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thmTheorem corollary[thm]Corollary lemma[thm]Lemma definition defi[thm]Definition remark[thm]Remark beweis IEEE Transactions on Software Engineering Shell et al.: Bare Demo of IEEEtran.cls for Computer Society JournalsANOVAANalysis Of VAriance ASTAbstract Syntax Tree AUCArea Under the ROC Curve CaAfferent Coupling CBOCoupling Between Objects CCACanonical Correlation Analysis CDCritical Distance CeEfferent Coupling CFSCorrelation-based Feature Subset CLAClustering and LAbeling CODEPCOmbined DEfect Predictor CPDPCross-Project Defect Prediction DBSCANDensity-Based Spatial Clustering DCVDataset Characteristic Vector DTBDouble Transfer Boosting fnfalse negative fpfalse positive GBGigaByte HLHosmer-Lemeshow ITSIssue Tracking System JITJust In Time LCOMLack of COhession between Methods LOCLines Of Code MDPMetrics Data Program MIMetric and Instances selection MODEPMultiObjective DEfect Predictor MPDPMixed-Project Defect Prediction NNNearest Neighbor PCAPrinciple Component Analysis RAMRandom Access Memory RFCResponse For a Class SCMSourceCode Management system SVMSupport Vector Machine TCATransfer Component Analysis tntrue negative tptrue positive RBFRadial Basis Function ROCReceiver Operating Characteristic UMRUnified Metric Representation VCBValue-Cognitive Boosting WPDPWithin-Project Defect PredictionUnfortunately, the article “A Comparative Study to Benchmark Cross-project Defect Prediction Approaches” has a problem in the statistical analysis which was pointed out almost immediately after the pre-print of the article appeared online. While the problem does not negate the contribution of the the article and all key findings remain the same, it does alter some rankings of approaches used in the study. Within this correction, we will explain the problem, how we resolved it, and present the updated results. cross-project defect prediction, benchmark, comparison, replication, correctionCorrection of “A Comparative Study to Benchmark Cross-project Defect Prediction Approaches” Steffen Herbold, Alexander Trautsch, Jens Grabowski S. Herbold, A. Trautsch, and J. Grabowski are with the University of Goettingen, Institute of Computer Science, Göttingen, Germany.E-mail: {herbold,grabowski}@[email protected] December 30, 2023 ============================================================================================================================================================================================================================================================================================ § INTRODUCTIONUnfortunately, the article “A Comparative Study to Benchmark Cross-project Defect Prediction Approaches” <cit.> has a problem in the statistical analysis performed to rank CPDP approaches. Prof. Yuming Zhou from Nanjing University pointed out an inconsistency in Table 8 of the the article. He noted that in some cases the rankscores are worse even if the mean values for performance metrics are better. While this is possible in theory, with the Friedman test <cit.> with post-hoc Nemenyi test <cit.>, such inconsistencies are unlikely. Therefore, we immediately proceeded to check our results. These checks revealed that the inconsistencies are due to a problem with our statistical analysis for the Research Question 1 (RQ1) “Which CPDP approaches perform best in terms of F-measure, G-measure, AUC, and MCC?”. None of the raw results of the benchmark, nor any of the other research questions are affected by the problem.We will describe the problem and how we solved in in Section <ref>. Then, we will show the updated results regarding RQ1 and discuss the changes in Section <ref>. Afterwards, we analyze the reasons for the changes in Section <ref> to determine if all changes due to the correction are plausible and the correction resolves the inconsistencies reported by Y. Zhou. In Section <ref>, we describe how we updated our replication kit as part of this correction. Finally, we will conclude in Section <ref>. Please note, that we assume that readers have read to the original article and are familiar with the terminology used. We do not re-introduce any of the terminology in this correction. § PROBLEM WITH THE NEMENYI TEST IMPLEMENTATIONOn July 15th 2017, Y. Zhou imformed us that he found an inconsistency between the results of CV-NET and CamargoCruz09-DT for the RELINK data for the performance metric AUC. He noted that the mean value for CV-NET was higher than for CamargoCruz09-DT, but the rankscore was lower. He went to the raw data provided as part of the replication kit <cit.> and confirmed that the mean values were correct, and the AUC for CV-NET was higher for all three products of the RELINK data. Based on this observervation, we re-checked our statistical analysis of the results. We found the problem in our implementation of the Nemenyi post-hoc test. §.§ Summary of the Friedman and Nemenyi tests To understand the problem, we briefly recap how the Friedman test with post-hoc Nemenyi test works. The Friedman test determines if there are stastical significant differences between populations. This is done using pair-wise comparisons between the rankings of populations. If the Friedman test determines significant differences, the Nemenyi post-hoc test compares the populations to each other to determine the statistically significantly different ones. The analysis with the Nemenyi test is based on two parameters: the CD and the average ranks of the populations in the pair-wise comparisons between all populations on each data set. Following the description by Demšar <cit.>, CD is defined asCD = q_α√(k(k+1)/6N)where q_α = qtukey(α, N, inf)/√(2) is the studentized range distribution with infinite degrees of freedom divided[For simplicity, we refer to the studentized range distribution as qtukey(α, N), following the name of the related method in R] by √(2), α the significance level, k the number of populations compared and N the number of data sets. We can thus rewrite CD asCD= qtukey(α, N, inf)/√(2)√(k(k+1)/6N)= qtukey(α, N, inf)√(k(k+1)/12N) If we now assume that R_i, R_j are the average ranks of populationi,j ∈{1,…,N}, the two populations are stastically significantly different if|R_i-R_j| > CD. In case a control population is available, it is possible to use a procedure like Bonferroni correction <cit.>. In this case, all populations are compared to the control classifier instead of to each other. This greatly reduces the number of pair-wise comparisons and can make the test more powerful. In this case, for each pair a z-value is computed asz = (R_i-R_j)/√(k(k+1)/6N).The z-values are then used to rank classifiers. Since we do not have a control classifier and have to do pair-wise comparisons with the CD and cannot make use of the z-values. However, the z-values play an important role when it comes to the problem with our analysis.§.§ z-values instead of Ranks Now that the concepts of the statistical tests are introduced, we can discuss the actual problem in our implementation. We used thefunction of the PMCMR package <cit.> to implement the test. As part of the return values, the function returns a matrix called . Without checking directly in the source code, we assumed these were the average ranks for each population, based on the documention of the package. However, these are actually the absolute z-values multiplied with √(2), i.e., z' = |R_i-R_j|/√(k(k+1)/6N·√(2)).Thus, when we compared ranks, we did not actually compare the average ranks, but the mean z'-values. This led to a wrong determination of ranks, which explain the inconsistencies found by Y. Zhou. §.§ The solutionTo resolve the problem, we adopted the code from the PMCMR package that determines the average ranks. We cross-checked our code with another implementation of the Nemenyi test <cit.>, to ensure that the new code solves the problem.[Both implementations of the test do not return the raw pair-wise comparisons and can, therefore, not be used directly.] We then used the mean averages from that code, instead of the z-values that were returned from the PMCMR package. As a result, the Nemenyi-test became much more sensitive, because the scale of the average ranks is different than the scale of the z-values. Let us consider the scales for our experiments with the JURECZKO data. Here, we have N=62 data sets and k=135 populations, i.e., CPDP approach and classifier combinations. The best possible average rank is 135 (always wins), the worst possible is 1 (always loses). Thus, the average ranks are on a scale from 1 to 135. In comparison, the highest possible z'-value isz = (135-1)/√(135·136/6·62)·√(2)≈ 26.97,i.e., the scale is just from 0 (no difference in average ranks) and 26.97. Thus, the scale of the z'-values has only about a fifth of the range the scale of the average ranks has. Basically, with z'-values, 135 populations are fit into the scale 0 to 26.97, with rankings in the scale 1 to 135. This means that the average distance between approaches is 0.2 with z'-values and 1 in case of average ranks. Considering that we have a CD ≈ 1.26 for α=0.95 in this example, this makes a huge difference. With z'-values, it is unlikely that two subsequently ranked approaches to have a distance greater than CD, because CD was more than 6.3 times higher than average distance expected on the scale. This changes if the real scale with ranks is used. If you have 135 cases with an average distance of 1, it is quite likely that a few of these distances will be greater than 1.26, i.e., the CD. We discuss this change in scales in this detail, because it requires a small change in the ranking of approaches based on the Nemenyi test. Before, we considered three distinct ranks for the calculation of the rankscore to achieve non-overlapping ranks: * The populations that are within the CD of the best average ranking population (top rank 1).* The populations that are within the CD of the worst average ranking population (bottom rank 3).* The populations that are neither (middle rank 2).This was the only way to deal with the small differences that resulted from using the z'-values. However, this approach breaks on the larger scale, because the distances now become larger, meaning fewer results are within the CD from the best/worst ranking. For example, for the JURECZKO data and the performance metric AUC, only two approaches would be on the first rank, i.e., only one approach is within the CD of the best approach. Similarly, only six approaches would be on the third rank, i.e., only five approaches are within the CD of the worst approach. This would leave us with 127 approaches on the middle rank. This ranking would be to coarse, and not show actual differences between approaches anymore. To deal with this larger scale of ranks, we use a simple and more fine-grained grouping strategy to create the ranks. We sort all approaches by their average ranking. Each time that the difference between two subsequently ranked approaches is larger than the CD, we increase the rank. Because the rank is only increased if the difference is larger than the CD, we ensure that each group only contains approaches that are statistically significantly different from the other groups. Afterwards, we calculate the normalized rankscore as before. Algorithm <ref> formalizes this strategy. This change in ranking increases the sensitivity of the test and makes the results more fine-grained in comparison to our original ranking procedure.§ RESULTSWe now show the corrected results for RQ1. We will directly compare the changes in the results with the originally published results. Figure <ref> shows the mean rankscore averaged over the four performance metrics F-Measure, G-Measure, AUC, and MCC and the five data sets JURECZKO, MDP, AEEEM, NETGENE, and RELINK. Table <ref> shows detailed results including the mean values and rankscores for each performance metrics and each data set. Figure <ref> is the correction of Figure 3 and Table <ref> the correction of Table 8 from the original publication. Table <ref> and Figure <ref> only report the results for the best classifier for each approach. In case these changed between the original results and our correction, you will not find the exact same rows. For example, for CamargoCruz09, we reported DT as best classifier in the original analysis, and now NB. This is because with the problem in the statistical analysis DT was ranked best for CamargoCruz09, but in the corrected version NB performs better. The reasons for these, and other changes are explained in Section <ref>The most important finding remains the same: the approach CamargoCruz09 still provides the best-ranking classification model with a mean rankscore of 0.917 for CamargoCruz09-NB. However, the rankscore is not a perfect 1.0 anymore. We attribute this to the more sensitive ranking due to the correction of the Nemenyi test. The differences to the next-ranking approaches are still rather small, though the group of approaches that is within 10% of the best ranking approach now only consists of CV-RF, Amasaki15, Peters15, and Ma12. The bottom of the ranking is nearly unaffected by the changes as well. The last seven ranked approaches are still the same. Additionally, our findings regarding the comparison of using ALL data, versus transfer learning approaches has not changed: ALL is still in the upper mid-field of approaches. With the corrected and more finegrained ranking, only six of the cross-project approaches actually outperform this baseline, whereas seventeen are actually worse.With respect to CV versus CPDP, we still observe that CPDP can outperform CV in case multiple performance criteria are considered because CV-RF is outperformed by CamargoCruz09-NB. Thus, we still note that this is possible, but far less conclusively than before, where CV was actually only in the mid-field of the approaches and not a close second.Due to these small overall small differences, we change our answer to RQ1 slightly: Answer RQ1: CamargoCruz09-NB performs best among the compared CPDP approaches and even outperforms cross-validation. However, the differences to other approaches are small. The baseline ALL-NB is ranked higher than seventeen of the CPDP approaches. § REASONS FOR CHANGESWe checked our raw results for the reasons for all changes in rankings. The problem with the statistical analysis actually led to two reasons for ranking changes: first, the z-values already consider differences in ranks. Thus, if the rank was very high, this could would lead to larger z-values, which would negatively impact the ranking. Second, because differences were downscaled with the z-values in comparison to differences in mean ranks, too many approaches were grouped together as not statistically significantly different. For approaches that are now better ranked than before, this means that they were often among the best performing approaches within a group. For those that are now ranking worse, they were at often near the bottom of their groups. For example, CV was often among the best approaches on the middle rank. Now, it is clearly distinguished from the others there, leading to the strong rise in the ranking. Others that were affected the same way, though to a lesser extend are Amasaki15, Peters15, YZhang15, and Herbold13. On the other hand Menzies11 and Watanabe08 were often at the bottom of their groups, leading to the big losses in rankings for both approaches.Another change in our results is that NB is often the best performing classifier, whereas before DT and LR were most often the best performing classifiers. We previously already noted in our discussion that “for many approaches the differences between the classifiers were rather small” <cit.>. Together with the reasons for ranking changes explained above, theses changes are not unexpected. Overall, all changes in the result are plausible. Moreover, our comparison of the results of the statistical analysis with both mean values, as well as the raw results of the benchmark did not reveal any inconsistencies of the type that Y. Zhou reported to us. Therefore, we believe that the problem was correctly resolved. § UPDATE OF THE REPLICATION KITWe updated the replication kit archived at Zenodo <cit.>. The changes two the replication kit are two-fold. * We corrected the problem with the statistical analysis in the generate_results.R script.* We updated the provided CD diagrams due to the changes in the Nemenyi test. The changes can be reviewed in detail in the commit to the GitHub archive of the replication kit[https://goo.gl/AbvSRj]. § CONCLUSIONA problem with the implementation of the Nemenyi post-hoc test led to incorrect results being published in our benchmark paper on cross-project defect prediction. The mistake only affected research question RQ1, the other three research questions were not affected. Within this correction paper, we explained the problem in the statistical test, how this problem affected our results, presented the corrected, and explained the changes that occoured. The major findings regarding RQ1 are not changed, including the best performing approach, the result that the naïve baseline of using all data outperforms most proposed transfer learning approaches, as well as that cross-validation can be outperformed by CPDP. Thus, the contributions of the article are still valid. Still, the correction leads to differences in the rankings which are properly corrected and discussed here. We apologize for this mistake and hope that this timely correction mitigates the potential negative impact the wrong results may have.§ ACKNOWLEDGEMENTS We want to thank Yuming Zhoufrom Nanjing University for pointing out the inconsistencies in the results to us so fast, as well as the editors of this journals who helped to determine how we should communicate this problem to the community within days.IEEEtran | http://arxiv.org/abs/1707.09281v1 | {
"authors": [
"Steffen Herbold",
"Alexander Trautsch",
"Jens Grabowski"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170727090623",
"title": "Correction of \"A Comparative Study to Benchmark Cross-project Defect Prediction Approaches\""
} |
[pages=1-last]AFSCNN_Final.pdf | http://arxiv.org/abs/1707.08630v2 | {
"authors": [
"Shizhong Han",
"Zibo Meng",
"Zhiyuan Li",
"James O'Reilly",
"Jie Cai",
"Xiaofeng Wang",
"Yan Tong"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170726201513",
"title": "Optimizing Filter Size in Convolutional Neural Networks for Facial Action Unit Recognition"
} |
1] Maria Rita Casali 2] Luigi Grasselli[1] Department of Physics, Mathematics and Computer Science, University of Modena and Reggio Emilia,Via Campi 213 B - 41125 Modena (Italy), [email protected][2] Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Via Amendola 2, Pad. Morselli - 42122 Reggio Emilia (Italy), [email protected] Combinatorial properties of the G-degree [ December 30, 2023 ======================================== A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the G-degree of the involved graphs, which drives the 1/N expansion in the tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension d≥ 4, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of (d-1)!.As a consequence, in even dimension,the terms ofthe1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context.In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of“associated" cyclic permutations,several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds. Keywords: edge-colored graph; PL-manifold; singular manifold; colored tensor model;regular genus; Gurau-degree. 2000 Mathematics Subject Classification:57Q15 - 57N13 - 57M15 - 83E99.§ INTRODUCTION It is well-known that regular edge-colored graphs may encode PL-pseudomanifolds, giving rise to a combinatorial representation theory (crystallization theory) for singular PL-manifolds of arbitrary dimension (see Section <ref>). In the last decade, the strong interaction between the topology of edge-colored graphs and random tensor models has been deeply investigated, bringing insights in both research fields.The colored tensor models theory arises as a possible approach to the study of Quantum Gravity: in some sense, its aim is to generalize to higher dimension the matrix models theory which, in dimension two, has shown to be quite useful at providing a framework for Quantum Gravity. The key generalization is the recovery of the so called 1/N expansion in the tensor models context. In matrix models, the 1/N expansion is driven by the genera of the surfaces represented by Feynman graphs; in the higher dimensional setting of tensor models the 1/N expansion is driven by the G-degree of these graphs (see Definition <ref>), that equals the genus of the represented surface in dimension two.If (ℂ^N)^⊗ d denotes the d-tensor product of the N-dimensional complex space ℂ^N, a (d+1)-dimensional colored tensor model is a formal partition function𝒵[N,{α_B}]:=∫_dTdT/(2π)^N^dexp(-N^d-1T· T + ∑_Bα_BB(T,T)),where T belongs to (ℂ^N)^⊗ d , T to its dual and B(T,T) are trace invariants obtained by contracting the indices of the components of T and T. In this framework, colored graphs naturally arise as Feynman graphs encoding tensor trace invariants.As shown in <cit.>, the free energy1/N^dlog𝒵[N,{t_B}] is the formal series1/N^dlog𝒵[N,{t_B}] = ∑_ω_G≥ 0N^-2/(d-1)!ω_GF_ω_G[{t_B}]∈ℂ[[N^-1, {t_B}]],where the coefficients F_ω_G[{t_B}] are generating functions of connected bipartite (d+1)-colored graphs with fixed G-degree ω_G.The 1/N expansion of formula (<ref>) describes the rôle of colored graphs (and of their G-degree ω_G) within colored tensor models theory and explains the importance of trying to understand which are the manifolds and pseudomanifolds represented by (d+1)-colored graphs with a given G-degree.A more detailed description of these relationships between Quantum Gravity via tensor models and topology of colored graphs may be found in <cit.>, <cit.>, <cit.>, <cit.>.A parallel tensor models theory, involving real tensor variables T∈ (ℝ^N)^⊗ d, has been developed, taking into account also non-bipartite colored graphs (see <cit.>): this is why both bipartite and non-bipartite colored graphs will be considered within the paper.Section <ref> contains a quick review of crystallization theory, including the ideaof regular embedding of edge-colored graphs into surfaces, which is crucial for the definitions of G-degree and regular genus of graphs (Definition <ref>).In Section <ref>, combinatorial properties concerning Hamiltonian decompositions of the complete graph allow to prove the main results of the paper.For eachbipartite (d+1)-colored graph (Γ,γ),with d even, d ≥ 4, ω_G (Γ) ≡ 0(d-1)!For each(bipartite or non-bipartite) (d+1)-colored graph (Γ,γ) representing a singular d-manifold, with d even, d≥ 4,ω_G (Γ) ≡ 0(d-1)! Note that the above results turn out to have specific importance in the tensor models framework. In fact Theorem <ref> implies that, in the d-dimensional complex context, with d even and d≥ 4,the only non-null terms in the1/N expansion of formula (<ref>) are the ones corresponding to even (integer) powers of 1/N.On the other hand, Theorem <ref> ensures that in the real tensor models framework, where also non-bipartite graphs are involved, the 1/N expansion contains colored graphs representing (orientable or non-orientable)singular manifolds - and, in particular, closed manifolds - only in the terms corresponding to even (integer) powers of 1/N. Both Theorems extend to arbitrary even dimension a result proved in <cit.> for graphs representing singular 4-manifolds.Section <ref> is devoted to the 4-dimensional case: in this particular situation, the general results ofSection <ref> allow to obtain interesting properties relating the G-degree with the topology of the associated PL 4-manifolds.In fact, the G-degree of a 5-colored graph is shown to depend only on the regular genera with respect to an arbitrary pair of“associated" cyclic permutations (Proposition <ref>). This fact yields relations between these two genera and the Euler characteristic ofthe associated PL 4-manifold (Proposition <ref> and Proposition <ref>); moreover, two interesting classes of crystallizations arise in a natural way,whose intersection consists in the known class of semi-simple crystallizations, introduced in <cit.> (see Remark <ref>).§ EDGE-COLORED GRAPHS AND G-DEGREEA singular d-manifold is a compact connected d-dimensional polyhedron admitting a simplicial triangulation where the links of vertices are closed connected (d-1)-manifolds, while the link of any h-simplex, for each h > 0, is a PL (d-h-1)-sphere.A vertex whose link is not a PL (d-1)-sphere is called singular. The class of singular d-manifolds includes the class of closed d-manifolds: in fact, a closed d-manifold is a singular d-manifold without singular vertices.Moreover, if N is a singular d-manifold, then a compact PL d-manifold Ň is obtained by deleting small open neighbourhoods of its singular vertices. Obviously, N=Ň if and only if N is a closed manifold; otherwise, Ň has a non-empty boundary without spherical components. Conversely, given a compact PL d-manifold M, a singular d-manifold M can be obtained by capping off each component of ∂ M by a cone over it.Note that, in virtue of the above correspondence, a bijection is defined between singular d-manifolds and compact PL d-manifolds with no spherical boundary components. A (d+1)-colored graph (d ≥ 2) is a pair (Γ,γ), where Γ=(V(Γ), E(Γ)) is a regular d+1 valent multigraph(i.e. multiple edges are allowed, while loops are forbidden)and γ: E(Γ) →Δ_d={0,…, d} is a map injective on adjacent edges, called coloration. For every ℬ⊆Δ_d let Γ_ℬ be the subgraph obtained from (Γ, γ) by deleting all the edges colored by Δ_d - ℬ. The connected components of Γ_ℬ are called ℬ-residues or, if #ℬ = h, h-residues of Γ; the symbol g_ℬ denotes their number. In the following, if ℬ ={c_1,…,c_h}, its complementary set Δ_d - ℬ will be denoted by ĉ_1…ĉ_h.Given a (d+1)-colored graph (Γ, γ), a d-dimensional pseudocomplex K(Γ) can be associated by the following rules: * for each vertex of Γ, let us consider a d-simplex and label its vertices by the elements of Δ_d;* for each pair of c-adjacent vertices of Γ (c∈Δ_d), let us glue the corresponding d-simplices along their (d-1)-dimensional faces opposite to the c-labeled vertices, so that equally labeled vertices are identified. |K(Γ)| turns out to be a d-pseudomanifold[In fact, |K(Γ)| is a quasi-manifold: see<cit.>.], whichis orientable if and only if Γ is bipartite,and (Γ, γ) is said to represent it.Note that, by construction, K(Γ) is endowed with a vertex-labeling by Δ_d that is injective on any simplex. Moreover, a bijective correspondence exists between the h-residues of Γ colored by any ℬ⊆Δ_d and the (d-h)-simplices of K(Γ) whose vertices are labeled by Δ_d - ℬ. In particular, for any color c∈Δ_d, each connected component of Γ_ĉ is a d-colored graph representinga pseudocomplex that is PL-homeomorphic to the link of a c-labeled vertex of K(Γ) in its first barycentric subdivision. As a consequence, |K(Γ)| is a singular d-manifold (resp. a closed d-manifold) iff for each color c∈Δ_d, all ĉ-residues of Γ represent closed (d-1)-manifolds (resp. the (d-1)-sphere). In virtue of the bijection described in Remark <ref>, a (d+1)-coloredgraph (Γ,γ) is said to represent a compact PL d-manifold M with no spherical boundary components if and only ifit represents the associated singular manifold M. A crystallization of a closed PL d-manifold M^d is a (d+1)-colored graph representing M^d, such that each d-residue is connected (i.e.g_î=1∀ i ∈Δ_d).The following theorem extends to singular manifolds a well-known result - due to Pezzana (<cit.>) - founding the combinatorial representation theory for closed PL-manifolds of arbitrary dimension via colored graphs (the so called crystallization theory).See also <cit.> and <cit.> for the 3-dimensional case.(<cit.>)Any singular d-manifold- or, equivalently, any compact d-manifold with no spherical boundary components -admits a (d+1)-colored graph representing it.In particular, each closed PL d-manifold admits a crystallization.It is well known the existence of a particular set of embeddings of a bipartite (resp. non-bipartite) (d+1)-colored graph into orientable (resp. non-orientable) surfaces.(<cit.>) Let (Γ,γ) be a bipartite (resp. non-bipartite) (d+1)-colored graph of order 2p. Then for each cyclic permutation ε = (ε_0,…,ε_d) of Δ_d, up to inverse, there exists a cellular embedding, called regular, of (Γ,γ) into an orientable (resp. non-orientable) closed surface F_ε(Γ) whose regions are bounded by the images of the {ε_j,ε_j+1}-colored cycles, for each j ∈ℤ_d+1. Moreover, the genus (resp. half the genus)ρ_ε (Γ) of F_ε(Γ) satisfies χ (F_ε(Γ)) =2 - 2ρ_ε(Γ)= ∑_j∈ℤ_d+1 g_ε_jε_j+1 + (1-d)p. No regular embeddings of (Γ,γ) exist into non-orientable (resp. orientable) surfaces. The Gurau degree (often called degree in the tensor models literature) and the regular genus of a colored graph are defined in terms of the embeddings of Theorem <ref>.Let (Γ,γ) be a (d+1)-colored graph. If {ε^(1), ε^(2), … , ε^(d!/2)} is the set of all cyclic permutations of Δ_d (up to inverse), ρ_ε^(i) (i=1, … , d!/2) is called the regular genus of Γ with respect to the permutation ε^(i). Then, the Gurau degree (or G-degree for short) of Γ, denoted byω_G(Γ), is defined asω_G(Γ)=∑_i=1^d!/2ρ_ε^(i)(Γ)and the regular genus of Γ, denoted by ρ(Γ), is defined asρ(Γ)=min {ρ_ε^(i)(Γ) / i=1,…,d!/2}. Note that, in dimension 2, any bipartite (resp. non-bipartite) 3-colored graph (Γ,γ) represents an orientable (resp. non-orientable) surface |K(Γ)| and ρ(Γ)= ω_G(Γ) is exactly the genus (resp. half the genus) of |K(Γ)|. On the other hand, for d≥ 3, the G-degree of any (d+1)-colored graph (resp. the regular genus of any (d+1)-colored graph representing a closed PL d-manifold) is proved to be a non-negative integer, both in the bipartite and non-bipartite case:see <cit.> (resp. <cit.>).§ PROOF OF THE GENERAL RESULTSWithin combinatorics,the problem of the existence of m-cycle decompositions of the complete graph K_n, or of the complete multigraph λ K_n (i.e. the multigraph with n vertices and with λ edges joining each pair of distinct vertices) is long standing: a survey result, for general m, n and λ,is given in <cit.>.Moreover, the following results hold, concerning Hamiltonian cycles (i.e. m=n) in K_n, both in the case n odd and in the case n even.<cit.>For all odd n ≥ 3 there exists a partition of all Hamiltonian cycles of K_n into (n-2)!Hamiltonian cycle decompositions of K_n. <cit.>For all even n ≥ 4 there exists a partition of all Hamiltonian cycles of K_n into (n-2)!/2 classes, so that each edge of K_n appears in exactly two cycles belonging to the same class.Figure 1 describes - as an example of Proposition <ref>- the six Hamiltonian cycle decompositions of K_5, each of them containing a pair of disjoint Hamiltonian cycles (given by the dashed and continuous edges respectively). Note that, by labelling the vertices of K_n with the elements of Δ_n-1, each Hamiltonian cycle in K_n defines a cyclic permutation of Δ_n-1 (together with its inverse), and viceversa. On the other hand, the following statement regarding the G-degree has been recently proved.<cit.>If (Γ, γ) is a (d+1)-colored graphof order 2p (d≥ 3), thenω_G(Γ)=(d-1)!/2·(d + p · (d-1) ·d/2-∑_r,s ∈Δ_d g_rs).As a consequence, the G-degree of any (d+1)-colored graph is a non-negative integer multiple of (d-1)!/2.The result of Proposition <ref>, which was originally stated in the bipartite case (see <cit.>), suggested the definition, for d≥ 3, of the (integer) reduced G-degreeω^'_G(Γ)=2/(d-1)!·ω_G(Γ),which is used by many authors within tensor models theory (see for example <cit.>).[In fact, the exponents of N^-1 in the1/N expansion of formula (<ref>) are all (non-negative) integers ω_G^'.] Actually, we are able to prove that, if d ≥ 4 is even, underrather weak hypotheses,the G-degree is multiple of (d-1)! (or, equivalently, the reduced G-degree is even). If d ≥ 4 is even, and(Γ, γ) is a (d+1)-colored graph such thateach 3-residue is bipartite and each d-residue is either bipartite or non-bipartite with integer regular genus with respect to any permutation,thenω_G(Γ) ≡ 0(d-1)! Let (Γ,γ) be a (d+1)-colored graph, with d ≥ 4, d even. Since n = d+1 is odd, Proposition <ref> implies that all d!/2 cyclic permutations (up to inverse) of Δ_d can be partitioned in (d-1)! classes, each containing d/2 cyclic permutations, ε̅^(1), ε̅^(2), … , ε̅^(d/2) say, so that∑_i=1^d/2[ ∑_j∈ℤ_d+1 g_ε̅^(i)_j,ε̅^(i)_j+1] = ∑_r,s∈Δ_d g_r,s.Hence, by Theorem <ref>,2 ·d/2 - 2 ·∑_i=1^d/2ρ_ε̅^(i) =∑_r,s ∈Δ_d g_rs + p · (1-d) ·d/2,and2 ·∑_i=1^d/2ρ_ε̅^(i) =d + p · (d-1) ·d/2-∑_r,s ∈Δ_d g_rs.This proves that the quantity∑_i=1^d/2ρ_ε̅^(i) is constant, for each class of the above partition; then,ω_G (Γ)=(d-1)! ·∑_i=1^d/2ρ_ε̅^(i)immediately follows.[In this way, one can reobtain - for d ≥ 4 even - relation (<ref>).] Now, the hypotheses on the 3-residues and d-residuesof Γ directly implies, in virtue ofTheorem <ref> andof <cit.> (originally proved in <cit.>), that∑_i=1^d/2ρ_ε̅^(i) is an integer, and hence ω_G (Γ) ≡ 0(d-1)! (or equivalently, ω_G^' (Γ) = 2 ·ω_G (Γ)/(d-1)! is even).Then, the results stated in Section <ref> trivially follow. Proofof Theorems 1 and 2. Both in the case of (Γ, γ) bipartite and in the case of (Γ, γ) representing a singular d-manifold,the residues of Γ obviously satisfy the hypotheses of Proposition <ref>.Hence, if d≥ 4 is even,ω_G(Γ) ≡ 0(d-1)! holds. Another particular situation is covered by Proposition<ref>, as the following corollary explains. Let (Γ,γ) be a (d+1)-colored graph, with d ≥ 4, d even. If (Γ,γ)is a non-bipartite (d+1)-colored graph such that eachd-residue is bipartite, thenω_G (Γ) ≡ 0(d-1)!-0.3truecmNote that there exist (d+1)-colored graphs, with d even, d ≥ 4 and odd reduced G-degree: of course, in virtue of Theorems 1 and 2, the represented d-pseudomanifold must be non-orientable and it can't be a singular d-manifold. As an example, for each d ≥ 4, the (d+1)-colored graph(Γ,γ) of Figure 2 represents the (d-2)-th suspension Σ of the real projective plane ℝℙ^2and ω_G^'(Γ)=d-1.It is easy to check that Γ is non-bipartite and Σ is not a singular manifold, since all d-residues of Γ, with the exception of Γ_0̂, do not represent a closed (d-1)-manifold.Moreover, its (unique) {0,1,c }-residue, for any c ∈{2, …, d}, is non-bipartite and representsthenon-orientable genus one surface ℝℙ^2.In the case d ≥ 3 odd, Proposition <ref> implies that all d!/2 cyclic permutations (up to inverse) of Δ_d can be partitioned in (d-1)!/2 classes, each containing d cyclic permutations, ε̅^(1), ε̅^(2), … , ε̅^(d) say, so that∑_i=1^d[ ∑_j∈ℤ_d+1 g_ε̅^(i)_j,ε̅^(i)_j+1] = 2 ∑_r,s∈Δ_d g_r,s.Hence, a reasoning similar to the one used to prove Proposition <ref> yields an alternative proof - for d ≥ 3 odd - of relation (<ref>):since2 · d - 2 ·∑_i=1^d ρ_ε̅^(i) = 2 ∑_r,s ∈Δ_d g_rs + p · (1-d) · dand∑_i=1^d ρ_ε̅^(i)=d + p · (d-1) ·d/2-∑_r,s ∈Δ_d g_rshold, thenω_G (Γ)= (d-1)!/2·∑_i=1^d ρ_ε̅^(i) = (d-1)!/2·(d + p · (d-1) ·d/2 -∑_r,s ∈Δ_d g_rs)directly follows. It is worthwhile to stress that, for d even (resp. odd), formula (<ref>) of Proposition <ref> (resp. formula (<ref>) of Remark <ref>) proves that the sum ∑_i=1^d/2ρ_ε̅^(i) (resp. ∑_i=1^d ρ_ε̅^(i)) of all regular genera with respect to the d/2 (resp. d) permutations belonging to the same class is half the reduced G-degree (resp. is the reduced G-degree)ω^'_G(Γ)=d + p · (d-1) ·d/2-∑_r,s ∈Δ_d g_rs,i.e. it is a constant which does not depend on the chosen partition class.Hence, the regular genus ρ(Γ) of the graph Γ is realized by the (not necessarily unique) permutationε which maximizes the differenceρ_ε̂(Γ) - ρ_ε(Γ),where ρ_ε̂(Γ) denotes the sum of the genera with respect to all other permutations of the same partition class.§ THE 4-DIMENSIONAL CASEIn the 4-dimensional setting, the above combinatorial properties allow to prove further results about the G-degree.In fact, it is easy to check that, for eachcyclic permutation ε = (ε_0, ε_1, ε_2, ε_3, ε_4) of Δ_4, we have{ (ε_j,ε_j+1)/j∈ℤ_5} ∪ { (ε_j, ε_j+2)/ j∈ℤ_5} ={ (i,j)/ i,j∈Δ_5, ij}. Let us denote by ε^'=(ε^'_0, ε^'_1, ε^'_2, ε^'_3, ε^'_4)the permutation (ε_0, ε_2, ε_4, ε_1, ε_3) of Δ_4, which may be said to be associated to ε.Note that, when d=4, the only partition of all 12 cyclic permutations of Δ_4 (up to inverse) is given by the six classes containing a given permutation ε and its associated ε^': see Figure 1, where each class of the Hamiltonian cycle decomposition of K_5 ensured by Proposition <ref>is shown to correspond to a pair (ε_i, ε_i^') of associated permutations, for i ∈ℕ_6. For example, the first class corresponds to the identical permutation ε_1 = (0,1,2,3,4) and its associated permutation ε_1^' = (0,2,4,1,3). Then, the following result holds.For each 5-colored graph(Γ, γ),and for each pair (ε, ε^') of associated cyclic permutations of Δ_4,ω_G(Γ) =6 (ρ_ε(Γ) + ρ_ε^'(Γ)). Equality (<ref>) directly yields∑_j∈ℤ_5 g_ε_j,ε_j+1 +∑_j∈ℤ_5 g_ε^'_j,ε^'_j+1= ∑_i,j∈Δ_5 g_i,j. As a consequence, the sum of all regular genera of Γ with respect to the 12 cyclic permutations (up to inverse) of Δ_4is six times the sum between the regular genera of Γ with respect to any pair ε, ε^' of associated permutations:ω_G(Γ) =6 (ρ_ε(Γ) + ρ_ε^'(Γ)).-0.6truecmBy Proposition <ref>, for any 5-colored graph the sum between the regular genera of Γ with respect to any pair ε, ε^' of associated cyclic permutationsis constant (see equality (<ref>) and Remark <ref>, for d=4):ρ_ε(Γ) + ρ_ε^'(Γ)=1/2ω'_G(Γ) =2+3p - 1/2∑_i,j∈Δ_5 g_i,j.Hence, the regular genus ρ(Γ) of the graph Γ is realized by the (not necessarily unique) permutation ε so that ρ_ε^'(Γ) - ρ_ε(Γ) is maximal. Moreover: (a) If(Γ, γ) is a 5-colored graph, then for each pair (ε, ε^') of associated cyclic permutations of Δ_4,2 (ρ_ε^'(Γ) - ρ_ε(Γ) ) =∑_j∈ℤ_5 g_ε_j,ε_j+1 - ∑_j∈ℤ_5 g_ε_j,ε_j+2 (b) If(Γ, γ) is a 5-colored graph representing a singular 4-manifold M^4,then for each pair (ε, ε^') of associated cyclic permutations of Δ_4,ρ_ε^'(Γ) - ρ_ε(Γ) =∑_j∈ℤ_5 g_ε_j,ε_j+1,ε_j+2 - ∑_j∈ℤ_5 g_ε_j,ε_j+2, ε_j+4 Statement (a) is an easy consequence of Theorem <ref>:2-2 ρ_ε(Γ) = ∑_j∈ℤ_5 g_ε_j,ε_j+1 -3pand2-2 ρ_ε^'(Γ) = ∑_j∈ℤ_5 g_ε^'_j,ε^'_j+1 -3p=∑_j∈ℤ_5 g_ε_j,ε_j+2 -3p.On the other hand, relation 2g_r,s,t = g_r,s + g_r,t+ g_s,t - p is known to be true for each order 2p 5-colored graph representing a singular 4-manifold (see <cit.>. As a consequence we have:2 ∑_j∈ℤ_5 g_ε_j,ε_j+1,ε_j+2= ∑_r,s∈ℤ_5 g_r,s + ∑_j∈ℤ_5 g_ε_j,ε_j+1 - 5p and2 ∑_j∈ℤ_5 g_ε_j,ε_j+2,ε_j+4= ∑_r,s∈ℤ_5 g_r,s + ∑_j∈ℤ_5 g_ε_j,ε_j+2 - 5p.By making the difference,2 (∑_j∈ℤ_5 g_ε_j,ε_j+1,ε_j+2 - ∑_j∈ℤ_5 g_ε_j,ε_j+2,ε_j+4) =∑_j∈ℤ_5 g_ε_j,ε_j+1 - ∑_j∈ℤ_5 g_ε_j,ε_j+2is obtained; so, statement (b) follows, via statement (a). Proposition <ref> enables to obtain the following improvement of<cit.>. Let (Γ,γ) be a 5-colored graph.Then:ω_G(Γ)=12 ·ρ (Γ)⟺ ∑_j∈ℤ_5 g_ε_j, ε_j+1 =∑_j∈ℤ_5 g_ε_j, ε_j+2 for each cyclic permutation ε of Δ_4. From <cit.>, it is known thatω_G(Γ)=12 ·ρ (Γ)⟹∑_j ∈ℤ_5 g_ε̅_j, ε̅_j+1 =∑_j∈ℤ_5 g_ε̅_j, ε̅_j+2,ε̅ being the cyclic permutation of Δ_4 such that ρ(Γ) = ρ_ε̅(Γ). So, ρ_ε̅^'(Γ) - ρ_ε̅(Γ) = 0 directly follows via Proposition <ref> (a). Now, since ρ_ε(Γ) + ρ_ε^'(Γ) is constant for each pair (ε, ε^') of associated cyclic permutations of Δ_4 (see Remark <ref>), then:|ρ_ε^'(Γ) - ρ_ε(Γ)| ≤ρ_ε̅^'(Γ) - ρ_ε̅(Γ).Hence, ρ_ε̅^'(Γ) - ρ_ε̅(Γ) = 0 implies ∑_j∈ℤ_5 g_ε_j, ε_j+1 =∑_j∈ℤ_5 g_ε_j, ε_j+2 (i.e. ρ_ε^'(Γ) - ρ_ε(Γ) = 0, in virtue of Proposition <ref> (a)) for each cyclic permutation ε of Δ_4.The reversed implication is straightforward, via Proposition <ref>(a).If (Γ, γ)is an order 2p 5-colored graph representing a singular 4-manifold M^4, then, for each pair (ε, ε^') of associated cyclic permutations of Δ_4:χ(M^4) =(ρ_ε(Γ) + ρ_ε^'(Γ)) -p + ∑_i∈Δ_4g_î -2.It is sufficient to applyProposition <ref>to the third equality of <cit.>. Let us now recall two particular types of crystallizations introduced and studied in <cit.> and <cit.>[Both semi-simple and weak semi-simple crystallizations generalize the notion of simple crystallizations for simply-connected PL 4-manifolds: see <cit.> and <cit.>.]: they are proved to be “minimal" with respect to regular genus, among all graphs representing the same PL 4-manifold. A crystallization of a PL 4-manifold M^4 with rk(π_1(M^4))= m ≥ 0 is called a semi-simple crystallization ifg_j,k,l = 1 + m ∀ j,k,l ∈Δ_4.A crystallization of a PL 4-manifold M^4 with rk(π_1(M^4))= m is called a weak semi-simple crystallization ifg_i, i+1, i +2 = 1 + m ∀ i ∈Δ_4 (where the additions in subscripts are intended in ℤ_5).According to <cit.>, for each order 2p crystallization(Γ, γ) of a closedPL 4-manifold M^4, with rk(π_1(M^4))= m0, let us setg_j,k,l= 1+m + t_j,k,l,with t_j,k,l∀ j,k,l ∈Δ_4.Semi-simple (resp. weak semi-simple) crystallizations turn out to be characterized by t_j,k,l = 0 ∀ j,k,l ∈Δ_4 (resp. t_i, i+1, i +2 = 0∀ i ∈ℤ_5).In <cit.>, the relationp= 3 χ(M^4)+ 5(2m -1) +∑_j,k,l ∈Δ_4 t_j,k,lis proved to hold; hence, p = p̅ + q follows, where q= ∑_j,k,l ∈Δ_4 t_j,k,l≥ 0 and p̅= 3 χ(M^4)+ 5(2m -1) is the minimum possible half order of a crystallization of M^4, which is attained if and only if M^4 admits semi-simple crystallizations. With the above notations, the following results can be obtained.Let (Γ,γ) be an order 2p crystallizationof a closed PL 4-manifold M^4, with rk(π_1(M^4))=m. Then, for each pair (ε, ε^') of associated cyclic permutations of Δ_4, with ρ_ε(Γ) ≤ρ_ε^'(Γ):ρ_ε^'(Γ) - ρ_ε(Γ) = q - 2 ∑_i∈ℤ_5t_ε_i,ε_i+2,ε_i+4 ≤ q.Moreover, The following statements are equivalent: (a) a cyclic permutation ε of Δ_4 exists, such that ρ_ε^'(Γ) - ρ_ε(Γ) =q;(b)Γ is a weak semi-simple crystallization;(c) ρ(Γ)= 2χ(M^4) + 5 m -4.[Note that, in this case, the permutation ε̅ such that ρ(Γ)= ρ_ε̅(Γ) coincides with the permutation ε of point (a); moreover, ρ(Γ)= 𝒢(M^4) holds, in virtue of the inequality𝒢(M^4) ≥ 2χ(M^4) + 5 m -4, proved in<cit.>.] In virtue of Proposition <ref>(b), an easy computation proves relation (<ref>):ρ_ε^'(Γ) - ρ_ε(Γ)=∑_i∈ℤ_5(m+1 + t_ε_i,ε_i+1,ε_i+2 ) - ∑_i∈ℤ_5 (m+1 + t_ε_i,ε_i+2,ε_i+4)= =∑_i∈ℤ_5t_ε_i,ε_i+1,ε_i+2 - ∑_i∈ℤ_5 t_ε_i,ε_i+2,ε_i+4 = =∑_j,k,l ∈Δ_4t_j,k,l - 2 ∑_i∈ℤ_5t_ε_i,ε_i+2,ε_i+4 = = q - 2 ∑_i∈ℤ_5t_ε_i,ε_i+2,ε_i+4 ≤q. Now, if Γ is a weak semi-simple crystallization, by definition itself a cyclic permutation ε of Δ_4 exists[ε turns out to be the permutation of Δ_4 associated to ε^'=(0,1,2,3,4).], so that, for each i ∈ℤ_5,g_ε_i,ε_i+2,ε_i+4= m+1, i.e. t_ε_i,ε_i+2,ε_i+4=0.So, ρ_ε^'(Γ) - ρ_ε(Γ) =q easily follows from relation (<ref>).On the other hand,ifa cyclic permutation ε of Δ_4 exists, so that ρ_ε^'(Γ) - ρ_ε(Γ) =q,relation (<ref>) yieldst_ε_i,ε_i+2,ε_i+4=0, i.e. g_ε_i,ε_i+2,ε_i+4= m+1, which is exactly - up to color permutation - the definition of weak semi-simple crystallization. Hence, (a) and (b) are proved to be equivalent.Then, by comparing Proposition <ref> (with the assumption g_î=1 ∀ i ∈Δ_4) and relation (<ref>), we obtain:ρ_ε^'(Γ) + ρ_ε(Γ) = 2 ( 2 χ(M^4) + 5m -4) + q.By making use of relation (<ref>),ρ_ε(Γ)= 2 χ(M^4) + 5m -4 + ∑_i∈ℤ_5t_ε_i,ε_i+2,ε_i+4easily follows, as well asρ_ε^'(Γ)= 2 χ(M^4) + 5m -4 + ∑_i∈ℤ_5t_ε_i,ε_i+1,ε_i+2.Relation (<ref>) directly yields the co-implication between statements (b) and (c). We conclude the paper with a list of remarks, which arise from the previous results. With the above notations, formulae (<ref>) and (<ref>) immediately give2 + ρ_ε(Γ)/2 - 5 m/2 - q/4 ≤ χ(M^4) ≤ 2 + ρ_ε^'(Γ)/2 - 5 m/2 - q/4for each crystallizationof a closed PL 4-manifold M^4. Another double inequality concerning the Euler characteristic of a closedPL 4-manifold M^4 and the regular genera of any order 2p crystallization of M^4 with respect to a pair of associated permutations may be easily obtained fromProposition <ref>, by making use of the assumptions ∑_i∈Δ_4g_î=5 and ρ_ε(Γ) ≤ρ_ε^'(Γ):2ρ_ε(Γ) -p +3≤ χ(M^4) ≤ 2 ρ_ε^'(Γ) -p +3.Note that such double inequalities assume a specific relevance in case of “low" difference between ρ_ε^'(Γ) and ρ_ε(Γ) (in particular ifρ_ε^'(Γ)= ρ_ε(Γ) occurs, possibly with ρ(Γ) < ρ_ε(Γ)).On the other hand, Proposition <ref> and formula (<ref>)(resp. formula (<ref>)) giveχ(M^4) =2 ρ_ε(Γ) -p + 3 + (q -2 ∑_i∈ℤ_5t_ε_i,ε_i+2,ε_i+4) (resp. χ(M^4) =2 + ρ_ε(Γ)/2 - 5 m/2 - 1/2∑_i∈ℤ_5 t_ε_i,ε_i+2,ε_i+4 ).Hence, the following double inequalities arise, too, both involving the regular genus with respect to only one cyclic permutation:2ρ_ε(Γ) -p +3≤ χ(M^4) ≤ 2 ρ_ε(Γ) - p + q +3; 2 + ρ_ε(Γ)/2 - 5 m/2 - q/4 ≤ χ(M^4) ≤ 2 + ρ_ε(Γ)/2 - 5 m/2 . Note that relation (<ref>) exactly corresponds, via Proposition <ref>, to <cit.>. Moreover, relation (<ref>)[Actually, relation (<ref>) corrects a trivial error in the proof (and statement) of <cit.>, not affecting the implications in order to prove the main result of that paper.] directly implies the inequality ρ_ε(Γ)≥ 2 χ(M^4) + 5m -4 (which is one of the upper bounds obtained in <cit.>) and ensures that, for each crystallization (Γ,γ) of a PL 4-manifold, the regular genus ρ(Γ) is realized by the (not necessarily unique) permutation ε so that ∑_i∈ℤ_5t_ε_i,ε_i+2,ε_i+4 (or, equivalently, ∑_i∈ℤ_5g_ε_i,ε_i+2,ε_i+4) is minimal. Finally, the equivalence between items (b) and (c) in the above Proposition <ref> gives a direct proof of <cit.>. Semi-simple crystallizations turn out to be the intersection (characterized by q=0) between the two classes ofweak semi-simple crystallizations and of crystallizations satisfying ω_G(Γ)=12 ·ρ (Γ) (and hence ρ_ε(Γ) = 2 χ(M^4) + 5m -4 + 1/2 q, for each cyclic permutation ε of Δ_4).Moreover, it is easy to check thatq≤ 2⟹Γis a weak semi-simple crystallization.In fact, if q = ∑_j,k,l ∈Δ_4 t_j,k,l≤ 2, at most two triads (j,k,l) of distinct elements in Δ_4 exist, so that g_j,k,l= 1+m + t_j,k,l > 1+m. This ensures the existence of a cyclic permutation ε of Δ_4 so that, for each i ∈ℤ_5,g_ε_i,ε_i+1,ε_i+2= m+1, which is exactly the requirement for a weak semi-simple crystallization. The formula obtained in <cit.> for bipartite (d+1)-colored graphs and extended to the general case in <cit.> gives, if d=4,ω_G(Γ)= 3 (p + 4 - ∑_i∈Δ_4g_î) + ∑_i∈Δ_4ω_G(Γ_î),where, for eachi∈Δ_4, ω_G(Γ_î) denotes the sum of the G-degrees of the connected components of Γ_î. Hence, ∑_i∈Δ_4ω_G(Γ_î) is always a multiple of 3 (recall Proposition <ref> and Theorem <ref>).Moreover, if (Γ,γ) represents a singular 4-manifold M^4,Proposition <ref> and Proposition <ref> imply:∑_i∈Δ_4ω_G(Γ_î) = 3 (2 χ(M^4) + p - ∑_i∈Δ_4g_î).Note that, if (Γ,γ) is a crystallization of a closed PL 4-manifold M^4 with rk(π_1(M^4))=m, relation (<ref>) gives:∑_i∈Δ_4ω_G(Γ_î) =3 [5 (χ(M^4) + 2m -2 ) + q ].In particular, if (Γ,γ) is semi-simple (i.e. q=0), ∑_i∈Δ_4ω_G(Γ_î)=15 (χ(M^4) + 2m -2) follows, as <cit.> trivially implies.§ ACKNOWLEDGMENTSThis work was supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INDAM) and by University of Modena and Reggio Emilia, projects“Colored graphs representing pseudomanifolds: an interaction with random geometry and physics" and “Applicazioni della Teoria dei Grafi nelle Scienze, nell'Industria e nella Societá". 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Phys., 129, 142-167 (2018). https://doi.org/10.1016/j.geomphys.2018.01.001 [Casali-Cristofori-Gagliardi JKTR 2015] Casali, M.R., Cristofori, P., Gagliardi, C.: PL 4-manifolds admitting simple crystallizations: framed links and regular genus, Journal of Knot Theory and its Ramifications, 25(1), 1650005 [14 pages] (2016). https://doi.org/10.1142/S021821651650005XCasali-Cristofori-Grasselli Casali, M.R., Cristofori, P.,Grasselli, L.: G-degree for singular manifolds, RACSAM,112 (3), 693-704 (2018).https://doi.org/10.1007/s13398-017-0456-xChiavacci-Pareschi Chiavacci, R., Pareschi, G.: Some bounds for the regular genus of closed PL manifolds, Discrete Math., 82, 165-180 (1990).Cristofori-Fomynikh-Mulazzani-Tarkaev Cristofori, P., Fomynikh, E., Mulazzani, M., Tarkaev, V.: 4-colored graphs and knot/link complements, Results in Mathematics, 72 (1-2), 471-490 (2017).https://doi.org/10.1007/s00025-017-0686-4Cristofori-Mulazzani Cristofori, P., Mulazzani, M.:Compact 3-manifolds via 4-colored graphs, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A, Matematicas, 110 (2), 395-416 (2015).https://doi.org/10.1007/s13398-015-0240-8vGagliardi 1979Gagliardi, C.: A combinatorial characterization of 3-manifold crystallizations, Boll. Un. Mat. Ital. A, 16,441-449 (1979).Gagliardi 1981 Gagliardi, C.:Extending the concept of genus to dimension n, Proc. Amer. Math. Soc.,81, 473-481 (1981).Gurau-book Gurau, R.:Random Tensors, Oxford University Press, New York (2016).Gurau-Ryan Gurau, R., Ryan, J.P.: Colored Tensor Models - a review, SIGMA, 8, 020 (2012). https://doi.org/10.3842/SIGMA.2012.020[Pezzana] Pezzana, M.: Sulla struttura topologica delle varietà compatte, Atti Semin. Mat. Fis. Univ. Modena, 23, 269-277 (1974).Witten Witten, E.: An SYK-Like Model Without Disorder, preprint 2016.ArXiv:1610.09758v2Zhao-Kang Zhao, H., Kang, Q.:Large sets of Hamilton cycle and path decompositions, Discrete Mathematics,308, 4931-4940 (2008). | http://arxiv.org/abs/1707.09031v2 | {
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doublecases { . TUM-HEP 1088/17SI-HEP-2017-17, QFET-2017-15LMU-ASC 44/17MPP-2017-154Sharp spectral features from light dark matter decay via gravity portals [6mm] Oscar Catà^1,2, Alejandro Ibarra^3,4, Sebastian Ingenhütt^3,5^1 Theoretische Physik 1, Universität Siegen,Walter-Flex-Straße 3, D-57068 Siegen, Germany^2 Ludwig-Maximilians-Universität München, Fakultät für Physik,Arnold Sommerfeld Center for Theoretical Physics, 80333 München, Germany^3 Physik-Department T30d, Technische Universität München,James-Franck-Straße, 85748 Garching, Germany^4 School of Physics, Korea Institute for Advanced Study, Seoul 02455, South Korea^5 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),[-0.05cm] Föhringer Ring 6, 80805 München, Germany ==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty 1.5cmWe investigate the phenomenology of dark matter decay assuming that it is induced by non-minimal coupling to gravity, when the dark matter mass is in the sub-GeV range, i.e. below the QCD confinement scale. We show that the decay of the singlet scalar dark matter candidate produces sharp features in the photon spectrum, in the form of lines, boxes, and also in the form of a novel spectral feature, characterized by the decay into e^+e^- γ through a contact interaction, with decay branching fractions depending only on a single parameter, namely the dark matter mass.We also derive upper limits on the strength of the gravity portal from the non-observation of sharp features in the isotropic diffuse gamma-ray spectra measured by COMPTEL, EGRET and Fermi-LAT, and the X-ray spectrum measured by INTEGRAL. Finally, we briefly comment on the impact of dark matter decay via non-minimal coupling to gravity on the reionization history of the Universe. § INTRODUCTIONCosmological observations reveal that approximately 16% of the matter density of our Universe is in the form of protons, while the remaining 84% is attributed to a non-luminous component <cit.>, dubbed dark matter, possibly constituted by new particles not contained in the Standard Model (for reviews, see <cit.>). Dark matter particles were produced in the very early stages of our Universe, therefore their presence today in galaxies, clusters of galaxies, and in the Universe at large scale requires their lifetime to be at least as long as the age of the Universe. A common strategy to make dark matter stable consists in postulating a new global symmetry, unbroken in the electroweak vacuum, under which the dark matter particle is charged while all particles in the observable sector remain neutral. This global symmetry is purported to forbid all operators leading to dark matter decay, rendering it absolutely stable. This discussion normally ignores the effects of gravity. However, it has long been argued that global symmetries cannot be preserved in the presence of gravitational interactions <cit.>. Since curved spacetime is the natural arena where dark matter models should be embedded, this generates a mechanism for dark matter decay into the Standard Model particles via a gravitational portal. In <cit.>, gravitationally-induced dark matter decay was explored for a class of operators linear in the dark matter field and coupled non-minimally to gravity through the Ricci scalar. In this framework, dark matter decays into Standard Model particles with a total decay rate suppressed by inverse powers of the Planck mass and with branching ratios which only depend on the dark matter mass. Refs. <cit.> covered the range with dark matter masses larger than ∼ 1 GeV, such that the Standard Model degrees of freedom (quarks, leptons, gauge bosons and the Higgs) provide a good description of the possible decay products. In contrast, for dark matter masses in the sub-GeV regime, the relevant degrees for freedom are the photon, the three neutrinos, the electron, the muon (and their antiparticles) from the electroweak sector, as well as the pions from the confinement of light quarks and gluons. The interactions of the latter are described by chiral perturbation theory, the theory of the strong interactions below the confinement scale(for reviews, see <cit.>). Hence, the phenomenology of dark matter decay in the sub-GeV regime is qualitatively different to the one in the supra-GeV regime and a dedicated analysis is in order.In this paper, we identify the dominant decay channels of a scalar singlet dark matter candidate with sub-GeV mass via the gravity portal, and we calculate the corresponding rates and branching ratios. Notably, we find that, for the whole mass range under consideration, the channels producing sharp spectral features in the photon spectrum have sizable branching fractions, thus providing a strong test on the size of the non-minimal couplings of dark matter to gravity. § GRAVITY PORTAL FOR LIGHT SCALAR DARK MATTER DECAYIn the absence of gravity effects, the total action for the observable and dark sector can be written as𝒮 = ∫ d^4x [ℒ^ eff_ obs(X) + ℒ_(ϕ,X)] ,where ϕ is the dark matter field and X generically denotes the dynamical degrees of freedom in the observable sector at the energy scale relevant for dark matter decay. ℒ^ eff_ obs(X) is the effective Lagrangian of the observable sector and ℒ_(ϕ,X) contains terms involving the dark matter field together with its possible interactions with the observable sector. We also assume, as commonly done in the literature, the existence of a stabilizing global symmetry, under which the fields of the observable sector transform trivially but the dark matter field does not. In the presence of gravitational interactions, the stabilizing symmetry remains unbroken provided the dark matter only couples minimally to gravity. Nevertheless, non-minimal coupling to gravity may break the global symmetry and therefore induce dark matter decay. The dominant operators will be the ones with the lowest dimension. In this work we focus on the singlet scalar dark matter candidate which, as discussed in <cit.>, can be non-minimally coupled to gravity through a gauge and Lorentz invariant operator in the action already at mass dimension 3,[Decay operators through non-minimal coupling to gravity for dark matter candidates with gauge charges and/or higher spin require higher dimensional operators in the action. As a result, these candidates are predicted to have naturally cosmologically long lifetimes, unless the dark matter mass is very large <cit.>.] namely:𝒮 = ∫ d^4x √(-g)[ -R /2 κ^2+ ℒ^ eff_ obs(X) + ℒ_(ϕ,X) -ξ M Rϕ] ,where g is the determinant of the metric tensor g_μν, κ=M_P^-1=√(8 π G) is the inverse (reduced) Planck mass, M is a mass scale and ξ is a dimensionless coupling. For dark matter heavier than the electroweak scale, ℒ^ eff_ obs(X) conservatively corresponds to the Standard Model Lagrangian (although it may be extended to account for the dynamics of new, still undiscovered, degrees of freedom). In contrast, for dark matter lighter than the GeV scale, which is the focus of this paper, ℒ^ eff_ obs(X) consists of terms describing the dynamics and interactions of the light degrees of freedom (the three pions, the photon, as well as the electron, the muon and the three neutrinos and their antiparticles)[Strictly, kaons and eta mesons are also dynamical at energy scales smaller than ∼ 1GeV. We will however not consider them explicitly in ℒ^ eff_ obs(X), since the decays of a scalar dark matter particle with mass below ∼ 1GeV into KK or ηη pairs is kinematically forbidden.], augmented with terms resulting from integrating out heavy particles from the action, which are off-shell at the energy scale relevant for dark matter decay (heavy fermions, W, Z and Higgs). The effect of these heavy fields at low energies is encapsulated in the Wilson coefficients of four-fermion operators, as well as in the vacuum polarization of the photon. We will neglect the former, since they give rise to four-body decays, which are heavily suppressed by phase-space factors. Then, the effective Lagrangian relevant for our analysis can be cast asL_obs^eff =∑_f=e,μ, ν_1,ν_2,ν_3(i/2f̅∇^⟷f-m_ff̅ f )+f_π^2/4g^μνTr[D_μU^†D_νU] +f_π^2m_π^2/2Tr[ U^†+U] -1/4Z_3^-1g^μνg^λρF_μλF_νρ .In the first line above we have defined ∇=γ^ae^μ_a∇_μ, where γ^a is a Dirac matrix, e^μ_a a vierbein and ∇_μ=D_μ -i/4e_ν^b (∂_μ e^ν c) σ_bc, with D_μ the gauge covariant derivative. In the second line we have included the dominant operators of chiral perturbation theory. The matrixU=exp[i τ⃗·π⃗/f_π] contains the pion fields, π^a,a=1,2,3 (τ^a are the Pauli matrices) and the covariant derivative is defined as D_μU=∂_μU+ieA_μ[Q,U], with Q=diag(2/3,-1/3,-1/3); the traces are taken over the flavor indices and f_π=93MeV is the pion decay constant. Finally, in the third line, Z_3 is the photon wavefunction renormalization constant, which receives contributions from all electrically charged degrees of freedom that have been integrated out at the cut-off scale of the theory, which in our case is ∼ 1GeV. These include the top, bottom and charm quarks, the tau lepton, the W boson, as well as all hadrons made of light quarks except for the pions, which are the only dynamical hadronic degrees of freedom below ∼ 1GeV.[In extensions of the Standard Model, effects of new charged particles should also be included in the calculation.] The hadronic contributions to the wavefunction renormalization constant involve scales which lie in the non-perturbative regime of the strong interactions, and are therefore difficult to estimate. These contributions are however expected to be modest, certainly not larger than the one from all other degrees of freedom combined. In this work we will neglect the contribution from the hadronic states to the wave function renormalization constant. In practice, this will translate into a theoretical uncertainty in the calculation of Z_3, which we expect to be at most of O(1). Under this assumption, we obtainZ_3^-1 ≈ 1-e^2/8π^2(∑_i= t,b,c,τ b_i logΛ/m_i+b_W logΛ/M_W) ,where b_t=b_c=-16/9, b_b=-4/9, b_τ=-4/3 and b_W=+7. We will see later on that current experimental bounds justify this strategy. Given the form of the non-minimal operator, the action in Eq. (<ref>) can be recast as𝒮 = ∫ d^4x √(-g)[ -R /2 κ^2Ω^2(ϕ,X) + ℒ_ obs^ eff + ℒ_] ,where Ω^2(ϕ,X) =1 + 2 κ^2 ξ Mϕ .Written in this form, it is clear that the non-minimal operator modifies the Einstein-Hilbert action. The pure gravitational sector can be brought to canonical form through the field rescaling (Weyl transformation)g_μν = Ω^2(ϕ,X) g_μν ,upon which the action gets transformed into the Einstein-frame form:𝒮=∫ d^4x √(- g)[ -R/2 κ^2 + 3/κ^2g_μν∇^μΩ∇^νΩ/Ω^2 + ℒ_ obs^ eff +ℒ_] .The main benefit of this field redefinition is that dark matter interactions with the light particles can now be read out in a straightforward manner. The transformation of Eq. (<ref>) brings the effective Lagrangian of the observable sector to the form:ℒ_ obs^eff =∑_f=e,μ, ν_1,ν_2,ν_3(i/2 Ω^3f̅∇^⟷f-m_f/Ω^4f̅ f )+f_π^2/4Ω^2g^μνTr[D_μU^†D_νU] +f_π^2m_π^2/2Ω^4Tr[ U^†+U] -1/4Z_3^-1g^μνg^λρF_μλF_νρ ,where hatted quantities denote that they are expressed in the Einstein frame. Note that the Weyl transformation modifies not just the metric and the coefficients of the operators, but also the photon wavefunction renormalization constant. This can be understood as a consequence of the rescaling that the kinetic and mass terms of the integrated-out particles also undergo when transforming the action from the Jordan to the Einstein frame.[Inside loops only the propagators are relevant, so our discussion will be restricted to the free fields.] Since the masses of the particles that have been integrated out are much larger than the momentum of the dark matter particle, the field ϕ is slowly-varying and effectively behaves as a constant. In this limit, the Lagrangian of the integrated-out W boson in the Einstein frame,L_W= -1/4g^μνg^λρW_μλW_νρ+1/2M_W^2/Ω^2g^μν W_μ W_ν+… ,can be understood as having a ϕ-dependent rescaled mass. Integrated-out fermions show a similar rescaling:L_f= i/2Ω^3f̅∇^⟷f-m_f/Ω^4f̅ f +. . . ≃i/2f̅∇^⟷f-m_f/Ωf̅f +… ,which can be transferred entirely to the mass term once the field is canonically normalized (last equality). The effect of the Weyl transformation therefore amounts to a ϕ-dependent change in the masses of the particles entering the wavefunction renormalization constant, which then reads:Z_3^-1(m_f, M_W)=Z_3^-1(m_f/Ω, M_W/Ω)≈ 1-e^2/8π^2(∑_i= t,b,c,τ b_i logΛ Ω/m_i+b_W logΛ Ω/M_W) .Finally, upon expanding Ω, one finds effective interaction terms between the dark matter field and the operator F^μνF_μν. This procedure is similar in spirit to the one presented in <cit.> to determine h→γγ using low-energy theorems.It is important to note that in the Einstein frame the field ϕ is not canonically normalized, as apparent from Eq. (<ref>). To bring the kinetic term of the dark matter field into the canonical form, we introduce the fieldϕ, defined from the condition thatL_ DM, kin = 1/2(1/Ω^2 +6 (∂_ϕΩ)^2/κ^2 Ω^2) g^μν (∂_μϕ)(∂_νϕ)= 1/2g^μν (∂_μϕ)(∂_νϕ) .The field ϕ is then related to ϕ by the transformationϕ=√(6)/κ{y-y_0 +1/2log[(1-y)(1+y_0)/(1+y)(1-y_0)]} ,withy=√(1+1+2 κ^2 ξ M ϕ/6κ^2ξ^2 M^2) , y_0=√(1+1/6κ^2 ξ^2 M^2) .Notice that when the dimensionless combination κξ M≪ 1, the effects of the canonical normalization can be safely neglected. However, as we will see below, these effects can have a significant impact on the phenomenology when κξ M≳ 1. The terms of the Lagrangian inducing dark matter decay can thus be identified by expanding Eq. (<ref>) in powers of ϕ, and keeping the linear terms. Using Eqs.(<ref>,<ref>), we obtain that for moderate values of κξ M (at the energies we are working κϕ≪ 1 is always fulfilled), the Weyl scaling factor can be approximated by:1/Ω^2≃ 1-2κ^2 ξ M ϕ/√(1+6 κ^2 ξ^2 M^2) .Furthermore, since we are interested in the dominant decay modes, we will expand the chiral field U to quadratic order in the pion fields. The part of the effective Lagrangian linear in the (canonically normalized) dark matter field ϕ reads:ℒ_ obs^ eff⊃ -2κ^2 ξ M ϕ/√(1+6 κ^2 ξ^2 M^2) [∑_f=e,μ, ν_1,ν_2,ν_3(3i/2f̅γ^μ∂_μ f-2 m_ff̅ f)-∑_f=e,μ(3e/2f̅γ^μ A_μ f) +1/2∂_μπ^a∂^μπ^a - 1/2m_π^2π^aπ^a+c_γγ F_μνF^μν] ,withc_γγ≈-e^2/8π^2(∑_i= t,b,c,τ b_i +b_W)=5e^2/24π^2 ,from where one can extract thedark matter decay vertices into electrons, muons, neutrinos, pions and photons.It is interesting to note that for κξ M≪ 1 the strength of the decay vertex is proportional to κξ M, while for large values, it becomes approximately constant, due to the effect of the canonical normalization of the dark matter field. The phenomenological consequences of this behavior will be discussed in the next section.§ DECAY RATES AND OBSERVATIONAL SIGNALS From the effective Lagrangian expressed in the Einstein frame, Eq. (<ref>), the dark matter partial decay rates can be calculated. In what follows we will drop the hats from the fields for ease of notation, bearing in mind that all fields are now canonically normalized.The rates for the decay modes with a fermion-antifermion pair or with pions in the final state can be straightforwardly calculated, the result being:Γ_f̅ f= m_ϕ^3/8π κ^4 ξ^2 M^2 /1+6κ^2 ξ^2 M^2 x_ f(1-4x_f)^3/2 ,Γ_f̅ f γ= α m_ϕ^3/16 π^2 κ^4 ξ^2 M^2 /1+6κ^2 ξ^2 M^2 g(x_f) ,Γ_π^+π^-= m_ϕ^3/16π κ^4 ξ^2 M^2 /1+6κ^2 ξ^2 M^2 (1+2x_π^+)^2 (1-4 x_π^+)^1/2 , Γ_π^0π^0= m_ϕ^3/32π κ^4 ξ^2 M^2 /1+6κ^2 ξ^2 M^2 (1+2x_π)^2 (1-4 x_π)^1/2 ,whereg(x)≡(1+2x+24x^2)√(1-4 x) -12 x^2 (3-4 x) log( 1-2 x +√(1- 4 x)/2 x) ,andx_a=m_a^2/m_ϕ^2. The rate for the decay into two photons requires a more careful analysis, sinceat order e^2 it receives contributions from the effective vertex ϕF_μν F^μν in Eq. (<ref>), as well as from loops of electrons, muons and pions. The result is:Γ_γγ= m_ϕ^3/16 π κ^4 ξ^2 M^2 /1+6κ^2 ξ^2 M^2 |F_ℓ(m_ϕ^2/4m_e^2) + F_ℓ(m_ϕ^2/4m_μ^2) + F_ pion +c_γγ|^2,where F_ℓ(x) is the leptonic form factor, given byF_ℓ(x) =e^2/8π^2[x + (x -1)f(x)]x^-2 ,withf(x)= arcsin ^2√(x)x≤1,-1/4[log1+√(1-x^-1)/1-√(1-x^-1) -iπ]^2x>1.,and F_ pion is the pionic form factor, which will be neglected along with the hadronic contribution to c_γγ. Therefore, and in contrast to the values of the decay rates into ff̅, ff̅γ and ππ, given in Eq. (<ref>), which are exact up to higher order corrections, the rate into γγ can be claimed to be accurate only up to an O(1) factor. This level of accuracy, on the other hand, will be sufficient for drawing our main conclusions.The decay widths for each of these final states depend on the dark matter mass and on the value of the parameter κξ M. Clearly, for smaller and smallerκξ M, the total rate decreases. However, for asymptotically large values, even for transplanckian scales κξ M ≫ 1, the total rate reaches a finite asymptotic value, which is a factor 7/6 larger than the corresponding rate for κξ M = 1. Notably, we see that due to the effect of the non-minimal coupling on the dark matter kinetic term, the decay rate cannot be significantly enhanced by taking κξ M ≫ 1.The inverse widths for the relevant channels are shown in Fig.<ref>, for m_ϕ between 400 keV and 700 MeV, assuming κξ M=1. Above 700 MeV, chiral perturbation theory is no longer valid and our approach is not applicable. For dark matter lighter than 2 m_e, only decays into photons or into neutrinos are kinematically accessible. For the latter, two-body decays are very suppressed by the conservation of angular momentum, hence we will only consider the former.[Other decays, such as2ν2ν̅, are suppressed by extra powers of the coupling constant and by the smaller phase space available in the decay, and can be safely neglected.] For dark matter masses between 2 m_e and 2 m_μ, the decay channels into e^+e^- and e^+e^-γ become kinematically accessible. Very close to the electron threshold, the decay into e^+e^- dominates, as Γ_e^+e^-γ is suppressed in this limit by an extra factor of α/π (1-4x_e). However, the rate for ϕ→ e^+e^-γ grows with the mass faster than the rate for ϕ→ e^+e^-. We find that the branching fraction for ϕ→ e^+e^-γ equals 5% for m_ϕ≃ 3 MeV, reaches 50% for m_ϕ≳ 14 MeV and becomes the dominant decay channel until decays into muons become kinematically accessible, for m_ϕ=2m_μ. Close to the muon threshold, ϕ→μ^+μ^- dominates overϕ→ e^+e^-, due to the enhancement in the rate by the factor m^2_μ/m^2_e, and over the decays ϕ→μ^+μ^-γ, e^+e^-γ, due to the extra coupling constant and smaller phase space. The latter two processes have a larger rate than ϕ→μ^+μ^- when m_ϕ≳ 3 GeV. However, for masses above the threshold of pion pair production, m_ϕ≳ 270 MeV, the decays ϕ→π^+π^-,π^0π^0 have rates larger than the processes with electrons or muons in the final state and become the dominant processes until m_ϕ∼ 700 MeV, which is the largest mass considered in this work. The gray dashed line in the Figure indicates the age of the Universe, τ_U=4× 10^17s. For dark matter masses above ∼ 270 MeV, decays are so fast that the abundance of dark matter today would be much smaller than the observed value. In this regime, the parameter κξ M is constrained to be smaller than 1, to ensure that the relic abundance of dark matter is compatible with present-day observations of our Universe. Stronger constraints on the parameter κξ M can be derived from the non-observation of photon fluxes generated in the dark matter decay over the astrophysical background. The prompt photon flux is dominated by the decay channels γγ, f f̅γ and π^0 π^0. The differential spectrum in each case reads:d N_γ^(γγ)/d y_γ =2 δ(y_γ-1/2) ,dN_γ^(ff̅γ)/d y_γ = 24/g(x_f)( 1+ 2 x_f - 2 y_γ)y_γ√(1-4 x_f/1-2 y_γ) , dN_γ^(π^0π^0)/d y_γ =8/√(1-4x_π^0)Θ(y_γ-y_-)Θ(y_+-y_γ) ,with y_γ≡ E_γ/m_ϕ, y_±≡ (1±√(1-4x_π^0))/4 andΘ(x) the Heaviside function.Notably, for most values of the dark matter mass, the decay via the gravity portal predicts a sharpfeature in the photon energy spectrum. For m_ϕ< 2m_e, the dominant decay channel is ϕ→γγ, which produces a line in the photon energy spectrum <cit.>. For 2 m_π^0< m_ϕ≲ 1GeV, the dominant decay channel is ϕ→π^0π^0, which produces a gamma-ray box from the decay in flight of the pions into two photons <cit.>. Finally, for 2 m_e≲ m_ϕ < 2 m_π^0, we find a new sharp spectral feature, which arises from the decay vertex ϕe̅γ^μ A_μ e. In Fig. <ref> we show the expected isotropic diffuse photon spectra from the decay of a dark matter particle with mass 5 MeV, 50 MeV and 500 MeV, assuming κξ M=1. The predicted flux includes contributions from the decay of the cosmological dark matter, with density Ω h^2≃ 0.12 as reported by Planck <cit.>, and from the decay of Galactic dark matter, assumed to be distributed following a Navarro-Frenk-White <cit.> profile with scale factor equal to 24 kpc <cit.>, normalized such that the dark matter density at the position of the Solar System is ρ_ loc=0.3GeV/ cm^3. The detector response was simulated by adopting a fixed 10% energy resolution over the whole range. The Figure also shows the isotropic diffuse X-ray spectrum in the energy range 5-100 keV, as determined by INTEGRAL <cit.>, and gamma-ray spectrum between 0.8 and 30 MeV, between 30 MeV and 50 GeV, and between 100 MeV and 820 GeV, as determined byEGRET <cit.>, COMPTEL <cit.> and Fermi-LAT <cit.>, respectively. As apparent from the Figure, when κξ M=1 the expected photon flux from dark matter decay can exceed the measured flux by many orders of magnitude. Furthermore, in all the cases, the energy spectrum presents a sharp fall-off close to the kinematic endpoint of the spectrum, which can be easily discriminated from the featureless astrophysical background. Therefore, the non-observation of sharp features in the isotropic diffuse photon spectrum translates into strong limits on the combination of parameters κξ M. In Fig. <ref> we show conservative limits on κξ M, calculated from requiring that in any energy bin of one of the four experiments under consideration, the photon flux from dark matter decay only does not exceed the measured flux by more than 2σ; for each experiment we used the appropriate energy resolution. We note that for very low dark matter mass the constraints disappear. The reason is that for every m_ϕ the decay rate has an upper limit, which is reached when κξ M≫ 1 (cf. Eq. (<ref>)). Therefore, this scenario will be unconstrained if the maximum flux predicted from dark matter decay is below the experimental sensitivity. We find numerically that this occurs for The observation of the spectral feature from dark matter decay via the gravity portal is subject to other phenomenological constraints, mainly from the modifications of the cosmic microwave background (CMB) anisotropy power spectrum induced by the injection of energy after recombination, which could significantly modify the ionization history of the Universe <cit.>. We have estimated the CMB constraints on our scenario using thecode <cit.>; the resulting limits are shown as a black line in Fig. <ref>. Interestingly, the CMB constraints are weaker than the ones from gamma-ray telescopes, thus opening the possibility of observing sharp spectral features from dark matter decay via the gravity portal in future missions, like e-ASTROGAM <cit.>. § CONCLUSIONSStabilizing global symmetries for dark matter do not need to be preserved by gravitational interactions. Therefore, in curved spacetime, which is the natural framework to describe dark matter, terms inducing dark matter decay through gravity are a logical possibility. In this paper we have investigated the impact of operators linear in the dark matter field and proportional to the Ricci scalar on the stability of a singlet scalar dark matter candidate with sub-GeV mass. In this energy regime, the final states kinematically accessible for dark matter decay contain pions, photons, electrons, muons and neutrinos (and their antiparticles). We have identified the dominant decay channels and calculated the corresponding decay widths in terms of known parameters of the Lagrangians of chiral perturbation theory and quantum electrodynamics. The only free parameters in our analysis are therefore the dark matter mass and the dark matter coupling strength to the Ricci scalar, given by κξ M, with ξ (M) a dimensionless (mass dimension) parameter and κ the inverse reduced Planck mass. We have shown that the decay via the gravity portal always produces a sharp feature in the isotropic diffuse photon spectrum, which could be readily discriminated from the featureless astrophysical backgrounds, thus providing a probe of the dark matter non-minimal coupling to gravity. Concretely, for m_ϕ< 2m_e, the dominant decay channel is ϕ→γγ, which produces a line in the photon energy spectrum; for 2 m_π^0< m_ϕ≲ 1GeV, the dominant decay channel is ϕ→π^0π^0, which produces a gamma-ray box from the decay in flight of the pions into two photons; and for 2 m_e≲ m_ϕ < 2 m_π^0, we have found a new sharp spectral feature, which arises from the decay vertex ϕe̅γ^μ A_μ e. The non-observation of sharp gamma-ray features in current experiments already places strong constraints on the dark matter non-minimal coupling to gravity, e.g. κξ M≲ 10^-6,10^-3 and 10^-1 for m_ϕ≃ 500,50 and 5MeV, respectively. Complementary limits on the parameter space stem from requiring that the decay products do not significantly modify the reionization history of the Universe. These limits are comparable or weaker than our conservative limits from the non-observation of sharp features in the photon spectrum. Planned X- and gamma-ray instruments will continue testing the possibility of dark matter decay via the gravity portal.§ ACKNOWLEDGEMENTSWe are grateful to Fred Jegerlehner for useful discussions. This work has been partially supported by the DFG cluster of excellence EXC 153 “Origin and Structure of the Universe” and by the Collaborative Research Center SFB1258. JHEP-mod | http://arxiv.org/abs/1707.08480v1 | {
"authors": [
"Oscar Catà",
"Alejandro Ibarra",
"Sebastian Ingenhütt"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170726145825",
"title": "Sharp spectral features from light dark matter decay via gravity portals"
} |
Deep Learning Models for Wireless Signal Classification with Distributed Low-Cost Spectrum SensorsSreeraj Rajendran, Student Member, IEEE, Wannes Meert, Member, IEEE Domenico Giustiniano, Senior Member, IEEE, Vincent Lenders, Member, IEEE and Sofie Pollin, Senior Member, IEEE. December 30, 2023 ===============================================================================================================================================================================================================================================§ ABSTRACT We introduce a directed, weighted random graph model, where the edge-weights are independent and beta-distributed with parameters depending on their endpoints. We will show that the row- and column-sums of the transformed edge-weight matrix are sufficient statistics for the parameters, and use the theory of exponential families to prove that the ML estimate of the parameters exists and is unique. Then an algorithm to find this estimate is introduced together with convergence proof that uses properties of the digamma function. Simulation results and applications are also presented.Keywords: exponential family, sufficient statistics, ML estimation, digamma function, successive approximationMSC2010: 62F10, 62B05.§ INTRODUCTION The theory of ML estimation in the following types of exponential family random graph models has frequently been investigated in the last decade, see, e.g., <cit.>. The graph has n vertices, and the adjacency relations between them are given by the n× n random edge-weight matrix =(w_ij ) of zero diagonal. Ifis symmetric, then we have anundirected graph; otherwise, our graph is directed, where w_ij is the nonnegative weight assigned to the i→ j edge according to the model. We assume that the edge-weights (above or out of the main diagonal) are completely independent (but their distribution usually depends on different parameters), and have an exponential family distribution _. So the likelihood function has the general formL_ ( )= e^< ,( ) > - Z ( ) · h( ) ,with the canonical parameter , log-partition (cumulant) function Z ( ), and canonical sufficient statistic . In these random graph models, components of = ( ) are the row-sums and/or column-sums ofor some -related matrix, i.e., they are vertex-degrees or in- and out-degrees of the observed undirected or directed, weighted or unweighted graph (in the weighted case, the edge-weights may undergo a suitable transformation). Also, h ( ) is usually 1 over the support of the likelihood function, indicating that given the canonical sufficient statistics,the joint distribution of the entries is uniform (microcanonical) in these models.To make inferences on the parameters, typically we have only one observation for the graph. It may seem that it is a one-element sample, but there are the adjacencies that form the sample; the number of them is n 2 in the undirected, and n(n-1) in the directed case. The number of parameters, contained in , is n in the undirected and 2n in the directed case. The parameters can be considered as affinities or potentials of the vertices to make ties in the undirected, and to emanate or adsorb edges in the directed case. It is important that we divide the components of the canonical parameterof the underlying distribution of the ij or i→ j edge between the connected vertices, like α_i +α_j in the undirected and α_i +β_j in the directed case (i j), see <cit.>.In regular exponential families (Θ is open), the ML equation ∇_ln L_ ( ) = is equivalent to∇_ Z( ) = .Since ∇_ Z( ) = _, the ML equation (<ref>) means that the canonical sufficient statistic is made equal to its expectation. But when is it possible? Now we briefly summarize existing theoreticalresults on this issue. Let M = {_:∈Θ} denote the so-called mean parameter space in the model; it is necessarily convex. LetM^0 denote its interior. When the canonical statistic is also complete, and hence, minimal sufficient, the representation (<ref>) is minimal (i.e., the model is not overparametrized). In exponential family, the gradient mapping ∇ Z:Θ→ M is one-to-one if and only if the exponential family representation is minimal. In a minimal exponential family, the gradient mapping ∇ Z is onto M^0.By Propositions <ref> and <ref>, any parameter in M^0 is uniquely realized by the _θ distribution for some θ∈Θ. Also, in a regular and minimal exponential family, M is an open set andis identical to M^0.As the ML estimate of is the solution of (<ref>), we have the following. Assume, the (canonical) parameter space Θ is open. Then there exists a solution ∈Θ to the ML equation ∇_ Z( ) = if and only if ∈ M^0; further, if such a solution exists, it is also unique. Note that in regular and minimal exponential families, M^0 is also the interior of T, which is the convex hull of all possible values of , see, e.g., <cit.>.In the case of discrete distributions, it frequently happens that the boundary of T has positive measure. For instance, the so-called threshold graphs are located on the boundary of the polyhedron, determined by the Erdős–Gallai conditions, in the model of <cit.> which uses Bernoulli distributed entries. However, in the case of an absolutely continuous _ distribution,the boundary of T has zero Lebesgue measure, and so, probability zero with respect to the_ measure. Therefore, in view of Proposition <ref>, the ML equation has a unique solution with probability 1. The organization of the paper is as follows. In Section <ref>, we introduce a model for directed edge-weighted graphs and prove that a unique ML estimate of the parameters exists. In Section <ref>, we define an iterative algorithm to find this solution, and prove its convergence with a convenient starting. In Section <ref>, the algorithm is applied to randomly generated and real-word data. In Appendix A, properties of the digamma function, whereas in Appendix B, the boundary of our M is discussed.The long proof of the main convergence theorem of the iteration algorithm, introduced in Section <ref>, is presented in Appendix C.We remark that edge-weighted graphs of uniformly bounded edge-weights are prototypes of real-world networks, see e.g., <cit.>. Without loss of generality, if the edge-weights are transformed into the [0,1] interval, the beta-distribution for them, with varying parameters, is capable to model a wide range of possible probability densities on them. This indicates the soundness of the model to be introduced in Section <ref>. § ARANDOM GRAPH MODEL WITH BETA-DISTRIBUTED EDGE-WEIGHTS Let =(w_ij) be the n× n (usually not symmetric) edge-weight matrix of a random directed graph on n vertices: w_ii =0 (i=1,… ,n) and w_ij∈ [0,1] is the weight of the i→ j edge (i j). Our model is the following: the i j weight obeys a beta-distributionwith parameters a_i >0 and b_j >0. The parameters are collected in =(a_1 ,… a_n ) and =̱ (b_1 ,… b_n ), or briefly, in = ( ,)̱. Here a_i can be thought of as the potential of the vertex i to send messages out, and b_i is its resistance to receive messages in.The likelihood function is factorized asL_ , ( )=∏_i jΓ (a_i+b_j )/Γ (a_i ) Γ (b_j )w_ij^a_i -1 (1-w_ij)^b_j -1 = C( ,)̱∏_i jexp[ (a_i -1 ) ln w_ij + (b_j -1) ln (1-w_ij) ]= exp[ ∑_i=1^n (a_i -1 ) ∑_j iln w_ij +∑_j=1^n (b_j -1 ) ∑_i jln (1-w_ij ) -Z ( , )̱],where C( ,)̱ is the normalizing constant, and Z ( , )̱ = -ln C( ,)̱ is the log-partition (cumulant) function. Since the likelihood function depends ononly through the row-sums of the n× n matrix = ( ) of general entry ln w_ij and the column-sumsof the n× n matrix =() of general entry ln (1-w_ij ), by the Neyman–Fisher factorization theorem, the row-sums R_1 ,… ,R_n ofand column-sumsC_1 ,… ,C_n ofare sufficient statistics for the parameters. Moreover, =( , ) = (R_1 ,… ,R_n , C_1 ,… ,C_n) is the canonical sufficient statistic, which is also minimal. Note thatcontains the log-weights of the original graph, whilecontains the the log-weights of the complement graph of edge-weight matrixwith entries 1-w_ij (i j ).The first factor in the Neyman–Fisher factorization (including gamma-functions) depends only on the parameters and on the sample through these sufficient statistics, whereas the seemingly not present other factor – which would merely depend on– is constantly 1, indicating that the conditional joint distribution of the entries, given the row- and column-sums of the log-weight and log-complement matrix is uniform (microcanonical) in this model. So under the conditions on the margins ofand , the directed graphs coming from the above model are uniformly distributed.The system of likelihood equations is obtained by making the derivatives of L_ , ( ) with respect to the parameters equal to 0:∂L_ , ( )/∂ a_i=∑_j iψ (a_i+b_j ) - (n-1) ψ (a_i) +R_i =0, i=1,… ,n;∂ L_ , ( )/∂ b_j=∑_i jψ (a_i+b_j ) - (n-1) ψ (b_j) +C_j =0 ,j=1,… ,n .Here ψ (x)=∂lnΓ (x)/∂ x = Γ' (x)/Γ (x) for x>0 is the digamma function. For its properties, see Appendix A.To apply the theory of Section <ref>, we utilize that the parameter space Θ⊂_+^2n is open, akin to the canonical parameter space, (-1 ,∞ )^2n. Note that the canonical parameteris, in fact, (' ,'̱ )=' = -, where ∈^2n is the vector of all 1 coordinates. With it, the log-partition functionisZ (', ' )=-∑_j iΓ ( a_i'+ b_j' +2 ) + ∑_j iΓ (a_i' +1) +∑_i jΓ (b_j' +1) .In view of (<ref>), the ML equation is equivalent to∂ Z (', ' )/∂a_i'=-∑_j iψ (a_i'+b_j' +2 ) + (n-1) ψ (a_i' +1)=R_i ,i=1,… ,n;∂ Z (', ' )/∂b_j'=-∑_i jψ (a_i'+b_j ' +2 ) + (n-1) ψ (b_j' +1)=C_j,i=1,… ,n.But this system of equations is the same as (<ref>), in terms of the parameter ' instead of .In view ofSection <ref>, the mean parameter space M consists of parameters (A_1 ,… A_n ,B_1 ,… ,B_n) obtained by the gradient mapping, that is,A_i = A_i ( ,)̱ =- ∑_j i[ ψ (a_i+b_j ) - ψ (a_i) ] ,i=1,… ,n;B_j = B_j ( ,)̱ =- ∑_i j[ ψ (a_i+b_j) -ψ (b_j ) ],j=1,… ,n .M is an open set, whose boundary is determined by the limit properties between the digamma and the log functions, see Appendix B for details. There we also find a correspondence between the points on the boundary of M and those on the boundary of theconvex hull T of the possible sufficient statistics =( , ) within _-^2n. It is interesting that while the boundary points of M do not belong to the open set M, the boundary points of T do belong to T, and can be realized as row- and column-sums of the ( ) and ( ) matrices with aof off-diagonal entries in(0,1). However, this boundary has 0 probability, and so, any canonical sufficient statisticof the observed graphis in M, with probability 1. Therefore, by Proposition <ref>,we can state the following.The system of the ML equations (<ref>) has a unique solution =(, ̱̂), with probability 1.Later we will use the following trivial upper bound for the sum of row- and column-sums (of theand matrices):∑_i=1^n R_i + ∑_j=1^n C_j =∑_i=1^n ∑_j iln w_ij+ ∑_j=1^n ∑_i jln (1-w_ij) = ∑_i jln [w_ij (1-w_ij )] ≤ -2ln 2 n (n-1)due to rearranging the terms and the relation w_ij (1-w_ij )≤ 1/4 for w_ij∈ [0,1] with equality if and only if w_ij=1/2 (i j). For finer estimates see Appendix B.Also note that the Hessian of the system of ML equations (consisting of the second order partial derivatives of L_) at ) does not contain thesufficient statistics any more, therefore the negative of it is the Fisher-information matrix at . Because of the regularity conditions, the information matrix is positive, and so, the Hessian is negative definite. This is also an indication of the existence of a unique ML estimate. § ITERATION ALGORITHM TO FIND THE PARAMETERS To use a fixed point iteration, now we rewrite the system of likelihood equations in the form =f( ), where = ( ,)̱, as follows:a_i= ψ^-1[ 1/n-1 R_i +1/n-1∑_j iψ(a_i+b_j ) ] =:g_i ( ,)̱,i=1,… ,nb_j= ψ^-1[ 1/n-1 C_j +1/n-1∑_i jψ(a_i+b_j ) ] =:h_j ( ,)̱,j=1,… ,n .Here g_i's and h_j's are the coordinate functions of f =(g,h) :^2n→^2n. Then, starting at ^(0), we use the successive approximation ^(it ) :=f (^(it -1) for it =1,2,…, until convergence. Now the the statement of convergence of the above iteration to the theoretically guaranteed unique(see Theorem <ref>) follows.Let =(,̱̂ ) be the unique solution of the ML equation (<ref>). Then the above mapping f=(g,h) is a contraction in some closed neighborhood K of , and so, starting at any ^(0)∈ K, the fixed point of the iteration ^(it ) =f (^(it -1)) exists and is .The prof of this theorem is to be found in Appendix C.Since K is only theoretically guaranteed, we need some practical considerations about the choice of ^(0 ), which should be adapted to the sufficient statistics. In the sequel, for two vectors =(x_1,…,x_n), =(y_1,…,y_n)we use the notation > if x_i>y_i for each i=1,…,n. Likewise, ≥ is the shorthand for x_i≥ y_i for each i=1,…,n.Recall that f=(g,h) is the mapping (<ref>) of the fixed point iteration, and =(,̱̂)>0 is the (only) solution of the equation f()=, where ∈^2n is the vector of all 0 coordinates.LetM:=max{max_i∈{ 1,…,n }(-R_i/n-1), max_i∈{ 1,…,n }(-C_i/n-1)}and ε>0 be the (only) solution of the equation ψ(2x)-ψ(x)=M. Then (,̱̂)≥ε. Proof. In view of(<ref>) we have that M≥ln 2. Since equality in (<ref>) is attained with probability 0, we have that M>ln 2 with probability 1. Therefore, by Lemma <ref> of Appendix A, there exists an ε, with probability 1, such that ψ(2ε )-ψ(ε )=M.Without loss of generality we can assume thatâ_i_0=min{min_i∈{ 1,…,n }â_i,min_i∈{ 1,…,n }b̂_i}.Then by the ML equation, the monotonicity of ψ, and Lemma <ref> of Appendix A, we get(n-1)M≥ -R_i_0=∑_j≠ i_0ψ(â_i_0+b̂_j)-( n-1)ψ(â_i_0)≥ (n-1)[ψ(2â_i_0)-ψ(â_i_0)] .Therefore, â_i_0≥ε, whence â_i,b̂_i≥ε holds for every i=1,…,n.□ With the solution ε of ψ(2x)-ψ(x)=M of (<ref>), we have f(ε )≥ε. Proof.g_i(ε ) =ψ^-1(ψ(2ε)+R_i/n-1)≥ψ^-1(ψ(2ε)-M)=ε . Likewise,h_i(ε ) =ψ^-1(ψ(2ε)+C_i/n-1)≥ψ^-1(ψ(2ε)-M)=ε .□It is also clear that we have the following.If (,)̱≥ (, )>, then f(,)̱≥ f(, ).With ε satisfying ψ(2ε )-ψ(ε )=Mof (<ref>), andstarting at ^(0) =ε, the sequence ^(it) of the iteration ^(it)=f(^(it-1)) for it→∞ converges at a geometric rateto the unique solution ( ,̱̂ ) of the ML equation. Proof. From Propositions <ref> and <ref> we obtain that the sequence ^(it) is coordinate-wise increasing. Moreover, it is clear that (^(it) ) is bounded from above by (,̱̂), due to Proposition <ref>. Therefore, the convergence of^(it) follows, andby the continuity of f, the limit is clearly a fixed point of f. However, in view of Section <ref>, the solution of the ML equation is a fixed point of f, and it cannot be else but the unique solution (,̱̂), guaranteed by Theorem <ref>. Further, from Theorem <ref> we get that the rate of convergence is (at least) geometric.□Therefore, a good starting can be chosen by these considerations. Also note that at the above ^(0) and possibly at its first (finitely many) iterates, f is usually not a contraction. It becomes a contraction only when some iterate ^(it_0 ) gets into the neighborhood K ofof Theorem <ref>, which is inevitable in view of the convergence of the sequence ^(it). So, Theorem <ref> is literally applicable only if we start the iteration at^(it_0 ). In practice, however, we do not know the theoretically guaranteed neighborhood K. The practical merit of Theorem <ref> is just that it offers a realizable starting.§ APPLICATIONS First we generated a random directed edge-weighted graph on n=100 vertices. The edge-weight matrixhad zero diagonal, and the off-diagonal entries w_ij's were independent. Further, for i j, the weight w_ij was generated according to beta-distributionwith parameters a_i >0 and b_j >0, where a_i's and b_j's were chosen randomly in the interval [1,5].Then we estimated the parameters based on , and plotted the a_i , â_i(i=1,… ,n)and b_j , b̂_j(j=1,… ,n) pairs.Figure <ref>shows a good fit between them.We also applied the algorithm to migration data between 34 countries. Here w_ij is proportional to the number of people in thousands who moved from country i to country j (to find jobs) during the year 2011, and it is normalized so that be in the interval (0,1).The estimated parameters are in Table <ref>.In this context, a_i's are related to theemigration and and b_i's to the counter-immigration potentials. When a_i is large, country i has a relatively large potential for emigration. On the contrary, when b_i is large, country i tends to have a relatively large resistance against immigration.It should be noted again that edge-weighted graphs of this type very frequently model real-world directed networks.§ APPENDIX§.§ A.Properties of the digamma function Though, we do not use it explicitly, the following approximation of the digamma functionψ (x)=∂lnΓ (x)/∂ x = Γ' (x)/Γ (x) (x>0) is interesting for its own right. ψ (x) = ln (x-1/2 ) +O( 1/x^2 ) for x>1.The statement of the lemma easily follows by Taylor expansion. 1/ψ'(x+y)>1/ψ'(x)+1/ψ'(y) for x,y>0. Proof. First we prove that the function u(x)=1/ψ'(x), x∈ (0,∞) is strictly convex. Indeed,one can easily see thatu”(x)=-ψ”'(x)[ψ'(x)]^2+2[ψ”(x)]^2ψ'(x)/[ψ'(x)]^4 ,and this is positive due to ψ' (x)>0 and the fact that[ψ”(x)]^2/ψ”'(x)ψ'(x)>1/2 .Latter one is a particular case of Corollary 2.3 in <cit.>.Now, in view of lim_x→0ψ'(x)=∞, we can extend u continuously to 0 by setting u(0)=0. Then u is still strictly convex, and therefore, for every x,y>0 we have u(x)=u(y/x+y· 0+x/x+y·(x+y))< y/x+yu(0)+x/x+yu(x+y). Consequently,u(x)< x/x+y u(x+y),and likewise,u(y)< y/x+yu(x+y).Adding (<ref>) and (<ref>) together, we get the statement of the lemma. □ The function ψ(2x)-ψ(x), x∈ (0,∞) is decreasing and its range is (ln 2,∞). Proof. It is easily seen by the identity ψ(2x)=1/2ψ(x)+1/2ψ(x+1/2)+ln 2 which can be found in <cit.>. □In the last lemma we collect some limiting properties of the digamma function and its derivative, see, e.g., <cit.> for details. The digamma function ψ is a strictly concave, smooth function on (0,∞) that satisfies the following limit relations:lim_x→ 0+ψ (x) =-∞, lim_x→∞ψ (x) =∞, lim_x→∞ψ' (x) =0, lim_x→∞ (ψ (x) -ln x ) =0, lim_x→ 0+ (ψ (2x) -ψ (x)) =∞ .§.§ B. Considerations on the boundary of the mean parameter space In Section <ref>, we saw that the mean parameter space M consists of 2n-tuples (A_1 ,… A_n ,B_1 ,… ,B_n) obtained from the parameters ( ,)̱ = (a_1 ,… a_n ,b_1 ,… ,b_n) of the underlying beta-distributions by Equations (<ref>).Denoting by L ( ,)̱ =(A_1 (, )̱ , … ,A_n ( ,)̱, B_1 ( ,)̱,… , B_n ( ,)̱) this dependence, i.e., the Θ→ M (one-to-one) mapping, a boundary point L̅ =(A̅_1 ,… ,A̅_n ,B̅_1 ,… ,B̅_n) of M can be obtained as L̅ =lim_k→∞ L (^k ,^̱k ), where ^k = (a^k_1 ,… ,a^k_n ), ^̱k = (b^k_1 ,… ,b^k_n ), andlim_k→∞ a^k_i = a̅_i ∈ [0,∞],i=1,… n;lim_k→∞ b^k_j = b̅_j ∈ [0,∞],j=1,… n .In view of Lemma <ref>, only the a̅_i,b̅_j =∞ cases have relevance.The sequence (^k ,^̱k ) can be chosen such thatlim_k→∞a^k_i/b^k_j =x_ijwith0<x_ij < ∞ , fori j.Then, using <ref>),A̅_i= -lim_k→∞∑_j i[ ψ (a^k_i+ b^k_j)- ψ (a^k_i ) ]=lim_k→∞∑_j i[ ln (a^k_i ) - ln (a^k_i+b^k_j) ] =∑_j ilnx_ij/1+x_ij, i=1,… ,n;B̅_j= -lim_k→∞∑_i j[ ψ (a^k_i+b^k_j) - ψ (b^k_j ) ]=lim_k→∞∑_i j[ ln (b^k_j ) - ln (a^k_i+b^k_j) ] = ∑_i jln1/1+x_ij , j=1,… ,n .These equations show that the boundary point L̅ of M contains – in its coordinates – the row- and column-sums of the matrices ( ) ad ( ) respectively (see Section <ref>), where the general off-diagonal entry of then× n edge-weight matrixis x_ij/1+x_ij.Observe that 2n-1 x_ij's can be chosen free, and all the others are obtainable from them. To see this, consider the complete bipartite graph on vertex classes (a_1 ,… ,a_n) and (b_1 ,… ,b_n), where to the edge connecting a_i and b_j we assign a_i /b_j. Choose a minimal spanning tree of this graph (it contains 2n-1 edges), and consider the sequence of (^k ,^̱k )'s satisfying condition (<ref>). Then, as k→∞, thex_ij's of the edges not included in the spanning tree can be obtained from the x_ij'sof the 2n-1 edges included inthe spanning tree. Therefore the row- and column-sums of theedge-weight matrixof entries x_ij/1+x_ij (i j) are on a (2n-1)-dimensional manifold in _-^2n, so they are on the boundary of the convex hull T of the possible sufficient statistics ( , ). However, this boundary has zero Lebesgue measure, and so, zero probability with respect to the underlying absolutely continuous distribution. §.§ C.Proof of Theorem <ref> It suffices to prove that some induced matrix norm of the matrix of the firstderivativesof f atis strictly less than 1. We prove this for the L_1-norm. From (<ref>) we obtain that∂ g_i/∂ a_i (, b̂ ) = 1/n-1∑_j≠ iψ'(â_i+b̂_j)/ψ'[ ψ^-1(1/n-1∑_j≠ iψ(â_i+b̂_j)+R_i/n-1)] .From (<ref>) we have 1/n-1∑_j≠ iψ(â_i+b̂_j)+R_i/n-1=ψ(â_i). Substituting it into (<ref>), we get∂ g_i/∂ a_j(,b̂)= {[ ∑_s≠ i1/n-1ψ'(â_i+b̂_s)/ψ'(â_i) j=i; 0j≠ i . ].Likewise,∂ g_i/∂ b_j(,b̂)= {[ 0 j=i; 1/n-1ψ'(â_i+b̂_j)/ψ'(â_i)j≠ i . ].Further,∂ h_i/∂ a_j(,b̂)= {[0j=i; 1/n-1ψ'(â_j+b̂_i)/ψ'(b̂_i) j≠ i ].and∂ h_i/∂ b_j ( ,̱̂ )= {[ ∑_s≠ i1/n-1ψ'(â_s+b̂_i)/ψ'(b̂_i)j=i;0 j≠ i . ].Observe that J ( ,b̂ ) has nonnegative entries. Therefore, its L_1-norm is the maximum of its column-sums. The jth column-sum of J ( ,b̂ ) is equal to∑_s≠ j1/n-1ψ'(â_j+b̂_s)/ψ'(â_j) + ∑_s≠ j1/n-1ψ'(â_j+b̂_s)/ψ'(b̂_s) =1/n-1∑_s≠ jψ'(â_j+b̂_s)( 1/ψ'(â_j)+1/ψ'(b̂_s))for j=1,… ,n; and likewise, the (n+j)th column-sum of J ( ,b̂ ) is1/n-1∑_s≠ jψ'(â_s+b̂_j)( 1/ψ'(â_s)+1/ψ'(b̂_j))for j=1,… ,n.As (<ref>) and (<ref>) are of similar appearance, it suffices to prove that the right hand side of(<ref>) is less than 1. But ψ'(â_j+b̂_s)( 1/ψ'(â_j)+1/ψ'(b̂_s))<1 holds by Lemma <ref>, and we have n-1 terms in the summation.Since f: ^2n→^2n is continuously differentiable in a neighborhood of =( ,̱̂), Theorem 3 of <cit.> implies that there is a closed neighborhood K ofsuch that f is a contraction on K. In particular, the fixed point iteration f (^(it -1) )= ^(it ) (it→∞ ) converges for every ^(0 )∈ K to , which is the unique solution of (<ref>). □ 1handbook Abramowitz, M., Stegun, I. A., eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th ed.). New York: Dover (1972), pp. 258-259.alzer Alzer, H., Wells, J., Inequalities for the polygamma functions, SIAM J. Math. Anal. 29 (6) (1998), 1459-1466.Bernardo Bernardo, J. M., Psi (digamma) function. Algorithm AS 103, Applied Statistics 25 (1976), 315–317.Bolla13 Bolla, M., Spectral clustering and biclustering. Wiley (2013).Bolla Bolla, M., Elbanna, A., Estimating parameters of a probabilistic heterogeneous block model via the EM algorithm, Journal of Probability and Statistics (2015), Article ID 657965.Chatterjee Chatterjee, S., Diaconis, P. and Sly, A.,Random graphs with a given degree sequence, Ann. Stat. 21 (2010), 1400–1435.Grasmair Grasmair, M., Fixed point iterations, https://wiki.math.ntnu.no/_media/ma2501/2014v/fixedpoint.pdfHillar13 C. J. Hillar, A. Wibisono, Maximum entropy distributions on graphs, arXiv:1301.3321v2 (2013).Lauritzen Lauritzen, S. L., Graphical Models. Oxfor Univ. Press (1995). Yan Yan, T., Leng, C., Zhu, J., Asymptotics in directed exponential random graph models with an increasing bi-degree sequence, Ann. Stat. (2016) 44, 31-57.Wainwright M. Wainwright,. M. I. Jordan, Graphical models, exponential families, and variational inference, Foundations and Trends in Machine Learning 1 (1-2), 1-305 (2008). | http://arxiv.org/abs/1707.08904v3 | {
"authors": [
"Marianna Bolla",
"Ahmed Elbanna",
"Jozsef Mala"
],
"categories": [
"math.ST",
"stat.TH",
"62F10, 62B05"
],
"primary_category": "math.ST",
"published": "20170727151204",
"title": "Estimating parameters of a directed weighted graph model with beta-distributed edge-weights"
} |
Robust Pricing and Hedging around the Globe [The authors thank David Hobson, Sigrid Källblad, Marcel Nutz, and Yavor Stoev for stimulating discussions.] Sebastian HerrmannDepartment of Mathematics, University of Michigan, email mailto:[email protected]@umich.edu.Florian StebeggDepartment of Statistics, Columbia University, email [email protected]@columbia.edu. December 30, 2023 ===================================================================================================================================================================================================================================================================================== We consider the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.Keywords Robust superhedging; Semi-static strategies; Martingale optimal transport; Duality.AMS MSC 2010 60G44,49N05,91G20. JEL Classification G12,G23,C61. § INTRODUCTIONThis paper studies the robust pricing and superhedging of derivative securities with a payoff of the formF(X,A)= f(∫_[0,T] X_tA_t).Here, f is a nonnegative Borel function, X is a càdlàg price process (realized on the Skorokhod space), and A is chosen by the buyer from a given setof exercise rights. More precisely,is a set of so-called averaging processes, i.e., nonnegative and nondecreasing adapted càdlàg processes A with A_T ≡ 1. Setting = {_τ, T : τ a [0,T]-valued stopping time} or = { t ↦ t/T } reduces (<ref>) to the relevant special cases of American- or Asian-style derivatives, respectively:f(X_τ) or f(1/T∫_0^T X_tt).Other relevant examples are Bermudan options and European options with intermediate maturity (cf. Examples <ref>–<ref>). Robust pricing problem. Let μ and ν be probability measures on . We denote by (μ,ν) the set of (continuous-time) martingale couplings between μ and ν, i.e., probability measures P under which X is a martingale with marginal distributions X_0P∼μ and X_TP∼ν. The value of the primal problem:=sup_P∈(μ,ν)sup_A∈[P]F(X,A)can be interpreted as the maximal model-based price for F over all models which are consistent with the given marginals.Ifis a singleton, then (<ref>) is a so-called (continuous-time) martingale optimal transport problem. This problem was introduced (for general payoffs) by Beiglböck, Henry-Labordère, and Penkner <cit.> in a discrete-time setting and by Galichon, Henry-Labordère, and Touzi <cit.> in continuous time; cf. the survey <cit.>. Semi-static superhedging problem. The formal dual problem to (<ref>) has a natural interpretation as a superhedging problem.[The primal problem (<ref>) can be viewed as an optimization over finite measures P with three constraints: two marginal constraints and the martingale constraint. Its formal dual problem is the Lagrange dual problem where suitable functions φ and ψ and a suitable process H are used as Lagrange multipliers for the marginal and martingale constraints, respectively.] Loosely speaking, a semi-static superhedge is a triplet (φ,ψ,H) consisting of functions φ,ψ and a suitable process H such that for every A ∈, the superhedging inequality holds:φ(X_0) + ψ(X_T) + ∫_0^T H^A_t- X_t ≥ F(X,A)pathwise.Here, the strategy H = H^A may depend in an adapted way on A (cf. Section 3.2 for a precise formulation). For the example of an American-style payoff, this means that at the chosen exercise time τ, the buyer communicates her decision to exercise to the seller, who can then adjust the dynamic part of his hedging strategy (cf. <cit.>). The left-hand side in (<ref>) is the payoff of a static position in two European-style derivatives on X plus the final value of a self-financing dynamic trading strategy in X. The inequality (<ref>) says that the final value of this semi-static portfolio dominates the payoff F for every choice of exercise right and “all” price paths. The initial cost to set up a semi-staticsuperhedge (φ,ψ,H) equals the price μ(φ)+ν(ψ) of the static part.[We use the common notation μ(φ) for the integral of φ against μ.] The formal dual problem to (<ref>),:= inf{μ(φ) + ν(ψ) : (φ,ψ,H)is a semi-static superhedge},amounts to finding the cheapest semi-static superhedge (if it exists) and its initial cost, the so-called robust superhedging price. Main objectives and relaxation of the dual problem. We are interested in strong duality, i.e., =, and dual attainment, i.e., the existence of a dual minimizer.Dual attainment requires a suitable relaxation of the dual problem. Indeed, for the discrete-time martingale optimal transport problem, Beiglböck, Henry-Labordère, and Penkner <cit.> show strong duality for upper semicontinuous payoffs but provide a counterexample that shows that dual attainment can fail even if the payoff function is bounded and continuous. Beiglböck, Nutz, and Touzi <cit.> achieve strong duality and dual attainment for general payoffs and marginals in the one-step case by relaxing the dual problem in two aspects. First, they only require the superhedging inequality to hold in the quasi-sure sense, i.e., outside a set which is a nullset under every one-step martingale coupling between μ and ν. The reason is that the marginal constraints may introduce barriers on the real line which (almost surely) cannot be crossed by any martingale with these marginals; this was first observed by Hobson <cit.> (see also Cox <cit.> and Beiglböck and Juillet <cit.>). These barriers partition the real line into intervals and the marginal laws into so-called irreducible components. Then strong duality and dual attainment can be reduced to proving the same results for each irreducible component <cit.>. Second, Beiglböck, Nutz, and Touzi <cit.> extend the meaning of the expression μ(φ) + ν(ψ) to certain situations where both individual integrals are infinite. For example, it can happen that μ(φ) = -∞ and ν(ψ) = ∞, but the price [P]φ(X_0) + ψ(X_T) of the combined static part is well-defined, finite, and invariant under the choice of P∈(μ,ν). In this situation, this price is still denoted by μ(φ)+ν(ψ). We employ both relaxations for the precise definition of the dual problem in Section <ref>.In continuous time, Dolinsky and Soner <cit.> show strong duality for uniformly continuous payoffs satisfying a certain growth condition. They use the integration by parts formula to define the stochastic integral ∫_0^T H_t- X_t pathwise for finite variation integrands H. However, dual attainment cannot be expected in this class in general. For our payoffs (<ref>), we need to allow integrands that are of finite variation whenever they are bounded but can become arbitrarily large or small on certain price paths. As the integrands are not of finite variation on these paths, the meaning of the pathwise integral needs to be extended appropriately.For the purpose of the introduction, we discuss our results and methodology in a non-rigorous fashion, ignoring all aspects relating to the relaxation of the dual problem. Main results. We prove strong duality and dual attainment for payoffs of the form (<ref>) under mild conditions on f andfor irreducible marginals (Theorem <ref>); all results can be extended to general marginals along the lines of <cit.>. The key idea is the reduction of the primal and dual problems to simpler auxiliary problems, which do not depend on the setof exercising rights. In particular, our results cover American-style derivatives f(X_τ) for Borel-measurable f and Asian-style derivatives f(1/T∫_0^T X_tt) for lower semicontinuous f and show that both derivatives have (perhaps surprisingly) the same robust superhedging prices and structurally similar semi-static superhedges. Methodology. Our methodology relies on two crucial observations which allow us to bound the primal problem from below and the dual problem from above by simpler auxiliary primal and dual problems, respectively. To obtain a primal lower bound, we show that for any law θ which is in convex order between μ and ν, there is a sequence (P_n)_n≥1⊂(μ,ν) such that the law of ∫_[0,T]X_t A_t under P_n converges weakly to θ if A is a suitable averaging process. This allows us to boundfrom below by the value of the auxiliary primal problem := sup_μ≤_cθ≤_cνθ(f).(The converse inequality also holds (cf. Lemma <ref>), so that in fact =.)Regarding the dual upper bound, we prove (modulo technicalities) that if φ is concave and ψ is convex such that φ + ψ≥ f, then (φ,ψ,H) is a semi-static superhedge, where the dynamic part H is given explicitly in terms of φ and ψ byH_t := φ'(X_0) - ∫_[0,t]{φ'(X_0) + ψ'(X_s) } A_s.This allows us to boundfrom above by the value of the auxiliary dual problem:= inf{μ(φ) + ν(ψ) : φ concave, ψ convex, φ + ψ≥ f}. As a consequence, strong duality and dual attainment forandfollow from the same assertions for the simpler problemsand , which we prove by adapting the techniques of <cit.>. Moreover, our reduction of the dual problem implies that if (φ,ψ) is optimal for , then it is also the static part of an optimal semi-static superhedge and the dynamic part H can be computed ex-post via (<ref>). This dramatically decreases the complexity of the superhedging problem: the optimization over two functions and a process satisfying an inequality constraint on the Skorokhod space is reduced to an optimization over two functions satisfying an inequality constraint on .Our methodology reveals that many derivatives have the same robust superhedging prices and semi-static superhedges. Indeed,anddo not depend on the setof exercise rights granted to the buyer, and this independence transfers toandunder mild conditions on f and . For example, if f is lower semicontinuous, then the Asian-style derivative f(1/T∫_0^T X_tt), the American-style derivative f(X_τ), and the European-style derivative f(X_T') (for a fixed maturity T' ∈ (0,T)) all have the same robust superhedging price (Remark <ref>). This invariance breaks down when more than two marginals are given. Related literature. Much of the extant literature on robust superhedging in semi-static settings is concerned with strong duality and dual attainment. The results vary in their generality and explicitness as well as their precise formulation. The semi-static setting, where call options are available as additional hedging instruments, dates back to Hobson's seminal paper <cit.> on the lookback option.[We note that the superhedging strategies described in <cit.> are actually dynamic in the call options.] Many other specific exotic derivatives (mostly without special exercise rights) have been analyzed in this framework in the past two decades; cf. the survey <cit.>.Securities with special exercise rights have been studied in the context of American-style derivatives in discrete-time settings. Bayraktar, Huang, and Zhou <cit.> obtain a duality result for a somewhat different primal problem (cf. <cit.>) and show that duality may fail in their setting if they formulate their primal problem in analogy to the present paper; see also <cit.> for related results with portfolio constraints. Hobson and Neuberger <cit.> (based on Neuberger's earlier manuscript <cit.>) resolve this issue by adopting a weak formulation for the primal problem: instead of optimizing only over martingale measures on a fixed filtered path space, the optimization there runs over filtered probability spaces supporting a martingale and thereby allows richer information structures and hence more stopping times. We also refer to <cit.> for recent developments in this regard. We note that all these papers permit significant restrictions on the set of possible price paths (e.g., binomial trees) while we allow all càdlàg paths. This difference may be the reason why strong duality holds in our setting without any relaxation of the primal problem.The case of an Asian-style payoff as in (<ref>) has been studied in the case of a Dirac initial law μ. For convex or concave f, Stebegg <cit.> shows strong duality and dual attainment. For nonnegative Lipschitz f, Cox and Källblad <cit.> obtain a PDE characterization of the maximal model-based price for finitely supported ν. Bayraktar, Cox, and Stoev <cit.> provide a similar, but not identical PDE for the corresponding pricing problem for American-style payoffs as in (<ref>). A consequence of our main duality result is that the Asian and American pricing problems are the same, so that both these PDEs have the same (viscosity) solution. Organization of the paper. The remainder of the article is organized as follows. In Section 2, we recall basic results on the convex order and potential functions, introduce the generalized integral of <cit.> and its relevant properties, and present the extension of the pathwise definition of the stochastic integral for finite variation integrands. Section <ref> introduces the robust pricing and semi-static superhedging problems and presents our duality result. The duality between the auxiliary problems, the structure of their optimizers, and their relation to the robust pricing and superhedging problem are treated in Section <ref>. In Section <ref>, we provide simple geometric constructions of primal and dual optimizers for risk reversals and butterfly spreads. Finally, some counterexamples are collected in Section <ref>. § PRELIMINARIESFix a time horizon T and let Ω = D([0,T];) be the space of real-valued càdlàg paths on [0,T]. We endow Ω with the Skorokhod topology and denote bythe corresponding Borel σ-algebra, by X = (X_t)_t∈[0,T] the canonical process on Ω, and by = (_t)_t∈[0,T] the (raw) filtration generated by X. Unless otherwise stated, all probabilistic notions requiring a filtration pertain to .For any process Y = (Y_t)_t∈[0,T] on Ω, we set Y_0- = 0, so that the jump of Y at time 0 is Δ Y_0 = Y_0.§.§ Martingale measures and convex orderLet μ and ν be finite[As in <cit.>, using finite measures as opposed to probability measures turns out to be useful.] measures onwith finite first moment. We denote by Π(μ,ν) the set of (continuous-time) couplings of μ and ν, i.e., finite measures P on Ω such that P ∘ X_0^-1 = μ and P ∘ X_T^-1 = ν. If, in addition, the canonical process X is a martingale under P (defined in the natural way if P is not a probability measure), then we write P∈(μ,ν) and say that P is a (continuous-time) martingale coupling between μ and ν.We also consider discrete-time versions of these notions. To wit, we denote by Π^d(μ,ν) the set of finite measures Q on ^2 with marginal distributions μ and ν and by ^d(μ,ν) the subset of measures Q under which the canonical process on ^2 is a martingale (in its own filtration). The sets Π^d(μ,θ,ν) and (μ,θ,ν) of finite measures on ^3 with prescribed marginal distributions are defined analogously.We write μ≤_c ν if μ and ν are in convex order in the sense that μ(φ) ≤ν(φ) holds for any convex function φ : →. In this case, μ and ν have the same mass and the same barycenter (μ) := 1/μ()∫ xμ( x).The potential function u_μ:→[0,∞] of μ is defined as u_μ(x):= ∫ |x-y|μ( y).We refer to <cit.> for basic properties of potential functions. In particular, the following relationship between the convex order, potential functions, and martingale measures is well known.Let μ and ν be finite measures with finite first moments and μ() = ν(). Then the following are equivalent: (i) μ≤_c ν, (ii) u_μ≤ u_ν, (iii) ^d(μ,ν) ≠∅, and (iv) (μ,ν)≠∅. An analogous result holds for three marginals μ, θ, ν, the corresponding potential functions, and the set ^d(μ,θ,ν).We recall the following definition from <cit.> (see also <cit.>).A pair of finite measures μ≤_c ν is called irreducible if the set I = {u_μ < u_ν} is connected and μ(I) = μ(). In this situation, let J be the union of I and any endpoints of I that are atoms of ν; then (I,J) is the domain of μ≤_c ν. We work with irreducible μ≤_c ν for the remainder of this article.§.§ Generalized integral Let μ≤_c ν be irreducible with domain (I,J). Beiglböck and Juillet <cit.> and Beiglböck, Nutz, and Touzi <cit.> appropriately extend the meaning of the expression μ(φ) + ν(ψ) to the case where the individual integrals are not necessarily finite. We present here a slight extension of their work in order to deal with intermediate laws μ≤_cθ≤_cν for which the pairs μ≤_cθ and θ≤_cν may not be irreducible.For the rest of this article, whenever we write μ≤_c ν for any two measures μ and ν, we implicitly assume that both measures are finite and have a finite first moment. Throughout this section, we fix μ≤_cθ_1≤_cθ_2≤_cν.Let χ:J→ be concave. Denote by -χ” the second derivative measure of the convex function -χ and by Δχ the possible jumps of χ at the endpoints of I. We set(θ_1-θ_2)(χ):= 1/2∫_I (u_θ_1-u_θ_2) χ” + ∫_J∖ I|Δχ| (θ_2 - θ_1) ∈ [0,∞].The right-hand side is well defined in [0,∞] because u_θ_1≤ u_θ_2 on I and θ_1({ b }) ≤θ_2({ b }) for b ∈ J∖ I. If θ_1 = μ and θ_2 = ν, then (<ref>) coincides with Equation (4.2) in <cit.> because μ is concentrated on I. As in <cit.>, there is an alternative representation of (θ_1-θ_2)(χ) in terms of an iterated integral with respect to a disintegration of a (one-step) martingale coupling on ^2:Let χ:J→ be concave and let Q∈^d(θ_1,θ_2). For any disintegration Q = θ_1 ⊗κ, we have(θ_1-θ_2)(χ)= ∫_J [ χ(x_1) - ∫_J χ(x_2) κ(x_1, x_2)] θ_1( x_1). The proof of <cit.> does not use that μ≤_c ν is irreducible. Moreover, for χ̅:J → concave and continuous, the same arguments as in the proof of <cit.> yield1/2∫_I (u_θ_1-u_θ_2) χ̅” = ∫_J [ χ̅(x_1) - ∫_J χ̅(x_2)κ(x_1, x_2) ] θ_1( x_1).(Note that ∫_Jχ̅(x_2)κ(x_1, x_2) = χ̅(x_1) for boundary points x_1 ∈ J∖ I because κ is a martingale kernel concentrated on J.)For a general concave χ:J→, we write χ = χ̅- |Δχ|_J∖ I with χ̅ continuous. Then we can replace χ̅ with χ on the left-hand side of (<ref>) and the integrand on the right-hand side reads asχ + |Δχ|_J∖ I - ∫_J χ(x_2) κ(·, x_2) - ∫_J∖ I|Δχ(x_2)| κ(·, x_2).Integrating this against θ_1 and using Fubini's theorem yields∫_J [ χ(x_1) - ∫_J χ(x_2) κ(x_1, x_2) ]θ( x_1) + ∫_J∖ I|Δχ|θ_1 - ∫_J∖ I|Δχ|θ_2.Together with (<ref>), this proves the claim. It can be shown as in <cit.> that (θ_1-θ_2)(χ) = θ_1(χ) - θ_2(χ) if at least one of the individual integrals is finite.We can now define the integral θ_1(φ) + θ_2(ψ) as in <cit.>.Let φ:J → and ψ:J→ be Borel functions. If there exists a concave function χ:J→ such that φ-χ∈ L^1(θ_1) and ψ + χ∈ L^1(θ_2), we say that χ is a concave moderator for (φ,ψ) with respect to θ_1≤_cθ_2 and setθ_1(φ) + θ_2(ψ):= θ_1(φ - χ) + θ_2(ψ + χ) + (θ_1-θ_2)(χ) ∈ (-∞,∞].As in <cit.>, the expression θ_1(φ) + θ_2(ψ) defined in (<ref>) does not depend on the choice of the concave moderator.We write L^c(θ_1,θ_2) for the space of pairs of Borel functions φ,ψ:J→ which admit a concave moderator χ with respect to θ_1≤_cθ_2 such that (θ_1-θ_2)(χ)<∞. We next present additional properties of the notions introduced above.Let (φ,ψ) ∈ L^c(θ_1,θ_2). * φ is finite on atoms of θ_1. If φ is concave, then φ < ∞ on J and φ > -∞ on the interior of the convex hull of the support of θ_1.* ψ is finite on atoms of θ_2. If ψ is convex, then ψ > -∞ on J and ψ < ∞ on the interior of the convex hull of the support of θ_2.* If a,b : → are affine, then (φ + a,ψ+b) ∈ L^c(θ_1,θ_2) andθ_1(φ+a) + θ_2(ψ+b)= {θ_1(φ)+θ_2(ψ) } + θ_1(a) + θ_2(b).We only prove (iii). Let χ be a concave moderator for (φ,ψ) with respect to θ_1≤_cθ_2. Then φ - χ∈ L^1(θ_1), ψ + χ∈ L^1(θ_2), and (θ_1-θ_2)(χ) < ∞. Being affine, a and b are θ_1- and θ_2-integrable. It follows that χ is also a concave moderator for (φ + a, ψ + b) with respect to θ_1 ≤_c θ_2 and that (φ + a, ψ + b) ∈ L^c(θ_1,θ_2). The last assertion is a direct computation. Recall that I is the interior of the convex hull of the support of ν and that J is the union of I and any endpoints of I that are atoms of ν. Hence, Lemma <ref> (ii) shows in particular, that if (φ,ψ)∈ L^c(θ_1,ν) with ψ convex, then ψ is finite on J. We conclude this section with a number of calculation rules for the integrals defined above when φ is concave and ψ is convex.Let μ≤_cθ_1≤_cθ_2≤_cθ_3≤_cν (where the pair μ≤_cν is irreducible) and let (φ,ψ)∈ L^c(θ_1,θ_3) be such that φ is concave and finite, ψ is convex and finite, and φ + ψ is bounded from below by a concave θ_3-integrable function. * φ and -ψ are concave moderators for (φ,ψ) with respect to θ_1≤_cθ_3.* (φ,ψ) ∈ L^c(θ_1,θ_2) ∩ L^c(θ_2,θ_3).* θ_1(φ) + θ_2(ψ) ≤θ_1(φ) + θ_3(ψ).* θ_2(φ) + θ_3(ψ) ≤θ_1(φ) + θ_3(ψ). Denote by ξ a concave θ_3-integrable lower bound for φ+ψ. By the concavity of ξ, we have θ_1(ξ) ≥θ_2(ξ) ≥θ_3(ξ)>-∞, so that ξ is also θ_1- and θ_2-integrable.(i): Regarding the concave moderator property of φ, it suffices to show that φ + ψ is θ_3-integrable. Denote by φ' the left-derivative of the concave function φ on I. Then for (x_1,x_3)∈ I× J,ξ(x_3)≤φ(x_3) + ψ(x_3) ≤φ(x_1) + ψ(x_3) + φ'(x_1)(x_3-x_1).Fix any Q ∈^d(θ_1,θ_3). Then (<ref>) also holds Q-a.e. on J× J (setting φ' = 0 on J∖ I for example); this uses that any mass in a point of J ∖ I stays put under a martingale transport plan. Since ξ is θ_3-integrable, the negative part of the right-hand side in (<ref>) is Q-integrable. Then it can be argued as in <cit.> that the right-hand side in (<ref>) is Q-integrable. It follows that φ+ψ is θ_3-integrable.Regarding the assertion about -ψ, it suffices to show that φ + ψ is θ_1-integrable. We have ξ(x_1)≤φ(x_1) + ψ(x_1)= [φ(x_1) + ψ(x_3) + φ'(x_1)(x_3-x_1)] + [ψ(x_1) - ψ(x_3) - φ'(x_1)(x_3-x_1)] Q-a.e. on J× J.By the above, the first term on the right-hand side is Q-integrable. Thus, the negative part of the second term is also Q-integrable. Hence, we may integrate the second term iteratively using Fubini's theorem as in <cit.>. The Q-integral equals -(θ_1-θ_3)(-ψ) ≤ 0. In particular, the right-hand side in (<ref>) is Q-integrable. It follows that φ+ψ is θ_1-integrable.(ii)–(iv): We know from the above that φ+ψ is θ_3-integrable. It follows that φ is a concave moderator for (φ,ψ) with respect to θ_2≤_cθ_3. Because θ_1≤_cθ_2, we have that u_θ_1≤ u_θ_2 and θ_1({ b }) ≤θ_2({ b }) for b ∈ J ∖ I. Thus, (θ_2 - θ_3)(φ) ≤ (θ_1-θ_3)(φ) < ∞ (cf. Definition <ref>). Hence, (φ,ψ) ∈ L^c(θ_2,θ_3) and θ_2(φ) + θ_3(ψ)= θ_2(φ-φ) + θ_3(φ+ψ) + (θ_2-θ_3)(φ)≤θ_1(φ-φ) + θ_3(φ+ψ) + (θ_1-θ_3)(φ)= θ_1(φ) + θ_3(ψ).One can show similarly that (φ,ψ)∈ L^c(θ_1,θ_2) and that θ_1(φ) + θ_2(ψ) ≤θ_1(φ) + θ_3(ψ). §.§ Pathwise stochastic integration For any -adapted càdlàg process H of finite variation, the integral H_-∙ X_T = ∫_(0,T] H_t- X_t can be defined pathwise, i.e., for each ω∈Ω individually, via integration by parts as follows:H_-∙ X_T:= X_T H_T - X_0 H_0 - ∫_(0,T] X_tH_t,where the integral on the right-hand side is the pathwise Lebesgue–Stieltjes integral. Setting H_0- = 0, so that Δ H_0 = H_0, we can recast(<ref>) asH_-∙ X_T= (X_T-X_0)H_0 + ∫_(0,T] (X_T - X_t)H_t.For any martingale measure P, if the (standard) stochastic integral of H_- with respect to X exists, then it is P-indistinguishable from the pathwise stochastic integral.We need to give a sensible meaning to the integral H_-∙ X_T for certain integrands H which are not necessarily of finite variation, but may diverge in finite time.The following example motivates our extension of the pathwise stochastic integral for finite variation integrands. Let μ = δ_0 and ν = 1/2δ_-1 + 1/2δ_1. Then μ≤_c ν are irreducible with domain (I,J) = ((-1,1),[-1,1]). Consider a payoff function f which is convex on [-1,1] and has infinite (one-sided) derivatives at -1 and 1, e.g., f(x) = 1-√(1-x^2)_[-1,1](x),A semi-static superhedge for the Asian-style derivative f(1/T∫_0^T X_tt) can be derived as follows. By Jensen's inequality and the convexity of f, for every path of X that evolves in [-1,1],f(1/T∫_0^T X_tt)≤∫_0^T f(X_t) t/T≤∫_0^T (f(X_T) - f'(X_t)(X_T-X_t)) t/T= f(X_T) - ∫_0^T (X_T-X_t) f'(X_t)t/T.Comparing this with (<ref>), a semi-static superhedge for the Asian-style derivative is obtained from a European-style derivative with payoff f(X_T) maturing at T and a dynamic trading strategy H with H_0 = 0 and dynamics H_t = -f'(X_t)t/T. Then H is of finite variation whenever X stays away from the boundaries of (-1,1). But, as X approaches -1 or 1, the derivative f'(X_t) becomes arbitrarily large (in absolute value), and H may fail to be of finite variation. It turns out, however, that the integral ∫_0^T (X_T-X_t)f'(X_t) t/T is still well defined on these paths. The reason is that when paths of X come arbitrarily close to 1, say, then for any martingale coupling P∈(μ,ν), X_T=1 P-a.s. on these paths (because J = [-1,1]), so that X_T-X_t becomes small and counteracts the growth of f'(X_t). We shall define a pathwise stochastic integral for -adapted càdlàg integrators X and integrands Ĥ_- of the formĤ_t= h_0 + ∫_(0,t] h_sY_sfor an -adapted càdlàg process Y = (Y_t)_t∈[0,T] of finite variation and an -adapted process h = (h_t)_t∈[0,T]—even in certain situations where the right-hand side of (<ref>) is not finite. The idea is to formally substitute (<ref>) into (<ref>), formally use the associativity of Lebesgue–Stieltjes integrals, and then employ the resulting expression as a definition for a pathwise stochastic integral. We first introduce a set of integrands for this integral.Let Ω' ⊂ D([0,T];). We denote by L(Ω') the set of pairs (h,Y) consisting of an -adapted process h and an -adapted càdlàg process Y of finite variation such that the process ((X_T - X_t)h_t)_t∈[0,T] is Y-integrable on (0,T] for each path in Ω'. If Y is an -adapted càdlàg process of finite variation, then (1,Y) ∈ L(Ω') for any Ω' ⊂ D([0,T];) (because any càdlàg function is bounded on compact intervals).We fix a set Ω' ⊂ D([0,T];) for the rest of this section.For H=(h,Y) ∈ L(Ω'), we setH ♢ X_T:= (X_T-X_0)h_0 + ∫_(0,T] (X_T - X_t) h_tY_tonΩ'.We note that the Lebesgue–Stieltjes integral on the right-hand side of (<ref>) is well defined and finite by the definition of L(Ω'). The following result shows that for pathwise bounded h, H♢ X_T coincides with Ĥ_-∙ X_T for Ĥ as in (<ref>). This motivates the interpretation of H♢ X_T as the gains from trading in X according to a self-financingtrading strategy Ĥ_-.Let H = (h,Y) ∈ L(Ω') and ω∈Ω'. If the function t↦ h_t(ω) is bounded on [0,T], then (H ♢ X_T)(ω)= (Ĥ_- ∙ X_T)(ω),where Ĥ = h_0 + ∫_(0,·] hY. If we set h_0 = Y_0 and h_t = 1 for t ∈ (0,T] for an -adapted càdlàg process Y of finite variation, then H=(h,Y)∈ L(Ω) and by Proposition <ref>,H ♢ X_T= Y_- ∙ X_TonΩ.So the integral H♢ X_T embeds all pathwise stochastic integrals Y_-∙ X_T. Since h(ω) is bounded on [0,T], Ĥ_t(ω) = h_0(ω) + ∫_(0,t] h_s(ω)Y_s(ω) is a well-defined càdlàg finite variation function on [0,T]. Thus, by (<ref>),Ĥ_- ∙ X_T= (X_T - X_0)h_0 + ∫_(0,T] (X_T-X_s) h_sY_s = H ♢ X_T. § ROBUST PRICING AND SUPERHEDGING PROBLEMS Throughout this section, we fix an irreducible pair μ≤_c ν with domain (I,J) and a Borel function f:→ which is bounded from below by a ν-integrable concave function.§.§ Pricing problem Our pricing and hedging duality applies to a wide range of exotic derivatives including American options, fixed strike Asian options, Bermudan options, and European options with an intermediate maturity. We now describe this class of derivative securities.A nonnegative -adapted nondecreasing càdlàg process A=(A_t)_t∈[0,T] is called an averaging process if A_T(ω)=1 for every ω∈Ω. If in addition A_0(ω) = 0 and Δ A_T(ω) = 0 for each ω∈Ω, then A is called an interior averaging process. If in addition there is t∈(0,T) such that A_t(ω) = 0 for each ω∈Ω, then A is called a strictly interior averaging process. Recall that we set A_0- = 0 and note that for each ω∈Ω, A(ω) can be identified with a Borel probability measure on [0,T]. If A is an interior averaging process, then this probability measure is supported on (0,T), and if A is a strictly interior averaging process then its support is (uniformly in ω) contained in [t,T) for some t ∈(0,T).Given a nonempty setof averaging processes, we consider a derivative security whose payoff at time T isf(∫_[0,T]X_tA_t),where A ∈ is chosen by the buyer and the seller observes (A_s)_s∈[0,t] at time t. Then the robust model-based price is defined as_μ,ν(f,)=sup_P ∈(μ,ν)sup_A ∈[P]f(∫_[0,T] X_tA_t).In other words, _μ,ν(f,) is the highest model-based price of the derivative security (<ref>) among all martingale models which are consistent with the given marginal distributions. One can show that for each P ∈(μ,ν) and each averaging process A, the law of ∫_[0,T] X_tA_t under P is in convex order between μ and ν; cf. Lemma <ref>. Because f is by assumption bounded from below by a ν-integrable concave function, the expectations in (<ref>) are well defined.Important special cases are obtained for specific choices of .[No special exercise rights]Setting = { A } deprives the buyer of any special exercise rights and reduces (<ref>) to the more familiar robust pricing problemsup_P∈(μ,ν)[P]Ffor the derivative security F = f(∫_[0,T]X_t A_t).* Asian options. Setting A_t = t/T recovers the Asian-style derivative f(1/T∫_0^T X_tt); this includes fixed strike Asian puts and calls, but not floating strike Asian options. This robust pricing problem is analyzed in <cit.>. * European options. Setting A_t = _[T',T](t) yields a European-style payoff f(X_T') with an intermediate maturity T' ∈ (0,T).[Special exercise rights] Fix a nonempty setof [0,T]-valued -stopping times, and consider = {_τ, T :τ∈}. Then (<ref>) reduces tosup_P∈(μ,ν)sup_τ∈[P]f(X_τ).* American options. Ifconsists of all [0,T]-valued -stopping times, then (<ref>) is the robust American option pricing problem analyzed in <cit.>. * Bermudan options. Bermudan options with exercise dates 0≤ T_1 < ⋯ < T_n ≤ T are covered by choosingto be the set of { T_1,…, T_n }-valued -stopping times. §.§ Superhedging problem In the case of robust semi-static superhedging of American options, it is well known that a pricing-hedging duality can in general only hold if the seller of the option can adjust the dynamic part of his trading strategy after the option has been exercised; cf. <cit.>. In other words, the buyer has to communicate her decision of exercising to the seller at the time of exercising. The analog in our setting is that the seller observes A_t at time t. That is, his trading strategy can be “adapted” to the averaging process chosen by the buyer.To make this precise, let Ω̂ be the cartesian product of Ω and the set of nonnegative, nondecreasing, càdlàg functions a:[0,T]→[0,1] with a(T) = 1. As Ω̂ is a subspace of the Skorokhod space D([0,T];×[0,1]), we can equip it with the subspace Skorokhod topology and denote bythe corresponding Borel σ-algebra. We write = (_t)_t∈[0,T] for the (raw) filtration generated by the canonical process on Ω̂. For any process Z on Ω̂ and any averaging process A (on Ω), we define the process Z^A on Ω byZ^A_t(ω)= Z_t(ω,A(ω)), ω∈Ω.Note that if Z is -adapted, then Z^A is -adapted, and if Z is càdlàg or of finite variation, then so is Z^A.Next, we define a suitable set of paths for the hedging problem. Let Ω_μ,ν⊂Ω denote the subset of paths which start in I, evolve in J, and are “captured” if they approach the boundary ∂ J:Ω_μ,ν := {ω∈Ω : ω_0 ∈ I, ω_t ∈ J for all t ∈ (0,T],if ω_t-∈∂ J, then ω_u =ω_t- for all u∈[t,T], andif ω_t ∈∂ J, then ω_u = ω_t for all u∈[t,T]}. One can show that every martingale coupling between μ and ν is concentrated on Ω_μ,ν:[The fact that μ and ν are concentrated on I and J, respectively, together with the martingale property implies that P-a.e. path has the first two properties in (<ref>). The other two properties can be shown similarly to the fact that nonnegative supermartingales are almost surely captured in zero (cf. <cit.>).]Ω_μ,ν∈ and P[Ω_μ,ν] = P[Ω] for every P∈(μ,ν). We are now ready to define the trading strategies for the robust superhedging problem. A semi-static trading strategy is a triplet (φ,ψ,H) consisting of a pair of functions (φ,ψ) ∈ L^c(μ,ν) and a pair H=(h_t,Y_t)_t∈[0,T] of -adapted processes on Ω̂ such thatH^A:= (h^A,Y^A) ∈ L(Ω_μ,ν) for every averaging process A.The portfolio value at time T of a semi-static trading strategy is given by the sum of the static part with payoffs φ(X_0) and ψ(X_T) and the gains H^A♢ X_T of the dynamic part:φ(X_0) + ψ(X_T) + H^A♢ X_T.The initial cost to set up this position is equal to the initial price of the static part:μ(φ) + ν(ψ). We now turn our attention to semi-static trading strategies which dominate the payoff (<ref>) of our derivative security foreach path in Ω_μ,ν and every averaging process in .A semi-static trading strategy (φ,ψ,H) is called a semi-static superhedge (for f and ) if for every A ∈,f(∫_[0,T] X_tA_t)≤φ(X_0) + ψ(X_T) + H^A ♢ X_TonΩ_μ,νand[P]φ(X_0) + ψ(X_T) + H^A ♢ X_T ≤μ(φ) + ν(ψ),P ∈(μ,ν).The set of semi-static superhedges for f andis denoted by _μ,ν(f,). The requirement (<ref>) is an admissibility condition. It demands that for every P∈(μ,ν), the portfolio value, consisting of both the static and the dynamic part, is a one-step P-supermartingale between the time at which the static part is set up and time T. In other words, the expectation of the terminal portfolio value (<ref>) is less than or equal to the initial portfolio value (<ref>). We define the robust superhedging price (for f and ) as the “minimal” initial capital required to set up a semi-static superhedge for f and :[We use the convention inf∅=∞.]_μ,ν(f,) = inf_(φ,ψ,H)∈_μ,ν(f,){μ(φ) + ν(ψ) }. §.§ Weak and strong duality Weak duality between the robust pricing and hedging problems is an immediate consequence of their definitions:Let f:→ be Borel and bounded from below by a ν-integrable concave function and letbe a nonempty set of averaging processes. Then_μ,ν(f,) ≤_μ,ν(f,). Let P ∈(μ,ν), A∈, and (φ,ψ,H) ∈_μ,ν(f,) (there is nothing to show if this set is empty). Taking P-expectations in (<ref>) and using (<ref>) shows that [P]f(∫_[0,T] X_tA_t)≤μ(φ) + ν(ψ).This proves the claim as P, A, and (φ,ψ,H) were arbitrary. With an additional mild assumption on eitheror f, we obtain strong duality and the existence of dual minimizers:Let μ≤_c ν be irreducible, let f:→[0,∞] be Borel, and letbe a set of averaging processes. Suppose that one of the following two conditions holds: * f is lower semicontinuous andcontains an interior averaging process;*contains a strictly interior averaging process.Then_μ,ν(f,)= _μ,ν(f,) ∈ [0,∞]and this value is independent ofas long as one of the two conditions above holds. Moreover, if _μ,ν(f,) < ∞, then there exists an optimizer (φ,ψ,H)∈_μ,ν(f,) for _μ,ν(f,).* For fixed f, the robust model-based price _μ,ν(f,) is invariant under the choice of the set(as long as the assumptions of Theorem <ref> hold). In particular, American, Bermudan, and European options with intermediate maturity (cf. Examples <ref>–<ref>) all have the same robust model-based price (because the corresponding setsall contain a strictly interior averaging process). If f is lower semicontinuous, this extends to the Asian-style option of Example <ref> (i). If more than two marginals are given, then the robust model-based prices of these derivatives typically differ; see Example <ref>. * Derivatives of the form (<ref>) that depend distinctly on X_0 and/or X_T such as f(1/2(X_0 + X_T)) are not covered by Theorem <ref> ( does not contain an interior averaging process). In these cases, the robust model-based price is still bounded above by the corresponding robust model-based price of, say, the European-style derivative f(X_T/2). However, the inequality is typically strict; see Example <ref>. * Theorem <ref> can be extended to non-irreducible marginals along the lines of <cit.>. * Strong duality continues to hold if we restrict ourselves to finite variation strategies; cf. Remark <ref> for an outline of the argument. It is an open question whether there is (in general) a dual minimizer (φ,ψ,H) with a dynamic part H of finite variation.We defer the proof of Theorem <ref> to the end of Section <ref>. The idea is as follows. We bound the pricing problem from below and the hedging problem from above by auxiliary maximization and minimization problems, respectively, and show that strong duality holds between those two auxiliary problems. Then all four problems have equal value and in particular strong duality for the pricing and hedging problems holds. Moreover, we show that the auxiliary dual problem admits a minimizer and that every element in the dual space of the auxiliary problem gives rise to a semi-static superhedge with the same cost. Then, in particular, the minimizer of the auxiliary dual problem yields an optimal semi-static superhedge for f and(which is independent of ). § AUXILIARY PROBLEMS Throughout this section, we fix an irreducible pair μ≤_c ν with domain (I,J) and a function f:→ which is bounded from below by a ν-integrable concave function.The auxiliary primal and dual problems are formally derived in Section <ref>. They are rigorously introduced in Sections <ref>–<ref> and proved to be lower and upper bounds of the robust model-based price and the robust superhedging price, respectively. Their strong duality is proved in Section <ref>. Finally, structural properties of primal and dual optimizers of the auxiliary problems are studied in Section <ref>.§.§ Motivation The key property of payoffs of the form (<ref>) is that the law of ∫_[0,T]X_t A_t under P∈(μ,ν) is in convex order between μ and ν. In this section, we explain this observation and how it can be used to estimate the robust pricing problem from below and the robust superhedging problem from above.Let P ∈(μ,ν) and let τ be a [0,T]-valued -stopping time. An application of the optional stopping theorem and Jensen's inequality shows that for any convex function ψ,μ(ψ) = [P]ψ(X_0) = [P]ψ([P]X_τ_0)≤[P]ψ(X_τ)and ν(ψ) = [P]ψ(X_T) ≥[P]ψ([P]X_T_τ) = [P]ψ(X_τ),so that the law of X_τ under P is in convex order between μ and ν.Using a time change argument and again Jensen's inequality and the optional stopping theorem, it can be shown that this property generalizes to the random variable ∫_[0,T]X_tA_t for an averaging process A.Let P∈(μ,ν) and let A be an averaging process. Then the law of ∫_[0,T]X_t A_t under P is in convex order between μ and ν. In the sequel, we write = _μ,ν(f,) and = _μ,ν(f,) for brevity. Lemma <ref> implies that≤sup_μ≤_cθ≤_cνθ(f) =: .We show in Section <ref> that also the converse inequality holds under mild assumptions on f and . Thus, = and one is led to expect that = for a suitable dual problemto .Let us thus formally derive the Lagrange dual problem for . Dualizing the constraint μ≤_cθ≤_cν suggests to consider the LagrangianL(θ,ψ_1,ψ_2):= θ(f) + (θ(ψ_1)-μ(ψ_1)) + (ν(ψ_2)-θ(ψ_2)),where convex functions ψ_1,ψ_2 are taken as Lagrange multipliers.[Note that the last two terms in (<ref>) are nonnegative for all convex ψ_1,ψ_2 if and only if the primal constraint μ≤_cθ≤_cν holds.] Then the Lagrange dual problem is=inf_ψ_1,ψ_2sup_θ L(θ,ψ_1,ψ_2) = inf_ψ_1,ψ_2sup_θ{θ(f+ψ_1-ψ_2) -μ(ψ_1) + ν(ψ_2)}where the infima are taken over convex functions and the suprema are taken over finite measures. Viewing the finite measure θ as a Lagrange multiplier for the constraint f ≤ -ψ_1 + ψ_2 and relabeling φ = -ψ_1 and ψ = ψ_2, we obtain= inf{μ(φ)+ν(ψ) : φ concave, ψ convex, and φ+ψ≥ f}.In the precise definition ofin Section <ref>, μ(φ)+ν(ψ) is understood in the generalized sense of Definition <ref> and the inequality φ + ψ≥ f is required to hold on J. We then show that each feasible element (φ,ψ) forentails an element (φ,ψ,H) ∈_μ,ν(f,) (Proposition <ref>), which implies that ≤.Combining the above with the weak duality inequality (Lemma <ref>) yields= ≤≤.Hence, strong duality and dual attainment for the robust pricing and superhedging problems reduce to the same assertions for the simpler auxiliary problems, which are proved in Section <ref>.§.§ Auxiliary primal problem Consider the auxiliary primal problem_μ,ν(f) = sup_μ≤_c θ≤_c νθ(f),where θ(f) is understood as the outer integral if f is not Borel-measurable. Under suitable conditions on f and , the primal value _μ,ν(f) is a lower bound for the robust model-based price (<ref>):Letbe a set of averaging processes. Suppose that one of the following two sets of conditions holds: *contains an interior averaging process and f is lower semicontinuous and bounded from below by a ν-integrable concave function φ:J→;*contains a strictly interior averaging process and f is Borel.Then_μ,ν(f) ≤_μ,ν(f,).The proof of Proposition <ref> is given at the end of this section. It is based on the following construction of measures in (μ,ν) under which the law of ∫_[0,T] X_tA_t equals (approximately or exactly) a given θ. This construction also highlights the importance ofcontaining an interior averaging process, which does not put any mass on the times 0 and T at which the marginal distributions of X are given; see Example <ref> for a counterexample.Let μ≤_c θ≤_c ν. * There is a sequence (P_n)_n≥1⊂(μ,ν) such that^P_n(∫_[0,T]X_tA_t) θweaklyfor every interior averaging process A.* If A is a strictly interior averaging process, then there is P ∈(μ,ν) (depending on A) such that ^P(∫_[0,T]X_tA_t) = θ. (i): By the two-step adaptation of Proposition <ref>, there exists a measure Q ∈^d(μ,θ,ν). For all n large enough, let ι^n:^3 →Ω be the embedding of ^3 in Ω which maps (y_1,y_2,y_3) to the piecewise constant path[0,T]∋ t ↦ y_1_[0,1/n)(t) + y_2_[1/n,T)(t) + y_3_{T}(t)(which jumps (at most) at times 1/n and T), and denote by P_n := Q ∘ (ι^n)^-1 the associated pushforward measure. Then P_n ∈(μ,ν) by the corresponding properties of Q. Moreover, denoting the canonical process on ^3 by (Y_1,Y_2,Y_3) and setting A^n = A ∘ι^n for an interior averaging process A, we have∫_[0,T] (ι^n)_tA^n_t - Y_2= Y_1 A^n_1/n- + Y_2 (A^n_T--A^n_1/n-) + Y_3 Δ A^n_T -Y_2A_T = (Y_1 - Y_2)A^n_1/n- +(Y_3 - Y_2)Δ A^n_T = (Y_1 - Y_2)A^n_1/n-on^3,where we use the properties A_T = 1 and Δ A_T=0 of an interior averaging process.By construction, the law of ∫_[0,T] (ι^n)_tA^n_t under Q coincides with the law of ∫_[0,T] X_tA_t under P_n and the law of Y_2 under Q is θ. It thus suffices to prove that the right-hand side in (<ref>) converges to zero in L^1(Q) as n→∞. To this end, note that |Y_1 - Y_2| ≤ |Y_1| + |Y_2| is Q-integrable because μ and θ have finite first moments. Thus, by dominated convergence, it is enough to show that A^n_1/n-→ 0 pointwise as n→0. So fix (y_1,y_2,y_3) ∈^3. Since A is -adapted, A_1/n-(ω) only depends on the values of the path ω on the interval [0,1/n). In view of the embedding (<ref>), this means that A^n_1/n-(y_1,y_2,y_3)= A_1/n-(ι^n(y_1,y_2,y_3)) = A_1/n-(y_1 _[0,T]),where y_1 _[0,T] denotes the constant path at y_1. Hence, the asserted pointwise convergence follows from the fact that A_0 = 0 and A is right-continuous.(ii): If A is a strictly interior averaging process, then the last expression in (<ref>) is identically zero for n large enough and setting P=P_n gives the desired result. Part (i) of Lemma <ref> remains true if we restrict ourselves to martingale measures with almost surely continuous paths. The analog of part (ii) for continuous martingales requires the additional assumption that there exists t < T such that A_t ≡ 1.The main ingredient for this assertion is <cit.>: for every discrete time-martingale {Y_n}_n ≥ 0, there is a continuous-time martingale {Z_t}_t ≥ 0 with continuous sample paths such that the processes {Y_n}_n ≥ 0 and {Z_n}_n ≥ 0 have the same (joint) distribution.Let μ≤_cθ≤_cν. Assume first that condition (ii) holds and let A be a strictly interior averaging process. Then by Lemma <ref> (ii), there is P∈(μ,ν) such that ^P(∫_[0,T]X_t A_t) = θ. Hence,θ(f) = [P]f(∫_[0,T]X A)≤_μ,ν(f,).As θ was arbitrary, the claim follows.Next, assume instead that condition (i) holds and let A be an interior averaging process and φ as in condition (i). By Lemma <ref> (i), there is a sequence (P_n)_n∈⊂(μ,ν) such thatθ_n:=^P_n(∫_[0,T]X_t A_t) →θ weakly. Define f_k = f ∨ (-k), k ≥ 1. Then f_k is bounded from below and lower semicontinuous, so lim inf_n→∞θ_n(f_k) ≥θ(f_k) by the Portmanteau theorem.Fix ε > 0. Choose first k large enough such that ν((φ+k)^-) ≤ε/2 and then N large enough such that θ_n(f_k)-θ(f_k) ≥ -ε/2 for all n ≥ N. Using that 0 ≤ f_k-f ≤ (φ + k)^- and that (φ + k)^- is convex, we obtain for n ≥ N,θ_n(f) - θ(f)= θ_n(f-f_k) + (θ_n(f_k) - θ(f_k)) + θ(f_k - f)≥ -θ_n((φ+k)^-)-ε/2≥ -ν((φ+k)^-) - ε/2≥ -ε.Thus, lim inf_n→∞θ_n(f) ≥θ(f). Now the claim follows fromθ(f)≤lim inf_n→∞θ_n(f) = lim inf_n→∞[P_n]f(∫_[0,T]X_t A_t)≤_μ,ν(f,).§.§ Auxiliary dual problem Consider the auxiliary dual problem_μ,ν(f) = inf_(φ,ψ)∈_μ,ν(f){μ(φ) + ν(ψ) },where _μ,ν(f) denotes the set of (φ,ψ) ∈ L^c(μ,ν) with concave φ:J→ and convex ψ:J→ such that φ + ψ≥ f on J.The dual value _μ,ν(f) is an upper bound for the robust superhedging price (<ref>):Let f:→[0,∞] be Borel. Then _μ,ν(f,) ≤_μ,ν(f). Proposition <ref> follows immediately from the next result (Proposition <ref>) which shows that every (φ,ψ)∈_μ,ν(f) gives rise to a semi-static superhedge for f and . More precisely, the semi-static superhedge is of the form (φ,ψ,H) and the dynamic part H can be explicitly written in terms of the “derivatives” of φ and ψ.Given a convex function ψ: J →, a Borel function ψ':I → is called a subderivative of ψ if for every x_0 ∈ I, ψ'(x_0) belongs to the subdifferential of ψ at x_0, i.e.,ψ(x) - ψ(x_0) ≥ψ'(x_0)(x-x_0), x ∈ J.Symmetrically, for a concave function φ: J→, a Borel function φ':I → is called a superderivative of φ if -φ' is a subderivative of -φ.If (φ,ψ)∈_μ,ν(f) and f > -∞ on J, then φ and ψ are both finite (so that sub- and superderivatives are well defined). Indeed, we already know from Remark <ref> that ψ is finite on J. Moreover, φ < ∞ on J by Lemma <ref> (i) and if f > -∞ on J, then φ≥ f-ψ > -∞, so that also φ is finite on J. Let f:→[0,∞] be Borel and let (φ,ψ)∈_μ,ν(f). Denoting the canonical process on Ω̂ by (X,A), define the -adapted process h = (h_t)_t∈[0,T] (on Ω̂) byh_0= φ'(X_0)(1-A_0) - ψ'(X_0)A_0,h_t= -φ'(X_0) - ψ'(X_t), t ∈ (0,T],where φ' is any superderivative of φ, ψ' is any subderivative of ψ, and we set φ'=ψ' = 0 on ∖ I. Set H = (h,A). Then (φ,ψ, H) ∈_μ,ν(f,) for any nonempty setof averaging processes. The proof of Proposition <ref>, given at the end of this section, relies on the following two technical lemmas. The definition of Ω_μ,ν in (<ref>) is crucial for the first one. We recall that (real-valued) càdlàg functions are bounded on compact intervals.Let ψ and ψ' be as in Proposition <ref>. For each ω∈Ω_μ,ν, the function [0,T] ∋ t ↦ (ω_T-ω_t)ψ'(ω_t) is bounded.Fix ω∈Ω_μ,ν and write I= (l,r) with l,r ∈. We consider three cases: (i) J = I, (ii) J=[l,r), and (iii) J = [l,r]. The case (l,r] is symmetric to (ii).(i): Suppose that J = I = (l,r). We claim that ω evolves in a compact (and hence strict) subset of I. Suppose for the sake of contradiction hat inf_t∈[0,T]ω_t = l ∈ [-∞,∞). Then there is a sequence (t_n)_n∈⊂ [0,T] such that lim_n→∞ω_t_n = l. Passing to a subsequence if necessary, this sequence may be chosen to be either (strictly) increasing or nonincreasing to a limit t^⋆∈ [0,T]. Then, as ω is càdlàg, ω_t^⋆- = l or ω_t^⋆ = l. But then ω_t^⋆=l in any case by the definition of Ω_μ,ν, a contradiction to ω_t^⋆∈ J = I. Thus, inf_t∈[0,T]ω_t > l and symmetrically sup_t∈[0,T]ω_t < r. This proves the claim. It follows that (ω_T-ω_t)ψ'(ω_t) is bounded over t∈[0,T] because the subderivative ψ' is bounded on compact subsets of I.(ii): Suppose that J = [l,r), i.e., ν has an atom in l > -∞. If ω evolves in I, then we can argue as in (i). We may thus assume that t^⋆ := inf{ t∈ [0,T] : ω_t = l }∈ (0,T]. Then, as ω is càdlàg and by the definition of Ω_μ,ν, we have ω_u = l for all u ∈ [t^⋆,T]). In particular, ω_T = l and it is enough to show that [0,t^⋆)∋ t ↦ (ω_T - ω_t)ψ'(ω_t) is bounded.We can argue similarly as in (i) that r':=sup_t∈[0,T]ω_t < r, so that the path ω evolves in the compact interval [l,r']. Because ψ is convex and finite on (l,r), ψ' is bounded from above on [l,r']. It follows that t↦(ω_T-ω_t)ψ'(ω_t) is bounded from below on [0,t^⋆). To show that this function is also bounded from above, we observe that by the convexity of ψ,(ω_T-ω_t)ψ'(ω_t) ≤ψ(ω_T) - ψ(ω_t) =ψ(l) - ψ(ω_t).Now ψ(l) is finite because ν has an atom at l, and ψ is bounded from below on [l,r'] because it is finite and convex on the compact interval [l,r']. Using this in (<ref>) shows the assertion.(iii): Suppose that J = [l,r], i.e., ν has atoms at l > -∞ and r<∞. As in (ii), we may assume that ω hits one of the endpoints of J before T. By symmetry, we may assume that ω hits l. By definition of Ω_μ,ν, the path ω is then bounded away from the right endpoint r (otherwise it would be captured in r), i.e., sup_t∈[0,T]ω_t < r. Now the same argument as in (ii) proves the assertion. The second technical lemma is an adaptation of <cit.> to our setting. It is used to show the admissibility condition (<ref>) of the semi-static trading strategy in Proposition <ref>.Let (φ,ψ) ∈ L^c(μ,ν) and let g_0,g_1:J → be Borel. Let τ be a [0,T]-valued -stopping time such thatφ(X_0) + ψ(X_T) + g_0(X_0)(X_τ - X_0) + g_1(X_τ)(X_T - X_τ)is bounded from below on Ω_μ,ν. Then for all P ∈(μ,ν),[P]φ(X_0) + ψ(X_T) + g_0(X_0)(X_τ - X_0) + g_1(X_τ)(X_T - X_τ) = μ(φ)+ν(ψ). Let χ be a concave moderator for (φ,ψ) with respect to μ≤_c ν and let θ be the law of X_τ. By optional stopping, μ≤_c θ≤_c ν. We expand (<ref>) to(φ - χ)(X_0)+ (ψ + χ)(X_T) + [χ(X_0) - χ(X_T) + g_0(X_0)(X_τ - X_0) + g_1(X_τ)(X_T - X_τ)],and observe that the first two terms are P-integrable. Then the assumed lower bound yields that the last term has a P-integrable negative part. We can therefore apply Fubini's theorem and evaluate its integral iteratively. To this end, let Q be the law of (X_0,X_τ,X_T) on the canonical space ^3 with a disintegration Q= μ( x_0) ⊗κ_0(x_0, x_1) ⊗κ_1(x_0,x_1, x_2)for martingale kernels κ_0 and κ_1. In view of the definition of μ(φ) + ν(ψ) in (<ref>), we have to show that the P-expectation of the last term in (<ref>) is (μ-ν)(χ).To this end, we observe that for μ⊗κ_0-a.e. (x_0,x_1) ∈ J^2,∫_J[ χ(x_0) - χ(x_2) + g_0(x_0)(x_1-x_0) + g_1(x_1)(x_2-x_1) ] κ_1(x_0,x_1, x_2)= ∫_J [χ(x_0) - χ(x_2) + g_0(x_0)(x_1-x_0) ] κ_1(x_0,x_1, x_2) = χ(x_0) - ∫_J χ(x_2) κ_1(x_0,x_1, x_2) + g_0(x_0)(x_1-x_0).Integrating the left-hand side of (<ref>) against μ⊗κ_0 gives the P-expectation of the last term in (<ref>). It thus remains to show that the corresponding integral of the right-hand side equals (μ-ν)(χ). Integrating the right-hand side of (<ref>) first against κ_0(x_0, x_1) yields for μ-a.e. x_0∈ J,χ(x_0) - ∫_Jχ(x_2)κ(x_0, x_2),where κ(x_0,·) = ∫_J κ_1(x_1,·) κ_0(x_0, x_1) is again a martingale kernel. Finally, the integral of (<ref>) against μ is∫_J [ χ(x_0) - ∫_Jχ(x_2)κ(x_0, x_2) ] μ( x_0).Noting that μ⊗κ is a disintegration of a one-step martingale measure on ^2 with marginals μ and ν, the last term equals (μ-ν)(χ) by Lemma <ref>.First, we show that (φ,ψ,H) is a semi-static trading strategy. As h and A are clearly -adapted and (φ,ψ)∈ L^c(μ,ν) by assumption, it remains to check condition (<ref>) (with Y^A replaced by A). So fix an averaging process A and note that H^A = (h^A,A). The only nontrivial part in proving H^A ∈ L(Ω_μ,ν) is to show that (X_T-X_t)h^A_t is A-integrable on (0,T] for each path in Ω_μ,ν. To this end, note that φ'(X_0) and ψ'(X_0) are finite because X_0 ∈ I. It thus suffices to show that (X_T-X_t)ψ'(X_t) is bounded on [0,T] for each path in Ω_μ,ν; this is the content of Lemma <ref>.Second, we show the superhedging property (<ref>). Fix an averaging process A and a path in Ω_μ,ν. To ease the notation, we write h instead of h^A in the following. Note, however, that h^A has the same formal expression as h in (<ref>), but with A being the fixed averaging process (and not the second component of the canonical process on Ω̂).Using the definitions of H♢ X_T and h as well as the fact that A_0 = Δ A_0, we obtainH ♢ X_T= (X_T-X_0)h_0 + ∫_(0,T](X_T-X_t) h_tA_t= (X_T-X_0)φ'(X_0) - ∫_[0,T] (X_T-X_t) (φ'(X_0) + ψ'(X_t))A_t.Then, using that A is a probability measure on [0,T], the concavity of φ and the convexity of ψ, and Jensen's inequality, we can estimateH ♢ X_T= ∫_[0,T]φ'(X_0)(X_t - X_0)A_t - ∫_[0,T]ψ'(X_t)(X_T - X_t)A_t ≥φ'(X_0) (∫_[0,T] X_tA_t - X_0) - ∫_[0,T]( ψ(X_T) - ψ(X_t))A_t ≥φ(∫_[0,T] X_tA_t) - φ(X_0) - ψ(X_T) + ∫_[0,T]ψ(X_t)A_t ≥φ(∫_[0,T] X_tA_t) - φ(X_0) - ψ(X_T) + ψ(∫_[0,T] X_tA_t).Rearranging terms and using that φ + ψ≥ f on J, we findφ(X_0) + ψ(X_T) + H ♢ X_T ≥ f(∫_[0,T] X_tA_t). Third, we show the admissibility condition (<ref>). Fix an averaging process A and P ∈(μ,ν). Define the family of -stopping times C_s, s∈(0,1), byC_s= inf{ t ∈ [0,T] : A_t > s }and note that 0 ≤ C_s ≤ T for s∈(0,1) because A_T =1. Then using the family C_s as a time change (cf. <cit.>) for the integral in (<ref>) yieldsφ(X_0) + ψ(X_T) + H♢ X_T=∫_0^1 {φ(X_0) + ψ(X_T) +φ'(X_0) (X_C_s - X_0) - ψ'(X_C_s)(X_T-X_C_s)} s.Now, suppose that the integrand in (<ref>) is bounded from below, uniformly over s∈(0,1) and ω∈Ω_μ,ν. Then by Lemma <ref>, the P-expectation of the integrand equals μ(φ) + ν(ψ) for each s∈(0,1). Using this together with Tonelli's theorem and (<ref>) gives[P]φ(X_0) + ψ(X_T) + H ♢ X_T = μ(φ) + ν(ψ),so that (<ref>) holds.It remains to show that the integrand in (<ref>) is uniformly bounded from below. This follows from concavity of φ and convexity of ψ together with the fact that φ + ψ≥ f ≥ 0 on J:φ(X_0)+ ψ(X_T) +φ'(X_0) (X_t - X_0) - ψ'(X_t)(X_T-X_t) ≥φ(X_t) + ψ(X_t) ≥ f(X_t) ≥ 0,t∈[0,T].This completes the proof.§.§ Duality We now turn to the duality between the auxiliary problems _μ,ν(f) and _μ,ν(f).Let μ≤_c ν be irreducible with domain (I,J) and let f: → [0,∞]. * If f is upper semianalytic, then _μ,ν(f) = _μ,ν(f) ∈ [0,∞].* If _μ,ν(f) < ∞, then there exists a dual minimizer (φ,ψ) ∈_μ,ν(f).A couple of remarks are in order. We only state the duality for one irreducible component. One can formulate and prove the full duality for arbitrary marginals μ≤_c ν in analogy to <cit.>. We omit the details in the interest of brevity. The lower bound on f in Theorem <ref> can be relaxed. Indeed, suppose that f:→ is upper semianalytic and bounded from below by an affine function g.We first consider the primal problem. Because g is affine and any μ≤_cθ≤_cν has the same mass and barycenter as μ,θ(f-g)= θ(f) - θ(g) = θ(f) - μ(g).Thus,_μ,ν(f-g)= _μ,ν(f) - μ(g). Regarding the dual problem, we note that (φ,ψ)∈_μ,ν(f-g) if and only if (φ+g,ψ)∈_μ,ν(f) and that by Lemma <ref> (iii),μ(φ) + ν(ψ)= {μ(φ+g) + ν(ψ) } - μ(g).Hence,_μ,ν(f-g)= _μ,ν(f) - μ(g).Because f-g is nonnegative, the left-hand sides of (<ref>)–(<ref>) coincide by Theorem <ref> (i). Therefore, _μ,ν(f) = _μ,ν(f) ∈ (-∞,∞].Moreover, if _μ,ν(f) < ∞, then also _μ,ν(f-g) < ∞ and a dual minimizer (φ,ψ)∈_μ,ν(f-g) for _μ,ν(f-g) exists by Theorem <ref> (ii). Now the above shows that (φ+g,ψ) ∈_μ,ν(f) is a dual minimizer for _μ,ν(f). The proof of Theorem <ref> is based on several preparatory results. We start with the crucial closedness property of the dual space in the spirit of <cit.>.Let μ≤_c ν be irreducible with domain (I,J), let f,f_n: J → [0,∞] be such that f_n → f pointwise, and let (φ_n,ψ_n) ∈_μ,ν(f_n) with sup_n {μ(φ_n) + ν(ψ_n)} < ∞. Then there is (φ,ψ) ∈_μ,ν(f) such that μ(φ) + ν(ψ) ≤lim inf_n →∞{μ(φ_n) + ν(ψ_n) }.Let h_n = φ_n':I→ be a superderivative of the concave function φ_n. Asφ_n(x) + ψ_n(y) + h_n(x)(y-x)≥φ_n(y) + ψ_n(y) ≥ f_n(y) ≥ 0,(x,y) ∈ I× J,(φ_n,ψ_n,h_n) is in the dual space ^c_μ,ν(0) of <cit.>. Hence, following the line of reasoning in the proof of <cit.> (which is based on Komlos' lemma; we recall that convex combinations of convex (concave) functions are again convex (concave)), we may assume without loss of generality that φ_n →φ̅μ-a.e.andψ_n →ψ̅ν-a.e.for some (φ̅,ψ̅)∈ L^c(μ,ν). Moreover, the arguments in <cit.> also show that μ(φ̅) + ν(ψ̅) ≤lim inf_n →∞{μ(φ_n) + ν(ψ_n) }.Now, define the functions φ,ψ:J→ by φ := lim inf_n→∞φ_n and ψ := lim sup_n→∞ψ_n. Then φ is convex, ψ is concave, φ = φ̅ μ-a.e., and ψ = ψ̅ ν-a.e. In particular, (φ,ψ) ∈ L^c(μ,ν) and μ(φ) + ν(ψ) ≤lim inf_n →∞{μ(φ_n) + ν(ψ_n) }. Furthermore, as φ_k + ψ_k ≥ f_k on J, we have for each n thatinf_k ≥ nφ_k + sup_k ≥ nψ_k≥inf_k ≥ n (φ_k + ψ_k) ≥inf_k ≥ n f_konJ.Sending n→∞ gives φ + ψ≥ f. In summary, (φ,ψ) ∈_μ,ν(f). We proceed to show strong duality for bounded upper semicontinuous functions. Let f: → [0,∞] be bounded and upper semicontinuous. Then _μ,ν(f) = _μ,ν(f). The proof is based on a Hahn–Banach separation argument similar to <cit.>. We first show the weak duality inequality. Let μ≤_cθ≤_cν and (φ,ψ)∈_μ,ν(f). In particular, φ+ψ is bounded from below. Then by Lemma <ref> (iii)–(iv), θ(f)≤θ(φ+ψ) = θ(φ) + θ(ψ) ≤θ(φ) + ν(ψ) ≤μ(φ) + ν(ψ),and the inequality _μ,ν(f) ≤_μ,ν(f) follows.The converse inequality is based on a Hahn–Banach argument, so let us introduce a suitable space. By the de la Vallée–Poussin theorem, there is an increasing convex function ζ_ν:_+→_+ of superlinear growth such that x ↦ζ_ν(|x|) is ν-integrable. Now, set ζ(x) = 1+ζ_ν(|x|), x ∈, and denote by C_ζ the space of all continuous functions f:→ such that f/ζ vanishes at infinity. We endow C_ζ with the norm f_ζ := f/ζ_∞. With this notation, the same arguments as in the proof of <cit.> show that the dual space C^*_ζ of continuous linear functionals on C_ζ can be represented by finite signed measures.Fix f ∈ C_ζ. Then-ζ(x) f _ζ ≤ f(x) ≤ζ(x) f _ζ, x ∈ J.Because ζ_ν is convex and x ↦ζ_ν(|x|) is ν-integrable, we have θ(ζ) ≤ν(ζ) < ∞ for all μ≤_cθ≤_cν. This together with (<ref>) shows that _μ,ν(f) is finite. Thus, adding a suitable constant to f, we may assume that _μ,ν(f) = 0. For the following Hahn–Banach argument, we consider the convex coneK:= {g ∈ C_ζ : _μ,ν(g) ≤ 0}.Proposition <ref> implies that K is closed.Suppose for the sake of contradiction that _μ,ν(f) > 0. Then, by the Hahn–Banach theorem, K and f can be strictly separated by a continuous linear functional on C_ζ. That is, there is a finite signed measure ρ such that ρ(f) > 0 and ρ(g) ≤ 0 for all g∈ K. For any compactly supported nonnegative continuous function g∈ C_ζ, we have _μ,ν(-g) ≤ 0. That is, -g ∈ K and hence ρ(-g) ≤ 0. This shows that ρ is a (nonnegative) finite measure. Multiplying ρ by a positive constant if necessary, we may assume that ρ has the same mass as μ and ν. Next, let ψ be convex and of linear growth. Then ψν() - ν(ψ)∈ K and -ψμ() + μ(ψ) ∈ K. Using that ρ≤0 for these two functions yields μ(ψ) ≤ρ(ψ) ≤ν(ψ). We conclude that μ≤_cρ≤_cν. But now ρ(f) > 0 contradicts _μ,ν(f) = 0. Thus, _μ,ν(f) ≤_μ,ν(f).Finally, let f be bounded and upper semicontinuous and choose f_n ∈ C_b() ⊆ C_ζ such that f_n ↘ f. By the above, we have _μ,ν(f_n) = _μ,ν(f_n) for all n. We show below that lim_n→∞_μ,ν(f_n) = _μ,ν(f). Using this and the monotonicity of _μ,ν, we obtain_μ,ν(f)≤lim_n→∞_μ,ν(f_n) = lim_n→∞_μ,ν(f_n) = _μ,ν(f) ≤_μ,ν(f).So strong duality holds for bounded upper semicontinuous functions.It remains to argue that lim_n→∞_μ,ν(f_n) = _μ,ν(f). We show more generally that _μ,ν is continuous along decreasing sequences of bounded upper semicontinuous functions. So let f_n ↘ f be a convergent sequence of bounded upper semicontinuous functions. Fix ε > 0 and set ℓ := lim_n→∞_μ,ν(f_n). Then for each n, ℓ≤_μ,ν(f_n) < ∞ and thus the setA_n:= {μ≤_c θ≤_c ν : θ(f_n) ≥ℓ - ε}is nonempty. Moreover, each A_n is a closed subset of the weakly compact set {θ : μ≤_c θ≤_c ν} and A_n+1⊆ A_n. Therefore, there exists a θ' in the intersection ∩_n ≥ 1 A_n. We then obtain by monotone convergence that_μ,ν(f)≥θ'(f) = lim_n→∞θ'(f_n) ≥ℓ - ε.This implies that _μ,ν(f) ≥ℓ as ε was arbitrary. The converse inequality follows from the monotonicity of _μ,ν. This completes the proof.(i): This is a consequence of Lemma <ref> and a capacitability argument that is almost verbatim to <cit.>. The same arguments can be found in <cit.>. We therefore omit these elaborations.(ii): Applying Proposition <ref> to the constant sequence f_n = f and a minimizing sequence (φ_n,ψ_n) ∈_μ,ν(f) of _μ,ν(f) yields a dual minimizer. We are now in a position to prove the duality between the robust pricing and superhedging problems. By Proposition <ref>, Lemma <ref>, and Proposition <ref>,_μ,ν(f)≤_μ,ν(f,) ≤_μ,ν(f,) ≤_μ,ν(f),and Theorem <ref> shows that _μ,ν(f) = _μ,ν(f). Hence,_μ,ν(f)= _μ,ν(f,) = _μ,ν(f,) = _μ,ν(f).In particular, the quantities in (<ref>) are all independent of the choice of(as long as one of the two conditions of Theorem <ref> holds).If _μ,ν(f,) < ∞, then _μ,ν(f) < ∞ and hence there is an optimizer (φ,ψ) ∈_μ,ν(f) for _μ,ν(f). Then Proposition <ref> provides an H = (h,A) such that (φ,ψ,H) ∈_μ,ν(f,). By (<ref>) and the definition of _μ,ν(f,), (φ,ψ,H) is an optimizer for _μ,ν(f,). Strong duality (without dual attainment) for the robust pricing and superhedging problems continues to hold if we restrict ourselves to trading strategies whose dynamic part is of finite variation.First, observe that the process Ĥ defined by (<ref>) is of finite variation when h is bounded. Recalling the definition (<ref>) of h in Proposition <ref>, we see that h is bounded on {ω : ω_T ∈ J^∘} as these paths are bounded in a compact subset of J, on which ψ' is bounded. This will more generally hold for almost all paths if ψ' is uniformly bounded on J. Therefore, strong duality (and dual attainment in strategies of finite variation) holds if J is open.Second, consider the case J = [a,b) for some -∞ < a < b ≤∞. Suppose that the assumptions of Theorem <ref> hold and that _μ,ν(f,) < ∞, and let (φ,ψ) ∈_μ,ν(f) be a dual auxiliary optimizer. Then ψ(a) < ∞ as ν has an atom at a (cf. Lemma <ref>). If ψ'(a) > -∞, then the same argument as above shows that the dynamic trading strategy constructed in Proposition <ref> is of finite variation. If ψ'(a) = -∞, then we construct a sequence of functionsψ_k(x) := ψ(x)forx ≥ a+1k,ψ(a) + k(x-a) (ψ(a+1k) - ψ(a))forx < a+1k.that approximates ψ by linear interpolation on the interval [a,a+1/k]. We then have ψ_k ↘ψ and μ(φ) + ν(ψ_k) ↘μ(φ) + ν(ψ) = _μ,ν(f) as k→∞. Since ψ'_k(a) > -∞, the associated process H^(k) is of finite variation almost surely. The cases J = (a,b] and J = [a,b] are analogous. §.§ Structure of primal and dual optimizers If a primal optimizer to the auxiliary problem exists, we can derive some necessary properties for the dual optimizer.Let μ≤_c ν be irreducible with domain (I,J) and let f: → [0,∞] be Borel. Suppose that _μ,ν(f) = _μ,ν(f), that μ≤_cθ≤_cν is an optimizer for _μ,ν(f), and that (φ,ψ)∈_μ,ν(f) is an optimizer for _μ,ν(f). Then * φ+ψ = f θ-a.e.,* φ is affine on the connected components of {u_μ < u_θ},* ψ is affine on the connected components of {u_θ < u_ν},* φ does not have a jump at a finite endpoint b of J if θ({b}) > 0, and* ψ does not have a jump at a finite endpoint b of J if θ({b}) < ν({b}). As in the proof of Lemma <ref>, we obtain (cf. (<ref>)) thatθ(f)≤θ(φ+ψ) ≤μ(φ) + ν(ψ).By the absence of a duality gap as well as the optimality of θ and (φ,ψ), all inequalities are equalities:θ(f)= θ(φ+ψ) = μ(φ) + ν(ψ). Now (i) follows from the first equality in (<ref>) and the fact that φ+ψ≥ f on J. Rearranging the second equality, we can write0= {μ(φ) + ν(ψ) } - θ(φ+ψ) = {μ(φ) + ν(ψ) } - {θ(φ) + ν(ψ)} + {θ(φ) + ν(ψ)}- θ(φ+ψ).Using the definition (<ref>) of the first three expressions (using φ as a concave moderator for the first two terms and -ψ for the third; cf. Lemma <ref> (i)), we obtain0= (μ - ν)(φ) - (θ - ν)(φ) + (θ - ν)(-ψ) = (μ - θ)(φ)+(θ-ν)(-ψ),where the last equality is a direct consequence of the definitions of (μ - ν)(φ) and (θ - ν)(φ) (cf. (<ref>)). Both terms on the right-hand side of (<ref>) are nonnegative by definition and hence must vanish:0= (μ - θ)(φ) = ∫_I (u_μ - u_θ) φ” + ∫_J ∖ I |Δφ| θand similarly for (θ-ν)(-ψ). This implies that φ” = 0 on {u_μ < u_θ} (which is assertion (ii)) and that |Δφ| = 0 for every endpoint of J on which θ has an atom (which is assertion (iv)). The proofs of (iii) and (v) are similar. The next result shows that for upper semicontinuous f, there is a maximizer for _μ,ν(f) which is maximal with respect to the convex order. We omit the proof in the interest of brevity.Let μ≤_c ν be irreducible and let f:→ [0,∞] be upper semicontinuous and bounded from above by a convex, continuous, and ν-integrable function. Furthermore, fix a strictly convex function g:→ with linear growth, and consider the “secondary” optimization problemsup_θ∈Θ(f)θ(g),where Θ(f) := {θ : μ≤_c θ≤_c νandθ(f) ≥_μ,ν(f) } is the set of optimizers of the auxiliary primal problem. * Θ(f) is non-empty, convex, and weakly compact and (<ref>) admits an optimizer.* Any optimizer θ of (<ref>) has the following properties: * θ is maximal in Θ(f) with respect to the convex order.* If O is an open interval such that O ⊆{ u_θ < u_ν} and f|_O is convex, then θ(O) = 0.* If K is an interval such that K^∘⊆{ u_μ < u_θ}, f|_K is strictly concave, and θ(K)>0, then θ|_K is concentrated in a single atom. The following example shows that the set optimizers for _μ,ν(f) can have multiple maximal or minimal elements with respect to the convex order; there is in general no greatest or least element for this partially ordered set. Let μ = δ_0 and ν = 1/3(δ_-1 + δ_0 + δ_1) and let f be piecewise linear with f(-1) = f(1) = 3, f(-1/2) = f(1/2) = 2, and f(0) = 0. We claim that there is no greatest or least primal optimizer.We construct candidate primal and dual optimizers as follows. On the primal side, set θ_1 = 23δ_-1/2 + 13δ_1 and θ_2 = 13δ_-1 + 23δ_1/2. On the dual side, set φ≡ 0 and let ψ be the convex function that interpolates linearly between ψ(-1) = ψ(1) = 3 and ψ(0) = 1. Direct computations yield θ_1(f) = θ_2(f) = 7/3 = ν(ψ) which shows that θ_1 and θ_2 are primal optimizers and that (φ,ψ) is a dual optimizer.First, we show that there is no primal optimizer which dominates both θ_1 and θ_2 in convex order. Indeed, one can check that u_ν = max(u_θ_1,u_θ_2), so that ν is the only feasible primal element which dominates both θ_1 and θ_2 in convex order. But ν(f) = 2 < 7/3 and therefore ν is not optimal.Second, we show that there is no primal optimizer which is dominated by both θ_1 and θ_2. Indeed, one can check that { u_μ < min(u_θ_1,u_θ_2)} = (-1/2,1/2), so that every feasible primal element that is dominated by both θ_1 and θ_2 must be concentrated on [-1/2,1/2]. But f ≤ 2 on [-1/2,1/2], so that no primal optimizer can be concentrated on this interval. We conclude this section with an example that shows that primal attainment does not hold in general if f is not upper semicontinuous.Let μ = δ_0, ν = 1/2(δ_-1+δ_1), and set f(x) := |x|_(-1,1)(x). Then μ≤_c ν is irreducible with domain ((-1,1),[-1,1]). Considering the sequence θ_n := 1/2(δ_-1+1/n + δ_1-1/n), one can see that _μ,ν(f) ≥ 1. But there is no μ≤_c θ≤_c ν such that θ(f) ≥ 1 because f < 1 on [-1,1].§ EXAMPLES Two common payoff functions are risk reversals and butterfly spreads. In this section, we provide solutions to the auxiliary primal and dual problems for these payoffs. Throughout this section, we fix irreducible marginals μ≤_c ν and denote their common total mass and first moment by m_0 and m_1, respectively.§.§ Risk reversalsThe payoff function of a risk reversal is of the formf(x)= -(a-x)_+ + (x-b)_+,for fixed a<b. The following result provides a simple geometric construction of the primal and dual optimizers in terms of the potential functions of μ and ν.[The authors thank David Hobson for the idea of this construction.] We recall that any convex function u lying between the potential functions u_μ and u_ν is the potential function of a measure θ which is in convex order between μ and ν (cf., e.g., <cit.>).Consider the line through the point (a,u_μ(a)) of maximal slope lying below (or on) the graph of u_ν; cf. Figure <ref>. This line is either (i) a tangent line to the graph of u_ν with a tangent point (z,u_ν(z)) for some z ∈ (a,∞) or (ii) the asymptote line for the graph of u_ν near +∞.[Note that case (ii) can only happen when (a,μ(a)) lies on the increasing part of the dashed potential function in Figure <ref>. In particular, in this case, μ is concentrated on the left of a.]In case (i), define the concave function φ and the convex function ψ byφ(x)=- α (x-a)_+, ψ(x)= x-a + α(x-(z∨ b))_+,where α = (b-a)/((z∨ b)-a). Moreover, let u be the unique convex function that coincides with u_μ on (-∞,a] and with u_ν on [z,∞) and is affine on [a,z] (i.e., u coincides on [a,z] with the tangent line considered above). Denote by θ the unique measure with potential function u_θ = u. In case (ii), set φ(x) = 0, ψ(x) = x-a, and θ = μ.Then, θ is an optimizer for the auxiliary primal problem _μ,ν(f), (φ,ψ) is an optimizer for the auxiliary dual problem _μ,ν(f), and the common optimal optimal value is given in terms of the potential functions of μ and ν by_μ,ν(f) = _μ,ν(f) = m_1 - a+b/2m_0 + b-a/2u_ν(z∨ b) - u_μ(a)/(z∨ b)-a in case (i),m_1 - a m_0in case (ii). We first note that θ and (φ, ψ) are admissible elements for the auxiliary primal and dual problems, respectively. Indeed, by construction, u_θ is convex and lies between u_μ and u_θ. Thus, the associated measure θ satisfies μ≤_c θ≤_c ν. Moreover, a straightforward computation shows that φ + ψ≥ f, and (φ,ψ) ∈ L^c(μ,ν) by Lemma <ref>.By the weak duality inequality (<ref>) (this also holds if f is bounded from below by an affine function; cf. Remark <ref>), θ(f) ≤μ(φ) + ν(ψ) holds for any admissible primal and dual elements. It thus suffices to show that θ(f) = μ(φ) + ν(ψ) for our particular choices for θ and (φ,ψ). Case (i): Using the identity (t-s)_+ = 1/2(| t-s | + t-s), the integrals θ(f), μ(φ), and ν(ψ) can be expressed in terms of the potential functions of μ,θ, and ν as follows:θ(f)= 1/2(u_θ(b) - u_θ(a)) + m_1 - a+b/2m_0, μ(φ)= -α/2(u_μ(a) + m_1 - am_0), ν(ψ)= m_1 - am_0 + α/2(u_ν(z ∨ b) + m_1 - (z∨ b)m_0).Substituting α = (b-a)/((z∨ b)-a) and simplifying givesμ(φ) + ν(ψ)= m_1 + 1/2((a - (z∨ b)) α - 2a) m_0 - α/2(u_μ(a) - u_ν(z∨ b))= m_1 - a+b/2m_0 + b-a/2u_ν(z∨ b) - u_μ(a)/(z∨ b)-a.Hence,θ(f) - (μ(φ) + ν(ψ))= b-a/2(u_θ(b) - u_θ(a)/b-a - u_ν(z∨ b) - u_μ(a)/(z∨ b)-a),and it suffices to show that the two quotients inside the brackets are equal. To this end, we distinguish two cases. On the one hand, if z ≤ b, it is enough to observe that u_θ(a) = u_μ(a) and u_θ(b) = u_ν(b) by the construction of u = u_θ. On the other hand, if z ≥ b, then the two quotients are the same because u = u_θ is affine on [a, z] ⊃ [a,b] and coincides with u_μ at a and with u_ν at z.Case (ii): On the one hand, since θ = μ and μ is concentrated on the left of a, we have θ(f) = μ(f) = m_1-am_0. On the other hand, μ(φ) + ν(ψ) = ∫ (x-a)ν( x) = m_1 - am_0.§.§ Butterfly spreadsThe payoff function of a butterfly spread is of the formf(x)= (x-(a-h))_+ - 2(x-a)_+ + (x-(a+h))_+,for fixed a and h > 0. We have the following analog to Proposition <ref>; we omit the proof.Consider the two lines l_+,l_- through the point (a,u_μ(a)) of maximal and minimal slope, respectively, lying below (or on) the graph of u_ν. We distinguish the cases (i+) l_+ is a tangent line with tangent point (z_+,u_ν(z_+)), (ii+) l_+ is an asymptote, (i-) l_- is a tangent line with tangent point (z_-,u_ν(z_-)), (ii-) l_- isan asymptote. In case (ii±), we set z_± = ±∞.Let u be the convex function that coincides with u_ν on (-∞,z_-] ∪ [z_+,∞) and is affine on [z_-,a] and on [a,z_+], and define the concave function φ and the convex function ψ byφ(x)= -(α+β)(x-a)_+, ψ(x)= α(x-(z_- ∧ (a-h)))_+ + β(x-(z_+ ∨ (a+h)))_+,where α = h/a-(z_- ∧ (a-h)) and β = h/(z_+ ∨ (a+h)) - a. Here, in the asymptote cases (ii±), φ,ψ need to be interpreted as the limiting functions that arise as z_±→±∞.[For instance, if z_- = -∞ and z_+ < ∞, then φ(x) = -β(x-a)_+ and ψ(x) = h + β(x-(z_+ ∨ (a+h)))_+.]Then, the intermediate law θ with potential function u_θ = u is an optimizer for the auxiliary primal problem _μ,ν(f), (φ,ψ) is an optimizer for the auxiliary dual problem _μ,ν(f), and the common optimal optimal value is given in terms of the potential functions of μ and ν by_μ,ν(f) = _μ,ν(f) = h2 (s_+ + s_-),wheres_+= u_ν(z_+ ∨ (a+h)) - u_μ(a)/(z_+ ∨ (a+h)) - a in case (i+),m_0in case (ii+), s_-= u_ν(z_- ∧ (a-h))-u_μ(a)/a-(z_- ∧ (a-h)) in case (i-),-m_0in case (ii-).§ COUNTEREXAMPLES In this section, we give four counterexamples. Example <ref> shows that strong duality for the auxiliary problems may fail for general (not necessarily irreducible) marginals if the dual elements φ,ψ are required to be globally concave and convex, respectively. Example <ref> shows that strong duality may fail if the dual elements φ and ψ are required to be μ- and ν-integrable, respectively. Example <ref> shows that the robust model-based prices of Asian- and American-style derivatives are typically not equivalent when more than two marginals are given. Example <ref> shows that the equality _μ,ν(f,)= _μ,ν(f) may fail when the assumptions of Proposition <ref> are violated.[Duality gap with globally convex/concave dual elements]Let μ = 1/2δ_-1 + 1/2δ_1, let ν be the uniform distribution on (-2,2), and set f(x) := | x |^-1/2, x ∈ (with f(0) = ∞).First, we show that _μ,ν(f) is finite. Fix any μ≤_cθ≤_cν. Computing the potential functions u_μ and u_ν shows that μ≤_c ν and that { u_μ < u_ν} = I_1 ∪ I_2 with I_1 = (-2,0) and I_2 = (0,2). Because ν does not have an atom at the common boundary 0 of I_1 and I_2, also θ cannot have an atom at 0. Thus, we can write θ = θ_1 + θ_2 with1/2δ_-1≤_c θ_1 ≤_c ν|_I_1and1/2δ_1≤_c θ_2 ≤_c ν|_I_2.Since f is convex when restricted to I_1 or I_2, we haveθ(f) = θ_1(f) + θ_2(f) ≤ν|_I_1(f) + ν|_I_2(f) = ν(f)< ∞.It follows _μ,ν(f) = ν(f) < ∞.Second, let φ be concave and ψ be convex such that φ + ψ≥ f. We show that then necessarily μ(φ) + ν(ψ) = ∞. To this end, we may assume that φ < ∞ on (μ) = { -1, 1}. Then φ < ∞ everywhere by concavity. Thus, evaluating φ + ψ≥ f at 0 implies that ψ(0) = ∞. Therefore, ψ = ∞ on (-∞,0] or on [0,∞) by the convexity of ψ. In both cases, we have μ(φ) + ν(ψ) = ∞. [Duality gap with individually integrable dual elements]We consider the marginalsμ := C ∑_n≥ 1 n^-3μ_n andν := C∑_n≥ 1 n^-3ν_n,where C := (∑_n≥ 1 n^-3)^-1, μ_n := δ_n and ν_n := 1/3(δ_n-1+δ_n + δ_n+1)for n≥ 1. These are the same marginals as in <cit.> where it is shown that μ≤_cν is irreducible with domain ((0,∞),[0,∞)). We now let f:_+→[0,1] be the piecewise linear function through the points given by f(n) = 0 and f(2n + 12) = 14 for n ≥ 0; cf. Figure <ref>.We proceed to construct candidates for optimizers for _μ,ν(f) and _μ,ν(f). For the primal problem, define the sequence (θ̅_n)_n ≥ 1 byθ̅_n= 1/3(δ_n-1 + 2δ_n+1/2)forneven,1/3(2δ_n-1/2 + δ_n+1)fornodd,and set θ̅:= C∑_n≥ 1 n^-3θ̅_n. One can check that μ_n ≤_c θ̅_n ≤_c ν_n and compute θ̅_n(f) = 1/6. Hence, μ≤_c θ̅≤_c ν (by linearity of potential functions in the measure) and θ̅(f) = 1/6.We now turn to the dual problem. Let φ̅ and ψ̅ be the unique concave and convex functions, respectively, with second derivative measures-φ̅” = ∑_n ≥ 0δ_2n+1/2andψ̅” = 1/2∑_n ≥ 1δ_nand φ̅(0) = ψ̅(0) = 0, φ̅'(0) = f'(0) = 1/2, and ψ̅'(0) = 0. The “initial conditions” are chosen such that f(0) = φ̅(0) + ψ̅(0) and f'(0) = φ̅'(0) + ψ̅'(0) and the choice of the second derivative measures ensures that φ and ψ pick up the negative and positive curvature of f, respectively. Thus, φ̅+ ψ̅= f on _+ by construction. We proceed to compute μ(φ̅) + ν(ψ̅) in the sense of Definition <ref>. (The individual integrals are infinite because φ̅ and ψ̅ have quadratic growth while μ and ν have no second moments.) To this end, we note that φ̅+ ψ̅= f vanishes on the support of ν. This implies that φ̅ is a concave moderator for (φ̅,ψ̅) with respect to μ≤_cν. We can then computeμ(φ̅) + ν(ψ̅)= μ(φ̅-φ̅) + ν(ψ̅+ φ̅) +(μ-ν)(φ̅) = (μ-ν)(φ̅) = C ∑_n ≥ 1 n^-3 (μ_n - ν_n)(φ̅).Fix n ≥ 1. Because φ̅ is continuous, we have(μ_n - ν_n)(φ̅) = 1/2∫_I (u_μ_n - u_ν_n) φ̅”.The difference u_μ_n - u_ν_n vanishes outside (n-1,n+1) and on this interval, φ̅” is concentrated on either n-1/2 (if n is odd) or on n+1/2 (if n is even) with mass 1. Therefore, the right-hand side of (<ref>) collapses to 1/2(u_μ_n - u_ν_n)(n±1/2) = 1/6. It follows that μ(φ̅) + ν(ψ̅) = 1/6 = θ̅(f). Hence, by (weak) duality, θ̅ and (φ̅,ψ̅) are primal and dual optimizers, respectively.We are now in a position to argue that no dual optimizer lies in L^1(μ) × L^1(ν). Suppose for the sake of contradiction that (φ,ψ) ∈ L^1(μ)× L^1(ν) is a dual optimizer and note that (θ̅) = { 0.5, 1, 2, 2.5, 3,…}. We have φ + ψ = f θ̅-a.e. by Proposition <ref> (i). One can show that the following modifications of (φ,ψ) do not affect its optimality nor the individual integrability of φ and ψ; we omit the tedious details. First, ψ is replaced by its piecewise linear interpolation at the atoms of ν. Second, φ is replaced by its piecewise linear interpolation at the kinks of f. Third, a suitable convex function is added to φ and subtracted from ψ (preserving their concavity and convexity, respectively) such that the second derivative measures -φ” and ψ” become singular.Because φ + ψ = f on (θ̅) and both sides are piecewise linear, we conclude that φ + ψ = f holds on [1/2,∞). As -φ” and ψ” are singular, φ and ψ must then account for the negative and positive curvature of f, respectively. It follows that both φ and ψ have quadratic growth. Since μ and ν do not have a second moment, we conclude that μ(φ) = -∞ and ν(ψ) = ∞, a contradiction. [Different robust model-based prices for Asian- and American-style derivatives for multiple marginals]For n ≥ 2 given marginals μ_0≤_c μ_1 ≤_c ⋯≤_cμ_n corresponding to the time points 0,1,…,n (say), the robust model-based price _μ_0,…,μ_n(f,) is defined analogously. But this robust model-based price now depends non-trivially on , as the following example shows. Fix a strictly convex function f. On the one hand, ifcorresponds to American-style derivatives, then one can check that _μ_0,…,μ_n(f,) = μ_n(f). On the other hand, for Asian-style derivatives, i.e., ' = {t ↦ t/n}, Jensen's inequality yieldsf(1/n∫_0^n X_tt) ≤1/n∑_i=0^n-1 f(∫_i^i+1 X_tt ),so that_μ_0,…,μ_n(f,') ≤1/n∑_i=1^n μ_i(f) ≤μ_n(f).For a generic choice of marginals, both inequalities are strict. Hence, the robust model-based price of an Asian-style derivative with a strictly convex payoff function is typically smaller than that of the corresponding American-style derivative.[Necessity of the assumptions of Proposition <ref>] * We show that _μ,ν(f,) = _μ,ν(f) may fail ifdoes not contain an interior averaging process. Set = { A } = {t ↦12 + 12_{ t = T }}, so that ∫_0^T X_t dA_t = (X_0 + X_T)/2, and consider f(x) = x^2. Then, using the martingale property of X under any P ∈(μ,ν), one can check that _μ,ν(f,) = (3μ(f) + ν(f))/4, whereas _μ,ν(f) = ν(f) since f is convex. Now, choose μ and ν such that μ(f) < ν(f) (f is strictly convex). Then, _μ,ν(f,) < _μ,ν(f). * We show that _μ,ν(f,) = _μ,ν(f)may fail ifcontains an interior averaging process but f is not lower semicontinuous. Set = {t ↦ t/T} and f(x) = _{| x |≥ 1 }, and choose μ = δ_0 and ν = (δ_1 + δ_-1)/2. On the one hand, since ν(f) = 1 and f ≤ 1, we have _μ,ν(f) = 1. On the other hand, we claim that _μ,ν(f,) = 0. To this end, fix P ∈(μ,ν). Since P-a.e. path of X starts in 0, evolves in [-1,1], and is right-continuous, |1/T∫_0^T X_tt | < 1 P-a.s. Thus, E^P[f(1/T∫_0^T X_tt)] = 0. Since P ∈(μ,ν) was arbitrary, _μ,ν(f,) = 0.amsplain | http://arxiv.org/abs/1707.08545v2 | {
"authors": [
"Sebastian Herrmann",
"Florian Stebegg"
],
"categories": [
"math.PR",
"math.OC",
"q-fin.MF",
"60G44, 49N05, 91G20"
],
"primary_category": "math.PR",
"published": "20170726171319",
"title": "Robust Pricing and Hedging around the Globe"
} |
Phase-matching-free parametric oscillators based on two dimensional semiconductors C. Conti^4,5 December 30, 2023 ================================================================================== empty § INTRODUCTION Magnetocaloric refrigeration has been proposed as a promising tool for more energy efficient cooling.<cit.> One of the materials under investigation is the Heusler alloy Ni-Mn-Ga-Co. This compound exhibits an inverse magnetocaloric effect around room temperature due to a first order phase transition from a ferromagnetic austenite to a weak magnetic martensite with an entropy change of up to 17 J/(kgK) in a magnetic field of 5 T.<cit.> However, one drawback of a first order transition is the narrow usable temperature range as the transition temperature of the material has to span the entire working region of the cooling device. The transition temperature of the Ni-Mn-based Heusler alloys highly depends on multiple parameters like the composition<cit.>, chemical order<cit.> and stress.<cit.> However, in many cases it is desired to control the transformation temperature of the final sample when the composition and chemical order cannot be changed anymore. Due to the structural change during the transition from the high to the low temperature phase, it is of high interest how this transition and the magnetic properties of the material can be influenced with mechanical stress.<cit.> Thin films can be easily strained when grown on a piezoelectric single crystal and are used as a model system to study the influence of mechanical stress on the functional properties. Significant attention has recently been given to this strain engineering approach, investigating various materials like ferromagnets,<cit.> semiconductors,<cit.> superconductors<cit.> and quantum dots<cit.> on Pb(Mg_1/3Nb_2/3)O_3-PbTiO_3 (PMN-PT) substrates and magnetocaloric materials like La_0.7Ca_0.3MnO_3<cit.> and FeRh<cit.> on BaTiO_3 substrates. Additionally, Ni_44Co_5.2Mn_36.7In_14.1 ribbons adhered to PMN-PT substrates have been studied.<cit.> The results obtained so far on these magnetocaloric materials are promising as the hysteresis accompanying the phase transition was reduced<cit.> or the transition temperature shifted.<cit.> Here, we will address the challenge of the relatively large hysteresis of a first order transformation, which often inhibits the reversibility of the phase transition in the low magnetic fields available in practical applications. For this, we will first analyse how the strain is transferred from the substrate to the film and estimate the limits of this approach. Then, we will experimentally probe the suitability to shift the transformation temperature in case of a large hysteresis. Though our analysis will show that and why this approach is not feasible, our experiments indicate for a novel mechanism not requiring a shift of transformation temperature, but utilizing the high sensitivity of magnetic properties on the interatomic distances in the metamagnetic Heusler alloys.§ RESULTS AND DISCUSSION§.§ Structural investigation First, we analysed the strain transfer from the ferroelectric substrate to the magnetocaloric film. For this, the in-plane and out-of-plane lattice parameters of the PMN-PT and the Ni-Mn-Ga-Co in the austenitic phase were determined with X-ray diffraction measurements (Philips X'Pert, Cu-K_ radiation). Therefore, reciprocal space maps (RSM) of the PMN-PT (013) and (013) as well as Ni-Mn-Ga-Co (026) and (026) reflections were measured for increasing applied electric fields from 0 kV/cm up to 12 kV/cm, depending on the temperature. The RSMs were recorded at 300 K, 320 K and 355 K. For the latter two temperatures, the Ni-Mn-Ga-Co was in the austenitic phase. The peaks were fitted with a 2D pseudo-Voigt-like function and the lattice parameters were calculated from the peak positions. An exemplary RSM and the exact fitting function is included in section 1 of the Supplementary online. Fig. <ref> summarises the change of the lattice parameters for the substrate and the thin film in dependence of the applied field for three different temperatures.Below 362 K, the PMN-PT substrate is in a monoclinic phase,<cit.> which can be approximated as pseudo-cubic. The same approximation is also possible for the austenitic Ni-Mn-Ga-Co film, exhibiting a small tetragonal distortion between the in-plane and out-of-plane lattice parameters at zero electric field (fig. <ref> (b,c)). The difference between the in-plane lattice parameters of substrate and thin film causes a growth with a 45^∘ rotation of their unit cells with respect to each other, which was already shown earlier by pole figure measurements.<cit.> An increasing electric field leads to a compression of the lattice in-plane (a-axis) and an expansion in the out-of-plane direction (c-axis). This opposing trend is expected for the conservation of volume and in accordance with previous investigations, where PMN-PT substrates were used for different thin film systems.<cit.> At 300 K (fig. <ref>(a)), there is the highest influence of the electric field on the PMN-PT lattice parameters with a linear in-plane compression of Δ a_PMN-PT = (a_PMN-PT( 12 kV/cm)-a_PMN-PT(0 kV/cm))/ a_PMN-PT(0 kV/cm) = -0.46 % and a linear out-of-plane elongation of Δ c_PMN-PT = +0.18 % for electric field changes of 12 kV/cm. However, at this temperature the Ni-Mn-Ga-Co is in the martensitic state. Therefore, we cannot give lattice parameters of Ni-Mn-Ga-Co in a pseudo-cubic approximation. In the non-modulated martensite, the spontaneous strain (c/a-ratio) typically is about 22 %,<cit.> which by far exceeds the maximum strain achieved by an electric field. At 320 K, the electric field induced in-plane strain is much smaller than at 300 K, but completely transferred from the substrate to the thin film with Δ a_PMN-PT = Δ a_Ni-Mn-Ga-Co = -0.06 % (Δ E = 10 kV/cm), and the Ni-Mn-Ga-Co is in the austenitic state. This direct coupling within the film plane is expected for a thin film on a thick substrate. As both in-plane directions are assumed to be equivalent, there is a biaxial compressive stress within the magnetocaloric film. For the perpendicular direction, the change of lattice constants is very similar for Ni-Mn-Ga-Co and PMN-PT with Δ c_Ni-Mn-Ga-Co = +0.12 % and Δ c_PMN-PT = +0.14 %. As in this direction the film is free to change its thickness, we estimated the Poisson's ratio from Δ a_Ni-Mn-Ga-Co and Δ c_Ni-Mn-Ga-Co<cit.> to beν = Δ c_Ni-Mn-Ga-Co/Δ c_Ni-Mn-Ga-Co - 2 Δ a_Ni-Mn-Ga-Co = 0.5.This is higher than the literature values of Ni_2MnGa in the austenitic phase ν = 1/3,<cit.> but a deviation between a bulk sample and a thin film on a substrate can be expected.To estimate the expected shift of transformation temperature caused by the strain, we compare the biaxial compressive stress in these films with uniaxial tensile stress, as this results in the same deformation of the unit cells. The stress σ in the sample, for the maximum strain ϵ = -0.06 %, can be calculated from the X-ray diffraction data using the relation<cit.>σ=ϵ· E/(1+ν)sin^2 (Δω)-2ν=7.06 MPawith the Poisson's ratio ν = 0.5 and the offset-angle Δω=-18.465^∘ of the (026)-reflection. For this estimation, we used E=10 GPa measured on Ni-Mn-Ga single crystals<cit.> as no values of Co alloyed samples are available. When using the Clausius-Clapeyron equation for the application of mechanical tensile stress along the Ni-Mn-Ga [001] direction<cit.>dσ/dT= 6.0 MPa/K ,we obtain a maximum change of transition temperature Δ T = +1.2 K.At 355 K, we observe non-linearities in the strain/electric field curves, untypical for this low field regime (compare, e. g. ref<cit.>). This can be attributed to the vicinity to the phase transition in the PMN-PT from a monoclinic to a tetragonal phase in this electric field-temperature region.<cit.> We observed that c > a for PMN-PT, which is a consequence of polarization. For the Ni-Mn-Ga-Co film, however, within the measurement accuracy both lattice parameters are equal at very low electric fields. This indicates an almost stress free state at 0 kV/cm. Presumably, poling of the substrate compensated the thermal and epitaxial strain commonly occurring also in epitaxial Heusler films.<cit.> When comparing Δ a_PMN-PT for the three different temperatures in fig. <ref>, the highest influence of the electric field is at 300 K (Δ a_PMN-PT = -0.42 %). Due to the higher temperature and the reduced maximum electric field, at 320 K Δ a_PMN-PT = -0.06 % is much smaller. Due to the phase transition in the substrate, the Δ a_PMN-PT = -0.15 % obtained at 355 K (for E=8 kV/cm) is larger than at 320 K. In our experiments, at relatively high temperatures, we were limited to 8 kV/cm as we observed irreversible shortcuts within the substrate in identically prepared samples when going to higher fields. We attribute this limit to the growth conditions required to form a metallic epitaxial film. Heating the PMN-PT-substrate under UHV conditions presumably results in some evaporation of Pb and loss of oxygen, which degrades the ferroelectric properties. §.§ Magnetic investigationFor the magnetic characterization, a SQUID setup (Quantum Design MPMS) with a custom built sample holder that provided electrical contacts was used. Fig. <ref> shows M(T) measurements at a magnetic field of _0 H = 0.1 T applied in-plane withand a temperature sweep rate of 3 K/min. The starting point of every M(T) loop was at high temperatures to ensure a fully austenitic state in the beginning. The M(T)-measurements at 0 and 2 kV/cm started at 385 K. For 4 kV/cm, the starting temperature was reduced to 380 K due to an increased risk of an electric breakthrough in the substrate at this higher electric field. For all potential values, we observe the common magnetisation changes of a metamagnetic Heusler alloy.<cit.> During cooling magnetisation first increases as expected below the Curie temperature of the austenitic phase. Below 340 K, magnetisation decreases due to the transition to a weak ferromagnetic martensite. Below 300 K, magnetisation increases again as some ferrimagnetic order within the martensite arises. During heating, the same trends are observed, but the hysteresis of the first order martensitic transition shifts the increase of magnetisation to substantially higher temperatures. In the austenite region, measurements under applied electric field show the appearance of steps in the magnetisation in the cooling as well as heating branch (see enlargement, fig. <ref>(b)). In the martensite region, there is no influence of the electric field at all. We do not observe a clear shift of the transformation temperature but some scattering of about ± 1 K, which we take as resolution limit of this setup. This, however, is higher than the maximum expected shift of +0.5 K from the Clausius-Clapeyron equation (<ref>) when considering that the maximum electric field in the M(T)-measurements was 4 kV/cm. It is interesting to put this Δ T in relation to the observed hysteresis of 24 K. Thus, the usable potential range is not large enough to overcome hysteresis in these epitaxial films. To understand the unexpected change of magnetisation in temperature loops under electric fields especially within the austenite (fig. <ref>(b)), M(E) measurements were performed at different fixed temperatures in the austenite, martensite and phase transition region. Fig. <ref> shows the resulting changes of the magnetisation for 0 ≤ E ≤ 8 kV/cm in the austenitic phase at 368 K. Prior to this measurement, the sample was cooled to 250 K to ensure being on the heating branch of the M(T)-loop seen in fig. <ref>.After a virgin effect during the first electric field sweep (shown in red in fig. <ref>), where M first increases and then decreases, a completely reversible change of the magnetisation with Δ M ≈ 7 % is observed. We attribute the virgin effect to the fact that these experiments are performed in a state, where still some martensite exists in the predominantly austenitic sample. This metastable state is evident from the still open hysteresis at 368 K in fig. <ref>(b). The reversible behaviour for all following cycles must have a different origin, as the maximum potential applicable is not sufficient to shift the transformation temperatures. As an alternative mechanism, we propose to consider the elastic straining of the lattice constants of the austenite (as observed in fig. <ref>). Following the analysis of the XRD measurements, an electric field of 8 kV/cm changes the tetragonality of the Ni-Mn-Ga-Co unit cell by around 0.24 %. Though this is low compared to the spontaneous strain of about 22 % occurring during the martensitic transition from austenite to non-modulated martensite, the same underlying mechanics may be relevant. For a complete metamagnetic transition, this spontaneous strain changes magnetisation by 100 %, which is commonly attributed to the Mn atoms being at an interatomic distance close to the crossover from ferromagnetic to antiferromagnetic coupling.<cit.> We propose that the high sensitivity of magnetism on the interatomic distances is also the underlying mechanism for the observed reversible change of magnetisation by 7 % at a biaxial in-plane compression of about 0.1 %. The strain-induced change of the magnetic coupling might also influence the Curie-temperature, but the limitation of the SQUID-device to T ≤ 395 K and the high risk of an electric breakthrough at such high temperatures hinders the direct measurement of T_C. Further density functional theory calculations, which are beyond the scope of this experimental paper, are required to confirm, if these slight changes of lattice parameters are sufficient to explain the substantial change of magnetisation.Similar M(E) measurements were also performed at temperatures corresponding to different states of the sample. Fig. <ref>(a) shows the M(E) measurements in the martensitic phase at 280 K on the cooling branch and 300 K on the heating branch. The electric field dependence of the magnetisation in the phase transition region for the transformation from austenite to martensite and vice versa at 320 K and 345 K, respectively, is shown in fig. <ref>(b). The maximum electric field was increased to 10 kV/cm as the breakthrough voltage increases with reduced temperature. Again, the first cycle is shown in red.In the martensitic state, no reversible change of M was observed and the Δ M almost vanished to a value below 0.1 % (see fig. <ref>(a)). We attribute this to the ability of the martensite to compensate the mechanical stress by a reorientation of the martensitic variants due to the highly mobile twin boundaries.<cit.> Therefore, the magnetisation in the low temperature phase is not changed. Within the transition region, a slight influence of the mechanical stress on the magnetisation is observed (fig. <ref>(b)). In this case, an initial change of the magnetisation in the order of 1 % was observed in the cooling (320 K) as well as the heating (345 K) branch of the magnetisation loop, when the electric field was applied for the first time. In all following cycles, no reversible change was observed. Noticeable is the different sign of this initial magnetisation change in both measurements. During the first cycle, the magnetisation increases in the heating branch while it decreases in the cooling branch. In other words, the mechanical stress narrows the hysteresis gap between the cooling and heating branch. This indicates that mechanical stress in this metastable coexistence region results in variant reorientation, which follows isothermal kinetics<cit.> towards the equilibrium state being in between both branches. All these measurements have been performed in a low magnetic field of 0.1 T, which may not be sufficient to saturate the sample completely in case of a high anisotropy. To exclude the role of magnetostriction, we performed additional experiments in a sufficiently high magnetic field of 2 T (see Supplementary Fig. S2 and S3 online) resulting in the same behaviour.§ SUMMERY To conclude, we investigated the influence of strain created by a ferroelectric substrate on the martensitic phase transition and magnetic properties of a Ni-Mn-Ga-Co thin film. It was demonstrated that the in-plane and out-of-plane lattice parameters of the substrate and austenitic thin film can indeed be tuned by the application of an electric field to the multiferroic stack. Although a relatively large reversible change of the magnetisation in the austenite phase was observed, the transformation temperature could not be varied. This is in contradiction to reports on Ni-Mn-In-Co ribbons adhered to PMN-PT, where a shift of the transition temperature for different applied electric fields was observed.<cit.> We attribute this to differences in the microstructure. In polycrystalline Ni-Mn-In-Co ribbons, the compensation of mechanical stress by twin boundary movement in the martensite is significantly limited due to grain boundaries.<cit.> In contrast, our single-crystalline films allow for an easy compensation of stress since twin boundary movement is possible. We attribute the reversible change of magnetisation, only observed within the austenitic state, to a different mechanism. We suggest that it originates from the change of interatomic distances, which have a strong influence on the ferromagnetic-antiferromagnetic coupling between Mn atoms. The advantage of this mechanism is its reversibility, avoiding the large hysteresis of the first order phase transition. Though the change of the magnetisation is significantly lower compared to the one of the martensitic transition, utilizing the intrinsic magnetic instability of metamagnetic Heusler alloys may become an interesting approach for a reversible tuning of magnetocaloric properties.§ METHODS As described previously,<cit.> epitaxial Ni-Mn-Ga-Co thin films were grown by DC magnetron sputter deposition on 0.3 mm thick single-crystalline PMN-PT substrates provided by Morgan Electroceramics. The Ni_43Mn_32Ga_20Co_5 films are 350 nm thick and a 20 nm Cr buffer layer was used to enhance epitaxial growth. The epitaxial relationship obtained by XRD-measurements is PMN-PT(001)[110]∥Cr(001)[100]∥Ni-Mn-Ga-Co(001)[100]. For the measurements with an applied electric voltage U, a Au/NiCr layer on the unpolished side of the substrate was used as bottom electrode whereas the conducting thin film was used as top electrode. More details on the sample preparation as well as first structural and magnetic characterizations can be found elsewhere.<cit.> For the stress-dependent measurements, a Keithley sourcemeter was used to apply an electric voltage to the multiferroic stack. In order to protect the samples from voltage breakthrough, the current was limited to 105 A and the maximum voltage applied was 300 V. For the magnetisation measurements in dependence of temperature T and voltage U, the temperature sweep rate was 3 K/min and the voltage step size used was 25 V. Since the electric field is the more physical and material(-thickness) independent parameter, the applied voltage was converted to the electric field E and for better readability rounded to full digits. Before every measurement series with applied electric field, the substrate was polarized along the [001] direction to eliminate the influence of the hysteretic behaviour of the PMN-PT substrates.<cit.> As we experienced cracking of one sample during demounting, we had to use two samples for structural analysis and another one for the magnetic characterization. All samples have been prepared under identical conditions and have the same composition within the measurement accuracy of about 1 at.% but differ about 40 K in the transition temperature. However, it is well known for Ni-Mn-based alloys that the transition temperature can drastically change even with small differences in composition<cit.> whereas all other properties are nonetheless practically equal.Data availability: The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.§ ACKNOWLEDGEMENTS Funding by DFG through SPP 1599 www.FerroicCooling.de grant no FA 453/11 and HU 1726/3 is gratefully acknowledged.§ AUTHOR CONTRIBUTIONS STATEMENT B.S., R.N. and S.F. conceived the experiments. B.S. conducted all experiments and analysed the SQUID measurements. R.N., S.S. and R.H. analysed the RSM measurements. B.S., S.F. and R.H. interpreted the results. L.S. and K.N. supervised the work of B.S. and S.S. 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"authors": [
"Benjamin Schleicher",
"Robert Niemann",
"Stefan Schwabe",
"Ruben Hühne",
"Ludwig Schultz",
"Kornelius Nielsch",
"Sebastian Fähler"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170727124520",
"title": "Reversible tuning of magnetocaloric Ni-Mn-Ga-Co films on ferroelectric PMN-PT substrates"
} |
Thomas Guyet et al.AGROCAMPUS-OUEST/IRISA-UMR6074 CHRU Brest/EA-7449 REPERES Rennes University 1/IRISA-UMR6074 Declarative sequential patternmining of care pathwaysThis research is supported by the PEPS project funded by the french agency for health products safety (ANSM) and the SePaDec project funded by Brittany Region. Thomas Guyet1 André Happe2 Yann Dauxais3 December 30, 2023 ====================================================================================================================================================================================================================== Sequential pattern mining algorithms are widely used to explore care pathways database, but they generate a deluge of patterns, mostly redundant or useless. Clinicians need tools to express complex mining queries in order to generate less but more significant patterns. These algorithms are not versatile enough to answer complex clinician queries.This article proposes to apply a declarative pattern mining approach based on Answer Set Programming paradigm. It is exemplified by a pharmaco-epidemiological study investigating the possible association between hospitalization for seizure and antiepileptic drug switch from a french medico-administrative database.§ INTRODUCTIONPharmaco-epidemiology applies the methodologies developed in general epidemiologyto answer questions about the uses of health products in the population in real condition.In pharmaco-epidemiology studies, people who share common characteristics are recruited. Then, a dataset is built from meaningful data (drug exposures, events or outcomes) collected within a defined period of time. Finally, a statistical analysis highlights the links (or the lack of link) between drug exposures and outcomes (adverse effects).The data collection of such prospective cohort studies is slow and cumbersome. Medico-administrative databases are readily available and cover a large population. They record, with some level of details, all reimbursed drug deliveries and all medical procedures, for insured people. Such database gives an abstract view on longitudinal care pathways. It has become a credible alternative for pharmaco-epidemiological studies <cit.>. However, it has been conceived for administrative purposes and their use in epidemiology is complex. Our objective is to propose a versatile pattern mining approach that extracts sequential patterns from care pathways. The flexibility of such new knowledge discovery tools has to enable epidemiologists to easily investigate various types of interesting patterns, frequent, rare, closed or emerging patterns, and possibly new ones.On the other hand, the definition of interesting patterns has to exploit in-depth the semantic richness of care pathways due to complex care event descriptions (units number, strength per unit, drugs and diagnosis taxonomies, etc.). By this mean, we expect to extract less but more significant patterns.This article presents the application of a declarative pattern mining framework based on Answer Set Programming (ASP) <cit.> to achieve care pathway analysis answering pharmaco-epidemiological questions.Answer Set Programming (ASP) is a declarative programming paradigm. It gives a description, in a first-order logic syntax, of what is a problem instead of specifying how to solve it. Semantically, an ASP program induces a collection of so-called answer sets. For short, a model assigns a truth value to each propositional atoms of the program. An answer set is a minimal set of true propositional atoms that satisfies all the program rules. ASP problem solving is ensured by efficient solvers. For its computational efficiency, we use <cit.> as a primary tool for designing our encodings. An ASP program is a set of rules of the form:a_0a_1,…,a_m, !not!a_m+1,…,!not!a_n,where each 𝚊_𝚒 is a propositional atom for 0≤𝚒≤𝚗 and !not! stands for default negation. In the body of the rule, commas denote conjunctions between atoms. If 𝚗=0, the rule body is empty, the rule is called a fact and the symbol “” may be omitted. Such a rule states that the atom a_0 has to be true. If 𝚊_0 is omitted, the rule head is empty, the rule represents an integrity constraint meaning that it must not be true.also includes several extensions to facilitate the practical use of ASP (variables, conditional literals and cardinality constraints). Recent researches has been focused on the use of declarative paradigms, including ASP, to mine structured datasets, and more especially sequences <cit.>. The principle of declarative pattern mining is closely related to the Inductive Logic Programming (ILP) <cit.> approach. The principle is to use a declarative language to model the analysis task: supervised learning for ILP and pattern mining for our framework. The encoding benefits from the versatility of declarative approaches and offersnatural abilities to represent and reason about knowledge.§ CONTEXT, DATA AND PHARMACO-EPIDEMIOLOGICAL QUESTIONIn this work, we exemplify our declarative pattern mining framework by investigating the possible association between hospitalization for seizure and antiepileptic drug switches, changes between drugs. The first step was to create a digital cohort of 8,379 patients with a stable treatment for epilepsy (stability criterion detailed in <cit.> have been used). This cohort has been built from the medico-administrative database, called SNIIRAM <cit.> which is the databaseof the french health insurance system. It is made of all outpatient reimbursed health expenditures. Our dataset represents 1,8M deliveries of 7,693 different drugs and 20,686 seizure-related hospitalizations.This dataset and background knowledge (ATC drugs taxonomy, ICD-10 taxonomy) are encoded as ASP facts. For each patient p⃗, drug deliveries are encoded with !deliv(p⃗,t,d,q)! atoms meaning that patient p⃗ got q deliveries of drug d at date t. Dates are day numbers starting from the first event date. We use french CIP, Presentation Identifying Code, as drug identifier. The knowledge base links the CIP to the ATC and other related informations (speciality group, strength per unit, number of units or volume, generic/brand-named status, etc). Each diagnosis related to an hospital stay is encoded with !disease(p⃗,t,d)! meaning that patient p⃗ have been diagnosed with d at date t. Data, 𝒟, and related knowledge base, 𝒦, represent a total of 2,010,556 facts. § SEQUENTIAL PATTERN MINING WITH ASPLet ℐ={i_1,i_2,…,i_|ℐ|} be a set of items. A temporal sequence s⃗, denoted by ⟨(s_j, t_j) ⟩_j ∈ [m] is an ordered list of items s_j ∈ℐ timestamped with t_j ∈ℕ.Let p⃗=⟨ p_j⟩_1≤ j≤ n, where p_j ∈ℐ be a sequential pattern. We denote by ℒ=ℐ^* the pattern search space. Given the pattern p⃗ and the sequence s⃗ with n ≤ m,we say that s⃗ supports p⃗, iff there exists n integers e_1<…<e_n such that p_k = s_e_k, ∀ k∈{1,…,n}. (e_k)_k∈[n] is called an embedding of pattern p⃗ in s⃗. ℰ_𝓅⃗^𝓈⃗={(e_k)_k∈[n]} denotes the set of the embeddings of p⃗ in s⃗. Let 𝒟={s⃗^k }_k ∈ [N], be a dataset of N sequences. We denote by 𝒯_p⃗ the sequence set supported by p⃗. Given a set of constraints 𝒞, the mining of sequential patterns consists in finding out all tuples ⟨p⃗,𝒯_p⃗,ℰ_p⃗⟩ satisfying 𝒞, where ℰ_p⃗=⋃_s∈𝒯_p⃗ℰ_p⃗^s⃗. The most used pattern constraint is the minimal frequency constraint, c_f_min:|𝒯_p⃗|≥ f_min, saying that the pattern support has to be above a given threshold f_min.Sequential pattern mining with ASP has been introduced by Guyet et <cit.>[Original encodings can be found here: <https://sites.google.com/site/aspseqmining/>]. It encodes the sequential pattern mining task as an ASP program that process sequential data encoded as ASP facts. A sequential pattern mining task is a tuple ⟨𝒮, 𝔐, 𝒞⟩, where 𝒮 is a set of ASP facts encodings the sequence database, 𝔐 is a set of ASP rules which yields pattern tuples from database, 𝒞 is a set of constraints (see <cit.> for constraint taxonomy). We have 𝒮∪𝔐∪𝒞{⟨p⃗,𝒯_p⃗,ℰ_p⃗⟩}.In our framework, the sequence database is modeled by !seq(s⃗,t,e)! atoms. Each of these atoms specifies that the event e∈ℐ occurred at time t in sequence s⃗.On the other hand, each answer set holds atoms that encode a pattern tuples. !pat(i,p_i)! atoms encode the pattern p⃗=⟨ p_i ⟩_i∈[l] where l is given by !patlen(l)!,!support(s⃗)! encodes 𝒯_s⃗and finally ℰ_p⃗ is encoded by !occ(s⃗,i,e_i)! atoms. § DECLARATIVE CARE PATHWAY MININGThe declarative care pathway mining task can be defined as a tuple of ASP rule sets ⟨𝒟, 𝔖, 𝒦, 𝔐, 𝒞⟩ where 𝒟 is the raw dataset and 𝒦 the knowledge base introduced in section <ref>; 𝔐 is the encoding of the sequence mining task presented in <cit.> and 𝒞 is a set of constraints. Finally, 𝔖 is a set of rules yielding the sequences database: 𝔖∪𝒟∪𝒦𝒮. Depending on the study, the expert has to provide 𝔖, a set of rules that specifies which are the events of interest and 𝒞, a set of constraints that specifies the patterns the user would like.In the following of this section, we give examples for 𝔖 and 𝒞 to design a new mining tasks inspired from a case-crossover study answering our clinical question <cit.>. For each patient, the 𝔖 rules generate two sequences made of deliveries within respectively the 3 months before the first seizure (positive sequence) and the 3 to 6 months before the first seizure (negative sequence). In this setting the patient serves as its own control. The mining query consists in extracting frequent sequential patterns where a patient is supported by the pattern iff the pattern appears in its positive sequence, but not in its negative sequence.A frequency threshold for this pattern is set up to 20 and we also constraint patterns 1) to have generic and brand-name deliveries and 2) to have exactly one switch from a generic to a brand-name anti-epileptic grugs – AED (or the reverse). Defining sequences to mine with 𝔖. Listing above illustrates the sequence generation of deliveries of anti-epileptic drug specialities within the 3 months (90 days) before the first seizure event. It illustrates the use of the knowledge base to express complex sequences generation. In this listing, !aed(i,c)! lists the CIP code i, which are related to one of the ATC codes for AED (N03AX09, N03AX14, etc.), and !firstseizure(p⃗,t)! is the date, t, of the first seizure of patient p⃗. A seizure event is a disease event with one of the G40-G41 ICD-10 code. The first seizure is the one without any other seizure event before. ASP enables to use a reified model of sequence where events are functional literals. !seq(P,T,deliv(AED,Gr,G))! designates that patient !P! was delivered at time !T! with a drug where !AED! is the ATC code, !Gr! identify the drug speciality and !G! indicates whether the speciality is a generic drug or a brand-named one. The same encoding can be adapted for sequences within the 3 to 6 months before the first seizure event. [numbers=none,basicstyle=] aed(CIP,AED):-cip_atc(CIP,AED),AED=(n03ax09;n03ax14;n03ax11;n03ag01;n03af01). firstseizure(P,T) :- disease(P,T,D), is_a(D,g40;g41), #countTp: disease(P,Tp,Dp), is_a(Dp,g40;g41), Tp<T=0.seq(P,T,deliv(AED,Gr,1)):- deliv(P,T,CIP,Q), aed(CIP,AED), grs(CIP,Gr),generic(CIP), T<Ts, T>Ts-90, firstseizure(P,Ts).seq(P,T,deliv(AED,Gr,0)):- deliv(P,T,CIP,Q), aed(CIP,AED), grs(CIP,Gr),not generic(CIP), T<Ts, T>Ts-90, firstseizure(P,Ts).Defining constraints on patterns. On the other side of our framework, 𝒞 enables to add constraints on patterns the clinician looks for. Lines 1-3 (see listing above) encode the case-crossover constraints. They select patterns (answer sets) that are frequently in the 3 months period but not in the 3 to 6 months period. The frequency threshold is set to 20. Finally, lines 5-6 illustrate a constraint on the shape of the pattern, that here must contains exactly one switch from a brand-name to a generic drug (or the reverse). [basicstyle=] discr(T) :- support(T), not neg_support(T). #const th=20. :-discr(T)< th. change(X) :- pat(X+1,deliv(AEDp,GRSp,Gp)), pat(X,deliv(AED,GRS,G)), Gp!=G. :- #countX:change(X)!=1. Results The solver extracts respectively 32 patterns and 21 patterns (against 4359 patterns with a regular sequential pattern mining algorithm). With such very constrained problem, the solver is very efficient and extracts all patterns in less than 30 seconds.The following pattern is representative of our results: ⟨ (N03AG01,438,1), (N03AG01,438,1), (N03AX14,1023,0), (N03AX14,1023,0) ⟩ is a sequence of deliveries showing a change of treatment from a generic drug of the speciality 438 of valproic acid to the brand-name speciality 1023 of levetiracetam. According to our mining query, we found more than 20 patients which have this care sequence within the 3 months before a seizure, but not in the 3 previous months preceding this period. These new hypothesis of care-sequences are good candidates for further investigations and possible recommendation about AE treatments.§ CONCLUSIONDeclarative sequential pattern mining with ASP is an interesting framework to flexibly design care-pathway mining queries that supports knowledge reasoning (taxonomy and temporal reasoning).We illustrated the expressive power of this framework by designing a new mining tasks inspired from case-crossover studies and shown its utility for care pathway analytics. We strongly believe that our integrated and flexible framework empowers the clinician to quickly evaluate various pattern constraints and that it limits tedious pre-processing phases. splncs 1martin2010pharmacoepidemiological Martin-Latry, K., Bégaud, B.: Pharmacoepidemiological research using french reimbursement databases: yes we can! Pharmacoepidemiology and drug safety 19(3) (2010)256–265Guyet_IJCAI2016 Gebser, M., Guyet, T., Quiniou, R., Romero, J., Schaub, T.: Knowledge-based sequence mining with ASP. In: Proceedings of IJCAI. (2016)1497–1504gekakaosscsc11a Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., Schneider, M.: Potassco: The Potsdam answer set solving collection. AI Communications 24(2) (2011)107–124Negrevergne15 Negrevergne, B., Guns, T.: Constraint-based sequence mining using constraint programming. In: Proceedings of International Conference on Integration of AI and OR Techniques in Constraint Programming, CPAIOR. (2015)288–305quiniou01 Quiniou, R., Cordier, M.O., Carrault, G., Wang, F.: Application of ILP to cardiac arrhythmia characterization for chronicle recognition. In: Proceedings of conference on Indictive Logic Programming. (2001) 220–227polard2015brand Polard, E., Nowak, E., Happe, A., Biraben, A., Oger, E.: Brand name to generic substitution of antiepileptic drugs does not lead to seizure-related hospitalization: a population-based case-crossover study. Pharmacoepidemiology and drug safety 24(11) (2015) 1161–1169 | http://arxiv.org/abs/1707.08342v1 | {
"authors": [
"Thomas Guyet",
"André Happe",
"Yann Dauxais"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170726094921",
"title": "Declarative Sequential Pattern Mining of Care Pathways"
} |
THE STRUCTURE OF A CHAOS OF STRANGE ATTRACTORS WITHIN A MATHEMATICAL MODEL OF THE METABOLISM OF A CELL] THE STRUCTURE OF A CHAOS OF STRANGE ATTRACTORS WITHIN A MATHEMATICAL MODEL OF THE METABOLISM OF A CELLBogolyubov Institute for Theoretical Physics,Nat. Acad. of Sci. of Ukraine 14b, Metrolohichna Str., Kiev 03680, Ukraine [email protected] National Taras Shevchenko University, Faculty of Cybernetics, Department of Computational Mathematics 64, Vladimirskaya Str., 01033, Kyiv, Ukraine [email protected] 05.45.-a, 05.45.Pq,05.65.+b[ I.V. Musatenko January 15, 2018 ==================== This work continues the study of the earlier constructed mathematical model of the metabolic process running in a cell.We will consider auto-oscillations arising on the level of enzyme-substrate interactions in the nutrient and respiratory chains, which leads to the self-organization in autocatalysis of the integral metabolic process in cells. The auto-oscillations organize themselves in the total metabolic process of cells at autocatalysis.The behavior of the phase-parametric characteristic under the high dissipation of a kinetic membrane potentialis analyzed. All possible oscillatory modes of the system and the scenario of formation and destruction of regular and strange attractors are studied. The bifurcations of the transitions "order-chaos", "chaos-order","chaos-chaos" and "order-order" are calculated. The total spectra of Lyapunov indices and the divergences for all types of attractors on a part of the phase-parametric characteristic under consideration are determined. For various types of strange attractors, their Lyapunov dimensions, KS-entropies, and "predictability horizons" are calculated. Some conclusions about the structure of the chaos of strange attractors and its influence on the stability of the metabolic process in a cell are drawn. § INTRODUCTION The study of the processes of self-organization in alive cells is one of the most important physical problems. Its solution will allow one to answer many questions about the physical nature of life. The most suitable object of studies is the metabolic processes, in which a complicated auto-oscillatory dynamics is manifested. Such modes were found in the processes of photosynthesis and glycolysis, the variation of the concentration of calcium in a cell, oscillations in heart muscle, etc. <cit.>-<cit.>.A complicated auto-oscillatory dynamics is also revealed in widely applied bacteria Arthrobacter globiformis. These bacteria are used in the decomposition of practically all hydrocarbons of oil, purification of waste waters, production of tannic extracts, neutralization of the toxic action of herbicides on plants, biotechnologies, medicine, etc. In laboratory studies, the researchers have found the unpredictable behavior of these bacteria. The intensity of their growth can vary without apparent reasons. The multistability of stationary states and the auto-oscillatory modes of growth are revealed.Earlier at G.K. Skryabin Institute of Biochemistry and Physiology of Microorganisms of the RAS, the studies aimed at the development of a biotechnological process of transformation of steroids by immobilized cells Arthrobacter globiformis in a bioreactor were carried out. By the results of those studies, V.P. Gachok and V.I. Grytsay constructed and studied a mathematical model of this.process. The stationary modes obtained within the model corresponded completely to experimental characteristics and were published in several works jointly with experimenters <cit.>-<cit.>. On the basis of the model, the appearance of various auto-oscillatory modes in granules with immobilized cells was theoretically established <cit.>.The first experimental confirmation of this phenomenon manifesting itself in the given population of cells was obtained later by researchers at the Institute of Microbiology of the RAS. The reason for its appearance was unknown to experimenters. For the quantitative interpretation of such unusual kinetics, they used a modified base synthetic chemostatic model (SCM) <cit.>. This model is based on some hypotheses about the presence of some functions of a certain form, which describe the inhibition and the inactivation of metabolic processes in cells, as well as about the storage of glucose in the polymeric form. Under such conditions, an oscillatory process arises in the metabolism of a cell.In 2002 at D.I. Mendeleev Russian Chemical-Technological University, a PhD dissertation was successively defended <cit.>, in which the author constructed some model of auto-oscillations arising in a population of the given cells and cited the works by V.P. Gachok and V.I. Grytsay. The article describing the model was published somewhat later <cit.>. In the article, it was stated that the mathematical model has a general character and can be applied to the description of oscillations of the biomass of various cells during their cultivation. The metabolic process running in a cell was described with some conditional intermediates. Oscillations in a population of cells appear due to the self-regulation of their number owing to feedbacks with delay under the loss of viability and the lysis of a part of cells due to the action of certain internal factors.The basic distinction of the model developed by us from the above mentioned one consists in that we consider a specific real metabolic process of transformation of steroids, rather than a conditional one. We study the dynamics of variations in the concentrations of real metabolites in a cell under the breaking of the stationary modes, which corresponds to the experiment. The purpose of our studies is the consideration of intracellular oscillations arising on the levels of substrate-enzyme interactions and the respiratory chain. These are intracellular oscillations in the metabolic process. By this example, the laws of self-organization of these oscillations and the appearance of chaos in the metabolic process in a cell are investigated. Just the manifestation of such oscillations in the external medium is registered in experiments. In the earlier above-mentioned models, namely these extracellular oscillations were studied, but the internal dynamics of the cell itself was not considered.By using the proposed model, we study a cell as an object of synergetics and as a nonlinear open self-organizing system. This approach allows us to consider the structural-functional connections inside a cell and to comprehend the physical laws of its vital activity.In <cit.>-<cit.>, the model was used in the numerical calculations of spatio-temporal dissipative and chaotic structures formed with the participation of immobilized cells Arthrobacter globiformis in granules of a bioreactor. The study showed that the oscillations observed in the external solution of a bioreactor are created due to intracellular oscillations in the metabolism. The diffusion instability changes only the form of spatio-temporal structures. Therefore, there appears the necessity to study the dynamics of the metabolic process itself in a cell in more details. The numerical calculations within the model involved a variation of the dissipation of a kinetic membrane potential. This allowed us to determine the intervals on the phase-parametric characteristic, where the periodic, quasiperiodic, and chaotic attractors, whose dimension depends on the dissipation of a kinetic membrane potential, are present. A regularity in the sequence of the appearances of attractors on a toroidal surface was revealed. The sections, where the chaos arises by Feigenbaum's scenario, and an intermittence of the first kind were determined. The Poincaré sections and maps were studied, and the strange attractors, whose structure is formed with the help of folds and funnels, were found. In what follows, we will carry out the further study of the dynamics of a metabolic process within the given model, as well as the regularities of scenarios of formation of strange attractors under a high dissipation of the kinetic membrane potential.§ MATHEMATICAL MODEL AND METHODS OF ITS STUDY A mathematical model of the given metabolic process is constructed with regard for the general scheme of the metabolic process in cells Arthrobacter globiformis at a transformation of steroids <cit.>-<cit.>:dG/dt=G_0/N_3 + G +γ _2ψ-l_1V(E_1)V(G)-α_3G, dP/dt=l_1V(E_1)V(G)-l_2V(E_2)V(N)V(P)-α_4P, dB/dt=l_2V(E_2)V(N)V(P)-k_1V(ψ)V(B)-α_5B, dE_1/dt=E_1_0G^2/β_1+G^2 ( 1- P+mN/N_1+P+mN )- -l_1V(E_1)V(G)+l_4V(e_1)V(Q)- α_1E_1, de_1/dt=-l_4V(e_1)V(Q)+l_1V(E_1)V(G)-α_1e_1, dQ/dt= 6lV( 2-Q)V(O_2)V^(1)(ψ)--l_6V(e_1)V(Q)-l_7V(Q)V(N), dO_2/dt=O_2_0/N_5+O_2-lV( 2- Q) × × V(O_2)V^(1)(ψ)-α_7O_2, dE_2/dt=E_2_0P^2/β_2+P^2N/β+N (1-B/N_2+B ) --l_10V(E_2)V(N)V(P)-α_2E_2, dN/dt= -l_2V(E_2)V(N)V(P)-l_7V(Q)V(N)+ +k_2V(B)ψ/K_10+ψ+N_0/N_4+N-α_6N, dψ/dt=l_5V(E_1)V(G)+l_8V(N)V(Q)-αψ . where, V(X)=X╱ ( 1+X); V^(1)(ψ)=1╱ ( 1+ψ^2 ); V(X) is a function describing the adsorption of the enzyme in the region of local coupling; andV^(1)(ψ) is a function characterizing the influence of the kinetic membrane potential on a respiratory chain.The variables in the equations are dimensionless <cit.>.We take the following values of parameters of the system: l=l_1=k_1=0.2; l_2=l_10=0.27; l_5=0.6; l_4=l_6=0.5; l_7=1.2; l_8=2.4; k_2=1.5; E_1_0=3; β_1=2; N_1=0.03; m=2.5; α=0.033; a_1=0.007; α_1=0.0068; E_2_0=1.2; β=0.01; β_2=1; N_2=0.03; α_2=0.02; G_0=0.019; N_3=2; γ_2=0.2; α_5=0.014; α_3=α_4=α_6=α_7=0.001; O_2_0=0.015; N_5=0.1; N_0=0.003; N_4=1; K_10=0.7.Equations (<ref> - <ref>) describe variables of the concentrations: Eq. (<ref>) – hydrocortisone (G); Eq. (<ref>) – prednisolone (P); Eq. (<ref>) – 20β-oxyderivative of prednisolone (B); Eq. (<ref>) – oxidized form of 3-ketosteroid– △ ' –dehydrogenase (E_1); Eq. (<ref>) – reduced form of 3-ketosteroid – △ '-dehydrogenase (e_1); Eq. (<ref>) – oxidized form of the respiratory chain (Q); Eq. (<ref>) – oxygen (O_2); Eq. (<ref>) – 20β – oxysteroid-dehydrogenase (E_2); Eq. (<ref>) – NAD · H(reduced form of nicotinamide adenine dinucleotide) (N).Eq. (<ref>) describes the variation of the kinetic membrane potential (ψ).The calculations are based on the Runge-Kutta-Merson method. The set accuracy is 10^-8. Prior to the approach to an attractor by the system, the duration of the transient initial phase was taken to be 1.000.000 in order to obtain the proper calculated values.To construct the phase-parametric characteristic, we used the method of sections. In the phase space with a trajectory of the system, we drew the cutting plane P=0.2. Such a choice is supported by the symmetry of oscillations relative to this point in multiple modes.The spectrum of Lyapunov indices was calculated, by using Benettin's algorithm with orthogonalization of the perturbation vectors within the Gram-Schmidt method <cit.>.Here, we consider the zero Lyapunov index to be a number, whose first significant number appears only in the fifth decimal place. By this, we identified the type of regular and strange attractors.To classify the geometric structures of strange attractors, we calculated their fractal dimensions. Strange attractors are fractal sets and possess the Hausdorff-Besicovitch fractional dimension. But its direct calculation is a very difficult problem, which has no standard algorithm. Therefore, we calculated the Lyapunov dimension of attractors, as a quantitative measure of fractality, by the Kaplan-Yorke formula <cit.>D_ Fr=m+∑ ^m_i=1λ_i/| λ _m+1| ,where m – number of the first Lyapunov indices in the decreasing order, whose sum ∑^m_i=1λ _i≥ 0; m+l - number of the first Lyapunov index, whose value λ _m+1<0.In addition, we studied the variation of a distance between close phase points of trajectories d(t)=|x_2(t)-x_1(t)| during the evolution of system (<ref>-<ref>). If the dynamics of the system is chaotic, then d(t) increases exponentially with time: d(t) ≈ d(0) e^kt. In this case, the mean rate of divergence of trajectories is defined as k= ln[d(t)/d(0)]/t. We consider alsoh= lim_d(0)→ 0 t→∞ln [d(t)/d(0) ]/t,which is called the Kolmogorov-Sinai entropy or KS-entropy <cit.>. With the use of the KS-entropy, we determined the conditions, under which the modes under study are chaotic or regular attractors. In particular, if the dynamics of the system is periodic or quasiperiodic, then the distance d(t) does not increase with time, and the KS-entropy is equal to zero (h=0). In the presence of a fixed point in the system, d(t)→ 0 and h<0. In the case of the chaotic dynamics of the system, the KS-entropy is positive (h>0).Since the values of characteristic Lyapunov indices determine the rates of divergence of trajectories in the m-dimensional phase space of the system, we use the spectrum of Lyapunov indices for the calculation of the value of h. By the Pesin theorem <cit.>, the KS-entropy corresponds to the sum of all positive Lyapunov characteristic indices:h=∑^m_i=1λ _i. The KS-entropy allows us to estimate the rate of loss of the information about the initial state of the system. The positiveness of the entropy is a criterion of the chaos. This gives the possibility to qualitatively evaluate properties of the local stability of attractors.The value reciprocal to the KS-entropy,t_min=h^-1,determines the time of mixing in the system and characterizes how rapidly the initial conditions will be forgotten. At t≪ t_min, the behavior of the system can be predicted with sufficient accuracy. At t>t_min, only the probabilistic description is possible. The chaotic mode is unforeseen due to the loss of the memory of initial conditions. The quantity t_min is called the Lyapunov index and characterizes the "predictability horizon" of a strange attractor.§ RESULTS OF STUDIES Earlier,we studied the part of the phase-parametric characteristic α∈ (0.032,0.32554) and established that, at α = 0.032554, regular attractor 10· 2^0 on a torus holds in the system. We now continue the study of oscillatory modes of the system under the variable dissipation of a kinetic membrane potential. Below in (Table <ref>), we present the calculated total spectra of Lyapunov indices and the divergences for the majority of modes under consideration.Letα increase. We observe that, at α = 0.0326735, strange attractor 10· 2^x appears instantly. Hence, we have the transition 10· 2^0→ 10 · 2^x of the "order-chaos" type. If the value of α grows, the given strange attractor transits gradually to strange attractor 9· 2^x (α = 0.03269) (Fig. <ref>,a). In this case, we observe the transition "chaos-chaos". Then strange attractor 9· 2^x shrinks to quasiperiodic cycle ≈ 9· 2^1 on a torus (α=0.032694).Letus consider the part of the phase-parametric characteristic for α∈ (0.032694,0.032706) (Fig. <ref>,b). By passing from right to left, we see that regular attractor 9·2^0 on a torus exists at α =0.032706. As the dissipation of a kinetic membrane potential in the interval α∈ (0.032705,0.0327036) decreases, the oscillations on a toroidal surface are destroyed, and the formation of the simple regular attractor 9· 2^1 with doubled period occurs (α =0.032704). The further decrease in α causes the renewal of cycle 9· 2^0 on a torus (α = 0.032703). At α=0.0327014, we observe the formation of attractor 9· 2^1 and the second appearance of the period doubling bifurcation on a torus. Then this cycle loses the stability, and quasiperiodic cycle ≈ 9· 2^1 is formed on a torodoidal surface (α =0.032697). Thechange of a section of the attractor of quasiperiodic cycle ≈9 · 2^1 in the interval α∈ (0.0327,0.032694) is shown in Fig. <ref>,b. At α = 0.03269609, we see the sudden appearance of regular attractor 9· 2^0 on a torus.As αincreases from 0.032706 (Fig. <ref>,b) to 0.032866 (Fig. <ref>,a), the oscillations on a toroidal surface of attractor 9· 2^0 cease, and the ordinary periodic 9-fold cycle is restored (Table <ref>). But, the further increase in α (Fig. <ref>,a) leads to its aperiodic fracture and to the formation of strange attractors 9· 2^x (α = 0.03287086). On the phase-parametric characteristic (Fig. <ref>,a), we observe the formation of the zones of stability and instability of a regular attractor with the appearance of the mutual transitions "order-chaos-order": 9· 2^0↔ 9 · 2^x↔ 9 · 2^0. The further increase in α causes the instability of strange attractors 9· 2^x and their self-organization in strange attractors 8 · 2^x.In Fig. <ref>,b, we show the projection of the phase portrait of a strange attractor at the point of the mutual transition "chaos-chaos": 9· 2^x↔ 8 · 2^x (α =0.032874). At α=0.032875, there occur the contraction of phase trajectories and the appearance of regular attractor 8· 2^0 on a torus. As α increases, the given regular attractor becomes unstable, and strange attractors 8· 2^x (α =0.0328765, Table <ref>) are formed. The further growth of α causes their aperiodic destruction and their alternation with regular attractors 8· 2^0 and 8· 2^1 on a torus. In the interval α∈ (0.03287980,0.0328808) (Fig. <ref>,a), regular attractors 8· 2^0 are conserved. Then, in the interval α∈ (0.0328809,0.0328828), strange attractors 8· 2^x alternating with regular attractors of the 8-fold period are formed. The transitions at the onset and the end of this interval, 8· 2^x↔ 8· 2^0↔ 8· 2^x in both directions, are realized through the period doubling bifurcation and the intermittence. Then, at α =0.032884, attractor 8· 2^0 arises again on a torus and conserves its stability up to α =0.033117. Then it contracts in an ordinary periodic 8-fold cycle (α =0.0331) (Table <ref>).The further variation of the multiplicity of an oscillatory process with increase in the dissipation of a kinetic membrane potential can be traced with the use of the whole phase-parametric characteristic (see Fig. <ref>,a and Table <ref>). Letus consider Fig. <ref>,a from right to left. At α =0.042, we observe a single periodic cycle 1· 2^0. As α decreases, it is deformed, and its period increases due to the bifurcation: 1· 2^0→ 2· 2^0. The further decrease in α causes a deformation of the new cycle and the appearance of 3-fold regular attractor 3· 2^0 from this cycle. Then the scenario is repeated. The appropriate cycle is deformed, and its multiplicity increases by 1 at the points of bifurcations. We see the successive increase in the multiplicity of regular attractors. In Fig. <ref>,b, we present the projections for some examples of regular attractors formed in the interval α∈(0.0328,0.042) with the following order of the growing multiplicity of periods:[ 1·2^0(α=0.042)→ 2·2^0(α=0.039)→; →3·2^0(α=0.0375)→ 4·2^0 (α=0.0348)→; →5·2^0 (α =0.0346)→6·2^0(α=0.0338)→; →7·2^0(α=0.0332)→8·2^0(α=0.0331). ] As a characteristic specific feature of such transitions, we mention the absence of strange attractors on the given part of the phase-parametric characteristic. As α decreases, the multiplicity of autooscillations in the metabolic process in a cell varies discretely due to the the self-organization. We observe the "order-order" transitions.At the increase of α from 0.0331 to 0.032884 (Fig. <ref>,a), regular attractor 8·2^0 holds.In addition, Fig. <ref>,a demonstrates a change of the dynamics of the metabolic process. In works <cit.>-<cit.>, this part of the phase-parametric characteristic was studied in detail, and the following scenario of variation of the multiplicity of attractors was found:[8·2^0(α=0.0331)→ 8·2^x(α=0.0328812)→;→9·2^x (α=0.03287086)→ 9·2^0 (α=0.032866)→; →9·2^x(α =0.03269)→10·2^x(α=0.0326735)→;→10·2^0(α=0.032554)→10·2^x(α=0.03254)→;→11·2^x(α =0.032517)→11·2^0(α=0.0325)→; →11·2^x(α=0.0324)→12·2^x(α=0.032387)→; →12·2^0(α =0.032386)→12·2^x(α=0.03229)→; →13·2^x(α=0.03227575)→13·2^0(α=0.032275)→;→13·2^x(α =0.03222)→14·2^x(α=0.03217)→; →≈ n·2^0(α=0.03215962)→14·2^0(α=0.0321596)→;→8·2^x(α=0.03211295)→8·2^0(α=0.0321107)→;→1·2^0(α=0.032). ]The transitions between multiple modes occur through the intermediate formation of strange attractors. Here, we observe the "order-chaos-chaos-order" transitions. At the limiting points of the interval covered by Fig. <ref>, a, regular attractors 1· 2^0 are established for the minimum and maximum dissipations. Inside the interval, we see the appearance of regular and strange attractors with various multiplicities as a result of bifurcations and the self-organization.We now consider the fractality of strange attractors. As an example, we take strange attractor (9↔8)·2^x (α=0.032874) for the transition between 9- and 8-fold cycles. By separating a small rectangular region of the projection of the phase space in Fig. <ref>,b, we map it onto Fig. <ref>,a. We now separate a small rectangular region in Fig. <ref>,a, which includes one of the phase curves and present it as Fig. <ref>,b. As is seen, the character of the geometric structure of the given strange attractor is repeated on small and large scales of the projection of the phase portrait. Each arisen curve of the projection of a chaotic attractor is a source of formation of new curves. Moreover, the geometric regularity of a structure of trajectories in the phase space is repeated. In Fig. <ref>,c, we show in detail that the geometric structure of the fractality is conserved also in the mixing funnel of the given strange attractor. This geometric structure reminds a two-scale Cantor parametric set. Since the direct calculation of the fractal dimension of an attractor is an extremely complicated problem, and no standard algorithms are available for its solution, we restrict ourselves only by the calculation of their Lyapunov dimension D_ Fr Eq.(<ref>).We obtain that the Lyapunov dimension of all regular attractors of simple periodic cycles is equal to 1 and that of the regular attractors on a toroidal surface, which correspond to quasiperiodic cycles, is equal to 2.We now calculate the fractal dimension for certain strange attractors (see Table <ref>). In addition, we calculate also the Kolmogorov–Sinai entropy hEq.(<ref>) and the Lyapunov index t_min Eq.(<ref>).Their phase portraits are presented in Fig. <ref>. At α=0.0328715, strange attractor 8· 2^x appears (Fig. <ref>,a). Inside itself, it forms a funnel, where we observe the mixing of diverging trajectories of this unstable period. The deterministic chaos is forming. The further decrease of α is accompanied by a complication of the structure of the strange attractor. For example, at α=0.032168 (Fig. <ref>,b), a more complicated strange attractor arises as a result of the intermittence of two chaotic processes: 14· 2^x and 7·2^x. Inside it, a funnel is located. The attractor is formed by two sources of chaotic autooscillatory processes: the transformation of steroids G→ P→ B and a change of the activity of a respiratory chain due to variations of the kinetic membrane potential ψ. According to the definition given by Pomeau and Manneville, such a transition is called an intermittence of the first kind <cit.>-<cit.>.At α=0.0321646 (Fig. <ref>,c), strange attractor 7·2^x appears. Its specific feature is the absence of a funnel. Nevertheless, the attraction regions of an unstable 7-fold cycle are clearly distinguished on the phase portrait.At α=0.03211295 (Fig. <ref>,d), we see strange attractor 8·2^x. It differs significantly from strange attractor 8·2^x (Fig. <ref>,a), has also no funnel, and the attraction regions of its trajectories are closed narrow strips with 8-fold period.In this case, the KS-entropies of these modes decrease in the following sequence: 0.000665 (Fig. <ref>,b); 0.000437 (Fig. <ref>,c ), 0.000385 (Fig. <ref>,a), and 0.000308 (Fig. <ref>,d). The KS-entropy indicates the value of unpredictability for the motion of the phase trajectory of a strange attractor and characterizes the value of chaoticity of its deterministic chaos. The higher the KS-entropy, the greater the exponential divergence of phase trajectories along the perturbation vector corresponding to λ_1 and λ_2. For the rest vectors corresponding to the negative values of λ_3-λ_10 (Table <ref>), the phase trajectories exponentially contract to the own attractor. As the KS-entropy increases, the structure of a chaos is complicated (compare Fig. <ref>,d,a,c,b). Among the modes under study, the mode shown in Fig. <ref>,d is the most ordered.In the same figures, we show the values of "predictability horizon" and the fractal dimension of strange attractors. The mode shown in Fig. <ref>,d (t_min=3247) turns out to be the most predictable as compared with those shown in Fig. <ref>,a (t_min=2597), Fig. <ref>,c (t_min=2288), and Fig. <ref>,b (t_min=1504). In these modes of deterministic chaos, the metabolic process is predictable only in the determined time intervals t_min.As distinct from the KS-entropy, the Lyapunov dimension of the given modes, which characterizes the fractality of these strange attractors, increases by a somewhat different sequence: Fig. <ref>,a (D_ Fr=2.073000), Fig. <ref>,d (D_ Fr=2.150759), Fig. <ref>,b (D_ Fr=2.216612), and Fig. <ref>,c (D_ Fr=2.323704). This can be explained by that the Lyapunov dimension for these modes is determined not only by λ_1 and λ_2, but also by |λ_3| Eq.(<ref>), which characterizes the deformation of an element of the phase volume along the relevant perturbation vectors. The deformation increases with λ_1 and with decrease in |λ_3|.In addition, the deformation of the total volume is also determined by the values of remaining negative Lyapunov indices λ_4 - λ_10 (Table <ref>). On the whole, the magnitude of such a change of the volume is determined by the divergence of a relevant mode (Table <ref>), namely: Λ=-0.898147 (Fig. <ref>,d), Λ=-0.901347 (Fig. <ref>,b), Λ=-0.903255 (Fig. <ref>,c) and Λ=-0.931419 (Fig. <ref>,a).The mode in Fig. <ref>,d is the most functionally stable for a cell. The metabolic process in a cell is self-organized, so that it possesses the least dissipation and the highest "predictability horizon". At the given coefficient of dissipation of the kinetic membrane potential, α=0.03211295, the insignificant variations of parameters of the system do not cause a change of the dynamics of the metabolic process.Analyzing the presented typical strange attractors, it is possible to indicate a certain regularity of the hierarchy of the chaotic behavior in the metabolism of a cell. The structures different in their characteristics replace one another. As a result, the cell adapts to varying conditions of the environment and conserves its functionality.§ CONCLUSIONS With the help of the mathematical model of a cell, we have performed the study of the dynamics of the metabolic process in the mode of oscillations under the enhanced dissipation of a kinetic membrane potential. The scenarios of formation and destruction of regular and strange attractors with various periods and types are determined. The boundaries of the phase-parametric characteristics of regions, where the bifurcations and the transitions "chaos-order","order-chaos", "chaos-chaos", and "order-order" arise, are given. The total spectra of Lyapunov indices and the divergences are calculated. For some typical types of strange attractors, we have determined Lyapunov dimensions of their fractality, KS-entropies, and "predictability horizons". The structure of the chaos of attractors, the hierarchy of their kinds, and the influence of the chaos on the stability of the metabolic process, and the adaptation and the functioning of a cell are studied.One of the purposesof the present work is to demonstrate a possibility to apply the mathematical apparatus of nonlinear dynamics to the study of the dynamics of metabolic processes within a specific model. This allows us to consider the structural-functional connections in a cell and the laws of its self-organization. These systems are the excellent field for applying the methods of nonlinear dynamics to the analysis of the multidimensional systems of nonlinear differential equations.The work is supported by the project N 0112U000056 of the National Academy of Scienses of Ukraine.331E.E. Selkov, Europ. J. Biochem. 4, 79 (1968).2B. Hess and A. Boiteux, Annu Rev Biochem. 40, 237 (1971).3A. Goldbeter and R. Lefer, Biophys J. 12, 1302 (1972).4A. Godlbeter and R. 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Pesin, Uspekhi Mat. Nauk, 32, No. 4, 55 (1977).30P. Berge, Y. Pomeau, C.H. Vidal, Order within Chaos, Wiley, New York, (1984).31P. Manneville, Y. Pomeau, Phys. Lett. A., 75, No. 1-2, 1 (1979).32P. Manneville, Y. Pomeau, Physica D.: Nonlinear Phenomena, 1, No. 2, 219 (1980).33P. Manneville, Y. Pomeau, Comm. Math. Phys., 74, No. 2, 189 (1980).Received 22.01.09-5mm вί ˲ ˲ ˲.. , .. ., -.... ."-", "-", "-""-".. , -ﳿ" ". . -5mm.. , .. .,-.... ."-", "-", "-""-".. , -" ". . | http://arxiv.org/abs/1707.08428v1 | {
"authors": [
"V. I. Grytsay",
"I. V. Musatenko"
],
"categories": [
"nlin.CD",
"q-bio.CB"
],
"primary_category": "nlin.CD",
"published": "20170726132605",
"title": "The structure of a chaos of strange attractors within a mathematical model of the metabolism of a cell"
} |
