDOMNESS In the next four sections, we apply the results above on the maximal guessing coupling problem to several informationtheoretic problems. First, we consider a new version of intrinsic randomness problem, named exact intrinsic randomness, and apply our results on maximal guessing coupling to this problem. The lossless source coding problem, intrinsic randomness problem, and source resolvability problem consist of three ingredients: [leftmargin=*] 1) a source distribution P X n , 2) a random variable M n ∈ [1 : e nR ], 3) and a mapping between them P X n |Mn or P Mn|X n . Define the uniform distribution as P U Mn := Unif[1 : e nR ]. In the lossless source coding problem, the source distribution P X n = P n X and X n is an asymptotic function of M n under the reconstruction mapping P X n |Mn ; in the intrinsic randomness problem, the source distribution P X n = P n X , M n is a function of X n under the randomness extractor P Mn|X n , and P Mn , P U Mn are asymptotically equal under some distance measure; and in the source resolvability problem, P Mn = P U Mn , X n is a deterministic function of M n under the resolvability code P X n |Mn , and P X n , P n X are asymptotically equal under some distance measure. However, we usually cannot find a joint distribution P MnX n such that P X n = P n X , P Mn = P U Mn , and X n is a function of M n or M n is a function of X n under P MnX n ; see Proposition 10. Therefore, in the traditional intrinsic randomness problem and source resolvability problem, we relax the constraint on marginal distributions, i.e., we do not constrain that P X n = P n X and P Mn = P U Mn , but require that P Mn , P U Mn or P X n , P n X are asymptotically equal under some distance measure. In this paper we define exact intrinsic randomness by relaxing the constraint on the mapping. Specifically, we require that P X n = P n X , P Mn = P U Mn , and M n is an asymptotic function of X n . Definition 10. Given a memoryless source P X and a uniform random variable M n with distribution P U Mn = Unif[1 : e nR ], define the exact intrinsic randomness rate S E (P X ) as the minimum rate needed to ensure there exists a code P Mn|X n such that P Mn = P U Mn , and M n is an asymptotic function of X n (lim n→∞ max fn P {M n = f n (X n )} = 1). That is, S E (P X ) : = sup R : ∃P Mn|X n : P Mn = P U Mn , lim n→∞ max fn P {M n = f n (X n )} = 1 ,(85) or equivalently, S E (P X ) : = sup R : ∃P X n Mn ∈ C(P n X , P U Mn ) : lim n→∞ max fn P {M n = f n (X n )} = 1 .(86) From Theorem 1, we know that the problems of exact and approximate intrinsic randomness are equivalent. Corollary 11 (Equivalence Between Exact and Approximate Intrinsic Randomness). Given a memoryless source P n X and a uniform distribution P U Mn , min P X n Mn ∈C(P n X ,P U Mn ) min fn P {M n = f n (X n )} = min fn |P U Mn − P fn(X n ) |.(87) Combining Corollary 11 and existing results on approximate intrinsic randomness, we completely characterize the exact intrinsic randomness rate. Theorem 5 (Exact Intrinsic Randomness). S E (P X ) = H(X). (88) Remark 9. It is easy to verify that Corollary 11 also holds for general sources. On the other hand, Vembu and Verdú [5] showed for a general source X, the intrinsic randomness rate for the approximate intrinsic randomness problem is H(X). Hence for a general source X, the intrinsic randomness rate S E (P X ) for the exact intrinsic randomness problem (defined similarly to the memoryless case) is S E (P X ) = H(X). Proof: For the approximate intrinsic randomness problem, Han [6, Theorem 1.6.1] showed there exists a code for the approximate intrinsic randomness problem if R < H(X) and only if R ≤ H(X). Invoking Corollary 11 completes the proof of Theorem 5. Theorem 6 (Second Order Rate). Given a memoryless source P n X , the optimal (maximum) code rate R * n generated under the condition that the output forms a uniform random variable, i.e., M n ∼ Unif[1 : e nRn ] and M n is an ε-asymptotic function of the output X n , i.e., lim sup n→∞ min fn P {M n = f n (X n )} ≤ ε, satisfies R * n = H(X) − V (X) n Q −1 (ε) + o 1 √ n ,(89) where Q is the complementary cumulative distribution function of a standard Gaussian and V (X) is the variance of ı X (X). Proof: Similarly to the proof of Theorem 5, we can prove Theorem 6 by the equivalence between maximal guessing coupling problem and source resolvability problem (which is also approximate intrinsic randomness for this case) (Theorem 1), and the second order rate results for the approximate intrinsic randomness given by Hayashi [43]. V. EXACT RESOLVABILITY The maximal guessing coupling problem through a channel defined in Section III-B is the minimization of the error probability of the channel output Y n and the target variable Z n . Theorem 3 shows this problem is equivalent to the traditional channel resolvability problem (with the TV distance measure). In this section, we consider a new channel (or source) resolvability problem, named exact channel (or source) resolvability problem. In this problem, we require that P Y n = P n Y , P Mn = P U Mn , and the channel input X n is an asymptotic function of M n (lim n→∞ max fn P {X n = f n (M n )} = 1). Definition 11. Given a uniform random variable M n with distribution P U Mn = Unif[1 : e nR ] a memoryless channel P Y |X , and a target distribution P Y , define the exact channel resolvability rate G E (P Y |X , P Y ) as the minimum rate needed to ensure there exists a code P X n |Mn such that P Y n = P n Y , and the channel input X n is an asymptotic function of M n (lim n→∞ max fn P {X n = f n (M n )} = 1). That is, G E (P Y |X , P Y ) : = inf R : ∃P X n |Mn : P Y n = P n Y , lim n→∞ max fn P {X n = f n (M n )} = 1 .(90) If the channel P Y |X is an identity channel, we define exact source resolvability rate G E (P X ) : = inf R : ∃P X n |Mn : P X n = P n X , lim n→∞ max fn P {X n = f n (M n )} = 1 ,(91) or equivalently, G E (P X ) : = inf R : ∃P MnX n ∈ C(P U Mn , P n X ) : lim n→∞ max fn P {X n = f n (M n )} = 1 . (92) Corollary 12 (Source Resolvability). Given a memoryless source P n X and a uniform distribution P U Mn , min P Mn X n ∈C(P U Mn ,P n X ) min fn P {X n = f n (M n )} = min fn |P n X − P fn(Mn) |.(93) Furthermore, G E (P X ) = H(X).(94) Remark 10. It is easy to verify that the equivalence (93) also holds for general sources. On the other hand, Han and Verdú [7] showed for a general source X, the resolvability rate for the approximate source resolvability problem is H(X). Hence for a general source X, the resolvability rate G E (P X ) for the exact source resolvability problem (defined similarly to the memoryless case) is G E (P X ) = H(X). Proof: The equivalence (93) follows from Theorem 1. Furthermore, Han and Verdú [7] showed there exists a code for the approximate source resolvability problem if R < H(X) and only if R ≤ H(X). Combining these two observations yields (94). Denote P(P Y |X , Q Y ) := P X : P Y |X • P X = Q Y ,(95) and assume P(P Y |X , Q Y ) = ∅. We are now are ready to establish the following multiletter characterization for the exact channel resolvability rate. The proof of Proposition 4 is given in Appendix F. Proposition 4 (Multiletter Characterization of G E (P Y |X , P Y )). G E (P Y |X , P Y ) = inf P X ∈P(P Y |X ,P Y ) H(X)(96) = lim n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n H(X n ),(97) where P(P Y |X , P Y ) := P X : P X n ∈ P(P n Y |X , P n Y ), ∀n . Remark 11. Unlike our definition, Li and El Gamal [44] defined an exact common information rate by considering variable-length coding. However, their exact common information rate has a similar characterization as (97), i.e., the exact common information rate C E (X; Y ) = lim n→∞ min P Wn |X n Y n :X n →Wn→Y n 1 n H(W n ). (98) Furthermore, we can bound G E (P Y |X , P Y ) as follows. Proposition 5. G TV (P Y |X , P Y ) ≤ G E (P Y |X , P Y ) ≤ G E (P Y |X , P Y ), (99) where G TV (P Y |X , P Y ) := min P X ∈P(P Y |X ,P Y ) I(X; Y ) denotes the channel resolvability rate under the TV distance measure, and G E (P Y |X , P Y ) := min P X ∈P(P Y |X ,P Y ) H(P X ). Proof: The upper bound is obtained by choosing X n in (97) such that P X n = P n X with P X ∈ P(P Y |X , P Y ). The lower bound is obtained by the following chain of inequalities: G E (P Y |X , P Y ) = lim n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n H(X n ) (100) ≥ lim inf n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n I(X n ; Y n ) (101) = lim inf n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n (H(Y n ) − H(Y n |X n )) (102) = lim inf n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n n i=1 (H(Y i ) − H(Y i |X i )) (103) = lim inf n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n n i=1 I(X i ; Y i ) (104) = lim inf n→∞ min P X n ∈P(P n Y |X ,P n Y ) I(X J ; Y J |J) (105) ≥ lim inf n→∞ min P X n ∈P(P n Y |X ,P n Y ) I(X J ; Y J ) (106) = min P X ∈P(P Y |X ,P Y ) I(X; Y ),(107) where in (105) J ∼ Unif[1 : n] denotes a time-sharing random variable, (105) follows from that Y J is independent of J since Y n are i.i.d. under P n Y , and in (107) X := X J and Y := Y J . Proposition 6. Neither the upper bound G E (P Y |X , P Y ) nor the lower bound G TV (P Y |X , P Y ) is tight in general, i.e., there exists P Y |X , P Y such that G E (P Y |X , P Y ) < G E (P Y |X , P Y )(108) and also there exists P Y |X , P Y such that G TV (P Y |X , P Y ) < G E (P Y |X , P Y ).(109) This proposition implies the exact and approximate channel resolvability are not equivalent. In general, the exact channel resolvability requires a larger rate. A. P Y -non-redundant Channel Although the upper bound G E (P Y |X , P Y ) is not tight in general, we will show it is tight for some special cases, e.g., full-rank channels and additive channels. Hence next, we focus on full-rank channels and additive channels, and prove G E (P Y |X , P Y ) = G E (P Y |X , P Y ) for these two classes of channels. Definition 12. We say P Y |X is a P Y -non-redundant channel if given P Y |X and P Y , the equation P Y |X P X = P Y has a unique solution P X . That is, there exists a unique distribution P X that induces P Y through P Y |X . Definition 13. We say P Y |X is a full-rank channel if rank(P Y |X ) = |X |. Definition 14. We say P X is a degenerate distribution if P X (x 0 ) = 1 for some x 0 and P X (x) = 0 for x = x 0 . Lemma 3. The following properties hold. [ leftmargin=*] 1) If P Y |X is a P Y -non-redundant channel, then either P Y |X is a full-rank channel or P X is a degenerate distribution. 2) For any n ∈ N, P n Y |X is a P n Y -non-redundant channel, if and only if P Y |X is a P Y -non-redundant channel. 3) Any additive channel Y = X + Z with Z independent of X, is a full-rank channel. 4) If P X n ∈ P(P n Y |X , P n Y ) for some n ∈ N, then n i=1 P Xi ∈ P(P n Y |X , P n Y ). Remark 12. In general, P X n ∈ P(P n Y |X , P n Y ) does not imply P X n must be a product distribution or that is uniquely defined. However if P Y |X is a P Y -non-redundant channel, it does imply that P X n must be a product distribution and that it is unique. Proof: Proof of Property 1): Consider the linear equation P Y |X Q = P Y where we do not constrain Q to a probability distribution, i.e., some components can be negative. We know that it must have no solution, a unique solution, or infinitely many solutions. If P X is a probability distribution and the linear equation P Y |X P X = P Y has a unique solution, then it means that the set of solutions of P Y |X Q = P Y and the probability simplex {P X : x P X (x) = 1, P X (x) ≥ 0} intersect at a single point. Hence either P Y |X Q = P Y has a single unique solution, or it has infinitely many solutions but they intersect with the probability simplex at the vertices points of the probability simplex. These two cases respectively correspond to the case rank(P Y |X ) = |X | and the case where the solution is P X (x 0 ) = 1 for some x 0 and P X (x) = 0 for x = x 0 . Property 2) follows from Property 1). Proof of Property 3): P Y |X = I |X | ⊗ P Z , where I |X | denotes the identity matrix with size |X |. Hence rank(P Y |X ) = rank(I |X | )rank(P Z ) = |X |. Property 4) is obvious. Theorem 7. If the channel P Y |X is a P Y -non-redundant channel, then 13 where P X is the unique distribution that induces P Y through P Y |X . G E (P Y |X , P Y ) = G E (P Y |X , P Y ) = H(P X ),(110) For an AWGN (additive white Gaussian noise) channel P Y |X and a Gaussian distribution P Y , we have that P X ∈ P P Y |X , P Y is also Gaussian and unique. So for this case, we get the following result. Proposition 7. For an AWGN channel P Y |X and a Gaussian distribution P Y , we have G E (P Y |X , P Y ) = G E (P Y |X , P Y ) = H(P X ) = ∞. (111) Remark 13. The exact channel resolvability rate is infinite, although the approximate channel resolvability rate G TV (P Y |X , P Y ) = 1 2 log N Y N Z is finite. This point is different from the exact common information. Li and El Gamal [44] showed the exact common information C E (X; Y ) satisfies I(X; Y ) ≤ C Wyner (X; Y ) ≤ C E (X; Y ) ≤ I(X; Y )+24 log 2. where C Wyner (X; Y ) is Wyner's common information, and C E (X; Y ) is the exact common information. Applying this result to two jointly Gaussian random variables shows that only a finite amount of common randomness is needed for simulating them in a distributed manner. Next we consider the second-order rate for the exact channel resolvability problem. Given a memoryless channel P Y |X , define R * n as the optimal (minimum) code rate needed to ensure the channel output follows distribution P n Y and X n is an ε-asymptotic function of the output M n , i.e., lim sup n→∞ min fn P {X n = f n (M n )} ≤ ε. Theorem 8 (Second Order Rate for P Y -non-redundant Channels). Given a memoryless P Y -non-redundant channel, we have R * n = H(X) + V (X) n Q −1 (ε) + o 1 √ n ,(112) where P X is the unique distribution that induces P Y through P Y |X . Proof: For P Y -non-redundant channels, the channel input distribution is unique and equal to P X . Hence for this case, the exact channel resolvability problem is equivalent to the exact source resolvability problem. On the other hand, by Corollary 12 we know that the exact source resolvability problem is also equivalent to the approximate source resolvability problem. Hence the exact channel resolvability problem is equivalent to the approximate source resolvability problem. Furthermore, for the latter problem, Nomura and Han [45, Theorem 1.6.1] showed that the optimal rate is as in (112). VI. CHANNEL CAPACITY WITH INPUT DISTRIBUTION CONSTRAINT Definition 15. Given a distribution P X , the channel capacity with input distribution constraint P X is defined as the maximum rate R such that there exists a sequence of codes (P X n |Mn , P Mn|Y n ) ∞ n=1 satisfying P X n = P n X and lim n→∞ P{M n = M n } = 1 with M n ∼ Unif[1 : e nR ]. That is, C (P X ) : = sup R : ∃(P X n |Mn , P Mn|Y n ) ∞ n=1 : P X n = P n X , lim n→∞ P M n = M n = 1 .(113) Theorem 9. C (P X ) = C GK (X; Y ), where C GK (X; Y ) denotes the GK common information between X and Y (under the distribution P X P Y |X ). Remark 14. C (P X ) ≤ I(X; Y ) ≤ C, where C denotes the traditional Shannon capacity (i.e., the channel capacity without the input distribution constraint). Proof: Assume W is a common part of X and Y (under distribution P X P Y |X ) (i.e., W = g(X) = h(Y ) a.s. for some functions g and h). If R < H(W ), then according to Theorem 2 there exists a maximal guessing coupling P MnW n such that P Mn = Unif[1 : e nR ] and max fn P {M n = f n (W n )} → 1. Assume f n is a maximizing function of max fn P {M n = f n (W n )}. Apply P W n |Mn (w n |m)P n X|W (x n |w n ) as the encoder, and P n W |Y (w n |y n )·1 { m = f n (w n )} as the decoder. Then P X n = P n X and lim n→∞ P{M n = M n } = 1. Hence C (P X ) ≥ C GK (X; Y ). On the other hand, we can convert a code for the problem of channel capacity with input distribution constraint P X into a code for the GK common information problem. For any code (P X n |Mn , P Mn|Y n ) satisfying P X n = P n X and lim n→∞ P{M n = M n } = 1, the induced joint distribution of X n and Y n is the product distribution P n XY . Hence (P Mn|X n , P Mn|Y n ) forms a code for the GK common information problem [42]. According to the converse for GK common information problem, we conclude that the code rate is not larger than C GK (X; Y ). Next we consider the second-order rate. Given a distribution P X , define R * n as the optimal (maximum) code rate needed to ensure that there exists a sequence of codes satisfying P X n = P n X and lim sup n→∞ P{M n = M n } ≤ ε. In order to present the bounds on the second-order rate on R * n , we need define some quantities. Given a distribution P X , let W be a common random variable of X and Y (under the distribution P X P Y |X ), i.e., W = f (X) = g(Y ) for some functions f and g achieving C GK (X; Y ) in (84) (where the sup is a max for finite-valued X and Y ). Let ρ m (X; Y |W ) denote the conditional maximal correlation [46], [47] between X and Y given the common random variable W defined as Denote ε * := − 1 3 + 1 + (1 − ρ m (X; Y |W )) −2 2 1 + 9 8 (1 − ρ m (X; Y |W )) −2 .(116) For θ ∈ (0, 1) such that ε θ < min{ε * , 1/3}, let ξ * ( ε θ ) is the unique solution on ( ε θ , 1/3) to the equation 2(1 − ρ m (X; Y |W )) ξ(1 − ξ)(ξ − ε θ )(1 − ξ + ε θ ) = ε θ (117) with ξ unknown. For 0 < ε < min{ε * , 1/3}, denote µ(ε) := inf max{3ε, ε ε * }≤θ<1 1 − (1 − θ) 1 − ξ * ( ε θ ) . (118) Theorem 10 (Second Order Rate). Given a distribution P X , we have for 0 < ε < 1, R * n ≥ C GK (X; Y ) − V (W ) n Q −1 (ε) + o 1 √ n ,(119) and for 0 < ε < min{ε * , 1/3}, R * n ≤ C GK (X; Y ) − V (W ) n Q −1 (µ(ε)) + o 1 √ n .(120) where W is a common random variable of X and Y , ε * is defined in (116), and µ(ε) is defined in (118). On the other hand, the legitimate user first recovers W n losslessly and then reconstructs M n as M n = f n (W n ). Hence (122) implies that lim sup n→∞ P{M n = M n } ≤ ε. That is, Y ), let W be a common random variable of X and Y . Let U, V be two random variables such that U → X → Y → V and P {U = V } ≤ ε(123) for some 0 < ε < min{ε * , 1/3}, where ε * is defined in (116). Then inf h:W→U P {U = h(W )} ≤ µ(ε),(124) where µ(ε) is defined in (118). For completeness, we provide the proof of Lemma 4 at the end of this proof. Applying this lemma to our setting by the identification (X n , Y n , M n , M n ) as (X, Y, U, V ), we have inf f P {M n = f (W n )} ≤ µ P M n = M n .(125) Taking limsup's, we have lim sup n→∞ inf f P {M n = f (W n )} ≤ lim sup n→∞ µ P M n = M n (126) ≤ µ(ε),(127) where (127) follows since µ(ε) is continuous and nondecreasing in ε. Then by the converse part of Theorem 6, we have that any achievable {R n } ∞ n=1 must satisfy lim sup n→∞ √ n(R n − H(W )) < − V (W )Q −1 (µ(ε)),(128) which completes the proof of the upper bound in (120). Proof of Lemma 4: We first make the following claim. Claim 1. If additionally, C GK (X; Y ) = 0, then p max := max u P U (u) ≥ 1 − ξ * (ε) where ξ * (ε) is the unique solution on (ε, 1/3) to the equation (117) with θ = 1. We now prove this claim. By assumption, C GK (X; Y ) = 0, i.e., any common random variables W are constant. Denote ϕ(ξ) := 2(1 − ρ m (X; Y |W )) ξ(1 − ξ)(ξ − ε)(1 − ξ + ε). (129) Obviously, ϕ(ξ) is continuous and increasing in ξ ∈ (ε, 1/3). Moreover, ϕ(ε) = 0 < ε and ϕ(1/3) > ε. The latter inequality follows by the assumption 0 < ε < min{ε * , 1/3}. Hence, there is a unique solution ξ * (ε) ∈ (ε, 1/3) to the equation ϕ(ξ) = ε. For brevity, we denote ξ * := ξ * (ε). Suppose instead that p max < 1 − ξ * . Then, there exists a set A ⊆ U such that ξ * < P U (A) < 1 − ξ * .(130) (Sort elements in U as u 1 , u 2 , ..., u m such that P U (u 1 ) ≥ P U (u 2 ) ≥ ... ≥ P U (u m ). If p max = P U (u 1 ) > ξ * , then A can be chosen as {u 1 }. If p max < ξ * , then A can be chosen as {u 1 , u 2 , ..., u k } for some 1 ≤ k ≤ m such that (130) holds. The existence of such k follows since ξ * < 1/3.) Observe that (123) implies that P {1 A (U ) = 1 A (V )} ≤ ε.(131) By (130) and (131), we get ξ * < P {1 A (U ) = 1} < 1 − ξ *(132) and ξ * − ε < P {1 A (V ) = 1} < 1 − ξ * + ε.(133) By [48,Theorem 2], 2(1 − ρ m (X; Y )) P {1 A (U ) = 1} P {1 A (U ) = 0} × P {1 A (V ) = 1} P {1 A (V ) = 0} ≤ P {1 A (U ) = 1 A (V )} ,(134)which implies that ϕ(ξ * ) < ε.(135) This contradicts with the assumption ϕ(ξ * ) = ε. Hence, p max ≥ 1 − ξ * (ε), i.e., Claim 1 holds. We now turn back to prove Lemma 4. Note that as assumed, C GK (X; Y ) > 0. For each w, denote ρ m (X; Y |W = w) = ρ m (X ; Y ) where (X , Y ) ∼ P XY |W =w . Then, ρ m (X; Y |W ) = max w ρ m (X; Y |W = w). Let θ be such that ε θ < min{ε * , 1/3}, which implies that ε θ < ε * w := − 1 3 + √ 1+(1−ρm(X;Y |W =w)) −2 2 1+ 9 8 (1−ρm(X;Y |W =w)) −2 for all w since ε * w ≤ ε * . Denote B as the set of w such that P {U = V |W = w} ≤ ε θ .(136) By definition, given W = w, C GK (X ; Y ) = 0 for (X , Y ) ∼ P XY |W =w . Then, applying Claim 1 to (X , Y ) ∼ P XY |W =w , we have p (w) max := max u P U |W (u|w) ≥ 1 − ξ * w ( ε θ ),(137)where ξ * w ( ε θ ) is the unique solution on ( ε θ , 1/3) to the equation ϕ w (ρ m (X; Y |W = w), ξ) = ε θ with ξ unknown and with ϕ w (s, ξ) := 2(1 − s) ξ(1 − ξ)(ξ − ε θ )(1 − ξ + ε θ ). (138) Since ϕ w (ρ m (X; Y |W = w), ξ) ≥ ϕ w (ρ m (X; Y |W ), ξ), we have ξ * w ( ε θ ) ≤ ξ * ( ε θ ). Therefore, p (w) max ≥ 1 − ξ * ( ε θ ).(139) On the other hand, observe that ε ≥ P {U = V } ≥ P {W ∈ B c } P {U = V |W ∈ B c } (140) ≥ P W (B c ) ε θ .(141) Hence, P W (B c ) ≤ θ,(142) i.e., P W (B) ≥ 1 − θ.(143) Therefore, sup h:W→U P {U = h(W )} = w P W (w) max u P U |W (u|w) (144) ≥ (1 − θ)(1 − ξ * ( ε θ )),(145) where the last line follows from (139). VII. PERFECT STEALTH AND SECRECY COMMUNICATION In this section, we apply the preceding results on exact resolvability to the perfectly stealthy (or covert) and secret communication over the discrete memoryless wiretap channel [49], [50]. Stealth or covert communication was studied by Hou and Kramer [13], Yu and Tan [11], Bash et al. [14], [15], Wang et al. [16], and Bloch [17], where the relative entropy and the Rényi divergence were used to measure the level of stealth (or covertness) of communication. In this paper, we consider a perfectly stealthy (or covert) and secret communication system, where the eavesdropper is forced to observe a channel output exactly, rather than approximately, following a target distribution and, at the same time, the secret part of transmitted messages is independent of the eavesdropper's observation. For this new problem, we aim at characterizing the rate region of secret and non-secret parts of the transmitted messages. Consider a discrete memoryless wiretap channel P Y Z|X , and two messages (M 0 , M 1 ) that are uniformly distributed over M 0 := [1 : e nR0 ] and M 1 := [1 : e nR1 ] respectively. A sender wants to transmit the pair (M 0 , M 1 ) to a legitimate user reliably, and, at the same time, ensure that M 1 is independent of the eavesdropper's observation Z n . Definition 16. An (n, R 0 , R 1 ) secrecy code is defined by two stochastic mappings P X n |M0M1 : M 0 × M 1 → X n and P M0 M1|Y n : Y n → M 0 × M 1 . Given a target distribution P Z , we wish to maximize the alphabet size (or rate) of M 1 such that the distribution P M1Z n induced by the code is equal to the target distribution P M1 P n Z and M 1 can be decoded correctly asymptotically when n → ∞. Definition 17. The tuple (R 0 , R 1 ) is P Z -achievable if there exists a sequence of (n, R 0 , R 1 ) secrecy codes with induced distribution P M0M1Z n M0 M1 such that [leftmargin=*] 1) Error constraint: lim n→∞ P (M 0 , M 1 ) = ( M 0 , M 1 ) = 0;(146) 2) Secrecy constraint: P M1Z n = P M1 P n Z .(147) Here we assume P Z satisfies P P Z|X , P Z = ∅ (P P Z|X , P Z is defined in (95)); otherwise, (147) cannot be satisfied by any secrecy code. Definition 18. The P Z -admissible region is defined as R(P Z ) := Closure {(R 0 , R 1 ) : (R 0 , R 1 ) is P Z -achievable} . (148) The perfect stealth (or perfect covertness) capacity is defined as C 0 (P Z ) := max (R0,R1)∈R(P Z ) R 0 .(149) The perfect stealth-secrecy capacity is defined as C 1 (P Z ) := max (R0,R1)∈R(P Z ) R 1 .(150) There are two reasons we assume M 0 , M 1 follow uniform distributions. Firstly, this assumption is consistent with the setting in traditional communication problems. Secondly, even if the sources (or messages) to be transmitted (denote them as S 0 , S 1 ) are not uniform, for example, they are memoryless and follow P S k , k = 0, 1, respectively, then by Theorem 5 we know that for k = 0, 1, there exists P S n k M k ∈ C(P n S k , P U M k ) such that lim n→∞ max fn P {M k = f n (S n k )} = 1 if the rate R k of M k satisfies R k > H(S k ). Hence using P M k |S n k , k = 0, 1, we transform the sources into two uniformly distributed messages. Moreover, for the error constraint, if the legitimate user can recover M 0 , M 1 , he can recover S 0 , S 1 as well since lim n→∞ max fn P {M k = f n (S n k )} = 1. For the secrecy constraint, P M1Z n = P M1 P n Z implies P S1Z n = P S1 P n Z . Therefore, the perfect stealth and secrecy communication of uniform messages implies the perfect stealth and secrecy communication of non-uniform messages if R k > H(S k ), k = 0, 1. Obviously, the converse holds if R k < H(S k ), k = 0, 1. Therefore, the perfect stealth and secrecy communication of non-uniform messages is feasible if and only if (H(S 0 ), H(S 1 )) is P Zachievable. This ensures that we only need to consider uniform messages. A. Main Result For full-rank channels, we completely characterize the admissible region. Theorem 11. If the wiretap channel P Z|X is of full-rank (including additive channels and identity channels), we have R(P Z ) = (R 0 , R 1 ) : R 0 ≤ C GK (X; Y ) R 1 = 0 ,(151) where P X is the unique distribution that induces the target distribution P Z . That is, C 0 (P Z ) = C GK (X; Y ) and C 1 (P Z ) = 0. Proof: The achievability part follows from the result on channel capacity with input distribution constraint (Theorem 9 in the previous section). Now we prove the converse part. Note that P ⊗n Z = P Z n |M1=m1 = P ⊗n Z|X P X n |M1=m1 for any m 1 , and P ⊗n Z|X is invertible. Hence P X n |M1=m1 = P ⊗n Z|X −1 P ⊗n Z = P −1 Z|X P Z ⊗n(152) for any m 1 . Note that (P −1 Z|X P Z ) ⊗n does not depend on m 1 , hence X n is independent of M 1 . On the other hand, M 1 → X n → Y n forms a Markov chain, hence Y n is independent of M 1 . That is, R 1 = 0. The converse part for R 0 ≤ C GK (X; Y ) follows from the converse part of Theorem 9. For general channels, we derive an upper bound and a lower bound for the perfect stealth-secrecy capacity. Theorem 12. The perfect stealth capacity and the perfect stealth-secrecy capacity are respectively bounded as sup k≥1 1 k max P X k ∈P(P k Z ) C GK (X k ; Y k ) ≤ C 0 (P Z ) (153) ≤ max P X ∈P(P Z ) I(X; Y ),(154) and max P U T X :U →T →Z,P X ∈P(P Z ) (156) I(U ; Y ) − I(U ; T ) ≤ C 1 (P Z )( Remark 15. The lower bound for C 1 (P Z ) can be further lower bounded by max P U X :U ⊥Z,P X ∈P(P Z ) I(U ; Y ). The upper bound for C 1 (P Z ) can be further upper bounded by max P U X :P X ∈P(P Z ) I(U ; Y ) − I(U ; Z). Remark 16. Wang et al. [16] proved that if the sender and the legitimate user share a sufficiently large rate of secret key, then the covert capacity C 0 (P Z ) = max P X ∈P(P Z ) I(X; Y ). Proof: The achievability part for C 0 (P Z ) follows from the result on channel capacity with input distribution constraint (Theorem 9 in the previous section). Conversely, C 0 (P Z ) ≤ 1 n I(X n ; Y n ) ≤ I(X Q ; Y Q ) ≤ max P X ∈P(P Z ) I(X; Y ), where Q ∼ Unif[1 : n] denotes a time-sharing random variable, independent of X n , Y n . The last inequality follows since P X Q ∈ P(P Z ). Next we prove the lower and upper bounds for C 1 (P Z ). Achievability for C 1 (P Z ): Suppose P U T X is a distribution such that U → T → Z, P X ∈ P(P Z ). Then we use the following scheme to obtain the inner bound. Codebook generation: Fix the conditional pmf P U |T and P X|U T and let R 1 > R 1 . For each message m 1 ∈ [1 : e nR1 ] generate a subcodebook C(m 1 ) consisting of e n( R1−R1) randomly and independently generated sequences u n (l), l ∈ [(m 1 − 1)e n( R1−R1) + 1 : m 1 e n( R1−R1) ], each according to n i=1 P U (u i ). Encoding: Generate a sequence t n according to n i=1 P T (t i ). Upon receiving message m 1 ∈ [1 : e nR1 ] and sequence t n , the encoder chooses a sequence u n (l) ∈ C(m 1 ) such that (u n (l), t n ) ∈ T (n) . If no such sequence exists, it picks l = 1. For brevity, denote U n = U n (L). Then upon U n = u n , T n = t n , the encoder generates x n according to n i=1 P X|U T (x i |u i , t i ) and transmits it. Decoding: Let > . Upon receiving y n , the decoder declares thatm 1 ∈ [1 : e nR1 ] is sent if it is the unique message such that (u n (l), y n ) ∈ T (n) for some u n (l) ∈ C(m 1 ); otherwise it declares an error. Analysis of Error Probability and Secrecy: If T n is considered as a side information, then the achievability scheme above is also a Gelfand-Pinsker code for the channel coding problem with non-causal side information at the transmitter. By Gelfand-Pinsker's proof [51, pp. 181], we have that if R 1 < I(U ; Y ) − I(U ; T ) then lim n→∞ P M 1 = M 1 = 0.(157) Furthermore, P Z n |M1 (z n |m 1 ) = u n ,t n P n Z|U T (z n |u n , t n )P n T (t n )P U n (L)|T n M1 (u n |t n , m 1 ) = u n ,t n P n Z|T (z n |t n )P n T (t n )P U n (L)|T n M1 (u n |t n , m 1 ) (159) = t n P n Z|T (z n |t n )P n T (t n ) (160) = P n Z (z n ),(161) where (159) follows from U → T → Z. Converse for C 1 (P Z ): Similar to the proof of Theorem 11, it can be shown that T n is independent of M with T i generated through a channel P Ti|XiZi such that P Zi|Ti is of full-rank and X i → T i → Z i . nR 1 ≤ I(Y n ; M ) (162) = I(Y n ; M ) − I(T n ; M ) (163) = n i=1 I(Y i ; M |V i ) − I(T i ; M |V i ) (164) = n i=1 vi P Vi (v i ) (I(Y i ; M |V i = v i ) − I(T i ; M |V i = v i )) (165) ≤ n max P U X :P X ∈P(P Z ) min P T |XZ :P Z|T is of full-rank, X→T →Z I(U ; Y ) − I(U ; T ),(166) where V i := Y i−1 T n i+1 , (164) follows from the standard steps in the weak converse proof for the wiretap channel [51, pp. 555], and (166) follows since P Ti|XiZi is arbitrary such that P Zi|Ti is of full-rank and X i → T i → Z i and (166) reduces to (165) if U is set to M . Definition 19. A function f (X) is said to be a sufficient statistic relative to P Z|X if X is independent of Z given f (X) for any distribution on X (i.e., for any distribution on X, X → f (X) → Z forms a Markov chain). The lower bound and upper bound in Theorem 12 coincide for full-rank sufficient statistic channels. Corollary 13 (Full-rank Sufficient Statistic Channel). If there exists a sufficient statistic f (X) relative to P Z|X such that P Z|f (X) is full-rank, then C 1 (P Z ) = max P U X :P f (X) ∈P(P Z ) I(U ; Y ) − I(U ; f (X)). (167) Remark 17. If P Z|X = P Z|X2 with X 2 = f (X) for some function f and for all input random variables X, and P Z|X2 is of full-rank, then the perfect secrecy capacity C 1 (P Z ) = max P U X :P X 2 ∈P(P Z ) I(U ; Y ) − I(U ; X 2 ). Remark 18. Corollary 13 is consistent with Theorem 11, since both of them imply C 1 (P Z ) = 0 for full-rank channels. As a special case of Corollary 13, we have the following result. Corollary 14 (Gaussian Wiretap Channel). If X = (X 1 , X 2 ), the channel satisfies Y = X 1 + X 2 + E 1 , Z = X 2 + E 2 , with E k ∼ N (0, N k ), k = 1, 2 and EX 2 1 ≤ P 1 , and P Z = N (0, N Z ) with N Z ≥ N 2 , then the perfect secrecy capacity C 1 (P Z ) = 1 2 log 1 + P 1 N 1 .(168) Similar to Definition 12, here we define P Z -redundant channel as follows. Definition 20. A channel P Y Z|X is a P Z -redundant channel if there exist two distributions Q X and Q X that induce the same P Z through P Z|X but induce two different distributions of Y through P Y |X . We give a sufficient and necessary condition for that the stealth-secrecy capacity is positive. The proof of the following theorem is provided in Appendix G. Theorem 13. C 1 (P Z ) > 0 if and only if the channel P Y Z|X is a P Z -redundant channel. VIII. CONCLUSION AND FUTURE WORK In this paper, we studied asymptotics of several coupling problems, including the problems of maximal coupling, minimum distance coupling, maximal guessing coupling, and minimum entropy coupling. We also applied these results to some information-theoretic problems, including the problems of exact intrinsic randomness, exact resolvability, and perfectly stealthy/covert and secret communication. Our results generalize or extend several classical and recent results. Firstly, our results on exact intrinsic randomness extend those by Vembu and Verdú [5] and Han [6] as we consider the scenario in which the output exactly follows a uniform distribution. Secondly, our resolvability results extend those by Han and Verdú [7], by Hayashi [8], [9], and by Yu and Tan [11] as we consider the scenario in which output exactly follows a target distribution. Finally, our results for the wiretap channel extend those by Hou and Kramer [13], by Yu and Tan [11], by Bash et al. [14], [15], by Wang et al. [16], and by Bloch [17], as we measure the stealth (or effective secrecy ) or covertness using an exact distribution constraint. A. Open Problems There are also some problems that remain to be solved. [leftmargin=*] 1) The optimal exponent of the minimum excess-distance probability coupling problem for the case in which d < D(P X , P Y ) has been solved in this paper. However, the optimal exponent for the case in which d > D(P X , P Y ) is still unknown. Besides, the minimum excess-distance probability and the corresponding optimal exponent for the case d = D(P X , P Y ) are still unknown. 2) In this paper, we characterized the limiting value of the maximal guessing coupling problem for the case H(X) = H(Y ). However, it is still open for the case H(X) = H(Y ). Furthermore, the optimal exponent for this problem is still unknown. The same comment applies to the optimal exponent for the minimum entropy coupling problem. 3) Under the assumption of uniform distributions, we provided the necessary and sufficient condition for the existence of a deterministic coupling or an asymptotically deterministic coupling for two product marginal distributions. However, the general case, stated in Conjectures 1 and 2, is still open. 4) An achievability result on the minimum Rényi (conditional) entropy coupling problem was provided in Corollary 9. Other minimum Rényi entropy coupling (or maximum Rényi mutual information coupling) problems are still open. 5) We only characterized the exact channel resolvability rate for full-rank channels. The complete characterization of the exact channel resolvability rate for general channels is still open. 6) We provided a sufficient and necessary condition in Theorem 13 for the scenario in which the stealth-secrecy capacity is positive. We also characterized the stealthsecrecy capacity for the full-rank sufficient statistic channel in Corollary 13. However, the complete characterization of the perfect stealth-secrecy capacity for general channels is still open. APPENDIX A PROOF OF PROPOSITION 2 Proof of Statement 1): Observe that max {D(Q X P X ), D(Q Y P Y )} is continuous in Q XY ∈ P(X × Y). By [11, Lem. 5], we know lim n→∞ min T XY ∈Pn(X ×Y): x,y T XY (x,y)d(x,y)≤d max {D(T X P X ), D(T Y P Y )} = min Q XY ∈P(X ×Y): x,y Q XY (x,y)d(x,y)≤d max {D(Q X P X ), D(Q Y P Y )} .(169) Hence to prove Statement 1), we only need to show the exponent E(d) is lim n→∞ min T XY ∈Pn(X ×Y): x,y T XY (x,y)d(x,y)≤d max {D(T X P X ), D(T Y P Y )} . (170) Next we prove this point. First we prove the converse part. P {d(X n , Y n ) ≤ d} = P x,y T X n Y n (x, y)d(x, y) ≤ d (171) = x n ,y n P X n Y n (x n , y n )1 x,y T x n y n (x, y)d(x, y) ≤ d (172) = T XY P X n Y n (T (T XY ))1 x,y T XY (x, y)d(x, y) ≤ d (173) ≤ T XY min{P X n (T (T X )), P Y n (T (T Y ))} × 1 x,y T XY (x, y)d(x, y) ≤ d(174)≤ (n + 1) |X Y| max T XY min{P X n (T X )), P Y n (T (T Y ))} × 1 x,y T XY (x, y)d(x, y) ≤ d(175). = max T XY min e −nD(T X P X ) , e −nD(T Y P Y ) × 1 x,y T XY (x, y)d(x, y) ≤ d(176) = e −n min T XY : x,y T XY (x,y)d(x,y)≤d max{D(T X P X ),D(T Y P Y )} The exponent E(d) is lower bounded by min T XY : x,y T XY (x,y)d(x,y)≤d max {D(T X P X ), D(T Y P Y )} . (178) Next we prove the achievability part. First we note that finding a coupling P X n Y n of P n X and P n Y that maximizes P {d(X n , Y n ) ≤ d} is equivalent to finding a "cou- pling" {P X n Y n (T (T XY ))} T XY of {P X n (T (T X ))} T X and {P Y n (T (T Y ))} T Y that maximizes T XY P X n Y n (T (T XY )) · 1 {E T XY d(X, Y ) ≤ d} .(179) This is because, on one hand, if we get a desired "coupling" {P X n Y n (T (T XY ))} T XY , and for each type T XY , let the sequences in the type class T (T XY ) uniformly share the total probability P X n Y n (T (T XY )), i.e., P X n Y n (x n , y n ) = P X n Y n (T (T XY )) |T (T XY )| , (x n , y n ) ∈ T (T XY ),(180) then the marginal distributions are also uniform in each type class. Moreover, the marginal distributions have the probabilities of the type classes {P X n (T (T X ))} T X and {P Y n (T (T Y ))} T Y . This two points ensure that the marginal distributions are respectively P n X and P n Y . Now we find a desired "coupling" {P X n Y n (T (T XY ))} T XY of {P X n (T (T X ))} T X and {P Y n (T (T Y ))} T Y . Denote T * XY as a type that achieves min T XY : x,y T XY (x,y)d(x,y)≤d max {D(T X P X ), D(T Y P Y )} .(181) Obviously, x,y T * XY (x, y)d(x, y) ≤ d. Without loss of generality, we only consider the case of D(T * X P X ) ≥ D(T * Y P Y ). We allocate P X n (T (T * X )) to P X n Y n (T (T * XY )), i.e., set P X n Y n (T (T * XY )) = P X n (T (T * X )) and P X n Y n (T (T XY )) = 0 for all T XY with T X = T * X but T XY = T * XY . On the other hand, there is no restriction for the probabilities of other joint types. Hence we set T XY ∈C(T X ,T Y ) P X n Y n (T (T XY )) T X ,T Y :T X =T * X to be any coupling of {P X n (T (T X ))} T X =T * X and {P Y n (T (T Y ))} T Y . Then for such a coupling {P X n Y n (T (T XY ))} T XY , we have P {d(X n , Y n ) ≤ d} = P x,y T X n Y n (x, y)d(x, y) ≤ d(182) = x n ,y n P X n Y n (x n , y n )1 x,y T x n y n (x, y)d(x, y) ≤ d (183) = T XY P X n Y n (T (T XY ))1 x,y T XY (x, y)d(x, y) ≤ d (184) ≥ P X n Y n (T (T * XY ))1 x,y T * XY (x, y)d(x, y) ≤ d(185) = P X n (T (T * X )) . = e −nD(T * X P X ) . By symmetry, for the case of D(T * X P X ) ≥ D(T * Y P Y ), we have P {d(X n , Y n ) ≤ d}≥ e −nD(T * Y P Y ) .(187) Therefore, E(d) ≤ max {D(T * X P X ), D(T * Y P Y )} (188) = min T XY : x,y T XY (x,y)d(x,y)≤d max {D(T X P X ), D(T Y P Y )} .(189) Invoking (169), we complete the proof of Statement 1). Proofs of Statements 2) and 3): Proof of the achievability by product couplings: For the product coupling P X n Y n = P n XY where P XY := arg min P XY ∈C(P X ,P Y ) Ed(X, Y ), by the large deviation theory, the exponents for the cases of Statement 1) and 2) are respectively max t≥0 −td − log Ee −td(X,Y ) , and max t≥0 td − log Ee td(X,Y ) , and for the case of Statement 3), by the central limit theorem, P {d(X n , Y n ) ≤ d} = 1 2 + O 1 √ n .(190) APPENDIX B PROOF OF THEOREM 2 Proof of Statement 1): From the soft-covering lemma or the distribution approximation problem [6, Theorem 2.1.1] we know that if H(X) > H(Y ), there exists a sequence of functions f n (x n ) such that |P fn(X n ) − P Y n | → 0 exponentially fast. On the other hand, by the equivalence between the maximal guessing coupling problem and the distribution approximation problem (Theorem 1), max P X n Y n ∈C(P n X ,P n Y ) max fn P {Y n = f n (X n )} → 1(191) at least exponentially fast as n → ∞. Furthermore, the lower bound in (53) But in the following, we prove it using the method of types, which gives us a different exponent. P {Y n = f n (X n )} = x n ,y n P X n Y n (x n , y n )1 {y n = f n (x n )} (192) = x n ,y n P X n Y n (x n , y n )1 {y n = f n (x n ), x n ∈ T n (P X )} + x n ,y n P X n Y n (x n , y n )1 {y n = f n (x n ), x n / ∈ T n (P X )} (193) ≤ x n ,y n P X n Y n (x n , y n )1 {y n ∈ A} + P X n ((T n (P X )) c ) (194) = P Y n (A) + P X n ((T n (P X )) c ) (195) = P Y n (A ∩ T n (P Y )) + P Y n (A ∩ (T n (P Y )) c ) + P X n ((T n (P X )) c ) (196) ≤ |A|e −n(1− )H(Y ) + P Y n ((T n (P Y )) c ) + P X n ((T n (P X )) c ) (197) ≤ e −n((1− )H(Y )−(1+ )H(X)) + e −nδ (P Y ) + e −nδ (P X )(198) where A := {f n (x n ) : x n ∈ T n (P X )} with T n (P X ) denoting the -typical set, and for a set B, B c denotes the complement of B. Hence if H(X) < H(Y ), and > 0 is elected to be sufficiently small such that (1− )H(Y )−(1+ )H(X) > 0, then in view of (30) it follows that G(P n X , P n Y ) → 0 exponentially fast as n → ∞. Proof of Statement 3): An optimal product coupling P X n Y n = P n XY with P XY achieving G(P X , P Y ) achieves the lower bound G n (P X , P Y ). That is, there exists a (asymptotically deterministic) coupling P X n Y n ∈ C(P n X , P n Y ) for which Y n is an asymptotic function of X n , if and only if there exists a (deterministic) coupling P XY ∈ C(P X , P Y ) for which Y is a function of X. APPENDIX C SOME SPECIAL CASES Remark 19. More explicitly, for the case that P X is uniform but P Y is not, we have G(P n X , P n Y ) ≤ α n where α n := 1 − 1 2 Φ − 1 n log 2 − η(Y ) nV 3 (Y )(199) with Φ(·) denotes the cumulative distribution function (cdf) of the standard Gaussian distribution, and V (Y ) := Var [log P Y (Y )] (200) η(Y ) := E P Y [| log P Y (Y ) + H(P Y )| 3 ];(201) and for the case that P Y is uniform but P X is not, we have G(P n X , P n Y ) ≤ β n where β n := 1 − sup γ≥1 1 2 1 − 1 γ Φ 1 n log γ − 1 + 1 γ η(X) nV 3 (X) .(202) Furthermore, lim n→∞ α n = lim n→∞ β n = 3 4 . Proof: If G(P X , P Y ) = 1, then G(P n X , P n Y ) = 1 for any n, regardless of whether P X is uniform or P Y is uniform. Next we focus on the other direction. Case 1 (P X is uniform): If G(P X , P Y ) < 1, then by the assumption H(X) = H(Y ) = log M , we know that P X is uniform but P Y is not. For this case, we have |P Y n − P fn(X n ) | ≥ 1 2 y n :P Y n (y n )< 1 2M n P Y n (y n ) (203) = 1 2 P Y n y n : P Y n (y n ) < 1 2M n (204) = 1 2 P Y n y n : − 1 n n i=1 log P Y (y i ) > H(Y ) + 1 n log 2 (205) ≥ 1 2 Φ − 1 n log 2 − η(Y ) nV 3 (Y ) ,(206) where (203) follows since P fn(X n ) (y n ) ≥ 1 M n or P fn(X n ) (y n ) = 0 for every y n ∈ Y n and thus |P Y n (y n ) − P fn(X n ) (y n )| ≥ P Y n (y n ) for every y n such that P Y n (y n ) < 1 2M n , and (206) follows from the Berry-Esseen theorem [52,Sec. XVI.5]. Hence G(P n X , P n Y ) = 1 − min fn |P Y n − P fn(X n ) | (207) ≤ 1 − 1 2 Φ − 1 n log 2 − η(Y ) nV 3 (Y )(208)→ 3 4 as n → ∞.(209) Case 2 (P Y is uniform): If G(P X , P Y ) < 1, then by the assumption H(X) = H(Y ) = log M , we know that P Y is uniform but P X is not. For this case, we have |P Y n − P fn(X n ) | ≥ sup γ≥1 1 2 y n :P fn(X n ) (y n )≥ γ M n P fn(X n ) (y n ) − 1 M n (210) ≥ sup γ≥1 1 2 x n :P X n (x n )≥ γ M n P X n (x n ) − 1 M n (211) ≥ sup γ≥1 1 2 Φ 1 n log γ − η(X) nV 3 (X) − 1 γ Φ( 1 n log γ) + η(X) nV 3 (X) (212) = sup γ≥1 1 2 1 − 1 γ Φ 1 n log γ − 1 + 1 γ η(X) nV 3 (X) ,(213) where (211) follows since to make (210) as small as possible, the function f n must be injective on the set x n : P X n (x n ) ≥ γ M n , (212) follows from the Berry-Esseen theorem [52,Sec. XVI.5] and Φ 1 n log γ + η(X) nV 3 (X) ≥ x n :P X n (x n )> γ M n P X n (x n ) (214) ≥ γ x n : P X n (x n ) ≥ γ M n M n .(215) Hence G(P n X , P n Y ) = 1 − min fn |P Y n − P fn(X n ) | (216) ≤ 1 − sup γ≥1 1 2 1 − 1 γ Φ 1 n log γ − 1 + 1 γ η(X) nV 3 (X) (217) → 3 4 as n → ∞.(218) This completes the proof. B. Two Special Cases of Conjecture 2 Proposition 9 (Entropy Criterion of Deterministic Coupling). [leftmargin=*] We have the following claims: 1) If H α (X) < H α (Y ) for some α ∈ [0, ∞] , then for any n, G(P n X , P n Y ) < 1. 2) If H α (X) = H α (Y ) for some α ∈ [0, ∞], then G(P n X , P n Y ) = 1 if and only if G(P X , P Y ) = 1. That is, there exists a deterministic coupling P X n Y n ∈ C(P n X , P n Y ) for which Y n is a function of X n , if and only if there exists a deterministic coupling P XY ∈ C(P X , P Y ) for which Y is a function of X. This is also equivalent to the fact that P X and P Y have the same set of probability values. Proof: We first prove Statement 1). Suppose G(P n X , P n Y ) = 1. Then by the definition (30), if G(P n X , P n Y ) = 1, then there exists a coupling of P n X , P n Y such that Y n is a deterministic function of X n . Therefore, we have nH α (X) = H α (X n ) (219) = H α (X n Y n ) (220) ≥ H α (Y n ) (221) = nH α (Y ).(222) This contradicts the assumption H α (X) < H α (Y ). We next prove Statement 2). Obviously if G(P X , P Y ) = 1, then G(P n X , P n Y ) = 1. Next we prove that if G(P n X , P n Y ) = 1 then G(P X , P Y ) = 1. Since in (222) we show that H α (X) ≥ H α (Y ), and as assumed, H α (X) = H α (Y ), the inequality in (221) is in fact an equality, i.e., H α (X n Y n ) = H α (X n ) = H α (Y n ). That is, Y n is a function of X n , and X n is also a function of Y n . Hence the mapping between X n and Y n is bijective, which further implies that P n X and P n Y have the same set of probability values. Since P n X and P n Y have the same number of positive probability values, the support sizes of P X and P Y are equal. Denote the size as k, i.e., k := |supp(P X )| = |supp(P Y )|. Suppose p 1 ≥ p 2 ≥ ... ≥ p k and q 1 ≥ q 2 ≥ ... ≥ q k are the positive probability values of P X and P Y , respectively, ordered in a non-increasing fashion. Then the positive probability values of P n X and P n Y must be p n 1 ≥ p n−1 1 p 2 ≥ ... ≥ p n−1 k p k−1 ≥ p n k and q n 1 ≥ q n−1 1 q 2 ≥ ... ≥ q n−1 k q k−1 ≥ q n k . Hence p 1 = q 1 , p 2 = q 2 , p k−1 = q k−1 , p k = q k . Next we prove p i = q i , i ∈ [3 : k − 2]. Remove Proposition 10 (Deterministic Coupling with Uniform P X ). If P X (x) = 1 M for all x ∈ [1 : M ] for some M ∈ N, then for any P Y , G(P n X , P n Y ) = 1 if and only if G(P X , P Y ) = 1. That is, there exists a deterministic coupling P X n Y n ∈ C(P n X , P n Y ) for which Y n is a function of X n , if and only if there exists a deterministic coupling P XY ∈ C(P X , P Y ) for which Y is a function of X. Proof: We split the proof into three cases. Case 1: If P Y (y 1 ) is irrational for some y 1 ∈ Y and P Y (y 2 ) is rational for other some y 2 ∈ Y, then G(P n X , P n Y ) = 1 only if P Y (y 1 ) = a 1 n 1 with a 1 rational. Consider the term P Y (y 1 ) (P Y (y 2 )) n−1 . It is irrational since P Y (y 1 ) is irrational and P Y (y 2 ) is rational. Hence for any n, P Y (y 1 ) (P Y (y 2 )) n−1 is not a multiple of the probability value P n X (x n ) = 1 M n . Case 2: If P Y (y) is irrational for all y ∈ Y, then G(P n X , P n Y ) = 1 only if for any y, P Y (y) = a 1 n y with a y rational. Consider the terms P Y (y i ) (P Y (y j )) n−1 . Next we prove that there must exist some (i, j) such that P Y (y i ) (P Y (y j )) n−1 is irrational. Suppose P Y (y i ) (P Y (y j )) n−1 is rational for any (i, j). Then P Y (yi) P Y (yj ) is rational since (P Y (y j )) n is rational. That is, P Y (y i ) = k i,j P Y (y j ) for some rational k i,j . Therefore, y∈Y P Y (y) = |Y| i=1 k i,1 P Y (y 1 ) = P Y (y 1 ) |Y| i=1 k i,1 is irrational, since P Y (y 1 ) is irrational and |Y| i=1 k i,1 is rational. However this contradicts the fact that y∈Y P Y (y) = 1 is rational. Therefore, P Y (y i ) (P Y (y j )) n−1 is irrational for some (i, j), and hence it cannot be composited by the probability values P n X (x n ) = 1 M n for any n. Case 3: If P Y (y) is rational for all y ∈ Y, then denote P Y (y) = a b with a, b coprime, and G(P n X , P n Y ) = 1 implies Hence G(P X , P Y ) = 1. On the other hand, it is obvious that G(P X , P Y ) = 1 implies G(P n X , P n Y ) = 1. Therefore, the theorem holds for the case where P Y (y) is rational for all y ∈ Y. Combining the above three cases completes the proof. APPENDIX D PROOF OF THEOREM 4 We first prove the upper bound in (67). To this end, we need the following one-shot achievability result due to Cuff. Lemma 5. [41, Theorem VII.1] Given a source distribution P W , codebook distribution P X|W , and channel P Y |W X , let C be a randomly generated collection of channel inputs x(w) ∈ X , w ∈ W, each drawn independently according to P X|W , and let P Y |C be the output distribution induced by applying the codebook. For any τ > 0, we have E C P Y |C − P Y ≤ P (A τ ) + 1 2 e τ /2 ,(223) where the expectation is with respect to the random codebook, and A τ := (w, x, y) : log P W (w)P Y |W X (y|w, x) P Y (y) > τ .(224) We have min f |P Y f − P Z | ≤ min P X|W E C P Y |C − P Z (225) ≤ min P X|W E C P Y |C − P Y + |P Y − P Z | (226) ≤ min P X|W {|P Y − P Z | + P (A τ )} + 1 2 e τ /2 ,(227) where (225) follows since min P X|W E C P Y |C − P Z ≥ min c P Y |C=c − P Z = min f |P Y f − P Z |, (226) follows from the triangle inequality, and (227) follows from Lemma 5. We next prove the lower bound in (67). Observe that P W Y f (w, y) : P W (w)P Y |W X (y|w, f (w)) P Y f (y) > 1 = P W Y f (w, y) : P W (w)P Y |W X (y|w, f (w)) w P W (w)P Y |W X (y|w, f (w)) > 1 (228) = 0.(229) We relax the deterministic function f to a random mapping P X|W . Then we get min f |P Z − P Y f | ≥ min P X|W :P W XY (A0)=0 |P Z − P Y | . (230) We finally prove the lower bound in (69). By the maximal coupling equality (Lemma 1), there exists a coupling P Y f Z ∈ C(P Y f , P Z ) such that P {Y f = Z} = |P Y f − P Z |.(231) Consider the joint distribution P W (w)P Y |W X (y|w, f (w))P Z|Y f (z|y). We have P {(W, Y f ) = (W, Z)} = P {Y f = Z} . (232) On the other hand, again by the maximal coupling equality, we have P {(W, Y f ) = (W, Z)} ≥ min P (W ,Y f ),(W,Z) ∈C(P W,Y f ,P W,Z ) P {(W , Y f ) = (W, Z)} (233) = |P W,Y f − P W,Z |.(234) Therefore, |P W,Y f − P W,Z | ≤ |P Y f − P Z |.(235) Observe that |P W,Y f − P W,Z | ≥ P W Z (w, y) : P W (w)P Y |W X (y|w, f (w)) P Y (y) > 1 − P W Y (w, y) : P W (w)P Y |W X (y|w, f (w)) P Y (y) > 1 (236) = P W Z (w, y) : P W (w)P Y |W X (y|w, f (w)) P Y (y) > 1 (237) = P P W (W )P Y |W X (Z|W, f (W )) P Z (Z) P Z (Z) P Y (Z) > 1 (238) ≥ P log P W (W )P Y |W X (Z|W, f (W )) P Z (Z) > τ, log P Z (Z) P Y (Z) > −τ (239) = P log P W (W )P Y |W X (Z|W, f (W )) P Z (Z) > τ − P log P W (W )P Y |W X (Z|W, f (W )) P Z (Z) > τ, log P Z (Z) P Y (Z) ≤ −τ (240) ≥ P log P W (W )P Y |W X (Z|W, f (W )) P Z (Z) > τ − P log P Z (Z) P Y (Z) ≤ −τ (241) ≥ P log P W (W )P Y |W X (Z|W, f (W )) P Z (Z) > τ − e −τ(242) ≥ min P W Z ∈C(P W ,P Z ) P W Z (w, z) : log P W (w)P Y |W X (z|w, f (w)) P Z (z) > τ − e −τ (243) ≥ min P X|W min P W Z ∈C(P W ,P Z ) P W XZ (B τ ) − e −τ ,(244) where (237) follows from (229), (243) follows since we relax the distribution P W Z to any coupling in C(P W , P Z ), and (244) follows since we relax the deterministic function f to a random mapping P X|W . Combining (235) and (244) gives us that for any f , |P Y f − P Z | ≥ min P W Z ∈C(P W ,P Z ) min P X|W P W XZ (B τ ) − e −τ .(245) This implies the lower bound in (69). Fig. 2. A standard maximal coupling P XY of (P X , P Y ) for which P Y |X keeps the region III unchanged and transfers probability mass from the region I to the region II [53]. APPENDIX E PROOF OF COROLLARY 9 We only need consider α < 1 case, since H α (Y n |X n ) is decreasing in α. By Theorem 1, we can construct a maximal guessing coupling of P n X and P n Y , which cascades a probability distribution approximation code f n (x n ) with a maximal coupling code P Y n |fn(X n ) . Here we adopt a standard maximal coupling code (see Fig. 2). The "diagonal" probabilities satisfy P Y n |fn(X n ) (y n |y n ) = 1, P fn(X n ) (y n ) ≤ P n Y (y n ); P n Y (y n ) P fn (X n ) (y n ) , P fn(X n ) (y n ) > P n Y (y n ),(246) for any y n ∈ Y n , while the "non-diagonal" probabilities can take on any value. Then by Theorem 2, we know G(P n X , P n Y ) → 1(247) at least exponentially fast as n → ∞. The optimal exponent is denoted as E (P X , P Y ). Denote Z n := f n (X n ), p := P Y n {z n : P Y n (z n ) < P Z n (z n )} , and (248) δ := |P Z n − P Y n |.(249) Then P Z n {z n : P Y n (z n ) < P Z n (z n )} = δ + p.(250) Therefore, we have e (1−α)Hα(Y n |X n ) ≤ e (1−α)Hα(Y n |fn(X n )) (251) = z n ,y n P Z n (z n )P α Y n |Z n (y n |z n ) (252) = z n P Z n (z n ) 1 {P Y n (z n ) ≥ P Z n (z n )} + P Y n (z n ) P Z n (z n ) α + y n =z n P α Y n |Z n (y n |z n ) × 1 {P Y n (z n ) < P Z n (z n )} (253) ≤ z n P Z n (z n ) 1 {P Y n (z n ) ≥ P Z n (z n )} + P Y n (z n ) P Z n (z n ) α + (|Y| n − 1) 1 − P Y n (z n ) P Z n (z n ) |Y| n − 1 α × 1 {P Y n (z n ) < P Z n (z n )} (254) = z n P Z n (z n )1 {P Y n (z n ) ≥ P Z n (z n )} + z n P Z n (z n ) P Y n (z n ) P Z n (z n ) α 1 {P Y n (z n ) < P Z n (z n )} + z n P Z n (z n ) (|Y| n − 1) 1−α 1 − P Y n (z n ) P Z n (z n ) α × 1 {P Y n (z n ) < P Z n (z n )} (255) ≤ P Z n {z n : P Y n (z n ) ≥ P Z n (z n )} + P Z n {z n : P Y n (z n ) < P Z n (z n )} × z n P Z n (z n )1 {P Y n (z n ) < P Z n (z n )} P Z n {z n : P Y n (z n ) < P Z n (z n )} P Y n (z n ) P Z n (z n ) α + |Y| (1−α)n P Z n {z n : P Y n (z n ) < P Z n (z n )} × z n P Z n (z n )1 {P Y n (z n ) < P Z n (z n )} P Z n {z n : P Y n (z n ) < P Z n (z n )} α × 1 − P Y n (z n ) P Z n (z n ) α (256) = P Z n {z n : P Y n (z n ) ≥ P Z n (z n )} + P 1−α Z n {z n : P Y n (z n ) < P Z n (z n )} × P α Y n {z n : P Y n (z n ) < P Z n (z n )} + |Y| (1−α)n P 1−α Z n {z n : P Y n (z n ) < P Z n (z n )} × z n (P Z n (z n )−P Y n (z n ))1 {P Y n (z n ) < P Z n (z n )} α (257) = 1 − (δ + p) + (δ + p) 1−α p α + |Y| (1−α)n (δ + p) 1−α δ α (258) = 1 − (δ + p) + (δ + p) p δ + p α + |Y| (1−α)n (δ + p) 1−α δ α (259) ≤ 1 + |Y| (1−α)n δ α(260)≤ 1 + |Y| (1−α)n e −αnE(P X ,P Y ) (261) → 1 as n → ∞,(262) where (254) follows since y n =z n P α Y n |Z n (y n |z n ) for α < 1 is maximized by the uniform distribution P Y n |Z n (y n |z n ) = 1 − P Y n (z n ) P Z n (z n ) |Y| n − 1(263) for y n = z n (this point is similar to the fact that the uniform distribution maximizes the Rényi entropy), and (256) follows since x α with 0 < α < 1 is concave in x. APPENDIX F PROOF OF PROPOSITION 4 According to the definition of G E (P Y |X , P Y ) and Remark 10, we have G E (P Y |X , P Y ) = inf P X ∈P(P Y |X ,P Y ) H(X). Next we prove G E (P Y |X , P Y ) = lim n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n H(X n ). First it is easy to lower bound G E (P Y |X , P Y ) as G E (P Y |X , P Y ) = inf P X ∈P(P Y |X ,P Y ) H(X)(264) ≥ lim sup n→∞ inf P X n ∈P(P n Y |X ,P n Y ) 1 n H(X n ) (265) where (265) follows since H(X) ≥ lim sup n→∞ 1 n H(P X n ) for any X with a finite alphabet X (see [6,Theorem 1.7.2]). Assume n = mk+l where l < k with a fixed number k. For the first mk symbols, we use the code f mk in [7] to exactly synthesize P m X k with P X k ∈ P(P k Y |X , P k Y ). By Corollary 12, we have that if the code rate R (1) > 1 k H(X k ) lim m→∞ P X mk = f mk (M (1) mk ) = 1,(266) where M (1) mk ∼ Unif[1 : e mkR (1) ]. On the other hand, for each of the last l symbols, we use a single-letter code f 1 to approximately synthesize P X with P X ∈ P(P Y |X , P Y ). Here we assume f 1 satisfies P X − P f1(M (2) ) ≤ |X |e −R (2) m where M (2) ∼ Unif[1 : e R (2) m ]. Obviously, there exists at least one code f 1 satisfying this condition. By the equivalence (93), we know that there exists a coupling P M (2) X ∈ C(P M (2) , P X ) satisfying min f1 P X = f 1 (M (2) ) ≤ |X |e −R (2) m . For this concatenated code, we have that the overall code rate is mkR (1) +lR (2) where M n = (M (1) mk , (M (2) ) l ). We choose R (1) , R m such that R (1) > 1 k H(X k ) for some P X k ∈ P(P k Y |X , P k Y ), lim n→∞ R This implies we get a channel resolvability code with rate min P X k ∈P(P k Y |X ,P k Y ) 1 k H(X k ). Since k is arbitrary, we have G E (P Y |X , P Y ) ≤ lim inf k→∞ min P X k ∈P(P k Y |X ,P k Y ) 1 k H(X k ).(271) Combining (265) and (271), we have G E (P Y |X , P Y ) = lim n→∞ min P X n ∈P(P n Y |X ,P n Y ) 1 n H(X n ). APPENDIX G PROOF OF THEOREM 13 Proof of "if": Suppose that Q X and Q X induce the same P Z through P Z|X but induce two different distributions of Y through P Y |X . Define P (B) X := BQ X + (1 − B)Q X , where B ∈ [0, 1] with distribution P B such that P B (0)P B (1) > 0. Consider a new wiretap channel P Y Z|B = P Y |B P Z . (272) The last inequality follows the following argument via contradiction. Suppose I(B; Y ) = 0, then B ⊥ Y . Hence P Y |B=0 = P Y |B=1 = P Y . This contradicts with the assumption that Q X and Q X induce two different distributions of Y through P Y |X . Proof of "only if": We prove this by contradiction. That is, we need to show if for any two distributions Q X and Q X that induce the same P Z through P Z|X , they must induce a same distribution of Y through P Y |X , then the perfect stealthsecrecy capacity is zero. Suppose P Z|X Q X = P Z has infinitely many solutions; otherwise, by Lemma 3, P Z|X is a full-rank channel or P Z|X Q X = P Z has a single unique solution P X which is a degenerate distribution. For the former case, by Theorem 11 we know that the perfect stealth-secrecy capacity is zero. For the latter case, since nR 1 ≤ I(Y n ; X n ) = 0, the perfect stealth-secrecy capacity is also zero. So we only need to consider the case that P Z|X Q X = P Z has infinitely many solutions. In addition, note that we also only need to consider the case that there exists a solution to P Z|X Q X = P Z which is an interior point of the probability simplex {P X : x P X (x) = 1, P X (x) ≥ 0}. This is because if all the solutions to P Z|X Q X = P Z are at the boundary of the probability simplex {P X : x P X (x) = 1, P X (x) ≥ 0}, then there exists a set X 0 such that the solutions satisfy Q X (x 0 ) = 0 for any x 0 ∈ X 0 . Hence remove the corresponding columns of P Z|X and the corresponding rows of P X , and denote the resulting matrix and vector as P Z|X and Q X respectively, then we get equation P Z|X Q X = P Z . For this new equation, there exists a solution which is an interior point of the probability simplex {P X ∈ P(X \X 0 ) : x P X (x) = 1, P X (x) ≥ 0}. Suppose P Y is the distribution induced by Q X through P Y |X where Q X is a distribution inducing P Z through P Z|X . By subtracting Q X from the solutions to P Z|X Q X = P Z and P Y |X Q X = P Y , we get the equation P Z|X Q = 0 and P Y |X Q = 0 (here Q denotes Q X − Q X ). Denote S Z as the set of solutions to P Z|X Q = 0 and S Y as the set of solutions to P Y |X Q = 0. Then by assumption, S Z ⊆ S Y . Note that the set of solutions to P Z|X Q X = P Z (with Q X constrained to be a probability distribution) is the intersection of the set of solutions to P Z|X Q = P Z without the probability constraint on Q and the probability simplex {P X : x P X (x) = 1, P X (x) ≥ 0}. If there exists a solution Q * X to P Z|X Q X = P Z which is an interior point of the probability simplex {P X : x P X (x) = 1, P X (x) ≥ 0}, then the subspace of R |X | spanned by the set S Z is the same to the orthogonal complement of the subspace of R |X | spanned by the rows of P Z|X , and also the same to the set of the solutions to P Z|X Q = 0 (without the probability constraint). Since S Z ⊆ S Y (or equivalently, S Z + Q X ⊆ S Y + Q X ), Q * X is also a solution to P Y |X Q X = P Y . Since Q * X is an interior point of the probability simplex, similarly, we have that the subspace of R |X | spanned by the set S Y is the same to the set of the solutions to P Y |X Q = 0 (without the probability constraint). Denote S Z as the set of solutions to P Z|X Q = 0 (without the probability constraint) and S Y as the set of solutions to P Y |X Q = 0 (without the probability constraint). Then S Z ⊆ S Y . A vector Q is a solution to P Z|X Q = 0 (without probability constraint) if and only if it lies in the orthogonal complement of the subspace of R |X | spanned by the rows of P Z|X . Hence S Z ⊆ S Y means that the orthogonal complement of the row space of P Z|X is a subset of that of the row space of P Y |X . It means that the row space of P Y |X is a subset of the row space of P Z|X . Hence every row of P Y |X is a linear combination of the rows of P Z|X . Thus, P Y |X = AP Z|X for some matrix A. On the other hand, observe that AP Z|X Q X = AP Z , AP Z|X = P Y |X , and P Y |X Q X = P Y . Hence AP Z = P Y . Now we prove the following property for any n: for all distributions Q X n that induce P n Z through P n Z|X , they must induce the same distribution of Y n through P n Y |X . Consider the equation P ⊗n Z|X Q X n = P ⊗n Z .(273) Multiply A ⊗n at both sides, then we get A ⊗n P ⊗n Z|X Q X n = A ⊗n P ⊗n Z which is equivalent to AP Z|X ⊗n Q X n = (AP Z ) ⊗n . Substituting AP Z|X = P Y |X and AP Z = P Y , we get P ⊗n Y |X Q X n = P ⊗n Y . Observe A, P Z are fixed, hence P Y is fixed as well. This means for all distributions Q X n that induce P n Z through P n Z|X , they must induce the same distribution P n Y through P n Y |X . Using on the property above, we return to proving that the perfect stealth-secrecy capacity is zero. Note that by the secrecy constraint, P n Z (·) = P Z n |M (·|m) = x n P n Z|X (·|x n )P X n |M (x n |m) (277) for any m. Hence for any m, P X n |M (·|m) is a distribution that induces P n Z through P n Z|X . By the property stated in (277), we have that for different m, P X n |M (·|m) induces the same distribution of Y n through P n Y |X , i.e., P Y n |M (·|m) = x n P n Y |X (·|x n )P X n |M (x n |m) does not depend on m. Consequently, Y n is independent of M , i.e., α (P X , P Y ) is monotonically decreasing in α since H α (Y |X) has this monotonicity property (the latter property was proved in [38, Proposition 4.6] and [39, Proposition 1]). Proposition 3 . 3[40, p. 223] The problem in (50) is NP-hard (more specifically, NP-complete). Fig. 1 . 1General source-channel resolvability. min f P {Y n = f (X n )}.Then Fano's inequality [24, Theorem 2.10.1] implies ρ m (X; Y |W ) := sup E[cov(g(X, W ), h(Y, W )|W )] E[var(g(X, W )|W )] E[var(h(Y, W )|W )](114)where the supremum extends over all functions g : X × W → R and h : Y × W → R satisfying E[var(g(X, W )|W )] > 0, and E[var(h(Y, W )|W )] > 0. Proof: Achievability (Lower Bound): Consider the coding scheme used in the proof of Theorem 9. By the achievability part of Theorem 6, we have that if lim sup n→∞ √ n(R n − H(W )) < − V (W )Q −1 (ε), (121) then there exists a maximal guessing coupling P MnW n such that P Mn = Unif[1 : e nRn ] and lim sup n→∞ min fn P {M n = f n (W n )} ≤ ε. 155) ≤ max P U X :P X ∈P(P Z ) min P T |XZ :P Z|T is of full-rank, X→T →Z I(U ; Y ) − I(U ; T ). Proposition 8 (Asymptotically Deterministic Coupling with Uniform P X or P Y ). Assume X = [1 : M ] and P X (x) = 1 M for all x ∈ [1 : M ] or Y = [1 : M ] and P Y (y) = 1 M for all y ∈ [1 : M ] for some M ∈ N, and H(X) = H(Y ) = log M . Then G(P n X , P n Y ) → 1 if and only if G(P X , P Y ) = 1. . They must be equal. Hence p 3 = q 3 . In the same way, we can showp i = q i , i ∈ [3 : k − 2]. n , i.e., M a b n = k. Hence b|M , otherwise, M a b n / ∈ N since M a b ∈ Q and M a b / ∈ N. Assume M = k b. Then P Y (y) = a b = k a M . P≤ {X n = f n (M n )} ≤ P X mk = f mk (M P X mk = f mk (M (1) mk ) + k|X |e −R (2) m , Then for fixed k, the overall rate lim n→∞ mkR (1) +lR (2) m mk+l = R (1) , and the overall minimum guessing error probability lim n→∞ min fn P {X n = f n (M n )} ≤ lim m→∞ P X mk = f mk (M C 1 ( 1P Z ) ≥ max P U X :U ⊥Z,P X ∈P(P Z ) I(U ; Y ) ≥ I(B; Y ) > 0. Associate Professor in the Department of Electrical and Computer Engineering and the Department of Mathematics at the National University of Singapore (NUS). He received the B.A. and M.Eng. degrees in Electrical and Information Sciences from Cambridge University in 2005 and the Ph.D. degree in Electrical Engineering and Computer Science (EECS) from the Massachusetts Institute of Technology (MIT) in 2011. His research interests include information theory, machine learning, and statistical signal processing. Dr. Tan received the MIT EECS Jin-Au Kong outstanding doctoral thesis prize in 2011, the NUS Young Investigator Award in 2014, the NUS Engineering Young Researcher Award in 2018, and the Singapore National Research Foundation (NRF) Fellowship (Class of 2018). He is also an IEEE Information Theory Society Distinguished Lecturer. He has authored a research monograph on "Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities" in the Foundations and Trends in Communications and Information Theory Series (NOW Publishers). He is currently an Associate Editor of the IEEE Transactions on Signal Processing. Unless explicitly stated, the logarithm base can be chosen arbitrarily. But regardless of the base, exp(x) or e x always denotes the inverse of log(x). AcknowledgmentsThe authors would like to thank Prof. Igal Sason for pointing out reference[22]. The authors also thank the reviewers and the editor for their suggestions to improve the quality of the paper. 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D A Levin, Y Peres, E L Wilmer, American Mathematical Soc107D. A. Levin, Y. Peres, and E. L. Wilmer. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. he was a postdoctoral researcher at the Department of Electronic Engineering and Information Science (EEIS), USTC. Currently, he is a research fellow at the. Lei Yu received the B.E. and Ph.D.Department of Electrical and Computer Engineering, National University of Singaporedegrees, both in electronic engineering, from University of Science and Technology of China (USTC) in 2010 and 2015, respectively. His research interests include information theory, probability theory, and securityLei Yu received the B.E. and Ph.D. degrees, both in electronic engineering, from University of Science and Technology of China (USTC) in 2010 and 2015, respectively. From 2015 to 2017, he was a postdoctoral researcher at the Department of Electronic Engineering and Information Science (EEIS), USTC. Currently, he is a research fellow at the Department of Electrical and Computer Engineering, National University of Singapore. His research interests include information theory, probability theory, and security.

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{'abstract': "A coupling of two distributions PX and PY is a joint distribution PXY with marginal distributions equal to PX and PY . Given marginals PX and PY and a real-valued function f of the joint distribution PXY , what is its minimum over all couplings PXY of PX and PY ? We study the asymptotics of such coupling problems with different f 's and with X and Y replaced by X n = (X1, . . . , Xn) and Y n = (Y1, . . . , Yn) where Xi and Yi are i.i.d. copies of random variables X and Y with distributions PX and PY respectively. These include the maximal coupling, minimum distance coupling, maximal guessing coupling, and minimum entropy coupling problems. We characterize the limiting values of these coupling problems as n tends to infinity. We show that they typically converge at least exponentially fast to their limits. Moreover, for the problems of maximal coupling and minimum excess-distance probability coupling, we also characterize (or bound) the optimal convergence rates (exponents). Furthermore, for the maximal guessing coupling problem we show that it is equivalent to the distribution approximation problem. Therefore, some existing results for the latter problem can be used to derive the asymptotics of the maximal guessing coupling problem. We also study the asymptotics of the maximal guessing coupling problem for two general sources and a generalization of this problem, named the maximal guessing coupling through a channel problem. We apply the preceding results to several new informationtheoretic problems, including exact intrinsic randomness, exact resolvability, channel capacity with input distribution constraint, and perfect stealth and secrecy communication.", 'arxivid': '1712.06804', 'author': [], 'authoraffiliation': [], 'corpusid': 35055050, 'doi': '10.1109/tit.2018.2857763', 'github_urls': [], 'n_tokens_mistral': 44807, 'n_tokens_neox': 40637, 'n_words': 25271, 'pdfsha': '854ef9975297d89ccbdf718804d61f33eff1dfda', 'pdfurls': ['https://arxiv.org/pdf/1712.06804v3.pdf'], 'title': ['Asymptotic Coupling and Its Applications in Information Theory', 'Asymptotic Coupling and Its Applications in Information Theory'], 'venue': []}

arxiv

Physics is the New Data Sergei V Kalinin Department of Materials Science and Engineering University of Tennessee Knoxville 37996KnoxvilleTennesseeUSA Maxim Ziatdinov Computational Sciences and Engineering Division Oak Ridge National Laboratory 37831Oak RidgeTennesseeUSA Center for Nanophase Materials Sciences Oak Ridge National Laboratory 37831Oak RidgeTennesseeUSA Bobby G Sumpter Center for Nanophase Materials Sciences Oak Ridge National Laboratory 37831Oak RidgeTennesseeUSA Andrew D White Department of Chemical Engineering University of Rochester 14534RochesterNew YorkUSA Physics is the New Data Notice: This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). The rapid development of machine learning (ML) methods has fundamentally affected numerous applications ranging from computer vision, biology, and medicine to accounting and text analytics. 1-3 Until now, it was the availability of large and often labeled data sets that enabled significant breakthroughs. However, the adoption of these methods in classical physical disciplines has been relatively slow, a tendency that can be traced to the intrinsic differences between correlative approaches of purely data-based ML and the causal hypothesis-driven nature of physical sciences. Furthermore, anomalous behaviors of classical ML necessitate addressing issues such as explainability and fairness of ML. We also note the sequence in which deep learning became mainstream in different scientific disciplines -starting from medicine and biology and then towards theoretical chemistry, and only after that, physics -is rooted in the progressively more complex level of descriptors, constraints, and causal structures available for incorporation in ML architectures. Here we put forth that over the next decade, physics will become a new data, and this will continue the transition from dot-coms and scientific computing concepts of 90ies to big data of 2000-2010 to deep learning of 2010-2020 to physics-enabled scientific ML. Neural networks and machine learning (ML) have been known since the early work on perceptrons in the 1950s. The introduction of the backpropagation algorithm in the 1980s laid the foundation for modern ML. However, the computational capabilities and lack of large labeled data sets limited the community to relatively shallow networks in applications such as molecular dynamics simulations, theory-experiment matching, and experimental data analysis. 4 The situation started to radically change after the 2000, when the availability of high-performance computing capabilities enabled large computational models and early experimental applications of neural networks were demonstrated. Similarly, the growth in internet technologies, including search engines and social networks, created a favorable environment for collecting and exploring large data sets, thus starting the deep learning revolution. The striking aspect of deep learning (DL) is that, from a certain perspective, the field is just around ten years old. Indeed, the development of ImageNet by Fei-Fei Li et al. in 2009 5 has demonstrated the utility of large-annotated data sets, setting the foundation of the big data revolution. The ground-breaking paper by Krizhevsky in 2012, 6 introducing the potential of deep neural networks, then became the harbinger of the deep learning revolution of the last decade. Generative Adversarial Networks (GANs) 7 and Variational Autoencoders (VAEs) 8 appeared in 2014 (and by now, these preprints have ~80k citations). The concepts such as attention layers and transformers appeared in 2017. 9 Significant reinforcement learning-based advances in robotics and chip design are happening now. 10 Throughout this past decade, much of this growth was fueled by big tech companies like Google, Facebook, Baidu, Amazon, Tencent, Uber, and Microsoft -and that in turn was predicated on their access to data and the necessity to derive actionable or monetizable insights from it. In many cases, the broad propagation of specific social media applications combined with the cell phone and mobile electronics ubiquity allows for centralized access to a significant fraction of all data in certain domains. However, equally important is that these advances started to strongly highlight the problems with data-based supervised learning methods. There are many aspects of these, ranging from adversarial attacks to the issues related to fairness and explainability -but they are ultimately linked to the fact that pure data-centric approaches are inherently limited in capturing the complexity of the real world. There are also limits to how far these can be scaled -simply as a matter of energy consumption, if nothing else. These limits have been noticed by the ML world. Causal ML, metric learning, and, more generally, methods aiming at avoiding out of distribution effects are growing in popularity. The same is true for physics-informed machine learning from Hamiltonian to equivariant neural networks. Similarly, there is a rapid growth of interest in the ML studies of dynamic data generation processes such as autonomous microscopy or physical characterization instead of the analysis of the static benchmark data sets. This is a very recent trend -led by the Flatiron institute, Google Accelerated Science, the Vector Institute, and similar organizations that typically combine strong connections to both the industry and academic world. A brief overview of the last ~3 years (at most) suggests that these developments can be a part of a more general trend -namely, ML methods progressively start to operate with concepts more abstract than just data sets. In other words, the ML field has evolved from "shallow" data-driven methods (think cats on the internet or hand-written digits) to methods utilizing more complex constructs such as graph and symbolic representations, invariances, positional embeddings -i.e., much deeper levels of scientific abstraction than simply data. This trend effectively brings in new concepts and connections to fundamental physics. Which brings us to a central concept of this perspective -if we know that big data is necessary but limited in potential, then what will be the new data? It seems that the existing (if not yet broadly realized) trend is the merger between physics and data sciences. It is also very remarkable that despite the tremendous developments in DL in biological, social, and economic sciences and theory, the progress in experimental sciences, including condensed matter physics, materials science, and chemistry, was relatively slow to develop. Indeed, much of the original insights and developments in ML came from biological areasstarting from Hebb and more recently to Sejnowski and many others. 11 This was due to scientific reasons, e.g., the perceived similarity between NN architectures and brain/vision. The second aspect is that neural networks (NNs) were adopted much earlier and more broadly in biology and medicine. While one should be careful with speculation as to why, some factors include that biological areas are generally less susceptible to analytical derivations than physics, necessitating the data focus. Furthermore, physical sciences generally operate with complex causal and hypothesis-driven paradigms. Some of the leading scientists in the field such as Judea Pearl, one of the fathers of causal inference methods, 12 was a professor of physics and worked with superconductors before moving into biostatistics. Interestingly, this community is still very heavily bio-based -and very few (so far) causal ML groups have moved into physics, even though physics is much more amenable to interpretation in terms of causal mechanisms than biology. Currently, the biological and medical areas are clearly the ones where classical ML methods can and are making major impact. However, these fields generally operate with relatively poorly defined entities and laws, making rigorous definitions difficult. As such, DL methods can be employed on this data directly. Comparatively, physical sciences offer rigorously defined entities, and often defined relationships between them structured via conservation laws, invariances, and causal relationships. In this case, ad hoc application of ML models based on data only may lead to unphysical results if the relationships are violated during training and fitting. In other words, while inference can violate relevant physical laws as a part of discovery (much like the introduction of complex numbers), the final results obtained need to comport to realistic physical constraints. Applications of machine learning in modeling and simulation provided some of the earliest impacts, with examples spanning prediction of structure/property relationships to molten salt phase diagrams to molecular energy transfer, to multi-dimensional potential landscape design to toxicity prediction alongside numerous process control/optimization and correlative studies for spectroscopy. However, computational methods application is facilitated by the lack of out of distribution drift, meaning that the data generation processes are similar. By the same token, the causal chains in theoretical models are well understood. Chemistry is a unique field because nearly all data are united upon the idea of molecular structure. Fields as far apart as protein structure and porous materials have a common underlying feature: atoms and bonds. Physics-informed ML is typically viewed as about the underlying model but in chemistry, the representations themselves imply equivariances. For example, features of molecular graphs and labels of atomic partial charge imply permutation equivariance. Features of Cartesian coordinates and labels of atomic partial charges imply translation and rotation invariance. Just choosing how to represent data is already connected with the physics of problems in chemistry. The most significant recent advances in chemistry and biology AI have been accomplished with equivariant neural networks. 13-15 These neural networks are built for specific symmetry groups, and this enables vast reductions in training data because the symmetry structure is explicit rather than learned. This depends on the underlying data being from a known symmetry group. As deep learning moves to less data-rich areas of physics, having symmetry group attributes can significantly reduce the training data by being a strong inductive bias. One of the strongest counterarguments the inductive biases of symmetry has been pretraining and other semi-supervised methods. For example, large language models (LLMs) appear to learn the structure of language without any explicit construction of grammar. LLMs are pretrained on large corpi, like billions of bioinformatics sequences or millions of natural language documents. The data is supposed to impart some "understanding" to the LLMs about language and then the LLMs have been quite successful on downstream tasks like predicting sentiment in a Tweet or predicting secondary structure in a protein. 16,17 It appeared that unlabeled, unstructured data can be a scalable approach to learning. However, research groups have recently shown that training an LLM music data, gave a similar performance on downstream tasks about English. This seems to counter the hypothesis that large corpi are responsible for the success of pretraining. Lipton et al. 18 even went further and showed that even pretraining on randomly generated nonsense gives good performance. They hypothesized that it is the structure of the pretraining algorithm and tasks, rather than the underlying large data, that are responsible for the success of LLMs. Semisupervised methods are being adapted to graphs 19 and point clouds, 20 so it remains to be seen if a purely data-driven approaches with enormous corpi can be successful. Another remarkable development over the last several years is the rapid growth of the symbolic regression methods. In many areas of science, the presence of closed-form symbolic representations is perceived as the understanding of underlying mechanisms. The most wellknown examples of these are equations of motions that stem from conservation laws of classical mechanics. However, in many scientific areas, the semiquantitative laws that define chemical reactivity of organic molecules, relate crystallographic structures to atomic radii, or define the direction of electron transfer between elements have been known for almost a century. The power of machine learning methods such as SISSO, 21 SinDy 22 and PySR 23 to derive these symbolic expressions from observational data is now opening new opportunities for scientific discovery. Particularly of interest is the transition of these methods towards active learning, in which case the ML algorithm interacts with the data generation process to effectively conduct scientific experiments. Finally, applications of DL in experimental physical sciences presents the ultimate challenge. While it is generally accepted that the observed structures and processes are governed by the causal physical laws, the observations and experiments necessarily yield biased representation of reality. In many cases, the property of the observed object and observing system cannot be decoupled, with examples ranging from microscopy to nanoindentation. In classical instrumental sciences, the calibration and quantification of the observations in terms of robust material-specific descriptors is at premium. Similarly, experiments almost inevitably contain observational biases and confounders, severely limiting interpretation. Indeed, one of the major limitations of modern DL models is their correlative nature. During the training stage, a DL model minimizes its loss objective by absorbing all the spurious correlations 24 -including confounding factors and selection biases -found in the training dataset. Because these correlations are not related to actual causal mechanisms, they can easily change between training and application domains. This commonly leads to poor performance of pre-trained models on data produced by distributions different from the training distribution (known as an out-of-distribution or OOD effect). For example, replacing a scanner in the electron microscopy experiment can drastically reduce the recognition capabilities of the DL model on the same sample. Incorporating causality into the DL frameworks is still in the early stages, including works on invariant risk minimization 25 and causal representation learning. 26 However, none of the suggested to-date approaches have demonstrated reliable performance on real-world OOD data. An alternative solution is to replace a model pre-trained on a static dataset with a model interacting actively with a data generation process. For example, in the active learning approach, a surrogate model -such as the Gaussian process or Bayesian neural network -takes the (sparsely) measured data as input and produces the expected values and associated uncertainties of the physical property of interest in the unmeasured parts of parameter space, which can then be used to derive the next measurement point. However, the standard active learning frameworks do not readily allow for incorporating different data modalities or prior physical knowledge. The recently introduced hypothesis learning framework 27 allowed a co-navigation of hypothesis and experimental spaces by incorporating probabilistic models of possible system behaviors and reinforcement learning policies into the active learning setup. It is based on the idea that a correct model of a system's behavior leads to a faster decay of the posterior predictive uncertainty and allows quickly learning the right model together with the overall data distribution using a relatively small number of measurements. Hypothesis learning can be extended to the active learning of structure-property relationships in the multi-modal experiments, where one can quickly learn the information channel with the best predictive capacity for the property of interest. However, being able to pose causative, interventional, and counterfactual questions as well as integration of abstract concepts, should probably start at the interface between physical and computational sciences where arguably the data would have higher veracity. The rise of deep learning in drug discovery. H Chen, O Engkvist, Y Wang, M Olivecrona, T Blaschke, Drug Discovery Today. 236Chen, H.; Engkvist, O.; Wang, Y.; Olivecrona, M.; Blaschke, T., The rise of deep learning in drug discovery. Drug Discovery Today 2018, 23 (6), 1241-1250. Deep learning. Y Lecun, Y Bengio, G Hinton, Nature. 20157553LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning. Nature 2015, 521 (7553), 436-444. Deep learning in neural networks: An overview. J Schmidhuber, Neural Netw. 61Schmidhuber, J., Deep learning in neural networks: An overview. Neural Netw. 2015, 61, 85-117. On the Design, Analysis, and Characterization of Materials Using Computational Neural Networks. B G Sumpter, D W Noid, Annu. Rev. Mater. Sci. 261Sumpter, B. G.; Noid, D. W., On the Design, Analysis, and Characterization of Materials Using Computational Neural Networks. Annu. Rev. Mater. Sci. 1996, 26 (1), 223-277. ImageNet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L J Li, L Kai, F.-F Li, IEEE Conference on Computer Vision and Pattern Recognition. Deng, J.; Dong, W.; Socher, R.; Li, L. J.; Kai, L.; Li, F.-F., ImageNet: A large-scale hierarchical image database. 2009 IEEE Conference on Computer Vision and Pattern Recognition 2009, 248-255. ImageNet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, Advances in Neural Information Processing Systems. 25Krizhevsky, A.; Sutskever, I.; Hinton, G. E., ImageNet classification with deep convolutional neural networks. Advances in Neural Information Processing Systems 25 (NIPS 2012) 2012. A. i. n. i. p. s., Generative adversarial nets. I Goodfellow, J Pouget-Abadie, M Mirza, B Xu, D Warde-Farley, S Ozair, A Courville, Y J Bengio, 27Goodfellow, I.; Pouget-Abadie, J.; Mirza, M.; Xu, B.; Warde-Farley, D.; Ozair, S.; Courville, A.; Bengio, Y. J. A. i. n. i. p. s., Generative adversarial nets. 2014, 27. Auto-encoding variational bayes. D P Kingma, M Welling, arXiv:1312.6114arXiv.org e-Print archiveKingma, D. P.; Welling, M., Auto-encoding variational bayes. 2013, arXiv:1312.6114. arXiv.org e-Print archive. https://arxiv.org/abs/1312.6114. . A Vaswani, N Shazeer, N Parmar, J Uszkoreit, L Jones, A N Gomez, Ł Kaiser, 30Polosukhin, I. J. A. i. n. i. p. s., Attention is all you need. 2017Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A. N.; Kaiser, Ł.; Polosukhin, I. J. A. i. n. i. p. s., Attention is all you need. 2017, 30. A graph placement methodology for fast chip design. A Mirhoseini, A Goldie, M Yazgan, J W Jiang, E Songhori, S Wang, Y.-J Lee, E Johnson, O Pathak, A Nazi, J Pak, A Tong, K Srinivasa, W Hang, E Tuncer, Q V Le, J Laudon, R Ho, R Carpenter, J Dean, Nature. 20217862Mirhoseini, A.; Goldie, A.; Yazgan, M.; Jiang, J. W.; Songhori, E.; Wang, S.; Lee, Y.-J.; Johnson, E.; Pathak, O.; Nazi, A.; Pak, J.; Tong, A.; Srinivasa, K.; Hang, W.; Tuncer, E.; Le, Q. V.; Laudon, J.; Ho, R.; Carpenter, R.; Dean, J., A graph placement methodology for fast chip design. Nature 2021, 594 (7862), 207-212. Neural networks for pattern recognition. C M Bishop, Oxford university pressBishop, C. M., Neural networks for pattern recognition. Oxford university press: 1995. Highly accurate protein structure prediction with AlphaFold. J Pearl, J Jumper, R Evans, A Pritzel, T Green, M Figurnov, O Ronneberger, K Tunyasuvunakool, R Bates, A Žídek, A Potapenko, A Bridgland, C Meyer, S A A Kohl, A J Ballard, A Cowie, B Romera-Paredes, S Nikolov, R Jain, J Adler, T Back, S Petersen, D Reiman, E Clancy, M Zielinski, M Steinegger, M Pacholska, T Berghammer, S Bodenstein, D Silver, O Vinyals, A W Senior, K Kavukcuoglu, P Kohli, D Hassabis, Causality: Models, Reasoning and Inference. Cambridge University Press13Pearl, J., Causality: Models, Reasoning and Inference. Cambridge University Press: 2009. 13. Jumper, J.; Evans, R.; Pritzel, A.; Green, T.; Figurnov, M.; Ronneberger, O.; Tunyasuvunakool, K.; Bates, R.; Žídek, A.; Potapenko, A.; Bridgland, A.; Meyer, C.; Kohl, S. A. A.; Ballard, A. J.; Cowie, A.; Romera-Paredes, B.; Nikolov, S.; Jain, R.; Adler, J.; Back, T.; Petersen, S.; Reiman, D.; Clancy, E.; Zielinski, M.; Steinegger, M.; Pacholska, M.; Berghammer, T.; Bodenstein, S.; Silver, D.; Vinyals, O.; Senior, A. W.; Kavukcuoglu, K.; Kohli, P.; Hassabis, D., Highly accurate protein structure prediction with AlphaFold. Nature 2021, 596 (7873), 583-589. OrbNet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features. Z Qiao, M Welborn, A Anandkumar, F R Manby, T F Miller, The Journal of Chemical Physics. 202012124111Qiao, Z.; Welborn, M.; Anandkumar, A.; Manby, F. R.; Miller, T. F., OrbNet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features. The Journal of Chemical Physics 2020, 153 (12), 124111. Deep-neural-network solution of the electronic Schrödinger equation. J Hermann, Z Schätzle, F Noé, Nat. Chem. 202010Hermann, J.; Schätzle, Z.; Noé, F., Deep-neural-network solution of the electronic Schrödinger equation. Nat. Chem. 2020, 12 (10), 891-897. Biological structure and function emerge from scaling unsupervised learning to 250 million protein sequences. A Rives, J Meier, T Sercu, S Goyal, Z Lin, J Liu, D Guo, M Ott, C L Zitnick, J Ma, R Fergus, 2020622803Rives, A.; Meier, J.; Sercu, T.; Goyal, S.; Lin, Z.; Liu, J.; Guo, D.; Ott, M.; Zitnick, C. L.; Ma, J.; Fergus, R., Biological structure and function emerge from scaling unsupervised learning to 250 million protein sequences. bioRxiv 2020, 622803. Language models are few-shot learners. T Brown, B Mann, N Ryder, M Subbiah, J D Kaplan, P Dhariwal, A Neelakantan, P Shyam, G Sastry, A Askell, Advances in neural information processing systems 2020. 33Brown, T.; Mann, B.; Ryder, N.; Subbiah, M.; Kaplan, J. D.; Dhariwal, P.; Neelakantan, A.; Shyam, P.; Sastry, G.; Askell, A., Language models are few-shot learners. Advances in neural information processing systems 2020, 33, 1877-1901. K Krishna, J Bigham, Z C Lipton, arXiv:2109.04953Does Pretraining for Summarization Require Knowledge Transfer? arXiv preprint. Krishna, K.; Bigham, J.; Lipton, Z. C., Does Pretraining for Summarization Require Knowledge Transfer? arXiv preprint arXiv:2109.04953 2021. Molecular contrastive learning of representations via graph neural networks. Y Wang, J Wang, Z Cao, A Barati Farimani, Nat. Mach. Intell. 20223Wang, Y.; Wang, J.; Cao, Z.; Barati Farimani, A., Molecular contrastive learning of representations via graph neural networks. Nat. Mach. Intell. 2022, 4 (3), 279-287. Self-supervised deep learning on point clouds by reconstructing space. J Sauder, B Sievers, Advances in Neural Information Processing Systems. 32Sauder, J.; Sievers, B., Self-supervised deep learning on point clouds by reconstructing space. Advances in Neural Information Processing Systems 2019, 32. SISSO: A compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates. R Ouyang, S Curtarolo, E Ahmetcik, M Scheffler, L M Ghiringhelli, Physical Review Materials. 2018883802Ouyang, R.; Curtarolo, S.; Ahmetcik, E.; Scheffler, M.; Ghiringhelli, L. M., SISSO: A compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates. Physical Review Materials 2018, 2 (8), 083802. PySINDy: A comprehensive Python package for robust sparse system identification. A A Kaptanoglu, B M De Silva, U Fasel, K Kaheman, J L Callaham, C B Delahunt, K Champion, J.-C Loiseau, J N Kutz, S L Brunton, arXiv:2111.08481arXiv preprintKaptanoglu, A. A.; de Silva, B. M.; Fasel, U.; Kaheman, K.; Callaham, J. L.; Delahunt, C. B.; Champion, K.; Loiseau, J.-C.; Kutz, J. N.; Brunton, S. L., PySINDy: A comprehensive Python package for robust sparse system identification. arXiv preprint arXiv:2111.08481 2021. Discovering symbolic models from deep learning with inductive biases. M Cranmer, A Sanchez Gonzalez, P Battaglia, R Xu, K Cranmer, D Spergel, S Ho, Advances in Neural Information Processing Systems. 33Cranmer, M.; Sanchez Gonzalez, A.; Battaglia, P.; Xu, R.; Cranmer, K.; Spergel, D.; Ho, S., Discovering symbolic models from deep learning with inductive biases. Advances in Neural Information Processing Systems 2020, 33, 17429-17442. Understanding the failure modes of outof-distribution generalization. V Nagarajan, A Andreassen, B Neyshabur, arXiv:2010.157752020arXiv preprintNagarajan, V.; Andreassen, A.; Neyshabur, B., Understanding the failure modes of out- of-distribution generalization. arXiv preprint arXiv:2010.15775 2020. Invariant risk minimization. M Arjovsky, L Bottou, I Gulrajani, D Lopez-Paz, arXiv:1907.02893arXiv preprintArjovsky, M.; Bottou, L.; Gulrajani, I.; Lopez-Paz, D., Invariant risk minimization. arXiv preprint arXiv:1907.02893 2019. Toward causal representation learning. B Schölkopf, F Locatello, S Bauer, N R Ke, N Kalchbrenner, A Goyal, Y Bengio, Proceedings of the IEEE 2021. the IEEE 2021109Schölkopf, B.; Locatello, F.; Bauer, S.; Ke, N. R.; Kalchbrenner, N.; Goyal, A.; Bengio, Y., Toward causal representation learning. Proceedings of the IEEE 2021, 109 (5), 612-634. M Ziatdinov, Y Liu, A N Morozovska, E A Eliseev, X Zhang, I Takeuchi, S V Kalinin, arXiv:2112.06649Hypothesis learning in an automated experiment: application to combinatorial materials libraries. arXiv preprintZiatdinov, M.; Liu, Y.; Morozovska, A. N.; Eliseev, E. A.; Zhang, X.; Takeuchi, I.; Kalinin, S. V., Hypothesis learning in an automated experiment: application to combinatorial materials libraries. arXiv preprint arXiv:2112.06649 2021.