Sectoring in Multi-cell Massive MIMO Systems Shahram Shahsavari, Parisa Hassanzadeh, Alexei Ashikhmin, and Elza Erkip S. Shahsavari, P. HassanzadehandE. Erkip are with the ECE Department of New York University, Brooklyn, NY. Email: {shahram.shahsavari,ph990, elza}@nyu.eduA. Ashikhmin is with Bell Labs, Nokia, Murray Hill, NJ, USA. Email:[email protected] December 30, 2023 ==========================================================================================================================================================================================================================================================================================================================================We present the first study of the evolution of the galaxy luminosity and stellar-mass functions (GLF and GSMF) carried out by the Dark Energy Survey (DES). We describe the COMMODORE galaxy catalogue selected from Science Verification images. This catalogue is made of ∼ 4× 10^6 galaxies at 0<z≲1.3 over a sky area of ∼155sq.deg with i-band limiting magnitude i=23mag. Such characteristics are unprecedented for galaxy catalogues and they enable us to study the evolution of GLF and GSMF at 0<z<1 homogeneously with the same statistically-rich data-set and free of cosmic variance effects.The aim of this study is twofold: i) we want to test our method based on the use of photometric-redshift probability density functions against literature results obtained with spectroscopic redshifts; ii) we want to shed light on the way galaxies build up their masses over cosmic time.We find that both the i-band galaxy luminosity and stellar mass functions are characterised by a double-Schechter shape at z<0.2. Both functions agree well with those based on spectroscopic redshifts. The DES GSMF agrees especially with those measured for the GAlaxy Mass Assembly and the PRism MUlti-object Survey out to z∼1.At 0.2<z<1, we find the i-band luminosity and stellar-mass densities respectively to be constant (ρ_ L∝ (1+z)^-0.12±0.11) and decreasing (ρ_ Mstar∝ (1+z)^-0.5±0.1) with z. This indicates that, while at higher z galaxies have less stellar mass, their luminosities do not change substantially because of their younger and brighter stellar populations. Finally, we also find evidence for a top-down mass-dependent evolution of the GSMF. galaxies: evolution – galaxies: formation – galaxies: photometry – astronomical methods: miscellaneous – astronomical data bases: surveys – astronomical data bases: catalogues § INTRODUCTION The most widely accepted structure-formation paradigm predicts that structures are generated from primordial density perturbations in the power spectrum <cit.> and form via gravitational collapse following dark matter clustering <cit.>. Within this picture, galaxies are thought to assemble their mass (including its baryonic component, e.g. gas and stars) over cosmic time following this hierarchical pattern. However, galaxy baryonic mass growth is the result of the interplay of several processes <cit.>, for example, to mention only some of them, star formation from accreted or in-situ gas (e.g., ), radiative cooling (e.g. ), supernova and Active Galactic Nuclei (AGN) feedbacks (e.g. ), mergers and galaxy interactions (e.g., ).Hence, the hierarchical picture portrayed for dark matter might not be fully applicable to baryonic matter, because the latter is not subject only to gravity. In fact, despite the general belief that galaxies form hierarchically [the majority of semi-analytic models are built on this premise (e.g., )], recent observational studies suggest a less significant role for the halo-scale environment in influencing galaxy formation and evolution, particularly for high-mass galaxies (e.g., see ). At the same time, the influence that the physics driving galaxy formation has on halo properties is still a matter of debate (e.g., <cit.>). As a consequence of this contrast, a number of fundamental issues remain unsolved. For instance, the latest galaxy formation models regard the time when galaxy stellar mass forms and assemble onto the main galaxy halo as two different stages in the life of a galaxy. These two stages might not coincide and they are both mass dependent (e.g. ). In general, the more massive the galaxy the earlier in time its stellar mass is formed, in the sense that at a given redshift, more massive galaxies will have formed a higher fraction of their final stellar mass than less massive galaxies. This is generally found both in observation- (e.g., ) and simulations-based studies (e.g., ). When focusing instead on the time when stellar mass is assembled, the current picture is blurred by the discrepancies between the findings of studies based on theoretical models applied to simulations and those based on observed data. In fact, while the former generally find that the stellar mass assembly happens in a hierarchical (bottom-up) fashion (e.g., ), the majority of the latter finds that the stellar mass assembly follows an anti-hierarchical (top-down) pattern (e.g., ). However there are also observation-based studies which show agreement with simulations, i.e. identify a hierarchical pattern in the galaxy mass assembly (see for instance ). As a result, galaxy stellar-mass assembly still remains a burning question for the scientific community.Galaxy luminosity and stellar mass functions (respectively GLF and GSMF) constitute some of the most useful observables for gaining insight into the mechanisms actually governing galaxy formation and evolution, hence are fundamental for replying to this burning question. In particular, studying the evolution over cosmic time of galaxy number, luminosity and stellar mass densities allows us to directly probe the galaxy mass growth process.Several studies in the literature focused on the investigation of these functions in relation to galaxy formation and evolution (e. g. ). The majority of such studies were either performed out to high redshift (z∼2 or higher) but in deep pencil-beam surveys (usually ≲ 2 deg^2 wide, e.g. ) or limited to low redshifts (z≲0.5) in relatively shallow but large-area surveys [e.g., SDSS and GAlaxy Mass Assembly (GAMA), ].Many of these studies made use of Spectral-Energy-Distribution (SED) fitting for deriving galaxy properties (such as stellar mass). The filter sets used generally listed more than 5 filters encompassing the spectral region from the UV-optical to the infrared. Variation of the filter sets and of the spectral coverage of galaxy SEDs is linked to variation in the precision of galaxy properties and hence of the GLF and GSMFs presented in these studies. In addition, especially when studying the GLF and GSMF out to z>1, the use of photometric redshifts was often adopted, leading to additional uncertainties on galaxy properties and on GLF and GSMF (e.g. ). Additional sources of uncertainties and systematics, like those due to evolutionary stellar population synthesis models, SED-fitting related assumptions (e. g.,model templates, initial mass function, metallicity, dust content and modeling) and others, were also treated in some of the studies of the GLF/GSMF in the literature (e.g., see ). All these sources of uncertainty must be properly taken into account in order to reliably measure spatial number densities and their evolution with cosmic time.In this paper we study the evolution of the GLF and GSMF out to z∼1 with the Dark Energy Survey, using data taken during its Science Verification (SV) phase over 5 broad-band filters (g, r, i, z, Y). Such data are characterised by a relatively high depth (∼ 23mag in i band) and a large area (∼ 155sq. deg), characteristics which allow us to better study the GLF and GSMF with respect to previous surveys. Our work is carried out by using photometric redshifts, which allow us to analyse a galaxy sample made of about 4× 10^6 galaxies out to redshift z∼1. The aim of our study is twofold. On one hand, we want to test if over a wide-field 5-band survey we can get consistent GLFs/GSMFs from photometric and spectroscopic redshifts (seefor a similar test but over a ≲ 1 deg^2 wide sky area). On the other hand, we want to shed light on the way galaxies build up their masses over cosmic time.Despite assessing the effects on our analysis due to uncertainties on photometry, photometric redshifts and to galaxy completeness, in our study we do not take into account sources of uncertainties due to evolutionary population synthesis models and SED-fitting assumptions [such as model templates, initial mass function (IMF), metallicity, dust content and modeling]. We present here the current study of GLF and GSMF on DES SV data as a proof of concept and build up on our current work to subsequently produce more precise estimates based on the survey full data set and taking into account all systematics and sources of uncertainty.The layout of the paper is the following: in Section <ref> we provide a description of the DES, the SV data-set, photometric redshifts and the selection of the galaxy catalogue (including galaxy completeness assessment). Section <ref> is dedicated to the determination of galaxy properties (both physical and detectability-related), while in Section <ref> the analysis carried out to measure the GLF and GSMF is presented. Section <ref> is dedicated to the description of our results in comparison with observational measurements in the literature based on spectroscopic redshifts. Finally, in Sections<ref> & <ref>we respectively discuss our results and draw our conclusions.Throughout the paper, we assume a flat Λ cold dark matter (Λ CDM) cosmology with H_0=70km s^-1 Mpc^-1 and Ω_m=0.286, use magnitudes in the AB system and, unless indicated otherwise, utilise a Salpeter IMF <cit.>. In addition, M^* stands for the cut-off value (either absolute magnitude or stellar mass) of the Schechter function, while M_* is used to refer to stellar mass. § DES SCIENCE VERIFICATION DATA The Dark Energy Surveyis a photometric survey carried out in 5 bands (g, r, i, z,Y) with the Dark Energy Camera (DECam, ) mounted on the 4-meter Blanco Telescope at Cerro Tololo Inter-American Observatory (CTIO). The survey started in August 2013 but from November 2012 to March 2013, DES carried out a Science Verification (SV) survey. These observations provide science-quality data for more than 250 sq. deg at close to the main survey's nominal depth (standard depth), the latter being achieved by the coaddition of 10 single-epoch images (see further down).The sky footprint over which these data were taken was chosen in order to contain a combination of large contiguous regions at standard depth (i-band 2-arcsec aperture magnitude=24), covering parts of the eastern (over ∼ 160sq.deg.) and western (over ∼ 35sq.deg.) areas of the South-Pole-Telescope (SPT, ) survey (referred to respectively as SPT-E and SPT-W fields), and smaller regions either at standard depth or deeper(limiting i-band 2-arcsec aperture magnitude=26). The former extend individually over <1sq.deg and are respectively centred on the well-known El Gordo, Bullet and RXJ2248 galaxy clusters (we will refer to these fields respectively as El Gordo, Bullet-cluster and RXJ2248 fields). The latter extend over a total area of ∼ 30sq.deg and we shall refer to them as Supernova (SN) fields. From now on we will generally refer to SN fields as Deep survey and to the remaining fields at the survey standard depth as Wide survey. In Figure <ref>, the footprint of the SV area is shown. §.§ The Gold SVA1 galaxy catalogue The data stored in the catalogue of the SV coadded imaging were created by the DES Data Management (DESDM) pipeline ( and Morganson et al. in prep), which used<cit.> as the main source extraction software. In order to make these data science ready, a team of DES scientists thoroughly tested and analysed them. In particular, the following steps were carried out:* Incorporating satellite trail and other artifact information to mask out specific areas.* Removing areas with colours severely affected by stray light in the images and areas with a small exposure count (at the borders of the footprint).* Applying an additional zero point correction via stellar locus regression (SLR), to tighten the calibration even further according to the distribution of star colours with respect to those of reference stars. * Removing the area (∼ 40sq. deg.) below declination of -61^∘, largely occupied by the Large Magellanic Cloud (LMC). This was done because our SLR tests showed that this area could not be accurately calibrated to the same scale as the rest of the survey. This cut has also the advantage of removing ∼ 5sq. deg contaminated by stray light from R Doradus, the second brightest star in the infrared sky.* Identifying a star/galaxy classifier (the `modest classifier') to perform star/galaxy separation (see Section <ref>).These additional steps led to the identification of a new photometric catalogue, called the SV Annual 1 (SVA1) Gold catalogue, containing 25,227,559 objects extending over an area ∼ 250 sq. deg.This catalogue is now publicly available[The SVA1 Gold catalogue, all related data products and documentation are publicly available at: <http://des.ncsa.illinois.edu/releases/SVA1>.]. §.§.§ Survey depth and systematics:mask and depth maps Despite the survey strategy being decided in such a way to minimize depth differences over the sky, variable observing conditions on different observing nights makes this difficult. In addition to variable depth over the sky, observations are also influenced by artificial effects like cosmic rays, airplane and satellite trails and by stray light from very bright stars. In order to take all these sky-region dependent effects into account, the creation of a mask via<cit.> software was implemented within the DESDM pipeline.takes into account properties of DECam CCDs and the sky during each night and gives an estimation of the 10σ-level depth for different regions (called Molygons). This depth is calculated for a 2arcsec aperture magnitude (). Despite the resulting mask having very high resolution (at a pixel size of 6.44arcsec), allowing to eliminate regions severely affected by the aforementioned effects and to take variable depth with the sky into account, aperture magnitudes such asdo not contain the total light of extended sources. Hence they are not typically used for the majority of galaxy studies, which mainly rely on integrated magnitudes. In addition, the depth value characterising a galaxy sample should be given in the same magnitude system used for galaxy selection. Hence, in our case, thedepth value should be converted into the total magnitude system used for galaxy selection. Because of this, a new mask was created for each total magnitude estimation algorithm [i.e., ,(similar to SDSS' modelMag) and(similar to SDSS' cmodelMag)] used during the derivation of the photometric catalogues. This was done by following the process described in <cit.> and resulted in new masks at a resolution of 0.18arcmin^2. These masks are used here for selecting a galaxy sample at 10σ level.§.§.§ Photometric Redshifts Reliably measuring photometric redshifts is a key step for carrying out cosmological measurements, for a proper determination of galaxy properties and for a correct study of GLF and GSMF. Keeping this in mind, for our study we make use of photometric redshifts measured with the “Trees for PHOTOZ”(TPZ) algorithm <cit.>, which provides individual photo-z probability density functions (pdfs) which fold in photometric uncertainty. We can then use such pdfs in our study to determine the effects on GLF and GSMF due to photometry and photo-z uncertainties. In particular, TPZ is a machine learning parallel algorithm which utilises prediction trees and random forest techniques to produce not only redshift pdfs but also ancillary information for a given galaxy sample. We refer the reader to <cit.> for details on this algorithm Our choice of using TPZ photo-z's was made also because the TPZ algorithm is one of the best performing methods in the study by <cit.>, who found that empirical methods using, for instance, artificial neural networks or random forests (as TPZ) yielded the best performance, achieving core photo-z resolutions σ_ 68∼ 0.08-0.1, defined as the 68 percentile width of Δ z around the median.We take into account photometry and photo-z uncertainties on galaxy properties (e.g. absolute magnitude and stellar mass) and on GLF and GSMF by performing Monte Carlo simulations. In particular we draw 100 photo-z's values from each galaxy pdf via inverse-cumulative-distribution-function resampling method, in order to construct 100 additional galaxy catalogues for our analysis. We carry out our analysis on each of these catalogues, from galaxy property estimation to GLF and GSMF measurements, so to derive uncertainties on galaxy properties and on number densities (see Sections <ref> and <ref>). We find (see ) that 100 draws are sufficient for producing unbiased resampled photo-z pdfs (both mean and standard deviation of the resampled photo-z pdfs are unbiased on average). Such resampling is also characterised by RMSE of pdf mean and standard deviation (respectively 0.0076 and 0.0082) larger than the typical width of the median photo-z pdf. However the precision quoted is sufficient for the study carried out here. We refer the reader to <cit.> for more details on the photo-z pdf resampling and to Sections <ref> and <ref> for the results of this Monte Carlo simulations on galaxy properties and on the study of GLF and GSMF.Note that in this paper we do not study the effect of using different photometric redshifts estimated via different methods, which are also available for the SVA1 Gold catalogue[<https://des.ncsa.illinois.edu/releases/sva1/doc/photoz>.] and are described in <cit.>. §.§.§ Star/galaxy separation The selection of galaxies from the Gold catalogue is carried out by using the so-called `modest classifier' (1 for galaxies and 2 for stars), which combines several SExtractor outputs in the i band: i) star/galaxy separation parameters, i.e.<cit.>,and its uncertainty<cit.>; ii) magnitude measurements, i.e.and ; iii) SExtractor internal flags (). The combination used to select galaxies is the following (see also ):= > 0.3<18.0 = +3* < 0.003 = > 30.0< 21.0 = > 3= The above combination ofparameters takes into account the relative magnitude regions whereandperform better and where star or galaxies are expected to dominate (hence the use of different magnitude types). In addition,internal flags serve for artifacts removal. The modest classifier is found to simultaneously achieve ≥ 90 per cent efficiency and purity for objects with i-band magnitude fainter that 19 mag. In addition to the `modest classifier', a probabilistic star/galaxy classifier is also provided by the TPZ algorithm[ (TPC), with values between 0 (for galaxy) and 1 (for star)]. We refer the reader to <cit.> for details on the TPC star/galaxy separation method, which will be also used in the next section.§.§ The COMMODORE galaxy catalogue In this section we describe the identification of a galaxy catalogue taken from Gold SVA1, with the purpose of a proper study of galaxy formation and evolution, in general, and of GLF and GSMF in particular.One of the difficulties with the SVA1 Gold catalogue is the sky variation of galaxy depth, which not only can affect the results of galaxy evolution studies but also makes galaxy completeness characterisation challenging. For these reasons, we selected the COMMODORE (COnstantDepth Originated REgion) galaxy catalogue, whose selection is described below.§.§.§ Sample selection The main requirement we apply for the identification of this catalogue is to have a galaxy catalogue selected over a region of constant limiting magnitude (depth). Such a requirement can be met by using the depth maps available for the SVA1 Gold catalogue. By using the latter in the i band, we can identify sky pixels with measured depth higher than a minimum value (_ i=23 mag) and select only them for the galaxy catalogue construction. These pixels do not have to be contiguous, an ideal characteristic for minimising Large-Scale-Structure effects and ensuring statistical independence over the sky for studies like ours for which contiguity is not a necessary requirement. Once these pixels are identified, only galaxies with magnitudes brighter than _ i=23 mag are kept. This enables us to identify a catalogue over a sky region with approximately homogeneous depth. We exclude SN fields for ensuring homogeneity of observing conditions and usebecause it performs better thanin estimating galaxy total fluxes, while the latter is more indicated for colour estimation. Hence, aiming at combining the most reliable information about galaxy SED shape and about galaxy total flux, we rescale galaxymagnitudes by the flux ratio (i.e., magnitude difference) between the latter and the i-bandvalues (see ). These magnitudes (hereafter ) are then used for galaxy properties estimation, as described in Section <ref>. The galaxy apparent magnitude depth of this catalogue in the various bands is then assessed via number counts drop. The values estimated, respectively for g,r,i,z and Y bands, are: 23.40, 22.98, 23.00 (by construction), 22.26 and 22.06. In addition to the described selection, we also apply the following cuts: i) _ i<0.11 ii) 1≤_( g-r) ≤4 &1≤_( i-z) ≤4iii)16≤_ i≤ 23 & 16≤_ i≤ 27iv) <0.00023. Criterion i) is used to select a galaxy sample at 10σ level since the SVA1 Gold depth maps provide limiting magnitude values at this detection level, while criterion ii) is used to make sure to exclude objects with strong colours characteristics of diffraction artifacts. Criterion iii) is used to make sure the final catalogue contains only galaxies with reliable values of apparent magnitude and surface brightness, the latter referred to the galaxy brightest pixel as measured by 's parameter . Finally, criterion iv) is used to identify a galaxy catalogue at ∼ 99 per cent galaxy purity but still at >90 per cent completeness level, a condition not ensured by the use of the `modest classifier' only in the apparent magnitude range selected with criterion iii). The conservativeness of the chosen galaxy purity level aims at making sure that contamination by stars does not affect our analysis, especially in low-number-count regimes as in the bright/massive ends of high-z GLFs/GSMFs. The newly-obtained catalogue counts 3,711,833 galaxies over a sky area of ∼ 155sq.deg. We point out that using our Monte Carlo simulations (see Section <ref>) we obtain 100 more such catalogues on which we repeat the entire analysis (from galaxy properties estimation to measuring GLF and GSMF). Figure <ref> shows the distribution of the median photo-z derived by the Monte Carlo simulations. §.§.§ Galaxy completeness as function of observed quantities A fundamental property for a galaxy sample is represented by its characteristic completeness as a function of apparent integrated magnitude and surface brightness. This is indispensable for studies of galaxy evolution with cosmic time like the one presented here. We study galaxy completeness of the COMMODORE catalogue by investigating a few fields located within the deeper SN fields.The reason why this study focuses only in regions within the SN fields is that in SVA1 data there is no overlap between the Wide and Deep surveys. As a consequence we have to rely on our SN fields for a study of galaxy completeness. In order to do so, single-epoch images in the tiles listed in Table <ref> were used to obtain new coadded images.The selected tiles used for this new set of coadded images were selected so to have limiting magnitude ∼ twice as deep as the Wide survey standard depth and to be far from footprint edges.Two new sets of coadded images are obtained for our completeness study. The first set is created such that to obtain coadded images equivalent to those of the Wide survey, hence matching the total exposure time and the standard limiting magnitude characterising them. We will refer to this set of coadded images as `shallow coadds'. The second set of coadded images is instead obtained by coadding all the single-epoch images available in the selected tiles, resulting in coadded images significantly deeper than (∼ twice as deep as) those obtained with the shallow coadds. This second set of coadded images will be hereafter referred to as `deep coadds'. Galaxy catalogues for both the shallow and deep coadds are then built up as done with SVA1 catalogues and selected following the same approach used for the SVA1 Gold catalogue.In order to study galaxy completeness as a function of total apparent magnitude and surface brightness, the ratio of the number of detected galaxies in both the deep and shallow coadds to those detected in the deep ones are studied as a function of these two quantities within 0.5mag x 0.5mag 2-dimensional bins and considering the deep coadds as truth. The identification of those galaxies detected in both the coadds is carried out by sky cross-matching the two galaxy catalogues with an angular radius of 0.5 arcsec.Before doing this, the 10-σ limiting i-band magnitude of the shallow-coadds galaxy catalogue was determined to be _ i=23 and the shallow- and deep-coadds galaxy catalogues were respectively deprived of galaxies with _ i>23.0 and _ i>23.5. The magnitude cut carried out on the deep-coadds catalogue was made in order to avoid mismatch of brighter galaxies in the shallow-coadds catalogue with fainter ones in the deep-coadds one.The results of our study of galaxy completeness are visualised in Figure <ref>, showing the completeness map as a function of i-band apparent magnitude and surface brightness, obtained by using the galaxy catalogues just described.This map is then used to assign a completeness factor to each galaxy contained in the COMMODORE galaxy catalogue according to theiri-band values of apparent magnitude and surface brightness (in our case ofand ). The results of this process are visualised in Figure <ref>, where the completeness factor is plotted against i-band(in this band equivalent to themagnitude) andfor the entire COMMODORE galaxy catalogue. These completeness factors will be then used in our study of the GLF and GSMF, as explained in Section <ref> and are also used in the study of the environmental dependence of the GSMF by <cit.>.§ GALAXY PROPERTIES DETERMINATION In order to measure galaxy properties such as absolute magnitudes, stellar ages, stellar masses and k-correctionsste, we carry out a two-step analysis. The two steps consist of: i) fitting observed spectral energy distributions (SEDs) with theoretical ones; ii) using the resulting best fit models to calculate star-formation-history dependent properties. §.§ SED fitting For carrying out step i), we use an adapted version of HYPERZ <cit.> (hereafter referred to as HYPERZSPEC), which allows to keep redshifts fixed at the measured values (in our case at the one measured photometrically, see Section <ref>). For each galaxy, the fitting procedure is performed by fitting different theoretical SEDs to observed ones and evaluating for each of them the reduced χ^2 (χ^2_ r), which is used as figure of merit to identify the best-fitting one. Once the latter is found, all the characteristics of the stellar populations generating the chosen model (i.e., SFH, age and metallicity) are assigned to the fitted galaxy SED. The SED fitting code is implemented with theoretical templates based on the evolutionary population synthesis models by <cit.>. The template setup used here (the same as the one utilised in&and referred to as wide template setup) is made of 32 theoretical spectra covering a broad range of SFHs: i) SSP (simple stellar population, corresponding to a single star burst); ii) exponentially declining star formation rate (SFR, τ-model with τ=0.1, 0.3, and 1 Gyr); iii) truncated SFR (step-like star formation, i.e. constant for a time interval t=0.1, 0.3 and 1 Gyr since galaxy formation, null afterwards); iv) constant SFR. Metallicity varies among four values (1/5, 1/2, 1 and 2 Z_⊙), while 221 values of age out to 15Gyr are investigated. The templates are calculated for a Salpeter IMF <cit.>. To avoid unrealistic solutions, galaxies are also constrained to have stellar ages younger than the age of the Universe at their redshifts, within the used cosmological model. In addition, we apply a low-age cutoff at age<0.1Gyr, so to avoid fitting unrealistically young ages <cit.>. In order to avoid age-dust degeneracy effects (for instance seefor a review of this problem), we do not use reddening in the SED fitting procedure. This is because this degeneracy increases when reddening is included as a free parameter, favouring dusty solutions with unlikely too young ages, as shown by <cit.> in their study of simulated galaxies. In practice, the inclusion of reddening was shown to produce well-recovered SEDs but significantly underestimated stellar masses, especially for old galaxies which have experienced a recent, small star-formation burst. For each galaxy, the best-fitting model is then used to calculate the remaining properties (see Section <ref>).§.§ K correction, physical and detectability-related galaxy properties The second step for determining the remaining galaxy properties consists of measuring k-correction, absolute magnitudes, stellar mass and the maximum (z_ max) and minimum (z_ min) detectability redshifts (i.e., the maximum and minimum redshifts at which each galaxy is detectable given the characteristics of the survey). These quantities are all estimated using the galaxy best-fit SEDs.Note that we do not apply evolutionary corrections to the obtained absolute magnitudes or to the calculation of z_ max and z_ min (see further below). Stellar masses are calculated by re-normalising the best-fitting model template SED to the observed one and allowing for mass loss (see& ), according the prescriptions of <cit.>.The actual calculation is carried out using a routine developed by E. Daddi and C. Maraston and already used in used <cit.> and <cit.>.The calculation of z_ max and z_ min values depends on the depth (m_ u) of the survey, its lower apparent magnitude limit (m_ l), the lower (z_ l) and upper (z_u) redshift limits of the surveyand k-correction. Once the absolute magnitude M^j (j=g,r,i,z,Y) and stellar age of a given galaxy are determined, we calculate what would be the redshift values at which the apparent magnitude of such a galaxy, with the same properties, will equalm_ u andm_ l (the apparent magnitude limits of the survey).We always make sure that the value of z_ max is such that the age of the galaxy considered is lower than the age of the Universe at this redshift within the cosmological model used. If this is not the case, we then correct the value of z_ max downward by the minimum redshift difference needed for the galaxy age to equal that of the Universe.As all galaxy properties will depend on uncertainties on photometric redshifts and photometry, we use our Monte Carlo simulations described in Sections <ref> and <ref> to estimate their uncertainties. Figure <ref> shows the median distributions of i-band absolute magnitude, logarithm of stellar mass and stellar age values (left-hand panels) and their uncertainties (right-hand panels). The latter are defined as half the difference between the 84^ th and 16^ th percentiles of each galaxy property (absolute magnitude, stellar mass and age) distributions. The range and median uncertainties are respectively 2.6 and 0.2 mag for absolute magnitudes, 1.3 and 0.1 dex for stellar masses and 3.2 and 0.5 Gyr for stellar age. As the median uncertainty due to photometric redshifts uncertainty and photometry is an order of magnitude smaller than the property range for both absolute magnitudes and stellar masses, these properties can be safely studied. Note that we find the median uncertainty on the logarithm of stellar mass to be lower than that on absolute magnitude. This is in agreement with the finding of <cit.>, who found error bars due to photometric redshift uncertainties on absolute magnitude and the logarithm of stellar mass to be respectively ∼ 0.3mag and ∼ 0.1dex, showing that the effect of photometric redshift uncertainties is significantly larger for absolute magnitudes than for stellar masses.More details on the study of the dependence of property uncertainties with redshifts, also in connection with environment, can be found in the companion paper by <cit.>, which also uses the COMMODORE catalogue. §.§ Sample physical-property completeness as a function of redshift When studying the luminosity and the stellar mass functions of a galaxy sample, it is important to estimate the completeness in absolute magnitude and stellar mass characterising the sample as a function of redshift, in addition to that in observed quantities like flux and surface brightness (see Section <ref>).In order to do so, we estimate the higher and lower ≳ 90 per cent completeness limits in absolute magnitude (respectively M_ faint^ j and M_ bright^ j) and stellar mass (respectively M_ *, high and M_ *, low) of our sample as a function of redshift.Such limits are shown in Figure <ref> (left panel for i-band absolute magnitudes, right panel for stellar masses). We will use these limits to identify absolute-magnitude and stellar-mass intervals complete at ≳ 90 per cent level to be used for evolutionary studies (see Sections <ref> and<ref>) and to exclude incomplete z-M_ i andz-M_ * 2-dimensional bins when calculating GLFs and GSMFs in our redshift bins.Obviously, the completeness limits identified depend on the individual galaxy photometric redshifts. As with other properties, the variation of these limits due to photometric redshifts' uncertainties is estimated via our Monte Carlo simulations.In Figure <ref> we show one realisation of the estimate of galaxy properties for the COMMODORE sample together with absolute magnitude (left-hand panel) and stellar mass (right-hand panel) completenesses as function of redshift. The latter are both identified as shaded regions, representing their total variation when taking into account all the 100 Monte Carlo realisations of the COMMODORE sample, as we calculate such completenesses for each of them in order to be able to properly carry out our 100 measurements of GLF and GSMF.§ GALAXY LUMINOSITY AND STELLAR MASS FUNCTIONS §.§ The method To estimate DES GLF (GSMF), we use the classic Schmidt-Eales (1/V_ max, ) method, which consists of determining the largest volume accessible to each galaxy <cit.>, given the galaxy absolute magnitude, the apparent magnitude depth of the survey (m_ u), its lower apparent magnitude limit (m_ l) and the solid angle Ω corresponding to the analysed sky area. We now briefly explain the method used only for the case of the GLF. However, the GSMF can be derived in the same way, as the majority of the quantities needed for such a calculation are only flux dependent (e.g., V_ max).The number density of galaxies within a considered absolute magnitude range (M_ l<M<M_ u) can be written as: ∫_M_l^M_uϕ(M) dM = ∑_s=1^N_obs1/C_ sV_max(s),V_max(s)≡∫_Ω∫_z_min,s^z_max,sd^2 V/dΩ dzdz dΩ, where z_ max,s and z_ min,s are the upper and lower redshift limits within which a galaxy with absolute magnitude M_ s can be detected in the survey (see Section <ref>), V_ max(s) being the maximum co-moving volume accessible to it and C_ s is the galaxy completeness factor (see Section <ref>), which takes into account incompleteness due to missed detection as a function of surface brightness and apparent magnitude. Note also that one has to take also into account that z_ l≤ z_ min,s < z_ max,s≤ z_ u, z_ l and z_ u being the redshift limits of the survey (or the redshift slice used for the calculation). In this way we are able to sum over the individual corrected inverse accessible volumes for each magnitude bin and obtain an estimate of the galaxy volume number density for each of them. This process allows us to obtain a discrete estimate of the luminosity and the mass functions. However, if these functions are obtained in relatively small redshift intervals (z_ l<z<z_ u), one has to make sure to removeabsolute magnitude (stellar mass) incomplete bins, in order to avoid to underestimate the number density. Since we do estimate GLF/GSMF in bins of z, we make sure that this requirement is satisfied. We do this by using the functions plotted in Figure <ref> and calculating the absolute magnitude (M^ j_ bright and M^ j_ faint) and stellar mass (M_ *, high and M_ *, low) completeness limits at z_ u and z_ l. Hence we make sure that within the considered redshift slice (z_ l<z<z_ u) only absolute magnitude/mass bins whose edges fall within these completeness limits are selected for the GLF/GSMF calculation (e.g., ).Uncertainties on number densities are derived by taking into account both the contribution from shot noise and from uncertainties in photometry and photometric redshifts. The first contribution is taken care of by determining the 84.13 % confidence Poisson upper and lower limits according to the <cit.> recipe in presence of shot noise, so to properly account for the low-counts regime (bright/massive end of the GLF/GSMF). In fact, in first approximation, the GLF (GSMF) must follow a scaled Poisson distribution and the scaling factor within each magnitude bin can be determined, following the approach of <cit.>, by introducing the effective weight W_ eff W_eff=[ ∑_s1/(CV_max)_s^2]/ [ ∑_s1/(CV_max)_s], which allows to write the effective number (N_ eff) of galaxies as N_eff=[∑_s1/(CV_max)_s]/ {W_eff}. The second contribution is estimated by our Monte Carlo simulations which allow us to repeat the entire analysis (SED fitting, galaxy properties derivation and derivation of GLF and GSMF) 100 times. This enables us to have number densities distributions for each investigated luminosity (stellar mass) bin on which we can identify 1σ confidence intervals. The two uncertainty contributions are kept separate due to their asymmetry. § RESULTS In this section we describe the results obtained for both the GLF and the GSMF.We point out that, as a result of our 100 Monte Carlo realisations, when measuring the GLF and GSMF, in each absolute magnitude and stellar mass bin we will refer to the median value of the obtained number density distributions.§.§ GLFIn Figure <ref> we show the i-band GLF in 5 different Δ z=0.2-sized redshift bins in a range between 0 and 1 (see Table <ref>).As a density dip and upturn at the faint end are evident, a double Schechter function is fitted to the data obtained in the lowest z bin. This fit (green line in the figure) is then also plotted in the higher z bin panels as a reference for highlighting redshift evolution. Such double Schechter function is defined as: ϕ(M)dM=0.4ln(10) [(ϕ_1^∗ 10^0.4(α_1+1)(M^⋆-M) ) . +.(ϕ_2^∗ 10^0.4(α_2+1)(M^⋆-M) )] e^-10^0.4(M^∗-M) dM, where M^∗ is the characteristic luminosity, α_1 and α_2 are the slopes of the two Schechter functions and ϕ_1^∗ and ϕ_2^∗ are their normalizations. The best fit values obtained are listed in Table <ref>. Focusing on the lowest redshift bin, we notice the presence of a dip at -20 ≲ M_ i≲ -18 mag followed by a new steep rise at M_ i≳ -18 mag, displaying the double-Schechter-like shape seen already at low redshift in the literature (e.g., ). This result is reassuring, as we use photometric redshifts, while the majority of the works in the literature are based on spectroscopic redshifts. Focusing on the bright end of the GLF, we notice that (as expected) the brightest (M_ i≲ -23.5mag) and rarest galaxies make their entrance in the GLF at higher redshifts (z>0.2) only. This is probably due to lack of volume in the lowest redshift bin.§.§.§ Comparison with the literature In this section we compare our results with those found in studies in the literature carried out on spectroscopic data in the local Universe. This is because we aim at understanding the reliability of our method based on photometric redshifts with respect to those based on spectroscopic ones. In particular, in our comparisons with other measurements of GLF (and of GSMF, see Section <ref> ) in the literature, we consider the results obtained for the GAMA survey <cit.> particularly relevant, as their measurements were carried out on a sky area (143sq.deg) almost equivalent to the one used by us (∼ 155sq.deg).We show such comparison in Figure <ref>. In this figure, we compare our z<0.2 i-band GLF with those obtained by <cit.> using SDSS Data-Release-2 (DR2) data (over an area of ∼ 2221sq.deg) at z<0.05 and by <cit.> using GAMA data at z<0.06. In general we find good agreement with spectroscopic GLFs from the literature, despite some differences that are also present. Our and GAMA GLFs turn out to be similar, despite the different approaches used to estimate galaxy properties and for estimating the actual GLF (differently from us,use a density-corrected V_ max method). We do notice, however, that our GLF shifts to higher number densities at -22.5 ≲ M_ i≲-19mag and to lower ones at fainter magnitudes compared to the GAMA one. Both our and GAMA GLFs are similar to the SDSS one at M_ i≲-19mag. At fainter magnitudes though, they both show lower densities than those measured by <cit.>.<cit.> associate part of these discrepancies at M_ i≳ -18mag (i.e., L_ i≲ 10^9L_⊙) to the particular way of estimating distances, i.e. whether using distances corrected for large-scale bulk flows using redshifts in the cosmic microwave background frame rather than standard heliocentric-redshift based distances or those referred to the Local Group (LG) frame. <cit.> did find the discrepancies with the results of <cit.> to decrease when using the SDSS DR2 catalogue with flow-corrected distances as done with GAMA (see Figure 9b in ), however such discrepancies remained significant. Part of these discrepancies might be due to the method used to correct for surface-brightness incompleteness, which in <cit.> is based on simulations, while in our case is based on data by relying on deeper DES fields. However, the source of discrepancy between the faint end of Blanton et al.'s GLF and those of DES and GAMA remains unclear to us and it would need to be further investigated.We notice that <cit.> estimated the GAMA GLF at z<0.1 also using Loveday et al.'s galaxy sample and method (step-wise maximum likelihood) but k-correcting galaxies at z=0 ( k-corrected galaxies to z=0.1 as they shifted their i-band filter to such z) and found good agreement with their estimate of the GAMA GLF, which is in turn in good agreement with the DES one.§.§.§ Evolution of GLF parameters and of luminosity density We identify an absolute magnitude range complete at ≳ 90 per cent out to z=1. We exclude from this exercise the lowest redshift bin (at z<0.2) so that we can explore the evolution with cosmic time also of the brightest galaxies (M_ i≲ -23.5mag), which are lacking at z<0.2. By doing so, such ≳ 90 per cent complete luminosity range is identified as -26<M_ i<-21.4mag over 0.2<z<1. As this luminosity range is relatively bright for being able to properly explore the evolution of the faint-end of the GLF, we then fit the GLF measured over this M_ i-range for the four z bins at z>0.2 with a single Schechter function and fix the faint-end slope at α=-1.2 (fits results are shown in Figure <ref>). In this way we are able to homogeneously investigate the evolution of M^∗ (left-hand panel of Fig. <ref>) and ϕ^∗ (middle panel of Fig. <ref>) with redshift. The chosen value of α is the one best fitting the GLFs of all the explored z bins. However, using any value between α=-1 and α=-1.2 (a sensible value range for the global population of galaxies, see, e.g., ) does not change the results shown in Figure <ref>.As expected, we find significant evolution (at >3σ) of both M^∗ and ϕ^∗ with redshift, with M^∗ and ϕ^∗ respectively getting brighter (by ∼0.9mag from z=0.2 to z=1) and decreasing (by a factor ∼4 from z=0.2 to z=1) with z. We also study the evolution of the i-band luminosity density ρ_ L with redshift, which we show in the right-hand panel of Figure <ref>. We estimate ρ_ L via two methods: i)by using the Schechter function fit, i.e. ρ_ L=∫_ L_ low^ L_ up L ϕ(L) dL=ϕ^∗L^∗[Γ(α+2,L_ low/L^∗)-Γ(α+2,L_ up/L^∗)], where the functions Γ(α+2,L/L^∗) are upper incomplete gamma functions; ii) by summing over the binned GLF, i.e. ρ_ L=∑^ j=n_ j=1 L_ jϕ(L_ j)Δ L_ j, where j is the luminosity bin number. The advantage of method ii) is its independence from the function used for fitting the binned data. However, both methods give consistent results. We find no statistically significant evolution of ρ_ L between z=0.2 and z=1. For reference, in the right-hand panel of Figure <ref> we also show the value of ρ_ L obtained for the GLF in our lowest redshift bin z<0.2 calculated with both of the methods described [method i) was carried out using the double-Schechter fit reported in Table <ref>]. However, we remind the reader that the values of ρ_ L shown at z<0.2 are affected by incompleteness due to lack of galaxies with M_ i≲-23.5 mag.We also compare our results on ρ_ L with those available in the literature. In addition to examining in contrast with the density measurements quoted in the studies we are comparing with, when GLFs are available, we also calculate such values ourselves in the same way as done with our data. This is done in order to be as conservative as possible in our comparison, using quantities measured as similarly as possible. Note that the same approach will be used in Sections <ref> and <ref> for GSMF, when studying the evolution of stellar mass (ρ_ Mstar) and number (ρ_ N) spatial densities with cosmic time in comparison with the literature.In Figure <ref> we compare our ρ_ L values with those derived for SDSS by <cit.> and for GAMA by <cit.> and by us via <cit.>'s binned GLF. Note thatthe ρ_ L values quoted by Loveday et al. and Bell et al. were measured over the entire GLF and that Loveday at al.'s measures refer to the i-band filter shifted to z=0.1. However, there is consistency within the uncertainties for the majority of the data points, excluding the two Loveday et al.'s measures at z∼ 0.275 and z∼ 0.35. Despite this, it is evident that Loveday et al.'s values are systematically higher than ours at comparable redshifts, which could be due to the different luminosity ranges used for estimating ρ_ L.Although we investigate luminosity density evolution only at z>0.2 and measure no such evolution out to z=1, when considering also our z∼0.1 value of ρ_ L (which is reported as reference but it has to be considered a lower limit as it is partially affected by incompleteness), we do see similar features between <cit.> 's measurements and ours, i.e., an increase ofρ_ L out to z∼0.1/0.2 followed by an average constancy of this quantity. Having said that, Loveday et al.'s values at z≳0.2 do seem quite sparse. The luminosity density in bands other than i (to our knowledge the evolution of ρ_ L in this band has been poorly studied) has already been seen to be approximately constant out to z∼ 1 in the literature (e.g., seefor measurements in rest-frame J and H obtained using IRAC channels). However, other studies also report an increase of the B-band ρ_ L with redshift (e.g., ). As the luminosity density is strictly bound to the dominant stellar population, which can vary at different wavebands, a dependence of the evolution of ρ_ L with cosmic time on waveband is plausible. §.§ GSMF We now describe the results of our analysis of the GSMF. As done with the GLF, in Figure <ref> we display the median GSMF for the galaxies in our catalogue (see Table <ref>), using the same redshift binning scheme and symbols as in Figure <ref>. Again, since a density dip and an upturn at the low-mass end are evident in the lowest redshift bin, we fit a double Schechter function to the data points obtained at z<0.2. Such function is defined as: ϕ(M)dM= e^-M/M^∗ [ϕ_1^∗ (M/M^⋆ )^α_1 + ϕ_2^∗ (M/M^⋆ )^α_2 ]dM/M^∗, whose parameters are the same as those described for equation <ref>. The best fitting values obtained for the parameters of such fitting are listed in Table <ref>. As for the GLF, a dip is noticeable at 8.5≲log(M/M_⊙)≲ 10 in the lowest-z-bin GSMF, followed by a steep rise at lower masses, displaying the classic double-Schechter-like shape seen at low z (e.g., ).Also here the most massive (log(M/M_⊙)≳11.3) and rarest galaxies enter into the GSMF only at z>0.2, again probably due to too small a volume to be able to survey these galaxies at lower redshifts.§.§.§ Comparison with the literature As previously done for the GLF, in this section we compare our results with those found in studies in the literature carried out on spectroscopic data (e.g., ), in order to assess the reliability of our method based on photometric redshifts with respect to those based on spectroscopic ones. Low Redshift In Figure <ref> we compare the GSMF obtained in our lowest redshift bin with that obtained by <cit.> at z<0.06 for the GAMA survey and with several other spectroscopic surveys, i.e. SDSS/2MASS (, ∼ 410sq.deg), SDSS (, 4681sq.deg) and SDSS/GALEX (, ∼ 2505sq.deg). As with the GLF, the comparison with GAMA GSMF shows an overall agreement, with DES SGMF showing somewhat larger densities at log(M/M_⊙)≳ 9 and vice versa at lower masses. Both the GAMA and the DES GSMFs show a double-Schechter-like shape. When comparing individually the double-Schechter function best-fitting parameters we obtain for the DES GSMF (see Table <ref>) with those obtained by <cit.> for GAMA, we find statistical consistency at levels between 1 and 3σ. At log(M/M_⊙)≳ 11, we estimate our number density values to be affected by stellar mass incompleteness. Furthermore, differently from the GAMA GSMF, we do not find any galaxy with log(M/M_⊙)≳ 11.5. We do note though, that the high-mass ends of the DES and GAMA GSMFs show good agreement and that the density values at log(M/M_⊙)≳ 11.5 in the latter are derived on galaxy number counts varying between 1 and 3 galaxies. When comparing our GSMF with those of the spectroscopic surveys mentioned above and shown in Figure <ref>, we find a general agreement, which is reassuring, given the variation of assumed IMF, stellar mass estimation methods [e.g., SED fitting and from M_*/L vs. colour (such as g-r) relations], ways of estimating galaxy integrated magnitudes (e.g., model magnitudes and Sersic surface-brightness fitting) and the fact that the DES GSMF is based on photometric redshifts rather than spectroscopic ones. However, some discrepancies, especially in specific stellar mass regimes, are also evident. For instance, the SDSS-2MASS GSMF <cit.> shows density values almost always higher than those obtained by the remaining surveys, especially at log(M/M_⊙)≲ 9.5, as also discussed by <cit.>.In addition, at log(M/M_⊙)≳ 10.8 our GSMF agrees better with the cmodel-magnitudes-based GSMF by <cit.> than with the Sersic-fit-magnitudes-based one by <cit.>. This is expected as we do not use Sersic fitting to estimate our galaxy magnitudes but use magnitudes which are more similar to the SDSS cmodel magnitudes. In fact, <cit.> showed that by using cmodel-like magnitudes leads to underestimating galaxy total fluxes and stellar masses, especially for massive galaxies with de-Vaucouleurs-like surface brightness profiles. This has the natural consequence of underestimating the spatial densities of such massive galaxies. Another important difference is that the DES GSMF does not show the presence of very massive galaxies (log(M/M_⊙)≳ 11.5). This is probably due to the significantly smaller area probed by us compared to the above-mentioned surveys other than GAMA. Intermediate/high Redshift We now focus on our results obtained in the redshift bins at z>0.2. A comparison with studies atz>0.2 is possible as the GSMF is probed out to z∼1 (when only considering spectroscopic surveys) in the literature.In Figure <ref> we compare the DES GSMF with that obtained by <cit.> at z>0.2. The latter measured the GSMF at 0.2<z<1 by using an i<23mag flux-limited sample of ∼ 40,000 galaxies from the PRism MUlti-object Survey (PRIMUS, ), over five fields (COSMOS, XMM-SXDS, XMM-CFHTLS, CDFS and ELAIS-S1) totaling ∼ 5.5sq. deg of sky area. The comparison with the results obtained with the PRIMUS galaxy catalogue is particularly useful, as this catalogue was selected using the same band utilised for the DES COMMODORE catalogue and because both these catalogues share the same depth. As shown in this figure, DES and the PRIMUS GSMFs at z<0.6 are consistent with each other. The agreement between these GSMFs is similar at z>0.6, but there are some discrepancies at log(M/M_⊙)≲ 11, where PRIMUS GSMFs show higher number densities. In particular, the latter is affected by incompleteness at log(M/M_⊙)≲ 10.5 (densities at such mass regimes are not publicly available so are not plotted) and seem to lack galaxies with log(M/M_⊙)≳ 11.8 (this at all redshifts), which are instead present in the DES GSMFs. As the COMMODORE and the PRIMUS galaxy catalogues share the same depth, these differences could be due to cosmic variance (as also discussed bythemselves). In fact, <cit.> showed that despite constructing an area-weighted average of all the 5, ≲1.5 sq. deg fields used in their study, the effect of cosmic variance was still significant, especially at z>0.6. In fact they found that large-scale overdensities still affect galaxy number densities at z>0.6 and at log(M/M_⊙)≳ 10.5 (see Figures 6-9 in ), i.e.exactly over the redshift and stellar mass regimes where we find the largest discrepancies.§.§.§ Evolution of GSMF parameters and of stellar mass density We now repeat for the GSMF the same exercise carried out for the GLF, now studying the evolution of M^∗, ϕ^∗ and ρ(M_ star) with redshift, within a stellar mass range complete at ≳ 90 per cent level, i.e. 10.2<log(M/M_⊙)<12. We show the results of this analysis in Figures <ref> and <ref> and find significant evolution at ≳ 3σ level of all these three quantities. In particular, log (M^∗) (Figure <ref>, left-hand panel) increases by ∼0.2 dex from z∼ 0.2 to z∼ 1, while both ϕ^∗ and log[ρ(M_ star)] (Figure <ref>, centre and right-hand panels) decrease with redshift, respectively by a factor ∼4 and ∼ 1.05 from z=0.2 to z=1. Right-hand panel of Figure <ref> also shows that there is an overall agreement between our values of log[ρ(M_ star)] and those based on the literature's studies we compare with <cit.>, especially when we are able to re-calculate the latter in the same way as done for ours. Some more significant discrepancies appear at z≳0.7. <cit.>'s measurement at z∼0.7 is significantly higher than ours (but this is known to be the z bin most affected by cosmic variance), while the one at z∼0.9 is partly affected by incompleteness, so despite consistent with ours, it has to be considered as a lower limit. Overall, taking Moustakas et al.'s values alone at z<0.8, would suggest constancy of ρ_ Mstar with redshift. However, as also mentioned for their GSMFs at z>0.6 in Figure <ref>, this could be due to large-scale structure effects due to the relatively small-area fields (≲ 1.5 sq. deg each) over which their measurements were performed.At low redshift, we notice that the estimate quoted by <cit.> (blue open diamond) is significantly higher. However this is probably due to its derivation from the GSMF fit over the entire spanned stellar mass values. In fact, when we use their binned GSMF and recalculate ρ_ Mstar in the appropriate ≳ 90-per-cent complete stellar mass range used for our measurements (see blue full diamond), their result becomes more similar to the rest of the values. Notice also that the value measured on the cmodel-based GSMF by <cit.> is closer to the other values than the one derived from the Sersic-fit-based GSMF by <cit.>. This is expected as all the GSMFs studied here were not based on the use of Sersic fitting for measuring integrated magnitudes. Overall, considering our data points and those of the literature (and considering also the possibility that the high-z values ofare affected by large-scale structure effects) a picture which displays ρ_ Mstar decreasing with redshift at 0.2<z<1 seems plausible. Such trend was also seen in several other studies in the literature, e.g. <cit.>. However constancy of ρ_ Mstar with z out to z=1 was also reported, for instance, by <cit.> and <cit.>.Our findings of a decreasing stellar mass density and an approximately constant luminosity density with redshift(respectively right-hand panels of Figures <ref> and <ref>) can be explained by the fact that while at higher redshift there is less stellar mass available in galaxies, their luminosities do not change substantially because of their stellar populations getting younger and brighter, as one would expect.§.§.§ Galaxy mass build up In order to shed some light on the evolution with cosmic time of galaxy spatial number density as a function of stellar mass, hence on how galaxies build their masses over cosmic time, we analyse and compare among them the cumulative distribution functions obtained in our five redshift bins. In Figure <ref> we show such cumulative mass functions (derived by summing densities starting from the massive end) in our five redshift bins (left-hand panel) and only in the four higher-z ones (right-hand panel). In the former case, we focus on stellar masses lower than log(M/M_⊙)=11, as at larger masses our lowest redshift GSMF is affected by stellar mass incompleteness. In the latter case instead, we extend this mass range out to log(M/M_⊙)=12. In both cases we notice that number densities values of lower-z cumulative functions are always higher than or comparable to (depending on the mass bin considered) those of higher-z ones. Such evolution is indeed mass dependent, as it is more pronounced at low masses, while there is virtually no evolution at the high masses. Such behaviour can be seen also in Figure <ref>, where we plot the total number densities obtained in four different mass ranges as a function of redshift, similarly to what done by <cit.>. We restrict ourselves to 10.2<log(M/M_⊙)<12, a mass range which is ≳ 90 per cent complete at 0.2<z<1. As a consequence, the points at z<0.2 in Figure <ref> have to be considered as lower limits. In particular, the number density of galaxies such that log(M/M_⊙)> 11.55 is consistent with no evoultion since z=1 at ≲ 3 σ level. From Figure <ref> the downsizing pattern is clearly visible, in agreement with the results by <cit.>. In this same figure, we also show additional points measured from <cit.> (for GAMA) and <cit.> (for PRIMUS). We do find agreement with the literature, especially for the highest mass bins (log(M/M_⊙)≳ 11.1). However, some differences are seen in bins with lower mass limit log(M/M_⊙)<11.1, where PRIMUS values at z>0.6 suggest constancy of spatial number density with redshift rather than a decreasing trend as indicated by our results. This issue, though, can be again the results of large-scale-structure effects. Overall we do find evidence for mass-dependent evolution of galaxy number spatial density as function of redshift, with densities of more massive galaxies remaining about constant at 0<z<1 or evolving less than densities of less massive galaxies. Similar results were already found in other studies (e.g., ), but never over such a large area and in combination with the use of photometric redshifts. However, results claiming a hierarchical-like mass-dependent evolution of galaxy number spatial densities were also found in other studies in the literature (see, for instance, ). We will further discuss this issue in Section <ref>.§ DISCUSSION Studying with precision and consistency the evolution of the GLF and the GSMF over a wide redshift range and over luminosity and stellar mass ranges accessible at all studied redshifts is a difficult task to carry out. This is because, in order to correctly identify or dismiss evolutionary trends able to shed light on galaxy formation (e.g., hierarchical or anti-hierarchical), we need to be able to access large cosmic volumes (i.e., via deep surveys over large sky areas) and measure distances with high precision (ideally via spectroscopy). This is because such approach allows to identify galaxy samples large enough to study with high precision even the extreme parts of the GLF and the GSMF. However, spectroscopy is too time consuming, hence we need to resort to photometric redshifts. In this work we investigate the GLF and the GSMFout to z=1 in comparison with spectroscopic measurements taken from the literature as we aim at validating our analysis methodology, in addition to inferring precious knowledge about galaxy formation and evolution. As our results show general agreement with similar analysis carried out with spectroscopic data over similar or narrower redshift ranges and out to similar or shallower flux depths, we can apply our analysis methodology to the full DES data-set (we currently use only ∼ 3 per cent of DES final sky coverage), once this becomes available. DES i-band GLF at z<0.2 shows the characteristic double-Schechter-function shape seen in several other studies of the GLF (e.g.,and ). Such shape, though, cannot be probed also at higher z as one would need data deeper than the ones currently used. However, by identifying a luminosity range complete at ≳ 90 per cent level at 0.2<z<1, we were able to study the evolution of the DES GLF homogeneously within such redshift range. Our results show brightening (of ∼ 0.9mag out to z=1) of M^*_ i, decrement (by a factor ∼ 4 out to z=1) of ϕ^* and ∼ constancy of the luminosity density ρ_ L with redshift. Such trends at 0≲ z≲ 1 were already seen in several studies of the GLF in the literature, also based on photometric redshifts (e.g, ). However, some other studies also found different trends. For instance, <cit.> found an increase of ρ_ L with redshift out to z=1. A direct comparison is particularly difficult for us as we study the i-band GLF, while all the studies that investigate the evolution of the GLF out to z∼1 use different wavebands. In fact, to our knowledge, ours is the first study of the evolution of the i-band GLF out to z>0.5 ever performed. When focusing on the DES GSMF at z<0.2, we can make the same considerations done for the local DES GLF, as a double-Schechter-function shape is also seen, in agreement with several studies in the literature (e.g., ). At higher z, we repeated the exercise carried out for the GLF, exploring the evolution of the GSMF at 0.2<z<1 of a ≳ 90-per-cent stellar-mass complete galaxy sample. Our results showed small increase(of ∼ 0.2dex out to z=1) of log M^* and decrement of both ϕ^* (by a factor ∼4 out to z=1) and stellar-mass density ρ_ Mstar (by a factor ∼ 1.05 out to z=1) with redshift. As the decrement of ϕ^* is more significant than the increment of log (M^*) with z, the evolution of ρ_ Mstar seems to be mainly due to the former. These results are consistent with several studies in the literature (e.g., ), however, some discrepancies can also be found. Note that we now extend the comparison also to studies based on photometric redshifts. For instance, <cit.> found constancy of both log M^* and ρ_ Mstar out to z∼1. Despite this, maybe the most important fact is that the majority of the studies do report results consistent with a mass-dependent evolution of the GSMF of the global galaxy population. Some claimed such dependence to be bottom-up (e.g., ), while others claimed it to be top-down (e.g., ). We do find evidence for a top-down galaxy formation scenario.In fact we do find that the densities of galaxies with log (M/M_⊙)≳11.1 remain constant over the analysed redshift range, while at smaller masses they decrease with redshift and they do it more the lower the masses included in the analysed mass bins (see Figure <ref>). This evidence is consistent with the results of <cit.> and of <cit.>, the latter when accounting also for the fact that some of their results were shown to be affected by cosmic variance. According to our analysis, most massive galaxies [log (M/M_⊙)≳11.1] appear to be already in place by z=1, while the mass build-up process is still ongoing from z=1 to z=0 for less massive galaxies.However, despite our findings discussed so far can point at a top-down galaxy formation scenario, we do also have to consider that in our work we have neglected some sources of uncertainties (described previously). Hence our analysis should be repeated taking all uncertainty sources into account and verify the robustness of our results. As an explanatory example of this issue, we note that <cit.> found a downsizing-like mass-dependent evolution of the GSMF when considering only random uncertainties and that such mass-dependent evolution was no longer robust when taking into account also systematic uncertainties due to SED-modelling assumptions. In addition, a more thorough study of the impact of systematics (e.g. seeing, airmass, sky brightness) on our measurements should be carried out as well, also using the aid of simulations <cit.>. Furthermore the effects ofphoto-z pdf uncertainty and estimation methods should also be studied in detail.Given that the ≳ 90-per-cent complete luminosity (-26<M_ i<-21.4mag) and stellar mass (10.2<log(M/M_⊙)<12) ranges used to explore the evolution of the GLF and GSMF at 0.2<z<1 do not allow us to study α in a homogeneous way over this redshift range, it is difficult to make reliable considerations on the evolution of the shapes of these functions with z. However, despite the fact that we fixed the value of α when fitting the GLFs and GSMFs at z>0.2, we have explored the effect of using different values, finding no major changes to the trends with z identified for M^*,ϕ^*, ρ_ L/Mstar, suggesting that, within the explored luminosity and stellar mass ranges, the effect of α is small (justifying also a posteriori our choice of fixing its value) and so that the shapes of the GLF and GSMF are not expected to change from z=1 to z=0.2 for the mass range studied here.This would suggest that the physical processes involved in the galaxy stellar-mass build up do not significantly change the shape of the GSMF since z=1. Hence one could also infer that the change in the GLF M^* may be purely due to stellar evolution, with galaxy stellar populations becoming younger and so brighter at higher z.§CONCLUSIONS We study the GLF and GSMF of galaxies within the DES COMMODORE catalogue out to z=1. This galaxy catalogue was selected within the DES SVA1 data, contains ∼ 4× 10^6 galaxies at 0<z≲1.3, covers ∼ 155sq. deg of sky area (only ∼ 3 per cent of the final DES footprint) and is characterised by an i-band depth of i=23mag. Such characteristics are unprecedented for galaxy catalogues and they enable us to study the evolution of GLF and GSMF between z=1 and z=0 homogeneously with the same statistically-rich data set and free of cosmic variance effects. This is the first time that such a statistically rich and deep galaxy catalogue, covering an area >20sq. deg, has ever been used for a comprehensive study of the evolution of GLF and GSMF at 0<z<1. All the previous studies were either based on deep pencil-beam surveys or large but relatively shallower surveys. Since we utilise photometric redshift pdfs, the aim of this paper is mainly testing our method in comparison with results from similar studies of GLF and GSMF based on spectroscopic redshifts, in addition to studying galaxy formation and evolution since z=1.We investigate the evolution of i-band GLF and GSMF, their Schechter-fitting parameters and of luminosity, stellar mass and galaxy number densities with cosmic time, attempting to shed some light on how galaxies build up their mass over time (hierarchically or anti-hierarchically?). We first identify the COMMODORE catalogue, then study galaxy completeness as function of surface brightness and apparent magnitude, estimate galaxy physical and detectability-related properties and finally estimate and study GLF and GSMF in five redshift bins. Our main results are as follows:* Our low-z estimates of GLF and GSMF show overall agreement with those obtained for GAMA. Our densities are somewhat higher at M_ i≲ -20mag/log(M/M_⊙)≳9 then those measured for GAMA and vice versa at fainter magnitudes/lower masses, but the shapes of DES and GAMA functions are similar. They both show a double-Schechter function shape, in agreement with what is found in the literature. When taken individually, all best-fitting parameters of DES and GAMA GSMFs are consistent within 1-3σ, depending on the parameter considered. This result is particularly reassuring as GAMA functions are based on spectroscopic redshifts and are measured over an area very similar to the one used by us. However, some discrepancies are seen when comparing also with other functions in the literature. In particular, DES GLF cannot reproduce the significant excess of faint galaxies seen by <cit.>. This could be due to the different methods for correcting for surface-brightness incompleteness. The discrepancies seen in the GSMF, are mainly due to lack of area for identifying very massive galaxies, different magnitude estimators and possibly to different ways of estimating stellar masses.* At higher redshift, there is no available study, to our knowledge, investigating the i-band GLF. However, we are able to compare DES GSMF with that measured for the PRIMUS survey <cit.>, finding very good agreement out to z=0.6. At higher redshifts, the effects of cosmic variance in the PRIMUS GSMFs (especially at z∼0.7) are significant and the agreement with the DES one degrades towards low masses. Overall, however, we consider our results promising and the agreement acceptable.* We investigate the evolution of DES GLF and GSMF at 0.2<z<1 by identifying luminosity and stellar mass ranges complete at ≳ 90 per cent level. Such ranges, however, do not allow to study the faint/low-mass end slope α, which we keep fixed at α=-1.2. For the GLF, we find M^* brightening and ϕ^* decreasing with z. Such effects approximately balance each other, producing a constant luminosity density ρ_ L with z (ρ_ L∝ (1+z)^-0.12±0.11). With regards to the GSMF,M^* is found to slightly increase with z, while ϕ^* to significantly decrease with z, resulting in a mildly decreasing stellar-mass density ρ_ Mstar with z (ρ_ Mstar∝ (1+z)^-0.5±0.1). These results suggest that, within the ≳ 90-per-cent complete stellar-mass range investigated, the physical processes regulating galaxy mass build-up do not modify the shape of the GSMF between z=1 and z=0.2. Finally, by investigating the galaxy number density ρ_ N in different mass intervals, we do find evidence for mass-dependent evolution of the DES GSMF, following a downsizing pattern. In particular, the number densities of galaxies with log (M/M_⊙>11.1) are found to be about constant from z∼1 to z∼0.2, while when including smaller masses such densities decrease with z and they do it more the smaller the masses. Our measurements are directly compared with those from PRIMUS and, despite some variation of the latter due to cosmic variance, are found to be in agreement.* Given our results and how our measurements compare with those based on spectroscopic data, we consider our method for studying GLF and GSMF based on photometric redshift satisfactory. We have only used ∼ 3 per cent of the final DES footprint and we plan to build on this study to take full advantage of DES full area when the full data set becomes available. In addition, we also plan to test our analysis on simulations tailored on DES and also to compare our observation-based results with those we obtained by analysing simulated galaxies in the same way. This will help us not only with assessing the robustness of our analysis method, but also with assessing how well theoretical models can reproduce observed GLFs and GSMFs. § ACKNOWLEDGMENTSThis paper has gone through internal review by the DES collaboration. DC thanks Maurizio Paolillo, Ivan Baldry, Guido Walter Pettinari, Lado Samushia, Joanne D. Cohn, Douglas L. Tucker, Chiara Spiniello and Maurizio Salaris for valuable scientific conversations and Gary Burton and Edd Edmondson respectively for providing help with code parallelisation and with IT-related matters.DC particularly thanks the following people for their outstanding moral support and the help provided throughout the past few years, without which this work would have not been possible: Angela Del Gaudio, Spartaco Capozzi, Alessandro Capozzi, Marcello Capozzi, Claudio Gino Capozzi, Margherita Lanzi, Davide Bianchi, Nora Siklodi, Matthew Withers, Hedda Gressel, Rob Baker, Robert Fairall, Milan Kreuschitz Markovič, Dida Markovič, Susana Sampaio Dias, Olivia Umurerwa Rutazibwa, James Dennis, Miguel Marraco Eibar, Jan Guthrie, Miguel Fernández López, Alex Panayides, Karen Claeys, Ambra Sottile, Tiziana La Piana, Emma Vittorio, Claudio Nicolò, Francesco Pace, Héctor Gil Marín, Claire Le Cras, Xan Morice-Atkinson, David Wilkinson, Jojo Pratt, Leonidas Christodoulou, Guido Walter Pettinari, Lucas Lombriser, Arna Karick, Spencer Craig, Tesla Jeltema, William Wester, Joshua Frieman, Rae Bull, Maurizio Paolillo, Ivan Baldry, Chris Collins, Giuseppe Longo, Alkistis Pourtsidou, Ben Bose, Alicia Bueno Belloso, Jennifer Pollack, Rossana Ruggeri, Deike Striez, Monica Rizzo, Magdalena Moszyńska, the staff from “WorkPlaceWellness” and the “Talking Change” NHS trust in Portsmouth.Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain,the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for SupercomputingApplications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago,the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico andthe Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas,Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh,the Eidgenössische Technische Hochschule (ETH) Zürich,Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC),the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe,the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth,SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium.The DES data management system is supported by the National Science Foundation under Grant Number AST-1138766. The DES participants from Spanish institutions are partially supported by MINECO under grants AYA2015-71825, ESP2015-88861, FPA2015-68048, SEV-2012-0234, SEV-2012-0249, and MDM-2015-0509, some of which include ERDF funds from the European Union. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.We are grateful for the extraordinary contributions of our CTIO colleagues and the DECam Construction, Commissioning and Science Verification teams in achieving the excellent instrument and telescope conditions that have made this work possible.The success of this project alsorelies critically on the expertise and dedication of the DES Data Management group.The STILTS and TOPCAT softwares by <cit.> were significantly used in the making of this work. mn2e§ AFFILIATIONS^1Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK^2SEPnet, South East Physics Network, (www.sepnet.ac.uk)^3Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, CA 94305, USA^4SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA^5Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas (CIEMAT), Madrid, Spain ^6Wisconsin IceCube Particle Astrophysics Center (WIPAC), Madison, WI 53703, USA^7Department of Physics, University of WisconsinMadison, Madison, WI 53706, USA^8Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, IL 61801, USA^9National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA^10Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA^11Aix Marseille Université, CNRS, LAM (Laboratoire D'Astrophysique de Marseille) UMR 7326, 13388, Marseille, France^12ESA/ESTEC, Noordwijk, The Netherlands^13Observatório Nacional Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil^14Laboratório Interinstitucional de e-Astronomia-LIneA, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil^15CNRS, UMR 7095, Institut d'Astrophysique de Paris, F-75014, Paris, France^16Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK^17Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut d'Astrophysique de Paris, F-75014, Paris, France^18Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA^19Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK^20Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK^21George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA^22Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa^23Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain^24Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA^25Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India^26Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA^27Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA^28Institut de Ciències de l'Espai, IEEC-CSIC, Campus UAB, Carrer de Can Magrans, s/n,08193 Bellaterra, Barcelona, Spain^29Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA^30Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain^31Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland^32Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195, USA^33Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile^34Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA^35Australian Astronomical Observatory, North Ryde, NSW 2113, Australia^36Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA^37Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil^38Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, OH 43210, USA^39Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA^40Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain^41Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA^42Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK^43School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, UK^44Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil^45Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 § TABLES§.§ Supporting InformationAdditional Supporting Information can be found in the online version of this article: Table <ref>: The binned GLFs obtained for the full galaxy sample in the five z bins studied in this paper and plotted in Figure <ref>. X=M_ i (mag) and Y=ϕ(M_ i) ( n Mpc^-3)/10^-4.Table <ref>: The binned GSMFs obtained for the full galaxy sample in the five z bins studied in this paper and plotted in Figure <ref>. X=log (M/M_⊙) and Y=ϕ[log (M/M_⊙)] ( n Mpc^-3)/10^-4. | http://arxiv.org/abs/1707.09066v1 | {
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"title": "Evolution of Galaxy Luminosity and Stellar-Mass Functions since $z=1$ with the Dark Energy Survey Science Verification Data"
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[][email protected][][email protected][][email protected] Universität München, Department Chemie, Lichtenbergstrasse 4,85747 Garching, GermanyWe study the tomography of multispin quantum states in the context of finite-dimensional Wigner representations. An arbitrary operator can be completely characterized and visualized using multiple shapes assembled from linear combinations of spherical harmonics [A. Garon, R. Zeier, and S. J. Glaser, Phys. Rev. A 91, 042122 (2015)]. We develop a general methodology to experimentally recoverthese shapes by measuring expectation values of rotatedaxial spherical tensor operators and provide an interpretation in terms of fictitious multipole potentials. Our approach is experimentally demonstratedfor quantum systems consisting of up to three spinsusing nuclear magnetic resonance spectroscopy. Wigner tomography of multispin quantum states Steffen J. Glaser January 15, 2018 ============================================= § INTRODUCTION Optical homodyne tomography can be applied to experimentally measure the quantum state of light <cit.>. One thereby recoversan infinite-dimensional Wigner function <cit.> as a classically motivated phase-space representation,providing a useful tool for the characterization and visualization ofquantum-optical systems <cit.>. This results in an advantageous dualism between measurement scheme and phase-space representation, which we would like to transfer to the case offinite-dimensional, coupled spin systems.One important representation of finite-dimensional quantum systemsrelies on discrete Wigner functions<cit.>. But we will restrict ourselves to continuous representations in order to naturally reflect the inherent rotational symmetries of spins. Individual spins are faithfully described by their magnetization vector (or Bloch vector), which, however, neglects relevant parts of the full density matrix in the case of multiple, coupled spins. These missing parts include zero- and multiple-quantum or antiphase coherenceas well as spin alignment <cit.>, which are partially characterized by visual approaches basedon single-transition operators <cit.>.We will follow the general strategy of Stratonovich <cit.> which specifies criteria for the definition ofcontinuous Wigner functions for finite-dimensional quantum systems.The case of single spins is widely studied in the literature <cit.>, and visualizations for multiple spins have been consideredin <cit.> with various degrees of generality. However, until very recently, it was not clear <cit.> if a general Wigner representation also exists for arbitrary, coupled spin systems, and even the case of three coupled spins 1/2 was open.Fortunately, a general Wigner representation forcharacterizing and visualizing arbitrary coupled spin systems has been developed in <cit.>. It is based on mapping arbitrary operators to a set of spherical functions which are denoted as droplets, while preserving crucial features of the quantum system. The characteristic shapes of these droplets can be interpreted as the result of an abstract mapping, but we also ask in this paper how they are related to experimentally measurable quantities. The general Wigner representation introduced in <cit.> is denoted as the representation (discrete representation of operators for spin systems),and its basics are recalled in Sec. <ref> where important properties are also summarized.In this paper, we theoretically developa tomography scheme for spherical functions of arbitrary multispin quantum states. We study experimental schemes to reconstructthe generalized Wigner representation of a given density operator(representing mixed or pure quantum states).Extensions to quantum process tomography <cit.> as given by the experimental reconstruction of entirepropagators (e.g., representing quantum gates) are, however, beyond the scope of this paper. Our scheme is particularly tailored to the Wigner representation of <cit.>, for whichan interpretation in terms of fictitious multipole potentials is provided. We will focus on systems consistingof spins 1/2, even though our approach is applicable to arbitrary spin numbers.We also provide explicit experimental protocols for our Wigner tomography scheme and demonstrate its feasibility usingnuclear magnetic resonance (NMR) experiments. Motivated by our experiments, mostof the discussed examples consider only the traceless part of the density matrix. This paper is organized as follows.A brief summary of theDROPS representation is presented in Sec. <ref>. Our general methodology for sampling spherical functions of multispin operators is introduced in Sec. <ref>, which also states the main technical results for the Wigner tomography. Section <ref> provides a physical interpretation of spherical functionsin terms of fictitious multipole potentials. The performed NMR experiments are summarized in Sec. <ref>, and Sec. <ref> discusses the use of temporal averaging. The precise experimental scheme and its implementation on a spectrometer are detailed in Secs. <ref> and <ref>. We conclude bysummarizing and discussing theoreticaland experimentalaspects, while also contrasting our paper with othertomography approaches. Further details are deferred to the Appendices.§ VISUALIZATION OF OPERATORS USING SPHERICAL FUNCTIONSWe summarize the approach of <cit.> to obtain a Wigner representation of arbitrary operators Ain coupled spin systemsusing multiple spherical functions, which is based on a general one-to-one mapping from spherical tensor operators to spherical harmonics. An operatorA = ∑_ℓ∈ L A^(ℓ)is decomposed according to a suitable set L of labels ℓ (i.e. quantum numbers) inducing a bijective mapping between the components A^(ℓ)and spherical functions f^(ℓ)=f^(ℓ)(θ,ϕ). These spherical functions can be plotted together as seen in the example of Fig. <ref>(a) where the corresponding mapping is highlighted. This provides a pictorial representation of the operator A, which conserves important propertiesand symmetriesdepending on the chosen label set L.footnote1 [footnote] The componentsT_jm ofvarying rank j∈{0,1,…,2s} andorder m∈{-j,…,j} form a complete orthonormalmatrix basis.In particular, one obtains the matrix basis T_00 =( [ 1 0; 0 1 ])/√(2), T_1,-1 =( [ 0 0; 1 0 ]), T_10 =( [10;0 -1 ])/√(2), T_11 =( [0 -1;00 ]) for a single spin with spin number s=1/2. tensorfootnote1 [footnote]Spherical harmonics Y_jm(θ,ϕ)=r(θ,ϕ) exp[iη(θ,ϕ)] (and spherical functions) are plottedthroughout this work by mapping their spherical coordinates θ and ϕ to the radialpart r(θ,ϕ) and phase η(θ,ϕ). sphericalThe components A^(ℓ)and the spherical functions f^(ℓ)=f^(ℓ)(θ,ϕ) can be further split up into their multipole contributionsA_j^(ℓ) andf_j^(ℓ)=f^(ℓ)_j(θ,ϕ) depending on the ranks j∈ J(ℓ) occuring for each label ℓ as shown in Fig. <ref>(b), i.e.,A^(ℓ)= ∑_j ∈ J(ℓ) A_j^(ℓ)and f^(ℓ)= ∑_j ∈ J(ℓ) f_j^(ℓ).Finally, the rank-j multipole contributions [see Fig. <ref>(c)]A^(ℓ)_j= ∑_m=-j^j c_jm^(ℓ) T_jm^(ℓ)and f^(ℓ)_j= ∑_m=-j^j c_jm^(ℓ) Y_jm,can be decomposed into components of irreducible spherical tensor operators T_jm^(ℓ) <cit.>and the corresponding spherical harmonicsY_jm <cit.> of order m with -j≤ m≤ j. Note the identical expansion coefficients c_jm^(ℓ) in Eq. (<ref>). The dualism in Eqs. (<ref>) and (<ref>) exploits the well-known correspondence betweenirreducible tensor operators and spherical harmonics <cit.>. In summary, an operator A is mapped to a set of spherical functions f^(ℓ), each of which is referred to as a droplet identified by ℓ. The whole representation (and its visualization) was introduced in <cit.> and is denoted as the representation (discrete representation of operators for spin systems), and it lendsitself to interactively exploring the dynamics of multispin systems, e.g., byuse of the free application <cit.>.footnote1 [footnote] Cartesian operators for single spins areI_x:=σ_x/2, I_y:=σ_y/2, and I_z:=σ_z/2, where the Pauli matrices are σ_x=( [ 0 1; 1 0 ]), σ_y=( [0 -i;i0 ]), and σ_z=( [10;0 -1 ]). For n spins, one has the operators I_k η := ⊗_s=1^n I_a_swhere a_s is equal to η for s=kand is zero otherwise; note I_0:= ([ 1 0; 0 1 ]). cartfootnote1 [footnote]Hermitian operators lead to positive and negative values whichare shown in red (dark gray) and green (light gray). colorsThe example presented in Fig. <ref>uses one particular version of this representationwhich relies on the tensor operator basis as defined in <cit.>, which is characterized by thelinearity of the basis operators, the involved subsystem,and auxiliary criteria, such as permutation symmetry.For coupled spins 1/2, operators are first decomposed in this basis according to the set of involved spins, e.g., one introduces the labels {k} and {kl} for linear and bilinear operatorsacting on a subset of one or two spins numbered by k and l, and so forth. Secondly, the LISA basis for operators acting on three or more spins needs to also distinguish symmetry properties under permutations, i.e.,combined labels such as “{klm},τ_p" are used, where the permutationsymmetry type τ_p is given by a Young tableau <cit.>. Finally,further ad hoc labels are necessary for operators involving six or more spins. Arbitrary operators of a coupled spin system can be uniquely represented using this tensor operator basis. Additional details for the visualization techniqueare given in <cit.> which also discusses alternative labeling approaches for representations. The presented Wigner representation can be applied to general mixedquantum states as represented by the density operator, and itis not limited to pure quantum states as given bya state function. In fact, it can be used torepresent arbitrary operators of spin systems: examples include Hermitian operatorsas Hamiltonians or density operators representing observables as well as non-Hermitian operators such as propagators or general quantum gates <cit.>.The Wigner representation using the basis is particularly attractive for thevisualization and analysis of quantum states in magnetic resonance spectroscopy<cit.> and quantum information processing <cit.> and its propertieshave been discussed in <cit.>.(a) The location of the dropletsinforms about which and how many spin operators are involved in a given quantum state and what symmetriesunder particle exchange are present [c.f. Fig. <ref>(a)].(b) The shape and colors of the droplets reflectspectroscopically important properties. For example,states with definedcoherence order p <cit.> can be recognized by their axially symmetricshape, and the magnitude as well as sign of the coherence order pare represented by the number and direction of “rainbows” perrevolution around thez axis [see Figs. <ref> and <ref>]. This also allows us torecognize characteristic states, such as inphase and anti-phasecoherences. (c) Furthermore, our representation directlydepicts information about reduced density matrices and thereby conveys information related to entanglementmeasures, which would have to be first computed from the density-matrix description via partial traces. In particular, the size of the droplets corresponding to linearterms [positioned in the vertices of the triangle in Fig. <ref>(a)]provides information on the amount of bipartite entanglement measured bythe concurrence (see Sec. IV E in <cit.>).This is an example of the fact that relevant information is often determined already by a subset of all droplets in the basis. The basisthereby offers a more structured picture than the density matrix,even though the number of droplets grows rapidly with increasing number of spins. However, as pointed out in <cit.>, this number grows less rapidly than the number of density matrix elements.(d) The droplets rotateundernon-selective pulses in a natural way. In combination with thecharacteristic droplet shapes, this property makes it in many cases possible to design experiments that transfera given initial state into a desiredtarget state without detailed calculations. Beyond merely interpretingthe occurring characteristic shapes as a result of an abstract mapping, itis interesting to ask whether they are connected to experimentallymeasurable quantities.§ SAMPLING SPHERICAL FUNCTIONS OF MULTI-SPIN OPERATORSWe explain now how the shape of spherical functionscan be characterized by suitable chosen spherical samples. This will be particularly relevant for spherical functions representing spin operators as discussed in Sec. <ref> for which these spherical samples can be experimentally measured. One obtains a reconstruction method for the quantum state in terms of spherical functions. In the general case, the associated rank-j components g_j(θ,ϕ)of an arbitrary spherical function g(θ,ϕ) are determined by its scalar product withrotated versions R_αβ Y_j0(θ,ϕ) of axial spherical harmonics Y_j0(θ,ϕ), which haverank j and order zero. Given two spherical functions h(θ,ϕ) and g(θ,ϕ),we recall the definition of their scalar product⟨ h(θ,ϕ)| g(θ,ϕ)⟩_L^2 := ∫_θ=0^π∫_ϕ=0^2 πh^∗(θ, ϕ) g(θ, ϕ) sinθdθdϕ. The rotation operator R_αβ acts on a spherical function h(θ,ϕ) by first rotating it around the y axis by a polar angle β and then rotating the result around the z axis by an azimuthal angle α, i.e.,R_αβ h(θ, ϕ):= h[R^-1_αβ(θ, ϕ)]=h(θ-β, ϕ-α).After these preparations, the mathematical result underpinning our reconstruction method states that the value of the rank-j component g_j(β, α)is proportional to the scalar product of R_αβ Y_j0(θ, ϕ) with g(θ, ϕ):Consider a spherical function g(θ,ϕ)=∑_j g_j(θ,ϕ). The rank-j components g_j(β,α) for angles β and α can be obtained from the scalar productsg_j(β, α) = s_j ⟨ R_αβ Y_j0(θ,ϕ) | g(θ,ϕ)⟩_L^2with s_j:=√((2j+1)/(4π)). Assuming that an operator A is represented by a set of spherical functionsf^(ℓ)(θ,ϕ), we can apply Result <ref> by settingg(θ,ϕ):= f^(ℓ)(θ,ϕ) for each label ℓ separately.We extend Result <ref> such that the spherical rank-j componentsf^(ℓ)_j(β, α) can also be recovered by comparing the operatorA directly with rotated axial tensor operators ℛ_αβ T_j0^(ℓ).Consequently, the values of the rank-j spherical components f^(ℓ)_j(β, α) can be experimentally measured for any combination of polar anglesβand azimuthal angles α. Here, ℛ_αβ C := ℜ_αβ C ℜ^-1_αβ describes the rotation of an n-spin operator C where the simultaneous rotation ℜ_αβ= e^- i α F_z e^- i β F_y of all spins is defined using the total spin operatorsF_z= ∑_k=1^n I_kz andF_y= ∑_k=1^n I_ky <cit.>. We recall the scalar product⟨ C| B⟩= ( C^† B) for operatorsC and B as well as the definition of the expectation value⟨ B ⟩_ρ=(ρ B) of an operator B if the state of the spin system is given by the density matrix ρ. Our result for recovering rank-j droplet components of an operator can now be stated as follows:Consider a multi-spin operator A which is represented by a set of spherical functions f^(ℓ)(θ,ϕ)=∑_j ∈ J(ℓ) f_j^(ℓ)(θ,ϕ). For each label ℓ, the rank-j component f_j^(ℓ)(β,α) can be experimentally measuredfor arbitrary angles β and α via the scalar productsf_j^(ℓ)(β, α) = s_j ⟨ℛ_αβ T_j0^(ℓ) | A⟩.If the density matrix ρ of a spin system can be prepared to be identical to the operator A, the rank-j droplet components are given by the expectation valuesf^(ℓ)_j(β, α)= s_j ⟨ℛ_αβ T^(ℓ)_j0⟩_ρ.The proofs of Results <ref> and <ref> are deferred toAppendices <ref> and <ref>. Equation (<ref>) implies that the rank-j droplet components f^(ℓ)_j(β, α)for a density matrix ρ can be calculated from the expectation values of rotated axial tensor operators ℛ_αβ T^(ℓ)_j0.Result <ref> shows that one can retrace the shapes of the spherical functions f^(ℓ)(β, α) representing any operator that can be mapped onto the density matrix if one experimentally measures f^(ℓ)(β, α) for sufficiently many angles β and α.§ DROPLETS AS MULTIPOLE POTENTIALSThe methodology of Wigner tomography as presented in Sec. <ref> can be motivated by relating spherical functions to physical multipole potentials. Section <ref> details connections to dipole potentials, which is then generalized to fictitious multipole potentials in Sec. <ref>. This allows us to interpret the proposed Wigner tomography as measuring a fictitious potential using axial multipole sensors (seeSec. <ref>).§.§ Spherical functions and dipole potentialsThe most direct physical interpretation of spherical functions is found for Hermitiansingle-spin terms <cit.>ρ_k =m_x I_kx+m_y I_ky+m_z I_kzof the density matrix with (possibly time-dependent) real coefficients m_x, m_y, and m_z. The corresponding spherical function f^{k}(θ, ϕ) is now related to a magneticdipole potential. The operator ρ_k associated with spin k is interpreted as amagnetization vector(or Bloch vector) M⃗= (m_x , m_y , m_z )^Tthe components of which areproportional to the expectation values of the spin operators I_kx,I_ky, and I_kz. An actual (time-dependent) magnetic dipoled⃗∝M⃗ creates a detectable signalin an NMR experiment by inducing a voltage in a detection coil. It is associatedwith a scalar dipole potential V_1(r⃗) at r⃗=r⃗ (sinθcosϕ,0.1emsinθsinϕ, 0.1em cosθ), whereθ and ϕ are polar and azimuthal angles, respectively. At a constant distance r⃗from the dipole, the potential V_1(r⃗) is proportional to the scalarproduct d⃗·r⃗ <cit.>. In the case of an axial dipole d⃗_ax∝ (0,0,1)^T oriented along the z axis,the dipole potential is proportional to the axial spherical harmonic Y_10(θ, ϕ)=√(3/(4 π))cosθ as detailed in Fig. <ref>(a)-(c). For a general dipole d⃗=R_αβ d⃗_ax∝ (sinβcosα, sinβsinα, cosβ)^T, the dipole potential V_1(r⃗) ∝R_αβY_10(θ, ϕ) is rotated accordingly as shown inFig. <ref>(d)-(f). Recall thatR_αβ denotes a rotationaround the y axisby a polar angle β followed by one around the z axis by an azimuthal angle α.A scalar dipole potential V_1(r⃗)=V_1 (θ, ϕ) can be indicated by itsvalues on the surface of asphere by encoding its sign by the color and its magnitude by the brightness [see Fig. <ref>(b) and (e)]. Alternatively, its magnitude can be representedby the distance from the origin as in Fig. <ref>(c) and (f), where dipole potentials are shown asa positive, red (dark gray) sphere and a negative, green (light gray) one which touch each other at the origin. This characteristic shape arises as V_1 (θ, ϕ) is proportionalto the projection of the dipole d⃗ onto r⃗ as depicted inFig. <ref>. Also, the vector from the center of the negative sphere to the positive one is collinear with d⃗. In summary, a single-spin axial spherical tensor operator T_10^{k} = √(2) I_kz is mapped to theaxial spherical harmonics Y_10(θ, ϕ), and ρ_k fromEq. (<ref>) is mapped tof^{k}(θ, ϕ) = M⃗ R_αβ Y_10(θ, ϕ) / √(2),where β=atan(m_z/√(m^2_x+m^2_y) ), α=atan (m_y/m_x), and M⃗=√(m^2_x+m^2_y+m^2_z). Although the direct correspondence between spherical functions and actual physical dipole potentials appears to be limited to the case of single-spin terms, it suggest the following interpretation for other spherical functions presented here. §.§ Fictitious multipole potentialsAny spherical function f^(ℓ)(θ, ϕ) can be regarded as thepotential V^(ℓ)(r⃗) of a fictitious charge distribution σ^(ℓ) (r⃗) localized in a small volume close to the origin, i.e., σ^(ℓ)(r⃗) isnon-zero onlyfor | r⃗| ≪ 1. At a radius of | r⃗| = 1, the potentialcan be expressedas a sum V^(ℓ)(θ, ϕ)=∑_j V^(ℓ)_j(θ, ϕ) of different 2^j-pole potentials V^(ℓ)_j(θ, ϕ). Although a large number of multipole potentialsmight be required in general, only a moderate number of components with different rank j appearfor up to three spins in the representation of Sec. <ref> <cit.>.Fictitious multipole potentials sufficient to completely describe the potential V^(ℓ)(θ, ϕ) are detailed in Table <ref>:one has monopoles (2^0=1), dipoles (2^1=2), quadrupoles (2^2=4), and octupoles (2^3=8). For a two-spin droplet with label “{kl}”, only ranks j of zero, one, and two occur which correspond to fictitious monopole, dipole, and quadrupole potentials, whereas the fully symmetric three-spindroplet with label “{ 123},τ_1” has only rank-1 and rank-3 components associated with dipole and octupole terms.§.§ Axial multipole sensors Based on the provided interpretation of droplet functions as fictitious multipole potentials, the results of Sec. <ref> on how toexperimentally measure spherical functions of spin operators can be mapped to the analogous problem of measuring an unknown electrostatic potential. This analogy is complete for Hermitian spin operators with real-valued spherical functions <cit.>. Suppose we would like to determine an unknown (real-valued) electrostaticpotential V(θ, ϕ) at a radius r⃗=1 that is created by an object located in the interior of a unit sphere. An electric point charge q at position r⃗ in an electric potential V(r⃗) has apotential energy U_pot(r⃗)=q V(r⃗). Given an electrostatic potential V(θ, ϕ), the electrostatic potential energy of a (nonconducting) unit sphere with the surface charge distribution σ(θ,ϕ) is given byU_pot(β, α)= ∫_θ=0^π∫_ϕ=0^2 π σ(θ,ϕ)V(θ, ϕ)sinθ dθ dϕ,which is equivalent to U_pot(β, α) =⟨σ(θ,ϕ) | V(θ, ϕ)⟩_L^2 for real valued σ(θ,ϕ) as in Eq. (<ref>) ofSec. <ref>. The 2^j-pole components V_j(θ,ϕ) of an unknownmultipole potential V(θ, ϕ) can be sampled by a set of axial 2^j-pole sensors,each consisting of a charge distributionσ_j0(θ,ϕ) proportional to the axial spherical harmonics Y_j0(θ, ϕ). Each individual sample V_j(β,α) can be determined using Eq. (<ref>)by measuring the potential energy V_j(β, α)∝U_pot(β,α) =⟨R_αβ σ_j0(θ, ϕ) | V(θ, ϕ)⟩of the axial 2^j-pole sensor rotated by the polar angle β aroundthe y axis followed by a rotation by the azimuthal angle α around the z axis.The full electrostatic 2^j-pole potential V_j(θ, ϕ) can be recovered by systematically incrementing β and α. In summary, the analogy between real-valued spherical functions and multipole potentials helps to better understand our results of Sec. <ref> on the measurement of spherical functions. It can also be extended in a straight-forward manner to non-Hermitian spin operatorsby considering complex, fictitious multipole potentials.§ SUMMARY OF NMR EXPERIMENTSBuilding on the previous sections, we demonstrate the Wigner tomography of various prepared density-matrix components in spin-1/2systems using nuclear magnetic resonance. Experimental details are deferred to Secs. <ref> and <ref> where the precise experimental scheme and its implementation on a spectrometer are discussed. The experiments were performed on one-, two-, and three-spin systems.The shapes of the spherical functions are recovered for theprepared Cartesian products operators listed in Table <ref>, where also their respective experimental reconstruction errors are given. Experimental and theoretical results for the reconstructionare visually compared for four examples in Fig. <ref>. For the rightmost example of 4I_1xI_2yI_3z in Fig. <ref>, multiple dropletscorresponding to different permutation symmetries are necessary to completely describethe quantum operator,as outlined in Sec. <ref> (see Fig. <ref> and <cit.>).§ TEMPORAL AVERAGING Re-using our experimental data as summarizedin Sec. <ref>, we can also highlight how temporal averaging <cit.> is used to emulate the preparation of quantum operators. The direct experimental preparation of Hermitian operators would be also possible, but we have chosen temporal averaging for its simplicity and convenience. The experimental values are shown in Fig. <ref>, while the corresponding theoretical predictions are given in Fig. <ref>. The Cartesian operators 2I_1xI_2x, 2I_1yI_2y, 2I_1xI_2y, and2I_1yI_2x had been sequentially measured and are now combined inFig. <ref>(a) to form thedouble quantum operators DQ_x := I_1xI_2x - I_1yI_2yandDQ_y := I_1xI_2y + I_1yI_2x.Their characteristic shapes reflect the fact that they havecoherence order p=2 and are invariant under non-selective rotations around the z axis by an integer multiple of 180 deg <cit.>. This is in contrast to single-quantum operators such as the linear operators I_x or I_y [see <ref>(c)]or the bilinear operator2 I_1z I_2x[see Fig. <ref>]whichare only invariant under non-selective rotations around the zaxis by an integer multiple of 360 deg <cit.>.In general, p-quantum operatorsare invariant under non-selective rotations around the z axis by an integer multiple of360/p deg, and their spherical functions illustrate this symmetry. Figure <ref>(b) exemplifies the invariance under 120 deg rotations aroundthe z axis in the case of p=3 for the triple-quantum operator TQ_x:= I_1xI_2xI_3x-I_1xI_2yI_3y-I_1yI_2xI_3y-I_1yI_2yI_3x, similar to the case ofTQ_y:=I_1yI_2xI_3x+I_1xI_2yI_3x+I_1xI_2xI_3y-I_1yI_2yI_3y.Up to experimental imperfections,only the spherical function f^τ_1contributes to the operator TQ_x.Finally, we also consider temporal averagingfor non-Hermitian operators which obviously cannotbe directly prepared in experiments (see also Sec. <ref>). Figure <ref>(c) presents the non-Hermitian operators I_1^+ = I_1x+iI_2y,I_1^+I_2^+ = DQ_x + i DQ_y, and I_1^+I_2^+I_3^+= TQ_x + i TQ_y. These operators have the respective coherence orders p of 1, 2, and 3 which results in donut-shaped spherical functions the colors of which cycle through one, two, or three rainbows <cit.>. § WIGNER TOMOGRAPHY USING NMRWe complement our results in Sec. <ref> and describethe experimental scheme for an NMR-based implementation of our Wigner tomography. Recall that Eq. (<ref>) of Result <ref>provides an approach for measuring an arbitrary operator. This can be translatedinto the diagram of Fig. <ref>(a): The operator A is decomposed into its components A_j^(ℓ) which are mapped by the Wigner transformation W to spherical samples f_j^(ℓ)(β,α). The spherical samples can be recovered using Eq. (<ref>). Very similarly, Fig. <ref>(b) depicts the equivalent measurement procedure for density matrices which relies on Eq. (<ref>) of Result <ref>. Before proceeding to the NMR-based scheme, it is importantto emphasize that only traceless operators can be measuredin NMR experiments <cit.>, which rules out theidentity component of a density matrix. However, the traceless part ofa density matrix covers most important features, and it is sufficientto calculate the time evolution andall relevant expectation values. Hence, we willconsider in the following only thetraceless part which isfor simplicity also denoted by ρ. Further complications arise from the fact that signatures ofCartesian product operators <cit.> that contain only a single transverse Cartesian operator I_ka with a∈{ x,y} can be measureddirectly; examples are I_ka, 2I_kaI_lz, and 4I_kaI_lz I_mz. This complication can be resolved in two steps: Firstly, any traceless operator can be decomposed into (Hermitian) Cartesian product operators C^(ℓ,n)_j. This decomposition of relevant axial tensors T^(ℓ)_j0=∑_n r^(ℓ,n)_j C^(ℓ,n)_jwith respect to real coefficients r^(ℓ,n)_j is given in Table <ref>. For example, the axial tensor operator T_10^{ k} acting on the k th spindecomposes directly into the Cartesian product operatorC_1^({k},1)=I_kz withthe coefficient r_1^({k},1)=√(2).Secondly, the Cartesian product operators have to be transformed into NMR-measurable onesM^(ℓ,n)_j= 𝒰^(ℓ,n)_j C^(ℓ,n)_j := U^(ℓ,n)_j C^(ℓ,n)_jU^(ℓ,n)†_j.The unitary operators U^(ℓ,n)_j can be experimentally realized using radio-frequency pulses and evolution periods under couplings as discussed in Sec. <ref> andtheir explicit form is detailed inApp. <ref>. Combining both steps leads to an indirect approach for measuring spherical functions of density operators,as schematically outlined in Fig. <ref>(c).The density matrix is equivalently rotated inversely in contrast toFig. <ref>(a)-(b) where the axial tensor operator T_j0^(ℓ) is rotated. The complete measurement scheme is formalizedalong the lines of Result <ref>:Consider a density operator ρ which is represented by a set of spherical functions f^(ℓ)(θ,ϕ)=∑_j ∈ J(ℓ) f_j^(ℓ)(θ,ϕ). For each label ℓ, the rank-jcomponent f_j^(ℓ)(β,α) can bemeasured for arbitrary angles β and α by determining the expectation values f^(ℓ)_j(β, α) = s_j ∑_n0.1em r^(ℓ,n)_j 0.1em ⟨ M^(ℓ,n)_j⟩_ρ̃̃̃^(ℓ,j,n)of suitable operators M^(ℓ,n)_j as in Eq. (<ref>), whereρ̃̃̃^(ℓ,j,n)= 𝒰^(ℓ,n)_j ρ̃= U^(ℓ,n)_j ρ̃U^(ℓ,n)†_j,ρ̃=R^-1_αβ ρ =ℜ^-1_αβ ρ ℜ_αβ,andℜ_αβ=exp(- i α∑_k=1^n I_kz) exp(- i β∑_k=1^n I_ky).A detailed derivation of Eq. (<ref>) is provided in Appendix <ref>.In summary, the rank-j components f^(ℓ)_j of spherical functions f^(ℓ)representing the density matrix ρ can be sampled in NMR experiments bytransforming the density operator ρ to the states ρ̃̃̃^(n) andthen measuring a set of expectation values of suitable operators ⟨ M^(ℓ,n)_j⟩_ρ̃̃̃^(ℓ,j,n). The explicit formof the Cartesian operators C^(ℓ,n)_j, theirNMR-measurable formsM^(ℓ,n)_j, andthe transformations U^(ℓ,n)_j for up to three spins is given in App. <ref>.The approach of Result <ref> can be extended to non-Hermitian operators, even though these cannot be prepared directly in an experiment. We apply temporal averaging <cit.> as already discussed in Sec. <ref>. Any operator A can be expressed as a complex linear combination A=∑_i c^(i)ρ^(i) of Hermitian operators ρ^(i). As the representation is linear, we can sample the tracelesspart of any operator A using the spherical functionsf^(ℓ)_j(β, α)= ∑_i c^(i) f^(i,ℓ)_j(β, α) = ∑_i c^(i) s_j ⟨ R_αβ T^(ℓ)_j0⟩_ρ^(i)=∑_i c^(i) s_j∑_n r^(ℓ,n)_j ⟨ M^(ℓ,n)_j ⟩_ρ̃̃̃^(ℓ,j,n,i).In an experiment,temporal averagingof Hermitian operators ρ^(i) can be implemented by sequentially measuring spherical functions for each operator ρ^(i) and linearly combining the results f^(i,ℓ)_j(β, α)=s_j ⟨ R_αβ T^(ℓ)_j0⟩_ρ^(i)= s_j∑_n r^(ℓ,n)_j 0.1em ⟨ M^(ℓ,n)_j ⟩_ρ̃̃̃^(ℓ,j,n,i) as illustrated in Fig. <ref>(d). § EXPERIMENTAL IMPLEMENTATION After outlining the experimental scheme for Wigner tomography in Sec. <ref>, we present now the details for the experimental implementation which results in the spherical functions in Figs. <ref>-<ref>. We start by describing the molecules and experimental settingin Sec. <ref>. We continue in Sec. <ref> with the experimental protocol, and we finally discuss experimental errors in Sec. <ref>. §.§ Molecules and experimental setting In order to simplify the experiments, the linear and bilinear Cartesian productoperators have been prepared and measured usingrespective single-spin and two-spin samples: The single-spin sample was prepared by dissolving 5% H_2O in D_2O, which resulted in a sample containing about 10% HDO, i.e.water molecules in which one of the ^1H spins is replaced bydeuterium (^2D) [see Fig. <ref>(a)]. In case of two spins, we have used a 10% sample of chloroform dissolved in fully deuterated DMSO-d6,where the ^1H spinand the ^13C spin of each chloroform moleculeform a system consisting of two coupled heteronuclear spins 1/2 [see Fig. <ref>(b)]. A three-spin sample consistingof 2-^13C-2-fluoromalonic-acid-diethyl-ester dissolved in CD_3CN [see Fig. <ref>(c)] was utilizedfor the preparation andreconstruction of trilinear operators.All liquid samples were measured in 5 mm Shigemi NMR tubes at room temperature (298 K) ina 14.1 T magnet using a Bruker Avance III 600 spectrometer.§.§ Experimental protocol Our experimental protocol is composed offive main building blocks (see Fig. <ref>). In the first block 𝒫, the desired density operator ρ is prepared starting from the initial thermal equilibrium density operator which in the high-temperature limit isproportional to <cit.> ρ_th =∑_k=1^Nγ_kI_kz, where γ_k denotes the gyromagnetic ratio of the kth nuclear spin.This requires unitary transformations which are created by pulses and evolution periods under theeffect of couplings and frequency offsets as well as non-unitary transformations which are implemented by pulsed magnetic-field gradients. The explicit pulse sequencesare discussed in App. <ref>. Table <ref> in Sec. <ref> summarizes all the different Cartesian product operators which have been experimentally prepared.The second block consists of the rotation ℛ^-1_α_rβ_r which rotates the prepareddensity operator ρinto ρ̃ in order to probe the corresponding spherical functionsf_j^(ℓ)(β_r, α_r) for differentpolar angles β_r and azimuthal angles α_r using axial tensor operators, i.e.,axial multipole sensors) (see Result <ref>). The rotation ℛ^-1_α_r β_ris implemented by rf pulses [β_r]_α_r-π/2 with flip angle β_r and phase (α_r-π/2) which are simultaneously applied to all spins.The unitary transformations 𝒰_j^(ℓ,n) [see Eq. (<ref>) of Result <ref>] are applied in the third block in order to transform the density matrix ρ̃ into directly detectable Cartesian product operators for the various linear, bilinear,and trilinear operators (see Table <ref> in Sec. <ref>). The specific experimental implementation of the unitary transformations 𝒰^(ℓ,n)_jconsists of rf pulses with flip angle π/2 as detailed in App. <ref>. In the fourth block, the NMR signal is measured in an acquisition period Acq which has a duration of 5.7 ms (one spin),11.4 ms (two spins), and 14.9 ms (three spins). In the last block, a relaxation delay RD with a duration of7 s (one spin), 10 s (two spins), and 15 s (three spins) recovers the initial equilibrium state ρ_th, In the tomography experiment, all blocks are repeated multiple times(see Fig. <ref>). The outer loop A runs over all possible droplets ℓ∈ L. Loop Bruns over allranks j contributing to the droplet ℓ. Loop Ccycles through all Cartesian product operatorsC^(ℓ,n)_j [see Eq. (<ref>)] appearing in the decomposition of the axial tensor operator T^(ℓ)_j0 (see Table <ref> inSec. <ref>). Finally, the discretized anglesβ_r∈{0,15, 30, … , 180} andα_r∈{0,15, 30, … , 360} (both in degrees) are incremented in the innermost loops D and E. Although not explicitly indicated in Fig. <ref>, one further loop is necessary for a temporal averaging scheme [seeEq. (<ref>)].The whole protocol allows us todetermine expectationvalues ⟨ M_j^(ℓ,n)⟩_ρ̃̃̃^(ℓ,j,n), which are normalized and range between 1 and -1. As illustrated in Fig. <ref>, the spherical functions f^(ℓ)(θ,ϕ) can be reconstructed by plotting the spherical samplesf^(ℓ)(β_r,α_r) for all anglesβ_r ∈{0,…,π}and α_r ∈{0,…,2π} at a distance | f^(ℓ)(β_r,α_r) | from the origin.The phaseφ^(ℓ) (β_r, α_r)=atan ( Im{ f^(ℓ)(β_r, α_r) } /Re{ f^(ℓ)(β_r, α_r) } )is color coded. For example, the spherical function of the Hermitian operator I_1x is given by a real function, and the positive andnegative values of f^(ℓ)(θ,ϕ) are indicated by the colors red(dark gray) and green (light gray),respectively.§.§ Experimental errorsA reasonable match between the experimentally reconstructed andtheoretical predicted spherical functions is found in Figs. <ref>-<ref>.Deviations are attributed to experimental imperfections, such as thefinite experimental signal-to-noise ratio, finite accuracy of pulse calibration,B_0 and B_1 inhomogeneity <cit.>,pulse shape distortions due to the amplifiers and the finite bandwidth of the resonator <cit.>, relaxation losses during the preparation and detection blocks, partial saturation of the signal due to a finite relaxation period between scans, radiation damping effects <cit.>, and truncation effects in the automated integration and comparison of the spectra. We quantify these deviations bythe root-mean-square difference between experiment and theory averaged over all measured angles β_r and α_r. The resulting errors for the prepared and measured Cartesian product operatorsare summarized in Table <ref> of Sec. <ref>. In order to minimize the effects of fieldinhomogeneities, Shigemi tubes were used in the experiments in order to reduce the sample volume.We have tested replacingthe simple rectangular pulses [β_r]_α_r-π/2in the implementation ofthe rotations ℛ^-1_α_r β_r withcomposite pulses <cit.>, which, however, did not result in an improved performance. § CONCLUSION We have theoretically developed and experimentally demonstrated a Wigner tomography scheme the mapping of whichmulti-spin operators and spherical functions is based on <cit.>. Our approach reconstructs the relevant spherical functions by measuring expectation values of rotated axial tensor operators, i.e.axial multipole sensors. It is universally applicable andnot restricted to NMR methodologies or particles with spin 1/2. A reasonable match between theoretical predictionsand NMR experiments was found. Our theoretical analysis provides a simple physical interpretation of the individualspherical functions in terms of fictitious multipole potentials which can be sampled locally. The objective was toexperimentally recover the three-dimensional shapes of the spherical functions for each of the prepared operators. In particular,we have not used any a priori information on what shapes to expect. A large number ofsampling points is necessary to recover the shapes in sufficient detail. As the simple rectangular grid of sampling points in the space of polarand azimuthal angles used here is highly anisotropic (i.e. more densely concentrated at the poles),a straight-forward improvement is to choosemore isotropic sampling strategies such as Lebedev grids <cit.>. Moreover, one could use interpolation methods or even adaptive sampling schemes which increase the sampling density in areas where the spherical function variesmore strongly. A more quantitative analysis of the complexity of our proposed tomography approach in terms of the number of individual measurements could rely first on a detailedempirical account on how the precision measured by the fidelity depends onthe sampling strategy and its density. From the theoretical side, it is clear that the number of measurements for any complete tomography of a quantum state will scale exponentially in the number of qubits (or spins). However, relevant information can be reconstructed in our approach even from a subsetof the droplets as discussed under (c) in Sec. <ref>. Amore accurate understanding of concrete sampling schemesand their optimization building onour reconstruction method will have to be addressed in future work.Beyond the optical tomographic methods mentioned at the start of theintroduction, tomographic approaches play obviously an important part inalmost any experimentin quantum information or quantum physicsin general. We will now shortly discuss some selected results from the literature. The proposed reconstruction procedure for spherical functions can be directly extended to other types of spherical representations such as multipole operators <cit.> or theso-called PROPS representation whichis based on products of single-spin representations <cit.>. Our scheme can also be comparedto <cit.> which introduces Wigner functions built fromproducts of single-spin representations, and a corresponding raster scan method in <cit.> utilizes theprobability to find the rotated system in each of the basis statesof a Stern-Gerlach-type experiment. In <cit.>, a similar tomographic methodbased on filtered back projections (in analogy to planar inverse Radontransforms used e.g. in medical imaging) is usedin Stern-Gerlach-type experiments.However, complementary to these tomographic reconstruction schemes for density matrices relying on Stern-Gerlach-type experiments, our scheme is based on projections of operators onto rotatedaxial tensor operators or experimentally accessible expectationvalues of transformed axial tensor operators. The measurement of spherical harmonics components of electromagnetic near-field radiation using specifically designed loopantennas as sensors <cit.> is closely related to our interpretation of droplets as multipole potentials (see Sec. <ref>).Further tomographic approacheshave been established in <cit.> with applications to molecular systems.In those works, the quantum state given asthe wave functions of an excited state is obtained by decomposing the wave function in a series of basis functionsand the expansion coefficients are acquired by calculating a set of Fourier integralsfrom the detected signal. Also, the wave packet isreconstructed in <cit.> by computing the overlap of the state with well-defined reference states for different time intervals.We want to also contrast our Wigner tomography approach to the so-called spherical tensor analysis (STA) method developed by Suter and Pearson <cit.> andvan Beek et al. <cit.>. Both Wigner tomography and STA experiments have asimilar structure. First, a preparation block (called an excitation sequence in STA)is used to prepare a density operator. In a second step,rotations around several axes are applied in a rotation block. Finally, the density matrix is transformed(by a reconversion sequence in STA) into a detectable basis and the signal isdetected during an acquisition period. Despite these similarities and the fact that both methods are based oncharacteristic properties of spherical tensor operators under rotations,the desired data and therefore also the details of the experimentsdiffer considerably.The goal of STA is not to measure a density operator represented byspherical functions, but to decompose the detectable signal at a later time into individual signal components depending onoccurring ranks j and orders m in the current density matrix. In the standard form of STA <cit.>, the detected signalof a density matrix is not characterized or decomposed in terms of additional quantum numbers or labels ℓ. While a single reconversion sequence is used in STA,different pulse sequences are appliedin the Wigner tomography in order to transform operators to directly detectable ones(seeFig. <ref>(c), (d) and Table <ref>). As a final difference, a rotation block in STA uses rotations forthree Euler angles α, β, and γ, whereas in the Wigner tomography only two Euler angles α and β are necessary.Lastly, our Wigner tomography can be seen as a stepping stone along the path to identifyingand characterizing operators in terms of expansion coefficients for a suitable chosen basis. One can consider various different bases such assimple matrix coefficients, a spherical tensor basis <cit.> (as the one used here),or a Cartesian product basis<cit.>. In this context of quantum state tomography, the shapes ofspherical functions recovered in the Wigner tomographyclearly contain highly redundant information, but they also provide information about random or systematic errors of the tomography process itself. One can also envision Wigner tomography as a componentof a more general approach where one would like to optimize the number and location of samples for achieving a desired fidelityand robustness against experimental errors <cit.>, or recover a physical density matrix and estimate experimental errors (see, e.g., <cit.> and references therein). This work was supported in part by the Excellence Network of Bavaria (ENB) through ExQM. R.Z.andS.J.G.acknowledgesupportfromtheDeutsche Forschungsgemeinschaft (DFG) through Grant No. Gl 203/7-2. We thank Raimund Marx for providing the samples.The experiments were performed at the Bavarian NMR Center at the Technical University of Munich. § PROOF OF RESULT <REF>We detail now the arguments leading to our reconstruction formula for a spherical function g(θ,ϕ) as stated in Result <ref>. This result relies on projections ofrotated spherical harmonics. We use the notation introduced in Sec. <ref>. The angles θ and ϕ indicate generic argument values of a spherical function g(θ,ϕ), but the angles β and α refer to specific argument values. First, the right hand sides_j ⟨ R_αβ Y_j0(θ,ϕ) | g(θ,ϕ)⟩_L^2 of Eq. (<ref>) in Result <ref> isrewritten ass_j ⟨ R_αβ Y_j0(θ,ϕ) | g_j(θ,ϕ)⟩_L^2, where the familiar orthogonality relation ⟨ Y_jm(θ,ϕ) |Y_j'm'(θ,ϕ)⟩_L^2 = δ_jj'δ_mm' of spherical harmonics (seep. 68 of <cit.>) implies the relation ⟨ g_j(θ,ϕ) |g_j'(θ,ϕ)⟩_L^2 = δ_jj' for the rank-j parts g_j(θ,ϕ) in the decomposition g(θ,ϕ)=∑_j g_j(θ,ϕ). Second,one obtains that s_j ⟨ R_αβ Y_j0(θ,ϕ) | g_j(θ,ϕ)⟩_L^2 =s_j ⟨Y_j0(θ,ϕ) | R_αβ^-1 g_j(θ,ϕ)⟩_L^2 holds, which can be deduced from the invariance ⟨ R_αβ Y_jm(θ,ϕ) |R_αβ Y_j'm'(θ,ϕ)⟩_L^2 = ⟨ Y_jm(θ,ϕ) |Y_j'm'(θ,ϕ)⟩_L^2 under rotations. The last relation is easily verified using the formula R_αβ Y_jm(θ,ϕ)= Y_jm(θ-β,ϕ-α) and a change of variables in the integral defining the scalar product (see Sec. <ref>). Finally,R_αβ^-1 g_j(θ,ϕ) is expanded into a linear combination∑_m'=-j^j c_jm'(α,β)Y_jm'(θ,ϕ)of spherical harmonics [Even though our arguments do not rely on explicitly knowing the expansion coefficients c_jm'(α,β), it might beinstructive to specify the expansion coefficients in terms of the widely used Wigner-D matrices D_m'm^j(α,β,γ)<cit.>. Let us alsoassume that the rank-j parts g_j(θ,ϕ)are expanded into g_j(θ,ϕ)=∑_m=-j^jc̃_jm Y_jm(θ,ϕ) using certain coefficients c̃_jm. The theory of Wigner-D matrices <cit.> implies that R_αβ^-1Y_jm(θ,ϕ)=∑_m'=-j^j D_m'm^j(0,-β,-α)Y_jm'(θ,ϕ). Hence, we obtain the formula R_αβ^-1g_j(θ,ϕ)=∑_m=-j^jc̃_jm×∑_m'=-j^j D_m'm^j(0,-β,-α)Y_jm'(θ,ϕ) which shows that c_jm'(α,β) = ∑_m=-j^jc̃_jm D_m'm^j(0,-β,-α).]. BL81,SR73 It follows thatthe right hand side of Eq. (<ref>) is given by s_j ⟨Y_j0(θ,ϕ) | ∑_m'=-j^j c_jm'(α,β)Y_jm'(θ,ϕ) ⟩_L^2=s_jc_j0(α,β).The left hand side of Eq. (<ref>) is transformed into g_j(β,α)=R_αβ^-1 g_j(0,0) = ∑_m'=-j^j c_jm'(α,β) Y_jm'(0,0) = s_jc_j0(α,β), wherethe formula Y_jm'(0,ϕ) = s_j δ_m'0(see p. 16 of <cit.>) has been applied. In summary, we have verified that both sides of Eq. (<ref>) agree, which completes the proof of Result <ref>.§ PROOF OF RESULT <REF> In this appendix, we demonstrate the tomography formulaas given in Result <ref> for an operator A byapplying the reconstruction formula of Result <ref>. The proof relies on mapping A to (a set of) spherical functions f^(ℓ)(θ,ϕ) as detailed in Sec. <ref>. By substituting g(θ,ϕ) withf^(ℓ)(θ,ϕ)in Result <ref> for each label ℓ separately, one obtains that f_j^(ℓ)(β, α)=s_j ⟨ R_αβ Y_j0(θ,ϕ) | f^(ℓ)(θ,ϕ)⟩_L^2. Note that ⟨ A^(ℓ) | B^(ℓ)⟩ =⟨ f_A^(ℓ) | f_B^(ℓ)⟩_L^2 for thespherical functions f_A^(ℓ) and f_B^(ℓ) corresponding to theoperators A and B, which can easily beverified by expanding thearguments into tensor operators and spherical harmonics and applying their orthonormality relations. Moreover, the correspondence between operators and spherical functions is covariant under rotations (see Proposition 1(d) in <cit.>), i.e., the operator ℛ_αβ T_j0^(ℓ) is mapped to R_αβ Y_j0(θ,ϕ)[ This relation can also be established using the theory of Wigner-D matrices <cit.> whereD^j_m'm(α,β,γ)= ⟨ Y_jm'(θ,ϕ) | R_αβγY_jm(θ,ϕ) ⟩_L^2 (see p. 41 and 276 of <cit.>) and D_m'm^j(α,β,γ) =⟨ T_jm'^(ℓ) | ℛ_αβγ T_jm^(ℓ)⟩ (see p. 45 of <cit.>). Here,ℛ_αβγ C := R_αβγ CR^-1_αβγ withR_αβγ:= e^- i α F_z e^- i β F_y e^- i γ F_z,cf. Sec. <ref> whereℛ_αβ = ℛ_αβ 0.]. The last two statements imply that f_j^(ℓ)(β, α)=s_j ⟨ℛ_αβ T_j0^(ℓ) | A^(ℓ)⟩ =s_j ⟨ℛ_αβ T_j0^(ℓ) | A⟩, where the last step follows as ⟨ A^(ℓ) | A^(ℓ')⟩ = 0 if ℓ≠ℓ' (which is a consequence of the orthonormality of the tensor operators T_jm^(ℓ)). Thiscompletes the proof of Eq. (<ref>) in Results <ref>, andEq. (<ref>) is then a direct consequence due to the fact that ℛ_αβ T_j0^(ℓ)= [ℛ_αβ T_j0^(ℓ)]^† is Hermitian. § DERIVATION OF EQ. (<REF>) Here, we derive the formula of Eq. (<ref>) given in Result <ref> starting from Result <ref>.In standard NMR experiments, only the signatures of Cartesian product operators <cit.>that contain a single transverse Cartesian operator I_ka with a∈{ x,y} (such as I_ka, 2I_kaI_lz, and 4I_kaI_lz I_mz) can be measureddirectly, and hence the expectation values of axial operators T^(ℓ)_j0 are not directlyaccessible.Nevertheless, these expectation values can be measured indirectly since the operators T^(ℓ)_j0=∑_n r^(ℓ,n)_j C^(ℓ,n)_j can always be expressed as real linear combinations of (Hermitian) standardCartesian product operators C^(ℓ,n)_j <cit.>. Thus,the tomography formula of Eq. (<ref>) can be rewritten asf^(ℓ)_j(β, α)=s_j ∑_nr^(ℓ,n)_j⟨ R_αβ C^(ℓ,n)_j⟩_ρ. One obtains that f^(ℓ)_j(β, α)=s_j ∑_nr^(ℓ,n)_j Tr{C^(ℓ,n)_jρ̃}where ρ̃:=ℜ^†_αβρℜ_αβ by exploiting the actionR_αβC^(ℓ,n)_j = ℜ_αβC^(ℓ,n)_jℜ^†_αβ via the operator ℜ_αβ:=exp(- i α∑_k=1^n I_kz) exp(- i β∑_k=1^n I_ky) and the fact that the trace is invariant under cyclic permutations. The operators C^(ℓ,n)_j can be transformed into measurable operatorsM^(ℓ,n)_j= U^(ℓ,n)_j C^(ℓ,n)_jU^(ℓ,n)†_j with unitary transformations U^(ℓ,n)_j, which can be realized experimentally using radio-frequency pulses andcoupling evolutions. Hence, the formula of Eq. (<ref>) is given by f^(ℓ)_j(β, α) =s_j∑_n r^(ℓ,n)_j Tr{M^(ℓ,n)_jρ̃̃̃^(ℓ,j,n)}, where immaterialcyclic permutations of the trace have again been applied and where ρ̃̃̃^(ℓ,j,n):= U^(ℓ,n)_jρ̃ U^(ℓ,n)†_j. Finally, the definition of the expectation value yieldsf^(ℓ)_j(β, α)= s_j ∑_n r^(ℓ,n)_j⟨ M^(ℓ,n)_j⟩_ρ̃̃̃^(ℓ,j,n).§ PREPARATION AND DETECTION SEQUENCES We detail the explicit form of the preparation and detection sequences used in the experiments in order to demonstrate our Wigner tomography. We denote a pulse with flip angle β and phase α that isapplied to spin k by [β]_α(I_k). Similarly, [β]_α(I_k, I_l) specifies two pulses both of flip angle β and phase α that aresimultaneously applied to spins k and l. We also use the notation𝒫^bil_x,y=[π/2]_y(I_2)-G-[π/2]_x,y(I_1)-t_a- [π]_y,x(I_1, I_2)-t_awhich represents a pulse sequence which is read from left to right. Here, G represents a pulsed magnetic field gradientthat dephases all present transverse spin operators and t_a refers to a time delay of length 1/(4J_12), whereJ_12 is the coupling constant between the first and second spin. The pulse sequences 𝒫^bil_x and 𝒫^bil_y create from I_z the bilinear product operators 2 I_1xI_2z and 2 I_1yI_2z, respectively. Moreover, the trilinear product operator4I_xzz:=4I_1xI_2zI_3z is obtained from I_z by applying the pulse sequence𝒫^tril_x= [π/2]_y(I_2, I_3)-G-[π/2]_y(I_1)-t_b-[π]_y(I_1, I_3)-t_c-[π]_y(I_2)-t_dwhere the time delays are t_b=1/(4J_13), t_c=1/(4J_13)-1/(4J_12), and t_d=1/(4 J_12). Using these notations, the preparation sequences are given inTable <ref>.For the detection, the Cartesian product operators C_j^(ℓ,n)have to be rotated intoNMR-measurable operators M_j^(ℓ,n). The relevant pairs of operators C_j^(ℓ,n) and M_j^(ℓ,n) are provided in Table <ref>. The rotation pulse sequencesare easily inferred, e.g.,one uses the pulse [π/2]_y(I_k) in order to rotate I_kz into I_kx. Similarly, [π/2]_-y(I_k), [π/2]_x(I_k), and [π/2]_-x(I_k) rotate respectively I_kx, I_ky, and I_kz into I_kz, I_kz, and I_ky. 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Riofrí, author D. Gross, author S. T. Flammia, author T. Monz, author D. Nigg, author R. Blatt,and author J. Eisert, 10.1038/ncomms15305 journal journal Nat. Commun. volume 8, pages 15305 (year 2017)NoStop [Suess et al.(2016)Suess, Rudnicki, and Gross]suess16 author author D. Suess, author Ł. Rudnicki, author T. O. Maciel,and author D. Gross, 10.1088/1367-2630/aa7ce9 journal journal New J. Phys volume 19, pages 093013 (year 2017) NoStop [Note5()]Note5 note Even though our arguments do not rely on explicitly knowing the expansion coefficients c_jm'(α ,β ), it might be instructive to specify the expansion coefficients in terms of the widely used Wigner-D matrices D_m'm^j(α ,β ,γ ) <cit.>. Let us also assume that the rank-j parts g_j(θ ,ϕ ) are expanded into g_j(θ ,ϕ )@ =@ ∑@ @ _m=-j^jtilde07Ec_jm Y_jm(θ ,ϕ ) using certain coefficients tilde07Ec_jm. The theory of Wigner-D matrices <cit.> implies that R_αβ^-1 +.1667em@ Y_jm(θ ,ϕ ) @ =@ ∑@ @ _m'=-j^j@ D_m'm^j(0,-β ,-α ) +.1667em@ Y_jm'(θ ,ϕ ). Hence, we obtain the formula R_αβ^-1 +.1667em g_j(θ ,ϕ ) =∑@ @ _m=-j^j@ tilde07Ec_jm@ ×∑@ @ _m'=-j^j@ D_m'm^j(0,-β ,-α ) +.1667em@ Y_jm'(θ ,ϕ ) which shows that c_jm'(α ,β ) = ∑@ @ _m=-j^jtilde07Ec_jm D_m'm^j(0,-β ,-α ).Stop [Steinborn and Rueddenberg(1973)]SR73 author author E. O. Steinborn and author K. Rueddenberg, 10.1016/S0065-3276(08)60558-4 journal journal Adv. Quantum Chem. volume 7, pages 1 (year 1973)NoStop [Note6()]Note6 note This relation can also be established using the theory of Wigner-D matrices <cit.> where D^j_m'm(α ,β ,γ ) ="426830AY_jm'(θ ,ϕ ) |R_αβγ Y_jm(θ ,ϕ )"526930B _L^2 (see p. 41 and 276 of <cit.>) and D_m'm^j(α ,β ,γ ) ="426830AT_jm'^(ℓ ) | ℛ_αβγ T_jm^(ℓ ) "526930B (see p. 45 of <cit.>). Here, ℛ_αβγ C := ℜ_αβγ C ℜ^-1_αβγ with ℜ_αβγ:= e^- i α F_z e^- i β F_y e^- i γ F_z (see Sec. <ref>) where ℛ_αβ = ℛ_αβ 0.Stop | http://arxiv.org/abs/1707.08465v2 | {
"authors": [
"David Leiner",
"Robert Zeier",
"Steffen J. Glaser"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170726143413",
"title": "Wigner tomography of multispin quantum states"
} |
Chiral transport along magnetic domain walls in the quantum anomalous Hall effect D. Goldhaber-Gordon December 30, 2023 ================================================================================= Français § INTRODUCTION Every smooth curve of genus two admits a unique degree-two hyperelliptic map to ℙ^1. The Riemann-Hurwitz formula forces such a map to have six ramification points called Weierstrass points; each non-Weierstrass point p exists as part of a conjugate pair (p,p') such that the images of p and p' agree under the hyperelliptic map.The locus of curves of genus two with ℓ marked Weierstrass points is codimension ℓ inside the moduli space _2,ℓ, and in <cit.> it is shown that the class of the closure of this locus is rigid and extremal in the cone of effective classes of codimension ℓ. Our main theorem extends their result to _2,ℓ,2m,n⊆_2,ℓ+2m+n,the locus of genus-two curves with ℓ marked Weierstrass points, m marked conjugate pairs, and n free marked points (see Definition <ref>). For ℓ,m,n≥ 0, the class _2,ℓ,2m,n, if non-empty, is rigid and extremal in the cone of effective classes of codimension ℓ+m in _2,ℓ+2m+n.In <cit.>, the authors show that the effective cone of codimension-two classes of _2,n has infinitely many extremal cycles for every n. Here we pursue a perpendicular conclusion: although in genus two ℓ≤ 6, the number of conjugate pairs and number of free marked points are unbounded, so that the classes _2,ℓ,2m,n form an infinite family of rigid and extremal cycles in arbitrarily-high codimension. Moreover, the induction technique used to prove the main result is genus-agnostic, pointing towards a natural extension of the main theorem to higher genus given a small handful of low-codimension cases.When ℓ + m ≥ 3, our induction argument (Theorem <ref>) is a generalization of that used in <cit.> to include conjugate pairs and free points; it relies on pushing forward an effective decomposition of one hyperelliptic class onto other hyperelliptic classes and showing that the only term of the decomposition to survive all pushforwards is the original class itself. This process is straightforward when there are at least three codimension-one conditions available to forget; however, when ℓ + m = 2, and in particular when ℓ = 2 and m = 0, more care must be taken. The technique used in <cit.> to overcome this problematic subcase relies on an explicit expression for [_2,2,0,0] which becomes cumbersome when a non-zero number of free marked points are allowed. Although adding free marked points can be described via pullback, pullback does not preserve rigidity and extremality in general, so we introduce an intersection-theoretic calculation using tautological ω-classes to handle this case instead.The base case of the induction (Theorem <ref>) is shown via a criterion (Lemma <ref>) given by<cit.> for rigidity and extremality for divisors; it amounts to an additional pair of intersection calculations. We utilize the theory of moduli spaces of admissible covers to construct a suitable curve class for the latter intersection, a technique which generalizes that used in <cit.> for the class of _2,1,0,0. §.§ Structure of the paper.We begin in §<ref> with some background on _g,n and cones of effective cycles. This section also contains the important Lemma <ref> upon which Theorem <ref> depends. In §<ref>, we prove Theorem <ref>, which establishes the base case for the induction argument of our main result, Theorem <ref>. Finally, we conclude in §<ref> with a discussion of extending these techniques for g≥ 3 and possible connections to a CohFT-like structure.§.§ Acknowledgments.The author wishes to thank Nicola Tarasca, who was kind enough to review an early version of the proof of the main theorem and offer his advice. The author is also greatly indebted to Renzo Cavalieri for his direction and support.§ PRELIMINARIES ON _G,N AND EFFECTIVE CYCLES §.§ Moduli spaces of curves, hyperelliptic curves, and admissible covers.We work throughout in _g,n, the moduli space of isomorphism classes of stable genus g curves with n (ordered) marked points. If 2g-2+n > 0 this space is a smooth Deligne-Mumford stack of dimension 3g-3+n. We denote points of _g,n by [C; p_1,…,p_n] with p_1,…,p_n∈ C smooth marked points. For fixed g, we may vary n to obtain a family of moduli spaces related by forgetful morphisms: for each 1≤ i ≤ n, the map π_p_i:_g,n→_g,n-1 forgets the ith marked point and stabilizes the curve if necessary. The maps ρ_p_i:_g,n→_g,{p_i} are the rememberful morphisms which are the composition of all possible forgetful morphisms other than π_p_i. Due to the complexity of the full Chow ring of _g,n, the tautological ring R^*(_g,n) is often considered instead <cit.> (for both rings we assume rational coefficients). Among other classes, this ring contains the classes of the boundary strata, as well as all ψ- and λ-classes. For 1≤ i ≤ n the class ψ_p_i is defined to be the first Chern class of the line bundle on _g,n whose fiber over a given isomorphism class of curves is the cotangent line bundle at the ith marked point of the curve; λ_1 is the first Chern class of the Hodge bundle. The tautological ring also includes pullbacks of all ψ- and λ-classes, including the ω-classes, sometimes called stable ψ-classes. The class ω_p_i is defined on _g,n for g,n≥ 1 as the pullback of ψ_p_i along ρ_p_i. Several other notable cycles are known to be tautological, including the hyperelliptic classes considered below (<cit.>).Hyperelliptic curves are those which admit a degree-two map to ℙ^1. The Riemann-Hurwitz formula implies that a hyperelliptic curve of genus g contains 2g+2 Weierstrass points which ramify over the branch locus in ℙ^1. For a fixed genus, specifying the branch locus allows one to recover the complex structure of the hyperelliptic curve and hence the hyperelliptic map. Thus for g≥ 2, the codimension of the locus of hyperelliptic curves in _g,n is g-2. In this context, requiring that a marked point be Weierstrass (resp. two marked points be a conjugate pair) is a codimension-one condition for genus at least two.We briefly use the theory of moduli spaces of admissible covers to construct a curve in _2,n in Theorem <ref>. These spaces are particularly nice compactifications of Hurwitz schemes. For a thorough introduction, the standard references are <cit.> and <cit.>. For a more hands-on approach in the same vein as our usage, see as well <cit.>.§.§ Notation.We use the following notation for boundary strata on _g,n ; all cycles classes are given as stack fundamental classes. For g≥ 1, the divisor class of the closure of the locus of irreducible nodal curves is denoted by δ_irr. By δ_h,P we mean the class of the divisor whose general element has one component of genus h attached to another component of genus g-h, with marked points P on the genus h component and marked points {p_1,…,p_n}\ P on the other. By convention δ_h,P = 0 for unstable choices of h and P.Restrict now to the case of g=2. We use W_2,P to denote the codimension-two class of the stratum whose general element agrees with that of δ_2,P, with the additional requirement that the node be a Weierstrass point. We denote by γ_1,P the class of the closure of the locus of curves whose general element has a genus-one component containing the marked points P meeting in two points conjugate under a hyperelliptic map a rational component with marked points {p_1,…,p_n}\ P (see Figure <ref>).The space Adm_2 0,t_1,…,t_6,u_1±,…,u_n± is the moduli space of degree-two admissible covers of genus two with marked ramification points (Weierstrass points) t_i and marked pairs of points (conjugate pairs) u_j+ and u_j-. This space comes with a finite map c to _0,{t_1,…,t_6,u_1,…,u_n} which forgets the cover and remembers only the base curve and its marked points, which are the images of the markings on the source. It comes also with a degree 2^n map s to _2,1+n which forgets the base curve and all u_j+ and t_i other than t_1 and remembers the (stabilization of the) cover.§.§ ω-class lemmas.The following two lemmas concerning basic properties of ω-classes prove useful in the last subcase of Theorem <ref>. The first is a unique feature of these classes, and the second is the ω-class version of the dilaton equation.Let g≥ 1, n≥ 2, and P⊂{p_1,…,p_n} such that |P|≤ n-2. Then for any p_i,p_j∉Pω_p_i·δ_g,P = ω_p_j·δ_g,Pon _g,n. This follows immediately from Lemma 1.9 in <cit.>. Let g,n≥ 2. Then on _g,n,π_p_i*ω_p_j = 2g - 2if i=j, and 0 otherwise. Let P = {p_1,…,p_n}. When i=j, the pushforward reduces to the usual dilaton equation for ψ_p_i on _g,{p_i}. If π is the morphism which forgets all marked points, the diagram [row sep=1cm, column sep=1cm]_g,n [d, "ρ_p_i"] [r, "π_p_i"] _g,P\{p_i} [d, "π"] _g,{p_i} [r, "π_p_i"] _gcommutes, so π_p_i*ω_p_i = π_p_i*ρ_p_i^*ψ_p_i = π^*π_p_i*ψ_p_i = (2g-2)1.If i≠ j, then π_p_i*ω_p_j = π_p_i*π_p_i^*ω_p_j = 0. §.§ Cones and properties of effective classes.For a projective variety X, the sum of two effective codimension-d classes is again effective, as is any ℚ_+-multiple of the same. This gives a natural convex cone structure on the set of effective classes of codimension d inside the ℚ vector space of all codimension-d classes, called the effective cone of codimension-d classes and denoted ^d(X). The closure of this cone (in the usual ℝ^n topology) is ^d(X), the pseudo-effective cone of codimension-d classes. Given an effective class E in the Chow ring of X, an effective decomposition of E is an equalityE = ∑_s=1^m a_sE_swith a_s > 0 and E_s irreducible effective cycles on X for all s. The main properties we are interested in for classes in the pseudo-effective cone are rigidity and extremality.Let E∈^d(X).E is rigid if any effective cycle with class rE is supported on the support of E.E is extremal if, for any effective decomposition of E, all E_s are proportional to E.When d=1, elements of the cone correspond to divisor classes, and the study of ^1(_g,n) is fundamental in the theory of the birational geometry of these moduli spaces. For example, _0,n is known to fail to be a Mori dream space for n≥ 10 (first for n≥ 134 in <cit.>, then for n≥ 13 in <cit.>, and the most recent bound in <cit.>). For n≥ 3 in genus one, <cit.> show that _1,n is not a Mori dream space; the same statement is true for _2,n by <cit.>. In these and select other cases, the pseudo-effective cone of divisors has been shown to have infinitely many extremal cycles and thus is not rational polyhedral (<cit.>).These results are possible due in large part to the following lemma, which plays an important role in Theorem <ref>. Here a moving curve 𝒞 in D is an irreducible effective curve 𝒞, the deformations of which cover a Zariski-dense subset of D.Let D be an irreducible effective divisor in a projective variety X, and suppose that 𝒞 is a moving curve in D satisfying ∫_X[D]·[𝒞] < 0. Then [D] is rigid and extremal. □ Using Lemma <ref> to show a divisor D is rigid and extremal in fact shows more: if the lemma is satisfied, the boundary of the pseudo-effective cone is polyhedral at D. We do not rely on this fact, but see <cit.> for further discussion. Lemma <ref> allows us to change a question about the pseudo-effective cone into one of intersection theory and provides a powerful tool in the study of divisor classes. Unfortunately, it fails to generalize to higher-codimension classes, where entirely different techniques are needed. Consequently, much less is known about ^d(_g,n) for d≥ 2. This paper is in part inspired by <cit.>, where the authors show that certain hyperelliptic classes of higher codimension are rigid and extremal in genus two. In <cit.>, the authors develop additional extremality criteria to show that in codimension-two there are infinitely many extremal cycles in _1,n for all n≥ 5 and in _2,n for all n≥ 2, as well as showing that two additional hyperelliptic classes of higher genus are extremal. These criteria cannot be used directly for the hyperelliptic classes we consider; this is illustrative of the difficulty of proving rigidity and extremality results for classes of codimension greater than one. § MAIN THEOREMIn this section we prove our main result, which culminates in Theorem <ref>. The proof proceeds via induction, with the base cases given in Theorem <ref>. We begin by defining hyperelliptic classes on _g,n. Fix integers ℓ,m,n≥ 0. Denote by _g,ℓ,2m,n the closure of the locus of hyperelliptic curves in _g,ℓ+2m+n with marked Weierstrass points w_1,…,w_ℓ; pairs of marked points +_1,-_1,…,+_m,-_m with +_j and -_j conjugate under the hyperelliptic map; and free marked points p_1,…,p_n with no additional constraints. By hyperelliptic class, we mean a non-empty class equivalent to some [_g,ℓ,2m,n] in the Chow ring of _g,ℓ+2m+n. Lemma <ref> allows us to establish the rigidity and extremality of the two divisor hyperelliptic classes for genus two, which together provide the base case for Theorem <ref>. For n ≥ 0, the class of _2,0,2,n is rigid and extremal in ^1(_2,2+n) and the class of _2,1,0,n is rigid and extremal in ^1(_2,1+n). Define a moving curve 𝒞 in _2,0,2,n by fixing a general genus-two curve C with n free marked points p_1,…,p_n and varying the conjugate pair (+,-). Since [_2,0,2,n] = π_p_n^*⋯π_p_1^*[_2,0,2,0], by the projection formula and the identity (see <cit.>) [_2,0,2,0] = -λ + ψ_+ + ψ_- - 3δ_2,∅ - δ_1,∅,we compute∫__2,2+n[_2,0,2,n]· [𝒞]= ∫__2,2[_2,0,2,0] ·π_p_1*⋯π_p_n*[𝒞] = 0 + (4-2+6) + (4-2+6) - 3(6) - 0 = -2.In particular, intersecting with λ is 0 by projection formula. Intersecting with either ψ-class can be seen as follows: pullback ψ_i from _2,1 to ψ_i - δ_2,∅, then use projection formula on ψ_i back to _2,1. This is just 2g-2, since ψ_i is the first Chern class of the cotangent bundle of C over i. The intersection with δ_2,∅ corresponds to the 2g+2 Weierstrass points. Finally, δ_1,∅ intersects trivially, since by fixing C we have only allowed rational tail degenerations. As _2,0,2,n is irreducible, it is rigid and extremal by Lemma <ref>.We next apply Lemma <ref> by constructing a moving curve ℬ which intersects negatively with _2,1,0,n using the following diagram. Note that the image of s is precisely _2,1,0,n⊂_2,1+n.[row sep=1cm, column sep=1cm]Adm_2 0,t_1,…,t_6,u_1±,…,u_n± [d, "c"] [r, "s"] _2,1+n _0,{t_1,…,t_6,u_1,…,u_n} [d, "π_t_6"] _0,{t_1,…,t_5,u_1,…,u_n}Fix a generic point [x_n] in _0,{t_1,…,t_5,u_1,…,u_n} corresponding to a smooth marked curve and the point [b_n] in _0,{t_1,…,t_5,u_1,…,u_n} corresponding to a chain of ℙ^1s with n+3 components and marked points as shown in Figure <ref> (if n=0, t_4 and t_5 are on the final component; if n=1, t_5 and u_1 are on the final component; etc.), and define [𝒳_n] = s_*c^*π_t_6^*[x_n] and [ℬ_n] = s_*c^*π_t_6^*[b_n] (with an additional relabeling of t_1 to w_1 and u_j_- to p_j). Now 𝒳_n is a moving curve in _2,1,0,n, and the deformations of 𝒳_n are parametrized by _0,{t_1,…,t_5,u_1,…,u_n}. Because the image of s is _2,1,0,n, the intersection [_2,1,0,n] · [𝒳_n] is not transverse, so we correct with minus the Euler class of the normal bundle of _2,1,0,n in _2,1+n restricted to 𝒳_n. Further, as all points in _0,{t_1,…,t_5,u_1,…,u_n} are equivalent, we may replace [𝒳_n] with [ℬ_n] in the intersection. In other words,∫__2,1+n[_2,1,0,n] · [𝒳_n]= ∫__2,1+n -π_p_n^*⋯π_p_1^*ψ_w_1·[𝒳_n] = ∫__2,1+n -π_p_n^*⋯π_p_1^*ψ_w_1·[ℬ_n] = ∫__2,1 -ψ_w_1·[ℬ_0].By passing to the space of admissible covers (see, for example, <cit.>), this integral is seen to be a positive multiple (a power of two) of∫__1,2 -ψ_w_1·[_1,2,0,0]= ∫__1,2 -ψ_w_1· (3ψ_w_1) = -1/8,where we have used the fact that [_1,2,0,0] = 3ψ_w_1 <cit.>. Therefore, by Lemma <ref>, _2,1,0,n is rigid and extremal. This establishes the base case for the inductive hypothesis in Theorem <ref>. The induction procedure differs fundamentally for the codimension-two classes, so we first prove the following short lemma to simplify the most complicated of those.The class W_2,{p_1,…,p_n} is not proportional to [_2,2,0,n] on _2,2+n. Let P = {p_1,…,p_n}. Note that in W_2,P the marked points w_1 and w_2 carry no special restrictions, and the class is of codimension two. Because the point w_1 is on a three-pointed rational component of the general element of W_2,P,∫__2,2+n W_2,P·ψ_w_1^n+3 = ∫__0,3ψ_w_1^n+3 = 0.However, using the equality[_2,2,0,0] = 6ψ_w_1ψ_w_2 - 3/2(ψ_w_1^2+ψ_w_2^2) - (ψ_w_1 + ψ_w_2)(21/10δ_1,{w_1} + 3/5δ_1,∅ + 1/20δ_irr)established in <cit.> and Faber's Maple program <cit.>, we compute∫__2,2+n[_2,2,0,n] ·ψ_w_1^n+3 = ∫__2,2+nπ_p_1^*⋯π_p_n^*[_2,2,0,0] ·ψ_w_1^n+3= ∫__2,2[_2,2,0,0] ·π_p_1*⋯π_p_n*ψ_w_1^n+3= ∫__2,2(6ψ_w_1ψ_w_2 - 3/2(ψ_w_1^2+ψ_w_2^2)- (ψ_w_1 + ψ_w_2)(21/10δ_1,{w_1} + 3/5δ_1,∅ + 1/20δ_irr)) ·ψ_w_1^3 = 1/384,so W_2,P is not a non-zero multiple of [_2,2,0,n]. We are now ready to prove our main result. The bulk of the effort is in establishing extremality, though the induction process does require rigidity at every step as well. Although we do not include it until the end, the reader is free to interpret the rigidity argument as being applied at each step of the induction.The overall strategy of the extremality portion of the proof is as follows. Suppose [_2,ℓ,2m,n] is given an effective decomposition. We show (first for the classes of codimension at least three, then for those of codimension two) that any terms of this decomposition which survive pushforward by π_w_i or π_+_j must be proportional to the hyperelliptic class itself. Therefore we may write [_2,ℓ,2m,n] as an effective decomposition using only classes which vanish under pushforward by the forgetful morphisms; this is a contradiction, since the hyperelliptic class itself survives pushforward.For ℓ,m,n≥ 0, the class _2,ℓ,2m,n, if non-empty, is rigid and extremal in ^ℓ+m(_2,ℓ+2m+n). We induct on codimension; assume the claim holds when the class is codimension ℓ+m-1. Theorem <ref> is the base case, so we may further assume ℓ+m ≥ 2. Now, suppose that[_2,ℓ,2m,n] = ∑_s a_s[X_s] + ∑_t b_t[Y_t]is an effective decomposition with [X_s] and [Y_t] irreducible codimension-(ℓ+m) effective cycles on _2,ℓ+2m+n, with [X_s] surviving pushforward by some π_w_i or π_+_j and [Y_t] vanishing under all such pushforwards, for each s and t.Fix an [X_s] appearing in the right-hand side of (<ref>). If ℓ≠ 0, suppose without loss of generality (on the w_i) that π_w_1*[X_s] ≠ 0. Since π_w_1*[_2,ℓ,2m,n] = (6-(ℓ-1))[_2,ℓ-1,2m,n]is rigid and extremal by hypothesis, π_w_1*[X_s] is a positive multiple of the class of _2,ℓ-1,2m,n and X_s⊆ (π_w_1)^-1_2,ℓ-1,2m,n. By the commutativity of the following diagrams and the observation that hyperelliptic classes survive pushforward by all π_w_i and π_+_j, we have that π_w_i*[X_s] ≠ 0 and π_+_j*[X_s] ≠ 0 for all i and j. [row sep=1cm, column sep=1cm]_2,ℓ,2m,n [d, "π_w_1"] [r, "π_+_j"] _2,ℓ,2(m-1),n+1 [d, "π_w_1"] _2,ℓ,2m,n [d, "π_w_1"] [r, "π_w_i"] _2,ℓ-1,2m,n [d, "π_w_1"] _2,ℓ-1,2m,n [r, "π_+_j"] _2,ℓ-1,2(m-1),n+1 _2,ℓ-1,2m,n [r, "π_w_i"] _2,ℓ-2,2m,n If ℓ = 0, suppose without loss of generality (on the +_j) that π_+_1*[X_s] ≠ 0. Then the same conclusion holds that [X_s] survives all pushforwards by π_+_j, sinceπ_+_1*[_2,ℓ,2m,n] = [_2,ℓ,2(m-1),n+1]is rigid and extremal by hypothesis, and π_+_1 commutes with π_+_j.It follows that for any ℓ+m≥ 2X_s ⊆⋂_i,j((π_w_i)^-1_2,ℓ-1,2m,n∩ (π_+_j)^-1_2,ℓ,2(m-1),n+1).We now have two cases. If ℓ+m ≥ 3, any ℓ + 2m - 1 non-free marked points in a general element of X_s are distinct Weierstrass or conjugate pair marked points, and hence all ℓ + 2m such non-free marked points in a general element of X_s are distinct Weierstrass or conjugate pair marked points. In this case we conclude that [X_s] is a positive multiple of [_2,ℓ,2m,n]. If ℓ+m = 2, we must analyze three subcases.If ℓ = 0 and m = 2, thenX_s⊆ (π_+_1)^-1_2,0,2,n+1∩ (π_+_2)^-1_2,0,2,n+1.The modular interpretation of the intersection leaves three candidates for [X_s]: W_2,P or γ_1,P for some P containing neither conjugate pair, or [_2,0,4,n] itself. However, for the former two, we have W_2,P≠π_+_1(W_2,P) and γ_1,P≠π_+_1(γ_1,P) for all such P, contradicting our assumption that the class survived pushforward. Thus [X_s] is proportional to [_2,0,4,n].If ℓ = 1 and m = 1, similar to the previous case, [X_s] could be [_2,1,2,n] or W_2,P or γ_1,P for some P containing neither the conjugate pair nor the Weierstrass point. However, if X_s is either of the latter cases, we have X_s ≠π_+_1(X_s), again contradicting our assumption about the non-vanishing of the pushforward, and so again [X_s] must be proportional to [_2,1,2,n].If ℓ = 2 and m = 0, as before, [X_s] is either [_2,2,0,n] itself or W_2,P or γ_1,P for P = {p_1,…,p_n}. Now W_2,P = π_w_iW_2,P, so the argument given in the other subcases fails (though γ_1,P is still ruled out as before). Nevertheless, we claim that W_2,P cannot appear on the right-hand side of (<ref>) for _2,2,0,n; to show this we induct on the number of free marked points n. The base case of n=0 is established in <cit.>, so assume that _2,2,0,n-1 is rigid and extremal for some n≥ 1. Suppose for the sake of contradiction that[_2,2,0,n] = a_0W_2,P + ∑_s a_s[Z_s]is an effective decomposition with each [Z_s] an irreducible codimension-two effective cycle on _2,2+n. Note thatW_2,P = π_p_n^*W_2,P\{p_n} - W_2,P\{p_n}.Multiply (<ref>) byω_p_n and push forward by π_p_n. On the left-hand side,π_p_n*(ω_p_n·[_2,2,0,n])= π_p_n*(ω_p_n·π_p_n^*[_2,2,0,n-1]) = π_p_n*(ω_p_n) ·[_2,2,0,n-1] = 2 [_2,2,0,n-1],having applied Lemma <ref>. Combining this with the right-hand side,2[_2,2,0,n-1]= a_0π_p_n*(ω_p_n·π_p_n^*W_2,P\{p_n} - ω_p_n· W_2,P\{p_n}) + ∑_s a_sπ_p_n*(ω_p_n·[Z_s]) = 2a_0W_2,P\{p_n} + π_p_n*(ω_p_n· W_2,P\{p_n}) + ∑_s a_sπ_p_n*(ω_p_n·[Z_s]).The term π_p_n*(ω_p_n· W_2,P\{p_n}) vanishes by Lemma <ref>:π_p_n*(ω_p_n· W_2,P\{p_n-1})= π_p_n*(ω_w_1· W_2,P\{p_n}) = π_p_n*(π_p_n^*ω_w_1· W_2,P\{p_n}) = ω_w_1·π_p_n* W_2,P\{p_n}= 0,where w_1 is the Weierstrass singular point on the genus-two component of W_2,P\{p_n}. Altogether, we have2[_2,2,0,n-1]= 2a_0W_2,P\{p_n} + ∑_s a_sπ_p_n*(ω_p_n·[Z_s]). <cit.> establishes that ψ_p_n is semi-ample on _2,{p_n}, so ω_p_n is semi-ample, and hence this is an effective decomposition. By hypothesis, _2,2,0,n-1 is rigid and extremal, so W_2,P\{p_n} must be a non-zero multiple of [_2,2,0,n-1], which contradicts Lemma <ref>. Therefore W_2,P cannot appear as an [X_s] in (<ref>).Thus for all cases of ℓ+m = 2 (and hence for all ℓ+m≥ 2), we conclude that each [X_s] in (<ref>) is a positive multiple of [_2,ℓ,2m,n]. Now subtract these [X_s] from (<ref>) and rescale, so that[_2,ℓ,2m,n] = ∑_t b_t[Y_t].Recall that each [Y_t] is required to vanish under all π_w_i* and π_+_j*. But the pushforward of [_2,ℓ,2m,n] by any of these morphisms is non-zero, so there are no [Y_t] in (<ref>). Hence [_2,ℓ,2m,n] is extremal in ^ℓ+m(_2,ℓ+2m+n).For rigidity, suppose that E:= r[_2,ℓ,2m,n] is effective. Since π_w_i*E = (6-(ℓ-1))r[_2,ℓ-1,2m,n] and π_+_j*E = r[_2,ℓ,2(m-1),n+1] are rigid and extremal for all i and j, we have that π_w_i*E is supported on _2,ℓ-1,2m,n and π_+_j*E is supported on _2,ℓ,2(m-1),n+1. This implies that E is supported on the intersection of (π_w_i)^-1[_2,ℓ-1,2m,n] and (π_+_j)^-1[_2,ℓ,2(m-1),n+1] for all i and j. Thus E is supported on _2,ℓ,2m,n, so [_2,ℓ,2m,n] is rigid.§ HIGHER GENUSThe general form of the inductive argument in Theorem <ref> holds independent of genus for g≥ 2. However, for genus greater than one, the locus of hyperelliptic curves in _g is of codimension g-2, so that the base cases increase in codimension as g increases.The challenge in showing the veracity of the claim for hyperelliptic classes in arbitrary genus is therefore wrapped up in establishing the base cases of codimension g-1 (corresponding to Theorem <ref>) and codimension g (corresponding to the three ℓ+m = 2 subcases in Theorem <ref>).In particular, our proof of Theorem <ref> relies on the fact that _2,0,2,n and _2,1,0,n are divisors, and the subcase ℓ = 2 in Theorem <ref> depends on our ability to prove Lemma <ref>. This in turn requires the description of _2,2,0,0 given by <cit.>. More subtly, we require that ψ_p_n be semi-ample in _2,{p_n}, which is known to be false in genus greater than two in characteristic 0 <cit.>. In genus three, <cit.> show that the base case _3,1,0,0 is rigid and extremal, though it is unclear if their method will extend to _3,1,0,n. Moreover, little work has been done to establish the case of a single conjugate pair in genus three, and as the cycles move farther from divisorial classes, such analysis becomes increasingly more difficult.One potential avenue to overcome these difficulties is suggested by work of Renzo Cavalieri and Nicola Tarasca <cit.>. They use an inductive process to describe hyperelliptic classes in terms of decorated graphs using the usual dual graph description of the tautological ring of _g,n. Such a formula for the three necessary base cases would allow for greatly simplified intersection-theoretic calculations, similar to those used in Theorem <ref> and Lemma <ref>. Though such a result would be insufficient to completely generalize our main theorem, it would be a promising start.We also believe the observation that pushing forward and pulling back by forgetful morphisms moves hyperelliptic classes to (multiples of) hyperelliptic classes is a useful one. There is evidence that a more explicit connection between marked Weierstrass points, marked conjugate pairs, and the usual gluing morphisms between moduli spaces of marked curves exists as well, though concrete statements require a better understanding of higher genus hyperelliptic loci. Although it is known that hyperelliptic classes do not form a cohomological field theory over the full _g,n, a deeper study of the relationship between these classes and the natural morphisms among the moduli spaces may indicate a CohFT-like structure, which in turn would shed light on graph formulas or other additional properties. X[ACV01]acv Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers, Comm. in Algebra 31 (2001), no. 8, 3547–3618.[BC18]bcomega Vance Blankers and Renzo Cavalieri, Intersections of ω classes in _g,n, Proceedings of Gökova Geometry–Topology Conference 2017 (2018), 37–52.[Cav06]cavalierihodgeint Renzo Cavalieri, Hodge-type integrals on moduli spaces of admissible covers, Geom. Topol. Monogr. 8 (2006), 167–194.[Cav16]cavalierihurwitz , Hurwitz theory and the double ramification cycle, Jpn. J. Math. 11 (2016), no. 2, 305–331.[CC14]chencoskun2014 Dawei Chen and Izzet Coskun, Extremal effective divisors on _1,n, Math. Ann. 359 (2014), no. 3-4, 891–908.[CC15]chencoskun2015 , Extremal higher codimension cycles on moduli spaces of curves, Proc. London Math. Soc. 111 (2015), no. 1, 181–204.[CT15]castravettevelev Ana-Maria Castravet and Jenia Tevelev, M_0,n is not a Mori dream space, Duke Math. J. 164 (2015), no. 8, 3851–3878.[CT16]chentarasca Dawei Chen and Nicola Tarasca, Extremality of loci of hyperelliptic curves with marked Weierstrass points, Algebra & Number Theory 10 (2016), no. 1, 1935–1948.[CT19]cavalieritarasca2017 Renzo Cavalieri and Nicola Tarasca, Classes of weierstrass points on genus 2 curves,Trans. Amer. Math. Soc. 372 (2019), no. 4, 2467–2492.[Fab]faberprogram Carel Faber, Maple program for computing hodge integrals.Available at http://math.stanford.edu/ vakil/programs. [FP05]faberpandharipande Carel Faber and Rahul Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. 7 (2005), no. 1, 13–49.[GK16]gonzalezkaru José Luis González and Kalle Karu, Some non-finitely generated Cox rings, Compos. Math. 152 (2016), no. 5, 984–996.[HKL18]hkl2016 Jürgen Hausen, Simon Keicher, and Antonio Laface, On blowing up the weighted projective plane, Mathematische Zeitschrift 290 (2018), 1339–1358.[HM82]harrismumford Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–88.[Kee99]keel99 Seán Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. 149 (1999), no. 1, 253–286.[Log03]logan2003 Adam Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no. 1, 105–138.[Mul17]mullane2017 Scott Mullane, On the effective cone of ℳ_g,n, Advances in Mathematics 320 (2017), 500–519.[Opi16]opie2016 Morgan Opie, Extremal divisors on moduli spaces of rational curves with marked points, Michigan Math. J. 65 (2016), no. 2, 251–285.[Rul01]rulla01 William Frederick Rulla, The birational geometry of moduli space M(3) and moduli space M(2,1), Ph.D. thesis, University of Texas at Austin, 2001. | http://arxiv.org/abs/1707.08676v5 | {
"authors": [
"Vance Blankers"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170727005312",
"title": "Hyperelliptic classes are rigid and extremal in genus two"
} |
Building Detection from Satellite Images on a Global Scale Keisuke Harigaya December 30, 2023 ==========================================================In the last several years, remote sensing technology has opened up the possibility of performing large scale building detection from satellite imagery.Our work is some of the first to create population density maps from building detection on a large scale.The scale of our work on population density estimation via high resolution satellite images raises many issues, that we will address in this paper.The first was data acquisition.Labeling buildings from satellite images is a hard problem, one where we found our labelers to only be about 85% accurate at.There is a tradeoff of quantity vs. quality of labels, so we designed two separate policies for labels meant for training sets and those meant for test sets, since our requirements of the two set types are quite different.We also trained weakly supervised footprint detection models with the classification labels, and semi-supervised approaches with a small number of pixel-level labels, which are very expensive to procure. § INTRODUCTIONWith the recent improvements in remote sensing technology, there has been a lot of work in building detection and classification from high resolution satellite imagery.However, we are the first to implement a system on a global scale.Other work uses handpicked features to define buildings <cit.> <cit.> which would not scale well across countries with very different styles of buildings. The work closest to ours is done by Yuan <cit.>, which also uses pixel level convolutional neural networks for building detection, but is only validated on a handful of cities in the US and likely would not transfer well to smaller settlements or other countries.In order to speed up our pipeline we need a fast bounding box proposal algorithm to limit the number of images that need to be run through our convolutional neural network.To maintain high recall, however, we need to be careful to not filter out too many candidates.We used a naive bounding box proposal algorithm, by performing straight edge detection to extract smaller masks to run through our classification network.This reduced the amount of landmass to process by 50%. The distribution of buildings is still very negatively skewed, where only 2% of proposals are positive.This also means we need to sample a large number of masks in order to have confident precision and recall numbers by country.We also use a weak building classifier to filter masks with over 0.3 IoU (intersection over union) by choosing the mask with the highest probability of containing a building in the center, since these overlapping masks are likely to contain the same building.Discovering systematic issues with our models is also a slow, manual problem that requires visualization of .kmz files, pinpointing large numbers of false positive or false negative areas, and debugging the causes. The problems encountered included noise, contrast issues, cloud cover, or just deficiencies in the model, and we set up a feedback loop to fix those problems. We will be open sourcing our population density results as well as our labeled dataset as a benchmark for future efforts.§ DATASET COLLECTION ISSUESWe have two goals for data collection, obtaining labels for training, and accuracy numbers on a country level.Obtaining accuracy numbers of the entire pipeline for a single country requires randomly sampling from all possible 64x64 masks.That distribution is incredibly skewed, and randomly sampling enough masks to obtain a reasonable confidence interval on accuracy is expensive.Instead, we measure how well our neural network performs building classification by randomly sampling from the distribution of masks generated by our bounding box proposal algorithm.The assumption is that the bounding box proposal algorithm only eliminates clear negatives, so reduces skew on the underlying distribution without affecting recall of the overall pipeline.This drops the number of labels we need by a factor of 10, because our new distribution now is 2%-5% positive.Collecting a training set went through several iterations because we want a more balanced dataset for training so the model can get enough samples of both the background and the building classes.We also employ simple active learning techniques by sampling from masks the network was "less sure" about, where the probability was closer to the threshold.§ GENERALIZING A GLOBAL MODELTraining a global building classification model has trade-offs.Buildings can look very different across different countries, but there is still a lot of information that can be transferred from country to country.We initially started with a model trained only on Tanzania, which when applied to a new country had a large drop in accuracy.However, we found that as we labeled data in more countries and re-trained our model with the new data, our new global model performed better on Tanzania than a Tanzania specific model.The generalizations learned from other countries made the model more robust.Another argument for training a global model is that building a large training set takes time, and the amount of data required to train a model from scratch for each country was prohibitive.The trade-off is that the global model doesn't work equally well on all countries, and we found it necessary to perform some amount of model specialization.We fine-tuned the global model with the same samples it had seen from the initial training, but only from a handful of countries that we wanted it to improve upon.We saw gains of 20-40% in precision and recall on the validation set using the extra fine-tuning step, but noticed there were trade-offs.The training and validation sets gave no evidence of overfitting, but we saw an increase in systematic false positives when visualizing the results on a country level, in certain countries. §.§ Building Classification ModelThe classification model we trained was a weakly supervised version of SegNet <cit.>, which is a fast yet accurate pixel classification network that uses deconvolution layers.We trained with weak “pixel level” labels, and generate a mask level probability using global average pooling on the final pixel level probabilities over the 64x64 mask. We have 500TB of satellite imagery, and being able to run the model over all these countries (multiple times) is crucial for fast iteration.It was a non-trivial task to develop a model that was large enough to capture the complex idea of what defines a building, while also being small enough to run quickly during inference time.SegNet performed well on this by saving the indices from the max pooling layers to perform non-linear upsampling in the deconvolution layers.§.§ Building Segmentation Model Finely pixel-wise labeled data is extremely time consuming to acquire, and errors will accumulate especially for small foreground objects. Instead of utilizing fully supervised semantic segmentation method such as FCN <cit.>, we investigated weakly supervised segmentation models relying on feedback neural network <cit.>, which utilizes the large amount of “cheap” weakly-supervised training data. Notably, to increase the efficiency of semantic segmentation, the classification model is composed to filter out negative candidate regions. By combining results from both models, the segmentation model successfully suppress false positives and generate best results, with an example shown in Figure <ref>§ DEALING WITH SYSTEMATIC ERRORS§.§ Finding Systematic Errors The precision and recall numbers we measure by randomly sampling from the mask candidates do not account for systematic errors arising from varying satellite image quality. To discover those systematic errors, we adopt both visually inspection and evaluation using external data.Intuitively, we visualize our results by construction KMZ files and overlaying with Google Earth to manually pinpoint areas of concern. We also use this strategy to sample ambiguous training data to fine-tune our model to reduce the chance of further systematic errors. Moreover, we also quantitatively measure systematic errors at a coarser scale by comparing our results with external datasets on those areas with adequate data coverage. However, it is still an open question to discover systematic errors on large scale with less manual work. Figure <ref> shows the comparison of an error with systematic error before and after fine-tuning.§.§ Data Quality One of the reasons for systematic errors is also issues with data quality. The satellite images are taken at various times of day, and pre-processed across multiple layers for the highest quality image.However, areas with a lot of cloud cover tend to have much fewer clear images taken, and so quality suffers.This has an impact on our model, since most of the data is randomly or semi-randomly sampled, and so it does not get a lot of exposure to these poorer quality images during training. We use geographical meta-information to further detect the cloud occlusion during deploying stage.Another key factor of low data quality comes from noise, which are introduced in either imaging or image enhancing phases. Traditional image denoising approach such as BM3D <cit.> is computationally expensive in handling large imagery files, and can only work for limited type of noises, such as white noise. To this end, we train a shallow neural network end-to-end by mimicking several kinds of noise existed in satellite images. The trained denoising model is appended as a transformer before imagery is fed to the classification network. Comparison of classification results of the same low data quality area before and after denoising is shown in Figure <ref>.§ RESULTSOverall the SegNet model by itself achieves a precision and recall of pr=0.9, re=0.89 on a global dataset where the imbalance is such that 93% of the randomly sampled testing data is not a building. Below we have some heat maps generated of building density in three countries: Mozambique, Madagascar, and India. So far we have released datasets for 5 countries: Haiti, Malawi, Ghana, South Africa, and Sri Lanka.The rest are pending validation with third party groups.Below we show precision recall curves and best F-score with confidence intervals for each of the countries released. The estimation of population density via settlement buildings as a proxy results in significant improvement compared with previous efforts. Figure <ref> shows the comparison of previous highest resolution estimation from Galantis and our own results. This gives a totally new perspective to various social / economic research. § CONCLUSIONWe have built one of the first building detection systems that can be deployed at a global scale.Future work includes reducing the amount of iteration required to achieve a robust model as we roll out to more countries, the biggest problem of which is detecting systematic errors.Detecting and solving these systematic issues in classification is still a work in progress.We are still looking into ways to automate the data validation process and data collection methods further, which will also shorten the length of each iteration required to improve our dataset accuracy.unsrt | http://arxiv.org/abs/1707.08952v1 | {
"authors": [
"Amy Zhang",
"Xianming Liu",
"Andreas Gros",
"Tobias Tiecke"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727175630",
"title": "Building Detection from Satellite Images on a Global Scale"
} |
Cascaded Scene Flow Prediction using Semantic Segmentation Zhile RenBrown [email protected] Deqing [email protected] [email protected] B. SudderthUC [email protected] ==============================================================================================================================================================================Given two consecutive frames from a pair of stereo cameras, 3D scene flow methods simultaneously estimate the 3D geometry and motion of the observed scene.Many existing approaches use superpixels for regularization, but may predict inconsistent shapes and motions inside rigidly moving objects. We instead assume that scenes consist of foreground objects rigidly moving in front of a static background, and use semantic cues to produce pixel-accurate scene flow estimates. Our cascaded classification framework accurately models 3D scenes by iteratively refining semantic segmentation masks, stereo correspondences, 3D rigid motion estimates, and optical flow fields. We evaluate our method on the challenging KITTI autonomous driving benchmark, and show that accounting for the motion of segmented vehicles leads to state-of-the-art performance. § INTRODUCTIONThe scene flow <cit.> is the dense 3D geometry and motion of a dynamic scene. Given images captured by calibrated cameras at two (or more) frames, a 3D motion field can be recovered by projecting 2D motion (optical flow) estimates onto a depth map inferred via binocular stereo matching. Scene flow algorithms have many applications, ranging from driver assistance <cit.> to 3D motion capture <cit.>.The problems of optical flow estimation <cit.> and binocular stereo reconstruction <cit.> have been widely studied in isolation. Recent scene flow methods <cit.> leverage 3D geometric cues to improve stereo and flow estimates, as evaluated on road scenes from the challenging KITTI scene flow benchmark <cit.>. State-of-the-art scene flow algorithms <cit.> assume superpixels are approximately planar and undergo rigid 3D motion.Conditional random fields then provide temporal and spatial regularization for 3D motion estimates. Those methods generally perform well on background regions of the scene, but are significantly less accurate for moving foreground objects. Estimating the geometry of rapidly moving foreground objects is difficult, especially near motion boundaries.Vehicles are particularly challenging because painted surfaces have little texture, windshields are transparent, and reflections violate the brightness constancy assumptions underlying stereo and flow likelihoods. However, accurate estimation of vehicle geometry and motion is critical for autonomous driving applications. To improve accuracy, it is natural to design models that separately model the motion of objects and background regions <cit.>. Several recent methods for the estimation of optical flow <cit.> have used semantic cues to improve accuracy. While motion segmentation using purely bottom-up cues is challenging, recent advances in semantic segmentation <cit.> make it possible to accurately segment traffic scenes given a single RGB image. Given segmented object boundaries, object-specific 3D motion models may then be used to increase the accuracy of optical flow methods.In this paper, we use instance-level semantic segmentations <cit.> and piecewise-rigid scene flow estimates <cit.> as inputs, and integrate them via a cascade of conditional random fields (CRFs) <cit.>. We define pixel-level CRFs relating dense segmentation masks, stereo depth maps, optical flow fields, and rigid 3D motion estimates for foreground objects. Due to the high dimensionality of these variables, we refine them iteratively using a cascaded classification model <cit.>, where each stage of the cascade is tuned via structural SVM learning algorithms <cit.>. We evaluate using previous scene flow annotations <cit.> of the challenging KITTI autonomous driving benchmark <cit.>, and improve on the state-of-the-art in two-frame scene flow estimation. Our work demonstrates the importance of semantic cues in the recovery of the geometry and motion of 3D scenes. § RELATED METHODS FOR SCENE FLOW ESTIMATIONVedula <cit.> first defined the scene flow as the dense 3D motion of all points in an observed scene, and recovered voxel-based flow estimates using 2D optical flow fields from several calibrated cameras.Huguet and Devernay <cit.> then proposed a variational approach and jointly solved for stereo and optical flow, while Wedel <cit.> decoupled the stereo and flow problems for efficiency. These classic algorithms only improve marginally over modern, state-of-the-art stereo and optical flow methods. Although scene flow algorithms require more input images than standardoptical flow or stereo reconstruction methods, the task is still challenging due to the high dimensionality of the output disparity and motion fields. To reduce the solution space, Vogel <cit.> introduced a piecewise rigid scene flow (PRSF) model and used superpixels to constrain scene flow estimation.For the first time, they showed that scene flow methods could outperform stereo and optical flow methods by a large margin on the challenging KITTI dataset <cit.>.In follow-up work <cit.> they extended their formulation to multiple frames and improved accuracy.However, because the PRSF model relies on bottom-up cues for superpixel segmentation, it tends to over-segment foreground objects such as cars. Over-segmented parts are allocated independent motion models, so global information cannot be effectively shared. Inspired by the success of Vogel <cit.>, Menze and Geiger <cit.> annotated a new KITTI dataset with dynamic foreground objects for scene flow evaluation.They proposed an object scene flow (OSF) algorithm that segments the scene into independently moving regions, and encourages the superpixels within each regionto have similar 3D motion.Although the performance of OSF improved on baselines, the “objects” in their model are assumed to be planar and initialized via bottom-up motion estimation, so physical objects are often over-segmented. The inference time required for the OSF method is also significantly longer than most competing methods. The successes of convolutional neural networks (CNNs) for high-level vision tasks has motivated CNN-based regression methods for low-level vision. Dosovitskiy <cit.> introduced a denoising autoencoder network, called FlowNet, for estimating optical flow. Mayer <cit.> extended the FlowNet to disparity and scene flow estimation with a large synthetic dataset. While CNN models generate scene flow predictions rapidly, networks trained on synthetic data are not competitive with state-of-the-art methods on the real-world KITTI scene flow benchmark <cit.>. Some related work integrates automatic motion segmentation with optical flow prediction <cit.>, but assumes large differences between the motion of objects and cameras, and requires multiple input frames. Exploiting the recent success of CNNs for semantic segmentation <cit.>, semantic cues have been shown to improve optical flow estimation <cit.>. Concurrent work <cit.> also shows that semantic cues can improve scene flow estimation. In this paper, we propose a coherent model of semantic segmentation, scene geometry, and object motion.We use a cascaded prediction framework <cit.> to efficiently solve this high-dimensional inference task. We evaluate our algorithm on the challenging KITTI dataset <cit.> and show that using semantic cues leads to state-of-the-art scene flow estimates.§ MODELING SEMANTIC SCENE FLOW Given two consecutive frames I, J and their corresponding stereo pairs I', J', our goal is to estimate the segmentation mask, stereo disparity, and optical flow for each pixel in the reference frame (Fig. <ref>).Let p_i = (d_i^(1), s_i^(1), m_i, f_i) denote the variables associated with pixel i in the reference frame, whered_i^(1)∈ℝ^+ is its disparity,s_i^(1)∈{0, 1, …} is a semantic label (0 is background, positive integers are foreground object instances), m_i ∈ SE(3) is its 3D rigid motion (translation and rotation), andf_i=[u_i, v_i] is its optical flow. We denote the disparity and semantic segmentation for each pixel in the second frame by q_i=(d_i^(2), s_i^(2)).We only use two frames to estimate scene flow, and thus need not explicitly model motion in the second frame.Existing scene flow algorithms make predictions at the superpixel level without explicitly modeling the semantic content of the scene <cit.>.Predictions inside each semantic object may thus be noisy or inconsistent. In this work, we assume that the scene contains foreground objects (vehicles, for our autonomous driving application) rigidly moving across a static background. Given an accurate semantic segmentation of some foreground object, the geometry of the pixels within that segment should be spatially and temporally consistent, and the optical flow should be consistent with the underlying 3D rigid motion.Due to the high dimensionality of the scene flow problem, we refine our estimates using a cascade of discriminative models <cit.>, with parameters learned via a structural SVM <cit.>.Every stage of the cascade makes a targeted improvement to one scene variable, implicitly accounting for uncertainty in the current estimates of other scene variables. We initialize our semantic segmentation 𝒮 using an instance-level segmentation algorithm <cit.>, and our disparities 𝒟 and optical flow fields ℱ using the PRSF method <cit.>. We discuss their cascaded refinement next.§.§ Refinement of Semantic Segmentation The initial single-frame segmentation is unreliable in regions with shadows and reflections. Given stereo inputs, however, our depth estimates provide a strong cue to improve the segmentation. Therefore for each segmentation instance, we define a CRF on the pixels in its enclosing bounding box B_i. We seek to estimate the foreground segmentation s given an initial noisy segmentation ŝ.Our data term encourages the inferred segmentation s to be close to the initial segmentation ŝ. The KITTI scene flow dataset <cit.> generates “ground truth” segmentations by aligning approximate CAD models, and these annotations are often inaccurate at object boundaries, thus violating that assumption that foreground and background objects typically have distinct color and geometry.To add robustness, we define a feature by computing the signed distance of pixel i to theoriginal segmentation border and using a sigmoid function to map these distances to [0, 1], denoted by ϕ_dist(i, ŝ). The data energy for our CRF model is thenE_seg^data(𝒮)=∑_i∈ B_i[λ_1 +λ_2 ϕ_dist(i, ŝ)]δ(s_i=0, ŝ_i=1) + [λ_3 +λ_4 ϕ_dist(i, ŝ)]δ(s_i=1,ŝ_i=0). We demonstrate the benefits of our signed distance feature ϕ_dist(i, ŝ) in Fig. <ref>. By allowing the CRF to reduce confidence in ŝ near boundaries, this feature allows other image-based cues to improve segmentation accuracy. To allow spatial regularization, we add edges ℰ to our CRF connecting each pixel to its 8 spatial neighbors:E_seg^space(𝒮)= ∑_(i,j) ∈ℰ[λ_5 + λ_6 ρ_img(I_i, I_j)+ λ_7 ρ_disp(d_i, d_j)]δ(s_i ≠ s_j).Here, ρ_img(I_i, I_j) = exp{-||I_i - I_j||/σ_img} measures RGB color similarity, and ρ_disp(d_i, d_j)=exp{-|d_i - d_j|/σ_disp} measures similarity of the current (approximate) disparity estimates.To learn the parameters λ = [λ_1, …, λ_7], we use a structured SVM <cit.> with loss equal to the average label error within bounding box B_i <cit.>. Feature bandwidths σ_img,σ_disp are tuned using validation data. To perform inference on E_seg^data + E_seg^space, we use an efficient implementation of tree-reweighted belief propagation <cit.>. Because pixel labels are binary, inference takes less than 0.5s. To apply our CRF model to the scene flow problem, we independently estimate the segmentation for each instance and frame.§.§ Estimation of Scene Geometry Given a disparity map 𝒟 and camera calibration parameters, a 3D point cloud representation of the scene may be constructed.Standard stereo estimation algorithms ignore semantic cues, and often perform poorly on surfaces that are shadowed, reflective, or transparent. As illustrated in Fig. <ref>, for autonomous driving applications the depth estimates for vehicle windshields are especially poor. Because inaccurate depth estimates lead to poor motion and flow estimates, we design a model that enforces local smoothness of depths within inferred segmentation masks.We define a CRF model of the pixels within each semantic segment previously inferred by our cascaded model. For each pixel i in the left camera with disparity hypothesis d_i, we denote its corresponding pixel in the right camera as P_d(i, d_i). The data term is defined to penalize the difference in smooth census transform between pixel i and P_d(i, d_i):E_geom^data(𝒟) = ∑_{i|s_i=s}ρ_CSAD(I_i, I'_P_d(i, d_i)).Here, ρ_CSAD(.,.) is the CSAD cost <cit.> for matched pixels in different images. The CSAD difference is a convex approximation of the census transform <cit.> that gives reliable pixel correspondences for many datasets <cit.>.We encourage piecewise-smooth depth maps by penalizing the absolute difference of neighboring pixel depths: E_geom^space(𝒟) = τ_1 ∑_(i, j) ∈ℰ_sρ_depth(d_i, d_j).Here ℰ_s contains neighboring pixels within segment s, ρ_depth(d_i, d_j) = |C/d_i - C/d_j|, and C is a camera-specific constant that transforms disparity d into depth C/d. We enforce consistency of pixel depths because the scale of disparities varies widely with the distance of objects from the camera.If naively applied to the full image, simple CRF models are often inaccurate at object boundaries <cit.>. However as illustrated in Fig. <ref>, although our stereo CRF uses standard features,it is effective at resolving uncertainties in challenging regions of foreground objects and it is much better able to capture depth variations within a single object. Moreover, because our pairwise distances depend only on the absolute value of depth differences,distance transforms <cit.> may be used for efficient inference in minimizing E_geom^data + E_geom^space. On average, it takes less than 5s to perform inference in a 200×200 region with 200 disparity candidates.We refine the disparities for each frame independently.§.§ Estimation of 3D MotionIf the segmentation mask and disparity estimates for each object instance were perfect, we could apply 3D rigid motion to the 3D point cloud for each segment, and project back to the image plane to recover the 2D optical flow. We let (x_i,y_i) denote the motion flow constructed in this way. Although our imperfect geometry estimates will cause the motion flow to differ from the true optical flow (u_i,v_i), each still provides valuable cues for the estimation of the other.For each detected segment, we let M=(R,t) denote its 3D relative motion between the first and second frames.The motion M has 6 degress of freedom: t is a translation vector, and R = (α,β,γ) is a rotation represented by three axis-aligned rotation angles.We match the rigid motion M to the current flow field estimate (u,v) by minimizing the following energy function: E_motion(M) = ν (ρ(α) + ρ(β) + ρ(γ))+ ∑_{i|s_i=s} |x_i(M,d_i) - u_i| + |y_i(M,d_i) - v_i|.where (x_i(M,d_i),y_i(M,d_i)) is the motion flow computed from disparity d_i, 3D motion M, and the camera calibration. We let ρ(a)=√(a^2 + ϵ^2) be the Charbonnier penalty, a smooth function similar to the L_1 penalty that provides effective regularization for motion estimation tasks <cit.>. We regularize R to avoid unrealistically large rotation estimates. We set the regularization constant ν using validation data, and use gradient descent to find the optimal value for M.We visualize an example motion flow map inFig. <ref>. §.§ Estimation of 2D Optical FlowThe estimated motion flow from the previous stage provides valuable cues for optical flow estimation. As in the example in Fig. <ref>, motion flow errors are primarily caused by imperfect geometries (or disparities).We thus seek a flow field f_i=(u_i, v_i) such that the corresponding pixel P_f(i, f_i) in the next frame matches pixel i, and f_i does not deviate too much from (x_i, y_i). We define a CRF model of the pixels within segment s in frame 1, with likelihoodE_flow^data(ℱ) = ∑_{i|s_i=s}ρ_CSAD(I_i, J_P_f(i, f_i)) + η_1 (|u_i - x_i| + |v_i - y_i|).We also encourage spatially smooth flow field estimates:E_flow^space(ℱ) = ∑_(i, j)∈ℰ_sη_2 (|u_i - u_j| + |v_i - v_j|). While many optical flow methods use superpixel approximations to make inference more efficient <cit.>, max-product belief propagation can be efficiently implemented for our pixel-level CRF using distance transforms <cit.>. As shown in Fig. <ref>, our refined optical flow improves the initial flow by smoothly varying across the segment, while simultaneously capturing details that are missed by the motion flow.To limit the memory consumption of our optical flow algorithm, we perform inference on a down-sampled image and then use the EpicFlow <cit.> algorithm to interpolate back to the full image resolution.Other recent optical flow algorithms have used a similar approximation <cit.>.Motion Estimation for Out-of-Frame PixelsWe notice that the EpicFlow interpolation tends to produce significant errors for pixels that move outside of the image border.Outside of the camera's field of view, optical flow can only be predicted using the known 3D rigid motion, and we thus propose a flow fusion CRF <cit.> to combine the estimated optical flow and motion flow for partially occluded objects.In particular, we use a binary CRF to determine whether the optical flow (u_i, v_i) or motion flow (x_i, y_i) provides a better estimate of the true flow (U_i,V_i) for each pixel i. Intuitively, for within-border pixels we should use the matching cost to compare flow fields, while out-of-border pixels should be biased towards the motion flow interpolation:E_fuse^data(ℱ) =ω_1 (|U_i - x_i| + |V_i - y_i|)δ[P_f(i, f_i)outside]+∑_f={(u, v), (x, y)}∑_{i|s_i=s}ρ_CSAD(I_i, J_P_f(i, f_i))δ[P_f(i, f_i)inside]. Spatial smoothness is encouraged for neighboring pixels:E_fuse^space(ℱ)= ∑_(i, j)∈ℰω_2 (|U_i - U_j| + |V_i - V_j|).We tune parameters ω_1, ω_2 using validation data, and minimize the energy using tree-reweighted belief propagation <cit.>. We show in Fig. <ref> that the fused flow estimate retains many details of the optical flow, while using the motion flow to better interpolate in occluded regions. We also apply our flow fusion technique to update the noisy background flow predictions.See Fig. <ref> for additional examples of our final optical flow estimates.§ CASCADED SCENE FLOW PREDICTIONThe CRF models defined in Sec. <ref> refine the various components of our scene model greedily, by estimating each one given the current best estimates for all others. However, this approach does not fully utilize the temporal relationships between the segmentation and geometry at different frames. Also, when the initial optical flow contains major errors, our motion flow estimates will be inaccurate. To better capture the full set of geometric and temporal relationships, we thus use multiple stages of cascaded prediction <cit.> to further refine our scene flow estimates. The inputs and outputs for each stage of our cascade are summarized by the directed graph in Fig. <ref>. Temporal Segmentation Consistency Rather than segmenting each video frame independently, in the second stage of our cascade, we use the inferred flow field f to encourage temporal consistency. Each pixel i in frame 1 is linked to matched pixel P_f(i, f_i) in frame 2:E_seg^time(𝒮) = λ_8 δ(s^(1)_i = 0, s^(2)_P_f(i, f_i) = 1) + λ_9 δ(s^(1)_i = 1, s^(2)_P_f(i, f_i) = 0) + ∑_i[λ_10 + λ_11ρ_CSAD(I_i, J_P_f(i, f_i))]δ(s^(1)_i = s^(2)_P_f(i, f_i)).We again use S-SVM learning of CRF parameters λ on E_seg^data+E_seg^space+E_seg^time, and infer segmentations using tree-reweighted belief propagation.Temporal Geometric Consistency As in our temporal segmentation model, we also extend the stereo CRF of Sec. <ref> to encourage smooth changes for the depths of pixels linked by our optical flow estimates:E_geom^time(𝒟) = τ_2 ∑_{i|s_i^(1)=s}ρ_depth(d_i(m_i), d_P_f(i, f_i)).Here, d_i(m_i) denotes the disparity value of pixel i in the second frame when rigid motion m_i is applied.The parameters τ are learned using validation data, and efficient distance transformation <cit.> is also used to solve E_geom^data+E_geom^space+E_geom^time. Fig. <ref> shows an example of the improved disparity and flow estimates produced across multiple stages of our cascade. Recovery from a Poor Optical Flow Initialization If the initial noisy optical flow is very inaccurate, our cascade cannot recover the correct 3D motions of objects because we assume motion flow should match optical flow.Since our updated semantic segmentation masks s^(1) and s^(2) are typically very accurate, when applying rigid motion M to pixels in s^(1),the shape of the new segmentation mask s(M) should be similar to s^(2). We measure this similarity via a cost defined on the second-frame bounding box B:1/|B|∑_i∈ Bα S(M)_i · C(S^(2)_i) + (1-α) C(S(M)_i) · S^(2)_i.Here, C(·) is the Chamfer difference map and α=0.5.This cost function is widelyused for human pose estimation <cit.>. If this cost exceeds 0.5, we replace the first term in Eq. (<ref>) with this silhouette cost. By optimizing this modified objective in Eq. (<ref>) using standard gradient-descent, we can recover from bad motion estimates. An illustration is in the supplementary material. Second Frame DisparitiesFor the KITTI scene flow dataset <cit.>, the ground truthdisparity for the second frame is represented as per-pixel disparity changes with respect to the first frame.To predict this quantity for evaluation, we apply our estimated 3D rigid motion for each pixel to its estimated geometry in the first frame. The accuracy of these disparity estimates is thus strongly dependent on the performance of our motion estimation algorithm.Global Energy Function The global energy function implicitly minimized by our cascade of CRFs can be constructed by adding all energy terms together. Our iterative optimization of subsets of variables (as in Fig. <ref>) can be seen as block coordinate descent, where the cascaded prediction framework refines the energy function to reflect the typical accuracy of previous stages.This cascaded framework enables efficient, adaptive discretization of a large state space for flow and disparity, and is a principled way of optimizing a limited number of inference iterations <cit.>. § EXPERIMENTS We test our semantic scene flow algorithm (SSF) with 3 iterations of cascaded prediction on the challenging KITTI 2015 benchmark <cit.>. We evaluate the performance of our disparity estimates for two frames (D1, D2), flow estimates (Fl) for the reference frame, and scene flow estimates (SF) for foreground pixels (fg), background pixels (bg), and all pixels (all). See Table <ref> for experimental results on all pixels, and Table <ref> for non-occluded pixels. We evaluate SSF cascades learned to refine PRSF <cit.> initializations (SSF-P) and also apply the learned parameters to OSF <cit.> initializations (SSF-O). Our cascaded approach is superior to the published two-frame scene flow algorithms with respect to all evaluation metrics. SSF-P is about 60% more accurate than the two-frame PRSF method; SSF-P is overall 2% more accurate than OSF, while 10 times faster. At the time of submission, the only published work that performed better than our SSF approach were the multi-frame PRSM <cit.> and OSFTC <cit.> methods, which require additional data. The concurrently developed ISF method <cit.> uses external training data for instance segmentation and disparity estimation, leading to further improvements over our approach at the cost of slower speed. We visualize the qualitative performance of our SSF-P method on training data in Fig. <ref>.In Table <ref>, we evaluate the performance gain provided by each stage of the cascade on the training set. There is an improvement at the first stageof the cascade when modeling segmentation and geometry independently at each frame, followed by another improvement at the second stage when temporal consistency is introduced. At the third stage, performance starts to saturate.Speed Scene flow estimation is a computationally demanding task, and efficient algorithms <cit.> usually sacrifice accuracy for speed.Although the number of variables in our scene flow representation is huge and we makepixel-level predictions, our cascaded algorithm with MATLAB/C++ implementation on a single-core 2.5 Ghz CPU remains efficient. The main reason is that we disentangle the output space, and utilize efficient message-passing algorithms <cit.> to solve each high-dimensional inference problem. Most of the computation time is spent on feature evaluation, and could be accelerated using parallelization.Failure Cases As shown in Fig. <ref>, in challenging cases where the semantic segmentation algorithm fails to detect vehicle boundaries, our scene flow estimates can be inaccurate. As previously studied for semantic optical flow methods <cit.>, we conducted an experiment using ground truth segmentation masks and witnessed a significant performance gain; see Table <ref>.Our framework for cascaded scene flow estimation will immediately benefit from future advances in semantic instance segmentation. § CONCLUSION In this paper, we utilize semantic cues to identify rigidly moving objects, and thereby produce more accurate scene flow estimates for real-world scenes. Our cascaded prediction framework allows computationally efficient recovery of high-dimensional motion and geometry estimates, and can flexibly utilize cues from sophisticated semantic segmentation algorithms. We improve on the state-of-the-art for the challenging KITTI scene flow benchmark <cit.>. While our experiments have focused on using vehicle detections to improve scene flow estimates for autonomous driving, our cascaded scene flow framework is directly applicable to any category of objects with near-rigid motion. Acknowledgements This research supported in part by ONR Award Number N00014-17-1-2094. Some work was completed by Zhile Ren during an internship at NVIDIA. ieee | http://arxiv.org/abs/1707.08313v2 | {
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Magnetism from intermetallics and perovskite oxides Richard Javier Caraballo Vivas December 30, 2023 ===================================================roman This is a version of the Thesis presented by RJCV atIntituto de Física at Universidade Federal Fluminense,Niterói-RJ Brazil at march 2017, on supervision of Dr. Mario de Souza Reis Jr. ([email protected]). For more information, you can contact the author by:to VanessaAcknowledgment * I thank the Brazilian institutions CAPES, CNPQ and FAPERJ for the financial assistance in the development of this work. * I thank to Programa de Pós-graduação do Instituto de Física da Universidade Federal Fluminense. * I thank for the access to laboratory: * Laboratório de Raios X da Universidade Federal Fluminense. * Laboratório Nacional de Luz Synchrotron. * Laboratório de espetroscopia Mössbauer da Universidade Federal Fluminense. * Laboratório de Materiais e Baixas Temperaturas da Universidade Estadual de Campinas. * I thank to my supervisor Mario de Souza Reis Jr. * I thank the several collaborators that contributed to the elaboration of this work.* I thank the components to experimental magnetism group of Intituto de Física da Universidade Federal Fluminense: * Prof. Dr. Mario de Souza Reis Jr. * Prof. Dr. Daniel Leandro Rocco. * Prof. Dr. Dalber Ruben Sanchez Candela. * Prof. Dr. Stéphane Serge Yves Jérôme Soriano. * Prof. Dr. Yutao Xing. * Prof. Dr. Wallace de Castro Nunes. * I thank to all people for the help. Family, friends and others. AbstractThe purpose of this thesis is to elaborate intermetallic alloys and perovskites oxides, in order to understand the magnetic properties of these samples. Specifically, the intermetallic alloys with boron YNi_4-xCo_xB samples and Co_2FeSi and Fe_2MnSi_1-xGa_x Heusler alloys were fabricated by arc melting furnace. The sol-gel method was implemented for the synthesis of perovskite oxides of cobalt Nd_0.5Sr_0.5CoO_3 and manganese La_0.6Sr_0.4MnO_3. YNi_4-xCo_xB was studied in order to explore the magnetic anisotropy originated in 3d sublattice of these samples.Here, we associate the anisotropy with the Co occupation, due to the non-magnetic nature of yttrium and the non-contribution to the anisotropy of Ni ions. From the occupation models for Co ions and magnetic measurements, we explained the magnetic behavior of these samples.We develop investigations on structural, magnetic and half-metallic properties of Co_2FeSi and Fe_2MnSi_1-xGa_x Heusler alloys. The Co_2FeSi is a promising material for spintronic devices, due to its Curie temperature above 1000 K and magnetic moment close to 6μ_B. However, these properties can be influenced by the atomic disorder. Through anomalous X-ray diffraction and Mösbauer spectroscopy, we obtained the atomic disorder of the Co_2FeSi, and the density functional theory calculations provided information about the influence of the atomic disorder in their half-metallic properties. On the order hand,we also explored the behavior of the magnetic and magnetocaloric properties with the increased of the valence electron number in the Fe_2MnSi_1-xGa_x Heusler alloys.In addition, we explored the structural and magnetic properties of perovskite oxides of the ABO_3 type, mainly theNd_0.5Sr_0.5CoO_3 cobaltite and La_0.6Sr_0.4MnO_3 manganite. For Nd_0.5Sr_0.5CoO_3, we focus in establishing its spin state configuration, for the Co^3+ and Co^4+, finding for both intermediate states. Also, we clarified their intrigue magnetic order, obtaining a ferrimagnetic material. The La_0.6Sr_0.4MnO_3 nanoparticles were synthesized in order to explore the effect of size on their properties. We found that the reduction of the nanoparticles size tends to broaden the paramagnetic to ferromagnetic transition, as well as promoting magnetic hysteresis and causing a remarkable change to the magnetic saturation. Keywords: Intermetallics, Perovskites, Half-metallic, Magnetocaloric effect, Sample preparation. empty arabic CHAPTER: INTRODUCTION Magnetism has been known for thousands of years. The manifestations in which it was formerly known are those corresponding to natural magnets or magnetic stones, such as magnetite (iron oxide). The ancient Greeks and Chinese were the first to have been known to use this mineral, which is the ability to attract other pieces of the same material and iron.Chronologically, in the 17th century, William Gilbert was the first to systematically investigate the phenomenon of magnetism using scientific methods. The first theoretical investigations were attributed to Carl Friedrich Gauss, while the first quantitative studies of magnetic phenomena were initiated in the 18th century by the French scientist, Charles Coulomb. The Danish physicist, Hans Christian Oersted, first suggested a link between electricity and magnetism, while the French scientist, Andre Marie Ampère, and the Englishman, Michael Faraday, performed experiments involving the interactions of magnetic and electric fields with each other. In the 19th century, James Clerk Maxwell provided the theoretical foundation of the physics of electromagnetism. The modern understanding of magnetic phenomena in condensed matter originates from the work of Pierre and Marie Curie, who examined the effect of temperature on magnetic materials and observed that magnetism suddenly disappeared above a certain critical temperature in materials such as iron.Pierre Weiss proposed a theory of magnetism based on an internal molecular field, proportional to the average magnetization that spontaneously aligns electronic micromagnets in magnetic matter. This is just to name a few historical examples. J. M. D. Coey <cit.> summarizes the history of magnetism in seven ages described in the table in Fig. <ref>. In each of these ages it can be seen how the understanding and functionality of magnetism has evolved, mainly owing to the use and development of new technologies based on phenomena and materials that are, in turn, based on magnetism. The greatest advance in technological development based on magnetism was in the 20th century with the manipulation of magnetic coercivity<cit.>, resulting in the combined control of magnetocrystalline anisotropy and microstructures. While, in the last years themagnetic technological development is oriented mainly atspintronics, whichis the study of the roles played by electron spin and the possible uses of their properties in order to develop devices in which it is not the electron charge but the electron spin that carries information. These devices, combining standard microelectronics with spin-dependent effects, arise from the interaction between the spin of the carrier and the magnetic properties of the material, as in the half-metallic ferromagnetic compounds <cit.>. The so-called half-metallic magnets have been proposed as good candidates for spintronic applications owing to their characteristic of exhibiting a hundred percent spin polarization at the Fermi level.Then, the development of materials is important for the magnetic phenomena and knowledge of their magnetic mechanisms. Thus, we synthetized several intermetallic alloys and perovskites oxides, in order to explore their structural and magnetic properties, in addition were made studies magnetic phenomenawhich provide information of these compounds for their use in technological devices. In Part <ref> of this thesis, we focus on basic magnetism , mainly describing several concepts to understand magnetic phenomena. The types of magnetic arrangement and their differences are explained in Chapter <ref>. Furthermore, we approach the magnetocaloric effect and main phenomenology in Chapter <ref>. Chapter <ref> is focused on sample preparation. Part <ref> of this Thesis, is focuses in the intermetallic compounds and are based our works in intermetallics with B (Chapter <ref> based on Ref. <cit.>), and Heusler alloys inChapters <ref>,<ref> and <ref> based on our investigation of Co_2FeSi and Fe_2MnSi_1-xGa_x <cit.>.Among the most studied intermetallic compounds that present magnetic hardness are the Nd-Fe-B system <cit.> and SmCo_5 <cit.>, the latter being in the family of intermetallic compounds with B (R_n+1M_5+3nB_2n, with R = rare earth, M = 3d transition metal and n=0, 1, 2, 3...∞). Magnetic anisotropy from the 3d sub-lattice is important for the magnetic hardness of these materials, since this strongly affects the shape of hysteresis loops which affect the design of most magnetic materials of industrial importance. Thus, we considered YNi_4-xCo_xB (n=1) alloys of the family of intermetallics with B (for n=1), and a non-magnetic rare-earth (yttrium) in order to be sure that the magnetic contributions are only due to the 3d sub-lattice, additionally Ni does not contribute to anisotropy. Thus, to explore these features, we developed a statistical and preferential model of Co occupation among the Wyckoff sites, in order to explore the 3d competition in the samples. We found that the preferential model agreed with the experimental measurements of the magnetization nature of 3d magnetic anisotropy. Thus, this result provides further knowledge in the area of hard magnets <cit.>. It is explored in details in Chapter <ref>. Chapter <ref> explores one of the most promising families of compounds with interesting materials for technological applications, the Heusler alloys. They have half-metallic properties, and such materials follow the Slater-Pauling rule that relates the magnetic moment to the valence electrons in the system. Generally, they are widely studied owing to their interesting structural and magnetic properties, such as magnetic shape memory ability, coupled magneto-structural phase transitions and half-metallicity. However, the atomic disorder produced by the interchange of X/Y atoms in Heusler alloys (X_2YZ) suppresses the half-metallic behavior of these compounds. For this,in the Chapter <ref>, we considered the Co_2-based Heusler alloys, mainly Co_2FeSi, which has the highest Curie temperature of these alloys at approximately 1100 K and a magnetic moment of approximately 6 μ_B <cit.>.The half-metallicity of the Heusler alloys has a close relationship with the valence electron number (N_v), which obeys the Slater-Pauling rule. Therefore, our approach seeks to understand the effect of N_v on the magnetic properties of the Si-rich side of the half-metallic series, Fe_2MnSi_1-xGa_x Heusler alloys, such as the structural, magnetic, and finally magnetocaloric potentials <cit.>. These ideas are developments in the Chapter <ref>.In Part <ref>, we explored the peroviskite oxides. As with intermetallic alloys, perovskite ceramics (ABO_3) are compounds that have contributed much to the development of magnetism. Among the advantages they possess is the fact that they are oxidized and, therefore, can be prepared in the air without the undesirable influences of oxygen, in addition to being more economical compared to the intermetallic compounds. Finally, the perovskite oxides have interesting properties for technological developments based on magnetism, such as: spintronics <cit.>, magnetic hyperthermia <cit.>, and magnetic refrigeration <cit.>, among others.Cobaltite (ACoO_3) compounds show interesting magnetic and transport properties, owing to the strong relationship between the crystal structure and magnetism. However, the spin state of Co ions has an additional degree of freedom due to competition between Hund couple and crystal field splitting. Nd_0.5Sr_0.5CoO_3 has no well-established spin configuration, thus, we developed magnetization measurements in order to understand this aspect in detail. Consequently , we found that Co^3+ and Co^4+ are in an intermediate spin state and the Co and Nd magnetic sub-lattices couple antiferromagnetically below the Curie temperature T_c=215 K, down to very low temperatures <cit.>. In the Chapter <ref> details of this study are given.On the other hand, the structural and magnetic properties of R_1-xT_xMnO_3 manganitesare determined by lattice distortion on the unit cell, since the Mn-O-Mn bond is very sensitive to structural changes. Meanwhile, when the particle size is reduced to a few nanometers, the broadening of the paramagnetic to ferromagnetic transition, which decreases the saturation magnetization value, increases the magnetic hysteresis and appearance of superparamagnetic (SPM) behavior at very low particle sizes. Thus, we synthesized La_0.6Sr_0.4MnO_3 nanoparticles to explore the effect of size on the structural and magnetic properties of these samples. We found that the reduction of the nanoparticles size tends to broaden the paramagnetic to ferromagnetic transition, as well as promoting magnetic hysteresis and a remarkable change on the magnetic saturation <cit.>. The Chapter <ref> is development these ideas.Finally, the main purpouse of this PhD thesis, isunderstand the magnetic mechanism of the sample explained above. For this, we implemented several experimental techniques for the study of structural and magnetic properties. The possibles technological applications such as magnetic refrigeration and spintronics, also are explored. General conclusions of all this thesis are in the Chapter <ref> PART:Background CHAPTER: CONCEPTS OF MAGNETISM This chapter discusses the basic concepts of magnetism, which will be addressed during the rest of the text. Here, the main objective is to introduce the fundamental ideas about magnetism to the reader to provide clear understanding of this work. § FUNDAMENTAL TERMS§.§ Magnetic moment and magnetic dipole In classical electromagnetism the magnetic moment (μ⃗) can be explained using theFig.<ref>, where we assuming a current aroundan infinitely small loop with an area of dA square meters (International System of Unit - SI). The corresponding magnetic moment dμ⃗, is equal to <cit.>,dμ⃗= I dA n̂, where I is the circulating current in amperes and n̂ is an unitary vector normal to the ring area, which comes of the relation between μ⃗ and I by the right-hand corkscrew rule <cit.>. The magnetic moments of this small loop allows us to calculate the total μ⃗, for a loop of a finite size:μ⃗ = I n̂∫ dA = IAn̂. The magnetic moment is measured in [Am^2] (SI), or [erg/G] (cgs). It is important to define that [erg/G] = [emu] (electromagnetic unit), because this quite common in the literature, mainly to express experimental results <cit.>.The magnetic dipole is equivalent to a magnetic moment of a current loop in the limit of a small area, but with a finite moment. The energy (E) of a magnetic moment is given by <cit.>:E = - μ_0 μ⃗·H⃗= -μ_0 μ H cosθ, where E is given by [J] ([erg]) for SI (cgs), θ being the angle between μ⃗ and an external magnetic field H⃗ and μ_0 = 4 π× 10^-7 N · A^-2 (SI) is the magnetic permeability of free space <cit.>. §.§ Magnetization The magnitude of the magnetization (M) can be defined as the total magnetic moment per unit volume <cit.>:M = ∑_i^Nμ_i/V, where M is given by [A/m] for SI or [emu/cm^3] for cgs. However, in the practice is more convenient to define the magnetization as the magnetic moment per mass.Since, we do not need to know the sample volume, only its mass, which can be easily obtained <cit.>. In this case the magnetization is given by[A m^2 kg^-1] for SI or [emu/g] for cgs <cit.>. §.§ Magnetic induction The magnetic response of a material when is applied an external magnetic field H⃗ is called the magnetic induction or magnetic flux density B⃗. The relationship between H⃗ and B⃗ is a characteristic property of the material itself. In the vacuum, we have a linear correlation between B⃗ and H⃗: B⃗ = μ_0 H⃗, considering SI. However, inside a magnetic material B⃗ and H⃗ may differ in magnitude and direction, due to of the magnetization M⃗ <cit.>. Considering SI, B⃗= μ_0 (H⃗+M⃗). In the following sections, we refer to both as the magnetic field due to the common usage in the literature. In every situation, it can be understood, which term is meant.§.§ Magnetic susceptibility If we consider that the magnetization M⃗ is parallel to an external magnetic field H⃗: χ = lim_H → 0∂ M/∂ H,with χ being the magnetic susceptibility, for this case the material is considered a linear material. In this situation, a linear relationship between B and H remains: B = μ_0 (1 + χ)H = μ_m H, where μ_m represent the magnetic permeability of the material.§ TYPES OF MAGNETIC ARRANGEMENT§.§ Diamagnetism Diamagnetism is intrinsic to all materials, it is manifest when the electrons are underan applied external magnetic field, then the precession around the nucleus changes the frequency to promote an extra magnetic field and shield the external one. Therefore, the diamagnetic susceptibility is negative <cit.>χ^= <0 A few examples of diamagnetic materials are:∘ Nearly all organic substances,∘ Metals like Hg or noble metals like Cu,∘ Superconductors below the critical temperature. These materials are ideal diamagnets. §.§ Paramagnetism Paramagnetism is a type of magnetism where there are no interactions between the magnetic moments. Thus, for creating an order of, is necessary the application of a magnetic field. Let us consider an ensemble of magnetic moments at a certain temperature, where they are directed randomlyand the magnetization is zero. The application of a magnetic field promotes a relative orientation of magnetic moments, increasing the value of magnetization. At high values of magnetization fields, all the magnetic moments are parallel to each other, and the magnetization reaches its maximum value (saturation M_S). For a certain value of the applied magnetic field, after decreasing temperature, this relative orientation of the magnetization and, at 0 K, the magnetization reaches its maximum value.The magnetic susceptibility, obtained for low values of the applied magnetic field, is given by Curie law: χ^ = C/T, where C is the Curie constant, given by <cit.>, C = N μ_0 μ_B^2 g^2 j(j+1)/3k_Bwhere N is the number of magnetic atoms per unit volume, g is the Landé g-factor, μ _B is the Bohr magneton, j is the total angular momentum and k_B is Boltzmann constant. Then, the inverse magnetic susceptibility is a straight line, which pass through zero at 0 K. §.§ Collective magnetism The other types of magnetic arrangements that we will discuss are the consequence of the interactions between the magnetic moments to obtain spontaneous magnetization, which results in collective magnetism (or cooperative systems). Thesusceptibility of the collective magnetismexhibits a functionality significantly more complicated compared to dia- and paramagnetism and consist in an interaction between permanent magnetic dipoles of the material <cit.>.For materials that present collective magnetism are characterized by a spontaneous magnetization bellow of the critical temperature T_c.Collective magnetism is divided into several subclasses, such as ferromagnetism, antiferromagnetism, ferrimagnetism, among others.§.§ Ferromagnetism This fact allows that the system reaches the magnetization saturation value for magnetic field relatively small values(those possible to be reached in a laboratory) depending on their magnetic anisotropy. Here, one magnetic moment depends on the neighbors to then creates the magnetic ordering; it is a long range interaction <cit.>. Two parameters characterize the ferromagnetic ordering: (i) the critical temperature T_C (Curie temperature), above which the system behaves like a paramagnetic system. Here, for a zero applied magnetic field, zero magnetization. Below T_C, the system has spontaneous magnetization, i.e., finite magnetization even without applied magnetic field. In the first approximation, T_C is the measure of how strong is the interaction between magnetic moments. (ii) The saturation value of the magnetization M_S (saturation magnetization) is analogously to the paramagnetic case <cit.>. Then, we expect the magnetization as a function of temperature curve to be a finite value of magnetization that decreases by increasing the temperature, to a critical value T_C, above which there is no longer spontaneous magnetization. By concerning the magnetization as a function of the magnetic field, there are two situations: the first one is for temperatures above T_C. For this case, as mentioned, the system behaves like a paramagnetic specie, and then, we expect a curve similar to the paramagnetic case. For temperatures below T_C, it has a spontaneous magnetization (see figure <ref>).The magnetic susceptibility is given by the Curie-Weiss law: χ = C/T- θ_p where θ_p is the paramagnetic Curie temperature (for the mean field model, θ_p = T_C). The inverse magnetic susceptibility is a straight line, and zero 1/χ meets T=θ_p>0 <cit.>. §.§ Antiferromagnetism Antiferromagnetism is a cooperative ordering that can be understood by considering two magnetic sub-lattices: M_A and M_B, of the same magnitude. Each one is ferromagnetic and behaves (approximately), according to the description mentioned above. The difference is that both sub-lattices are oriented in opposition, i.e., the total magnetization vanishes <cit.> M = M_A + M_B = 0.The paramagnetism like behavior appears above the critical temperature T_N (Neel temperature), below which these sub-lattices are spontaneously ordered in opposition <cit.>. This ordering is not a simple addition of two ferromagnetic sub-lattices, aligned in an antiparallel configuration, there is an interaction between these two sub-lattices, making this system a bit more complex that described here.The magnetization as a function of themagnetic field, at T<<T_N, one sub-lattice (say, M_A), is aligned with the external magnetic field and then does not change by increasing the field. The other sub-lattice (M_B), is opposite to the field and then will be flipped due to the increase of the magnetic field. For the case without external applied magnetic field, each sub-lattice has a ferromagnetic like dependence with temperature and the total magnetization is then zero. The magnetic susceptibility is also given by the Curie-Weiss law, however, 1/χ is zero in the T axis at θ_p < 0 <cit.>. This behavior is shown in Fig.<ref>. §.§ Ferrimagnetism Ferrimagnetism is quite similar to antiferromagnetic cooperative ordering, however, for the present case, the two sub-lattices have different values of the magnetization in opposition <cit.>, |M_A|≠|M_B|. The behavior of the magnetization as a function of the magnetic field (for low values of the temperature), is the same as before; however, the difference in the values of the magnetization of each sub-lattice, it is not zero at a zero magnetic field. Therefore,M = M_A + M_B ≠ 0.Thus, these two sub-lattices are different, can cross themselves for a certain value of the temperature T_, and then promote a compensation, where the total magnetization is zero. The system loses the spontaneous ordering above T_N, analogously to the ferromagnetic case. Finally, the magnetic susceptibility only follows the Curie-Weiss law for a very high temperature. Close to T_N the inverse magnetic susceptibility loses its linearity and assumes a hyperbolic-like behavior, with a downturn to zero 1/χ <cit.>.Fig.<ref> shows the behaviors of all cooperative and non-cooperative systems mentioned above, from Ref. <cit.>.§ HYSTERESIS CYCLES When a ferromagnetic material is magnetized in one direction until the saturation magnetization, it will not relax back to a zero magnetization when the imposed magnetizing field is removed. The amount of magnetization retained at a zero applied magneticfield is called remanence. Then, it must be driven back to zero by a field in the opposite direction; the amount of reverse applied magneticfield required to driving to zero the magnetizationit is called coercivity. The reverse magnetic field is applied until the saturation magnetization, then it is reversed again until saturation at first direction of applied magnetic field; thus its magnetization will trace out a loop called a hysteresis loop <cit.> (see Fig. <ref>).The absence of reversibility of the magnetization curve is the property called hysteresis and it can be related to several factors including the sample shape, surface roughness, microscopic defects and thermal history <cit.>.This property of ferromagnetic materials is useful as a magnetic memory. Some compositions of ferromagnetic materials will retain an imposed magnetization indefinitely and are useful as permanent magnets <cit.>. The main cause of this phenomenon is magnetic anisotropy and will be explained in the next section.§ MAGNETIC ANISOTROPY The magnetic anisotropy is defined as the energy of the rotation of the magnetization direction from the easy into the hard direction <cit.>. Magnetic anisotropies may be generated by the electric field of a solid or crystal, by the shape of the magnetic body, or by mechanical strain or stress, all of which are characterized by polar vectors <cit.>. Hence, they cannot define a unique direction of the magnetization, which is an axial vector. This is why no unique anisotropy direction can exist, but only a unique axis. Therefore, the energy density E_ connected with the magnetic anisotropy must be constant when the magnetization is inverted, which requires that it be an even function of the angle θ enclosed by M⃗ and the magnetic axes <cit.>, E_= K_0 + K_1 sin^2θ + K_2 sin^4θ + K_3 sin^6θ + ... where K_i (i =1, 2, 3,...) are the anisotropy constants in the series expansion.These are not usually defined in theoretical terms, but rather through measurements, depending on the magnetic material. The values of the constants are affected by the magnetic behavior of system and depended on the symmetry of the lattice <cit.>. Table <ref> shows the different symmetry systems with their respective expressions of anisotropy energies (E_). The cubic case is more complex because of its high symmetry. For uniaxial symmetry, when K_1 is positive, the easy direction is an axis (z for instance), while, when has negative value, the perpendicular plane is the easy direction. As hexagonal, tetragonal and rhombohedral systems are considered, K_1 and K_2 constants play a relevant role in the anisotropy energy density. Here are distinguished cases for them <cit.>: (i) For K_1=K_2=0, the system is an isotropic ferromagnet. (ii) For K_1>0 and K_2+K1>0, we have an easy axis of magnetization for θ = 0. (iii) For K_1>0 (K_1< 0) and K_2+K_1<0 (2K_2+K_1>0), the perpendicular plane to the z axis is the easy magnetization plane. (iv) For 2K_2<K_1<0, the easy axis will be reached for a θ value given by, sin^2 θ = -K_1/2K_2. The determination of K_1 and K_2 constants is possible by different methods, such as, measurement of the anisotropy magnetic field, area method, torque method (all explained in Ref. <cit.>), Sucksmith-Thompson fit for single crystal samples <cit.> andSucksmith-Thompson fit modified version proposed by Ram and Gaut for powder samples <cit.>. This last method was used by Dung and co-workers <cit.> and Kowalczyk<cit.> to determine the anisotropic constants of YCo_4B, in order to determine the anisotropic energy by crystallographic site in this compound.Sucksmith-Thompson fit modified consist in the construction of a graph of H/ (μ_0M) as a function of (μ_0M)^2 from a powder sample oriented at small applied magnetic field, and easy direction. It is results in a linear relationship, where K_1 can be estimated by the vertical interception and K_2by the slope. An example of this construction is shown in the Fig.<ref>. CHAPTER: FUNDAMENTALS OFMAGNETOCALORIC EFFECTIn this chapter, we introduce basic concepts of the magnetocaloric effect (MCE). Here, the processes that define the MCE are described qualitatively and quantitatively. In addition, we will describe some aspects of how magnetic transitions affect the MCE. § MCE PHENOMENOLOGY The MCE is an intrinsic thermodynamic property of magnetic materials and it was discovered in 1881 by Warburg <cit.>, when he observed that iron absorbedand emitted heat under the influence of an external magnetic field. However, Smith <cit.> proposed that MCE was observed first by Weisse-Piccard <cit.>. In simple term, MCE is the temperature change or heat exchange of a magnetic material with application of an external magnetic field. This effect can be observed in either an adiabatic or an isothermal process, due to a change of the applied magnetic field. We considering a magnetic material in an adiabatic process, and in the presence of a variable external magnetic field. The increase in the magnetic field causes magnetic dipoles to align themselves,leading to a decrease in magnetic entropy. However, the total systementropy should be constant, and in consequence, the lattice entropy increases causing an increase in the system temperature.From an isothermal process, the material is in thermal equilibrium with a reservoir. Then, a variable external magnetic field is applied to align the magnetic dipoles and the magnetic entropy is changed. As it is an isothermal process, the internal energy of the system does not change, and the material must be in continuous heat exchange with the reservoir. To clarify these processes, see the Fig. <ref>. § GENERAL THERMODYNAMIC APPROACH For the description of magnetothermal effects in magnetic materials, we considered theGibbs free energy G as a function of the temperature (T), pressure (p) and magnetic field (H), we can write <cit.>: G= U-TS+pV-MH, where U is the internal energy and S is the entropy. Considering an isobaric system, variations of G are given by<cit.>:dG= - SdT - MdH. To find the internal parameters S and M, we use the equations of state, S(T,H,p)= - (∂ G/∂ T)_p,H, M(T,H,p)= - (∂ G/∂ H)_T,p. From Eqs. <ref> and <ref>, we can find the so-called Maxwell equation, (∂ S/∂ H) = (∂ M/∂ T). Thus, we obtain the expression for the magnetic entropy change (Δ S) from an initial magnetic field (H_i) to the final magnetic field (H_f), Δ S(T,Δ H) = ∫_H_i^H_f( ∂ M(T,H)/∂ T)_H dH. It can be easily observed that this quantity will be maximized around large variations in magnetization with temperature, as those that happen around the Curie temperature (T_C).Experimentally is realized a mapping of the magnetization measurement as a function of the magnetic field around T_C, for then calculate of magnetic entropy changes. InFig.<ref>.(a) and (b) is shown the magnetization mapping and magnetic entropy changes of Gd_5(Si_2Ge_2) from Ref. <cit.>.On the other hand, we can calculate the specific heat of a system with the second derivative of the Gibbs free energy and Eq. <ref>, C_H =-T ( ∂^2 G/∂ T^2)_H,p = T ( ∂ S/∂ T)_H. By considering the entropy as a function of the temperature and magnetic field, S= S(T,H): dS= ( ∂ S/∂ T)_H dT + ( ∂ S/∂ H)_T dH.Then, in an adiabatic process (dS=0), using Eqs.<ref> and <ref>, we obtain: C/T dT = - (∂ M/∂ T) dH. Consequently, we find the change in the adiabatic temperature, Δ T_ad= ∫_H_i^H_f( T/C(T)_H) ( ∂ M(T,H)/∂ T)_H dH, and from the magnetic entropy <cit.>, Δ T_ad (T) ≊T/C(T)_HΔ S(T)_Δ H The calculation of Delta T_ad requires the experimental measurements of the magnetization with the magnetic field and specific heat data. Thus, in the practice, it is mathematically and experimentally more difficult.Δ T_ad results for Gd_5(Si_2Ge_2) sample compared with Gd sample in different Δ H is in Fig. <ref>.(c). § MAGNETIC-PHASE TRANSITIONS AND MCE Order phase transitions can be caused by either varying the temperature or applying a magnetic field, and they have been extensively studied in the context of magnetocaloric materials <cit.>. For, a material undergoes the transition of a first-order, then the first-order derivatives of the thermodynamic potential change discontinuously, and values such as entropy, volume and magnetization display a jump at the point of transition and characterized by the existence of latent heat. In the second order transitions, the derivatives of thermodynamic potential are continuous, they are no associated latent heat, and the second derivatives are discontinuous.As mentioned above, the MCE is maximized at the transition temperature in magnetic systems, for example T_C in the ferro-paramagnetic transition. A way to approach this phenomena is by Landau theory <cit.>, which an analytical function, such as free energy (F), is taken and realized a potential expansion over the order parameter. The latter can be the magnetization for ferro-paramagnetic transitions, and then F is expressed around the magnetic transition temperature as: F= F_0 + 1/2 g_2(T) M^2 + 1/4g_4(T) M^4, where g_2(T) = α(T-T_C) is a null parameter at T=T_C and g_4 is constant <cit.>. Then, from the minimization of F, we can determine the equilibrium condition to obtain: M=0,M_± = ±( α(T-T_C)/2g_4)^1/2, For M=0, the system is at T> T_C dominant disorder state, while M_± is taken when T,T_C represents the minimum values of F and in the ferromagnetic magnetization is proportional to (T_C - T)^1/2.When a magnetic field (H) is applied, an additional term is added. F= F_0 + 1/2 g_2(T) M^2 + 1/4g_4(T) M^4 - HM. From minimized F for T=T_C, this result can be reached, H/M= β M^2,where β = 4g_4 is the parameter that determines the magnetic transition type. From experimental construction of H/M vs M^2 graph, it is possible found β, which is β > 0 in a second-order transition, while that β <0 is first-order transition. This, is known as the Banerjee criterion <cit.>. CHAPTER: SAMPLE PREPARATIONS The development of many experimental works is based on the quality of the samples studied, which allows us to study efficiently their physicalproperties. For this reason, sample preparation plays an important role in the development of this Thesis. Thus, the goal of thischapter is to describe the techniques used for sample synthesis, which were implemented in the Laboratório de preparação de amostras of the Instituto de Física da Universidade Federal Fluminense (UFF).§ INTERMETALLIC SYNTHESIS BY ARC MELTING FURNACE §.§ Arc melting furnace Arc melting furnace is extensively used both in industries and in research laboratories as a mean of producing samples of intermetallic compounds. The furnace used by us was fabricated at Universidade de Campinas and consists of a vacuum chamber, incorporated internally by tungsten electrode and a copper crucible cooled with water. Additionally, it has externally pressure valves, a vacuum pump, a high voltage source and other components that are shown in Figs. <ref> (a) and (b).The fusion of the elements is carried out by a voltage source, which generates a potential difference between the tungsten electrode (cathode) and the copper crucible cooled with water (anode), where is achieved temperatures from 2000 ^∘C to 3000 ^∘C, Fig. <ref>(c) illustrated the process scheme. The process begins when the compounds are placed into the copper crucible, for later, close the chamber and we realize the vacuum and subsequent purges. Then, these materials are melted by the potential difference, and a red mass (live red) is obtained due to the high temperature achieved. This mass is cooled by a water flux, and a new sample is formed, as can be seen in Fig.<ref>(d). §.§ Compounds used For the preparation of intermetallic samples by the arc furnace, the compounds usedshould be treated carefully for improving the efficiency of the fusion process and to obtain samples of good quality. Here, we enunciate some aspects to consider:Purity:the purity of the components used is very important for obtaining samples of high quality. However, some of the compounds have not the same purity. In this case, the quality of the sample will be evaluated from the component with minor purity. Actually, to our knowledge, the most widely used method to check that the samples crystallized in a single phase isthe X-rays diffraction. The purity of the components used in the preparations of the compounds is shown in Table <ref>.Melting and boiling point: the research on elements to be used in an arc furnace shall consider their melting and boiling points, the temperature achieved in the furnace. An example of this is phosphorus (P), which has a boiling temperature of 550 K (277 ^∘C) <cit.>. This is much lower than the temperatures of the arc furnace, and might cause total evaporation of this material. As a result it is virtuallyimpossible to synthesize materials with phosphorus in an arc furnace. The melting and boiling points of elements used in the synthesis of the intermetallic samples in this Thesis are presented in Table <ref>.Additional mass: it is important to add a suitable amount of mass of some of the elements for obtaining the desired stoichiometry. However, it is necessary to add additional mass due to evaporation of the elements. This is caused by thetemperature achieved by the arc, which is higher than the boiling point of the material. To avoid the loss of stoichiometry, we recommend to find the additional massbefore fusion. For example in our work <cit.> on YNi_4-xCo_xB, we measured the yttrium mass before and after the meltingand found that the requiredadditional mass of yttrium was 7%. While that for Fe_2MnSi_1-xGa_x <cit.>3% of manganese was added.§.§ Sample annealing The next step after the fusion is sample annealing. The annealing of a sample at high temperatures is important to obtain homogeneous samples without spurious phases. The main variables for carrying out annealing of samples are: (i) the temperature used, and (ii) the annealing time. Therefore, the information obtained from the literature before of sample annealing is important. Temperature and time of annealing can be seen in Table <ref>.To carry out the annealing process first, we need to wrap the samples in tantalum (Ta) foils in order to prevent contact between them. Ta foils are used since their melt temperature is3290 K <cit.> and do not contaminate the samples. Then, the samples are encapsulated within a quartz tube filled with argon (Ar) to avoid oxidation and promote uniformity of the annealing temperature. Finally, the samples are placed inside the annealing furnace (see the Fig. <ref>), and subsequently, they are subjected to quenching in water for maintaining the crystalline structure obtained at high temperatures.§ PECHINI METHOD (SOL-GEL) The experimental technique used for sample preparation of perovskites (manganites and cobaltites) was the Pechini method,also known as the sol-gel method. This method was developed by Maggio P. Pechini in 1967 <cit.>, mainly motivated by difficulties in obtaining oxide compounds of high quality by other methods, such as solid state reaction or mechanically-ground mixture.The Pechini method uses a chemical route for the production of compounds, and consists of separating oxygen from their cations using an acidic solution. From this solution, and with a polymerizing agent, a gel (sol-gel) with the desired stoichiometry is obtain. This process is schematically illustrated in Fig. <ref>. §.§ Synthesis of the nanoparticlesFor the synthesis of perovskite nanoparticles, we have used the reagents listed in Tab. <ref>, and taking the following steps:∙ The quantities of reagents (oxides or nitrates) used were calculated by their molecular weights and percentages of each compound to ensure the desired stoichiometry.∙ Dissociation of metal: when the samples are synthesized from nitrate reagents, we dissolved these reagents ina solution of citric acid (or another organic acid) and deionized water (alcohol can also be used) at room temperature, with high acid pH (≈ 1). Cases when the reagents are dissociate individually or together happens, but all have to be placed in the same solution.When the samples are synthesized from oxide reagents the process is more complicated since the dissociation can take several hours. In this case, the reagents are dissolved separately in the solutions with 100 ml of deionized water, nitric acid, and citric acid to obtain homogeneous solutions. Then, these solutions are mixed, and from here, the following two processes are similar. ∙ Polymerization : in this process a polymerizing agent such as ethylene glycol, which produces organic chains, is added, thus capturing the dissociated metals from the previous step. For the pH control (≈ 5), diamine ethylene should be added. This solution is stirred at 70 ^∘C for approximately 4 h, to finally obtain a gel.∙ Evaporation and annealing: because of the various compounds that can be synthesized by this method, a large number of evaporation and annealing processes are required to obtain the materials. Our case depends on the reagents used in the synthesis: oxides (see Fig. <ref>-top ) or nitrates (see Fig. <ref>-bottom).- From oxide reagents for synthesis of La_0.6Sr_0.4MnO_3 the evaporation process is as follows: The samples are heated for 150 min from the room temperature to 150 ^∘C. After 30 minutes at 150^∘C, it is heated for 50 min up to 200 ^∘C, and waiting for 30 min at 200 ^∘C. Then, the sample is heated for 100 min up to 400 ^∘C and the temperature is maintained for 240 min. This process is for the evaporation of acid and other liquid compounds. The final product of this evaporation is a brown-red powder, which it is not the final compound of interest. To obtain the manganite phase, the powder is treated at 400 ^∘C for 4 hs. Then, the powder is separated in portions for annealing at 700 ^∘C (973 K), 800 ^∘C (1073 K), 900 ^∘C (1173 K) and 1000 ^∘C (1273 K). - In the case of nitrates used from precursor reagents, only one process that includes both two evaporation and annealing treatments is necessary. The gel is placed in an oven at room temperature and is heated at a rate of 5 ^∘C/min to 360 ^∘C, where it remains during 4 hs to promote the evaporation. Next, it is warmed up to the annealing temperature, and finally remains at this temperature for 6 hs (depending on the sample the temperature and the annealing time are different).The scheme in Fig. <ref> shows the process for oxide reagents (here for La_0.6Sr_0.4MnO_3 samples), while that of nitrate reagents is shown in Fig. <ref> (here is Nd_0.5Sr_0.5CoO_3 sample). Both processes result in little particles of the order of nanometers or hundreds of nanometers. In the case of manganite samples, we obtain nanoparticles with a diameter between 5 nm and 100 nm of high quality. They will be explored in the following chapters. For Nd_0.5Sr_0.5CoO_3 cobaltite, we obtain a powder of paticles with diameters ranging between 500 nm and 1000 nm, which is pilled for getting a bulk sample shown in Fig. <ref> (c).§ SAMPLE CHARACTERIZATION BY X-RAY DIFFRACTION X-rays diffraction (XRD) is the most common technique used to (among other purposes) characterization of the samples, because their wavelength λ is typically the same order of magnitude (1-100 Å) as the spacing d between planes in the crystal.Physicist W.L. Bragg considered a family of parallel planes separated by a distance d. The path difference between the rays reflected by neighboring planes is 2d sinθ, where θ is the angle of incidence. The rays reflected by the different planes interfere constructively, when the path difference is equal to an integer n of wavelengths λ, that is, when <cit.>, 2dsinθ =n λ .This is Bragg's law <cit.>, illustrated in Fig. <ref>. Among other parameters, the intensity of the reflections in the diffraction patterns depends on the structure factor (F_hkl), which depends on how the radiation in the crystallographic planes of the material is scattered. This quantity depends of the atomic scattering factor, which is defined asf = f_0(θ)+f^'(E)+if^''(E), where f_0 is known as the normal scattering factor, while f^' and f^'' are dispersive and absorption terms, respectively. The last two terms are not taken into account in conventional XRD, however, they are very important in the anomalous X-ray diffraction (AXRD) and their values may be found in the International Tables for Crystallography <cit.>. XRD conventionaldata of the polycristal sampleswere obtained at Laboratório de Raios X at Universidade Federal Fluminense and at room temperature, using a Bruker AXS D8 Advance diffractometer with CuK_α radiation (λ = 1.54056 Å), 40 kV and 40 mA.Synchrotron radiation was used for obtain AXRD. It is made of X-ray beams generated by the accelerated electrons confined in acircular loop using magnetic fields. AXRD patterns were obtained at room temperature using a MYTHEN 24K system, from Dectris^ at XRD1 beamline at Laboratório Nacional de Luz Síncrotron. In addition, a NIST SRM640d standard Si powder was used to determine the X-ray wavelengths with precision. For analysis of XRD, the Rietveld Powder Cell <cit.> refinement program was used. The experimental diffractograms are adjusted by calculated model generated by the program from the crystallographic data of the compound. In Fig. <ref> we see an example of how the data fits and provides us with a theoretical model to characterize. The AXRD data was analyzed byusing FULLPROF suite software <cit.> in the same way; however, it takes the correction factors for the data fit.§ MAGNETIC MEASUREMENT The magnetic measurements as a function of magnetic field and temperature were carried out in order of explore the magnetic properties of all samples. The equipment used were a commercial vibrating sample magnetometer (VSM) and a commercial superconducting quantum interference device (SQUID) at Universidade Estadual de Campinas and Quantum Condensed Matter Division, Oak Ridge National Laboratory, in the temperature range between 4 K to 320 K and magnetic field between 0 and 70 kOe. In general, these measurement equipment operate based on themagnetic flux change of the sample inside a detector coil. The most commonly used magnetization measurement protocols are zero field cooling (ZFC), field cooling (FC) and warming field cooling (WFC). ZFC is a measurement type of magnetization as a function of the temperature, which consists incooling the sample without an external magnetic field applied, to lowest measurement temperature. Then, the magnetic field is switched on, and the data is acquired during the temperature rise process. FC measurement is realized during the cooling process of the sample with applied magnetic field. WFC differs from FC because it is performed when the sample is in the heating process after being cooled with applied external magnetic field.From these magnetic data, we can obtain the susceptibility χ as a function of the temperature (as in Chapter <ref>), andinverse susceptibility as a function of the temperature. From lineal fit at paramagnetic regime and Curie-Weiss law, we can obtain parameters such asthe effective magnetic moment, the Curie-Weiss constant and the Curietemperature. PART:Intermetallic alloys CHAPTER: INTERMETALLICS WITH B: COMPETING ANISOTROPIES ON 3D SUB-LATTICE OF YNI_4-XCO_XB COMPOUNDS In this chapter, we focus on competing anisotropies on 3d sub-lattice of YNi_4-xCo_xB alloys, which plays an important role on the overall magnetic properties of hard magnets. Intermetallic alloys with boron (for instance, R-Co/Ni-B) belong to those hard magnets family and are useful objects to help to understand the magnetic behavior of 3d sub-lattice, specially when the rare earth ions R are not magnetic, for instance YCo_4B. Interestingly, YNi_4B is a paramagnetic material and the Ni ions do not contribute to the magnetic anisotropy. Here, we focused our attention on YNi_4-xCo_xB series, with x=0, 1, 2, 3 and 4. For this purpose, we development two model for Co occupation into the crystallographic sites in the CeCo_4B type hexagonal structure.X-ray powder diffraction data were obtained at UFF and at room temperature, the magnetic measurements were carried out using a commercial vibrating sample magnetometer (VSM) and a commercial superconducting quantum interference device (SQUID) at Unicamp, and in order to determine composition and topology of the samples, we carried out scanning electron microscopy (SEM) measurements at IF Sudeste MG.§ BRIEF SURVEY OF R_N+1CO_3N+5B_2N FAMILY ALLOYS Since the 70's intermetallic alloys with boron, like Nd_2Fe_14B <cit.> and SmCo_4B <cit.>, have been very much studied due to their permanent magnets properties. Some of these materials were inspired by SmCo_5 <cit.>, from the substitution of Co by B into the R_n+1Co_3n+5B_2n family (R= rare earth), with n=1, 2, 3 and ∞<cit.>. It is well known that the magnetic anisotropy is an important property that rules the magnetic hardness of the material, specially the anisotropy from the 3d sub-lattice.The aim of this effort is to provide further knowledge about the 3d magnetic anisotropy of intermetallic alloys with boron. To this purpose, we consider a non-magnetic rare-earth (yttrium), in order to be sure that the magnetic contributions are coming only from the 3d sub-lattice. In addition, we considered two transition metals: Ni and Co. From one side, Ni ions do not contribute to the magnetic anisotropy <cit.>, while Co ions are extremely anisotropic <cit.>.Thus, YNi_4-xCo_xB alloys were synthesizedin an arc furnace under argon atmosphere with appropriate amounts of cobalt, nickel, boron, and yttrium , with x=0, 1, 2, 3 and 4. YNi_4B (x=0) is a paramagnetic material and does not present signatures of anisotropy <cit.>, while YCo_4B (x=4) has its Curie temperature at 380 K and spin reorientation (due to a strong anisotropy competition), at 150 K <cit.>. Therefore, it isclear that the anisotropy of YNi_4-xCo_xB alloys depends on the Co content x. To explore these features, we develop a statistical model of Co occupation among the crystallographic (Wyckoff) sites (2c with axial anisotropy and, 6i with planar anisotropy <cit.>), in which predicts a strong competing anisotropy among these two sites and spin reorientation for all samples of this series. On the other hand, a preferential model, in which Co ions go into a preferential position into Wyckoff sites, is developed and predicts that only x=2 and x=4 samples would have strong competing anisotropies with spin reorientation. Experimental measurements of magnetization on those samples verify that this last model successfully describes the nature of 3d magnetic anisotropy of this family. This preferential occupation of Co into 3d sites has a simple physical meaning: maximization of Co-Co distances. Indeed, this kind of approach was already experimentally verified with neutron diffraction measurements in other samples, likePrNi_5-xCo_x <cit.> and YCo_4-xFe_xB <cit.>. § CRYSTALLOGRAPHY OF YNI_4-XCO_XB ALLOYS R_n+1Co_3n+5B_2n structures with n = 1,2,3 and ∞are possible due to the replacement of Co by B in every second layer of RCo_5 (n=0) <cit.>, as illustrated in Fig. <ref>. More precisely, RCo_4B (n=1) compound consist of two crystallographic sites for rare-earth: 1a and 1b; two crystallographic sites for Co (or 3d ions): 2c and 6i; and only one site for B ions: 2d <cit.>. This can be seen in the figure <ref>, where the RCo_5 (n=0) case is also shown for comparison.These compounds have the CeCo_4B type structure, with space group P6/mmm (ISCD n^∘191) <cit.>. The first YNi_4B alloy was reported by Niihara <cit.> with the same structure as above. Later, Kuz'ma and Khaburskaya <cit.> reported a superstructure with lattice constant a=3a_0 and c=c_0, where a_0 and c_0 are the lattice constant of the original structure found by Nihara. This superstructure was also found on YNi_4-xCo_xB series by Isnard and co-workers <cit.>.X-ray diffraction on our samples show that all those crystallize in a single phase, similar to CeCo_4B structure (see figure <ref>-a), without extra picks in the diffractograms associated to the superstructure reported by Isnard et al.<cit.>. The lattice parameters a and c were determined using the standard pattern matching method of the Powder Cell software <cit.>; and thesechange almost linearly as a function of Co content, as can be seen in figure <ref>-b. A similar behavior was found by Chacon on YNi_4-xCo_xB <cit.> and Ail et al. on PrNi_4-xCo_xB <cit.>. SEM measurements also show that the stoichiometry of the experimental composition of the samples are according with nominal composition (see Fig. <ref>). § COMPETING ANISOTROPIES IN 3D SUB-LATTICE The magnetic anisotropy of YNi_4-xCo_xB compounds is due to the presence of Co ions, since Ni ions do not contribute to the anisotropy <cit.>. To understand the anisotropy of these compounds is necessary know from the crystallographic point of view, the mechanism of Ni/Co substitution. Let us first consider the YCo_4B compound,where 2c and 6i sites are fully filled of Co. The anisotropy/ion, at 0 K, are known <cit.> ϵ_2c=47.6514 × 10^-23 J/ion in 2c siteand ϵ_6i=-15.9804 × 10^-23 J/ion in 6i site; and therefore the total anisotropy for each site reads as: E_2c=1ϵ_2c= 47.6514 × 10^-23 J and E_6i=3ϵ_6i=-47.9412 × 10^-23 J <cit.>. Note the pre-factor are the corresponding occupation factor of the Wyckoff sites (3/4 for 6i and 1/4 for 2c), and we are considering only one formula unit, i.e., YCo_4B. On the other hand, 2c site has its magnetic moment with axial anisotropy, while 6i site has planar anisotropy (result from Mössbauer measurement <cit.>). These facts lead therefore to a strong competition of anisotropies, since the magnitude of those two contributions are the same, but the directions are different. The consequence is simple: a minor energy addition to the system (either thermal or magnetic, for instance), is able to unbalance this fragile equilibrium; and indeed it occurs: a spin reorientation from the plane to the axis happens at 150 K. Note these competing anisotropies lead to an almost vanishing overall anisotropy energyE_a = E_2c+ E_6i.To understand the magnetic anisotropy of the proposed series, we need analyses the mechanism of Ni/Co substitution. Thus, for a given compound of the YNi_4-xCo_xB series let us considerP_k (x)= x!/k!(x-k)! p^k (1-p)^x-k, as the probability of finding k Co ions in the 6i site, for a given Co content x. It simply considers 6i site has a weight of p=3/4, due to its bigger size. Based on this distribution probability, let us focus on two different models: one with statistical distribution, in which all probabilities distribution are taken into account; and a second model, in which only the most probable distribution is considered. This latter represents a preferential site occupancy for the Ni/Co substitution, and its hypothesis has been verified in PrNi_5-xCo_x <cit.> and YCo_4-xFe_xB <cit.>.The first model considers all possibilities of occupancy to obtain the anisotropy energy for each site. Thus, it is straightforward to write:E_6i = ∑^x_k=0 P_k(x) k ϵ_6i,E_2c = ∑^x_k=0 P_k(x) (x-k) ϵ_2c.Evaluation of the above energies leads to the result shown in top panel of Fig. <ref>. This model predicts that the anisotropy energy of 2c and 6i sites are the same in magnitude for all samples, and therefore a strong anisotropy competition would be observed with further spin reorientation on all of them.In a different fashion as before, the preferential occupation model considers that the Co ions are distributed among those two Wyckoff sites in a preferential fashion. To evaluate this idea, we considered only the most probable element of the set {P_k (x)} and save its corresponding k value, named as k_max, i.e.: P_k_= { P_k(x)}. Thus, the anisotropy energy of each site can be written as E_6i= k_ϵ_6i,andE_2c= (x-k_)ϵ_2c. The most probable distributions, for each value of Co content x, are shown on Table <ref>. Note the physical meaning of this preferential occupation model: Co ions try to keep the maximum distance of each other. The bottom panel of Fig. <ref> summarizes the results of the of latter model. For x=1, it predicts that the anisotropy energy of the 2c site is zero (there are no Co ions in this site for such Co concentration), while the anisotropy energy of 6i site is finite. As a consequence, there is not a competing anisotropy and the magnetic moment of the 6i site stays in the basal plane. Obviously, without anisotropy competition there is no spin reorientation. This analysis is similar toassociated with the casex=3, i.e., there is neither anisotropy competition no spin reorientation. The scenario is different for the samples with x=2 and x=4. For these two samples, the anisotropy energy of each site is comparable, leading to a strong anisotropy competition and, therefore, to a spin reorientation. Summarizing, the present model considers a preferential occupation of the Wyckoff sites, given by the most probable value of the distribution considered in equation <ref>. The physical roots of this model, interestingly, is to maximize the Co-Co distances. As a consequence, we found magnetic anisotropies for all samples of the series. However, competing anisotropies with a consequent spin reorientation is found only for samples with x=2 and x=4. It is important take account that this kind of model was experimentally verified previously in similar materials: PrNi_5-xCo_x <cit.> and YCo_4-xFe_xB <cit.>.§ MAGNETISM IN YNI_4-XCO_XB ALLOYS Nagarajan et al. <cit.> studied the YNi_4B compound and observed that it behaves as a paramagnet from room temperature down to 12 K. Below this threshold temperature, they found a superconducting behavior. Later <cit.>, this superconducting behavior was ascribed to be from an additional phase containing carbon. On the other hand, YCo_4B is a ferromagnetic material withT_c=380 K, exhibiting spin reorientation at 150 K due to the competition between the two crystallographic sites of Co <cit.>. The magnetic properties of the RNi_4-xCo_xB compounds were studied for R = Sm <cit.>, Pr <cit.>, Nd <cit.> and La <cit.>. In all these systems the saturation magnetization (M_s) and the Curie temperature (T_c) increase monotonically with Co content. In our samples, we measured magnetization as a function of magnetic field at 4K (see Fig. <ref>-a). The YNi_4B sample shows no hysteresis, and has a quite small value of magnetization. Increasing the Co content, the hysteresis width becomes larger, with the maximum width occurring for x=2. This fact is in accordance with both models, since the anisotropy energy of each Wyckoff site promotes this hysteresis. Note also that the saturation value of magnetization for these samples increases with the Co content. The temperature dependence of the magnetization was also measured, and the results exrepessed as the magnetic moment per formula unitary are presented inFig. <ref>-b. YNi_4B is indeed paramagnetic with a possible superconducting behavior below 20 K, in accordance with references<cit.>. By adding Co, the compound become ferromagnet, for example, Isnard <cit.> showed that the sample with x=1 has a ferro-paramagnetic phase transition at T_c=180 K, without a spin reorientation phenomena. Forx=2, we observed strong drop of magnetization at 150 K as a spin reorientation,and a much higher Curie temperature of 310 K. This series is able to receive more Co ions, for x=3 sample, we observe the Curie temperature at 307 K, in accordance with reference <cit.>. Finally, the last sample of our series exhibits a spin reorientation at 150 K and the para-ferromagnetic Curie temperature at 380 K, in agreement with Refs. <cit.>. These remarkable temperatures are exhibits in Table <ref>.It is worth to note that our experimental result are in excellentagreement with the model of preferencial occupancy, which predicts spin reorientation forthe samples with x=2 and x=4 only (see Fig <ref> and Table <ref>).§ CONCLUDING REMARKS ON YNI_4-XCO_XB ALLOYS Two possible occupation models for Co ions in the 3d sub-lattice of the YNi_4-xCo_xB samples has been analyzed. Oneconsiders a statistical distribution of Co/Ni ions, while the other considers a preferential occupation for Co ions. The former predicts strong anisotropy competition among the two 3d possible Wyckoff sites (2c and 6i) with spin reorientation for all Co contents. In contrast, the second model predicts that both sites have strong anisotropies, however, the competition (|E_2c|=|E_6i|) and spin reorientation arises only for x=2 and x=4. Our experimental data of magnetization as a functionof magnetic field and temperature show that only x=2 and x=4 compositions exhibits spin reorientation in agreement with the preferential occupation model. Similar results have been previously obtain for other compounds <cit.>. From the physical point view, this preferential occupation model interestingly mimics the case in which Co-Co distances are maximized. CHAPTER: HEUSLER ALLOYS: GENERAL CONSIDERATION The Heusler alloys family have attracted great scientific and technological interest mainly in the fields of spintronics and magnetocaloric effect, among others. In this chapter, we review some general properties of Heusler alloys, which will help us tothe following chapters that deal with these compounds.§ INTRODUCING HEUSLER ALLOYS In 1903 Fritz Heusler discovered that it was possible to make ferromagnetic alloys entirely from non ferromagnetic elements such as copper-manganese bronze alloyed with tin, aluminum, arsenic, antimony, bismuth, or boron <cit.>. Then, the constituent elements of the Heusler compounds cover almost the whole periodic table, as shown in Fig. <ref>.We can subdivide the Heusler alloys into the following categories, according with the stoichiometric function, nomenclature, and ion positions:Half-Heusler alloys (or semi-Heusler alloys) have the form XYZ, where X, Y and Z are mainly transition metals or are replaced by rare earth metals, metalloids, or other elements like Li, Be, and Mg. Thus, the nomenclature order in the literature varies greatly, ranging from sorting the elements alphabetically, according to their electronegativity, or randomly, and then all three possible permutations can be found according to the convenience of the authors. This semi-Heusler type crystallize in the C1_b cubic structure.Full-Heusler alloys (or simply Heusler alloys) are intermetallics with the X_2YZ form, where X and Y are transition metals, and metalloids are mainly used for Z. Thus, the nomenclature of these compounds generally has two parts of the first metal followed by other metal element of minor quantity and finally by a metalloid, e.g., Co_2FeSi, Fe_2MnGa, and many others. The first Heusler alloys studied crystallized in the L2_1 structure, which consists of 4 interpenetrating fcc sublattices in an A, B, C and D order. Here, the sequence of the atoms is X-Y-X-Z, in aCu_2MnAl-type structure<cit.>. In this thesis, we focus on the study of this Heusler alloys type.Inverse Heusler alloys are compounds that also have the chemical formula of X_2YZ, but in their case, the valence of an X transition metal atom is smaller than the valence of an Y transition metal atom. As a consequence, inverse Heusler compounds crystallize in the so-called XA or X_α structure, where the sequence of atoms is X-X-Y-Z and the prototype is Hg_2TiCu-type <cit.>. This is energetically preferred to the L2_1 structure of the usual full-Heusler compounds. Several inverse Heusler alloys have been studied using first-principles electronic structure calculations <cit.>. Inverse Heusler alloys are interesting for applications since they combine coherent growth of films on semiconductors with large Curie temperatures (e.g. Cr_2CoGa <cit.>). Quaternary Heusler alloys are compounds with the chemical formula of XX^'YZ, where X, X^', and Y are transition metal atoms. The valence of X^' is lower than the valence of X, and the valence of the Y element is lower than the valences of both X and X^'. The sequence of the atoms in the fcc structure is X-Y-X^'-Z which is energetically the most stable <cit.>. A large series of such compounds has recently been studied <cit.>.Fig. <ref> shows the positions of the constituent atoms in each Heusler alloy type. In all cases, the lattice consists of four interpenetrating fcc lattices, except for half-Heusler alloys, where the C sublatticeis not occupied.§ CRYSTAL STRUCTURE OF HEUSLER ALLOYS The family of full-Heusler compounds X_2YZ crystallize in the cubic space group Fm3̅m (space group no. 225 ICSD) in a Cu_2MnAl (L2_1) structure type <cit.>. X atoms occupy Wyckoff position 8c (1/4,1/4, 1/4), Y and Z atoms are located at 4a (0, 0, 0) and 4b (1/2,1/2, 1/2), respectively. This structure consists of four interpenetrating fcc sublattices, two of which are equally occupied by X and the remaining atoms occupy the other two sublattices, as illustrated in the Fig. <ref>.However, various variants of the L2_1 structure can be formed, if X and/or Y atoms are intermixed at the respective crystallographic positions, leading to different (local) symmetries and structure types <cit.>. We describe the most common types of structures in what follows. ∙ B2-type structureIf Y and Z atoms are randomly intermixed at their crystallographic positions, a B2-type structure is obtained, in which Y and Z sites become equivalent. This structure may also be described using a CsCl lattice, and as a result of this intermixing, the CsCl lattice with X at the center of the cube randomly surrounded by Y and Z atoms is obtained (see Fig. <ref>). The symmetry is reduced, and the resulting space group is Pm3̅m. All X atoms are at the 1b Wykhoff position, Z and Y atoms are randomly distributed at the 1a position. ∙ A2-type structureA completely random intermixing at the Wykhoff 2a position in X_2YZ Heusler compounds between all sites results in the A2-type structureIm3̅m with reduced symmetry. The X, Y, and Z sites become equivalent leading to a body-centered cubic lattice, also known as the tungsten (W) structure-type (see Fig. <ref>). ∙ DO_3-type structureThe space group Fm3̅m is kept, but if X and Y atoms are mixed at their crystallographic positions, a DO_3-type structure is obtained; the corresponding ICSD notation is BiF_3 structure type (see Fig. <ref>).The different structure types as described above will lead to the formation of different local environments for each atom. This is known as the atomic disorder and causes modifications in the intrinsic properties of these materials. In many cases, X-ray diffraction with a CuK_α source is not enough for experimental determination of the disorder, andspecial measurement setups such as anomalous X-ray diffraction (AXRD), Nuclear Magnetic Resonance Spectroscopy or Mössbabuer spectroscopy are required.In Chapter <ref>, this problemis discussed in detail.§ HALF-METALLIC FERROMAGNETISM HEUSLER ALLOYS The first attractive property of Heusler compounds comes from their magnetic characteristics.F. Heusler found that the Cu_2MnSn alloy is ferromagnetic although it is comprised of nonferromagnetic elements at room temperature <cit.>. Despite this, the impact of these compounds in the scientific community was not high for several decades. Until the eighties, the electronic structure of several Heusler compounds were investigated, and an unexpected result was found: depending on the spin direction, certain Heusler materials showed metallic as well as insulating properties at the same time; a feature called half-metallic ferromagnetism <cit.>, which can be found in other materials such as half-metallic oxides (Sr_2FeMoO_6, Fe_3O_4, La_0.7Sr_0.3MnO_3) <cit.>, diluted magnetic semiconductors <cit.>, for instance.Half-metallic ferromagnets are metals with 100 % spin-polarized electrons at the Fermi energy, i.e., electron with one spin direction behave as an insulator o semiconductor, while those with opposite spin are metallic. These compounds can be used for spintronic devices such as spin filters <cit.>, and tunnel junctions <cit.>, among other. Formally, the complete spin polarization of charge carriers in half-metallic ferromagnets is only reached in the limiting case of zero temperature and vanishing spin-orbit interactions. Since most of the Heusler compounds containing only 3d elements do not show significant spin-orbit coupling, they are ideal candidates to exhibit half-metallic ferromagnetism <cit.>. The half-metallic properties of Heusler alloys can be verified, for example,by calculations of density of states (DOS) at the Fermi level, using first-principle methods (see Fig. <ref>-top). The other method is by the total magnetic moment of the compound, which has to obey the generalized Slater-Pauling rule.Slater and Pauling discovered that the magnetic moment m of 3d elements and their binary alloys can be estimated based on the average valence electron number (N_V) per atom. The materials are divided into two groups depending on m(N_V) <cit.>. The first group has low valence electron concentrations (N_V ≤ 8) and localized magnetism. Here, mostly bcc and bcc-related structures are found. The second group has high valence electron concentrations (N_V ≥ 8) and itinerant magnetism. Here, systems with closed packed structures (fcc and hcp) are found. Iron is located at the border line between localized and itinerant magnetism. The magnetic moment, measured in unit of the magnetons (μ_B) is given by <cit.>:m = N_V - 2n_↓, where 2n_↓ denotes the number of electrons in the minority states. In the case of X_2YZ Heusler material, the magnetic moment per formula unit can be written as <cit.>:m = N_V - 24. The bottom panel in Fig. <ref> shows the magnetization as a function of the number of valence electrons per formula unit.§ MAGNETOCALORIC EFFECT IN HEUSLER ALLOYS The magnetocaloric effect is present in Heusler alloys. In addition, they may exhibit structural transitions under the influence of the temperature or external magnetic field, andsome of these materials are shape-memory alloys <cit.>. The most famous compounds of this family with these characteristics are those based on Ni-Mn-Ga<cit.>, because of the occurrence of coupledmagneto-structural transition near the room temperature. This has motivated several scientists to optimize these alloys for applications in magnetic refrigeration. Fig. <ref>-a shows the low-field ac magnetic susceptibility as a function the temperature of several samples of Ni-Mn-Ga Heusler alloys, where both the structural and magnetic transition changes can be observed, depending on the Mn concentration, from Ref. <cit.>. The entropy changes as a function of Mn concentration are shown in Fig. <ref>-b <cit.>.In Chapter <ref>, we provide more information on the magnetocaloric effect in Heusler alloys, and discuss how it is affected by the increase of the valence electron number. CHAPTER: INTERMIXED DISORDER EFFECT IN CO_2FESI HEUSLER ALLOY Co_2-based Heusler alloys as half-metallic compounds present promising properties such as T_C above 1000 K and large saturation magnetization. However, atomic disorder can affect the half-metallic properties of these materials and decreases their potential use in spintronic devices. Here, our aim is to evaluate the role played by atomic disorder in Co_2FeSi, which has a Curie temperature at 1100 K and magnetic moment of 6 μ_B according to the Slater-Pauli rule <cit.>. We have synthesized samples of Co_2FeSi following the process described in the Chapter <ref>, and annealed, then for 0, 3, 6 and 15 days at 1323 K with subsequent quenching in water. Here, the samples will be called 0d, 3d, 6d, and 15d based on the annealing times. Anomalous X-ray diffraction were obtained at room temperature at Laboratório Nacional de Luz Síncrotron. The Mössbabuer spectroscopy measurements at room temperature were performed at Laboratório de Mössbabuer at UFF. Density functional theory (DFT) calculations were carried out in collaboration with the University of Aveiro in Portugal, using the SPR-KKR (spin polarized relativistic Korringa-Kohn-Rostoker) package <cit.>, which implements the KKR-Green's function formalism. It is known from previous studies <cit.> that the local density approximation (LDA)fails to reproduce the half-metallic behavior of Co_2FeSi; therefore, the LDA+U approximation (where U is the calculation of the Hubbard) <cit.> was used. The U values (U_=3.8, U_=3.75 ) were chosen to obtain the expected behavior of the ordered system and were then fixed for the disordered system calculations. To simulate the atomic disorder, the Coherent Potential Approximation (CPA) <cit.> was used and the lattice parameters were kept fixed at the experimental values for Co_2FeSi. The angular momentum cut-off was set at l=3, and 2119 k-vectors in the irreducible Brillouin zone was used in all calculations, which were carried out in a relativistic approach. § CO_2FESI ATOMIC DISORDERFull-ordered Co_2FeSi crystallizes in the Cu_2MnAl-type structure, where atoms are localized in8c site (Co), 4a site (Fe) and 4b site (Si)(see Fig.<ref>.(a). However, it is possible to occur interchanges between Co-Fe atoms in Co_2FeSi (see Fig. <ref>.(b), and observed the variations in the magnetic and thermodynamic properties that affect the half-metallic behavior of this system. We show that the potential of Co_2FeSi for spintronics decreases with the increase in atomic disorder, and new ways for optimizing the production process of these materials are required to improve their performance. Co_2FeSi crystallized in the full-ordered L2_1 structure, the Co atoms occupy the 8c Wyckoff position at (1/4,1/4,1/4), the Fe atoms are located at the 4a sites (0,0,0) and the Si atoms occupy the 4b site (1/2,1/2,1/2). Nevertheless, the Co and Fe atoms interchanges their positions, leading to what is known as DO3 disorder.X-ray diffraction is commonly used to evaluate chemical disorder in Heusler alloys <cit.>; however, there are difficulties using this techniquewith a CuK_αsource. The main problem ofX-ray diffraction with a CuK_α source is the fact that the atomic scattering factors (f_hkl) of Co and Fe ions have very close values (see Fig. <ref>.a ). Making this technique inefficient for detection of the atomic disorder in Co_2FeSi. Other measurements setups, such as AXRD and Mössbabuer spectroscopy are required to evaluate the atomic disorder these systems. §.§ Anomalous X-ray difraction For Co_2FeSi,the AXRD data were collected close to the absorption edge energies of Co (7709 eV) and Fe (7112 eV), where the atomic scattering factors of Co and Fe are well distinguishable( see Figs.<ref>.b and Fig.<ref>.c ), allowing us to evaluate theDO3 disorder. AXRD of all the annealed samples of Co_2FeSi are presented in Fig.<ref>.a. A suitable q-range was chosen to show details of the first three peaks, which are the most sensitive to change in the energy of the focused beam. The differences in the intensitiesare due to the structure factors of each crystallographic plane: F_111∝ f_- f_ for (1,1,1) plane and F_200∝ 2f_ - f_+ f_ for (2,0,0) plane, where f_, f_ and f_ are the atomic scattering factors of Co, Fe and Si respectively.The full q-range for the 15d sample is shown in Fig <ref>.bfor three wavelengths: 1.60942 Å (Co edge absorption), 1.74389 Å (Fe edge absorption) and 1.8999 Å. The data analysis was realized using the FULLPROF software <cit.>, and was conducted simultaneously for three diffraction patterns of each sample. We found that all samples crystallize in the single phase Cu_2MnAl structure type, and the lattice parameters a= 5.645 Ådoes not change. However, the occupation of Co and Fe ions are different in all sample; and this is due to the influence of the annealing time. Thus, we have Co_2-xFe_xFe_1-xCo_xSi samples, where x is the disorder degree, for x=1 the sample iscompletely ordered, while, x=1 represents thecompletely disordered sample, i.e., the 4a site is occupied completely by Co ions. The 0d sample (non-annealed) presents the larger disorder degree in comparison with other samples (3d, 6d and 15d), which have disorder degree between x ≈ 0.12 and 0.18. The Fig. <ref> shows these results in comparison with Mössbauer spectroscopic results. §.§ Surrounding of the Fe ions by Mössbauer spectroscopy Fe^57 Mössbauer spectroscopy was used by us, for investigate the local environment of the Fe ions in the samples. This technique based on the Mössbauer effect discovered by Rudolf Mössbauer in 1958 <cit.>, and consists in energy level transitions of the nuclei, which can be associated with the emission or absorption of a γ-ray.These changes in the energy levels can provide information about local environment of an Fe-ions withinof the material <cit.>. Therefore, we performance Mossbauer spectroscopy measurements and determine the probability of the Fe ions has hyperfine magnetic field, which are characteristic of the Fe sites (stoichiometric and disordered) in the samples. For Co_2FeSi full orderedL2_1 structure, the Mössbauer spectrum at room temperature should show a single sextet, corresponding to the Fe ions occuping the 4a sites, while an additional sextets is observed in a disordered structure when the Fe ions occupy the Co Co position (8c sites) <cit.>. The Mössbauer spectra at room temperature of all samples are shown in Fig.<ref>-left. We found two sextets for each sample, indicating the presence of atomic disorder. The main sextet due to Fe ions in stoichiometric 4a site being surrounded by eight magnetic Co ions <cit.>, and therefore the hyperfine magnetic field of these is between B_^= 31.3 and 32.2 T <cit.>. The additional sextet is due to Fe ions occupying the disordered 8c site, is surrounded by four magnetic Fe ions (at 4a sites) and four non-magnetic Si ions (at 4b sites ) as the nearest neighbors <cit.>. In this case, the hyperfine magnetic field from disordered ions is in the range B_^ = 28.0 and 31.1 T.From this information, we obtain the probability that a Fe atom has a hyperfine field between the characteristic ranges of each crystallographic site. The Fig. <ref> shown these results, it is possible observedthat the samples have larger probability of possessing high value of hyperfine magnetic field, which indicate that they have greater portions of Fe in the stoichiometric site.Fig.<ref> shows the DO3 disorder parameter (x) ofCo_2-xFe_xFe_1-xCo_xSi samples from the measurements Mössbauer spectroscopy in comparison with AXRD results (discuss above). Using both techniques, we found that the 0d sample is more atomically disordered with x ≈ 0.46, whichis expected since this sample was not subject to annealing. Mössbauer spectroscopy observes Fe ions directly, and is able to identify their positions with this technique, we obtain disorder coefficients between x ≈ 0.21 and 0.33. In contrast with AXRD technique, we found that the disorder changes more that 10 % with the time annealing. This result can be explained by the fact the elements required more time the annealing for arrangement in their stoichiometric position. § HALF METALLIC PROPERTIES OF DISORDERED CO_2FESIDOS for Co_2-xFe_xFe_1-xCo_xSi system were obtain by DFT calculation for atomic disorder from x=0 to 0.5, is shown in the Fig. <ref>. The number of states at the Fermi energy (E_F) for the spin-up and spin-down band are plotted as a function of the disorder in Fig.<ref>.a. The increase in disorder augments the number of states in the spin-down band, while the spin-up band remains almost constant. This leads to a decreasingpolarization(P = D^↑_E_F-D^↓_E_F/D^↑_E_F+D^↓_E_F) as a function of disorder, as shown in Fig.<ref>.b. Since there is a small amount of states present in the calculated spin-down band of the x=0 case, the simulated system does notstrictly display a half-metallic behavior (P=1). However, the polarization is high (P=0.8) and the variation of P with x shows that the characteristic half-metallic properties decrease as the disorder increases. § CONCLUDING REMARKS ON CO_2FESI ATOMIC DISORDER From a single phase sample of Co_2FeSi , we explored experimentally the influence of annealing on atomic disorder in the sample, and showed that disorder suppresses the half-metallic properties of this Heusler alloy. DFT calculations provided theoretical information about the electronic states at the Fermi level in this compound.The polarization P is smaller than the on obtained for a totally ordered compound. This work provides further knowledge on new strategies to help optimization developments of Heusler alloys for spintronics devices. CHAPTER: EFFECTS OF GA SUBSTITUTION ON THE STRUCTURAL, MAGNETIC AND MAGNETOCALORIC PROPERTIES OF HALF METALLIC FE_2MNSI HEUSLER COMPOUND. Fe_2MnSi is a Heusler compound that crystallizes in Cu_2MnAl-type cubic structure and T_C = 224 K, and it does not follow the Slater-Pauling rule (explained in <ref>). On the other hand, Fe_2MnGa crystallizes inPt_2CuIn tetragonal-type, or Cu_3Au type, and T_C= 800 K. Despite the fact that these two compounds crystallize in different structures, we found single phase samples for the substitution of Si by Ga in Fe_2MnSi_1-xGa_x only for the Si-rich side until x=0.5. Here, we shall explorethe substitution of Si by Ga in Fe_2MnSi_1-xGa_x compounds to see how it affects their structural and magnetic properties. This substitution reflects the change of the valence electron number, and therefore change the half-metallic properties andthe magnetic entropy of the samples. § CHARACTERIZATION AND CRYSTAL STRUCTURE Fe_2MnSi Heusler compound order in the full Heusler L2_1 Cu_2MnAl-type structure, while Fe_2MnGa can be found in two distinct structures: Pt_3Au tetragonal-type or the Cu_3Au type L2_1 phase, presenting half-metallic features only in the L2_1 phase. In Fe_2MnSi_1-xGa_x, the Fe atoms occupy the 8c Wyckoff position at (0.25, 0.25 0.25); Mn atoms are located in the 4a site (0, 0, 0) and the Ga and Si atoms are assumed to occupy randomly the 4b site (0.5, 0.5, 0.5). The energy dispersive X-ray spectroscopy (EDS) was used to determine the sample compositions. We performed measurements at several points on the polished surface of each sample. The average values we have found are in very good agreement with the nominal compositions.Figure <ref>-top exhibits the X-ray diffratograms of Fe_2MnSi_1-xGa_x measured at room temperature. The characteristic reflections in the diffractograms obtained of the samples are in accordance with the L2_1 phase. Evidences of secondary phases were not detected. The powder X-ray diffraction data were refined by Rietveld method with the PowderCell software, using the Fe_2MnSi structural data (ICSD code number 186061) as basis. The results show that the cell parameter a tends to increase with the substitution of Si by Ga, from a=5.6627 Å (x = 0) to 5.7359 Å (x = 0.5). The reason for the increase of the cubic cell parameter is directly related with the atomic radii of Ga and Si, since the atomic radius of Ga is larger than Si.This tendency can be seen in Figure <ref>-bottom, where the lattice parameter a obtained from the Rietveld refinement is shown as a function of Ga content. By increasing the Ga content we observed that the reflections are shifted to lower angles.The same behavior was observed earlier in Fe_2MnSi, where Si was replaced by Ge <cit.>. From the figure, we can see that the cell parameter a increases linearly with the Ga content within the concentration range considered.We note that there is no evidence of changes in the crystal structure for Si substitution up to x = 0.50. The value of the cell parameter a for x = 1.0 (Fe_2MnGa) estimated by linear extrapolation using the concentration dependence is approximatelly equal to 5.8053 Å. This result is very close to the value of 5.808 Å reported by Kudryavtsev and co-workers <cit.>.§ MAGNETIC PROPERTIES Fig. <ref>-top shows the temperature dependence of the magnetic susceptibility (χ=M/H) for the Fe_2MnSi_1-xGa_x compounds for an applied magnetic field of 200 Oe. These compounds exhibit a magnetic transition from a paramagnetic state at high temperature to a ferromagnetic ordered state, and the transition temperature dependent on the Ga content. The transition does not present detectable thermal hysteresis, and has a second order character, despite being sharp for low magnetic field.From our data, we have determined how T_C varies with the Ga concentration for the Fe_2MnSi_1-xGa_x compounds, and the results are shown in Fig. <ref>-bottom. The Curie temperatures were obtained from the first derivative of the magnetization at 200 Oe. Atx = 0, T_C is 224 K. It is clear from Fig. <ref>-top and bottom that the ferro-paramagnetic transition temperature initially decreases (almost linearly) with the increase of Ga content, down to a minimum of 112 K for x = 0.50.The paramagnetic Curie temperatures (θ_p) were calculated for all samples from the fitting of the linear segment of the inverse susceptibility 1/χ data as a function of temperature. The obtained values are shown in Fig. <ref>-bottom.Fig. <ref> shows the magnetic field dependence of the magnetization up to 50 kOe for x = 0.02, 0.12 and 0.50 samples at 4 K. There is a clear tendencyof magnetization reduction with increasing the Ga content, but the saturation is not reachedfor the field values up to 50 kOe. However, the magnetic saturation can be estimated from M vs. 1/H curves (not shown) and are 2.11, 1.98 and 1.45 μ_B/F.U. for x = 0.02, 0.12 and 0.50 samples, respectively (see inset of Fig. <ref>). § HALF-METALLIC PROPERTIES Fe_2MnSi has a total of (2 × 8) + 7 + 4 = 27 valence electrons in the unit cell and, accordingly, Fe_2MnGa has 26 (Ga contributes with 3 valence electrons); for this reason the magnetic moment is expected to vary linearly from 3 μ_B to 2 μ_B by increasing Ga content in the Fe_2MnSi_1-xGa_x. As mentioned earlier, the magnetic moments estimated from M vs. 1/H curves at 4 K and 50 kOe for x = 0.02, 0.12 and 0.50 samples are 2.11, 1.98 and 1.45 μ_B per unit formula, respectively. These values are far from the expected ones according to the Slater Pauling rule (see section <ref>), of 2.98, 2.88 and 2.50 μ_B/F.U., respectively, but the decrease of the magnetic moment with the Ga concentration is clearly visible, as can be seen in the inset of Fig. <ref>. The difference between the estimated values and the experimental ones is quite large. Such discrepancies may be attributed to a partial atomic disorder in the structure, since the formed structures do not constitute a superlattice, as confirmed by the low intensity of the (111) and (200) X-ray diffraction peaks. The same behaviour was observed by Nakatani and co-authors<cit.>, who studied the magnetic and structural properties of the Co_2FeAl_xSi_1-x Heusler alloy. In their work, the authors report spin polarization and saturation magnetization dependences on the Al content. Nevertheless, the half-metallicity of the compound is preserved even for a partially disordered state, although the saturation magnetization values do not follow the Slater-Pauling rule. In this way, atomic disorder effects may indicate the Slater-Pauling rule is not followed by our samples, but our system still remains half-metallic, since both parent compounds Fe_2MnSi and Fe_2MnGa exhibit half-metal behavior.According to a work conducted by Graf and co-authors <cit.>, there is a linear dependence of the Curie temperature T_C with the magnetic moment on half metallic Heusler alloys. In the same work, the authors also showed that T_C increases for half-metallic compounds with the valence electron number. The decrease of T_C with x is probably related to the reduction of the magnetic moment caused by changes in the number of valence electrons in the system <cit.>. This tendency was observed in our Fe_2MnSi_1-xGa_x series, as can be seen in Fig. <ref>, where the magnetic transition temperature T_C presents an almost linear dependence as a function of the number of valence electrons. Such behaviour reinforces the half-metallic behaviour of the Fe_2MnSi_1-xGa_x series, despite the fact that it does no follow the Slater-Pauling rule due to a minor structural disorder in the system.Despite this latter fact, it is possible to observe the linear dependence of saturation magnetization and T_C with the N_V. Other studies connecting T_c and N_V confirm the linear growth tendency of these quantities for half-metallic Heusler alloys<cit.>. Thus, we provide a multifunctional Heusler alloy with enhanced magnetocaloric properties ruled by N_V and half-metallicity.The figure <ref> the extrapolation the Curie temperature to 300 K and verify that N_V = 27.44 would bring the ferromagnetic transition up to room temperature (a desired feature expected to optimize magnetocaloric materials). Therefore, one should increase the valence electrons number N_V to further optimize the magnetocaloric properties of half-metal Heusler alloys. To achieve this goal, either Si or Ga may be replaced by other element (or elements) that can contribute with more electrons; i.e., those elements belonging to, group 15 of the periodic table, such as P or As, for instance. These elements contribute with 5 electrons, and can increase the overall valence electrons number of the system. On the other hand, a substitution by elements of group 14, such as Ge and Sn, does not increase the overall valence electrons number for this series, because they have only 4 valence electrons (the same valence electron number of Si). In fact, Zhang <cit.> found values for T_C varying between 243 and 260 K only by replacing Si by Ge in parental Fe_2MnSi compound.We expect that Fe_2MnSi_0.56P_0.44 may be an interesting system. Since Kervan and Kervan <cit.> conducted an ab initio calculations for Fe_2MnP and confirmed the half-metallic features of this system. § MAGNETOCALORIC EFFECT AND THE VALENCE ELECTRONS NUMBER We performed magnetization measurements as a function of magnetic field for several temperatures around T_c (see Fig. <ref>-left). We have chosen a few compositions (x=0.50, 0.12 and 0.02), among those depicted in Fig. <ref>. Our results are shown in Fig. <ref>-left, thermo-magnetic curves are presented in Fig. <ref>-center. The magnetic entropy change Δ S(T,Δ H) calculated by: Δ S(T,Δ H)=∫^H_0(∂ M(T,H)/∂ T)_HdH,and the results are shown in Fig. <ref>-right for Δ H = 10, 20, 30, 40 and 50 kOe. An interesting result is found by increasing N_V (replacing Ga by Si), showing that the maximum magnetic entropy change also increases. As mentioned above, the Curie temperature increases when N_V rises (see Fig. <ref>), and therefore, a shift towards higher temperatures on the magnetic entropy change peak is expected as observed. At this point it is important to explore the character of the magnetic transition; and, to this purpose, we have generated Arrott plots for the sample with x=0.02, as shown in Fig. <ref>. The curves present a positive slope (B parameter of the Landau expansion <cit.>), for low values of magnetization, which indicates a second order magnetic transition according to Banerjee's criterion <cit.>. Similar results were obtained for the other samples.The relative cooling power RCP[Defined as the maximum magnetic entropy change times the full width at half maximum] (at 50 kOe), ranges from 47 J/kg (for N_V=26.50), up to 76 J/kg (for N_V=26.98); but we will focus our attention on the maximum magnetic entropy change |Δ S|_max, depicted on Fig. <ref>. It shows a linear behavior as a function of N_V. For N_V=27.44, it is expected to achieve 1.2 J/kg.K@20 kOe, that would indeed be comparable to standard metallic Gd (4 J/kg.K@20 kOe). Thus, the Heusler alloy with N_V=27.44 would optimize the Curie temperature, shifting |Δ S|_max towards room temperature, and, in addition, will enhance the magnetocaloric properties (see figure <ref>). § CONCLUDING REMARKS ON FE_2MNSI_1-XGA_X The substitutional series of the Heusler compound Fe_2MnSi_1-xGa_x has been synthesized and investigated experimentally. The phase obtained crystallizes in the cubic L2_1 structure and the lattice parameter a increases linearly with the increase Ga content. The Curie temperature changes significantly with Ga content in the 0 ≤ x ≤ 0.5 range, but the Ga doping interval chosen was not enough to bring T_C to room temperature. The compounds do not follow the Slater-Pauling rule, probably due to a minor degree of disorder in the system, but the almost linear dependence of the magnetic transition temperature with the valence electron number reinforce the half-metallic behavior of the compound. Samples with x > 0.5 were prepared, but x-ray diffraction analysis revealed the appearance of additional peaks, indicating the existence of secondary phases or structural changes in these samples. Furthermore, we explored the Si-rich side of Fe_2MnSi_1-xGa_x Heusler alloys, and concluded that the valence electron number N_V plays an important rule on their Curie temperature and magnetic entropy change. Increasing N_V (equivalent to increase the Si content in Fe_2MnSi_0.5Ga_0.5 compound), leads to a linear increase of both quantities. Our conclusion is that N_V=27.44 would bring the Curie temperature of the compound to room temperature, as well as promote an increase in the maximum magnetic entropy change. We also propose to substitute Ga by a group 15 element, like P; and we expect that Fe_2MnSi_0.56P_0.44 would have a Curie temperatures close to 300 K, with an enhanced magnetocaloric effect. These ideas can indeed lead to new multifunctional materials, opening doors for further researches on this topic.PART:Perovskite oxides CHAPTER: GENERAL CONSIDERATION ON PEROVSKITE OXIDE STRUCTURE Perovskite-type oxides belong to a big family that includes A_n+1B_nO-type perovskites, and they form a type of structure that is very versatile for developments in several technological areas. Perovskites are functional materials that exhibit a range of stoichiometries and crystal structures, and they continue to be actively studied because of their special magnetic and electronic properties. This chapter, we shall focus on the ABO_3 perovskite structure type, and explore some of its magnetic properties. § INTRODUCTION TO THE PEROVSKITE STRUCTURE PerovskiteABX_3 (ABO_3 for us) is the name of a structural family, which the ion A has a larger size than ion B, and most of the metallic ions in the periodic table can be used to create ABX_3 perovskites <cit.>. Although the majority of the perovskite compounds are oxides or fluorides (for instance KBF_3 with B= Mn,Fe and Zn <cit.>), other forms like heavier halides, sulfides, hydrides, cyanides, oxyfluorides and oxynitrides are also reported <cit.>. Perovskites refer to the cubic crystal structure Pm3m, which consist of a three-dimensional arrangement of corner-sharing BO_6 octahedrons; however, ABO_3 perovskitescan crystallize also in other structures.An A-site cation fills 12 coordinate cavities formed by the BO_3 network and is surrounded by 12 equidistant anions <cit.>, as seen in Fig.<ref>. According to Lufaso et al. <cit.>, a cubic perovskite can transform into other crystal structures through the distortion of the octahedral BO_6, andthe resulting structures are closely related to the cubic perovskite.In general, the stability of perovskites is often studied in terms of the tolerance factor (t), which was introduced by Goldschmidt in the 1920s <cit.>, t = r_A+r_O/√(2) (r_B+r_O). Here, r_A andr_B are the ionic radii of cation A andanion B, respectively, while r_O is the ionic oxygen radius. When t is approximately 1, we have a cubic structure. However, if r_A and consequently t decrease, the crystal structure changes to rhombohedral (0.96 < t < 1) or orthorhombic (t < 0.96) structures <cit.>. In fact, variations in the t values define the limits of the crystal structure. § DISTORTION EFFECTS ON THE PEROVSKITE OXIDE STRUCTURE The observed magnetic and electric properties of perovskite oxidesare directly related to their structure , and constituent elements. As mentioned above, site B can be occupied by a3d transition metal (for example M = Mn and Co ), and site A can be occupied by a trivalent rare earth (R) such asLa^3+, Nd^3+, and Pr^3+ or by a divalent alkaline earth (T) like Sr^2+, Ca^2+ and Ba^2+. The valence of the 3d transitionmetal depends directly on what material occupies the site A (R or T). When it is occupied by a trivalent element (R), the 3d transition metal becomes trivalent (R^3+M^3+O_3). However, if A is occupied by a divalent element (T), then 3d transition metal becomes tetravalent (T^2+M^4+O_3). For compounds withpartial substitution, A=(1-x)R + x T, the M valence is a mixture of 3+ and 4+, characterized by:R_1-x^3+ T_x^2+ M_1-x^3+ M_x^4+O_3. This mixture of valence leads to the so-called double exchange interaction, as proposed by Zener using the manganites (where M = Mn) <cit.>. This mechanism of exchange has its origin in the itinerant character of the electron that occupies the e_g orbital of Mn^3+, and the hybridization between theMn d-orbital and the oxygen p-orbital. The left panel of Fig. <ref> illustrated this process. When the initial state has the {Mn^3+-O-Mn^4+} configuration, one electron passes from the Mn^3+ to the oxygen, and another (previously in O^2-) simultaneously migrates to Mn^4+, leading to the final state {Mn^4+-O-Mn^3+}. The double exchange occurs when there is a ferromagnetic coupling between Mn ions. The ferromagnetic arrangement observed in these oxides is closely related to the electron movement. Another important distortion of these perovskites is due to the Jahn-Teller effect <cit.>, which consists in a distortion caused by the presence of Mn^3+ ions (manganites case) inside the oxygen octahedrons.It happens because of the interaction between the 3d transition metal orbitals (e_g and t_2g) and the oxygens at vertices of the octahedral. Electrostatic repulsion induces a distortion of the octahedron in order to decrease the total energy of the system. Therefore, depending on the degree of such lattice distortions, cubic (ideal), orthorhombic and even rhombohedral symmetries may occur <cit.>. This rearrangement of octahedron atoms breaks the degeneracy of the e_g and t_2g orbitals, so that the t_2g orbitals split into three levels of lower energy, d_xy, d_yz, and d_zx, the orbitals into two levels (d_z^2 and d_x^2-y^2) with higher energy <cit.>, as illustrated in the right panel of Fig.<ref>.The splitting occur due to interaction between the transition metal and oxygen orbitals <cit.>. A Mn^4+ ion has three electrons at the 3d level so Hund's strong coupling favors population of the orbital e_g, and as consequence gives a spin value of 3/2. Since, a Mn^3+ ion has four electrons at the 3d level, an additional electron can occupy either a place in orbital t_2g (with spin antiparallel to the others), or in orbital e_g with spin parallel to the others. The latter is energetically more favorable because of the reduction in the Coulomb interaction, leading to a total spin value equal to `2'. The important consequence of the spin separation in different energy levels in the 3d orbitals complex is the strong coupling between the e_g conduction electron and those in thet_2g orbitals. In the case of cobaltites (M = Co) this does not always occur, because the energies of the Hund coupling and the crystalline field energy are comparable. This gives to cobaltites an additional degree of freedom, which depends on the spin state of the material; this property will be further explored in Chapter <ref>.§ PEROVSKITES IN TECHNOLOGY Owing to simple preparation, low cost of production, and several interesting physical properties, perovskite oxides are one of the most versatile family of compounds for technological developments. Some of the technologies that are currently being developed that are based on oxides with perovskite structure are the following: * Solar cells: based on organic-inorganic perovskite structured semiconductors that exhibit high charge-carrier mobilities, together with high charge-carrier lifetimes. The light-generated electrons and holes can move relativity long distances, being extracted as current instead of losing their energy as heat within the cell <cit.>. * Magnetic hyperthermia: the magnetic properties of perovskites nanoparticles immersed in a fluid (ferrofluid)and subjected to external ac external magnetic field, make them useful for local heat treatments. It is possible to heat specific zones affected by cancer and kill the undesired cell. Combined with traditional treatment, magnetic hyperthermia is an alternative tool to help the cure of certain types of cancer <cit.>. * Spintronics: several properties of perovskite compounds, such as giant magneto-resistance, colossal magneto-resistance, and their dependence on the electronic charge and spin degrees of freedom, are exploited for efficient use of spintronic devices, and may be efficiently used in advanced technology <cit.>. * Magnetic refrigeration: as we explained in Chapter <ref>, the change in the magnetic entropy is one of the qualities that quantify the magnetocaloric effect, and how good a material is for magnetic refrigeration. The magnetic entropy change of perovskite-type manganese oxides is larger than that of Gd (which is the most studied material for this type of application), and thereforeperovskite-type oxides appear to be more suitable candidates for magnetic refrigeration at a high temperature, especially near room temperature <cit.>. In the two following chapters, we will explore the magnetic properties of these compounds and focus on half-doped cobaltite (Nd_0.5Sr_0.5MnO_3) and manganites (La_0.6Sr_0.4MnO_3). CHAPTER: SPIN STATE AND MAGNETIC ORDERING OF HALF-DOPED ND_0.5SR_0.5COO_3 COBALTITE Cobaltites show intriguing magnetic and transport properties. When compared with manganites, for instance, they exhibit an additional degree of freedom: the spin state of the Co ions. For Nd_0.5Sr_0.5CoO_3 this spin stateconfiguration is not well-established, as well as the magnetic ordering below the Curie temperature. In this Chapter, we report magnetization measurements that we have performed in order to understand various aspects of the half-doped Nd_0.5Sr_0.5CoO_3 cobaltite. Our results show that the Co and Nd magnetic sub-lattices couple antiferromagnetically below the Curie temperature T_c=215 K, down to very low temperatures. They clarify the presence of the plateau observed at 80 K in the M(T) curve, which in the literature is erroneously attributed, to the onset of an antiferromagnetic ordering. Our magnetization data also clearly shows that Co^3+ and Co^4+ are in an intermediate spin state. In addition, we have investigated the magnetic entropy changes in this system. Our experimental results and theoretical analysis allowed a consistent description of this magnetic behavior. Finally, a magnetic phase diagram for Nd_0.5Sr_0.5CoO_3 was built, bored on our results.Room temperature synchrotron powder diffraction measurements were carried out at beamline 17-BM at the Advanced Photon Source using a 100 μm monochromated x-ray beam at a wavelength of λ = 0.727750 Å. Magnetization measurements were carried out as a function of temperature and magnetic field using a commercial Superconducting Quantum Interference Device (SQUID) from Quantum Design^. § BRIEF SURVEY ON COBALTITE COMPOUNDSPhysical properties of cobaltites, as well as manganites, depend on the interplay between charge, spin, orbital and lattices degrees of freedom. However, cobaltites have an additional degree of freedom as Co^3+ and Co^4+ can occur in three different spin configurations <cit.>: low-spin LS (t^6_2ge_g^0 for Co^3+ and t^5_2ge_g^0 for Co^4+), intermediate-spin IS (t^5_2ge_g^1 for Co^3+ and t^4_2ge_g^1 for Co^4+) and high-spin HS (t^4_2ge_g^2 for Co^3+ and t^3_2ge_g^2 for Co^4+). This unusual flexibility arises from the comparable energies between the Hund's coupling and the crystal-field splitting <cit.>.Those intriguing magnetic and transport properties are mainly ruled by the doping level x inR_1-xT_xCoO_3, which induces lattice distortions and changes the valence of the Co ions. Compounds with x = 0 contain only Co^3+ (3d^6), while those with x=1 have only Co^4+ (3d^5); and, obviously, intermediate values of x lead to mixed valence compounds. Rao et al. <cit.> investigated R_1-xT_xCoO_3 (with R = Eu, Nd, Pr, Sm and Gd and T = Sr, Ca and Ba) and show that, in general, lower values of x lead to a very high electrical resistivity with no magnetic ordering; and above a critical value of x, a ferromagnetic ordering occurs. The Curie temperature, for this heavy doped region of the phase diagram, decreases with decreasing the size of the rare-earth ion. La_1-xSr_xCoO_3 <cit.> and Pr_1-xSr_xCoO_3 <cit.> series are the most studied cobaltites, and the latter are more interesting due to the non-zero magnetic moment of Pr. Some compounds of Pr_1-xSr_xMnO_3 series, for example, exhibit a broad peak in the susceptibility data around 90 K, whose originremainsan open question <cit.>. Some authors claiming that it is due to a state spin transition from the intermediate to low-spin state <cit.>, while others argue that it is caused by a change of the ferromagnetic state associated with orbital ordering <cit.>.Nd_1-xSr_xCoO_3 compounds have also been intensely investigated. Fondado and co-workers <cit.>, for example, studied the compounds in the composition range 0 ≤x ≤ 0.4. The authors found that, for x >0.2, the system is ferromagnetic, but for x =0 (NdCoO_3), the material is paramagnetic due to the zero paramagnetic effective magnetic moment of Co ions, which indicates that they are in low spin configuration. However, for T > 250 K, a deviation from linearity appears in χ^-1(T) indicating an onset of spin transition in the Co ions. This behavior is in fact observed for samples with x ≤ 0.1. Transport measurements <cit.> show that as x increases from 0.1 to 0.4, the activation energy also increases, which would induce different spin-states in the Co ions for the Nd_1-xSr_xCoO_3 compounds. However, the authors were not able to define if the spin state transition occurs due to thermal energy or crystal distortion as x changes. For x = 0.33, Paraskevopoulos et al. <cit.> showed that for temperatures above T_C (∼ 173 K) the Co^3+ and Co^4+ are in the IS and LS configurations, respectively, which confirms the change of the spin-state of the Co ions as x increases. For samples of half-doped Nd_1-xSr_xCoO_3, the spin-state is not yet well defined and, the broad maximum around 80 K is also an open question in the literature.Here we investigated the magnetic investigation of half-doped Nd_0.5Sr_0.5CoO_3 cobaltite. Magnetization measurements were made in order to determine the magnetic arrangement of Co and Nd sub-lattices, as well as the spin-state of the Co ions. In addition, magnetic entropy change was also explored, and those results are also consistent with our other findings. We close this contribution with a complete magnetic phase diagram for half-doped cobaltite.§ MAGNETOSTRUCTURAL RELATIONSHIP Crystal structure and magnetism are strongly related. In the present case, the Co ions are able to assume low, intermediate and high spin configurations, and identification of this relationship is of great importance. This section discusses the crystal structure of Nd_0.5Sr_0.5CoO_3 cobaltite (see Fig. <ref>). §.§ Crystal structure Rietveld analysis ofthe Nd_0.5Sr_0.5CoO_3 diffractogram shows a single phase, as can be seen in Fig. <ref>. The system crystallizes in an orthorhombic structure with space group Imma <cit.>. In this structure, the rare-earth metal, Sr and one of the oxygens ions (O1 - apical) occupy the Wyckoff position 4e (0,1/4,x), Co occupies the 4b site (0,0,1/2) and the other oxygen (O2 - equatorial) occupies the 8g (3/4,y,1/4) positions.The lattice parameters of our sample are presented in Table <ref>, and are in very good agreement with other result reported in the literature<cit.>. The table also shows the interatomic distances, and it is possible to see that O1 - Co distance (apical distance of the octahedron) is larger than the O2-Co basal distance, which indicates that, at least at room temperature, the octahedron is mostly elongated. In other words, the equatorial oxygens (O2) are symmetrically arranged around the cobalt ion. at a distance of 1.898 (Å), while the Co-O1 distance (apical oxygen) is of 1.986 (Å), almost 4.5 % of difference. These values of the Co-O distances are larger than those reported previously <cit.> for the Nd_0.67Sr_0.33CoO_3 compound, where the Co ions are assumed to be in a low-spin configuration. Considering that the ionic radius of high spin Co can be 10 % larger than the one for the low spin Co <cit.>, this indicate that the Co spin state is either on intermediate or high spin configuration. As we shall see, this is an important point in the discussion of the magnetic properties of this system.§ MAGNETIC PHASE DIAGRAM As discussed in section <ref>, the Nd and Co magnetic arrangement, as well asthe Co spin state of the Co ion in cobaltites remains as open questions. In order to draw a reliable phase diagram and be able to specify the Co spin state, we have performed magnetization measurements in this system. To support our findings, we have also investigated (both experimentally and teoretically) the magnetocaloric aspect in this compound. Our results allowed us to achieve our goals of determining the Co spin state and build up a magnetic phase diagram for Nd_0.5Sr_0.5CoO_3. §.§ Cobalt spin state Magnetization as a function of temperature was obtained under applied magnetic field of 1000 Oe, as shown on Fig. <ref> (left axis). In that curve it is possible to see a clear transition at T_c=215 K and a broad maximum peaking at T_k=80 K, consistent with previous reports <cit.>.The reciprocal magnetic susceptibility was also be obtained, and is presented on Fig. <ref> (right axis). The linear behavior of this quantity above T_c is a clear indication that the system reached the paramagnetic phase and therefore satisfies the Curie-Weiss law. However, a slight deviation from the genuine linear behavior was detected and a better fit is obtain with the theoretical reciprocal molar magnetic susceptibility given byχ(T)^-1=[χ_0+C/T-θ_P]^-1,where C=1/2(C_Nd+C_Co^3++C_Co^4+) is the overall molar Curie constant, θ_p is the paramagnetic Curie constant <cit.> and χ_0 was added to take into account some diamagnetic contribution, perhaps from the sample holder. The fitting of the experimental data with Eq. <ref> returns χ_0= 6.2× 10^-7 emu/Oe FU, θ_p= 215 K and p_eff= 4.3 μ_B/FU. The latter was calculated considering the contribution of all magnetic atoms (Nd and Co).The experimental molar Curie constant is depicted in Fig. <ref>, together with the theoretical values for several Co spin configurations. The theoretical Curie constants cwere determined the Landé factor for both Co sub-lattices g_Co=2 and, for the Nd sub-lattice, g_Nd=8/11 <cit.>, while J_Nd=9/2 and J_Co=S_Co, assuming that there is no orbital contribution. The experimental Curie constant matches the theoretical values obtained for both Co ions (Co^3+ and Co^4+) are in an intermediate state; and, have with g=2. These results are consistent with our precious hypothesis that the distortions occur in such way that the orbital moment is quenched, and the Co^3+ octahedral environment becomes elongated, and Co^4+ octahedral environment is compressed. However, no such distinction is apparent in the diffraction data directly, most likely because it probes long range order and there are no reports in the literature of charge (or orbital) ordering in this family of materials.To understand the magnetic behavior of this system, Reis et al. <cit.> considered a ferrimagnetic arrangement, between two sub-lattices and a mean field approach to calculate the magnetization, shown in the top panels of Fig. <ref>. Sub-lattice `1' is associated with the Nd sites and sub-lattice `2' with Co sites. M_1 andM_2 refer to corresponding sub-lattices magnetization. In the Fig. <ref>, the calculated values of M_1, M_2 and M_T=M_1+M_2 are plotted as function of the temperature. The `2' sub-lattice rules the critical temperature, as can be seen on figure <ref>-top. In the high temperature limit both M_1 and M_2 goes to zero, and do goes M_T. On the other hand, as the temperature decreases, the absolute value of M_1, due to the low intra-lattice exchange interaction, and as a result, the value of M_T decreases, leading to value of peak atT_k. T_k is defined as temperature at which M_T (T) is maximum when dM_T/dT=0 or, equivalently, when dM_1/dT=-dM_2/dT, as the bottom panel of Fig. <ref>, clearly illustrated. We note that the temperature dependence of the total magnetization is dominated by M_1, at low temperatures and by M_2 at high temperatures close to and above T_C.At this point we can compare the theoretical (figure <ref>-top) and experimental (figure <ref>) magnetic moment curve. Indeed, Nd sub-lattice has a ferromagnetic intra-lattice arrangement, as well as the Co sub-lattice. On the other hand,Nd-Co sub-lattices aligns in an antiparallel to each other; leading to an overall ferrimagnetic arrangement. It is important to note here that previous reports in the literature <cit.> ascribe T_k as the onset of antiferromagnetism, i.e., the Nd-Co ordering is antiparallel only below T_k. However, from the above theoretical results, it is clear that the magneticzation of theNd-Co sub-lattices are always aligned antiparallel for T<T_c. §.§ The magnetocaloric effect We have discussed general aspect of the magnetocaloric effect (MCE) in the Chapter <ref>. Here, wemeasured themagnetization of Nd_0.5Sr_0.5CoO_3 as a function of an applied magnetic field, for several temperatures. The result are depicted in Fig. <ref>(a). From the full set of data that we have collected, only a few selected temperatures are shown in this figure. It is noteworthy that the 4 K isothermal curve exhibitsa negative value of magnetization at low values of magnetic field,characteristic behavior of a ferrimagnetic material <cit.>.In Fig. <ref>(b), we show the magnetization as a function of temperature, for several applied magnetic fields. This data presents another important signature of a ferrimagnetic arrangement: a compensation temperature T_comp, in which |M_1|=|M_2|. From our data, it is possible to extract the magnetic entropy change as a function of temperature, for several values of the applied magnetic field change, and the result are depicted in Fig. <ref>(c). They display some interesting aspects: (i) a peak at T_c, similarly to what is observed in ferromagnetic materials, and (ii) a zero crossing at T_k. The literature <cit.> claims that the system is ferromagnetic above T_k and antiferromagnetic below this temperature, due to the signal (either positive or negative) of the magnetic entropy change<cit.>. However, in Ref. <cit.> is our theoretical approach, shows that the system is an ferrimagnetic material and is compatible with the experimental data. Following the stpdf described in Ref. <cit.>, it is possible to obtain the zero field magnetic entropy and, at H=50 kOe; shown i the top panel of the Fig. <ref>. Note that there is a crossing of the curves for H=0 and 50 kOe exactly at T_k, emphasizing that this temperature is indeed of extreme importance to the whole magnetic phenomena that occurs in this sample. The difference of these curves, i.e., the magnetic entropy change, way also be evaluated for several values of applied magnetic field and the result are depicted in the bottom panel of the Fig. <ref>. This result qualitatively agree with the ones obtained experimentally and shown in Fig. <ref>(c). It ratifies our claim that this material is indeed ferrimagnetic for the whole temperature range and T_k is the temperature at which the magnetic entropy curves cross, and not the one which the Nd-Co becomes antiparallel, as claimed in the literature <cit.>.§.§ Phase diagram and concluding remarks Based on the previous results and analysis, it is possible to construct a phase diagram of Nd_0.5Sr_0.5CoO_3, as depicted on Fig. <ref>. This material has a paramagnetic phase above 215 K, below which the system changes to a ferrimagnetic ordering, with the Nd sub-lattice magnetization in opposition to ones associated with both Co sub-lattices (Co^3+ and Co^4+). Due to the difference between the intra-exchange interaction (Co-Co interaction is much stronger than Nd-Nd interaction), the Co sub-lattice dominated the magnetization behavior close to the Curie temperature. At low temperatures, however the Nd sub-lattice controls the total magnetization behavior. T_k in phase diagram is defined as the temperature at which the total magnetization becomes maximum. Finally, at even lower temperatures and low field, due to the ferrimagnetic arrangement, it is convenient to specify a compensation temperature T_comp, which is also shown in the phase diagram. X-ray diffraction using synchrotron radiation with Rietveld refinement shows that the Nd_0.5Sr_0.5CoO_3 sample has a single phase, and O1-Co (apical distance) is, on average, larger than O2-Co (equatorial distance). The Curie constant indicates that the system in on intermediate spin state for both, Co^3+ and Co^4+, without orbital contribution and, consequently, a Landé factor g=2, in accordance with the theoretical model of Ref. <cit.>. Magnetization data shows a critical temperature T_c=215 K and a broad maximum at T_k=80 K. While previous published reports <cit.> associate these features to a ferromagnetic transition and the onset below which the Nd and Co sub-lattices become antiparallel. We show that those temperatures are, respectively, the ferrimagnetic transition and the onset below which Nd sub-lattice (negative) magnetization rules the total magnetization. In addition, T_k is the temperature in which (i) the temperature derivative of Nd and Co magnetizations become equal to each other and (ii) the field and zero-field magnetic entropy cross each other. The magnetic entropy change was also experimentally calculated and these results are consistent withferrimagnetic system. Finally, from all these results, the magnetic phase diagram for Nd_0.5Sr_0.5CoO_3 cobaltite could be drawn. CHAPTER: MAGNETIC AND STRUCTURAL INVESTIGATIONS ON NANOSTRUCTURED LA_0.6SR_0.4MNO_3This Chapter presents the structural and magnetic properties of La_0.6Sr_0.4MnO_3 nanoparticles with sizes distribution ranging from 21 to 106 nm. They have been prepared using the sol-gel method (see Chapter <ref>). The reduction of the nanoparticles size tends to broaden the paramagnetic to ferromagnetic transition, as well as to promote magnetic hysteresis and a remarkable change on the magnetic saturation. The XRD measurements were carried out at Laboratório de Raios X da UNiversidade Federal Fluminense,the transmission electron microscopy (TEM) technique was carried out at Inmetro, and the magnetic measurements were carried out using a commercial superconducting quantum interference device (SQUID) at Unicamp. § CONSIDERATIONS ON MANGANITES NANOPARTICLES The physics of nano-materials has been an exciting research field since the 90s.The magnetic properties of these systems have attracted special attention to fundamental and technological point view <cit.>. They find applications on medicine<cit.>, catalysis processes <cit.> and magnetic refrigeration <cit.>, for instance. When the size of the magnetic nanoparticles decreases to a few nanometers, these materials exhibit different magnetic behaviors, such as surface spin-glass, superparamagnetism, large coercivities, low-field saturation magnetization, and low Curie temperature, as compared to their bulk counterparts <cit.>. The magnetic behavior of the particle's surface usually differs from that of its to the core, because of the distinct atomic coordination, compositional gradients, concentration and nature of the defects present in both regions <cit.>. The surface gives rise to a large magnetocrystalline anisotropy due to its low symmetry and may induces different magnetic arrangements due to weakening of exchange interaction <cit.>.The magnetic properties of manganite nanoparticles synthesized by sol-gel method present novel features compared with the bulk materials, prepared by conventional solid-state reaction method <cit.>. When the particle size is reduced to few nanometers, are observed effects as the broadening of the paramagnetic to ferromagnetic transition, decreasing of the saturation magnetization value, increase the magnetic hysteresis <cit.> and appearance of superparamagnetic (SPM) behavior to very low particle size (< 17 nm) <cit.>. Generally, the magnetization of ferromagnetic nanoparticles is lower that bulk samples, this is explained usually by the consideration of a not magnetic layer (in the shell) around to magnetic nucleus (core) <cit.>. In the following sections, the structural and magnetic properties of La_0.6Sr_0.4MnO_3 nanoparticles produced by sol-gel method will be discuss. X-ray diffraction and transmission electron microscope measurements provide the average diameter of nanoparticles ranges from 21 nm to 106 nm. Magnetic measurements showed a dependence of the nanoparticle size magnetic saturation and transition. § CRYSTAL STRUCTURE AND MORPHOLOGY X-ray diffraction dataat room temperature and pressure of the samples submitted to calcination temperatures of 973 K, 1073 K, 1173 K and 1273 K compared with bulk sample is shown in Fig. <ref>. Rietveld method using the FullProf software <cit.> was applied in all La_0.6Sr_0.4MnO_3. We found that crystallized in perovskite structure space group R3̅c (group 167) in a rhombohedral crystal system, as reported by Shen <cit.>. One example of fit is shown in the bottom of Fig.<ref>, for sample calcined at 973 K. The crystallographic parameters and reliability factors obtained from refinement can be seen in Table <ref> (only presented for nanoparticles).The convergence factors R_p, R_wp and R_F obtained from Rietveld analysis point to the good quality of the refinement.From XRD data, it was possible to estimate the average particle size D using the Scherrer equation, which we found the average diameter of nanoparticles using the size broadening of the peaks from the relation<cit.>, D = k λ / β_Lcosθwhere k is a dimensionless constant (close to one), λ is the X-ray wavelength, θ is the diffraction angle for the most intense peaks and β_L = √(U tan^2θ + V tanθ + W)is related to the full width at half maximum (FWHM) of the peaks. The broadening of the peaks indicates the formation of a nanocrystalline compound <cit.>. When the calcination temperature arises, the diffraction peaks become sharper, resulting in larger particle sizes <cit.>. The particle sizes estimated from powder X-ray data are shown in table <ref>. In order to confirm the average grain size, we used the TEM images showed in Fig.<ref>. It is possible to verify that the average nanograin size decreases from approximately 100 nm down to 20 nm for calcination temperatures from 1273 down to 973 K. These results show a very good agreement between the observed particle size from TEM images and the one from X-ray diffraction analysis. The TEM images also show that the particles (nanograins) exhibit a spherical morphology and that are slightly connected to each other.§ MAGNETIC PROPERTIES Magnetic moment per formula unitary as a function of applied magnetic field, at 4 K, for all studied samples are presented in Fig. <ref>. We found that the magnetic moment of saturation is strongly depends on the nanoparticle size. The bulk sample has no hysteresis, while, the nanoparticle samples presents increased of remanence with the decrease of the nanoparticle size (see inset of Fig.<ref>). This is a common result <cit.> and it is due to the large surface anisotropy. On the other hand, we can estimate the Landé factor considering the saturation as M_S=g(0.6×2+0.4×3/2) μ_B. The bulk sample has a magnetic saturation close to 3.9 μ_B, and the Landé factor as g=2.17, which is a reasonable result for manganites when the quench of the angular moments is not perfect and a really minor orbital contribution takes place <cit.>. For manganite nanoparticles, the magnetic saturation ranges from 2.9 μ_B with g=1.6(106 nm), down to 2.1 μ_B with g=1.16 (21 nm), which are non-acceptable Landé factor for manganites without orbital contribution. Thus, other mechanisms are reducing the magnetic saturation of those nanoparticles by almost half. See Fig. <ref> for further details.ZFC and FC magnetization measurements the all samples are shown in Fig. <ref>. It is possible observe the influenced of the surface on the magnetization curves, where a large irreversibility Δ M at 4 K between the ZFC and FC magnetization can be seen. Fig. <ref> shows Δ Mdecreases as the nanoparticles size increases (in order to include the bulk value we considered 1/D, instead of D). The bulk sample shows no irreversibility in the M(T) curves, as well as hysteresis in the M(H) isotherms. To further understand the magnetic behavior of those manganite nanoparticles, in the Ref. <cit.> is considered a simple model, based on a ferromagnetic core (since the bulk sample is ferromagnetic), and a ferrimagnetic shell (the only way to justify those values of magnetic saturation for these manganites). Is based in that, the core has two sub-lattices (Mn^3+ with s=2 and 3/5 of molar fraction; andMn^4+ with s=3/2 and 2/5 of molar fraction) aligned in a parallel fashion; and the shell has the same kind of two sub-lattices, however, aligned in an antiparallel way.This model considers single-domain particles (and thus works for the smaller nanoparticles); and increasing the nanoparticle size, magnetic domains appear and the model loses validity. Indeed, M(T) of 106 nm nanoparticle does not follow the tendency of the other nanoparticles and starts to approach to the bulk behavior, that, on its turn, has magnetic domains. Some works <cit.> also propose that the single/multi-domain crossover lies close to 70 nm. § CONCLUDING REMARKS ON LA_0.6SR_0.4MNO_3 From standard sol-gel techniques we produced La_0.6Sr_0.4MnO_3 nanoparticles by controlling the annealing temperature. From the X-ray diffraction and Scherrer equation we obtained the nanoparticle size that ranges from 21 nm up to 106 nm; and these values were confirmed from transmission electron microscopy. Magnetization measurements show that those nanoparticles have a remarkable different behavior compared with bulk sample. CHAPTER: GENERAL CONCLUSIONSWe synthesized several intermetallic alloys by arc melting furnace in order to explore the structural and magnetic properties of YNi_4-xCo_xB alloys and X_2YZ Heusler compounds, obtaining the following conclusions:∙ Intermetallic alloys with BBy XRD, we found that the YNi_4-xCo_xB samples crystallize in CeCo_4B type structure (space group P6/mmm), and the lattice parameters of these compounds change almost linearly as a function of Co content. These structures contain two Wyckoff sites (2c and 6i) for 3d metal transition in anisotropy competitions by Co substitution. Thus, we developed a preferential and statistical occupation models for Co ions into the 3d sub-lattice of the YNi_4-xCo_xB samples. The experimental data of magnetization as a function of magnetic field and temperature indicate that our model of preferential occupation fits the physical mechanism that rules the 3d magnetic anisotropies. ∙ Heusler alloys- For Co_2FeSi, we performed AXRD measurements and Mössbauer spectroscopy in order to found the atomic disorder in this sample. The annealing time in the sample preparation induces different atomic disorder degrees in this compound. Finally, DFT calculations demonstrated that theatomic disorder affects the half-metallic properties of Co_2FeSi.- Fe_2MnSi_1-xGa_x crystallize in the cubic L2_1 structure and the lattice parameter a increases linearly with increasing Ga content. Through magnetization as a function of magnetic field and temperature, we found thatthe valence electron number (N_V) plays an important role on the magnetic parameters of these compounds, such as Curie temperature, saturation magnetization and magnetic entropy change. Increasing N_V (equivalent to increase the Si content from Fe_2MnSi_0.5Ga_0.5 compound), leads to a linear increase of those magnetic quantities. In addition, we also propose to substitute Ga by a group 15 element, like P; and we expect that Fe_2MnSi_0.56P_0.44 would have its Curie temperature close to 300 K, with an enhanced magnetocaloric effect.Another research line which we developed were the studies on the perovskite oxides, and here are the concluding remarks:∘ Nd_0.5Sr_0.5CoO_3 cobaltiteWe synthesized a single phase sample of Nd_0.5Sr_0.5CoO_3 by sol-gel method with nitrate reagents, and it crystallized in an orthorhombic structure with space group Imma, critical temperature T_C=215 K and a broad maximum of magnetization at T_k=80 K. Through of the magnetization as a function of magnetic field and temperature, we found that these temperatures are the ferrimagnetic transition (T_C) and the temperature (T_k) below which Nd sub-lattice (negative) magnetization rules the total magnetization. Also, the Curie constant indicates that the system is in an intermediate spin state for both Co^3+ and Co^4+.∘ La_0.6Sr_0.4MnO_3 manganitesFrom standard sol-gel method with oxides reagents, we produced La_0.6Sr_0.4MnO_3 nanoparticleswith sizes that range from 21 nm up to 106 nm,due to different annealing temperatures.These sizes were determinated byXRD measurement andtransmission electron microscopy. Magnetization measurements show that those nanoparticles have a remarkably different behavior compared with bulk sample, due to increase of the ratio surface/volume.In general, we explored several magnetic phenomena, through synthesis, and study ofstructural and magnetic properties of intermetallic alloys and perovskite oxides. From the intermetallic alloys, the phenomena studied include the competition of anisotropies in materials with 3d elements, the magnetocaloric effect and half-metallicity. From the perovskite oxides, we were able to study the spin state in the cobaltiteand finally observe the effect of nanostructured materials of manganite compounds.CHAPTER: LIST OF PUBLICATIONS §ASSOCIATED WITH THE THESIS 1 R. J. Caraballo Vivas, D. L. Rocco, T. Costa-Soares, L. Caldeira, A. A. Coelho, M. S. Reis. Competing anisotropies on 3d sub-lattice of YNi_4-xCo_xB compounds. Journal of Applied Physics, v. 116, p. 063907, 2014. 2 R. J. Caraballo Vivas, N. R. Checca, J. C. G. Tedesco, N. M. Fortunato, J. N. Goncalves, R. D. Candela, A. A. Coelho, A. Magnus. G. Carvalho, M. S. Reis. Experimental and theoretical evidences that atomic disorder suppress half-metallic porperties of Heusler alloys. In redaction (2017). 3 S. S. Pedro, R. J. Caraballo Vivas, V. M. Andrade, C. Cruz, L. S. Paixão, C. Contreras, T. Costa-Soares, L. Caldeira, A. A. Coelho, A. M. G. Carvalho, D. L. Rocco, M. S. Reis. Effects of Ga substitution on the structural and magnetic properties of half metallic Fe2MnSi Heusler compound. Journal of Applied Physics, v. 117, p. 013902, 2015. 4 R.J. Caraballo Vivas, S. S. Pedro, C. Cruz, J. C. G. Tedesco, A. A. Coelho, A. M. G. Carvalho, D. L. Rocco, M. S. Reis. Experimental evidences of enhanced magnetocaloric properties at room temperature and half-metallicity on Fe_2MnSi-based Heusler alloys. Materials Chemistry and Physics, v. 174, p. 23-27, 2016. 5 M.S. Reis, D.L. Rocco, R.J. Caraballo Vivas, B. Pimentel, N.R. Checca, R. Torrão, L. Paixão, A.M. dos Santos. Spin state and magnetic ordering of half-doped Nd_0.5Sr_0.5CoO_3 cobaltite Journal of Magnetism and Magnetic Materials , v. 422, p. 197-203, 2017. 6 V. M. Andrade, R. J. Caraballo-Vivas, T. Costa-Soares, S. S. Pedro, D. L. Rocco, M. S. Reis, A. P. C. Campos, A. A. Coelho. Magnetic and structural investigations on La_0.6Sr_0.4MnO_3 nanostructured manganite: Evidence of a ferrimagnetic shell. Journal of Solid State Chemistry, v. 219, p. 87-92, 2014. § NOT ASSOCIATED WITH THE THESIS7 R. Ribeiro-Palau, R. Caraballo, P. Rogl, E. Bauer, I. Bonalde. Strong-coupling BCS superconductivity in noncentrosymmetric BaPtSi_3: a low-temperature study. Journal of Physics. Condensed Matter, v. 26, p. 235701, 2014. 8 V. M. Andrade, R.J. Caraballo Vivas, S. S. Pedro, J. C. G. Tedesco, A. L Rossi, A. A. Coelho, D. L. Rocco, M. S. Reis. Magnetic and magnetocaloric properties of La_0.6Ca_0.4MnO_3 tunable by particle size and dimensionality. Acta Materialia, v. 102, p. 49-55, 2016. 9 V. M. Andrade, S. S. Pedro, R.J. Caraballo Vivas, D. L. Rocco, M. S. Reis, A. P. C. Campos, A. A. Coelho, M. Escote, A. Zenatti, A. L Rossi. Magnetocaloric functional properties of Sm_0.6Sr_0.4MnO_3 manganite due to advanced nanostructured morphology. Materials Chemistry and Physics, v. 172, p. 20-25, 2016. 10 J. C. G. Tedesco, S. S. Pedro, R.J. Caraballo Vivas, C. Cruz, V. M. Andrade, A.M. dos Santos, A. M. G. Carvalho, M. Costa, P. Venezuela, D. L. Rocco, M. S. Reis. Journal of Physics. Condensed Matter, v. 28, p. 476002, 2016. unsrt | http://arxiv.org/abs/1707.09868v2 | {
"authors": [
"R. J. Caraballo-Vivas"
],
"categories": [
"cond-mat.mtrl-sci",
"cond-mat.other"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170726193511",
"title": "Magnetism from intermetallics and perovskite oxides"
} |
[email protected]@univ-lyon1.frUniversité Lyon 1, F-69622 Villeurbanne, France, UMR5306 CNRS, Institut Lumiere Matiere, PRES-Université de Lyon, F-69361 Lyon, France A predictive model for nanoparticle nucleation has not yet been successfully achieved. Classical nucleation theory fails because the atomistic nature of the seed has to be considered since geometrical structure as well as stoichiometry do not always match the bulk values. We present a fully microscopic approach based on a first-principle study of aluminium oxide clusters. We have calculated stable structures of Al_xO_y and their associated thermodynamic properties. From these data, the chemical composition of a gas composed of aluminium and oxygen atoms can be calculated as a function of temperature, pressure, and aluminium to oxygen ratio. We demonstrate the accuracy of this approach in reproducing experimental results obtained with time resolvedspectroscopy of a laser induced plasma from an Al_2O_3 target. We thus extended the calculation to lower temperatures, i.e. longer time scales, to propose a scenario ofcomposition gas evolutionleading to the first alumina seeds. Atomistic Mechanisms for the Nucleation of Aluminium Oxide Nanoparticles Abdul-Rahman Allouche January 15, 2018 ========================================================================§ INTRODUCTION Although the nucleation of nanoparticles is of crucial interest for a wide range of applications, such as the nanoparticle synthesis or the fight against atmospheric nanoparticles, a predictive model of its mechanisms is not yet achieved. Nucleation consists on the formation of a more stable seed within a metastable phase. This phenomenon can be observed in any first-order phase transition especially with vapour-phase condensation, liquid-to-vapour boiling, solid-state precipitation and binary separations <cit.>. Since the beginning of the 19th century, classical nucleation theory (CNT) has been used for intuitive descriptions <cit.>. The CNT describes the formationof a spherical particle immersed in a mother phase as a competition between the volume and the surface energies. While Δ G_v the standard Gibbs free energy difference per unit volumebetween the two phases favors the growth, the surface term, proportional to the surface tension σ, has an opposite effect. As a result, CNT states the existence of a minimum radius r^⋆ allowing the new phase to grow for radii larger than r^⋆. r^⋆ is defined by <cit.>: r^⋆=-2σ/Δ G_v,CNT appears to be successful for liquid nucleation in a supersaturated vapour <cit.>. However, this approach remains controversial regarding quantitative results <cit.>, and its conceptual limitations have been raised by several authors <cit.>. On the one hand, CNT assumes that the seed is homogeneous and isostructural to the bulk crystal. On the other hand, the Otswald rule <cit.> states that the first growing phase is not always the most stable. During the crystal growth, the geometric structure may indeed reorganise with transient states potentially having a different structure from the bulk analog <cit.>.A second limitation of CNT arises from the capillary approximation which assumes that nucleus and bulk materials have the same thermodynamic properties, and particularly the same surface tension. As an illustration, one may consider the case of Al_2O_3. For both crystallographic phases (α or γ), σ is almost 1.5 J.m^-2 for nanoparticles <cit.> and Δ G_v is almost -62 kJ.cm^-3 at 298 K <cit.>. CNT predicts a critical radius of about the Bohr radius.Lastly, the nucleation rate is driven by the probability of a seed reaching r^⋆ by addition of molecules <cit.>. This approach involves parameters difficult to quantify <cit.>. Consequently, the CNT usually fails to predict the nucleation rate, with tens of orders of magnitude discrepancy between theory and experiment <cit.>.Over and beyond the CNT's limitations, the need to explore crystal nucleation becomes increasingly crucial since it may provide a control on the crystal structure and size distribution for nanoparticle synthesis <cit.>. Experimentally, it is challenging to probe nucleation processes since it involves time and length scales that are usually too fast and too small in most experiments.For computer simulations, two trends can be identified in the literature. On the one hand, molecular dynamics calculations are used with hard-sphere <cit.>, Lennard-Jones <cit.> or more refined potentials <cit.> to investigate the nucleation kinetics. On the other hand, first principle calculations are used to study with precision clusters for particular systems such as zinc <cit.>, silicium <cit.>, and titanium <cit.> oxides. Apart from the work of Loschen et al. <cit.> who tested different stoichiometries of cerium oxides only the bulk stoichiometry is generally considered in simulations. In addition, very few works compared these computational results with experimental measurements <cit.>. Our work is carried out in the context of laser ablation, including pulsed laser deposition <cit.> or pulsed laser ablation in liquids <cit.>. But it can be generalized to other methods such as cluster sources <cit.> and gas-phase combustion <cit.>. For all these techniques, an atomic vapour or a plasma,is generated at temperatures higher than a thousand Kelvins. Then, the system is quenched down to room temperature, leading to nucleation and growth of the nanoparticles. We investigate nucleation processes through an original use of different tools from quantum chemistry combined with experimental measurements obtained from laser induced plasma spectroscopy <cit.>.We choose to work with a model system made of aluminium and oxygen atoms. We first compute the lowest energy structures of Al_xO_y molecules. Thereafter, we calculate the gas-phase equilibrium composition as a function of the temperature, the pressure and the initial ratio of aluminium to oxygen atoms. For the highest temperatures, the computed composition is compared to experimental measurements obtained from the laser ablation of an alumina target (α-Al_2O_3). Moreover, plasma spectroscopy allows the probing of short time scales, when the plasma is hot and optically active.§ NUMERICAL CALCULATIONS Figure <ref> reviews schematically the computational algorithm used for the structural investigation. For each value of x and y, we start from a set of 1000 geometries where atoms are randomly disposed with interatomic distances corresponding to the covalent bond distances. The system is then relaxed via a PM3 semi-empirical method <cit.>. The remaining geometries are then optimized using Density Functional Theory (DFT) calculation. Two sets of bases are used successively, B3LYP/6-31G* and B3LYP/6-311+G(3df,3pd). The Gibbs energies were determined using MP2 and B3LYP according to the following steps. The harmonic frequencies were calculated by B3LYP/6-311+G(3df,3pd) using the structures optimized. MP2 thermochemistry was determined by adding B3LYP thermal correction factors to the MP2 single-point energies and is reported as MP2/6-311++G(3df,3pd)//B3LYP/6-311+G(3df,3pd).The dissociated geometries are removed and only geometries whose energy is at most 2 eV higher than the ground state are kept as the others are not relevant for the temperatures studied .For the biggest clusters (Al_2O_3)_3 and (Al_2O_3)_4, the structures published by Sharipov et al. <cit.> were used as inputs in our optimization process. DFT and MP2 calculations were performed with the Gaussian09 D01 revision.<cit.>We investigated all the molecular formulas following (x,y) ∈ [1;4]. In addition, two stoichiometric trends were followed, (AlO)_n and (Al_2O_3)_n. (AlO)_n with n ⩽ 8 were chosen because Patzer et al. demonstrated that for clusters with x and y smaller than 4, this stoichiometry is the most stable <cit.>.(Al_2O_3)_n with n ⩽ 4 were investigated because it corresponds to the bulk stoichiometry.The figure <ref> shows the most stable structures we obtained. Point-zero energies and geometrical parameters can be found in the Supporting Information (see SI.1). The method we used provides a systematic procedure to select the different stable chemical structures for a given molecular formula. Indeed, for some of the molecules, the results are consistent with various works published previously <cit.>. Nevertheless, for others such as AlO_4 and Al_2O_3, we were able to find isomers that are more stable than those obtained earlier.The first principle calculations provide dissociation energy, vibrational and rotational constants for all Al_x O_y^(i) molecules, where i denotes an isomer of the molecular formula Al_x O_y.The formation Gibbs free energy Δ_f G_Al_x O_y^(i) of each Al_xO_y^(i) is then computed as a function of the temperature T and the pressure P_∘. The Gibbs free energy calculation is described in the Supporting Information SI.2. Based on the model proposed by Patzer et al. <cit.>, we computed the composition of a gas fed with aluminium and oxygen atoms as a function of the temperature, the initial proportion of elements (λ≡ N_Al/N_O) and the pressure. We considered the set of reactions which correspond to the formation of the Al_xO_y^(i) molecules from an atomic gas:x Al + y O ⇄Al_x O_y^(i)The gibbs free energy of reactions Δ_r G_Al_x O_y^(i) is deduced from the Δ_f G_Al_x O_y^(i) <cit.>.We improved the Patzer's model by taking into account the temperature and pressure dependences of Δ_r G_Al_x O_y^(i)and the contribution of all isomers for each molecules (see Supporting Information SI.2).§ EXPERIMENTAL DATA The experimental measurements used for this work have been published previously <cit.> and are sketched in figure <ref>. To summarize, we performed the spectroscopic characterisation of a plasma induced by the laser ablation of an Al_2O_3 target in ambient air. The emission intensity of aluminium monoxide molecules and aluminium atoms was measured as a function of time. The electronic temperature T_e was also measured <cit.>. Using the formalisms developed in our previous article <cit.>, we deduced the density ratio N_AlO/N_Al. In the meantime, we measured a temporal evolution of the rotational temperature T_rot. This can account for the temperature used in the thermochemistry model because the rotational temperature probes the kinetic temperature of atoms <cit.>.N_AlO/N_Al is then reported in figure <ref>(a) as a function of the time (bottom axis), but also as a function of the temperature T_rot (top axis). These experimental data will be used to validate the thermochemistry model.Theλ ratio is expected to be between two extreme values. λ_max=2/3 corresponds to a plasma only composed by the ablated matter. λ_min corresponds to the ablated matter combined with the ambient air in the same plasma plume volume. The amount of matter ablated is obtained from the crater shape. The crater depth is measured using Alpha-Step D100 profiler from Tencor.The crater depth measured after 5 pulses is 1.5 μm ± 500 nm. The crater diameter is measured with an optical microscope.The crater diameter is 500 μm ± 100 μm. Assuming an α-Al_20_3 density of 3.95 g.cm^-3, a molar mass of 101.96 g.mol^-1, we have obtained 7× 10^15 ± 70% atoms ablated per pulse.Considering the size of the plasma, 2 mm in diameter <cit.>, and assuming an ideal gas, the number of O_2 molecules contained in the same volume of air is 2× 10^16.The ratio between the number of aluminium atoms provided by the target and the number of oxygen atoms provided by the target and the air leads to a λ_min of 0.06 (see Supporting Information SI.3).§ DISCUSSION In figure <ref>(b), the experimental measurements from 10 μs to 24 μs are compared with our calculations using the Pearson's cumulative statistic test(χ ^2 = ∑( X_th - X_exp) ^2). We emphasis the valley of highest likelihood which exhibits a minimum of χ ^2 for P_∘=4 bars and λ=0.6. The λ value is consistent with the expected one. The deduced pressure is also consistent with previous plasma spectroscopy measurements <cit.>. Assuming these values, the theoretical ratio N_AlO/N_Al is computed as a function of the temperature and compared to the experimental data in figure <ref>(a).The agreement between theoretical calculation and experiment data relies on the idea that the plasma follows a quasi-static cooling in the considered time-range. The cooling rate measured is 25 K.μs^-1. The known kinetics of the chemical reactions leading to oxide molecules <cit.> are fast enough to follow this cooling rate. This result is also consistent with the absence of an energy barrier for the reactions Al+O→AlO and Al+O_2 →AlO+O <cit.>. However, in order to assert equilibrium in the reacting gas, it could be relevant to take into account the homogeneity and the diffusion properties of the gas. Based on the consistency between experimental and numerical results, we extended the thermochemistry calculation to lower temperature considering P_∘=4 bars and λ=0.6. Figure <ref> shows the gas composition as a function of the temperature. Here, the partial pressure P_Al_xO_y for a given molecular formula corresponds to the summation over all isomers partial pressure. As expected, at high temperature (T≳ 5000 K) i.e. short times, the gas is mainly composed of the smallest species i.e. Al, O, O_2, AlO and Al_2O. Later, the amount of Al_2O and O simultaneously decrease leading to the two derivatives Al_2O_2 and Al_3O_3. We emphasize that Al_2O_3 never seems to emerge at this stage.The stoichiometry of alumina, corresponding to Al_6O_9 and Al_8O_12 molecules, emerges only for temperatures lower than 2000 K. We observe a drastic dependence of the final composition for λ varying around 2/3. It indicates that a small excess of oxygen is required to favor the (2:3) stoichiometry. This result is consistent with the general idea that to synthesize oxides, it is required to have a gas sursaturated in oxygen.For example, plasma enhanced chemical vapor deposition (PECVD) uses an oxygen plasma to grow layers of oxides <cit.>. Pulsed laser deposition (PLD) of Al_2O_3 has been performed in an oxygen gas atmosphere to enhance the stoichiometry of the layer <cit.>. O_2 is injected during the growth of ZnO nanoparticles in low energy cluster beam deposition experiment (LECBD) <cit.>. § CONCLUSIONS In summary, first principle calculations were employed to investigate aluminium oxide clusters at different stoichiometries. A systematic optimization approach was followed to obtain the stable structures. We find more stable isomers than the previously published ones for AlO_4 and Al_2O_3molecules <cit.>. Temperature evolution of the composition of a gas made of aluminium and oxygen atoms was also calculated using these molecular properties.Although for high temperature, (AlO)_n is the most preponderant structure, the bulk aluminium oxide stoichiometry (2:3) starts to exceed the (1:1) stoichiometry for temperature lower than 2000 K. Besides the equilibrium considerations, the question of the kinetic of chemical reactions is not addressed here. Indeed, the occurrence of an equilibrium condition has to be combined with reasonable time scales of reaction kinetics to ensure the molecule formation. Especially, at low temperature, plasma spectroscopy can no longer provide the chemical composition since the system does not emit light. Nevertheless, the theoretical absorption and emission spectra can be deduced from our first-principles calculations. Laser induced fluorescence will be performed to probe the gas. Finally, the high level of theory employed in our calculations prevents addressing bigger clusters. A complementary work could consist on using the clusters we obtained to parametrize a semi-empirical model and perform molecular dynamics simulations. We were nevertheless able to predict the requirements for an oxygen rich gas for synthetizing the desired oxide stoichiometry. § SUPPORTING INFORMATIONThe geometry of each isomer is reported in the Supporting Information file Supplementary-xyz.zip (See SI.1 for content description). SI.2 gives the formulas used to calculate the Gibbs free energy and the gas composition. SI.3 describes λ_min calculation.The ratio N_AlO/N_Al is computed for different values of λ and P_∘ in SI.4. The full curve corresponding to the Figure <ref> is shown in SI.5.§ ACKNOWLEDGEMENTThis work was granted access to the HPC resources of the FLMSN, "Fédération Lyonnaise de Modélisation et Sciences Numériques", partner of EQUIPEX EQUIP@MESO. The authors are grateful to Nora Abdellaoui for her participation on target crater measurements. | http://arxiv.org/abs/1707.08434v1 | {
"authors": [
"Julien Lam",
"David Amans",
"Christophe Dujardin",
"Gilles Ledoux",
"Abdul-Rahman Allouche"
],
"categories": [
"physics.chem-ph",
"cond-mat.mes-hall",
"physics.atm-clus",
"physics.comp-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170726133718",
"title": "Atomistic Mechanisms for the Nucleation of Aluminium Oxide Nanoparticles"
} |
§ INTRODUCTION Even in the era of giant telescopes, small telescopes – telescopes with aperture size ≲ 0.5 m – are still valuable observational facilities. They can cover a large field of view thanks to short focal lengths, and they can be dedicated for a few specific research projects due to the availability of the telescope time. Furthermore, small telescopes can be relatively easily deployed for robotic operations and save observation time for astronomers. Therefore, monitoring observations and rapid follow-up observations of objects like active galactic nuclei (AGNs), Gamma Ray Bursts (GRBs) and supernovae (SNe) are routinely carried out with small telescopes (Drake et al. 2009; Klotz et al. 2008; Shappee et al. 2014). The major disadvantage of small telescopes is the lack of light gathering power. Currently, the best we can do to augment the lack of light gathering power of a single small telescope is to use CCD cameras with high Quantum Efficiency (QE). Yet, many small telescopes are equipped with commercially available font-illuminated CCD cameras that have very low QEs at short (∼ 450 nm) and long (> 800 nm) wavelengths.Recently, we began operating a robotic 0.43 m telescope, the Lee Sang Gak Telescope (LSGT; Im et al. 2015a), and carrying out a monitoring survey of nearby galaxies with it. To maximize the observing efficiency of LSGT, we have assembled a new CCD camera system, Seoul National University CCD Camera II (SNUCAM-II), by utilizing a commercially available CCD camera with a deep depletion chip that has QEs at around 80% over 400 – 900 nm, and a 20 position filter wheel that can house 13 medium-band filters and five broad-band filters. The camera's QE represents improvement in QE by factors of 1.5 to 3 at 400 nm and 3 to 20 at 900 nm, in comparison to commonly available CCDs such as KAI-16000 and KAF-3200ME (see Figure 3 in Im et al. 2015a) making this telescope as one of the most sensitive small telescopes in the world in its class. The camera is named as a successor of SNUCAM (Im et al. 2010), a sensitive 4k×4k CCD camera, and SNUCAM-II improves upon SNUCAM in sensitivity at long wavelengths out to Y-band regime, making it possible for this instrument to study high redshift quasars and GRBs (e.g., Choi et al. 2012).Another unique aspect of SNUCAM-II is the availability of many medium-band filters. The use of medium-band filters makes it possible to trace the SEDs of objects of interest, and this is especially advantageous for monitoring observation of AGNs where the time lag of the broad emission lines and the continuum flux can be traced efficiently for black hole mass measurement. We have demonstrated the power of the medium-band observation on a moderate-sized telescope using the SED Camera for Quasars in Early Universe (SQUEAN; Kim et al. 2016) on the 2.1 m telescope at the McDonald observatory where 9 medium-band filters are installed. Specifically, we have shown that medium-band observation can select high redshift quasars effectively (Jeon et al. 2016), and we have been using the instrument to survey faint quasars at z ∼ 5. Many other applications of the medium-band imaging are possible by tracing SEDs of various objects such as asteroids, stars, galaxies, to name a few. In this paper we describe the overall characteristics of the SNUCAM-II system, and its performance that are derived from the laboratory testing and the on-sky observation using LSGT. § SYSTEM Here, we describe each component of SNUCAM-II. Those are CCD camera, band pass filters and a filter wheel. Software to control LSGT system and typical procedure of remote observation are explained here to help the potential observer. §.§ CCD Camera Due to a wide variety of the applications we expect for SNUCAM-II, we employed a CCD camera with good sensitivity over a broad wavelength region from 300 nm to 1100 nm. We adopted the ANDOR[<http://www.andor.com>] iKon-M DU934P BEX2-DD model CCD camera that uses a deep depletion, back illuminated and fringe suppression E2V CCD chip with extended range dual Anti Reflection coating on it. Its QE is about 90% from 400 nm to 800 nm and even at longer wavelength 1000nm, QE is over 30% thanks to the deep depletion chip (the black dashed line in Figure 2). Each pixel has a full well depth of 130,000 e^-. The CCD chip has 1024×1024 pixels and the physical size of each pixel is 13 μm × 13 μm, which translates into 0.92 arcsec pixel scale at the f/6.8 focal plane of LSGT. We use a model with thermoelectric cooling that can cool the CCD chip to -80^∘C. For our nominal operation, we set its cooling temperature as -70^∘C to achieve a reasonable balance between the QE at long wavelength and the reduction in dark current (see Park et al. 2012). The camera offers 4 different readout rates (5.0, 3.0, 1.0, 0.05 MHz) as a default, and users can choose parameters of Vertical Shift Speed (VSS), Horizontal Shift Speed (HSS) and Preamp Gain (PG). VSS and HSS can be set to one of 2.25, 4.25, 8.25, 16.25, 32.25, 64.25 μsec. PG has three kinds of values as ×1, ×2 and ×4. Also binning can be configured as one of following setting, 1×1, 2×2, 4×4, 8×8 and 16×16. Shutter speed is at the default value of 30 msec at opening and closing and adjustable. For mechanical connection to the filter wheel, C mount is installed on top of CCD camera, so we used a custom made adapter for connecting CCD camera to the filter wheel. As for the setting of the parameters, we use the following values as default: PG of ×4 (gain ∼ 1.15 ± 0.03 e^-/ADU), 1 MHz readout rate, and the cooling temperature of -70^∘C. VSS, HSS, and the shutter speed are set at 4.25 μsec, 2.25 μsec and 30 msec respectively as recommended by the manufacturer. §.§ Filters and Filter Wheel The SNUCAM-II uses a default set of 18 circular filters with 25 mm diameter each: SDSS ugriz filters and 13 medium band filters. The filter transmission curves and the transmission curves multiplied by the CCD QE and the throughput of the telescope optics are presented in Figures 2 and 3. Table 1 shows the filter names, their effective wavelengths and FWHM (Full Width at Half Maximum) values. The SDSS ugriz filters are purchased from the Astrodon company (their Generation 2 Sloan filters)[<http://www.astrodon.com>]. The 13 medium band width (50 nm) filters spanning from 400nm to 1050nm are standard products from Edmund Optics.[<http://www.edmundoptics.co.kr>] We named them by adding the central wavelength in units of nm to the initial m meaning `medium band filter' (e.g., m425 for the medium band filter with the central wavelength at 425 nm). These 18 filters are mounted on the Finger Lake Instrumentation LLC[<http://www.flicamera.com>] (FLI) CenterLine-1-20 color filter wheel. The filter wheel has dual filter wheels Wheel 0 and Wheel 1, with 10 slots each. Keeping the first slot of each wheel as blank position, 18 filters can be installed. After installation of the filters, the physical positions of the filters and logical filter names used in programs are connected by naming in `FLIfilters' program provided by the manufacturer. §.§ Software and Data Collection For the camera system control, we use commercial softwares: `MaximDL pro 5'[<http://diffractionlimited.com>] and `FLIfilter' programs. These programs and the telescope operating system software are integrated into the ACP Observatory Control Software for the command tasks necessary for the observations (observation program reservation, remote communication, actual performance of the system and the roof control). A web page is set up for system status monitoring, the observation plan, the observation program reservation and the weather information of SSO. An observer can configure all the parameters of the actual observation on the webpage and monitor activities of LSGT and SNUCAM-II in real time. A common mode of the observation uses the `plan' document that defines the target coordinate, the number of images to take, the filter, and the binning and the exposure times. This `plan' document is uploaded to the web-based reservation system for robotic observation of the target. When the observation plan is finished, the system sends an observation summary e-mail to observer along with the weather data figure, logs and the list of the obtained data. The images are uploaded on the data storage server where an observer can access with ID and password and download them. Currently, these systems run on the Window 7 operating system. The schematic diagram of the system is shown in Figure 4. The telescope and the computer system is under management of iTelescope.net. § CHARACTERISTICS OF SNUCAM-II In this section, we characterize the properties of SNUCAM-II camera. The characteristics of SNUCAM-II are summarized in Table 2. §.§ Dark Current We took bias and flat images in the laboratory and on the sky. By combining the bias frame values and the flat images taken at two different epochs (B_1 and B_2 for bias, F_1 and F_2 for flats), we calculated the gain and the readout noise values using the equations below (Howell et al. 2006). Gain = (F̅_1 + F̅_2) - (B̅_1 + B̅_2)/σ^2_F_1 - F_2 - σ^2_B_1 - B_2Read Noise = Gain ·σ_B_1 - B_2/√(2) We find that the gain and the readout noise are 1.15 ± 0.03 e^-/ADU (Analog-to-Digital Unit) and 6.0 ± 0.1 e^-, consistent with the test report values from the manufacturer.§.§ Bias, Dark, and Flat The dark current values were measured in the laboratory and in the dome during the daytime in dark condition with the dome light turned off. The temperature settings were varied from -80^∘C to -30^∘C, and the mean values of the frames from 300 sec exposures were recorded after bias subtraction. The result is shown in Figure 5, where the blue circles are measured values and the green circles are values from test report of the manufacturer. The laboratory values are consistent with the test report, with the dark current of 0.2 e^-/s/pixel at -70^∘C setting. We checked the variability of the bias image, dark current and pixel-to-pixel variation of flat images. Compared to the readout noise of 6.6 e^- of the optimal setting from the test report, the measured value from multiple test bias data is 6.0 e^-, consistent with the test report results. And its variation over different locations in a single image is up to 0.5%. There is a small amount of intra-night variability which may originate from electrical instability. We examined the bias variation over several hours and found that the mean bias fluctuation level is lower than 4 ADUs which is less than the readout noise value but not negligible. We also compared the flat images taken at different nights, and examined the pixel-to-pixel variation in normalized flat images. The peak-to-peak variation of flat images is less than 5% for all the filters. These calibration frames are taken regularly every month. So far, there is no noticeable variation in these frames for 6 months since the regular use of SNUCAM-II started. §.§ Linearity We examined the linearity of the detector in the laboratory. In a dark room, a light source is placed to create a diffuse background light. Then, the mean values of the image are recorded by varying the exposure time. The result is plotted in Figure 6. The mean values are well fitted with a linear relation with a constant slope up to 60000 ADU level. Even at a low signal level of <1000 ADU taken from sub second exposures, the detector shows good linearity at well within 2%. §.§ Shutter Pattern To examine at what exposure time the shutter pattern starts to show in the images, we obtained short exposure light images from 0.1 sec to a few sec. Figure 7 shows the light images taken with short exposures. One of the images is the division of the 0.1 sec images by the 6.0 sec image, which shows no significant change in the image pattern from 0.1 to 6.0 sec exposures. Considering the shutter speed is set at 30 msec for the open and close time each, shutter pattern may show up in exposure times well below 0.1 sec. So far, we can conclude that images taken with the exposure time as short as 0.1 sec is not affected by the shutter pattern.§ PERFORMANCE§.§ Standard Star Observation, Zero Points,and AB Offsets We derived photometric zero points from standard star observations and AB magnitude offset. Standard stars were observed in dark clear nights of seeing condition of FWHM ∼ 2.5 arcsec. We observed an A0V star HIP 114918 (V = 9.05 mag) from the HIPPARCOS catalog (Perryman et al. 1997) in all the 18 filters on 2016 Oct 31, Nov 02 and 21. First, we calculated synthetic magnitudes in our filter system using the Vega model spectra of Kurucz (1993) based on equation (3), where F(λ) is the specific flux of Vega model in λ, and R(λ) is the throughput of the SNUCAM-II system consisting of filter transmission, CCD QE, and the telescope optics throughput. The AB magnitudes of a zero magnitude star were estimated to find the AB offset from Vega magnitude by substituting F(λ) =10 ^48.6/2.5c/λ^2 for Vega model flux, where c is the speed of light. m =-2.5 log∫ F(λ)R(λ)dλ/∫ R(λ)dλ +constM =m + κ (X-1) + C We measured the magnitudes of the standard star using Source Extractor (Bertin & Arnouts 1996) auto-mag following equation (4), where M is the apparent magnitude of the star, m is the instrumental magnitude, κ is the atmospheric extinction coefficient, X is airmass term (Z, where Z is the zenith distance) and C is the zero point at airmass 1. We measured the atmospheric extinction coefficients and the zero point of all the filters and they are presented in Table 3 along with AB offsets. The errors indicate the root mean square (rms) values of three nights data. Note that there was considerable variation in the atmospheric coefficient values in each night, and the presented values should serve only as a rough measure. Figure 8 shows the zero point of each filter at X = 1 along with the overall efficiency of the SNUCAM-II, including the telescope throughputs. The most sensitive band filter is g where the overall throughput of the system is the highest and the zero points generally follows the overall throughput of the system. The zero point of the long wavelength region is shallower than that of central wavelength region depending on throughput.§.§ NGC6902 Observation: On-Sky Magnitude Calibration and Magnitude Limits To test the on-sky performance of SNUCAM-II, we took images of NGC6902 in all the filters. Figure 9 shows the images of NGC 6902 in all the 18 SNUCAM-II filters taken with 180 sec single exposure. They were taken one after another on the same night with the `filter offset' option in a clear night with seeing FWHM ∼ 3 arcsec, where `filter offset' means that the focus changes among various filters were made using a preset amount of the focus shift values between filters. We also derived the zero-point of the images and the image depths from the data. To derive the zero point, we downloaded the BVgri photometric data of APASS (Henden et al. 2016) and J, Hmagnitudes of stars in the field of view of SNUCAM-IIimages, and selected stars at 12 < r < 15 mag as photometric reference. The spectra of 175 stars from the stellar spectral library (Gunn & Stryker 1983) are fitted to the APASS+2MASS (Skrutskie et al. 2006) photometric data points (up to J-band), and the best-fit stellar spectra was chosen with a proper offset to the observed magnitude of the stars in the NGC 6902. Then, the photometric zero point of each filter was derived for each reference star, by taking the mean of the zero points from the stars. The error in the zero point is estimated as the rms scatter of the zero points from different stars. The 5-σ point source detection limits are calculated assuming a seeing condition of 3 arcsec which is slightly worse than the median seeing of LSGT (Im et al. 2015a), the aperture diameter of 3 arcsec with the aperture correction included, and at the airmass as indicated. The zero points and the detection limits are shown in Table 4, and also plotted in Figure 8. The derived values agree with the standard star data result within 0.15 mag. The rms scatters in the zero points indicate that the photometric zero-points can be derived to an accuracy of 0.011 to 0.081 mag from the stars in the science images alone by using the APASS+2MASS data. § SCIENTIFIC PROGRAMS By virtue of its many filters and increased sensitivity, LSGT is now more powerful than before for various scientific observations. The 13 medium band filter photometry can provide low resolution spectroscopy of R ∼ 15 (Kim et al. 2016). Remote observation of the southern hemisphere has shown its promises for the transient observation of SNe, GRB afterglow (Choi et al. 2016) and gravitational wave sources. LSGT has been one of the main observational facilities of IMSNG (Intensive Monitoring Survey of Nearby Galaxy), which aims to discover newly appearing SNe and detect early shock-heated emission after explosion that can constrain the size of the SN progenitor star (Im et al. 2015b). AGN monitoring study can benefit from medium band photometry that can sample broad emission lines for the reverberation mapping study of AGN black hole masses, and the medium-band reverberation mapping of several AGNs is ongoing. SNUCAM-II on LGST has been used in small research projects for graduate student classes. Examples of the small projects include the SED study of asteroids, SN remnants, variable stars, and stellar clusters.§ SUMMARY We have presented the characteristics and the performance of the SNUCAM-II system that is installed on LSGT at the Siding Spring Observatory in Australia. This upgraded system is more powerful than the front-illuminated CCD camera systems that have been in use earlier, with QE of >80% from 400–900 nm, factors of a few to tens improvement at short and long wavelengths. SNUCAM-II also boasts 18 filters, ugriz and 13 medium band pass filters from 400 nm to 1100 nmhaving 50 nm band width for the characterization of SEDs of many different kinds of sources. Under the adopted operation parameters, the CCD gain is 1.15±0.03 e^-/ADU, the readout time is 0.9 sec for the 1k × 1k frame. The readout noise is 6.0 e^-, and the dark current is 0.2 e^-/sec at -70^∘C. The SNUCAM-II system shows a good linearity (better than 98% at the currently measurable limit) ranging from tens of ADU ∼ 60000 ADU. The shutter pattern was also examined, and we find no visible shutter pattern in images even with exposure time as short as 0.1 s. Photometric calibration parameters were derived from the analysis of the data of a standard star and reference stars in the vicinity of NGC 6902, showing that SNUCAM-II on LSGT can reach the magnitude limit of g= 19.91 AB mag and z=18.20 AB mag at 5-σ with 180 sec exposure time for point source detection. With its high sensitivity at short and long wavelengths, the availability of many medium-band filters, and the robotic operation capability, SNUCAM-II on LSGT can be used to perform unique scientific projects such as photometric reverberation mapping of AGNs and intensive monitoring of galaxies to catch the early light curve of SNe. This work was supported by the Creative Initiative program, No. 2017R1A3A3001362, and the grant program No. 2016R1D1A1B03934815 of the National Research Foundation of Korea (NRFK) funded by the Korean government (MSIP). Authors thank to Brad Moore and Pete Lake of iTelescope.net for their help and support. We also thank Won-kee Park for useful discussion.[Bertin & Arnouts(1996)]Bertin1996 Bertin, E., & Arnouts, S. 1996, SExtractor: Software for Source Extraction, A&AS, 117[Choi et al.(2012)]Choi2012 Choi, C., Im, M., Jeon, Y., & Ibrahimov, M. 2012, A Y-Band Look of the Sky with 1-m Class Telescopes, JKAS, 45, 7[Choi & Im(2016)]Choi2016 Choi, C., & Im, M. 2016, LSGT Observation of GRB 160425A, GRB Coordinates Network, 19358, 1[Drake et al.(2009)]2009ApJ...696..870D Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009, First Results from the Catalina Real-Time Transient Survey, ApJ, 696, 870[Fukugita et al.(1996)]Fukugita1996 Fukugita, M., Ichikawa, T., Gunn, J. 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Laboratoire Interfaces & Fluides Complexes, Université de Mons, 20 Place du Parc, B-7000 Mons, Belgium.Laboratoire Interfaces & Fluides Complexes, Université de Mons, 20 Place du Parc, B-7000 Mons, Belgium.Laboratoire Interfaces & Fluides Complexes, Université de Mons, 20 Place du Parc, B-7000 Mons, Belgium. Laboratoire de Physico-Chimie Théorique, UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 10 rue Vauquelin, 75005 Paris, France.Laboratoire de Physico-Chimie Théorique, UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 10 rue Vauquelin, 75005 Paris, France. Univ. Bordeaux, CNRS, LOMA, UMR 5798, F-33405 Talence, France. Global Station for Soft Matter, Global Institution for Collaborative Research and Education, Hokkaido University, Sapporo, Japan.PMMH, ESPCI Paris/CNRS-UMR 7636/Univ. Paris 6 UPMC/Univ. Paris 7 Diderot, PSL Research Univ., 10 rue Vauquelin, 75231 Paris cedex 05, [email protected] Laboratoire Interfaces & Fluides Complexes, Université de Mons, 20 Place du Parc, B-7000 Mons, Belgium. Granular chain packings exhibit a striking emergent strain-stiffening behavior despite the individual looseness of the constitutive chains. Using indentation experiments on such assemblies, we measure an exponential increase in the collective resistance force F with the indentation depth z, and with the square root of the number 𝒩 of beads per chain. These two observations are respectively reminiscent of the self-amplification of friction in a capstan or in interleaved books, as well as the physics of polymers. The experimental data are well captured by a novel model based on these two ingredients. Specifically, the resistance force is found to vary according to the universal relation: log F ∼μ√(𝒩)Φ^11/8z/ b, where μ is the friction coefficient between two elementary beads, b is their size, and Φ is the volume fraction of chain beads when semi-diluted in a surrounding medium of unconnected beads. Our study suggests that theories normally confined to the realm of polymer physics at a molecular level can be used to explain phenomena at a macroscopic level.This class of systems enables the study of friction in complex assemblies, with practical implications for the design of new materials, the textile industry, and biology.Emergent Strain-Stiffening in Interlocked Granular Chains Pascal Damman December 30, 2023 =========================================================In nature and architecture, a wide variety of stable structures are formed from dense assemblies of randomly-distributed objects <cit.>, as beautifully illustrated by the various shapes of bird nests <cit.>. The aggregates can be made of elementary components with arbitrarily complex and sometimes living shapes, as in the case of fire ants <cit.>. Most physical studies focus on simpler but highly anisotropic objects, e.g. rods, that ensure a solid-like collective behavior and thus the great stability of their assembly <cit.>, even under large compressive stresses. However, there is no topological constraint between the constitutive objects that could hinder their motion within the aggregate. The mechanical stability of rod aggregates is thus intimately related only to the packing <cit.> and the solid friction <cit.> at the contact points. Further works have been devoted to the mechanics of assemblies made of more complex objects, such as stars <cit.>, or U- and Z-shaped particles <cit.>. In those cases, in contrast to rods, it is considered that topological constraints are the key ingredient for the observed increase in rigidity. Apart from using rods or rigid non-convex objects, a third approach to produce cohesive assemblies from non-interacting elements is to aggregate soft slender objects, such as fibers <cit.> or granular chains <cit.>. In the latter case, a strain-stiffening phenomenon was reported experimentally in the absence of dilution, and was qualitatively related to the formation of loops between chains <cit.>. However, a quantitative theoretical mechanism was still lacking, despite the fact that the jamming of these systems might have some fundamental connexion with the glass transition of polymeric materials. Besides, while the packing of rigid rods is drastically affected by their aspect ratio, granular chains exhibit less impressive changes. For very thin rods, packing fractions as low as 0.2 have been reported and theoretically described through the random-contact equation <cit.>, while for granular chains the lowest packing fractions range between 0.4 and 0.5 <cit.>. This reduced influence of the shape anisotropy might be related to the high flexibility of granular chains.In this Letter, we investigate the emergent mechanical rigidity of granular chain assemblies. Performing indentation experiments, we observe an exponential growth of the resistance force with the product of the indentation depth by the square root of the number of beads per chain. The first factor is reminiscent of the self-amplification of friction in a capstan <cit.>, as well as in interleaved books <cit.>, where the increase in resistance is created by, and proportional to, the force exerted by the operator. The second factor points towards the central role played by inter-chain topological constraints, as in polymer physics <cit.>. Therefore, we propose a new interlocking model based on these two ingredients, and confront it to the experimental data. We find a very good agreement, including for the extended situation of semi-dilute chains within a surrounding medium of unconnected beads. Our experiments are conducted on large, monodisperse, assemblies of granular chains, each chain being made of 𝒩 connected steel beads of diameter b=2 mm, with 2≤𝒩≤ 50. The random packing of the ensemble is achieved through sharp taps until a constant packing density is reached, in agreement with <cit.>. Using a ProbeTack system (Fig. <ref>(a)), the force needed to indent the assembly in its cylindrical container is measured as a function of the indentation depth z. We use a cylindrical indentor with diameter d=1.27 cm, that is displaced at constant velocity. Low velocities, in the 0.1-1 mm s^-1 range, are used in order to ensure a quasistatic regime and to avoid velocity-dependent drag forces <cit.>. We employ floppy containers, molded with soft crosslinked Sylgard, in order to limit vaulting and arching, and thus prevent the Janssen effect <cit.>. Semi-dilute systems are also studied, and prepared by mixing monodisperse granular chains (𝒩=30) with unconnected beads, resulting in a volume fraction Φ=V_chains/(V_chains+V_unconnected) of chain beads, where V_chains and V_unconnected are the total volumes of chain beads and unconnected beads respectively, before mixing (we neglect the volume of mixing).We first start with Φ=1, i.e. with pure granular chain assemblies. As qualitatively shown in Fig. <ref>(b), these assemblies exhibit a behavior similar to rod aggregates <cit.>: when initially packed in a cylindrical container, short-chain assemblies collapse after removal of the container and form a conical pile, like bare sand, while long-chain assemblies are surprisingly able to retain their cylindrical shape despite the looseness of each individual chain. To understand this striking emergent rigidity, we systematically study the influence of the depth z and the number 𝒩 of beads per chain, on the force F needed to indent such an assembly. The resulting motion of the indentor involves local plastic-like structural rearrangements <cit.>, characterized by small force discontinuities, that we do not discuss further here. Figure <ref>(a) summarizes our raw results. The indentation force continuously increases with the depth. For unconnected beads (𝒩=1) and small indentation depths, a nearly-linear force-displacement behavior is observed, in agreement with <cit.>. A sudden increase of the force has been however reported when the indentor gets very close to the bottom of the container <cit.>.For 𝒩≥2, the force increase is not compatible anymore with an effective Hookean behavior. Indeed, the stiffness, given by the slope dF/dz, increases with z thus confirming the strain-stiffening scenario previously reported <cit.>. In addition, the force and stiffness both increase sharply with the number 𝒩 of beads per chain, indicating the predominant influence of chain connectivity. As shown in Fig. <ref>(b), the experimental curves are consistent with an exponential growth of the force with the indentation depth. Furthermore, the dependence in the number of beads per chain appears through the slope d log(F)/dz given in Fig. <ref>(b) which can be fitted by the power law ∼𝒩^0.46. To summarize, the observed mechanical stiffness of the granular chain assembly can thus essentially be described by the relation log F ∝ z √(𝒩). Hereafter, we propose an interlocking model based on two main ingredients that allows to rationalize these observations.First, since we operate in a quasistatic regime there is no drag force, and the resistance to indentation should be only related to internal friction forces hindering the motion of the constitutive objects <cit.>. Previous studies about self-amplified friction in complex assemblies have shown that the frictional resistance essentially grows exponentially with the relevant control parameter, e.g. the angle in a capstan <cit.>, or the Hercules number <cit.> in interleaved books. This general behaviour is intimately rooted in the fact that the increase in the force exerted by the operator is proportional to that force itself and to an effective coefficient of friction induced by the specific geometry. Indeed, the applied force generates normal loading on a certain amount of locking points in the system, which in turn induces a resistive friction force according to Amontons-Coulomb law at the onset of motion. For our granular chains, the motion of the indentor becomes possible as soon as the locking points on one chain are released, and we thus postulate a geometrically-induced self-amplification of friction described by the equation:dF/dz∼μ F M_z ,where μ is the friction coefficient between two elementary beads, and where M_z is the number of locking points on one chain per unit length of indentation.Secondly, inspired by Edwards' general conjecture for athermal granular matter <cit.>, we assume that randomly-packed granular chains behave as thermally equilibrated flexible polymers. In addition, we focus only on the large-𝒩 asymptotics.In this framework, the chains follow the ideal random-walk statistics <cit.> with a typical radius of gyration R_0∼√(𝒩) b. We stress that self-avoiding effects indeed cancel in polymer melts (Φ=1) due to the screening of excluded-volume interactions. Since the chain assembly is dense, the volume v∼𝒩^3/2 b^3 pervaded by a given chain is in fact occupied by v/(𝒩 b^3)∼√(𝒩) chains, on average. As a corollary, non-bonding contacts between beads of the assembly are rarely (probability ∼ 1/√(𝒩)) intrachain contacts and thus mostly interchain contacts. Since all, and only, the non-bonding contacts are frictional locking points, quantifying M_z simply amounts to estimating the number of interchain contacts – the so-called interlocking points from now on – per chain and per unit length of indentation. As each of the 𝒩 beads of a given chain corresponds to ∼ 1 interlocking point, one has:M_z∼𝒩/R_0∼√(𝒩)/b . Then, inserting Eq. (<ref>) in the solution of Eq. (<ref>) leads to:log(F/F_0)∼μ√(𝒩)z/b ,which is the observed form in Fig. <ref>(b), with F_0 = 1 N an arbitrary reference force. Since we remained at the level of scaling, we missed a prefactor in the right-hand side of Eq. (<ref>). However, this prefactor can be estimated from the fit of Fig. <ref>(b), knowing b=2 mm and μ. As no specific cleaning procedure was applied to our chains, a thin layer of lubricant could likely remain between beads, and μ should thus be close to 0.1, bringing the missing prefactor around 0.6. We have thus demonstrated that our interlocking model, based on both self-amplified friction and polymer chain statistics, is compatible with the 𝒩≥2 data in the pure case (Φ=1). To avoid any coincidence, and validate the model further, we finally test the polymer analogy by considering the dilution of granular chains (𝒩=30) with unconnected beads. Depending on the volume fraction Φ, polymer solutions can be ineither <cit.>: (i) the dilute regime (i.e. very low volume fractions), where the chains are isolated from each other. There, the relevant length scale is the size of a swelled chain, given by Flory's radius R_F∼𝒩^3/5b. We stress that the 3/5 exponent reflects the importance of self-avoiding effects in dilute solutions, in contrast to the previous melt-like case; or (ii) the semi-dilute regime (i.e. high volume fractions), where the chains start to interpenetrate. This regime is characterized by a new length scale ξ(Φ), the blob size, with b≤ξ≤ R_F. The transition between the two regimes occurs at a critical volume fraction Φ^*∼𝒩b^3/R_F^ 3∼𝒩^-4/5. We specifically prepare our system in the semi-dilute regime, in order to maintain some interchain contacts, and the associated self-amplification of friction. For 𝒩=30 granular chains, this requires that Φ> Φ^*≈ 0.06.Essentially, the self-amplified friction mechanism is maintained while the dilution of chains reduces the number of interlocking points. Thus, Eq. <ref> remains valid provided the number of interlocking points on one chain per unit length M_z is adjusted. It must range from the expression in Eq. <ref> at the melt-like volume fraction Φ∼1, down to ∼1/R_F at the critical volume fraction Φ∼Φ^*. According to the physics of semi-dilute polymer solutions in a good – athermal – solvant <cit.>, each chain can be viewed as a succession of 𝒩/g independent blobs containing a subset of g connected beads each. In such a renormalized picture, the whole solution can be viewed as a dense melt of ideal blob chains, which implies Φ∼ gb^3/ξ^3. The portion of a chain inside a blob is in a purely dilute state (i.e. with only unconnected beads around) and thus has a self-avoiding conformation leading to ξ∼ g^3/5b. As each of the 𝒩/g blobs of a given ideal chain of blobs corresponds to ∼ 1 interlocking point, Eq. <ref> must be replaced by:M_z ∼√(𝒩/g)ξ∼√(𝒩)Φ^11/8 b ,that has the correct limiting values mentioned above. Then, inserting Eq. (<ref>) in the solution of Eq. (<ref>) leads to the universal relation:log(FF_0) ∼μ√(𝒩)Φ^11/8z/ b ,that is valid for all Φ such that Φ^*<Φ<1, and for which Eq. (<ref>) is a limiting case when Φ→1. The reference force F_0 and the missing prefactor have already been discussed after Eq. (<ref>) and are not modified here. As shown in Fig. <ref>(a), the measured indentation force F in the semi-dilute regime strongly increases with the volume fraction Φ of chain beads, while keeping the previous exponential behavior with indentation depth z. Similarly to Fig. <ref>(b), we plot the corresponding slope d log(F)/dz as a function of Φ in Fig. <ref>(b), and find a best power-law fit of d log(F)/dz∼Φ^1.1, with an exponent close to the theoretical value 11/8. Following the prediction of Eq. (<ref>), all the experimental data of Figs. <ref>(b)(inset) and <ref>(a) should belong to a single master curve. The universal collapse observed in Fig. <ref> corroborates the proposed interlocking mechanism. In conclusion, the emergent strain-stiffening behavior in granular chain assemblies, as observed through indentation experiments, seems to originate from a self-amplification of friction due to polymer-like interlocking contacts. Interestingly, this system exhibits a new self-amplification exponent (1/2, in chain length) with respect to previous exponents for capstan (1, in angle) <cit.> and interleaved books (2, in number of sheets) <cit.>. However, even if the developed interlocking model seems to capture well the experimental data, we would like to stress that it implicitly assumes sufficiently long chains for proper conformational statistics to be achieved, which is only approximate in our experiments. Beyond the intrinsic glassy-polymer-mimetic feature of dense granular chain assemblies, this study embodies a new illustration of friction in complex assemblies, with practical implications for new materials, textiles <cit.>, and biology <cit.>.This work was supported by the ARC Mecafood project at UMONS and the PDR project “Capture biomimétique de fluides” of the FRS-FNRS. T.S. acknowledges funding from the Global Station for Soft Matter, a project of Global Institution for Collaborative Research and Education at Hokkaido University. P.D. acknowledges the Joliot and Total chairs from ESPCI Paris. 10 insect C. Anderson, G. Theraulaz, and J.-L. Deneubourg. Self-assemblages in insect societies. Insectes soc. 49, 99-110 (2002). archi K. Dierichs, and A. Menges. Towards an aggregate architecture: designed granular systems as programmable matter in architecture Granular matter 18, 25 (2016). nest M. Hansell. Animal Architecture (Oxford Animal Biology Series, Oxford 2005). ant M. Tennenbaum, Z. Liu, D. Hu, and A. Fernandez-Nieves. Mechanics of fire ant aggregations Nature Mat. 15, 54-59 (2016). philipse_1996 A.P. Philipse. The random contact equation and its implication for (colloidal) rods in packings, suspensions, and anisotropic powders Langmuir 12, 1127-1133 (1996). trepanier_2010 M. Trepanier, and S.V. Franklin.Column collapse of granular rods Physical Review E 82, 011308 (2010).nagel S. R. Nagel. Experimental soft-matter science Rev. Mod. Phys. 89, 025002 (2017). friction_book D. Dowson and D. Dowson.History of Tribology (Longman, London 1979). star Y. Zhao, K. Liu, M. Zheng, J. Barés, K. Dierichs, A. Menges, and R.P. Behringer. Packings of 3D stars: stability and structure Granular Matter 18, 24 (2016). U_Z_particle1 N. Gravish, S.V. Franklin, D.L. Hu, and D.I. Goldman. Entangled Granular Media Physical Review Letters 108, 208001 (2012). U_Z_particle2 K.A. Murphy, N. Reiser, D. Choksy, C.E. Singer, and H.M. Jaeger. 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Self-Amplification of Solid Friction in Interleaved Assemblies Physical Review Letters 116, 015502 (2016). Dalnoki_2016 K. Dalnoki-Veress, T. Salez, and F. Restagno. Why can't you separate interleaved books ? Physics Today 69, 74-75 (2016). degennes P.G. de Gennes.Scaling Concepts in Polymer Physics, (Cornell University Press, Cornell 1979). durian H. Katsuragi and D.J. Durian.Drag force scaling for penetration into granular media Physical Review E 87, 052208 (2013). andreotti B. Andreotti, O. Pouliquen, Y. Forterre. Granular Media: Between Fluid and Solid, (Cambridge University Press, Cambridge 2013). Peng_2009 Z. Peng, X. Xu, K. Lu, and M. Hou.Depth dependence of vertical plunging force in granular medium Physical Review E 80, 021301 (2009). expo_bead M. B. Stone, R. Barry, D. P. Bernstein, M. D. Pelc, Y. K. Tsui, and P. Schiffer. Local jamming via penetration of a granular medium Phys. Rev E 70, 041301 (2004). edwards A. Baule, F. Morone, H.J. Herrmann, H.A. Makse.Edwards Statistical Mechanics for Jammed Granular Matter arXiv:1602.04369v3. ghosal S. Ghosal. Capstan Friction Model for DNA Ejection from Bacteriophages Physical Review Letters 109, 248105 (2012). ward A. Ward, F. Hilitski, W. Schwenger, D. Welch, A.W. C. Lau, V. Vitelli, L. Mahadevan, and Z. Dogic. Solid friction between soft filaments Nature Materials 14, 583-588 (2015). | http://arxiv.org/abs/1707.08547v2 | {
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figures/ arrows,positioning,automata,decorations,fit,backgrounds,calc shapes.geometricshapes.geometricdecorations.pathmorphingdecorations.pathmorphingdecorations.textdecorations.textmatrix main #1#2 < g r a p h i c s > #1#2file=#1.epsli en1ex myitemize ∙ myenumctr myenumerate(myenumctr)myenumctr =0ptmyoenumctr myoenumerate (Omyenumctr)myenumctr =0pt . exampleExample ="2D ASDefdef: Defndefn = DefnCustomdefn Letlet Halthalt KwInInput ForEachforeach Oror AndandIfElseIfElseifelse ifelseendif Matchmatch Casecase MapTo notes | http://arxiv.org/abs/1707.08876v2 | {
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[pages=1-38]final_arxiv.pdf | http://arxiv.org/abs/1707.08299v1 | {
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Equivariant extensions of 𝔾_a-torsors over punctured surfaces Takashi Kishimoto ============================================================= The existence of a coalition strategy to achieve a goal does not necessarily mean that the coalition has enough information to know how to follow the strategy. Neither does it mean that the coalition knows that such a strategy exists. The paper studies an interplay between the distributed knowledge, coalition strategies, and coalition “know-how" strategies. The main technical result is a sound and complete trimodal logical system that describes the properties of this interplay. Equivariant extensions of 𝔾_a-torsors over punctured surfaces Takashi Kishimoto =============================================================§ INTRODUCTION An agent a comes to a fork in a road. There is a sign that says that one of the two roads leads to prosperity, another to death. The agent must take the fork, but she does not know which road leads where. Does the agent have a strategy to get to prosperity? On one hand, since one of the roads leads to prosperity, such a strategy clearly exists. We denote this fact by modal formula _a p, where statement p is a claim of future prosperity. Furthermore, agent a knows that such a strategy exists. We write this as _a_a p. Yet, the agent does not know what the strategy is and, thus, does not know how to use the strategy. We denote this by _a p, where know-how modality _a expresses the fact that agent a knows how to achieve the goal based on the information available to her.In this paper we study the interplay between modality , representing knowledge, modality , representing the existence of a strategy, and modality , representing the existence of a know-how strategy. Our main result is a complete trimodal axiomatic system capturing properties of this interplay. §.§ Epistemic Transition Systems In this paper we use epistemic transition systems to capture knowledge and strategic behavior. Informally, epistemic transition system is a directed labeled graph supplemented by an indistinguishability relation on vertices. For instance, our motivational example above can be captured by epistemic transition system T_1 depicted in Figure <ref>. In this system state w represents the prosperity and state w' represents death. The original state is u, but it is indistinguishable by the agent a from state v. Arrows on the diagram represent possible transitions between the states. Labels on the arrows represent the choices that the agents make during the transition. For example, if in state u agent chooses left (L) road, she will transition to the prosperity state w and if she chooses right (R) road, she will transition to the death state w'. In another epistemic state v, these roads lead the other way around. States u and v are not distinguishable by agent a, which is shown by the dashed line between these two states. In state u as well as state v the agent has a strategy to transition to the state of prosperity: u⊩_a p and v⊩_a p. In the case of state u this strategy is L, in the case of state v the strategy is R. Since the agent cannot distinguish states u and v, in both of these states she does not have a know-how strategy to reach prosperity: u⊮_a p and v⊮_a p. At the same time, since formula _a p is satisfied in all statesindistinguishable to agent a from state u, we can claim that u⊩_a_a p and, similarly, v⊩_a_a p. As our second example, let us consider the epistemic transition system T_2 obtained from T_1 by swapping labels on transitions from v to w and from v to w', see Figure <ref>. Although in system T_2 agent a still cannot distinguish states u and v, she has a know-how strategy from either of these states to reach state w. We write this as u⊩_a p and v⊩_a p. The strategy is to choose L. This strategy is know-how because it does not require to make different choices in the states that the agent cannot distinguish.§.§ Imperfect RecallFor the next example, we consider a transition system T_3 obtained from system T_1 by adding a new epistemic state s. From state s, agent a can choose label L to reach state u or choose label R to reach state v. Since proposition q is satisfied in state u, agent a has a know-how strategy to transition from state s to a state (namely, state u) where q is satisfied. Therefore, s⊩_a q. A more interesting question is whether s⊩_a_a p is true. In other words, does agent a know how to transition from state s to a state in which she knows how to transition to another state in which p is satisfied? One might think that such a strategy indeed exists: in state s agent a chooseslabel L to transition to state u. Since there is no transition labeled by L that leads from state s to state v, upon ending the first transition the agent would know that she is in state u, where she needs to choose label L to transition to state w. This argument, however, is based on the assumption that agent a has a perfect recall. Namely, agent a in state u remembers the choice that she made in the previous state. We assume that the agents do not have a perfect recall and that an epistemic state description captures whatever memories the agent has in this state. In other words, in this paper we assume that the only knowledge that an agent possesses is the knowledge captured by the indistinguishability relation on the epistemic states. Given this assumption, upon reaching the state u (indistinguishable from state v) agent a knows that there exists a choice that she can make to transition to state in which p is satisfied: s⊩_a_a p. However, she does not know which choice (L or R) it is: s⊮_a_a p.§.§ Multiagent Setting So far, we have assumed that only agent a has an influence on which transition the system takes. In transition system T_4 depicted in Figure <ref>, we introduce another agent b and assume both agents a and b have influence on the transitions. In each state, the system takes the transition labeled D by default unless there is a consensus of agents a and b to take the transition labeled C. In such a setting, each agent has a strategy to transition system from state u into state w by voting D, but neither of them alone has a strategy to transition from state u to state w' because such a transition requires the consensus of both agents. Thus, u⊩_a p∧_b p∧_a q∧_b q. Additionally, both agents know how to transition the system from state u into state w, they just need to vote D. Therefore, u⊩_a p∧_b p.In Figure <ref>, we show a more complicated transition system obtained from T_1 by renaming label L to D and renaming label R to C. Same as in transition system T_4, we assume that there are two agents a and b voting on the system transition. We also assume that agent a cannot distinguish states u and v while agent b can. By default, the system takes the transition labeled D unless there is a consensus to take transition labeled C. As a result, agent a has a strategy (namely, vote D) in state u to transition system to state w, but because agent a cannot distinguish state u from state v, not only does she not know how to do this, but she is not aware that such a strategy exists: u⊩_a p∧_a p ∧_a_a p. Agent b, however, not only has a strategy to transition the system from state u to state w, but also knows how to achieve this: u⊩_b p.§.§ Coalitions We have talked about strategies, know-hows, and knowledge of individual agents. In this paper we consider knowledge, strategies, and know-how strategies of coalitions. There are several forms of group knowledge that have been studied before. The two most popular of them are common knowledge and distributed knowledge <cit.>. Different contexts call for different forms of group knowledge.As illustrated in the famous Two Generals' Problem <cit.> where communication channels between the agents are unreliable, establishing a common knowledge between agents might be essential for having a strategy. In some settings, the distinction between common and distributed knowledge is insignificant. For example, if members of a political fraction get together to share all their information and to develop a common strategy, then the distributed knowledge of the members becomes the common knowledge of the fraction during the in-person meeting.Finally, in some other situations the distributed knowledge makes more sense than the common knowledge. For example, if a panel of experts is formed to develop a strategy, then this panel achieves the best result if it relies on the combined knowledge of its members rather than on their common knowledge.In this paper we focus on distributed coalition knowledge and distributed-know-how strategies. We leave the common knowledge for the future research.To illustrate how distributed knowledge of coalitions interacts with strategies and know-hows, consider epistemic transition system T_6 depicted in Figure <ref>. In this system, agents a and b cannot distinguish states u and v while agents b and c cannot distinguish states v and u'. In every state, each of agents a, b and c votes either L or R, and the system transitions according to the majority vote. In such a setting, any coalition of two agents can fully control the transitions of the system. For example, by both voting L, agents a and b form a coalition {a,b} that forces the system to transition from state u to state w no matter how agent c votes. Since proposition p is satisfied in state w, we write u⊩_{a,b} p, or simply u⊩_a,b p. Similarly, coalition {a,b} can vote R to force the system to transition from state v to state w. Therefore, coalition {a,b} has strategies to achieve p in states u and v, but the strategies are different. Since they cannot distinguish states u and v, agents a and b know that they have a strategy to achieve p, but they do not know how to achieve p. In our notations, v⊩ S_a,bp∧_a,bS_a,bp ∧_a,b p. On the other hand, although agents b and c cannot distinguish states v and u', by both voting R in either of states v and u', they form a coalition {b, c} that forces the system to transition to state w where p is satisfied. Therefore, in any of states v and u', they not only have a strategy to achieve p, but also know that they have such a strategy, and more importantly, they know how to achieve p, that is, v⊩_b,c p. §.§ Nondeterministic Transitions In all the examples that we have discussed so far, given any state in a system, agents' votes uniquely determine the transition of the system. Our framework also allows nondeterministic transitions. Consider transition system T_7 depicted in Figure <ref>. In this system, there are two agents a and b who can vote either C or D. If both agents vote C, then the system takes one of the consensus transitions labeled with C. Otherwise, the system takes the transition labeled with D. Note that there are two consensus transitions starting from state u. Therefore, even if both agents vote C, they do not have a strategy to achieve p, i.e., u⊮_a,bp. However, they can achieve p∨ q. Moreover, since all agents can distinguish all states, we have u ⊩_a,b(p∨ q).§.§ Universal Principles In the examples above we focused on specific properties that were either satisfied or not satisfied in particular states of epistemic transition systems T_1 through T_7. In this paper, we study properties that are satisfied in all states of all epistemic transition systems. Our main result is a sound and complete axiomatization of all such properties. We finish the introduction with an informal discussion of these properties.Properties of Single Modalities Knowledge modality K_C satisfies the axioms of epistemic logic S5 with distributed knowledge. Both strategic modality S_C and know-how modality _C satisfy cooperation properties <cit.>:_C(ϕ→ψ)→(_Dϕ→_C∪ Dψ),C∩ D=∅,_C(ϕ→ψ)→(_Dϕ→_C∪ Dψ),C∩ D=∅.They also satisfy monotonicity properties_Cϕ→_Dϕ,C⊆ D, _Cϕ→_Dϕ,C⊆ D.The two monotonicity properties are not among the axioms of our logical system because, as we show in Lemma <ref> and Lemma <ref>, they are derivable. Properties of InterplayNote that w⊩_Cϕ means that coalition C has the same strategy to achieve ϕ in all epistemic states indistinguishable by the coalition from state w. Hence, the following principle is universally true:_Cϕ→ K_C_Cϕ.Similarly, w⊩_Cϕ means that coalition C does not have the same strategy to achieve ϕ in all epistemic states indistinguishable by the coalition from state w. Thus,_Cϕ→ K_C_Cϕ.We call properties (<ref>) and (<ref>) strategic positive introspection and strategic negative introspection, respectively.The strategic negative introspection is one of our axioms. Just as how the positive introspection principle follows from the rest of the axioms in S5, the strategic positive introspection principle is also derivable (see Lemma <ref>).Whenever a coalition knows how to achieve something, there should exist a strategy for the coalition to achieve. In our notation,_Cϕ→_Cϕ.We call this formula strategic truth property and it is one of the axioms of our logical system.The last two axioms of our logical system deal with empty coalitions. First of all, if formula _∅ϕ is satisfied in an epistemic state of our transition system, then formula ϕ must be satisfied in every state of this system. Thus, even empty coalition has a trivial strategy to achieve ϕ:_∅ϕ→_∅ϕ.We call this property empty coalition principle. In this paper we assume that an epistemic transition system never halts. That is, in every state of the system no matter what the outcome of the vote is, there is always a next state for this vote. This restriction on the transition systems yields property _C.that we call nontermination principle.Let us now turn to the most interesting and perhaps most unexpected property of interplay. Note that _∅ϕ means that an empty coalition has a strategy to achieve ϕ. Since the empty coalition has no members, nobody has to vote in a particular way. Statement ϕ is guaranteed to happen anyway. Thus, statement _∅ϕ simply means that statement ϕ is unavoidably satisfied after any single transition.For example, consider an epistemic transition system depicted in Figure <ref>. As in some of our earlier examples, this system has agents a and b who vote either C or D. If both agents vote C, then the system takes one of the consensus transitions labeled with C. Otherwise, the system takes the default transition labeled with D. Note that in state v it is guaranteed that statement p will happen after a single transition. Thus, v⊩_∅ p. At the same time, neither agent a nor agent b knows about this because they cannot distinguish state v from states u and u' respectively. Thus, v⊩_a_∅ p ∧_b_∅ p. In the same transition system T_8, agents a and b together can distinguish state v from states u and u'. Thus, v⊩_a,b_∅ p.In general,statement _C_∅ϕ means that not only ϕ is unavoidable, but coalition C knows about it. Thus, coalition C has a know-how strategy to achieve ϕ:_C_∅ϕ→_Cϕ.In fact, the coalition would achieve the result no matter which strategy it uses. Coalition C can even use a strategy that simultaneously achieves another result in addition to ϕ: _C_∅ϕ∧_Cψ→_C(ϕ∧ψ).In our logical system we use an equivalent form of the above principle that is stated using only implication:_C(ϕ→ψ)→(_C_∅ϕ→_Cψ).We call this property epistemic determinicity principle. Properties (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), together with axioms of epistemic logic S5 with distributed knowledge and propositional tautologies constitute the axioms of our sound and complete logical system.§.§ Literature Review Logics of coalition power were developed by Marc Pauly <cit.>, who also proved the completeness of the basic logic of coalition power.Pauly's approach has been widely studied in the literature <cit.>. An alternative logical systemwas proposed by More and Naumov <cit.>. Alur, Henzinger, and Kupferman introduced Alternating-Time Temporal Logic (ATL) that combines temporal and coalition modalities <cit.>. Van der Hoek and Wooldridge proposed to combine ATL with epistemic modality to form Alternating-Time Temporal Epistemic Logic <cit.>. They did not prove the completeness theorem for the proposed logical system. Ågotnes and Alechina proposed a complete logical system that combines the coalition power and epistemic modalities <cit.>. Since this system does not have epistemic requirements on strategies, it does not contain any axioms describing the interplay of these modalities.Know-how strategies were studied before under different names. While Jamroga and Ågotnes talked about “knowledge to identify and execute a strategy" <cit.>,Jamroga and van der Hoek discussed “difference between an agent knowing that he has a suitable strategy and knowing the strategy itself" <cit.>. Van Benthem called such strategies “uniform" <cit.>. Wang gave a complete axiomatization of “knowing how" as a binary modality <cit.>, but his logical system does not include the knowledge modality.In our AAMAS'17 paper, we investigated coalition strategies to enforce a condition indefinitely <cit.>. Such strategies are similar to “goal maintenance" strategies in Pauly's “extended coalition logic" <cit.>. We focused on “executable" and “verifiable" strategies. Using the language of the current paper, executability means that a coalition remains “in the know-how" throughout the execution of the strategy. Verifiability means that the coalition can verify that the enforced condition remains true. In the notations of the current paper, the existence of a verifiable strategy could be expressed as _C_Cϕ.In <cit.>, we provided a complete logical system that describes the interplay between the modality representing the existence of an “executable" and “verifiable" coalition strategy to enforce and the modality representing knowledge. This system can prove principles similar to the strategic positive introspection (<ref>) and the strategic negative introspection (<ref>) mentioned above.In the current paper, we combine know-how modalitywith strategic modalityand epistemic modality . The proof of the completeness theorem is significantly more challenging than the one in <cit.>. It employs new techniques that construct pairs of maximal consistent sets in “harmony" and in “complete harmony", which are discussed in the full version of this paper <cit.>.§.§ Paper Outline This paper is organized as follows. In Section <ref> we introduce formal syntax and semantics of our logical system. In Section <ref> we list axioms and inference rules of the system. Section <ref> provides examples of formal proofs in our logical systems. Section <ref> concludes the paper.The proofs of the soundness and the completeness can be found in the full version of this paper <cit.>. The key part of the proof of the completeness is the construction of a pair of sets in complete harmony.§ SYNTAX AND SEMANTICS In this section we present the formal syntax and semantics of our logical system given a fixed finite set of agents 𝒜. Epistemic transition system could be thought of as a Kripke model of modal logic S5 with distributed knowledge to which we add transitions controlled by a vote aggregation mechanism. Examples of vote aggregation mechanisms that we have considered in the introduction are the consensus/default mechanism and the majority vote mechanism. Unlike the introductory examples, in the general definition below we assume that at different states the mechanism might use different rules for vote aggregation. The only restriction on the mechanism that we introduce is that there should be at least one possible transition that the system can take no matter what the votes are. In other words, we assume that the system can never halt.For any set of votes V, by V^𝒜 we mean the set of all functions from set 𝒜 to set V. Alternatively, the set V^𝒜 could be thought of as a set of tuples of elements of V indexed by elements of 𝒜. A tuple (W,{∼_a}_a∈𝒜,V,M,π) is called an epistemic transition system, where * W is a set of epistemic states,* ∼_a is an indistinguishability equivalence relation on W for each a∈𝒜,* V is a nonempty set called “domain of choices", * M⊆ W× V^𝒜× W is an aggregation mechanism where for each w∈ W and each 𝐬∈ V^𝒜, there is w'∈ Wsuch that (w,𝐬,w')∈ M,* π is a function that maps propositional variables into subsets of W. A coalition is a subset of 𝒜. Note that a coalition is always finite due to our assumption that the set of all agents 𝒜 is finite. Informally, we say that two epistemic states are indistinguishable by a coalition C if they are indistinguishable by every member of the coalition. Formally, coalition indistinguishability is defined as follows:For any epistemic states w_1,w_2∈ W and any coalition C, let w_1∼_C w_2 if w_1∼_a w_2 for each agent a∈ C.Relation ∼_C is an equivalence relation on the set of states W for each coalition C.By a strategy profile {s_a}_a∈ C of a coalition C we mean a tuple that specifies vote s_a∈ V of each member a∈ C. Since such a tuple can also be viewed as a function from set C to set V, we denote the set of all strategy profiles of a coalition C by V^C: Any tuple {s_a}_a∈ C∈ V^C is called a strategy profile of coalition C. In addition to a fixed finite set of agents 𝒜 wealso assume a fixed countable set of propositional variables.The language Φ of our formal logical system is specified in the next definition.Let Φ be the minimal set of formulae such that * p∈Φ for each propositional variable p,* ϕ,ϕ→ψ∈Φ for all formulae ϕ,ψ∈Φ,* _Cϕ,_Cϕ,_Cϕ∈Φ for each coalition C and each ϕ∈Φ.In other words, language Φ is defined by the following grammar:ϕ := p | ϕ | ϕ→ϕ | _Cϕ | _Cϕ | _Cϕ. Bywe denote the negation of a tautology. For example, we can assume thatis (p→ p) for some fixed propositional variable p. According to Definition <ref>, a mechanism specifies the transition that a system might take for any strategy profile of the set of all agents 𝒜. It is sometimes convenient to consider transitions that are consistent with a given strategy profile 𝐬 of a give coalition C⊆𝒜. We write w→_𝐬u if a transition from state w to state u is consistent with strategy profile 𝐬. The formal definition is below. For any epistemic states w,u∈ W, any coalition C, and any strategy profile 𝐬={s_a}_a∈ C∈ V^C, we write w→_𝐬u if (w,𝐬',u)∈ M for some strategy profile 𝐬'={s'_a}_a∈𝒜∈ V^𝒜such that s'_a=s_a for each a∈ C. For any strategy profile 𝐬 of the empty coalition ∅, if there are a coalition C and a strategy profile 𝐬' of coalition C such that w→_𝐬' u, then w→_𝐬u. The next definition is the key definition of this paper. It formally specifies the meaning of the three modalities in our logical system. For any epistemic state w∈ W of a transition system (W,{∼_a}_a∈𝒜,V,M,π) and any formula ϕ∈Φ, let relation w⊩ϕ be defined as follows * w⊩ p if w∈π(p) where p is a propositional variable,* w⊩ϕ if w⊮ϕ,* w⊩ϕ→ψ if w⊮ϕ or w⊩ψ,* w⊩_Cϕ if w'⊩ϕ for each w'∈ W such that w∼_C w',* w⊩_Cϕ if there is a strategy profile 𝐬∈ V^C such thatw→_𝐬 w' implies w'⊩ϕ for every w'∈ W,* w⊩_Cϕ if there is a strategy profile 𝐬∈ V^C such that w∼_C w' and w'→_𝐬 w” imply w”⊩ϕ for all w',w”∈ W.§ AXIOMS In additional to propositional tautologies in language Φ, our logical system consists of the following axioms.* Truth: _Cϕ→ϕ,* Negative Introspection: _Cϕ→_C_Cϕ,* Distributivity: _C(ϕ→ψ)→(_Cϕ→_Cψ),* Monotonicity: _Cϕ→_Dϕ, if C⊆ D,* Cooperation: _C(ϕ→ψ)→(_Dϕ→_C∪ Dψ), where C∩ D=∅.* Strategic Negative Introspection: _Cϕ→_C_Cϕ,* Epistemic Cooperation: _C(ϕ→ψ)→(_Dϕ→_C∪ Dψ), where C∩ D=∅,* Strategic Truth: _Cϕ→_Cϕ,* Epistemic Determinicity:_C(ϕ→ψ)→(_C_∅ϕ→_Cψ),* Empty Coalition: _∅ϕ→_∅ϕ,* Nontermination: _C.We have discussed the informal meaning of these axioms in the introduction. In the full version of this paper <cit.>, we formally prove the soundness of these axioms with respect to the semantics from Definition <ref>.We write ⊢ϕ if formula ϕ is provable from the axioms of our logical system usingNecessitation, Strategic Necessitation, and Modus Ponens inference rules:ϕ_Cϕϕ_Cϕϕ,ϕ→ψψ.We write X⊢ϕ if formula ϕ is provable from the theorems of our logical system and a set of additional axioms X using only Modus Ponens inference rule.§ DERIVATION EXAMPLES In this section we give examples of formal derivations in our logical system. In Lemma <ref> we prove the strategic positive introspection principle (<ref>) discussed in the introduction.⊢_Cϕ→_C_Cϕ. Note that formula _Cϕ→_C_Cϕ is an instance of Strategic Negative Introspection axiom. Thus, ⊢_C_Cϕ→_Cϕ by the law of contrapositive in the propositional logic. Hence, ⊢_C(_C_Cϕ→_Cϕ) byNecessitation inference rule. Thus, byDistributivity axiom and Modus Ponens inference rule, ⊢_C_C_Cϕ→_C_Cϕ. At the same time, _C_Cϕ→_Cϕ is an instance of Truth axiom. Thus, ⊢_Cϕ→_C_Cϕ by contraposition. Hence, taking into account the following instance of Negative Introspection axiom _C_Cϕ→_C_C_Cϕ, one can conclude that ⊢_Cϕ→_C_C_Cϕ. The latter, together with statement (<ref>), implies the statement of the lemma by the laws of propositional reasoning. In the next example, we show that the existence of a know-how strategy by a coalition implies that the coalition has a distributed knowledge of the existence of a strategy. ⊢_Cϕ→_C_Cϕ. By Strategic Truth axiom, ⊢_Cϕ→_Cϕ. Hence, ⊢_C(_Cϕ→_Cϕ) by Necessitation inference rule. Thus, ⊢_C_Cϕ→_C_Cϕ by Distributivity axiom and Modus Ponens inference rule. At the same time, ⊢_Cϕ→_C_Cϕ by Lemma <ref>. Therefore, ⊢_Cϕ→_C_Cϕ by the laws of propositional reasoning. The next lemma shows that the existence of a know-how strategy by a sub-coalition implies the existence of a know-how strategy by the entire coalition. ⊢_Cϕ→_Dϕ, where C⊆ D. Note that ϕ→ϕ is a propositional tautology. Thus, ⊢ϕ→ϕ. Hence, ⊢_D∖ C(ϕ→ϕ) by Strategic Necessitation inference rule. At the same time, by Epistemic Cooperation axiom, ⊢_D∖ C(ϕ→ϕ)→(_Cϕ→_Dϕ) due to the assumption C⊆ D.Therefore, ⊢_Cϕ→_Dϕ by Modus Ponens inference rule. Although our logical system has three modalities, the system contains necessitation inference rulesonly for two of them. The lemma below shows that the necessitation rule for the third modality is admissible.For each finite C⊆𝒜, inference rule ϕ_Cϕ is admissible in our logical system. Assumption ⊢ϕ implies ⊢_Cϕ by Strategic Necessitation inference rule. Hence, ⊢_Cϕ by Strategic Truth axiom and Modus Ponens inference rule. The next result is a counterpart of Lemma <ref>. It states that the existence of a strategy by a sub-coalition implies the existence of a strategy by the entire coalition. ⊢_Cϕ→_Dϕ, where C⊆ D. Note that ϕ→ϕ is a propositional tautology. Thus, ⊢ϕ→ϕ. Hence,⊢_D∖ C(ϕ→ϕ) by Lemma <ref>. At the same time, by Cooperation axiom, ⊢_D∖ C(ϕ→ϕ)→(_Cϕ→_Dϕ) due to the assumption C⊆ D.Therefore, ⊢_Cϕ→_Dϕ by Modus Ponens inference rule. § CONCLUSION In this paper we proposed a sound and complete logic system that captures an interplay between thedistributed knowledge, coalition strategies, and how-to strategies. In the future work we hope to explore know-how strategies of non-homogeneous coalitions in which different members contribute differently to the goals of the coalition. For example, “incognito" members of a coalition might contribute only by sharing information, while “open" members also contribute by voting.eptcs | http://arxiv.org/abs/1707.08759v1 | {
"authors": [
"Pavel Naumov",
"Jia Tao"
],
"categories": [
"cs.AI",
"cs.LO"
],
"primary_category": "cs.AI",
"published": "20170727075329",
"title": "Together We Know How to Achieve: An Epistemic Logic of Know-How (Extended Abstract)"
} |
1Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA; [email protected] 2Earth and Planetary Science Department & Planetary Geosciences Institute, University of Tennessee, Knoxville, TN 37996, USAThe Hilda asteroids are primitive bodies in resonance with Jupiter whose origin and physical properties are not well understood. Current models posit that these asteroids formed in the outer Solar System and were scattered along with the Jupiter Trojans into their present-day positions during a chaotic episode of dynamical restructuring. In order to explore the surface composition of these enigmatic objects in comparison with an analogous study of Trojans <cit.>, we present new near-infrared spectra (0.7–2.5 μm) of 25 Hilda asteroids. No discernible absorption features are apparent in the data. Synthesizing the bimodalities in optical color and infrared reflectivity reported in previous studies, we classify 26 of the 28 Hildas in our spectral sample into the so-called less-red and red sub-populations and find that the two sub-populations have distinct average spectral shapes. Combining our results with visible spectra, we find that Trojans and Hildas possess similar overall spectral shapes, suggesting that the two minor body populations share a common progenitor population. A more detailed examination reveals that while the red Trojans and Hildas have nearly identical spectra, less-red Hildas are systematically bluer in the visible and redder in the near-infrared than less-red Trojans, indicating a putative broad, shallow absorption feature between 0.5 and 1.0 μm. We argue that the less-red and red objects found in both Hildas and Trojans represent two distinct surface chemistries and attribute the small discrepancy between less-red Hildas and Trojans to the difference in surface temperatures between the two regions. § INTRODUCTION The Hilda asteroids are a large population of minor bodies located in the 3:2 mean-motion resonance with Jupiter. These objects orbit within a relatively narrow range of heliocentric distances around 4.0 AU, between the outer edge of the Main Belt (roughly 3.3 AU) and Jupiter's orbit (5.2 AU). and Early theories of solar system formation posited that the Hildas originated in the middle Solar System and were captured into their present-day orbits during a period of smooth migration <cit.>. However, recent advances in our understanding of solar system evolution have placed Hildas in a new light. Current models describe a scenario in which the gas giants crossed a mutual mean-motion resonance sometime after the era of planet formation, triggering chaotic alterations throughout the Solar System <cit.>. These models predict that a significant fraction of planetesimals that formed beyond the primordial orbits of the ice giants was scattered inward during the period of dynamical instability and now resides in the middle Solar System <cit.>.In particular, simulations carried out within this dynamical framework have demonstrated that the present-day Hildas and Jupiter Trojan, both resonant asteroid populations in the vicinity of Jupiter, should be comprised almost exclusively of objects from the outer Solar System <cit.>. Such a scenario also points toward a common progenitor population for the Kuiper Belt objects, Trojans, and Hildas, presenting the enticing possibility that studying the surface properties of the more readily accessible Hildas could give insight into the composition of all three populations. The compositional characterization of Hildas and comparison with the properties of Trojans serve as a powerful observational test for probing the veracity of current dynamical instability models.At present, the composition of Hildas, and indeed that of all other middle and outer solar system minor bodies, remains a subject of much speculation. The general compositional outline for these asteroids posits a mixture of water ice, anhydrous silicates, and organics <cit.>. Observational studies of Hildas have hitherto revealed largely featureless, reddish spectra in the optical and near-infrared <cit.> and very low albedos averaging around 4% <cit.>. Taxonomically, Hildas are predominantly D- and P-type asteroids, with a small fraction of C-type objects <cit.>. Further into the infrared, a handful of published Hilda spectra reveal a broad, rounded feature centered at around 3 μm, which has been interpreted to be due to a thin layer of water frost coating a dark-grained regolith <cit.>. A study of a similar feature on Trojans shows that this feature could also be consistent with ammonia ice or irradiation products thereof <cit.>; such a surface chemistry would naturally point toward an outer solar system origin. Analysis of optical colors derived from Sloan Digital Sky Survey photometry and infrared reflectivities measured by the Wide-field Infrared Survey Explorer reveals a strong bimodality among both Hildas <cit.> and Trojans <cit.>. This result demonstrates that Hildas and Trojans are comprised of two spectrally distinct sub-populations and signifies a new dimension in the study of these objects. Comparative spectroscopy promises to expand our understanding of the surface composition, the underlying cause(s) of the sub-populations' bimodal features, possible divergent evolutionary signatures between Hildas and Trojans, and ultimately, the origin of these objects in the broader context of theories of solar system formation and evolution.<cit.> analyzed near-infrared spectra of 68 Trojans and uncovered a highly robust bifurcation in spectral slope that corresponds with the previously described bimodality in optical colors. In this paper, we describe the results from an analogous near-infrared spectroscopic survey of Hildas. These observations were undertaken with the objective of compiling a significant body of high-quality Hilda spectra to enable detailed comparison with the earlier results of <cit.>, as well as continuing the search for spectral features that may help further constrain the surface composition of these poorly-understood objects. § OBSERVATIONS AND DATA REDUCTION We carried out five observing runs at the NASA Infrared Telescope Facility (IRTF) throughout 2016. The spectra were obtained with the medium-resolution spectrograph and imager SpeX <cit.>. We used the LoRes prism mode with a 0.8× 15 arcsec slit, which produces single-order spectra spanning the wavelength range 0.7–2.5 microns. A total of 25 Hildas were observed, of which 5 were observed on two nights.Our observing strategy closely mirrored the methods described in <cit.>. Objects were dithered 7.5 arcsec along the slit between pairs of observations. In order to minimize readout time, while reducing the effect of atmospheric variability (in particular, the OH emission at these wavelengths), single-exposure integration times were set at 120 s for all of our observations. The telescope tracked each object using the automatic guider, which measures the spillover of the object outside of the slit and corrects the pointing to center the object in the slit at a rate of several times a minute. Solar analog G-dwarfs within 5^∘ of the asteroid were regularly observed — typically every ∼30 minutes, or whenever the airmass of the object changed by 0.10–0.15. The effect of differential refraction across the wavelength range was minimized by aligning the slit with the parallactic angle (±20^∘) for all asteroid and star observations. Flat-field and argon lamp wavelength calibration frames were taken at the beginning or end of each observing run. Details of our Hilda observations are listed in Table <ref>.The prism data were reduced following standard procedures for near-infrared spectra. We utilized the IDL-based spectral reduction program Spextool <cit.> in our data reduction. For each pair of dithered frames, this program handles flat-field removal, wavelength calibration, non-linearity correction, background subtraction, and spectral extraction through a graphical user interface. To correct for telluric absorptions as well as the solar spectrum, each extracted asteroid spectrum was divided by the corresponding calibration star spectrum that was obtained closest in time and airmass. The effects of instrument flexure on the wavelength calibration and telluric correction were addressed by shifting each asteroid spectrum relative to the corresponding calibration star spectrum to minimize the variability within the water absorption regions. Bad pixels and other significant outliers were manually pruned from the extracted spectra in Spextool before the individual dithered-pair spectra for each object were combined into a single reflectance spectrum. § RESULTS AND DISCUSSION The normalized reflectance spectra of the 25 Hildas targeted in our observing runs are shown in the Appendix. Here, and in the subsequent analysis, we have included additional spectra of three Hildas (153 Hilda, 190 Ismene, and 361 Bononia) obtained using IRTF/SpeX and published in <cit.>. The signal-to-noise ratio (S/N) of the 28 Hilda spectra in K-band (2.22 μm) range from 20 to 300.As is the case with the 0.7–2.5 micron Trojan spectra published in <cit.>, none of the Hilda spectra show any absorption features to within the noise level in the data. Therefore, while the surface composition of Hildas is typically assumed to be similar to that of Trojans, i.e., rich in water ice, organics, and crystalline silicates <cit.>, no evidence for these materials are found in the near-infrared spectra. §.§ Classification into Sub-populationsSeveral previous studies have revealed that Hildas are bimodal with respect to visible color and infrared reflectivity. Using these results, we can classify objects into sub-populations and examine their average near-infrared spectra separately.Analysis of photometry for Hildas contained in the Sloan Digital Sky Survey Moving Object Catalog (SDSS-MOC) demonstrates that the distribution of spectral slopes at visible wavelengths is strongly bimodal, indicative of two sub-populations with visible colors centered at 4.0 and 9.3, in units of %/1000 Å <cit.>. Following previous works, we refer to these two sub-populations as less-red (LR) and red (R). Ten of the 28 Hildas in our sample have SDSS-MOC photometry; an additional eight objects have earlier published visible spectral slopes <cit.>. We note that there is significant overlap between the two modes in the visible color distribution, so some objects with intermediate spectral slopes cannot be definitively categorized as LR or R. The sub-population classifications based on visible colors are listed in Table <ref>. We classify 8 objects as LR, 6 objects as R, and 4 objects as intermediate.The reflectivity of Hildas in the W1 (3.4 μm) and W2 (4.6 μm) bands of the Wide-field Infrared Survey Explorer (WISE) space telescope also allows for classification of Hildas into the LR and R sub-populations. Specifically, the distribution of relative infrared reflectance in the W1/W2 bands with respect to visible albedo shows two clearly separated clusters <cit.>, with one group systematically more reflective at infrared wavelengths than the other. Selecting for Hildas with reflectance measurements in both W1 and W2 bands, we are able to categorize 18 objects (9 LR and 9 R). The sub-population classifications based on infrared reflectivity are listed in Table <ref>.Comparing the classifications for objects with both visible spectral slope and infrared reflectivity data, we see that the classifications from these two independent methods are consistent, i.e., objects categorized as LR via visible spectral slope are also categorized as LR based on infrared reflectivity, and likewise for R objects. This same pattern was observed for Trojans <cit.> and indicates that LR and R objects differ systematically and predictably in both the visible and the infrared. Combining the two sets of classifications, we are able to categorize 22 out of 28 Hildas in our near-infrared spectral sample: 11 LR and 11 R. We can classify an additional 4 objects (3 R and 1 LR) in our spectral sample, which do not have independent published optical or infrared observations, based on their relative positions in near-infrared color space (see Section <ref>). This brings the total classification count to 12 LR and 14 R.Figure <ref> plots the average of all spectra in the two color sub-populations. The main observation is that the average LR and R Hilda spectra are highly distinct, with the difference in shape concentrated primarily at the shorter end of the near-infrared range (λ < 1.5). While the R Hilda spectrum is concave down throughout most of the near-infrared, the LR Hilda spectrum is mostly straight, with only a slight downturn at the shortest wavelengths. This indicates that the LR and R Hilda sub-populations have distinct surface properties, in agreement with the bimodality previously seen at both visible and infrared wavelengths. §.§ Comparison with Jupiter Trojans In order to compare the absolute reflectivity spectra of LR and R Hildas and Trojans, we compiled all previously published visible spectra <cit.> as well as broadband SDSS-MOC photometry of Hildas within our near-infrared spectral sample.[We do not include the visible spectra from <cit.> and <cit.>, as those spectra are not published with uncertainties on the individual data points.] All five Hildas with visible spectra (153 Hilda, 190 Ismene, 361 Bononia, 1180 Rita, and 1754 Cunningham) are LR objects. The visible spectral and photometric data are matched to the near-infrared reflectance spectra at 0.75 and 0.913 (Sloan z-band) μm, respectively. The absolute reflectivity level is set by the average visible (0.55 μm) geometric albedo: 0.04 <cit.>. Figure <ref> shows the average reflectivity spectra of LR and R Hildas, along with analogous data for Trojans taken from <cit.> and <cit.>. We note that the Hilda and Trojan visible spectral samples are a subset of the corresponding near-infrared spectral samples; moreover, there is little overlap between the visible spectral and SDSS-MOC photometric samples. Therefore, some mismatch between the visible spectral and photometric data can be expected.The main observation from the comparison plot is that the corresponding sub-populations in the Hildas and Trojans have notably similar spectra. Looking at the R objects separately, we find that R Hildas and R Trojans display largely identical spectral slopes across the visible and near-infrared wavelengths. This observation is corroborated by the reported average visible spectral slopes: 9.3 %/1000 Å for R Hildas <cit.>, and 9.6 %/1000 Å for R Trojans <cit.>. A slight difference in spectral slope is apparent between 0.7 and 0.9 μm. While this disparity between the two populations may be real, this region corresponds to the lower end of the IRTF/SpeX wavelength range, where the transmission function rises steeply. As such, data in this wavelength region are especially susceptible to residual signals from telluric correction, as, for example, in the case of uncorrected nonlinear effects from instrument flexure (Section <ref>).The average LR Hilda and Trojan spectra likewise demonstrate similar overall shapes. However, a closer look reveals the possible presence of a very broad and shallow absorption feature in the region 0.5–1.0 μm. We note that the sharp inflection point at 0.7 μm is likely to be primarily caused by the concatenation of different data sets for the visible and near-infrared average spectra, with the former being a subset of the latter. We note that none of the individual Hilda spectra spanning both visible and near-infrared wavelengths shows a discernible feature in this wavelength region above the level of the uncertainties.Nevertheless, both visual comparison and color indices (see below) demonstrate that LR Hildas are noticeably redder on average (i.e., have steeper spectral slopes) than LR Trojans throughout much of the near-infrared (0.7–2.0 μm).In contrast, at visible wavelengths, LR Hildas and somewhat bluer than LR Trojans, as evidenced by their average spectral slopes: 4.0 %/1000 Å for LR Hildas <cit.>, and 5.3 %/1000 Å for LR Trojans <cit.>. Together, these disparities imply an overall concave-up shape in the average LR Hilda spectrum relative to the LR Trojan spectrum extending from the visible into the near-infrared. The implications of such an absorption feature are discussed in Section <ref>.§.§ Near-infrared Colors To study the spectra more quantitatively, we calculated near-infrared colors. Following the methods described in <cit.>, we derived color indices from the normalized reflectance at four wavelengths: 0.85 μm, J (1.25 μm), H (1.65 μm), and K (2.22 μm). The color m_λ1-m_λ2 corresponding to a given reflectance ratio R_λ2/R_λ1 was calculated using the relation m_λ1-m_λ2 = 2.5log(R_λ2/R_λ1). Standard error propagation was used to calculate the color uncertainties. The measured colors for all the Hildas in our sample are listed in Table <ref>.Figure <ref> shows the distribution of Hilda colors (colored points) in the J-K vs. 0.85-J space, along with the corresponding distribution of Trojan colors (black points), taken from <cit.>. The colors of the LR and R Hildas are denoted by blue and red. The overall Trojan color distribution is robustly bimodal, with the locations of the LR and R Trojans clearly identifiable. In contrast, while the LR and R Hildas in our near-infrared spectral sample occupy disparate regions in the J-K vs. 0.85-J space, these regions overlap, resulting in an overall Hilda color distribution that lacks the bimodality seen in the Trojan data. Taking the positions of the 6 hitherto unclassified Hildas on the two-color plot into consideration, we are able classify one (2067 Aksnes) as LR and three (3561 Devine, 5661 Hildebrand, and 14669 Beletic) as R, with the remaining two Hildas unclassifiable due to their having near-infrared color indices that lie in the overlap of the LR and R color regions. The final classification count in our spectral sample is 12 LR and 14 R. Examining the color indices more closely, we see that the R Trojans and R Hildas have comparable near-infrared colors, in line with the observation from our comparison of their average spectra in Section <ref>. Meanwhile, the LR Hildas have redder (i.e., higher) color indices than the LR Trojans. The relative shift in the cluster of LR Hildas relative to the LR Trojans is particularly noticeable along the 0.85-J axis, in agreement with the visual spectroscopic comparison in Figure <ref>, which shows that the LR Hildas have steeper spectral slopes than LR Trojans in the short wavelength end of IRTF/SpeX wavelength range.§.§ Implications In the context of general characteristics shared by Hildas and Trojans, e.g., reddish colors, near-identical optical albedos, and bimodalities in visible color and infrared reflectivity, the large-scale similarity in near-infrared spectra offers further support to the idea of a common origin for these two minor body populations. While the overall near-infrared color distribution of Hildas does not display bimodality, as in the case of Trojans, Hildas classified as LR or R via visible color and/or infrared reflectivity cluster around two separate near-infrared color centers. The systematic spectral differences between LR and R objects within the Hilda and Trojan populations, which are manifested from the visible to the infrared, are indicative of distinct surface compositions <cit.>. <cit.> developed a hypothesis within the framework of current dynamical instability models to explain the origin of the sub-populations attested in the Hildas, Trojans, and similarly-sized Kuiper Belt objects <cit.>. In short, the precursor objects to these three populations initially formed in a planetesimal disk spanning a wide range of heliocentric distances in the outer solar system beyond the orbits of the ice giants with roughly cometary compositions rich in volatile ices such as ammonia, methanol, hydrogen sulfide, etc. Subsequent insolation-driven sublimation loss led to the development of distinct surface compositions over the course of the first few hundred million years after the end of planet formation, prior to the onset of dynamical stability that scattered these bodies into their present-day locations.From our modeling, we predict hydrogen sulfide ice (H_2S) to be the crucial volatile ice species responsible for the development of two distinct spectral types. Objects in this outer solar system primordial disk would have been split into two groups — the objects situated closer to the Sun experienced higher surface temperatures and became depleted of H_2S on their surfaces, while objects farther out retained H_2S. Previous experimental work has shown that irradiation of volatile ices leads to a general reddening of the optical color <cit.>. We posit that the presence of H_2S, which is known to induce a strong reddening of the surface upon irradiation <cit.>, would contribute additional reddening relative to the case where H_2S was absent. In both cases, the result of irradiation would have been the formation of a refractory mantle on the surface of the objects, with the presence or absence of sulfur-bearing minerals yielding a color bimodality among objects in the primordial planetesimal disk. Objects that retained H_2S on their surfaces would develop R colors, while those that became depleted in H_2S would develop LR colors. The subsequent dynamical instability spread these objects across the present-day minor body populations, which therefore inherit this primordial color bimodality.Our present analysis has shown that LR Hildas have relatively steeper spectral slopes in the near-infrared when compared to LR Trojans; meanwhile, R Hildas and Trojans have near-identical visible and near-infrared colors. In the context of the aforementioned color bimodality hypothesis, which posits that both LR and R Hildas and Trojans were scattered into their present-day orbits from the same progenitor population in the outer Solar System, this disparity between LR Hildas and Trojans is likely explained by their different present-day environments. Hildas orbit significantly closer to the Sun than Trojans and thereby experience higher surface temperatures. It follows that the surface chemistries (sulfur-bearing or sulfur-less) of LR and R objects react differently to heating. While R objects appear to be stable to the temperature change between the Trojan and Hilda regions, LR objects develop redder spectral slopes in the near-infrared and bluer optical colors when heated.The putative broad and shallow absorption feature in the average LR Hilda spectrum between 0.5 and 1.0 μm (Section <ref>) is notable in that it suggests the first spectroscopic signature to be found at visible or near-infrared wavelengths on Hildas. Future study of more precise spectra of Hildas with continuous wavelength coverage across the visible and near-infrared is needed to confirm and/or characterize this feature. Nevertheless, comparing the Hildas with other nearby minor body populations offers a clue to its possible origin. Looking inward, we find some clearer examples of this feature. Combined visible and near-infrared spectra of two Cybele group asteroids — 76 Freia and 107 Camilla — show a pronounced concave-up shape throughout the 0.5–1.0 μm region within otherwise featureless spectra <cit.>. These objects lie in the 7:4 mean-motion resonance with Jupiter, with orbital semimajor axes of 3.41 and 3.49 au, respectively, and, similar to Hildas and Trojans, are presumed to originate in the outer solar system within the dynamical instability scenario <cit.>. Both of these Cybele asteroids have LR visible colors and, when considered alongside the LR Hildas and Trojans, point to an intriguing trend: the depth of the broad 0.5–1.0 μm feature appears to increase with decreasing heliocentric distance.This broad absorption feature between 0.5 and 1.0 μm in asteroidal spectra is generally interpreted as indicating hydrated phyllosilicate minerals on the surface <cit.>. The possible presence of hydrated minerals on the surface of Hildas has major implications for their evolutionary history, since it would require melting of surficial water ice, which does not occur at the present-day surface temperatures of Hildas. Within the framework of dynamical instability models, the aforementioned trend between the depth of the 0.7 μm feature and heliocentric distance among LR objects could be explained if the Cybeles and Hildas were scattered onto orbits that passed systematically closer to the Sun than Trojans during the period of dynamical instability, thereby experiencing higher surface temperatures and possible melting of water ice. The additional observation that R Hildas do not display this absorption signature may indicate that the presence of sulfur-bearing components on the surface blocks the development of hydrated mineral absorption features. Further spectral modeling and dynamical instability orbital simulations are needed to determine the specific chemical species responsible for the absorption feature and how it figures into the origin and evolution of these minor body populations.§ CONCLUSIONIn this paper, we presented new near-infrared spectra of 25 Hilda asteroids. As in the case of analogous spectra of Trojans <cit.>, we did not detect any absorption features within the wavelength range covered by the IRTF/SpeX spectrograph (0.7-2.5 μm). Classifying the Hildas into less-red and red sub-populations based on their previously published visible color and/or infrared albedo, we found that the average less-red and red Hilda spectra have very distinct shapes. Taken together, the systematic differences in spectroscopic/photometric properties from the visible through the infrared firmly establish that the two Hilda sub-populations possess distinct intrinsic surface compositions, in agreement with the conclusions of earlier studies of Trojans.Combining our near-infrared spectra with visible spectra from the literature and comparing the results with analogous data for Trojans, we observed that the corresponding sub-populations within the Trojans and Hildas have similar overall spectral shapes. This lends support to the idea that the less-red and red Trojans and Hildas are each sourced from a single progenitor population, as is posited by recent dynamical instability theories of solar system evolution. Upon closer inspection, we uncovered a notable difference between the less-red and red objects in the Hilda and Trojan populations. Whereas the red Hildas and Trojans have nearly identical spectra, the less-red Hildas are significantly redder than less-red Trojans in the near-infrared, while being somewhat bluer than their Trojan counterparts at visible wavelengths. From this observation, we inferred the presence of a very broad and shallow absorption feature in the average less-red Hilda spectrum between 0.5 and 1.0 μm, suggesting the possible presence of hydrated minerals on the surface. In the context of our hypothesis regarding the origin of the two sub-populations, we proposed that the discrepancy between less-red Hilda and Trojan spectra is a consequence of their different surface temperatures. Meanwhile, objects in the red spectral type are stable to the temperature gradient between the Hilda and Trojan regions.Pinning down a more detailed picture of the surface (and bulk) composition of Hildas and Trojans remains a crucial and unfulfilled step in the overarching objective of understanding the origin of these minor body populations and validating current models of solar system evolution. A potentially fruitful avenue of further study is analyzing new Hilda spectra covering the 3 μm region. Previous study of Trojans has revealed a significant absorption feature centered at around 3.1 μm, consistent with both water frost and ammonia ice <cit.>. Intriguingly, the depth of this feature is strongly correlated with spectral type, with less-red Trojans having systematically deeper absorption than red Trojans. This absorption signal has already been detected in the three extant Hilda spectra covering this wavelength range <cit.>; however, the three objects all belong to the less-red sub-population. Additional spectroscopic observations targeting both less-red and red Hildas will provide a more detailed look into this feature in relation to the two sub-populations and will aid in the continued exploration of environmental effects on the surface properties of these primordial bodies.§ APPENDIX Plotted here are the final reduced near-infrared spectra of Hildas presented in this paper. Of these, 25 are derived from the IRTF/SpeX observations described in Section <ref>, and three (153 Hilda, 190 Ismene, and 361 Bononia) are previously published in <cit.>. All spectra are normalized to unity at 2.2 μm and shifted in intervals of 0.5 for clarity. The gray bars denote the main wavelength regions of telluric water vapor absorption. Residual signals within these bands are caused by variable atmospheric conditions during and/or between observations of asteroids and solar analogs stars used for telluric correction. urlstyle[Brown(2016)]brownBrown, M. E. 2016, AJ, 152, 159[Brunetto et al.(2006)Brunetto, Barucci, Dotto, and Strazzulla]brunetto Brunetto, R., Barucci, M. A., Dotto, E., & Strazzulla, G. 2006, ApJ, 644, 646[Bus & Binzel(2002)]bus Bus, S. J., & Binzel, R. P. 2002, Icar, 158, 106[Carlson et al.(2007)]carlson Carlson, R. W., Kargel, J. S., Doute, S., Soderblom, L. A., & Brad Dalton, J. 2007, in Io After Galileo, ed. R. M. C. Lopes, & J. R. Spencer (Chichester, UK: Praxis Publishing Ltd), 194[Cushing et al.(2004)]cushing Cushing, M. C., Vacca, W. D., & Rayner, J. T. 2004, PASP, 116, 362[Dahlgren & Lagerkvist(1995)]Hildaspectra Dahlgren, M., & Lagerkvist, C.-I. 1995, A&A, 302, 907[Dahlgren et al.(1997)]Hildaspectra2 Dahlgren, M., Lagerkvist, C.-I., Fitzsimmons, A., Williams, I. 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F., Tsiganis, K., & Gomes, R. 2005, Natur, 435, 462[Rayner et al.(2003)]rayner Rayner, J. T., Toomey, D. W., Onaka, P. M., et al. 2003, PASP, 115, 362[Rivkin et al.(2015)]rivkin Rivkin, A. S., Campins, H., Emery, J. P., et al. 2015, in Asteroids IV, ed. P. Michel, F. E. DeMeo, & W. F. Bottke (Tucson, AZ: University of Arizona Press), 65[Roig & Nesvorný(2015)]roig2 Roig, F., & Nesvorný, D. 2015, AJ, 150, 186[Roig et al.(2008)Roig, Ribeiro, and Gil-Hutton]roig Roig, F., Ribeiro, A. O., & Gil-Hutton, R. 2008, A&A 483, 911[Ryan & Woodward(2011)]Hildaalbedo Ryan, E. L., & Woodward, C. E. 2011, AJ, 141, 186[Takir & Emery(2012)]takiremery Takir, D., & Emery, J. P. 2012, Icar, 219, 641[Vilas et al.(1994)]vilas Vilas, F., Jarvis, K. S., & Gaffey, M. J. 1994, Icar, 109, 274[Wong & Brown (2016)Wong, Brown]wong3 Wong, I., & Brown, M. E. 2016, AJ, 152, 90[Wong & Brown (2017a)Wong, Brown]wong4 Wong, I., & Brown, M. E. 2017a, AJ, 153, 69[Wong & Brown (2017b)Wong, Brown]wong5 Wong, I., & Brown, M. E. 2017b, AJ, 153, 145[Wong et al.(2014)Wong, Brown, and Emery]wong Wong, I., Brown, M. E., & Emery, J. P. 2014, AJ, 148, 112[Yang & Jewitt(2007)]yangjewitt Yang, B., & Jewitt, D. 2007, AJ, 134, 223 | http://arxiv.org/abs/1707.09064v1 | {
"authors": [
"Ian Wong",
"Michael E. Brown",
"Joshua P. Emery"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170727223847",
"title": "$0.7-2.5~μ$m spectra of Hilda asteroids"
} |
𝐱 ŁL IEEE Transactions on Multimedia Shi et al.: Structure-Preserving Image Super-resolution via Contextualized Multi-task Learning Yukai Shi, Keze Wang, Chongyu Chen, Li Xu, and Liang LinYukai Shi, Keze Wang, Chongyu Chen and Liang Lin are with School of Data and Computer Science, Sun Yat-sen University, Guangzhou, China. Keze Wang is also with Dept. of Computing, The Hong Kong Polytechnic University, Hong Kong. Li Xu is with SenseTime Group Limited, China. Contact: [email protected]. Liang Lin is also with the Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, Sun Yat-sen University, China.======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= Single image super resolution (SR), which refers to reconstruct a higher-resolution (HR) image from the observed low-resolution (LR) image, has received substantial attention due to its tremendous application potentials. Despite the breakthroughs of recently proposed SR methods using convolutional neural networks (CNNs), their generated results usually lack of preserving structural (high-frequency) details. In this paper, regarding global boundary context and residual context as complimentary information for enhancing structural details in image restoration, we develop a contextualized multi-task learning framework to address the SR problem. Specifically, our method first extracts convolutional features from the input LR image and applies one deconvolutional module to interpolate the LR feature maps in a content-adaptive way. Then, the resulting feature maps are fed into two branched sub-networks. During the neural network training, one sub-network outputs salient image boundaries and the HR image, and the other sub-network outputs the local residual map, i.e., the residual difference between the generated HR image and ground-truth image. On several standard benchmarks (i.e., Set5, Set14 and BSD200), our extensive evaluations demonstrate the effectiveness of our SR method on achieving both higher restoration quality and computational efficiency compared with several state-of-the-art SR approaches. [The source code and some SR results can be found at: <http://hcp.sysu.edu.cn/structure-preserving-image-super-resolution/>]Structure-preserving Image super-resolution; Convolutional network;Context learning; Multi-task learning;§ INTRODUCTION Image super-resolution (SR) is a fundamental problem in image processing. Single image SR approaches, which aim at restoring a high-resolution (HR) image only from a single low-resolution (LR) image, have been applied to manyimage and video analysis tasks, such as video surveillance <cit.>, image-based medical analysis <cit.>,and image/video streaming <cit.>.Common techniques for single image SR can be roughly categorized into reconstruction-, example- and interpolation- based approaches. Reconstruction-based approaches <cit.>, which restore HR images by deconvolutional methods <cit.> with a global blur degradation model, usually introduce ringing artifacts around salient structures <cit.> due to inaccurate blurring kernels in the inverse problem. Example-based approaches <cit.> boost the amplification factor by using internal or external patch data to guide the image restoration. Recently, Huang et al. <cit.>proposed to exploit self-similarity for single image SR, which greatly expands the internal patch searching space. Hu et al. <cit.> proposed a cascaded linear regression technique to model the relationship between HR and LR images. Interpolation-based approaches can achieve acceptable trade-off between performance and efficiency with a pre-defined kernel. However, pre-defined kernels use fixed weights for interpolation, which will inevitably cause blur when the weight definition is inconsistent with image structures. To address issue, various adaptive interpolations <cit.> are proposed. But the improvements in restoration quality are still limited.The success of deep convolutional neural network (CNN) in computer vision tasks has inspired novel trends in low-level image restoration researches, such as rain/dirt removal <cit.>, noise removal <cit.>, face hallucination <cit.>, hashing <cit.> and image inpainting <cit.>. Focusing on learning an end-to-end mapping between the LR images and their corresponding HR images, several CNN-based methods <cit.> have been proposed to perform image SR in a pure data-driven manner. That is, they directly minimize the mean squared error (MSE) between the predicted and ground-truth images in the training stage. Although the restoration performance is significantly improved, the structural inconsistency between the LR input and HR output still exists.This is because human visual system is more sensitive to structural changes, which are difficult to be exploited from MSE-based loss functions. Recent advances in image SR try to address this issue <cit.> by introducing feature-based perceptive loss functions to the training stage. However, unwanted artifacts and unreal details are also introduced, which make their SR results look unrealistic. Considering single image SR is an ill-defined problem, it is necessary to exploit the priors of natural image to further improve the SR performance. Motivated by recent advances in deep learning researches that exploit priors in the form of context information in designing neural networks <cit.>, in this work, we propose to design neutral networks to investigate two types of image structural information, i.e., global structural information which corresponds to salient boundaries in a global perspective and residual structural information which contains noticeable details that are critical to visual quality.The success of multi-task learning framework inspires us to leverage such structural information in a unified manner. For instance, Yang et al. <cit.> proposed to utilize the common knowledge (e.g., feature selection functions) of multiple tasks as supplementary information to facilitate decision making. Considering aforementioned structural information are usually considered as complementary context rather than common knowledge, in this work, we concentrate on complimentary contextualized multi-task learning for structure-preserving single image SR. In particular, we propose a deep joint contextualized multi-task learning framework, where three types of image components are imposed as complimentary contexts and jointly learned, i.e., the base image content, the boundary map, and the residual map.Besides a convolutional network that learns content-adaptive interpolations to produce the intermediate base image, we impose an auxiliary task to back-propagate the global boundary structural context.Meanwhile, an independent sub-network is introduced to explicitly model the noticeable details to provide residual structural context.The major contribution of this work is the proposed contextualized multi-task learning framework, which is the first attempt to incorporate joint learning of local, global, and residual contexts into CNNs for single image SR. Other contributions mainly come from the proposed content-adaptive interpolation and the sub-networks for capturing complementary image contents, which enables better trade-off between restoration quality and the number of network parameters. Extensive experiments on several benchmarks datasets (e.g. Set5, Set14, BSD500) demonstrate that the proposed framework shows superior performance to most learning-based approaches in the perspective of both visual quality and quantitative metrics, which facilitates the real-time image SR process.We would like to point out that a preliminary version of this work is reported in <cit.>, which coarsely concatenates content-adaptive interpolation and holistic edge context. In this paper, we inherit the idea of preserving structures and refining the network architecture. A simple yet powerful sub-network is further employed to capture noticeable image details for better visual quality. The whole framework is re-interpreted from the aspect of joint context learning and multi-task learning.Besides, more comparisons with state-of-the-art approaches and more detailed analyses of the proposed modules are added to further verify our statements. The rest parts of this paper are organized as follows. Section <ref> briefly reviews existing machine learning-based SR approaches which motivate this work. Section <ref> presents the details of the proposed framework, with thorough analysis of every component. Section <ref> demonstrates the experimental results on several public benchmarks, comparing with state-of-the-art alternatives. Finally, Section <ref> concludes this paper. § RELATED WORK§.§ Interpolation-based image super-resolution Interpolation-based approaches typically start from evenly placing the pixels of LR image to the HR grid (the integral coordinates in the HR image domain). The basic idea of these approaches is to estimate the unknown pixel values in the HR grid by weighted average of surrounding known pixels. Considering common pixel changes in a local region can be approximated by continuous functions,people have proposed various weight definitions for image interpolation. For example, bilinearinterpolation is proposed to utilize local linearity, and bicubic interpolation is proposed to exploit the high-order continuity <cit.>.However, there are plenty of pixel changes that cannot be described by these pre-defined functions, especially for regions with rich image structures. In this case, structures will be blurred due to improper pixel averaging. To address this problem, various adaptive interpolation <cit.> are proposed. For instance, Walt et al. <cit.> proposed to express polygonal pixel overlap as a linear operator to improve the interpolation performance. But the improvements are still limited. §.§ Multi-task learning in image super-resolutionDecades of researches on multi-task learning have demonstrated that learning multiple correlated tasks simultaneously can significantly improve the performance of the main task <cit.>.In single image SR, there is also a trend of utilizing multi-task learning. For example, Yang et al. <cit.> proposed a multi-task K-SVDlearning for image SR, in which example image patches are divided into different groups and K-SVD is applied to every group. It is shown that simultaneous learning multiple dictionaries can lead to better SR quality. Liang et al. <cit.> proposed a multi-task learning framework that jointly considers image SR process and the image degeneration process. These works claim that the multi-task learning framework is a feasible way of utilizing priors in learning-based image SR.§.§ Deep learning in image super-resolutionRecently, deep learning has achieved significant quality improvements in image SR. For example, Dong et al. <cit.> utilized a three-layer fully convolutional network to learn the non-linear mapping between HR and LR patches, which has a close relationship to sparse coding.Ren et al. <cit.> introduced Shepard CNNs to facilitate translation variant interpolation, which gives a solution to both inpainting and SR. Wang et al. <cit.> proposed a sparse coding based network for image SR. Based on learned iterative shrinkage and thresholding algorithm(LISTA) <cit.>, they employ a set of neural networks to restore images. Zeng et al. <cit.> proposed a deep autoencoder for SR, which explores the consistent representations of HR and LR images anddemonstrate a superior efficiency compared to similar methods based on sparse representation.Kumar et al. <cit.> studied on several factors that affect the training phase to facilitate learning-based SR with fewer training samples. The models of these methods, although being proposed from different aspects, are trainedto minimize the squared error w.r.t. the ground-truth HR image, which is not necessarily correlated to good perceptual quality. Bruna et al. <cit.> referred this problem as regression to mean. Their proposed solution is a conditional generative model, which demonstrates improvement over visual quality, but with high time cost in both training and testing.More recently, researchers notice the importance of image details and make various of attempts for exploration. Kim et al. <cit.> further improved the SR quality by different network architectures such as very deep and recursive network structures. However, these methods heavily rely on very deep networks with plenty of parameters. e.g., a 20-layer convolutional neural network <cit.>. In addition, perceptual losses have been proposed for CNNs <cit.>, which conduct the loss from the image space to high-level feature space of a pre-trained VGG-net <cit.>.At the same time, Ledig et al. <cit.> proposed to apply adversarial network to the task of SR, which results in more image details but lower PSNR score. More related to our work, there are several attempts to accelerate image SR. By developing a sub-pixel convolutional layer, Shi et al. <cit.> used a single model to handle real-time image SR. Similarity, Dong et al. <cit.> applied convolutional layers on LR image and upscaled it with deconvolution. They both promise low computational complexity, but there still exists plenty of room for performance improvement. § CONTEXTUALIZED MULTI-TASK LEARNING In this section, we present the details of our framework. As sketched in Fig. <ref>, the proposed framework includes three components: feature extraction, content-adaptive interpolation, and multi-task estimation. §.§ Feature ExtractionInspired by the Pyramid-Net <cit.>, we design a pyramid network structure for feature extraction. That is, there are 3 convolutional layers with 16, 32 and 128 kernels, respectively. Detailed setup is summarized in Table <ref>. The first layer with kernel size 5×5 is designed as a spacious receptive field to capture as much image information as possible, as illustrated in <cit.>.The other two layers with 3×3 kernel are adopted for better efficiency as <cit.>. Note that we focus on extracting featuresfrom original LR images instead of the interpolated images. Thanks to the decreased computations of convolutional operations caused by the small size of feature maps, the proposed feature extraction can significantly accelerate the speed without obvious quality drop. Since the LR image has been represented as high-dimension feature maps through the first 3 layers, the computation cost may become pretty high if we import the high-dimension feature maps to content-adaptive interpolation directly.Therefore, we apply a shrinking layer with 8 kernels of size 1×1 to reduce the feature dimension. Note that the kernel number is empirically chosen for a reasonable trade-off between effectiveness and efficiency. Benefitting from the shrinking layer, our model not only avoids parameter explosion but also promotes the restoration efficiency.§.§ Content-adaptive InterpolationThe second component is one deconvolutional layer, which is used to interpolate the LR feature maps in a content-adaptive way. The deconvolutional layer has 8 kernels of size n × n.Note that in this work, n is determined by the upscaling factor, which follows the principles of bicubic interpolation. That is, the kernel should be large enough to cover the second pixel around the anchor pixel in the HR grid. For example, the deconvolutional kernel is of size 8×8, 11×11, and 16×16 for the upscaling factors of 2, 3 and 4, respectively.In this way, the deconvolutional layer can be regarded as a neural network implementation of standard image interpolation. Let 𝐲 be the HR image with a HR grid. We construct another HR image 𝐱 by evenly placed the LR image inthe HR grid with identical pixel intervals. Then, standard interpolation can be written as:𝐲_j = ∑_i ∈Ω_j𝐱_i ω_ji,where i and j are the pixel indices in the HR grid,Ω_j represents the subset of n× n neighbouring pixels around pixel j, and ω_ji is the pre-defined weight for interpolation.Note that 𝐱_i is non-zero only when it comes from a pixel in the LR image.With these definitions, we re-formulate the interpolation process as a basic component of a deconvolutional layer, i.e.,𝐲_j = δ (∑_i∈Ω_j𝐱_i W(i') + b)where δ(·) represents the activation function, W is the deconvolutional kernel, i' represents the pixel of W that contributes to pixel j, and b is the bias.In the proposed content-adaptive interpolation, we use multiple deconvolutional kernels in a similar fashion.That is, we evenly place the LR image in the HR grid to construct 𝐡^l. Then,𝐡^l+1_k = δ ( 𝐡^l⊗ W_k + b_k ),where the subscript k represents the kernel index, “⊗” represents the convolutional operator, and 𝐡^l+1 is the output image of the l^th layer.In this way, content-adaptive image interpolation can be accomplished via a deconvolutional layer, whose kernels are learned from sufficient training data. Note that the deconvolutional layer is in the middle of the proposed network, which is different from other CNN-based SR methods <cit.> that use deconvolution as the last layer. It is shown empirically that the proposed network can achieve nice restoration quality with reasonably increasing network parameters.To compare the proposed network with the bicubic interpolation, we construct a small network which only has one deconvolutional layer to learn an adaptive kernel, taking BSD300 as training data and bicubic interpolation parameters for initialization. The intensity changes of bicubic and our learned kernels are visualized in Fig. <ref>, which illustrates that the learned kernel contains more high-frequency components. Meanwhile, the restoration results also indicate that the learned kernel leads to a superior restoration quality with more recovered details compared to the bicubic kernel. Thus, the effectiveness of the proposed adaptive interpolation is verified. §.§ Contextualized Multi-task Learning In spired by the multi-task learning principles, we make an attempt to introduce auxiliary knowledge to SR issue.Global Boundary Context: We develop a Boundary Context sub-Network (BCN) to preserve salient boundaries that represent global image structures. BCN consists of two convolutional layers with 3×3 kernels, where one layer is with 12 kernels and the other layer is with 2 kernels. In the training phase of BCN, we propose to exploit salient image boundaries by regarding edge detection as a joint task of HR image restoration. In particular, we introduce an auxiliary term into the objective function, which computes the error between predicted and human-labeled edge/boundary maps. These boundary maps are from Berkeley Segmentation Dataset (BSD) <cit.>. Note that there are multiple boundary maps in BSD500 data set, we use their summation for better visualization and show the examples in Fig.<ref>. With the two tasks of image restoration and edge detection, image components and structural features are firstly extracted and enlarged by content-adaptive interpolation before being fed into the BCN. Several representative samples of the extracted feature maps are shown in Fig. <ref>, in which the top row and bottom row show image-like and edge-like features, respectively. This implies that these layers simultaneously extract redundant components and features, making it possible to produce base image and boundary maps in the HR image domain. Through joint optimization in an end-to-end manner, feature extraction, content-adaptive interpolation and BCN can provide complimentary context information to each other. In this way, structure-aware feature representations can be learned with the content-adaptive interpolation. Residual Context: As a result of paying close attention to generating the HR image with salient boundaries, the concatenated BCN might fail to restore some subtle but noticeable structures. Motivated by the recent residual learning paradigm <cit.>, we make an attempt to address this issue by employing a Residue Context sub-Network (RCN). The objective of the RCN is to synthesize a residual image, which is defined as the difference between the interpolated HR image and the ground-truth HR image. In contrast to using the bicubic interpolated HR image as in <cit.> and <cit.>, our model uses the intermediate HR image provided by BCN. This can bring us two benefits: i) Higher image SR performance. As the HR image provided by BCN achieves comparable performance to the state-of-the-art methods, RCN can focus on remedying the overlooked information for higher SR quality; ii) A lightweight network architecture for RCN. Our used interpolated image contains significantly richer information than the bicubic one. Hence, compared with <cit.> and <cit.>, the synthesization of residual images is much easier. As illustrated in Fig. <ref>, the architecture of RCN is the same as that of the concatenated BCN. For the joint optimization of content-adaptive interpolation, BCN and RCN, we develop a fusion layer to merge the intermediate output of RCN and BCN in a data-driven way. In particular, the final HR image 𝐲 of our framework is obtained by:𝐲=𝐟⊗𝐈_interHR + 𝐈_r,where 𝐟 denotes a 3×3 convolutional filter, 𝐈_interHR is the intermediate HR image provided by BCN, and 𝐈_r is the residue image synthesized by RCN. In this way, the parameters of 𝐟 can be adaptively updated during the learning process.§ FRAMEWORK TRAININGThe proposed framework is jointly optimized on a set of “LR image, HR image and HR edge map[In BSD data sets, more than one boundary maps are provided for every image, which are all used in our training process. Since multiple boundary maps are used in the same way, in this subsection, we focus on the case of one boundary map for simplicity.]” triplets. For convenience, we use 𝐈_l,𝐈_h and 𝐈_b to represent the LR image, HR image and boundary map, respectively. Given the input 𝐈_l, the objective of our model is to reconstruct a HR image similar to 𝐈_h and predict a boundary map similar to 𝐈_b.The parameter 𝐖 of our model can be divided into 4 disjoint parts, i.e., 𝐖={𝐖_s, 𝐖_h, 𝐖_b, 𝐖_d}, where 𝐖_s and 𝐖_d denote the parameters of content-adaptive interpolation and RCN, respectively. We denote the parameter of feature extraction stage has combined into content-adaptive interpolation part. For BCN, we use 𝐖_h and 𝐖_b to represent the specific weights for generating the intermediate HR image and the boundary maps, respectively. Since the parameters are separable, we propose to train our model in three iterative steps. First, we jointly train content-adaptive interpolation and BCN until their convergence; Second, fixing the parameters of content-adaptive interpolation and BCN, we update the parameters of RCN. Third, we jointly optimize content-adaptive interpolation, BCN and RCN. Specifically, content-adaptive interpolation and BCN are trained according to the following objective function:[ L(𝐈_l,𝐈_h, 𝐈_b, 𝐖)= L_h(𝐈_l,𝐈_h, 𝐖_s, 𝐖_h)+; α· L_b (𝐈_l,𝐈_b, 𝐖_s, 𝐖_b), ]where L_h and L_b represent the HR image reconstruction objective and the boundary prediction objective, respectively. The balance weight α is used to control the importance of L_h and L_b, which is empirically set to 1 in all our experiments. Both L_h and the L_b are in the form of mean squared error (MSE), i.e.,L_h = 1/N∑_i=1^N ( 𝐈_h^i - f_h( 𝐖_s, 𝐖_h, 𝐈_l^i)) ^ 2,and L_b = 1/N∑_i=1^N ( 𝐈_b^i - f_b( 𝐖_s, 𝐖_b, 𝐈_l^i)) ^ 2, where f_h(·) andf_b(·)denote the reconstructed HR image and the predicted boundary map, respectively, i represents the sample index, and N is the number of training triplets. For simplicity, we use 𝐈_ω to denote f_b( 𝐖_s, 𝐖_b, 𝐈_l).Note that when multiple boundary maps are available, there will be more edge prediction objectives.The loss function for training RCN is defined as:L_d =1/N∑_i=1^N(𝐈_h^i-𝐈_ω^i-f_d(𝐖_s,𝐖_d, 𝐈_l^i))^2. Finally, the whole framework is optimized by employing the standard back propagation algorithm, i.e., L = 1/N∑_i=1^N (𝐈_h^i - y)^2,where y, the output of fusion layer, is the final HR image in the testing phase.The whole training phase is summarized as Algorithm <ref>,which accords with the pipeline of our proposed framework in Fig. <ref>. § EXPERIMENTS §.§ Experiment SettingDatasets: All experiments are evaluated on three challenging benchmarks, i.e., Set5 <cit.>, Set14 <cit.> and BSD500 <cit.>. The BSD500 dataset consists of 500 natural images and human annotations for corresponding boundaries. We use the 300 images from its training and validation set for training. The rest of 200 images in BSD500 dataset form a widely used benchmark called BSD200. Besides, the Set5 and Set14 datasets are also adopted as testing sets in other state-of-the-art methods such as <cit.>. Thus, we conduct experiments on the three benchmarks. Implementation details: In the training phase, we first convert the original color image to grayscale image by extracting the luminance component in YCbCr color space. Then, we downscale the training images by requested scaling factors (e.g., 2, 3, and 4) to obtain the LR images. The LR images are cropped into a set of patches with a stride of 4.The size of patches is set to be same as receptive field. The corresponding HR images and boundary maps are cropped with respect to the scaling factors. Before training, we initialize the network parameters by a zero-mean Gaussian distribution with a standard deviation of 1×10^-4.For the pre-training of the proposed model, we use the 91-images <cit.> and PASCAL VOC2012 <cit.> datasets, which totally contain 13,487 clear images. Specifically, the model using LR and HR image pairs is pre-trained following the same strategy as <cit.>. Since the feature extraction stage employ pyramid structure, we speed it up with the help of Factorized CNN <cit.>. In the training on BSD300 dataset, The learning rate of the last layer is set to 1×10^-5, while the rest layers are using a fixed learning rate of 1×10^-4.To increase the number of training samples, we also employ data augmentation for BSD300 dataset, as reported in <cit.>.Methods and metrics: We compare our model with several recent state-of-the-art methods, including a three-layer CNN (SRCNN) <cit.>, super-resolution forest (SRF) <cit.>, sparse coding-based network (SCN) <cit.>, anchored neighborhood regression (A+) <cit.>, shepard interpolation neural network (ShCNN) <cit.>, very deep convolutional network (VDSR) <cit.>, and fast convolutional network for SR (FSRCNN) <cit.>. For fair comparisons, we employ the popular PSNR and SSIM metrics for evaluation. To evaluate the structure-preserving capability, we introduce a new metric called “EPSNR”, which can be formulated as:EPSNR = 10log_10(MAX_I^2/1/|E|∑_i ∈ E(G_i-P_i)^2),where MAX_I=255 is used for 8-bit images, G and P denote the ground-truth and the produced HR images, respectively, E indicates the pixels whose distances to their closest boundary are less than 2 pixels, and i is the pixel index. It is believed that EPSNR can better exploits image fidelity on edge regions.We have also investigated the model complexity from the aspect of parameter number. Two profiles of our model are used, i.e., the common model (denoted as “ours”) used in the above comparisons, and the model with a much deeper architecture (denoted as “deeper ours”). In the “deeper ours” profile, we only increase the convolutional layer number of feature extraction stage from 4 to 18. Thus our model has similar number of parameters compared to VDSR. Both profiles can be accelerated by cuDNN <cit.>. All the CNN-based methods are compared using the Set5 dataset with a scaling factor of 3. The results illustrated in Table <ref> demonstrate that the performance of our model keeps increasing as the parameter number increases. Using comparable network parameters, our model can achieve a PSNR gain of 0.14 dB compared to VDSR.Since fewer parameters can benefit both the training and testing phases, we recommend our model with the common profile. Fig. <ref> illustrates the efficiency of all the compared methods using the “time-quality” diagram. It is demonstrated that our model with common profile runs nearly 2 times faster than VDSR while maintaining the second best SR performance, which is quite suitable for lightweight and fast implementation on consumer-grade devices. For applications that require extremely high SR quality, deeper ours will be a nice choice.Some promising examples are visualized in Fig. <ref> and Fig. <ref>. For better viewing, we interpolate the chrominance components by the bicubic method to generate color images. To clearly demonstrate the difference, we choose onepatch from each image and attach them below. Compared to other methods, our model can produce images with sharper and clearer boundaries. Visual Comparison with SRGAN: We compare our method with the super-resolution generative adversarial network (SRGAN) <cit.>. Because of their proposed adversarial loss, SRGAN has obtained promising performance. However, it still has problems in recovering real details, which is verified by the comparisons shown in Fig. <ref>. It is shown in the enlarged patches of Fig. <ref> (c) and (d) that some waterdrops exist in the ground-truth image disappear, which are produced by SRGAN methods. But these waterdrops are captured by our method and ShCNN.As pointed out in <cit.>, SRGAN tends to bring in similar textures instead of recovering real details.Therefore, our proposed framework performs better than SRGAN on recovering more accurate details. Discussion on real-world cases: To justify the effectiveness of our method, we move one step forward to deal with images from video surveillance and mobile device. Specifically, we apply our model on real-world images with a scaling factor of 3. As reported in Fig. <ref>, “Original” indicates the original images and “Proposed” represent the images processed with our model. As one can observe from results shown in Fig. <ref>, “Proposed” have fewer artifacts compared with “Original”. This demonstrates the robustness of our method towards real-world challenges.§.§ Ablation Study In this subsection, we conduct detailed analyses on the proposed modules, i.e., content-adaptive interpolation, BCN and RCN, for better understanding of our framework.We hope such analysis can lead to new insights into image restoration researches.Content-adaptive interpolation: One of the major differences between our model and SRCNN <cit.> is the employment of the deconvolutional layer. To demonstrate the superiority of our design, we train several fully convolutional networks (FCNs) with various layer numbers for comparisons. Specifically, we increase the number of middle layers from 5 to 16, resulting in FCN-5, FCN-9, FCN-12, and FCN-16. These FCNs follow the bicubic upsampling strategy as in SRCNN <cit.>. Our content-adaptive interpolation consist of 5 convolutional layers and one deconvolutional layer, which contain feature extraction stage, content-adaptive interpolation and BCN. We remove the task of boundary objective to address the effectiveness of content-adaptive interpolation. By comparing content-adaptive interpolation with these FCNs on Set5 dataset with a scaling factor of 3, we obtain the results shown in Table <ref>.It is indicated in these results that although the SR performance of FCN keeps increasing as the network depth increases, it still cannot outperform content-adaptive interpolation even when there are 16 layers. Nevertheless, our content-adaptive interpolation network, which only has 6 layers, surpasses these FCNs by a clear margin. More specifically, content-adaptive interpolation network outperforms FCN-16 by 0.32 dB. This explicitly verifies the superiority of the content-adaptive interpolation.Global Boundary Context: The proposed BCN is motivated by the paradigm of mult-task learning, which incorporates edge estimation as a co-task of HR image generation. Therefore, its analysis is conducted by comparing the SR performance between with and without the edge prediction objective.Since the BSD200 dataset contains manually labeled boundary maps, based on which we can easily compute the EPSNR. We compare two profiles of our model on this dataset with a scaling factor of 3 using both PSNR and EPSNR metrics.By removing the boundary prediction objective, we degrade BCN into single-task learning and denote it as “ours w/o boundary”. As illustrated in Table <ref>, the PSNR and EPSNR gains indicate the benefit of multi-task learning.Because the boundaries only occupy a small portion of the whole image, the improvement on overall PSNR is minor. However, the large improvement on EPSNR verifies the effectiveness of BCN. Another benefit of incorporating boundary prediction objective is the acceleration of training process. As shown in the PSNR curves of Fig. <ref>, the edge prediction objective not only accelerates the convergence, but also contributes to a higher restoration quality.Local Residue Context: We design RCN to provide complementary information for image SR. Therefore, the SR performance of our model will be degraded if RCN is removed. To verify our statement, we use another profile named “ours w/o RCN”, which is very similar to the previous version of this work <cit.>,to conduct more comparisons on the aforementioned datasets with a scaling factor of 3. Table <ref> reports the comparison results. It is shown that, although content-adaptive interpolation and BCN can produce HR image of high quality, the SR performance can still be further improved.The improvement on PSNR is minor because PSNR is a squared error-based metric, which is difficult to reveal subtle structure differences. In contrast, because SSIM concentrates on structure similarity, the improvement on SSIM is more significant.§ CONCLUSION AND FUTURE WORK In this paper, to address single image super-resolution, we have proposed a novel contextualized multi-task deep learning framework. Our neural network model incorporates global boundary context and residual context to super-resolve images while well preserving their structural details. Moreover, we have introduced “content-adaptive interpolation", which leverages a set of filters that are adaptive to the training samples. Different from the kernel estimation in blind image SR which usually employs only a single filter, our proposed content-adaptive interpolation has more filtering parameters and better convenience of being embedded into CNNs. Our extensive experiments suggest that the proposed method outperforms other leading image super-resolution approaches, and achieves state-of-the-art performances on both popular evaluation metrics and visual quality comparison. There are several directions to extend our method. First, we are considering to introduce a perceptual loss into the multi-task optimization, aiming to better capture realistic and meaningful image details. Second, we shall generalize this framework to adapt to video data by taking spatio-temporal coherency into consideration. Third, considering that additional common knowledge in deep neural networks would be an interesting trial, we intend to utilize complementary spatial-temporal contexts as privileged information for video SR, as suggested by Yang et al. <cit.>. § ACKNOWLEDGEMENTSThis work is partially supported by NSFC (No. 61602533), The Fundamental Research Funds for the Central Universities, in part by Hong Kong Scholars Program and Hong Kong Polytechnic University Mainland University Joint Supervision Scheme. We are grateful to acknowledge NVIDIA for GPU donations. IEEEbib [ < g r a p h i c s > ]Yukai Shi received his B.E. degree from Heilongjiang University. He is currently working toward the Ph.D. degree with the School of Data and Computer Science, Sun Yat-Sen University. His research interests including computer vision and machine learning. [ < g r a p h i c s > ]Keze Wang received his B.S. degree in software engineering from Sun Yat-Sen University, Guangzhou, China, in 2012. He is currently pursuing the dual Ph.D. degree at Sun Yat-Sen University and Hong Kong Polytechnic University, advised by Prof. Liang Lin and Lei Zhang . His current research interests include computer vision and machine learning. More information can be found in his personal website http://kezewang.com[ < g r a p h i c s > ]Chongyu Chen is a distinguished associate research fellow of Sun Yat-sen University. He received his B.S. and Ph.D. degrees from Xidian University, Xi'an, China, in 2008 and 2014, respectively. From 2015 to 2017, he was a post-doctoral fellow at the Hong Kong Polytechnic University. His research interests include image restoration, 3D computer vision, signal separation, etc.[ < g r a p h i c s > ]Li Xu received the BS and MS degrees in computer science and engineering from Shanghai JiaoTong University (SJTU) in 2004 and 2007, respectively, and the PhD degree in computer science and engineering from the Chinese University of Hong Kong (CUHK) in 2010. He joined Lenovo R & T Hong Kong in Aug 2013, where he leads the imaging & sensing group in the Image & Visual Computing (IVC) Lab. Li received the Microsoft Research Asia Fellowship Award in 2008 and the best paper award of NPAR 2012. His major research areas include motion estimation, motion deblurring, image/video analysis and enhancement. He is a member of the IEEE. [ < g r a p h i c s > ]Liang Lin is a full Professor of Sun Yat-sen University. He received his B.S. and Ph.D. degrees from the Beijing Institute of Technology (BIT), Beijing, China, in 2003 and 2008, respectively, and was a joint Ph.D. student with the Department of Statistics, University of California, Los Angeles (UCLA). From 2008 to 2010, he was a Post-Doctoral Fellow at UCLA. From 2014 to 2015, as a senior visiting scholar he was with The Hong Kong Polytechnic University and The Chinese University of Hong Kong. His research interests include Computer Vision, Data Analysis and Mining, and Intelligent Robotic Systems, etc. He has authorized and co-authorized on more than 100 papers in top-tier academic journals and conferences. He has been serving as an associate editor of IEEE Trans. Human-Machine Systems. He was the recipient of the Best Paper Runners-Up Award in ACM NPAR 2010, Google Faculty Award in 2012, Best Student Paper Award in IEEE ICME 2014, Hong Kong Scholars Award in 2014 and The World's First 10K Best Paper Diamond Award in IEEE ICME 2017. More information can be found in his group website http://hcp.sysu.edu.cn. | http://arxiv.org/abs/1707.08340v1 | {
"authors": [
"Yukai Shi",
"Keze Wang",
"Chongyu Chen",
"Li Xu",
"Liang Lin"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170726094803",
"title": "Structure-Preserving Image Super-resolution via Contextualized Multi-task Learning"
} |
IV. XMM-Newton and NuSTAR spectra dominated by twotemperature (warm, hot) Comptonization processes.Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, [email protected] Group, School of Physical & GeographicalSciences, Keele University, Keele ST5 5BG, UKCSST, University of Maryland Baltimore County, 1000Hilltop Circle, Baltimore, MD 21250, USA Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, via della Vasca Navale 84, 00146, Roma, Italy INAF-Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125, Firenze, ItalyINAF-Osservatorio Astronomico di Brera, via Bianchi 46, 23807, Merate (LC), ItalyCenter for Relativistic Astrophysics, School of Physics,Georgia Institute of Technology, 837 State Street, Atlanta,GA 30332-0430, USA Space Sciences Laboratory, University of California,Berkeley, CA 94720, USADTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, DK-2800 Lyngby, DenmarkDr Karl Remeis-Observatory and Erlangen Centre for Astroparticle Physics, Sternwartstr. 7, D-96049 Bamberg, Germany Department of Physics, Virginia Tech, Blacksburg, VA 24061, USACahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USAHarvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Columbia Astrophysics Laboratory, Columbia University, New York, New York 10027, USAJet Propulsion Laboratory, California Institute ofTechnology, Pasadena, CA 91109, USAINAF-IASF Bologna, Via Gobetti 101, I-40129 Bologna, Italy NASA Goddard Space Flight Center, Code 662, Greenbelt, MD 20771, USA [Abstract Truncated] We perform an X-ray spectral analysis ofthe brightest and cleanest bare AGN known so far, Ark 120, in order to determine the process(es) at work in thevicinity of the SMBH. We present spectral analysis of data from an extensivecampaign observing Ark 120 in X-rays with XMM-Newton (4×120 ks, 2014 March 18–24), and NuSTAR (65.5 ks, 2014 March 22). During this very deep X-ray campaign, the source was caught in ahigh flux state similar to the earlier 2003 XMM-Newton observation, and about twice as bright as the lower-flux observation in 2013. The spectral analysis confirms the “softer when brighter” behaviour of Ark 120.The four XMM-Newton/pn spectra are characterized by the presence of a prominent soft X-ray excess and a significant Fe Kα complex. The continuum is very similar above about 3 keV, while significant variability is present for the soft X-ray excess. We find that relativistic reflection from a constant-density, flat accretion disk cannot simultaneously produce the soft excess, broad Fe Kα complex, and hard X-ray excess.Instead, Comptonization reproduces the broadband (0.3–79 keV) continuum well, together with a contribution from a mildly relativistic disk reflection spectrum.During this 2014 observational campaign, the soft X-ray spectrum of Ark 120 below ∼0.5 keV was found to be dominated by Comptonization of seed photons from the disk by a warm (kT_ e∼0.5 keV), optically-thick corona (τ∼9).Above this energy, the X-ray spectrum becomes dominated by Comptonization from electrons in a hot optically thin corona, while the broad Fe Kα line and the mild Compton hump result from reflection off the disk at several tens of gravitational radii. A deep X-ray view of the bare AGN Ark120. D. Porquet 1J. N. Reeves2,3G. Matt4 A. Marinucci4 E. Nardini5 V. Braito6 A. Lobban2D. R. Ballantyne7S. E. Boggs8F. E. Christensen 9T. Dauser10D. Farrah11 J. Garcia12,10,13C. J. Hailey14F. Harrison12D. Stern15A. Tortosa4 F. Ursini16 W. W. Zhang17 Received , 2017; accepted , 2017 ====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION In the standard picture, the emission of an active galactic nucleus (AGN) stems from an accretion disk around a supermassive black hole (SMBH) with mass spanning from a few millions to billions of solar masses. X-ray spectra offer a unique potential to probe matter very close to the black hole and to measure the black hole spin.The X-ray spectra of AGN usually exhibit one or more of the following components: a soft excess below 2 keV, a power-law continuum up to about 10 keV,a Fe Kα line complex near 6.4 keV, and a Compton scattering humpnear 20–30 keV. Recent studies using XMM-Newton have shown that the soft X-ray excess component is commonly seen in AGN,and that for most AGN (the exception possibly being low-mass Narrow Line Seyfert 1 galaxies; ) this soft excess is not the hard tail of the big blue bumpobserved in the UV coming from blackbody emission of the accretion disk <cit.>.Different explanations have been proposed:e.g., photo-ionized emission blurred by relativistic motion in the accretion disk <cit.>; Comptonization of soft (extreme UV) seed photons by the electrons of a corona above the disk <cit.>; and an artifact of strong, relativistically smeared, partiallyionized absorption <cit.>.These models can give a good representation of the soft excess, though the current simulations of line-driven AGN accretion disk winds cannot reproduce the smooth, soft X-ray excess <cit.>. Therefore, the origin of the soft excess is still anopen issue. Besides, the study of the Fe Kα complex emission can be usedto provide fundamental diagnostics of the physical and dynamical conditions of the AGN central engine from the inner part of the accretion disk to the far-away molecular torus. Relativistic (or broad) Fe Kα lines have been studiedin numerous AGN <cit.>; especially MCG-06-30-15 <cit.>, andother particular AGN thanks notably to XMM-Newton and/or NuSTAR data(NGC 1365:and ; Mrk 335: ;and SWIFT J2127.4+5654: ).Yet, the physical interpretation of the observed broad Fe Kα lines has been disputed due to the common presence of a warm absorber <cit.>,which has been proposed to explain in part the broadness of the lines<cit.>. Indeed, the presence of this hot gas on the line-of-sight couldseverely complexify the X-ray data analysis,in particular by distortingthe underlying continuum of the Fe K line.However, arguments against this interpretation have been invoked by <cit.>. Likewise, several origins for the hard X-ray spectrum above 10 keVfor type 1 Seyferts have been proposed, such as relativistic reflection, complex absorption, and Comptonization, or a combination thereof<cit.>. Therefore, AGN showing no (or very weak) presence of any X-ray warm absorber – so-called “bare AGN” – are the best targets to directly investigate the process(es) at work in the vicinity of SMBHs. Ark 120 (z=0.033, M_ BH=1.50±0.19×10^8 M_⊙[Black hole mass determined via reverberation mapping <cit.>.])is the brightest and cleanest bare AGN known so far, i.e. displaying neither intrinsic reddening in its IR continuumnor evidence for absorption in UV and X-rays<cit.>.The first XMM-Newton observation performed in August 2003 (∼80 ks pn net exposure time) confirmed that its spectrum is warm absorption-free, with the smallest upper limit to the column density found among AGN (∼3×10^19 cm^-2; ).Ark 120 thereforerepresents the best target to have the “purest” view of the properties of the accretion disk and of the black hole spin in AGN.This object displays a prominent soft excess observed down to 0.3 keV and a significantFe Kα line complex <cit.>. The width of the broad component of the Fe Kα line (FWHM∼30 000 km s^-1) was much larger than that of the broad optical lines from the broad line region (BLR), with FWHM(H_β)=5 850±480 km s^-1 (). Ark 120 wasalso observed with Suzaku in April 2007 (∼100 ks) confirming the presence of a large soft excess and of a significant broad Fe Kα line<cit.>.Recently, <cit.> reported the first spectral analysis of a simultaneous XMM-Newton and NuSTAR observation performed in February 2013. They found that the smooth soft excess was more likely explained by Comptonization.In contrast with the 2003 XMM-Newton observation <cit.> and the 2007 Suzaku observation <cit.>, while a significant soft excess was present too,no obvious signature for relativistic reflection was found in the 2013 observation. It is worthmentioning that in 2013 the flux of Ark 120 was lower by abouta factor of two than during the 2003 observation.Both this lower flux and the lack of any relativistic signature in this2013 observation (“low-flux state”) may be explained by the presence of an extendedoptically-thick corona which hides most of the relativitic reflection from theaccretion disk, while in 2003 (“high-flux state” ) and in 2007 (“intermediate-flux state”) this corona was likely less thick and/or less extended <cit.>. An extensive X-ray observational campaign was performed from 2014 March 18 to March 24 to study Ark 120 in orderto directly probe the accretion disk properties and the SMBHspin in this moderate Eddington ratio AGN (Ṁ≳0.05 Ṁ_ Edd; ). This campaign combined a deep XMM-Newton observation (480 kssplit into four consecutive 120 ks observations from 2014 March 18 to March 24; PI: D. Porquet)with a simultaneous 120 ks Chandra/HETG observation[This was the first Chandra observation of Ark 120. The observation was split in threeconsecutive sequences as described in Reeves et al. (2016) and Nardini et al. (2016).] (PI: D. Porquet). Furthermore, a NuSTAR observation (65 ks; PI: NuSTAR AGN team) was performed during the third XMM-Newtonobservation, i.e., on 2014 March 22. In <cit.> (hereafter Paper I), we reported on the analysis of the soft X-ray spectrum using the480 ks XMM-Newton/RGS and 120 ks Chandra/HETG spectra.We confirmed that there were no detectable absorption lines due to the warm absorber in the deep RGS spectra,and that Ark 120 is the cleanest bare nucleus AGN known so far.Only absorption lines from the interstellar medium of our Galaxy were found.Interestingly, several soft X-ray emission linesfrom the He-like and H-like ions of N, O, Ne and Mg were revealed for the first timethanks to this very high signal-to-noise (S/N) RGS spectrum.As a consequence, Ark 120 is not intrinsically bare since substantial X-ray emitting gas is present out of the direct line of sight towards this AGN. This result is very important in the framework of the unified scheme of AGN, which invokes the existence of wide scale obscuring and emitting gas <cit.>. In <cit.> (hereafter Paper II)we took advantage of the unprecedented depth of the new data sets tostudy the properties of the composite emission complex due to iron fluorescence at6–7 keV. The most prominent feature peaks around 6.4 keV, and can be plainlyidentified with the Kα transition from neutral iron. The profile of the narrow Fe Kα core is resolved in the Chandra/HETG spectrum with a FWHM of 4 700^+2700_-1500 km s^-1, consistent with a BLR origin (as was also found for the soft X-ray emissionlines; Paper I). Excess components are systematically detected redwards (6.0–6.3 keV) andbluewards (6.5–7.0 keV) of the narrow Fe Kα core. The energy and equivalent widthof the red wing rule out an interpretation in the form of the Compton shoulder of the 6.4-keV Kα feature. Moreover, its variability over timescales of about one year (February 2013 to March 2014) hints at the presence of an emission component from the accretion disk. Excess emission maps and time-resolved spectra based on the four consecutive orbits of XMM-Newton monitoring show that both the red and blue features arehighly variable on short timescales (30–50 ks) but appear to be disconnected.Such a timescale suggests an origin for these two components at a few tens of gravitational radii from the central supermassive black hole, potentially from discrete hot spots on the disk surface.In <cit.> (submitted; hereafter Paper III), we presented the spectral/timing properties of Ark 120 using all available XMM-Newton data (including this XMM-Newton Large Programme),a recent ∼6-month Swift monitoringcampaign <cit.>, and data from RXTE obtained between1998 and 2006.The spectral decomposition was investigated through fractional rms, covariance and difference spectra, where we found that the mid- to long-timescale (∼day-year) variability is dominated by a relatively smooth, steep component, which peaks in the soft X-ray band.Additionally, we found evidence for a variable component of Fe Kα emission on the red side of the near-neutral Fe Kα core on long timescales, consistent with Paper II.We also measured the power spectrum and searched for frequency-dependent Fourier lags, obtaining the first detection of a high-frequency soft X-ray lag in this source. Finally, we found well-correlated optical/UV/X-ray variations with the Swift UVOT and searched for multi-wavelength time delays, finding evidence for the optical emission lagging behind the X-rays with a time delay of τ = 2.4 ±1.8 days. In this Paper IV, we report on the X-ray spectral analysisof the four 120 ks XMM-Newton/pn time-averaged spectra performed in March 2014,which represent the deepest and longest elapsed time X-ray observation for a bare AGN. We also report on a NuSTAR observation that was simultaneous with the third 2014 XMM-Newton observation (see Table <ref> for details).In section <ref>, we describe the observations, thedata reduction, and the spectral analysis method.The spectral analysis of the four 2014 XMM-Newton/pn spectra is presented in section <ref>,and the broad-band X-ray spectrum (XMM-Newton and NuSTAR) of the 2014 March 22 observation in section <ref>.In section <ref> our main results are summarized, before the discussion and conclusions in section <ref>. § OBSERVATIONS,DATA REDUCTION AND ANALYSIS§.§ XMM-Newton and NuSTAR data reductionDuring this observational campaign Ark 120 was observed by XMM-Newton over four consecutive orbits between 2014 March 18 and March 24(Table <ref>). As reported in Paper II, the event files were reprocessed with the Science Analysis System (SAS) v14.0, applying the latest calibrations available in 2015 February. Due to the high source brightness, the EPIC instruments were operated in Small Window mode.However, this observation mode was not sufficient to prevent pile-up in the MOS cameras, and therefore only the EPIC/pn <cit.>data are taken into account (selecting the event patterns 0-4, i.e., single and double pixels). The four pn spectra were extractedfrom a circular region centered on Ark 120, with a radius of 30 to avoid the edge of the chip. The background spectra were extracted from a rectangular region in the lower part of the small window that contains no (or negligible) source photon.The latest part of each orbit was not used due to high background flaring level.After the correction for dead time and background flaring, the total net exposure was about 330 ks. Redistribution matrices and ancillary response files for the four pn spectra were generated with the SAS tasks rmfgen and arfgen. As shown in Paper III, there is a significant flux variability during each of the four XMM-Newton observations and between them. However, the spectral variability within any single orbit is slow and moderate, so we areable to use time-averaged spectra for each of the four observations.As detailed in Paper II, a gain shift has to be applied to take into account the known inaccuracy of the EPIC/pn energy scale likely due to inaccuracies in the long-term charge transfer (CTI) calibration[See http://xmm2.esac.esa.int/docs/documents/CAL-SRN-0300-1-0.pdf and <cit.>.]. The corresponding values for the xspec gain function are reported in Paper II. The 0.3–10 keV pn spectra were binned to give 50 counts per bin.NuSTAR <cit.> observed Ark 120 with its two co-aligned X-ray telescopes with corresponding Focal Plane Modules A(FPMA) and B (FPMB) starting on 2014 March 22 for a total of ∼131 ks of elapsed time.The Level 1 data products were processed with the NuSTAR Data Analysis Software (NuSTARDAS) package (v. 1.6.0). Cleaned event files (level 2 data products) were produced and calibrated using standard filtering criteria with the nupipeline task and the calibration files available in the NuSTAR calibration database (CALDB: 20170222). Extraction radii for both the source and the background spectra were 1.25 arcmin. After this process, the net exposure time for the observation was about 65 ks, with most of the time lost to Earth occultations. The pair of 3.5–79 keV NuSTAR spectra were binned in order to over-sample the instrumental resolution by at least a factor of 2.5 and to have a S/N ratio greater than 5 in each spectral channel.The summary log of the Ark 120 X-ray observations used in this work from the 2014 observational campaign are reported in Table <ref>. §.§ Spectral analysis method The xspec v12.9.0g software package <cit.> was used for the spectral analysis.The Galactic column density is assumed to be N_ H=9.78×10^20 cm^-2as inferred from the weighted average N_ H value of the LAB Survey of Galactic Hi <cit.>. Since there can be some additional contribution associated with molecular hydrogen <cit.>, we allow the value of Galactic N_ H to vary slightly (except for the fits above 3 keV, where the value is fixed to 9.78×10^20 cm^-2).However, we do not allow for any intrinsic absorption in the rest frame ofArk 120, since, as found in Paper I from the deep RGS spectrum, none is observed.We used the X-ray absorption model tbnew (v2.3.2) from <cit.>, assuming throughout their interstellar medium (ISM) elemental abundances and the cross-sections from <cit.>. As reported in Paper II, the narrow neutral core of the Fe Kα emission complex is consistent with being associated with the BLR, and so is somecontribution to the H-like line of iron. Consequently, throughout this workwe take into account the contribution from the BLR to the Fe K complex using three Gaussian lines:the Fe Kα_ BLR (E fixed at 6.40 keV) plus itsassociated Fe Kβ_ BLR line (E fixed at 7.05 keV), and the H-like iron line (E fixed at6.97 keV).The normalization of Fe Kβ_ BLR is set to 0.135 timesthat of Fe Kα_ BLR <cit.>.The widths of these three lines are fixed to the value inferred in Paper II for the Fe Kα narrow core, i.e., 43 eV, as determined from the simultaneous Chandra/HETG spectrum (Paper II).These three BLR emission lines are called hereafter “3 zgaussians(BLR)”.Throughout this work, when considering relativistic reflection modelling, we use the relxill package(v0.4c[This package version is more recent than the one used in Paper II (v0.4a), however the fit differences are negligible.]; released in May 2016).These reflection models calculate the proper emission angle of the radiation at each point on the accretion disk, and then take the corresponding reflection spectrum into account. This angle-dependentmodel connects self-consistently the xillver<cit.> reflection models with the relativistic blurring code relline <cit.>.In this version 0.4c the cosmological redshift is properly taken into accountin the cut-off energy as well.The models used in this work assume a constant density profile of the accretion disk with n_ e=10^15 cm^-3 (except when mentioned otherwise, see <ref>), and an exponentially broken power-law as the intrinsic continuum shape.Two main geometries are possible: the coronal one (relxill) and the lamppost one (relxilllp). Detailed descriptions of these models and their corresponding parameters are reported in Appendix <ref>. We use χ^2 minimization throughout, quoting 90 percent errors for one interesting parameter (Δχ^2=2.71) unless otherwise stated. Default values of H_ 0=70 km s^-1 Mpc^-1, Ω_ m=0.27, and Ω_Λ=0.73 are assumed.All figures are displayed in the AGN rest-frame. § SPECTRAL ANALYSIS OF THE FOUR 2014 XMM-NEWTON PN OBSERVATIONSIn order to characterize the main X-ray components of the spectra, we fit the four XMM-Newton/pn spectra between 3–5 keV using a simple absorbed power-law model. The absorption column density has been fixed to the Galactic one, i.e. 9.78×10^20 cm^-2.The power-law index is tied between the four spectra, while the normalization is allowed to vary between the observations (χ^2/d.o.f.=1656.8/1591).We find a photon index of 1.87±0.02, which is typical of those found for radio-quiet quasars <cit.>. The unabsorbed flux values between 3 and 5 keV are about 1.8-2.0×10^-11 erg cm^-2 s^-1and are similar to that found forthe 2003 XMM-Newton observation, and about twice as bright as the 2013 observation.Then, we extrapolate over the whole 0.3–10 keV energy range.As illustrated in Fig. <ref> (top panel), the four observations show a significantsoft X-ray excess below 2 keV that is variable between the observations (see also Paper III) with the first observation (2014 March 18) exhibiting the largest soft X-ray excess. The prominent Fe K line profile is consistently seen in each of the observations (Fig. <ref>, bottom panel), though as shown in Paper II the red and blue sides of the Fe Kα complex are variable on time-scale of about 10-15 hours, i.e faster than the total duration of each observation.§.§ Spectral analysis above 3 keV In this section, we aim to characterize the disk component(s) above 3 keV, i.e., without any bias from the soft excess contribution. This has been already investigated in Paper II, but here more general relativistic reflection configurations are probed (e.g, non-solar iron abundance, free inclination angle).We use the baseline reflection model 𝒜 defined as tbnew×[relxill+3×zgaussians(BLR)]. The photon index (Γ), the reflection fraction (ℛ), and thenormalization (norm) of the underlying continuum of therelativistic reflection componentare allowed to vary between each observation.The high-energy cut-off (E_ cut) is fixed to 1000 keV since it cannot be constrained from the pn energy range. We first consider a coronal geometry assuming a singlepower-law disk emissivity index q (ϵ∝ R^-q), tiedbetween the four observations, and fixing the inner radius of the reflection component (R_ in) atthe innermost stable circular orbit (ISCO) which is self consistently determined from the spin value in the relxill models.We find a good fit (χ^2/d.o.f.=4461.7/4568) and infer a very flat emissivity index of ≤1.1 (see Table <ref>).The photon indices (Γ∼1.85–1.92) are typical of what is found in type-1 AGN<cit.>.The extrapolation of the fit down to0.3 keV shows that the soft excess is not accounted for by this model, which leaves a large positive residual below 2 keV (Fig. <ref>).If, instead, we fix the disk emissivity index to the standard value of 3, the spin value to 0, and allow R_ in (expressed in R_ ISCO units) to vary, we also find a good fit in the 3–10 keV energy range and derive R_ in= 17.8^+32.7_-8.6 R_ ISCO (see Table <ref>). If we fix the spin value to the maximal ones, we find R_ in≥56 R_ ISCO and R_ in=11.9^+22.3_-5.6 R_ ISCO, for a=0.998 and a=-0.998, respectively.This suggests that whatever the spin value is, the reflectioncomponent does not arise in the veryinner part of the accretion disk, in agreement with the results discussed in Paper II. The extrapolation of the fit down to 0.3 keV shows that the soft excess is not accounted for, with a similarly large positive residual.The reflection fraction is found to be rather low, with ℛ=0.4–0.5in both cases. Then we adopt a lamppost geometry[The “lamppost” model represents a X-ray source geometry in AGN,where the primary source of radiation is modelled by a point source on the rotational axis of the black hole.]using the relxilllp model. The reflection fraction is calculated directly from the lamppost geometry with the parameter fixReflFrac fixed to 1.We find that the height of the X-ray source above the accretion disk is rather high (h=93^+29_-25 R_ g, see Table <ref>).This suggests that the disk illumination is not centrally concentrated. As for the coronal geometry, a strong positive residual is found below 3 keV when the best fit is extrapolated down to 0.3 keV. To summarize, good fits are found using a combination of a primary power law and a relativistic reflection component when considering the 3–10 keV energy range,but they indicate that reflection does not arise in the very inner partof the accretion disk (flat disk emissivity index, large R_ in or h),and has a moderate reflection fraction of about 0.4–0.5 (coronal geometry). These results are similar to those found in Paper II.Moreover, in all cases the softexcess is not accounted for, meaning that the soft excess mayoriginate from another physical process. §.§ Investigation of relativistic reflection modelling over the 0.3–10 keV energy range We now consider the full 0.3–10 keV energy rangeto investigate whether reflection modelscan after all reproduce both the soft excess and the hard X-ray emission up to 10 keV. We use model 𝒜, but here we allow for a broken power-law disk emissivity index (q_1, q_2, and R_ br).We find a good fit statistic (χ^2/d.o.f.=7246.6/6728, χ^2_ red = 1.08) over the 0.3–10 keV energy range (Fig. <ref>, top panel), though there are positive residuals in the Fe K complex energy range (χ^2_ red = 1.16 when considering only the 6–7 keV energy range). Since the fit is driven by the smooth soft X-ray emission, very high values for the spin and the disk emissivity indices, with a low inclination angle, are required to reproduce it (see Table <ref>).Large values (R∼7–10) for the reflection fraction are required as well. This would correspond to a scenario of a compact corona very close to the black hole, leading to strong gravitational light bending <cit.>.Moreover the primary photon index needed to reproduce the soft X-rayexcess is much steeper (Γ∼2.4) than that associated to the Fe Kα features when fitting above 3 keV (Γ∼1.9).These extreme parameters are incompatible with those found to reproduce the 3–10 keV spectra (see Table <ref>).Moreover, as shown in Figure <ref> (top panel), there are stillpositive residuals with moderately broad line profiles from about 6–6.3keV (red wing) and 6.6–6.9 keV (blue wing)in all the four spectra (AGN rest-frame), which cannot be reproduced by the extreme and very fine-tuned parameter values required to reproduce the smooth soft excess.The two emission features correspond to the red and blue Fe Kα features reported in Paper II, where the energy–time map showed that they both arisefrom the accretion disk.We therefore add two relativistic line components (relline, ) that enable us to model these features for a small inclination angle[We noticethat we find similar results considering a single relativistic line, but in such case a larger inclination angle of about 30 degrees is required (see Paper II, and Table <ref>).] as found for the soft excess (see Table <ref>). The inclination angle and the spin values are linked to those of the broadband blurred reflection component, but we allow the disk emissivity index to vary (R_ in is fixed to the ISCO). We find a statistical improvement of the fit compared to the baseline model 𝒜(Δχ^2∼-238for five additional parameters).We infer line energies of 6.47±0.01 keV and 6.76^+0.02_-0.03 keV, and a disk emissivity index of 1.6^+0.2_-0.1, much flatter than the ones found for the baseline reflection model 𝒜, i.e., 7.5±0.5 and 4.5±0.4 for the 2014 March 18 observation (similar values are found for the three other sequences). If, instead, we force the disk emissivity index to be the same of theblurred component and allow the inner radius to vary, we also find a good fit(Δχ^2∼-240 for five additional parameters) with R_ in=25.7^+5.0_-4.3 R_ g and lineenergies[(p) means that the value ispegged at the maximum or minimum ones allowed by the model.] of 6.45^+0.10_-0.05(p) keV and 6.97^+0.00(p)_-0.05 keV.Hence, in both cases, a flat disk emissivity index (q∼1.6) or a large inner radius (R_ in∼25 R_ g) arein conflict with the values required to account for the soft excess. This implies that this reflection model cannot self-consistently produce both the soft excess and the mildly relativistic Fe Kα line(s). We also notice that, in order to reproduce these Fe Kαfeatures by means of another broadband reflection component (relxill), not only q (or R_ in) and the disk inclination must be untied, but also the Γ value, whichmust be much harder, i.e. below about 1.9. So it is worth investigating whether these accretion disk features(soft excess and Fe Kα residuals) can originate from a more complex disk emissivityshape such as a twice broken power-law, which allows for an intermediate regionwith a flat emissivity index as could be found for aco-rotating continuum point source or an extended corona (see Figure 7 in ). However, even with this reflection model we cannot simultaneously reproduce these emission features. A disk ionization gradient (log ξ ∝ R^-index) does not work either. The inferred parameters for these fits are reported in Table <ref>. Then, we test a lamppost geometry, but a less good fit is found compared to the extended corona model (relxill), and significant residuals near 6.35 keV and 6.7 keV are still present and cannot be removed by allowing foran ionization gradient (assuming an α accretion disk ora powerlaw density profile (density ∝ R^ -index)).See Table <ref> for the inferred fit parameters.In conclusion, these high S/N spectra of Ark 120 show that relativisticreflection models from a constant-density, geometrically flat accretion disk,while leading to reasonable χ^2/d.o.f value over 0.3–10 keV, arephysically unsatisfactory, since they cannot simultaneously account for both the soft excess and the Fe Kα lines. Note that such residual Fe K features could have been missed or readily neglected in lower S/N spectra, thus leadingto an interpretation of a relativistic reflection-dominated spectrum, with rather extreme and tuned parameters in terms of spin and emissivity.§.§ Comptonization process as the origin of the soft excess In this section, we assume that the soft X-ray excess originates from the Comptonization of seed photons from the accretion disk by warm electrons from the corona (here using comptt), as found by <cit.> for the February 2013observation. For this, we use the baseline model ℬ defined astbnew×[comptt+zpo+3×zgaussians(BLR)].The comptt model <cit.> is characterized by the input soft photon temperature (expressed in keV), the plasma temperature (kT_ e expressed in keV), the plasma optical depth (τ) and the geometry assumed (disk, sphere, analyticalapproximation). We assume a disk geometry, andan input soft photon temperature[We would like to notice that the mean value of the input soft photon temperature (around 15 eV) has a negligible impact on the plasma temperature and optical depth values.] of 15 eV according to the black hole mass and the mean accretion rate of Ark 120. The power-law component (zpo) is used to readily reproduce Comptonization by the hot electrons of the corona.We obtain an overall good fit with χ^2/d.o.f.=7384.1/6735(χ^2_ red=1.10) (see Table <ref>, column 2), except for the positive residuals atthe Fe Kα complex energy range (see below and Fig. <ref> top panel), as already found in the previous reflection-based models.We derive for the Comptonized plasma low temperature values of about 0.6–0.8 keV, and high optical depth values of about 7–8for the four observations (see the mean values and their associated errors bars for each observation in Table <ref>).For illustration purpose, we display in Fig. <ref> the 2D contour plot of the plasma temperature (kT_ e in keV) versus the plasma optical depth for the third XMM-Newton observation. The temperature value of the four observations seems to follow the soft excess strength, but the values between the four observations are consistent with each other within their errors bars. The primary photon indices of about1.74–1.79 are much harder than those required to producethe soft excess from relativistic reflection modelling (Γ∼2.4). Not surprisingly with such a power-law continuum shape above 3 keV, the same red and blue components of Fe Kα residuals are still present.Indeed, they are known to be associated with the accretion disk (Paper II).Thanks to the high S/N of the present spectra and the ability to detectunambiguously these red and blue emission features,we are thus able to establish that even in this scenario,where the soft excess originates from Comptonization, a relativistic reflection component is still required.We therefore add a relativistic line profile (relline), fixing itsemissivity index to the standard value of 3 and the inclination angle to30 degrees (see Paper II). If R_ in is allowed to vary, then the fit is significantly improved(see Table <ref>, column 3), and R_ in=45.3^+13.1_-11.0 R_ g is inferred.The relative contribution of the different model components is displayed in Fig. <ref> (unfolded spectrum for illustration purposes only; bottom panel), and showsthat Comptonization of seed disk photons by warm electrons of the corona is the dominant process below about 0.8 keV. In summary, the spectral analysis shows that Comptonization by a low temperature (kT_ e∼0.6–0.8 keV) optically-thick (τ∼7–8) corona can reproduce well the soft excess below 1 keV. Above this energy, the power law component dominates the continuum, which may represent Comptonization by the hot electrons in a thin corona <cit.>. We note that a contribution from disc reflection originating at tens of R_ g is still required to account for a broad component of the FeK line, which was not the case for the low flux 2013 observation of Ark 120 <cit.>.§ BROAD-BAND X-RAY VIEW OF ARK 120 OBSERVED ON 2014 MARCH 22In this section, we investigate the simultaneous broadband XMM-Newton/pn andNuSTAR observations of Ark 120 performed on 2014 March 22.First, we investigate the two NuSTAR spectra (FPMA and FPMB) by fitting them using a simple absorbed power-law model in the 3–10 keV energyrange excluding the 5.5–7.5 keV energy range. We fix the absorption column density to 9.78×10^20 cm^-2, and tie the power-law parameters between both NuSTAR spectra. We allow for possible cross-calibration uncertainties between these two NuSTAR spectra. The resulting data/model ratio extrapolated up to 79 keV is reported in Fig. <ref>, where a significant hard X-ray excess is present in addition to the prominent Fe Kα complex.§.§ Investigation of the relativistic reflection scenario §.§.§ Spectral analysis above 3 keVWe start the simultaneous fit of the XMM-Newton/pn andof the two NuSTAR spectra in the 3–10 keV bandusing model 𝒜. We allow for cross-calibration uncertainties between the two NuSTAR spectra and the XMM-Newton/pn spectrum by including in the fit a cross-normalization constant corresponding toC_ NuSTAR A and C_ NuSTAR B for NuSTAR FPMA and FPMB spectra, respectively (see values in Table <ref>), related to the XMM-Newton/pn spectrum. The absorption column density has been fixed to the Galactic one, i.e. 9.78×10^20 cm^-2.The fit parameters are similar (see Table <ref>, column 2) to those found when fitting simultaneously the four XMM-Newton/pn spectra over this energy range (see Table <ref>), showing a good match between XMM-Newton/pn and NuSTAR data. Only the inferred disk emissivity index q is larger and the inclination angle is lowerdue to the lower spectral resolution of the NuSTARcamera which broadens the apparent Fe K profile.We then extrapolate this fit up to 79 keV, and find that the data/model ratio is rather good, although the model slightly overpredicts the emission in the 10–40 keV energy range (see Fig. <ref>, top panel). The χ^2/d.o.f. is 1698.1/1519 (Δχ^2_ red=1.12) without any refitting. Now, we refit over the entire3–79 keV energy range and find a very good fit (Fig. <ref>, bottom panel)with parameter values that arevery similar to those found for the 3–10 keV energy range, except for the iron abundance which has increased from ≤0.8 to 3.3^+1.5_-1.3 in order to better adjust the 10–40 keV emission (see Table <ref>). We find an upper limit of 0.6 for the spin value (at 90% confidence level), but we note that after the calculation of a 2D contour plot of the spin versus the inclination angle, the spin is actually unconstrained at the 90% confidence level.A power-law photon index of about 1.86represents well the underlying continuum over the 3–79 keV energy range. This value is softer than that found for the 2013 observation (Γ∼1.73), when the source was in a low-flux state <cit.>. Adding the possible contribution from a molecular torus[Forthis, we use the unblurred reflection xillver model fixinglog ξ to zero. The inclination of the blurred and unblurredreflection components, as well as their cut-off energies, are tied together,and solar iron abundance for the unblurred reflectioncomponent is assumed. ]does not improve the fit at all. As illustrated in Fig. <ref> (top panel), the extrapolation of the XMM-Newton/pn spectrum down to 0.3 keV shows that this model is not able to account for the soft X-ray excess, as might be expected given that the best fitis characterized by similar parameters as when considering the3–10 keV band alone.§.§.§ Spectral analysis over the 0.3–79 keV energy range First, we note that, if we try to reproduce the hard X-ray excess with a contribution from the torus using xillver,[Same results are found if, instead of xillver, we use mytorus <cit.> or pexmon <cit.>.] then the normalization of the Fe Kα line emitted by the BLR is consistent with 0.This, is at odds with analysis of the Chandra/HETG spectra, where the narrow profile is resolved with a width compatible with the BLR (Paper II). We can therefore conclude, from both the3–79 keV and 0.3–79 keV analysis, that any contribution from the torus is not significant, and is not considered from now on. However, we checked that the following results do not depend on the modeling of the narrow Fe Kα core either from the BLR (as established) in paper II or by the common modeling of a molecular torus.Using model 𝒜 (and allowing for a single broken power-law emissivity index),we are able to find a satisfactory fit only up to 30 keV, with the parameters reported inTable <ref> (χ^2/d.o.f.=2483.2/2058, χ^2_ red=1.21). Indeed, as shown in Figure <ref> (bottom panel), abovethis energy there is a significant hard X-ray excess. Of course, as for the four XMM-Newton/pn spectra, there are still residuals present at Fe Kα, but these appear less apparent on the model/ratio plot due to the very significant positive residual observed above about 30 keV.As also found for the 0.3–10 keV spectral analysis, a large reflection fraction and a very steep disk emissivity shape (q_1≳8 or h∼2 R_g) are required,as well as a very high (and strongly fine-tuned) value for the spin.Moreover, the inferred photon index is significantly steeper (Γ∼2.2) than that found considering only data above 3 keV (i.e., Γ∼1.9) explaining the presence of the hard X-ray excess residual seen above about30 keV.We also notice that this hard X-ray excess residual cannot be accounted for by any other alternative relxill models (i.e., the ones tested in <ref>).The lamppost geometrical configuration leads to theworst fit of the data (see Table <ref>)We check whether such unsatisfactory fits are due to the specific modelcomponents and assumptions, still using the baseline model𝒜 by alternatively:* Allowing the primary photon index of the NuSTAR spectra to bedifferent from that of the XMM-Newton/pn spectrum in order to compensate forany possible calibration issues. * Replacing the relxill model with kyreflionx, which is a model combining the relativistic smearing <cit.>and X-ray ionized reflection models reflionx <cit.> or xillver <cit.>. * Allowing the incident continuum of the relativisticcomponent to be different from the direct, observed one (see appendix <ref> for details)* Assuming a larger accretion disk density of 10^19 cm^-3(see appendix <ref> for details)Yet, none of these reflected dominated scenarios allows us to account simultaneously for the 3 main components (soft excess, broadFe Kα lines and Compton hump), or to obtain physically meaningful fit results.Furthermore, limiting the analysis to one or another energy range would lead to erroneous results on the physical condition of the disk/corona system (see a comparison of Table <ref> and Table <ref>).To summarize, during this 2014 observational campaign of Ark 120, we can safely rule out relativistic reflection as the origin of both the soft and hardX-ray excesses, and the red/blue relativistic Fe Kα features. §.§ Model combining Comptonization and relativistic reflectionHere we investigate if a combination of soft and hard Comptonizationand mildly relativistic reflection can explain the whole 0.3–79 keV continuum shape, as found for the four XMM-Newton/pn spectra. To do this, we use model 𝒞 defined astbnew×[comptt+cut-off PL+relxill+3×zgaussians(BLR)]. The cut-off power-law component is used here in order to merely parametrize Comptonization from hot electrons of the thin corona.This continuum shape is also the one used as underlying hard X-ray continuum for the relativistic reflection component.Since the mildly relativistic Fe K component(s) do not appear to be formed in the very inner part of the accretion disk, as found previously in this work (see also Paper II), we allow R_ in to vary and fix the emissivity index to 3. We find a good fit (χ^2/d.o.f.=2197.6/2058, χ^2_ red=1.07), as illustrated inFig. <ref> (top panel).The inferred fit parameters are reported in Table <ref>.We notice that the excess found above about 8 keV in the residuals for the pn spectrum is likely due to calibration issue between pn spectrum using the small window mode and the NuSTAR spectra, and is particularly prominent for high S/N pn spectrum[We notethat this pn excess above 8 keV (when fitting simultaneous NuSTAR spectra) is present even with the latest SAS version (16.0.0) and calibrations available in April 2017.].However, removing pn data above 8 keV leads to compatible fit parameters within their error bars (except for Γ=1.91^+0.02_-0.01, the discrepancy, however, is marginal), with χ^2/d.o.f.=1972.9/1876(χ^2_ red=1.05). We confirm that the Comptonizing plasma responsible for the soft excess has a low temperature (kT_ e∼0.5 keV) and a high optical depth (τ∼9). Another interesting result is that during this 2014 observation,the source spectrum above 3 keV shows a softer power-law index (Γ∼1.87)compared to the 2013 one (Γ∼1.73; ),in agreement with the recent study based on the Swift monitoring of Ark 120, which suggests that the source has a “steeper when brighter” behaviour<cit.>. The value of R_ in of 25.5^+40.9_-8.0 R_ g is much larger than the innermost stable orbit even for a maximal retrograde spinof the black hole, i.e a=-0.998.This is another hint that the observed Fe K lines are not formed in thevery inner part of the accretion disk. The relative contribution of the different model components is displayed in Fig. <ref> (unfolded spectrum for illustration purposes only; bottom panel), and showsthat the Comptonization process (including both warm and hot electrons of the corona) is the dominant one over the broadband X-ray range. § SUMMARY This paper is the fourth of a series of articles reporting on the study of an extensive X-ray (and also optical and UV) observationalcampaign of a bare AGN, targeting Ark 120.Here, we first performed the simultaneous spectral analysis of the four 120 ks XMM-Newton/pn spectra obtained in March 2014, and then we analysed the broadband X-ray spectrum – combining pn and NuSTAR spectra – obtained on 2014 March 22.The main results are summarized below: (i) The four XMM-Newton observations of Ark 120 in 2014 reveal both a strong and variable soft excess and complex Fe Kα emission. Above 3 keV, both the continuum (with Γ∼1.9) and the iron-line complex are similarwith limited variability between the XMM-Newton sequences.(ii) Above 3 keV, the four 2014 pn spectra can be well fitted with a disk reflection (relxill) model with moderate reflection fraction (ℛ∼0.4–0.5), which requires a flat emissivity profile or a large height of the X-ray source above the disk. Both of these interpretations suggest that the disk reflection emission originates much further away than the ISCO, at typical radii of tens of R_ g. (iii) The extrapolation of the above reflection models leaves strong residuals due to the soft excess below 3 keV. To account for this, thereflection models tend to extreme, finely tuned values requiring a high degree of blurring and nearly maximal black hole spin, plus a steep (Γ∼2.4) continuum. However, such models produce a very smooth broadband spectrum, and do not account for the red/blue wings of the Fe Kα line, whichrequire a flat emissivity index (or large R_ in) and a harder continuum (Γ∼1.9).(iv) Likewise, when a reflection dominated model is applied to the 2014 NuSTAR data above 10 keV, simultaneous with the third XMM-Newton observation, it cannot fit the spectrum in the highest energy range, leaving a large excess ofresidual emission above 30 keV. Hence,reflection-only models from a constant-density, geometrically flat accretion disk cannot simultaneously account for the soft excess, iron line and hard excess in the broadband 0.3–79 keV spectrum.(v) Instead, the X-ray broadband spectrum can be readily accounted for by a Comptonization model, whereby the soft and hard continuum components arise from a two temperature (warm, hot) disk corona. The warm part of the corona (kT_ e∼0.5 keV) produces the low energy part of the soft X-ray excess and is optically thick (τ∼9). A disk reflection component is still required, but it is less strong and originates at tens of gravitational radii from the ISCO of the black hole.§ DISCUSSION AND CONCLUSION During this extensive X-ray observational campaign of the bare AGN Ark 120,carried out in March 2014, the source was caught in a high flux state similar to the 2003 XMM-Newton observation <cit.>, and about as bright as the 2013 low-flux observation <cit.>.Based on the long-term Swift monitoring of Ark 120 <cit.>,this large XMM-Newton program and the 2013 observation cover thetypical high- to low-flux range observed in this source.Our spectral analysis confirms the “softer when brighter” behaviour of Ark 120 <cit.>, as commonly found in AGN <cit.>and black-hole binaries systems <cit.> with accretion ratesabove 0.01.From the analysis of the four XMM-Newton/pn spectra (2014 March 18–24) and the March 22 XMM-Newton and NuSTAR observations, we find that relativistic reflection models froma constant-density, geometrically flat accretion diskcannot self-consistently reproduce the soft excess, the mildly relativistic red and blue Fe Kαfeatures, and the hard X-ray excess. We note that this shortcoming with relativistic reflection models is found in all four 120 ks XMM-Newton/pn spectra separately, thanks to the high-S/N spectra that can be obtained for a brightsource like Ark 120. In order to form the large, smooth soft X-ray excess, extreme and tunedparameters are found for the “pure” relativistic reflection scenario: a maximally rotatingblack hole (a∼0.998), avery centrally peaked disk emissivity, a soft primary photon index (Γ∼2.4), and a very large reflection fraction (ℛ∼10), as reported in Table <ref>.By contrast, the red/blue Fe Kα features require just a moderatereflection fraction (ℛ∼0.3), a flat emissivity index(q∼1.6) or large R_ in (a few tens of R_g), and a harderpower-law index (Γ∼1.9).The former case (reflection-dominated soft X-ray excess) would correspondto a compact corona located very close to the black hole, while the latter case (disk origin of the Fe Kα features) would correspond to an extendedcorona or a lamppost geometry with a large height of the X-ray source above the disk.Both conditions cannot be therefore explained Instead, the whole 0.3–79 keV spectrum can be readily explained by a combination of Comptonization (dominating process) of the thermal optical-UV seed photons from the accretion disk by a warm (kT_ e∼0.5 keV) optically thick plasma (τ∼9) below about 0.5 keV, by a hot optically thin corona above 0.5 keV, and mildly relativistic reflection at a few ×10 R_ g. As shown in <cit.>, such a high optical depth of the warm corona could mean that either a strong magnetic field or vertical outflows to stabilize the system are required. The in-depth investigation ofthe physical properties of the warm and hot corona will be performed in a forthcoming paper (Tortosa et al.in preparation). Interestingly, such a soft X-ray excess origin (i.e., enabling to rule out a relativistic reflection scenario too) is similar to that found, for example, insome non-bare AGN like Mrk 509 <cit.> andNGC 5548 <cit.>, from deep X-ray (and multi-wavelength)observational campaigns. Such an origin has also been found for the lower-mass SMBH AGN NGC 4593, fromhigh-energy monitoring with XMM-Newton and NuSTAR <cit.>.Contrary to the 2013 observation, a relativistic reflection component is still required to explain part of the Fe Kα complex.This could be the signature that the optically thick corona is hiding partly or totally the inner accretion disk, as proposed by <cit.> to explain the X-ray characteristics of the 2013 observation, which is a factor of 2 lower in hard X-ray flux and does not appear to show a broad Fe Kα line component.In March 2014, the optically thick part of the corona may have been less extended and/or have displayed a lower covering factor,allowing us to detect a larger part of the relativistic reflection emission.For example, as demonstrated by <cit.>, for a covering factor below about 85%, the blurred reflection features become more detectable. Moreover, aComptonizing corona that covers the inner regions of the accretion disk can have substantial impact on the observed reflection spectrum <cit.>.Furthermore, if the corona covers a sufficient fraction of the inner accretion disk so as to Comptonize the reflected emission, a low reflection fraction can be measured, as found during these March 2014 observations (ℛ∼0.3). This could support the presence of an extended corona in this object.The case of a receding, full covering corona between February 2013 and March 2014 will be investigated through the fitting of the spectral energy distribution from optical/UV to hard-X-rays in a forthcoming paper (Porquet et al., in preparation). The alternative scenario of a patchy corona will be tested in a future work too (Wilkins et al.in preparation). In conclusion, the great advantage of a source like Ark 120 is thatits “bare” properties remove any fit degeneracy with warm absorption contributions. Thanks to bothXMM-Newton (4×∼120 ks) and NuSTAR (∼65 ks), we are able to discriminate between Comptonization and relativistic reflection for the soft X-ray excess origin, as well as emission above about 2 keV. As revealed in paper I thanks to the very deep 2014 RGS spectrum, a substantial amount of X-ray emitting warm gas is present out of the direct line of sight.The presence of this warm gas (warm absorber) is seen in a very large amount of type I AGN <cit.> via mainly absorption lines, but also via emission lines <cit.>.Such result demonstrates that Ark 120 is not a peculiarsource but merely a source where the line-of-sight does not intercept the warm gas, and then broadly fits into the AGN unified scheme.Therefore, since Ark 120 has typical AGN properties, such as mass andaccretion rate, it can be used as a prototype to perform an in-depth study of the X-ray corona and of its physical and geometrical properties over time,and of its possible impact on reflection spectra.The authors would like to deeply thank the anonymous referee for useful and constructive comments. Based on observations obtained with the XMM-Newton, and ESA science mission with instruments and contributions directly funded by ESA member states and the USA (NASA). This work made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by NASA.This research has made use of the NuSTARData Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center and the California Institute of Technology. D.P. would like to acknowledge financial support from the French “Programme National Hautes Energies” (PNHE). Part of the work was supported by the European Union Seventh Framework Program under the grant agreement No. 312789 (DP, GM, AM, AF). JNR acknowledges financial support via Chandra grant number GO4-15092X and NASA grant NNX15AF12G. 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N., Done, C., Smith, D., & McNaron-Brown, K. 1995, , 438, L63§ DESCRIPTION OF THE RELATIVISTIC REFLECTION PACKAGE RELXILL§.§ Coronal geometryThe relxill models are characterized by the following parameters: - the photon index of the illuminating radiation (identical for both the intrinsic cut-off power law and the relativistic reflection spectrum): Γ;- the black hole spin: a;- the disk inclination angle: θ;- the inner and outer radii of the disk: R_ in and R_ out, respectively;- the broken power-law disk emissivity index: q_1 (for R<R_ br), q_2 (for R>R_ br) andthe radius (R_ br) where emissivity changes from q_1 to q_2; - the reflection fraction as defined in <cit.>: ℛ; - the ionization parameter (erg cm s^-1, in log units) at the surface of the disk(i.e., the ratio of the X-ray flux to the gas density): log ξ;- the iron abundance relative to the solar value (here ): A_ Fe; - the high-energy cut-off (identical for both the intrinsic power law and the relativistic reflection spectrum): E_ cut. The relxill_ion model is similar to the relxill onebut allows us to calculate the reflection from the disk with several zones of different ionization (log ξ ∝ R^ -index).The number of the zones (which has to be high enough for a good physicalrepresentation, here taken at 15) and the ionization gradient versus the radius can be specified directly in the model. §.§ Lamppost geometry The relxill_lp models are defined for the lamppost geometry.The parameters q_1, q_2, and R_ br are replacedby the height of the primary source h, and the reflection fraction value is self-consistently determined by the the lamppost geometry (fixReflFrac=1).The relxill_lp_ion model takes into accountan ionization gradient of the accretion disk as for the relxill_ion model,while the relxill_lp_alpha model calculates self-consistently the radialdependence of the ionization from the irradiation of the disk, using a certain mass accretion rate (ṁ)and assuming the density profile of an α accretion disk <cit.>.§ REFLECTION MODELS FOR THE FOUR 2014 XMM-NEWTON/PN SPECTRA Table <ref> reports the spectral fits of the four 2014 XMM-Newton/pn spectra using the relxill models (see <ref>).§ REFLECTION MODELLING FOR THE X-RAY BROAD-BAND SPECTRA IN 2014 MARCH 22§.§ Investigation of possible difference between the observed and incident photon indicesFollowing the results from <cit.> for the X-ray binary GX 339-4 (in low-luminosity, hard states), we then investigate if the inability of the relativistic reflection models to account for the broad-band spectrumcould be explained by a moderate difference in the spectral powerlaw indices. For GX 339–4 a ΔΓ∼0.3 were found[However, we note that <cit.> proposed that this hardeningof reflection spectrum can be explained by Compton scattering in the corona.]. For the coronal geometry, we allow the blurred reflector to see a different continuum (Γ) with respect tothe primary one. Since in the relxill model the intrinsiccontinuum is included in the model, not enabling us to have a different continuum shape for the direct and the reflected one, we modify the baseline model 𝒜 as tbnew×(cutoffpl + relxill + 3 zgaussians), with the reflection fraction of the relxill model set to negative values to allow only the reflection component to be returned.cutoffpl is a power law with a high-energy exponential cut-off.The fit statistic improves significantly with Δχ^2∼-287 for only one additional parameter. However, this would mean an unlikely scenariowith very different continua with ΔΓ∼1.0.We then investigate a scenario with two relativistic reflection models with different continuum shapes.This improves significantly the fit statistic compared to the single reflection component, with Δχ^2∼176 for threeadditional parameters. We find ΔΓ∼0.4, but a hard X-ray excess residual is still present above 35 keV. We perform the same tests as above assuming a lamppost geometry,but we are unable to find satisfactory fits, even consideringa vertically extended primary continuum source on the rotation axis (relxill_lp_ext⊗xillver) or a moving continuum source(relxilllp_velo).Of course in all above fits, the Fe Kα red and blue residuals are still present. §.§ High-density reflection modelling The reflection models used in this work are calculated for a density of 10^15 cm^-3, for which it is assumed that the ionization state of the gas is identical for a given ξ value.But, as reported very recently by <cit.>, higher densities “are most relevant for low mass accreting black holes and when the coronal fraction is high”. Therefore, they computed the reflected spectra for densities larger than the commonly assumed density of 10^15 cm^-3, i.e., up to n=10^19 cm^-3.They showed that, for such higher density models,a very significant increase in the continuum flux at energies below about 2 keV occurs due to a large increase of thermal emission at soft X-ray energies. In other words, a high-density accretion disk leads to a larger soft excess compared tolower-density disks (such as 10^15 cm^-3). Therefore, since such a high density may be appropriate for Ark 120,a full grid of reflection models for n=10^19 cm^-3with high-energy cut-off fixed at 1000 keV has been produced(J. Garcia, private communication). We apply this high-density model to the 2014 March 22 XMM-Newton/pn and NuSTAR spectra, for a coronal geometry. We find an unsatisfactory fit of the X-ray broad-band energy (χ^2/d.o.f.=2513.3/2058; χ_ red=1.22). A weaker – though still significant – hard X-ray excess is foundsince a harder photon powerlaw index (Γ=1.76±0.01) can produce a larger softexcess emission, while a softphoton index is required for lower disk density(Γ=2.22±0.01, see Table <ref>). In addition, there are still significant model/data deviations below about 1 keV and in the Fe Kα energy range (see Fig. <ref>).In conclusion, even a high-density disk cannot reproduce the broad-band X-ray spectrum. | http://arxiv.org/abs/1707.08907v1 | {
"authors": [
"D. Porquet",
"J. N. Reeves",
"G. Matt",
"A. Marinucci",
"E. Nardini",
"V. Braito",
"A. Lobban",
"D. R. Ballantyne",
"S. E. Boggs",
"F. E. Christensen",
"T. Dauser",
"D. Farrah",
"J. Garcia",
"C. J. Hailey",
"F. Harrison",
"D. Stern",
"A. Tortosa",
"F. Ursini",
"W. W. Zhang"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170727152643",
"title": "A deep X-ray view of the bare AGN Ark120. IV. XMM-Newton and NuSTAR spectra dominated by two temperature (warm, hot) Comptonization processes"
} |
http://arxiv.org/abs/1707.08430v2 | {
"authors": [
"Yucheng Wang",
"Jian-Jian Miao",
"Hui-Ke Jin",
"Shu Chen"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170726132843",
"title": "Exact solution to an interacting dimerized Kitaev model at symmetric point"
} |
|
1]Mykhailo Potomkin 2]Andreas Kaiser 1]Leonid Berlyand 1,3,*]Igor Aranson[1]Department of Mathematics, Pennsylvania State University, University Park, 16803, USA [2]Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA [3]Department of Biomedical Engineering, Pennsylvania State University, University Park, 16803, USA[*]corresponding author: [email protected] Focusing of Active Particles in a Converging Flow [ December 30, 2023 ================================================= We consider active particles swimming in a convergent fluid flow in a trapezoid nozzle with no-slip walls.We use mathematical modeling to analyze trajectories of these particles inside the nozzle. By extensive Monte Carlo simulations, we show that trajectories are strongly affected by the background fluid flow and geometry of the nozzle leading to wall accumulation and upstream motion (rheotaxis). In particular, we describe the non-trivial focusing of active rods depending on physical and geometrical parameters. It is also established that the convergent component of the background flow leads to stability of both downstream and upstream swimming at the centerline. The stability of downstream swimming enhances focusing, and the stability of upstream swimming enables rheotaxis in the bulk.§ INTRODUCTION Active matter consists of a large number of self-driven agents converting chemical energy, usually stored in the surrounding environment, into mechanical motion <cit.>.In the last decade various realizations of active matter have been studied including living self-propelled particles as well as synthetically manufactured ones. Living agents are for example bacteria <cit.>, microtubules in biological cells <cit.>, spermatozoa <cit.> and animals <cit.>. Such systems are out-of-equilibrium and show a variety of collective effects, from clustering <cit.>to swarming, swirling and turbulent type motions <cit.>, reduction of effective viscosity <cit.>, extraction of useful energy <cit.>,and enhanced mixing <cit.>.Besides the behavior of microswimmers in the bulk the influence of confinement has been studied intensively in experiments <cit.> and numerical simulations <cit.>.There are two distinguishing features of swimmers confined by walls and exposed to an external flow: accumulation at the walls and upstream motion (rheotaxis). Microorganisms such as bacteria <cit.> andsperm cells <cit.> are typically attracted by no-slip surfaces. Such accumulation was also observed for larger organisms such as worms <cit.> and for synthetic particles <cit.>. The propensity of active particles to turn themselves against the flow (rheotaxis) is also typically observed. While for larger organisms, such as fish, rheotaxis is caused by a deliberate response to a stream to hold their position<cit.>, for micron sized swimmers rheotaxis has a pure mechanical origin <cit.>. These phenomena observed in living active matter can also be achieved using synthetic swimmers, such as self-thermophoretic <cit.> and self-diffusiophoretic <cit.> micron sized particles as well as particles set into active motion due to the influence of an external field <cit.>.Using simple models we describe the extrusion of a dilute active suspension through a trapezoid nozzle.We analyze the qualitative behavior of trajectories of an individual active particle in the nozzle and study the statistical properties of the particles in the nozzle. The accumulation at walls and rheotaxis are important for understanding how an active suspension is extruded through a nozzle. Wall accumulation may eliminate all possible benefits caused by the activity of the particles in the bulk.Due to rheotaxis active particles may never reach the outlet and leave the nozzle through the inlet, so that properties of the suspension coming out through the outlet will not differ from those of the background fluid. The specific geometry of the nozzle is also important for our study. The nozzle is a finite domain with two open ends (the inlet and the outlet) and the walls of the nozzle are not parallel but convergent, that is, the distance between walls decreases from the inlet to the outlet. The statistical properties of active suspension (e.g., concentration of active particles) extruded in the infinite channel with parallel straight or periodic walls are well-established, see e.g., <cit.> and <cit.>, respectively. The finite nozzle size leads to a “proximity effect", i.e., the equilibrium distribution of active particles changes significantly in proximity of both the inlet and the outlet. The fact that the walls are convergent, results in a “focusing effect", i.e., the background flow compared to the pressure driven flow in the straight channel (the Poiseuille flow) has an additional convergent component that turns a particle toward the centerline.Specifically, in this work it is shown that due to this convergent component of the background flow both up- and downstream swimming at the centerline are stable. Stability of the upstream swimming at the centerline is somewhat surprising since from observations in the Poisueille flow it is expected that an active particle turns against the flow only while swimming towards the walls, where the shear rate is higher. This means that we find rheotaxis in the bulk of an active suspension.§ MODEL To study the dynamics of active particles in a converging flow, two modeling approaches are exploited. In both, an active particle is represented by a rigid rod of length ℓ swimming in the xy-plane. In the first - simpler - approach, the rod is a one-dimensional segment which cannot penetrate a wall, whereas in the second - more sophisticated - approach we use the Yukawa segment model <cit.> to take into account both finite length and width of the rod, as well as a more accurate description of particle-wall steric interaction.The active particle's center location and its unit orientation vector are denoted by r=(x,y) and p=(cosφ,sinφ), respectively. The active particles are self-propelled with a velocity directed along their orientation v_0 p.The active particles are confined by a nozzle, see Fig. <ref>, which is an isosceles trapezoid Ω, placed in the xy-plane so that inlet x=x_in and outletx=x_out are bases and the y-axis is the line of symmetry:Ω={x_in<x<x_out,α^2 x^2 -y^2>0}.The nozzle length, the distance between the inlet and the outlet, is denoted by L, i.e., L=|x_out-x_in|. The width of the outlet and the inlet are denoted by w_out and w_in, respectively, and their ratio is denoted by k=w_out/w_in.Furthermore, the active particles are exposed to an external background flow. We approximate the resulting converging background flow due to the trapezoid geometry of the nozzle byu_BG( r)=(u_x(x,y),u_y(x,y))=(-u_0 (α^2 x^2-y^2)/x^3, -u_0 y (α^2x^2-y^2)/x^4),where u_0 is a constant coefficient related to the flow rate and α is the slope of walls of the nozzle.Equation (<ref>) is an extension of the Poiseuille flow to channels with convergent walls[In order to recover the Poiseuille flow (for channels of width 2H) from Eq. (<ref>), take x=H/α, u_0=H^3/α^3 and pass to the limit α→ 0. Note that the walls of the nozzle are placed so that they intersect at the origin, so in the limit of parallel walls, α→ 0, both the inlet and the outlet locations, x_in and x_out, go to -∞.].Active particles swim in the low Reynolds-number regime. The corresponding overdamped equations of motion for the locationsr and orientations p are given by:d rdt= u_BG( r)+v_0 p, d pdt =(I- p p^T)∇_ r u_BG( r) p +√(2D_r) ζe_φ.Here (<ref>) is the Jeffery's equation <cit.> for rods with an additional term due to random re-orientation with rotational diffusioncoefficient D_r; ζ is an uncorrelated noise with the intensity ⟨ζ(t),ζ(t')⟩=δ(t-t'), e_φ=(-sinφ, cosφ).Equation (<ref>) can also be rewritten for the orientation angle φ:dφdt=ω+ ν sin 2φ + γ cos 2φ+√(2D_r) ζ.Here ω=12(∂ u_y∂ x-∂ u_x∂ y), ν=12(∂ u_y∂ y-∂ u_x∂ x)=∂ u_y∂ y=-∂ u_x∂ x, and γ=12(∂ u_y∂ x+∂ u_x∂ y) are local vorticity, vertical expansion (or, equivalently, horizontal compression; similar to Poisson's effect in elasticity) and shear.The strength of the background flow is quantified by the inverse Stokes number, which is the ratio between the background flow at the center of the inlet and the self-propulsion velocity v_0. Specifically, σ = u_x(x_in,0)v_0=u_0α^2v_0|x_in|,where (x_in,0) denotes the location at the center of the inlet. In the first modeling approach we include the particle wall interaction in the following way: an active particle is not allowed to penetrate the walls of the nozzle. To enforce this, we require that both the front and the back of the particle, r(t)±(ℓ/2)p, are located inside the nozzle. In numerical simulations of the system (<ref>)-(<ref>) this requirement translates into the following rule: if during numerical integration of (<ref>)-(<ref>) a particle penetrates one of the two walls, then this particle is instantaneously shifted back along the inward normal at the minimal distance, so its front and back are again located inside the nozzle while its orientation is kept fixed.Unless mentioned otherwise, in this modeling approach we consider a nozzle whose inlet width w_in=0.2 mm and outlet width w_out=0.1 mm are fixed. The following nozzle lengths are considered:L=0.2 mm, L=0.5 mm and L=1.0 mm. The length of the active particles is ℓ = 20 μm, they swim with a self-propulsion velocity v_0=10 μm s^-1 and their rotational diffusion coefficient is given by D_r=0.1 s^-1. All active particles are initially placed at the inlet, x(0)=x_in, with random y-component y(0) and orientation angle φ(0). The probability distribution function for initial conditions y(0) and φ(0) is given by Ψ∝ 1 (uniform). The trajectory of an active particle is studied until it leaves the nozzle either through the inlet or the outlet. To gather statistics we use 96,000 trajectories. We use the second approach to describe the particle-wall interactions and the torque induced by the flow more accurately. For this purpose each rod, representing an active particle, of length ℓ, width λ and the corresponding aspect ratio a=ℓ/λ is discretized into n_r spherical segments with n_r = ⌊ 9 a /8 ⌉ (⌊ x ⌉ denotes the nearest integer function). The resulting segment distance is also used to discretize the walls of the nozzle into n_w segments in the same way. Between the segments of different objects a repulsive Yukawa potential is imposed. The resulting total pair potential is given by U = U_0∑_i=1^n_r∑_j=1^n_wexp [-r_ij / λ]/r_ij, where λ is the screening length defining the particle diameter, U_0 is the prefactor of the Yukawa potential and r_ij = | r_i -r_j| is the distance between segment i of a rod and j of the wall of the nozzle, see Fig. <ref>. The equations of motion (<ref>) and (<ref>) are complemented by the respective derivative of the total potential energy of a rod along with the one-body translational and rotational friction tensors for the rods f_ T and f_ R which can be decomposed into parallel f_∥, perpendicular f_⊥ and rotational f_ R contributions which depend solely on the aspect ratio a <cit.>. For this approach we measure distances in units of λ, velocities in units of v_0=F_0/f_∥ (here F_0 is an effective self-propulsion force), and time in units of τ = λ f_∥ / F_0. While the width of the outlet w_out is varied, the width of the inlet w_in as well as the length of the nozzle L is fixed to 100λ in our second approach. Initial conditions are the same as in the first approach.To avoid that a rod and a wall initially intersect each other, the rod is allowed to reorient itself during an equilibration time t_e = 10 τwhile its center of mass is fixed.Furthermore, we use the second approach to study the impact of a finite density of swimmers. For this approach we initialize N active rods in a channel confinement which is connected to the inlet of the nozzle, see Fig. <ref>. Inside the channel we assume a regular (non-converging) Poiseuille flow <cit.>. We restrict our study to a dilute active suspension with a two dimensional packing fraction ρ=0.1. To maintain this fraction, particles which leave the simulation domain are randomly placed at the inlet of the channel confinement.§ RESULTS§.§ Focusing of outlet distributionHere we characterize the properties of the particles leaving the nozzle at either the outlet or the inlet. Specifically, our objective is to determine whether particles accumulate at the center or at walls when they pass through the outlet or the inlet. We start with the first modeling approach. Figure <ref> shows the spatial distribution of active particles leaving the nozzle at the outlet for various inverse Stokes number σ and three different lengths L of the nozzle, while the width of the inlet and the outlet are fixed.For small inverse Stokes number σ, the background flow is negligible compared to the self-propulsion velocity. Active particles swim close to the walls and peaks at walls are still clearly visible for σ=0.5 for all nozzle lengths L, see Fig. <ref>(a). For σ=1, the self-propulsion velocity and the background flow are comparable; in this case the histogram shows a single peak at the center of the outlet, see Fig. <ref>(b). Further increasing the inverse Stokes number from σ=1 to σ=9 leads to a broadening of the central peak and then to the formation of two peaks with a well in the center of the outlet, see Fig. <ref>(c)-(e). Finally, for an even larger inverse Stokes number σ,the self-propulsion velocity is negligible and the histogram becomes close to the one in the passive (no self-propulsion, v_0 = 0) case, see Fig. <ref>(f). Here the histogram for a nozzle length L=0.2 mm is uniform except at the edges where it has local peaks due to accumulation at the walls caused by steric interactions.Histograms for both the y-component and the orientation angle φ of the active particles reaching the outlet are depicted in Fig. <ref>(a)-(c). While active particles leave the nozzle with orientations away from the centerline for small inverse Stokes number, σ = 0.5, they are mostly oriented towards the centerline for larger values of the inverse Stokes number. In Fig. <ref>(c), one can observe that the histogram is concentrated largely for downstream orientations φ≈ 0 and slightly for upstream orientations φ≈±π. These local peaks for φ≈±π away from walls are evidence of rheotaxis in the bulk. These peaks are visible for large inverse Stokes numbers only and the corresponding active particles are flushed out of the nozzle with upstream orientations. Due to rotational diffusion and rheotaxis it is possible that an active particle can leave the nozzle through the inlet. We compute the probability of active particles to reach the outlet. This probability, as a function of the inverse Stokes number σ for the three considered nozzle lengths L, is shown in Fig. <ref>(a), together with selected trajectories, see insets in Fig. <ref>(a).The figure shows that the probability that an active particle eventually reaches the outlet monotonically grows with the inverse Stokes number σ. Note that a passiveparticle always leaves the nozzle through the outlet. By comparing the probabilities for different nozzle lengths L it becomes obvious that an active particle is less likely to leave the nozzle through the outlet for longer nozzles. Due to the larger distance L between the inlet and the outlet an active particle spends more time within the nozzle, which makes it more likely to swim upstream by either rotational diffusion or rheotaxis. In Fig. <ref>(b)-(d) histograms for active particles leaving the nozzle through the inlet are shown. In the case of small inverse Stokes number, σ=0.5, the majority of active particles leaves the nozzle at the inlet. Specifically, most of them swim upstream due to rheotaxis close to the walls, but some active particles leave the nozzle at the inlet close to the center. These active particles are oriented upstream due to random reorientation. By increasing the inverse Stokes number σ≥ 1, active particlesare no longer able to leave the nozzle at the inlet close to the center.Let us now consider specific examples of active particles' trajectories, seeFig. <ref>. The first trajectory (red) starts and ends at the inlet.Initially the active particle swims downstream and collides with the upper wall due to the torque induced by the background flow. Close to the wall it exhibits rheotactic behavior, but before it reaches the inlet it is expelled towards the center of the nozzle due to rotational diffusion, similar to bacteria that may escape from surfaces due to tumbling <cit.>.Eventually, the active particle leaves the nozzle at the inlet.As for the other depicted trajectory (blue), the active particle manages to reach the outlet. Along its course through the nozzle it swims upstream several times but in the end the active particle is washed out through the outlet by the background flow. For larger flow rates the trajectories of active particles are less curly, since the flow gets more dominant, see insets of Fig. <ref>(a). Next we present results of the second modeling approach which is based on the Yukawa-segment model.So far we have concentrated on fixed widths of the inlet and outlet. Here we consider nozzles with fixed length L and inlet width w_in and varynozzle ratio k. We study the behavior of active rods with varied aspect ratio a.As shown in Fig. <ref>, neither the aspect ratio a, see Fig. <ref>(a),nor the nozzle ratio k,see Fig. <ref>(b),have a significant impact on the probability P_out which measures how many active rods leave the nozzle at the outlet. However, the aspect ratio a is important for the location where the active rods leave the nozzle at the inlet and the outlet, see Fig. <ref>. For short rods (a=2) and small inverse Stokes numbers (σ≤ 1) the distribution of active particles shows just a single peak located at the center. This peak broadens if the inverse Stokes number increases, which is in perfect agreement with the results obtained by the first approach, cf. Fig. <ref>.It is more likely for short rods than for long ones to be expelled towards the center due to rotational diffusion.Hence the distribution of particles at the outlet for long rods (a=10) shows additional peaks close to the wall. These peaks become smaller if the inverse Stokes number increases. The distribution of particles leaving the nozzle at the inlet is similar to our first approach. While the distribution is almost flat for small inverse Stokes numbers, increasing this number makes it impossible to leave the nozzle close to the center at the inlet. Similar to the outlet the wall accumulation at the inlet is more pronounced for longer rods.By comparing the orientation of the particles at the outlet, the influence of the actual length of the rod becomes visible, see Fig. <ref>. As seen before for short rods, a=2, for small inverse Stokes numbers σ there is no wall accumulation. Hence most particles leave the nozzle close to the center and are orientated in the direction of the outlet. This profile smears out if the inverse Stokes number is increased to σ = 1. Forlarger inverse Stokes numbers the figures are qualitatively similar to the one obtained by the first approach, cf. Fig. <ref>(c). Particles in the bottom half of the nozzle tend to point upwards and particles in the top half tend to point downwards. The same tendency is seen for long rods a=10 and small inverse Stokes number. However for long active rods, this is because they slide along the walls. The bright spots close to the walls for long rods and large inverse Stokes numbers indicate that particles close to the wallsare flushed through the outlet by the large background flow even if they are oriented upstream. In addition, there are blurred peaks away from the walls for large inverse Stokes numbers σ. The corresponding particles crossed the outlet with mostly upstream orientations. This is similar to Fig. <ref>(c), where particles exhibiting in-bulk rheotactic characteristics were observed at the outlet of the nozzle.By comparing the results for individual active rods, see again Fig. <ref>, with those for interacting active rods at a finite packing fraction ρ = 0.1, see Fig. <ref>, we find that wall accumulation becomes more pronounced. Mutual collisions of the rods lead to a broader distribution of particles. For long rods, a=10, the peaks at φ≈ 0 and φ≈±π remain close to the walls and the blurred peaks at the center vanish. §.§ Optimization of focusingHere we study the properties of the active particles in more detail and provide insight into the nozzle geometry, the background flow and the size of the swimmers that should be used in order to optimize the focusing at the outlet of the nozzle. For this purpose we study three distinct quantities. The averaged dwell time ⟨ T⟩, the time it takes for an active particle to reach the outlet, the mean alignment of the particles measured by ⟨cosφ_out⟩ and the mean deviation from the center y=0 at the outlet ⟨ |y_out|⟩. As depicted in Fig. <ref>, for increasing inverse Stokes number the probability for active particles to reach the outlet increases. However they are spread all over the outlet. This is quantified by the ⟨ |y_out|⟩. Small values of ⟨ |y_out|⟩ correspond to a better focusing. If particles leave the nozzle with no preferred orientation, their mean orientation vanishes, ⟨cosφ_out⟩ = 0; in case of being orientated upstream we obtain ⟨cosφ_out⟩ = -1 and finally ⟨cosφ_out⟩ = 1 if the particles are pointing in the direction of the outlet. Obviously in an experimental realization a fast focusing process and hence small dwell times T would be preferable.The numerical results obtained by the first modeling approach are depicted in Fig. <ref>. While the dwell time hardly depends on the size ratio k of the nozzle, obviously the strength of the background flow has a huge impact on the dwell time and large inverse Stokes numbers σ lead to a faster passing through the nozzle of the active particles, see Fig. <ref>(a). The alignment of the active particles, ⟨cosφ_out⟩, becomes better if the nozzle ratio k is large and the flow is slow, see Fig. <ref>(b). The averaged deviation from the centerline ⟨ |y_out|⟩ increases with increasing nozzle ratio k since the width of the outlet becomes larger. As could already be seen in Fig. <ref>, the averaged deviation from the centerline is non-monotonic as a function of the inverse Stokes number and shows the smallest distance from the centerline for all nozzle ratios if the strength of the flow is comparable to the self-propulsion velocity of the swimmers, σ=1.Let us now study how these three quantities depend on the aspect ratio of the swimmer. To this end, we use the second modeling approach. We consider all three parameters as a function of the inverse Stokes number σ. Longer rods have a shorter dwell time so that they reach the outlet faster, see Fig. <ref>(a). Increasing the flow velocity obviously leads to a decreasing dwell time. The same holds for the mean alignment –it decreases for increasing inverse Stokes number, see Fig. <ref>(b). Moreover, for small inverse Stokes numbers, σ≤ 2, the mean alignment is better for long rods. For large inverse Stokes numbers,long rods a=10 are washed out with almost random orientation, however short rods a=2 are slightly aligned with the flow. Short rods are focused better for small inverse Stokes numbers, σ≤ 2, see Fig. <ref>(c),due to wall alignment and wall accumulation of longer rods. For larger inverse Stokes numbers, it is the other way around – long rods are better focused. Comparing various nozzle ratios k with fixed simmers' aspect ratio a, we obtain that smaller ratios k lead to smaller dwell times [Fig. <ref>(d)] and better alignment [Fig. <ref>(e)]. For narrow outlets (small k) the active particles leave the outlet closer to the center, see Fig. <ref>(f).§ DISCUSSIONWe discuss the stability of particles around the centerline y=0 in the presence of a background flow and confining walls if they are converging with a non-zero slope α. This stability is in contrast to a channel with parallel walls, where an active particle swims away from the centerline provided that its orientation angle φ is different from nπ, n=0,±1,±2,….Indeed, in the case of a straight channel, α = 0, the background flow is defined as u_x=u_0 (H^2-y^2), u_y=0 (Poiseuille flow; u_0 is the strength of the flow, 2H is the distance between the walls).Then the system (<ref>)-(<ref>) reduces to φ̇ =u_0 y (1-cos 2φ)ẏ =v_0sinφ. Here we omit the equation for x(t) due to invariance of the infinite channel with respect to x and neglect orientation fluctuations, that is D_r=0. The phase portrait for this system is depicted in Fig. <ref>(a). Dashed vertical lines φ=nπ, n=0,±1,±2,… consist of stationary solutions: if an active particle is initially oriented parallel to the walls, it keeps swimming parallel to them. If initially φ is different from nπ, then the active particle swims away from the centerline, y(t)→±∞ as t →∞. When the walls are converging, α > 0, the y-component of the background flow is non-zero and directed towards the centerline. For the sake of simplicity we take u_y=-α y, α>0 and u_x as in the Poiseuille flow, u_x=u_0 (H^2-y^2). In this case, the system (<ref>)-(<ref>) reduces to φ̇ =-(α/2)sin 2φ + u_0 y (1-cos 2φ)ẏ =-α y+ v_0sinφ. The corresponding phase portrait for this system is depicted in Fig. <ref>(b). Orientations φ=nπ represent stationary solutions only if y=0. In contrast to the Poiseuille flow in a straight channel, see Eqs. (<ref>) and (<ref>),these stationary solutions (φ=π n, y=0) are asymptotically stable with a decay rate α (recall that α is the slope of walls). In addition to these stable stationary points there are pairs of unstable (saddle) points with non-zero y (provided that v_0>0). In these saddle points, the distance from centerline |y| does not change, sincea particle is oriented away from centerline, so the propulsion force moves the particle away from the centerline and this force is balanced by the convergent component of the background flow, u_y, moving the particle toward the centerline. The orientation angle φ does not change since the torque from the Poiseuille component of the background flow, u_x, is balanced by the torque from the convergent component, u_y.We also draw the phase portrait for the converging flow u_BG=(u_x,u_y) introduced in Section <ref>, Fig. <ref>(c).One can compare the phase portraits Fig. <ref>(b) and Fig. <ref>(c) around the stationary point (φ=0, y=0) to see that the qualitative picture is the same: this stationary point is stable and it neighbors with two saddle points.The asymptotic stability of(φ=0, y=0) means that if a particle is close to the centerline and its orientation angle is close to 0 (particle is oriented towards the outlet), it will keep swimming at the centerline pointing toward the outlet, whereas in Poiseuille flow the particle would swim away. The asymptotic stability of (φ=±π, y=0) isevidence of that in the converging flow there is rheotaxis not only at walls but also in the bulk, specifically at the centerline. Another consequence of this stability is the reduction of effective rotational diffusion of an active particle in the region around the centerline, that is the mean square angular displacement ⟨Δφ^2⟩ is bounded in time due to the presence of restoring force coming from the converging component of the background flow(cf. diffusion quenching for Janus particles in <cit.>).Finally, we note that the nozzle has a finite length L and thus, the conclusions of the stability analysis are valid if the stability relaxation time, 1/α s, does not exceed the average dwell time ⟨ T ⟩. We introduce a lower bound T̃ for the dwell time ⟨ T⟩ as the dwell time of an active particle swimming along the centerline oriented forward, φ=0:T̃=Lk/(σ v_0 (1-k))ln |1+σ (1-k)/(k(σ+1))|.Our numerical simulations show that T̃ underestimates the average dwell time by a factor larger than two. Using this lower bound, we obtain the following sufficient condition for stability: k w_inσ v_0ln|1+σ(1-k)k(σ+1)|≥ 1.§ CONCLUSIONIn this work we study a dilute suspension of active rods in a viscous fluid extruded through a trapezoid nozzle.Using numerical simulations we examined the probability that a particle leaves the nozzle through the outlet - which is the result of the two counteracting phenomena. On the one hand, swimming downstream together with being focused by the converging flow increases the probability that an active rod leaves the nozzle at the outlet. On the other hand, rheotaxis results in a tendency of active rods to swim upstream.Theoretical approaches introduced in this paper can be used to design experimental setups for the extrusion of active suspensions through a nozzle.The optimal focusing is the result of a compromise. While for large flow rates it is very likely for active rods to leave the nozzle through the outlet very fast, their orientation is rather random and they pass through the outlet close to the walls. The particles are much better aligned with the flow for small flow rates and focused closer to the centerline of the nozzle, however the dwell time of the particles becomes quite large. Based on our findings the focusing is optimal if the velocity of the background flow and the self-propulsion velocity of the active rods are comparable. To reduce wall accumulation, the rods should have a small aspect ratio. We find that rheotaxis in bulk is possible for simple rigid rodlike active particles. We also established analytically the local stability of active particle trajectories in the vicinity of the centerline. This stability leads to the decrease of the effective rotational diffusion of the active particles in this region as well as the emergence of rheotaxis away from walls. Our findings can be experimentally verified using biological or artificial swimmers in a converging flow.§ ACKNOWLEDGEMENTSThe work was supported by NSF DMREF grant DMS-1628411. A.K. gratefully acknowledges financial support through a Postdoctoral Research Fellowship (KA 4255/1-2) from the Deutsche Forschungsgemeinschaft (DFG). § AUTHOR CONTRIBUTIONS STATEMENT Simulations have been performed by M.P. and A.K., the research has been conceived by L.B. and I.S.A. and all authors wrote the manuscript. | http://arxiv.org/abs/1707.08665v1 | {