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{'abstract': 'The rapid development of machine learning (ML) methods has fundamentally affected numerous applications ranging from computer vision, biology, and medicine to accounting and text analytics. 1-3 Until now, it was the availability of large and often labeled data sets that enabled significant breakthroughs. However, the adoption of these methods in classical physical disciplines has been relatively slow, a tendency that can be traced to the intrinsic differences between correlative approaches of purely data-based ML and the causal hypothesis-driven nature of physical sciences. Furthermore, anomalous behaviors of classical ML necessitate addressing issues such as explainability and fairness of ML. We also note the sequence in which deep learning became mainstream in different scientific disciplines -starting from medicine and biology and then towards theoretical chemistry, and only after that, physics -is rooted in the progressively more complex level of descriptors, constraints, and causal structures available for incorporation in ML architectures. Here we put forth that over the next decade, physics will become a new data, and this will continue the transition from dot-coms and scientific computing concepts of 90ies to big data of 2000-2010 to deep learning of 2010-2020 to physics-enabled scientific ML.', 'arxivid': '2204.05095', 'author': ['Sergei V Kalinin \nDepartment of Materials Science and Engineering\nUniversity of Tennessee Knoxville\n37996KnoxvilleTennesseeUSA\n', 'Maxim Ziatdinov \nComputational Sciences and Engineering Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n\nCenter for Nanophase Materials Sciences\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n', 'Bobby G Sumpter \nCenter for Nanophase Materials Sciences\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA\n', 'Andrew D White \nDepartment of Chemical Engineering\nUniversity of Rochester\n14534RochesterNew YorkUSA\n'], 'authoraffiliation': ['Department of Materials Science and Engineering\nUniversity of Tennessee Knoxville\n37996KnoxvilleTennesseeUSA', 'Computational Sciences and Engineering Division\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA', 'Center for Nanophase Materials Sciences\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA', 'Center for Nanophase Materials Sciences\nOak Ridge National Laboratory\n37831Oak RidgeTennesseeUSA', 'Department of Chemical Engineering\nUniversity of Rochester\n14534RochesterNew YorkUSA'], 'corpusid': 248085105, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7223, 'n_tokens_neox': 6211, 'n_words': 3899, 'pdfsha': 'f69b661e4f562df5d6db19a3dbb75f2d938b37da', 'pdfurls': ['https://arxiv.org/pdf/2204.05095v1.pdf'], 'title': ['Physics is the New Data', 'Physics is the New Data'], 'venue': []}

arxiv

Structure Prediction of Epitaxial Organic Interfaces with Ogre, Demonstrated for TCNQ on TTF Saeed Moayedpour Imaneul Bier Department of Materials Science and Engineering Carnegie Mellon University 15213PittsburghPAUSA Wen Wen Derek Dardzinski Department of Materials Science and Engineering Carnegie Mellon University 15213PittsburghPAUSA Olexandr Isayev Noa Marom Department of Materials Science and Engineering Carnegie Mellon University 15213PittsburghPAUSA Department of Physics Carnegie Mellon University 15213PittsburghPAUSA Department of Chemistry Carnegie Mellon University 15213PittsburghPAUSA Structure Prediction of Epitaxial Organic Interfaces with Ogre, Demonstrated for TCNQ on TTF (Dated: 19 January 2023) Highly ordered epitaxial interfaces between organic semiconductors are considered as a promising avenue for enhancing the performance of organic electronic devices including solar cells, light emitting diodes, and transistors, thanks to their well-controlled, uniform electronic properties and high carrier mobilities. Although the phenomenon of organic epitaxy has been known for decades, computational methods for structure prediction of epitaxial organic interfaces have lagged far behind the existing methods for their inorganic counterparts. We present a method for structure prediction of epitaxial organic interfaces based on lattice matching followed by surface matching, implemented in the open-source Python package, Ogre. The lattice matching step produces domain-matched interfaces, where commensurability is achieved with different integer multiples of the substrate and film unit cells. In the surface matching step, Bayesian optimization (BO) is used to find the interfacial distance and registry between the substrate and film. The BO objective function is based on dispersion corrected deep neural network interatomic potentials, shown to be in excellent agreement with density functional theory (DFT). The application of Ogre is demonstrated for an epitaxial interface of 7,7,8,8-tetracyanoquinodimethane (TCNQ) on tetrathiafulvalene (TTF), whose electronic structure has been probed by ultraviolet photoemission spectroscopy (UPS), but whose structure had been hitherto unknown [Organic Electronics 48, 371 (2017)]. We find that TCNQ(001) on top of TTF(100) is the most stable interface configuration, closely followed by TCNQ(010) on top of TTF(100). The density of states, calculated using DFT, is in excellent agreement with UPS, including the presence of an interface charge transfer state. I. INTRODUCTION Organic-organic heterojunctions, i.e., interfaces between different organic materials, are at the heart of organic electronic devices, including organic photovoltaics (OPV), 1,2 organic light emitting diodes (OLEDs), [3][4][5][6] and organic field effect transistors (OFETs). [7][8][9][10][11] Essential device functionalities take place at interfaces. In organic solar cells, charge separation of bound excitons into free charge carriers occurs at donor-acceptor interfaces, where the holes go into the highest unoccupied molecular orbital (HOMO) of the donor and the electrons go into the lowest unoccupied molecular orbital (LUMO) of the acceptor. Conversely, in OLEDs, electrons and holes recombine at a donor-acceptor interface, leading to photon emission. 6 In more sophisticated multilayer OLED designs, additional functions may be separated into different materials. A multilayer OLED may include a hole-injection layer, a hole-transport layer, an electron-blocking layer, an emission layer, a hole-blocking layer, an electron-transport layer, and an electron-injection layer, forming multiple active interfaces. 3 In OFETs, the gate stack contains, at the minimum, an organic semiconductor interfaced with the gate dielectric and the source and drain electrodes. In some cases, more than one organic semiconductor is included in the gate stack to create an n-channel and a p-channel. Additional organic layers may be included, e.g., to control the morphology of the organic semiconductor(s) and the contact resistance at a) Electronic mail: nmarom@andrew.cmu.edu the electrode interfaces. 8 Hence, the structure and electronic properties of organic interfaces are critical to the performance of organic electronic devices. Often, the films used in organic devices are amorphous and the interfaces between them are disordered. Disorder may lead to non-uniform electronic and optical properties that are averaged over multiple configurations. 12,13 In addition, various defects, as well as grain boundaries in polycrystalline films, 14 may form traps and provide scattering sites for charge carriers. This is detrimental to charge transport and lowers the carrier mobility, which is a key performance parameter for electronic devices. Molecular order produces well-controlled properties and higher mobilities, which may lead to improved device performance. 11,[15][16][17][18][19][20][21][22] In analogy with inorganic materials, high-quality crystalline organic heterojunctions may be grown by molecular beam epitaxy. [23][24][25][26][27][28][29][30][31][32][33] In contrast to inorganic epitaxy, molecular materials are bound by weak dispersion interactions and organic substrates do not have dangling bonds on the surface. Moreover, there is competition between adsorbate-substrate and adsorbate-adsorbate intermolecular interactions. This leads to less strict latticematching requirements between the substrate and film for organic epitaxial growth. 27,[32][33][34] Epitaxial interfaces between various organic semiconductors have been grown, including α-quaterthiophene on rubrene 35 , pentacene on C 60 36 , rubrene on tetracene 25 , C 60 on pentacene [37][38][39][40][41] or rubrene 21,42 , perfluoropentacene on pentacene 43 or diindenoperylene (DIP) 44 , DIP on copper-hexadecafluorophthalocyanine (F 16 CuPc) 45 , bis(trifluoromethyl)dimethylrubrene on rubrene 46 , tetraazanaphthacene on pentacene, 47 48 . Band transport has been observed in organic epitaxial interfaces. 21,43 Moreover, epitaxial interfaces with a well-defined structure and uniform properties are conducive to spectroscopic characterization of the interface band alignment, interface charge transfer (CT) states, and exciton dynamics. 39,41,[46][47][48][49] Computer simulations can help explore the vast configuration space of possible substrate and film combinations to direct fabrication efforts to systems likely to result in robust epitaxial growth of high-quality films. Furthermore, simulations can help the characterization of epitaxial organic interfaces by assigning spectroscopic signatures to putative interface structures. The structure prediction capabilities for organic interfaces lag far behind the existing methods for inorganic interfaces. Several computational tools and codes exist for constructing models of inorganic epitaxial interfaces. [50][51][52][53][54][55][56] Because these tools are designed to work with spherical atoms, not molecules, which are significantly more complex, they do not work out of the box for organic interfaces. Recently, progress has been achieved in the development of algorithms for structure prediction of organic monolayers on inorganic substrates. [57][58][59][60][61] . Organic-organic interfaces have been investigated mainly by molecular dynamics simulations, based on classical force fields, which enable studying the effect of disorder. [62][63][64][65] However, to our knowledge, to date, no computational tools have been developed for structure prediction of epitaxial organic-organic interfaces. Here, we introduce a new version of the open source Python package, Ogre, with new functionality of predicting the structure of epitaxial organic interfaces by lattice and surface matching. We note that Ogre generates ideal interfaces with no disorder and does not take into account growth conditions and kinetics. The first version of Ogre was designed for modelling molecular crystal surfaces, calculating surface energies, and predicting Wulff shapes. 66 In the second version, the capability of performing structure prediction for epitaxial inorganic interfaces was added. 51 Similar to the previous version of Ogre, the workflow for organic interfaces begins by using Zur and McGill's lattice matching algorithm 67 and proceeds to perform surface matching. The lattice matching step identifies potential domain-matched epitaxial interfaces, where commensurability is achieved with different integer numbers of the substrate and film unit cells. The lattice misfit tolerances for domain matching have been updated for the case of weak van der Waals epitaxy. In addition, a streamlined calculation of the surface energies of different substrate facets may be performed to prioritize the most stable orientations for an interface Miller index search. In the surface matching step, Bayesian optimization (BO) is utilized to find the most stable interface configurations by exploring the three-dimensional space of interfacial distance and registry. For surface matching, Ogre constructs commensurate surface unit cells of the substrate and film, ensuring that no molecules are broken when the surface is cleaved. 66 The film is then moved with respect to the substrate in order to determine the optimal configuration. Owing to the large domain size and the number of atoms per molecule, even the smallest models of organic epitaxial interfaces, with only one layer of the substrate and the film, may contain hundreds to thousands of atoms. Therefore, it is imperative to use a computationally efficient method for evaluating the relative stability of interface configurations. The geometric score function we developed for inorganic interfaces 51 is not appropriate for organic interfaces because it is designed to work with spherical atoms, whose radii are determined by Hirshfeld partitioning 68 of the charge density of the bulk materials calculated with density functional theory (DFT). Instead, we use the Accurate Neural Network Engine for Molecular Energies (ANI) generalpurpose machine learned interatomic potentials. 69 The ANI deep neural network potentials have been trained on a large and diverse database of first principles calculations for small molecule conformations 70 using active learning 71 . They have been shown to be transferable across chemical space, achieving a DFT level of accuracy on a large set of organic molecules while being six orders of magnitude faster. Because the ANI potentials were trained on isolated molecules they do not inherently contain a description of intermolecular dispersion (van der Waals) interactions. Therefore, the Grimme D3 dispersion correction is added to the ANI energies. [72][73][74] We show that ANI+D3 is in close agreement with DFT for the interfacial distance and registry. Preliminary ranking of the interface energies of the surface-matched configurations of all domainmatched interfaces is performed with ANI+D3. The ANI+D3 ranking is in reasonably good agreement with the DFT ranking and the most stable structures are identified. In the final stage, DFT is used to calculate the electronic properties of a small subset of the most promising candidate structures. The application of Ogre is demonstrated for the epitaxial interface of TCNQ grown on top of TTF. 48 TTF and TCNQ are a quintessential donor/acceptor system. Both TTF and TCNQ are wide band gap semiconductors. On their own, they are insulators but when they are paired, electrons can transfer from the HOMO of TTF to the LUMO of TCNQ. 48,[75][76][77][78] Hence, TTF-TCNQ co-crystals and interfaces may exhibit metallic behavior. The conducting layer, which forms at the interface, has been reported to exhibit a high carrier density, exceeding 10 14 cm −2 . 48,[79][80][81] The interface formed by TTF and TCNQ single crystals has been found to exhibit 2D metallic conductivity, which may be useful for organic transistors and sensing devices. [82][83][84][85][86][87][88][89] Kattel et al. have grown an epitaxial interface of TCNQ on top of TTF. 48 The electronic structure of the interface was probed by ultraviolet photoemission spectroscopy (UPS) as a function of the thickness of the TCNQ film. A charge transfer (CT) state was observed at the interface, whose signature decayed when the TCNQ film thickness exceeded 2 nm. However, the interface was not structurally characterized, leaving the orientation of the TTF and TCNQ and their relative positions at the interface hitherto unknown. Using Ogre, we find that TCNQ(001) on top of TTF(100) is the most stable interface configuration, closely followed by TCNQ(010) on top of TTF (100). The resulting density of states (DOS) of the TTF substrate, a TCNQ film, and an interface with one layer of each material are in excellent agreement with the UPS experiment of Kattel et al.. A CT state is found at the interface, consistent with their observations. II. COMPUTATIONAL DETAILS A. DFT The all-electron electronic structure code FHI-aims 90 was used to perform DFT calculations. Computationally efficient light numerical settings and tier 1 basis sets were used throughout, owing to the large system sizes with several hundred atoms. Geometry optimizations, surface energy evaluations, reference DFT calculations for surface matching, and interface energy evaluations were conducted using the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE) 91,92 with the Tkatchenko-Scheffler (TS) pairwise dispersion method. 93 Structural relaxations were performed for the bulk crystal structures of TTF and TCNQ until the residual force per atom was less than 10 −2 ev/Å. Relaxation was performed with fixed unit cell angles of β = 101.43 • for TTF 94 and β = 98.351 • for TCNQ. 95 The number of k-points in each direction was determined by calculating the closest integer greater than 24 divided by the norm of the respective lattice parameter. The relaxed lattice parameters of TTF were a = 7.44 Å, b = 4.28 Å, and c = 13.68 Å, compared with the experimental values of a = 7.35 Å, b = 4.01 Å, and c= 13.901 Å. 94 The relaxed lattice parameters of TCNQ were a = 8.98 Å, b = 7.31 Å, and c = 16.40 Å, compared with the experimental values of a = 8.8834 Å, b = 6.9539 Å, and c = 16.3948 Å. 95 The interface models were constructed based on the relaxed bulk geometry with one layer of the TTF substrate, one layer of the TCNQ film, and a vacuum region of 40 Å. The electronic structure calculations for the the final lowest energy interfaces were performed using the range-separated hybrid functional of Heyd-Scuseria-Ernzerhof (HSE). 96,97 We note that the size of the systems calculated here is very large. For example, the TCNQ(001)/TTF(100) interface model with a single layer of each material contained 1298 atoms. B. ANI neural network potentials The ANAKIN-ME (ANI) 69 transferable neural network molecular potentials are used for calculations of organic interfaces. The representation used by the ANI-2x model 98 is the Smith symmetry functions (SSFs), − → G i Z , a modified version of the Behler and Parrinello symmetry functions. A separate neural network model is used for each element. The SSFs encode N inter-atomic distances between the i th atom and its neighbors within a cutoff radius R c into an invariant fixed-length atomic environment vector (AEV), − → G i Z = {G 1 , G 2 , G 3 , ..., G M }. The elements G M encodes specific regions of an individual atom's radial and angular chemical environment. Each − → G i Z for the i th atom with atomic number Z is then used as input into a single neural network (NNP). With an invariant AEV, − → G i Z , the total energy of a molecule m is expressed as: E total (m) = all atoms ∑ i NNP Z i ( − → G i Z )(1) The ANI-2x potentials have been trained on a large and diverse dataset of first-principles calculations for small organic molecules selected by an active learning procedure. 71 Test cases have demonstrated the ANI potentials to be chemically accurate compared to reference DFT calculations. The Grimme D3 dispersion correction 99 is added post hoc to ANI-2x calculations to account for intermolecular van der Waals interactions. All functionality is implemented in the open source TorchANI 100 package (https://github.com/aiqm/torchani), which is written in python using the PyTorch library. TorchANI is designed to take advantage of the modularity and simplicity of PyTorch to provide a framework for fast iterative improvements in architecture and training schedules for ML models, while also allowing for experimentation and easy integration into existing software, such as Ogre. III. RESULTS AND DISCUSSION A. Workflow Overview Ogre is written in Python 3. It employs several modules from the Python Materials Genomics (pymatgen) 101 and the Atomic Simulation Environment (ASE) 102 libraries. The package can be downloaded from www.noamarom.com under the terms of a BSD-3 license. The inputs of Ogre are the bulk structures of the substrate and film materials, as well as a configuration file with user-specified settings for directing the workflow of Ogre. Standard input structure file formats such as the POSCAR format of the Vienna ab initio Simulations Package's (VASP) 103? -106 , crystallographic information files (CIF), and the geometry.in format of the FHI-aims code 90 are supported by Ogre. Figure 4 shows an overview of the workflow of interface structure prediction with Ogre. Ogre's workflow includes four main steps: surface selection (Section III B), lattice matching (Section III C), surface matching (Section III D), and ranking (Section III E). Finally, the electronic properties may be calculated for the interface(s) predicted to be the most stable (Section III F) As a preliminary step, Ogre may be used for substrate and surface selection. For organic epitaxy, substrates that form platelets are desirable. 107 Ogre's morphology prediction capability 66 could be used for substrate screening. If the substrate has already been chosen, as in the case of the TCNQ/TTF interface studied here, but the orientation is unknown, Ogre may be used to calculate the surface energy of all the symmetrically unique facets up to a selected Miller index. 66 The lowest energy surfaces form the largest facets and are best suited to serve as the substrate for epitaxial growth. Lattice matching may then be performed for the chosen facets. The lattice matching step for organic interfaces is similar to inorganic interfaces. 51 orientations. The maximum interface area, the misfit tolerance, and the Miller indices for the substrate and film are the input parameters for lattice matching. The lattice matching step generates a list of structures that are sorted by super-cell area misfit values. As explained above, owing to the weak nature of dispersion interactions in organic materials, robust epitaxial growth can occur at higher interface misfit values compared to inorganic interfaces. 26,109 Therefore, a looser default tolerance of 5% has been adopted for the supercell area misfit, lattice vector length misfit, and angle misfit (the user may choose to alter the misfit tolerance). All the domain-matched interfaces within the misfit tolerance proceed to the surface matching step. In the surface matching step, a search is performed to find the optimal distance in the z direction and registry in the xy plane between the substrate and film. Similar to the case of inorganic interfaces, 51 Bayesian optimization (BO) is used to find the most stable interface configuration by shifting the film in the x,y, and z directions on top of the substrate. For inorganic interfaces the BO objective function is a geometric score function, based on the overlap and empty space between atomic spheres. Because this score function is not suitable for molecules, organic interfaces require a different approach for fast evaluation of the relative stability of different interface configurations. Here, we use the ANI deep neural network interatomic potentials 69 coupled with the Grimme D3 disper-sion correction 99 as the BO objective function. To this end, Ogre uses the TorchANI 100 package and the DFTD3 calculator class of the ASE package. Subsequently, the interface energy calculated with ANI+D3 is used for preliminary ranking of the optimized structures. The user can select a percentage of the generated interfaces to output. Finally, DFT may be used to accurately rank a small number of the best candidate interfaces and calculate their electronic properties. Ogre automates the construction of interface models with user-defined number of layers and vacuum space. If the substrate orientation is unknown, as in the case of the TTF substrate used by Kattel et al., 48 surface energy calculations may be performed by Ogre. We assume that the most stable facets are the most likely substrate orientations because they are the largest facets in the Wulff shape. 66 The workflow of the surface selection step is shown in Figure 2. In order to streamline the calculation of surface energies, Ogre is accompanied by the OgreSWAMP utilities 66 . OgreSWAMP determines the Miller indices of all unique surfaces up to a userdefined maximum Miller index based on the space group symmetries of a given a molecular crystal structure. Automated DFT calculations are performed to obtain the total energy of surface slab models with an increasing number of layers. The surface energy is then calculated using the linear method 55,110 : E Slab = NE Bulk + 2Aγ(2) where E slab is the total energy of the surface slab, N, is the number of layers in the slab, and A is the slab surface area. The energy of the bulk crystal, E bulk , and the surface energy, γ, are extracted from the slope and intercept, respectively. Figure 3 shows the convergence of the surface energy with the number of layers for the (111), (001), and (100) facets of TTF. All other unique surfaces are shown in the SI. The stability of molecular crystal surfaces may be attributed to the chemical groups exposed on the surface. 66 The (111) facet, shown in Figure 3a, cleaves through the strong interactions in stacking direction of the TTF molecules 111 and exposes the electron-rich π-system and the sulfur atoms of TTF on the surface. Therefore, this facet has a relatively high surface energy. In contrast, the (001) and (100) planes, shown in Figure 3b,c, cleave through a direction nearly perpendicular to the stacking direction of the TTF molecules, exposing hydrogen atoms on the surface. This produces more stable surfaces with lower surface energies. Table I Figure 4 show the workflow of Ogre's lattice matching step. Lattice matching is controlled by the following input parameters: the substrate and film Miller indices, the maximum interface super-cell area, and the misfit tolerance for the (surface) lattice parameters, surface area, and angles. For organic interfaces, all misfit tolerances are set at 5% by default, and the maximum super-cell area is set at 1000 Å 2 . ASE is used to obtain the basis vectors of the substrate and film surfaces by cleaving the bulk crystal structures along the specified Miller planes. In order to compare substrate and film two-dimensional lattices using a unique representation, the surface basis vectors are reduced to a pair of primitive basis vectors using Pymatgen. Next, all transformation matrices that would produce lattice matched super-cells within the user-defined tolerance of lattice vector length and angle misfit are constructed using the Pymatgen substrate analyzer mod-ule. Finally, all unique commensurate interfaces of a particular system are identified using a reduction scheme and the interface structure is generated. The user may choose to apply strain to either the substrate or the film, however in order to simulate an epitaxial growth experiment, the substrate's lattice parameter is fixed by default and the film layer is strained to conform to the substrate. In order to build a lattice matched interface, Ogre generates the matching substrate and film super-cells and aligns the atomic coordinates. The required parameters for creating the interface model, including the interfacial distance, the initial shift in the xy plane, the number of layers, and the length of the vacuum region can be determined by the user. By default, the interfacial distance and vacuum region and are set to 2 Å and 40 Å, respectively. The initial interface structure has no xy shifts applied by default. The user may determine the substrate and film thickness by specifying the number of layers or a range of values to calculate. Ogre can detect all possible surface terminations for each Miller plane and automatically generate the corresponding interfaces. The models constructed by Ogre are subsequently used for the surface matching step. The user may specify the substrate and film Miller indices. If multiple substrate orientations are feasible, and/or the film orientation is unknown Ogre may conduct a Miller index scan, where for each combination of substrate and film Miller indices lattice matching is performed. The input of the Miller index search module is the maximal single index. Ogre enumerates all possible symmetrically unique Miller indices as described in Ref. 66 . For example, the results of a Miller index scan for a TCNQ/TTF interface with a maximal Miller index of 1, a misfit tolerance of 5%, and a maximum area of 1000 Å 2 are shown in Figure 5. Panel (a) displays a histogram of the number of domain-matched interfaces generated for each combination of Miller indices. In total, 965 candidate interfaces are generated with these settings. Additional candidate interfaces may be generated by selecting higher misfit tolerances or a larger maximum area, however, they are likely to be less stable. Panel (b) shows the minimum misfit percentage obtained for each interface orientation. Out of 81 possible Miller index combinations, 57 have a misfit under 5%. To narrow down the number of candidates, only the three most stable facets of the TTF substrate, (100), (001), and (011), and only TCNQ facets that lead to an interface with a misfit lower than 2.5 % are considered further. These leave 9 candidate interfaces that proceed to surface matching. D. Surface Matching Domain-matched interfaces proceed from the lattice matching step to the surface matching step, whose workflow is shown in Figure 6. The optimal position of the film above the substrate is determined using Bayesian optimization (BO) to efficiently search the 3D parameter space of the film position on top of the substrate in the x, y, and z directions. To evaluate the BO objective function, the user may choose DFT or ANI+D3. Ogre provides a grid search option that can generate potential energy surfaces or binding energy curves. Surface matching produces a list of structures ranked by the ANI+D3 interface energy. The final optimized interface structures are exported in the appropriate geometry file format (e.g., geometry.in for FHI-aims). Performance of ANI+D3 To validate the performance of ANI+D3, we compare its results to DFT, using PBE+TS, for a representative interface structure of TCNQ(110)/TTF(011), illustrated in Figure 7. This structure was selected because it has the lowest area mismatch of 0.23% , as shown in Figure 5. Figure 7 shows a comparison of the potential energy curves as a function of the interfacial distance, obtained with ANI+D3 and PBE+TS. Both curves are referenced to their respective minima. Excellent agreement is demonstrated between ANI+D3 and PBE+TS for the position of the minimum and the shape of the curve. This is possibly because the interfacial distance is dominated by dispersion interactions, whose description by the TS and D3 pairwise methods is similar. Figure 8 shows contour plots of the energy as a function of the position of the TCNQ film on top of the TTF substrate in the xy plane at a fixed interface distance of 1.4 Å, obtained with ANI+D3 compared with PBE+TS. ANI+D3 agrees well with the positions of the extrema and the features of the DFT potential energy surface. The in-plane registry depends mainly on the local electrostatic and exchange-correlation contributions because the dispersion contribution does not vary significantly at a fixed interfacial distance (a comparison of ANI to PBE without dispersion corrections is provided in the Fig. S3. Hence, the good agreement between ANI+D3 and PBE+TS may be attributed to the fact that ANI is trained to reproduce DFT data for local, intramolecular bonds. Bayesian Optimization Bayesian optimization is a machine learning algorithm designed to find the extremum of a "black box" objective function with a minimal number of function queries. 112 Based on the sampled points, a Gaussian process is typically used to construct a statistical model of the objective function, known as the prior. 113 The following point to be sampled is then de- The BO objective function for surface matching is defined as the negative of the total energy of the interface, calculated using either ANI+D3 or DFT, such that the energy is minimized by maximizing the objective function. Ogre uses the upper confidence bound acquisition function [113][114][115] : r n+1 = argmax(µ n ( r) + κσ n ( r))(3) where µ and σ are the mean and standard deviation at each sampled point. The trade-off between exploitation and exploration is balanced by the hyperparameter κ. The higher the value of κ, the more exploration of regions with a high uncertainty is performed. A lower value of κ accelerates convergence by favoring exploitation of promising regions, however if κ is too low the BO might converge prematurely to a local extremum. Figure 9 illustrates the effect of the choice of κ on the BO behavior for the two-dimensional surface of the TCNQ (110) surface. The Gaussian process predicted mean bears little resemblance to the true ANI+D3 PES, shown in Figure 8a, and the local minimum at (1.25,3.0) is not found. With κ=5 the BO algorithm performs more exploration of the PES, while still converging relatively quickly to the global minimum. The Gaussian process predicted mean appears more similar to the true PES and the region of the local minimum is also sampled. With κ=10 significant exploration of the PES is performed, including the regions of the global and local maxima. As a result, the Gaussian process predicted mean appears very similar to the true PES. However, because computer time is wasted on exploring irrelevant regions of the PES, the convergence to the global minimum is slower. For the κ values of 1 and 5 the BO started to exploit the global minimum region after 11 and 18 iterations, respectively. In comparison, for the κ value of 10 it took over 25 iterations for the exploitation of the global minimum region to begin, although most of the sampled points were from the lower energy regions. Thus, κ=5 provides an optimal balance between exploration and exploitation. The bayesian-optimization Python package 116 is used by Ogre to perform BO. The bounds for shifts in the x, y,and z directions define the parameter space to be searched. These bounds have default values of (0, a), (0, b), and (d − 1 Å, d + 1 Å), respectively, where a and b are the interface lattice parameters in the xy plane and d is the initial interface distance. The default values for the number of iterations, N, and κ are 100 and 5, respectively. The surface matching settings file enables the user to input different parameters than the default values. The most stable interface structure and any structures whose total energy is within a user-defined tolerance of the minimum are output by Ogre once the maximum number of BO iterations has been reached. BO is significantly more efficient than a grid search because it samples fewer points while maximizing the amount of information learned about the objective function. For example, for the TCNQ(110)/TTF(011) interface with a lattice parameter of 19.4 Å, a grid search with a step size of 0.2 Å and a 1 Å range for the z axis would require evaluating more than 47000 points. In comparison, the BO algorithm converges to the optimal interface configuration within 300 iterations. The computational cost of performing BO amounts to the number of sampled points multiplied by the cost of the objective function evaluation, which is the cost of an energy calculation with either ANI+D3 or DFT. ANI+D3 is faster than DFT by orders of magnitude. For example, single point energy evaluations for the TCNQ(110)/TTF(011) interface took about 7 s using ANI+D3, compared to about 4000 s using DFT (PBE+TS). E. Ranking Every interface structure passed from the lattice matching step is surface matched, which could result in a significant number of candidate structures. Fast preliminary ranking is performed with ANI+D3 in order to down-select a smaller group of the most promising candidate structures for the final evaluation with DFT. The interface energy, σ , is defined as the energy required to eliminate two surfaces and form an interface: 50,51,117,118 σ = γ sub + γ f ilm −W ad(4) where, W ad is the interface adhesive energy and γ sub/ f ilm are the surface energies of the substrate and film, calculated using Eq. 2, 66 The adhesive energy of an interface is defined as: [119][120][121][122] W ad = 1 A (E sub + E f ilm − E int )(5) where E sub , E f ilm , and E int are total energies of the substrate surface slab, the film surface slab, and the interface, respectively. E sub , E f ilm , and E int may be evaluated using either ANI+D3 or DFT. For inorganic interfaces, we have proposed a linear method for converging the interface energy as a function of the number of layers. 50,51 For organic interfaces it is not feasible to calculate the interface energy for models with more than one layer of the substrate and one layer of the film, owing to the large system size. With only one layer of substrate and film, the smallest interface studied here, TCNQ(110)/TTF(011), contains 777 atoms and the largest interface studied here, TCNQ(111)/TTF(001), contains 1655 atoms. In addition, the thickness of these organic interface models ranges from 93.58 Å for TCNQ(110)/TTF(011) to 109.15 Å for TCNQ(100)/TTF(001), which is roughly equivalent to an inorganic interface with 40 atomic layers. Owing to the weak nature of the dispersion interactions at organic interfaces and their large size, it is therefore a reasonable approximation to calculate the interface energy for models with only one layer of the substrate and one layer of the film. In figure 10 the interfaces energies obtained with ANI+D3 are compared to DFT interface energies obtained with PBE+TS for the TCNQ/TTF interfaces with the lowest substrate surface energies and interface area misfit. Because the ANI potentials have been trained on isolated molecules and have never "seen" an interface, we do not expect the interface energy values to be in close agreement with DFT values. Nevertheless, the ranking obtained with ANI+D3 is in reasonable agreement with the DFT ranking in the sense that the group of most stable structures is correctly identified. Hence we have confidence in the preliminary raking with ANI+D3 as a method of narrowing down the number of candidate structures to be calculated with DFT. F. Electronic Structure We now proceed to study the electronic structure of the two most stable interface configurations of TCNQ(001)/TTF (100) and TCNQ(010)/TTF (100). Previously, Beltrán et al. conducted DFT simulations of the TCNQ/TTF interface, whose structure was derived from a TTF-TCNQ co-crystal. 77 However, the structure of the co-crystal is significantly different than the bulk structure of either material. Therefore, such a model is not appropriate for the epitaxial growth of TCNQ on top of TTF by Kattel et al. 48 Beltrán et al. used the local density approximation (LDA), with an empirical correction of the interface energy level alignment to compensate for the severe underestimation of band gaps by the LDA. Here, we use the HSE hybrid functional, which contains 25% of exact (Fock) exchange. Atalla et al. have shown for an isolated dimer of TTF and TCNQ molecules that a larger fraction of exact exchange is required to reproduce the correct energy level alignment. 78,123 However, in extended systems the energy level alignment changes owing to band dispersion and polarization-induced gap narrowing. 124 Hence, the HSE functional provides a correct description of the energy level alignment and the charge transfer at the interface of TTF and TCNQ, as we demonstrate below. Kattle et al. conducted UPS experiments for TCNQ films ranging in thickness from 0.3 nm to 10 nm, deposited on a single crystal of TTF. 48 In our model of the of TCNQ(001)/TTF(100) interface with a single layer of each material the thickness of the TCNQ film is 1.653 nm. Therefore, we compare our results to the UPS result of the 1 nm thick TCNQ film. The UPS results of bulk TTF and TCNQ are compared to calculations conducted for the respective crystals. Figure 11a shows the comparison of the computed density of states (DOS) with UPS experiments. The absolute energies of the Kohn-Sham orbitals from ground state DFT calculations using approximate exchange-correlation functionals do not strictly correspond to ionization energies. 125 Therefore, the computed DOS are shifted to align the first peak with the first UPS peak. In addition Gaussian broadening of 0.1 eV is applied to the DOS to simulate the resolution of the experiment. The comparison between theory and experiment is focused only on peak positions. We do not consider the secondary electron background, which leads to the increase of the spectral intensity towards lower binding energies, 48 vibrational effects, or intensity variations due to the scattering cross-section and penetration depth. 126 The positions of the peaks in the DOS of the TTF crystal, the TCNQ crystal, and the TCNQ/TTF interface are in a good agreement with the position of the peaks in the UPS spectra. The DOS of the TTF(100)/TCNQ(001) and the TCNQ(010)/TTF(100) interface structures are similar with the TTF(100)/TCNQ(001) DOS being in slightly better agreement with experiment. An experiment with a finer resolution may be required to unambiguously determine the interface structure. In the spectrum of the interface with the 1 nm thick TCNQ film, the peak observed around -1.5 eV was attributed by Kattle et al. to a CT state, which is likely responsible for the high conductivity of the interface. Indeed, in our DFT simulations, the interface exhibits metallic behavior. Owing to the number of molecules included in the interface models, several orbitals contribute to each peak in the computed DOS. Within a 0.01 eV energy range of the peaks centers, 7 orbitals contribute to the first peak and 9 orbitals contribute to the second peak. The orbitals with energies closest to the peak centeres are depicted in 11 The first peak, labeled CT, originates from a state that is delocalized over the interface and is distinct from the HOMO of TTF, as shown in Figure 11. We attribute this interface state to ground state CT from TTF to TCNQ, rather than a CT excitation, in agreement with experiment. Kattle et al. estimated the spatial extent of the CT state to be 1-2 nm or less, which is in agreement with our simulation. The second peak in the DOS of the interface, labeled CT2, arises from a delocalized state across the interface, comprising mixed contributions from the TTF HOMO-1 and the TCNQ HOMO, which are close in energy. The charge transfer at the TCNQ(001)/TTF(100) interface is given by the difference between the DFT charge density of the interface and the charge densities of isolated substrate and bulk slabs with the same geometry 127 : C net (z) = C inter f ace −C substrate −C f ilm(6) where the charge is averaged over the xy plane at each position along the z-axis. Figure 11f-g shows the charge transfer as a function of position across the interface. The net charge is negative at the interfacial layer of TTF (100) and positive at the TCNQ(001) layer, which confirms that the charge is transferred from the TTF substrate to the TCNQ film. IV. CONCLUSION We have presented a method for structure prediction of epitaxial organic interfaces by lattice and surface matching, implemented in the Ogre open-source Python package. The lattice matching step identifies all potential domain-matched interface super-cells within user-defined tolerances for interface area and lattice mismatch. The most stable facets of the substrate may be pre-selected by calculating the surface energies. If several substrate and/or film orientations are possible, a Miller index scan may be performed to find the combinations that result in the best candidate interface structures. In the surface matching step, Bayesian optimization (BO) is utilized to determine the optimal configuration of each domain-matched interface in terms of the interfacial distance in the z direction and the registry in the xy plane. The BO objective function is based on the energy of the interface evaluated using the ANI deep neural network inter-atomic potentials with the Grimme D3 dispersion correction. ANI+D3 successfully reproduces the DFT results for the energy as a function of the interfacial distance and the features of the DFT potential energy surface in the xy plane at a fraction of the computational cost. In addition, the preliminary ranking of interface structures with ANI+D3 is in reasonable agreement with the DFT ranking and the most stable interface structures are correctly identified. DFT simulations can be performed for a small number of the most promising candidate interface structures. Ogre streamlines the construction of surface and interface slab models. The application of Ogre has been demonstrated for an epitaxial interface of TCNQ on TTF, which is of interest for organic electronics. The electronic structure of the TCNQ/TTF interface had been probed by UPS, however its structure had not been experimentally characterized. The three facets of the TTF substrate with the lowest surface energies were found to be (001), (100), and (011). The stability of these surfaces may be attributed to the H-terminated edges of the TTF molecules being exposed on the surface, rather than the π system in the plane of the rings. A Miller index scan was performed to identify the TCNQ film orientations that lead to interfaces with the lowest lattice mismatch. Nine interfaces with different Miller index combinations were selected for surface matching. In the final ranking, TCNQ(001)/TTF(100) was found to be the most stable interface configuration followed by TCNQ(010)/TTF (100). Electronic structure calculations were performed for both of these interfaces and compared with UPS experiments. Excellent agreement was obtained between the computed DOS and UPS measurements in terms of the relative peak positions, although experiments with higher resolution would be required to unam- biguously distinguish between the TCNQ(001)/TTF(100) and TCNQ(010)/TTF(100) interface configurations. For the most stable TCNQ(001)/TTF(100) interface configuration, we have also investigated the charge transfer. The first peak of the spectrum of the interface was found to be associated with an orbital delocalized over the interface, rather than states belonging to either the TTF or TCNQ. Ground state charge transfer from TTF to TCNQ was observed in agreement with experiment. In conclusion, we have shown that Ogre can contribute to advancements in the understanding of the structure and properties of epitaxial organic interfaces. Ogre can help interpret experiments by predicting the most likely interface configurations and correlating the structures with observed spectral features. Furthermore, Ogre may guide experimental growth efforts of epitaxial organic interfaces by identifying material combinations that are likely to form high-quality interfaces. In the future, Ogre may be integrated into automated materials discovery workflows to enable the computational design and discovery of epitaxial organic interfaces for high-performance organic electronic devices. Ogre uses the algorithm of Zur and McGill 108 to identify all domain-matched supercells of the substrate and film within the user-defined tolerances for interface area and lattice misfit. A Miller index scan can be performed if the substrate and/or film have multiple possible Import configuration file and bulk structures Export optimized interface models to geometry files Perform lattice matching Output domain-matched interfaces Perform surface matching Output optimized structures sorted by interface energy Call OgreSwamp to optimize number of interface layers Perform surface selection Output Miller indices with the lowest surface energies FIG. 1. workflow of interface structure prediction with Ogre. The blue boxes represent the input and output of the code. Code modules are represented by green boxes. The orange boxes represent module outputs that serve as inputs of the subsequent module. to generate all unique surface slab structures Perform automated energy calculations using OgreSWAMP Extract surface energies for each facet using the linear method Output facets with the lowest surface energies for lattice matching FIG. 2. The workflow of surface selection module in Ogre. The blue boxes represent the input and output of the surface selection step. Code modules are represented by green boxes. summarizes the surfaces energies obtained for all nine unique facets of TTF with a maximal Miller index of 1. The three facets with the lowest surface energies, (001), (100), and (011), were selected for the surface matching step. FIG. 3. Representative surface energy convergence plots obtained using PBE+TS for the (a) (111), (b) (100), and (c) (001) surfaces of TTF. The corresponding structures are also illustrated. C, S, and H atoms are colored in brown, yellow, and light pink, respectively. FIG. 5 . 5Results of a Miller index scan for the TCNQ/TTF interface with a maximal Miller index of 1, maximum interface area of 1000 Å 2 and area misfit tolerance of 5%. (a) Histogram of the number of domain-matched interfaces generated for each combination of Miller indices. (b) Heat map plot of the lowest area misfit obtained for each combination of Miller indices. Miller index combinations for which no structures with an area misfit below 5% were found are represented by white cells. Import surface matching settings and interfacesSearch 3D shift space using Bayesian optimization Calculate interface energies for optimized interfaces Export interfaces sorted by interface energies Call ANI + D3 FIG. 6. The workflow of surface matching in Ogre. The blue boxes represent the inputs and outputs of the surface matching step. The yellow box represents the external codes that are called by Ogre for efficient calculation of total energies as the BO objective function. The green boxes represent different code modules. FIG. 7 . 7Energy as a function of the interfacial distance, obtained with ANI+D3 (red) and PBE+TS (black) for the TCNQ(110)/TTF(011) interface. Both curves are referenced to their respective minima. An illustration of the interface is also shown. C, S, N, and H atoms are colored in brown, yellow, blue, and light pink, respectively. termined by an acquisition function. The prior is subsequently updated based on the newly learned information to create a new surrogate model, known as the posterior. The posterior serves as the new prior for the subsequent iteration and so on, until convergence is reached or a predetermined number of steps have been performed. /TTF(011) interface registry in the xy plane at a fixed interfacial distance of 1.4 Å. With κ=1 the BO algorithm converges to the global minimum very fast. Nearly all the points sampled are in the vicinity of the minimum and there is very little information on the remainder of the potential energy FIG. 8. Energy as a function of the registry in the xy plane obtained with (a) ANI+D3 and (b) PBE+TS for the TCNQ(110)/TTF(011) interface at a fixed interfacial distance of 1.4 Å. The ANI+D3 and PBE+TS energies are referenced to the respective minima. The illustration on the bottom shows the top view of the initial interface structure without any shifts applied in the xy plane. C, S, N, and H atoms are colored in brown, yellow, blue, and light pink, respectively. The TTF substrate and TCNQ film unit cells are shown in blue and red, respectively. FIG. 9 . 9BO predicted mean energy as a function of the registry in the xy plane obtained with ANI+D3 using κ values of (a) 1 (b) 5 (c) 10. all potential energy surfaces are referenced to their respective minima. FIG. 10 . 10ANI+D3 interface energies compared with DFT interface energies calculated using PBE+TS for the nine most stable interface structures of TCNQ/TTF. FIG. 11 . 11Electronic structure of the TCNQ/TTF interface: (a) Computed density of states of bulk TTF (red dashed line), bulk TCNQ (black dashed line), the TCNQ(001)/TTF(100) interface (green dashed line), and the TCNQ(010)/TTF(100) interface (blue dashed line) compared to UPS spectra of the TTF substrate (red solid line), a TCNQ thick film (gray solid line), and a 1 nm film of TCNQ on top of TTF (green solid line), reproduced from Ref. 48 . Visualizations of (b) the HOMO and (c) HOMO-1 of TTF, (d) the HOMO of TCNQ, and orbitals corresponding to (e) the first and (f) the second DOS peaks of the TCNQ(001)/TTF(100) interface, attributed to charge transfer (CT) states. Charge transfer at the TCNQ(001)/TTF(100) interface: (g) Net charge transferred as a function of the distance from the interface for TTF(100)/TCNQ(001) interface, where the interface center is defined as z = 0. (h) Visualization of the charge density difference between the interface and the isolated substrate and film slabs as a function of position across the interface with an isosurface value of 0.004 e/Å 3 . C, S, N, and H atoms are colored in brown, yellow, blue, and light pink. FIG. 4. The workflow of lattice matching module in Ogre. Different code modules are represented by green boxes. The red diamond indicates a decision whether a Miller index scan should be performed.Substrate (TTF) Miller index Surface energy (mJ/m 2 ) 001 137.901 100 144.843 011 148.198 101 150.802 111 153.019 010 153.754 110 164.103 111 164.117 101 182.249 TABLE I. Surface energy values for all symmetrically unique surface orientations of TTF obtained with OgreSWAMP module C. Lattice Matching Read lattice matching settings Scan Miller indices Known surface orientations? 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Density functional theory study of Al/NbB2 heterogeneous nucleation interface. Z Zhuo, H Mao, H Xu, Y Fu, Applied Surface Science. 456Z. Zhuo, H. Mao, H. Xu, and Y. Fu, "Density functional theory study of Al/NbB2 heterogeneous nucleation interface," Applied Surface Science 456, 37-42 (2018). First-principles calculations on electronic structure and interfacial stability of Mg/NbB2 heterogeneous nucleation interface. J Wang, Y Li, R Xu, Surface Science. 691121487J. Wang, Y. Li, and R. Xu, "First-principles calculations on electronic structure and interfacial stability of Mg/NbB2 heterogeneous nucleation interface," Surface Science 691, 121487 (2020). Understanding the charge transfer at the interface of electron donors and acceptors: TTF-TCNQ as an example. C Park, V Atalla, S Smith, M Yoon, 10.1021/acsami.7b04148ACS Applied Materials & Interfaces. 9C. Park, V. Atalla, S. Smith, and M. Yoon, "Understanding the charge transfer at the interface of electron donors and acceptors: TTF-TCNQ as an example," ACS Applied Materials & Interfaces 9, 27266-27272 (2017). Phenylated acene derivatives as candidates for intermolecular singlet fission. X Wang, X Liu, R Tom, C Cook, B Schatschneider, N Marom, 10.1021/acs.jpcc.8b12549The Journal of Physical Chemistry C. 123X. Wang, X. Liu, R. Tom, C. Cook, B. Schatschneider, and N. Marom, "Phenylated acene derivatives as candidates for intermolecular singlet fis- sion," The Journal of Physical Chemistry C 123, 5890-5899 (2019). Gas-phase valence-electron photoemission spectroscopy using density functional theory. L Kronik, S Kümmel, 10.1007/128_2013_522First Principles Approaches to Spectroscopic Properties of Complex Materials. C. Di Valentin, S. Botti, and M. CococcioniBerlin Heidelberg; Berlin, HeidelbergSpringerL. Kronik and S. Kümmel, "Gas-phase valence-electron photoemission spectroscopy using density functional theory," in First Principles Ap- proaches to Spectroscopic Properties of Complex Materials, edited by C. Di Valentin, S. Botti, and M. Cococcioni (Springer Berlin Heidelberg, Berlin, Heidelberg, 2014) pp. 137-191. Electron spectroscopy of functional organic thin films: Deep insights into valence electronic structure in relation to charge transport property. N Ueno, S Kera, 10.1016/j.progsurf.2008.10.002Progress in Surface Science. 83N. Ueno and S. Kera, "Electron spectroscopy of functional organic thin films: Deep insights into valence electronic structure in relation to charge transport property," Progress in Surface Science 83, 490-557 (2008). Topological properties of SnSe/EuS and SnTe/CaTe interfaces. S Yang, C Wu, N Marom, 10.1103/PHYSREVMATERIALS.4.034203/FIGURES/6/MEDIUMPhysical Review Materials. 434203S. Yang, C. Wu, and N. 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{'abstract': 'Highly ordered epitaxial interfaces between organic semiconductors are considered as a promising avenue for enhancing the performance of organic electronic devices including solar cells, light emitting diodes, and transistors, thanks to their well-controlled, uniform electronic properties and high carrier mobilities. Although the phenomenon of organic epitaxy has been known for decades, computational methods for structure prediction of epitaxial organic interfaces have lagged far behind the existing methods for their inorganic counterparts. We present a method for structure prediction of epitaxial organic interfaces based on lattice matching followed by surface matching, implemented in the open-source Python package, Ogre. The lattice matching step produces domain-matched interfaces, where commensurability is achieved with different integer multiples of the substrate and film unit cells. In the surface matching step, Bayesian optimization (BO) is used to find the interfacial distance and registry between the substrate and film. The BO objective function is based on dispersion corrected deep neural network interatomic potentials, shown to be in excellent agreement with density functional theory (DFT). The application of Ogre is demonstrated for an epitaxial interface of 7,7,8,8-tetracyanoquinodimethane (TCNQ) on tetrathiafulvalene (TTF), whose electronic structure has been probed by ultraviolet photoemission spectroscopy (UPS), but whose structure had been hitherto unknown [Organic Electronics 48, 371 (2017)]. We find that TCNQ(001) on top of TTF(100) is the most stable interface configuration, closely followed by TCNQ(010) on top of TTF(100). The density of states, calculated using DFT, is in excellent agreement with UPS, including the presence of an interface charge transfer state.', 'arxivid': '2301.07594', 'author': ['Saeed Moayedpour ', 'Imaneul Bier \nDepartment of Materials Science and Engineering\nCarnegie Mellon University\n15213PittsburghPAUSA\n', 'Wen Wen ', 'Derek Dardzinski \nDepartment of Materials Science and Engineering\nCarnegie Mellon University\n15213PittsburghPAUSA\n', 'Olexandr Isayev ', 'Noa Marom \nDepartment of Materials Science and Engineering\nCarnegie Mellon University\n15213PittsburghPAUSA\n\nDepartment of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA\n', '\nDepartment of Chemistry\nCarnegie Mellon University\n15213PittsburghPAUSA\n'], 'authoraffiliation': ['Department of Materials Science and Engineering\nCarnegie Mellon University\n15213PittsburghPAUSA', 'Department of Materials Science and Engineering\nCarnegie Mellon University\n15213PittsburghPAUSA', 'Department of Materials Science and Engineering\nCarnegie Mellon University\n15213PittsburghPAUSA', 'Department of Physics\nCarnegie Mellon University\n15213PittsburghPAUSA', 'Department of Chemistry\nCarnegie Mellon University\n15213PittsburghPAUSA'], 'corpusid': 255998305, 'doi': None, 'github_urls': ['https://github.com/aiqm/torchani),'], 'n_tokens_mistral': 32323, 'n_tokens_neox': 26559, 'n_words': 14688, 'pdfsha': 'e3016550d37a5bd044ad1a453da96f9644b9a24d', 'pdfurls': ['https://export.arxiv.org/pdf/2301.07594v1.pdf'], 'title': ['Structure Prediction of Epitaxial Organic Interfaces with Ogre, Demonstrated for TCNQ on TTF', 'Structure Prediction of Epitaxial Organic Interfaces with Ogre, Demonstrated for TCNQ on TTF'], 'venue': []}

arxiv

Multi-Agent Reinforcement Learning for Cooperative Air Transportation Services in City-Wide Autonomous Urban Air Mobility Chanyoung Park Gyu Seon Kim Soohyun Park Member, IEEESoyi Jung Senior Member, IEEEJoongheon Kim Multi-Agent Reinforcement Learning for Cooperative Air Transportation Services in City-Wide Autonomous Urban Air Mobility 1 The development of urban-air-mobility (UAM) is rapidly progressing with spurs, and the demand for efficient transportation management systems is a rising need due to the multifaceted environmental uncertainties. Thus, this paper proposes a novel air transportation service management algorithm based on multi-agent deep reinforcement learning (MADRL) to address the challenges of multi-UAM cooperation. Specifically, the proposed algorithm in this paper is based on communication network (CommNet) method utilizing centralized training and distributed execution (CTDE) in multiple UAMs for providing efficient air transportation services to passengers collaboratively. Furthermore, this paper adopts actual vertiport maps and UAM specifications for constructing realistic air transportation networks. By evaluating the performance of the proposed algorithm in data-intensive simulations, the results show that the proposed algorithm outperforms existing approaches in terms of air transportation service quality. Furthermore, there are no inferior UAMs by utilizing parameter sharing in CommNet and a centralized critic network in CTDE. Therefore, it can be confirmed that the research results in this paper can provide a promising solution for autonomous air transportation management systems in city-wide urban areas.Index Terms-Urban-Air-Mobility (UAM), Air transportation service, Multi-agent deep reinforcement learning (MADRL), Centralized training and distributed execution (CTDE) Preliminary version of this paper was accepted I. INTRODUCTION C OUNTLESS hours are squandered daily due to road congestion in numerous global metropolises. Inhabitants of cities like Los Angeles and Sydney experience an annual commuting duration equivalent to seven full working weeks, during which two weeks are consumed by traffic delays [2]. The resulting time inefficiency contributes to decelerated economic expansion, diminished productivity, and unwarranted carbon emissions. To address this issue, expanding mobility from the horizontal plane to the vertical dimension is imperative. Vertical transportation modalities, such as urban-airmobility (UAM), promise to mitigate these pervasive urban dilemmas [3]. UAM has garnered considerable interest in contemporary times, owing to its capacity for mitigating traffic bottlenecks and facilitating effective transportation services [4]- [7]. At present, a multitude of companies worldwide are engaged in the development of UAMs to commercialize them, such as the Volocopter 2X by e-Volo GmbH, Neva AirQuadOne by Neva Aerospace, Boeing PAV by Aurora Flight Sciences, Joby S2 VTOL / Cruise Configuration by Joby Aviation, and Opener Blackfly by Opener [8]. Nonetheless, there remains a need for more concrete evidence regarding the feasibility of air transportation in relation to the possibilities and demands, both in the presence and absence of autonomous vehicles. As for delivering telecommunications, cellular networks emerge as viable options, owing to their widespread accessibility and substantial capacity, particularly in urban environments [8]. Furthermore, the 5G standard offers services such as ultra-low latency reliable communications and vehicle-to-everyting (V2X) communication, which serve as intriguing components for constructing cellular-based drone operations [9], [10]. Next, from the perspective of feasibility in implementing pricing policies to convert existing transportation users to UAM users, the work in [11] used actual ground access transportation data collected from Incheon International Airport (ICN) and a multinomial logit model (MNL) to esti-mate the fare range for UAM services between Seoul Station and ICN. Assuming that UAM services can reduce travel time by 30-40 minutes compared to conventional ground taxis, the fare range was estimated to be between 96 and 108 US dollars. When comparing these estimated fares with those from expert institutions, it was demonstrated that such pricing policies are reasonable. Lastly, the work in [12] demonstrates that UAM aircraft presents a viable option for air transportation networks, offering significant advantages with respect to vehicle payload and certification risks. Although the required runway lengths for UAMs range from 100 to 300 feet, which may appear short, they are achievable with near-future technology. In fact, a 300foot runway length can be easily attained for UAM aircraft even with current technology, making it feasible from an urban infrastructure development perspective as well. The primary advantage of UAM compared to traditional ground-based transportation lies in its potential to significantly reduce overall travel durations. Additionally, UAMs address the issue of air contamination resulting from greenhouse gas emissions (GGE), which adversely affects the well-being of proximate inhabitants [13]. This is achievable because UAM operates on the basis of an eco-friendly electric propulsion system [14]. Despite the potential benefits, the integration of UAMs within prevailing transportation frameworks encounters a multitude of difficulties, such as the synchronization and collaboration between multiple UAMs [4]. Therefore, this paper puts forth a novel multi-agent deep reinforcement learning (MADRL) strategy with the objective of devising a trustworthy and effective aerial transportation system. Within the suggested MADRL-focused methodology, a communication network (CommNet) [15] is employed, incorporating centralized training and decentralized execution (CTDE) [16]. CommNet facilitates inter-communication among multi-UAM to achieve coordination, while CTDE employs a centralized critic to enhance the training efficacy of all actors commensurate with the agent's policy. The proposed MADRL-centric method tackles numerous obstacles in UAM functioning, including collision circumvention, trajectory formulation, and passenger routing. The employment of a multi-agent framework facilitates optimized utilization of airspace and resources, concurrently guaranteeing the security of passengers and UAM vehicles. This paper evaluates the suggested MADRL-centric approach using data-driven simulations within realistic environment settings. A diverse set of findings demonstrates that the proposed technique surpasses current methodologies with respect to efficiency while maintaining safety. This proposed strategy holds the potential to substantially advance the development of a resilient and effective air transportation framework. A. Contributions The main contributions of this work are as follows. • Multi-UAM Cooperation and Coordination using MADRL. This paper utilizes a novel CommNet/CTDEbased MADRL strategy to manage air transportation networks efficiently under the concept of the cooperation and coordination of multiple UAMs. This advanced training structure helps multiple UAMs perform given common tasks more efficiently without explicit central coordination rules after the policy training phase. • Realistic City-Wide Environment Design and Performance Evaluation. A vertiport map and UAM model are designed with the actual reference model to construct a realistic autonomous air transportation system. The energy charging/discharging model is accurately modeled with the specifications of the adopted UAM model, and even the passenger boarding and alighting system are designed in detail. Therefore, the justification of the proposed air transportation system becomes more potent than when an arbitrary system is assumed. Furthermore, in order to validate the MADRL supremacy of the proposed algorithm, data-intensive performance evaluation is conducted with the latest DRL algorithms from various aspects. Furthermore, this paper analyzes the results in detail with the subdivided groups according to the learning strategy. B. Organization The remainder of the paper is structured as follows. Sec. II reviews related work in air transportation management systems and Sec. III explains the system model considered in this paper. Sec. IV presents the proposed management algorithm for efficient autonomous air transportation networks employing CommNet/CTDE-based MADRL. Sec. V demonstrates the performance of the proposed algorithm via various dataintensive evaluations. Finally, Sec. VI concludes this paper. II. RELATED WORK Self-governing vehicular have increasingly garnered interest from scholarly and commercial domains due to their potential to address enduring issues in transportation, such as safety enhancement, traffic alleviation, energy conservation, and other related concerns [17]- [19]. Advanced computational methodologies, such as deep learning (DL) and machine learning (ML) approaches, are employed to augment the efficacy of autonomous mobility systems [20], [21]. In urban settings, traffic data exhibit consistent characteristics concerning location and time within logistical systems. Consequently, learning-based algorithms demonstrate resilience, as they are capable of capturing the recurrence of similar patterns occurring at identical locations and times. A joint K-mean clustering and Gaussian mixture model-based expectation maximization [22] and the attention mechanism are utilized for trajectory optimization [23]. However, the strategies utilized in the aforementioned traditional optimization and rule-based research, including those in [24], [25] that suggest methods for optimizing trajectories in autonomous aerial vehicles, predominantly concentrate on centralized optimization concerns. This limited scope results in hindrances when attempting to provide real-time solutions for complex, dynamic, interrelated, and widespread transportation systems [26]. Furthermore, their approaches in massive environment may cause pseudo-polynomial computational complexity based on dynamic programming [27]. In the context of deep reinforcement learning (DRL) algorithms, the computational complexity does not exhibit a general pseudopolynomial time complexity; instead, it primarily depends on factors such as the size of the state space and the size of the action space. In addition, imitation learning with advanced sensors is also employed for smart cruise control and lanekeeping systems [28]. However, imitation learning necessitates the provision of expert demonstrations and may suffer from covariate shift or suboptimal performance due to the limited exploration. Besides, among various DL and ML algorithms, DRL techniques have demonstrated the most effective performance by adaptively executing sequential decision-making processes in dynamic autonomous driving environments [20], [29]. Furthermore, the scalability of RL allows for the application of MADRL to govern the operation of multiple mobilities. Among MADRL algorithms, the Q-Mix algorithm incorporates a comprehensive action-value function derived from the amalgamation of individual agents' action-value functions [30]. This algorithm is applied for the trajectory optimization of multiple electric vertical takeoff and landings (eVTOLs) [3] in the context of aerial drone-taxi applications [31]. In addition, there are MADRL algorithms founded upon their inherent communication protocols, which utilize a neural network structure designed to facilitate inter-agent data exchange. Examples of these algorithms include differentiable inter-agent learning (DIAL) [32], bilateral complementary network (BiCNet) [33], and CommNet. DIAL-based agents use differentiable communication channels within their neural networks to learn cooperative strategies through end-to-end backpropagation, leading to improved coordination and cooperation among agents in complex, dynamic environments. Next, BiCNet introduces a cutting-edge approach to address complex coordination challenges in MADRL by utilizing a twin neural network architecture, which enables agents to learn complementary policies collaboratively. Finally, as elucidated in [15], agents employing the CommNet framework acquire communication capabilities in tandem with a unified, centralized single deep neural network to process local observations for multiple agents. Subsequently, each agent's decisions are influenced by its individual observations and the mean of other agents' observations [34]. The CommNet architecture is extensively applied in diverse multi-agent systems in a distributed manner, such as management for electric vehicle charging station [35], charging scheduling in UAV networks [36], and autonomous surveillance system [37]. While the previous study in [1] also suggests the CommNetcentric MADRL algorithm for establishing an autonomous multi-UAM network, several distinctions exist between the present and aforementioned works. Primarily, the algorithm proposed herein further utilizes centralized critic, in contrast to only focusing on CommNet in [1]. Moreover, this work evaluates the performance of the proposed MADRL algorithm while considering various communication statuses compared to previous studies in [1]. Additionally, an additional inference step is conducted to validate the effectiveness of the learned policy. A. Realistic Air Transportation Environment Uber will provide UAM service to various metropolitan areas such as Sao Paulo and Los Angeles within 10 years [2]. Dallas, Texas, a central US metropolitan area, is no exception. Uber Air will begin commercial operations in Dallas in 2023, making it the first city to provide flights. The vertiport environment with the highest possibility of realization is Dallas, Texas. The actual background of the vertiport map used in the experiment corresponds to a movement zone connecting Downtown Dallas, Texas, USA, and Bedtown Frisco to the north. The actual detailed vertiport map information is illustrated in Fig. 2 [38]. B. Realistic UAM Model It is vital to take the energy consumption/remains of UAM devices into account because they are energy-constrained and power-hungry [4], [39]. In contrast to the aircraft powered by internal combustion engines, which produce energy by burning fossil fuels, UAM is an electrified aircraft that relies on batteries. Accordingly, the energy model of UAM can be represented by aerodynamic power calculations independent to the specific fuel consumption (SFC). UAM's hovering power expenditure P h when take-off or landing with passengers can be mathematically expressed as follows [40], P h = C d 8 ρsAΩ 3 R 3 blade profile, Po + (1 + k) W 3 2 √ 2ρA induced, Pi ,(1) where C d is the drag coefficient, which is a dimensionless coefficient that quantifies the drag force of a body in a fluid, ρ is the density of air that decreases exponentially with altitude, and s is the rotor solidity that means the ratio of the rotor blade area to the rotor disk area. In addition, A, Ω, R, W stand for the rotor disc area, blade angular velocity, rotor radius, and total weight considering passenger payload, respectively. k is an induced drag coefficient that is inversely proportional to the efficiency factor (e) and aspect ratio (AR). UAM must rotate the rotor to overcome the drag increased by the induced drag, so k must be considered in (1). When a UAM carrying a passenger rises to an altitude of 600m by eVTOL and propels forward, the force on the x-axis is added. Thus, the propulsion power consumption P p must be considered. Here, the energy expenditure during a round trip or propulsion can be mathematically expressed as follows [41], P p = P i 1 + v 4 4v 4 0 − v 2 2v 2 0 0.5 induced + P 0 1 + 3v 2 U 2 tip bladeprofile + 1 2 d 0 ρsAv 3 parasite ,(2) where v is the cruising flying speed of UAM at which UAM reaches an altitude of 600 m and cruises after picking up passengers. v 0 is mean rotor-induced velocity which is the flow's average speed as caused by the wing tip vortex. U tip is tip speed of the rotor blade and d 0 is fuselage drag ratio. The values of the aerodynamic parameters used in (1) and (2) are for JOBY AVIATION's S4 and are displayed in Table I [42]. According to Joby Aviation's analyst day presentation [42], S4 ascends to an altitude of 600 m in around 30 seconds after carrying passengers. Since the turnaround time is about 6 minutes, UAM landing on a vertiport must unload passengers and recharge its battery as much as possible in the meantime. Here, the realistic actual charging time is only 5 minutes, excluding the time to turn on/off the electric engine and plug in/out the charging cap. However, S4 can charge its battery quickly via a charger plugged directly behind the inboard tilt electric motor nacelle. It is equipped with four battery packs, two on both sides of the main wing inboard and the others in the nacelle at the rear of the main wing inboard electric motor. To charge each battery with 30 kW h in 5 minutes, the charger's supply power must be at least 360 kW . Under this condition, S4's current rate (C-rate) indicating the battery charge/discharge rate becomes 2.4 per hour when a 150 kW h battery is charged with a 360 kW charger. Finally, the state of charge (SOC) that is the time to charge the battery to 100% takes 25 minutes, and S4 can charge 20% of the total battery capacity per journey. Table II summarizes a detailed description of the battery specifications. IV. ALGORITHM DESIGN A. RL Formulation for UAM Networks In order to consider the physical capabilities of UAM, this paper formulates UAM networks as a decentralized partially observable MDP (Dec-POMDP) [43], [44]. According to the nature of RL-based algorithms, each UAM with Dec-POMDP can sequentially make action decisions founded on partial environmental information. The considering reference air transportation model consists of J, N , and Ξ numbers of UAMs, vertiports, and passengers, respectively. Thus, the sets of UAMs, vertports, and passengers are defined as ∀u j ∈ U where U ≜ {u 1 , · · · , u j , · · · , u J }, ∀ϱ ξ ∈ G where G ∈ {ϱ 1 , · · · , ϱ ξ , · · · , ϱ Ξ }, and ∀ν n ∈ V where V ∈ {ν 1 , · · · , ν n , · · · , ν N }. Here, the air transportation service provided by UAMs in this paper can be defined as the passenger delivery using UAM from one vertiport to the other target vertiport. The POMDP of the proposed air transportation networks with J UAMs can be modeled as ⟨J, S, O, A, R, P, Z, γ⟩ where s ∈ S is a set of ground truth states. Next, o j ∈ O j and a j ∈ A j stand for a set of j-th UAM's observations and actions, respectively. Note that these two sets can be jointly denoted as O j ⊂ O and A j ⊂ A. At every time step, each UAM gets reward r j with reward function R(s, a, s ′ ) by selecting joint action a while observing joint observation information o based on conditional observation probability Z(s ′ , a, o) = S × A → O. Then, the global state s is transited to the next state s ′ with state transition probability function P (s ′ | s, a) = S×A×S → S. Lastly, γ is a discount factor that weights current rewards versus future rewards. The below subsections materialize the MDP delineating observation, state, action, reward, and objective of the proposed UAM networks via mathematical description. 1) Observation: Due to the physical limitation of the considered UAM model, every UAM can recognize other UAMs or vertiports within its coverage. Here, the eyesight of the j-th UAM is dependent on its absolute position denoted as p j ∈ {x j , y j , z j } corresponding to Cartesian coordinates. The j-th agent can recognize distances with other UAMs and vertiports in its observation scope D th , which information can be mathematically represented as follows, d(u j , u j ′ ) = ∥u j − u j ′ ∥ 2 , if. ∥u j − u j ′ ∥ 2 ≤ D th , −1, (otherwise),(3)d(u j , ν n ′ ) = ∥u j − ν n ∥ 2 , if. ∥u j − ν n ∥ 2 ≤ D th , −1, (otherwise),(4) where d(·) and ∥ · ∥ 2 = ( N i=1 | · | 2 ) 1 2 stand for the function that outputs the distance between two inputs and L2norm, respectively. UAMs can also carry up to Λ passengers, and also know the status of their seats denoted as Φ j = {ϕ 1 j , · · · , ϕ λ · · · , ϕ Λ j }, where ϕ λ j is defined as follows, ϕ λ j = ϱ ξ , if. ϱ ξ boards the λ-th seat on u j , −1, (otherwise),(5) Lastly, each j-th UAM has to observe its energy state e j to avoid the battery's full discharge for a safe air transportation system denoted as follows, e j = e max − (P p × |A h | 2 (vt) 2 + P h × |Av| 2 (vt) 2 ), if. t = 0, max (e j − (P p × |A h | 2 (vt) 2 + P h × |Av| 2 (vt) 2 ), (otherwise),(6) where the P h and P p stand for power consumption when UAM takes-off or lands on the vertiport defined in Sec. III-B. In addition, A h and A v are the action UAM takes, which are specified in Sec. IV-A2. In brief, the partial information that j-th UAM can observe in the environment is connoted as follows, O j ≜ {p j , J j̸ =j ′ {d(u j , u j ′ )}, J j̸ =j ′ {d(u j , ν n )}, Φ j , e j }. (7) 2) State: The ground truth state includes the overall air transportation service information consisting of the number of passengers serviced by each UAM represented as Ξ ξ=1 {1 j (ϱ ξ )}, where the 1(·) is an indicator function differentiating serviced (one) or not (zero). The type of vertiports each UAM landed on is also contained in the ground truth state denoted as N n=1 {1 j (ν n )}, which represents visited (one) or not (zero). Additionally, the distance to the target vertiport of the passenger boarding the j-th UAM is one of the components of the state space, which is between u j and ν λ target dependent on seating state can be defined as follows, d ′ (u j , ν target n ). Here, the distance ! " (a) Top View. Take-off Landing (b) Forward View.d ′ (u j , ν λ target ) = ∥u j − ν λ target ∥ 2 , if. ϕ λ j ̸ = −1, −1, (otherwise).(8) To recap, the ground truth state can be organized as follows, S ≜ { J j=1 Ξ ξ=1 {1 j (ϱ ξ )}, J j=1 N n=1 {1 j (ν n )}, J j=1 Λ λ=1 {d ′ (u j , ν λ target )}.(9) 3) Action: In every time step t, every UAM can take two types of actions; i) horizontal and ii) vertical moving, where the set of actions is A ≜ {A h , A v } as illustrated in Fig. 3. Note that A h and A v have an alternate relationship, thus A h is zero when A v is selected by UAM (vice versa). The first type of action is horizontal moving from one vertiport to another vertiport in ordinal directions expressed in vector form as follows, A h ∈ {⟨±vt, 0, 0⟩, ⟨0, ±vt, 0⟩, ⟨± 1 √ 2 vt, ± 1 √ 2 vt, 0⟩}. (10) The other is moving for vertical take-off and landing in any vertiport, which can be indicated in vector form as follows, A v ∈ {⟨0, 0, vt⟩, ⟨0, 0, −vt⟩}.(11) 4) Reward: Every UAM takes action at every time step, and then it gets a reward from reward function R(s, a, s ′ ) when the state s is transited to the next state s ′ . There is a common goal of UAMs in MADRL, this paper divides the reward function into two elements; i) individual reward, ii) common reward. Thus, the reward function can be configured as R(s, a, s ′ ) = J j=1 (R j Indiv (s, a, s ′ )) + R Comm (s, a, s ′ ) . First, each UAM receives individual rewards based on its ac-tions. Accordingly, all UAMs have different individual reward values from each other, which are denoted as follows, R j Indiv (s, a, s ′ ) = J j̸ =j ′ 1(d(u j , u j ′ ) ≥ C th ) × Ξ ξ=1 1 j (ϱ ξ )+ N n=1 1 j (ν n )− Λ λ=1 d ′ (u j , ν λ target ) Γ/2 + e j e max ,(12) where C th and Γ are the minimum distance to occur collision and considered environment size in this paper, respectively. Next, all UAMs have the same common reward value simultaneously which helps them to cooperatively achieve a shared objective without preemption or competition. The common reward is established as follows, R Comm (s, a, s ′ ) = J j=1 1 j (ϱ ξ ) J .(13) As a result, as seen in the above definition of the reward function, all UAMs try to maximize not only the quality components of air transportation service but also consider safety conditions. By shaping appropriate reward functions, UAMs can learn the intelligence to select the suitable strategy concerning shared objectives in any given state. 5) Objective: All UAMs' main objectives can be mathematically expressed as follows, π * θ = argmax θ E s∼E, a∼π θ T t=1 γ t−1 ·R (s, a, s ′ ) ,(14) where θ, E, and T correspond to the parameters of actornetwork, environment, and episode length, respectively. In summary, it is obvious that the main objective of UAMs is to find the optimal policy maximizing the expected cumulative reward over a given finite step T . B. Information Sharing by CommNet The CommNet algorithm provides mutual communication between multiple UAMs with a communication step when multiple actors proceed to learn their hidden variables as depicted in Fig. 4. This process of sharing observation information with each other is particularly effective when multiple agents try to achieve a common goal in a Dec-POMDP environment where agents cannot observe the global state. First of all, every UAM explores the environment consisting of state and observation in Fig. 4(a). Their experiences are encoded into hidden variables of the first layer h 1 j with encoder function in Fig. 4(b) as follows, h 1 j = Encoder (s, o j ).(15) When hidden variables are fed into the deeper hidden layers, the communication variables are entered simultaneously. The communication variable of j-th UAM in i-th hidden layer is acquired by averaging hidden variables of other UAMs in Fig. 4(c) as follows, c i j = 1 J − 1 J j̸ =j ′ h i j ′ .(16) At the input of every i-th layer except for the first layer, h i j and c i j are feed-forwarded to the next layer with the single-agent module f i (·), which returns output vector h i+1 j as follows, h i+1 j = f i (h i j , c i j ),(17)subject to f i (·) = Activ(Concat(h i j , c i j )),(18) where Activ(·) and Concat(·) stand for a non-linear activation function (e.g., ReLU, hyperbolic tangent, or sigmoid) and concatenate function. The information sharing by intercommunication between UAMs comes about when averaging hidden variables of different UAMs. Finally, the action probabilities of j-th UAM are obtained by decoding the output of the last layer in Fig. 4(d) as follows, p θj (A j ; o j ) = Decoder (h K j ),(19) where p(·) is the action distribution. To sum it up, the above sequential process of feed-forwarding can be implied as follows, p θj (A j ; o j ) = Q(o j , a ; [θ 1 , · · · , θ j , · · · , θ J ]),(20) where Q(o j , a) is commensurate with the action-value function in Dec-POMDP. It is noteworthy that the output probabilities of j-th UAM are also dependent on the network parameters of other UAMs. C. CTDE-based Parameterized Policy Training As motivated in [45], multiple actors distributedly get experiences by exploring environments, and the centralized critic evaluates every global state's value in Fig. 4(f). The centralized critic may be a central server such as a control tower observed in Fig. 1. This strategy helps all UAMs symmetrically to learn near-optimal policies. Additionally, below introduces the multi-agent policy gradient (MAPG) based on the temporal difference (TD) actor-critic method [46] with Bellman optimality equation to prevent the occurrence of high variance. Centralized Critic. To evaluate the value of parameterized policies of decentralized actors, the centralized critic tries to learn its network parameters ϕ to approximate the optimal joint state-value function which is configured as follows, V ϕ (s) = E s∼E, a∼π θ T u=t γ u−t · R(s u , a u , s u+1 ) .(21) With (21), the centralized critic learns its network parameters to minimize the loss function which is leveraged as follows, ∇ ϕ L(ϕ) = T t=1 ∇ ϕ δ t ϕ 2 ,(22) subject to δ t ϕ = R(s t , a t , s t+1 ) + γV ϕ (s t+1 ) TD Target −V ϕ (s t ),(23) where δ t ϕ is the TD error based on Bellman optimality equation in time step t. As seen in (23) Communication Step of the summation of current and future reward values. Based on (23), the centralized critic trains its network parameters in the direction of minimizing the loss function by gradient descent as follows, Module & ℎ ' & , & = ℎ ' &#(ϕ t+1 ≈ ϕ t + α critic × [ δ t ϕ · ∇ ϕ V ϕ (s t ) ],(24) where α critic stands for a learning rate of the centralized critic network that decides the inclination of updating neural network parameters by policy gradient. Multiple Actors. Actors correspond to UAMs providing air transportation service to passengers in environments. They learn policy parameters θ to approximate optimal policy for providing efficient air transportation service. At every time step t, they make sequential decision-making based on their parameterized strategy function as follows, a t j = argmax a t π θj (a t | o t j ).(25) It can be seen that each UAM selects an action with high probability among all possible actions as presented in Fig. 4(e). Here, j-th UAM's parameterized policy π θj is defined as follows, π θj (a t | o t j ) ≜ softmax(p θj (A j | o t j )),(26) subject to softmax(x) ≜ e x1 N i=1 e xi , · · · , e x N N i=1 e xi ,(27) where softmax(·) stands for exponential softmax distribution function to activate normalization of action probabilities. Finally, the objective function that dispersed actors need to maximize is mathematically constituted as follows, ∇ θj J(θ j ) = E oj ∼E T t=1 δ t ϕ ·∇ θ log π θj (a t j | o t j ) .(28) Using (28), UAMs learn their parameters toward maximizing the objective function by gradient ascent as follows, θ t+1 j ≈ θ t j + α actor × [ δ t ϕ ·∇ θ log π θj (a t j | o t j ) ],(29) where α actor is a learning rate of the actor network. The details of the CTDE-based policy training and inference are organized in Algorithm 1 in consecutive order. V. PERFORMANCE EVALUATION A. Experimental Setting The air transportation area has a size of (2 × Γ) 2 m 2 , illustrated in Fig. 2. In that area, J-UAMs autonomously provide air transportation service to passengers by transporting them from one vertiport to the destination vertiport. As mentioned in Sec. III-B, the realistic UAM model, named JOBY AVIATION's S4, transports passengers at an altitude of 600 m. In addition, this UAM model can carry up to four passengers based on the first-in-first-out (FIFO). UAMs start an episode at random vertiports, and destinations/departures of passengers also change from episode to episode. In addition, UAMs can recharge their batteries for five minutes at the vertiport while dropping off and picking up passengers as mentioned in Sec. III-B. Lastly, note that this paper considers a total of five vertiports, as depicted in Fig. 2. With merely five vertiport locations, the network effectively represents a citywide, large-scale, and intricate system capable of realistically accommodating a significant number of passengers. In [38], Uber has established five vertiports in the Dallas Metropolitan area, which is the closest region to the implementation of UAM-based air transportation service. Since Uber has already determined the number and location of vertiports based on actual urban traffic demand, a configuration with only five vertiports still results in a relatively complex and practical map. The overall system parameter notations and corresponding values are arranged in Table III. B. Benchmarks This paper conducts data-intensive experiments focusing on the validation of the proposed algorithms' performance, which coincided with i) CommNet and ii) CTDE. For this purpose, this paper divides benchmarks into the following two groups. 1) Benchmarks for CommNet: To scrutinize the performance of CommNet in MADRL, an ablation study is conducted according to the number of UAMs participating interagent communications. • CommNet (Proposed summarize the final reward convergence value in both POMDP and FOMDP environments. Note that FOMDP is unrealistic due to UAM model's physical limitations. Firstly, among benchmarks of the first group, UAMs in CommNet get the fastest reward-increasing rate in Fig. 5, and the most enormous reward value of 0.647 in POMDP and 0.602 in FOMDP as summarized in Table IV. Notably, only the proposed algorithm allows UAMs to get a higher reward in POMDP than FOMDP, regardless of information loss. On the other side, rewards of UAMs in Hybrid and DNN converge to 0.438 and 0.148, respectively. It means that these schemes are obviously susceptible to information loss in POMDP by getting 3.94 % and 56.1% lower reward values than in FOMDP. In the case of DNNs utilizing only DNN-based UAMs, policy training failed with smaller rewards than Monte Carlo. Indeed, the presence of a CommNet-based UAM helps serve robust air transportation service with limited environmental information in MADRL. Besides, in Fig. 6 and Table V According to [48], a limitation of utilizing DQN-based DRL in multiagent settings is the reduced effectiveness of experience replay. Unlike single-agent scenarios, the same action executed in an identical state may yield varying outcomes, contingent on other agents' actions. Consequently, agents that have trained neural network parameters through DQN in MARL may struggle to optimize their parameters efficiently. This occurrence leads to agents collectively exhibiting alike inappropriate behavior when encountering similar environmental information. As a result, adopting random actions, as observed in Monte Carlo, could yield higher rewards than DQN due to the wider action decisions of agents. Next, among DRL algorithms that successfully trained policies, only CTDE results in higher reward values in POMDP than in FOMDP. In a nutshell, the training performance of the proposed algorithm with CommNet and CTDE is corroborated by showing the powerful reward convergence ability despite information loss. 2) Trained Trajectories: This section intuitively investigates the CommNet's training performance with Fig. 9, which exhibits the trajectories of UAMs trained by CommNet and DNN benchmarks in the POMDP environment. At the last training epoch, it can be seen that CommNet-based UAMs do not deviate much from the considered system map. In contrast, DNN-based UAMs have unnecessary trajectories in terms of air transportation service provision (Agents 4 and 10) or are isolated in a limited area (Agent 3). Some agents are even isolated in areas where vertiports do not exist (Agents 2, 5-6, and 7-9). In addition, it can be seen that UAMs have non-linear trajectories for transporting passengers to target vertiports. UAM's linear trajectories are likely to provide more ideal air transport services than non-linear trajectories, but not because of 'safety' [49]. In aviation systems, there are horizontal and vertical separations to prevent collisions between aircraft [50]. All aircraft must maintain a safe distance to the separation criteria. Especially in unmanned aerial system (UAS) operating without a pilot, the plane must detect and resolve potential collisions on its own. In the case of UAM, since many UAMs fly simultaneously at low altitudes in urban areas, separation must be managed more strictly than normal aircraft. In particular, near congested vertiport where many UAMs take off and landing, since they must fly while avoiding other UAMs, non-linear trajectories are more ideal for transporting large numbers of passengers with safety considerations. In Fig. 8, transportation services are relatively concentrated in A, C, D, and E except for B (FORT WORTH). This is the result of considering actual passenger demand. In Fig. 2, the traffic circle connecting Dallas, Texas and Frisco is very busy. In 2018, 27.2 million people visited Dallas, and 40% of Frisco Collin's residents commute there [38]. So traffic can achieve economies of scale, and there are DFW Airport (A) and Dallas Love Field Airport(D), which are the hubs of air traffic in the heart of the United States, in the metropolitan area. Demand is concentrated at airports with high traffic volumes or at FRISCO between DOWNTOWN DALLAS, where commuting volume is high. In the vertiport map in Fig. 2, the vertiport layout is designed based on these actual passenger demands, so the experimental results shown in Fig .8 are also very reasonable. More exact values are organized in Fig. 10, where the total type of vertiport CommNet-based UAMs landed on is twice as DNN-based UAMs. In addition, every CommNet-based UAM landed on various vertiports on average 2.3 times more than a DNN-based UAM. It is also confirmed that all CommNetbased UAMs successfully transported passengers by visiting at least more than two different vertiports, with the maximum types of vertiports being visited even reaching four (Agents 5 and 6). However, in DNN, only three UAMs (Agents 1, 5, and 8) landed on more than two different vertiports, and the maximum type of visited vertiport is less than the CommNet. 3) Equity in Policy Training: A MARL algorithm is meaningless if at least one agent fails to learn the policy, even if it goes through the training process and achieves a high total reward value. Therefore, to ensure that all UAMs have equally well-trained their policies, Fig. 7 provides the convergence behavior of every UAM trained by the proposed CommNet/CTDE-based MADRL algorithm. All UAMs show similar tendencies in learning policies, starting with an average reward value of 0.0203 ranging from 0.0119 to 0.0274. At the end of the training, all UAMs get reward values of 0.0348-0.0765. Here, the average value is 0.0564, which is 53.01 % higher than the average reward value of Monte Carlo (≈ 0.0265) summarized in Tables IV and V. As a result, it is confirmed that all UAMs trained by the proposed algorithm have equitable training performance. D. Feasibility of the Proposed Air Transportation System This section evaluates the feasibility of the proposed air transportation network in versatile aspects with 100 inference times in the POMDP environment. For feasibility studies, this paper adopts the quality of air transportation service, UAMs' unbiased performance, and energy management as evaluation indicators. 1) Service Quality: The air transportation service quality considered in this paper encompasses the number of serviced passengers and the number/types of landing vertiports. Suc- CommNet Hybrid DNN Monte Carlo : Transportation Service Quality in Inference Phase in POMDP Benchmark Fig.11(a) Fig.11(b) Fig.11 cessfully transporting many passengers to their destination vertiports is the most crucial element of air transportation services. In addition, the above quality factors are correlated since the more diverse vertiports UAM visits, the more passengers it can provide air transportation services to. Fig. 11 presents the service quality of the first benchmark group. It can be seen that UAMs in CommNet outperform regarding air transportation service quality, except for less than two UAMs. Furthermore, it is noteworthy that among all UAMs trained by other benchmark algorithms that outperform UAMs in the proposed MADRL algorithm, CommNet-based UAMs in Hybrid are unique, corresponding to agent 2 in Fig. 11(a) and agents 1, 2 in Figs. 11(b)-(c). The other DNNbased UAMs in Hybrid and DNN show inferior validation performance to CommNet-based UAMs in CommNet and Hybrid. This result definitely verifies that inter-agent communications by CommNet can help UAMs learn optimal policies to serve autonomous air transportation services. Regarding the second benchmark group's service quality, UAMs in CTDE outperform others in all service quality factors except for agents 2 and 6 according to Fig. 12. Even though these two UAMs provided the second-highest quality of service, they served similar quality of air transportation service to other IAC agents. Even though these two UAMs Fig. 12(a), UAMs cannot serve air transportation service to any passenger similar to the training process in Fig. 6. Finally, Table VI summarizes the average validation performance of all UAMs trained with each benchmark represented in Figs. 11-12. It can be seen that the proposed training scheme with CommNet and CTDE algorithms shows the best air transportation performance in terms of the number of serviced passengers and the number/types of vertiports on which UAMs land. The benchmark with the second-highest air transportation performance is IAC consisting entirely of CommNet-based UAMs. Next, Hybrid follows with the third highest air transportation performance. The other learning benchmarks, which correspond to DNN and DQN, failed to train parameterized policies of UAMs by serving inferior performance than Monte Carlo which is not a learning algorithm. In summary, it is confirmed that environmental information sharing by CommNet and training strategy based on CTDE are suitable for building efficient autonomous air transportation networks. 2) Service Fairness: This section inspects in detail the quality of air transportation services of all UAMs. Fig. 13 provides the records of serving air transportation services in POMDP of benchmarks that outperform Monte Carlo in Table VI, which corresponds to the proposed, Hybrid, and IAC algorithms, with 100 inference times. As shown in Fig. 13(a), agent 1 in the proposed algorithm has served the largest number of services, providing air transportation service to 8 passengers. Additionally, agent 2 has visited all vertiports in the environment as shown in Fig. 13(g). In addition to these best records, UAMs in the proposed algorithm have shown the highest and fairest air transportation service than the other benchmarks without an inferior UAM. However, there are UAMs with inferior performance to other benchmarks, such as agents 7-10 in Hybrid and agents 3, 5, and 7-9 in IAC. Note that DNN-based UAMs (Agents 6-10) perform relatively inferior to CommNet-based UAMs (Agents 1-5) in Hybrid. Table VII shows the variance of all service quality factors represented in Fig. 13. Notably, it can be confirmed that UAMs trained with the proposed algorithm have the smallest variance than Hybrid and IAC while achieving the highest service quality as known in Fig. 13. These results demonstrate that all UAMs in the proposed MADRL strategy cooperate unbiasedly to construct a high-quality autonomous air transportation network. 3) Energy Management: The reward function is designed in the direction of preventing the battery from being completely discharged with energy-related observation information when UAM learns its policy in Sec. IV-A4. Fig. 14 shows the energy state of all UAMs in the progress of the episode. It can be observed that UAMs trained by the proposed MADRL algorithm consume their energy within the maximum energy capacity limit for a given episode length T . In particular, agents 6 and 8 prevent complete discharging by reducing their energy consumption near the end of the episode. Hence, it is confirmed that UAMs successfully optimized their policies in both service quality and energy management. E. Discussions This section analyzes the above training and inference results in detail by describing the effects of CommNet and CTDE, the proposed training methods in this paper. 1) Effect of CommNet: As mentioned, the justification for using CommNet lies in achieving a common objective through mutual communication between UAMs. To evaluate the effect of CommNet, this paper conducted an ablation study according to the number of CommNet-based UAMs. The supremacy of this inter-communication scheme is confirmed in Fig. 5 and Table IV by accomplishing the highest reward between all benchmarks in the training phase. In particular, it is meaningful for agent-to-agent communication through CommNet because UAMs in the proposed algorithm obtained a higher reward than FOMDP despite information loss in POMDP. Furthermore, CommNet-based information sharing helps UAMs learn out-of-range environmental information, allowing them to transport passengers efficiently to diversified vertiports as illustrated in Fig. 9. Next, the performance of the air transportation service quality provided after the learning phase is presented in Fig. 11. In the inference phase, the proposed strategy consisting of only CommNet-based UAMs served the best quality of service, as in the learning phase. Furthermore, parameter-sharing while training policies can make all policies optimal without any inferior UAMs as observed in Figs. 7 and 13, and Table VII. 2) Effect of CTDE: In addition to CommNet, this subsection investigates the effect of utilizing a centralized critic for CTDE. Since the critic network evaluating the value of every state also needs to be trained to proceed with the episode, so using one centralized critic can serve as a solid standard for multiple actors to learn the policy with fewer episodes than when utilizing independent actor-critic strategy [46]. The advantage of CTDE in both training and inference processes is well represented in Fig. 6 and Table V, where training performance is different depending on the learning method if the same number of CommNet-based UAMs are employed. Indeed, since the centralized critic network can learn Jtimes more according to the progress of training epochs and establish more firm criteria for judging the state value than each independent critic network, the fastest convergence was achieved with the highest reward as depicted in Fig. 6. In addition, UAMs trained with a centralized critic network outperform other benchmarks in the inference phase in Fig. 12. Finally, the existence of a firm criterion for evaluating state values ensures that all multiple actors have uniformly superior policies as shown in Figs. 7 and 13, and Table VII. VI. CONCLUSIONS AND FUTURE WORK This work aims to efficiently manage autonomous air transportation systems utilizing CommNet and CTDE-based MADRL algorithm. In particular, this paper conducts a realistic evaluation by adopting an actual vertiport map and considering the POMDP environment. Through extensive numerical results, it is demonstrated that the proposed MADRL algorithm has the most superior performance among conventional DRL algorithms in terms of air transportation service quality by helping multiple UAMs cooperatively learn their policies to coordinate their actions. In addition, all UAMs trained with the proposed strategy serve unbiased air transportation service without any inferior UAMs. In other words, the proposed MADRL algorithm has the most robust policy training performance for information loss in POMDP, effectively reflecting the physical limitations of UAM model. Since UAM-based air transportation services are not yet commercialized, the scale outlined in this paper is suitable for preliminary air transportation service considerations. Nevertheless, once UAM materializes in the future, it would be advantageous to take into account a larger number of UAMs. Indeed, as demonstrated in [51], the MARL-based approach has the potential for scalability and the capacity to handle a considerable number of agents. Lastly, in order to improve the quality of the autonomous air transportation service, it is also a promising direction to consider incorporating factors that take passenger comfort into account in our reward function. Fig. 1 . 1Reference air transportation service management network. Fig. 2 . 2Vertiport map considered in this paper for realistic design[38].III. REALISTIC CITY-WIDE AUTONOMOUS AIR TRANSPORTATION SYSTEM DESIGNThis paper constructs a realistic air transportation environment and UAM model based on actual vertiports and aircraft. This realistic design suggests a direction in which UAM can be realized in various bustling metropolitan areas, including Dallas, Texas, and Bedtown Frisco, USA. Fig. 2 shows the central transportation network of the United States, which connects downtown Dallas, Texas, with Frisco, centered on Dallas Fort Worth International(DFW) Airport. In this transportation network, Uber will build a total of five vertiports at DFW AIRPORT (A), FORT WORTH (B), DOWNTOWN DALLAS (C), LOVE FIELD (D), and FRISCO (E). The black number means the distance between vertiports. The length of the bar on the top left represents about 10 km in real space. The scale is 1 : 1, 230, 000, thus 1 cm in Fig. 2 corresponds to 12.3 km. Fig. 3 . 3Types of actions UAM can take. Fig. 4 . 4Overall CTDE-based neural network training architecture of CommNet algorithm. 1▷ 10 ▷ 11 ▷▷ 16 ▷ 17 ▷▷ 10111617Initialize parameters of actor networks and centralized critic network which are denoted as [θ1, · · · , θj, · · · , θJ ] and ϕ; 2 Initialize replay buffer D = {} and mini-batch B = {}; Select the action aj based on its policyπ θ j (a t j | o t j ) at time step t; t → s t+1 , o t → o t+1 with the reward r t ; Set ξ = {s t , o t , a t ,r t , s t+1 , o t+1 }; Update Get V ϕ by sampling mini-batch B from D; Update ϕ by gradient descent to loss function of the centralized critic network: ∇ ϕ L(ϕ); Update θj by gradient ascent to objective function of j-th actor network: ∇ θ j J(θj); Select the action aj based on its policy π θ j (a t j | o t j ) at time step t; t → s t+1 , o t → o t+1 ; Fig. 5 . 5Training performance in POMDP and FOMDP over training epochs. Fig. 6 . 6Training performance in POMDP and FOMDP over training epochs. Fig. 7 . 7Every agent's learning progress with the proposed algorithm in the POMDP environment over training epochs. 0.440 and 0.131 in POMDP. Especially, DQN fails to learn UAMs in both MDP environments by attaining lower reward values of 0.131 and 0.111 than Monte Carlo. Fig. 8 . 8The trained trajectories of UAMs trained with CommNet in the progress of episodes at the end of the policy training. (a) Agent 1. (b) Agent 2. (c) Agent 3. (d) Agent 4. (e) Agent 5. (f) Agent 6. (g) Agent 7. (h) Agent 8. (i) Agent 9. (j) Agent 10. Fig. 9 . 9The trained trajectories of UAMs trained with DNN in the progress of episodes at the end of the policy training. Fig. 10 . 10Type of vertiport where each UAM lands inFig. 9. Fig. 11 . 11Average validation performance in 100 inference processes of the first benchmark group. Fig. 12 . 12Average validation performance in 100 inference processes of the second benchmark group. Number of landings (Proposed). Number of landings (IAC). Type of vertiport (Proposed). Type of vertiport (Hybrid). Type of vertiport (IAC). Fig. 13 . 13Total air transportation service quality of all UAMs trained with the proposed, Hybrid, and IAC algorithms in the inference process. Fig. 14 . 14Energy state of all UAMs in the progress of the episode. TABLE I SPECIFICATION IOF UAM MODEL.[42] Notation Value Maximum number of passengers, Λ 4 Flight speed, v 73.762 [m/s] Aircraft mass including battery and propellers, m 1,815 [kg] Aircraft weight including battery and propellers, W = mg 17,799 [N] Rotor radius, R 1.45 [m] Rotor disc area, A = πR 2 6.61 [m 2 ] Number of blades , b 5 Rotor solidity, s = 0.2231b πR 0.2449 Blade angular velocity, Ω 78 [radius/s] Tip speed of the rotor blade , Utip = ΩR 2 112.776 [m/s] Air density, ρ 1.225 [kg/m 3 ] Fuselage drag ratio, d0 = 0.0151 sA 0.01 Mean rotor-induced velocity in hovering, v0 = W sρA 26.45 [m/s] Profile drag coefficient, C d 0.045 Incremental correction factor to induced power, k 0.052 TABLE II SPECIFICATION IIOF S4'S BATTERY.[42] Notation Value Battery capacity 150 [kW h] Battery charge capacity per journey 30 [kW h] Charging time per journey 5 [min] Charger supply power 360 [kW ] C-rate 2.4 [per hour] SOC 25 [min] Charge rate per journey 20 % , the TD target is composedEnvironment UAM 1 UAM UAM Centralized Critic Distributed Actors Experience Transition Evaluation Module Activate Hidden + 1 Hidden Hidden 1 Encoding Softmax Hidden Module Activate Hidden + 1 Hidden Hidden 1 Encoding Softmax HIdden Module Activate Hidden + 1 Hidden Hidden 1 Encoding Softmax HIdden # # # Averaging hidden variables without itself State Encoding Action Decision State Observe TABLE III SYSTEM IIIPARAMETERS FOR PERFORMANCE EVALUATIONNotation Value Size of environment, Γ 32, 000 [m] Length of episode, T 60 [min] Number of UAMs, J 10 Number of vertiports, N 5 Size of state space, |S| 67 Size of action space, |A| 15 Size of mini-batch, |B| 32 Size of experience replay buffer, |D| 5 × 10 4 Discount factor, γ 9.8 × 10 −1 Initial value of epsilon, ϵ init 0.275 Minimum value of epsilon, ϵ min 0.01 Annealing epsilon 5 × 10 −5 Hidden layer dimension of actor 64 Hidden layer dimension of centralized critic 256 Learning rate of actor, αactor 1 × 10 −2 Learning rate of centralized critic, α critic 2.5 × 10 −3 Training epochs 5, 500 Activation function ReLU Optimizer Adam Hybrid. This benchmark has half CommNet-based UAMs and half DNN-based UAMs, where 'DNN' stands for the conventional neural network architecture. In other words, only half of UAMs communicate environmental information with each other. • DNN. There are no mutual communications betweenUAMs. In other words, they behave as in a single-agent DRL environment. • Monte Carlo. Since UAMs in this benchmark have no policies, they take actions randomly without sequential decision-making. Although it is not an ML/DRL algorithm, it can serve as a standard for evaluating the performance of other learning algorithms. 2) Benchmarks for CTDE: Benchmarks in the other group differ in their training methodology. The performance of the proposed CTDE-based training approach is compared with conventional DRL algorithms. Monte Carlo. This benchmark is for the Monte Carlo simulation as described above.). All UAMs in this benchmark learn their policies with rich experiences (i.e., hid- den variables) using information sharing via inter- communications. • • CTDE (Proposed). A central server utilizes centralized critic when evaluating the ground truth state made by decentralized actors. • IAC. Instead of a central server, every UAM has its independent actor-critic (IAC) networks. UAMs in this benchmark learn their policies based on the existing TD actor-critic algorithm [46]. • DQN. Deep Q-network (DQN) algorithm [47] only ap- proximates Q-function without separate neural network approximating optimal state-value function (i.e., critic). • C. Training Performance 1) Reward Convergence: Figs. 5-6 plot UAMs' obtained reward value while training their policies, and Table IV-V CommNet Hybrid DNN Monte Carlo TABLE IV REWARD IVCOMPARISON TO INVESTIGATE THE EFFECT OF COMMNET.1 : Reward Convergence in POMDP and FOMDP Algorithm POMDP FOMDP CommNet (Proposed) 0.647 0.602 Hybrid 0.438 0.456 DNN 0.148 0.337 Monte Carlo 0.265 0.265 CTDE IAC DQN Monte Carlo , CTDE outperforms the other DRL benchmarks of the second group in POMDP/FOMDP. UAMs trained by IAC and DQN receive reward values of TABLE V REWARD VCOMPARISON TO INVESTIGATE THE EFFECT OF CTDE.2 : Reward Convergence in POMDP and FOMDP Algorithm POMDP FOMDP CTDE (Proposed) 0.647 0.602 IAC 0.440 0.481 DQN 0.131 0.111 Monte Carlo 0.265 0.265 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Agent 7 Agent 8 Agent 9 Agent 10 TABLE VI PERFORMANCE VICOMPARISON FOR VALIDATION OF TRAINED POLICY. 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Nolan, "Fundamentals of air traffic control-5th edition," Delmar Cengage Learning, 2010. Multi-agent game abstraction via graph attention neural network. Y Liu, W Wang, Y Hu, J Hao, X Chen, Y Gao, Proc. AAAI Conference on Artificial Intelligence (AAAI). AAAI Conference on Artificial Intelligence (AAAI)New York, NY, USA34Y. Liu, W. Wang, Y. Hu, J. Hao, X. Chen, and Y. Gao, "Multi-agent game abstraction via graph attention neural network," in Proc. AAAI Conference on Artificial Intelligence (AAAI), vol. 34, no. 5, New York, NY, USA, February 2020, pp. 7211-7218. Her research focuses include deep learning applications, quantum machine learning algorithms and their software engineering methodologies, big-data platforms, and autonomous networking. She was a recipient of the IEEE Vehicular Technology Society (VTS) Seoul Chapter Award. Seoul, Republic of Korea; Seoul, Republic of KoreaElsevierDepartment of Electrical and Computer Engineering, Korea University, Seoul, Republic of KoreaBest Reviewer AwardShe received the Ph.D. degree in electrical and computer engineering from Korea University. IEEE Seoul Section Student Paper Content Award (2020), and ICT ExpressSoohyun Park has been a postdoctoral scholar at the Department of Electrical and Computer Engineering, Korea University, Seoul, Republic of Korea, since September 2023. She received the Ph.D. degree in electrical and computer engineering from Korea Uni- versity, Seoul, Republic of Korea, in August 2023. She received the B.S. degree in computer science and engineering from Chung-Ang University, Seoul, Republic of Korea, in February 2019. Her research focuses include deep learning applications, quantum machine learning algorithms and their software en- gineering methodologies, big-data platforms, and autonomous networking. She was a recipient of the IEEE Vehicular Technology Society (VTS) Seoul Chapter Award (2019), IEEE Seoul Section Student Paper Content Award (2020), and ICT Express (Elsevier) Best Reviewer Award (2021). Her current research interests include network optimization for autonomous vehicles communications, distributed system analysis, big-data processing platforms, and probabilistic access analysis. She received her B.S., M.S., and Ph.D. degrees in electrical and computer engineering from Ajou University. Irvine, CA, USA; Seoul, Republic of Korea; Institute, Gwacheon, Republic of Korea; Suwon, Republic of KoreaKTRDepartment of Electrical of Computer Engineering, Ajou University, Suwon, Republic of Korea ; Before joining Ajou University, she was an assistant professor at Hallym University, Chuncheon, Republic of Korea ; Korea UniversityKorea Testing and Research. ICT Paper Contest Award by Electronic Times (2019), and IEEE ICOIN Best Paper AwardSoyi Jung (Member, IEEE) has been an assistant professor at the Department of Electrical of Com- puter Engineering, Ajou University, Suwon, Repub- lic of Korea, since September 2022. Before joining Ajou University, she was an assistant professor at Hallym University, Chuncheon, Republic of Korea, from 2021 to 2022; a visiting scholar at Donald Bren School of Information and Computer Sciences, Uni- versity of California, Irvine, CA, USA, from 2021 to 2022; a research professor at Korea University, Seoul, Republic of Korea, in 2021; and a researcher at Korea Testing and Research (KTR) Institute, Gwacheon, Republic of Korea, from 2015 to 2016. She received her B.S., M.S., and Ph.D. degrees in electrical and computer engineering from Ajou University, Suwon, Republic of Korea, in 2013, 2015, and 2021, respectively. Her current research interests include network optimization for autonomous vehicles communications, distributed system analysis, big-data processing platforms, and probabilistic access analysis. She was a recipient of Best Paper Award by KICS (2015), Young Women Researcher Award by WISET and KICS (2015), Bronze Paper Award from IEEE Seoul Section Student Paper Contest (2018), ICT Paper Contest Award by Electronic Times (2019), and IEEE ICOIN Best Paper Award (2021). respectively; and the Ph.D. degree in computer science from the University of Southern California (USC). Joongheon Kim (M'06-SM'18) has been with Korea University, Seoul, Korea, since 2019, and he is currently an associate professor. He received the B.S. and M.S. degrees in computer science and engineering from Korea University. Seoul, Korea; Los Angeles, CA, USA; Seoul, Korea; San Diego, CA, USA; Santa Clara in Silicon Valley, CA, USA; Seoul, KoreaIEEE Communications Society (ComSocChung-Ang UniversityIEEE Systems CouncilJoongheon Kim (M'06-SM'18) has been with Ko- rea University, Seoul, Korea, since 2019, and he is currently an associate professor. He received the B.S. and M.S. degrees in computer science and engineering from Korea University, Seoul, Korea, in 2004 and 2006, respectively; and the Ph.D. degree in computer science from the University of Southern California (USC), Los Angeles, CA, USA, in 2014. Before joining Korea University, he was with LG Electronics (Seoul, Korea, 2006-2009), InterDigital (San Diego, CA, USA, 2012), Intel Corporation (Santa Clara in Silicon Valley, CA, USA, 2013-2016), and Chung-Ang University (Seoul, Korea, 2016-2019). He serves as an editor for IEEE TRANSACTIONS ON VEHICULAR TECH- NOLOGY, IEEE TRANSACTIONS ON MACHINE LEARNING IN COMMUNICA- TIONS AND NETWORKING, IEEE COMMUNICATIONS STANDARDS MAGA- ZINE, Computer Networks (Elsevier), and ICT Express (Elsevier). He is also a distinguished lecturer for IEEE Communications Society (ComSoc) (2022- 2023) and IEEE Systems Council (2022-2024). Intel Corporation Next Generation and Standards (NGS) Division Recognition Award. IEEE ComSoc Multimedia Communications Technical Committee (MMTC) Outstanding Young Researcher Award. IEEEHe was a recipient of Annenberg Graduate Fellowship with his Ph.D. admission from USCHe was a recipient of Annenberg Graduate Fellowship with his Ph.D. admission from USC (2009), Intel Corporation Next Generation and Standards (NGS) Division Recognition Award (2015), IEEE SYSTEMS JOURNAL Best Paper Award (2020), IEEE ComSoc Multimedia Communications Technical Committee (MMTC) Outstanding Young Researcher Award (2020), IEEE He also received numerous awards from IEEE conferences including. Ict Express, IEEE Vehicular Technology Society (VTS) Seoul Chapter Awards for APWCS. ElsevierBest Editor AwardIEEE ICTC Best Paper AwardComSoc MMTC Best Journal Paper Award (2021), ICT Express (Elsevier) Best Special Issue Guest Editor Award (2022), and ICT Express (Elsevier) Best Editor Award (2023). He also received numerous awards from IEEE conferences including IEEE ICOIN Best Paper Award (2021), IEEE Vehicular Technology Society (VTS) Seoul Chapter Awards for APWCS (2019, 2021, 2022), and IEEE ICTC Best Paper Award (2022).