"authors": [
"Mykhailo Potomkin",
"Andreas Kaiser",
"Leonid Berlyand",
"Igor Aranson"
],
"categories": [
"cond-mat.soft",
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.soft",
"published": "20170726234323",
"title": "Focusing of Active Particles in a Converging Flow"
} |
Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation ApproachHuajin Wang12, Jianhui Li1, Zhihong Shen1 and Yuanchun Zhou11Computer Network Information Center, Chinese Academy of Sciences, China Email: {wanghj, lijh, bluejoe, zyc}@cnic.cn 2University of Chinese Academy of Sciences, ChinaThis work was supported by the National Key Research Program of China (2016YFB1000600, 2016YFB0501900). Corresponding author is Jianhui Li.December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================= Compared to basic fork-join queues, a job in (n,k) fork-join queues only needs its k out of all n sub-tasks to be finished. Since (n,k) fork-join queues are prevalent in popular distributed systems, erasure coding based cloud storages, and modern network protocols like multipath routing, estimating the sojourn time of such queues is thus critical for the performance measurement and resource plan of computer clusters. However, the estimating keeps to be a well-known open challenge for years, and only rough bounds for a limited range of load factors have been given. In this paper, we developed a closed-formfor jointly-identical random variables: An order statistic can be represented by a linear combination of maxima. This brand-new technique is then used to transform the sojourn time of non-purging (n,k) fork-join queues into a linear combination of the sojourn times of basic (k,k), (k+1,k+1),...,(n,n) fork-join queues. Consequently, existing approximations for basic fork-join queues can be bridged to the approximations for non-purging (n,k) fork-join queues. The uncovered approximations are then used to improve the upper bounds for purging (n,k) fork-join queues. Simulation experiments show that this linear transformation approach is practiced well for moderate n and relatively large k. § INTRODUCTIONThe performance of fork-join queues is a highly focused research topic for many years for the ubiquitousness of fork-join queues in both real-life workflows and computing systems. In a fork-join queueing system, a job is forked into n sub-tasks when it arrives at a control node, and each sub-task is sent to a single node to be conquered. Results of finished sub-tasks are summarized at a central join node. When the job arrival rate λ is high, a sub-task may have to wait for service in the sub-queue of its hosting node in a first-come-first-serving order. A basic fork-join queue considers a job is done after all results of the job have been received at the join node (see Fig. <ref> (a)).In Big Data era, more and more mainstream computing infrastructures become distributively deployed, and inevitably recruit fork-join queues to facilitate the storing and processing of large-scale datasets. For example: 1) Cassandra<cit.> and Dynamo<cit.>, two popular key-value data stores, use fork-join queues to concurrently perform read and write operations on all the replicas of the target key-value pairs; 2) The client of an (n,k) MDS (maximum distance separable) erasure coding based cloud storage system only needs to retrieve any k out of all n blocks of a file to reconstruct the file; 3) The data transmission process of multipath routing protocols can generally be simplified as a multi-stage fork-join queueing process.Latency is commonly a critical concern in building and optimizing Big Data clusters. For example, in Amazon's cloud platform, services commonly have latency requirements which are in general measured at the 99.9^th percentile of the distribution <cit.>. The Dynamo storage system must be capable of meeting such stringent SLAs. In this scenario, basic fork-join queues may cause serious performance issues when the number of data replicas are large, since they require all the sub-tasks of a job to be finished before making the job's response. By contrast, (n,k) fork-join queues, as named in <cit.>, only require the job's any k out of n sub-tasks to be finished, and thus have performance advantages in such scenarios. For example, a write request in Casandra can either be responded when a quorum of replicas have been successfully written, or just get responded once the fast answer from all touched replicas is acknowledged when there is a need to pursue high throughputs.As depicted in Fig. <ref>, there are mainly two versions of (n,k) fork-join queues: The purging one removes all the remaining sub-tasks of a job from both sub-queues and service stations once it receives the job's k^th answer. The file retrieval process from an MDS coded cloud storage system is such an example. As a contrast, the non-purging one keeps queuing and executing remaining sub-tasks. For example, a write operation in Cassandra needs to update all the replicas of the target key-value pair, while can response to user as soon as a quorum of replicas have been successfully written. The State-of-the-Art Research on Basic Fork-Join QueuesThe popularity of fork-join systems has drawn great attentions from database/OS/networking communities to the performance analyses of fork-join queues for a rather long period of time. Unfortunately, there is still no exact closed-form solution of the sojourn time of the job in n≥ 3 basic fork-join queues. The difficulty lies in the fact that the sojourn times of a job's sub-tasks are not independent, as their hosting sub-queues share the same sub-task arrival process. Since most of existing exact analysis techniques are developed for independent and identical (iid) random variables, it is very hard to trace the sojourn time distribution for fork-join queues.For n≥ 3 fork-join queues under Poisson job arrival process and with iid exponential service time distributions, Nelson et al. <cit.> proposed an initiative approximation technique which is based on the fact that the sojourn times X_1 k of sub-tasks 1,2,...,k are associated variables, whose maximum can be bounded by the maximum of their iid equivalents <cit.>: P(X_1 n≤ t)≥∏_i=1^nP(X^IID_i≤ t). According to that, the upper bounds and closed-form approximations of the sojourn time were given in this work. Simulation experiments in <cit.> showed that Nelson's approximation is still the most reliable one, compared to following works such as <cit.> and <cit.>. The State-of-the-Art Research and Open Challenges on (n,k) Fork-Join QueuesDespite the popularity of (n,k) fork-join queues in Big Data systems and many other fields, there are even no practical approximations on the sojourn time of (n,k) fork-join queues: Unlike the maximum, the k^th order statistic cannot be bounded by using associated variables' property, which makes the sojourn time of (n,k) fork-join queues more hard to analyze, compared to basic fork-join queues.Currently, there are only exact quantity analyses for purging (n,1) fork-join queues <cit.>, because such a queue is equivalent to a single queue with n times the sub-queue's service rate. For general purging (n,k) fork-join queues, there are only rough bounds have been given: Joshi et al. <cit.> resort to the split-merge queue model (see Fig. <ref>) to find proper upper and lower bounds. Compared to purging (n,k) fork-join queues, all empty sub-queues in the split-merge model are blocked and cannot serve subsequent tasks until k sub-tasks of the current job are completed, which makes the split-merge model much easier to trace. However, these split-merge based bounds tend to be extremely loose when increasing k or the load factor ρ, as we depict in Section <ref>. Since non-purging (n,k) fork-join queues cannot be reduced to the split-merge model, they are more difficult to analyze, even including (n,1) queues. Recently, Fidler et al. <cit.> gave non-asymptotic statistical bounds on the sojourn times for non-purging fork-join queues. However, no reasonable approximations have been proposed. Methodology and Contributions This paper aims at fixing the lack of proper approximations for non-purging (n,k) fork-join queues and tackling the uncontrollability of bounds for purging (n,k) fork-join queues. To achieve these objectives, we trace fork-join queues in a fundamental way: The linear relationship between (n,k) fork-join queues and their basic (k,k), (k+1,k+1),...,(n,n) equivalents is depicted for the first time; This relationship is then used to bridge the existing approximations for basic fork-join queues to the approximations and bounds for (n,k) fork-join queues.Our innovations and contributions are highlighted as follows: * A brand-new closed-formfor jointly-identical random variables, by which order statistics can be transformed into a closed-form linear combination of maxima. Besides, there is no need to assume the independence of variables.* The first reasonable and practical method to approximate the expected sojourn time of non-purging (n,k) fork-join queues with general service time distributions. This method relies on the cooperation between the linear transformation technique and the existing approximations for basic fork-join queues.* Improvements over the upper bounds on the expected sojourn time of purging (n,k) fork-join queues, which are gained by resorting the bounds to that of the non-purging equivalent (n,k) fork-join queues. This paper is organized as follows: The linear transformation technique is developed in Section <ref>; This technique is then employed in Section <ref> to find proper approximations for non-purging (n,k) fork-join queues; The flaws of existing bounds for purging (n,k) fork-join queues and our improvements over upper bounds are depicted in Section <ref>; In Section <ref>, we discuss the limitation of this linear transformation technique; Related works are reviewed in Section <ref>; We conclude this work and point out some promising future research directions in Section <ref>.§ PRELIMINARIES: LINEAR TRANSFORMATIONS OF ORDER STATISTICSIn this section, we consider a family of rvs (random variables) X_1, X_2,...,X_n (denoted as X_1 n) defined on a probability space, and let X_(n,k) denotes their k^th order statistic, P_k denotes the possibility P(X_1≤ t, X_2≤ t,...,X_k≤ t) and P_n,k denotes the possibility P(X_1≤ t, X_2≤ t,...,X_k≤ t, X_k+1>t, X_k+2>t,...,X_n>t). Obviously, P_k is the distribution of the maximum of X_1 k. For n identically distributed rvs X_1 n and ∀ k∈ [1 n], if any k arbitrarily chosen rvskeep the same joint probability distribution, these n identical rvs are named as jointly-identical rvs. For n jointly-identical rvs X_1 n,P_n,k=∑_i=k^nA^n,k_iP_i, 1≤∀ k ≤ n, where the const coefficient A^n,k_i can be calculated by the following recurrence:A^n,k_i={[ 1i=k,; -∑_j=1^i-kn-i+jjA^n,k_i-jk+1≤ i≤ n. ].Let P_n-k|k denotes P(X_k+1>t,X_k+2>t,...,X_n > t|X_1≤ t,X_2≤ t,...,X_k≤ t) andP_n-i,i-k|k denotes P(X_i+1>t,X_i+2>t,...,X_n > t, X_k+1≤ t, X_k+2≤ t, X_i≤ t|X_1≤ t,X_2≤ t,...,X_k ≤ t), k+1≤ i ≤ n. Certainly, we haveP_n-k|k=P_n,k/P_k , P_n-i,i-k|k=P_n,i/P_k.As X_1 n are jointly-identical rvs, the following equation holds:P_n-k|k=1-∑_i=k+1^nn-ki-kP_n-i,i-k|k.By insertion of Eq. <ref> into Eq. <ref>, the following recurrence holds:P_n,k=P_k-∑_i=k+1^nn-ki-kP_n,iExpanse Eq. <ref>, we complete the proof of Lemma <ref>. §.§ Linear Transformation of Order Statistics For n jointly-identical rvs X_1 n, there exists a linear transformation from maxima to order statistics:[ X_(n,1); X_(n,2); ⋮; X_(n,n) ] = [ W_1^n,1 W_2^n,1 … W_n^n,1; 0 W_2^n,2 … W_n^n,2; ⋮ ⋮ ⋱ ⋮; 0 0 … W_n^n,n; ][ X_(1,1); X_(2,2); ⋮; X_(n,n) ].Namely, X_(n,k)=∑_i=k^nW_i^n,kX_(i,i), 1≤∀ k ≤ n, where the const coefficientW_i^n,k=∑_j=k^injA^n,j_i Let F_n,k≡ P(X_(n,k)≤ t) be the probability distribution of the k^th order statistic. Equivalently, we need to proveF_n,k=∑_i=k^nW_i^n,kP(X_(i,i)≤ t)=∑_i=k^nW_i^n,kP_i. As F_n,k=∑_i=k^nniP_n,i and P_n,k=∑_i=k^nA^n,k_iP_i (Lemma <ref>), we derive the following recurrence:F_n,k={[ P_nk=n,; F_n,k+1+nk∑_i=k^nA^n,k_iP_i1≤ k≤ n-1. ].Expanse Eq. <ref>, we complete the proof ofEq. <ref>. There is no need to assume the independence of the identical rvs X_1 n. There may exist other linear transformations of order statistics than the one given by Theorem <ref>. For any possible linear transformations from maxima to order statistics, the corresponding const coefficient W_i^n,k of X_(i,i) is named as a W coefficient.The calculation of the W coefficient given by Eq. <ref> is not straightforward, as it consists of many items. We use a simple solver (see Appendix) to give W coefficient values. §.§ Linear Transformation of ExpectationsLet 𝔼_n,k≡ E[X_(n,k)] be the expectation of the k^th order statistic of rvs X_1 n. Specially, we use 𝔼_n to denote 𝔼_n,n. Then the following theorem holds. For n jointly-identical rvs X_1 n, there exists a linear transformation from the expectations of maxima to the expectations of order statistics:𝔼_n,k=∑_i=k^nW_i^n,k𝔼_i, 1≤∀ k ≤ n. E[X_(n,k)]=E[∑_i=k^nW_i^n,kX_(i,i)]=∑_i=k^nW_i^n,k𝔼_i§ APPROXIMATIONS FOR NON-PURGING (N,K) FORK-JOIN QUEUES We consider a homogenous cluster consisting of n nodes, where each node i (1≤ i≤ n) has the same service time distribution when processing sub-tasks of the same job. Each node owns a first-come-first-serving sub-queue q_i (1≤ i≤ n) with the assumption of unlimited queue capacity. These n sub-queues constitute a homogenous fork-join queue.Each incoming job consists of n tasks. Let t_i^j be the sojourn time of the j^th job's sub-task assigned to node i. Then, the stable sojourn time of a sub-task in the sub-queue q_i is t_i≡lim_j→∞t_i^j, and the sojourn time of a job in the (n,k) fork-join queue is the k^th order statistic t_(n,k) consequently. For an (n,n) fork-join queue, the sub-queues' stable sojourn times t_1 n constitute a family of jointly-identical rvs. Recall that all the sub-queues have an unlimited capacity, then the sub-task sojourn time distribution of a sub-queue depends only on the job arrival process and the sub-queue's service time distribution. As these sub-queues are under the same job arrival process and have the same service time distribution, the sub-queues' stable sojourn times t_1 n are identical rvs.By the constitution methodology of fork-join queues, an (n,n) fork-join queue is symmetrical, which means its sub-queues are interchangeable and thus any k arbitrarily chosen sub-queues keep the same joint probability distribution. By definition the stable sojourn times t_1 n are jointly-identical rvs. Those basic fork-join queues, purging/non-purging (n,k) fork-join queues and split-merge queues that are under the same job arrival process and with the same sub-queue's service time distribution are called (λ,μ)-equivalent queues to each other, where λ and μ are the job arrival rate and the sub-queue's service rate respectively.§.§ Approximations for General Fork-Join QueuesThis paper uses the termto denote the fork-join queues with identically and generally distributed sub-queues' service times and job inter-arrival times.The sojourn time of a general non-purging (n,k) fork-join queue can be represented by a linear combination of the sojourn times of the (λ,μ)-equivalent basic fork-join queues:t_(n,k)=∑_i=k^nW_i^n,kt_(i,i) where W_i^n,k is the corresponding W coefficient. The only difference between the (λ,μ)-equivalent non-purging (n,k) queue and basic (n,n) queue lies in the job departure process, which has no influence over the job arrival process, the sub-queue's service time distribution and therefore the distribution of the sub-queue length. Consequently, the target non-purging (n,k) queue and its (λ,μ)-equivalent (n,n) fork-join queue have the same family of jointly-identical rvst_1 n, and therefore the same order statistic t_(n,k).According to Lemma <ref> and Theorem <ref>, t_(n,k)=∑_i=k^nW_i^n,kt^sub_(i,i), where t^sub_(i,i) is the maximum stable sojourn time of i arbitrarily chosen sub-queues from the target non-purging (n,k) queue. By the constitution methodology of fork-join queues, the i chosen sub-queues can constitute an (i,i) queue which is (λ,μ)-equivalent to the target non-purging (n,k) queue. Therefore, t_(i,i)=t^sub_(i,i) and t_(n,k)=∑_i=k^nW_i^n,kt^sub_(i,i)=∑_i=k^nW_i^n,kt_(i,i) hold.The expected sojourn time ℕ𝕋_n,k≡ E[t_(n,k)] of a general non-purging (n,k) fork-join queue can be represented by a linear combination of the expected sojourn times of the (λ,μ)-equivalent basic fork-join queues:ℕ𝕋_n,k=∑_i=k^nW_i^n,k𝕋_iwhere 𝕋_i is the expected sojourn time of the (λ,μ)-equivalent basic (i,i) fork-join queue and W_i^n,k is the corresponding W coefficient. E[t_(n,k)]=E[∑_i=k^nW_i^n,kt_(i,i)]=∑_i=k^nW_i^n,k𝕋_iThe independence of the service times of the sub-queues is not required for Theorem <ref> and <ref> to hold. To put Theorem <ref> into practices, we need to find the existing computing methods of 𝕋_i.§.§ Approximations for Exponential Fork-Join Queues This paper uses the termto denote the fork-join queues under a Poisson job arrival process and with iid exponential sub-queues' service time distributions.For exponential (n,n) fork-join queues, there exist two reliable approximation methods: Nelson's approximation<cit.> and Varma's approximation<cit.>. Accordingly, we can propose two approximation methods for exponential non-purging (n,k) fork-join queues: Thebased on Nelson's approximation and thebased Varma's approximation.§.§.§ The Nelson-LT Approximation For exponential (n,n) fork-join queues, Nelson et al. <cit.> proposed the following approximations:𝕋_1=1/μ(1-ρ), 𝕋_2=12-ρ/8𝕋_1 𝕋_n≃[H_n/H_2+4/11(1-H_n/H_2)ρ]𝕋_2, n≥ 2,where λ and μ are respectively the job arrival rate and the sub-queue's service rate, ρ≡λ/μ is called the load factor of the queue, 𝕋_n is the expected sojourn time of (n,n) basic fork-join queue, and H_n=∑_i=1^n1/i is called the harmonic number. Consequently, our approximations can be specified in the following theorem. According to Nelson's approximation, the expected sojourn time ℕ𝕋_n,k of an exponential non-purging (n,k) fork-join queue can be approximated as follow:ℕ𝕋_n,k≃{[ [ n/μ(1-ρ)+12-ρ/88μ(1-ρ)×; ∑_i=2^nW_i^n,1[11H_i+4ρ(H_2-H_i)/H_2] ] k=1,; 12-ρ/88μ(1-ρ)∑_i=k^nW_i^n,k[11H_i+4ρ(H_2-H_i)/H_2]k≥ 2. ].where λ and μ are respectively the job arrival rate and the sub-queue's service rate, and ρ≡λ/μ is the load factor of the target queue. Specially, we replace any negatively approximated ℕ𝕋_n,k with 0.We examine above linear-transformation approximations against the mean sojourn times of jobs sampled from various simulated exponential non-purging fork-join queues (details of simulations in this paper can be found in Appendix). The value of W coefficients used in Eq. <ref> are given by Eq. <ref>. The results depicted in Fig. <ref> confirmed the validity of our technique under a moderate value of n (n≤ 50) and a relatively large value of k (compared to n). The relative errors are calculated as APP/SIM-1.We notice that when k is relatively small, the approximation tends to be uncontrollable, which can be due to the fact that the smaller k is, the more items in Eq. <ref> are summed, and consequently, the more relative errors introduced by Nelson's approximations are accumulated.These results also confirmed the high-precision merit of Nelson's approximations, since W coefficients tend to be very large with the increase of n, for example W^25,9_16=13146544125. As a result, the relative error introduced by Nelson's approximation has to be amplified by the large value of the corresponding W coefficient.§.§.§ The Varma-LT ApproximationFor exponential non-purging (n,n) fork-join queues, Varma et al. <cit.> gave another well-known approximation method based on the so-called light traffic interpolation technique. The expected sojourn time is approximated as𝕋_n≃[H_n+(V_n-H_n)λ/μ]1/λ-μ, 0≤λ<μwhere V_n= ∑_r=1^nnr(-1)^r-1∑_m=1^rr m(m-1)!/r^m+1. As our linear transformation technique is orthogonal to the concrete approximation methods for basic fork-join queues, we replace Nelson's approximation with the above Varma's approximation to try to avoid the approximations' uncontrollability appeared in Theorem <ref>. Consequently, the new approximations can be specified in the following theorem. According to Varma's approximation, the expected sojourn time of an exponential non-purging (n,k) fork-join queue can be approximated as follow:ℕ𝕋_n,k≃∑_i=k^nW_i^n,k[H_i+(V_i-H_i)λ/μ]1/λ-μwhere λ and μ are respectively the job arrival rate and the sub-queue's service rate. We examine Eq. <ref> against the mean sojourn times of jobs sampled from various simulated non-purging fork-join queues. The employed W coefficients are calculated from Eq. <ref>. The results depicted in Fig. <ref> showed that the new approximations are fairly good when n≤ 10 and ρ is not too extreme (ρ≤ 0.9): The relative error is generally less than 10%, which is much more controllable than the approximations given by Theorem <ref>.However, as the Varma's approximation itself becomes out of control when n≥ 55, Theorem <ref> is more valuable in general cases.§ BOUNDS FOR PURGING (N,K) FORK-JOIN QUEUESUnlike in non-purging (n,k) fork-join queues, the sojourn time distribution of a sub-task in purging (n,k) fork-join queues changes when either n or k varies, and thus differs from the sojourn time distribution of a sub-task in the (λ,μ)-equivalent basic fork-join queues. As a result, we cannot build similar linear-transformation approximations for purging queues. However, the expected sojourn times of non-purging queues can serve as the upper bounds of the (λ,μ)-equivalent purging queues' expected sojourn times. §.§ The Naive Upper BoundsThe expected sojourn time ℙ𝕋_n,k of a purging (n,k) fork-join queue can be upper bounded as follow:ℙ𝕋_n,k≤∑_i=k^nW_i^n,k𝕋_iwhere 𝕋_i is the expected sojourn time of the (λ,μ)-equivalent basic fork-join queue.The right side of Eq. <ref> is the expected sojourn time of the (λ,μ)-equivalent non-purging (n,k) fork-join queue. As the expected sub-queue length of a stable purging (n,k) fork-join queue is no longer than that of the (λ,μ)-equivalent stable non-purging queue, the expected sojourn time of the purging (n,k) fork-join queue is thus no larger than that of its non-purging (λ,μ)-equivalent queue. Comparing with Existing Stat-of-the-Art Upper Bounds For purging (n,k) fork-join queues with iid service time distributions and under a Poisson job arrival process, Existing state-of-the-art upper bounds on the expected sojourn time are the split-merge upper bounds given by Joshi et al. <cit.>:ℙ𝕋_n,k≤ E[X_(n,k)]+ λ E[X_(n,k)^2]/2(1-λ E[X_(n,k)]) where λ is the job arrival rate and X_(n,k) is the k^th order statistic of the iid service time rvs X_1 n. The right side of Eq. <ref> is the expected sojourn time of the (λ,μ)-equivalent (n,k) split-merge queue.Split-merge upper bounds become much looser than naive upper bounds when E[X_(1,1)]< E[X_(n,k)] and λ→ [1/E[X_(n,k)]]^-.When λ→ [1/E[X_(n,k)]]^-, the bounds given by Eq. <ref> approach +∞. Moreover, the bounds become meaningless when λ≥1/E[X_(n,k)]. As a contrast, our naive bounds are finite meaningful values, as long asλ < [μ≡1/E[X_(1,1)]]. Besides, μ>1/E[X_(n,k)]. Apparently there is a range of load factors ρ∈ [E[X_(1,1)]/E[X_(n,k)],1) cannot be bounded by Eq. <ref>, and the larger k is, the smaller bounded-able ρ range becomes, while the naive bounds are applicable as long as ρ<1. Naive upper bounds become much looser than split-merge upper bounds when k → 1.When k → 1, more and more sub-tasks are purged from both the sub-queues and the service stations when the k^th finished sub-task is acknowledged by the purging (n,k) queue, as a result of which, the expected sub-queue length becomes shorter and shorter than that of the (λ,μ)-equivalent non-purging (n,k) queue. On the contrary, the expected sub-queue length of the target purging (n,k) fork-join queue becomes closer and closer to that of the (λ,μ)-equivalent (n,k) split-merge queue. At last, the purging (n,1) fork-join queue equates to the (n,1) split-merge queue, which gives us the following exact closed-form solution: ℙ𝕋_n,1= E[X_(n,1)]+ λ E[X_(n,1)^2]/2(1-λ E[X_(n,1)]). On the other side of Corollary <ref>, when k → n, the expected sub-queue length of the purging (n,k) fork-join queue becomes closer and closer to that of the (λ,μ)-equivalent non-purging queues. At last, the purging (n,n) fork-join queue equates to the (λ,μ)-equivalent non-purging (n,n) fork-join queue. §.§ The Refined Upper Bounds For a purging (n,k) fork-join queue with iid service time distributions and under a Poisson job arrival process, the expected sojourn time ℙ𝕋_n,k can be upper bounded as follows:ℙ𝕋_n,k≤{[ ∑_i=k^nW_i^n,k𝕋_i λ≥1/E[X_(n,k)],; min(∑_i=k^nW_i^n,k𝕋_i,[E[X_(n,k)]+; λ E[X_(n,k)^2]/2(1-λ E[X_(n,k)]) ])otherwise. ].where 𝕋_i is the expected sojourn time of the (λ,μ)-equivalent (i,i) fork-join queue, λ is the job arrival rate, and X_(n,k) is the k^th order statistic of the iid service time rvs X_1 n.According to Corollary <ref> and <ref>, and excluding the split-merge bounds when λ≥1/E[X_(n,k)], we derive Eq. <ref>. Although Eq. <ref> has extended the bounded-able range of ρ from [0,E[X_(1,1)]/E[X_(n,k)]) to [0,1), there is still an untamed range of ρ, since purging (n,k) fork-join queues may still keep stable even when ρ≥ 1. Upper Bounds for Exponential Queues Specially, we give the refined upper bounds for exponential purging (n,k) fork-join queues. For exponential purging (n,k) fork-join queues, the expected sojourn time ℙ𝕋_n,k can be upper bounded as follows:ℙ𝕋_n,k≤{[[ 12-ρ/88μ(1-ρ)∑_i=k^nW_i^n,k[11H_i+4ρ(H_2-H_i)/H_2] ]_Naive; when k≥ 2and ρ≥1/H_n-H_n-k,; ; min(Naive,[H_n-H_n-k/μ+; ρ[(H_n^2-H_(n-k)^2)+(H_n-H_(n-k))^2]/2μ[1-ρ(H_n-H_n-k)] ]^Split-Merge);otherwise. ].where λ and μ are respectively the job arrival rate and the sub-queue's service rate, ρ≡λ/μ, and H_n^2=∑^n_i=11/i^2.The split-merge part of the Eq. <ref> is already given by Theorem 2 of <cit.>. According to Theorem <ref> and <ref>, we derive Eq. <ref>. We make numerical comparisons between naive upper bounds and split-merge upper bounds, and examine refined upper bounds against the mean sojourn times of jobs sampled from various simulated purging fork-join queues. The value of W coefficients used in Eq. <ref> are given by Eq. <ref>. We find that: * The split-merge upper bounds become extremely pessimistic with the increase of k, but tend to be much tighter than naive bounds when k is small (see Fig. <ref> (a)). These results are consistent with Corollary <ref> and <ref>.* There is still plenty of room of improving the upper bounds when k is relatively large (see Fig. <ref> (b)). §.§ Lower BoundsTo complete our work, we review and compare the state-of-the-art lower bounds for purging (n,k) fork-join queues.For purging (n,k) fork-join queues with iid service time distributions and under a Poisson job arrival process, the state-of-the-art lower bounds are the split-merge lower bounds given in <cit.>:ℙ𝕋_n,k≥ E[X_(n,k)]+λ E[X_(n,1)^2]/2(1-λ E[X_(n,1)]) where λ is the job arrival rate and X_(n,k) is the k^th order statistic of the iid service time rvs X_1 n.For exponential purging (n,k) fork-join queues, there is another staging analysis based lower bound <cit.>:ℙ𝕋_n,k≥H_n-H_n-k/μ+ρ(H_n(n-ρ)-H_(n-k)(n-k-ρ))/μwhere λ and μ are respectively the job arrival rate and the sub-queue's service rate, ρ≡λ/μ, and H_n(n-ρ)=∑_i=1^n1/i(i-ρ). This staging lower bound is adapted from the staging lower bound for basic fork-join queues proposed in <cit.>, which requires a memory-less property of the service time distribution. Accordingly, this bound cannot be applied to general purging queues. The staging lower bounds are tighter than the split-merge lower bounds.The exact form of Eq. <ref> for exponential queues can be transformed into:ℙ𝕋_n,k≥H_n-H_n-k/μ+ρ (H_n(n-ρ)-H_(n-1)(n-1-ρ))/μ.As H_(n-k)(n-k-ρ))< H_(n-1)(n-1-ρ)) when k>1, we derive Theorem <ref>. We examine bounds for exponential purging queues against simulations. Fig. <ref> depicts the large gap between the upper bounds and the lower bounds when k is relatively large, due to which, we can hardly find reasonable approximations of the expected sojourn time of purging (n,k) fork-join queues. § DISCUSSION Currently, there is an unnegligible limitation when put the new proposed linear transformation technique into practices: The value of W coefficient given by Eq. <ref> increases explosively with the increase of n, for example W^40,37_37=4037=9880, W^50,37_37=5037=354860518600, and W^100,37_37=10037=3.42002954749393 × 10^27, as a result of which, the original negligible relative error of 𝕋_i in Theorem <ref> will be amplified into a huge deviation of the approximated ℕ𝕋_n,k. Consequently, the practicability of the linear transformation technique depends on whether we can find high-precision approximated or simulated 𝕋_i. For example, when we use simulated 𝕋_i to estimate ℕ𝕋_n,k, the results are far from acceptable (see Table <ref>), and also far behind the Nelson-LT approximations (see Fig. <ref>). These surprising results can be due to the fluctuation of the simulated 𝕋_i (see Fig. <ref>). Fig. <ref> has depicted that the accuracy of the Nelson-LT approximation is acceptable when n≤ 50 and k is relatively large (for example, k> 37 when n=50). However the approximations are similarly unacceptable when k is relatively small (see Table <ref>). The fundamental solution of this problem is to scale down W coefficients,a promising research direction needs further efforts. Anyway, this linear transformation technique is capable of estimating/bounding the performance of most of practical non-purging/purging (n,k) fork-join queueing systems, where the replication factor (n) rarely exceeds 10, a result of cost-effective tradeoff. For example, the replication factor of either Dynamo or Cassandra is commonly 3. Under such configurations, a write operation in Dynamo/Cassandra will be forked into 3 copies exactly. For such n≤ 10 cases, we have proposed the fairly good Varma-LT approximations (Theorem <ref>).From another perspective, the linear transformation technique can be used to check simulators' precision and to find better closed-form approximations for basic fork-join queues.§ RELATED WORKS Order Statistics Bertsimas et al. <cit.> gave some tight bounds on the expectation of the k^th order statistic given the first and second moment information on n real-valued rvs. We gave exact linear transformations for k^th order statistic instead of bounds. Shi et al. <cit.> proposed a dynamic programming algorithm to compute the order statistics of a family of correlated rvs whose distributions are not required to be identical. This algorithm relies on the existence of computing methods for the distributions of both minimum and maximum of target rvs. As a contrast, our work is more formal and easier to practice for the reveal of the closed-form linear transformation, and only relies on the existence of computing methods for the maximum's distribution. Basic Fork-Join Queues For n=2 fork-join queues under a Poisson job arrival process: 1) Flatto et al. <cit.> gave the queue length distribution for exponential queues in stable state; 2) Baccelli <cit.> extended Flatto's work to queues with general service time distributions; 3) Nelson et al. <cit.> proposed the exact closed-form solution of the expected sojourn time for exponential queues in stable state.For n≥ 3 exponential fork-join queues, the most influential approximation work was proposed by Nelson et al. in 1988 <cit.>, which is based on the fact that the sojourn times X_1 k of sub-tasks 1, 2,...,k are associated rvs <cit.>, whose maximum can be bounded by the maximum of their iid equivalents. The lower bound is obtained by neglecting queueing effects. The approximation is a linear mixture of the upper and lower bounds. Parameters of the mixture are learned based on the mean sojourn times of jobs sampled from simulated basic fork-join queues. Varki et al. <cit.> improved the lower bound by using an staging analysis technique <cit.> based on the memory-less property of the exponential service time distribution, and use the mean value of Nelson's upper bound and the staging lower bound as the approximation. According to experiments in <cit.>, Nelson's approximation is still the most reliable one for exponential queues, compared to following works including <cit.> and <cit.>.Varma et al. <cit.> extended Nelson's approximation to general service time distributions using a light traffic interpolation technique. Thomasian et al. <cit.> employed linear regression over the statistics of simulated fork-join jobs to find the parameters of their approximation equation for the expected sojourn time. However any change in service time distributions will require for re-simulations and re-regressions. Recently, Rizk et al. <cit.> proposed the first computable bounds on waiting and sojourn time of fork-join queues with general service time distributions by using martingales. However the upper bound is looser than Nelson's when it comes to the exponential service time distribution. Fidler et al. <cit.> considered the multi-stage nature of many fork-join queue networks, and proposed their end-to-end delay bounds.We refer readers to <cit.> for a more comprehensive survey on fork-join queuing systems. To conclude, our work is orthogonal to existing approximation methods for basic fork-join queues. Purging (n,k) Fork-Join QueuesThere are some exact quantity analyses <cit.> for purging (n,1) fork-join queues, as it is equivalent to a single queue with n times the service rate. Gardner et al. <cit.> gave comprehensive research on purging (n,1) fork-join queues, with considerations on multi-class jobs, interferences from un-forked jobs and heterogeneous service time distributions. Lee et al. <cit.> take the purging overheads into consideration, since the cancellation of running jobs typically incurs unnegligible delays in practice.For purging (n,k>1) fork-join queues, there are even no applicable approximations currently. Joshi et al. <cit.> extended the staging analysis to exponential (n,k) fork-join queues to find the lower bounds. Bounds for queues with general service time distributions are given by <cit.> and <cit.>, by resorting the fork-join queue to the split-merge queue model, where all empty sub-queues are blocked until any k sub-tasks of the current job are completed. As depicted in Fig. <ref> (a), the proposed upper bounds tend to be very rough when increasing k or the load factor ρ.Non-Purging (n,k) Fork-Join Queues A typical use case of non-purging (n,k) fork-join queues is the writing process in Cassandra <cit.>. Fidler et al. <cit.> gave non-asymptotic statistical bounds on the sojourn time of non-purging (n,k) fork-join queues. As a contrast, we give proper approximations instead of bounds.§ CONCLUSION AND FUTURE WORK Despite the popularity of (n,k) fork-join queues, there were no practical existing approximations of their expected sojourn times. Only some rough bounds have been given, which tend to be extremely loose when increasing k or the load factor ρ. This paper gave the first applicable approximation method for non-purging (n,k) fork-join queues and tackled the uncontrollability of the bounds for purging (n,k) fork-join queues: * A brand-new closed-form linear transformation technique is developed for jointly-identical rvs, which provides a bridge to reduce the sojourn time approximation problem of non-purging (n,k) fork-join queues to that of basic fork-join queues.* Improvements over upper bounds on the expected sojourn time of purging (n,k) fork-join queues are also gained by resorting the purging queues to their non-purging (λ,μ)-equivalents. Above innovations are examined by simulation experiments and numerically compared to the stat-of-the-arts. Results show that this linear transformation approach is practiced well for exponential (n,k) fork-join queues with moderate n and relatively large k. However, as currently found W coefficients (coefficients of the linear combination) increase explosively with the increase of n, there is an uncontrollable deviation in new proposed approximations when n is large and k is relatively small. Fortunately, approximations for real-life fork-join systems are unlikely influenced by this problem.In the future, more efforts should be put into: Scaling down W coefficients, improving the approximations for basic fork-join queues with the help of the linear transformation technique, and evaluating the performance of real-life (n,k) fork-join systems as complement to existing experimental methods <cit.>.The simulator employed by this work is Forkulator-p: https://github.com/excelwang/forkulator-p, which is modified from Forkulator <cit.> with additional features of simulating purging queues. For each (n,k) pair, the sub-queue's service rate μ is set to 1, simulated jobs are sampled at a rate of 1%, and the mean sojourn times are calculated on 10000 samples.The solver for currently found W coefficients and some pre-calculated values can be accessed from: https://github.com/excelwang/WCoefficients. We give some frequently used W^n,k_i coefficients bellow. =2.5pt1.8533405[W^3,1_1: 3W^6,3_4: -45W^8,1_7: 8W^9,2_6: 420W^10,2_3: -240; W^3,1_2: -3 W^6,3_5: 36 W^8,1_8: -1 W^9,2_7: -216 W^10,2_4: 630;W^3,1_3: 1W^6,3_6: -10 W^8,2_2: 28 W^9,2_8: 63 W^10,2_5: -1008;W^3,2_2: 3 W^6,4_4: 15 W^8,2_3: -112 W^9,2_9: -8W^10,2_6: 1050; W^3,2_3: -2W^6,4_5: -24W^8,2_4: 210 W^9,3_3: 84W^10,2_7: -720;W^3,3_3: 1 W^6,4_6: 10 W^8,2_5: -224 W^9,3_4: -378 W^10,2_8: 315;W^4,1_1: 4W^6,5_5: 6W^8,2_6: 140W^9,3_5: 756 W^10,2_9: -80; W^4,1_2: -6 W^6,5_6: -5W^8,2_7: -48 W^9,3_6: -840W^10,2_10: 9;W^4,1_3: 4W^6,6_6: 1W^8,2_8: 7W^9,3_7: 540 W^10,3_3: 120; W^4,1_4: -1W^7,1_1: 7 W^8,3_3: 56 W^9,3_8: -189W^10,3_4: -630;W^4,2_2: 6W^7,1_2: -21 W^8,3_4: -210 W^9,3_9: 28W^10,3_5: 1512; W^4,2_3: -8 W^7,1_3: 35W^8,3_5: 336W^9,4_4: 126 W^10,3_6: -2100;W^4,2_4: 3W^7,1_4: -35 W^8,3_6: -280 W^9,4_5: -504W^10,3_7: 1800;W^4,3_3: 4 W^7,1_5: 21W^8,3_7: 120W^9,4_6: 840W^10,3_8: -945; W^4,3_4: -3 W^7,1_6: -7W^8,3_8: -21 W^9,4_7: -720 W^10,3_9: 280;W^4,4_4: 1W^7,1_7: 1 W^8,4_8: 35W^9,4_8: 315W^10,3_10: -36;W^5,1_1: 5 W^7,2_2: 21 W^8,4_4: 70W^9,4_9: -56 W^10,4_4: 210;W^5,1_2: -10W^7,2_3: -70 W^8,4_5: -224 W^9,5_8: -315 W^10,4_5: -1008; W^5,1_3: 10W^7,2_4: 105W^8,4_6: 280 W^9,5_9: 70W^10,4_6: 2100; W^5,1_4: -5W^7,2_5: -84 W^8,4_7: -160W^9,5_5: 126 W^10,4_7: -2400;W^5,1_5: 1 W^7,2_6: 35W^8,5_8: -35 W^9,5_6: -420W^10,4_8: 1575; W^5,2_2: 10 W^7,2_7: -6 W^8,5_5: 56W^9,5_7: 540W^10,4_9: -560;W^5,2_3: -20 W^7,3_3: 35 W^8,5_6: -140W^9,6_8: 189 W^10,4_10: 84; W^5,2_4: 15 W^7,3_4: -105W^8,5_7: 120W^9,6_9: -56 W^10,5_5: 252; W^5,2_5: -4W^7,3_5: 126 W^8,6_8: 21 W^9,6_6: 84 W^10,5_6: -1050; W^5,3_3: 10W^7,3_6: -70 W^8,6_6: 28 W^9,6_7: -216W^10,5_7: 1800;W^5,3_4: -15 W^7,3_7: 15W^8,6_7: -48W^9,7_8: -63 W^10,5_8: -1575;W^5,3_5: 6 W^7,4_4: 35 W^8,7_8: -7 W^9,7_9: 28 W^10,5_9: 700;W^5,4_4: 5W^7,4_5: -84W^8,7_7: 8 W^9,7_7: 36 W^10,5_10: -126; W^5,4_5: -4 W^7,4_6: 70W^8,8_8: 1W^9,8_8: 9 W^10,6_8: 945;W^5,5_5: 1W^7,4_7: -20W^9,1_1: 9 W^9,8_9: -8W^10,6_9: -560;W^6,1_1: 6 W^7,5_5: 21W^9,1_2: -36W^9,9_9: 1W^10,6_10: 126;W^6,1_2: -15W^7,5_6: -35 W^9,1_3: 84W^10,1_1: 10 W^10,6_6: 210; W^6,1_3: 20 W^7,5_7: 15 W^9,1_4: -126 W^10,1_2: -45W^10,6_7: -720;W^6,1_4: -15W^7,6_6: 7W^9,1_5: 126 W^10,1_3: 120W^10,7_8: -315;W^6,1_5: 6 W^7,6_7: -6W^9,1_6: -84W^10,1_4: -210 W^10,7_9: 280; W^6,1_6: -1W^7,7_7: 1 W^9,1_7: 36 W^10,1_5: 252W^10,7_10: -84; W^6,2_2: 15W^8,1_1: 8 W^9,1_8: -9W^10,1_6: -210 W^10,7_7: 120;W^6,2_3: -40W^8,1_2: -28W^9,1_9: 1 W^10,1_7: 120W^10,8_8: 45; W^6,2_4: 45 W^8,1_3: 56 W^9,2_2: 36 W^10,1_8: -45 W^10,8_9: -80;W^6,2_5: -24W^8,1_4: -70 W^9,2_3: -168W^10,1_9: 10 W^10,8_10: 36;W^6,2_6: 5 W^8,1_5: 56W^9,2_4: 378 W^10,1_10: -1W^10,9_9: 10; W^6,3_3: 20W^8,1_6: -28 W^9,2_5: -504W^10,2_2: 45 W^10,9_10: -9; ] IEEEtran | http://arxiv.org/abs/1707.08860v7 | {
"authors": [
"Huajin Wang",
"Jianhui Li",
"Zhihong Shen",
"Yuanchun Zhou"
],
"categories": [
"cs.PF",
"cs.DC",
"cs.NI",
"stat.AP"
],
"primary_category": "cs.PF",
"published": "20170727133513",
"title": "Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation Approach"
} |
Analysis of the behavior of singlet pairs in inorganic crystal XOCl(X=Ti,Bi) Chen-Huan Wu [email protected] of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China December 30, 2023 ========================================================================================================================================= This article explore the behavior of singlet pairs in Inorganic crystals and take the Ti-Ti and Bi-Bi dimer for example, including the transition in critical temperture by directly or indirectly. Through the analysis, It is proposed that with the decrease of temperature, the strength of spin-orbit coupling (SOC) increases and the phase difference also increases. In the one-dimensional spin 1/2 chain system of TiOCl and BiOCl crystals, the possibility of reversible parameter modulation is proposed bythe calculation of the first principle analyzing the process and analysis of the structural phase transition process. It's shown that the different sturcture transition is relate to the difference of the dimer structure and the fluctuation of orbital orderthrough the comparative study of TiOCl and BiOCl. § INTRODUCTIONThe critical temperature of phase transition of the inorganic crystal TiOCl has been observed as T_c1=67 k and T_c2=90 k , and the two transitions are one order and two order transition, respectively <cit.>.The former is accomplished by the two-fold superstructure of Ti-Ti singlet pairs in a layer-shape SU(N) square lattice, and since it's the incommensurate spin-Peierls state or the commensurate dimerized state for N≥ 5 <cit.>, there is dominated by the spin-Peierls state in low temperature through the first order transition in T_C1and resulting in the existence of spin gap in intermedia phase,the gap is becomes the pseudospin with the increase of temperature.The first order transition is accompany with a huge change of magnetic behavior which produce a magnetic gap below T_c1 and dominated by a low symmetrymonoclinic P2_1/m phase <cit.>. The transition to the commensurate dimerized state is happen within the T_c1 and T_c2. These behavior is indeed the Ti-Ti dimerization modulated by the temperature. As for the semiconductor BiOCl which has the similar propertice in chemical and structure with TiOCl but with more higher symmetry phase P4/nmm,we can also see the important role of the Bi-Bi dimer in the structural transition. The singlet pairs of Ti-Ti dimer and Bi-Bi dimer is horizontal presented in the Fig.1, they are the view of ab plane with the top-to-down layer-struture,by increase the degrees of spin-orbit coupling (SOC), the amplitude of bond <cit.> is enlarged and therefore the collective dynamics is more severe <cit.>. Under this condition, the in-phase array singlet pairs is possible to transition to the out-of-phase arrangment, this is the resonce why it's a good platform tostudy the propertices of solid material. I also indicate and compare the dimers behavior of TiOCl and BiOCl to see whether the Bi-Bi dimers in BiOCl have the samepropertices as TiOCl. The reason why using layer-shape TiOCl and square-shape BiOCl is these two crystals have similar electron configuration (in ab plane),and they show a similar nature in dimers, magnetic frustraction, and the fluctuation effects <cit.>. They have the similar peak-position of the Raman spectra in roomtemperature<cit.> as well as the similar band gap (see below). § MODEL AND ANALYSIS Unlike the VBS state which is temperature-independent if non-doping <cit.>, the spin-Peierls transition which relate to the obvious changes is found in TiOCl at low temperature ( 65 k ∼ 67 k, <cit.>, 26 k∼ 100 k, <cit.> ) even if subtract the Curie tail. Fig.1(a) show the layered struture of TiOCl, we can see three layers stack up and down along the a axis, and the distance between each layer is 1.8915Å acording to data of Ref.<cit.>.Fig.1(b) show the irregular octahedron formed in TiOCl crystal, and (c) is the irregular octahedron decahedron formed in BiOCl crystal <cit.>. According to the data of Ref.<cit.>, the nearest Ti-Ti ions pair is the one belong to different layer which mearsured as 3.21 Å <cit.> (i.e., the Ti ions in site 1 and site 2 which showed in Fig.1(a)).In the model shown in the Fig.2, there are two kinds of interaction, one of it is the superexchange interaction (orbit d_xz) along the a axis which via the oxygen orbitals and another is the directly exchange interaction from the chains in layer plana and formed by Ti-Ti singlet pairs along the b axis (orbit d_xy) which take a key role in the process of crystal sturcture transition. Both these two exchange interactiontogether to complete the phase transition process, i.e., containing both the interchain and intrachain interaction. Since the crystal field (see Fig.1(a)) split the d level into the excited one d_xy, d_yz, and d_xz <cit.>,it's conscious for what exhibited in Fig.3.Despite this, the variation of distence within Ti-Ti (along b axis) is nearly three times large than that of the nearest Ti-Ti<cit.>. That also reflect that there is a great difference in the strength of the exchange between these two interaction.The ab plane of TiOCl crystals which showed in Fig.2 is consist of the Ti-Ti dimers, the Fig.2 reflect the existence of tight-binding chains in different layers with almost the same distance. Fig.2(b) shows the ab plane ofBiOCl, it's obviously that the crystal of BiOCl is present as a square-shape due to it's tetragonal phase, and the distence of Bi-Bi spin pair which consisted of the majority and minority spin is not much difference. It's shown thatBiOCl is more stable that TiOCl due to its' superlattice order suggested by the XRD result<cit.>. Spacing of Ti-Ti dimer is shorter than that ofBi-Bi dimer according to the crystal structure models I using here (The spacing of Ti-Ti is 3.4653Å and for Bi-Bi is 3.9032Å here which are close to 3.3415Åand 3.980Å from Ref.<cit.> and Ref.<cit.> respectively). The distance withineach singlet pair and that between two pairs is not equal, but since the difference is within 0.1Å∼ 0.2Åin most case, so it's ignorable. So Hamiltonian of the Heisenberg ladder model which shown in the Fig.1 can be written as H=-J_1∑_⟨ i,j ⟩ S_i· S_j-J_2∑_⟨ i,j ⟩⟨ k,l ⟩( S_i· S_j)( S_k· S_l)-J_3∑_⟨ i,j ⟩⟨ k,l ⟩⟨ m,n ⟩( S_i· S_j)( S_k· S_l)( S_m· S_n) Where J_1, J_2, and J_3 is the coupling constant in this two-, four-, and six-spin configuration and S is the spin operator of different site, and the minus sign represents antiferromagnetic.In fact it's the zigzag chains if takes the view of ac plane of TiOCl. The spin operators in (···) describe the correlation of singlet pair. For this S=1/2 system H_ik the singlet projection is⟨ S_i· S_j⟩=1/4- S_i· S_j <cit.>, and the strength of interaction between singlet pairs ⟨ i,j ⟩ is U_ij=2C∫|ϕ_i(x)|^2|ϕ_j(x)|^2d^3x where C is interaction constant and ϕ is the wave function of these two sites <cit.>. It's the similar form for orther singlet pairs. This ladder model(Equ.(1)) conclude the effect of itinerant electrons which is not localized and have contribute to the spin-Peierls transition, while the localized electron occupy the degenerate orbit states. The S_i in Equ.(1) has S_i=1/2∑ c_iσ^†ρ^μc_iσ where σ denotes the spin andρ^μ is the pauli matrix with μ=x,y,z. c_iσ^† and c_iσ are the creation operator and annihilate operator in site i respectively which has c_qσ=1/√(N)∑_ie^iqrc_iσ and c_qσ^†=1/√(N)∑_ie^-iqrc_iσ^†<cit.> where q stands the momentum space of Brillouin zone,The repretation of this three singlet pairs system in TiOCl using the hopping parameter is H=-t_1(c^†_iσc_jσ+c^†_jσc_iσ)-t_2(c^†_kσc_lσ+c^†_lσc_kσ)-t_3(c^†_mσc_nσ+c^†_nσc_mσ)+U∑_r=i∼ n n_rσ_↑n_rσ_↓ where the last term is the site antiferromagnetic interaction (Coulomb repulsion), and this term equal to zero for the zero modes. Here ignore the weak interaction of ferromagnetic between different layers and the third particle in same layer. according to Ref,<cit.> the hopping t of TiOCl for nearest neighbor atoms (Fig.2(a)) in one layer is -0.21 which is about seven times of the hoppingbetween first atom and third atoms and also the same multiple compare to hopping along a axis. Further, since the special tructure of BiOCl which exhibit a square shape in the view of ab plane and a hexagon shape in the bc plane,the caculation of square lattice model and honeycomb lattice model has been presented in the Ref.<cit.> and Ref.<cit.>, respectively.The ab plane shown in Fig.2(a) only exhibit the directli interaction intrachains, the interchains one is ont shown.Since the coupling strength J (discrebe the exchange interaction) fot TiOCl of intrachain is much large than that of interchain's which is about 300 times large than the latter one <cit.>, and for BiOCl this multiple is bigger (according to the DOS pattern). The coupling J=2t^2/U for antiferromagnetic exchange<cit.> where t is the hopping between nearest spin antiferromagnetic pairs. According to this we can know that the on site interaction U for Ti-Ti dimer which along the b axis is an order of magnitude smaller than the Ti-Ti in in adjacent layers. It's the two-fold superstructure which appear below T_c1 for TiOCl. Since the superstructure parameter can be separated into basic structure parameter <cit.>, and the modulation parameters can be applied in adjustion in amplitude and interaction strength of each atom in this dimers system. This two fold structure can lead to two independent atoms in supercell <cit.>, i.e., the spin singlet pair, and it's found that the interaction strength of the pairs atoms is doubled in the first transition point for TiOCl <cit.>and indicate that the two phases which dominate before and after the phase transition are direct coexistence, that's consistent with the characteristics offirst order transition <cit.>. § SIMULATION RESULTS AND DISCUSSION The simulation and the first-principle calculations are performed use the method ofVienna ab initio simulation package (VASP). The split of crystal field U-term in LDA+U caculation (LDA, local-density-approximate) is relate to the spin arrangement and the occupation of bands,to caculate the antiferromagnetic interaction between the singlet pairs, the U-term can be set in the orbitals which occupying by the interacting atoms (i.e., the d orbital here), and that will lead to split of d_xy band along the b axis (not shown here). That's clearly reflected in the transition ofin-phase pattern and out-of-phase pattern (Fig.2) as well as the peak distributi on density of states (DOS) graph (Fig.3). By analyzing Fig.3, it's obvious that the density of total state is simplely the summation of the each orbital state's density, and the peak of Bi-d's state density is much large than that of Ti-d's. That means the interaction strength within the pairing atoms of Bi-Bi is much large than that of Ti-Ti.In Fig.3(c) and (d), we can see that the peaks of O-p and Cl-p is in the left side and far away from the peak of Ti-d, and for BiOCl the peaks of O-p and Cl-p is in theright side and far away from the peak of Bi-d (This phenomenon is also reflected in the bound struture of Fig.4).And the band gap width of BiOCl is also much large than the TiOCl's due to semiconductor characteristics of BiOCl. To indicate the different position of these two mainly peak in Fig.3(c) and (d), I present the DOS of Bi_1-xTi_xOCl with Bi:Ti=1:1. In this graph, the peaks of all lines have fallen compare to above ones, and the peaks of O-p,Cl-p, and Ti-d are slightly moving to the Fermi level. This exhibit the reduction of the energy gap and the interaction between O-p,Cl-p, and Ti-d is increase. In Fig.4, the LDA band structure of BiOCl is shown, the two subgraphs in the left side with band gap of 2.186 eV is the case of without spin-orbit coupling (SOC), and the two subgraphs in the right side with band gap of 2.107 eV is the case with SOC. The band sturcture has already presented in the Ref.<cit.> and I don'trepeat here. It's easy to find that the band sturcture of both the TiOCl and BiOCl have no band touching point, and the band gaps of these two crystal is almost equal. For a stable coupling, the weak SOC corresponding to Gaussian distribution of atoms and the strong one corresponding to non-Gaussian distribution <cit.>, and the simulation results from Ref.<cit.> also show that Gaussian distribution is broken with the increase of strength of SOC, and the system becomeout-of-phase, and the spin polarization ratio of these two spin in a singlet pair μ_s=n_↑-n_↓/n_totis also increse if the strength of SOC is large enough, where n_↑ and n_↓ is the majority spin atom (here is the up-spin) and the minority spin (here is the down-spin) atoms' number and n_tot is the total number of atoms, and a violent high-frequency vibration signal appeared<cit.>. Since the appearence of antiferromagnetic spin interaction is due to the change of electron structure essentially, the disturbution of density of electron also reflect the crystal sturcture and play a key role in the spin, charge, and magnetic order as well as the phase transition, and that's also how crystal field Accroding to analysis of TiOCl's transition, I expect that the strength of SOC of TiOCl is increase as the temperature decrease, and therefore the order properties is also weakened as the temperature decrease. That means in low temperature the spin polarization ratio is more large and therefore the spin long range order is hard to establish, a evident is thecredible signal of nuclear magnetic resonance (NMR) under low temperature <cit.>.The orbital fluctuation rise with the temperature rise, which consistent with the previous inferences that too much orbital fluctuationis not conducive to the increasement of SOC, will gives rise to entropy, and enhance the spin ferromagnetic interaction between adjacent layers along the a direction (see Fig.1) <cit.>. The fluctuation also gives rise to a pseudo spin gap above T_c2 which no shown in the Fig.3<cit.>. The broad NMR signal also reflect the incommensurate propertice of spin-peierls state, and for TiOCl, according to Ref.<cit.>, a spin gap of430K which is observed from NMR<cit.> is exist as a intermediate phase separated from the undimerized paramagnetic state by the critical temperature T_c2 <cit.>,To explore the difference of these two dimers in Fig.2, the X-ray drffraction (XRD) pattern of doped crystals Bi_1-xTi_xOCl is presented in the Fig.5,the different lines corresponding to different doping ratio (here using the wt.%). Note that evry lines in the XRD pattern was raised by ten units for more convenient to theoretical description. Roughly, with the increase in Ti content, the peak value of the XRD is also increased, e.g., the highest peak of ratio 48.20 wt.% is near one hundred, and we can see that the peaks are mainly concentrated in two regions of 10 deg. and 30 deg.. In conclusion, since for both the TiOCl and BiOCl the singlet pairs in incommensurate structure phase can be modulated by the single modulation <cit.> (include both the temperature modulate and pressure modulate<cit.>),In principle, the structure distort effect of SOC is through the way of enhance the phase difference and change the electron configuration. It provide a good platform to indicate the SOC effect by the behaviours of singlet pairs with spin antiferromagnetic consist if the up-spin and down-spin (or the majority and minority spin). This effect which is obviously in the TiOCl crystal structure provide a new way of structure transition modulation by the phase difference modulation through the adjustion of harmonic potential in the laboratory. 99 Shaz MShaz M, van Smaalen S, Palatinus L, et al. Physical Review B, 2005, 71(10): 100405. Harada KHarada K, Kawashima N, Troyer M. Néel and Spin-Peierls Ground States of Two-Dimensional S U (N) Quantum Antiferromagnets[J]. 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Zhang K LZhang K L, Liu C M, Huang F Q, et al. Study of the electronic structure and photocatalytic activity of the BiOCl photocatalyst[J]. Applied Catalysis B: Environmental, 2006, 68(3): 125-129. Beach K S DBeach K S D, Alet F, Mambrini M, et al. SU (N) Heisenberg model on the square lattice: A continuous-N quantum Monte Carlo study[J]. Physical Review B, 2009, 80(18): 184401. Li Y QLi Y Q, Ma M, Shi D N, Zhang F C, Physical review letters, 1998, 81(16): 3527. Krimmel AKrimmel A, Strempfer J, Bohnenbuck B, et al. Incommensurate structure of the spin-Peierls compound TiOCl in zero and finite magnetic fields[J]. Physical Review B, 2006, 73(17): 172413. Khaliullin GKhaliullin G, Horsch P, Oleś A M. Spin order due to orbital fluctuations: Cubic vanadates[J]. Physical review letters, 2001, 86(17): 3879. Ma FMa F, Ji W, Hu J, et al. First-principles calculations of the electronic structure of tetragonal α-FeTe and α-FeSe crystals: evidence for a bicollinear antiferromagnetic order[J]. Physical review letters, 2009, 102(17): 177003. Lemmens PLemmens P, Choi K Y, Caimi G, et al. Giant phonon softening in the pseudogap phase of the quantum spin system TiOCl[J]. Physical Review B, 2004, 70(13): 134429. Biswas ABiswas A, Das R, Dey C, Banerjee R, Poddar P, Ligand-free one-step synthesis of 001 faceted semiconducting BiOCl single crystals and their photocatalytic activity[J]. Crystal Growth and Design, 2013, 14(1): 236-239. Elcoro LElcoro L, Etxebarria I, Perez-Mato J M. Modulation parameters in incommensurate modulated structures with inflation symmetry[J]. Journal of Physics: Condensed Matter, 2000, 12(6): 841. Blanco-Canosa SBlanco-Canosa S, Rivadulla F, Pineiro A, et al. Enhanced dimerization of TiOCl under pressure: spin-Peierls to peierls transition[J]. Physical review letters, 2009, 102(5): 056406. Hubbard JHubbard J. Electron correlations in narrow energy bands[C]//Proceedings of the royal society of london a: mathematical, physical and engineering sciences. The Royal Society, 1963, 276(1365): 238-257. Lemmens PLemmens P, Choi K Y, Valentí R, et al. Spin gap formation in the quantum spin systems TiOX, X= Cl and Br[J]. New Journal of Physics, 2005, 7(1): 74. Zhou WZhou W, Jones R H, Thomas J M, et al. Ordering of lanthanum (3+) ions in the bismuth (3+) sublattice of layered oxychloride catalysts[J]. Chemistry of Materials, 1990, 2(3): 215-216. Baskaran GBaskaran G, Mandal S, Shankar R. Exact results for spin dynamics and fractionalization in the Kitaev model[J]. Physical review letters, 2007, 98(24): 247201. | http://arxiv.org/abs/1707.09033v1 | {
"authors": [
"Chen-Huan Wu"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170727202208",
"title": "Analysis of the behavior of singlet pairs in inorganic crystal XOCl(X=Ti,Bi)"
} |
red 1School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China 2Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023, China 3National Radio Astronomy Observatory, Socorro, NM 87801, USA 4Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China 5Key Laboratory for Research in Galaxies and Cosmology, Chinese Academy of Sciences, China 6Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, China [email protected] The Andromeda galaxy (M 31) hosts one of the nearest and most quiescent super-massive black holes, which provides a rare, but promising opportunity for studying the physics of black hole accretion at the lowest state. We have conducted a multi-frequency, multi-epoch observing campaign, using the Karl G. Jansky Very Large Array (VLA) in its extended configurations in 2011-2012, to advance our knowledge of the still poorly known radio properties of M 31*. For the first time, we detect M 31* at 10, 15 and 20 GHz, and measure its spectral index, α≈ -0.45±0.08 (S_νν^α), over the frequency range of 5-20 GHz.The relatively steep spectrum suggests that the observed radio flux is dominated by the optically-thin part of a putative jet, which is located at no more than a few thousand Schwarzschild radii from the black hole. On the other hand, our sensitive radio images show little evidence for an extended component, perhaps except for several parsec-scale “plumes", the nature of which remains unclear. Our data also reveal significant (a few tens of percent) flux variation of M 31* at 6 GHz, on timescales of hours to days.Furthermore, a curious decrease of the mean flux density, by ∼50%, is found between VLA observations taken during 2002-2005 and our new observations, which might be associated with a substantial increase in the mean X-ray flux of M 31* starting in 2006. § INTRODUCTIONThe advent of multi-wavelength surveys in the past two decades has led to the consensus that the vast majority of super-massive black holes (SMBHs) in the local Universe are radiatively quiescent (e.g., ), with bolometric luminosities only a small fraction of their Eddington limit (≲10^-3L_ Edd).Direct probes of these so-called low-luminosity active galactic nuclei (LLAGNs; see review by ) prove to be challenging and often require high-resolution, high-sensitivity observations.Nevertheless, studies of LLAGNs are crucial for our comprehensive understanding of the physics of SMBH accretion and feedback over cosmic time <cit.>. Lying at the extreme faint end of the LLAGN family is the Galactic center black hole, Sgr A* <cit.>. At the radio band, in which it was first discovered, Sgr A* is a compact synchrotron source with an inverted spectrum extending into the millimeter wavelengths (e.g., ), and exhibits moderate flux variations on timescales from hours to years <cit.>. The variability of Sgr A* is much stronger at higher frequencies <cit.>. In particular, X-ray flares have been detected on average once per day, with amplitudes reaching ∼100 times the quiescent level <cit.>. Despite this remarkable (and still not well understood) variability, Sgr A* is exceptionally underluminous, showing a bolometric luminosity only ∼ 10^-8L_ Edd. Nevertheless, Sgr A* occupies a key position in the development of theories for LLAGNs. It is now widely thought that LLAGNs are powered by a radiatively inefficient, advection-dominated accretion flow (ADAF), which is probably coupled with outflows in the form of jets and/or winds (see review by Yuan & Narayan 2014).This ADAF-jet paradigm gains support from radio interferometric surveys of nearby galactic nuclei, in which compact cores with high brightness temperatures, and sometimes with elongated components, are frequently detected and best interpreted as synchrotron radiation from the magnetized, relativistic jets (e.g., Nagar et al. 2000, 2005). Ironically, the ADAF-jet paradigm is not without controversy with Sgr A* itself.Despite its virtue of proximity, Sgr A* is persistently seen as a compact source (i.e., detailed morphology not detected) under the currently best available resolution afforded by VLBI observations,down to the vicinity of its presumed event horizon(Bower et al. 2004; Shen et al. 2005; Doeleman et al. 2008), thus leaving serious doubt on the existence of the putative jet (but see Li et al. 2013). The Andromeda galaxy (M 31) hosts the second nearest SMBH, known as M 31*, which has a dynamical mass of 1.4^+0.9_-0.3×10^8 M_ inferred from stellar kinematics <cit.>.M 31* has been detected only in the radio and X-rays to date. Crane et al. (1992) first identified a compact 8.4 GHz source at the nucleus of M 31 using the Very Large Array (VLA). Its physical association with the putative SMBH was reinforced by a follow-up 8.4 GHz observation that found mild variability (∼ 30-40 μJy; ).Due to its intrinsic dimness and contamination from neighboring X-ray sources, the X-ray counterpart of M 31* was not firmly established until a much later time (Garcia et al. 2010). The bolometric luminosity of M 31* is estimated to be ∼ 10^-8L_ Edd, making it only the second known LLAGN, after Sgr A*, to exhibit such a low state. Interestingly, M 31* has exhibited flaring X-ray emission since early 2006, with amplitudes similar to those seen in Sgr A* (Li et al. 2011).The proximity of M 31*, as well as its similarities with Sgr A*, holds promise for exploring the still poorly understood physics of SMBHs accreting at very sub-Eddington rates.We have carried out a systematic observing campaign of M 31*, using the Karl G. Jansky Very Large Array (JVLA[The official acronym for the Karl G. Jansky Very Large Array is VLA. Here and in the following we use the acronym “JVLA" to avoid potential confusion with the observations taken in the pre-JVLA era.]) in multi-epoch and multi-frequency.These observations of unprecedented sensitivity and resolution allow us to detect M 31* at high frequencies, probe any extended component, constrain its instantaneous spectrum, and study its variability on various timescales.We describe the observations and data reduction procedure in Section 2. The results are presented in Section 3, followed by a discussion in Section 4 and a summary in Section 5.We adopt a distance of 780 kpc for M 31* <cit.>.Thus 1^'' corresponds to 3.8 pc, or ∼2.8×10^5 R_ Sch (the Schwarzschild radius for a black hole mass of 1.4×10^8 M_). Throughout this work, errors are quoted at the 1 σ confidence level, unless otherwise noted.§ OBSERVATIONS AND DATA REDUCTION§.§ JVLA Observations We have observed M 31* with JVLA at the C, X, Ku and K bands, in a total of 19 epochs between May 2011 and December 2012 (Project ID: 11A-178, SD0487, 11A-137 and 12B-002). The array configurations included A, B and BnA, which are well suited for detecting a compact source.The central frequencies were 5.0, 6.0 or 6.6 GHz for C-band, 8.5 or 10 GHz for X-band, 15 GHz for Ku-band and 20 GHz for K-band. We note that these observations were carried out during several stages of commissioning of the new receiver systems, hence the different projects have different parameters (especially bandwidth and central frequency).In 17 of the 19 epochs, a dual-frequency was employed, including C-band and one of the other three higher-frequency bands, which allows us to measure an instantaneous spectral index. The remaining two epochs were taken with the C-band only. In total, we obtained 19 C-band, 7 X-band, 2 Ku-band and 8 K-band observations, which together provide a good sampling of the potential variability in M 31* on timescales ranging from hours to weeks and months.Table <ref> presents details of these observations (each assigned an ID), including observing date, central frequency, integration time, array configuration, bandwidth, beam size and RMS noise. In all observations, the phase center was placed at the nucleus of M 31, i.e., [RA, DEC] (J2000) = [00^h42^m44329, +41160842]. We used 3C48 as the flux density calibrator for the amplitude scale and instrumental bandpass, with the exception of Project 11A-137, in which 3C147 was used.The Perley-Butler 2010 flux density scale was applied.The on-target scans, 6-10 minutes each, were sandwiched between scans of the nearby gain calibrator J0038+4137. Since the flux density of M 31* is too weak to be self-calibrated, we relied on the cycle time being short enough to properly compensate for the temporal atmospheric phase fluctuations. Each individual visibility data was flagged, calibrated, imaged and restored using the Common Astronomy Software Applications package[http://casa.nrao.edu] (CASA, version 4.3) in standard procedures. We inspected all visibility data by eye, and manually flagged bad data (from RFI and instrumental effects).The Stokes I images were then produced using the CLEAN task, with the Multi-frequency Synthesis mode, nterms=1 and natural weighting.The flux densities of M 31*, whenever detected, were measured by the IMFIT task, fitting an elliptical Gaussian to the Stokes I image. To maximize the signal-to-noise ratio, we used the CONCAT task to combine the individual visibility data of same central frequency into a concatenated visibility data. The CLEAN task was employed again to make Stokes I images at different bands. Details of the concatenated images, including central frequency, total integration time, beam size, RMS level and the individual observations involved, are given in Table <ref>. We note that Projects 11A-178 and SD0487 were taken in coordination with simultaneous Chandra observations. The relation between the radio and X-ray emission from M 31* will be studied in a forthcoming paper. §.§ Historical VLA observationsTo extend our temporal baseline for variability study, we revisit historical VLA observations of M 31*, conducted by one of us (L.O.S.) during 2002–2005.These ten observations, taken in the C-band (5 GHz) under A- or B-configuration, are summarized in Table <ref> (IDs 37-46). We reduced the data also using CASA, and following the standard “old" VLA tutorial[https://casaguides.nrao.edu/index.php/Calibrating_a_VLA_5_GHz_continuum_survey], to minimize systematics when compared to the JVLA data. The primary flux density calibrator was 3C48 and the gain calibrator was J0038+4137. All Stokes I images were synthesized with natural weighting and fitted with the IMFIT task. We also used the CONCAT task to combine the ten observations to obtain a concatenated 5 GHz image. § RESULTS§.§ Morphology Figure <ref>a-d display the concatenated JVLA images in the four bands, enclosing the central 4×4 (∼15.2 pc×15.2 pc) region of M 31. These images have RMS noise of 1.9, 2.0, 2.0 and 2.0 μJy at 6, 10, 15 and 20 GHz, respectively, and corresponding beam size of 058×050, 025×023, 017×016 and 025×024.M 31* is detected as a compact source at its putative position at all four frequencies, in particular, at 10, 15 and 20 GHz for the first time.On the other hand, M 31* was not detected in either of the two observations at 8.5 GHz, nor in their concatenated image (hence not shown here).A closer look at the 6 GHz image (Figure <ref>a) indicates substructures on a scale of ≲ 1^'', immediately around an otherwise point-like source. Two small “plumes” seem to protrude toward the north and the south, reminiscent of a bipolar outflow or jets from the central black hole. Complicating this picture, another “plume" reaching out to the west is also hinted.We further show in Figure <ref>e the concatenated C-band image including the 11 A-array JVLA observations only.The above substructures persist in this image of higher resolution (beam size of 051×042). This remains true in further tests in which a subset of data is taken out from the concatenated image. A similar morphology appears in the 15 GHz image, but the source is virtually unresolved in the 10 and 20 GHz images.Finally, the concatenated image of the historical VLA observations (Figure <ref>f) shows a markedly different morphology of M 31*, but this is likely affected by the poorer resolution and sensitivity in this image.We caution that the “plumes" are only detected at a ∼3 σ level. The reality and physical meaning of these substructures will be further discussed in Section <ref>. §.§ VariabilityWe have measured the flux density (or upper limit, in the case of non-detection) of M 31* in a uniform fashion for both the JVLA and historical VLA images.The peak flux density, Gaussian size and integrated flux density are listed for each image (ID) in Table <ref>.Figure <ref> presents the light curves of M 31* derived from the 19 JVLA epochs, with different color-coded symbols representing different bands.To facilitate the comparison among the slightly different C-band central frequencies, we have converted the 5.0 or 6.6 GHz flux density into a 6.0 GHz flux density, assuming a spectral index of α = -0.45 (S_ν∝ν^α; sec Section <ref>). From the C-band light curve (red circles), which covers all epochs, we identify two significant variations: the peak flux density reached a maximum of 48.6±8.0 μJy/beam on Jun 05, 2011, having increased by ∼71% (∼2 σ significance) in about eight days; On December 22, 2012, the peak flux density increased by ∼44% (∼3 σ significance) within just six hours. Otherwise the C-band light curve appears flat, exhibiting a modest rms fluctuation of ∼15% about its mean (28.3±1.9 μJy/beam). The light curves at the other bands are much more sparse and show no significant variability, either.The mean peak flux densities of M 31* are 20.2±2.0, 16.6±2.0 and 15.0±2.0 μJy/beam, derived from the concatenated images at 10, 15 and 20 GHz, respectively. More quantitatively, we calculate for each band the variability index <cit.>, VI = 1/N∑_i|S_i-S̅/σ̅|,where S_i the flux density at the ith epoch, S̅ the mean flux density, and σ̅ the mean error over N epochs.A source is considered variable in the case of VI > 1.0, i.e., its flux density varies more than the mean measurement error.We find that in the C-band VI ≈ 1.1, whereas VI < 1.0 in the other bands.This reinforces our conclusion that during 2011-2012, M 31* shows marginal variability at 6 GHz, but is consistent with no significant variability at higher frequencies, due partially to the limited coverage.A comparison with the historical VLA observations allows to probe the long-term variability in M 31*.We measure a mean peak flux density of 60.0±10.0 μJy/beam at 5 GHz between 2002-2005, which is consistent with that reported in Garcia et al. (2010).Compared to the mean value of 28.3±1.9 μJy/beam at 6 GHz between 2011-2012, this indicates a ∼50%(∼3 σ significance) decrease over about a decade. We note that the 5 GHz flux density of our calibrator, 3C48, exhibits no significant variability (to within 0.2%) in the mean time. A long-term decrease is also evident at a higher frequency: Crane et al. (1992, 1993) measured 8.4 GHz flux densities of 28 ± 5 and 39 ± 5 μJy, while our observations set a 3 σ upper limit of 22 μJy at 8.5 GHz.§.§ Spectral Index In 9 of the 17 dual-frequency epochs, M 31* is detected (i.e., > 3 σ) in both frequencies. We derive a quasi-instananeous spectral index for each of these 9 epochs and find values ranging from -1.09±0.16 to -0.31±0.28 (see lower panel of Figure <ref>).We also obtain a time-averaged spectral index over the longest possible frequency range (Figure <ref>), based on the peak flux densities at 5, 6.0, 6.6, 10, 15 and 20 GHz from the concatenated images.This leads to α = -0.45±0.08 (if instead the integrated flux densities were adopted, the resultant spectral index is α = -0.44±0.10), which is consistent with synchrotron radiation.The upper limit at 8.5 GHz is compatible with this average spectrum.Notably, the average spectrum predicts a flux density of 12.5 μJy at 30 GHz and 11.0 μJy at 40 GHz, inviting future JVLA observations at these higher frequencies.§ DISCUSSION Our JVLA observations reveal unprecedented details about the radio emission from M 31*. This provides important clues to its otherwise underluminous radiation, and in turn sheds light to the nature of LLAGNs in general, as addressed below. §.§ Temporal behavior: LLAGN in contextThe 19 C-band observations taken during 2011-2012 provide insights on the variability in M 31* arising from a physical scale of ≲10^5 R_ Sch.In particular, the rms fluctuation is found to be only modest (∼15%) on weekly to monthly timescales. This is comparable to that in Sgr A* (∼17% at 5 GHz; Zhao et al. 1989).It is often considered that interstellar scintillation contributes substantially to the observed variability in Sgr A* at low frequencies (≲ 5 GHz; e.g., ). Since M 31* has a much lower line-of-sight column density (≲ 10^21 cm^-2; Li et al. 2009), the 15% fluctuation can be considered intrinsic, which might be due to a stochastic accretion rate or plasma instability in the course of jet propagation.It remains to be explored whether such fluctuations are typical of LLAGNs. Significant variations are also caught on short timescales (Section 3.2). In particular, the ∼40% increase in the 6 GHz flux density, which occurred within 6 hrs on December 22, 2012, places a strong constraint on the size of the emitting region, ≲16 R_ Sch. The amplitude and physical scale of this event are similar to some of the radio flares in Sgr A*.We point out, however, the 15 GHz flux density measured at the same epochs showed no significant variability (Figure <ref>).The comparison between the JVLA and historical VLA measurements further reveals that M 31* has decreased its mean flux density by about 50% over about a decade (Section 3.2; Figure 3).On such timescales, M 31* might have experienced a substantial adjustment in its accretion rate, leading to a subsequent change in the jet power.Interestingly, the mean X-ray flux of M 31* had shown an opposite trend in the past decade.M 31* was essentially undetected in any Chandra observation taken before 2006 (with a mean X-ray luminosity ≲ 10^36 erg s^-1), but alleviated its X-ray emission by a factor of ∼5 on average, apparently after an outburst on January 6, 2006 (Li et al. 2011). We note that such a trend has not been found in the X-ray emission from Sgr A*, although the two most underluminous LLAGNs both currently exhibit flaring X-ray emission.A long-term decrease in the radio flux of Sgr A* has not been reported, either. Unfortunately, there was no useful VLA observation of M 31* between March 2005 and April 2011, i.e., the period during which the mean radio flux presumably experienced a substantial (and perhaps abrupt) drop. Whether there is a causal relation between the increase in X-rays and decrease in radio remains an intriguing possibility to investigate. We defer such a study, to be assisted with standard ADAF-jet models (Yuan et al. 2009), to a future work.To our knowledge, long-term radio variability has been reported for few LLAGNs, due partially to the lack of extensive monitoring observations.One exception is M 81*, an archetype LLAGN with a black hole mass of 7×10^7 M_ and a bolometric Eddington ratio of ∼10^-5 <cit.>. <cit.> reported a long-term radio (5.0 and 8.4 GHz) flare in M 81* from 1997 to 2001, which climbed to a peak more than twice the quiescent level. This suggests that decade-long monitoring observations be crucial to obtaining a clear picture of the variability in LLAGNs. §.§ Spectrum and morphology: insights on the putative jetWe have derived an average spectral index of α≈-0.45±0.08 over the available frequency range of 5-20 GHz (Figure <ref>). While none of our 19 epochs covers more than two frequencies, this average slope is expected to represent the true slope, in view of the very modest variability among these epochs. Indeed, the six epochs with a C/K combination show an instantaneous spectral index ranging between -0.31±0.28 to -0.72±0.26. This relatively steep spectrum is to be contrasted with that of Sgr A*, which is observed to be α≈ 0.2-0.3 over a similar frequency range <cit.>.In the case of M 81*, in which substructures are clearly resolved, <cit.> measured α0 near the core and α≈-0.7 in the apparently one-sided jet at ∼1 milli-arcsec (∼ 3×10^3 times the Schwarzschild radius of M 81*) away from the core. These are understood in the standard jet models (e.g., ), which predict a negative gradient in the spectral index along the jet path as the result of decreasing synchrotron opacity.In this regard, the spectral index of M 31* (∼-0.45) suggests that the observed radio emission over 5-20 GHz is dominated by the optically-thin part of the jet. We note that the physical scale corresponding to the synthesized beam of our JVLA images, is ∼ (2-7)×10^4R_ Sch. This places an upper limit in the distance of the emitting region from the black hole. The short-timescale variability might take place much closer to the black hole (Section 4.1). It is natural to look for a trace of the putative jet on larger scales.In this regard, the three “plumes" seen primarily in the concatenated JVLA C-band image (Figure 1a and e) deserve some remarks. These faint (∼3 σ) features are the only hint for an extended component around an otherwise unresolved core. Their combined morphology, however, is not easily compatible with a canonical bipolar jet/outflow.On the other hand, the vicinity of M 31* on a scale of few arcsecs is known to be devoid of cold gas and young, massive stars (Li et al. 2009).Therefore, it is also unlikely that these “plumes" arise from circumnuclear free-free emission.We speculate, on the basis of their combined morphology, the “plumes" might be tracing the synchrotron radiation from a hot wind launched from the inner part of the accretion flow of M 31*. Such a wind/outflow is a generic prediction by the theories and numerical simulations of hot accretion flows in LLAGNs (Blandford & Begelman 1999; Yuan, Bu & Wu 2012; Narayan et al. 2012; Yuan et al. 2015).These theoretical works have gained increasing observational support, from LLAGNs (e.g., Wang et al. 2013; Tombesi et al. 2014) to the hard state of black hole X-ray binaries (Homan et al. 2016). The predicted winds have a wide opening angle as compared to the highly collimated jet, as theyoriginate from the corona region of the hot accretion flow, where frequent magnetic reconnection should occur and can efficiently accelerate electrons (Yuan et al. 2009b). Therefore, relativistic electrons should be present in the winds and emit radio emission via synchrotron radiation.Recently, we detected the radio counterpart of the central SMBH in M 32 (named M 32*), a dwarf elliptical companion of M 31, using deep JVLA C-band observations (Yang et al. 2015). M 32* has a 6 GHz flux density similar to that of M 31*, and also shows a pc-scale, extended morphology marginally resolved by the synthesized beam. This source could be another case of synchrotron radiation from the wind of a LLAGN.A more meaningful test of our speculation awaits further work. § SUMMARYThanks to the unprecedented sensitivity afforded by our JVLA observations, we have significantly advanced our knowledge about the radio properties of M 31*, one of the nearest and most underluminous SMBHs. Our main results include: * Detection of M 31* at 10, 15 and 20 GHz for the first time;* Constraints on the short- and long-term variability in M 31*. In particular, a curious decrease of the mean flux density, by ∼50%, is found between the JVLA and historical VLA observations separated by nearly a decade, which might be associated with an observed increase in the mean X-ray flux of M 31*; * Measurement of the spectral index covering 5-20 GHz. The relatively steep spectrum (α≈ -0.45) is best interpreted as dominated by the optically-thin part of a putative jet, which is located at no more than a few thousand Schwarzschild radii from the black hole. However, no clear evidence is provided in our images for an extended component tracing this putative jet.These findings clearly invite further high-resolution interferometric observations, as well as dedicated numerical models, to deepen our understanding of the formation and evolution of jets and outflows from M 31*.The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work is supported by the National Natural Science Foundation of China under grants 11473010, 11573051 and 11633006. We thank Mike Garcia for the effort in preparing Project SD0487. Z.L. acknowledges support from the Recruitment Program of Global Youth Experts. F.Y. acknowledges support from the Key Research Program of Frontier Sciences of CAS (No. QYZDJ-SSW-SYS008) and the Ministry of Science and Technology of China (No. 2016YFA0400704). ccccccccccccObservation Log and M 31* Measurements in Individual Images0pt 3cIMFIT Gaussian 10-12 IDProjectDateνTimeConfig.BandwidthBeam sizeRMSI_ peakImage component sizeS_ int(GHz)(hr)(MHz)(×,)(μJy/beam)(μJy/beam)(×,)(μJy)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) 1 11A-1782011 May 275.02.5BnA2561.52×0.67, -68 6.0 34.9 1.58×0.78, 113 42.1±5.52 11A-1782011 May 278.5 2.5BnA2560.91×0.42, -6611.0 < 33.0 - -3 11A-1782011 May 285.0 2.5BnA2561.52×0.62, -68 7.030.8 1.66×0.91, 129 49.0±9.64 11A-1782011 May 288.5 2.5BnA2560.95×0.38, -65 8.0< 24.0 - -5 11A-1782011 Jun 05 5.02.5 BnA2560.71×0.50, -87 8.052.8 0.79×0.53, 96 62.2±7.96 11A-1782011 Jun 05 8.52.5 BnA2560.45×0.30, 89 10.0< 30.0 - -7 11A-1782011 Jul 015.0 2.5A2560.50×0.45, -74 7.037.9 0.50×0.45, -74 37.9±7.08 11A-1782011 Jul 018.52.5 A2560.30×0.27, -73 8.0< 24.0 - -9 11A-1782011 Jul 275.02.5 A2560.59×0.44, -786.5 32.0 0.59×0.44, -78 32.0±6.510 11A-1782011 Jul 278.52.5 A2560.38×0.27, -717.5 < 23.0 - -11 11A-1782011 Aug 275.02.5 A2560.46×0.41, -611.0 < 33.0 - -12 11A-1782011 Aug 278.52.5 A2560.28×0.24, 129.0 < 27.0 - -13 SD0487 2011 Sep 025.06 A2560.45×0.43, -4 5.033.0 0.62×0.46, 113 48.6±5.714 11A-1372011 Sep 046.61 A20480.36×0.33, -73 4.524.4 0.36×0.33, -73 24.4±4.515 11A-1372011 Sep 04204.5 A20480.12×0.11, -5 4.5 18.1 0.12×0.11, -5 18.1±4.516 11A-1372011 Sep 066.61 A20480.37×0.33, -64 11.0< 33.0 - -17 11A-1372011 Sep 06204.5 A20480.11×0,10, -10 4.524.9 0.11×0.10, -10 24.9±4.518 SD0487 2011 Sep 075.06 A2560.57×0.44, -40 6.019.4 0.75×0.66, 159 38.2±6.619 SD0487 2012 Jun 016.61.5 B20481.20×1.08, -86 6.035.2 1.20×1.08, -86 35.2±6.020 SD0487 2012 Jun 01204.5B20480.35×0.35, 12 3.021.6 0.42×0.33, 94 24.2±1.221 SD0487 2012 Jun 066.61.5 B20481.35×1.11, 537.0 32.4 1.59×1.44, 97 49.0±11.022 SD0487 2012 Jun 06204.5B20480.40×0.35, 383.0 15.3 0.40×0.35, 38 15.3±3.023 SD0487 2012 Jun 126.61.5 B20481.22×1.18, 808.0 24.0 2.42×1.46, 168 58.7±5.224 SD0487 2012 Jun 12204.5B20480.39×0.35, -26 3.515.8 0.39×0.35, -26 15.8±7.025 SD0487 2012 Jul 016.61.5 B20481.23×1.05, -89 4.524.8 1.50×0.98, 55 28.3±4.726 SD0487 2012 Jul 01204.5B20480.35×0.34, -544.5 16.0 0.50×0.31, 24 20.4±4.527 SD0487 2012 Aug 146.6 1.5 B20481.24×1.10, 764.5 30.3 1.24×1.10, 176 30.3±4.528 SD0487 2012 Aug 14204.5B20480.37×0.35, -76 7.0< 21.0 - -29 SD0487 2012 Oct 296.61.5 A20480.39×0.33, -805.0 23.3 0.60×0.48, 29 52.6±5.730 SD0487 2012 Oct 29204.5A20480.16×0.13, -224.5 17.0 0.19×0.15, 158 22.4±4.631 12B-002 2012 Dec 226.00.9A40000.36×0.36, 31 3.029.1 0.45×0.32, 100 31.1±3.432 12B-002 2012 Dec 2215.02.7A60000.16×0.14, 352.0 17.5 0.19×0.16, 26 23.2±2.833 12B-002 2012 Dec 226.00.9 A40000.45×0.36, 80 3.041.9 0.53×0.48, 102 65.9±5.134 12B-002 2012 Dec 2215.02.7A60000.18×0.17, 73 2.015.4 0.18×0.18, 96 16.9±1.735 12B-002 2012 Dec 306.00.9A40000.39×0.36, -74 3.027.3 0.39×0.36, -74 27.3±3.736 12B-002 2012 Dec 3010.02.7A 4000 0.25×0.23, -20 2.021.0 0.25×0.25, 11 22.7±2.237 AJ0289 2002 Jul 05 5 8 B 100 1.52×1.46, 83 7.0 25.8 1.52×1.46, 83 25.8±7.038 AJ0289 2002 Jul 06 5 8 B 100 1.52×1.46, 85 7.0 32.8 1.52×1.46, 85 32.8±7.039 AJ0289 2002 Jul 29 5 8 B 100 1.53×1.48, -88 7.0 48.7 1.53×1.48, -88 48.7±7.040 AJ0289 2002 Aug 12 5 2 B 100 1.69×1.43, -80 12.0 37.8 1.69×1.43, -80 37.8±12.041 TLS17 2002 Aug 14 5 1 B 100 1.53×1.40, -53 15.0 66.7 1.53×1.40, -53 66.7±15.042 AS0819 2004 Oct 27/28 5 6.3 A 100 0.66×0.44, -71 18.0 65.0 0.72×0.31, 51 65.0±18.043 S63142004 Dec 06/07512 A1000.64×0.51, 68 8.0 54.4 0.64×0.51, 6854.4±8.044 S63142004 Dec 27/28512A1000.56×0.44, -88 9.0 54.0 0.56×0.44, -88 54.0±9.045 S63142005 Jan 28/29512 AB1001.27×0.67, 80 9.0 65.0 1.27×0.67, 80 65.0±9.046 S63142005 Feb 21/22512 B1001.83×1.41, -798.0 67.9 1.83×1.41, -79 67.9±8.0(1) Image ID; (2) Project ID; (3) Date of observation; (4) The central frequency; (5) Total integration time; (6) Array and configuration; (7) Total bandwidth; (8) Synthesized beam FWHM of major and minor axes, and position angle of the major axis; (9) Image RMS level; (10) Peak flux density of M 31*; (11) major axis, minor axis and position angle of the fitted elliptical Gaussian; (12) integrated flux density.ccccccccccMean Flux Densities of M 31*0pt 3cIMFIT Gaussian8-10Reference imageID Config. ν Time Beam size RMS I_ peak Image component size S_ int (GHz) (hr) (×, ) (μJy/beam) (μJy/beam) (×, ) (μJy)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 4*Fig. 1(a) 1, 3, 5, 7, 9, 11, 13,4*A, B, BnA 4*6.0 4*40.7 4*0.58×0.50, -78 4*1.9 4*28.3 4*0.64×0.57, 125 4*35.5±4.4 14, 16, 18, 19, 21,23, 25, 27, 29, 31,33, 35Fig. 1(b) 36 A 10 2.7 0.25×0.23, -14 2.0 20.2 0.26×0.26, 39 23.9±3.5Fig. 1(c) 32, 34 A 15 5.4 0.17×0.16, 46 2.0 16.6 0.18×0.17, 19 20.7±2.12*Fig. 1(d) 15, 17, 20, 22, 2*A, B 2*20 2*27 2*0.25×0.24, -3 2*2.0 2*15.0 2*0.27×0.22, 9 2*17.0±3.4 24, 26, 28, 30 2*Fig. 1(e) 7, 9, 11, 13, 14, 16, 2*A 2*6.0 2*22.2 2*0.41×0.38, -81 2*1.2 2*28.7 2*0.51×0.42, 114 2*39.7±3.2 18, 29, 31, 33, 35Fig. 1(f) 37-46 A, B 5 81.3 0.88×0.66, 80 10.0 60.0 0.79×0.65, 87 62.0±7.6(1) Concatenated image in Figure <ref>; (2) ID as in Table 1; (3) Configuration; (4) Central frequency; (5) Total integration time; (6) Synthesized beam size, including major and minor axis, and position angle of the major axis; (7) Image RMS noise; (8) Peak flux densities of M 31*; (9) Major and minor axes, and position angle of the fitted elliptical Gaussian; (10) Integrated flux density. 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"authors": [
"Yang Yang",
"Zhiyuan Li",
"Loránt O. Sjouwerman",
"Feng Yuan",
"Zhi-Qiang Shen"
],
"categories": [
"astro-ph.GA",
"astro-ph.HE"
],
"primary_category": "astro-ph.GA",
"published": "20170726083221",
"title": "Very Large Array multi-band monitoring observations of M 31*"
} |
^1Oxford Astrophysics, Department of Physics, Keble Road, Oxford, OX1 3RH, UK Photometric redshift surveys map the distribution of matter in the Universe through the positions and shapes of galaxies with poorly resolved measurements of their radial coordinates. While a tomographic analysis can be used to recover some of the large-scale radial modes present in the data, this approach suffers from a number of practical shortcomings, and the criteria to decide on a particular binning scheme are commonly blind to the ultimate science goals. We present a method designed to separate and compress the data into a small number of uncorrelated radial modes, circumventing some of the problems of standard tomographic analyses. The method is based on the transform (KL), and is connected to other 3D data compression bases advocated in the literature, such as the Fourier-Bessel decomposition. We apply this method to both weak lensing and galaxy clustering. In the case of galaxy clustering, we show that the resulting optimal basis is closely associated with the Fourier-Bessel basis, and that for certain observables, such as the effects of magnification bias or primordial non-Gaussianity, the bulk of the signal can be compressed into a small number of modes. In the case of weak lensing we show that the method is able to compress the vast majority of the signal-to-noiseÀ into a single mode, and that optimal cosmological constraints can be obtained considering only three uncorrelated KL eigenmodes, considerably simplifying the analysis with respect to a traditional tomographic approach. Science-driven 3D data compression David Alonso^1 December 30, 2023 ==================================§ INTRODUCTIONAstronomical data is inherently three-dimensional: the main observable is the intensity of the electromagnetic emission in the sky as a function of wavelength and line-of-sight direction, determined by two angles. An idealized cosmological analysis would therefore use, as a data vector, the full cube I(λ,θ,ϕ) probed over sufficiently well resolved angular and frequency scales <cit.>. However, the operational costs of obtaining such a dataset imply that we can realistically only access a compressed version of it, where the compression method comes in different flavours: * We can decrease the measurement noise by integrating the sky intensity over large frequency bands. This approach has been used for instance in CMB observations <cit.>.* Angular resolution can also be sacrificed for wider sky and frequency coverage, as has been proposed for future intensity mapping experiments <cit.>.* The size of the dataset can also be reduced by collecting only the flux associated with the brightest extra-galactic objects. An accurate measurement of their spectra then allows a determination of their redshift, producing the well-known spectroscopic redshift surveys <cit.>.* Measuring individual spectra is a costly operation, however, and can usually only be done for a small subsample of all available sources. This problem can be mitigated by inferring the source's redshift from their emission in a small number of wider frequency bands, in what is known as a photometric redshift survey <cit.>. The redshifts thus determined are far less precise than their spectroscopic counterparts, and usually only an imperfect estimation of the redshift probability distribution for each galaxy is accessible. Even after this first compression stage, the size of the dataset makes a direct analysis of it as a data vector a computationally intractable problem. Typically this should not be an issue in terms of information loss, since large portions of the data are usually dominated by measurement noise, contaminated by sources of systematic uncertainty (both observational and theoretical) or contain only redundant information. An efficient data compression method will therefore identify these sections of data space and eliminate them, or collect them into summary statistics, while minimizing the loss of meaningful cosmological information. An example of this is the standard analysis of cosmological datasets in terms of their two-point statistics <cit.>. However, even in this case the resulting data vector can be large enough to present an important computational challenge in terms of likelihood evaluation and covariance estimation. The complexity of latter problem, in particular, scales with the square of the data vector size, and can become an important drain of computational resources <cit.>.In this work we will concern ourselves with the topic of 3D data compression: the problem of identifying the uncorrelated angular and radial modes of the data that optimally contain the maximum amount of information. This problem has been previously addressed in the literature <cit.>, and a number of approaches have been proposed depending on the definition of uncorrelatedness used, and on the type of information one wishes to preserve. Here we will present a method to derive a set of uncorrelated radial eigenmodes that are manifestly optimal in terms of information compression for any quantity, such as individual cosmological parameters or the amplitude of the cosmological signal over any set of known contaminants. The method is based on the well-understood transform <cit.>, and is similar in spirit to the derivation of optimal weighting schemes for the analysis of spectroscopic surveys <cit.>. Although we will focus here on the case of photometric redshift surveys, the method can be applied to any set of cosmological datasets.The article is structured as follows: Section <ref> describes the (KL hereupon) transform and its applicability in the context of 3D data compression. Section <ref> shows the performance of the method in a number of science cases, such as the derivation of optimal radial bases for galaxy clustering (Sec. <ref>) and weak lensing (Sec. <ref>) observations and the use of the KL eigenmodes to measure the effects of primordial non-Gaussianity (Sec. <ref>) and lensing magnification (Sec. <ref>) with a small number of modes. Finally Section <ref> summarizes our findings and dicusses the advantages and shortcomings of the method.§ METHOD §.§ The Karhunen-Loeve transformThe idea behind the transform, as developed within the field of cosmological data analysis in e.g. <cit.>, is to compress a given data vector into a small set of modes containing most of the useful information on a particular parameter (or set of parameters). Let x be a data vector of dimension N_s, and let θ be a particular parameter we want to measure. Under the assumption that x is Gaussianly distributed with mean 0 and covariance C, a set of linear combinations y_p≡ e_p^†x can be found such that the y_p are white and uncorrelated (⟨ y_py_q^*⟩=δ_pq), and such that the first m<N_s combinations contain most of the information about θ. This is done by solving the generalized eigenvalue problem <cit.>:∂_θ Ce_p=λ_pCe_p,where ∂_θ≡∂/∂_θ.Although the transform can be used to compress the information on any particular parameter, it has been most commonly used to separate signal-dominated and noise-dominated modes by optimizing for the amplitude of the signal, as we explore below. Before moving on, however, it is worth noting that a generalized eigenvalue problem such as Eq. <ref> can always be recast as a standard eigenvalue problem of the form A ẽ_p=λ_p ẽ_p, whereA≡ C^-1/2 (∂_θ C)C^-1/2,ẽ_p≡ C^1/2 e_p,and we have made use of the fact that C is positive-definite (and therefore C^1/2 is well-defined and invertible). §.§.§ The KL transform for the signal-to-noiseLet us decompose the data vector x into uncorrelated signal and noise components x= s+ n where, in this context, the signal is the part of the data containing any information of cosmological interest, and the noise is any contaminant preventing us from accessing it[n could include, for instance, the contribution of foregrounds in intensity mapping experiments, which motivates the use of the KL transform as a foreground cleaning method <cit.>.]. In this particular case, the data covariance matrix can be split into their independent contributions C= S+ N.The KL transform has traditionally been used to design an eigenbasis that maximizes the overall signal-to-noise ratio (e.g <cit.>). This can be done by defining a fictitious parameter ρ multiplying the signal part of the data with fiducial value ρ=1 (i.e. x=ρ s+ n). In this case, after some trivial manipulations, the eigenvalue equation (Eq. <ref>) takes the form:( S+ N) e_p=λ_p N e_p,where we have redefined 2/(2-λ_p)→λ_p. This can be cast into a standard eigenvalue equation using the Cholesky decomposition of the noise covariance matrix N= L L^†:[ L^-1 C ( L^-1)^†] ẽ_p=λ_pẽ_p,where ẽ_p≡ L^† e_p.At this point it is worth noting that the generalized eigenvalue problem in Eq. <ref> can be understood as the problem of diagonalizing C under a non-standard dot product ∘ given by the inverse noise covariance matrix (i.e. a∘ b≡ a^† N^-1 b). Under this dot product, an eigenbasis F≡( f_1, f_2,..., f_N_s) can be found such that F is orthonormal F∘ F= I, and the covariance of the transformed data vector y≡ F∘ x is diagonal:⟨ yy^†⟩= F^† N^-1 C N^-1 F=Λ≡ diag(λ_1,...,λ_N_s).Using the orthonormality of F (with respect to the non-standard dot product), this can be cast into the same form as Eq. <ref>, where f_p= Lẽ_p= N e_p.Finally, note that, because both S and N are positive-definite matrices, their eigenvalues will also be positive. Since the eigenvalues of N under the KL transform are, by construction, 1, the elements of Λ above will all be greater than 1, and converging to 1 for the noise-dominated modes.§.§.§ The KL transform with correlated contaminantsLet us now consider a more general case in which we further split the noise into two parts n→ n+ m, where m is a contaminant with a non-zero correlation with the signal. The covariance matrix of the data is then given by:⟨ xx^†⟩=ρ^2 S+2ρ M_s+ M+ N,where M_s≡(⟨ ms^†⟩+⟨ sm^†⟩)/2, M≡⟨ mm^†⟩ and we have kept the fictitious parameter ρ defined in the previous section. Eq. <ref> then reads:( S+ M_s)e_p= λ_p/2 Ce_p. Unfortunately, in this case the manipulation that lead us to Eq. <ref> cannot be performed. If we were to do so, the matrix remaining on the right hand side of this equation would not be positive-definite, and the corresponding generalized eigenvalue problem would be ill-defined. This is not a problem, since the solutions to Eq. <ref> still separate the modes with the highest signal. The separation of the noise-dominated modes becomes less obvious, however, since the resulting eigenvalues cannot be simply compared with 1, corresponding to noise-dominated modes in the previous section.The eigenvector solutions to the generalized eigenvalue problem in Eq. <ref> can be collected as columns of a matrix E that simultaneously satisfies the equations:E^†( S+ M_s) E=Λ,E^† C E= I,where I is the identity and Λ= diag(λ_1,...,λ_N_s). Since the second equation implies C E≡( E^†)^-1, the original vector x can be recovered from the coefficients y≡(y_1,...,y_N_s) as x= CEy. More interestingly, one can identify the principal eigenvectors of the Eq. <ref> (e.g. those with associated eigenvalues λ_p above a given threshold λ_ thr) and project out the remaining modes, which are presumably more contaminated by m. This procedure defines a filter W≡ CEPE^†, where P is a projection matrix with 1s in the diagonal elements corresponding to the principal eigenmodes and zeros everywhere else. The filtered data vector is therefore x̃= W x. §.§ Application to tomographic datasetsThe standard method to draw cosmological constraints from photometric redshift surveys is to divide the galaxy sample into bins in photo-z space and use the information encoded in all the relevant auto- and cross-correlations between different bins <cit.>, making use of various calibration methods in order to estimate the true redshift distribution of each bin. Several criteria can be followed in order to select these redshift bins, such as minimizing the correlation between non-neighbouring bins or preserving a roughly constant number density on all bins. Other approaches (<cit.>) involve projecting the main observable (e.g. galaxy overdensity or shear) onto the Fourier-Bessel eigenbasis. None of these schemes are manifestly optimal from the point of view of S/N, final cosmological constraints or contaminant deprojection, however. This section presents an alternative slicing scheme addressing these shortcomings, based on the KL transform.§.§.§ Tomographic analysesLet us start by assuming that we have split the galaxy sample into N_s subsamples. As mentioned above, we will think of each of these subamples as some kind of redshift binning (e.g. binning galaxies in terms of their maximum-likelihood redshift), but the formalism applies to any set of subsamples. Let a^α() be the a field on the sphere at the angular positionand defined in terms of the properties of the sources in the α-th sample (e.g. the cosmic shear field γ^α or the galaxy overdensity δ^α), and let ϕ^α(z) be the redshift distribution of these sources. Finally, let a^α_ℓ m be the spherical harmonic coefficients of a^α[Spin-2 fields, such as the cosmic shear, will be decomposed in spin-2 spherical harmonics, however the discussion below holds for fields of arbitrary spin.]. The power spectrum for our set of subsamples is defined as the two-point correlator of a^α_ℓ m:⟨ a_ℓ ma^†_ℓ' m'⟩≡δ_ℓℓ'δ_mm' C_ℓ,where we have packaged a^α_ℓ m as a vector for each (ℓ,m): a_ℓ m≡(a^1_ℓ m,...,a^N_s_ℓ m). Usually the observed field can be decomposed into uncorrelated signal and noise components a= s+ n, with a similar decomposition in the power spectrum, C_ℓ= S_ℓ+ N_ℓ.Once the choice of subsamples α is made, the standard analysis method would proceed by performing a likelihood evaluation of the two-point statistics of these subsamples. While this procedure is relatively simple, it suffers from a number of drawbacks, an incomplete list of which is: * It is not clear what the optimal strategy should be to define the sub-samples. The brute-force solution to make sure one exploits all of the information present in the data would be to use a large number of very narrow redshift bins, and let the likelihood evaluation pick up the information encoded in them.* C^αβ_ℓ is a N_s× N_s× N_ℓ data vector. Thus increasing N_s will increase the computational time required for each likelihood evaluation like N_s^2 and number of elements of the covariance matrix of C^αβ_ℓ like N_s^4, with the corresponding increase in complexity needed to estimate this covariance. Although this can be partially alleviated by considering only correlations between neighbouring redshift shells, the amount of information lost by neglecting all correlations beyond a given neighbouring order is not clear a priori.* Estimating the redshift distribution for a large number of subsamples can be inaccurate, depending on the method used to do so, on the quality of the photometric redshift posterior information and on the statistics of the available spectroscopic sample.§.§.§ Optimal radial eigenbasisFollowing the description in Section <ref>, it is straightforward to derive an optimal set of radial, uncorrelated eigenmodes. * We start by assuming that the field a has been measured in a number of narrow redshift bins, and by defining the inverse-variance weighted field ã_ℓ m≡ N^-1_ℓa_ℓ m.* Let us consider a set of linear combinations of the weighted field measured on narrow redshift bins:b_ℓ m= F_ℓ^†·ã_ℓ m≡ F_ℓ∘ a,where F_ℓ is a yet-unspecified matrix and, as in Section <ref>, we have let N^-1_ℓ define the non-standard dot product v_ℓ∘ w_ℓ≡ v^†_ℓ· N^-1_ℓ· w_ℓ. The power spectrum for this new observable would then simply be given by:D_ℓ≡⟨ b_ℓ mb^†_ℓ m⟩= F_ℓ^†∘ C_ℓ∘ F_ℓ. * Requiring that the new modes be uncorrelated, we can identify Eq. <ref> with the generalized eigenvalue equation <ref>, which defines the KL eigenbasis F_ℓ by additionally requiring that it be orthonormal (F_ℓ∘ F_ℓ= I). Note that, after this transformation and without any further optimization, some of the practicalities of the original problem are already simplified, since we can now focus on the diagonal elements of the new power spectrum and its covariance.* The data can be further compressed by assuming that we are interested in measuring a set of cosmological parameters Θ≡{θ_1,...}. The information regarding this set of parameters encoded in a given data vector x can be quantified in terms of its Fisher matrix (the expectation value of the Hessian of the log-likelihood with respect to Θ), which assuming ⟨ x⟩=0 readsF_ij≡⟨∂_i∂_j L⟩=1/2 Tr(∂_i CC^-1∂_j CC^-1),where C≡⟨ xx^†⟩ is the covariance matrix of the data. We can thus rank the eigenvectors ( F_ℓ)^p_α in terms of their information content (in a Fisher-matrix sense). In the simplest scenario one may be interested in maximizing the overall signal-to-noise (S/N), in which case each mode contributes independently to the Fisher matrix element of the signal amplitude.* The final set of uncorrelated modes can then be truncated to the first M defined by this procedure, which will contain the bulk of the information needed to constrain Θ.Besides the elegance of this method in defining a natural set of radial basis functions for the particular dataset under study, analogous to the Fourier-Bessel basis in a translationally-invariant system (see Section <ref>), its merits are better evaluated in terms of data compression. This strategy allows one to reliably and significantly reduce the dimensionality of the data vector from N_s^2× N_ℓ to M× N_ℓ while minimizing the loss of information. This can lead, for instance, to a substantial reduction of the computational costs of likelihood sampling and covariance estimation.Note that, although the method is based on an initial thin-slicing of the galaxy distribution, the fact that the final datased comprises only a small set of samples means that the method is not penalized in terms of photometric redshift uncertainties. Once the KL eigenmodes F_ℓ are found for a fiducial cosmological model, they can be directly applied as weights to all the objects in the survey to generate the b^p modes. These modes are be characterized by their own window function:ϕ̃_ℓ^p(z)=∑_α( F_ℓ)^p_α ϕ^α(z)/N^αα_ℓ,where we have assumed a diagonal noise power spectrum for simplicity. The same methods used to calibrate photo-z uncertainties in the standard tomographic analysis can be applied on b^p to calibrate ϕ̃^p with minor modifications (e.g. weighed and ℓ-dependent stacking of photo-z pdfs, or cross-correlations of the b^p maps with a spectroscopic survey in the case of clustering redshifts). Furthermore, using F_ℓ for the fiducial cosmology as model-agnostic weights and inserting them in Eq. <ref>, the theoretical prediction for the power spectrum of each mode D^p_ℓ can be computed in a model-independent way.Finally, the method outlined in this section is based on the KL decomposition that maximizes the amplitude of the signal under study. This is the main application advocated in this article, since it is plausible that the set of modes containing the bulk of the cosmological signal will also drive the constraints on any comprehensive set of cosmological parameters (we explore this in more detail in Section <ref>). However, we must note that, for individual parameters, the optimal degree of data compression is achieved by solving the general KL eigenvalue problem (<ref>), which can lead to substantial improvements with respect to the S/N-optimal basis. We explore one particular example of this in Section <ref>.§ PERFORMANCE AND PARTICULAR EXAMPLESThis Section explores the performance of the KL decomposition in a number of specific science cases. §.§ Special case: the harmonic-Bessel basisLet us start by considering a simplified case where the field a is the overdensity field of a non-evolving galaxy population for which we neglect the effects of redshift-space distortions. Let us further assume that we have perfect redshift information, such that we can split the sample into thin radial slices of equal width δχ, which we label by their comoving radius χ. The noise in the measurement of a is given purely by shot noise, and since (as per our initial assumptions) the number density of sources does not change with χ, the noise power spectrum is diagonal and scales like N_ℓ(χ,χ')∝δ_χ,χ' χ^-2. Thus, the dot product is just given by:b^†∘ c∝∫ dχ χ^2 b(χ)^* c(χ).In this case, the cross-shell signal power spectrum is given by,S_ℓ^χχ'=2/π∫_0^∞ dk k^2 P_k j_ℓ(kχ)j_ℓ(kχ'),and it is trivial to show that the KL eigenmodes are simply given by the spherical Bessel functions: ( F_ℓ)^k_χ∝√(2/π)j_ℓ(kχ): D_ℓ^kk' ≡∑_χ,χ'(F_ℓ)^k_χ(F_ℓ)^k'_χ'S_ℓ^χχ'∝2/π∫ dχ χ^2∫ dχ' χ'^2 j_ℓ(kχ)j_ℓ(k'χ') S_ℓ^χχ'=∫ dq q^2P_q[2/π∫ dχ χ^2 j_ℓ(qχ)j_ℓ(kχ)][2/π∫ dχ' χ'^2 j_ℓ(qχ')j_ℓ(k'χ')]=∫ dq q^2P_qδ(k-q)/q^2δ(k'-q)/q^2=P_kδ(k-k')/k^2=P_k/k^2Δ kδ_k,k' This choice of basis defines the so-called harmonic-Bessel (or Fourier-Bessel) decomposition, and has been used as a data-compression method for the analysis of photometric redshift datasets (e.g. <cit.>). In any realistic scenario – e.g. in the presence of redshift uncertainties, redshift-space distortions or in the analysis of weak lensing data – this basis is non-optimal (among other things different k-modes will be correlated), as opposed to the KL basis described in the previous section. §.§ Galaxy clustering - Bessel-like eigenfunctions The assumptions used in the previous section are an ideal limit of the data collected by a photometric survey. In a more realistic (although still idealized) scenario, the information about the radial position of a given source is encoded in its posterior photo-z distribution p(z|α), where α is a continuous variable determining the properties of the photo-z (e.g. the mean of the posterior). The cross-power spectrum of two samples with photo-z properties α and β is given byC_ℓ^αβ=S^αβ_ℓ+N^αβ_ℓ,S_ℓ^αβ=2/π∫_0^∞ dk k^2 Δ_ℓ^α(k) Δ_ℓ^β(k),N_ℓ^αβ=δ(α-β)/n_t p(α),where n_t is the total angular number density of sources andΔ^α_ℓ(k)≡∫ dz p(z|α) Ψ_ℓ(k,z) √(P(k,z)),Ψ_ℓ(k,z)=b^α(z)j_ℓ(k χ(z))-f(z)j_ℓ”(k χ(z)).Here b^α(z) is the linear galaxy bias, f(z)=dlogδ/dlog a is the growth rate of structure, P(k,z) is the matter power spectrum at redshift z, p(α) is the probability that a source has photo-z properties α, and p(z|α) is the conditional redshift distribution of these sources (we have labelled this quantity ϕ^α(z) in previous sections). Note that, for simplicity, we have kept the contribution of redshift-space distortions at linear order and neglected the effect of magnification (this will be studied in Section <ref>).For a continuous variable α, the generalized eigenvalue problem in Eq. <ref> becomes a homogeneous Fredholm integral equation of the second kind:∫ dβ C^αβ_ℓ e^p_ℓ(β)=λ_p∫ dβ N^αβ_ℓ e^p_ℓ(β)⇒ ⇒∫ dβ n_t p(α) S^αβ_ℓ e^p_ℓ(β)=(λ_p-1)e^p_ℓ(α).In the limit of perfect photo-zs (p(z|α)=δ(z-α)), and in the absence of redshift-space distortions, the solution to this equation are the spherical Bessel functions, as proven in the previous section. For general kernels, however, no analytical solution to the homogeneous Fredholm equation can usually be found, and the standard procedure to solve it is through discretization, which is equivalent to taking finite bins in α. We will use this method here to find the KL eigenmodes that maximize the signal content for galaxy clustering.To do so, we have considered a specific strawman photometric survey targeting a sample of red galaxies, characterized by their higher bias and better photo-z uncertainties than their blue counterparts (making them better suited for clustering analyses). The sample we consider is compatible with what could be observed by the Large Synoptic Survey Telescope <cit.>, characterized by the redshift distribution shown in the left panel of Fig. <ref> (full details can be found in <cit.>). We assume a photo-z uncertainty of σ_z=0.02 (1+z) and split the sample into redshift bins in photo-z space with z_ ph>0.5 and a width given by the photo-z uncertainty at the bin centre. The auto-power spectra for our set of 23 bins are shown in the right panel of Fig. <ref>. The large overlap between bins implies that a choice of thinner slices is unlikely to unveil significantly more information, and we have verified that the results shown below do not change after doubling the number of bins. All power spectra were computed using a modified version of the code presented in <cit.>.Using the prescription described in Section <ref>, we find the KL eigenmodes and associated power spectra, and rank them according to their contribution to the total signal-to-noise ratio (defined here as the Fisher matrix element of the signal amplitude). The power spectra of the resulting KL modes are shown in Figure <ref>. Unlike the case of weak lensing, explored in Section <ref>, the information encoded in the galaxy overdensity is local in redshift, and thus the correlation between different bins decays rapidly with redshift separation. The signal-to-noise is therefore spread over ∼15 signal-dominated modes, and the noise-dominated modes can be thought of as the radial scales filtered out by the finite photo-z uncertainty (as we mentioned in Section <ref>, the noise power spectrum gets mapped into 1 under the KL transform). The relative contribution of each mode to the total signal-to-noise is shown in the top panel of Fig. <ref>. 90% of the total constraining power can be achieved by considering the first 13 eigenvectors. The form of the first 7 of these eigenvectors for ℓ=30 are shown in the right panel of Fig. <ref>. The eigenmodes are sinusoids with increasing frequencies, in agreement with the expectation that, in the limit of σ_z→0 and no background redshift dependence, the KL decomposition is achieved by the spherical Bessel functions. A Fourier-Bessel decomposition is therefore probably a near-optimal analysis method for galaxy clustering, although the KL decomposition allows a more precise determination of the truly orthogonal radial modes. §.§ Galaxy clustering - optimal basis for f_ NLIt is expected that future large-scale photometric surveys will make the search for primordial non-Gaussianity one of their main science cases. This can be achieved by measuring the excess power on large scales caused by a non-zero value of f_ NL[The reader is referred to <cit.> for a thorough review of non-Gaussianity and a definition of f_ NL.] generates in the two-point statistics of biased tracers of the matter distribution <cit.>. Since the signal is most relevant on large scales, we can expect the bulk of it to be concentrated in a small number of radial modes, which makes the general KL decomposition outlined in Section <ref> an ideal analysis method. Similar approaches have been explored in the literature to devise optimal weights for spectroscopic galaxy surveys <cit.>.We again consider the red galaxy sample used in the previous section, but now estimate the KL basis of eigenmodes that optimize the information content on f_ NL instead of the overall signal amplitude. I.e. we solve the generalized eigenvalue problem in Eq. <ref> where θ=f_ NL. We compare the performance of this basis with other choices of radial modes as follows: for a given number of modes, we estimate the associated uncertainty on f_ NL, σ(f_ NL), by summing the contributions to the corresponding Fisher matrix element of those modes, and compute the excess of σ(f_ NL) with respect to the best achievable constraint σ_ best(f_ NL). The results are shown in Fig. <ref> for three choices of radial functions: * The KL eigenbasis resulting from optimizing the information content on f_ NL discussed in this section (in red).* The KL eigenbasis resulting from optimizing the overall signal-to-noise of the galaxy clustering signal, as discussed in the previous section (in gray).* Photo-z tomography: the result of dividing the galaxy sample into a number of top-hat photo-z bins of equal width (in blue).As demonstrated by this figure, for a fixed number of modes the optimal KL basis always outperforms any other data compression prescription. In particular, the constraints on f_ NL are only degraded by ∼30% when considering only the first principal eigenmode, and almost 90% of the total constraining power is contained in the first three. Interestingly, a naive tomographic approach achieves the same uncertainty on f_ NL with a smaller number of modes (redshift bins) than the KL eigenbasis for the S/N. However, since the tomographic bins are not orthogonal, unlike the KL modes, for a fixed σ(f_ NL) both KL bases typically outperform the tomographic approach in terms of the size of the associated power spectrum. In any case, this example serves to stress the fact that the optimal radial basis in terms of overal S/N is not necessary optimal in terms of final constraints for cosmological parameters that depend on specific features of the power spectrum.§.§ Galaxy clustering - magnification bias Gravitational lensing of the observed galaxy positions alters their clustering pattern. This appears as an extra term in the galaxy clustering transfer function (Eq. <ref>):Δ^M,α_ℓ(k)=-2ℓ(ℓ+1)∫ dχ W^M,α(χ)j_ℓ(kχ)/k^2a(χ)√(P(k,z(χ))),W^M,α(χ)=3H_0^2Ω_M/2∫_z(χ)^∞ dz' ϕ^α(z')2-5 s/2 χ(z')-χ/χ(z')χ,where s is the tilt in the number counts of sources as a function of magnitude limit. This effect, commonly labeled “magnification bias” <cit.>, can be used as an alternative measurement of gravitational lensing, through galaxy positions instead of shapes. The contribution of the magnification term is, however, weak in comparison with the density and RSD terms (Eq. <ref>), and therefore its measurement can be hampered by the cosmic variance contribution of these terms.One can therefore think of the density and RSD terms as correlated contaminants of the magnification signal, and use the KL formalism described in Section <ref> to devise an optimal basis of radial eigenmodes containing the bulk of its signal-to-noise.To test this approach we consider, as in the previous section, an LSST-like survey.Since lensing magnification is an integrated effect, it is less hampered by poor photo-z uncertainties, and it is most easily measured by cross-correlating high-redshift and low-redshift data <cit.>. For this reason, in this case we consider a sample of blue galaxies, with inferior photo-z errors but wider redshift support. Full details can be found in <cit.>. In summary, we consider a sample with ∼40 objects per arcmin^2 with the redshift distribution shown in Figure <ref>. We also approximate the photo-z distributions as Gaussians with a scatter σ_z=0.05(1+z), and divide the sample into 16 top-hat bins in photo-z space with z_ ph<0.5 and widths given by the value of σ_z at the bin center (again, we verified that our conclusions did not change after decreasing the width by a factor 2).A key property of the magnification bias effect is the fact that, since gravitational lensing is caused by the integrated matter distribution between source and observer, the magnification signals in widely separated redshift bins can be tightly correlated. This is shown explicitly in Figure <ref>. The figure shows the correlation coefficients between the 16 redshift bins, defined as R^αβ_ℓ=C^αβ_ℓ/√(C^αα_ℓ C^ββ_ℓ), at ℓ=400, with (bottom panel) and without (top panel) the magnification bias effect. Although the contribution of lensing magnification to the correlation between neighbouring bins is subdominant, it produces noticeable correlations between distant ones.This property is particularly interesting in the context of the KL decomposition: a signal that is tightly correlated across samples will contribute significantly only to a small set of eigenmodes. To explore this possibility, we follow the prescription outlined in Section <ref> for correlated contaminants. The contribution of each eigenmode to the total signal-to-noise of the magnification bias (in a Fisher-matrix sense) is shown in Figure <ref>. As expected, most of the signal (>80%) is contained in the first eigenvalue, with the practical totality of it concetrated in the first three modes.We finish this section by noting that this approach is similar to the “nulling” method of <cit.>, and that an analogous treatment could be carried out on the cosmic shear field to separate the lensing and intrinsic alignment contributions <cit.>.§.§ Weak lensing The effects of gravitational lensing can be measured directly by studying the correlation it induces on the shapes and orientation of galaxy images. This effect, labeled “cosmic shear” is arguably the most promising observational probe for photometric redshift surveys, and therefore we will discuss the KL analysis of this signal in particular detail.As in the case of lensing magnification, and unlike the dominant galaxy clustering terms, the cosmic shear signal is correlated between widely separated redshift bins due to the integrated nature of gravitational lensing. Thus we can expect that a KL transform should be able to compress most of the signal to noise into a small set of radial eigenmodes. To quantify this we consider the same survey configuration used in Section <ref>. The signal part of the cross-power spectrum between the cosmic shear measurements made in two different redshift shells is given again by Eq. <ref>, where now the transfer functions Δ^α_ℓ take the form:Δ^α_ℓ(k)≡√((ℓ+2)!/(ℓ-2)!)∫ dχ W^α(χ)j_ℓ(kχ)/k^2a(χ)√(P(k,z(χ))), W^α(χ)≡3H_0^2Ω_M/2∫_z(χ)^∞ dz ϕ^α(z')χ(z')-χ/χ(z')χ.The noise power spectrum is white and simply given by the intrinsic ellipticity scatter weighed by the angular number density of sources in each redshift bin n̅^α:N^αβ_ℓ=δ_αβσ_γ^2/n̅^α,with n̅^α in units of srad^-1 and σ_γ=0.28 <cit.>. The lensing auto-power spectra (both signal and noise) for these bins are shown in Figure <ref>.We compute the KL modes for this setup and rank them according to their contribution to the total lensing signal (in a Fisher matrix sense). The power spectra of the resulting set of modes are shown in Figure <ref>. Comparing against Fig. <ref> we can see that the KL decomposition effectively separates the signal-dominated and noise-dominated modes, with all modes p>3 dominated by noise. The fractional contribution of each mode to the total signal-to-noise is shown in the top panel of Figure <ref>. Most of the signal (∼95%) is contained within a single mode, and the first two modes are able to recover more than 99% of the total. The eigenvectors corresponding to the first three principal modes for different values of ℓ are shown in the right panel of the same figure. We observe that the eigenvectors preserve roughly the same shape for all ℓ, and converge to the same shape at large ℓ. The first eigenvector upweights the parts of the redshift range with the highest signal-to-noise, penalizing the low-z regime due to its poor lensing signal and the high-z bins due to their high shot noise. The second and third eigenmodes then recover part of this information by marginally upweighting these regions. The dashed black line in the same figure shows the weighting scheme associated with the measurement of the lensing signal integrated over a single bin encompassing the redshift range covered by the 16 bins in Fig. <ref>. These weights are similar to the principal KL eigenmode, and thus the KL decomposition determines, broadly speaking, that the bulk of the signal-to-noise is mostly concentrated in the redshift-integrated signal, and extra information regarding the growth of structure can be picked up by up- or down-weighting the contributions in different sections of the redshift range.As we have discussed in the previous sections, the principal KL eigenmodes that optimize the recovery of the cosmological signal are not necessarily optimal in terms of encoding cosmological information, although it is plausible to expect so in general. In order study this further we have performed a Fisher-matrix forecast of the final constraints on cosmological parameters achievable by collecting the information encoded in the first M principal eigenmodes, and compared them with the best possible constraints coming from the use of the full set of 16 redshift bins (or, equivalently, all of the KL eigenmodes). We do so following the approach described in Section 3 of <cit.> and using, as observables, the corresponding set of KL modes b^p_ℓ m. For these forecasts we considered a set of 9 parameters: the relative density of cold dark matter ω_c, the relative contribution of baryons ω_b, the normalized local expansion rate h, the amplitude A_s and spectral index n_s of primordial scalar perturbations, the sum of neutrino masses Σ m_ν, the equation of state of dark energy w and two parameters, log_10M_c and η_b, parametrizing the contribution of baryonic effects in the matter power spectrum as described in <cit.>.Figure <ref> shows the results of this analysis in terms of 1σ contours in the Σ m_ν-w plane marginalized over all other parameters. The results are shown for the set of 1, 2 and 3 principal KL eigenmodes in red, blue and yellow respectively, while the best achievable constraints using all of the modes are shown as a solid black ellipse. We have removed the axis labels to focus the reader's attention on the relative improvement of the constraints with the number of modes. Even though the first eigenmode contains the vast majority of the lensing signal, as evidenced by the top panel of Fig. <ref>, the extra information contained in the second and third eigenmodes is necessary in order to break the degeneracies between cosmological parameters. Three modes are however sufficient to recover the full constraining power with negligible loss of information.To finalize, we would like to emphasize the fact that, as shown in the bottom panel of Fig. <ref>, the three principal eigenmodes preserve roughly the same shape as a function of multipole order, converging to the same curve for large ℓ. An ℓ-independent basis of radial functions would be advantageous from the point of view of data analysis since, for instance, the radial window functions associated with each mode (see Eq. <ref>) would only have to be calibrated once (independently of ℓ). It is therefore interesting to explore the cosmological constraints achievable by using the radial functions associated with the KL eigenmodes at high ℓ for all ℓ, even though, for a fixed multipole order, the corresponding set of modes will not be exactly orthogonal. We have verified that, doing so for the first three KL eigenmodes, the final constraints on either w or Σ m_ν degrade by less than 0.5%. This is a reasonable result, given the larger statistical weight of the small-scale (large-ℓ) fluctuations.§ DISCUSSIONNext-generation cosmological observations will gather their constraining power from a variety of observables, and will therefore have to deal with enormous data vectors. This will present a computational challenge, both from the point of view of likelihood evaluation and in the estimation of the covariance matrix. An efficient data compression scheme would be able to not only alleviate these problems, but also to separate the most significant and less contaminated modes in the data.In this article we have studied the problem of 3D data compression in the context of photometric redshift surveys, and presented a method, based on the KL transform, to derive a basis of orthogonal radial functions that optimally separate the data into uncorrelated modes, where optimality can be defined in terms of overall signal to noise ratio, information content on a particular cosmological parameter or separability between clean signal and contaminants. This basis is a general and natural extension of the well-know harmonic-Bessel (or Fourier-Bessel) decomposition of spherically-symmetric and translationally-invariant systems, adapted to the particular properties of the dataset under study. Even though the definition of this basis requires prior knowledge of some of these properties, including uncertain ones such as a model for the photo-z distributions, once the radial eigenfunctions are selected, the analysis of the resulting data eigenmodes can proceed as usual, including any calibration of these properties. Thus, a suficiently well-educated model of the survey parameters should preserve the near-optimality of the associated eigenbasis, while not hampering the robustness of the analysis.We have shown that, for the study of galaxy clustering in an idealized spectroscopic survey, the optimal set of eigenmodes corresponds to the standard harmonic-Bessel basis, and that this would not be the case in the presence of redshift uncertainties, RSDs or in the analysis of weak lensing observables. For the study of galaxy clustering in a photometric redshift survey, we have shown that the KL basis that maximizes the recovery of the cosmological signal is Bessel-like, although more optimal compression schemes can be derived to optimize the measurement of individual cosmological parameters. In particular, in the case of f_ NL we have shown that the bulk of the constraining power is concentrated in ∼3 radial modes. We have also extended the method to maximize the recovery of a particular signal in the presence of correlated contaminants, and shown that it could be used to simplify the measurement of the effect of lensing magnification as a subdominant contribution to the statistics of the galaxy distribution.In the case of cosmic shear measurements we have shown that, due to the integrated nature of the gravitational lensing effect, the bulk of the signal (∼95%) is concentrated in a single radial mode, qualitatively equivalent to the measurement of the weak lensing effect over the full redshift range of the survey. The next subdominant modes are however needed in order to break degeneracies between different parameters, and we have shown that three modes are enough to recover the best achievable cosmological constraints.Further work is needed in order to alleviate some of the practical shortcomings of the method: the KL decomposition is arguably less connected to real-space, directly observable quantities (although not less so than standard Fourier-space methods). Some of the usual methods for systematics calibration thus need to be adapted for a KL-based analysis, and this is particularly relevant for photo-z systematics. In the case of weak lensing, however, we have shown that the shape of the radial eigenfunctions converges to the same curve on large multipole orders, and that the use of ℓ-independent eigenfunctions would have a negligible impact on the final cosmological constraints. In this case, photo-z calibration methods could be used in exactly the same manner as in the standard tomographic analysis.It is also worth emphasizing that, as is the case for the standard harmonic decomposition of fields defined on the sphere, the KL radial eigenmodes are no longer uncorrelated in the presence of an incomplete sky coverage, and a standard pseudo-C_ℓ analysis reveals non-zero coupling between different multipole orders (ℓ,ℓ')as well as different KL indices (p,p') (see Appendix <ref> and <cit.>). The impact of these correlations on the performance of the KL decomposition should be studied in more detail, and well-understood contaminant-deprojection techniques, implemented in standard power spectrum methods <cit.>, should be adapted for this analysis.Finally, although we have explored the applicability of this method to independent galaxy clustering and weak lensing measurements, current and upcoming photometric redshift surveys will draw cosmological constraints from a joint analysis of both observables <cit.>. The direct application of this method to the joint data vector would in general produce eigenmodes that mix both signals. Alternatively a joint analysis of the KL modes of each observable, taken individually, could be performed, and the merits and drawbacks of each approach should be studied in detail.§ ACKNOWLEDGEMENTSThe author would like to thank Justin Alsing, Pedro Ferreira, Alan Heavens, Boris Leistedt, Jason McEwen, Anže Slosar and David Spergel for useful comments and discussions, and the Center for Computational Astrophysics, part of the Simons Foundation, for their hospitality. He also aknowledges support from the Science and Technology Facilities Council and the Leverhume and Beecroft Trusts. § PSEUDO-C_ℓ ESTIMATION OF THE KL MODESOne of the standard methods to estimate the angular power spectrum of any two quantities in the cut sky is the so-called pseudo-C_ℓ estimator <cit.>. This method can be directly applied to the two-point statistics of the KL eigenmodes, and reveals the correlation between radial modes generated by an incomplete sky coverage <cit.>.The standard pseudo-C_ℓ method is based on computing the spherical harmonic coefficients of the masked field:â^α_ℓ m=∫ d a^α() w^α(),where w^α is the weights map characterizing the mask of the field a^α. One then estimates the power spectrum of this object by averaging over m for each ℓ:Ĉ^αβ_ℓ≡∑_mâ^α_ℓ mâ^β *_ℓ m/2ℓ+1.This object is then related to the true underlying power spectrum through a mode-coupling matrix M^αβ_ℓℓ' such that⟨Ĉ^αβ_ℓ⟩=∑_ℓ'M^αβ_ℓℓ'C^αβ_ℓ', M^αβ_ℓℓ'≡∑_ℓ”(2ℓ'+1)(2ℓ”+1)/4πW^αβ_ℓ”( [ℓ ℓ' ℓ”;000 ])^2where the coupling matrix M depends solely on the power spectrum of the masks W^αβ_ℓ≡(2ℓ+1)^-1∑_mw^α_ℓ mw^β *_ℓ m.The extension of this estimator to the power spectrum of the KL modes is straightforward: we project the masked harmonic coefficients â^α over the KL eigenvectors F_ℓ (i.e. b̂_ℓ m≡ E_ℓ∘â_ℓ m) and compute their power spectra by averaging over m. The resulting estimator takes the form D̂^p_ℓ=∑_ℓ'M_ℓℓ'^pp'D^p'_ℓ', where the new mode-coupling matrix is given by:M^pp'_ℓℓ'≡∑_αβM^αβ_ℓℓ'[∑_α'( F_ℓ)^p_α( N^-1)_αα'( F_ℓ')^p'_α'][∑_β'( F_ℓ)^p_β( N^-1)_ββ'( F_ℓ')^p'_β'] =M_ℓℓ'[∑_αβ( F_ℓ)^p_α( N^-1_ℓ)_αβ( F_ℓ')^p'_β]^2where the second equality holds only if all the maps a^α_ℓ share the same mask w. Note that, for full-sky coverage M_ℓℓ'=δ_ℓℓ', and using the orthonormality of F we recover M^pp'_ℓℓ'=δ_ℓℓ'δ_pp'. | http://arxiv.org/abs/1707.08950v1 | {
"authors": [
"David Alonso"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170727174552",
"title": "Science-driven 3D data compression"
} |
Perfect diode in quantum spin chains Dario Poletti December 30, 2023 ==================================== Dynamical properties of spherically symmetric galaxy models where both the stellar and total mass density distributions are described by the Jaffe (1983) profile (with different scale-lenghts and masses), are presented.The orbital structure of the stellar component is described by Osipkov–Merritt anisotropy, and a black hole (BH) is added at the center of the galaxy; the dark matter halo is isotropic.First, the conditions required to have a nowhere negative and monothonically decreasing dark matter halo density profile, are derived. We then show that the phase-space distribution function can be recovered by using the Lambert-Euler W function, while in absence of the central BH only elementary functions appears in the integrand of the inversion formula. The minimum value of the anisotropy radius for consistency is derived in terms of the galaxy parameters.The Jeans equations for the stellar component are solved analytically, and the projected velocity dispersion at the center and at large radii are also obtained analytically for generic values of the anisotropy radius. Finally, the relevant global quantities entering the Virial Theorem are computed analytically, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, even though highly idealized, represent a substantial generalization of the models presentd in Ciotti et al. (2009), and can be useful as starting point for more advanced modeling the dynamics and the mass distribution of elliptical galaxies. celestial mechanics – galaxies: kinematics and dynamics – galaxies: elliptical and lenticular, cD § INTRODUCTION Spherically symmetric galaxy models, despite their simplicity, are useful tools for theoretical and observational works in Stellar Dynamics, and for the modelization of stellar systems (e.g., Bertin 2000, Binney & Tremaine 2008). Quite obviously spherical symmetry is an oversimplification when considering the vast majority of stellar systems, and a useful spherical model must compensate this limitation with other features, that make its use preferred or even recommended, especially in preliminary investigations. Among the important features of a useful spherical model here we list analytical simplicity, structural and dynamical flexibility, i.e., possibility to add to the stellar component a dark matter halo with adjustable density profile, or alternatively to specify the total density profile, to include the dynamical effects of a central black hole, to control orbital anisotropyFor example, the density profile of the stellar distribution of the model, once projected, should be similar to that of early-type galaxies, i.e. to the de Vaucouleurs (1948) R^1/4 law, or better, to its generalization, the so-called R^1/m law (Sersic 1963). Unfortunately the R^1/m law doesn't allow for an explicit deprojecton in terms of elementary functions, however the so-called γ models (Dehnen 1993, Tremaine et al. 1994) in projection are well fitted over a large radial range, by the R^1/m law. This is especially true for the Jaffe (1983) and Hernquist (1990) models.Another important feature of a useful spherical model is the possibility to reproduce the large scale observational properties of the total density profile of early-type galaxies.In fact, analysis of stellar kinematics (e.g. Bertin et al. 1994, Rix et al. 1997, Gerhard et al. 2001), as well as several studies combining stellar dynamics and gravitational lensing support the idea that the dark and the stellar matter in elliptical galaxies are distributed so that their total mass profile is described by a density distribution proportional to r^-2 (e.g., see Treu & Koopmans 2002, 2004; Rusin et al. 2003; Rusin & Kochanek 2005; Koopmans et al. 2006; Gavazzi et al. 2007; Czoske et al. 2008; Dye et al. 2008, Nipoti et al. 2008, see also Shankar et al. 2017). It is clear that simple dynamical models of two-component galaxies can be useful as starting point of more sophisticated investigations based on axysimmetric or triaxial galaxy models (e.g., Cappellari et al. 2007, van den Bosch et al. 2008). Simple models with flat rotation curve have been in fact constructed (e.g. Kochaneck 1994, Naab & Ostriker 2007). In particular we recall the family of two-component galaxy models whose total mass density is proportional to r^-2, while the visible (stellar) mass is described by the γ models (Ciotti et al. 2009, hereafter CMZ09; see also the double power-law models of Hiotelis 1994). These latter models have been used in hydrodynamical simulations of accretion onto the central supermassive black hole (hereafter, BH) in elliptical galaxies (Ciotti & Ostriker 2012, and references therein).We notice that other models built with the same approach have been recently applied for the interpretation of observations (Poci et al. 2017).We also remark that the approach used to build these models is different from the standard one, where a dark matter halo (herefater, DM) is added to the stellar distribution (e.g. Ciotti & Renzini 1993; Ciotti et al. 1996, hereafter CRL96; Ciotti 1996, 1999; Sect. 4.4 in CMZ09 ).A third important feature of a useful spherical model, strictly related to the previous point, is the possibility to easily compute the dynamical properties of the stellar component in presence of a central BH, and possibly to be proved dynamically consistent (see Sect. 3.1). In fact, supermassive BHs with a mass of the order of ≃ 10^-3 are routinely found at the center of the stellar sferoids of total mass(e.g., see Magorrian et al. 1988, Kormendy & Ho 2013).Following the arguments above, this paper builds on the CMZ09 model, and present an even more general (and realistic) class of models, containing the CMZ09 model as a limit case. On one side, we maintain the assumption of a Jaffe profile for the stellar distribution, but now the total density profile is described by another Jaffe law (instead of a pure r^-2 law), so that the total mass of the models (that we call JJ models) is finite. At the same time, the scale-lenght of the total density is a free parameter and so we can reproduce an r^-2 profile over an arbitrary large radial range. Finally, a central BH of arbitrary mass (missing in CMZ09 models) is considered when solving the dynamical equations. For JJ models we show that the Jeans equations for the stellar component with Osipkov-Merritt (Osipkov 1979, Merritt 1985a, hereafter OM) radial anisotropy can be solved analytically, and the projected velocity dispersion at the center and at large radii can be expressed by means of extremely simple formulae for generic values of the model parameters. The positivity of the phase-space density distribution function of the stellar component (hereafter DF), the so-called consistency, is easily investigated by using a remarkable property of JJ models, i.e. the fact that the radial coordinate can be written in terms of the total potential in terms of the so-called Lambert-Euler W function. By using this property, we determine the maximum amount of radial anisotropy allowable for consistency as a function of the galaxy parameters. These results add to the large amount of phase-space information already available about one and two-component γ models (e.g., Carollo et al. 1995, Ciotti 1996, 1999; Baes et al. 2005, Buyle et al. 2007, Ciotti & Morganti 2009).As a byproduct of our analysis we also found that the one-component Jaffe model, at variance with statements in the literature, cannot be supported by purely radial orbits.We note that the W function also appears in the recenty discovered analytical solution of the isothermal Bondi accretion problem in Jaffe galaxies with central BH (Ciotti & Pellegrini 2017), and this fact suggests a first natural application of JJ models outside the field of Stellar Dynamics, namely in the field of BH accretion and AGN feedback.The paper is organized as follows. In Section 2 the main structural properties of the models are presented. In Section 3 an investigation of the phase-space properties of the models is carried out both from the point of view of necessary and sufficient conditions for consistency, and from direct inspection of the DF. In Section 4 the solution of the Jeans equation with OM radial anisotropy is presented, together with the projection of the velocity dispersion profile at small and large radii.In Section 5 the important properties related to the Virial Theorem and global energetic are explicitly calculated, and the maximum amount of radial anisotropy that can be sustained by the model without developing Radial Orbit Instability is estimated. The main results are summarized in Section 6, while more technical details are given in the Appendix.§ THE MODELSAs anticipated in the Introduction, the present models are characterized by a total density distribution (stars plus DM)described by a Jaffe (1983) profile; the stellar density distributionis also described by a Jaffe profile, in general with a different scale radius. For future use we recall that the Jaffe density of total mass and scale lengthis given by(r)= 4π r^2( + r)^2.The cumulative mass contained within the sphere of radius r, and the associated gravitational potential (with the natural condition of vanishing at infinity, pertintent to systems of finite mass), are given by (r)= r+r,(r)=Glnr+r. Moreover the Jaffe model belongs to the family of γ-models(r)=(3-γ) 4π r^γ(+r)^4-γ, 0≤γ <3,whereis the total mass,is a scale-length, and eq. (1) is obtained for γ=2. The cumulative mass within the sphere of radius r is given by(r)=×(r +r)^3-γ,so that the half-mass (spatial) radius is =/(2^1 3-γ-1), and = for the Jaffe model.For generic values of γ the projected density at radius R in the projection plane is given by(R)=2∫_R^∞ (r)rdr√(r^2-R^2),(e.g., Binney & Tremaine 2008), but unfortunately it cannot be expressed in terms of elementary functions.However, for γ=2(R)=^2×1 4η + √(1-η^2)-(2-η^2)arcsech(η) 2π (1-η^2)^3/2, 0<η <1 ;1 4 - 2 3π,η=1;1 4η- √(η^2-1)+(η^2-2) arcsec(η) 2π(η^2-1)^3/2, η >1;where η≡ R/. In the central and in the very external regions the projected density profile behave like a power law, with(R)∼^2×1 4η, R→ 0;1 8η^3, R→∞,respectively.Finally, an important structural property that we will consider in the following is the projected mass (R) contained within the cylinder of radius R. It can be proved that for spherical systems of finite total mass(R)≡2π∫_0^R Σ (R)RdR =M-4π∫_R^∞ρ(r)r√(r^2-R^2)dr.It follows that the projected mass of the Jaffe model is given by (R)=× g(η), where g(η)=η×π 2-ηarcsech(η)√(1-η^2), 0<η <1 ;π 2-1,η=1;π 2-ηarcsec (η)√(η^2-1), η >1.In particular, the effective radiusof the Jaffe profile (i.e., the radius in the projection plane encircling half of the total mass), where g(η_ e)=1/2, is ≃ 0.7447 (in the Jaffe original paper the slightly erroneous value of 0.763 is reported).§.§ Stellar and total mass distributionWe denote our family of models as “JJ” models, to indicate that it is a two-component Jaffe model, even though constructed in a different way with respect to other two-components Jaffe models in the literature (CLR96, Ciotti 1996, 1999). The properties of the stellar component are obtained with = and = in eqs. (1)-(9), while for the galaxy total density distribution (stars plus DM) = and =.We adoptandas the natural mass and length scales, and we defines≡r, ξ≡,≡ = +1.From the request that the DM component has a non-negative total massit follows that ≡/≥ 0, and so ≥ 1. It is important to note that the request of a non-negativedoes not prevent the possibility of an unphysical, locally negative DM density. This case will be excluded with the introduction of an additional constraint, determined in Sect. 2.2. We also define≡ 4π^3,≡G,as the natural density and potential scales. With these conventions, eqs. (1) and (2) for the galaxy model become(r)= ξ s^2(ξ+s)^2,and(r)= sξ+s, (r)=ξlnsξ+s.We note here an important connection of the JJ models with the models in CMZ09. In fact, the total galaxy density profile in CMZ09 (eq. [6] therein) can be written as^CMZ(r)=^2 4π G r^2= s^2,whereis the constant circular velocity. As the total mass associated with eq. (14) diverges, the parameter =^2/ is not the ratio of the total-to-stellar mass as in JJ models. An elementary integration shows that =()/, i.e., it is the total mass contained within the half mass radius of the Jaffe stellar density profile, normalized to the total stellar mass.From eqs. (12) and (14) it follows that the total density distribution (and the associated quantities, such as the cumulative and the projected mass profiles, and the force field) of CMZ09 models can be obtained from JJ models with the substitution=ξin the corresponding quantities, and then considering the limit for ξ→∞. Some care is needed for the case of the potential. In fact JJ models have finite mass and vanishing potential at infinity, while the logarithimic potential of CMZ09 models^CMZ(r) =^2ln s,diverges for s→∞.The proper way to reobtain ^CMZ from eq. (13) is to apply the substitution (15) to JJ scaled potential +(lnξ)/ξ, and then to take the limit ξ→∞.§.§ The dark matter distribution: positivity and monotonicityBefore studying the dynamical properties of the models, it is important to determine the conditions for the positivity and radial monotonicity of the density distribution of the DM halo. While as anticipated in Sect. 2.1 the request of positivity is natural, a brief comment is in order to justify the requirement of monotonicity. In fact, it can be shown that monotonicity of the density as a function of the potential is necessary for the positivity of the phase-space distribution function. From the second Newton's Theorem, the gravitational potential of a spherical system is necessarily radially monotone, so that the density profile must be a monotone function of radius (Ciotti & Pellegrini 1992, hereafter CP92, see also Sect. 3).As already found in the simpler two component model of CMZ09, also in JJ models not all values ofand ξ are compatible with a nowhere negative DM distribution . Curiously, it is possible to obtain analytically the positivity condition for the more general family of two-component γ models, built with the same approach of JJ models. For γγ models the DM distribution can be written as:(r)=(3-γ) s^γ[ξ (ξ+s)^4-γ - 1 (1+s)^4-γ]:note thatof γγ models is not a γ model, unless the stellar and total length scale are equal, so that in general the local DM-to-stellar mass ratio (r)/ (r) depends on r. It is easy to verify that the total DM mass associated withis =4π^3 ( -1).In Appendix A we determine, for given 0≤γ <3, the conditions onand ξ to have ≥ 0 for r≥ 0. In the case of the JJ models (i.e. γγ models with γ=2), the positivity condition (A2) reduces to≥(ξ)= max(1ξ,ξ);a DM halo of a model with = is called a minimum halo. From equation above it follows that more and more DM is needed for a total density distributionmore and more contracted, or more and more expanded than the stellar distribution . The minimum value =1 can be only adopted when ξ=1, i.e. when the stellar and total density are proportional, and socan vanish everywhere. The situation is illustrated in Fig. 1 (left panel).As anticipated the positivity ofis just a first condition for the viability of the model.A second request is the monotonicity ofas a function of radius, and this reduces to the determination of the minimum valueso that d /dr ≤ 0. The explicit discussion of this additional restriction is given in Appendix A, for the whole family of γγ models. In particular we found that for 1≤γ<3 (the range containing JJ models, or the analogous two-component Hernquist models), the positivity and monotonicity conditions forcoincide, i.e. (ξ) =(ξ).Equation (18) allows to discuss the relative trend of DM and stars in JJ models, both at large radii and near the center, as a function ofand ξ.For r→∞ and ξ >1 it is easy to show that ∼ (ξ -1), and so in the outskirts DM and stars are proportional. When ξ <1 instead the situation is more complicated: while in non minimum halo models ∼ (/ξ -1) and so DM and stars distributions are again proportional, in the minimum halo case ∼ 2(1-ξ)/s∝ r^-5, so that the galaxy is baryon-dominated in the external regions. The situation inverts for r→ 0. In fact, in this case for ξ <1 we have ∼ (/ξ -1) so that DM and stars mass are locally proportional, but for ξ >1, while in non minimum halo models ∼ (/ξ-1), in the minimum-halo models ∼ 2(1-1/ξ) s∝ r^-1, so that these models are centrally baryon-dominated.It can be of interest for applications to evaluate the relative amount of dark and visible mass within a prescribed (spatial or projected) radius. The minimum value for this quantity is easily calculated from eqs. (2) and (13),(r)(r)≥(ξ)(1+s)/ξ+s-1,where (r)=(r)-(r). In Fig. 1 (middle panel) the mass ratios corresponding to three representative values of r are shown as a function of ξ. For example in the case of a sphere of radius equal to a half mass radius of the stellar distribution (i.e. r=), the minimum value / is less than unity for ξ >1: this is a significant improvement of JJ models with respect to the models of CMZ09, where this ratio can not be less than unity (see Fig.3 therein).A similar behavior is obtained for the ratio of projected DM-to-visible mass within some prescribed aperture R, and from the eq. (9) it is easy to show thatM_pDM(R) M_ p*(R)≥(ξ) g(η/ξ) g(η)-1.In Fig. 1 (right panel) we plot this quantity as a function of ξ for three representative values of the aperture radius, i.e. /2,, and 2. Again the qualitative trend is the same as in the other panels, with minimum value well below unity for ξ >1. Note that for R= and considering the limit of eq. (20) for ξ→∞, we obtain for the mass ratio the value ≃ 1.43, in perfect agreement with the analogous result for CMZ09 models.It is interesting to comparethe DM halo profile of JJ models in eq. (17) with the NFW profile (Navarro et al. 1997), that we rewrite for r < (the so-called truncation radius) as(r)=( -1) f(c)s( +s)^2, f(c)=ln(1+c) -c 1+c,where ≡ r_ NFW/ is the NFW scale-lenght in units of , and c≡/r_ NFW: note that in equation above we impose that the total halo massis the same as in eq. (17). From the asymptotic expansion ofwe already know thatandat small and large radii cannot in general be similar. Hovever, in the case of minimum halo with ξ≥ 1, near the centerincreases as 1/r, so thatand can be made indentical for r→ 0 with the additional choice=√(ξ 2f(c)).Therefore once a specific JJ minimum halo model is considered and a radial range fixed, eqs. (21)-(22) allow to determine the best-fit NFW profile with same total mass and central density profile ofby tuning the value of c. For example, after a simple “trial-and-error” exploration, we found that over a range extending out to ≃ 4-8, a “best-fit” NFW profile can be made to agree with a minimum halowith ξ in the range ≃ 2-5, with deviations < 10-20% (at large radii), and < 5% inside ≃ 4R_ e, adopting c in the range ≈ 10-20, and resulting r_ NFW in the range ≈ 0.9 - 1.5. § THE PHASE-SPACEDISTRIBUTION FUNCTIONHaving established the structural limitations of the models, before solving the Jeans equations, it is useful to discuss some basic property of the phase-space distribution function (hereafter DF) of JJ models, in order to exclude dynamically inconsistent combinations of parameters (i.e., choices that would correspond to a somewhere negative DF). Fortunately, as discussed extensively in CP92 (see also Ciotti 1996, 1999), it is possible to obtain lower bounds for the OM anisotropy radius as a function of the density slope and the total mass profile, without actually recovering the DF, which is in general impossible in terms of elementary functions.More specifically, in CP92 a simple theorem was proved regarding the necessary and sufficient limitations onin multi-component OM models. We also recall that the CP92 result has been shown to be just a very special case of a class of important and more general inequalities connecting the local density slope and the anisotropy profile in consistent spherical models (the so-called Global Density Slope - Anisotropy Inequality, GDSAI, e.g., see de Bruijne et al. 1996, An & Evans 2006, Ciotti & Morganti 2009, 2010ab, van Hese et al. 2011).Thus, following the standard nomenclature (e.g., Binney & Tremaine 2008), we assume for the stellar component a DF with the OM parameterizationf=f(Q), Q≡ - J^2 2^2,where =-v^2/2 and J are the relative energy and angular momentum modulus of each star (per unit mass), respectively, and Ψ =-Φ is the relative potential; moreover the DF is truncated as f(Q)=0 for Q<0.As a central BH of massis added at the center of the galaxy, the total (relative) gravitational potential is =+G/r, and from eq. (13) (r)≡ψ(s) =μ s+ξlnξ+s s, μ=.As well known the radial () and tangential () components of the velocity dispersion tensor in OM models are related asβ(r)≡1-^2(r)2^2(r)=r^2r^2+^2,so that the fully isotropic case is obtained for →∞, while for =0 the galaxy is supported by pure radial orbits.For finite values of , the velocity dispersion tensor becomes isotropic for r→ 0 (in practice for r<), and fully radially anisotropic for r→∞ (in practice for r>). Introducing the augmented densityϱ(r)≡(r)(1+r^2^2),the phase-space DF of the stellar component can be recovered from the inversion integralf(Q) = 1√(8)π^2d dQ∫_0^Qdϱ dd√(Q-) = 1√(8)π^2∫_0^Qd^2ϱ d^2d√(Q-);an analogous expression holds for the DF of the isotropic DM halo, obtained by using ϱ=, and=∞ in eq. (23). In the integral above it is intended that ϱ is expressed interms of , and the second identity follows from integration byparts when considering spatially untruncated profiles such those of JJmodels. Note that the OM inversion for the CMZ09 model is somewhat different (see eqs. [19]-[28] therein, and relative discussion), because for these latter models Q is not defined in terms of the relative potential (the potential in eq. [16] is purely logaritmic and so diverges both r→ 0 and r→∞, making the introduction of the relative potential useless), and f(Q) is not truncated as a function of Q.In Sects. 3.1 and 3.2, after a general discussion about the limitations on theimposed by the request of phase-space consistency, i.e., f(Q)≥ 0 over the accessible phase-space, we will see how far we can proceed analytically in the recovery of the DF of JJ models with central BH.§.§ Necessary and sufficient conditions for consistencyFollowing CP92 a necessary condition for the positivity of the DF of each of the mass components of JJ models (stars or DM) in the total (galaxy plus central BH) potential is thatdϱ (r) dr≤ 0[NC]:this condition is independent of the behavior of the other density components of the system. A weak sufficient condition for consistency is obtained by requiring that thederivative inside the last integral in eq. (27) be positive. Also this condition can be expressed as a function of radius asd dr[dϱ(r) drr^2 (r)]≥ 0,[WSC],where the total mass profile is given by(r)=(r)+,and (r) is given in eq. (13).Therefore, a model failing eq. (28) is certainly inconsistent, while a model obeying eq. (29) is certainly consistent.It follows that the true boundary in the parameter space separating consistent and inconsistent models - that in general can be only determined by direct inspection of the DF - is “bracketed” by the NC and WSC limits. Before embarking in the analysis of JJ models, some preliminary consideration is in order.First, about the effect of the central BH on consistency. From eqs. (29) and (30) it follows quite easily that if 1) the component is consistent for =0, and 2) d (r^2 dϱ/dr)/dr≥ 0, then the model with central BH is certainly consistent. Note that point 2) is nothing else than the WSC for the considered density profile interpreted as a tracer in the gravitational field of the central BH itself; we will use this result in the following discussion. A second consideration is about the effect of anisotropy. When dealing with OM anisotropic systems, the investigation of the NC and WSC, and the study of the DF positivity, lead to cosider inequalities of the kindF+G^2≥ 0, ≡,that must hold over the domainspanned by the functions' argument.In practice, the functions F and G are functions of r (in the case of the NC and WSC) or functions of Q (in the case of the DF). From inequality (31) it follows that all OM models can be divided in two families.When F is nowhere negative over(e.g., in the case of a consistent isotropic DF), consistency in the anisotropic case is obtained for≥≡√( max[0, sup_(-G F)]).If G is also positive over , then =0 and the system can be supported by radial orbits only.In the second case F is positive only over some proper subsetof , and negative (or zero) over the complementary subset .If G <0 somewhere[ In Ciotti (1999) and Ciotti (2000) it is erroneously stated that the model is inconsistent if G<0 everywhere on .All the results presented therein are however correct.]on , then the condition (31) cannot be satisfied and the model is inconsistent.If G ≥ 0 onone must consider not only the lower limitin eq. (32) evaluated over , but also the condition≤=√( inf_(-G F)),and consistency is possible only if <.Summarizing, if F≥ 0 then ≥ for consistency. If F≤ 0 overand G≥ 0 there, then the inequality ≤≤ must be verified. Finally, if overthe function G <0 somewhere, or <, then inequality (31) cannot be satisfied and, in case of a DF analysis, the model must be rejected as inconsistent.The first application of eqs. (28)-(29) to JJ models concerns the consistency of the DM halo.Following the similar analysis in CMZ09, for simplicity we restrict to the isotropic case, and then eq. (28) shows the equivalence of the request of monotonicity of(Sect. 2.2) with the NC for a consistent DM halo.Of course, the restriction to isotropic case is quite arbitrary, as the virialized end-states of N-body collapses are invariably characterized by some amount of radial anisotropy (e.g., van Albada 1982; Nipoti, Londrillo & Ciotti 2006), but for the present illustrative purposes this assumption is fully justified.The WSC for a fully isotropic DM halo is worked out analytically in Appendix A.In particular, when restricting to the case of no central BH (μ = 0) we found, quite surprisingly, that the condition imposed by the WSC to the halo is nothing else than the limit (18) imposed by positivity and monotonicity.It remains to discuss the effect of a central BH. Following the argument after eq. (30), it is not difficult to show (Appendix A) that the addition of the central BH in case of isotropy reinforces consistency, i.e., a DM halo that is consistent in absence of central BH, it is certainly consistent when a BH is added. Taken together, the two results above and those in Sect. 2.2 shows that the isotropic DM halo of JJ models with central BH, once positivity only ofis assured, automatically satisfies the NC and WSC conditions, and so it is supported by a nowhere negative phase-space DF.We now move to the more interesting case of the NC and WSC for the stellar component of OM anisotropic JJ models. First, we recall that NC of Jaffe models just reduces to have ≥ 0 while, from the solution of a cubic equation the WSC for the one-component Jaffe model gives ≥≃ 0.1068 (Ciotti 1999), marked by the red solid red circle in Fig. 2.Second, in Appendix A we show that the WSC of the stellar component of JJ models is always in the case described by eq. (32), i.e. onlyexists. However, the function at the r.h.s. of eq. (32) in the general case is sufficiently complicated that only a numerical study is feasible. In any case, as in the next Section we will determine the exact limit onobtained from the DF, here we just restrict to the case μ=0. The resulting eq. (A8) is much simpler than the general one, and in particular =(ξ), i.e. when μ =0 the limit on anisotropy is independent of(dotted red line in Fig. 2).The red triangle at ≃ 0.0487 marks the position of the WSC limit for the CMZ09 model obtained by solving a cubic equation, and that as expected is in accordance with the value of the red line for ξ→∞. At the opposite limit we have the BH dominated case (see Appendix A), with ≃ 0.31 marked by the red square, coincident with the value of the red line for ξ→ 0, when the total potential becomes that of a central point mass.In practice, from the arguments after eq. (30), we have now proved that the stellar component of JJ models with central BH and OM anisotropy is certainly consistent for > 0.31, independently of the mass of the central BH and of the DM halo total mass and scale-lenght.We conclude this introductory analysis by noticing the fact that for JJ models, the presence of a diffuse halo appears to increase the model ability to sustain radial anisotropy, while for concentrated halos the consistency of the stellar distribution requires a more isotropic velocity dispersion tensor, as already found in other two-component OM models (Ciotti 1996, 1999, CMZ09).§.§ Explicit phase-space DFWith the introduction of the dimensionless potential ψ=/ and augmented density ϱ̃=ϱ/ from eqs. (24) and (26), respectively, eq. (27) writesf(q) = √(8)π^2^3/2∫_0^qd^2ϱ̃ dψ^2d ψ√(ψ -q)=√(8)π^2^3/2[U(q)+V(q)^2],where q≡ Q/. From eqs. (23)-(24) it follows that 0≤ q≤∞.In Appendix B we show that it is possible to invert eq. (24) and express analytically the radius as a function of the relative total potentialby using the Lambert-Euler W function, obtainings(ψ)= ξ Wμ-1,W=W(0, μ e^ξψ +μ),where W(0,z) is one of the two branches of the real determination of the complex function W. In absence of the central BH we have = and it can be shown that eq. (35) reduces to the elementary functions(ψ)=ξ e^ψξ/-1,in agreement with the solution of eq. (24) with μ =0.With the substitution (35) in eq. (26) we finally obtain the expression for ϱ̃(ψ) to be used in eq. (27). The derivatives inside the integral are evaluated from the exact relation in eq.(B3). We note that the field of application of the W function to physical problems is rapidly expanding (e.g. see Valluri et al. 2000, Cranmer et al. 2004, Ciotti & Bertin 2005 for an application to self-consistent toroidal structures, Veberic 2012, Waters & Proga 2012, Herbst 2015, Ciotti & Pellegrini 2017 for the solution of isothermal accretion on BHs at the center of galaxies).In CMZ09 it is shown that for the stellar Jaffe model embedded in a total singular isothermal density profile, and in absence of the central BH, the functions U and V can be expressed as simple linear combinations of exponentials and polylogarithms.Here, not surprisingly, the functions U and V cannot be expressed in terms of known functions, even in absence of the central BH. However, it is interesting to notice that in case of a dominant central BH (in practice, sufficiently near to the center), the function f(q) can be expressed by using simple functions (Appendix C).We now determine numerically the lower limit for consistency ofby inspection of the functions U and V.Note that in absence of the central BH (μ=0), the variable q in eq. (34) can be further scaled as =q/, while a factor of ^-3/2 appears in the functions U and V, as shown in eq. (C4). In particular, for these models without BH the position of the maximum in eq. (32) depends on(and so in terms of q scales linearly with ), but the value ofis independent of . The same situation occurs in the CMZ09 models, and in the extreme case of a BH dominated JJ model, where =q/μ (Appendix C, eqs. [C4]-[C5]), and the scaling arguments above apply to the DF withreplaced by μ.It is numerically found that U≥ 0, so that eq. (32) applies and onlyexists: the solid line in Fig. 2 shows (ξ) determined by the DF in absence of the central BH, for comparison with the other curves presented.Notice how the shape of the critical consistency curve parallels the WSC condition (red dotted line), and how there are consistent models failing the WSC. The black circle at ξ =1 marks the value of the minimum value ≃ 0.02205 for the OM one-component Jaffe model (Appendix C).From the figure it i apparent how the effect of a concentrated DM halo reduces the ability of the stellar component to sustain radial orbits, while the opposite happens for models with ξ >1. As an independent test of the derived DF, the black triangle indicates the limit value of ≃ 0.0141 obtained in CMZ09 by numerical inspection of the DF (coincident, as expected, with the limit value of the curve for ξ→∞), and the black square the value of the BH dominated case ≃ 0.082, coincident with the limit of the curve for ξ→ 0.In Fig. 3 the DF of the stellar component of a selection of representative JJ models is presented, in the isotropic (top panel) and anisotropic (bottom panel, =0.1) cases. In both cases the DFs are shown with and without the effect of the central BH (with μ = 10^-3), and for illustration, also the BH dominated DF (green dashed line) is shown. It is clear how at high (relative) energies the DF of JJ models with central BH is perfectly described by the BH dominated DF. Also, it is apparent how at high energies the isotropic and anisotropic DFs for models with the same structure are almost identical, a property of OM anisotropy parameterization leading to almost isotropic models in the central regions. It is also important to note how the DFs of models without the central BH are higher at high energies than in the analogous models with the central BH. Also, notice how models with heavier and more extended halos and so with higher velocity dispersions at large radii (Fig. 4, top panel) at low relative energies have a lower DF. The physical reason of this behavior is due to the fact that, qualitatively, the phase-space DF is inversely proportional to the cube of velocity dispersion (because the integral over the velocity space, at fixed position, must reproduce the same value of the local density), so that, empirically, high velocity dispersions corresponds to low values of the DF. This is particularly apparent in the BH dominated case, with a low DF at high energies and a high DF at low energies. This is also confirmed by the low-energy tail of the DF, which is higher in the anisotropic cases. In fact, from eqs. (25) and (35) it follows that for r→∞, the total velocity dispersion profile is proportional to ( A+^2 I)/r^2, i.e., it is lower for smaller values of . Finally, notice how orbital anisotropy produces a drop of the DF at intermediate energies, with a depression that would be of increasing depth for decreasing values of , finally leading to an inconsistent DF. The curves relative to the anisotropic cases are very similar to the analogous curves in Ciotti & Lanzoni (1997, Fig. 2), and C99 (Figs. 2 and 3) and CMZ09 (Fig. 3), revealing the common qualitative behavior of OM anisotropic DFs near the consistency limit, i.e. the fact that the inconsistency manifests itself in general at intermediate energies (see also Ciotti & Morganti 2008 for a discussion). § JEANS EQUATIONS WITH OM ANISOTROPYThe Jeans equations for spherical systems with general (radial or tangential) anisotropy has been discussed in Binney & Mamon (1982), and in the OM case the solution can be written as(r)^2(r) = G r^2 +^2∫_r^∞(r) (r) (1+^2 r^2)dr = A(s)+ ^2 I(s) s^2+^2,where (r) is given in eq. (30),and the two radial functionsI =(s) +μ(s), A = (s)+μ(s),are the isotropic and purely radial anisotropic components of the velocity dispersion tensor, respectively.In the formula above the dimensionless mass factors,and μ, have been explicitely factorized. For →∞ we obtain the solution of the fully isotropic case, while for =0 we obtain the purely radial case.§.§ The velocity dispersion profileThe integration of eq. (37) is elementary. In fact it is formally equivalent to an integration already performed, for the different class of two component Jaffe models in CLR96. where the OM Jeans equation is integrated for a stellar Jaffe distribution, superimposed to Jaffe DM halo of total mass , and length scale r_DM. Therefore in CLR96 the combined contribution of the stars and of the DM potential to the stellar velocity dispersion profile is given by the sum of two different expressions. Here, instead, only one integration is required because the total potential is assigned and, in practice, with a suitable renaming of parameters, the formula in CLR96 for the DM halo contribution could be used.However, as we now consider also the effect of the central BH, not included in the models in CLR96 and CMZ09, we give the full set of formulae in homogeneous notation.For the isotropic component=ln(ξ+s)ξ^3(ξ-1)^2+(3ξ-4)ln(1+s)(ξ-1)^2-(3ξ^2+2ξ+1)ln sξ^3-2s^2(3ξ^2-ξ-1)+s(3ξ+2)(ξ-1)-ξ(ξ-1) 2ξ^2(ξ-1)s^2(1+s),-(6s^2+6s-1)(2s+1) 2s^2(1+s)-6 lns 1+s,where the first expression holds for ξ 1, and the second for ξ=1. As expected the two expressions agree with eqs. (A11) and (A5) in CLR96, respectively[Due to a typo, the sign of the terms inside the square brackets of eq. (A11) in CLR96 should be, from left to right, plus, plus, minus, minus.].The contribution of the BH to the stellar isotropic velocity dispersion profile is given by= 12s^3+6s^2-2s+1 3s^3(1+s)+4 lns 1+s.Note that this expression could be formally obtained also by considering the limit for ξ→ 0 of the function , because from eq. (2)for fixed r and → 0, becomes the potential of a point mass.For the anisotropic part we have=ln(ξ+s)ξ (ξ-1)^2+(ξ-2)ln(1+s) (ξ-1)^2-ln sξ-1 (1+s)(ξ-1),-2s+3 2(1+s)^2-lns 1+s,where the first expression holds for ξ 1 and the second for ξ =1, and they agree with eqs. (A10) and (A4) in CLR96, respectively. The contribution of the central BH to the anisotropic stellar velocity dispersion profile is=1+2s s(1+s)+2lns 1+s,and again it is simple to prove that At= for ξ→ 0. Following eq. (15), we also verified eqs. (39) and (41) considering the limit for ξ→∞ of the functions ξ and ξ, and recovering eqs. (C2)-(C3) in CMZ09 evaluated for γ=2.An insight of the behavior ofcan be obtained by considering the expansion for r→∞ and r→ 0 of the obtained formulae. We begin with the outer galaxy regions.A simple expansion of the functions A and I shows that for r→∞ (in practice, for r >>) the leading order term is the same for the galaxy as for the BH, with∼∼1 5s^5+O(s^-6), ∼∼1 3s^3+O(s^-4).The coincidence of the leading term is just due to the fact that for r→∞ the cumulative mass profile in eq. (13) converges to the total galaxy mass, and for the Newton's theorem this leads to the same contribution to the velocity dispersion as that of a central mass .Following the same approach adopted in CMZ09 we combine eqs. (43)-(44), and the leading term ofin eq. (37) for r→∞ is obtained, for arbitrary value of , by retaining the leading order term of the expansion of the much simpler expression(r) ^2(r) ∼ (+μ) 5s^2+3^2 15s^5 (s^2+^2).In the case of finitewe have ^2 ∝ 1/(3s), while in the fully isotropic case ^2 ∝ 1/(5s): as expected, the isotropicis lower than in case of finite , when the outer regions become populated by radial orbits only. As expected eq. (45) agrees with the analogous expression obtained for the two-component model briefly discussed in Sect. 4.4 of CMZ09 (eq. [40] therein, for γ=2 and for a dominant DM halo). This is at variance with the behavior of the genuine CMZ09 model, where for r→∞(r)^2(r)∼^22s^2+^2 4s^4(s^2+^2).Therefore, although the full velocity profile of the CMZ09 model is recovered from the the limit procedure in eq. (15) applied to eqs. (39)-(41), the limit procedure applied to eq. (45) does not converge to the asymptotic expansion of the velocity dispersion profile in CMZ09. This is due to the fact that for ξ→∞ and r→∞ the integral (37) is not uniform in the variables ξ and r, so that the two limits cannot be in general exchanged.The other important region for observational and theoretical works, is the galaxy center: here the velocity dispersion profile is dominated by the BH contribution. In fact, at the leading order∼1 2ξ s^2+O(s^-1),∼ -ln sξ+O(1),so that for r→ 0 the galaxy contribution to the stellar velocity dispersion profile is given by(r)^2(r) ∼ξ s^21 2,>0 ;-ln s, =0.In particular, if =0, the central velocity dispersion diverges as ^2 ∝ -ln s, while for all values >0 the central velocity dispersion converges to a finite value, coincident with that of the isotropic case^2 (0)= 2ξ.This is relevant from the modelistic point of view, as it is well known that self-gravitating isotropic γ models present a depression of their velocity dispersion near the center with (0)=0, except for the γ=0 and γ=2 models (e.g., see Bertin et al. 2002 for a general discussion of this phenomenon; see also Binney & Ossipkov 2001). Notice that the value of the central velocity dispersion, in the minimum halo model with ξ≥ 1 is, according to eq. (18), independent of ξ, and coincident with that of the purely stellar Jaffe model. This shows the danger of a “blind” use of (0) as a robust indicator of the actual depth and shape of the galaxy potential well.For the BH we obtain∼1 3s^3+O(s^-2),∼1 s+O(ln s),and the formula analogous to eq. (48) is(r) ^2(r) ∼μ s^31 3, >0 ;1, =0.As expected, ^2 diverges as μ /r for r→ 0, and with a factor of 3 of difference between the fully radially anisotropic case, and all the other cases with >0 in agreement with the general property ofin the central regions of γ models with a BH (e.g., see C96, Baes & Dejonghe 2004, Baes et al. 2005). We conclude by noticing that eqs. (48), (49) and (51) are also in accordance with the analogous quantities for the CMZ09 model (eqs. [C5]-[C6]) and the two-component models in Sect. 4.4 there (eq. [44]), and with the results in the spherical (isotropic) limit of one and two-component oblate power-law models with central BH in Ciotti & Bertin (2005, eq. [C3]) and in Riciputi et al. (2005, eq. [A4]), evaluated for γ=2All the relevant properties ofdescribed in this Section are illustrated in Fig. 4 (top panel) by a selection of representative JJ models. In particular, the effects of the central BH, of the DM halo, and of orbital anisotropy, can be clealry seen near the center and at large radii.§.§ Projected velocity dispersionThe projected velocity dispersion profile associated with a general anisotropy function β(r) is given byΣ_*(R)^2(R)= 2∫_R^∞ [1-β(r)R^2r^2 ](r)^2(r) r√(r^2-R^2)dr,(e.g., Binney & Tremaine 2008), and in the OM case β (r) is given in eq. (25).Unsurprisingly the projection integral cannot be evaluated analytically for JJ models in terms of elementary functions.However, as for the spatial velocity dispersion profile interesting informations can be obtained outside the core radius and near the center.In practice, in the external regions the stellar and total density profiles can be approximated as a pure power-law of slope -4. In this region the projection integral can be evaluated for generic values ofand in analogy with eq. (45) the asymptotic trend with radius of the projected profile can be obtained by retaining the leading order term of the expansion of^2(R) ∼ 8( +μ) 15πη[1+η^4 2^2(^2+η^2).- .η^4(2^2+η^2)archsinh(/η) 2^3(^2+η^2)^3/2],where η≡ R/. The expression in square parentheses converges to 1 in the isotropic case, and to 1/3 for all finite values of .The analogous formula for the CMZ09 limit models is^2(R)∼^2(^2+η^2)^5/2-η^3(2^2+η^2)4^2(^2+η^2)^3/2,and the same considerations made after eq. (46)hold.The case of the central regions is more complicated.In fact, both the integral (52) and the projected surface density Σ_* (see eq. [7]) are asymptotically dominated by their integrands for r→ 0, so that ^2 can be properly defined only as the limit for R→ 0 of the ratio of two diverging quantities. For what concerns the galaxy contribution, a simple calculation shows that for >0(0)=(0),where (0) is given by eq. (49), again in agreement with eq. (33) in CMZ09 for γ=2. For =0 instead the central projected velocity dispersion diverges.Therefore, for the stellar component of JJ models and >0, the projected central velocity dispersion coincides with the central radial component of the isotropic velocity dispersion. In presence of the central BH,is dominated by the BH contribution, and so it is the projected velocity dispersion. With some care, from eqs. (51)-(52) it can be shown that, from eqs. (51) and (52) and independently of the value of ≥ 0,^2(R)∼2μ 3πη.All the relevant properties ofexpressed by the formulae in this Section can be noticed in Fig. 4 (bottom panel), where we show the projected velocity dispersion profiles for the same JJ models in the top panel.In particular Fig. 4 shows a well known consequence of the OM parameterization, i.e., the fact that the isotropicprofiles (black lines) in the outer regions are below those in the corresponding radially anisotropic cases (red lines), while the opposite holds for the sigp profiles, due to projection effects on radial orbits in the outer regions, where the l.o.s. direction is almost perpendicular to the stellar orbits.We conclude this Section by noticing that CMZ09 (eq. [39]) briefly commented on the spatial and projected velocity dispersion of a two-component galaxy model made by the superposition of a stellar distribution described by a γ model, and a DM halo described by a Jaffe model.Of course, when γ=2 this family reduces to JJ models in CLR96: in turns it is easy to check the perfect correspondance of eqs. (49) and (55) with eq. (42) in CMZ09 by assuming there →∞ and β=ξ, when the model becomes formally identical (in the limiting case of a DM halo “infinitely massive”) to JJ models (without central BH).The formulae (55) and (56) also agree, as expected, with the projection formulae in the spherical limit of the ellipsoidal models with γ=2 in Ciotti & Bertin (2005, eqs. [C1] and [C7] therein). § VIRIAL, POTENTIAL, ANDKINETIC ENERGIESAmong the several global quantities that are associated with a stellar system, those entering the Virial Theorem (hereafter VT) are certainly the most interesting for many observational and theoretical studies (e.g., Ciotti 2000, Binney & Tremaine 2008). For the stellar component of JJ models we have2≡ - =--,where =2π∫_0^∞ (^2+^2)r^2dr is the total kinetic energy of the stars,= -∫ <𝐱 ,∇>d^3𝐱 = - 4π G∫_0^∞ r (r)(r)dr,is the interaction energy of the stars with the gravitational field of the galaxy (stars plus DM), and finally=-4π G ∫_0^∞ r(r) dr,is the interaction energy of the stars with the central BH. For a Jaffe galaxydiverges, because the stellar density profile diverges near the origin as r^-2; instead, this quantity converges for γ models with 0≤γ<2. Therefore, the VT implies that also the volume integral of σ_*BH^2 diverges near the origin for a Jaffe galaxy, as can be seen by direct integration of eq. (51)The contribution of the total galaxy potential to= + (whereis due to the self-interaction of the stellar distribution, andto the effect of the DM halo) is finite, with the remarkably simple result=-ξ-1-lnξ (ξ-1)^2,ξ 1;1 2,ξ=1,and taking the limit as in eq. (15), =-G^2, in accordance with eq. (33) in CMZ09. More generally it can be shown thatis a finite quantity for the stellar component of γγ models, provided 0≤γ < 5/2 (e.g., for two component Hernquist model, obtained for γ=1).It follows that for this class of models it is possible to define the (3-dimensional) galactic virial velocity dispersion as σ^2_ V=-/: moreover, from eqs. (60), (55) and (49) the value of σ_ V^2 is proportional to the value of the central projected velocity dispersion ^2(0), and the proportionality constant is a function of ξ only: for ξ=1, σ_ V=(0). We also notice the interesting behavior ofas a function of ξ in the minimum halo case. While for increasing ξ≥ 1 it follows that ==ξ increases correspondingly to arbitrarily large values, the dimensionless coefficient in eq. (60) just increases from 1/2 for ξ=1 to 1 for ξ→∞, due to the fact that in minimum-halo case, more massive halos are necessarily more and more extended, with a compensating effect on the depth of the total potential.As well known, in multi component systems the virial energy W of a given component is not the gravitational energy of the component itself in the total potential. For this reason we now calculate explicitly the different contributions to the potential energyof the stellar component of JJ models, and we also show how to obtain the expression ofandin a simple way.As for the interaction energy , also for the potential energyholds the decomposition=+,where =+=1 2∫ d^3𝐱+∫ d^3𝐱,and=∫ d^3𝐱=-4π G ∫ r (r) dr=.Thereforediverges as . From a well known result, the self-gravitational energy and the virial self energy of each density component of a multi-component system coincide, and in our casefrom eq. (60) with =1 and ξ=1,== - 2,so that we can compute = - without performing additional integrations.The evaluation ofis slightly more complicated, because in principle it would require to substitute =- in the second integral in eq. (60), and therefore compute two integrals. But we adopt a different strategy, and we compute the integral≡∫ d^3𝐱=-lnξξ-1,ξ 1;1,ξ=1.so that from eq. (60)= -2,and finallyis obtained by adding .Note thatis not the gravitational energyof the stars in the galaxy total potential, . Yet,is not just an useful mathematical quantity, but it has an important physical interpretation, and togetherplays a fundamental role in the theory of galactic winds and X-ray emission of early-type galaxies. In fact, the energy per unit time to be provided to the ISM of early-type galaxies (for example by supernova explosion, thermalization of stellar winds, and AGN feedback) required to steadily extract the mass losses of stars, injected over the galaxy body at the rate ρ̇_inj=α(t) is given by L_grav=α(t)|| (e.g, see Pellegrini 2011, 2012, Posacki et al. 2013). A nice feature of JJ models is thatis finite and given by a remarkably simple expression, at variance with the situation of CMZ09 models, where this quantity would diverge, or other two-component models, whereis given by quite cumbersome formulae. Therefore JJ models provide a very simple framework to estimate the energetic of galactic gas flows hosted by X-ray emitting early-type galaxies.§.§ StabilityAnother particularly relevant application of the VT is in the field of model stability, i.e. the determination of the conditions required to prevent the onset of the so-called Radial Orbit Instability (hereafter, ROI). In fact, it is well known that stellar systems supported by a large amount of radial orbits are in general unstable (e.g., Fridman & Polyachenko 1984, and references therein).A stability analysis is obviously well beyond the task of this work, but we can obtain some quantitative information by investigating the value, as a function of the model parameters, of the stability indicatorΞ≡2= -4 2+/,whereand = - are the total kinentic energes of the stellar component of JJ models, associated with the radial and tangential components of the velocity dispersion tensor, respectively, and the last expression is obtained by evaluatingfrom the VT. Of course, we exclude the effect of the central BH, due to the formal divergence of the kinetic energy K_ *BH discussed in previous Section.From its definition Ξ→1 for →∞ (globally isotropic models), while Ξ→∞ for → 0 (fully radially anisotropic models).Numerous investigations of one-component systems have confirmed that the onset of ROI is in general prevented by the empirical requirement that Ξ < 1.7± 0.25; the exact value of the limit is model dependent (see, e.g., Merritt & Aguilar 1985; Bertin & Stiavelli 1989; Saha 1991, 1992; Bertin et al. 1994; Meza & Zamorano 1997; Nipoti, Londrillo & Ciotti 2002).Here we are considering two-component systems, however N-body simulations have shown that the presence of a DM halo does not change very much the situation with respect to the one-component systems (e.g., see Stiavelli & Sparke 1991, Nipoti et al. 2002). In our case, we assume as a fiducial maximum value for stability 1.7. Note that from eq. (60) and volume integration of eq. (37) with μ =0, eq. (67) shows that Ξ is independent of .Unfortunatelycannot be expressed by using elementary functions, so that we explore numerically the fiducial stability condition Ξ (,ξ)=1.7. In Fig. 2 with green dashed curve we plot the resulting lower bound for stability (ξ).It is apparent that the critical value ofincreases with ξ, and this is due to the fact that a spatially extended DM halo increases the contribution to the kinetic energy of the velocity dispersion in the outer parts that, in the OM case, are radially anisotropic. Therefore, in order to guarantee stability in presence of an extended DM halo, the permitted amount of radial orbits must correspondingly reduced, and larger values ofare needed. As a limit case the green triangle marks the position of the stability indicator for the limit models in CMZ09, with a limit value of ≃ 1.78.The opposite situation occurs when the DM halo is more concentrated than the stellar component, because in this case the velocity dispersion is increased preferentially in the central regions, that in the OM case are in practice isotropic, and so a larger amount of radial orbits can be supported. All these trends nicely agree with those found for different famillies of one and two-component γ models (Ciotti 1996, 1999, see also Carollo et al. 1995).We finally notice how the stability criterion requires minimum anisotropy radii appreciably larger than those obtained from the consistency analysis (see Sect. 3.2), and so it is likely that the maximally radially anisotropic models with positive DF, would be prone to develop ROI. § CONCLUSIONSThe family of spherical, two-component galaxy models with the stellar density distribution described by the Jaffe profile, embedded in a DM halo such that the total density distribution is also a Jaffe profile, is presented.The DM halo is defined as the difference between the total and the stellar density distributions.A BH is added at the center of to the system, and the dynamics of the stellar component is described by the Osipkov-Merritt anisotropy profile. The models are fully determined once the total stellar mass () and scale length () are assigned, together with the total-to-stellar mass ratio (), the total-to-stellar scale length ratio (ξ), the BH-to-stellar mass ratio (μ), and finally the anisotropy radius () of the stellar distribution.These models represent a generalization of the CMZ09 models, where the total density profile was fixed at r^-2 at all radii. In fact JJ models, while retaining interesting properties such as a realistic stellar density profile and a total density profile that can be described an arbitrarily large radial range by a r^-2 profile, have a finite total mass, and a central BH.At the same time, they still allow for an almost complete analytical treatment, and several quantities of interest in observational and theoretical works have remarkably simple explicit expressions.The main results can be summarized as follows. * After providing a summary of the structural quantities of observational interest for JJ models, for the more general family of two-component γγ models, we derive analitically the constraints onand ξ needed to assure positivity and monotonicity of the DM halo density distribution.For a given value of ξ, the model corresponding to the minimum value allowed foris called minimum halo model.In JJ models (in which the positivity and monotonicity limits coincide), ≥max(ξ, 1/ξ).Near the origin the density profile of the DM halo diverges as ∝ r^-2, but in the minimum halo model with ξ >1 the models are centrally “baryon dominated”, with ∝ r^-1. * It is shown that the models presented in CMZ09 are limit cases JJ models (in absence of the central BH), and we provide the framework to derive all the structural and dynamical quantities of the CMZ09 models from those of JJ models. * The minimum value of anisotropy radius , corresponding to a dynamically consistent stellar component (i.e., characterized by a nowhere negative DF), is first estimated by using the necessary and sufficient conditions given in CP92. It is shown that in absence of the central BH the minimum value ofso determined is a function of ξ only. The criticaldecreases for increasing ξ, i.e., as already found in other two-component models, a DM halo more extended than the stellar distribution increases the ability of the stellar component to sustain radial anisotropy. On the contrary, more concentrated DM halos (and in particular a central BH), require a more isotropic orbital distribution. The preliminary consistency analysis is also performed for the DM halo, and it is proved that for isotropic DM halos in JJ models with μ=0 the conditions of positivity, monotonicity, and phase-space consistency coincide; the addition of a central BH reinforces consistency. * We then moved to study the phase-space DF for the stellar component as given by OM inversion.We found that for JJ models it is possible to express analytically the dependence of radius on the total potential in terms of the Lambert-Euler W function, allowing for a fast and accurate recovery of the DF. In case of no BH (μ=0) the resulting expression reduces to elementary functions, and in the limit case of a dominat BH (or a very concentrated DM halo corresponding to ξ→ 0), the DF itself can be obtained in terms of elementary functions. After presenting a few representative cases of DFs, corresponding to different choices of , ξ, and , we determined numerically the (minimum) critical value ofas a function of the model parameters, and we found that the obtained curve nicely parallels the bound given by the sufficient condition in CP92. We showed that in absence of the central BH, and in the case of a dominant BH, the criticaldepends only on ξ, and it is independent ofand μ, respectively. In general, we confirmed that DM halos more extended than the stellar component increase the amount of radial anisotropy that can be supported by a positive DF, while the opposite happens in case of concentrated halos (or in presence of a central BH), again in accordance with previous findings relative to different two-component OM models. Quite unexpectedly, from the inspection of the analytical DF, and by independent numerical verification, we found that the single component Jaffe model cannot support purely radial orbits in the OM formulation, as detailed in Appendix C. * The Jeans equations for the stellar component are solved explicitely for generic values of the model parameters in terms of elementary functions.The asymptotic expansions ofandfor r→ 0 and r→∞ are obtained, and in particular it is shown that when μ=0 and for all values of >0 (isotropic case included) ^2(0)=/(2ξ). In this case, by asymptotic expansion of the projection integral with >0, it is also shown that independently of the value of the anisotropy radius, (0)=(0). In presence of the BH, in the central regions ^2∝ r^-1 with a coefficient which is different for =0 or >0. In projection, due to a compensating effect, ^2(R)∼ 2μ/(3π R) for ≥ 0. * Finally, the analytical expressions of relevant quantities entering the Virial Theorem, such as the stellar kinetic energy, the virial energy interactions, the potential energies, are derived as a functions of the model parameters. With the aid of the obtained formulae we determined the minimum value ofcorresponding to a value of ≃ 1.7 of the Friedmann-Poliachenko-Shuckman instability indicator, so that more anisotropic models are prone to the onset of Radial Orbit Instability. Again, in line with previous results, the minimumfor stability increases for increasing ξ, and in absence of the central BH its value depends only on ξ, being independent of .We conclude by noting that JJ models, albeit highly idealized, suggest a few interesting remarks of observational and theoretical character. For example, after having fixed the properties of the models by using available observational constraints (e.g., see Negri et al. 2014), one could use JJ models to investigate how the so called sphere of influence of the BH depends on the galaxy properties and how its definition is affected by orbital anisotropy. Following a preliminary study (Ziaee Lorzad 2016), it is natural to define the radius of the sphere of influence as the distance from the galaxy center where the quantityΔσ^2 ≡σ_ *g^2+σ_ *BH^2-σ_ *g^2σ_ *g^2= μ (r) (r),reaches some prescribed value (for example 20%, 50%, 100%) as a function structural and dynamical properties of the galaxy itself.JJ models could also be used to obtain some preliminary estimate of structural/dynamical properties of high-redshift galaxies (e.g., see Sect. 4.4.1 in Vanzella et al. 2017), thanks to the very simple expressions of their virial quantities.Another interesting application of JJ models is in the field of BH accretion because, as shown in Ciotti & Pellegrini (2017), it is possible to solve analytically the generalized isothermal Bondi accretion problem in Jaffe (or Hernquist) potentials with a central BH. As the total density profile of JJ models is a Jaffe law, it follows that for these models we can solve both the accretion problem for the gas and the Jeans equations for the stellar component. Moreover, JJ models allow for the computation of the stellar kinetic energy, a quantity strictly related to the average temperature of the ISM in early-type galaxies.As the gas temperature determines the location of the Bondi radius, JJ models represent a fully analytical family of self consistent stellar dynamical-hydrodynamical models, that will allow to compare the relative position of the sonic radius and the radius of the sphere of influence as a function of the galaxy properties. § ACKNOWLEDGMENTS We thank the anonymous Referee for useful comments, and John Magorrian and Silvia Pellegrini for interesting discussions on the models. L.C. thanks G. Bertin, J. Binney, T. de Zeeuw, W. Evans, D. Lynden-Bell, D. Merritt and S. Tremaine for useful comments on Appendix C.A.Z. acknowledges the Department of Physics and Astronomy of Padua University, where a preliminary study of JJ models has been the subject of her Master Thesis.§ POSITIVITY AND MONOTONICITY OF THE DARK MATTER HALO IN ΓΓ MODELSThe condition for the positivity of the DM halo density profilein γγ-models with 0≤γ <3 is established from eq. (17) as≥(ξ+s)^4-γξ (1+s)^4-γ,s≥ 0.Therefore,must be greater than or equal to the maximum (ξ,γ) of the radial function at r.h.s.: note thatis the minimum value ofin order to have a nowhere negative DM halo. Simple algebra shows that the maximum is attained at infinity for ξ <1, and at the origin for ξ >1, while for ξ=1 the radial function is identically equal to 1.From eq. (A1) it follows that≥(ξ,γ)=max(1ξ, ξ^3-γ),and for γ=2 we obtain eq. (18).The monotonicity condition foris obtained by requiring that d/dr≤ 0, i.e.≥(ξ+s)^5-γ(γ+4s) (1+s)^5-γξ(ξγ+4s), s≥0.Again we must determine the maximum (ξ,γ) of the r.h.s. of the equation above. It is easy to show that for γ=0(ξ,0) =max(1ξ,ξ^4),while for 0<γ<1(ξ,γ) =max(1ξ,f_+),where f_+ is the value of the r.h.s. of eq. (A3) at the critical points_+(ξ,γ)= √(γ[γξ^2+ξ(5-3γ)+γ]) - γ (1+ξ) 10:f_+ →ξ^4 and f_+ →ξ^2 for γ→ 0^+ and γ→ 1^-, respectively. Finally for 1≤γ <3 (and so in particular for JJ models, or for two-component Hernquist models that could be constructed by using the same approach of JJ models) it can be shown, quite surprisingly, that the monotonicity condition coincides with the positivity condition, and sois given by eq. (A2).The application of the WSC to the isotropic DM halo is obtained from eq. (26) with →∞, i.e. ϱ=. The condition in absence of the central BH (μ =0) reduces to≥2(s+ξ)^3[6s^3 + 4(1+2ξ)s^2+(1+7ξ)s +2ξ]4 ξ (1+s)^4 (3s^2 +3ξ s +ξ^2),s≥ 0.For ξ =1 the r.h.s. equals 1 independently of s. For ξ≠ 1 the determination of the maximum leads to study a fifth degree equation. Fortunately, it can be proved by inspection that the resulting expression with s≥ 0 is negative for ξ >1 (and thus the maximum of eq. [A7] is reached at s=0), and positive for 0<ξ <1 (and so the maximum is reached for s→∞). In the two limits eq. (A7) evaluates to ξ and 1/ξ, respectively, and so we conclude that the isotropic DM halo of JJ models (in absence of central BH) is certainly consistent whensatisfies the positivity and monotonicity condition in eq. (18). We are now in position to consider the effect of the central BH. A direct analysis would lead to a cumbersome expression, to be explored numerically. However, by using the considerations after eq. (30), it is simple to show that the additional term due to the BH is positive, and so it reinforces the WSC when the positivity condition onis verified.The application of the WSC to the OM anisotropic stellar component of JJ models leads to the study of a seventh degree equation, and shows that we are in the conditions pertinent to eq. (32). In absence of the central BH the dependence ondisappears,^2≥ - s^3 [s^2 +2 (ξ -1)s -ξ] 6s^3 +4 (1+2ξ)s^2 + (1+7ξ)s +2ξ,s≥ 0,and we should solve a fifth degree equation when searching for the maximum of the r.h.s. In Sect. 3.1 we present the results obtained by numerical inspection of equation above. Restricting further to the case ξ =1 (i.e., reducing to the one-component Jaffe model), the equation to be solved becomes cubic, with ≃ 0.1068 (Ciotti 1999). Finally, the case obtained for ξ→ 0 is formally coincident with the case of a dominant central BH (i.e., onlyis retained in eqs. [29]-[30]), and for this limiting case the WSC reduces again to a cubic equation, with solution ≃ 0.31.§ THE LAMBERT-EULER W FUNCTION As discussed in Sect.3, for JJ models with central BH it is possible to invert eq. (24) and express the radius r in terms of the relative total potential by using the Lambert-Euler W function.The integrand in the inversion integral (34) is then obtainedin explicit and easily tractable form, without resorting to complicate numerical procedures, because the W function is now fully implemented in the most used computer algebra systems. The function W(z) (see, e.g., Corless et al. 1996) is a multivalued complex function defined implicitly by the identityW e^W=z,and the two real branches W(0,z) and W(-1,z) for real values of z are shown in Fig. B1.With the transformation of variables W=(1+ξ/s)μ/, eq. (24) can be rewritten as W+ln W = ξψ +μ +lnμ,so that eq. (35) is obtained by exponentiation of eq. (B2) and comparison with eq. (B1). It is immediate to conclude that for the present problem the relevant branch is given by W(0,z), restricting to points beyond point A.In fact, when ψ→∞, the argument in eq. (B2) is infinite, W(0,z)→∞, and from eq. (35) s→ 0. Instead, when ψ→ 0, the argument tends to μ/ e^μ/, W →μ /, and so from eq. (35) s→∞ from eq. (35), as it should [By definition of W, it follows that W(ye^y)=y.] .Finally, note that the derivatives inside the integral (34) can be expressed analytically in terms of W itself, because from eq. (B1) it follows thatdW dz=W z(1+W).§ DF OF OM JAFFE MODEL WITH DOMINANT CENTRAL BHWe report the explicit phase-space DF of a Jaffe model with OM anisotropy and with dominant central BH (i.e., the gravitational field is produced by the BH only, and the stellar distribution is only a tracer). The resulting expression can be interpreted as the asymptotic limit of the DF at high relative energies, i.e. for galactic regions sufficiently near the central BH. By using the nomenclature in eq. (34), it is easy to show that the functions U and V can be written asU(q)=U()μ^3/2, V(q)=V()μ^3/2,≡qμ,whereU()=(16^3+40^2+18+9)√() 4 (1+q)^3 -3(3+8) arcsenh√() 4 (1+)^7/2,andV()=(13-2)√() 4 (1+)^3+ 3(1-4) arcsenh√() 4 (1+)^7/2.The function U is nowehere negative in the range 0≤ <∞, so the BH dominated Jaffe models are always in the first case discussed in Sect. 3.1, and onlyexists: a numerical evaluation shows that ≃ 0.082, in agreement with the trend of the solid line in Fig. 2 for ξ→ 0, when the DM halo “collapses” to a central point mass.For completeness we also report the explicit DF for the stellar component of JJ models with ξ =1 and in absence of the central BH, when the resulting expression reduces to the one-component DF in the OM case. From eq. (34) we now haveU(q)=U()^3/2, V(q)=V()^3/2,≡q,with 0≤ <∞ andU() 4√(2)=(√(2))+ (√(2)) - √(2)[ (√()) +(√())], V() 4√(2)= (√(2)) -(√())√(2),where (x)=e^-x^2∫_0^xe^t^2dt is the Dawson's function, and (x)=e^x^2∫_0^xe^-t^2dt=√(π)e^x^2 Erf(x)/2. The functions above, when combined according to eq. (34), are in perfect agreement with those given by Merritt (1985b, eq. [6]) and Binney & Tremaine (2008).The function U in eq. (C5) is positive for alla values of , as shown by the WSC, but the function V in eq. (C6) becomes negative for admissible values of , so thatcannot be arbitrarily small.Numerical evaluation of eq. (32) shows that for consistency ≥≃ 0.02205, in perfect agreement with the solid line in Fig. 2 (obtained from the general DF) for ξ =1. From this result one could conclude that the purely radial model does not exist. However the situation is not so simple. In fact, the DF of a purely radial model can be written in all generality as f=δ(J^2)h(), so that for a finite mass, spatially untruncated modelρ() =2π r^2∫_0^h() d√( -),(e.g., Ciotti 2000), and the inversion formula can be immediately found (e.g., see Richstone & Tremaine 1984, Oldham & Evans 2016)h() = 1√(2)π^2d d∫_0^ϱd√(-) = 1√(2)π^2∫_0^dϱ dd√(Q-),where ϱ =r^2ρ is expressed in terms of , and the second identity follows from integration by parts when considering spatially untruncated profiles such those of JJ models.As shown by Merritt (1985b, eq. 8) and Evans et al. (2015, eq. 31), for the purely radial one-component Jaffe model h()= 2^2π^2√()[√(2) (√()) -(√(2))],where ≡/. The function is positive at all energies, thus showing that the purely radial Jaffe model is consistent. These two seamingly contradictory results indicate that the purely radial case, at least for the Jaffe model, is a singular limit for the OM parameterization. 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"authors": [
"L. Ciotti",
"A. Ziaee Lorzad"
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"published": "20170727114154",
"title": "Two-component Jaffe models with a central black hole. I: the spherical case"
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unsrtnat outdir=./ #1_#1 | http://arxiv.org/abs/1707.08838v1 | {
"authors": [
"Alexander F. Schäffer",
"Hermann A. Dürr",
"Jamal Berakdar"
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"published": "20170727123911",
"title": "Ultrafast nanoscale magnetic switching via intense picosecond electron bunches"
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theoremTheorem lemma[theorem]Lemma corollary[theorem]Corollaryproposition[theorem]Proposition./Figs/. | http://arxiv.org/abs/1707.08343v1 | {
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"Peter Varkonyi",
"Alan Champneys"
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"published": "20170726095210",
"title": "Dynamics beyond dynamic jam; unfolding the Painlevé paradox singularity"
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[NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ====================== In a controlled experiment of sequence-to-sequence approaches for the task of sentence correction, we find that character-based models are generally more effective than word-based models and models that encode subword information via convolutions, and that modeling the output data as a series of diffs improves effectiveness over standard approaches. Our strongest sequence-to-sequence model improves over our strongest phrase-based statistical machine translation model, with access to the same data, by 6 M^2 (0.5 GLEU) points. Additionally, in the data environment of the standard CoNLL-2014 setup, we demonstrate that modeling (and tuning against) diffs yields similar or better M^2 scores with simpler models and/or significantly less data than previous sequence-to-sequence approaches.§ INTRODUCTION The task of sentence correction is to convert a natural language sentence that may or may not have errors into a corrected version. The task is envisioned as a component of a learning tool or writing-assistant, and has seen increased interest since 2011 driven by aseries of shared tasks<cit.>. Most recent work on language correction has focused on the data provided by the CoNLL-2014 shared task <cit.>, a set of corrected essays by second-language learners. The CoNLL-2014 data consists of only around 60,000 sentences, and as such, competitive systems have made use of large amounts of corrected text without annotations, and in some cases lower-quality crowd-annotated data, in addition to the shared data. In this data environment, it has been suggested that statistical phrase-based machine translation (MT) with task-specific features is the state-of-the-art for the task <cit.>, outperforming word- and character-based sequence-to-sequence models <cit.>, phrase-based systems with neural features <cit.>, re-ranking output from phrase-based systems <cit.>, and combining phrase-based systems with classifiers trained for hand-picked subsets of errors <cit.>.We revisit the comparison across translation approaches for the correction task in light of the Automated Evaluation of Scientific Writing (AESW) 2016 dataset, a correction dataset containing over 1 million sentences, holding constant the training data across approaches. The dataset was previously proposed for the distinct binary classification task of grammatical error identification. Experiments demonstrate that pure character-level sequence-to-sequence models are more effective on AESW than word-based models and models that encode subword information via convolutions over characters, and that representing the output data as a series of diffs significantly increases effectiveness on this task. Our strongest character-level model achieves statistically significant improvements over our strongest phrase-based statistical machine translation model by 6 M^2 (0.5 GLEU) points, with additional gains when including domain information. Furthermore, in the partially crowd-sourced data environment of the standard CoNLL-2014 setup in which there are comparatively few professionally annotated sentences, we find that tuning against the tags marking the diffs yields similar or superior effectiveness relative to existing sequence-to-sequence approaches despite using significantly less data, with or without using secondary models. All code is available at <https://github.com/allenschmaltz/grammar>.§ BACKGROUND AND METHODS TaskWe follow recent work and treat the task of sentence correction as translation from a source sentence (the unedited sentence) into a target sentence (a corrected version in the same language as the source). We do not make a distinction between grammatical and stylistic corrections.We assume a vocabularyof natural language word types (some of which have orthographic errors). Given a sentence 𝐬 = [s_1 ⋯ s_I], where s_i ∈ is the i-th token of the sentence of length I, we seek to predict the corrected target sentence 𝐭 = [t_1 ⋯ t_J], where t_j ∈ is the j-th token of the corrected sentence of length J. We are given both 𝐬 and 𝐭 for supervised training in the standard setup. At test time, we are only given access to sequence 𝐬. We learn to predict sequence 𝐭 (which is often identical to 𝐬). Sequence-to-sequence We explore word and character variants of the sequence-to-sequence framework.We use a standard word-based model (Word), similar to that of luong-pham-manning:2015:EMNLP, as well as a model that uses a convolutional neural network (CNN) and a highway network over characters (CharCNN), based on the work of KimEtAl-2016-CharLM, instead of word embeddings as the input to the encoder and decoder. With both of these models, predictions are made at the word level. We also consider the use of bidirectional versions of these encoders (+Bi). Our character-based model (Char+Bi) follows the architecture of the Word+Bi model, but the input and output consist of characters rather than words. In this case, the input and output sequences are converted to a series of characters and whitespace delimiters. The output sequence is converted back to 𝐭 prior to evaluation. The Word models encode and decode over a closed vocabulary (of the 50k most frequent words); the CharCNN models encode over an open vocabulary and decode over a closed vocabulary; and the Char models encode and decode over an open vocabulary.Our contribution is to investigate the impact of sequence-to-sequence approaches (including those not considered in previous work) in a series of controlled experiments, holding the data constant. In doing so, we demonstrate that on a large, professionally annotated dataset, the most effective sequence-to-sequence approach can significantly outperform a state-of-the-art SMT system without augmenting the sequence-to-sequence model with a secondary model to handle low-frequency words <cit.> or an additional model to improve precision or intersecting a large language model <cit.>. We also demonstrate improvements over these previous sequence-to-sequence approaches on the CoNLL-2014 data and competitive results with JiEtal2017arXiv-NestedAttention, despite using significantly less data. The work of schmaltz-EtAl:2016:BEA11 applies Word and CharCNN models to the distinct binary classification task of error identification. Additional ApproachesThe standard formulation of the correction task is to model the output sequence as 𝐭 above. Here, we also propose modeling the diffs between 𝐬 and 𝐭. The diffs are provided in-line within 𝐭 and are described via tags marking the starts and ends of insertions and deletions, with replacements represented as deletion-insertion pairs, as in the following example selected from the training set:“Some key points are worth del emphasiz /del ins emphasizing /ins .”. Here, “emphasiz” is replaced with “emphasizing”. The models, including the Char model, treat each tag as a single, atomic token. The diffs enable a means of tuning the model's propensity to generate corrections by modifying the probabilities generated by the decoder for the 4 diff tags, which we examine with the CoNLL data. We include four bias parameters associated with each diff tag, and run a grid search between 0 and 1.0 to set their values based on the tuning set.It is possible for models with diffs to output invalid target sequences (for example, inserting a word without using a diff tag). To fix this, a deterministic post-processing step is performed (greedily from left to right) that returns to source any non-source tokens outside of insertion tags. Diffs are removed prior to evaluation. We indicate models that do not incorporate target diff annotation tags with the designator –diffs.The AESW dataset provides the paragraph context and a journal domain (a classification of the document into one of nine subject categories) for each sentence.[The paragraphs are shuffled for purposes of obfuscation,so document-level context is not available.] For the sequence-to-sequence models we propose modeling the input and output sequences with a special initial token representing the journal domain (+dom).[Characteristics of the dataset preclude experiments with additional paragraph context features. (See Appendix A.)] § EXPERIMENTS Data AESW <cit.> consists of sentences taken from academic articles annotated with corrections by professional editors used for the AESW shared task. The training set contains 1,182,491 sentences, of which 460,901 sentences have edits. We set aside a 9,947 sentence sample from the original development set for tuning (of which 3,797 contain edits), and use the remaining 137,446 sentences as the dev set[The dev set contains 13,562 unique deletion types, 29,952 insertion types, and 39,930 replacement types.] (of which 53,502 contain edits). The test set contains 146,478 sentences.The primary focus of the present study is conducting controlled experiments on the AESW dataset, but we also investigate results on the CoNLL-2014 shared task data in light of recent neural results <cit.> and to serve as a baseline of comparison against existing sequence-to-sequence approaches <cit.>. We use the common sets of public data appearing in past work for training: the National University of Singapore (NUS) Corpus of Learner English (NUCLE) <cit.> and the publicly available Lang-8 data <cit.>. The Lang-8 dataset of corrections is large[about 1.4 million sentences after filtering] but is crowd-sourced[derived from the Lang-8 language-learning website] and is thus of a different nature than the professionally annotated AESW and NUCLE datasets. We use the revised CoNLL-2013 test set as a tuning/dev set and the CoNLL-2014 test set (without alternatives) for testing. We do not make use of the non-public Cambridge Learner Corpus (CLC) <cit.>, which contains over 1.5 million sentence pairs.Evaluation We follow past work and use the Generalized Language Understanding Evaluation (GLEU) <cit.> and MaxMatch (M^2) metrics <cit.>. Parameters All our models, implemented with OpenNMT <cit.>, are 2-layer LSTMs with 750 hidden units. For the Word model, the word embedding size is also set to 750, while for the CharCNN and Char models we use a character embedding size of 25. The CharCNN model has a convolutional layer with 1000 filters of width 6 followed by max-pooling, which is fed into a 2-layer highway network. Additional training details are provided in Appendix A. For AESW, the Word+Bi model contains around 144 million parameters, the CharCNN+Bi model around 79 million parameters, and the Char+Bi model around 25 million parameters. Statistical Machine TranslationAs a baseline of comparison, we experiment with a phrase-based machine translation approach (SMT) shown to be state-of-the-art for the CoNLL-2014 shared task data in previous work <cit.>, which adds task specific features and the M^2 metric as a scorer to the Moses statistical machine translation system. The SMT model follows the training, parameters, and dense and sparse task-specific features that generate state-of-the-art results for CoNLL-2014 shared task data, as implemented in publicly available code.[SRI International provided access to SRILM <cit.> for running junczysdowmunt-grundkiewicz:2016:EMNLP2016] However, to compare models against the same training data, we remove language model features associated with external data.[We found that including the features and data associated with the large language models of junczysdowmunt-grundkiewicz:2016:EMNLP2016, created from Common Crawl text filtered against the NUCLE corpus, hurt effectiveness for the phrase-based models. This is likely a reflection of the domain specific nature of the academic text and LaTeX holder symbols appearing in the text. Here, we conduct controlled experiments without introducing additional domain-specific monolingual data.] We experiment with tuning against M^2 (+M^2) and BLEU (+BLEU). Models trained with diffs were only tuned with BLEU, since the tuning pipeline from previous work is not designed to handle removing such annotation tags prior to M^2 scoring. § RESULTS AND ANALYSIS: AESW Table 1 shows the full set of experimental results on the AESW development and test data. The Char+Bi+dom model is stronger than the Word+Bi+dom and CharCNN+dom models by 2.9 M^2 (0.2 GLEU) and 3.3 M^2 (0.3 GLEU), respectively.The sequence-to-sequence models were also more effective than the SMT models, as shown in Table <ref>. We find that training with target diffs is beneficial across all models, with an increase of about 5 M^2 points for the Word+bi model, for example. Adding +dom information slightly improves effectiveness across models.We analyzed deletion, insertion, and replacement error types. Table <ref> compares effectiveness across replacement errors. We found the CharCNN+Bi models were less effective than CharCNN variants in terms of GLEU and M^2, and the strongest CharCNN models were eclipsed by the Word+Bi models in terms of the GLEU and M^2 scores. However, Table <ref> shows CharCNN+dom is stronger on lower frequency replacements than Word models. The Char+Bi+dom model is relatively strong on article and punctuation replacements, as well as errors appearing with low frequency in the training set and overall across deletion and insertion error types, which are summarized in Table <ref>.Errors never occurring in trainingThe comparatively high Micro F_0.5 score (18.66) for the Char+Bi+dom model on replacement errors (Table<ref>) never occurring in training is a result of a high precision (92.65) coupled with a low recall (4.45). This suggests some limited capacity to generalize to items not seen in training. A selectively chosen example is the replacement from “discontinous” to “discontinuous”, which never occurs in training. However, similar errors of low edit distance also occur once in the dev set and never in training, but the Char+Bi+dom model never correctly recovers many of these errors, and many of the correctly recovered errors are minor changes in capitalization or hyphenation. Error frequency About 39% of the AESW training sentences have errors, and of those sentences, on average, 2.4 words are involved in changes in deletions, insertions, or replacements (i.e., the count of words occurring between diff tags) per sentence. In the NUCLE data, about 37% of the sentences have errors, of which on average, 5.3 words are involved in changes. On the AESW dev set, if we only consider the 9545 sentences in which 4 or more words are involved in a change (average of 5.8 words in changes per sentence), the Char+Bi model is still more effective than SMT+BLEU, with a GLEU score of 67.21 vs. 65.34. The baseline GLEU score (No Change) is 60.86, reflecting the greater number of changes relative to the full dataset (cf. Table 1). Re-annotation The AESW dataset only provides 1 annotation for each sentence, so we perform a small re-annotation of the data to gauge effectiveness in the presence of multiple annotations. We collected 3 outputs (source, gold, and generated sentences from the Char+Bi+dom model) for 200 randomly sampled sentences, re-annotating to create 3 new references for each sentence. The GLEU scores for the 200 original source, Char+Bi+dom, and original gold sentences evaluated against the 3 new references were 79.79, 81.72, and 84.78, respectively, suggesting that there is still progress to be made on the task relative to human levels of annotation. § RESULTS AND ANALYSIS: CONLL Table <ref> shows the results on the CoNLL dev set, and Table <ref> contains the final test results.Since the CoNLL data does not contain enough data for training neural models, previous works add the crowd-sourced Lang-8 data; however, this data is not professionally annotated. Since the distribution of corrections differs between the dev/test and training sets, we need to tune the precision and recall.As shown in Table <ref>, Word+Bi effectiveness increases significantly by tuning the weights[In contrast, in early experiments on AESW, tuning yielded negligible improvements.] assigned to the diff tags on the CoNLL-2013 set[The single model with highest M^2 score was then run on the test set. Here, a single set is used for tuning and dev.]. Note that we are tuning the weights on this same CoNLL-2013 set. Without tuning, the model very rarely generates a change, albeit with a high precision. After tuning, it exceeds the effectiveness of Word+Bi–diffs. The comparatively low effectiveness of Word+Bi–diffs is consistent with past sequence-to-sequence approaches utilizing data augmentation, additional annotated data, and/or secondary models to achieve competitive levels of effectiveness.Table <ref> shows that Word+Bi is within 0.2 M^2 of JiEtal2017arXiv-NestedAttention, despite using over 1 million fewer sentence pairs, and exceeds the M^2 scores of XieEtAl.2016-arxiv-NLCwithCharAttention and yuan-briscoe:2016:N16-1 without the secondary models of those systems. We hypothesize that further gains are possible utilizing the CLC data and moving to the character model. (The character model is omitted here due to the long training time of about 4 weeks.) Notably, SMT systems (with LMs) are still more effective than reported sequence-to-sequence results, as in JiEtal2017arXiv-NestedAttention, on CoNLL.[For reference, the reported M^2 results of the carefully optimized SMT system ofjunczysdowmunt-grundkiewicz:2016:EMNLP2016 trained on NUCLE and Lang-8, with parameter vectors averaged over multiple runs, with a Wikipedia LM is 45.95 and adding a Common Crawl LM is 49.49. We leave to future work the intersection of a LM for the CoNLL environment and more generally, whether these patterns hold in the presence of additional monolingual data.] § CONCLUSION Our experiments demonstrate that on a large, professionally annotated dataset, a sequence-to-sequence character-based model of diffs can lead to considerable effectiveness gains over a state-of-the-art SMT system with task-specific features, ceteris paribus. Furthermore, in the crowd-sourced environment of the CoNLL data, in which there are comparatively few professionally annotated sentences in training, modeling diffs enables a means of tuning that improves the effectiveness of sequence-to-sequence models for the task. emnlp_natbib§ SUPPLEMENTAL MATERIAL Additional Model Training and Inference Details We provide additional replication details for our experiments here. Our code and related materials are available at the following url: <https://github.com/allenschmaltz/grammar>.The training and tuning sizes of the AESW dataset are those after dropping sentences exceeding 126 tokens on the source or target side (in source sequences or target sequences with diff annotation tags) from the raw AESW dataset. All evaluation metrics on the development and test set are on the data without filtering based on sentence lengths.As part of preprocessing, the sentences from the AESW XML are converted to Penn Treebank-style tokenization. Case is maintained and digits are not replaced with holder symbols for the sequence-to-sequence models. For the SMT models, the truecasing[Here, the truecase language model is created from the training 𝐭 sequences (or where applicable, the target with diffs).] and tokenization pipeline of the publicly available code is used. For consistency, all model output and all reference files are converted to cased Moses-style tokenization prior to evaluation. For the Char model, the L_2-normalized gradients were constrained to be ≤ 1 (instead of ≤ 5 with the other models), and our learning rate schedule started the learning rate at 0.5 (instead of 1 for the other models) for stable training. The maximum sequence length of 421 was used for models given character sequences, which was equivalent to the maximum sequence length of 126 used for models given word sequences. The maximum sequence lengths were increased by 1 for the models with the +dom features. The training and tuning set sizes cited in Section 3 are the number of sentences from the raw dataset after dropping sentences exceeding these maximum sequence lengths.In practice, we were able to train each of the purely character-based models (e.g., the Char+Bi+dom model) with a single NVIDIA Quadro P6000 GPU with 24 GB of memory in about 3 weeks with a batch size of 12.For the sequence-to-sequence models, the closed vocabularies were restricted to the 50,000 most common tokens, and a single special 𝗎𝗇𝗄 token was used for all remaining low frequency tokens. An 𝗎𝗇𝗄 token generated in the target sentence by the Word and CharCNN models was replaced with the source token associated with the maximum attention weight. The “open” vocabularies were only limited to the space of characters seen in training.For the phrase-based machine translation baseline model from the work of junczysdowmunt-grundkiewicz:2016:EMNLP2016, for dense features, we used the stateless edit distance features and the stateful Operation Sequence Model (OSM) of durrani-EtAl:2013:Short[The OSM features use the SRI Language Modeling Toolkit (SRILM) <cit.>. ]. Since for our controlled data experiments we removed the language model features associated with external data, we did not use the word-class language model feature, so for the sparse features, we used the set of edit operations on “words with left/right context of maximum length 1 on words” (set “E0C10” from the original paper), instead of those dependent on word classes.The training and tuning splits for the phrase-based machine translation models were the same as for the sequence-to-sequence models. For tuning, we used Batch-Mira, setting the background corpus decay rate to 0.001, as in previous work. As in previous work, we repeated the tuning process multiple times (in this case, 5 times) and averaged the final weight vectors.The sequence-to-sequence models were decoded with a beam size of 10. Decoding of the SMT models used the same approach of junczysdowmunt-grundkiewicz:2016:EMNLP2016 (i.e., the open-source Moses decoder run with the cube pruning search algorithm).In our experiments, we do not include additional paragraph context features, since the underlying AESW data appears to have been collected such that nearly all paragraphs (including those containing a single sentence) contain at least one error; thus, modeling paragraph information provides additional signal that seems unlikely to reflect real-world environments. CoNLL-2014 Shared Task For training, we used the copy of the Lang-8 corpus distributed in the repo for the code of junczysdowmunt-grundkiewicz:2016:EMNLP2016: <https://github.com/grammatical/baselines-emnlp2016>. We filtered the Lang-8 data to remove duplicates and target sentences containing emoticon text, informal colloquial words (e.g., “haha”, “lol”, “yay”), and non-ascii characters. Target sentences not starting with a capital letter were dropped, as were target sentences not ending in a period, question mark, exclamation mark, or quotation mark. (Target sentences ending in a parenthesis were dropped as they often indicate informal additional comments from the editor.) In the combined NUCLE and Lang-8 training set, source sentences longer than 79 tokens and target sentences longer than 100 tokens were dropped. This resulted in a training set with 1,470,992 sentences. Diffs were created using the Python class difflib.SequenceMatcher.For tuning on the dev set[Previous work, such as junczysdowmunt-grundkiewicz:2016:EMNLP2016, also used the CoNLL-2013 set for tuning.], a coarse grid search between 0 and 1.0 was used to set the four bias parameters associated with each diff tag. (Training was performed without re-weighting.) The bias parameter (in this case 0.7) yielding the highest M^2 score on the decoded dev set was chosen for use in evaluation of the final test set. The M^2 scores across the tuning runs on the dev set for the Word+Bi model are shown in Table <ref>.For future comparisons to our work on the CoNLL-2014 shared task data, we recommend using the preprocessing scripts provided in our code repo (<https://github.com/allenschmaltz/grammar>). Table 2 The seven columns of Table 2 appearing in the main text are Micro F_0.5 scores for the errors within each frequency grouping. There are a total of 39,916 replacement changes. The replacements are grouped in regard to the changes within the opening and closing deletion tags and subsequent opening and closing insertion tags, as follows: (1) whether the replacement involves (on the deletion and/or insertion side) a single punctuation symbol (comma, colon, period, hyphen, apostrophe, quotation mark, semicolon, exclamation, question mark); (2) whether the replacement involves (on the deletion and/or insertion side) a single article (a, an, the); (3) non-article, non-punctuation grouped errors with frequency greater than 100 in the gold training data; (4) non-article, non-punctuation grouped errors with frequency less than or equal to 100 and greater than or equal to 5; (5) non-article, non-punctuation grouped errors with frequency less than 5 and greater than or equal to 2; (6) non-article, non-punctuation grouped errors with frequency equal to 1; (7) non-article, non-punctuation grouped errors that never occurred in the training data. Note that the large number of unique instances occurring for the “punctuation” and “articles” classes are a result of the large number of errors that can occur on the non-article, non-punctuation side of the replacement. The Micro F_0.5 scores are calculated by treating each individual error (rather than the agglomerated classes here) as binary classifications. | http://arxiv.org/abs/1707.09067v1 | {
"authors": [
"Allen Schmaltz",
"Yoon Kim",
"Alexander M. Rush",
"Stuart M. Shieber"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170727225055",
"title": "Adapting Sequence Models for Sentence Correction"
} |
^1The Department of Physics, Princeton University^2The Department of Chemistry, Princeton University We present a Monte Carlo method for computing the renormalized coupling constants and the critical exponents within renormalization theory.The scheme, which derives from a variational principle, overcomes critical slowing down, by means of a bias potential that renders the coarse grained variables uncorrelated. The 2D Ising model is used to illustrate the method. Valid PACS appear here A Variational Approach to Monte Carlo Renormalization Group Yantao Wu^1 and Roberto Car^1, 2 December 30, 2023 =========================================================== Since the introduction of renormalization group (RG) theory <cit.>, there has been strong interest in methods to compute the renormalized coupling constants and the critical exponents in a non-perturbative fashion.This goal has been achieved with the Monte Carlo (MC) RG approach of Swendsen.In 1979, he introduced a method to compute the critical exponents, which did not require explicit knowledge of the renormalized Hamiltonian <cit.>.A few years later, he solved the problem of calculating the renormalized coupling constants, using an equality due to Callen <cit.> to write the correlation functions in a form explicitly depending on the couplings.By imposing that the standard MC expression of a correlation function and its corresponding Callen form be equal, he derived equations whose iterative solution led to the coupling constants <cit.>.Finding the renormalized Hamiltonian is an example of inverse statistical mechanical problem <cit.>.MCRG has been used successfully in many applications but difficulties related to sampling efficiency may be severe.Typically, the evaluation of the correlation functions near a critical point suffers from critical slowing down and is affected by large sampling errors in large systems.This difficulty can be alleviated with ingenious cluster algorithms <cit.>, which, however, are limited to specific models. Here we present an MCRG framework based on a variational principle for a biasing potential acting on the coarse grained degrees of freedom of a RG transformation.In our approach, the coupling constants and the critical exponents derive from the same unifying principle.Swendsen's formulae emerge as a special case, but our scheme also leads to formulations exempt from critical slowing down.In addition, it permits to estimate variationally the effect of truncating the Hamiltonian.Although the approach is rather general, here we limit ourselves, for concreteness, to lattice models with discrete spin degrees of freedom, {σ}.A generic Hamiltonian has the formH(σ) = ∑_α K_α S_α(σ),where the K_α are coupling constants and the S_α are operators acting on the spins σ, such as sums or products of spins or combinations thereof.RG considers a flux in the space of Hamiltonians (<ref>) under scale transformations that reduce the linear size of the original lattice by a factor b.The rescaled degrees of freedom take the same discrete values of the original spins, to which they are related by a coarse graining transformation, σ' = τ(σ).For example, τ can be the block spin transformation of Kadanoff <cit.>. The distribution of the σ' is obtained from the distribution of the σ by tracing out the original degrees of freedom while keeping the σ' fixed:p(σ') = ∑_σδ_τ(σ), σ' e^-H(σ)/Z =e^-H'(σ')/Z' .Here δ is the discrete Kroneker-delta function,Z and Z' are partition functions that ensure the normalization of the corresponding distributions.While the partition function Z' is invariant under RG transformations, the renormalized Hamiltonian H' is not, except at fixed points of the RG flow:Z = ∑_σ e^-H(σ) = ∑_σ' e^-H(σ') = Z'and H'(σ') = -log∑_σδ_τ(σ), σ' e^-H(σ)Repeated at infinitum, the RG transformations generate a flux in the space of Hamiltonians, in which all possible coupling terms appear, unless forbidden by symmetry.For example, in an Ising model with no magnetic field, only even spin products appear.The space of the coupling terms is, in general, infinite.However, perturbative and non-perturbative calculations suggest that only a finite number of couplings should be sufficient for a given degree of accuracy. In the proximity of a critical point, the distribution (<ref>) of the block spins σ' displays a divergent correlation length, originating critical slowing down of local MC updates.This can be avoided by modifying the distribution of the σ' by adding to the Hamiltonian H'(σ') a biasing potential V(σ') to force the biased distribution of the block spins, p_V(σ'), to be equal to a chosen target distribution, p_t(σ'). For instance, p_t can be the constant probability distribution. Then the σ' have the same probability at each lattice site and act as uncorrelated spins, even in the vicinity of a critical point. It turns out that V(σ') obeys a powerful variational principle that facilitates the sampling of the Landau free energy <cit.>. In the present context, we define the functional Ω[V] of the biasing potential V(σ') by: Ω [V] = log∑_σ' e^-[H'(σ') + V(σ')]/∑_σ' e^-H'(σ')+ ∑_σ' p_t (σ') V(σ'),where p_t(σ') is a normalized known target probability distribution. As demonstrated in <cit.>, the following properties hold: * Ω [V] is a convex functional with a lower bound. * The minimizer, V_min(σ'), of Ω is unique up to a constant and is such that: H'(σ') = - V_min(σ') - log p_t (σ') + constant * The probability distribution of the σ' under the action of V_min is: p_V_min(σ') = e^-(H'(σ') + V_min(σ'))/∑_σ' e^-(H'(σ') + V_min (σ')) =p_t(σ') The above three properties lead to the following MCRG scheme.First, we approximate V(σ') with V_J⃗(σ'), a linear combination of a finite number of terms S_α(σ') with unknown coefficients J_α, forming a vector J⃗ = {J_1, ..., J_α, ..., J_n}.V_J⃗(σ') = ∑_α J_α S_α(σ')Then the functional Ω[V] becomes a convex function of J⃗, due to the linearity of the expansion, and the minimizing vector, J⃗_min, and the corresponding V_min(σ') can be found with a local minimization algorithm using the gradient and the Hessian of Ω: ∂Ω(J⃗)/∂ J_α = - ⟨S_α(σ')|_⟩V_J⃗ + ⟨S_α(σ')|_⟩p_t ∂^2 Ω(J⃗)/∂ J_α∂ J_β = ⟨S_α(σ') S_β(σ')|_⟩V_J⃗ - ⟨S_α(σ')|_⟩V_J⃗⟨S_β(σ')|_⟩V_J⃗Here ⟨·|_⟩V_J⃗ is the biased ensemble average under V_J⃗ and ⟨·|_⟩p_t is the ensemble average under the target probability distribution p_t.The first average is associated to the Boltzmann factor exp{-(H'(σ') + V(σ'))} = ∑_σδ_τ(σ), σ'exp(-H(σ)) exp(-V(τ(σ))) and can be computed with MC sampling.The second average can be computed analytically if p_t is simple enough. ⟨·|_⟩V_J⃗ always has inherent random noise, or even inaccuracy, and some sophistication is required in the optimization problem.Following <cit.>, we adopt the stochastic optimization procedure of <cit.>, and improve the statistics by running independent MC simulations, called multiple walkers, in parallel.For further details, consult <cit.> and theSupplementary Material (SM) <cit.>. The renormalized Hamiltonian H'(σ') is given by Eq. <ref> in terms of V_min(σ'). Taking a constant p_t, we have modulo a constant: H'(σ') = -V_min(σ') = ∑_α (-J_min, α) S_α(σ')In this finite approximation the renormalized Hamiltonian has exactly the same terms of V_min(σ') with renormalized coupling constants K'_α = -J_min, α.The relative importance of an operator S_α in the renormalized Hamiltonian can be estimated variationally in terms of the relative magnitude of the coefficient J_min, α.When J_min, α is much smaller than the other components of J⃗_min, the corresponding S_α(σ') is comparably unimportant and can be ignored. The accuracy of this approximation could be quantified by measuring the deviation of p_V_min(σ') from p_t(σ').To illustrate the method, we present a study of the Ising model on a 2D square lattice in the absence of a magnetic field.We adopt 3 × 3 block spins with the majority rule.26 coupling terms were chosen initially, including 13 two-spin and 13 four-spin products.One preliminary iteration of variational RG (VRG) was performed on a 45× 45 lattice starting from the nearest-neighbor Hamiltonian. The coupling terms with renormalized coupling constants smaller than 0.001 in absolute value were deemed unimportant and dropped from further calculations.13 coupling terms, including 7 two-spin and 6 four-spin products, survived this criterion and were kept in all subsequent calculations <cit.>.Each calculation consisted of 5 VRG iterations starting with nearest-neighbor coupling, K_nn, only.All the subsequent iterations used the same lattice of the initial iteration.Standard Metropolis MC sampling <cit.> was adopted, and the calculations were done at least twice to ensure that statistical noise did not alter the results significantly. In Fig. <ref>, results are shown for a 300 × 300 lattice with two initial K_nn, equal to 0.4355 and to 0.4365, respectively. When K_nn = 0.4365, the renormalized coupling constants increase over the five iterations shown, and would increase more dramatically with further iterations.Similarly, they decrease when K_nn = 0.4355.Thus, the critical coupling K_c should belong to the window 0.4355-0.4365.The same critical window is found for the 45×45, 90× 90, 150× 150, and 210× 210 lattices <cit.>. Because each iteration is affected by truncation and finite size errors, less iterations for the same rescaling factor would reduce the error. For example, 4 VRG iterations with a 2×2 block have the rescaling factor of a 16 × 16 block.The latter is computationally more costly than a calculation with 2× 2 blocks, but can still be performed with modest computational resources. Indeed, with a 16 × 16 block, RG iterations on a 128 × 128 lattice gave a critical window 0.4394-0.4398 <cit.>, very close to the exact value, K_c ∼0.4407, due to Onsager <cit.>.The statistical uncertainty of the renormalized couplings from the variational method is small.Using the standard approach, Ref. <cit.> found a renormalized nearest neighbor coupling equal to 0.408 ± 0.002 after the first RG iteration on a 36× 36 lattice using a 3 × 3 block spin, starting with K_nn = 0.4407.This result required 5.76 × 10^5 MC sweeps.With our method, applied to a 300 × 300 lattice, starting with K_nn = 0.4365, we found a renormalized nearest-neighbor coupling equal to 0.38031 ± 0.00002 after 3.398 × 10^5 MC sweeps. The standard error in our case was computed with the block averaging method <cit.>. Because <cit.> used only seven coupling terms and a different initial K_nn, the renormalized couplings should not be expected to be the same in the two calculations, but a comparison of the corresponding statistical uncertainties should be meaningful. According to theory <cit.>, the critical exponents are obtained from the leading eigenvalues of ∂ K'_α/∂ K_β, the Jacobian matrix of the RG transformation, at a critical fixed point.In order to find ∂ K'_α/∂ K_β near a fixed point, we need to know how the renormalized coupling constants K'_α from a RG iteration on the Hamiltonian H = ∑_β K_β S_β, change when K_β is perturbed to K_β + δ K_β, for fixed target probability p_t and operators S_α. The minimum condition, Eq. <ref>,implies dΩ/d J_α = 0, i.e. for all γ:∑_σ S_γ(σ') e^- ∑_β (K_β S_β(σ) - K'_β S_β(σ'))/∑_σ e^- ∑_β (K_β S_β(σ) - K'_β S_β(σ'))= ⟨S_γ(σ')|_⟩p_t,and∑_σ S_γ(σ') e^- ∑_β ((K_β + δ K_β) S_β(σ) - (K'_β + δ K'_β) S_β(σ'))/∑_σ e^- ∑_β ((K_β + δ K_β) S_β(σ) - (K'_β + δ K'_β)S_β(σ'))= ⟨S_γ(σ')|_⟩p_t.Expanding Eq. <ref> to linear order in δ K'_α and δ K_β, we obtain (<cit.>) A_βγ = ∑_α∂ K'_α/∂ K_β·B_αγ,where A_βγ = ⟨S_β(σ) S_γ(σ')|_⟩V - ⟨S_β(σ)|_⟩V⟨S_γ(σ')|_⟩V,and B_αγ = ⟨S_α(σ') S_γ(σ')|_⟩V - ⟨S_α(σ')|_⟩V⟨S_γ(σ')|_⟩V.Here ⟨·|_⟩V denotes average under the biased Hamiltonian, H = ∑_β K_β S_β(σ) - K'_β S_β(σ').If we require the target average of S_γ(σ') to coincide with the unbiased average under H = ∑_β K_β S_β, K' would necessarily vanish and Eqs. <ref>-<ref> would coincide with Swendsen's formulae <cit.>. If we use a uniform target probability, the σ' at different sites would be uncorrelated, and critical slowing down would be absent.In practice, in order to compute the critical exponents, we first need to locate K_c.From the above calculations on the 45 × 45, 90 × 90, and 300 × 300 lattices with a 3 × 3 block spin, we expect that K_c = 0.436 should approximate the critical nearest-neighbor coupling in our model.Indeed an RG iteration starting from this value gives couplings that remain essentially constant, as illustrated in Figs. S11-S13 of the SM <cit.>. Then, we use Eqs. <ref>-<ref> to compute the Jacobian of the RG transformation by setting K_c = 0.436.The renormalized coupling constants after the first RG iteration represent K_α, and those after the second RG iteration represent K'_α.The results for biased and unbiased ensembles are shown in Table <ref>, which reports the leading even (e) and odd (o) eigenvalues of ∂ K'_α/∂ K_β when including 13 coupling terms for the three L× L lattices with L = 45, 90, and 300. As seen from the table, biased and unbiased calculations give slightly different eigenvalues, as one should expect, given that the respective calculations are different embodiments of the truncated Hamiltonian approximation.For L = 300 the results are well converged in the biased ensemble. By contrast, we were not able to obtain converged results for this lattice in the unbiased ensemble on the time scale of our simulation. The absence of critical slowing down in the biased simulation is demonstrated in Fig. <ref>, which displaystime decay of a correlation function in the biased and unbiased ensembles. See also Figs. S14-S15 of the SM <cit.>. The fixed point used for Table <ref> is approximate, and we did not make any effort to fine tune the approximation.Refinements could be done iteratively using Eqs. <ref>-<ref>, as we will discuss in a future paper.There is an important benefit in knowing accurately the location of the fixed point, because then a single RG iteration, instead of multiple implicit iterations would suffice to compute the Jacobian. Moreover, one could use small block spins, having a smaller statistical uncertainty than larger block spins. In summary, we have unified the calculation of critical exponents and renormalized couplings within the same framework. A key feature of our approach is that we adopt a biased ensemble, ⟨·|_⟩V, for the averages.This not only simplifies the algorithm, but also enhances the sampling.In fact, the original motivation for the variational principle <cit.> was to overcome the long correlation time in first-order phase transitions.The bias potential constructed by optimizing the functional acquires a history-dependence that discourages the sampling of previously visited configurations <cit.>, thereby breaking the long correlation time of the unbiased simulation.In the RG context, enhanced sampling eliminates critical slowing down. We expect that it should be also helpful in systems with deep local free energy minima, as the variational method was originally designed to deal precisely with such systems. The finite size of the numerical samples is a source of error. If the RG iterations are carried out on a single L× L lattice, the coarse grained lattice will have size L/b×L/b.Then, as noted in <cit.>, the calculated renormalized couplings will have different size errors on the L × L and L/b×L/b lattices.A better way, as suggested in <cit.>, would be to perform calculations on two lattices, L × L and L/b×L/b, so that the coarse grained lattice rescaled by b^n, at the nth iteration starting from L × L, would coincide with the lattice rescaled by b^n-1, at the (n-1)th iteration starting from L/b×L/b.In this way, two successive RG iterations have the same lattice size, with a significant cancellation of finite size errors.We plan to discuss in a future paper how this idea could be implemented within VRG.In the present paper we have used a constant probability distribution p_t, but there is no reason to always do so.For example, in systems with continuous and unbounded degrees of freedom, like molecular systems or lattice field theory, it may be convenient to use a Gaussian distribution for p_t. Finally, we note that a regular term g(K) always appears as the inhomogeneous part of a RG transformation <cit.>:exp[H'(K'; σ') + Ng(K)] = ∑_σδ_τ(σ), σ'exp[H(K; σ)]The g(K) in this equation is precisely the thermodynamic free energy per site in the biased ensemble ⟨·|_⟩V, as shown in the SM <cit.>. It is then interesting, and somewhat surprising, that the information on the critical behavior is fully contained in the statistical behavior of ⟨·|_⟩V, even though g(K) is a regular function and ⟨·|_⟩V does not show singular behavior. All the codes used in this project were written in C, and would be available upon request. The authors would like to thank C. Castellani and L. Pietronero for discussions. Partial support for this work was provided by the Department of Energy under Grant no. DE-FG02-05ER46201.apsrev17 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL[Wilson(1971)]wilson_rg authorK. G. Wilson, journalPhys. Rev. 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Moulines, in booktitleAdvances in Neural Information Processing Systems 26, edited by editorC. J. C. Burges, editorL. Bottou, editorM. Welling, editorZ. Ghahramani, and editorK. Q. Weinberger (publisherCurran Associates, Inc., year2013), pp. pages773–781.[sm()]sm noteSupplementary Material.[Metropolis et al.(1953)Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller]metropolis authorN. Metropolis, authorA. W. Rosenbluth, authorM. N. Rosenbluth, authorA. H. Teller, and authorE. Teller, journalThe Journal of Chemical Physics volume21, pages1087 (year1953).[Onsager(1944)]onsager authorL. Onsager, journalPhys. Rev. volume65, pages117 (year1944).[Swendsen(1984b)]mcrg2dising authorR. H. Swendsen, journalPhys. Rev. B volume30, pages3866 (year1984b).[Flyvbjerg and Petersen(1989)]block_method authorH. Flyvbjerg and authorH. G. Petersen, journalThe Journal of Chemical Physics volume91, pages461 (year1989).[Wilson(1975)]wilsonkondo authorK. G. Wilson, journalRev. Mod. Phys. volume47, pages773 (year1975).[Landau and Swendsen(1981)]two_lattice authorD. P. Landau and authorR. H. Swendsen, journalPhys. Rev. Lett. volume46, pages1437 (year1981).[Nauenberg and Nienhuis(1974)]nauenberg authorM. Nauenberg and authorB. Nienhuis, journalPhys. Rev. Lett. volume33, pages1598 (year1974). | http://arxiv.org/abs/1707.08683v1 | {
"authors": [
"Yantao Wu",
"Roberto Car"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170727015953",
"title": "A Variational Approach to Monte Carlo Renormalization Group"
} |
To Be DeterminedA Locally Adapting Technique for Boundary Detection using Image Segmentation Marylesa Howard [email protected] Processing and Applied Mathematics Nevada National Security Site P.O. Box 98521, M/S NLV078, Las Vegas, NV 89193-8521, USAMargaret C. Hock [email protected] Processing and Applied Mathematics Nevada National Security Site P.O. Box 98521, M/S NLV078, Las Vegas, NV 89193-8521, USADepartment of Mathematical Sciences University of Alabama in Huntsville 301 Sparkman Drive, 258A, Huntsville, AL 35899, USAB. T. Meehan [email protected] Signal Processing and Applied Mathematics Nevada National Security Site P.O. Box 98521, M/S NLV078, Las Vegas, NV 89193-8521, USALeora Dresselhaus-Cooper [email protected] Department of Physical Chemistry, Institute for Soldier Nanotechnology Massachusetts Institute of Technology 77 Massachusetts Avenue, NE47-593, Cambridge, MA 02139, USA December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Rapid growth in the field of quantitative digital image analysis is paving the way for researchers to make precise measurements about objects in an image. To compute quantities from an image such as the density of compressed materials or the velocity of a shockwave, object boundaries must first be determined, where images containing regions that each have a spatial trend in intensity are of particular interest here. We present a supervised, statistical image segmentation method that incorporates spatial information to locate boundaries between regions with overlapping intensity histograms. The segmentation of a pixel is determined by comparing its intensity to distributions from nearby pixel intensities. Because of the statistical nature of the algorithm, we use maximum likelihood estimation to quantify uncertainty about each boundary. We demonstrate the success of this algorithm at locating boundaries and providing uncertainty bands on a radiograph of a multicomponent cylinder and on an optical image of a laser-induced shockwave.boundary detection, image processing, statistical segmentation, discriminant analysis, maximum likelihood estimation § INTRODUCTION Quantitative digital image analysis goes beyond qualitatively discriminating between materials or phenomena in a scene and instead allows for precise measurements to be made about objects. Boundary detection is a vital part of extracting information encoded in images, allowing for the computation of quantities of interest including density, velocity, pressure, etc. For example, <cit.> calculates the volume of fruit by identifying fruit boundaries from magnetic resonance images and <cit.> monitors changes in breast tissue location and density to infer breast cancer risk. We present a supervised image segmentation method that is designed to identify boundaries for images with spatially varying intensity, including those with high noise and low contrast. As the purpose of this method is boundary identification, we assume that the user will provide training data for a majority of the image. To assign each pixel, our algorithm considers only locally occurring classes to build a statistical segmentation model. The user defines local with two parameters that reduce the amount of training data considered for the identification of the pixel's class. Our algorithm determines the most probable class for each pixel by comparing its intensity with intensity distributions for each class present in the local training data. To enhance its utility for applications, we combine our method for boundary identification with a statistical analysis to determine uncertainty bands about identified boundaries to provide additional quantitative information describing the objects in the image. In this work, we measure the uncertainty in the segmented boundary using two different spatially resolved metrics. Because of the statistical nature of our algorithm, we employ maximum likelihood estimation to quantify uncertainty about each boundary and analysis of variance to identify the separability of classes at the boundaries. Both analyses provide pixel-by-pixel tests that produce uncertainty (p-value) maps for the segmentation. The boundary uncertainties determined in this way can then be propagated through calculations, giving error bars on quantities. We provide a novel method to objectively obtain statistically justified error bars on image quantities that are measured by segmentation boundary identification. §.§ BackgroundComputational boundary detection methods typically fall into two main categories: edge detection and segmentation. Classical edge detection methods include the Sobel edge detector <cit.> and gradient methods <cit.> and are often most successful when distinguishing between regions that have high contrast <cit.>. Image segmentation determines boundaries between regions by partitioning an image into separate classes or materials <cit.>.Advancements in computing technology have enabled the development of many sophisticated segmentation and edge detection methods, ranging in computational and mathematical complexity, including statistical discriminant analysis models <cit.>, support vector machines that downselect training data <cit.>, fuzzy logic edge detection <cit.>, gradient-based image segmentation <cit.>, and deep convolutional neural networks <cit.>. Many advanced methods are developed to solve specific application problems, yet the value of each method lies within the particular data sets used and the experience of the analyst to extract the best performance from each method <cit.>. In fact, to compute quantities from images, communities withinmedicine and remote sensing manually determine object boundaries due to image qualities such as low contrast, heteroskedasticity, and objects whose intensities vary spatially <cit.>. Manual analysis is time consuming, and makes it difficult to determine mathematically justified errors and uncertainties on manual segmentations.In addition to identifying class and boundary locations, applications benefit from understanding associated uncertainties, with typical measures of error including k-fold crossvalidation, confusion matrices, and Kappa statistics. These statistical measures provide an overall assessment of the analysis, but recent literature highlights the need for methods that spatially identify uncertainty in image segmentation <cit.>. For neural networks, <cit.> produces a confidence map that indicates the number of voting networks that agree on each pixel's predicted label and <cit.> presents a method for estimating a pixel-scale confidence map when using boosting. Work by <cit.> takes an alternative approach to developing uncertainty maps by using the spectral domain rather than the spatial domain. <cit.> presents an information-based criterion for computing a thematic uncertainty measure that describes the overall spatial variation of the segmentation accuracy.The paper is organized as follows. Our locally-adaptive image segmentation algorithm is detailed in Section <ref>, with corresponding uncertainty quantification for boundary identification in Section <ref>. Numerical results for two real world images are presented in Section <ref>, and the conclusions follow in Section <ref>. § LOCALLY ADAPTIVE DISCRIMINANT ANALYSISGlobal image segmentation methods do not account for spatial variation in intensity within a class. The mean and variance of a class may vary spatially within an image. By restricting our knowledge to local information about a pixel, we will have a better sense of the pixel's true class, without being misled by information on the other side of the image. For example, Figure <ref>(a) is a radiograph from the Nevada National Security Site of a radially symmetric cylinder turned on its side, with a graphic representation given in Figure <ref>(b). The cylinder is hollow in its center (top of image (a)) with varying bore widths, and the adjacent concentric layers are copper, aluminum, and teflon, followed by air outside the cylinder (bottommost region in image (a)). The varying thicknesses of the materials affect the intensity on the radiograph as it is a function of areal density, and an objective in the quantitative analysis of this image is to identify boundaries between materials in order to compute material density <cit.>. The purple, blue, teal, and green rectangles in image (a) all represent the air class, and the intensity of the pixels contained in those colored rectangles are shown in the respectively colored histograms given in Figure <ref>(c). While each of those regions of pixels come from the same class, they are completely separated in feature space, indicating a spatial relationship between intensity and class. Furthermore, the yellow rectangle in image (a) corresponds to a region of copper, and its corresponding intensity values are plotted in image (c). The copper intensities completely overlap with two of the air intensity histograms, suggesting that a global approach to image segmentation may lead to issues with non-separable data. We present our method, locally adaptive discriminant analysis (LADA), which restricts the training data for each pixel of interest to the locally-occurring classes based on two user-selected parameters and builds a trainer based on the Gaussian assumption of discriminant analysis.LADA is, therefore, a local implementation of discriminant analysis to adequately separate classes with spatially-varying intensities and is appropriate for images with shadows, with heterogeneous illumination, or of areal density, including those with ow contrast and/or high noise. In addition to the user-defined classes, LADA allows for a bonus class to be selected for a pixel when there are not enough, or no, local training data to represent a class. The algorithm is described here, with pseudocode in Algorithm <ref>.Given an image X, a pixel x_ij, and the set of training data T⊂ X, we are interested in determining the most likely class to which x_ij belongs, for each x_ij∈ X. The classes, ω_c, are defined by the training data, with c=1,…,C, for a total of C known classes occurring in X. Let T_ω_c⊂ T be the set of all training pixels for class ω_c.Rather than considering the entire set T to build a trainer for x_ij, we reduce the training data via two user-selected parameters: d and n. Given the distance parameter d, we define the subimage S_ij⊂ X about pixel x_ij to be S_ij ={x_kl | √((i-k)^2+(j-l)^2)≤ d },such that S_ij is the set of all pixels within radius d to pixel x_ij. All known regions (training data) outside S_ij are temporarily ignored and will have no effect on the segmentation of x_ij. Further reducing the trainer's view of local, for each class ω_c, we define the local training data, T_ω_cij⊂ S_ij, to be the set of, at most, n nearest (Euclidean distance) training pixels to x_ij, where ties are broken via lexicographical ordering. Together, d and n serve as our definition of local for LADA.A simplified visual example of the restriction of the local training data for parameters d=3 and n=4 is given in Figure <ref>. The two training data classes that make up T are shown in pink (class 1) and blue (class 2) colored pixels, with the center pixel identified as x_ij. All pixels in subimage S_ij (within radius d=3) are shaded, with elements of T_ω_1ij being the pink, shaded pixels with demarcation of n_4 and elements of T_ω_2ij being the shaded, blue pixels with demarcation of n_4. Notice this selection of four nearest training points for the blue class is not unique but obeys the lexicographical ordering of left to right, top to bottom. Each class ω_c with |T_ω_cij|≥3 is considered a potential class for x_ij such that we can compute statistics on the locally occurring training data. A set with |T_ω_cij|<3 is possible even with n≥ 3 if not enough training pixels of class c occur within S_ij, and, in such a case, that class is not considered within the computations for local. We assume the local training data for class ω_c are drawn from a Gaussian distribution, with mean μ_ω_cij and standard deviation σ_ω_cij. By restricting to local training data, a Gaussian distribution assumption is often more reasonable than on a global scale where the full training data set often violates such an assumption in images with spatially-varying intensities. This will be demonstrated in Section <ref>.The class ω_c to which x_ij most likely belongs, given these distributional assumptions, is defined to beG(x_ij) =ω_c{p(x_ij|μ_ω_cij,σ_ω_cij) > p(x_ij|μ_ω_bij,σ_ω_bij)∀ ω_b≠ω_c},where, without loss of generality,x_ij∼𝒩(μ_ω_cij,σ_ω_cij),and G(·) is the function mapping a pixel into its segmented class. It is possible that, via choice of d, there are too few training points within subimage S_ij to reliably compute a standard deviation (i.e., |T_ω_cij|≤2,∀c). In such a case, we place x_ij into the bonus class, G(x_ij) = C+1,indicating there was not enough local information to identify to which of the C classes it belongs. In general, if a significant portion of the image is being placed into the bonus class, the analyst might consider choosing more training data if more are known or choosing a larger distance parameter d.Furthermore, it should be noted that as d approaches the bounds of the diagonal distance of the image and n is increased to the magnitude of the training data sets for each class, the effect of looking at local training data diminishes and the focus becomes global. In such a case, this algorithm simply becomes quadratic discriminant analysis <cit.>. The algorithm may be restricted further to perform similarly to linear discriminant analysis if σ_ω_cij = σ_ij for all considered classes ω_c.For images with strong spatial variation even in local regions, the values of d and n should be chosen with smaller magnitudes if significant training data can be provided to avoid abundant bonus class assignment. There can be a tradeoff between a priori knowledge and the selection of parameters d and n. Since LADA is an edge detection method, we assume the majority of the image is known, sans the boundaries.§ UNCERTAINTY ON BOUNDARY DETECTIONWe build two uncertainty maps corresponding to the segmentation that describe, first, our confidence in the selected segmentation based on local information, and second, our ability to discriminate between classes at boundaries. For the former, we apply maximum likelihood estimation, and, for the latter, we apply a standard analysis of variance (ANOVA) test.§.§ Maximum likelihood estimation p-valueGiven a LADA segmentation for a pixel, we wish to quantify the probability that the pixel belongs to that class. Consider the example in Figure <ref>(a): given the Gaussian distributions for two, hypothetical local classes and the intensity observation marked by the black star, the pixel would be segmented into class 1, given by the blue line, since it has greater probability density at that observation. However, because the observation is on the tail of the class 1 distribution, it is not well represented by that class either.The estimation method by which LADA determines the segmented class, c, is considered a maximum likelihood estimator (MLE). In terms of segmentation, MLE is a technique for determining the class that maximizes the probability distribution for an observed pixel intensity <cit.>. For a grayscale image, we assume the local population mean μ_ω_cij and variance σ_ω_cij are known for the class ω_c to which pixel x_ij is segmented, obtained via the local training data. The class was selected using the optimizationω_c = ω_cmax_(μ_ω_cij,σ_ω_cij) f(x|μ_ω_cij,σ_ω_cij),where x ∼𝒩(μ_ω_cij,σ_ω_cij).From this, we compute the p-value, the probability of observing the pixel x_ij or something more extreme, given the segmented class ω_c and its associated parameters:p = {[ P(X ≥ x_ij|μ_ω_cij,σ_ω_cij)ifx_ij>μ_ω_cij; P(X ≤ x_ij|μ_ω_cij,σ_ω_cij) otherwise ]}.Assuming pixel x_ij had equal probability of having observed either positive or negative noise, we multiply the p-value by two, for a two-sided p-value, which is demonstrated visually in Figure <ref>(b) for a distribution with mean μ_ω_cij = 0.5 and σ_ω_cij = 0.38, and observation x_ij = 0.8. For an image segmented by LADA, we can produce a corresponding image of p-values from the statistical inference given here. This concept can be extended to multibanded images (non-grayscale) using a multivariate normal distribution on the colored pixel x_ij. §.§ Analysis of variance p-valueAnalysis of variance (ANOVA) provides a hypothesis test as to whether local classes are equal in mean intensity or not by analyzing the variance of the local classes <cit.>. We model the random variable pixel intensity asX_ij = μ_ω_cij + ϵ_ω_cij,where X_ij may be of any length. Thus each pixel from a given class has a known mean, μ_ω_cij, and some noise, ϵ_ω_cij, which makes up the observed intensity. Assuming the ϵ_ω_cij are iid Gaussian with zero mean, equal variances, and zero covariances, the null hypothesis states that all the class means are equal and the alternative states that at least one mean is different. Rather, we are interested in knowing if any two means of local classes are evidenced to be equal, so we perform multiple ANOVA's to compare only two classes at a time. In our case, the hypotheses of interest are H_0:μ_ω_cij = μ_ω_bij,H_a:μ_ω_cij≠μ_ω_bij,for all classes ω_c and ω_b local to pixel x_ij.The resulting p-value of the ANOVA test quantifies evidence for or against the null hypothesis. A large ANOVA p-value indicates that we have evidence in favor of the null hypothesis which states that the two classes have the same mean and are undifferentiable with such a measure. We compute all ANOVA paired p-values and consider the largest value, which provides the most evidence that there are at least two classes with equal means, which can be visualized with an image corresponding to each pixel of the original image. Thus, the ANOVA image is used to indicate “problem areas” within an image that has data which are especially difficult to discriminate between classes, or may be improved upon by editing the training data. In the case of multibanded images, multivariate analysis of variance (MANOVA) is used. § NUMERICAL RESULTS In this section, we include results on grayscale images from two physics applications. In demonstrating the utility of LADA, we highlight issues seen in images typical to different fields of science that pose problems for conventional segmentation methods. The first image is a radiograph of a static, multi-layered cylindrical object. The second image is of a laser-induced converging shockwave, collected using optical imaging <cit.>. §.§ Static data: multi-material cylinder A radiograph from the Cygnus X-ray machine at the U1a underground facility at the Nevada National Security Site is shown in Figure <ref>(a), and was presented in Section <ref>. It is half of a radially symmetric cylinder with the axis of rotation at the top of the image. The cylinder is hollow, with varying bore widths, and has concentric cylinders of copper, aluminum, and teflon. The final material (darkest horizontal layer in the image) is outside the cylinder, i.e. air. A prime objective in the quantitative analysis of this image is to identify boundaries between materials for subsequent calculations of material density.Boundary identification for the concentric cylinders is made difficult by the intensity gradient across each class corresponding to a change in areal density. The training data are provided in Figure <ref>(c), superimposed on the radiograph, and comprise 88% of the image. A large amount of training data is provided to accommodate the locally-adaptive nature of LADA, which requires training data throughout the image. It is simple to provide a large quantity of training data since the only aspect of the image that is unknown is the boundaries. Classical segmentation methods have difficulty correctly segmenting the class representing air, which has a very wide, multimodal distribution as it appears at the top (light to medium gray) and bottom (black) regions of the image, as shown in the training data histograms in Figure <ref>(d). The histograms demonstrate overlap between class probability densities, which is the driving reason why classical threshold methods like quadratic discriminant analysis (QDA) do not work well on data such as these with spatial gradients in classes. In addition, the Gaussian assumption of QDA is not appropriate for the training data histograms. The LADA segmentation is provided in Figure <ref>(a), based on the training data in Figure <ref>(c), with parameters d = 25 and n=25. The segmentation captures the large-scale features of three materials, surrounded on either side by air, and has very few “noisy” segmented pixels, i.e. a pixel whose class is unlike its immediate neighbors. The implied boundaries between classes are likely smoother in actuality, with the roughness likely due to image quality or parameter selection. Reducing the view of local may help alleviate segmenting errors due to noise.Figure <ref>(b) visualizes the MLE p-values that describe how probable the chosen class is, given the observed pixel intensity. The bright yellow regions indicate pixels that are not well represented by the local training data in the available classes. The thickness of the yellow lines is influenced by the quality of the image and how close to the boundaries the user provides training data. In this image, having a pixel near a boundary between classes (with the closest training data being many pixels away) can lead to that pixel not looking similar to either training subset. For example, consider a pixel near the boundary of air and teflon at (49,551) in the lower left-hand corner of Figure <ref>(a) and its corresponding local training data presented in Figure <ref>(d) [We note the local training data shown here pass the Shapiro-Wilk test for normalcy]. In this case, the closest training data are at least ten pixels away for either class, and the observed pixel is not well represented by either class due to the strong intensity gradient in this region. The yellow regions of the MLE p-value map indicate all of these such circumstances. In providing training data, the user implies a level of uncertainty about the boundary location through the width of the void between classes. The regions of uncertainty about the material boundaries provided by LADA (p-value < 0.05 in Figure <ref>(b)) are thinner than the corresponding regions in Figure <ref>(c). For example, the horizontal region of uncertainty centered about the y-axis value of 365 has a width of roughly ten pixels, whereas the corresponding void in training data of Figure <ref>(c) has a width of roughly eighteen pixels. LADA has reduced the a priori user uncertainty by nearly half.The ANOVA p-value map is given in Figure <ref>(c). The white regions correspond to areas in which only one local class is present within S_ij and are thus far away enough from the boundary such that no other class is considered local. The dark blue regions (low p-value) indicate that the local training data of the classes are well separable. Lighter regions, mostly absent in this example, indicate that the local training data for at least two of the classes are not well separable. Thus, our training data demonstrate that the classes at the boundaries are well separable with our definition of local parameters. For comparison, the classical segmentation via QDA is provided in Figure <ref>(e). It is visibly clear that the global assumptions of QDA fail to provide physically correct boundaries between classes, and the method misidentifies large regions of the image, completely missing two of the hollow bores at the top center and right regions in the image. In addition, the QDA segmentation incorrectly suggests a higher pitched angle of the cylinders from horizontal than LADA, as if the image has been rotated slightly, and an inclusion of air (purple) between each of the layers.From the segmentation given in Figure <ref>(a), the boundaries between classes are obtained by taking the gradient of the segmentation and fitting lines to the horizontal boundaries and logistic curves to the vertical boundaries, which are reasonable assumptions as we know the geometry of the calibration object. The final, fitted boundaries are given by the black lines in Figure <ref>. To compute regions of uncertainty about the boundaries, we choose a significance level of α=0.05 on the p-value map from Figure <ref>(b). All pixels with a p-value <α that are nearby a detected boundary are highlighted in purple in Figure <ref>, indicating a region of uncertainty. Note that the regions of uncertainty are not required to be centered about each boundary.§.§ Dynamic data: cylindrically converging shock waves When shock waves travel through a material, the material is irreversibly changed. Temporal quantitative analysis of wave dynamics requires all measurements, (in this case, images) to be collected from a single shock experiment. As a result, the signal-to-noise ratio is limited and the images contain high noise content, especially for high spatiotemporal resolution <cit.>.When strong shock waves travel through a material, they can separate into multi-wave structures, beginning with a one-dimensional elastic wave where the material is reversibly compressed and followed by a series of plastic waves corresponding to the irreversible changes that occur upon fast dynamic compression. In an image, the multi-wave structure can appear as a relatively uniform material with bright, narrow attributes that separate the regions. In some cases, there are changes in pixel intensity or in texture between the different regions of the shock, but in most cases these classes are difficult to separate with current boundary detection techniques because the various physics classes (e.g. shock wave) have very little to distinguish them from the other classes (e.g. are overlapping in feature space). Figure <ref>(a) is an image of a cylindrically converging shock wave traveling through a thin layer of water that is between thick glass substrates <cit.>. The shock wave is generated from the interaction of a 200 micrometer diameter laser ring with an absorber to produce a shock wave that travels within the sample plane, perpendicular to the incident laser. A 180 femtosecond duration pulse from the same laser collects a shadowgraph image in transmission, which gives an image that is a spatial map of the second derivative of the density for the material <cit.>. In this experiment, six images were taken in a single experiment at 5 nanosecond intervals in order to visualize the convergence and subsequent divergence of the shock, which depicts the two-dimensional physics of the complex system. To obtain quantifiable data from these images, the shock must be precisely and accurately located with clear understanding of any error from that measurement. Identifying the location of the shock wave throughout a series of images will enable the researcher to compute quantities such as velocity.Figure <ref>(b) displays the training data selected for the shock wave image, with the classes beginning at the center of the image and moving out: unshocked water, shock front, shocked water, laser ring, outside region. 85% of the image is represented in the training data, and Figure <ref>(c) displays a histogram of the global training data, of which all classes have significant overlap. Traditional methods fail to provide adequate identification of boundaries on this image, but the underlying cause of this failure is exactly the motivation for why a spatial segmentation technique that adapts to local intensity variations is appropriate for this image.The LADA segmentation is given in Figure <ref>(a) for d=25 and n=20, with a smaller n than in the previous example to provide a more narrowed local view of the data. With large gaps in the training data and a small value of d, some pixels lack local training data and are segmented into the bonus class, located near the outer boundary of the laser ring at the bottom of the image. A classical method such as QDA, for comparison, produces a nonsensical segmentation, based on the physics, and is not displayed here. MLE p-values are given in Figure <ref>(b) and indicate regions of uncertainty in the segmentation (bright yellow regions). Note that p-values are not computed for pixels placed in the bonus class. The ANOVA p-values are given in Figure <ref>(c) and indicate regions in which the local classes were difficult to discriminate between (large p-value). Notice the less disparate regions in the ANOVA p-values are radial shapes, not unlike the texture in the image.With the segmentation in Figure <ref>(a), the boundaries between classes are obtained by taking the gradient. Given circular assumptions on the laser ring and shock fronts, we fit circles to each boundary. To compute bands of uncertainty on the class boundaries, we take a Neyman-Pearson approach and choose a significance level of α = 0.05 on the p-value map in Figure <ref>(c). From the p-values < α, we determine the smallest and largest radius of uncertainty about each boundary and use that as a uniform uncertainty band about each boundary. Both the fitted circles and the corresponding uncertainty bands are shown in Figure <ref>, superimposed upon the original image. Note that there are no requirements that the uncertainty band be symmetric about its fitted circular boundary.§ CONCLUSIONS This work presents a locally-adaptive segmentation method for boundary identification, designed for images that contain spatial trends in intensity, low contrast, or heteroskedasticity. Building on the classical image segmentation technique of discriminant analysis, one of our novel contributions is an algorithm that determines the class for each pixel by comparing probability distributions constructed from local training data. In recognizing spatial dependence of pixels, we shift from the discriminant analysis paradigm of viewing training data globally and focus on local training data that are restricted by two user-specified parameters. We assume the local training data are Gaussian, which we demonstrate is more reasonable than a global Gaussian assumption on a radiograph produced at the Nevada National Security Site. Understanding uncertainty in the segmentation and region boundaries is provided through two statistical tests, the application of which is our second novel contribution. Given a pixel's segmentation and the statistics estimated from local training data, we use maximum likelihood estimation (MLE) to compute the probability of observing the pixel's intensity or something more extreme. For small p-values, this suggests that either the segmentation may not be the true class, even if it was most probable of the locally available classes, or that the local training data are all too far from the pixel to be truly representative. We apply MLE, paired with physics geometric assumptions, to create uncertainty bands for the segmentation boundaries in the radiograph and in an image of a converging shock wave. Finally, an analysis of variance (ANOVA) describes the ability to discriminate between local classes based on the training data, with small p-values indicating well-separable classes. The ANOVA p-value map ultimately describes class contrasts and associated variances and may be used to guide adjusting the local parameters or training data to provide better separability between classes. The authors are grateful to Dongsheng Wu at the University of Alabama in Huntsville and Aaron Luttman at the Nevada National Security Site for their advice and support. The authors also acknowledge Keith A. Nelson at MIT for his support in the shock work. This manuscript has been authored in part by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the U.S. Department of Energy, National Nuclear Security Administration, NA-10 Office of Defense Programs, and supported by the Site-Directed Research and Development Program. 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 content of this manuscript, or allow others to do so, for United States Government purposes. The U.S. 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). DOE/NV/25946--3282. The authors would like to acknowledge the Office of Naval Research for funding that supported one of the authors from this work on grant numbers N00014-16-1-2090 and N00014-15-1-2694.0.2in | http://arxiv.org/abs/1707.09030v1 | {
"authors": [
"Marylesa Howard",
"Margaret C. Hock",
"B. T. Meehan",
"Leora Dresselhaus-Cooper"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170727200652",
"title": "A Locally Adapting Technique for Boundary Detection using Image Segmentation"
} |
Benkart]Georgia Benkart [G. Benkart]Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr. Madison, WI 53706, U.S.A. [email protected]]Laura Colmenarejo [L. Colmenarejo]Department of Mathematics and Statistics,York University, 4700 Keele Street, Toronto,Ontario M3J 1P3, Canada [email protected] http://www.yorku.ca/ lcolme/Harris]Pamela E. Harris [P. E. Harris]Mathematics and Statistics, Williams College, Bascom House, Rm 106C Bascom House, 33 Stetson Court, Williamstown, MA 01267, [email protected] https://math.williams.edu/profile/peh2/Orellana]Rosa Orellana [R. Orellana]Mathematics Department, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, U.S.A. [email protected] https://math.dartmouth.edu/ orellana/Panova]Greta Panova [G. Panova]Mathematics Department, University of Pennsylvania,209 South 33rd St, Philadelphia, PA 19104, U.S.A [email protected] https://www.math.upenn.edu/ panova/Schilling]Anne Schilling [A. Schilling]Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A. [email protected] http://www.math.ucdavis.edu/anneYip]Martha Yip [M. Yip]Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506-0027, U.S.A. [email protected] http://www.ms.uky.edu/ myip/[2000]Primary 05A19; Secondary 05A18, 05E05, 05E10, 20G42, 17B37We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture.This conjecture was stated by Haglund, Remmel and Wilson as a generalizationof the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connectedcomponents of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobeniusseries of the generalization R_n,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra R_n. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major indexstatistic on ordered multiset partitions. A minimaj-preserving crystal on ordered multiset partitions [ December 30, 2023 =========================================================== § INTRODUCTIONThe Shuffle Conjecture <cit.>, now a theorem due to Carlsson and Mellit <cit.>, provides an explicit combinatorial description of the bigraded Frobenius characteristic of the S_n-module ofdiagonal harmonic polynomials. It is stated in terms of parking functions and involves two statistics, 𝖺𝗋𝖾𝖺and 𝖽𝗂𝗇𝗏.Recently, Haglund, Remmel and Wilson <cit.> introduced a generalization of the Shuffle Theorem, coined the Delta Conjecture. The Delta Conjecture involves two quasisymmetric functions 𝖱𝗂𝗌𝖾_n,k(𝐱;q,t) and 𝖵𝖺𝗅_n,k(𝐱;q,t), which havecombinatorial expressions in terms of labelled Dyck paths. In this paper, we are only concerned with the specializationsq=0 or t=0, in which case <cit.> and <cit.> show𝖱𝗂𝗌𝖾_n,k(𝐱;0,t) = 𝖱𝗂𝗌𝖾_n,k(𝐱;t,0) = 𝖵𝖺𝗅_n,k(𝐱;0,t) = 𝖵𝖺𝗅_n,k(𝐱;t,0).It was proven in <cit.> that𝖵𝖺𝗅_n,k(𝐱;0,t) = ∑_π∈𝒪𝒫_n,k+1t^(π)𝐱^(π),where 𝒪𝒫_n,k+1 is the set of ordered multiset partitions of the multiset {1^ν_1,2^ν_2,…}into k+1 nonempty blocks and ν=(ν_1,ν_2,…) ranges over all weak compositions of n. The weak composition ν is also called the weight of π, denoted (π)=ν.In addition, (π) is the minimum value of the major index of the set partition π over all possible ways toorder the elements in each block of π. The symmetric function 𝖵𝖺𝗅_n,k(𝐱;0,t) isknown <cit.> to be Schur positive, meaning that the coefficients are polynomials in twith nonnegative coefficients.In this paper, we provide a crystal structure on the set of ordered multiset partitions 𝒪𝒫_n,k. Crystal bases are q→ 0 shadows of representations for quantum groups U_q(𝔤) <cit.>, though they can also be understood from a purely combinatorialperspective <cit.>. In type A, the character of a connected crystal component with highest weight element of highest weight λ is the Schur function 𝗌_λ. Hence, having a type A crystal structure on a combinatorial set (in our case on 𝒪𝒫_n,k) naturally yields the Schur expansion of the associated symmetric function. Furthermore, if the statistic (in our case ) is constant on connected components, then the graded character can also be naturally computed using the crystal.Haglund, Rhoades and Shimozono <cit.> introduced a generalization R_n,k for k⩽ nof the coinvariant algebra R_n, with R_n,n=R_n. Just as the combinatorics of R_n is governed by permutationsin S_n, the combinatorics of R_n,k is controlled by ordered set partitions of {1,2…,n} with k blocks. The graded Frobenius series of R_n,k is (up to a minor twist) equal to 𝖵𝖺𝗅_n,k(𝐱;0,t). It is still an open problem to find a bigraded S_n-module whose Frobenius image is 𝖵𝖺𝗅_n,k(𝐱;q,t). Our crystal provides another representation-theoretic interpretation of 𝖵𝖺𝗅_n,k(𝐱;0,t) as a crystal character.Wilson <cit.> analyzed various statistics on ordered multiset partitions, including 𝗂𝗇𝗏, 𝖽𝗂𝗇𝗏, , and . In particular, he gave a Carlitz type bijection, which proves equidistributivity of 𝗂𝗇𝗏, 𝖽𝗂𝗇𝗏,on 𝒪𝒫_n,k. Rhoades <cit.> provided a non-bijective proof that these statistics are also equidistributed with . Using our new crystal, we can give a bijective proof of the equidistributivity of the statistic and thestatistic on ordered multiset partitions.The paper is organized as follows. In Section <ref> we define ordered multiset partitions andtheandstatistics on them. In Section <ref> we provide a bijectionφ from ordered multiset partitions to tuples of semistandard Young tableaux that will be used inSection <ref> to define a crystal structure, which preserves . We conclude inSection <ref> with a proof that theandstatistics are equidistributed usingthe same bijection φ.§.§ AcknowledgmentsOur work on this group project began at the workshop Algebraic Combinatorixx 2 atthe Banff InternationalResearch Station (BIRS) in May 2017.“Team Schilling,”as our group of authors is known, would like to extend thanksto the organizers of ACxx2,to BIRS for hosting this workshop, and to the Mathematical Sciences Research Institute (MSRI)for sponsoring a follow-up meeting of some of the group members at MSRI in July 2017 supported by the National Science Foundation under Grant No. DMS-1440140. We would like to thank Meesue Yoo for early collaboration and Jim Haglund, Brendon Rhoades and Andrew Wilson forfruitful discussions. This work benefited from computations and experimentations in Sage <cit.>.P. E. Harris was partially supported by NSF grant DMS–1620202.R. Orellana was partially supported by NSF grant DMS–1700058.G. Panova was partially supported by NSF grant DMS–1500834. A. Schilling was partially supported by NSF grantDMS–1500050.M. Yip was partially supported by Simons Collaboration grant 429920. § ORDERED MULTISET PARTITIONS AND THE MINIMAJ AND MAJ STATISTICS We consider ordered multiset partitions of order n with k blocks. Given a weak composition ν = (ν_1, ν_2, …) of n into nonnegative integer parts, which we denote ν n,let 𝒪𝒫_ν,k be the set of partitions of the multiset{i^ν_i| i ⩾ 1}into k nonempty ordered blocks, such that the elements within each block are distinct. For each i⩾ 1, the notation i^ν_i should be interpreted as saying that the integer i occurs ν_i timesin such a partition.The weak composition ν is also called the weight (π) ofπ∈𝒪𝒫_ν,k. Let𝒪𝒫_n,k=⋃_ν n𝒪𝒫_ν,k.It should be noted that in the literature 𝒪𝒫_n,k is sometimes used for ordered set partitions rather than ordered multiset partitions (that is, without letter multiplicities).We now specify a particular reading order for an ordered multiset partition π = (π_1|π_2 |…|π_k) ∈𝒪𝒫_n,k with blocks π_i. Start by writing π_k in increasing order. Assume π_i+1 has been ordered,andlet r_i be the largest integer in π_i that is less than or equal to the leftmost element of π_i+1. If no such r_iexists, arrange π_i in increasing order.When such an r_i exists, arrange the elements of π_i in increasing order,and then cycle them so that r_i is the rightmost number. Continue with π_i-1, …, π_2, π_1 until all blocks havebeen ordered. This ordering of the numbers in π is defined in <cit.> and is called the minimaj order.lf π = (157 | 24 | 56 | 468 | 13 | 123) ∈𝒪𝒫_15,6,then the minimaj order of π is π = (571 | 24 | 56 | 468 | 31 | 123). For two sequences α,β of integers,we write α < β to mean that each element of α is less than every element of β.Suppose π∈𝒪𝒫_n,k is in minimaj order. Then each block π_i of πis nonempty andcan be written in the form π_i = b_i α_i β_i, where b_i ∈_>0, and α_i,β_i are sequences(possibly empty) of distinct increasing integers such that either β_i < b_i< α_i or α_i=∅.Inequalities with empty sets should be ignored. With the above notation, π∈𝒪𝒫_n,k is in minimaj order if the following hold: (1) π_k = b_kα_k with b_k<α_k and β_k = ∅;(2)for 1 ⩽ i < k,either (a)α_i = ∅,π_i = b_i β_i,and b_i < β_i ⩽ b_i+1,or (b)β_i ⩽ b_i+1 < b_i < α_i.A sequence or word w_1 w_2 ⋯ w_n has a descent in position 1⩽ i<n if w_i>w_i+1. Let π∈𝒪𝒫_n,k be in minimaj order. Observe that a descent occurs in π_i only in Case 2 (b) of Lemma <ref>, and such a descent is either between the largest and smallest elements of π_i or between the last element of π_i and the first element of π_i+1. Continuing Example <ref> with π = (571 | 24 | 56 | 468 | 31 | 123), we have[b_1 = 5, α_1 =7, β_1 = 1b_2 = 2, α_2 = ∅, β_2 =4b_3 = 5, α_3 = 6, β_3 =∅;b_4 = 4,α_4 = 68, β_4 =∅b_5 = 3,α_5 = ∅, β_5 = 1 b_6 = 1,α_6 = 23,β_6 = ∅. ] Suppose that π in minimaj order has descents in positions(π) = {d_1, d_1+d_2, …, d_1+d_2 + ⋯ + d_ℓ}for some ℓ∈ [0,k-1](ℓ= 0 indicates no descents). Furthermore assume that these descents occur in the blocks π_i_1, π_i_1+i_2, …,π_i_1+i_2+⋯ + i_ℓ, wherei_j >0 for 1⩽ j ⩽ℓ andi_1+i_2+⋯+i_ℓ <k.Assume d_ℓ+1 and i_ℓ+1 are the distances to the end, that is, d_1+d_2 + ⋯+d_ℓ + d_ℓ+1 = n and i_1+i_2+⋯+ i_ℓ + i_ℓ+1 = k.The minimaj statistic (π) of π∈𝒪𝒫_n,k as given by <cit.> is(π) = ∑_d ∈(π) d=∑_j=1^ℓ (ℓ+1-j) d_j.The descents for the multiset partitionπ = (57.1 | 24 | 56. | 468. | 3.1 | 123)occur at positions(π)={2,7,10,11} and are designated with periods. Hence ℓ=4, d_1 = 2, d_2 = 5, d_3 = 3, d_4 = 1and d_5 = 4, and(π) = 2 + 7 + 10 + 11 = 30.The descents occur in blocks π_1, π_3, π_4, and π_5,so that i_1 = 1, i_2 = 2, i_3 = 1, i_4 = 1, and i_5 = 1. To define the major index of π∈𝒪𝒫_n,k, we consider the word w obtained by ordering eachblock π_i in decreasing order, called the major index order <cit.>. Recursively construct a word v by setting v_0=0 and v_j = v_j-1+ χ(jis the last position in its block) for each 1⩽ j ⩽ n. Here χ(True)=1 and χ(False)=0. Then(π) = ∑_jw_j>w_j+1 v_j.Continuing Example <ref>, note that the major index order of π = (157 | 24 | 56 | 468 | 13| 123) ∈𝒪𝒫_15,6 is π = (751 | 42 | 65 | 864 | 31 | 321). Writing the word vunderneath w (omitting v_0=0), we obtainw= 751 | 42 | 65 | 864| 31 | 321 v= 001 | 12 | 23 | 334 | 45 | 556,so that (π) = 0+0+1+2+3+3+4+4+5+5=27. Note that throughout this section, we could have also restricted ourselves to ordered multiset partitions withletters in {1,2,…, r} instead of _>0. That is, let ν=(ν_1,…,ν_r) be a weak composition of n and let 𝒪𝒫^(r)_ν,k be the set of partitions of the multiset {i^ν_i| 1⩽ i ⩽ r} into k nonempty ordered blocks, such that the elements within each block are distinct.Let𝒪𝒫^(r)_n,k=⋃_ν n𝒪𝒫^(r)_ν,k.This restriction will be important when we discuss the crystal structure on ordered multiset partitions. § BIJECTION WITH TUPLES OF SEMISTANDARD YOUNG TABLEAUX In this section, we describe a bijection from ordered multiset partitions to tuples of semistandard Young tableaux that allows us to impose a crystal structure on the set of ordered multiset partitions in Section <ref>.Recall that a semistandard Young tableau T is a filling of a (skew) Young diagram (also called the shape of T) with positive integers that weakly increase across rows and strictly increase down columns. The weight of T isthe tuple (T)=(a_1,a_2,…), where a_i records the number of letters i in T. The set of semistandard Youngtableaux of shape λ, where λ is a (skew) partition, is denoted by 𝖲𝖲𝖸𝖳(λ). If we want torestrict the entries in the semistandard Young tableau from _>0 to a finite alphabet {1,2,…,r}, we denote the set by 𝖲𝖲𝖸𝖳^(r)(λ).The tableaux relevant for us here are of two types: a single column of boxes with entries that increase from top to bottom, or a skew ribbon tableau.If γ =(γ_1,γ_2, …, γ_m) is a skew ribbon shape withγ_j boxes in the j-th row starting from the bottom, the ribbon condition requires that row j+1 starts in thelast column of row j. This condition is equivalent to saying that γ is connected and contains no 2 × 2block of squares.For example boxsize=1.1em,,corresponds to γ = (2,1,3). Let 𝖲𝖲𝖸𝖳(1^c) be the set of semistandard Young tableaux obtained by filling a column oflength c and 𝖲𝖲𝖸𝖳(γ) be the set of semistandard Young tableaux obtained by filling the skew ribbon shape γ.To state our bijection, we need the following notation. For fixed positive integers n and k, assumeD= {d_1,d_1+d_2,…,d_1+d_2+⋯+d_ℓ}⊆{1,2,…,n-1} andI = {i_1,i_1+i_2,…,i_1+i_2+⋯+i_ℓ}⊆{1,2,…, k-1} are sets of ℓ distinct elementseach. Define d_ℓ+1 := n-(d_1+⋯ +d_ℓ) and i_ℓ+1 := k - (i_1+⋯ +i_ℓ).For fixed positive integers n and k and sets D and I as above, letM(D,I) = {π∈𝒪𝒫_n,k|(π) = D,and the descents occur in π_i for i ∈I}.Then the following map is a weight-preserving bijection:φM(D,I) →𝖲𝖲𝖸𝖳(1^c_1) ×⋯×𝖲𝖲𝖸𝖳(1^c_ℓ) ×𝖲𝖲𝖸𝖳(γ) π ↦ T_1 ×⋯× T_ℓ× T_ℓ+1where (i)γ = (1^d_1-i_1, i_1, i_2, …, i_ℓ+1) and c_j = d_ℓ+2-j - i_ℓ+2-j for1⩽ j ⩽ℓ.(ii)The skew ribbon tableau T_ℓ+1 of shape γ is constructed as follows: ∙ The entries in the first column of the skew ribbon tableau T_ℓ+1 beneath the first box are the firstd_1-i_1 elements of π in increasing order from top to bottom, excluding any b_j in that range.∙ The remaining rows d_1-i_1+j of T_ℓ+1 for 1⩽ j ⩽ℓ+1 are filled withb_i_1+… + i_j-1 + 1, b_i_1+… + i_j-1 + 2, …, b_i_1+… + i_j.(iii) The tableau T_j for 1⩽ j ⩽ℓ is the column filled with the elements of π from the positionsd_1+d_2+⋯ + d_ℓ-j+1+1 through and including position d_1+d_2+⋯ + d_ℓ-j+2, but excluding anyb_i in that range.Note that in item (ii), the rows of γ are assumed to be numbered from bottom to top and are filled starting with rowd_1-i_1+1 and ending with row d_1-i_1+ℓ+1 at the top.Also observe that since the bijection stated in Proposition <ref> preserves the weight, it can be restricted to abijection φM(D,I)^(r)→𝖲𝖲𝖸𝖳^(r)(1^c_1) ×⋯×𝖲𝖲𝖸𝖳^(r)(1^c_ℓ) ×𝖲𝖲𝖸𝖳^(r)(γ),where M(D,I)^(r) = M(D,I)∩𝒪𝒫^(r)_n,k.Before giving the proof, it is helpful to consider two examples to illustrate the map φ. When the entries of π∈𝒪𝒫_n,k in minimaj order are increasing, then ℓ = 0. In this case, d_1 = n and i_1 = k. The mapping φ takes π to the semistandard tableau T =T_1 that is of ribbon-shape γ = (1^n-k,k). Theentries of the boxes in the first column of the tableau T areb_1, followed by the n-k numbers in the sequences β_1,β_2,…, β_k-1,α_k from top to bottom. (The fact that π has no descents means that all the α_i = ∅ for 1⩽ i <k and we are in Case 2 (a) of Lemma <ref> for 1⩽ i<k and Case 1 for i=k.)Columns 2through k of T_1 are filled with the numbers b_2,…,b_k respectively, and b_2 ⩽ b_3 ⩽⋯⩽ b_k.The result is a semistandard tableau T_1 of hook shape. For example, consider π = (12 | 2 | 234) ∈𝒪𝒫_6,3. Then γ=(1^3,3) andT_1 = 122,2,3,4 . Now suppose that T is such a hook-shape tableau with entries b_1,b_2,…,b_k from left to right in its top row,and entries b_1, t_1, …, t_n-k down its first column. The inverse φ^-1maps T to the set partition πthat has as its first block π_1= b_1β_1, where β_1=t_1, …, t_m_1, and t_1 < … < t_m_1⩽ b_2, but t_m_1+1 > b_2 so that β_1 is in the interval (b_1,b_2]. The second block of π is given byπ_2 = b_2 β_2, where β_2 = t_m_1+1,…,t_m_2, and t_m_1+1< t_m_1+2< … <t_m_2⩽ b_3,but t_m_2+1 > b_3 and β_2⊆ (b_2,b_3]. Continuing in this fashion, we set π_k = b_k α_k, where α_k = t_m_k-1+1,…, t_n-k and α_k ⊆ (b_k,+∞). Thenφ^-1(T) = π = (π_1|π_2 |⋯|π_k), where the ordered multiset partition π has no descents. The ordered multiset partition π = (124 | 45. | 3 | 46.1| 23.1| 1 | 25) ∈𝒪𝒫_15,7has the following data: [ b_1 = 1,α_1 =∅, β_1 = 24 b_2 = 4,α_2 = 5, β_2 =∅ b_3 = 3, α_3 = ∅, β_3 =∅; b_4 = 4, α_4 = 6, β_4 =1 b_5 = 2, α_5 = 3,β_5 = 1 b_6 = 1, α_6 = ∅,β_6 = ∅;b_7=2, α_7 = 5, β_7 = ∅ ;]and ℓ=3, d_1=5, d_2=d_3=3, d_4=4 and i_1=i_2=2, i_3=1, i_4=2. Thenboxsize=1.1emπ = (124|45. |3|46.1|23.1|1|25) ↦1,5×1,3×6×12,2,34,14,2,4,5 .It is helpful to keep the following picture in mind during the proof of Proposition <ref>, where the map φis taking the ordered multiset partition π to the collection of tableaux T_ias illustrated below. We adopt the shorthandnotation η_j :=i_1+⋯+i_j for 1⩽ j ⩽ℓ, where we also set η_0=0 and η_ℓ+1=k:π = (b_1 β_1 | b_2 β_2 | ⋯ |b_η_1α_η_1. β_η_1|b_η_1+1β_η_1+1 |⋯|b_η_jα_η_j.β_η_j| b_η_j+1β_η_j+1 | ⋯ | b_k α_k)boxsize=2.9emT_ℓ+1-j = β_η_j,β_η_j+1,⋮,β_η_j+1-1,α_η_j+1for 1⩽ j⩽ℓ, T_ℓ+1 =b_η_ℓ+1⋯b_η_ℓ+1, ⋮, b_η_j-1+1⋯b_η_j, ⋮,b_1⋯b_η_1,β_1,⋮,β_η_1-1,α_η_1 .Since the entries of π are mapped bijectively to the entries of T_1 × T_2 ×⋯× T_ℓ+1, the map φ preserves the total weight (π)=(p_1,p_2,…) ↦ (T), where p_i is the number ofentries i in π for i ∈ℤ_>0. We need to show that φ is well defined andexhibit its inverse. For this, we can assume that ℓ⩾ 1, as the case ℓ = 0 was treated in Example <ref>. Observe first that there are d_j entries in π which are between two consecutive descents,and among these entries there are exactly i_j entries that are first elements of a block, since descents happen i_j blocks apart. This implies that the tableaux have the shapes claimed. To see that the tableaux are semistandard, consider firstT_ℓ+1, and let η_j = i_1+⋯+i_j as above.A row numbered d_1-i_1+j for 1 ⩽ j ⩽ℓ+1 is weakly increasing, because the lack of a descent ina block π_i means b_i ⩽ b_i+1, and this holds for i in the intervalη_j-1 +1, …, η_j betweentwo consecutive descents.The leftmost column is strictly increasing because it consists of the elements b_1 < β_1< β_2 < ⋯ <β_η_1-1 <α_η_1 (the lack of a descent before π_η_1 implies thatα_i=∅ for i<η_1 and b_i <β_i ⩽ b_i+1< β_i+1 byCase 2 (a) of Lemma <ref>).The rest of the columns of T_ℓ+1 contain elements b_i, where b_η_j-1+1 is the first element in rowd_1-i_1+j and b_η_j is the last, and b_η_j+1 is the first element in the row immediately above it.We haveb_η_j > b_η_j+1,since there is a descent in block π_i_j which implies this inequality by the orderingcondition in Case 2 (b) of Lemma <ref>.The strict inequalities for the column tableaux T_1,…,T_ℓ hold for the same reason that they hold for the firstcolumn in T_ℓ+1. That is,the columns consist of the elements β_η_j < β_η_j+1 <⋯ < β_η_j+1-1 <α_η_j+1, where all the α_i for η_j⩽ i<η_j+1 are in fact∅, since we are in Case 2 (a) of Lemma <ref> here. Next, to show that φ is a bijection, we describe the inverse map of φ. For D = {d_1,d_1+d_2, …, d_1+d_2+⋯+d_ℓ}⊆{1,2,…,n-1} and I = {i_1,i_1+i_2,…,i_1+i_2+⋯+i_ℓ} = {η_1,η_2,…,η_ℓ}⊆{1,2,…,k-1} with ℓ distinct elements each, suppose d_ℓ+1 and i_ℓ+1 are such that d_1+d_2+⋯+d_ℓ+1 = n andη_ℓ+1=i_1+i_2+⋯+i_ℓ+1=k.Assume T_1 ×⋯× T_ℓ× T_ℓ+1∈𝖲𝖲𝖸𝖳(1^c_1) ×⋯×𝖲𝖲𝖸𝖳(1^c_ℓ)×𝖲𝖲𝖸𝖳(γ), where γ = (1^d_1-i_1,i_2,…,i_ℓ+1) and c_j = d_ℓ+2-j - i_ℓ+2-jfor 1⩽ j ⩽ℓ. We construct π by applying the following algorithm.Read off the bottom d_1-i_1 entries of the first column of T_ℓ+1. Let b_1 be the element immediately abovethese entries in the first column of T_ℓ+1, and note that b_1 is less than all of them. Let b_2,…, b_i_1 be theelements in the same row of T_ℓ+1 as b_1, reading from left to right.Assignb_η_1+1,…, b_η_2 to theelements in the next higher row, and so forth, until reaching row d_1-i_1+ℓ+1 (the top row) of T_ℓ+1and assigning b_η_ℓ+1,…, b_η_ℓ+1=b_k to its entries. The elements in β_1,…,β_η_1-1,α_η_1 are obtained by cutting the entries inthe first column of T_ℓ+1 above b_1, so that β_i lies in the interval (b_i, b_i+1], andα_η_1lies in the interval(b_η_1,∞).Now for 1⩽ j⩽ℓ, we obtain β_η_j,β_η_j+1,…,β_η_j+1-1,α_η_j+1 bycutting the elements in T_ℓ+1-j into sequences as follows: β_η_j = T_ℓ+1-j∩ ( -∞, b_η_j+1 ],β_η_j+m = T_ℓ+1-j∩ (b_η_j+m+1, b_η_j+m+2] and α_η_j+1 = T_ℓ+1-j∩(b_η_j+1,+∞). The inequalities are naturally forced from the inequalities in the semistandard tableaux, and the descents at thegiven positions are also forced,because by construction α_η_j > b_η_j > b_η_j+1⩾β_η_j.This process constructs the b_i, α_i, and β_i for each i=1,…,k, where we assume that sequences thathave not been defined by the process are empty.Then φ^-1(T_1 × T_2 ×⋯× T_ℓ+1) = π = (π_1 |π_2 |⋯|π_k), where π_i = b_i α_i β_i.For a (skew) partition λ, the Schur function 𝗌_λ(𝐱) is defined as𝗌_λ(𝐱) = ∑_T ∈𝖲𝖲𝖸𝖳(λ)𝐱^(T).Similarly for m ⩾ 1, the m-th elementary symmetric function 𝖾_m(𝐱) is given by𝖾_m(𝐱) = ∑_1 ⩽ j_1 < j_2 < ⋯ < j_m x_j_1 x_j_2⋯ x_j_m.As an immediate consequence of Proposition <ref>, we have the following symmetric function identity.Assume D⊆{1,2,…,n-1} and I⊆{1,2,…, k-1} are sets of ℓdistinct elements each and let M(D,I), γ and c_j for 1⩽ j ⩽ℓ be as inProposition <ref>. Then ∑_π∈M(D,I)𝐱^(π) = 𝗌_γ(𝐱)∏_j=1^ℓ𝖾_c_j(𝐱). § CRYSTAL ON ORDERED MULTISET PARTITIONS §.§ Crystal structure Denote the set of words of length n over the alphabet {1,2,…,r} by 𝒲^(r)_n. The set 𝒲_n^(r) can be endowed with an 𝔰𝔩_r-crystal structure as follows. The weight (w) of w∈𝒲_n^(r) is the tuple (a_1,…,a_r), where a_i is the number of letters i in w. The Kashiwara raising and lowering operatorse_i, f_i 𝒲_n^(r)→𝒲_n^(r)∪{0}for 1⩽ i <rare defined as follows. Associate to each letter i in w an open bracket “)” and to each letter i+1 in w a closed bracket “(”. Then e_i changes the i+1 associated to the leftmost unmatched “(” to an i; if thereis no such letter, e_i(w)=0. Similarly, f_i changes the i associated to the rightmost unmatched “)” to an i+1; if there is no such letter, f_i(w)=0.For λ a (skew) partition, the 𝔰𝔩_r-crystal action on 𝖲𝖲𝖸𝖳^(r)(λ) is induced by the crystal on 𝒲_|λ|^(r), where |λ| is the number of boxes in λ. Consider therow-reading word 𝗋𝗈𝗐(T) of T∈𝖲𝖲𝖸𝖳^(r)(λ), which is the word obtained from T byreading the rows from bottom to top, left to right. Then f_i(T) (resp. e_i(T)) is the RSK insertion tableau of f_i(𝗋𝗈𝗐(T)) (resp. e_i(𝗋𝗈𝗐(T))). It is well known that f_i(T) is a tableau in 𝖲𝖲𝖸𝖳^(r)(λ) with weightequal to (T)-ϵ_i+ϵ_i+1, where ϵ_i is i-thstandard vector in ^r. Similarly, e_i(T) ∈𝖲𝖲𝖸𝖳^(r)(λ), and e_i(T) has weight (T)+ϵ_i-ϵ_i+1. See forexample <cit.>.In the same spirit, an 𝔰𝔩_r-crystal structure can be imposed on 𝖲𝖲𝖸𝖳^(r)(1^c_1,…,1^c_ℓ,γ) := 𝖲𝖲𝖸𝖳^(r)(1^c_1) ×⋯×𝖲𝖲𝖸𝖳^(r)(1^c_ℓ) ×𝖲𝖲𝖸𝖳^(r)(γ)by concatenating the reading words of the tableaux in the tuple. This yields crystal operatorse_i,f_i 𝖲𝖲𝖸𝖳^(r)(1^c_1,…,1^c_ℓ,γ) →𝖲𝖲𝖸𝖳^(r)(1^c_1,…,1^c_ℓ,γ) ∪{0}.Via the bijection φ of Proposition <ref>, this also imposes crystal operators on ordered multiset partitionsẽ_i,f̃_i 𝒪𝒫_n,k^(r)→𝒪𝒫_n,k^(r)∪{0}as ẽ_i = φ^-1∘ e_i ∘φ and f̃_i = φ^-1∘ f_i ∘φ.An example of a crystal structure on 𝒪𝒫_n,k^(r) is given in Figure <ref>.The operators ẽ_i, f̃_i, andimpose an 𝔰𝔩_r-crystal structure on 𝒪𝒫_n,k^(r). In addition, ẽ_i and f̃_i preserve thestatistic.The operators ẽ_i, f̃_i, andimpose an 𝔰𝔩_r-crystal structure by construction since φ is a weight-preserving bijection. The Kashiwara operators ẽ_i and f̃_i preserve thestatistic, since by Proposition <ref>, the bijection φ restricts to M(D,I)^(r)which fixes the descents of the ordered multiset partitions in minimaj order. §.§ Explicit crystal operators Let us now write down the crystal operator f̃_i 𝒪𝒫_n,k→𝒪𝒫_n,kof Theorem <ref> explicitly on π∈𝒪𝒫_n,k in minimaj order.Start by creating a word w from right to left by reading the first element in each block of π from right to left,followed by the remaining elements of π from left to right. Note that this agrees with 𝗋𝗈𝗐(φ(π)). For example, w=513165421434212 for π in Example <ref>. Use the crystal operator f_i on words to determinewhich i in w to change to an i+1. Circle the corresponding letter i in π. The crystal operator f̃_ion π changes the circled i to i+1 unless we are in one of the following two cases:⋯i| if̃_i⟶⋯| ii+1 ,|ii+1f̃_i⟶ i+1 |i+1 . Here “⋯” indicates that the block is not empty in this region. In Figure <ref>, f̃_2(31 2| 2) = (31 | 23) is an example of (<ref>). Similarly, f̃_1(31 |1 2) = (312 |2) is an example of (<ref>).The above explicit description for f̃_i is well defined and agrees with the definition of Theorem <ref>.The word w described above is precisely 𝗋𝗈𝗐(φ(π)) on which f_i acts. Hence the circled letter i is indeed the letter changed to i+1. It remains to check how φ^-1 changes the blocks. We will demonstrate this for the cases in (<ref>) as the other cases are similar.In case (<ref>) the circled letter i in block π_j does not correspond to b_j in π_j as it is not at the beginning of its block. Hence, it belongs to α_j or β_j. The circled letter is not a descent. Changing it to i+1 would create a descent. The map φ^-1 distributes the letters in α_j and β_j to preserve descents, hence the circled i moves over to the next block on the right and becomes acircled i+1. Note also that i+1 ∉π_j+1, since otherwise the circled i would have been bracketed in w,contradicting the fact that f_i is acting on it.In case (<ref>) the circled letter i in block π_j corresponds to b_j in π_j. Again, φ^-1 now associates the i+1 ∈π_j to the previous block after applying f_i. Note thati+1 ∉π_j-1 since it would necessarily be b_j-1. But then the circled i would have been bracketed in w, contradicting the fact that f_i is acting on it.§.§ Schur expansion The character of an 𝔰𝔩_r-crystal B is defined asch B = ∑_b∈ B𝐱^(b).Denote by B(λ) the 𝔰𝔩_∞-crystal on 𝖲𝖲𝖸𝖳(λ) defined above. This is a connected highest weight crystal with highest weight λ, and the character is the Schurfunction 𝗌_λ(𝐱) defined in (<ref>)chB(λ) = 𝗌_λ(𝐱).Similarly, denoting by B^(r)(λ) the 𝔰𝔩_r-crystal on 𝖲𝖲𝖸𝖳^(r)(λ), its character is the Schur polynomialchB^(r)(λ) = 𝗌_λ(x_1,…,x_r).Let us define𝖵𝖺𝗅^(r)_n,k(𝐱;0,t) = ∑_π∈𝒪𝒫^(r)_n,k+1t^(π)𝐱^(π),which satisfies 𝖵𝖺𝗅_n,k(𝐱;0,t) = 𝖵𝖺𝗅_n,k^(r)(𝐱;0,t) for r⩾ n, where 𝖵𝖺𝗅_n,k(𝐱;0,t) is as in (<ref>).As a consequence of Theorem <ref>, we now obtain the Schur expansion of𝖵𝖺𝗅_n,k^(r)(𝐱;0,t). We have𝖵𝖺𝗅_n,k-1^(r)(𝐱;0,t) = ∑_π∈𝒪𝒫^(r)_n,k ẽ_i(π) = 0 ∀ 1⩽ i <r t^(π)𝗌_(π). When r⩾ n, then by <cit.> and <cit.> this is also equal to𝖵𝖺𝗅_n,k-1(𝐱;0,t) = ∑_λ⊢ n∑_T ∈𝖲𝖸𝖳(λ) t^(T) + n-k2 -(n-k) 𝖽𝖾𝗌(T)[ [ 𝖽𝖾𝗌(T);n-k ]] 𝗌_λ(𝐱),where 𝖲𝖸𝖳(λ) is the set of standard Young tableaux of shape λ (that is, the elements in 𝖲𝖲𝖸𝖳(λ) of weight (1^|λ|)), 𝖽𝖾𝗌(T) is the number of descents of T, (T) is the major index of T (or the sum of descents of T), and the t-binomial coefficients in the sum are defined using the rule[ [ m; p ]] = [m]!/[p]! [m-p]!where [p]! = [p][p-1] ⋯ [2][1] and[p] = 1 + t + ⋯ + t^p-1.The crystal 𝒪𝒫_4,2^(3), displayed in Figure <ref>, has four highest weight elements with weights (2,1,1), (2,1,1), (2,1,1), (2,2) from left to right. Hence, we obtain the Schur expansion𝖵𝖺𝗅^(3)_4,1(𝐱;0,t) = (1+t+t^2) 𝗌_(2,1,1)(𝐱) + t𝗌_(2,2)(𝐱).§ EQUIDISTRIBUTIVITY OF THE MINIMAJ AND MAJ STATISTICS In this section, we describe a bijection ψ_n,k→_n,k in Theorem <ref> with the property that (π) = (ψ(π)) for π∈_n,k. This proves the link betweenandthat was missing in <cit.>. We can interpret ψ as a crystal isomorphism, where _n,k on the left is thecrystal of Section <ref> and _n,k on the right is viewed as a crystal of k columns with elements written in major index order.The bijection ψ is the composition of φ of Proposition <ref> with a certain shift operator. When applying φ to π∈_n,k, we obtain the tuple T^∙=T_1 ×⋯× T_ℓ+1in (<ref>). We would like to view each column in the tuple of tableaux as a block of a new ordered multiset partition. However, note that some columns could be empty, namely if c_j=d_ℓ+2-j-i_ℓ+2-j in Proposition <ref> is zero for some1⩽ j ⩽ℓ. For this reason, let us introduce the set of weak ordered multiset partitions𝒲𝒪𝒫_n,k, where we relax the condition that all blocks need to be nonempty sets.Let T^∙ = T_1 ×⋯× T_ℓ+1 be a tuple of skew tableaux. Define (T^∙) to be theweak ordered multiset partition whose blocks are obtained from T^∙ by reading the columns from theleft to the right and from the bottom to the top; each column constitutes one of the blocks in (T^∙). Note that given π= (π_1 | π_2| ⋯ | π_k) ∈_n,k in minimaj order, (φ(π)) is a weak ordered multiset partition in major index order. Let π = (1| 56.| 4.| 37.12| 2.1| 1| 34) ∈_13,7, written in minimaj order.We have (π)=22. Thenboxsize=1.1emT^∙ = φ(π) = 1,4×1,2×7×∅× 13, 2, 3, 4,15,6and π'=(T^∙) = (4.1| 2.1| 7.|∅| 6.1| 5.4.3.2.1 | 3). Let ℐ={(φ(π)) |π∈_n,k}⊆𝒲𝒪𝒫_n,k, π' = (φ(π)) ∈ℐ, and b_i the first elements in each block of π in minimaj order as in Lemma <ref>. Then π' has the following properties: * The last k elements of π' are b_1,…,b_k, and b_i and b_i+1 are in different blocks if and onlyif b_i ⩽ b_i+1.* If b_1,…,b_k are contained in precisely k-j blocks, then there are at least j descents in the blockscontaining the b_i's.Let π∈_n,k, written in minimaj order. Then by (<ref>),π'=(φ(π)) is of the formπ'= (α^rev_η_ℓ+1β^rev_η_ℓ+1-1⋯β^rev_η_ℓ| ⋯| α^rev_η_1β^rev_η_1-1⋯β^rev_1b_1 ⋯|⋯| b_η_1b_η_1-1⋯|⋯|⋯ b_k),where the superscript rev indicates that the elements are listed in decreasing order (rather than increasing order). Since the rows of a semistandard tableau are weakly increasing and the columns are strictly increasing, the blocksof π'=(φ(π)) are empty or in strictly decreasing order. This implies that b_i and b_i+1 are in differentblocks of π' precisely when b_i⩽ b_i+1, so a block of π' that contains a b_i cannot have a descentat its end. This proves (1).In a weak ordered multiset partition written in major index order, any block of size r⩾ 2 has r-1 descents.So if b_1,…, b_k are contained in precisely k-j blocks, then at least j of these elements are contained in blocksof size at least two, so there are at least j descents in the blocks containing the b_i's. This proves (2).Let π'∈𝒲𝒪𝒫_n,k be in major index order such thatthere are at least k elements after the rightmost occurrence of a block that is either empty or has a descent at its end. In this case, there exists a skew tableau T^∙ such that π'=(T^∙). In fact, this characterizesℐ := im (∘φ). The mapis invertible.Suppose π' ∈𝒲𝒪𝒫_n,k is in major index order such that there are at least k elements after therightmost occurrence of a block that is either empty or has a descent at its end. Since there are no occurrences of anempty block or a descent at the end of a block amongst the last k elements of π', the blocks of π' containingthe last k elements form the columns of a skew ribbon tableau T∈𝖲𝖲𝖸𝖳(γ), and the remaining blocksof π' form the column tableaux to the left of the skew ribbon tableau, sois invertible. We are now ready to introduce the shift operators.We define the left shift operation Ł on π'∈ℐ = {(φ(π)) |π∈𝒪𝒫_n,k}as follows. Suppose π' has m ⩾ 0 blocks π_p_m',…, π_p_1' that are either empty or havea descent at the end, and 1 ⩽ p_m < ⋯ <p_2 < p_1<k. SetŁ(π') = Ł^(m)(π'),where Ł^(i) for 0⩽ i⩽ m are defined as follows: * Set Ł^(0)(π')=π'.* Suppose Ł^(i-1)(π') for 1⩽ i ⩽ m is defined. By induction, the p_i-th block of Ł^(i-1)(π') is π'_p_i. Let S_i be the sequence of elements starting immediately to the right of block π'_p_i in Ł^(i-1)(π')up to and including the p_i-th descent after the block π_p_i'. Let Ł^(i)(π') be the weak ordered multisetpartition obtained by moving each element in S_i one block to its left. Note that all blocks with index smaller than p_i in Ł^(i)(π') are the same as in π'.Continuing Example <ref>, we haveπ'=(4.1| 2.1| 7.|∅|6.1|5.4.3 .2.1 | 3), which is in major index order. We have m=2 with p_2=3<4=p_1, S_1=61543,S_2=6154 andŁ^(1)(π')= (4.1| 2.1| 7. |6.1|5.4.3.| 2.1| 3), Ł(π') = Ł^(2)(π')= (4.1| 2.1| 7.6.1 | 5.4.| 3.| 2.1| 3).Note that (π')=28, (Ł^(1)(π'))=25, and (Ł(π')) = 22 = (π). The left shift operation Łℐ→𝒪𝒫_n,k is well defined.Suppose π'∈ℐ has m ⩾ 0 blocks π_p_1',…, π_p_m' that are either empty or have adescent at the end, and 1⩽ p_m < ⋯ < p_2 < p_1 < k. If m=0, then Ł(π')=π' ∈𝒪𝒫_n,k and we are done.We proceed by induction on m. Note that Ł^(1) acts on the rightmost block π_p_1'. Notice that π_p_1' cannot contain any of the b_i's by Lemma <ref> (1). Hence, since there are at least k elements in the k-p_1 blocks following π_p_1',by Lemma <ref> (2), there are at least p_1 descents after π_p_1', so Ł^(1) can be appliedto π'.Observe that applying Ł^(1) to π' does not create any new empty blocks to the right of π_p_1', becausecreating a new empty block means that the last element of S_1, which is a descent, is at the end of a block.This cannot happen, since the rightmost occurrence of an empty block or a descent at the end of its block was assumedto be in π_p_1'.However, note that applying Ł^(1) to π' does create a new block with a descent at its end,and this descent is given by the p_1-th descent after the block π_p_1' (which is the last element of S_1).Now suppose Ł^(i-1)(π') is defined for i ⩾ 2. By induction, there are at least p_1>p_i descents followingthe block π_p_i', so the set S_i of Definition <ref> exists and we can move the elements in S_i left one block to construct Ł^(i)(π') from Ł^(i-1)(π'). Furthermore, Ł^(i)(π') does not have anynew empty blocks to the right of π_p_i'. To see this, note that the number of descents in S_i is p_i, so thenumber of descents in S_i is strictly decreasing as i increases. This implies that the i-1 newly created descentsat the end of a block of Ł^(i-1)(π') occurs strictly to the right of S_i, and so the last element of S_i cannotbe a descent at the end of a block of Ł^(i-1)(π').Lastly, Ł(π') = Ł^(m)(π')∈_n,k, since it does not have any empty blocks, and every block ofŁ(π') is in decreasing order because either we moved every element of a block into an empty block or we movedelements into a block with a descent at the end. We define the right shift operationon μ∈𝒪𝒫_n,k in major index order as follows. Suppose μ has m⩾ 0 blocksμ_q_1, …, μ_q_m that have a descent at the end andq_1 < q_2 < ⋯ < q_m. Set(μ) = ^(m)(μ),where ^(i) for 0⩽ i ⩽ m are defined as follows: * Set ^(0)(μ)=μ.* Suppose ^(i-1)(μ) for 1⩽ i ⩽ m is defined. Let U_i be the sequence of q_i elementsto the left of, and including, the last element in the q_i-th block of ^(i-1)(μ).Let ^(i)(μ) be theweak ordered multiset partition obtained by moving each element in U_i one block to its right. Note that all blocks to the right of the (q_i+1)-th block are the same in μ and ^(i)(μ).Note thatcan potentially create empty blocks. Continuing Example <ref>, let μ = Ł(π') = (4.1| 2.1| 7.6.1 | 5.4.| 3.| 2.1| 3). We have m=2 with q_1=4<5=q_2, U_1=6154, U_2=61543 and^(1)(μ)= (4.1| 2.1| 7. | 6.1 | 5.4.3.| 2.1| 3), (μ) = ^(2)(μ)= (4.1| 2.1| 7. |∅| 6.1 | 5.4.3.2.1| 3),which is the same as π' in Example <ref>. The right shift operationis well defined and is the inverse of Ł.Suppose μ∈𝒪𝒫_n,k in major index order has descents at the end of the blocksμ_q_1,…, μ_q_m. If m=0, then (μ) = μ∈𝒪𝒫_n,k⊆𝒲𝒪𝒫_n,kand there is nothing to show.We proceed by induction on m. The ordered multiset partition μ does not have empty blocks, so there are atleast q_1 elements in the first q_1 blocks of μ, and ^(1) can be applied to μ.Now suppose ^(i-1)(μ) is defined for i⩾2. By induction, there are at least q_i-1+1 elements inthe first q_i-1+1 blocks of ^(i-1)(μ).Since the blocks μ_q_i-1+2,…, μ_q_i in μare all nonempty, there are at least q_i-1+1+(q_i-(q_i-1+1)) = q_i elements in the first q_i blocks of ^(i-1)(μ),so the set U_i of Definition <ref> exists and we can move the elements in U_i one block to theright to construct ^(i)(μ) from ^(i-1)(μ).Furthermore, every nonempty block of (μ) is in decreasing order because the rightmost element of each U_i isa descent. So (μ)∈𝒪𝒫_n,k remains in major index order. This completes the proof thatis well defined.Next we show thatis the inverse of Ł. Observe that if π' ∈ℐ has m occurrences of either anempty block or a block with a descent at its end, then μ=Ł(π') has m blocks with a descent at its end. Hence it suffices to show that ^(m+1-i) is the inverse operation to Ł^(i) for each 1⩽ i ⩽ m.The property that the last element of S_i cannot be a descent at the end of a block of Ł^(i-1)(π') in the proof of Proposition <ref> similarly holds for every element in S_i. Therefore, if the last element of S_i is in ther_i-th block of Ł^(i-1)(π'), then |S_i| = p_i + (r_i-1-p_i) = r_i-1 because the blocks are decreasing and none of theelements in S_i can be descents at the end of a block. Since the last element of S_i becomes a descent at the end of the (r_i-1)-th block of Ł^(i)(π), this impliesr_i-1 = q_m-i+1, so U_m-i+1 = S_i for every 1⩽ i ⩽ m. As the operation Ł^(i) is a left shiftof the elements of S_i by one block and the operation ^(m+1-i) is a right shift of the same set of elementsby one block, they are inverse operations of each other. For what follows, we need to extend the definition of the major index to the set 𝒲𝒪𝒫_n,k of weak orderedmultiset partitions of length n and k blocks, in which some of the blocks may be empty. Givenπ' ∈𝒲𝒪𝒫_n,k whose nonempty blocks are in major index order, if the block π_j'≠∅, thenthe last element in π_j' is assigned the index j, and the remaining elements in π_j' are assigned the index j-1 for j=1,…, k. Then (π') is the sum of the indices where a descent occurs. This agreeswith (<ref>) in the case when all blocks are nonempty.Let π'∈ℐ. With the same notation as in Definition <ref>, we have for 1⩽ i ⩽ m(Ł^(i)(π')) = (Ł^(i-1)(π'))-p_i+1,if π_p_i'=∅, (Ł^(i-1)(π'))-p_i,if π_p_i' has a descent at the end of its block.Assume π_p_i'=∅. In the transformation from Ł^(i-1)(π') to Ł^(i)(π'), the index of each of the firstp_i-1 descents in S_i decreases by one, while the index of the last descent remains the same, since it is not at theend of a block in Ł^(i-1)(π'), but it becomes the last element of a block in Ł^(i)(π'). The indices of elementsnot in S_i remain the same, so (Ł^(i)(π'))=(Ł^(i-1)(π'))-p_i+1 in this case.Next assume that π_p_i' has a descent at the end of the block. In the transformation from Ł^(i-1)(π') to Ł^(i)(π'),the indices of the descents in S_i change in the same way as in the previous case, but in addition, the index of the lastdescent in π_p_i' decreases by one, so (Ł^(i)(π'))=(Ł^(i-1)(π'))-p_i in this case.Let ψ_n,k→_n,k be the map defined byψ(π) = Ł((φ(π))) for π∈_n,k in minimaj order.Then ψ is a bijection that maps ordered multiset partitions in minimaj order to ordered multiset partitions inmajor index order. Furthermore, (π) = (ψ(π)). By Proposition <ref>, φ is a bijection. By Lemma <ref>, the mapis invertible, and by Proposition <ref> the shift operation Ł has an inverse. This implies that ψ is a bijection.It remains to show that (π) = (ψ(π)) for π∈𝒪𝒫_n,k in minimaj order.First suppose that π' = (φ(π)) has no empty blocks and no descents at the end of any block. In this case Ł(π')=π', so that in fact π' = ψ(π). Using the definition of major index (<ref>) andthe representation (<ref>) (where the columns in the ribbon are viewed as separate columns dueto ), we obtain(π') = ∑_j=1^ℓ (ℓ+1-j) ( d_j - i_j -1) + ℓ + ∑_j=1^ℓ ( ℓ+η_j-j),where d_j,i_j,η_j = i_1 + ⋯ + i_j are defined in Proposition <ref> for π. Here, the first sum in the formula arises from the contributions of the first ℓ blocks and the summand ℓcompensates for the fact that b_1 is in the ℓ-th block. The second sum in the formula comes from thecontributions of the b_i's. Comparing with (<ref>), we find(π') = (π) - ℓ+12 - ∑_j=1^ℓ (ℓ+1-j) i_j + ℓ+12 + ∑_j=1^ℓη_j = (π),proving the claim.Now suppose that π' = (φ(π)) has a descent at the end of block π'_p. This will contribute an extra p compared to the major index in (<ref>). If π'_p=∅, then c_p = d_ℓ+2-p - i_ℓ+2-p = 0 and the term j=ℓ+2-p in (<ref>) should be (ℓ+1-j)(d_j-i_j) instead of (ℓ+1-j)(d_j-i_j-1) yielding a correction term of ℓ+1-j = ℓ+1-ℓ-2+p=p-1. Hence, with the notation of Definition <ref>, we have(π') = (π) + ∑_i=1^m p_i - e,where e is the number of empty blocks in π'. Since ψ(π) = Ł(π'), the claim follows byLemma <ref>.alpha | http://arxiv.org/abs/1707.08709v2 | {
"authors": [
"Georgia Benkart",
"Laura Colmenarejo",
"Pamela E. Harris",
"Rosa Orellana",
"Greta Panova",
"Anne Schilling",
"Martha Yip"
],
"categories": [
"math.CO",
"05A19 (Primary), 05A18, 05E05, 05E10, 20G42, 17B37 (Secondary)"
],
"primary_category": "math.CO",
"published": "20170727051238",
"title": "A minimaj-preserving crystal on ordered multiset partitions"
} |
[email protected]@lanl.gov Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, [email protected] Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Sud, Université Paris-Saclay, 91406 Orsay, France Department of Physics, University of Arizona, Tucson,Arizona 85721, [email protected] Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859 Campinas SP, BrazilThe properties of cold Bose gases at unitarity have been extensively investigated in the last few years both theoretically and experimentally. In this paper we use a family of interactions tuned to two-body unitarity and very weak three-body binding to demonstrate the universal properties of both clusters and matter.We determine the universal properties of finite clusters up to 60 particles and, for the first time, explicitly demonstrate the saturation ofenergy and density with particle number and compare with bulk properties. At saturation in the bulk we determine the energy, density, two- and three-body contacts and thecondensate fraction. We find that uniform matter is more bound than three-body clusters by nearly two orders of magnitude, the two-body contact is very large in absolute terms, and yet the condensate fraction is also very large, greater than 90%. Equilibrium properties of these systems may be experimentally accessible through rapid quenching of weakly-interacting boson superfluids.Ground-state Properties of Unitary Bosons: From Clusters to Matter S. A. Vitiello December 30, 2023 ==================================================================Introduction: Strongly-interacting fermionic cold atoms have been the subject of a great deal of study both theoretically and experimentally across the BEC to BCS transition, and especially at unitarity, where the two-body system has nearly a zero-energy bound state <cit.>. These systems are universal in that all properties, including ground-state energy, superfluid pairing gaps, superfluid transition temperatures, etc., are obtained as a set of universal dimensionless parameters multiplied by the Fermi energy or momentum of a free Fermi gas at the same density.Studies of bosonic superfluids, however, have concentrated on the weakly-interacting regime described by the Gross-Pitaevski mean-field equation. These systems are comparatively simple to study as they were the first to be cooled to very low temperatures and their properties can be described in a mean-field picture.It has been known for some time that short-range two- and three-body interactions can be used to describe the low-energy properties of small clusters of bosons. To obtain universal properties, the two-body interaction can similarly be taken to generate a zero-energy dimer, but a three-body interaction is required <cit.> to avoid the so-called “Thomas collapse” <cit.> of three or more particles. The resulting discrete scale invariance leads to geometric towers of states in systems with three <cit.> and more<cit.>bosons. Many atomic and nuclear few-body systems fall into this universality class <cit.>. In this paper we demonstrate that large clusters and bulk matter are stable with such interactions, and similarly to the fermionic case described by a fairly simple set of universal parameters. We provide the first estimates for the universal parameters describing the ground-state energy, the equilibrium density, two- and three-body contacts, and the condensate fraction of such a system.Our calculations are the analog of those carried out for fermions inRefs. <cit.>, but here the universal parameters are directly related to the properties of the three-body system, i.e. its energy andradius.These bosonic universal properties may be accessible through cold-atom experiments, including those studying rapid quenching from weakly-interacting Bose condensates.Interaction and Method: The Hamiltonian we consider isH=-ħ^2/2m∑_i ∇_i^2+ ∑_i<j V_ij + ∑_i<j<k V_ijk,where the first term is the non-relativistic kinetic energy, the second the attractive short-range interaction tuned to infinite scattering length, and the last term is a repulsive three-body contact interaction tuned to produce a weakly-bound trimer.For zero-range interactions universality has been demonstrated in Ref. <cit.>.For this study we employ finite-range two- and three-body interactions, keeping the range of these interactions much smaller than the size of the weakly-bound trimer. For unitarity bosons this restriction is very stringent, as we shall see.The interaction must also be much shorter ranged than the average interparticle spacing in the bulk, which is an order of magnitude smaller than the three-body cluster size.Here we employ Gaussian two- and three-body interactions:V_ij =V_2^0ħ^2/mμ_2^2exp [ - (μ_2 r_ij)^2/2 ], V_ijk =V_3^0ħ^2/m(μ_3/2)^2exp [ - (μ_3R_ijk/2)^2/2],where r_ij=r_i-r_j is the relative distance between bosons i and j, and R_ijk=(r_ij^2+r_ik^2+r_jk^2)^1/2. The strength V_2^0 is tuned to unitarity, and V_3^0 is tuned to reproduce aweakly-bound three-particle state with a binding energy -E_3 and an associated radius R̅_3 ≡ (-2m E_3 / ħ^2)^-1/2. The introduction of both two- and three-body range parameters μ_2,3 allows us to produce arbitrarily weakly-bound trimers for a given set of interactionranges, which is essential to extract universal physics in the deeply-bound many-body system. Specific details of the interaction are not relevant as long as they are very short-ranged and the ground state can be tuned to a shallow trimer. In any physical system, the geometric towerof Efimov states at unitarity is truncated from below due tothe range of the interaction.The binding energy of the would-be next deeper trimer is ≃ (22.7)^2 larger than that of the calculated ground-state trimer,hence the shape of our potentials should produce small effects for μ_2,3R̅_3≫ 23 <cit.>. Corrections due to the physical interaction range can be includedthrough a two-body potential with two derivatives <cit.>.We use Variational and Diffusion Monte Carlo (VMC, DMC) methods for the solution of the Schrödinger equation.The trial-state wave functions are of the formΨ_T=∏_i f^(1)(r_i) ∏_i<j f^(2)(r_ij)∏_i<j<k f^(3)(R_ijk),withf^(1)(r) = exp(-α r^2),f^(2)(r) = Ktanh(μ_J r)cosh(γ r)/r, and f^(3)(R) = exp[u_0exp(-R^2/(2 r_0^2))]. The parameters K and γ are chosen to have f^(2)(d)=1 andf^(2)'(d)=0 at the “healing distance” d. The variational parameters α, μ_J, d, u_0 and r_0 are optimized at the VMC level for each system and interaction as describedin Ref. <cit.>, and α=0 to simulate uniform matter.The VMC wave function is then used as input for exact DMC calculations, see for example Ref. <cit.>. The calculated energies are exact subject to statistical andtime-step errors that can be made arbitrarily small.Results for the energy are independent of the trial wave function, though statistical errors may be large for poor choices. Other properties are extrapolated from the VMC and DMC results,which we have tested using different trial wave functions. The extrapolation errors are very small, on the order of a few percent or less, similar or smaller than the reported statistical errors.Clusters: Clusters with six or fewer bosons have been studied extensively in the literature with an emphasis on Efimov physics<cit.>, for a review see Ref. <cit.>. Slightly larger clusters with similar interactions have also been consideredpreviously <cit.>. Universal behavior was found for small clusters up to N ≤ 15.Non-universal behavior beyond this point was attributed tofinite-range effects.For sufficiently small range, it is expected that clusters will be universal and have abinding energy per particleE_N/N= ξ_B (N) E_3/3,where ξ_B (N) is a universal function of N.In Fig. <ref> we show results for clusters of up to 60 bosons for Hamiltonians with μ_2 R̅_3= 46 and 65, and compare to those of Ref. <cit.> for N ≤ 15. These yield a trimer rms radius⟨ r_3^2 ⟩ ^1/2≈ 0.61R̅_3for our finite-range Hamiltonians. We consider three-body interactions with different ratios of two- to three-body interactions ranges,X_μ≡μ_3 / μ_2 = 0.5, 0.75 and 1.0. Finite-range interactions will show non-universal effects when the range of two- or three-particle interactions becomes significant compared to the average interparticle distance.This can be seen in the results of Refs. <cit.>around N=15, and also in our results corresponding to the more bound trimers(open symbols with μ_2 R̅_3 = 46 inFig. <ref>) for smaller X_μ. For μ_2 R̅_3 = 65 the three sets of points withX_μ = 0.5, 0.75, 1.0agree within statistical errors.For N=4 our result (3E_4/(4E_3)=3.5(1) for μ_2 R̅_3 = 65 and X_μ = 1.0) also agrees very well with the precise calculation ofRef. <cit.> (3E_4/(4E_3)=3.46),suggesting that Efimov-related few-body physics is properly captured by our potential.Studies of unitary bosons commonly employ a zero-range two-body interaction with three-body hard-core interaction of radius R_0. That interaction has a fixed value of R̅_3/ R_0 ≈ 15.3 <cit.>, which can be compared to ourμ_2 R̅_3 = 65 and μ_3 R̅_3 = 32, 49, 65 for X_μ = 0.5, 0.75, 1.0.The zero-range two-body plushard-core interaction can reproduce universal physics for small clusters butthe three-body hard core is not small compared to typical near-neighbor separations for larger clusters (N > 15) or matter, as discussed below.For small N the binding energy per particle increasesapproximately linearly with N,and by N > 7 it is an order of magnitude larger than the trimer's. Since we have tuned the trimer energy to be very smallwe can find universal behavior up to N=60 clusters,as shown by the solid points (μ_2 R̅_3 = 65) in Fig. <ref>. For a 60-particle cluster thebinding per particle is approximately 50 times that of the trimer. Naive dimensional arguments would suggest that the repulsive three-body interaction will become more important for large N, resulting in saturation to a constant binding energy per particle similarto what is observed in atomic nuclei. The energies per particle for large clusters are beginning to saturate to a constant value as shown inFig. <ref>.Similar behavior has been seen in finite-temperature simulations in a trap <cit.>. We have also calculated the single-particle densities andradii of the N-particle clusters. Radii are also expected to scale with a universal ratioof the trimer rms radius: ⟨ r^2_N ⟩^1/2 = β (N) ⟨ r^2_3 ⟩^1/2. Results are shown in Fig. <ref>. The upper panel shows that the cluster radius reaches a minimum around N=5-7, and then increases as saturation sets in. For larger clusters one would expect the radius to increase as N^1/3 for a system saturating to an equilibrium density.The lower panel shows single-particle densities for different particle numbers N and demonstrates the saturation of the single-particledensity near the center of the clusters at a valueindependent of cluster size.Matter: We have also computed the properties of the bulk Bose liquid atunitarity for these same interactions using periodic boundary conditions.We expected very small finite-sizeeffects, and confirmed this by comparing results for 20, 40 and 60 particles. Results for different N at the same density are equivalent within statistical errors. We find a universal equation of state (EOS) with an equilibrium ground-state energy per particleof 87 ± 5 times that of the trimer, and a saturation density of ρ_0 4 πR̅_3^3 /3 = 275 ± 20. The results are summarized in Fig. <ref>. Near saturation density they are well described by 3E_N (ρ)/N|E_3| |_N→∞= ξ_B (N→∞) [-1+ κ(ρ-ρ_0/ρ_0)^2],with the dimensionless compressibility κ= 0.42 (5). The curves in Fig. <ref> are fits to the EOS calculations with two different X_μ.The calculations of the liquid are consistent with those obtained by extrapolating the cluster results. A liquid-drop extrapolation of the cluster binding energies, E_N/N= E_B (N→∞) ( 1 + η N^-1/3 + … ), is consistent with the energies found for the bulk.Fitting results for N > 30, we find that the universal energy parameter ξ_B (N →∞) = 90 ± 10.The surface energy scaled by the volume energy E_B (N→∞) is η = - 1.7 ± 0.3, but has relatively large statistical errors. Similarly, the single-particle densitynear thecenter of the drops shown in Fig. <ref> is consistent with the equilibrium density of matter. Despite the growth in energy with N, the liquid can be considered universal: the interparticle separation at equilibrium,[3/(4πρ_0)]^1/3≃R̅_3/6.4, is almost four times larger than the distance scale set by thenext deeper Efimov trimer in the universal system without cutoffs. It is also 5-10 times larger than the two- and three-body interaction ranges, in contrast to ∼ 2for a zero-range two-body plus three-body hard-core interaction at the same density.It is interesting to compare these results to liquid ^4He, which has a large two-body scattering length and, for small N, weakly-bound clusters that can be described by short-rangeinteractions.Per particle,the binding energy of liquid ^4He is -7.14 K <cit.>, which is about 180 times thatof the ^4He trimer, -0.0391 K <cit.>. The scaled surface energy is ≈ - 2.7 <cit.> and thedimensionless compressibility is ≈ 1.9 <cit.>. For small N the helium clusters are universal <cit.>,but for large N the interaction range is comparable to the interparticle separation and hence not universal. Nevertheless, the ratio of binding energiesξ_B (N→∞)and the scaled surface energy η arewithin a factor of 2 of unitary bosons.We have also examined the two- and three-body contact parameters C_2,3 for the unitary Bose fluid at equilibrium density. These contact parameters impact various properties of the system, and relate the short-distance behavior to the high-momentum tail of the momentum distribution, see for exampleRefs. <cit.>. The two- and three-body distribution functions are shown in Fig. <ref>, normalizedto one at large distances (differing by a factor of ρ N! fromthe g_N defined in Refs. <cit.>).In the universal regime outside the range of the interaction, the two-body distribution g_2 (r), with r≡ r_ij, is expected to be proportional to 1/r^2. The upper lines in the top panel show 32 π^2 ρ^2/3 r^2 g_2 (r) / 10 for the different simulations, and the dashed line is a quadratic fit to results in the universal regime that can be extrapolated to r=0 to give the dimensionlesstwo-body contact α_2, with C_2 = N α_2 ρ^4/3 <cit.>. From the extrapolation of r^2 g_2we find α_2 = 17(3). More accurate results may be achievable through simulations at different scattering lengths with fixed E_3. This result is larger but qualitatively comparable to those obtained in moreapproximate approaches <cit.> or those obtained with zero-range two-body plus hard-core three-body interactions <cit.>,and quite similar to those extracted through rapid experimental quenches<cit.>.Similarly, in the bottom figure the dashed line is a fit to g_3 (r), with r≡ R_ijk. In the universal regime, extrapolating to r=0 gives the three-body contact. It is more accurate to extract the dimensionless three-body contactβ_3, with C_3= Nβ_3 ρ^2/3 <cit.>, from the derivative of the energy with respect to R̅_3 at constant scattering length. Using the equilibrium properties calculated in Fig. <ref> we obtain β_3 = 0.9(1). The density dependence of β_3 around equilibrium can be extracted from Eq. (<ref>). Further simulations could yield the density dependence of α_2, and also the asymptotic behavior of the momentum distribution g_3.With these strong correlationsand the large binding and small radiirelative to the trimer, one might expect that the condensatefraction may be reduced in the bulk.In fact we find quite a large condensate fraction at equilibrium density, withn (k=0)= 0.93 (1), compared to a value of unity for a weakly-interacting Bose fluid. One can also compare to liquid ^4He which has a condensate fraction of0.0725(75) at equilibrium density <cit.>.The large condensate fraction implies that it is reasonable to access equilibrium properties of the universal Bose fluid as a function of density through experiments with rapid quenching of a weakly-interactinggas <cit.>. The universal properties of the unitary Bose fluid are difficult to measure using standard techniques because of losses to deeply-bound three-body states that occur in cold atoms but are absent in our simulations. These loss mechanisms can lead to a trap lifetime smaller than that needed to reach full equilibrium, and presently available studies investigate this dynamics of the rapid quenching of the free-to-unitary transition. Our results indicate that a rapid quench from a weakly-interacting Bose fluidat the appropriate density may enable one to obtain the equilibrium properties. The relatively large overlap of the two states should lead to a rapid ejection of particles through high-energy two- and three-body processes, leading to a rapid cooling of the system.Quantifying this energy loss could lead to an experimental verification of the universal properties of the unitary Bose fluid in thermal equilibrium.Summary: We havedemonstrated the universal nature of bosons at unitarity using short-range interactions tuned to unitarity in the two-body system and weak binding (Efimov) trimers in the three-body system. We have determined many of the universal properties of the unitary Bose fluid, including the energies and radii of clusters of up to 60 bosons and calculated the universal saturation and contacts of the unitary Bose fluid. We find a ground-state energy per particle of approximately90 times that of the trimer at an associated high density. We find a large two-body contact parameter, yet the condensate fraction in the bulk is greater than 90%. We also calculate the three-body contact parameter for the first time. Further experimental and theoretical studies of theunitary Bose fluid will be very intriguing.Many new properties can be studied, including those described above, collective effects and the static and dynamic response of the system.Acknowledgments: We thank Daekyoung Kang for many valuable discussions. The work of J.C. and S.G. was supported by the NUCLEI SciDAC program, and by the U.S. DOE under contract DE-AC52-06NA25396. The work of U.vK. was supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under award number DE-FG02-04ER41338, andby the European Union Research and Innovation program Horizon 2020 under grant agreement no. 654002.S.V. thanks the hospitality and financial support from LANL and the facilities offered by CENAPAD-SP. Computational resources have been provided by Los Alamos Open Supercomputing. 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"authors": [
"J. Carlson",
"S. Gandolfi",
"U. van Kolck",
"S. A. Vitiello"
],
"categories": [
"cond-mat.quant-gas",
"nucl-th",
"physics.atm-clus",
"quant-ph"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170726171747",
"title": "Ground-state properties of unitary bosons: from clusters to matter"
} |
[ V. Grytsay December 30, 2023 ===================== Determining semantic textual similarity is a core research subject in natural language processing. Since vector-based models for sentence representation often use shallow information,capturing accurate semantics is difficult. By contrast, logical semantic representations capture deeper levels of sentence semantics, but their symbolic nature does not offer graded notions of textual similarity. We propose a method for determining semantic textual similarity by combining shallow features with features extracted fromnatural deduction proofsof bidirectional entailment relations between sentence pairs.For the natural deduction proofs, we use ccg2lambda, a higher-order automatic inference system,which convertsCombinatory Categorial Grammar (CCG) derivation trees into semantic representations andconducts natural deduction proofs. Experiments show that our systemwas able to outperform other logic-based systems and that features derived from theproofs are effective for learning textual similarity.§ INTRODUCTION Determining semantic textual similarity (STS) is one of the most critical tasks in information retrieval and natural language processing.Vector-based sentence representation models have been widely used to compare and rank words, phrases or sentences using various similarity and relatedness scores <cit.>. Recently, neural network-based sentence representation models <cit.> have been proposed for learning textual similarity. However, these vector-based models often use shallow information, such as words and characters, andwhether they can account for phenomena such as negation and quantification is not clear. Consider the sentences: Tom did not meet some of the players and Tom did not meet any of the players.If functional words such as some or any are ignored or represented as the same vector, then these sentences are to be represented by identical vectors. However, the first sentenceimplies that there is a player who Tom did not meet,whereas the second sentence means that Tomdid not meet anyone, so the sentences have different meanings.Conversely, logic-based approacheshave been successful in representingthe meanings of complex sentences, having had a positive impactfor applications such as recognizing textual entailment <cit.>. However, purely logic-based approaches onlyassess entailment or contradiction relations between sentences and do not offer graded notions of semantic similarity.In this paper, we propose to leverage logic cues to learn textual similarity. Our hypothesis is that observingproof processes when testing the semantic relations is predictive of textual similarity. We show that our approachcan be more effectivethan systems that ignore these logic cues.§ RELATED WORKVector-based models of semantic composition have been widely studied with regards to calculating STS. <cit.> proposed a sentence vector model involving word vector addition or component-wise multiplication. Addition and multiplication are commutative and associative and thus ignore word order. <cit.> proposed a discourse-based sentence vector model considering extra-intra sentential context. Also, a categorical compositional distributional semantic model has been developed for recognizing textual entailment and for calculating STS <cit.>. However, these previous studies are mostly concerned with the structures of basic phrases or sentences and do not address logical and functional words such as negations and connectives. Neural network-based models of semantic composition <cit.> have also been proposed.Although these models achieve higher accuracy, their end-to-end nature introduces challenges in the diagnosis of the reasons that make two sentences to be similar or dissimilar to each other. These diagnosis capabilities may play an important role in making the system explainable and also to guide future system improvements in a more precise manner. Our approach presented in this paper is partially inspired by the latter two objectives.Meanwhile, some previous studies have proposed logic systems for capturing the semantic relatednessof sentences. The Meaning Factory <cit.>uses both shallow and logic-based features for learning textual similarity. In this system, the overlap of predicates and entailment judgments are extracted as logic-based features. UTexas <cit.>uses Probabilistic Soft Logic for learning textual similarity. In this system, each ground atom in the logical formulas has a probability based on distributional semantics of a word. The weights of the logical formulas are calculated from the probabilities of their ground atoms and are extracted as features. These previous studies improved the accuracy by using logic-based features derived fromthe entailment results of first-order theorem proving in addition to using shallow features such as sentence lengths. In our study, wedetermine the semantic similarity of sentences based on the conception of proof-theoretic semantics <cit.>.The key idea is that not only the entailment results but also the theorem proving process can be considered as features for learning textual similarity. That is, by taking into account not only whether a theorem is proved but also how it is proved, we can capture the semantic relationships between sentence pairs in more depth. Another difference between our study and previous logic systems is that we use higher-order predicate logic. Higher-order predicate logic is able to represent complex sentence semantics such as generalized quantifiers more precisely than first-order predicate logic. In addition, higher-order predicate logic makes the logical structure of a sentence more explicit than first-order predicate logic does, so it can simplify the process of proof search <cit.>.§ SYSTEM OVERVIEWFigure 1 showsan overview of the system which extracts features for learning textual similarity fromlogical proofs. To produce semantic representations of sentences and prove them automatically, we use ccg2lambda <cit.>, which is a semantic parser combined with an inference system based on natural deduction.First, sentences are parsed into syntactic trees based on Combinatory Categorial Grammar (CCG) <cit.>. CCG is a syntactic theory suitable for semantic compositionfrom syntactic structures. Meaning representations are obtained based on semantic templates and combinatory rulesfor the CCG trees. Semantic templates are defined manually based on formal semantics. Combinatory rules specify the syntactic behaviors of words and compositional rulesfor the CCG trees. In ccg2lambda, two wide-coverage CCG parsers, C&C <cit.> and EasyCCG <cit.>, are used for converting tokenized sentences into CCG trees robustly. According toa previous study <cit.>, EasyCCGachieves higher accuracy. Thus,when the output of both C&C and EasyCCG can be proved, we use EasyCCG's output for creating features.Second, the meanings of wordsare describedusinglambda terms. Semantic representations are obtained by combining lambda terms in accordance with the meaning composition rules specified in the CCG tree. The semantic representations are based on Neo-Davidsonian event semantics <cit.>, in which every verb is decomposed into a predicate over events and a set of functional expressions relating the events. Adverbs and prepositions are also represented as predicates over events.Third, we attempt to prove entailment relations between sentence pairs. For this purpose, we use Coq <cit.>, which can be used for efficient theorem-proving for natural language inference using both first-order and higher-order logic <cit.>. Coq's proof calculus is based on natural deduction <cit.>, a proof system based on inference rules called introduction and elimination rules for logical connectives. The inference system implemented in ccg2lambdausing Coqachieves efficient automatic inference by feeding a set of predefined tactics and user-defined proof-search tactics to its interactive mode.The natural deduction system is particularly suitable for injecting external axioms during the theorem-proving process <cit.>.Finally, features for learning textual similarity are extracted from the proofs produced by ccg2lambda during the theorem-proving process. In this study, we experimented with logistic regression, support vector regression and random forest regression, finding that random forest regression was the most effective.We therefore chose random forest regression for learning textual similarity, with its hyperparameters being optimized by grid search. The mean squared error (MSE) was used to measure the prediction performance of our system.§ PROOF STRATEGY FOR LEARNING TEXTUAL SIMILARITY §.§ Overview of the proof strategySentence similarity depends on complex elements,such as word overlaps and semantic relations.We capture the similarity between the sentence pair (A, B) as a function of the provability of bidirectional entailment relations for (A, B) and combine it with shallow features.After obtaining logical formulas A' and B' from A and B, we attempt to prove the bidirectional entailment relations, A' ⇒ B' and B' ⇒ A'.If the initial natural deduction proofs fail, were-run the proof, adding relevant external axioms or skipping unproved sub-goals until the proof iscompleted. After that, features for learning textual similarity are extracted by quantifying the provability of the bidirectional entailment relations.The details of the procedure are as follows. First, we attempt a natural deduction proof without using external axioms, aiming to prove entailment relations, A' ⇒ B' and B' ⇒ A'. If both fail, then we checkwhether A' contradicts B', which amounts to proving the negation of the original conclusion, namely A' ⇒ B' and B' ⇒ A'.The similarityof a sentence pair tends to be higher when the negation of the conclusion can be proved,compared with the casewhere neither the conclusion nor its negation can be proved. In the SICK (Sentences Involving Compositional Knowledge) dataset <cit.> (see Section 6.1 for details),70% of the sentence pairs annotated ascontradictory are assigned a relatedness score in [3, 5). Next, if we fail to prove entailment or contradiction, that is,we cannot prove the conclusion or its negation,we identify an unproved sub-goalwhich is not matched by any predicate in the premise.We then attempt to prove A' ⇒ B' and B' ⇒ A'using axiom injection, following the method introduced in <cit.>. In axiom injection, unproved sub-goals are candidates to form axioms. We focus only onpredicates that share at least one argument with both the premise and the conclusion.This means that an axiom can be generated only if there is a predicate p in the pool ofpremises and a predicate q in a sub-goal and p and q share a variable in an argument position, possibly with the same case (e.g., Subject or Object). In generating axioms, the semantic relationships between the predicates in the premise and those in the conclusion are checkedusing lexical knowledge. In this study, we use WordNet <cit.> as the source of lexical knowledge. Linguistic relations between predicates are checked inthe following order: inflections, derivationally related forms, synonyms, antonyms, hypernyms, similarities, and hyponyms. If any one of these relations is found in the lexical knowledge, an axiom can be generated. Again, if the proof fails, we attempt to prove the negation of the conclusionusing the axiom injection mechanism.If the proof by axiom injection failsbecause of a lack of lexical knowledge, we obtain sentence similarity information from partial proofs by simply accepting the unproved sub-goals andforcibly completing the proof. After the proof iscompleted, information about the generated axioms and skipped sub-goals is used to create features.§.§ Proving entailment relations As an illustration of how our natural deduction proof works, consider the case of proving entailmentfor the following sentence pair: A: A man is singing in a bar. B: A man is singing.The sentences A and B are mapped onto logical formulas A' andB' based on event semantics via CCG-based semantic composition, as follows. 0.9A': ∃ e_1 x_1 x_2(man(x_1) ∧sing(e_1) ∧ (subj(e_1) = x_1)0.9∧ bar(x_2) ∧in(e_1, x_2))0.9B': ∃ e_1 x_1(man(x_1) ∧sing(e_1)∧ (subj(e_1) = x_1)) First, we attempt a natural deduction proof of A' ⇒ B', setting A' as the premise and B' as the goal of the proof. Then A' and B' are decomposed according tothe inference rules.Figure <ref> shows the major inference rules we use in the proofs. Inference rules in natural deduction are divided into two types: introduction rules and elimination rules. Introduction rules specify how to prove a formula in the goal,decomposing a goal formula into smaller sub-goals. Elimination rules specify how to use a premise, decomposing a formula in the pool of premises into smaller ones. The proof processfor A' ⇒ B' is shown in Figure <ref>. Here A' is initially set to the premise P_0 and B' to the goal G_0. P_0 and G_0 are then decomposed using elimination rules (∧-Elim, ∃-Elim) and introduction rules (∧-Intro, ∃-Intro). Then we obtain a set of premise formulas 𝒫 = P_2, P_3, P_4, P_5, P_6, and a set of sub-goals 𝒢 = G_2, G_3, G_4. The proof is performed by searching for a premise P_i whose predicate and arguments match those of a given sub-goal G_j. If such a logical premise is found, the sub-goal is removed. In this example, the sub-goals G_2, G_3, and G_4 match the premises P_2, P_3, and P_4, respectively. Thus, A' ⇒ B' can be proved without introducing axioms.Second, we attempt the proof inthe opposite direction, B' ⇒ A', by switching P_0 and G_0 in Figure <ref>. Again, by applying inference rules, we obtain the following sets of premises 𝒫 and sub-goals 𝒢: 𝒫 = {P_2: man(x_1), P_3: sing(e_1), P_4: subj(e_1) = x_1} 𝒢 = {G_2: man(x_1),G_3: sing(e_1),G_4: subj(e_1) = x_1,G_5: bar(x_2), G_6: in(e_1, x_2))}Here, the two sub-goals G_5 and G_6 do not match any of the premises, so the attempted proof of B' ⇒ A' fails. We therefore attempt to inject additional axioms, but in this case no predicate in 𝒫 shares the argument x_2 of the predicates bar(x_2) and in(e_1,x_2) in 𝒢. Thus, no axiom can be generated. To obtain information from a partial proof,we forcibly complete the proof of B' ⇒ A' by skipping the unproved sub-goals bar(x) and in(e_1,x_2). §.§ Proving the contradiction The proof strategy illustrated here can be straightforwardly applied to proving the contradiction.In natural deduction,a negative formula of the form A can be defined as A →False (“the formula A implies the contradiction”), by using a propositional constant False to encode the contradiction. Thus, the inference rules for negation can be taken as special cases of implication rules, as shown in Figure <ref>.As an illustration, let us consider the following sentence pair:A: No man is singing.B: There is a man singing loudly.Figure <ref> showsthe proof process. The sentences A and B are mapped to P_0 and P_1, respectively, via compositional semantics and the goal G_0 is set to False. By decomposing P_1 using elimination rules and then by combining P_2, P_3, and P_4, we can obtain P_6. From P_0 and P_6 we can then derive the contradiction.These proofs are performed by an automated prover implemented on Coq, using tactics for first-order theorem proving. When a proof is successful,Coq outputs the resulting proof (a proof term), from which we can extract detailed information such as the number of proof steps andthe types of inference rules used. In addition to the entailment/contradictionresult, information aboutthe proof process is used to createfeatures.§ DESCRIPTION OF THE FEATURESTo maximize accuracy when learning textual similarity, we adopt a hybrid approachthat uses both logic-based features extracted from the natural deduction proof and other, non-logic-based features.All features are scaled to the [0, 1] range. §.§ Logic-based FeaturesWe propose 15 features consisting of nine different types of logic-based features. Six of these feature types are derived from the bidirectional natural deduction proofs: six features are extracted from the direct proof (A' ⇒ B') and another six from the reverse proof (B' ⇒ A'). The remaining three feature types are derived from semantic representations of the sentence pairs. The feature types are as follows. Logical inference result. As stated in Section 4, weinclude features to distinguish the case where either the conclusion or its negation can be proved from theone where neither can be proved. If the conclusion can be proved, the feature is set to 1.0. If the negation of the conclusion can be proved, the feature is set to 0.5. If neither can be proved, the feature is set to 0.0.Axiom probabilities. The probability of an axiom and the number of axioms appearing in the proof are usedto create features. The probability of an axiom is defined as the inverse of the length of the shortest path that connects the senses in the is-a (hypernym/hyponym) taxonomy in WordNet. When multiple axioms are used in the proof, the average of the probabilities of the axiomsis extracted as a feature. If the proofcan be completed without using axioms, the feature is set to 1.0.Proved sub-goals. Given thatproofs can be obtained either by proving all the sub-goals or skipping unproved sub-goals, we use the proportion ofproved sub-goals as a feature. Our assumption is that if there are more unproved sub-goals then the sentence pair is less similar. When there are m logical formulas in the premise pool and n proved sub-goals, we set the feature to n/m. If the theorem can be proved without skipping any sub-goals, the feature is set to 1.0.It may be the case that the number of sub-goals is so large that some sub-goals remain unproved even after axiom injection.Since the proportion of unproved sub-goals is decreased by axiom injection,we use the proportion of unproved sub-goals both with and without axiom injection as features.Cases in unproved sub-goals. Subject or object wordscan affect the similarity of sentence pairs. Therefore, the number of each case in unproved sub-goals,like subj(e_1) in Figures <ref> and <ref>, is used as a feature.Here, we count subjective, objective, and dative cases.Proof steps. In general, complex theorems are difficult to prove and in such cases the sentence pairs are considered to be less similar. We therefore use the number of Coq's proof steps, namely the number of inference rule applications in a given proof, as a feature.Inference rules. The complexity of a natural deduction proof can be measuredin terms of the inference rules usedfor each proof step. We therefore extract the relative frequency with which each inference rule is used in the proof as a feature. We check seven inference rules for natural deduction using Coq (cf.Figure <ref>): introduction and elimination rules for conjunction (∧-Intro, ∧-Elim),implication (→-Intro, →-Elim), and existential quantification (∃-Intro, ∃-Elim), and the elimination rule for equality (=-Elim).Predicate overlap. Intuitively, the more predicates overlap between the premise and the conclusion, the more likely it is that the inference can be proved. We therefore use the proportion of predicates that overlap between the premise and the conclusion as a feature.Semantic type overlap. Each semantic representation in higher-order logic has a semantic type, such asfor entities andfor propositions. As with predicates, we use the degree of semantic type overlap between the premise and the conclusion as a feature.Existence of negative clauses. Whether or not the premise or conclusion contain negative clauses is an effective measure of similarity. In semantic representations, negative clauses are represented by the negation operator , so we check for negation operators in the premise and the conclusion and set this feature to 1.0 if either contains one.§.§ Non-logic-based FeaturesWe also use the following eight non-logic-based features.Noun/verb overlap. We extract and lemmatize all nouns and verbs from the sentence pairs and use the degrees of overlap of the noun and verb lemmas as features.Part-of-speech overlap. We obtain part-of-speech (POS) tags for all words in the sentence pairs by first tokenizing them with the Penn Treebank Project tokenizer[ftp://ftp.cis.upenn.edu/pub/treebank/public_html/tokenization.html] and then POS tagging them with C&C POS tagger <cit.>. The degree of overlap between the sentences' POS tags is used as a feature.Synset overlap. For each sentence in the pair, we obtain the set containing all the synonym lemmas (the synset) for the words in the sentence. The degree of overlap between the sentences' synsets is used as a feature.Synset distance. Foreach word in the first sentence, we compute the maximum path similarity between its synset and the synset of any other word in the second sentence. Then, we use the average of maximum path similarities as a feature.Sentence length. If the conclusion sentence is long, there will possibly be many sub-goals in the proof. We therefore use the average of the sentence lengths and the difference in length between the premise and the conclusion sentences as features.String similarity. We use the similarity of the sequence of characters within the sentence pairs as a feature.The Python Difflib[https://docs.python.org/3.5/library/difflib.html] function returns the similarity between two sequences as a floating-point value in[0, 1]. This measure is given by 2.0*M / T, where T is the total number of elements in both sequences and M is the number of matches. This feature is 1.0 if the sequences are identical and 0.0 if they have nothing in common.Sentence similarity from vector space models. We calculate sentence similarity by using three major vector space models, TF-IDF, latent semantic analysis (LSA) <cit.>, and latent Dirichlet allocation (LDA) <cit.>. We use these cosine similarities as features.Existence of passive clauses. Passive clauses have an influence on similarity. In CCG trees, passive clauses are representedusing the syntactic category S_pss\ NP. We check for the occurrence of passive clauses in the premise and conclusion, and if either of them contains a passive clause then the feature is set to 1.0.§ EXPERIMENTS AND EVALUATION §.§ Experimental ConditionsWe evaluated our system[Available at https://github.com/mynlp/ccg2lambda.] using two datasets: the SemEval-2014 version of the SICK dataset <cit.> and the SemEval-2012 version of the MSR-paraphrase video corpus dataset (MSR-vid) <cit.>. The experimental conditions were as follows.§.§.§ The SICK datasetThe SICK dataset is a dataset for studyingSTS as well as for recognizing textual entailment (RTE). It was originally developed for evaluating compositional distributional semantics, so it contains logically challenging expressions such as quantifiers, negations, conjunctions and disjunctions. The dataset contains 9927 sentence pairs with a 5000/4927 training/test split. These sentence pairs are manually annotated with three types of labelsyes (entailment), no (contradiction), or unknown (neutral) as well as a semantic relatedness scores in[1, 5] (see Table <ref> for a sample).In this dataset, sentence pairs whose gold entailment labels areno tend to be scored a little more highly than the average, whereasthose whose labels are unknown have a wide range of scores. Thus, we set the baseline of the relatedness score to 5 when the gold entailment label was yes and to 3 when the label was no or unknown.We compared our systemwith the following systems: the state-of-the-art neural network-based system <cit.>; the best system <cit.> from SemEval-2014; and two of the logic-based systems stated in Section 2: namely The Meaning Factory <cit.> and UTexas <cit.>. The Pearson correlation coefficient γ, Spearman's rank correlation coefficient ρ, and the MSE were used as the evaluation metrics.§.§.§ The MSR-vid datasetThe MSR-vid dataset is our second dataset for the STS task and contains 1500 sentence pairs with a 750/750 training/test split.All sentence pairs are annotated with semantic relatedness scores in the range [0, 5]. We used this dataset to compare our system with the best system from SemEval-2012 <cit.> and the logic-based UTexas system <cit.>. We used the Pearson correlation coefficient γ as the evaluation metric. §.§ Results Table <ref> shows the results of ourexperiments with the SICK dataset. Although the state-of-the-art neural network-based system yielded the best results overall, our system achieved higher scores than SemEval-2014 submissions, including the two logic-based systems (The Meaning Factory and UTexas), in terms of Pearson correlation and Spearman's correlation. The main reason for our system's lower performance in terms of MSE is that some theorems could not be provedbecause of a lack of lexical knowledge. In the current work, we only consider word-level knowledge (word-for-word paraphrasing); we may expand the knowledge base in the future by using more external resources.As we mentioned above, the sentence pairs annotated as unknown produced a wide range of scores. The Pearson correlation of the unknown portion of the SICK dataset was 0.766, which suggests that our logic-based system can also be applied to neutral sentence pairs. Table <ref> shows the results of ourexperiments with the MSR-vid dataset. These results also indicate that our logic-based system achieved higher accuracy than the other logic-based systems.Table <ref> shows evaluation results for each feature group in isolation,showing that inference rules and predicate overlaps are the most effective features.Compared with the non-logic-based features, the logic-based features achieved a slightly higher accuracy, a point that will be analyzed in more detail in the next section.Overall, our results show that combining logic-based features with non logic-based ones is an effective method for determining textual similarity. §.§ Positive examples and error analysisTable <ref> shows someexamples for which the prediction score was better when using logic-based features than when using non-logic-based ones.For IDs 642 and 1360, one sentence contains a passive clause while the other sentence does not. In such cases, the sentence pairs are not superficially similar.By using logical formulas based on event semantics we were able to interpret the sentence containing the passive clause correctly and judge that the passive and non-passive sentences are similar to each other.In ID 891,one sentence contains a negative clausewhile the other does not.Using shallow features, the word overlap is small and the prediction score was much lower than the correct score. Our logic-based method, however, interpreted the first sentence as a negative existential formula of the form ∃ x 𝒫(x) and the second sentence as an existential formula ∃ x 𝒫'(x). Thus, it could easily handle the semantic difference between the positive and negative sentences.In ID 1158, by contrast, the proportion of word overlap is so high that the prediction score with non-logic-based features was much higher than the correct score. Our method, however, was able toprove the contradiction using an antonym axiom of the form ∀ x (remove(x) →add(x)) from WordNet and thus predict the score correctly.In ID 59, the proportion of word overlap is low, so the prediction score with non-logic-based features was lower than the correct score. Our method, however, was able toprove the partial entailment relations for the sentence pair and thus predict the score correctly. Here the logic-based method captured the common meaning of the sentence pair: both sentences talk about the kids playing in the leaves.Finally, in ID 71, the prediction score with non-logic-based features was much higher than the correct score. There are two reasons for this phenomenon:negations tend to be omitted in non-logic-based features such as TF-IDFand the proportion of word overlap is high. However, as logical formulas andproofs can handle negative clauses correctly,our method was able to predict the score correctly.Table <ref> showsexamples where using only logic-based features produced erroneous results. In ID 3974, the probability of axiom ∀ x (awaken(x) →up(x)) was low (0.25) and thus the prediction score was lower than the correct score.Likewise, in ID 4833, the probability of axiom ∀ x (file(x) →do(x)) was very low (0.09) and thus the prediction score was negatively affected. In these cases, we need to consider phrase-level axioms such as ∀ x (awaken(x) →wake_up(x)) and ∀ x (file_nail(x) →do_manicure(x)) using a paraphrase database. This, however, is an issue for future study. In ID 1941, the system wrongly proved the bidirectional entailment relations by adding external axioms, so the prediction score was much higher than the correct score.Setting the threshold for the probability of an axiom may be an effective way of improving our axiom-injection method.§ CONCLUSIONWe have developed a hybrid method for learning textual similarity by combining features based on logical proofs of bidirectional entailment relations with non-logic-based features. The results of our experiments on two datasets show that our system was able to outperform other logic-based systems. In addition, the results show that information about the natural deduction proof process can be used to create effective features for learning textual similarity. Since these logic-based features provide accuracy improvements that arelargely additive with those provided by non-logic-based features, neural network-based systems may also benefit from using them.In future work, we will refine our system so that it can be applied to other tasks such as question answering. Compared with neural network-based systems, our natural deduction-based system can not only assess how similar sentence pairs are, but also explain what the sources of similarity/dissimilarity are by referring to information about sub-goals in the proof. Giventhis interpretative ability, we believe that our logic-based systemmay also be of benefit to other natural language processing tasks, such as question answering and text summarization.§ ACKNOWLEDGMENTSWe thank the three anonymous reviewers for their detailed comments. This work was supported by JST CREST Grant Number JPMJCR1301, Japan. emnlp_natbib | http://arxiv.org/abs/1707.08713v1 | {
"authors": [
"Hitomi Yanaka",
"Koji Mineshima",
"Pascual Martinez-Gomez",
"Daisuke Bekki"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170727054951",
"title": "Determining Semantic Textual Similarity using Natural Deduction Proofs"
} |
Journal ofClass Files, Vol. 14, No. 8, August 2015 S. Sadrizadeh and S. Kiani : Simultaneous Sparse ApproximationIterative method for simultaneous sparse approximation Sahar Sadrizadeh, Shahrzad Kiani, Mahdi Boloursaz, Student Members, IEEE, and Farokh Marvasti, Senior Member, IEEE S. Sadrizadeh, S. Kiani, M. Boloursaz and F. Marvasti are with the Advanced Communication Research Institute (ACRI), Electrical Engineering Department, Sharif University of Technology, Tehran, Iran (email: [email protected]; [email protected]; [email protected]). December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================This paper studies the problem of Simultaneous Sparse Approximation (SSA). This problem arises in many applications which work with multiple signals maintaining some degree of dependency such as radar and sensor networks. In this paper, we introduce a new method towards joint recovery of several independent sparse signals with the same support.We provide an analytical discussion on the convergence of our method called Simultaneous Iterative Method with Adaptive Thresholding (SIMAT). Additionally, we compare our method with other group-sparse reconstruction techniques, i.e., Simultaneous Orthogonal Matching Pursuit (SOMP), and Block Iterative Method with Adaptive Thresholding (BIMAT) through numerical experiments.The simulation results demonstrate that SIMAT outperforms these algorithms in terms of the metrics Signal to Noise Ratio (SNR) and Success Rate (SR). Moreover, SIMAT is considerably less complicated than BIMAT, which makes it feasible for practical applications such as implementation in MIMO radar systems. Simultaneous Sparse Approximation; Iterative Method; Adaptive Thresholding; Joint Recovery.§ INTRODUCTIONSparsesignal processing has recently been exploited in various fields of communication, due to fact that sparse signals can be approximated by only a few nonzero coefficients and hence sub-Nyquist sampling and Compressed Sensing (CS) <cit.>. The general CS problem is formulated as follows:|l| 𝐱_0𝐲=𝐀𝐱+𝐯where x is the main sparse signal, y is the measurement vector, A is the sensing matrix, and v is the additive noise vector. Two main models are considered in CS for reconstruction of sparse signals. Models with one measurement vector are referred to as Single Measurement Vector (SMV) models, while the other models with at least two measurement vectors are called Multiple Measurement Vector (MMV) models. The problem investigated in MMV models, known as SSA, aims to jointly recover sparse representation of the measurement vectors. The SSA applications may be encountered in various fields such as sensor networks <cit.>, Electroencephalography and Magnetoencephalography (EEG and MEG) <cit.>, source localization <cit.>, and distributed MIMO radar systems <cit.>. <cit.> investigates the theory of MMV models. Some algorithms have been developed by extending the general SMV model into the MMV model to solve the SSA problems. Orthogonal Matching Pursuit (OMP)<cit.> as a greedy algorithm is one of the very first algorithms used for sparse recovery. At each iteration of this algorithm, the best local improvement to the current approximations is found in hope of obtaining a good overall solution. The extension of the OMP algorithm to The MMV paradigm, Simultaneous OMP or SOMP, has been developed in <cit.>. The Iterative Method with Adaptive Thresholding (IMAT) algorithm was originally proposed for sparse signal reconstruction from missing samples <cit.>. The Block Iterative Method with Adaptive Thresholding (BIMAT) <cit.> as an extension of IMAT is employed for block sparse recovery for distributed MIMO radar systems.In this paper, we propose SIMAT as a generalization of IMAT for simultaneous reconstruction of jointly sparse signals from their missing samples.§.§ Paper Overview The rest of this paper is structured as follows. In Section <ref>, we first provide the description of SSA model. Then the proposed method is introduced and its convergence is analyzed. Numerical experiments of our method in comparison with the SOMP algorithm are presented in Section <ref>. SIMAT is then demonstrated as a simple decoding algorithm for MIMO radar systems, and its performance is compared with BIMAT by means of simulation. Finally, the paper is concluded in Section <ref>.§.§ NotationsScalar variables, vectors, and matrices are denoted by italic lower-case, boldface lower-case, and boldface upper-case, respectively. The elements of a vector are denoted by subscript, i.e., x_i is the i-th element of the vector 𝐱. |𝐱| calculates the absolute value of each entries of the vector 𝐱. The pseudoinverse of matrix 𝐀 is represented by 𝐀^†.Finally, the output of the thresholding operator TH( 𝐱,thr) is defined as a diagonal matrix whose diagonal entries are determined as follows:TH( 𝐱,thr)_ii = {[ 1. | x_i| ≥ thr; 0. | x_i| < thr ].§ THE PROPOSED METHOD §.§ Problem StatementIn this section we provide the formulation of SSA problems. Assume that 𝐱^1, 𝐱^2, ..., 𝐱^L are L equal-length signals that share the same sparsity support in a specific transform domain. These signals are random-sampled in another domain by independent sampling masks 𝐬^1, 𝐬^2, ..., 𝐬^L. These masks are binary and each element of them is generated independently based on a Bernoulli distribution , i.e., 𝐬^i_n∼ Bernoulli(p), 1≤ i ≤ L, and 0≤ p ≤ 1 is the sampling probability. The random-sampled signals 𝐲^1, 𝐲^2, ..., 𝐲^L are derived as follows:y^i_n = s^i_n ×x^i_n The problem is to simultaneously reconstruct the original sparse signals from their random-sampled versions by employing the additional information that the signals share a common support in a specific transform domain.§.§ AlgorithmOne can find the SIMAT algorithm in Algorithm <ref>. Let n , m and k denote the lengths of the original signal, the length of the observed vectors, and the sparsity number, respectively. Moreover, let 𝐱^j_i represent the reconstruction of the j-th signal after i-th iteration. It should be noted that the measurement matrices, i.e., [𝐀^1, 𝐀^2, ..., 𝐀^L], can be calculated by multiplying the transformation matrix, which maps the signals to their sparsity domain, and diagonal matrices whose diagonal entries are equal to the elements of𝐬^1, 𝐬^2, ..., 𝐬^L. This algorithm gradually extracts the sparse components of the signals by thresholding the estimated signals iteratively. Each iteration involves two different steps of thresholding and projection. The thresholding step provides an approximation of the common support of the signals by hard-thresholding the summation of the absolute values of the approximated signals. The projection step projects each of the estimated signals onto the convex set defined by the support vector approximated in the previous step. In this algorithm λis the relaxation parameter and controls the convergence speed. The threshold value is decreased exponentially by βe^ - α( k - 1), where k is the iteration number. The algorithm performance is less dependent on the parameters λ, β,and α; however, these parameters are optimized empirically to achieve a faster convergence. §.§ Analytical DiscussionIn <cit.>, it is proved that the IMAT method converges to the sparset solution of the random sample SMV problem if the threshold value is less than ε^2/k, where k is the sparsity number, and ε is the minimum distance between the subspace induced by the random samples and all the subspaces induced by a specific support which do not intersect with the previously mentioned subspace. Now, we show that the probability of finding the support of signals with the SIMAT is more than that of the IMAT. Before proving this statement, we note the following points.We assume that the original signals are sparse in the time domain, and each non-zero coefficient has a Gaussian distribution. Additionally, the signals are random-sampled and polluted by additive white Gaussian noise in the frequency domain. Since the signals are estimated by line 16 of Algorithm <ref> in each iteration, each coefficient of the estimated signals has a Gaussian distribution.The distribution of the absolute value ofa random variable with Gaussian distribution N(0,σ^2) is Half-Normal with the following Probability Density (PDF) and Cumulative Distribution (CDF):P_Y( y;σ) =√(2)/√(σ ^2×π)exp(- y^2/2 ×σ ^2)y ≥ 0Q_Y( y;σ) = erf( y/√(σ ^2× 2))In the thresholding step of each iteration of our algorithm, the absolute values of the coefficients are calculated. Hence, these variables have Half-Normal distribution. We define the variable z as the absolute value of the estimation of a non-zero coefficient of a signal and assume that it has variance σ ^2_1 and mean μ _1. Similarly, we define the variable w associated with a zero-coefficient of a signal and assume that its variance and mean are σ ^2_0 and μ _0, respectively. We also assume that μ _1 > μ _0 since the variance of the additive noise is small. Due to the fact that the mean of L signals is calculated for the SIMAT, the variables z_L and w_L can be defined as the mean of the variables mentioned above.The variance and the mean of z_L and w_L are σ ^2_1/L, μ _1, σ ^2_0/L and μ _0, as a result of the independency of signals. We prove the superiority of SIMAT over IMAT through Lemma <ref> and Lemma <ref>. As a general idea of these lemmas, according to the law of large numbers, the coefficients of the signal in the SIMAT become nearer to their mean when L, the number of signals, go to infinity. In fact, the none-zero coefficients tend to μ _1 and the zero coefficients tend to μ _0. Therefore, it would be easier to find the non-zero coefficients of the original signals.If the number of signals L satisfies the following inequalityL ≥σ ^2_0/( ε- μ _0)( 1 - erf( ε/√(σ ^2_0 × 2))) > 0,then P( w ≥ε) ≥ P( w_L≥ε), ∀ε≥ 2 ×μ _0, where P is the probability sign. w and w_L are the variables representing the estimated zero coefficients of a signal in each iteration of IMAT and SIMAT, respectively.As a result of ε≥ 2 ×μ _0 and positivity of w_L, one can easily show the following equality: P( w_L≥ε) =P( | w_L - μ _0 | ≥ε- μ _0) According to the Chebyshev's theorem, we have P( | w_L - μ _0| ≥ε- μ _0) ≤σ ^2_0/L/ε- μ _0 If we define Q_w( ε) as the CDF of P( w ), we get P( w ≥ε) = 1 - Q_w( ε) = 1 - erf( ε/√(σ ^2_0 × 2)) Since 1 - erf( ε/√(σ ^2_0 × 2)) ≥ 0 and ( ε- μ _0) ≥ 0, the lemma is proved for the number of signals specified by (<ref>).If the number of signals L satisfies the following inequalityL ≥σ ^2_1/( μ _1 - ε)( erf( ε/√(σ ^2_1 × 2))) > 0,then P( z ≥ε) ≤ P( z_L≥ε), ∀μ _1≥ε≥ 2 ×μ _0, where P is the probability sign, and the estimated non-zero coefficients of a signal in each iteration of IMAT and SIMAT are denoted by z and z_L, respectively.It is not too difficult to derive the following inequality P( z_L≥ε) ≥ 1 -P( | z_L - μ _1| ≥μ _1 - ε) According to the Chebyshev's theorem, we have P( | z_L - μ _1| ≥μ _1 - ε) ≤σ ^2_1/L/μ _1 - ε⇒ P( z_L≥ε) ≥ 1 -σ ^2_1/L/μ _1 - ε If we define Q_z( ε) as the CDF of P(z), then we get P( z ≥ε) = 1 - Q_z( ε) = 1 - erf( ε/√(σ ^2_1 × 2)) Since erf( ε/√(σ ^2_1 × 2)) ≥ 0 and ( μ _1 - ε) ≥ 0 for μ _1≥ε≥ 2 ×μ _0, one can easily prove the lemma for the L indicated in (<ref>) If the number of signals L satisfies the following inequality∀ μ _1 ≥ε≥ 2 ×μ _0,L ≥max(σ ^2_1/( μ _1 - ε)( erf( ε/√(σ ^2_1 × 2))) ,σ ^2_0/( ε- μ _0)( 1 - erf( ε/√(σ ^2_0 × 2)))),then ∀μ _1≥ε≥ 2 ×μ _0, the probability of finding the support with SIMAT is higher than IMAT.According to Lemma <ref>, if L satisfies (<ref>), the probability of mistaking a zero coefficient for a support in SIMAT is less than that of IMAT. According to Lemma <ref>, if L satisfies (<ref>), the probability of finding a none-zero coefficients in SIMAT is higher than IMAT.Therefore, if L satisfies (<ref>), it is more probable to find the support with the SIMAT than with the IMAT. One can find the minimum value of L which satisfies (<ref>) as shown in Table <ref>. This table shows that the minimum number of signals L that guarantees superiority of SIMAT over IMAT is very low, and hence there is no need to have a large number of signals to benefit from SIMAT.§ SIMULATION RESULTSIn each trial of our simulation, we generate L number of K-sparse signals. We choose K components out ofN=256 randomly, and set them to a random number in the interval [-1,1]. Then the noisy signals are random-sampled by a sampling rate of M/N. By the law of algebra, the number of samples needed to specify the sparsity profile of the signals is at least twice the sparsity number, hence K ≤ M/((2× N)). We optimize the parameters of the algorithm in each trial.Table <ref> compares the average reconstruction SNR (dB) of three algorithms, IMAT, SIMAT and SOMP, for different density numbers and input SNRs. As observed in Table <ref>, the simultaneous reconstruction methods outperform IMAT, especially for higher sparsity numbers. In the case of not knowing the density rate, SIMAT yields the best results both in low and high density rate, and in noisier channels. In noiseless channels, the signal can be perfectly reconstructed by SIMAT, when the sparsity number is small. In the case of knowing the density rate, SOMP and SIMAT exhibit similar performances in all the cases. However, based on the results of Table <ref>, SIMAT outperforms SOMP in terms of the complexity measured by the run-time. The success rate of IMAT, SIMAT and SOMP algorithms for different density rates is depicted in Fig. <ref>. A reconstruction is considered to be successful if the output SNR is more than 20dB. As seen in this figure, all curves experience a sudden knee-like fall as the density rate increases. This fall is considered as the boundary between successful and unsuccessful reconstruction. The simulation results reveal that for the SIMAT, success rate falls around 20% density rate. While, the knee-like fall happens in 18% and 12% density rate for IMAT and SOMP, respectively. This indicates that SIMAT can successfully reconstruct the signals with higher sparsity number in comparison with SOMP and IMAT algorithms.Fig. <ref> shows the effect of the number of signals on the performance of SOMP and SIMAT.As observed in this figure, by increasing the number of signals, we get better results. Additionally, the SIMAT algorithm can reconstruct the signals better than the SOMP when the number of signals is small. The reconstruction SNR for different sampling rates is depicted in Fig. <ref>. According to this figure, the SNR values can be improved by increasing the sampling rate. Although the SOMP algorithm has better results for a small interval in the high sampling rates, SIMAT outperforms SOMP in most cases. Table <ref> lists the SNR of two reconstruction methods extended from IMAT. Simulation results demonstrate the superiority of SIMAT over BIMAT in terms of the output SNR and the complexity. Therefore, the SIMAT can be used instead of the BIMAT in many applications such as distributed MIMO radar systems.§ CONCLUSIONIn this paper a novel method, SIMAT, was introduced for SSA problems. The proposed method is an extension of IMAT into the MMV models, and the idea of this extension is that the summation of sparse vectors, sharing the same support, in the thesholding step of each iteration can enhance the probability of reconstruction. Indeed, the theoretical analysis with simulation results prove that SIMAT outperforms IMAT with respect to the SNR metric. We compared the proposed method with SOMP, as a well-known algorithm in the MMV models. 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Juan-Llacer, “Comparison between measurements and simulations of conventional and distributed mimo system,” IEEE Antennas and Wireless Propagation Letters, vol. 7, pp. 546–549, 2008.mohammadi2017sampling E. Mohammadi, A. Fallah, and F. Marvasti, “Sampling and distortion tradeoffs for indirect source retrieval.”1em plus 0.5em minus 0.4emIEEE, 2017, accepted for publication.tropp2007signal J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Transactions on information theory, vol. 53, no. 12, pp. 4655–4666, 2007.determe2017noise J.-F. Determe, J. Louveaux, L. Jacques, and F. Horlin, “On the noise robustness of simultaneous orthogonal matching pursuit,” IEEE transactions on signal processing, vol. 65, no. 4, pp. 864–875, 2017.tropp2006algorithms J. A. Tropp, A. C. Gilbert, and M. J. 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Marvasti, Nonuniform sampling: theory and practice.1em plus 0.5em minus 0.4emSpringer Science & Business Media, 2012.abtahi2016iterative A. Abtahi, M. Azghani, J. Tayefi, and F. Marvasti, “Iterative block-sparse recovery method for distributed mimo radar,” in Communication and Information Theory (IWCIT), 2016 Iran Workshop on.1em plus 0.5em minus 0.4emIEEE, 2016, pp. 1–4.esmaeili2016iterative A. Esmaeili, E. Asadi, and F. Marvasti, “Iterative null-space projection method with adaptive thresholding in sparse signal recovery and matrix completion,” arXiv preprint arXiv:1610.00287, 2016. | http://arxiv.org/abs/1707.08310v2 | {
"authors": [
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"Shahrzad Kiani",
"Mahdi Boloursaz",
"Farokh Marvasti"
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"published": "20170726074607",
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μ ^∘ .3ex>-14mu0.6ex∼ .3ex<-14mu0.6ex∼ μ_p G_E^p / G_M^p G_E^p / G_M^p ≥G_E G_M G_E^p G_M^p G_E^n G_M^n σ_R et al.maxmax cutGeV | http://arxiv.org/abs/1707.09063v2 | {
"authors": [
"Zhihong Ye",
"John Arrington",
"Richard J. Hill",
"Gabriel Lee"
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"published": "20170727223154",
"title": "Proton and neutron electromagnetic form factors and uncertainties"
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Department of Physics, Princeton University, Princeton, New Jersey 08544, USAWe use microwave radiation generated by a semiconductor double quantum dot (DQD) micromaser for charge state detection. A cavity is populated with n_c ∼ 6000 photons by driving a current through an emitter DQD. These photons are used to sense the charge state of a target DQD that is located at the opposite end of the cavity. Charge dynamics in the target DQD influence the output power and emission frequency of the maser. Three different readout mechanisms are compared. The detection scheme requires no cavity input field and may potentially be used to improve the scalability of semiconductor and superconducting qubit readout technologies. 73.21.La, 73.23.Hk, 84.40.lk On-Chip Quantum Dot Light Source for Quantum State Readout J. R. Petta December 30, 2023 ==========================================================§ INTRODUCTIONQuantum state readout is a crucial component of any quantum computing architecture. For semiconductor quantum dots, charge state readout has been performed using quantum point contacts <cit.> and quantum dots <cit.> as detectors. Electronic spin states can also be resolved using spin-to-charge conversion, which relies on spin selective tunneling and sensitive charge state detection<cit.>. To increase measurement bandwidths, conventional dc transport measurement approaches have to a large extent been replaced by rf and microwave frequency reflectometry setups <cit.>. In particular, the circuit quantum electrodynamics (cQED) architecture allows for dispersive readout of superconducting qubits <cit.>, as well as semiconductor charge and spin qubits <cit.>.Both rf-reflectometry and cQED measurement implementations rely on costly room temperature microwave sources, rf components, and coaxial lines that occupy a significant amount of space in a dilution refrigerator. As one scales to many qubits, the resource requirements will increase dramatically. Moreover, to suppress the room temperature microwave background, a typical attenuation of 60–70 dB is required in the coax lines connecting the signal generator to the quantum device (∼ 10 mK). To reduce the experimental complexity, the source would ideally be isolated from the 300 K environment. Over the past 10 years it has been shown that a variety of voltage-biased quantum devices generate microwave frequency photons. For example, voltage-biased Cooper pair boxes and superconducting quantum intereference devices embedded in superconducting cavities have been shown to mase <cit.>. Cavity-coupled semiconductor double quantum dots (DQD) can serve as an electrically tunable maser gain medium <cit.>. These devices are fabrication compatible with other qubits and they can be integrated on the same chip.It is therefore of interest to determine if these devices, which already operate at millikelvin temperatures, can be utilized as microwave frequency sources in quantum computing experiments <cit.>.In this Letter we show that microwave frequency photons generated by a cavity-coupled DQD can be used to sensitively readout the charge state of a second DQD that is located in the same cavity. A source-drain bias is applied across an emitter DQD (e-DQD) and results in above-threshold masing. The photons generated by the e-DQD are used to sense a target DQD (t-DQD) that is located in the same cavity. Charge dynamics in the t-DQD influence the maser emission, changing its output power and emission frequency, allowing for charge state readout.We explore three different readout approaches. In the first approach the total power emitted by the cavity is measured and used to reconstruct the t-DQD charge stability diagram. In the second approach, the e-DQD emission frequency, which is dependent on the charge state of the t-DQD, is used to measure the t-DQD charge stability diagram by mapping out the maser emission frequency as a function of the t-DQD gate voltages. In the third approach, we measure the power emitted into a narrow band centered around the free-running maser frequency. Shifts in the emission frequency significantly change the narrow-band power, allowing us to detect t-DQD interdot charge transitions. While the target qubit in these experiments is a semiconductor DQD, it could in principle be replaced with any other cavity-coupled qubit, such as a superconducting transmon qubit. No external cavity drive is required in our experiments, further supporting the use of a DQD maser as a light source for quantum state readout in quantum computation architectures.§ DEVICE DESCRIPTION Figure 1(a) captures the important elements of our experimental setup. The e-DQD and t-DQD are coupled to a microwave cavity. To serve as a microwave source, the e-DQD is source-drain biased, which results in single electron tunneling events and microwave frequency photon emission into the cavity mode <cit.>. These cavity photons are used to sense the charge state of the t-DQD.An optical micrograph of the device is shown in Fig. <ref>(b). The cavity consists of a λ/2 superconducting resonator with resonance frequency f_ c = 7596 MHz and linewidth κ_ tot/2π = 1.77 MHz. The cavity is designed to have output (input) coupling rates κ_ out/2π = 0.8 MHz (κ_ in/2π = 0.04 MHz). The InAs nanowire DQDs are located at opposite ends of the cavity near the voltage anti-nodes. The confinement potential of the t-DQD is created by voltage biasing the five bottom gates labeled as B^T_L, L^T, C^T, R^T, and B^T_R in Fig. 1(c). Independent control of the e-DQD is achieved using a separate set of bottom gates. We further define source and drain ohmic contacts using electron beam lithography. In contrast with our previous work, the source contacts to the e-DQD and t-DQD are electrically decoupled such that the source-drain bias voltages can be independently controlled <cit.>. The drain electrode of each DQD is connected to the microwave resonator. Coupling between a charge trapped in the DQD confinement potential and the electric field of the cavity leads to an electric dipole interaction with strength g_c/2π≈ 30 – 40 MHz <cit.>.§ CHARACTERIZATION OF THE EMITTER DOUBLE QUANTUM DOT We first characterize the microwave field emitted by the e-DQD. For these measurements, the t-DQD source drain bias V^ T_ SD = 0 and the gates are tuned such that the t-DQD is in Coulomb blockade, where the charge number is fixed. In this configuration the t-DQD will not have any impact on the cavity field. The e-DQD is configured to emit photons by applying a finite source drain bias V^ E_ SD = 2 mV, which results in single electron tunneling events. The interdot charge transition leads to photon emission and, in a high quality factor cavity, a transition to a masing state <cit.>. To measure the emitted radiation, the cavity output field is amplified using a high electron mobility transistor (HEMT) amplifier and detected with a spectrum analyzer. Figure <ref>(a) shows the power spectral density S(f) of the radiation emitted from the cavity, along with a fit to a Lorentzian. The best fit parameters yield the emission frequencyf_ e = 7595.68 MHz and FWHM = 8 kHz. We obtain a total output power P_ out = 0.16 pW by integrating over S(f). The emission power yields an intra-cavity photon number n_ c = P_ out/(hf_eκ_ out)≈ 6000 given κ_ out/2π = 0.8 MHz. The FWHM is 200 times narrower than the bare cavity linewidth, which is suggestive of masing.The output field can be examined in more detail by measuring (I,Q) histograms. To acquire the histograms, the cavity output field is first amplified with a HEMT and then demodulated into the in-phase (I) and quadrature-phase (Q) components by a local reference set to a frequency f_ lo = f_ c <cit.>. Figure <ref>(b) shows an (I,Q) histogram obtained by accumulating 1.7×10^7(I,Q) samples at a rate of 12.3 MHz. The histogram has a ring shape that is consistent with coherent emission <cit.>. Combined, these data sets show that the voltage-biased e-DQD can serve as a coherent source that populates the cavity with approximately 6000 photons. These photons may be used to read out the charge state of the t-DQD, as will be demonstrated in the following sections of the paper.§ TARGET DOUBLE QUANTUM DOT CHARGE STATE DETECTION In this section we compare several different approaches for measuring the charge stability diagram of the t-DQD. We first measure the stability diagram using standard cavity input-output readout, where an external tone is used to populate the cavity with photons. These data are then compared with charge stability diagrams that are obtained by measuring the total power emitted from the cavity when it is populated with e-DQD photons. Two additional transduction methods are examined that are based on the effect that charge dynamics in the t-DQD have on the emission properties of the e-DQD. Specifically, we show that the t-DQD charge stability diagram can be reconstructed by measuring the emission frequency of the e-DQD and the narrow band power emitted by the e-DQD. §.§ Charge state readout through measurements of the cavity transmission The conventional cavity input-output readout approach is illustrated in Fig. <ref>(a). Here the cavity is driven by an input tone of frequency f_ in and power P_ in≈ -112 dBm that results in approximately n_ c≈10 intra-cavity photons. The resulting cavity output is amplified with a HEMT and demodulated by a local reference having a frequency f_ lo = f_ in. Both the phase shift Δϕ and power gain G = CP_ out/P_ in can be extracted from the cavity transmission. Here the constant C is set such that G = 1 with f_ in = f_ c and both DQDs in Coulomb blockade <cit.>. Figure <ref>(c) plots G as a function of the t-DQD gate voltages with f_ in = f_ c and V^ T_ SD = 0. For this data set the e-DQD is in idle mode, with V^ E_ SD = 0 and the gate voltages tuned to Coulomb blockade. These measurements reveal the t-DQD charge stability diagram, consistent with previous measurements of cavity-coupled InAs nanowire DQDs <cit.>. §.§ Charge state readout through measurements of the total cavity output power To make a comparison with cavity input-output readout we now turn off the cavity input tone and configure the e-DQD in the “on state," such that it is emitting coherent radiation as shown in Fig. 2. We then measure the output power P_ out and plot it as a function of V^ T_ L and V^ T_ R in Fig. <ref>(d). Writing the cavity output field complex amplitude as α = I + iQ, P_ out is determined from measurements of ⟨α^*α⟩ = ⟨ I^2+Q^2⟩. The (I,Q) data are processed using a digital filter of 2.6 MHz bandwidth that covers the entire cavity linewidth and therefore ⟨ I^2+Q^2⟩ captures the total emitted power <cit.>. The scenario is equivalent to a power meter measuring over a wide bandwidth as illustrated in Fig. <ref>(b). The data in Fig. <ref>(d) show that measurements of P_ out can be used to extract the t-DQD charge stability diagram. §.§ Impact of charge dynamics in the t-DQD on the emission properties of the e-DQD We now more carefully examine the readout mechanism by studying the effect that the t-DQD charge configuration has on the emission properties of the e-DQD. Figure <ref>(a) shows a high resolution measurement of P_ out near one of the t-DQD interdot charge transitions in Fig. <ref>(d). These data were acquired in the absence of a cavity input tone and with the e-DQD emitting photons. The left dot and right dot charge transitions are visible in these data, while the visibility of the interdot charge transition is significantly less than in the data shown in Fig. 3(c). To better understand what sets the visibility of the charge transitions in these data, we measure S(f) of the emitted radiation with the gate voltages of the t-DQD tuned to different regions of the t-DQD charge stability diagram. Figure <ref>(b) shows measurements of S(f) with the t-DQD configured in Coulomb blockade, at the interdot charge transition, and at a left dot charge transition. With the t-DQD configured in Coulomb blockade the emission peak in S(f) is centered at f_ e^0 = 7595.68 MHz. When the t-DQD is configured to a left dot charge transition, the emission peak shifts down in frequency by 214 kHz, the peak power is reduced by approximately a factor of 10, and the peak in S(f) is significantly broader. In comparison, with the t-DQD configured at the interdot charge transition the emission peak is only shifted down in frequency by 37 kHz. The emission peak has a height and width that is comparable to the data acquired with the t-DQD in Coulomb blockade. Therefore, it is difficult to resolve the interdot charge transitions in measurements of the total emitted power P_ out. However, since the emission peak shifts by an amount that is much greater than the FWHM ∼ 8 kHz of the emission peak, measurements of the emission frequency may be used to reconstruct the t-DQD charge stability diagram.By fitting S(f) to a Lorentzian at every point in the t-DQD charge stability diagram we can extract f_ e as a function of V^ T_ L and V^ T_ R. A plot of the extracted f_e is shown in Fig. <ref>(c) and is much more sensitive to the interdot charge transition. Therefore a measurement of f_ e can in principle be used to readout the device. The approach is similar to cQED readout of transmon qubits, where the state-dependent dispersive shift of the cavity is used for readout <cit.>.It is important to note here that in general we do not know the phase of the maser emission, and that previous work showed that the coherence time of the maser is only on the order of 10 μs <cit.>. Even with a long coherence time, f_e = f_e(V^ T_ L, V^ T_ R) is t-DQD dependent and thus the phase shift in the maser output Δϕ(t) = ∫Δ f_e(t) dt where f_e(t) =f_e(V^ T_ L(t), V^ T_ R(t)) is dependent on the “path" of (V^ T_ L(t), V^ T_ R(t)). Δϕ is then not a well defined variable. Therefore we cannot simply measure the dispersive shift Δϕ, as is commonly achieved with phase-sensitive measurement approaches in cQED. §.§ Charge state readout through narrow-band measurements of the total cavity output power The previous section demonstrated that measurements of the emission frequency f_ e can be used to reconstruct the t-DQD charge stability diagram. However, extracting f_ e from measurements of S(f) is too time consuming (3–4 seconds per spectrum) to allow for efficient charge state readout. The challenge of devising a practical measurement that quickly extracts the state-dependent frequency shift has been solved in the standard readout schemes. For example, in cQED systems, state dependent shifts in the resonance frequency of the cavity can be measured by driving the cavity with a weak input tone at f_ in = f_ c and detecting the large phase shift Δϕ = arctan(I,Q) of the cavity output field using heterodyne demodulation techniques. As a demonstration of the standard readout approach, Fig. <ref>(a) plots the phase shift Δϕ as a function of V^ T_ L and V^ T_ R. Single dot transitions associated with the left and right dots, as well as the interdot charge transition, are clearly visible in the phase response.Phase readout is not feasible when e-DQD emission is used to populate the cavity with photons since f_ e exhibits fluctuations that randomize the phase. Moreover, since f_ e is a quantity that depends on the t-DQD configuration the phase shift is not a well-defined quantity.Instead, a quantity analogous to the phase shift can be measured for fast readout. The emission spectrum [Fig. 4(b)] shifts in response to the charge state of the t-DQD, allowing us to simply measure the output power P_ out within a narrow resolution bandwidth (RBW), as schematically illustrated in Fig. <ref>(b). The frequency range over which the power is integrated f_ e^0 ± RBW/2 should be smaller than the expected state-dependent shift in f_ e, yet large enough to tolerate the drift in f_ e caused by charge fluctuations in the emitter <cit.>. We operate with RBW > FWHM of the e-DQD emission spectrum to tolerate the drift in f_ e, and RBW <|f_ e-f_ e^0| to allow sensitivity to changes in the emission spectrum due to the t-DQD charge state. Figure <ref> (b) shows the output power P_ out measured around f_ e^0 = 7595.68 MHz with a 30 kHz RBW. The state-dependent shift in the emission center frequency at a t-DQD interdot charge transition leads to a factor of 100 change in P_ out within the measured bandwidth.§ SUMMARY AND OUTLOOK In summary, we have shown that a voltage-biased DQD can be used as a light source for qubit readout in the cQED architecture. Readout based on measurements of the total output power, emission center frequency, and narrow-band output power were compared. While the total output power is sensitive to single dot charge transitions, it does not have sufficient sensitivity to resolve interdot charge transitions. Measurements of the emission center frequency reveal both single dot and interdot charge transitions, but this approach is slow and not well-suited for single shot readout. The narrow-band power measurement approach yields high sensitivity to both single dot and interdot charge transitions. In some applications, it may be desirable to place the e-DQD in a separate cavity. In the masing state, the e-DQD generates a large intra-cavity photon number n_ c∼ 6000, which may cause saturation effects and broaden the linewidth of the target transition. Separating the emitter from the target qubit would more easily allow the emitted field to be attenuated. Lastly, previous work has shown that the maser can be switched on and off rapidly <cit.>. A switchable maser could be turned off during quantum control sequences and then rapidly activated for high power readout of the qubit state <cit.>. We hope that this study will motivate further applications of nanoscale emitters in quantum computing readout architectures.We thank M. J. Gullans and J. M. Taylor for helpful discussions and acknowledge support from the Packard Foundation, the National Science Foundation Grant No. DMR-1409556, and the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4535. Devices were fabricated in the Princeton University Quantum Device Nanofabrication Laboratory.apsrev_lyy2017 | http://arxiv.org/abs/1707.08906v1 | {
"authors": [
"Y. -Y. Liu",
"J. Stehlik",
"X. Mi",
"T. Hartke",
"J. R. Petta"
],
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"published": "20170727152140",
"title": "On-Chip Quantum Dot Light Source for Quantum State Readout"
} |
label1,label2]B. Blank[1],label1]G. Canchel, label1]F. Seis[2], label3]P. Delahaye [label1]Centre d'Etudes Nucléaires de Bordeaux Gradignan, 19 Chemin du Solarium, CS10120, F-33175 Gradignan Cedex, France[label2]ISOLDE/CERN, EP Department, CH-1211 Geneve 23, Switzerland [label3]Grand Accélérateur National d'Ions Lourds, Bd Henri Becquerel, BP 55027,14076 CAEN Cedex 05, France Calculated fusion-evaporation cross sections from five different codes are compared to experimental data. The present comparison extents over a large range of nuclei and isotopic chains to investigate the evolution of experimental and calculated cross sections. All models more or less overestimate the experimental cross sections. We found reasonable agreement by using the geometrical average of the five model calculations and dividing the average by a factor of 11.2. More refined analyses are made for example for the ^100Sn region.fusion-evaporation reactions comparison experiment - calculations[1]e-mail address: [email protected] [2]Summer student at CENBG § INTRODUCTION On Earth, 255 stable nuclides are available for nuclear physics studies. In addition, 31 quasi stable nuclides having a half-life comparable to or longer than the age of the Earth exist. All other nuclei must be created in order to be usable for experimental studies. Different types of nuclear reactions exist to produce these unstable and radioactive nuclei. Two methods can be used to create basically all bound or quasi bound (i.e. bound for ashort laps of time) nuclei: spallation or fragmentation. Spallation reactions are usually induced by light particles (protons or neutrons)on heavier stable nuclei. In these spallation reactions, the incident light projectile ejects nucleons from the target nucleus by nucleon-nucleoncollisions and the excited fragment (often called pre-fragment) evaporates light particles (protons, neutrons, α particles) to get rid of excitation energy. With e.g. incident proton energies of a few hundred MeV up to 1 or 2 GeV, basically all nuclei, bound or quasi bound, but lighter than the target nucleus itself, can be produced. However, as these spallation reactions are basically always "thick-target" reactions, the reaction products have to diffuse out of the target to become useful. As this takes some time and depends very sensitively on the chemistry of the element of interest, short-lived nuclides of condensable elements are very difficult to produce by this means.Fragmentation reactions employ heavy-ion induced reactions on different heavy-ion targets. Therefore, target as well as projectile fragmentation can be used. Target fragmentation suffers from the same problem as spallation reactions: the products have to diffuse from the target itself. Therefore, this process is again limited to relatively volatile isotopes with sufficiently long half-lives. In projectile fragmentation reactions, one can use "thin targets" which allows the products to recoil out of the target due to the incident projectile energy. This approach is basically universal and allows all nuclides to be produced. However, there are at least two drawbacks of projectile fragmentation: i) it needs high-energy heavy-ion accelerators and ii) the beam quality of these fragment beams is rather bad.In deep-inelastic or transfer reactions, two heavy nuclei interact with each other at energies around the Fermi energy (typically 20-60 MeV/A)and nucleons are transferred from one nucleus to the other producing thus more or less neutron-rich or neutron-deficient isotopes. However,as the number of nucleons transferred is limited, only nuclei relatively close to stability can be produced.In nuclear fission, a very heavy nucleus, e.g. ^238U or ^252Cf, fissions by creating two medium-mass nuclides. This fission process can be induced (e.g. by proton, neutron or γ-ray impact) or spontaneous. Due to the curvature of the nuclear valley of stability, the heavy fissioning nuclei have always an excess of neutrons compared to lighter nuclei. Therefore, nuclear fission always produces neutron-rich isotopes in the mass range of A ≈ 50 - 170.Finally, neutron-deficient nuclides can be produced by fusing two lighter nuclei. In this case, the situation is reversed compared to fission. The light stable nuclei that interact are proton-rich compared to the heavier nuclei in the valley of stability. For example, the reaction ofa stable ^40Ca nucleus with a stable ^58Ni nucleus produces as the compound nucleus, i.e. the sum of all nucleons, ^98Cd, a nucleus which is 8 neutrons more neutron-deficient than the most neutron-deficient stable isotope of the element cadmium. From this list of possible reactions, it is evident that the experimenter has some choice to use the reaction best suited for the production of the nucleus of interest. However, evidently this choice depends also strongly on the accelerator available, the separation possibilities and much more. For each type of reaction, parameters like the reaction partners and the incident energy have to be optimized in order to achieve the highest production rates of the isotope of interest. For spallation, fragmentation, and deep-inelastic reactions, it is most often advantageous to use a stable nucleus close to the desired final nucleus to enhance the production rate. This choice basically does not exist of fission because only a few quasi stable fissioning nuclei exist. For these reactions, analytical codes have been developed which have a rather good predictive power for the reaction cross sections. Let us mention the EPAX code <cit.> for projectile fragmentation, the SPACS code <cit.> for spallation reactions or the GRAZING model <cit.> for deep-inelastic reactions. The ABRABLA <cit.> code deals with fission, fragmentation, and spallation.For fusion-evaporation, the situation is different in the sense that all nuclei can be produced with different combinations of projectile, target andincident energy. Therefore, an optimization of these three parameters is needed for any nucleus to be produced. To do so, different codes are available,some of them being analytical, others being of the Monte-Carlo type. In the present work, we have used five codes to calculate fusion-evaporation cross sections: CASCADE <cit.>, HIVAP <cit.>, POTFUS+ABLA called CNABLA <cit.>, PACE <cit.>, andPOTFUS+GEMINI++ <cit.>. All codes have advantages and draw-backs and we could not decide a priori which code would perform betterover a wide range of nuclei.The original reason for the present work was to determine production rates for SPIRAL2 where fusion-evaporation reactions were foreseen as a tool to produce neutron-deficient isotopes from mass 20 or so to the heaviest nuclei in the super-heavy element region by means of a target - ion-source ensemble in the production building. However, due to financial constraints, the construction of the SPIRAL2 production building was put on hold. The same work was used in the mean time to predict production rates for the S^3 separator <cit.> or at other facilities.For this purpose, we performed a literature research of all fusion-evaporation reactions used to produce proton-rich nuclei. Using the projectile-target combination and the energy given in the literature, cross sections were calculated with the five codes. To predict SPIRAL2 production rates, the in-target yields were determined using the predicted primary-beam intensities and the extracted yields were obtained by means of release functions found in the literature. In this way production rates could be predicted for more than 700 proton-rich nuclei.In order to evaluate the performance of the fusion-evaporation codes and the quality of the production rate predictions, we have performedcalculations with all five codes and compared the results to either fusion-evaporation cross sections found in the literature or to productionrates from the GSI on-line separator <cit.>. The former values constitute a more direct comparison, however, in most casesthe authors had to use transmissions of their separators which contain quite some uncertainties. For the second data, release efficienciesare needed in order to determine in-target production rates and thus production cross sections. To compare our cross-section calculations with these values we will use release data collected in the frame work of the SPIRAL2 facility <cit.> where these release functions are needed for fusion as well as fission products.The purpose of the present paper is to describe the results of this comparison between calculated cross sections or production rates and experimental data for fusion-evaporation reactions. The general outcome is that the different codes overestimate the experimental data by about a factor of 10. Therefore,for e.g. planning an experiment using a fusion-evaporation reaction, the predictions deduced from calculations using fusion-evaporation codes should be reduced by this factor in order to obtain a realistic estimate of the production rates to be expected.§ EXPERIMENTAL DATA In this section, we summarize the experimental data used for the comparison with the theoretical predictions. Table <ref> gives the experimental cross sections used in the present work [The authors are eager to increase the present data base of experimental cross sections and encourage readers to communicate other experimental fusion-evaporation cross sections to us.].In general, relatively few fusion-evaporation cross sections are found in the literature and those found have often large error bars or, even worse, no uncertainties at all. This is to a large part due to the fact that the cross sections are often determined at ISOL facilities whereinformation of effusion and diffusion is scarce and induce large uncertainties. Other cross sections are determined by means of massseparators or velocity filters where the transmissions are not well known. Another problem with a comparison of experimental cross sections and calculated values is that it is often not clear whether the beam energy givenis the one at the entrance or in the center of the target. We always use the energy given in the paper for the calculations.If the energy is the one at the target entrance and thus too high compared to the energy in the center of the target, we believe this in not a problem.The maximum of the cross sections is reached at a certain incident energy. At higher energies, the cross sections fall off slowly, whereas at lowerenergies there is a threshold effect to overcome the Coulomb repulsion which makes that the cross sections fall off much faster on the low-energy side.Therefore, taking in some cases a slightly higher beam energy is somehow on the "safe" side.Experimental production rates can be found in a number of publications from the former GSI on-line separator <cit.>. They are summarized in table <ref>.§ SIMULATION CODES In this section, we give a short overview of the fusion-evaporation codes used to calculate the theoretical cross sections. In total,five codes were used: i) CASCADE, ii) HIVAP, iii) CNABLA, iv) PACE, and v) GEMINI++.These codes use a two-step scenario for the reaction: projectile and target nuclei completely fuse and then decay according to a statistical model approach of compound nucleus reactions. They take into account competition between different decay channels like proton, neutron, and α emission as well asγ decay and fission. All codes give a variety of decay information like the particles emitted, their energy and angular distribution etc. In the present work, we only use the production cross section of the isotope of interest. All programs used the Atomic Mass Evaluation data base from2012 <cit.>. §.§ The CASCADE code The program CASCADE was originally written by F. Pühlhofer <cit.>. The original version of the program was modified by different persons (e.g. E.F. Garman, F. Zwarts and M.N. Harakeh) to perform calculations for specialstates of good spin and parity, to include isospin and parity properly in the statistical decay as well as to include the electric quadrupole decay.CASCADE is an analytic program which is quite fast and thus convenient to optimize projectile-target combinations and the beam energy. In the present work, we use a version of CASCADE provided by D. R. Chakrabarty <cit.>. §.§ The HIVAP code HIVAP is a statistical evaporation code written by W. Reisdorf <cit.>. Several improvements were introduced later <cit.>. We used a version provided to us by F. Hessberger <cit.>. Like CASCADE, HIVAP is an analytical program being thus very fast. §.§ The CNABLA code CNABLA is a program which combines the POTFUS fusion code <cit.> for the first step of the reaction with the ABLA part from the ABRABLA code <cit.> for the evaporation. POTFUS is a quite successfully used fusion code and allows us to prepare an input file with a predefined number of events with four parameters: the mass and the charge of the complete-fusion product, its excitation energy and its spin. These events are then used with a special version of ABRABLA <cit.> to perform the evaporation part by means of a Monte-Carlo technique. §.§ The PACE code PACE is probably the most widely used fusion-evaporation code. It was originally written by A. Gavron <cit.>. This Projection Angular-momentum Coupled Evaporation (PACE) code is again based on the statistical model and uses the Monte-Carlo approach for the de-excitation of the compound nucleus. Only the equilibrium part of the decay is treated, no pre-equilibrium emission is considered. §.§ The GEMINI++ code The GEMINI++ code <cit.> is the C++ version of the original GEMINI code <cit.> written by R.J. Charity. In addition to light particle emission and symmetric fission,it allows for all binary decays to occur. This new version cures problems with heavier systems in the original code. The complete fusion compound nuclei are again produced by the POTFUScode <cit.> and read into GEMINI++ where a Monte-Carlo procedure is used to perform the de-excitation step.§.§ Averages from calculations In order to compare the experimental results to the theoretical predictions from the five codes, some averaging of the calculations is needed. This task is not so easy because the calculations can differ by one or two orders of magnitude from one code to another. A standard average would favor the larger cross sections (e.g. the average of 1 mb and 100 mb being about 50 mb). Therefore, we decided to use thegeometrical average yielding for the example above an average of 10 mb. As the uncertainty range we used the maximum and minimum value from all codes.In general, not all codes give results for all isotopes or projectile, target, and energy combinations. The average is therefore made with the results available. § RESULTS AND DISCUSSION §.§ Comparison with experimental cross sections Figure <ref> gives an overview of all experimental data compared with the results of the individual codes and the averages of these calculations as explained above. As indicated on the figure <ref>a to get the best match between the average of the simulations and the experimental data, we had toreduce the results of the calculations by a scale factor of 11.2. The parameter called deviation is a measure for the scatter of the calculated cross sections,after scaling,around the experimental ones. Again due to large differences between the calculated values from different models, we used a logarithmic differencedefined as:deviation = 10^**[1/n ∑_n abs(log_10(σ_cal/sf/σ_exp))]where n is the number of data points, σ_cal and σ_exp are the calculated and the experimental cross sections, respectively, and sf is the scale factor mentioned above. Therefore, this deviation is the average factor by which the calculations deviate from the experimental value: the smaller this value is, the better the model calculation, once scaled by a constant factor, agrees with experimental data.From figure <ref>a, we conclude that the average of the five model calculations corrected by a scale factor of 11.2 deviate onaverage by a factor of close to 5 for individual value. As can be seen from the left-hand side of the figure <ref>, the scatter between the models and the experimental data is much better for lighter nuclei and gets worse when moving to heavier nuclei. The right-hand side of figure <ref> gives an analysis of the results as a function of the model used to calculate the cross sections. From a first glance, it seems that PACE is the best model, because the scale factor is the smallest of all. However, the scatter of the data is the largest of all models. Overall we believe that the GEMINI++ model coupled to the POTFUS fusion program gives the most convincing answer for fusion-evaporation cross sections. As in the other cases, the agreement is better for the low-mass region (A < 90) with a scale factor of 3.4 and a deviation parameter of 1.9 and for the medium mass region (90 < A < 130) with values of 2.8 and 1.9.Interestingly the models which need a large scale factor to match the experimental data, CASCADE and CNABLA, have a reasonably small scatter of the data. This is in particular true for the CASCADE model. HIVAP has a reasonably small scale factor but a very large scatter of the data.In the ^100Sn region, a lot of experiments have been performed and experimental cross sections determined, notably atthe former GSI on-lineseparator <cit.>. Therefore, this region allows for a detailed comparison of experimental data and calculations. If we use the overall scale factor of 11.2, we obtain a rather good match between experimental data and calculations for the most exotic nuclei (see figure <ref>). However, closer to stability the experimental data are underestimated by the thus scaled calculations. This statement is valid for all elements from silver (Z=47) to barium (Z=56).An interesting question is certainly, which model predicts best cross sections in the ^100Sn region. If we compare the large body of experimental data from A=94 to A=117 to the different models, we get scale factors and deviations as given in Table <ref>. In this region, HIVAP and POTFUS+GEMINI++ give the best results with small scale factors and small deviations.§.§ Comparison with on-line production rates Another possibility to compare predictions and experimental rates is to use production ratesachieved in experiments and compare them to calculated rates. This comparison is possiblewith production rates published from the former GSI on-line separator (see Table <ref>). However, in such a comparison the uncertainties are expected to be even larger because, inorder to calculate these rates, one has to make assumptions about release and ionizationefficiencies. This is a rather difficult task, because it involves a lot of chemistry and the on-line rates are known to fluctuate from one run to the other dueto often apparently minor differences of the experimental conditions of differentexperiments. Nevertheless, we have attempted to predict production rates for the future SPIRAL2 facilityat GANIL, be it for neutron-induced fission of ^238U or fusion-evaporation reactionsfor proton-rich nuclei <cit.>. For this purpose, we have collectedexperimental parameters of two types: (i) empirical parameterizations of the releasefractions based on measured data at different facilities or (ii) parameters from diffusion andeffusion laws which then allow the determination of the total release efficiency, as wasestablished by Kirchner et al. studying the performances of the UNILAC target ion sourcesystems <cit.>. The latter approach has been used for the present study, where thediffusion and effusion coefficients were mostly obtained from measurements atUNILAC <cit.>, CERN and Dubna <cit.>. Because of a lack of data inthe case of Ga and In, we used diffusion coefficients of the neighboring Ge and Snelements, respectively. The FEBIAD ionization efficiencies were estimated fromefficiencies measured at ISOLDE for rare gases <cit.>. For the metallic elements ofinterest, an interpolation in mass gives results which are compatible with the order ofmagnitude of the efficiencies quoted by Kirchner for UNILAC (30 - 50 % <cit.>).Figure <ref> shows the results of this comparison. The in-target yields wereestimated from the cross-section averages as described in the previous section. The extractedyields are in-target yields multiplied by the diffusion, effusion and ionization efficiencies,and have to be compared to the experimental production rates measured at UNILAC. As in the case ofthe production cross sections, the production rates also scatter a lot. However, with the crosssection scale factor for the low-mass region of 7.3 (figure <ref>a) and 4.6 for themass A=100 region (figures <ref>b-e), we reach a reasonable agreement whichseems to indicate that a reduction of the calculated cross section is also needed for thiscomparison.We note that some of the less exotic isotopes have not been produced in ideal conditions, but experimenters set their apparatus for a short while on these nuclei to start their experiment. As for these nuclei the beam energy was therefore certainly not optimized, the simulation codes may have even larger deficiencies.§.§ Excitation function of fusion-evaporation cross sections As mentioned above the body of experimental data for production cross sections is quite scarce. This is even worse in terms of excitation functions where the production cross sections are measured as a function of the energy of the incident beam. We have found one example where sufficient data are available to make a meaningful comparison. In the Bi - Po region <cit.>, a few cross sections have beenmeasured as a function of the incident beam energy, however, only over a short range. In figure <ref>, we compare this excitation function to the different models used in the present work. Interestingly, if we exclude the CNABLA model for the two A=200 nuclei, the maximum of the calculated values is rather close for the different models. It is difficult to say whether the experimental trend is reproduced by the model predictions. For such a statement, more data over a wider range of energies would be needed. The figure also evidences that, in case of doubt, a slightly higher energy is more convenient to move away from the threshold effect at low energies. § SUMMARY We have performed a detailed study of fusion-evaporation cross sections and production rates.Our first finding was that there is a rather limited number of experimental data available in the literature. In addition, these data are most likely subject to large uncertainties keeping in mind that for most of these data no experimental error bars are given in the literature. Therefore, in order to improve the basis for this kind of studies, experimenters need to make efforts to extract cross sections or production rates with experimental uncertainties.We found that all codes that we tested over-estimate the experimental production cross sections or rates with factors of 4 or more. 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Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, FranceInstitut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France We show that in a central nucleus-nucleus collision, the variation of the mean transverse mass with the multiplicity isdetermined, up to arescaling, by the variation of the energy over entropy ratio as a function of the entropy density, thus providing a direct link between experimental data and the equation of state.Each colliding energy thus probes the equation of state at an effective entropydensity, whose approximate value is 19 fm^-3 for Au+Au collisions at 200 GeV and 41 fm^-3 for Pb+Pb collisions at2.76 TeV, corresponding to temperatures of 227 MeV and 279 MeV if the equation of state is taken from lattice calculations. The relative change of the mean transverse mass as a function of thecolliding energy gives a direct measure of the pressure over energy density ratio P/ϵ, at the corresponding effective density.Using RHIC and LHC data, we obtain P/ϵ=0.21± 0.10, in agreement with the lattice value P/ϵ=0.23 in the corresponding temperature range.Measurements over a wide range of colliding energies using a single detector with good particle identification would help reducing the error.Constraining the equation of state with identifiedparticle spectra Jean-Yves Ollitrault December 30, 2023 ====================================================================== § INTRODUCTION One of the motivations for studyingnucleus-nucleus collisions at high energies is to probe experimentally the equation of state of QCD matter <cit.>.Ultrarelativistic collisions probe the phase diagram at vanishing chemical potential: at high temperatures, hadrons merge into a quark-gluon plasma.It was originally hoped that this change occurred through a first-order phase transition <cit.>. However, it was progressively understood that it is a smooth, analyticcrossover <cit.>, and that a phase transition, if any <cit.>, can only take place at high baryon density <cit.>.The equation of state of baryonless QCD matter is now knownprecisely from lattice simulations withphysical quark masses <cit.>.The goal of this paper is to understand the imprints of the equation of state on heavy-ion data, in particular transverse momentum spectra. Relativistic hydrodynamics <cit.> plays a central role in our understanding of heavy-ion observables in the soft sector. Its simplest version is ideal hydrodynamics <cit.>, which describes most of the qualitative features seen in transverse momentum spectra, elliptic flow, and interferometry radii <cit.>. This simple description can be refined by taking into account finite-size corrections due to viscosity <cit.> which are important for azimuthal anisotropies <cit.>.The equation of state lies at the core of the hydrodynamic description, and the vast majority of modern hydrodynamic calculations <cit.>, which give a satisfactory description of soft observables,use as an input an equation of state from lattice QCD calculations. While the success of hydrodynamics suggests thatequilibration takes place to somedegree <cit.>,most dynamical calculations predict that the system produced in the early stages of a heavy ion collision is far from chemical equilibrium, typically with overpopulation in gluon numbers <cit.> and underpopulation in quark numbers <cit.>. The resulting effective equation of state might differ significantly from that calculated in lattice QCD, and it is important to understand what experimental data tell us about the equation of state, beyond a comparison between different lattice results <cit.>. It has been recently shown that a simultaneous fit of several observables to hydrodynamic calculations constrains the equation of state to some extent <cit.>. However, this recent study uses a systematic, Bayesian framework, and the nature of therelationships between model parameters and observables remains obscure.Further Bayesian studies have shown <cit.> that interferometry radii and transverse momentum spectra are theobservables which are most sensitive to the equation of state, but they are still unable to provide a simple picture of how this dependence takes place. Another related approach isto use a deep learning method to distinguish the crossover and first-order phase transitions in equations of state from heavy-ion particlespectra <cit.>.We show that for central collisions, the variation of the mean transverse mass per particle as a function of the multiplicity density dN/dy (which itself depends on the collision energy √(s))reproduces, up to proportionality factors, the variation of energy over entropy ratio ϵ/s as a function of the entropy density s <cit.>. We illustrate our point by discussing an ideal experiment in Sec. <ref>.We then carry out detailed hydrodynamic simulations using a variety of equations of state.The equations of state are presented in Sec. <ref>.Results from hydrodynamic calculations are discussed in Sec. <ref>.Calculations are compared with experimental data from RHIC and LHCin Sec. <ref>.§ AN IDEAL EXPERIMENT In order to illustrate our picture, we first describe a simpleideal experiment: the fluid is initially at rest in thermal equilibrium at temperature T_0 in a container of arbitrary shape, and large volumeV. At t=0, the walls of the container disappear and the fluid expands freely into the vacuum. If V is large enough,this expansion follows the laws of ideal hydrodynamics.At some point, the fluid transforms into N particles. We assume for simplicity that this transformation occurs at a single freeze-out temperature T_f <cit.>.The thermodynamic properties at the initial temperature T_0 can be easily be reconstructed by measuring the energy E and the number of particles N at the end of the evolution, provided that the initial volume V is known.The total energy E is conserved throughout the evolution, hence the initial energy density is: ϵ(T_0)=E/V.For simplicity, we assume throughout this paper that the net baryon number is negligible (which corresponds to high-energy collisions) so that the energy density depends solely on the temperature. The initial entropy density can be inferred from the final number of particles N.Ideal hydrodynamics conserves the total entropy S.The fluid is transformed into particles at the freeze-out temperature T_f, and the multiplicity N is directly proportional to the entropy.[Both the multiplicity N and the entropy S are scalar quantities, hence, the entropy per particle only depends on the freeze-out temperature T_f, not on the fluid velocity.]Therefore, the initial entropy density is related to the final multiplicity through the relation:s(T_0)= (S/N)_T_fN/V,The volume dependence cancels in the energy per particle: ϵ(T_0)/s(T_0)=(N/S)_T_fE/N,One can repeat the experiment for several values of the initialdensity, and plot the energy per particle E/N as a function of N/V.One thus obtains a plot of ϵ/s versus s,which gives access to the equation of state.Note that Eqs. (<ref>) and (<ref>) do not involve the fluid velocity pattern,which depends on the shape of the initial volume.Hydrodynamic modeling only enters through the entropy per particle at freeze-out (S/N)_T_f. This ideal experiment thus allows one to measure the equation of state for temperatures larger than T_f.Based on a similar picture, Van Hove <cit.> argued that the transition from a hadronic gas to a quark-gluon plasma should result in a flattening of the mean transverse momentum ⟨ p_T⟩ as a function of the multiplicity. It has been recently attempted to extract an approximate equation of state from recent pp and pp̅ collision dataon this basis <cit.>.The little liquid produced in an ultrarelativistic nucleus-nucleus collision has similarities with this ideal experiment if one cuts a thin slice perpendicular to the collision axis and looks at its evolution in the transverse plane.The initial transverse velocity is initially zero, and the fluidexpands freely into the vacuum right after the collision takes place.The two main differences are: * The initial temperature profile is not uniform in a box but has a non-trivial transverse structure.* The slice expands in the longitudinal direction and its energy decreases as a result of the work of the longitudinal pressure <cit.>exerted by neighboring slices: dE=-PdV.As we shall see, both effects can be taken care of by appropriately redefining the volume V and the temperature T_0, and replacing the energy per particle E/N with the mean transverse mass, where the transverse mass is defined by m_T=√(p_T^2+m^2).Eqs. (<ref>) and (<ref>) are replaced with: s(T_ eff) =a 1/R_0^3dN/dy,ϵ(T_ eff)/s(T_ eff) = b⟨ m_T⟩,where R_0 is a measure of the transverse radius, which will be defined in Sec. <ref>, T_ eff is an effectivetemperature taking into account the longitudinal cooling(T_ eff<T_0), and dN/dy is the multiplicity per unit rapidity, and a and b are dimensionless parameters whose values are independent of the equation of state and of the colliding energy.Their values will be determined in Secs. <ref> using hydrodynamic calculations, which take into account thelongitudinal cooling and the inhomogeneity of the initial profile. By measuring the mean transverse mass and the multiplicity density in a given system at different colliding energies, one obtains the variation of ⟨ m_T⟩ as a function of dN/dy.Neglecting the energy dependence of the transverse size R_0(this will be justified in Sec. <ref>),the slope of this curve in a log-log plot is the ratio of pressure over energy density, P(T_ eff)/ϵ(T_eff) <cit.>.Using Eqs. (<ref>), one obtainsdln⟨ m_T⟩/dln dN/dy= .dlnϵ-dln s/dln s|_T_ eff= .P/ϵ|_T_ eff, where we have used the thermodynamic identities dϵ=Tds and ϵ+P=Ts.Note that the dependence on the unknown coefficients a and b cancels in this expression.One thus obtains a measure of the ratio P/ϵ ofthe quark-gluon matter produced in the collision from data alone.The entropy density s(T_ eff) at which this ratio is measured, however, depends on thecoefficient a, which can only be obtained through detailed hydrodynamic simulations. These will be carried out in Sec. <ref>. § EQUATIONS OF STATEThe equation of state of QCD is characterized by a transitionfrom a hadronic, confined system at low temperatures to a phase dominated by colored degrees of freedom at high temperatures. It has been determined precisely through lattice calculations <cit.>.Lattice calculations are carried out at zero baryon chemical potential, and the matter produced at central rapidity in high-energy collisions also has small net baryon number. We therefore choose toneglect net baryon density in the present study. In lattice calculations, one first computes the trace anomaly I≡ϵ-3Pas a function of the temperature T, where ϵ is the energy density and P the pressure. Other quantities are then determined through the thermodynamic relations:P/T^4 = ∫_0^T I/T^5 dT ,ϵ = I+3P , s = ϵ+P/T.The equation of state used in hydrodynamic calculations is constrained, on the low-temperature side, by the condition that it matches that of the hadron resonance gas created at the end of the evolution <cit.>.All the equations of state used in this paper match the hadron resonance gas for temperatures smaller than 140 MeV, which is the freeze-out temperature of our hydrodynamic calculation.We choose to vary the high-temperature part along two different directions: either by varying the high-temperature limit of P/T^4, which is proportional to the number of degrees of freedom of the quark-gluon plasma (denoted as equation of state (EOS) A, B, L and C in Fig. <ref> (a) where EOS L corresponds to the lattice QCD-based equation of state), or by varying the temperature range over which the transition occurs (denoted as equation of state (EOS) D, E, L and F in Fig. <ref> (b)). The parameterization is explicated in Appendix <ref>.We thus span a range of equations of state around the lattice value.Note that the error on P/T^4 from lattice calculations is smaller than 0.1 for all T <cit.>. We explore a much wider range of equations of state.According to the picture outlined in Sec. <ref>,heavy-ion collisions measure the variation of theenergy over entropy ratio as a function of the entropy density.This variation is displayed in Fig. <ref> for the various equations of state displayed in Fig. <ref>.Note that the ratio ϵ/s is closely related to the temperature <cit.>:3T/4<ϵ/s<T,where the lower bound corresponds to the ideal gas limit P=ϵ/3 and the upper bound to P=0.Thus, the variation ofϵ/s as a function of s is essentially the variation of the temperature with the entropy density.In the high-temperature phase, s∝ν T^3, where ν is the effective number of degrees of freedomof the quark gluon plasma.More degrees of freedom implies a smaller temperature, for the same entropy density, which explains why the order of the curves is inverted in Fig. <ref> compared to Fig. <ref>.§ HYDRODYNAMIC CALCULATIONSIn this section, we carry out hydrodynamical simulations in order to determine the mapping between observables and the equation of state according to Eq. (<ref>).We model the evolution of the fluid near midrapidity and assume boost invariance in the longitudinaldirection <cit.>.We solve the transverse expansion numerically using a (2+1)-dimensionalcode <cit.>.The initial transverse velocity is assumed to be zeroat the proper time τ_0=0.4 fm/c at which the hydrodynamic expansion starts.This small value of τ_0 accounts for the early transverse expansion <cit.>,irrespective of whether or not hydrodynamics is applicable at earlytimes <cit.>. Initial conditions are defined by the initial transverse density profile.The most important quantity involving initial conditions in this study is the effective radius R_0 defined by: R_0^2≡ 2(⟨ | x|^2⟩-|⟨ x⟩|^2),where x is the position in the transverse plane, andangular brackets denote an average value weighted with theinitial entropy density: ⟨ F( x)⟩≡∫ F( x) s( x,τ_0)d^2 x/∫ s( x,τ_0)d^2 x.The normalization factor 2 in Eq. (<ref>) ensures that one recovers the correct result for a uniform entropy density profile within a circle of radius R_0. In the ideal experiment described in Sec. <ref>, the mapping between observables and the equation of state is independent of the shape of the initial volume.For this reason, one expects that most of the dependence on the shape of the initial density profile is through the radius R_0.This has been checked in detail in studies of transverse momentum fluctuations <cit.>, where it was shown that the mean transverse momentum in hydrodynamics is sensitive to initial state fluctuations only through fluctuations of R_0.We have checked it independently by comparing two standard models ofinitial conditions, the Monte Carlo Glauber model <cit.> and the MCKLN <cit.> model, as will be explained below.The default setup of our hydrodynamic calculation uses a Monte Carlo Glauber simulation of 0-5% most central Au+Au collisionswhere the energy density is a sum of contributions of binary collisions, and the contribution of each collision is a Gaussian of width 0.4 fm centered half way between the colliding nucleons. The resulting density profile is centered, and then averaged over a large number of events in order to obtain a smooth profile <cit.>.The normalization of the density profile determines the multiplicity dN/dy. We run each calculation with 5 different normalizations spanning a range which covers the LHC and RHIC data which will be used in Sec. <ref>.§.§ Ideal hydrodynamics We first carry out ideal hydrodynamic simulationsfor all the equations of state displayed in Fig. <ref>.The fluid is converted into hadrons through the standard Cooper-Frye freeze-out procedure <cit.> at a temperature T_f=140 MeV.We include all hadron resonances with M<2.25 GeV,and compute ⟨ m_T⟩ and dN/dy directly at freeze-out, before resonances decay. Our goal here is to mimic as closely aspossible the ideal experiment outlined in Sec. <ref>. The symbols in Fig. <ref> correspond to the right-hand sideof Eq. (<ref>), where the dimensionless parameters a and b have been fittedto achieve the best possible agreement with theleft-hand side.There are 5 points for each equation of state, which correspond to different initial temperatures.The overall agreement is excellent, and shows that the variation of⟨ m_T⟩ as a function of (1/R_0^3)(dN/dy) is determined by the equation of state. In order to test that this mapping is independent of initial conditions, we have carried out a calculation with MCKLN initial conditions.While both models give values of R_0 that differ by 5%, they yield the same value of ⟨ m_T⟩ when compared at the same value of (1/R_0^3)dN/dy.Let us now comment on the order of magnitude of the fit parameters a and b.First, compare Eq. (<ref>) and the second line of Eq. (<ref>).The entropy per particle at freeze-out before decays is (S/N)_T_f=6.5 in this calculation. The transverse mass of a particle is smaller than its energy, since it does not include the longitudinal momentum p_z.The relevant longitudinal momentum here is that relative to the fluid, which cannot be measured, since data are integrated over allfluid rapidities.The value of b=0.202is slightly larger than (N/S)_T_f=0.154,and thus compensates for the loss of longitudinal momentum. We now discuss the order of magnitude of a. The main difference between the ideal experiment described in Sec. <ref> and the real experiment is that the energy of the fluid slice decreasesas a result of the work done by the longitudinal pressure.In ideal hydrodynamics, this cooling is only significant atearly times: After the transverse expansion sets in, the pressure decreases very rapidly, the work becomes negligible and the energy stays constant.A rough, but qualitatively correct, picture is that the expansion is purely longitudinal during a time τ_ eff and that the energy is conserved for τ>τ_ eff <cit.>.For dimensional reasons, τ_ eff=fR_0, where f is of order unity.The volume at τ_ eff is V=π R_0^2 τ_ eff=π fR_0^3. Inserting this value into Eq. (<ref>) and identifying the right-hand side with the first line of Eq. (<ref>), one obtains f≃ 0.5, in agreement with the value obtained in previous calculations <cit.>.Ideal hydrodynamics thus probes the equation of state at a time τ_ eff∼ 0.5R_0, which is the typical time at which transverseflow and elliptic flow develop <cit.>. §.§ Resonance decaysThe largest correction to the naive ideal fluid picture comes from decays occurring through strong or electromagnetic interactions,which occur after freeze-out, but before thedaughter particles reach thedetectors. We compute particle spectra after strong andelectromagnetic decays, but before weak decays. Decays are treated in Ref. <cit.>, by assuming that the decay rate is proportional to the invariant phase space.After decays, the only remaining particles are pions, kaons, nucleons and strange baryons.In this preliminary study, we neglect strange baryons, which are a small fraction of the total number of particles, and areidentified in separate analyses <cit.>.We therefore evaluate the multiplicity dN/dy and the mean transverse mass including only pions, kaons, and (anti)nucleons, both charged and neutral.As shown in Fig. <ref>,decays increase the multiplicity by 40%.They also conserve the total energy, so that ⟨ m_T⟩ decreases, while the product ⟨ m_T⟩ dN/dy only changesby a few percent. Since the increase of dN/dy due to decays depends solely on the freeze-out temperature, but is independent of the colliding energy and the equation of state,decays amount to further rescalings of ⟨ m_T⟩ anddN/dy. They can be taken into account by modifying the values of the coefficients a and b in Eq. (<ref>). We again determine the values of a and b through a simultaneous least-square fit to all equations of state. The result is shown inFig. <ref>, where only the equations of state of Fig. <ref> (a) are shown.After rescaling, the effective entropy density of the fluid is unchanged: locations of symbols in Fig. <ref> (a) and Fig. <ref> are identical to within less than 0.5%.The fact that they are identical confirms that Eqs. (<ref>) reconstruct thermodynamic properties of the fluid. A more realistic description of the hadronic stage should include not only decays, but also rescatterings, for instance by coupling hydrodynamics to a transport code <cit.>.It has been recently shown <cit.> that transverse momentum spectra are remarkably independent of the temperature at which one switches from the hydrodynamic to the transport description, which implies that our results would be unchanged if we switched from a hydrodynamic description to a transport calculation at a temperature larger than 140 MeV. Below 140 MeV, effects of hadronic scatterings are suppressed due to the lower density. Our choice of T_f allows us to roughly reproduce observed particle ratios, in agreement withRef. <cit.>. This is important as the mean m_T, averaged over all particle species, strongly depends on particle ratios.§.§ ViscosityWe finally study viscous corrections to the ideal fluid picture. We use “minimal” shear viscosity η/s=1/4π <cit.> and bulk viscosity ζ/s = 2 (1/3 - c_s^2) η/s <cit.> based on the gauge-string correspondence, where c_s is the sound velocity. The relaxation times are also conjectured in the holographic approach <cit.>. Viscosity modifies the equations of motion of the fluid <cit.>,and the momentum distribution of particles at freeze-out <cit.>.We show both effects separately in Fig. <ref>. The main effect of viscosity is toincrease the multiplicity for a given initial condition, which is a consequence of the entropy increase due to dissipative processes.On the other hand, the value of ⟨ m_T⟩ changes little,which is due to a partial cancellation between effects of shear viscosity (which increases ⟨ m_T⟩) and bulk viscosity(which decreases ⟨ m_T⟩) <cit.>. The values of ⟨ m_T⟩ and dN/dy can again be matched to the equation of state through Eqs. (<ref>).We again determine the values of a and b which give the best simultaneous fit to all equations of state.The result is displayed in Fig. <ref>, where only the equations of state of Fig. <ref> (a) are shown.This figure shows that viscous corrections do not alter qualitatively the ideal fluid picture, and that the variation of the mean transverse mass with the multiplicity density is still driven by the equation of state in the presence of viscosity.Comparison with Fig. <ref> shows that symbols are shifted to the right, which means that for the same initial temperature, viscous hydrodynamics results in a higher effective entropy density.The reason is that entropy is produced in the off-equilibrium processes. The actual value of the shear and bulk viscosity are not known precisely. Since a and b depend slightly on the viscosity,the uncertainty on the viscosity translate into an uncertainty on the mapping of experimental data onto the equation of state through Eq. (<ref>).The upper bound on constant η/s from heavy-ion data is typically 0.2 <cit.>. It has been recently noted that the inclusion of bulk viscosity tends to lower the preferred valueof the shear viscosity <cit.>, so that η/s<0.2 seems conservative.We assume that viscous corrections are proportional to the viscosity,therefore the uncertainty can be inferred from the difference between our viscous and ideal calculations.The uncertainty on a is 7% and amounts on an uncertainty on the effective entropy density s_ eff.The uncertainty on b is 11% and is essentially an uncertainty on the corresponding temperature.Note, however, that the dependence on a and b cancels in the logarithmic slope, Eq. (<ref>), and the ratio P/ϵ can be determined precisely even if transport coefficients are not precisely determined. § COMPARISON WITH DATA We now discuss to what extent existing data constrain the equation of state.Both dN/dy and ⟨ m_T⟩ require spectra of pions, kaons and protons.Such data have been published by STAR <cit.> and PHENIX <cit.> at the Relativistic Heavy Ion Collider (RHIC) and by ALICE <cit.> at the Large Hadron Collider (LHC). PHENIX and ALICE data for protons are corrected for the contamination from weak Λ decays, while STAR data are not.We correct STAR data by assuming that a fraction 35%± 10% of protons come from Λ decays, as determined by the PHENIX analysis <cit.>.Particles are only identified within a limited p_T range, which depends on the experiment, and spectra must be extrapolated in order to obtain dN/dy and ⟨ m_T⟩.These extrapolations are discussed in Appendix <ref>.The data we use are for charged particles, and we need ⟨ m_T⟩and dN/dy for all hadrons, including neutral ones.Yields of neutral particles are obtained assuming isospin symmetry.The resulting values of ⟨ m_T⟩ and dN/dy are given in Table <ref>.For 200 GeV, we include both STAR and PHENIX measurements, which are slightly different, but compatible within errors. In order to convert the multiplicity dN/dy into a density, one needs an estimate of the initial transverse size R_0.This quantity, which represents the mean square radius of the initial density profile, is not measured and can only be estimated in a model. As we shall see, it turns out to be the largest source of uncertainty when constraining the equation of state from data.In particular, the uncertainty from R_0 is larger than the uncertainty from transport coefficients. We discuss how we estimate R_0.Note that the transverse size fluctuates event to event, even in a narrow centrality window <cit.>. Ideally, we would like to estimate the average value over events of (1/R_0^3)dN/dy. Since the input available from experiment is an average of dN/dy, for the sake of simplicity, we estimate the average value of R_0 over many events to divide dN/dy for our analyses. We use the same Monte Carlo Glauber model as in our hydrodynamic calculation.The resulting values, averaged over many events, are given in Table <ref>. TheMCKLN model <cit.> gives values 5% smaller, which implies that the density is 15% larger.This shows that the uncertainty on the transverse size is significant.However, the variation of R_0 with colliding energy for a given system is small, so that the evolution of the density is mostly driven by the increase in the multiplicity dN/dy.Therefore, uncertainties on R_0 cancelwhen comparing two different collision energies.The variation of the mean transverse mass with dN/dy directly gives the ratio P/ϵ, as shown by Eq. (<ref>).As pointed out in Sec. <ref>,uncertainties from the viscosity also cancel in thisenergy dependence. Using PHENIX and ALICE data, which span a wide range of dN/dy, and taking into account the different sizes of Au and Pb nuclei, Eq. (<ref>) gives.P/ϵ|_T_ eff=0.21± 0.10,where the error is solely from experiment. The only significant theoretical uncertainty is on the effective temperature T_ eff at which this ratio is measured. We provide in Table <ref> the values of the effective entropy density s_ eff given by Eq. (<ref>), where a is given by our viscous hydrodynamic calculation.The value at 5.02 TeV, where identified particle spectra are not yet published, is obtained by assuming that the relative increase in dN/dy from 2.76 TeV equals that of dN_ch/dη, that is, 20% <cit.>.As discussed in Sec. <ref>, the uncertainty on s_ eff from transport coefficients is 7%, and thatfrom the transverse size R_0 is at least 15%. The value of the temperature T_ eff corresponding to s_ effcan only be obtained if the equation of state is known.The values in the last column of Table <ref> correspond to the lattice equation of state.Lattice calculations give P/ϵ=0.23 for a temperature half-way between the values of T_ eff corresponding to 200 GeV and 2.76 TeV. The experimental value, Eq. (<ref>), is compatible with the lattice result.Experiments at √(s)=5.02 TeV, for which identified particle spectra are yet unpublished, will probe the equation of state at a temperature close to 300 MeV.Note that the theoretical uncertainty of ≃ 20% on s_ eff translates into an uncertainty∼ 15 MeV on the effective temperature at the LHC, which is dominated by the uncertainty on the initial transverse radius R_0. Figure <ref> shows the comparison between experimental data and the values obtained from the equation of state through Eqs. (<ref>), where a and b are taken from our viscous hydrodynamic calculation (see Fig. <ref>).With the minimal viscosity chosen in this calculation, LHC data slightly favor the equation of state C, which has a larger pressure than the lattice equation of state. With a higher viscosity, however, the lattice equation of state would be preferred.Equations of state A and B are ruled out: as already well known, heavy-ion data favor a soft equation of state.Note that current experiments only probe the equation of state up to T∼ 300 MeV (see Table <ref>).§ CONCLUSIONSWe have shown that in central nucleus-nucleus collisions, the variation of the meantransverse mass as a function of the multiplicity density is, up to rescaling factors, driven by the variation of the energy over entropy ratio ϵ/s as a function of the entropy density s.Each collision energy probes the equation of state at a different entropy density s_ eff, which correspondsroughly to the average density at a timeτ_ eff∼ 3 fm/c. RHIC and LHC experimentsprobe the equation of state for temperatures up to ∼ 300 MeV.The largest source of uncertainty at the theoretical level is the initial transverse size R_0.The uncertainty from unknown transport coefficients (shear and bulk viscosity) is twice smaller.These theoretical uncertainties cancel if one measures the evolution of the meantransverse mass as a function of collision energy, which gives direct access to the pressure over energy density ratio P/ϵ of the quark-gluon plasma. This analysis requires precise experimental data on identified particle spectra.One could think of replacing the transverse mass with the transverse momentum, and the rapidity by the pseudorapidity, which was the original idea of Van Hove <cit.>, and would allow to work with unidentified particles. However, we have checked that the mapping onto the equation of state is not as good in this case. Thevalue of P/ϵ obtained from the evolution of spectra from RHIC to LHC energies is compatible with the lattice equation ofstate, but with large errors. Carrying out an energy scan at the LHC with a single detector would greatly improve the quality of the measurement. § ACKNOWLEDGEMENTSWe thank Matt Luzum and Michele Floris for discussions and Jean-Paul Blaizot for useful comments on the manuscript.AM is supported by JSPS Overseas Research Fellowships.§ VARYING THE EQUATION OF STATEThe equation of state is constructed by connecting the traceanomaly of the hadron resonance gas model smoothly tothat of lattice QCD <cit.>. To systematicallygenerate variations of the equation of state, modificationis made through two factors c_w and c_h in the QGPphase for our analyses. The expression reads:I(T)= 1/2[ 1 - tanh(T-T_s/Δ T_s) ] I_HRG(T) + c_h/2[ 1 + tanh(T-T_s/Δ T_s) ] I_lat(T_w),where T_w = T_s + c_w(T-T_s). c_w and c_h areassociated with the width and the magnitude of I(T) in theQGP phase, respectively. c_w = 1 and c_h = 1recover the lattice QCD result. The hadronic equation of stateis left untouched because, as mentioned earlier, theCooper-Frye formula requires that kinetic theory reproducesthe equation of state used in the hydrodynamic model atfreeze-out for energy-momentum conservation. When onechooses T_s = 160 MeV and Δ T_s = 0.1 T_s,this is satisfied at and below T = 140 GeV.The pressure is obtained through the thermodynamic relations (<ref>). Since the trace anomaly is integrated, c_wand c_h have to be modified simultaneously to shift the pseudo-criticaltemperature and change the effective number of degreesof freedom in the pressure or the entropy density (Fig.<ref>).We first consider a set of equation of state with different numbersof QGP degrees of freedom by choosing (c_w,c_h) =(2,0.5), (1.5, 0.75), (1,1), and (0.5,1.25). They are labeledas EOS A, B, L and C, respectively. The normalized pressureas a function of the temperature for each equation of state is plottedin Fig. <ref> (a). It is note-worthy that we consider anequation of state which exceeds the Stefan-Boltzmann limitwith the last parameter set (0.5,1.25). We also vary thepseudo-critical temperature by setting the parameters to(c_w,c_h) = (2,1.5), (1.5, 1.25), (1,1), and (0.5,0.75) asshown in Fig. <ref> (b), which are labeled as EOS D, E, L and F.The equation of state becomes harder for larger T_cbecause it is fixed on the hadronic side. § IDENTIFIED PARTICLE SPECTRA AT RHIC AND LHC In order to estimate the mean transverse mass per particle from experimental data, we use as input p_T spectra ofidentified charged hadrons in the central rapidity region.More specifically, we use data for charged pions, charged kaons, protons and antiprotons, which are shown as symbols in Fig. <ref>. These plots show the probability distribution of p_T near midrapidity, dN/dp_Tdy.Experimental data are shown as symbols.Pion and kaon yields increase smoothly with collision energy as expected.This does not appear to hold for proton and antiprotons, but the reason is simply that STAR data for protons and antiprotons include, in addition to primary particles, secondary products of weak Λ and Λ̅ decays.Apart from this difference, PHENIX and STAR data at 200 GeV are compatible within error bars.The effect of the net baryon number becomes visible at the lower energies:it results in more protons than antiprotons at midrapidity, and also slightly more K^+ than K^- becausethe strangeness chemical potential is non-vanishing inthe presence of the net baryon chemical potential μ_B owing tothe strangeness neutrality condition.While the differences between particles and antiparticles are linear in μ_B, the total multiplicities are even functions of μ_B, hence effects of net baryon number only appear to order μ_B^2. We assume that they are negligible down to 62.4 GeV. Particles are identified only in a limited p_T range which depends on the experiment. In order to evaluate the mean m_T, we need toextrapolate the measured spectrum to the whole p_T range.These extrapolations are done with blast-wave fits <cit.>.For ALICE data, we fit each particle speciesindependently, as in the experimental paper <cit.>.The resulting values of dN/dy and ⟨ p_T⟩ are given in Tables <ref> and <ref>. They are very close to the values in the experimental paper. The small differences, which are much smaller thanerror bars, can be ascribed to different fitting algorithms.For sake of consistency, we also use blast-wave fits to extrapolate PHENIX data <cit.>.The resulting values of dN/dy and ⟨ p_T⟩ differ somewhat from the experimental values which use a different extrapolation scheme, but are compatible within error bars.For STAR data, the p_T range is too limited to fit each particle species independently: therefore, we follow the recommendation of the experimental paper <cit.> and carry out a simultaneous fit for kaons and (anti)protons.For pions, however, we carry out an independent blast-wave fit as forPHENIX data.Agreement between STAR and PHENIX pion yields at 200 GeV is much better than in the corresponding experimental papers, which suggests that the differences were mostly due to the different extrapolationmethods. Finally, the values of ⟨ m_T⟩, which are needed in this paper, are listed in Table <ref>. 99Stoecker:1986ciH. 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"authors": [
"Akihiko Monnai",
"Jean-Yves Ollitrault"
],
"categories": [
"nucl-th",
"hep-lat",
"hep-ph",
"nucl-ex"
],
"primary_category": "nucl-th",
"published": "20170726143521",
"title": "Constraining the equation of state with identified particle spectra"
} |
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, USA Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC) Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA [E-mail:][email protected][E-mail:][email protected] We investigate the quarkonium production mechanisms in jets at the LHC, using the Fragmenting Jet Functions (FJF) approach. Specifically, we discuss the jet energy dependence of the J/ψ production cross section at the LHC. By comparing the cross sections for the different NRQCD production channels (^1S_0^[8],^3S_1^[8],^3P_J^[8], and ^3S_1^[1]), we find that at fixed values of energy fraction z carried by the J/ψ, if the normalized cross section is a decreasing function of the jet energy, in particular for z > 0.5, then the depolarizing ^1S_0^[8] must be the dominant channel. This makes the prediction made in [Baumgart et al., JHEP 1411, 003 (2014)] for the FJF's also true for the cross section.We also make comparisons between the long distance matrix elements extracted by various groups. This analysis could potentially shed light on the polarization properties of the J/ψ production in high p_T region.Quarkonium polarization and the long distance matrix elements hierarchies using jet substructure Prashant Shrivastava December 30, 2023 ================================================================================================ 3.3ex§ INTRODUCTION Analyzing quarkonium production in jets provides a new way of probing the physics involved in their production. Recent developments include the LHCb measurements of J/ψ production in jets <cit.> and the related analyses <cit.>. A factorization theorem based on Non-Relativistic QCD (NRQCD)[3.0ex NRQCD is an effective theory with a double expansion in the relative velocity v of the heavy quark and anti-quark bound state and the strong coupling constant α_s <cit.>.] can be used to calculate the cross section for J/ψ production <cit.>. Due to the large mass of the charm quark (m_c), the short distance production of the cc pair can be calculated perturbatively while the non-perturbative physics of the hadronization into a J/ψ is captured by the long distance matrix elements (LDMEs) of the relevant production channels (^1S_0^[8],^3S_1^[8],^3P_J^[8], and ^3S_1^[1]). The predictive power of the theory is then predicated on our knowledge of these LDMEs. Different groups have extracted these matrix elements by using various fits to the data <cit.> but have arrived at very different values. Currently the NRQCD factorization theorem can consistently fit the unpolarized J/ψ production cross section.The cc pair produced by the fragmentation of a nearly on-shell gluon[3.0ex For J/ψ production via gluon fragmentation in NRQCD, the ^3S_1^[1] contribution is leading order in the v expansion since the color octet channels are suppressed by v^4. But the ^3S_1^[1] is suppressed relative to the ^3S_1^[8] channel by power of α_s^2. The matching onto ^3P_J^[8] and ^1S_0^[8] is down by α_s compared to ^3S_1^[8] but their LDMEs are of the same order as ^3S_1^[8] in v. An alternate power counting for charmonium production is formulated in Ref. <cit.>.] should inherit the transverse polarization of the gluon. Due to the spin symmetry of the leading order NRQCD Lagrangian, this polarization remains intact during the non-perturbative hadronization process (up to power corrections) <cit.>. At leading order in α_s, only the ^3S_1^[8] channel for the gluon contributes among the octet channels and since the color octet contribution is expected to dominate at high p_T <cit.>, the J/ψ meson should be produced with significant polarization at high p_T. However this prediction of NRQCD is at odds with the measurements of the J/ψ polarization <cit.>. Understanding this polarization puzzle is one of the most important challenges in quarkonium physics <cit.>.A method based on jet substructure techniques to study the different production mechanisms of the J/ψ was proposed in Ref. <cit.>. By using the properties of the Fragmenting Jet Functions (FJF) <cit.>, it is predicted in Ref. <cit.> that for a jet of energy E and cone size R, containing a J/ψ with energy fraction z (z=E_J/ψ/E), if the FJF is a decreasing function of the jet energy, then the dominant contribution to the J/ψ production at high p_T should be the depolarizing ^1S_0^[8] channel and hence, if confirmed by the data, this would resolve the polarization puzzle.In this work, we investigate how the predictions of the diagnostic tool introduced in Ref. <cit.> are affected by inclusion of the hard scattering effects. To do this, we calculate the total production cross section for the J/ψ. This should make the comparison of theory with experiments much simpler since the cross section can be directly measured. In order to make the distinction between various production channels, we calculate the cross section normalized in two different ways. In one case we normalize by summing over the contribution of all the channels and integrating over z while in the other case we normalize by using the 1-jet inclusive cross section. Additionally we also make comparisons between the LDMEs extracted by various groups. The main result of our paper is that the prediction made in Ref. <cit.>, regarding the shapes of the FJF's, is also true for the cross section. By using a combination of differently normalized cross sections, we can break the degeneracy of the production channels and isolate the dominant contribution to the J/ψ production at high p_T. Our results show that if the normalized cross section is a decreasing function of the jet energy at large z, in particular for z>0.5, then the ^1S_0^[8] channel dominates at high p_T and this prediction should be easily verifiable with the LHC data. A recent work <cit.> also proposed using observables similar to ours to probe the J/ψ production mechanisms.[3.0ex Ref. <cit.> differentiates between the NRQCD global fits based on inclusive J/ψ cross section and suggests using the polarization measurements of J/ψ meson produced in the jets as a way of constraining the heavy quarkonium production mechanisms.]§ THE FRAGMENTING JET FUNCTIONS We briefly review the factorization theorem for the production of J/ψ <cit.> before moving onto our main results in the next section. We consider the process pp → dijets at √(s) = 13 TeV and integrate over one of the jets, assuming that the other jet contains an identified J/ψ. The dijet cross section <cit.> with one jet of energy E, cone size R and a J/ψ in the jet carrying an energy fraction z, is schematically of the formdσ/dEdz=∑_a,b,i,j H_ab→ ij⊗ f_a/p⊗ f_b/p⊗ J_j⊗ S ⊗𝒢^ψ_i(E,R,z,μ),where H_ab→ ij is the hard process, f_a/p and f_b/p are the parton distribution functions (PDF), J_j is the jet function for the jet not containing the J/ψ, S is the soft function and 𝒢^ψ_i(E,R,z,μ) is the FJF for the jet containing the J/ψ. The parton i can be a gluon, charm or an anti-charm (contributions of the other partons are suppressed). We are interested in the E and z dependence of the cross section, which comes from the hard function (including PDFs) and the FJF. We integrate over the jet originating from the parton j so the jet function J_j enters the cross section multiplicatively. The soft function S does not affect 𝒢^ψ_i(E,R,z,μ), R, E and z (up to power corrections) <cit.> and so it also enters the cross section multiplicatively. Hence both the jet function J_j and the soft function S give an overall normalization to the cross section and are ignored in the rest of our analysis. In Ref. <cit.>, the hard function was not included but here we calculate the normalized cross section, including both the charm quark and gluon contributions, and account for its E dependence.The FJF can be further factorized <cit.> into perturbatively calculable coefficients 𝒥_ij(E,R,z,μ) and the fragmentation function D_j→ψ: 𝒢_i^ψ(E,R,z,μ)=∫_z^1dy/y𝒥_ij(E,R,y,μ)D_j→ψ(z/y,μ)(1+𝒪(m^2_ψ/4E^2tan^2(R/2))).The collection of NRQCD based fragmentation functions D_j→ψ used in this paper can be found in Ref. <cit.>.Large logarithms in 𝒥_ij(E,R,z,μ) are minimized at the scale μ=2Etan(R/2)(1-z) and can be easily resummed using the jet anomalous dimension <cit.>. But we do not consider this resummation in this work since for us, 1-z ∼𝒪(1) <cit.>. Instead we evaluate the PDFs and 𝒥_ij(E,R,z,μ) at the jet scale μ_J=2Etan(R/2) and evolve the fragmentation function from 2m_c to the scale μ_J using the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation, μ∂/∂μD_i(z,μ)=α_s(μ)/π∑_j∫_z^1 dy/yP_i→ j(z/y,μ)D_j(y,μ),where P_i→ j(z/y,μ) are the QCD splitting functions. We consider mixing between the charm quark and gluon splitting functions only for the ^3S_1^[1] channel.[3.0ex The charm quark fragmentation into a J/ψ is dominated by the ^3S_1^[1] channel because the color singlet and octet contributions start at same order in α_s but the color octet channels are suppressed in the v expansion.] To leading order in α_s, it can be shown that <cit.>𝒢_i^ψ(E,R,z,μ_J)/2(2π)^3→ D_i→ψ(z,μ_J)+𝒪(α_s(μ_J)). Later in <ref>, we will also consider the 1-jet inclusive cross section. This is calculated by replacing the FJF in Eq. (<ref>) with the jet function for a cone-type algorithm <cit.>. The FJFs are defined in Ref. <cit.> so that the sum over all possible fragmentations of a parton into hadrons equals the inclusive jet function.J_i(E,R,μ)=1/2∑_h∫dz/(2π)^3z𝒢^h_i(E,R,z,μ).For further details about these calculations we refer the reader to Ref. <cit.>. Throughout this paper we choose m_c=1.4 GeV and R=0.4.§ DISCUSSION OF THE J/Ψ PRODUCTION MECHANISMSIn this section, we discuss the predictions for J/ψ production in jets using the LDMEs extracted by various groups and reveal some generic features that are independent of these extractions. The LDMEs we use in this paper are summarized in Table <ref>. Refs. <cit.> use a global fit to 194 data points from 26 data sets and predict significant polarization of the J/ψ in the high p_T region, which contradicts the measurements at the Tevatron <cit.> and the LHC <cit.>. The extractions in Refs. <cit.> focus on the high p_T region and attempt to solve the polarization puzzle. §.§ Normalized J/ψ production cross sectionTo discuss the dependence of J/ψ production on the associated jet energy, we use a normalized differential cross section defined asdσ̃_i/dEdz≡dσ_i/dEdz/∑_i∫_z_min^z_maxdz dσ_i/dEdz,anddσ̃/dEdz≡∑_i dσ̃_i/dEdz,where i denotes different J/ψ production channels (i.e., for the gluon initiated jets i ∈ {^1S_0^[8], ^3S_1^[8], ^3P_J^[8], ^3S_1^[1]} and for the charm initiated jets i=^3S_1^[1]), and dσ_i/dEdz is defined in Eq. (<ref>). In Eq. (<ref>), z_min (z_max) should not be too close to 0 (1) where the factorization breaks down. The motivation for studying this normalized cross section is that we want to isolate the properties of quarkonium fragmentation in jets from the hard process that generates the jet initiating parton's. Fig. (<ref>) shows the energy distributions of the hard process for gluon and charm jets at the LHC[3.0ex We consider leading order partonic cross sections convoluted with PDF <cit.>, which includes the following processes: gg→ gg, gq(q) → gq(q), qq→ gg, gg → cc, gc(c) → gc(c), cc → cc, c c→c c, cq(q) → cq(q), cq(q) →cq(q), qq→ cc, cc→ cc.]. For all the figures in this paper, we fix the center of mass energy to be √(s)=13 TeV. Fig. (<ref>) shows the comparison of the normalized (Eq. (<ref>)) and unnormalized cross sections (Eq. (<ref>)), where the LDMEs from Ref. <cit.> are used with z_min=0.3 and z_max=0.8. Corresponding plots for the LDMEs of Ref. <cit.> and Ref. <cit.> are shown in appendix <ref> and <ref> respectively. We would like to emphasize the fact that both the unnormalized and normalized cross sections are directly measurable in experiments, although the normalized cross section has a better resolving power than the unnormalized cross section. In particular, the unnormalized cross section is a decreasing function of E for all the production channels due to the decreasing nature of the hard process, while the normalized cross section can be an increasing function for certain production channels due to the properties of their FJF's. A measurement of the normalized cross section (Eq. (<ref>)) for z>0.5, can help identify both the dominant channel and the favored set of LDMEs. From Fig. (<ref>), we can see that if dσ̃_i/dEdz turns out be a decreasing function of the jet energy for z>0.5, then the depolarizing ^1S_0^[8] should be the dominant channel. We find this result to be true for LDME extractions of Ref. <cit.> as well (see appendix <ref>).In Fig. (<ref>), we show the jet energy dependence of the total normalized cross sections (Eq. (<ref>)) based on different LDME extractions. The error bands are purely due to the LDME uncertainties, that is, we consider the uncertainty due to each LDME and sum by quadrature to obtain the total uncertainties[3.0ex To obtain the error bands corresponding to the extraction from Bodwin et al., we have used the error correlation matrix not shown in the original paper <cit.>.]. It can be seen in Fig. (<ref>) that as z goes from 0.4 to 0.6, the shapes change from an increasing function to a decreasing function. However since different extractions have distinct slopes, this observable has the potential power to test these extractions at the LHC. A different choice of (z_min,z_max) does not change our arguments as we demonstrate in appendix <ref>.We also consider the possibility that for z>0.3, the contribution of the ^3S_1^[1] channel to the J/ψ production is negligible for the p_T range considered here <cit.>. We test this by ignoring the ^3S_1^[1] channel contribution to thenormalization and arrive at the same conclusion of ^1S_0^[8] being the dominant contribution if the normalized cross section decreases with jet energy for z>0.5 (see appendix <ref>). §.§ Normalization using 1-jet inclusive cross section We now normalize the cross section in such a way that the denominator is independent of the LDMEs. This allows us to make a direct comparison of our results to those of Ref. <cit.>. The normalization is defined as dσ̂_i/dEdz≡dσ_i/dEdz/dσ_J/dE,anddσ̂/dEdz≡∑_i dσ̂_i/dEdz,where dσ_i/dEdz is the same as that in Eq. (<ref>) and dσ_J/dE[3.0ex This includes the contributions of gluon, light quarks, charm and bottom jets.] is the 1-jet inclusive cross section[3.0ex The definition of Eq. (<ref>) is essentially the same as the jet fragmentation function introduced in Ref. <cit.>, except that we have integrated the jet pseudorapidity over the region |η_J|<1.2 for the denominator and numerator.]. Note that the z-dependence of Eq. (<ref>) comes only from the 𝒢_i^J/ψ(E,R,z,μ) in Eq. (<ref>).Fig. (<ref>) shows the total J/ψ production cross section based on Eq. (<ref>). The key feature of this plot is that the arguments given Ref. <cit.> based on the FJFs are also true for the cross section (see Fig. 6 in Ref. <cit.>)[3.0ex To facilitate direct comparison of our Fig. (<ref>) to Fig. (6) in Ref. <cit.>, we make plots for z=0.3, 0.5 and 0.8.].Specifically, when z>0.5, the shapes of the curves are very different for the extraction based on a global fit (black curves) and the other two based on fit to high p_T region (red and blue curves). Since the extractions from the global fit and high p_T fit give rise to different slopes for the J/ψ production cross section, one can test which set of the LDME extractions are preferred by measuring these slopes. Note that because our results are for the cross section, all the curves have positive values, in contrast to the gluon FJF for the LDMEs of Ref. <cit.> (shown in Fig. (6) of Ref. <cit.>) which became negative at large energies. In Fig. (<ref>), we plot the E dependence of the individual J/ψ production channels for the different LDMEs using Eq. (<ref>). We find that if the measurements of the observable defined in Eq. (<ref>) results in a cross section which is a decreasing function of the jet energy for z>0.5, then the ^1S_0^[8] channel should have an anomalously large contribution to the J/ψ production. The fourth row in Fig. (<ref>), with the curves normalized to unit area, clearly shows that only ^1S_0^[8] channel is a decreasing function of jet energy for z>0.5.Hence a verification of our results in this and the previous section will give strong evidence in favor of the depolarizing ^1S_0^[8] channel being dominant at high p_T and provide a clear explanation for the lack of polarization in the J/ψ production at high p_T. Note that in the fourth row of Fig. (<ref>), the LDME dependence gets canceled due to normalization to unit area and so the prediction for ^1S_0^[8] channel being dominant at high p_T is independent of any specific LDME extractions. To conclude this section, we mention a few things about the normalization conventions in Eq. (<ref>) and Eq. (<ref>). First of all, both the normalizations can be directly tested in experiments. Also since both the numerator and denominator of Eq. (<ref>) depend on the LDMEs, they are statistically correlated and hence the width of error bands in Fig. (<ref>) is reduced. However, Eq. (<ref>) does not have such a correlation since the jet cross section used for the normalization is independent of the LDMEs. Indeed, if we look at Bodwin et al.'s extraction near z=0.5 and E=100 GeV, the ratio of the width of error band to the center value is ∼ 4% in Fig. (<ref>) and ∼ 30% in Fig. (<ref>). On the other hand, in both Fig. (<ref>) and Fig. (<ref>), the shapes of blue and red curves (high p_T fit) are in contrast to the black curve (global fit).§ CONCLUSIONIn this paper, we have looked at the total cross section for J/ψ production at the LHC by using the FJF approach. We make comparisons between the different NRQCD production channels for the J/ψ. We show that if for z>0.5 the normalized cross section is a decreasing function of jet energy, then the depolarizing ^1S_0^[8] should be the dominant production channel at high p_T. We find this to be true for two sets of normalized cross sections. Our results confirm that the prediction made in Ref. <cit.> regarding the decreasing nature (with E) of the FJF for ^1S_0^[8] channel, does not change by inclusion of the hard scattering effects. Using our normalized cross sections, one can also test which set of the LDME extractions are favored. The authors would like to thank Adam Leibovich and Ira Rothstein for their guidance and comments on the manuscript. We would also like to thank James Russ for useful discussions. LD was supported in part by NSF grant PHY-1519175.§ UNNORMALIZED AND NORMALIZED CROSS SECTIONS FOR BODWIN ET.ALFig. (<ref>) shows the unnormalized (Eq. (<ref>)) and normalized cross section (Eq. (<ref>)) for Bodwin et al.'s LDME extractions <cit.>. The ^3P_J^[8] channel contribution is negative, which is a feature of these LDMEs as it leads to a cancellation between the ^3S_1^[8] and ^3P_J^[8] channels, making the depolarizing ^1S_0^[8] the dominant production channel of J/ψ for z>0.5.§ UNNORMALIZED AND NORMALIZED CROSS SECTIONS FOR CHAO ET.ALFig. (<ref>) shows the unnormalized (Eq. (<ref>)) and normalized cross section (Eq. (<ref>)) for Chao et al.'s LDME extractions <cit.>. Similar to Bodwin et al., these LDMEs result in a cancellation between the ^3S_1^[8] and ^3P_J^[8] channels.§ INSENSITIVITY TO Z_MIN AND Z_MAXComparison of the normalized cross sections (Eq. (<ref>)) for different values of z_min and z_max is shown. This confirms that the discussion in section <ref> is not sensitive to (z_min,z_max) since the shapes of different LDMEs do not change. For validity of the factorization formula Eq. (<ref>), we don't pick z_min too close to 0 and z_max too close to 1.§ NORMALIZATION USING ONLY COLOR OCTET CHANNELSFig. (<ref>) shows the cross section for the different J/ψ production channels based on the LDMEs in Ref. <cit.> and Ref. <cit.> with the contribution of ^3S_1^[1] channel ignored in Eq. (<ref>), i.e., setting ⟨ O^J/ψ(^3S_1^[1]) ⟩ to 0. Since ^1S_0^[8] channel (green curves) has very different slopes for the two LDMEs, if the ^1S_0^[8] channel dominates at high p_T, then one can distinguish between these two extractions. We don't include Chao et al.'s extractions <cit.> because it gives rise to a negative total cross section and so one can not ignore the color singlet contribution.§ LOWER Z PLOTSFig. (<ref>) shows the J/ψ production cross section (Eq. (<ref>)) at lower z values for all the three LDME extractions <cit.> used in this paper.apsrev4-1 | http://arxiv.org/abs/1707.08629v2 | {
"authors": [
"Lin Dai",
"Prashant Shrivastava"
],
"categories": [
"hep-ph",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170726201151",
"title": "Quarkonium Polarization and the Long Distance Matrix Elements Hierarchies using Jet Substructure"
} |
Technische Universität Carolo-Wilhelmina zu Braunschweig and ADVA Optical Networking [email protected], [email protected] Smart optical networks are the next evolution of programmable networking and programmable automation of optical networks, with human-in-the-loop network control and management. The paper discusses this evolution and the role of Artificial Intelligence (AI). § INTRODUCTION Smart systems encompass a wide area of applications in computing and sensing that have permeated multiple industrial sectors, such as (smart) cities, transportation, energy and homes. While the term smart alone may have a broad meaning, the key differentiator for modern compute and sensing systems over any previous generation is their ability to connect over a network with cloud and storage systems. In this way, large data collection, processing, exchange and analysis have become an integral part of a smart system, unlike stand-alone systems of the past. This has naturally led to the emergence of artificial intelligence (AI) as an integral part. AI enables a computing system to implement cognitive functions akin to humans, and its innate abilities to provide solutions to problems such as self-configuration, self-healing, and self-optimization, have propelled it to one of the most promising concepts in networking today.Two evolving capabilities, namely programmability and elasticity, can make the integrating of AI with optical networks transformative.Programmablity of optical network technologies today leverage the basic concepts of software-defined networking, which decouple the network control plane from the underlying hardware, or data plane. While many systems in the past pioneered the SDN conceptually in the research communities <cit.>, it is not until SDN gained wide industrial acceptance, that the optical layer embraced true software programmability, including open source development projects, such as the Open Network Operating System (ONOS) <cit.>, OpenDaylight platform (ODL) <cit.>, Open Roadm<cit.>, and OpenConfig<cit.>. Elasticity of optical networks is another important technology evolution towards flexible spectrum management. Elastic or flexi-grid optical networks divide optical spectrum into flexible grids (slices), offering flexible and just-enough spectrum to variable bandwidth demands which previous generation of optical networks based on Wavelength Division Multiplexing (WDM) could not offer. This trend has led to the emergence of Sliceable Bandwidth Variable Transponder (SBVT) technologies that on its own provides programmability deeper in the photonics, making it possible to choose among various modulation schemes, spectrum slice, including tunability of physical layer parameters. It is intuitive to envision that adding the AI capabilities to the concepts of elasticity and programmability is expected to result in a transformative ability of optical network to serve future applications. The paper discusses the evolution of optical networking towards smart optical networks enabling a seamless integration of Artificial Intelligence (AI). We first revisit the concepts of programmable networking and postulate that its evolution towards programmable automation of optical networks is the first step towards AI. We then present the vision of smart optical networks with their integral concepts of programmability, elasticity and AI. § FROM PROGRAMMABLE NETWORKING TO PROGRAMMABLE AUTOMATION OF NETWORKSOptical network systems have been traditionally designed to enable Internet traffic to grow smoothly, without major aspirations to support fully automated IP-optical or other cross-layer programmability.Figure 1a depicts a state-of-the-art management system for optical networks. Network control orchestration is the primary focus of programmability in current networks, with a special focus on automation of network control operations such as provisioning of services. Approaches that focus on the control of individual devices, typical to IP networks, have not gained traction in optical networks, due to the challenges with modeling the physical properties of light propagation while working with abstracted device representations. As a result, programmable network control has focused on definition of operations at the network scope <cit.>, while leaving the control of individual domains to vendor-specific softwares. At the same time, automation of network management operations is a relatively unexplored field. The primary challenge in the context of optical network management is related to the complexity associated with different physical components involved in optical networking (AWGs, amplifiers, WSSs, lasers) in different configurations and the vendor specific representations for these components. There has been work to employ AI to interpret network management data to perform root-cause analysis of failures and even predict failures in the network, but has typically been siloed within vendor implementations and is not accurate enough to date. As a result, these tools serve as a potential diagnostic tool, but network management operations still require interpretation and intervention from experienced network operators. The operation of a network also includes significant interaction with other software subsystems which is has a strong institutional dependency, and is typically a part of the Operations Support System (OSS). While there have been significant advances in programmable networking, full automation of network operations has not achieved widespread adoption. The primary challenge has been the lack of mechanisms to incorporate humans-in-the-loop, which is an intrinsic requirement.Also, as seen in Fig. 1a, reliance on humans to perform diagnostics and analysis during management operations, and take appropriate actions has limited the scope of automation to network control. Finally, most techniques for automation have not addressed the challenge of integrating these systems with numerous external subsystems (billing, reporting etc.) which is typically under the purview of the OSS.Figure 1b illustrates the evolution which can address the challenges of programmable networking, as we envision it. Orchestration is a key component in this architecture, and has been used to reduce the number of human operations, and consequently the operational complexity, of typical operations. Numerous methods have been proposed to create programmable workflows that reflect basic services in infrastructure management: Programmable workflows allow users to define how to orchestrate network operations, with the syntax of the workflow defined using traditional programming languages <cit.> or as abstract workflows <cit.>. The so-called network operating systems <cit.> have also experimented with the use of intent-based interfaces, providing users a syntax to define what is desired from the network, and translating these commands into workflows to orchestrate network operations. As seen in Figure 1b, orchestration is envisioned as a key enabler for incorporating programmable automation, while facilitating human intervention if desired. Cognitive management systems will observe and learn from human operators to understand the fundamental network-centric as well as other business-related operations and eventually automate these operations while still potentially allowing for human intervention and correction. The network operators, as Figure 1b illustrates, are still an integral part of the future systems, and are now able to chose the level of automation, or override the compilers of intent-based instruction, which otherwise would run fully automatically. AI-based cognitive network and management is envisioned, which can to talk to humans in human (or at least intent-based) languages, while addressing the problems at scale that only machines can. At the same time, automation of network control will be enhanced by AI-based techniques to perform optimization and traditional network management operations requiring operator intervention, primarily for diagnostics, will be replaced by cognitive techniques.§ THE ROLE OF AI: CHALLENGES AND OPPORTUNITIES To evolve towards the vision of AI-enabled optical networks shown in Figure 2b, we see three main areas of influence.§.§ Network Control and Optimization Network control has been extensively researched, with areas focusing primarily on interface specifications for programmatic control and optimization for basic network control operations. With the introduction of automation in the network, and growing demand for capacity, network control will evolve to incorporate complex network optimization operations. Genetic Algorithms (GA) are one of the popular algorithms in optical networking among many evolutionary algorithms discovered and designed for optimizing complex systems, which has been used extensively and in a myriad network design problems. Swarm based systems have also been used to solve network optimization problems: <cit.> proposed a heuristic method based on ant colony optimization to reduce network energy footprint, whereas <cit.>presented a comparative study among three multi-objective evolutionary algorithms (MOEAs) based on swarm intelligence to solve the RWA problem in optical networks. We expect that the increasing bandwidth requirements will demand network optimization as a core network feature, while the introduction of flex-grid optics, sliceable variable bandwidth transponders and other emerging optical technologies will further increase the computational complexity of the optimization problem. AI-based techniques will play a major role for optimization mechanisms to support the scale and flexibility demanded from the next generation of programmable photonic infrastructure. §.§ Network ManagementResearch in the use of AI for Network Management operations has focused on the identification/classification of problems related to the optical channels.Machine Learning algorithms have been used to address the issue of nonlinearity in optical fiber channel, and also in extracting information about the optical signal detection <cit.>. Neural networks can be used OSNR estimation and modulation format classification <cit.>. A loss classification technique for OBS networks based on machine learning techniques was proposed in <cit.>. This approach used both a supervised learning technique (hidden Markov model (HMM)) and an unsupervised learning technique (expectation maximization (EM) clustering) on the observed losses and classified them into a set of states (clusters) after which an algorithm differentiates between the congestion and contention losses. Machine learning algorithms based on an artificial neural network were also used to provide robust and adaptive traffic models and cognitive receiver design <cit.>. One of the big challenges with network management is the collection and interpretation of data from vendor specific devices. Data collection in optical networks is primarily based on SNMP which has vendor specific representations and is inefficient for collecting measurement data. To this end, projects like OpenROADM which attempt to define management models for the various entities used in an optical network and concepts like streaming telemetry, introduced in OpenConfig, are key enablers for efficient data collection, which in turn will make AI-based algorithms more portable in multi-vendor environments. In conjunction, AI based techniques will become more prevalent in replacing human operators for interpreting data, and applications like fault management, failure prediction and intrusion detection will be likely driven by cognitive systems. It is also clear that the models of choice for network control and management today also have limited overlap: models that abstract complexity of device configuration are favored for network control operations, while a more detailed model is desired for management and monitoring operations. Automation of network operations will also require interpreting cues from the network (e.g. failure prediction) which might potentially lead to re-configuration of the network, and consequently, mechanisms to maintain and correlate multiple views of the network based on different models will be essential for automation of network operations. §.§ Intent-based API and OrchestrationOrchestration of operations requires significant operator involvement in current networks, and we see this as the prime target to be enhanced using techniques derived from AI. The future orchestration subsystem is illustrated in Figure 2. Requests to the orchestration engine will be defined via an intent-based interface, and integration over other interfaces (including voice based, text based or even s/w based) would involve subsystems that can interpret the intent of an incoming request, much like the techniques used in Natural Language Processing to date.After receiving an intent, the orchestration will try and serve an intent, which could be performed via pre-programmed intent compilers. In case a suitable pre-programmed compiler is not available, we envision the use of AI to generate workflows to serve an intent on the fly. Research in computer science has explored the use of AI techniques to generate code for a defined problem <cit.>. With advances in this area, we envision that such algorithms would be able to break-down complex requirements defined in an intent and create workflows for the same. At the same time, based on the experience with AI-based techniques to date, there will likely be no single mechanism to generate the best workflows for all situations, and we expect that a number of such techniques will work in parallel and give multiple potential solutions.Intent negotiation, seen in Figure 2a, provides the mechanism to choose from one of the potential solutions generated. The negotiation framework receives an incoming intent request, and broadcasts it to all suitable components, and has the capability to choose from a set of potential workflows for a given intent. The intent negotiation block also provides a uniquely suitable point for human intervention, allowing an operator to run the orchestration to run in a fully automated fashion, review and choose from potential workflows proposed, or even override with a human-defined workflow. Figure 2b illustrate a human operator talking to a AI-enabled assistant about scheduling a secure city cloud service for a major city marathon. The network operator is part of the AI-enabled system embedded into the network and cloud infrastructure of the future. The network operator is "gone fishing", while asking the network operator (avatar) to "send configuration and logs to her mobile phone".§ CONCLUSIONWe expect the automation of optical network operations to be replaced by full AI-based cognitive network and management, which can to talk to humans in human (or at least intent-based) languages, while addressing the problems at scale that only machines can.The human-in-the-loop systems will retain human network operators as an integral part of photonic systems, able to chose the level of automation, or override the compilers of intent-based instruction, which otherwise would run fully automatically. We also expect optimization and management operations to empoy AI-based techniques to adapt to the demands of the next generation of programmable photonic infrastructure.99 oscars On-Demand Secure Circuits and Advance Reservation System (OSCARS), https://www.es.net/engineering-services/oscars/. onos ONOS: Open Network Operating System, http://onosproject.org. odl OpenDaylight, Open Source Software Defined Network (SDN) Controller, https://www.opendaylight.orgopenroadmOpenROADM multi-source agreement, http://openroadm.org openconfig OpenConfig, vendor neutral, model driven network management designed by users, http://openconfig.net tapi Functional Requirements for Transport API, ONF TR-527, https://www.opennetworking.org/images/stories/downloads/sdn-resources/technical-reports/TR-527_TAPI_Functional_Requirements.pdf, June 2016 bpel M. Chamania,et al. “Using BPEL Workflow Processing for Cross-Layer Orchestrations in IP-over-Optical Networks: A Proof of Concept”, IEEE/IFIP Network Operations and Management Symposium (NOMS), 2012Kyriakopoulos2016C. A. Kyriakopoulos,“Energy-Efficient Lightpath Establishment in Backbone Optical Networks Based on Ant Colony Optimization,” Journal of Lightwawe Technology, vol. 34, no. 23, Dec 2016. Largo2012 A. R. Largo et al,“A Comparative Study on Multiobjective Swarm Intelligence for the Routing and Wavelength Assignment Problem,” IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, vol. 42, no .6, Nov 2012 Zibar2017D. Zibar et al, “Machine Learning Techniques in Optical Communication,” Journal of Lightwave Technology, vol. 34, no. 6, Mar. 2016.Thrane2017 J. Thrane et al., “Machine Learning Techniques in Optical Performance Monitoring From Directly Detected PDM-QAM Signals,” Journal of Lightwave Technology,vol. 35, no. 4, Feb 2017.Jayaraj2008 A. Jayaraj et al., “Loss Classification in Optical Burst Switching Networks using Machine Learning Techniques: Improving the Performance of TCP,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 6, Aug 2008.Morales2017F. Morales et al., “Virtual Network Topology Adaptability Based on Data Analytics for Traffic Prediction,” Journal of Optical Communications and Networking, vol. 9, no. 1, Jan 2017.Borkowski2015 R. Borkowski et al., “Cognitive Optical Network Testbed: EU Project CHRON [Invited],” Journal of Optical Communications and Networking, vol. 7, no. 2, Feb 2015.deepcoderM. Balog, et al, “DeepCoder: Learning to Write Programs” 5th International Conference on Learning Representations, 2017 | http://arxiv.org/abs/1707.09032v1 | {
"authors": [
"Admela Jukan",
"Mohit Chamania"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170727201632",
"title": "Evolution towards Smart Optical Networking: Where Artificial Intelligence (AI) meets the World of Photonics"
} |
Very Large Array multi-band monitoring observations of M 31* Yang Yang1,2, Zhiyuan Li1,2, Loránt O. Sjouwerman3, Feng Yuan4,5, Zhi-Qiang Shen4,6=======================================================================================A variety of representation learning approaches have been investigated for reinforcement learning; much less attention, however, has been given to investigating the utility of sparse coding. Outside of reinforcement learning, sparse coding representations have been widely used, with non-convex objectives that result in discriminative representations. In this work, we develop a supervised sparse coding objective for policy evaluation. Despite the non-convexity of this objective, we prove that all local minima are global minima, making the approach amenable to simple optimization strategies.We empirically show that it is key to use a supervised objective, rather than the more straightforward unsupervised sparse coding approach.We compare the learned representations to a canonical fixed sparse representation, called tile-coding, demonstrating that the sparse coding representation outperforms a wide variety of tile-coding representations.§ INTRODUCTION For tasks with large state or action spaces,where tabular representations are not feasible, reinforcement learning algorithms typically rely onfunction approximation. Whether they are learning the value function, policy or models, the success of function approximation techniques hinges on the quality of the representation. Typically, representations are hand-crafted, with some common representations including tile-coding, radial basis functions, polynomial basis functions and Fourier basis functions <cit.>.Automating feature discovery, however, alleviates this burden and has the potential to significantly improve learning.Representation learning techniques in reinforcement learninghave typically drawn on the large literature in unsupervised and supervised learning.Common approaches include feature selection, including ℓ_1 regularization on the value function parameters <cit.> and matching pursuit <cit.>; basis-function adaptation approaches <cit.>; instance-based approaches, such as locally weighted regression <cit.>, sparse distributed memories <cit.>, proto-value functions <cit.> and manifold learning techniques <cit.>; and neural network approaches, including more standard feedforward neural networks <cit.> as well as random representations <cit.>, linear threshold unit search <cit.>, and evolutionary algorithms like NEAT <cit.>. Surprisingly, however, there has been little investigation into using sparse coding for reinforcement learning.Sparse coding approaches have been developed to learn MDP models for transfer learning <cit.>; outside this work, however, little has been explored. Nonetheless, such sparse coding representations have several advantages, including that they naturally enable local models, are computationally efficient to use, are much simpler to train than more complicated models such as neural networks and are biologically motivated by the observed representation in the mammalian cortex <cit.>.In this work, we develop a principled sparse coding objective for policy evaluation. In particular, we formulate a joint optimization over the basis and the value function parameters, to provide a supervised sparse coding objective where the basis is informed by its utility for prediction. We highlight the importance of using the Bellman error or mean-squared return error for this objective, and discuss how the projected Bellman error is not suitable.We then show that,despite being a nonconvex objective, all local minima are global minima, under minimal conditions. We avoid the need for careful initialization strategies needed for previous optimality results for sparse coding<cit.>,using recent results for more general dictionary learning settings <cit.>, particularly by extending beyond smooth regularizers using Γ-convergence.Using this insight, we provide a simple alternating proximal gradient algorithmand demonstrate the utility of learning supervised sparse coding representations versus unsupervised sparse coding and a variety of tile-coding representations.§ BACKGROUND In reinforcement learning, an agent interacts with its environment,receiving observations and selecting actions to maximize a scalarreward signal provided by the environment.This interaction is usually modeled by aMarkov decision process (MDP).An MDP consists of (, , P, R) whereis the set of states; is a finite set of actions;P: ××→ [0,1], the transition function, which describes the probability of reaching a states' from a given state and action (s,a); and finally the reward functionR: ××→, which returns a scalar value for transitioning from state-action(s,a) to state s'. The state of the environment is said to beMarkov if Pr(s_t+1 | s_t , a_t) = Pr(s_t+1| s_t , a_t, …, s_0, a_0). One important goal in reinforcement learning is policy evaluation:learning the value function for a policy. A value function : → approximates the expected return. The return G_t from a state s_t is the total discounted future reward, discounted by γ∈ [0,1),for following policy π: ×→ [0,1] G_t= ∑_i=0^∞γ^i R_t+1+i = R_t+1 + γ G_t+1 where (s_t) is the expectation of this return from state s_t.This value function can also be thought of as a vector of values ∈^ satisfying the Bellman equation =+ γ where(s,s')= ∑_a ∈π(s, a) P(s,a,s') (s)= ∑_a ∈π(s, a) ∑_s' ∈ P(s,a,s') R(s,a,s') Given the reward function and transition probabilities, the solution can be analytically obtained:= ( - γ)^-1.In practice, however, we likely have a prohibitively large state space. The typical strategy in this setting is to use function approximation to learn (s) from a trajectory of samples: a sequence of states, actions, and rewards s_0, a_0, r_0, s_1, a_1, r_1, s_2, r_2, a_2 …, where s_0 is drawn from the start-state distribution,s_t+1∼ P(· | s_t, a_t) and a_t ∼π(· | s_t). Commonly, a linear function is assumed, (s) ≈(s)^⊤for ∈^ a parameter vector and : →^ a feature function describing states. With this approximation, however, typically we can no longer satisfy the Bellman equation in (<ref>), because there may not exist asuch thatequals + γfor ∈^×. Instead, we focus on minimizing the error to the true value function. Reinforcement learning algorithms, such as temporal difference learning and residual gradient, therefore focus on finding an approximate solution to the Bellman equation, despite this representation issue. The quality of the representation is critical to accurately approximatingwith , but also balancing compactness of the representation and speed of learning. Sparse coding, and sparse representations, have proven successful in machine learning and in reinforcement learning, particularly as fixed bases, such as tile coding, radial basis functions and other kernel representations.A natural goal, therefore, and the one we explore in this work, is to investigate learning these sparse representations automatically. § SPARSE CODING FOR REINFORCEMENT LEARNING In this section, we formalize sparse coding for reinforcement learning as a joint optimization over the value function parameters and the representation. We introduce the true objective over all states, and then move to the sampled objective for the algorithm in the next section.We begin by formalizing the representation learning component. Many unsupervised representation learning approaches consist of factorizing input observations[This variablecan also be a base set of features,on which the agent can improve or which the agent can sparsify.] ∈^× into a basis dictionary ∈^× andnew representation ∈^×. The rows ofform a set of bases, with columns inweighting amongst those bases for each observation (column) in . Though simple, this approach encompasses a broad range of models, including PCA, CCA, ISOMAP, locally linear embeddings and sparse coding <cit.>. The (unsupervised) sparse coding objective is <cit.>min_∈^×, ∈^× - _D^2 + _F^2 + _D,1 where _F^2 = ∑_ij_ij^2 is the squared Frobenius norm; ∈^× is a learned basis dictionary; , > 0 determine the magnitudes of the regularizers; ∈ [0,1]^× is a diagonal matrix giving a distribution over states, corresponding to the stationary distribution of the policy : → [0,1]; and ||||^2_D = ^⊤ is a weighted norm. The reconstruction error- _D^2 = ∑_s ∈(s) (s,:)- (s,:) _2^2 is weighted by the stationary distributionbecause states are observed with frequency indicated by .The weighted ℓ_1_D,1 = ∑_s ∈(s)∑_j=1^ |(s,j)|promotes sparsity on the entries of , preferring entries into be entirely pushed to zero rather than spreading magnitude across all of .The Frobenius norm regularizer onensures thatdoes not become too large. Without this regularizer, all magnitude can be shifted to , producing the same , but pushing _D,1 to zero and nullifying the utility of its regularizer.Optimizing this sparse coding objective would select a sparse representationforeach observationsuch thatapproximately reconstructs .Further, however, we would like to learn a new representation that is alsooptimized towards approximating the value function.Towards this aim, we need to jointly learnand , whereprovides the approximate value function. In this way, the optimization must balance between accurately recreatingand approximating the value function . For this, we must choose an objective for learning .We consider two types of objectives: fixed-point objectives and squared-error objectives.Two common fixed-point objectives arethe mean-squared Bellman error (MSBE), also called the Bellman residual <cit.>- ( + γ) ^2_Dand mean-squared projected BE (MSPBE) <cit.>- Π ( + γ) ^2_D where ∈ [0,1]^× is a diagonal matrix giving a distribution over states, corresponding to the stationary distribution of the policy; ||||^2_D = ^⊤ is a weighted norm; and the projection matrix for linear value functions is Π =(^⊤)^-1^⊤.The family of TD algorithms converge to the minimum of the MSPBE,whereas residual gradient algorithms typically use the MSBE (see <cit.> for an overview).Both have useful properties <cit.>, though arguably the MSPBE is more widely used. There are also two alternative squared-error objectives, that do not correspond to fixed-point equations:the mean-squared return error (MSRE) and the Bellman error (BE).For a trajectory of samples {(_i, r_i+1, _i+1)}_i=0^-1, BE is defined as ∑_i=0^-1 r_i+1 + γ_i+1^⊤ - _i^⊤_2^2and the MSRE as∑_i=0^-1 g_i+1 - _i^⊤_2^2where g_i+1 = ∑_j=i^-1γ^j-i r_j+1 is a sample return.In expectation, these objectives are, respectively ∑_s ∈(s) 𝔼[ ( r(S_t,A_t,S_t+1) + γ(S_t+1)^⊤ - (S_t)^⊤)^2 | S_t = s ]∑_s ∈(s) 𝔼[(∑_i=0^∞γ^i r(S_t+i,A_t+i,S_t+1+i) - (s)^⊤)^2 | S_t = s ] where the expectation is w.r.t. the transition probabilities and taking actions according to policy π.These differ from the fixed-point objectives because of the placement of the expectation. To see why, consider the MSBE and BE. The expected value of the BEis the expected squared error between the prediction from this state and the reward plus the value from a possible next state. The MSBE, on the other hand,is the squared error between the prediction from this state and the expected reward plus the expected value for the next state. Though the MSPBE and MSBE constitute the most common objectives chosen for reinforcement learning, these squared-error objectives have also been shown to be useful particularly for learning online <cit.>.For sparse coding, however,the MSPBE is not a suitable choice—compared to the MSBE, BE and MSRE—for two reasons. First, the MSBE, BE and MSRE are all convex in , whereas the MSPBE is not.Second,because of the projection onto the space spanned by the features, the MSPBE can be solved with zero error for any features . Therefore, because it does not inform the choice of , the MSPBE produces a two stage approach[This problem seems to have been overlooked in two approaches for basis adaptation based on the MSPBE: adaptive bases algorithm for the projected Bellman error (ABPBE) <cit.> andmirror descent Q(λ) with basis adaptation <cit.>. For example, for ABPBE, it is not immediately obvious this would be a problem, because a stochastic approximation approach is taken. However, if written as a minimization over the basis parameters and the weights, one would obtain a minimum error solution (i.e., error zero) immediately for any basis parameters. The basis parameters are considered to change on a slow timescale, and the weights on a fast timescale, which is a reflection of this type of separate minimization. <cit.> avoided this problem by explicitly using a two-stage approach, using MSPBE approaches for learning the parameters and using other score function, such as the squared Bellman error, to update the bases. This basis learning approach, however, is unsupervised.Representation learning strategies for the MSPBE have been developed, by using local projections <cit.>. These strategies, however, do not incorporate sparse coding.], wherefeatures are learned in a completely unsupervised way and prediction performance does not influence .The final objective for loss L(, ) set to either MSBE, BE or MSRE is min_∈^,∈^×, ∈^× L(, ) +- _D^2 + _2^2 + _F^2 + _D,1 § ALGORITHM FOR SPARSE CODING We now derive the algorithm for sparse coding for policy evaluation: . We generically consider either the BE or MSRE.For a trajectory of samples {(_i, r_i+1, _i+1)}_i=0^-1,the objective is min_∈^,∈^+1 ×, ∈^×1/∑_i=0^-1 ( y_i + γ̅_i+1^⊤ - _i^⊤ )^2 + /∑_i=0^_i- _i_2^2 + _F^2+ _2^2 + /∑_i=0^_i ^p_1 . for BE, y_i = r_i+1 and γ̅ = γ and for MSRE, y_i = ∑_j=i^γ^j-i r_j+1 and γ̅ = 0.We consider two possible powers for the ℓ_1 norm p = 1 or 2, where the theory relies on using p=2, but in practice we find they perform equivalently and p=1 provides a slightly simpler optimization.The loss is averaged by , to obtain a sample average, which in the limit converges to the expected value under .This averaged loss is also more scale-invariant—in terms of the numbers of samples—to the choice of regularization parameters.consists of alternating amongst these three variables, , and , with a proximal gradient update for the non-differentiable ℓ_1 norm. The loss in terms ofandis differentiable; to solve for(or ) with the other variables fixed, we can simply used gradient descent.To solve forwith theandfixed, however, we cannot use a standard gradient descent update because the ℓ_1 regularizer is non-differentiable.The proximal update consists of stepping in the direction of the gradient for the smooth component of the objective—which is differentiable—and then projecting back to a sparse solution using the proximal operator: a soft thresholding operator.The convergence of this alternating minimization follows from results on block coordinate descent for non-smooth regularizers <cit.>. To apply the standard proximal operator for the ℓ_1 regularizer, we need to compute an upper bound on the Lipschitz constant for this objective.The upper bound is 2(1+γ̅^2)_2^2 + 2_sp^2, computed by finding the maximum singular value of the Hessian of the objective w.r.t. _i for each i. We will provide additional details for this calculation, and implementation details, in a supplement. §.§ Local Minima Are Global Minima In this section, we show that despite nonconvexity, the objective forhas the nice property that all localminima are in fact global minima. Consequently, though there may be many different local minima, they are in fact equivalent in terms of the objective.This result justifies a simple alternating minimization scheme, where convergenceto local minima ensures an optimal solution is obtained. We need the following technical assumption. It is guaranteed to be true for a sufficiently large ≤ (see<cit.>).Assumption 1 For the given ≥, the following function is convex in 𝐙∈^+1 ×+1 min_, , , 𝐙 = _F^2 + _2^2 + /∑_i=0^_i ^2_1 For the objective in equation (<ref>) with p = 2, 1. under Assumption 1, all full-rank local minima are global minima; and2. if a local minimum (, , ) has _:i = (i.e., a zero column) and _i = 0, _i: = (i.e., a zero row) for some 1 ≤ i ≤, then it is a global minimum.For the first statement, we construct a limit of twice-differentiable functions f_n that Γ-converge to theobjective f. With this, we can then show that all minimizers of the sequence converge to minimizers of f, and vice-versa <cit.>.Because all local minimizers of the twice-differentiable functions f_n are global minimizers from <cit.>, we can conclude that all corresponding minimizers of f are global minimizers. We use the pseudo-Huber loss <cit.>, which is twice-differentiable approximation to the absolute value: |x|_μ = √(μ^2 + x^2) - μ. Let = (, , ). The sequence of functions f_n are defined with μ_n = 1/n, as f_n() = L() + /∑_ij√(μ_n^2 + _ij^2) - μ_n where L() equals the equation in (<ref>), but without the ℓ_1^2 regularizer on . Part 1: All local minima of f_n for all n are global minima. To show this, we show each f_n satisfies the conditions of <cit.>.Part 1.1We can rewrite the loss in terms of1/∑_i=0^-1 y_i + γ̅_i+1^⊤ - _i^⊤_2^2 + /∑_i=0^_i- _i_2^2 = 1/ -_2^2 +1/𝐲 -(_0:-1-γ̅_1:)_2^2 where _1:∈^×+1 a diagonal matrix of all ones with the first diagonal entry set to zero, and _0:-1 with the last diagonal entry set to zero.This loss is convex in the joint variablebecause equation (<ref>) is the composition of a convex function (squared norm) and an affine function (multiplication by γ̅_1: - _0:-1 and addition of 𝐲).Part 1.2 The regularizer on must be a weighted Frobenius norm, with weightings on each column; here, we have weighting using regularization parametersfor the firstcolumns (corresponding to ) and regularization parameterfor the last column (corresponding to ).Part 1.3 The inner dimension >, which is true by assumption and the common setting for sparse coding.Part 1.4 The pseudo-Huber loss, on the columns of , is convex, centered and twice-differentiable. Part 2: The sequence f_n converges uniformly to f. To see why, recall the definition of uniform convergence. A sequence of functions { f_n } is uniformly convergent with limit f if for every ϵ > 0, there exists N ∈ such that for all ∈ all n ≥ N, |f_n() - f()| < ϵ.Further recall that for any complete metric space, if f_n is uniformly Cauchy, then it is uniformly convergent. The sequence is uniformly Cauchy if for all n,m ≥ N, |f_n() - f_m()| < ϵ.Take any ϵ > 0 and let N = ⌈4 (+1)ϵ⌉. Then|f_n() - f_m()| =/|( ∑_ij√(μ_n^2 + _ij^2) - μ_n)-(∑_ij√(μ_m^2 + _ij^2) - μ_m )|≤/∑_ij(| √(μ_n^2 + _ij^2) - √(μ_m^2 + _ij^2)| + | μ_n -μ_m | ) The upper bound of the first component is maximized when _ij = 0, and so we get |f_n() - f_m()|≤2(+1)| μ_n -μ_m | ≤2(+1)| 1n -1m|≤4(+1) N≤ϵ . Part 3: Asymptotic equivalence of minimizers of f_n and f.Because f is continuous, and so lower semi-continuous, and f_n uniformly converges to f, we know that f_n Γ-converges to f: f_nf <cit.>. By the fundamental theorem of Γ-convergence, if the { f_n } is an equi-coercive family of functions, then the minimizers of f_n converge to minimizers of f.A sequence of functions { f_n } is equi-coercive iff there exists a lower semi-continuous coercive function ψ: →∪{ -∞, ∞} such that f_n ≥ψ onfor every n ∈ <cit.>. A function is coercive if ψ(θ) →∞ as θ→∞. For ψ(θ) = L(), it is clear that ψ is coercive, as well as lower semi-continuous (since it is continuous). Further, f_n() ≥ L() = ψ(), because the regularizer onis non-negative. Therefore, the family { f_n } is equi-coercive, and so the minimizers of f_n converge to minimizers of f.For the other direction, if a local minimumof f is an isolated local minimum, then there exists a sequence _n → with _n a local minimizer of f_n for μ_n sufficiently small <cit.>. Because we have Frobenius norm regularizers on ,, which are strongly convex, the objective is strictly convex with respect to ,. Further, becauseis full rank,- _F^2 is a strictly convex function with respect to .Therefore, locally the objective is strictly convex with respect to . We therefore know that local minima of f are isolated, and so there exists an N such that for all n > N, _n are local minimizers of f_n. Since these local minimizers are global minimizers,and they converge to , this meansis a global minimum of f.For the second statement, we use <cit.>. Because we already showed above that our loss can be cast as factorization, it is clear our loss and regularizers are positively homogenous, of order 2. A minimum is guaranteed to exist for our objective, because the loss function is continuous, bounded below (by zero) and goes to infinity as the parameters go to ±∞. § EXPERIMENTAL RESULTS We aim to address the question: can we learn useful representations using ? We therefore tackle the setting where the representation is first learned, and then used, to avoid conflating incremental estimation and the utility of the representation. We particularly aim to evaluate estimation accuracy, as well as qualitatively understanding the types of sparse representations learned by .Domains.We conducted experiments in three benchmark RL domains - Mountain Car, Puddle World and Acrobot <cit.>.All domains are episodic, with discount set to 1 until termination.The data inis generated using the standard energy-pumping policy policy with 10% randomness.The data inis generated by a policy that chooses to go North with 50% probability, and East with 50% probability on each step, with the starting position in the lower-left corner of the grid, and the goal in the top-right corner.The data inis generated by a near-optimal policy.Evaluation. We measure value function estimation accuracy usingmean absolute percentage value error (MAPVE), with rollouts to compute the true value estimates. MAPVE = 1/t_test∑_s∈ X_test|V̂(s) - V^*(s)|/|V^*(s)|, whereX_test is the set of test states, t_test=5000 is the number of samples in the test set,V̂(s) is the estimated value of state sand V^*(s) is the true value of state s computed using extensive rollouts.Errors are averaged over 50 runs.Algorithms. We compare to using several fixed tile-coding (TC) representations. TC uses overlapping grids on the observation space. It is a sparse representation that is well known to perform well for , , and . We varied the granularity of the grid-size N and number of tilings D,where D is the number of active features for each observation. The grid is either N×N forandor N^4 for . We explore (D=4, N=4), (D=4,N=8), (D=16,N=4), (D=16,N=8), (D=32,N=4), (D=32,N=8); a grid size of 16 performed poorly, and so is omitted.Forand the number of features respectivelyare 64, 256, 256, 1024, 512, 2048, then hashed to 1024 dimensions; for , the number of features are 1024, 16384, 4096, 65536, 8192, 131072, then hashed to 4096. Both of these hashed sizes are much larger than our chosen = 100. For consistency, once therepresentation is learned, we use the samebatch gradient descent update on the MSRE for all the algorithms, with line search to select step-sizes. The regularization weightsare chosen from {1^-5, …, 1^-1, 0}, based on lowest cumulative error. For convenience,is fixed to be the same as .For learning therepresentations, regularization parameters were chosen using5-fold cross-validation on 5000 training samples, with = 0.1 fixed to give a reasonable level of sparsity. This data is only used to learn the representation; for the learning curves, the weights are learned from scratch in the same way they are learned for TC.The dimension = 100 is set to be smaller than for tile coding, to investigate ifcan learn a more compact sparse representation. We tested unsupervised sparse coding, but the error was poor (approximately 10× worse). We discuss the differences between the representations learned by supervised and unsupervised sparse coding below.Learning curves. We first demonstrate learning with increasing number of samples, in Figure <ref>. The weights are recomputed using the entire batch up to the given number of samples. Across domains,results in faster learning and, inand , obtains lowest final error.Matching the performance of TC is meaningful, as TC is well-understood and optimized for these domains. For , it's clear a larger TC is needed resulting in relatively poor performance, whereascan still perform well with a compact, learned sparse representation.These learning curves provide some insight that we can learn effective sparse representations with ,but also raise some questions. One issue is thatis not as effective inas some of the TC representations, namely 4-4 and 16-4. The reason for this appears to be that we optimize MSRE to obtain the representation, which is a surrogate for the MAPVE. When measuring MSRE instead of MAPVE on the test data,consistently outperforms TC. Optimizing both the representation and weights according to MSRE may have overfitting issues; extensions to MSBE or BE, or improvements in selecting regularization parameters, may alleviate this issue. Learned representations. We also examine the learned representations, both for unsupervised sparse coding and , shown in Figure <ref>. We draw two conclusions from these results: the structure in the observations is not sufficient for unsupervised sparse coding, and the combination of supervised and unsupervised losses sufficiently constrain the space to obtain discriminative representations.For these two-dimensional and four-dimensional observations,it is relatively easy to reconstruct the observations by using only a small subset of dictionary atoms (row vectors ofin equation (<ref>)). The unsupervised representations, even with additional non-negativity constraints to narrow the search space,are less distributed, with darker and thicker blocks, and more frequently pick less features.For the supervised sparse coding representation, however, the sparsity pattern is smoother and more distributed: more features are selected by at least one sample, but the level of sparsity is similar. We further verified the utility of supervised sparse coding, by only optimizing the supervised loss (MSRE), without including the unsupervised loss;the resulting representations looked similar to the purely unsupervised representations. The combination of the two losses, therefore, much more effectively constrains or regularizesthe space of feasible representations and improves discriminative power. The learning demonstrated forhere is under ideal conditions. This was intentionally chosen to focus on the question: can we learn effective sparse representationsusing theobjective? With the promising results here, future work needs to investigate the utility of jointly estimating the representation and learning the value function, as well as providing incremental algorithms for learning the representations and setting the regularization parameters.§ CONCLUSION In this work, we investigated sparse coding for policy evaluation in reinforcement learning. We proposed a supervised sparse coding objective, for joint estimation of the dictionary, sparse representation and value function weights.We provided a simple algorithm that uses alternating minimization on these variables, and proved that this simple and easy-to-use approach is principled. We finally demonstrate results on three benchmark domains, Mountain Car, Puddle World and Acrobot, against a variety of configurations for tile coding. This paper provides a new view of using dictionary learning techniques from machine learning in reinforcement learning. It lays a theoretical and empirical foundation for further investigating sparse coding, and other dictionary learning approaches, for policy evaluation and suggests that they show some promise.Formalizing representation learning as a dictionary learning problem facilitates extending recent and upcoming advances in unsupervised learning to the reinforcement learning setting. For example, though we considered a batch gradient descent approach for this first investigation, the sparse coding objective is amenable to incremental estimation, with several works investigating effective stochastic gradient descent algorithms <cit.>. The generality of the approach and easy to understand optimization make it a promising direction for representation learning in reinforcement learning. named | http://arxiv.org/abs/1707.08316v1 | {