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{'abstract': 'The development of urban-air-mobility (UAM) is rapidly progressing with spurs, and the demand for efficient transportation management systems is a rising need due to the multifaceted environmental uncertainties. Thus, this paper proposes a novel air transportation service management algorithm based on multi-agent deep reinforcement learning (MADRL) to address the challenges of multi-UAM cooperation. Specifically, the proposed algorithm in this paper is based on communication network (CommNet) method utilizing centralized training and distributed execution (CTDE) in multiple UAMs for providing efficient air transportation services to passengers collaboratively. Furthermore, this paper adopts actual vertiport maps and UAM specifications for constructing realistic air transportation networks. By evaluating the performance of the proposed algorithm in data-intensive simulations, the results show that the proposed algorithm outperforms existing approaches in terms of air transportation service quality. Furthermore, there are no inferior UAMs by utilizing parameter sharing in CommNet and a centralized critic network in CTDE. Therefore, it can be confirmed that the research results in this paper can provide a promising solution for autonomous air transportation management systems in city-wide urban areas.Index Terms-Urban-Air-Mobility (UAM), Air transportation service, Multi-agent deep reinforcement learning (MADRL), Centralized training and distributed execution (CTDE) Preliminary version of this paper was accepted', 'arxivid': '2306.04137', 'author': ['Chanyoung Park ', 'Gyu Seon Kim ', 'Soohyun Park ', 'Member, IEEESoyi Jung ', 'Senior Member, IEEEJoongheon Kim '], 'authoraffiliation': [], 'corpusid': 259095850, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 23514, 'n_tokens_neox': 20630, 'n_words': 12597, 'pdfsha': '48f0ec7ab56b309faf205d2a4054b54f52959d6f', 'pdfurls': ['https://export.arxiv.org/pdf/2306.04137v1.pdf'], 'title': ['Multi-Agent Reinforcement Learning for Cooperative Air Transportation Services in City-Wide Autonomous Urban Air Mobility', 'Multi-Agent Reinforcement Learning for Cooperative Air Transportation Services in City-Wide Autonomous Urban Air Mobility'], 'venue': []}

arxiv

Generalised shape theory via pseudo-Wishart distribution 16 Sep 2010 José A Díaz-García Francisco J Caro-Lopera Department of Statistics and Computation Department of Basic Sciences Universidad Autónoma Agraria Antonio Narro 25350Buenavista, SaltilloCoahuilaMéxico Universidad de Medellín MedellínColombia Generalised shape theory via pseudo-Wishart distribution 16 Sep 2010 The non isotropic noncentral elliptical shape distributions via pseudo-Wishart distribution are founded. This way, the classical shape theory is extended to non isotropic case and the normality assumption is replaced by assuming a elliptical distribution. In several cases, the new shape distributions are easily computable and then the inference procedure can be studied under exact densities. An application in Biology is studied under the classical gaussian approach and two non gaussian models. Introduction With the introduction of several innovative statistical and mathematical tools for highdimensional data analysis, now the classical multivariate analysis have a new and modern image. Developments as generalised multivariate analysis, latent variable analysis, DNA microarray data, pattern recognition, multivariate analysis nonlinear, data mining, manifold learning, shape theory, etc., open a range of potential applications in many areas of the knowledge. As consequence of these new statistical and mathematical tools a new theory can be considere from the conjunction between generalised multivariate analysis and the statistical shape theory is termed Generalised Shape Theory, in which the methodology developed in the shape theory under Gaussian models is extended to a general class of distributions, the elliptically contoured densities. γ : K × 1 (the translation), 1 N : N × 1, 1 N = (1, 1, . . . , 1) ′ , and β > 0 (the scale). Thus, in this context, the shape of a matrix X is all the geometrical information about X that is invariant under Euclidean similarity transformations, see Goodall and Mardia (1993) and Dryden and Mardia (1998). In the classical statistical shape theory is assumed that X has the isotropic matrix multivariate Gaussian distribution with mean µ X , see Goodall and Mardia (1993), i.e. X ∼ N N ×K (µ X , σ 2 I N ⊗ I K ). In the context of the generalised shape theory, it is assumed that X ∼ E N ×K (µ X , Σ X ⊗ Θ, h). Thus, two fundamental extensions of classical shape theory are provided, namely: • The generalised theory assumes a matrix multivariate elliptical distribution for the landmark data instead of considering a matrix multivariate Gaussian distribution. • Also, the usual isotropic Gaussian condition is replaced by assuming a non isotropic elliptical model. Two important advantages are obtained: first, the errors are correlated among landmarks, this is considered with the introduction of Σ, a N ×N definite positive matrix; and second, the errors are correlated among coordinates of landmarks, this condition is noticed with the introduction of Θ, a K × K definite positive matrix. The shape coordinates denoted as u of X can be constructed by several ways in terms of QR decomposition, see Goodall and Mardia (1993); and singular value decomposition (SVD), see Goodall (1991), Le and Kendall (1993) and Goodall and Mardia (1993). For example, in terms of the QR decomposition, shape coordinates u of X are constructed in several steps summarised in the expression LXΘ −1/2 = LZ = Y = TH = rWH = rW(u)H,(1) Observe that µ Z = µ X Θ −1/2 and the QR shape coordinates of µ Z are defined analogously. The matrix L : (N − 1) × N has orthonormal rows to 1 = (1, . . . , 1) ′ . L can be a submatrix of the Helmert matrix, for example. Now, let be n = min(N − 1, K) and p = rank µ. In (1), Y = TH is the QR decomposition, where T : (N − 1) × n is lower triangular with t ii > 0, i = 1, . . . , min(n, K − 1), and H : n × K, H ∈ V n,K = {H ∈ ℜ n×K |HH ′ = I n }, the Stiefel manifold. Note that T is invariant to translations and rotations of Z. The matrix T is referred as the QR size-and-shape and their elements are the QR size-and-shape coordinates of the original landmark data Z. Typically in shape analysis there are more landmarks than dimensions (N > K). H acts on the right to transform ℜ K instead of acting on the left as in the multivariate analysis. In our case we see the landmarks as variables and the dimensions as observations, then the transposes of our matrices Z and Y can be seen as classical multivariate data matrices. Now, if we divide T by its size, the centroid size of Z, r = T = √ tr T ′ T = Y . we obtain the so-termed QR shape matrix W in (1). Note that W = 1, the elements of W are a direction vector for shape, and u comprises m = (N − 1)K − nK + n(n + 1)/2 − 1 generalised polar coordinates. Observe that, if Θ 1/2 is the positive definite square root of the matrix Θ, i .e. Θ = (Θ 1/2 ) 2 , with Θ 1/2 : K × K, Gupta and Varga (1993, p. 11), and noting that Gupta and Varga (1993, p. 20). And we arrive at the classical starting point in shape theory where the original landmark matrix is replaced by Z = XΘ −1/2 . Then we can proceed as usual, removing from Z, translation, scale, rotation and/or reflection in order to obtain the shape of Z (or X) via the QR decomposition, for example. XΘ −1 X ′ = X(Θ 1/2 Θ 1/2 ) −1 X ′ = XΘ −1/2 (XΘ −1/2 ) ′ = ZZ ′ , where Z = XΘ −1/2 , then Z ∼ E N ×K (µ Z , Σ X , I K , h) with µ Z = µ X Θ −1/2 , see Let be µ = Lµ X , then Y : (N − 1) × K is invariant to translations of the figure Z, and Y ∼ E N −1×K (µΘ −1/2 , Σ ⊗ I K , h), where Σ = LΣ X L ′ . As suggest Goodall and Mardia (1993), the density of YY ′ essentially is the refection size-and-shape distribution of Y, moreover, it is invariant to orientation and reflection. Recall that for a given Y : N −1×K, n = N −1 < K, then V = YY ′ has the noncentral Wishart distribution with respect to Lebesgue measure on the subspace of definite positive matrices V > 0. However, the density of V = YY ′ when, n ≥ K, exist on the (nK − K(K − 1)/2)dimensional manifold of rank-K positive semidefinite N − 1 × N − 1 matrices with K distinct positive eigenvalues, which is termed pseudo-Wishart distribution, see Uhlig (1994), Díaz-García and González-Farías (2005) and Díaz-García and Gutiérrez-Jáimez (2006). Therefore, alternatively to (1) we propose the following steeps for obtain the shape coordinates LXΘ −1/2 = LZ = Y ⇒ V = rW = rW(u),(2) where V = YY ′ and W = V/r, with r = ||V||. In this work the size and shape distribution for any elliptical model in terms of pseudo-Wishart distribution is derived in section 2. Then the shape density is obtained in section 3. The central case of the shape density is studied in section 4, and is established that the central QR reflection shape density is invariant under the elliptical family. Some particular shape densities are derived in section 5 in order to perform inference on exact distributions; i.e. a subfamily of shape distributions generated by Kotz distributions including the Gaussian is obtained and applied. Finally in section 6, two elements of that class (the Gaussian and a non Gaussian model) are applied to an existing publish data, the mouse vertebra study. Some test for detecting shape differences are gotten and the models are discriminated by the use of a dimension criterion such as the modified BIC * criterion. 2 Pseudo-Wishart size-and-shape distribution Let V = YY ′ . In general (n = N − 1 < K or n ≥ K), the matrix V can be written as V ≡    V 11 n×n V 12 n×(N −1)−n V 21 (N −1)−n×n V 22 (N −1)−n×(N −1)−n    with rank of V 11 = n, such that, the number of mathematically independent elements in V are m = (N − 1)K − nK + n(n + 1)/2 corresponding to the mathematically independent elements in V 11 > 0 if n = N − 1 < K or to the mathematically independent elements of V 12 , and V 11 > 0 if n ≥ K. Recall that V 11 > 0, in such a way that V 11 has n(n + 1)/2 mathematically independent elements, therefore, (dV) ≡            (dV 11 ) = n i≤j dv ij , if n = N − 1 < K; (dV 11 ) ∧ (dV 12 ) = n i=1 (N −1) j=i dv ij , if n ≥ K. Formally, the measure (dV) is the Hausdorff measure defined on subspace of positive semidefinite matrices, see Billingsley (1986), Díaz-García and , and Díaz-García and González-Farías (2005). Explicit forms for (dV) can be obtained under diverse factorisations of the measure (dV). For example, by using the Cholesky decomposition V = TT ′ , where T : (N − 1) × n is lower triangular with t ii > 0, i = 1, . . . , min(n, K − 1) (dV) = 2 n n i=1 t N −i ii (dT).(3) Alternatively, under the nonsingular part of the spectral decompositions V = W ′ 1 DW 1 , W 1 ∈ V n,N −1 and D = diag(d 1 , . . . , d n ), d 1 > · · · > d n > 0 , then (dV) = 2 −n |D| N −1−n n i<j (d i − d j )(dD)(W ′ 1 dW 1 ).(4) Alternative explicit form for (dV) are given in Díaz-García and González-Farías (2005). Theorem 2.1. The pseudo-Wishart size-and-shape density is dF V (V) = π nK/2 |V * | (K−N )/2 Γ n [K/2] |Σ| K/2 ∞ t=0 κ h (2t) [tr(Σ −1 V + Ω)] 1 2 K κ C κ (ΩΣ −1 V) t! (dV),(5) where (dV) is defined in (3) or (4) (among many others), Ω = Σ −1 µΘ −1 µ ′ , C κ (B) are the zonal polynomials of B corresponding to the partition κ = (t 1 , . . . t α ) of t, with α i=1 t i = t; and (a) κ = i=1 (a − (j − 1)/2) tj , (a) t = a(a + 1) · · · (a + t − 1) , are the generalized hypergeometric coefficients and Γ s (a) = π s(s−1)/4 s j=1 Γ(a − (j − 1)/2) is the multivariate Gamma function, see James (1964) and Muirhead (1982). And h (j) (v) is the j-th derivative of h with respect to v. The matrix V * is given as, V * = V 11 , under Cholesky decomposition; D, under spectral decomposition. Proof. See Díaz-García and González-Farías (2005). Observe that the density functions (5) with respect to corresponding Hausdorff measure (3) or (4) are not unique, moreover, the Hausdorff measures (3) or (4) are also not unique; however, from a practical point of view, for example, the maximum likelihood estimation of the unknown parameters is invariant under different choices of measures (3) or (4) and their corresponding density functions (5), see Khatri (1968, p. 275) and Rao (1973, p. 532). Pseudo-Wishart shape distribution Observe that for V : N − 1 × N − 1, of rank n = min(N − 1, K), hence the matrix V contains (N − 1)K − nK + n(n + 1)/2 mathematically independent pseudo-Wishart coordinates (v ij ). Let vecw V a vector consisting of mathematically independent elements of V, taken column by column. Then the pseudo-Wishart shape matrix W can be written as vecw W = 1 r vecw V, r = ||V|| = √ tr V 2 = tr(Y ′ Y) 2 , then by Muirhead (1982, Theorem 2.1.3, p.55), (d vecw V) = r m m i=1 sin m−i θ i m i=1 dθ i ∧ dr, with m = (N − 1)K − nK + n(n + 1)/2 − 1. Denoting u = (θ 1 , . . . , θ m ) ′ , (du) = m i=1 dθ i and J(u) = m i=1 sin m−i θ i , with r > 0, 0 < θ i ≤ π (i = 1, . . . , m − 1), 0 < θ m ≤ 2π, then (dV) = r m J(u)(du) ∧ dr. Theorem 3.1. The pseudo-Wishart reflection shape density is dF W (W) = π nK/2 |W * | (K−N )/2 J(u) Γ n [K/2] |Σ| K/2 ∞ t=0 κ C κ (ΩΣ −1 W) t! 1 2 K κ × ∞ 0 r m−n(K−N )/2+t h (2t) [r tr Σ −1 W + tr Ω](dr)(du),(6) where W * = V * /r. Proof. The density of V is dF V (V) = π nK/2 |V * | (K−N )/2 Γ n [K/2] |Σ| K/2 ∞ t=0 κ h (2t) [tr(Σ −1 V + Ω)] 1 2 K κ C κ (ΩΣ −1 V) t! (dV). Making the change of variables W(u) = V/r, the joint density function of r and u is f r,W (r, W) = π nK/2 |rW * | (K−N )/2 Γ n [K/2] |Σ| K/2 ∞ t=0 κ h (2t) [tr(rΣ −1 W + Ω)] 1 2 K κ × C κ (rΩΣ −1 W) t! r m J(u)dr ∧ (du). Now, note that • C κ (rΩΣ −1 W) = r t C κ (ΩΣ −1 W). • |rW * | (K−N )/2 = r n(K−N )/2 |W * | (K−N )/2 . • h (2t) [tr(rΣ −1 W + Ω)] = h (2t) [r tr Σ −1 W + tr Ω]. Finally, collecting powers of r by r m+n(K−N )/2+t , the marginal of W is obtained integrating with respect to r. When Σ = σ 2 I, then Ω = µΘ −1 µ ′ /σ 2 , |Σ| K/2 = σ M , M = (N − 1)K and r tr Σ −1 W = r tr W/σ 2 , thus Theorem 3.1 becomes. Corollary 3.1. The isotropic pseudo-Wishart reflection shape density is dF W (W) = π nK/2 |W * | (K−N )/2 J(u) Γ n 1 2 K σ M ∞ t=0 κ C κ 1 σ 2 ΩW t! 1 2 K κ × ∞ 0 r m−n(K−N )/2+t h (2t) [r tr W/σ 2 + tr Ω](dr) ∧ (du).(7) 4 Central case The central case of the preceding sections can be derived easily. Corollary 4.1. The central pseudo-Wishart reflection size-and-shape density is given by dF V (V) = π nK/2 |V * | (K−N )/2 Γ n [K/2] |Σ| K/2 h[tr Σ −1 V](dV). Proof. It is straightforward from Theorem 2.1 just take µ = 0 and recall that h (0) [tr ·] = h[tr ·]. Similarly: Corollary 4.2. The central pseudo-Wishart reflection shape density is given by dF W (W) = π nK/2 |W * | (K−N )/2 J(u) Γ n [K/2] |Σ| K/2 ∞ 0 r m−n(K−N )/2 h[r tr Σ −1 W](dr)(du). Proof. Just take µ = 0 and h (0) [tr ·] = h[tr ·] in Theorem 3.1. Observe that it is possible to obtain an invariant central shape density, i.e. the density does not depend on function h(·) Let h be the density generator of Y ∼ E N −1, K (0, I ⊗ I, h), i.e. f Y (Y) = h(tr YY ′ ), then by Fang and Zhang (1990, eq. 3.2.6, p.102), ∞ 0 r (N −1)K−1 h(r 2 )dr = Γ[(N − 1)K/2] 2π (N −1)K/2 . Taking s = r 2 with dr = ds/(2 √ s) ∞ 0 s (N −1)K/2−1 h(s)dr = Γ[(N − 1)K/2] π (N −1)K/2 . Hence, if s = (tr Σ −1 W)r, ds = (tr Σ −1 W)(dr), then ∞ 0 r m−n(K−N )/2 h[r tr Σ −1 W](dr) = ∞ 0 s (tr Σ −1 W) m−n(K−N )/2 h(s) ds (tr Σ −1 W) = (tr Σ −1 W) n(K−N )/2−m−1 ∞ 0 s (2m−n(K−N ))/2+1−1 h(s)ds = (tr Σ −1 W) n(K−N )/2−m−1 Γ[m − n(K − N )/2 + 1] π m−n(K−N )/2+1 . Thus: Corollary 4.3. When µ = 0 the pseudo-Wishart reflection shape density is invariant under the elliptical family and it is given by Corollary 4.4. When µ = 0 and Σ = σ 2 I the pseudo-Wishart reflection shape density is invariant under the elliptical family and it is given by dF W (W) = π nK−m+n(K−N )/2−1 Γ[m − n(K − N )/2 + 1] Γ n [K/2] |Σ| K/2 |W * | (K−N )/2 ×J(u)(tr Σ −1 W) n(K−N )/2−m−1 (du).dF W (W) = π nK−m+n(K−N )/2−1 Γ[m − n(K − N )/2 + 1] 2Γ n [K/2] (σ 2 ) n(K−N )/2+M/2−m−1 |W * | (K−N )/2 (tr W) n(K−N )/2−m−1 × J(u)(du). Some particular models Finally, we give explicit shapes densities for some elliptical models. The Kotz type I model is given by h(y) = R T −1+ K(N −1) 2 Γ K(N −1) 2 π K(N −1)/2 Γ T − 1 + K(N −1) 2 y T −1 exp{−Ry},(8) Then, the corresponding k-th derivative d k [y T −1 exp{−Ry}] dy k , is (−R) k y T −1 exp{Ry} 1 + k m=1 k m m−1 i=0 (T − 1 − i) (−Ry) −m .(9) It includes the Gaussian case, i.e. when T = 1 and R = 1/2, here the derivation is straightforward from the general density. The required derivative follows easily, it is, h (k) (y) = R M/2 π M/2 (−R) k exp(−Ry) and ∞ 0 r m−n(K−N)/2+t h (2t) [r tr Σ −1 W + tr Ω]dr = π −M/2 R −m+t+ 1 2 (−2+M +n(K−N)) (tr Σ −1 W) −1−m−t+n(K−N)/2 × etr (−RΩ) Γ 1 + m + t + 1 2 n(−K + N ) . So, we have proved that Corollary 5.1. The Kotz type I (T = 1) Pseudo-Wishart reflection shape density is dF W (W) = π (nK−M)/2 |W * | (K−N )/2 J(u) etr (−RΩ) R m− 1 2 (−2+M+n(K−N )) Γ n [K/2] |Σ| K/2 × ∞ t=0 Γ [1 + m + t + n(−K + N )/2] t!(tr Σ −1 W) 1+m+t−n(K−N )/2 κ C κ (RΩΣ −1 W) 1 2 K κ . where M = (N − 1)K. Finally, for the Kotz type I model (8) and the given 2t-th derivative, we can prove easily that Corollary 5.2. The pseudo-Wishart reflection shape density based on the Kotz type I model is given by dF W (W) = π nK/2 |W * | (K−N )/2 J(u) Γ n [K/2] |Σ| K/2 ∞ t=0 κ C κ (ΩΣ −1 W) t! 1 2 K κ I(W(u), r) (du)(10) where This density seems uncomputable but it easy to see that it has the form of a generalised hypergeometric functions (see next section). These series can be determined by suitable modifications of the algorithms given by Koev and Edelman (2006) for 0 F 1 and at the same computational costs. Moreover, if the parameter T > 0 is an integer, the series are simplified substantially. For example, we can prove that the shape density associated to a Kotz model with T = 3, R = 1/2 and the isotropic assumption (Σ = σ 2 I N −1 and Θ = I K ), is given by: I(W(u), r) = ∞ 0 r m−n(K−N)/2+t h (2t) [r tr Σ −1 W + tr Ω](dr)(du) = G e −RB A −a−1 ∞ u=0 (u!) −1 R 2t−1−a−u B T −1−u Γ[1 + a + u] u−1 s=0 (T − 1 − s) + 2t v=1 2t v v−1 i=0 (T − 1 − i) ∞ u=0 (u!) −1 (−1) −v R 2t−1−a−u−v B T −1−u−v ×Γ[1 + a + u]dFW(W) = π (nK−M )/2 |W * | (K−N)/2 J(u) etr −µ ′ µ/2σ 2 2 −3−m+(M +n(K−N))/2 M (M + 2)Γn [K/2] (11) × ∞ t=0 [(B − 2t) 2 − 2t]Γ [a] + 2(B − 2t)Γ [a + 1] + Γ [a + 2] t!σ M −2−2m+n(K−N) (tr W) a κ Cκ( 1 2σ 2 µ ′ Wµ) 1 2 K κ . where M = (N − 1)K, B = tr µ ′ µ/2σ 2 and a = 1 + m + t + n(−K + N )/2. Other examples shall be considered in the next section, when T = 1 and T = 2. More complex densities in the context of affine shape theory were computed by using the same idea, see Caro-Lopera et al. Example This problem is studied in detail by Dryden and Mardia (1998) under a number of approaches (see also Mardia and Dryden (1989)). The experiment considers the second thoracic vertebra T2 of two groups of mice: large and small. The mice are selected and classified according to large or small body weight, respectively; in this case, the sample consists of 23 large and small bones (the data can be found in Dryden and Mardia (1998, p. 313-316)). It is of interest to study shape differences between the two groups. The vertebras are digitised and summarised in six mathematical landmarks which are placed at points of high curvature, see figure 1; they are symmetrically selected by measuring the extreme positive and negative curvature of the bone. See Dryden and Mardia (1998) for more details. Here we study three models, the Gaussian shape, and two shape Kotz type I models with T = 2 and T = 3. First, the isotropic Gaussian shape density is obtained from corollary 5.1 when we set R = 1 2 , Σ = σ 2 I N −1 , Θ = I K , Ω = Σ −1 µΘ −1 µ ′ = σ −2 µµ ′ , namely Corollary 6.1. The Pseudo-Wishart reflection shape density based on the isotropic Gaussian is given by dFW(W) = π (nK−M )/2 |W * | (K−N)/2 J(u) etr −µ ′ µ/2σ 2 2 −m+(−2+M +n(K−N))/2 Γn [K/2] × ∞ t=0 Γ [1 + m + t + n(−K + N )/2] t!σ M −2−2m+n(K−N) (tr W) m−n(K−N)/2+t+1 κ Cκ( 1 2σ 2 µ ′ Wµ) 1 2 K κ . where M = (N − 1)K. A second shape distribution that we will use follows from corollary 5.2 by taking R = 1/2, T = 2, i.e. dF W (W) = π (nK−M)/2 |W * | (K−N )/2 J(u) etr −µ ′ µ/2σ 2 2 −2−m+(M+n(K−N ))/2 M Γ n [K/2] × ∞ t=0 (B − 2t)Γ [a] + Γ [a + 1] t!σ M−2−2m+n(K−N ) (tr W) a κ C κ ( 1 2σ 2 µ ′ Wµ) 1 2 K κ . where M = (N − 1)K, B = tr µ ′ µ/2σ 2 and a = 1 + m + t + n(−K + N )/2. And the third shape model of this example corresponds to the isotropic Kotz distribution with T = 3 and R = 1/2, see (11). In order to select the best elliptical model, a number of dimension criteria have been proposed. We shall consider a modification of the BIC * statistic as discussed in Yang and Yang (2007), and which was first achieved by Rissanen (1978) in a coding theory framework. The modified BIC * is given by: BIC * = −2L( µ, σ 2 , h) + n p (log(n + 2) − log 24), where L( µ, σ 2 , h) is the maximum of the log-likelihood function, n is the sample size and n p is the number of parameters to be estimated for each particular shape density. Now, if the goal of the shape analysis searches the best elliptical distribution, among a set of proposed models, the modified BIC * criterion suggests to choose the model for which the modified BIC * receives its smallest value. In addition, as proposed by Kass and Raftery (1995) and Raftery (1995), the following selection criteria have been employed in order to compare two contiguous models in terms of its corresponding modified BIC * . X ∼ E N ×K (µ X , Σ X ⊗ Θ, h), where µ = Lµ X , then Y ∼ E N −1×K (µΘ −1/2 , Σ ⊗ I K , h), with Σ = LΣ X L ′ . In the mouse vertebra experiment, we want to find the maximum likelihood estimators (MLE) of the mean shape µ =       µ 11 µ 12 µ 21 µ 22 µ 31 µ 32 µ 41 µ 42 µ 51 µ 52       , and the scale parameter σ 2 defined in the isotropy assumption Θ = I K and Σ = σ 2 I N −1 , (in order to accelerate the computations of this example we fix the variance of the process as 50 -the maximum median variance of the two samples-). This optimisation is applied in the two independent populations, the small and large groups; first by assuming a Gaussian model and afterwards by considering two Kotz models indexed by T = 2 and T = 3. The general procedure is the following: Let L( µ, σ 2 , h) be the log likelihood function of a given group-model. The maximisation of the likelihood function L( µ, σ 2 , h), is obtained in this paper by using the Nelder-Mead Simplex Method, which is an unconstrained multivariable function using a derivative-free method; specifically, we apply the routine fminsearch implemented by the sofware MatLab. As the reader can check, the shape densities are series of zonal polynomials of the form ∞ t=0 f (t, tr X) t! κ C κ (X) (a) κ ,(12) which has hypergeometric series ∞ t=0 1 t! κ C κ (X) (a) κ , as a particular case; these series were non computable for decades. The work of Koev and Edelman (2006) solved the problem and it let the computation of the hypergeometric series by truncation of the series until the coefficient for large degrees are zero under certain tolerance. The cited algorithm gives the coefficients of the series, then, we can modified the algorithm for hypergeometric series to compute the shape densities with the same computational costs, multiplying each coefficient of the series by the required function f (t, tr X). At this point the log likelihood can be computed, then we use fminsearch for the MLE's. The initial value for the algorithm is the sample mean of the elliptical matrix variables Y ∼ E N −1×K (µΘ −1/2 , Σ⊗I K , h). However, we need to deal with an open problem proposed by Koev and Edelman (2006), the relationship between the convergence and the truncation of the series. Concretely, how many terms we need to consider in the series (12) in order to reach some fixed tolerance for convergence. A numerical solution consists of optimising the log likelihood, by increasing the truncation until, the MLE's and the maximum of the function, reach an equilibrium, which depends on the standard accuracy and tolerance of the routine fminsearch. We tried the truncations 20, 40, 60, 80, 100, 110, 120, 140 and 160, and we note that after the truncation 120 the solutions stabilise. the maximum likelihood estimators for location parameters associated with the small and large groups under the Gaussian, Kotz T = 2 and Kotz T = 3 models, are summarized in tables 2-7, respectively. Tables also show the modified BIC * value, the number of iterations for obtaining the convergence and the time in seconds for each optimisation. The computations were performed with a processor Intel(R) Corel(TM)2 Duo CPU, E7400@2.80GHz, and 2,96GB of RAM. Figures 2 show the behavior of the maximum of the log likelihood when the number of iterations is increased. In this case we use a truncation of 160, and again, we note that the log likelihood is bounded for a very small number of iterations in each particular model. According to the modified BIC criterion, we can order the models in the large and small 0 A final comment, for any elliptical model we can obtain the SVD reflection model, however a nontrivial problem appears, the 2t-th derivative of the generator model, which can be seen as a partition theory problem. For the general case of a Kotz model (s = 1), and another models as Pearson II and VII, Bessel, Jensen-logistic, we can use formulae for these derivatives given by Caro-Lopera et al. (2009). The resulting densities have again a form of a generalised series of zonal polynomials which can be computed efficiently after some modification of existing works for hypergeometric series, see Koev and Edelman (2006), thus the inference over an exact density can be performed, avoiding the use of any asymptotic distribution, and the initial transformation avoids the invariant polynomials of Davis (1980), which at present are not computable for large degrees. Now, if Σ = σ 2 I, then (tr Σ −1 W) n(K−N )/2−m−1 = (1/σ 2 ) n(K−N )/2−m−1 (tr W) n(K−N )/2−m−1 , and |Σ| K/2 = (σ 2 ) M/2 , thus: ( T − 1 − v − s) , with M = (N − 1)K, G = π −M/2 R T −1+M/2 Γ[M/2]/Γ [T − 1 + M/2], A = tr Σ −1 W, B = tr Ω and a = m − n(K − N )/2 + t. Figure 1 : 1Mouse vertebra Table 1 : 1Grades of evidence corresponding to values of the BIC * difference.BIC * difference Evidence 0-2 Weak 2-6 Positive 6-10 Strong > 10 Very strong Now, recall that for a general density generator h(·) Table 2 : 2The maximum likelihood estimators for the small group under the Gaussian modelTrunc. µ11 µ12 µ21 µ22 µ31 µ32 µ41 20 30.40 -8.13 5.73 9.47 4.01 17.34 -2.70 40 -0.47 -44.69 15.04 -4.54 25.27 0.60 4.88 60 -2.10 -54.84 18.31 -6.09 31.03 -0.12 6.17 80 -0.70 -63.48 21.37 -6.46 35.89 0.83 6.94 100 -2.61 -71.03 23.73 -7.85 40.19 -0.10 7.98 110 -0.54 -74.58 25.13 -7.50 42.16 1.14 8.12 120 -3.41 -77.44 25.88 -8.71 43.94 -0.37 8.79 140 -3.41 -77.44 25.88 -8.71 43.94 -0.37 8.79 160 -3.41 -77.44 25.88 -8.71 43.94 -0.37 8.79 Trunc. µ42 µ51 µ52 BIC * Time Iter. 20 4.24 -6.82 -20.65 -3538.26 317 4103 40 5.21 -30.81 2.11 -4155.34 281 1881 60 6.23 -37.75 3.63 -4659.16 417 1923 80 7.40 -43.77 3.01 -5110.98 426 1455 100 8.08 -48.90 4.63 -5532.79 742 2025 110 8.72 -51.43 3.34 -5735.95 607 1507 120 8.76 -53.43 5.37 -5914.74 721 1640 140 8.76 -53.43 5.37 -5914.74 721 1640 160 8.76 -53.43 5.37 -5914.74 721 1640 Table 3 : 3The maximum likelihood estimators for the small group under the Kotz T = 2 modelTrunc. µ11 µ12 µ21 µ22 µ31 µ32 µ41 20 -6.06 -32.06 10.23 -5.19 18.24 -2.80 4.18 40 4.42 -46.00 15.97 -3.02 25.91 3.39 4.45 60 -0.53 -56.62 19.07 -5.73 32.01 0.80 6.18 80 -1.88 -65.36 21.89 -7.04 36.97 0.20 7.28 100 -0.04 -73.09 24.68 -7.18 41.31 1.40 7.90 110 -1.62 -76.63 25.72 -8.06 43.34 0.57 8.47 120 -1.84 -79.60 26.76 -8.39 45.13 0.56 8.83 140 -1.84 -79.60 26.76 -8.39 45.13 0.56 8.83 160 -1.84 -79.60 26.76 -8.39 45.13 0.56 8.83 Trunc. µ42 µ51 µ52 BIC * Time Iter. 20 3.12 -21.88 5.46 -3584.58 311 1957 40 5.89 -31.91 -1.22 -4203.67 627 2052 60 6.61 -39.04 2.62 -4709.27 890 1986 80 7.49 -45.02 3.90 -5162.67 951 1566 100 8.60 -50.43 2.94 -5585.92 1468 1978 110 8.84 -52.81 4.17 -5789.74 1160 1386 120 9.18 -54.98 4.37 -5969.11 1464 1656 140 9.18 -54.98 4.37 -5969.11 1464 1656 160 9.18 -54.98 4.37 -5969.11 1464 1656 Table 4 : 4The maximum likelihood estimators for the small group under the Kotz T = 3 modelTrunc. µ11 µ12 µ21 µ22 µ31 µ32 µ41 20 -2.37 -33.66 11.14 -4.10 19.07 -0.68 3.91 40 -10.75 -46.42 14.62 -8.18 26.44 -5.17 6.28 60 -0.29 -58.24 19.64 -5.81 32.92 0.97 6.32 80 -1.69 -67.11 22.50 -7.15 37.96 0.35 7.45 100 -1.31 -74.91 25.17 -7.79 42.36 0.72 8.24 110 -1.33 -78.51 26.38 -8.15 44.39 0.77 8.64 120 -1.75 -81.49 27.42 -8.54 46.20 0.65 9.03 140 -1.75 -81.49 27.42 -8.54 46.20 0.65 9.03 160 -1.75 -81.49 27.42 -8.54 46.20 0.65 9.03 Trunc. µ42 µ51 µ52 BIC * Time Iter. 20 3.71 -23.13 2.97 -3625.80 101 2083 40 4.30 -31.60 9.27 -4247.02 185 2067 60 6.82 -40.17 2.52 -4754.54 273 2050 80 7.72 -46.23 3.84 -5209.68 322 1816 100 8.68 -51.63 3.88 -5634.52 329 1449 110 9.10 -54.11 4.05 -5839.09 440 1776 120 9.41 -56.30 4.38 -6019.10 461 1688 140 9.41 -56.30 4.38 -6019.10 461 1688 160 9.41 -56.30 4.38 -6019.10 461 1688 Table 5 : 5The maximum likelihood estimators for the large group under the Gaussian modelTrunc. µ11 µ12 µ21 µ22 µ31 µ32 µ41 20 -19.04 -22.88 5.37 -8.26 15.82 -10.82 3.84 40 -29.85 -29.93 6.54 -12.38 20.99 -17.32 5.62 60 -15.90 -49.41 14.07 -9.87 32.64 -7.20 5.30 80 -41.34 -43.55 9.73 -17.35 30.42 -23.86 7.93 100 -66.69 -8.40 -3.88 -21.92 9.43 -42.24 8.53 110 -40.91 -57.46 14.17 -18.57 39.32 -22.74 8.83 120 -32.30 -65.98 17.67 -16.67 44.17 -16.68 8.38 140 -32.30 -65.98 17.67 -16.67 44.17 -16.68 8.38 160 -32.30 -65.98 17.67 -16.67 44.17 -16.68 8.38 Trunc. µ42 µ51 µ52 BIC * Time Iter. 20 1.42 -18.06 15.31 -3540.51 155 2075 40 1.52 -23.59 23.97 -4159.47 259 1824 60 4.80 -39.19 13.02 -4665.18 274 1295 80 2.35 -34.35 33.21 -5118.90 300 1044 100 -3.59 -6.20 53.12 -5542.62 978 2753 110 4.04 -45.42 32.97 -5746.72 449 1143 120 5.65 -52.16 26.15 -5926.64 509 1172 140 5.65 -52.16 26.15 -5926.64 509 1172 160 5.65 -52.16 26.15 -5926.64 509 1172 Table 6 : 6The maximum likelihood estimators for the large group under the Kotz T = 2 modelTrunc. µ11 µ12 µ21 µ22 µ31 µ32 µ41 20 -21.67 -21.97 4.83 -9.01 15.40 -12.56 4.10 40 -36.15 -24.57 4.23 -13.84 17.94 -21.68 6.00 60 -32.77 -42.35 10.19 -14.52 29.14 -18.44 6.82 80 -31.52 -53.20 13.74 -15.19 36.02 -16.98 7.42 100 -31.91 -61.32 16.28 -16.10 41.25 -16.74 8.02 110 -38.06 -61.70 15.79 -18.09 41.86 -20.66 8.78 120 -42.11 -62.61 15.65 -19.44 42.61 -23.16 9.32 140 -42.11 -62.61 15.65 -19.44 42.61 -23.16 9.32 160 -42.11 -62.61 15.65 -19.44 42.61 -23.16 9.32 Trunc. µ42 µ51 µ52 BIC * Time Iter. 20 1.13 -17.32 17.40 -3586.83 339 2165 40 0.44 -19.28 28.94 -4207.80 476 1594 60 2.80 -33.46 26.38 -4715.29 519 1133 80 4.18 -42.10 25.47 -5170.59 706 1144 100 5.12 -48.56 25.83 -5595.75 1143 1528 110 4.74 -48.81 30.73 -5800.52 1105 1349 120 4.58 -49.41 33.92 -5981.01 1320 1459 140 4.58 -49.41 33.92 -5981.01 1320 1459 160 4.58 -49.41 33.92 -5981.01 1320 1459 Table 7 : 7The maximum likelihood estimators for the large group under the Kotz T = 3 model 24.01 -49.58 13.33 -12.45 33.24 -12.39 6.27 80 -32.14 -54.75 14.17 -15.53 37.05 -17.29 7.60 100 -41.95 -57.11 13.96 -18.87 39.15 -23.43 8.93 110 -35.24 -65.37 17.23 -17.55 44.04 -18.Trunc. µ11 µ12 µ21 µ22 µ31 µ32 µ41 20 -31.74 3.30 -4.16 -9.72 -0.19 -20.55 3.56 40 -41.25 -18.15 1.70 -14.83 14.14 -25.34 6.17 60 -63 8.70 120 -39.44 -66.43 17.11 -18.97 44.89 -21.22 9.27 140 -39.44 -66.43 17.11 -18.97 44.89 -21.22 9.27 160 -39.44 -66.43 17.11 -18.97 44.89 -21.22 9.27 Trunc. µ42 µ51 µ52 BIC * Time Iter. 20 -2.58 2.86 25.23 -3628.05 73 1494 40 -0.67 -14.14 32.95 -4251.15 125 1411 60 4.26 -39.27 19.47 -4760.57 163 1239 80 4.32 -43.32 25.97 -5217.61 197 1110 100 3.93 -45.13 33.79 -5644.35 299 1345 110 5.37 -51.75 28.52 -5849.87 304 1246 120 5.21 -52.46 31.83 -6031.00 387 1457 140 5.21 -52.46 31.83 -6031.00 387 1457 160 5.21 -52.46 31.83 -6031.00 387 1457 AcknowledgmentsThis research work was supported by University of Medellin (Medellin, Colombia) and Universidad Autónoma Agraria Antonio Narro (México), joint grant No. 469, SUMMA group. Also, the first author was partially supported by IDI-Spain, Grants No. FQM2006-2271 and MTM2008-05785 and the paper was written during J. A. Díaz-García's stay as a visiting professor at the Department of Statistics and O. R. of the University of Granada, Spain. Finally, F. Caro thanks to the project No. 105657 of CONACYT, México.This order can be seen infigure 3, which compares the log-likelihood of the two groups under the three models in terms of the algorithm iteration when the truncation is set in 160.Modified BIC * of both groups shows a very strong difference (seetable 1) between the best model (1) and the classical Gaussian (3).In both cases, the true models of the data maybe have tails that are weighted more or less than Gaussian model or that the shape distribution present grater or smaller degree of kurtosis than the Gaussian model.Remark 6.1. We have used this example from the literature to illustrate the generalised shape theory; moreover, based on the modified BIC * , we found that the Kotz distribution (with T = 3) is the best model in this experiment. However, suppose the expert in the area of application knows that the landmarks have a Gaussian distribution, then we must apply the classical theory of shape (based on normality). Alternatively, if the expert in the application area suspects that the landmarks do not have a Gaussian distribution, so we can apply the generalized theory proposed here. In this case the expert has the necessary tools to choose an elliptical model (as an alternative to the Gaussian distribution), according to the characteristic of the sample which reveal and/or support a non Gaussian distribution, i.e. to select a distribution with more or less heavy tails, or more or less kurtosis than the Gaussian density; among many others possible characteristics.Once the best models are selected for the small and large groups, we can test equality in mean shape between the two independent populations. In this experiment we have: two independent samples of 23 bones and 10 population shape parameters to estimate for each group. Namely, if L(µ s , µ l ) is the likelihood, where µ s , µ l , represent the mean shape parameters of the small and large group, respectively, then we want to test: H 0 : µ s = µ l vs H a : µ s = µ l . Then −2 log Λ = 2 sup H1 log L(µ s , µ l ) − 2 sup H0 log L(µ s , µ l ), and according to Wilk's theorem −2 log Λ ∼ χ 2 10 under H 0 . Using fminsearch with a truncation of 160 we obtained that: −2 log Λ = 2(3999.1273) − 2(3990.3601) = 17.5344, this is the same result when the series were truncated at 120 and 140. Since the p-value for the test is P (χ 2 10 ≥ 17.5344) = 0.0633 we have some evidence that the small and large mouse vertebrae are different in mean shape.Mardia and Dryden (1989)studied this problem with a Gaussian model and Bookstein coordinates (see alsoDryden and Mardia (1998)) and they obtained for the same test an approximate p-value of zero (P (χ 2 8 ≥ 127.75). Our test also rejects the equality of mean shape based on a better non Gaussian model but without an strong evidence as the Gaussian model suggests.Note that the MLE's given by tables 2-4 correspond to the matrix µ in Y ∼ E N −1×K (µ, Σ⊗ I K , h), we can use this information and the transformations(with V = YY ′ and W = V/r) to estimate the different means at each step, i.e.: the original elliptical mean µ X , the size-and-shape mean µ V and the shape mean µ W .This example deserves a detailed study about some important facts, i.e. the distribution of −2 log Λ for small samples, the truncation of the series, global optimisation methods, etc. These problems shall be considered in a subsequent work. . P References, Billingsley, Probability and Measure. John Wiley & SonsReferences P. Billingsley, Probability and Measure, John Wiley & Sons, New York, 1986. Noncentral elliptical configuration density. F J Caro-Lopera, J A Díaz-García, G González-Farías, J. Multivariate Anal. 1011F. J. Caro-Lopera, J. A. Díaz-García and G. González-Farías, Noncentral elliptical configu- ration density, J. Multivariate Anal. 101(1) (2009), 32-43. Invariant polynomials with two matrix arguments, extending the zonal polynomials. A W Davis, Multivariate Analysis V. Krishnaiah, P. R.North-HollandA. W. Davis, Invariant polynomials with two matrix arguments, extending the zonal poly- nomials, in: Multivariate Analysis V, (Krishnaiah, P. R. ed.), North-Holland, 1980. Proof of the conjectures of H. Uhlig on the singular multivariate beta and the jacobian of a certain matrix transformation. J A Díaz-García, R Gutiérrez-Jáimez, Ann. Statist. 25J. A. Díaz-García, and R. Gutiérrez-Jáimez, Proof of the conjectures of H. Uhlig on the singular multivariate beta and the jacobian of a certain matrix transformation, Ann. Statist., 25, (1997) 2018-2023. Wishart and Pseudo-Wishart distributions and some applications to shape theory. J A Díaz-García, R Gutiérrez-Jáimez, K V Mardia, J. Multivariate Anal. 63J. A. Díaz-García, R. Gutiérrez-Jáimez, and K. V. Mardia, Wishart and Pseudo-Wishart distributions and some applications to shape theory, J. Multivariate Anal. 63 (1997) 73-87. Singular random matrix decompositions: Distributions. J A Díaz-García, G González-Farías, J. Multivariate Anal. 1941J. A. Díaz-García and G. González-Farías, Singular random matrix decompositions: Distri- butions, J. Multivariate Anal. 194(1) (2005), 109-122. Wishart and Pseudo-Wishart distributions under elliptical laws and related distributions in the shape theory context. J A Díaz-García, R Gutiérrez-Jáimez, J. Stat. Plan. Inference. 13612J. A. Díaz-García and R. Gutiérrez-Jáimez, Wishart and Pseudo-Wishart distributions under elliptical laws and related distributions in the shape theory context, J. Stat. Plan. Inference 136(12) (2006), 4176-4193. Statistical shape analysis. I L Dryden, K V Mardia, John Wiley and SonsChichesterI. L. Dryden and K.V. Mardia, Statistical shape analysis, John Wiley and Sons, Chichester, 1998. K T Fang, Y T Zhang, Generalized Multivariate Analysis. BeijingSpringer-VerlagK. T. Fang, and Y. T. Zhang, Generalized Multivariate Analysis, Science Press, Springer- Verlag, Beijing, 1990. Procustes methods in the statistical analysis of shape (with discussion). C G Goodall, J. Roy. Statist. Soc. Ser. B. 53C. G. Goodall, Procustes methods in the statistical analysis of shape (with discussion), J. Roy. Statist. Soc. Ser. B, 53 (1991) 285-339. Multivariate Aspects of Shape Theory. C R Goodall, K V Mardia, Ann. Statist. 21C. R. Goodall, and K. V. Mardia, Multivariate Aspects of Shape Theory, Ann. Statist. 21 (1993) 848-866. Elliptically Contoured Models in Statistics. A K Gupta, T Varga, Kluwer Academic PublishersDordrechtA. K. Gupta, and T. Varga, Elliptically Contoured Models in Statistics, Kluwer Academic Publishers, Dordrecht, 1993. Distributions of matrix variate and latent roots derived from normal samples. A T James, Ann. Math. Statist. 35A. T. James, Distributions of matrix variate and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964) 475-501. Bayes factor. R E Kass, A E Raftery, J. Amer. Statist. Soc. 90R. E. Kass, and A. E. Raftery, Bayes factor, J. Amer. Statist. Soc. 90 (1995) 773-795. Some results for the singular normal multivariate regression models, Sankhyā A. C G Khatri, 30C. G. Khatri, Some results for the singular normal multivariate regression models, Sankhyā A 30 (1968) 267-280. The efficient evaluation of the hypergeometric function of a matrix argument. P Koev, A Edelman, Math. Comp. 75P. Koev and A. Edelman, The efficient evaluation of the hypergeometric function of a matrix argument, Math. Comp. 75 (2006) 833-846. The Riemannian structure of Euclidean spaces: a novel environment for statistics. H L Le, D G Kendall, Ann.Statist. 21H. L. Le, and D. G. Kendall, The Riemannian structure of Euclidean spaces: a novel envi- ronment for statistics, Ann.Statist. 21 (1993) 1225-1271. The Statistical Analysis of Shape Data. K V Mardia, I L Dryden, Biometrika. 762K. V. Mardia and I. L. Dryden, The Statistical Analysis of Shape Data, Biometrika, 76(2) (1989) 271-281 Aspects of multivariate statistical theory. R J Muirhead, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, IncR. J. Muirhead, Aspects of multivariate statistical theory, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc. 1982. Bayesian model selection in social research. A E Raftery, Sociological Methodology. 25A. E. Raftery, Bayesian model selection in social research, Sociological Methodology, 25 (1995) 111-163. C R Rao, Linear Statistical Inference and its Applications. New YorkJohn Wiley & Sons2nd ed.C. R. Rao, Linear Statistical Inference and its Applications (2nd ed.), John Wiley & Sons, New York, 1973. Modelling by shortest data description. J Rissanen, Automatica. 14J. Rissanen, Modelling by shortest data description, Automatica, 14 (1978) 465-471. On singular Wishart and singular multivariate Beta distributions. H Uhlig, Ann. Statist. 22H. Uhlig, On singular Wishart and singular multivariate Beta distributions, Ann. Statist. 22 (1994) 395-405. Separating latent classes by information criteria. Ch, Ch, Ch Yang, Ch, Yang, J. Classification. 24Ch. Ch. Yang and Ch. Ch. Yang, Separating latent classes by information criteria, J. Clas- sification 24 (2007) 183-203.

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{'abstract': 'The non isotropic noncentral elliptical shape distributions via pseudo-Wishart distribution are founded. This way, the classical shape theory is extended to non isotropic case and the normality assumption is replaced by assuming a elliptical distribution. In several cases, the new shape distributions are easily computable and then the inference procedure can be studied under exact densities. An application in Biology is studied under the classical gaussian approach and two non gaussian models.', 'arxivid': '1009.3168', 'author': ['José A Díaz-García ', 'Francisco J Caro-Lopera ', '\nDepartment of Statistics and Computation\nDepartment of Basic Sciences\nUniversidad Autónoma Agraria Antonio Narro\n25350Buenavista, SaltilloCoahuilaMéxico\n', '\nUniversidad de Medellín\nMedellínColombia\n'], 'authoraffiliation': ['Department of Statistics and Computation\nDepartment of Basic Sciences\nUniversidad Autónoma Agraria Antonio Narro\n25350Buenavista, SaltilloCoahuilaMéxico', 'Universidad de Medellín\nMedellínColombia'], 'corpusid': 88512607, 'doi': '10.1007/s13171-013-0024-1', 'github_urls': [], 'n_tokens_mistral': 17997, 'n_tokens_neox': 14754, 'n_words': 7106, 'pdfsha': '22ef000767b6429dda19c7c7ffff695501c68586', 'pdfurls': ['https://arxiv.org/pdf/1009.3168v1.pdf'], 'title': ['Generalised shape theory via pseudo-Wishart distribution', 'Generalised shape theory via pseudo-Wishart distribution'], 'venue': []}