"authors": [
"Lei Le",
"Raksha Kumaraswamy",
"Martha White"
],
"categories": [
"cs.AI",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.AI",
"published": "20170726082304",
"title": "Learning Sparse Representations in Reinforcement Learning with Sparse Coding"
} |
TUMGarching]Lukas HeinrichTUMGarching,ICB]Johannes MüllermycorrespondingauthorTUMWeihen]Aurélien TellierTUMWeihen]Daniel Živković[TUMGarching]Center for Mathematics, Technische Universität München, 85748 Garching, Germany [TUMWeihen]Section of Population Genetics, Center of Life and Food Sciences Weihenstephan, Technische Universität München, 85354 Freising, Germany [ICB]Institute for Computational Biology, Helmholtz Center Munich, 85764 Neuherberg, Germany [mycorrespondingauthor]Corresponding author Population genetics models typically consider a fixed population size anda unique selection coefficient. However, population dynamics inherentlygenerate noise in numbers of individuals and selection acts on variouscomponents of the individuals' fitness. In plant species with seed banks,the size of both the above- and below-ground compartments present noisedepending on seed production and the state of the seed bank. We investigateif this noise has consequences on 1) the rate of genetic drift,and 2) the efficacy of selection. We consider four variants of two-alleleMoran-type models defined by combinations of presence and absence of noisein above-ground and seed bank compartments. Time scale analysis and dimensionreduction methods allow us to reduce the corresponding Fokker-Planck equationto a one-dimensional diffusion approximation of a Moran model.We first show that ifthe above-ground noise classically affects the rate of genetic drift,below-ground noise reduces the diversity storage effect of the seed bank.Second, we consider that selection can act on four different componentsof the plant fitness: plant or seed death rate, seed production or seedgermination. Our striking result is that the efficacy of selection for seeddeath rate or germination rate is reduced by seed bank noise, whereasselection occurring on plant death rate or seed production is not affected.We derive the expected site-frequency spectrum reflecting this heterogeneityin selection efficacy between genes underpinning different plant fitnesscomponents. Our results highlight the importance to consider the effectof ecological noise to predict the impact of seed banks on neutral and selectiveevolution. Diffusion Moran model seed bank selection site-frequency spectrum[2010] 92D10, 39A50,60H10 § INTRODUCTION Genetic drift and natural selection are prominent forces shaping the amount of geneticdiversity in populations. In diploid dioecious organisms, natural selection can bedecomposed in different components: 1) viability selection as the differentialsurvival of the genotypes from zygotes to adults, 2) fecundity (or fertility)selection as the differential zygote production, 3) sexual selection as the differentialsuccess of the genotypes at mating, and 4) gametic selection as the distorted segregationin heterozygotes <cit.>. In effect, population genetic models withdiscrete generations or with Malthusian parameter ignoring age-structure lump thesecomponents into one unique selection parameter. Experimental or genomic populationstudies describing thus changes in allele frequencies often fail to describeand dissect the respective effects of these selective modes. Several theoreticalstudies on fertility <cit.> or on sexual selection <cit.> as well as experimental work on animal <cit.> and plant <cit.>populations have attemptedto disentangle the respective influence of these selection components.In age-structured populations, however, genetic drift and selectioncan act differently than predicted by models without age structure(overview in the book by ).The first type of age-structured models are simply obtainedby individuals' life span and reproduction overlapping severalgenerations. Here, genetic drift acts equally on all individualsof all age classes at any generation. The magnitude of genetic driftis defined by the population size which can be fixed, or fluctuatingfollowing a logistic dynamic size constrained by the population carryingcapacity. As an extreme type of overlapping generation model, the Moranmodel exhibits a rate of random genetic drift all but the same asthe Wright-Fisher (WF) model up to a rescaling of the populationsize (see, e.g., the book by ). Meanwhile, in age-structuredpopulations, selection for fecundity andviability can show different outcomes, such as time to allele fixation and/or maintenanceof alleles, compared to discrete models. This occurs if fecundity or longevity actat different ages of the structured population and under a logistic population sizedynamic <cit.>.An interesting questionarising from the current increasing availability of genomic data is whetherin age-structured populations selection for fecundity can be disentangled from thatof viability using population genomics statistics (such as the site-frequencyspectrum, SFS).A second type of age-structured model is obtained when considering that individuals mayremain as dormant/quiescent structures spanning several generations. Quiescencein reproductive structure is in fact wide-spread, such that seeds or eggs can be persistentstates that allow to buffer a variable environment <cit.>. The time at which offspring germinates or hedges can be variable, such thatonly some of the offspring live in detrimental periods, and most likely at least some ina beneficial environment. Bacterial spores or lysogenic states of temperate phages are alsoexamples of such a bet-hedging strategy. Seedbanks represent thus a storage of geneticdiversity decreasing the probability of population extinction <cit.>, diminishing the effect of genetic drift <cit.>, slowing down theaction of natural selection <cit.>and favouring balancing selection <cit.>.The strength of the seed bank effect clearly depends on the organism under consideration.Dormant seeds and diapausing eggs have a somehow short live span compared tothe average coalescent time,and we call these seed banks “weak”. Dormant states of bacteria, however, can lasta longer time (many generations) even than the average coalescent time. These seed banksare called “strong”, and modelled ina similar way as weak selection: the time scaleof the quiescent state is scaled by the inverse of the population size. In particular<cit.> investigated in a series of papers strongseed banks, and find mathematically appealing results as deviations from the Kingmancoalescent. In the present paper, we focus on weak seed banks aiming at applications to plantor invertebrate species. In a seminal paper <cit.>investigated the effect of weak seed banks on the coalescent.The key parameter here is G that denotes the average number of plant generationsa seed rests in the soil.<cit.> have obtainedthe Kingman n-coalescent rescaled by (1+G)^-2 (the nomenclature in the paperof <cit.> is somewhat different, as they consider non-overlappinggenerations in a discrete time setting, while we formulate the result already for a time continuous model with overlapping generations).It was subsequently shownthat1+G can be estimated using polymorphism data and information on the census sizeof populations <cit.>. Furthermore, neglecting seed banks may yielddistorted results for the inference of past demography using for example theSFS <cit.>. Interestingly, the effectof (weak) selection is enhanced by the slow-down of the time scale due to seedbanks <cit.>. The effect of genetic driftandweak natural selection on allele frequencies can be computed ina diffusion frameworkin a Moran model with deterministic seed bank <cit.>. Whilethe diffusion term, defining genetic drift is scaled by (1+G)^-2, matchingthe backward coalescent result of <cit.>, the coefficientof natural selection in the drift term is multiplied only by (1+G)^-1.In biologicalterms, this means that the strength of selection, as defined by a uniqueselectivecoefficient, is enhanced by the seed bank compared to the effectof genetic drift,even though the time to reach fixation for an allele isincreased <cit.>. We investigate two additions to this current body of theoretical literature on weak seedbanks.First, we compute the effect of realistic models relaxing the hypotheses of a fixed sizefor the population above ground, and of deterministically large for the seed bankcompartment. By doing so, we generate noise in the population size above or below ground.This extends the classic Moran or Wright-Fisher models, as the importance of the noiseeffect on population dynamics on the rate of genetic drift or selection is beingrecognized <cit.>. However, this approach is faced with an additionaldifficulty: if two alleles are present in a population of a constant size,it is sufficient to keep track of the number of individuals for one allele only.In the case of a fluctuating population size with logistic dynamics constrainedby the carrying capacity, the state space becomes essentially two-dimensional.This more general case can still be examined via a time scale analysis and a dimensionreduction by singular perturbation approaches. These methods are applicablein the case where the population size becomeslarge <cit.>. Though in those papers(as in the present one) the arguments are used in a formal way, the validityof this approach is proven <cit.>. This latter analysis revealsthat the dynamics are well described by appropriate scaleddiffusion approximation of a Moran models. Second, we dissect the fitness of plants into four components, which can be possiblyaffected by genetic drift occurring above ground and in the seed bank. We computeclassic population genetics results for neutral and selected alleles and derivethe expected SFS for the alleles under different fitness components.The analysisreveals the effect of noise in the above-ground population (plants) as well asin the below-ground population (seeds). In particular, we find that genetic driftand selection are differently influenced by above and below-ground noise. Additionally,the selection coefficient of alleles involved in seed death rate or germination rateare reduced by seed bank noise, but not by above-ground noise, while the two othersare not affected (plant death rate and seed production). Below-ground noise alsoreduces the seed bank storage effect of neutral genetic diversity.§ METHODSThe aim of the present study is to investigatethe effect of above-ground/below-ground population noise on evolutionin presence of seed banks. We extend the results obtainedin <cit.>,where a seedbank model with deterministic seedbank, fixedabove ground population size and weak selection in seed survivaland seed productionhas been considered. In the present paper, we consider more sourcesof variability and selection also in plant death rate and seed germination,and compare the results of four different models(Figure <ref>):fluctuations/no fluctuations in the total above ground population, intrinsic fluctuations/no fluctuations in the seed bank.Noise in population size can be located eitherabove ground (plant population), or below ground (seed bank or seed population).Traditionally, the plant above-ground population is assumed to consist of N individuals, N being fixed. A noisy substitute of thisassumption is a logistic (fluctuating) populationmodel <cit.>. For seeds, we find inthe literature either the assumption of a finite, fixed number <cit.> of seeds, or, a deterministic infinite seed density <cit.>.In the latter, the assumption is that the number of seeds per plantis large enough,such that intrinsic noise is negligible. We suggest, here, an alternativemodel,in which each plant produces single seeds at time points that aredistributed accordingto a Poisson process. The seeds also die (or loose their ability to germinate) after a random time.Throughout the paper, we only consider Markovian processes, that is,all waiting times are exponentially distributed.The “fluctuatingseedbank” assumption assumesstochastically varying seed bank size. We therefore consider four modelsdefinedby all possible combinations of fixed or logistic above-ground populationand deterministic or fluctuating seed bank.Notation:X_1,t denotes the number of allele-A plants. In a model with fixed above-groundpopulation size N, the number of allele-B plants reads X_2,t=N-X_1,t;in the logistic fluctuating version, we set up a stochastic process for X_1,tand X_2,t. In that case, N does denote the carrying capacity of the population(i.e., the maximal possible size). The average population size (conditioned on non-extinction)is κ N for some κ∈(0,1).Y_t (Z_t) always refers to the amount of allele A (allele B) seeds in thebank. For the deterministic seed bank, Y_t (Z_t) are realnumbers that follow (conditioned on X_1,t, X_2,t) an ordinary differentialequation (ODE). In the case of stochastically fluctuating seed banks,Y_t (Z_t) are non-negative integers, that follow a stochasticbirth-death process.We allow for weak natural selection.The rates for allele B individuals slightly differ from those for alleleA individuals on a scale of 1/N, as it is usual for weakeffects. If σ_i>0, allele B has a disadvantage in comparisonwith allele A in the respective process (where σ_1 addresses the death of a plant, σ_2 the death of seeds,σ_3 the production of seeds, and σ_4 the germination of seeds). Of course, the signsof σ_i can be chosen in an arbitrary way to considera genotype B that has an advantage above genotype A (σ_i<0),or for example a situation where B has a disadvantageabove ground (σ_1, σ_3>0) and anadvantage below ground (σ_2, σ_4<0).The parameters of our models are summarized in Table <ref>. meaning symbol (Allele A) symbol (Allele B)death rate ofplants ζ ζ (1+σ_1/N)death rate of seeds μ μ (1+σ_2/N)production rate of seeds β β (1-σ_3/N)germination rate of seeds (log. pop. only) γγ (1-σ_4/N) tableParameters of the models. §.§ Fixed population size and deterministic seed bank This model has been developed and discussed before <cit.>. The totalabove-ground population has size N, and the transitions for theallele-A plant population X_1,t∈{0,…,N} given theseed densities Y_t and Z_t∈_+ are presented in Table <ref>.E.g., an A-plant dies at rate ζ X_1,t.In the standard Moran model, it is instantaneouslyreplaced by a B-individual withprobability 1-X_t,1/N. In our case, the seeds determine the probabilityfor a B-individual, where this probability is given by Z_t/(Y_t+Z_t). In the same way we obtainthe rate at which a B-individual dies (the rate is ζ (1+σ_1/N) (N-X_1,t)) and is replaced byan A-individual (the probability is Y_t/(Y_t+Z_t)).event offsetratedeath of A, birth of BX_1,t↦ X_1,t-1 ζ X_1,tZ_t/(Y_t+Z_t)death of B, birth of AX_1,t↦ X_1,t+1 ζ (1+σ_1/N) (N-X_1,t)Y_t/(Y_t+Z_t) tablePossible transitions and their rates. In this model we assume that the number of seeds a plant producesis large (basically infinitely large), such that the seed densityin the soil follows a deterministic process, given the history ofthe above ground population. The dynamics of seeds follows a Davis' piecewisedeterministic process <cit.>, where plants produce seedsat rate β (resp. β(1-σ_3/N)) and seeds die at rateμ (resp. μ(1+σ_4/N))Ẏ_t = β X_1,t-μ Y_t,Ż_t = β(1-σ_3/N) (N-X_1,t)-μ(1+σ_2/N) Z_t.§.§ Logistic population dynamics and deterministic seed bank For the logistic model, we do not couple death and birth of a plant asit is usually done in Moran-type models to keep the total populationsize constant. We generalize the logistic dynamics as investigated, e.g.,by <cit.>, or <cit.>for the situation at hand and separate birth and death events. Ifthe seed densities Y_t,Z_t∈_+ are given, the transitionsfor X_1,t, X_2,t∈{0,…,N}, X_1,t+X_2,t≤ N,read as summarized in Table <ref>. event offset rate death of A X_1,t→ X_1,t-1 ζ X_1,t death of B X_2,t→ X_2,t-1 (1+σ_1/N)ζ X_2,t birth of A X_1,t→ X_1,t+1 γ(1-(X_1,t+X_2,t)/N)Y_t birth of B X_2,t→ X_2,t+1 (1-σ_4/N)γ(1-(X_1,t+X_2,t)/N)Z_t tablePossible transitions and their rates. Conditioned on X_1,t and X_2,t, the dynamics of seeds areagain deterministic, Ẏ = β X_1 - μ Y,Ż = (1-σ_3/N)β X_2 - (1+σ_2/N)μ Z. §.§ Fixed population size and fluctuating seed bank For the above-ground population we return to a fixed population size,s.t. it is sufficient to follow X_1,t asX_2,t=N-X_1,t.In the present model we address the noise in the number of seeds, Y_t,Z_t∈_0.The seeds follow a stochastic birth-death process, where the death rateis kept constant, and the birth rate is proportional to the number ofcorresponding above-ground plants. We obtain the transitions summarized in Table <ref>.event offset rate death of A, birth of A (X_1,t,Y_t)→ (X_1,t,Y_t-1) ζ X_1,t Y_t/(Y_t+Z_t) death of A, birth of B (X_1,t,Z_t)→ (X_1,t-1,Z_t-1) ζ X_1,t Z_t/(Y_t+Z_t) death of B, birth of A (X_1,t,Y_t)→ (X_1,t+1,Y_t-1) (1+σ_1/N)ζ (N-X_1,t) Y_t/(Y_t+Z_t) death of B, birth of B (X_1,t,Z_t)→ (X_1,t,Z_t-1) (1+σ_1/N)ζ (N-X_1,t) Z_t/(Y_t+Z_t) birth of A-seed Y_t → Y_t+1 β X_1,t death of A-seed Y_t → Y_t-1 μ Y_t birth of B-seed Z_t → Z_t+1 (1-σ_3/N)β (N-X_1,t) death of B-seed Z_t → Z_t-1 (1+σ_2/N)μ Z_ttablePossible transitions and their rates. §.§ Logistic population dynamics and fluctuating seed bankThe last model incorporates logistic growth and a stochastically fluctuatingseed bank. This model is an obvious combination of the last two models. event offset ratedeath of A X_1,t→ X_1,t-1 ζ X_1,t death of B X_2,t→ X_2,t-1 (1+σ_1/N)ζ X_2,t birth of A (X_1,t,Y_t)→ (X_1,t+1,Y_t-1) γ(1-(X_1,t+X_2,t)/N)Y_tbirth of B (X_2,t,Z_t)→ (X_2,t+1,Z_t-1) (1-σ_4/N)γ(1-(X_1,t+X_2,t)/N)Z_tbirth of A-seed Y_t → Y_t+1 β X_1,t death of A-seed Y_t → Y_t-1 μ Y_t birth of B-seed Z_t → Z_t+1 (1-σ_3/N)β X_2,t death of B-seed Z_t → Z_t-1 (1+σ_2/N)μ Z_t tablePossible transitions and their rates.§.§ Strategy for the analysis of the models The aim of the analysis is the reduction of the four models toa one-dimensional diffusion approximation of a Moran model representing the fraction of allele-A individuals withinthe population. As the details of the analysis are tedious, we present them in detail in the appendix and only outline the basic idea in the present section.The strategy of the analysis differs for the first model(fixed population size, deterministic seed bank) and the other three models.The reason is that there is one single stochastic statevariable X_1,t in the first model, so that if we know the history of X_1,t,the state of the seed bank is known. No dimensionreduction method is thus required. We basically can use the approach of <cit.>to derive the diffusion approximation of a Moran model. However, we outline in appendix <ref>a heuristic argument based on a small-delay approximation, as this route seems to providean appealingshort-cut (though to our knowledge for this approximation,that is used in theoreticalphysics, no hard convergence theorem is available): if ecologicaltime t is not considered but rather the evolutionarytime τ=t/N (population size N large), the delay ofa weak seed bank is small. In this case,the solution can be expanded w.r.t. the delay.As a result, the seed bank can be removed fromthe stochastic process and replaced by appropriatelyrescaled parameters. Since for the time beeing, the short delay approximation is only a heuristic approach, we formulate also a proof for the result based on time scale arguments (as explained next) in appendix <ref>. The three other models have two, three or four stochastic state variables.Methods of dimension reduction are required to obtain a one-dimensionaldiffusion approximation of aMoran model. The key insight here is that any realization saddles faston a one-dimensional manifold. If we consider, e.g., the deterministic versionof the logistic population dynamics with stochastically fluctuating seed bank,we find, according to arguments by, e.g., <cit.>, for the deterministic limit as N→∞(with x_i(t)=X_t,i/N, y(t)=Y_t/N, z(t)=Z_t/N) ẋ_̇1̇ = γ(1-x_1-x_2)y -ζ x_1ẋ_̇2̇ = γ(1-x_1-x_2)z -ζ x_2ẏ = β x_1 - μ y ż = β x_2 - μ zIt is straightforwardto show that a line of stable equilibria,the so-called coexistence line, exists:Let ϑ := (β-ζ)/μ>0, κ := (γϑ-ζ)/(γϑ) ∈ [0,1]. Then, there is a line of stationary points in [0,κ]^2×ℝ^2_+ given by(x_1,x_2,y,z)=(x,κ-x,ϑ x,ϑ(κ-x)), x∈[0,κ].The line of stationary points is transversally stable(locally and globally in the positive cone).It turns out that the stochastic process rapidly approaches this line of equilibria, and performs a random walk close toit (see Figure <ref>). The analysis reveals thatthe distribution on a transversal cut is just a normal distribution witha variance of O(1/N). Along the line of stationary points, however,the realizations will moveaccording to a one-dimensional diffusion approximation of a Moran process.This approximative process is a combinationof one component of the full process parallel to this line,and a second component that results from an interaction between a componentperpendicular to this line with the deterministic vector field directedtowards this line.In order to reveal this structure, we firstuse a large population limit (Kramers-Moyal expansion) to obtain a Fokker-Planck/Kolmogorovforward equation for the full process. In the second step we apply singularperturbation methods as described, e.g., in <cit.> or <cit.> to perform the dimensionreduction to the one-dimensional Moran model. § RESULTS§.§ Timescales for different seed bank models For all of our models, the resulting one-dimensional Fokker-Planck equationassumes the form of a diffusion approximation of a Moran-model with weak selection, ∂_τu = -σ a∂_{(1-)u} +1/2b∂_^2{(1-)u}where a and b describe the speed of selection and genetic drift, respectively. The termσ represents selective coefficients and is a generic parameter for notation including σ_i, i=1,…,4. In order to formulatethe results for a and b, let us introduce three composite parameters:G=ζ/μ is the number of plant generations a seed survives on average;Y=β/ζ is the average number of seeds produced by a plant;κ already defined in Proposition <ref> is the averagefraction of theabove-ground population size in the logistic model,in comparison with the maximal possiblepopulation size N. Furthermore, for deterministic and fluctuating seed banks, we respectively denote (1+G)^-1 and (1+(1-1/Y)G)^-1 as 𝒢, which can be seen as the number of plant generations that seeds survive on average corrected by the size of the seed bank. Using these abbreviations, the parameters a, band σ for our four models are summarized in Table <ref>. h]llll model/scale term a(selection) b(genetic drift) σ (selection coeff.)fix. pop., det. s.b.ζ 𝒢^-1 2ζ 𝒢^-2 σ_1+σ_2+σ_3fix. pop., fluct. s.b. ζ 𝒢^-1 2ζ 𝒢^-2 σ_1+(1-1/Y)σ_2+σ_3log. pop., det. s.b.ζ 𝒢^-1 2ζ 𝒢^-2 κ^-1 σ_1+σ_2+σ_3+σ_4log. pop., fluct. s.b.ζ 𝒢^-1 2ζ 𝒢^-2 κ^-1σ_1+σ_2+σ_3+(1-1/Y)σ_4 tableDrift, diffusion, and selection coefficients for the different population/seed bank models.The basic seed bank model (fixed population size, deterministic seed bank)and the standard Moran model without seed bankcan be used as reference models.A seed bank slows down the time scale of selection as well as that of genetic drift <cit.>,where selection is less affected (by a factor of (1+G)^-1) than genetic drift (by a factor of (1+G)^-2).We find that fluctuations in the above-ground population and in the seeds have different effects.Fluctuations in the seed number reduce the storage effect of seed banks. The additional noise yields a reduction of the effective time a seed spends within the seed bank, and thus increases the rate of genetic drift. For Y→∞(noise in seed bank tends to zero), we obtain the result for the deterministicseed bank, for Y→ 1 (noise is maximized), the model converges towardsthe standard Moran model without seed bank. Note that Yis not the average number of seeds per plant directly measured but the effective number of seeds per plants.For example, a certain fraction of seeds might be getting lost due to other environmental reasons (abiotic or biotic factors) than their intrinsic mortality.These seeds do not contribute to the bank. The noise in the above-ground population only affects genetic drift and does not appear in the selection term.This result reflects thatthe actual competition between alleles A and B only happens above ground.Nonlinear terms in the transition rates only appear in the birth term of the plants.By increasing solely genetic drift, the above-ground noise can counteract the amplification of selection by seed banks. The scaling of selection by 𝒢^-1 and thatofgenetic drift by 𝒢^-2 is somehow expected.All mutations are affected in the sameway by the above-ground noise. Our result concerning the lumpedselection coefficient σ, however, is unexpected:Mutations for some fitness components (mortality of seeds, σ_2,and germination ability, σ_4) show reduced selection while this isnot the case for selective coefficients of other fitness components(σ_1 andσ_3).This means that if the number of seeds per plant is not too large,beneficial mutations in the mortality of seeds (σ_2) havea reduced chance to reach fixation compared witha beneficial mutation for the production of seeds (σ_3). §.§ Site-frequency spectrum (SFS) The SFS is a commonly used statistic for the analysis of genomewide distributed SNPs. It is defined as the distribution of the number of times a mutation is observed in a population or a sample of n sequences conditional on segregation.Herein, this distribution is taken over numerous unlinked sites and mutations occur only on previously monomorphic ones <cit.> at rate θ/2 per N generations.Each mutant allele that arises from the wildtype at such an independent site is assumed to marginally follow the diffusion model specified in (<ref>) so that in particular all mutants have equivalent selective effects among sites. Mutations are allowed to occur in plants and seeds, but the following results can be easily adapted to the scenario, where mutations may only arise in plants. The proportion of sites at equilibrium, where the mutant frequency is in (y, y + dy), is routinely obtained as(e.g., )f̂(y)=θ/by (1-y)1-exp{-2 a/b σ (1-y)}/1-exp{-2 a/b σ},where a, b and σ≠0 are given in Table <ref>. The sample SFSat equilibrium can be immediately obtained from (<ref>) via binomialsampling as f̂_n,k=θn/bk(n-k)1-_1F_1(k;n;2 a/b σ)e^-2 a/b σ/1-e^-2 a/b σ,where _1F_1 denotes the confluent hypergeometric function of the firstkind <cit.>. The neutral versionsof (<ref>) and (<ref>) are respectively givenby f̂(y)=θ/(b y) and f̂_n,k=θ/(b k). Fig. <ref> indicates the striking effects of the seed bank noisefor the model with logistic population dynamics and fluctuating seed banks.When the noise is low (Y high) the number of segregating sites is expectedto be high, and mutations involved in the selection coefficients σ_1and σ_4 lead to similar SFS. The SFS show the typical U-shape expectedunder positive pervasive selection (left panel). If, however, Y becomes small, the numberof segregating sites decreases, and the selection coefficientσ_1 showsa U-shaped SFS while mutations under σ_4 do not (right panel). § DISCUSSION We considered four models to investigate the effect of combined noisein the above-ground population and below-ground seed bank. In all four cases time scale arguments allow us a reduction toa diffusion approximation of a Moran model. Our results extendthe findings that a seed bank without noise yields a change oftime scale in selection and genetic drift that amplifies the effect of weak selection <cit.>.The first main result of this present work is that there is no directinteraction between the noise above ground and below ground.The above-ground noise increases the effect of genetic driftcompared to a fixed above-ground population size, but doesnot affect selection. One can propose a new definition ofthe effective population size N_e =κ 𝒢^2 N/2describing the change in genetic drift due to this noise(according to <cit.>.This allows us to redefine the evolutionary time scaleT = ζ t/N_e.As a result of this procedure the parameter κ,which represents the reduction of the averagetotal population size and indicates the increase of the above-groundnoise by the logistic population dynamics, appears as a factorin front of the selectionterm in the Moran model. Note that in this notation we find that the terms for below-ground(𝒢^2) and above-ground (κ) noiseare multiplied indicating their independence. The second result is that the below-ground noise affects the scaling of timefor both the selection and thegenetic drift term. We introduce the concept of the mean effective number ofseeds per plant in a similar definition to the effective populationsize <cit.>.If this average seed number tends to infinity, we recover the effect ofa deterministic seed bank (no noise), and if this number tends to one,the noise of the seed bank accelerates the time scale such thatthe seed bank has no effect at all. The magnitude of noise in the seed bank isthus tuned between the two extreme cases of “no seed bank” (minimal effect)and “deterministic seed bank” (maximal effect).The third result of the present study isthe insight that below-ground noise may affect the four plants' fitness componentsof viability and fecundity differentially. If thebelow-ground noise is large, the selective effect ofmutations involved in seed death and germination may even be cancelled out.Other mutations involved in fitness traits of the above-ground population, suchas plant death or seed production are not affected. In other words, while finitesize and noise above ground do not affect selection, noise and finite size of theseed bank do change the selection coefficients. In biological terms we interpret this result as follows.Above ground, the fate of an allele under positive selectionis classically determined by the strength of genetic drift, whichdepends on the population size (the diffusion term in the Fokker-Planck equation)compared to the strength of selection which depends on the selectioncoefficient (the drift term in the forward model). Under a deterministicseed bank (as in the present models and in <cit.>), selectionis efficient because it occurs on plants, when they are above ground, with a probability (1+G)^-1 and genetic drift occurs on a coalescent scale of (1+G)^-2.Any change in the allele frequencies above ground translates directly into the deterministicseed bank, just with a small time delay. However, when the seed bank hasa fluctuating finite size, the strength of selection on seed fitness (thecoefficients σ_2 and σ_4) is decreased by the noise in theseed compartment. This occurs because selection and genetic drift in the bankoccur at every generation and not only when seeds germinate.This observation implies that competition experiments, which measure andcompare the effect of mutations and/or allow to determine selection coefficientson the ecological time scale can hardly be used to extrapolate to the evolutionarytime scale. Indeed, even if two mutations seem to have an equivalent value forthe plant fitness in a competition experiment, the presence of seed bank noise maylead to different evolutionary outcomes.We finally discuss two ways of using genome polymorphism data to estimate seed bankparameters as well as the selection coefficients.We can attempt to infer the seed bank parameters based on neutral genetic diversity under the idealized conditions that the sample SFS reached an equilibrium as f̂_n,k (i.e., there is no recent demographic impact) and that we can measure or estimate the population mutation rate θ and the death rate of plants ζ. It turns out that 𝒢 (the number of plant generations a seed survives on average) and κ (the average above-groundpopulation size) can then be identified, so that it is possible to disentangle the effectsof the seed bank from those of the above-ground noise on neutral evolution by utilizing f̂_n,k. However,we cannot identify the below-ground noise, as Gand Y only appear in the composite parameter 𝒢. Note that if we only have information about the relativesample SFS f̂_n,k/∑_if̂_n,i, the multiplicativeconstant in f_n,· cancels out, and therefore only κ 𝒢 and the combined effect of above- and below-ground noise can be estimated.Extending the work by <cit.>, we suggest that the number of plantgenerations a seed survives on average can be estimated from the absolute SFS using for example a Bayesian inference method withpriors on the census size of the above-ground population and the death rate of plants.We may also aim to infer the selection coefficients underpinning the variousplants' fitness components, namely the fecundity and viability of plants or seeds.With the abundance of gene expression, molecular and Gene Ontology data, it becomes feasibleto group genes by categories of function or pathways, for example to know the genes involved in seedgermination or seed integrity, viability and seed dormancy (e.g., ). These functional groups of genes actually underlie the different plants' fitness components investigated in this study. Using genome-wide polymorphism data of several individuals, the SFS for each functional groups of genes can be computed and used to draw inference of selection, for example as the distribution of selective effects (e.g., the method by ). Our prediction is thus that the SFS would reflect the differential selection on thesefitness components and can be observed over the genes involved in the different functional groups. The limitation in current data lies so far on the functional side, as more gene expression study are needed to assigne genes to functional network and to different plant fitness components. As an extension, our expected SFS shows that the behavior of the seed fitness coefficients can be affectedby the below-ground noise. So, we predict that populations with a small sized seed bankshould exhibit less selection signatures on genes relatedto seed fitness compared to populations with a larger seed bank compartment. The same analysis as above can be conducted, but now comparing the SFS and inferred selection for the different functional groups of genes across several populations with know ecological set-ups which define the seed bank size. We further suggest that such procedure can be applied to disentangle selection for fecundity from that for viability in age-structured populations, if functional groups of genes can be assigned to these traits (for example using gene expression at different life stages and ages).Our results differ thus from those of classic age-structured populations by the overlapof generations, as the seed bank can present its own rate of genetic drift. Moreover,selection acts differently above ground and below ground on the different plants' fitnesscomponents, which may allow us to disentangle their effect on the overall selection coefficient.§ ACKNOWLEDGEMENTSThis research is supported in part by Deutsche Forschungsgemeinschaft grants TE 809/1 (AT), STE 325/14 from the Priority Program 1590 (DZ),and MU2339/2-2 from the Priority Program 1617 (JM).§ REFERENCESelsarticle-harv§ ANALYSIS OF THE MODELSWe present the conceptions for the analysis of the model of a fixed population size and a deterministic seed bank as well as forlogistic population dynamics and a deterministic seed bank in detail. Note that the computations for the remaining two models are similar but even more extensive, so that we do not present them in full length but only mention the results of themain steps. The computations for the dimension reduction have been checked usingthe computer algebra package MAXIMA <cit.> (see the available supplementary files). §.§ Fixed population size and deterministic seed bank To keep the demonstration short, we present a nice but heuristic argument using theidea of a short delay approximation <cit.>. To our knowledge no rigorousapproximation theorem is available, therefore wealso give an alternative approach, based on time scale analysis, at the end of the appendix (appendix <ref>). Both approaches yield identical results. Using the variation-of constant formula, we find Y_t = β∫_0^∞ e^-μ sX_t-sds,Z_t = β(1-σ_3/N)∫_0^∞ e^-μ(1+σ_2/N)s(N-X_t-s)ds.Let x_t=X_1,t/N and ε^2=1/N. Then, X_1,t→ X_1,t+1ε^-2(1+σ_1ε^2) ζ (1-x_t) ∫_0^∞ e^-μ sx_t-sds/∫_0^∞ e^-μ sx_t-sds+(1-σ_3 ε^2)∫_0^∞ e^-μ(1+σ_2 ε^2)s(1-x_t-s)ds, X_1,t→ X_1,t-1ε^-2 ζ x_t (1-σ_3^2)∫_0^∞ e^-μ(1+σ_2 ε^2)(t-s)(1-x_t-s)ds/∫_0^∞ e^-μ sx_t-sds+(1-σ_3 ε^2)∫_0^∞ e^-μ(1+σ_2 ε^2)s(1-x_t-s)ds.With standard arguments, we obtain a stochasticdelay differential equation (SDDE) at the evolutionarytime scale τ = t ε^2 asdx_τ = (ε^-4(1+σ_1ε^2) ζ (1-x_τ) ∫_0^∞ e^-μ s/^2x_τ-sds/^-2∫_0^∞ e^-μ s/^2x_τ-sds+^-2(1-σ_3 ε^2)∫_0^∞ e^-μ(1+σ_2 ε^2) s/^2(1-x_τ-s)ds - ε^-4 ζ x_τ ∫_0^∞ e^-μ(1+σ_2 ε^2)s/^2(1-x_τ-s)ds/^-2∫_0^∞ e^-μ s/^2x_τ-sds+^-2(1-σ_3 ε^2)∫_0^∞ e^-μ(1+σ_2 ε^2)s/^2(1-x_τ-s)ds)dτ+ (^-2 (1+σ_1ε^2)ζ (1-x_τ) ∫_0^∞ e^-μ s /^2x_τ-sds/^-2∫_0^∞e^-μ s/^2x_τ-sds+^-2 (1-σ_3 ε^2)∫_0^∞e^-μ(1+σ_2 ε^2)s/^2 (1-x_τ-s)ds + ^-2 ζ x_τ ∫_0^∞ e^-μ(1+σ_2 ε^2)s/^2(1-x_τ-s)ds/^-2∫_0^∞ e^-μ s/^2x_τ-sds+^-2(1-σ_3 ε^2)∫_0^∞ e^-μ(1+σ_2 ε^2)s/^2(1-x_τ-s)ds)^1/2 dW_τ.We aim a small delay approximation. Therefore, we note that for a function Φ(t), which is sufficiently smooth and bounded, we have (for μ>0)^-2 μ ∫_0^∞ e^-μ(τ-s)/^2Φ(τ-s)ds= ^-2∫_0^∞μ e^-μ s/^2(Φ(τ)-sΦ'(τ)+ O(s^2))ds= Φ(τ)-^2μ^-1Φ'(τ)+ O(^4).Thus, at a formal level, lim_→ 0ζ(^-4 μ ∫_0^∞ e^-μ s/^2x_τ-sds-^-2 x_τ)dτ = -ζ/μdx_τ = -Gdx_τ,where G=ζ/μ.Note that this equation has to be interpreted interms of the Euler-Maruyama approximation of an SDDE,where differential quotients are replaced by difference quotients.We again emphasize that this approach is onlymend to be formal, as to our knowledge no rigorous approximation theoremsfor small delay approximations in the context of SDDE are available. The result isconsistent with the time scale analysis of thepresent model (appendix <ref>); we conjecture that the short delay approach yields a valid approximation under suited (rather general) conditions.If we add - ζ(^-4 μ ∫_0^∞ e^-μ s/^2x_τ-sds- ^-2 x_τ)dτ/^-2μ ∫_0^∞ e^-μ s/^2x_τ-sds+(1-σ_3 ε^2) ^-2 μ ∫_0^∞ e^-μ(1+σ_2 ε^2)s/^2(1-x_τ-s)ds to both sides of the SDDE and let → 0, we obtain(1+G)dx_τ = ζ(σ_1+σ_2+σ_3)x_τ(1-x_τ)dτ + (2ζ x_τ(1-x_τ))^1/2dW_τ.This equation yields the desired result withσ=σ_1+σ_2+σ_3. §.§ Logistic population dynamics and deterministic seed bank Let p_i,j(k,l,t)=ℙ(X_1,t=i,X_2,t=j,Y_t∈(k,k+dk),Z_t∈(l,l+dl)) be thejoint probability density of the resulting stochastic process (with discrete i,j and continuous k,l). The corresponding master equation reads:ṗ_i,j(k,l,t) + ∇[[ β i-μ k; β (1-σ_3/N) j-μ (1+σ_2/N) l ]p_i,j(k,l,t)] =-[ζ (i + (1+σ_1/N)j) + γN-i-j/N(k+(1-σ_4/N)l]p_i,j(k,l,t) + ζ(i+1)p_i+1,j(k,l,t) + (1+σ_1/N)ζ(j+1)p_i,j+1(k,l,t) + γ(N-i-j+1)k/N p_i-1,j(k,l,t) + (1-σ_4/N)γ(N-i-j+1)l/N p_i,j-1(k,l,t),where the operator ∇ acts with respect to continuous state space variables k,l.Standard arguments yield the Fokker-Plank-approximationfor large populations, where_̣tu(x_1,x_2,y,z,t) =∂_x_1{[ζ x_1-γ(1-x_1-x_2)y]u(x_1,x_2,y,z,t)}+∂_x_2{[(1+σ_1/N)ζ x_2-(1-σ_4/N)γ(1-x_1-x_2)z]u(x_1,x_2,y,z,t)}+_̣y{[(μ y -β x_1)]u(x_1,x_2,y,z,t)}+_̣z{[(1+σ_2/N)μ z -(1-σ_3/N)β x_2]u(x_1,x_2,y,z,t)}+1/2N_̣x_1^2{[ζ x_1+γ(1-x_1-x_2)y]u(x_1,x_2,y,z,t)}+1/2N_̣x_2^2{[(1+σ_1/N)ζ x_2+(1-σ_4/N)γ(1-x_1-x_2)z]u(x_1,x_2,y,z,t)}.Note that the second order noise terms are solelydue to x_1 and x_2; no noise is added by theseed bank variables y and z. §.§.§ Deterministic modelThe corresponding deterministic model(drift terms only)yields the ODEsẋ_̇1̇ = γ(1-x_1-x_2)y -ζ x_1,ẋ_̇2̇ = γ(1-x_1-x_2)z -ζ x_2,ẏ = β x_1 - μ y, ż = β x_2 - μ z.The lifetime reproductive success (or basic reproduction number) of a plant readsR_0=β γ/(μζ). If R_0>1, the plant population can persist. Under this condition, there is a line of stationary solutions:Assume R_0>0.Let ϑ := β/μ, κ := (γϑ-ζ)/γϑ. Then, κ∈[0,1], and there is a line of stationary points in [0,κ]^2×ℝ^2_+ given by(x_1,x_2,y,z)=(κ x,κ (1-x),ϑ κ x,ϑ κ (1-x)), x∈[0,1].The line of stationary points is transversally stable (locally and globally).Proof:It is straightforward to check that the line indicated aboveconsists indeed of stationary points with non-negative values. The local stability is a consequence of Hartman-Grobmanand the analysis of the Jacobian (due to the block structureof this matrix the eigenvalues can be stated explicitly).Note that one eigenvalue necessarily is zero with an eigenvector pointing in the direction of the line of stationary points. Now, to show global stability, we prove that the system will approach the equilibrium line from any starting point. We first respectively denote P=x_1+x_2 and S=y+z as the total plant and seed populations and consider the resulting reduced systemṖ = γ(1-P)S -ζ P, Ṡ = β P - μ S.Here we use that we only consider weak selection: allselection effects tend to zero for N→∞, and hence the ODE describes the neutral case. The divergenceof this system is negative, and thus the combination of thetheorems of Bendixon-Dulac and Pointcaré-Bendixon implythat trajectories (P,S) tend to stationary points.This observation yields the desired global stability. Note that in equilibrium, x_1+x_2=κ. That is, κNrepresents the average above-ground population size of the modelconditioned on non-extinction. §.§.§ Dimension reduction by time scale analysisThe computations in this section followclosely the calculations in the paper of <cit.> to perform a dimension reduction by a time scale analysis. First, new local variables for the boundary layer around the equilibrium line are defined asx_1 = κ+ε/2 ,x_2 = κ (1-) +ε/2 ,y= κ ϑ +ε/2 (++ϑ ),z =κ ϑ (1-) +ε/2 (-+ϑ ),where=x_1-x_2+κ/2κ,= ε^-1(x_1+x_2-κ),= ε^-1(y+z-ϑ(x_1+x_2)),= ε^-1(y-z-ϑ(x_1-x_2),and ε^2=1/N.For the transformed density ρ(,,,,t;ε)=u(x_1,x_2,y,z,t), we find_̣tρ =ε_̣{[-ζ/2ϑκ-γϑ/2(1-2)]ρ} + ε^2_̣{[γ/2κ+ζ/2(-σ_1-σ_4)(1-)]ρ}+ _̣{[-ζ/ϑ+γϑκ]ρ} + ε_̣{[γ+γϑ^2+(σ_4+σ_1)ζκ(1-)]ρ} + ε^2_̣{…}+ _̣{[μ+ζ-γϑ^2κ]ρ}+ ε_̣{[-γϑ-γϑ^2^2+((-σ_1-σ_4)ζϑ+(σ_3+σ_2)β)κ(1-)]ρ} + ε^2_̣^2{[…]}+ _̣{[μ+ζ+γϑ^2κ(1-2)]ρ}+ ε_̣{[-γϑ+((σ_4+σ_1)ζϑ+(-σ_2-σ_3)β)κ(1-)]ρ} + ε^2_̣^2{[…]}+ ε^2_̣^2{[ζ/4κ]ρ} + ε_̣_̣{[-ζ(1-2)]ρ} + ε^2_̣_̣{[…]}+ ε_̣_̣{[ζϑ(1-2)]ρ} + ε^2_̣_̣{…} + ε_̣_̣{[-ζϑ]ρ} + ε^2_̣_̣{[…]}+ _̣^2{[ζκ]ρ} + ε_̣^2{…} + _̣^2{[ζϑ^2κ]ρ} + ε_̣^2{[…]}+ _̣^2{[ζϑ^2κ]ρ} + ε_̣^2{[…]ρ}+ _̣_̣{[-2ζϑκ]ρ} + ε_̣_̣{…} + _̣_̣{[2ζϑκ(1-2)]ρ} + ε_̣_̣{[…]}+ _̣_̣{[-2ζϑ^2κ(1-2)]ρ} + ε_̣_̣{[…]ρ}, where terms that will not be needed later on are represented by “…”. The reparametrized Fokker-Planck equation in operator notation is_̣tρ(,,,,t;ε) = (Ł+ε + ε^2L^(2))ρ(,,,,t) + 𝒪(ε^3), with linear differential operatorsL^(0)ρ =_̣{[-ζ/ϑ+γϑκ]ρ}+_̣{[(μ+ζ)-γϑ^2κ]ρ}+_̣{[(μ+ζ)+γϑ^2κ(1-2)]ρ}+_̣^2{ζκρ} +_̣^2{ζϑ^2κρ} + _̣^2{ζϑ^2κρ}+ _̣_̣{-2ζϑκρ} + _̣_̣{2ζϑκ(1-2)ρ} + _̣_̣{-2ζϑ^2κ(1-2)ρ},L^(1)ρ=_̣{[-ζ/2ϑκ-γϑ/2(1-2)]ρ}+_̣{[γ+γϑ^2+(σ_4+σ_1)ζκ(1-)]}+_̣{[-γϑ-γϑ^2^2-(σ_1+σ_4)ζϑκ(1-)+(σ_3+σ_2)βκ(1-)]ρ}+_̣{[-γϑ+(σ_4+σ_1)ζϑκ(1-)-(σ_2+σ_3)βκ(1-)]ρ}+_̣_̣{-ζ(1-2)ρ} + _̣_̣{ζϑ(1-2)ρ}+ _̣_̣{-ζϑρ}+_̣^2{…} +_̣^2{…} + _̣^2{…} +_̣_̣{…} +_̣_̣{…} + _̣_̣{…},L^(2)ρ = _̣{[γ/2κ-(σ_1+σ_4)ζ/2(1-)]ρ}+_̣^2{ζ/4κρ}+_̣{…}+_̣{…}+_̣{…}.We employ a time scale separation and focus on a solution evolving on the slow time τ = ε^2t=t/N using the Ansatz ρ(,,,,t) = ρ^(0)(,,,,ε^2t) + ερ^(1)(,,,,ε^2t) + ε^2ρ^(2)(,,,,ε^2t)+𝒪(ε^3). Plugging this into (<ref>) and comparing same order terms, we have L^(0)ρ^(0)=0, L^(0)ρ^(1)=-L^(1)ρ^(0), L^(0)ρ^(2)=_̣τρ^(0)-L^(1)ρ^(1)-L^(2)ρ^(0), which indicates that ρ^(0) can be written as ρ^(0)(,,,,τ) = f(,τ)ρ̂(,,;),with a time independent normal distribution ρ̂(,,;) in ,, that satisfies L^(0)ρ̂=0.The function f(,τ) modifies ρ̂(,,;)and represents the time evolution of ρ^(0) along the line of stationary points. Integrating the last equation of (<ref>) from -∞ to ∞ with respect to ,, - the left hand side is a total derivative w.r.t. variables of integration and becomes zero - yields the evolution equation_̣τf = ∫ρ^(1)d(,,) + ∫ L^(2)d̊(,,).We start by computing the second integral. Since all terms that are full derivatives w.r.t. ,orvanish upon integration, we have∫ L^(2)ρ^(0) d(,,) = _̣{∫[γ/2κ+(-σ_1-σ_4)ζ/2(1-)]ρ^(0)d(,,)} + _̣^2{∫ζ/4κ ρ^(0)d(,,)}= _̣{∫[γ/2κ]ρ^(0)d(,,)} + _̣{(-σ_1-σ_4)ζ/2(1-)f} + _̣^2{ζ/4κf}.We take a closer look at the first integral.We aim to write =(L^(0))^+h^+-g()for suitable h^+, g, where (L^(0))^+ is the adjoint of L^(0).If we can do so, then∫ρ^(0)d(,,) = ∫ h^+L^(0)ρ^(0)d(,,)-g()f(,τ) =-g()f(,τ) .To identify h^+ and g, we reduce theproblem to linear algebra.Define the finite-dimensional vector space (for k∈ℕ)H_k:={P:ℝ^4→ℝ, P polynomial homogenous of degree k (w.r.t. variables ,,)},so that, e.g., γ/2κ ∈ H_2, while γ/2κ/;+c∉ H_2 for c∈ℝ∖{0}. Examining (Ł)^+, we find(Ł)^+h^+ = 0,for h^+∈ H_0.(Ł)^+h^+ = h∈ H_1,forh^+∈ H_1.(Ł)^+h^+ = h+g∈ H_2 ⊕ H_0,forh^+∈ H_2.In particular the last observation (Ł)^+H_2→ H_2 ⊕ H_0, h^+↦(h,g)allows to define an operatorM:H_2→ H_2, h^+↦ h.To simplify notation, we identify vectors w.r.t. a given basisin H_k and the corresponding polynomials. W.r.t. the canonical basis(^2,^2,^2,,,), the operator M has therepresentationM = [-2γϑκ00γϑ^2κ-γϑ^2κ(1-2)0;0-2(μ+ζ)0ζ/ϑ00;00-2(μ+ζ)000; 2ζ/ϑ 2γϑ^2κ0 -(μ+ζ+γϑκ)0-γϑ^2κ(1-2);00 -2γϑ^2κ(1-2)0 -(μ+ζ+γϑκ)γϑ^2κ;0000ζ/ϑ-2(μ+ζ) ],and is invertible, ifall parameters are positive.Let h=(a_1(),…,a_6())^T∈ H_2.Then, (L^(0))^+(h) = M h + g(),where g()∈ H_0 is uniquely defined by g()=Ĝ()h and Ĝ() denoting the row-vectorĜ() = (2ζκ, 2ζϑ^2κ, 2ζϑ^2κ,- 2ζϑκ, 2ζϑκ(1-2), - 2ζϑ^2κ(1-2)). For h̃∈ H_2, we find (L^(0))^+(M^-1h̃) = h̃ + g()with g()=Ĝ()M^-1h̃, so that∫h̃ ρ(,,,,τ) d(,,)=-g()f(,τ).Proof:It is straightforward to obtain the representation of Mand g by applying Ł to the elements of the basis given above. Using, e.g.,Gauß-elimination, we find that M is invertible if the parametersare positive. In order to obtain the i''th component of Ĝ(),consider the i'th entry of the canonical basis b_i, compute (L^(0))^+b_i, and identify the component that is in H_0. E.g., for i=1 we findb_1=^2, and(L^(0))^+^2= -2 [(-ζ/ϑ)+γϑκ]+2ζκ,where -2 [(-ζ/ϑ)+γϑκ]∈ H_2 and2ζκ∈ H_0.Hence, (Ĝ())_1=2ζκ. The equation (L^(0))^+(M^-1h) = h + g() with g()=Ĝ()M^-1 h implies∫ h ρ(,,,,τ) d(,,)=∫ [(L^(0))^+(M^-1h)- g()] ρ(,,,,τ) d(,,)= 0 - g()f(,τ). Solving the system for h(,,,)=γ/2κ, we obtaing() = -γζϑ/2(μ+γϑ)(1-2)by using the computer algebra package MAXIMA <cit.>. In summary, we have∫ L^(2)ρ^(0)d(,,)= _̣{( γζϑ/2(μ+γϑ) +(-σ_1-σ_4)ζ/2) (1-2)f(,τ)} + _̣^2{ζ/4κf(,τ)}.Now, we turn to the computation of the first integral in (<ref>). Withh_0^+ := ζ/2ϑκ(μ+ζ)[+(1-2)+(β+ζϑ)/ζ(1-2)],we obtain (Ł)^+h_0^+ = -ζ/(2ϑκ)-γϑ/2(1-2). Remember that full derivatives w.r.t. ,orvanish upon integration. So,∫ρ^(1)d(,,) = _̣∫[-ζ/2ϑκ-γϑ/2(1-2)]ρ^(1)d(,,) = _̣∫[(Ł)^+h_0^+]ρ^(1)d(,,) =_̣∫ h_0^+ [Łρ^(1)]d(,,) = _̣∫-h_0^+[ρ^(0)]d(,,).To handle ∫-h_0^+[ρ^(0)]d(,,),we use partial integration to move all derivatives in L^(1)w.r.t. ,, from ρ^(0) to h_0^+ (note thath_0^+ is linear in these variables, s.t. all second derivativesin these variables vanish). Let us consider one of these termsoccurring in ∫-h_0^+[ρ^(0)]d(,,):∫[+(1-2)+(β+ζϑ)/ζ(1-2)] ∂_{…}d(,,) = ∫∂_[+(1-2)+(β+ζϑ)/ζ(1-2)] {…}d(,,)= 0. It is not possible to use the sameprocedure for , as we do not integrate w.r.t. . Here we use the product rule, e.g.,∫[+(1-2)+(β+ζϑ)/ζ(1-2)] _̣{[-ζ/2ϑκ-γϑ/2(1-2)]ρ}d(,,) =_̣x ∫[+(1-2)+(β+ζϑ)/ζ(1-2)] {[-ζ/2ϑκ-γϑ/2(1-2)]ρ}d(,,) - ∫_̣[+(1-2)+(β+ζϑ)/ζ(1-2)] {[-ζ/2ϑκ-γϑ/2(1-2)]ρ}d(,,).In this way, we obtain ∫ρ^(1)d(,,)= ζ/2ϑκ(μ+ζ)_̣^2∫[ [t]γϑ^2(μ+ζ)/2ζ(1-2)^2^2+ζ/2ϑκ^2+γϑ/2(1-2)^2+μ+γϑ/2κ(1-2)+ζ/2ϑκ(1-2)]ρ^(0)d(,,) =.5pt-.9pta +ζ/2ϑκ(μ+ζ)_̣∫[γϑ^22μ+ζ/ζ(1-2)^2+γϑμ+ζ/ζ(1-2)+μ+ζ-γϑκ/κ+ζ/ϑκ]ρ^(0)d(,,)=.5pt-.9ptb +_̣∫[(σ_4+σ_1)ζ/2(1-)-(σ_4+σ_1+σ_3+σ_2)ζμ/μ+ζ(1-)]ρ^(0)d(,,)+ζ/2ϑκ(μ+ζ)_̣∫ -β(1-2)_̣{(1-2)ρ^(0)}-_̣{ζϑρ^(0)}d(,,) =.5pt-.9ptc.We will treat all three terms separately starting with .5pt-.9ptc..5pt-.9ptc = _̣{[(σ_4+σ_1)ζ/2(1-)-(σ_4+σ_1+σ_3+σ_2)ζμ/μ+ζ(1-)]f}-ζ/2ϑκ(μ+ζ)_̣^2{[β(1-2)^2+ζϑ]f}-ζ/2ϑκ(μ+ζ)_̣{2β(1-2)f}.To compute .5pt-.9ptb, we proceed as in the computations for ∫ L^(2)ρ^(0)d(,,), that is, we solveM[ a_1(); a_2(); a_3(); a_4(); a_5(); a_6() ] = [ γϑ^22μ+ζ/ζ(1-2); 0; 0;γϑμ+ζ/ζ(1-2); μ+ζ-γϑκ/κ;ζ/ϑκ ]_=̂h,and find.5pt-.9ptb = ζ/2ϑκ(μ+ζ)_̣{-g()f(,τ)} = ζ/2ϑκ(μ+ζ)_̣{[2β-γϑ^2κ(μ+ζ)/μ+γϑ](1-2)f(,τ)}. For .5pt-.9pta, this can be done similarly, where the system to be solved is M[ a_1(); a_2(); a_3(); a_4(); a_5(); a_6() ] = [ γϑ^2(μ+ζ)/2ζ(1-2)^2; 0; ζ/2ϑκ; γϑ/2(1-2)^2;μ+γϑ/2κ(1-2);ζ/2ϑκ(1-2) ]_=̂h, and as a result, we have.5pt-.9pta = ζ/2ϑκ(μ+ζ)_̣^2{-g()f(,τ)}= ζ/2ϑκ(μ+ζ)_̣^2{[β+ζϑ/2(1-2)^2-ζ^2ϑ/2(μ+ζ)[(1-2)^2-1]]f(,τ)}.With∫ρ^(1)d(,,) = .5pt-.9pta +.5pt-.9ptb + .5pt-.9ptcand _̣τf = ∫ρ^(1)d(,,)+∫ L^(2)ρ^(0)d(,,),we obtain_̣τf = ζ/(1+G)^2_̣x^2{(1-)f} + ζ(-σ_1-σ_4-σ_2-σ_3)/1+G_̣x{(1-)f}.§.§ Fixed population size and fluctuating seed bank We only present the main steps for this and the following model, as the computations are lengthy and resemble that of the last model. Let p_i,j,k(t)=ℙ(X_t=i,Y_t=j,Z_t=k) be the probability distribution of the resulting stochastic process. The corresponding master equation reads: ṗ_i,j,k(t)=-[ζ N + σ_1/Nζ (N-i) + β N - σ_3/Nβ (N-i) + μ (j+k) + σ_2/Nμ k]p_i,j,k(t) + [ζi(j+1)/j+k+1]p_i,j+1,k(t) + [ζ(i+1)(k+1)/j+k+1]p_i+1,j,k+1(t) + [(1+σ_1/N)ζ(N-i+1)(j+1)/j+k+1]p_i-1,j+1,k(t) + [(1+σ_1/N)ζ(N-i)(k+1)/j+k+1]p_i,j,k+1(t) + [β i]p_i,j-1,k(t) + [μ (j+1)]p_i,j+1,k(t) + [(1-σ_3/N)β (N-i)]p_i,j,k-1(t) + [(1+σ_2/N)μ (k+1)]p_i,j,k+1(t). We now obtain the Fokker-Planck equation for the approximating diffusion process as described before. The first step is to transform the system to a quasi-continuous state space by scaling with N^-1, i.e., defining x=i/N, y=j/N, z=k/N, h=1/Nand the quasi-continuous density u(x,y,z,t)=p_i,j,k(t). The resulting PDE reads _̣tu(x,y,z,t)= _̣x[(ζ x-ζy/y+z-σ_1/Nζ(1-x)y/y+z)u(x,y,z,t)] + _̣y[(ζy/y+z+(μ y -β x) +σ_1/Nζ(1-x)y/y+z)u(x,y,z,t)] + _̣z[(ζz/y+z+(μ z -β (1-x)) +σ_1/Nζ(1-x)z/y+z + σ_3/Nβ(1-x) +σ_2/Nμ z )u(x,y,z,t)] + 1/2N_̣x^2[(ζxz/y+z+(1+σ_1/N)ζ(1-x)y/y+z)u(x,y,z,t)]+ 1/2N_̣y^2[(ζxy/y+z+(1+σ_1/N)ζ(1-x)y/y+z+β x + μ y)u(x,y,z,t)]+ 1/2N_̣z^2[(ζxz/y+z+(1+σ_1/N)ζ(1-x)z/y+z+(1-σ_3/N)β (1-x) + (1+σ_2/N)μ z)u(x,y,z,t)]+ 1/N_̣x_̣y[-(1+σ_1/N)ζ(1-x)y/y+zu(x,y,z,t)] + 1/N_̣x_̣z[ζxz/y+zu(x,y,z,t)].For N→∞, we obtain a deterministic model, governed by the ODEsẋ = -ζ x + ζy/y+z = -ζ xz/y+z + ζ(1-x)y/y+z,ẏ = β x - μ y - ζy/y+z,ż = β(1-x) - μ z - ζz/y+z,which is a neutral competition model since the selection terms vanish in the limit.Let ϑ = (β-ζ)/μ>0. Then, there is a line of stationary points for (<ref>) in [0,1]×ℝ^2_+ given by(x,y,z)=(x,ϑ x,ϑ(1-x)),x∈[0,1].The line of stationary points is transversally stable. The eigenvectors perpendicular to the line of stationary points (together with the eigenvalues) readX_1 = [ 0; x; 1-x ],λ_1 = -μ,X_2 = [ζ; -β;β ],λ_2 = -ζ-β/ϑ. The proof of this proposition is straight forward, along the lines of the proof of Proposition <ref>.To formulate the inner solution, we introduce local coordinates =x,= ε^-1(y+z-ϑ),= ε^-1(y-z-ϑ(2x-1),ρ(,,,t)=u(x,y,z,t), where ϑ=(β-ζ)/μ again and ε^2=1/N. Alternatively formulated, we have x = , y = ε/2(+)+ϑ x,z = ε/2(-)+ϑ(1-x).can be thought of as measuring the deviation of the total amount of seeds from its deterministic value ϑ andas measuring the deviation of the allele ratio in seeds from the allele ratio in plants. Both are scaled by ε^-1 so we expect them to be of order 𝒪(1). By transforming derivatives, we have_̣x= /x̣_̣ + /x̣_̣ + /x̣_̣ = _̣ - 2ε^-1ϑ_̣,_̣y = ε^-1(_̣+_̣),_̣z= ε^-1(_̣-_̣). We now approximate _̣tρ(,,,t) by transforming all terms on the r.h.s. of (<ref>) and ignoring terms of 𝒪(ε^3). We find_̣tρ(,,,t)=(L^(0)+ε L^(1)+ε^2 L^(2))ρ(,,,t)+𝒪(ε^3), with linear differential operators L^(0), L^(1) and L^(2) that take the formL^(0)ρ = _̣{μρ}+_̣{[μ+ζ(1+1/ϑ)(+(1-2))]ρ}+_̣^2{βρ} -_̣_̣{2β(1-2)ρ} + _̣^2{[β+4ζϑ(1+ϑ)(1-)]ρ},L^(1)ρ= -_̣{ζ/2ϑ(+(1-2))ρ} +_̣{[σ_1ζ(1-)+σ_3β(1-)+σ_2μϑ(1-)]ρ}-_̣{[ζ/ϑ(1+1/ϑ)(+(1-2)) -σ_1ζ (2(ϑ+1)-1) (1-)+σ_3β(1-)+σ_2μϑ(1-)]ρ}+_̣^2{μ/2ρ} +_̣^2{[ζ(1+ϑ)(+(1-2))-4ζ(ϑ+1)(1-)+μ/2- 2ζ(-)(1+ϑ) ]ρ}-_̣_̣{2(2ϑ+1)ζ(1-)ρ} +_̣_̣{(μ+ζ(1+1/ϑ)(+(1-2)))ρ},L^(2)ρ = _̣{[ζ/2ϑ^2(+(1-2)) -σ_1ζ(1-)]ρ}+_̣^2{ζ(1-)ρ}+_̣{…}+_̣{…}.Before we proceed, we define the functionsh_0(,;) = -[+(1-2)]/μ+ζ(1+1/ϑ),h_1(,;) = -[(+(1-2))]/2μ+ζ(1+1/ϑ),h_2(,;) = ((1-2))^2ζ(1+1/ϑ)/2(2μ+ζ(1+1/ϑ))(μ+ζ(1+1/ϑ)- (1-2)μ/(2μ+ζ(1+1/ϑ))(μ+ζ(1+1/ϑ) - ^2/2(μ+ζ(1+1/ϑ)), g() = 4((1-))β+ζϑ(1+ϑ)/μ+ζ(1+1/ϑ),and note that(L^(0))^+h_0(,;)= (+(1-2)),(L^(0))^+h_1(,;)= (+(1-2)),(L^(0))^+h_2(,;) + g()= (+(1-2)).That is, h_0 and h_1 are eigenfunctions of (L^(0))^+.Observing the system on slow time τ=ε^2t=t/N,we see rapid dynamics for τ close to zero, τ∈𝒪(1/N)to be precise. This is the new boundary layer. For τ∈𝒪(1),only slow drift effects along the equilibrium line should remain.Being interested in the outer solution that develops on the slow time scale,we make the Ansatzρ(,,,t;ε) = ρ^(0)(,,,ε^2t) + ερ^(1)(,,,ε^2t) + ε^2ρ^(2)(,,,ε^2t)+𝒪(ε^3), assuming _̣tρ=𝒪(ε^2) to focus on the outer solution and neglect the boundary layer. We plug this Ansatz into (<ref>), compare same order terms on both sides and obtain L^(0)ρ^(0)=0, L^(0)ρ^(1)=-L^(1)ρ^(0), L^(0)ρ^(2)=_̣τρ^(0)-L^(1)ρ^(1)-L^(2)ρ^(0). We have ∫(+(1-2))ρ d(,) =∫(L^(0))^+ h^+_1ρ d(,)=0, and hence∫ L^(2)ρ^(0) d(,)= _̣∫[ζ/2ϑ(+(1-2))-σ_1ζ(1-)]ρ^(0) d(,) +_̣^2∫ζ(1-)ρ^(0) d(,)= -σ_1 ζ _̣ ((1-) f(τ,)) + ζ_̣^2(ζ(1-)f(τ,) ).Using the same procedure as in Appendix <ref>,consider (<ref>)-(<ref>), to handle∫ L^1ρ^1 d(,,), and by applying (<ref>) - (<ref>), we find ∫ L^1ρ^1 d(,,) = T_a+T_b+T_c,whereT_a= _̣-ζ^2/2ϑ^2((ϑ+1)ζ+μϑ)∫(+(1-2))ρ^(0) d(,,) = 0,T_b= _̣(-ζ(1-) (-ζσ_1ϑ - ζσ_2 + β(σ_2+σ_3))/(ϑ+1)ζ+μϑ∫ρ^(0) d(,,)), T_c= ^̣2_( ζ^2/4ϑ^2((1+1/ϑ) ζ+μ)∫ (+(1-2)) + (+(1-2))(1-2)) ρ^(0) d(,,))- ^̣2_( ζ^2/ϑ ((1+1/ϑ) ζ+μ)∫ (2ϑ+1)(1-) ρ^(0) d(,,))= ^̣2_( -ζ^2 [(ϑ+1)^2ζ+μϑ(1+2ϑ)+β)] (1-) /(ϑ((1+1/ϑ)ζ+μ))^2∫ ρ^(0) d(,,)). With_̣τf = ∫ρ^(1)d(,,)+∫ L^(2)ρ^(0)d(,,),we find_̣τf(,τ) =- ζ[σ_1+(1-1/Y)σ_2+σ_3]/(1+(1-1/Y)G)_̣{(1-)f(,τ)} + ζ/(1+(1-1/Y)G)^2_̣^2{(1-) f(,τ)}.§.§ Logistic population dynamics and fluctuating seed bankLet p_i,j,k,l(t)=P(X_1,t=i,X_2,t=j,Y_t=k,Z_t=l) be the probability distribution of the resulting stochastic process. The corresponding master equation reads: ṗ_i,j,k,l(t)= -[ [t]ζ (i + (1+σ_1/N)j) + γN-i-j/N(k+(1-σ_4/N)l) + μ (k + (1+σ_2/N)l)+ β (i + (1-σ_3/N)j)]p_i,j(k,l,t) + ζ(i-1)p_i-1,j,k,l(t) + (1--σ_1/N)ζ(j-1)p_i,j-1,k,l(t) + γ (1-(i+1)/N-j/N) (k+1) p_i-1,j,k+1,l(t) + (1-σ_4/N)γ (1-i/N-(j+1)/N) (l+1) p_i,j-1,k,l+1(t) + β i p_i,j,k-1,l(t) + (1-σ_3/N)β j p_i,j,k,l-1(t)+ μ(k+1)p_i,j,k+1,l(t)+ (1+σ_2/N) μ(l+1)p_i,j,k,l+1(t). We transform the system to the quasi-continuous state space with rescaled parameters x_1=i/N, x_2=j/N, y=k/N, z=l/N, h=1/N and quasi-continuous density u(x_1,x_2,y,z,t)=p_i,j,k,l(t). After expanding about (x_1,x_2,y,z) in terms up to order 𝒪(ε^2), we have a 4-dimensional Fokker-Planck equation again: _̣tu=_̣x_1{[ζ x_1-γ(1-x_1-x_2)y]u}+_̣x_2{[(1+σ_1/N)ζ x_2-(1-σ_4/N)γ(1-x_1-x_2)z]u}+_̣y{[μ y -β x_1 + γ(1-x_1-x_2)y]u}+_̣z{[(1+σ_2/N)μ z -(1-σ_3/N)β x_2+ (1-σ_4/N)γ(1-x_1-x_2)z]u}+1/2N_̣x_1^2{[ζ x_1+γ(1-x_1-x_2)y]u}+1/2N_̣x_2^2{[(1+σ_1/N)ζ x_2 +(1-σ_4/N)γ(1-x_1-x_2)z]u}+1/2N_̣y^2{[μ y+β x_1+ γ(1-x_1-x_2)y]u}+1/2N_̣z^2{[ (1+σ_2/N)μ z+ (1-σ_3/N)β x_2 + (1-σ_4/N)γ(1-x_1-x_2)z]u} -1/N_̣x1_̣y{[γ(1-x_1-x_2)y]u}-1/N_̣x2_̣z{[(1-σ_4/N)γ(1-x_1-x_2)z]u}. The limiting deterministic system for N→∞ has dynamicsẋ_̇1̇ = γ(1-x_1-x_2)y -ζ x_1,ẋ_̇2̇ = γ(1-x_1-x_2)z -ζ x_2,ẏ = β x_1 - μ y - γ(1-x_1-x_2)y, ż = β x_2 - μ z - γ(1-x_1-x_2)z.As before, we can show that a line of stable equilibria exists: Let ϑ := (β-ζ)/μ>0,κ := (γϑ-ζ)/γϑ∈ [0,1].Then, there is a line of stationary pointsin [0,κ]^2×ℝ^2_+ given by(x_1,x_2,y,z)=(x, κ-x, ϑ x, ϑ(κ-x)), x∈[0,κ]. The line of stationary points is transversally stable (locally and globally). Since the proof is similar to the proof of Proposition <ref>,it is omitted here.§.§.§ Perturbation approximationAs before, new local variables for the boundary layer around the equilibrium line are defined:x_1 = κ+1/2ε , x_2 = κ (1-) +1/2ε , y= κ ϑ +1/2ε (++ϑ ), z =κ ϑ (1-) +1/2ε (-+ϑ ),where= x_1-x_2+κ/2κ,= ε^-1(x_1+x_2-κ), = ε^-1(y+z-ϑ(x_1+x_2)),= ε^-1(y-z-ϑ(x_1-x_2)),and ε^2=1/N.The considerations for the global behavior are the same as for model 2. We choose the same local variables for the boundary layer around the equilibrium line and transform (<ref>) up to terms of 𝒪(ε^3) and higher.The resulting reparametrized Fokker-Planck equation in the local variables is_̣tρ(,,,,t) = (Ł+ε + ε^2L^(0))ρ(,,,,t) + 𝒪(ε^2),with linear differential operators L^(0)ρ = _̣{[-ζ/ϑ+ γϑκ ] ρ}+ _̣{[(β+ζϑ)/ϑ - γϑ (1-ϑ)κ] ρ}+ _̣{[ (β+ζϑ)/ϑ + γϑ (1+ϑ)κ (1-2)] ρ}+ ^̣2_v{ζ κρ}+ ^̣2_{[ κ (β + ζϑ (1+ϑ))] ρ}+ ^̣2_{[κ (β + ζϑ(1+ϑ))] ρ}- _̣ _̣{ζ (1+2ϑ)κρ}+_̣ _̣{ζ(1+2ϑ)κ (1-2)ρ}-_̣ _̣{[2 (1+ϑ)ϑζ +2 β] κ (1-2)ρ},L^(1)ρ= - _̣{[ ζ/2ϑκ+γϑ/2(1-2)]ρ}+ _̣{[ γ (ϑ +) + (σ_4+σ_1)ζκ(1-) ]ρ}+_̣{[- γ(1+ϑ)(ϑ+) + (σ_2+σ_3)βκ(1-) - (σ_4 (1+1/ϑ)+σ_1)(1-)ζϑκ]ρ}+_̣{[ -γ(1+ϑ) - (σ_2+σ_3)βκ(1-) + (σ_4 (1+1/ϑ)+σ_1)(1-)ζϑκ]ρ} - _̣_̣{ζ (1-2)ρ} + _̣_̣{ζ(1-2)(ϑ+1/2)ρ} - _̣_̣{ζ(ϑ +1/2)ρ} +_̣^2{…} +_̣^2{…} + _̣^2{…} +_̣_̣{…} +_̣_̣{…} + _̣_̣{…}, L^(2)ρ =_̣{[γ/2κ-(σ_1+σ_4)ζ/2(1-)]ρ}+_̣^2{ζ/4κρ}+_̣{…}+_̣{…}+_̣{…}. To handle the integrals below, we define an operatorM:H_2→ H_0 resp. Ĝ:H_2→ H_0 inasimilar way as above (Proposition <ref>).Recall that we use(^2,^2,^2,,,) as the basis in H_2. The operator M has therepresentationM = [ -2γϑκ 0 0γϑ(1+ϑ)κ-γϑ(1+ϑ)κ(1-2) 0; 0-2(β+ζϑ)/ϑ 0 ζ/ϑ 0 0; 0 0-2(β+ζϑ)/ϑ 0 0 0;2ζ/ϑ 2γϑ(1+ϑ)κ 0 -γϑκ-(β+ζϑ)/ϑ 0-γϑ(1+ϑ)κ(1-2); 0 0 -2γϑ(1+ϑ)κ(1-2) 0 -γϑκ-(β+ζϑ)/ϑγϑ(1+ϑ)κ; 0 0 0 0 ζ/ϑ-2(β+ζϑ)/ϑ ].If we define g via g()=Ĝ()h, we findĜ() = κ ( 2ζ,2(ϑ(ϑ+1)ζ+β), (2(ϑ(ϑ+1)ζ+β), - (1+2ϑ)ζ, (1+2ϑ)ζ (1-2), -2(ϑ(1+ϑ)ζ+β)(1-2) ).As before, we employ time scale separation and focus on a solution evolving on the slow time τ = ε^2t=t/N, using the Ansatz ρ(,,,,t) = ρ^(0)(,,,,ε^2t) + ερ^(1)(,,,,ε^2t) + ε^2ρ^(2)(,,,,ε^2t)+𝒪(ε^2).If we define h_0^+ by h_0^+= ζ/2κ(β+ζϑ)[+(1-2)+β+ζϑ/ζ (1-2) ],we find(L^(0))^+h_0^+ = ζ/2ϑκ+γϑ/2(1-2).By now, we have all ingredients together to go along the same route as in Appendix <ref>.We start with ∫ L^(2)ρ^(0)d(,,).If we use γ/2 κ= (L^(0))^+ M^-1( γ/2 κ ) +γζ(β+ζϑ-βϑ)/2μ(β+γϑ^2)(1-2),we obtain ∫ L^(2)ρ^(0)d(,,) = _̣{[γζ(β+ζϑ-βϑ)/2μ(β+γϑ^2)(1-2) - (σ_1+σ_4)ζ(1-)/2] f } +^̣2_{ζ/4κf }.Next, we turn to ∫ L^(1)ρ^(1)d(,,) =-∫ h_0^+ L^(1)ρ^(0)d(,,) withh_0^+=ζ(+(1-2)+(β+ζϑ)(1-2)/ζ)/(2κ(β+ζϑ)).We integrate by parts w.r.t. all derivatives_̣, _̣, _̣, and respectively move the derivatives_̣ in front of the integral by means of the chain rule, so that we obtain -∫ h_0^+ L^(1)ρ^(0)d(,,) = _̣(T_a+T_b+T_c+T_d,1)+^̣2_T_d,2,withT_a= ∫ρ^(0) ζ/2κ(β+ζϑ) (γϑ(β+ζϑ)/2ζ(1-2)^2^2+ ζ^2/2ϑκ+ γϑ/2 (1-2)^2+ β+γϑ^2/2ϑκ(1-2)+ ζ/2ϑκ (1-2)) d(,,),T_b= ∫ζρ^(0)/2κ(β+ζϑ)(γϑ(2β+ζϑ-ζ)/ζ(1-2)^2+ γϑ (μ+ζ)/ζ(1-2)+ β+ζϑ-γϑ(1+ϑ)κ/ϑκ+ ζ/ϑκ )d(,,),T_c= ζ ∫(1/2 (σ_1+σ_4) (1-) -β (σ_1+σ_4(1-ζ/β) -σ_2-σ_3)/β+ζϑ(1-))ρ^(0)d(,,),T_d,1 = ∫( ζ/2κ(β+ζϑ) (ζ-2β) (1-2)ρ^(0))d(,,),T_d,2 =- ∫(-ζ^2(2(-1)-ϑ) -ζβ (4(1-)-1)/2κ(β+ζϑ)) ρ^(0) d(,,).In particular, the integrals in T_a and T_b can be transformedusing the operators M and Ĝ, T_a= ζ/2κ(β+ζϑ) ∫( β+ζϑ/2 (1-2)^2- 2 ζ(β+ζϑ(1+ϑ))/2(β+ζϑ) (1-) ) ) ρ^(0) d(,,),T_b= ζ/2κ(β+ζϑ) ∫( 2β-ζ+γκ (β+ζϑ-βϑ) (β+ζϑ)/μ(β+γϑ^2)))(1-2)ρ^(0) d(,,).Recall that in the present variant of the modelϑ=(β-ζ)/μ,G=ζ/μ and Y=β/ζ. With_̣τf = ∫ρ^(1)d(,,)+∫ L^(2)ρ^(0)d(,,),we find_̣τf = ζ/(1+(1-1/Y)G)^2_̣x^2{(1-)f} + ζ(-σ_1-(1-1/Y)σ_4-σ_2-σ_3)/1+(1-1/Y)G_̣x{(1-)f}. §.§ Fixed population size and deterministic seedbank - Singular perturbation approachLet p_i(k,l,t)=ℙ(X_t=i, Y_t∈(k,k+dk), Z_t∈(l,l+dl)). The master equationsare given by ṗ_i(k,l,t) +∇[ ([ β i-μ l; β (1-σ_3/N)(N-i)-μ(1+σ_2/N) k ])p_i(k,l,t) ]= - [ζ i l/k+l+ζ(1+σ_1/N)(N-i)k/k+l] p_i(k,l,t) + [ζ(i+1)l/k+l] p_i+1(k,l,t) + [ζ (1+σ_1/N)(N-i+1)k/k+l] p_i-1(k,l,t)The usual expansion yields the corresponding Fokker-Planck equation(x=i/N, y=k/N, z=l/N)∂_t u= ∂_x{[ζxz/y+z-ζ(1+σ_1/N)(1-x)y/y+z]u} - ∇_y,z{[[ β x-μ y; β (1-σ_3/N)(1-x)-μ(1+σ_2/N) z ]]u} + 1/2 N∂_x^2{[ ζxz/y+z+ζ(1+σ_1/N)(1-x)y/y+z]u} §.§.§ Deterministic model The drift term of the Fokker-Planck equationdefine the ODE modelẋ =-ζxz/x+z+ζ(1-x)y/y+z ẏ = β x -μ y ż = β(1-x)-μ zwith the line of stationary points(x,y,z) = (y,ϑ x, ϑ(1-x)),ϑ = β/μ, x∈[0,1]. §.§.§ Dimension reduction by time scale analysisWe introduce new coordinates, x = x̃, y = ϑ+1/2(+), z = ϑ (1-) + 1/2(-),ρ(t,x̃,ỹ,z̃)=u(t,x,y,z)where, as before, ϑ= β/μ and ^2=1/N. With=x,= ^-1(y+z-ϑ),= ^-1(y-z+ϑ (1-2x)),we obtain∂_x= ∂_-2^-1ϑ∂_,∂_y= ^-1(∂_ +∂_), ∂_z = ^-1(∂_ -∂_).We transform the Fokker-Planck equation, neglecting terms of O(^3).For ρ(,,,t;) we obtain ∂_tρ=L^(0)ρ+ L^(1)ρ+^2L^(2)ρ with L^(0)ρ =∂_[(μ )ρ]+ ∂_[(μ+ζ(+(1-2)))ρ]+ ∂_^2[(4ζϑ^2(1-)))ρ] L^(1)ρ = - ∂_[( ζ(+(1-2))/(2ϑ))ρ] - ∂_[( - σ_3β(1-ß) - σ_2μϑ(1-))ρ] - ∂_[(ζ(1/ϑ)(+(1-2))- 2ζϑσ_1(1-))+β(σ_2+σ_3)(1-) )ρ] + ∂_^2[(ϑζ(+(1-2)) -4ζϑ(1-))ρ] - ∂_∂_[(4ϑζ(1-) )ρ] L^(2)ρ =∂_[(ζ (+(1-2))/(2ϑ^2) - σ_1ζ(1-) )ρ]+∂_^2 [( ζ(1-) )ρ] + ∂_[(⋯)ρ] + ∂_[(⋯)ρ]As before, we introduce τ = ε^2t=t/N, expand ρ w.r.t. ,ρ(,,,t) = ρ^(0)(,,,ε^2t) + ερ^(1)(,,,ε^2t) + ε^2ρ^(2)(,,,ε^2t)+𝒪(ε^3) and obtain L^(0)ρ^(0)=0, L^(0)ρ^(1)=-L^(1)ρ^(0), L^(0)ρ^(2)=_̣τρ^(0)-L^(1)ρ^(1)-L^(2)ρ^(0).The reduced Fokker-Planck equation is given by_̣τf = ∫ρ^(1)d(,) + ∫ L^(2)d̊(,).In the following computations we use that L^(0)+[+(1-2)^2))]=-(2μ+ζ)( +(1-2) )L^(0)+(+(1-2)))=-(μ+ζ) (+(1-2))) L^(0)+[(+(1-2))^2]= -2(μ+ζ)(+(1-2))^2+8ζϑ^2(1-)Therewith we find∫ L^(2)ρ^(0) d(,zt)= ζ/2ϑ^2 ∂_ ∫ (+(1-2)) ρ^(0)d(,) - σ_1ζ∂_[((1-) )f(,τ)] +∂_^2 [( ζ(1-) )f(,τ)]=- σ_1ζ∂_[((1-) )f(,τ)]+∂_^2 [( ζ(1-) )f(,τ)].Furthermore, ∫ L^(1)ρ^(1)d(,)= -ζ/2ϑ∂_∫(+(1-2)) ρ^(1)d(,)= ζ/2ϑ(μ+ζ)∂_∫L^(0)+(+(1-2)) ρ^(1)d(,)= -ζ/2ϑ(μ+ζ)∂_∫(+(1-2)) L^(1)ρ^(0)d(,)= -ζ/2ϑ(μ+ζ)∂_( T_1+T_2+T_3+T_4+T_5 )where T_1,…,T_5 are given byT_1 = ∫(+(1-2)) (- ∂_)[( ζ(+(1-2))/(2ϑ) )ρ^(0) ]d(,)=- ζ/2ϑ[∂_x ∫ (+(1-2))^2ρ^(0)d(,) +2 ∫(+(1-2))ρ^(0)d(,)]=- ∂_ζ/2ϑ 4ϑ^2ζ(1-)/ζ+μ f = -2ϑζ^2 /ζ+μ∂_((1-) f)T_2= ∫(+(1-2)) (- ∂_)( - σ_3β(1-ß) - σ_2μϑ(1-) )ρ^(0)d(,) =(1-2)( - (σ_2+σ_3)β(1-ß))f T_3= ∫(+(1-2)) ( - ∂_)( ζ(1/ϑ)(+(1-2))- 2ζϑσ_1(1-) +β(σ_2+σ_3)(1-) )ρ^(0)d(,) =- 2ζϑσ_1(1-)f +β(σ_2+σ_3)(1-)fT_4= ∫(+(1-2)) (∂_^2)( ϑζ(+(1-2)) -4ζϑ(1-))ρ^(0)d(,) = 0T_5= ∫ ∂_( 4ϑζ(1-))ρ^(0)d(,)= ∂_( 4ϑζ(1-) f )Adding up the corresponding terms yields the reduced Fokker-Planck equationf_τ = -ζ (σ_1+σ_2+σ_3)/1+ζ/μ ∂_[(1-)f] +ζ/(1+ζ/μ)^2 ∂_^2[(1-)f]. | http://arxiv.org/abs/1707.08284v2 | {
"authors": [
"Lukas Heinrich",
"Johannes Müller",
"Aurélien Tellier",
"Daniel Zivković"
],
"categories": [
"q-bio.PE",
"92D10, 39A50, 60H10"
],
"primary_category": "q-bio.PE",
"published": "20170726041612",
"title": "Effects of population- and seed bank noise on neutral evolution and efficacy of natural selection"
} |
[email protected] Dipartimento di Fisica, Università di Cagliari Cittadella Universitaria, 09042 Monserrato, ItalyINFN, Sezione di Cagliari, Italy [email protected] of Physics,Indian Institute of Technology Kanpur,Kanpur 208016, INDIA [email protected] Department of Physics,Indian Institute of Technology Kanpur,Kanpur 208016, [email protected] Department of Physics,Indian Institute of Technology Kanpur,Kanpur 208016, INDIAWe propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in (1+1)-dimensional dual conformal field theories through the AdS/CFT correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature (1+1)-dimensional holographic conformal field theories dual to the bulk AdS_3 vacuum and the Euclidean BTZ black hole respectively. Our holographic conjecture exactly reproduces the conformal field theory results obtained through the replica technique, in the large central charge limit.We briefly elucidate the corresponding issue for the AdS_d+1/CFT_d scenario. Holographic entanglement negativity conjecture for adjacent intervals inAdS_3/CFT_2Gautam Sengupta December 30, 2023 ======================================================================================§ INTRODUCTION Quantum entanglement in recent times has impacted an expansive list of theoretical issues from condensed matter physics to quantum gravity through the holographic AdS/CFT correspondence <cit.>. This geometric connection has allowed the characterization of quantum entanglement in extended systems like holographic conformal field theories. For bipartite quantum systems in a pure state this involves the entanglement entropy which is defined as the von Neumann entropy of the reduced density matrix. In a series of interesting communications Calabrese et al advanced a comprehensive procedure to compute the entanglement entropy of (1+1)-dimensional conformal field theories (CFT_1+1) <cit.> utilizing the replica technique. Following <cit.>, in a seminal work Ryu and Takayanagi proposed a holographic characterization of the entanglement entropy in d-dimensional conformal field theories (CFT_d), involving bulk dual AdS_d+1 geometries through the AdS/CFT correspondence <cit.> ( for an extensive review see <cit.>). According to the Ryu and Takayanagi ( RT) conjecture the universal part of the entanglement entropy of a subsystem in a dual CFT_d was described by the area of a co-dimension two bulk AdS_d+1 static minimal surface homologous to the subsystem. For the AdS_3/CFT_2 scenario the static minimal surface reduces to a space like geodesic in the bulk AdS_3 geometry anchored on the appropriate spatial interval in the dual CFT_1+1. The holographic entanglement entropy obtained from the RT conjecture exactly reproduces the corresponding CFT_1+1 results obtained through the replica technique in the large centralcharge limit. It is well known however in quantum information theory that the entanglement entropy ceases to be a valid measure for the characterization of mixed state entanglement where it receives contributions from irrelevant correlations. This is a complex issue in quantum information theory and necessitates the introduction of suitable entanglement measures for the distillable entanglement. In a seminal communication Vidal and Werner <cit.> addressed this issue and proposed a computable measure termed entanglement negativity which provides an upper bound on the distillable entanglement whose non convexity was subsequently demonstrated by Plenio in <cit.>. In the recent past Calabrese et al in <cit.> utilized an alternative replica technique to compute the entanglement negativity for mixed states in both zero and finite temperature CFT_1+1. Naturally the above developments lead to the significant issue of a holographic description for the entanglement negativity involving the bulk geometry for dual conformal field theories in the AdS/CFT frame work <cit.>. In a recent interesting communication two of the present authors (VM and GS) in the collaborations <cit.>(CMS),proposed a holographic conjecture for the entanglement negativity of mixed states in holographic CFT_ds. The conjecture involves a specificalgebraic sum of the areas of co-dimension two bulk extremal surfaces ( geodesic lengths in AdS_3/CFT_2 ) anchored on the corresponding subsystems. Note that the proposed conjecture was in relation to the entanglement negativity for a singly connected subsystem in an infinite system. Remarkably, this conjecture precisely characterizes the upper bound onthe distillable entanglement through the elimination of the thermal contributions for finite temperature mixed states. Furthermore it could be shown that the holographic entanglement negativity was proportionalto the sum of the holographic mutual information for the relevant partitioning of the system.In this article motivated by the CMS conjecture, <cit.> we advance an independent holographic entanglement negativity conjecture for the mixed states of adjacent intervals in bothzero and finite temperature CFT_1+1. In the context of the AdS_3/CFT_2 scenario we establish that the holographic entanglement negativity for this configuration is described by a specific algebraic sum of the bulk geodesic lengths which is proportional to the holographic mutual information between the intervals. Note that our case is distinct from <cit.> where the entanglement entropy characterizing the entanglement between the intervals and the rest of the system was considered. On the other hand here we consider the entanglement negativity which characterizes the entanglement between these adjacent intervals. Our computation for the entanglement negativity is expected to have significant applications to various entanglement issues in diverse fields.This article is organized as follows. In section <ref> we briefly review thecharacterization of entanglement negativity in quantuminformation theory and describe the computation of this quantity for mixed states of adjacent intervals in a CFT_1+1. In the subsequent section <ref> we establish our holographic conjecture for the entanglement negativity which exactly reproduces the replica technique results for the CFT_1+1 in the large central charge limit. We then summarize our results in the last section <ref>. § ENTANGLEMENT NEGATIVITYIn this section we review the characterization of entanglement negativity in quantum information theory and the computation of this entanglement measure for mixed state configuration of adjacent intervals in a CFT_1+1 for both zero and finite temperatures as well as for a finite size system. §.§ Entanglement negativity in quantum information theoryWe begin by brieflyreviewing the definition of entanglement negativity in quantum information theory as described in <cit.>. In this context it is required to consider a tri-partition of a quantum system into the subsystems A_1, A_2 and B, with A=A_1∪ A_2 and B=A^c representing the rest of the system. The Hilbert space ℋ for the bipartite system A may be written as a direct productℋ=ℋ_1 ⊗ℋ_2 of the corresponding Hilbert spaces of the individual subsystems A_1 and A_2 respectively. The reduced density matrix of the subsystem A is given as ρ_A=Tr_A^c(ρ). The partial transpose of this reduced density matrix with respect to A_2 may be defined as follows⟨ e^(1)_ie^(2)_j|ρ_A^T_2|e^(1)_ke^(2)_l⟩ =⟨ e^(1)_ie^(2)_l|ρ_A|e^(1)_ke^(2)_j⟩,where |e^(1)_i⟩ and |e^(2)_j⟩ are the bases for the respective Hilbert spaces ℋ_1 andℋ_2. The entanglement negativity between the subsystems A_1 and A_2 is then described as followsℰ = lnTr|ρ_A^T_2|,where the trace norm Tr|ρ_A^T_2| is given by the sum of absolute eigenvalues of the partially transposed reduced density matrix ρ_A^T_2. §.§ Entanglement negativity in CFT_1+1 Here we briefly review the computation of the entanglement negativity for mixed states of two disjoint intervals in a CFT_1+1 as described in <cit.>. To this end the authors considered the tri-partition of the system as depicted in Fig. (<ref>), where A_1 describes the interval (A_1∈[u_1,v_1]) of length l_1 and A_2 is the interval (A_2∈[u_2,v_2]) of length l_2 while B=A^c represents the rest of the system.The entanglement negativity between the subsystems A_1 and A_2 is obtained through a suitable replica technique which involves the construction of the quantity Tr(ρ_A^T_2)^n_e for even n_e and considering the analytic continuation of the even sequence to n_e→1 as [Note that the explicit construction of this analytic continuation is complex and remains an open issue except for some simple conformal field theories <cit.> (See also <cit.>).] ℰ = lim_n_e → 1 lnTr(ρ_A^T_2)^n_e.It was shown that the quantity Tr(ρ_A^T_2)^n_e in the above expression may be expressed as a four point function of the twist fields on the complex plane as followsTr(ρ_A^T_2)^n_e =⟨𝒯_n_e(u_1)𝒯_n_e(v_1)𝒯_n_e(u_2)𝒯_n_e(v_2)⟩_ℂ. §.§ Adjacent intervals in vacuumThe formulation described above was utilized to characterize theentanglement negativity for mixed states of adjacent intervals in <cit.>. This configuration is obtained through the limit v_1→ u_2 for Fig. (<ref>) which reduces to the configuration depicted in Fig. (<ref>).The four point function eq. (<ref>) in this limit effectively reduces to a three point function of the twist fields as followsTr(ρ_A^T_2)^n_e =⟨𝒯_n_e(-l_1)𝒯^2_n_e(0)𝒯_n_e(l_2)⟩,The entanglement negativity for the adjacent intervals at zero temperature is obtained by employing eqs. (<ref>) and (<ref>), which may be expressed asℰ = c/4ln[l_1l_2/(l_1+l_2)a]+ const,where a and c are the UV cut off and the central charge of the corresponding CFT. Note that the first term in the above expression is universal whereas the second `const' term depends on the detailed operator content of the CFT.§.§ Adjacent intervals in vacuum for finite sizeFor a finite sized system of length L with a periodic boundary condition, the entanglement negativity for the adjacent intervals may be computed from the correspondingthree point functioneq. (<ref>) on a cylinder<cit.>. This is obtained through the conformal mapz → w = iL/2πln z, from the complex plane to the cylinder with the circumferenceL. The entanglement negativity may then be expressed asℰ = c/4ln[(L/π a)sin(π l_1/L)sin(π l_2/L)/sinπ (l_1+l_2)/L] +const.As earlier the `const' term in the above expression is non universal. §.§ Adjacent intervals at finite temperatureAn analysis as above may be performed to obtain the entanglement negativity for the mixed state of adjacent intervals in a finite temperature CFT_1+1 which is defined on a cylinder of circumference β=1/T which is the inverse temperature. This is related to the CFT_1+1 on the complex plane through the conformalmap z → w = β/2πln z. Using the above transformation andeq. (<ref>), the entanglement negativity is given asℰ = c/4ln[(β/π a)sinh(π l_1/β)sinh(π l_2/β)/sinhπ (l_1+l_2)/β] +const,where the `const' as earlier is non universal. §.§ Large central charge limitHere we review the large central charge analysis for the entanglement negativity of the mixed state configuration of adjacent intervals at zero temperature as described in <cit.>. According to the AdS_3/CFT_2 correspondence, the holographic limit in which the bulk spacetime is semiclassical ( Newton's constant G_N^(3)→0) is equivalent to the large central charge limit c→∞ in the dual CFT_1+1 as these quantities are related through the Brown-Henneaux formula c=3 R/2 G_N^(3). In <cit.>, such a large central charge analysis for the four point correlator in eq. (<ref>) was obtained by employing the monodromy technique to determine the conformal blocks providing the dominant contribution. The authors demonstrated that in the limit v_1→ u_2, which corresponds to the case of adjacent intervals depicted in Fig. (<ref>),the dominant contribution to the four point function in eq. (<ref>) arises from the conformal block corresponding to the intermediate operator 𝒯^2. This ledto the following expression for the large central charge limit of the entanglement negativity asE=c/4ln[l_1l_2/(l_1+l_2)a].Note that this matches exactly with the universal part of the entanglement negativity of the adjacent intervals given in eq. (<ref>), implying that the non-universal constant term in eq. (<ref>) is sub leading in the large central charge limit[ Note that the monodromy technique was also utilized earlier to determine the large central charge limit of the entanglement entropy for multiple disjoint intervals in <cit.> (and through a different technique in <cit.>). The authors demonstrated that in this limit the entanglement entropy is universal and matched exactly with the predictions of the holographic Ryu-Takayanagi conjecture.]. The authors also demonstrated numerically that the entanglement negativity vanishes E→ 0 in the limit of disjoint intervals placed far apart. In the subsequent sections we will establish a holographic conjecture which exactly reproduces the large-c results for the case of the adjacent intervals from the corresponding dual bulk geometry. § HOLOGRAPHIC ENTANGLEMENT NEGATIVITY FOR ADJACENT INTERVALSIn this section we establish our holographic entanglement negativity conjecture involving the bulk AdS_3 geometry for the mixed state configuration of adjacent intervals in a dual CFT_1+1. As described in (<ref>) the large central charge analysis for the entanglement negativity clearly indicates the plausibility of a holographic characterization for the entanglement negativity in a dual CFT_1+1 through the AdS_3/CFT_2 correspondence. To this end we consider the adjacent intervals A_1 and A_2 of lengths l_1 and l_2 respectively as depicted in Fig. (<ref>), where the subsystem A_1∪ A_2 is in a mixed state. As described in (<ref>) for this mixed state configuration in the CFT_1+1, the entanglement negativity involves the three point twist correlator in eq. (<ref>) whose form is fixed from conformal invariance as follows⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ = c_n_e^2 C_𝒯_n_e𝒯_n_e^2𝒯_n_e/|z_12|^Δ_𝒯^2_n_e |z_23|^Δ_𝒯^2_n_e|z_13|^2Δ_𝒯_n_e-Δ_𝒯^2_n_e,where c_n are normalization constants and C_𝒯_n_e𝒯_n_e^2𝒯_n_e is the structure constant for the relevant twist field OPEs. The scaling dimensions Δ_𝒯_n_e andΔ_𝒯_n_e^2 of the twist fields 𝒯_n_e and 𝒯_n_e^2 are respectively given as <cit.>Δ_𝒯_n_e =c/12(n_e - 1/n_e), Δ_𝒯_n_e^2 =2Δ_𝒯_n_e/2= c/6(n_e/2- 2/n_e). In this connection we note here that the two point twist correlators in the CFT_1+1 are given as as follows <cit.>< T_n_e(z_k) T_n_e(z_l)>_ℂ=C_^(1)/|z_kl|^Δ_𝒯_n_e, < T^2_n_e(z_k) T^2_n_e(z_l)>_ℂ=C_^(2)/|z_kl|^Δ_𝒯^2_n_e,where C^(1), C^(2) are normalization constants. A close examination of the form of the above two point twist correlator now clearly indicates that the three point twist correlator in eq. (<ref>) may be factorized as follows ⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ=C < T_n_e(z_1) T_n_e(z_3)>_ℂ(< T^2_n_e(z_1) T^2_n_e(z_2)>_ℂ< T^2_n_e(z_2) T^2_n_e(z_3)>_ℂ/< T^2_n_e(z_1) T^2_n_e(z_3)>_ℂ)^1/2,where the constant C is given as followsC=c_n_e^2 C_𝒯_n_e𝒯_n_e^2𝒯_n_eC_^(1)( C_^(2))^1/2.Interestingly, the leading universal part of the three point function eq. (<ref>), which is dominant in the large central charge limit may be expressed as follows⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ=C < T_n_e(z_1) T_n_e(z_3)>_ℂ < T_n_e/2(z_1) T_n_e/2(z_2)>_ℂ< T_n_e/2(z_2) T_n_e/2(z_3)>_ℂ/< T_n_e/2(z_1) T_n_e/2(z_3)>_ℂ.Note that in the above expression in eq. (<ref>) we have used the following factorization in a CFT_1+1 as described in <cit.>⟨ T^2_n_e(z_i) T^2_n_e(z_j)⟩_ℂ=< T_n_e/2(z_i) T_n_e/2(z_j)>^2_ℂ .We have ignored the non-universal constants c_n and C_𝒯_n_e𝒯_n_e^2𝒯_n_ein eq. (<ref>), in the large central charge limit as mentioned in (<ref>).It is well known from the AdS_3/CFT_2 correspondence that the two point function of the twist fields located at z_i and z_j in a CFT_1+1 is related to the length of the dual bulk space like geodesic ℒ_ijanchored on the corresponding interval as follows <cit.>< T_n_e(z_k) T_n_e(z_l)>_ℂ∼ e^-Δ_n_e L_kl/R,< T_n_e/2(z_i) T_n_e/2(z_j)>_ℂ∼ e^-Δ_n_e/2 L_ij/R,where R is the AdS_3 radius. The three point twist correlator in eq. (<ref>), upon employing eqs. (<ref>) and (<ref>) may be expressed as ⟨𝒯_n_e(z_1)𝒯_n_e^2(z_2)𝒯_n_e(z_3)⟩_ℂ =exp[-Δ_𝒯_n_eℒ_13-Δ_𝒯_n_e/2(ℒ_12+ℒ_23-ℒ_13)/R].Here the corresponding points defining the adjacent intervals are (z_1=-l_1, z_2=0,z_3=l_2). In the replica limit[Note that the large central charge limit has to be taken prior to the replica limit to arrive at eq.(16). This order of limits is critical and is also true for the case of the entanglement entropy where the scaling dimension of the twist fieldτ_n vanishes in the replica limit and has to be understood in the sense of an analytic continuation (see <cit.>). In a similar fashion the negative scaling dimension of the twist field τ_n_e/2 in the replica limit is in the sense of the non-trivial analytic continuation involved in the replica definition for the entanglement negativity.](n_e→ 1) the scaling dimensions Δ_𝒯_n_e→ 0 and Δ_𝒯_n_e/2→-c/8. Hence by utilizing eqs. (<ref>) and (<ref>) in eq. (<ref>), the holographic entanglement negativity for the mixed state of the adjacent intervals in a dual CFT_1+1 may be expressed in terms of a specific algebraic sum of the lengths of the bulk space like geodesics ℒ_ij (i,j∈ 1,2,3) anchored on appropriate intervals as followsℰ = 3/16G^3_N(ℒ_12+ℒ_23-ℒ_13),where we have used the Brown-Henneaux formula c=3R/2G^3_N<cit.>. Upon employing the Ryu-Takayanagi conjecture <cit.> for the holographic entanglement entropythe eq. (<ref>) above may be expressed in terms of a specific algebraic sum of the holographic entanglement entropies as follows ℰ =3/4(S_A_1+S_A_2-S_A_1∪ A_2)=3/4 I(A_1, A_2).This algebraic sum of the holographic entanglement entropies is precisely the holographic mutual information I(A_1,A_2)between the two intervals for the mixed state configuration under consideration.It is important to note that the mutual information and the entanglement negativity are entirely distinct entanglement measures in quantum information theory. Negativity is an upper bound on the distillable entanglement and the mutual information is the upper bound on the total correlation in the bipartite system. However the universal parts of these two quantities which are dominant in the holographic (large-c) limit are proportional for the particular mixed state configuration, although this is not the case for other configurations where a more intricate relation between these quantities are valid in the holographic limit [ Interestingly this proportionality has also been observed in the context of global and local quench in a CFT_1+1 <cit.> for the same configuration utilizing the replica technique. However note that our results are distinct and pertain to a holographic description ( distinct from the replica technique) of entanglement negativity for the mixed state of a dual CFT_1+1 without quench.].As explained in the introduction the entanglement negativity in our case characterizes the entanglement between the adjacent intervals for the mixed state configuration of a CFT_1+1 in question. It is important to mention here that eq. (<ref>) provide an elegant holographic construction involving the dual bulk AdS_3 geometry for the entanglement negativity of this mixed state in terms of a specific algebraic sum of the lengths of space like geodesics anchored on the intervals and their union. Interestingly as shown in eq. (<ref>) this may be expressed as an algebraic sum of the corresponding entanglement entropies which reduces to the holographic mutual information for the mixed state configuration in question. Our construction characterizes a significant progress for the critical issue of a holographic description of entanglement negativity in a generic AdS_d+1/CFT_d scenario started by two of the present authors in <cit.>. The present work is an important sequel towards further elucidation of this crucial issue and provides critical insight into the structure of holographic entanglement negativity for generic mixed states in CFT_ds. Our construction for the mixed state configuration under consideration constitutes a significant examplefor the holographic description of entanglement negativity and indicatesan elegant geometrical technique which may generalized for application to other mixed state configurations in a CFT_1+1.The construction described here suggests a possible higher dimensional generalization of the holographic negativity conjecture for mixed states of adjacent subsystems in a generic AdS_d+1/CFT_d scenario. The entanglement negativity in this case would involve a similar algebraic sum of the areas of bulk co-dimension two static minimal surfaces anchored on the appropriate subsystems, which may be expressed as followsℰ = 3/16G^(d+1)_N(𝒜_1+𝒜_2-𝒜_12)=3/4 I(A_1, A_2),where 𝒜_i denotes the area of the bulk co dimension two static minimal surface anchored on the subsystem A_i. Note that the above expression is also proportional to the holographic mutual information between the two subsystems. However the higher dimensional generalization requires substantiation through specific examples and a possible proof from a bulk perspective which constitute an open issue [ Note that the higher dimensional holographic entanglement negativity construction suggested above has been implemented for mixed state configurations of adjacent subsystems described by rectangular strip geometries in CFT_ds dual to bulk pure AdS_d+1 space time, AdS_d+1-Schwarzschild and RN-AdS_d+1 black holes in <cit.>. The results reproduce certain universal features of entanglement negativity for CFT_1+1s which are consistent with quantum information theory expectations and constitute significant consistency checks for the higher dimensional conjecture.].§.§ Adjacent intervals in the vacuum The holographic conjecture established above in eq. (<ref>) which describes the entanglement negativity for the mixed state configuration of adjacent intervals in terms of the lengths of the bulk space like geodesics anchored on appropriate intervals may now be applied to specific examples. In this context we first obtain the holographic entanglement negativity for the mixed state configuration of adjacent intervals at zero temperature in a CFT_1+1 dual to a bulk pure AdS_3 geometry. The bulk metric in the Poincaré coordinates is given asds^2 = -(r^2/R^2)dt^2 +(r^2/R^2)^ -1 dr^2 + r^2/R^2 dx^2,where R is the AdS_3 radius and the coordinate x ∈ℝ. The bulk spacelike geodesic length L_γ anchored to the interval γ of length l_γin the CFT_1+1 is given as follows <cit.>ℒ_γ = 2R lnl_γ/a,where a is the UV cut off. The holographic entanglement negativity for the adjacent intervals of lengths l_1 and l_2 may now be expressed by employingeq. (<ref>) as followsℰ = c/4ln[l_1l_2/(l_1+l_2)a],where we have used the Brown-Henneaux formula. Remarkably, the holographic entanglement negativity eq. (<ref>) matchesexactly with the corresponding replica technique result in eq. (<ref>) at the large central charge limit.§.§ Adjacent intervals for finite sized systems in vacuum We now proceed to compute the holographic entanglement negativity for mixed states of adjacent intervals in a finite sized system of length L at zero temperature in a dual CFT_1+1 from our conjecture described in eq. <ref>. In this case as explained in (<ref>) the corresponding CFT_1+1 is defined on an infinite cylinderwith the spatial direction compactified on a circle of circumference L. The bulk dual configuration for this case is the AdS_3 vacuum in global coordinates with the following metric <cit.> ds^2= R^2(-cosh^2 ρ dt^2 + dρ^2 + sinh^2ρ d ϕ^2 ),where ϕ is periodic with a period of 2π. The length of a bulk space like geodesic anchored on the subsystemγ of length l_γ in theboundary CFT_1+1 is given as <cit.>ℒ_γ = 2R ln(L/π asinπ l_γ/L).The holographic entanglement negativity for the adjacent intervals of lengths l_1 and l_2 may now be computed by substituting the respective geodesic lengths given by eq. (<ref>) in eq. (<ref>), which leads to the following expressionℰ = c/4ln[(L/π a)sin(π l_1/L)sin(π l_2/L)/sinπ (l_1+l_2)/L],where we have utilized the the Brown-Henneaux formula. Interestingly, the holographic entanglement negativity obtained in eq. (<ref>) exactly reproduces the replica technique results eq. (<ref>)in the large central charge limit.§.§ Adjacent intervals at finite temperatureWe now proceed to compute the holographic entanglement negativity for the mixed state ofadjacent intervals in a dual CFT_1+1 at a finite temperature utilizing our conjecture described in eq. (<ref>). The corresponding CFT_1+1 is defined on a spatially infinite cylinder with the Euclidean time direction compactified on a circle of circumference β. The appropriate bulk dual configuration is then described by a Euclidean BTZ black hole <cit.>at a Hawking temperature T=1/β with the ϕ direction uncompactified[Note that this configuration is actually a BTZ black string.]. The metric for this is given asds^2 = (r^2 - r_h^2)/R^2dτ^2 +R^2/(r^2 - r_h^2)dr^2 + r^2/R^2dϕ^2 ,where r=r_h denotes the event horizon and τ is the Euclidean time. The length of a bulk spacelike geodesic anchored on a subsystem γ is given as <cit.>ℒ_γ = 2R ln(β/π asinhπ l_γ/β),where β=2π R^2/r_h is the inverse Hawking temperature. It is now possible to compute the holographic entanglement negativity for the mixed state of adjacent intervals of lengths l_1 and l_2 in the finite temperature CFT_1+1 from our conjecture by utilizing the eqs. (<ref>) and (<ref>), which leads to the following expressionℰ = c/4ln[(β/π a)sinh(π l_1/β)sinh(π l_2/β)/sinhπ (l_1+l_2)/β],where once again we have employed the Brown-Henneaux formula. As earlier the holographic entanglement negativity eq. (<ref>) exactly reproduces the replica technique result eq. (<ref>) for the finite temperatureCFT_1+1 in the large central charge limit. This clearly constitutes compelling evidence for the validity of our holographic entanglement negativity conjecture. § SUMMARY AND DISCUSSIONTo summarize we have established a holographic entanglement negativity conjecture for mixed states of adjacent intervals in zero and finite temperature dual CFT_1+1in the AdS_3/CFT_2 scenario. Note that this configuration is a mixed state as the degrees of freedom of the rest of the system are traced over and hence the corresponding entanglement negativity characterizes the entanglement between the adjacent intervals. Our conjecture involves a specific algebraic sum of the lengths of bulk space like geodesics anchored on respective intervals in the dual holographic CFT_1+1 which reduces to the holographic mutual information between the intervals. In the AdS_3/CFT_2 scenario, the bulk dual configuration for the zero temperature CFT_1+1 is the pure AdS_3 vacuum expressed in terms of the Poincare coordinates for the infinite sized system and in the global coordinates for the finite sized system respectively. The finite temperature case on the other hand involves a bulk Euclidean BTZ black hole (black string). In both the cases we have clearly demonstrated that our conjecture exactly reproduces the corresponding replica technique results for both zero and finite temperature dual CFT_1+1s in the large central charge limit. Note however that our holographic conjecture in this article is specific to the mixed state configuration of adjacent intervals. From this perspective a more general holographic entanglement negativity conjecture for the mixed state of disjoint intervals is desirable. This constitutes an interesting open issue for a future investigation.Naturally the holographic conjecture proposed here in the context of the AdS_3/CFT_2 scenario suggests a possible generalization in a AdS_d+1/CFT_d scenario. The mixed state in this case would be defined by the configuration of adjacent subsystems in the dual holographic CFT_d. The holographic entanglement negativity would involve a specific algebraic sum of the areas of bulk co dimension two extremal surfaces anchored on the respective subsystems, which would be proportional to the holographic mutual information between them. However as such a higher dimensional generalization would require substantiation through specific examples and a possible proof froma bulk perspective.We emphasize here that our holographic conjecture provides a simple and elegant method to compute the entanglement negativity for such mixed states in holographic CFTs both at zero and finite temperatures. Naturally this is expected to have significant applications in diverse areas like strongly coupled theories in condensed matter physics, topological phases, quantum phase transitions and critical issues in quantum gravity. These are interesting open questions for the future.§ ACKNOWLEDGEMENTParul Jain would like to thank Prof. Mariano Cadoni for his guidance andthe Department of Physics, Indian Institute of Technology Kanpur, India for their warm hospitality. Parul Jain's work is financially supported by Università di Cagliari, Italy and INFN, Sezione di Cagliari, Italy.utphys | http://arxiv.org/abs/1707.08293v5 | {
"authors": [
"Parul Jain",
"Vinay Malvimat",
"Sayid Mondal",
"Gautam Sengupta"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170726060123",
"title": "Holographic entanglement negativity conjecture for adjacent intervals in $AdS_3/CFT_2$"
} |
firstpage–lastpage The Caccioppoli UltrafunctionsVieri Benci, Luigi Carlo Bersellicorresponding author, and Carlo Romano Grisanti====================================================================================Star formation rate density (SFRD) has not been constant throughout the history of the Universe. The rate at which stars form greatly affects the evolution of the Universe, but the factors which drive SFRD evolution remainuncertain. There must be sufficient amount of gas to fuel the star formation, either as a reservoir within a galaxy, or as inflow from the intergalactic medium (IGM). This work explores how the gas accretion rate onto galaxies over time has affected star formation rate. We propose a novel method of measuring cosmic gas accretion rate. This involves comparing the comoving densities of available Hi and _2 gas and the densities of existing stars at different redshifts. We constrained gas accretion until z = 5, and we found that the gas accretion rate density (GARD) is relatively constant in the range from z = 5 to z = 0. This constancy in the GARD is not reflected by the SFRD, which declines significantly between z = 1.0 and z = 0. This work suggests that the decline is not due to a reduction in GARD. galaxies: evolution, intergalactic medium, galaxies: ISM, galaxies: star formation § INTRODUCTIONOne of the most important issues of the evolution of the Universe is how galaxies acquire gas which fuels star formation. Numerical galaxy formation models require significant gas inflows from the intergalactic medium (IGM) to fuel star formation <cit.>, and indeed the current gas reservoirs in many galaxies are too low to sustain the current level of star formation, even for normal galaxies like the Milky Way <cit.>. This is reinforced by simulations of <cit.>, who found that most galaxies assemble their mass through steady gas accretion, and only most massive galaxies grow predominately due to dramatic merger events. However, such inflow process cannot be studied in details because it is very difficult to directly detect it. Only for a few galaxies was the observational indication of gas inflow obtained <cit.>, and indirect evidence was reviewed by <cit.>. <cit.> provide evidence for gas inflow directly fueling star formation. This was achieved by reporting the first 21 cm line observations for long gamma-ray burst galaxies. These observations implied high levels of Hi gas in areas of recent star formation, whereas low molecular gas content was reported for GRB hosts (, but see ). This in turn suggests that star formation is either directly fueled by Hi gas, or that there is a very efficient conversion process between Hi and _2 underway <cit.>. As reported by <cit.> within the Milky Way itself, rate of star formation has been very constant over the course of its life. This clearly demonstrates that the gas being used up in star formation is somehow being replaced. Whilst it is obvious that gas accretion must be happening for almost all galaxies, it is very unclear how much is taking place. Whilst there is a wide range of theories regarding the evolution of star formation rate, there is little observation based evidence available to study the details of this process. Observations suggest that it is the accretion of metal-poor gas that drives the formation of disc galaxies <cit.>. There has been a lot of work done to measure the densities of Hi and _2 mass in galaxies at varying redshifts. In order to detect _2 gas, the CO molecule is generally used as a tracer <cit.>.The aim of this paper is to provide the first observational measurement of the cosmic gas accretion rate density (GARD) by applying a novel method, and comparing it to the measurement of the star formation rate density (SFRD). A cosmological model with h = 0.7, Ω_m= 0.3, Ω_Λ = 0.7 is assumed.§ DATAThe compilations of ρ_HI and ρ_H_2 from <cit.> and <cit.> were used. Significantly more ρ_HI data was available than ρ_H_2, over a wider redshift range. We used the compilation ofρ_stellar (including stellar remnants) from <cit.>. The Hi densities were obtained by a variety of methods over the redshift range. These included spectral stacking of the 21-cm Hydrogen emission line and damped Lyman-α (DLA) absorbers. The complete compilation of ρ_HI measurements is presented in table <ref>. All density values are densities per unit of comoving volume.The values from <cit.> and <cit.> were for the local universe, at z = 0. The <cit.> value was based on the Arecibo Legacy Fast ALFA (ALFALFA) survey, which at that time had completed source extraction for 40% of the total sky area, allowing the value to be calculated from a sample of 10 119 galaxies. In comparison, the <cit.> value was based on theHi Parkes All Sky Survey (HIPASS) which was based on 4315 extra galactic emission-line detections. As such, it was reasonable to assume that the <cit.> value is superior to the <cit.> value. The <cit.> value was given relative to z = 0, so required a conversion using an established value of ρ_HI at z = 0. The <cit.> was based on observations from the Giant Metrewave Radio Telescope (GMRT) in order to measure ρ_HI at z = 0.24. The <cit.> values were based on a combination of detected sources and the spectral stacking technique. Contrary to previous estimates, they also suggested that ρ_HI evolution over the last 1 Gyr was minimal. The <cit.> values were obtained using Hi signal stacking technique.The <cit.> values also relied upon the detection of hydrogen 21-cm emission with Arecibo Ultra Deep Survey. <cit.> used the Hi intensity mapping based on the DEEP2 optical galaxy redshift survey. The <cit.> value was obtained by cross-correlating hydrogen 21-cm information with data from the WiggleZ Dark Energy Survey.The <cit.>, <cit.>, <cit.>, <cit.>, and <cit.> values were all obtained using data from surveys of damped Lyman-α (DLA) absorbers. The ρ_H_2 values were sourced from <cit.> for z = 0, and <cit.> for other redshifts.§ METHOD: GAS ACCRETION RATE DENSITY (GARD) ESTIMATEIn the calculations presented below care must be taken in consistently treating the extent of the gas and stellar components which are considered to lie inside or outside a galaxy. Here we broadly define a galaxy extent as the size of the atomic gas disk, which is usually a few times larger than the stellar disk, i.e. extends for a few tens of kpc. In this definition the hot gas in the dark matter halo is not considered.Absorption measurements of ρ_HI in principle could trace the gas outside galaxies, but the DLA systems used for these studies are dense enough to safely assume that they are associated with galaxies. Indeed, DLAs have SFRs and velocity dispersions characteristic for galaxies, and metallicities higher than that of the IGM <cit.>. Moreover within the error bars there is no step change from Hi-line- to DLA-derived ρ_HI values (Fig. <ref> and Table <ref>), which would be expected if DLAs probed additional neutral gas component in the IGM. Finally, the sizes of the absorbing gas of DLAs are ∼10–20 kpc <cit.>, safely within our limit of a few tens of kpc. There are some known DLAs with impact parameters from the background quasar of 50–100 kpc <cit.>, but these DLAs are rare <cit.>, so their impact on the measured ρ_HI is minor. Moreover, they all have low identification confidence. Similarly, simulations show that the hydrogen density required to classify as a DLA (2×10^20 ^-2; ) is only present within ∼20 kpc from the galaxy centre ( fig. 7 and 18; Shen et al., in prep.).Calculating the GARD involves the following steps. The total matter density inside galaxies for each redshift bin was calculated as: ρ_total =ρ_HI + ρ_H_2 + ρ_stel. Over time this density changes only by inflows and outflows: ρ_a = ρ_b + ρ_inflow - ρ_outflow Where a and b represent total ρ at different epochs. We do not distinguish different mechanisms of inflow and outflow, so these terms include all processes that add to and remove gas from galaxies, respectively. The inflow processes include cold and hot mode accretion, as long as the gas ends up inside galaxies (as defined above) during the relevant time span. It also includes gas expelled to the galactic halo during star-formation episodes which is subsequently re-accreted. Galactic fountains eject cold high-metallicity gas into the corona, and by mixing, cool the low-metallicity gas of the corona sufficiently that fountain clouds form and inflow back into the galactic disc <cit.>. Gas accreted from fountain clouds is considered inflow. Accretion of gas into intra-cluster medium is not included, as such gas does not end up inside galaxies, and is not included in the Hi and H_2 measurements. The outflows include supernova and AGN feedback, gas stripping, etc.The average density of gas accreted by galaxies between two epochs (i.e. the difference between the gas flowing in and out) can be calculated as: ≡ρ_inflow - ρ_outflow = ρ_a - ρ_bBy calculating the difference in ρ_total between consecutive epochs, represented in equation <ref> by ρ_a and ρ_b, only the gas remaining within the galaxy — either free or as stars — is accounted for. This quantity was labelled the gas accretion density (GAD). From there, the GARD could then be calculated using: =/Δ. Where Δtime was the time elapsed between neighboring redshift bins.§ RESULTS The binned values for all three data sets are presented in Table <ref> and can be seen in Figure <ref> extending between z = 0 to z = 5. The first six bins cover points where ρ_HI, ρ_H_2 and ρ_stel data were all available. The seventh and final bin centred at z = 4.6 does not include ρ_H_2 data, as none was available at that redshift. It seemed legitimate to continue to seven bins, as the contribution to ρ_tot from ρ_H_2 was already very minimal in the sixth bin, relative to those of ρ_HI and ρ_stel. There is a clearly decreasing trend in ρ_tot. This is primarily due to the ρ_steldata dominating at z < 2. Both GAD and GARD are presented in Table <ref>. The value of GAD (see Equation <ref>) represents the density present at a lower redshift that had not been present previously, either as a fresh gas reservoir or newly formed star. For plots of GAD against redshift, z see Appendix <ref>. Figure <ref> compares the GARD with SFRD. It can be seen that the GARD values seem to by relatively constant. This does not tally with the clear drop in SFRD observed from between z = 1.5 to 0.The errors in GAD and GARD were calculated by propagating the errors of individual densities in a standard manner, whereas the errors of these densities are the errors of the means of the densities in relevant redshift ranges reported in other studies. § DISCUSSION We did not find significant variations in GARD. Indeed, a straight line fit to the GARD vs. redshift plot resulted in a slope of -0.009±0.013 M_⊙yr^-1Mpc^-3, consistent with no evolution. However, argument could be made for an increasing trend in GARD from z = 2 to the present. This is an unexpected result as it does not tally up with the SFRD which increases relatively steadily from z = 5 to z = 1.5, and then declines from around z = 1.5 to z = 0. Therefore, the decline in SFRD observed cannot be attributed to a change in gas accretion rate. It seems that galaxies in the earlier universe used up their gas supplies faster than the accretion of fresh gas could maintain (SFRD > GARD). SFRD now has dropped beneath the GARD, so the current SFRD is now sustainable. This drop in SFRD is therefore not due to decrease in gas supply. It could be because average density of gas in galaxies dropped, leaving significant amounts of gas below the star-formation threshold. In order to compare our unexpected results with those from a well established simulation Figure <ref> compares GARD from this work with that obtained from a smoothed particle hydrodynamics simulation including dark matter, gas, and stars from <cit.>. Simulated GARD shows a distinctive decline from z < 4 which more closely echoes SFRD than the GARD from this work, however, there is a convincing order of magnitude consistency between the values. There are a few feasible explanations for the discrepancies between the simulation and the results from this work. It is possible that the Hi densities based on DLA data led to underestimations at higher redshifts for GARD. It is also possible that the simulation did not properly account for gas which inflows but is then expelled shortly after, so is not present inside galaxies at later epochs.Finally, the simulated GARD denote the gas accreted onto dark matter halos, whereas we measure gas accretion onto galaxies, as explained in Section <ref>. Assuming that some gas accretes onto halos, but does not end up inside galaxies, it is not surprising that the simulated GARD is higher than the measured one.This work could be considerably improved upon by a wider range of data. The lack of _2 data at higher redshifts might have affected the gas accretion determination. This will be improved by the Atacama Large Millimeter Array (ALMA) and the Northern Extended Millimeter Array (NOEMA) which will perform CO-line scans leading to the determination of ρ_H_2 <cit.>. On the other hand the Square Kilometre Array (SKA) will deliver direct measurements of ρ_HI at least at z < 1, which will make it possible to test the result of a constant GARD in this regime.We do not include the ionized medium in eq. <ref> because, even though its filling factor is large <cit.>, its mass fraction is minor. In the Milky Way by mass the fraction of the ionized ISM ranges from ∼2% <cit.> to ∼23% <cit.>. If this is common among galaxies, then the contribution of the ionized medium to the changes of the total densities we measure is therefore likely smaller than the uncertainties involved.§ CONCLUSIONSStar formation rate has not been constant throughout the history of the Universe. This work sought to demonstrate whether the gas accretion onto galaxies over time has affected star formation rate. GARD in this work represents a simply calculated value that shows how much gas mass density has been accreted by galaxies over a certain redshift interval. 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W., Sargent M., Sheth K., van der Wel A., van der Werf P., Wagg J., 2016, ApJ, 833, 67 [Wolfe et al.2005]Wolfe05Wolfe A. M., Gawiser E., Prochaska J. X., 2005, ARA&A, 43, 861 [Wolfire et al.2003]Wolfire03Wolfire M. G.; McKee, Christopher F., Hollenbach, D., Tielens, A. G. G. M., 2003, ApJ, 587, 278 [Zafar et al.2013]Zafar13Zafar T., Péroux C., Popping A., Milliard B., Deharveng J.M., Frank S., 2013, A&A, 556, A141 [Zwaan et al.2005]Zwaan05Zwaan M. A., Meyer M. J., Staveley-Smith L., Webster R. L., 2005, MNRAS, 359, 30 § FURTHER PLOTSWe plot here the complete data sets for ρ_HI and ρ_stel, over-plotted with the binned data. The ρ_HI data followed an increasing trend from z = 0 to 1.5, after which it seems to remain constant.§ Ρ_HI DATA COMPILATIONThe ρ_H_2 data came from <cit.> and <cit.>. The full ρ_stel compilation was sourced from <cit.>. Our complete ρ_HI compilation is presented in full in table <ref>. | http://arxiv.org/abs/1707.08877v1 | {
"authors": [
"Eleanor F. Spring",
"Michał J. Michałowski"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727142629",
"title": "Observational Evidence For Constant Gas Accretion Rate Since z = 5"
} |
In this article we provide a systematic way of creating generalized Moran sets using an analogous iterated function system (IFS) procedure.We use a step-wise adjustable IFS to introduce some variance (such as non-self-similarity) in the fractal limit sets.The process retains the computational simplicity of a standard IFS procedure. In our construction of the generalized Moran sets, we relax the second Moran Structure Condition so that the limit set is not necessarily self-similar. We also weaken the fourth Moran Structure Condition so that there are no limitations on the ratios of the diameters of the sets across a generation. Moreover, we provide upper and lower bounds for the Hausdorff dimension of the fractals created from this generalized process.Specific examples (Cantor-like sets, Sierpinski-like Triangles, etc) with the calculations of their corresponding dimensions are studied. [2010]28A80 Bullen's parameter as a seismic observable for spin crossovers in the lower mantle Adrian C. Murza December 30, 2023 ==================================================================================§ INTRODUCTIONThe Moran construction is a typical way to generate self-similar fractals, and has been studied extensively in the literature (e.g. <cit.>,<cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>, and references therein).In this paper, we extend ideas from iterated function systems (IFS) and Moran constructions by describing a new process that allows for the functions to be updated at every iteration while still maintaining the computational simplicity of an IFS.This process provides more variance in the limit sets (such as non-self-similarity) using an analogous approach to an IFS procedure.We also give estimates of the Hausdorff dimension of the limit sets created from such a process, and provide concrete examples.The classic construction of Moran sets was introduced in <cit.>.We reproduce the definition here with a more current interpretation to introduce notations. Let {n_k}_k≥1 be a sequence of positive integers for k ≥ 1.Here k will represent the generation, and n_k will be the number of children in generation k that each parent set from generation k-1 has.For any k ∈, define D_k = { (i_1, i_2, ⋯, i_k) : 1 ≤ i_j ≤ n_k, 1 ≤ j ≤ k} andD = ⋃_k ≥ 0 D_k withD_0=∅. Let σ = (σ_1, ⋯, σ_k) ∈ D_k and τ = (τ_1, ⋯, τ_m) ∈ D_m, then denoteσ * τ = (σ_1, ⋯, σ_k, τ_1, ⋯, τ_m) ∈ D_k+m.Using this notation, we may expressD_k ={σ * j |σ∈ D_k-1, 1 ≤ j ≤ n_k}to emphasize the process of moving between generations.Suppose 𝒥 := {J_σ : σ∈ D} is a collection of subsets of ^N. Set E_k = ⋃_σ∈ D_k J_σ,andF = ⋂_k ≥ 0 E_k.We call F the limit set associated with the collection 𝒥. Suppose that J ⊂^N is a compact set with nonempty interior.Let {n_k}_k ≥ 1 be a sequence of positive integers, and {Φ_k}_k ≥ 1 be a sequence of positive real vectors with Φ_k = (c_k,1, c_k,2, …, c_k,n_k), ∑_1 ≤ j ≤ n_k c_k,j≤ 1, k ∈. Suppose that := {J_σ : σ∈ D} is a collection of subsets of ^N, where D is given in (<ref>).We say that the collectionfulfills the Moran Structure provided it satisfies the following Moran Structure Conditions (MSC): MSC(1) J_∅ = J. MSC(2) For any σ∈ D, J_σ is geometrically similar to J.That is, there exists a similarity S_σ: ^N →^N such that J_σ = S_σ(J). MSC(3) For any k ≥ 0 and σ∈ D_k, J_σ⋆ 1, … , J_σ⋆ n_k are subsets of J_σ, and int(J_σ⋆ i) ∩ int(J_σ⋆ j) = ∅ for i ≠ j. MSC(4) For any k ≥ 1 and σ∈ D_k-1, 1 ≤ j ≤ n_k, diam(J_σ⋆ j)diam(J_σ) = c_k,j. For the collectionfulfilling the MSC, the limit set F given in (<ref>) is a nonempty compact set. This limit set F is called the Moran set associated with the collection .This Moran set is self-similar, and has been studied extensively by many authors with various approaches (e.g. <cit.>, <cit.>, <cit.>, <cit.>, <cit.>). The fact that there are four conditions to create a Moran set makes the area ripe for generalizations or restrictions.Note that in condition MSC(2), the sets in the new generation are geometrically similar, which is a rather strong condition.In MSC(3) the interiors of the next generation may not overlap, but says nothing else of the spacing between these sets.Condition MSC(4) requires that the sets in the new generation all have the same pattern of ratios for each iteration.There is even a hidden condition in MSC(2) that one may want to do away with in that the locations of the sets J_σ *j are completely determined by the similarities used on J_σ.Often times in the literature one may define a self-similar set to be a set satisfying the MSC as well as the fact that the set of similarities {S_σ} has finite cardinality, the ratios described in MSC(4) do not change over generations (i.e. c_k,j = c_j), and that the system is deterministic.In this case, the dimensions (Hausdorff, Box, Packing etc.) of the Moran set are known to coincide, see <cit.>.However, the dimensions may differ when we begin to modify the MSC.Several approaches have been used to relax MSC in order to create more general limit sets.There are many generalizations for MSC(2).For a self-similar set, one could change MSC(2) to use conformal maps <cit.> or affine maps <cit.> instead of similarities.In this setting, however, calculations of the dimension of limit sets can become particularly difficult.One could also study the limit sets generated by infinitely many similarities, as in <cit.>. In <cit.>, the authors removed MSC(2), but required int(J_σ) = J_σ in their construction, and studied the dimension of the resulting fractals.In <cit.>, Holland and Zhang studied a construction that replaced similarity maps in MSC(2) with a more general class of functions that are not necessarily contractions.In <cit.>, Pesin and Weiss removed the requirement for similarities from MSC(2), but also relaxed MSC(3) from non-intersecting basic sets to non-intersecting balls contained in the basic sets.In particular they pursued sufficient conditions for which the box dimension and Hausdorff dimensions coincide.For more examples of modifications to the Moran set definition, see <cit.>, <cit.> and the references therein. A special case of Moran sets can be constructed from an iterated function system (IFS).An iterated function system {S_1, S_2, ⋯, S_m} is a finite family of similarities for a fixed natural number m≥2 (see <cit.> for more details and applications).In MSC(2), define n_k=m and set S_σ = S_i_k∘ S_i_k-1∘⋯∘ S_i_1 for σ = (i_1, i_2, ⋯, i_k) ∈ D.Then the resulting Moran set is self-similar and agrees with the attractor of the IFS {S_1, S_2, ⋯, S_m}.The dimension of the limit set can be quickly calculated from the Moran-Hutchinson formula in <cit.>.Using iterated function systems is a popular way to construct fractals, and has been used to great effect (e.g. <cit.>, <cit.>, <cit.>, <cit.> ). A natural question arises: Can we construct more general fractals (e.g. non-self-similar Moran type sets) using an analogous approach while preserving the computational simplicity of the IFS?In this paper, we present a method to do so. We first make the following observations about the general construction of a Moran set.Note that in the construction of a Moran set described in (<ref>),J_σ * i = S_i (J_σ),for alli=1, ... m,and σ∈ D. Suppose that there is a tuning parameter in the expression of the function S_i (e.g. the coefficients a_i, b_i in a linear function S_i(x) = a_i x+ b_i).One can tune the values of the parameter to get a comparable function.When J_σ is given, applying the comparable function to J_σ, as in equation (<ref>), will not significantly change the computational complexity of constructing J_σ * i. The advantage of doing this at each iteration is that we introduce some variance into the limit set.Another observation is about which space the functions are defined. In classical IFS constructions, the functions are usually defined on all of the ambient space ^n (as in <cit.>, the functions are C^1+α diffeomorphisms on ^n).For our construction, we wish to relax the condition MSC(2) as well.Instead of restricting our attention to functions of higher regularity defined on the whole ambient space ^n, we use maps from a collection of subsets to itself.This article is organized as follows. In Section <ref> we find bounds for the Hausdorff dimension of the limit sets in a general metric space setting of a collection of bounded sets, not necessarily satisfying the MSC conditions. In particular, we introduce the concept uniformcoveringcondition in Definition <ref> for the purpose of studying the lower bound of the Hausdorff dimension of the limit set. Then in Section <ref> we formulate the general setup for the construction of Moran-type limit sets using the ideas from a modified IFS procedure, as discussed in the previous paragraph.In our construction we relax MSC(2) so that the limit set is not necessarily self-similar.More importantly, we drop MSC(4) from the construction process so that there are no limitations on the ratios of the diameters of the sets.Specifically, the ratio diam(J_σ * j)diam(J_σ) in (<ref>) is not limited to depend on just k and j, but varies with σ.This change allows us to produce a mosaic of possible fractals.An important observation is that the computational complexity of generating these fractals is the same as using an analogous, standard IFS.In Section <ref> we give estimates of the Hausdorff dimension of the limit sets created from the general construction.In Section <ref> we apply the results to specific examples, including modifications of the Cantor set, the Sierpinski triangle, and the Menger sponge.We also give a remark to discuss similarities and differences of this construction with V-variable fractals created by Barnsley, Hutchinson, and Stenflo in <cit.>, <cit.>. In section <ref>, we explore the sufficient conditions needed for a fractal to satisfy the uniform covering condition, which plays a vital role in computing a lower estimate for the Hausdorff dimension of a fractal.§ HAUSDORFF DIMENSION OF THE LIMIT SETSIn this section we investigate the Hausdorff dimension _H(F) of the fractals F defined in (<ref>), which do not necessarily satisfy all the MSC conditions.To start, we determine an upper bound for the dimension of the limit set F by considering the step-wise relative ratios between the diameters of sets. Suppose 𝒥 := {J_σ : σ∈ D} is a collection of bounded subsets of a metric space (X,d), and s>0. Let E_k = ⋃_σ∈ D_k J_σ,andF = ⋂_k ≥ 0 E_k be defined as in (<ref>).If there exists a sequence of positive numbers {c_k}_k=1^∞ such that lim inf_k→∞∏_i=1^k c_i =0 and ∑_j=1^n_k ((J_σ*j))^s ≤ c_k ( (J_σ))^s, for all σ∈ D_k-1 and all k=1,2,⋯, then dim_H(F) ≤ s. We prove by using mathematical induction that for k=1,2,⋯, ∑_σ∈ D_k((J_σ))^s ≤(∏_i=1^k c_i ) (diam(J_∅))^s. When k=1, (<ref>) follows from (<ref>). Now assume (<ref>) is true for some k≥ 1.Then by (<ref>), (<ref>), and (<ref>), ∑_σ∈ D_k+1 ((J_σ))^s= ∑_σ∈ D_k(∑_j=1^n_k+1 ((J_σ*j))^s)≤c_k+1∑_σ∈ D_k((J_σ))^s ≤ (∏_i=1^k+1 c_i )((J_∅))^s as desired. By the induction principle, (<ref>) holds for all k=1,2,⋯. For each k, set δ_k=max{(J_σ): σ∈ D_k}>0. Then, by (<ref>),δ_k≤(∏_i=1^k c_i )^1/s(J_∅).Moreover, by (<ref>) ℋ_δ_k^s(F) ≤ℋ_δ_k^s(E_k)≤∑_σ∈ D_kα(s)((J_σ)/2)^s ≤(∏_i=1^k c_i )α(s)((J_∅)/2)^s. Since lim inf_k→∞∏_i=1^k c_i =0, there exists a sequence {k_t}_t=1^∞ such that lim_t→∞∏_i=1^k_t c_i =0.Thus, lim_t→∞δ_k_t= 0, ℋ^s(F) =lim_t→∞ℋ_δ_k_t^s(F)=0, and hence _H(F) ≤ s. Conversely, a lower bound on the Hausdorff dimension of the limit set F can also be obtained as follows. Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of a metric space (X, d), and F be the limit set of 𝒥 as given in (<ref>).𝒥 is said to satisfy the uniform covering condition if there exists a real number γ>0 and a natural number N such that for all closed ball B in X, there exists a subset D_B⊂ D with cardinality of D_B at most N, B∩ F ⊆⋃_σ∈ D_B J_σ and diam(B)≥γ∑_σ∈ D_Bdiam(J_σ).Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of a metric space (X, d) with diam(J_∅)>0, and F be the limit set of 𝒥 as given in (<ref>).If 𝒥 satisfies the uniform covering condition, and if for some s>0, ∑_j=1^n_k(J_σ*j)^s ≥(J_σ)^s for all σ∈ D_k-1 and all k=1,2,⋯, then dim_H(F)≥ s. We first show that under condition (<ref>), there exists a probability measure μ on X concentrated on F such that for each σ∈ D, μ(J_σ)≤((J_σ)/(J_∅))^s. Let μ(J_∅)=1, and for each σ∈ D_k for k>0 and i=1, ⋯, n_k, we inductively set μ(J_σ *i) =(J_σ*i)^s/∑_j=1^n_k(J_σ *j)^sμ(J_σ). For any Borel set A in X, define μ(A) = inf{∑_i=1^∞μ(J_σ_i) : A ∩ F ⊂⋃_i=1^∞ J_σ_i andJ_σ_i∈𝒥}. One can check that μ defines a probability measure on X, concentrated on F. To prove (<ref>) for J_σ, ∀σ∈ D, we proceed by using induction on k when σ∈ D_k. It is clear for k=0.Now assume that (<ref>) holds for each σ∈ D_k for some k.Then by induction assumption and (<ref>), for each i=1,⋯, n_k+1, μ(J_σ*i)=(J_σ*i)^s/∑_j=1^n_k(J_σ*j)^sμ(J_σ) ≤(J_σ*i)^s/∑_j=1^n_k(J_σ*j)^s((J_σ)/(J_∅))^s≤((J_σ*i)/(J_∅))^s. This proves inequality (<ref>). Now, for any δ>0, let {B_i} be any collection of closed balls with (B_i)≤δ and F⊆∪_i B_i. For each i, let D_B_i be the subset of D corresponding to B_i as given in equation (<ref>). Note that F⊆⋃_i B_i∩ F ⊆⋃_i⋃_σ∈ D_B_iJ_σ=⋃_σ∈D̃J_σ, where D̃:=∪_i=1^∞ D_B_i⊆ D. Let C(s) := max{∑_i=1^N (x_i)^s: (x_1,x_2,⋯, x_N)∈ [0,1]^Nwith ∑_i=1^N x_i=1}= N^1-s, if 0<s<1 1, if s≥ 1. and c(s)=α(s)/C(s)(γ(J_∅)/2)^s>0. Then, by (<ref>) and (<ref>), ∑_iα(s)((B_i)/2)^s ≥ ∑_iα(s)/2^s(γ∑_σ∈ D_B_i(J_σ))^s ≥ ∑_iα(s)/2^s C(s)γ^s∑_σ∈ D_B_i((J_σ))^s ≥ α(s)/2^s C(s)γ^s∑_σ∈D̃((J_σ))^s ≥ α(s)/2^s C(s)γ^s((J_∅))^s∑_σ∈D̃μ(J_σ) ≥c(s)μ(∑_σ∈D̃J_σ) ≥ c(s)μ(F) = c(s). Thus, ℋ^s(F) =lim_δ→ 0ℋ_δ^s(F)≥ c(s)>0, and hence _H(F) ≥ s. In section <ref> we will explore sufficient conditions for 𝒥 to satisfy the uniform covering condition.§ GENERAL SETUP OF -LIMIT SETS We now formalize the ideas stated in the Introduction to give a description of the construction of generalized fractals.We concentrate on the maps in order to take advantage of the computational nature of an IFS, but allow for the maps to be updated and changed at each iteration.Let 𝒳 be a collection of nonempty compact subsets of a metric space. A mapping f: 𝒳→𝒳 is called a compression on 𝒳 if f(E) ⊆ E for each E ∈𝒳.For each natural number m, let𝒞_m(𝒳) = { (f^(1), f^(2), …, f^(m)): f_iis a compression on 𝒳, i=1, …, m}. Let ℳ be a nonempty set.A mapping ℱ:ℳ→𝒞_m(𝒳) k → f_k = (f_k^(1), f_k^(2), ⋯ , f_k^(m))is called a marking of 𝒞_m(𝒳) by ℳ. Each element k ∈ℳ is called the marker of f_k. Given a markingand an initial set E_0 ∈𝒳, we will construct a generalized Moran set from any sequence of markers in ℳ.Note that any sequence {k_ℓ}_ℓ = 0^∞ in ℳ can be represented as a mapping from the ordered set D to ℳ. Let ℱ be a marking of 𝒞_m(𝒳) by ℳ, let E_0 be any element in 𝒳, and D be as in (<ref>).Suppose k⃗ : D →ℳ is a map sending σ to k_σ. For each σ∈ D and 1 ≤ j ≤ m, we recursively define J_∅ = E_0 and J_σ * j = f__̨σ^(j) (J_σ), where f_k_σ is given by ℱ as in (<ref>). The limit set F = ⋂_k≥ 1⋃_σ∈ D_k J_σ associated with 𝒥(k⃗)={J_σ: σ∈ D} is called the ℱ-limit set generated by k⃗ with the initial set E_0. We now make two observations relating the concepts of an -limit set with the attractor of an IFS.First we observe that the attractor of an IFS { S_1, S_2, …, S_m} on a closed subset Δ of ^n can be viewed as an -limit set as follows. Let 𝒳 = { E : Eis a non-empty compact subset of Δ,S_i(E) ⊆ E,for alli}. Since each S_i is a contraction on Δ, the set E_r := Δ∩B(0,r) is a non-empty compact subset of Δ, and S_i(E_r) ⊆ E_r for each i when r is sufficiently large.In other words, E_r ∈𝒳 for sufficiently large r.Also, each contraction map S_i acting on Δ naturally determines a map f^(i) :𝒳→𝒳 given by f^(i)(E) = S_i(E) := { S_i(x) | x ∈ E ⊆Δ} for each E ∈𝒳.Since f^(i)(E) = S_i(E) ⊆ E, f^(i) is a compression for each i.Setf=(f^(1), f^(2), … , f^(m) ). For any non-empty set ℳ, define the markingof 𝒞_m(𝒳) to be the constant function (k)=f for all k ∈ℳ. Thus, for each σ∈ D_k and i=1, … ,m, we have that J_σ * i = S_i(J_σ) from (<ref>). As a result, for any map k⃗: D →ℳ, the collection 𝒥(k⃗) = { J_σ : σ∈ D} is independent of the choice of k⃗.Thus, the associated ℱ-limit set F = ⋂_k≥ 1⋃_σ∈ D_k J_σagrees with the attractor of the given IFS { S_1, S_2, …, S_m}. Conversely, let ℱ be a marking of 𝒞_m(𝒳) by ℳ where 𝒳 is a collection of non-empty compact subsets of Δ.Suppose there is a mapping k⃗ : D →ℳ such that the sequence { f_k_σ}_σ∈ D is constant in 𝒞_m(𝒳) (i.e. there exists an f ∈𝒞_m(𝒳) such that f_k_σ =f for all σ∈ D)and for each i=1,2,…,m, there exists a contraction S_i on Δ such that equation (<ref>) holds for each E ∈𝒳. Then the ℱ-limit set F generated by k⃗ is the attractor of the IFS {S_1, S_2, …, S_m }. Therefore, choosing k⃗: D →ℳ to be a constant map will result in a limit set F that is the attractor of an IFS. In the above sense, our approach is a generalization of the standard IFS construction.An important observation is that replacing {k_σ}_σ∈ D by another sequence{k̃_σ}_σ∈ D in (<ref>) will not change the computational complexity of the construction of 𝒥(k⃗).Thus, generating the limit set F will have a similar computational complexity as generating the attractor of a comparable IFS.In the following section we will compute the Hausdorff dimension of the constructed -limit sets.In section 5 we will provide examples along with their dimensions.§ HAUSDORFF DIMENSIONS OF -LIMIT SETSAs indicated in Propositions <ref> and <ref>, the relative ratio between the diameters of the sets plays an important role in the calculation of the dimension of the limit set.Therefore, we introduce the following definition. For any compression g: 𝒳→𝒳, define U(g)=sup_E∈𝒳(g(E))/(E),andL(g)=inf_E∈𝒳(g(E))/(E). Note that, for each E∈𝒳,L(g)·(E)≤(g(E)) ≤ U(g) ·(E). For any 𝐤∈ℳ and f_𝐤 = (f_^(1), f_^(2), ⋯ , f_^(m)) ∈𝒞_m(𝒳), define𝐔_𝐤=( U(f_𝐤^(1)), ⋯,U(f_𝐤^(m))) ∈ℝ^m,and 𝐋_𝐤=( L(f_𝐤^(1)), ⋯,L(f_𝐤^(m))) ∈ℝ^m. Also, for each x=(x_1, ⋯, x_m)∈ℝ^m and s>0, denote||x||_s=(∑_i=1^m |x_i|^s)^1/s. These notations, Proposition <ref> and Proposition <ref> motivate our main theorem. Let F be the -limit set generated by a sequence {_̨σ}_σ∈ D with initial set J_∅, and s> 0. (a) If F satisfies the uniform covering condition (<ref>) and inf_σ∈ D{||𝐋__̨σ||_s}≥ 1, then _H(F) ≥ s. (b) If sup_σ∈ D{||𝐔__̨σ||_s}<1, then _H(F)≤ s. (a)By (<ref>) and (<ref>), for all σ∈ D, ∑_j=1^m(J_σ * j)^s =∑_j=1^m(f__̨σ^(j)(J_σ))^s ≥∑_j=1^m(L(f__̨σ^(j)))^s (J_σ)^s ≥(J_σ)^s. Thus, by Proposition <ref>, _H(F)≥ s. (b) Similarly, for all σ∈ D, ∑_j=1^m(J_σ*j)^s ≤∑_j=1^m(U(f__̨σ^(j)))^s (J_σ)^s ≤ c ·(J_σ)^s, where c:=sup_σ{(||𝐔__̨σ||_s)^s}<1. By Proposition <ref>, _H(F)≤ s. For practical reasons, we find that it is more convenient to represent the mapping k⃗ : D →ℳ by a sequence {k_ℓ}_ℓ=0^∞⊆ℳ.For each σ=(i_1, i_2, …, i_k) ∈ D_k, let ℓ(σ) = ∑_p=0^k-1 m^p i_k-p be the ordering of σ in the ordered set D.Using this notation, we can rewrite Definition <ref> as follows. Letbe a marking of 𝒞_m(𝒳) by ℳ, let {_̨ℓ}_ℓ=0^∞ be a sequence in ℳ, and E_0 ∈𝒳 be a starting set.For each ℓ =0,1, 2,⋯ and j=1,2,⋯, m, we iteratively denote the set E_mℓ+j=f__̨ℓ^(j)(E_ℓ) ∈𝒳, where f__̨ℓ is given by ℱ as in (<ref>). Let (0)=0 and for n ≥ 1, (n)=m+m^2+⋯+m^n=m^n+1-m/m-1 denote the number of sets in the n^th generation, i.e. the cardinality of D_n. The limit setF=⋂_n=1^∞⋃_ℓ=(n-1)+1^(n) E_ℓ is called the -limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0). In the following, we will use the notation from Definition <ref> to describe the construction of the -limit sets.Clearly, using this notation, Theorem <ref> simply says that if F satisfies the uniform covering condition (<ref>) and inf_ℓ{||𝐋__̨ℓ||_s}≥ 1, then _H(F) ≥ s, and if sup_ℓ{||𝐔__̨ℓ||_s}<1, then _H(F)≤ s. When both {||𝐋__̨ℓ ||_s}_ℓ=0^∞ and {||𝐔__̨ℓ ||_s}_ℓ =0^∞ are convergent sequences, the following corollary enables us to quickly estimate the dimension of F. Let F be the limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0). (a) Let s_* := sup{s: lim inf_ℓ→∞{||𝐋__̨ℓ ||_s}>1 }.Then _H(F) ≥s_*, provided F satisfies the uniform covering condition (<ref>). (b) Let s^* := inf{s: lim sup_ℓ→∞{||𝐔__̨ℓ||_s}<1 }.Then _H(F)≤s^*. For any 0< s< s_*, by the definition of s_*, lim inf_ℓ→∞{||𝐋__̨ℓ ||_s}>1. Thus, when ℓ_*∈ℕ is large enough, inf_ℓ≥ℓ_*{||𝐋__̨ℓ||_s }≥ 1, i.e. inf_ℓ≥ 0{||𝐋__̨ℓ_*+ℓ||_s}≥ 1. Since F∩ E_ℓ_* is the set generated by the triple (, {_̨ℓ_*+ℓ}_ℓ =0^∞ , E_ℓ_*),by Theorem <ref>, it follows that _H(F∩ E_ℓ_*) ≥ s for any ℓ_* large enough. This implies that _H(F) ≥ s for any s<s_* and hence _H(F) ≥s_*. Similarly, we also have _H(F)≤s^*. In the following corollaries, we will see that bounds of the dimension of F can also be obtained from corresponding bounds on 𝐋__̨ℓ and 𝐔__̨ℓ. Notation. For any two points x=(x_1,⋯, x_m) and y=(y_1,⋯, y_m) in ℝ^m, we say x ≤ y if x_i ≤ y_i for each i=1,⋯, m. Let 𝐭=(t_1,⋯, t_m)and 𝐫=(r_1,⋯, r_m) be two points in (0,1)^m ⊂ℝ^m. Let s_* and s^* be the solutions to ||𝐭||_s_*=1, and ||𝐫||_s^*=1 respectively,i.e. t_1^s_*+t_2^s_*+⋯+t_m^s_*=1, andr_1^s^*+r_2^s^*+⋯+r_m^s^*=1. (a) If 𝐋__̨ℓ≥𝐭 for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)≥ s_*. (b)If 𝐔__̨ℓ≤𝐫 for all ℓ, then _H(F) ≤ s^*. (c) If 𝐋__̨ℓ=𝐫 =𝐔__̨ℓ for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)=s^*. (a) Let 0< s<s_*. Then, inf_ℓ{||𝐋__̨ℓ||_s}≥ ||𝐭||_s ≥ ||𝐭||_s_*= 1. Thus, by Theorem <ref>, _H(F)≥ s for any s<s_*, and hence _H(F)≥ s_*. (b) Similarly,let 0<s^*<s. Then, sup_ℓ{||𝐔__̨ℓ||_s}≤ ||𝐫||_s < ||𝐫||_s^*= 1. Thus, by Theorem <ref>,_H(F)≤ s for any s>s^*, and hence _H(F)≤ s^*. (c) follows from (a) and (b). A special case of Corollary <ref> gives the following explicit formulas for the bounds on the dimension of F. Let F be the limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0).Let 𝐭=(t, ⋯, t)and 𝐫=(r, ⋯, r), for some 0<t, r<1. (a) If 𝐋__̨ℓ≥𝐭 for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)≥logm/-log t. (b)If 𝐔__̨ℓ≤𝐫 for all ℓ, then _H(F) ≤logm/-log r. (c) If 𝐋__̨ℓ=𝐫 =𝐔__̨ℓ for all ℓ and F satisfies the uniform covering condition (<ref>), then _H(F)=logm/-log r. Other types of bounds on 𝐋__̨ℓ and 𝐔__̨ℓ can also be used to provide bounds on _H(F), as indicated by the following result. Let F be the limit set generated by the triple (, {_̨ℓ}_ℓ =0^∞ , E_0). (a) If F satisfies the uniform covering condition (<ref>) and w:=inf_ℓ{||𝐋__̨ℓ||_1}≥ 1, then _H(F) ≥log(m)/log(m)-log(w). (b) If u:=sup_ℓ{||𝐔__̨ℓ||_1} < 1, then _H(F) ≤log(m)/log(m)-log(u). (a). In this case, for s = log(m)/log(m)-log(w)≥ 1, we have ∑_j=1^m(L(f__̨ℓ^(j)) )^s/m≥(∑_j=1^mL(f__̨ℓ^(j))/m)^s ≥(w/m)^s for each ℓ. Thus, inf_ℓ{||𝐋__̨ℓ||_s}≥ m^1/sw/m = 1, then by Theorem <ref>, _H(F) ≥ s. (b).In this case, for any 1 ≥ s > log(m)/log(m)-log(u), we have ∑_j=1^m(U(f__̨ℓ^(j)))^s/m≤(∑_j=1^mU(f__̨ℓ^(j))/m)^s ≤(u/m)^s for each ℓ. Thus, sup_ℓ{||𝐔__̨ℓ||_s}≤ m^1/su/m <1. By Theorem <ref>, _H(F) ≤ s. Hence, _H(F) ≤log(m)/log(m)-log(u). Note that this corollary generally provides better bounds on _H(F) than those obtained from directly applying Theorem <ref>.§ EXAMPLES OF -LIMIT SETS In this section we describe the construction of both classical fractals and generalized Moran sets in the language of Section <ref>, and calculate the dimension using the results from Section <ref>.Cantor-Like Sets We first consider Cantor-like sets.Let𝒳 = { [a,b] : a,b ∈}be the collection of closed intervals, m=2, and let ℳ = [0,1]^2 ⊆. For each 𝐤=(k^(1),k^(2)) ∈ℳ, we consider the following two maps, f^(1)_𝐤 :𝒳 →𝒳[a,b]↦[a, k^(1)(b-a)+a] f^(2)_𝐤 :𝒳 →𝒳 [a,b]↦ [k^(2)(a-b)+b, b]. Note that both f^(1)_$̨ andf^(2)_are compression maps for any∈̨ℳ.Thus, this defines a marking :ℳ →𝒞_2(𝒳) ↦f_= (f^(1)_, f^(2)_). Here, for each𝐤=(k^(1), k^(2)) ∈ℳ, one can clearly see that(f^(i)_𝐤([a,b]))=k^(i)·([a,b]). Thus,L (f^(i)_𝐤 )=k^(i)=U(f^(i)_𝐤), and hence𝐋_𝐤=𝐤=𝐔_𝐤. LetE_0 = [0,1] ∈𝒳be fixed. For any sequence{_̨ℓ}_ℓ=0^∞ ∈ℳ, we define the following: E^(0) = E_0E^(1) = f^(1)_𝐤_0( E_0 ) ∪f^(2)_𝐤_0( E_0 ) =: E_1∪ E_2E^(2) = f^(1)_𝐤_1( E_1) ∪f^(2)_𝐤_1( E_1) ∪f^(1)_𝐤_2( E_2) ∪f^(2)_𝐤_2( E_2):= E_3∪E_4∪E_5∪E_6⋮E^(n) = ⋃_i=2^n-1-1^2^n-2(f__̨i^(1)(E_i) ∪ f__̨i^(2)(E_i) ):=⋃_i=2^n-1-1^2^n-2(E_2i+1∪ E_2i+2)=⋃_ℓ= 2^n-1^2(2^n-1) E_ℓ. Note that when𝐤_ℓ=(1/3, 1/3)for allℓ,E^(n) is then^th-generation of the Cantor setandF =lim_n →∞ E^(n) = ⋂_n E^(n) =. Observe that the process of constructing the sequence{E^(n)}_n=0^∞is independent of the values of{_̨ℓ}_ℓ=0^∞. To allow for more general outcomes, we can update the linear functionsf^(1)_$̨ and f^(2)_$̨ simply by changing the value of$̨ at each stage of the construction, which does not change the computational complexity of the process.Using this idea, we now construct some examples of Cantor-like sets by choosing suitable sequences {_̨ℓ}_ℓ=0^∞. Let 𝐤_ℓ = (ℓ+1/4ℓ+6, 2ℓ +5/8ℓ+16) for ℓ≥ 0, and let F be the -limit set generated by the triple (, {_̨ℓ}_ℓ=0^∞, E_0).In Figure <ref> we plot the usual Cantor set(in blue) below the set F (in red) to illustrate the comparison. We can see that the set F has the same basic shape as the Cantor set , but is no longer strictly self-similar. In order to compute the Hausdorff dimension of the new Cantor-like set F, we apply Corollary <ref>.Note that by equation (<ref>), lim_ℓ→∞||𝐋_𝐤_ℓ||_s=lim_ℓ→∞||𝐤_ℓ ||_s=2^1/s/4. So, s_* =sup_s{lim inf_ℓ→∞||𝐋_𝐤_ℓ||_s>1} = sup_s {2^1/s/4>1}=1/2. Similarly, we also have s^*=1/2. By Corollary <ref>, _H(F)=1/2.Here, F satisfies the uniform covering condition (<ref>) sincesup{k_ℓ^(1)+k_ℓ^(2): ℓ=0,1,2, ⋯}=1/2<1,according to Proposition <ref>. In the next example, we will construct a random Cantor-like set as follows. For each ℓ≥ 0, we take 𝐤_ℓ=(q_ℓ, 1/2-q_ℓ) where q_ℓ is a random number between 1/8 and 3/8. Let F be the corresponding -limit set generated by the triple (, {_̨ℓ}_ℓ=0^∞, E_0).We plot the first few generations in Figure <ref>.In this example, the total length of the n^th generation E^(n) is chosen to be (1/2)^n, while the scaling factors of the left subintervals at each stage are randomly chosen. We now estimate the dimension of F.By (<ref>), (1/8,1/8)≤𝐋_𝐤_ℓ =𝐤_ℓ=𝐔_𝐤_ℓ≤(3/8,3/8). By Corollary <ref>, log(2)/-log(1/8)≤_H(F) ≤log(2)/-log(3/8). That is, 1/3≤_H(F) ≤log(2)/log(8/3)≈ 0.7067. Note that due to Proposition <ref>, F satisfies the uniform covering condition (<ref>) since q_ℓ + 1/2-q_ℓ = 1/2 <1 for each ℓ≥ 0. In this example, we create a sequence {_̨ℓ}_ℓ=0^∞ that results in a limit set with a given measure, e.g. 1/3. Of course, the classic example of such a limiting set is the fat Cantor set.For a different approach, let ∑_n=0^∞ a_n be any convergent series of positive terms with limit L.We consider a sequence {_̨ℓ}_ℓ=0^∞ defined in the following way. Let n≥1 be the generation of the construction and for each ℓ with 2^n-1-1 ≤ℓ≤ 2^n-2, define 𝐤_ℓ=(b_n, b_n) whereb_1:= 3/2L-a_02(3/2L) andb_n := 3/2L-∑_i=0^n-1 a_i2(3/2L-∑_i=0^n-2a_i) forn≥ 2. With this sequence {_̨ℓ}_ℓ =0^∞, one can find that the length of each interval in the n^th generation isb_1 b_2 ⋯ b_n=3/2L-∑_i=0^n-1 a_i2^n ·3/2L. Thus, the total length of the n^th generation is3/2L-∑_i=0^n-1 a_i3/2L=1-2/3L∑_i=0^n-1 a_iwhich converges to 1/3 as desired.As an example, we take the convergent series ∑_n=0^∞1n! = e and use it to create the -limit set F with measure 1/3.The first few generations are shown in Figure <ref>. Sierpinski Triangle The Sierpinski triangle is another well known fractal. Following the general setup in Section <ref>, we take𝒳 = { (A,B,C) | A,B,C ∈^2}representing the collection of all triangles Δ ABC in ^2, m=3, and ℳ = [0,1]^6 ⊆^6.For each 𝐤=(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ℳ and i=1,2,3 we can define affine transformations f_^(i) : 𝒳→𝒳 as f^(1)_ (A,B,C)=(A,A+k^(1)(B-A), A+k^(2)(C-A)) f^(2)_ (A,B,C)=(B+k^(4)(A-B),B, B+k^(3)(C-B)) f^(3)_ (A,B,C)=(C+k^(5)(A-C),C+k^(6)(B-C), C) for every (A,B,C) ∈𝒳. Note that each f^(i)_$̨ is a compression map fori=1,2,3andany∈̨ℳ.Thus, this defines a marking :ℳ →𝒞_3(𝒳) ↦f_= (f^(1)_, f^(2)_, f^(3)_). Of course, to prevent overlaps we can require thatk^(1) + k^(4) ≤1, k^(2) +k^(5) ≤1, k^(3)+k^(6) ≤1. When each of the inequalities are strict, the images off^(i)_are three disconnected triangles, as illustrated in Figure <ref>. When all equalities hold, the images are connected, as illustrated in Figure <ref>. In the case of the connected sets, the values of𝐤=(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) are determined byk^(1), k^(2), k^(3)sincek^(4) = 1-k^(1),k^(5)=1-k^(2),k^(6)=1-k^(3).In this case, we may also view=̨(k^(1), k^(2), k^(3))as a vector in[0,1]^3 ⊆^3. To create the normal Sierpinski triangle, we choose E_0 = [ -1/21/20;00 √(3)/2 ], the equilateral triangle of unit side length, and_̨ℓ ∈ℳto be the constant sequence_̨ℓ ==̨(1/2, 1/2, 1/2, 1/2, 1/2, 1/2)so that each iteration maps a triangle to three triangles of half the side length with the desired translation.In this case the-limit set generated by(, {_̨ℓ}_ℓ=0^∞, E_0)corresponds to the standard Sierpinski Triangle as seen in Figure <ref>. To generate Sierpinski-like fractals, we now adjust the values of the marking parameters{_̨ℓ}_ℓ=0^∞.For each𝐤=(k^(1), k^(2),⋯, k^(6))∈ℳand1≤i ≤3, U(f_𝐤^(i))=sup_(A,B,C)∈𝒳diam (f_𝐤^(i)(A,B,C))/diam((A,B,C)) =max{k^(2i-1), k^(2i)}, and L(f_𝐤^(i))=inf_(A,B,C)∈𝒳diam (f_𝐤^(i)(A,B,C))/diam((A,B,C)) =min{k^(2i-1), k^(2i)}. When$̨ is bounded, i.e. if λ≤ k^(j)≤Λ<1 for all j=1,⋯ ,6, then 𝐔_𝐤≤𝐫:=(r,⋯,r)and 𝐋_𝐤≥𝐬:=(s,⋯,s), where r=max{1-λ, Λ} and s=min{1-λ, Λ}. Following our general process, we construct some random Sierpinski-like sets by introducing randomness into the choice of the sequence {_̨ℓ}_ℓ=0^∞. Let {_̨ℓ}_ℓ=0^∞ ={ (k_ℓ^(1), k_ℓ^(2), k_ℓ^(3) )}_ℓ=0^∞ be a sequence in [0,1]^3 with each k_ℓ^(i) a random number between given numbers λ and Λ for each i=1,2,3.Let F be the -limit set generated by (, {_̨ℓ}_ℓ=0^∞, E_0).Then the 6^th generation of the construction results in images like Figure <ref>. Here, in Figure <ref>, λ=1/4 and Λ=3/4;while in Figure <ref>, λ=0.45 and Λ=0.55. Note that the sets are no longer self-similar. In Figure <ref>, we pick λ=0.45 and Λ=0.55. By Corollary <ref>, log(m)/-log(s)≤_H(F) ≤log(m)/-log(r), where m=3, r=0.55 and s=0.45. That is, 1.3758 ≤_H(F)≤ 1.8377,provided F satisfies the uniform covering condition (<ref>). As in Example <ref>, but replacing E_0 withẼ_̃0̃ = [ 0 1 0; 0 0 1 ], the 7^th generation of the construction results in an image like Figure <ref>, when λ=1/4 and Λ=3/4. For each ℓ=0,1, ⋯, let 𝐤_ℓ=(k_ℓ^(1), k_ℓ^(2), ⋯, k_ℓ^(6)) where k_ℓ^(1) = 1/2+a_ℓ/√(ℓ+1),k_ℓ^(2)=1-k_ℓ^(1), k_ℓ^(3) = 1/2+b_ℓ/√(ℓ+1),k_ℓ^(4)=1-k_ℓ^(3), k_ℓ^(5) = 1/2+c_ℓ/ℓ+1,k_ℓ^(6)=1-k_ℓ^(5). for random numbers a_ℓ, b_ℓ, c_ℓ∈ [-1/3, 1/3].Let F be the -limit set F generated by (, {_̨ℓ}_ℓ=0^∞, E_0). Then the seventh generation of the construction of F results in an image like Figure <ref>. In this case, we can calculate the exact value of the Hausdorff dimension of F.Indeed, by Corollary <ref>, lim_ℓ→∞ (||𝐔_𝐤_ℓ||_s)^s=3/2^s=lim_ℓ→∞ (||𝐋_𝐤_ℓ||_s)^s. Thus, _H(F)=log(3)/log(2), provided F satisfies the uniform covering condition (<ref>). Menger Sponge Let𝒳 = { (O,A,B,C) | O,A,B,C ∈^3 }representing the collection of all rectangular prisms (OABC) in ^3, m=20, and ℳ = {(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ [0,1]^6 :k^(1)≤ k^(2),k^(3)≤ k^(4), k^(5)≤ k^(6)}. For each ∈̨ℳ and i =1, 2, … , 20, we can define affine transformations f^(i)_ : 𝒳→𝒳 as follows.For any =̨(k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ℳ, define T = [ 0 k^(1) k^(2) 1 ],R= [ 0 k^(3) k^(4) 1 ],S= [ 0 k^(5) k^(6) 1 ]. Let I = { (a,b,c) | 1 ≤ a,b,c≤3witha,b,c ∈, and no two of a,b,c equal to 2}. For each (a,b,c) ∈ I and ∈̨ℳ, define M_(a,b,c)=[ 1-( T(a) + R(b) +S(c)) T(a) R(b) S(c); 1-( T(a+1) + R(b) +S(c)) T(a+1) R(b) S(c); 1-( T(a) + R(b+1) +S(c)) T(a) R(b+1) S(c); 1-( T(a) + R(b) +S(c+1)) T(a) R(b) S(c+1);]. Note that the set I contains 20 elements, so we can express it asI = { (a_i, b_i, c_i) | 1 ≤ i ≤ 20}. For each ∈̨ℳ and 1 ≤ i ≤ 20, we consider the affine transformation f^(i)_ : 𝒳→𝒳 given by f^(i)_ (O,A,B,C)= M_ (a_i,b_i,c_i)[ O; A; B; C ] for every (O, A, B, C) ∈𝒳. Note that for i=1, …, 20 and ∈̨ℳ, f^(i)_ is a compression. Thus, we can define a marking : ℳ→𝒞_20(𝒳) by sending ↦̨f_= (f^(1)_, … , f^(20)_). Using this, for any starting rectangular prism E_0 = (O,A,B,C) ∈𝒳, we can generate a sequence of sets that follows a similar construction to the Menger Sponge. Let E_0 = [ 0 1 0 0; 0 0 1 0; 0 0 0 1 ] be the cube of unit side length and choose _̨ℓ∈ℳ to be the constant sequence _̨ℓ ==̨(1/3, 2/3, 1/3, 2/3, 1/3, 2/3).Then the -limit set F generated by the triple (, {_̨ℓ}_ℓ =0^∞, E_0 ) is the classical Menger sponge, as seen in Figure <ref>. Now we consider variations of Menger Sponge. For each 𝐤=(k^(1), k^(2), ⋯, k^(6))∈ℳ and 1≤ i ≤ 20, U(f_𝐤^(i)) = sup_(O,A,B,C)∈𝒳diam (f_𝐤^(i)(O,A,B,C))/diam((O,A,B,C))= sup_(O,A,B,C)∈𝒳diam (M_𝐤(a_i, b_i, c_i)[O,A,B,C]')/diam((O,A,B,C))= max{T(a_i+1)-T(a_i), R(b_i+1)-R(b_i), S(c_i+1)-S(c_i)}. Similarly, L(f_𝐤^(i))=min{T(a_i+1)-T(a_i), R(b_i+1)-R(b_i), S(c_i+1)-S(c_i)}. When k^(2j)=1-k^(2j-1) for each j=1, 2, 3, it is easy to check that ∑_i=1^20U(f_𝐤^(i))^s = ∑_i=1^20max{T(a_i+1)-T(a_i), R(b_i+1)-R(b_i), S(c_i+1)-S(c_i)}^s= 8max{k^(1), k^(3), k^(5)}^s+4max{1-2k^(1), k^(3), k^(5)}^s+4max{k^(1), 1-2k^(3), k^(5)}^s +4max{k^(1), k^(3), 1-2k^(5)}^s. LetẼ_̃0̃ = [ 0 3 0 0; 0 0 1 0; 0 0 0 2 ].Let (k^(1), k^(2), k^(3), k^(4), k^(5), k^(6)) ∈ℳ where each k^(i) is a random number in [0,1], but still satisfying the condition k^(1)≤ k^(2), k^(3)≤ k^(4), k^(5)≤ k^(6).Then the first generation E^(1) of the construction results in a set like Figure <ref>. Let _̨ℓ =(k_ℓ^(1), k_ℓ^(2), k_ℓ^(3), k_ℓ^(4), k_ℓ^(5), k_ℓ^(6)) ∈ℳ with each k_ℓ^(2j-1) a random number between given parameters λ and Λ and k_ℓ^(2j)=1-k_ℓ^(2j-1) for each j=1,2,3.Let F be the -limit set generated by (, {_̨ℓ}_ℓ =0^∞, E_0). Then the third iteration of the construction of F results in images like Figure <ref>. Here, in Figure <ref> the parameters λ=0 and Λ=1/2, while in Figure <ref> the parameters λ=0.32 and Λ=0.35. We now calculate the dimension of the limit fractal F illustrated by Figure <ref> in Example <ref>. Note that in general, when λ≤ k^(2j-1)≤Λ for each j=1, 2, 3, it follows that (||𝐔_𝐤||_s)^s=∑_i=1^20U(f_𝐤^(i))^s ≤ 8Λ^s +12max{1-2λ, Λ}^s. Similarly, (||𝐋_𝐤||_s)^s ≥ 8λ^s +12min{1-2Λ, λ}^s. In particular, when λ=0.32 and Λ=0.35,for any s>2.901, (||𝐔_𝐤||_s)^s≤8Λ^s+12max{1-2λ, Λ}^s ≤ 8*0.35^s+12*0.36^s <8*0.35^2.901+12*0.36^2.901≈ 1.000. By Theorem <ref>, _H(F)≤ 2.901. Similarly, for any s ≤ 2.546, (||𝐋_𝐤||_s)^s ≥8λ^s+12min{1-2Λ, λ}^s ≥8*0.32^s+12*0.3^s ≥ 8*0.32^2.546+12*0.3^2.546≈ 1.000. By Theorem <ref> again, _H(F)≥ 2.546, provided F satisfies the uniform covering condition (<ref>). As a result, 2.546 ≤_H(F)≤ 2.901. For each ℓ≥ 0, let 𝐤_ℓ=(k_ℓ^(1), k_ℓ^(2), ⋯, k_ℓ^(6)) where k_ℓ^(1) = 1/3+(-1)^ℓ/12(ℓ+1)^2,k_ℓ^(2)=1-k_ℓ^(1), k_ℓ^(3) = 1/3-(-1)^ℓ/6(ℓ+1)^2,k_ℓ^(4)=1-k_ℓ^(3), k_ℓ^(5) = 1/3+(-1)^ℓ/18(ℓ+1)^2,k_ℓ^(6)=1-k_ℓ^(5). Let F be the -limit set generated by (, {_̨ℓ}_ℓ =0^∞, E_0).Then the third generation of the construction of F leads to an image like Figure <ref>. In this case, we can still calculate the exact Hasudorff dimension of F. By direct computation, lim_ℓ→∞ (||𝐔_𝐤_ℓ||_s)^s=20/3^s=lim_ℓ→∞ (||𝐋_𝐤_ℓ||_s)^s. Thus, by Corollary <ref>, _H(F)=log(20)/log(3)≈ 2.7268, since F satisfies the uniform covering condition according to Example <ref>. Here, we discuss similarities and differences of this construction with V-variable fractals created by Barnsley, Hutchinson, and Stenflo in <cit.>, <cit.>. These authors have described a similar approach to creating more generalized fractals that can take on a prescribed amount of randomness.In <cit.> and <cit.>, they describe a generating process for some fractals along with calculations of their dimensions.In essence, a V-variable fractal set has at most V ∈ number of distinct patterns in each generation of the construction.This is done through the following process.Let (X,d) be a metric space, Λ an index set, F^λ = {f^λ_1, f^λ_2, …, f^λ_m} an IFS for each λ∈Λ, and P a probability distribution on some σ-algebra of subsets of Λ.Then denote 𝐅 = {(X,d), F^λ, λ∈Λ, P} to be a family of IFSs (with at least two functions in each IFS) defined on (X,d).Assume that the IFSs F^λ are uniformly contractive and uniformly bounded, that is, for some 0<r<1,sup_λmax_m d ( f^λ_m (x), f^λ_m(y) ) ≤ r d(x,y), sup_λmax_m d( f^λ_m(a),a )< ∞for all x,y ∈ X and some a ∈ X.A tree code is a map ω from the set of all finite sequences {1, … ,m } to Λ.A tree code is V-variable if for each positive integer k, there are at most V distinct tree codes in the tree truncated at the k^th generation.For example, consider the Sierpinski triangle.We let F be the IFS that maps the triangle to three copies of 1/2 the size, as usual.Let G be the IFS that maps the initial triangle to three triangles that are 1/3 the size, with the vertices shared with the initial set being the fixed points of the maps.See Figure <ref> for the image of the initial step of each.Thus, 𝐅 = { (^2, d), {F,G}, P=(1/2, 1/2)} is the family {F,G} with probability function uniformly choosing 1/2 for each IFS.Using these IFSs, three V-variable pre-fractals are given in Figure <ref>, being 1-variable, 2-variable, and 3-variable respectively.Now, we express V-variable fractals in terms of -limit sets.Let 𝒳, ℳ,ℱ and E_0 be as in Section <ref>.Let F = ( 1/2, 1/2, 1/2, 1/2,1/2,1/2) ∈ℳ and G=( 1/3, 1/3, 1/3, 1/3, 1/3, 1/3) ∈ℳ.We will use F and G to denote terms in the sequence { k_ℓ}_ℓ =1^∞.Consider the third generation examples in Figure <ref>.Then from left to right we have the following: [V=1{ k_ℓ} _ℓ =1^13= { F, F, F, F, G, G, G, G, G, G, G, G, G}; V=2 { k_ℓ} _ℓ =1^13= { F, G, F, G, F, G, F, F, F, G, F, G, F};V=3{ k_ℓ} _ℓ =1^13 = { F, F, F, G, F, F, G, F, G, G, G, G, G}. ]From these examples, we can see that if we want to create a V-variable fractal, for each generation we should choose at most V distinct triples from the set { (A,B,C) | A,B,C ∈{F,G}} and repeat those triples in any order.When V < ∞, there are at most V distinct tree codes in the address of point in the set.We can create such a situation from our construction described earlier in section <ref> by choosing blocks of {k_ℓ} that repeat across generations.In the case that V = ∞, the fractal is based off of a probability distribution for applying specific IFSs.In our construction we also can use a probability distribution to determine the contraction ratios within a generation (as in Examples <ref>, <ref> and <ref>), but we do not require such a choice. We allow for deterministic sequences that also do not repeat any blocks, thus not falling into the category of V-variable.§ UNIFORM COVERING CONDITION In previous sections, we have seen that the uniform covering condition (<ref>) plays a vital role in computing a lower estimate for the Hausdorff dimension of a fractal.In this section we explore the sufficient conditions needed for a fractal to satisfy the uniform covering condition. Let (X, d) be a metric space with the following property: For any ϵ>0, there exists a natural number N_ϵ such that for any ρ>0, any closed ball in X of diameter ρ contains at most N_ϵ many disjoint balls of diameter ϵρ. Clearly, any Euclidean space satisfies this property. Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of (X, d), and F be the limit set of 𝒥 as given in (<ref>).Suppose that 𝒥 satisfies the following conditions: * there exists a number r∈(0,1] such that for any k∈ℕ and for each σ∈ D_k, rc_k≤ diam(J_σ)≤c_k/r where c_k:=min{ diam(J_σ̅): σ̅∈ D_k-1}. * there exists a number τ∈ (0, 1] such that for each σ∈ D, the convex hull of J_σ contains a closed ball W_σ such that diam(W_σ)≥τ· diam(J_σ) and for each k∈ℕ, the collection {W_σ: σ∈ D_k} are pairwise disjoint. Then F satisfies the uniform covering condition (<ref>). For any closed ball B in X, let k be the number such that c_k+1≤ diam(B) <c_k where by convention, we set c_0=∞. Let D_B:={σ∈ D_k: B∩ F∩ J_σ≠∅}. Note that B∩ F = B ∩ F ∩⋃_σ∈ D_k J_σ⊆⋃_σ∈ D_B J_σ. Also for any σ∈ D_B, since diam(J_σ)≤c_k/r and B∩ J_σ≠∅, it follows that J_σ⊆B̅(x_0, r+2/2r c_k), where x_0∈ X is the center of the ball B.Thus, W_σ⊆B̅(x_0, r+2/2r c_k). Let ρ=r+2/rc_k and ϵ =r^2 /r+2τ, then diam(W_σ)≥τ· diam(J_σ) ≥τ r c_k = ϵρ. Since {W_σ: σ∈ D_B} are pairwise disjoint, the cardinality of D_B is at most N:=N_ϵ. On the other hand, for γ =r^2/N, it holds that diam(B)≥ c_k+1≥ rc_k= γ N c_k/r≥γ∑_σ∈ D_Bc_k/r≥γ∑_σ∈ D_Bdiam(J_σ). As a result, 𝒥 satisfies the condition (<ref>) as desired. We now discuss some specific sufficient conditions concerning the types of examples provided in section 3.4.To start, let's first consider Cantor-like constructions.Let X be the family of closed intervals described in (<ref>), m=2, and ℳ = [0,1]^2 ⊆. Let {_̨ℓ}_ℓ=0^∞ be a sequence in ℳ with sup{k_ℓ^(1)+k_ℓ^(2): ℓ=0,1,2, ⋯}<1, and F be the -limit set generated by the triple (, {_̨ℓ}_ℓ=0^∞, E_0). Then F satisfies the uniform covering condition (<ref>). Let N=1 and γ=inf_ℓ{1-k_ℓ^(1)-k_ℓ^(2)}∈ (0,1] . For any closed interval B in ℝ with B∩ F ≠∅, consider the set ℒ:={ℓ(σ): B∩ F ⊆ J_σ, σ∈ D}, where ℓ(σ) is given in (<ref>). Note that ℒ is nonempty because B∩ F ⊆ J_0 implies that ℓ(0)∈ℒ. If ℒ is an infinite set, then since diam(J_σ)→ 0 as ℓ(σ)→∞, there exists σ^*∈ D such that ℓ(σ^*)∈ℒ and diam(B)≥ diam(J_σ^*)≥γ· diam(J_σ^*). If ℒ is finite, let ℓ(σ^*) be the maximum number in ℒ for some σ^*∈ D. Then,ℓ(σ^*)∈ℒ but ℓ(σ^* * j)∉ℒ for each j=1,2. This implies thatB∩ J_σ^* *j ≠∅ for both j=1, 2 because J_σ^* = J_σ^* *1 ∪J_σ^* *2. Since B is an interval, the gap J∖ (J_σ^* * 1∪ J_σ^* * 2) between J_σ^* * 1 and J_σ^* * 2 is contained in B, which yields that diam(B) ≥diam(J∖ (J_σ^* * 1∪ J_σ^* * 2) )=diam(J)-diam (J_σ^* * 1)-diam (J_σ^* * 2)≥ diam(J_σ^*)(1-k_ℓ(σ^*)^(1)-k_ℓ(σ^*)^(2) )≥γ· diam(J_σ^*). As a result, in both cases, the uniform covering condition (<ref>) holds.Motivated by Proposition <ref>, we now consider a generalization of the above result. Let n ≥ 1 and ℋ be a collection of subsets of a metric space (X,d).Define ρ_n(ℋ)=inf{r: There exists a ball B in X of radius r that intersectsat leastn+1elements in ℋ}. Here ρ_n(ℋ) is a quantity describing the “gap" between n+1 elements of ℋ.Let 𝒥={J_σ : σ∈ D} be a collection of compact subsets of a metric space (X,d), and n ≥ 1.Defineγ_n(𝒥):=inf{ρ_n({J_σ*i: σ∈ R_k, i=1,2,⋯, m})/∑_σ∈ R_kdiam(J_σ):for some k. .andR_k⊆ D_kwith1 ≤ |R_k|≤ n }, where |R_k| denotes the cardinality of the set R_k. Here γ_n(𝒥) is a quantity describing the relative size of the “gap" between n+1 children of a generation and the size of the parent sets. Now we give some examples of calculations of these two quantities. Let 𝒥 be the collection of closed intervals used in the construction of a Cantor-like set given in (<ref>).Thenγ_1(𝒥)= inf{ρ_1({J_σ*i: σ∈ R_k, i=1,2})/diam(J_σ):for some k andR_k⊆ D_kwith |R_k|= 1}= inf{ρ_1({J_σ*1, J_σ*2}) /diam(J_σ) :for σ∈ D }= inf{diam(J_σ) - diam(J_σ*1) - diam(J_σ*2)/diam(J_σ): σ∈ D }= inf{1-diam(J_σ * 1)/diam(J_σ) - diam(J_σ * 2)/diam(J_σ): σ∈ D },which agrees with the γ in (<ref>), see Figure <ref>. Let 𝒥 be the collection of triangles used in (<ref>).In the following figures, we plot the smallest ball that intersects a certain number of children.The children that have non-empty intersection with the ball are colored red, while those that have empty intersection are light blue. First note that for any σ∈ D, ρ_1({J_σ*1, J_σ*2, J_σ*3})=0 since any pair of children share a vertex.At the intersection of the two children of J_σ one can construct a ball of arbitrarily small radius.See Figure <ref>. Moreover, ρ_2({J_σ*1, J_σ*2, J_σ*3})>0 because the radius of any ball that intersects all three children of J_σ is bounded below by the radius of the inscribed circle of the removed center triangle.In other words, ρ_2({J_σ*1, J_σ*2, J_σ*3}) is equal to the radius of the inscribed circle.See Figure <ref> for illustration. Now we may compute γ_n(𝒥) as follows. Note that for n=1, γ_1(𝒥)= inf{ρ_1({J_σ*i: σ∈ R_k, i=1,2,3})/diam(J_σ):for some k andR_k⊆ D_kwith |R_k|= 1}= inf{ρ_1({J_σ*1, J_σ*2, J_σ*3}) /diam(J_σ) :for σ∈ D }=0. On the other hand, when n=2, we haveγ_2(𝒥)= inf{ρ_2({J_σ*i: σ∈ R_k, i=1,2,3})/diam(J_σ):for some k andR_k⊆ D_kwith |R_k|≤ 2}.When |R_k|=1, this is reduced to the same case as Figure <ref>.When |R_k|=2, we use two parent triangles, and must find the ball with smallest radius that intersects three or more children. See Figure <ref> for a few candidates for the ball with smallest radius.For each R_k ⊆ D_k with |R_k| ≤ 2,ρ_2({J_σ*i: σ∈ R_k, i=1,2,3}) >0.For some nice 𝒥, one may expect γ_2(𝒥) to also be positive. Let 𝒥 := {J_σ : σ∈ D} be a collection of compact subsets of (X, d) satisfying MSC(3) and lim_k→∞max{diam(J_σ): σ∈ D_k }=0,and let F be the limit set of 𝒥 as given in (<ref>).If there exists an N such that γ_N(𝒥)>0, then F satisfies the uniform covering condition (<ref>). Let γ=γ_N(𝒥)>0. For any closed ball B in X with B∩ F≠∅, let g(k) be the number of elements σ in D_ksuch that B∩ F∩ J_σ≠∅. Then g:ℕ∪{0}→ℕ is monotone increasing with g(0)=1. Case 1: If g(k)≤ N for all k=0,1,2,⋯, that is,for each k, there exists an index set I_k with |I_k|≤ Nsuch that B∩ F⊆⋃_i∈ I_kJ_σ_i^(k) for some σ_i^(k)∈ D_k. Thus when k is large enough, diam(B)>γ·∑_i∈ I_kdiam(J_σ_i^(k)) due to the fact that 0≤lim_k→∞∑_i∈ I_kdiam(J_σ_i^(k)) ≤ N·lim_k→∞max{diam(J_σ): σ∈ D_k}=0. Hence, equation(<ref>) holds for B. Case 2: There exists k^* ≥ 0 such that g(k^*)≤ N but g(k^*+1)>N. Since g(k^*) ≤ N, there are g(k^*) many elements σ∈ D_k^* such that B ∩ F ∩ J_σ≠∅.That is, there exists R_k^*⊆ D_k^* with |R_k^*| ≤ N such that B ∩ F ⊆⋃_σ∈ R_k^* J_σ.On the other hand, since g(k^*+1)>N, B ∩ F intersects at least N+1 elements of D_k^*+1.Since B∩ F ⊆⋃_σ∈ R_k^* J_σ, all of these N+1 elements must be children of {J_σ : σ∈ R_k^*}.Then, by the definition of ρ_N in (<ref>), diam(B) ≥ρ_N( {J_σ*i: σ∈ R_k^*, i=1,2,… m}) ≥γ·∑_σ∈ R_k^* diam(J_σ). As a result, F satisfies the uniform covering condition (<ref>). To show an application of Theorem <ref>, we now consider some examples provided in section <ref>.Let {_̨ℓ}_ℓ=0^∞ be a sequence in ℳ as defined in (<ref>) and F be the ℱ-limit set generated by (ℱ, {_̨ℓ}_ℓ=0^∞, E_0 ) associated with 𝒥(k⃗)={J_σ: σ∈ D} as defined in Definition <ref>. Let ℋ⊆𝒥_k := {J_σ: σ∈ D_k} for some k ≥0, and consider ρ_8(ℋ). We now make the following observation: Suppose there exists a ball B that intersects at least 9 elements of ℋ.Then diam(B) is greater than or equal to the smallest edge length of the elements in ℋ.Indeed, by considering the projections to the three coordinate axes, one can see that at least one coordinate contains three non-identical projected images of these 9 elements.As a result the ball B intersected with these 9 elements will have a diameter at least the length of the smallest side of the three projected images.This proves our observation. Letm_ℓ = min{ k_ℓ^(1), k_ℓ^(2)-k_ℓ^(1),1-k_ℓ^(2), k_ℓ^(3), k_ℓ^(4)-k_ℓ^(3),1-k_ℓ^(4), k_ℓ^(5), k_ℓ^(6)-k_ℓ^(5),1-k_ℓ^(6)}and M_ℓ = max{ k_ℓ^(1), k_ℓ^(2)-k_ℓ^(1),1-k_ℓ^(2), k_ℓ^(3), k_ℓ^(4)-k_ℓ^(3),1-k_ℓ^(4), k_ℓ^(5), k_ℓ^(6)-k_ℓ^(5),1-k_ℓ^(6)}.For any σ∈ D, direct calculation shows thatm_ℓ(σ)≤diam(J_σ*i)diam(J_σ)≤ M_ℓ(σ)where ℓ(σ) is given in (<ref>). Thus, for any σ = (i_1, i_2, …, i_k) ∈ D_k, we havem_ℓ((i_1)) m_ℓ((i_1,i_2))⋯ m_ℓ((i_1,…,i_k))≤diam(J_σ)diam(J_∅)≤ M_ℓ((i_1)) M_ℓ((i_1,i_2))⋯ M_ℓ((i_1,…,i_k)). Let R_k⊆ D_k for some k.Suppose |R_k|≤8.Then for any σ∈ R_k, by the observationρ_8({J_σ*i: σ∈ R_k , i=1,2, … 20})∑_σ∈ R_k diam(J_σ)≥smallest diameter of J_σ*i8 ·max{ diam(J_σ): σ∈ R_k}≥ min_σ=(i_1, i_2, … , i_k) ∈ R_k, i_k+1=1, … , 20{m_ℓ((i_1)) m_ℓ((i_1,i_2))⋯ m_ℓ((i_1,…,i_k+1))diam(J_∅)M_ℓ((i_1)) M_ℓ((i_1,i_2))⋯ M_ℓ((i_1,…,i_k)) diam(J_∅)}≥ 18 (∏_i=1^∞ m_iM_i) lim inf_i →∞ m_i,where the last inequality follows from 0 ≤ m_i ≤ M_i for each i. Using this observation, we show that the -limit set in Example <ref> satisfies the uniform covering condition.In this example, m_ℓ = a_ℓ ℓ even b_ℓ ℓ odd and M_ℓ = b_ℓ ℓ even a_ℓ ℓ oddwhere a_ℓ = k_ℓ^(3)= 1/3 - (-1)^ℓ/6(ℓ +1)^2 andb_ℓ= 1-2 k_ℓ^(3)= 1/3 + (-1)^ℓ/3(ℓ+1)^2. One may show that the product ∏_i=1^∞ m_iM_i is convergent, whose numerical value is 0.369761… and lim inf_i →∞ m_i =1/3.Thus γ_8(𝒥) >0.Therefore, by Theorem <ref>, F satisfies the uniform covering condition. 7 barnsleyBarnsley, M.F., Fractals Everywhere, Academic Press Professional, Inc. San Diego, CA, USA (1988) vvar1 Barnsley, M.F., Hutchinson, J.E., Stenflo, O.,V-variable fractals: fractals with partial similarity. Adv. Math. 218, 2051–2088 (2008) vvar2 Barnsley, M.F., Hutchinson, J.E., Stenflo, O.,V-variable fractals: dimension results. Forum Math. 24, 445–470 (2012) BV Barnsley, M.F., Vince, A., Developments in fractal geometry. Bull. Math. Sci. 3, 299-348 (2013) Beardon Beardon, A. On the Hausdorff dimension of general Cantor sets. Mathematical Proceedings of the Cambridge Philosophical Society, 61(3), 679-694 (1965) Bedford Bedford, T. Applications of dynamical systems to fractal-A study of cookie-cutter Cantor sets. Fractal Geometry and Analysis, Canada: Kluwer, 1-44 (1991) hutch Hutchinson, J., Fractals and Self-Similarity Indiana University Math. J. 30 (5), 713-747 (1981) inhom Holland, M. and Zhang, Y., Dimension results for inhomogeneous Moran set constructions. Dyn. Syst., 28(2):222–250, (2013) Falconer1Falconer, K. Random fractals. Math. Proc. Cambridge Philos. Soc. 100 (3), 559-582, (1986) Falconer Falconer, K. Fractal Geometry: Mathematical Foundations and Applications, (2014) Li Li, W., Xiao, D., A note on generalized Moran set. Acta Math. Sci. (Suppl.) 18 88-93 (1998) Mandelbrot Mandelbrot, B. B. The fractal geometry of nature. San Francisco, California: W. H. Freeman and Co. (1982) mauldin Mauldin, R. D., Urbanski, M., Dimensions and Measures in Infinite Iterated Function Systems, Proc. London Math. Soc., 73(3):105 (1996) mcmullen McMullen, Curt. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96, 1–9. (1984) Moran Moran P.A.P. Additive functions of intervals and Hausdorff measure. Proc. Camb. Philos. Soc, 42, 15-23 (1946) peresPeres, Y., Solomyak B. Problems on self-similar sets and self-affine sets: An update, Fractal Geometry and Stochastics II, Progress in Probability, 46, 95-106, (2000) PWPesin, Y., Weiss, H., On the Dimension of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, and the Eckmann-Ruelle Conjecture. Comm. Math. Phys., 182: 105. (1996) Su Hua, S., Rao, H., Wen, Z. et al. On the structures and dimensions of Moran Sets, Sci. China Ser. A-Math. 43: 836 (2000) Wen Wen, Z. Moran Sets and Moran Classes, Chinese Science bulletin, 46 (22), 1849-1856 (2001) | http://arxiv.org/abs/1707.08634v3 | {
"authors": [
"Tynan Lazarus",
"Qinglan Xia"
],
"categories": [
"math.CA",
"math.DS",
"nlin.PS",
"28A80"
],
"primary_category": "math.CA",
"published": "20170726204023",
"title": "Generalized Moran sets Generated by Step-wise Adjustable Iterated Function Systems"
} |
[ [ December 30, 2023 =====================Humans use various social bonding methods known as social grooming, e.g. face to face communication, greetings, phone, and social networking sites (SNS). SNS have drastically decreased time and distance constraints of social grooming. In this paper, I show that two types of social grooming (elaborate social grooming and lightweight social grooming) were discovered in a model constructed by thirteen communication data-sets including face to face, SNS, and Chacma baboons. The separation of social grooming methods is caused by a difference in the trade-off between the number and strength of social relationships. The trade-off of elaborate social grooming is weaker than the trade-off of lightweight social grooming. On the other hand, the time and effort of elaborate methods are higher than lightweight methods. Additionally, my model connects social grooming behaviour and social relationship forms with these trade-offs. By analyzing the model, I show that individuals tend to use elaborate social grooming to reinforce a few close relationships (e.g. face to face and Chacma baboons). In contrast, people tend to use lightweight social grooming to maintain many weak relationships (e.g. SNS). Humans with lightweight methods who live in significantly complex societies use various social grooming to effectively construct social relationships.Keywords: Social Networking Site; Primitive Communications; Modern Communications; Social Grooming; Weak Ties; Social Relationship From§ INTRODUCTIONThe behaviour of constructing social relationships is called “social grooming,” which is not limited to humans but widely observed in primates <cit.>. Humans use different social grooming methods according to their strength of social relationships <cit.> (see also Fig. 1 on electronic supplementary materials (ESM)), e.g. primitive methods (face to face communications) and modern methods (E-mails and social networking sites (SNS)). Social grooming gives different impressions and has different effects on its recipients depending on the time and effort involved <cit.>. Face to face communication and on video calls get more satisfaction than communication in phone and text <cit.>. On Facebook, personal messages give more happiness than 1-click messages (like) and broadcast messages <cit.>. In other words, humans favor social grooming by elaborate methods (time-consuming and space constrained). Additionally, people in a close relationship tend to do these elaborate methods <cit.>. Furthermore, its positive effect in close relationships is larger than in weak social relationships <cit.>.Humans face cognitive constraints <cit.> (for example, memory and processing capacity) and time constraints (that is, time costs) in constructing and maintaining social relationships. These time costs are not negligible, as humans spend a fifth of their day in social grooming <cit.> and maintaining social relationships <cit.>. Therefore, the mean strength of existing social relationships has a negative correlation with the number of social relationships <cit.>. The trade-off between the number N and mean strength m of social relationships on online communications (SNS, mobile phones, and SMS) is described as C=Nm^a where a>1 <cit.>, i.e. total communication cost C obeys Nm^a. C which represents the amount of investment for social relationships differs depending on individuals. This suggests that social grooming behaviour depends on the strength of social relationships and the strength of this trade-off (a).Humans construct and maintain diverse social relationships within the constraints of this trade-off. These relationships provide various advantages to them in complex societies. Close social relationships lead to mutual cooperation <cit.>. On the other hand, having many weak social relationships, i.e. weak ties, help in obtaining information, which is advantageous because weak social relationships where people rarely share knowledge often provide novel information <cit.>. As a result, social relationship forms (distributions of social relationship strengths) often show a much skewed distribution <cit.> (distributions following a power law <cit.>). This skewed distribution has several hierarchies called circles. The sizes of these circles (the number of social relationships on the inside of each circle) are 5, 15, 50, 150, 500, and 1500, respectively <cit.>. That is, the ratios between neighboring circles of social relationships are roughly three irrespective of social grooming methods, e.g. face to face, phone, Facebook, and Twitter <cit.>.I aim to explore how and why humans use various social grooming methods and how those methods affect human behaviour and social relationship forms. For this purpose, I analyze the strength of the trade-off between the number and mean strength of social relationships as a key feature of social grooming methods. For this analysis, I extend the model <cit.> for C=Nm^a from a>1 to a>0 by introducing individuals' strategies about the amount of social grooming behaviour. This model explains not only online communication but also offline communication. The key feature restricts social relationship forms (size and distributions of strengths). Therefore, humans should change social grooming strategies depending on the trade-off, i.e. they tend to use several social grooming methods for constructing various strengths of social relationships. My model is supported by the common features of thirteen diverse communication data-sets including primitive human communication (face to face and communication in a small community constructed by kin and friends), modern communication (phone calls, E-mail, SNS, and communication between unrelated people), and non-human primate communication (Chacma baboons). The model connects social behaviour and social relationship forms with a trade-off between the number and mean strength of social relationships.§ DATA ANALYSIS I found two types of social grooming methods based on the trade-off between the number and strength of social relationships (Fig. 1). One was “elaborate social grooming,” which was face to face and by phone (Face to face (Pachur) <cit.>[<https://doi.org/10.5061/dryad.pc54g>] and Phone (Pachur) <cit.>[<https://doi.org/10.5061/dryad.pc54g>]), in kin and friends (Mobile phone (friends and family) <cit.>[<http://realitycommons.media.mit.edu/realitymining4.html>] and Short Message Service (SMS (friends and family)) <cit.>[<http://realitycommons.media.mit.edu/realitymining4.html>]), and Chacma baboon social grooming (Baboon group A and B) <cit.>[<https://doi.org/10.5061/dryad.n4k6p/1>]. This should be nearer to primitive human communications than the others. That is, these communications tend to bind individuals due to time and distance constraints or in primitive groups constructed by kin and friends. Another one was “lightweight social grooming” which was by SNS and E-mail (Twitter <cit.>[<https://archive.org/details/twitter_cikm_2010>], 755 group chat <cit.>[<https://doi.org/10.6084/m9.figshare.3395956.v3>], 755 wall communication <cit.>[<https://doi.org/10.6084/m9.figshare.3395956.v3>], Ameba Pigg <cit.>[<https://doi.org/10.6084/m9.figshare.3395956.v3>], and E-mail/Letter (Pachur) <cit.>[<https://doi.org/10.5061/dryad.pc54g>]), and in relationships between unrelated people (Mobile phone (dormitory) <cit.>[<http://realitycommons.media.mit.edu/socialevolution.html>] and SMS (dormitory) <cit.>[<http://realitycommons.media.mit.edu/socialevolution.html>]), which has appeared in the modern age. These communications tend to unbind humans from time and distance constraints. This tended to be used with unrelated people. Details of data-sets noted in brackets are shown in the Data-Sets section on ESM.Both were divided by parameter a on C_i=N_i m_i^a model (Fig. 2), where a showed strengths of the trade-off between N_i and m_i, and individual i's total social grooming cost was C_i, N_i was i's number of social relationships, m_i was i's mean strength of their social relationships (m_i = ∑_j=1^N_i d_ij / N_i), and d_ij was the total number of the days on which i did social grooming to individual j (the strength of social relationships between i and j). C_i represents i's available social capital on each social grooming method which varied widely among individuals (see Fig. 3). I estimated statistically parameter a of the data-sets by using a regression model log N ∼ Normal(-a log m + b log u, σ), where u was the number of days of participation for each person and σ was a standard deviation, that is, this model assumed that a user's total social grooming costs were equal to the b-th power of the number of days for which they had participated in the activity (C = u^b) <cit.>. u^b was entered as a covariate to control the usage frequency of social grooming methods. Table 1 shows the details of this regression result. A few p-values of a were larger than 0.05, but social grooming methods overall seem to be divided by parameter a.a ≠ 1 suggests that social grooming behaviour depends on the strength of social relationships because a will be 1 when social grooming behaviour does not relate to the strength of social relationships (C_i=N_i m_i =∑_j=1^N_i d_ij). a<1 shows that people have stronger social relationships than when a=1 because the effect of strong relationships (large m) to cost C is smaller than when a=1 (C_i=N_i m_i^a < ∑_j=1^N_i d_ij when m>1). In contrast, a>1 shows that people have weaker social relationships than when a=1 because of the effect of strong relationships (large m) to cost C is larger than when a=1 (C_i=N_i m_i^a > ∑_j=1^N_i d_ij when m>1). I call the social grooming methods of a>1 (i.e. modern methods) lightweight social grooming (Fig. 2a-g) and the methods of a<1 (i.e. primitive methods) elaborate social grooming (Fig. 2h-m). That is, the trade-off of elaborate social grooming between the number and strength of social relationships is smaller than that of lightweight social grooming. On the other hand, the time and effort of lightweight methods are less than elaborate methods <cit.>.This trade-off parameter a affected human social grooming behaviour. People invested more time in their closer social relationships, i.e. the amount of social grooming between individuals increased with their strength of social relationships. Additionally, a changed people's trends of social relationship constructions. People having limited and deep social relationships tended to do frequent social grooming (amount of social grooming was large) when a<1. On the other hand, people having expanded and shallow social relationships tended to do frequent social grooming when a>1. I show these in the following.Fig. 4, 5, and the previous work <cit.> show that the amount of social grooming from individual i to individual j tended to increase with the density of social grooming w_ij, where w_ij=d_ij/t and t was the number of elapsed days from the start of observation, i.e. the amount of social grooming did not depend on t. I modeled this phenomenon as linear increase which was the simplest assumption, that is, v(w_ij) = α w_ij + 1, where v(w_ij) was the amount of social grooming from individual i to individual j and α was a parameter.This assumption and the definition of C_i=N_i m_i^a suggest a relationship between individual social relationship trends and their total amount of social grooming. m_i shows individual i's sociality trend (mean limitation and depth of its social relationships) by the trade-off between N and m. The total amount of social grooming per day to reinforce social relationships isG(a, α; C, m) = α C (m^1-a- m^-a)/T,where T is the number of days of the data periods. m_i=N_i^-1∑_j=1^N_i d_ij(T) = T N_i^-1∑_j=1^N_i w_ij based on the definition of m and w. Therefore, the total amount of social grooming V_i is ∑_j^N_iv(w_ij) = α m_i N_i T^-1 + N_i. G(a, α; C, m) is acquired by subtracting the total cost for creating social relationships (V_0 = α + T) from V_i (see the Development of Eq. 1 section on ESM). α was decided by the following simulation where an individual-based model was fitted to the data-sets. This equation shows that people who have large m (i.e. limited and deep social relationships) have a large amount of social grooming when doing elaborate methods (a<1; the orange line in Fig. 13d). On the other hand, people who have small m (i.e. expanded and shallow social relationships) have a large amount of social grooming when doing lightweight methods (a>1; the green line in Fig. 13d). That is, G(a, α; C, m) shows that social grooming methods were used depending on the strength of social relationships (elaborate social grooming used for strong social relationships and lightweight social grooming used for weak social relationships). The threshold of both social grooming methods was a=1. The amount of social grooming for construction of new social relationships G_0 does not depend on a (G_0 ≃ N; see Development of Eq. 1 section on ESM for details).a changed the peak of a total amount of social grooming G depending on sociality trend m, nevertheless, the number of social relationships for each social grooming method did not show clear relationships to a (Fig. 6). Additionally, their number of social relationships which were smaller than the general number (about 150 <cit.>).There may be the problem for evaluating the number of social relationships due to the differences in data gathering on the data-sets. There may be not a difference of the number of social relationships among the data-sets because the previous studies <cit.> showed that the number of social relationships did not particularly depend on social grooming methods.a affected the ratio of very weak social relationships. Fig. 7 shows the size ratio between neighboring hierarchies of social relationships, i.e. H_k/H_k+1, where H_k is the number of social relationships when d ≥ k. That is, H_1 is the number of all social relationships, H_2 is the number of social relationships excluding relationship d=1, and H_k with large k is the number of close relationships. The trade-off parameter a only affected H_k/H_k+1 when k was very small. Thus, the strength of trade-off a related only to very weak social relationships which seemed to be the circle of acquaintances <cit.>.§ INDIVIDUAL-BASED SIMULATIONS §.§ Model In the previous section, I found the threshold a=1 on social grooming behaviour (Eq. 1). In contrast, a threshold was not observed on social relationship forms due to the differences of data gathering and sampling on the data-sets. In this section, I conducted individual-based simulations to analyze changes of social relationship forms depending on a under the same conditions.I constructed an individual-based model to explore the effects of the trade-off parameter a to social relationship forms based on the monotonic increasing of v(w_ij) and the difference of the peak of G(a, α; C_i, m_i) depending on a (Eq. <ref>). That is, individual i does social grooming to individual j, the amount is proportional to w_ij. i's total amount of social grooming for reinforcing all social relationships is G(a, α; C_i, m_i) Additionally, I assumed the Yule–Simon process on social grooming partner selection, because people basically do act this way <cit.> (see also ESM Fig. 2). In the Yule–Simon process, which is one of the generating processes of power law distributions <cit.>, individuals select social grooming partners in proportion to the strength of their social relationships, that is, the individuals reinforce their strong social relationships. In the model, individuals construct new social relationships and reinforce existing social relationships, where they pay their limited resources R=G(a, α; C_i, m_i) for the reinforcement. This model is an extension of the individual-based model of the previous study <cit.> in which G is introduced. The Source Code of the Individual-based Simulations section on ESM shows a source code of this model.I consider two types of individuals, groomers and groomees. Groomers construct and reinforce social relationships using their limited resources R (that is, time), based on these assumptions and the Yule–Simon process. I use the linear function v(w_ij) =α w_ij + 1 as the amount of social grooming from groomer i to groomee j as with the above section.I conducted the following simulation for T days to construct social relationships d_ij in simulation experiments 1 and 2. Individuals have a social relationship where strength is 1 as the initial state. On each day t ∈ [1, T], groomer i repeats the following two-processes for its resource R_i > 0. R_i is reset to an initial value G(a, α; C_i, m_i) before each day t. Each i spends R_i reinforcing its social relationships. Creating new social relationships Each i creates social relationships with strangers (groomees). The strength of a new social relationship with j (d_ij) is 1. The number of new relationships obeys a probability distribution Poisson(p_i), where p_i is (N_i-1)/T. Therefore, i is expected that it has N_i social relationships until day T, because the relationship between N and a was unclear in the previous section, the expected value of N does not depend on a and is constant. This setting should be natural because the previous studies <cit.> showed independence of N from social grooming methods. Creating new relationships does not spend R. Reinforcing existing social relationships i also reinforces its social relationships. Each i selects a social grooming partner j depending on a probability proportional to the strength of the social relationships between i and j, then i adds 1 to d_ij (that is, the Yule–Simon process) and spends the amount of social grooming v(w_ij) from R_i (if R_i < v(w_ij), then i adds R_i/v(w_ij) to d_ij and R_i becomes 0). Each i does not perform the act of social grooming more than once with the same groomees in each day t. Therefore, selected groomees are excluded from the selection process of a social grooming partner j on each day t. §.§ Simulation Experiment 1: Checking the Model Consistency In this experiment, I confirmed a consistency between C=Nm^a and two assumptions of social grooming behaviour (G(a, α; C_i, m_i) and v(w_ij) =α w_ij + 1). Therefore, I fitted my individual-based model to the data-sets optimized by unknown parameter α. I used actual values of the data-sets as a, T, and C_i in each simulation, where a was the values in Fig. 2 and Table 1, T was the period for each data-set, and C was the 75th percentile of u^b. N_i was equally divided in a logarithmic scale (N ∈ [1, T]). Unknown parameter α was calculated by the optimization which decreased error values of simulations (e=∑_i=1^M {(log N_i - log N'_i)^2 + (log m_i - log m'_i)^2 }/ M), where m_i = (C/N_i)^(1/a), M was the number of individuals (M=30), and N'_i and m'_i were calculated by simulation results (social relationship strengths d of each individual).Next, I calculated social relationship forms (d_ij) in each a by using actual settings (T, C_i, N_i) and the optimized α, where T was the period for each data-set, C_i was individual i's u_i^b, and N_i was i's number of social relationships in each data-set.This model fit all data-sets (Fig. 8 and Table 2). Their distributions of social relationships d_ij were roughly similar to actual distributions excluding Face to face (Pachur) (Fig. 9). Additionally, the amount of social grooming predicted by Eq.1 with the optimized α showed a high correlation with the actual amount of social grooming in each data-set (Table 3). That is, this model roughly has an explanation capacity for generating the process of social relationships, depending on human social grooming behaviour, regarding the trade-off constraint. The difference between the simulation result and Face to face (Pachur) data-set may have been because the approximations of this model did not work with small a.§.§ Simulation Experiment 2: Effects of Social Grooming Methods In this experiment, I analyzed the effect of parameter a on the structure of social relationships by using the model. First, I calculated error value e_aα in each a and α, where a was {0.50, 0.55, …, 2.00} and α was {1.00, 1.02, …, 3.00}. That is, the number of combinations is 31 × 101 = 3,131. In each simulation, T was the period for the Twitter data-set, C_i was the 75th percentile of u^b in the Twitter data-set, N_i was equally divided in a logarithmic scale, and M=30. Each e_aα was calculated fifty times, i.e. there are 5,050 results on each a. I used α which was ranked in the lowest twenty of e_aα of each a.Next, I calculated social relationship forms (the distributions of social relationship strengths d_ij) in each a by using actual settings (T, C_i, N_i) and α, where T was the period for the Twitter data-set, C_i was individual i's u_i^b, N_i was i's number of social relationships in the Twitter data-set, and M was the number of people in the Twitter data-set.Firstly, I evaluated a power law coefficient ϕ as overall effects of a on social relationship forms because social relationship forms (distributions of social relationship strengths d_ij) follow power-law distributions <cit.>. Secondly, I analyzed the ratio between neighboring circles of weak social relationships which depended on a in the previous section (Fig. 7).I found that a changed the social relationship forms ϕ and social behaviour parameter α where the threshold was around a=0.8 (Fig. 10) because a linear regression model with the threshold at a=0.8 was more accurate than a model without the threshold, i.e. the former had smaller AIC than the latter. The former is ϕ∼ Normal(β_1 a f + β_2 a (1 - f) + β_3 f + β_0, σ), where f=1 when a ≥ 0.8 otherwise f=0 and σ is standard deviations (AIC: -1452.6; see Table 4). The latter is ϕ∼ Normal(β_1 a + β_0, σ) (AIC: -1400.9). That is, the changes of a in a<0.8 (β_2) had a smaller effect on powerlaw coefficients ϕ of social relationship forms than in a≥0.8 (β_1). This shows that strong social relationships decreased in a≥0.8 because individuals having expanded and shallow social relationships have more of the amount of social grooming than individuals having limited and deep social relationships (G(a, α; C, m)). This threshold seems to be due to the threshold of G(a, α; C, m). The difference between this threshold (a=0.8) and the threshold of G(a, α; C_i, m_i) (a=1) may have been because of the approximations of this model.Additionally, α in a<0.8 was larger than α in a≥0.8 excluding a=0.8.Interestingly, α also drastically changed in the range of 0.8≤ a≤ 1.3. That is, individuals in a≥0.8 decreased the amount of social grooming v(w_ij) with close social relationships as compared to a<0.8.a affected the ratio of very weak social relationships. Fig. 11 shows the size ratio between neighboring hierarchies of social relationships, i.e. H_k/H_k+1, where H_k is the number of social relationships when d > k. That is, H_1 includes all social relationships excluding relationships of d ≤ 1 (one time interactions), H_2 is the number of social relationships excluding relationship d ≤ 2, and H_k with large k is the number of close relationships. There is the difference of the definition between this H and H in the Data Analysis section due to the difference of the definition between d in this section (positive real number) and d in the Data Analysis section (natural number). The ratio between neighboring hierarchies of very weak social relationships H_1/H_2 significantly changed around a=0.8. On the other hand, H_k/H_k+1 on k ≥ 2 gradually increased with a. This was due to the fact that H_1 increased with the increase a and H_k when k ≥ 2 decreased with the increase a (Fig. 12). Thus, very weak social relationship forms were especially affected by a compared with strong social relationships.As a result, the social relationship forms were expanded and shallow in a≥0.8. This suggests that societies with lightweight social grooming had different properties when compared to societies with elaborate social grooming.§ DISCUSSION I constructed a model of social relationship forms depending on human behaviour restricted by a trade-off between the number and strength of social relationships depending on social grooming methods. This model was supported by common features of thirteen diverse communication data-sets including primitive human communication, modern communication tools, and non-human primates. By analyzing the model, I found two types of social grooming (elaborate social grooming and lightweight social grooming). They made different social relationship forms. This was caused by people's social grooming behaviour depending on the different trade-off between the number and strength of social relationships. Both methods were separated by trade-off parameter a on C=Nm^a model (a<1: elaborate social grooming, a>1: lightweight social grooming). This separation was due to the total amount of social grooming G having the threshold a=1. The model from the previous study <cit.> was expanded by adding G.People tended to use elaborate social grooming in face to face communication and communication in small communities made up of kin and friends, i.e. the communities should be near primitive groups. Additionally, Chacma baboons also showed a similar trend. They tended to use this social grooming to reinforce close social relationships. That is, elaborate social grooming is a primitive method (i.e. a priori). This may be used in non-human primates, primitive human societies, and close relationships of modern humans, i.e. these may not have a qualitative difference.On the other hand, people tended to use lightweight social grooming in SNS, E-mail, and communication in communities made up of unrelated people, i.e. the communities should be non-primitive groups. That is, this social grooming is posterior. People tended to use these methods to construct many weak social relationships. As a result, social relationship forms may have changed significantly when people have used lightweight social grooming. Therefore, human societies have become expanded and shallow. Fig. 13 shows the relationship between the trade-off parameter of social grooming methods, human social behaviour, and social relationship forms.Due to these differences, people use both social grooming methods depending on the strengths of social relationships <cit.> (Fig. 1 on ESM). A typical person who has various strengths of social relationships <cit.> uses elaborate social grooming (a<1) for constructing a few close relationships and lightweight social grooming for having many weak social relationships. This is caused by the change of the peak of the function G around threshold a=1.The function G represents the total amount of social grooming to all relationships depending on sociality trend m and trade-off parameter a of social grooming. That is, individuals' total amount of social grooming are limited, and these limits (G(a, α; C, m)) depend on m and a. Thus, some individuals opt for lightweight social grooming (a>1) a consequence of the fact that they want to have a larger network (small m and large N). In other words, two types of social grooming have emerged in consequence of social grooming strategies, which are how individuals distribute their limited resources by using several social grooming methods depending on trade-off parameter a of the social grooming.This qualitative difference between the two types of social grooming may be caused by the number of very weak social relationships an individual wants to create. The most affected relationships according to the difference of social grooming methods were the ratio between a hierarchy of social relationships including very weak social relationships and its neighboring hierarchy. On elaborate social grooming, the trade-off parameter a has an insignificant effect on this ratio. In contrast, on lightweight social grooming, an increase of the trade-off parameter a increases the ratio. Humans may have acquired lightweight social grooming by the necessity to create many very weak social relationships (acquaintances). It seems to have been due to the increase in the number of accessible others or community sizes.Some online social grooming methods (Mobile phone (friends & family), SMS (friends & family), and Phone (Pachur)) showed a<1. These might have been caused by using for constructing non-weak social relationships. That is, the subjects of Mobile phone (friends & family) and SMS (friends & family) were members of a young family living in a residential community which was constructed by kin and neighbors. Phone (Pachur) was used for communication closer relationships than E-mails when people use phone and E-mails (ESM Fig. 1). This could be achieved by comparing diverse communication data-sets gathered by similar conditions.Both social grooming methods also differ from a cost and effect perspective. The trade-off of interacting in close social relationships using elaborate methods is weaker than that of lightweight methods. On the other hand, the time and effort of lightweight methods are less than elaborate methods <cit.>. Social grooming with time and effort (elaborate social grooming) is effective to construct close social relationships <cit.>. Therefore, elaborate methods are suited to maintain a few close relationships. In addition, lightweight methods make it easier for people to have many weak social relationships <cit.>.The two types of social grooming methods have different roles. The role of elaborate methods should be to get cooperation from others. Humans tend to cooperate in close friends <cit.> because cooperators cannot cooperate with everyone <cit.>. The role of lightweight social grooming should be to get information from others. Weak social relationships tend to provide novel information <cit.>.Thus, it should be effective for people to use elaborate social grooming to close relationships while expecting cooperation from these relationships. They use widely lightweight social grooming in weak relationships while expecting novel information. As a result, the number of close relationships before and after SNS has not changed much <cit.>. Weak relationships after the appearance of SNS have been maintained effectively <cit.>.An advantage of having information would have increased with the changes of societies. As a result, lightweight social grooming has been necessary, and humans have had expanded and shallow social relationship forms. Humans probably have acquired this social grooming in the immediate past. This consideration will become clearer by analyzing various data-sets, e.g. other non-human primates, social relationship forms in various times and cultures, and other communication systems.The way of using both methods may also depend on people's extroversion/introversion. In general, introverts have limited deep social relationships and extroverts have expanded shallow social relationships <cit.>. The diversity of the amount of social grooming C on each social grooming method suggests that usage strategies of the two types of social grooming methods differ for each person, e.g. introverts tend to do elaborate social grooming, in contrast, extroverts tend to do lightweight social grooming.Primitive humans also used lower-cost social grooming methods (e.g. gaze grooming <cit.> and gossip <cit.>) than fur cleaning in non-human primates. These methods have evolved more for larger groups than that of non-human primates because these grooming methods enable humans to have several social relationships and require less time and effort <cit.>. However, my model does not distinguish these social grooming methods from social grooming in non-human primates; nevertheless, the model separates modern social grooming from primitive human social grooming. This may suggest that an appearance of lightweight social grooming significantly affects human societies nearly as much as the changes between non-human primates and primitive humans. § DATA ACCESSIBILITYAll data needed to evaluate the conclusions in the paper are present in the paper and the electronic supplementary materials. § COMPETING INTERESTSMasanori Takano is an employee of CyberAgent, Inc. There are no patents, products in development or marketed products to declare. This does not alter the authors' adherence to all Royal Society Open Science policies on sharing data and materials, as detailed online in the guide for authors.§ FUNDINGI received no funding for this study.§ RESEARCH ETHICS This study did not conduct human experiments (all data-sets were published by previous studies).§ ANIMAL ETHICS This study did not conduct animal experiments (all data-sets were published by previous studies).§ PERMISSION TO CARRY OUT FIELDWORK This study did not conduct fieldwork.§ ACKNOWLEDGEMENTSI are grateful to associate professor Genki Ichinose at Shizuoka University, Dr. Vipavee Trivittayasil at CyberAgent, Inc., Dr. Takuro Kazumi at CyberAgent, Inc., Mr. Hitoshi Tsuda at CyberAgent, Inc. and Dr. Soichiro Morishita at CyberAgent, Inc. whose comments and suggestions were very valuable throughout this study.10 urlstyle Kobayashi1997 Kobayashi H, Kohshima S, 1997 Unique morphology of the human eye. 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Evolutionary Anthropology 6, 178–190 | http://arxiv.org/abs/1707.08517v2 | {
"authors": [
"Masanori Takano"
],
"categories": [
"cs.SI"
],
"primary_category": "cs.SI",
"published": "20170726160620",
"title": "Two Types of Social Grooming Methods depending on the Trade-off between the Number and Strength of Social Relationships"
} |
ChulaAddress]Vitara Pungpapong mycorrespondingauthor [email protected] [mycorrespondingauthor]Corresponding author. PurdueAddress]Min Zhang PurdueAddress]Dabao Zhang[ChulaAddress]Department of Statistics, Faculty of Commerce and Accountancy, Chulalongkorn University, Bangkok, Thailand [PurdueAddress]Department of Statistics, Purdue University, West Lafayette, IN, USA High-dimensional linear and nonlinear models have been extensively used to identify associations between response and explanatory variables. The variable selection problem is commonly of interest in the presence of massive and complex data. An empirical Bayes model for high-dimensional generalized linear models (GLMs) is considered in this paper. The extension of the Iterated Conditional Modes/Medians (ICM/M) algorithm is proposed to build up a GLM. With the construction of pseudodata and pseudovariances based on iteratively reweighted least squares (IRLS), conditional modes are employed to obtain data-drive optimal values for hyperparameters and conditional medians are used to estimate regression coefficients. With a spike-and-slab prior for each coefficient, a conditional median can enforce variable estimation and selection at the same time. The ICM/M algorithm can also incorporate more complicated prior by taking the network structural information into account through the Ising model prior. Here we focus on two extensively used models for genomic data: binary logistic and Cox's proportional hazards models. The performance of the proposed method is demonstrated through both simulation studies and real data examples. The implementation of the ICM/M algorithm for both linear and nonlinear models can be found in theR package which is freely available on CRAN.Empirical Bayes variable selection; High-dimensional data; Generalized linear models; Binary logistic model; Cox's model. § INTRODUCTIONDue to the advanced development of computing technology, high-dimensional data are widely found in many fields ranging from genomics to marketing and finance. Such data are known as large p small n data where the number of variables p is relatively larger than the sample size n. Generalized linear models (GLMs) have been extensively used to identify associations between response and explanatory variables. Similar to a normal linear regression model, the classical GLMs approach also suffers in large p small n scenario due to collinearity among covariates. Furthermore, regression coefficients are usually assumed to be sparse. In biomedical science, variable selection plays an important role in analysis of high-throughput genotype. A typical microarray data have several thousands predictors. For genome-wide association studies (GWAS), the data may contain thousand to million Single nucleotide polymorphisms (SNPs). However, the sample size is usually only in hundreds or thousands. Collinearity problem occurs especially for those genes or SNPs resided in the same biological pathway. It is also believed that only few genes or SNPs are associated with the phenotype of interest. In addition to continuouse phenotype, discrete or censored outcomes frequently appear in the field of biomedical science. Binary logistic and Cox's proportional hazards models with high-dimensional covariates have received considerable attention over the past decade.One popular approach in fitting high-dimensional GLMs is through a penalized likelihood method. The most basic and popular penalized likelihood estimator is lasso (<cit.>) where ℓ_1-penalty is employed to produce sparse coefficients. Many works has been proposed to efficiently compute the regularization paths for GLMs with lasso penalty. <cit.> introduced the predictor-corrector method to determine the entire path of lasso for GLMs. <cit.> proposed a fast algorithm exploiting cyclical coordinate descent to compute a regularization path for GLMs which can shorten computing time considerably. An important extension of the lasso method is adaptive lasso (<cit.>) which allows weights of predictors in the penalty term to be varied. <cit.> presented that the adaptive lasso enjoys the oracle properties in linear model and GLMs under mild condition. This means that it has variable selection consistency and the ability to correctly select the nonzero coefficients with probability converging to one. Much efforts have been devoted in investigating theoretical properties of adaptive lasso. <cit.> presented that the adaptive lasso has oracle property even when the number of covariates is much larger than the sample size in linear model. Other recent work including <cit.> and <cit.> also demonstrated the oracle property of the adaptive lasso in high-dimensional setting for GLMs. Another advantage of the adaptive lasso method is that it can be solved by the same efficient algorithm for solving lasso.Alternatively, Spike-and-slab approaches to Bayesian variable selection has gained increasing attention recently. The spike part concentrates its mass at the values near zero and the slab component has a density spread over parameter spaces. There has been much work on hierarchical linear models with absolutely continuous spikes (i.e., mixture normal priors). Examples include <cit.>, <cit.>, <cit.>, <cit.>. MCMC stochastic search has been extensively used to fit the model with such Bayesian formulation in large-scale data analysis. However, it is computational expensive when the number of predictors p is large. Instead of using MCMC, <cit.> proposed an EMVS algorithm to fit a linear model with mixture normal prior. With conjugate spike-and-slab prior, the EMVS has closed form solutions for both E- and M- steps which can save computational time over MCMC. Although an absolutely continuous spike component has ability to determine the amount of shrinkage adaptively, regression coefficients are not sparse in the exact sense. To enforce some regression coefficients to be exactly zero,a dirac spike using a point mass at zero should be considered instead. Nonetheless, EMVS algorithm is not in the closed form anymore. Moreover, extensions of the EMVS algorithm to GLMs are not straightforward. <cit.> presented a contribution of empirical Bayes thresholding (<cit.>) to select variables in a linear regression framework which can identify and estimate sparse predictors efficiently. A dirac spike-and-slab prior, a mixture prior of an atom of probability at zero and a heavy-tailed density, is put on each regression coefficient. An iterated conditional modes/medians (ICM/M) was also proposed for fast and easy-to-implement algorithm for empirical Bayes variable selection. Similar to an iterated conditional modes algorithm proposed by <cit.>, conditional modes are used to obtain hyperparameters and parameters other than regression coefficients. With a dirac spike-and-slab posterior for each regression coefficient, a conditional median is employed to enforce the variable selection. As demonstrated in <cit.>, empirical Bayes variable selection can also handle the case when the information about structural relationship among predictors is available through the Ising prior. The ICM/M algorithm can be easily implemented for such complicated prior.The aim of this paper is to generalize empirical Bayes variable selectionto more class beyond linear model. The general framework for extension of empirical Bayes variable selection to GLMs will be described. The details on the implementation of ICM/M algorithm for binary logistic model and Cox's proportional hazards model will also be discussed here. The rest of of this paper is organized as follows. The next section presents a Bayesian model formulation for high-dimensional GLMs. Section 3 introduces the ICM/M algorithm for GLMs. In Section 4, we introduce how to quantify variable importance and select variables based on false discovery rate. Section 5 and Section 6 focus on the implementation of ICM/M algorithm for binary logistic model and Cox's proportional hazards model respectively. Numerical results based on simulation studies are also be shown in Section 5 and Section 6. Application to real data sets are presented in Section 7 and 8. We then conclude this paper with a discussion in Section 9.§ A BAYESIAN MODEL FORMULATION FOR HIGH-DIMENSIONAL GLMSConsider a generalized linear model (GLM) where each response variable Y_i is assummed to be independent and has distribution belonging to exponential family taken the formf_Y_i(y_i|θ_i, ϕ) = exp{y_iθ_i-b(θ_i)/a(ϕ)+c(y_i, ϕ)}where a(.),b(.), and c(.) are functions which vary according to distributions. θ=(θ_i, ..., θ_n) is known as the canonical parameter and ϕ is a dispersion parameter assummed to be known here. A link function g(.) is used to connect the linear predictors η_i = β_0+𝐗_iβ to the mean μ_i=E[Y_i]. That is, g(μ_i)=η_i=β_0+𝐗_iβ.With an n × 1 matrix of responses 𝐘 and n × p matrix containing values of p predictors, the maximum likelihood estimators (MLEs) are obtained to estimate β in low-dimensional setting (n > p). An iteratively reweighted least squares (IRLS) algorithm is typically used to estimate such coefficients. Based on current parameter estimates (β̂_0, β̂), a quadratic approximation to the likelihood is employed to construct pseudodata 𝐙 = (Z_1,..,Z_n)^t and pseudovariances Σ=diag{σ_i^2} as follows:𝐙 = η̂+ (𝐘-μ̂)(dη/dμ|η=η̂) Σ = diag{σ_i^2} = diag{ Var[Y_i] (dη/dμ|η=η̂)^2}where η̂ = β̂_0+𝐗β̂. <cit.> showed that the update of β in IRLS is the result of a weighted least squares regression of 𝐙 on 𝐗 where the weighted matrix is Σ^-1. Indeed, the underlying distribution of pseudodata 𝐙 is approximated by N(β̂_0+𝐗β̂, Σ).When n≪ p, a unique solution for the classical MLEs does not exist due to the fact that the design matrix 𝐗 is not full rank. Thus, the regression coefficients β cannot be updated using a weighted least squares in IRLS procedure. Furthermore, typical high-dimensional data analysis usually requires that only a subset of important variables are used in the modeling process. Variable selection to get sparse regression coefficients is a crucial task when analyze such massive data.To introduce sparsity in the modeling process, an independent mixture of an atom of probability at zero and a distribution describing non-zero effect can be put on each of the regression coefficient β_j as prior in the Bayesian framework. With the approximated distribution of pseudodata 𝐙 and assuming that all other parameters except β_j are known, β_j has a sufficient statistic (X_j^tΣ^-1X_j)^-1(X_j^tΣ^-1Z̃_j) where Z̃_j = 𝐙 - η̂ + X_jβ_j and the approximated distribution of the sufficient statistics is N(β_j, (X_j^tΣ^-1X_j)^-1). Following <cit.>, the prior of β_j is in the formβ_j | ω∼ (1-ω)δ_0(β_j) + ωγ(β_j).Under the prior distribution in (<ref>), each β_j is zero with probability (1-ω) and β_j is drawn from the nonzero part of prior γ(β_j) with probability ω. Laplace distribution is employed for γ(β_j), that is,γ(β_j) = α(X_j^tΣ^-1X_j)^-1/2/2exp{ -α(X_j^tΣ^-1X_j)^1/2 |β_j|},where α > 0 is a scale parameter. When the information of structural relationships among predictors is available, an indicator variable τ = (τ_1,...,τ_p)^t where τ_j = I_{β_j0} is introduced and the underlying relationships represented by an undirected graph is put on τ. Specifically, given τ_j, the prior of β_j isβ_j | τ_j ∼ (1-τ_j) δ_0(β_j)+τ_j γ(β_j)Again, the Laplace distribution is employed for γ(β_j). To model the relationship among τ under an undirected graph G=(V,E) comprising a set V of vertices and a set E of edges, the following Ising model is considered:P(τ) = 1/Z(a,b)exp{a∑_j τ_j + b ∑_<j,l>∈ Eτ_j τ_l}where a and b are parameters and Z(a,b) is a normalizing constant, that is, Z(a,b) = ∑_τ∈{0,1}^pexp{a∑_j τ_j + b ∑_<j,l>∈ Eτ_j τ_l}When b>0, the interaction between nearest neighboring nodes is called ferromagnetic, i.e., neighboring τ_j and τ_l tend to have the same value. When b<0, the interaction is called anitiferromagnetic, i.e., neighboring τ_j and τ_l tend to have different values. When b=0, the prior gets back to independent and identical Bernoulli distribution. The value of a+b indicates the preferred value of each τ_j. In fact, τ_j tends to be one when a+b>0 and τ_j tends to be zero when a+b<0.§ THE ITERATED CONDITIONAL MODES/MEDIANS ALGORITHM FOR GLMS<cit.> presented an iteratated conditional modes/medians (ICM/M) algorithm for fast computation of empirical Bayes variable selection in a linear model. Data-driven optimal values for hyperparameters and auxiliary parameters are obtained as the modes of their full conditional distribution functions. Each regression coefficient is obtained as the median of its full conditional distribution function. With a spike and slab prior on β as in (<ref>) and(<ref>), the regression coefficients obtained from conditional medians are sparse which therefore enforce variable selection and estimation simultaneously. The iterative procedure for updating regression coefficients and other parameters is carried out until convergence.Adopting the idea of IRLS, we can extend the ICM/M algorithm to GLMs. Based on the approximated distribution of the current pseudo data, the ICM/M algorithm is applied to update all parameters. The procedure consists of the outer and inner loop. The outer loop is taken place to update pseudodata and pseudovariances based on current parameter estimates. The inner loop is where the ICM/M is employed to cycle through all parameters update. Proposition 1With current values of β_0 and β, pseudodata and pseudovariances {𝐙, Σ} can be calculated and a sufficient statistic for β_j is (X_j^t Σ^-1 X_j)^-1(X_j^t Σ^-1Z̃_̃j̃) w.r.t. the approximated distribution of 𝐙 which is N(β_0+𝐗β, Σ). Then, the iterative conditional median of β_j in the ICM/M algorithm can be constructed as the posterior median of β_j in the following Bayesian analysis, {[ (X_j^t Σ^-1 X_j)^-1/2(X_j^t Σ^-1Z̃_j) | β_j ∼ N( (X_j^t Σ^-1 X_j)^1/2, 1); β_j | ω∼ (1-ω)δ_0(β_j) +ωα (X_j^t Σ^-1 X_j)^1/2/2exp{-α (X_j^t Σ^-1 X_j)^1/2 |β_j| }. ].With the independent prior as in (<ref>), the details of the ICM/M algorithm is demonstrated in Algorithm 1. Algorithm 1* Initialize (β_0^(0), β^(0)) and set k=0.* Set k = k+1.* Update pseudodata and pseudovariances (𝐙^(k), Σ^(k)).* Update weight ω as the mode of its full conditional distribution function.ω^(k) = mode P(ω | 𝐘, 𝐗, 𝐙^(k), Σ^(k), β_0^(k), β^(k)) * For j=1,...,p, update β_j as the posterior median. β_j^(k) = median P(β_j | 𝐘, 𝐗, 𝐙^(k), Σ^(k), ω^(k), β_0^(k-1), β_1:(j-1)^(k), β_(j+1):p^(k-1)) * Update β_0 as follows:β_0^(k) = Z̅_w - X̅_w β^(k)where Z̅_w and X̅_w are the weighted means of 𝐙^(k) and 𝐗 respectively with weight for each observation being σ_i^-2.* Iterate between 2 - 6 until convergence. To incorporate the information of structural relationships among predictors when it is available, the Ising prior (<ref>) is employed. Since the normalizing constant of the Ising model Z(a,b) is often computationally expensive, the values of hyperparameters a and b can be computed by maximizing the pseudo-likelihood instead of the original prior likelihood. Specifically, the pseudo-likelihood is as follows:L_p(a,b) = ∏_j=1^p P ( τ_j | {τ_l: <j,l> ∈ E}, a, b ) = ∏_j=1^p exp{τ_j (a+b∑_<j,l>∈ Eτ_l)}/1+exp{a+b∑_<j,l> ∈ Eτ_l }Indeed, a and b are the logistic regression coefficients when the binary variable τ_j is regressed on ∑_<j,l> ∈ Eτ_l for j=1,...,p. Thus, the conditional median of β_j can be constructed on the basis of Proposition 2 and the details of the algorithm can be found in Algorithm 2. Proposition 2 With current values of β_0 and β, pseudodata and pseudovariances {𝐙, Σ} can be calculated and a sufficient statistic for β_j is (X_j^t Σ^-1 X_j)^-1(X_j^t Σ^-1Z̃_̃j̃) w.r.t. the approximated distribution of 𝐙 which is N(β_0+𝐗β, Σ). Then, the iterative conditional median of β_j in the ICM/M algorithm can be constructed as the posterior median of β_j in the following Bayesian analysis, {[ (X_j^t Σ^-1 X_j)^-1/2(X_j^t Σ^-1Z̃_j) | β_j ∼ N( (X_j^t Σ^-1 X_j)^1/2, 1); β_j | ϖ_j ∼ (1-ϖ_j)δ_0(β_j) +ϖ_j α (X_j^t Σ^-1 X_j)^1/2/2exp{-α (X_j^t Σ^-1 X_j)^1/2 |β_j| }, ].where the probability ϖ_j is specified as follows,ϖ_j^-1 = 1+exp{-a-b ∑_l:<j,l> ∈ Eτ_l}Algorithm 2* Initialize (β_0^(0), β^(0)) and set k=0.* Set k = k+1.* Update pseudodata and pseudovariances (𝐙^(k), Σ^(k)).* Update τ^(k) = (τ_1^(k), ..., τ_p^(k)) where τ_j^(k) = I_{β_j^(k-1) 0}.* Update hyperparameters (a,b) as the mode of its pseudo-likelihood function.(a^(k), b^(k)) = mode{∏_j=1^p P (τ_j^(k) | τ_l^(k) : <j,l> ∈ E ) } * For j=1,...,p, update β_j as the posterior median. β_j^(k) = median P(β_j | 𝐘, 𝐗, 𝐙^(k), Σ^(k), τ^(k), a^(k), b^(k)β_0^(k-1), β_1:(j-1)^(k), β_(j+1):p^(k-1)) = median P(β_j | 𝐘, 𝐗, 𝐙^(k), Σ^(k), ϖ_j^(k), β_0^(k-1), β_1:(j-1)^(k), β_(j+1):p^(k-1))where the probability ϖ_j^(k) =[ 1+exp{-a^(k)-b^(k)∑_l:<j,l>∈ Eτ_l^(k)} ]^-1.* Update β_0 as follows:β_0^(k) = Z̅_w - X̅_w β^(k)where Z̅_w and X̅_w are the weighted means of 𝐙^(k) and 𝐗 respectively with weight for each observation being σ_i^-2.* Iterate between 2 - 7 until convergence. § VARIABLE IMPORTANCE AND FALSE DISCOVERY RATEAlthough the ICM/M algorithm with a spike-and-slab prior can enforce variable estimation and selection simultaneously, it is often of interest to quantify the importance of variables. <cit.> proposed a local posterior probability to evaluate the importance of variables. Specifically, given all other parameter estimates except for β_j, a local posterior probability of j-th predictor is defined asζ_j= P(β_j0 | 𝐘, 𝐗, β̂_0, β̂_1:(j-1), β̂_(j+1):p, .) = ∫ P(β_j | 𝐘, 𝐗, β̂_0, β̂_1:(j-1), β̂_(j+1):p, .).However, with the Bayesian formulation for GLMs, the local posterior probability in (<ref>) is not in the closed form and it is natural to estimate it by using the pseudodata in the last iteration (𝐙, Σ) and its approximated distribution. That is, ζ_j ≈∫ P(β_j | 𝐙, Σ, 𝐗, β̂_0, β̂_1:(j-1), β̂_(j+1):p, .).Such probability in (<ref>) has a closed form and can be easily computed. With the local posterior probability ζ and true β, a true false discovery rate (FDR) given the data can be computed asFDR = ∑_j=1^p 1_{β_j=0} 1_{ζ_j>κ}/∑_j=1^p 1_{ζ_j>κ},0 ≤κ < 1Following <cit.>, the expected FDR given the data in Bayesian scheme is defined asFDR = ∑_j=1^p (1-ζ_j) 1_{ζ_j>κ}/∑_j=1^p 1_{ζ_j>κ}We then can select a set of important predictors based on the local posterior probability and FDR. By controlling FDR at prespecified level, κ can be chosen and a set of important predictors lists all predictors having the local posterior probability greater than κ. It has been shown that the local posterior probability is a good indicator to quantify the importance of variables.§ THE ICM/M ALGORITHM FOR BINARY LOGISTIC REGRESSION §.§ Impplementation detailsWhen the response variable is binary taking values {0,1}, the logistic regression is commonly used. The logistic regression assumes Y_i is independently Bernoulli distributed with mean E[Y_i] = P(Y_i=1) = π(𝐗_i) and variance Var[Y_i] = π(𝐗_i)(1-π(𝐗_i)) whereπ(𝐗_i) = e^β_0+𝐗_iβ/1+e^β0+𝐗_iβ.Equivalently, the logistic regression model can be written aslog( π(𝐗_i)/1-π(𝐗_i)) = β_0+𝐗_iβ. The pseudodata and pseudovariance for binary logistic regression can be constructed based on the current parameter estimates (β̂_0, β̂) asZ_i= β̂_̂0̂+𝐗_iβ̂ + Y_i -π̂(𝐗_i)/π̂(𝐗_i)(1-π̂(𝐗_i)) σ_i^2= 1/π̂(𝐗_i)(1-π̂(𝐗_i)) where π̂(𝐗_i) = e^β̂_0+𝐗_iβ̂/1+e^β̂_0+𝐗_iβ̂ for i=1,...,n. Some cautions need to be considered to avoid any divergence issues. The details in the implementation of ICM/M algorithm are discussed here to prevent divergence problem: * When a probability is within the range of ϵ =10^-5 of 0 or 1, we set it to 0 and 1 respectively.* When a probability is close to 1. Numeric difficulty arises in calculating pseudodata and pseudovariances due to the term π̂(𝐗_i)(1-π̂(𝐗_i)) in the denominator. Hazard or tail functions are employed to improve numerical stability in IRLS (<cit.>). We defineF(η_i) = e^η_i/1+e^η_i = 1/1+e^-η_i, h_+(η_i) = F(η_i) andh_-(η_i) = 1 - F(η_i).The pseudodata and pseudovariance can be computed asZ_i= {[ η̂_i + 1/h_+(η̂_i), ifY_i = 1; η̂_i - 1/h_-(η̂_i), ifY_i = 0 ] .σ^2_i= 1/h_-(η̂_i)h_+(η̂_i).In addition, to avoid the overflows, when η̂_i < -30 and η̂_i>30, η̂_i is set to -30 and 30 respectively. Care should also be taken to select the appropriate form of the function in each tail. Specifically, the form involving e^-η should be used for positive value of η and the form involving e^η should be used for negative value of η.* The idea of active-set convergence is adopted to check convergence in each iteration. The algorithm stops when a complete cycle to update the coefficients in the inner loop does not change the active set of predictors - those with nonzero coefficients. Active set convergence is also mentioned in <cit.>, <cit.>, and <cit.>.§.§ Simulation studiesSimulation studies were conducted to evaluate the performance of our proposed logistic empirical Bayes variable selection via the ICM/M algorithm. Large p small n data sets were simulated from the model (<ref>). We compare the performance of our approach with other two popular methods: the regularized logistic regression with lasso (Lasso) and adaptive lasso (ALasso) penalties. A tuning parameter for both methods was chosen based on 10-fold cross-validation. Using lasso fits as initial values, the ICM/M algorithm was carried out. Three cases of prior on structured predictors were considered here. Case 1 assumed that all predictors are mutually independent. For Case 2 and Case 3, the information of structural relationships among predictors was assumed to be known and the ICM/M with the Ising prior was applied in these two cases. Average misclassification rates (MR) calculated on the test data sets, number of false positives (FP), number of false negatives (FN), model sizes (MS), ∥β-β̂∥_1 = ∑_j-1^p |β_j-β̂_j|, and ∥β-β̂∥_2^2=∑_j=1^p |β_j-β̂_j|^2 were calculated among 100 simulated data sets. §.§.§ Case 1: Independent priorThe data for case 1 was generated with n=250 and p=1,000. The intercept term β_0 was set to zero and there are 10 non-zero regression coefficients includingβ_1=...=β_5=10 and β_11=...=β_15=-5. The covariates were partitioned into 10 blocks, where each block containing 100 covariates were serially correlated at the same level ρ. The values of ρ were {0, 0.3, 0.5, 0.7, 0.9}.From Table <ref>, misclassification rates among the three methods were comparable and they tended to get smaller when the correlation among predictors was high.In order to assess the ability to select the correct variables, average number of false positives and false negatives are also reported in Table 1. Lasso produced large number of false positives across all values of ρ. Adaptive lasso had relatively lower false positive rates than lasso. ICM/M had the smallest number of false positive rates among the three methods. However, it had relatively higher number of false negatives. Furthermore, we also observed that the number of false positives got smaller and the number of false negatives got bigger for higher value of ρ. While the true model contained only 10 non-zero regression coefficients, the average model size from lasso was five to nine times larger than those in the true model. Adaptive lasso selected moderate numbers of non-zero regression coefficients. The average model sizes of ICM/M was closest to the true model. However, several true positives are missed especially when the correlation among predictors was high (ρ = 0.9). For estimation errors, ICM/M outperformed the other two methods in terms of both ∥β-β̂∥_1 and ∥β-β̂∥_2^2 except when ρ=0.9 that adaptive lasso was better than ICM/M. In order to evaluate the importance of variables through the local posterior probability obtained from ICM/M, the true and estimated FDR were plot against κ in Figure <ref> . We first noticed that the estimated FDR was inflated from the true FDR. Furthermore, the FDR was well approximated when the correlation among covariates got higher. By controlling the FDR at level 0.05, the threshold of κ was chosen and all variables having local posterior probabilities suppassed the threshold were selected. Table <ref> shows the results on the value of κ, number of false positives, number of false negatives, and number of important variables for Case 1 simulation. We observed that the value of κ was decreasing for higher value of ρ causing lower number of false positives and higher number of false negatives. Overall, the results in Table <ref> are better than those in Table <ref>.Thus, the local posterior probability and estimated FDR was a suitable tool to select important variables. §.§.§ Case 2: Linear chain priorFor case 2, we again fixed n=250 and p=1000. The intercept term β_0 was set to zero and the values of regression coefficients β depend on an indicator variable τ which follows a Markov linear chain model with transition matrix:T =τ_j+1=0τ_j+1=1 τ_j=0 0.99 0.01 τ_j=1 0.5 0.5And τ_1 ∼π, where π = (0.50.5). The effect size for non-zero coefficients were drawn from Uniform[3,10]. The covariates 𝐗 were generated from AR(1) with different value of ρ in {0, 0.3, 0.5, 0.7, 0.9}.As shown in Table <ref>, ICM/M had smallest misclassification rates when predictors were independent or had mild correlation among them while adaptive lasso had smallest misclassification rates for mild/high correlation among predictors. Again, we saw the same pattern as in Case 1 simulation that misclassification rates were lower when correlation among predictors was higher for all three methods. ICM/M had much smaller number of false positives comparing to lasso and adaptive lasso across values of ρ. Lasso and adaptive lasso generally selected large number of non-zero predictors as the true model in Case 2 simulation contains 21 non-zero coefficients. ICM/M produced relatively larger number of false negatives and smaller model sizes especially for high correlation among predictors.When comparing estimation errors, ICM/M yielded the smallest ∥β-β̂∥_1 while adaptive lasso yielded the smallest ∥β-β̂∥_2^2 across all values of ρ. Thus, it depended on the circumstances to choose between ICM/M and adaptive lasso based on estimation errors. If we would like to avoid large errors, adaptive lasso is preferred since it penalized large deviation more heavily. But if we prefer a model that the estimates are generally close to the actual values but miss badly in some of the coefficients then ICM/M is more appropriate than adaptive lasso.With the Ising prior, the estimated FDR curves obtained from ICM/M were underestimated making the chosen value of κ to be smaller than it supposed to be. When FDR was controlled at level 0.05, the results in Table <ref> had smaller number of false positives and slightly higher number of false negatives. To reduce the number of false negatives, we can increase the FDR level.§.§.§ Case 3: Pathway structure prior To assess the performance of our proposed method in pathway-based analysis, the data was simulated based on the genotype from publicly available Parkinson's disease (PD) dataset (dbGaP study accession number: phs000089.v3.p2). Parkinson metabolic pathway and other six pathways related to PD were obtained from KEGG database. For each genetic region in these pathways, SNPs resided in the genetic region were mapped to the gene based on their physical locations. With this procedure, most genes had at least one affiliated SNPs and there were few genes with more than thousand associated SNPs. To make the simulated data manageable, we selected all SNPs for genes having 5 or less than 5 associated SNPs. For those genes having more than 5 associated SNPs, only 5 SNPs representing each gene were randomly chosen. This process yielded p=1,152 SNPs representing 341 genes left for analysis. The phenotype 𝐘 was generated using logistic regression with these 1,152 SNPs as covariates. The regression coefficients for all 46 SNPs affiliated to 15 genes resided in Parkinson metabolic pathway were chosen to have non-zero effect size. The location of non-zero coefficients were fixed but the effect sizes were random and drawn from Uniform[1,10]. There were 1,741 individuals in the PD dataset which we randomly selected 871 individuals to be served as training data and 870 individuals as test data. The simulation was run for 100 times. The results from Case 3 simulation were consistent with simulation studies in Case 1 and Case 2. ICM/M outperformed the othere two methods in terms of number of false positives although it has higher number of false negatives. ICM/M also had lowest misclassificationr rate in this case. For estimation errors, ICM/M and adaptive lasso also had similar ∥β-β̂∥_1 but adaptive lasso yielded smaller ∥β-β̂∥_2^2.As shown in Figure <ref>, FDR curve was well approximated in this case. In addition, the results in Table <ref> were similar to the results in Table <ref> when FDR was controlled at level 0.05.To sum up the simulation section for binary logistic model, it is worth noticing that ICM/M with lasso fits as initial values improved significantly over lasso in terms of number of false positives. When correlation among predictors was very high, ICM/M might miss some important variables. However, it was interesting to see that ICM/M was able to reduce model size from lasso dramatically while increasing predictive ability at the same time. Furthermore, instead of relying on only regression coefficients from ICM/M, the local posterior probability along with the plot of estimated FDR can be used as a tool to select important variable.§ THE ICM/M ALGORIHM FOR COX'S PROPORTIONAL HAZARDS MODEL §.§ Implementation detailsthe classical survival analysis framework, the observed data consists of (Y_i, 𝐗_i, δ_i) where Y_i is the ovserved time for subject i, 𝐗_i is a p-dimensional vector of covariates, and δ_i is 1 if Y_i is the actual survival time and 0 for right-censoring. The Cox's regression (<cit.>) models survival times through hazard functionh(Y_i|𝐗_i) = h_0(Y_i)e^𝐗_iβwhere h(Y_i|𝐗_i) is the hazard for subject i and h_0(Y_i) is the baseline hazard function (i.e., the hazard function when all covariates 𝐗_i are 0). The Cox's proportional hazard model assumes that covariates are time-independent and the baseline hazard h_0(Y_i) is an unspecified function. Let t_1<t_2<...<t_m be the increasing list of distinct failure time and define cumulative baseline hazard at time t_j for subject i to beH_0(Y_i) = ∑_j:t_j≤ Y_iΔ H_0(t_j),where Δ H_0(t_j) is the increment of the cumulative baseline hazard at time t_j. Following <cit.> and <cit.>, the regression coefficients can be found by maximizing the extended likelihood taking the formL(β) = ∏_i=1^n[ {Δ H_0(Y_i)e^𝐗_iβ}^δ_iexp{-H_0(Y_i)e^𝐗_iβ} ].Given value of β, the increment of the cumulative baseline hazard that maximizes the likelihood isΔĤ_0(t_j) = ∑_i:Y_i=t_jδ_i/∑_i: Y_i≥ t_j e^𝐗_iβ.Hence,Ĥ_0(Y_i) = ∑_j:t_j≤ Y_iΔĤ_0(t_j).The explicit form of the pseudodata and pseudovariace can be computed based on current coefficients β̂ as Z_i = 𝐗_i β̂ + 1/Ĥ_0(Y_i) e^𝐗_i β ( δ_i - Ĥ_0(Y_i) e^𝐗_i β̂ ) σ_i^2 = 1/Ĥ_0(Y_i) e^𝐗_iβ̂.The approximated distribution of pseudodata 𝐙 is N(𝐗β̂, Σ) where Σ = diag{σ_i^2}. The ICM/M algorithm can now be used to cycle through parameters update and the active-set convergence is employed as stopping criterion. §.§ Simulation studiesIn this section, three cases in simulation were conducted to show the performance of the ICM/M algorithm for Cox's proportional hazards model. Here we compare the results with lasso and adaptive lasso for Cox's model in large p small n settings. In all three scenarios, we fixed n=250 and p=1,000. Survival times were simulated from a Cox model with the baseline hazard function of a Weibull distribution with a shape parameter ν=10 and a scale parameter λ=1. The censoring times were generated randomly to achieve censoring rate of 50%.Case 4 simulation assumed independence among regression coefficients and ICM/M with independent prior in (<ref>) was carried out. Case 5 and Case 6 assumed that regression coefficients followed some prior distribution representing the relationship among them. The ICM/M with the Ising prior was applied for Case 5 and Case 6 simulations. Here we report average number of false positives (FP), number of false negatives (FN), model sizes (MS), ∥β-β̂∥_1 = ∑_j-1^p |β_j-β̂_j|, and ∥β-β̂∥_2^2=∑_j=1^p |β_j-β̂_j|^2 calculated among 100 simulated data sets for each case.§.§.§ Case 4: Independent priorThe covariates were simulated the same way as in Case 1 with n=250 and p=1,000. Among 1,000 predictors, the failure times were determined by a linear combination of 20 non-zero coefficients: β_1 = ...=β_10=5 and β_101=...=β_110=2.Table <ref> shows that ICM/M performed the best regarding the ability to select true important variables across all levels of correlation among covariates. The average numbers of both false positives and negatives were close to 0. Moreover, the average model size was close to the size of true model which was 20. Although lasso also had small numbers of false negatives, it turned out that lasso tended to select much larger number of non-zero coefficients which was more than 100 variables. This caused large numbers of false positives. Adaptive lasso selected moderate size of predictors but its numbers of false positives and negatives were relatively large comparing to those produced by ICM/M.In a comparison of ∥β-β̂∥_1, ICM/M surpassed the other two methods for ρ=0, 0.3, and 0.5. For ρ=0.7 and 0.9, adaptive lasso performed slightly better than ICM/M. When comparing estimation errors in terms of ∥β-β̂∥_2^2, adaptive lasso performed well when correlation among covariates was high while the performance of lasso and ICM/M were better than adaptive lasso for small value of ρ. Furthermore, we also observed that lasso and ICM/M yielded similar values of ∥β-β̂∥_2^2.Figure <ref> shows that the estimated and true FDR curves were very close to each other. In addition, the results in Table <ref> and Table <ref> were consistent. The average number of false positives and negatives were all close to zero making the ICM/M was a competitive method for high-dimensional variable selection in Cox's model.§.§.§ Case 5: Linear chain priorCovariates with n=250 and p=1,000 were generated the same way as in Case 2. The location of non-zero coefficients follows a Markov linear chain with the same transition matrix as in Case 2. The effect sizes of those non-zero coefficients were drawn from Uniform[0.5,5]. Ability to select variables correctly were assessed in 3 criteria: number of false positives, false negatives, and model size. As you can see in Table 5, ICM/M was the best method to select correct variables here due to small numbers of false positives and false negatives across all value of ρ.Average model sizes were also close to the true model size containing 19 non-zero coefficients. We also noticed that ICM/M gave slightly higher number of false positives when ρ increased. In contrast, lasso and adaptive lasso yielded lower number of false positives when ρ increased. Although lasso and adaptive lasso performed reasonable well with regard to number of false negatives, both methods always selected large number of unimportant variables resulting in high false positive rates. Besides the variable selection ability, ICM/M overall outperformed the other two methods regarding to the two criteria of estimation errors including ∥β-β̂∥_1 and ∥β-β̂∥_2^2. Numbers of important variables in Table <ref> when controlling FDR at level 0.05 were slightly higher than the model size in Table <ref> but still very close to the true number of important variables. §.§.§ Case 6: Pathway structure priorFor Case 6, gene expression data within an assumed network were simulated. The network consisted of 10 disjoint pathways. Each of which contained 100 genes resulting in p=1,000 in total. Ten regulated genes were assumed in each pathway. The gene expression values were generated from a standard normal distribution. For those regulated genes in the same pathway, the expression values were generated from normal distribution with a correlation of ρ=0.7 among those 10 regulated genes. Eighteen regulated genes from three pathways were chosen to have non-zero coefficients that were drawn from Uniform[0.5, 5].As you can see in Table <ref>, ICM/M outperformed the other two methods in terms of the ability to select true relevant genes. Both average numbers of false positives and negatives were close to 0 with the average model size being 17.86 which is close to 18. Lasso and adaptive lasso selected much more variables into the model resulting in large number of false positives. We also observed that adaptive lasso had smallest avarage values for ∥β-β̂∥_1 and ∥β-β̂∥_2^2. Similar to other simulation studies for Cox's model, the true FDR curve was well approximated by the estimated FDR curve. Moreover, the results based on the regression coefficients from ICM/M method in Table <ref> were similar to variable selection procedure at FDR level = 0.05 in Table <ref>.To conclude the simulation results for Cox's proportional hazards model, ICM/M gave the best variable selection performance. Lasso and adaptive lasso tended to select much larger number of variables into the model yielding large value of false positives though they might have lower number of false negatives than ICM/M. This suggested the substantial improvement of ICM/M with lasso coefficients as initial values over lasso. In addition, we also noticed that the results based on ICM/M coefficients were comparable to variable selection based on the local posterior probability when FDR is being controlled at level 0.05.§ ILLUSTRATION: PARKINSON'S DISEASE GENOME-WIDE ASSOCIATION STUDYAs an application in genomics, a GWAS was carried out to identify a set of susceptible genetic markers to Parkinson's disease (PD). This PD dataset is publicly available (dbGaP study accession number: phs000089.v3.p2). The dataset consisted of 940 PD patients and 801 controls resulting in 1,741 samples in total. All individuals were Americans with European ancestry. The genotyping assays were derived from Illumina Infinium HumanHap300 and HumanHap500 SNP chips. There were 310,860 SNPs in common between these two arrays which we focused in this analysis. The following GWAS preprocessing criteria were applied to the data: (i) missingness per individual < 10%, (ii) missingness per marker < 10%, (iii) minor allele frequency (MAF) ≥ 5%, and (iv) Hardy-Weinberg test at significant level = 0.001. In addition, K-nearest neighbor (KNN) method was employed to impute missing genotype. Parkinson metabolic pathway and other six pathways related to PD were acquired from the KEGG database. There were 26,101 SNPs representing 341 genes in the seven pathways. To make the computation more manageable, the univariate tests were applied and only 2,664 SNPs having p-values surpassed the significant threshold at 0.1.Assume known pathway information among SNPs from the KEGG database, ICM/M with Ising prior was applied to fit a high-dimensional binary logistic regression. Again, lasso fits were used as initial values of coefficients for ICM/M algorithm. ICM/M was then compared with lasso and adaptive lasso. From Table <ref>, Lasso performed relatively poor due to large misclassification rate and model size. Adaptive lasso had the smallest misclassification rate. Similar to results in simulation studies, ICM/M tended to select smaller number of variables into the model comparing to lasso and adaptive lasso. There was also an improvement of ICM/M in terms of misclassification rate over lasso.Figure <ref> demonstrates the number of SNPs identified by the three methods. There were 24 SNPs identified by all three methods. The details of these overlapping SNPs are shown in Table <ref>. We noticed that the signs of β̂ for each SNPs were the same for three methods. Overall, ICM/M tended to produce larger effect sizes for regression coefficients than the other two methods. This might due to the thresholding of ICM/M to screen out the small regression coefficients.For ICM/M, FDR obtained from the local posterior probability was estimated. When controlling FDR at level 0.05, κ was chosen to be 0.86 yielding 114 important variables. If FDR level was increased to 0.10, the corresponding κ = 0.64 yielding 157 important variables. Due to large number of identified SNPs, we further investigated our analysis. For comparison purposes, we only focused on the results of ICM/M based on the fitted model for the rest of this section although one might consider using FDR and the local posterior probability to select important variables instead.Since the correlation among SNPs resided in the same genetic region tended to be high, it was possible that different variable selection methods select different sets of SNPs representing the same gene. We therefore obtained genetic markers by grouping SNPs affiliated to the same gene. The genetic marker for gene g was defined as 𝐗̃_g = 𝐗_g β̂_g where 𝐗_g was a design matrix consisting of all SNPs resided in gene g and β̂_g was a vector of corresponding estimated coefficients obtained from variable selection method when considered all SNPs in original data as predictors. With the construction of genetic markers, it was easy to check which genes revealed genetic risk to PD. For each method, the logistic regression model with disease status as a response and genetic markers as covariates was fitted to obtain the p-values. The numbers of identified SNPs within the genetic regions are reported here. It was possible that some SNPs were affiliated to more than one gene. Table <ref> provides the results for genes identified by ICM/M (Full table can be found in Supplementary Table A).There were 28, 35, and 24 genes identified by lasso, adaptive lasso, and ICM/M respectively (See Supplementary Table A). Among 13 genes detected for all three methods, 4 of them including MAPT, PARK2, PIK3CD, and SNCA had been well studied and known as PD susceptible genes in a number of literatures (see e.g., <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>). There were another 6 genes that had been reported in Harmonizome database (see <cit.>) to have association to PD based on GWAS and other genetic association datasets from the GWASdb SNP-Disease Associations dataset. These 6 genes included BIRC6, COX7B2, PRKAR2B, TNPO3, and TRIP12. Among these 6 genes, lasso, adaptive lasso, and ICM/M can identify 5, 4, and 4 genes respectively. In addition, UCHL1 had also been reported to be involved with PD risk in recent literatures (<cit.> and <cit.>). Here, only ICM/M and adaptive lasso are able to detect UCHL1 gene.We observed that there was a relationship between p-values and number of identified SNPs resided in the genetic region. Indeed, the genes having larger number of identified SNPs tended to have smaller p-values. This phenomenon was not unusual due to the fact that genes with larger number of identified SNPs were accounted for more variations to the response. Interestingly, we noticed that the p-value for ICM/M method tended to be smaller than the other two methods despite the number of identified SNPs resided in a gene. § ILLUSTRATION: LUNG ADENOCARCINOMA MICROARRAY ANALYSISWe illustrate our proposed methodology for Cox's model to the lung cancer data from <cit.>. The microarray data consisting of p=7,129 genes were from n=86 patients with primary lung adenocarcinomas. The survival time of patients in early-stage lung adenocarinomas were recorded and the censoring rate in this data set is 27.9%. Three pathways related to lung adenocarcinoma were obtained from KEGG database. Genes in the data set were then mapped to genes in these three pathways to generate an undirectedg graph. This resulted in a gene network consisting of 55 nodes and 109 edges. The rest of the genes were either isolated or not in one of these three pathways. Using lasso coefficients as initial values, the ICM/M algorithm with pathway structure prior was applied to the data set. For comparison purposes, the results from lasso and adaptive lasso were also reported.Lasso , adaptive lasso, and ICM/M selected 24, 23, and 12 genes respectively. Figure <ref> shows the number of overlapping genes that were identified by the three methods. The numbers of selected genes by lasso and adaptive lasso were similar and there were 21 overlapping genes between these two methods. On the other hands, ICM/M selected less genes than the other two methods. There were 9 genes identified by all three methods. Among these 9 genes, 4 of them including BAG1, FXYD3, HPIP, and SLC2A1 were among the top 100 genes related to survival of patients reported in Beer et. al. (2002) (see Table <ref>). Furthermore, we found that another 3 genes were reported to have some relation with lung cancer in literatures. This included INSL4 (<cit.>), PRKACB (<cit.>), and SPRR1B(<cit.>).In addition to 9 genes in Table <ref>, ICM/M also identified 3 more genes including NME2, STMN1, and ZFP36 (see Table <ref>). In fact, NME2, identified by both Lasso and ICM/M, was among the top 100 genes related to survival of patients reported in Beer et. al. (2002). For STMN1 and ZFP36, only ICM/M was able to identify these 2 genes. Interestingly, both genes were related to lung cancer in recent literatures: STMN1 (<cit.>) and ZFP36 (<cit.>). For ICM/M method, estimated FDR were calculated and κ was chosen as 0.81 which corresponded to FDR level 0.05. This yielded 6 genes having local posterior probabilities surpass κ=0.81. If one would like to increase the number of selected genes, FDR level can be increased to any desired level. For example, by controlling FDR level at 0.10, the κ was chosen to be 0.66 yielding 8 important genes. To select all 12 genes, the corresponding FDR level was 0.15 in this case. § DISCUSSION<cit.> proposed an empirical Bayes methods in selecting massive variables for linear model. A apike-and-slab prior is employed to introduce sparsity in regression coefficient and the Ising prior is used to capture structural relationship among predictors. The ICM/M algorithm is also introduced to cycle through each coordinate of the parameters in the model. This paper intends to extend Empirical Bayes variable selection for GLMs. We here focus on two important models arised in analyzing genomic data including the binary logistic and Cox's proportional hazards models. A main challenge of such an extension has emerged as how to generalize the framework from linear to nonlinear models. Here we borrow the idea of IRLS like in fitting classical GLMs to the ICM/M algorithm. This simple-to-implement algorithm achieves fast computation even in high-dimensional data analysis. The ICM/M algorithm for both linear and nonlinear models is implemented in theR package.The lasso is the most widely used method in fitting high-dimensional GLMs. Its popularity stems from its computational feasibility and availability in various packages. Our results show that ICM/M with lasso fits as initial values improves significantly over lasso in terms of number of false positives for both logistic and Cox's proportional hazards model. It can reduce the model size from lasso substantially by thresholding property of the ICM/M algorithm. For binary logistic model, we observed that the predictive ability of our proposed method was also better than lasso. For Cox's model, ICM/M outperformed lasso and adaptive lasso in terms of the ability to select correct variables into the model. ICM/M is also compuational feasible even for large p.In biomedical field, most researchers would like to integrate expert knowledge such as gene regulartory networks in data analysis. While lasso and adaptive lasso do not utilize any prior information on strutural information among predictors, an important feature of ICM/M is that it can also incorporate more complicated prior easily. Here we demonstrated how to integrate a Markov random field representing relationships among predictors through an Ising model for binary logistic and Cox's models. Simulation studies and real data show a considerably improvement of ICM/M from lasso.Another advantage of our proposed methodology over lasso and adaptive lasso is that our method does not only provide regression coefficients but also a local posterior probability. A local posterior probability for each predictor is proposed to quantify the importance of variable in linear model (<cit.>). For Bayesian variable selction for GLMs, the local posterior probability is estimated using the pseudodata in the last iteration of IRLS. FDR in Bayesian scheme can also be computed and employed to select the final set of important variables. Therefore, our method provides flexibility to select varibles while controlling FDR at a desired level rather than relying on only regression coefficients.Finally, we would like to note that our method seems to work better with survival data than binary data in terms of false negative rate. This might be due to the fact that binary data is less informative than continuous data even though the censoring rate is quite high (50%).§ ACKNOWLEDGEMENT § REFERENCES § SUPPLEMENTARY TABLEl | c c c | c c c *Table A: Results from PD data analysis: details on identified genes (Bolded genes were identified by all methods). 2*Gene (Location) 3|c|Number of identified SNPs 3|cp-value Lasso ALasso ICM/M Lasso ALasso ICM/M *Table A: Results from PD data analysis: details on identified genes (Bolded genes were identified by all methods) (continued). 2*Gene (Location) 3|c|Number of identified SNPs 3|cp-value Lasso ALasso ICM/M Lasso ALasso ICM/M 7rContinue on next page ATP5F1 (1p13.2) 1 0 1 0.0088 11.01e-6 NDUFS2 (1q23) 0 0 1 1 1 0.0010 PIK3CD (1p36.2) 1 1 1 5.93e-8 0.0037 1.14e-8 CDC20 (1p34.1) 1 0 0 0.0372 1 1 AKT3 (1q44) 0 1 0 1 0.0113 1 LCLAT1 (2p23.1) 1 0 0 0.0150 1 1 BIRC6 (2p22.3) 2 0 1 5.23e-8 1 2.13e-9 TRIP12 (2q36.3) 1 1 1 0.0352 0.0638 4.51e-5 NDUFB4 (3q13.33) 159 141 62 1.16e-70 3.2e-58 8.69e-66 HGD (3q13.33) 159 141 62 1.16e-70 3.2e-58 8.69e-66 PRKAR2A (3p21.3-p21.2) 0 0 1 1 1 0.9480 IL1RAP (3q28) 2 01 0.0021 1 3.17e-5 COX7B2 (4p12) 1 1 0 0.153 0.0091 1 UCHL1 (4p14) 0 1 1 1 0.0317 0.0146 SNCA (4q21) 1 1 1 0.0043 0.0029 0.0020 ATP6V1G2 (6p21.3) 48 38 17 2.41e-28 5.99e-24 1.26e-38 TNF (6p21.3) 48 38 17 2.41e-28 5.99e-24 1.26e-38 PARK2 (6q25.2-q27) 5 5 3 3.29e-8 2.42e-5 1.34e-17 DDC (7p12.2) 0 2 0 1 0.8540 1 UBE2D4 (7p13) 1 1 0 0.0786 0.0049 1 PRKAR2B (7q22) 1 0 1 6.00e-4 1 7.01e-07 TNPO3 (7q32.1) 0 1 0 1 0.0950 1 ATP6V0A4 (7q34) 2 1 1 0.0098 0.0934 0.0002 TCEB1 (8q21.11) 0 1 0 1 0.0149 1 TNFRSF10B (8p22-p21) 1 0 1 0.0205 1 0.0183 TYRP1 (9p23) 1 1 0 0.1880 0.0062 1 CUL2 (10p11.21) 1 1 1 0.0339 0.0034 0.0005 ATP5C1 (10p15.1) 0 1 0 1 0.1770 1 FAS (10q24.1) 0 0 1 1 1 0.0044 LHPP (10q26.13) 0 1 0 1 0.4230 1 PRMT3 (11p15.1) 1 2 1 3.97e-5 0.0001 2.58e-7 PSMD13 (11p15.5) 0 1 0 1 0.0068 1 TYR (11q14-q21) 0 1 0 1 0.1320 1 NDUFA12 (12q22) 1 0 0 0.3050 1 1 UBE2N (12q22) 0 1 0 1 0.0128 1 UBE3A (15q11.2) 0 1 0 1 0.0214 1 UBE2Q2 (15q24.2) 0 1 0 1 0.0205 1 DET1 (15q25.3) 1 2 0 0.1450 0.02791 NDUFB10 (16p13.3) 0 1 0 1 0.02701 MAPT (17q21.1) 107 51 72 7.11e-52 2.78e-28 6.17e-66 ANAPC11 (17q25.3) 107 51 72 7.11e-52 2.78e-28 6.17e-66 NEDD4L (18q21) 0 1 0 1 0.9180 1 ECH1 (19q13.1) 0 1 0 1 0.0252 1 AKT2 (19q13.1-q13.2) 0 1 0 1 0.6110 1 CBLC (19q13.2) 0 1 0 1 0.0813 1 NDUFA11 (19p13.3) 1 0 1 0.0118 1 8.15e-5 ATP4A (19q13.1) 1 1 0 0.0682 0.0973 1 UBE2G2 (21q22.3) 2 0 0 0.3360 1 1 ATP5O (21q22.11) 0 0 1 1 1 7.98e-9 | http://arxiv.org/abs/1707.08298v2 | {
"authors": [
"Vitara Pungpapong",
"Min Zhang",
"Dabao Zhang"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20170726063552",
"title": "Variable Selection for High-dimensional Generalized Linear Models using an Iterated Conditional Modes/Medians Algorithm"
} |
LTCI, Télécom ParisTech, Université Paris-Saclay IMT, Université de ToulouseMax K-armed bandit: On the ExtremeHunter algorithm and beyond Mastane Achab1 Stephan Clémençon1 Aurélien Garivier2 Anne Sabourin1 Claire Vernade1 December 30, 2023 ======================================================================================= This paper is devoted to the study of the max K-armed bandit problem, which consists in sequentially allocating resources in order to detect extreme values. Our contribution is twofold. We first significantly refine the analysis of the ExtremeHunter algorithm carried out in <cit.>, and next propose an alternative approach, showing that, remarkably, Extreme Bandits can be reduced to a classical version of the bandit problem to a certain extent. Beyond the formal analysis, these two approaches are compared through numerical experiments.§ INTRODUCTION In a classical multi-armed bandit (MAB in abbreviated form) problem, the objective is to find astrategy/policy in order to sequentially explore and exploit K sources of gain, referred to as arms, so as to maximize the expected cumulative gain. Each arm k∈{1, …,K} is characterized by an unknown probability distribution ν_k. At each round t≥ 1, a strategy π picks an arm I_t=π((I_1, X_I_1,1),…,(I_t-1, X_I_t-1, t-1)) and receives a random reward X_I_t, t sampled from distribution ν_I_t. Whereas usual strategies aim at finding and exploiting the arm with highest expectation, the quantity of interest in many applications such as medicine, insurance or finance may not be the sum of the rewards, but rather the extreme observations (even if it might mean replacing loss minimization by gain maximization in the formulation of the practical problem). In such situations, classical bandit algorithms can be significantly sub-optimal: the "best" arm should not be defined as that with highest expectation, but as that producing the maximal values. This setting, referred to as extreme bandits in <cit.>, was originally introduced by <cit.> by the name of max K-armed bandit problem. In this framework, the goal pursued is to obtain the highest possible reward during the first n≥ 1 steps. For a given arm k, we denote byG_n^(k)=max_1≤ t≤ nX_k, tthe maximal value taken until round n≥ 1 and assume that, in expectation,there is a unique optimal armk^∗= _1≤ k≤ K𝔼[G_n^(k)] .The expected regret of a strategy π is here defined as𝔼[R_n]= 𝔼[G_n^(k^∗)] - 𝔼[G_n^(π)],where G_n^(π)=max_1≤ t≤ nX_I_t, t is the maximal value observed when implementing strategy π. When the supports of the reward distributions (i.e. the ν_k's) are bounded, no-regret is expected provided that every arm can be sufficiently explored, refer to <cit.> (see also <cit.> for a PAC approach). If infinitely many arms are possibly involved in the learning strategy, the challenge is then to explore and exploit optimally the unknown reservoir of arms, see <cit.>. When the rewards are unbounded in contrast, the situation is quite different: the best arm is that for which the maximum G_n^(k) tends to infinity faster than the others. In <cit.>, it is shown that, for unbounded distributions, no policy can achieve no-regret without restrictive assumptions on the distributions. In accordance with the literature, we focus on a classical framework in extreme value analysis. Namely, we assume that the reward distributions areheavy-tailed. Such Pareto-like laws are widely used to model extremes in many applications, where a conservative approach to risk assessment might be relevant (e.g. finance, environmental risks). Like in <cit.>, rewards are assumed to be distributed as second order Pareto laws in the present article. For the sake of completeness, we recall that a probability law with cdf F(x) belongs to the (α, β, C, C')-second order Pareto family if, for every x≥ 0,|1-Cx^-α-F(x)|≤ C'x^-α(1+β),where α, β, CandC' are strictly positive constants, see e.g. <cit.>. In this context, <cit.> have proposed the ExtremeHunter algorithm to solve the extreme bandit problem and provided a regret analysis.The contribution of this paper is twofold. First, the regret analysis of the ExtremeHunter algorithm is significantly improved, in a nearly optimal fashion. This essentially relies on a new technical result of independent interest (see Theorem <ref> below), which provides a bound for the difference between the expectation of the maximum among independent realizations X_1, …,X_T of a(α, β, C, C')-second order Pareto distribution, 𝔼[max_1≤ i≤ T X_i] namely, and its rough approximation (TC)^1/αΓ(1-1/α). As a by-product, we propose a more simple Explore-Then-Commit strategy that offers the same theoretical guarantees asExtremeHunter. Second, we explain how extreme bandit can be reduced to a classical bandit problem to a certain extent. We show that a regret-minimizing strategy such as Robust-UCB (see <cit.>), applied on correctly left-censored rewards, may also reach a very good performance. This claim is supported by theoretical guarantees on the number of pulls of the best arm k^∗ and by numerical experiments both at the same time. From a practical angle, the main drawback of this alternative approach consists in the fact that its implementation requires some knowledge of the complexity of the problem (i.e. of the gap between the first-order Pareto coefficients of the first and second arms). In regard to its theoretical analysis, efficiency is proved for large horizons only. This paper is organized as follows. Section <ref> presents the technical result mentioned above, which next permits to carry out a refined regret analysis of the ExtremeHunter algorithm in <ref>. In <ref>, the regret bound thus obtained is proved to be nearly optimal: precisely, we establish a lower bound under the assumption that the distributions are close enough to Pareto distributions showing the regret bound is sharp in this situation. In <ref>, reduction of the extreme bandit problem to a classical bandit problem is explained at length, and an algorithm resulting from this original view is then described. Finally, we provide a preliminary numerical study that permits to compare the two approaches from an experimental perspective. Due to space limitations, certain technical proofs are deferred to the Supplementary Material. § SECOND-ORDER PARETO DISTRIBUTIONS: APPROXIMATION OF THE EXPECTED MAXIMUM AMONG I.I.D. REALIZATIONS In the extreme bandit problem, the key to controlling the behavior of explore-exploit strategies is to approximate the expected payoff of a fixed arm k∈{1, …,K}. The main result of this section, stated in Theorem <ref>, provides such control: it significantly improves upon the result originaly obtained by <cit.> (see Theorem 1 therein). As shall be next shown in Section <ref>, this refinement has substantial consequences on the regret bound. In <cit.>, the distance between the expected maximum of independent realizations of a (α,β, C,C')-second order Pareto and the corresponding expectation of a Fréchet distribution (TC)^1/αΓ(1-1/α) is controlled as follows:|𝔼[max_1≤ i≤ TX_i]-(TC)^1/αΓ(1-1/α)| ≤4D_2C^1/α/T^1-1/α+2C'D_β+1/C^β+1-1/αT^β-1/α +(2C'T)^1/(1+β)α .Notice that the leading term of this bound is (2C'T)^1/((1+β)α) as T→ +∞. Below, we state a sharper result where, remarkably, this (exploding) term disappears, the contribution of the related component in the approximation error decomposition being proved as (asymptotically) negligible in contrast. (Fréchet approximation bound) IfX_1, …,X_T are i.i.d. r.v.'s drawn from a (α, β, C, C')-second order Pareto distribution with α>1 and T≥ Q_1, where Q_1 is the constant depending only on α, β, C and C' given in <ref> below, then, |𝔼[max_1≤ i≤ TX_i]-(TC)^1/αΓ(1-1/α)|≤4D_2C^1/α/T^1-1/α+2C'D_β+1/C^β+1-1/αT^β-1/α +2(2C'T)^1/(1+β)αe^-HT^β/β+1= T→∞o (T^1/α), where H=C(2C')^1/(α(1+β))/2. In particular, if β≥ 1, we have: |𝔼[max_1≤ i≤ TX_i]-(TC)^1/αΓ(1-1/α)| = o(1)asT→ +∞. We emphasize that the bound above shows that the distance of 𝔼[max_1≤ i≤ TX_i] to the Fréchet mean (TC)^1/αΓ(1-1/α) actually vanishes as T→∞ as soon as β≥ 1, a property that shall be useful in Section <ref> to study the behavior of learning algorithms in the extreme bandit setting. Assume that T≥ Q_1, where Q_1 = 1/2C'max{(2C'/C)^(1+β)/β, (8C)^1+β}. As in the proof of Theorem 1 in <cit.>, we consider the quantity B=(2C'T)^1/((1+β)α) that serves as a cut-off between tail and bulk behaviors. Observe that | 𝔼[max_1≤ i≤ TX_i] - (TC)^1/αΓ(1-1/α) | ≤ | ∫_0^∞{ 1-ℙ(max_1≤ i≤ TX_i≤ x)-1+e^-TCx^-α}dx |≤|∫_0^B{ℙ(max_1≤ i≤ TX_i≤ x) -e^-TCx^-α}dx | + |∫_B^∞{ℙ(max_1≤ i≤ TX_i≤ x)-e^-TCx^-α}dx| . For p∈{2, β+1}, we set D_p=Γ(p-1/α)/α. Equipped with this notation, we may write |∫_B^∞{ℙ(max_1≤ i≤ TX_i≤ x)-e^-TCx^-α}dx| ≤4D_2C^1/α/T^1-1/α+2C'D_β+1/C^β+1-1/αT^β-1/α. Instead of loosely bounding the bulk term by B, we write |∫_0^B{ℙ(max_1≤ i≤ TX_i≤ x)-e^-TCx^-α}dx| ≤ B ℙ(X_1≤ B)^T + ∫_0^Be^-TCx^-αdx . First, using (<ref>) and the inequality C'B^-(1+β)α≤ CB^-α/2 (a direct consequence of <ref>), we obtain ℙ(X_1≤ B)^T ≤(1-CB^-α+C'B^-(1+β)α)^T≤(1-1/2CB^-α)^T ≤ e^-1/2TCB^-α= e^-HT^β/(β+1). Second, the integral in <ref> can be bounded as follows: ∫_0^B e^-TCx^-αdx ≤ Be^-TCB^-α = (2C'T)^1/((1+β)α)e^-2HT^β/(β+1).This concludes the proof. § THE EXTREMEHUNTER AND EXTREMEETC ALGORITHMSIn this section, the tighter control provided by Theorem <ref> is used in order to refine theanalysis of the ExtremeHunter algorithm (Algorithm <ref>) carried out in<cit.>. This theoretical analysis is also shown to be valid for ExtemeETC, a novel algorithm we next propose, that greatly improves upon ExtremeHunter, regarding computational efficiency. §.§ Further Notations and Preliminaries Throughout the paper, the indicator function of any event ℰ is denoted by 1{ℰ} and ℰ̅ means the complementary event of ℰ. We assume that the reward related to each arm k∈{1, …,K} is drawn from a (α_k, β_k, C_k, C')-second order Pareto distribution. Sorting the tail indices by increasing order of magnitude, we use the classical notation for order statistics: α_(1)≤…≤α_(K). We assume that α_(1)> 1, so that the random rewards have finite expectations, and suppose that the strict inequality α_(1)<α_(2) holds true. We also denote by T_k,t the number of times the arm k is pulled up to time t. For 1≤ k≤ K and i≥ 1, the r.v. X_k, i is the reward obtained at the i-th draw of arm k if i≤ T_k, n or a new r.v. drawn from ν_k independent from the other r.v.'s otherwise.We start with a preliminary lemma supporting the intuition that the tail index α fully governs the extreme bandit problem. It will allow to show next that the algorithm picks the right arm after the exploration phase, see Lemma <ref>. (Optimal arm) For n larger than some constant Q_4 depending only on (α_k, β_k, C_k)_1≤ k≤ K and C', the optimal arm for the extreme bandit problem is given by: k^∗ = _1≤ k≤ Kα_k = _1≤ k≤ K V_k, where V_k = (nC_k)^1/α_kΓ(1-1/α_k). We first prove the first equality. It follows from <ref> that there exists a constant Q_2, depending only on {(α_k, β_k, C_k)}_1≤ k≤ K and C', such that for any arm k∈{1, …,K }, |𝔼[G_n^(k)]-V_k |≤ V_k/2.Then for k≠ k^∗ we have, for all n>Q_2, V_k/2 ≤𝔼[G_n^(k)]≤𝔼[G_n^(k^∗)]≤ 3V_k^∗/2. Recalling that V_k is proportional to n^1/α_k, it follows that α_k^∗=min_1≤ k≤ Kα_k. Now consider the following quantity:Q_3 = max_k≠ k^∗[2C_k^1/α_kΓ(1-1/α_k)/C_k^∗^1/α_k^∗Γ(1-1/α_k^∗)]^1/(1/α_k^∗-1/α_k) .For n>Q_4=max(Q_2, Q_3), we have V_k^∗>2V_k for any suboptimalarm k≠ k^∗, which proves the second equality.From now on, we assume that n is large enough for<ref> to apply.§.§ The ExtremeHunter algorithm <cit.>Before developing a novel analysis of the extreme bandit problem in Section <ref> (see Theorem <ref>), we recallthe mainfeatures of ExtremeHunter, and in particular the estimators andconfidence intervals involved in the indices of this optimistic policy. <ref> states that for any arm k∈{1, …,K}, 𝔼[G_n^(k)]≈ (C_k n)^1/α_kΓ(1-1/α_k). Consequently, the optimal strategy in hindsight always pulls the arm k^∗=_1≤ k≤ K{(nC_k)^1/α_kΓ(1-1/α_k)}. At each round and for each arm k∈{1, …,K}, ExtremeHunter algorithm <cit.> estimates the coefficients α_k and C_k (but not β_k, see Remark 2 in <cit.>). The corresponding confidence intervals are detailed below. Then, following the optimism-in-the-face-of-uncertainty principle (see <cit.> and references therein), the strategyplays the arm maximizing an optimistic plug-in estimate of (C_kn)^1/α_kΓ(1-1/α_k). To that purpose, Theorem 3.8 in <cit.> and Theorem 2 in <cit.> provide estimators α_k(T) and C_k(T) for α_k and C_k respectively, after T draws of arm k. Precisely, the estimate α_k(T) is given byα_k(T)=log( ∑_t=1^T1{X_t>e^r}/∑_t=1^T1{X_t>e^r+1}),where r is chosen in an adaptive fashion based on Lepski's method, see <cit.>, while the estimator of C_k considered isC_k(T)=T^-2b/(2b+1)∑_i=1^T 1{X_k,i≥ T^h_k(T)/(2b+1)},whereh_k(T)=min(1/α_k(T), 1) .The authors also providefinite sample error bounds for T≥ N, whereN= A_0(log n)^2(2b+1)/b,with b a known lower bound on the β_k's (b≤ min_1≤ k≤ Kβ_k), and A_0 a constant depending only on (α_k, β_k, C_k)_1≤ k≤ K and C'. These error bounds naturally define confidence intervalsof respective widths Λ_1 and Λ_2 at level δ_0 defined byδ_0 = n^-ρ,whereρ=2α_k^∗/α_k^∗-1.More precisely, we haveℙ(|1/α_k-h_k(T)| ≤Λ_1(T), |C_k-C_k(T)|≤Λ_2(T) ) ≥ 1-2δ_0,whereΛ_1(T)=D√(log(1/δ_0))T^-b/(2b+1) and Λ_2(T)=E√(log(T/δ_0))log(T)T^-b/(2b+1), denoting by D and E some constants depending only on (α_k, β_k, C_k)_1≤ k≤ K and C'. When T_k, t≥ N, denote by h_k, t=h_k(T_k, t) and C_k, t=C_k(T_k, t) the estimators based on the T_k, t observations for simplicity. ExtremeHunter's index B_k, t for arm k at time t, the optimistic proxy for 𝔼[G_n^(k)], can be then written asB_k, t=Γ(1-h_k, t-Λ_1(T_k, t)) ((C_k, t+Λ_2(T_k, t))n)^h_k, t+Λ_1(T_k, t),where Γ(x)=Γ(x) if x>0 and +∞ otherwise. On computational complexity. Notice that after the initialization phase, at each time t>KN, ExtremeHunter computes estimators h_I_t, t and C_I_t, t, each having a time complexity linear with the number of samples T_I_t, t pulled from arm I_t up to time t. Summing on the rounds reveals that ExtremeHunter's time complexity is quadratic with the time horizon n. §.§ ExtremeETC: a computationally appealing alternativeIn order to reduce the restrictive time complexity discussed previously, we now propose the ExtremeETC algorithm, an Explore-Then-Commit version of ExtremeHunter, which offers similar theoretical guarantees. After the initialization phase, the winner arm, which has maximal index B_k, KN, is fixed and is pulled in all remaining rounds. Then ExtremeETC's time complexity, due to the computation of h_k, KN and C_k, KN only, is 𝒪(KN)=𝒪((log n)^2(2b+1)/b), which is considerably faster than quadratic time achieved by ExtremeHunter. For clarity, Table <ref> summarizes time and memory complexities of both algorithms.Due to the significant gain of computational time, we used the ExtremeETC algorithm in our simulation study (Section <ref>) rather than ExtremeHunter. Controlling the number of suboptimal rounds. We introduce a high probability event that corresponds to the favorable situation where, at each round, all coefficients (1/α_k, C_k)_1≤ k ≤ K simultaneously belong to the confidence intervals recalled in the previous subsection. The event ξ_1 is the event on which the bounds |1/α_k-h_k(T)|≤Λ_1(T)|C_k-C_k(T)|≤Λ_2(T)hold true for any 1≤ k≤ K and N≤ T≤ n. The union bound combined with (<ref>) yieldsℙ(ξ_1)≥ 1-2Knδ_0. For n>Q_5, where Q_5 is the constant defined in (<ref>), ExtremeETC and ExtremeHunter always pull the optimal arm after the initialization phase on the event ξ_1. Hence, for any suboptimal arm k≠ k^∗, we have on ξ_1: T_k, n = N and thus T_k^∗, n = n - (K-1)N. Here we place ourselves on the event ξ_1. For any arm 1≤ k≤ K, Lemma 1 in <cit.> provides lower and upper bounds for B_k, t when T_k, t≥ N V_k ≤ B_k, t≤ V_k(1+Flog n √(log(n/δ_0))T_k, t^-b/(2b+1)), where F is a constant which depends only on (α_k, β_k, C_k)_1≤ k≤ K and C'.Introduce the horizon Q_5, which depends on (α_k, β_k,C_k)_1≤ k≤ K and C'Q_5 = max(e^(F√(1+ρ)A_0^-b/(2b+1))^2, Q_4).Then the following <ref>, proved in <ref>, tells us that for n large enough, the exploration made during the initialization phase is enough to find the optimal arm, with high probability. If n>Q_5, we have under the event ξ_1 that for any suboptimal arm k≠ k^∗ and any time t>KN that B_k, t<B_k^∗, t . Hence the optimal arm is pulled at any time t> KN. The following result immediately follows from <ref>. For n larger than some constant depending only on (α_k, β_k, C_k)_1≤ k≤ K and C' we have under ξ_1 T_k^∗, n≥ n/2.Upper bounding the expected extreme regret. The upper bound on the expected extreme regret stated in the theorem below improves upon that given in <cit.> for ExtremeHunter. It is also valid for ExtremeETC. For ExtremeETC and ExtremeHunter, the expected extreme regret is upper bounded as follows 𝔼[R_n] = 𝒪((log n)^2(2b+1)/bn^-(1-1/α_k^∗)+n^-(b-1/α_k^∗)),as n→ +∞. If b≥ 1, we have in particular 𝔼[R_n]= o(1) as n→ +∞. The proofof <ref> is deferred to<ref>.It closely follows that of Theorem 2 in <cit.>, the main difference being that their concentration bound (Theorem 1 therein) can be replaced by our tighter bound (see Theorem <ref> in the present paper). Recall that in Theorem 2 in <cit.>, the upper bound on the expected extreme regret for ExtremeHunter goes to infinity when n→ +∞:𝔼[R_n]= 𝒪(n^1/(1+b)α_k^∗). In contrast, in <ref> when b≥ 1, the upper bound obtained vanisheswhen n→ +∞. In the case b<1, the upper bound still improves upon <ref> by a factor n^(α_k^∗ b(b+1)-b)/((b+1)α_k^∗)>n^b^2/(2α_k^∗). § LOWER BOUND ON THE EXPECTED EXTREME REGRETIn this section we prove a lower bound on the expected extreme regret for ExtremeETC and ExtremeHunter in specific cases. We assume now that α_(2)>2α_k^∗^2/(α_k^∗-1) and we start with apreliminary result on second order Pareto distributions, proved in <ref>. If X is a r.v. drawn from a (α, β, C, C')-second order Pareto distribution and r is a strictly positive constant, the distribution of the r.v. X^r is a (α/r, β, C, C')-second order Pareto.In order to prove the lower bound on the expected extreme regret, we first establish that the event corresponding to the situation where the highest reward obtained by ExtremeETC and ExtremeHunter comes from the optimal arm k^∗ occurs with overwhelming probability. Precisely, we denote by ξ_2 the event such that the boundmax_k≠ k^∗max_1≤ i≤ NX_k,i≤max_1≤ i≤ n-(K-1)NX_k^∗,i.holds true. The following lemma, proved in <ref>, provides a control of its probability of occurence. For n larger than some constant depending only on (α_k, β_k, C_k)_1≤ k≤ K and C', the following assertions hold true. * We have: ℙ(ξ_2)≥ 1-Kδ_0,where δ_0 is given in <ref>. * Under the event ξ_0=ξ_1∩ξ_2, the maximum reward obtained by ExtremeETC and ExtremeHunter comes from the optimal arm: max_1≤ t≤ nX_I_t, t = max_1≤ i≤ n-(K-1)NX_k^∗, i . The followinglower bound shows that the upper bound(<ref>)is actually tight in the case b≥ 1. If b≥ 1 and α_(2)>2α_k^∗^2/(α_k^∗-1), the expected extreme regret of ExtremeETC and ExtremeHunter are lower bounded as follows 𝔼[R_n] = Ω((log n)^2(2b+1)/bn^-(1-1/α_k^∗)) . Here, π refers to either ExtremeETC or else ExtremeHunter. In order to bound frombelow 𝔼[R_n]=𝔼[G_n^(k^∗)]-𝔼[G_n^(π)], we start with bounding 𝔼[G_n^(π)] as follows 𝔼[G_n^(π)] = 𝔼[max_1≤ t≤ nX_I_t, t] = 𝔼[max_1≤ t≤ nX_I_t, t1{ξ_0}]+𝔼[max_1≤ t≤ nX_I_t, t1{ξ̅_0 }]≤ℙ(ξ_0)𝔼[max_1≤ t≤ nX_I_t, t |ξ_0] +∑_k=1^K𝔼[max_1≤ i≤ T_k, nX_k, i1{ξ̅_0 }], whereX_k, i has beendefined in <ref>. From (ii) in <ref>, we have 𝔼[max_1≤ t≤ nX_I_t, t | ξ_0] = 𝔼[max_1≤ i≤ n-(K-1)NX_k^∗, i |ξ_0] . In addition, in the sum of expectations on the right-hand-side of <ref>, T_k,n may be roughly bounded from above by n. A straightforward application ofHölder inequality yields ∑_k=1^K𝔼[max_1≤ i≤ T_k, nX_k, i1{ξ̅_0 }] ≤∑_k=1^K(𝔼[max_1≤ i≤ nX_k, i^α_k^∗+1/2])^2/α_k^∗+1ℙ(ξ̅_0 )^α_k^∗-1/α_k^∗+1 . From (i) in <ref> and <ref>, we have ℙ(ξ̅_0 )≤ K(2n+1)δ_0. By virtue of <ref>, the r.v. X_k, i^(α_k^∗+1)/2 follows a (2α_k/(α_k^∗+1), β_k, C_k, C')-second order Pareto distribution. Then, applying <ref> to the right-hand side of (<ref>) and using the identity (<ref>), the upper bound (<ref>) becomes 𝔼[G_n^(π)] ≤𝔼[max_1≤ i≤ n-(K-1)NX_k^∗, i1{ξ_0}]+ ∑_k=1^K((n C_k)^α_k^∗+1/2α_kΓ(1-α_k^∗+1/2α_k) +o(n^α_k^∗+1/2α_k))^2/α_k^∗+1(K(2n+1)δ_0)^α_k^∗-1/α_k^∗+1≤𝔼[max_1≤ i≤ n-(K-1)NX_k^∗, i] + 𝒪(n^-(1-1/α_k^∗)), where the last inequality comes from the definition of δ_0. Combining <ref> and(<ref>) we finally obtain the desired lower bound 𝔼[R_n] = 𝔼[G_n^(k^∗)]-𝔼[G_n^(π)]≥Γ(1-1/α_k^∗)C_k^∗^1/α_k^∗(n^1/α_k^∗-(n-(K-1)N)^1/α_k^∗) +𝒪(n^-(1-1/α_k^∗))= Γ(1-1/α_k^∗)C_k^∗^1/α_k^∗/α_k^∗(K-1)Nn^-(1-1/α_k^∗) +𝒪(n^-(1-1/α_k^∗)),where we used a Taylor expansion of x↦ (1+x)^1/α_k^∗ at zero for the last equality. § A REDUCTION TO CLASSICAL BANDITSThe goal of this section is to render explicit the connections between the max K-armed bandit considered in the present paper and a particular instance of the classical Multi-Armed Bandit (MAB) problem. §.§ MAB setting for extreme rewardsIn a situation where only the large rewards matter, an alternative to the max k-armed problem would be to consider the expected cumulative sum of the most `extreme' rewards, that is, those which exceeds a given high threshold u.For k∈{1, …,K} and t∈{1, …,n}, we denote by Y_k, t these new rewardsY_k, t=X_k, t1{X_k, t>u} .In this context, the classical MAB problemconsists in maximizing the expected cumulative gain𝔼[G^MAB] = 𝔼[ ∑_t=1^n Y_I_t,t]. It turns out that for a high enough threshold u, the unique optimal arm for this MAB problem, _1≤ k≤ K𝔼[Y_k, 1], is also the optimal arm k^∗ for the max k-armedproblem. We still assume second order Pareto distributions for the random variables X_k, t and that all the hypothesis listed in Section <ref> hold true. The rewards {Y_k, t}_1≤ k≤ K, 1≤ t≤ T are also heavy-tailed so that it is legitimate to attack this MAB problem with theRobust UCB algorithm <cit.>, which assumes that the rewards have finite moments of order 1+ϵmax_1≤ k≤ K𝔼[|Y_k, 1|^1+ϵ]≤ v,where ϵ∈(0, 1] and v>0 are known constants. Given our second order Pareto assumptions, it follows that <ref> holds with 1+ϵ<α_(1). Even if the knowledge ofsuch constants ϵ and v is a strong assumption, it is still fair to compare Robust UCB to ExtremeETC/Hunter, which also has strong requirements. Indeed, ExtremeETC/Hunter assumes that b and n are known and verify conditions depending on unknown problem parameters (e.g. n≥ Q_1, see <ref>).The following Lemma, whose the proof is postponed to <ref>, ensures that the two bandit problems are equivalent for high thresholds. If u>max(1, (2C'/min_1≤ k≤ KC_k)^1/min_1≤ k≤ Kβ_k, (3max_1≤ k≤ KC_k/min_1≤ k≤ KC_k)^1/α_(2)-α_(1)) , then the unique best arm for the MAB problem is _1≤ k≤ Kα_k=k^∗. Tuning the threshold u based on the data is a difficult question, outside our scope. A standard practice is to monitor a relevant output (e.g. estimate of α) as a function of the threshold u and to pick the latter as low as possible in the stability region of the output. This is related to the Lepski's method, see e.g. <cit.>, <cit.>, <cit.>.§.§ Robust UCB algorithm <cit.> For the sake of completeness, we recall below the main feature of Robust UCBand make explicit its theoretical guarantees in our setting. Thebound statedin the following proposition is a direct consequence of the regret analysis conducted by <cit.>. Applying the Robust UCB algorithm of <cit.> to our MAB problem, the expected number of times we pull any suboptimal arm k≠ k^∗ is upper bounded as follows 𝔼[T_k, n] = 𝒪(log n) . See proof of Proposition 1 in <cit.>.Hence, in expectation, Robust UCB pulls fewer times suboptimal arms than ExtremeETC/Hunter. Indeed with ExtremeETC/Hunter, T_k, n≥ N=Θ((log n)^2(2b+1)/b). <ref> may be an indication that the Robust UCB approach performs better than ExtremeETC/Hunter. Nevertheless, guarantees on its expected extreme regret require sharp concentration bounds on T_k, n (k≠ k^∗), which is out of the scope of this paper and left for future work. § NUMERICAL EXPERIMENTSIn order to illustrate some aspects of the theoretical results presented previously, we consider a time horizon n=10^5 with K=3 arms and exact Pareto distributions with parameters given in <ref>. Here, the optimal arm is the second one (incidentally, the distribution with highest mean is the first one).We have implemented Robust UCB with parameters ϵ=0.4, which satisfies 1+ϵ<α_2=1.5, v achieving the equality in <ref> (ideal case) and a threshold u equal to the lower bound in <ref> plus 1 to respect the strict inequality. ExtremeETC is runned with b=1<+∞=min_1≤ k≤ Kβ_k. In this setting, the most restrictive condition on the time horizon, n > KN ≈ 7000 (given by <ref>), is checked, which places us in the validity framework of ExtremeETC. The resulting strategies are compared to each other and to the random strategy pulling each arm uniformly at random, but not to Threshold Ascent algorithm <cit.> which is designed only for bounded rewards. Precisely, 1000 simulations have been run and Figure <ref> depicts the extreme regret (<ref>) in each setting averaged over these 1000 trajectories. These experiments empirically support the theoretical bounds in <ref>: the expected extreme regret of ExtremeETC converges to zero for large horizons. On the log-log scale (<ref>), ExtremeETC's extreme regret starts linearly decreasing after the initialization phase, at n > KN≈ 7000, which is consistent with <ref>. The corresponding linear regression reveals a slope ≈ -0.333 (with a coefficient of determination R^2≈ 0.97), which confirms <ref> and <ref> yielding the theoretical slope -(1-1/α_k^∗)=-1/3. § CONCLUSION This paper brings two main contributions. It first provides a refined regret bound analysis of the performance of the ExtremeHunter algorithm in the context of the max K-armed bandit problem that significantly improves upon the results obtained in the seminal contribution <cit.>, also proved to be valid for ExtremeETC, a computationally appealing alternative we introduce. In particular, the obtained upper bound on the regret converges to zero for large horizons and is shown to be tight when the tail of the rewards is sufficiently close to a Pareto tail (second order parameter b≥ 1).On the other hand, this paper offers a novel view of this approach, interpreted here as a specific version of a classical solution (Robust UCB) of the MAB problem, in the situation when only very large rewards matter.Based on these encouraging results, several lines of further research can be sketched. In particular, future work will investigate to which extent the lower bound established for ExtremeETC/Hunter holds true for any strategy with exploration stage of the same duration, and whetherimprovedperformance is achievable with alternative stopping criteria for the exploration stage.§ ACKNOWLEDGMENTSThis work was supported by a public grant (Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH) and by the industrial chair Machine Learning for Big Data from Télécom ParisTech.apalike§ APPENDIX §.§ Proof of <ref>For n>Q_3 (defined in <ref>), one has V_k^∗>2max_k≠ k^∗V_k, which implies that max_k≠ k^∗V_k/(V_k^∗-V_k) < 1. Hencemax_k≠ k^∗e^(F√(1+ρ)A_0^-b/(2b+1)V_k/V_k^∗-V_k)^2<e^(F√(1+ρ)A_0^-b/(2b+1))^2≤ Q_5 .Then, as T_k, t≥ N and by definitions of N (<ref>) and δ_0 (<ref>), we have for n>Q_5 that for any suboptimal arm k≠ k^∗(C_k n)^1/α_kΓ(1-1/α_k) (1+Flog n √(log(n/δ_0))T_k, t^-b/(2b+1))<(C_ k^∗ n)^1/α_k^∗Γ(1-1/α_k^∗),which implies, using <ref>, that under ξ_1: B_k, t<B_k^∗, t for t>KN.§.§ Proof of <ref> We want to upper bound 𝔼[R_n]=𝔼[G_n^(k^∗)]-𝔼[G_n^(π)]. To do so, we lower bound 𝔼[G_n^(π)] as follows 𝔼[G_n^(π)] =𝔼[max_t≤ nX_I_t, t] ≥𝔼[max_{t≤ n, I_t=k^∗}X_I_t, t]=𝔼[max_{i≤ T_k^∗, n}X_k^∗, i]. Thus 𝔼[G_n^(π)] ≥𝔼[max_i≤ n-(K-1)NX_k^∗, i1{ξ_1}], where we used that under ξ_1, T_k^∗, n=n-(K-1)N. Now we call the following result (<ref>, proved in <ref>), giving a lower bound on the expected maximum of i.i.d. second order Pareto r.v. given some event. Let X_1, ..., X_T be i.i.d. samples from an (α, β, C, C')-second order Pareto distribution. Let ξ be an event of probability larger than 1-δ. If δ<1/2 and T≥max(4c, (4c)^1/βlog(2)C(2C')^1/β, 8log^2(2)) for a given constant c depending only on β, C and C', we have 𝔼[max_1≤ i≤ TX_i1{ξ}] ≥ (TC)^1/αΓ(1-1/α) -(4+8/α-1)(TC)^1/αδ'^1-1/α-2(4D_2C^1/α/T^1-1/α+2C'D_β+1/C^β+1-1/αT^β-1/α +2(2C'T)^1/(α(1+β))e^-HT^β/(β+1)). Then, applying <ref> with ξ=ξ_1 and δ=δ_0 we obtain after simplification 𝔼[R_n] ≤ H'n^1/α_k^∗{ 1/n+1/n^b+K/n(log n)^2(2b+1)/b +δ_0^1-1/α_k^∗+n^1/(α_k^∗(1+β_k^∗))e^-H_k^∗(n/2)^β/(β+1)}, where H_k^∗=1/2C_k^∗(2C')^1/(α_k^∗(1+β_k^∗)) and H' is a constant depending only on (α_k, β_k, C_k)_1≤ k≤ K and C'. The definition of δ_0 concludes the proof. §.§ Proof of <ref>We follow the proof of Lemma 2 in <cit.> except that we use <ref> instead of their Theorem 1. Let x_δ be such that ℙ(max_1≤ t≤ TX_t≤ x_δ)=1-δ. Then we have𝔼[max_1≤ t≤ TX_t1{ξ}] = 𝔼[max_1≤ t≤ TX_t]-𝔼[max_1≤ t≤ TX_t1{ξ̅}] = 𝔼[max_1≤ t≤ TX_t] - ∫_0^x_δℙ(max_1≤ t≤ TX_t1{ξ̅}>x) dx- ∫_x_δ^∞ℙ(max_1≤ t≤ TX_t1{ξ̅}>x) dx≥𝔼[max_1≤ t≤ TX_t] - δ x_δ - ∫_x_δ^∞ℙ(max_1≤ t≤ TX_t1{ξ̅}>x) dx,where the inequality comes from ℙ(max_1≤ t≤ TX_t1{ξ̅}>x)≤ℙ(ξ̅)≤δ. Since T≥log(2)max(C(2C')^1/β, 8log(2)) and δ<1/2, we have from Lemma 3 in <cit.>|ℙ(max_1≤ i≤ TX_i ≤ (TC/log(1/(1-δ)))^1/α)-(1-δ)|≤ (1-δ)(4/T(log1/1-δ)^2 +2C'/C^1+β(log1/1-δ)^1+β)≤4/T(2δ)^2+2C'/C^1+β(2δ)^1+β≤ cδmax(δ/T, δ^β/T^β) ≤ cδmax(1/T, 1/T^β),where c is a constant that depends only on C, C' and β. As we have cmax(T^-1, T^-β)≤ 1/4, this impliesx_-=(TC/log(1/(1-2δ)))^1/α≤ x_δ≤ (TC/log(1/(1-δ/2)))^1/α= x_+ .It follows𝔼[max_1≤ t≤ TX_t1{ξ}] ≥𝔼[max_1≤ t≤ TX_t] - δ x_+ - ∫_x_-^∞ℙ(max_1≤ t≤ TX_t>x) dx .From <ref> we deduce𝔼[max_1≤ t≤ TX_t1{ξ}] ≥𝔼[max_1≤ t≤ TX_t] - δ x_+ - ∫_x_-^∞ (1-e^-TCx^-α) dx-(4D_2C^1/α/T^1-1/α+2C'D_β+1/C^β+1-1/αT^β-1/α +2(2C'T)^1/(α(1+β))e^-HT^β/(β+1)).From the proof of Lemma 2 in <cit.> we have for δ small enough∫_x_-^∞ (1-e^-TCx^-α) dx ≤8/α-1(TC)^1/αδ^1-1/αandδ x_+≤ 4(TC)^1/αδ^1-1/α .<ref> concludes the proof. §.§ Proof of <ref> Let F and F_r be respectively the cumulative distribution functions of X and X^r. For x≥ 0, F_r(x)=ℙ(X^r≤ x) = ℙ(X≤ x^1/r) = F(x^1/r) . As X follows an (α, β, C, C')-second order Pareto distribution we have |1-Cx^-α/r-F_r(x)| =|1-Cx^-α/r-F(x^1/r)| ≤ C'x^-(α/r)(1+β), which concludes the proof. §.§ Proof of <ref> We first state the following result (<ref>, proved in <ref>), yielding high probability lower and upper bounds for the maximum of i.i.d. second order Pareto r.v. For X_1, ..., X_T i.i.d samples drawn from an (α,β,C,C')-second-order Pareto distribution we define high probability lower and upper bound ℓ(T,δ)= (TC/2log1/δ)^1/αand L(T,δ)= (4TC/log1/1-δ)^1/α, where δ∈(0,1) can depend on T and is such that lim_T→∞ℓ(T,δ)=∞ and lim_T→∞L(T,δ)=∞. For T large enough such that Cℓ(T,δ)^-α≥ 2C'ℓ(T,δ)^-α(1+β), CL(T,δ)^-α≥ C'L(T,δ)^-α(1+β) and L(T,δ)^-α≤1/4C we have ℙ(max_1≤ i≤ TX_i≤ℓ(T,δ))≤δandℙ(max_1≤ i≤ TX_i≥ L(T,δ))≤δ. With the notations of <ref>, we respectively denote by ℓ_k and L_k the high probability lower and upper bounds for any arm k. Using <ref> we have by a union bound that with probability higher than 1-Kδ_0 max_1≤ i≤ n-(K-1)NX_k^∗,i≥ℓ_k^∗(n-(K-1)N,δ_0), and for any suboptimal arm k≠ k^∗ max_1≤ i≤ NX_k,i≤ L_k(N, δ_0) . Under this event, using the definition of the confidence level δ_0 we observe for n larger than some constant that for any suboptimal arm k≠ k^∗, L_k(N,δ_0)≤ℓ_k^∗(n-(K-1)N,δ_0), which concludes the proof.§.§ Proof of <ref>For the high probability lower bound we write:ℙ(max_1≤ i≤ TX_i≤ℓ(T,δ)) =ℙ(X_1≤ℓ(T,δ))^T≤(1-Cℓ(T,δ)^-α+C'ℓ(T,δ)^-α(1+β))^T≤(1-1/2TCℓ(T,δ)^-α)^T ≤ e^-1/2TCℓ(T,δ)^-α =δ.And for the high probability upper bound:ℙ(max_1≤ i≤ TX_i≤ L(T,δ)) =ℙ(X_1≤ L(T,δ))^T≥(1-CL(T,δ)^-α-C'L(T,δ)^-α(1+β))^T≥ (1-2CL(T,δ)^-α)^T ≥ e^-4TCL(T,δ)^-α =1-δ .§.§ Proof of <ref> From <ref>, we have for any arm k∈{1, …,K}, 𝔼[X_k, 11{X_k, 1>u}] ≤∫^∞_0 ℙ(X_k, 11{X_k, 1>u}≥ x)dx= u(1-F_k(u)) + ∫^∞_u (1-F_k(x))dx ≤ M_k + Δ_k, where M_k=(C_kα_k/(α_k-1))u^-α_k + 1 and Δ_k=(C'α_k(1+β_k)/(α_k(1+β_k)-1))u^-α_k(1+β_k)+1. Similarly, we have 𝔼[X_k, 11{X_k, 1>u}] ≥ M_k-Δ_k. For u large enough, we want to prove that M_k^∗-Δ_k^∗>M_k+Δ_k for any arm k≠ k^∗, which would prove that _1≤ k≤ K𝔼[Y_k, 1]=k^∗. First, we observe for u>max(1, (2C'/min_1≤ k≤ KC_k)^1/min_1≤ k≤ Kβ_k) that Δ_k < 1/2M_k. Then, for u>(3max_1≤ k≤ KC_k/min_1≤ k≤ KC_k)^1/(α_(2)-α_(1)), we have that 1/2M_k^∗>3/2M_k for any arm k≠ k^∗, which concludes the proof. | http://arxiv.org/abs/1707.08820v1 | {
"authors": [
"Mastane Achab",
"Stephan Clémençon",
"Aurélien Garivier",
"Anne Sabourin",
"Claire Vernade"
],
"categories": [
"stat.ML",
"cs.LG"
],
"primary_category": "stat.ML",
"published": "20170727112655",
"title": "Max K-armed bandit: On the ExtremeHunter algorithm and beyond"
} |
[email protected] [email protected] School of Physics and Astronomy, Rochester Institute of Technology, Rochester, New York 14623, USA.05.40.Jc, 87.16.aj, 87.16.dp, 87.16.KaInside cells, cargos such as vesicles and organelles are transported by molecular motors to their correct locations via active motion on cytoskeletal tracks and passive, Brownian diffusion. During the transportation of cargos, motor-cargo complexes (MCC) navigate the confining and crowded environment of the cytoskeletal network and other macromolecules. Motivated by this, we study a minimal two-state model of motor-driven cargo transport in confinement and predict transport properties that can be tested in experiments. We assume that the motion of the MCC is directly affected by the entropic barrier due to confinement if it is in the passive, unbound state, but not in the active, bound state where it moves with a constant bound velocity. We construct a lattice model based on a Fokker Planck description of the two-state system, study it using a kinetic Monte Carlo method and compare our numerical results with analytical expressions for a mean field limit. We find that the effect of confinement strongly depends on the bound velocity and the binding kinetics of the MCC. Confinement effectively reduces the effective diffusivity and average velocity, except when it results in an enhanced average binding rate and thereby leads to a larger average velocity than when unconfined. Active and Passive Transport of Cargo in a Corrugated Channel: A Lattice Model Study Moumita Das December 30, 2023 ==================================================================================== § INTRODUCTION Intracellular transport of cargos by molecular motors is critical to development, maintenance, and homeostasis in most eukaryotic cells <cit.>. There exist several types of motors that use ATP, the energy currency of the cell, to move cargo through the cell using cytoskeletal tracks. The motors kinesin-1 and cytoplasmic dynein transport cargo using microtubules <cit.> while myosin-5 and -6 do so via actin filaments <cit.>. Some motors have direction bias; some carry larger, and some smaller cargos. Examples of intracellular cargo include organelles such as mitochondria, and dysfunctional or damaged protein aggregates that occur in disease states <cit.>. While the transport of the former is essential to proper functioning of the cell, the latter need to be cleared out of the cell to prevent cell damage and disease progression. Understanding the mechanistic principles underlying intracellular cargo transport will provide insights into the proper functioning of cells, and aid in the creation of new drugs or agents to help regain function in disease states. Microtubules and actin filaments which provide the pathway for the motors to walk during intracellular transport, are semiflexible biopolymers that are found throughout the cell interior <cit.>. Over the past two decades, there have been many studies, both experimental and theoretical, on cargo transport by motors on single microtubules in-vitro. The speed and travel distance of molecular motors on surface-immobilized microtubules is very well understood <cit.>; examples include the molecular motor kinesin pulling fluid membranes on a microtubule <cit.>, multiple motors transporting a single cargo <cit.>, and motors carrying cargos to multiple targets in neurons <cit.>. Within cells, however, microtubules and actin rarely exist as individual filaments. Instead, they are found as networks of filaments and have very interesting mechanical structure-function properties. Despite decades of studies of motors moving and carrying cargo on single cytoskeletal tracks, cargo transport in complex and dynamic architectures in cells is not well understood. In fact, experimental <cit.> and theoretical studies <cit.> have only recently begun to investigate how intracellular transport is affected by the physical properties of the cytoskeletal network and crowded cellular environments. Notable experimental studies in this area include in-vitro experiments that have studied how crowding of motors <cit.> and organization of microtubules within bundles <cit.> affect the efficiency of cargo transport. In particular, in <cit.> Conway et al. found that the motion of the cargo being transported is inhibited in a bundle of randomly oriented, closely packed microtubules. Several theoretical models with active and passive transport have also investigated the collective transport properties and the spatial organizations of motors and cargos on single microtubules <cit.> and inhomogeneous cytoskeletal networks <cit.>. Despite these advances, there remain many open questions.Here we ask: How does confinement due to the cytoskeletal network affect motor-driven cargo transport?We address this question by developing a minimal two-state model that describes cargo transport in the presence of confinement.The two states are: (i) an active state when the motor-cargo complex (MCC) is attached to the microtubule and moves with a constant speed, and (ii) a passive state when it is unattached and undergoes diffusive motion.Such two-state models have been useful in elucidating active transport of Brownian particles in confined geometries <cit.>, specifically how the cooperative rectification between geometric constraints and Brownian ratchets impacts net particle motion. The interplay between passive and active transport and confinement, as is common in intracellular transport, however, remains poorly understood. In this paper, we combine a Fokker Planck description with a lattice model framework to study how confinement, and motor dynamics and binding kinetics interact to modify directed transport of cargos by motors. The paper is organized as follows. We write down the Fokker Planck Equations (FPE) for the two-state model and propose a lattice model that can capture the physics described by the FPE and reduces to the FPE in the continuum limit. We simulate the lattice model using a kinetic Monte Carlo method, and show that it reproduces known analytical results for passive (diffusive) transport in confinement. Thereafter we investigate the full two state problem in confinement, and calculate transport properties such as mean squared displacement (MSD), average velocity, and effective diffusivity for the MCC, and discuss the implications of our results.We want to note that, in this paper, we use the term “active” to refer to the driven motion of cargo fueled by ATP-hydrolysis of kinesin motors via a “Brownian ratchet” mechanism <cit.>, and implemented as constant velocity motion of an MCC on a microtubule (see Sec. <ref>). It should not be confused with self-propelled motion in soft matter literature. The confinement effect on the motion of a self-peopelled particle studied in <cit.>.§ MODEL AND METHOD We model and study the active and passive transport of cargo by a motor moving unidirectionally on a microtubule track and confined in a corrugated channel as schematically shown in Fig. <ref>.The motor can move micrometer-long distances along the microtubule before detaching. Kinesin motors, a well-characterized family of motor proteins thatmove organelles (e.g. mitochondria) and macromolecules (e.g. RNA) in many cell types <cit.> are good examples of such motors. While the confinement faced by an MCC in a live cell is heterogeneous and dynamic, for simplicity, we consider an effective confining channel described by w(x) = a sin(2π x/L) + b, where L is the periodicity, a and b control the effective width of the channel, and the effective bottleneck width is given by 2(b-a) <cit.>. The effective channel width and periodicity are set by the length scales associated with localized cages and network mesh sizes.The main ingredients of the model are as follows: * Two State Transport: Over long time scales, the MCC alternates between two states: (i) An active state which phenomenologically represents the MCC being bound to a microtubule and moving along the microtubule in the forward direction with a speed v_b, and (ii) an passive state where the MCC is detached from the microtubule and undergoes overdamped Brownian motion in the viscus medium of the cytoplasm with a free diffusion constant D_0.* Confinement: The MCC only encounters physical confinement due to the channel walls while undergoing diffusive motion in a viscous medium, i.e. in the passive state. Our model, therefore, incorporates an explicit, microscopic description of the physical confinement due to cytoskeletal networks. This is in contrast to a more coarse-grained, continuum description where interior of the cell is treated as a viscoelastic medium <cit.>. The channel is symmetric, and hence we do not expect confinement-induced symmetry-breaking for purely diffusive motion as in some Brownian Ratchet models <cit.>.* Binding Kinetics: The binding rate, k_ on(x), of the motor can be constant or it can vary inversely with the width of the confining channel in Fig. <ref>. The latter represents the case where tighter confinement leads to greater likelihood of the motor attaching to the microtubule due to increased proximity to binding sites. The unbinding rate, k_ off(x), of the motor is assumed to be constant.We note that the actual width of the confining channel is larger than the size of the MCC. In our formulation, w(x) is the effective channel width (depicted by the dashed curved lines in Fig. <ref>), and the MCC can be thought of as a point particle, with the actual size of the MCC incorporated through the free diffusivity. With this description, our model is valid for a wide range of cargo sizes as long as the bound state is unaffected by the confining channel.The model predicts three distinct regimes, a diffusive regime at early times, an intermediate sub-diffusive regime, and a ballistic regime at large times, as discussed in detail in the Results section. The timescale for the crossover from diffusive to sub-diffusive motion is set by the diffusion time of the MCC in the corrugated channel before it starts to experience the impact of confinement, while the timescale for the crossover to the ballistic regime is set by the interplay of the binding kinetics and the driven motion of the MCC when bound to the microtubule. §.§ Fokker Planck Description In our model, the motion of the MCC is directly affected by the confining wall of the channel when it is undergoing Brownian motion in the unbound (passive) state.First, let us discuss the motion of an overdamped Brownian particle in a 2D confining channel, with the channel axis along the x direction. The 2D motion inside the channel can be described by a 1D Fokker-Planck equation, known as the Fick-Jacobs equation <cit.>,∂ P(x,t)/∂ t =D_0 ∂/∂ x( e^-β𝒜(x)∂/∂ x e^β𝒜(x)P(x,t) ).Here, P(x,t) represents the probability density at a given position x along the direction of the channel at time t,D_0 is the diffusion coefficient in the absence of confinement, and β=1/k_BT, where k_B is the Boltzmann constant and T is the temperature. To derive the above equation, rapid equilibration is assumed in the transverse direction of the channel. This implies that the time scale for longitudinal (axial) motion is very large compared to the equilibration time scale in the transverse direction. Under this assumption, one can successfully integrate out the transverse variable and recast the two-dimensional motion into the above Fick-Jacobs equation <cit.>. The confinement is incorporated though an effective free energy 𝒜(x)=𝒱(x) - T S(x), where S(x) is the entropy barrier due to confinement and 𝒱(x) is an external energy barrier. The entropy due to the confining wall w(x) is S(x) =k_B log(2 w(x)/w_ave), where w_ave=2∫_0^L w(x) dx is the average width of the channel. In the absence of any external potential, the free energy is purely entropic, 𝒜(x)=-k_B T log(2 w(x)/w_ave). Now, we return to the problem of two-state transport. The MCC walks with a velocity v_b when it is bound, diffuses with a free diffusion constant D_0 when unbound, and alternates between the two states with rates k_ off(x) and k_ on(x), respectively. The Fokker-Planck equation for the probability densities for the bound state P_ b (x,t) and unbound state P_ ub(x,t) are given by:∂ P_ b(x,t)/∂ t =k_ on(x) P_ ub(x,t) - k_ off(x) P_ b(x,t) -v_b ∂ P_ b(x,t)/∂ x,∂ P_ ub(x,t)/∂ t =-k_ on(x) P_ ub(x,t) + k_ off(x) P_ b(x,t) + D_0 ∂/∂ x(e^-β𝒜(x)∂/∂ x e^β𝒜(x)P_ ub(x,t)).The first two terms in Eqs. <ref> and <ref> correspond to binding and unbinding transitions respectively. The third term in Eq. <ref> represents active motion of the MCC, while the third term in Eq. <ref> describes passive motion of the MCC under confinement. Given kinesin is a highly processive motor and can take over a hundred steps along a microtubule before dissociating <cit.>, we neglect diffusion in the active state.However, one can easily incorporate diffusive behavior for other motor types by adding a diffusion term in Eq. <ref>.The analytical solutions of Eq. <ref> are difficult, and have closed form expressions only in the passive limit <cit.> and in a mean field limit for two-state transport discussed later in the paper. We, therefore, construct the corresponding lattice model which reduces to Eq. <ref> in the continuum limit and evolve the system using a kinetic Monte Carlo method as discussed below.§.§ Lattice Model We study the dynamics of a two-state MCC described by the continuum Fokker-Plank equation (Eq. <ref>) using an equivalent lattice model. The model is schematically shown in Fig. <ref>, and consists of an MCC on a one-dimensional lattice. The MCC can switch between a bound and an unbound state. The bound MCC can further hop to its forward neighboring site while the unbound MCC can hop to both its backward and forward neighboring sites. The spacing between neighboring lattice sites is ℓ.Consider that the MCC is at the lattice site at position x at time t in a particular state. The transition rates from the unbound state to bound state is k_ on(x) and from the bound state to unbound state is k_ off(x). The MCC in the bound state can either hop to its forward neighbor (x+ℓ) with rate λ_v(x), or it can switch to the unbound state with rate k_ off(x). The MCC in the unbound state can hop either to its forward neighbor (x+ℓ) with rateλ^ ub_+(x), backward neighbor (x-ℓ) with rateλ^ ub_-(x), or switch to the bound state with rate k_ on(x). The master equations describing the time evolution of the probability densities for the bound state P_ b(x,t) and the unbound state P_ ub(x,t) for this process are ∂ P_ b(x,t)/∂ t =k_ on(x) P_ ub (x, t) - k_ off(x) P_ b (x, t) + λ_v(x-ℓ) P_ b (x - ℓ, t) - λ_v(x) P_ b (x, t), ∂ P_ ub(x,t)/∂ t =-k_ on(x) P_ ub (x, t) + k_ off(x) P_ b (x, t) + λ^ ub_+ (x - ℓ) P_ ub (x - ℓ, t)+ λ^ ub_-(x + ℓ) P_ ub (x + ℓ, t) - (λ^ ub_+(x) + λ^ ub_-(x)) P_ ub (x, t). As the bound velocity v_b in our model is independent of position, the bound state hopping rate λ_v(x) is also position independent and is given by λ_v(x) = v_b/ℓ. We incorporate the effect of confinement using position dependent hopping rates λ^ ub_±(x) for the unbound state. The hopping rates λ^ ub_±(x) depend on the free-energy 𝒜(x) which has a contribution from the entropic barrier due to confinement. In the presence of an external potential, it also has an energy contribution. The hopping rates are given by λ^ ub_±(x) = (D_0/ℓ^2) e^-β(𝒜(x±ℓ) -𝒜(x))/2. The factor 1/2 in the exponent ensures local detailed balance condition. With these choice of rates, for ℓ→ 0, the Eq. <ref> reduces to Eq. <ref> (see Appendix. <ref> for details).We use a kinetic Monte Carlo algorithm to evolve the system. For the MCC at the lattice site x at time t, we choose an event out of all possible events at random with a probability proportional to its rate, and increase the time by δ t = 1/Γ(x), where Γ(x) is the total rate. For the bound MCC, the event space consists of a forward hopping event with probability λ_v/Γ, and a transition to the unbound state with probability k_ off(x)/Γ, where the total rate in the bound state Γ=k_ off(x) + λ_v(x).For the unbound MCC, the event space consists of a hopping event in the forward direction, a hopping event in the backward direction, and a transition to the bound state, with probabilities λ^ ub_+(x)/Γ, λ^ ub_-(x)/Γ, and k_ on(x)/Γ respectively, where the total rate in the bound state is Γ=k_ on(x) + λ^ ub_+(x) + λ^ ub_-(x). For simplicity and efficiency, the mean of the exponential distribution Γ (x) exp(-δ tΓ (x)) is used as the time step in our simulations.Although a time step drawn at random from the exponential distribution would have been more appropriate, we have checked that the choice of the mean does not change any of our results, while it makes the simulation more efficient. This was also verified by one of the authors in a study of a lattice model for ballistic aggregation in <cit.>. It has been shown that the introduction of a position dependent diffusivity, D(x)=D_0/(1+w^'(x)^2)^α (with α=1/3 for 2D and =1/2 for 3D), increases the numerical accuracy considerably for larger amplitude w(x) <cit.>. The qualitative behavior of the results do not change if constant diffusivity D_0 is considered <cit.>. Here, for simplicity, we study our lattice model with constant diffusivity. However, it can be easily extended to incorporate x-dependent diffusivity by choosing λ^ ub_±(x) = (D(x)/ℓ^2) e^-β(𝒜(x±ℓ) -𝒜(x))/2.§.§ Simulation Details and Parameters Throughout this study, the lengthscales associated with the corrugated channel are taken to be L=1μ m, a=1/(2π) μ m, and b=1.02/(2π) μ m. Our choice of effective widths a and b implies fairly strong confinement (effective bottleneck width =2(b-a)=0.02/(2π) μ m). The parameter values for the two-state motion (k_ on, k_ off, D_0, v_b, and ℓ) of the MCC are informed by experiments on kinesin motors carrying cargos or pulling membranes <cit.>.The value of the step size or lattice spacing ℓ=8 nm. The simulations are performed with the free diffusion constant of the unbound MCC D_0=0.64μ m^2s^-1 and the off-rate k_ off= 0.42s^-1, unless otherwise specified. To explore extended parameter space, the on-rate and bound velocity are varied over wide ranges, k_ on= 0.05-50 s^-1 andv_b=0.04-1.6 μ m s^-1. The experimental values of k_ on (∼ 4.7s^-1) and v_b (∼0.8 μ m s^-1) lie well within the range.For the binding (on) rate, we study two cases: (i) k_ on=k_ on^0, and (ii) k_ on(x)∝ k_ on^0/(w(x)). In the latter case, we further investigate two situations – when the spatial average of k_ on, in the interval L is k_ on^0, to allow for comparison with (i), and when it is greater than k_ on^0. The simulations are performed with open boundary condition, meaning that the channel can be thought of as extending to infinity in both directions. All the data presented in this paper are averaged over 25000 or more realizations. § RESULTS We characterize the motor-driven cargo transport in our model by the mean squared displacement, ⟨δ x^2(t) ⟩=⟨ (x(t+t_0)-x(t_0))^2 ⟩,the average velocity, ⟨ẋ⟩, and the effective diffusivity, D_ eff of the MCC. The last two quantities are defined in the asymptotic limit as ⟨ẋ⟩ = lim_t→∞⟨ (x(t) -x(0) ⟩/t and D_ eff = lim_t→∞⟨ x^2(t) ⟩ -⟨ x(t)⟩^2/2 t, where x(t) is the position of the particle at time t and ⟨·⟩ represents ensemble averages <cit.>. This definition of effective diffusivity allows for a more accurate estimate than inferring it from the MSD.While the MSD may or may not grow linearly with time depending on context, the fluctuations around the mean position of the MCC in our systems grow linearly with time at large times and therefore D_ eff is independent of time. §.§ Passive Transport in a Confining ChannelWe first study the passive, diffusive transport of a particle in confinement usingkinetic Monte Carlo simulations of the lattice model. We demonstrate that the lattice model correctly incorporates hopping rates through the entropic barrier dependent free energy, and discuss properties which will be used to compare and understand the results of two-state transport in the next section. In this case, a particle at the lattice site x can hop to one of its neighboring sites (x±ℓ) with rate λ_± = (D_0/ℓ^2)e^-β(𝒜(x±ℓ) -𝒜(x))/2. For Brownian motion under constant force F, 𝒜(x)=-F x -T S(x), which givesλ_±(x) = (D_0/ℓ^2)e^±β a F/2√(w(x±ℓ)/w(x)). In the absence of any confinement, the rates of hopping then become, λ_±(x) = (D_0/ℓ^2) e^±βℓ F/2, i.e. independent of x. Confinement makes the hopping probabilities x dependent, which are given by λ_±(x)/Γ(x), where Γ(x) = λ_+(x) +λ_-(x).In Fig. <ref>(a) and (b), we present the lattice model results for the scaled mobility μ_ eff and scaled diffusion coefficient D_ eff and compare them with the corresponding analytical predictions. The latter are obtained by solving the Fick-Jacobs equation (Eq. <ref>) for a particle undergoing2D overdamped Brownian motion under an external driving force F in a corrugated channel w(x) with periodicity L <cit.> and given by μ_ eff := ⟨ẋ⟩/F = D_0/k_BT(1-e^-f)/∫_0^Ldx/L I(x,f) f^-1,and D_ eff/D_0 = ∫_0^L dx/L∫_x-L^x dz/L e^𝒜(x)/k_BT/ e^𝒜(z)/k_BT I^2(z,f) ×[ ∫_0^L I(x,f) dx/L]^-3, whereI(x,f) :=e^𝒜(x)/k_BT∫_x-L^x dy/L e^-𝒜(y)/k_BT depends on the dimensionless force (or Peclet number) f:=FL/k_BT. Please note that the above expressions are nonlinear in f. In the absence of any geometric confinement, the effective mobility and diffusion coefficient reduce to μ_ eff=D_0/k_BT=μ_0 and D_ eff=D_0, by substituting 𝒜(x)=-Fx in the Eq. <ref>. The numerical results are in very good agreement with the analytical predictions, demonstrating that our lattice model is an accurate representation of the Fick-Jacobs equation (Eq.<ref>), and suggesting that this method can be used to studya wide range of systems with entropic barriers.We now discuss the results shown in Fig. <ref> in more detail. The scaled mobility shown in Fig. <ref>(a) is always less than 1,approaching 1 asymptotically as f is increased. This suggests that a symmetric confinement without any rectification mechanism cannot enhance the mobility of a purely diffusive system. The behavior of the scaled effective diffusivity D_ eff/D_0 in Fig. <ref>(b) is non-monotonic with a peak at a critical value of f, suggesting that while at small f confinement causes the effective diffusivity to decrease, at large f the interplay of the force and confinement leads to enhanced diffusivity. The value of the critical force depends on the modulation of the confining wall as discussed in ref <cit.>. Confinement constrains diffusive motion of the particle by reducing the available space for movement; for f=0, it leads to a significant decrease in the MSD compared to the unconfined case at intermediate and large times as seen in Fig. <ref>(c); the particle's motion changes from free diffusion with D_0 at early times to an intermediate sub-diffusive regime, and finally to effective diffusive behavior with D_ eff<D_0 at large times. The intermediate sub-diffusive regime presumably emerges due to the slowing down of motion near the neck of the channel. For large Peclet numbers f>>1, (Fig. <ref> (d)), where the particle motion is largely driven rather than diffusive, confinement, has much smaller impact compared to (c). The motion changes from free diffusion at very early times to force-driven ballistic motion at large times, and as in (c) there is a visible slowing down in at intermediate times for the confined case, but the gap between the two asymptotic MSDs is much smaller. As we will see in the next section, the intermediate slowing down due to confinement plays a critical role in two-state cargo transport. §.§ Active and Passive Transport in a Confining ChannelWe now discuss the transport properties, namely MSD, average velocity, and effective diffusivity of the MCC for the two state model with confinement, and compare them with the results for the unconfined case.Where appropriate, we also compare our results with corresponding steady state values in the mean field limit without any confinement. The following sections we refer to this limit as the Mean Field No Confinement (MFNC) limit.In this limit, the probabilities of bound and unbound states are given by P̃_b=k_ on/(k_ on+k_ off) and P̃_ub=k_ off/(k_ on+k_ off), respectively, and the average velocity and effective diffusivity can be written as,V_l= P̃_b v_b = k_ onv_b/k_ on + k_ off,D_l= P̃_ub D_0 + D_act,ub = k_ offD_0/k_ on + k_ off + k_ on k_ off v_b^2/(k_ on + k_ off)^3. In Eq. <ref>, the velocity V_l does not depend on the probability of the unbound state since there is no net directed movement during passive motion. In Eq. <ref>, the first term is due to the diffusive motion in the unbound state and the second term accounts for an additional contribution due to the stochastic transition between active and passive motion. The analytical expression for the latter can be found in <cit.>. Maximizing this equation provides the condition for the occurrence of the peaks in effective diffusivity as well as their positions, as observed in (Fig. <ref>(a)). The above mean field analytical expressions are exact for the unconfined case if the transition rates and the diffusivity of the motion are independent of position. As the confinement makes the diffusivity position dependent (through position dependent hopping rates), these expressions are no longer expected to hold true for the confined case. However, comparing results with these expressions are very useful for understanding the role of the confinement.Mean Squared Displacements (MSD): In Fig. <ref>, we present the MSD for small and large bound velocities v_b for two different binding ratesk_ on. At very small time t, the MSD behaves as ⟨δ x^2(t) ⟩≃ 2 D_0 t. This behavior suggests that below a crossover time scale, say t_c1, the effect of binding/unbinding kinetics and confinement are negligible such that the MCC can diffuse freely. For t>t_c1, it shows a transition from free diffusion to “sub-diffusion”. In the absence of confinement, the sub-diffusive behavior is due to the time spent by the MCC alternately transitioning between bound and unbound states, and the crossover time scale, t_c1, mainly depends on the transition rates. In the confined case, in addition to transition events, the motion of the MCC slows down even further due to the strong entropic barrier in the diffusive state close to a neck of the channel. Consequently, the crossover time t_c1 becomes even smaller for the confined case than that without any confinement. There is a second crossover from sub-diffusive to ballistic behavior i.e., for t>t_c2, ⟨δ x^2(t) ⟩≃⟨ẋ⟩^2 t^2.A representative case of the locations of t_c1 and t_c2 are shown in Fig. <ref> (a). From the Fig. <ref>, it is clear that confinement impacts the motion of the MCC more strongly for small v_b – it spends longer times in the intermediate sub-diffusive regime leading to smaller asymptotic velocities at large times compared to the unconfined case.For larger v_b, the sub-diffusion regime shrinks and the MCC has the same asymptotic velocities with and without confinement. The observed MSDs in Fig. <ref> (c) and (d) are similar to (a) and (b) respectively, but with the crossover to ballistic motion occurring at much larger times (t_c2).The motion of the MCC and how its MSD will scale with time is determined by the interplay of four timescales: the inverse of the binding rate k_ on, the inverse of the unbinding rate k_ off, the diffusion time L̃^2/D_0, and the drift timescale L̃/v_b, where L̃ is the characteristic length scale in the system and is equal to L in the confined case. The precise dependence of t_c1 and t_c2 on these timescales is nontrivial and will be studied in future work. Here we make the following qualitative observations for our confining channel. For a given k_ off, the value of the first crossover timescale t_c1,C seems to scale as D_0/L^2 and seems independent of v_b and k_ on, while the second crossover timescale t_c2,C decreases with increasing v_b and k_ on.Average velocity:In Fig. <ref>(a) and (b), we show the results for the scaled average velocity for various values of v_b, without and with confinement, respectively.In the absence of confinement, for large separation between the times spent by the MCC in the unbound and bound states (Eq. <ref>), the average velocity approaches constant values– for k_ on≫ k_ off,⟨ẋ⟩ approachesv_b and motion of the MCC is predominantly ballistic, while for for k_ on≪ k_ off, it approaches zero suggesting diffusive (or no) motion. In the intermediate regime, their is an interplay between ballistic and diffusive motion, and the scaled average velocities grow monotonically with k_ on. Fig. <ref> (a) captures this behavior, with the data for all v_b following Eq. <ref>, as expected. Confinement reduces the average velocity of the MCC for small bound velocities v_b, while for large v_b the results follow the MFNC limit (Fig. <ref>(b)). For small v_b (0.04, and 0.1μ m s^-1), confinement leads to a reduction in the scaled average velocity and it stays below the MFNC limit described by Eq. <ref>. However, the data can be fit to Eq. <ref> with an effective k^ eff_ off, and an effective v^ eff_b. We find that k^ eff_ off>k_ off and v^ eff_b < v_b suggesting that confinement renormalizes the bound velocity and the unbinding rate to values lower than without confinement, and therefore effectively reduces the processivity of the motor. For large v_b (0.5, 0.8,and, 1.6 μ m s^-1), the scaled average velocities are unaffected by confinement. These results are consistent with MSDs discussed earlier.Effective Diffusivity : Confinement has a much more striking impact on the effective diffusivity of the MCC. Unlike for the average velocity, we do not find a data collapse in this case.We show the scaled diffusion coefficient as a function of k_ on for the unconfined case in Fig. <ref>(a). For small k_ on, the motion of the MCC is dominated by passive diffusion and D_ eff is large.While for large k_ on, as the motion is largely ballistic,D_ eff become smaller. As expected, the data for the unconfined case follow the analytical prediction given by Eq. <ref>. The condition for a maximum to exist is predicted by Eq. <ref>: v_b > √(k_ off/D_0) (= 0.81 for parameters used here). We observed that the scaled diffusion coefficient shows a maximum for v_b=1.6 μ m s^-1 at k_ on≃ 0.2 s^-1. In the case of confinement (Fig. <ref>(b)), the effective diffusion coefficient becomes smaller than for the case without confinement, but they are well separated with prominent peaks, and the presence of a maximum at k_ on≃0.2 s^-1 is now seen for small v_b. For v_b=0.8 μ m s^-1 we observe a peak here which is not possible for the unconfined case. The lowering of effective diffusivity suggests that the confinement reduces the noise in the motion by limiting the availability of space for the motion. The observed maximum in diffusivity is reminiscent of similar behavior for a particle undergoing diffusion in a tiled washboard potential <cit.>, orin a periodic confined profile <cit.>. There the appearance of a peak in the diffusivity is associated with a “locked-to-running” transition <cit.>. In the locked state, the particle shows no net movement over a significant amount of time, while in the running state the particle has a net drift velocity. Transitions between the two states can be induced via occasional large kicks due to noise.We observe peaks in the two-state model even without any confinement, with the passive (unbound) and active (bound) states corresponding to the locked and running states respectively. The transition between the two states are induced by the binding kinetics of the motors to the microtubule. Confinement makes the peaks much more pronounced, and leads to greater separation between the scaled effective diffusivity curves for different v_b. The qualitative behavior of the average velocity and effective diffusivity does not depend on the free diffusion constant D_0 and the effective widths of the channel. We have checked this by studying systems with D_0=0.064μ m^2 s^-1 and b=1.2/(2π) μ m (results not presented here). §.§ Active and Passive Transport in a Confining Channel with Spatially Varying Binding Rates Next we study cargo transport by motors with a spatially varying binding rate that depends on the local width of the confining channel.We have studied the following two cases: (i) k_ on(x) = k^0_ on√(b^2-a^2)/(L w(x)), where the binding rate is normalized to ensure that the spatial average ⟨ k_ on(x) ⟩= k^0_ on, and(ii) k_ on(x) = k^0_ on/w(x), where the binding rate is not normalized, and its spatial average⟨ k_ on(x) ⟩= k^0_ onL/√(b^2-a^2). For both (i) and (ii),the unbinding rate k_ off is assumed to be independent of the spatial variation of the channel width. Let us first consider the case (i) with the normalized spatially varying binding rate. We present the scaled velocities against k^0_ on for two different values of unbinding rates k_ off=0.42 and 0.0955s^-1 in Fig. <ref>(a). For each value of the unbinding rate, we consider three different bound velocities v_b. For each unbinding rate, the average velocity data collapse onto a single curve, suggesting that unlike for constant binding rates the scaled velocities do not depend on v_b. We further find that for both unbinding rates, the average velocities of the MCC stay below the corresponding velocities V_lin the MFNC limit given by Eq. <ref>. The decrease in the scaled velocity suggests that the effective unbinding rate k^ eff_ off has increased, and the collapse to a single curve suggests that k^ eff_ off is independent of bound velocity v_b (see Fig. <ref>(c)), unlike the case with constant binding rate. Next we compare case (i) with case (ii), where the binding rate is not normalized so that the average ⟨ k_ on(x) ⟩ is not equal to k^0_ on and is, in fact, larger than k^0_ on.In Fig. <ref>(b), we show the scaled velocities against k^0_ on for case (i) and case (ii), for a given unbinding rate k_ off=0.42 s^-1 along with the corresponding MFNC prediction (Eq. <ref>). We observe that for both cases, the data for all bound velocities show a good collapse as in Fig. <ref>(a). More interestingly, the average velocity of the MCC for case (ii) is greater than the MFNC velocity V_l, unlike case (i) where it is always less thanV_l. This enhancement in the average velocity is because of ⟨ k_ on(x) ⟩ being larger than k^0_ on by a factor of by L/√(b^2-a^2); multiplying k^0_ on by this factor can collapse both data onto a single curve which stays below V_l (Fig. <ref>(c)). Nevertheless this suggests thatif confinement were to cause an enhancement of the average binding rate ⟨ k_ on(x) ⟩, it would lead to larger average velocities of the MCC. In fact, in a study similarly to (ii) but for Brownian ratchets in confined media, the authors found an enhancement of the net particle for non-processive motors with a confinement dependent binding rate <cit.>. The qualitative behavior of the effective diffusion coefficient for an MCC with spatially varying binding rates was observed to be similar to that for an MCC with constant binding rates (results not shown here). §.§ Reduced Probability Density for Bound and Unbound States To understand the behavior of the scaled average velocity, in particular the observed data collapsed, we examine the probability densities of the bound and unbound states.Given the periodic nature of the corrugated confining channel <cit.>, we study the reduced probability densities <cit.> for the bound state P̂_ b(x,t) and unbound state P̂_ b(x,t)defined as follows:P̂_ b(x,t)= ∑_n=-∞^n=∞ P_ b(x + nL,t), P̂_ ub(x,t)= ∑_n=-∞^n=∞ P_ ub(x + nL,t), with ∫_0^Ldx[P̂_ b(x,t)+P̂_ ub(x,t)]=1. We have checked numerically that these probability densities reach their steady state values P̂^st_ b(x) and P̂^st_ ub(x) at large times.In Fig. <ref>, we have shown P̂^st_ b(x) and P̂^st_ ub(x) for three cases: (i) unconfined (Fig. <ref>(b)), (ii) confinement with constant binding rate with k_ on=k_ on^0 (Fig. <ref>(c)), and (iii) confinement with spatially varying binding rate with k_ on(x) = k_ on^0√((b^2-a^2))/(L w(x)) (Fig. <ref>(d)). In Fig. <ref>(a), we plot the spatial profile of the confining wall w(x) and its entropic barrier. For case (i), P̂^st_ b(x) and P̂^st_ ub(x) are uniform in x, independent of v_b, and follow the corresponding analytical predictions as expected. Interestingly, for case (ii), the P̂^st_ b(x) and P̂^st_ ub(x) are not uniform but modulate with the same wavelength as that associated with the spatial variation of the confining channel. The probability density P̂^st_ ub(x) is independent of v_b, while P̂^st_ b(x) vary with v_b, approaching to a constant value at large v_b. For case (iii), P̂_ ub^st follows the spatial variation of the channel in x, while P̂_ b^st is uniform– this is because the normalization of the binding rate involves scaling by w(x); both these reduced probability densities are independent of v_b. It is important to note that for the latter case the value of P̂_ ub^st (P̂_ b^st) has significantly increased (decreased) (Fig. <ref>(c)) compared to that for the confined MCC with constant binding rate (fig. <ref>(b)). This enhancement of P̂_ ub^st and decrease in P̂_ b^st leads to smaller scaled velocity in the case of spatially varying (normalized) binding rate. In summary, when the reduced probability densities are independent of v_b, the corresponding scaled velocity data will collapse onto a single curve, and vice versa. This also explains why for the case of confinement with constant binding rate, scaled velocity data collapse was only observed for large v_b, but not for small v_b. Furthermore, the increase in unbinding probability density in the case of spatially varying binding rate explains the decrease in the scaled velocity (in Fig. <ref>). § SUMMARY AND DISCUSSION We have studied the role of confinement in two-state cargo transport in a two-dimensional corrugated channel using the Fick-Jacobs formalism, and an equivalent one-dimensional lattice model. The effect of confinement is incorporated through a position dependent entropic barrier. At any given time, the MCC can be in one of two states: an active state where it moves on a microtubule track with a constant speed, and a passive state when it is detached from the microtubule and undergoes diffusive motion. We assumed small cargo sizes such that while the diffusive motion is impaired by confinement, the bound state directed motion is not. The results from the lattice model exactly match known analytical results for purely diffusive motion in confinement, demonstrating that the Arrhenius description for hopping rates works for our system and other similar systems with entropic barriers. Moreover, the lattice based approach and simple evolution rules make our model computationally more efficient for simulating two state transport in complex confinement profiles than numerical simulations of the corresponding 2D Langevin equations.In order to understand and quantify how confinement impacts transport properties, we computed and compared the MSD, as well as the average velocity and effective diffusivity of the MCC with and without confinement. The MSD of the confined MCC shows three distinct dynamical regimes corresponding to diffusive motion at small times, ballistic motion at large times, and sub-diffusive motion at intermediate times. The crossover timescale (∼ 10^-2 s) from diffusive to sub-diffusive motion is determined by the interplay between passive or diffusive motion and confinement, and suggests a mesh size ∼ 100 nm for the parameters used in our study if the confinement were due to a cytoskeletal network. The crossover from sub-diffusive to the ballistic motion is dictated by the motor properties, specifically the binding kinetics and the speed of the motor when bound. Confinement significantly reduces the crossover time from diffusive to sub-diffusive behavior, and also leads to a significant intermediate sub-diffusive regime. For unconfined MCCs, this intermediate regime is either absent or much smaller than for confined MCCs.We also found that confinement effectively enhances the motor unbinding rate and thus reduces the average velocity when the bound velocity is small, but has a negligible effect otherwise. The impact of confinement on the effective diffusivity is more remarkable. In the absence of any confinement, for less active MCCs (v_b ≲ 0.8 μ m s^-1 ),an increase in the binding rate leads to a decrease in the effective diffusivity because of the comparatively less time spent in the unbound state; for more active MCCs(v_b ≳ 1.6 μ m s^-1), however, the diffusivity initially increases with the binding rate reaching a peak, and then decreases. This can be attributed to locked-to-running transitions in the two state model. While confinement leads to smaller diffusivities, the peaks now start appearing at smaller v_b ∼ 0.8 μ m s^-1 and are more prominent. Since kinesin-1 motors have an in vitro speed of0.8 μ m s^-1 and an in vivo speed of 2.0μ m s^-1 <cit.>, the peaks should be readily observed in experiments in live cells. In vitro, the predictions of our model can be tested in experiments on kinesin-based microtubule transport in enclosed microfluidic channels <cit.>. An exception to the above confinement induced slowing down of the MCC is observed when confinement enhanced the average rate of binding of the MCC to the microtubule, thereby leading to an enhancement in the average velocity.This suggests that the impact of confinement on cargo transport strongly depends on if and how it modulates the binding kinetics of the motors. Its impact on binding kinetics can be obtained in enclosed microchannel experiments <cit.> by measuring the MCC residence times in the bound and unbound states for different channel widths. The same experimental set up can be used to obtain the scaled average velocity, and thus test the predictions of our study. In addition to studying motor driven cargo transport in confinement, such microfabricated enclosed channels can be potentially used to deliver specific proteins or to separate DNA or RNA strands from a complex mixture by binding them to microtubules and transporting them to desired locations. Our results therefore may not only be useful in understanding cargo transport in cells, but also may help in advancing the nanoscale drug delivery system within cells and sequencing techniques for DNA and RNA. Our model can also be easily extended to study bidirectional cargo transport <cit.>. Finally, for completeness, we comment on the effect of hydrodynamic coupling between the wall and the cargo in the light of a recent experimental study <cit.>.In our work, we study only the effect of entropic barrier ignoring the hydrodynamic coupling between the confining channel and the diffusing particle, as the latter is not taken into account in the Fick-Jacobs approach <cit.>.A recent experiment on colloidal diffusion in corrugated micro-channels found that confinement can increase the hydrodynamic drag which is not captured by the Fick-Jacobs theory using free diffusivities <cit.>. However, the authors have demonstrated that this theory can be used to explain their results if it is reformulated in terms of the experimentally measured diffusion coefficients. It may be interesting to study how hydrodynamic effects impact our system; while it is outside the scope of our current study, we will pursue this in future work. Such effects have been found to be important for microswimmers which, unlike diffusive particles, create and use hydrodynamic flow fields for their propulsion <cit.>. The authors would like to thank Jennifer Ross, Megan Valentine, and Ajay Gopinathan for illuminating and helpful discussions, and acknowledge helpful suggestions from anonymous reviewers. SD would also like to thank Dibyendu Das for useful discussions. This research is funded in part by the Gordon and Betty Moore Foundation through Grant GBMF5263.02 to MD. MD and KC were also partially supported bya Cottrell College Science Award from Research Corporation for Science Advancement.§ CONTINUOUS LIMIT OF THE LATTICE MODELAs discussed in the main text, the master equations describing the time evolution of the probability densities of the bound (active) and unbound (passive) state are given by, ∂ P_ b(x,t)/∂ t =k_ on(x) P_ ub (x, t) - k_ off(x) P_ b (x, t) + λ_v P_ b (x - ℓ, t) - λ_v P_ b (x, t) ∂ P_ ub(x,t)/∂ t =-k_ on(x) P_ ub (x, t) + k_ off(x) P_ b (x, t) + λ^ ub_+ (x - ℓ) P_ ub (x - ℓ, t) + λ^ ub_-(x + ℓ) P_ ub (x + ℓ, t) - (λ^ ub_+(x) + λ^ ub_-(x)) P_ ub (x, t),where k_ on(x) and k_ off(x) are the transition rates for the bound and unbound state respectively, λ_v=v_b/ℓ is the hopping rate in the forward direction when the motor-cargo complex is in the bound state, and λ^ ub_±(x) = (D_0/ℓ^2) e^-β(𝒜(x ±ℓ) -𝒜(x))/2 is the unbound state hopping rates for the forward and backward direction respectively. Using Taylor's expansion forP_ b,ub (x ±ℓ, t) and λ^ ub_+ (x ±ℓ) around x and keeping the terms up to 2^nd order in ℓ, we get ∂ P_ b(x,t)/∂ t =k_ on P_ ub - k_ off P_ b - v_b ∂ P_ b/∂ x + D_v ∂^2 P_ b/∂ x^2 + 𝒪(ℓ^3), ∂ P_ ub(x,t)/∂ t =-k_ on P_ ub + k_ off P_ b + ℓP_ ub[ - dλ^ ub_+/dx + dλ^ ub_-/dx + ℓ/2(dλ^ ub_+/dx - dλ^ ub_-/dx) ] + ℓ ∂ P_ ub/∂ x[ -λ^ ub_+ + λ^ ub_- + ℓ ( dλ^ ub_+/dx + dλ^ ub_-/dx ) ] + ℓ^2/2∂^2 P_ ub/∂ x^2( λ^ ub_+ + λ^ ub_-) + 𝒪(ℓ^3),where D_v≡ℓ^2 λ_v/2=ℓ v_b/2. In ℓ→ 0 limit, neglecting the term with D_v in Eq. <ref> we recover the continuum Fokker-Planck equation for the bound state (Eq. (2a) in the main text)∂ P_ b(x,t)/∂ t =k_ on P_ ub - k_ off P_ b - v_b ∂ P_ b/∂ x.Considering the leading order contributions for the coefficients of P_ ub and dP_ ub/dx in Eq. <ref> we get∂ P_ ub(x,t)/∂ t =-k_ on P_ ub + k_ off P_ b + ℓP_ ub[ - dλ^ ub_+/dx + dλ^ ub_-/dx]+ ℓ ∂ P_ ub/∂ x[ -λ^ ub_+ + λ^ ub_-] + ℓ^2/2∂^2 P_ ub/∂ x^2( λ^ ub_+ + λ^ ub_-).For ℓ→ 0 limit, λ^ ub_±(x) = (D_0/ℓ^2)e^∓βℓ/2d𝒜/dx.In this limit, the coefficients of P_ ub, ∂ P_ ub/∂ x, and ∂^2 P_ ub/∂ x^2 are given by,ℓ( -dλ^ ub_+/dx + dλ^ ub_-/dx) ≃ D_0 βd^2𝒜/dx^2, ℓ(-λ^ ub_+ + λ^ ub_-) ≃D_0 βd𝒜/dx, and ℓ^2 (λ^ ub_+ + λ^ ub_-)/2 ≃ D_0 respectively. Using these expressions in Eq. <ref> we recover the continuum Fokker-Planck equation for the unbound state (Eq. 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"authors": [
"Supravat Dey",
"Kevin Ching",
"Moumita Das"
],
"categories": [
"cond-mat.soft",
"q-bio.SC"
],
"primary_category": "cond-mat.soft",
"published": "20170726232954",
"title": "Active and Passive Transport of Cargo in a Corrugated Channel: A Lattice Model Study"
} |
^1Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland^2Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland^3New Technologies-Research Center University of West Bohemia, Plzeň, Czech Republic^4National Technical University, Kharkiv Polytechnic Institute, Frunze Str. 21, 61002 Kharkiv, Ukraine^5Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany^6Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland^7Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland^8Institut für Halbleiter-und Festkörperphysik, Johannes Kepler Universität, A-4040 Linz, AustriaThe control of the electron spin by external means is a key issue for spintronic devices. Using spin- and angle-resolved photoemission spectroscopy (SARPES) with three-dimensional spin detection, we demonstrateoperando electrostatic spin manipulation in ferroelectricand multiferroic . We not only demonstrate for the first time electrostatic spin manipulation in Rashba semiconductors due to ferroelectric polarization reversal, but are also able to follow the switching pathway in detail, and show a gain of the Rashba-splitting strength under external fields. In multiferroicoperando SARPES reveals switching of the perpendicular spin component due to electric field induced magnetization reversal. This provides firm evidence of effective multiferroic coupling which opens up magnetoelectric functionality with a multitude of spin-switching paths in which the magnetic and electric order parameters are coupled through ferroelastic relaxation paths. This work thusprovides a new type of magnetoelectric switching entangled with Rashba-Zeeman splitting in a multiferroic system.Operando imaging of all-electric spin texture manipulation in ferroelectric and multiferroic Rashba semiconductors J. Krempaský^1, S. Muff^1,2, J. Minár^3 , N. Pilet^1, M. Fanciulli^1,2, A.P. Weber^1,2,V.V. Volobuiev^4,8, M. Gmitra^5,C.A.F. Vaz^1, V. Scagnoli^6,7, G. Springholz^8, J. H. Dil^1,2December 30, 2023 ===============================================================================================================================================================================================§ INTRODUCTION Manipulating the spin texture of ferroelectrics (FE) through electric fields and of multiferroics through both magnetic and/or electric fields is a key requirement for programmable spintronic devices <cit.>. The recent discovery of giant Rashba splitting in ferroelectricprovides a promising candidate for such devices and raises the question of whether the spin texture in such a material can be modulated by electric fields by controlling the inversion asymmetry and thus also the Rashba splitting<cit.>. Such devices would strongly benefit from the highest bulk Rashba spin splitting ofat room temperature<cit.>. This spin splitting is exemplified in Fig. <ref>a that shows theRashba bands near the Z-point of the Brillouin zone where the Rashba-splitting is most pronounced <cit.>. Ideally, electric field induced ferroelectric switchingwill change the spin orientation of these band as sketched in Fig. <ref>b, thus allowing a reprogramming of spin currents by external means. is the simplest known binary ferroelectric semiconductor with a narrow band gap<cit.>. Below T_C≈700 K it assumes a non-centrosymmetric rhombohedral structure in which an electric dipole is formed due to a relative Ge/Te sublattice displacement along the [111] direction (see Fig. <ref>c). For moderate Mn-dopings,becomes ferromagnetic as well <cit.>, which opens up new spin-based functionalities because this multiferroicity entangles the Rashba and Zeeman effects within a three-dimensional system <cit.>. In this new class of MUltiFErroic Rashba Semiconductors (MUFERS), we already showed that external magnetic fields switch the bulk spin texture<cit.>. In this work, we demonstrate that spin manipulation is also possible by electric fields, meaning that the magnetoelectric coupling so far induced only by magnetic fields<cit.> expands even to all-electric control of magnetism. Thus, from an application view,fulfils all criteria for mutual control of magnetism and ferroelectricity via magnetoelectric coupling effects <cit.>. Our epitaxial films grown by molecular beam epitaxy <cit.> on conducting InP (111) substrates were studied by means of spin- and angle-resolved photoemission under applied voltage (operando SARPES) at 20 K, which was complemented by piezoresponse force microscopy (PFM) investigations (see Appendix A and Ref.GeTe_Gruverman,Calarco_nano,Liebmann_GeTe,JK_PRB). To understand the functional electronic properties ofwe need to consider its FE lattice structure in which the Ge atoms are displaced along [111] with respect to the six neighboring Te-atoms as sketched in Fig. <ref>c. The displacement, indicated by the green arrow, is very large reaching a value almost 10% of the unit cell <cit.>.An important issue for switching of thepolarization is that, due to the four-fold degeneracy of the rhombohedral lattice distortion, the ferroelectricity may form in eight different domains with individual polarization vectors pointing along differentdirections. This multi-domain structure is independent of the substrate epitaxial registry because thin films grown on InP, BaF_2 or Si(111) develop the same major and secondary FE-domains<cit.>. In such a multidomain structure, polarization reversal may involve intermediate steps via oblique [1̅11] domains (purple arrow in Fig. <ref>d), typically leading to a whole realm of FE-fatigue effects <cit.>. § EXPERIMENTAL METHODS AND SETUPSSARPES allows one to directly determine the spin texture of the electronic band structure and therefore probe in operando the reversal of the Rashba spin-polarization as a function of the applied electric field. The layout of the novel device structure developed for such measurements is shown in Fig. <ref>e and consists of a protective Se-cap on top of the , respectively,epilayers onto which a Au-mesh is placed as a top gate (see Appendix A). Two strategies were employed in the device assembly. In the first (Fig. <ref>f), the in situ desorbed Se-cap sticks to the bottom-side of the Au-mesh providing a quasi-insulating contact between the Au-gate and the semiconducting . The second assembly (Fig. <ref>g) employs an Al_2O_3 layer predeposited on the Au mesh to provide better dielectric insulation. Consequently, higher bias voltages can be applied and the setup can withstand several annealing treatments to rejuvenate the sample after electric cycling. Both setups have similar capacitor-like structures and yield consistent results.§.§ Electric control of spin orientation Figure <ref>a-c shows ARPES spectra measured from our device with an applied bias voltage of -1.6 V. In agreement with our earlier studies on bare <cit.>, near the Z-point (=22 eV) we discern the valence band maximum with the giant Rashba splitting beneath a narrow gap (≈90 meV). Compared to Fig. <ref>a, the quality of the spectra is affected by the top Au-gate electrode, yet the data is clear enough to locate the bulk bands and their surface-resonance replica (SR)<cit.>, sketched in black and red in Fig. <ref>d. In the following, we focus on the effect of the electric field on the Rashba states in spin-detection. For this purpose, we visualize the SARPES data as spin-resolved momentum (MDC) and energy (EDC) distribution curves recorded at selected bias voltages, measured along the cuts indicated by the dashed lines in Fig. <ref>b within an experimental geometry sketched in panel (e). For clarity, panels f-g summarize the in-plane operando modulation of the spin orientations. In accordance with the theoretical model in Refs. JK_PRB,Schoenhense_GeTe, the validity of the spin vectors is confirmed by the fact that the bulk bands and their resonance replica show always an almost perfect anti-parallel spin arrangement. The SARPES vectorial spin fitting (see Appendix B) is detailed Figs. <ref> and <ref>, respectively. SARPES data from first device type in Fig. <ref> identify the SR and bulk bands as peaks 1 and 2, respectively.As summarized in Fig. <ref>f, both show an almost perfect spin reversal in the 0→-1.6 V transition, and an incomplete spin rotation in the -1.6→+1.6 V transition. This clearly demonstrates that the spin texture can be electrically manipulated by the bias applied. Similar electric spin manipulation is observed for the second type of device at higher bias voltages summarized in Fig. <ref>g, and detailed in Fig. <ref>. The data again shows the Rashba splitting of the bulk bands B-D and their surface resonance replica A-C with anti-parallel spins <cit.>. For clarity the intersection of the band A-D with the measured EDC path is sketched in Fig. <ref>d, Fig. <ref>e summarizes the spin vectorial fitting for all the bands including thecomponents.The spin rotation of bulk bands B-D appear to not fully follow the changes of applied electric poling field, presumably because each bulk band is screened by its resonance replica. This means that electric spin manipulation becomes blocked after the second cycling, an effect referred to as fatigue.However, besides this unipolar FE-fatigue also domain wall pinning should be considered. It typically occurs when the system relaxes from a relatively high bias voltage in which the back-switching is activated by charge agglomerates at the domain walls<cit.>. We find that this is exactly the case for the sudden, almost 180 spin rotation in the +20→0 V relaxation transition denoted by dashed arrows in Fig. <ref>g, thus demonstrating that also ferroelastic effects have to be considered in our operando SARPES measurements.We note that the observed ferroelastic effects are probably related with the (metavalent) resonant-bonding mechanism of GeTe<cit.> which leads to a decrease of the relative stability of the FE-order. §.§ FE domains probed by PFM and operando SARPES To elucidate the polarization reversal induced by applied electric fields and the reduced spin rotation after the initial poling cycle, PFM studieswere performed on the samples as shown in Fig. <ref>a-d.In this technique, by excitation of the sample via an AC voltage applied between surface and probe tip the piezoelectric effect induces an oscillation of the cantilever deflection. Its amplitude is a measure for the absolute magnitude of the FE polarization, and its phase corresponds to the polarization direction. Fig. <ref>a shows the hysteresis loop of the PFM phase as a function of DC tip voltage, evidencing the reversal of the polarization direction in thefilms induced by the electric field, which corresponds exactly to the spin-reversal that we observe by SARPES. However, the shift of the hysteresis loop, similar to that in Ref.Liebmann_GeTe, suggests a preferential built-in polarization direction related to the surface termination.Such experimental observation calls for a more accurate investigations of the PFM data, as well as theoretical models reflecting the FE-domain switching in thesurface region. Local FE poling ofby applied DC voltages is demonstrated by Fig. <ref>c-d, which presents the PFM amplitude image of differently poled regions written in the sequence 1-3 from 0→+4 V→-4 V, indicated by the green, yellow and blue boxes, respectively. The corresponding integrated PFM amplitudes depicted in Fig. <ref>d reveal a significantly reduced PFM amplitude signal in the second poling cycle displayed in blue, similar to what we observe in SARPES by measuring only partial spin reorientation after the second poling. This suggests the existence of depolarization fields due to unipolar FE-fatigue near domains and/or grain boundaries by charge agglomerates<cit.>. The characteristicthin film topography (Fig. <ref>b) originating from stacking faults<cit.> are primary candidates for such carrier agglomeration, also reflected in theintrinsic p-type transport properties <cit.>. Charge agglomeration is expected to occur in the sub-surface region and screens the bulk bands, hence hindering the full spin reversal in repetitive electric cycles in agreement with our operando SARPES results. Based on the fact that PFM-phase switching seen in Fig. <ref>a remains unchanged after several hours also in domain mapping<cit.>, we rule out charging artifacts induced by PFM in the FE-switching. Due to the small escape depth of photoelectrons,SARPES is a surface sensitive technique and therefore the role of film surface must be taken into account. To this end, we performed ab initio calculations based on density functional theory (see Appendix C) to assess the equilibrium surface structure as a function of applied electric field. In the calculations, the bulk FE switching was implemented by switching the inner electric field in the bulk region (dashed rectangle in Fig. <ref>e), which results in a substantial structural rearrangement of the surface region as shown by Fig. <ref>f. Given that the Te-terminated surface is preferred for <cit.>, the strong electronegativity of the surface Te-atoms imposes an unipolar FE-polarization at the surface. The surface relaxation associated with FE switching induces a sub-surface FE domain wall (similar to the rock salt β-GeTe crystal structure), underneath of which the bulkFE-order is reversed (yellow area in Fig. <ref>f).This triple-layer domain wall separating oppositely oriented domains is an essential ingredient in the FE-switching mechanism, which involves swapping between the longer and shorter Ge-Te bonds<cit.>. Near the surface the formation of the domain wall gives rise to a depolarization field P_z' and asymmetric hysteresis loops such as seen in Fig. <ref>a.A way to test the contribution of unipolar FE-fatigue to the switching behavior is thermal annealing below the FE Curie point<cit.>. Indeed upon re-annealing of the samples to temperatures around 250-280the spin rotation inside the first cycle observed by SARPES is restored to the initial value of the as grown samples. This is consistent with unipolar FE-fatigue due to charge agglomeration atgrain boundaries and support the theoretical model depicted in Fig. <ref>f, as well as the asymmetric PFM hysteresis loop observed in Fig. <ref>a. §.§ Control of Rashba coupling constant by electric fields Inspection of the spin-fits of Fig. <ref>a-c reveals significant changes in the energy splittings ΔE_1,2,3 of the bulk Rashba bands when the samples are poled by the applied electric fields. Because the spin-fits of the bulk bands B-D (black lines in Fig. <ref>a-c) essentially coincide with the total EDC intensity in Fig. <ref>c, we can evaluate these changes from the EDCs (purple dots in Fig. <ref>a-c). Compared with the measurements of the as-grown films we find that the ±10 V bias variation changes the energy splitting of the bulk Rashba bands B-D by ±8%. This is a significant change of the Rashba parameter α_R due to poling of the material. The 8% increase in α_R implies that the initial poling field is able to align differently oriented FE domains present in the as-grown state, thus increasing the macroscopic polarization, consistent with the increase of PFM amplitude after the first poling process (yellow bar in Fig. <ref>d). Indeed, this effect can explain the deviation between the experimental Rashba constant measured for as grownfilms of ≈4.2 eVÅ<cit.>, and the theoretically predicted value of ≈5 eVÅ<cit.>. § ELECTRIC FIELD-INDUCED MAGNETIZATION REVERSAL IN GE_1-XMN_XTE. We now turn to electric spin manipulation of multiferroic . In comparison to ferroelectrics, in MUFERs there is an additional spontaneous magnetization superimposed over the FE polarization<cit.>. Even more, because of the magnetoelectric coupling between magnetization and polarization, field-induced switching of polarization can also induce a switching of magnetization, thus providing additional means of spin control. A particularly interesting feature ofis that the easy axis of magnetization (M_z) and the electric dipole moment (P_z) are colinear and perpendicular to the surface <cit.>. As a result, the Rashba bands are Zeeman-split and thereby assume an additional out-of-plane spin polarization p_z around the Z-point<cit.>. As illustrated by Fig. <ref>a and verified in Ref.JK_GMT, this out-of-plane spin component p_z is directly linked to the magnetization direction. Thus, the switching of M_z can be directly detected by SARPES. Figure <ref>d-f presents the measured p_z polarization of thedevice across the Zeeman gap in normal emission under different bias conditions. Clearly, the characteristic p_z(E) dependence switches sign upon bias reversal of the gate, indicating an induced reversal of the magnetization by the applied electric field. This experimental result presents the first unambiguous and direct evidence for the existence of a strong magnetoelectric coupling between the electric dipoles and the magnetic moments in this material.Contrary to , where the spin gate control becomes blocked after the first poling cycle, SARPES ofis found to change for each change of the external electric field. This is demonstrated by Fig. <ref>d-f, where the p_z data, collected at each step of the colour-coded electric field sequence, is presented in comparison with that recorded at +10V (blue symbols). Evidently, each electric field change induces a significant change of the spin polarization, indicating a significantly weaker pinning of the FE polarization compared to . This is explained by the fact that the lattice distortion and off-center displacement of the Ge atom with respect to the Te atoms indecreases with increasing Mn content<cit.>. This evidently reduces the energy barriers for switching of the atomic positions in the FE reorientation and thus leads to a softening of the FE properties. As illustrated in Fig. <ref>c, moreover, the coupling to the magnetization can give rise to complex switching paths that will result in unconventional spin texture evolutions as a function of biasing sequence. § CONCLUSING REMARKS In summary, we have demonstrated all-electric spin manipulation inferroelectric and multiferroic Rashba semiconductor devices using operando spin-resolved photoemission spectroscopy.The results not only give direct evidence for tuning of the spin texture by electrostatic gates, but also reveal a sizeable increase of the Rashba coupling strength achieved by poling of the material. Moreover, by the out-of-plane spin polarization we demonstrated magnetisation reversal in multiferroicby applying electric fields, which provides unambiguous evidence for the strong magnetoelectric coupling between the ferroelectric and magnetic orders in this system. Our experimental findings open up a promising path toward robust and programmable semiconductor-based spintronics with functionally coupled electronic and magnetic properties. However, ferroelectric fatigue and ferroelasticity play a crucial role in determining the device performance. Therefore, robust all-electric spin switching between remnant polarization states requires further improvements in domain stabilization and sample growth.§ ACKNOWLEDGEMENTSConstructive discussions with V. N. Strocov are gratefully acknowledged. This work was supported by the Swiss National Science Foundation Project PP00P2_144742 1 and the Austrian Science Funds Project SFB F2504-N17 IRON. M.G. acknowledges the German funding agency DFG SFB 689 and J.M. the CENTEM Project, Reg.No.CZ.1.05/2.1.00/03.0088, CENTEM PLUS (LO1402) and COST LD15147. § APPENDIX §.§ A: Experimental techniquesExperiments were performed on 200 nm thickandfilms grown by molecular beam epitaxy on semiconducting InP(111) substrates. Samples were capped by a protective stack of amorphous Te and Se layers which were removed by annealing in-situ just before the photoemission experiments. SARPES experiments were performed at the COPHEE end-station of the Surfaces and Interfaces Spectroscopy beamline at the Swiss Light Source (SLS) synchrotron radiation facility, Paul Scherrer Institute, Switzerland, using p-polarized, =22 eV photons. The Omicron EA125 hemispherical energy analyzer is equipped with two orthogonally mounted classical Mott detectors<cit.>. The whole setup allows simultaneous measurements of all three spatial components of the spin-polarization vector for each point of the band structure. The SARPES data were measured with the sample azimuthsoraligned perpendicular to the scattering plane as denoted in Fig. <ref>e. The angular and combined energy resolution were 1.5 and 60 meV, respectively. In spin-integrated mode these resolutions were set to 0.5 and 20 meV. All data were collected with the sample kept at a temperature of 20 K. Temperature was measured by a Si diode placed near the sample. The InP substrate was grounded during the SARPES measurements and a bias was applied to the Au-mesh. Care was taken to avoid a direct grounding of the front of the sample. The measured kinetic energies were adjusted according to the applied voltage.Piezo-force microscopy was performed at the NanoXAS endstation at the SLS using a plain platinum tip at room temperature<cit.>. The sample topography and PFM channels (amplitude and phase) are measured simultaneously. The PFM is measured close to resonance; no cross-talking with topography is visible which validate the quality of our PFM data seen in Fig. <ref>c.The electro-formed Au-mesh used in the poling device had a nominal aperture of 64 μm, thickness 4 μm and 70% open area. The nominal aperture is much larger compared to domain wall size, which, in first approximation, is around 1 micron according to the PFM topography in Fig. <ref>b. As the beam spot size is ca. 100 microns, and the Au-mesh hole is much larger compared to the domain wall size, we do not expect artefacts in operando SARPES due to domain wall motions. §.§ B: SARPES vectorial analysisBy means of the two orthogonal Mott detectors of the COPHEE experimental station it is possible to simultaneously measure the two in-plane directions and the out-of-plane direction, constituting a three-dimensional (3D) spin-resolved data set. Such a typical SARPES data set as presented in the main text consists of a total intensity spectrum I_t and polarization spectra along the three spatial directions P_x,y,z described in Fig. <ref>e. From this data set the spin resolved spectra can be obtained by projecting the results on these spatial directions. The measured polarization and total intensity yield spin-resolved intensity spectra calculated as I_x,y,z^up = (1+P_x,y,z)I_t/2 I_x,y,z^down = (1-P_x,y,z)I_t/2 In a well established fitting routine the photoemission spectrum is first dissected into individual peaks and background and afterwards all polarization directions are fitted simultaneously to obtain spin vectors<cit.>. Figure <ref> shows how the total intensity I_t of the MDC is fitted with Voigt functions and a linear background. Similarly, Figure <ref> shows the vectorial fitting of the energy-distribution curve (EDC) by Voigt functions and a background for a momentum near 0.1 1/Å along thedirection indicated in Fig. <ref>b. We note that in order to asses the spin manipulations upon gate control in . the full 3D spin analysis is mandatory because theRashba-splitting manifest a canted spin arrangement also in the in-plane spin texture<cit.>.The polarization curves in Fig. <ref> (Fig. <ref>) are modeled until the best fit is reached by simultaneously fitting the MDC (EDC) intensity (purple dots) and the polarizations , and(green lines). All the MDC SARPES fits in Fig. <ref> unambiguously identify the surface resonance as peak 1 in red and the bulk as peak 2 in black.Concerning the operando SARPES we emphasize that during the 0 V→-1.6 V transition in Fig. <ref>a-b, i.e. during the first poling of the as-grownfilm, there is almost an ideal switching of all the spin polarization components, well observed also in the raw spin polarization curves P_x,y. These experimental observations are consistent with complete reversal of the spin vectors as depicted in Fig. <ref>b. §.§ C: First principles calculationsTo calculate structural properties associated with the ferroelectric switching in α-GeTe, we perform density functional theory calculations employing the Quantum ESPRESSO package <cit.>. We used projector augmented wave pseudopotentials with kinetic energy cutoff of 40 Ry for wave functions and 240 Ry for charge density. For the exchange-correlation functional we used the generalized gradient approximation <cit.>. 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Lett. volume 77, pages 3865 (year 1996)NoStop | http://arxiv.org/abs/1707.08431v1 | {
"authors": [
"J. Krempaský",
"S. Muff",
"J. Minár",
"N. Pilet",
"M. Fanciulli",
"A. P. Weber",
"V. V. Volobuiev",
"M. Gmitra",
"C. A. F. Vaz",
"V. Scagnoli",
"G. Springholz",
"J. H. Dil"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170726133018",
"title": "Operando imaging of all-electric spin texture manipulation in ferroelectric and multiferroic Rashba semiconductors"
} |
ALMA OBSERVATIONS OF THE PROTOSTAR L1527 IRS: PROBING DETAILS OF THE DISK AND THE ENVELOPE STRUCTURES Hsi-Wei Yen December 30, 2023 ===================================================================================================== Using the recently introduced declination function, we estimate the net number of seats won in the US House of Representatives due to asymmetries in vote distributions. Such asymmetries can arise fromcombinations of partisan gerrymandering and inherent geographic advantage. Our estimates show significant biases in favor of the Democrats prior to the mid 1990s and significant biases in favor of Republicans since then. We find net differences of 28, 20 and 25 seats in favor of the Republicans in the years 2012, 2014 and 2016, respectively. The validity of our results is supported by the technique of simulated packing and cracking. We also use this technique to show that the presidential-vote logistic regression model is insensitive to the packing and cracking by which partisan gerrymanders are achieved.§ INTRODUCTION The partisan composition of the US House of Representatives is the result of a number of factors: the economy, social issues, who is president, party platforms, voter ID laws, propaganda, campaign finance laws, the characteristics and incumbency statuses of individual candidates, and district plans, to name a few. Our focus in this paper is on district plans. A partisan gerrymander is, by definition, a district plan that enables a party to win more seats than it would have under a “neutral” district plan. It is sometimes easy to detect a partisan gerrymander. For example, a North Carolina legislator openly admitted that his committee planned to redraw districts with a goal of partisan advantage <cit.>. Other times the convoluted shapes of districts provide evidence that is almost as compelling. In fact, significant energy has been devoted to developing geometric methods of identifying gerrymandered districts (see <cit.> for an overview). But none of these geometric approaches provide a direct way to determine the number of extra seats that have been won through the gerrymander — the fundamental purpose of a partisan gerrymander.In the first part of this paper we use the declination function, δ, introduced by the second author in <cit.>, to estimate the effect of asymmetries in how district plans treat the two major parties on the number of House seats won by each. To do so, we explore more thoroughly the scaled function 5Nδ/12 which we refer to in this paper as the S-declination. (The variable, N, denotes the number of districts in the election; “S” is intended to remind the reader that this scaling counts seats.) We argue that the S-declination provides a good estimate of the number of extra seats won through these partisan asymmetries. In <cit.> we consider a similar scaling of Nδ/2. While simpler, this appears to slightly overcount the number of seats in a given state and year. The effect is not significant when single-state US House elections are considered as is done in <cit.>, but in the typically larger state legislatures or when considering national effects, the difference between the two scalars is non-trivial. Table <ref> presents the results of applying the S-declination to the US House elections since 1972. The data indicate that, on a national level, while the Democrats consistently benefited through the early 1990s from partisan asymmetry in district plans, the situation has been reversed since the late 1990s. There have been a number of previous attempts to estimate the net effect of gerrymandering on a national level (see, e.g., the references in <cit.>). The most recent of which we are aware is a report <cit.> by Royden and Li that considers House elections since 2012. Our estimate for the year 2012 of 28 extra republican seats falls within 25–36, the narrower of the two ranges provided by Royden and Li.However, our estimate differs markedly from the estimate of approximately one seat arrived at in a recent paper <cit.> by Chen and Cottrell.(See also, <cit.> whose results are intermediate to those of the mentioned studies.) In the second part of this paper we attempt to explain the discrepancy between the Chen-Cottrell estimate and the two other estimates. Our main tool will be a simulated packing and cracking technique that we introduce in Section <ref>. We first use this technique to validate the S-declination as a means of counting extra seats. We then use it to show that the logistic-regression approach of Chen and Cottrell is insensitive to the main technique of partisan gerrymandering. The structure of this paper is as follows. In Section <ref> we introduce the necessary background and terminology on both partisan gerrymandering as well as on packing and cracking. We also describe a simple framework for simulated packing and cracking that we use in later sections to gauge how faithfully various methods register partisan gerrymandering efforts. In Section <ref> we review the declination function and show using the packing and cracking technique that the declination faithfully registers partisan gerrymandering. In Table <ref> we record the net number of seats in House elections since 1972 that, according to the declination, should be attributed to partisan asymmetry in the vote distribution. In Section <ref> we show that the model utilized in <cit.> is insensitive to , thereby explaining the discrepancy between their estimate and the estimates of both this paper and <cit.>. We end with our conclusion. § BACKGROUNDPartisan gerrymanders have historically been recognized by identifying individual districts with contorted boundaries. The primary advantage of this approach is that wild shapes are immediately convincing as evidence of nefariousness. Unfortunately, there are several disadvantages. First, there are many valid reasons for a district to have a strange shape (e.g., geographic constraints, the Voting Rights Act, and county boundaries). Second, gerrymandering can occur without particularly unusual boundaries. Third, the shapes of the districts do not directly tell one anything about how effective the gerrymander is in winning additional seats for one party. An alternative approach to the study of district shapes is to analyze the distribution of votes among the various districts. In this section we introduce the notation and terminology necessary to work with these distributions and to see the effect partisan gerrymandering has on them. We assume that each district has the same number of total votes.We define an N-district election to be a weakly increasing sequence = (ℓ_1,ℓ_2,…,ℓ_N) in which ℓ_i indicates the democratic fraction of the two-party (legislative) vote in district i. We visualizeby plotting a point v_i = (i/N-1/2N,ℓ_i) for each i. Figure <ref> illustrates plots offor three hypothetical elections.While researchers are still working on robust tests both for identifying gerrymanders and for assessing the effects of gerrymandering, how partisan gerrymandering occurs is well understood: Parties create partisan gerrymanders by “packing and cracking” votes. Suppose the Democrats are in control of redistricting and the Republicans are poised to win district X. In packing, Republicans are moved from X to other districts in which the Republicans already have enough strength to win. These votes are effectively wasted in the new districts while district X falls to the Democrats. Cracking works similarly, except now the Republicans are spread among districts that they have no chance of winning. Once the cracking occurs, the recipient districts are lost by the Republicans by smaller margins, but they are still lost. Figure <ref> illustrates the effect ofon the vote distribution illustrated in Figure <ref>.A.§.§ Simulated packing and crackingIn order to validate the S-declination (see Section <ref>) as a measure of the number of extra seats won under a partisan gerrymander, we will examine how perturbations of a vote distribution by packing and/or cracking affect the value of the S-declination. The simple technique we will use is that of simulated packing and cracking (). In short, we manually modify a vote distribution by packing or cracking so as to flip a single district from one party to the other. There are four possible composite choices: whether we are packing or cracking and whether it is the Republicans or the Democrats who are in charge of the gerrymander. In reality, the votes from the flipped district could be distributed among other districts by a combination of packing and cracking, but we do not attempt to model this. We focus below on the case in which the Republicans are flipping a single democratic district to republican control using cracking. The other three cases are treated similarly.In practice, the details of a gerrymander will depend on many factors. One such factor will be the geography of the state. If a given district is being cracked so as to turn it from a democratic district to a republican district, the surplus democratic voters will have to be allocated to adjacent districts. Of course, this process can be iterated by swapping other democratic voters for republican voters in second-order neighbors of the original district. Nonetheless, there are obvious geographic constraints that may be significant. Another factor is how risk averse the gerrymandering party is. For example, if the Democrats wish to maximize their potential gain in seats (albeit at a high risk of the plan backfiring) they can crack republican districts by creating districts that are (say) 49% republican. On the other hand, if the Democrats feel the political winds will be against them in the upcoming decade, they may prefer to pack Republicans into districts so that the democratic districts are no more than (say) 35% republican.For our model, we make the following conventions for how the gerrymander is achieved. * The flipped district is the democratic district that is won by the narrowest margin.* The gerrymander does not create any new republican-majority districts with a democratic vote fraction of greater than 0.45. We choose this value on the basis that a 45–55 split is frequently considered the threshold for a race to be competitive (see, for example, <cit.>). Any republican-majority district with a democratic vote fraction higher than this before the cracking is allowed to remain at such a level.* The modified democratic vote fraction in the flipped district is chosen according to a linear regression of the democratic vote fractions among the republican districts. If the linear regression yields a new democratic vote fraction in the flipped district greater than 0.45, then the value is set to exactly 0.45.* The democratic votes shifted from the flipped district are distributed evenly among the republican-majority districts with a democratic vote fraction of at most 0.45. In order to avoid violating the second convention, this process may need to be iterated (see Example <ref> below). In order to illustrate the method in practice we present the following example of flipping a district from democratic to republican by cracking.While we use hypothetical data in this example, all subsequent applications of simulated packing and cracking in this paper involve vote distributions from actual elections. Consider a 10-district election = (0.37,0.40,0.43,0.46,0.60,0.63,0.66,0.69,0.72,0.75).By Convention <ref>, we flip the fifth district. A linear regression of the four points{i/10-1/2· 10,ℓ_i}_i=1^4 = {(0.05,0.37), (0.15,0.40), (0.25,0.43), (0.35,0.46)}yields a line with intercept 0.355 and slope 0.3. Thus, according to the linear regression line, the flipped district should be switched from a democratic vote fraction of 0.6 to one of 0.49. However, by Convention <ref> we instead choose a value of 0.45. In order to maintain the same statewide democratic vote fraction, there must be a net increase of 0.15 among the first three districts (note that the fourth district is not included since its democratic vote fraction already exceeds 0.45). Convention <ref> instructs us to distribute these democratic votes evenly among the three districts. The resulting vote distribution is(0.42,0.45,0.48,0.46,0.45,0.63,0.66,0.69,0.72,0.75).However, following Convention <ref>, we iterate the process by redistributing the excess fraction of 0.03=0.48-0.45 from the third district evenly among the first two districts. Since the second district is already at a value of 0.45, the amount is entirely distributed to the first district. This yields a final vote distribution of= (0.45,0.45,0.45,0.46,0.45,0.63,0.66,0.69,0.72,0.75).We utilizefor two purposes in this paper. In each instance, we analyze a vote distribution using some function both before and after the . We have rerun the analyses in this paper with different values of the maximum democratic vote fraction equal to 0.40 and 0.49. We have also analyzed the effect of a “greedy” distribution of votes in which the extra votes from the flipped district are distributed to the districts whose democratic and republican votes are distributed most unevenly. The resulting data and figures corresponding to Figures <ref>, <ref> and <ref> are qualitatively very similar. § THE DECLINATIONThe declination function was introduced in <cit.> as a way of identifying potential gerrymanders by quantifying partisan asymmetry in the vote distribution. A number of other vote-distribution functions created for this purpose can be found in the literature (several of these are mentioned in Section <ref>).The declination is based on two observations. The first is that a constitutionally manageable standard for partisan gerrymandering could conceivably be based on some measure of partisan asymmetry, that is, a measure of how the district plan treats the parties differently. The second observation, which we make in <cit.>, is that there should be nothing special about the 50% vote threshold in individual districts. Combining these observations leads us to compute the ratio of “average winning margin” to “fraction of seats won” for each party. The declination is simply a comparison of these two ratios. If the 50% threshold is truly not special, then the resulting ratios for each party should be approximately equal. The declination can be computed geometrically as follows (see <cit.> for details).Let v_i denote the point (i/N-1/2N,ℓ_i) for each 1≤ i≤ N. Place a point F at the center of mass of the points v_i corresponding to the districts the Democrats lose; a point H at the center of mass of the points v_i the Democrats win; and a third point G at (k/N,1/2), where k is the number of districts the Democrats lose. See Figure <ref>.If the district plan treats the parties symmetrically, we would expect the point G to lie on the line FH. As such we define the declination, δ, to be 2/π times the angle between the lines FG and GH (using the convention that a counterclockwise angle from FG to GH is measured positively). Multiplication by 2/π converts from radians to fractions of 90 degrees. Therefore, possible values of the declination are between -1 and 1. Positive values indicate a republican advantage while negative values indicate a democratic advantage.§.§ The S-declination The declination metric was introduced for the purpose of measuring the degree to which district plans result in partisan asymmetry in the vote distribution. It is essentially an angle associated to each election and it is not obvious that it can be used to provide a good estimate of how many extra seats have been won due to the asymmetry. While the declination itself does not count seats, we propose that the S-declination, which we set equal to 5Nδ/12, does.To support this claim we useon the vote distributions from a number of elections and examine the effect on the S-declination. If the distribution is modified by flipping one district from democratic to republican through packing or cracking, the S-declination should increase by about one. If the flip is from republican to democratic, it should decrease by about one. To this end, we consider the data set from <cit.> consisting of all state-wide elections to the US House of Representatives in presidential-election years since 1972. (Note that a multilevel model is used in <cit.> to impute the democratic vote fraction for uncontested elections.) For any such election we can attempt to pack or crack it in favor of either party. In Figure <ref> we illustrate the change in S-declination for those instances in which theis successful.The instances ofplotted in Figure <ref> are restricted in two ways. First, we require that each party win at least one seat both before and after the packing/cracking. This is necessary for the S-declination of each distribution to be defined. Second, we require that there be at least three districts into which to distribute the votes from the flipped district. For example, when a seat is being flipped from democratic to republican by cracking, we require that there be at least three republican seats in the original distribution. Together, these restrictions exclude states in which there are four or fewer congressional districts. In the current apportionment cycle there are 21 such states:AK, AR, DE, HI, IA, ID, KS, ME, MS, MT, ND, NE, NH, NM, NV, RI, SD, UT, VT, WV, and WY.There were 352 state-year pairs in which there were at least five districts and each party won at least one seat. Given the four possible combinations of packing/cracking and pro-republican/pro-democratic, this offers 1408 possible applications of . However, for 415 of these, either there was not enough room for the chosen packing/cracking or one of the constraints was not satisfied.When flipping a seat from democratic to republican, we find that 95% of the time the S-declination changes by an amount between 0.69 and 1.20. For a flip from republican to democratic, the corresponding range is -1.26 to -0.74. We conclude that the declination is a reasonably good recorder of .We estimate how many net seats have been won in each year on the basis of partisan asymmetry in the vote distribution as follows: For a given year add together the values of the S-declination (rounded to the nearest integer) for each state. The results are shown in Table <ref>. Unfortunately, the S-declination simply provides an estimate without error bounds. Note that a given state-year does not contribute to the sum from that year if one party wins all of the seats in the state that year. However, we do not place the additional constraints used for the generation of Figure <ref> as the values in Table <ref> rely only on actual vote distributions and not on values derived from .§ TWO RECENT STUDIESWe now turn our attention to two recent studies that have also attempted to answer the question of the net result of gerrymandering in the House.§.§ Royden and Li study The most recent study, a Brennan Center <cit.> report by Royden and Li, considers House elections since 2012. The authors' approach is to use functions (akin to the declination) that identify gerrymanders by quantifying asymmetries in the vote distribution among districts. They consider three such functions from the literature: the efficiency gap <cit.>, the seats-votes curve <cit.> and the median-mean difference <cit.>. The first two of these functions are able to provide estimates of the number of seats won as a result of gerrymandering. For the 2012 congressional elections, these functions both come to very similar conclusions: the Republicansgained between 25 and 37 extra seats. The estimate from Table <ref> falls at the lower end of this range. An important caveat to their methodology is that it does not account for any inherent democratic disadvantage owing to geographic clustering. This is true of the declination as well. However, the authors find a high correlation between states with single-party redistricting control and large numbers of extra seats, thereby suggesting that geography fails to account for a significant portion of the asymmetry. §.§ Chen and Cottrell study In <cit.>, Chen and Cottrell take what is perhaps a more intuitive approach to evaluating the net effect of gerrymandering. For each state, they use 200 computer-generated “neutral” district plans as a standard to which the results of the enacted district plan are compared. Utilizing aggregated precinct-level presidential vote data, they estimate the probability that each simulated district in each simulated plan elects a democratic representative. These probabilities, over all districts, immediately yield an expected number of democratic representatives in the House under each such simulated district plan. They find that the expected number of democratic seats in the enacted plans is only one less than the average of the expected number of seats in the simulated plans, leading them to conclude that the net effect of gerrymandering in the House is trivial. Note that a putative advantage is that by incorporating the geographic distribution of voters, this approach has the potential to account for geographic clustering. Choosing simulated district plans from an appropriate distribution is a notoriously difficult problem that is not even well defined. Issues one must contend with include how to deal with constraints imposed by the Voting Rights Act; how to recognize communities of interest; the extent to which one should respect existing political boundaries; how to define compactness; how strict compactness constraints should be made; and how to weigh all of these disparate factors. Even once one makes such decisions, it is not necessarily clear how one chooses uniformly from the space of remaining “acceptable” district plans. Finally, there remains the nebulous problem of how to relate the choices made to what human map drawers do, or should be doing, when they create maps. We do not address these issues as they arise in <cit.> as they are nuanced and beyond the scope of this paper.The declination discussed in Section <ref> and the functions utilized by Royden and Li directly count extra seats won as a result of partisan asymmetry in the vote distribution. However imperfect these measures may be, we have no more logically direct way to count such seats. As mentioned above, the approach of Chen and Cottrell has the advantage that it more directly addresses inherent geographic advantages. However, while this approach should, a priori, be able to count extra seats, the connection is less direct. In the remainder of this section we argue, in fact, that the logistic regression model used fails to capture gerrymandering to any appreciable degree. In the computations that follow, we will omit uncontested districts: Our goal is not to accurately estimate the exact number of seats a given party would win, but simply to see how the estimate changes under . We now present the notation required for the model. Let p_i denote the presidential vote in district i. Following Chen and Cottrell, we consider a simple logistic-regression model for estimating the expected number of democratic seats won in district i as a function of p_i.According to their logistic model, the expected number of democratic seats in district i is F(β_0+β_1 p_i) where F(x)=(1+e^-x)^-1 and (β_0,β_1) are regression coefficients that are estimated from the data. For an election with presidential vote = (p_1,p_2,…,p_N), the expected number of democratic seats is thenE()=∑_i=1^N F(β_0+β_1 p_i).Chen and Cottrell used presidential data from 2008 and 2012 along with the four House elections from 2006 to 2012 to estimate the parameters β_0 and β_1. In order to use election data from 1972 to 2012, we replaced β_0 and β_1 with random year effects β_0[j] and β_1[j] with j indexing the year. Under this model (suppressing the dependence of E() on j), the expected number of democratic seats becomesE()=∑_i=1^N F(β_0[j]+β_1[j] p_i).Estimates of the intercept and slope parameters β_0[j] and β_1[j] using maximum likelihood are listed in Table <ref>.Our approach to analyzing this model is analogous to our validation of the S-declination: We applyto historical vote distributions and observe the effect on the predicted number of seats each party will win. However, there is an added complication. When packing and cracking, the vote of greatest interest in district i is not the presidential vote p_i, but the legislative vote ℓ_i. Unfortunately,provides us only with a modified legislative vote. We will need a mechanism for estimating district-level presidential votes from district-level legislative votes.If p_i and ℓ_i directly determined each other, there would be no need for a logistic regression in (<ref>). You would be able to determine who won the seat from the value of p_i without any appeal to probability. As shown in Figure <ref> (see also Table <ref>), there is some justification for asserting a linear relation between the presidential vote and the legislative vote. (This relation is certainly much stronger in some years than others.) One could also choose to fit the three subsets corresponding to elections in which there is a democratic incumbent, a republican incumbent, or no incumbent, however we do not pursue this variation.In the remainder of this subsection we will model p_i as a linear function, g, of ℓ_i. Letting j index the year, we set p_i = g(ℓ_i) := γ_0[j] + γ_1[j]ℓ_i for some coefficients γ_0[j] and γ_1[j]. The function g is dependent on the year j, however we will suppress this from the notation. For the remainder of this section we will reserve = (_1,…,_N) and = (_1,…,_N) for the historical values given to us in our data set. We reserve = (_1,…,_N) for the legislative vote after applying one of the four variations ofto . The presidential vote associated to _i will be written _i = g(_i). (Note, however, that g(_i) does not equal _i.)§.§.§ The simplest model: γ_0[j] = 0, γ_1[j]=1 for all j.We begin with the simplest reasonable parameters: γ_0[j] = 0 and γ_1[j] = 1 for all j. While there are a number of obvious objections to assuming the presidential and legislative votes are equal, there are reasons to believe that it is a best-case scenario for the Chen-Cottrell approach. That is, if their approach does not recordunder this model, then it will not do so under a more tenuous relation between the two votes. Since E(g()) is the expected number of democratic seats initially, E(g())-E(g()) is the change in the expected number of democratic seats after . In Figure <ref>, a vertical coordinate of 1 indicates one more seat for the Republicans. For consistency, therefore, we consider the negative, E(g()))-E(g()), the expected change in the number of republican seats. By (<ref>) and our choice of γ_0[j]=0, γ_1[j]=1, we then have that the expected change in republican seats is given byE(g())-E(g()) = ∑_i=1^N [F(β_0[j]+β_1[j] g()_i) - F(β_0[j]+β_1[j] g()_i)]= ∑_i=1^N [F(β_0[j]+β_1[j] (0 + 1·_i) - F(β_0[j]+β_1[j] (0 + 1·_i)]= ∑_i=1^N [F(β_0[j]+β_1[j]_i) -F(β_0[j]+β_1[j]_i))].Note that we do not usein the above computation. No matter how the function g biases the answer, we at least want this bias to be consistent both before and after the .Figure <ref> depicts values of the expression in (<ref>) under the flipping of one seat via . This figure is analogous to Figure <ref>, which shows the corresponding results for the S-declination.If this model faithfully recorded the effects of gerrymandering — and hence packing and cracking — we would see the red squares vertically clustered around +1 and the blue circles vertically clustered around -1. However, as seen in Figure <ref>, this is not at all the case. §.§.§ Qualitative explanation for why Chen-Cottrell model fails. We believe there is a simple reason for the relative invisibility of packing and cracking to the logistic model. When districts are redrawn so that, say a 55% democratic district becomes 45% democratic, the map drawers are doing so with a detailed understanding of the partisan composition of each affected district. While the gerrymander will degrade over time and may be susceptible to wave elections, the probability of the particular district electing a democratic legislator goes from essentially one to essentially zero. However, the probability of that district being democratic as determined by the fitted logistic curve will change by a much smaller amount. Furthermore, the decrease in probability for the given district will be at least partially offset by increased probabilities in the districts to which the democratic voters are now allocated.Figure <ref> illustrates how cracking or packing a single district could lead to at most a small change in the expected number of seats as determined by a sum over the probabilities obtained from the logistic regression. Since we are setting p_i = 0 + 1·ℓ_i = ℓ_i, there is only a single vote fraction associate to each district, which we denote by v_i. We write q_i for F(β_0 + β_1 v_i), the probability of a democratic win in district i. The corresponding values afterare denoted by v^*_i and q^*_i, respectively. Suppose District 1 is being flipped from republican to democratic and suppose District 2 received some of the reallocated democratic votes. In Figure <ref>.A, we see that the reduction in probability from q_1 to q^*_1 is almost exactly offset by a corresponding increase from q_2 to q^*_2. In Figure <ref>.B, there is almost no change from q_2 to q^*_2, leading to a net decrease in the expected number of democratic seats. However, the net decrease is still much less than one. §.§.§ Choosing γ_0[j] and γ_1[j] through linear regression.The choice of (γ_0[j],γ_1[j]) = (0,1) for all j is useful for understanding the structural reasons for why the presidential-vote logistic model does not record partisan gerrymandering. However, an ordinary linear regression fitted to the presidential and legislative votes should yield a quantitatively superior model. If we perform an ordinary linear regression using a random effects model, we obtain the estimates of the coefficients γ_0[j] and γ_1[j] displayed in Table <ref>. But these more defensible choices of γ_0[j] and γ_1[j] do little to improve the correspondence betweenand the difference E(g()) - E(g()). In Figure <ref> we display the values ofE(g())-E(g()) =∑_i=1^N F(β_0[j] + β_1[j](γ_0[j] + γ_1[j]_i)) -∑_i=1^N F(β_0[j] + β_1[j] (γ_0[j] + γ_1[j]_i))as j indexes the years from 1972 to 2012. While there is better separation between the two populations, the absolute change in the expected number of seats is only about a tenth of a seat, even worse than when we conflated legislative and presidential votes.§.§.§ Noise in the linear model.One possibility worth examining is that the failure of the Chen-Cottrell model in our simulations ofis due to our modeling of the relationship between presidential vote and legislative vote as linear. The linear relationship is much truer in some years than others and, with the exception of 2012, is never completely convincing. But arguing against this possibility is the fact that data points from 2012 are included in Figures <ref> and <ref> and there is no subpopulation clustered around ± 1. In fact, the sensitivity does seem better for 2012 with a median (absolute-value) change of just over one half. Regardless, any “noise” added to the relationship will merely impede the ability of the presidential-vote logistic regression model to record packing and cracking.A model that postulates a linear relation between the proportion of legislative and presidential vote in a district induces a probability model for the probability of a democratic win in a district given the presidential vote.It can be shown that the effect of presidential vote in the probability model decreases to zero as the correlation between legislative and presidential vote decreases to zero.In other words, if there is a significant amount of noise in the relation between legislative and presidential vote, presidential vote will be a weak predictor of the winner of a district, further degrading the ability of the logistic model to capture the effects of gerrymandering. Incumbent US House Representatives typically win upwards of 90% of their races. In particular, the fitted logistic regression curves will be very different for the classes of 1) democratic incumbents, 2) republican incumbents and 3) no incumbents. There is no reason incumbency cannot be incorporated into the Chen-Cottrell model, but there is one very important issue: The assumptions made for the distribution of incumbents will have a significant impact on the conclusions. Any historical gerrymander will change the relative proportion of incumbents which, in turn, will affect the future probabilities for who wins which seats. § CONCLUSIONIf the net impact of partisan gerrymandering on the US democracy is understood to be minimal on a national level, there are likely to be only piecemeal efforts to mitigate its influence. If, on the other hand, its effect is shown to be large, there is likely to be a greater political and judicial will to take steps to counter it on a national level. Unfortunately, as detailed in <cit.>, there are manifest difficulties in directly measuring the net effect of gerrymandering. The Chen-Cottrell approach inarguably addresses some of these difficulties. For example, by using contemporaneous electoral data, it can remove year-to-year effects. And, assuming the simulations draw from the space of all districts in an appropriate manner, it can account for geographic clustering. (Unfortunately, there is no objectively “correct” distribution to draw from. Even showing that the draws are sufficiently random is a difficult matter.)However, as we have attempted to show above, performing a logistic regression on presidential data is fatally flawed for this particular purpose. Not only do our simulations indicate that it does not effectively record packing and cracking, but we believe there are convincing theoretical reasons for concluding it does not.The S-declination, defined and studied in this paper, provides a measure of the number of seats gained/lost through partisan gerrymandering. The validity of the S-declination for this purpose is strongly supported by simulated packing and cracking (Figure <ref>).Neither we nor Royden and Li take into account geographic clustering that has been shown to exist in <cit.>. Nonetheless, their results and our Table <ref> suggest that the net number of seats won by gerrymandering is likely to be significant. And, as noted in <cit.>, it is not clear to what degree district plans should be allowed to exacerbate (or mitigate) inherent geographic distributions. Regardless, if there is only a minimal net effect of gerrymandering on a national level, we find no evidence in <cit.> to support this position.§ ACKNOWLEDGMENTS The US congressional data through 2014 was provided by <cit.>. The election data was analyzed using the python-based SageMath <cit.>. See <cit.> for packages used for, and details of, the imputation of votes. Additional Python packages employed in this paper were Matplotlib <cit.> for plotting and visualization and SciPy <cit.> for statistical methods. Estimated coefficients for the linear and logistic regressions were computed using the lme4 package <cit.> in R <cit.>. We thank Jordan Ellenberg for suggesting a secondalgorithm to check.alpha | http://arxiv.org/abs/1707.08681v2 | {
"authors": [
"Jeffrey S. Buzas",
"Gregory S. Warrington"
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"categories": [
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"primary_category": "stat.AP",
"published": "20170727014628",
"title": "Gerrymandering and the net number of US House seats won due to vote-distribution asymmetries"
} |
⟨ ⟩ (())[[ ]] | http://arxiv.org/abs/1707.08499v2 | {
"authors": [
"Ada Altieri",
"Maria Chiara Angelini",
"Carlo Lucibello",
"Giorgio Parisi",
"Federico Ricci-Tersenghi",
"Tommaso Rizzo"
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"categories": [
"cond-mat.dis-nn"
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"primary_category": "cond-mat.dis-nn",
"published": "20170726153420",
"title": "Loop expansion around the Bethe approximation through the $M$-layer construction"
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RankIQA: Learning from Rankings for No-reference Image Quality Assessment Xialei LiuComputer Vision Center Barcelona, [email protected] van de WeijerComputer Vision CenterBarcelona, [email protected] Andrew D. Bagdanov MICC, University of Florence Florence, [email protected] ================================================================================================================================================================================================================================================================We propose a no-reference image quality assessment (NR-IQA) approach that learns from rankings(RankIQA). To address the problem of limited IQA dataset size, we train a Siamese Network to rank images in terms of image quality by using synthetically generated distortions for which relative image quality is known. These ranked image sets can be automatically generated without laborious human labeling. We then use fine-tuning to transfer the knowledge represented in the trained Siamese Network to a traditional CNN that estimates absolute image quality from single images. We demonstrate how our approach can be made significantly more efficient than traditional Siamese Networks by forward propagating a batch of images through a single network and backpropagating gradients derived from all pairs of images in the batch. Experiments on the TID2013 benchmark show that we improve the state-of-the-art by over 5%. Furthermore, on the LIVE benchmark we show that our approach is superior to existing NR-IQA techniques and that we even outperform the state-of-the-art in full-reference IQA (FR-IQA) methods without having to resort to high-quality reference images to infer IQA.§ INTRODUCTION Images are everywhere in our life. Unfortunately, they are often distorted by the processes of acquisition, transmission, storage, and external conditions like camera motion. Image Quality Assessment (IQA) <cit.> is a technique developed to automatically predict the perceptual quality of images. IQA estimates should be highly correlated with quality assessments made by a range of very many human evaluators (commonly referred to as the Mean Opinion Score (MOS) <cit.>). IQA has been widely applied to problems where image quality is essential, like imagerestoration <cit.>, image super-resolution <cit.>, and image retrieval <cit.>.IQA approaches are generally divided into three categories based on whether the undistorted image (called reference image) or information about it is available: full-reference IQA (FR-IQA), reduced-reference IQA (RR-IQA), and no-reference IQA (NR-IQA). Research has mostly focussed on the more realist scenario of NR-IQA where the image quality of an image without any reference image has to be estimated. In NR-IQA, many methods focus on a specific distortion <cit.>, which limits the applicability of these methods. Other methods consider a range of distortions <cit.>.Convolutional Neural Networks (CNNs) are having an enormous impact on computer vision research and practice. Though they have been around for decades <cit.>, it wasn't until 2012, when Krizhevsky et al. <cit.> achieved spectacular results with a CNN in the ImageNet competition, that they achieved wide attention and adoption in the broader computer vision community. The architectures of networks are getting deeper and deeper with respect to the original AlexNet, with ResNet being an example of very deep network architecture <cit.>. The result of this trend is that state-of-the-art CNNs like AlexNet and ResNet have hundred of millions of parameters and require massive amounts of data to train from scratch (without overfitting). The success of CNNs encouraged research exploring their potential application to the NR-IQA problem. This research resulted in significant improvements compared to previous hand-crafted approaches <cit.>. The main problems these papers had to address is the absence of large datasets for IQA . Especially as networks grow deeper and wider, the number of parameters increases dramatically. As a consequence, larger and larger annotated datasets are required for training. However, the annotation process for IQA image datasets requires multiple human annotations for every image, and thus the collection process is extremely labor-intensive and costly. As a results, most available IQA datasets are too small to be effective for training CNNs.We propose an approach to address the absence of large datasets. The main idea (see Fig. <ref>) is that while human-annotated IQA data is difficult to obtain, it is easy to generate images that are ranked according to their image quality. That is, we can generate image sets in which, though we do not have an absolute quality measure for each generated image, for any pair of images we know which is of higher quality. We call this learning from rankings approach RankIQA, and with it we learn to rank image in terms of quality using Siamese Networks, and then we transfer knowledge learned from ranked images to a traditional CNN fine-tuned on IQA data in order to improve the accuracy of IQA. The idea to learn IQA features from distorted reference images was proposed by Zhang et al. in a patent <cit.>. In this paper we go beyond this patent in that we provide a detailed description of our method and experimentally verify the usefulness of pre-training networks using ranked datasets.As a second contribution we propose a method for efficient backpropagation in Siamese networks. The method forwards a batch of images through a single network and then backpropagates gradients derived from all pairs in the batch. In extensive experiments on established IQA datasets we show that learning from rankings significantly improves results, and that our efficient backpropagation algorithm allows to train these networks better and faster than other training protocols, like hard-negative mining. The supplementary material and project page are available at urlhttps://github.com/xialeiliu/RankIQA.§ RELATED WORKWe briefly review the literature related to our approach. We focus on distortion-generic NR-IQA since it is more generally applicable than the other IQA research lines. Traditional NR-IQA approaches.Most traditional NR-IQA can be classified into Natural Scene Statistics (NSS) methods and learning-based methods. In NSS methods, the assumption is that images of different quality vary in the statistics of responses to specific filters. Wavelets <cit.>, DCT <cit.> and Curvelets <cit.> are commonly used to extract the features in different sub-bands. These feature distributions are parametrized, for example with the Generalized Gaussian Distribution <cit.>. The aim of these methods is to estimate the distributional parameters, from which a quality assessment can be inferred. The authors of <cit.> propose to extract NSS features in the spatial domain to obtain significant speed-ups. In learning-based methods, local features are extracted and mapped to the MOS using, for example, Support Machine Regression or Neural Networks <cit.>. The codebook method <cit.> combines different features instead of using local features directly. Datasets without MOS can be exploited to construct the codebook by means of unsupervised learning, which is particularly important due to of the small size of existing datasets. Saliency maps <cit.> can be used to model human vision system and improve precision in these methods. Deep learning for NR-IQA.In recent years several works have used deep learning for NR-IQA <cit.>. One of the main drawbacks of deep networks is the need for large labeled datasets, which are currently not available for NR-IQA research. To address this problem Kang et al. <cit.> consider small 32 × 32 patches rather than images, thereby greatly augmenting the number of training examples. The authors of <cit.> follow the same pipeline. In <cit.> the authors design a multi-task CNN to learn the type of distortions and image quality simultaneously. Bianco at al. <cit.> propose to use a pre-trained network to mitigate the lack of training data. They extract features from a pre-trained model fine-tuned on an IQA dataset. These features are then used to train an SVR model to map features to IQA scores. In our paper, we propose a radically different approach to address the lack of training data: we use a large number of automatically generated rankings of image quality to train a deep network. This allows us to train much deeper and wider networks than other methods in NR-IQA which train directly on absolute IQA data. Learning to rank. These approaches learn a ranking function from ground-truth rankings by minimizing a ranking loss <cit.>. This function can then be applied to rank test objects. The authors of <cit.> adapt the Stochastic Gradient Descent method to perform pairwise learning to rank. This has been successfully applied to large datasets. Combining ideas from ranking and CNNs, the Siamese network architecture achieves great success on the face verification problem <cit.>, and in comparing image patches <cit.>. The only other work which applies rankings in the context of NR-IQA is <cit.> in which they combine different hand-crafted features to represent image pairs from the IQA dataset.Our approach is different in that primarily we are not aiming to learn rankings. Instead we use learning from rankings as a data augmentation technique: we use easily obtainable datasets of ranked images to train a large network, which is then fine-tuned for the task of NR-IQA.Hard-negative mining for Siamese network training. It is well known that a naive approach to sampling pairs to training Siamese networks is suboptimal. To address this problem several approaches to hard-negative mining have been proposed. In <cit.>, they propose a hard positive and hard-negative mining strategy to forward-propagate a set of pairs and sample the highest loss pairs with back-propagation. However, hard mining comes with a high computational cost (they report an increase of up to 80% of total computation cost). In <cit.> they propose semi-hard pair selection, arguing that selecting hardest pairs can lead to bad local minima. The batch size used is around 1800 examples, which again leads to a considerable increase in computational cost. In <cit.> the authors take a batch of pairs as input and choose the four hardest negative samples within the mini-batch. To solve for a bad local optimum, <cit.> optimize a smooth upper bound loss function to take advantage of all possible pairs in the mini-batch.In contrast with these works, we propose a method for efficient Siamese backpropagation which does not depend on hard-negative selection. Instead, it considers all possible pairs in the mini-batch. This has the advantage that the main computational bottleneck in training deep networks, namely the forward-propagation of images through the network, is optimally exploited. § LEARNING FROM RANKINGS FOR NR-IQAIn this section we describe our approach to exploiting synthetically generated rankings for NR-IQA. We first lay out a general framework for our approach, then describe how we use a Siamese network architecture to learn from rankings. Finally, in section <ref> we show how backpropagation for training Siamese networks from ranked samples can be made significantly more efficient. §.§ Overview of our approachThe lack of large IQA datasets motivates us to propose a new strategy to take advantage of large, unlabelled databases from which we can generate images ranked by image quality. Our approach is based on the observation that, given a set of arbitrary reference images, it is very easy to apply image distortions to generate a ranking image dataset. As an example, given a reference image we can apply various levels of Gaussian blur. The set of images which is thus generated can be easily ranked because we do know that adding Gaussian blur (or any other distortion) always deteriorates the quality score. Note that in such set of ranked images we do not have any absolute IQA scores for any images – but we do know for any pair of images which is of higher quality.After learning on these ranked images, we can use fine-tuning on small image quality datasets in order to address the IQA problem. The difference between our approach and the straightforward, classical approach <cit.> is shown in Fig. <ref>. The standard approach trains a shallow network directly on the IQA dataset to estimate IQA score from images. Due to the limited data only few layers can be used, which limits accuracy. Since we have access to much larger datasets with ranked images, we can now train deeper and wider networks to learn a distance embedding. Next we follow this by fine-tuning for domain adaptation to the absolute IQA problem. The overall pipeline of our approach is:* Synthesize ranked images. Using an arbitrary set of images, we synthetically generate deformations of these images over a range of distortion intensities. The absolute distortion amount for each image is not used in subsequent steps, but within each deformation type we know, for any pair of images, which is of higher quality. See section <ref> for a description of the datasets used for generating ranked images and the distortions applied.* Train Siamese network for ranking. Using the set of ranked images, we train a Siamese network described in the next section using the efficient Siamese backpropagation technique proposed in section <ref>. The result is a Siamese network that ranks images by image quality. * Fine-tune on IQA data. Finally, we extract a single branch of the Siamese network (we are interested at this point in the representation learned in the network, and not in the ranking itself), and fine-tune it on available IQA data. This effectively calibrates the network to output IQA measurements.§.§ Siamese networks for rankingHere we introduce the Siamese network <cit.> to learn from image rankings, which is a network with two identical network branches and a loss module. The two branches share weights during training. Pairs of images and labels are the input of the network, yielding two outputs which are passed to the loss module. The gradients of the loss function with respect to all model parameters are computed by backpropagation and updated with the stochastic gradient method. Specifically, given an image x as the input of the network, the output feature representation of x, denoted by f(x;θ), is obtained by capturing the activation in the last layer. Here θ are the network parameters, and we will use y to denote the ground truth value for the image which for image quality assessment is its quality score. Consequently, in our Siamese networks the output of the final layer is always a single scalar which we want to be indicative of the image quality.Since our aim is to rank the images, we apply the pairwise ranking hinge loss:L(x_1, x _ 2 ;θ) = max(0,f(x_2;θ)-f( x_ 1 ;θ)+ε )where ε is the margin. Here we assume without loss of generality that the rank of x_1 is higher than x_2. The gradient of the loss in Eq. <ref> is given by: 0.8!∇_θL =0 iff(x _ 2 ;θ) - f(x _ 1 ;θ) + ε≤ 0∇_θf(x _ 2 ;θ) -∇_θf(x _ 1 ;θ) otherwise.In other words, when the outcome of the network is in accordance with the ranking, the gradient is zero. When the outcome of the network is not in accordance we decrease the gradient of the higher and add the gradient of the lower score. Given this gradient of L with respect to model parameters θ, we can train the Siamese network using Stochastic Gradient Descent (SGD). §.§ Efficient Siamese backpropagationOne drawback of Siamese networks is the redundant computation. Consider all possible image pairs constructed from three images. In a standard implementation all three images are passed twice through the network, because they each appear in two pairs. Since both branches of the Siamese network are identical, we are essentially doing twice the work necessary since any image need only be passed once through the network. It is exactly this idea that we exploit to render backpropagation more efficient. In fact, nothing prevents us from considering all possible pairs in the mini-batch, without hardly any additional computation. We add a new layer to the network that generates all possible pairs in a mini-batch at the end of the network right before computing the loss. This eliminates the problem of pair selection and boosts efficiency. To approximate the speed-up of efficient Siamese backpropagation consider the following. If we have one reference image distorted by n levels of distortions in our training set, then for a traditional implementation of the Siamese network we would have to pass a total of n ^ 2 -n images through the network – which is twice the number of pairs you can generate with n images. Instead we propose to pass all images only a single time and consider all possible pairs only in the loss computation layer.This reduces computation to just n passes through the network. Therefore, the speed-up is equal to: n ^ 2 -n / n=n -1. In the best scenario n=M, where M is equal to the number of images in the mini-batch, and hence the speed-up of this method would be in the order of the mini-batch size. Due to the high correlation among the set of all pairs in a mini-batch, we expect the final speedup in convergence to be lower. To simplify notation in the following, we let ŷ_i = f(x_i;θ), where f(x_i;θ) is the output of a single branch of the Siamese network on input x_i. Now, for one pair of inputs the loss is:L(x_1,x_2,l_12; θ) = g(ŷ_1,ŷ_2,l_12),where l_12 is a label indicating the relationship between image 1 and 2 (for example, for ranking it indicates whether x_1 is of higher rank than x_2), and θ = {θ_1, θ_2, …, θ_k} are all model parameters in the Siamese network. We omit the θ dependency in g for simplicity. The gradient of this loss function with respect to the model parameter θ is:∇_θ L= ∂ g(ŷ_1,ŷ_2,l_12) /∂ŷ_1 ∇_θŷ_1 + ∂ g(ŷ_1,ŷ_2,l_12) /∂ŷ_2 ∇_θŷ_2.This gradient of L above is a sum since the model parameters are shared between both branches of the Siamese network and ŷ_1 and ŷ_2 are computed using exactly the same parameters.Considering all pairs in a mini-batch, the loss is:L = ∑_i=1^M∑ _ j > i ^ Mg(ŷ_i,ŷ_j,l_ij) The gradient of the mini-batch loss with respect to parameter θ can then be written as: ∇_θ L= ∑_ i=1 ^ M ∑_ j > i ^M∂ g(ŷ_i,ŷ_j, l_ij) /∂ŷ_i∇_θŷ_i+∂ g(ŷ_i,ŷ_j,l_ij) /∂ŷ_j ∇_θŷ_jWe can now express the gradient of the loss function of the mini-batch in matrix form as:∇ _θL = [ [ ∇ _θŷ_1 ∇ _θŷ_2 … ∇ _θŷ_M; ]]P1_Mwhere 1_M is the vector of all ones of length M. For a standard single-branch network, we would average the gradients for all batch samples to obtain the gradient of the mini-batch. This is equivalent to setting P to the identity matrix in Eq. <ref> above. For Siamese networks where we consider all pairs in the mini-batch we obtain Eq. <ref> by setting P to: =1.5pt P = [0∂ g(ŷ_1,ŷ_2,l_12) /∂ŷ_1⋯ ∂ g(ŷ_1,ŷ_M,l_1M)/∂ŷ_1∂ g(ŷ_1,ŷ_2,l_12) /∂ŷ_2 0⋯ ∂ g(ŷ_2,ŷ_M,l_2M) /∂ŷ_2⋮ ⋮ ⋱ ⋮ ∂ g(ŷ_1,ŷ_M,l_1M) /∂ŷ_M⋯ ⋯0 ] The derivation until here is valid for any Siamese loss function of the form Eq. <ref>. For the specific case of the ranking hinge loss replace g in Eq. <ref> with Eq. <ref> and obtain:P =[0 a _ 12⋯a _ 1M a _ 21 0⋯a _ 2M⋮ ⋮ ⋱ ⋮a _ M1⋯a _ M(M-1) 0 ],wherea_ij =0ifl_ij( ŷ_j -ŷ_i ) + ε≤0 l_ij otherwiseand l_ij∈{ - 1,0,1} where 1 (-1) indicates that y_i > y_j (y_i < y_j), and 0 that y_i = y_j or that they cannot be compared as is the case with images corrupted with different distortions. The above case considered a single distortion. Suppose instead we have D types of distortions in the training set. We can then compute the gradient of the loss using a block diagonal matrix as:∇ _θL = [ [ ∇ _θŶ^1 …∇ _θŶ^D;]][ [ A^1 0 ⋯ 0; 0 A^2 ⋯ 0; ⋮ ⋮ ⋱ ⋮; 0 0 ⋯ A^d; ]]1_M whereA ^ d= [0 a _ 12 ^ d⋯a _ 1M^d ^ d a _ 21 ^ d 0⋯a _ 2M^d ^ d⋮ ⋮ ⋱ ⋮a _ M^d1 ^ d⋯a _ M^d(M^d-1) ^ d 0 ],∇ _θŶ^d = [ ∇ _θŷ_d1 ⋯ ∇ _θŷ_dM ]. where ŷ_ mn refers to the network output of the n^th image with the m^th distortion, d ∈{1, 2, …, D}, and M = ∑M^d where M^d is the number of images with distortion d in the mini-batch. In the definition of A^d above, the a^d_ij are the gradient coefficients as in Eq. <ref>.§.§ Fine-tuning for NR-IQAAfter training a Siamese network to rank distorted images, we then extract a single branch from the network for fine-tuning. Given M images in mini-batch with human IQA measurements, we denote the ground truth quality score of the i-th image as y_i, and the predicted score from the network is ŷ_i, as above. We fine-tune the network using squared Euclidean distance as the loss function in place of the ranking loss used for the Siamese network:L(y_i, ŷ_i) =1 / M ∑ _ i=1 ^ M(y_i-ŷ_i)^2§ EXPERIMENTAL RESULTSIn this section we report on a number of experiments designed to evaluate the performance of our approach with respect to baselines and the state-of-the-art in IQA.§.§ Datasets We use two types of datasets in our experiments: generic, non-IQA datasets for generating ranked images to train the Siamese network, and IQA datasets for fine-tuning and evaluation.IQA datasets.We perform experiments on two standard IQA benchmark datasets.The LIVE <cit.> consists of 808 images generated from 29 original images by distorting them with five types of distortion: Gaussian blur (GB), Gaussian noise (GN), JPEG compression (JPEG), JPEG2000 compression (JP2K) and fast fading (FF). The ground-truth Mean Opinion Score for each image is in the range [0, 100] and is estimated using annotations by 161 human annotators.The TID2013 <cit.> dataset consists of 25 reference images with 3000 distorted images from 24 different distortion types at 5 degradation levels. Mean Opinion Scores are in the range [0, 9]. Distortion types include a range of noise, compression, and transmission artifacts. See the original publication for the list of specific distortion types.Datasets for generating ranked pairs. To test on the LIVE database, we generate four types of distortions which are widely used and common : GB, GN, JPEG, and JP2K. To test on TID2013, we generate 17 out of a total of 24 distortions (apart from #3, #4,#12, #13, #20, #21, #24). For the distortions which we could not generate, we apply fine-tuning from the network trained from the other distortions. This was found to yield satisfactory results. We use two datasets for generating ranked image pairs.The Waterloo dataset consists of 4,744 high quality natural images carefully chosen from the Internet. Additionally, we use the validation set of the Places2 <cit.> dataset of 356 scene categories. There are 100 images per category in the validation set, for a total 36500 images. After distortion, we have many more distorted images for learning an image quality ranking embedding. The aim of using this dataset is to demonstrate that high-quality ranking embeddings can be learned using datasets not specifically designed for the IQA problem. §.§ Experimental protocols We investigate a number of network architectures and use standard IQA metrics to evaluate performance. Network architectures. We evaluate three typical network architectures varying from shallow to deep. We refer to them as: Shallow, AlexNet <cit.>, and VGG-16 <cit.>. The shallow network has four convolutional layers and one fully connected layer.For AlexNet and VGG-16 we only change the number of outputs since our objective is to assign one score for each distorted image. Strategy for training and testing. We randomly sample sub-images from the original high resolution images. We do this instead of scaling to avoid introducing distortions caused by interpolation or filtering. The size of sampled images is determined by each network. However, the large size of the input images is important since input sub-images should be at least 1/3 of the original images in order to capture context information. This is a serious limitation of the patch sampling approach that samples very small 32 × 32 patches from the original images. In our experiments, we sample 227 × 227 and 224 × 224 pixel images, depending on the network. We use the Caffe <cit.> framework and train using mini-batch Stochastic Gradient Descent (SGD) with an initial learning rate of 1e-4 for efficient Siamese network training and 1e-6 for fine-tuning. Training rates are decreased by a factor of 0.1 every 10K iterations for a total of 50K iterations. For both training phases we use ℓ_2 weight decay (weight 5e-4). During training we sample a single subimage from each training image per epoch. When testing, we randomly sample 30 sub-images from the original images as suggested in <cit.>. The average of the outputs of the sub-regions is the final score for each image.Evaluation protocols. Two evaluation metrics are traditionally used to evaluate the performance of IQA algorithms: the Linear Correlation Coefficient (LCC) and the Spearman Rank Order Correlation Coefficient (SROCC). LCC is a measure of the linear correlation between the ground truth and the predicted quality scores. Given N distorted images, the ground truth of i-th image is denoted by y_i, and the predicted score from the network is ŷ_i. The LCC is computed as:LCC = ∑_ i=1 ^ N (y _ i - y) ( ŷ_ i -ŷ )/√(∑ _i^N (y _ i - y)^ 2 )√(∑ _ i ^ N( ŷ_ i -ŷ)^2)wherey andŷ are the means of the ground truth and predicted quality scores, respectively.Given N distorted images, the SROCC is computed as:SROCC = 1 -6∑ _ i=1 ^ N (v _ i - p _ i )^ 2 / N(N ^ 2 -1 ),where v_i is the rank of the ground-truth IQA score y_i in the ground-truth scores, and p_i is the rank of ŷ_i in the output scores for all N images. The SROCC measures the monotonic relationship between ground-truth and estimated IQA.§.§ Learning NR-IQA from rankingsWe performed a number of experiments to evaluate the ability of Siamese networks to learn to capture image distortions from a large dataset of image quality rankings.In addition, we measure the impact of the efficient Siamese backpropagation approach described in section <ref>. Siamese networks and IQA discrimination.To demonstrate the ability of our ranking networks to discriminate image quality, we trained our Siamese network on the Places2 validation set (without applying fine-tuning on IQA data) corrupted with five levels of a single distortion. We then used that network to predict image quality for synthetically-distorted images from the Waterloo dataset corrupted using the same five levels of the same distortion. The network outputs are plotted as histograms in Fig. <ref> for the JPEG distortion[Graphs for the other distortions are in the supplementary material.]. In the plot, we divide the observations according to the true distortion level (indicated by the color of the histogram). The model discriminates different levels of distortions on Waterloo, even though the acquisition process and the scenes of the two datasets are totally different.Efficient Siamese backpropagation.The objective of this experiment is to evaluate the efficiency of our Siamese backpropagation method.We compare our method to both standard random pair sampling, and a hard-negative mining method similar to <cit.>.[We experimented with several hard-negative mining methods and found this to work best.]For standard random pair sampling we randomly choose 36 pairs for each mini-batch from the training sets. For the hard negative mining strategy we start from 36 pairs in a mini-batch, and gradually increase the number of hard pairs every 5000 iterations. For our method we pass 72 images in each mini-batch. With these settings the computational costs of all three methods is equal, since at each iteration 72 images are passed through the network. We use AlexNet for this experiment. The comparison of convergence rates on JPEG[Results for the other distortions are in the supplementary material.] is shown in Fig. <ref>. The efficient Siamese backpropagation not only converges much faster, but also converges to a considerably lower loss.Network performance analysis. Here we evaluate whether we can learn a useful image representation from large image ranking datasets. We randomly split on the original, high-quality images before distortion from the LIVE dataset into 80% training and 20% testing samples and compute the average LCC and SROCC scores on the testing set after training to convergence. This process is repeated ten times and the results are averaged. In Table <ref> we compare results for three different networks: Shallow, AlexNet and VGG-16. We obtain the best results with the VGG-16 network, which is also the deepest network. This indicates learning from ranking makes it possible to train very deep networks efficiently without overfitting. These results are obtained by training from scratch, however we found that initializing the weights with a network pre-trained on ImageNet further improved the results. In the remainder of the experiments we thus use the VGG-16 network initialized with a pre-trained weights to train the ranking network in the following experiments. Baseline performance analysis. In this experiment, we evaluate the effectiveness of using rankings to estimate image quality. We compare tree methods: fine-tuning the VGG-16 network initialized from ImageNet to obtain the mapping from images to their predicted scores (called Baseline), our method to train VGG-16 (initialized from ImageNet) on ranking database using all ranking dataset we generate (called RankIQA), and finally our RankIQA approach fine-tuned on the TID2013 database after training using ranked pairs of images (called RankIQA+FT). We follow the experimental protocol used in HOSA <cit.>. The entire TID2013 database including all types of distortions is divided into 80% training images and 20% testing images according to the reference images. Thus, the same image can never appear in both training and test sets.The results are shown in Table <ref>, where ALL means testing all distortions together. All the experiments are performed 10 times and the average SROCC is reported[LCC results are provided in supplementary material.].From Table <ref>, we can draw several conclusions. First, it is hard to obtain good results by training a deep network directly on IQA data. This is seen in the Baseline results and is due to the scarcity of training data. Second, our RankIQA method achieves superior results on almost all individual distortions even without ever using the TID2013 dataset – which strongly demonstrates the effectiveness of training on ranking data. Slightly better results are obtained on ALL without comparing among different distortions during training the ranking network. The RankIQA-trained network alone does not provide accurate IQA scores (since it has never seen any) but does provide high correlation with the IQA scores as measured by SROCC.After fine-tuning on the TID2013 database (RankIQA+FT), we considerably improve the ALL score, and improve the baseline by 16%. However, in the fine-tuning process to optimize the ALL score the network balances the various distortions, and results decrease for several distortions. §.§ Comparison with the state-of-the-art We compare the performance of our method using the VGG-16 network with state-of-the-art methods.We perform experiments on the TID2013 and LIVE dataset.[Results on CSIQ <cit.> and MLIVE <cit.> are in supplementary material.] Evaluation on TID2013. Table <ref> also includes results of state-of-the-art methods. We see that for several very challenging distortions (14 to 18), where all other methods fail, we obtain satisfactory results. For individual distortions, there is a huge gap between our RankIQA method and other IQA methods on most distortions. The state-of-the-art method HOSA performs slightly better than our methods on 6 out of 24 distortions. For all distortions, our method RankIQA+FT achieves about 5% higher than HOSA. Our methods perform well on distortions which are not included when training the ranking network, which indicates that different distortions share some common representation and training the network jointly on all distortions.Evaluation on LIVE.As done in <cit.>, we randomly split the reference images on LIVE dataset into 80% training samples and 20% testing, and compute the average LCC and SROCC scores on the testing set after training to convergence. This process is repeated ten times and the results are averaged. These results are shown in Table <ref>. The best method for each dataset is indicated in bold.The column indicated with ALL means we combine all five distortions together on LIVE dataset to train and test the model. For fair comparison with the state-of-the-art, we train our ranking model on four distortions except FF, but we fine-tune our model on all five distortions in the LIVE dataset to compute ALL. Our approach achieves about 1% better than the best results reported on ALL distortions for LCC. Similar conclusions are obtained for SROCC. This indicates that our method outperforms existing work including the current state-of-the-art NR-IQA method SOM <cit.> and DNN <cit.>, and also state-of-the-art FR-IQA method DCNN <cit.>. To the best of our knowledge this is the first time that an NR-IQA method surpasses the performance of FR-IQA methods (which have access to the undistorted reference image) on all LIVE distortions using the LCC and SROCC evaluation method. §.§ Independence from IQA training data This final experiment is to demonstrate that our framework can be also trained on non-IQA datasets. In the previous experiment the network is trained from high-quality images of the Waterloo dataset. Instead here we use the validation set of the Places2 dataset to generate ranked images in place of the Waterloo dataset. The Places2 dataset is of lower quality than Waterloo and is not designed for IQA research.As in the previous experiment, the final image quality scores are predicted by fine-tuning on the LIVE dataset.The performance of this model is compared with the results trained on Waterloo in Table <ref>. The SROCC and LCC values are very similar, demonstrating that our approach can be learnt from arbitrary, non-IQA data.§ CONCLUSIONSTo address the scarcity of IQA data we have proposed amethod which learns from ranked image datasets. Since this data can be generated in abundance we can train deeper and wider networks than previous work. In addition, we have proposed a method for efficient backpropagation in Siamese networks which circumvents the need for hard-negative mining. Comparison with standard pair sampling and hard-negative sampling shows that our method converges faster and to a lower loss.Results on LIVE and TID2013 datasetsshow that our NR-IQA approach obtains superior results compared to existing NR-IQA techniques and even FR-IQA methods. Acknowledgements We acknowledgetheSpanish project TIN2016-79717-R, the CHISTERA project M2CR (PCIN-2015-251) and the CERCA Programme / Generalitat de Catalunya. Xialei Liu acknowledges the Chinese Scholarship Council (CSC) grant No.201506290018. We also acknowledge the generous GPU donation from NVIDIA. ieee | http://arxiv.org/abs/1707.08347v1 | {
"authors": [
"Xialei Liu",
"Joost van de Weijer",
"Andrew D. Bagdanov"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170726100240",
"title": "RankIQA: Learning from Rankings for No-reference Image Quality Assessment"
} |
Label1]Johannes Pekkilä [email protected] Label2]Miikka S. Väisälä [email protected],Label1]Maarit J. Käpylä Label4,Label1,Label3]Petri J. Käpylä Label5,Label1]Omer Anjum[Label1]ReSoLVE Centre of Excellence, Department of Computer Science, Aalto University, PO Box 15400, FI-00076 Aalto, Finland [Label2]Department of Physics, Gustaf Hällströmin katu 2a, PO Box 64, FI-00014 University of Helsinki, Finland [Label3]Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, D-37077 Göttingen, Germany [Label4]Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, D-11482 Potsdam, Germany [Label5]Nokia Solutions and Networks, Finland We focus on implementing and optimizing a sixth-order finite-difference solver for simulating compressible fluids on a GPU using third-order Runge-Kutta integration.Since graphics processing units perform well in data-parallel tasks, this makes theman attractive platform for fluid simulation. However, high-order stencil computation is memory-intensive with respect to both main memory and the cachesof the GPU. We present two approaches for simulating compressible fluids using 55-point and 19-point stencils. We seek to reduce the requirements for memory bandwidth and cache size in our methods by using cache blocking and decomposing a latency-bound kernel into several bandwidth-bound kernels. Our fastest implementation is bandwidth-bound and integrates 343 million grid points per second on a Tesla K40t GPU, achieving a 3.6 × speedup over a comparable hydrodynamics solver benchmarked on two Intel Xeon E5-2690v3 processors. Our alternative GPU implementation is latency-bound and achieves the rate of 168 million updates per second. ©2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license, <http://creativecommons.org/licenses/by-nc-nd/4.0/> Publisher DOI: 10.1016/j.cpc.2017.03.011Computational techniques: fluid dynamics Finite difference methods in fluid dynamics Hydrodynamics: astrophysical applications Computer science and technology47.11.-j 47.11.Bc 95.30.Lz 89.20.Ff§ INTRODUCTIONThe number of transistors in a microprocessor has been doubling approximately every two years and as a result, the performance of supercomputers measured infloating-point operations per second (FLOPS) has been following a similar increase.However, since increasing the clock frequencies of microprocessors to gain better performanceis no longer feasible because of power constraints, this has lead to a change in their architectures from single-core to multi-core. While modern central processing units (CPUs) utilize more cores and wider SIMD units, they are designed to perform well in general tasks where low memory access latency is important.On the other hand, graphics processing units (GPUs) are specialized in solving data-parallel problems found in real-time computer graphics and as a result, house more parallel thread processors and use higher-bandwidth memory than CPUs. With the introduction of general-purpose programming frameworks, such as OpenCL and CUDA, GPUs can now also be programmed to do general purpose tasks using a C-like language instead of using a graphics application-programming interface (API), such as OpenGL. In addition, APIs such as OpenACC can be used to convert existing CPU programs to work on a GPU. For these reasons, GPUs offer an attractive platform for physical simulations which can be solved in a data-parallel fashion.In this work we concentrate on investigating sixth-order central finite-difference scheme implementations on GPUs, suitable for multiphysics applications. The justification for the use of central differences withexplicit time stepping, a configuration which is not ideal concerning its stability properties, comes from the fact that, even though someamount of diffusion is required for stability, they provide very good accuracy and are easy to implement (see, e.g. <cit.>).In addition, the various types of boundary conditions and grid geometries needed in multiphysics codes such as the Pencil Code[1] are easy to implement with central schemes. Moreover, the problem has the potential to exhibit strong scaling with the number of parallel cores in the optimal case. [1] http://github.com/pencil-code There are astrophysical hydro- and magnetohydrodynamic solvers already modified to take advantage of accelerator platforms (i.e. <cit.>, <cit.>, <cit.>), that most often use low-order discretization. As an example of a higher-order scheme for cosmological hydrodynamics, we refer to <cit.>. We also note that more theoretical than application-driven work on investigating higher-order stencils on GPU architecture exists in the literature, see e.g. <cit.>. There are many scientific problems, such as modeling hydromagnetic dynamos, where long integration times are required, either to reach a saturated state (see e.g. <cit.>), or toexhibit non-stationary phenomena and secular trends (see e.g. <cit.>). Therefore, it is highly desirable to find efficient ways to accelerate the methods, GPUs offering an ideal framework. The accelerated codes typically employ lower-order conservative schemes, in which case the halo region to be communicated to compute the differences is small, and does not pose the main challenge for the GPU implementation. High-order schemes of similar type as presented here exist for two-dimensional hydrodynamics (e.g. <cit.>); in this paper, we deal with a 3D implementation of a higher-order finite-difference solver. Such schemes are much less diffusive and they are more suitable for accurate modeling of turbulence, which is, on the other hand, crucial for e.g. investigating various types of instabilities in astrophysical settings. One mundane example, which isthe solar dynamo, is responsible for all the activity phenomena on the Sun, driving the space weather and climate that affect life on Earth <cit.>. The accurate modeling of turbulence is also important in understanding suchphenomena as the structure of interstellar medium <cit.>and star formation <cit.>. We make the following contributions in this work. First, we describe, implement and optimize two novel methods for simulating compressible fluids on GPUs using sixth-order finite differences and 19- and 55-point stencils. The current implementation is for simulations of isothermal fluid turbulence. The bigger picture is that it uses the same core methods as the Pencil Code. Thus the current code development works as a pilot project in the conversion of the Pencil Code to use GPUs.Our implementations perform 1.7 × and 3.6 × fasterthan a state-of-the-art finite difference solver, Pencil Code, used for scientific computation on HPC-clusters. Second, we present an optimization technique called kernel decomposition, which can beused to improve the performance of latency-bound kernels. Currently our code, called Astaroth, supports isothermal compressive hydrodynamics, but it will be expanded in the future to include more complex physics, in the end supporting the full equations ofmagnetohydrodynamics (MHD). In this paper, we present the physical motivation (Sect. <ref>) behind our implementations, and the technical justification and background (Sect. <ref>). The details of our implementations and the Astaroth code are presented in Sect. <ref>. In Sect. <ref> we present the performance of our GPU implementationsand compare the results with physical test cases in Sect. <ref>. Finally, in Sect. <ref>, we discuss our results and conclude the paper.§ PROBLEM SPECIFICATIONHere we describe the basic equations and numerical methods featured in thecurrent implementation of the Astaroth code, which were also used in testingthe different optimizations. For simplicity, the code is limited to the domainof hydrodynamics. We consider the fluid to be isothermal and compressible,and we include the full formulation of viscosity to the momentum equation.This allows for testing our methods with reasonable enough physics whileavoiding overt complexity during the development of the methods. §.§ Governing equations In a compressible system, the conservation of mass can be expressed as the rate equation for density ρ, called the continuity equation: D lnρ/D t = - ∇·𝐮. Here D / Dt is the convective derivative∂ / ∂ t + (𝐮·∇ ) and density isexpressed in logarithmic form lnρ and 𝐮 is athree-dimensional velocity vector.The logarithmic form of density helps to avoid numerical errors that can occur with large stratifications or erroneously negative values of density.Momentum conservation in a viscous fluid is modelled by a rate equation commonly known as the Navier-Stokes equation. In the case ofisothermal viscous hydrodynamics featured in the Astaroth code it is given by: D 𝐮/D t = -c_s^2 ∇lnρ + 𝐟 +ν( ∇^2𝐮 +1/3∇(∇·𝐮) + 2 𝐒·∇lnρ), where -c_s^2 ∇lnρ is the pressure termin the isothermal case where the pressure is given by p=ρ c_s^2 with c_s being the constant sound speed, 𝐟 is an external body force, such as an external gravity field or a forcing function (see Sect. <ref>), ν is the kinematic viscosity coefficient, which is assumed constant, and 𝐒 is the traceless rate-of-strain tensor: S_ij = 1/2( ∂ u_i/∂ x_j +∂ u_j/∂ x_i - 2/3δ_ij∇·𝐮).§.§ Non-dimensional units and system parameters While solving the equations, we assume that the variables are described indimensionless manner. Depending on the nature of the actual physical question,the result can be scaled during the data analysis phase into relevant physicaldimensions. In this way, we can avoid using numerically unsound parametervalues. The dimensionless physical units are defined as ρ = ρ_phys/M_u/L_u^3,𝐮 = 𝐮_phys/L_u/T_u,ν = ν_phys/L_u^2/T_u, c_s = c_s,phys/L_u/T_u, k = k_phys/2π / L_u where L_u, M_u and T_u denote chosenunit scaling of length, mass and time respectively.In addition, we use an important dimensionless measure, the Reynolds number, Re = u L/ν where u and L represent characteristic velocities and length scales in the system. §.§ The finite difference methodWe discretize the fluid volume (L_x,L_y,L_z) onto an equidistant grid of (N_x, N_y,N_z) points, the distance between neighbouring grid points in each dimension being δ_i=L_i/N_i.With these, the approximation of the first derivative of function f on the grid point i with respect to the x-direction when using sixth-order central differences can be written as follows <cit.>.∂/∂ x f_i = 1/60δ_x( - f_i-3 + 9 f_i-2 - 45 f_i-1+ 45 f_i+1 - 9 f_i+2 + f_i+3) + O(δ_x^6). The second derivatives can be approximated with ∂^2/∂ x^2 f_i = 1/180δ_x( 2f_i-3 - 27f_i-2 + 270f_i-1 - 490f_i+270f_i+1 - 27f_i+2 + 2f_i+3) + O(δ_x^6). Additionally, mixed derivatives with respect to any two arbitrary directions can be approximated by using the following bidiagonal scheme <cit.>, here with respect to x- and y-directions. ∂^2/∂ x ∂ y f_i,j = 1/720δ_xδ_y [ 270( f_i+1, j+1 - f_i-1, j+1 + f_i-1, j-1 - f_i+1, j-1 )- 27 ( f_i+2, j+2 - f_i-2, j+2 + f_i-2, j-2 - f_i+2, j-2 )+ 2( f_i+3, j+3 - f_i-3, j+3 + f_i-3, j-3 - f_i+3, j-3 ) ]+ O(δ_x^6, δ_y^6) . Finally, we can approximate second and mixed derivatives using Eq. (<ref>) as follows ∂/∂ x(∂/∂ y f_i,j) = 1/60δ_x[ -(∂/∂ yf_i-3, j) + 9(∂/∂ yf_i-2, j) - 45(∂/∂ yf_i-1, j)+ 45(∂/∂ yf_i+1, j) - 9(∂/∂ yf_i+2, j) + (∂/∂ yf_i+3, j) ]+ O(δ_x^6).Computing the derivatives of a grid point requires information from its neighboringgrid points. This data access pattern is called a k-point stencil, where data from k input points isread in order to update the output point. We use R to denote the radius of the stencil,which is the Chebychev distance from the output point to the farthest input point of the stencil. Forcentral differences, the relation between the radius of the stencil R and the order of the finitedifference method n is thus n = 2R. Additionally, a function solving only the first- andsecond-order derivatives in d ≥ 1 dimensions using an nth-order central finite difference methoduses a (dn + 1)-point stencil to update a grid point, whereas a function, which computes also themixed derivatives requires a (dn + 2d2n + 1)-point stencil for d ≥ 2. The stencils used in suchfunctions are shown in Fig. <ref>.§.§ Runge-Kutta integration Our implementations are based on an explicit third-order Runge-Kutta formula,which is written as a 2N-storage scheme <cit.>. In this approach, only one set of intermediate values have to be memory-resident during integration.Let 𝐮 be a vector field the integration is performed on and 𝐮 be the field containing the intermediate values. Additionally, let𝐮^(s) be the value of 𝐮 during integration substep s. Finally,α^(s) and β^(s) are coefficients whose values depend on the chosen 2N-RK3 scheme and δ_t is the length of the time step.We can now write the 2N-RK scheme as 𝐮^(s+1) = α^(s)𝐮^(s) + ∂/∂ t𝐮^(s)δ_t𝐮^(s+1) = 𝐮^(s) + β^(s)𝐮^(s+1) . The pseudocode for a naïve integration with this scheme is shown in Algorithm <ref>. Here ρ_i and 𝐮_i are the intermediate results for density and velocity of a grid point at index i. Handling of the out-of-bound indices depends on the chosen boundary condition scheme. For the first step, α^(1) must be set to 0. § GPU ARCHITECTUREIn this section, we review GPU architecture using NVIDIA's CUDA as the framework of choice and discuss the challenges of high-order stencil computation on GPUs. While we use terminology associated with CUDA, the ideas represented here are also analogous with those found in OpenCL and computer architecturein general. We denote the alternative terminology in the footnotes. Throughout this work, we use NVIDIA'scompute capability 3.5 GPUs as the baseline architecture. Graphics processing units operate in a multi-threaded SIMD fashion and are designed to perform well in data-parallel tasks. In order to maximize throughput in thesetypes of tasks, GPUs employ a large number of parallel thread processors[1] and usespecialized GDDR SGRAM to increase memory bandwidth with the cost of increased access latency. ModernGPUs also employ small L1 and L2 caches to reduce pressure to the on-device memory. See Table <ref> for the detailed specifications of a Tesla K40t accelerator card used in this work. In order to hide pipeline and memory access latencies, GPUs rely mainly on multithreading a large number of threads on theirprocessors in a fine-grained fashion. Alternatively, in certain problems instruction-level parallelism (ILP) can beused for the same latency-hiding effect <cit.>.Modern NVIDIA GPUs consist of Streaming Multiprocessors[2] (SMX), which executewarps[3] of CUDA threads[4]. In current NVIDIA architectures, awarp is composed of 32 threads. The threads of a warp are executed in lockstep on the thread processors ofan SMX. Finally, sets of warps form thread blocks[5], which are distributed among SMXs by a thread block scheduler. For a more detailed description of the architecture of GPUs, we refer thereader to <cit.> and <cit.>.[1]Analogous terms: Processing Element, SIMD lane, CUDA core. [2]Analogous terms: Compute Unit, multi-threaded SIMD processor. [3]Analogous terms: Wavefront, SIMD thread. [4]Analogous terms: Work item, instruction stream of a SIMD thread. [5]Analogous terms: Work group.§.§ Related work and optimization techniques Optimization of GPU programs is often non-trivial and requires careful tuning to attain the highest throughput. Techniques for optimizing low-order stencil computation have been studied extensively in literature, e.g. by <cit.> <cit.> <cit.> <cit.> <cit.>, while work on higher-order stencil computation ismore limited <cit.> <cit.> and focuses on stencils, where only the axis-aligned elements are usedduring an update. To our knowledge, no previous work has been published on optimizing 55-point stencil computationusing stencil elements of size 16 bytes and larger.Memory bandwidth is a common bottleneck in stencil computation because of the low bandwidth-to-compute ratio incurrent microprocessor architectures. Previous work has suggested several techniques for reducing bandwidthrequirements, notably spatial cache blocking techniques which aim to reduce memory fetches by reusing as much ofthe data as possible before evicting it out of caches. We also adopted this idea in our implementations we discuss indetail in Sect. <ref> and <ref>.However, with higher-order stencils more data is required to be cache-resident in order to improve performance withcache blocking. Using large amounts of shared memory in a kernel reduces its occupancy, which in turn results in decreasedability to hide latencies if increasing ILP in the kernel is not possible. In current GPU architectures, the cache size isinsufficient for housing large three-dimensional blocks of 16-byte-sized grid points. § GPU IMPLEMENTATIONSHere we give an overview of our GPU implementations and the optimizations performed upon bothimplementations. First, we discretize the simulation domain into a grid as described in Sect. <ref>. The grid consists of a computational domain, which is surrounded by a ghost zone. We update the grid points in the computational domain using third-order Runge-Kutta integration, while the ghostzone is used to simplify integration near the boundaries. After each integration step, a number of grid pointsare copied from the computational domain into the ghost zone according to the boundary conditions. With this approach, we do not have to compute the boundary conditions within the integration kernel and can thus update all grid points in the computational domain using the same code. We use periodic boundary conditions throughout this work.We store density ρ, velocity 𝐮 and the intermediate result arrays ρ and 𝐮 into global memory as a structure of arrays. Arrays ρ and 𝐮include the ghost zone and are padded manually, such that the first grid point of the computational domainis stored in a memory address which is a multiple of 128 bytes. By padding, we seek to reduce the numberof memory transactions required to update the grid, which is discussed more in detail in Sect. <ref>. In order to avoid a race condition during integration, we allocate memory for arrays ρ, 𝐮, ρ^out and 𝐮^out, such that separate arrays are used for reading and writing. Arrays ρ and 𝐮 are passed to the integration kernels using theintrinsic, which enables these arrays to be read through the read-only texture cache. Additionally, we store all constants discussed in Sect. <ref> into constant memory. Finally, in both our integration kernels, we compute each derivative only once and store each value, which is used more than once, into either shared memory or registers.§.§ 55-point integration methodWe solve the problem on a GPU using a modified version of Algorithm <ref>. Thedifference is, that we use separate arrays for reading and writing and thus can update the computational domain in one pass over the grid points. At the end of each integration substep, the arrays are swapped efficiently using pointers.In our modified integration algorithm, we update a grid point by solving the continuity and Navier-Stokes equations, Eq. (<ref>) and (<ref>), within a single kernel. As wesolve the derivatives in these equations with the finite-difference equations (<ref>),(<ref>) and (<ref>), the integration kernel requires a 55-point stencil in order to update agrid point. Therefore we call this approach the 55-point method. The pseudocode for the 55-point method is shown in Algorithm <ref> using the notationintroduced in Sections <ref> and <ref>.The algorithm is implemented in CUDA as follows.Let τ_x, τ_y and τ_z be the dimensions of a thread block and R the radius of the stencil as defined in Sect. <ref>. We perform the integration by decomposing thecomputational domain into τ_x ×τ_y ×τ_z-sized blocks, where each grid point isupdated by a CUDA thread. Since the stencils used to update nearby grid points overlap, we can reduce globalmemory fetches by fetching the data used by the threads in a thread block into shared memory.The block of grid points stored into shared memory per a thread block is shown in Fig. <ref>. We call the area surrounding the shared memory block a halo in order todistinguish it from the ghost zones discussed in Sect. <ref>. Unlike ghost zones, theboundary conditions are not applied to the halo and the grid points in the halo are solely used for updating thegrid points near the boundaries of the thread block. For simplicity, we fetch a total of (τ_x+2R) ×(τ_y+2R) × (τ_z+2R) grid points into shared memory per a thread block, even when some of thegrid points are used only by a single thread or none at all. This approach avoids branching in the integrationkernel, as all threads follow the same execution path with the cost of additional memory fetches. We discuss these redundant memory fetches more in detail in Sect. <ref>.In order to reduce the number of memory transactions from global memory, we adopted the idea of a rollingcache. In this approach, part of the data in shared memory is reused for updating multiple grid pointsalong the z-axis. We implemented cache blocking for Alg. <ref> as follows. Initial step:(a) Assign a block of grid points in the decomposed computational domain to a CUDA thread block.(b) Fetch the data required for updating this block of grid points from global memory into shared memory[(τ_x+2R) × (τ_y+2R) × (τ_z+2R) grid points]. (c) Update the block of grid points using the data stored in shared memory. Subsequent steps: While a thread block has updated less than E_z blocks of grid points, do the following:(d) Assign the next block of grid points in the z-axis to the thread block.(e) Since the halos of nearby blocks overlap, part of the data obtained in the previous step can also be used to update the current block of grid points[(τ_x + 2R) × (τ_y + 2R) × 2R grid points]. Hold this data in shared memory. Load rest of the required data from global memory into shared memory[(τ_x + 2R) × (τ_y + 2R) ×τ_z grid points].(f) Update the assigned block of grid points using the data in shared memory.In our implementation of the rolling cache, we avoid copying data around shared memory by using a counter indicating the currentmid-point in the shared memory block. After updating a grid point, we increment this counter by τ_z inmodulo τ_z+2R. During differentiation, any out-of-bound indices encountered when accessing shared memory are also wrapped around modulo τ_z+2R.Additionally, let N_x, N_y and N_z be the dimensions of the computational domain andE_z the number of grid points updated by a CUDA thread. The total number of thread blocks required toupdate the grid is thus γ = N_x/τ_x·N_y/τ_y·N_z/τ_z E_z , and the number of grid points fetched from global memory is Γ_55p = (τ_x + 2R) × (τ_y + 2R) × (τ_zE_z + 2R). With the 55-point method, we perform the following number of read-writes RW_55p to global memory when N_xyz = N_xN_yN_z.Here we read from four arrays which require the halo (ρ and 𝐮) and four intermediate value arrays (ρ and 𝐮), and write the result back to eight arrays (ρ^out, 𝐮^out, ρ and 𝐮). RW_55p = 4 Γ_55pγ + 4 N_xyz_Reads + 8 N_xyz_Writes . However, the main problem with the 55-point method is low occupancy caused by the large amount of shared memory required by a thread block in order to benefit from cache blocking. This is the casealso for small thread blocks. For example, when using thread blocks of size τ_x = τ_y = τ_z = 8, R=3 andstoring 16 bytes of information per grid point, then a thread block requires 43904 bytes of the 49 152 bytes available shared memory on a Tesla K40t GPU. Therefore only onethread block can run on the GPU at a time. This is not enough to hide the latencies in our integration kernel,which becomes latency-bound. Moreover, instruction-level parallelism cannot be used extensively to hide thelatencies in our approach, since the data in shared memory can be updated only after the threads of a threadblock have updated their currently assigned grid points. §.§ 19-point integration methodTo alleviate the problem with high shared memory usage in our 55-point method, we represent an alternativeintegration method which uses an axis-aligned 19-point stencil to update a grid point. This is achieved bycomputing the gradient of divergence in Eq. <ref> in two passes over the grid. The benefit of thisapproach is, that stencil computation on GPU with axis-aligned stencils is extensively studied and efficient cacheblocking methods for such stencils are well known <cit.> <cit.> <cit.>. However,the disadvantages of this approach compared with our 55-point method are three-fold: first, we have to performmore floating-point arithmetic in order to update a grid point, which introduces a slight error. We show in Sect. <ref> that this error is negligible. Second, as the grid is updated in two steps, more memory transactions are requiredto complete a single integration step. Third, when solving the system with multipleGPUs, part of the divergence field has to be communicated between the nodes. The 19-point method works as follows. First, we reformulate the Navier-Stokes equation in such form, that mixedderivatives do not have to be solved in order to compute the gradient of divergence. This is achieved by dividing asubstep of a full integration step into two passes, where the divergence field is solved during the first pass andduring the second pass, thegradient of divergence is solved using the precomputed divergence field. The completereformulation of the Navier-Stokes equation is shown in Appendix A. Using the notation from Sect. <ref> and 𝐮_partial^(s) and 𝐮_partial^(s) to denote thepartially computed Navier-Stokes equation, we can write the calculations done during the first pass as follows.Here s denotes some substep of a full integration step. For RK3, s ∈{1, 2, 3}. ∇·𝐮^(s) = ∂/∂ x u + ∂/∂ y v + ∂/∂ z w𝐮_partial^(s+1)= α^(s)𝐮^(s) + δ_t[ -(𝐮^(s)·∇)𝐮^(s)-c_s^2 ∇lnρ^(s)+ ν( ∇^2𝐮^(s) + 2 𝐒·∇lnρ^(s))] 𝐮_partial^(s+1) = 𝐮^(s) + β^(s)𝐮_partial^(s+1) ,where 𝐮=(u,v,w).Then, with the second pass we complete the integration step by computing𝐮^(s+1) and 𝐮^(s+1) using the previously computed divergence field and the partial results. 𝐮^(s+1) = 𝐮_partial^(s+1) + δ_tν/3∇(∇·𝐮^(s))𝐮^(s+1) = 𝐮_partial^(s+1) + β^(s)δ_tν/3∇(∇·𝐮^(s)).The pseudocode for this approach is shown in Algorithm <ref>. We implemented the 19-point method in CUDA using the idea of 2.5D cache blocking <cit.> <cit.> to reduce the number of global memory transactions. In this approach, we set τ_z = 1 and store a 2-dimensional slab of data into shared memory, shown in Fig. <ref>. For simplicity, weallocate shared memory for (τ_x + 2R) × (τ_y + 2R) grid points, where the four R^2-sized corners ofthe slab are unused. Since the shared memory slab contains only grid points in the xy-plane, the rest of the stencil points required to solve the derivatives with respect to the z-axis are defined as local variables, which are placed into registers by the compiler. Similarly as in our 55-point method, each thread of a threadblock then updates multiple grid points along the z-axis. Cache blocking works in our implementation as follows.Initial step:(a) Assign a 2-dimensional block of grid points to a thread block.(b) Fetch the data required for solving the the derivatives in the xy-plane into shared memory[(τ_x + 2R) × (τ_y + 2R) - 4R^2 grid points] and the data required for solving the derivatives in the z-axis into the registers of each thread[2R grid points per thread].(c) Update the block of grid points using the data in shared memory and registers. Subsequent steps: While a thread block has updated less than E_z blocks of grid points, do the following:(d) Assign the next block of grid points in the z-axis to the thread block.(e) Update the non-halo area[(τ_x ×τ_y) grid points] of the shared memory slab using data stored in the registers of the threads and update the halo[(τ_x + 2R) × (τ_y + 2R) - (τ_x ×τ_y) - 4R^2 grid points] from global memory. For each thread, hold part of the data obtained in the previous steps in registers[2R - 1 grid points], but update the local variable corresponding to the stencil point furthest along the z-axis from global memory.(f) Update the assigned block of grid points using the data in shared memory and registers. Using the same notation as in Sect. <ref> and τ_z = 1, the total number of grid pointsfetched from global memory by a thread block is Γ_19p = τ_xτ_yE_z_Computational area + 2R(τ_xτ_y + τ_xE_z + τ_yE_z)_Halo area . Additionally, by using Eq. <ref> to solve the number of thread blocks γrequired to update the grid, we can now write the number of read-writes performed in the first and second pass asfollows. During the first pass, we read-write to the same arrays as with the 55-point method in Eq. (<ref>) with the addition of writing the divergence field to global memory. In the second pass, we read from one array including the halo (∇·𝐮^(s)) and six partial result arrays (𝐮_partial and 𝐮_partial), and write to six arrays (𝐮 and 𝐮). RW_19p, 1st pass = 4 Γ_19pγ + 4 N_xyz_Reads+ 9 N_xyz_Writes andRW_19p, 2nd pass = Γ_19pγ + 6 N_xyz_Reads+ 6 N_xyz_Writes .§ GPU PERFORMANCEIn this section, we present the results of our GPU implementations described in Sections <ref> and <ref>. Additionally, we compare the performance of our implementations with the Pencil Code <cit.>, which is a high-order finite-difference solver for compressible magnetohydrodynamic flows and is developed to run efficiently on multi-CPU hardware.We strived to benchmark both our GPU implementations in a comparable way by using equally optimized versions of both approaches. Additionally we compared the performance of our GPU implementations with the Pencil Code using equally modern hardware sold at similar price points.The performance of our GPU implementations is not compared with any other GPU finite-difference solver, because to our knowledge, no previous work has been done on simulating compressible fluids on GPUs using sixth-order finite-differences.We generated the benchmarks for the implementations by running a test case, which simulated compressible hydrodynamic flow by using sixth-order finite differences and third-order Runge-Kutta integration. The benchmarks were run using single-precision floating-point numbers unless otherwise mentioned. Forcing was disabled in all performance tests. In order to get a fair comparison, we used grid sizes that are multiples of 12 for generatingthe CPU results, since the workload in this case is divided more evenly on the 12 cores of an Intel Xeon E5-2690v3 processor. In contrast, the optimal grid sizes shared by both of our GPU implementations are multiples of 32, which we used to generate the GPU results.Diverging from our GPU implementations, Pencil Code uses a 2N-storage Runge-Kutta integration method which we described in Sect. <ref>. Therefore our 55-point method gives the same error as the Pencil Code, but our 19-point methoddoes not. Additionally, we do not know whether the single-pass approach we used in our 55-point method would also be suitable for CPUs, and how it would affect the performance if used within the Pencil Code. We tested our GPU implementations on an NVIDIA Tesla K40t accelerator card, based on a single 875-MHz Kepler GK110BGL GPU (15 SMXs, 192 CUDA cores per SMX, 745 MHz base clock rate). The on-device memory has a bus width of 384 bits and consists of a total of 12288 MiB of GDDR5-3004 SDRAM (24 × 256MiB chips in clamshell mode), of which 11520 MiB is usable as global memory. Tests were performed with ECC enabled. A compute node consists of two 2.6-GHz Intel Xeon E5-2620-v2 Ivy Bridge CPUs (2.1 GHz base clock rate, 6 cores per CPU) with 32 GiB of DDR3-1600 memory and two NVIDIA Tesla K40t accelerator cards, which are connected via a 16x PCI Express 3.0 bus. We compiled the program with CUDA 6.5 and Intel 14.0.1 compiler (the only Intel compiler supported in CUDA 6.5) for compute capability 3.5 invokingwith flags ,and .The test case for Pencil Code was run on a compute node consisting of two 12-core 2.6-GHz Intel Xeon E5-2690v3 processors based on Haswell microarchitecture. Each core has 32 KiB of L1 cache and 768 KiB of L2 cache while a 30-MiB L3 cache is shared by the cores of the CPU.The main memory of a node consists of 8 × 16 GiB DDR4-2133 DIMMs. We used a revision of Pencil Code fetched on 2015-08-27 [Revision fdf3802edcd2d9036005534c94f288cd124a069a]. This build was compiled with a Fortran 90 compiler for MPI programs invoked by thecommand using , , ,andflags. We used Intel compiler version 15.0.2 and Intel MPI library version 5.0.2. The performance comparison of our GPU implementations and the Pencil Code is shown in Fig. <ref>. We achieved the rate of 343 million updates per second using the 19-point method, which was 2.0 times faster than integration with the 55-point method, which achieved the update rate of 168 million updates per second.The Pencil Code achieved an update rate of 51 million updates per second using one CPU, while the update rate using two CPUs was 96 million updates per second. With 256^3-sized grids and single-precision, we achieved the best performance for the 19-point method by updating 16 and 64 grid points per thread in the first and second half of the algorithm, respectively. With double-precision and the 55-point method, we had to decrease the size of a thread block to 4 × 4 × 4 threads in order to fit the required data into shared memory. In this case, the best performance was achieved when a total of 32 grid points were updated by each thread. With double-precision, we achieved the rate of 25 and 154 million elements integrated per second with our 55-point and 19-point implementations, respectively.Fig. <ref> shows the optimizations performed upon our GPU implementations.Notably, the performance decreases during the optimization step, where more work is added per thread. This is caused by overusing registers in the loop, which handles updating multiple grid points per thread,which in turn limits the occupancy of the integration kernel. As the next optimization step, we limited register usage with theintrinsic, which causes any additional local variable above the limit to spill into L1 cache, but which in turn results in better performance because of the increased occupancy. Without adding more work per thread, we could not increase the performance by limiting register usage nor using the texture cache for reading. As the final optimization step, we moved the device functions used to compute derivatives from separately compiled modules to the same source file with the integration kernel, which resulted in a large boost in performance. The reason for this is, that when compiling a source, the compiler cannot optimize any calls to functions in separately compiled units, and must replace them with expensive calls which adhere to the application binary interface used in CUDA. With this step, we measured a 1.8× and 1.9× increase in performance of our 55-point and 19-point methods, respectively.We further improved the implementation of our 55-point method by setting the shared memory configuration to use eight-byte addressing mode, which resulted in a speedup of 34%.This step reduced the bandwidth requirement for L1 and shared memory, which was previously the limiting factor in the kernel. With the 19-point method, we did not see any notable difference in performance. The integration rate using double-precision is 2.1× slower for the 19-point method and 6.3× slower for the 55-point method. Because of the increased shared memory requirements, the latency issues in the 55-point method are accentuaded. In order to fit all the required data into shared memory, smaller thread blocks have to be used, which reduces both occupancy and the amount of data which can be reused. This causes the performance to degrade more than 2 × from the single-precision results. The 19-point method is still bound by memory bandwidth, even though the increased shared memory requirements limit the occupancy of the first half to 25% or less regardless of the thread block dimensions.Tables <ref> and <ref> show the resource usage of our integration kernels. The 55-point method is limited by low occupancy and is latency-bound, achieving the memory bandwidth of 72 GB/s. The 19-point method achieved 72% and 79% of the theoretical maximum bandwidth during the first and second pass, respectively. As the performance of the kernels used in the 19-point method is mostly limited by the available bandwidth, the 19-point method is bandwidth-bound. Occupancy of the 19-point method is limited by both shared memory and register usage. § PHYSICS TESTSWe chose three simple test cases to validate the code physics-wise. In the first test case we initialized a smooth sine-wave velocity profile with varying wavenumber into the domain, and observed its decay due to diffusion, for which an analytical solution can be derived. If weassume an initial condition where 𝐮 = u_0 sin(kx) ê_y and a constantdensity, the velocity profile decays as a function of time into𝐮(x,t) =u_0 e^-ν t k^2sin(kx) ê_ywhich can be compared with a snapshot calculated by the code. We performed the test both with the 19- and 55-point methods, and chose a high wavenumber, k=13, to have a challenging test case for the finite difference scheme.Our results are shown in Fig. <ref>, where we plot the difference between the numerical and analytical solutions as a function of δ_x, and make a power-law fit to compare the trend with the expected one for sixth-order finite differences, ∝δ_x^6, as the resolution is decreased. With both methods, the measured deviation from the analytical solution is very close to the theoretical discretization error, the average change in the deviation when halving the amount of grid points being 2^5.7.Between the two highest resolution cases the wave is becoming too well resolved, and the single grid point deviation is approaching the floating point accuracy, and therefore the powerlaw starts showing hints of breaking down. However, including double-precision, this breakdown is avoided at the highest resolution. These results clearly show a satisfactory accuracy of the finite-difference scheme with respect to the theoretical expectation.The second test case was a radial kinetic explosion. The sphericalsymmetry of the test made it possible to check the symmetry of the computational operation within the code, the rate-of-strain tensor,Eq. (<ref>), in particular. The kinetic energy was released in a small volume with a Gaussian profile, directedradially outward from the centre of the computational domain. Inspherical coordinates, this initial condition was defined as 𝐮(𝐫) = u_amplexp( (r - r_shell)^2/2d_shell^2)ê_r,where r_shell is the radius of the shell at the peak, u_ampl the peak velocity and d_shell the width parameter. The Gaussian profile helped to avoid numerical instabilities due to discontinuous initial conditions.In a 64^3, 128^3- and 256^3-sized grids we kept u_ampl = 1, d_shell = 0.2, r_shell = 0.8, ρ_0 = 1 and variedRe (Fig. <ref>) by changing the kinematic viscosity values.The tests were performed with both the 55-point and the 19-point method.We measure the degree of spherical symmetry on a Cartesian grid by comparing the values of the quantities along 7-directional axeswith each other. Three were thex-, y-, z-axes and four diagonals which went from all corners to their opposites. As we could not match the coordinates of Cartesian and the diagonal axes point by point, we integrated the sum of values of each axis ( ∑ρ)_i and ( ∑ |u| )_i. In the ideal spherically symmetric case, the resultsfor all axes should be the same, therefore we estimated the relativeerrors between axes by taking standard deviationsΔ( ∑ρ) and Δ( ∑ |u| ). The results are shown in Fig. <ref> for the 19-point stencil method. Δ( ∑ |u| ) is largest for the smallest Reynolds numbers within a certain set with the same resolution. There is a strong decrease when the Reynolds number is increased, and at the highest Reynolds numbers investigated, the difference between the diagonal and off-diagonal elements no longer changes for the highest resolution cases. This is as expected, as the numerical solution is more likely to deviate from the analytic one at small Reynolds numbers where the viscous effects are stronger. For the lowest resolution case no convergence is seen, indicating that this test is too demanding to be performed with that grid spacing. There is a weak, but opposite, trend in Δ( ∑ρ). This is likely to be due to the fact that the decrease of viscosity enhances the effects of the non-conservative nature of the discretized equations. As the 55-point stencil method produces results that practically coincide with the 19-point method, the results are shown in Fig. <ref> only for the 19-point method. This test shows that we can satisfactorily re-produce spherically symmetric structures with the numerical scheme.The third test case explored is a typical forced non-helicalturbulence setup. We switch on the external force term in the momentum equation, and use a non-helical forcing function, that can be expressed as 𝐟(𝐱,t) = Re{ N 𝐟_𝐤(t)exp(i 𝐤(t) ·𝐱 + i ϕ(t) ) }, whereN = f_0 c_s √(|𝐤| c_s/δ t)and𝐟_𝐤(t) = 𝐤×𝐞/√(|𝐤|^2 - (𝐤·𝐞)^2). Here f_0 scales the magnitude of the forcing and 𝐞 is an arbitrary unit vector perpendicular to forcing wave vector 𝐤 <cit.>. An isotropic set of𝐤-vectors, with k_min≤ |𝐤| ≤ k_max, is generated at thebeginning of a computational run, from which they are picked randomly at every timestep. We denote the mean value of the set of vectors |𝐤| as k_f,and chose sets with k_f = 1.53, 2.23, 3.13, 4.12 and 10.0. With the default domain size (2π, 2π, 2π) the box size corresponds to k = 1. We performed the forcing tests using the 19-point stencil method with a 256^3 grid, with ν = 1.4 · 10^-4,which was the lowest viscosity still stable with this resolution.The results of the forced turbulence test are shown asspectra in Fig. <ref>, which describe the relative distribution of turbulent energy for different wavenumbers k.The peak of turbulent energy was situated at the wavenumberof the forcing k_f and cascaded intosmaller scales or higher wavenumbers. In addition, the spectra show that turbulence behaves isotropically. Following the turbulence theory of Kolmogorov the kinetic energy is expected to scale as E(k) ∝∼ k^-5/3. However, the inertial range of the turbulence, where the energy distributionfollows well the Kolmogorov theory, is quite limited at the Reynolds numbers achievable with the current resolution and viscosity scheme. To increase the effective Reynolds numbers, the implementation of dynamic numerical diffusion schemes, such as shock and hyperviscosities <cit.> would be needed. However, they are special features which are not relevant to the focus ofthis study. § DISCUSSION AND CONCLUSIONSWe have implemented two methods based on 19-point and 55-point stencils for simulating compressible fluids on GPUs usingthird-order Runge-Kutta integration and sixth-order finite differences. Our study is meant as a proof of concept for further developments with multi-GPU systems. The physical problem currently solved for is that of isothermal compressible hydrodynamics. We show that integration kernels, which operate on 55-point stencils and where each stencil point holds several variables, three-dimensional cache blocking is inefficient because of the limited amount of shared memory available on current GPUs. While the performance of these kernels can be improved with further optimizations, the latencies caused by arithmetic and memory operations are difficult to hide with occupancy without introducing redundant global memory fetches. Using instruction-level-parallelism to hide latencies is similarly problematic, since large amounts of data are required to be cache resident when processing several grid points at once.We propose a reformulation of the problem, which can be used to improve the performance of the 55-point method by updating the simulation grid in two passes using 19-point stencils.Our results show that the increased occupancy and simpler memory access pattern in our 19-point method outweighs the penalty of performing additional memory transactions.Moreover, we show that the difference in accuracy of our 19-point and 55-point methods is minimal. Other studies have shown that it is possible to achieve the bandwidth-bound limit in stencil computation <cit.> <cit.> <cit.>, which we also achieve with our 19-point implementation. We report the speedup of 1.7 × and 3.6 × between a 24 CPU core node and a single GPU with the 55-point and the 19-point methods, respectively.Between a 12-core CPU and a GPU, the corresponding speedups are 3.3 × and 6.8 ×. We consider this as a success as typical supercomputer nodes have at least two GPUs, while the most recent offerings of NVIDIA (DGX-1) have up to eight GPUs per node. Although the final outcome of the efforts depends crucially on how well the code scales to multiple GPUs, the current performance is in our view clearly worth the invested time and effort.We devote the rest of this section for explaining our design decisions anddiscussing the lessons learnt for future development.There are at least three ways to reduce the cache requirements in order to improve latency-hiding in the 55-point method.First, smaller thread blocks can beused, which require fewer grid points to be cache-resident. However, since the size of the halo stays the same, thisapproach increases the ratio of memory transfers required for updating a grid point. In a 128^3-sized grid, we reached the update rate of only 85 million points per second using (32, 4, 1)-sized thread blocks which updated 128 points per thread and alternatively the update rate of 72 million points per second with (4, 4, 4)-sized blocks updated 16 points per thread.Second, part of the grid points can be stored into local memory while using 2.5-dimensional cache blocking.However, this in turn complicates the program when using 55-point stencils and introduces a large number of redundantfetches from the on-device memory, since only the grid points stored in registers within a warp can be shared with theshuffle instruction. For this reason, we did not explore this approach further.Finally, the integration kernel can be decomposed into a number of kernels, which use a smaller amount of resourcesthan the initial kernel. As we show with our 19-point method, this optimization technique can be used to transform alatency-bound kernel into a number of kernels, which exhibit higher occupancy. Instead of decomposing the integrationkernel of the 55-point method with respect to each dimension, we chose to solve the gradient of divergence separately,which in turn allowed us to use smaller stencils and 2.5-dimensional blocking efficiently, resulting in two bandwidth-bound kernels. However, the exact speedup gained from kernel decomposition is difficult to predict. If we assume that the time spenton computation in the initial kernel is negligible and that we achieve high enough occupancy with the decomposedkernels to be bandwidth-bound, we can approximate the potential maximum speedup of kernel decomposition with thefollowing equations. Let BW_max be the maximum effective bandwidth and BW_initial the bandwidth achieved with the latency-bound kernel. Additionally, letM_initial be the number of bytes transferred by the initial kernel and M_i, where i ∈ N, be the number of bytes transferred by the ith kernel of N decomposed kernels.With these definitions, we can write the time spent on transferring data in the initial kernel and the time spent on transferring data in the decomposed kernels as T_initial = M_initial/BW_initial ,T_decomposed = ∑_i=1^NM_i/BW_max Furthermore, we can now write the theoretical maximum speedup asT_initial/T_decomposed = M_initial/M_1 + M_2 + ... + M_n·BW_max/BW_initial . However, both our implementations perform more than the optimal amount of memory transaction to updatethe simulation grid. Since we store the grid as a structure of arrays and Kepler architecture GPUs servicememory transactions from global memory and L2 cache in segments of 32 bytes, memory bandwidth is wasted ontransferring irrelevant bytes when fetching data to the halos on the left and right side of the shared memoryblock in the x-axis. For R=3 and using floating-point precision, one such memory transaction wastesbandwidth on transferring 24 additional bytes. By using an array of structures to store the grid, we wouldexpect a reduction of 25.0% and 3.4% in the number of transactions used to read data with our 55-point andthe first half of the 19-point methods, respectively. In addition, shared memory and memory bandwidth is wasted in our implementation of 55-point method,since we fetch (τ_x+2R) × (τ_y+2R) × (τ_z+2R) grid points into shared memory during the initial update even when some of the data is used by only one thread, or none at all. We chose this approach for its simplicity and to avoidbranching execution paths in the integration kernel. The optimal size for the shared memory block when using 55-point stencils is hard to formulate, but with a numerical test we found thatwhen τ_x = τ_y = τ_z = 8 and R=3 we use approximately34% more shared memory with our 55-point method than the amount required for storing onlythose grid points, which are used by more than one thread. However, we do not know if using less shared memory wouldincrease performance, as in addition to introducing branching to the algorithm, all gridpoints except the R^3-sized corners of the shared memory block fall on a memory segment,which is serviced from global memory or L2 cache with a single memory transaction in both approaches regardless of whether the point is stored in shared memory.Additionally, our 19-point implementation is not compute-bound. Nguyen et. al <cit.> suggest, that it is possible to be compute-bound in stencil computation by using temporalblocking, where a thread block updates a block of grid points over several timesteps and therefore larger amount of data in shared memory can be reused. However, they also state that temporal blocking requires a large cache to increase performance instencil computation where R and the size of a grid point is high <cit.>. In future work, we are planning to explore this approach as well as storing the grid as an array of structures. Using single-precision and 256^3-sized grids, we saw a 4% improvement in performance when integrating significantly more grid points per thread than in our optimal solution for 128^3-sized grids. A contributing factor was the following. While increasing the workload per thread reduces the number of thread blocks running on the GPU, with larger grids thenumber of thread blocks is sufficiently large to saturate the GPU with work even when integrating a much larger number of grid points per thread. Solving more grid points per thread increases the reuse factor, which in turn reduces the relative number of global memory fetches needed to integrate a grid point.While the focus of this work was to accelerate integration using single-precision,we showed that the 19-point method scales well also to double-precision with minimal changes.With the 55-point method, we show that three-dimensional cache blocking becomes less efficient if the data required to integrate a grid point is further increased. Whether further decomposition could improve the performance over our current best solution is a matter of future research. Our tests without the rate-of-strain tensor, 𝐒, indicate that the performance of the first half of the 19-point method can potentially be improved by increasing occupancy of the kernel by reducing the requirements for shared memory and registers, which in turn pushes the bandwidth achieved closer to the hardware maximum. However, in order to avoid introducing a large number of additional global memory fetches, as much of the data used for integrating a grid point should be retained in caches until the data is no longer needed.Also one possible way to reduce shared memory requirements is to decouple the computation with the velocity components and solve the results with respect to one axis at a time. Instead of defining four arrays in shared memory as in the current kernel implementation, we could limit their number to two. In order to improve latency hiding, we could also interleave compute and memory instructions by using a technique called Ping-Pong buffering, where the data is being processed in the 'Ping' buffer while new data is being fetched into the 'Pong' buffer. However, a thorough investigation is required to determine whether these approaches can be used to increase the performance of our solver.We are also planning to extend our implementations to support the full MHD equations. These equations would require additional arrays to be stored in global memory, such as the thermal energy and magnetic fields of the fluid.Since the addition of these new arrays increase the size of a grid point, special care must be taken in order to avoid using large amounts of shared memory. With a single-pass approach, such as our 55-point method, we would expect the occupancy to degrade further if more data is required to be cache-resident during an integration step. Thus, we would expect kerneldecomposition to provide speedups also with the full MHD equations,since less data need to be stored in caches at a time in a multi-pass approach.On the other hand, the latest GPU architectures geared towards scientific computing employ larger caches. For example, a Tesla K80 contains 114 688 bytes of shared memory. In our current 55-point implementation, this would allow two (8, 8, 8)-sized thread blocks to be multithreaded on a SIMD processor instead of only one on a Tesla K40t. However, we did not have access to a Tesla K80 and do not know if the increased occupancy is high enough to hide the latencies in our 55-point method and to outperform our 19-point method.Additionally, a natural follow-up to our implementations is to extend them to work with multi-node GPUs. We expect inter-node communication to become the bottleneck, because of the comparably slow PCIe bus and communication required for solving boundary conditions. However, an important benefit of a multi-node implementation is that it would allow us to handle larger grids than the current maximum of 512^3 on a Tesla K40t. § The reformulation of the Navier-Stokes equations for the 19-point method is done as follows. For the intermediate result holds that 𝐮^(s+1) = α^(s)𝐮^(s) + δ_t∂/∂ t𝐮^(s)= α^(s)𝐮^(s)+ δ_t[ -(𝐮^(s)·∇)𝐮^(s) -c_s^2 ∇lnρ^(s)+ ν( ∇^2 𝐮^(s)+ 1/3∇(∇·𝐮^(s)) + 2 𝐒·∇lnρ^(s)) ] = α^(s)𝐮^(s)+ δ_t[ -(𝐮^(s)·∇)𝐮^(s) -c_s^2 ∇lnρ^(s)+ ν( ∇^2𝐮^(s) + 2 𝐒·∇lnρ^(s))] + δ_tν/3∇(∇·𝐮^(s)). When we set𝐮_partial^(s+1) = α^(s)𝐮^(s)+ δ_t[ -(𝐮^(s)·∇)𝐮^(s) -c_s^2 ∇lnρ^(s) + ν( ∇^2𝐮^(s) + 2 𝐒·∇lnρ^(s))], it follows that𝐮^(s+1) =𝐮_partial^(s+1) + δ_tν/3∇(∇·𝐮^(s)). Likewise for the final result 𝐮^(s+1) = 𝐮^(s) + β^(s)𝐮^(s+1)= 𝐮^(s) + β^(s)( 𝐮_partial^(s+1) + δ_tν/3∇(∇·𝐮^(s)) ) = 𝐮^(s) + β^(s)𝐮_partial^(s+1) + β^(s)δ_tν/3∇(∇·𝐮^(s)). When𝐮_partial^(s+1) = 𝐮^(s) + β^(s)𝐮_partial^(s+1) , it follows that𝐮^(s+1)= 𝐮_partial^(s+1) + β^(s)δ_tν/3∇(∇·𝐮^(s)). § ACKNOWLEDGEMENTS M.V. thanks financial support from Jenny and Antti Wihuri Foundation and Finnish Cultural Foundation grants. J.P. thanks Aalto University for financial support. Financial support from the Academy of Finland through the ReSoLVE Centre of Excellence (JP, MJK, PJK; grant No. 272157) and the University of Helsinki research project `Active Suns' (MV, MJK, PJK) is acknowledged. We thank Dr. Matthias Rheinhardt for his helpful comments on this paper and Prof. Petteri Kaski for general guidance.We acknowledge CSC – IT Center for Science Ltd., who are administered by the Finnish Ministry of Education, for the allocation of computational resources. This research has made use of NASA’s Astrophysics Data System.elsarticle-num | http://arxiv.org/abs/1707.08900v1 | {
"authors": [
"Johannes Pekkilä",
"Miikka S. Väisälä",
"Maarit J. Käpylä",
"Petri J. Käpylä",
"Omer Anjum"
],
"categories": [
"physics.comp-ph",
"astro-ph.IM",
"cs.DC",
"physics.flu-dyn"
],
"primary_category": "physics.comp-ph",
"published": "20170727150510",
"title": "Methods for compressible fluid simulation on GPUs using high-order finite differences"
} |
DESY 17-112 gobble Towards de Sitter from 10D Jakob Moritz^1, Ander Retolaza^2 and Alexander Westphal^3 Deutsches Elektronen-Synchrotron, DESY, Notkestraße 85, 22607 Hamburg, Germany Abstract Using a 10D lift of non-perturbative volume stabilization in type IIB string theory we study the limitations for obtaining de Sitter vacua. Based on this we find that the simplest KKLT vacua with a single Kähler modulus stabilized by a gaugino condensate cannot be uplifted to de Sitter. Rather, the uplift flattens out due to stronger back-reaction on the volume modulus than haspreviously been anticipated, resulting in vacua which are meta-stable and SUSY breaking, but that are always AdS.However, we also show that setups such as racetrack stabilization can avoid this issue. In these models it is possible to obtain supersymmetric AdS vacua with a cosmological constant that can be tuned to zero while retaining finite moduli stabilization. In this regime, it seems that de Sitter uplifts are possible with negligible backreaction on the internal volume. We exhibit this behavior also from the10D perspective. July 26, 2017 ^1mailto:[email protected]@desy.de ^2mailto:[email protected]@desy.de ^3mailto:[email protected]@desy.de arabic § INTRODUCTION The cosmological constant (c.c.) problem stands as our perhaps most drastic disagreement between theory and observation. When applying quantum field theory (QFT) to various stages of symmetry breaking and to the Einstein field theory limit of gravity given by general relativity (GR), we arrive at a prediction which is off by some 120 orders of magnitude from the value of the c.c. inferred from, for example, precision observations of the cosmic microwave background (CMB).Attempting to resolve this tension is difficult given a no-go theorem by Weinberg <cit.> which essentially states that dynamic adjustment mechanisms based on a fully local field theory only must fail. This leaves us with few options. One possibility is to give up locality, motivated by the fact that a true c.c. needs the full 4-volume of the universe for aproper detection, which makes it borderline non-local. Recent work has shown that a combination of protective scale and shift symmetries with the introduction of certain effectively global variables encoded via 4-form field strengths can `sequester' the QFT vacuum energy from its gravitating effects <cit.>. Another possibility, already foreseen by Linde, Weinberg and others <cit.>, is to replace theQFT-limit of GR with a full candidate theory of quantum gravity, and assume that it will render the quantum vacuum energy finite and generically large, but produce an enormous number of different solutions (vacua) with differing finite c.c.'s. If we couple this with a cosmological population mechanism such as eternal inflation <cit.>, then a form of weak anthropic argument can explain the observed c.c. value as natural given certain observed features such as structure formation, and provided the number of different vacua is larger than O(Λ^-4)= O(10^120) (and the distribution of vacuum energies shows no strong features).Crucial for this line of argument is having in hand such a candidate quantum gravity with an enormous `landscape' of discrete vacua with varying c.c. values <cit.>.One of the first serious and still one of the best candidates for this task is the setup by Kachru, Kallosh, Linde and Trivedi (KKLT) <cit.> which uses string theory as a candidate quantum gravity to realize a version of the Bousso-Polchinski proposal, a toy model of the landscape solution to the c.c. problem.KKLT propose to generate de Sitter vacua in string theory in a three-step process: * Step 1: Stabilize complex structure moduli and the axio-dilaton perturbatively using three-form fluxes <cit.>. * Step 2: Stabilize Kähler moduli using non-perturbative effects. The result is a supersymmetric AdS vacuum <cit.>. * Step 3: Break supersymmetry and lift to de Sitter space by including an anti-brane at the bottom of a warped throat <cit.>. In looking at these three steps we observe the following situation. The 4D effective supergravity description of step 1 is derived via dimensional reduction from the warped CY compactifications with ISD 3-form fluxes, and its critical points match with solutions of the full 10D equations of motion, the 10D Einstein equations and Bianchi identities supplemented by O3/O7 planes and D3/D7 branes as local sources. The volume modulus ρ is a flat direction at this level <cit.>.The 4D effective description of step 2 is based on the 4D effective theory of gaugino condensation <cit.>. However, the presence of unbroken supersymmetry in the ensuing AdS vacuum of ρ as well as the suppression of the ρ mass scale with the control parameter |W_0| given by the fluxes provides justification for the choice of the form of the 4D EFT of the ρ modulus, as it is more or less dictated by holomorphy of the superpotential, supersymmetry of the setup, non-perturbative gauge dynamics, and the relation between the holomorphic gauge kinetic function of the D7-brane gauge theory and the 4-cycle volume modulus chiral multiplet f_D7(Σ_4^i)∼ρ_i. Moreover, all scales of the volume stabilization process lie far below those of the flux stabilization of step 1 <cit.>.There is by now very strong evidence that step 3, inserting in particular a single anti-D3-brane at the bottom of the warped throat of step 1 + step 2, is still a well-controlled meta-stable solution of the full system with spontaneously broken supersymmetry. Its non-perturbative instability arises from the Kachru-Pearson-Verlinde (KPV) flux-brane annihilation process <cit.>. Describing this system beyond the probe approximation for the anti-D3-brane <cit.> while respecting a well-defined EFT description of the anti-D3 backreaction onto the warped throat geometry, with correct matching onto the string scale physics <cit.> provides clear arguments that the KPV process is the microscopic non-perturbative instability of the setup with all perturbative directions (the anti-D3 position moduli) frozen.However, we observe a more subtle peculiarity of this uplift effect in the KKLT setting. Namely, the amount of uplift necessary to reach de Sitter is tied to the mass scale of the ρ modulus from its stabilization in step 2. To reach de Sitter we require validity of a particular form of the 4D EFT. Modifications to the EFT that are compatible with our current knowledge of the supersymmetric moduli stabilization as well as the classical D3-brane potential can lead to substantial backreaction effects on the volume modulus. In the extreme case these modifications could lead to a flattened uplift that never reaches a positive vacuum potential. `Flattening' means that the backreacted uplift turns out to be smaller than the estimate without backreaction which scales linear with the anti-brane tension.We note here, that such flattening effects from moduli backreaction were observed before in a related context: namely, if we replace the control parameter of the uplifting vacuum energy, the anti-brane tension, by the potential energy of a slow-rolling scalar field driving inflation, the same backreaction effects lead to flattened scalar potentials for inflation <cit.>, in complete analogy with the uplift flattening we observe here.Indeed such modifications to the EFT can be argued to arise through the same effects that lead to the lifting of the moduli space of D3-brane position moduli by non-perturbative effects. However, the very same reasoning suggests that if the racetrack stabilization of Kallosh and Linde <cit.> is assumed, de Sitter uplifts can be realized because the AdS depth of the supersymmetric potential can be parametrically decoupled from the mass scale of the Kähler modulus.Consequently, the questions we would like to ask and partially answer in this paper in the course of sections <ref> through <ref> are as follows. Is the volume stabilization of the original KKLT proposal rigid enough to allow for a de Sitter uplift or is racetrack stabilization a minimal requirement? If the latter is true, can we verify the stabilization mechanism also from a 10D perspective?We find indeed that back-reaction effects play asubstantially stronger role than has previously been anticipated. The uplift flattening we observe is strong enough that uplifts reaching de Sitter are not possible using the simplest version of KKLT. Instead the solutions for volume stabilization with a single gaugino condensate are meta-stable SUSY breaking AdS vacua. However, assuming certain properties of the compactification manifold, we are able to give a 10D picture of racetrack stabilization that strongly indicates the possibility to uplift to de Sitter in this case. We use these 4D insights to build a modified single-condensate moduli stabilization scheme by including α ' corrections. This allows to decouple the mass scale of the Kähler modulus from the AdS depth, although not parametrically.We then discuss our findings and conclude in section <ref>. § MODULI STABILIZATION AND DE SITTER UPLIFTS IN TYPE IIB STRING THEORY: THE 4D PERSPECTIVEThe best understood constructions of de Sitter vacua in type IIB string theory start from Calabi-Yau (CY) compactifications to 4D Minkowski and orientifolds thereof. These compactificationscome with many massless moduli which need to be stabilized in order to make them phenomenologically interesting and typically require some extra structure/ingredient to give rise to4Dde Sitter. This program of `moduli stabilization' and de Sitter uplift usually is carried out in three steps: * Stabilization of complex structure moduli and the axio-dilaton at tree-level by introduction of three-form fluxes <cit.>. * Stabilization of Kähler moduli. Different approaches exist to stabilize them, probably the most common ones being the following: in the so-called KKLT approach <cit.>,onestabilizes them using non-perturbative effects, typically leadingto asupersymmetric anti de Sitter (AdS) geometry in 4D. In the large volume scenario <cit.>, instead, they are stabilized using an interplay of α' corrections with non-perturbative corrections, such that the 4D vacuum is also AdS but non-supersymmetric. In what follows we will mainly focus on the former approach, the only exception being section <ref>, where we will combine α ' corrections with a gaugino condensate to stabilize the Kähler modulus. * The final step consists of uplifting the 4D geometry to de Sitter space. Many proposals exist to achieve this goal (see e.g.<cit.>), the most common one beingthe inclusion of a supersymmetry breaking source, most prominently an D3-brane, at the bottom of a warped throat <cit.>.These steps are usually followed through entirely within the framework of 4D supergravity, the low energy effective field theory describing the4D fields after the compactification. In this section we will quickly review some of its aspects.Before doing so, let us give the main point of this section right away: the first two steps can be followed with reasonable amount of control purely within the 4D effective field theory. However, in many models of moduli stabilization <cit.> the uplift can not be parametrically decoupled from Kähler moduli stabilization. In these cases, although we believe that there is nothing wrong with the use of 4D supergravity as a matter of principle, the question whether or not a proposed uplift can be successful heavily depends on the detailed moduli dependence of the uplift ingredient. We will argue that at least for D3-brane uplifts the required knowledge about the moduli dependence is hard to acquire from 4D effective field theory reasoning alone. In other words, guessing the correct effective field theory is not as obvious as it might seem.§.§ Complex Structure Moduli StabilizationThe effective 4D supergravity that governs the light moduli of type IIB O3/O7 CY-orientifolds is well known. Fora single Kähler modulus (i.e. h_1,1^+=1) the Kähler potential is given by𝒦(ρ,z_α,τ)=-3ln (-i(ρ-ρ̅))-ln(-i(τ-τ̅))-ln(-i∫_M_6Ω∧Ω),to leading order in the α' expansion. Since we are considering a unique Kähler modulus ρ, this is the (complexified) volume modulus (Im(ρ)=𝒱^2/3). Also, τ=C+ie^-ϕ is the axio-dilaton and z_α are the h_2,1^- complex structure moduli that appear in the Kähler potential implicitly via the holomorphic 3-formΩ=Ω(z_α).Dilute three-form fluxes (such that warping effects are negligible) generate a superpotential, the Gukov-Vafa-Witten (GVW) superpotential <cit.>W_GVW=√(2/π)∫_M_6 G_3∧Ω,whereG_3=F_3-τ H_3 is the complexified 3-form flux. This superpotential depends holomorphically on the axio-dilaton and the complex structure moduli. It was derived by comparing the tension 𝒯^DW of BPS domain walls that separate flux vacua with different three-form fluxes (i.e. D5/NS5 branes that wrap internal three-cycles) with the change in superpotential Δ W across the domain wall and demand the standard 4D SUSY relation𝒯^DW=2e^𝒦/2|Δ W|.The SUSY conditions D_iW=0 (where i runs over the complex structure moduli and the axio-dilaton) require that G_3 is of Hodge type (2,1)⊕ (0,3). This should be read as an equation for the complex structure moduli and the axio-dilaton, which obtain masses of the orderm_CS∼α'/R^3,where R is a typical scale of the internal geometry. Thus, at large volume they are much lighter than the Kaluza-Klein (KK) modes of the internal geometry. If the superpotential is entirely determined by the GVW superpotential, which does not depend on the Kähler modulus ρ, the 4Dpotential vanishes at the minimum of the complex structure moduli and the axio-dilaton due to the no-scale relation𝒦^ρρ̅_ρ𝒦_ρ̅𝒦=3. This means that the volume modulus remains massless and the final SUSY condition0=D_ρW=(_ρ𝒦)Wcannot be solved at finite volume unless the non-generic condition W=0 happens to be met, i.e. G_3 is of Hodge type (2,1).The upshot is that for generic fluxes all but the single Kähler modulus are stabilized and SUSY is broken. Hence, at low energies one is left with a single modulus ρ and a constant superpotential W=W_0.§.§ Kähler Moduli Stabilization: KKLTIn order to stabilize the volume modulus KKLTargued for a ρ-dependent correction of the superpotential <cit.>. Because the superpotential must depend holomorphically on the volume modulus and the real part of ρ enjoys a continuous shift symmetry to all orders in perturbation theory, the only possible corrections to the superpotential are non-perturbativeδ W_np =∑_n A_n e^ia_nρ,and break the continuous shift symmetry of Re(ρ) to a discrete one. There are two kinds of non-perturbative effects that can generate the desired exponential terms: * Euclidean D3 braneswrappinginternal 4-cycles<cit.>. * Gaugino condensation on a stack of N D7-branes <cit.>.In what follows we will focuson (b): a stack of N D7-branes wrapping a rigid 4-cycle Σ, i.e. the position moduli of the D7's are massive. Assuming that no further branes intersect the 7-brane stack (i.e. light flavors are absent) the low energy effective theory isSU(N) SYM. At low energies this theory confines and develops a non-perturbative superpotential via gaugino condensation,⟨λλ⟩=-32π^2Λ^3, ⇒δ W_np=-N/32π^2⟨λλ⟩=NΛ^3,where Λ is the dynamical scale of the gauge theory which determines the perturbative running of the holomorphic gauge coupling τ_YM=Θ_YM/2π+i 4π/g_YM^2via τ_YM=3N/2π iln(Λ/μ),i.e.Λ=μ e^2π i/3Nτ_YM.By dimensionally reducing the D7 brane action to 4D one can infer that classically the holomorphic gauge coupling is identified with the Kähler modulus ρ. In the quantum theory this equation should be read as the renormalization condition that matches the low energy effective theory to its UV completion. In other wordsΛ=μ_0 e^2π i/3Nρ,where μ_0 is a high scale. In summary, there is a non-perturbative correction of the superpotentialδ W_np(ρ)=Nμ_0^3 e^2π i/Nρ,that we may associate to a non-vanishing expectation value of the gaugino condensate ⟨λλ⟩. Combining this with the constant flux superpotential W_0, at energies below the non-perturbative scale of the gauge theory the Kähler and superpotential reads𝒦=-3ln(-i(ρ-ρ̅)), W=W_0+Ae^iaρ.Now the F-term equation D_ρW=0 is solved byW_0=-(1+2a Im(ρ)/3)Ae^iaρ,and at the minimum of the potential the value of the vacuum energy is negative,V=-3e^𝒦|W|^2=-a^2|A|^2/12Imρe^-2a Imρ<0.Control over the instanton expansion requires aρ>0 and control over the α' expansion of string theory requires ρ≫ 0. Both can be satisfied if due to a sufficient tuning of the flux configuration |W_0|≪ 1.An essential feature of this stabilization mechanism is that the value of the cosmological constant and the mass of the volume modulus are related via|V_AdS | ∼ M_ P^2 m_ρ^2. §.§ The upliftThe perhaps best understood way to uplift SUSY AdS-vacua of type IIB string theory to de Sitter is by the inclusion of an anti-D3 brane at the bottom of a warped throat. This way of uplifting is attractive because in ISD flux backgrounds the anti-brane is driven to strongly warped regions where its associated potential energy is naturally small <cit.>. Moreover such SUSY-breaking vacua are known to be connected to the SUSY ISD vacua via the non-perturbative KPV transition <cit.>. Arguably, no perturbative decay channel exists <cit.>. In this paper we assume perturbative stability of such a configuration. Because the 10D warped throat supergravity solution is dual to the Klebanov-Strassler (KS) gauge theory <cit.>, one should be able to equivalently describe the anti-brane as a state of the KS gauge theory that breaks supersymmetry spontaneously rather than explicitly <cit.>. If this is the case, it has been argued that at very low energies the only degrees of freedom are the nilpotent goldstino multiplet S (nilpotency means S^2=0, for constrained superfields see e.g. <cit.>) and the volume modulus ρ <cit.>. The proposals for theKähler- and the superpotential wereK=-3ln (-i(ρ-ρ̅)-SS̅) and W=W_0+Ae^iaρ+e^2𝒜_0μ^2S.Here, exp(2𝒜_0)μ^2 parametrizes the strength of supersymmetry breaking with exp(𝒜_0) being the warp factor at the tip of throat, while μ is related to the unwarped tension of the anti-D3 brane as |μ|^4∼ T_3.In deriving the scalar potential one should treat S as a usual chiral multiplet and in the end set S=0. For real parameters W_0 and A, the scalar potential readsV(ρ)=aAe^-a Im(ρ)/6 Im(ρ)^2[Ae^-a Im(ρ)(a Im(ρ)+3)+3W_0cos(a Re(ρ))]+e^4𝒜_0 |μ|^4/12 Im(ρ)^2.The expectation of this being the correct way to describe the anti-brane state comes from the fact that in the limit of vanishing non-perturbative stabilization A⟶ 0 one recovers the known runaway potential that is easily read off of the anti-brane DBI+CS action, while in the limit μ⟶ 0 one recovers the supersymmetric KKLT potential. The corresponding potential is simply the sum of the (would-be) runaway D3 potential and the (would-be) supersymmetric KKLT potential.Tuning the tension contribution of the anti-brane (by tuning the fluxes that determine the warp factor e^𝒜_0) one can obtain de Sitter vacua with tunably small cosmological constant <cit.>. Interestingly, within this description of the D3-induced uplift, the potential energy of the D3 adds on top of the negative potential at the supersymmetric AdS minimum to good approximation until a maximum uplift of about δ V ∼ 2× |V_AdS|. Beyond that we encounter run-away behavior ρ→∞. However, we would like to point out that matching the KPV form of the D3-induced uplift does not uniquely determine the form of the superpotential. The route determining the dependence of W on S described above constitutes only one possible outcome of matching KPV. The ambiguity arises from the 4D point of view by the appearance of a new mass scale due to non-perturbative Kähler moduli stabilization, which was absent in the reduction leading to the KPV form of the D3-induced scalar potential. To see this in detail, we shall repeat the steps needed to match a candidate form of W=W(S) to the KPV form of the uplift: * The classical potential for an anti-brane is determined by dimensional reduction over a classical no-scale GKP background <cit.>. * The SUSY-breaking anti-brane can be described within the 4D effective supergravity of volume stabilization in the KKLT setup by a nilpotent multiplet S which incorporates the non-linear SUSY of the anti-brane into the supergravity description. * The 4D volume stabilization generates a new scale m_ρ^2 M_ P^2 . * In this situation, the classical potential of the anti-brane carries over from the no-scale GKP background only in the limit δ V≫ m_ρ^2 M_P^2 (i.e. comparatively weak warping) while a constant uplift potential is appropriate in the limit δ V≪ m_ρ^2 M_P^2 where backreaction on the volume modulus can be neglected. * Matching with the no-scale reduction does not uniquely fix where the nilpotent superfield S appears in the superpotential W(S,ρ). * The sign of the vacuum energy as a function of increasing uplifting depends on the detailed form of W(S,ρ).The most general superpotential to leading order in the (fractional) instanton expansion isW=W_0+b· S+A(1+c· S)e^iaρ.The special choice c=0, b=e^2𝒜_0μ^2 corresponds to the standard KKLT potential. The other extreme case would be to set b=0. In this case the SUSY KKLT vacuum corresponding to c=0 cannot be uplifted to de Sitter for any value of c. Obviously there exist infinitely many intermediate situations.[Note that this is not the most general way to parametrize the antibrane potential (see e.g. <cit.>): More generally, starting from eq. (<ref>) one may rescale S⟶ e^-2𝒜_0S such that the warp factor appears in the Kähler rather than in the superpotential as in <cit.>. The leading non-perturbative corrections to the anti-brane potential could then be encoded by replacing e^4𝒜_0⟶ e^4𝒜_0+α(ρ,ρ). Only for special choices of the real function α(ρ,ρ) this can be transformed into a superpotential correction. We thank Susha Parameswaran for bringing this to our attention.]Of course it would be rather surprising if the term proportional to c were completely absent for the following reason. It is well known that if mobile D3 branes are present, the coefficient of e^iaρ in the superpotential is a holomorphic function of their position moduli <cit.>. Their moduli space is therefore lifted by the same non-perturbative effects that lead to volume stabilization. If D3-branes modify the coefficient of the exponential term in the superpotential, one would expect an D3-brane to do so as well.However, this modification has to be quite drastic in order to have any important effect. In other words, because the coefficient b is associated to the squared warp factor at the tip of the throat e^2𝒜_0≪ 1 and the coefficient c is multiplied by the gaugino condensate, the effect of the term Se^iaρ will be negligible if it is further suppressed by warping. At first glance this seems unreasonable as the anti-brane sits in a strongly warped region of space-time. So, why should there exist any unwarped contributions to the superpotential? First of all, there is no 4D EFT reason that would forbid such a term so in the spirit of a Wilsonian effective action we should include it. If coefficients in the effective action are small, there should better exist a good reason for their smallness. We have such a reason, namely warping, only for the classical term. On the contrary, we have no a priori reason to assume smallness also of the S-coupling to the non-perturbative term. Still, an unsuppressed coupling of the S field parametrizing a warped anti-D3-brane may sound quite counterintuitive. While we must leave a fuller discussion to later when we have introduced the full 10D description, we note here that the presence of the non-perturbative term signals the presence of a new mass scale ∼⟨λλ⟩ in the effective physics. Absent any protective symmetries or dynamical coupling suppression, such a new mass scale will communicate itself generically to all other scalar couplings, in particular the warped KK-scale. This would in turn lead one to naively expect couplings between S and ⟨λλ⟩ of 𝒪(1) following the spirit of Wilsonian EFT.Given this, to us it appears plausible that any energy density, be it in a warped throat or in the bulk Calabi-Yau, effectively couples to the non-perturbative effects without any further warp. Hence, to us the following superpotential seems to adequately parametrize the interplay of an anti-brane with non-perturbative volume stabilization,W=W_0+e^2𝒜_0·μ^2· S+A(1+ c(μ^2)· S)e^iaρ. Again, |μ|^4 is associated to the unwarped tension of the anti-brane. e^2𝒜_0 is the IR warp factor of the classical ISD background while c(μ^2) is an unknown function of the anti-brane tension that we believe is not suppressed by powers of the warp factor. Whether or not an uplift to de Sitter is possible depends on the magnitude and functional form of c(μ^2).[We consider an uplift to de Sitter successful if it does not reduce the mass-scale of the lightest modulus as compared to the pre-uplifted configuration by factors that control the perturbative expansion of string theory such as cycle volumes or the dilaton. Only with this restriction, a truncation to the lowest order Kähler potential necessary for moduli stabilization is consistent in presence of the uplift.]The observation that the fate of the uplift depends so heavily on the details of the moduli potential can be embedded into a more systematic approach towards a de Sitter uplift: in the 4D effective field theory one may expand the scalar potential (before any attempt to uplift it) around its SUSY AdS (or Minkowski) minimum,V=V_0+∑_i 1/2m_i^2u_i^2+⋯,with squared masses m_i^2 (assuming canonical normalization).Including an uplift means adding a further term δ V(u_i) to the potential,V(u_i)⟶Ṽ(u_i)=V(u_i)+δ V(u_i).Although in general in order to determine the value of the potential at the new minimum u^1_i one has to minimize the new potential Ṽ with respect to all moduli u_i one easily sees that the resulting shift in potential energy can be expanded in powers of δ V asṼ(u_i^1)-V(u_i^0)=δ V(u_i^0)(1-∑_i1/2(M_ P_u_iδ V|_u_i^0)^2/M_ P^2m_i^2 δ V(u_i^0)+𝒪((δ V/m_i^2M_ P^2)^2)).Evidently, for sufficiently small uplift potential δ V the back-reaction on the fields u^i can be neglected and the extra potential energy simply adds up to the pre-existing one. However, in the simplest examples of moduli stabilization <cit.> the value of the cosmological constant at the supersymmetric minimum V_AdS is tied to the mass-scale m_ρ of the lightest modulus|V_AdS|∼ m_ρ^2M_ P^2.Clearly, in order to uplift such vacua to de Sitter, an uplift potential δ V has to be added that is at least of the order of |V_AdS|. In this case the expansion in (<ref>) is not under parametric control and the strength of back-reaction effects heavily depends on the detailed moduli dependence of δ V. For the uplifted KKLT potential a strong back-reaction effect does not occur due to the proposed power-law dependence of δ V≡ |μ^4|/12 Im(ρ)^2. In this case the first correction in (<ref>) is suppressed by a volume factor (a Im(ρ))^-2≪ 1. In contrast, for an exponential dependence δ V ∼ e^-2a Im(ρ) corrections in (<ref>) are unsuppressed. This is precisely what a sufficiently dominant term ∝ c(μ^2) in (<ref>) would entail.We conclude that if the vacuum potential of the SUSY AdS_4 vacua can be parametrically decoupled from the mass-scale of the lightest modulus,|V_AdS|≪ m_ρ^2 M_ P^2,the required uplift can be small enough such that backreaction effects are negligible and thus, de Sitter uplifts are generically possible. Conversely, if this is not the case detailed knowledge about the moduli dependence of δ V(u_i) is required in order to determine the uplift in potential energy beyond parametric estimates. Finally, we summarize the upshot of this section and set the goal for the rest of the paper: * In order to determine if an anti-brane can uplift the SUSY KKLT vacua to de Sitter we need more information about the interplay of the anti-brane with non-perturbative effects. * In the remainder of the paper we will determine only the sign of the cosmological constant in the presence of an anti-brane uplift using a 10D description. This will allow us to constrain the coefficients of W(S,ρ) only to the extend that the sign of the 4D vacuum energy as determined from W(S,ρ) must be matched to the 10D result. In particular we will not determine the form of the scalar potential V(ρ) away from its minimum. This important task that would uniquely fix the coefficients in the superpotential we leave for future work.§ SOME HIGHER DIMENSIONAL CONSIDERATIONS In view of the surprising difficulties that one usually encounters when trying to construct consistent de Sitter vacua in string theory we find it worthwhile to investigate if the seemingly conspirative modification of the 4D effective field theory that would prevent an uplift to de Sitter indeed occurs.Clearly the cleanest way to do this would be to derive the correct effective field theory of the volume modulus together with all its SUSY breaking states from first principles. Due to the obvious difficulty of this approach we will opt for another one. Instead of deriving the off-shell 4D effective potential we will constrain it using a 10D on-shell description of the non-perturbative effects as well as the uplift. Before turning to the 10D setup westudy some general aspects of compactifications that will later be relevant and use them to explain this notion of an on-shell description. This concept will be clarified in terms of a toy model that will also be useful to develop a physical intuition that is applicable to the case of type IIB string theory.§.§ On-shell vs off-shell potentialsLet us first explain what we mean by on-shell and off-shell potentials. We refer to the usual 4D potential that is determined by straightforward dimensional reduction as the off-shell potential. It is easily determined from the higher dimensional Einstein equations as follows: for a theory of D=d+4 dimensional gravity and further fields compactified to 4D de Sitter, Minkowski or anti de Sitter, the most general D-dimensional metric ansatz isds^2=e^2𝒜(y)g̃^4_μν(x)dx^μdx^nu+g^d_mn(y)dy^mdy^n,with warp factor e^𝒜, 4D coordinates x^μ and internal coordinates y^m. The higher dimensional Einstein equationsR_MN-1/2g_MNR=T_MNdetermine the effective 4D potential in terms of the internal curvature R_d and the higher dimensional energy momentum tensor T_MN,V· M_ P^-4 =14 R̃_4· M_P^-2 =𝒱_w^-2M^d∫ d^d y√(g^d)e^4𝒜(- 1/4T^μ_μ-1/2R_d)/M^2,where M is the D-dimensional Planck mass, 𝒱_w=M^d∫ d^d y√(g^d)e^2𝒜 is the warped volume and M_ P^2=M^2𝒱_w is the 4D Planck mass. 4D vacua of the higher-dimensional theory correspond to local minima of this potential which encodes the value of the cosmological constant as well as the scalar mass-spectrum.In many instances it is hard to determine this potential in full explicitness and it may be very useful to use an alternative expression for the 4D potential that is only valid when all geometric moduli are assumed to be at their minimum. One may obtain such an expression by considering the trace-reversed Einstein equations that relate the internal space Ricci scalar R_d to the energy momentum tensor. The result isV· M_ P^-4=𝒱_w^-2M^d∫ d^d y √(g^6)e^4𝒜(D-6)T^μ_μ-4T^m_m/4(D-2)M^2.It is important to realize that in writing this expression one has integrated out all geometric moduli, that is the potential is only valid at the minimum of the off-shell potential (<ref>) where the two coincide <cit.>.This has several consequences: for example,if a higher-dimensional source contributes an energy momentum tensor δ T_MN with(D-6)δ T^μ_μ-4δ T^m_m< 0,one must not conclude that the 4D cosmological constant decreases when the source is included since all other terms in the full energy momentum tensor T_MN will be perturbed even to leading order in δ T_MN. Although this means that any on-shell potential has to be treated carefully one can derive powerful statements from it. For example, in the absence of any sourcesatisfying(D-6)T^μ_μ-4T^m_m≥ 0,the 4D vacuum energy can never be positive. Using this, one may show that if one only allows for p-form fluxes with 1≤ p ≤ D-1 and localized objects of positive tension and co-dimension ≥ 2, de Sitter solutions are ruled out <cit.>. Further details are given in Appendix <ref>.This means that in many interesting examples the on-shell potential encodes subtle back-reaction effects that correspond to the correction terms in (<ref>): whenever a source of positive higher-dimensional energy is turned on that does not satisfy the condition (<ref>) in a compactification that is stabilized by sources that also do not satisfy the condition, higher order corrections in (<ref>) conspire to keep the overall potential energy negative.§.§ A simple example: Freund-Rubin compactificationWe will now demonstrate this behavior using the well known Freund-Rubin compactification. This is a6D theory compactifiedon a S^2, with the S^2 stabilized by 2-form fluxes <cit.>. The 6D action isS_6=M^4/2∫(*R_6-1/2F_2∧*F_2),with a 2-form field-strength F_2=dA_1. The equations of motion/Bianchi identity areR_MN=1/2F_MPF_N^P-1/8g_MN|F_2|^2, dF_2=0=d*F_2.These admit a solution where the 6D geometry is a product AdS_4× S^2 and the S^2 is threaded by N units of 2-form flux F_2=N/2qω_2. Here, ω_2 is the volume form of the S^2 normalized to ∫_S^2ω_2=4π and q is theU(1) charge of the particle that couples electrically to A_1 with smallest charge. The S^2 radius is fixed atL_0^2=3N^2/32q^2,and in agreement with (<ref>) the 4D vacuum energy reads V· M_ P^-4=𝒱^-2M^2∫ d^2 y √(g^S^2)(-2|F_2|^2)/16M^2=-(12π M^4L_0^4)^-1.One may try to uplift the four dimensional vacuum energy by adding a number N_3 of three-branes of positive tension T_3 smeared over the internal two-sphere. Clearly they are a source ofenergy density and therefore the expectation is that they give rise to an increase of the 4 dimensional vacuum energy. However, their energy momentum tensor satisfies (D-6)T^μ_μ-4T^m_m=0and so there is no new contribution under the integral in(<ref>). Even without knowing the full solution to the 6D equations of motion we can hence express the 4D vacuum energy as a function of the a priori unknown size of the two-sphere:V· M_ P^-4=𝒱^-2M^2∫ d^2 y √(g^S^2)(-2|F_2|^2)/16M^2=-M^2L_0^2/12π M^6 L_1^6,where L_0 is given in eq. (<ref>) and L_1 is the adjusted length-scale of the two-sphere. Here, the wonders of on-shell potentials manifest themselves explicitly for the first time: The source of uplift in potential energy, the three-brane tension, has apparently disappeared from the integrand of the on-shell potential. It does however appear implicitly through the dependence of the S^2-radius on the three-brane tension. This way of parameterizing the 4D vacuum energy in terms of a single unknown quantity (in this case the radius) should be kept in mind as it will play an important role in the remainder of the paper. From eq. (<ref>) one may already follow that no matter how much three-brane tension is added, the vacuum energy cannot increase beyond zero but at most approaches zero from below. In this simple 6D case one can do better and from the internal Einstein equations determine L_1 as a function of the three-brane tension:L_1^2=(1-𝒯_3)^-1L_0^2, with𝒯_3≡N_3T_3/4π M^4.Plugging this into eq. (<ref>) we see that the vacuum energy adjusts and as expected to leading order in the dimensionless tension 𝒯_3 the change in potential energy corresponds precisely to the three-brane tensionδ(V· M_ P^-4)=𝒯_3/4π L_0^4M^4+𝒪(𝒯_3^2)=N_3 T_3/M_ P^4+𝒪(𝒯_3^2).Of course, as we increase the three-brane tension such that 𝒯_3⟶ 1 the 4D vacuum energy increases, butthe higher order terms become relevant and will prevent an uplift to de Sitter. Therefore,the 4D vacuum energy will approachzero from below, the limit being 𝒯_3=1, wherethe S^2 decompactifies. Note that as predicted by (<ref>) the expansion parameter that controls back-reaction is given by δ V/m_KK^2M_ P^2 because the KK-scale is the mass of the lightest modulus.One might be concerned that three-branes and two-form fluxes share an intrinsic property making them unsuitable uplifting ingredients because they appear with the wrong sign under the integral of (<ref>). Of course, if one only includes these ingredients in the compactification de Sitter solutions are ruled out <cit.>. But as we now demonstrate it is enough to include also a positive 6D c.c., or equivalently a five-brane of tension T_5≡𝒯_5 M^6, for an uplift to de Sitter by three-branes or fluxes to be possible (see also <cit.> for related conclusions). In this case the size of the 2-sphere is bounded from above vial^2≡ L^2M^2=1-𝒯_3/𝒯_5(1-√(1-3/16𝒯_5n^2/(1-𝒯_3)^2))≤1-𝒯_3/𝒯_5,where n≡ N· M/q corresponds to the number of two-form flux quanta. When n^2>n_max^2≡16/3𝒯_5^-1(1-𝒯_3)^2 the sphere decompactifies. Thus, in order for the curvature of the sphere to be sub-planckian we need both a small positive 6D cosmological constant 𝒯_5≪ 1 as well as a large number of two-form fluxes n.The on-shell potential according to (<ref>) readsV· M_ P^-4=1/16π[2𝒯_5/l^2-n^2/8l^6]. Clearly the no-go theorem of Maldacena and Nunez <cit.> is evaded and de Sitter vacua are possible although not generic.[As an aside we note that the de Sitter vacua with largest possible cosmological constant lie along the line of marginal stability n=n_max for which l^2=l_max^2≡1-𝒯_3/𝒯_5. The value of the c.c. is given by V_max· M_ P^-4=𝒯_5/12π𝒯_5/1-𝒯_3_=l_max^-2≪ 1≲ l_max^-4. It is of order the higher dimensional c.c. times the usual volume suppression factor. Note however that the required tuning of the 6D c.c. amounts to a further volume suppression V_max≲ l^-4.] We give a concrete example in Figure <ref>. For completeness we also give the off-shell potential (plotted in Figure <ref>)V· M_ P^-4(l)=1/16π[4𝒯_5/l^2 +n^2/4l^6-4(1-𝒯_3)/l^4].Evidently, the flattening behavior observed for the case with only fluxes and three-branes does not exhibit any intrinsic feature of branes and fluxes but is merely a property of the simple scheme of moduli stabilization. By including a positive 6D c.c. it was possible to decouple the lightest modulus mass from the value of the 4D c.c. such that a small perturbation could uplift it to 4D de Sitter.§ THE 10D PERSPECTIVEWe would now like to study moduli stabilization and the uplift to de Sitter space from a ten-dimensional point of view. The classical part is well understood: the Gukov-Vafa-Witten superpotential can be lifted to the ten-dimensional three-form potential of type IIB supergravity <cit.> and the 4D SUSY conditions that determine the three-form fluxes to be of Hodge-type (2,1) lift to the 10D SUSY conditions of B-type <cit.>. Furthermore the 4D scalar potential is minimized precisely when the 10D equations of motion are solved by the imaginary self-dual (ISD) solutions of <cit.>.An analogous 10D ⟷ 4D correspondence of Kähler moduli stabilization is somewhat harder to establish, both conceptually as well as technically:[We thank Arthur Hebecker for a very helpful discussion concerning this point.]the dynamical origin of the exponential superpotential is the condensation of gaugino bilinears in the 4D SYM gauge theory (or euclidean D3 brane instantons). The scale below which the condensation occurs is the dynamical scale of the gauge-theory which typically lies far below the Kaluza-Klein scale. So, how can it be possible even in principle to include the non-perturbative effects in a higher-dimensional setup? First, there certainly exist geometrical setups compatible with the correct order of scales: an example isthat of an `anisotropic' Calabi-Yau space in which the four-cycle that the 7-branes wrap is much smaller than the typical length-scale of the transverse space <cit.>. In this case the non-perturbative scale of gaugino condensation can lie far below the Kaluza-Klein scale of the four-cycle and at the same scale as the transverse Kaluza-Klein scale. Anothersituation of this type corresponds to a compactification space that is equipped with warped throats of significant warping. In this case the warped Kaluza-Klein scale lies exponentially below the bulk KK-scale.There has however been crucial progress in recent years in establishing a far more general ten dimensional picture of gaugino condensation <cit.>. First, note that if a mobile D3-brane is present, the classical moduli space of the world-volume scalars is identified with the compactification geometry. In the absence of non-perturbative effects there is no potential for the world-volume scalars and the internal geometry can thus be probed at arbitrarily small energies. Thus, even if non-perturbative effects generate a potential for the world volume scalars one may probe the (quantum-deformed) internal geometry at scales that lie far below the KK-scale. With this in mind one should be able to effectively describe the SUSY vacua with non-perturbative Kähler stabilization by the 10D equations of motion, corrected at order of the value of the gaugino condensate ⟨λλ⟩.So, how is the 10D supergravity corrected in order to account for the non-perturbative effects? Remarkably, the following simple prescription advocated by the authors of <cit.> seems to capture at least the leading order effects: * Use the classical type IIB supergravity together with the DBI and CS actions for localized objects to quadratic order in the worldvolume fermions. * Solve the 10D equations of motion, assuming that the fermion bilinear that corresponds to the 7-brane gaugino is non-vanishing.Clearly this approach needs to be justified. For this let us consider the non-perturbative lifting of the D3-brane position moduli space.This can be studied from different angles. The first is the standard 4D perspective. In compactifications with both D7-branes and mobile D3 branes the gauge-kinetic function of the D7 brane gauge theory depends on the open-string D3-brane position moduli z^i via one-loop open string threshold corrections which were calculated explicitly for a T^4/ℤ_2× T^2 orientifold of type IIB string theory <cit.>. Then, at low energies the non-perturbative superpotential is a function of the position moduli z^i which obtain a potential. The open string calculation of <cit.> was perfectly matched with a dual closed string calculation in <cit.> as follows: a mobile D3 brane treated as a classical localized source in the 10D supergravity induces corrections to the volume of the 4-cycle that the D7-branes wrap which determines the gauge-coupling of the D7 gauge theory.Again, the D3 position moduli enter the non-perturbative superpotential in the 4D EFT[See also <cit.> for a derivation using the language of generalized complex geometry.]. The closed string computation is particularly useful as it readily generalizes beyond simple toroidal orientifolds. In particular, for a stack of N D7-branes with holomorphic embedding equation h(z)=0, the gauge-kinetic function f(ρ,z^i) of the D7 gauge theory depends on the volume modulus ρ as well as the D3 position moduli z^i<cit.>f(ρ,z)=ρ+ln h(z)/2π i.Using this dependence of the gauge-kinetic function f on the D3-brane position moduli one may determine the 4D non-perturbative superpotential to beW∝ e^2π i/Nf=h(z)^1/Ne^2π i/Nρ.So far, classical 10D physics has been used only to obtain the gauge kinetic function (<ref>) while the generation of a non-trivial potential for the D3-brane moduli is determined entirely within 4D effective field theory. Crucially these two steps could be separated because the classical back-reaction of a D3-brane on the classical 10D supergravity solution is finite. This is clearly not the case for an D3-brane, so the quantum corrected 10D action is needed. So how can it be deduced? The key points were derived in<cit.>, where the authors analyzed the generation of a non-trivial classical potential for the position moduli of D3 branes in ISD backgrounds subject to harmonic non-ISD perturbations. Crucially, it was shown that in conifold backgrounds every superpotential that can be written down for the position moduli in the 4D effective field theory can be matched to a non-compact classical 10D supergravity solution such that the scalar potentials coincide. Hence, the quantum corrected 10D supergravity that reproduces the correct D3 brane potential is only corrected by terms that are localized away from the warped throat. Such localized terms are necessary because the entirely uncorrected type IIB supergravity equations do not admit static non-ISD perturbations in the compact case due to the global constraints of<cit.>. It is tempting to identify these localized terms with the terms in the 7-brane action that are proportional to the gaugino bilinear ⟨λλ⟩. Indeed, the superpotential (<ref>) can be encoded in so-called series I three-form flux(G_3)_ij̅k̅∝⟨λλ⟩∇_i ∇_l Re(ln h(z) )g^lm̅Ω̅_m̅j̅k̅,where Ω is the holomorphic three-form of the Calabi-Yau <cit.>. This is precisely the perturbation of three-form fluxes that is sourced by the fermionic bilinear term in the D7-brane action (extrapolated to the non-abelian case). Guided by thisnon-trivial consistency check we believe that the relevant details of non-perturbative volume stabilization are indeed captured by the classical 10D supergravity action assuming a non-vanishing expectation value of the gaugino bilinear. While (by construction) the 10D and 4D pictures can be used equivalently to determine the non-perturbative D3-brane position moduli potential (i.e. the back-reaction of the D3-brane on the 4-cycle size), the 10D approach allows us to also incorporate the back-reaction of an D3-brane unambiguously (for a diagrammatic representation of the argument, see Figure <ref>). Based on this conclusion, we will investigate whether or not the non-perturbative AdS vacua of KKLT can be uplifted to de Sitter. The approach is to compactify the (quantum corrected) 10D theory and identify the terms in the corresponding on-shell potential to find a 10D picture of KKLT moduli stabilization, followed by the (partial) uplift. §.§ Stabilization of Complex Structure Moduli: ISD solutionsFlux compactifications of type IIB string theory were pioneered by Giddings, Kachru and Polchinski (GKP) <cit.>. They showed that in the limit of dilute three-form fluxes the Gukov-Vafa-Witten superpotential descends from the 10D potential of type IIB string theory by straightforward dimensional reduction. Moreover, even for non-dilute fluxes, the 4D solutions obtained from minimizing the GVW superpotential lift to full 10D solutions, the imaginary self-dual (ISD) solutions. These feature exponential warping thus realizing the proposal of Randall and Sundrum <cit.> to naturally generate large hierarchies. Following their discussion one may derive an on-shell expression for the 4D potential that will prove extremely useful (see appendix <ref> for details). Starting from the 10D Einstein frame action of Type IIB supergravity, we make a warped compactification ansatz for the metricds^2=e^2𝒜(y)g̃^(4)_μν(x)dx^μdx^ν+ g^6_mn(y)dy^mdy^n,and write the self dual 5-form flux as F_5=(1+⋆_10)dα (y)∧√(-g̃^(4))d^4x. The corresponding trace-reversed Einstein equations allow to write the 4D Ricci scalar R̃_4 in terms of the matter content on the compactification. The result can be combined with the Bianchi identity for F_5 such that <cit.>∇̃^2 Φ^-= R̃_4D+e^2𝒜/Im(τ)|G_3^-|^2+e^-6𝒜|Φ^-|^2+ e^2𝒜Δ^loc/2π,whereG_3^±≡1/2(*_6± i)G_3,Φ^±≡ e^4𝒜±α, andΔ^loc≡1/4(T^m_m-T^μ_μ)^loc- T_3ρ_3^loc.HereG_3 is the complexified three-form F_3-τ H_3. Moreover T_MN^loc and T_3ρ_3^loc are the energy momentum tensor and D3-brane charge density of localized objects. We work in units such that (M_ P,10D)^8=4π. By assuming the existence of a solution and demanding its global consistency one obtains an expression for the on-shell potential by integrating the above equation over the internal spaceV· M_ P^-4=∫d^6 y√(g^6)/16π𝒱_w𝒱̃_w[- e^8𝒜Δ/2π-|Φ^-|^2], where Δ ≡ 2π|G_3^-|^2/Im(τ)+Δ^loc,𝒱̃_w≡∫_M_6d^6y√(g^6)e^6𝒜,and𝒱_w≡∫_M_6d^6y√(g^6)e^2𝒜. From the on-shell potential (<ref>) it follows immediately that as long as all localized sources satisfy Δ^loc≥ 0 the unique classical Minkowski solutions of type IIB string theory are the ISD solutions,G_3^-=Δ^loc=R̃_4D=Φ^-=0 .Under the same assumption de Sitter solutions are ruled out as well. Therefore, a necessary condition for realizing 4D de Sitter solutions is that there exists at least one localized object that satisfies Δ^loc<0.It is important to note that the ISD solutions also match the 4D no-scale behavior: the volume modulus remains unfixed and corresponds to an overall rescaling of regions of weak warping, leaving strongly warped regions approximately invariant <cit.>. Therefore, the inclusion of any further sources of positive potential energy cannot lead to a stable solution but must rather lead to decompactification. Therefore, in order to discuss the inclusion of further sources of positive energy, one first needs to incorporate Kähler moduli stabilization.§.§ KKLT Kähler moduli stabilization: The 10D perspectiveAs explained in the introduction of this section we will determine the effective on-shell potential (<ref>) by dimensionally reducing the 10D action of type IIB supergravity in the presence of a non-vanishing value of the gaugino condensate. In order to do so the following quantities have to be evaluated, * The value of Δ^loc induced by the non-vanishing gaugino bilinear that appears in the D7 brane action. * The back-reaction on the three-form fluxes G_3 and Φ^-.We evaluate these quantities only in the bulk Calabi-Yau where the effects of warping and three-form fluxes are volume suppressed. We will neglect these effects and thus work to leading order in an inverse volume expansion. Moreover we assume the 7-brane configuration to be in the Sen-limit <cit.>, i.e. 4 D7-branes on top of an O7-plane. In this case there is no classical 10D back-reaction on the Calabi-Yau geometry and the axio-dilaton is constant. The gauge group is SO(8).The calculation is a somewhat tedious but straightforward exercise that has been partially done by the authors of <cit.>. We have provided the detailed derivation inappendix <ref> and merely quote the result here: the piece in the action responsible for the perturbation of the ISD background is[Note that we extrapolate the abelian D7-brane term to the non-abelian case. λλ will be shorthand for λ^αλ_α. We would like to stress that the most powerful argument for the validity of this approach comes not from the fact that this term can be obtained from the (non-abelian) seven-brane action but rather from the 10D⟷ 4D matching of the D3-brane position moduli potential. Therefore the extrapolation of the non-abelian 7-brane action is no further reason of concern for us here.]S_D7⊇∫_M_10πδ^(0)_De^ϕ/2e^-4𝒜λ̅λ̅/16π^2 G_3∧ *Ω +c.c.The action (<ref>) acts as a source for G_3 and a particular solution to the equations of motion is given by <cit.>G^λλ_3=G_3^++G_3^-,with imaginary self-dual (ISD) componente^4𝒜G_3^+=1/πe^-ϕ/2⟨λλ⟩/16π^2(g^ij̅∇_i ∇_j̅Ψ)Ω,and imaginary anti-self-dual (IASD) component(e^4𝒜G_3^-)_ij̅k̅=-i/π e^-ϕ/2⟨λλ⟩/16π^2(∇_i ∇_l Ψ)g^lm̅Ω̅_m̅j̅k̅,where Ψ is determined by ∇^2Ψ=2πδ(Σ), and δ(Σ) is the scalar delta function that localizes on the four-cycle that the 7-branes wrap[Strictly speaking, Ψ is the solution to ∇^2 Ψ= 2π(δ(Σ)-Vol(Σ)/𝒱) <cit.>. We will be interested in the behavior of supergravity fields in the vicinity of localized objects. In this regime the constant correction to the source of Ψ only gives rise to small (i.e. volume suppressed) corrections which we shall neglect consistently.]. Note that Ψ is identified with Relog h(z^i), where h(z^i)=0 is the holomorphic embedding equation of the 7-brane divisor <cit.>. The flux profile (<ref>) is only a particular solution to the equations of motion and is completed by the global harmonic fluxes.Building on these results we now proceed. In order to match the 4D description of KKLT we should be able to determine the value of the volume modulus at the supersymmetric minimum from the value of the GVW superpotential. In order to do this in general one would have to derive the full off-shell potential (<ref>) while we derive only the on-shell potential (<ref>). Fortunately if we assume a relation between the condensate and the 4-cycle volume ⟨λλ⟩∼ e^ia ρ we may deduce the value of ρ by demanding that the quantum deformed 10D SUSY conditions are fulfilled. These were derived in <cit.