arxiv

Introduction to dynamical mean-field theory of generic random neural networks May 17, 2023 Wenxuan Zou School of Physics PMI Lab Sun Yat-sen University 510275GuangzhouPeople's Republic of China Haiping Huang huanghp7@mail.sysu.edu.cn School of Physics PMI Lab Sun Yat-sen University 510275GuangzhouPeople's Republic of China Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices Sun Yat-sen University 510275GuangzhouPeople's Republic of China Introduction to dynamical mean-field theory of generic random neural networks May 17, 2023SciPost Physics Lecture Notes Submission Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of generic random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.SciPost Physics Lecture NotesSubmission B Static cavity method for recurrent dynamics 19 C Stability analysis for the ReLU transfer function 21 D Derivation of fluctuation-dissipation theorem in equilibrium 22References 24 Introduction Stochastic differential equations (SDEs) have a very wide range of applications in physics [1], biology, neuroscience and machine learning (see many examples in understanding the brain [2]). Recently, as the world-wide brain projects are being promoted, and the artificial intelligence starts the fourth industrial revolution, understanding how cognition arises both for a natural brain or an artificial algorithm (like Chat GPT) becomes increasingly important. Better understanding leads to precise predictions, which is impossible without solid mathematical foundation. Stochastic noise inherent in the neural dynamics can either stem from the algorithm details (e.g, in stochastic gradient descent of a deep learning cost function [3], the noise is anisotropic and changes with the training) or stem from unreliable synaptic transmission and noisy background inputs [4]). Therefore, SDEs provide a standard tool to model the complex dynamics common in complex systems. In particular, one can write the SDE into a Langevin dynamics equation, and put the dynamics into the seminal Onsager-Machlup formalism [5], which introduces the concept of action in a pathintegral framework [6]. However, the form of the Onsager-Machlup action is not amenable for calculating the quenched disorder average as commonly required in studying typical behaviors of random neural networks. To overcome this drawback, a response field is introduced and thus the correlation (spontaneous fluctuation) as well as the response function can be easily derived (see below). This new field-theoretical approach is called the Martin-Sigga-Rose-De Dominics-Janssen (MSRDJ) formalism [7][8][9]. The MSRDJ formalism has been used to analyze the recurrent neural networks, e.g., studying the onset of chaos [10][11][12], and to analyze deep neural networks in recent years [13][14][15]. We will next provide a detailed explanation of this field-theoretical formalism. The MSRDJ formalism bears concise mathematics, resulting in the mean-field description of the original high dimensional stochastic dynamics. The same equation can also be derived using a physically more transparent method, namely the dynamical cavity method [16][17][18]. In essence, a neuron is added into the system, and the associated impacts on other neurons are seft-consistently derived using the linear response approximation. The dynamics of this additional neuron bears the same characteristics with other neurons, thereby being a representative of the original high dimensional dynamics. In addition, the static version of this method can be used to derive the fixed point solution of the dynamics [16,18]. By numerically solving the intergro-differential equations involving the correlation and response functions, one can further probe the fundamental fluctuation-dissipation relation in equilibrium dynamics, which we shall show in the last section of this tutorial. All technical details are given in the Appendices. Generic random neural networks We consider a random neural network composed of N fully-connected neurons. The state of each neuron in time t is characterized by the synaptic current x i (t), i = 1, . . . , N , which obeys the following non-linear dynamical equation, d x i (t) d t = −x i (t) + g N j=1 J i j φ j (t) + σξ i (t),(1) where g is the coupling strength and φ j (t) = φ x j (t) is the transfer function that transforms current to firing rate. A Gaussian white-noise ξ i (t) of zero mean and unit variance ξ i (t)ξ j (t ) = δ i j δ(t−t ) is introduced to model the stochastic nature of neural wiring (e.g., in cortical circuits [2]). The parameter σ serves as the noise strength. Each element J i j of the connection matrix is drawn from a Gaussian distribution with zero mean and variance J 2 i j = 1/N . In fact, J i j may correlate with J ji for each pair of neurons. This asymmetric correlation is thus characterized as follows, J i j J ji = η N ,(2) where η ∈ [−1, +1] describes the degree of asymmetry. In particular, the connections are fully symmetric when η = 1 and fully asymmetric when η = 0 [19]. Besides, we set J ii = 0 to remove all the self-coupling interaction. The dynamical mean-field theory (DMFT) equation of this generic model is easy to acquire when the asymmetric correlation is absent i.e., η = 0 [10,20]. In this case, when we consider the limit N → ∞, the input current each neuron receives via the coupling g N j=1 J i j φ j (t) converges to a Gaussian field according to the central limit theorem. Therefore, the complex network dynamics can be simplified to an effective single-neuron dynamics, x(t) = −x(t) + γ(t) + σξ(t),(3) where γ(t) is the effective Gaussian noise with covariance 〈γ(t)γ(t )〉 = g 2 C(t, t ) + σ 2 δ(t − t ). The auto-correlation of firing rates C(t, t ) is self-consistently defined as C(t, t ) = 〈φ(t)φ(t )〉,(4) thereby closing the DMFT equation. Note that · (or [·] in the following) and 〈·〉 represent the quenched-disorder average and thermal average (over different noise trajectories), respectively. However, when the asymmetric correlation between connections is present, the mean-field description can not be obtained directly from the central limit theorem. This is because in the summation of the afferent currents, a non-negligible correlation between J i j and φ j (t) emerges via J ji when η = 0, and thus the central limit theorem breaks down. In the following sections, we introduce two powerful physics tools to tackle this challenge and derive the DMFT equation for generic random neural networks of an arbitrary asymmetric correlation level. Dynamical mean-field theory Generating functional formalism We first introduce the generating functional formalism and then unfold the derivation following the standard procedure. The main idea of this method is to recast the original high-dimensional dynamical equation to the path integral formalism. Here, we consider the MSRDJ path integral. The moment-generating functional with a specific action corresponding to the dynamical equation can be written out explicitly, which is helpful to reduce the dynamics to a low-dimensional meanfield description. First, we add a perturbation j i (t) to the original dynamical equation [Eq. (1)], x i (t) = −x i (t) + g j=1 J i j φ j (t) + j i (t) + σξ i (t), i = 1, . . . , N ,(5) which will be useful in the following derivation. Then, we discretize the dynamical equations under the Ito convention [21], x i [t] − x i [t − 1] = −x i [t − 1]h + g j=1 J i j φ j [t − 1]h + j i [t − 1]h + σξ i [t],(6) where h is a time interval between two consecutive time steps, and [t] indicates the discrete time index. The white noise ξ i [t] = t t−h ξ i (s) ds becomes a Wiener process with the following statistics, ξ i [t]ξ j [t ] = t t−h t t −h ξ i (s)ξ j (s ) ds ds = t t−h t t −h σ 2 δ i j δ(s − s ) ds ds = δ i j δ t t σ 2 h,(7) where δ i j is a Kronecker delta function. Because of the Markovian property, we introduce the joint distribution of the currents {x (t)} T t=1 across time and space by Dirac delta functions, P {x (t)} T t=1 = i,t p(ξ i [t]) dξ i [t] δ x i [t + 1] − x i [t] + x i [t]h − g j=1 J i j φ j [t]h − j i [t]h − σξ i [t] ,(8) where the initial current state x [0] can be arbitrarily chosen, which does not influence the derivation, and T denotes the length of the trajectory. We next represent these delta functions by their Fourier integral as δ( x) = 1 2πi i∞ −i∞ dx e −x x , P {x (t)} T t=1 = i,t p (ξ i [t]) dξ i [t] i∞ −i∞ dx i [t] 2πi exp −x i [t] x i [t + 1] − x i [t] + x i [t]h − g j=1 J i j φ j [t]h − j i [t]h − σξ i [t] = i,t i∞ −i∞ dx i [t] 2πi exp −x i [t] x i [t + 1] − x i [t] + x i [t]h − g j=1 J i j φ j [t]h − j i [t]h − σ 2 2x i [t]h ,(9) where Eq. (7) is used to derive the last equality. Hence, we can formally define the momentgenerating functional of the stochastic dynamics, Z[j,j|J] = i,t ∞ −∞ dx i [t] exp j i [t]x i [t]h P {x (t)} T t=1 = i,t ∞ −∞ dx i [t] i∞ −i∞ dx i [t] 2πi exp j i [t]x i [t]h + j i [t]x i [t]h exp      −h i,tx i [t] x i [t + 1] − x i [t] h + x i [t] − g j=1 J i j φ j [t] − σ 2 2x i [t]      ,(10) wherej and j are two types of source fields, whose physical meaning would be clear below. The source j could be an external perturbation to which the response is measured by the response fieldx, which allows one to compute the linear response function by taking the correlation with x (see below). Taking the continuous limits of T → ∞ and h → 0 at the same time, we obtain h T t=0 f (t) = f (t) dt, and lim h→0 x i [t+1]−x i [t] h =ẋ i (t). We also introduce the notations i,t dx i [t] h→0 → Dx and i,t dx i [t] 2πi h→0 → Dx for simplicity. Under the continuous limit, the moment generating functional reads, Z[j,j|J] = Dx (t)Dx exp −S[x ,x |J] + N i=1 j i (t)x i (t)d t + N i=1 j i (t)x i (t)d t ,(11) where the action of the dynamical equation is naturally introduced as, S[x ,x |J] = N i=1 x i (t) ẋ i (t) + x i (t) − g N j=1 J i j φ j (t) − σ 2 2x i (t) d t.(12) It is easy to verify that when N → ∞, the out-of-equilibrium behavior is independent of the realization of the disorder [22]. We thus focus on the typical behavior of the self-averaging dynamical partition function Z [j,j|J]. This partition function is simpler compared to its equilibrium counterpart, as the zero source generating functional is identical to one. In the dynamical setting, taking the average of Z[j,j|J] over P(J) is sufficient to get the thermal and disorder averaged two-point functions, such as correlation and response. This is in contrast to the equilibrium spin glass theory where a replica trick is commonly applied to obtain the disorder anverage of the free energy function [20]. In fact, computing the average of J Z[j,j|J] reduces to computing J exp (−S[x ,x |J]). To proceed, we decompose the connection into symmetric and asymmetric parts [19], J i j = J s i j + kJ a i j ,(13) where J s i j = J s ji and J a i j = −J a ji , both of which follow the centered Gaussian distribution with the same variance, J s i j J s i j = J a i j J a i j = 1 N 1 1 + k 2 .(14) Under this decomposition, it is easy to derive that, J i j J i j = 1 N , J i j J ji = 1 N 1 − k 2 1 + k 2 ,(15) SciPost Physics Lecture Notes Submission which gives k 2 = (1 − η)/(1 + η). Now, we can deal with the term involving J i j , i = jx i (t)J i j φ j (t) = i = jx i (t) J s i j + kJ a i j φ j (t) = i< j J s i j x i (t)φ j (t) +x j (t)φ i (t) + kJ a i j x i (t)φ j (t) −x j (t)φ i (t) .(16) Then, carrying out the average over J s i j and J a i j leads to J s ,J a exp dt i< j J s i j x i (t)φ j (t) +x j (t)φ i (t) + kJ a i j x i (t)φ j (t) −x j (t)φ i (t) = exp g 2 2N i = j x i (t)φ j (t)x i (t )φ j (t ) + η x i (t)φ j (t)x j (t )φ i (t ) dt dt ≈ exp   g 2 2N   ix i (t)x i (t ) j φ j (t)φ j (t ) + η ix i (t)φ i (t ) j φ j (t)x j (t )   dt dt   . (17) Note that, we have added back the negligible diagonal term (i = j) to arrive at the last equality. Then, we define Z[j,j] = J Z[j,j|J], the average moment-generating functional is given by Z[j,j] = Dx (t)Dx exp −S 0 [x ,x ] + σ 2 2x ·x +j · x + j ·x + g 2 2N   ix i (t)x i (t ) j φ j (t)φ j (t ) + η ix i (t)φ i (t ) j φ j (t)x j (t )   dt dt ,(18)where S 0 [x ,x ] =x · [ẋ + x ] is called the free action. f · g = N i=1 f i (t)g i (t) dt is introduced for compactness. From Eq. (18), we have to introduce two auxiliary overlaps, Q 1 (t, t ) = g 2 N j φ j (t)φ j (t ), Q 2 (t, t ) = g 2 η N j φ j (t)x j (t ),(19) which converges to (scaled) Gaussian fields due to the central limit theorem when N is sufficiently large. Thus, we can insert these order parameters into Eq. (18) by the Fourier integral representation of Dirac delta functions, δ − N g 2 Q 1 (t, t ) + j φ j (t)φ j (t ) = 1 2π DQ 1 (t, t ) exp   Q 1 (t, t ) − N g 2 Q 1 (t, t ) + j φ j (t)φ j (t ) dt dt   , δ − N g 2 Q 2 (t, t ) + η j φ j (t)x j (t ) = 1 2π DQ 2 (t, t ) exp   Q 2 (t, t ) − N g 2 Q 2 (t, t ) + η j φ j (t)x j (t ) dt dt   .(20) Finally, we can re-express the averaged moment-generating functional as Z[j,j] = DXDQ exp − N g 2Q 1 · Q 1 − N g 2Q 2 · Q 2 − S 0 [x ,x ] + σ 2 2x ·x +j · x + j ·x + 1 2 jx j (t)Q 1 (t, t )x j (t ) dt dt + 1 2 jx j (t)Q 2 (t, t )φ j (t ) dt dt + j φ j (t)Q 1 (t, t )φ j (t ) dt dt + η j φ j (t)Q 2 (t, t )x j (t ) dt dt ,(21) where DX ≡ Dx Dx , and DQ ≡ N 2πg 2 2 DQ 1 (t, t )DQ 1 (t, t )DQ 2 (t, t )DQ 2 (t, t ) , and we also introduce new notations,Q 1 · Q 1 = Q 1 (t, t )Q 1 (t, t ) dt dt , Q 2 · Q 2 = Q 2 (t, t )Q 2 (t, t ) dt dt .(22) We can now remark that the averaged moment-generating functional is completely factorized over neurons, which implies that the original complex dynamics with N interacting neurons is captured by a mean-field one-neuron system subject to a correlated Gaussian noise. More compactly, we recast the averaged moment-generating functional as Z[ j,j] = DQ exp N f (Q,Q, x,x) , f (Q,Q, x,x) = − 1 g 2Q 1 · Q 1 − 1 g 2Q 2 · Q 2 + logZ[ j,j], Z[ j,j] = DX exp L Q,Q, x,x , L Q,Q, x,x = − S 0 [x,x] + σ 2 2x ·x +j · x + j ·x + 1 2x T Q 1x + 1 2x T Q 2 φ + φ TQ 1 φ + ηφ TQ 2x ,(23) whereZ[ j,j] is the effective moment generating functional for one-neuron system, which will be mathematically clear at the end of the derivation. Thus, x,x, j,j, X are the mean-field counterpart of their original meaning in the high dimensional space. Accordingly, we have f ·g = f (t)g(t)d t. In addition, we define the new notation involving {Q,Q} in terms of the quadratic form as f T Q g = f (t)Q(t, t )g(t) In N → ∞, we estimate asymptotically the averaged dynamical partition function by applying the Laplace method, Z[ j,j] = DQ exp N f (Q,Q, x,x) ≈ exp N f (Q ,Q , x,x) ,(24) where {Q ,Q } maximizes the dynamical action f . We thus have, δ f (Q,Q, x,x) δQ 1 (t, t ) = 0 → Q 1 (t, t ) = g 2 φ(t)φ(t ) L , δ f (Q,Q, x,x) δQ 1 (t, t ) = 0 →Q 1 (t, t ) = g 2 2 x(t)x(t ) L , δ f (Q,Q, x,x) δQ 2 (t, t ) = 0 → Q 2 (t, t ) = g 2 η φ(t)x(t ) L , δ f (Q,Q, x,x) δQ 2 (t, t ) = 0 →Q 2 (t, t ) = g 2 2 x(t)φ(t ) L ,(25) where, 〈O〉 L = O(X ) exp[L(X )]DX Z[ j,j] .(26) This average can be seen as the dynamical mean field measure provided byZ [ j,j] in the oneneuron system, in analogy with the replica analysis in equilibrium spin glass theory [20]. In the following text, we will omit the subscript L for simplicity. Now we come to the physical meaning of the dynamics order parameters. First, it is easy to find that Q 1 (t, t ) is related to the auto-correlation function, which is C(t, t ) = 1 N i 〈φ i (t)φ i (t )〉 → 〈φ(t)φ(t )〉,(27) and (10)]. In other words, these response fields do not propagate. Finally, Q 2 (t, t ) and Q 2 (t, t ) bear exactly the same physical meaning, which relates to the response function, Q 1 (t, t ) = g 2 C(t, t ). Second,Q 1 (t, t ) will always vanish because δ n δ j(t 1 )···δ j(t n ) Z[j, 0]| j=0 = 0 [see Eq.R(t, t ) = 1 N i δ〈φ i (t)〉 δ j i (t ) j=0 ,(28) where the average is taken under the path probability, i.e., R(t, t ) = 1 N i δ〈φ i (t)〉 δ j i (t ) j =0 = 1 N i 〈φ i (t)x i (t )〉 → 〈φ(t)x(t )〉.(29) This relation bears the similarity with the linear response relation (e.g., susceptibility and fluctuation) in equilibrium statistical physics. Thus, Q 2 (t, t ) = g 2 ηR(t, t ) andQ 2 (t, t ) = g 2 2 R(t , t). Moreover, the response function R(t, t ) will vanish once t < t because of the causality that perturbations in a later time do not affect the present and past states. In addition, the equal time response function R(t, t) also vanishes under the Ito convention [21]. It is now clear that the term Q 2 ·Q 2 vanishes because of the causality and the Ito convention (note that R(t, t )R(t , t) = 0). Finally, we achieve the final form of the moment generating functional, Z[ j,j] = N iZ [ j,j] = Z [ j,j] N , Z[ j,j] ∝ DX exp −S 0 [x,x] +j · x + j ·x + 1 2x T Γx + ηg 2x T Rφ = DX exp −S[x,x] +j · x + j ·x ,(30) where Γ (t, t ) = g 2 C(t, t ) + σ 2 δ(t − t ) , and the effective action (decomposed into free and interation parts) reads, S[x,x] =x · [ẋ + x] − 1 2x T Γx − ηg 2x T Rφ.(31) The first line of Eq. (30) clearly illustrates that the N -neuron interactive system degrades to N factorized one-neuron effective systems. And equation (31) further suggests that the dynamical mean-field description of the N -neuron dynamics exists [from Eq. (12)], i.e., x(t) = −x(t) + ηg 2 t 0 R(t, t )φ(t ) dt + γ(t),(32) where 〈γ(t)γ(t )〉 = g 2 C(t, t ) + σ 2 δ(t − t ). When η = 0, Eq. (32) reduces to Eq. (3). It is interesting that the spatially correlated asymmetric connections between neurons in the Nneuron system are transformed into accumulated interactions of dynamics history in the oneneuron (mean-field description) system. The underlying physics is more transparent in the cavity framework introduced later. The MSDRJ formalism allows one to derive integro-differential equations involving response and correlation functions. For example, we could compute the dynamical equations of mean- field correlation function ∆(t, t ) = 〈x(t)x(t )〉 and response function χ(t, t ) = δ〈x i (t)〉 δ j i (t ) j =0 for currents. First, we consider two identities, δx(t ) δx(t) = 0, δx(t ) δx(t) = δ(t − t ).(33) Then, we take the path average over the probability defined by Eq. (30), δx(t ) δx(t) = DX δx(t ) δx(t) exp (−S[x,x]) = DX x(t ) δS[x,x] δx(t) exp (−S[x,x]) = x(t ) ẋ(t) + x(t) − Γ (t, s)x(s) ds − ηg 2 R(t, s)φ(s) ds = 0,(34) where the integral by parts is used to derive the second equality. This relation will immediately give rise to, ∂ ∂ t ∆(t, t ) = −∆(t, t )+ g 2 t 0 χ(t , s)C(t, s)ds+σ 2 χ(t , t)+ηg 2 t 0 R(t, s) x(t )φ(s) ds. (35) Similarly, the other identity in Eq. (33) leads to, ∂ ∂ t χ(t, t ) = −χ(t, t ) + δ(t − t ) + ηg 2 t t R(t, s)R(s, t )ds.(36) These integro-differential equations are particularly difficult to solve, e.g., no closed form solutions exist except at η = 0. In general, an perturbative expansion of the non-linear transfer function may be required [12]. Dynamical cavity approach In this section, we introduce the dynamical cavity approach, which is more physically transparent (like its static counterpart [20]). The dynamical cavity approach gives exactly the same DMFT equation that is based on the moment generating functional method. Our starting point is still the N -neuron stochastic dynamics, x i (t) = −x i (t) + g j=1 J i j φ j (t) + j i (t) + σξ i (t), i = 1, . . . , N ,(37) where the Gaussian noise ξ i (t) has the variance ξ i (t)ξ j (t ) = δ i j δ(t − t ). Connections J i j are drawn from the centered Gaussian distribution with the variance 1 N as well as the covariance J i j J ji = η N . First, we add a new neuron into the original system, such that we have a new synaptic current x 0 (t) together with the corresponding connections (J 0i , J i0 ), for i = 1, . . . , N . As a result, all the neurons in the original system will be affected by this new neuron. We regard this impact as a small perturbation in the large network limit. We can thus apply the linear response theory as follows,φ i (t) = φ i (t) + N k=1 t 0 δφ i (t) δ j k (s) j=0 j k (s) ds = φ i (t) + N k=1 t 0 R ik (t, s) [g J k0 φ 0 (s)] ds ,(38) where R ik (t, s) = δφ i (t) δ j k (s) j =0 defines the linear response function, and the small perturbation is given by j k (s) = g J k0 φ 0 (s). Then, we can write down the dynamical equation of x 0 (t), x 0 (t) = −x 0 (t) + g j =0 J 0 jφ j (t) + j 0 (t) + σξ 0 (t) = −x 0 (t) + g N j=1 J 0 j φ j (t) + N k=1 t 0 R jk (t, s) [g J k0 φ 0 (s)] ds + j 0 (t) + σξ 0 (t) = −x 0 (t) + g N j=1 J 0 j φ j (t) + σξ 0 (t) + g 2 t 0 jk J 0 j R jk (t, s)J k0 φ 0 (s) ds + j 0 (t),(39) where the fourth term captures how the asymmetric correlation affects the current state of the new neuron through the response function. The bare field without the effects of synaptic correlation is separated out as follows, γ 0 (t) = g N j=1 J 0 j φ j (t) + σξ 0 (t),(40) which becomes the centered Gaussian field whose variance is given by 〈γ 0 (t)γ 0 (t )〉 = g 2 C(t, t ) + σ 2 δ(t − t ),(41) where C(t, t ) = 1 N j φ j (t)φ j (t ) is the population averaged auto-correlation function. The computation of the fourth term in Eq. (39) requires us to estimate jk J 0 j R jk (t, s)J k0 which is subject to central limit theorem by construction. We first consider the diagonal part j J 0 j R j j (t, s)J j0 , which will converge asymptotically to its mean because of the negligible variance (of the order 1/ N ), J j J 0 j R j j (t, s)J j0 = η N j R j j (t, s).(42) Then, we turn to the non-diagonal part j =k J 0 j R jk (t, s)J k0 , whose mean is zero due to J 0 j J k0 = 0, and we should thus consider the fluctuation given by J j =k j =k J 0 j J 0 j J k0 J k 0 R jk (t, s)R j k (t, s) = j =k J 2 0 j J 2 k0 R 2 jk (t, s) ∼ 1 N ,(43) where we assume that R jk (t, s) is of the order O(1/ N ) for j = k, as in the equilibrium limit, the response function has the exactly the same magnitude order with the correlation function (O(1/ N ) in fully connected mean-field models), and a proof for a dynamical system is shown in Ref. [16]. Therefore, the contribution from the non-diagonal part can be neglected when N is large, and the dynamical equation of x 0 (t) is thus simplified as, x 0 (t) = −x 0 (t) + γ(t) + g 2 η t 0 R(t, s)φ 0 (s) ds + j 0 (t),(44) where the population averaged response function R(t, s) = 1 N i δφ i (t) δ j i (s) j=0 . The added neuron x 0 (t) is not special, and its dynamics is a representative of the typical behavior of other neurons. Therefore, we could omit the subscript 0, and write down the meanfield dynamics as follows,ẋ (t) = −x(t) + γ(t) + g 2 η t 0 R(t, s)φ(s) ds ,(45) where γ(t) is the effective noise with the temporally correlated variance, 〈γ(t)γ(t )〉 = g 2 C(t, t ) + σ 2 δ(t − t ).(46) Finally, to close this self-consistent equation, we further assume that in the large N limit, the population average of the correlation and response functions converge to their path average (with respect to the noise trajectories and the initial conditions). More precisely, C(t, t ) = 1 N j φ j (t)φ j (t ) = 〈φ(t)φ(t )〉, R(t, t ) = 1 N i δφ i (t) δ j i (t ) j =0 = δφ(t) δ j(t ) j =0 ,(47) which leads to the same Eq. (32) that has been previously derived from the moment generating functional. Note that the last equality in the above response function is equivalent to the definition in Eq. (28) [23]. Numerical and theoretical analysis Numerical solution of the DMFT equations In general, the DMFT equation does not have a closed-form solution (e.g., for η = 0). Therefore, we have to solve the equation numerically. In fact, solving the self-consistent DMFT equation [Eq. (32)] is more challenging compared to the counterpart of an equilibrium system. The main reason is that the self-consistent iteration involves time-dependent function (C(t, t ) and R(t, t )) rather than scalar variables like overlaps in spin glass theory. Note that the two-point correlation could relax towards the Edwards-Anderson order parameter in mean-field spin glass models [24]. Following the previous work [16], we give the numerical implementation details of solving the DMFT equations for the generic random neural networks below. Codes are available at the Github link [25]. The iteration scheme works in discrete time steps, and we must set a duration T (ms) as well as a time interval ∆t(ms) for discretization. The time is measured in units of millisecond. In the beginning of the iteration, we initialize the self-consistent function matrix C[t, t ] and R[t, t ], whose dimensions are both (T /∆t) × (T /∆t). In each iteration, we carry out the following steps: 1. Draw M samples of noise trajectories {γ a [t]} M a=1 from the multivariate Gaussian distribution N (0, g 2 C[t, t ] + σ 2 /∆t), where the emergence of ∆t is the result of discretization for the Dirac delta function. 2. For these noise trajectories, run M corresponding current trajectories independently by a direct discretization, x a [t + 1] = (1 − ∆t)x a [t] + γ a [t]∆t + g 2 η∆t 2 t s=0 R[t, s]φ a [s].(48)χ a [t + 1, t ] = (1 − ∆t)χ a [t, t ] + δ t,t + ηg 2 ∆t 2 t s=t R iter [t, s]R a [s, t ],(49) where the superscript a is the trajectory index, and the response function is computed by R[t, t ] = φ (x[t])χ[t, t ]. Here R iter [t, s] refers to the response function estimated from An alternative way to compute the response function is using the Novikov's theorem (see Appendix A [26]). We do not apply this formula in the iteration, as it needs much more trajectories for the convergence. We compare the observables obtained from the direct simulation of N -neuron dynamics [Eq. (1)] and the mean-field solution for the one-neuron dynamics [Eq . (32)] to check the effectiveness of the DMFT. Besides the correlation function and response function, we also compare the observable of mean firing-rate m(t) = 〈φ(t)〉, which is also an important quantity of the current system. The curves show an excellent agreement for each observable [ Figure. (1)], where the curve for the temporal integration is computed from 100 independent runs. For the response function, the argument is selected to be the time difference, as R(t − t ) is our focus in the steady state where the time-translation invariance holds. Note that R(0) = 0 is a direct result of the Ito convention. The inset shows the relative differences between two types of observables, computed by, m − m 2 m 2 , Ĉ − C F C F , R − R F R F ,(50) where hated variables represent the simulation results, while the non-hated ones are DMFT results, and the subscripts 2 and F means the 2 norm and Frobenius norm respectively. The relative differences decrease as N grows, which meets our expectation that the DMFT equation predicts the typical behavior of the dynamics under the large network limit. Analysis of fixed point solutions In this section, we derive the fixed point solution of the DMFT equation. Under a special choice of model parameters, we could even obtain the analytic fixed point solution in the mean-field description. We then focus on the noise-free case of σ = 0 with the dynamics: x(t) = −x(t) +γ(t) + g 2 η t 0 R(t, t )φ(t ) dt + j(t),(51) where j(t) is a perturbation andγ(t) is the noise-free effective field with the variance, 〈γ(t)γ(t )〉 = g 2 C(t, t ).(52) We assume that the dynamics converges to a fixed point (ẋ(t) = 0). In the steady state, we get R(t, t ) = R(t − t ) to simplify the integral term, t 0 R(t, t )φ(t ) dt = t 0 R(u)φ(t − u)du t→∞ −→ ∞ 0 R(u)φ(∞)du = R int φ * ,(53) where R int = ∞ 0 R(u)du is the integrated response function and * indicate the steady state. Then, we could obtain the fixed point relation, x * =γ * + wφ * + j,(54) where w = g 2 ηR int , and 〈(γ * ) 2 〉 = g 2 C. (55) However, the fixed point relation is not closed, and we must evaluate R int by its definition. In essence, the integrated response function R int could be computed by 〈 δφ * δ j 〉, setting j to zero later. An equivalent derivation is given in the Appendix B. One can generate a series of noise sampleŝ γ * from N (0, g 2 C iter ), where the superscript iter denotes the value from the last iteration step. Second, the new observables from these samples are computed as, C = (φ * ) 2 γ * , R int = φ [γ * + wφ * ] 1 + wR iter int γ * .(56) These equations can be iteratively solved, requiring a high computational complexity due to the noise sampling and fixed point searching for each noise sample. To achieve an analytic fixed point solution, we choose the ReLU function φ(x) = xΘ(x), which is commonly used in machine learning and theoretical neuroscience studies. We could thus recast Eq. (54) to the following form, φ * = φ(γ * + wφ * ),(57) where j is erased. With the help of ReLU function, we can write φ * as a function ofγ * , φ * =γ * Θ(γ * ) 1 − w ≡ ψ(γ * ),(58) and the response function becomes, Therefore, we can derive the self-consistent equations of C, R int as well as m, R int = δφ * δ j = δφ * δγ * = Θ(γ * ) 1 − w .(59C = 〈(φ * ) 2 〉 = γ * Θ(γ * ) 1 − w 2 p(γ * ) dγ * = g 2 C 2(1 − w) 2 , R int = Θ(γ * ) 1 − w = Θ(γ * ) 1 − w p(γ * ) dγ * = 1 2(1 − w) , m = 〈φ * 〉 = γ * Θ(γ * ) 1 − w p(γ * ) dγ * = g 2 C 2π(1 − w) ,(60) where the integral range covers the entire real value region, and w = g 2 ηR int . In the following, we omit the subscript of R int . These relations will give the analytic fixed point (observables), m = 0, C = 0, R = 1 − 1 − 2g 2 η 2g 2 η .(61) Note that we discard the other root for R because of the divergence in the limit η → 0 (keeping g finite). Equation (61) implies that g 2 /(2(1 − w) 2 ) = 1 and 1 − 2g 2 η > 0 (see Appendix C). We compare the fixed point solution obtained directly from Eq. (51) to the analytic fixed point given by Eqs. (60, 56). The fixed point iteration becomes difficult when φ = tanh in the current model. This fixed point is a well-known result of m = C = 0, which is hard to achieve numerically, because numerical errors always make it impossible to get a perfect zero value. We must set a very small value for the initialization of C, or we can just set C to zero and then get R int . In spite of this numerical error, we observe a perfect match (Figure 2 Fluctuation-dissipation theorem The fluctuation-dissipation theorem (FDT) relates the linear response function to the correlation function in equilibrium, which establishes a model independent relationship connecting the statistics of spontaneous fluctuation to the response to perturbations [22]. A static counterpart is the linear response theory that relates the fluctuation and susceptibility. FDT allows one to predict the mean response to external perturbations without applying any perturbation, and instead, by analyzing the time-dependent correlations. FDT holds particularly in a stochastic system subject to conservative forces and the dynamics bears a steady state [19,22]. We discuss the relevance of FDT in the context of random neural networks in this subsection. As we know, dynamical systems tend to be more difficult to study than equilibrium systems, because we have no prior knowledge of the steady state (if any) in a general context. In the simplest case, we consider a Langevin dynamics, λẋ i (t) = − ∂ H(x ) ∂ x i (t) + η i (t),(62) where λ is a friction coefficient, η i (t) is a Gaussian white noise, and 〈η i (t)η j (t )〉 = 2T λδ i j δ(t−t ). The temperature bridges the relationship between the noise strength and the friction in the steady state. Here, we assume that the dynamics could be interpreted as moving particles in a potential H(x ) (or gradient dynamics). This Langevin dynamics, in the long time limit, is able to reach the thermal equilibrium that is captured by the Gibbs-Boltzmann distribution, P(x ) ∼ exp − H(x ) T ,(63) where the Hamiltonian is exactly the potential function that drives the dynamics through Eq. (62). This precise probability measure can be derived from the Fokker-Planck equation by setting the probability current to zero [22]. We have assumed k B = 1 without loss of generality. Next, we consider a linear and full-symmetric network whose dynamics is governed by d x i (t) d t = −x i (t) + g N j=1 J i j x j (t) + σξ i (t),(64) where J i j = J ji . By comparing this equation with the Langevin dynamics Eq. (62), we can directly write down the Hamiltonian H(x ) = − 1 2 i x 2 i − 1 2 g i = j J i j x i x j , and the temperature is determined by T = σ 2 /2. We remark that non-gradient dynamics (e.g., η = 1) may have a nonequilibrium steady state for which FDT breaks. In this simple gradient dynamics, the equilibrium can be reached and the FDT holds as follows, χ(t, t ) = − 1 T ∂ t ∆(t, t )Θ(t − t ),(65) where the instantaneous response function χ(t, t ) = 1 N i ∂ 〈x i (t)〉 ξ ∂ j i (t) and the time-dependent fluc- tuation ∆(t, t ) = 1 N i 〈x i (t)x i (t )〉 ξ . These functions also bear the time-translation invariance due to the steady state condition. In addition, we can prove that FDT is valid in the dynamical system of Eq. (64), and we leave the proof to the Appendix D. An experimentally measurable quantity is the integrated response function calculated by χ int (t, t ) = t t ds χ(s, t ).(66) Then we rescale the integrated response by the equal time correlation function, and get Thus, when t ≥ t , we have the relation as, χ int (t, t ) = χ int (t, t )/∆(t , t ),∆(t, t ) = ∆(t, t )/∆(t , t ).(67χ int (t, t ) = 1 T 1 −∆(t, t ) .(68) Equation (68) establishes an easy way to measure the temperature determined by the slope of the parametric plotχ int (t + t w , t w ) versus∆(t + t w , t w ) where t w is a waiting time for reaching the steady state. This temperature is called the effective temperature [24], which may be a constant or changes with the time difference. If the Gibbs-Boltzmann measure exists, the effective temperature coincides with the thermodynamic temperature. However, these two temperatures are not equal in general. Even in aging systems where the the decay of the correlation and response functions depend on the waiting time (how long the system is prepared), the effective temperature may be different for different ranges of the correlation function (displaying multiple relaxation time scales as in mean-field glass models [24]). We verify the fluctuation-dissipation theorem in the rate model via solving the DMFT equation and provide insights on the long time behavior. We focus on the influence of the nonlinear function and asymmetric connections on the FDT. The results are summarized in Figure (3). For the case where η = 1 and φ is linear, the FDT must be valid (see the Appendix D), the slope obtained by the linear fitting indicates a temperature closer to the ground truth compared to other non-gradient dynamics. The slight deviation is caused by the numerical errors from simulations. Interestingly, the nonlinear function and the other asymmetry correlation levels do not yield a strongly-violated FDT, because a linear fitting with a larger effective temperature is observed. This suggests that the out-of-equilibrium steady state with unknown probability measure may be approximated by an equilibrium FDT with a larger effective temperature, which offers an interesting perspective to bridge the non-gradient dynamics (commonly observed in recurrent neural networks) to an effective thermodynamic behavior. Conclusion In this lecture note, we briefly introduce the path integral framework, from which the dynamical mean-field theory of generic stochastic dynamics in high dimensional systems is derived. We also introduce a complementary cavity method to derive the exactly same results. Considering the long time limit of the dynamics, we analyze the fixed point solution of the dynamics by a direct deduction of the DMFT equation [16] and by a static cavity analysis [18]. The FDT is also discussed in the context of generic random neural networks. Based on these theoretical descriptions, it is interesting to detect the fundamental relationship between spontaneous fluctuations and response functions to external perturbations, especially in the gradient dynamics commonly observed in deep learning [3,27]. The fluctuation dissipation relation is also studied in a recent work for spiking neural networks [28], and the path integral framework can also find applications in revealing inner workings in recurrent network models of memory and decision making [29,30]. We hope this tutorial will expand the cutting-edge researches of learning in neural networks, inspiring more fundamental physics laws governing high dimensional complex neural dynamics and novel algorthimic designs. A Novikov's theorem Novikov's theorem characterizes a useful identity to estimate the dynamic response function. In this section, we give a derivation for both the original rate model Eq.(1) as well as the DMFT equation [Eq. (32)]. We consider a general form of dynamical equations, x i (t) = F i (x ) + cΞ i (t) + j i (t), i = 1, 2, . . . , N ,(69) where F is a general functional of x and c is a constant. Note that, Novikov's theorem is valid only if we consider a Gaussian noise here. Thus, we set a general Gaussian noise with the covariance structure as 〈Ξ i (t)Ξ j (t )〉 = D i j (t, t ). For the above dynamics, the response function is defined by R i j (t, t ) = δ〈φ i (t)〉 δ j j (t ) j =0 .(70) The path average 〈O〉 can be defined by Dx DΞOP(x |Ξ)P(Ξ), where P(x |Ξ) = i,t δ(ẋ i (t) − F i (x ) − cΞ i (t) − j i (t)) , P(Ξ) = 1 Z Ξ exp − 1 2 i, j dt dt Ξ i (t)D −1 i j (t, t )Ξ j (t ) ,(71) where Z Ξ is a normalization constant. With this definition, the response function could therefore be calculated as, R i j (t, t ) = δ δ j j (t ) Dx DΞP(Ξ)P(x |Ξ)φ i (t) = Dx DΞφ i (t)P(Ξ) δ cδΞ j (t ) P(x |Ξ) = − Dx DΞφ i (t)P(x |Ξ) δ cδΞ j (t ) P(Ξ) = 1 c Dx DΞφ i (t)P(x |Ξ) k ds D −1 jk t , s Ξ k (s) P(Ξ) = 1 c φ i (t) k ds D −1 jk t , s Ξ k (s) ,(72) where we have used the property that P(x |Ξ) is symmetric with respect to Ξ j (t ) and j j (t ), and we have applied the integral by parts in the third equality. For the original N -neuron model [Eq.(1)], c = σ, Ξ i (t) = ξ i (t) and D i j (t, t ) = δ i j δ(t − t ). We thus have, R i j (t, t ) = 1 σ φ i (t) k ds δ −1 jk δ −1 (t − s)ξ k (s) = 1 σ φ i (t)ξ j (t ) ,(73) where we have used the identity δ −1 (t − s)δ(s − t ) ds = δ(t − t ). For the DMFT equation [Eq. (32)] of one-neuron system, c = 1, Ξ(t) = γ(t) and D(t, t ) = Γ (t, t ) = g 2 C(t, t )+σ 2 δ(t−t ). Therefore, the response function becomes, R(t, t ) = φ(t) ds Γ −1 (t , s)γ(s) .(74) B Static cavity method for recurrent dynamics In this section, we introduce the static cavity method [18] to derive the self-consistent equations for the fixed point of dynamics. We consider the rate model with the ReLU transfer function φ(x) = xΘ(x). The procedure starts from the fixed-point condition of the noise-free rate model [Eq. (1)], which is x i = g N j=1 J i j v j + j i ,(75) where we set v j = φ(x j ) for simplicity. Note that asymmetric connections incorporate correlation between J i j and v j , and thereby the central limit theorem in the summation does not apply. To overcome this barrier, we can remove the contribution of J ji and consider this deletion as a small perturbation. Therefore, the sum in Eq. (75) can be separated into two parts as, x i = g N j=1 J i j v j→i + g N j=1 J i j δv j + j i ,(76) where v j→i denotes the firing rate of neuron j in the absence of J ji , and δv j is the perturbation caused by the presence of J ji . According to the central limit theorem, the first term on the right hand side can now be treated as a Gaussian field, which we denote asγ i and compute the variance, 〈(γ i ) 2 〉 = g 2C ,(77) where· indicates the cavity quantity. Note that,C = 〈v 2 j→i 〉 is the self-consistent cavity variance function. As for the second term, we compute the perturbation by a linear response approximation, δv j = N k=1 R jk η k ,(78) where R jk = δv j δ j k is the linear response function. The small perturbation η k here is actually the contribution from neuron i to neuron k through J ki , which is exactly J ki v i . Therefore, the second term in the r.h.s. of Eq. (76) becomes, g N j=1 J i j δv j = g N j=1 N k=1 J i j R jk J ki v i ≈ gη 2 Rv i ,(79) where R = 1 N j R j j . Note that, the approximation in the last equality is exactly the same as what we did in the dynamical cavity approach [see Eqs.(42,43)]. Finally, we recast the fixed point equation ( j i = 0) as x i =γ i + wv i ,(80) where w = gη 2 R and 〈γ 2 i 〉 = g 2C . The above equation can be transformed to v i = φ(γ i + wv i ) and then written as a function ofγ i as, v i =γ i Θ(γ i ) 1 − w ≡ ψ(γ i ).(81) We then compute the cavity variance function and the linear response function as follows, C = 〈v 2 i 〉 = γ i Θ(γ i ) 1 − w 2 p(γ i )dγ i = g 2C 2(1 − w) 2 , R = δv i δγ i = Θ(γ i ) 1 − w p(γ i )dγ i = 1 2(1 − w) .(82) We remark thatC, under the limit of N → ∞, can be replaced by the full variance functioñ C = 〈v 2 i 〉. The above results are now exactly the same with Eq. (60). C Stability analysis for the ReLU transfer function The stability of the trivial fixed point can be linked to the connectivity spectrum. To be more precise, we take the rate model whose transfer function is φ = tanh as an example. We can first linearize the neural dynamics Eq. (1) (in the absence of white noise) around the fixed point (x * = 0),∆ x i (t) = j D i j ∆x j (t) → ∆x (t) = exp (Dt)∆x (0),(83) where, D i j = −δ i j + g J i j φ (x * j ) = −δ i j + g J i j(84) is the local Jacobian matrix at the fixed point. The eigenvalues of this Jacobian matrix determine the stability of the local dynamics. In particular, if the eigenvalues with the largest real part cross zero along the real axis, the dynamics becomes chaotic in our current model. In general, the instability is not a sufficient but necessary condition for the transition to chaos [6]. Moreover, the spectrum of J i j is actually a well-known elliptic law [31], ρ(λ) = 1 π(1−η 2 ) , x 1+η 2 + y 1−η 2 < 1, 0, otherwise ,(85) where λ is the eigenvalue of complex value, while x and y are the coordinates on the real-and imaginary-axis. The special case of η = 0 gives the circular law in random matrix theory. Thus, the eigenvalue with the largest real part of D i j is g(1+η)−1. Consequently, the stability condition is given by g(1 + η) < 1. However, the Jacobian is ill-defined when φ = ReLU. Instead, we rely on the static cavity method [18]. For the sake of clarity, we follow the same notations introduced in Appendix B. Our starting point is the relation of v j = ψ(γ j ), which states that the firing-rate of neuron j could be seen as a function of its cavity input. From the cavity idea, the presence of neuron i contributes to a perturbation δv j , i.e., δv j = ψ (γ j )   g k =i J jk δv k + g J ji v i   ,(86) where a linear expansion atγ j is used when the effect of the cavity operation is small. Note that the term in the bracket denotes the deviation of the cavity input to neuron j between with and without the neuron i. It is reasonable that if the fixed point is stable, the variance of δv j must be finite and positive [18]. Taking the average over the network statistics, we get 〈δv j 〉 = 0. To compute the variance, we square both sides of Eq. (86) and take the disorder average, which results in, 〈(δv) 2 〉 = g 2 χ φ 〈(δv) 2 〉 + g 2 N χ φ v 2 i ,(87) where χ φ = (ψ (γ j )) 2 . This equation leads to, N 〈(δv) 2 〉 = g 2 χ φ 1 − g 2 χ φ v 2 i .(88) where the bold functions χ(t, t ) and C (t, t ) refers to N × N matrices. Next, we calculate the mean auto-correlation function [32], ∆(t, t ) ≡ 1 N Tr ∆(t, t ) = σ 2 2 −2 dk 2π 4 − k 2 min(t,t ) 0 dt e (−1+k)(t+t −2t ) = σ 2 min(t,t ) 0 dt I 1 (2(t + t − 2t )) t + t − 2t e −(t+t −2t ) = σ 2 t+t |t−t | dw I 1 (2w) 2w e −w ,(96) where the trace used in the second line can be seen as an integral over the eigenvalues. Wigner semi-circle law is used here for the eigenvalue spectrum of fully-symmetric matrix J [20]. In the second line, a substitution k = 2 cos θ is used and the modified Bessel function of the first kind is introduced, i.e., I 1 (τ) τ = 1 π π 0 dθ (sin θ ) 2 e τ cos θ . The final step also involves a substitution of w = t + t − 2t . In the long time limit, the upper limit of integral tends to ∞ and the mean auto-correlation function becomes time translation invariant. Follow the same procedure, we can compute the mean response function, χ(t, t ) ≡ 1 N Tr χ(t, t ) = Θ(t − t ) I 1 2 t − t t − t e −(t−t ) .(97) Finally, it would be easy to verify the following FDT using Eq. (95) and Eq. (96), χ(t, t ) = − 1 T ∂ t ∆(t, t )Θ(t − t ),(98) where T = σ 2 /2 represents the thermodynamic temperature. Performing the Fourier transform, we can also recast the FDT in the frequency domain ∆(ω) = 2T ω Imχ(ω). Figure 1 : 1Comparison between observables obtained from direct simulation (color curves) and iterative mean-field solution (dash curves).The parameters set {g, η, σ, φ, ∆t, t } is {0.2, 0.5, 0.1, tanh, 0.1(ms), 10(ms)}. Inset: relative differences vs N .the last iteration step. After running the dynamics, we compute the new response function χ[t, t ] = 1/M a χ a [t, t ]. Figure 2 : 2Convergence of observables to analytic fixed point solutions during iteration of the DMFT equation. The parameters set {g, η, σ, ∆t} is {0.2, 0.5, 0.1, 0.1(ms)}. (a) Transfer function φ = tanh. (b) Transfer function φ = ReLU. ) as the iteration step of the DMFT equation [Eq. (51)] increases. Figure 3 : 3Effective temperature of the system given different asymmetry correlation levels and nonlinearities. The waiting time t is fixed at 15s and the color of points becomes lighter as t increases. The dash line indicates FDT for the linear system with symmetric connection η = 1, whose thermodynamic temperature is T = σ 2 /2 = 0.5 (indicated by a black circle in the inset).(a) Comparison among different asymmetry correlation levels in the linear system. For η = [−1, 0, 1], the effective temperatures obtained by a linear fitting are T eff = [0.551, 0.538, 0.524], respectively (inset). (b) Comparison among different nonlinear functions when η = 1. For φ selected to be ReLU, Tanh and linear, the effective temperatures obtained by a linear fitting are T eff = [0.527, 0.526, 0.524], respectively (inset). Funding informationThis research was supported by the National Natural Science Foundation of China for Grant number 12122515, and Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices (No. 2022B1212010008), and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023B1515040023). AcknowledgementsWe thank all PMI members for inspiring discussions of the dynamical mean-field theory and path integral formalism.To ensure 〈(δv) 2 〉 physical, the condition 1 − g 2 χ φ > 0 must be satisfied. To proceed, we first compute χ φ ,Note that w = g 2 ηR and R = 1 2(1−w) , which implies thatwhere ∆ = 1 − 2g 2 η. The stability thus requires that g 2 2(1−w) 2 < 1, which finally leads to the condition by using Eq. (90),The stability condition further implies that 1 − 2g 2 η > 1 − 2g 2 ( 2/g − 1) ≥ 0.D Derivation of fluctuation-dissipation theorem in equilibriumIn this section, we give a proof of fluctuation-dissipation theorem for the model [Eq.(1)] with linear transfer function and fully-symmetric connections. We begin the derivation by rewriting the model in the vector form,ẋwhere J = J T and 〈ξ(t) T ξ(t )〉 = N ×N δ(t − t ) by construction. 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{'abstract': 'Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of generic random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.SciPost Physics Lecture NotesSubmission B Static cavity method for recurrent dynamics 19 C Stability analysis for the ReLU transfer function 21 D Derivation of fluctuation-dissipation theorem in equilibrium 22References 24', 'arxivid': '2305.08459', 'author': ["Wenxuan Zou \nSchool of Physics\nPMI Lab\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China\n", "Haiping Huang huanghp7@mail.sysu.edu.cn \nSchool of Physics\nPMI Lab\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China\n\nGuangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China\n"], 'authoraffiliation': ["School of Physics\nPMI Lab\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China", "School of Physics\nPMI Lab\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China", "Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices\nSun Yat-sen University\n510275GuangzhouPeople's Republic of China"], 'corpusid': 258686674, 'doi': '10.48550/arxiv.2305.08459', 'github_urls': [], 'n_tokens_mistral': 19614, 'n_tokens_neox': 17278, 'n_words': 10402, 'pdfsha': 'afdbb62e689af6cf6e219171bec39a07d5db6777', 'pdfurls': ['https://export.arxiv.org/pdf/2305.08459v2.pdf'], 'title': ['Introduction to dynamical mean-field theory of generic random neural networks', 'Introduction to dynamical mean-field theory of generic random neural networks'], 'venue': []}

arxiv

Sensing Aided Uplink Transmission in OTFS ISAC with Joint Parameter Association, Channel Estimation and Signal Detection May 2023 Xi Yang Hang Li Senior Member, IEEEQinghua Guo Senior Member, IEEEJ Andrew Zhang Senior Member, IEEEXiaojing Huang Member, IEEE.Zhiqun Cheng Sensing Aided Uplink Transmission in OTFS ISAC with Joint Parameter Association, Channel Estimation and Signal Detection May 20231 In this work, we study sensing-aided uplink transmission in an integrated sensing and communication (ISAC) vehicular network with the use of orthogonal time frequency space (OTFS) modulation. To exploit sensing parameters for improving uplink communications, the parameters must be first associated with the transmitters, which is a challenging task. We propose a scheme that jointly conducts parameter association, channel estimation and signal detection by formulating it as a constrained bilinear recovery problem. Then we develop a message passing algorithm to solve the problem, leveraging the bilinear unitary approximate message passing (Bi-UAMP) algorithm. Numerical results validate the proposed scheme, which show that relevant performance bounds can be closely approached.Index Terms-ISAC, joint channel estimation and signal detection, (unitary) approximate message passing, OTFS. I. INTRODUCTION Recently, the integrated sensing and communications (ISAC) based intelligent transportation has received tremendous attention, as it can potentially enable various applications such as autonomous driving, traffic management, and Internet of Vehicles (IoV) [1], [2]. Sensing the states (e.g., locations and speeds) of vehicles and surrounding objects improves the safety of IoV effectively, as well as the reliability of communications aided by sensing. In order to meet the needs of high-mobility, orthogonal time frequency space (OTFS) modulation has been employed in ISAC systems [3]- [6]. In this work, we focus on OTFS uplink transmission in an ISAC vehicular network. Various OTFS signal detectors have been proposed in the literature, e.g., the message passing based detectors [7]- [9]. These detectors require accurate channel state information, which can be acquired using different ways. The channel can be estimated before data transmission [7]- [9], which occupies channel coherence time and leads to substantial overhead. One can also separate the pilot symbols and data symbols in the delay-Doppler (DD) domain using the guard intervals [10], which, however, results in large overhead. To spectral loss, pilot symbols can also be superimposed with data symbols, where the interference between data and pilot needs to be carefully handled with joint channel estimation and signal detection [11]. In [12], sensing-aided communications in IoV was proposed to reduce the delay and overhead, and improve the communication performance. However, the parameter association issue is not considered. In this work, we propose a novel scheme for uplink transmission with sensing-assisted joint channel estimation and signal detection in an ISAC OTFS vehicular network. To achieve this, the sensing parameters acquired by the roadside unit (RSU) through downlink sensing need to be first associated with the vehicles in the network. To tackle this challenging problem, we propose joint parameter association, channel estimation and signal detection (PACESD). We show that the joint PACESD can be formulated as a constrained bilinear recovery problem, where parameter association and channel estimation lead to a sparse vector that needs to be jointly recovered with a discrete-valued symbol vector. Then, leveraging the bilinear UAMP (Bi-UAMP) algorithm [13], we develop a message passing based Bayesian algorithm to solve this problem. Numerical results validate the proposed scheme and show that the proposed algorithm can achieve performance close to the relevant bounds. Notations: Unless otherwise specified, we use a boldface lowercase letter, a boldface capital letter, and a calligraphy letter to denote a vector, a matrix, and a set, respectively; the C denotes the complex number field; vec(·) and ⊗ denote the vectorization and the Kronecker product operator, respectively; the superscripts (·) T , (·) H and (·) * denote the transpose, the conjugate transpose and conjugate operations, respectively; F N and I N denote N -dimensional discrete Fourier transform (DFT) matrix and identity matrix, respectively; 1 and 0 denote the all-ones vector and all-zeros vector, respectively; δ (·) denotes the Dirac delta function; f (x) ∝ g(x) denotes the relation f (x) = cg(x) for some positive constant c; | · | 2 denotes the element-wise magnitude squared operation; · denotes the l 2 norm; a · b and a./b denote the elementwise product and division of the two vectors, respectively; < f (x) > g(x) denotes the expectation of f (x) with respect to probability density function g(x); M a→b (x) denotes a message passed from node a to b ,which is a function of x; b(x) denotes the belief of x. II. ISAC VEHICULAR NETWORK AND OTFS MODULATION A. ISAC Vehicular Network As shown in Fig. 1, we consider an ISAC vehicular network, where the RSU provides communication services to multiple vehicles, while sensing the targets in the area. Assume that the RSU is equipped with a uniform linear array with N BS antennas and each vehicle has a single antenna. We focus on uplink transmission, i.e., vehicles transmit signals to the RSU, which is assisted by the sensing function of the network. 1 The workflow is as follows. 1) State estimation: The RSU first sends broadcast signals, which are reflected by the targets/vehicles within the service area and received by the RSU. With the received echo signals, the RSU estimates the sensing parameters including the arrival angles, time delays, and Doppler frequencies, based on which the RSU can determine the locations and speeds of the targets. 2) State prediction: With the estimated parameters, the RSU predicts the states of targets/vehicles in the next timeslot for uplink communication. 3) Sensing aided uplink transmission and parameter association: Then, with time division multiple access (TDMA), the vehicles transmit communication signals to the RSU [12], and the RSU performs channel estimation and signal detection with the aid of sensing parameters acquired in Step 2. Parameter association is jointly conducted in this process. It is noted that sensing parameter acquisition and prediction in Steps 1 and 2 have been investigated in the literature, e.g., in [5], [12]. This work focuses on Step 3 for joint PACESD. B. OTFS Modulation Let X ∈ C M×N denote the transmitted data symbol matrix in the DD domain, where M and N denote the numbers of subcarriers and time slots, respectively. Let ∆f and T be the subcarrier spacing and time slot duration, respectively, where T ∆f = 1. Then, the transmitted signal S ∈ C M×N in the time-delay (TD) domain is given by [14] S = G tx F H M Heisenberg transform × F M XF H N ISFFT ,(1) where the inverse symplectic finite Fourier transform (ISFFT) is equivalent to an M -point FFT of the columns and an N -ponit IFFT of the rows of X, G tx =diag[g tx (0), g tx (T /M ), . . . , g tx ((M − 1)T /M )] ∈ C M×M , and g tx (t) is the pulse-shaping waveform. In this work, we employ the rectangular waveform, i.e., G tx is an identity matrix I M . The transmitted signal vector s ∈ C MN ×1 can be represented by vectorizing the TD domain signal matrix S, i.e., s = vec I M XF H N = F H N ⊗ I M x,(2) where x = vec (X). Assuming that the number of independent resolvable paths between the RSU and Vehicle i is P i , the TD domain channel can be represented as [12] h i (τ, ν) = Pi p=1 h i,p b (θ i,p ) δ (τ − τ i,p ) δ (ν − ν i,p ) ,(3) where h i,p , τ i,p , ν i,p and θ i,p denote the channel gain, the delay, the Doppler frequency, and the angle relative to the RSU for the p-th path of vehicle i, respectively, and b (θ i,p ) = 1/ √ N BS [1, e jπsinθi,p , · · · , e jπ(NBS−1)sinθi,p ] T is the receive steering vector. Assuming a cyclic prefix is used in each OTFS block, we define the channel matrix [15] H i = Pi p=1 h i,p b (θ i,p ) Π li,p ∆ ki,p ,(4) where the permutation matrix Π is obtained by shifting the first column of an identity matrix to the last column (see (9) in [15]). ∆ = diag 1, e The RSU uses a bank of receive beamformers f p ∈ C NBS×1 to receive the signals transmitted by vehicle i, and the received signal in the TD domain can be expressed as [12] r i = Pi p=1 h i,p f H p b (θ i,p ) Π li,p ∆ ki,p s i + q i ,(5) where q i denotes the additive white Gaussian noise (AWGN) vector in the TD domain. With (2) and (5), the received signal in the DD domain can be written as y i = Pi p=1 h i,p f H p b (θ i,p ) (F N ⊗ I M ) Π li,p ∆ ki,p F H N ⊗ I M x i + n i ,(6) where n i = (F N ⊗ I M ) q i denotes the noise vector in the DD domain. III. JOINT CHANNEL ESTIMATION, SIGNAL DETECTION AND SENSING PARAMETER ASSOCIATION A. Problem Formulation We consider uplink transmission of vehicle i. Defining (6) can be rewritten as G i,p = f H p b (θ i,p ) (F N ⊗ I M ) Π li,p ∆ ki,p F H N ⊗ I M ∈ C MN ×MN ,y i = Pi p=1 h i,p G i,p x i + n i .(7) We have the following remarks: • According to the workflow of the vehicular network in Section II.A, the RSU possesses the sensing parameters from Steps 1 and 2, including time delays, arrival angles, and Doppler frequencies of the paths for all objects in the sensing area. These parameters can be used to construct G i,p in (7), so that the RSU only needs to estimate h i,p , (p = 1, . . . , P i ) and x i . So, the uplink transmission can be assisted by sensing. • However, we note that the sensing parameters at the RSU are yet to be associated with the vehicles. Model (7) is only available when the parameters have been associated with the vehicles. In this work, we propose a novel method to achieve joint PACESD. Let P ′ = K i=1 P i , where K is the number of targets/vehicles. We can construct the following model y i = P ′ p ′ =1 h p ′ G p ′ x i + n i .(8) Regarding the model, we note the following: • As the RSU does not know which sensing parameters belong to vehicle i, G p ′ (p ′ = 1, . . . , P ′ ) is constructed using all sensing parameters available at the RSU. • We can see that h p ′ should be nonzero if the corresponding G p ′ belongs to vehicle i, otherwise h p ′ = 0. Hence h i = [h 1 , . . . , h P ′ ] T is a sparse vector. • To achieve joint PACESD with model (8), we need to recover both h i and x i based on y i at the same time. This is a bilinear recovery problem. Also, we note that the constraints on the bilinear recovery, i.e., h i is sparse and the entries of x i are discrete-valued as they are transmitted symbols of vehicle i. Next, leveraging the Bi-UAMP algorithm [13], we develop a message passing algorithm as the core of the PACESD algorithm. B. Probabilistic Representation and Message Passing Algorithm Design Following the Bi-UAMP algorithm, we define G = [G 1 , . . . , G P ′ ] ∈ C MN ×MN P ′ , and rearrange the order of the columns of G to get Ψ = [Ψ 1 , . . . , ΨṀ ] ∈ C MN ×MN P ′ , so that (8) can be rewritten as y i = Ψc i + n i ,(9) with the auxiliary vector c i = x i ⊗ h i = x 1 h T i , . . . , xṀ h T i T ,(10) where c i = [c 1,1 , . . . , c 1,P ′ , . . . , cṁ ,p ′ , . . . , c MN, P ′ ] T = c T 1 , . . . , c Ṫ m , . . . , c TṀ T with cṁ ,p ′ = xṁh p ′ , 1 ≤ṁ ≤Ṁ = M N . To enable the use of UAMP [16], [17], we perform singular value decomposition (SVD) of the matrix Ψ , i.e., Ψ = UΛV H , and carry out a unitary transformation with U H on (9), yielding r i = Φc i + ω i ,(11) where r i = U H y i ,Φ = ΛV H and ω i = U H n i is still AWGN with the precision β. We assume that β is unknown, which Algorithm 1 Bi-UAMP based algorithm for joint PACESD Define: Φ = [Φ1, . . . , ΦṀ ], φṁ = |Φṁ| 2 1 P ′ , and ci = c T 1 , . . . , c TṀ T ,ṁ ∈ [1,Ṁ ] and p ′ ∈ [1, P ′ ]. Initialization: vxṁ = 1, vcṁ = 1,ĉṁ = 0, z = 0, s = 0,β = 1. Repeat: Param. assoc. and chan. est.: Lines 1-15 (18) and (19) 15: 1: vp = ṁ φṁvcṁ 2: p = ṁ Φṁĉṁ − vp · z 3: v ζ = vp./(1 +βvp) 4:ζ = (βvp · ri + p)./(1 +βvp) 5:β =Ṁ /( r −ζ 2 + 1 T v ζ ) 6: vz = 1./(vp +β −1 1 P ′ ) 7: z = vz · (r − p) 8: ∀ṁ : vqṁ = 1/ |Φ Ḣ m | 2 vs 9: ∀ṁ : qṁ =ĉṁ + vqṁ Φ Ḣ m z 10: ∀ṁ : v hṁ = 1 P ′ vqṁ ./(|xṁ| 2 + vxṁ ) 11: ∀ṁ : hṁ = qṁ ·x * ṁ ./(|xṁ| 2 + vxṁ ) 12: v h i = 1 P ′ ./( ṁ 1 P ′ ./ v hṁ ) 13: hi = v h i · ṁ ( hṁ./ v hṁ ) 14: computev h i =< [v h 1 , . . . , v h P ′ ] > 1 P ′ ,ĥi = [ĥ1, . . . ,ĥ P ′ ] T Sig. det.: Lines 16-21 16: needs to be estimated as well. With (10) and (11), we aim to recover h i and x i . Inspired by the sparse Bayesian learning technique, we use sparsity-promoting hierarchical Gaussian-Gamma distribution for the entries in h i , i.e., ∀ṁ : vxṁ = vqṁ 1 P ′ ./(|ĥ| 2 + v h ) 17: ∀ṁ : xṁ = qṁĥ * /(|ĥ| 2 + v h ) 18: ∀ṁ : vxṁ = (1 T P ′ (1 P ′ ./ vxṁ )) −− v xṁ = (vxṁ vxṁ )./( vxṁ − vxṁ 1 P ′ ) 23: ∀ṁ : ← − xṁ = (xṁ vxṁ − vxṁ xṁ)./( vxṁ − vxṁ 1 P ′ ) 24: ∀ṁ : ← − v hṁ = (1./v h i − 1./ v hṁ ) −1 25: ∀ṁ : ← − hṁ = ← − v hṁ · (ĥi./v h i − hṁ./ v hṁ ) 26: ∀ṁ : ← − cṁ = ← − xṁ · ← − hṁ 27: ∀ṁ : ← − v cṁ = | ← − xṁ| 2 · ← − v hṁ + ← − v xṁ · | ← − hṁ| 2 + ← − v xṁ · ← − vp(h i |γ) = p ′ p(h p ′ |γ p ′ ) = p ′ N (h p ′ ; 0, γ −1 p ′ ),(12) where the precisions γ p ′ are Gamma distributed, i.e., p(γ p ′ ) = Ga (γ p ′ ; ǫ, η) , ∀p ′(13) with ǫ and η being the shape and rate parameters. The prior of x i can be expressed as p(x i ) = 1/|A| ṁ |A| a=1 δ(xṁ − α a ),(14) where A denotes the alphabet of the symbols in DD domain, i.e., α a ∈ A = α 1 , . . . , α |A| . Define an auxiliary variable ζ = Φc i . The joint conditional distribution of the unknown variables can be expressed as p(h i , x i , c i , ζ, β|r) ∝ p(r|ζ, β)p(ζ|c i ) × p(c i |h i , x i )p(h i )p(x i )p(β),(15) where p(r|ζ, β) = N ζ; r, β −1 I , p(ζ|c i ) = δ (ζ − Φc i ), p(c i |h i , x i ) = δ (c i − x i ⊗ h i ) and p(β) ∝ β −1 . Following ' ǫ = 1 2 log( 1 P ′ P ′ p ′ =1γ p ′ ) − 1 P ′ P ′ p ′ =1 logγ p ′ .(22) The above results lead to Line 14 of the algorithm. Following Bi-UAMP, performing the average operations on v h p ′ in (18) and arranging (17) in a vector form lead to Line 15 of the algorithm. According to Bi-UAMP, Lines 16-19 generate xṁ and v xṁ . The message from variable node xṁ to function node f xṁ in Fig. 2 also follows a Gaussian distribution, i.e., M xṁ→fxṁ (xṁ) = N (xṁ; xṁ, v xṁ ), where xṁ and v xṁ are theṁ-th element of x i and v xi , respectively. Hence, we have the following pseudo observation model xṁ = xṁ + ̟ṁ, ∀ṁ(23) where ̟ṁ denotes a model Gaussian noise with mean 0 and variance v xṁ . According to Bi-UAMP, with the prior (14) and model (23), we compute the a posteriori mean and variance of xṁ, which are given aŝ xṁ = |A| a=1 α a ̺ṁ ,a , ∀ṁ(24) and v xṁ = |A| a=1 ̺ṁ ,a |α a −xṁ| 2 , ∀ṁ(25) where ̺ṁ ,a = τṁ ,a / |A| a=1 τṁ ,a and τṁ ,a = exp(− v −1 xṁ |α a − xṁ| 2 ). In the last iteration, hard decisions are made to the symbols based on the a posteriori means {xṁ}. The above operations correspond to Line 20. Lines 21-30 are obtained following Bi-UAMP. It is noted that there is an inherent ambiguity problem with a bilinear problem. To mitigate the ambiguity, we assume a very small fraction (1/128) of the symbols in x i is known at the RSU. This only leads to an overhead about 0.78%. Algorithm 1 requires an SVD for pre-processing, and the complexity is O(N M 2 P ′ log(M P ′ )) with a modern SVD algorithm [18]. The proposed algorithm does not involve matrix inversion, and the complexity of each iteration is dominated by the matrixvector products, which is O(Ṁ P ′ ) + O(|A|) per symbol. IV. SIMULATION RESULTS In the simulation, we set N BS = 128, M = 128, N = 32, carrier frequency f c = 4 GHz and subcarrier spacing ∆f = 15 kHz. We assume the number of targets/vehicles K = 3, the number of channel paths between each target/vehicle and RSU is 6, the maximum Doppler index k max = 6, and the maximum delay index l max = 6. QPSK modulation is used. Fig. 3 shows the BER performance of the proposed algorithm. To the best of our knowledge, parameter association has not been considered in the literature. To facilitate the comparison, we assume perfect CSI and parameter association for the coventional MMSE detector, the MP based detector in [9], and the UAMP-based detector [8] (serving as a BER lower bound). It can be seen from Fig. 3 that our proposed algorithm achieves performance close to the bound, and outperforms the other two detectors. Fig. 4 shows the parameter association performance in terms of hit rate (i.e., the rate that all non-zero elements in h i are correctly detected) and false alarm rate (i.e., the rate that zero elements in h i are detected as non-zero elements). It can be seen that when the SNR is larger than 5 dB, the proposed algorithm can almost achieve a 100% hit rate, and zero false alarm rate for parameter association. Fig. 5 compares the normalized mean squared error (NMSE) performance for channel estimation. An oracle performance bound is also included, which is obtained by performing MMSE estimation with perfect parameter association and known transmitted symbols x i . It can be seen that the performance of our proposed Algorithm approaches that of the UAMP-SBL [16] with known transmitted symbols x i , and also the oracle bound with the increase of SNR. V. CONCLUSION In this paper, we have proposed a novel scheme for sensingassisted uplink transmission in ISAC OTFS vehicle networks. To enable to exploit the acquired sensing parameters, joint PACESD is formulated as a bilinear recovery problem, which is solved by developing a message passing algorithm. Simulation results show the excellent performance of the scheme with the proposed algorithm. Fig. 1 . 1Illustration of an ISAC vehicle network. characterizing the Doppler influence, l i,p = τ i,p M ∆f , 0 ≤ l i,p ≤ M − 1 and k i,p = ν i,p N T , 0 ≤ k i,p ≤ N − 1 denote the indices of delay and Doppler associated with the p-th path of vehicle i, respectively. 1 19 : 19∀ṁ : xṁ = vxṁ 1 T P ′ ( xṁ./ vxṁ ) 20: compute (25) and (26) 21: vx i = [vx 1 , . . . , vxṀ ] 1Ṁ ,xi = [x1, . . . ,xṀ ] T 22: ∀ṁ : ← hṁ 28: ∀ṁ : vcṁ = (1/vqṁ 1 P ′ + 1./ ← − v cṁ ) −1 29: ∀ṁ :ĉṁ = vcṁ · (1/vqṁ qṁ + ← − cṁ./ ← − v cṁ ) 30: ∀ṁ : vcṁ = vcṁ Until terminated Fig. 4 . 4Performance of parameter association. avoid the Xi Yang, Hang Li and Zhiqun Cheng are with the School of Electronics and Information, Hangzhou Dianzi University, Hangzhou 310018, China (e-mail: yancy486@163.com; hangli@hdu.edu.cn; zhiqun@hdu.edu.cn). Qinghua Guo is with the School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, NSW 2522, Australia (e-mail: qguo@uow.edu.au). J. Andrew Zhang and Xiaojing Huang are with the Global Big Data Technologies Centre, University of Technology Sydney, Ultimo, NSW 2007, Australia (e-mail: andrew.zhang@uts.edu.au; Xiaojing.Huang@uts.edu.au). Although pilots can be used to estimate channel state information (CSI) for signal detection, sensing is still necessary for dynamic network monitoring and vehicle safety improvement. the Bi-UAMP algorithm, we can compute the (approximate) a posteriori distributions p(xṁ|r i ) and p(h p ′ |r i ), so that their estimates in terms of the a posteriori means can be obtained.It is noted that Bi-UAMP in[13]is algorithmic framework, which does not specify the priors of the variables to be recovered. Hence, we need to derive the concrete message updating rules related to the priors of h i and x i . The relevant part of the factor graph is shown inFig. 2. The developed algorithm is summarized in Algorithm 1, with major steps detailed below.According to the derivation of Bi-UAMP, running Lines 1-13 produces a mean vector h i and variance vector v hi . The message from variable node h p ′ to function node f h p ′ inFig. 2follows a Gaussian distribution, i.e.,(12)), we obtain the belief ofwhereĥwhere the computation ofγ p ′ is given in (21). According to the mean field rule, message M f h p ′ →γ p ′ (γ p ′ ) can be expressed asCombining it with the message M fγ(13)), we can get the belief ofandγ p ′ =< γ p ′ > b(γ p ′ ) = (2ǫ + 1)/(|ĥ p ′ | 2 + v h p ′ + 2η), (21) where η is set to 0. For the shape parameter ǫ, we use the automatic tuning rule proposed in[16] Enabling joint communication and radar sensing in mobile networks-A survey. J A Zhang, IEEE Commun. Surveys Tuts. 2411st Quart.J. A. Zhang et al., "Enabling joint communication and radar sensing in mobile networks-A survey," IEEE Commun. Surveys Tuts., vol. 24, no. 1, pp. 306-345, 1st Quart., 2022. Joint radar and communication design: Applications, state-of-the-art, and the road ahead. F Liu, C Masouros, A P Petropulu, H Griffiths, L Hanzo, IEEE Trans. Commun. 686F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, "Joint radar and communication design: Applications, state-of-the-art, and the road ahead," IEEE Trans. Commun., vol. 68, no. 6, pp. 3834-3862, Jun. 2020. Orthogonal time frequency space (OTFS) modulation based radar system. P Raviteja, K T Phan, Y Hong, E Viterbo, Proc. IEEE Radar Conf. (RadarConf). IEEE Radar Conf. (RadarConf)P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, "Orthogonal time frequency space (OTFS) modulation based radar system," in Proc. IEEE Radar Conf. (RadarConf), Apr. 2019, pp. 1-6. On the effectiveness of OTFS for joint radar parameter estimation and communication. L Gaudio, M Kobayashi, G Caire, G Colavolpe, IEEE Trans. Wireless Commun. 199L. Gaudio, M. Kobayashi, G. Caire, and G. Colavolpe, "On the effective- ness of OTFS for joint radar parameter estimation and communication," IEEE Trans. Wireless Commun., vol. 19, no. 9, pp. 5951-5965, Sep.2020. A novel ISAC transmission framework based on spatiallyspread orthogonal time frequency space modulation. S Li, IEEE J. Sel. Areas Commun. 406S. Li et al., "A novel ISAC transmission framework based on spatially- spread orthogonal time frequency space modulation," IEEE J. Sel. Areas Commun., vol. 40, no. 6, pp. 1854-1872, Jun. 2022. Orthogonal time frequency space modulation. R Hadani, Proc. IEEE Wireless Commun. Netw. Conf. (WCNC). IEEE Wireless Commun. Netw. Conf. (WCNC)R. Hadani et al., "Orthogonal time frequency space modulation," in Proc. IEEE Wireless Commun. Netw. Conf. (WCNC), Mar. 2017, pp. 1-6. Receiver design for OTFS with a fractionally spaced sampling approach. Y Ge, Q Deng, P C Ching, Z Ding, IEEE Trans. Wireless Commun. 207Y. Ge, Q. Deng, P. C. Ching, and Z. Ding, "Receiver design for OTFS with a fractionally spaced sampling approach," IEEE Trans. Wireless Commun., vol. 20, no. 7, pp. 4072-4086, Jul. 2021. Iterative detection for orthogonal time frequency space modulation with unitary approximate message passing. Z Yuan, F Liu, W Yuan, Q Guo, Z Wang, J Yuan, IEEE Trans. Wireless Commun. 212Z. Yuan, F. Liu, W. Yuan, Q. Guo, Z. Wang, and J. Yuan, "Iterative detection for orthogonal time frequency space modulation with unitary approximate message passing," IEEE Trans. Wireless Commun., vol. 21, no. 2, pp. 714-725, Feb. 2022. Interference cancellation and iterative detection for orthogonal time frequency space modulation. P Raviteja, K T Phan, Y Hong, E Viterbo, IEEE Trans. Wireless Commun. 1710P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, "Interference can- cellation and iterative detection for orthogonal time frequency space modulation," IEEE Trans. Wireless Commun., vol. 17, no. 10, pp. 6501- 6515, Oct. 2018. Embedded pilot-aided channel estimation for OTFS in delay-Doppler channels. P Raviteja, K T Phan, Y Hong, IEEE Trans. Veh. Technol. 685P. Raviteja, K. T. Phan, and Y. Hong, "Embedded pilot-aided channel estimation for OTFS in delay-Doppler channels," IEEE Trans. Veh. Technol., vol. 68, no. 5, pp. 4906-4917, May 2019. Data-aided channel estimation for OTFS systems with a superimposed pilot and data transmission scheme. W Yuan, S Li, Z Wei, J Yuan, D W K Ng, IEEE Wireless Commun. Lett. 109W. Yuan, S. Li, Z. Wei, J. Yuan, and D. W. K. Ng, "Data-aided channel estimation for OTFS systems with a superimposed pilot and data transmission scheme," IEEE Wireless Commun. Lett., vol. 10, no. 9, pp. 1954-1958, Sep. 2021. Integrated sensing and communication-assisted orthogonal time frequency space transmission for vehicular networks. W Yuan, Z Wei, S Li, J Yuan, D W K Ng, IEEE J. Sel. Topics Signal Process. 156W. Yuan, Z. Wei, S. Li, J. Yuan, and D. W. K. Ng, "Integrated sensing and communication-assisted orthogonal time frequency space transmission for vehicular networks," IEEE J. Sel. Topics Signal Process., vol. 15, no. 6, pp. 1515-1528, Nov. 2021. Approximate message passing with unitary transformation for robust bilinear recovery. Z Yuan, Q Guo, M Luo, IEEE Trans. Signal Processing. 69Z. Yuan, Q. Guo, and M. Luo, "Approximate message passing with unitary transformation for robust bilinear recovery," IEEE Trans. Signal Processing, vol. 69, pp. 617-630, 2021. A simple two-stage equalizer with simplified orthogonal time frequency space modulation over rapidly time-varying channels. L Li, H Wei, Y Huang, Y Yao, W Ling, G Chen, P Li, Y Cai, abs/1709.02505CoRR. L. Li, H. Wei, Y. Huang, Y. Yao, W. Ling, G. Chen, P. Li, and Y. Cai, "A simple two-stage equalizer with simplified orthog- onal time frequency space modulation over rapidly time-varying channels," CoRR, vol. abs/1709.02505, 2017. [Online]. Available: http://arxiv.org/abs/1709.02505 Practical pulse-shaping waveforms for reduced-cyclic-prefix OTFS. P Raviteja, Y Hong, E Viterbo, E Biglieri, IEEE Trans. Veh. Technol. 681P. Raviteja, Y. Hong, E. Viterbo, and E. Biglieri, "Practical pulse-shaping waveforms for reduced-cyclic-prefix OTFS," IEEE Trans. Veh. Technol., vol. 68, no. 1, pp. 957-961, Jan. 2019. Unitary approximate message passing for sparse bayesian learning. M Luo, Q Guo, M Jin, Y C Eldar, D Huang, X Meng, IEEE Trans. Signal Process. 69M. Luo, Q. Guo, M. Jin, Y. C. Eldar, D. Huang, and X. Meng, "Unitary approximate message passing for sparse bayesian learning," IEEE Trans. Signal Process., vol. 69, pp. 6023-6039, 2021. Approximate message passing with unitary transformation. Q Guo, J Xi, abs/1504.04799CoRR. Q. Guo and J. Xi, "Approximate message passing with unitary transformation," CoRR, vol. abs/1504.04799, 2015. [Online]. Available: http://arxiv.org/abs/1504.04799 Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. N Halko, P G Martinsson, J A Tropp, 10.1137/090771806SIAM Review. 532N. Halko, P. G. Martinsson, and J. A. Tropp, "Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions," SIAM Review, vol. 53, no. 2, pp. 217-288, 2011. [Online]. Available: https://doi.org/10.1137/090771806

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{'abstract': 'In this work, we study sensing-aided uplink transmission in an integrated sensing and communication (ISAC) vehicular network with the use of orthogonal time frequency space (OTFS) modulation. To exploit sensing parameters for improving uplink communications, the parameters must be first associated with the transmitters, which is a challenging task. We propose a scheme that jointly conducts parameter association, channel estimation and signal detection by formulating it as a constrained bilinear recovery problem. Then we develop a message passing algorithm to solve the problem, leveraging the bilinear unitary approximate message passing (Bi-UAMP) algorithm. Numerical results validate the proposed scheme, which show that relevant performance bounds can be closely approached.Index Terms-ISAC, joint channel estimation and signal detection, (unitary) approximate message passing, OTFS.', 'arxivid': '2305.11548', 'author': ['Xi Yang ', 'Hang Li ', 'Senior Member, IEEEQinghua Guo ', 'Senior Member, IEEEJ Andrew Zhang ', 'Senior Member, IEEEXiaojing Huang ', 'Member, IEEE.Zhiqun Cheng ', 'Xi Yang ', 'Hang Li ', 'Senior Member, IEEEQinghua Guo ', 'Senior Member, IEEEJ Andrew Zhang ', 'Senior Member, IEEEXiaojing Huang ', 'Member, IEEE.Zhiqun Cheng '], 'authoraffiliation': [], 'corpusid': 258822967, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9234, 'n_tokens_neox': 8235, 'n_words': 4711, 'pdfsha': '681b89db8883f646c7684abea1ced6c664eb5842', 'pdfurls': ['https://export.arxiv.org/pdf/2305.11548v1.pdf'], 'title': ['Sensing Aided Uplink Transmission in OTFS ISAC with Joint Parameter Association, Channel Estimation and Signal Detection', 'Sensing Aided Uplink Transmission in OTFS ISAC with Joint Parameter Association, Channel Estimation and Signal Detection', 'Sensing Aided Uplink Transmission in OTFS ISAC with Joint Parameter Association, Channel Estimation and Signal Detection', 'Sensing Aided Uplink Transmission in OTFS ISAC with Joint Parameter Association, Channel Estimation and Signal Detection'], 'venue': []}

arxiv

Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-polynomial time 19 Aug 2018 Jatin Batra Department of Computer Science and Engineering IIT Delhi Naveen Garg Department of Computer Science and Engineering IIT Delhi Amit Kumar Department of Computer Science and Engineering IIT Delhi Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-polynomial time 19 Aug 2018 In the weighted flow-time problem on a single machine, we are given a set of n jobs, where each job has a processing requirement p j , release date r j and weight w j . The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudo-polynomial time constant approximation algorithm for this problem. The algorithm also extends directly to the problem of minimizing the ℓ p norm of weighted flow-times. The running time of our algorithm is polynomial in n, the number of jobs, and P , which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multi-cut problem on trees, which we call Demand MultiCut problem. Even though we do not give a constant factor approximation algorithm for the Demand MultiCut problem on trees, we show that the specific instances of Demand MultiCut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of DP table. Introduction Scheduling jobs to minimize the average waiting time is one of the most fundamental problems in scheduling theory with numerous applications. We consider the setting where jobs arrive over time (i.e., have release dates), and need to be processed such that the average flow-time is minimized. The flow-time, F j of a job j, is defined as the difference between its completion time, C j , and release date, r j . It is well known that for the case of single machine, the SRPT policy (Shortest Remaining Processing Time) gives an optimal algorithm for this objective. In the weighted version of this problem, jobs have weights and we would like to minimize the weighted sum of flow-time of jobs. However, the problem of minimizing weighted flow-time (WtdFlowTime) turns out to be NP-hard and it has been widely conjectured that there should a constant factor approximation algorithm (or even PTAS) for it. In this paper, we make substantial progress towards this problem by giving the first constant factor approximation algorithm for this problem in pseudo-polynomial time. More formally, we prove the following result. Theorem 1.1. There is a constant factor approximation algorithm for WtdFlowTime where the running time of the algorithm is polynomial in n and P . Here, n denotes the number of jobs in the instance, and P denotes the ratio of the largest to the smallest processing time of a job in the instance respectively. We obtain this result by reducing WtdFlowTime to a generalization of the multi-cut problem on trees, which we call Demand MultiCut. The Demand MultiCut problem is a natural generalization of the multi-cut problem where edges have sizes and costs, and input paths (between terminal pairs) have demands. We would like to select a minimum cost subset of edges such that for every path in the input, the total size of the selected edges in the path is at least the demand of the path. When all demands and sizes are 1, this is the usual multi-cut problem. The natural integer program for this problem has the property that all non-zero entries in any column of the constraint matrix are the same. Such integer programs, called column restricted covering integer programs, were studied by Chakrabarty et al. [7]. They showed that one can get a constant factor approximation algorithm for Demand MultiCut provided one could prove that the integrality gap of the natural LP relaxations for the following two special cases is constant -(i) the version where the constraint matrix has 0-1 entries only, and (ii) the priority version, where paths and edges in the tree have priorities (instead of sizes and demands respectively), and we want to pick minimum cost subset of edges such that for each path, we pick at least one edge in it of priority which is at least the priority of this path. Although the first problem turns out to be easy, we do not know how to round the LP relaxation of the priority version. This is similar to the situation faced by Bansal and Pruhs [4], where they need to round the priority version of a geometric set cover problem. They appeal to the notion of shallow cell complexity [8] to get an O(log log P )-approximation for this problem. It turns out the shallow cell complexity of the priority version of Demand MultiCut is also unbounded (depends on the number of distinct priorities) [8], and so it is unlikely that this approach will yield a constant factor approximation. However, the specific instances of Demand MultiCut produced by our reduction have more structure, namely each node has at most 2 children, each path goes from an ancestor to a descendant, and the tree has O(log(nP )) depth if we shortcut all degree 2 vertices. We show that one can effectively use dynamic programming techniques for such instances. We show that there is a near optimal solution which has nice "smoothness" properties so that the dynamic programming table can manage with storing small amount of information. Related Work There has been a lot of work on the WtdFlowTime problem on a single machine, though polynomial time constant factor approximation algorithm has remained elusive. Bansal and Dhamdhere [1] gave an O(log W )-competitive on-line algorithm for this problem, where W is the ratio of the maximum to the minimum weight of a job. They also gave a semi-online (where the algorithm needs to know the parameters P and W in advance) O(log(nP ))-competitive algorithm for WtdFlowTime, where P is the ratio of the largest to the smallest processing time of a job. Chekuri et al. [10] gave a semi-online O(log 2 P )-competitive algorithm. Recently, Bansal and Pruhs [4] made significant progress towards this problem by giving an O(log log P )-approximation algorithm. In fact, their result applies to a more general setting where the objective function is j f j (C j ), where f j (C j ) is any monotone function of the completion time C j of job j. Their work, along with a constant factor approximation for the generalized caching problem [5], implies a constant factor approximation algorithm for this setting when all release dates are 0. Chekuri and Khanna [9] gave a quasi-PTAS for this problem, where the running time was O(n Oǫ(log W log P ) ). In the special case of stretch metric, where w j = 1/p j , PTAS is known [6,9]. The problem of minimizing (unweighted) ℓ p norm of flow-times was studied by Im and Moseley [12] who gave a constant factor approximation in polynomial time. In the speed augmentation model introduced by Kalyanasundaram and Pruhs [13], the algorithm is given (1 + ε)-times extra speed than the optimal algorithm. Bansal and Pruhs [3] showed that Highest Density First (HDF) is O(1)-competitive for weighted ℓ p norms of flow-time for all values of p ≥ 1. The multi-cut problem on trees is known to be NP-hard, and a 2-approximation algorithm was given by Garg et al. [11]. As mentioned earlier, Chakrabarty et al. [7] gave a systematic study of column restricted covering integer programs (see also [2] for follow-up results). The notion of shallow cell complexity for 0-1 covering integer programs was formalized by Chan et al. [8], where they relied on and generalized the techniques of Vardarajan [14]. Preliminaries An instance of the WtdFlowTime problem is specified by a set of n jobs. Each job has a processing requirement p j , weight w j and release date r j . We assume wlog that all of these quantities are integers, and let P denote the ratio of the largest to the smallest processing requirement of a job. We divide the time line into unit length slots -we shall often refer to the time slot [t, t + 1] as slot t. A feasible schedule needs to process a job j for p j units after its release date. Note that we allow a job to be preempted. The weighted flow-time of a job is defined as w j · (C j − r j ), where C j is the slot in which the job j finishes processing. The objective is to find a schedule which minimizes the sum over all jobs of their weighted flow-time. Note that any schedule would occupy exactly T = j p j slots. We say that a schedule is busy if it does not leave any slot vacant even though there are jobs waiting to be finished. We can assume that the optimal schedule is a busy schedule (otherwise, we can always shift some processing back and improve the objective function). We also assume that any busy schedule fills the slots in [0, T ] (otherwise, we can break it into independent instances satisfying this property). We shall also consider a generalization of the multi-cut problem on trees, which we call the Demand MultiCut problem. Here, edges have cost and size, and demands are specified by ancestordescendant paths. Each such path has a demand, and the goal is to select a minimum cost subset of edges such that for each path, the total size of selected edges in the path is at least the demand of this path. In Section 2.1, we describe a well-known integer program for WtdFlowTime. This IP has variables x j,t for every job j, and time t ≥ r j , and it is supposed to be 1 if j completes processing after time t. The constraints in the IP consist of several covering constraints. However, there is an additional complicating factor that x j,t ≤ x j,t−1 must hold for all t ≥ r j . To get around this problem, we propose a different IP in Section 3. In this IP, we define variables of the form y(j, S), where S are exponentially increasing intervals starting from the release date of j. This variable indicates whether j is alive during the entire duration of S. The idea is that if the flow-time of j lies between 2 i and 2 i+1 , we can count 2 i+1 for it, and say that j is alive during the entire period [r j +2 i , r j +2 i+1 ]. Conversely, if the variable y(j, S) is 1 for an interval of the form [r j +2 i , r j +2 i+1 ], we can assume (at a factor 2 loss) that it is also alive during [r j , r j +2 i ]. This allows us to decouple the y(j, S) variables for different S. By an additional trick, we can ensure that these intervals are laminar for different jobs. From here, the reduction to the Demand MultiCut problem is immediate (see Section 4 for details). In Section 5, we show that the specific instances of Demand MultiCut obtained by such reductions have additional properties. We use the property that the tree obtained from shortcutting all degree two vertices is binary and has O(log(nP )) depth. We shall use the term segment to define a maximal degree 2 (ancestor-descendant) path in the tree. So the property can be restated asany root to leaf path has at most O(log(nP )) segments. We give a dynamic programming algorithm for such instances. In the DP table for a vertex in the tree, we will look at a sub-instance defined by the sub-tree below this vertex. However, we also need to maintain the "state" of edges above it, where the state means the ancestor edges selected by the algorithm. This would require too much book-keeping. We use two ideas to reduce the size of this state -(i) We first show that the optimum can be assumed to have certain smoothness properties, which cuts down on the number of possible configurations. The smoothness property essentially says that the cost spent by the optimum on a segment does not vary by more than a constant factor as we go to neighbouring segments, (ii) If we could spend twice the amount spent by the algorithm on a segment S, and select low density edges, we could ignore the edges in a segment S ′ lying above S in the tree. An integer program We describe an integer program for the WtdFlowTime problem. This is well known (see e.g. [4]), but we give details for sake of completeness. We will have binary variables x j,t for every job j and time t, where r j ≤ t ≤ T . This variable is meant to be 1 iff j is alive at time t, i.e., its completion time is at least t. Clearly, the objective function is j t∈[r j ,T ] w j x j,t . We now specify the constraints of the integer program. Consider a time interval I = [s, t], where 0 ≤ s ≤ t ≤ T , and s and t are integers. Let l(I) denote the length of this time interval, i.e., t − s. Let J(I) denote the set of jobs released during I, i.e., {j : r j ∈ I}, and p(J(I)) denote the total processing time of jobs in J(I). Clearly, the total volume occupied by jobs in J(I) beyond I must be at least p(J(I)) − l(I). Thus, we get the following integer program: (IP1) min j t∈[r j ,T ] w j x j,t(1)j∈J(I) x j,t p j ≥ p(J(I)) − l(I) for all intervals I = [s, t], 0 ≤ s ≤ t ≤ T(2) x j,t ≤ x j,t−1 for all jobs j, and time t, r j < t ≤ T x j,t ∈ {0, 1} for all j, t It is easy to see that this is a relaxation -given any schedule, the corresponding x j,t variables will satisfy the constraints mentioned above, and the objective function captures the total weighted flow-time of this schedule. The converse is also true -given any solution to the above integer program, there is a corresponding schedule of the same cost. Theorem 2.1. Suppose x j,t is a feasible solution to (IP1). Then, there is a schedule for which the total weighted flow-time is equal to the cost of the solution x j,t . Proof. We show how to build such a schedule. The integral solution x gives us deadlines for each job. For a job j, define d j as one plus the last time t such that x j,t = 1. Note that x j,t = 1 for every t ∈ [r j , d j ). We would like to find a schedule which completes each job by time d j : if such a schedule exists, then the weighted flow-time of a job j will be at most t≥r j w j x j,t , which is what we want. We begin by observing a simple property of a feasible solution to the integer program. Claim 2.2. Consider an interval I = [s, t], 0 ≤ s ≤ t ≤ T . Let J ′ be a subset of J(I) such that p(J ′ ) > l(I) . If x is a feasible solution to (IP1), then there must exist a job j ∈ J ′ such that x j,t = 1. Proof. Suppose not. Then the LHS of constraint (2) for I would be at most p(J(I) \ J ′ ), whereas the RHS would be p( J ′ ) + p(J(I) \ J ′ ) − l(I) > p(J(I) \ J ′ ), a contradiction. It is natural to use the Earliest Deadline First rule to find the required schedule. We build the schedule from time t = 0 onwards. At any time t, we say that a job j is alive if r j ≤ t, and j has not been completely processed by time t. Starting from time t = 0, we process the alive job with earliest deadline d j during [t, t + 1]. We need to show that every job will complete before its deadline. Suppose not. Let j be the job with the earliest deadline which is not able to finish by d j . Let t be first time before d j such that the algorithm processes a job whose deadline is more than d j during [t − 1, t], or it is idle during this time slot (if there is no such time slot, it must have busy from time 0 onwards, and so set t to 0). The algorithm processes jobs whose deadline is at most d j during [t, d j ] -call these jobs J ′ . We claim that jobs in J ′ were released after t -indeed if such a job was released before time t, it would have been alive at time t − 1 (since it gets processed after time t). Further its deadline is at most d j , and so, the algorithm should not be processing a job whose deadline is more than d j during [t − 1, t] (or being idle). But now, consider the interval I = [t, d j ]. Observe that l(I) < p(J ′ ) -indeed, j ∈ J ′ and it is not completely processed during I, but the algorithm processes jobs from J ′ only during I. Claim 2.2 now implies that there must be a job j ′ in J ′ for which x j ′ ,d j = 1 -but then the deadline of j ′ is more than d j , a contradiction. A Different Integer Program We now write a weaker integer program, but it has more structure in it. We first assume that T is a power of 2 -if not, we can pad the instance with a job of zero weight (this will increase the ratio P by at most a factor n only). Let T be 2 ℓ . We now divide the time line into nested dyadic segments. A dyadic segment is an interval of the form [i · 2 s , (i + 1) · 2 s ] for some non-negative integers i and s (we shall use segments to denote such intervals to avoid any confusion with intervals used in the integer program). For s = 0, . . . , ℓ, we define S s as the set of dyadic segments of length 2 s starting from 0, i.e., T ]}. Clearly, any segment of S s is contained inside a unique segment of S s+1 . Now, for every job j we shall define a sequence of dyadic segments Seg(j). The sequence of segments in Seg(j) partition the interval [r j , T ]. The construction of Seg(j) is described in Figure 1 (also see the example in Figure 2). It is easy to show by induction on s that the parameter t at the beginning of iteration s in Step 2 of the algorithm is a multiple of 2 s . Therefore, the segments added during the iteration for s belong to S s . Although we do not specify for how long we run the for loop in Step 2, we stop when t reaches T (this will always happen because t takes values from the set of end-points in the segments in ∪ s S s ). Therefore the set of segments in Seg(j) are disjoint and cover [r j , T ]. Algorithm FormSegments(j) 1. Initialize t ← r j . 2. For s = 0, 1, 2, . . . , (i) If t is a multiple of 2 s+1 , add the segments (from the set S s ) [t, t + 2 s ], [t + 2 s , t + 2 s+1 ] to Seg(j) update t ← t + 2 s+1 . (ii) Else add the segment (from the set S s ) [t, , t + 2 s ] to Seg(j). update t ← t + 2 s . For a job j and segment S ∈ Seg(j), we shall refer to the tuple (j, S) as a job-segment. For a time t, we say that t ∈ (j, S) (or (j, S) contains t) if [t, t + 1] ⊆ S. We now show a crucial nesting property of these segments. Lemma 3.1. Suppose (j, S) and (j ′ , S ′ ) are two job-segments such that there is a time t for which t ∈ (j, S) and t ∈ (j ′ , S ′ ). Suppose r j ≤ r j ′ , and S ∈ S s , S ∈ S s ′ . Then s ≥ s ′ . Proof. We prove this by induction on t. When t = r j ′ , this is trivially true because s ′ would be 0. Suppose it is true for some t ≥ r j ′ . Let (j, S) and (j ′ , S ′ ) be the job segments containing t. Suppose S ∈ S s , S ′ ∈ S s ′ . By induction hypothesis, we know that s ≥ s ′ . Let (j ′ ,S ′ ) be the job-segment containing t + 1, and letS ′ ∈ Ss′ (S ′ could be same asS ′ ). We know thats ′ ≤ s ′ + 1. Therefore, the only interesting case is s = s ′ ands ′ = s ′ + 1. Since s = s ′ , the two segments S and S ′ must be same (because all segments in S s are mutually disjoint). Since t ∈ S, t + 1 / ∈ S, it must be that S = [l, t + 1] for some l. The algorithm for constructing Seg(j ′ ) adds a segment from S s ′ +1 after adding S ′ to Seg(j ′ ). Therefore t + 1 must be a multiple of 2 s ′ +1 . What does the algorithm for constructing Seg(j) do after adding S to Seg(j)? If it adds a segment from S s+1 , then we are done again. Suppose it adds a segment from S s . The right end-point of this segment would be (t + 1) + 2 s . After adding this segment, the algorithm would add a segment from S s+1 (as it cannot add more than 2 segments from S s to Seg(j)). But this can only happen if (t + 1) + 2 s is a multiple of 2 s+1 -this is not true because (t + 1) is a multiple of 2 s+1 . Thus we get a contradiction, and so the next segment (after S) in Seg(j) must come from S s+1 as well. We now write a new IP. The idea is that if a job j is alive at some time t, then we will keep it alive during the entire duration of the segment in Seg(j) containing t. Since the segments in Seg(j) have lengths in exponentially increasing order (except for two consecutive segments), this will not increase the weighted flow-time by more than a constant factor. For each job segment (j, S) we have a binary variable y(j, S), which is meant to be 1 iff the job j is alive during the entire duration S. For each job segment (j, S), define its weight w(j, S) as w j · l(S) -this is the contribution towards weighted flow-time of j if j remains alive during the entire segment S. We get the following integer program (IP2): Observe that for any interval I, the constraint (5) for I has precisely one job segment for every job which gets released in I. Another interesting feature of this IP is that we do not have constraints corresponding to (3), and so it is possible that y(j, S) = 1 and y(j, S ′ ) = 0 for two job segments (j, S) and (j, S ′ ) even though S ′ appears before S in Seg(j). We now relate the two integer programs. Proof. Suppose we are given a solution x for (IP1). For every job j, let d j be the highest t for which x jt = 1. Let the segments in Seg(j) (in the order they were added) be S 1 , S 2 , . . .. Let S i j be the segment in Seg(j) which contains d j . Then we set y(j, S i ) to 1 for all i ≤ i j , and y(j, S i ) to 0 for all i > i j . This defines the solution y. First we observe that y is feasible for (IP2). Indeed, consider an interval I = [s, t]. If x jt = 1 and j ∈ J(I), then we do have y(j, S) = 1 for the job segment (j, S) containing t. Therefore, the LHS of constraints (2) and (5) for I are same. Also, observe that S∈Seg(j) y(j, S)w(j, S) = i j i=1 w j · l(S i ) ≤ w j 4l(S i j ), where the last inequality follows from the fact that there are at most two segments from any particular set S s in Seg(j), and so, the length of every alternate segment in Seg(j) increases exponentially. So, i j i=1 l(s i ) ≤ 2 l(S i j ) + l(S i j −2 ) + l(S i j −4 ) + · · · ≤ 4 · l(S i j ). Finally observe that l(S i j ) ≤ 2(d j − r j ). Indeed, the length of S i j−1 is at least half of that of S i j . So, l(S i j ) ≤ 2l(S i j −1 ) ≤ 2(d j − r j ). Thus, the total contribution to the cost of y from job segments corresponding to j is at most 8w j (d j − r j ) = 8w j t≥r j x j,t . This proves the first statement in the lemma. Now we prove the second statement. Let y be a solution to (IP2). For each job j, let S i j be the last job segment in Seg(j) = {S 1 , S 2 , . . .} for which y(j, S) is 1. We set x j,t to 1 for every t ≤ d j , where d j is the right end-point of S i j , and 0 for t > d j . It is again easy to check that x is a feasible solution to (IP1). For a job j the contribution of j towards the cost of x is w j (d j − r j ) = w j · i j i=1 l(S i ) ≤ 4w j · l(S i j ) ≤ 4 · (j,S)∈Seg(j) w(j, S)y(j, S). The above lemma states that it is sufficient to find a solution for (IP2). Note that (IP2) is a covering problem. It is also worth noting that the constraints (5) need to be written only for those intervals [s, t] for which a job segment starts or ends at s or t. Since the number of job segments is O(n log T ) = O(n log(nP )), it follows that (IP2) can be turned into a polynomial size integer program. Reduction to Demand MultiCut on Trees We now show that (IP2) can be viewed as a covering problem on trees. We define the covering problem, which we call Demand Multi-cut(Demand MultiCut) on trees. An instance I of this problem consists of a tuple (T , P, c, p, d), where T is a rooted tree, and P consists of a set of ancestor-descendant paths. Each edge e in T has a cost c e and size p e . Each path P inP has a demand d(P ). Our goal is to pick a minimum cost subset of vertices V ′ such that for every path P ∈ P, the set of vertices in V ′ ∩ P have total size at least d(P ). We now reduce WtdFlowTime to Demand MultiCut on trees. Consider an instance I ′ of WtdFlowTime consisting of a set of jobs J. We reduce it to an instance I = (T , P, c, p, d) of Demand MultiCut. In our reduction, T will be a forest instead of a tree, but we can then consider each tree as an independent problem instance of Demand MultiCut. We order the jobs in J according to release dates (breaking ties arbitrarily) -let ≺ J be this total ordering (so, j ≺ J j ′ implies that r j ≤ r j ′ ). We now define the forest T . The vertex set of T will consist of all job segments (j, S). For such a vertex (j, S), let j ′ be the job immediately preceding j in the total order ≺ J . Since the job segments in Seg(j ′ ) partition [r j ′ , T ], and r j ′ ≤ r j , there is a pair (j ′ , S ′ ) in Seg(j ′ ) such that S ′ intersects S, and so contains S, by Lemma 3.1. We define (j ′ , S ′ ) as the parent of (j, S). It is easy to see that this defines a forest structure, where the root vertices correspond to (j, S), with j being the first job in ≺. Indeed, if (j 1 , S 1 ), (j 2 , S 2 ), . . . , (j k , S k ) is a sequence of nodes with (j i , S i ) being the parent of (j i+1 , S i+1 ), then j 1 ≺ J j 2 ≺ J · · · ≺ J j k , and so no node in this sequence can be repeated. For each tree in this forest T with the root vertex being (j, S), we add a new root vertex r and make it the parent of (j, S). We now define the cost and size of each edge. Let e = (v 1 , v 2 ) be an edge in the tree, where v 1 is the parent of v 2 . Let v 2 correspond to the job segment (j, S). Then p e = p j and c e = w e · l(S). In other words, picking edge e corresponds to selecting the job segment (j, S). Now we define the set of paths P. For each constraint (5) in (IP2), we will add one path in P. We first observe the following property. Fix an interval I = [s, t] and consider the constraint (5) corresponding to it. Let V I be the vertices in T corresponding to the job segments appearing in the LHS of this constraint. Proof. Let j 1 , . . . , j k be the jobs which are released in I arranged according to ≺ J . Note that these will form a consecutive subsequence of the sequence obtained by arranging jobs according to ≺ J . Each of these jobs will have exactly one job segment (j i , S i ) appearing on the LHS of this constraint (because for any such job j i , the segments in Seg(j i ) partition [r j i , T ]). All these job segments contain t, and so, these segment intersect. Now, by construction of T , it follows that the parent of (j i , S i ) in the tree T would be (j i−1 , S i−1 ). This proves the claim. Let the vertices in V I be v 1 , . . . , v k arranged from ancestor to descendant. Let v 0 be the parent of v 1 (this is the reason why we added an extra root to each tree -just in case v 1 corresponds to the first job in ≺ J , it will still have a parent). We add a path P I = v 0 , v 1 , . . . , v k to P -Lemma 4.1 guarantees that this will be an ancestor-descendant path. The demand d(P ) of this path is the quantity in the RHS of the corresponding constraint (5) for the interval I. The following claim is now easy to check. = (v 1 , v 2 ) ∈ E where v 2 = (j, S) is the child of v 1 , we set y(j, S) = 1. For rest of the job segments (j, S), define y(j, S) to be 0. Since the cost of such an edge e is equal to w(j, S), it is easy to see that the two solutions have the same cost. Feasibility of (IP2) also follows directly from the manner in which the paths in P are defined. This completes the reduction from WtdFlowTime to Demand MultiCut. This reduction is polynomial time because number of vertices in T is equal to the number of job segments, which is O(n log(nP )). Each path in P goes between any two vertices in T , and there is no need to have two paths between the same pair of vertices. Therefore the size of the instance I is polynomial in the size of the instance I ′ of WtdFlowTime. Approximation Algorithm for the Demand MultiCut problem In this section we give a constant factor approximation algorithm for the special class of Demand MultiCut problems which arise in the reduction from WtdFlowTime. To understand the special structure of such instances, we begin with some definitions. Let I = (T , P, c, p, d) be an instance of Demand MultiCut. The density ρ e of an edge e is defined as the ratio c e /p e . Let red(T ) denote the tree obtained from T by short-cutting all non-root degree 2 vertices (see Figure 3 for an example). There is a clear correspondence between the vertices of red(T ) and the non-root vertices in T which do not have degree 2. In fact, we shall use V (red(T )) to denote the latter set of vertices. The reduced height of T is defined as the height of red(T ). In this section, we prove the following result. We say that a (rooted) tree is binary if every node has at most 2 children. Theorem 5.1. There is a constant factor approximation algorithm for instances I = (T , P, c, p, d) of Demand MultiCut where T is a binary tree. The running time of this algorithm is poly(n, 2 O(H) , ρ max /ρ min ), where n denotes the number of nodes in T , H denotes the reduced height of T , and ρ max and ρ min are the maximum and the minimum density of an edge in T respectively. Remark: In the instance I above, some edges may have 0 size. These edges are not considered while defining ρ max and ρ min . Before we prove this theorem, let us see why it implies the main result in Theorem 1.1. Proof of Theorem 1.1: Consider an instance I = (T , P, c, p, d) of DemandMultiCut obtained via reduction from an instance I ′ of WtdFlowTime. Let n ′ denote the number of jobs in I ′ and P denote Figure 3: Tree T and the corresponding tree red(T ). Note that the vertices in red(T ) are also present in T , and the segments in T correspond to edges in red(T ). The tree T has 4 segments, e.g., the path between r and u. r u v w x r u x w v T red(T ) the ratio of the largest to the smallest job size in this instance. We had argued in the previous section that n, the number of nodes in T , is O(n ′ log P ). We first perform some pre-processing on T such that the quantites H, ρ max /ρ min do not become too large. • Let p max and p min denote the maximum and the minimum size of a job in the instance I ′ . Each edge in T corresponds to a job interval in the instance I. We select all edges for which the corresponding job interval has length at most p min . Note that after selecting these edges, we will contract them in T and adjust the demands of paths in P accordingly. For a fixed job j, the total cost of such selected edges would be at most 4w j p min ≤ 4w j p j (as in the proof of Lemma 3.2, the corresponding job intervals have lengths which are powers of 2, and there are at most two intervals of the same length). Note that the cost of any optimal solution for I ′ is at least j w j p j , and so we are incurring an extra cost of at most 4 times the cost of the optimal solution. So we can assume that any edge in T corresponds to a job interval in I ′ whose length lies in the range [p min , n ′ p max ], because the length of the schedule is at most n ′ p max (recall that we are assuming that there are no gaps in the schedule). • Let c max be the maximum cost of an edge selected by the optimal solution (we can cycle over all n possibilities for c max , and select the best solution obtained over all such solutions). We remove (i.e., contract) all edges of cost more than c max , and select all edges of cost at most c max /n (i.e., contract them and adjust demands of paths going through them) -the cost of these selected edges will be at most a constant times the optimal cost. Therefore, we can assume that the costs of the edges lie in the range [c max /n, c max ]. Therefore, the densities of the edges in T lie in the range [ cmax npmax , cmax p min ]. Having performed the above steps, we now modify the tree T so that it becomes a binary tree. Recall that each vertex v in T corresponds to a dyadic interval S v , and if w is a child of v then S w is contained in S v (for the root vertex, we can assign it the dyadic interval [0, T ]). Now, consider a vertex v with S v of size 2 s and suppose it has more than 2 children. Since the dyadic intervals for the children are mutually disjoint and contained in S v , each of these will be of size at most 2 s−1 . Let S 1 v and S 2 v be the two dyadic intervals of length 2 s−1 contained in S v . Consider S 1 v . Let w 1 , . . . , w k be the children of v for which the corresponding interval is contained in S 1 v . If k > 1, we create a new node w below v (with corresponding interval being S 1 v ) and make w 1 , . . . , w k children of v. The cost and size of the edge (v, w) is 0. We proceed similarly for S 2 v . Thus, each node will now have at most 2 children. Note that we will blow up the number of vertices by a factor 2 only. We can now estimate the reduced height H of T . Consider a root to leaf path in red(T ), and let the vertices in this path be v 1 , . . . , v k . Let e i denote the parent of v i . Since each v i has two children in T , the job interval corresponding to e i will be at least twice that for e i+1 . From the first preprocessing step above, it follows that the length of this path is bounded by log(n ′ P ), where P denotes p max /p min . Thus, H is O(log(n ′ P )). It now follows from Theorem 5.1 that we can get a constant factor approximation algorithm for the instance I in poly(n, P ) time. We now prove Theorem 5.1 in rest of the paper. Some Special Cases To motivate our algorithm, we consider some special cases first. Again, fix an instance I = (T , P, c, p, d) of Demand MultiCut. Recall that the tree red(T ) is obtained by short-cutting all degree 2 vertices in T . Each edge in red(T ) corresponds to a path in T -in fact, there are maximal paths in T for which all internal nodes have degree 2. We call such paths segments (to avoid confusion with paths in P). See Figure 3 for an example. Thus, there is a 1-1 correspondence between edges in red(T ) and segments in T . Recall that every vertex in red(T ) corresponds to a vertex in T as well, and we will use the same notation for both the vertices. Figure 4: The left instance represents a segment confined instance whereas the right one is a segment spanning instance. Segment Confined Instances The instance I is said to be segment confined if all paths in P are confined to one segment, i.e., for every path P ∈ P, there is a segment S in T such that the edges of P are contained in S. An example is shown in Figure 4. In this section, we show that one can obtain constant factor polynomial time approximation algorithms for such instances. In fact, this result follows from prior work on column restricted covering integer programs [7]. Since each path in P is confined to one segment, we can think of this instance as several independent instances, one for each segment. For a segment S, let I S be the instance obtained from I by considering edges in S only and the subset P S ⊆ P of paths which are contained in S. We show how to obtain a constant factor approximation algorithm for I S for a fixed segment S. x e ∈ {0, 1} for all e ∈ S Note that this is a covering integer program (IP) where the coefficient of x e in each constraint is either 0 or p e . Such an IP comes under the class of Column Restricted Covering IP as described in [7]. Chakrabarty et al. [7] show that one can obtain a constant factor approximation algorithm for this problem provided one can prove that the integrality gaps of the corresponding LP relaxations for the following two special class of problems are constant: (i) 0-1 instances, where the p e values are either 0 or 1, (ii) priority versions, where paths in P and edges have priorities (which can be thought of as positive integers), and the selected edges satisfy the property that for each path P ∈ P S , we selected at least one edge in it of priority at least that of P (it is easy to check that this is a special case of Demand MultiCut problem by assigning exponentially increasing demands to paths of increasing priority, and similarly for edges). Consider the class of 0-1 instances first. We need to consider only those edges for which p e is 1 ( contract the edges for which p e is 0). Now observe that the constraint matrix on the LHS in (IP3) has consecutive ones property (order the paths in P S in increasing order of left end-point and write the constraints in this order). Therefore, the LP relaxation has integrality gap of 1. Rounding the Priority Version We now consider the priority version of this problem. For each edge e ∈ S, we now have an associated priority p e (instead of size), and each path in P also has a priority demand p(P ), instead of its demand. We need to argue about the integrality gap of the following LP relaxation: min e∈S c e x e(9) e∈P :pe≥p(P ) x e ≥ 1 for all paths P ∈ P S (10) x e ≥ 0 for all e ∈ S We shall use the notion of shallow cell complexity used in [8]. Let A be the constraint matrix on the LHS above. We first notice the following property of A. Claim 5.2. Let A ⋆ be a subset of s columns of A. For a parameter k, 0 ≤ k ≤ s, there are at most k 2 s distinct rows in A ⋆ with k or fewer 1's (two rows of A ⋆ are distinct iff they are not same as row vectors). Proof. Columns of A correspond to edges in S. Contract all edges which are not in A ⋆ . Let S ⋆ be the remaining (i.e., uncontracted) edges in S. Each path in P S now maps to a new path obtained by contracting these edges. Let P ⋆ denote the set of resulting paths. For a path P ∈ P ⋆ , let E(P ) be the edges in P whose priority is at least that of P . In the constraint matrix A ⋆ , the constraint for the path P has 1's in exactly the edges in E(P ). We can assume that the set E(P ) is distinct for every path P ∈ P ⋆ (because we are interested in counting the number of paths with distinct sets E(P )). Let P ⋆ (k) be the paths in P ⋆ for which |E(P )| ≤ k. We need to count the cardinality of this set. Fix an edge e ∈ S ⋆ , let S ⋆ (e) be the edges in S ⋆ of priority at least that of e. Let P be a path in P ⋆ (k) which has e as the least priority edge in E(P ) (breaking ties arbitrarily). Let e l and e r be the leftmost and the rightmost edges in E(P ) respectively. Note that E(P ) is exactly the edges in S ⋆ (e) which lie between e l and e r . Since there are at most k choices for e l and e r (look at the k edges to the left and to the right of e in the set S ⋆ (e)), it follows that there are at most k 2 paths P in P ⋆ (k) which have e as the least priority edge in E(P ). For every path in P ⋆ (k), there are at most |E ⋆ | = s choices for the least priority edge. Therefore the size of P ⋆ (k) is at most sk 2 . In the notation of [8], the shallow cell complexity of this LP relaxation is f (s, k) = sk 2 . It now follows from Theorem 1.1 in [8] that the integrality gap of the LP relaxation for the priority version is a constant. Thus we obtain a constant factor approximation algorithm for segment restricted instances. Segment Spanning Instances on Binary Trees We now consider instances I for which each path P ∈ P starts and ends at the end-points of a segment, i.e., the starting or ending vertex of P belongs to the set of vertices in red(T ). An example is shown in Figure 4. Although we will not use this result in the algorithm for the general case, many of the ideas will get extended to the general case. We will use dynamic programming. For a vertex v ∈ red(T ), let T v be the sub-tree of T rooted below v (and including v). Let P v denote the subset of P consisting of those paths which contain at least one edge in T v . By scaling the costs of edges, we will assume that the cost of the optimal solution lies in the range [1, n] -if c max is the maximum cost of an edge selected by the optimal algorithm, then its cost lies in the range [c max , nc max ]. Before stating the dynamic programming algorithm, we give some intuition for the DP table. We will consider sub-problems which correspond to covering paths in P v by edges in T v for every vertex v ∈ red(T ). However, to solve this sub-problem, we will also need to store the edges in T which are ancestors of v and are selected by our algorithm. Storing all such subsets would lead to too many DP table entries. Instead, we will work with the following idea -for each segment S, let B opt (S) be the total cost of edges in S which get selected by an optimal algorithm. If we know B opt (S), then we can decide which edges in S can be picked. Indeed, the optimal algorithm will solve a knapsack cover problem -for the segment S, it will pick edges of maximum total size subject to the constraint that their total cost is at most B opt (S) (note that we are using the fact that every path in P which includes an edge in S must include all the edges in S). Although knapsack cover is NP-hard, here is a simple greedy algorithm which exceeds the budget B opt (S) by a factor of 2, and does as well as the optimal solution (in terms of total size of selected edges) -order the edges in S whose cost is at most B opt (S) in order of increasing density. Keep selecting them in this order till we exceed the budget B opt (S). Note that we pay at most twice of B opt (S) because the last edge will have cost at most B opt (S). The fact that the total size of selected edges is at least that of the corresponding optimal value follows from standard greedy arguments. Therefore, if S 1 , . . . , S k denote the segments which lie above v (in the order from the root to v), it will suffice if we store B opt (S 1 ), . . . , B opt (S k ) with the DP table entry for v. We can further cut-down the search space by assuming that each of the quantities B opt (S) is a power of 2 (we will lose only a multiplicative 2 in the cost of the solution). Thus, the total number of possibilities for B opt (S 1 ), . . . , B opt (S k ) is O(log k n), because each of the quantities B opt (S i ) lies in the range [1, 2n] (recall that we had assumed that the optimal value lies in the range [1, n] and now we are rounding this to power of 2). This is at most 2 O(H log log n) , which is still not polynomial in n and 2 O(H) . We can further reduce this by assuming that for any two consecutive segments S i , S i+1 , the quantities B opt (S i ) and B opt (S i+1 ) differ by a factor of at most 8 -it is not clear why we can make this assumption, but we will show later that this does leads to a constant factor loss only. We now state the algorithm formally. Dynamic Programming Algorithm We first describe the greedy algorithm outlined above. The algorithm GreedySelect is given in Figure 5. For a vertex v ∈ red(T ), define the reduced depth of v as its at depth in red(T ) (root has reduced depth 0). We say that a sequence B 1 , . . . , B k is a valid state sequence at a vertex v in red(T ) with reduced depth k if it satisfies the following conditions: • For all i = 1, . . . , k, B i is a power of 2 and lies in the range [1, 2n]. If S 1 , . . . , S k is the sequence of segments visited while going from the root to v, then B i will correspond to S i . Consider a vertex v ∈ red(T ) at reduced depth k, and a child w of v in red(T ) (at reduced depth k + 1). Let Λ v = (B 1 , . . . , B k ) and Λ w = (B ′ 1 , . . . , B ′ k+1 ) be valid state sequences at these two vertices respectively. We say that Λ w is an extension of Λ v if B i = B ′ i for i = 1, . . . , k. In the dynamic program, we maintain a table entry T [v, Γ v ] for each vertex v in red(T ) and valid state sequence Γ v at v. Informally, this table entry stores the following quantity. Let S 1 , . . . , S k be the segments from the root to the vertex v. This table entry stores the minimum cost of a subset E ′ of edges in T v such that E ′ ∪ G(v) is a feasible solution for the paths in P v , where G(v) is the union of the set of edges selected by GreedySelect in the segments S 1 , . . . , S k with budgets B 1 , . . . , B k respectively. The algorithm is described in Figure 6. We first compute the set G(v) as outlined above. Let the children of v in the tree red(T ) be w 1 and w 2 . Let the segments corresponding to (v, w 1 ) and (v, w 2 ) be S 1 k+1 and S 2 k+1 respectively. For both these children, we find the best extension of Γ v . For the node w r , we try out all possibilities for the budget B r k+1 for the segment S r k+1 . For each of these choices, we select a set of edges in S r k+1 as given by GreedySelect and lookup the table entry for w r and the corresponding state sequence. We pick the choice for B r k+1 for which the combined cost is smallest (see line 7(i)(c)). We will not analyze this algorithm here because it's analysis will follow from the analysis of the more general case. We would like to remark that for any v ∈ red(T ), the number of possibilities for a valid state sequence is bounded by 2 O(H) · log n. Indeed, there are O(log n) choices for B 1 , and given B i , there are only 7 choices for B i+1 (since B i+1 /B i is a power of 2 and lies in the range [1/8, 8]). Therefore, the algorithm has running time polynomial in n and 2 O(H) . Table : Input: A node v ∈ red(T ) at reduced depth k, and a state sequence Λ v = (B 1 , . . . , B k ). 0. If v is a leaf node, set D[v, Λ v ] to 0, and exit. 1. Let S 1 , . . . , S k be the segments visited while going from the root to v in T . Fill DP 2. Initialize G(v) ← ∅. 3. For i = 1, . . . , k (i) Let G i (v) be the edges returned by GreedySelect(S i , B i ). (ii) G(v) ← G(v) ∪ G i (v)G(v) ∪ G k+1 (w r ) exit this loop (c) M r ← min(M r , cost of G k+1 (w r ) + D[w r , Γ wr ]). 7. D[v, Λ v ] ← M 1 + M 2 . General Instances on Binary Trees We now consider general instances of Demand MultiCut on binary trees. We can assume that every path P ∈ P contains at least one vertex of red(T ) as an internal vertex. Indeed, we can separately consider the instance consisting of paths in P which are contained in one of the segments -this will be a segment confined instance as in Section 5.1.1. We can get a constant factor approximation for such an instance. We will proceed as in the previous section, but now it is not sufficient to know the total cost spent by an optimal solution in each segment. For example, consider a segment S which contains two edges e 1 and e 2 ; and e 1 has low density, whereas e 2 has high density. Now, we would prefer to pick e 1 , but it is possible that there are paths in P which contain e 2 but do not contain e 1 . Therefore, we cannot easily determine whether we should prefer picking e 1 over picking e 2 . However, if all edges in S had the same density, then this would not be an issue. Indeed, given a budget B for S, we would proceed as follows -starting from each of the end-points, we will keep selecting edges of cost at most B till their total cost exceeds B. The reason is that all edges are equivalent in terms of cost per unit size, and since each path in P contains at least one of the end-points of S, we might as well pick edges which are closer to the end-points. Of course, edges in S may have varying density, and so, we will now need to know the budget spent by the optimum solution for each of the possible density values. We now describe this notion more formally. Algorithm Description We first assume that the density of any edge is a power of 128 -we can do this by scaling the costs of edges by factors of at most 128. We say that an edge e is of density class τ if it's density is 128 τ . Let τ max and τ min denote the maximum and the minimum density class of an edge respectively. Earlier, we had specified a budget B(S) for each segment S above v while specifying the state at v. Now, we will need to store more information at every such segment. We shall use the term cell to refer to a pair (S, τ ), where S is a segment and τ is a density class 1 . Given a cell (S, τ ) and a budget B, the algorithm GreedySelect in Figure 7 describes the algorithm for selecting edges of density class τ from S. As mentioned above, this procedure ensures that we pick enough edges from both end-points of S. The only subtlety is than in Step 4, we allow the cost to cross 2B -the factor 2 is for technical reasons which will become clear later. Note that in Step 4 (and similarly in Step 5) we could end up selecting edges of total cost up tp 3B because each selected edge has cost at most B. Algorithm GreedySelect: Input: A cell (S, τ ) and a budget B. As in the previous section, we define the notion of state for a vertex v ∈ red(T ). Let v be a node at reduced depth k in red(T ). Let S 1 , . . . , S k be the segments encountered as we go from the root to v in T . If we were to proceed as in the previous section, we will store a budget B(S i , τ ) For each cell (S i , τ ), i = 1, . . . , k, τ ∈ [τ min , τ max ]. This will lead to a very large number of possibilities (even if assume that for "nearby" cells, the budgets are not very different). Somewhat surprisingly, we show that it is enough to store this information at a small number of cells (in fact, linear in number of density classes and H). To formalize this notion, we say that a sequence C v = σ 1 , . . . , σ ℓ of cells is a valid cell sequence at v if the following conditions are satisfied : (i) the first cell σ 1 is (S k , τ max ), (ii) the last cell is of the form (S 1 , τ ) for some density class τ , and (iii) if σ = (S i , τ ) is a cell in this sequence, then the next cell is either (S i , τ − 1) or (S i−1 , τ + 1). To visualize this definition, we arrange the cells (S i , τ ) in the form of a table shown in Figure 8. For each segment S i , we draw a column in the table with one entry for each cell (S i , τ ), with τ increasing as we go up. Further as we go right, we shift these columns one step down. So row τ of this table will correspond to cells (S k , τ ), (S k−1 , τ + 1), (S k−2 , τ + 2) and so on. With this picture in mind, a a valid sequence of cells starts from the top left and at each step it either goes one step down or one step right. Note that for such a sequence C v and a segment S i , the cells (S i , τ ) which appear in C v are given by (S i , τ 1 ), (S i , τ 1 + 1), . . . , (S i , τ 2 ) for some τ 1 ≤ τ 2 . We say that the cells (S i , τ ), τ < τ 1 , lie below the sequence C v , and the cells (S i , τ ), τ > τ 2 lie above this sequence (e.g., in Figure 8, the cell (S 2 , 6) lies above the shown cell sequence, and (S 4 , 2) lies below it). Besides a valid cell sequence, we need to define two more sequences for the vertex v: • Valid Segment Budget Sequence: This is similar to the sequence defined in Section 5.1.2. This is a sequence Λ (i) For a cell σ j = (S i , τ ) in C v , the quantity B cell j ≤ B seg i . Again, the intuition is clear -the first quantity corresponds to cost of density class τ edges in S i , whereas the latter denotes the cost of all the edges in S i (which are selected by the optimal solution). 2 Informally, the idea behind these definitions is the following -for each cell σ j in C v , we are given the corresponding budget B cell j . We use this budget and Algorithm GreedySelect to select edges 2 During the analysis, B seg i will be the maximum over all density classes τ of the density class τ edges selected by the optimal solution from this segment. But this inequality will still hold. corresponding to this cell. For cells σ = (S i , τ ) which lie above C v , we do not have to use any edge of density class τ from S i . Note that this does not mean that our algorithm will not pick any such edge, it is just that for the sub-problem defined by the paths in P v and the state at v, we will not use any such edge (for covering a path in P v ). For cells σ = (S i , τ ) which lie below C v , we pick all edges of density class τ and cost at most B seg i from S i . Thus, we can specify the subset of selected edges from S 1 , . . . , S k (for the purpose of covering paths in P v ) by specifying these sequences only. The non-trivial fact is to show that maintaining such a small state (i.e., the three valid sequences) suffices to capture all scenarios. The algorithm for picking the edges for a specific segment S is shown in Figure 9. Note one subtlety -for the density class τ 1 (in Step 4), we use budget B seg i instead of the corresponding cell budget. The reason for this will become clear during the proof of Claim 5.12. Algorithm SelectSegment: Input: A vertex v ∈ red(T ), State(v) := (C v , Λ Seg v , Λ cell v ) , a segment S i lying above v. 1. Initialize a set G to emptyset. 2. Let (S i , τ 1 ), (S i , τ 1 + 1), . . . , (S i , τ 2 ) be the cells in C v corresponding to the segment S i . 3. For τ = τ 1 + 1, . . . , τ 2 do (i) Add to G the edges returned by GreedySelect((S i , τ ), B cell j ), where j is the index of (S i , τ ) in C v . 4. Add to G the edges returned by GreedySelect((S i , τ 1 ), B seg i ). 5. Add to G all edges e ∈ S i of density class strictly less than τ and for which c e ≤ B seg i . 6. Return G. In other words, the two sequences agree on segments S 1 , . . . , S k . • Recall that the first cell of C w is (S k+1 , τ max ). Let τ 1 be the smallest τ such that the cell (S k+1 , τ 1 ) appears in C w . Then the cells succeeding (S k+1 , τ 1 ) in C w must be of the form (S k , τ 1 + 1), (S k−1 , τ 1 + 2), . . . , till we reach a cell which belongs to C v (or we reach a cell for the segment S 1 ). After this the remaining cells in C w are the ones appearing in C v . Pictorially (see Figure 8), the sequence for C w starts from the top left, keeps going down till (S k+1 , τ 1 ), and then keeps moving right till it hits C v . After this, it merges with C v . • The sequences Λ cell v and Λ cell w agree on cells which belong to both C v and C w (note that the cells common to both will be a suffix of both the sequences). Having defined the notion of extension, the algorithm for filling the DP table for D[v, State(v)] is identical to the one in Figure 6. The details are given in Figure 10. This completes the description of the algorithm. Table : Input: A node v ∈ red(T ) at reduced depth k, Fill DP State(v) = (C v , Λ Seg v , Λ cell v ). 0. If v is a leaf node, set D[v, State(v)] to 0, and exit. 1. Let S 1 , . . . , S k be the segments visited while going from the root to v in T . 2. Initialize G(v) ← ∅. 3. For i = 1, . . . , k (i) Let G i (v) be the edges returned by Algorithm SelectSegment(v, S i , State(v)). (ii) G(v) ← G(v) ∪ G i (v). 4. Let w 1 , w 2 be the two children of v in red(T ) and the corresponding segments be S 1 k+1 , S 2 k+1 . 5. Initialize M 1 , M 2 to ∞. 6. For r = 1, 2 (go to each of the two children and solve the subproblems) (i) For each extension State(w r ) of State(v) do (a) Let G k+1 (w r ) be the edges returned by SelectSegment(w r , S r k+1 , State(w r )). (b) If any path in P v ending in the segment S r k+1 is not satisfied by G(v) ∪ G k+1 (w r ) exit this loop (c) M r ← min(M r , cost of G k+1 (w r ) + D[w r , State(w r )]). 7. D[v, Λ v ] ← M 1 + M 2 . Algorithm Analysis We now analyze the algorithm. Running Time We bound the running time of the algorithm. First we bound the number of possible table entries. Proof. The length of a valid cell sequence is bounded by (τ max − τ min ) + 2H. To see this, fix a vertex v at reduced depth k, with segments S 1 , . . . , S k from the root to the vertex v. Consider a valid cell sequence σ 1 , . . . , σ ℓ . For a cell σ j = (S i , τ ), define a potential Φ j = j + 2i + τ . We claim that the potential Φ j = Φ j+1 for all indices j in this sequence. To see this, there are two options for σ j+1 = (S i ′ , τ ′ ) : • S i = S i ′ , τ ′ = τ − 1: Here, Φ j+1 = j + 1 + 2i + τ − 1 = j + 2i + τ = Φ j . • S i ′ = S i−1 , τ ′ = τ + 1: Here Φ j+1 = j + 1 + 2(i − 1) + τ + 1 = Φ j . Therefore, ℓ + τ min ≤ Φ l = Φ 0 ≤ 2H + τ max . It follows that ℓ ≤ 2H + τ max − τ min . Given the cell σ j , there are only two choices for σ j+1 . So, the number of possible valid cell sequences is bounded by 2 2H+τmax−τ min ≤ 2 2H · ρ max /ρ min . Now we bound the number of valid segment budget sequences. Consider such a sequence B [1/8, 8]. Therefore, the number of such sequences is at most O(log n) · 7 k ≤ O (7 H log n). Similarly, the number of valid cell budget sequences is at most O(log n) · 7 ℓ , where ℓ is the maximum length of a valid cell sequence. By the argument above, ℓ is at most 2H + τ max − τ min . Combining everything, we see that the number of possible states for v is bounded by a constant times 2 2H · ρ max /ρ min · 7 H log n · log n · 7 2H · ρ max /ρ min , which implies the desired result. We can now bound the running time easily. Feasibility We now argue that the table entries in the DP correspond to valid solutions. Fix a vertex v ∈ red(T ), and let S 1 , . . . , S k be the segments as we go from the root to v. Recall that T (v) denotes the sub-tree of T rooted below v and P(v) denotes the paths in P which have an internal vertex in T (v). For a segment S and density class τ , let S(τ ) denote the edges of density class τ in S. Proof. We prove this by induction on the reduced height of v. If v is a leaf, then P(v) is empty, and so the result follows trivially. Suppose it is true for all nodes in red(T ) at reduced height at most k − 1, and v be at height k in red(T ). We use the notation in Figure 10. Consider a child w r of v, where r is either 1 or 2. Let the value of M r used in Step 7 be equal to the cost of G k+1 (w r ) + D[w r , State(w r )] for some State(w r ) given by (C wr , Λ ). Let G(v) and G k+1 (w r ) be as in the steps 3 and 6(i)(a) respectively. We ensure that G(v) ∪ G k+1 (w r ) covers all paths in P(v) which end before w r . The following claim is the key to the correctness of the algorithm. Claim 5.6. Let G(w r ) be edges obtained at the end of Step 3 in the algorithm in Figure 10 when filling the DP table entry D[w r , State(w r )]. Then G(w r ) is a subset of G(v) ∪ G k+1 (w r ). Proof. Let S r k+1 be the segment between v and w r . By definition, G k+1 (w r ) and G(w r ) ∩ S r k+1 are identical. Let us now worry about segments S i , i ≤ k. Fix such a segment S i . We know that after the cells corresponding to the segment S r k+1 , the sequence C wr lies below C v till it meets C v . Now consider various case for an arbitrary cell σ = (S i , τ ) (we refer to the algorithm SelectSegmentin Figure 9): • The cell σ lies above C wr : G(w r ) does not contain any edge of S i (τ ). • The cell σ lies below C wr : The cell will lie below C v as well, and so, G(v) and G(w) will contain the same edges from S i (τ ) (because B seg i = B seg ′ i are same). • The cell σ lies on C wr : If it also lies on C v , then the fact that B cell and B cell ′ values for this cell are same implies that G(v) and G(w r ) pick the same edges from S i (τ ) (in case τ happens to be the smallest indexed density class for which (S i , τ ) ∈ C wr , then the same will hold for C v as well). If it lies below C v , then the facts that B seg i = B seg ′ i , and B seg i ≥ B cell j = B cell ′ j ′ , where j and j ′ are the indices of this cell in the two cell sequences respectively, imply that G(v) will pick all the edges fof cost at most B seg i from S i (τ ), whereas G(w r ) will pick only a subset of these edges. We see that G(v) ∩ S i contains G(w r ) ∩ S i . This proves the claim. By induction hypothesis, there is a subset Y (w r ) of edges in the subtree T (w r ) of cost equal to D[w r , State(w r )] such that Y (w r ) ∪ G(w r ) satisfies all paths in P(w r ). We already know that G(v) ∪ G k+1 (w r ) covers all paths in P(v) which end in the segment S r k+1 . Since any path in P(v) will either end in S 1 k+1 or S 2 k+1 , or will belong to P(w 1 ) ∪ P(w 2 ), it follows that all paths in P(v) are covered by ∪ 2 r=1 (Y (w r ) ∪ G(w r ) ∪ G k+1 (w r )) ∪ G(v) . Now, the claim above shows that G(w r ) ⊆ G k+1 (w r )∪G(v). So this set is same as Y (w 1 )∪Y (w 2 )∪G k+1 (w 1 )∪G k+1 (w 2 )∪G(v) (and these sets are mutually disjoint). Recall that M r is equal to the cost of G k+1 (w r )∪Y (w r ), it follows that the the DP table entry for v for these parameters is exactly the cost of Y (w 1 )∪Y (w 2 )∪G k+1 (w 1 )∪G k+1 (w 2 ). This proves the lemma. For the root vertex r, a valid state at r must be the empty set. The above lemma specialized to the root r implies: Corollary 5.7. Assuming that D[r, ∅] is not ∞, it is the cost of a feasible solution to the input instance. Approximation Ratio Now we related the values of the DP table entries to the values of the optimal solution for suitable sub-problems. We give some notation first. Let OPT denote an optimal solution to the input instance. For a segment S, we shall use S opt to the denote the subset of S selected by OPT. Similarly, let S opt (τ ) denote the subset of S(τ ) selected by OPT. Let B opt (S, τ ) denote the total cost of edges in S opt (τ ), and B opt (S) denote the cost of edges in S opt . We first show that we can upper bound B opt (S, τ ) by values B ⋆ (S, τ ) values such that the latter values are close to each other for nearby cells. For two segments S and S ′ , define the distance between them as the distance between the corresponding edges in red(T ) (the distance between two adjacent edges is 1). Similarly, we say that a segment is the parent of another segment if this relation holds for the corresponding edges in red(T ). Let cells(T ) denote the set of all cells in T . Lemma 5.8. We can find values B ⋆ (S, τ ) for each cell (S, τ ) such that the following properties are satisfied: (i) for every cell (S, τ ), B ⋆ (S, τ ) is a power of 2, and B ⋆ (S, τ ) ≥ B opt (S, τ ), (ii) (S,τ )∈cells(T ) B ⋆ (S, τ ) ≤ 16 · (S,τ )∈cells(T ) B opt (S, τ ) and (iii) (smoothness) for every pair of segments S, S ′ , where S ′ is the parent of S, and density class τ , 8B ⋆ (S, τ + 1) ≥ B ⋆ (S, τ ) ≥ B ⋆ (S, τ + 1)/8, and 8B ⋆ (S ′ , τ ) ≥ B ⋆ (S, τ ) ≥ B ⋆ (S ′ , τ )/8. Proof. We define B ⋆ (S, τ ) := i≥0 S ′ ∈N i (S) j B opt (S ′ , τ + i + j) 4 i+|j| , where i varies over non-negative integers, j varies over integers and the range of i, j are such that τ + i + j remains a valid density class; and N i (S) denotes the segments which are at distance at most i from S. Note that B ⋆ (S, τ ) is a not a power of 2 yet, but we will round it up later. As of now, B ⋆ (S, τ ) ≥ B opt (S, τ ) because the term on RHS for i = 0, j = 0, is exactly B opt (S, τ ). We now verify the second property. We add B ⋆ (S, τ ) for all the cells (S, τ ). Let us count the total contribution towards terms containing B opt (S ′ , τ ′ ) on the RHS. For every segment S ∈ N i (S ′ ), and density class τ ′ − i − j, it will receive a contribution of 1 4 i+|j| . Since |N i (s ′ )| ≤ 2 i+1 (this is where we are using the fact that T is binary), this is at most i≥0 j 2 i+1 4 i+|j| ≤ i≥0 2 i+2 4 i ≤ 8. Now consider the third condition. Consider the expressions for B ⋆ (S, τ ) and B ⋆ (S ′ , τ ) where S ′ is the parent of S. If a segment is at distance i from S, its distance from S ′ is either i or i ± 1. Therefore, the coefficients of B opt (S ′′ , τ ′′ ) in the expressions for B ⋆ (S, τ ) and B ⋆ (S ′ , τ ) will differ by a factor of at most 4. The same observation holds for B ⋆ (S, τ ) and B ⋆ (S, τ + 1). It follows that 4B ⋆ (S, τ + 1) ≥ B ⋆ (S, τ ) ≥ B ⋆ (S, τ + 1)/4, and 4B ⋆ (S ′ , τ ) ≥ B ⋆ (S, τ ) ≥ B ⋆ (S ′ , τ )/4. Finally, we round all the B ⋆ (S, τ ) values up to the nearest power of 2. We will lose an extra factor of 2 in the statements (ii) and (iii) above. We will use the definition of B ⋆ (S, τ ) in the lemma above for rest of the discussion. For a segment S, define B ⋆ (S) as the maximum over all density classes τ of B ⋆ (S, τ ). The following corollary follows immediately from the lemma above. Proof. Let B ⋆ (S) be equal to B ⋆ (S, τ ) for some density class τ . Then, B ⋆ (S) = B ⋆ (S, τ ) Lemma 5.8 ≤ 8B ⋆ (S ′ , τ ) ≤ 8B ⋆ (S ′ ). The other part of the argument follows similarly. The plan now is to define a valid state State ⋆ (v) = (C ⋆ v , Λ ⋆seg v , Λ ⋆cell v ) for each of the vertices v in red(T ). We begin by defining a critical density τ ⋆ (S) for each segment S. Recall that S(τ ) denotes the edges of density class τ in S. Let S(≤ τ ) denote the edges class of density class at most τ in S. For a density class τ and a budget B, let S(≤ τ, ≤ B) denote the edges in S(≤ τ ) which have cost at most B. Define τ ⋆ as the smallest density class τ such that the total cost of edges in S(≤ τ, ≤ B ⋆ (S)) is at least 4B ⋆ (S) + τ ′ ≤τ B opt (S, τ ′ ) (if no such density class exists, set τ to τ max ). Intuitively, we are trying to augment the optimal solution by low density edges, and τ ⋆ (S) tells us the density class till which we can essentially take all the edges in S (provided we do not pick any edge which is too expensive). Having defined the notion of critical density, we are now ready to define a valid state State ⋆ (v) for each vertex v in red(T ). Let v be such a vertex at reduced depth k and let S 1 , . . . , S k be the segments starting from the root to v. Again, it is easier to see the definition of the cell sequence C ⋆ v pictorially. As in Figure 11, the cell sequence starts with (S k , τ max ) and keeps going down till it reaches the cell (S k , τ ⋆ (S k )). Now it keeps going right as long as the cell corresponding to the critical density lies above it. If this cell lies below it, it moves down. The formal procedure for constructing this path is given in Figure 12. For sake of brevity, let τ ⋆ i denote τ ⋆ (S i ). Construct Sequence C ⋆ v : Input: A node v ∈ red(T ) at depth k, integers τ ⋆ 1 , . . . , τ ⋆ k 1. Initialise C ⋆ v to empty sequence, and i ← k, τ ← τ max 2. While (i ≥ 1) Claim 5.11. (i) Add the cell (S i , τ ) to C ⋆ v . (ii) If τ > τ ⋆ i , τ ← τ − 1 (iii) Else i = i − 1, τ ← τ + 1.(S j ,τ )∈Dom(S i ) p(S opt j (τ )) ≤ 128 −τ ⋆ i B ⋆ (S i , τ ⋆ i ). Proof. Fix a segment S j . For sake of brevity, letτ denote τ ⋆ i + (i − j). Recall that for any pair (S, τ ), B ⋆ (S, τ ) ≥ B opt (S, τ ) (Lemma 5.8). Therefore, terms in the above sum corresponding to S j add up to τ ≥τ p(S opt j (τ )) ≤ τ ≥τ 128 −τ B ⋆ (S j , τ ). By repeated applications of Lemma 5.8, B ⋆ (S j , τ ) ≤ 8 τ −τ ⋆ i · 8 i−j B ⋆ (S i , τ ⋆ i ). Therefore, τ ≥τ p(S opt j (τ )) ≤ τ ≥τ 128 −τ · 8 τ −τ ⋆ i · 8 i−j B ⋆ (S i , τ ⋆ i ) = 128 −τ ⋆ i τ ≥τ 16 −(τ −τ ⋆ i ) · 8 i−j B ⋆ (S i , τ ⋆ i ) ≤ 128 −τ ⋆ i · 2 · 8 i−j 16τ −τ ⋆ i · B ⋆ (S i , τ ⋆ i ) = 128 −τ ⋆ i · 2 8 i−j · B ⋆ (S i , τ ⋆ i ) Summing over all j < i now implies the result. Let G ⋆ i (v) denote the set of edges selected by the Algorithm SelectSegment in Figure 9 for the vertex v and segment S i when called with the state State ⋆ (v). Let G ⋆ i (τ, v) be the density τ edges in G ⋆ i (v). The following claim shows that the total size of edges in it is much larger than the corresponding quantity for the optimal solution. Claim 5.12. For any segment S i , τ ≤τ ⋆ i p(G ⋆ i (τ, v)) − τ ≤τ ⋆ i p(S opt i (τ )) ≥ 128 −τ ⋆ i B ⋆ (S i , τ ⋆ i ). Proof. Recall that for a segment S, density class τ and budget B, S(τ, ≤ B) denotes the edges in S(τ ) which have cost at most B. The quantity S(≤ τ, ≤ B) was defined similarly for edges of density class at most τ in S. Consider the Algorithm SelectSegment for S i with the parameters mentioned above. Note that τ ⋆ i is same as τ 1 in the notation used in Figure 9. Clearly, for τ < τ ⋆ i , the algorithm ensures that G ⋆ i (τ, v) contains S opt i (τ ) (because it selects all edges in S i (τ, ≤ B ⋆ (S i )). Since each egde in S opt i (τ ) has cost at most B opt (S i , τ ) ≤ B ⋆ (S i ), this implies that S i (τ, ≤ B ⋆ (S i )) contains S opt i (τ )). For the class τ 1 , note that the algorithm tries to select edges of total cost at least 4B ⋆ (S i ). Two cases arise: (i) If it is able to select these many edges, then the fact that the optimal solution selects edges of total cost at most B ⋆ (S i ) from S i (τ 1 ) implies the result, or (ii) The total cost of edges in S i (τ 1 , ≤ B ⋆ (S i )) is less than 4B ⋆ (S i ): in this case the algorithm selects all the edges from S i (≤ τ 1 , ≤ B ⋆ (S i )), and so, G ⋆ i (τ, v) contains S opt i (τ ) for all τ ≤ τ ⋆ i . The definition of τ ⋆ i implies that the total cost of edges in ∪ τ ≤τ ⋆ i G ⋆ i (τ, v) \ S opt i (τ ) is at least 4B ⋆ (S i ), and so, the result follows again. We are now ready to prove Lemma 5.10. Let P be a path in P(v) which ends in the segment S r k+1 . Suppose P starts in the segment S i 0 . Note that P contains the segments S i 0 +1 , . . . , S k , but may partially intersect S i 0 and S k+1 . Proof. Fix a segment S i which is intersected by P . P contains the lower end-point of this segment S i . If S opt i (τ ) ⊆ G ⋆ i (τ, v) , there is nothing to prove. Else let e be the first edge in S opt i (τ ) \ G ⋆ i (τ, v) as we go up from the lower end-point of this segment. It follows that during Step 5 of the Algorithm GreedySelect in Figure 7, we would select edges of total cost at least 2B opt (S i , τ ) (because B opt (S i , τ ) ≤ B ⋆ (S i , τ )). The claim follows. Clearly, if (S i , τ ) lies below the cell sequence C ⋆ (v), G ⋆ i (τ, v) contains S opt i (τ ) (because G ⋆ i (τ, v) is same as S i (τ, ≤ B ⋆ (S i )) and B ⋆ (S i ) ≥ B opt (S i )). Let Above(C ⋆ (v)) denotes the cells lying above this sequence, and Below(C ⋆ (v)) the ones lying below it. The above claim now implies that τ :(S i 0 ,τ )∈C ⋆ (v)∪Below(C ⋆ (v)) p(S opt i 0 (τ ) ∩ P ) ≤ τ :(S i 0 ,τ )∈C ⋆ (v)∪Below(C ⋆ (v)) p(G ⋆ i 0 (τ, v) ∩ P ) Note that any cell (S i , τ ), i ≥ i 0 , lying above C ⋆ (v) must be dominated by one of the cells (S i ′ , τ ⋆ i ′ ) for i ′ = i 0 + 1, . . . , k. Therefore, Claim 5.11 shows that (S i ,τ )∈Above(C ⋆ (v)),i 0 ≤i≤k p(S opt i (τ )) ≤ i 0 <i≤k 128 −τ ⋆ i B ⋆ (S i , τ ⋆ i )(13) Further, for cells lying on or below C ⋆ (v), we get using Claim 5.12 and Claim 5.13 (S i ,τ )∈C ⋆ (v)∪Below(C ⋆ (v)),i 0 <i≤k p(S opt i (τ )) ≤ (S i ,τ )∈C ⋆ (v)∪Below(C ⋆ (v)),i 0 <i≤k G ⋆ i (τ, v) − 2 i 0 <i≤k 128 −τ ⋆ i B ⋆ (S i , τ ⋆ i )(14) Adding the three inequalities above, we see that k i=i 0 p(S opt i ∩ P ) is at most k i=i 0 p(G ⋆ i (v)). It remains to consider segment S r k+1 . In an argument identical to the one in Claim 5.13, we can argue that p(G ⋆ k+1 (w r ) ∩ P ) ≥ p(S r,opt k+1 ∩ P ), where S r,opt k+1 denotes the edges in S r k+1 selected by the optimal solution. This completes the proof of the technical Lemma 5.10. Rest of the task is now easy. We just need to show that DP table entries corresponding to these valid states are comparable to the cost of the optimal solution. For a vertex v ∈ red(T ), we shall use the notation cells(T (v)) to denote the cells (S, τ ), where S lies in the subtree T (v). Lemma 5.14. For every vertex v, the table entry D[v, State ⋆ (v)] is at most 20 (S,τ )∈cells(T (v)) B ⋆ (S, τ ). Proof. We prove by induction on the reduced depth of v. If v is a leaf, the lemma follows trivially. Now suppose v has children w 1 and w 2 . Consider the iteration of Step 6 in the algorithm in Figure 10, where we try the extension State ⋆ (w r ) of the child w r . Lemma 5.10 shows that in Step 6 (b), we will satisfy all paths in P(v) which end in the segment S r k+1 . We now bound the cost of edges in G r k+1 (w r ) defined in Step 6(a). Claim 5.15. The cost of G r k+1 (w r ) is at most 20 τ B ⋆ (S r k+1 , τ ). Proof. We just need to analyze the steps in the algorithm SelectSegment in Figure 9. For sake of brevity, let τ ⋆ denote τ ⋆ (S r k+1 ), and B ⋆ denote B ⋆ (S r k+1 ). The definition of τ ⋆ shows that the total cost of edges in S r k+1 (≤ τ ⋆ , ≤ B ⋆ ) is at most τ ≤τ ⋆ B opt (S r k+1 , τ )+4B ⋆ ≤ τ ≤τ ⋆ B ⋆ (S r k+1 , τ )+4B ⋆ . For the density class τ ⋆ , the set of edges selected would cost at most 6B ⋆ , because the algorithm in Figure 7 will take edges of cost up to 3B ⋆ from either ends. Similarly, for density classes τ more than τ ⋆ , this quantity is at most 6B ⋆ (S, τ ), Summing up everything, and using the fact that B ⋆ = B ⋆ (S, τ ) for some density class τ gives the result. The lemma now follows by applying induction on D[w r , State ⋆ (w r )]. Applying the above lemma to the root vertex r, we see that D[r, ∅] is at most a constant time (S,τ ) B ⋆ (S, τ ), which by Lemma 5.8, is a constant times the optimal cost. Finally, Lemma 5.7 shows that this entry denotes the cost of a feasible solution. Thus, we have shown the main Theorem 5.1. Discussion We give the first pseudo-polynomial time constant factor approximation algorithm for the weighted flow-time problem on a single machine. The algorithm can be made to run in time polynomial in n and W as well, where W is the ratio of the maximum to the minimum weight. The rough idea is as follows. We have already assumed that the costs of the job segments are polynomially bounded (this is without loss of generality). Since the cost of a job segment is its weight times its length, it follows that the lengths of the job segments are also polynomially bounded, say in the range [1, n c ]. Now we ignore all jobs of size less than 1/n 2 , and solve the remaining problem using our algorithm (where P will be polynomially bounded). Now, we introduce these left out jobs, and show that increase in weighted flow-time will be small. Further, the algorithm also extends to the problem of minimizing ℓ p norm of weighted flow-times. We can do this by changing the objective function in (IP2) to ( j s (w(j, S)) p y(j, S)) 1/p and showing that this is within a constant factor of the optimum value. The instance of Demand MultiCut in the reduction remains exactly the same, except that the weights of the nodes are now w(j, S) p . We leave the problem of obtaining a truly polynomial time constant factor approximation algorithm as open. Figure 1 : 1Forming Seg(j). Figure 2 : 2The dyadic segments S 1 , . . . , S 4 and the corresponding Seg(j 1 ), Seg(j 2 ) for two jobs j 1 , j 2 j, S)p j ≥ p(J(I)) − l(I) for all intervals I = [s, t], 0 ≤ s ≤ t ≤ T (5) y(j, S) ∈ {0, 1} for all job segments (j, S) Lemma 3 . 2 . 32Given a solution x for (IP1), we can construct a solution for (IP2) of cost at most 8 times the cost of x. Similarly, given a solution y for (IP2), we can construct a solution for (IP1) of cost at most 4 times the cost of y. Lemma 4 . 1 . 41The vertices in V I form a path in T from an ancestor to a descendant. Claim 4 . 2 . 42Given a solution E to the Demand MultiCut instance I, there is a solution to (IP2) for the instance I ′ of the same objective function value as that of E.Proof. Consider a solution to I consisting of a set of edges E. For each edge e Let the edges in S (in top to down order) be e 1 , . . . , e m . The following integer program (IP3) captures the Demand MultiCut problem for I x e ≥ d(P )for all paths P ∈ P S Algorithm GreedySelect: Input: A segment S in T and a budget B. 1. Initialize a set G to emptyset. 2. Arrange the edges in S of cost at most B in ascending order of density. 3. Keep adding these edges to G till their total cost exceeds B. 4. Output G. Figure 5 : 5Algorithm GreedySelect for selecting edges in a segment S with a budget B. • For any i = 1, . . . , k − 1, B i /B i+1 lies in the range [1/8, 8]. . 4 . 4Let w 1 , w 2 be the two children of v in red(T ) and the corresponding segments be S 1 k+1 , S 2 k+1 . 5. Initialize M 1 , M 2 to ∞. 6. For r = 1, 2 (go to each of the two children and solve the subproblems) (i) For each extension Γ wr = (B 1 , . . . , B k , B r k+1 ) of Γ v do (a) Let G k+1 (w r ) be the edges returned by GreedySelect(S r k+1 , B r k+1 ). (b) If any path in P v ending in the segment S r k+1 is not satisfied by Figure 6 : 6Filling a table entry D[v, Λ v ] in the dynamic program. 1 . 1Initialize a set G to emptyset. 2. Let S(τ ) be the edges in S of density class τ and cost at most B. 3. Arrange the edges in S(τ ) from top to bottom order. 4. Keep adding these edges to G in this order till their total cost exceeds 2B.5. RepeatStep 4 with the edges in S(τ ) arranged in bottom to top order. 6. Output G. Figure 7 : 7Algorithm GreedySelect for selecting edges in a segment S of density class τ with a budget B. 1 , . . . , B seg k ), where B seg i corresponds to the segment S i . As before, each of these quantities is a power of 2 and lies in the range [1, 2n]. Further, for any i, the ratio B seg i /B seg i+1 lies in the range [1/8, 8]. Figure 8 : 8w is a vertex at reduced depth 6 and v is the parent of v in red(T ). The segments above w are labelled S 1 , . . . , S 6 (starting from the root downwards). The cells are arranged in a tabular fashion as shown -the density classes lie in the range {2, 3, . . . , 7}. The solid line shows a valid cell sequence for v. The dotted line shows a valid cell sequence for w which is also an extension of the cell sequence for v -note that once the dotted line meets the solid line (in the cell (S 3 , 5), it stays with it till the end. • Valid Cell Budget Sequence: Corresponding to the valid cell sequence C v = σ 1 , . . . , σ ℓ and valid budget sequence (the cell σ j . Each of the quantities B cell j lies in the range [1, 2n]. Further. the ratio B cell j /B cell j+1 lies in the range [1/8, 8]. Intuitively, B seg i is supposed to capture the cost of edges picked by the optimal solution in S i , whereas B cell j , where σ j = (S i , τ ), captures the cost of the density class τ edges in S i which get selected by the optimal solution. A valid state State(v) at the vertex v is given by the triplet (C v , Λ seg v , Λ cell v ) which in addition satisfies the following properties: Figure 9 : 9Algorithm SelectSegment for selecting edges in a segment S as dictated by the state at v. The notations B seg and B cell are as explained in the text. Before specifying the DP table, we need to show what it means for a state to be an extension of another state. Let w be a child of v in red(T ), and let S k+1 be the corresponding segment joining v and w. Given states State(v) := (C v , Λ Seg v , Λ cell v ) and State(w) := (C w , Λ Seg w , Λ cell w ), we say that State(w) is an extension of State(v) if the following conditions are satisfied: i = 1, . . . , k. Figure 10 : 10Filling a table entry D[v, State(v)] in the dynamic program. Lemma 5 . 3 . 53For any vertex v, the number of possible valid states is O (log n) 2 · 2 O(H) · (ρ max /ρ min ) 2 . seg 1 , . . . , B seg k . Since B seg 1 ∈ [1, 2n] and it is a power of 2, there are O(log n) choices for it. Given B seg i , there are at most 7 choices for B seg i+1 , because B seg i+1 /B seg i is a power of 2 and lies in the range Lemma 5 . 4 . 54The running time of the algorithm is polynomial in n, 2 H , ρ max /ρ min .Proof. To fill the table entry for D[v, Γ(v)], where v has children w 1 , w 2 , the Algorithm inFigure 10cycles through the number of possible extensions of Γ(v) for each of the two children. Since any valid extension of Γ(v) for a child w r is also a valid state at w r , the result follows from Lemma 5.3. Lemma 5 . 5 . 55Consider the algorithm in Figure 10 for filling the DP entry D[v, State(v)], and let G(v) be the set of vertices obtained after Step 3 of the algorithm. Assuming that this table entry is not ∞, there is a subset Y (v) of edges in T (v) such that the cost of Y (v) is at equal to D[v, State(v)] and Y (v) ∪ G(v) is a feasible solution for the paths in P(v). Corollary 5 . 9 . 59Let S and S ′ be two segments in T such that S ′ is the parent of S. Then B ⋆ (S) and B ⋆ (S ′ ) lie within factor of 8 of each other. Figure 12 : 12Construction of the path C ⋆ v . right quadrant with respect to it if we arrange the cells as shown in the figure. For a segment S i , let Dom(S i ) be the set of cells dominated by (S i , τ ⋆ i ). The following claim shows why this notion is useful. For a set E of edges , let p(E) denote e∈E p e . Recall that S opt (τ ) denotes the set of edges in S(τ ) selected by the optimal solution. Claim 5 . 13 . 513For a cell (S i , τ ) on the cell sequence C ⋆ (v), p(G ⋆ i (τ, v) ∩ P ) ≥ p(S opt i (τ ) ∩ P ). For technical reasons, we will allow τ to lie in the range [τmin, τmax + 1] AcknowledgementThe authors would like to thank Sungjin Im for noting the extension to ℓ p norm of weighted flow-times.Further, let w be a child of v in red(T ). It is again it is to see that State ⋆ (w) is an extension of State ⋆ (v). The procedure for constructing C ⋆ w ensures that this property holds: this path first goes down till (S k+1 , τ ⋆ (S k+1 )), where S k+1 is the segment between v and w. Subsequently, it moves right till it hits C ⋆ v (seeFigure 11for an example). The following crucial lemma shows that is alright to ignore the cells above the path C ⋆ v . Let w 1 and w 2 be the children of v in red(T ). We consider the algorithm inFigure 10for filling the DP entry D[v, State ⋆ (v)]. Let G ⋆ (v) be the edges obtained at the end of Step 3 in this algorithm. Further, let G ⋆ k+1 (w r ) be the set of edges obtained in Step 6(i)(a) of this algorithm when we use the extension State ⋆ (w r ).Lemma 5.10. For r = 1, 2, any path in P(v) which ends in the segment S r k+1 is satisfied byThis is the main technical lemma of the contribution and is the key reason why the algorithm works. We will show this by a sequence of steps. We say that a cell (S i , τ ) dominates a cell (S j , τ ′ ) if j < i and τ ′ − τ > j − i. As inFigure 11, a cell (S, τ ) dominates all cells which lie in the upper Minimizing weighted flow time. Nikhil Bansal, Kedar Dhamdhere, ACM Trans. Algorithms. 3439Nikhil Bansal and Kedar Dhamdhere. Minimizing weighted flow time. ACM Trans. Algorithms, 3(4):39, 2007. On capacitated set cover problems. Nikhil Bansal, Ravishankar Krishnaswamy, Barna Saha, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques -14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011. Princeton, NJ, USANikhil Bansal, Ravishankar Krishnaswamy, and Barna Saha. On capacitated set cover prob- lems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques -14th International Workshop, APPROX 2011, and 15th International Work- shop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. Proceedings, pages 38-49, 2011. Server scheduling in the weighted l p norm. Nikhil Bansal, Kirk Pruhs, LATIN 2004: Theoretical Informatics, 6th Latin American Symposium. Buenos Aires, ArgentinaNikhil Bansal and Kirk Pruhs. Server scheduling in the weighted l p norm. In LATIN 2004: Theoretical Informatics, 6th Latin American Symposium, Buenos Aires, Argentina, April 5-8, 2004, Proceedings, pages 434-443, 2004. The geometry of scheduling. Nikhil Bansal, Kirk Pruhs, SIAM J. Comput. 435Nikhil Bansal and Kirk Pruhs. The geometry of scheduling. SIAM J. Comput., 43(5):1684- 1698, 2014. A unified approach to approximating resource allocation and scheduling. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, Baruch Schieber, J. ACM. 485Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. J. ACM, 48(5):1069- 1090, 2001. Approximation algorithms for average stretch scheduling. Michael A Bender, S Muthukrishnan, Rajmohan Rajaraman, J. Scheduling. 73Michael A. Bender, S. Muthukrishnan, and Rajmohan Rajaraman. Approximation algorithms for average stretch scheduling. J. Scheduling, 7(3):195-222, 2004. On column-restricted and priority covering integer programs. Deeparnab Chakrabarty, Elyot Grant, Jochen Könemann, Integer Programming and Combinatorial Optimization, 14th International Conference, IPCO 2010. Lausanne, SwitzerlandDeeparnab Chakrabarty, Elyot Grant, and Jochen Könemann. On column-restricted and prior- ity covering integer programs. In Integer Programming and Combinatorial Optimization, 14th International Conference, IPCO 2010, Lausanne, Switzerland, June 9-11, 2010. Proceedings, pages 355-368, 2010. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. Timothy M Chan, Elyot Grant, Jochen Könemann, Malcolm Sharpe, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012Kyoto, JapanTimothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capaci- tated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1576-1585, 2012. Approximation schemes for preemptive weighted flow time. Chandra Chekuri, Sanjeev Khanna, Proceedings on 34th Annual ACM Symposium on Theory of Computing. on 34th Annual ACM Symposium on Theory of ComputingMontréal, Québec, CanadaChandra Chekuri and Sanjeev Khanna. Approximation schemes for preemptive weighted flow time. In Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 297-305, 2002. Algorithms for minimizing weighted flow time. Chandra Chekuri, Sanjeev Khanna, An Zhu, Proceedings on 33rd Annual ACM Symposium on Theory of Computing. on 33rd Annual ACM Symposium on Theory of ComputingHeraklion, Crete, GreeceChandra Chekuri, Sanjeev Khanna, and An Zhu. Algorithms for minimizing weighted flow time. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 84-93, 2001. Primal-dual approximation algorithms for integral flow and multicut in trees. Naveen Garg, Vijay V Vazirani, Mihalis Yannakakis, Algorithmica. 181Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Primal-dual approximation algo- rithms for integral flow and multicut in trees. Algorithmica, 18(1):3-20, 1997. Fair scheduling via iterative quasi-uniform sampling. Sungjin Im, Benjamin Moseley, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete AlgorithmsBarcelona, Spain, Hotel Porta FiraSungjin Im and Benjamin Moseley. Fair scheduling via iterative quasi-uniform sampling. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2601-2615, 2017. Speed is as powerful as clairvoyance. Bala Kalyanasundaram, Kirk Pruhs, J. ACM. 474Bala Kalyanasundaram and Kirk Pruhs. Speed is as powerful as clairvoyance. J. ACM, 47(4):617-643, 2000. Weighted geometric set cover via quasi-uniform sampling. R Kasturi, Varadarajan, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010. the 42nd ACM Symposium on Theory of Computing, STOC 2010Cambridge, Massachusetts, USAKasturi R. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. 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{'abstract': 'In the weighted flow-time problem on a single machine, we are given a set of n jobs, where each job has a processing requirement p j , release date r j and weight w j . The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudo-polynomial time constant approximation algorithm for this problem. The algorithm also extends directly to the problem of minimizing the ℓ p norm of weighted flow-times. The running time of our algorithm is polynomial in n, the number of jobs, and P , which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multi-cut problem on trees, which we call Demand MultiCut problem. Even though we do not give a constant factor approximation algorithm for the Demand MultiCut problem on trees, we show that the specific instances of Demand MultiCut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of DP table.', 'arxivid': '1802.07439', 'author': ['Jatin Batra \nDepartment of Computer Science and Engineering IIT Delhi\n\n', 'Naveen Garg \nDepartment of Computer Science and Engineering IIT Delhi\n\n', 'Amit Kumar \nDepartment of Computer Science and Engineering IIT Delhi\n\n'], 'authoraffiliation': ['Department of Computer Science and Engineering IIT Delhi\n', 'Department of Computer Science and Engineering IIT Delhi\n', 'Department of Computer Science and Engineering IIT Delhi\n'], 'corpusid': 3507217, 'doi': '10.1109/focs.2018.00079', 'github_urls': [], 'n_tokens_mistral': 26879, 'n_tokens_neox': 25076, 'n_words': 17449, 'pdfsha': 'c58a8c6430ec717219d92f8a71f2ce6fc4beb0e9', 'pdfurls': ['https://arxiv.org/pdf/1802.07439v3.pdf'], 'title': ['Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-polynomial time', 'Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-polynomial time'], 'venue': []}

arxiv

"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra (2008 Re-Release) 18 Apr 2008 Eberhard H Gerbracht "E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra (2008 Re-Release) 18 Apr 2008Dedicated to the memories of Prof. Dr.-Ing. Ernst-Helmut Horneber (1946-2001) and Prof. Giuseppa Carrà Ferro (1952-2007)(semi-)state equationssystems of nonlinear ODEsnon- linear circuitsdifferential algebraMaple Mathematics Subject Classification (2000) Primary 34A09; Sec- ondary 12H0565W3094C05 In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit.In the linear case, we apply basic "symbolic" methods from linear algebra to the system of differential equations which is used to model the analog circuit. For nonlinear circuits and their corresponding nonlinear differential equations, we show how to employ computer algebra tools implemented in Maple, which are based on differential algebra. I. INTRODUCTION Usually the input-output response of a linear time-invariant circuit is described in the frequency domain by its transfer function, i.e. a single rational function. This translates directly into a linear differential equation with constant coefficients in the time domain. The advantage of this approach is, that in any guise, only one (differential) equation is needed to completely describe the quantity a designer is interested in. This method fails miserably, when a transformation from the time to the frequency domain is not possible, e.g., when nonlinear circuits have to be examined, which lead to systems of nonlinear differential equations. Even though most of the times, these can be given in symbolic terms, any single quantity in the circuit usually is described by a "waveform", which results from assigning a numerical value to each symbol, followed by a numerical calculation using computer simulation. The advantage of having only one describing equation seems to have been irrevocably lost, when nonlinear This article first appeared in: Proceedings of the 7th International Workshop on Symbolic Methods and Applications to Circuit Design, SMACD 2002, Sinaia, Romania, October [10][11]2002. Bukarest 2002, pp. 65-70. Due to the low distribution of these proceedings, the author has decided to make the article available to a wider audience through the arXiv. At the time of origin of this paper the author was with the Institut für Netzwerktheorie und Schaltungstechnik, Technische Universität Braunschweig, D-38106 Braunschweig, Germany. His current (April 17th, 2008) address is Bismarckstraße 20, D-38518 Gifhorn, Germany. Current e-mail: e.gerbracht@web.de circuits are considered. Thus, up until now, nonlinear circuits seemed nearly inaccessible to most symbolic approaches. This problem is not a new one, and it is not limited to the area of analog circuits alone. Several years ago, researchers in nonlinear control theory have proposed to use constructive methods from differential algebra to tackle their problems. At the same time -and inspired in part by this proposal -mathematicians started to implement algorithms from differential algebra, which had already been formulated in the 1950s. These programmes became part of the MAPLE computer algebra system. At the SMACD-meeting in 1998 G. Carrà Ferro 1 gave examples of how to transform systems of nonlinear differential equations containing certain transcendental functions, that arise from analog circuits, into systems of nonlinear "algebraic" differential equations [1], and thus brought the area of constructive differential algebra in contact with the area of symbolic circuit analysis and design. In this paper we will take this approach several steps further and show, how single equations for any quantity in a circuit can be derived from these systems. We will be able to give a new and easy algorithm for linear circuits, which works in the time domain, and uses only differentiation and Gaussian elimination. For nonlinear circuits we will resort to the algorithms already implemented in MAPLE. We will comment on how to apply them and produce several examples. II. LINEAR CIRCUITS AND LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS To get a first flavour of things, we start our discussion with linear circuits and their corresponding systems of linear differential equations. But, instead of working in the frequency domain, using the Laplace-transform to deduce the transfer function for a sought-after quantity and thus its characterizing differential equation, we will remain in the time domain. A. Linear State Equations In this section we will present a new algorithm, that, starting from a set of state equations, only uses repeated differentiation and, finally, Gaussian elimination, to compute the single differential equation for any given quantity. We will restrict our presentation to just three state variables x 1 , x 2 , x 3 , but it is easy to extend the method, shown below, to any number of state variables. So, let us suppose, that a given linear circuit can be described by a set of linear state equations. We ask for one single differential equation describing, without loss of generality, the state variable x 1 . Again it is easy to handle other state variables or any output variable y, which is a linear combination of state variables and inputs, in an analogous manner. Let the system be given bẏ x 1 (t) = a 11 · x 1 (t) + a 12 · x 2 (t) + a 13 · x 3 (t) + e 1 (t), x 2 (t) = a 21 · x 1 (t) + a 22 · x 2 (t) + a 23 · x 3 (t) + e 2 (t), x 3 (t) = a 31 · x 1 (t) + a 32 · x 2 (t) + a 33 · x 3 (t) + e 3 (t),(1) where x 1 , x 2 , x 3 denote the state variables and e 1 , e 2 , e 3 represent linear combinations of the inputs (and eventually their derivatives). Clearly, if x 1 , x 2 , x 3 satisfy (1), their derivatives 2 will satisfÿ x 1 (t) = a 11 ·ẋ 1 (t) + a 12 ·ẋ 2 (t) + a 13 ·ẋ 3 (t) +ė 1 (t), x 2 (t) = a 21 ·ẋ 1 (t) + a 22 ·ẋ 2 (t) + a 23 ·ẋ 3 (t) +ė 2 (t), x 3 (t) = a 31 ·ẋ 1 (t) + a 32 ·ẋ 2 (t) + a 33 ·ẋ 3 (t) +ė 3 (t). (2) If n > 3 state variables are given, we have to repeat the above procedure n − 1 times. It is a well known observation (which will be shown as a byproduct of our algorithm), that n linear first order state equations lead to one nth order differential equation for any single quantity. Thus, in our example, we need one further equation for the third derivative ... x 1 of x 1 , which we get by differentiating once again the first equation of (2). ... x 1 (t) = a 11 ·ẍ 1 (t) + a 12 ·ẍ 2 (t) + a 13 ·ẍ 3 (t) +ë 1 (t). (3) Next, we write down all of the above equations into one system, where the vector of variables is given by all the derivatives of all state variables, that have been produced by the above procedure. ¼ ½¾ ½¿ ½ ½½ ¼ ¼ ½¾ ½¿ ½ ½½ ½ ¼ ¾¾ ¾¿ ¼ ¾½ ¼ ½ ¿¾ ¿¿ ¼ ¿½ ¼ ¼ ½¾ ½¿ ½ ½½ ½ ¼ ¾¾ ¾¿ ¼ ¾½ ¼ ½ ¿¾ ¿¿ ¼ ¿½ ½ ¡ ¼ ܾ´Øµ We point out the fact, that the variables should be arranged according to the orderingẍ 2 (t) >ẍ 3 (t) >ẋ 2 (t) >ẋ 3 (t) > x 2 (t) > x 3 (t) > ... x 1 (t) >ẍ 1 (t) >ẋ 1 (t) > x 1 (t). For our algorithm to work, it is essential, that the last entries of the vector are the derivatives of the variable, for which we want to deduce the differential equation, in decreasing order. The final step is to use Gaussian elimination to convert the system into one in upper triangular form ¼ ½ £ £ £ £ £ £ £ £ £ ½ £ £ £ £ £ £ £ £ ½ £ £ £ £ £ £ £ ½ £ £ £ £ £ £ ½ £ £ £ £ £ ½ £ £ £ £ ½ « ¾ « ½ « ¼ ½ ¡ ¼ Ü ¾´Ø µ Ü ¿´Ø µ Ü ¾´Ø µ Ü ¿´Ø µ Ü ¾´Ø µ Ü ¿´Ø µ ººº Ü ½´Ø µ Ü ½´Ø µ Ü ½´Ø µ Ü ½´Ø µ ½ ¼ £ £ £ £ £ £ ©´Øµ ½ (5) where Ψ(t) is a linear combination of the functionsë 1 (t), e 1 (t), e 1 (t),ė 2 (t), e 2 (t),ė 3 (t), and e 3 (t). Since we had (n−1)·n+1 (obviously) linearly independent equations for n 2 + 1 variables, the last row will give one equation for the last n + 1 variables, which according to our special ordering are all derivatives of x 1 . Thus we have deduced the differential equation for x 1 , which we were looking for. This is the same equation, which we would have found, if we had worked in the frequency domain and had calculated the transfer function for the Laplace transform L(x 1 ) of x 1 . B. Linear SemiState Equations It might happen, that a linear time-invariant circuit does not possess a description by state equations. Nevertheless it may be describable by so-called semistate equations, i.e. equations of the form (6) in which A, B, C, D and E are constant matrices, E being singular, x(t) denotes the semistate vector, u(t) the vector of inputs and y(t) the vector of outputs. Eẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), The algorithm given above can be adapted to this situation, but we have to keep in mind, that even if n semistate equations are given, the differential equation for any quantity might be of degree strictly less than n. We will see this effect in the example given below, which recently appeared in [2]. Example 1: (OTA-Circuit) + − x (t) 1 + − 4 1 x (t)=e (t) C g 1 2 x (t) 2 3 x (t) m4 g m3 The OTA circuit shown above can be described by the semistate system   C −C 0 −C C 0 0 0 0   ·   ẋ 1 (t) x 2 (t) x 3 (t)    = =   −g m4 g m4 1 −g m3 0 0 1 0 0   ·    x 1 (t) x 2 (t) x 3 (t)    +   0 0 −e 1 (t)   , where x 1 (t) and x 2 (t) denote the node to ground voltages of the respective nodes and x 3 (t) denotes the current into node 1 '! &" %# $ . As shown in [2], the transfer function H(s) := X 3 (s) E 1 (s) is given by H(s) = g m3 Cs + g m4 Cs .(7) We will deduce the corresponding differential equation for x 3 (t) in the time domain, using the above system of semistate equations and a slight modification of the algorithm for state equations.This algorithm is closer to the one, that will be used for nonlinear circuits. It is based on two main principles: 1) The variables x 1 , x 2 , x 3 and their derivatives, are supposed to be ordered by x 3 <ẋ 3 <ẍ 3 < · · · < x 2 <ẋ 2 <ẍ 2 < . . . < x 1 <ẋ 1 <ẍ 1 < . . . When a term is to be eliminated, we always choose the term of highest order. 2) Equations are only differentiated when "necessary". So, let us get into details; the system was given by Cẋ 1 − Cẋ 2 = −g m4 x 1 + g m4 x 2 + x 3 (8) −Cẋ 1 + Cẋ 2 = −g m3 x 1 (9) 0 = x 1 − e 1(10) The term of highest order appearing in (8) -(10) is x 1 . Equation (10) gives x 1 = e 1(11) This can be used to eliminate x 1 from (8) and (9). Furthermore, after differentiating (11), we geṫ x 1 =ė 1 .(12) Thus we are able to remove all instances of x 1 in the first two equations. Consequently we are led to: Cẋ 2 = Cė 1 − g m3 e 1(13)Cẋ 2 = Cė 1 + g m4 e 1 − g m4 x 2 − x 3(14) Clearly, these two equations imply the equality − g m3 e 1 = g m4 e 1 − g m4 x 2 − x 3 ,(15) which gives x 2 = g m3 + g m4 g m4 e 1 − x 3 g m4 (16) andẋ 2 = g m3 + g m4 g m4ė 1 −ẋ 3 g m4 (17) Putting this into (14), we arrive at Cė 1 − g m3 e 1 = C g m4 · (g m3 + g m4 )ė 1 − C ·ẋ 3 g m4 ,(18) which finally results in C ·ẋ 3 = C · g m3 ·ė 1 + g m3 · g m4 · e 1 .(19) This is the time domain equivalent of the transfer function (7), which we wanted to deduce. III. CONSTRUCTIVE DIFFERENTIAL ALGEBRA AND THE diffalg-PACKAGE IN maple This is not the time and the space to give even a cursory treatment of those parts of differential algebra, which are needed to understand the sometimes subtle generalization to the nonlinear case of the algorithms shown above. For our purposes it is enough to know, that, cum grano salis, all the mechanisms are already visible in the example of the OTA circuit. Fortunately there already exist several implementations of the necessary algorithms within the computer algebra system MAPLE. We will show by way of the above example the workings of one of these, the diffalg-package, created by F. Boulier [3], [4] and improved by E. Hubert [5] et al. A more detailed description (suitable for beginners with a mathematical background) can be found in the world wide web [6], [7]. After starting a MAPLE-session, one first has to load the package diffalg: > with(diffalg); [Rosenf eld Groebner, belongs to, delta leader, . . . ] (The second line in slanted notation represents the output produced by diffalg.) After initialization, one has to enter the set of differential equations under consideration. This has to be done in form of so-called differential polynomials. These are polynomials in the unknown functions x 1 , . . . , x m , their (time) derivatives x (α) i := D (α) x i := d α dt α x i , 1 ≤ i ≤ m, α ∈ N, the excitations e 1 , . . . , e k and their derivatives, again. In our example, we get three differential polynomials, which read in MAPLE-notation: > p 1 := C * diff(x 1(t), t) − C * diff(x 2(t), t) + g m4 * x 1(t) − g m4 * x 2(t) − x 3(t); > p 2 := −C * diff(x 1(t), t) + C * diff(x 2(t), t) + g m3 * x 1(t); > p 3 := x 1(t) − e 1(t); Next we have to tell the programme, which symbols it has to treat as constants. This is done with the command field extension. diffalg assumes, that we work over the rational numbers as ground field, where any further constants are considered as lying in a transcendental field extension of Q (i.e. we are allowed to divide by constants different from 0, and constants do not satisfy any algebraic relations). If we work with symbols, e.g. for capacitors, resistors etc., this poses no problem. If we work with real coefficients (e.g. floating point numbers or algebraic numbers like √ 2) major problems may arise. In our case we define > K := field extension(transcendental elements = [C, g m3, g m4]); K := ground f ield Finally we define a so-called differential ring, which is supposed to contain all the objects of interest (i.e. differential polynomials and constants), and in which we are allowed to do the following operations 1) multiply a differential polynomials with a constant; 2) add and multiply differential polynomials; 3) differentiate differential polynomials (if a constant is differentiated, the result is 0). As we have seen above, it is very important, that we define an ordering on differential monomials. This is needed to control the elimination process. For this purpose, diffalg asks for a ranking of the time dependent variables, from which it produces the obvious "elimination ordering". The variable, for which we want to know the differential equation, should be the last before the excitations. > R := differential ring(ranking = [x 1, x 2, x 3, e 1], derivations = [t], field of constants = K, notation = diff); R := ODE ring The command Rosenfeld Groebner lies at the heart of diffalg. It produces minimal sets of differential polynomials generating the differential polynomials, we have entered. > GE := Rosenfeld Groebner({p 1, p 2, p 3}, R); GE := [regular] GE is a list and may contain several components 3 . These components contain the sought-after differential equations, which can be listed with the help of the rewrite rules command; in our example this gives > rewrite rules(GE [1]); [x 1(t) = e 1(t), x 2(t) = −x 3(t) + e 1(t)g m3 + e 1(t)g m4 g m4 , ∂ ∂t x 3(t) = g m3 e 1(t)g m4 + C ∂ ∂t e 1(t) C     Mark, that the last entry is the differential equation for x 3 , which we had deduced in example 1. IV. NONLINEAR CIRCUITS AND THEIR CORRESPONDING DIFFERENTIAL EQUATIONS Now, that we know, how the MAPLE package diffalg can be used to deduce the time equivalent of the transfer function from any system of linear differential equations (with constant coefficients), in this section we will apply diffalg in an analogous manner to several systems of nonlinear equations, which come from nonlinear circuits. Example 2: (Damped resonant circuit with a nonlinear inductivity) As our first nonlinear example we have chosen a well known circuit from the literature, which leads to the Duffing equation (cp. [8], example 11-1, or [9], chapter 1. 3.2.). v (t) o v C v R C R L i L v L The resistor and the capacitor are assumed to be linear, i.e., they are described by i C (t) = Cv C (t) and v R (t) = R i R (t).(20) The inductor is assumed to be nonlinear, being described by v L (t) =Ψ(t),(21) where the current i L (t) is approximated by the cubic i L (t) = a · ψ(t) + b · ψ(t) 3 .(22) Finally, Kirchhoff' s equation lead to i R (t) = i L (t) = i C (t) and v R (t) + v L (t) + v C (t) = −v 0 (t). (23) Equations (20)-(23) are translated into their corresponding differential polynomials and are used as input for the diffalgroutine. This produces as part of the output of the subroutine rewrite rules: ∂ 2 ∂t 2 ψ(t) = − aψ(t) + bψ(t) 3 + ∂ ∂t v 0 (t) C C − CRa ∂ ∂t ψ(t) + 3CRb ψ(t) 2 ∂ ∂t ψ(t) C ,(24) which is formula 1.30 in [9] and is an equivalent of the Duffing equation: d 2 dt 2 ψ = −(a + 3b · ψ 2 ) · R d dt ψ − aψ C − bψ 3 C − d dt v 0 (t) (25) Example 3: (Chua's Circuit) Our next example is Chua's circuit, the first example of a physically realizable circuit, showing chaotic behavior [10]. This circuit consists of two linear capacitors, one linear inductor, one linear resistor and one nonlinear resistor, as shown below. v nl v L i L C C R nl v i C1 v 1 2 C2 R1 1 nl R Since it is easy to write down the equations for the linear elements and the Kirchhoff equations, we concentrate on the mathematical description of the nonlinear resistor. The classic description of R nl , using piecewise linear functions, is unsuitable for our purposes, because it does not satisfy the right differentiability conditions. Thus we use the one, presented e.g. in [11], where the negative arctangent is used to produce the nonlinear v-i characteristic of R nl . This leads to i nl (t) = −I 0 · arctan v nl (t) V 0 .(26) We are now confronted with a new problem: the arctangent is a transcendental function, thus (26) would not give a differential polynomial, as needed. One procedural solution to this problem was presented before in [1]. In the present paper, we are satisfied with the fact, that (26) implies yet another algebraic differential equation d dt i nl (t) = − I 0 V 0 · 1 + u nl (t) V 0 2 −1 · d dt u nl (t) (27) which obviously is given by a differential polynomial and which we can use as input for diffalg instead of (26). From this equation together with the Kirchhoff equations and the characterizing equations of the linear elements, diffalg produces the differential equation x (4) = − 1 C 1 C 2 RL · (C 1 + C 2 )L ... x + C 1 Rẍ +ẋ − I 0 · V 0 x 2 + V 2 0 [C 2 LR ... x + Lẍ + Rẋ] + 2 I 0 · V 0 (x 2 + V 2 0 ) 2 · L xẋ · [3 C 2 Rẍ +ẋ] − I 0 · V 0 (6x 2 − 2V 2 0 ) (x 2 + V 2 0 ) 3 C 2 LRẋ 3 ,(28) where x(t) denotes the voltage v C1 (t) through the capacitor C 1 . It has to be said that the final result given by diffalg looks slightly different, since it is given in expanded form, i.e. the numerator consists of 31 summands. Formula (28) has been reached at, only after some laborous post-processing. Example 4: (Simple Model of a Peak Rectifier Circuit) Now we are going to show how to handle diodes (and consequently by way of the Ebers-Moll model the large signal behaviour of BJTs) in electric circuits. As an example we have chosen a simple model of a peak rectifier circuit as seen in [12], chapter 3.7 pp. 185ff. v (t) o v C v a R C v D R a e D i Again we concentrate on the only nonlinear element in the circuit -the diode. It is well known, that the v-i characteristic of a nonideal diode can be approximately described by i D (t) = I s · exp v D (t) V T − 1 ,(29) where I s is the saturation current and V T is the thermal voltage -quantities, which we consider constant during the course of our analysis. As before we have to translate a transcendental equation into a differential polynomial. This can be done easily by differentiating (29) once, which due to the chain rule d dt i D (t) = d dvD i D · d dt v D (t) results in the equation d dt i D (t) = 1 V T d dt v D (t) · [i D (t) + I s ].(30) This time diffalg produces the following second order differential equation for the output voltage v a (t) : v a = − R a · [V Tva − (v 0 −v a ) · v x ] + [v a · R e v x ] CR a · (V T R a + R e v x ) ,(31) where we have set v x := (CR ava + v a + R a I s ) . For the purpose of comparing this result to that appearing in the literature, we give the differential equation in case of an ideal voltage source, i.e. R e = 0Ω. It is given bÿ v a = 1 CR a −v a + 1 V T [v 0 −v a ] · [CR ava + v a + R a I s ] .(32) V. FURTHER EXAMPLES Example 5: (Diode Circuit with LC-Load and Nonideal Voltage source) v (t) o v C D i R C v D L v L i L The above circuit may not be of much practical interest. Nevertheless it is a good test of the power of our approach (and the capabilities of diffalg), since it slightly generalizes example 4 and we increase the number of dynamic elements in our circuit. Again, the diode is assumed to be nonideal, given by (30). The output voltage x(t) := v C (t) is described by the equation ... x = − 1 CLẋ + x + CLẍ x −v 0 · 1 V T · CL · R L (x + CLẍ) + (ẋ −v 0 ) 2 − V T (ẍ −v 0 ) .(33) In the course of our investigations, we have tried a number of larger circuits, which in principle are accessible to our approach. We met two main obstacles, which are natural in the "business" of symbolic methods: 1) the combinatorial explosion, resulting in a larger and larger number of terms contained in the final differential equation, and 2) the massive increase in time, needed by diffalg to produce this equation. In the sequel we give a short report on these experiments: Example 6: (Peak Rectifier with Power Transformer) v (t) o D i v /ü 1 R C L L h s C v D a R a v a v 1 R + R 0 g1 i i s h This kind of circuit is described in the introduction to chapter 3.7 in [12]. As shown above, we have used the model given by Horneber in his PhD-thesis [13], section 14.1. There, it is the smaller of two examples, the other being the "Ring Modulator", which has become a benchmark in the numerical analysis of initial value problems [14]. From our point of view, we can tell, that diffalg, although needing substantial more time than in the previous examples (1-2 minutes instead of only seconds), is able to produce a fifth order differential equation for the output voltage v a . Unfortunately, we are not able to reproduce this result here, since the initial output even after some simplifications contains more than 600 summands. Thus some more "post processing" is needed to get an intelligible result. Even with the help of other facilities of MAPLE this work has not been finished, yet 4 . Finally we have tried our approach on a "simple" single-stage common-emitter amplifier ([12], chapter 4.11) as modelled in [15] and on the above mentioned ring modulator of Horneber. In both cases, up until now, even though we have used several days of computing time, we were not able to produce any results. Although the latter -very ambitious -example (which presumably will lead to an differential equation of order 18) may be beyond the scope of any computer algebra system for some time, the first should be within our grasp and should be attacked further. VI. CONCLUSION In this paper we have shown, how, using constructive methods from differential algebra and one of their realizations -the package diffalg of the computer algebra system MAPLElinear and nonlinear circuits can be described by a single differential equation. In the future it will be necessary to further examine the power of this approach, i.e. to find more and larger circuits, which can be treated this way. Furthermore, if the number of these circuits is large enough, methods have to be found, that allow a fast and "easy" analysis of the resulting equations, analogous to the analysis of linear circuits by way of their transfer functions. NOTE ADDED TO THE ELECTRONIC VERSION In this electronic document, some small typographical errors of the printed version were corrected. This especially refers to formulas (18) and (19). Furthermore, for the convenience of the reader the abstract has been rewritten, and keywords, an MSC classification, and a short CV according to IEEE standards have been added. URLs have been checked again, and, where necessary, have been updated. Finally the dedication has been expanded. The main body of the article, however, remains unchanged. (April 17th, 2008) During that time he was also appointed lecturer for several courses on digital circuit design at the University of Applied Sciences Braunschweig/Wolfenbüttel, Germany. From 2001 to 2002 he was appointed lecturer for a two-semester course in linear circuit analysis at the TU Braunschweig. After a two-year stint as a mathematics and computer science teacher at a grammar school in Braunschweig and a vocational school in Gifhorn, Germany, he is currently working as advisor, and independent researcher in various areas of mathematics. His research interests include combinatorial algebra, C*-algebras, the history of mathematics in the 19th and early 20th century and applications of computer algebra and dynamical geometry to graph theory, calculus, and electrical engineering. Dr. Gerbracht is a member of the German Mathematical Society (DMV), the German Society for Didactics of Mathematics (GDM), and founding member of the society "Web Portal: History in Braunschweig -www.gibs.info". Eberhard H.-A. Gerbracht received a Dipl.-Math. degree in mathematics, a Dipl.-Inform. degree in computer science, and a Ph.D. (Dr. rer.nat.) degree in mathematics from the Technical University Braunschweig, Germany, in 1990, 1993, and 1998, respectively. From 1992 to 1997 he was a Research Fellow and Teaching Assistant at the Institute for Geometry at the TU Braunschweig. From 1997 to 2003 he was an Assistant Professor in the Department of Electrical Engineering and Information Technology at the TU Braunschweig. Note added in 2008: Between the time of this article's first publication and the publication of the electronic version, Giuseppa Carrà Ferro passed away on March 22nd, 2007. Thus a dedication to her memory was added to this electronic version. In the sequel wherever necessary, we will assume that all expressions are differentiable over a suitable extension field of the real numbers. This is one of the subtleties of working with diffalg, the details of which we do not want to present here. Note added in 2008: Meanwhile these calculations have been done. The end result still is to unwieldy to be presented here. Furthermore the collecting and combining of fully symbolic terms by hand has turned out to be so errorprone that, in the opinion of the author, some kind of additional "plausibility measures" need to be introduced. Computer Algebra. Carrà Giuseppa, Ferro, Proceedings of the 5th International Workshop on Symbolic Methods and Applications in Circuit Design -SMACD '98. Ralf Sommer, and Eckhard Hennigthe 5th International Workshop on Symbolic Methods and Applications in Circuit Design -SMACD '98Kaiserslautern; KaiserslauternDieter Prätzel-WoltersNonlinear Differential Equations and Analog Circuit DesignGiuseppa Carrà Ferro, "Computer Algebra, Nonlinear Differential Equations and Analog Circuit Design," in Proceedings of the 5th International Workshop on Sym- bolic Methods and Applications in Circuit Design - SMACD '98, Kaiserslautern, October 8-9,1998, Dieter Prätzel-Wolters, Ralf Sommer, and Eckhard Hennig, Eds., Kaiserslautern, 1998, pp. 49-56. Semistate Theory and Analog VLSI Design. Angela M Hodge, Robert W Newcomb, IEEE Circuits and Systems Magazine. 22Second QuarterAngela M. Hodge and Robert W. Newcomb, "Semistate Theory and Analog VLSI Design," IEEE Circuits and Systems Magazine, vol. 2, no. 2, pp. 30-51, Second Quarter 2002. Representation for the radical of a finitely generated differential ideal. F Boulier, D Lazard, F Ollivier, M Petitot, Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, ISSAC '95. M. Leveltthe 1995 International Symposium on Symbolic and Algebraic Computation, ISSAC '95Montreal, Canada; New YorkACM PressF. Boulier, D. Lazard, F. Ollivier, and M. Petitot, "Re- presentation for the radical of a finitely generated dif- ferential ideal," in Proceedings of the 1995 Interna- tional Symposium on Symbolic and Algebraic Compu- tation, ISSAC '95, Montreal, Canada, July 10-12, 1995., A.H.M. Levelt, Ed., New York, 1995, pp. 158-166, ACM Press, https:// hal.archives-ouvertes.fr/ hal-00138020/en, viewed April 17th, 2008. François Boulier, Etude et implantation de quelques algorithmes en algèbre différentielle. Université des Sciences et Technologies de LillePh.D. thesisFrançois Boulier, Etude et implantation de quelques algorithmes en algèbre différentielle, Ph.D. the- sis, Université des Sciences et Technologies de Lille, 1994, http://tel.archives-ouvertes.fr/ tel-00137866/en, viewed April 17th, 2008. Evelyne Hubert, Étude Algébrique et Algorithmique des Singularités des Equations Différentielles Implicites. Institut National Polytechnique GrenoblePh.D. thesisEvelyne Hubert,Étude Algébrique et Algorithmique des Singularités des Equations Différentielles Implicites, Ph.D. thesis, Institut National Polytechnique Grenoble, 1997, http://www-sop.inria.fr/cafe/ The diffalg package. Evelyne Hubert, Evelyne Hubert, "The diffalg package," http:// www-sop.inria.fr/cafe/Evelyne.Hubert/ diffalg, viewed April 17th, 2008. Basic concepts in constructive differential algebra and their representation in the diffalg package. Evelyne Hubert, Evelyne Hubert, "Basic concepts in constructive differential algebra and their representation in the diffalg package," http://www-sop.inria.fr/ cafe/Evelyne.Hubert/diffalg/diffalg/ differential − algebra1.html, viewed April 17th, 2008. Thomas E Stern, Theory of Nonlinear Networks and Systems. An Introduction. Reading, MassachusettsAddison-WesleyThomas E. Stern, Theory of Nonlinear Networks and Systems. An Introduction, Addison-Wesley, Reading, Massachusetts, 1965. Analyse nichtlinearer dynamischer Systeme der Elektrotechnik. Eugen S Philippow, Wolfgang G Büntig, Hanser-VerlagMünchenEugen S. Philippow and Wolfgang G. Büntig, Analyse nichtlinearer dynamischer Systeme der Elektrotechnik, Hanser-Verlag, München, 1992. The Genesis of Chua's Circuit. Leon O Chua, Int. J. Electron. Commun. (AEÜ). 464Leon O. Chua, "The Genesis of Chua's Circuit," Int. J. Electron. Commun. (AEÜ), vol. 46, no. 4, pp. 250-257, 1992. Behavioral Modeling and Transient Analysis with Analog Insydes. Thomas Halfmann, Eckhard Hennig, Manfred Thole, Proceedings of the 5th International Workshop on Symbolic Methods and Applications in Circuit Design -SMACD '98. Ralf Sommer, and Eckhard Hennigthe 5th International Workshop on Symbolic Methods and Applications in Circuit Design -SMACD '98Kaiserslautern; KaiserslauternDieter Prätzel-WoltersThomas Halfmann, Eckhard Hennig, and Manfred Thole, "Behavioral Modeling and Transient Analysis with Ana- log Insydes," in Proceedings of the 5th International Workshop on Symbolic Methods and Applications in Circuit Design -SMACD '98, Kaiserslautern, October 8-9,1998, Dieter Prätzel-Wolters, Ralf Sommer, and Eck- hard Hennig, Eds., Kaiserslautern, 1998, pp. 49-56. Adel S Sedra, Kenneth C Smith, Microelectronic Circuits. New YorkOxford University Press4Adel S. Sedra and Kenneth C. Smith, Microelectronic Circuits, Oxford University Press, New York, 4 1998. Analyse nichtlinearer RLCÜ-Netzwerke mit Hilfe der gemischten Potentialfunktion mit einer systematischen Darstellung der Analyse nichtlinearer dynamischer Netzwerke. Ernst-Helmut Horneber, Fachbereich Elektrotechnik der Universität KaiserslauternPh.D. thesisErnst-Helmut Horneber, Analyse nichtlinearer RLCÜ- Netzwerke mit Hilfe der gemischten Potentialfunktion mit einer systematischen Darstellung der Analyse nichtlin- earer dynamischer Netzwerke, Ph.D. thesis, Fachbereich Elektrotechnik der Universität Kaiserslautern, 1976. Test set for initial value problem solvers. M Walter, Jacques J B Lioen, De Swart, Release 2.1. Also available asWalter M. Lioen and Jacques J.B. de Swart, "Test set for initial value problem solvers," http://db.cwi.nl/ rapporten/abstract.php?abstractnr=796, 1999, Release 2.1. Also available as http:// ftp.cwi.nl/IVPtestset/, viewed April 17th, 2008. Symbolic Methods in Analog Circuit Design. Eckhard Hennig, Ralf Sommer, 2nd IMACS Conference on Applications of Computer Algebra. Eckhard Hennig and Ralf Sommer, "Symbolic Methods in Analog Circuit Design," in 2nd IMACS Conference on Applications of Computer Algebra, Linz 17-20 July 1996, 1996.

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{'abstract': 'In this paper we describe by a number of examples how to deduce one single characterizing higher order differential equation for output quantities of an analog circuit.In the linear case, we apply basic "symbolic" methods from linear algebra to the system of differential equations which is used to model the analog circuit. For nonlinear circuits and their corresponding nonlinear differential equations, we show how to employ computer algebra tools implemented in Maple, which are based on differential algebra.', 'arxivid': '0804.2992', 'author': ['Eberhard H Gerbracht ', 'Eberhard H Gerbracht '], 'authoraffiliation': [], 'corpusid': 16720538, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10367, 'n_tokens_neox': 9060, 'n_words': 5687, 'pdfsha': '9d893bc735a6343c192885cdd345f722a53e18c2', 'pdfurls': ['https://arxiv.org/pdf/0804.2992v1.pdf'], 'title': ['"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra (2008 Re-Release)', '"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra (2008 Re-Release)', '"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra (2008 Re-Release)', '"E pluribus unum" or How to Derive Single-equation Descriptions for Output-quantities in Nonlinear Circuits using Differential Algebra (2008 Re-Release)'], 'venue': []}

arxiv

Prepared for submission to JHEP Vortices in a generalized Maxwell-Higgs model with visible and hidden sectors 18 May 2018 D Bazeia bazeia@fisica.ufpb.br Departamento de Física Universidade Federal da Paraíba 58051-970João PessoaPBBrazil L Losano losano@fisica.ufpb.br Departamento de Física Universidade Federal da Paraíba 58051-970João PessoaPBBrazil M A Marques Departamento de Física Universidade Federal da Paraíba 58051-970João PessoaPBBrazil R Menezes rmenezes@dce.ufpb.br Departamento de Física Universidade Federal da Paraíba 58051-970João PessoaPBBrazil Departamento de Ciências Exatas Universidade Federal da Paraíba 58297-000Rio TintoPBBrazil Prepared for submission to JHEP Vortices in a generalized Maxwell-Higgs model with visible and hidden sectors 18 May 20181 Corresponding author.Solitons Monopoles and Instantons, Field Theories in Lower Dimensions We investigate the presence of vortices in generalized Maxwell-Higgs models with a hidden sector. The model engenders U (1) × U (1) symmetry, in a manner that the sectors are coupled via the visible magnetic permeability depending only on the hidden scalar field. We develop a first order framework in which the hidden sector decouples from the visible one. We illustrate the results with two specific examples, that give rise to the presence of vortices with internal structure. Introduction Topological structures appear in high energy physics in a variety of dimensions [1][2][3]. Among the mostly known ones, there are kinks, vortices and monopoles, which are static configurations that appear in (1, 1), (2,1) and (3, 1) spacetime dimensions, respectively. These structures have been investigated for more than 40 years, and they find a myriad of applications in high energy and condensed matter physics [1][2][3][4]. Vortices, in particular, were firstly investigated by Helmholtz in Ref. [5] and are commonly found in fluid mechanics [6]. They also appear in condenser matter, in the study of type II superconductors under specific conditions [7] when one deals with the Ginzburg-Landau theory of superconductivity [8]. As it is known, the Ginzburg-Landau theory is non-relativistic and the first relativistic model that supports vortex configurations was introduced by Nielsen and Olesen in Ref. [9]. It consists of a Higgs field coupled to a gauge field with the Maxwell dynamics under a local U (1) symmetry. These vortices are electrically neutral but have quantized magnetic flux. They were largely investigated in several different contexts, including the case of generalized models; see Refs. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In Refs. [10,11], for instance, the presence of a generalized magnetic permeability in a model of the Maxwell-Higgs type allowed to simulate properties of Chern-Simons vortices [29]. By using a similar model in Ref. [24], we have found the presence of compact vortices, which seems to map the magnetic field of an infinitely long solenoid. Due to the aforementioned works with generalized kinematics, we have introduced a first order formalism to describe vortices in generalized models in Ref. [27] and a procedure to find analytic solutions in Ref. [28]. More recently, in Ref. [30], we have enlarged the U (1) symmetry to accommodate a Z 2 symmetry, with the addition of a single neutral scalar field that interacts with the gauge field via the magnetic permeability. In this case, it was shown that the neutral field acts as a source to generate the vortex. In the past, a similar idea was developed, but with an U (1) × U (1) symmetry engendered by two gauge fields and two complex scalar fields interacting through an extension of the standard Higgs-like potential [31]. This model is of interest in the study of superconducting strings, and has been used in several contexts, in particular in Refs. [32][33][34] and in references therein. For instance, in Ref. [33] the authors describe the presence of novel solutions for non-Abelian cosmic strings and in [34] the study focuses on the critical behavior of magnetic field field in a superconductor coupled to a superfluid, intending to simulate the core of neutrons stars, where superconducting protons are supposed to couple with superfluid neutrons. Among other possibilities, the U (1) symmetry may be enlarged to accommodate other fields. In particular, in Refs. [35][36][37] the U (1) symmetry is enlarged to become U (1) × SO(3), in which the SO(3) symmetry is governed by the addition of some neutral scalar fields. Furthermore, in the recent work [38] we considered the symmetry SU (2) × Z 2 , extending the study of monopoles in the non Abelian model considered Refs. [39,40] to include a new neutral scalar field to produce magnetic monopoles with internal structure in the three-dimensional space. Models with the U (1) × U (1) symmetry are useful to include the so-called hidden sector, which seems to be of current interest [41,42] and may play a role in the study of dark matter [43][44][45][46]. This hidden (or dark) sector may be coupled to the visible one via the Higgs fields, as it appears in the Higgs portal [47,48], in a way similar to the coupling that appeared before in Refs. [31][32][33]. Another possibility is to add the interaction of the gauge field strengths as in [49][50][51][52][53], with the two strength tensors coupled to each other. We can also enlarge the symmetry and consider SU (2) × U (1), as in [33], for instance, and also SU (3) × U (1), as investigated very recently in [54], to search for color-magnetic structures in dense quark matter, compatible with the interior of compact stars and able to produce detectable gravitational waves. In this work, however, we follow a different direction and consider the U (1) × U (1) symmetry, choosing the magnetic permeability of the visible sector to be driven by the scalar field of the hidden sector, without the coupling between the electromagnetic strength tensors. This is a novel possibility, which leads to results of current interest. In order to introduce and investigate the U (1) × U (1) model, we organize the paper as follows: in Sec. 2, we present the model and develop the Bogomol'nyi procedure [55], to find first order differential equations which are very relevant to describe stable vortex configurations. For the proposed model, we also show that the first order framework acts to decouple the hidden sector from the visible one. This is an interesting result, and we can then use the hidden sector as a source to describe the visible sector. In Sec. 3, we illustrate the results with two examples, which give rise to vortices with features of current interest. We end the work in Sec. 4, where we add conclusions and perspectives for future works. The general model We work in (2, 1) flat spacetime dimensions with the action S = d 3 xL, where the Lagrangian density is L = − 1 4 P (|χ|)F µν F µν − 1 4 Q(|χ|)F µν F µν + |D µ ϕ| 2 + |D µ χ| 2 − V (|ϕ|, |χ|). (2.1) Here ϕ and A µ are the visible complex scalar and gauge fields, and the corresponding hidden fields are χ and A µ . As usual, F µν = ∂ µ A ν − ∂ ν A µ is the visible electromagnetic strength tensor, and D µ = ∂ µ + ieA µ stands for the covariant derivative. Their equivalent hidden counterparts are F µν = ∂ µ A ν − ∂ ν A µ and D µ = ∂ µ + iqA µ . The potential is denoted by V (|ϕ|, |χ|) and may present terms that couple both the visible and hidden scalar fields. In principle, the explicit form of the potential is unknown, and the generalized magnetic permeabilities are controlled by P (|χ|) and Q(|χ|), which are non negative functions that depend exclusively on the field χ. The above gauge invariant Lagrangian density engenders the U (1) × U (1) symmetry. The inclusion of functions of the scalar field multiplying the Maxwell term is also present in models related to holography; see, e.g., [56,57] for two distinct possibilities, used to describe an holographic insulator model with nonsingular zero temperature infrared geometry [56], and the electric charge transport in a strongly coupled quark-gluon plasma [57]. In our model, the visible and hidden sectors are coupled through P (|χ|), i.e., the hidden scalar field controls the magnetic permeability of the visible sector. This is a different approach from the one considered in Ref. [53], where the coupling was done with the electromagnetic strength tensors, through the term F µν F µν , which we are not considering in the current work. Here, we use the metric tensor η µν = (1, −1, −1) and the natural units = c = 1. By varying the action with respect to the fields, we get that the equations of motion associated to the Lagrangian density (2.1) are D µ D µ ϕ = − ϕ 2|ϕ| V |ϕ| , (2.2a) D µ D µ χ = − χ 2|χ| P |χ| 4 F µν F µν + Q |χ| 4 F µν F µν + V |χ| , (2.2b) ∂ µ (P F µν ) = J ν , (2.2c) ∂ µ (QF µν ) = J ν , (2.2d) where the currents are J µ = ie ϕ D µ ϕ − ϕ D µ ϕ and J µ = iq χ D µ χ − χ D µ χ . Also, we have used the notation V |ϕ| = ∂V /∂|ϕ|, V |χ| = ∂V /∂|χ| and so on. By setting ν = 0 in Eqs. (2.2c) and (2.2d) and considering static field configurations, one can show the Gauss' laws are identities for A 0 = 0 and for A 0 = 0. In this case, the solutions are electrically neutral since the electric charges vanish. Invariance under spacetime translations x µ → x µ + b µ , with b µ constant, lead to the conserved energy-momentum tensor T µν = P F µλ F λ ν + QF µλ F λ ν + D µ ϕD ν ϕ + D ν ϕD µ ϕ + D µ χD ν χ + D ν χD µ χ − η µν L. (2.3) To search for vortexlike solutions, we consider static configurations and the usual ansatz ϕ = g(r)e inθ , χ = h(r)e ikθ , A =θ er [n − a(r)], A =θ qr [k − c(r)],(2.4) in which n and k are nonvanishing integer numbers that control the vorticity of the visible and hidden solutions, respectively. The functions a(r), c(r), g(r) and h(r) obey the boundary conditions g(0) = 0, h(0) = 0, a(0) = n, c(0) = k, (2.5a) g(∞) = v, h(∞) = w, a(∞) = 0, c(∞) = 0. (2.5b) Here, v and w are parameters that control the asymptotic behavior of the functions g(r) and h(r). Considering Eqs. (2.4), the visible and hidden magnetic fields are giving by, respectively, B = −F 12 = − a er and B = −F 12 = − c qr , (2.6) where the prime stands for the derivative with respect to r. By using this, one can show the magnetic fluxes are quantized Φ (B) = 2π rdrB = 2π e n and Φ (B) = 2π rdrB = 2π q k. (2.7) We use Eqs. (2.4) to rewrite the equations of motion (2.2) in the form 1 r rg − a 2 g r 2 − 1 2 V |ϕ| = 0, (2.8a) 1 r rh − c 2 h r 2 − 1 2 P |χ| a 2 2e 2 r 2 + Q |χ| c 2 2q 2 r 2 +V |χ| = 0, (2.8b) r P a er − 2eag 2 = 0, (2.8c) r Q c qr − 2qch 2 = 0. (2.8d) Using the above Eqs. (2.4), the nonvanishing components of energy-momentum tensor (2.3) become T 00 = P a 2 2e 2 r 2 + Q c 2 2q 2 r 2 + g 2 + a 2 g 2 r 2 + h 2 + c 2 h 2 r 2 + V, (2.9a) T 12 = g 2 − a 2 g 2 r 2 + h 2 − c 2 h 2 r 2 sin(2θ), (2.9b) T 11 = P a 2 2e 2 r 2 + Q c 2 2q 2 r 2 + g 2 + h 2 2 cos 2 θ − 1 + a 2 g 2 r 2 + c 2 h 2 r 2 2 sin 2 θ − 1 − V, (2.9c) T 22 = P a 2 2e 2 r 2 + Q c 2 2q 2 r 2 + g 2 + h 2 2 sin 2 θ − 1 + a 2 g 2 r 2 + c 2 h 2 r 2 2 cos 2 θ − 1 − V. (2.9d) In the above equations, we identify the energy density as ρ = T 00 and the stress tensor as T ij . The equations of motion (2.8) present coupling between the functions and are of second order. In order to simplify the problem, it is of interest to find a first order formalism for the model. This can be done by using Eq. (2.9a) to develop the Bogomol'nyi procedure [55]. After some algebraic manipulations, we can write ρ = P 2 a er + e v 2 − g 2 P 2 + Q 2 c qr + q w 2 − h 2 Q 2 + g − ag r 2 + h − ch r 2 + V − e 2 2 v 2 − g 2 2 P + q 2 2 w 2 − h 2 2 Q − 1 r a v 2 − g 2 + c w 2 − h 2 . (2.10) As we stated before, the potential is in principle arbitrary. Nevertheless, the Bogomol'nyi procedure requires it to have the form V (|ϕ|, |χ|) = e 2 2 v 2 − |ϕ| 2 2 P (|χ|) + q 2 2 w 2 − |χ| 2 2 Q(|χ|) . (2.11) In this case, the energy can be written as E = 2π ∞ 0 r dr P 2 a er + e v 2 − g 2 P 2 + 2π ∞ 0 r dr Q 2 c qr + q w 2 − h 2 Q 2 + 2π ∞ 0 r dr g − ag r 2 + 2π ∞ 0 r dr h − ch r 2 + E B ,(2.12) where E B = 2π ∞ 0 dr a v 2 − g 2 + c w 2 − h 2 = 2π |n|v 2 + |k|w 2 . (2.13) In Eq. (2.12), the first four terms are all non-negative, so we see that the energy is bounded to the value E = E B , if the solutions obey the first order equations g = ag r , (2.14a) − a er = e v 2 − g 2 P (h) , (2.14b) for the visible sector, and h = ch r , (2.15a) − c qr = q w 2 − h 2 Q(h) , (2.15b) for the hidden sector. Therefore, we have obtained four first order equations to study the problem. One can show they satisfy the equations of motion (2.8) with the potential (2.11). An interesting fact is that they are compatible with the stressless condition, T ij = 0, which assures the stability of the solutions under rescaling; see Refs. [27,58,59]. Notice that the first order equations (2.15) are decoupled from the ones for the visible sector in Eq. (2.14). Therefore, we firstly solve the hidden sector to see how it modifies the visible one by choosing P (|χ|). Before doing that, however, we notice that the presence of the first order equations (2.15) and (2.14) allows us to write the energy density (2.9a) in the form ρ = ρ hidden +ρ visible , where ρ hidden = Q(h) c 2 q 2 r 2 + 2h 2 , (2.16a) ρ visible = P (h) a 2 e 2 r 2 + 2g 2 , (2.16b) indicate the contribution of the visible and hidden sectors, respectively. We also highlight the fact that, from Eq. (2.13), the energy of the visible and hidden sectors are independent and fixed, despite the presence of the functions P (|χ|) and Q(|χ|). For the visible sector, we have E visible = 2πv 2 |n| and for the hidden one, E hidden = 2πw 2 |k|. Some specific models Let us now illustrate our findings with two distinct examples. Before doing so, we can simplify the problem by rescaling the fields as ϕ → vϕ, χ → vχ, A µ → vA µ and A µ → vA µ ,(3.1) and taking r → r/ev and L → e 2 v 4 L. This leads us to work with dimensionless quantities. We also consider e = 1 and v = 1 and understand the parameter q and w as the ratio of the respective constants (q/e) and (w/v) of the hidden and visible sectors. First example Considering the hidden sector to be controlled by Q(|χ|) = 1, its first order equations (2.15) take the form h = ch r , (3.2a) − c qr = q w 2 − h 2 , (3.2b) which admits vortexlike solutions of the Nielsen-Olesen type. Unfortunately, the analytical solutions a(r) and h(r) are currently unknown. Therefore, one has to use numerical methods to find them. Since their profiles are well known (see Refs. [9,60]), we will not display them here. We then go on and investigate how the hidden fields modify the visible sector. We take the choice P (|χ|) = 1 (1 − β|χ| 2 ) 2 , (3.3) where β is a non negative, dimensionless parameter. Notice that β controls how strongly the hidden scalar field modifies the magnetic permeability of the visible sector. The case β = 0 leads to P (|χ|) = 1, which decouples the sectors. The above function makes the potential in Eq. (2.11) to become V (|ϕ|, |χ|) = 1 2 1 − |ϕ| 2 2 1 − β|χ| 2 2 + q 2 2 w 2 − |χ| 2 2 . (3.4) From Eq. (2.15), we get that the first order equations for the visible sector are g = ag r , (3.5a) − a r = 1 − g 2 1 − βh 2 2 . (3.5b) It is straightforward to see that the case β = 0 leads to Nielsen-Olesen solutions, which are very similar to the ones of the hidden sector, except for the constants q and w. The parameter β plays an important role in the model. One can show that a (r) vanishes for h 2 = 1/β. We then expect to see a region in which a(r) is approximately uniform. Since 0 ≤ h(r) < 1, this feature appears only for β > 1, which is the range that we will consider here. In Fig. 1, we plot the solutions of the above equations, the visible magnetic field (2.6) and the energy density (2.9a) for some values of β and q, w = 1. As β increases, the solutions, the magnetic field and the energy density shrink. The solution a(r), in particular, presents a region in which it is approximately uniform, as expected. This region is very wide for β ≈ 1 and tends to shrink and to become closer to the origin as we increase β. Due to this behavior in the solution, the magnetic field present a splitting. This also happens, in a more subtle manner, with the energy density. Therefore, the presence of the hidden sector generates an internal structure to the vortex engendered by the visible sector of the model. It is worth commenting that these features do not affect the magnetic flux and the energy of the vortex, which remain unchanged, given by Eqs. (2.7) and (2.13). The presence of a internal structure in the quantities related to the visible sector motivated us to plot the magnetic field and the energy density in the plane. It can be seen in Fig. 2. Second example We now make a modification in the magnetic permeability of the hidden sector and suggest that it is driven by the function Q(|χ|) = 1 2 1 − |χ| 2 w 2 . (3.6) According to the boundary conditions in Eq. (3.7b) In this case, we have been able to find their analytical solutions. They are given by c(r) = 1 1 + (qwr) 2 and h(r) = qw 2 r 1 + (qwr) 2 . (3.8) The above expression for c(r) allows us to calculate magnetic field associated to the hidden gauge field from Eq. (2.6), which is B = 2qw 2 (1 + (qwr) 2 ) 2 . (3.9) The energy density of the hidden sector in Eq. (2.16a) may be also found explicitly ρ hidden = 2q 2 w 4 3 + (qwr) 2 (1 + (qwr) 2 ) 4 . We then investigate how the solutions (3.8) modify the visible sector by considering its magnetic permeability to be governed by the same function in Eq. (3.3). By doing so, the potential in Eq. (2.11) can be written as V (|ϕ|, |χ|) = 1 2 1 − |ϕ| 2 2 1 − β|χ| 2 2 + q 2 w 2 w 2 − |χ| 2 3 . (3.11) In this case, the first order equations for the visible sector are g = ag r , (3.12a) − a r = 1 − g 2 1 − βq 2 w 4 r 2 1 + q 2 w 2 r 2 2 . (3.12b) The solutions of the above equations, the visible magnetic field (2.6) and energy density (2.9a) are depicted in Fig. 3. The parameter β, as in the previous example, plays an important role in the model. Here, we highlight that the presence of the analytical solutions in Eq. (3.8) allows us to show that the derivative of the solution a(r) vanish at the point r mod = 1/(qw √ β − 1). This makes the solution approximately uniform in the neighborhood of this point. As before, we only take β > 1, since this is the range in which this feature appears. Note that, as β increases, r mod approaches the origin. The visible magnetic field also feels the modification introduced by the hidden fields. It has a valley at the point r mod , presenting an internal structure that becomes more and more apparent and nearer the origin as β increases. A similar, although more subtle, behavior occurs in the energy density of the visible sector. This model resembles the previous one in a qualitative manner, presenting only quantitative differences. However, the presence of analytical solutions in the hidden sector helps us to better understand the internal structure of the quantities of the visible sector. The new features presented by our models motivated us to depict the magnetic field and the energy density for the solutions of Eqs. (3.12) of the visible sector in the plane. It can be seen in Fig. 4. As it was mentioned before, the presence of the internal structure is more strong in the magnetic field than in the energy density. Conclusions In this work, we studied vortex structures in generalized Maxwell-Higgs models with visible and hidden sectors. The models considered in this work contain two additional functions, P (|χ|) and Q(|χ|), that control the magnetic permeability of the visible and hidden sectors, respectively. The interaction between the two sectors are controlled by the hidden scalar field which appears in the function P (|χ|). We have chosen an specific form of the potential, which allowed for the Bogomol'nyi procedure to work out, and for the presence of first order differential equations that solve the equations of motion. The procedure has shown that the first order equations of the hidden sector decouple from the visible one and could be solved independently. In this sense, the hidden sector acts as a source for the solutions in the visible sector. By taking specific forms for the magnetic permeabilities, we have found that the hidden charged scalar field affects the visible vortex configuration, generating an internal structure to it. The magnetic fields present an apparent valley which seems to connect two separated structures. The effect is less evident in the energy density, although it is also there. Surprisingly, the energies and fluxes of the hidden and visible vortices are fixed by the boundary conditions, and they are independent from each other. Moreover, they do not depend on the specific form we choose for the magnetic permeabilities to construct the system. As a particularly interesting result, we could find a specific function Q(|χ|) which allowed us to describe the vortex solution of the hidden sector, and the corresponding magnetic field and energy density, analytically. In this paper, we have discarded the coupling between the two electromagnetic strength tensors, so an issue to be further examined would be to add this kind of coupling. Another possibility is to generalize the covariant derivative terms, as suggested before in [28]. Since the model examined above attains the Bogomol'nyi bound, it appears to be the bosonic portion of a lager, supersymmetric theory, and this is presently under consideration, to see how supersymmetry is working to lead us to the first order equations. We are also thinking of enlarging the model to accommodate other symmetries, such as the SU (2) × U (1) symmetry, to explore the presence of solutions within the non-Abelian context examined before in [33], and also the SU (3) × U (1) symmetry, to investigate colormagnetic structures in the dense quark matter scenario explored recently in [54]. The presence of non-Abelian symmetries makes the problem much harder, so the search for first order differential equations that solve the equations of motion is of current interest and would be welcome. We hope to report on some of these issues in the near future. Figure 1 . 1The solutions a(r) (descending lines) and g(r) (ascending lines) of Eqs. (3.5) (left), the magnetic field (middle) and the energy density (right) of the visible sector for q, w = 1 and β = 4, 20 and 50. The thickness of the lines increase with β. Figure 2 . 2The magnetic field (left) and the energy density (right) of the solutions of Eqs. (3.5) for the visible sector, with q, w = 1 and β = 20. The darkness of the colors is directly related to the intensity of the quantities. (2.5), |χ| ∈ [0, w]. Thus, Q(|χ|) is non negative where the solution exists. From Eq. (2.15), we get the following first order equations for the hidden sector profiles of the functions in Eqs. (3.8)-(3.10) are very similar to the well known Nielsen-Olesen case, they are not depicted here, although they are found analytically and easy to display. Figure 3 . 3The solutions a(r) (descending lines) and g(r) (ascending lines) of Eqs. (3.12) (left), the magnetic field (middle) and the energy density (right) of the visible sector for q, w = 1 and β = 4, 20 and 50. 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{'abstract': 'We investigate the presence of vortices in generalized Maxwell-Higgs models with a hidden sector. The model engenders U (1) × U (1) symmetry, in a manner that the sectors are coupled via the visible magnetic permeability depending only on the hidden scalar field. We develop a first order framework in which the hidden sector decouples from the visible one. We illustrate the results with two specific examples, that give rise to the presence of vortices with internal structure.', 'arxivid': '1805.07369', 'author': ['D Bazeia bazeia@fisica.ufpb.br \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil\n', 'L Losano losano@fisica.ufpb.br \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil\n', 'M A Marques \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil\n', 'R Menezes rmenezes@dce.ufpb.br \nDepartamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil\n\nDepartamento de Ciências Exatas\nUniversidade Federal da Paraíba\n58297-000Rio TintoPBBrazil\n'], 'authoraffiliation': ['Departamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil', 'Departamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil', 'Departamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil', 'Departamento de Física\nUniversidade Federal da Paraíba\n58051-970João PessoaPBBrazil', 'Departamento de Ciências Exatas\nUniversidade Federal da Paraíba\n58297-000Rio TintoPBBrazil'], 'corpusid': 119301433, 'doi': '10.1155/2019/3187289', 'github_urls': [], 'n_tokens_mistral': 12869, 'n_tokens_neox': 10812, 'n_words': 6390, 'pdfsha': 'd5b83133451e621fed7956577350eb6801f713b2', 'pdfurls': ['https://arxiv.org/pdf/1805.07369v2.pdf'], 'title': ['Prepared for submission to JHEP Vortices in a generalized Maxwell-Higgs model with visible and hidden sectors', 'Prepared for submission to JHEP Vortices in a generalized Maxwell-Higgs model with visible and hidden sectors'], 'venue': []}

arxiv

η-INVARIANT AND MODULAR FORMS 9 Jul 2014 Han Fei Weiping And Zhang η-INVARIANT AND MODULAR FORMS 9 Jul 2014 We show that the Atiyah-Patodi-Singer reduced η-invariant of the twisted Dirac operator on a closed 4m − 1 dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight 2m up to an integral q-series. We prove this result by combining our construction of certain modular characteristic forms associated to a generalized Witten bundle on spin c -manifolds with a deep topological theorem due to Hopkins. Introduction and statement of results Let X be a smooth manifold. Let T C X be the complexification of the tangent bundle T X. One defines the Witten bundle on X ( [15]) as follows, (1.1) Θ q (T X) = ∞ u=1 S q u (T C X − C dimX ) ⊗ ∞ v=1 Λ −q v− 1 2 (T C X − C dimX ), where S t (·) (resp. Λ t (·)) denotes the symmetric (resp. exterior) power and q = e 2π √ −1τ with τ ∈ H, the upper half-plane. Let g T X be a Riemanian metric on T X and ∇ T X the associated Levi-Civita connection. If we write (1.2) Θ q (T X) = B 0 (T X) + B 1 (T X)q 1 2 + B 2 (T X)q + · · · , then each B i (T X) carries a Hermitan metric as well as a Hermitian connection ∇ B i (T X) canonically induced from g T X and ∇ T X . In this way, ∇ T X induces a Hermitian connection ∇ Θq(T X) on the Witten bundle Θ q (T X). Now assume that X is closed, spin and of dimension 4m. Let S(T X) = S + (T X) ⊕ S − (T X) be the corresponding Hermitian bundle of spinors. For each i, let D B i (T X) X,+ : Γ(S + (T X) ⊗ B i (T X)) → Γ(S − (T X) ⊗ B i (T X)) be the corresponding twisted Dirac operator. It is an important and well-known fact (cf. [16]) that the q-series which by the Atiyah-Singer index theorem [2] equals to the elliptic genus 1 X A T X, ∇ T X ch Θ q (T X), ∇ Θq(T X) = ∞ i=0 q i 2 X A T X, ∇ T X ch B i (T X), ∇ B i (T X) ,(1.4) is an integral modular form of weight 2m over Γ 0 (2), where Γ 0 (2) is the index 2 modular subgroup of SL 2 (Z) defined by Γ 0 (2) = a b c d ∈ SL 2 (Z) b ≡ 0 (mod 2) . It is natural to look at what would happen if X is a 4m − 1 dimensional closed spin manifold. In this case, let E be a Hermitian vector bundle over X carrying a Hermitian connection ∇ E . Let D E X : Γ(S(T X) ⊗ E) → Γ(S(T X) ⊗ E) be the associated twisted Dirac operator, which is formally self-adjoint. Following [1], for any Re(s) >> 0, set (1.5) η(D E X , s) = λ∈Spec(D E X )\{0} Sgn(λ) |λ| s . By [1], one knows that η(D E X , s) is a holomorphic function in s with Re(s) > dimX 2 . Moreover, it extends to a meromorphic function over C, which is holomorphic at s = 0. The η invariant of D E X , in the sense of Atiyah-Patodi-Singer [1], is defined by η(D E X ) = η(D E X , 0), while the reduced η invariant is defined and denoted by η(D E X ) = dim(kerD E X ) + η(D E X ) 2 . It is the aim of this paper to study the modularity of the q-series (1.6) η D Θq(T X) X = ∞ i=0 η D B i (T X) X q i 2 , which is a spectral invariant depending on g T X . Assume temporarily that X is the boundary of a 4m dimensional spin manifold Y . Let g T Y be a Riemannian metric on T Y which is of product structure near ∂Y = X and restricts to g T X on X. By the Atiyah-Patodi-Singer index theorem established in [1], one has Y A T Y, ∇ T Y ch Θ q (T Y ), ∇ Θq(T Y ) − η D Θq(T X) X ∈ Z[[q 1 2 ]]. (1.7) 1 We refer to [11,Section 2.1] and [18,Chapter 1] for the notations of the corresponding characteristic forms appearing below. The term of integration over Y in (1.7) is a modular form of weight 2m over Γ 0 (2) (similar to the modularity mentioned above for the elliptic genus in (1.4), c.f. [13]), although it is not necessary to be an integral modular form anymore. Therefore, from (1.7), one sees that if X bounds a spin manifold, then for any Riemannian metric on T X, η(D Θq(T X) X ), up to an integral q-series, is a modular form of weight 2m over Γ 0 (2). Now the natural question is whether this modularity property for the reduced η-invariants holds for any 4m − 1 dimensional closed spin manifold. The main difficulty of this problem lies in the fact that given a 4m − 1 dimensional closed spin manifold, it may happen that it does not bound a spin manifold. Indeed, it is a well-known fact in cobordism theory that there is a positive integer k such that k disjoint copies of X bound a spin manifold Y . In this case, one has the following analogue of (1.7), On the other hand, if g is another Riemannian metric on T X with ∇ T X being its Levi-Civita connection and D Θq(T X) X being the corresponding twisted Dirac operator, then by the variation formula for the reduced η invariant (cf. [1] and [4]), one has Y A T Y , ∇ T Y ch Θ q (T Y ), ∇ Θq(T Y ) − k η D Θq(T X) X ∈ Z[[q(1.9) η D Θq(T X) X −η D Θq(T X) X = X CS Φ (∇ T X , ∇ T X , τ ) mod Z[[q 1 2 ]], where CS Φ (∇ T X , ∇ T X , τ ) is the Chern-Simons transgression form associ- ated to Φ(∇ T X , τ ) = A T X, ∇ T X ch Θ q (T X), ∇ Θq(T X) (4m) . It is easy to see that X CS Φ (∇ T X , ∇ T X , τ ) is a modular form of weight 2m over Γ 0 (2) (cf. [9]). Thus the variation of η D Θq(T X) X has mod Z modularity property. It turns out to be an interesting open problem that whether η D Θq(T X) X is by itself a modular form of weight 2m over Γ 0 (2) up to an element in Z[[q 1/2 ]]. The purpose of this short note is to give an answer to this question. Our main result can be stated as follows. Here meromorphic modular form is a weaker notion than modular form without requiring holomorphicity but only meromorphicty on the upper half plane. To prove Theorem 1.1, instead of using the cobordism result as above, we make use of a result due to Hopkins (cf. [12,Section 8]) which asserts that for any complex vector bundle V over X, there is a nonnegative integer s such that X × CP 1 × · · · × CP 1 (s-copies of CP 1 ) bounds a spin manifold Y and V ⊠H s on X ×CP 1 ×· · ·×CP 1 extends to Y , where H denotes the Hopf hyperplane bundle on CP 1 . We then apply the modular characteristic forms, which is associated to a generalized Witten bundle we have constructed in [11], on the bounding manifold, as well as the Atiyah-Patodi-Singer index theorem [1] to get the modularity of the reduced η-invariant in question. It remains a challenge to find a purely analytic proof of Theorem 1.1 without using the deep topological results as the above mentioned Hopkins' theorem. Theorem 1.1 immediately implies that the quantity in (1.8) is a meromorphic modular form up to an element in kZ[[q 1 2 ]], where k is the positive integer such that k disjoint copies of X bounds Y as explained before (1.8). Observe that in (1.8) each q-coefficient mod k is a mod k index studied by Freed and Melrose in [10]. It is a topological invariant and the main result in [10] provides a topological interpretation of it. Therefore, as an application of Theorem 1.1, we have For completeness, we would like to point out what happens in dimension 4m+1. Actually, when X is an 8n+5 dimensional closed spin manifold, since can be identified with Ochanine's beta invariant β q (X), the modularity of which has been shown in [14]. This paper is organized as follows. In Section 2, we briefly recall our construction (in [11]) of the modular form associated to a generalized Witten bundle involving a complex line bundle. In Section 3, we combine our modular form and the Hopkins boundary theorem to prove Theorem 1.1. In Section 4 we propose a possible refinement of Theorem 1.1 in 8n + 3 dimension. for each i, η D B i (T X) X = 0 and dim kerD B i (T X) X is even (c.f. [1]), we have η D Θq(T X) X = 0 mod Z[[q Complex Line Bundles and Modular Forms In this section, we briefly review our construction (in [11]) of a modular form, which is associated to a generalized Witten bundle involving a complex line bundle. Let M be a 4l dimensional Riemannian manifold. Let ∇ T M be the associated Levi-Civita connection. Let ξ be a complex line bundle over M . Equivalently, one can view ξ as a rank two real oriented vector bundle over M . Let ξ carry a Euclidean metric and also a Euclidean connection ∇ ξ , let c = e(ξ, ∇ ξ ) be the Euler form associated to ∇ ξ (cf. [18,Section 3.4]). Let ξ C be the complexification of ξ. If E is a complex vector bundle over M , set E = E − dim E ∈ K(M ). Following [11, (2.5)], set It is shown in [11] that P (T M, ξ, τ ) can be expressed by using the formal Chern roots of (T C M, ∇ T C M ) and c through the Jacobi theta functions, which are defined as follows (cf. [8] and [11, Section 2.3]): Θ q (T M, ξ) = ∞ u=1 S q u ( T C M ) ⊗ ∞ v=1 Λ −q v− 1 2 ( T C M − 2 ξ C ) ⊗ ∞ r=1 Λ q r− 1 2 ( ξ C ) ⊗ ∞ t=1 Λ q t ( ξ C ),(2.θ(v, τ ) = 2q 1/8 sin(πv) ∞ j=1 (1 − q j )(1 − e 2π √ −1v q j )(1 − e −2π √ −1v q j ) , θ 1 (v, τ ) = 2q 1/8 cos(πv) ∞ j=1 (1 − q j )(1 + e 2π √ −1v q j )(1 + e −2π √ −1v q j ) , θ 2 (v, τ ) = ∞ j=1 (1 − q j )(1 − e 2π √ −1v q j−1/2 )(1 − e −2π √ −1v q j−1/2 ) , θ 3 (v, τ ) = ∞ j=1 (1 − q j )(1 + e 2π √ −1v q j−1/2 )(1 + e −2π √ −1v q j−1/2 ) . The theta functions are all holomorphic functions for (v, τ ) ∈ C × H, where C is the complex plane and H is the upper half plane. Let {±2π √ −1x i } be the formal Chern roots for (T C M, ∇ T C M ) and c = 2π √ −1u, we have (2.3) P (T M, ξ, τ ) = 2l i=1 x i θ ′ (0, τ ) θ(x i , τ ) θ 2 (x i , τ ) θ 2 (0, τ ) θ 1 (u, τ ) θ 1 (0, τ ) θ 2 2 (0, τ ) θ 2 2 (u, τ ) θ 3 (u, τ ) θ 3 (0, τ ) (4l) . By using the transformation laws of theta functions (cf. [8] and [11, Section 2.3]), one sees as in [11, Proposition 2.6] that P (T M, ξ, τ ) is a modular form of weight 2l over Γ 0 (2). Proof of the Main Theorem In this section, we will prove our main result Theorem 1.1. The topological tool we will use is the following boundary theorem of Hopkins (cf. [12,Section 8]). Hopkins). Let X be a compact, odd dimensional spin manifold and V → X a complex vector bundle over X. Then there is an integer s such that the vector bundle V ⊠ (⊠ s j=1 H j ) → X × (CP 1 ) s is a boundary, where H j denotes the Hopf hyperplane bundle on the j-th copy of CP 1 . In other words, there is a spin manifold Y with a complex vector bundle W on Y such that W | ∂Y = V ⊠ (⊠ s j=1 H j ), In what follows, we will combine this Hopkins boundary theorem with the modular characteristic form constructed in Section 2 to give a proof of Theorem 1.1. Proof of Theorem 1.1: Without loss of generality, for the 4m−1 dimensional closed spin manifold X, in view of the Hopkins boundary theorem, we take an even integer s so that the complex line bundle p * (⊠ s j=1 H j ) → X × (CP 1 ) s bounds, where p : X × (CP 1 ) s → (CP 1 ) s is the natural projection. This means that there is a spin manifold Y and a complex line bundle ζ over Y such that ∂Y = X × (CP 1 ) s and ζ| X×(CP 1 ) s = p * (⊠ s j=1 H j ). Let g T X be any Riemmnian metric on X. Equip CP 1 's with arbitrary Riemannian metrics and the H j 's with arbitrary Euclidean metrics and Euclidean connections. Let g T Y be a metric on T Y such that it is of product structure near X × (CP 1 ) s and restricts to the product metric on X × (CP 1 ) s . Let ∇ T Y be the Levi-Civita connection associated to g T Y . Let g ζ be an Euclidean metric on ζ (viewed as an oriented real plane bundle) such that g ζ is of product structure near X × (CP 1 ) s and restricts to the Euclidean metric on p * (⊠ s j=1 H j ) on X × (CP 1 ) s . Let ∇ ζ be an Euclidean connection of g ζ which is of product structure near X × (CP 1 ) s and restricts to the canonically induced Euclidean connection on p * (⊠ s j=1 H j ) on X × (CP 1 ) s . Let c = e(ζ) and z j = c 1 (H j ) π √ −1 , 1 ≤ j ≤ s. By applying the Atiyah-Patodi-Singer index theorem [1] to the twisted Dirac operator D Θq(T Y,ζ 2 )⊗ζ Y , in noting that Θ q (T Y, ζ 2 ) ⊗ ζ X×(CP 1 ) s = Θ q T X × (CP 1 ) s , p * (⊠ s j=1 H j ) 2 ⊗p * ⊠ s j=1 H j , one finds that there exist integers a i 's such that η D Θq(T (X×(CP 1 ) s ),(p * (⊠ s j=1 H j )) 2 )⊗p * (⊠ s j=1 H j ) X×(CP 1 ) s = Y A T Y, ∇ T Y ch Θ q (T Y, ζ 2 ) ⊗ ζ, ∇ Θq(T Y,ζ 2 )⊗ζ − ∞ i=0 a i q i 2 = Y A T Y, ∇ T Y e c ch Θ q (T Y, ζ 2 ), ∇ Θq(T Y,ζ 2 ) − ∞ i=0 a i q i 2 = Y A T Y, ∇ T Y cosh(c) ch Θ q (T Y, ζ 2 ), ∇ Θq(T Y,ζ 2 ) − ∞ i=0 a i q i 2 , (3.1) where the last equality follows from the fact that s is an even integer. Let r : X × (CP 1 ) s → X be the natural projection. For bundles E → X and F → (CP 1 ) s , by separation of variables, we have η D (r * E)⊗(p * F ) X×(CP 1 ) s = η(D E X ) · Ind(D F (CP 1 ) s ,+ ). So we have η D (r * E)⊗(p * F ) X×(CP 1 ) s = η(D E X )·Ind(D F (CP 1 ) s ,+ )+dim(kerD E X )dim(ker(D F (CP 1 ) s ,− )). From the above formula, we can see that there are integers b i 's such that η D Θq(T (X×(CP 1 ) s ),(p * (⊠ s j=1 H j )) 2 )⊗p * (⊠ s j=1 H j ) X×(CP 1 ) s − ∞ i=0 b i q i 2 =η D Θq(r * T X⊕p * T (CP 1 ) s ,(p * (⊠ s j=1 H j )) 2 )⊗p * (⊠ s j=1 H j ) X×(CP 1 ) s − ∞ i=0 b i q i 2 =η D r * Θq(T X)⊗p * (Θq(T (CP 1 ) s ,(⊠ s j=1 H j ) 2 )⊗⊠ s j=1 H j ) X×(CP 1 ) s − ∞ i=0 b i q i 2 =η D Θq(T X) X · Ind D Θq(T (CP 1 ) s ,(⊠ s j=1 H j ) 2 )⊗⊠ s j=1 H j (CP 1 ) s ,+ =η D Θq(T X) X · (CP 1 ) s A(T (CP 1 ) s , ∇ T (CP 1 ) s )e c 1 (H 1 )+···+c 1 (Hs) ch Θ q (T (CP 1 ) s , (⊠ s j=1 H j ) 2 ) =η D Θq(T X) X · (CP 1 ) s   s j=1 z j θ ′ (0, τ ) θ(z j , τ ) θ 2 (z j , τ ) θ 2 (0, τ )   θ 1 ( s j=1 z j , τ ) θ 1 (0, τ ) θ 2 2 (0, τ ) θ 2 2 ( s j=1 z j , τ ) θ 3 ( s j=1 z j , τ ) θ 3 (0, τ ) =η D Θq(T X) X · (CP 1 ) s θ 1 ( s j=1 z j , τ ) θ 1 (0, τ ) θ 2 2 (0, τ ) θ 2 2 ( s j=1 z j , τ ) θ 3 ( s j=1 z j , τ ) θ 3 (0, τ ) ,(3.2) where the last equality holds due to the fact that x θ(x,τ ) and θ 2 (x, τ ) are both even functions about x and CP 1 z n j = 0 if n > 1. Since s is an even integer, from the knowledge about the modular form P (T M, ξ, τ ) constructed in Section 2, we know that f s (τ ) := (CP 1 ) s θ 1 ( s j=1 z j , τ ) θ 1 (0, τ ) θ 2 2 (0, τ ) θ 2 2 ( s j=1 z j , τ ) θ 3 ( s j=1 z j , τ ) θ 3 (0, τ ) is an integral modular form of weight s over Γ 0 (2). Moreover, since (CP 1 ) s A T (CP 1 ) s , ∇ T (CP 1 ) s e c 1 (H 1 )+···+c 1 (Hs) = 1, we see that f s (τ ) has constant term 1. Therefore f −1 s (τ ) ∈ Z[[q 1 2 ]]. From (3.1) and (3.2), we have η D Θq(T X) X =f −1 s (τ ) · Y A T Y, ∇ T Y cosh(c) ch Θ q (T Y, ζ 2 ), ∇ Θq(T Y,ζ 2 ) − f −1 s (τ ) · ∞ i=0 (a i + b i ) q i 2 . (3.3) Still by the modularity of P (T M, ξ, τ ) constructed in Section 2, we know that Y A T Y, ∇ T Y cosh(c) ch Θ q (T Y, ζ 2 ), ∇ Θq(T Y,ζ 2 ) is a modular form of weight 2m + s over Γ 0 (2). So f −1 s (τ ) · Y A T Y, ∇ T Y cosh(c) ch Θ q (T Y, ζ 2 ), ∇ Θq(T Y,ζ 2 ) is a meromorphic modular form of weight 2m over Γ 0 (2). Therefore from (3.3), we see that η(D Θq(T X) X ) = f −1 s (τ )· Y A T Y, ∇ T Y cosh(c) ch Θ q (T Y, ζ 2 ), ∇ Θq(T Y,ζ 2 ) mod Z[[q 1 2 ]], a meromorphic modular form of weight 2m over Γ 0 (2). The proof of Theorem 1.1 is complete. Q.E.D. In the proof of Theorem 1.1, we may take any even number s ∈ H(X) and denote the corresponding Y and ζ by Y s and ζ s . Then the proof of Theorem 1.1 tells us that, up to an element in Z[[q 1 2 ]], η(D Θq(T X) X ) = f −1 s (τ )· Ys A T Y s , ∇ T Ys cosh(e(ξ s )) ch Θ q (T Y s , ζ 2 s ), ∇ Θq(T Ys,ζ 2 s ) . Clearly, if h(X) = 0, one gets (1.7). Therefore, for every even number s ≥ 2[ h(X)+1 2 ] , one can construct a meromorphic modular form of weight 2m over Γ 0 (2) of above form, that is equal to η(D Θq(T X) X ) up to an element in Z[[q 1 2 ]]. The poles of these meromorphic modular forms are just the zeros of the modular forms f s (τ ). We hope that further study of the modular forms f s (τ ) will bring better understanding of modularity of η(D Θq(T X) X ). Remark 3.3. We refer to [6] for an alternative approach to the modularity of η(D Θq(T X) X ), which is shown to be not only a meromorphic modular form but also a modular form using the theory of universal η-invariant. The cases of dimension 8n + 3 In this section, we discuss the case of dimension 8n+3. In this dimension, it is known that η(D Θq(T X) X ) is mod 2Z[[q 1 2 ]] smooth. That is, in the right hand side of (1.9), the term mod Z[[q 1 2 ]] can be replaced by mod 2Z[[q 1 2 ]]. Therefore it is natural to propose the following conjecture whose statement refines Theorem 1.1 in this case. Recall that a mod 2k refinement of the Freed-Melrose mod k index for real vector bundles over 8n + 4 dimensional manifolds has been defined in [17,Section 3]. In view of this, one can propose a refinement of Corollary 1.1, in the case of dim Y = 8n + 4, as follows. Conjecture 4.2. Let Y be an 8n + 4 dimensional spin Z/k-manifold in the sense of Sullivan (cf. [10]). Then the mod 2k index associated to the Witten bundle Θ q (T Y ) can be represented by a meromorphic modular form of weight 4n + 2 over Γ 0 (2). By the method of this paper, in order to prove Conjectures 4.1 and 4.2, one perhaps needs a kind of Hopkins boundary theorem for real vector bundles. Or, one may try to develop a direct analytic approach, which, even for Theorem 1.1, is a challenging problem as we indicated in Section 1. Theorem 1 . 1 . 11Let X be a 4m − 1 dimensional closed spin Riemannian manifold. Then the reduced η-invariant η D Θq(T X) X of the twisted Dirac operator D Θq(T X) X is a meromorphic modular form of weight 2m over Γ 0 (2), up to an element in Z[[q 1 2 ]]. Corollary 1 . 1 . 11Let Y be an 4m dimensional spin Z/k-manifold in the sense of Sullivan (cf.[10]). Then the mod k index associated to the Witten bundle Θ q (T Y ) can be represented by a meromorphic modular form of weight 2m over Γ 0 (2).On the other hand, in view of[7, (25)], which corresponds to the case of k = 1 in (1.8) for the category of stable almost complex manifolds, Theorem 1.1 might become a starting point of a kind of tertiary index theory, in the sense of[7, Theorem 4.2], for spin manifolds. Recently Ulrich Bunke informed us that Theorem 1.1 can be given an alternative proof by using the theory of the universal η invariant ([6], Lemma 3.1) and a spin version of the f -invariant has also been constructed in ([6], Definition 13.2). of the Atiyah-Singer mod 2 index theorem, η D Θq(T X) X ]. As before, ∇ T M and ∇ ξ induce a Hermitian connection ∇ Θq(T M,ξ) on Θ q (T M, ξ).Let P (T M, ξ, τ ) ∈ Ω 4l (M ) be the characteristic form defined by (2.2)P (T M, ξ, τ ) := A(T M, ∇ T M ) cosh c 2 ch Θ q (T M, ξ), ∇ Θq(T M,ξ) (4l) . Conjecture 4 . 1 . 41Let X be an 8n + 3 dimensional closed spin Riemannian manifold. Then the reduced η-invariant η(D Θq(T X) X ) of the twisted Dirac operator D Θq(T X) X is a meromorphic modular form of weight 4n + 2 over Γ 0 (2), up to an element in 2Z[ Acknowledgement This work was started when both of us were participating an international conference at UC Santa Barbara organized by Xianzhe Dai in July, 2012. We would like to thank Xianzhe Dai for the invitation as well as the kind hospitality. We would also like to thank Siye Wu for bringing our attention to[12]. We are grateful to Ulrich Bunke for the discussion on the topic.The work of F. H. was partially supported by a start-up grant and AcRF R-146-000-163-112 from National University of Singapore. The work of W. Z. was partially supported by NNSFC and MOEC.Remark 3.1. The modular form f s (τ ) in the above proof can be explicitly expressed by theta functions and their derivatives. For example, we haveand Spectral asymmetry and Riemannian geometry I. M F Atiyah, V K Patodi, I M Singer, Proc. Camb. Philos. Soc. 77M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry I. Proc. Camb. Philos. Soc. 77 (1975), 43-69. The index of elliptic operators. M F Atiyah, I M Singer, III, Ann. Math. 87M. F. Atiyah and I.M. Singer, The index of elliptic operators, III, Ann. Math. 87 (1968), 546-604. The index of elliptic operators. M F Atiyah, I M Singer, Ann. Math. 93M. F. Atiyah and I.M. Singer, The index of elliptic operators, V, Ann. Math. 93 (1971), 139-149. The analysis of elliptic families. J.-M Bismut, D S Freed, Comm. Math. Phys. IIJ.-M Bismut and D.S. Freed, The analysis of elliptic families, II, Comm. Math. Phys. 107, 1986, 103-163. U Bunke, arXiv:1103.4217On the topological contents of eta invariants. U. Bunke, On the topological contents of eta invariants, arXiv: 1103.4217. U Bunke, arXiv:1403.2030The universal eta-invariant for manifolds with boundary. U. Bunke, The universal eta-invariant for manifolds with boundary, arXiv: 1403.2030. The f -invariant and index theory. U Bunke, N Naumann, Manuscripta Math. 132U. Bunke and N. Naumann, The f -invariant and index theory. Manuscripta Math. 132 (2010), 365-397. Elliptic Functions. K Chandrasekharan, Springer-VerlagK. Chandrasekharan, Elliptic Functions. Springer-Verlag, 1985. Elliptic genera, transgression and loop space Chern-Simons forms. Q Chen, F Han, Comm. Anal. Geom. 17Q. Chen and F. Han, Elliptic genera, transgression and loop space Chern-Simons forms, Comm. Anal. Geom. 17 (2008), 73-106. A mod k index theorem. D S Freed, R B Melrose, Invent. Math. 107D. S. Freed and R. B. Melrose, A mod k index theorem. Invent. Math., 107 (1992), 283-299. Modular invariance, characteristic numbers and η invariants. F Han, W Zhang, J. Diff. Geom. 67F. Han and W. Zhang, Modular invariance, characteristic numbers and η invariants. J. Diff. Geom. 67 (2004), 257-288. K R Klonoff, An Index Theorem in Differential K-Theory. Univ. Texas at AustinPh. D. ThesisK. R. Klonoff, An Index Theorem in Differential K-Theory. Ph. D. Thesis, Univ. Texas at Austin, 2008. Download address: http://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf Modular invariance and characteristic numbers. K Liu, Commun. Math. Phys. 174K. Liu, Modular invariance and characteristic numbers. Commun. Math. Phys. 174 (1995), 29-42. Elliptic genera, modular forms over KO * and the Brown-Kervaire invariant. S Ochanine, Math. Z. 206S. Ochanine, Elliptic genera, modular forms over KO * and the Brown-Kervaire in- variant. Math. Z. 206 (1991), 277-291 The index of the Dirac operator in loop space. E Witten, Elliptic Curves and Modular Forms in Algebraic Topology (Proceedings. P.S. LandweberPrincetonSpringerE. Witten, The index of the Dirac operator in loop space, in P.S. Landweber, ed., Elliptic Curves and Modular Forms in Algebraic Topology (Proceedings, Princeton 1986), Lecture Notes in Math., 1326, pp. 161-181, Springer, 1988. D Zagier, Elliptic Curves and Modular Forms in Algebraic Topology (Proceedings. P.S. LandweberPrincetonSpringerNote on the Landweber-Stong elliptic genusD. Zagier, Note on the Landweber-Stong elliptic genus, in P.S. Landweber, ed., Ellip- tic Curves and Modular Forms in Algebraic Topology (Proceedings, Princeton 1986), Lecture Notes in Math., 1326, pp. 216-224, Springer, 1988. On the mod k index theorem of Freed and Melrose. W Zhang, J. Diff. Geom. 43W. Zhang, On the mod k index theorem of Freed and Melrose. J. Diff. Geom. 43 (1996), 198-206. W Zhang, Lectures on Chern-Weil Theory and Witten Deformations. SingaporeWorld Scientific4W. Zhang, Lectures on Chern-Weil Theory and Witten Deformations , Nankai Tracts in Mathematics Vol. 4, World Scientific, Singapore, 2001. . F Han, Singapore, Block S17, 10 Lower Kent Ridge Road; SingaporeDepartment of Mathematics, National University ofmathanf@nus.edu.sgF. Han, Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076 (mathanf@nus.edu.sg) . W Zhang, P.R. China. 300071Chern Institute of Mathematics & LPMC, Nankai Universityweiping@nankai.edu.cnW. Zhang, Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China. (weiping@nankai.edu.cn)

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{'abstract': 'We show that the Atiyah-Patodi-Singer reduced η-invariant of the twisted Dirac operator on a closed 4m − 1 dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight 2m up to an integral q-series. We prove this result by combining our construction of certain modular characteristic forms associated to a generalized Witten bundle on spin c -manifolds with a deep topological theorem due to Hopkins.', 'arxivid': '1312.7494', 'author': ['Han Fei ', 'Weiping And ', 'Zhang '], 'authoraffiliation': [], 'corpusid': 118215899, 'doi': '10.1093/qmath/hau022', 'github_urls': [], 'n_tokens_mistral': 9457, 'n_tokens_neox': 8222, 'n_words': 4850, 'pdfsha': 'e1a58c56ffc643c9270b922fdb86a64e776cd77e', 'pdfurls': ['https://arxiv.org/pdf/1312.7494v2.pdf'], 'title': ['η-INVARIANT AND MODULAR FORMS', 'η-INVARIANT AND MODULAR FORMS'], 'venue': []}

arxiv

SEMI-SUPERVISED MULTI-DOMAIN MULTI-TASK TRAINING FOR METASTATIC COLON LYMPH NODE DIAGNOSIS FROM ABDOMINAL CT Saskia Glaser Gabriel Maicas Sergei Bedrikovetski Tarik Sammour Department of Surgery Colorectal Unit Royal Adelaide Hospital Australia Gustavo Carneiro Australian Institute for Machine Learning Faculty of Health and Medical Science School of Medicine The University of Adelaide Australia The University of Adelaide Australia SEMI-SUPERVISED MULTI-DOMAIN MULTI-TASK TRAINING FOR METASTATIC COLON LYMPH NODE DIAGNOSIS FROM ABDOMINAL CT Index Terms-semi-supervisedmulti-taskmultiple do- mainROI annotationsweak annotationscolon cancerlymph node diagnosisabdominal CT The diagnosis of the presence of metastatic lymph nodes from abdominal computed tomography (CT) scans is an essential task performed by radiologists to guide radiation and chemotherapy treatment. State-of-the-art deep learning classifiers trained for this task usually rely on a training set containing CT volumes and their respective image-level (i.e., global) annotation. However, the lack of annotations for the localisation of the regions of interest (ROIs) containing lymph nodes can limit classification accuracy due to the small size of the relevant ROIs in this problem. The use of lymph node ROIs together with global annotations in a multi-task training process has the potential to improve classification accuracy, but the high cost involved in obtaining the ROI annotation for the same samples that have global annotations is a roadblock for this alternative. We address this limitation by introducing a new training strategy from two data sets: one containing the global annotations, and another (publicly available) containing only the lymph node ROI localisation. We term our new strategy semisupervised multi-domain multi-task training, where the goal is to improve the diagnosis accuracy on the globally annotated data set by incorporating the ROI annotations from a different domain. Using a private data set containing global annotations and a public data set containing lymph node ROI localisation, we show that our proposed training mechanism improves the area under the ROC curve for the classification task compared to several training method baselines. INTRODUCTION Colon cancer is one of the most frequently diagnosed cancers in the world, with nearly 1 million new cases and 551,269 deaths in 2018 [1]. Patients with cancer have a high risk for nodal metastasis, therefore oncologically adequate surgery consists of segmental colectomy with lymph node * Supported by Australian Research Council through grant DP180103232 and by SA Health eHealth Innovation Grants Program (eIGP) dissection followed by adjuvant treatment [2]. However approximately 20% to 30% of patients develop another cancer months later, thus alternative treatment strategies including neoadjuvant therapy prior to surgery are currently being investigated [3]. Under these circumstances, accurate preoperative metastatic lymph node diagnosis is crucial in determining the eligibility of neoadjuvant treatment and to avoid the over treatment of patients. However, diagnosing lymph nodes is a challenging task that is prone to interobserver variability [4]. As a result, computer-aided diagnosis systems are being designed to assist radiologists in staging lymph node metastasis on cross-sectional imaging. State-of-the-art deep learning classification models [5] could in principle produce relatively accurate diagnosis of metastatic lymph nodes. However, these models are trained with weakly annotated data sets (i.e. image-level, or global, labels), and fail to produce precise diagnosis in situations where the regions of interest (ROIs) that explain such diagnosis occupy a small portion of the input image, which is the case for metastatic lymph node diagnosis from abdominal CT scans. The classification accuracy of models which perform diagnosis that depends on relatively small ROIs can be improved by incorporating localization information to the training process [6], [7], [8]. The main goal is that the model learns to focus on the relevant ROIs to perform classification. For these models, the training process becomes a multi-task training, where the aims are to classify the entire image and to localize the ROIs in image. However, such training process is based on a strongly annotated training set that contains local ROI and global diagnosis annotations, which are costly and time consuming to obtain. Aiming to reduce the burden of obtaining such ROI and global annotations for the single-domain multi-task training mentioned above, we hypothesize that global classification accuracy can be improved if we use separate data sets (i.e., multi-domain), each one containing one particular type of annotation (i.e., multi-task). This setup introduces the new challenge of having a multi-domain multi-target training mechanism, where each domain (i.e., data set) is annotated for a different task -this motivates us to propose a new semi-supervised learning approach that transfers the multi-task labels between multiple domains. In this paper, we introduce a new semi-supervised training strategy to train multi-domain multi-task models where different domains have different types of annotations. Our strategy consists of jointly training a multi-task model with semi-supervised learning on the unlabelled task of the other domains. More precisely, we design a training strategy that improves image classification in one domain by including a ROI localization task from a different domain. We evaluate our training method on the problem of metastatic colon lymph node diagnosis from abdominal CT, using a private data set that only contains global labels (RAH data set) and a public data set that only has lymph node ROI annotations (CI data set) [9]. Results on the RAH data set show that our semi-supervised multi-domain multi-task training strategy outperforms other training strategies such as semi-supervised or multi-task learning without semi-supervision. RELATED WORK Diagnosing metastatic lymph nodes from abdominal CT scans has traditionally been tackled with the design of hand-crafted features [10]. The main disadvantage of these methods lies in the sub-optimality of the features as there is no guarantee that these features are optimal to perform diagnosis, yielding relatively low accuracy scores. Feature sub-optimality has been addressed by the computer vision community by using deep learning classifiers that learn optimal features while being trained to perform diagnosis [5]. However, the performance of such classifiers decreases in problems where the regions of interest in the image to be classified are relatively small. The medical image analysis community focused on including the localization of regions of interest in the training process [6], [7], [8] to increase classification performance, in a single-domain multi-task approach. The training process in such cases does not only optimise image classification, but also ROI localisation. Although successful, this increase in performance comes at the expense of strongly annotating training sets with ROIs and classification labels, a costly process for many medical image analysis problems. Multi-task learning from multi-domain data [11] was proposed to learn from data sets that have been labelled for different but related tasks. The main aim is to improve the performance during inference on each of the labelled tasks in the data sets. However, these approaches require the multitask annotations in all data sets to generalize well in each of the tasks across data sets, which is costly to be acquired. Moreover, multi-domain multi-task methods do not use all the information available in the data sets as they do not exploit, during training, un-labelled data for a given task. Semi-supervised learning has been proposed to incorporate un-annotated data for a given task into the learning process. Initially proposed for incorporating unlabelled data from the same task and domain [12], [13], it has recently been successfully explored in a multi-task setup [14], [15]. Contrary to previous semi-supervised multitask approaches [14], [15], we propose a training method that not only deals with tasks that have been labelled in the same domain, but also where each task is labelled in a different domain. METHODS Data Sets Let D (1) = v (1) i , y (1) i i∈{1,...,|D (1) |} be a weakly annotated data set, where v (1) i : Ω → R represents the abdominal CT scan of the i th patient in the data set D (1) and Ω ∈ R 3 is the volume lattice, and y : Ω → R represents the abdominal CT scan of the j th patient in the data set D (2) , and s (2) j : Ω → {0, 1} is the voxel-wise ROI annotation of the presence of a lymph node in the CT scan of patient j. Assume that each data set comes from a different domain, implying that the data set distributions (v, y) ∼ P (1) (2) ) are different. Note that each data set is labelled for a different task: D (1) for image classification and D (2) for lymph node ROI detection. We split each data set into training, validation and testing sets in a patient-wise manner, forming the sets T (v, y) (for (v, y) ∈ D (1) ) and (v, s) ∼ P (2) (v, s) (for (v, s) ∈ D test and T (2) test for data sets D (1) and D (2) . Semi-supervised Multi-domain Multi-task training Our model is composed of three modules: an encoder f θe , a classification branch σ θc and a detection branch g θ d with parameters θ e , θ c , θ d respectively. The model receives as input a CT scan v and forms a feature embedding o = f θe (v) using the encoder. The embedding o is used as input by the classification and detection branches. The classification branch returnsỹ = σ θc (o) representing the binary classification of the input scan v. The detection branch produces an ROI maps = g θ d (o) that estimates the probability that each voxel represents a lymph node. A binary ROI masks ζ is generated by thresholdings(ω) > ζ, for ω ∈ Ω. See Fig. 1 for a summary of the architecture of our method. The training consists of two alternating stages: 1) label propagation between the two data sets sets [16], and 2) multidomain multi-task training using the real and propagated labels. For label propagation between the two data sets [16], we replace the original D (1) by D For the multi-domain multi-task stage, we jointly minimise the classification loss with θ * c , θ * e = arg max θc,θe E P (1) (v,y) [ C (σ θc (f θe (v)), y)],(1) where C denotes the binary cross entropy loss, and the ROI detection loss with θ * d , θ * e = arg max θ d ,θe E P (2) (v,s) [ D (g θ d (f θe (v)), s)}],(2) where D denotes a loss that sums the voxel-wise cross entropy loss and the Dice loss [17]. We only use the subsets T (1) train , T (2) train ,T (1) val , T(2) val from D (1) and D (2) during the training process. The inference consists of feeding the model with an input scan v that is encoded to obtain the feature embedding and forwarded through the classification branch σ θ * c (f θ * e (v)) to yield the probability of metastatic lymph nodes. EXPERIMENTS AND RESULTS We assess our proposed method on the problem of diagnosing the presence of metastatic lymph nodes from abdominal CT scans. For the weakly labelled data set D (1) we use a private data set from the Royal Adelaide Hospital (RAH data set) that contains 123 scans from 123 patients. Weak labels indicating the presence of any metastatic lymph node are obtained from pathology reports and clinical notes. There are 57 scans labelled with the presence of metastatic lymph nodes and 66 scans labelled with the absence of metastatic lymph nodes. Note that no lymph node localization is provided in this data set. The data set D (2) that provides lymph node localization information is publicly available [9] (CI data set). There are 595 ROI localisation annotations of lymph nodes from 85 scans of 85 patients. Note that no scan-level label about metastatic lymph node is provided with this data set. The RAH data set is randomly divided into training (90 patients, 43 of them with metastatic lymph nodes and 47 without), validation (10 patients, 4 with metastatic lymph nodes, 6 without), and testing (23 patients, 10 with metastatic lymph nodes, 13 without). The CI data set is randomly divided into training (62 patients with 439 annotated lymph nodes), validation (7 patients with 54 annotated lymph nodes), and testing (16 patients with 102 lymph nodes). We train our method with the training set from RAH and the entire CI data set. We utilize the validation set of the RAH data set to perform model selection and we report classification results on the testing set of the RAH data set. We pre-process each image in both data sets by subtracting the mean and dividing by the standard deviation of the training set of the corresponding data set. The encoder f θe (.) is represented by a 3-D Densenet [5] consisting of 5 dense blocks. The input volume v size is 512×512×256 and the feature vector embedding o = f θe (v) is of size 16 × 16 × 8. The classifier σ θc (.) is composed of two fully connected layers and outputs the probability of presence of metastatic lymph nodes in v. The decoder g θ d (.) is also composed of 5 dense blocks, and outputs an ROI map s of the same size as the input scan and is thresholded at ζ = 0.8 to obtains ζ . The training process runs for 500 epochs and optimizes the parameters θ e , θ c , θ d with Adam optimizer (learning rate of 0.05). We evaluate our proposed semi-supervised multi-domain multi-task training method with the metastic lymph node classification performance, measured with the area under the ROC curve (AUC). We compare our proposed training strategy against other training procedures: 1) supervised baseline: a DenseNet [5] classifier trained on the RAH data set and composed of the encoder and classification branch; 2) semi-supervised: the same DenseNet classifier from (1), trained with the RAH training set and including the CI data set trained with propagated classification labels; and 3) supervised multi-domain multi-task: the proposed architecture trained with the RAH and CI data sets, where the encoder and classifier are trained with the RAH data set and the encoder and detector branches are trained with CI data set. We present quantitative classification results in terms of AUC on Table I. DISCUSSION AND CONCLUSION The experimental results presented in Table I show that our proposed training method outperforms several baseline training strategies for metastatic lymph node diagnosis from abdominal CT scans. As explained in Sec. 1, the baseline Table I: Classification AUC on the RAH Test Set obtained by the classifier trained with our proposed training strategy and baseline methods. classifier achieves the lowest AUC score, which can be in part due to the lack of lymph node localization. Interestingly, propagating the classification labels from RAH to CI data set (without including any localization in the training process) yielded better results for classification than the supervised multi-domain multi-task baseline. We believe this is due to difficulty of integrating multiple tasks from different domains into the training process without any semi-supervision. Finally, our proposed semi-supervised multi-domain multitask training outperformed all baseline methods. As we hypothesized in Sec. 1, jointly semi-supervising each of the labelled tasks from a different domain results in a more accurate model, probably due to the extra-supervision that can facilitate the addition of data from a different domain. In conclusion, we proposed a new semi-supervised multidomain multi-task training, where we semi-supervised the detection task of the data set containing global annotations and the classification task on the data set containing ROI annotations. Results on diagnosing the presence of metastatic lymph nodes from CT scans showed that our method successfully incorporates lymph node localization information from a different domain to improve classification results in the original domain. We leave for future work the evaluation of the lymph node ROI localization task in each data set. 1} is the scan-level label indicating the presence (y ( 1 ) 1new = D (1) D (1) , with D (1) = {(v,ỹ)|v ∈ D(2) ,ỹ = σ θc (f θe (v))} and the original D (2) by D(2) new = D (2) D (2) , with D (2) = {(v,s)|v ∈ D (1) ,s = g θ d (f θe (v))}. Fig. 1 : 1Architecture of our method. The model receives a scan v as input and the encoder builds its feature representation o = f θe (v), which is used by the classification branch σ θc (o) and the ROI detection branch g θ d (o). The ROI binary map is obtained by thresholding the output from the ROI detection branch. Fig. 2 : 2Visual results produced by our proposed method on the RAH data set. Image 2a shows the positive classification of an scan containing a metastatic lymph node (marked in red). Image 2b contains the negative classification of an image with non-metastatic lymph nodes. Training Method for the classifier Data Sets AUC Supervised Multi-domain Multi-Task RAH + CI 0.82 Semi-Supervised Multi-domain Multi-Task RAH + CI 0.86Supervised Baseline RAH 0.81 Semi-Supervised RAH + CI 0.84 Global cancer statistics 2018: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries. Freddie Bray, Jacques Ferlay, CA: a cancer journal for clinicians. Freddie Bray, Jacques Ferlay, and et al., "Global cancer statistics 2018: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries," CA: a cancer journal for clinicians, 2018. How reliable is ct scan in staging right colon cancer?. M Laura, Albert Fernandez, Parlade, Diseases of the Colon & Rectum. Laura M Fernandez, Albert Parlade, and et al., "How reliable is ct scan in staging right colon cancer?," Diseases of the Colon & Rectum, 2019. Potential image-based criteria of neoadjuvant chemotherapy for colon cancer: multireaders diagnostic performance. Hyungsik Park, Hyun Seung, Cho, Abdominal Radiology. Hyungsik Park, Seung Hyun Cho, and et al., "Potential image-based criteria of neoadjuvant chemotherapy for colon cancer: multireaders diagnostic performance," Abdominal Radiology, 2019. Accuracy of computed tomography in nodal staging of colon cancer patients. H Audrey, Rebecca A Choi, Nelson, WJGS. Audrey H Choi, Rebecca A Nelson, and et al., "Accu- racy of computed tomography in nodal staging of colon cancer patients," WJGS, 2015. Densely connected convolutional networks. Gao Huang, Zhuang Liu, CVPR. Gao Huang, Zhuang Liu, and et al., "Densely con- nected convolutional networks," in CVPR, 2017. Identification of metastatic lymph nodes in mr imaging with faster region-based convolutional neural networks. Yun Lu, Qiyue Yu, Cancer research. Yun Lu, Qiyue Yu, and et al., "Identification of metastatic lymph nodes in mr imaging with faster region-based convolutional neural networks," Cancer research, 2018. Thoracic disease identification and localization with limited supervision. Zhe Li, Chong Wang, CVPR. Zhe Li, Chong Wang, and et al., "Thoracic disease identification and localization with limited supervi- sion," in CVPR, 2018. End-toend diagnosis and segmentation learning from cardiac magnetic resonance imaging. Gerard Snaauw, Dong Gong, ISBI. Gerard Snaauw, Dong Gong, and et al., "End-to- end diagnosis and segmentation learning from cardiac magnetic resonance imaging," in ISBI, 2019. A new 2.5 d representation for lymph node detection using random sets of deep convolutional neural network observations. R Holger, Le Roth, Lu, MICCAIHolger R Roth, Le Lu, and et al., "A new 2.5 d repre- sentation for lymph node detection using random sets of deep convolutional neural network observations," in MICCAI, 2014. Development and validation of a radiomics nomogram for preoperative prediction of lymph node metastasis in colorectal cancer. Y Q Huang, C H Liang, Journal of clinical oncology. YQ Huang, CH Liang, and et al., "Development and validation of a radiomics nomogram for preoperative prediction of lymph node metastasis in colorectal can- cer.," Journal of clinical oncology, 2016. Multitask, multi-domain learning: application to semantic segmentation and pose regression. Damien Fourure, Rémi Emonet, Neurocomputing. Damien Fourure, Rémi Emonet, and et al., "Multi- task, multi-domain learning: application to semantic segmentation and pose regression," Neurocomputing, 2017. Signet ring cell detection with a semi-supervised learning framework. Jiahui Li, Shuang Yang, IPMI. Jiahui Li, Shuang Yang, and et al., "Signet ring cell detection with a semi-supervised learning framework," in IPMI, 2019. A semi-supervised cnn learning method with pseudo-class labels for atherosclerotic vascular calcification detection. Jiamin Liu, Jianhua Yao, ISBI. Jiamin Liu, Jianhua Yao, and et al., "A semi-supervised cnn learning method with pseudo-class labels for atherosclerotic vascular calcification detection," in ISBI, 2019. Semi-supervised multi-task learning with chest x-ray images. Abdullah-Al-Zubaer Imran, Demetri Terzopoulos, MICCAIw -MLMIAbdullah-Al-Zubaer Imran and Demetri Terzopoulos, "Semi-supervised multi-task learning with chest x-ray images," in MICCAIw -MLMI, 2019. Multi-task attention-based semi-supervised learning for medical image segmentation. Shuai Chen, Gerda Bortsova, MICCAIShuai Chen, Gerda Bortsova, and et al., "Multi-task attention-based semi-supervised learning for medical image segmentation," in MICCAI, 2019. Learning from labeled and unlabeled data with label propagation. Xiaojin Zhu, Zoubin Ghahramani, Xiaojin Zhu and Zoubin Ghahramani, "Learning from labeled and unlabeled data with label propagation," . V-net: Fully convolutional neural networks for volumetric medical image segmentation. Fausto Milletari, Nassir Navab, Seyed-Ahmad Ahmadi, 3Fausto Milletari, Nassir Navab, and Seyed-Ahmad Ah- madi, "V-net: Fully convolutional neural networks for volumetric medical image segmentation," in 3DV, 2016.

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{'abstract': 'The diagnosis of the presence of metastatic lymph nodes from abdominal computed tomography (CT) scans is an essential task performed by radiologists to guide radiation and chemotherapy treatment. State-of-the-art deep learning classifiers trained for this task usually rely on a training set containing CT volumes and their respective image-level (i.e., global) annotation. However, the lack of annotations for the localisation of the regions of interest (ROIs) containing lymph nodes can limit classification accuracy due to the small size of the relevant ROIs in this problem. The use of lymph node ROIs together with global annotations in a multi-task training process has the potential to improve classification accuracy, but the high cost involved in obtaining the ROI annotation for the same samples that have global annotations is a roadblock for this alternative. We address this limitation by introducing a new training strategy from two data sets: one containing the global annotations, and another (publicly available) containing only the lymph node ROI localisation. We term our new strategy semisupervised multi-domain multi-task training, where the goal is to improve the diagnosis accuracy on the globally annotated data set by incorporating the ROI annotations from a different domain. Using a private data set containing global annotations and a public data set containing lymph node ROI localisation, we show that our proposed training mechanism improves the area under the ROC curve for the classification task compared to several training method baselines.', 'arxivid': '1910.10371', 'author': ['Saskia Glaser ', 'Gabriel Maicas ', 'Sergei Bedrikovetski ', 'Tarik Sammour \nDepartment of Surgery\nColorectal Unit\nRoyal Adelaide Hospital\nAustralia\n', 'Gustavo Carneiro ', '\nAustralian Institute for Machine Learning\nFaculty of Health and Medical Science\nSchool of Medicine\nThe University of Adelaide\nAustralia\n', '\nThe University of Adelaide\nAustralia\n'], 'authoraffiliation': ['Department of Surgery\nColorectal Unit\nRoyal Adelaide Hospital\nAustralia', 'Australian Institute for Machine Learning\nFaculty of Health and Medical Science\nSchool of Medicine\nThe University of Adelaide\nAustralia', 'The University of Adelaide\nAustralia'], 'corpusid': 204837949, 'doi': '10.1109/isbi45749.2020.9098372', 'github_urls': [], 'n_tokens_mistral': 5927, 'n_tokens_neox': 5053, 'n_words': 3418, 'pdfsha': '326b13ddc4b6a16074c5b49b36f374b445b7b5c1', 'pdfurls': ['https://arxiv.org/pdf/1910.10371v1.pdf'], 'title': ['SEMI-SUPERVISED MULTI-DOMAIN MULTI-TASK TRAINING FOR METASTATIC COLON LYMPH NODE DIAGNOSIS FROM ABDOMINAL CT', 'SEMI-SUPERVISED MULTI-DOMAIN MULTI-TASK TRAINING FOR METASTATIC COLON LYMPH NODE DIAGNOSIS FROM ABDOMINAL CT'], 'venue': []}

arxiv

KORN-MAXWELL-SOBOLEV INEQUALITIES FOR GENERAL INCOMPATIBILITIES 26 Dec 2022 Franz Gmeineder Peter Lewintan Patrizio Neff KORN-MAXWELL-SOBOLEV INEQUALITIES FOR GENERAL INCOMPATIBILITIES 26 Dec 2022 We establish a family of coercive Korn-type inequalities for generalised incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work [31], where we focussed on the case p = 1 and incompatibilities governed by the matrix curl, the case p > 1 considered in the present paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalisations of recently proved inequalities by the last two authors and MÜLLER[43]in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher order scenarios as well, our approach equally gives the first and sharp inequalities involving KRÖNER's incompability tensor inc .Date: December 27, 2022. 2020 Mathematics Subject Classification. 35A23, 26D10, 35Q74/35Q75, 46E35. INTRODUCTION Korn-Maxwell-Sobolev inequalities. Let Ω ⊂ R n be an open and bounded set with Lipschitz boundary. A key device in the study of variational principles or non-linear partial differential equations from elasticity or fluid mechanics are the so-called Korn inequalities. Such inequalities allow us to control the L q -norms of the full gradients by merely controlling their symmetric parts (cf. e.g. [16,17,21,38,60]). There are different forms of such inequalities, and in the following we shall focus on two such core inequalities that are distinguished by zero or non-zero boundary values of the admissible competitors. Specifically, for any 1 < q < ∞ there exists a constant c = c(n, q) > 0 such that we have for all u ∈ W 1,q 0 (Ω; R n ) Du L q (Ω) ≤ c sym Du L q (Ω) . This inequality persists when Ω is replaced by R n , in which case (K1) can be reduced to classical Calderón-Zygmund estimates. In the sequel, we shall refer to (K1) and variants thereof to as Korn inequalities of the first kind. It is easily seen that inequality (K1) does not hold true when considering Sobolev maps u ∈ (W 1,q \ W 1,q 0 )(Ω; R n ). Indeed, the set of rigid deformations R (i.e., maps of the form u(x) = Ax+b with a skew-symmetric matrix A ∈ R n×n , which we also express by writing A ∈ so(n), and b ∈ R n ) are contained in the nullspace of the symmetric, but not of the full gradients. To arrive at a valid inequality, the corresponding estimates must keep track of the rigid deformations. Such bounds are provided by the Korn-type inequalities of the second kind: For any connected open and bounded set Ω ⊂ R n with Lipschitz boundary and any 1 < q < ∞, there exists a constant c = c(n, q, Ω) > 0 such that inf Π∈R Du − Π L q (Ω) ≤ c sym Du L q (Ω) (K2) holds for all u ∈ W 1,q (Ω; R n ). In many applications -so for instance in elastoplasticity models, fluid mechanical problems and their numerical approximation through finite element methods, see Section 2 for an overview -the underlying differential nature of Du or sym Du is not available. This requires refinements of (K1) and (K2) which, by now, has been accomplished in different but only special situations [6,7,29,31,32,43,44,45,46,47,55,56] and still lacks a unifying perspective. To motivate such inequalities in their easiest form, let us note that there are no constants c, c ′ > 0 such that P L q (Ω) ≤ c sym P L q (Ω) , inf A∈so(n) P − A L q (Ω) ≤ c ′ sym P L q (Ω) (1.1) hold for all P ∈ L q (Ω; R n×n ). For (1.1) 1 this can be easily seen by considering maps that take values in so(n). The failure of (1.1) 2 can be established similarly: Pick an arbitrary continuous function ζ : Ω → R with ζ ≡ 0 and mean value (ζ) Ω = 0, so that in particular ζ L q (Ω) > 0. Then we have for all α ∈ R: 0 < ζ L q = ζ − (ζ) Ω L q ≤ 2 ζ − α L q , so 0 < inf α∈R ζ − α L q . (1.2) Considering the matrix field P : Ω → R n×n defined by · · · · · · · · · · · · 0 P (x) :=   0 −ζ(x) ζ(x)          L q (Ω) (1.1) 2 ≤ 0. In light of the failure of (1.1), we see that it is indeed the gradient structure of the specific matrix fields P = Du that make inequalities (K1) and (K2) work. Recalling that on simply connected domains Ω ⊂ R n a sufficiently smooth map P : Ω → R n×n is a gradient -hence is compatible -if and only if it satisfies Curl P = 0, it is natural to ask for substitutes of (1.1) that account for the lack of curl-freeness of arbitrary matrix fields P . This is achieved by the Korn-Maxwell-Sobolev inequalities (for brevity, KMS-inequalities). In such estimates, the non-valid inequalities (1.1) are modified by an additive Curl-term on their right-hand sides as a corrector, taking into account the non-Curl-freeness of generic competitor maps. Such inequalities, see e.g. [46,47], assert the existence of c = c(n, q) > 0 and c ′ = c ′ (n, q, Ω) > 0 P L q (Ω) ≤ c sym P L q (Ω) + Curl P L p (Ω) , P ∈ C ∞ c (Ω; R n×n ), inf A∈so(n) P − A L q (Ω) ≤ c ′ sym P L q (Ω) + Curl P L p (Ω) , P ∈ C ∞ (Ω; R n×n ). (1.3) Note that q determines the range of possible exponents p (and vice versa), and that the optimal p for given q is given by the corresponding Sobolev-or Morrey conjugate exponent (also see Section 1.2 below). These inequalities are the starting point for the present paper. Aiming to generalise (1.3) in a basically optimal way and thereby to provide a unifying framework for a wealth of coercive inequalities used in applications, we proceed by giving the detailled underlying set-up of the desired inequalities first. Generalised incompatibilities. As discussed at length in [43] (also see Section 2 below), applications from elasticity such as the so-called relaxed micromorphic model require estimates that hinge on strictly weaker quantities than Curl P . To provide a unifying approach to the matter, we shall thus consider more general (i) parts A[P ] than the symmetric part as in (1.3), and (ii) differential operators B than the matrix curl as in (1.3). As such, the corresponding variants of (1.3) shall be referred to as generalised KMS-inequalites. Specifically, let V, W, V be three finite dimensional real inner product spaces, k ∈ N and let B be a k-th order, linear and homogeneous differential operator on R n from V to W . By this we understand that B has a representation B = |α|=k B α ∂ α (1.4) with fixed linear maps B α : V → W for α ∈ N n 0 . Towards (i), we let A : V → V be a linear part map. Let 1 < p < ∞. Aiming to generalise (K1), we first work on the entire R n and wish to classify all parts A, differential operators B and exponents q for which we have validity of the generalised Korn-Maxwell-Sobolev inequality of the first kind: P X k,p (R n ) ≤ c A[P ] X k,p (R n ) + BP L p (R n ) , P ∈ C ∞ c (R n ; V ).(KMS1) Here, the function spaces X k,p (and consequently their norms · X k,p (R n ) ) are chosen in a way such that (KMS1) scales suitably. Specifically, if we choose k = 1 and work with Lebesgue spaces, then validity of the inequality P L q (R n ) ≤ c A[P ] L q (R n ) + BP L p (R n ) , P ∈ C ∞ c (R n ; V ), with 1 < p < n directly determines q to equal np n−p by considering maps P λ (x) := P ( x λ ) for λ > 0. More generally and hence for k > 1, suitable choices of X k,p are given by 1 X k,p =Ẇ k−1, np n−p (homogeneous Sobolev spaces) if 1 < p < n or X k,p =Ċ k−1,1− n p (homogeneous Hölder spaces) if p > n. Also, validity of inequality (KMS1) immediately implies the corresponding inequality P X k,p (Ω) ≤ c A[P ] X k,p (Ω) + BP L p (Ω) , P ∈ C ∞ c (Ω; V ), (1.5) for any open and bounded set Ω ⊂ R n . This can be seen by trivially extending maps P ∈ C ∞ c (Ω; V ) to the entire R n . By the link to the classical Korn-type inequalities (K1), we shall put some emphasis on inequalities involving Lebesgue norms. One retrieves the known Korn-Maxwell-Sobolev inequality (1.3) 1 by specifying V = V = R n×n , A = sym and B = Curl. Other inequalities that appear as special cases and arise in concrete models shall be addressed in Section 2 below. However, to explain some of the mechanisms underlying inequalities of the form (1.5), let us note that there is some coupling between A and B: Heuristically, the stronger B becomes, the weaker we may assume A to be, and vice versa. As an important instance of this effect, let us compare the classical div-curl-complex with the SAINT VENANT or elasticity complex: C ∞ (R n ; R n ) C ∞ (R n ; R n×n ) C ∞ (R n ; R n× n(n−1) 2 ) C ∞ (R n ; R n ) C ∞ (R n ; Sym(n)) C ∞ (R n ; Sym( n(n−1) 2 )), D Curl sym D Curl(Curl ⊤ ) (1.6) with the symmetric (n × n)-matrices Sym(n); see CIARLET et al. [2,3,19] for more on the complex (1.6) 2 and [4,5,18,20,59] on its role in (in)compatible elasticity. Similarly as in (1.6) 1 , where the exactness at the mid vector space implies that an R n×n -valued map on R n is a gradient if and only if it is Curl-free, (1.6) 2 expresses the fact that a Sym(n)-valued map is a symmetric gradient if and only if it is Curl(Curl) ⊤ -free. However, if a map into the symmetric matrices is already Curl-free, then it is already a symmetric gradient. Hence, the compatibility condition inc P := Curl((Curl P ) ⊤ ) = 0 is weaker than Curl P = 0 on the Sym(n)-valued 1 Here we adopt the conventionẆ 0,q := L q for 1 < q < ∞. maps P ; inc is also referred to as KRÖNER's strain incompatibility tensor [39] (see Sections 2, A for this terminology). There are two borderline cases that we wish to address explicitly: If B is elliptic (see Section 3.1 for this terminology), then (1.5) is satisfied even for A ≡ 0. On the other hand, if B ≡ 0, then (1.5) is only satisfied if A is injective 2 . In between the borderline cases, a non-trivial obstruction to estimates (1.5) is given by the following examples: Example 1.1. For P ∈ R n×n , we define the deviatoric part dev P := P − tr P n · 1 n (with the unit matrix 1 n ∈ R n×n ). For ϕ ∈ C ∞ c (R 3 ), we put P ϕ = ϕ · 1 3 , so that Curl P ϕ =   0 ∂ 3 ϕ −∂ 2 ϕ −∂ 3 ϕ 0 ∂ 1 ϕ ∂ 2 ϕ −∂ 1 ϕ 0   . We consider (A, B) given by • A = dev, B = sym Curl. Then dev P ϕ = 0, sym Curl P ϕ = 0 but P ϕ ≡ 0, so that (KMS1) or (1.5) yield an immediate contradiction. • A = dev sym and B = sym Curl or B = dev sym Curl. The requisite contradictions directly follow from the previous item. (3) is the canonical identification map (see the appendix for more detail). Using NYE's formula (see (A.8) in the appendix) Example 1.2. Consider P ψ = Anti(∇ψ) for ψ ∈ C ∞ c (R 3 ), where Anti : R 3 → soCurl P ψ = ∆ψ · 1 3 − D∇ψ ∈ Sym(3) (1.7) and for (A, B) = (sym, skew Curl) or (A, B) = (dev sym, skew Curl) we would obtain a contradiction to the validity of (KMS1) or (1.5). These examples show that the operator B might turn out non-elliptic on elements for which the parts A vanish. As such, for inequalities (1.5) to hold, B must be elliptic on precisely such elements. In Theorem A, to be stated and proved in Section 3.2, we will establish that this is also sufficient. Especially, we recover all such KMS inequalities that are known so far for 1 < p < ∞ and generalise them in an optimal way. The preceding inequalities require zero boundary values in a suitable sense, as do the Korn inequalities of the first kind (cf. (K1)). Therefore, our second focus is on generalised KMS inequalities of the second kind, thereby dealing with the situation on domains. Working from (K2) or (1.3) 2 , respectively, we let Ω ⊂ R n be an open, bounded and connected Lipschitz domain (e.g. with Lipschitz boundary ∂Ω), K be a fixed subspace of the V -valued polynomials on R n and consider, for a given part map A and differential operators B as introduced above, validity of the inequality inf Π∈K P − Π L q (Ω) ≤ c A[P ] L q (Ω) + BP L p (Ω) (KMS2) for all P ∈ C ∞ (Ω; R n×n ). Here, 1 < p < ∞, which in turn determines the possible range of q depending on the order k of B and the underlying space dimension n. Different from the entire space case, we will then find that it does not suffice anymore for B to behave elliptically on maps P for which A[P ] vanishes identically. Indeed, as will be made precise in our second main result, Theorem B below, such inequalities require to rule out a certain bad boundary behaviour. As we shall establish in 3.3, this is equivalent to B behaving like a so-called C-elliptic operator along maps for which A[P ] vanishes; we refer the reader to Section 3.1 for this terminology. Remark 1.3 (p = 1). In contrast to the precursor [31] of the present paper, which focussed on KMS-inequalities with B = Curl but particularly addressed the case p = 1 for the sharp class of part maps A, we here allow for joint maximal flexibility for both the part maps A and differential operators B, however, concentrate on the case 1 < p < ∞. For the integrability regime p = 1, the techniques displayed in this paper allow to obtain inequalities involving the homogeneous Hardy space H 1 (cf. Corollary 3.9). Yet, this does not suffice to obtain inequalities with the mere L 1 -norms of BP , see Remark 3.10, and we shall pursue this systematically in future work. 1.3. Structure of the paper. Away from this introduction, the paper is organised as follows: After fixing notation, Section 2 discusses models for which the results of the present paper provide a unifying perspective on the underlying inequalities. In Section 3 we then state and prove the generalised KMS inequalities alluded to above. After discussing selected function space implications in Section 3.4, we discuss in Section 4 how the results in the present paper let us finally retrieve and extend previously known KMS-type inequalities. NOTATION Our notation is fairly standard, but we wish to comment on certain aspects. Throughout, 1 n denotes the (n × n)-unit matrix and V, W finite dimensional real inner product spaces. Given x 0 ∈ R n and r > 0, the open ball of radius r centered at x 0 is denoted B r (x 0 ) := {x ∈ R n | |x − x 0 | < r}. The n-dimensional Lebesgue and the (n − 1)-dimensional Hausdorff measure are denoted L n and H n−1 , respectively, and we set ω n−1 := H n−1 (∂B 1 (0)). For a bounded measurable set A ⊂ R n with L n (A) > 0 and u ∈ L 1 loc (R n ; V ), we write A u dx := 1 L n (A)ˆA u dx. We moreover use the dot notation to denote homogeneous function spaces; e.g., we writė W k,p (R n ; V ) to indicate the homogeneous Sobolev space of order (k, p) of V -valued maps, so the closure of C ∞ c (R n ; V ) for the k-th order L p -gradient norm D k · L p (R n ) . This notation also carries over to other space scales such as e.g. homogeneous Triebel-Lizorkin spaces, where we consequently stick to the conventions of TRIEBEL [67,Chpt. 5]. Inner products on a linear space V will be denoted ·, · V , and if it is clear from the context, we shall omit the subscript. For m ∈ N and a finite dimensional inner product space V , the symmetric, m-multilinear V -valued maps on R n are denoted SLin m (R n ; V ). For completeness, let us recall that the symmetric or skew-symmetric (n × n)-matrices are denoted Sym(n) or so(n), respectively. We shall occasionally also apply some notation on underlying identification maps and algebraic identities that we, for the convenience of the reader, have concisely gathered in the Appendix, Section A. Lastly, we write c > 0 for a constant that might change from line to line, and shall only be specified in case its precise value is required. MODELS, CONTEXT AND PREVIOUSLY KNOWN RESULTS AS SPECIAL CASES For a wealth of specific constellations (A, B, p, q), the generalised KMS-inequalities (KMS1) and (KMS2), to be addressed in Theorems A and B below, provide inequalities that play instrumental roles in the well-posedness theory for a multitude of mathematical models. We now proceed to contextualise the unifying approach of the present paper with previous contributions, and how it extends and recovers key coercive inequalities in several recently studied models. 2.1. Contextualisation. KMS-inequalities and related inequalities have been studied in particular situations; see [7,43,44,45,46,47,55,56] for a non-exhaustive list. Slightly more systematically, in the specific case where 1 < p < ∞ and B = Curl in n = 3 dimensions, [32] establishes the equivalence (2.1) The latter condition can be expressed algebraically via ξ∈R 3 \{0} ker(A[· ⊗ ξ]) = {0} (⇔ (∀ξ = 0 : A[v ⊗ ξ] = 0) ⇒ v = 0). (2.2) The equivalence (2.1) extends to arbitrary dimensions n ≥ 2, cf. [31]. In these contributions, the underlying KMS-inequalities are approached by performing an analytic split embodied by Helmholtz decompositions of generic maps P . Decomposing a map P into its divand curlfree parts then allows to use the fractional integration [1, §3] or Sobolev embedding theorem on the former and Calderón-Zygmund theory on the latter. In particular, this split is dictated by the differential operator B = Curl which is then viewed as the central object, hence the name analytic split. Here we introduce a different approach and perform an algebraic split; see the proof of Theorem A below. Namely, it is now the part map A that motivates the pointwise decomposition P = Π ker(A) [P ] + Π ker(A) ⊥ [P ]. Contrary to Π ker(A) ⊥ [P ], which is a priori controllable by A[P ], it is now the part Π ker(A) [P ] that requires elliptic estimates and Sobolev's embedding. In contrast to the analytic one, this split is not dictated by the differential operator B = Curl but the purely algebraic part map which is then viewed as central. Let us note, though, that while the algebraic split approach in the present paper yields the optimal result for the regime 1 < p < ∞, we do not see how it allows to obtain results in the borderline case p = 1 which has been resolved recently [31] by the authors in the case of B = Curl employing the analytic split. In the following, we now proceed to discuss three illustrative models from continuum or fluid mechanics where generalised KMS inequalities as described in Section 1.2 play a pivotal role. As a key point, up to now these inequalities required single approaches, requiring different intricate algebraic identities or analytic estimates each, whereas they now appear as special cases of the results of the present paper. Throughout, let now Ω ⊂ R 3 be open and bounded. 2.2. Gradient plasticity models. The modelling of plastic deformations in the geometrically linear framework is based on the additive decomposition of the displacement gradient Du = e + P into the non-symmetric elastic distortion e and the plastic distortion P . With the presence of plastic deformations the (free) elastic energy describing the elastic response of the material is of the form F [u, P ; Ω] :=ˆΩ W elastic (sym(Du − P )) + W plastic (sym P, Curl P ) dx,(2.3) whereby dislocations are modelled by the control of Curl P , which enters the plastic energy, see [22,27,29,34,43,50,51,63] for an incomplete list. For X ∈ R 3×3 , we consider the orthogonal decomposition X = dev sym X + 1 3 tr(X)1 3 + skew X ∈ (sl(3) ∩ Sym(3)) ⊕ R1 3 ⊕ so(3),(2.4) where sl(3) = {X ∈ R 3×3 | tr X = 0}. Applying this pointwisely to Du for a displacement u : Ω → R 3 , (2.4) decomposes Du into a shear part dev sym Du capturing the shape change of the material, whereas 1 3 tr(Du) reflects purely volumetric infinitesimal changes and skew Du describes rotations of the material; the latter two constituents hence do not give rise to shape change of the material. The additive decomposition (2.4) is also meaningful for X = Curl P , cf. LAZAR [40,41] and NEFF et al. [54]. Specifically, the diagonal entries of Curl P display screw dislocations and the off-diagional entries describe edge dislocations. The single constituents in (2.4) then describe, in this order, symmetric edge dislocations combined with single screw edge dislocations, screw dislocations and skew-symmetric edge dislocations. In a phenomenologically simple model, the plastic energy density can account for this particular split via an additive description W plastic (sym P, Curl P ) = W (1) plastic (sym P ) + W (2) plastic (Curl P ), where for suitable µ 1 , µ 2 , µ 3 ≥ 0 and q, p 1 , p 2 , p 3 > 1 W (1) plastic (sym P ) = | sym P | q−2 C hard sym P, sym P , (2.5) W(2) plastic (Curl P ) = µ 1 | dev sym Curl P | p1 + µ 2 | tr Curl P | p2 + µ 3 | skew Curl P | p3 , whereas the elastic energy density in (2.3) is given by the quadratic form W elastic (sym(Du − P )) = C elastic sym(Du − P ), sym(Du − P ) . (2.6) In (2.5) or (2.6), C elastic , C hard ∈ R (3×3)×(3×3) are positive definite elasticity or plastic hardening tensors; the reader might notice that for q = 2, (2.5) 1 reduces to the usual PRAGER backstress term [25]. The choice of parameters µ 1 , µ 2 , µ 3 leads to different models for certain material aspects, and admitting exponents p 1 , p 2 , p 3 = 2 is natural (see e.g. WULFINGHOFF et al. [69] or CONTI & ORTIZ [22] for a similar appearance of non-quadratic curl-terms). Coercivity of the functionals (and hereafter existence of minimizers) then relies on suitable KMS-inequalities of type (1.3). Specifically, as discussed by the second named authors and MÜLLER [43], if µ 2 , µ 3 = 0, then the specific KMS-type inequality inf Π∈K P − Π L q (Ω) ≤ c sym P L q (Ω) + dev sym Curl P L p (Ω) (2.7) for P ∈ C ∞ (Ω; R 3×3 ) is key to the coercivity of the associated energy functionals. On the other hand, if µ 1 = 0 and so only screw dislocations and edge dislocations are reflected in the model, coercivity for the forced functional (for some suitably integrable force f : Ω → R 3 ) F f [u, P ; Ω] := F [u, P ; Ω] −ˆΩ f, u dx subject to zero Dirichlet conditions necessitates a KMS-inequality that replaces the part map dev sym in (2.7) by the part map R 3×3 ∋ X → skew X + tr(X)1 3 . As a consequence of Theorem A (see Section 4.1.3 and Figure 2), the requisite coercive KMS-inequalities are available for zero Dirichlet data and suitable choices of exponents p 2 , p 3 , but not immediately for non-zero Dirichlet data on P . More generally, recalling that ( and we refer the reader to Section 4 for more background on such constellations. 2.3. The relaxed micromorphic model. Another key application of KMS-type inequalities is given by (extended continuum) micromorphic models. Here, a key modelling assumption is the attachment of a microstructure to each single point of the material, with this microstructure in turn deforming elastically. In analogy with the discussion in Section 2.2, in the relaxed micromorphic model the plastic distortion is replaced by the micro distortion [53,54]. Specifically, under suitable side constraints on the displacement field u : Ω → R 3 and the (in general nonsymmetric) micro distortion P : Ω → R 3×3 , one then aims to find minimizers of functionals F [u, P ; Ω] :=ˆΩ W elastic (sym(Du − P )) + W micro (sym P ) (2.9) + W coupling (skew(Du − P )) + W curv (Curl P ) + f, u dx =: I + ... + V. In (2.9), term I is as in (2.6), whereas the other parts II-IV are of the form Cz, z . As to term II, C = C micro for a suitable elasticity tensor C micro : Sym(3) → Sym(3), whereas C = C c for a rotational coupling tensor on so(3). Term IV then is a curvature energy term, and a physically meaningful reduction of the complexity of the tensor C = C curv then yields equality of IV with the right-hand side of (2.5) 2 with p 1 = p 2 = p 3 = 2, see [43,53,54]. If the rotational coupling W coupling is absent it is then clear that, similarly as in Section 2.2, generalised KMStype inequalities are instrumental for the existence of minimizers, where different part maps A, B as in (2.8) then model different material aspects. 2.4. Pseudostress-velocity formulation for stationary Stokes. The stationary Stokes system for incompressible fluids can be recast in a pseudostress formulation. To be more precise, let Ω ⊂ R 3 be open and bounded with Lipschitz boundary. Subject to f : Ω → R 3 and suitable boundary conditions, one aims to find a velocity function u : Ω → R 3 , a corresponding pressure function p : Ω → R and, in addition, a stress function σ : Ω → R 3×3 such that the following first order system is satisfied:                σ − µ sym Du + p1 = 0 in Ω, Div σ = f in Ω, div u = 0 in Ω, u = 0 on Γ ν ⊂ ∂Ω, σ × ν = 0 on Γ τ ⊂ ∂Ω,(2.10) where Γ ν , Γ τ are relatively open subsets of ∂Ω and (2.10) 5 is understood in the row-wise sense. Taking the row-wise divergencence of both sides of (2.10) 1 , one recovers the usual stationary Stokes system for incompressible fluids. Weak solutions, in a suitable yet canonical sense, for (2.10) can be obtained variationally as the minimizer of the functional F [σ, u; Ω] :=ˆΩ | dev σ − µ sym Du| 2 + | Div σ − f | 2 dx over σ ∈ H(Div; Γ ν , Ω), u ∈ H(Grad; Γ τ , Ω), and the pressure function in (2.10) is then re-introduced via p = − 1 3 tr(σ). The underlying spaces H(Div; Γ ν , Ω) and H(Grad; Γ τ , Ω) are particular instances of the spaces W A,q,B,p 0,Γ (Ω) introduced in Section 3.4 below, see (3.60) below. The formulation (2.10) originally appeared in CAI et al. [11] and has been studied from numerical perspectives in [12,30]. As discussed at length in BAUER et al. [7], the coercivity of the functional F and hence the existence of minimizers in turn is based on the KMS-type inequality P L 2 (Ω) ≤ c dev P L 2 (Ω) + Div P L 2 (Ω) , P ∈ H(Div; Γ ν , Ω), (2.11) which we recover as a special case of Proposition 3.22 below. Inequalities (2.11) with nonquadratic integrabilities also arise naturally in the realm of Non-Newtonian fluids, where the viscosity of the fluid depends on the velocity. Imposing CARREAU's law, the term µ sym Du then is replaced by (1 + |sym Du| 2 ) q−2 2 sym Du and then corresponds to shear thickening (q > 2) or shear thinning fluids (q < 2); see [7,28] and [48,49] for recent numerical results. Miscallaneous other models and applications. Variations of KMS-inequalities also enter in different applications, and we here gather some of such results. First, a variant of inequality (2.11) had been studied by ARNOLD et al. [6] in view of developing a mixed higher order finite element method for dealing with planar elasticity and the corresponding error analysis; in this context, see also BOFFI et al. [10] and CARSTENSEN et al. [13,14]. Different from (2.11), in this situation it is natural to not require maps to vanish on parts of the boundary but to satisfy a certain normalisation condition. Here, such inequalities arise as special cases of Corollary 3.19 below; also see Remark 3.20 for their link to the aforementioned contributions. Moreover, BOTTI et al. [9] apply KMS inequalities of the second type to establish an adequate discrete version of a Poincaré-Korn type inequality, which itself is used to justify a reduction of mesh face degrees of freedom through serendipity techniques. It is worth mentioning that their presented Korn type inequalities already follow from the C-ellipticity of the appearing differential operators on the right-hand side of the estimates in [9,Prop. 25], see the definition in the corresponding equation [9, Eq. (2.1)] and, in turn, [23,37]. Lastly, in incompatible elasticity, the symmetric strain field ε instead of the displacement field is viewed as the primary quantity (see e.g. CIARLET [18,20] or the non-variational models from AMSTUTZ & VAN GOETHEM [4,5]) and is not a priori assumed to be a symmetric gradient of a displacement field. In terms of the complex (1.6) 2 ff., this is expressed via inc ε = 0, in which case ε = sym Du for any displacement field u. The use of inc-based gradient plasticity models is based on an additional invariance condition beyond isotropy [26,66]. The associated kinematic problems then naturally lead to Sobolev-type spaces defined in terms of inc , which then are a special case of the general spaces introduced in Section 3.4. However, the corresponding sharp KMS inequalities in Theorems A, B, Corollary 3.19, Proposition 3.22 and the results from Section 4.3 in turn show that inc is often too weak to give suitable control over lower order quantities which has only been conjectured so far. This also indicates that including lower order Curl-terms might be necessary to obtain well-posedness in certain fourth order gauge invariant variational plasticity models for polycrystals, cf. EBOBISSE et al. [26]. MAIN RESULTS We now proceed to state and prove our main results, Theorem A and B, in Sections 3.2 and 3.3 below. Beforehand, we start by introducing the underlying terminology. Here we tacitly adopt the conventions gathered in Section 1.2. 3.1. Algebraically compatible ellipticities. In order to introduce the notions that are necessary and sufficient for the generalised KMS inequalities (KMS1) and (KMS2), we start by recalling some general terminology for vectorial differential operators. Thus, let A be an l-homogeneous, linear, constant coefficient differential operator on R n between V and W (two finite-dimensional real inner product spaces), meaning that Au := |α|=l A α ∂ α u, u : R n → V, (3.1) for linear maps A α : V → W and multi-indices α ∈ N n 0 with |α| = l. With this operator we associate the symbol map A[ξ] : V → W, A[ξ]v := |α|=l ξ α A α v, ξ ∈ R n , v ∈ V, (3.2) where ξ α = ξ α1 1 · · · ξ αn n for α = (α 1 , . . . , α n ) ∈ N n 0 . An operator of the form (3.1) is then called elliptic (in the sense of HÖRMANDER or SPENCER [35,65]) or R-elliptic if ker R (A[ξ]) = {0} for all ξ ∈ R n \{0}, (3.3) and is said to be C-elliptic if ker C (A[ξ]) = {0} for all ξ ∈ C n \{0}. (3.4) Specifically, as A α is a map between real vector spaces, let us point out that (3.4) has the interpretation that, for any ξ ∈ C n \ {0} and all v 1 , v 2 ∈ V , |α|=l ξ α A α v 1 + i |α|=l ξ α A α v 2 = 0 =⇒ v 1 , v 2 = 0. (3.5) The following lemma is essentially due to SMITH [64]; also see KAŁAMAJSKA [37]. dim{P ∈ D ′ (Ω; V )| AP = 0} < ∞. Moreover, the nullspace in (c) consists of V -valued polynomials and is independent of Ω. 3.2. Generalised KMS inequalities of the first kind. We directly start by displaying the first main result of the present paper, providing the classification of all constellations (A, B, q) with 1 < p < ∞ that lead to validity of the corresponding generalised KMS-inequalities (KMS1) of the first kind. For expository reasons, we start with the following basic version involving Lebesgue spaces. This result can be generalised and then appears as a special case of a more general statement on Triebel-Lizorkin spaces, cf. Theorem 3.8 below. Theorem A (Generalised KMS-inequalities of the first kind). Let 1 < p < n and suppose that the part map A and the k-th order differential operator B are as in Section 1.2. Then the following are equivalent: (a) There exists a constant c = c(A, B, p) > 0 such that the inequality P Ẇ k−1, np n−p (R n ) ≤ c A[P ] Ẇ k−1, np n−p (R n ) + BP L p (R n ) (3.6) holds for all P ∈ C ∞ c (R n ; V ). (b) B is reduced elliptic (relative to A), meaning that ker(A) ∩ Λ B = ξ∈R n \{0} ker(A) ∩ ker(B[ξ]) = {0}, (3.7) where Λ B := ξ∈R n \{0} ker(B[ξ]) denotes the wave cone of B. Equation (3.7) expresses the fact that B behaves like an elliptic operator on maps whose image is contained in ker(A). This also shows the naturality of the condition, letting us control the part of the field P which is contained in ker(A) by the operator B. .7) is equivalent to A[D·] being an elliptic operator, so that we recover the previously known conditions for the validity of KMS-inequalities with B = Curl, cf. [32,31], but only in the case p > 1. The following lemma is almost trivial, but since it is crucial for our purposes, we give the quick proof. Lemma 3.4. Let (X, · X ), (Y, · Y ) be two finite dimensional real, normed vector spaces. A linear map A : X → Y is injective if and only if there exists λ > 0 such that λ x X ≤ Ax Y holds for all x ∈ X. (3.8) Proof. It is clear that (3.8) implies the injectivity of A. Now suppose that A is injective. The unit sphere S X := {z ∈ X| z X = 1} is compact and, since z → Az is continuous, min z∈SX Az Y =: λ is attained at some z 0 ∈ S X . Since A is injective, this value cannot be equal to zero. We then use the linearity of A to deduce (3.8). Proof of Theorem A. We start by proving '(b)⇒(a)' and put p * := np n−p . Let Π ker(A) : V → ker(A) be the orthogonal projection onto ker(A) and accordingly Π ker(A) ⊥ : V → ker(A) ⊥ be the orthogonal projection onto ker(A) ⊥ . For the following, let P ∈ C ∞ c (R n ; V ) be arbitrary but fixed. Denoting the norms on V, V by · V , · V , let us note that it is possible to pointwisely bound Π ker(A) ⊥ [P ] V against A[P ] V . To see this, note that A| ker(A) ⊥ : ker(A) ⊥ → V is injective, whereby we may employ the Lemma 3.4. We thus have for any x ∈ R n (where the constant λ > 0 is provided by Lemma 3.4) and any α ∈ N n 0 : λ ∂ α Π ker(A) ⊥ [P (x)] V = λ Π ker(A) ⊥ [∂ α P (x)] V ≤ A[Π ker(A) ⊥ [∂ α P (x)]] V = A[Π ker(A) ⊥ [∂ α P (x)] + Π ker(A) [∂ α P (x)]] V = A[∂ α P (x)] V = ∂ α A[P (x)] V ,(3.9) noting that ∂ α acts componentwisely and Π ker(A) , A are linear; note that (3.9) 3 holds because of A[Π ker(A) [∂ α P (x)]] = 0. We next claim that B, viewed as a differential operator on the ker(A)-valued maps B : C ∞ c (R n ; ker(A)) → C ∞ c (R n ; W ) is elliptic. (3.10) Indeed, let v ∈ ker(A) and let ξ ∈ R n \ {0} be arbitrary. Then, using (b), B[ξ]v = 0 ⇐⇒ v ∈ ker(B[ξ]) ∩ ker(A) (3.7) =⇒ v = 0. By classical elliptic regularity, if L is a linear, homogeneous elliptic differential operator on R n between the finite dimensional vector spaces V 1 and V 2 of order k ∈ N, we have for any 1 < p < ∞ u L p (R n ) ≤ c Lu Ẇ− k, p (R n ) for all u ∈ C ∞ c (R n ; V 1 ), (3.11) where c = c( p, L) > 0 is a constant. Since Π ker(A) [P ] is ker(A)-valued, we thereby obtain by (3.11) for all α ∈ N n 0 with |α| = k − 1 ∂ α Π ker(A) [P ] L np n−p (R n ) = Π ker(A) [∂ α P ] L np n−p (R n ) (3.11) ≤ c BΠ ker(A) [∂ α P ] Ẇ −k, np n−p (R n ) = c B(∂ α P − Π ker(A) ⊥ [∂ α P ]) Ẇ −k, np n−p (R n ) ≤ c ∂ α BP Ẇ −k, np n−p (R n ) + c Π ker(A) ⊥ [∂ α P ] L np n−p (R n ) (3.9) ≤ c BP Ẇ −1, np n−p (R n ) + c A[∂ α P ] L np n−p (R n ) ≤ c BP L p (R n ) + c ∂ α A[P ] L np n−p (R n ) , (3.12) where we have used that L p (R n ; W ) ֒→Ẇ −1, np n−p (R n ; W ). In fact, this is equivalent tȯ W 1, np np−n+p (R n ; W ) ֒→ L p ′ (R n ; W ) , and since np np−n+p < n, its Sobolev conjugate exponent is well-defined and given by p ′ . To conclude, we estimate ∂ α P L np n−p (R n ) ≤ ∂ α Π ker(A) [P ] L np n−p (R n ) + ∂ α Π ker(A) ⊥ [P ] L np n−p (R n ) , employ (3.12) on ∂ α Π ker(A) [P ] L np n−p (R n ) , (3.9) on ∂ α Π ker(A) ⊥ [P ] L np n−p (R n ) and then sum the resulting inequalities over all α ∈ N n 0 with |α| = k − 1 to conclude (3.6). This settles the sufficiency in the general case. For the necessity, we have to show that the restricted op- erator B : C ∞ c (R n ; ker(A)) → C ∞ c (R n ; W ) is elliptic as this is clearly equivalent to (3.7) (cf. (3.10)ff.). Applying inequality (KMS1) to maps P ∈ C ∞ c (R n ; ker(A)), we obtain P Ẇ k−1, np n−p (R n ) ≤ c BP L p (R n ) , so that the requisite reduced ellipticity, and thus (b), follows from VAN SCHAFTINGEN [68,Cor. 5.2]. This completes the proof. Remark 3.6. Assuming (3.7), the same proof as for Theorem A can be employed to obtain lower order estimates. For instance, if j ∈ N 0 and 1 < p < ∞ are such that j < k and p(k − j) < n, then we obtain D j P L np n−(k−j)p (R n ) ≤ c D j A[P ] L np n−(k−j)p (R n ) + BP L p (R n ) (3.13) for all P ∈ C ∞ c (R n ; V ). One imitates the steps with the obvious modifications until (3.12), which is then employed with α ∈ N n 0 with |α| = j. The only modification then takes place in (3.12) 4 , leading to ∂ α Π ker(A) [P ] L np n−(k−j)p (R n ) ≤ c ∂ α BP Ẇ −k, np n−(k−j)p (R n ) + c Π ker(A) ⊥ [∂ α P ] L np n−(k−j)p (R n ) (3.9) ≤ c BP Ẇ j−k, np n−(k−j)p (R n ) + c A[∂ α P ] L np n−(k−j)p (R n ) ≤ c BP L p (R n ) + c ∂ α A[P ] L np n−p (R n ) , using that L p (R n ; W ) ֒→Ẇ j−k, np n−(k−j)p (R n ; W ) under the given assumptions. Let us now briefly address other space scales. If we allow the exponent p to be larger than n, n < p < ∞, then scaling suggests to work with the spaceĊ k−1,s (R n ; V ) with s = 1 − n p . One may then realise this Hölder space as the corresponding Besov space,Ċ k−1,s (R n ; V ) ≃ B k−1+s ∞,∞ (R n ; V ), cf. [67, §5]. Put t := k − 1 + s in the sequel. Given P ∈ C ∞ c (R n ; V ), we then estimate theḂ t ∞,∞ (R n ; V )-norm of P by splitting P into Π ker(A) [P ] and Π ker(A) ⊥ [P ]. As in the proof of Theorem A, we then only have to suitably bound Π ker(A) [P ] Ḃt ∞,∞ (R n ) . To conclude the requisite inequality in this case, we record as a substitute for (3.11) u Ḃt ∞,∞ (R n ) ≤ c Lu Ḃ t− k ∞,∞ (R n ) , u ∈ C ∞ c (R n ; V 1 ),(3.14) and, moreover, L p (R n ) ֒→Ḃ t−k ∞,∞ (R n ). (3.15) We now briefly explain the validity of (3.14) and (3.15). Inequality (3.14) is folklore, and one may argue rigorously as follows: Since in the framework of (3.11) the k-th order operator L is assumed elliptic, the Fourier multiplier operator T m1 (g) := F −1 [m 1 (ξ) g] := F −1 [ |ξ| k (L * [ξ]L[ξ]) −1 L * [ξ] g], is well-defined on maps g ∈ C ∞ c (R n ; V 2 ), and defining for h ∈ S ′ (R n ; V 1 ) T m2 (h) := F −1 [m 2 (ξ) h] := F −1 [|ξ| − k h], we have T m2 T m1 Lu = u for all u ∈ C ∞ c (R n ; V 1 ). We then invoke TRIEBEL [67, Thm. 1, §5.2.3] to find that T m2 :Ḃ t− k ∞,∞ (R n ; V 1 ) →Ḃ t ∞,∞ (R n ; V 1 ) boundedly. On the other hand, the symbol of T m1 is componentwisely smooth off zero and homogeneous of degree zero. Under these assumptions, [24,Thm. 4.13] implies that there exists z 0 ∈ L (V 2 ; V 1 + i V 1 ) and a C ∞ -function Θ : S n−1 → L (V 2 ; V 1 ) with zero mean for H n−1 S n−1 such that we have the representation T m1 g = z 0 g + p.v. Θ( x |x| ) |x| n * g for all g ∈ C ∞ c (R n ; V 2 ). (3.16) This particularly implies that the kernel K(x) := |x| −n Θ( x |x| ) satisfies for some constants A 1 , A 2 , A 3 > 0 sup 0<R<∞ 1 RˆB R (0) |K(x)| |x|dx ≤ A 1 (size condition), sup y∈R n \{0}ˆR n \B 2|y| (0) |K(x − y) − K(x)|dx ≤ A 2 (Hörmander's condition), (3.17) sup 0<R1<R2<∞ ˆB R 2 (0)\BR 1 (0) K(x)dx ≤ A 3 (cancellation conditionT m1 :Ḃ t− k ∞,∞ (R n ; V 2 ) →Ḃ t− k ∞,∞ (R n ; V 1 ) boundedly. Summarising, u Ḃt ∞,∞ (R n ) = T m2 (T m1 (Lu)) Ḃt ∞,∞ (R n ) ≤ c T m1 (Lu) Ḃ t− k ∞,∞ (R n ) ≤ c Lu Ḃ t− k ∞,∞ (R n ) ,(3.L p (R n ) ≃Ḟ 0 p,2 (R n ) ֒→Ḃ 0 p,∞ (R n ) ֒→Ḃ t−k ∞,∞ (R n ) upon recalling the definition of t and s in terms of k, p and n. Recalling thatĊ k−1,s (R n ; V ) ≃ B k−1+s ∞,∞ (R n ; V ) and using Lemma 3.4 on Π ker(A) ⊥ [P ] in the second inequality, we have Π ker(A) ⊥ [P ] Ḃs ∞,∞ (R n ) ≤ c |α|=k−1 Π ker(A) ⊥ [∂ α P ] Ċ0,s (R n ) (3.19) ≤ c λ |α|=k−1 A[∂ α P ] Ċ0,s (R n ) = c λ |α|=k−1 ∂ α A[P ] Ċ0,s (R n ) so that estimates (3.14) and (3.15) combine to the corresponding KMS-type inequality 3 D k−1 P Ċ0,s (R n ) ≤ c D k−1 A[P ] Ċ0,s (R n ) + BP L p (R n ) (3.20) for all P ∈ C ∞ c (R n ; V ). This scheme of proof also persists for other Besov spaces and Triebel-Lizorkin spaces, and it is possible to directly argue via Fourier multipliers and avoid (3.19). Since they provide us with a limiting case of independent interest, we focus on theḞ γ p,q -spaces in the sequel. Here, the modification of our above approach hinges on the following Mihlin-Hörmander multiplier-type and embedding result: (a) Let α, γ ∈ R and 0 < p < ∞. Given ℓ ∈ N with ℓ > max{ n p , n q } + n 2 , let m ∈ C ℓ (R n \ {0}) be a function that satisfies the generalised Hörmander condition, meaning that we have for all σ ∈ N n 0 with |σ| ≤ ℓ sup R>0 R −n+2α+2|σ|ˆ{ R<|ξ|<2R} |∂ σ ξ m(ξ)| 2 dξ ≤ C σ < ∞. (3.21) Then the Fourier multiplier operator T m u := F −1 (m u), originally defined on C ∞ c (R n ), extends to a bounded linear operator T m :Ḟ γ p,q (R n ) →Ḟ α+γ p,q (R n ). (3.22) (b) Let 0 < p 1 < p 2 < ∞ and −∞ < s 2 < s 1 < ∞. Then we havė F s1 p1,q (R n ) ֒→Ḟ s2 p2,q (R n ) provided s 1 − n p 1 = s 2 − n p 2 . Note that in the previous theorem, (b) is actually a consequence of (a), but we prefer to state the theorem in this way to facilitate future referencing. We now have Theorem 3.8 (Generalised KMS-inequalities of the first kind in the TL-scales). Let 0 < p 1 < p 2 < ∞ and −∞ < s 2 < s 1 < ∞ be such that s 1 − s 2 ≤ k and s 1 − n p 1 = k + s 2 − n p 2 ,(3. 23) Moreover, let the part map A and the k-th order differential operator B as introduced in Section 1.2 satisfy Then for any 0 < q < ∞ there exists a constant c = c(s 1 , p 1 , q, A, B) > 0 such that P Ḟ s 1 p 1 ,q (R n ) ≤ c A[P ] Ḟ s 1 p 1 ,q (R n ) + BP Ḟ s 2 p 2 ,q (R n ) . (3.25) Proof. Let P ∈ C ∞ c (R n ; V ). Similarly as in the proof of Theorem A, we start with the algebraic split P = Π ker(A) [P ] + Π ker(A) ⊥ [P ]. Now consider the operator B which, by (3.24), is elliptic on C ∞ c (R n ; ker(A)) just as in the proof of Theorem A. Now consider the Fourier multiplier operator T m Q := F −1 (B[ξ] * B[ξ]) −1 B * [ξ] Q , the symbol of which is C ∞ in R n \ {0} and homogeneous of degree (−k). Therefore, all its β-th partial derivatives, β ∈ N n 0 , are homogeneous of degree (−k − |β|). Thus, (3.21) is fulfilled for any ℓ ∈ N. We now apply We then estimate the homogeneous Triebel-Lizorkin norm of the key term as follows: Π ker(A) [P ] Ḟ s 1 p 1 ,q (R n ) ≤ c( BΠ ker(A) [P ] Ḟ s 1 −k p 1 ,q (R n ) ≤ c( BP Ḟ s 1 −k p 1 ,q (R n ) + BΠ ker(A) ⊥ [P ] Ḟ s 1 −k p 1 ,q (R n ) ) ≤ c( BP Ḟ s 1 −k p 1 ,q (R n ) + Π ker(A) ⊥ [P ] Ḟ s 1 p 1 ,q (R n ) ). (3.27) Condition (3.23) implies that s 1 − k ≤ s 2 and, in particular, F s2 p2,q (R n ) ֒→Ḟ s1−k p1,q (R n ). (3.28) We thus obtain P Ḟ s 1 p 1 ,q 1 (R n ) ≤ c Π ker(A) ⊥ [P ] Ḟ s 1 p 1 ,q 1 (R n ) + BP Ḟ s 2 p 2 ,q 1 (R n ) , and, realising that Id ker(A) ⊥ = (A * A| ker(A) ⊥ ) −1 (A * A| ker(A) ⊥ ), we obtain Π ker(A) ⊥ [P ] Ḟ s 1 p 1 ,q 1 (R n ) ≤ c A[P ] Ḟ s 1 p 1 ,q 1 (R n ) ,(3.29) and thereby conclude (3.25). The proof is complete. The sufficiency part of Theorem A can then be retrieved from Theorem 3.8 as follows. For the Lebesgue scale, we set s 1 = k − 1, s 2 = 0 (whereby (3.23) 1 is fulfilled), and then put p 2 = p, letting us compute p 1 via (3.23) 2 as p 1 = np n−p provided p < n. Realising thatḞ 0 p,2 (R n ) ≃ L p (R n ) for 1 < p < ∞, we recover the sufficiency part of Theorem A in the Lebesgue scale. If p = 1, we have insteadḞ 0 1,2 (R n ) ≃ H 1 (R n ) for p = 1 with the (homogeneous) Hardy space H 1 (R n ) (see, e.g. [33,Rem. 6.5.2] ). This observation gives us the following borderline inequality: Corollary 3.9 (Inequalities involving H 1 ). Let the part map A and the first order differential operator B as introduced in Section 1.2 satisfy (3.24). Then there exists a constant c > 0 such that we have P L n n−1 (R n ) ≤ c A[P ] L n n−1 (R n ) + BP H 1 (R n ) for all P ∈ C ∞ c (R n ; V ). (3.30) Remark 3.10. The Hardy norm appearing on the right-hand side of (3.30) is basically the best which one can obtain by the general Fourier multiplier techniques employed in this paper, and cannot be improved to be taken as the L 1 -norm. Specifically, as discussed by the authors in [31, Ex. 2.2], if one takes n = 2, V = R 2×2 and B = Curl, then condition (3.24) is tantamount to ellipticity of the differential operator A[Du] acting on u ∈ C ∞ c (R 2 ; R 2 ). This is satisfied by A = dev sym, but by [31, Ex. 2.2] the resulting KMS-inequality is false. We finally provide a variant of Theorem A for open and bounded sets. Here, we stick to the Lebesgue scale for simplicity; based on the above arguments, it is clear that analogous results can be obtained for other space scales. Corollary 3.11. Let 1 < p < n, 1 < q ≤ p * and Ω ⊂ R n be open and bounded. Moreover, let the part map A and the k-th order differential operator B as defined in Section 1.2 satisfy (3.7). Then there exists a constant c = c(p, q, Ω, A, B) > 0 such that we have P W k−1,q (Ω) ≤ c A[P ] W k−1,q (Ω) + BP L p (Ω) for all P ∈ C ∞ c (Ω; V ).(3. 31) If Ω = B r (x 0 ) for some x 0 ∈ R n and r > 0, then we have Br (x0) | D k−1 P | q dx 1 q ≤ c Br (x0) | D k−1 A[P ]| q dx 1 q + r Br(x0) |BP | p dx 1 p (3.32) for all P ∈ C ∞ c (B r (x 0 ); V ) with a constant c = c(p, q, A, B) > 0. Proof. If q = p * , this is immediate from Theorem A by extending C ∞ c (Ω; V ) trivially to R n . Hence let 1 < q < p * . As in the proof of Theorem A, we split P ∈ C ∞ c (Ω; V ) as P = Π ker(A) ⊥ [P ] + Π ker(A) [P ]. It is then clear that we have to only control the L q (Ω)-norm of Π ker(A) [P ]. Extending P trivially to R n , we replace the Sobolev exponent p * = np n−p in (3.12) 1 -(3.12) 4 by q. Since supp(BP ) ⊂ Ω, we only require L p (Ω; W ) ֒→ W −1,q (Ω; W ) to conclude, but this is a direct consequence of q < p * (if q > n n−1 , we use Sobolev's embedding theorem and the John-Nirenberg inequality otherwise). Now, (3.32) follows from (3.31) by scaling and the proof is complete. Clearly, the scaled variant (3.32) also holds for more general domains, but we confine ourselves to balls here for simplicity. The exponent restriction on q, however, is strict: Remark 3.12. We cannot allow q = 1 in the previous inequality in general. This is visible best in the situation where B = Curl, V = R n×n and so the B-free fields are precisely gradient fields. Following Example 3.3, we may send r → ∞ in (3.32) and recover the Korn-type inequality Du L q (R n ) ≤ c A[Du] L q (R n ) for all u ∈ C ∞ c (R n ; R n ). It is well-known that this inequality only persists for q = 1 if one has the pointwise inequality |Du| ≤ c|A[Du]| (cf. [58,36]), and the latter is clearly not the case subject to the sole assumption (3.7). Generalised KMS inequalities of the second kind. Opposed to the inequalities studied in the previous paragraph, we now turn to generalised KMS inequalities on domains. In particular, we drop the assumption of our competitor maps vanishing (to some order) at the boundary. Our main result then is as follows: Theorem B (Generalised KMS-inequalities of the second kind). Let p > 1, j ∈ N 0 , k ∈ N with j < k satisfy (k − j)p < n and let q ∈ (1, min Π∈K D j (P − Π) L q (Ω) ≤ c D j A[P ] L q (Ω) + BP L p (Ω) ,(3. 33) holds for all P ∈ C ∞ (Ω; V ). + i v 2 ∈ V + i V | A[v 1 ] + i A[v 2 ] = 0}. The notion of reduced C-ellipticity as displayed in Theorem B is actually already implicitly contained in the very general situation considered by SMITH [64], and we only use this terminology to stress its difference to the (full) C-ellipticity. Other space scales are equally possible, but we stick to the present formulation of Theorem B for ease of exposition. For its proof, we recall the following Lemma 3.14 (Nečas-Lions estimate, [52]). Let Ω ⊂ R n be a bounded Lipschitz domain, m ∈ Z and 1 < q < ∞. Denote by D l f the collection of all distributional derivatives of order l. Then f ∈ D ′ (Ω; R d ) and D l f ∈ W m−l, q (Ω; SLin l (R n ; R d )) imply f ∈ W m, q (Ω; R d ). Moreover, f W m, q (Ω) ≤ c f W m−1, q (Ω) + D l f W m−l, q (Ω) ,(3. 35) with a constant c = c(m, q, d, Ω) > 0. holds for all ϕ ∈ C ∞ (Ω; R M ). Applying this to ϕ = a, we then obtain for an arbitrary β ∈ N n The second ingredient is 0 with |β| ≤ k − 1 D k+ℓ ∂ β Π ker(A) [P ] W −k−ℓ,q (Ω) = ∂ β D k+ℓ (Ta) W −k−ℓ,q (Ω) (3.37) = ∂ β LB(Ta) W −k−ℓ,q (Ω) = L(∂ β B)(Ta) W −k−ℓ,q (Ω) ≤ c ∂ β BTa W −k,q (Ω) ≤ c ∂ β BP W −k,q (Ω) + B∂ β Π ker(A) ⊥ [P ] W −k,q (Ω) ≤ c BP W −k+|β|,q (Ω) + ∂ β Π ker(A) ⊥ [P ] L q (Ω) ,(3.38) where the constant c > 0 only depends on A, B and q (and hence implicitly on L, ℓ and k). Invoking the Nečas-Lions Lemma 3.14, we consequently arrive at ∂ β Π ker(A) [P ] L q (Ω) ≤ c ∂ β Π ker(A) [P ] W −1,q (Ω) + D k+ℓ ∂ β Π ker(A) [P ] W −k−ℓ,q (Ω) (3.38) ≤ c ∂ β Π ker(A) [P ] W −1,q (Ω) + ∂ β Π ker(A) ⊥ [P ] L q (Ω) + BP W −k+|β|,q (Ω) (3.39) and thus, summing over all multi-indices β ∈ N n 0 with |β| := j ≤ k − 1, conclude Since B • T is a C-elliptic operator and Ω is connected, we obtain from Lemma 3.1 (c) that m 0 := dim K < ∞ and, in particular, K is contained in a fixed finite dimensional subspace of polynomials. We now claim that we have D j P L q (Ω) ≤ c D j Π ker(A) [P ] W −1,q (Ω) + D j Π ker(A) ⊥ [P ] L q (Ω) + BP W −k+j,q (Ω) .min Π∈K D j (P − Π) L q (Ω) ≤ c D j A[P ] L q (Ω) + BP W −k+j,q (Ω) ,(3.42) for which, by a similar argument as in (3.9), it suffices to establish min Π∈K D j (P − Π) L q (Ω) ≤ c D j Π ker(A) ⊥ [P ] L q (Ω) + BP W −k+j,q (Ω) . (3.43) To see (3.43), we first note that that K ⊂ L 2 (Ω; V ) and D j K ⊂ L 2 (Ω; SLin j (R n ; V )) so that we may choose an orthonormal basis {e 1 , . . . , e m }, m ≤ m 0 , of D j K for the usual L 2 -inner product ·, · L 2 (Ω) ; it is then easy to see that we have e j = D j f j for all j ∈ {1, . . . , m} for suitable f 1 , . . . , f m ∈ K. We then put (D j K) ⊥ := D j Π | Π ∈ W j,1 (Ω; V ) and D j Π, e j L 2 (Ω) = 0 for all j ∈ {1, . . . , m} and subsequently claim that there exists a constant c > 0 such that D j P L q (Ω) ≤ c D j Π ker(A) ⊥ [P ]| L q (Ω) + BP W −k+j,q (Ω) + m j=1 ˆΩ D j P, e j dx (3.44) holds for all P ∈ C ∞ (Ω; V ). To this end, we assume towards a contradiction that (3.44) does not hold. We then are provided with a sequence (P i ) i∈N ⊂ C ∞ (Ω; V ) such that D j P i L q (Ω) = 1 and D j Π ker(A) ⊥ [P i ] L q (Ω) + BP i W −k+j,q (Ω) + m j=1 ˆΩ D j P i , e j dx < 1 i (3.45) for all i ∈ N. Since we assume 1 < q < ∞, the Banach-Alaoglu theorem provides us with (a here non-relabeled) subsequence and some Q ∈ L q (Ω; SLin j (R n ; V ))) such that D j P i ⇀ Q in L q (Ω; SLin j (R n ; V )) as i → ∞. Note that we may write Q = D j P for some P ∈ W j,q (Ω; V ). In fact, by connectedness of Ω, ∂Ω being Lipschitz, and hereafter the Poincaré inequality, for each i ∈ N, there exists a polynomial P i : Ω → V such that, for some constant c > 0 independent of i, D j P i = 0 and |γ|≤j−1 ∂ γ (P i − P i ) L q (Ω) ≤ c D j P i L q (Ω) . We then conclude that (F i ) i∈N := (P i − P i ) i∈N is bounded in W j,q (Ω; V ), whereby we may pass to another subsequence (F i l ) l∈N such that, for some F ∈ W j,q (Ω; V ), we have F i l ⇀ F in W j,q (Ω; V ) as l → ∞. Then we especially have D j P i l = D j F i l ⇀ D j F weakly in L q (Ω; SLin j (R n ; V )). Since D j P i l ⇀ Q weakly in L q (Ω; SLin j (R n ; V )), we conclude Q = D j F by uniqueness of weak limits. We set P := F in the sequel. The convergence D j P i ⇀ D j P in L q (Ω; SLin j (R n ; V )) already suffices to conclude BP i * ⇀ BP in W −k+j,q (Ω; W ). Indeed, let ϕ ∈ W k−j,q ′ 0 (Ω; W ). We write BP = B[D k P ] with some B ∈ L (SLin k (R n ; V ); W ). Since B is homogeneous of order k and j ≤ k − 1, we have BP i = 0 in Ω and thus, recalling div k−j B * [ϕ] ∈ L q ′ (Ω; SLin j (R n ; V )) with div k−j being the formal L 2 -adjoint of D k−j , Ω BP i , ϕ dx =ˆΩ B[D k P i ], ϕ dx =ˆΩ D k−j D j P i , B * [ϕ] dx = (−1) k−jˆΩ D j P i , div k−j B * [ϕ] dx → (−1) k−jˆΩ D j P, div k−j B * [ϕ] dx = BP, ϕ W −k+j,q (Ω)×W k−j,q ′ 0 (Ω) as i → ∞. By routine lower semicontinuity results for weak*-convergence, we then use (3.45) 2 to deduce BP = 0 as an equality in W −k+j,q (Ω; W ). Let β ∈ N n 0 be arbitrary with |β| = j. Since Π ker(A) ⊥ : V → ker(A) ⊥ is linear and bounded, the convergence ∂ β P i ⇀ ∂ β P weakly in L q (Ω; V ) implies ∂ β Π ker(A) ⊥ [P i ] ⇀ ∂ β Π ker(A) ⊥ [P ] weakly in L q (Ω; V ), whereby (3.45) 2 gives D j Π ker(A) ⊥ [P ] = 0 again by lower semicontinuity of norms for weak convergence and arbitrariness of |β| = j. Moreover, e j ∈ L q ′ (Ω; SLin j (R n ; V )) ∩ D j K for all j ∈ {1, . . . , m}, and so D j P i ⇀ D j P weakly in L q (Ω; SLin j (R n ; V )) yields that P ∈ (D j K) ⊥ by (3.45) 2 . Summarising, we have BP = 0, (3.46a) D j Π ker(A) ⊥ [P ] = 0, (3.46b) P ∈ (D j K) ⊥ .[P ] ∈ D j K. But then D j P ∈ D j K ∩ (D j K) ⊥ = {0}. In particular, D j Π ker(A) [P i ] ⇀ (∂ β Π ker(A) [P ]) |β|=j = (Π ker(A) [∂ β P ] ) |β|=j = 0 weakly in L q (Ω; SLin j (R n ; V )). Since the embedding L q (Ω; SLin j (R n ; V )) ֒→ W −1,q (Ω; SLin j (R n ; V )) is compact, we deduce that (again for a non-relabeled subsequence) D j Π ker(A) [P i ] → 0 strongly in W −1,q (Ω; SLin j (R n ; V )). Inserting P i into (3.40), we then arrive at the desired contradiction for sufficiently large i and thus have established (3.44). The passage from (3.44) to (3.43) is then accomplished as follows: We set Π P := m j=1 D j P, e j L 2 (Ω) f j , whereby clearly Π P ∈ K and apply (3.44) min Π∈K P − Π L q (Ω) ≤ c sym P L q (Ω) + Curl P L p (Ω) , P ∈ C ∞ (Ω; R 3×3 ) as established in [47]. Then K = so (3), which can be seen by taking P to be gradients. Then Theorem B yields precisely the same set of correctors K: Namely, in this case we have j = 0 so that, letting P belong to K given by (3.41), we may write P = Ta with BTa = 0, where now T = Anti : R 3 → so (3) is the canonical identification map (see (A.6) in the appendix for the details). By NYE's formula (see (A.8) 2 in the appendix), Curl Anti a = 0 is equivalent to Da = 0. Since Ω is connected, a is constant, and hence K from (3.41) reduces to so(3) indeed. The following corollary is obtained by scaling the inequalities from Theorem B and can be obtained in the same way as Corollary 3.11 follows from Theorem A: Corollary 3.18 (Scaled inequalities). Let p > 1, j ∈ N 0 , k ∈ N with j < k satisfy (k − j)p < n and let q ∈ (1, np n−(k−j)p ]. Moreover, let the part map A and the k-th order differential operator B as defined in Section 1.2 satisfy (3.34). Then there exists a finite dimensional space K of V -valued polynomials and a constant c = c(j, p, q, A, B) > 0 such that we have min Π∈K r j Br(x0) | D j (P − Π)| q dx 1 q ≤ c r j Br (x0) | D j A[P ]| q dx 1 q + r k Br (x0) |BP | p dx 1 p for all x 0 ∈ R n , r > 0 and P ∈ C ∞ (B r (x 0 ); V ). We conclude this subsection with a result on alternative normalisation conditions. To connect this to the statements alluded to in Section 2.5, see Remark 3.20 below, we here choose a slightly different statement involving negative norms on BP that still can be extracted from the corresponding proofs of Theorems A and B. D j P L q (Ω) ≤ c D j Π ker(A) ⊥ [P ] L q (Ω) + BP W −k+j,q (Ω) (3.49) for all P ∈ C ∞ (Ω; V ) subject to the normalisation condition´Ω D j Π ker(A) [P ]dx = 0. Proof. By (3.48) we deduce that if both Π ker(A) ⊥ [Π] = 0 and BΠ = 0 hold, then Π is a polynomial of a fixed maximal degree. In particular, taking T as in Lemma 3.15(c), BT is C-elliptic. From the proof of Theorem B we then infer that we have min Π∈K D j (P − Π) L q (Ω) ≤ c D j Π ker(A) ⊥ [P ] L q (Ω) + BP W −k+j,q (Ω) (3.50) for all P ∈ C ∞ (Ω; V ), where K is given by (3.41). Now suppose that such a map P satisfieś Ω ∂ β Π ker(A) [P ]dx = 0 for all β ∈ N n 0 with |β| = j. Then we have ∂ β Π ker(A) [P ] L q (Ω) = ∂ β Π ker(A) [P ] − ffl Ω ∂ β Π ker(A) [P ]dx L q (Ω) ≤ c(Ω, q) ∂ β (Π ker(A) [P ] − Π) L q (Ω) for all Π ∈ K, where we have used that ∂ β Π ≡ const for all Π ∈ K. We then conclude ∂ β P L q (Ω) ≤ ∂ β Π ker(A) [P ] L q (Ω) + ∂ β Π ker(A) ⊥ [P ] L q (Ω) ≤ c ∂ β (Π ker(A) [P ] − Π) L q (Ω) + ∂ β Π ker(A) ⊥ [P ] L q (Ω) = c Π ker(A) [∂ β (P − Π)] L q (Ω) + Π ker(A) ⊥ [∂ β (P − Π)] L q (Ω) ≤ c ∂ β (P − Π) L q (Ω) for all Π ∈ K. Summing over all multi-indices β with |β| := j we obtain D j P L q (Ω) ≤ c D j (P − Π) L q (Ω) for all Π ∈ K. (3.51) Thus, taking the minimum over all Π ∈ K we conclude (3.49) in view of (3.50). Remark 3.20. In case (n, q, j, A, B) = (3, 2, 0, sym, Curl) the normalisation condition for gradient fields P = Du reads´Ω skew Du dx = 0 and is equivalent to´Ω curl u dx = 0 the condition which was imposed also originally by KORN [38] so that we recover his statement: Du L 2 (Ω) ≤ c sym Du L 2 (Ω) for all u satisfyingˆΩ curl u dx = 0. For the particular choice (n, q, j, A, B) = (n, 2, 0, dev, Div) we recover the required estimate needed for error analysis in higher order mixed finite element methods for plane elasticity, cf. Moreover, a higher order relation, namely in the case (n, q, j, A, B) = (3, q, 1, dev, sym inc ) is discussed in the examples Section 4.3. 3.4. Implications for function spaces. The inequalities obtained in the previous paragraphs directly translate to embeddings for function spaces. Let 1 < p, q < ∞. Given a part map A and a k-th order differential operator B as in Section 1.2 and ℓ ∈ N 0 with ℓ ≤ k − 1, we define for an open set Ω ⊂ R n the Sobolev-type spaces (Ω) corresponds to certain combinations of partial derivatives vanishing along ∂Ω in a suitable sense. However, applications from elasticity and material science (see e.g. [27,43,54]) sometimes necessitate to incorporate partial boundary conditions for which (3.54) proves insufficient. This can be made precise by virtue of trace operators which, for ease of exposition, shall be executed for first order operators B in the sequel; see Proposition 3.22 below. In this situation, ℓ = 0, and we put W q,B,p (Ω) := W Id,0,q,B,p (Ω) for brevity. We begin with Lemma 3.21. Let B be a first order, linear and homogeneous differential operator on R n from V to W . Moreover, let Ω ⊂ R n be open and bounded with Lipschitz boundary ∂Ω and let q > n n − 1 , p > 1 and 1 p ≤ 1 q + 1 n or 1 < q ≤ n n − 1 and p > 1. for any P ∈ C 1 (Ω; V ). Here, ·, · ∂Ω denotes the dual pairing between W -valued W −1/q,qand W 1/q,q ′ -maps on the boundary, and ν is the outer unit normal to ∂Ω. Proof. We start by recalling that there exists a bounded, linear, surjective trace operator tr ∂Ω : W 1,q ′ (Ω; W ) → W 1−1/q ′ ,q ′ (∂Ω; W ) with a corresponding bounded, linear right-inverse E : W 1−1/q ′ ,q ′ (∂Ω; W ) → W 1,q ′ (Ω; W ). Now consider the map tr B ∂Ω : W q,B,p (Ω) → W −1/q,q (∂Ω; W ) defined by tr B ∂Ω (P ), Q ∂Ω :=ˆΩ BP, EQ W + P, B * (EQ) V dx (3.57) for Q ∈ W 1/q,q ′ (∂Ω; W ). Noting that ((q ′ ) * ) ′ ≤ p by condition (3.55) if q > n n−1 , we have that L p (Ω; W ) ֒→ L ((q ′ ) * ) ′ (Ω; W ) and so, by Sobolev's embedding theorem, | tr B ∂Ω (P ), Q ∂Ω | ≤ BP L p (Ω) EQ L (q ′ ) * (Ω) + P L q (Ω) EP W 1,q ′ (Ω) ≤ c(p, q, Ω) P W q,B,p (Ω) EQ W 1,q ′ (Ω) ≤ c(p, q, Ω) P W q,B,p (Ω) Q W 1/q,q ′ (∂Ω) . The same conclusion remains valid for 1 < q ≤ n n−1 , then invoking the John-Nirenberg theorem since then W 1,q ′ (Ω; W ) ֒→ L p ′ (Ω; W ). This implies that tr B ∂Ω (P ) ∈ W −1/q,q (∂Ω; W ). Now let P ∈ C 1 (Ω; V ). Then the Gauß-Green theorem implies that for all Q ∈ C 1 (Ω; W ) and hence, by density, for all Q ∈ W 1,q ′ (Ω; W ). Since the trace operator tr ∂Ω is surjective as a map W 1,q ′ (Ω; W ) → W 1−1/q ′ ,q ′ (∂Ω; W ), this suffices to conclude (3.56), and the proof is complete. As the reader might notice, this construction is completely analogous to the definition of weak normal or tangential traces for the spaces H 1 (div) or H 1 (curl) (see, e.g. [10,Chapter 2]). Now let Ω be as in the preceding lemma and Γ ⊂ ∂Ω be relatively open. In the situation of the previous lemma, we may then define W q,B,p 0,Γ (Ω) := P ∈ W q,B,p (Ω) | tr B ∂Ω (P )| Γ = 0 ,(3.58) where we say as usual that F ∈ W −1/q,q (∂Ω; W ) ≃ (W 1/q,q ′ (∂Ω; W )) ′ vanishes on Γ, in formulas F | Γ = 0, if F, ϕ W −1/q,q × W 1/q,q ′ = 0 for every ϕ ∈ W 1/q,q ′ (∂Ω; W ) with supp(ϕ) ⊂ Γ. By continuity of tr B ∂Ω : W q,B,p (Ω) → W −1/q,q (∂Ω; V ), cf. Lemma 3.21, we have that W q,B,p 0,Γ (Ω) is a closed subspace of W q,B,p (Ω). (3.59) The best known special instances of such spaces are given by H(Div; Γ ν ; Ω) := u ∈ L 2 (Ω; R 3×3 ) | Div u ∈ L 2 (Ω; R 3 ), tr Div Γν (u)| Γν = 0 , H(Curl; Γ τ ; Ω) := u ∈ L 2 (Ω; R 3×3 ) | Curl u ∈ L 2 (Ω; R 3×3 ), tr Curl Γτ (u)| Γτ = 0 ,(3.60) for relatively open sets Γ ν , Γ τ ⊂ ∂Ω, the indices indicating vanishing of the (weak) normal or tangential traces, respectively. Using the spaces W q,B,p 0,Γ (Ω), we may now formulate the main result of this section: By the different function space setting, the preceding lemma is not directly applicable to our objectives; for this, we record the following consequence of Lemma 3.23: Proof. Let P be an element of the left-hand side of (3.62) and pick a parametrising isomorphism T : R M → ker(A) from Lemma 3.15, so that P = Ta for some polynomial a. Since P belongs to the left-hand side of (3.62), we have BTa = 0 and since B • T is C-elliptic, Lemma 3.1 implies that a, and thus Ta, is a polynomial of a fixed maximal degree. Especially, a and so Ta have classical traces, for which the condition P ∈ W q,B,p 0,Γ (Ω) implies by virtue of (3.56) 0 = B[ν]Ta = n j=1 ν j B j Ta = (B • T)[ν] a H n−1 -a.e. on Γ. Since BT is Cand thus, in particular, R-elliptic, we conclude by ν = 0 H n−1 -a.e. on ∂Ω that a = 0 H n−1 -a.e. on Γ. On the other hand, since P is a polynomial and thus trivially belongs B 1 (0) R n (a) (b) {f = 1} {f = 0} {f = 0} L Ω R n FIGURE 1. The geometric situation of Remark 3.26. to W 1,p (Ω; V ), this implies that Ta ∈ W 1,p 0,Γ (Ω; V ) and so a ∈ W 1,p 0,Γ (Ω; R M ). In consequence, Lemma 3.23 gives us a = 0 and so P = Ta = 0. This completes the proof. We now come to the Proof of Proposition 3.22. We work from (3.40) and subsequently claim that P L q (Ω) ≤ c Π ker(A) ⊥ [P ] L q (Ω) + BP L p (Ω) (3.63) holds for all P ∈ W q,B,p 0,Γ (Ω). Suppose that (3.63) does not hold. Similarly as in the proof of Theorem B, we then find (P i ) ⊂ W q,B,p 0,Γ (Ω) such that P i L q (Ω) = 1, Π ker(A) ⊥ [P i ] L q (Ω) + BP i L p (Ω) < 1 i . (3.64) By routine techniques, it is clear that C ∞ (Ω; V ) is dense in W q,B,p (Ω) and that, as (3.34) is in action, (3.33) also holds for W q,B,p (Ω)-maps. In the present situation, the set K from Theorem B, cf. (3.41), is given by the finite dimensional space K = {Ta | BTa = 0 in D ′ (Ω; W )}, and in conjunction with (3.64) 2 this now implies that for each i ∈ N there exists Π i ∈ K such that P i − Π i L q (Ω) < c i with c > 0 independent of i. Combining this with (3.64) 1 , we infer that (Π i ) ⊂ K is bounded in L q (Ω; V ) and so, recalling that dim K < ∞, there exists Π ∈ K and a (non-relabeled) subsequence such that Π i → Π strongly in L q (Ω; V ). This, in turn, yields that P i → Π strongly in L q (Ω; V ). We now establish that Π = 0, since then P i → 0 strongly in L q (Ω; V ) (for a suitable subsequence) and this clearly contradicts (3.64) 1 . Since BΠ = 0, we conclude P i − Π L q (Ω) + B(P i − Π) L p (Ω) ≤ P i − Π L q (Ω) + BP i L p (Ω) → 0, i → ∞. On the other hand, W q,B,q 0,Γ (Ω) is a closed subspace of W q,B,q 0,Γ (Ω) by (3.59). We hence conclude Π ∈ W q,B,p 0,Γ (Ω) ∩ K = {0} by Corollary 3.24, and the proof is complete. We conclude this section by discussing the assumptions underlying Proposition 3.22. Remark 3.25. We first address the standing of assumption (3.34) for the validity of (3.61). To this end, note that C-ellipticity of A is not necessary for the conclusion of Lemma 3.23. To see this, we identify C ≃ R 2 via ι : R 2 ∋ (x 1 , x 2 ) → x 1 + i x 2 ∈ C. Given an open set Ω ⊂ R 2 , we put dev sym(Du) := sym Du − 1 2 tr(Du)1 2 for u ∈ C 1 (Ω; R 2 ), leading to an operator which fails to be C-elliptic (see, e.g., [8,Ex. 2.2(c)]). We have u = (u 1 , u 2 ) ⊤ ∈ ker dev sym D ⇔ u 1 + i u 2 : ι(Ω) → C is holomorphic (3.65) by virtue of the Cauchy-Riemann equations. Let Γ be an arc in ∂D := {z ∈ C | |z| = 1} and suppose that the holomorphic function u 1 + i u 2 : D → C vanishes identically on Γ. Then the Schwarz reflection principle and the maximum principle imply that u 1 +i u 2 vanishes identically on D. As such, (3.65) implies that the conclusion of Lemma 3.23 persists without C-ellipticity of A. In a different language, this has also been observed in [7,62]. Because of this, it is not clear to us whether assumption (3.34) is necessary for Proposition 3.22 to hold true. Remark 3.26. We only discussed the first order case in Proposition 3.22, as the tools for the higher order case are beyond the scope of this paper. The main reason for this is Lemma 3.23, which does not extend to the higher order scenario even for C-elliptic operators and necessitates higher order conditions for the (normal) traces to be satisfied. This can be seen (a) for the C-elliptic operator A = D 2 (the Hessian) acting on u : R n → R, for which every affine-linear map f : R n → R is in its nullspace. In consequence, f might vanish on an (n − 1)-dimensional hyperplane L. Especially, if a non-empty, relatively open subset Γ of L is contained in ∂Ω, then we find a map f vanishing on Γ but not vanishing identifically in Ω (see Figure 1 (a)). Such a behaviour clearly can be ruled out by posing additional conditions on certain combinations of normal and tangential traces, which shall pursued in a future work. Let us note, though, that there is an interplay with the geometry of ∂Ω, since for instance there are no non-trivial affine linear maps that vanish on relatively open sets of spheres. (b) for the non-C-elliptic operator ∆ (the Laplacian) acting on u : R n → R, for which we may take any continuous f : Figure 1 (b)) and solve the corresponding homogeneous Dirichlet problem with boundary data f through Poisson's formula: ∂B 1 (0) → R with H n−1 ({f = 0}) > 0, H n−1 ({f = 1}) > 0 (seeu(x) = 1 ω n−1ˆ∂B 1(0) 1 − |x| 2 |x − ζ| n f (ζ)dH n−1 (ζ). Clearly, u is non-constant, satisfies u| {f =0} = 0 and ∆u = 0 in B 1 (0). EXAMPLES: OLD AND NEW INEQUALITIES In this concluding section we discuss how specific constellations, among others underlying the models sketched in Section 2, can be retrieved from the general theory outlined above. Moreover, we obtain several new inequalities and non-inequalities that we proceed to outline now. The principal finding in case V = R 3×3 for Curl-based operators are gathered in Figure 2 4.1.2. The first interesting case is A = dev. Then, ker(A) = {α · 1 | α ∈ R}. In this case we consider part maps of Curl(ζ · 1) = − Anti(∇ζ). Since tr Curl(ζ · 1) ≡ 0 the operator B[Curl(· 1)] with B ∈ {Id, dev, skew + tr, skew} behaves like the usual gradient ∇ζ and, thus, is C-elliptic in these cases. On the contrary for B ∈ {sym, dev sym, tr} the corresponding operator B[Curl(· 1)] is not elliptic, since we also have sym Curl(ζ · 1) ≡ 0. We display exemplarily two KMS-type inequalities both of first and second type which hence follow by Proposition 3.22 and Theorem B: P L 3p 3−p dev P L 3p 3−p + dev Curl P L p for all P ∈ W Γ , A B Id dev sym dev sym skew + tr skew tr Id dev sym R- C- dev sym R- C- skew + tr R- C- R- C- skew tr FIGURE 2. Overview when B[Curl]| C ∞ c (R n ;ker(A)) is (C-)elliptic, where means C-ellipticity, denotes non-ellipticity, R-/ Cmeans R-ellipticity but no C-ellipticity. but also min Π∈K1 P − Π L 3p 3−p dev P L 3p 3−p + dev Curl P L p for all P ∈ W, these estimates can also be deduced from any of [31,32,44]. However, we also retrieve the following new inequalities, which, to the best of our knowledge, have not been observed in the literature so far: P L 3p 3−p dev P L 3p 3−p + skew Curl P L p for all P ∈ W Γ , but also min Π∈K2 P − Π L 3p 3−p dev P L 3p 3−p + skew Curl P L p for all P ∈ W. By our theorems we cannot replace skew Curl by sym Curl here, also see Example 1.1. Here we have K 1 = K 2 = {γ · 1 | γ ∈ R} so that the normalised KMS-inequalities hold for j = 0 in these cases, cf. Corollary 3.19, so e.g. we have: P L q (Ω) ≤ c ( dev P L q (Ω) + skew Curl P W −1,q (Ω) ) for all P subject to´Ω tr P dx = 0. We discuss the situation of each part map B separately, whereby we assume that ellipticity properties of gradient based operators are well known: • For B = Id we consider the full operator Curl • Anti above, which is C-elliptic: On the symbol level we have: ξ, a · 1 − ξ ⊗ a ! = 0 tr(·) =⇒ 2 ξ, a = 0 ⇒ ξ, a = 0 ξ =0 =⇒ ξ⊗a=0 a = 0, and we recover the Korn-Maxwell-Sobolev inequalities from (1.3). • For B = dev we obtain −(dev Da) ⊤ = div a 3 − (Da) ⊤ a C-elliptic operator. Indeed, since ξ = 0 there exists an index i such that ξ i = 0. Then considering the i-th column we obtain that a j = 0 for all j = i. Hence, in the (j, j)th entry it remains 1 3 a i ξ i = 0 so that also a i = 0. Thus, we can apply Proposition 3.22 and Theorem B to conclude P L 3p 3−p sym P L 3p 3−p + dev Curl P L p for all P ∈ W Γ , but also min Π∈K3 P − Π L 3p 3−p sym P L 3p 3−p + dev Curl P L p for all P ∈ W, estimates which also follow from each of the papers [31,32,44], whereby for the expression of K 3 we refer the reader to [44,Lemma 11 (b)]. • With B = sym we consider div a · 1 − sym Da a C-elliptic operator (same argument as in the penultimate item) and the discussion here is postponed to the next item. • For B = dev sym we obtain − dev sym Da a C-elliptic operator (we are in three space dimensions), thus with this and the previous item we recover the results from [43]: P L 3p 3−p sym P L 3p 3−p + dev sym Curl P L p for all P ∈ W Γ , but also min Π∈K4 P − Π L 3p 3−p sym P L 3p 3−p + dev sym Curl P L p for all P ∈ W. Since for the latter combination the kernel K 4 consists of special quadratic polynomials, cf. [43, Lemma 2.9], the normalised KMS-inequalities hold for j = 2 here, cf. Corollary 3.19: D 2 P L q (Ω) ≤ c ( D 2 sym P L q (Ω) + dev sym Curl P W 1,q (Ω) ), for all P satisfying´Ω D 2 skew P dx = 0. • The situation changes for B = skew + tr. The corresponding operator here reads 2 div a · 1 + skew Da, which is related to the div + curl-operator, both are R-elliptic but not C-elliptic: For the ellipticity, let v ∈ R 3 , ξ ∈ R 3 \ {0} and consider on the symbol level: 2 v, ξ · 1 + skew(v ⊗ ξ) ! = 0 ⇔ skew(v ⊗ ξ) = 0 and v, ξ = 0 ⇔ v × ξ = 0 and v, ξ = 0. (4.1) Since, the cross-product satisfies the area property we obtain: 0 = |v × ξ| 2 = |v| 2 |ξ| 2 − v, ξ 2 = |v| 2 |ξ| 2 ξ∈R 3 \{0} =⇒ v = 0, meaning that the corresponding operator is (R-)elliptic. This operator is not C-elliptic: 2   1 i 0   ,   − i 1 0   · 1 + skew     1 i 0   ⊗   − i 1 0     = 0. Hence, only the Korn-Maxwell-Sobolev inequality of the first kind (cf. Theorem A) applies for this combination: for all P ∈ W ∂Ω it holds P L 3p 3−p sym P L 3p 3−p + skew Curl P L p + tr Curl P L p . The combination of both Curl-terms is needed on the right hand side, cf. the next items. • For B = skew we obtain skew Da which behaves like curl a and both are not elliptic. The case B = skew is also covered by the above Example 1.2. • Finally, for B = tr it remains only div a which is not elliptic. 4.1.4. To obtain trace-free symmetric Korn-type inequalities we consider A = dev sym. Then ker(A) = {Anti a + α · 1| a ∈ R 3 , α ∈ R} and the corresponding operator has the form Curl [Anti a + ζ · 1] = div a · 1 − (Da) ⊤ − Anti(∇ζ). We distinguish the part maps B: • With B = Id we have the full operator, which is C-elliptic (consider its symmetric and skew-symmetric parts to this end). Hence, we recover from any of [31,32,44]: • For B = dev we obtain −(dev Da) ⊤ − Anti(∇ζ) also a C-elliptic operator (again consider its symmetric and skew-symmetric parts). Thus, it holds P L 3p 3−p dev sym P L 3p 3−p + Curl P L p for all P ∈ W Γ ,P L 3p 3−p dev sym P L 3p 3−p + dev Curl P L p for all P ∈ W Γ , but also min Π∈K6 P − Π L 3p 3−p dev sym P L 3p 3−p + dev Curl P L p for all P ∈ W, whereby both results also follow from any of the three articles [31,32,44] and K 6 consists of special affine linear polynomials, cf. [44,Lemma 11 (c)]. • When B ∈ {sym, dev sym} we obtain div a · 1 − sym Da and − dev sym Da, respectively, which are both not elliptic since they do not see the operation on ζ, so that there are no KMS inequalities for these combinations, see also Example 1.1. • For B = skew + tr we consider the operator 2 div a · 1 + skew Da − Anti(∇ζ) = 2 div a · 1 + Anti curl a 2 − ∇ζ . We show that it is R-elliptic but not C-elliptic. Indeed, considering first the symmetric part on the symbol level we obtain a, ξ = 0. Then, for the skew-symmetric part on the symbol level we have a 2 × ξ + α · ξ ! = 0 ξ =0 ⇐⇒ a,ξ =0 α = 0, a = 0 over R, whereby over C we have with a = (2 + 2 i, 0, 0) ⊤ , ξ = (0, 1, i) ⊤ , α = i −1: 2 a, ξ · 1 − Anti a 2 × ξ + α · ξ = 0. Hence, only Theorem A applies here, i.e. for all P ∈ W ∂Ω the following new estimate holds: P L 3p 3−p dev sym P L 3p 3−p + skew Curl P L p + tr Curl P L p . Note that the combination of both Curl-terms is needed on the right hand side, cf. the next items. • With B = skew we have skew Da − Anti(∇ζ) = Anti curl a 2 − ∇ζ which is not elliptic (set ζ ≡ 0, the non-ellipticity follows from the non-ellipticity of the usual curl). Also our Example 1.2 gives the non-inequality in this case. • For B = tr we obtain 2 div a which is not elliptic. 4.1.5. An intricate constellation appears for A = skew + tr mapping P → skew P + tr P · 1. Its kernel consists of trace-free symmetric matrices. For the corresponding operator we have on the symbol level:   δ α β α ǫ γ β γ −δ − ǫ     0 −ξ 3 ξ 2 ξ 3 0 −ξ 1 −ξ 2 ξ 1 0   =   αξ 3 − βξ 2 βξ 1 − δξ 3 δξ 2 − αξ 1 ǫξ 3 − γξ 2 γξ 1 − αξ 3 αξ 2 − ǫξ 1 γξ 3 + (δ + ǫ)ξ 2 −βξ 3 − (δ + ǫ)ξ 1 βξ 2 − γξ 1   =: P . We show that ellipticity occurs only for B = Id or B = dev: • We start with B = Id and consider P = 0. Subtracting the (1, 3)th from the (3, 1)th entry we obtain 0 = γξ 3 + ǫξ 2 + αξ 1 . Hence, multiplying with ξ 1 , ξ 2 and ξ 3 we have over R: 0 = γξ 1 ξ 3 + ǫξ 1 ξ 2 + αξ 2 1 P 23 =0,P 22=0 = α|ξ| 2 ξ =0 =⇒ α = 0, 0 = γξ 2 ξ 3 + ǫξ 2 2 + αξ 1 ξ 2 P 21 =0,P 23=0 = ǫ|ξ| 2 ξ =0 =⇒ ǫ = 0, 0 = γξ 3 3 + ǫξ 2 ξ 3 + αξ 1 ξ 3 P 21 =0,P 22=0 = γ|ξ| 2 ξ =0 =⇒ γ = 0, and the condition P = 0 becomes   −βξ 2 βξ 1 − δξ 3 δξ 2 0 0 0 δξ 2 −βξ 3 − δξ 1 βξ 2   = 0 ⇔ ξ 1 −ξ 3 ξ 3 ξ 1 β δ = 0, δξ 2 = 0, βξ 2 = 0 which for ξ = 0 only has β = δ = 0 as solution over R, meaning that the corresponding operator is R-elliptic. On the contrary, the induced operator is not C-elliptic:   − i 0 1 0 0 0 1 0 i     0 − i 0 i 0 −1 0 1 0   = 0. • For B = dev note that tr P = 0 so that we can argue as in the last item. Thus, only the Korn-Maxwell-Sobolev inequality of the first kind, Theorem A, holds true here: P L 3p 3−p skew P L 3p 3−p + tr P L 3p 3−p + dev Curl P L p for all P ∈ W ∂Ω , an estimate which also follows from either of [31,32]. This is the best possible result involving the part map A = skew + tr, cf. the following items. 4.2. Div-operator in all dimensions. Let n ≥ 2, k = 1, V = V = R n×n and consider the operator B = Div. We will see that the most interesting part map for this constellation is A = dev. Then, ker(A) = {α · 1 n | α ∈ R} and we have Div(ζ · 1 n ) = ∇ζ which is a C-elliptic operator, so that we even strengthen the result from [7] for 1 < p < n: ) is (C-)elliptic, where means C-ellipticity, denotes non-ellipticity, R-/ Cmeans R-ellipticity but no C-ellipticity. We have K 7 = {γ · 1 n | γ ∈ R} so that with the normalised KMS-inequality in case j = 0, cf. Corollary 3.19, we recover [10, Proposition 9.1.1]: P L q (Ω) ≤ c ( dev P L q (Ω) + Div P W −1,q (Ω) ) for all P satisfying´Ω tr P dx = 0. Considering now A = sym, then ker(A) = so(n). In case n = 2 the corresponding operator is C-elliptic: Div 0 ζ −ζ 0 = ∂ 2 ζ −∂ 1 ζ , but in two space dimensions the divergence is just a rotated curl. For n ≥ 3 the corresponding operator is not elliptic. Indeed, on the symbol level we have  . We will see that the only nontrivial case where ellipticity plays a role is for the part map A = dev. 4.3.1. With A = dev we have ker(A) = {α · 1 | α ∈ R}, so that the corresponding operator becomes inc (ζ · 1) = ∆ζ · 1 3 − D∇ζ ∈ Sym(3). Thus, • For B ∈ {Id, sym} we obtain a C-elliptic operator. Indeed, on the symbol level we consider α ξ, ξ · 1 3 − α ξ ⊗ ξ ! = 0 ξ∈C\{0} =⇒ α = 0. For the corresponding kernel we consider inc (ζ · 1) = ∆ζ · 1 3 − D∇ζ = 0, thus, taking the trace 2∆ζ = 0, so that we have D∇ζ = 0 and we obtain here for the kernel K = {( a, x + α) · 1 3 | a ∈ R 3 , α ∈ R}. Corollary 3.19 is applicable with j = 1: DP L q (Ω) ≤ c ( Ddev P L q (Ω) + sym inc P W −1,q (Ω) ) for all P that satisfy´Ω D(tr P )dx = 0. • With B ∈ {dev, dev sym} we consider also a C-elliptic operator. Indeed, again on the symbol level we conclude 1 3 α ξ, ξ · 1 3 − α ξ ⊗ ξ ! = 0 ξ∈C\{0} =⇒ α = 0. Thus, with 1 < p < 3 the strongest estimates among the previous combinations read: for all P ∈ C ∞ c (R 3 ; R 3×3 ) it holds P Ẇ 1, 3p and to the best of our knowledge, these estimates are new. • If B = skew then skew inc (ζ · 1) ≡ 0 and the operator is not elliptic. 3−p (R 3 ) dev P Ẇ 1, • For B ∈ {skew + tr, tr} the corresponding operator is R-elliptic but not C-elliptic, since on the symbol level we have: 2α ξ, ξ ! = 0, which over R only has the trivial solution, whereby over C we can take ξ = (1, i, 0) ⊤ . Thus, only Theorem A applies here, so that for all 1 < p < 3 and all P ∈ C ∞ c (R 3 ; R 3×3 ) we have In this paragraph we briefly revisit some differential operators and the underlying algebraic identities that have been employed in the main part of the paper. In order to elaborate on potential links to applications, we thus put a special emphasis on the three-dimensional case. P Ẇ 1, 3p 3−p (R 3 ) dev P Ẇ 1, Let us start with the general case n ≥ 2. To define for P : R n → R n×n the matrix curl Curl P , we recall from [42,45] the inductive definition of the generalised cross product × n : R n × R n → R n(n−1) 2 via a × n b := a × n−1 b b n · a − a n · b ∈ R n(n−1) 2 with a 1 a 2 × 2 b 1 b 2 := a 1 b 2 − a 2 b 1 . (A.1) for a = (a, a n ) ⊤ ∈ R n and b = (b, b n ) ⊤ ∈ R n where a, b ∈ R n−1 . Due to the linearity in the second component of the generalised cross product a × n · it can be expressed by a multiplication with a matrix, which we denote a ×n ∈ R n(n−1) 2 ×n , so that a × n b =: a ×n b for all b ∈ R n . (A.2) Thus, for a vector field a : R n → R n or a matrix field P : R n → R r×n , with r ∈ N, we define curl a and row-wise Curl P by curl a := a × n (−∇) = ∇ ×n a, Curl P := P × n (−∇) = P ∇ ⊤ ×n , (A. 3) meaning that the corresponding symbol maps read for ξ ∈ R n : − a × n ξ = ξ ×n a, and − P × n ξ = P ξ ⊤ ×n , (A.4) for a ∈ R n and P ∈ R r×n . Furthermore, for square matrix fields P : R n → R n×n the incompatibility operator is defined by inc P := Curl [Curl P ] ⊤ = ∇ ×n P ⊤ ∇ ⊤ ×n = −∇ × n P ⊤ × n ∇ (A. 5) and is also referred to as Curl Curl ⊤ in the literature; see KRÖNER [39] for the continuum mechanical background. Let us now draw particular attention to the three-dimensional case n = 3. Here, this construction is usually related to the classical cross product, so that for P = a b c ⊤ with a, b, c : R 3 → R 3 , we here denote Curl the row-wise classical curl, so Curl P := curl a curl b curl c ⊤ . However, it follows from our main theorems below, that in three dimensions it does not matter which curl operator (i.e. related to the classical cross product or to the generalised one) we require on the right-hand side, since both matrix curl operators have the same wave cone. For the following, it is useful to define the linear map Anti : R 3 → so(3) via Anti : (α, β, γ) ⊤ →   0 −γ β γ 0 −α −β α 0   . (A.6) This special identification was chosen in such a way that it is related to the usual cross product. Thus, the matrix Curl has as corresponding symbol map that arises as the multiplication with Anti(−ξ) from the right: − P Anti ξ, for ξ ∈ R 3 , P ∈ R 3×3 . (A.7) We then have, for differentiable maps a : R 3 → R 3 , NYE's formulas [57] Curl Anti a = div a · 1 3 − Da ⊤ , Da = tr Curl Anti a 2 · 1 3 − (Curl Anti a) ⊤ . (A. 8) Especially, this implies the identity: dev sym Curl Anti a = − dev sym Da. (1. 5 ) 5⇐⇒ A induces an elliptic operator via Au := A[Du]. Lemma 3. 1 . 1Let A be an operator of the above form (3.1). The following are equivalent: (a) A is C-elliptic. (b) There exists another homogeneous, linear, constant coefficient differential operator L on R n and a number d ∈ N such that D d = L • A. (c) For any open and connected set Ω ⊂ R n we have Example 3 . 2 . 32In the case (A, B) = (sym, Curl) we have ker(A) = so(n) and for all ξ = 0 that ker(B[ξ]) = {a ⊗ ξ| a ∈ R n }. Since there are no non-trivial skew-symmetric rank-1 matrices the condition (3.7) is fulfilled and we recover (1.3) 1 .Remark 3.3. For B = Curl the condition (3 Remark 3 . 5 . 35In many applications, A : V → V is directly given by a (orthogonal) projection. For instance, if we let V = R n×n and let • A[P ] := sym P together with V = Sym(n), or • A[P ] := dev sym P together with V being the trace-free symmetric matrices, then we have A 2 = A in each of the cases. One may then directly work with Π ker(A) ⊥ = A and Π ker(A) = Id − A. Lemma 3.7 ([15, Thm. 5.1], [67, Thm. 2.7.1]). Let 0 < q ≤ ∞. Then the following hold: ξ∈R n \{0} ker(A) ∩ ker(B[ξ]) = {0}.(3.24) Lemma 3.7 (a) to γ := s 1 − k and α := k which, by virtue of Π ker(A) [P ] = T m [BΠ ker(A) [P ]], gives us Π ker(A) [P ] Ḟ s 1 p 1 ,q (R n ) = T m [BΠ ker(A) [P ]] Ḟ s 1 p 1 ,q (R n ) ≤ c BΠ ker(A) [P ] Ḟ s 1 −k p 1 ,q (R n ) . speaking, inequality(3.20) should be referred to as a generalised Korn-Maxwell-Morrey inequality; note that, if we put A ≡ 0 and B := D k , then (3.7) is certainly fulfilled and the resulting inequality is just MORREY's inequality underlying the embeddingẆ 1,p (R n ) ֒→Ċ 0,1−n/p (R n ) for p > n. p ]. Moreover, suppose that the part map A and the k-th order differential operator B are as in Section 1.2. Then the following are equivalent: (a) There exists a finite dimensional subspace K of the V -valued polynomials such that for any open, bounded and connected subset Ω ⊂ R n with Lipschitz boundary ∂Ω there exists a constant c = c(p, q, A, B, Ω) > 0 such that the inequality (b) B is reduced C-elliptic (relative to A), meaning that ξ∈C n \{0} ker C (A) ∩ ker C (B[ξ]) = {0}.(3.34)Here, we have set ker C (A) = {v 1 Remark 3.13. For B = Curl the condition (3.34) is equivalent to A[D·] being a C-elliptic operator. Lemma 3 . 15 . 315Let A and B as in Section 1.2. Then the following are equivalent: (a) Condition (3.34) holds. (b) There exist M ∈ N 0 and an isomorphism T : R M → ker(A) such that the operator C ∞ (R n ; R M ) ∋ a → BTa is C-elliptic. (c) There exists M ∈ N 0 such that, for any isomorphism T : R M → ker(A), the operator C ∞ (R n ; R M ) ∋ a → BTa is C-elliptic. Proof. We first establish '(a)⇒(b)'. Let M := dim(ker(A)) and choose an arbitrary R-linear isomorphism T : R M → ker(A). Let D = BT and suppose that, forv = v 1 + i v 2 ∈ C M and ξ ∈ C n \ {0}, . Then Tv 1 + i Tv 2 ∈ ker C (A) ∩ ker C (B[ξ]), so that Tv 1 + i Tv 2 = 0 by (a). Since T is an isomorphism, we conclude v 1 , v 2 = 0 and hence D is C-elliptic. Ad (b)⇒(c). If T isthe isomorphism provided (b) and T 1 another such isomorphism, we write BT 1 = BT(T −1 T 1 ), from where (c) follows at once. Ad (c)⇒(a). Put M := dim(ker(A)) and suppose that there exists an isomorphism T : R M → ker(A) for which D := BT is not C-elliptic. Then there exists ξ ∈ C n \ {0} and v = v 1 + i v 2 ∈ C M \ {0} for which (3.36) holds. It then suffices to note that 0 = Tv = Tv 1 + i Tv 2 belongs to the union on the left-hand side of (3.34). Thus, (a) follows and the proof is complete. Proof of Theorem B. We begin by establishing that (b) implies (a). Let P ∈ C ∞ (Ω; V ) and consider the decomposition P = Π ker(A) [P ] + Π ker(A) ⊥ [P ]. Now fix a parametrising isomorphism T : R M → ker(A) as in Lemma 3.15, so that we may write P = Π ker(A) ⊥ [P ] + Ta for some suitable a ∈ C ∞ (Ω; R M ). By our assumption (3.34), the operator a → BTa is C-elliptic by Lemma 3.15, whereby Lemma 3.1 (b) implies the existence of a number ℓ ∈ N 0 and a linear, homogeneous, constant coefficient differential operator L of order ℓ such that D k+ℓ (Tϕ) = (LB)(Tϕ) (3.37) define K := {Π ∈ L q (Ω; V ) | Π ker(A) ⊥ [Π] = 0 a.e. and BΠ = 0 in D ′ (Ω; W )} = {Ta ∈ L q (Ω; V ) | BTa = 0 in D ′ (Ω; W )}.(3.41) 3.46b) and the connectedness of Ω, Π ker(A) ⊥ [P ] is a polynomial P of degree j − 1 ≤ k − 2, whereby BP = 0. Since P = P+Π ker(A) [P ], we conclude 0 = BP = BΠ ker(A) [P ] by (3.46c). Therefore, Π ker(A) [P ] ∈ K and so D j Π ker(A) to P − Π P . Then we have Π ker(A) ⊥ [P − Π P ] = Π ker(A) ⊥ [P ], B(P −Π P ) = BP and, by the orthonormality of the e j = D j f j 's, the third term on the right-hand side of (3.44) vanishes. In consequence, the proof of (3.43) is complete. To finally deduce (KMS2) from (3.43), we realise that, under the exponent condition1 < q ≤ np n−(k−j)p , we have L p (Ω; W ) ֒→ W −k+j,q (Ω; W ). This completes the proof of direction '(b)⇒(a)'.We now turn to the direction '(a)⇒(b)', and hence suppose that(3.33) holds. Letting M := dim ker(A), we pick an arbitrary isomorphism T : R M → ker(A). Since Π ker(A) [Ta] = 0, we obtain by applying (KMS2) to P = Ta min Π∈K D j (Ta − Π) L q (Ω) ≤ c BTa L p (Ω) for all a ∈ C ∞ (Ω; R M ).Now suppose that BTa = 0. Then the previous inequality implies that D j Ta and so Ta and hence, writing a = T −1 (Ta), a is a polynomial too. Hence the nullspace of B • T is finite dimensional and consists of polynomials of a fixed maximal degree on R n . Thus, by Lemma 3.1 (c) we conclude the C-ellipticity of B • T, and the proof is complete.Remark 3.16. Let us briefly explain the consistency of the preceding theorem with previous results. To this end, consider the particular inequality Remark 3 . 17 . 317Under the assumptions of Theorem B, the same proof with the obvious modifications yields equivalence of (3.34) to validity of the inequalitymin Π∈K P − Π W j,q (Ω) ≤ c A[P ] W j,q (Ω) + BP L p (Ω) ,(3.47)for all j ∈ {0, . . . , k − 1} and p > 1 with (k − j)p < n and q ∈ (1, np n−(k−j)p ], where K is a suitable finite dimensional nullspace of the V -valued polynomials. Corollary 3 . 319 (Generalised normalised KMS-inequalities). Let 1 < q < ∞, k ∈ N, and the part map A and the k-th order differential operator B be as in Section 1.2. Moreover, let Ω ⊂ R n be an open, bounded and connected subset with Lipschitz boundary ∂Ω. If for a j ∈ N 0 with j ≤ k − 1 we have the conclusion (Π ker(A) ⊥ [Π] = 0 and BΠ = 0) ⇒ D j Π = const., (3.48) then there exists a constant c = c(q, A, B, Ω) > 0 such that we have [ 6 , 6Lemma 3.1] or [10, Proposition 9.1.1]. WIf A,ℓ,q,B,p (Ω) := {P ∈ L 1 loc (Ω; V ) | A[P ] ∈ W ℓ,q (Ω; V ) and BP ∈ L p (Ω; W )}. (3.52) Note thatP W A,ℓ,q,B,p (Ω) := A[P ] W ℓ,q (Ω) + BP L p (Ω) (3.53)defines a seminorm on W A,ℓ,q,B,p (Ω) for which W A,ℓ,q,B,p (Ω) is closed. Let us note that, subject to (3.7), (3.53) is a norm on C ∞ c (Ω; V ) by Corollary 3.11. Hence, if we define W A,ℓ,q,∂Ω is sufficiently regular, it is clear that membership in W A,ℓ,q,B,p 0 ( 3 . 55 ) 355Then there exists a bounded, linear trace operator tr B ∂Ω : W q,B,p (Ω) → W −1/q,q (∂Ω; W ) such that we have tr B ∂Ω (P ), Q ∂Ω =ˆ∂ Ω B[ν]P, Q W dH n−1 (3.56) tr B ∂Ω (P ), tr ∂Ω (Q) ∂Ω =ˆ∂ Ω B[ν]P, tr ∂Ω (Q) W dH n−1 Proposition 3 . 322 (Partially vanishing boundary conditions). Let the part map A and the first order differential operator B be as in Section 1.2. Moreover, let 1 < p < n and 1 < q ≤ p * = np n−p . Then for any connected, open and bounded set Ω with Lipschitz boundary and any relatively open, non-empty subset Γ ⊂ ∂Ω there exists a constant c = c(A, B, Ω, Γ, p, q) > 0 such that we haveP L q (Ω) ≤ c A[P ] L q (Ω) + BP L p (Ω)for all P ∈ W q,B,p 0,Γ (Ω).(3.61) For the proof of Proposition 3.22, we require the following additional ingredient: Lemma 3.23 (POMPE, [61, Thm. 2.4]). Let 1 < p < ∞ and let A be a first order, linear C-elliptic differential operator of the form (3.1) and let Ω ⊂ R n be an open, bounded and connected set with Lipschitz boundary. If Γ is a non-empty, relatively open subset of ∂Ω and u ∈ W 1,q 0,Γ (Ω) := {v ∈ W 1,p (Ω; V ) | tr ∂Ω (v) = 0 H n−1 -a.e. on Γ} is such that Au = 0, then u = 0. Here, tr ∂Ω denotes the trace operator on W 1,p (Ω; V ) as usual. Corollary 3. 24 . 24In the situation of Lemma 3.21, suppose moreover that Ω is connected, A is as in Section 1.2 and that B satisfies (3.34). Then we have {P ∈ W q,B,p 0,Γ (Ω) | A[P ] = 0 and BP = 0} = {0}. (3.62) . 4 . 1 . 41Curl-based operators in three space dimensions. In this section we consider the constellation n = 3, k = 1, V = V = R 3×3 for operators of the form B = B[Curl]. This is not only the most interesting constellation from the point of view of applications but also from an algebraic one, since only in three space dimensions the matrix Curl of a (3 × 3)-matrix returns again a (3 × 3)-matrix. Furthermore, we always have here the restriction 1 < p < 3 and display the results on open, bounded and connected Lipschitz domains Ω ⊂ R 3 . To abbreviate notation in the following, let us set W := W3p 3−p ,B, p (Ω; R 3×3 ) and W Γ :; R 3×3 ) for any relatively open and non-empty Γ ⊂ ∂Ω. Let A, B : R 3×3 → R 3×3 with the possible choices 4 A, B ∈ {Id, dev, sym, dev sym, skew + tr, skew, tr}. Applying Lemma 3.15, we investigate in the (C-)ellipticity of B[Curl] : C ∞ c (R n ; ker(A)) → C ∞ c (R n ; R 3×3 ). 4.1.1. For the sake of completeness we start with the trivial case A = Id. Here, we have ker(A) = 0, so that (3.7) and (3.34) are trivially fulfilled. 4.1.3. The most prominent Korn-type inequalities focus on part maps A = sym. Here, ker(A) = so(3) = {Anti a | a ∈ R 3 }, so that we have to investigate the (C-)ellipticity of Curl Anti a = (div a) · 1 − (Da) ⊤ . + Curl P L p for all P ∈ W, and the elements of K 5 are described in[44, Lemma 11 (a)]. ••• For B = sym or B = dev sym. The corresponding operator is not elliptic. Indeed, on the symbol level we have Also for B = skew or B = skew + tr the induced operator is not elliptic. Indeed, on the symbol level we have Finally, for B = tr the corresponding operator is not elliptic since we always have tr P = 0.4.1.6. For the part map A = skew we have ker(A) = Sym(3) and already in the case B = Id the induced operator is not elliptic since on the symbol level we have: we cannot replace the symmetric part in (1.3) only by the skew-symmetric part.4.1.7. The linear part map A = tr maps P → tr(P ) · 1, so that the kernel consists of trace-free matrices. Again, already in the case B = Id the corresponding operator is not elliptic since on the symbol level we have: FIGURE 3 . 3p + Div P L p for all P ∈ W np n−p ,Div,p 0,Γ (Ω; R n×n ), p + Div P L p for all P ∈ W np n−p ,Div,p (Ω; R n×n ). Overview when B[inc ]| C ∞ c (R n ;ker(A) . inc-based operators in three space dimensions. Finally, let us focus on some higher order operators and for presentation reasons remain in three space dimensions, let n = 3, k = 2, V = V = R 3×3 and consider the inc -based operators B = B[inc ] 3p 3 3p−p (R 3 ) + dev sym inc P L p (R 3 ) ,and for all P ∈ C ∞ (Ω; sym inc P L p (Ω) , • inc P L p (R 3 ) .4.3.2. The corresponding operators in case of the part mapsA ∈ {sym, dev sym, skew + tr, skew, tr} are all non-elliptic:• If A = sym then ker(A) = so(3) = {Anti a | a ∈ R 3 }, and the corresponding operator becomesinc (Anti a) = − Anti(∇ div a),which is not elliptic, since the divergence is already not elliptic.• For A = dev sym we have ker(A) = {Anti a+α·1 3 | (a, α) ⊤ ∈ R 4 } and the operator is inc (Anti a + ζ · 1 3 ) = − Anti(∇ div a) + ∆ζ · 1 3 − D∇ζ,which is not elliptic.• If A = skew + tr then ker(A) consists of trace-free symmetric matrices and already in the case B = Id the induced operator is not elliptic since on the symbol level we have When A ∈ {skew, tr} the non-ellipticity of the induced operator follows with the same examples as in (4.2) and (4.3), respectively. A. APPENDIX: DIFFERENTIAL OPERATORS AND ALGEBRAIC IDENTITIES If A is not injective, take P ∈ ker(A) \ {0} and consider Pϕ := ϕP for ϕ ∈ C ∞ c (R n ) \ {0}. 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{'abstract': "We establish a family of coercive Korn-type inequalities for generalised incompatible fields in the superlinear growth regime under sharp criteria. This extends and unifies several previously known inequalities that are pivotal to the existence theory for a multitude of models in continuum mechanics in an optimal way. Different from our preceding work [31], where we focussed on the case p = 1 and incompatibilities governed by the matrix curl, the case p > 1 considered in the present paper gives us access to substantially stronger results from harmonic analysis but conversely deals with more general incompatibilities. Especially, we obtain sharp generalisations of recently proved inequalities by the last two authors and MÜLLER[43]in the realm of incompatible Korn-type inequalities with conformally invariant dislocation energy. However, being applicable to higher order scenarios as well, our approach equally gives the first and sharp inequalities involving KRÖNER's incompability tensor inc .Date: December 27, 2022. 2020 Mathematics Subject Classification. 35A23, 26D10, 35Q74/35Q75, 46E35.", 'arxivid': '2212.13227', 'author': ['Franz Gmeineder ', 'Peter Lewintan ', 'Patrizio Neff '], 'authoraffiliation': [], 'corpusid': 255125280, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 43756, 'n_tokens_neox': 38502, 'n_words': 21862, 'pdfsha': '5ab4b96ea56fb1affe635eaa155b2634fe11391e', 'pdfurls': ['https://export.arxiv.org/pdf/2212.13227v1.pdf'], 'title': ['KORN-MAXWELL-SOBOLEV INEQUALITIES FOR GENERAL INCOMPATIBILITIES', 'KORN-MAXWELL-SOBOLEV INEQUALITIES FOR GENERAL INCOMPATIBILITIES'], 'venue': []}

arxiv

Prophet Inequality with Correlated Arrival Probabilities, with Application to Two Sided Matchings Van-Anh Truong vatruong@ieor.columbia.edu Department of Industrial Engineering and Operations Research Columbia University New YorkNYUSA Xinshang Wang Department of Industrial Engineering and Operations Research Columbia University New YorkNYUSA Prophet Inequality with Correlated Arrival Probabilities, with Application to Two Sided Matchings The classical Prophet Inequality arises from a fundamental problem in optimal-stopping theory. In this problem, a gambler sees a finite sequence of independent, non-negative random variables. If he stops the sequence at any time, he collects a reward equal to the most recent observation. The Prophet Inequality states that, knowing the distribution of each random variable, the gambler can achieve at least half as much reward in expectation, as a prophet who knows the entire sample path of random variables(Krengel and Sucheston 1978). In this paper, we prove a corresponding bound for correlated non-negative random variables. We analyze two methods for proving the bound, a constructive approach, which produces a worstcase instance, and a reductive approach, which characterizes a certain submartingale arising from the reward process of our online algorithm.We apply this new prophet inequality to the design of algorithms for a class of two-sided bipartite matching problems that underlie online task assignment problems. In these problems, demand units of various types arrive randomly and sequentially over time according to some stochastic process. Tasks, or supply units, arrive according to another stochastic process. Each demand unit must be irrevocably matched to a supply unit or rejected. The match earns a reward that depends on the pair. The objective is to maximize the total expected reward over the planning horizon. The problem arises in mobile crowd-sensing and crowd sourcing contexts, where workers and tasks must be matched by a platform according to various criteria. We derive the first online algorithms with worst-case performance guarantees for our class of two-sided bipartite matching problems. Introduction The classical Prophet Inequality arises from a fundamental problem in optimal-stopping theory. In this problem, a gambler sees a finite sequence of independent, non-negative random variables. If he stops the sequence at any time, he collects a reward equal to the most recent observation. The Prophet Inequality states that, knowing the distribution of each random variable, the gambler can achieve at least half as much reward in expectation, as a prophet who knows the entire sample path of random variables (Krengel and Sucheston 1978). The classical prophet inequality with independent random variables was proved by Krengel and Sucheston (1977). Its importance arises from its role as a primitive in a wide range of decision problems. Since the appearance of the first result, various versions of the prophet inequality has been proved. Hill and Kertz (1983) study the inequality for independent, uniformly bounded random variables. Rinott et al. (1987) prove a version for bounded negatively-dependent random variables. Samuel-Cahn (1991) obtain general results for negatively dependent random variables, and provide some examples for the case of positively dependent variables. In this paper, we study a version of prophet inequality where the sequence of random variables are modeled as a customer arrival process and can be arbitrarily correlated. The specification of our prophet inequality will be made clear in the problem formulation. We apply this new prophet inequality to the design of algorithms for a class of two-sided bipartite matching problems underlying online task assignment problems (OTA). In these problems, demand units of various types arrive randomly and sequentially over time according to some stochastic process. Tasks, or supply units, arrive according to another stochastic process. Each demand unit must be irrevocably matched to a supply unit or rejected. The match earns a reward that depends on the pair. The objective is to maximize the total expected reward over the planning horizon. The problem arises in mobile crowd-sensing and crowd-sourcing contexts, where workers and tasks that arrive randomly overtime and must be matched by a platform according to various criteria. For example, the marketplaces Upwork, Fiverr, and Freelancer match providers with customers of professional services. Walmart evaluated a proposal to source its own customers to deliver orders (Barr and Wohl 2013). The mobile platforms Sensorly, Vericell, VTrack, and PIER outsource the task of collecting analyzing data, called sensing, to millions of mobile users. Enabled by information technology, these crowd-sourcing and crowd-sensing businesses are revolutionalizing the traditional marketplace. For example, Freelancers constitute 35% of the U.S. workforce and have generated a trillion dollars in income as of 2015 (Pofeldt 2016). A survey found that 73% of freelancers have found work more easily because of technology (Pofeldt 2016). The two-sided matching problem that underlies many examples of OTA is very difficult to solve optimally, due to three main reasons. First, given the many characteristics of both demand and supply types and their importance in determining the quality of a match, the decision problem must keep track of a vast amount of information, including the current state of supply and future demand and supply arrivals. Second, both demand and supply processes may change over time, so that the decision-making environment might be constantly changing. Third, demand units tend to be time-sensitive and unmatched supply units might also leave the system after a time, so that they have a finite period of availability. Our contributions in this paper are as follow: • We prove the first prophet inequality for a class of arbitrarily correlated non-negative random variables. We analyze two methods for proving the bound, a constructive approach, which produces a worst-case instance, and a reductive approach, which characterizes a certain submartingale arising from the reward process of our online algorithm. • We formulate a new model of bipartite matching with non-homogenous Poisson arrivals for both demand and supply units. Supply units can wait a deterministic amount of time, whereas demand units must be matched irrevocably upon arrival. Decisions are not batched and must be made for one demand unit at at time. Our model underlies an important class of online task assignment problems for crowd-sourcing and crowd-sensing applications. • We derive the first online algorithms with worst-case performance guarantees for our class of two-sided bipartite matching problems. We prove that our algorithms have expected reward no less than 1 4 times that of an optimal offline policy, which knows all demand and supply arrivals upfront and makes optimal decisions given this information. • We provide numerical experiments showing that despite the conservative provable ratio of 1/4, our online algorithm captures about half of the offline expected reward. We propose improved algorithms that in the experiments capture 65% to 70% of the offline expected reward. Moreover, the improved algorithms outperform the greedy and the bid-price heuristics in all scenarios. These results demonstrate the advantage of using our online algorithms as they have not only optimized performance in the worst-case scenario, but also satisfactory performance in average-case scenarios. Literature Review We review five streams of literature that are most closely related to our problem class. Static matching A variety of matching problems have been studied in static settings, for example, college-admissions problems, marriage problems, and static assignment problems. In these problems, the demand and supply units are known. The reward of matching each demand with each supply unit is also known. The objective is to find a maximum-reward matching. See Abdulkadiroglu and Sönmez (2013) for a recent review. Our setting differs in that demand units arrive randomly over time and decisions must be made before all the units have been fully observed. Dynamic assignment Dynamic-assignment problems are a class of problems in which a set of resources must be dynamically assigned to a stream of tasks that randomly arrive over time. These problems have a long history, beginning with Derman, Lieberman and Ross (1972). See Su and Zenios (2005) for a recent review of this literature. Spivey and Powell (2004) study a version of the dynamic assignment problem in which the resources may arrive randomly over time. They develop approximate-dynamic-programming heuristics for the problem. They do not derive performance bounds for their heuristics. Anderson et al. (2013) study a specialized model in which supply and demand units are identical, and arrivals are stationary over time. They characterize the performance of the greedy policy under various structures for the demand-supply graph, where the objective is to minimize the total waiting time for all supply units. Akbarpour et al. (2014) analyze a dynamic matching problem for which they derive several broad insights. They analyze two algorithms for which they derive bounds on the relative performance under various market conditions. Their work differs from ours in four ways. First, they assume that arrivals are stationary where as we allow non-stationary arrivals. Second, they assume that demand units are identical except for the time of arrivals and supply units are also identical except for the time of arrival, whereas we allow heterogeneity among the units. Finally, they study an unweighted matching problem in which each match earns a unit reward, whereas we study a more general weighted matching problem. Finally, their results hold in asymptotic regimes, where the market is large and the horizon is long, whereas our results hold in any condition. More recently Hu and Zhou (2015) study a dynamic assignment problem for two-sided markets similar to ours. They also allow for random, non-stationary arrivals of demand and supply units. They derive structural results for the optimal policy and asymptotic bounds. We depart from both of the above papers in focusing on providing algorithms with theoretical performance guarantees on all problem instances. Baccara et al. (2015) study a dynamic matching problem in which demand units can wait, and there is a tradeoff between waiting for a higher-quality match, and incurring higher waiting costs. Their setting is limited to just two types of units (demand or supply), whereas we allow arbitrarily many types. They also assume stationary arrivals whereas we allow non-stationary arrivals. Online Matching In online matching, our work fundamentally extends the class of problems that have been widely studied. In existing online matching problems, the set of available supply units is known and corresponds to one set of nodes. Demand units arrive one by one, and correspond to a second set of nodes. As each demand node arises, its adjacency to the resource nodes is revealed. Each edge has an associated weight. The system must match each demand node irrevocably to an adjacent supply node. The goal is to maximize the total weighted or unweighted size of the matching. When demands are chosen by an adversary, the online unweighted bipartite matching problem is originally shown by Karp, Vazirani and Vazirani (1990) to have a worst-case relative reward of 0.5 for deterministic algorithms and 1 − 1/e for randomized algorithms. The weighted this problem cannot be bounded by any constant (Mehta 2012). Many subsequent works have tried to design algorithms with bounded relative reward for this problem under more regulated demand processes. Three types of demand processes have been studied. The first type of demand processes studied is one in which each demand node is independently and identically chosen with replacement from a known set of nodes. Under this assumption, (Jaillet and Lu 2013, Manshadi et al. 2012, Bahmani and Kapralov 2010, Feldman et al. 2009) propose online algorithms with worst-case relative reward higher than 1 − 1/e for the unweighted problem. Haeupler, Mirrokni, Vahab and Zadimoghaddam (2011) study online algorithms with worst-case relative reward higher than 1 − 1/e for the weighted bipartite matching problem. The second type of demand processes studied is one in which the demand nodes are drawn randomly without replacement from an unknown set of nodes. This assumption has been used in the secretary problem (Kleinberg 2005, Babaioff, Immorlica, Kempe, andKleinberg 2008), adwords problem (Goel and Mehta 2008) and bipartite matching problem (Mahdian andYan 2011, Karande, Mehta, andTripathi 2011). A variation to the second type of demand processes studied is one in which each demand node requests a very small amount of resource. This assumption, called the small bid assumption, together with the assumption of randomly drawn demands, lead to polynomial-time approximation schemes (PTAS) for problems such as ad-words (Devanur 2009), stochastic packing (Feldman, Henzinger, Korula, Mirrokni, and Stein 2010), online linear programming (Agrawal, Wang and Ye 2009), and packing problems (Molinaro and Ravi 2013). Typically, the PTAS proposed in these works use dual prices to make allocation decisions. Devanur, Jain, Sivan, and Wilkens (2011) study a resource-allocation problem in which the distribution of nodes is allowed to change over time, but still needs to follow a requirement that the distribution at any moment induce a small enough offline objective value. They then study the asymptotic performance of their algorithm. In our model, the amount capacity requested by each customer is not necessary small relative to the total amount of capacity available. The third type of demand processes studied are independent, non-homogenous Poisson processes. Alaei, Hajiaghayi and Liaghat (2012), Wang, Truong and Bank (2015) and Stein, Truong and Wang (2017) propose online algorithms for online allocation problems. We depart from these papers in two major ways. The algorithms in these papers consist of two main steps. In the first step, they solve a deterministic assignment LP to find the probabilities of routing each demand to each supply unit. Given this routing, in the second step, they make an online decision to determine whether to match a routed demand unit to a supply unit at any given time. In contrast, in the first step, we find conditional probabilities of routing demand to supply units, given the set of supply units that have arrived at any given time. We approximate these conditional probabilities because they are intractable to compute directly. In the second step, after routing demands to supply units according to these conditional probabilities, we design an admission algorithm based on the solution of our prophet inequality that, unlike existing admission techniques, deals with correlated demand arrivals. Online Task Assignment This is a subclass of online matching problems that has seen an explosion of interest in recent years. Almost all of these works model either the tasks or the workers as being fixed. Ho and Vaughan Revenue Management Our work is also related to the revenue management literature. We refer to Talluri and Van Ryzin (2004) for a comprehensive review of this literature. Our work is related to the still limited literature on designing policies for revenue management that are have worst-case performance guarantees. Ball and Queyranne (2009) analyze online algorithms for the single-leg revenue-management problem. Their performance metric compares online algorithms with an optimal offline algorithm under the worst-case instance of demand arrivals. They prove that the competitive ratio cannot be bounded by any constant when there are arbitrarily many customer types. Qin, Zhang, Hua and Shi (2015) study approximation algorithms for an admission control problem for a single resource when customer arrival processes can be correlated over time. They prove a constant approximation ratio for the case of two customer types, and also for the case of multiple customer types with specific restrictions. They allow only one type of resource to be allocated. Gallego, Li, Truong and Wang (2015) study online algorithms for a personalized choice-based revenue-management problem. They allow multiple customer types and products, and non-stationary independent demand arrivals. They allow customers to select from assortments of offered products according to a general choice model. They prove that an LP-based policy earns at least half of the expected revenue of an optimal policy that has full hindsight. Prophet Inequality with Correlated Arrivals Problem Formulation Throughout this paper, we let [k] denote the set {1, 2, . . . , k} for any positive integer k. Consider a finite planning horizon of T periods. There are I customer types and one unit of a single resource that is managed by some platform. In each period t, depending on exogenous state information S t that is observable by time t, a customer of type i will arrive with some probability p it (S t ). Upon an arrival of a customer, the platform can either sell the resource to the customer, or irrevocably reject the customer. The reward earned by the platform for selling the resource to a customer of type i ∈ [I] in period t ∈ [T ] is r it ≥ 0. The goal of the platform is to maximize the expected total reward collected over the planning horizon. Unlike existing research assuming independent or stationary arrival distributions, we allow the sequence of arrival probabilities (p 1t (S t ), p 2t (S t ), . . . , p It (S t )) t=1,...,T to be a correlated stochastic process, which depends on the sample path (S 1 , S 2 , . . . , S T ) that is realized. We assume that we know the joint distribution of this stochastic process of arrival probabilities, that is, the distribution of {S t }, and the distribution of arrivals conditional upon {S t }. As a simple example, S t may represent the weather history at times {1, . . . , t}. Our model would capture any correlation in the weather forecast, and assumes that the customer arrival probabilities p it (S t ) in each period t are determined by the weather history S t up to time t. Let X it ∈ {0, 1} be the random variable indicating whether a customer of type i ∈ [I] will arrive in period t ∈ [T ], in state S t . We must have E[X it |S t ] = p it (S t ). We assume the time increments are sufficiently fine so that, similar to many standard models in revenue management (van Ryzin and Talluri 2005), we can assume that at most one customer arrives in any given period. More precisely, we apply the common practice in revenue management assuming that i∈[I] p it (S t ) ≤ 1 almost surely for all t ∈ [T ], and X it = 1{u t ∈ [ i−1 k=1 p kt (S t ), i k=1 p kt (S t ))},(1) where u t is an independent [0, 1] uniform random variable associated with period t. As a result, we have i∈[I] X it ≤ 1 almost surely for all t ∈ [T ]. In each period t ∈ [T ], events take place in the following order: 1. The platform observes S t , and thus knows p it (S t ) for all i ∈ [I]. 2. The arrivals (X 1t , . . . , X It ) are realized according to (1). 3. The platform decides whether to sell the resource to the arriving customer if any. Let S t denote the support of S t , for all t ∈ [T ]. Without loss of generality and for ease of notation, we define S t as the set of external information in all the periods 1, 2, . . . , t. As a result, given any S t , the path (S 1 , . . . , S t ) is uniquely determined. We call a realization of S T a sample path. We assume that we know the joint distribution of (S 1 , . . . , S T ) in the sense that we are able to simulate the sample paths, and able to estimate the expected values of functions of the sample paths. We can use a tree structure to represent the process (S t ) t∈ [T ] . Let S 0 be a dummy root node of the tree. Every realization of S 1 is a direct descendant of S 0 . Recursively, for any tree node that is a realization of S t , its direct descendants are all the different realizations of S t+1 conditional on the value of S t . Definition of Competitive Ratios We will state the prophet inequality with correlated arrivals by proving the competitive ratio of an algorithm used by the platform. Specifically, define an optimal offline algorithm OFF as an algorithm that knows (X it ) i∈[I];t∈[T ] at the beginning of period 1 and makes optimal decisions to sell the resource given this information. By contrast, the platform can use an online algorithm to make decisions in each period t ∈ [T ] based on only S t and (X it ) i∈[I];t ∈[t] . We use V OFF to denote the reward of OFF, and V ON the reward of an online algorithm ON. Definition 1. An online algorithm ON is c-competitive if E[V ON ] ≥ c E[V OFF ], where the expectation is taken over both S T and the random arrivals (X it ) i∈[I];t∈ [T ] . Offline Algorithm and Its Upper Bound We first show that the expected reward of the offline algorithm can be bounded from above by the expected total reward collected from the entire sample path. Proposition 1. E[V OFF ] ≤ E[ T t=1 I i=1 r it p it (S t )]. Proof. Suppose the resource has infinitely many units, so that the platform sells one unit of the resource to every arriving customer. The resulting expected total reward E[ i∈[I] t∈[T ] r it X it ] = i∈[I] t∈[T ] r it E[X it ] = i∈[I] t∈[T ] r it E[p it (S t )] is clearly an upper bound on E[V OFF ]. Online Algorithm In this section, we propose a simulation-based threshold policy (STP) for the model of prophet inequality with correlated arrivals and prove its performance guarantee. Conditioned on any sample path S T , define T (S T ) := i∈[I] t∈[T ] p it (S t ) as the total expected number of customer arrivals. STP needs to know a uniform upper boundT on T (·) (i.e.,T ≥ T (S T ) with probability one). Given such aT , STP computes the threshold h(S t ) := E T t =t+1 I i=1 r it p it (S t ) 1 +T − t t =1 I i=1 p it (S t ) S t(2) for deciding whether to sell the resource in period t. Specifically, upon an arrival of a type-i customer in period t, if the resource is still available, STP sells the resource to the customer if r it ≥ h(S t ) and rejects the customer otherwise. Note that h(·) can be computed for each scenario S t by simulating the sample paths that potentially arise conditional upon S t . In particular, by Proposition 1 and the definition of h(·), we have (recall that S 0 is a (deterministic) dummy variable) h(S 0 ) = E T t =1 I i=1 r it p it (S t ) 1 +T S 0 = E[ T t =1 I i=1 r it p it (S t )] 1 +T ≥ E[V OFF ] 1 +T .(3) Given that E[V OFF ] can be upper-bounded by h(S 0 ) with a multiplicative factor, the goal of our analysis is to establish a relationship between h(S 0 ) and the expected reward E[V STP ] of our online algorithm. Performance Guarantee In this section, we provide two methods for proving the prophet inequality, i.e., the competitive ratio of STP, under correlated arrival probabilities. The first method is constructive in the sense that it reasons about the structure of a worst-case problem instance, and exhibits this structure explicitly. The second method is deductive in that it proves the existence of the bound without shedding light on the worst-case problem instance. 4.1.1. Constructive method for proving performance bound. With the constructive method, we focus on proving the competitive ratio of STP for the caseT = 1. Since all the decisions made by an online algorithm are based on the realized value of S 1 , we can assume without loss of generality that S 1 is deterministic (i.e., the competitive ratio holds when S 1 is set to be any of its possible realizations). Then we fix the upper bound R := T t=1 I i=1 E[r it p it (S t )|S 1 ](4) on V OFF . We will transform problem data progressively, each time making the expected total reward of STP smaller on this instance. Then we will show that at some point, the expected total reward of STP is easily bounded below by a constant. Conditioned on S t and the event that the resource has not been sold by the beginning of period t, let V STP (S t ) be the expected reward earned from assigning it to a demand unit during periods from t to T . We can express V STP (S t ) explicitly by the following recursion: V STP (S t ) = I i=1 p it (S t )1(r it ≥ h(S t )) r it − E[V STP (S t+1 )|S t ] + E[V STP (S t+1 )|S t ],(5) and V STP (S T +1 ) = 0. We will work with the tree representation of the stochastic process S 1 , S 2 , . . . , S T . We call a node S t in the tree a terminal node if p it (S t ) > 0 for some i but p it (S t ) = 0 for all i = 1, 2, ..., I and all descendants S t of S t . We will arrive at our bound by proving two sets of structural results for the worst-case instance of the problem. The first set of structural results concern the reward process. Lemma 1. Assume that the given problem instance achieves the worst-case ratio V STP /V OFF . Then without loss of generality, 1. T t =1 I i=1 p it (S t ) =T almost surely. 2. The reward is scenario dependent. That is, demand unit (i, t) has reward r it (S t ) at each scenario S t . 3. In each scenario S t , there is at most one customer type with positive arrival probability. We thus use p(S t ) and r(S t ) to denote the arrival probability and the reward of that customer type. 4. r(S t ) = h(S t ) or r(S t ) = (h(S t )) − for all non-terminal nodes S t . Proof. The first property is easy to see, since we can always add nodes with 0 reward and positive arrival probabilities to paths in the tree to ensure that the property holds. This transformation does not change either R or the outcome of STP. By adding demand types if necessary, we can assume without loss of generality that the reward is scenario dependent. That is, demand unit (i, t) has reward r it (S t ) at each scenario S t . We can split up each period into several periods if necessary, such that each scenario S t has at most one arrival with probability p(S t ) and reward r(S t ). This change preserves R and decreases the expected reward for h according to (5), if the rewards are chosen to be increasing with time. In the tree representation, if there is some highest-level non-terminal node S t for which r(S t ) > h(S t ) then conditioned on S t , we can decrease r(S t ) and scale up r(S t ) by some factor for all scenarios S t that descend from S t , using the same factor for all S t , such that the value of T t =t E[r(S t )p(S t )|S t ] is unchanged. As a result, the equality in (4) is maintained. Do this until r(S t ) = h(S t ). We claim that this change decreases V STP (S t ), hence V STP (S 1 ). To see the claim, note that according to (5), the change reduces the immediate reward given S t by some amount ∆ and increases the future reward given S t by no more than ∆. Thus the net effect is to reduce V STP (S t ). Also, since the value of h(S s ) and the rewards stay the same for every node S s preceding S t , V STP (S 1 ), is reduced. In the tree representation, if there is some highest-level non-terminal node S t for which r(S t ) < h(S t ) then conditioned on S t , we can increase r(S t ) by a small amount and scale down r(S t ) by some factor for all scenarios S t descending from S t , using the same factor for all S t , such that the equality in (4) is maintained and all rewards remain non-negative. Do this until r(S t ) = (h(S t )) − , where (h(S t )) − denotes a value infinitessimally smaller than h(S t ). It is easy to see that this change decreases V STP (S t ). Hence V STP (S 1 ) is decreased as we argued just above. Repeat the previous transformations until at all non-terminal nodes S t , we have r(S t ) = h(S t ) or r(S t ) = (h(S t )) − . Lemma 1 implies that T (S T ) =T in the worst-case instance. We will assume for the rest of the subsection that our worst-case data has the structure imposed by Lemma 1. Therefore, for the rest of this subsection, we write the threshold function h in the following alternative way h(S t ) = E T t =t+1 I i=1 r it p it (S t ) 1 + T t =t+1 I i=1 p it (S t ) S t .(6) Our second set of structural results concern the arrival probabilities. Lemma 2. LetT = 1. Assume that the data achieves the worst-case ratio V STP /R. Without loss of generality, for T ≥ 2, the followings hold: 1. There is a unique path S 1 , S 2 , . . . , S T with positive arrival probabilities. 2. At every node S t , t = 1, . . . , T − 1, r(S t ) = h(S T ) = E[r(S T )p(S T )|S t ]; 3. p(S t ) = 0, t = 2, . . . , T ; 4. V STP (S 1 ) ≥ R 2 . Proof. We will prove the theorem by induction on T . First, we prove the result for T = 2. By the tree reward simplifications, r(S 1 ) ≈ h(S 1 ) = u∈S 2 P(S 2 = u|S 1 ) r(u)p(u) 1 + p(u) ≤ u∈S 2 P(S 2 = u|S 1 )r(u)p(u) = E[V STP (S 2 )|S 1 ]. Therefore r(S 1 ) = h(S 1 ) to make V STP (S 1 ) as small as possible. Define c 2 = u∈S 2 P[S 2 = u]p(u), R 2 = u∈S 2 P[S 2 = u]p(u)r(u), c = c 2 + p(S 1 ). Notice that by definition of R we have R = R 2 + p(S 1 )r(S 1 ). Fix R, R 2 , and c. Consider what happens when we scale down c 2 by a factor α, scale up r(u) for leaf nodes u to maintain R 2 constant, and scale down r(S 1 ) to maintain p(S 1 )r(S 1 ) = R − R 2 constant. We argue that we will reduce V STP (S 1 ) while keeping R constant. Indeed, for α = 1, p(S 1 ) =T − αc 2 . This implies that V STP (S 1 ) = p(S 1 )r(S 1 ) + (1 − p(S 1 )) u P[S 2 = u]p(u)r(u), = R − R 2 + (1 −T + αc 2 )R 2 . This is where we make p(S 1 )r(S 1 ) = R − R 2 . Hence, ∂V STP (S 1 ) ∂α = c 2 R 2 ≥ 0. Therefore, V STP /R is minimized when α = 0, or p(S 2 ) = 0 for all S 2 ∈ S 2 . Therefore, the base case is proved. Assume the theorem holds for T − 1. Fix c + = T t=2 P(S t |S 1 )p(S t ) and R + = T t=2 P(S t |S 1 )p(S t )r(S t ). Let the immediate successors of S 1 be S k 2 , k = 1, . . . , K. Let R k = E[ T t=2 r(S t )p(S t )|S k 2 ], k = 1, . . . , K. Since the instance that minimizes V STP (S 1 ) must minimize E[V ( S 2 )|S 1 ] subject to c + and R + , we have by the induction hypothesis, that E[V STP (S 2 )|S 1 ] ≥ K 1 P(S k 2 ) R k 2 = R + 2 . This lower bound is attained when K = 1, T t=2 P(S t |S K 2 )p(S t ) = 1, p(S t ) = 0 for all t = 3, . . . , T , and P(S K 2 |S 1 ) = c − p(S 1 ). By the induction hypothesis, V STP (S 2 ) = E[p(S T )r(S T )|S 2 ] = h(S 1 ). We also know that r(S 1 ) ≈ h(S 1 ) by the tree reward simplifications, we conclude that r(S 1 ) = h(S 1 ), since the impact on V STP (S 1 ) is the same in either case. Thus, V STP (S 1 ) = E[p(S T )r(S T )|S 1 ]. We know that R = p(S T )r(S T )(P(S T |S 1 ) + T −1 s=1 P(S s |S 1 )P(S T |S s )p(S s )) = p(S T )r(S T )(P(S T |S 1 ) + P(S K 2 |S 1 )P(S T |S K 2 )p(S K 2 ) + P(S T |S 1 )p(S 1 )) = p(S T )r(S T )P(S T |S 1 )(1 + p(S K 2 ) + p(S 1 )). This implies that V STP (S 1 ) = p(S T )r(S T )P(S T |S 1 ) = R 1 + p(S K 2 ) + p(S 1 ) ≥ R 2 , with the lower bound being realizable when p(S K 2 ) = 0 and p(S 1 ) = 1. By induction, the lemma holds for all T . Lemmas 1 and 2 and Proposition 1 combine to give us the competitive ratio of STP directly: Theorem 1. ForT = 1, we have V STP (S 1 ) ≥ R 2 = 1 2 T t=1 I i=1 E[r it p it (S t )|S 1 ] ≥ 1 2 E[V OFF ].(7) 4.1.2. Reductive martingale method for proving performance bound. In this section, we provide an alternative, reductive proof for the performance bound of STP that works for a more general case, whenT > 0. Define τ ∈ [T + 1] as the random period in which the resource is sold to a customer under STP. If the resource is not sold at the end of the last period T , we set τ = T + 1. In this way, τ is a stopping time bounded from above by T + 1. We define p i,T +1 (S T +1 ) = 0 for all i ∈ [I]. Define a stochastic process {Z(S t )} t=0,1,...,T +1 as Z(S t ) = h(S t ) + t t =1 I i=1 p it (S t )(r it − h(S t )) + . Proposition 2. E[V STP ] = E[Z(S τ )]. Proof. Recall that in any period t ∈ [T ], at most one customer can arrive, i.e., I i=1 X it ≤ 1 with probability one. For any t ∈ [T ], conditioned on τ = t, i.e., STP sells the resource in period t, the following two conditions must hold: 1. Exactly one customer arrives in period t, i.e., I i=1 X it = 1. 2. The type i of the customer who arrives in period t must satisfy the threshold condition r it ≥ h(S t ) (so that STP sells the resource), or more precisely, I i=1 X it (r it − h(S t )) ≥ 0. Altogether, using the fact that X it 's are indicators, we can obtain I i=1 X it (r it − h(S t )) = I i=1 X it (r it − h(S t )) + .(8) The expected reward of STP is E[V STP ] =E T t=1 I i=1 X it r it 1(τ = t) =E T t=1 I i=1 X it (r it − h(S t )) + I i=1 X it h(S t ) 1(τ = t) x =E T t=1 I i=1 X it (r it − h(S t )) + h(S t ) 1(τ = t) y =E T t=1 I i=1 X it (r it − h(S t )) + + h(S t ) 1(τ = t) =E T t=1 I i=1 X it (r it − h(S t )) + 1(τ = t) + E T t=1 h(S t )1(τ = t) z =E T t=1 I i=1 X it (r it − h(S t )) + 1(τ = t) + E[h(S τ )]. Above, x is because i∈[I] X it = 1 conditioned on τ = t ∈ [T ]; y is by equation (8); z is because the definition of h(·) naturally gives h(S T +1 ) = 0. If τ > t, i.e., STP does not sell the resource in period 1, 2, . . . , t, then any customer who arrives in period t must not satisfy the threshold condition. Thus, conditioned on τ > t, we must have I i=1 X it (r it − h(S t )) + = 0. Consequently, I i=1 X it (r it − h(S t )) + 1(τ > t) = 0 =⇒ I i=1 X it (r it − h(S t )) + 1(τ = t) = I i=1 X it (r it − h(S t )) + 1(τ ≥ t). Then the expected reward can be further written as E[V STP ] =E T t=1 I i=1 X it (r it − h(S t )) + 1(τ = t) + E[h(S τ )] =E T t=1 I i=1 X it (r it − h(S t )) + 1(τ ≥ t) + E[h(S τ )] =E T t=1 E I i=1 X it (r it − h(S t )) + 1(τ ≥ t) S t , {X i t } i =1,2,...,I;t =1,2,...,t−1 + E[h(S τ )] =E T t=1 E I i=1 X it (r it − h(S t )) + S t , {X i t } i =1,2,...,I;t =1,2,...,t−1 1(τ ≥ t) + E[h(S τ )] (the event τ ≥ t depends only on the information from periods 1 to t − 1) =E T t=1 E I i=1 X it (r it − h(S t )) + |S t 1(τ ≥ t) + E[h(S τ )]. Finally, we use p it (S t ) = E[X it |S t ] and p i,T +1 (S T +1 ) = 0 to obtain E[V STP ] =E T t=1 E I i=1 X it (r it − h(S t )) + |S t 1(τ ≥ t) + E[h(S τ )] =E T t=1 I i=1 p it (S t )(r it − h(S t )) + 1(τ ≥ t) + E[h(S τ )] =E τ t=1 I i=1 p it (S t )(r it − h(S t )) + + E[h(S τ )] =E[Z(S τ )]. Lemma 3. For any b ≥ 1, a ≥ 0, r 1 , ..., r n ≥ 0 and p 1 , ..., p n ≥ 0, a + n i=1 p i r i b + n i=1 p i ≤ a b + n i=1 p i (r i − a b ) + . Proof. Let I ⊆ {1, 2, ..., n} be the set such that r i ≥ a/b for all i ∈ I. a b + n i=1 p i (r i − a b ) + = a b + i∈I p i (r i − a b ) = a b + i∈I p i (r i − a b ) (b + i∈I p i ) b + i∈I p i = a + i∈I p i r i + (b − 1 + i∈I p i )( i∈I p i (r i − a/b)) b + i∈I p i ≥ a + i∈I p i r i b + i∈I p i ≥ a + n i=1 p i r i b + n i=1 p i . The last inequality follows from the fact that for any j ∈ I, r j < a b ≤ a + i∈I p i r i b + i∈I p i . Proposition 3. The process {Z(S t )} t≥0 is a sub-martingale with respect to S t . Proof. For any t ≥ 1, by definition of Z(S t ) and Z(S t−1 ), we can obtain E[Z(S t )|S t−1 ] =E[h(S t ) + t t =1 I i=1 p it (S t )(r it − h(S t )) + |S t−1 ] =E[h(S t ) − h(S t−1 ) + I i=1 p it (S t )(r it − h(S t )) + |S t−1 ] + Z(S t−1 ). It suffices to prove that in expectation, h(S t−1 ) ≤ h(S t ) + I i=1 p it (S t )(r it − h(S t )) + . We can derive h(S t−1 ) =E T t =t I i=1 r it p it (S t ) 1 +T − t−1 t =1 I i=1 p it (S t ) S t−1 =E T t =t+1 I i=1 r it p it (S t ) + I i=1 r it p it (S t ) 1 +T − t t =1 I i=1 p it (S t ) + I i=1 p it (S t ) S t−1 =E E T t =t+1 I i=1 r it p it (S t ) + I i=1 r it p it (S t ) 1 +T − t t =1 I i=1 p it (S t ) + I i=1 p it (S t ) S t S t−1 =E   E T t =t+1 I i=1 r it p it (S t )|S t + I i=1 r it p it (S t ) 1 +T − t t =1 I i=1 p it (S t ) + I i=1 p it (S t ) S t−1   ≤E    E T t =t+1 I i=1 r it p it (S t )|S t 1 +T − t t =1 I i=1 p it (S t ) + I i=1 p it (S t )   r it − E T t =t+1 I i=1 r it p it (S t )|S t 1 +T − t t =1 I i=1 p it (S t )   + S t−1    =E   E T t =t+1 I i=1 r it p it (S t ) 1 +T − t t =1 I i=1 p it (S t ) |S t + I i=1 p it (S t ) r it − E T t =t+1 I i=1 r it p it (S t ) 1 +T − t t =1 I i=1 p it (S t ) |S t + S t−1   =E h(S t ) + I i=1 p it (S t ) (r it − h(S t )) + S t−1 , where the inequality follows from Lemma 3 and the fact thatT is an upper bound on the expected total number of arrivals on any sample path: T ≥ t t =1 I i=1 p it (S t ) =⇒ 1 +T − t t =1 I i=1 p it (S t ) ≥ 1. With Propositions 2 and 3 established, we apply the optional stopping theorem, to obtain our main result, namely the prophet inequality under correlated arrival probabilities: Theorem 2. E[V STP ] ≥ E[h(S 0 )] = E[ T t=1 I i=1 r it p it (S t )] 1 +T ≥ E[V OFF ] 1 +T . Proof. E[V STP ] = E[Z(S τ )] ≥ E[Z(S 0 )] = E[h(S 0 )] = E[ T t=1 I i=1 r it p it (S t )] 1 +T ≥ E[V OFF ] 1 +T , where the first inequality is by the optimal stopping theorem for sub-martingales, and the last equality is given by (3). Application to Two-sided Matching Problems In this section, we describe how our results can be applied to design algorithms for a basic matching problem in two-sided markets. Model Again consider a finite planning horizon of T periods. There are I types of demand units and J We assume that supply units can wait but demand units cannot. At the end of each period t, after arrivals of demand and supply units have been observed, the demand unit that arrives in period t, if any, must be matched immediately to an existing supply unit or rejected. Note that if a supply unit (j, s) can only wait a finite amount of time, then we can require that r ijts = 0 for any t that is sufficiently large compared to s. Thus, regardless of the availability of supply units, the demand arrival processes are not correlated a priori. However, for each particular supply unit, the best demand unit that should be assigned to it must depend on the availability of other supply units. When there are many supply units, some of them might not even be assigned to any demand unit. By contrast, when there are very few supply units, each of them can be matched to some demand unit. Let Λ it ∈ {0, We also assume that the time increments are sufficiently fine, so that at most one demand unit and one supply unit arrive in any period t ∈ [T ]. More precisely, we require i∈[I] λ it ≤ 1 and j∈[J] µ js ≤ 1 for all t, s ∈ [T ]. Also similar to (1), the arrival events can be defined as Λ it = 1{u t ∈ [ i−1 k=1 λ kt , i k=1 λ kt )}, M js = 1{v s ∈ [ j−1 k=1 µ ks , j k=1 µ ks )},(9) where u 1 , . . . , u T and v 1 , . . . , v T are mutually independent [0, 1] uniform random variables. In any period t, the platform first observes Λ 1t , . . . , Λ It and M 1t , . . . , M Jt . Then, if there is any arriving demand unit in period t, the platform uses an online algorithm to make an assignment decision. In other words, a demand unit can be matched to any supply unit arriving in the same period or earlier. The objective of the problem is to match demand and supply units in an online manner to maximize the expected total reward earned over the horizon. We do not allow fractional matchings. That is, each demand unit must be matched in whole to a supply unit. Offline Algorithm and Its Upper Bound In the above LP, the variable x ijts (Λ, M ) encapsulates the probability that both demand unit (i, t) and supply unit (j, s) arrive and (i, t) is assigned to (j, s). The fourth constraint requires that a demand unit in period t cannot be matched to any supply unit arriving later than t. The competitive ratio c of an online algorithm ON is similarly defined as c = E[V ON ]/E[V OFF (Λ, M )], where V ON is the total reward of the online algorithm, and the expectation is taken over (Λ, M ). Note that (10) cannot be solved without a priori access to the realizations of (Λ, M ). Thus, we are interested in finding an upper bound on the expected optimal offline reward E[V OFF (Λ, M )] when we do not have such a priori access. The following LP solves for the total probabilities x ijts of having demand unit (i, t) arrived and assigned to (j, s). max x ijts i,j,t,s x ijts r ijts s.t. i,t x ijts ≤ µ js , ∀j ∈ [J]; s ∈ [T ], j,s x ijts ≤ λ it , ∀i ∈ [I]; t ∈ [T ], x ijts ≤ λ it µ js , ∀i ∈ [I]; j ∈ [J]; t, s ∈ [T ], x ijts = 0, ∀i ∈ [I]; j ∈ [J]; t ∈ [T ]; s = t + 1, . . . , T, x ijts ≥ 0, ∀i ∈ [I]; j ∈ [J]; t, s ∈ [T ].(11) The constraints above are derived from those of (10). Theorem 3. The optimal objective value of (11) is an upper bound on E[V OFF (Λ, M )]. Proof. Let x * ijts (Λ, M ) be an optimal solution to LP (10). Define a solution to LP (11) as x ijts := E[x * ijts (Λ, M )]. Since i,t x * ijts (Λ, M ) ≤ M js , j,s x * ijts (Λ, M ) ≤ Λ it , and x * ijts (Λ, M ) ≤ Λ it M js are required in (10), we must have i,tx ijts = i,t E[x * ijts (Λ, M )] ≤ E[M js ] = µ js , j,sx ijts = j,s E[x * ijts (Λ, M )] ≤ E[Λ it ] = λ it , x ijts = E[x * ijts (Λ, M )] ≤ E[Λ it M js ] = λ it µ js . Also, x * ijts (Λ, M ) = 0 for all s > t impliesx ijts = 0 for all s > t. Thus,x ijts is a feasible solution to LP (11). It follows that the optimal value of LP (11) is an upper bound on i,j,t,sx ijts r ijts = i,j,t,s E[x * ijts (Λ, M )]r ijts = E[V OFF (Λ, M )]. Online Algorithm for Two-Sided Matching In this section, we describe and analyze a matching algorithm. The algorithm is composed of two simpler sub-routines, a Separation Subroutine and an Admission Subroutine. The Separation Subroutine randomly samples a supply unit for each incoming demand unit. This sampling splits the demand arrivals into separate arrival streams, each coming to a separate supply unit. Subsequently, for each supply unit independently, the Admission Subroutine uses the algorithm in Section 4 to control the matching of at most one among all incoming demand units to it. Let S t := {0, 1} J×t . Define S t ∈ S t as the information set that records the arrivals of supply units up to period t. That is, For convenience, let S 0 be a dummy constant. In our analysis, S T = M is the sample path of scenarios defined in Section 3.1. S t = ({M j1 } j=1 The matching algorithm first needs to compute an optimal solution x * to LP (11). Then, the Separation Subroutine calculates a probability p ijts (S t ) := min(λ it , J j =1 t s =1 M j s x * ij ts µ j s ) J j =1 t s =1 M j s x * ij ts µ j s · M js x * ijts µ js(12) of choosing (j, s) as a candidate supply unit to be matched to (i, t). Note that if s > t, then we must have x * ijts = 0 (see LP (11)) and thus p ijts (S t ) = 0. That is, our algorithm never tries to match a demand unit in period t to a supply unit arriving later than t. When applying the prophet inequality theory developed in previous sections, we will fix a supply unit (j, s), and think of (p ijts (S t )) i,t as the probabilities that demand units "arrive" at (j, s). We first establish some important properties regarding the arrival probabilities p ijts (·). Proposition 4. 1. I i=1 T t=1 p ijts (S t ) ≤ 1, for all j ∈ [J] and s ∈ [T ]. 2. J j=1 t s=1 p ijts (S t ) ≤ λ it , for all i ∈ [I] and t ∈ [T ]. Proof. I i=1 T t=1 p ijts (S t ) = I i=1 T t=1 min(λ it , J j =1 t s =1 M j s x * ij ts µ j s ) J j =1 t s =1 M j s x * ij ts µ j s · M js x * ijts µ js ≤ M js I i=1 T t=1 x * ijts µ js ≤ M js ≤ 1, where the second inequality is given by the first constraint of LP (11). We can then derive J j=1 t s=1 p ijts (S t ) = J j=1 t s=1 min(λ it , J j =1 t s =1 M j s x * ij ts µ j s ) J j =1 t s =1 M j s x * ij ts µ j s · M js x * ijts µ js = min(λ it , J j =1 t s =1 M j s x * ij ts µ j s ) J j =1 t s =1 M j s x * ij ts µ j s · J j=1 t s=1 M js x * ijts µ js = min(λ it , J j =1 t s =1 M j s x * ij ts µ j s ) ≤λ it . Online Matching Algorithm: • (Initialization) Solve (11) for an optimal solution x * . • Upon an arrival of a demand unit (i, t) in period t: 1. (Separation Subroutine) Randomly pick a supply unit (j, s) with probability p ijts (S t )/λ it , for all j ∈ [J], s ∈ [t] (recall that we assume λ it to be strictly positive). Notice that by definition of p ijts (·), only those supply units that have arrived (i.e., satisfy M js = 1 and s ≤ t) can have a positive probability to be picked. Also, since Proposition 4 gives j,s p ijts (S t )/λ it ≤ 1, if the inequality is strict (i.e., j,s p ijts (S t )/λ it < 1), then it is possible that no supply unit is picked. In such a case, reject the demand unit directly. 2. (Admission Subroutine) Let X ijts be the indicator of whether demand unit (i, t) arrives and the Separation Routine picks supply unit (j, s). We have E[X ijts |S t ] = P(Λ it = 1) · p ijts (S t )/λ it = λ it · p ijts (S t )/λ it = p ijts (S t ). For the supply unit (j, s) picked by the Separation Subroutine (i.e., X ijts = 1), we apply algorithm STP by viewing (p ijts (S t )) i,t as the sequence of correlated arrival probabilities. Specifically, match (i, t) to (j, s) if (j, s) is still available and r ijts ≥ h js (S t ), where h js (S t ) := E T t =t+1 I i=1 r ijt s p ijt s (S t ) 2 − t t =1 I i=1 p ijt s (S t ) S t . Notice that we have chosenT = 1 (see (2)) for this two-sided online matching problem. This is because the first property of Proposition 4 guarantees I i=1 T t=1 p ijts (S t ) ≤ 1 =T . Performance of the Online Algorithm We first establish an approximation bound for the Separation Subroutine, which relates the arrival probabilities p ijts (·) to the LP upper bound (11). We start with a technical lemma. Lemma 4. For any λ > 0 and x > 0, min(λ, x) x ≥ 1 − 1 4λ x. Proof. For x ≤ λ, min(λ, x)/x = 1 ≥ 1 − 1 4λ x. For x > λ, min(λ, x) x − (1 − 1 4λ x) = λ x − 1 + 1 4λ x = λ x + 1 4 · x λ − 1 ≥2 · λ x · 1 2 x λ − 1 ≥0. Using the above lemma and the definition of p ijts (·), we are ready to prove the proximity of p ijts (·) relative to x * . Theorem 4. E[p ijts (S t )] ≥ 0.5x * ijts . Proof. Fix any i, j, t, s, but M (and thus S t ) is random. For any supply unit (j , s ), define Y j s ≡ M j s · x * ij ts /µ j s . Note that E[Y j s ] = x * ij ts /µ j s · P(M j s = 1) = x * ij ts /µ j s · µ j s = x * ij ts . We can then deduce E[p ijts (S t )] =E[ min(λ it , j ,s Y j s ) j ,s Y j s · Y js ] ≥E[(1 − 1 4λ it j ,s Y j s ) · Y js ] (by Lemma 4; if j ,s Y j s = 0, we have Y js = 0 so the inequality still holds) =E[1 − 1 4λ it j =j,s =s Y j s ]E[Y js ] − 1 4λ it E[Y 2 js ] =(1 − 1 4λ it j =j,s =s x * ij ts )x * ijts − 1 4λ it x * ijts · x * ijts /µ js ≥(1 − 1 4 )x * ijts − 1 4 x * ijts (because the constraints of LP (11) requires j s x * ij ts ≤ λ it and x * ijts ≤ µ js λ it ) = 1 2 x * ijts . Now, we tie together the above approximation bound with the prophet inequality established in Section 4: Theorem 5. The total reward V ON of our matching algorithm satisfies E[V ON ] ≥ 1 4 E[V OFF (Λ, M )]. Proof. Fix any (j, s) for j ∈ [J] and s ∈ [T ]. A demand unit (i, t) is matched to (j, s) if and only if X ijts = 1 and r ijts ≥ h js (S t ). This is exactly the single-resource problem presented in Section 3.1. Therefore, by Theorem 2, the expected total reward earned from (j, s) is at least (recall that we chooseT = 1 for this two-sided online matching problem) E[h js (S 0 )] = E[ T t=1 I i=1 r ijts p ijts (S t )] 2 . We then use Theorem 4 to obtain E[h js (S 0 )] = E[ T t=1 I i=1 r ijts p ijts (S t )] 2 = T t=1 I i=1 r ijts E[p ijts (S t )] 2 ≥ T t=1 I i=1 r ijts x * ijts 4 . Consequently, the total expected reward summed over all supply units is at least j∈[J] s∈[T ] E[h js (S 0 )] ≥ j∈[J] s∈[T ] t∈[T ] i∈[I] r ijts x * ijts 4 ≥ 1 4 E[V OFF (Λ, M )], where the final inequality follows from Theorem 3. Note that the ratio of 1/4 above results from a loss of a factor of 1/2 from the solution of Prophet Inequalities, and another factor of 1/2 from the tractable approximation to the upperbound deterministic LP. Since 1/2 is an upper bound on the competitive ratio for the standard prophet inequality (which is a special case of our prophet inequality with correlated arrival probabilities), any improvement to the bound of the online algorithm must come from refining the solution to the LP. Numerical Studies In this section, we conduct numerical experiments to explore the performance of our algorithms. We model our experiments on applications that match employers with freelancers for short-term projects, such as web design, art painting, and data entry. We assume there are 30 employer (demand) types and 30 worker (supply) types. We set the reward r ijts according to the formula r ijts = s ij · f ts · g ij , where we use s ij , f ts and g ij to capture three different aspects of a matching: • Ability to accomplish tasks. s ij represents the ability of workers of type j to work for employers of type i. We randomly draw s ij from a normal distribution N (0, 1) for each pair (i, j). In particular, if s ij < 0, the reward of the matching will be negative, and thus no algorithm will ever match worker type j to employer type i. • Idle time of workers. It may be wise to limit the total time that a worker is idle in the system before being assigned a job. Thus, we set f ts = 1 − α + αe −(t−s)/τ so that the reward of a matching is discounted by α when the idle time of the worker exceeds τ . • Geographical distance. Certain freelance jobs may require a short commute distance between workers and employers. For demonstration purpose, we assume that each worker type and employer type is associated with a random zip code in Manhattan, with probability proportional to the total population in the zip code zone. Let d(i, j) be the Manhattan distance between the centers of zip code zones of worker type j and employer type i. We assume that g ij = 1 − β + βe −d(i,j)/ω , so the reward is discounted by β when the commute distance exceeds ω. We consider a horizon of 60 periods. Depending on the application, one period may correspond to a day or a 10-minute span. In any period, a random number of workers may sign in to be ready to provide service. The type of a worker is uniformly drawn from all the worker types. Let µ(t) be the rate at which workers appear in the system. Similarly, when an employer arrives, the type of the employer is uniformly drawn from all the employer types. Let λ(t) be the arrival rate of employers. We randomly generate multiple test scenarios. In each scenario, we independently draw µ(t) and λ(t), for every period t, from a uniform distribution over [0,1]. Given the rates µ(t) and λ(t), we further vary other model parameters by first choosing the base case to be α = 0.5, τ = 10 (periods), β = 0.5, ω = 0.05 • , and then each time varying one of these parameters. We test the following algorithms: • (ON) Our online algorithm without resource sharing. • (ON 1 + ) A variant of our online algorithm with resource sharing. When ON 1 + rejects a customer in the admission subroutine, ON 1 + offers another resource with the largest non-negative margin E T t =t+1 I i=1 r it p it (S t ) 2 − t t =1 I i=1 p it (S t ) S t − r ijts . • (ON 2 + ) A variant of our online algorithm with resource sharing. When ON 2 + rejects a customer in the admission subroutine, ON 2 + offers another resource with the largest non-negative margin 130% × E T t =t+1 I i=1 r it p it (S t ) 2 − t t =1 I i=1 p it (S t ) S t − r ijts . • (ON 3 + ) A variant of our online algorithm with resource sharing. When ON 3 + rejects a customer in the admission subroutine, ON 3 + offers another resource with the largest non-negative margin 160% × E T t =t+1 I i=1 r it p it (S t ) 2 − t t =1 I i=1 p it (S t ) S t − r ijts . • (ON 4 + ) A variant of our online algorithm with resource sharing. When ON 4 + rejects a customer in the admission subroutine, ON 4 + offers another resource with the largest non-negative margin 200% × E T t =t+1 I i=1 r it p it (S t ) 2 − t t =1 I i=1 p it (S t ) S t − r ijts . • A greedy algorithm that always offers a resource with the highest reward. • A bid-price heuristic based on the optimal dual prices of LP (10) We report numerical results in Tables 1 to 4, where the performance of each algorithm is simulated using 1000 replicates. For our online algorithms, in each period we compute the threshold h js (S t ) by simulating 100 future sample paths. We find that, despite the 1/4 provable ratio, the algorithm ON captures about half of the offline expected reward, and the improved algorithms ON 1 + and ON 2 + capture 65% to 70% of the offline expected reward. Moreover, the improved algorithms outperform the greedy and the bid-price heuristics in all scenarios. These results demonstrate the advantage of using our online algorithms as they have not only optimized performance in the worst-case scenario, but satisfactory performance on average as well. ( 2012 ) 2012,Assadi et al. (2015),Hassan and Curry (2014),Manshadi et al. (2012) study variations of OTA problems.Singer and Mittal (2013) consider both pricing and allocation decisions for OTA.Singla and Krause (2013) study both learning and allocation decisions for OTA. Zhao et al. (2014), Subramanian et al. (2015) study auction mechanism for OTA. Tong et al. (2016) study OTA when the arrivals of both workers and tasks are in random order. Their algorithms achieve a competitive ratio of 1/4. Concurrent with our work, Dickerson et al. (2018) study a similar model with two-sided, i.i.d. arrivals. They prove that a non-adaptive algorithm achieves a competitive ratio of 0.295. Further, they show that no online algorithm can achieve a ratio better than 0.581, even if all rewards are the same. Note that both the models of Tong et al. (2016) and Dickerson et al. (2018) are more restrictive than ours. In a model with time-varying arrivals such as ours, non-adaptive algorithms such as the one proposed by Dickerson et al. (2018), or a greedy algorithms such as one of the algorithms proposed by Tong et al. (2016), are unlikely to perform well. types of supply units. Both demand and supply units arrive randomly over the T periods. The demand unit of type i ∈ [I] that arrives at time t ∈ [T ] if any, can be identified using the pair (i, t).Similarly, the supply unit of type j ∈ [J] that arrives at time s ∈ [T ] if any, can be identified with the pair (j, s).Each demand unit (i, t) has a known non-negative reward r ijts when matched with a supply unit (j, s). This reward can capture how far apart the units are in time and how compatible their respective types are. If types i and j are incompatible, then the reward r ijts could be very small or 0. If i and j are compatible, then r ijts can decrease with the length of the interval [s, t] to capture the diminishing value of the match when the supply unit must wait for a long time for the demand unit. 1} be a random indicator of whether demand unit (i, t) arrives, and M js ∈ {0, 1} a random indicator of whether supply unit (j, s) arrives. We assume that all the random indicatorsΛ it , ∀i ∈ [I], t ∈ [T ], and M js , ∀j ∈ [J], s ∈ [T ], are mutually independent. The arrival probabilities λ it := E[Λ it ] and µ js := E[M js ] are deterministic and known to the platform a priori. To avoid trivialities in the analysis, we assume all the λ it and µ js are strictly positive, but their values can be arbitrarily small. An optimal offline algorithm OFF can see the arrivals of all the demand and supply units (Λ, M ) at the beginning of period 1. Given (Λ, M ), The maximum offline reward V OFF (Λ, M ) is equal to the value of the following maximum-weight matching problem. V OFF (Λ, M ) = max x ijts (Λ,M ),i∈[I];j∈[J];t,s∈[T ] i,j,t,s x ijts (Λ, M )r ijts s.t. i,t x ijts (Λ, M ) ≤ M js , ∀j ∈ [J]; s ∈ [T ], j,s x ijts (Λ, M ) ≤ Λ it , ∀i ∈ [I]; t ∈ [T ], x ijts (Λ, M ) ≤ Λ it M js , ∀i ∈ [I]; j ∈ [J]; t, s ∈ [T ], x ijts (Λ, M ) = 0, ∀i ∈ [I]; j ∈ [J]; t ∈ [T ]; s = t + 1, . . . , T, x ijts (Λ, M ) ≥ 0, ∀i ∈ [I]; j ∈ [J]; t, s ∈ [T ]. , 2 , 2...,J , {M j2 } j=1,2,...,J , ..., {M jt } j=1,2,...,J ). Table 1 1Scenario 1. Performance of different algorithms relative to LP (10). Base 49.8% 62.2% 63.7% 66.1% 67.9% 67.9% 67.5% α = 0 48.0% 56.9% 63.4% 65.3% 67.6% 67.9% 67.4% α = 0.2 48.6% 58.8% 64.0% 65.7% 67.8% 67.9% 67.5% α = 0.8 51.8% 64.9% 62.9% 66.1% 67.6% 67.6% 67.2% α = 1 53.4% 65.3% 64.0% 65.7% 66.9% 66.8% 66.4% τ = 2 50.3% 62.8% 65.7% 66.9% 68.8% 69.0% 68.3% τ = 5 50.3% 63.1% 64.8% 66.6% 68.2% 68.5% 67.9% τ = 20 49.3% 60.8% 63.2% 65.8% 67.5% 67.8% 67.4% τ = 30 49.1% 60.0% 63.2% 65.6% 67.6% 67.7% 67.1% β = 0 49.9% 62.6% 63.6% 66.1% 68.0% 68.1% 67.5% β = 0.2 49.9% 62.6% 63.7% 66.1% 68.0% 67.9% 67.7% β = 0.8 49.9% 60.4% 63.2% 65.3% 67.0% 67.1% 66.6% β = 1 49.6% 56.9% 62.4% 64.0% 65.7% 65.8% 65.4% ω = 0.005 49.7% 61.1% 63.0% 65.6% 67.4% 67.4% 66.9% ω = 0.02 49.8% 61.6% 63.5% 65.8% 67.6% 67.6% 67.1% ω = 0.08 49.9% 62.5% 63.8% 66.2% 68.0% 68.1% 67.5% ω = 0.15 50.1% 62.7% 63.7% 66.1% 68.0% 68.2% 67.6%ON Greedy BPH ON 1 + ON 2 + ON 3 + ON 4 + Table 2 2Scenario 2. Performance of different algorithms relative to LP (10). Base 50.3% 65.4% 66.6% 67.7% 69.6% 69.4% 69.2% α = 0 48.2% 61.2% 67.3% 67.9% 70.1% 70.3% 70.0% α = 0.2 49.2% 63.1% 67.3% 67.8% 69.9% 70.1% 69.7% α = 0.8 51.7% 66.5% 65.1% 67.0% 68.7% 68.4% 68.2% α = 1 52.5% 66.5% 65.2% 66.1% 67.7% 67.5% 67.1% τ = 2 51.3% 67.4% 69.3% 69.4% 71.5% 71.3% 71.0% τ = 5 50.9% 66.6% 67.8% 68.3% 70.1% 70.0% 69.7% τ = 20 49.7% 64.1% 66.3% 67.4% 69.6% 69.5% 69.2% τ = 30 49.4% 63.5% 66.3% 67.6% 69.6% 69.5% 69.4% β = 0 50.5% 66.5% 67.1% 68.6% 70.5% 70.4% 69.9% β = 0.2 50.5% 66.4% 66.9% 68.3% 70.3% 70.0% 69.8% β = 0.8 50.1% 62.4% 65.9% 66.2% 68.1% 67.8% 67.7% β = 1 50.4% 58.2% 64.8% 64.5% 66.5% 66.3% 66.1% ω = 0.005 50.2% 64.6% 66.0% 67.1% 68.9% 68.7% 68.6% ω = 0.02 50.2% 64.9% 66.2% 67.3% 69.3% 69.0% 68.8% ω = 0.08 50.4% 65.8% 66.7% 68.0% 69.8% 69.7% 69.5% ω = 0.15 50.3% 66.1% 67.0% 68.2% 70.2% 70.1% 69.8%ON Greedy BPH ON 1 + ON 2 + ON 3 + ON 4 + Table 3 3Scenario 3. Performance of different algorithms relative to LP (10). Base 51.2% 63.5% 65.2% 67.0% 68.2% 67.7% 67.1% α = 0 48.6% 58.1% 65.8% 66.7% 68.2% 67.7% 66.8% α = 0.2 49.7% 60.2% 66.1% 66.8% 68.1% 67.6% 66.9% α = 0.8 53.2% 65.9% 63.8% 66.3% 67.6% 67.1% 66.6% α = 1 54.2% 66.2% 64.7% 65.4% 66.5% 66.0% 65.6% τ = 2 52.1% 64.9% 68.1% 68.2% 69.5% 69.0% 68.5% τ = 5 52.0% 64.8% 66.7% 67.4% 68.7% 68.2% 67.6% τ = 20 50.4% 61.9% 64.7% 66.6% 68.1% 67.4% 66.9% τ = 30 50.1% 61.0% 64.9% 66.7% 68.0% 67.5% 66.9% β = 0 51.0% 64.4% 65.3% 67.4% 68.9% 68.2% 67.6% β = 0.2 51.4% 64.4% 65.1% 67.4% 68.6% 68.1% 67.5% β = 0.8 50.8% 60.5% 64.8% 64.9% 66.3% 65.8% 65.0% β = 1 50.8% 56.3% 64.0% 63.4% 64.7% 64.2% 63.4% ω = 0.005 50.6% 62.8% 65.2% 66.4% 67.7% 67.2% 66.5% ω = 0.02 51.0% 62.9% 65.0% 66.4% 67.8% 67.3% 66.5% ω = 0.08 51.4% 63.9% 65.2% 67.0% 68.3% 67.8% 67.4% ω = 0.15 51.3% 64.2% 65.1% 67.2% 68.6% 68.2% 67.4%ON Greedy BPH ON 1 + ON 2 + ON 3 + ON 4 + Table 4 4Scenario 4. Performance of different algorithms relative to LP (10). Base 50.4% 64.1% 66.5% 68.6% 69.9% 69.5% 68.8% α = 0 48.1% 59.5% 66.4% 68.8% 70.0% 69.9% 69.1% α = 0.2 49.0% 61.2% 67.0% 68.6% 70.1% 69.7% 68.9% α = 0.8 51.9% 65.9% 65.3% 67.9% 69.1% 68.8% 68.1% α = 1 53.2% 65.7% 65.3% 67.2% 68.1% 67.8% 67.3% τ = 2 50.9% 65.5% 69.2% 70.0% 71.4% 70.8% 70.3% τ = 5 50.9% 65.2% 67.5% 69.0% 70.4% 70.0% 69.4% τ = 20 49.9% 62.6% 66.0% 68.3% 69.6% 69.2% 68.6% τ = 30 49.5% 61.9% 65.9% 68.3% 69.7% 69.4% 68.7% β = 0 50.4% 64.5% 66.4% 68.8% 69.9% 69.6% 68.9% β = 0.2 50.3% 64.4% 66.5% 68.7% 70.0% 69.6% 69.1% β = 0.8 50.2% 62.2% 66.4% 67.7% 68.8% 68.4% 67.8% β = 1 50.3% 59.1% 65.6% 66.4% 67.5% 67.3% 66.5% ω = 0.005 50.3% 63.8% 66.4% 68.5% 69.7% 69.2% 68.7% ω = 0.02 50.7% 63.9% 66.3% 68.4% 69.7% 69.5% 68.9% ω = 0.08 50.3% 64.1% 66.6% 68.6% 69.9% 69.6% 68.9% ω = 0.15 50.3% 64.3% 66.6% 68.8% 69.9% 69.6% 69.0% Alaei, Saeed, MohammadTaghi Hajiaghayi, Vahid Liaghat. 2012. Online prophet-inequality matching with applications to ad allocation. 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{'abstract': 'The classical Prophet Inequality arises from a fundamental problem in optimal-stopping theory. In this problem, a gambler sees a finite sequence of independent, non-negative random variables. If he stops the sequence at any time, he collects a reward equal to the most recent observation. The Prophet Inequality states that, knowing the distribution of each random variable, the gambler can achieve at least half as much reward in expectation, as a prophet who knows the entire sample path of random variables(Krengel and Sucheston 1978). In this paper, we prove a corresponding bound for correlated non-negative random variables. We analyze two methods for proving the bound, a constructive approach, which produces a worstcase instance, and a reductive approach, which characterizes a certain submartingale arising from the reward process of our online algorithm.We apply this new prophet inequality to the design of algorithms for a class of two-sided bipartite matching problems that underlie online task assignment problems. In these problems, demand units of various types arrive randomly and sequentially over time according to some stochastic process. Tasks, or supply units, arrive according to another stochastic process. Each demand unit must be irrevocably matched to a supply unit or rejected. The match earns a reward that depends on the pair. The objective is to maximize the total expected reward over the planning horizon. The problem arises in mobile crowd-sensing and crowd sourcing contexts, where workers and tasks must be matched by a platform according to various criteria. We derive the first online algorithms with worst-case performance guarantees for our class of two-sided bipartite matching problems.', 'arxivid': '1901.02552', 'author': ['Van-Anh Truong vatruong@ieor.columbia.edu \nDepartment of Industrial Engineering and Operations Research\nColumbia University\nNew YorkNYUSA\n', 'Xinshang Wang \nDepartment of Industrial Engineering and Operations Research\nColumbia University\nNew YorkNYUSA\n'], 'authoraffiliation': ['Department of Industrial Engineering and Operations Research\nColumbia University\nNew YorkNYUSA', 'Department of Industrial Engineering and Operations Research\nColumbia University\nNew YorkNYUSA'], 'corpusid': 119314431, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 26583, 'n_tokens_neox': 23054, 'n_words': 13507, 'pdfsha': 'b3420c373e325175276aa6c2e4a7b3ba5863acb9', 'pdfurls': ['https://arxiv.org/pdf/1901.02552v1.pdf'], 'title': ['Prophet Inequality with Correlated Arrival Probabilities, with Application to Two Sided Matchings', 'Prophet Inequality with Correlated Arrival Probabilities, with Application to Two Sided Matchings'], 'venue': []}

arxiv

Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage Alexandre Anahory Simoes Asier López-Gordón Anthony Bloch Leonardo Colombo Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage 10.13039/501100011033 Forced variational integrators are given by the discretization of the Lagrange-d'Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem. with foot slip, giving rise to motion equations with fewer degrees of freedom than [6]. Trajectory optimization algorithms aim to find an input trajectory that minimizes a cost function subject to a set of constraints on the system's states and inputs. Trajectory optimization has been implemented extensively for systems with continuous-time dynamics, but many applications in control theory and robotics include impacts and friction contacts making the dynamics non-smooth. In this paper, we develop a trajectory optimization policy for a passive walker experiencing foot slip by introducing controls into the passive walker and by defining geometric integrators for a class of hybrid mechanical systems-that is, (smooth) dynamical systems together with a discrete transition (impact map)-by using discrete (geometric) mechanics techniques. Variational integrators are a class of geometric integrators for Lagrangian systems derived from a discrete variational principle as discussed e.g. by [7] and [8]. These integrators retain some of the main geometric properties of continuous systems, such as symplecticity and momentum conservation (as long as the symmetry survives the discretization procedure), and good (bounded) behavior of the energy associated to the system. This class of numerical methods has been applied to a wide range of problems including optimal control [9], [10], constrained systems [11], [12], power systems [13], nonholonomic systems [14], [15], [16], multi-agent systems [17], [18], [19], and systems on Lie groups [20], [21]. Variational integrators for hybrid mechanical systems were used in [22] and [23]. However, these works do not consider the problem of trajectory generation. Such a problem is considered in [24] but for the compass gait biped, while in this work we consider passive walkers under foot slippage. The main contributions of this work are summarized as follows: • We introduce simple hybrid holonomically constrained forced Lagrangian systems and we construct forced variational integrators for this class of hybrid mechanical system. • We introduce a reduced dynamical model for walking with foot slip and derive variational integrators for the proposed model. • We present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems and illustrate the method with a passive walker experiencing foot slip. • We design discrete-time sub-optimal trajectories to reach a desired configuration via discrete mechanics and optimal control as the solution to a constrained nonlinear optimization problem. The remainder of the paper is structured as follows. In Section II the constrained Lagrange-d'Alembert principle is used to derive forced Euler-Lagrage equations for mechanical systems subject to holonomic constraints. After defining simple hybrid holonomically constrained forced Lagrangian system in III-A, this is applied to construct a model for a passive walker with foot slip in Section III-B. In Section IV we construct variational integrators for a passive walker experiencing foot slip. Finally, in Section V we introduce controls into the model and we employ a variational integrator for the uncontrolled system, together with a suitable discretization of the cost function associated to an optimal control problem, to derive optimal control policies for trajectory generation in a tracking problem. II. CONSTRAINED LAGRANGE-D'ALEMBERT PRINCIPLE Let Q be an n-dimensional differentiable manifold with local coordinates (q A ), 1 ≤ A ≤ n, the configuration space of a mechanical system. Denote by T Q its tangent bundle with induced local coordinates (q A ,q A ). Given a Lagrangian function L : T Q → R, its Euler-Lagrange equations are d dt ∂L ∂q A − ∂L ∂q A = 0, 1 ≤ A ≤ n.(1) In general, Eqs. (1) determine a system of implicit secondorder differential equations. If we assume that the Lagrangian is hyper-regular, that is, the n × n matrix M AB = ∂ 2 L ∂q A ∂q B is non-singular, the local existence and uniqueness of solutions is guaranteed for any given initial condition. Assume L : T Q → R is a hyper-regular Lagrangian and that q(t) satisfies Euler-Lagrange equations (1). Then, there is a smooth Lagrangian vector field f L on T Q, associated with L, that is, there is a dynamical system associated to the Lagrangian. For t ∈ [t 0 , t f ] we say that γ(t) = (q(t),q(t)) is a solution of f L with initial condition γ(t 0 ) = (q(t 0 ),q(t 0 )) ifγ(t) = f L (γ(t)). The Lagrangian vector field f L associated with L : T Q → R takes the form f L (q A ,q A ) = q A , M AB ∂L ∂q A − ∂L ∂q A ∂q Bq B (2) where M AB denotes the inverse matrix of M AB . A constrained Lagrange-d'Alembert principle (or principle of virtual work) for systems subject to external nonconservative forces and holonomic constraints (i.e., constraints of the form Φ(q) = 0 with Φ : Q → R a smooth function and 0 a regular value of Φ), establishes that the natural motions of the system are those paths q : [0, T ] → Q × R m satisfying δ T 0 (L(q,q) + λΦ(q)) dt + T 0 F (q,q)δq dt = 0 (3) for null boundary variations δq(0) = δq(T ) = 0 and δλ ∈ R m . Here λ(t) ∈ R m represents the vector of timedependent Lagrange multipliers. The first term in Eq. (3) is the action variation augmented with a Lagrange multiplier to ensure the dynamics satisfy the constraint, while the second is known as virtual work since F (q,q)δq is the virtual work done by the force field F with a virtual displacement δq. Denoting the Jacobian of the constraints as G(q A ) = ∂Φ ∂q A , the constrained Lagrange-d'Alembert principle, leads to the constrained forced Euler-Lagrange equations d dt ∂L ∂q A − ∂L ∂q A − F (q i ,q i ) + G T (q A )λ = 0,(4) together with Φ(q) = 0. Note that the term −G T (q A )λ ∈ T * Q is the force imposing the system to remain in the constraint submanifold defined by Φ(q) = 0. Now, consider the augmented LagrangianL : T Q×R m → R given byL(q,q, λ) = L(q,q)+λΦ(q). If the Lagrangian L is hyperregular, it induces a well defined map, the Lagrangian flow for the augmented Lagrangian, F t : T Q × R m → T Q × R m by F t (q 0 ,q 0 , λ 0 ) := (q(t),q(t), λ(t)), where (q, λ) ∈ C 2 ([0, T ], Q × R m ) is the unique solution of the constrained Euler-Lagrange equation with initial condition (q 0 ,q 0 , λ 0 ) ∈ T Q×R m . Nevertheless, this approach requires one to introduce a new equation for the Lagrange multiplier, incrementing the computational costs of the optimal control problem we want to solve in Section V. To overcome this issue, alternatively one can consider the submanifold N = {q ∈ Q | Φ(q) = 0} ⊆ Q and suppose that the force F on the restriction to N , F | N : N → T * N , is well-defined. Then the unconstrained Lagrange-d'Alembert principle associated with the restricted Lagrangian L N and the forces F N gives the same trajectories as the previous construction on the ambient manifold Q. In that sense we can define the flow for the holonomically constrained forced Lagrangian system as f N (q 0 ,q 0 ) := (q(t),q(t)) ∈ T N , where q ∈ C 2 ([0, T ], N ) is the unique solution of the constrained forced Euler-Lagrange equation with initial condition (q 0 ,q 0 ) ∈ T N . III. A PASSIVE WALKER WITH FOOT SLIP Next, we will examine a simple case of a passive walker where the base is allowed to slide and we will formulate this system as a simple hybrid Lagrangian system. This model is inspired by [6]. In comparison with that model ours avoids incrementing the dimension of the configuration space to include Lagrange multipliers. In our approch we reduce the dynamics of the system to the constraint submanifold N . Before modeling of the passive walker with foot slip we introduce the basics of simple hybrid holonomic forced Lagrangian systems. The dynamics associated with a hybrid systems corresponds to an autonomous system with impulse effects. We denote by Σ H the simple hybrid dynamical system generated by H, that is, Σ H : ẋ(t) = f (x(t)), x − (t) / ∈ S x + (t) = ∆(x − (t)) x − (t) ∈ S(5) with x : I ⊂ R → D and x − , x + the states just before and after the moments when integral curves of f intersects S. Remark 1: A solution of a simple hybrid system may experience a Zeno state if infinitely many impacts occur in a finite amount of time [27], [29], [30], [31]. However, by considering the class of hybrid systems given by mechanical systems with impulsive effects as in [26], we exclude such behavior by considering that the set of impact times is closed and discrete, meaning that there is no chatering about an impact point and therefore excluding Zeno behavior. Necessary and sufficient conditions for the existence of Zeno behavior in the class of simple hybrid Lagrangian systems have been explored in [32] and [30]. Consider D = T Q and a hyper-regular Lagrangian L : T Q → R. Associated with the dynamics generated by L, there exists a Lagrangian vector field f L as in (2). Note that ∆ : S → T Q is continuous. If we denote the closure of ∆(S) by ∆(S), then we must assume ∆(S) ∩ S = ∅ and, therefore, an impact does not lead immediately to another impact (see Section 4.1 [26] for more details). We further assume that S = ∅ and there exists an open subset U ⊂ T Q and a differentiable function h : U → R such that S = {x ∈ U | h(x) = 0} with ∂h ∂x (s) = 0 for all s ∈ S (that is, S is an embedded submanifold of T Q with co-dimension 1) and the Lie derivative of the vector field f L with respect to h does not vanish on T Q, that is L f L h(w) = 0, ∀w ∈ T Q. A trajectory γ : [0, T ] → T Q crosses the switching surface S at t − i = inf{t > 0|γ(t) ∈ S}. We allow the trajectory γ(t) to be continuous but nonsmooth at t − i . That is, the velocity before the impactq − is different from the velocity q + after the impact at S, namely,q(t − i ) =q(t + i ). Definition 2: A simple hybrid system H = (D, f, S, ∆) is said to be a simple hybrid holonomically constrained forced Lagrangian system if it is determined by H L N := (T N, f N , S N , ∆ N ), where f N : T N → T (T N ) is the flow for the holonomically constrained forced Lagrangian system as described in Section III-A, and S N and ∆ N are the switching surface and impact maps as described above restricted to submanifolds N and T N , respectively. The simple hybrid Lagrangian dynamical system generated by H L N is given by Σ H L N : ẋ(t) = f N (x(t)), if x − (t) / ∈ S N , x + (t) = ∆ N (x − (t)), if x − (t) ∈ S N , where x(t) = (q(t),q(t)) ∈ T N . That is, a trajectory of a simple hybrid holonomically constrained forced Lagrangian system is determined by the restricted forced Lagrangian dynamics until the instant when the state attains the switching surface S N . We refer to such an instant as the impact time. The impact map ∆ N gives new initial conditions from which f N evolves until the next impact occurs. Solutions for the simple hybrid holonomically constrained forced Lagrangian system H L N , are considered right continuous and with finite left and right limits at each impact with S N . B. Modeling passive walking with foot slip We model a passive walker as a two-masses inverted pendulum. The mass of the foot is denoted by m 1 and the hip by m 2 . The length of the leg is given by . The angles of the leg are restricted to θ ∈ [−a, a] ⊂ R (when θ hits the boundary, −a, a new step is taken and θ is reset to a). The coordinates of the center of mass will be given by (x, y) and the coordinates of the foot are (x, y) (see Figure 1). , r = m2 m , where I is the moment of inertia about the center of mass and r is the distance from the foot to the center of mass, which is kept constant along the motion. Also, note that the coordinates of the center of the mass satisfy x = x + r sin θ, y = y + r cos θ, so that the center of the mass is located along the leg at some point between the foot and the hip. In addition, we impose the constraint y = 0 which means that the foot will not leave the floor. With this notation, the constraint implies that y − r cos θ = 0. Note that for this model, θ is constrained to be in [−a, a]. If θ crosses the negative boundary, we say a new step occurs and θ is reset to a. If θ crosses the positive boundary, specifically if θ = π 2 (i.e., x =x), we say that a crash has occurred. In this case, the model stops walking and we report a failure. This also implies that falling forwards is not permitted; the only way to crash is by falling backwards. Before deriving the hybrid dynamics, we first need to determine the switching surface S. Assume that the leg takes symmetric steps of angle a, that is, θ ∈ [a, −a]. When θ = −a, the angle is reset to a (corresponding to a new step taking place and the swing legs switching). Therefore, we will take the switching surface S, to be S = {θ = −a}. The continuous dynamics is determined by a Lagrangian function L : T Q → R corresponding to a planar rigid body, where Q = R 2 × S 1 is the configuration space locally described by the coordinates q = (x, y, θ), and L(q,q) = K(q,q) − V (q), where K(q,q) = m 2 (ẋ 2 +ẏ 2 ) + Iθ 2 2 , V (q) = mgr cos θ, together with the (holonomic) constraint y − r cos θ = 0 defining the submanifold N which may be seen as diffeomorphic to R × S 1 . The restricted Lagrangian L N defined on coordinates (x, θ,ẋ,θ) is given by the restricted kinetic energy K N = m 2 (ẋ 2 + r 2 sin 2 θθ 2 ) + Iθ 2 2 minus the restricted potential function which remains the same under the restriction to N . We assume that the friction forces of the foot with the ground are non-conservative forces (conservative forces might be included into the potential energy V ), which are determined by a fibered map F : T Q → T * Q. The forces exerted from the friction on the foot in the configurations q = (x, y, θ) are given by F x = −κẋ = −κ(ẋ + rθ cos θ), F y = 0, F θ = −κẋ(r cos θ) = −κr cos θ(ẋ + rθ cos θ). This force is well-defined on the restriction to N . At a given position and velocity, the force will act against variations of the position (virtual displacements) and the dynamics should also satisfy the holonomic constraint Φ(q) = y − r cos θ = 0. Euler-Lagrange equations (4) for the restricted Lagrangian L N and forces F N are given by mẍ = −κ(ẋ + rθ cos θ) (6) θ(I + mr 2 sin 2 θ) = −κr cos θ(ẋ + rθ cos θ) + rm sin θ(g − rθ 2 cos θ) on the submanifold N . The last step to describe the hybrid dynamics for the simple hybrid holonomically constrained forced Lagrangian system is to find the impact map ∆ N . We assume as in [33] a rigid hip, that is, the horizontal position and velocity of the foot do not change at impacts (see Figure 2), namelȳ x + =x − andẋ + =ẋ − . Additionally, we assume that the angular momentum is conserved in the impact. Under these assumptions (see [33], [3] for the case without foot slip and horizontal ground), the impact map is defined as the map ∆ N : S → T N ⊆ T Q, where S N = {θ = −a}, with ∆ N (x − , −a,ẋ − ,θ − ) = (x + , θ + ,ẋ + ,θ + ) given by x + − r sin θ + = x − − r sin(−a) θ + = θ − + 2ȧ x + − rθ + cos θ + =ẋ − − rθ − cos(−a) θ + = cos(2a)θ − . (8) x y (x + , y + ) (x − , y − ) (x, y) −a a IV. FORCED VARIATIONAL INTEGRATOR FOR A PASSIVE WALKER EXPERIENCING FOOT SLIP A discrete Lagrangian is a differentiable function L d : Q× Q → R, which may be considered as an approximation of the action integral defined by a continuous regular Lagrangian L : T Q → R. That is, given a time step h > 0 small enough, L d (q 0 , q 1 ) ≈ h 0 L(q(t),q(t)) dt, where q(t) is the unique solution of the Euler-Lagrange equations with boundary conditions q(0) = q 0 and q(h) = q 1 . Construct the grid T = {t k = kh | k = 0, . . . , N }, with N h = T and define the discrete path space P d (Q) := {q d : {t k } N k=0 → Q}. We identify a discrete trajectory q d ∈ P d (Q) with its image q d = {q k } N k=0 , where q k := q d (t k ). The discrete action A d : P d (Q) → R for this sequence of discrete paths is calculated by summing the discrete Lagrangian on each adjacent pair, and it is defined by A d (q d ) = A d (q 0 , ..., q N ) := N −1 k=0 L d (q k , q k+1 ).(9) The discrete variational principle [7], states that the solutions of the discrete system determined by L d must extremize the action sum given fixed points q 0 and q N . Extremizing A d over q k with 1 ≤ k ≤ N −1, we obtain the following system of difference equations D 1 L d (q k , q k+1 ) + D 2 L d (q k−1 , q k ) = 0.(10) These equations are usually called the discrete Euler-Lagrange equations. Given a solution {q * k } k∈N of Eq. (10) and assuming the regularity hypothesis, i.e., the matrix (D 12 L d (q k , q k+1 )) is regular, it is possible to define implicitly a (local) discrete flow Υ L d : U k ⊂ Q × Q → Q × Q by Υ L d (q k−1 , q k ) = (q k , q k+1 ) from (10), where U k is a neighborhood of the point (q * k−1 , q * k ). A. Forced variational integrators for holonomically constrained forced Lagrangian systems The key idea of variational integrators is that the variational principle is discretized rather than the resulting equations of motion. As we explained before, we discretize the state space T Q as Q × Q and consider a discrete Lagrangian L d : Q × Q → R and, in addition, we consider discrete "external forces" F ± d : Q × Q → T * Q approximating the continuous-time action and non-conservative external forces given by (12) Alternatively, we can directly work with a discretized version of the submanifold N . Here, the restricted discrete Lagrangian L d N : N × N → R and discrete "external forces" F ± N,d : N × N → T * N are approximating the continuous time restricted Lagrangian and force map, respectively. t k+1 t k L(q(t),q(t)) dt L d (q k , q k+1 ) (11) t k+1 t k F i (q(t),q(t))δq dt F − d (q k , q k+1 )δq k + F + d (q k , q k+1 )δq k+1 . Note that, physically speaking, F ± d are not external forces. They are in fact momentum, since F ± d are defined by a discretization of the work done by the force F . The idea behind the ± is that one needs to combine the two discrete forces to give a single one-form F d : Q × Q → T * (Q × Q) defined by F d (q 0 , q 1 )(δq 0 , δq 1 ) = F + d (q 0 , q 1 )δq 1 + F − d (q 0 , q 1 )δq 0 .0 =D 1 L d N (q k , q k+1 ) + D 2 L d N (q k−1 , q k ) (13) + F − N,d (q k , q k+1 ) + F + N,d (q k−1 , q k ).(14) B. Constrained forced variational integrator for a passive walker under foot slip Next, consider the midpoint (second order) discretization rule, that is, q(t) q k +q k+1 2 ,q(t) q k+1 −q k h and define the discrete Lagrangian L d : R 3 × R 3 → R as L d (q k , q k+1 ) = hL q k + q k+1 2 , q k+1 − q k h , with h > 0 denoting the time step and q k = (x k , y k , θ k ) for k = 0, . . . , N . In our model, the discrete Lagrangian L d : (R × S) × (R × S) → R is given by L d = m 2h (x k+1 − x k ) 2 + 1 2h I + mr 2 sin 2 θ k+1 + θ k 2 × (θ k+1 − θ k ) 2 − hmgr cos θ k + θ k+1 2 . The discrete external forces are given by F + d = h 2 F q = q k−1 + q k 2 ,q = q k − q k−1 h ,(15)F − d = h 2 F q = q k + q k+1 2 ,q = q k+1 − q k h .(16) where F + d is evaluated in (q k−1 , q k ) and F − d is evaluated in (q k , q k+1 ). Note that the restricted discrete force maps have the same expression. The discrete Euler-Lagrange equations with forces are then 0 = m h (2x k − x k+1 − x k−1 ) + hmg sin α + F − d,x + F + d,x , 0 = I h (2θ k − θ k−1 − θ k+1 ) + mr 2 2h sin θ k + θ k−1 2 cos θ k + θ k−1 2 (θ k − θ k−1 ) 2 + mr 2 2h sin θ k+1 + θ k 2 cos θ k+1 + θ k 2 (θ k+1 − θ k ) 2 + F + d,θ +F − d,θ − mghr 2 sin( θ k + θ k−1 2 ) − sin(α − θ k + θ k+1 2 ) . V. DISCRETE MECHANICS AND OPTIMAL CONTROL FOR A CONTROLLED WALKER UNDER FOOT SLIP Next, we add control forces to the previous formalism. The equations of motion are now given by d dt ∂L N ∂q A − ∂L N ∂q A = u a Y a A + (F N ) A ,(17) where Y a = Y a A (q)dq A , 1 ≤ a ≤ m < n are the control forces, u(t) = (u 1 (t), ..., u m (t)) ∈ U are the control inputs, and U is an open subset of R m , the set of admissible controls. Note the previous equations give a model of an affine control system of the form q = f N (q,q) + g(q,q)u,(18) where g = (M AB P, 0 (n−a)×(n−a) ) T and P is a matrix mapping u to the system's generalized forces. In a typical optimal control problem, one whishes to find a trajectory and a control minimizing a cost function of the form J (q, u) = T 0 C(q(t),q(t), u(t)) dt verifying a control equation such as (17) and, in addition, some boundary conditions giving information about the initial and terminal states of the system. Let us suppose now that the control force is given byȲ 1 = dx andȲ 2 = dθ. Hence, we have the following controlled equations of motion on N mẍ = −κ(ẋ + rθ cos θ) + u x ,(19) mr 2 (θ sin 2 θ +θ 2 cos θ sin θ) + Iθ = −κr cos θ(ẋ + rθ cos θ) + u θ ,(20) as long as θ = −a. Remark 3: Note that in the restricted configuration space R × S 1 the system is fully actuated but in the ambient space R 2 × S 1 the system is underactuated. Suppose that we would like to follow a known reference trajectory γ r : [0, T ] → Q denoted by γ r (t) = (x r (t), θ r (t)). We want to find a control strategy minimizing the cost functional J (q, u) = 1 2 ε u 2 + η γ r − γ 2 + ρ γ r −γ 2 dt, with γ satisfying the control equations (19) and (20). The parameters ε, η and ρ are the weights of the control inputs, the trajectory-tracking and the velocity-tracking terms, respectively, We may transpose the optimal control problem to a nonlinear constrained optimization problem using a discretization of the principle above. Indeed, after applying the discretization procedure we come down to the problem of minimizing J d (q d , u d ) = N −1 k=0 C d (q k , q k+1 , u k , u k+1 ), subject to the discrete dynamics 0 =D 1 L d N (q k , q k+1 ) + D 2 L d N (q k−1 , q k ) + F − N,d (q k , q k+1 ) + F + N,d (q k−1 , q k )+u k−1 + u k ,(21) with the boundary values q 0 , q N given. Notice that Eq. (21) correspond to the forced discrete Euler-Lagrange equations (14) with F ± N,d,u (q k , q k+1 , u k ) = F ± N,d (q k , q k+1 ) + u k . Next, we discretize the optimal control problem. Fixing a time step h > 0, we discretize the cost function so that C d (q k , q k+1 , u k , u k+1 ) ≈ (k+1)h kh C (q(t),q(t), u(t)) dt. Thus we set C d (q k , q k+1 , u k , u k+1 ) = hC q k + q k+1 2 , q k+1 − q k h , u k , where u k = u t k +t k+1 2 . The problem is subjected to the discrete dynamics 0 = m h (2x k − x k+1 − x k−1 ) + F − d,x + F + d,x + u x,k + u x,k−1 , 0 =mr 2gh sin θ k−1 + θ k 2 + sin θ k + θ k+1 2 + r h (θ k−1 − θ k ) 2 sin(θ k−1 + θ k ) +2(θ k−1 − θ k ) cos(θ k−1 + θ k ) +(θ k − θ k+1 ) 2 sin(θ k + θ k+1 ) +2(θ k+1 − θ k ) cos(θ k + θ k+1 )]} − 2 h (θ k−1 − 2θ k + θ k+1 ) 2I + mr 2 + 4(F − d,θ + F + d,θ + u θ,k−1 + u θ,k ),(22) and to the boundary conditions q 0 = (x 0 , θ 0 ) and q N = (x N , θ N ) fixed. In addition, we have the following conditions on the initial and final velocities: FL(q 0 ,q 0 ) = F F − N,d,u L d (q 0 , q 1 , u 0 ), FL(q N ,q N ) = F F + N,d,u L d (q N −1 , q N −1 , u N −1 ),(23) where FL denotes the continuous Legendre transform and F F ± N,d,u denotes the forced discrete Legendre transform (see [7] for instance), i.e., D 2 L(q 0 ,q 0 ) + D 1 L d (q 0 , q 1 ) + F − d (q 0 , q 1 ) + u 0 = 0, (24) D 2 L(q N ,q N ) − D 2 L d (q N −1 , q N ) − F + d (q N −1 , q N ) + u N −1 = 0.(25) Remark 4: If we discretize the reference trajectory by evaluating it at discrete time γ r h 2k+1 2 = x r h 2k+1 2 , θ r h 2k+1 2 ≡ (x r,k , θ r,k ), then the midpoint discrete cost function reads C d (q k ,q k+1 , u k , u k+1 ) = h 2 εu 2 k + η x k+1 + x k 2 − x r,k 2 +η θ k+1 + θ k 2 − θ r,k 2 + ρ x k+1 − x k h −ẋ r,k 2 +ρ θ k+1 − θ k h −θ r,k 2 . The discrete optimal control problem consists on finding a discrete trajectory {(x k , θ k , u k )} solution of the problem (22) boundary conditions (23) (26) Next, we incorporate impacts in the variational setting by finding a discretization of the impact set S d ⊆ Q × Q and of the impact map ∆ d : min N −1 k=0 C d (q k , q k+1 , u k , u k+1 ) discrete equationsS d → Q × Q. Let S d = {(x 0 , θ 0 , x 1 , θ 1 )|θ 0 = a} and ∆ d (x − 0 , −a, x − 1 , θ − 1 ) is given by the discretization of Eqs. (8) via the midpoint rule: x + 0 + x + 1 2 = x − 0 + x − 1 2 − r sin(−a) + r sin θ + 0 + θ + 1 2 , θ + 0 = 2a + θ − 0 , x + 1 − x + 0 − r(θ + 1 − θ + 0 ) cos θ + 0 + θ + 1 2 = x − 1 − x − 0 −r(θ − 1 − θ − 0 ) cos(−a), θ + 1 − θ + 0 = cos(2a)(θ − 1 − θ − 0 ) , that is, x + 0 = x − 0 − 1 2 r(θ − 0 − θ − 1 ) (cos a − cos(2a) cos ψ) + r(sin a + sin ψ), x + 1 = x − 1 + 1 2 r(θ − 0 − θ − 1 )(cos a − cos(2a) cos ψ) + r(sin a + sin ψ), θ + 0 = 2a + θ − 0 , θ + 1 = cos(2a)(θ − 1 − θ − 0 ) + a, where ψ = a + 1 2 cos(2a)(θ − 1 − θ − 0 ) . Note that the energy of the system is not preserved between impacts. Indeed, E L = m 2 (ẋ 2 + r 2 sin 2 θθ 2 ) + Iθ 2 2 + mgr cos θ, and then, E L • ∆ = · · · + I(θ + ) 2 2 = · · · + Iθ 2 2 cos 2 (2a) = E L . We have performed a Python numerical simulation with N = 80 steps, time step h = 0.1, parameters g = 9.8, α = 0, κ = 0.2, r = 1, m = 1, I = 0.5, a = π 6 , ε = 0.1, η = 100, ρ = 1; initial values x 0 = 0, θ 0 = π 6 ,ẋ 0 = 1, andθ 0 = 0.1. The reference trajectory is given by γ r (t) = (x r (t) + r cos(θ r (t)), θ r (t)) for t i−1 < t < t i , wherē x r (t) =x r,i−1 +ẋ r,i−1 t and θ r (t) = a +θ r,i−1 (t − t i−1 ). The values of the parameters are t 0 = 0,x r,0 = 0, θ r,0 = a,ẋ r,0 = 1,θ r,0 = −0.08 and t i for i ≥ 1 is the instant of the i-th impact (determined by the equation θ(t i ) = −a). The parameters x r,i , θ r,i ,ẋ r,i ,θ r,i are defined by the impact map (8). The evolution of the xand θ-coordinates of the center of mass are plotted in Figs. 3 and 4, respectively; comparing them with the reference trajectory. The curves that the center of mass, the foot, the leg and the reference trajectory describe on the xy-plane are represented in Fig. 5. One can clearly observe how the trajectory of the foot approaches the reference one. The evolution of the control inputs is represented in Fig. 6. VI. CONCLUSIONS We have introduced simple hybrid holonomically constrained forced Lagrangian systems and we have constructed forced variational integrators for this class of hybrid system subject to holonomic constraints. In particular, we applied the discretization to a model of a passive walker with foot slip. This discretization is employed in a trajectory generation problem, together with a suitable discretization of a cost function, in a trajectory tracking task. This study sheds light on how to identify in the model when the walker falls. It also indicates how to design controllers based on momentum balance [34] in order to avoid falls while tracking where leg amplitudes are equal. Fig. 1 : 1Leg and foot: The coordinates of the foot are given by (x, y), the center of mass are (x, y) and θ is the angle between the leg of length and the vertical axis. Let us denote by m = m 1 + m 2 , I = 2 m1m2 m Fig. 2 : 2Depiction of the impact. The position of the foot is continuous at the impact. 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{'abstract': "Forced variational integrators are given by the discretization of the Lagrange-d'Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem.", 'arxivid': '2209.14255', 'author': ['Alexandre Anahory Simoes ', 'Asier López-Gordón ', 'Anthony Bloch ', 'Leonardo Colombo '], 'authoraffiliation': [], 'corpusid': 252568339, 'doi': '10.48550/arxiv.2209.14255', 'github_urls': [], 'n_tokens_mistral': 13594, 'n_tokens_neox': 11864, 'n_words': 7220, 'pdfsha': '7c0b4799109706220ba5bfd35d5fd0f0ec8e07ec', 'pdfurls': ['https://export.arxiv.org/pdf/2209.14255v1.pdf'], 'title': ['Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage', 'Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage'], 'venue': []}