> and we merely quote their result: to leading order in the gaugino condensate, the three-form flux G_3 is given byG_3=G_3^(2,1)+G_3^λλ,where G_3^(2,1) is harmonic and of Hodge type (2,1). Thus, the (0,3) component of G_3 localizes completely on the 7-brane divisor and is related to the value of the gaugino condensate via the G_3 equations of motion. As a consequence, just as in the 4D EFT description the value of the Gukov-Vafa-Witten superpotential determines the value of the condensate,W_0∼∫_M_6 G_3∧Ω∼e^-ϕ/2⟨λλ⟩∼ e^iaρ.Apart from the harmonic (2,1) component the G_3 flux profile is thus fully determined. Furthermore one may show that the field Φ^- is not perturbed to leading order in the condensate. Δ^loc is given byΔ_loc=-3π/8 e^ϕ/2e^-4𝒜⟨λ̅λ̅⟩/16π^2 Ω· G_3 δ^(0)_D+c.c.This means that all the ingredients of the on-shell potential are gathered and after some algebra (see appendix <ref>) we arrive atV· M_ P^-4=-∫_M_6d^6 y √(g)/32π^2𝒱_w𝒱̃_w(α-β) |⟨λλ⟩/16π^2∇_i∇_j Ψ|^2.Here, the positive but otherwise unspecified numbers α,β are related to the fact that the integral in eq. (<ref>) receives contributions of opposite sign from bulk-fluxes near the position of the seven-branes, as well as fluxes that are fully localized on the branes. Both contributions are UV-divergent (see appendix <ref> for details)[We note that these divergences are very similar to the divergent self-energy of an anti-brane at the bottom of a warped throat which is explained in <cit.>.]. We expect that imposing a string-scale cutoff gives the right result up to regulator dependent 𝒪(1) factors α,β[In an earlier version of this paper, the stronger claim was made that the ratio α/β could be determined uniquely which was based on an ad-hoc regularization of UV-divergences. However, as pointed out to us by the authors of <cit.> this would be inconsistent with the assumption that supersymmetric KKLT vacua exist. Hence, the constraint α>β must be imposed by hand, rather than being a prediction of the 10D calculation itself.]. Assuming the existence of supersymmetric KKLT vacua we conclude that α>β, so that the cosmological constant is negative and proportional to the strength of the gaugino condensate. One can readily generalize this expression to the case of n stacks of 7 branes that wrap different holomorphic representatives of the same divisor,V· M_ P^-4=-∫d^6 y √(g) /32π^2𝒱_w𝒱̃_w(α|∑_a=1^n⟨λλ⟩_a/16π^2∇_i∇_j Ψ_a |^2-β∑_a=1^n|⟨λλ⟩_a/16π^2∇_i∇_j Ψ_a |^2).Crucially, the sign of the cosmological constant depends on the relative phases between the condensates ⟨λλ⟩_a. This suggests that for two condensates one may be able to obtain a cosmological constant that is parametrically smaller than the strength of the individual condensates that set the mass-scale of the lightest modulus. For more than two condensates it may even be possible to obtain de Sitter solutions.This fits nicely with the effective 4D description of multiple condensates: the case of two condensates was studied by Kallosh and Linde <cit.> who show that it is possible to tune the cosmological constant to zero supersymmetrically while retaining a finite volume stabilization.Before we proceed one more comment is in order: the integrals (<ref>) and (<ref>) are quadratically UV-divergent. Imposing a UV-cutoff Λ_UV one finds|V|· M_ P^-4∼Vol(Σ_4)/𝒱^2|⟨λλ⟩/16π^2|^2Λ_UV^2+finite.Since we are not interested in overall order one coefficients we leave a proper EFT treatment of this divergence for future work and simply cut-off the integrals at the string scale[Recall that we work in 10D Einstein frame in units M_ P,10D^8=4π.] Λ_UV^2∼ e^ϕ/2. §.§ The uplift In order to determine if an anti-brane (or in fact any other source of SUSY breaking) at the bottom of a warped throat can uplift to de Sitter in the presence of only a single condensate, we have to take a closer look at the on-shell potential (<ref>). The anti-brane perturbs it in two ways (see figure <ref>), * It sources all local 10D supergravity fields in the throat. Away from the brane their field profiles are harmonic and fall off towards the bulk-CY. * It pulls on the volume modulus, the lightest degree of freedom in the problem. Because we assume an exponential relation between the condensate and the volume modulus, the value of the condensate is changed as well. Let us focus on the local back-reaction first. The naive approach to include the effect of the D3-brane would be to compute its corresponding contribution to the on-shell potential. The outcome is Δ^loc_D3=2T_3>0 (T_3 is the tension of the brane), so the inclusion of the D3-brane implies a new negative contribution to (<ref>). Then, how is the anti-branegoing to provide an uplift? As we already saw in the toy model, the use of on-shell potentials comes with certain peculiarities: The dominant source of uplift (the three-brane tension) did not appear explicitly in the on-shell potential. Rather its uplifting effect is contained implicitly in the induced shift in the volume modulus by which the uplift can be efficiently parametrized even when the back-reacted solution is not known. Now, we are in a similar situation. The explicit appearance of the anti-brane via Δ^loc and in fact all the details of the back-reacted warped throat geometry are suppressed by eight powers of the warp factor.[Note that close to the anti-D3 brane the potential Φ^- is sourced at O(e^4𝒜_0). Hence, the integrand of the on-shell potential is dressed with an overall factor of e^8𝒜.] We know from the off-shell potential (<ref>) that the leading order uplift should be of order e^4𝒜_0 T_3, and therefore, if we only work to order e^4𝒜_0 we can completely neglect the local physics of the throat, making the sign of Δ^loc_D3 irrelevant. Instead, as in the toy example, the leading order uplift is parametrized implicitly by the induced back-reaction on the bulk Calabi-Yau on which we will focus in the following.Before doing so let us set a precision goal for the upcoming analysis. Because the minimal uplifting potential must compete with the AdS depth of the SUSY KKLT minimum one should have e^4𝒜_0≳ |⟨λλ⟩ |^2 as a minimal requirement. Guided by this in the following we will work to leading order ine^(4-2q)𝒜_0|⟨λλ⟩|^q,for any q and consistently neglect higher combined powers of the condensate and the warp factor.The next step is to estimate how the local physics of the bulk is affected by the SUSY breaking source in the strongly warped region. Of course one cannot use on-shell methods to do this. But since the supergravity solution corresponding to a warped throat is known in full detail, this estimate can be done explicitly by perturbing the IR-end of the throat and deriving how fast the corresponding field profiles fall off towards the UV. Building on the work of <cit.> we have provided a treatment of this question in appendix <ref>. The outcome is simple and intuitive: all field profiles fall off exponentially towards the bulk such that for an IR perturbation of order one the corresponding perturbation in the UV is suppressed by powers of the IR warp factor e^𝒜_0. If we neglect dependencies on the CY volume which we assume to be only moderately large it turns out that to leading order in e^𝒜_0 all local supergravity fields receive corrections of order 𝒪(e^p𝒜_0· T_3), p∈{3,4}. Because both G_3^- as well as Φ^- appear quadratically in (<ref>) we may neglect their adjustment that is sourced directly by the anti-brane.Also the localized contribution in (<ref>) contains an overall factor of ⟨λλ⟩. Hence, local field profiles would have to be corrected at least at order e^2𝒜_0 to enter our discussion. No such field profiles can be sourced directly by the antibrane.Hence, we need only consider the adjustment of the universal Kähler modulus.However, calculating the value of the volume modulus is not possible using our on-shell methods and we lack the required off-shell methods to explicitly determine back-reaction effects in the bulk. The only reason that we were able to deduce its value at the supersymmetric point and relate it to the value of the GVW superpotential was that the 10D SUSY conditions gave us enough constraints. At no point did we explicitly minimize an effective off-shell potential of the volume modulus. Let us argue why we are able to proceed despite this shortcoming of the on-shell methods. First, the validity of the on-shell methods as such does not require supersymmetry. We assume that once the antibrane is included all fields adjust to their new respective minima. Hence once the system has found its energetically most favorable configuration the outcome must be consistent with the on-shell potential (<ref>). The only technical difference between the supersymmetric and non-supersymmetric configuration is that we are no longer able to determine the shifted value of the volume modulus. For our purposes this is not a problem. We simply assume that the new minimum lies at a different volumeρ⟶ρ +δρ, and express all the quantities that need to be evaluated as functions of the new (unknown) value of the volume modulus. When the volume modulus assumes a new value, all seven-brane sourced effects adjust accordingly. Discarding effects that are sub-leading in inverse volume (to be addressed shortly), this means that in configurations with one or more gaugino condensates stabilizing the Kähler modulus, upon the inclusion of the anti-brane, equations (<ref>) and (<ref>) still hold albeit as a function of the (unknown) shifted value of the volume modulus.Hence, from (<ref>) it is easy to see that in the case of a single gaugino condensate even in the presence of the anti-brane the on-shell potential is manifestly negative. Ultimately, the reason that the contribution of a single gaugino condensate to V is always negative rests on the fact that its contribution to the integrand of the on-shell potential is always negative[The reason for this is that the 7-brane contribution to Δ≡ 2π|G_3^-|^2/Im(τ)+Δ^loc is positive.] (see appendix <ref> for details). Although there are two 7-brane induced contributions to the integrand of either sign from the induced bulk and brane-localized fluxes, the negative contribution of the 7-brane-induced bulk piece G_3^- overcompensates the 7-brane-localized G_3^+ piece. This happens for all values of ρ as long as both G_3^+ and G_3^- are sourced by a unique 7-brane world volume coupling of the condensate ⟨λλ⟩ to G_3. We have shown in appendix <ref>, that in absence of any other fermion condensates besides ⟨λλ⟩ the coupling ⟨λλ⟩ G_3·Ω is unique at 𝒪(⟨λλ⟩)[We use the straightforward non-abelian generalization of the single D7-brane outcome, guided by our expectation that gauge-symmetry and supersymmetry will uniquely fix this.]. This statement holds at the level of the 10D equations of motion and does not use background supersymmetry for the derivation. Hence, in the end the situation here is analogous to the Freund-Rubin toy model (without the 5-branes) and also here it follows immediately that vacua that are stabilized bya single condensing non-abelian gauge group (in a regime where perturbative corrections can be neglected, see section <ref>) cannot be uplifted to de Sitter.As we have explained we cannot in principle derive the adjusted value of the volume modulus and the corresponding change in the cosmological constant from the on-shell potential. The on-shell potential (<ref>) only reveals the overall sign of the resulting cosmological constant but does not allow to compute its value. For this one would need the full off-shell potential which we currently lack. Nevertheless, we can attempt a parametric estimate (neglecting volume powers) based on generic assumptions about the coupling of an uplift parametrized by the warped tension e^4𝒜_0T_3 to the volume modulus ρ (see appendix <ref>) δρ∼e^4𝒜_0 T_3m_ρ^2∼e^4𝒜_0· T_3|⟨λλ⟩|^2. This estimate is easy to understand: the weaker the stabilization the stronger the volume modulus will react to a small IR perturbation. Therefore for e^4𝒜_0 T_3 ≪ |⟨λλ⟩|^2 we can plug this estimate into the on-shell potential (<ref>) to obtainδ V∼V_∼ |⟨λλ⟩|^2·δρ_∼ e^4𝒜_0· T_3/|⟨λλ⟩|^2∼ e^4𝒜_0· T_3, which describes the naive result of adding energy contributions. Note that this regime corresponds precisely to the region of parametric control where the uplift is small compared to the initial vacuum energy δ V ≪ |V_AdS| and thus the final state is also AdS.When the warped tension is not much smaller then |V_AdS| it is not possible even to estimate the magnitude of the uplift sinceδρ is no longer small. From the on-shell potential we only know that the final state will be 4D AdS and the inclusion of the anti-brane should increase the potential energy. From here we conclude that its magnitude will be flattened out due to strong backreaction on the volume modulus and thus the final energy is not just the sum of V_AdS and the warped anti-brane tension. It is now interesting to go back tothe 4D superpotential (<ref>) to note that our 10D analysis is compatible with the extreme case b → 0, suggesting the existence and significant strength of the superpotential term describing the interaction between the anti-brane and the gaugino condensate (see figure <ref>). We will comment further on this in section <ref>.For completeness, we would like to comment here on an interesting effect of the volume shift caused by the D3-brane.We are assuming an exponential relation between the condensate ⟨λλ⟩ and the volume modulus, and so the condensate must be reduced in magnitude due to the shift. Since the profile G_3^λλ is uniquely determined by the strength of the gaugino condensate, the situation is so far analogous to the supersymmetric case except that we have to allow for a global harmonic (0,3) component in the three-formG_3=G_3^(2,1)+G_3^(0,3)+G_3^(λλ).The global (0,3) component enters the on-shell potential (<ref>) via Δ^loc and thus gives rise to a further term in the on-shell potential. The magnitude of the global (0,3) component is determined as a function of the shifted volume by demanding the value of ∫ G_3 ∧Ω to be conserved.[Here we assume that back-reaction on complex structure moduli is negligible.] Since the localized (0,3) piece is reduced in comparison with the supersymmetric setup, part of it is converted into the global (0,3) component. This harmonic (0,3) component is interesting because it determines soft masses on branes <cit.> of order the warped tension of the anti-brane. For our purposes, it enters the on-shell potential (<ref>) via Δ^loc only with a relative volume suppression with respect to e.g. the localized flux contribution on the D7-brane stack (see appendix <ref> for details). We work to leading order in 1/𝒱 andtherefore neglect this contribution.The obstruction towards reaching de Sitter in the single condensate configuration canbe evaded in a rather simple generalization of the setup: in racetrack configurationswith at least two condensing non-abelian gauge groups the multi condensate potential (<ref>) confirms from a 10D perspective the possibility to decouple the lightest modulus mass from the AdS-depth in the supersymmetric vacuum by giving the individual condensates opposite phases. In a limiting case it may even allow for Minkowski vacua after stabilizing the Kähler modulus. This situation would correspond to the case when the two terms in (<ref>) are of the same magnitude upon integration and thus compensate each other. Note that in this case there exists the required positive contribution to the on-shell potential, allowing to evade the Maldacena-Núñez theorem, in analogy with the Freund-Rubin toy model when the 5-brane ispresent. Therefore, the inclusion of the D3-brane on the bottom of a warpedthroat in this configuration could well provide the necessary uplift to de Sitter (see Figure <ref>).While we cannot prove the full 10D consistency of such racetrack stabilization mechanisms we see no sign of failure in an anticipated fully fledged back-reacted 10D solution. If the required tuning can be realized it would seem that such vacua can be uplifted to de Sitter generically (as explained in section <ref>), i.e. by any SUSY breaking energy density that is red-shifted sufficiently strongly to suppress back-reaction on the volume modulus (i.e. e^4𝒜_0≪ |⟨λλ⟩_a|^2). Thus, it seems that such two-condensate (racetrack) stabilization mechanisms are ideal backgrounds for uplifting scenarios, independently of the details of the uplifting mechanism, and the details of its couplings to the lightest moduli. However, we would also like to point out that the mere existence of warped throats with warping in the regime e^4𝒜_0≪ |⟨λλ⟩_a|^2 is not obvious. §.§ Towards an interpretation of an unsuppressed S⟨λλ⟩ coupling We would like to conclude this section with an attempt of a physical interpretation of our result. We have constrained the anticipated fully fledged backreacted solution using on-shell methods. Although these methods are powerful enough to constrain the sign of the cosmological constant they reveal very little about its physical origin. So, let us speculate about it:We are assuming that near the bottom of the warped throat the local description of a warped throat à la Klebanov-Strassler still holds, while at the same time we believe that the 4D EFT description of the anti-brane state using a nilpotent superfield is valid. Putting these two expectations together, and recalling the superpotential of eq. (<ref>), we are led to the interpretation that the IR warp factor itself receives corrections of order |⟨λλ⟩|, i.e.e^2𝒜_0⟶ e^2𝒜_0+𝒪(|⟨λλ⟩|).We have asked ourselves if this expectation is reasonable in view of considerable effort that has been put into determining the position moduli potential of D3-branes <cit.> and D3-branes <cit.>. In principle, the quantum corrected warp factor can be reconstructed by adding the D3- and D3 potentials. In <cit.> it was found that the D3-brane moduli potential as induced by the non-perturbative bulk effects is of order e^𝒜_cl,0|⟨λλ⟩|^2 and is hence suppressed by a classical warp factor. However, to the best of our knowledge, the tools used in <cit.> are only sensitive to the force that acts on the D3-brane, that is only the position-dependent part of the D3-brane moduli potential is determined. Their results are hence compatible with the correction that we have suggested.If this correction indeed occurred, an interesting consequence would be that the IR warp factor receives significant corrections in the regime e^4𝒜_0∼ |⟨λλ⟩|^2. Nevertheless, the local description of the IR-region of the warped throat would stay intact as long as the weaker requirement |⟨λλ⟩|^2≪ e^3𝒜_0 is fulfilled.[If this requirement is violated local IASD field strengths would become dominant <cit.>.] A local throat observer would notice the effect we are suggesting merely as a change of Newton's constant.An interesting prospect for future research would be to validate or falsify our interpretation by finding the explicit back-reacted warped throat solution. The tools needed to do this have been laid out in <cit.> who have put the type IIB supergravity equations of motion into 'triangular' form suitable for determining the response of the throat to a source term in a systematic fashion. Following their strategy one could expand the fields Φ^- and Φ^+ that determine the backreacted warp factor asΦ^-=Φ^-|_ISD_=0+δΦ^-|_ℐℋ+δΦ^-|_ℋ,(Φ^+)^-1=(Φ^+)^-1|_ISD_=1/2e^-4A|_ISD+δ(Φ^+)^-1|_ℐℋ+δ(Φ^+)^-1|_ℋ,where the subscripts ℐℋ and ℋ denote the inhomogeneous respectively homogeneous part. Integrating the equations of motion would determine Φ^- as well as Φ^+ from which the warp factor can be determined uniquely.§ APPLYING THE LESSONS:DS IN KKLT WITH Α'-CORRECTIONS? Let us use the lessons learned from the previous analysis to speculate about alternatives to the racetrack setup in the language of 4D EFT. From the type IIB analysis in <cit.> as well as more recent lifts to F-theory in <cit.> we know that the R^4-terms and their supersymmetric completion in 10D induce the leading correction to the 2-derivative kinetic terms of the Kähler moduli at O(α'^3). The results of <cit.> demonstrated that other α'-corrections involving higher powers of the RR and/or NSNS p-form field strengths produced 4D contributions suppressed by additional inverse powers of the compactification volume, while the results of <cit.> show that the same is true for string loop corrections due to the extended no-scale structure present in type IIB Calabi-Yau compactifications. As a result, the Kähler potential of the volume moduli acquires at leading order an O(α'^3) correctionK=-2ln(𝒱+ξ/2)+K_c.s. with:ξ∼ - g_s^-3/2χ_CYwhere χ_CY=2 (h^11-h^21) is the Euler characteristic of the Calabi-Yau in question. We again assume supersymmetric flux stabilization of the c.s. moduli and axio-dilaton. We then proceed to use the superpotential of KKLT W=W_0+A e^iaρ. The resulting scalar potential is (see <cit.>) V_F = e^K [K^ρρ̅(a^2 A^2 e^ia (ρ-ρ̅)+iaA(e^iaρW̅ K_ρ̅-e^-iaρ̅ W K_ρ))+ 3ξξ^2+7ξ𝒱+𝒱^2/(𝒱-ξ) (2𝒱+ξ)^2|W|^2] = e^K_c.s./8 σ^3[4σ^2/3(a^2A^2 e^-2a σ+2aA e^-a σIm(e^-ia Re ρW K_ρ)) . . + 3ξξ^2+7ξσ^3/2+ σ^3/( σ^3/2-ξ) (2 σ^3/2+ξ)^2|W|^2]where we defined σ≡ Im ρ = 1/2𝒱^2/3. We plot this potential as a function of σ for the parameter choice of the original numerical example in KKLT: a=0.1, A=1, W_0=-10^-4 in figure <ref>. We see that the α'-correction only starts to modify the vacuum energy of the SUSY KKLT vacuum significantly once ξ is large enough to violate ξ≪𝒱. Looking at the figure, we observe the following. Firstly, m_ρ is determined roughly by the whole potential difference between the minimum and the barrier top. Secondly, by arranging for ξ→𝒱 we can get a finite hierarchy between the AdS vacuum energy of the minimum and the total potential difference between the minimum and the barrier top |V_AdS|≳ 1/3 Δ V_tot< Δ V_tot where Δ V_tot=V_barrier-V_top. Putting this together, this implies that we can arrange for a finite hierarchy of the necessary dS upliftδ V∼ |V_AdS|∼ 1/3 Δ V_tot<Δ V_tot∼ m_ρ^2M_ P^2. As this hierarchy was the condition for perturbatively small uplifting to de Sitter which we learned from the preceding discussion, it is possible that this simple modification of KKLT with a single gaugino condensate for each volume modulus might allow forperturbatively controlled uplifts to de Sitter. Note however, that for the hierarchy m_ρ^2M_ P^2 > |V_AdS| to emerge, we need to have ξ≃𝒱. This is a regime where we might worry about further, higher α'-corrections no longer being smaller than the leading one, so further analysis is necessary to clarify the viability of this modification of KKLT to get to de Sitter. Given these observations, it is tempting to speculate about the situation in models where the α'-correction to K is under parametrical control, such as the Large Volume Scenario (LVS) <cit.>. In its simplest version it involves a CY with two Kähler moduli ρ_b and ρ_s and a volume of swiss cheese form 𝒱= σ_b^3/2-σ_s^3/2 where we denote the 4-cycle volumes as σ_i =Im ρ_i. Furthermore, the only relevant non-perturbative contribution to W will wrap σ_s W= W_0+A_s e^i a_sρ_s. For this setup there exists for generic values of |W_0|∼𝒪(1) a minimum for both 𝒱 and σ_s∼ξ^2/3∼𝒪(10) at exponentially large volume 𝒱∼ |W_0| e^a_s σ_s≫σ_s^3/2. The interesting observation for our purpose here is, that this minimum is a SUSY breaking AdS minimum with the following scales |V_AdS|∼|W_0|^2/𝒱^3∼ m_ V^2 M_ P^2∼ m_σ_b^2 M_ P^2 ≪ m_σ_s^2 M_ P^2 ∼|W_0|^2/𝒱^2. Hence, in the LVS setup the amount of anti-brane uplift δ V ∼ |V_AdS| ∼𝒱^-3≪ m_σ_s^2 M_ P^2 is much smaller than the mass scale of the only 4-cycle volume directly controlling the scale of a non-perturbative effect.Naively, this seems to satisfy the upshot of our preceding 4D and 10D analysis, namely the controlled-uplift condition δ V ≪ m_ϕ_i^2 M_ P^2 for all moduli ϕ_i involved in non-perturbative moduli stabilization dynamics. However, LVS setups involve always at least two volume moduli, and their mechanism use the α'-correction to produce one volume modulus with exponentially enhanced VEV via an exponential lever arm in terms of a small, but heavy blow-up volume modulus. At this point it is therefore far from clear, that the naive scale separation between the necessary amount of uplift in LVS models and the parametrically heavier blow-up 4-cycle controlling the leading non-perturbative effect, will shield the light overall volume direction sufficiently from the backreaction effects of uplift to conclude a higher level of safety of LVS compared to KKLT models with a single non-perturbative effect per 4-cycle.Another important point pertains to the fact, that the inclusion of the α'-correction is so far only possible in 4D EFT, given that its effect on the scalar potential only emerges after compactification. It is an open question how to include such α'-corrections from higher-curvature terms at the level of the 10D description directly. Hence, we leave the analysis of the 10D stability of scenarios like LVS under uplifting as a tantalizing problem for the future. § CONCLUSIONIn this paper we have reported progress towards a 10D description of the steps involving volume moduli stabilization without and with anti-D3-brane uplifting as part of the KKLT mechanism of generating a landscape of controlled meta-stable de Sitter vacua in type IIB string theory. We started by outlining a simple bottom-up 4D EFT argument indicating that models which require an uplift of the order δ V≳ m_ρ^2M_ P^2 (such as the simplest version of KKLT) are highly sensitive to the functional form of δ V(ρ) with ρ being the lightest modulus. We have argued that with our current knowledge of non-perturbative volume stabilization and D3-brane potentials the existence of SUSY breaking de Sitter vacua within the simplest version of KKLT is hard to establish. It is for this reason that we have turned to a 10D description of the required non-perturbative effects that is powerful enough to decide the fate of the original version of the KKLT uplift as well as its racetrack improvement.In order to do so, we have relied on two central methodological steps. Firstly,we were interested in the energetics of the volume modulus in a full 10D description. Assuming a solution to the full 10D system exists, we were interested in the on-shell potential of this solution, and its scaling under changes of the stabilized CY volume. Conveniently this on-shell scalar potential incorporates all geometric back-reaction effects by expressing them via the matter contributions to the theory – provided the backreacted solution exists. For this purpose it was crucial that we could determine this on-shell potential from the matter sources alone by combining the trace-reversed 10D Einstein equations with the 10D Bianchi identities <cit.>.We have analyzed a simple toy model in 6D consisting of quantized 2-form flux on a positively curved S^2 and 3-branes, where fully backreacted solutions are easy to establish. We showed that sufficient uplifting by the 3-branes to naively reach de Sitter led to significant flattening of the uplift: by this we mean that the increase of the S^2 radius R with increasing uplift occurred such that the total vacuum energy remained AdS while approaching zero from below, supporting the 4D EFT worry. Without including the backreaction on R the uplift would have been linear in the tension of the 3-branes which would have naïvely allowed for de Sitter minima.We then extended this toy model by adding a positive 6D cosmological constant. This allowed us to somewhat decouple the scale of the AdS vacuum energy from m_ρ^2 prior to adding the 3-branes for uplifting. In this case, the flattening from volume increase becomes less pronounced, and it is possible to reach de Sitter. From here we concluded that for models where reaching de Sitter requires δ V_D3≳ m_ρ^2M_ P^2 according to the off-shell potential, the details of the method of volume stabilization are essential.Based on this, we needed to re-analyze volume stabilization in KKLT itself in 10D. We started by re-deriving the back-reaction onto the flux geometry caused by the gaugino condensation process on a D7-brane stack in the KKLT setup based on <cit.> and extracting for the first time the full contribution from the D7-gaugino condensate to the matter field side of the trace-reversed Einstein equations. This allowed us to calculate an on-shell potential for the ρ-modulus in the simplest KKLT setup. In turn, this enabled us to understand the minimization of the ρ-potential into a SUSY AdS minimum while matching the step 2 4D effective description outlined above. A crucial result of this analysis is that the (0,3) component of the 3-form flux generating the small |W_0| VEV of the Gukov-Vafa-Witten flux superpotential in step 1 becomes effectively localized at the D7-brane stack due to the backreaction to the formation of the gaugino condensation on the 7-branes.Finally, we added back an anti-D3-brane at the bottom of the throat to complete the KKLT recipe. We observed that the in general complicated field profiles that the anti-brane sources in the warped throat fall off too rapidly towards the bulk Calabi-Yau to cause relevant effects in the on-shell potential. In contrast the on-shell potential preserves its functional form but is to be evaluated at the adjusted value of the Calabi-Yau volume and the condensate. In other words we have calculated the on-shell potential for ρ as a function of the anti-D3 induced ρ-increase.Because for a single gaugino condensate arising from a single D7-brane stack the on-shell potential remains manifestly negative (at leading order in an expansion in inverse compactification volume) the 4D vacuum energy flattens out as a function of increasing anti-D3-brane tension sufficiently strongly to prevent a successful uplift to de Sitter. The simplest KKLT vacua are therefore meta-stable SUSY breaking AdS vacua always. For sufficiently small warped anti-brane tension the uplift simply adds to the AdS vacuum energy as predicted by the usual off-shell potential. We match this behavior to the effective 4D description of the anti-D3-brane by a nilpotent superfield S appearing in the superpotential. The flattening of the uplift then requires the presence of an unsuppressed S⟨λλ⟩ coupling in the superpotential.However, we then proceed to replace the single non-perturbative effect driving SUSY stabilization of ρ by a racetrack superpotential arising from a product gauge group with at least two condensing subgroups (on one or several D7-brane stacks). In this case we can tune the supersymmetric AdS vacuum energy before uplifting arbitrarily close to zero, thus breaking the single-condensate KKLT relation m_ρ^2M_ P^2∼ |V_AdS| into m_ρ^2M_ P^2≫ |V_AdS|. While this tuning possibility was long known in the context of the effective 4D supergravity description of the gaugino condensates, we provide a 10D description of this situation, and identify the 10D origin of the tuning possibility.Relying on this tuning, the amount of uplifting δ V∼ |V_AdS|≪ m_ρ^2 M_ P^2 is now perturbatively small. We can thus treat ρ as rigid, which seems to make uplifting to de Sitter possible in the racetrack case both from the 4D and 10D perspective.Summing up our results, we believe that they enable a 10D understanding of the KKLT mechanism including its racetrack improvement. This relaxes the necessity of relying on the purely 4D methods employed so far for the steps of volume stabilization and uplifting in the KKLT framework.§ ACKNOWLEDGEMENTS We thankWilfried Buchmüller, Markus Dierigl, Ben Freivogel, Fridrik Freyr Gautason, Vincent Van Hemelryck, Luis Ibáñez, Fernando Marchesano, Paul Oehlmann,SushaParameswaran, Thomas Van Riet, Irene Valenzuela, Timo Weigand, and especially Francisco Pedro, Eva Silverstein and Angel Uranga for various illuminating discussions. We are deeply grateful to Arthur Hebecker and Liam McAllister for carefully reading the manuscript and providing many very useful comments and extensive discussions. We are grateful to the SITP in Stanford, the IFT UAM/CSIC, and the PASCOS 2017 conference in Madrid, as well as the Lorentz Center Workshop “Theoretical Approaches to Cosmic Acceleration” in Leiden for their warm hospitality during various stages of this work. This work is supported by the ERC Consolidator Grant STRINGFLATION under the HORIZON 2020 grant agreement no. 647995.§ TYPE IIB AND THE NO-GO THEOREM OF GKPFor flux compactifications of type IIB string theory the no-go theorem of <cit.> was extended significantly by Giddings, Kachru and Polchinski (GKP) <cit.>: following them, we start with the Einstein frame action of type IIB Supergravity, S_IIB=2π∫_M_10(*R_10-dτ∧ *dτ/2(Im(τ))^2-G_3∧ *G_3/2Im(τ)-F_5∧ *F_5/4-iC_4∧ G_3∧G_3/4Im(τ))+S_loc,where τ=C_0+ie^-Φ is the axio-dilaton, G_3=F_3-τ H_3 is the complexified three-form field-strength, and F_5=dC_4-1/2C_2∧ H_3+1/2B_2∧ F_3.S^loc is the action of localized objects and we have set (2π)^2 α'=1.Let us take the following ansatz for the 10D metricds^2=e^2𝒜(y)g̃^(4)_μν(x)dx^μdx^ν+e^-2𝒜(y)g̃^(6)_mn(y)_≡ g^6_mn(y)dy^mdy^n,where g̃^4_μν is a maximally symmetric 4D metric. Then, the most general ansatz for F_5 that respects the symmetries of the 4D space-time isF_5=dC_4+ℱ_5,where ℱ_5 is purely internal and satisfies dℱ_5=iG_3∧G_3/2Im(τ)+T_3/2πρ_D3^loc,with D3-brane charge T_3=2π and D3-charge density of localized objects ρ_D3^loc. C_4 can be expressed asC_4=α(y)√(-g̃^4) dx^0∧ dx^1∧ dx^2∧ dx^3,with a real function α(y) and is subject to the self-duality constraint *dC_4=ℱ_5.The 4D components of the 10D Ricci tensor can be expressed asR_μν=R̃_μν-1/4g_μνe^-4𝒜∇^2e^4𝒜,where R̃_μν is the 4D Ricci tensor of the metric g̃^4_μν. Therefore, the trace over the 4D components of Einsteins equations can be expressed asR̃_4-∇̃^2_6 e^4𝒜=-e^2𝒜/2Im(τ)|G_3|^2-e^-6𝒜((α)^2+( e^4𝒜)^2)-1/8πe^2𝒜(T^m_m-T^μ_μ)^loc,where T_MN^loc=-2/√(-g)δ S_loc/δ g^MN is the energy momentum tensor of localized sources and R̃_4D is the Ricci-scalar of the metric g̃^4.For the ansatz (<ref>), the F_5 Bianchi identity can be expressed as∇̃^2_6 α=ie^2𝒜/2Im(τ)G_3· *_6 G̅_̅3̅+2e^-6𝒜(α)·( e^4𝒜)+T_3/2π e^2𝒜ρ_3^loc,and by taking the difference between (<ref>) and (<ref>) one obtains <cit.>∇̃^2 Φ^-= R̃_4D+e^2𝒜/Im(τ)|G_3^-|^2+e^-6𝒜|Φ^-|^2+ e^2𝒜Δ^loc/2π,where R̃_4D is the 4D Ricci scalar and we have definedG_3^±≡1/2(*_6± i)G_3,Φ^±≡ e^4𝒜±α, andΔ^loc≡1/4(T^m_m-T^μ_μ)^loc- T_3ρ_3^loc.Integrating this expression over the compact manifold it follows that in the absence of localized sources that violate the BPS-like boundΔ^loc≥ 0,the 4D vacuum energy V· M_ P^-4=1/4M_ P^-2R̃_4D is negative semi-definite,V· M_ P^-4=∫d^6 y√(g^6)/16π𝒱_w𝒱̃_w[- e^8𝒜Δ/2π-|Φ^-|^2]≤ 0,where Δ ≡ 2π|G_3^-|^2/Im(τ)+Δ^loc,𝒱̃_w≡∫_M_6d^6y√(g^6)e^6𝒜,and𝒱_w≡∫_M_6d^6y√(g^6)e^2𝒜.In this notation the solutions of <cit.> correspond toG_3^-=Δ^loc=R̃_4D=Φ^-=0 .It is important to note that these solutions leave at least one modulus unfixed, the volume modulus. Clearly the inclusion of any further sources of positive energy cannot lead to a stable solution but rather lead to decompactification.This is true in particular for D3 branes: the fact that they give a negative contribution to Δ^loc has the following simple interpretation: without Kähler moduli stabilization the presence of an D3 brane leads to decompactification. § CONTRIBUTIONS OF FLUXES AND LOCALIZED OBJECTS TO THE ON-SHELL POTENTIALIn this appendix we would like to summarize the contributions of localized objects of tension T_p and p-form fluxes to the on-shell potential (<ref>) following <cit.>.§.§ The Contribution of p-form Fluxes For definiteness, we consider the case of a higher-dimensional p-form field strength, with actionS_p-form=M^D-2/2∫(-1/2F_p∧ *F_p).Then, the energy momentum tensor readsT_MN=1/21/(p-1)!F_MO_2⋯ O_pF_N^O_2⋯ O_p-1/4g_MN|F_p|^2.Let us now write F_p=F_p,ext+F_p,int, where F_p,ext threads the 4 non-compact directions if p≥ 4 and vanishes if p<4 while F_p,int threads internal directions only. It follows thatT_P^P=2p-D/4(|F_p,ext|^2+|F_p,int|^2) and T_μ^μ=|F_p,ext|^2-|F_p,int|^2,and the on-shell potential (<ref>) is proportional to a (weighted) integral over the expression (D-6)T^μ_μ-4T^m_m=2(D-p-1)|F_p,ext|^2_≤ 0+2(1-p)|F_p,int|^2_≥ 0.It follows that internal components make a positive contribution only in the case p=0 and do not contribute when p=1 <cit.>. External components make a positive contribution for the top-form p=D and a vanishing contribution for p=D-1. The latter two are of course equivalent to the former two by Hodge-duality.§.§ The Contribution of Localized ObjectsIn addition, let us see how localized objects of spatial dimension p and tension T_p contribute. For this, consider a DBI-like actionS_loc=-T_p∫ d^p+1ξ √(-(P[g])),where P[g] is the pullback of the ambient space metric on the localized objects world-volume. The energy momentum tensor isT_MN=-T_p/M^D-2Π_MN(Σ)·δ(Σ),where Π(δ) is the projector on the p-3 cycle Σ that the object wraps and δ( Σ) is the transverse δ function. Hence, they contribute with(D-6)T^μ_μ-4T^m_m=-4(D-p-3)T_p/M^D-2δ(Σ).For positive tension T_p one immediately sees that positive contributions to the on-shell potential (<ref>) come only from space time filling or co-dimension one objects. Also, negative tension objects of spatial dimension smaller than D-3 give a positive contribution, while co-dimension two objects do not contribute directly. § GAUGINO CONDENSATES AND THE COSMOLOGICAL CONSTANTIn this appendix we would like to provide the detailed derivation of the formulas used in section <ref>. The perturbed G_3 profile has previously been calculated in <cit.>. We provide also the contribution of all the terms under the integrand of (<ref>).We are interested in the effects of the gaugino condensate in the bulk Calabi-Yau where (by definition) the back-reaction of fluxes on the geometry is negligible. Thus, in the following we work in the constant warp factor approximation where R_mn= 0. Hence, all results are valid to leading order in an inverse volume expansion.§.§ The λ̅λ̅G_3 couplingWe will follow appendix A of <cit.> using results of appendix A of <cit.>. For vanishing worldvolume flux, the fermionic part of the κ-symmetric D7-brane action reads <cit.>S_D7^ferm.=iπ∫_Σ_8e^-ϕ√(- g)θ̅(1-Γ)(Γ^α𝒟_α-1/2𝒪)θ,where α,β... are 8d indices and θ is the two-component GS spinor. The operators 𝒟_α and 𝒪 are the pullbacks of the 10D operators𝒟_Mθ=∇_Mθ+1/4H_M[10;0 -1 ]θ +1/16e^ϕ[0 FΓ_M; -σ(F)Γ_M0 ]θ, 𝒪θ=ϕθ +1/2H[10;0 -1 ]θ+ 1/16e^ϕ[0 Γ^M FΓ_M; -Γ^M σ(F)Γ_M0 ]θ,where Γ_M are 10D gamma matrices. Furthermore,Γ=-iσ_2/√(-g)ϵ_α_0...α_7/8!Γ^α_0...α_7,and the map σ reverses the order of indices. The barred spinor is defined as θ̅=θ^†Γ^0. Underlined tensors are contracted with gamma matrices. The action (<ref>) is written in the (RR-)democratic formulation of type IIB supergravity. This means that F=F_1+...+F_9 and the equations of motion have to be supplemented by the duality constraintFΓ_*=F,in particular F_7Γ_*=F_3. Equivalently, one may work with F_1,F_3 and F_5 only, while doubling the contribution of F_1 and F_3 in (<ref>) and (<ref>). In this case one only imposes the usual self-duality constraint F_5Γ_*=F_5. We will do so in the following.Imposing the κ-fixing gaugeθ̅Γ=-θ̅,and keeping only three form fluxes with purely internal components the action can be written asS_D7^ferm. =2π i∫ d^8 σe^-ϕ√(-g)(θ̅Γ^α(∇_α-1/2_αϕ)θ..+θ̅[1/4(Γ^αH_α-H)σ_3+e^ϕ/8(Γ^αF_3Γ_α-1/2Γ_M F_3Γ^M)σ_1]θ).This action is transformed to 10D Einstein frame usingg_αβ⟶ e^ϕ/2g_αβ,⇒Γ^M⟶ e^-ϕ/4Γ^M,and the redefinitionΘ⟶ e^-3ϕ/8Θ.The result isS_D7^ferm. =2π i∫ d^8 σ √(-g)θ̅Γ^α∇_αθ+e^ϕ/2θ̅[e^-ϕ/4(Γ^αH_α-H)σ_3+1/8(Γ^αF_3Γ_α-1/2Γ_M F_3Γ^M)σ_1]θ.Let us further massage this coupling of the schematic form θ̅θ G_3. Two expressions need to be simplified,I:Γ^αH_α-H, II:Γ^αF_3Γ_α-1/2Γ_M F_3Γ^M.Let us decompose the three-form fluxes as follows:F_3=F_3^(0)+F_3^(1)+F_3^(2), H_3=H_3^(0)+H_3^(1)+H_3^(2),where the upstairs index denotes the number of indices transverse to the divisor Σ_4 the 7-brane wraps. By commuting Γ matrices one can show thatΓ^αFΓ_α-1/2Γ_M FΓ^M=-2F_ijk/3!(P_⊥(Γ^i)Γ^jΓ^k+Γ^iP_⊥(Γ^j)Γ^k+Γ^iΓ^jP_⊥(Γ^k)).Here, P_⊥(Γ^i)=Γ^i if the index i is transverse to Σ_4 while it vanishes otherwise. Under the decomposition (<ref>) this readsΓ^αFΓ_α-1/2Γ_M FΓ^M=-2F^(1)-4F^(2).Similarly, one shows thatΓ^αH_α-H=2H-3H_ijk/3!P_⊥(Γ^i)Γ^jΓ^k=2H^(0)+H^(1).Thus, the θ̅θ G_3 coupling readsS_D7⊇π/2 i ∫ d^8 σ √(-g)e^ϕ/2θ̅(e^-ϕ(2H^(0)+H^(1))σ_3-(F^(1)+2F^(2))σ_1)θ.To perform the dimensional reduction one decomposes the 10D Γ-matrices asΓ^μ=e^-𝒜γ̃^μ⊗𝕀,μ=0,...,3, Γ^i=γ̃_*⊗γ_6D^i-3, i=4,...,9with 4D respectively 6D gamma matrices γ̃^μ and γ^i-3_6D.The dimensional reduction ansatz for the gaugino isθ_1=1/4π e^-2𝒜λ_D ⊗η_1+ c.c. , θ_2=-1/4π e^-2𝒜λ_D ⊗η_2+ c.c.Here, λ_D is a 4D Dirac spinor of positive chirality, and c.c. denotes charge conjugation. Moreover, η_1 and η_2 are the 6D Weyl spinors that appear in the 10D SUSY parameters:ϵ_1=ξ⊗η_1 +c.c.,ϵ_2=ξ⊗η_2 +c.c.For O3/O7 orientifolds with ISD fluxes and D3/D7-branes, the relation between η≡η_1 and η_2 is <cit.>η_2=i η_1. Thus, we may writeθ_1=ψ+ψ^c, θ_2=-i(ψ-ψ^c), ψ≡1/4πe^-2𝒜λ_D⊗η,where the superscript ^c denotes charge conjugation.With this one calculates thate^4𝒜θ̅Γ_ijkσ_3θ=2e^4𝒜ψ^cΓ_ijkψ-c.c.=-i/8π^2λ_D^cλ_D(η^c)^†γ_ijkη-c.c.,ande^4𝒜θ̅Γ_ijkσ_1θ=-2ie^4𝒜ψ^cΓ_ijkψ-c.c.=-1/8π^2λ_D^cλ_D(η^c)^†γ_ijkη-c.c.Here, the 4D barred spinor is the usual Dirac conjugate λ_D=iλ_D^†γ̃^0.Let us defineG_3^(i,j):=F_3^(i)-ie^-ϕH_3^(j), 𝒢_3:=G_3^(1,1)+2G_3^(2,0). Then we may write the result asS_D7 ⊇∫_M_10d^10x√(-g)ℒ^loc_λλ≡∫_M_4×Σ_4 d^8 σ√(-g)e^-4𝒜e^ϕ/2iλ_D^cλ_D/16π Ω·𝒢_3+c.c.=∫_M_4×Σ_4 d^8 σ√(-g)e^-4𝒜e^ϕ/2 λ̅_α̇λ̅^α̇/16π Ω·𝒢_3+c.c.=∫_M_10δ^(0)_De^-4𝒜e^ϕ/2λ̅_α̇λ̅^α̇/16π𝒢_3∧ *Ω +c.c..where we have also used that the holomorphic three-form Ω is given in terms of the 6D spinor η asΩ_ijk=(η^c)^†γ_ijkη,and we have written the action as a 10D integral by introducing the scalar delta-function δ_D^(0) that localizes on the divisor Σ_4. It satisfies <cit.>2πδ^(0)_D= ∇^2 Relog h=2g^ii̅∇_i∇_i̅Relog h,where the holomorphic function h defines the divisor that the seven-brane wraps via h=0.Moreover, in the second line of (<ref>) we have chosen the 4D Weyl-representation where the spinor λ_D takes the form (0,λ̅^α̇)^T. Finally, we used that λ_D^cλ_D=λ_D^TCλ_D=-i λ̅_α̇λ̅^α̇. From now on we will write λ̅λ̅=λ̅_α̇λ̅^α̇ and λλ= (λ̅λ̅)^*.One further simplification can be made: Ω·𝒢_3 projects on the (0,3) piece of 𝒢_3. Because the submanifold that the 7-branes wrap is complex <cit.> (see also <cit.> for flux compactifications beyond GKP) the components of F_3 and H_3 that have precisely zero or two legs transverse to the brane are of Hodge type (2,1)⊕ (1,2) which implies that 𝒢_3·Ω=G_3·Ω. Thus, we finally arrive atS_D7⊇∫_M_10πδ^(0)_De^ϕ/2e^-4𝒜λ̅λ̅/16π^2 G_3∧ *Ω +c.c.. §.§ The G_3 equations of motion and their solutionIn the presence of a non-vanishing expectation value ⟨λλ⟩, the equations of motion and Bianchi identity of G_3 readdΛ -dτ/τ-τ̅∧ (Λ+ Λ̅)=dX-dτ/τ-τ̅∧ (X+X̅), dG^+-dτ/τ-τ̅∧(G^++G̅^̅+̅)=dG^–dτ/τ-τ̅∧(G^-+G̅^̅-̅),where Λ≡ e^4𝒜*_6 G_3-iα G_3 and X is defined viad^4x ∧(dX-dτ/τ-τ̅∧ (X+X̅))=-i/2π(τ d(ℒ^loc_λλ/ dC_2)+d(ℒ^loc_λλ/ dB_2)),which determinesX=e^-ϕ/2⟨λλ⟩/16π^2δ_D *_6 Ω̅.The ISD solutions correspond to Λ=e^4𝒜G^-=0.We expand the above equation to linear order in ⟨λλ⟩. Equation (<ref>) implies that to this order Φ^-=0. Therefore[We thank the authors of <cit.> for pointing out an erroneous factor of two that we have corrected in what follows.],Λ=Φ^+ G_3^-+Φ^- G_3^+=2e^4𝒜G_3^-+𝒪(|⟨λλ⟩|^2),and henced(e^4𝒜G_3^-)=1/2dX.Note that it is not possible to simply set e^4𝒜G_3=1/2X because by definition G_3^- is IASD while X is ISD. In the flat Calabi-Yau limit R_ij=0 the solution was derived in <cit.>. It is shown that bulk IASD fluxes of Hodge type (1,2) are sourced:(e^4𝒜G_3^-)_ij̅k̅=-i/2π e^-ϕ/2⟨λλ⟩/16π^2(∇_i ∇_l Re(log h))g^lm̅Ω̅_m̅j̅k̅, ⇒ 2π e^ϕ|G_3^-|^2=8/πe^-8𝒜|⟨λλ⟩/16π^2|^2 |∇_i∇_j Relog h|^2,using |Ω|^2≡1/3!Ω_ijkΩ̅^ijk=8. Eq. (<ref>) only determines the IASD flux. The ISD part of G_3 follows from imposing the Bianchi-identity (<ref>), i.e.dG_3^+=dG_3^-= 1/2e^-4𝒜dX,which is solved bye^4𝒜G_3^+=1/2X=1/2πe^-ϕ/2⟨λλ⟩/16π^2(g^ij̅∇_i ∇_j̅Re(log h))Ω.This localized ISD flux was computed in <cit.>.§.§ The contributions to the on-shell potentialWe are now ready to derive the contribution of the fermionic action (<ref>) to Δ_loc. For this one has to vary the action with respect to the vielbein e^M_a defined byg^MN=e^M_a e^N_b η^ab.There are two contributions: the first is due to the variation of the volume form. Because the D7 has co-dimension two, this will drop out in Δ_loc. The second contribution comes from varying the curved space Γ-matrices Γ^M=e^M_a Γ^a with constant matrices Γ^a. Then, by using g^mn/ g^mn=1/2e^m_c/ e^m_c it is straightforward to obtain the desired negative contribution to Δ_loc,Δ_loc ⊇Δ_loc^λλ≡ -1/4e^m_cℒ^loc_λλ/ e^m_c=-3π/8 e^ϕ/2e^-4𝒜⟨λ̅λ̅⟩/16π^2 Ω· G_3 δ^(0)_D+c.c.∼-e^-8𝒜|⟨λλ⟩/16π^2|^2 |g^ij̅∇_i ∇_j̅Re(log h)|^2<0.Bulk IASD fluxes (<ref>) contribute to Δ withΔ_bulk^λλ∼e^-8𝒜|⟨λλ⟩/16π^2|^2 |∇_i∇_j Relog h|^2>0.Thus finally,Δ⊇Δ^λλ=e^-8𝒜|⟨λλ⟩/16π^2|^2 [α|∇_i∇_j Relog h|^2-β|g^ii̅∇_i∇_i̅Relog h|^2].Here, we have introduced two undetermined positive 𝒪(1) factors α,β for the following reason[In an earlier version of this paper the "smoothing regularization" was implemented too literally, i.e. specified numerical values of the α,β were given with α/β=4/3. This is problematic for two reasons: First, the regularization scheme is simply too ad-hoc for such a stringent prediction. Second, due to an erroneous factor of two that was pointed out to us by the authors of <cit.>, one would actually have α/β=2/3 which would be inconsistent with the existence of supersymmetric AdS vacua of KKLT type. We thank the authors of <cit.> for discussions about this point.]. The integral of both terms in eq. (<ref>) over the internal manifold diverges toward the position of the seven-brane stacks, and has to be regularized. We expect generically that cutting off the integrals at the string-scale (or smoothing Relog h over an order l_s interval transverse to the seven-branes) gives the correct answer up to uncertainties of order unity. We parametrize this regulator dependence by the 𝒪(1) factors α,β.For a set of n stacks of seven-branes that wrap divisors Σ_a, a=1,...,n, the contribution to Δ becomesΔ^λλ=e^-8𝒜[α|∑_a=1^n⟨λλ⟩_a/16π^2∇_i∇_j Relog h_a|^2-β∑_a=1^n|⟨λλ⟩_a/16π^2 g^ii̅∇_i∇_i̅Relog h_a|^2],where upon integration regularization is implicit as explained above.Because the contribution to the cosmological constant is determined by integrating this expression over the internal manifold we may partially integrate∫_M_6 *_6 |∇_i∇_j g|^2=∫_M_6 *_6 g^ii̅g^jj̅∇_i ∇_jg∇_i̅∇_j̅gp.I.=-∫_M_6 *_6 (g^jj̅(g^ii̅∇_i∇_i̅∇_jg) ∇_j̅g)=-∫_M_6 *_6 (g^jj̅g^ii̅(∇_j ∇_i∇_i̅ g+1/2[∇^2,∇_j]g_=R_jk^k g=0)∇_j̅g)p.I.=∫_M_6 *_6 |g^ii̅∇_i∇_i̅g|^2.for any real function g.Therefore, we may writeΔ^λλ=e^-8𝒜[α|∑_a=1^n⟨λλ⟩_a/16π^2∇_i∇_jRelog h_a|^2-β∑_a=1^n|⟨λλ⟩_a/16π^2∇_i∇_j Relog h_a|^2].Crucially, for a single gaugino condensateΔ^λλ=(α-β)e^-8𝒜|⟨λλ⟩/16π^2∇_i∇_j Relog h|^2,which implies that if gaugino condensation is the only ISD-breaking source, the 4D potential energy is given byV· M_ P^-4=-α-β/32π^2𝒱_w𝒱̃_w∫ d^6 y √(g)|⟨λλ⟩/16π^2∇_i∇_j Relog h|^2.Demanding the existence of supersymmetric KKLT AdS-vacua implies that α-β>0.For two condensates the first term in (<ref>) can be made small by giving the condensates opposite phases. In this case the two terms in (<ref>) cancel each other at least partially. Of course, for a SUSY vacuum the final result can never be positive but we find it very conceivable that a small SUSY breaking source (like an D3 brane) can lift such a vacuum to de Sitter.§ A VOLUME SUPPRESSED CONTRIBUTION Here we would like to address an example of a volume suppressed contribution to the on-shell potential (<ref>) that we must neglect for consistency of our expansion scheme. Before uplifting the flux configuration is as followsG_3=G_3^(2,1)_harmonic+G_3^λλ_particular solution,where G_3^λλ is the profile that has been computed in <cit.>. Therefore, the localized (0,3) piece in G_3^λλ is in the same cohomology class as the harmonic G_3 piece of Hodge type (0,3) that would have existed without non-perturbative stabilization. The value of the gaugino condensate is tied to the profile G_3^λλ via the equations of motion. Thus, the condensate will dynamically find its correct value such that the localized (0,3) piece in G_3^λλ can account for all of W_0=∫ G_3∧Ω.We start with this supersymmetric situation and deform it by including a SUSY breaking source at the bottom of a warped throat. Such a source will push the value of the volume modulus towards larger values and thereby reduce the magnitude of the condensate. ⟨λλ⟩_0⟶⟨λλ⟩_1.Because the field profile G_3^λλ is still tied to the value of the condensate, in order for this to be consistent with flux quantization and a negligible pull on the complex structure moduli there must develop a harmonic (0,3) piece in G_3,G_3⟶ G_3+g^(0,3)Ω,such that ∫ G_3∧Ω/||Ω|| remains constant. Using the identities||Ω||^2=∫_M_6Ω∧Ω=8𝒱 and∫_M_6δ_DΩ∧Ω=8Vol(Σ_4),this fixes g^(0,3) to beg^(0,3)=e^-4𝒜|_Σe^-ϕ/2Vol(Σ)_1/𝒱_1(⟨λλ⟩_0/16π^2Vol(Σ)_0/Vol(Σ)_1√(𝒱_1/𝒱_0) -⟨λλ⟩_1/16π^2).Here the subindices 0 and 1 denote pre- respectively post uplift situations. Clearly this constant flux piece contributes to Δ^loc and therefore contributes a term in the on-shell potential (<ref>). However, to leading order in the inverse volume expansion this term must be neglected. This is clear intuitively because only the value of G_3^(0,3) at the position of the 7-brane divisor Σ_4 enters the on-shell potential (<ref>). If part of the localized term is traded for a constant term to leading order in the inverse transverse volume only a reduction in the localized G_3^(0,3) component is seen. Let us show this explicitly: the constant flux contributes a term to the on-shell potentialV_g^(0,3)· M_ P^-4≡3/16πVol(Σ)_1^2/𝒱_1^3⟨λ̅λ̅⟩_1/16π^2[ ⟨λλ⟩_0/16π^2Vol(Σ)_0/Vol(Σ)_1√(𝒱_1/𝒱_0) -⟨λλ⟩_1/16π^2]+c.c.Because all volumes are pushed towards larger values and the value of the condensate decreases exponentially with the volume we have ⟨λλ⟩_1<⟨λλ⟩_0. Hence this contribution is positive.If a warped tension δ V/M_ P^4=e^4𝒜_0T_3/(4π)^2𝒱^2 is added to the setup it is clear that if the warp factor e^𝒜_0 is sufficiently small, the change in potential is given precisely by this term. Using this information and our knowledge about the off-shell potential we can deduce how far the volume modulus shifts. We assume the usual exponential dependence|⟨λλ⟩| ∼ e^-aVol(Σ),and choose to calculate to leading order in 1/Vol(Σ). In this approximation we need only consider the shift of ⟨λλ⟩ itself and not of explicit volume powers that appear in the above formulas. The shift in the potential in equation eq. (<ref>) induced by a small shift δVol(Σ) isδ V· M_ P^-4∼ a δVol(Σ)· |V_0· M_ P^-4|∼ a δVol(Σ)·Vol(Σ)/𝒱^2|⟨λλ⟩|^2,whileV_g^(0,3)· M_ P^-4∼ a δVol(Σ)·Vol(Σ)^2/𝒱^3|⟨λλ⟩|^2.Hence the positive contribution coming from g^(0,3) is suppressed by the transverse volume and must be neglected. The shift in the volume modulus is related to the warped tension viaa δVol(Σ)∼e^4𝒜_0 T_3/Vol(Σ)|⟨λλ⟩|^2,which determines the value of g^(0,3) to beg^(0,3)∼ e^-ϕ/2e^4𝒜_0 T_3/𝒱.Therefore, similarly to the situation of non-supersymmetric classical ISD solutions the value g^(0,3) parametrizes SUSY breaking for instance by inducing gaugino mass-terms on D7 and D3 branes <cit.>.§ RADIAL PROFILES OF THROAT PERTURBATIONS Our analysis relies on the fact that perturbations that are sourced at the bottom of a warped throat fall-off towards the bulk Calabi-Yau fast enough to have no significant effect on the integrand of the on-shell potential (<ref>). Therefore, in this appendix we provide details about the radial scaling of throat modes. For our purpose it is sufficient to determine the radial scaling of IR perturbations.§.§ Throat perturbations sourced in the IRIn order to be able to determine how (<ref>) is affected by a SUSY breaking source at the bottom of a warped throat, we need to know the approximate radial dependence of field profiles ϕ(r) that correspond to perturbations of the KS throat that are sourced in the IR. We do not need to know their field profiles in full detail but only need to estimate the ratio ϕ(r_UV)/ϕ(r_IR) in terms of powers of the warp factor in the IR e^𝒜_0≡.e^𝒜|_IR and the ρ-modulus mass m_ρ. This makes the analysis much easier as for this purpose one may approximate parts of the throat by AdS_5× T^1,1. In this approximation the field profiles can be determined analytically.In more detail: away from the tip of the throat the geometry is well approximated by the Klebanov-Tseytlin (KT) solution <cit.>ds^2=e^2𝒜(r)dx^2+e^-2𝒜(dr^2+r^2ds^2_T^1,1).withe^-4𝒜=L^4ln(r/r_s)/r^4.The KT solution is valid for r≫ r_s. For our purposes the logarithmic running will be irrelevant and we approximate the throat by AdS_5× T^1,1, i.e. e^-4𝒜⟶ L^4/r^4.For this case the analysis of radial profiles of throat perturbation was done for instance in <cit.>. §.§ Scalar perturbationsSince we are interested in perturbations that are sourced only deep in the IR, only the homogeneous modes of perturbations need to be studied. For scalar perturbations these are solutions to Laplace' equation∇^2 ϕ=0.This is solved by a product ansatz ϕ(r,Ψ)=∑_iϕ_i(r)Y_i(Ψ), where the functions Y_i(Ψ) are eigenfunctions of the Laplacian of T^1,1∇_T^1,1^2Y_i=-λ_i^2 Y_i.Because the 10D Laplacian reads ∇^2=e^2𝒜[_r^2+5/r_r+1/r^2∇_T^1,1^2], the remaining radial equation of motion reads_r^2ϕ_i+5/r_rϕ_i-λ_i^2/r^2ϕ_i=0,and is solved by ϕ_i(r)=∑_ia_ir^Δ_i-4+b_ir^-Δ_i,Δ_i ≡ 2+√(4+λ_i^2).For an infinite throat perturbations that are sourced in the IR are the normalizable ones. Because Δ_i≥ 4, the normalizable mode corresponds to ϕ∝ r^-Δ_i. The KK zero mode of T^1,1 (i.e. the mode with λ_0=0) falls off least rapidly towards the UV, i.e.ϕ_0(r_UV)/ϕ_0(r_IR)∼(r_UV/r_IR)^-4= e^4𝒜_0.In the following we would like to see how this behavior is changed when the throat is finite. In this case the would-be non-normalizable mode is not frozen and can in principle play a substantial role as well. This is particularly clear for the KK-zero mode: if it is sourced in the IR it will simply adjust its VEV along the whole throat. In this case its radial profile is determined entirely by the non-normalizable mode.To discuss this we have to know the UV boundary conditions. Although we do not know these precisely we can encode them in a 'UV-potential' in the effective 5d theory. First we would like to discuss the KK-modes with λ_i≠ 0. For now we set the UV-potential to zero, i.e. impose _rϕ_i|_r=r_UV=0. In this case the solution to the equations of motion can be expressed in terms of the value ϕ_IR≡ϕ(r_IR) as followsϕ(r)=(Δ-4)e^(4-2Δ)𝒜_0r^-Δ+Δ r^Δ-4/(Δ-4)e^(4-2Δ)𝒜_0+Δϕ_IR,which impliesϕ_UV/ϕ_IR=(2Δ-4) e^Δ𝒜_0/e^(2Δ-4)𝒜_0Δ+(Δ-4)=2 e^Δ𝒜_0 for 1≪λ^2, ∼ e^Δ𝒜_0,forλ∼ 1, 4e^4𝒜_0/4e^4𝒜_0+λ^2/4 forλ^2≪ 1.Thus, up to factors of order unity the dependence on the warp factor e^𝒜_0 can be correctly determined by retaining the normalizable mode only as long as λ^2≫ e^4𝒜_0. For all but the zero mode this is the case. Clearly, including a UV-potential can only further reduce ϕ_UV/ϕ_IR.Next we consider those fields that have no bulk potential (i.e. KK-zero modes of T^1,1) but are stabilized by global effects of the bulk CY, like the axio-dilaton in generic flux compactifications. For these fields the UV potential determines the field profile entirely so we cannot neglect it. We approximate the UV-potential around its minimum byV_UV(ϕ)=m_UV^2/2kϕ^2.Now the UV-boundary condition reads r_r ϕ|_r=r_UV=-m_UV^2/k^2ϕ(r_UV) and the boundary values ϕ_UV and ϕ_IR are related byϕ_UV/ϕ_IR=4k^2e^4𝒜_0/m_UV^2(1-e^4𝒜_0)(1+4k^2e^4𝒜_0/m_UV^2(1-e^4𝒜_0))^-1Again, for stabilization m_UV^2/k^2≫ e^4𝒜_0 one obtainsϕ_UV/ϕ_IR≈4k^2e^4𝒜_0/m_UV^2.For the UV-potential we may not assume that m_UV∼ k. Thus, by only keeping the normalizable mode one would miss a factor of k^2/m_UV^2. In the setting of type IIB flux compactification with only a single Kähler-modulus, all fields but the universal Kähler modulus are stabilized perturbatively. This means that all massless bulk fields have UV-masses m_UV^2/k^2 that are at most suppressed by the CY-volume which we assume to be only moderately large. Therefore, we may estimate the UV-tails of perturbations by keeping the normalizable mode only for all throat perturbations.The universal Kähler modulus ρ plays a very different role: the volume modulus corresponds to a rescaling of the bulk CY where warping is weak. Regions of strong warping are not affected <cit.>. This means that in the RS1 language the universal Kähler modulus is a UV-brane field stabilized with non-perturbative mass scale m_ρ^2≪ k^2 <cit.>. It couples universally to all UV-perturbations δϕ_UV,ℒ_UV⊃1/2m_ρ^2ρ^2+ Λ^3/2δϕ_UVρ+...,where the coupling Λ^3/2δϕ_UVρ is perturbative and we have omitted higher order couplings. Then, a shift δϕ_UV leads to a shift in δρ of orderδρ∼Λ^3/2/m_ρ^2δϕ_UVLet us work in the regime of moderately large CY-volume, strong warping e^𝒜_0≪ 1 and weak non-perturbative stabilization m_ρ^2/k^2≪ 1. Let us further neglect dependencies on the CY-volume and only retain the dependence on e^𝒜_0 and m_ρ which we take to be parametrically small. Then,δρ∼e^4𝒜_0/m_ρ^2δϕ_IR.For the non-perturbative volume stabilization of KKLT and uplift with an D3-brane of tension T_3 this implies the (parametric) estimateδρ∼e^4𝒜_0 T_3/|⟨λλ⟩|^2. §.§ Three-form perturbationsFor the three-form field strength G_3 the analogous analysis was done in <cit.>. Setting the dilaton constant for simplicity, the linearized three-form equation of motion and Bianchi identity ared(e^4𝒜G^-)=0=dG_3 .They are solved by a product ansatzG_3=∑_id((a_ir^δ_i-4+b_ir^-δ_i)Ω_2^(i)),where the Ω^(i)_2 are eigen-forms of the Laplace-Beltrami operator *_5 d of T^1,1,*_5dΩ_2^(i)=iδ_i Ω_2^(i),with imaginary eigenvalues iδ_i.In <cit.> it is found that the ISD/IASD pieces are given byG^+=G_0^++i/2∑_i≠ 0(a_i/δ_ir^δ_i-4+2b_i r^-δ_i)(dΩ_2^(i)-δ_idr/r∧Ω_2^(i)), G^-=G_0^–i∑_i≠ 0a_iδ_i-2/δ_ir^δ_i-4(dΩ_2^(i)+δ_idr/r∧Ω_2^(i)),where the zero mode pieces are given byG_0^±=-2a_0r^-4(*_5Ω_2^(0)± i dr/r∧Ω_2^(0)).Only those eigen two-forms with δ_i<2 give rise to normalizable IASD perturbations. The lowest eigenvalues of the Laplace-Beltrami operator are listed in tables 3-5 of <cit.>. It can be read off that the zero mode falls off least rapidly towards the UV, namely as r^-4. For ISD perturbations the story is a bit different as also those modes that correspond to the coefficients b_i contribute. Those that feature δ_i>2 are normalizable. The one that falls off least rapidly towards the UVcorresponds to δ=3 and has a radial profile that is proportional to r^-3 <cit.>. This corresponds to the two-form Ω_2^(δ=3)=(g^1+ig^4)∧(g^2-ig^3),using the basis of one-forms of T^1,1 used in <cit.>. Far from the tip, the KS solution can be viewed as a perturbation of the KT solution and the normalizable δ=3 mode precisely encodes the leading perturbation that is sourced by the deformation at the tip <cit.>. Hence, it is of Hodge type (2,1) and does not enter (<ref>). As a consequence, the perturbation of 3-form fluxes on (<ref>) is irrelevant to the order of precision we are interested in, analogous to the scalar case.§.§ Metric perturbationsIn <cit.> the radial scaling of metric modes was derived. After gauge fixing, the only non-vanishing modes are those that correspond to transverse traceless two-tensors of T^1,1,δ g_ij∼∑_n(a_n r^Δ_n(g)-4+b_n r^-Δ_n(g))Y^(n)_ij,Δ_n(g)≡ 2+√(λ̂_n^2-4),where the Y^(n)_ij are the transverse traceless eigentensors of the Lichnerowicz operator of T^1,1 with eigenvalue λ̂^2_n,∇^2 Y^(i)_ij-2∇^k ∇_(iY^(i)_j)k=-λ̂_n^2Y^(i)_ij.In <cit.> it is found that the lowest eigenvalues are λ̂^2=4,5,36-8√(7),65/4,20,... corresponding toΔ_n(g)=2,3,5.29,11/2,6....The dominant normalizable mode thus falls off as r^-3 and corresponds to the deformation of the conifold. All further perturbations fall off at least as r^-5.29.JHEP | http://arxiv.org/abs/1707.08678v2 | {
"authors": [
"Jakob Moritz",
"Ander Retolaza",
"Alexander Westphal"
],
"categories": [
"hep-th",
"astro-ph.CO",
"hep-ph"
],
"primary_category": "hep-th",
"published": "20170727011407",
"title": "Towards de Sitter from 10D"
} |
Instituto de Física da Universidade de São Paulo,05314-970 São Paulo, BrazilInstituto de Física, Universidade Federal de Goiás, Caixa Postal 131, 74001-970, Goiânia, [email protected] Instituto de Física da Universidade de São Paulo,05314-970 São Paulo, BrazilCentre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen's University Belfast, Belfast BT7 1NN, United Kingdom Thermodynamic irreversibility is well characterized by the entropy production arising fromnon-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into aterm that only depends on population unbalances, and one that is underpinned by quantum coherences.This allows us to identify a genuine quantum contribution to the entropy production in non-equilibrium quantum processes.We discuss how these features emerge both in Lindblad-Daviesdifferential maps and finite maps subject to the constraints of thermal operations.We also show how this separation naturally leads to two independent entropic conservation laws for the global system-environment dynamics, one referring to the redistribution of populations between system and environmentand the other describing how the coherence initially present in the system is distributed into local coherences in the environment and non-local coherences in the system-environment state. Finally, we discuss how the processing of quantum coherences and the incompatibility of non-commuting measurementsleads to fundamental limitations in the description of quantum trajectories and fluctuation theorems. The role of quantum coherence in non-equilibrium entropy production Mauro Paternostro December 30, 2023 =================================================================== § INTRODUCTION Irreversible processes undergone by an open quantum system are associated with a production of entropythat is fundamentally different from the possible flow of entropy resulting from the mutual coupling of the system with itsenvironment. Such inevitable contribution to the entropy change of the state of a system is termed entropy production. If entropy is labelled as S, we describe its rate of change as S/ t = Π - Φ,where Φ is the entropy flux rate from the system to the environment and Π is the entropy production rate.According to the second law of thermodynamics, we should have Π≥ 0, with Π = 0 if and only if the system is at equilibrium.The characterization of the degree of irreversibility of a process, and thus the understanding of entropy production is both fundamentally relevant and technologically desirable. On one hand, such understanding would provide the much needed foundations to the emergence of time-symmetry breaking entailed by irreversibility and epitomized, for instance, by seminal results such as Onsager's theory of irreversible currents <cit.>. On the other hand, a characterization of irreversible entropy could help us designing thermodynamically efficient quantum technologies <cit.>. In general, the open dynamics of a quantum system gives rise to two processes.The firstcorresponds to the transitions between energy levels of the system, which will cause the populations to adjust to the values imposed by the bath.Measures of the entropy production for this kind of processes have been known for many years, particularly in the context of Fokker-Planck equations <cit.> and Pauli master equations <cit.>.The second process undergone by an open quantum system is the loss of coherence in the energy eigenbasis.Coherence is an essential resource for quantum processes <cit.>, likely representing the ultimate feature setting quantum and classical worlds apart.Only very recently have steps towards the formulation of a unified resource theory of coherence been made <cit.>.While the role of quantum coherences in thermodynamics is yet to be fully understood (both qualitatively and quantitatively), it is known that coherence plays a role in the allowed transitions generated by thermal operations <cit.>. Moreover, it also affects the performance of non-adiabatic work protocols <cit.>. Understanding the interplay between population dynamics and loss of coherence represents a pressing problem in the field of thermodynamics of quantum systems.In particular, one is naturally led to wonder how entropy production is underpinned by the dynamics of quantum coherences in irreversible open dynamics. Shedding light on such an intimate relation is the main motivation of this paper, where we put forward a formal description of entropy production in terms of two contributions,one clearly related to the dynamics of populations and the other depending explicitly on the coherence within the state of the system undergoing the open process. We first discuss how these features emerge in the case of Davies-type master equations, for which the formulation is transparent.We then address more general dynamical maps satisfying the constraints of thermal operations <cit.>.In this case, we show how our result can be connected to recent resource theoretical developments <cit.>. Afterwards, we address the main physicalimplications of these two contributions to the entropy production.First, we discuss how it enables one to construct two independent entropic conservation laws for the global system-environment dynamics.The first is entirely classical and relates to the redistribution of populations between system and enviroment.The second, on the other hand, dictateshow the coherence initially prepared in the system is distributed among local coherences in the environment andnon-local coherences in the global system-environment state.Lastly, we address the issue of how to access entropy production by means of quantum measurements andquantum trajectories.We show that even in the presence of coherence, it is possible to construct measurement protocols which satisfy fluctuation theorems.However, unlike in the case of closed systems <cit.>, it is not possible to consider a single measurement protocol in which the fluctuation theorems are satisfied for both contributions of the entropy production individually.Several advances inthe past decade have consistently shown that it is possible to engineersystems in which thermodynamicscoexists with quantum effects.It is our hope that the framework put forth in thispaper may contribute for the development of a unified theory describing both thermal and quantum resources. § RESULTS§.§Entropy production due to the processing of coherence Coherence is a basis dependent concept and, in principle, no preferred basis exists.Here we adopt the viewpoint according to which a preferred basis only emerges when it is imposed by the environment <cit.>, a perspective that is typically referred to as einselection. There are several scenarios in which a preferred basis may emerge. We shall focus on two of them. The first are Davies maps <cit.>, which make use of the weak-coupling approximation, and the second are the so-called thermal operations <cit.>.Both scenarios lead to energy conservation so that the energy eigenbasesare naturally selected as preferred bases.However, while in the weak-coupling scenario, energy conservation is taken as an approximation midway through the calculations, in the case of thermal operations this is imposed from the start, as an assumption of the model.Consequently, thermal operations are much more versatile, while simultaneously offering a much clearer physical interpretation. We also assume, for simplicity, that the energy eigenvalues of the system are non-degenerate, since this would cause the preferred basis to depend on additional details of the system-environment interaction, which one seldom has access to.§.§.§ Lindblad-Davies mapsWe begin by analyzing the so-called Lindblad-Davies maps, which generally describe the weak contact of a system with a thermal environment. They have the form<cit.>ρ_S/ t = - i [H_S,ρ_S] + D(ρ_S),where ρ_S is the density matrix of the system, H_S is the Hamiltonianand D(ρ_S) is a Lindblad dissipator having the Gibbs state ρ^eq_S=e^-β H_S/Z_S as a fixed point; i.e. D(ρ_S^eq) = 0(here Z_S =e^-β H_S is the partition function and β the inverse temperature of the bath). Davies maps are known to leadto a separation between the evolution of populations, p_n = ⟨ n | ρ_S | n ⟩, of the density matrix and theoff-diagonal coherences p_nm = ⟨ n | ρ_S | m ⟩. Here, we have called {|n⟩} the elements of the basis imposed by the specific environment being considered. In the specific case of a Davies map, such basis is that of the energy eigenstates, whose set of eigenvalues we call {E_n}.Following Eq. (<ref>), the populations evolve following the Pauli master equationp_n/ t = ∑_m[ W(n|m) p_m - W(m|n) p_n],where W(n|m) are the transition rates from the energy level E_n to level E_m. They satisfy the detailed-balance condition W(n|m)/W(m|n) = p_n^eq/p_m^eq = e^-β (E_n - E_m),where p_n^eq=⟨ n | ρ^eq_S | n ⟩. As for the coherences, provided that the Bohr frequencies ω_nm=E_n-E_m are non-degenerate, they evolve independently of each other according to the equationp_nm/ t = - {i ω_nm -1/2∑_k [ W(k|n) + W(k|m) ]} p_nm.As the second term in the right-hand side can be seen as an effective (generally temperature-dependent) damping term, the evolution of the coherences can be interpreted as that resulting from damped oscillations.We now definethe non-equilibrium free energy as F(ρ_S) =(H_S ρ_S) +T (ρ_S lnρ_S), where T is the temperature of the bath. While Eq. (<ref>) reduces to the usual expression F_eq = - T ln Z_S at equilibrium, for general non-equilibrium states, we can writeF(ρ_S) = F_eq + T S(ρ_S||ρ_S^eq),whereS(ρ||σ) = (ρlnρ - ρlnσ) is the quantum relative entropy. As S(ρ_S||ρ_S^eq) ≥ 0, we have that F(ρ_S) ≥ F_eq.This condition thus defines the equilibrium state of a system as the state that minimizes the free energy <cit.>.Moreover, it establishes that, during relaxation, the free energy is a monotonically decreasing function of time whose value is determined by the distance, in state space, between the instantaneous state of the system ρ_S and its equilibrium version ρ_S^eq. Hence, one is naturally led to define the entropy production rate as<cit.>Π = - 1/T F(ρ_S)/ t,which ensures that Π≥ 0 andΠ = 0 iff ρ_S = ρ_S^eq. Next we notice thatin terms of the eigenbasis of H_S, we can always split S(ρ|| ρ_eq) as S(ρ_S|| ρ_S^eq) =S(p_S||p_S^eq) +C(ρ_S).Here, S(p_S||p_S^eq) = ∑_n p_n ln p_n/p_n^eq is the Kullback-Leibler divergence of the classical probability distribution entailed by the populations p_S={p_n} from that of the equilibrium state p_S^eq={p_n^eq}. Moreover, we have introduced the relative entropy of coherence <cit.>C(ρ_S) = S(Δ_H_S(ρ_S))- S(ρ_S),where Δ_H_S(ρ) is the dephasing map, acting on the density matrix ρ_S, which removes all coherences from the various energy eigenspaces of H_S.With this at hand, Eq. (<ref>) becomesF(ρ_S) = F_eq + TS(p_S||p_S^eq) + TC(ρ_S).This is one of the central results of this work: It shows that quantum coherence is actually a part of the non-equilibrium free energy, and thus contributes significantly to the determination of the non-equilibirum thermodynamics stemming from Eq. (<ref>). The second term in F(ρ_S) quantifies the increase in free energy due to population imbalance with respect to the equilibrium configuration and, as such, is a purely classical term. The last term, which is of a genuine quantum nature, determines the surplus in free energy that a non-equilibrium state with quantum coherences offers with respect to its diagonal (and thus classical) counterpart.Let us now use the formal splitting in Eq. (<ref>) to recast the entropy production rate in Eq. (<ref>) asΠ = Π_d + Υ.The first term is written as Π_d= - / t S(p_S|| p_S^eq)= 1/2∑_n,m[ W(n|m) p_m - W(m|n) p_n ] lnW(n|m) p_m/W(m|n) p_n.and is then exactly the classical result derived in <cit.>. The second contribution reads Υ = -C(ρ_S)/ t,which thus shows that the rate of loss of coherence that might ensue from the dynamics of the system enters quantitatively into the entropy production: the rate at which entropy is produced in a process where quantum coherences are destroyed as a result of the coupling with an environment surpasses the corresponding classical version.Clearly both Π_d and Υ are non-negative and null only for ρ_S = ρ_S^eq.Finally, let us address the entropy flux defined inEq. (<ref>). Using Eq. (<ref>) we find Φ = Φ_E/T = - 1/T∑_n E_n^Sp_n/ t,where Φ_E is the energy flux from the system to the environment. In deriving this equation we have used p_n^eq = exp[- β E_n^S]/Z_S. The entropy flux has thus no contribution arising from quantum coherences: entropy (and energy) will only flow due to imbalances in the populations.Any loss of coherence contributes only to the entropy production rate and has no associated flux. It is important to emphasize that this is a feature of the present type of master equation.In other situations, such as strong coupling dynamics, the coherences in the system may play a relevant role in the entropy and heat fluxes.§.§.§Thermal Operations We now address the case of more general maps with the aim of gaining access to the environmental degrees of freedom, hence enhancing our understanding of the two contributions to the entropy production from the perspective of the joint system-environment properties.We consider explicitly thermal operations, which have been the subject of numerous recent investigations in the context of resource theories <cit.>.A thermal operation is physically described as the interaction of the system with an arbitrary environment, initially prepared in equilibrium ρ_E^eq = e^-β H_E/Z_E, through a unitary U which conserves the total energy, that is an operation such that [U, H_S+H_E]=0.In this sense, the thermal operation hypothesis reminds us of the framework set by the weak-coupling approximation.However, it allows us to go significantly beyond the limitations ofweak-coupling, and thus extend our analysis to a larger set of physically meaningful cases <cit.>. We also call attention to the fact that in thermal operations the energy conservation condition is only imposed on the global unitary U, irrespectiveof how this is generated.One way to do so is by means of a time-dependent interaction that is turned on and off.Another, much simpler approach, is to simply have any potential V which satisfies [V, H_S+H_E] = 0. This will then generate a time-independent unitary U = e^- (H_S+H_E+V)t which will be energy conserving. This type of thermal operation was recently used in Ref. <cit.> to study the heat exchange between two qubits in a magnetic resonance setup. The state of the composite system after the evolution in a thermal operation will be ρ_SE' = U (ρ_S ⊗ρ_E^eq) U^†.We label the environmental energies and eigenstates as {E_μ^E} and {|μ⟩}, respectively. We also call q_μ = e^-β E_μ^E/Z_Eits initial thermal populations.Energy conservation then implies that ⟨ m,ν| U |n,μ⟩∝δ(E_n^S + E_μ^E - E_m^S - E_ν^E).Tracing out the environment one obtains the Kraus mapfor the systemρ_S' = _E[ U(ρ_S ⊗ρ_E^eq) U^†] = ∑_μ,ν M_μ,νρ_S M_μ,ν^†where M_μ,ν = √(q_μ)⟨ν | U | μ⟩.Clearly, the Gibbs state ρ_S^eq is a fixed point of this equation.Moreover, the Davies maps studied in the previous Section correspond to particular Markovian limits of Eq. (<ref>).The energy conservation condition implies that, when the eigenvalues of H_S are non-degenerate, Eq. (<ref>) is an incoherent operation in the sense of Ref. <cit.>. That is, defining theenergy eigenstates as the set of incoherent states,this process always maps incoherent states into incoherent states. This in turn allows for an independent processing of both populations and coherences.The diagonal entries will, in particular, evolve according to the classical Markov chain p_m' = ∑_n Q(m|n) p_n,where Q(m|n) = ∑_μ,ν |⟨ m |M_μ,ν| n ⟩ |^2 is the transition probability from state n to state m, a quantity playing the role ofthe transition rate W(m|n) in Eq. (<ref>). The processing of the coherences, on the other hand, takes place independently of the changes in populations.In particular, if the Bohr frequencies ω_mn are non-degenerate, this processing takes the simple form p_n,m' = α_n,m p_n,m, α_n,m = ∑_μ,ν⟨ n | M_μ,ν | n ⟩⟨ m |M_μ,ν^† |m⟩. As shown in Ref. <cit.>, the processing of coherence is not independent of the population changes, but must satisfythe inequality|α_n,m|^2 ≤ Q(n|n) Q(m|m),which provides a bound to the maximum amount of coherence that may be lost in a thermal operation.We now turn to the analysis of the entropy production in this scenario.Unlike the previous Section, as the dynamics in Eq. (<ref>) is in general non-Markovian and we only have access to the global map, it is not possible to address the rate of entropy production Π, but only the total entropy Σ produced in the process.In this case, using the contractive property of the relative entropy <cit.>, we haveS(ρ_S'|| ρ_S^eq) ≤ S(ρ_S|| ρ_S^eq). Consequently, the free energyEq. (<ref>) remains a non-increasing function, thus justifying the following definition of total entropy productionΣ = - Δ F/T=S(ρ_S|| ρ_S^eq) - S(ρ_S'|| ρ_S^eq) ≥ 0.This expression may be taken as a basic postulate in our framework, motivated by physical consistency arguments that can be even reinforced by a quantum trajectory point of view, as discussed e.g. in Refs. <cit.>. Other approaches have also been used elsewhere <cit.>.As Σ is also based on the quantum relative entropy, a splitting akin to Eq. (<ref>) into non-negative population-related and a coherence-related terms is in order, and we have Σ = Σ_d + Ξ,whererClΣ_d=S(p_S||p_S^eq)- S(p_S'||p_S^eq), Ξ = C(ρ_S) - C(ρ_S'). The non-negativity of Σ_d follows immediately from the fact that in thermal operations diagonal elements evolve independently of coherences. The positivity of Ξ, on the other hand, follows from the fact that a thermal operation is incoherent <cit.>.In the limit where the Davies maps are recovered,Σ_d and Ξ become respectively the integrated versions of Π_d and Υ in Eq. (<ref>). This demonstrates the generality, under suitable and reasonable assumptions on the form of the system-environment coupling, of the central result of our investigation.§.§ Implications of the central results We now explore which considerationscan be drawn in light of the formal splitting of the entropy production demonstrated above. §.§.§ Entropic conservation lawsThe structure of thermal operationsand Eq. (<ref>) imply a series of conservation rules for the processing of populations and coherences.First, energy conservation implies that the total entropy production in Eq. (<ref>) can be written as <cit.>Σ = S(ρ_E'||ρ_E^eq) + ℐ(ρ_SE'),where ρ_E'=_S(ρ'_SE)and ℐ(ρ_AB) = S(ρ_A) + S(ρ_B) - S(ρ_AB) is the mutual information of a bipartite system.This gives an interesting interpretation of Σ as being related to the change in the environmental state, measured by the first term, and the total degree of correlations created by the thermal operation, measured by the mutual information. As discussed in Ref. <cit.>, Eq. (<ref>) also provides a clear interpretation of how irreversibility emerges from a global unitary dynamics, ascribing it to two reasons. One is the creation of correlations between system and environment, which are never recovered once one traces out the environment (hencegiving rise to an irretrievable loss of information).The second is related to the fact that the system pushes the environment away from equilibrium.As shown in Ref. <cit.>, if ρ_E' - ρ_E^eq∼ϵ then S(ρ_E'|| ρ_E^eq) ∼ϵ^2, whereas S(ρ_E') - S(ρ_E^eq) ∼ϵ.Thus, when the reservoir is large, the first term becomes negligible and the main contribution to the entropy production will come from the total correlations created between system and environment.Next we note that as the map (<ref>) is unitary, it follows that S(ρ_SE') = S(ρ_SE).However, as [U,H_S+H_E]=0,the dephasing operation Δ_H_S+H_E commutes with the unitary evolution so that, in addition to the total entropy being conserved, the same is also true for the dephased entropiesrClS(Δ_H_S+H_E(ρ_SE'))=S(Δ_H_S(ρ_S)) + S(ρ_E^eq).This result reflects how the changes in population in the system and environment affect theinformation content of the diagonal elements of ρ_SE'.Note that the left-hand side contemplates, at most, coherences in the degenerate subspaces of H_S+H_E, which are not resources from the perspective of this operation. From Eqs. (<ref>) and (<ref>), it follows that a similar law must also hold for the relative entropy of coherenceC(ρ_SE') =C(ρ_S).Where C(ρ_SE')=S(Δ_H_S+H_E(ρ_SE'))-S(ρ_SE'). Thus, we see that the reduction in the coherence of the system after the map is due to a redistribution of this coherence over the global system-environment state. Substituting Eq. (<ref>) into Eq. (<ref>) shows that the contribution of the entropy production due to quantum coherences may be written as Ξ =C(ρ_SE') -C(ρ_S'). Thus, the entropy production due to quantum coherences can be seen as the mismatch between the global coherences in the correlated system-environment state and the local coherences in the final state. We can also relate Ξ to the notion of correlated coherence, introduced recently in <cit.> and defined asC_cc(ρ_SE') =C(ρ_SE') -C(ρ_S') -C(ρ_E') ≥ 0,where C(ρ_E')=S(Δ_H_E(ρ_E'))-S(ρ_E'). This quantity therefore represents the portion of coherence that is contained within the correlations between system and environment.Combining the Eqs. (<ref>) and (<ref>), it is then possible to writeΞ =C(ρ'_E) + C_cc(ρ'_SE).This has the same form as Eq. (<ref>), with the first term representing the local coherences developed in the environment and the second term representing the non-local contribution.Thus, we may conclude from this result that entropy production has a clearly local contribution, related to the creation of coherences in the environment, and a non-local contribution related to the creation of shared coherences in the system-environment state. §.§.§ Stochastic trajectories and fluctuation theoremsLastly, let us consider the stochastic version of the entropy production arising from quantum trajectories.In order to correctly treat the coherences present in the system, we adopt the following procedure. In the forward protocol, the environment is prepared in the thermal state ρ_E = ∑_μ q_μ |μ⟩⟨μ|, whereq_μ = e^-β E_μ^E/Z_E. The system, on the other hand, is taken to be in an arbitrary state ρ_S = ∑_α p_α |ψ_α⟩⟨ψ_α|, which in general contain coherences, so that the basis |ψ_α⟩ isincompatible with the energy basis |n⟩.As the first step in the protocol, we then perform local measurements in the bases |ψ_α⟩ and |μ⟩ of S and E, obtaining the state |ψ_α,μ⟩with probability p_α q_μ.Next we evolve both with a joint unitary U. Finally, in the third step we measure only the environment, again in the energy basis |ν⟩. Due to the measurement backaction, the state of the system then collapses to the pure state|Φ_F|αμν⟩ = ⟨ν | U | ψ_α, μ⟩/√(P_F(ν|αμ)),where P_F(ν|αμ) = ||⟨ν | U | ψ_α, μ⟩||^2.For a discussion on the effects of choosing different bases for the second measurement in the environment, see Ref. <cit.>.The final stateρ_S' of the system will then be given by an ensemble average over all possible final states (<ref>), weighted by the probability of the stochastic trajectory (α, μ, ν); viz., ρ_S' = ∑_α, μ, ν P_F(ν|α,μ) p_α q_μ |Φ_F|αμν⟩⟨Φ_F|αμν|.One may directly verify that this state is indeed equal to the unmeasured final state ρ_S' in Eq. (<ref>). The states (<ref>), however, are not the eigenstates of ρ_S' and in factdon't even form a basis. The diagonal structure of ρ_S' will thus be of the form ρ_S' = ∑_β p_β' |ψ_β'⟩⟨ψ_β'|, where |ψ_β'⟩ is a new basis set that is not trivially related to neither |ψ_α⟩ nor |Φ_F|αμν⟩.This is the key difference that appears due to the presence of coherences (if the initial state of the system were diagonal, the same would be true for the final state, since this is a thermal operation).The probabilities p_β' will be given by p_β' = ∑_α, μ, ν p_β|α, μ, νP_F(ν|α, μ) p_αq_μ,where p_β|α, μ, ν = |⟨ψ_β' | Φ_F|αμν⟩|^2 is the conditional probability of finding the system in |ψ_β'⟩ given that it ended the forward protocol in |Φ_F|αμν⟩. The stochastic trajectory generated by the measurement outcomes is specified by the three quantum numbers (α, μ, ν).However, following Ref. <cit.>, we may augment the trajectory (an idea first introduced by Dirac <cit.>) by introducing β as an additional quantum number, so that the trajectory 𝒳 = (α, μ, β, ν) is defined by the probability 𝒫_F[𝒳] = p_β|α, μ, νP_F(ν|α, μ) p_αq_μ .Indeed, using the definition (<ref>), one readily finds thatp_β|α, μ, νP_F(ν|α, μ)= |⟨ψ_β', ν | U | ψ_α, μ⟩ |^2 = P(β,ν | α, μ),which is nothing but the transition probability of observing a transition from |ψ_α, μ⟩ to |ψ_β', ν⟩.Next we define the backward protocol.The initial state of the system is drawn from one of the possibleeigenstates |ψ_β'⟩ of ρ_S' <cit.>, whereas the environment is taken to be in equilibrium and is again measured in the energy basis |ν⟩. This yields the state |ψ_β', ν⟩ with probability p_β' q_ν.We then apply the time-reversed unitary U^† and, in the end, measure E in the basis |μ⟩.As a consequence the system collapses to |Φ_B|βνμ⟩ = ⟨μ | U^† | ψ_β', ν⟩/√(P_B(μ|β,ν)),where P_B(μ|β,ν) = ||⟨μ | U^† | ψ_β', ν⟩||^2.The backward trajectory is specified by the quantum numbers (β, ν, μ).However, as in the forward case, we can define the augmented trajectory 𝒳 = (α, μ, β, ν) by introducing the conditional probability p_α|β, ν,μ = |⟨ψ_α | Φ_B|βνμ⟩|^2.The probability for the augmented backward trajectory will then be given by 𝒫_B[𝒳] = p_α|β, ν,μP_B(μ|β,ν) p_β' q_ν .With the path probabilities (<ref>) and (<ref>), we can now define the entropy production in the usual way, as σ[𝒳] = ln𝒫_F[𝒳] /𝒫_B[𝒳] .By construction, σ satisfies a detailed fluctuation theorem <cit.>⟨ e^-σ[𝒳] ⟩ = 1.Moreover, similarly to Eq. (<ref>), it follows that p_α|β, ν,μP_B(μ|β,ν) = P(β,ν|α, μ), so that σ[𝒳] = lnp_β|α, μ, νP_F(ν|α, μ) p_αq_μ/p_α|β, ν,μP_B(μ|β,ν) p_β' q_ν = lnp_α q_μ/p_β' q_ν.Thus, we see that the conditional terms cancel out.Physically, this means that there is no additional entropic cost in introducing the augmented trajectories, which is a consequence of the fact that the augmentation was done using the eigenstates |ψ_β'⟩ of ρ_S'.One may also directly verify that⟨σ[𝒳]⟩ = Σ is the average entropy production in Eq. (<ref>).Next we address the question of how to define stochastic quantities for the two contributions to the entropy production inEq. (<ref>). That is, we wish to separate σ[𝒳] = σ_d[𝒳] + ξ[𝒳],such that ⟨σ_d[𝒳]⟩ = Σ_d and ⟨ξ[𝒳]⟩ = Ξ. This can be accomplished by augmenting the trajectory once again to include the populations of the system in the energy basis. That is, we define 𝒳̃ = {α, μ, n, β, ν, m},with associated path probabilities 𝒫_F(B)[𝒳̃] = 𝒫_F(B)[𝒳] p_n|α p'_m|β, wherewe defined theconditional probabilities p_n|α = |⟨ n |ψ_α⟩|^2 and p_m|β = |⟨ m | ψ_β'⟩|^2.We then define the stochastic quantities rClσ_d[𝒳̃]= ln( p_n q_μ/p_m' q_ν), ξ[𝒳̃]= ln( p_αp_m'/p_β' p_n),where p_n = ⟨ n | ρ_S | n ⟩ and p_m' = ⟨ m |ρ_S' |m ⟩ are the populations in the energy eigenbasis at the initial and final states [cf. Eq. (<ref>)].Summingthese two contributions immediately yields Eq. (<ref>). Moreover, one may also verify that ⟨σ_d[𝒳̃]⟩ = Σ_d and ⟨ξ[𝒳̃]⟩ = Ξ. Hence, these quantities do indeed represent the stochastic counterparts of the two contributions to the entropy production.A second glance at Eq. (<ref>) reveals that, on the stochastic level, the entropy production ξ due to the loss of coherence,is nothing but the change ininformation betweenincompatible bases, a quantity sometimes referred to as information gain <cit.>. Hence, we arrive at the important conclusion that the incompatibility between classical and quantum entropy production can be traced back, at the stochastic level, to the basis incompatibility of the quantum rules. Indeed, if werewrite thefluctuation theorem (<ref>) as⟨ e^-σ_d[𝒳̃]- ξ[𝒳̃]̃⟩ = 1, we can clearly see that due to the presence of coherences, the classical fluctuation theorem, that one finds for diagonal initial states, is not satisfied.Instead, it must be corrected by the information gain.It is also possible to draw an alternative interpretation in terms of the entropy production due to quantum measurements, as studied for instance in Refs. <cit.>.When a measurement is performed in a basis which commutes with the system's density matrix, no entropy is produced.Non-commuting measurements, on the other hand, have an associated entropy production related to the loss of coherence.This is precisely the content of Eq. (<ref>). No such additional entropy production was generated for the first augmentation that led us to Eq. (<ref>), as in this case there is no information gain since the basis |ψ_β'⟩ is the basis that diagonalizes ρ_S'.§ DISCUSSION We have addressed the role played by quantum coherence in determining the behaviour of the entropy production, a fundamental quantifier of thermodynamic irreversibility. By making physically reasonable assumptions on the form of the dynamics undergone by a quantum system and its environment, we have been able to single out the contribution that quantum coherences, a genuine non-classical feature of the state of a given dynamical system, play in quantifying the rate of irreversible entropy production. Such contribution appears to be fully distinct from the one arising from unbalances in the energy eigenbasis of the state of the system, which brings about a classical flavour. Moreover, it can be interpreted in a physically transparent manner as the thermodynamic cost that one has to pay for the destruction of coherences that were seeded in the state of the system itself. In turn, our results have interesting consequences for the interpretation of the process of producing entropy as a result of the dynamical generation of correlations (or equivalently coherences) between a quantum system and its environment.We believe that the theory put forth in this paper may provea useful step forward towards the setting up of a self-contained framework for the interpretation of thermodynamic irreversibility at the quantum nanoscale, which is still sorely missing despite the key role that entropy production will play in the quantification of the thermodynamic fingerprint of managing quantum dynamics.For instance, it could serve as a starting point for the development of a theory of quantum entropy production in non-equilibrium steady-states of systems connected to multiple reservoirs. Or, as a tool for quantifying the contribution of loss of coherence in the operation of finite-time quantum heat engines.§ DATA AVAILABILITY STATEMENT No data sets were generated or analyzed during the current study.§ ACKNOWLEDGMENTSGTL acknowledges the funding from the São Paulo Research agency, under grant number 2016/08721-7. MP acknowledges funding from the SFI-DfE Investigator Programme, the EU H2020 Collaborative Project TEQ, the Leverhulme Trust, and the Royal Society.JPS and GTL acknowledge the Brazilian funding agency CAPES. § ADDITIONAL INFORMATION Competing interests: The authors declare no competing interests.§ AUTHOR CONTRIBUTIONS JPS developed the detailed formal calculations in collaboration with GTL; LCC, GTL, and MP conceived the original idea and shaped it with the significant help of JPS.All authors contributed to the research and the preparation of the manuscript. 50Onsager1931 Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. 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Inf. 3, 9 (2017), arXiv:1607.02404.Crooks1998 Crooks, G. E. Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems. J. Stat. Phys. 90, 1481 (1998).Groenewold1971 Groenewold, H. J. A problem of information gain by quantal measurements. Int. J. Theor. Phys. 4, 327 (1971). | http://arxiv.org/abs/1707.08946v5 | {
"authors": [
"Jader P. Santos",
"Lucas C. Céleri",
"Gabriel T. Landi",
"Mauro Paternostro"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170727173735",
"title": "The role of quantum coherence in non-equilibrium entropy production"
} |
APS/123-QED Defect Engineering and Substrate Effects in Ion-Irradiated Monolayer 2D Materials]Defect Sizing, Separation and Substrate Effects in Ion-Irradiated Monolayer 2D Materials School of Physics, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Material Science and Engineering, Nanchang University, Youxun W Rd, Xinjian Qu, Nanchang Shi, Jiangxi Sheng, People's Republic of China National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, Jiangsu Province, People's Republic of China AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland School of Physics, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, Ireland School of Chemistry, Trinity College Dublin, Dublin 2, Ireland Institute of Physics, EIT 2, Faculty of Electrical Engineering and Information Technology, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany Corresponding author E-mail: [email protected] School of Physics, Trinity College Dublin, Dublin 2, Ireland AMBER Centre, CRANN Institute, Trinity College Dublin, Dublin 2, IrelandPrecise and scalable defect engineering of 2D nanomaterials is acutely sought-after in contemporary materials science.Here we present defect engineering in monolayer graphene and molybdenum disulfide (MoS_2) by irradiation with noble gas ions at 30 keV. Two ion species of different masses were used in a gas field ion source microscope: helium (He^+) and neon (Ne^+). A detailed study of the introduced defect sizes and resulting inter-defect distance with escalating ion dose was performed using Raman spectroscopy. Expanding on existing models, we found that the average defect size is considerably smaller for supported than freestanding graphene and that the rate of defect production is larger. We conclude that secondary atoms from the substrate play a significant role in defect production, creating smaller defects relative to those created by the primary ion beam. Furthermore, a similar model was also applied to supported MoS_2, another promising member of the 2D material family. Defect yields for both ions were obtained for MoS_2, demonstrating their different interaction with the material and facilitating comparison with other irradiation conditions in the literature. [ Hongzhou Zhang December 30, 2023 ===================== § INTRODUCTION [lines=2,lhang=0]In recent years, the extraordinary properties and tunability of 2D materials have been repeatedly demonstrated, heralding a new era of materials science <cit.>. Their physical properties (electrical, thermal, etc.) are highly distinguished from their bulk counterparts due to the evolution of band structure with decreasing layer number <cit.>. The ideas and methodologies developed from the investigation of graphene have been extended to many other 2D materials, including transition metal dichalcogenides (TMDs) such as molybdenum disulfide <cit.>. With the demands of modern semiconductor technology, precise nanoscale control of 2D material properties is fervently pursued. The alteration of nanoscale geometry and the introduction of structural defects or strain to these materials can be used to modulate their electronic and optical properties <cit.>. Ion irradiation has an illustrious record over many decades in the scalable and precise defect engineering of materials <cit.>. Modern ion irradiation techniques with sub-nanometre probe sizes have demonstrated sub-10 nm precision in fabricating nanoribbons from various materials <cit.>.Ultimate modification precision is the convolution of the impact probe and the interaction volume. Given the superlative confinement of 2D materials in the out-of-plane direction, and the confinement of modern ion beams in the in-plane direction, it is possible to restrict the ion-specimen interaction to an exceptionally small volume. Such methods have been used to control doping by implantation and to introduce precise quantities of defects <cit.>. He^+ irradiation of graphene encapsulated in hexagonal boron nitride has been used to introduce n-type doping <cit.> while dose-dependent irradiation of TMDs has introduced pseudo-metallic phases in the monolayer limit <cit.>.This paper reports a novel experimental comparison of two ion bombardment species, He^+ and Ne^+, at high energy. It is also one of very few reports to carry out explicit comparisons between supported and freestanding 2D materials. The introduction of defects by ion irradiation is shown to be highly precise, and Raman spectroscopy proves to be a highly versatile and sensitive characterisation method for these defective monolayer materials. §.§ Defect Engineering by Ion Irradiation in 2D MaterialsDefects in 2D materials such as a graphene and MoS_2 have been introduced by a range of energetic species. For the application of such ion irradiation methods to defect engineering, the defect yield per ion and the types of defect introduced must be known. The mass and energy of the primary ion species both play a substantial role in determining the average defect yield and the nature of the dominant defect type. With increased mass of energetic noble gas ions, both the defect yield and size in graphene are expected to increase <cit.>. For high energy ions the defect yield decreases with increasing energy while the defect size increases.However, defects induced in supported 2D materials are created not just by the primary ion beam. While a substrate is expected to lower the damage probability per ion for low energy ions, the converse is true for high energy ions as the substrate facilitates backscattering of the primary beam and thus provides a source of energised secondary particles <cit.>.found that irradiation damage was enhanced in supported graphene (compared to freestanding) for energies above 5 keV for Ar^+ (M=18) and 3 keV for Si^+ (M=14) <cit.>. Excited substrate particles have a much lower energy than the primary ions which suggests that they would create defects of greater size. used low energy (90 eV) Ar^+ ions to alter the average distance between defects, L_D, in graphene <cit.>. In these studies L_D was calculated based on the expected density of defects, σ, which is estimated from the irradiation dose, S, in ions per unit area. The approximation that σ≃ S was used which assumes a random distribution of ions, yielding:<cit.>L_D = 1/√(σ)This is a valid assumption for ions with a suitable cross-section for creating a single carbon vacancy defect. The 90 eV Ar^+ ions applied to graphene byare one such example due to their relatively large mass and low energy. The vast majority of those ions do indeed interact with surface carbon atoms but due to the low energy can only remove one carbon atom each. §.§ Local Activation Model for Raman Spectra of Defective 2D Materials The Raman spectra of graphene have been extensively studied <cit.> with labelled characteristic peaks: 2D at ∼2640 cm^-1 and G at ∼1583 cm^-1. In the Raman spectra of defective graphene, the disruption to normal selection rules also allows the detection of two additional peaks: D at ∼1322 cm^-1, and D' at ∼1600 cm^-1 <cit.>. Furthermore, increasing structural disorder also causes the G peak to consistently broaden. L_D in graphene has been related to the ratio of the intensity of the D peak (I_D) to the intensity of the G peak (I_G) and a version of that relationship is given by: I_D/I_G = C_A (L^2_S+2r_SL_S)/(L^2_S+2r_SL_S-r^2_S)[e^-π r^2_S/L_D^2 -e^-π (L^2_S+2r_SL_S)/L_D^2]+ C_S[1-e^-π r^2_S/L_D^2]where r_S and r_A represent the radii of a structurally disordered area created by an ion and the radii of the outer D band-activated area respectively.L_S=r_A-r_S is the Raman relaxation length for the resonant Raman scattering. The intensity of the D peak is proportional to the total area of crystalline graphene that is activated by local defects. Thus, as L_D becomes low (i.e. the defect density increases and the material becomes less crystalline) the D band intensity falls due to the overlapping of the disordered areas and the decrease of the total D-activated area. The dispersive effect of the excitation energy, E_l, on the ratio of excitation of the D and G bands, is included in the C_A parameter, being a measure of the maximum possible value of the I_D/I_G ratio <cit.>. Where E_l is stated in eV, C_A has been given experimentally by: <cit.> C_A = (160± 48)× E_l^-4 The C_S parameter is the value of the I_D/I_G ratio in the highly disordered limit and it is important in the large defect density regime, L_D ≤ r_S <cit.>. Typically, three stages are discussed in the evolution of the relationship described by equation <ref>. The first stage begins with pristine graphene. As isolated defects initially appear in the crystalline lattice a rising D peak is observable, increasing I_D/I_G. The second stage features red-shifting and continued broadening of the G peak and a now diminishing D peak. It is reached when defects coalesce and carbon valence declines <cit.>.The third stage is marked by the transition of the specimen to amorphous carbon bearing limited resemblance to the original graphene.Raman spectroscopy has been employed extensively in the characterisation of MoS_2 in various forms <cit.> including monolayer (of polytype 1H, point group D_3H) which has the labelled characteristic peaks: E' at ∼383 cm^-1 and A'_1 at ∼401 cm^-1. In defective material, an additional peak, the LA(M) at ∼227 cm^-1 is found <cit.>. The E' peak comes from the intralayer, in-plane motion of Mo and S atoms with respect to each other and the A'_1 peak comes from the intralayer, out of plane motion of S atoms <cit.>.The LA(M) peak appears in nanoparticle/multi-layer samples but exhibits no intensity in pristine monolayer MoS_2 <cit.>. However, it intensifies quickly with increased defect density <cit.>.Since it is defect-activated,draw an analogy between the LA(M) peak in monolayer MoS_2 and the D peak in graphene as both represent a good measure of disorder when normalized <cit.>. The intensity of the LA(M) peak, I(LA), normalized to that of either the E' peak, I(LA)/I(E'), or the A'_1 peak, I(LA)/I(A'_1), is related to the inverse square of L_D by:I(LA)/I(X)=C(X)/L_D^2In the case where the Raman spectrum is acquired with a 532 nm laser, the following constants were reported from the fitting of experimental data: C(E') = 1.11±0.08 nm^2 and C(A'_1) = 0.59±0.03 nm^2. X = E' or A'_1 depending on the peak studied. During defect engineering, the increase of these intensity ratios is attributed to two concomitant factors: (i) an increase in the absolute intensity of the defect-activated LA(M) peak, and (ii) a decrease in the intensity of the E' and A'_1 peaks attributed to the ablation of the specimen <cit.>.§ EXPERIMENT §.§ Preparation of Monolayer 2D Materials Chemical vapour deposition (CVD) was used to grow the graphene sample on copper foil <cit.>. The graphene was transferred onto a Si substrate using a polymer-assisted process as outlined previously <cit.>. The Si substrate had arrays of holes with a diameter of ∼2 μm and depths of >10 μm as pictured in the supplementary information.MoS_2 was also prepared using a CVD technique <cit.>. MoO_3 substrates were placed face-up in a ceramic boat with a blank SiO_2 substrate face-down on top. This was situated in the centre of the heating zone of a quartz tube furnace, and ramped to 750 ^∘C under 150 SCCM of Ar flow. Sulfur (S) vapour was then produced by heating S powder to ∼120 ^∘C in an independently controlled upstream heating zone of the furnace, and carried downstream to the MoO_3 for a duration of 20 min. After this, the furnace was held at 750 ^∘C for 20 min, then cooled down to room temperature. Monolayer flakes of MoS_2 with a typical triangular shape could then be identified on the SiO_2 surface by optical contrast.§.§ Irradiation with 30 keV He^+ and Ne^+The Zeiss ORION NanoFab microscope was used to irradiate arrays of 5 × 5 μ m^2 regions in graphene and MoS_2 with He^+ and Ne^+ at an energy of 30 keV and an angle of incidence of 0^∘. These regions received doses ranging from 1.5×10^11 to 1×10^16 Ne^+ cm^-2 or 1×10^13 to 1×10^17 He^+ cm^-2.The beam was defocused (∼10s of nm) to ensure a uniform distribution of ions and the sample was irradiated at the desired dose. 1 pA beam current and 10 nm pixel spacing were used. The beam dwell time at each pixel and/or the number of repeats at each position were varied to achieve the desired dose. The chamber pressure was of the order 3 × 10^-7 Torr.§.§ Raman Spectroscopy Raman spectroscopy was carried out on graphene with a Horiba Jobin-Yvon system (633 nm laser) with a 1200 lines/mm diffraction grating and a 100× objective aperture (NA=0.66) (laser spot size was ∼0.7 μm). These spectra were comprised of 10 acquisitions, each of 1 s duration at a single point for each irradiated region. Raman spectroscopy was carried out on MoS_2 using a WITec Alpha 300R system (532 nm laser) with a 1800 lines/mm diffraction grating and a 100× objective (NA=0.95) (laser spot size was ∼0.3 μm). Raman maps were generated by taking four spectra per μm in both x and y directions over large areas <cit.>. The acquisition time was 0.113 s. The spectra from a desired region were acquired by averaging.For both materials, the laser power was ∼1 mW to minimise sample damage. Peaks in the Raman spectra were fitted with Lorentzian functions for graphene. For MoS_2, Gaussian functions were used for the E' and A'_1 peaks and Lorentzians for the region around the LA(M) peak (demonstrated in the supplementary information). Error bars, where used and unless otherwise stated, are the largest of either the instrumental (1 cm^-1) or the fitting error as acquired from the fityk software package, which uses a weighted sum of squared residuals to measure agreement between the fit and the data <cit.>.§ RESULTS & DISCUSSION §.§ Graphene In Fig. <ref>, four sets of Raman spectra are presented, labelled as follows: (a) He^+ irradiated freestanding graphene, (b) He^+ irradiated supported graphene, (c) Ne^+ irradiated freestanding graphene and (d) Ne^+ irradiated supported graphene. The spectra obtained from the non-irradiated regions of both supported and freestanding samples are shown in black and agree with the literature for monolayer graphene <cit.>. In each subplot, multiple spectra are shown with dose increasing in ascent from the bottom. The first spectrum (black) represents the non-irradiated graphene. In the second spectrum (red), the defect-activated D peak (at ∼1322 cm^-1) can be observed to have increased in intensity relative to the G band intensity. The third spectrum (green) shows a very intense D peak. Finally, in the fourth spectrum (blue) the material is amorphous carbon with little to no remaining crystallinity. The width of the G peak (Γ_G) increases as a function of ion dose as shown in Fig. <ref>(a) for both He^+ and Ne^+. For each type of sample and choice of ion this represents an unequivocal increase in structural disorder. While the trends for both ions are similar, the incidence of the Ne^+ species causes changes in the G peak to occur at ion doses which are lower by between one and two orders of magnitude than those of He^+. This is due to the increased mass of Ne^+ ions and an enhanced milling capability at the incident surface compared to He^+ <cit.>. It should be noted that the difference between the effects of ion irradiation on Γ_G for supported and freestanding graphene is very small here and difficult to separate from the direct effect of the substrate/suspension on the G peak <cit.>. The fit to the data is of the form: Γ _G = Γ _G0 +bS^c Where Γ _G0 is the width of the G peak in pristine graphene, b is a fitting parameter related to the defect yield, S is the ion dose and c is a fitting parameter related to the effect of defect density on the G peak. c is found to be less than 1 in all cases, suggesting that the relationship between dose and Γ _G is sub-linear.In Fig. <ref>(b), the evolution of the I_D/I_G ratio against dose is displayed for both He^+ and Ne^+ and for both freestanding and supported graphene. Progression through the three previously discussed stages is observed. I_D/I_G of the supported graphene is noted to rise faster and reach a much higher maximum than the freestanding material. The maximum I_D/I_G ratio is also observed to be lower for freestanding Ne^+ irradiated graphene than He^+ irradiated graphene but larger for supported Ne^+ irradiated graphene than freestanding Ne^+ irradiated graphene. This is discussed in relation to defect sizes later.§.§.§ Defect ProbabilitiesFor lighter and/or higher energy ions such as those used in this work, the assumption made bythat one ion produces one defect may no longer be valid. In those cases, the average distance between defects becomes:L_D = 1/√(α S)Where α is the defect per ion yield and is distinguished from the sputtering yield, γ, because not all defects causing local activation of Raman modes need necessarily be a vacancy. The sputtering yields of graphene were calculated using stopping and range of ions in matter (SRIM) simulations (γ _C(SRIM)) as detailed in the supplementary information <cit.>. In calculating the yield of carbon atoms from graphene, SRIM accounts for those carbon atoms removed by primary ions, backscattered ions and secondary particles.The four scenarios applied in experiment were simulated: (i) freestanding, irradiated with He^+, (ii) supported, irradiated with He^+, (iii) freestanding, irradiated with Ne^+, and (iv) supported, irradiated with Ne^+. These values are presented in table <ref> alongside values calculated using the molecular dynamics derived simulation package of<cit.>. The sputtering yields calculated using SRIM for each ion are about 10 times larger in supported than freestanding graphene. This indicates that secondary particles from the substrate have a significant impact on the rate of defect introduction to the graphene layer.The process of fitting equation <ref> to our data involved using initial values from previous studies. The process is described in detail in the supplementary information. Equation <ref> was fitted to the four data sets in Fig. <ref>(a)(b), showing the close agreement of the experimental data and the fitted equation.The values for α _C obtained by fitting, α_C(fit), are shown in table <ref> alongside their corresponding seed values, the sputtering yield found using SRIM, γ_C(SRIM). For comparative purposes, two further values are included which are applicable to freestanding graphene only. γ_C(MD) is the sputtering yield and α _C(MD) is the defect per ion value calculated using the molecular dynamics-derived online simulation of<cit.>. The sputtering yield of He^+ on freestanding graphene has also been measured experimentally before using a single pixel exposure to completely mill through a graphene layer. The value reported byis γ _C = 0.007, which is in close agreement with atomistic simulations <cit.>. Given that this omits defects that do not involve sputtering, it is expectedly smaller than the value of α _C(fit) obtained in this work. Similarly, in the case of Ne^+ on freestanding graphene, we obtain a value of α _C(fit) which is more than twice the value of γ _C(MD).The obtained result, that α _C is consistently larger for supported rather than freestanding graphene for these irradiation conditions, is thus in keeping with expectations discussed in the introduction. A point to note is that for both He^+ and Ne^+ in freestanding graphene the experimentally derived α _C(fit) is smaller than the computationally derived α _C(MD), a discrepancy which is worthy of future study beyond this work. §.§.§ Defect SizesOther results of the fitting process are summarised in table <ref>. It is noted that the maximum I_D/I_G value is considerably higher for supported graphene than for freestanding graphene. It is observed from equation <ref> and displayed in Fig. S1 that there is a close relationship between r_S and the maximum of the I_D/I_G ratio. Depending on many factors, but principally the irradiation species, energy and angle, we can generally expect r_S values between approximately 0.8 and 2.5 nm <cit.>.It has been established that larger incident species usually produce larger defects <cit.>. However, in the freestanding case, r_S is unexpectedly not found to be larger for Ne^+ than He^+, despite the very different ion mass. This seemingly anomalous He^+ behaviour may be related to a similarly unexpected experimental finding by<cit.>. Also observable are the substantially higher values for r_S for freestanding than supported graphene.found that the size of defects introduced by energetic particles typically increases with energy <cit.>. While the high energy interaction of the direct beam is expected to cause large defects, it also excites substrate particles which have lower energies than the incident ion. These lower energy atoms can produce numerous, smaller defect sites in the graphene at the surface. A comparison of ion species is provided in table <ref>. For supported graphene irradiated with 30 keV He^+, r_S has been found to be ∼0.8-1 nm <cit.>. The value obtained in this work for He^+ is slightly smaller than previous reports <cit.>. Since the mass of Ne^+ is between those of He^+ and Ga^+, by a simple argument, it might be expected that the corresponding r_S value would be similarly intermediate. Although the result for He^+ irradiated freestanding graphene seems to be anomalous as previously discussed, this is indeed true for our supported graphene. Ga^+, also incident on supported graphene and also at 30 keV, has been reported to create defects of r_S=1.6 nm <cit.>. Thus the trend for supported graphene is clearly one of increasing defect size with increasing ion mass, m_a.We propose that the larger ion transfers energy more efficiently to substrate atoms (the masses of Si and Ne are very close), and it is these particles, being more efficiently energised than their He^+-induced counterparts, which create larger defects in the graphene layer. The variety of r_S values obtained in this work suggests a variety of defect types with different weighting in the four experimental scenarios. Such defects may include single vacancies, double vacancies, complex defects or amorphisation <cit.>. This underscores the importance of choosing ion and substrate carefully for both nanofabrication and defect engineering.In Fig. <ref> (a)(b), the characteristic E' and A'_1 peaks of monolayer MoS_2 are marked by dark grey dashed lines. In the pristine spectra (in black) they are in good agreement with the literature and the small separation of these two peaks (∼18 cm^-1) is indicative of monolayer MoS_2. With increasing ion dose, quenching and broadening of these two characteristic peaks are observed, reflecting the growing disorder which the ion beams create in the material. The emergence of the LA(M) peak at ∼227 cm^-1(dashed grey line), particularly at high doses of He^+ and Ne^+, is evident in Fig. <ref>. The evolutions of width and position of the E' and A'_1 peaks are shown in Fig. <ref> (a) and (b) respectively.Both peaks are observed to broaden with increasing disorder and the peak positions shift as expected from previous reports <cit.>. Broadening begins at a substantially lower dose for Ne^+ than for He^+ as expected. The E' peak red-shifts and this downward shift in energy is attributed to the introduction of defects causing lattice distortion, similarly to tensile strain <cit.>. The A'_1 peak also blue-shifts for some of the higher doses used, as previously reported <cit.>. Fig. <ref> (a) shows the evolution of the intensity ratios extracted from the spectra as a function of dose. With increasing disorder introduced by both ion beams, a sharp increase in the intensity of the LA(M) peak normalized to both the E' and A'_1 peaks is observed. Once again, the increased defect yield of Ne^+ compared to He^+ is highlighted. used equation <ref> to relate these intensity ratios directly to the average interdefect distance <cit.>. We highlight some caveats to this approach in addition to those present for graphene. Since MoS_2 is non-monoatomic, changes in stoichiometry may cause more complex defect-dependent behaviour. Also,made an implicit assumption that each ion causes one defect which we do not consider to be a safe assumption for either the 25 keV Mn^+ used in their work or indeed the lighter 30 keV ions used in this work <cit.>. Nonetheless, this form of the defect-activation model and the constants provided previously were applied to our data to calculate the average displacement between defects, L_D(M), for both peaks and for both He^+ and Ne^+ irradiation. In order to calculate the defect yield per ion, α_M, the initial defect level is accounted for in an adjusted version of equation <ref>:L_D(M) = 1/√(α_M S + σ_i)Where σ_i is the defect density in non-irradiated MoS_2. These L_D(M) values are presented as a function of ion dose in Fig. <ref> (b). Even the non-irradiated MoS_2 has a somewhat low L_D(M), not unusual for CVD-grown material.It is notable that in the high L_D(M) range, where the ion dose is small, there is a discrepancy between the values given by the two peaks. Using I(E')/I(LA(M) to calculate L_D(M) yields a consistently higher value than using I(A'_1)/I(LA(M). However, the two values approach each other as L_D(M) decreases which suggests that the nature of the initial defects may be different to those introduced by ion irradiation. Equation <ref> allows for the extraction of the σ_i and α_M values for both peaks and both ions as presented in table <ref>. The values for σ_i are in good agreement between the two ions, though as mentioned already do not agree between the two peaks. The values for α_M do agree well between peaks but, as expected are very different for the two ions of such different masses.§ CONCLUSIONIn this study, we explained the effects of both He^+ and Ne^+ irradiation at 30 keV on graphene and MoS_2 using models of their Raman spectra. For both materials, doses above 5×10^15 cm^-2 and 5×10^14 cm^-2 for the respective ion species have resulted in severe changes in the spectra relative to the starting material. For both material systems, this severe breakdown occurs at a Ne^+ dose which is between one and two orders of magnitude less than that of He^+. We believe this represents the first accurate experimental comparison of defect sizes produced in 2D materials by different noble gas ion probes. The dose dependence of irradiation species and relationships to interdefect distance have been established for graphene. A clear comparison to the literature is also reported for MoS_2. In addition, we studied the effects of both primary ion species and secondary particles on the sizes of the defects produced in graphene. The role of substrate particles on defect production and size are also highlighted as a concern for nanofabrication methodologies due to the clear secondary atom effect.These results will allow a more informed and precise defect engineering of the investigated monolayer materials.§ ACKNOWLEDGEMENTSWe thank the staff at both the Photonics and the Advanced Microscopy Laboratories (AML), CRANN, Trinity College Dublin and at the National Laboratory of Solid State Microstructures, Nanjing University. We acknowledge support from the following grants: Science Foundation Ireland [grant numbers: 12/RC/2278, 11/PI/1105, 07/SK/I1220a, 15/IA/3131 and 08/CE/I1432]. | http://arxiv.org/abs/1707.08893v3 | {
"authors": [
"Pierce Maguire",
"Daniel S. Fox",
"Yangbo Zhou",
"Qianjin Wang",
"Maria O'Brien",
"Jakub Jadwiszczak",
"Conor P. Cullen",
"John McManus",
"Niall McEvoy",
"Georg S. Duesberg",
"Hongzhou Zhang"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170727144622",
"title": "Defect Sizing, Separation and Substrate Effects in Ion-Irradiated Monolayer 2D Materials"
} |
A Jointly Learned Deep Architecture for Facial Attribute Analysis and Face Detection in the Wild Keke He, Yanwei Fu, Xiangyang XueFudan University {kkhe15, yanweifu, xyxue}@fudan.edu.cnDecember 30, 2023 =================================================================================================§ INTRODUCTION Light-pulse atom interferometers (AIs) <cit.> use the recoil momentum from photon-atom interactions to coherently split and recombine matter waves. They have been used for measuring gravity <cit.>, the gravity gradient <cit.>, rotation <cit.>, fundamental constants <cit.>, and for testing fundamental laws of physics <cit.>. Since the laser wavelength defines the photon momentum with high precision, AIs are accurate. Thus they are ideal candidates for inertial sensing or navigation. For this purpose, AIs need to be simple, reliable, and sensitive to multiple axes of acceleration and rotation. Even transportable single-axis AIs, however, require several lasers and laser amplifiers for atom trapping, interferometry, and detection <cit.>. So far, the only AI with six-axis sensingutilized two parabolically launched atom clouds and a complex combination of separate interferometry setups <cit.>. It achieved a sensitivity of 22 μrad/s/√( Hz) and 16 μm/s^2/√( Hz) for rotation and acceleration, respectively. Two-axis of rotations and one-axis of acceleration have also been demonstrated in an atomic fountain interferometer with atomic point sources and spatially resolved detection <cit.>. An atomic sensor using Bose-Einstein condensates has simultaneously measured gravity and magnetic field gradients <cit.>. A dual-axis accelerometer and gyroscope atom interferometer has been built by launching and recapturing two cold ensembles toward each other <cit.>. These examples illustrate that multiaxis AIs are more complex than single-axis ones. Additionally, other advances towards field operations, such as cold atom pyramidal gravimeters <cit.>, atom interferometers with short integration time <cit.>, atom interferometers with optical lattice <cit.>, and atom interferometry in an optical cavity <cit.> or a warm vapor <cit.>, and atom-chip gravimeters <cit.> have been demonstrated as well. However, multiaxis operation and simplicity have yet to come together in AIs.Generally, the laser system contributes the most complexity. Magneto-optical traps (MOTs) require six orthogonal beams and matter-wave splitters need relatively high laser intensity and low phase noise. Besides, specific laser frequencies are demanded for different procedures. In order to construct simple and reliable laser systems, fiber lasers <cit.> and integrated diode lasers <cit.> are developed. However, to our knowledge, AIs have never been operated based on a single diode laser without optical amplifiers. Laser systems with a single diode laser and pulsed modulators can avoid frequency-locking or phase-locking between different lasers and thus improve robustness. Without optical amplifiers, laser systems will also gain simplicity and power efficiency. Here, we demonstrate a multiaxis AI based on a single diode laser and a pyramidal MOT.The pyramidal geometry requires only a single laser beam to trap atoms and form a vertical atom interferometer. Additional beams, orthogonal to the pyramidal faces, allow for a total of five AIs along different axes. Using the Mach-Zehnder geometry and the butterfly geometryallows for measuring acceleration and rotation separately. A single diode laser serves the multiaxis AI to maintain simplicity. With efficient two-photon Raman transitions and zero differential AC Stark shift, high-contrast fringes have been achieved using a μK-sample without velocity selection. As a demonstration, we achieve a sensitivity of 6 μm/s^2/√( Hz), 300 μrad/s/√( Hz), and 4 μrad/√( Hz) for acceleration, rotation, and inclination, respectively, limited by vibrational noise. This work offers a path towards building simple, precise and multiaxis AIs. § MULTIAXIS ATOM INTERFEROMETRYFigure <ref>(a) shows the principle of the multiaxis atom interferometry in a pyramid. The pyramid consists of four orthogonal reflection faces. A MOT is created inside the pyramid by irradiating one laser beam vertically toward the entire pyramid, where six orthogonal trapping beams can be generated by the reflections <cit.>. Utilizing the incidence and its reflections from either the whole pyramid or individual pyramidal faces as matter-wave splitters, we can buildone vertical AI as well as four angled AIs along different axes. The matter-wave splitter of our AI is based on Doppler-sensitive two-photon Raman transitionsbetween the F=3 and F=4 hyperfine ground states of cesium atoms <cit.>. An atom, initially in the state |F=3, p=0⟩, is transferred to a state |F=4, p=2ħ k⟩, where ħ k is the photon momentum. To make a beam splitter, a π/2 pulse places the atom in a superposition of the two states. A mirror is formed by a π-pulse, which has a 100% probability of changing the state. As shown in Fig. <ref>(b), Mach-Zehnder interferometry is performed by a π/2 -π-π/2 pulse sequence and is used for measuring acceleration. We assume a⃗·Ω⃗ T_0 and Ω⃗·v⃗_0 are negligible compared to the acceleration a⃗ and the gravity g⃗, where Ω⃗ is the rotation, T_0 is the sequence time, and v⃗_0 is the initial velocity of the atom cloud at the first laser pulse. This condition is fulfilled, e.g., in stationary operation or aboard moving vehicles. The phase shift <cit.> caused by a⃗ and g⃗ is expressed as ϕ_a = k⃗· (a⃗ + g⃗ )T^2,where k⃗ is the effective wave vector of two counterpropagating photons and T is the pulse separation time. A vertical interferometer (addressed by the vertical laser beam and its reflection from the pyramid) and at least two angled ones (formed by beams aimed directly at a pyramid face) allows us to measure the full acceleration vector a⃗=(a_x, a_y, a_z).To measure rotation independent of acceleration, we use the butterfly geometry with a π/2-π-π-π/2 pulse sequence, as shown in Fig. <ref>(c). The rotation-induced phase shift <cit.> is ϕ_ =1/2k⃗· [(a⃗ +g⃗)×]T^3.Since g⃗ is along the z axis, two components of rotation (_x, _y) can be measured using laser beams pointing at two different pyramid faces. Additionally, measurement of _z can be achieved by applying appropriate acceleration in the xy plane to the interferometer. This allows us to measure the full rotation vector =(_x, _y, _z). § SINGLE-DIODE ATOM INTERFEROMETRY§.§ Single-diode laser system Atom interferometers consist of three procedures: atom cloud preparation, interferometry, and population detection. Only one diode laser is used for all the functions. The laser system is shown in Fig. <ref>(a). All laser radiation originates from a 240-mW distributed Bragg reflector diode laser (Photodigm, PH852DBR240T8). A sample of its power is sent to Doppler-free polarization spectroscopy, frequency stabilizing (“locking") the laser to the cesium F=4 → F'=4/5 D_2 crossover transition at 852 nm. An acousto-optical modulator (AOM 1) shifts the sample so that the light reaching the atoms can be at the MOT frequency (about 10 MHz red from the 4-5 transition) or the detection frequency (resonant with 4-5). Adding an offset voltage at the servo input jumps the lock point to the F=4→ F'=4 transition and generates the large detuning necessary for polarization gradient cooling. The timing sequences of our atom interferometry is shown in Fig. <ref>(b). The MOT cooling light is the undeflected beam after AOM 3; a repumping frequency is generated by sending a sample of the laser through a fiber electro-optical modulator (EOM). To avoid instability resulting from interference with the MOT light, the EOM is driven such that the carrier frequency is nulled. A liquid crystal retarder is placed after the fiber to convert the linear polarization to the circular polarization, so that counterpropagating σ^+/σ^- polarization pairs are formed inside the pyramid. Before interferometry, a microwave pulse followed by a blow-away laser pulse (resonant with 4-5) selects atoms into the magnetically insensitive state.The Raman frequency pairs for the interferometer are formed by the carrier and the first-order sidebands from the EOM. The Raman pulses have high intensity in the fiber EOM, but their pulse duration is too short to cause photo-refractive damage to the crystal. To minimize phase noise, the EOM is driven by a phase-locked dielectric resonator oscillator. The pyramidal geometry enables lin ⊥ lin polarization in the vertical AI, and σ^+/σ^+ or σ^-/σ^- polarization in the angled AIs. The vertical laser to the pyramid is blocked by a shutter when the angled AIs are operated. For detecting the F=3 and F=4 populations at the interferometer output ports, a pushing beam, slightly red-detuned to the F=4 → F'=5 transition, horizontally separates atoms in the two hyperfine ground states <cit.>. Both the cooling beam and repumping beam are then used for fluorescence imaging. A camera images both populations simultaneously, which makes the interference fringe immune to the fluctuation of atom number and imaging laser power. §.§ Zero differential AC Stark shift with small detuning In order to drive Raman transitions with modest laser intensity, the performance with a small single photon detuning is investigated. Figure <ref>(a) shows the energy levels and laser frequencies involved in the Raman transitions. The transitions must satisfy several requirements. Rapid Raman transitions (π-pulse time 10-20 μs) are needed in order to address all atoms from the thermal velocity distribution of the MOT efficiently, but it requires high laser intensity and/or small single-photon detuning Δ. For high accuracy and fringe contrast, the AC Stark shift of the F=3 and F=4 states needs to be equal, so that it cancels out of the interferometer phase <cit.>. We calculate the effective two-photon Rabi frequency , the differential AC Stark shift , and the single-photon scattering rate . To do so, we define A_n=√(I/)Γ J_n(β)/√(2) to describe the amplitude of each EOM sideband, where I is the total laser intensity,is the saturation intensity, Γ is the linewidth, J_n is the Bessel function of order n and β is the modulation index of the EOM. With this, we have= ∑_F'=2^5∑_n=-∞^∞M_3,0^F',- A_n M_4,0^F',+ A_n+1/2Δ_3,=∑_F',n(|M_3,0^F',-A_n|^2/4Δ_3-|M_4,0^F',+A_n|^2/4Δ_4),where M_F, m_F^F',± = ⟨ F, m_F|F', m_F± 1 ⟩ are the cesium D_2 dipole matrix elements for σ^± transitions, expressed as multiples of ⟨ J = 1/2||er||J' = 3/2⟩ <cit.>, Δ_F=n+_F^F'+Δ is the effective detuning of F= 3 or 4, -ħ_4^F' is the energy of the |F'⟩ excited state relative to the |F'=5⟩ state, and _3^F'=_4^F'-. The ground state hyperfine splittingis also the EOM driving frequency, and Δ is the detuning of the carrier relative to the F=4→ F'=5 transition. The scattering rates for atoms starting in F=3 or F=4 are^F=∑_F',nΓ(M_F,0^F',-A_n)^2/Γ^2+2(M_F,0^F',-A_n)^2+4(Δ_F)^2. From these, the scattering probability for an entire interferometer can be calculated.A suitable detuning Δ/(2π) ≃ -160MHz (slightly dependent on β), can be found for whichvanishes and both ,are acceptable. This detuning can easily be reached with an AOM. In this case, the theoretical limit on the contrast of Mach-Zehnder fringes from single-photon scattering is approximately 40%. Figure <ref>(b) shows a comparison of theory to experiment for the two-photon Rabi frequency, differential AC Stark shift, and single-photon scattering as function of detuning. If we modulate the fiber EOM with an index of about 1, the differential AC Stark shift is zeroed at a red single photon detuning of 158 MHz. The zero differential AC Stark shift is verified with 30-Hz accuracy by measuringwith optical Ramsey interferometry. For the Doppler-sensitive Raman transition, the width of a π-pulse is as short as 12 μs. Since Doppler-sensitive Raman transitions can only transfer atoms distributed within a certain velocity bandwidth, the efficiency of Raman transitions can be increased by use of a faster Rabi flopping frequency. Without velocity selection, a π pulse transfers as many as 60% of all atoms. § EXPERIMENT AND RESULTS§.§ ExperimentThe pyramid is in a glass cube of 25.4× 25.4× 25.4mm^3, dielectrically coated for equal phase shift at two orthogonal polarizations at 45^∘. The vertical AI (and MOT and detection) beam has a waist of 15 mm (1/e^2 radius) and a power of ∼ 60mW before the pyramid. The diagonal beams have waists of 12 mm (1/e^2 radius) and power of ∼ 40mW.We capture approximately 5 million atoms from background cesium vapor in 1 s. With increased laser detuning of -160 MHz and decreased (∼ 1/4) laser power, polarization gradients cool the atoms to about 2 μK in 5 ms. The cooling beam is turned off 1 ms before the repumping beam to ensure that all the atoms stay in F=4. As the atoms freely fall, a microwave π-pulse (100 μs and +20 dBm) followed by a blow-way laser pulse selects ∼ 5× 10^5 atoms from |F=4, m_F=0⟩ into |F=3, m_F=0⟩ with a bias field of 500 mG. During interferometry, the magnetic quantization axis is aligned with the direction of the Raman pairs in order to enhance the Raman transition between |F=3, m_F=0⟩ and |F=4, m_F=0⟩.As a proof of multiaxis atom interferometry, we demonstrate three AIs on a passive vibration isolation platform (minusK, 150BM-1) placed on an optical table. One AI uses the vertical Raman pair, and the other two use angled Raman pairs. The vertical AI is operated with π/2 -π-π/2 sequences for gravity measurement. The angled AIs are operated with π/2 -π-π/2 sequences and π/2-π-π-π/2 sequences to measure the projected gravity with an angle of approximately 45^∘ and the earth's rotation rate.§.§ Acceleration measurementFigure <ref> shows fringes measured in Mach-Zehnder geometry.In the vertical AI, the frequency difference between the Raman frequency pair is linearly ramped at approximately 23 MHz/s to compensate the time-varying Doppler shift of the free-falling atoms. The ramp rate of the angled AI, at about 45^∘ to the vertical, is about 16 MHz/s, which is a factor of √(2) smaller than that of the vertical one. While scanning the phase of the last interferometer pulse, the fringes are obtained by counting the atom populations in the two hyperfine ground states. In particular, the pulse sequence time of the vertical AI is constrained by the geometry of the apparatus and that of the angled AIs is limited by the Raman beam waist. The vertical AI has a fringe constrast of 18% with a sequence time of 80 ms. The angled AIs have a fringe constrast of 22% with a sequence time of 40 ms. As the sequence time is as short as 2 ms, the fringe contrast is improved to 30%. The decreasing contrast with longer sequence time can be explained by inhomogeneous Rabi flopping of the three Raman pulses. The top angle of pyramid is 90^∘±1^', which leads to a systematic error of 42 parts per billion for the vertical AI. For the angled AI, the accuracy of the top angle produces a negligible error in the alignment. However, the error of the projection angle results in systematic errors to the accelerations along x or y axis. Given an angle error of 1^' at 45^∘, the systematic error would be about 2 mm/s^2.Figure <ref>(a) shows the Allan deviation of the gravity measurement by the vertical AI, with the tide variation subtracted from the model. The sensitivity is 6 μm/s^2/√( Hz). Measuring gravity continuously during 4 days, the tide variation has beenobserved, as shown in Fig. <ref>(b). The systematic error of the pyramidal top angle is negligible compared to the current sensitivity. Using a commercial tilt sensor (Jewell Instruments, 756-1326) to monitor the alignment between the Raman beam and the gravity axis, the long-term drift of the platform is calibrated. The sensitivity of the vertical AI is competitive to the state-of-the-art compact AIs <cit.>.The sensitivity can be further improved with a faster cycling rate, better vibration isolation, and longer sequence time. Our current cycling time is limited by the computer control software. Once this is overcome, the cycling rate can further be increased by shortening the MOT loading time. Additionally, the vibrational noise can be decreased by better vibration isolation, such as an active feedback system <cit.>. Finally, since the sensitivity scales with T^2, it can be improved by longer sequence time. For example, a freely-falling time of ∼ 300 ms, corresponding to a length of ∼ 0.5 m, can improve the current sensitivity by one order of magnitude. §.§ Inclination measurementThe Mach-Zehnder AIs along the diagonal axes measure the projected gravity with an angle of approximately 45^∘, which makes them sensitive to the angle variation between the Raman beam and the gravity axis. As the fiber ports of the Raman beams are also put on the vibration isolation platform, the angled AIs work as atomic inclinometers.For long-term measurement, we observe that the fluctuation of the projected gravity is correlated with the tilt sensor, as shown in Fig. <ref>.This fluctuation is the drift of the platform, which has a period of half an hour. It also indicates that the sensitivity of the vertical AI is limited by vibration noise. The sensitivity of the angled AI is about 25 μm/s^2/√( Hz). According to the projection angle, gravity variation of about 150 μm/s^2 corresponds to the tilt of 21 μrad. The sensitivity of our atomic inclinometer is 4 μrad/√( Hz), compared to 800 μrad/√( Hz) in previous work <cit.>. §.§ Rotation measurementFigure <ref>(a) shows interference fringes of a symmetric butterfly interferometer (i.e., with pulse separation times T/2, T, T/2) for rotation measurements. With a sequence time of 40 ms, we achieve a fringe contrast of 11%. The symmetric configuration is necessary to fully cancel the phase contribution from constant acceleration <cit.>. An asymmetric configuration could be used to suppress parasitic interferometers, further enhancing contrast <cit.>. Although the relative AC Stark shift is small, itwould still result in systematics for the rotation measurement. In order to cancel even this residual relative AC Stark shift, the rotation-sensitive AI is operated with opposite effective wave vectors +k and -k.As shown in Fig. <ref>(b), the phase sensitivity of the butterfly AI is 40 mrad for a single fringe measurement, which corresponds to the rotation rate sensitivity of 170 μrad/s. By operating the AI on the phase-sensitive slopes of the fringes, we achieve a sensitivity of about 300 μrad/s/√( Hz) for rotation measurement. The Earth's rotation rate is measured by the gyroscope. Using the angled Mach-Zehnder AI, we measure the absolute angle between the laser wave vector and the gravity axis. During the rotation rate measurement, the projection angle is monitored by the tilt sensor. We then obtain the Earth's rotation rate of 5(1)× 10^-5 rad/s after 20 min of averaging time. The expected rate at Berkeley (latitude 37.78^∘) is 57.9 μrad/s corresponding to a phase shift of 24 mrad. The phase sensitivity of our gyroscope is similar to that achieved by the state-of-art atom interferometry gyroscopes <cit.>. The rotation rate sensitivity is constrained by the total phase accumulated from the rotation, which can be improved by longer sequence time.§ CONCLUSION In conclusion, we have demonstrated multiaxis AIs with a single-diode laser system and a pyramidal MOT. Efficient Doppler-sensitive Raman transitions are achieved using a small single photon detuning and modest laser intensity. With zero differential AC Stark shift and insignificant single photon scattering, high-contrast fringes are obtained. Gravity as well as the tilt of the platform are measured in Mach-Zehnder geometry with a sensitivity of 6 μm/s^2/√( Hz) and 4 μrad/√( Hz), respectively. Rotation is measured using the butterfly geometry with a sensitivity of 300 μrad/s/√( Hz). Being simple, precise, and capable of multiaxis operation, our multiaxis and single-diode AI has the potential to become a versatile atomic sensor in rough environments, such as drones, submarines or satellites. Comparing with classical inertial sensors, AIs are more accurate and have better long-term stability. Generally, AIs are too sensitive to operate in environments with strong vibration noise. In order to overcome this problem, either measuring vibration with another sensor or simultaneous multiple AIs are feasible. One example is that AIs have been operated on an airplane by monitoring the vibration noise with a mechanical accelerometer <cit.>. 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Bresson, “New concepts of inertial measurements with multi-species atom interferometry,” arXiv:1710.06289 (2017).gratingMOT M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Laser cooling with a single laser beam and a planar diffractor,” Opt. Lett. 35(20), 3453-3455 (2010).chipMOT C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nat. Nanotechnol. 8, 321–324 (2013).SubDoppler J. P. McGilligan, P. F. Griffin, R. Elvin, S. J. Ingleby, E. Riis, and A. S. Arnold, “Grating chips for quantum technologies,” Sci. Rep. B 7, 834 (2017).tetrahedralMOT M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Single-laser, one beam, tetrahedral magneto-optical trap,” Opt. Express 17(16), 13601-13608 (2009). | http://arxiv.org/abs/1707.08693v4 | {
"authors": [
"Xuejian Wu",
"Fei Zi",
"Jordan Dudley",
"Ryan J. Bilotta",
"Philip Canoza",
"Holger Müller"
],
"categories": [
"physics.atom-ph",
"physics.ins-det",
"physics.optics"
],
"primary_category": "physics.atom-ph",
"published": "20170727032145",
"title": "Multiaxis atom interferometry with a single diode laser and a pyramidal magneto-optical trap"
} |
, [email protected] Institute of Biochemistry and Biophysics, P.O.B. 30, Kazan 420111, Russia. Tel. 8-843-231-90-37. e-mail: [email protected] one-dimensional Schrödinger equation with symmetric trigonometric double-well potential is exactly solved via angular prolate spheroidal function. Although it is inferior compared with multidimensional counterparts and its limitations are obvious nevertheless its solution is shown to be analytic rather than commonly used numerical or approximate semiclassical (WKB) one. This comprises the novelty and the merit of the present work. Our exact analytic description of the ground state splitting can well be a referee point for comparison of the accuracy of numerous WKB formulas suggested in the literature. The approach reasonably well suits for the inversion mode in the ammonia molecule NH_3 and thus yields a new theoretical tool for its description. The results obtained provide good quantitative description of relevant experimental data on microwave and IR spectroscopy of NH_3.Schrödinger equation, confluent Heun's equation, spheroidal function.§ INTRODUCTIONAmmonia molecule NH_3 is a very important object both in itself and for the development of physics. In particular it played a cornerstone role in the development of radiospectroscopy and quantum electronics as the basis for the first maser <cit.>. Up to now it remains a subject of intensive researches (see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and refs. therein). Besides from the very beginning of quantum mechanics it is a "work horse" object for verification of analytic and numerical methods of treating Schrödinger equation (SE) with a double-well potential <cit.>, <cit.>, <cit.>, <cit.>. For ammonia molecule a pertinent degree of freedom (inversion mode) in the vibrational spectrum of NH_3 corresponds to the motion of the N atom relative to the H_3 symmetrical triangle <cit.> (physically three hydrogen atoms undergo simultaneous tunneling from one side of the nitrogen atom to the other <cit.> although the nitrogen also slightly moves to provide that the position of the center of mass remains constant). As a result a fictitious quantum particle with the reduced massM=3m_Hm_N/3m_H+m_Nmoves under the influence of a double-well potential (DWP) along the coordinate corresponding to the distance of the N atom from the H_3 plane <cit.>. DWP results from the Coulomb repulsion between the nitrogen nucleus and the three protons. This picture is somewhat simplified and more complex paths for atoms movement at inversion are suggested <cit.>. However these subtleties are not important for the present analysis. We remain within the framework of an approach in which NH_3 inversion description is reduced to the solution of one-dimensional SE with a pertinent double well potential and parameters chosen to suit corresponding experimental data of microwave and IR spectroscopy. We fully aware that our SE is inferior compared with multidimensional counterparts and its limitations are obvious. Nevertheless its solution is shown to be exact analytic rather than commonly used numerical or approximate semiclassical (WKB) one. This comprises the novelty and the merit of the present work.SE with a DWP finds many well-known applications in physics and chemistry beginning from the above noted inversion of NH_3 and ending by heterostructures, Bose-Einstein condensates and superconducting circuits (see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and refs. therein). We mention here only smooth double-well potentials and leave aside numerous models with rectangular wells or sewing together two single-well potentials (harmonic, Morse, etc.) that pervade textbooks and pedagogical-style articles on quantum mechanics. There is noticeable advance in obtaining quasi-exact (i.e., exact for some particular choice of potential parameters) <cit.>, <cit.>, <cit.> and exact (those for an arbitrary set of potential parameters) <cit.>, <cit.> solutions for such SE by their reducing to the confluent Heun's equation (CHE). A lot of potentials for SE are shown to be solvable via the confluent Heun's function (CHF) <cit.>. The CHF is a well described special function tabulated in Maple <cit.>, <cit.>, <cit.>. The latter makes its usage to be a routine procedure. This fact renders the obtained solution of SE to be very convenient for applications. Recently the exact solution of the Smoluchowski equation for reorientational motion in Maier-Saupe DWP was obtained via CHF <cit.>, <cit.>. The method yields the probability distribution function in the form convenient for application to nuclear spin-lattice relaxation <cit.>. Here we apply similar approach to SE with a practically important type of DWP.In the present article we modify the trigonometric DWP suggested in <cit.> for the description of intrinsically asymmetric hydrogen bonds to that including the symmetric case as a particular one. The trigonometric DWP is a particular case of some general potential from <cit.> (N2 with m_1,2=(1/2,1/2) from Table.1). However the specific form of the potential investigated in <cit.> can not be directly applied to the symmetric case. Let us denote the dimensionless coordinate as y. The form in <cit.>U(y)=h tan^2 y+√(h)sin y/cos^2 y-bsin^2 y+asin yis intrinsically asymmetric. Even if we make the depths of the wells to be equal by an appropriate choice of a there is still an asymmetry in barrier shape. Here we consider the formU(y)=h tan^2 y-bsin^2 y+asin yThis potential is truly symmetric at a=0.The aim of the article is to solve exactly SE with the latter potential, to determine the quantization rule for obtaining energy levels and to provide analytic representation of the wave functions for these energy levels. We show that the wave functions are expressed via Coulomb (generalized) spheroidal function (CSF). In the particular case of symmetric potential (a=0) the latter is reduced to an ordinary spheroidal function that is realized in Mathematica. We apply our results to the inversion of the ammonia molecule NH_3 that is a classical example of the process with a symmetric DWP.The physics of NH_3 (and more generally of tunneling phenomena including energy levels splitting) is very well understood by now as a result of qualitative arguments, semiclassical (WKB) analysis and numerical solution of SE with DWP. The stringent analytic results for the exactly solvable trigonometric DWP are complementary to this understanding. The results obtained agree well with previous knowledge and supplement it with precise calculations within the framework of the particular potential (trigonometric DWP). Also it should be noted that in the present article we do not touch upon the problem of the rotation-vibration interaction, i.e.,we assume NH_3 molecule to be on the lowest J=1; K=1 rotational level (the ground state of para-ammonia <cit.>).The paper is organized as follows.In Sec. 2 the problem under study is formulated.In Sec. 3 and 4 the solution of SE is presented via CHF and CSF respectively. In Sec. 5 the case of symmetric potential is treated. In Sec.6 the results for NH_3 are discussed and the conclusions are summarized. In Appendix 1 and Appendix 2 some minor technical details are presented.§ SCHRÖDINGER EQUATION AND THE POTENTIALWe consider the one-dimensional SEd^2 ψ (x)/dx^2+2M/ħ^2[E-U(x)]ψ (x)=0where U(x) is a DWP that is infinite at the boundaries of the finite interval x=± L. We introduce the dimensionless energy ϵ and the dimensionless distance yϵ=8ML^2E/ħ^2 π^2y=π x/2Lso that -π/2≤ y ≤π/2. The dimensionless SE with the potential (<ref>) takes the formψ”_yy (y)+[ϵ-h tan^2 y+bsin^2 y-asin y]ψ (y)=0where h is the barrier width parameter, b is the barrier height parameter and a is the asymmetry parameter (see Appendix). An example of this potential is depicted in Fig.1. § SOLUTION OF THE SCHRÖDINGER EQUATION VIA THE CONFLUENT HEUN'S FUNCTIONWe introduce a new function φ (y) by the relationshipψ (y)=cos^1/2 y exp(-√(b)sin y )[ tan(π/4+y/2)]^√(h+1/4)φ (y)and a new variablez=1+sin y/2The equation for φ (z) isz(z-1)φ”_zz(z)+[-4√(b) z^2+2(2√(b)+1)z-(√(h+1/4)+1)]φ'_z(z)+ [2(a-2√(b))z+2√(b)(√(h+1/4)+1)+1/4-ϵ-h-b-a]φ (z)=0It belongs to the type of CHE. For the infinite at the boundaries potential the wave function must be zero there. As a result the solution of our SE consistent with this requirement has the formψ (y)=cos^1/2 y exp(-√(b)sin y )[ tan(π/4+y/2)]^√(h+1/4)× HeunC(-4√(b),√(h+1/4),-√(h+1/4),2a, 3/8-a-b-h/2-ϵ;1+sin y/2)At y=-π/2 the wave function is automatically zero due to cos and tan. However at y=π/2 the function tan diverges and cos can not cope with this disaster at h>0 without the help from the side of HeunC. The above mentioned requirement can be satisfied only by a specific constraint imposed on HeunC and thus yields the boundary condition for determining the energy levels. As a result we obtain the equation for eigenvalues by setting y=π/2HeunC(-4√(b),√(h+1/4),-√(h+1/4),2a, 3/8-a-b-h/2-ϵ;1)=0Its solutions form the spectrum of eigenvalues ϵ_n where n=0,1,2, ... for the energy ϵ.Maple enables one to solve easily and efficiently (<ref>) and to plot the wave function (<ref>). However even the normalization of the wave function (let alone the calculation of matrix elements with its help) requires dealing with integrals containing HeunC. Unfortunately Maple carries out this task extremely inefficiently (see <cit.> where some drawbacks of HeunC realization in Maple are expertly discussed). Attempts to circumvent this difficulty lead to tiresome numerical calculations that practically cancel out all the advantages of our analytic solution compared with a numerical solution of SE. For this reason we further look for another form of the solution that gives much more convenient computational tool for the case of symmetric potential. We remind that namely the latter is necessary for the description of inversion mode in NH_3.§ SOLUTION OF THE SCHRÖDINGER EQUATION VIA THE COULOMB SPHEROIDAL FUNCTIONWe introduce a new variables=2z-1where -1≤ s ≤ 1 and a new function v(s) by the relationshipφ (s)=(1-s/1+s)^√(h+1/4)/2exp(s√(b))v(s)The equation for v(s) is obtained from (<ref>)d/ds[(1-s^2)dv(s)/ds]+ [b+ϵ+h-1/4-√(b (h+1/4))-b(1-s^2)-as-h+1/4/1-s^2]v(s)=0We denoteh=m^2-1/4If m is integer then (<ref>) belongs to the type of Coulomb (generalized) spheroidal equations <cit.> and its solution isv(s)=Ξ̅_mq(√(b), -a;s)where q=0,1,2,... and Ξ̅_mq(√(b), -a;s) is CSF. The energy levels are determined from the relationshipϵ_q=λ_mq(√(b), -a)+1/2-b-m^2where λ_mq(√(b), -a) is the spectrum of eigenvalues for Ξ̅_mq(√(b), -a;s). CSF is normalized by the requirement <cit.>∫_-1^1ds Ξ̅_mq'(√(b), -a;s)Ξ̅_mq(√(b), -a;s)=δ_qq' As a result the wave function takes the formψ_q (y)=cos^1/2 y [ tan(π/4+y/2)]^m(1-sin y)^m/2× (1+sin y)^-m/2Ξ̅_mq(√(b), -a;sin y)The formal proof that the product of the Coulomb spheroidal function appearing in (<ref>) with the divergent trigonometric functions is zero at y=π/2 is given in Appendix 2. Unfortunately the CSF is realized neither in Maple nor in Mathematica at present and the solution (<ref>) is practically inapplicable for calculations.§ SYMMETRIC POTENTIALFor the symmetric potential (a=0) CSF is reduced to an angular prolate spheroidal function <cit.>Ξ̅_mq(√(b), 0;s)=S̅_m(q+m)(√(b);s)where m is integer and q=0,1,2,.... The latter is realized inMathematica as S̅_m(q+m)(√(b);s)≡ SpheroidalPS[(q+m),m,i√(b),s] and is normalized by the requirement <cit.>∫_-1^1ds S̅_m(q+m)^2(√(b);s)=1 Thus we have in this caseψ_q (y)=cos^1/2 y [ tan(π/4+y/2)]^m(1-sin y)^m/2× (1+sin y)^-m/2S̅_m(q+m)(√(b);sin y)The formal proof that the product of the angular prolate spheroidal function appearing in (<ref>) with the divergent trigonometric functions is zero at y=π/2 is given in Appendix 2. The energy levels are determined from the relationshipϵ_q=λ_m(q+m)(√(b))+1/2-b-m^2where λ_m(q+m)(√(b)) is the spectrum of eigenvalues for S̅_m(q+m)(√(b);s). It is realized in Mathematica as λ_m(q+m)(√(b))≡ SpheroidalEigenvalue[(q+m),m,i√(b)].The formulas (<ref>) and (<ref>) provide a highly efficient and convenient tool for calculating the wave functions and the energy levels of SE with symmetric trigonometric potential (<ref>) with the help of Mathematica.§ RESULTS AND DISCUSSIONFig.1 shows that the parameters of the potential (<ref>) can be chosen to provide good description of the energy levels structure for a set of specific experimental data. In Fig.1 the energy levels for the inversion of the ammonia molecule NH_3 (experimental data determined with help of microwave and IR spectroscopy are taken from <cit.>, <cit.>) are presented. The energy levels for the inversion mode of NH_3 form a pair of doublets within the wells. For the ground-state splitting the values E_1-E_0=0.66÷0.8cm^-1 are given in the literature <cit.>, <cit.>, <cit.>. To be specific we choose the value E_1-E_0=0.76cm^-1. For the transition frequency of the upper state E_2-E_0 there seems to be a consensus E_2-E_0=932.4cm^-1 <cit.>, <cit.>, <cit.>. However for the transition frequency of the upper state E_3-E_0 also a range of values exists E_3-E_0=968.4÷ 977.3cm^-1 (E_3-E_2=35.7÷ 45.1cm^-1) <cit.>, <cit.>, <cit.>. We obtain the best description by taking the value E_3-E_0=977.3cm^-1 (E_3-E_2=45.1cm^-1) <cit.>.The distance between the minima of the potential 2l≈ 0.76 Å is known from the geometry of NH_3 <cit.>, <cit.>. These experimental values are obtained from our dimensionless ones if we take m=321 (h≈ 103041), b=120017.49, a=0 (with taking into account that for hydrogen m_H=1amu and for nitrogen m_N=14amu so that (<ref>) yields M≈ 2.47amu). From (<ref>) we have the relationship of the position for the dimensional minimum l≈ 0.38 Å <cit.>, <cit.> with our dimensionless one y_miny_min=l/2ħ√(%s/%s)8M( E_1-E_0)ϵ_1-ϵ_0From our value y_min≈ 0.27 it follows that for the conversion coefficient we should set the value L≈ 2.23Å.The normalized wave functions are obtained from (<ref>) as followsψ_i(y)^normalized=ψ_i(y){∫_-π/2^π/2dy (ψ_i(y))^2}^-1/2 They are depicted in Fig.2. All these functions are highly delocalized as it should be for a symmetric DWP. The functions ψ_0(y)^normalized and ψ_1(y)^normalized correspond to the split ground state. The ground state splitting is very small (0.76cm^-1 in dimensional units). The functions ψ_0(y)^normalized and ψ_2(y)^normalized are even while those ψ_1(y)^normalized and ψ_3(y)^normalized are odd. The functions ψ_2(y)^normalized and ψ_3(y)^normalized describe excited states. It should be stressed that the latter are rather close to the barrier top (see Fig.1). Thus the functions ψ_2(y)^normalized and ψ_3(y)^normalized can not be described in semiclassical approximation (WKB). The exact analytic representation of these wave functions is the merit of the present approach. In contrast the functions of the split ground state can be treated by WKB method (see <cit.>, <cit.> for review). In recent years there is noticeable progress in developing this method (see <cit.>, <cit.>, <cit.>, <cit.> and refs. therein) but different variants of ordinary semiclassical approximation along with instanton approach are suggested in the literature. Our exact analytic result can well be a referee point for comparison of the accuracy of suggested formulas.Both CSF and ordinary spheroidal functions are defined for integer m <cit.>. As a result we have to limit ourselves by a discrete set of the values for the parameter h=m^2-1/4(<ref>). For these reason we do not pretend to provide a very precise description of NH_3 with the help of spheroidal functions. The application of CHF (making use of (<ref>) and (<ref>)) circumvents this difficulty (i.e., the approach is valid for arbitrary values of h) and hence is able to provide more accurate description. However as was mentioned above the calculations of integrals containing CHF lead to tiresome numerical manipulations that deprive our approach of its advantage compared with a numerical solution of SE. In contrast Mathematica tackles the integrals containing spheroidal functions very efficiently. Thus we conclude that the formulas (<ref>) and (<ref>) are more convenient for calculations and provide reasonably good accuracy.In the present article we speak about NH_3 to be specific. It is obvious however that our approach is applicable to ND_3 or NT_3 with corresponding definition of the reduced mass in (<ref>). Moreoverour approach remains valid for any specific problem where one has to deal with a one-dimensional DWP in SE. Three parameters in the trigonometric DWP (<ref>) enable one to model experimental data of IR spectroscopy or results of quantum chemical calculations of the potential surface for such problem with reasonable accuracy.One can conclude that the Schrödinger equation with symmetric trigonometric double-well potential can be exactly solved via angular prolate spheroidal function. Our exact analytic description of the ground state splitting can well be a referee point for comparison of the accuracy of numerous WKB formulas suggested in the literature. The approach reasonably well suits for the inversion mode in the ammonia molecule NH_3. Thus it yields a new theoretical tool for the description of this important molecule. The results obtained provide reasonably good quantitative description of relevant experimental data on microwave and IR spectroscopy of NH_3.§ APPENDIX 1For the symmetric DWP (a=0) the parameters h and b can be related with the barrier height B=-U(y_min) and barrier width Δ=y_min^(1)-y_min^(2) as followsΔ=2arccos(h/b)^1/4 B=(√(h)-√(b))^2Inversely one obtainsb=B/{1-[cos(Δ/2)]^2}^2 h=B[cos(Δ/2)]^4/{1-[cos(Δ/2)]^2}^2 § APPENDIX 2Here we provide the formal proof that the product of both the Coulomb and the angular prolate spheroidal functions appearing in (<ref>) and (<ref>) respectively with the divergent trigonometric functions is zero at y=π/2. For both Ξ̅_mq(√(b), -a;sin y) and S̅_m(q+m)(√(b);sin y) the point y=π/2 is a regular singular one. In this regular point they behave as <cit.>Ξ̅_mq(√(b), -a;sin y)∼(1-sin^2 y)^m/2 S̅_m(q+m)(√(b);sin y)∼(1-sin^2 y)^m/2As a result it is enough to prove that the following limit is zerolim_y→π/2cos^1/2 y [ tan(π/4+y/2)]^m [(1-sin y)/(1+sin y)]^m/2(1-sin^2 y)^m/2=0Making use of the substitution y=π/2-ϵ we obtain after straightforward calculationlim_ϵ→ 0ϵ^1/2+m=0for any positive m that proves the above assertion.The behavior of both (<ref>) and (<ref>) results from the requirement that for the infinite at the boundaries potential the wave function must be zero thereψ_q (y) | [ ; boundary;]. =0This boundary condition does not contain the derivatives of the wave function and hence is of the so-called Dirichlet type. The so-called Neumann boundary condition (containing only the derivatives) or Cauchy one (that of mixed type) can arise in a number of physical systems <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> where a hyperbolic DWP tends to zero at the boundaries which are at ±∞. For instance the well known Manning potential <cit.> (which was applied to the ammonia molecule long ago) is of that type. It was shown to be a particular case of the general hyperbolic DWP considered in <cit.>. However for NH_3 the atoms are bound by covalent bonds and as a result (unless the latter are broken) our fictitious quantum particle discussed in the Introduction can not go to infinity. This situation is physically described by a DWP which is infinite at the finite boundaries. The trigonometric DWP considered in the present article is just the one of the required type. As was mentioned in the text the formulas (<ref>) with spheroidal function and (<ref>) with its spectrum of eigenvalues very efficiently provide calculations by means of Mathematica. The formula (<ref>) yields the result for the energy of a q-th level immediately (at a click). In contrast the usage of CHF requires numerical solution of (<ref>) with the help of Maple. Although the latter treats CHF efficiently the search of the required root involves tedious manipulations as compared with the automated call of (<ref>) in Mathematica. The difference in computational time in that event is insignificant and the advantage of the present approach compared with previous one is only a matter of convenience. However this is not the case for calculation of integrals containing the wave functions. The usage of (<ref>) provides considerable advantage in this regard. As was discussed in <cit.> Maple is practically unable to calculate integrals with wave functions containing CHF. To circumvent this difficulty one has to resort to tedious manipulations with the series representation of CHF involving the problem of its proper truncation and subsequent verification of required accuracy. In contrast Mathematica treats the integrals with (<ref>) directly and efficiently. For instance the calculation of a normalization integral in (<ref>) proceeds within several tens of seconds with the help of (<ref>) on a PC of low to average power and this is achieved by direct call of the automated sequenceSpheroidalPS in Mathematica. One can achieve comparable computational time with the help of (<ref>) but this includes a lot of preliminary programming for the series representation of CHF with all mentioned above problems. One can conclude that the usage of spheroidal function is not merely a matter of convenience but provides a qualitatively higher level of computational facilities. Unfortunately it has been gained by present only for the symmetric case of trigonometric DWP. However there is no doubt that CSF will be realized in a mathematical software package sooner or later and then the formulas (<ref>) and (<ref>) will provide the same level of facilities for the asymmetric case.Acknowledgements. The author is grateful to Dr. Yu.F. 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II Infrared and Raman spectra of polyatomic molecules, Van Nostrand, 1945. Gar00 A. Garg, Tunnel splittings for one dimensional potential wells revisited, Am. J. Phys. 68 (2000) 430-437. Son08 D.-Y. Song, Tunneling and energy splitting in an asymmetric double-well potential, Annals of Physics 323 (2008) 2991-2999. Ras12 G. Rastelli, Semiclassical formula for quantum tunneling in asymmetric double-well potentials, Phys.Rev. A86 (2012) 012106. Son15 D.-Y. Song, Localization or tunneling in asymmetric double-well potentials, Annals of Physics 362 (2015) 609-620. | http://arxiv.org/abs/1707.08432v3 | {
"authors": [
"A. E. Sitnitsky"
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"categories": [
"physics.chem-ph",
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"primary_category": "physics.chem-ph",
"published": "20170726133423",
"title": "Analytic description of inversion vibrational mode for ammonia molecule"
} |
Swimming in spacetime: the view from a Fermi observer Eric Poisson December 30, 2023 ===================================================== Far-UV photons (FUV, E < 13.6 eV) from hot massive stars regulate, or at least influence, the heating, ionization, and chemistry of most of the neutral interstellar medium ( and H_2 clouds). Investigating the interaction between FUV radiation and interstellar matter (molecules, atoms and grains) thus plays an important role in astrochemistry.The Orion Bar, an interface region between the Orion A molecular cloud and theregion around the Trapezium cluster, is a textbook example of a strongly illuminated dense PDR (photodissociation region).The Bar is illuminated by aFUV field of a few 10^4 times the mean interstellar radiation field. Because of its proximity and nearly edge-on orientation, it provides a very good template to investigate the chemical content, structure, and dynamics of a stronglyirradiated molecular cloud edge.We have used ALMA to mosaic a small field of the Bar where the critical transition from atomic to molecular gas takes place. These observations provide an unprecedented sharp view of this transition layer (≲ 1” resolution or ≲ 414 AU). The resulting images (so far in the rotational emission of CO, HCO^+, H^13CO^+, SO^+, SO, and reactive ions SH^+ and HOC^+) show the small-scale structure in gas density and temperature, andthe steepabundance gradients.Theimages reveal a pattern of high-density substructures, photo-ablative gas flows and instabilities at the edge of the molecular cloud. These first ALMA images thus show a more complex morphology than the classical clump/interclump static model of a PDR.In order to quantify the chemical content in strongly FUV-irradiated gas, we havealso used the IRAM-30 m telescope to carry out a complete line-survey of the illuminated edge of the Bar in the millimeter domain. Our observations reveal the presence of complex organic molecules (and precursors) that were not expected in such a harsh environment. In particular, we havereported the first detection of the unstable cis conformer of formic acid (HCOOH) in the ISM. The energy barrier to internal rotation (the conversion from trans to cis) is approximately 4827 cm^-1 (≈7000 K).Hence, its detection is surprising. The low inferred trans-to-cis abundance ratio of 2.8±1.0supports a photoswitching mechanism: a given conformer absorbs a FUV stellar photon that radiatively excites the molecule to electronic states above the interconversion barrier. Subsequent fluorescent decay leaves the molecule in a different conformer form. This mechanism, which we have specifically studied with ab initio quantum calculations, was not considered so far in astrochemistry although it can affect the structure of a variety of molecules in PDRs.§ INTRODUCTION: THE ROLE OF FAR-UV STELLARRADIATION ON THE ISM Far-UV photons (FUV: E <13.6 eV) from hot massive O and B stars regulate, or at least greatly influence, the dynamics, heating, ionization, and chemistry of the neutralinterstellar medium (ISM). From a global perspective, the photo-induced processes that take place in FUV-illuminated environments reflect the radiative feedback from such stars. This feedback takes place at very different spatial scales: in star-forming regions of the Milky Way, the ISM of starburst galaxies, and back in cosmic history: see the rapidly increasing number of ALMA detections of the [] 158 μm fine-structure line at(<cit.>. This far-IR emission line ofionized carbon (C^+) was first detected in the 80s (<cit.>) and is often the brightest and most important line coolant of the neutral ISM (hydrogen atoms in neutral form).Investigating the interaction between FUV radiation and interstellar baryonic matter (atoms, molecules and dust grains) ultimately contributes to a better understanding of a varity of FUV-irradiated objects and environments of the galaxy: the diffuse interstellar clouds, the interfaces between regions near young massive stars and their natal clouds, reflection and planetary nebulae around evolved stars, and the externally irradiated protoplanetary disks (proplyds). All these environments in which the underlying physics and chemistry are driven by the presence of FUV photons are generically defined as photodissociation regions (PDRs; see the review paper by <cit.>). FUV photons with wavelengths longer than 911 Å do not ionize H atoms (as in regions) but do dissociate molecules and ionize atoms with ionization potental below 13.6 eV (C, S, Si, Fe, P, etc.). Hence, PDRs are essentialy neutral, with ionization fractions as high as [e^-]/[H]≃10^-4 when most of the electrons are produced by the ionization of carbon atoms, a key process in the ISM (<cit.>). PDRs are very relevant because they host the critical conversion from atomic to molecular ISM (the H^+/H/H_2 and C^+/C/CO transition zones, seeFigure <ref>). This stratification occurs as the column density of gas and dust increases deeper inside clouds and the FUV photon flux is attenuated (see <cit.> and references therein). In addition, PDRs emit most of the IR radiation from the ISM of star-forming galaxies: mid-IR bands from polycyclic aromatic hydrocarbons (PAHs), IR H_2 ro-vibrational lines,[] 158 μm and [] 63 μm fine-structure lines, and warm dust continuum: after absorbing a FUV photon, grains are heated and radiation is reemitted in the mid- and far-IR. At the same time, photoelectrons ejected from grains and PAHs heat the gas. The theoretical study and first thermo-chemical models of dense PDRs started to strongly develop more than 30 years ago. With the above broad definition, PDRs represent a significant fraction of the neutral atomic and molecular material in the Milky Way, perhaps more than 90%: all the gas and dust in the ISM between A_V≈0.1 and 7 mag (<cit.>). At deeper cloud depths, the flux of stellar FUV photons is almost completely attenuated, gas-phase molecules and atoms freeze, and a much slower chemistry and weak ionization take place driven bycosmic-ray particles (see Figure <ref>).In addition to being excellent tracers of the gas physical conditions (density, temperature, FUV field strenght, etc.),PDR observational diagnostics ([] 158 μm, PAH bands,H_2 lines, and far-IR continuum emission) can be used to estimate the star-formation rate in galaxies. More locally, PDR diagnostics can be used to determine the radiative feedback of massive stars on their natal clouds. Understanding how molecular clouds are evaporated by a strong FUV radiation field is important to determine their lifetimes, and how the star-formation process can be quenched in a given region. On the other hand, the dynamical effects induced by a strong stellar FUV field (e.g. radiation pressure), by the winds of massive stars, or by Supernova explosions could trigger, even regulate, the formation of a new generation of low-mass stars (<cit.>). In this contributed paper we summarize our latest results regarding the study of the Orion Bar PDR, an excellent astrochemical laboratory to investigate the interaction of interstellar matter with a strong flux of FUV photons from nearby massive stars. The Orion Bar: Owing to its nearly edge-on orientation, “the Bar” (an interface region between the Orion A molecular cloud and the region around the Trapezium cluster, see Figure <ref>) is a prototypical strongly irradiated PDR with G_0≃2×10^4 (G_0=1.7 is roughly the mean interstellar FUV field in Habing units, equivalent to a few 10^8 photons cm^-2 s^-1 between 13.6 and 6 eV). Its closeness (∼414 pc), high gas temperatures (T_ k≃150-300 K) and thus bright molecular lines, makes it an ideal target for high signal-to-noise spectral-imaging at different wavelengths. In the last years, our team has led several efforts to constrain the molecular content of the Orion Bar by using the Herschel Space Observatory and theIRAM-30 m telescope, and more recently to study the small-scale density and temperature structures with ALMA. From these works we highlight:(i) A line survey of the Bar edge in the 3 to 0.8 mm bands, complemented with ∼2'×2'maps at ∼7” resolutionwith the IRAM-30m telescope (<cit.>,<cit.>, <cit.>). Once thought to be a harsh environment to develop a rich chemistry, our observations show that PDRs host unique chemical features: reactive ions (SH^+ or HOC^+; <cit.>, <cit.>), hydrocarbon ions (C_3H^+; <cit.>;<cit.>), and unstable isomers of organic species (cis-HCOOH, <cit.>) that are not seen, or that are not abundant, in regions shielded from FUV radiation.(ii) Herschel/HIFIlarge-scale (7.5'×11.5') velocity-resolved maps of Orion A (including the Bar) in FUV-radiation diagnostics such as [] 158 μm (<cit.>).(iii) Pilot ALMA mosaics of a small field (∼50”×50”) of the Bar in several molecular line diagnostics at ∼1” resolution (Figures <ref> and <ref>) (<cit.>, <cit.>).§ MORPHOLOGY OF THE PDR: SMALL-SCALE STRUCTURES AND FLOWSSurprisingly, our pilot ALMA images show that there is no appreciable offset between the peak of the H_2 vibrational emission at 2.2 μm (delineatingthe H_2 dissociation front)and the edge of the observed CO and HCO^+ rotational emission (Figures <ref> and <ref>). This implies that the H/H_2 and C^+/C/CO transition zones are very close, much closer than stationary PDR model predictions. In addition, our ALMA mosaics reveal a fragmented ridge of high-density substructures, photoablative gas flows and instabilities at the irradiated molecular cloud surface (Figure <ref>; see <cit.> for details). These results suggest that the cloud edge has been compressed by a high-pressure wave that is moving into the molecular cloud, demonstrating that dynamical effects aremore important than previously thought.Whether the newly detected small-scale structures could be the seed of future star-forming clumps (e.g., by merging into more massive collapsing clumps) is uncertain and is subject of study. Gravitational collapse is not yet apparent from their density distribution nor from their low masses. However, our images resolve theemission from proplyd 203-506 (Figure <ref>a and <cit.>). Interestingly, its location in the atomic PDR (A_V=0 to 1 mag,thus in-between the ionization and dissociation fronts) suggests that this proplyd (a low-mass protostar and protoplanetary disk) could have emerged from the molecular cloud after surviving the passage of the dissociation front. Even on its early cycles (the antena array was not yet completed at the time of our observations) ALMA clearly goes a step beyond in our understanding of the closest, and most iconic, massive star forming region. The results obtained from these pilot ALMA mosaics were unexpected and challenge many years of PDR modelling. However, they were obtained by imaging a small fraction of theBar (≈10-20% of its size; see Figure <ref>). In order to support and better understand our findings, it is mandatory to obtain ALMA images of the entire Bar, and of course, to observe other (more distant) massive star forming regions, i.e., at different stages of evolution and having different stellar populations, thus different number of ionizing massive stars. § THE SHORT, BUT INTERESTING, LIFE OF REACTIVE IONS Reactiveions (CH^+, SH^+, CO^+, HOC^+, ...)are transient species for which the timescale of reactive collisions with H_2, H, or e^- (leading to a chemical reaction, and thus molecule destruction) is comparable to, or shorter than, that of inelastic collisions (<cit.>). The formation ofreactive ions such as SH^+, for example, depends on the availability of S^+ ions (i.e., on the availabilityof FUV photons that can ionize sulphur atoms) and on the presence of excited H_2 (eitheror hot and thermally excited).These two ingredients allow overcoming the high endothermicity (and sometimes energy barrier) of some of the key initiating chemical reactions(<cit.> for a review). The reaction , in particular, is endothermic byif v=0, and only becomes exothermic when v≥2 (<cit.>). Despite their short lifetimes, from a few years in diffuse gas to only a few hours (!) indense PDR gas, reactive ions can be detected and probe energetic processes in FUV-irradiated warm molecular gas.We have used the ALMA-ACA array to obtain high angular resolution (5”×3”) images of some reactive ions toward theBar (<cit.>). The observed SH^+ and HOC^+ emission is restricted to a narrow layer of 2”- to 10”-widthdepending on the assumed PDR geometry) that follows the vibrationally excited H_2^* emission (Figure <ref>). Both ions efficiently form very close to thetransition zone, at a depth of A_ V≲1 mag into the neutral cloud, where abundant C^+, S^+, and H_2^* coexist. The observedions have low rotational temperatures() and narrow line-widths, a factor of ≃2 narrower that those of the lighterreactive ion CH^+ (<cit.>). In <cit.> we show that thisis consistent with the higherreactivity[For the physicalconditions at the edge of the Bar, the destruction time scales (τ_ D) of CH^+, HOC^+, and SH^+ are ≃5 h, ≃25 h and ≃50 h respectively. Comparing with the timescales for collisional and radiative excitation showsthatCH^+ molecules are excited by radiation many times during their short lifetime, but not by collisions.Hence, CH^+ can remain rotationally warm (owing to formation pumping; <cit.>, <cit.>) while it emits.On the other hand, comparing with the timescales for collisional and radiative excitation shows that HOC^+ molecules (and likely the other heavier ions as well) are excited by collisions several times during their lifetime. Inelastic collisions thus can drive their rotational populations to lower T_ rot relatively fast(T_ rot(HOC^+) ≪ T_ rot(CH^+)≈150 K; <cit.>).]and faster radiative pumping rates ofCH^+compared to the heavier ions, which are driven relatively morequickly toward smaller velocity dispersion by elastic collisions and toward lower T_ rot by inelastic collisions (see also <cit.>).Regardless of the excitation details, the ALMA-ACA images show that SH^+ and HOC^+ clearly trace the most exposed layers of the FUV-irradiated molecular cloud surface. § YES, COMS ARE SEEN IN STRONGLY FUV-IRRADIATED GAS Observations towardsUV-shielded cold cores (e.g. TMC 1, L1689B, or B1-b) have revealed molecules once considered to be present only in hot cores and corinos(<cit.>). In these environments, complex organic molecules (COMs) are thought to form on the surface of grains and to be releasedthrough non-thermal desorption processes, chemical desorption, direct desorption by cosmic rays impacts, or by secondary FUV-photon induced processes. Until recently, observational studies ofenvironments illuminatedby stellar FUV photons were more scarce.Guzmán et al. (<cit.>) presented theunexpected detection of trans-HCOOH, CH_2CO, CH_3CN, CH_3OH, CH_3CHO, and CH_3CCH at the edge of the Horsehead (a lowPDR with ). They found enhanced abundances compared to a nearby cold and dense core shieled from external FUV radiation.Guzmán et al. (<cit.>) proposed that owing to the cold dust temperatures in this PDR (T_ d≲30 K, i.e., grains should be coated by ices), ice-mantle photodesorption processes dominate the formation of gaseous COMs. In lower density translucent clouds (G_0≈1), only H_2CO has been unambiguously detected (<cit.>). These observations might suggest that the presence of COMs diminishesas G_0/n_ H increases, and hence, that COMs might not be present in strongly FUV-irradiated gas.Despite being a much harsher environment, wedetected more than 250 lines from COMs and related precursors toward the irradiated edge of the Orion Bar: H_2CO, CH_3OH, HCO, H_2CCO, CH_3CHO, H_2CS, HCOOH, CH_3CN, CH_2NH, HNCO, H_2^13CO, and HC_3N (in decreasing order of abundance, <cit.>). Taking into account the elevated gas (≈200 K) and dust (≈60 K) temperatures, we suggested the following scenarios for the formation of COMs in the Orion Bar: (i) hot gas-phase reactions not included in current chemical models; (ii) bare (little ice) warm grain surface chemistry; or (iii) the PDR dynamics is such that COMs previously formed in cold icy grains deeper inside the molecular cloud desorb and advect into the PDR(<cit.>). §.§ Trans-cis photoisomerization of HCOOH Conformational isomerism refers to isomers (molecules with the same formula but different chemical structure) having the same chemical bonds but different geometrical orientations around a single bond. Such isomers are called conformers. An energy barrier often limits the isomerization. This barrier can be overcome by light. Photoisomerization (or photoswitching) has been studied in ice IR-irradiation experiments, in biological processes, and, for large polyatomic molecules, in gas-phase experiments. HCOOH is the simplest organic acid and has two conformers (trans and cis) depending on the orientation of the hydrogen single bond. The most stable trans conformer was the first acid detected in the ISM. Gaseous trans-HCOOH shows moderate abundances towards hot cores and hot corinos, in cold dark clouds, and in cometary coma. Solid HCOOH is present in interstellar ices and in chondritic meteorites. The energy barrier to internal conversion from trans- to cis-HCOOH is much higher than the thermal energy available in molecularclouds[The ground-vibrational state of cis-HCOOH ishigher in energy than that of the trans conformer. The energy barrier to internal rotation (the conversion from trans to cis) isapproximately 4827 cm^-1 (≈7000 K in temperature units).]. Thus, only the most stable conformer (trans) is expected to exist in detectable amounts. In Cuadrado et al. (<cit.>) we reported the first interstellar detection of cis-HCOOH. Its presence in the Orion Bar (but not in the hot core of Orion nor in the cold core of Barnard 1-b) with avery lowabundance ratio of(>100 and >60 in the Orion hot core and inrespectively), supports a photoswitching mechanism: a given conformer absorbs a FUV photon with energies around 5 eV (precisely in the λ≃2300-2800 Å range, higher energy photons dissociate the molecule) that radiatively excites the molecule to electronic states above the interconversion barrier. Subsequent fluorescent decay leaves the molecule in a different conformer form. This mechanism, which we specifically studied with ab initio quantum calculations (to compute the ∼5 eV-photon absorption cross–sections and the trans-to-cis and cis-to-trans photoswitching probabilities) was not considered in the ISM before, but likely induces structural changes of a variety of molecules in PDRs. In Cuadrado et al. (<cit.>) we used the Meudon PDR code to estimate the flux of FUV photons at different positons of the Bar. The well-known “2175 Å bump” of the dust extinction curve (produced by the absorption of FUV photons by PAHs and small carbonaceous grains)reduces the number of HCOOH dissociating photons relative to those producing fluorescence ( right). We determine that at a cloud depth ofA_V≈ 2-3 mag (for the Orion Bar conditions), the cis conformer should be detectable with a trans-to-cis abundance ratio of 3.5-4.1. These ratios are remarkably close to the observed values.Therefore, both the detection of , and the lowabundance ratio in the Bar (much lower than in environments shielded from FUV radiation) would agree with the photoswitching scenario. To summarize,the presence of transient reactive ions such as CH^+, SH^+ or HOC^+ is an unambiguous indication of FUV-irradiated warm molecular gas. The presence of COMs, however, is more widespread, and nowincludes harsh environments such asstrongly FUV-irradiated gas (where conformers can photoisomerize). COM formation reflects the complicated interplay between gas and grainchemistry. Hence, more theory and laboratory experiments of the possible formation routes are needed.ALMA and soon JWST (formolecules, H_2 lines, PAHs and dust features) will open the study of PDRs to sub-arcsecond resolution spectral-imaging. Exciting times ahead for PDR research.[Black (1998)]Black_1998 Black, J. H. 1998, Faraday Discussions, 109, 257 [Capak et al. (2015)]Capak_2015 Capak, P. L., Carilli, C., Jones, G., et al. 2015, Nature, 522, 455 [Cernicharo et al.(2012)]Cerni_2012 Cernicharo, J., Marcelino, N., Roueff, E., et al. 2012, ApJL, 759, L43 [Champion et al. (2017)]Champion_2017Champion, J., Berné, O., Vicente, S., et al. 2017, accepted in A&A, arXiv:1702.00251 [Cuadrado et al. (2015)]Cuadrado_2015 Cuadrado, S., Goicoechea, J. R., Pilleri, P., et al. 2015, A&A, 575, A82 [Cuadrado et al. (2016)]Cuadrado_2016 Cuadrado, S., Goicoechea, J. R., Roncero, O., et al. 2016, A&A, 596, L1 [Cuadrado et al. (2017)]Cuadrado_2017 Cuadrado, S., Goicoechea, J. R., Cernicharo, J., et al. 2017, A&A,603, A124[Dalgarno & McCray (1972)]Dalgarno_1972 Dalgarno, A., & McCray, R. A. 1972, ARAA, 10, 375 [Fuente et al.(2003)]Fuente_2003 Fuente, A. et al.2003, A&A, 406, 899[Gerin et al. (2016)]Gerin_2016 Gerin, M., Neufeld, D. A., & Goicoechea, J. R. 2016, ARAA, 54, 181 [Godard & Cernicharo (2013)]Godard_2013 Godard, B., & Cernicharo, J. 2013, A&A, 550, A8 [Goicoechea & Le Bourlot (2007)]Goico_2007 Goicoechea, J. R., & Le Bourlot, J. 2007, A&A, 467, 1 [Goicoechea et al. (2015)]Goico_2015 Goicoechea, J. R., Teyssier, D., Etxaluze, M., et al. 2015, ApJ, 812, 75 [Goicoechea et al. (2016)]Goico_2016 Goicoechea, J. R., Pety, J., Cuadrado, S., et al. 2016, Nature, 537, 207 [Goicoechea et al. (2017)]Goico_2017 Goicoechea, J. R., Cuadrado, S., Pety, J., et al. 2017, A&A, 601, L9 [Guzmán et al. 2014]Vivi_2014 Guzmán, V. V., Pety, J., Gratier, P., et al. 2014, Faraday Discussions, 168, 103[Hollenbach & Tielens (1999)]Hollenbach_1999 Hollenbach, D. J., & Tielens, A. G. G. M. 1999, Reviews of Modern Physics, 71, 173 [Liszt et al. (2006)]Liszt_2006 Liszt, H. S., Lucas, R., & Pety, J. 2006, A&A, 448, 253 [Nagy et al. (2013)]Nagy_2013 Nagy, Z., Van der Tak, F. F. S., Ossenkopf, V., et al. 2013,A&A, 550, A96[Pety et al. (2012)]Pety_2012 Pety, J., Gratier, P., Guzmán, V., et al. 2012, A&A, 548, A68 [Russell et al. (1980)]Russell_1980 Russell, R. W., Melnick, G., Gull, G. E., & Harwit, M. 1980, ApJL, 240, L99[Walmsley et al. (2000)]Walmsley_2000 Walmsley, C. M., Natta, A., Oliva, E., & Testi, L. 2000, A&A, 364, 301[Zanchet et al. (2013)]Zanchet_2013 Zanchet, A., Agúndez, M., Herrero, V. J., Aguado, A., & Roncero, O. 2013, AJ, 146, 125 | http://arxiv.org/abs/1707.08869v1 | {
"authors": [
"Javier R. Goicoechea",
"S. Cuadrado",
"J. Pety",
"A. Aguado",
"J. H. Black",
"E. Bron",
"J. Cernicharo",
"E. Chapillon",
"A. Fuente",
"M. Gerin",
"C. Joblin"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170727140412",
"title": "The ALMA view of UV irradiated cloud edges: unexpected structures and processes"
} |
Zero-temperature dynamics in the dilute Curie–Weiss model]Zero-temperature dynamics in the dilute Curie–Weiss model R. GheissariCourant InstituteNew York University251 Mercer StreetNew York, NY 10012, USA. [email protected] C. M. NewmanCourant InstituteNew York University251 Mercer StreetNew York, NY 10012, USA. [email protected] D. L. SteinCourant InstituteNew York University251 Mercer StreetNew York, NY 10012, USA. [email protected] We consider the Ising model on a dense Erdős–Rényi random graph, 𝒢(N,p), with p>0 fixed—equivalently, a disordered Curie–Weiss Ising model with (p) couplings—at zero temperature. The disorder may induce local energy minima in addition to the two uniform ground states.In this paper we prove that, starting from a typical initial configuration, the zero-temperature dynamics avoids all such local minima and absorbs into a predetermined one of the two uniform ground states. We relate this to the local MINCUT problem on dense random graphs; namely with high probability, the greedy search for a local MINCUT of 𝒢(N,p) with p>0 fixed, started from a uniform random partition, fails to find a non-trivial cut. In contrast, in the disordered Curie–Weiss model with heavy-tailed couplings, we demonstrate that zero-temperature dynamics has positive probability of absorbing in a random local minimum different from the two homogenous ground states.[ Daniel L. Stein=================== § INTRODUCTIONThe disordered Curie–Weiss (CW) model is a mean-field random ferromagnet defined as follows: consider the complete graph on N vertices and for every edge (i,j), assign a random coupling value J_ij=J_ji i.i.d. according to some non-negative distribution μ (i.e., (μ) ∈ [0,∞)). Define the Hamiltonian H(σ) for σ∈{± 1}^N by H(σ)= - 1/N∑ _1≤ i<j≤ N J_ijσ_i σ_j .If μ were instead symmetric on (-∞,∞), this would be proportional to the Hamiltonian of the Sherrington–Kirkpatrick spin glass; in our setup, where μ is non-negative, it is a mean-field analogue of a random ferromagnet. Specifically, when μ is (p), this corresponds to the Ising model on a dense Erdős–Rényi random graph, referred to as the dilute Curie–Weiss model. In the latter case, the thermodynamics of the model behaves similarly to that of the homogenous Curie–Weiss model, but the coupling disorder induces a more complex energy landscape, whose influence may only manifest itself in the zero-temperature dynamics. The goal of the present paper is to more precisely understand these effects (see Theorem <ref>) and relate them to the random constraint satisfaction problem of finding local MINCUTs of a random graph (see <ref>).The zero-temperature limit of the Glauber dynamics <cit.> is called the zero-temperature dynamics and in discrete time, is the Markov chain (X_t)_t∈ℕ with transition matrix given by the following: consider a configuration σ∈{± 1}^N; for every i, if σ^(i) is the configuration with σ^(i)_j=σ_j for all j≠ i and σ^(i)_i=-σ_i, thenP(σ,σ^(i))= 1/N [ 1{H(σ^(i))<H(σ)}+1/2 1{H(σ^(i))=H(σ)} ] .The zero-temperature dynamics is a random walk on the hypercube {± 1}^N, that only assigns positive transition rates to moves that do not increase the energy. If the zero-temperature dynamics stops (i.e., is absorbed) in some particular state, the only possible such absorbing states are local minima and uniform ground states of H. For any non-negative (ferromagnetic) coupling distribution μ, the uniform ground states of the disordered CW model are σ=(1,...,1) and σ=(-1,...,-1). We call a state σ∈{± 1}^N a local minimum of H if it is not a uniform ground state ± (1,...,1) and is such that for any σ'∈{±1}^N with Hamming distance d(σ,σ')=1, it satisfies H(σ')≥ H(σ). Notice that if μ has atoms, there could be connected sets of local minima with the same energy, amongst which the zero-temperature dynamics jumps for all sufficiently large time, in which case we call the set an absorbing set of configurations.Henceforth, we mostly restrict our attention to the dilute Curie–Weiss model, whose analysis ends up being more straightforward than the general J_ij≥ 0 case, by letting J_ij∼(p) for p∈ (0,1)—the case p=1 corresponds to the classical Ising Curie–Weiss model. The Ising model on sparse random graphs (p=p_N≪log N/N) is an extensively studied model with rich relations to random optimization and the theory of spin glasses (e.g. <cit.> and for a more extensive overview, <cit.>); in the context of zero-temperature dynamics, when the random graph has disjoint components, there are many non-trivial absorbing states, and in particular, the zero-temperature dynamics will have positive probability of absorbing into a local minimum (see e.g., <cit.> for a rigorous analysis of zero-temperature dynamics for the Ising model on sparse random graphs). On dense random graphs, the thermodynamics of the Ising model is essentially the same as that of the classical (homogenous) Curie–Weiss model. However, the zero-temperature dynamics is particularly sensitive to small changes in the energy landscape as it can absorb in any local minimum it encounters.Let ℙ_σ(0) be the product measure over initial configurations, ℙ_𝒥 be the product measure over couplings, and ℙ_ω be the distribution over the evolution of the Markov chain, with corresponding expectations, 𝔼_σ(0), 𝔼_𝒥, and 𝔼_ω; we will also sometimes write ℙ_𝒥,ω=ℙ_𝒥⊗ℙ_ω etc. In the present paper we show that while we believe there exist local minima in the energy landscape of the disordered CW model with general ferromagnetic μ (so that the disorder does indeed make the energy landscape nontrivial—see Question <ref>), in the case of the dilute CW model, the zero-temperature dynamics avoids them with high probability, i.e., with probability going to 1 as N→∞ (w.h.p.). For ϵ>0 sufficiently small, the zero-temperature dynamics (X_t)_t≥ 0 of the Ising model on the Erdős–Rényi random graph 𝒢(N,p) with p>0 fixed satisfies the following: for every initial configuration X_0 with magnetization ∑_i X_0(i) ≥ N^1/2-ϵ, lim_N→∞ℙ_𝒥,ω (lim_t→∞ X_t=(1,...,1) )=1 .Note that by heuristic considerations, one would suspect the same should hold for any non-negative coupling distribution μ that has, for instance, all exponential moments finite. For such distributions, it seems the main obstacle in adapting our argument to give the same result is obtaining some conditional concentration for sums of couplings via an analogue of Proposition <ref>, though we often make use of the convenience of dealing with bounded random variables.In contrast, if {J_ij}_ij are i.i.d. heavy-tailed random variables we prove that with probability bounded away from 0, the dynamics gets stuck in the exponentially many non-trivial local minima. Results for heavy-tailed couplings are found in <ref>, Theorem <ref>.In order to study the zero-temperature dynamics from a physical point of view, a zero-temperature dynamical order parameter was introduced <cit.> and has been examined numerically in a number of models with and without disorder <cit.>. Let σ_i(t) be the i'th spin value of X_t, i.e. X_t(i). The quantityq_D(N)= 𝔼_σ(0),𝒥 [(𝔼_ω[σ_1(∞)])^2]is designed to capture how much the absorbing state depends on the initial state and how much it depends on the realized evolution of the dynamics. In the above, and throughout the paper, σ_i(∞) denotes lim_t→∞σ_i(t) if it exists and similarly with X_∞(i). Another way to view the dynamical order parameter q_D(N) is to fix a coupling configuration, and consider a replicated dynamics wherein two replicas σ(t),σ'(t) undergo zero-temperature dynamics independently (ω,ω') from the same initial configuration; one can then ask about the average (under ℙ_ω,ω') evolution of their overlap. The dynamical order parameter is the expectation (under ℙ_σ(0),𝒥) of this average overlap:𝔼_σ(0),𝒥[𝔼_ω,ω'[N^-1∑ _j=1^N σ_j(∞) σ_j'(∞)]] = q_D(N) .In the dilute CW model, the dynamical order parameter (<ref>) has lim_N→∞ q_D(N)=1 .In fact, we have the stronger result that,lim_N→∞𝔼_σ(0)[ (𝔼_𝒥,ω(σ_1(∞)))^2]=1 . We briefly mention that in the above literature regarding this dynamical order parameter, different graphs have also been considered. In <cit.>, the zero-temperature dynamics of the disordered Curie–Weiss model and random ferromagnet on ℤ^d were studied numerically and heuristically (with light-tailed coupling distributions). There, it was predicted that the zero-temperature dynamics of the disordered Ising model on (ℤ/N ℤ)^d starting from a single state ends up randomly in one of many almost-orthogonal local minima in whose basins of attraction it lies. The numerics suggest that on d-dimensional torii, lim_d→∞lim_N→∞ q_D (N)=0; combined with Corollary <ref>, this suggests the existence of a singularity in the d→∞ behavior of q_D(N). We also note that in the physics literature there has been recent interest in dynamics of the Ising model on networks at low and zero temperatures (e.g., <cit.> study, at the level of physics, absorption and persistence in a densely-connected small world network).Over the last several years, significant progress has been made in understanding random optimization problems like extremal cuts on random graphs and random instances of constraint satisfaction problems (see e.g., <cit.>) via both heuristic and rigorous connections to spin glasses and other models with quenched disorder. We now discuss a different perspective, related to such random optimization problems, on the fundamental questions underlying Theorem <ref>, which may be of independent interest. §.§ The local MINCUT problem Consider a dense Erdős–Rényi random graph G∼(N,p) for p>0 fixed; for any subset A⊂{1,...,N}, define _G(A) as the number of edges between A and A^c. A local MINCUT is a partition (A,A^c) of {1,...,N} such that for every A' which consists of the addition or removal of one vertex to or from A (in which case we say the Hamming distance d(A,A')=1), _G(A') ≥_G (A). A nontrivial local MINCUT is one in which both A and A^c are nonempty. Consider G∼𝒢(N,p) with p>0 fixed. Is it the case that lim_N→∞ℙ(∃ A:A≠∅, A^c ≠∅, _G(A))=1?Note that since the Erdős–Rényi graph is dense, all vertices are connected and have high degrees of dependence; if we were considering a graph with multiple connected components for instance it is obvious how to construct nontrivial local MINCUTs. The requirement that a set A be a local MINCUT is equivalent to demanding that every v∈ A have more edges to vertices in A than in A^c, and similarly for vertices in A^c.Viewed from this perspective, Theorem <ref> can be restated in terms of a greedy search for a local MINCUT defined as follows: start from a uniformly randomly chosen partition (A,A^c) and at every iteration, select a vertex uniformly at random and move it either to A or A^c depending on which move has the lower _G value (if the cut-value is unchanged flip a coin to determine whether to move it). With probability going to 1 as N→∞, the greedy search for a local MINCUT of G∼(N,p) with p>0 fixed, started from a uniformly random partition of {1,...,N} terminates in the trivial partition (∅, {1,...,N}).This suggests the interesting situation where there exist nontrivial local MINCUTs while, with high probability, they are not found by a greedy search algorithm. In fact, there is some numerical evidence in this direction to appear in <cit.>. We call such metastable states, invisible local minima as they typically do not affect the natural dynamics, even at zero-temperature.Of course, as with the dilute CW model, the local MINCUT problem can be presented in greater generality by assigning edges of the complete graph i.i.d. random weights w_ij and asking the analogous questions about _w (A) = ∑_i∈ A,j∉ A w_ij. In the case where w_ij are symmetric, this corresponds to finding local energy minima of the canonical Sherrington–Kirkpatrick spin glass.This is an extensively studied question, both at the physics level <cit.> and more rigorously recently in the related problem of multiple peaks <cit.>: there the energy landscape is expected to be complex with exponentially many local minima in the system size— <cit.> developed a complete understanding of the critical points and complexity of the energy landscape in the case where the state space is relaxed to the sphere in dimension N. In the above cases, the rugged energy landscape arises due to frustration, a phenomenon that does not exist in the ferromagnetic setup. We also note that in a similar setup to ours, the algorithmic complexity of the local MAXCUT problem has been studied (most recently in <cit.>), though that problem again has a very different flavor due to the absence of dominant trivial ground states.If we instead restrict ourselves to ferromagnetic disorder (w_ij are a.s. non-negative), we expect that Corollary <ref> and the techniques of this paper extend to the general case when w_ij have light (e.g., exponential or Gaussian) tails; there is again some numerical evidence in this direction <cit.>. In contrast, our results in <ref> on heavy-tailed disorder imply an affirmative answer to Question <ref> while showing that there, with strictly positive probability, the greedy search terminates in a non-trivial local MINCUT.§.§ NotationWe introduce some notation that we will use throughout the paper. We say two sequences f_N and g_N are such thatf_N≲ g_N if there exists C>0 such that f_N ≤ Cg_N for all N and we say f_N ≍ g_N if f_N≲ g_N≲ f_N. Finally, we write f_N = O(g_N) if f_N≲ g_N and f_N = o(g_N) if f_N ≤ c g_N for large enough N for every c>0. For readability, we will, throughout the paper, omit floors and ceilings, though all our variables will be integer-valued. We will also assume N is sufficiently large.The discrete-time zero-temperature dynamics chain is alternately denoted by (X_t) and σ(t)= (σ_1(t),...,σ_N(t)) where t is always understood to be integer—clearly, the results of Theorem <ref> and Corollary <ref> would also hold for the analogously defined continuous-time zero-temperature dynamics. The magnetization at time t is given byM_t =∑ _i=1^N σ_i(t) .The effective field on site i at time t is given bym_i(t)= ∑ _j≠ i, j=1,...,N J_ijσ_j(t) .It will be notationally useful to define the related m̅_i(t)=(m_i(t)) so that m̅_i(t)∈{± 1,0} where m̅_i(t)=0 if m_i(t)=0.Then we let {𝒮_i(t)}_i=1^∞ be the set of satisfaction random variablesS_i(t)=m̅_i(t)σ_i(t) ,so that when the dynamics chooses a site i to update, σ_i(t)=σ_i(t-1) with probability 1 if S_i(t)=S_i(t-1)=1, probability 1/2 if S_i(t)=0, and probability 0 if S_i(t)=-1.We will use the probability measure ℙ to denote the product measure ℙ_𝒥⊗ℙ_ω, since we will always be fixing X_0=σ(0) and sometimes averging over 𝒥,ω at once. §.§ Proof approachHere we give an overview of our approach to proving Theorem <ref>. To avoid the difficulties present in analyzing systems with quenched disorder, particularly with non-Gaussian disorder, our analysis of the zero-temperature dynamics reveals only partial information about the couplings as the dynamics proceeds. This gradual “revealing scheme" may be of independent interest in analyzing the short-time (t≪ N) dynamics of other systems with quenched disorder. Here it allows us to bound the drift of the magnetization chain (M_t)_t∈ℕ conditioned on this partial information from below, and compare (M_t)_t∈ℕ to a random walk with positive drift. More precisely,* In 2.1, we define a revealing scheme to see the evolution of the zero-temperature Markov chain as measurable w.r.t. the σ-algebra ℱ_t generated by the sequence (Y_t)_t of updated sites, their satisfactions at update time (S_Y_t(t))_t, and the couplings ({J_Y_t,Y_k}_k=t+1,...,T)_t for T=N^1/2+δ. To gradually reveal this information, we first fix the update sequence {Y_1,...,Y_T} then sequentially jointly reveal the satisfaction and aforementioned couplings of the next site to update. * In 2.2, we compare the joint distribution of ({J_Y_t,Y_k}_k=t+1,...,T)_t given S_Y_t(t) and _t-1 to a product measure. In particular, for all short times t=O( N^1/2+δ) we show that the conditional joint law of the couplings revealed at time t dominates i.i.d. (p-O(N^1/2+2δ)) and is dominated by i.i.d. (p+O(N^1/2+2δ)). * In 2.3, we show that for all times t≤ N^1/2+δ, as long as M_t ≥ N^1/2-δ holds, the chain (M_t)_t≥ 0 has a positive drift of at least cN^-δ/2, so that, (M_t)_t≥ 0 typically stochastically dominates a random walk with drift of c N^-δ/2.* By this comparison, at time T=N^1/2+δ, M_T ≥ N^1/2+δ/2 and all sites that have not yet been updated have a positive effective field. This can then be boosted in 2.4 to show that by time T'=N^2/3 all sites have a positive field and the dynamics will thereafter quickly absorb into the all-plus ground state. § THE DILUTE CURIE–WEISS MODEL§.§ Random mapping representationDenote by Y_k∈{1,...,N} the site chosen at time step k to be updated so that for every k∈ℕ, (Y_k)=({1,...,N}). Then consider the sequence of update sites and their satisfactions, (Y_k, S_Y_k(k))_k.In the dilute Curie–Weiss model, μ is atomic and there may be zero-energy flips with positive probability, so (X_t)_t≥ 0 is not measurable with respect to the sigma-algebra generated only by X_0 and (Y_k,S_Y_k(k))_k≤ t. Thus define a sequence of i.i.d. random variables (B_k)_k≥ 0 with B_1∼ 2(1/2)-1 which will determine the spin at site Y_k in the case that m_Y_k(k)=0. Then the history of the chain (X_k)_k≤ t is fully determined by X_0=σ(0) and the sequence (Y_k,𝒮_Y_k(k),B_k)_k≤ t. The grand coupling is the coupling of two independent dynamics with different X_0 such that both dynamical realizations use the same random variable sequence (Y_k)_k≥ 1 and (B_k)_k≥ 0 as well as the same couplings {J_ij}. The grand coupling has the added feature that it preserves monotonicity, so that if X_0≽ X_0' then X_k≽ X_k' for all t≥ 0. Moreover, by permutation invariance of the measure ℙ_,ω, we can begin by permuting all initial configurations so that if M_0 ≥ M_0', X_0 ≽ X_0' and identify initial configurations only with their magnetization, then apply the grand coupling of ,ω.§.§ Preliminary estimates We first estimate the probability of the magnetization of X_0 being atypical. The following is a consequence of e.g., Berry–Esseen theorem. For every ϵ>0, we haveℙ_σ(0) (X_0:|M_0|≥ N^1/2-ϵ)=1-O(N^-ϵ) .By the grand coupling and permutation symmetry of the model, it suffices to prove Theorem <ref> for a fixed X_0 with M_0=N^1/2-ϵ; for every coupling and dynamical realization in which the chain with that initial configuration absorbs in X_∞ =(1,...,1), via the grand coupling, so does every chain with more positive initial configuration. Moreover, in order to simplify our considerations, by monotonicity we may assume that for all k≥ 1, we always have B_k=-1; via the grand coupling of the dynamics, it suffices to show that this chain absorbs into the all-plus configuration w.h.p. to prove Theorem <ref>. Abusing notation, (X_k)_k will henceforth refer to this new chain with B_k=-1 for all k. In conjunction with this change, whenever m_Y_k(k)=0, we set S_Y_k(k)=-σ_Y_k(k-1). Fix T=N^1/2 +3ϵ and for every t∈ℕ, let {ℱ_t}_t be the filtration of σ-algebras generated by (Y_1,...,Y_T∨ t), (S_Y_k(k))_k≤ t,(J_Y_k Y_l)_k ≤ t, l≤ TThe chain X_t is measurable w.r.t. the σ-algebra generated by (Y_k, S_Y_k(k))_k=1,...,t and X_0 (and therefore measurable w.r.t. ℱ_t and X_0). It will be crucial to understand the conditional distribution of J_ij given ℱ_t when i∈{Y_k}_k≤ t but j∉{Y_k}_k≤ t, as well as the joint law of such {J_ij}_j. (This is the main reason we restrict ourselves to Bernoulli J_ij, where the distribution is determined by the mean, instead of general light-tailed random variables like half-normal distributed random variables. To extend Theorem <ref> to that setting, the main technical hurdle is obtaining appropriate analogues to Proposition <ref> for exponential moments in order to obtain conditional concentration of sums of couplings, namely (<ref>).)We will need the following notation: for any sequence of order updates, denote by ℛ_t the set of vertices whose clocks have rung more than once before time t, i.e., ℛ_t= {i:∑_k≤ t 1{Y_k=i}>1} . Let t ≤ T=N^1/2+3ϵ, suppose that j=Y_t, j∉ℛ_t, that ℓ∉{Y_k}_k=1^t, and suppose that |M_0| ≤ N^1/2+3ϵ. Then we have that, |𝔼_[J_ℓ jℱ_t-1, S_j(t), {J_ja}_a∈{Y_k}_t+1^T - ℛ_T-{ℓ}] -p| = O(N^-1/2+4ϵ) .First fix the update sequence {Y_k}_k=1^T and fix any such t and ℓ.Let 𝔼̃ [ · ] = 𝔼_[ ·|ℱ_t-1, {J_ja}_a∈{Y_k}_k=t+1^T-ℛ_T-{ℓ}] .Now we can expand 𝔼̃[m_j(t)| S_j(t)]-𝔼_ [m_j(0)]= σ_ℓ(0) 𝔼̃[J_ℓ j-p | S_j(t)]+ ∑_i ∈{Y_k}_k=1^T-{ℓ}[σ_i(t)𝔼̃[J_i j| S_j(t)] -pσ_i(0)]+ ∑_i∉{Y_k}_k=1^T∪{ℓ}σ_i(0)𝔼̃[J_i j-p | S_j(t)] + O(|_T|) .Since |ℛ_T| ≤ T, we can replace O(_T) by O(T); moreover the second sum consists of at most T terms and is thus bounded in absolute value by 2T. Now suppose without loss of generality that m_j(t) > 0 and σ_ℓ (0)=+1—the same argument carries through in the other cases. We claim, first of all that for every m∉{Y_k}_k=1^T with σ_m(t)=-1,𝔼̃ [ J_m j| m_j(t)>0] ≤𝔼̃ [ J_m j]=p .To see this, we write by Bayes' Theorem, the left hand side above as ℙ̃(m_j(t)>0 | J_m j=1) ℙ̃ (J_m j = 1) ℙ̃ (m_j(t)>0)^-1 .Observe that since ℙ̃ doesn't condition on m_j nor on m_m at any time, J_m j is independent of the σ-algebra conditioned on under ℙ̃, and thus its conditional distribution is (p) so that 𝔼̃[J_m j]=p. Moreover, expanding out m_j(t)>0, we see that under ℙ̃, all the summands except σ_m(0)J_mj are conditionally independent of J_mj, so that because σ_m(t)=-1, conditioning also on J_mj=1 only decreases m_j(t), implying (<ref>). Analogously, if σ_m(0)=1, and m∉{Y_k}_k=1^T, then 𝔼̃[J_mj| m_j(t)>0] ≥ p. Therefore, every summand in the third term in the right-hand side of (<ref>) is nonnegative.As a result, we have0 ≤σ_ℓ(0) 𝔼̃[J_ℓ j -p| S_j(t)]+ ∑_i∉{Y_k}_k=1^Tσ_i(0) 𝔼̃[J_ij - p| S_j(t)] ≤ |𝔼̃ [m_j(t) | S_j(t)]|+ |𝔼[m_j(0)]|+O(T) . We now upper bound the right-hand side of (<ref>). Writing each term out, we see that under ℙ̃, except for at most t=O(N^1/2+3ϵ) summands, m_j(t) is distributed as a difference of two binomial random variables with mean that is M_0 ± O(N^1/2+3ϵ) and variance O(N). (Under the measure ℙ_, m_j(0) is of course just a difference of two binomials.) In that case, standard lower and upper tail estimates, via Chernoff bounds for binomial random variables, along with the fact that |M_0|≤ N^1/2 +3ϵ imply that 𝔼̃ [m_j(t) | S_j(t)]=O(N^1/2 +4ϵ) , 𝔼_[m_j(0)]=O(N^1/2 +4ϵ) . Returning to (<ref>), by (<ref>) we see that when σ_ℓ(0)=+1, we have𝔼̃[J_ℓ j - p| S_j(t)] + ∑_i∉{Y_k}_k=1^T, σ_i(0) = +1𝔼̃[J_ij - p| S_j(t)] = O(N^1/2+4ϵ) .There are deterministically at least N/3 terms in the sum above, so that in order to conclude, it suffices to show that for an i∉{Y_k}_k=1^T with σ_i(0)=1, we have that 𝔼̃ [J_ℓ j| S_j(t)]= 𝔼̃[J_ij| S_j(t)]. (One would then divide both sides by the number of terms in the sum above, and obtain the desired.) If ℓ∉{Y_k}_k=1^T, this is evident by symmetry. Else, write, 𝔼̃ [J_ℓ j| m_j(t)>0] = ℙ̃(J_ℓ j=1)ℙ̃(m_j(t)>0| J_ℓ j=1)/ℙ̃(m_j(t)>0) .The conditioning in ℙ̃ is independent of J_ℓ j as well as J_ij so the first term in the numerator is the same under ℓ↦ i. Likewise, for the other two terms, expanding out m_j(t), one sees that since ℙ̃ conditions on J_kj for every k ∈{Y_1,...,Y_t-1}, the other couplings conditioned on under ℙ̃ do not affect the distribution of m_j(t); moreover σ_i(t)=σ_ℓ(t)=σ_i(0)=σ_ℓ(0), so that those remaining two terms are also unchanged under ℓ→ i. §.§ Short time dynamicsIn this section we study the evolution of (X_t)_t∈ℕ started from X_0 such that M_0 =N^1/2 -ϵ until T= N^1/2 +3ϵ. We begin by fixing the update sequence Y_1,...,Y_T, then for each realization of the update sequence, we bound probabilities of evolutions of the chain under ℙ_𝒥.Recall the definition of ℛ_k from (<ref>). We define the following good events for the update sequence {Y_k}_k=1^T:Γ_ω,t^1:={|ℛ_t| < N^ϵ∨2t^2/N}Γ_ω,t^2:={|∑_k=1^tσ_Y_k (0)|< t^1/2+ϵ} .The dynamical good event is then defined as Γ_ω= ⋂_t=N^ϵ^TΓ_ω,t^1 ∩Γ_ω,t^2.Let X_0 be such that M_0 = N^1/2 -ϵ. There exists c(ϵ)>0 such that ℙ_ω ( Γ_ω^c) ≲ Te^-cN^2ϵ^2 . We union bound over the (Γ_ω,t^i)^c for i=1,2 and N^ϵ≤ t≤ T. The bound for i=1 follows from a union bound over all t≤ T and the following. For each t≤ T, the probability of selecting a site that has already been updated at time t is at most t/N≤ N^-1/2 +3ϵ.Therefore, |ℛ_t| ≼ (t, t/N) so that by the Chernoff inequality, ℙ_ω(|ℛ_t|≥max{N^ϵ, 2t^2 N^-1}) ≤ 2e^-cN^ϵ ,for some c>0. For the i=2 bound, because the update order (Y_k)_k≤ T is independent of σ(0) and t≤ N^1/2 +3ϵ, by Hoeffding's inequality,ℙ_ω(|∑_k=1^tσ_Y_k (0)|≥ t^1/2+ϵ)≤ 2e^-ct^2ϵ ,because uniformly in t≤ T, the probabilities of σ_Y_t(0)=± 1are within M_0/N of 1/2 and M_0= N^1/2-ϵ. Union bounding over all N^ϵ≤ t≤ T yields the desired.By Lemma <ref>, without loss, we can now restrict our attention to realizations of ω such that the sequence Y_1,...,Y_T satisfies Γ_ω.We also define a good coupling event measurable with respect to ℱ_t which we will restrict our attention to. Fix a sequence (Y_1,...,Y_T). Begin by definingZ_++(t)= {j∈{Y_k}_k=1^t-1 - ℛ_T : σ_j (0)=+ 1, σ_j(t)=+1}and the analogously defined Z_–,Z_+-,Z_-+, measurable with respect to (Y_k,S_Y_k(k))_k< t. Then for N^ϵ≤ t ≤ T, letΓ_𝒥,t^++ = ⋂_s=N^ϵ^t{|∑_j∈ Z_++(s) [J_Y_s j -p]| ≤ s^1/2+ϵ} . Then let Γ_𝒥,t= Γ^++_𝒥,t∩Γ^+-_𝒥,t∩Γ^-+_𝒥,t∩Γ^–_𝒥,t. Fix an initial configuration X_0 such that M_0 = N^1/2 - ϵ. There exists c(p)>0 such that for every sequence Y_1,...,Y_T, and everyN^ϵ≤ t≤ T,ℙ_𝒥 ( Γ^c_𝒥,t) ≲ te^-cN^2ϵ^2 .By a union bound over all N^ϵ≤ s ≤ t, it suffices to show that there exists c(p)>0 such that for every such s,ℙ_𝒥( | ∑_j∈ Z_++(s) [J_Y_s j-p]| ≥ s^1/2 +ϵ (Y_k,S_Y_k(k))_k<s)≲exp(-cs^2ϵ)and similarly for Z_+-(s), Z_-+(s), Z_–(s), then average over all possible realizations of (S_Y_k(k))_k<s. We use Proposition <ref> to jointly sample the couplings {J_Y_s j}_j∈ Z_++(s) and S_Y_1(1),...,S_Y_s-1(s-1) according to the revealing scheme defined in Definition <ref>. One sees by Proposition <ref> that under this revealing process, independently of the couplings that have already been revealed and the information obtained from (S_Y_k(k))_k<s, the distribution of J_j Y_s dominates (p-cN^-1/2+4ϵ) and is dominated by (p+cN^-1/2+4ϵ).Therefore, conditional on any actualization of (S_Y_k(k))_k<s and therefore Z_++(s), the joint distribution of {J_Y_s j}_j∈ Z_++(s) is dominated by a product measure of (p+cN^-1/2+4ϵ) and dominates a product measure of (p-cN^-1/2 +4ϵ). At that point, using Chernoff–Hoeffding inequality, we see that (<ref>) holds.The main estimate on the effective fields at short times t≤ T is the following. Fix X_0 with magnetization M_0= N^1/2 -ϵ. Suppose for t≤ T, we are on a set in ℱ_t-1 such that Γ_ω∩Γ_𝒥,t holds and M_t-1≥ M_0-N^2ϵ. Suppose also that Y_t ∉ℛ_t. There exists c(p)>0 such that for sufficiently small ϵ>0, we haveℙ_𝒥(m_Y_t(t)>0|ℱ_t-1)≥1/2 +cN^-ϵ .In particular, this estimate holds independent of σ_Y_t(0)=σ_Y_t(t-1).For ease of notation, let i=Y_t.By definition, the σ-algebra ℱ_t-1 reveals information about the couplings {J_i Y_k}_k=1^t-1 and is independent of any other couplings to i.Using the fact that if j∉{Y_k}_k<t, σ_j(0)=σ_j(t), the event Γ_ω,t^2 implies∑_j∉{Y_k}_k<tσ_j(0) = ∑_j∉{Y_k}_k<tσ_j(t)≥ N^1/2-ϵ-N^1/4+3ϵ . Now expanding m_i(t), we obtainm_i(t)=∑_j∈{Y_k}_k< tσ_j(t) J_ij+∑_j∉{Y_k}_k< tσ_j(0) J_ij .We first consider the second sum in (<ref>). As remarked earlier, because j∉{Y_k}_k<t, conditional on ℱ_t-1, these J_ij are distributed as i.i.d. (p). Thus, given Γ_ω,t^2 holds, we have the following conditional (on _t-1) stochastic domination:∑_j∉{Y_k}_k<tσ_j(0) J_ij ≽(N2+(1-ϵ2)N^ 12-ϵ,p)- (N2-(1-ϵ2)N^ 12-ϵ,p) .By Berry–Esseen Theorem, we thus obtain for some c(p)>0,ℙ_(∑_j∉{Y_k}_k<tσ_j(0)J_ij≥p/2 N^1/2 -ϵ) ≥1/2 +cN^-ϵ + O(N^-1/2) .We now control the contribution of the first term of (<ref>). Consider separately the case t≤ N^2ϵ and t≥ N^2ϵ. If t≤ N^2ϵ, then clearly the first sum in (<ref>) is bounded above by t ≤ N^2ϵ. Now consider the case t≥ N^2ϵ.Recall the definitions of Z_++, Z_+-, Z_-+ and Z_– from (<ref>). Then we can expand∑_j∈{Y_k}_k<tσ_j(t) J_ij=∑ _j∈ℛ_tσ_j(t) J_ij + ∑_j∈ Z_++(t) J_ij +∑_j∈ Z_-+(t)J_ij - ∑_j∈ Z_+-(t)J_ij -∑_j∈ Z_–(t)J_ij .First of all, by the fact that Γ_ω,t^1 holds, |ℛ_t| and in turn the first sum on the right-hand side of (<ref>), are bounded above by N^7ϵ independently of t≤ T.For the latter four sums in (<ref>), we use the fact that Γ_𝒥,t holds.First we bound how many summands there are in each of the four terms. Since Γ_ω,t^1,Γ_ω,t^2 both hold and M_t-1≥ M_0-N^2ϵ, ||Z_++(t)| + |Z_+-(t)|-|Z_–(t)|-|Z_-+(t)||≤ N^1/4 +3ϵ+N^7ϵ|Z_+-(t)|-|Z_-+(t)|≤ N^2ϵ+N^7ϵ ,so that |Z_++(t)|+|Z_-+(t)| ≥ |Z_–(t)|+|Z_+-(t)|-N^1/4+3ϵ- 2N^ 7ϵ-N^2ϵ. Now we observe that by definition of Γ_𝒥,t, we have | ∑_j∈ Z_++(t) [J_Y_t j-p]| ≤ t^1/2 +ϵand likewise when summing over Z_+-(t),Z_-+(t) and Z_–(t).If Z_+(t) = Z_++(t) + Z_-+(t) and Z_-(t) = Z_–(t) + Z_+-(t), by applying (<ref>) separately to Z_+(t) and Z_-(t), we see that under the event Γ_ω∩Γ_𝒥,t, by (<ref>) and the fact that N^2ϵ≤ t≤ N^1/2+3ϵ, for ϵ small,∑_j∈{Y_k}_k<tσ_j(t) J_ij≥ p[Z_+(t)-Z_-(t)]-4t^1/2+ϵ≥ -2N^1/4+3ϵ ,and if t≤ N^2ϵ the same bound trivially holds. Combined with (<ref>), this implies thatℙ_𝒥(m_Y_t(t)≤ 0 |ℱ_t-1) ≤ℙ_𝒥(∑_j∉{Y_k}_k<tσ_j(0)J_ij≤ 3 N^1/4 +3 ϵℱ_t-1) ≤1/2 -cN^-ϵ+O(N^-1/2) ,yielding the desired for all sufficiently small ϵ>0. The following shows the implications of Proposition <ref> for the magnetization chain.Fix X_0 with magnetization M_0=N^1/2-ϵ. Suppose that t≤ T, that Γ_ω∩Γ_𝒥,t holds, and that M_t-1≥ M_0 - N^2ϵ. Suppose also that Y_t ∉ℛ_t; if σ_Y_t(0)=-1,ℙ_𝒥 (M_t-M_t-1 = +1|ℱ_t-1) = 1-ℙ_𝒥 (M_t-M_t-1 = 0|ℱ_t-1) ≥1/2+cN^-ϵ , for some uniform c(p)>0 and if σ_Y_t(0)=+1,ℙ_𝒥 (M_t-M_t-1 = 0|ℱ_t-1) = 1-ℙ_𝒥 (M_t-M_t-1 = -1|ℱ_t-1) ≥1/2+cN^-ϵ .We now use Corollary <ref> to lower bound the magnetization chain (M_t)_t≤ T. Define for ϵ,t,θ>0, the following subset of the filtration ℱ_t:ℬ_ϵ,t,θ={ M_k > M_0 - N^2ϵ+θ k N^-ϵ k≤ t} ,and observe that ℬ_ϵ,t,θ is anincreasing event in the magnetization.Fix X_0 such that M_0= N^1/2-ϵ. Suppose Γ_ω∩Γ_𝒥,T holds for the realization of ℱ_t. Then for all θ>0 sufficiently small, there exists a constant d(p,θ)>0 such thatℙ_𝒥(ℬ^c_ϵ,T,θ)≤ T e^-dN^2ϵ^2 . We fix any t≤ T and show ℙ_𝒥 (_ϵ,t,θ^c) ≤ te^-dN^2ϵ^2, implying in particular, the desired. In order to do so, we consider two random walk chains, ℳ^+_t which lower bounds the change in magnetization over times when σ_Y_k(0)=+1 and ℳ^-_t which does the same over times when σ_Y_k(0)=-1. The chains ℳ^+_t and _t^- are defined as follows: let B_i^+, B_i^- be i.i.d. (1/2+cN^-ϵ) for c(p)>0 given by Corollary <ref>,ℳ^+_t+1-^+_t = 1-B_t^+ , ^-_t+1-^-_t = B_t^- ,and _0^+ = _0^-=0. Define also t_+= ∑_j=1^t1{σ_Y_j(0)=+1} and analogously t_-. Clearly t_+ + t_- =t. As long as _ϵ,t-1,θ holds for some θ≥ 0, by Corollary <ref>,M_t ≽ℳ_t-|ℛ_t|_t:= M_0+ℳ_t_+^+ + ℳ_t_-^- . Letbe the probability distribution over (B_i^+,B_i^-)_i and therefore the random walks _t^+ and _t^-. Moreover, let 𝒜_ϵ,t,θ be the analogue of ℬ_ϵ,t,θ for the random walk ℳ_t-|_t| (note that _t is fully determined by the sequence (Y_k)_k≤ T). We prove that (_t - |_t| ∈_ϵ,t,θ^c) ≤ t e^-dN^2ϵ^2 for sufficiently small θ>0 for some c(p,θ)>0; this would imply the desired since M_t and _t can be coupled through all realizations of _ϵ,t,θ in a way that M_t≥_t, implying that _ϵ,t,θ also holds. To prove this, note first that for all t≤ N^2ϵ, _ϵ,t,θ holds trivially for every θ≥ 0. Thus consider t≥ N^2ϵ. Observe that since Γ_ω holds, for every N^ϵ≤ t≤ T,|t_+-t_-| ≤ t^1/2+ϵ , |_t| ≤ N^ϵ∨ 2t^2/N .By Hoeffding's inequality, there exists d>0 such that for every δ>0,(|_t_++t_+( 12-cN^-ϵ)|≥ t_+^1/2+δ)≤ 2e^-dt_+^2δ ,(|_t_--t_-( 12+cN^-ϵ)|≥ t_-^1/2+δ) ≤ 2e^-dt_-^2δ .Combining the above with (<ref>), we see that for some different d>0,(|_t_++_t_- - (t_–t_+2 + tcN^-ϵ)| ≥ t_+^1/2+δ+t_-^1/2+δ) ≤ 4e^-dt^2δand in particular, letting ϵ=δ, since |t_+ - t_-| ≤ t^1/2+ϵ,(_t ≤ M_0-3t^1/2+ϵ+tcN^-ϵ)≤ 2e^-dt^2ϵ .Now note that by the bound on |ℛ_t| in (<ref>), for all N^ϵ≤ t≤ T, M_0 -3t^1/2+ϵ-(N^ϵ∨ 2t^2/N)+c t N^-ϵ≥ M_0 - N^2ϵ + θ t N^-ϵ for, say, θ=c/2; this implies by a union bound and (<ref>), that for every t≤ T,ℙ_(M_t∈_ϵ,t,θ^c)≤(_t - |_t| ∈_ϵ,t,θ^c) ≤ te^-dt^2ϵ^2 . §.§ Long time dynamicsUsing the bounds in <ref> on the zero-temperature dynamics Markov chain, we can deduce the following conditions at time T=N^1/2+3ϵ.If M_0 = N^1/2-ϵ and T=N^1/2+3ϵ, there exists θ(p)>0 such that lim_N→∞ℙ(⋂_i∉{Y_k}_k<T{ m_i(T)>ϵ p θ N^1/2 +2ϵ}) =1 .We consider a fixed i∉{Y_k}_k<T and prove the proposition using the decomposition of m_i used in the proof of Proposition <ref> and then union bound over all such i∉{Y_k}_k<T. First note that by a union bound with Lemmas <ref>–<ref> and Proposition <ref>, if θ is as in Proposition <ref>, there exists c(p)>0 so thatℙ (M_T∈ℬ_ϵ,T-1,θ,Γ_ω,Γ_,T)≥ 1-O(Te^-cN^2ϵ^2) .By a union bound, it suffices to prove that there exists c>0 such that for every i∉{Y_k}_k<T, there exists θ>0 such thatℙ_(m_i(T)≤ϵ pθ N^1/2+2ϵ|ℱ_T-1,Γ_ω,Γ_,T, _ϵ,T-1,θ) ≤ e^-cN^ϵ .Now suppose we are on the intersection of ℬ_ϵ,T-1,θ, Γ_ω and Γ_,T. The couplings between i and sites not in {Y_k}_k<T are independent of ℱ_T-1 and are handled identically to the proof of Proposition <ref>, whence by Hoeffding's inequality applied to the difference of the binomial random variables in (<ref>), there exists c(p,θ)>0 such thatℙ_(∑_j∉{Y_k}_k<Tσ_j(0)J_ij≤ -p4θ N^1/2+2ϵ) ≤ e^-cN^ϵ .We now bound the contribution from couplings to sites j∈{Y_k}_k<T-_T. Under the event ℬ_ϵ,T-1,θ, the magnetization has M_T-1≥M_0+ θ N^1/2 +2ϵ-N^2ϵ. Recalling the sets Z_++(T),Z_+-(T),Z_-+(T),Z_–(T), as before, because Γ_ω holds, we have||Z_++(T)| + |Z_+-(T)|-|Z_–(T)|-|Z_-+(T)||≤ N^1/4 +3ϵ+N^7ϵ|Z_-+(T)|-|Z_+-(T)|≥θ N^1/2+2ϵ -N^2ϵ-N^7ϵ ,for some θ(p)>0, implying that|Z_++(T)|+|Z_-+(T)|-|Z_–(T)|-|Z_+-(T)| ≥θ N^1/2+2ϵ-o(N^1/2) .In order to reveal jointly the couplings {J_Y_k i}_k<T conditional on the history of the chain (Y_k,S_Y_k(k))_k<T, we use the revealing procedure defined in Definition <ref>, taking i to be Y_T; we know by Proposition <ref> that under this revealing procedure, for every realization of (S_Y_k)_k<T, the joint distribution of {J_l i}_l∈ Z_++(T) is dominated by a product measure of (p+N^-1/2+4ϵ) and dominates independent (p-N^-1/2+4ϵ) and the same holds for {J_l i}_l∈ Z_+-(T), and likewise when l∈ Z_-+(T) and l∈ Z_–(T).Then letting Z_+=Z_++(T)+Z_-+(T) and Z_-=Z_–(T)+Z_+-(T), by Hoeffding's inequality,ℙ_(∑_j∈{Y_k}_k<T-_T J_ijσ_j(T)≤p2 θN^1/2+2ϵ_T-1) ≤ℙ_(∑_j∈ Z_+ J_ij≤ p|Z_+|-p4 θ N^1/2+2ϵ_T-1) + ℙ_(∑_j∈ Z_- J_ij≥ p|Z_-| +p4 θ N^1/2 +2ϵ_T-1) ≤ 2e^-c N^ϵ ,for some c(p,θ)>0. Together with the bound of N^7ϵ on ∑_j∈ℛ_T J_ijσ_j(T),and a union bound, we obtain for some c(p,θ)>0, under _ϵ,T-1,θ∩Γ_ω∩Γ_,T,ℙ_(⋃_i∉{Y_k}_k<T{m_i(T)≤ p6 θ N^1/2+2ϵ}) ≲ N e^-cN^ϵ . If M_0=N^1/2-ϵ and T'= N^2/3, we have lim_N→∞ℙ(⋂_i=1^N { m_i(T')>0}) =1 .First, consider the update sequence Y_T+1,...,Y_T'. Dominating the number of updates there that are in {Y_k}_k≤ T by (T',T/N), we see that the probability of that being at most N^1/2 is 1-O(e^-c√(n)). At the same time, since T'=o(N) and M_0=N^1/2-ϵ, with probability at least 1-O(e^-cN^2/3) there are at least N^2/3-ϵ distinct sites i∈{Y_k}_k=T+1^T'-{Y_k}_k≤ T that have σ_i(0)=-1. Since both of these happen with ℙ_ω-probability going to 1 as N→∞ and are independent ofand _T, suppose we are on the intersection of these events (and also Γ_ω) and fix any such an update sequence. Then let _1,...,_n∈{Y_k}_k=T+1^T'-{Y_k}_k≤ T be the sequence of updates in Y_T+1,...,Y_T' with initial spin value -1 and not in _T (observe that n≥ N^2/3-ϵ).Assume also, by Proposition <ref> that every site i∉{Y_k}_k<T has m_i(T)≥ϵ p θ N^1/2 +ϵ, as this occurs w.h.p. Now consider sites i∈{Y_k}_k<T. We will need the following lower bound on m_i(T): if T=N^1/2 +3ϵ and Γ_ω holds,there exists c(p,θ)>0 such thatlim_N→∞ℙ_ (⋃_i ∈{Y_k}_k≤ T{m_i(T) ≤ -N^1/2 + 4ϵ}) ≲ Te^-cN^2ϵ .We now define a new ℙ_ event Γ^2_ as Γ^2_ =⋂_i∈{Y_k}_k≤ T{∑_j∈_1,...,_m J_ij≥ (p-ϵ)m, |∑_j∉{Y_k}_k≤ Tσ_j(0)J_ij| ≤ N^1/2+ϵ} . By a union bound and standard applications of Hoeffding's inequality as done before, along with the fact that Γ^2_ω,T holds, we see that for some c>0,ℙ_ ((Γ_^2)^c) ≤ 2Te^-cN^2ϵ ,so without any loss, we also assume we are on the event Γ_^2. Recall that for every j∉{Y_k}_k≤ T, σ_j(0)=σ_j(T). At the same time, we can assume a worst case bound on the couplings between site i and other sites j∈{Y_k}_k≤ T, which is to say we take every such J_ijσ_j(T)=-1, contributing at most -N^1/2 +3ϵ to m_i(T). Putting these together, we see that for update sequences in Γ_ω, under the event Γ_^2 we have deterministically that for every i∈{Y_k}_k≤ T,m_i(T) ≥ -N^1/2+ϵ-N^1/2+3ϵ =-o(N^1/2+4ϵ)so that (<ref>) holds. Under the intersection of the events in (<ref>) and (<ref>), we claim that by time T', deterministically, every site will have positive effective field. Note that our update sequence Y_T+1,...,Y_T' is such that the dynamics only selects at most N^1/2 sites in {Y_k}_k≤ T between times T+1 and T' and as a result, by (<ref>), for every i∉{Y_k}_k≤ T, for every T+1≤ t ≤ T', its field satisfies m_i(t)≥ m_i(T) -N^1/2 m_i(t)>0(only sites with nonpositive field can flip from +1 to -1 and decrease the field on i). At the same time, every time an update on a site _1,...,_n occurs, that spin has positive field by the above, and so it flips from -1 to +1. Then since n≥ N^2/3-ϵ and Γ_^2 holds, for every i∈{Y_k}_k≤ T, its field satisfiesm_i(T')≥ m_i(T)-N^1/2+ (p-ϵ)N^2/3-ϵ m_i(T')>0 ,whenever ϵ is sufficiently small, concluding the proof of the proposition. By Proposition <ref>, by time T'=N^2/3, with high probability, every site i has positive field. In that case by attractivity of the dynamics, whenever a negative site is selected to be updated, it flips to plus and those are the only possible spin flips, so that by the time every site has been updated again after time T', the zero-temperature dynamics will have absorbed into the all-plus state.Putting this together with the grand coupling of chains starting from every possible M_0 implies that if ϵ>0 is sufficiently small, for every configuration with M_0≥ N^1/2 -ϵ, lim_N→∞ℙ ( M_∞=N)=lim_N→∞ℙ (lim_t→∞ M_t =N)=1 .By spin flip symmetry, when M_0≤ -N^1/2-ϵ, with high ℙ_,ω-probability, M_∞=-N. Thus, for every i=1,...,N,𝔼_σ(0)[(𝔼_ω,𝒥[σ_1(∞)])^2] = 𝔼_σ(0)[(𝔼_ω,𝒥[σ_1(∞)])^2( 1{|M_0|≥ N^1/2-ϵ}+ 1{|M_0|< N^1/2-ϵ})] ≥ℙ_σ(0)(|M_0| ≥ N^1/2-ϵ) min_σ(0):|M_0|≥ N^1/2-ϵ (𝔼_ω,𝒥[σ_1(∞)])^2.Upper bounding the left-hand side by 1 and taking limits as N→∞ on both sides, we obtain by Fact <ref> and Theorem <ref> thatlim_N→∞𝔼_σ(0)[ (𝔼_ω,𝒥 [σ_1(∞)])^2] = 1 .By Jensen's inequality, q_D(N)= 𝔼_σ(0) [𝔼_𝒥 [(𝔼_ω[σ_i(∞)])^2]] ≥𝔼_σ(0)[(𝔼_ω,𝒥 [ σ_1 (∞)])^2]and trivially upper bounding q_D(N)≤ 1 implies lim_N→∞ q_D(N)=1. § DIFFERENT BEHAVIOR WITH HEAVY TAILS§.§ Heavy-tailed disordered CW modelIn this section, we demonstrate that the behavior above is not a general consequence of a mean-field disordered ferromagnetic system. We show that when we consider heavy-tailed coupling distributions, a completely different picture, regarding the structure of local-minima and the probability of ending up in them, emerges. One intuition for this comes from the fact that if we had considered an Erdős–Rényi graph with p=λ/N for λ>0 fixed, the results of <ref> on the dilute CW model would no longer hold as the underlying random graph would have many disconnected clusters of influence, and the dynamics would absorb in one of the many possible independent assignments of {± 1} to the disconnected clusters. The arguments here are straightforward but we provide them to emphasize the contrast to the situation with light-tailed couplings.If J_ij∼μ is non-negative, we say it is heavy-tailed if ℙ(J_ij≥ x)=x^-αL(x)for some 0<α<1 and some slowly varying L(x) (i.e., for every a>0, L satisfies lim_x→∞ L(ax)/L(x)=1).As before, for σ∈{±1}^N define the Hamiltonian of this disordered Curie–Weiss model by H(σ)= -1/N∑_i,j=1^N J_ijσ_iσ_j.If {J_ij}_i,j are non-negative heavy-tailed i.i.d. couplings with 0<α<1, then H has at least one non-trivial local minimum on the hypercube {± 1}^N with high probability. Moreover, the zero-temperature dynamics of the corresponding disordered Curie–Weiss model satisfies ϵ<q_D(N)<1-ϵ for some ϵ>0 uniform in N.One can also glean from the proof (specifically (<ref>)) that on average, the number of local minima grows exponentially in N as seen in many spin-glass models. While we expect that in the light-tailed setup, also with high probability there exist non-trivial local minima, there we guess the number in fact grows sub-exponentially.First recall the following classical theorems concerning sums and maxima of i.i.d. random variables with power-law tails (see e.g., <cit.>): Let Y_i be i.i.d. random variables satisfying the following conditions: there exists 0<α<1 such that x^αℙ(|Y_1| ≥ x)=L(x) is slowly varying and for some β∈ [-1,1], as x→∞,ℙ (Y_1 ≥ x)/ℙ(|Y_1| ≥ x) ⟶ (1+β)/2.Then there exists a sequence a_n given by a_n^-αL(a_n)=n^-1(∫_0^∞ x^-αsin x dx)^-1 so that∑_i=1^n Y_i/a_n⟶ Z_α,β ,where Z_α,β is a standard α-stable random variable with asymmetry parameter β. If Y_1,...,Y_n are i.i.d. with ℙ(Y_i≥ x)=x^-α L(x) for 0<α<1 and L(x) is slowly varying, then for each x, as n→∞,ℙ(max_i=1,..,n Y_i≤ b_n x ) ⟶Φ_α(x)=e^-x^-α1{x> 0}where b_n is the smallest sequence such that b_n^-α L(b_n) = n^-1. We prove that with high probability, the Hamiltonian H has non-trivial local minima. In order to proceed we need to define bully bonds.For a coupling realization {J_ij}_1≤ i<j≤ N, the coupling J_ij is a bully bond if J_ij > max{∑ _k≠ j |J_ik|,∑_k≠ i |J_jk|} ,and we define the event, E_ij={J_ij >max{∑_k≠ j |J_ik|, ∑_k≠ i |J_jk|}}. We begin by proving that with high probability, there exist non-trivial local minima. Notice that if, with probability going to 1 as N→∞, there exist at least two bully bonds, then H has non-trivial local minima: if E_ij, E_kl hold for k,l∉{i,j}, both σ_i=σ_j=+1, σ_k=σ_l=-1 and σ_i=σ_j=-1,σ_k=σ_l=+1 are satisfied independent of σ_V-{i,j,k,l} and neither combination is possible in a ground state of H. By symmetry between 1≤ i<j ≤ N/2 and N/2<k<l≤ N, and a union bound, it suffices to prove that lim_N→∞ℙ_(∑_i,j=1^N/2 1{E_ij}>0)=1 .We show this by considering the probability that max_1≤ i<j≤ N/2 J_ij is a bully bond: ℙ_(J_12 >max{∑_k≠ 1 J_1k,∑_k≠ 2 J_2k} J_12=max_1≤ i<j≤ N/2 J_ij) ≥ 1-ℙ_(max_1≤ i<j≤ N/2 J_ij≤ N^2/α-ϵ) - 2ℙ_(∑_k=3^N J_1k≥ N^2/α-ϵ)for some small ϵ(α)>0. In the inequality, we used a union bound and the fact that the coupling distribution of J_1k is decreased by conditioning on J_1k not being the maximum coupling in a set. Observe that the first probability above is o(1) by Theorem <ref> and the fact that it is a maximum over order N^2 i.i.d. heavy-tailed random variables, and the second probability is o(1) by Theorem <ref> and the fact that the sum is over O(N) i.i.d. heavy-tailed random variables. By symmetry this implies that with high probability the maximum over all couplings J_ij for 1≤ i<j ≤ N/2 is a bully bond, implying that with high probability there exist non-trivial local minima.It remains to prove that q_D(N) is uniformly bounded away from 0 and 1. To do so, we begin by computing the expected number of bully bonds, 𝔼_[∑_ij 1{E_ij}]= N(N-1)/2ℙ_(E_12) .Let Z_α=Z_α,β with β=1 and let the sequence a_N be defined as in Theorem <ref>; by definition of a_N and independence,ℙ_(E_12)≥ℙ_ ( J_12 > a_2N)ℙ_(∑_k≠ 1 J_1k+∑_k≠ 2 J_2k≤a_2N-2)≥c/2N(ℙ_(Z_α≤ 1)+o(1))≥c'/N ,for constants c(α),c'(α)>0. Note that this implies that 𝔼_[∑_ij 1{E_ij}] ≥ρ N for some ρ(α)>0. Moreover, ∑_ij 1{E_ij} is bounded above by N/2 since every vertex can be adjacent to at most one bully bond. As a consequence, we have that ℙ_(∑_1≤ i<j≤ N 1{E_ij}≥ρ N) ≥ 2 ρ .We claim that this implies q_D(N) is uniformly bounded away from 0 and 1. If with probability greater than ϵ>0, there are at least δ N sites adjacent to bully bonds, then the probability of a single site i being adjacent a bully bond J_ij is at least ϵδ. The contribution to q_D on that event is 1/2; this is because with ℙ_σ(0),𝒥-probability 1/2, that bully bond is initially satisfied (σ_i(0)=σ_j(0)) and otherwise, it is equally likely that the dynamics absorbs with σ_1=σ_j=+1 as σ_1=σ_j=-1—this is completely determined by whether σ_i or σ_j is updated first. The definition of q_D(N) then implies that ϵδ/2<q_D(N)<1-ϵδ/2 as desired.§.§ Heavy–tailed spin glassesUsing the above proof, one can derive similar estimates for heavy–tailed spin glasses, where J_ij is now distributed as a symmetric heavy-tailed random variable with ℙ(|J_ij|≥ x)=x^-α L(x) for L slowly varying and 0<α<1. The proof carries through as in the proof of Theorem <ref>, with the sums becoming sums of absolute values of such random variables, which are in the same basin of attraction of fully asymmetric α-stable random variables.Therefore, a straightforward adaptation of the proof of Theorem <ref> shows that for such coupling distributions, the heavy-tailed spin glass has local minima with high probability, and its zero-temperature dynamical order parameter is uniformly bounded away from 0 and 1.abbrv | http://arxiv.org/abs/1707.08875v1 | {
"authors": [
"Reza Gheissari",
"Charles M. Newman",
"Daniel L. Stein"
],
"categories": [
"math.PR",
"math-ph",
"math.MP"
],
"primary_category": "math.PR",
"published": "20170727142244",
"title": "Zero-temperature dynamics in the dilute Curie-Weiss model"
} |
We study the most luminous known supernova (SN) associated with a gamma-ray burst (GRB), SN 2011kl. The photospheric velocity of SN 2011kl around peak brightness is 21,000±7,000 km s^-1. Owing to different assumptions related to the light-curve (LC) evolution (broken or unbroken power-law function) of the optical afterglow of GRB 111209A, different techniques for the LC decomposition, and different methods (with or without a near-infrared contribution), three groups derived three different bolometric LCs for SN 2011kl. Previous studies have shown that the LCs without an early-time excess preferred a magnetar model, a magnetar+^56Ni model, or a white dwarf tidal disruption event model rather than the radioactive heating model. On the other hand, the LC shows an early-time excess and dip that cannot be reproduced by the aforementioned models, and hence the blue-supergiant model was proposed to explain it.Here, we reinvestigate the energy sources powering SN 2011kl. We find that the two LCs without the early-time excess of SN 2011kl can be explained by the magnetar+^56Ni model, and the LC showing the early excess can be explained by the magnetar+^56Ni model taking into account the cooling emission from the shock-heated envelope of the SN progenitor, demonstrating that this SN might primarily be powered by a nascent magnetar. 1School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China; [email protected] 2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, China 3Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA 4Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, E-18008, Granada, Spain 5Astroparticle Physics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 6Miller Senior Fellow, Miller Institute for Basic Research in Science, University of California, Berkeley, CA 94720, USA§ INTRODUCTION The optical spectra of Type Ic supernovae (SNe Ic) lack hydrogen and helium absorption lines (see, e.g.,for reviews). A small fraction of SNe Ic have spectra with broad absorption troughs indicating large photospheric velocities and were dubbed “broad-lined SNe Ic (SNe Ic-BL)" <cit.>. Some SNe Ic-BL have very large kinetic energies (≳ 10^52 erg; e.g., ) and were coined “hypernovae" <cit.>. Hypernovae are usually rather luminous (with peak luminosities ∼ 10^43 erg s^-1), indicating that they must have synthesized ∼ 0.5 M_⊙ of ^56Ni if they can be explained by the radioactive heating model <cit.>. Moreover, some SNe Ic-BL are associated with long-duration gamma-ray bursts (GRBs; e.g., ) and have been called “GRB-SNe" (see , and references therein). While most GRB-SNe are Type Ic-BL, not all SNe Ic-BL are associated with GRBs. The most luminous SNe Ic are Type Ic superluminous supernovae (SLSNe; ); their peak luminosities are ≳ 7×10^43 erg s^-1 <cit.>, tens of times greater than those of ordinary SNe.In this paper, we study a very luminous SN Ic, SN 2011kl, which is associated with an ultralong (T_90≈ 10^4 s) GRB 111209A (see, e.g., ) at redshift z=0.677 <cit.>. By using WMAP ΛCDM cosmology <cit.> and adopting H_0 =71 km s^-1 Mpc^-1, Ω_M = 0.27, and Ω_Λ= 0.73, <cit.> derived a luminosity distance (D_L) of 4076.5 Mpc for GRB 111209A/SN 2011kl. With a peak luminosity ∼ 3.63^+0.17_-0.16× 10^43 erg s^-1 <cit.>[<cit.> found a slightly lower peak luminosity (∼ 2.8^+1.2_-1.0× 10^43 erg s^-1) since they ignored the near-infrared contribution.], SN 2011kl is the most luminous GRB-SN yet detected and is significantly more luminous than all other GRB-SNe, whose average peak luminosity is 1 × 10^43 erg s^-1 with a standard deviation of 0.4 × 10^43 erg s^-1 <cit.>. The time of peak brightness of SN 2011kl is ∼ 14 days, slightly larger than the average (13.0 days with a standard deviation of 2.7 days; ) of other GRB-SNe.The peak luminosity of SN 2011kl is comparable to that of some not-quite-superluminous “SLSNe" (e.g., PTF11rks, PTF10hgi, iPTF15esb, iPTF16bad, and PS15br) and commensurate with the SN-SLSN gap transients observed by <cit.>; moreover, <cit.> pointed out that the spectrum of SN 2011kl is very different from those of GRB-SNe, but consistent with that of SLSNe. So, the energy sources powering its LC might be similar to those powering the SLSN LCs instead of the ordinary SNe Ic. The most popular prevailing models for explaining SLSNe are the magnetar model <cit.> and the ejecta plus circumstellar medium (CSM) interaction model <cit.>. [Almost all SLSNe (except maybe pair-instability SLSNe — e.g., SN 2007bi ; but seeand ) cannot be explained by the radioactive heating model <cit.>. Some SLSNe I exhibiting late-time rebrightening and Hα emission lines <cit.> might be powered by a hybrid model containing contributions from magnetars and ejecta-CSM interaction <cit.>.]There have been several papers modeling the LCs of SN 2011kl. The LC obtained by <cit.> (hereafter the G15 LC) cannot be explained by the ^56Ni model but can be by the magnetar model <cit.> [<cit.> showed that GRB 111209A and SN 2011kl can be explained purely by the magnetar model, both for the afterglow and SN-powered phases.], the magnetar+^56Ni model <cit.>, the white dwarf tidal disruption event (WD TDE) model <cit.>, and the collapsar model <cit.> involving a stellar-mass black hole and a fallback accretion disk <cit.>. By subtracting the contribution from the ultraviolet (UV)/optical/near-infrared (NIR) afterglow <cit.> of GRB 111209A, <cit.> obtained a new bolometric LC (hereafter the K16 LC) and modeled it with the radioactive heating model, finding 2.27 ± 0.64 M_⊙ of ^56Ni and 6.79_-2.84^+3.67 M_⊙ of ejecta, suggesting that this model is disfavored. The LC derived by <cit.> (hereafter the I16 LC) has an early-time excess and a dip [This early-time excess arose from the assumption that a single power law could describe the evolution of the optical LC during the afterglow phase. A broken power law showed no such excess (e.g., ), so this excess is model-dependent. Moreover, a single power law was used by <cit.>, and no such excess was observed, making it inconclusive.] which cannot be reproduced by the models mentioned above, but it can be fit by the blue supergiant (BSG) model with explosive energy injection (“the BSG model"; ). However, it is premature to exclude the magnetar model in explaining the I16 LC since the cooling emission from the shock-heated extended envelope <cit.> or the emission from magnetar shock breakout <cit.> can reproduce the early-time excess.Thus, the issue of the energy source of SN 2011kl deserves additional attention. In this paper, we reinvestigate the possible mechanisms powering the LC of SN 2011kl and discuss their implications. In Section <ref> , we use the radioactive heating model, the magnetar model, the magnetar+^56Ni model, and the magnetar+^56Ni+cooling model to fit the LCs of SN 2011kl. Our discussion and conclusions are presented in Sections <ref> and <ref>, respectively.§ MODELING THE LIGHT CURVES OF SN 2011KL Extracting the LC of a SN associated with a GRB is difficult since this process requires a decomposition of the observations to account for the contributions from the optical afterglow, the SN, and the underlying host galaxy (e.g., ); moreover, it depends on the assumptions adopted for modeling the GRB optical afterglow. By assuming that the LC of the optical afterglow of GRB 111209A can be described by a broken power-law function and subtracting the contributions from the optical afterglow and the host galaxy, <cit.> obtained the LC of SN 2011kl (the G15 LC). The G15 LC did not include the NIR contribution and is a pseudobolometric LC. By adding the NIR contribution, <cit.> obtained a genuine bolometric LC (the K16 LC)[The K16 LC must be brighter than the pseudobolometric LC (the G15 LC) without the NIR contribution. However, the median values of the luminosities of the first two points of the K16 LC are dimmer than that of the G15 LC. This fact is puzzling, perhaps indicating that the uncertainties are rather large.]. However, based on the assumption that the optical afterglow of GRB 111209A can also be described by a single (unbroken) power-law function, <cit.> derived another LC (the I16 LC) for SN 2011kl. This LC has an early-time excess and a dip. Hence, SN 2011kl has three different LCs (see Figure <ref>) that must be modeled.The LCs of SLSNe PTF11rks and PTF10hgi <cit.> are also plotted in Figure <ref>. By comparing the K16 LC of GRB-SN 2011kl, whose peak luminosity L_peak,bol is ∼ 3.6 × 10^43 erg s^-1, to the LCs of SLSNe PTF11rks and PTF10hgi, we find that the peak of the LC of SN 2011kl is as luminous as that of these two SLSNe. Furthermore, we point out that other SLSNe also have similar values of L_peak,bol— e.g., iPTF15esb (L_peak,bol≈ 4× 10^43 erg s^-1; ), iPTF16bad (L_peak,bol≈ 4× 10^43 erg s^-1; ), and PS15br (L_peak,bol≈ 4.15× 10^43 erg s^-1; ).[SN 2011kl, along with PTF11rks, PTF10hgi, iPTF15esb, iPTF16bad, and PS15br, are dimmer than the strict threshold for SLSNe (L_peak>7× 10^43 erg s^-1 and the peak absolute magnitudes M_peak<-21 mag in any band; ).]In this section, we use some models to fit these LCs. For the G15 and K16 LCs, we adopt the Markov Chain Monte Carlo (MCMC) method to minimize the values of χ^2/dof and get the best-fit parameters and the range. For the I16 LC, we don't use the MCMC method because its data have no known error bars. §.§ The Radioactive Heating Model Based on their derived LC (the G15 LC), <cit.> fit the ^56Ni model and found that the required mass of ^56Ni is ∼ 1 M_⊙ if the LC was powered by ^56Ni decay. <cit.> suggested that their bolometric LC (the K16 LC), which included the NIR contribution, needed 2.27 ± 0.64 M_⊙ of ^56Ni to reproduce its luminosity if the LC was powered by radioactive heating.Here we employ the semi-analytic ^56Ni model <cit.> to repeat the fits for the G15 LC, the K16 LC and I16LC. The free parameters of the ^56Ni model are the optical opacity κ, the ejecta mass M_ej, the ^56Ni mass M_Ni, the initial scale velocity of the ejecta[The scale velocity of the ejecta v_sc is the velocity of the surface of the ejecta. The radius of the ejecta can be calculated from R(t) = R(0)+v_sct <cit.>.] v_sc0, and the gamma-ray opacity of ^56Ni-cascade-decay photons κ_γ,Ni. The value of κ is rather uncertain and has been assumed to be 0.06 cm^2 g^-1 (e.g., ), 0.07 cm^2 g^-1 (e.g., ), 0.07±0.01 cm^2 g^-1 (e.g., ), 0.08 cm^2 g^-1 (e.g., ), 0.10 cm^2 g^-1 (e.g., ), and 0.20 cm^2 g^-1 (e.g., ). Assuming that the dominant opacity source is Thomson electron scattering and the temperature of the SN ejecta consisting of carbon and oxygen is not very high (≲ 10,000 K), we adopted 0.07 cm^2 g^-1 to be the value of κ. The value of κ_γ,Ni is fixed to be 0.027 cm^2 g^-1 (e.g., ).The photospheric velocity of SN 2011kl inferred from the spectra is ∼ 21,000±7,000 km s^-1 <cit.>. We assume that the moment of explosion (t_expl) for SN 2011kl is equal to that of GRB 111209A.The LCs reproduced by the radioactive heating models A1, A2, and A3 are shown in Figure <ref>, and the best-fit parameters are listed in Table <ref>. While the best-fit model parameters for the G15 LC are approximately equal to that derived by <cit.>, our inferred ^56Ni mass (1.42_-0.04^+0.04 M_⊙) for the K16 LC is smaller than the value derived by <cit.> (2.27±0.64 M_⊙). Moreover, the value of the ejecta mass (4.57_-1.03^+0.80 M_⊙) is also smaller than that derived by <cit.> (6.79_-2.84^+3.67 M_⊙).Another method for estimating the value of ^56Ni is to use the “Arnett law" <cit.>, which says that the ^56Ni energy input LC intersects the peak of the SN LC. According to the Arnett law and Eq. 19 of <cit.>, the mass of ^56Ni is [Eq. 19 of <cit.> is valid only in the case of full gamma-ray trapping. For partial gamma-ray trapping (κ_γ,Ni = 0.027 cm^2 g^-1), however, the inferred ^56Ni mass is equal to that corresponding to the case of full gamma-ray trapping, because the gamma-ray leakage doesn't influence the luminosity before and around the LC peak.]M_Ni=(L_peak/10^43 erg s^-1) [6.45 e^-t_rise/8.8 days +1.45 e^-t_rise/111.3 days]^-1 M_⊙,since the value of t_rise of SN 2011kl is ∼ 14 days and the peak luminosity is 3.63^+0.17_-0.16× 10^43 erg s^-1 (for the K16 LC) or 2.8^+1.2_-1.0× 10^43 erg s^-1 (for the G15 LC), the ^56Ni mass is 1.40^+0.07_-0.06 M_⊙ (for the K16 LC) or 1.08^+0.46_-0.39 M_⊙ (for the G15 LC). §.§ The Magnetar Model <cit.> analyzed the spectrum of SN 2011kl and pointed out that the spectral features imply that the ^56Ni mass must be significantly smaller than 1 M_⊙; they excluded the ^56Ni model for explaining the LC of SN 2011kl. Alternatively, <cit.> employed the magnetar model developed by <cit.> to fit the G15 LC and obtained a rather good result. But the K16 LC has not yet been fit with the magnetar model.Here, we use the semianalytic magnetar model developed by <cit.> and <cit.> that considers the leakage, photospheric recession, and acceleration effects to fit the K16 LC, and we repeat the fits for the G15 LC. The free parameters of the magnetar model are κ, M_ej, the magnetic strength of the magnetar B_p, the initial rotation period of the magnetar P_0, v_sc0, and the gamma-ray opacity of magnetar photons κ _γ ,mag. The value of κ _γ ,mag depends mainly on the energy of the photons (E_photons) emitted from the nascent magnetars, varying between ∼ 0.01 and 0.2 cm^2 g^-1 for γ-ray photons (E_γ≳ 10^6 eV) and between ∼ 0.2 and 10^4 cm^2 g^-1 for X-ray photons (10^2 eV ≲ E_X≲ 10^6 eV); see Fig. 8 of <cit.>.The LCs reproduced by the magnetar models (B1, B2, and B3) are shown in Figure <ref> and the best-fit parameters are listed in Table <ref>. We find that the K16 LC can be powered by a magnetar having B_p = 5.99_-5.55^+1.75×10^14 G and P_0 = 10.83_-6.94^+0.49 ms, while the G15 LC can be powered by a magnetar with B_p = 9.72_-4.13^+3.23×10^14 G and P_0 = 12.07_-2.52^+1.15 ms (the values of B_p and P_0 derived by <cit.> are ∼ (6–9) ×10^14 G and ∼ 13.4 ms, respectively). §.§ The Magnetar Plus ^56Ni Model The contribution of ^56Ni cannot be ignored when modeling some luminous SNe Ic <cit.>. <cit.> instead used a hybrid model containing contributions from ^56Ni and a magnetar to fit three luminous SNe Ic-BL whose peak luminosities are approximately equal to that of SN 2011kl. <cit.> and <cit.> used the same model to fit SN 2011kl, suggesting that 0.2 M_⊙ of ^56Ni must be added so that a better fit can be obtained. Therefore, we used the hybrid model (the ^56Ni+magnetar model) to fit the K16 LC and the G15 LC. The free parameters of this model are κ, M_ej, M_Ni, B_p, P_0, v_sc0, κ_γ,Ni, and κ_γ,mag.Since an energetic SN explosion can synthesize ∼ 0.2 M_⊙ of ^56Ni <cit.>, we assumed that the range of ^56Ni is 0–0.2 M_⊙ (for the G15 and K16 LCs with error bars, which allow us to use the MCMC method and determine the best-fit parameters) or 0.2 M_⊙ (for the I16 LC without error bars). The LCs reproduced by the ^56 Ni+magnetar model (C1, C2, and C3) are shown in Figure <ref> and the best-fit parameters are listed in Table <ref>. We find that this hybrid model can also reproduce good fits, and that almost all parameters (except for P_0) adopted by the ^56Ni+magnetar model are the same as that of the magnetar models. Moreover, the masses of ^56Ni determined by the MCMC are 0.11_-0.07^+0.06 M_⊙ (for the K16 LC) and 0.10_-0.07^+0.06 M_⊙ (for the G16 LC). §.§ The magnetar plus ^56Ni plus cooling model The I16 LC has an early-time excess which cannot be reproduced by the ^56Ni model, the magnetar model, or the magnetar+^56Ni model (A3, B3, and C3, respectively; see Figure <ref>). The WD-TDE model proposed by <cit.> also cannot explain the I16 LC (see Figure 1 of ). <cit.> suggested that the I16 LC can be explained by the BSG model.Here, we use another model to explain the I16 LC. We suggest that the early-time excess might be due to the cooling emission from the shock-heated extended envelope <cit.>, while the main peak of the LC might be powered by a magnetar or a magnetar plus a moderate amount of ^56Ni. In this scenario, the progenitor of the SN is surrounded by an extended, low-mass envelope which is heated by the SN shock and produces a declining bolometric LC when it cools. At the same time, sources at the center of the SN release energy and produce a rising LC until it peaks. By combining the magnetar(+^56Ni) model developed by <cit.> and <cit.> and the cooling model developed by <cit.>, a LC with an early-time excess and a dip can be produced. In this model, two additional parameters must be added: the mass of the extended envelope (M_env) and the radius of the extended envelope (R_env). [The energy of the SN, E_SN, must also be added into the modeling. However, it can be supposed that E_SN=E_K= 0.3 M_ejv_sc^2, so this parameter is determined by M_ej and v_sc, while v_sc is determined by v_sc0, B_p, and P_0 (the acceleration effect).] The LCs reproduced by the cooling model and the magnetar+^56Ni+cooling model (D1–D3) are shown in Figure <ref> and the corresponding parameters are listed in Table <ref>. We find thatthe magnetar+^56Ni+cooling models (D2 and D3) can well fit the entire I16 LC. In the scenario containing the contribution from the cooling emission, the progenitor of SN 2011kl was supposed to be surrounded by an extended envelope whose mass and radius are ∼ 0.63 M_⊙ (or ∼ 0.45 M_⊙) and ∼ 51.4 R_⊙ (or ∼ 103 R_⊙), respectively. Owing to the parameter degeneracy, there must be other choices of these two parameters.One advantage of the magnetar-dominated model is that it can explain the I16 LC if the cooling emission from the extended envelope is added, and it can explain the K16 LC and the G15 LC if the progenitor was not surrounded by an extended envelope. Another advantage is that the LC reproduced by the model with cooling emission can trace the entire LC (including the early excess, the dip, the peak, and the post-peak) while the LC reproduced by the BSG model is a smooth, monotonically decreasing LC that is brighter than the dip and dimmer than the peak. Although the uncertainties of the early-time data indicate that the LC reproduced by the BSG model might fit the data, the magnetar+^56Ni+cooling model can give a better match. §.§ Summary In summary, the K16, G15, and I16 LCs cannot be powered solely by the ^56Ni model since (a) ∼ 1.00–1.42 M_⊙ of ^56Ni is inconsistent with the spectral analysis performed by <cit.>, and (b) the ratio of ^56Ni masses (∼ 1.0–1.4 M_⊙) to the ejecta mass (∼ 2–5 M_⊙) is ∼ 0.3, which is significantly larger than the upper limit (∼ 0.20; ). Alternatively, the K16 and G15 LCs can be explained by the magnetar model and the ^56Ni+magnetar model; the values of M_ej, B_p, and P_0 can be found in Table <ref>.Owing to the degeneracy of model parameters, the values of these parameters have many choices that can also give the best-fit LC. For example, <cit.> obtained P_0 = 3.5 ms, B_p=1.95×10^15 G, M_ej = 2.5 M_⊙ if κ = 0.2 cm^2 g^-1, and M_Ni = 0.2 M_⊙; <cit.> derived P_0 ≈ 2 ms, B_p≈ 4×10^14 G, M_ej = 3 M_⊙ if κ = 0.2 cm^2 g^-1, and M_Ni = 0.2 M_⊙; and <cit.> derived P_0 = 11.5–13.0 ms, B_p= (1.1–1.3) ×10^15 G, and M_ej = 5.2 M_⊙ if κ = 0.07 cm ^2 g^-1.§ DISCUSSION§.§ What Determines the Peak Luminosities of SNe and SLSNe? The inferred best-fit values of P_0 and B_p of the magnetar powering the LCs of SN 2011kl are ∼ 9–14 ms and ∼ (6–14) × 10^14 G, respectively. The values of P_0 obtained by <cit.>, <cit.>, and <cit.> are 3.5 ms, 2 ms, and 11.5–13.0 ms (respectively), while the respective values of B_p derived by these groups are 1.95×10^15 G, ∼ 4×10^14 G, and (1.1–1.3) ×10^15 G. The spin-down timescale of a magnetar (τ_p) is determined by the values of P_0 and B_p, τ_p = 1.3 (B_p/10^14 G)^-2(P_0/10 ms)^2 yr <cit.>. We note that other SLSNe Ic whose peak luminosities are larger than ∼ 10^44 erg s^-1 must be powered by magnetars with P_0 ≈ 1–10 ms and 10^13–10^15 (see, e.g.,), indicating that the values of P_0 and B_p play a crucial role in determining the peak luminosities of SLSNe. When P_0 ≳ 10 ms and B_p is a few 10^14 G, the magnetar-powered SNe are luminous <cit.>; when P_0 ≲ 10 ms and B_p is a few 10^16 G, the magnetars can explain the LCs of some SNe Ic-BL (; see also ), while <cit.> suggested that most GRB-SNe are powered by radioactive heating.<cit.> emphasized the role of P_0 while <cit.> highlighted the role of B_p. We suggest that the more reasonable scheme discussing the factors influencing the peak luminosities and the shapes of magnetar-powered SNe and SLSNe is to take into account both of these parameters. However, it should be noted that the values of τ_m (or M_ej) also influence the peak luminosities of magnetar-powered SNe and SLSNe <cit.>. Hence, the luminosities of SNe and SLSNe are determined by all of these parameters (P_0, B_p, and τ_m, which is primarily determined by M_ej) and cannot solely be influenced by any single one. Similarly, the peak luminosities of ^56Ni-powered SNe and SLSNe are determined by the values of M_Ni and τ_m. §.§ Correlations Between the Opacity and Other Parameters In Section <ref>, we assumed that the value of κ is 0.07 cm^2 g^-1. As mentioned above, however, the value of κ is rather uncertain (0.06–0.20 cm^2 g^-1). Therefore, it is necessary to explore the correlations between the opacity and all other parameters involved in the modeling.Assuming κ = 0.10 cm^2 g^-1 and 0.20 cm^2 g^-1, and using the ^56Ni model, the magnetar model, and the magnetar plus ^56Ni model, we repeated the fits for the K16 LC and the G15 LC, obtaining the best-fit parameters (see Table <ref>).By comparing the values of the parameters corresponding to different values for κ, we found that while the values of the ejecta masses are significantly influenced by the values of κ and M_ej = aκ^-1 +b (a and b are constants; see Figure <ref>, and a similar figure can be found in ), all other parameters are slightly affected by the values of κ and no correlation between them and κ has been found.We did not study the correlations between the opacity and other parameters for the I16 LC, since its data lack error bars, and the value of M_ej is proportional to that of κ ^-1 while the values of all other parameters are completely independent of that of M_ej. §.§ The Initial Kinetic Energy of SN 2011kl The multidimensional simulations (see , and references therein) for neutrino-driven SNe suggest that the value of E_K provided by the neutrinos can be up to ∼ (2.0–2.5) × 10^51 erg. Provided the value of v_sc0 is 14,000 km s^-1 (then the ejecta mass is lowered to 2.4 M_⊙ for the K16 LC or 1.4 M_⊙ for the I16 LC), the origin of E_K,0 of SN 2011kl (E_K,0=0.3 M_ejv_sc0^2≈ 2.4× 10^51 erg) can be comfortably explained by the neutrino-driven model and other processes are not required to provide additional E_K,0.According to the equation E_rot,0≈ (1/2)IΩ_0^2 ≈ 2× 10^52(P_0/1 ms)^-2 erg (I ≈ 10^45 g cm^2 is the rotational inertia of the magnetar), we find that the initial rotational energy of the magnetar (P_0=11.4–14.2 ms) powering the LCs of SN 2011kl is ∼ 1.5 × 10^50 erg ( obtained E_rot,0= (1.2–1.6) ×10^50 erg), significantly smaller than their E_K,0. Even if the initial rotational energy of this magnetar is all converted to the E_K,0 of the ejecta, the ratio of final E_K to E_K,0 is ∼ 1.1. Therefore, the acceleration effect is rather small and can be neglected. §.§ The Ejecta Mass and Binarity Assuming that κ = 0.07 cm^2 g^-1 and according to the results of our work based on the magnetar models (the ^56Ni models have been excluded), the inferred ejecta mass of SN 2011kl is 3.83_-1.46^+2.06 M_⊙ (for the K16 LC), 4.76_-1.78^+3.05 M_⊙ (for the G15 LC), or ∼ 2.12 M_⊙ (for the I16 LC). If κ = 0.10 cm^2 g^-1 or 0.20 cm^2 g^-1 were adopted, the inferred masses would decrease by a factor of about 2–3, ∼1–2 M_⊙.By adopting κ = 0.04 cm^2 g^-1 and v (v_sc0) = 20,000 km s^-1, <cit.> concluded that the mass of the ejecta is 2 M_⊙ if SN 2011kl is powered by a magnetar. Assuming that κ = 0.07 cm^2 g^-1 and v (v_sc0) = 21,000 km s^-1, the ejecta mass must be 2 M_⊙×0.04/0.07 ×21,000/20,000 =1.323 M_⊙, significantly smaller than the value derived here (2.12 M_⊙).This difference is caused by the fact that we adopted the equation τ_m=(2κM_ej/βv_scc)^1/2 while <cit.> adopted the equation τ_m=(3κM_ej/4πv_scc)^1/2. The ratio of the masses derived from these two equations is 3β/8 π=3×13.8/8×3.1416=1.647. By multiplying this factor by 1.323 M_⊙, we obtain 2.18 M_⊙, consistent with the value derived here (2.12 M_⊙). The ejecta mass would be 1.45 M_⊙ if κ = 0.1 cm^2 g^-1 and τ_m= (2κM_ej/βv_scc)^1/2.<cit.> and <cit.> argued that the ejecta mass of SN 2011kl inferred from the magnetar model is too low to be produced by a core-collapse SN whose remnant harbors a magnetar. However, it must be pointed out that such a small value is not disfavored by the explosion of a SN Ic and magnetar formation, since the massive hydrogen and helium envelopes of the progenitors of SNe Ic had been stripped by their companions or winds, and the mean value of the ejecta mass of SNe Ic is 2.9 M_⊙ with a standard deviation of 2.2 M_⊙ (see Table 6 of ). The inferred ejecta masses of Type Ic-BL SNe 2002ap, 2006aj, 2009bb, and 2010bh are 2.0_-0.7^+0.8 M_⊙, 1.4_-0.2^+0.4 M_⊙, 1.9_-0.5^+0.6 M_⊙, and 0.9_-0.2^+0.2 M_⊙, respectively (see Table 5 of ). Note, especially, that SN 2006aj with M_ej=1.4_-0.2^+0.4 M_⊙ (or ∼ 2 M_⊙; ) is a SN Ic-BL associated with an X-ray flash (a “soft GRB") that has long been believed to be powered by a nascent magnetar <cit.>.Such small ejecta masses indicate that the progenitor of SN 2011kl might be located in a binary system and most of its mass had been stripped by its companion before the explosion. Although the mass-transfer process reduced the mass of the progenitor, it can enhance the angular momentum and angular velocity of the progenitor, making it easier for the nascent neutron stars left by the SN explosions to be millisecond (P_0 ≈ 1–10 ms) magnetars.§ CONCLUSIONS In this paper, we modeled the LCs of SN 2011kl, which is the most luminous ( L_peak = 3.63^+0.17_-0.16× 10^43 erg s^-1) GRB-SN identified thus far. By using the bolometric LC of SN 2011kl derived by <cit.> and assuming that this SN is powered by ^56Ni cascade decay, we find that the required ^56Ni mass is 1.42_-0.04^+0.04 M_⊙, which is smaller than that (2.27±0.64 M_⊙) inferred by <cit.>. The ^56Ni model can be excluded from explaining the LCs of SN 2011kl, since the spectral features indicate that the amount of ^56Ni must be significantly smaller than ∼ 1 M_⊙ <cit.>.Alternatively, we used the magnetar model and the magnetar+^56Ni model to fit the K16 LC and the G15 LC, finding that both of the models can well explain these LCs. It is noteworthy that <cit.> argue that the fact that the location of GRB 111209A-SN 2011kl is near the nucleus of the host galaxy favors the WD-TDE model. However, the position of the SN does not necessarily mean that this event must be a TDE rather than a SN. The magnetar model is still possible, explaining the K16 LC and the G15 LC, while the TDE model is also a plausible one for the G15 LC and the K16 LC.Whereas the magnetar model and the magnetar+^56Ni model can account for the K16 LC and the G15 LC, they cannot explain the I16 LC because these models cannot produce the early-time excess and the dip, although the presence of the early-time excess is entirely model-dependent (i.e., the parameters of the optical-afterglow component in the LC decomposition technique). To solve this problem, we added the contribution of the cooling emission from a low-mass (M_env≈ 0.5 M_⊙), extended (R_env≈ 50–100 R_⊙) SN progenitor envelope, and found that the magnetar+cooling model and the magnetar+^56Ni+cooling model can account for the I16 LC. In these models, cooling emission from the shock-heated envelope powers the early-time excess, while the magnetar or magnetar+^56Ni power the main peak of the LC, and the sum of these two produces the dip.[It should be mentioned that the model in which the progenitors have a nonstandard structure <cit.> has been employed to explain some SLSNe whose bolometric LCs exhibit an early-time excess <cit.>.] Hence, we conclude that the BSG model is not unique for explaining the I16 LC even if the LC of SN 2011kl has an early-time excess, since a progenitor surrounded by a low-mass, extended envelope can also power a LC with an early excess.It seems that the magnetar+^56Ni+cooling and the magnetar+^56Ni models are more reasonable than the magnetar and magnetar+cooling models for the LCs of SN 2011kl, since core-collapse SNe must synthesize a moderate amount of ^56Ni. However, discriminating between the LCs reproduced by the models with and without ^56Ni is very difficult because the contribution of a moderate amount of ^56Ni (∼ 0.1-0.2 M_⊙) is significantly smaller than that of the magnetar.Provided that the initial velocity of the ejecta of SN 2011kl is ∼ 14,000 km s^-1 (the lower limit of the ejecta velocity; see ), the inferred values of the initial kinetic energy of this SN is E_K,0≈ 2.0 × 10^51 erg , indicating that the neutrino-driven mechanism <cit.> is able to provide the E_K,0 for this SN. But larger velocities require other mechanisms to provide additional E_K,0.Furthermore, we used a MCMC method for the G15 and the K16 LCs to constrain the range of the model parameters (we did not perform MCMC for the I16 LC owing to the absence of the error bars); see Table <ref>. By adopting different values of κ, we found that while the inferred mass is significantly influenced by the values of κ (M_ej = aκ^-1 +b (a and b are constants), all other parameters are only slightly affected by the values of κ and no correlation between them and κ has been found.According to these results, we suggest that the magnetar and the magnetar+^56Ni models, with or without the cooling effect, can reproduce the LCs of SN 2011kl.[We caution that the question of how the magnetar powers an ultralong GRB is still unsolved. <cit.> suggested that a magnetar with P_0 ≈ 2 ms and B_p≈ 4×10^14 G can power the ultralong GRB 111209A and SN 2011kl, but <cit.> demonstrated that it is difficult for a magnetar to produce an ultralong GRB. This issue is beyond the scope of this paper and needs additional research. 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"authors": [
"Shan-Qin Wang",
"Zach Cano",
"Ling-Jun Wang",
"Wei-Kang Zheng",
"Zi-Gao Dai",
"Alexei V. Filippenko",
"Liang-Duan Liu"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170726160353",
"title": "Modeling The Most Luminous Supernova Associated with a Gamma-Ray Burst, SN 2011kl"
} |
^1Dipartimento di Fisica “E. Fermi”, Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy ^2Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy [email protected] Understanding the nature of the turbulent fluctuations below the ion gyroradiusin solar-wind turbulence is a great challenge.Recent studies have been mostly in favor of kinetic Alfvén wave (KAW) type of fluctuations, but other kinds of fluctuations with characteristics typical of magnetosonic, whistler andion Bernstein modes, could also play a role depending on the plasma parameters. Here we investigate the properties of the sub-proton-scale cascadewith high-resolution hybrid-kinetic simulations of freely-decaying turbulence in 3D3V phase space, including electron inertia effects. Two proton plasma beta are explored: the “intermediate” β_p=1 and “low” β_p=0.2 regimes, both typically observed in solar wind and corona. The magnetic energy spectum exhibits k_⊥^-8/3 and k_^-7/2 power laws at β_p=1, while they are slightly steeper at β_p=0.2. Nevertheless, both regimes develop a spectral anisotropy consistent with k_∼ k_⊥^2/3 at k_⊥ρ_p>1, and pronounced small-scale intermittency. In this context, we find that the kinetic-scale cascade is dominated by KAW-likefluctuations at β_p=1, whereas the low-β case presents a more complex scenario suggesting the simultaneous presence of different types of fluctuations. In both regimes, however, a possible role of ion Bernstein type of fluctuations at thesmallest scales cannot be excluded.Kinetic cascade in solar-wind turbulence: 3D3V hybrid-kinetic simulations with electron inertia Francesco Califano^1 December 30, 2023 =============================================================================================== § INTRODUCTION Nearly all astrophysical and space plasmas are in a turbulent state.In this context, the solar wind (SW) represents an ideal environment for studying collisionless plasma turbulence from the magnetohydrodynamic (MHD)range down to kinetic scales <cit.>. Increasingly accurate in-situ measurements of SW turbulencedown to electron scales have been available over the past years <cit.>, showing the presence of breaks in the electromagnetic fluctuations at kinetic scales. In the proton kinetic range, for instance, the typical slope for the magnetic energy spectrum is found to be between -2.5 and -3, i.e., steeper than the correspondent spectrum at MHD scales, while the electric spectrum becomes simultaneously shallower below the proton gyroradius scale.A wide number of theoretical models <cit.> and numerical investigations <cit.>have been exploited in order to explain the observed behavior of SW turbulent spectra,mostly in terms of the properties of fluctuations derived from wave physics. In this context, the observed spectra at kinetic scales are usually interpreted asa cascade of kinetic Alfvén waves (KAWs) and/or of higher frequency waves,such as magnetosonic (MS), whistler waves (WWs) and/or ion Bernstein (IB) modes.Most of the SW observations points towards a cascade of KAW-like fluctuationsat β∼1 <cit.>, where β is the ratio between thermal and magnetic pressures,although also whistler-like turbulence have been observed <cit.>.In fact, theoretical arguments suggest that different kinds of fluctuations could coexist and interact, depending on the plasma parameters <cit.>.This idea has been recently explored via 2D numerical simulations that suggested an increasingly KAW-like turbulence as β increases,whereas a more complex scenario - i.e., a mixture of different kind of fluctuations, including KAW-like ones -seems to emerge in the low-β regimes <cit.>. However, interpreting the turbulent cascade only in terms of wave physics is perhaps limiting and unsatisfactory <cit.>. Recently, the idea that magnetic reconnection can play a fundamental rolein the formation of the small-scale spectrum has emerged <cit.>. These interpretations are somewhat at odds with the picture of turbulencemade solely by a cascade of waves, as pointed out also by the intermittent behaviorof SW turbulence <cit.>. In this Letter, we present high-resolution 3D3V simulations of the turbulent cascade below the proton gyroradius within a hybrid Vlasov-Maxwell (HVM) model of plasmaincluding finite electron inertia (m_p/m_e=100). Here, we focus on the spectral and intermittent properties of kinetic-scale turbulencein order to address the question of a possible dependenceof the physics of such cascade on the plasma beta parameter.We remind that our hybrid approach, although not retaining all the electron kinetic effects,fully captures the ion kinetic physics and allows for both KAWs,magnetosonic, whistlers and ion Bernstein fluctuations to be present. We want to stress that here we analyze the properties of the turbulent fluctuationsand we relate them to the characteristic features of the corresponding linear modes,but in doing this we are not assuming that turbulence is made by a sea of linear waves: the aim of the analysis is to understand and classify the characteristics of turbulent fluctuations in analogy with those derived via linear theory. § THE HVM MODEL AND SIMULATIONS SETUP The HVM model couples fully-kinetic protons to fluid electronsthrough a generalized Ohm's law <cit.>.The model equations, normalized with respect to the proton characteristic quantities(mass m_p, gyrofrequency Ω_p and inertial length, d_p) and to the Alfvén speed v_A, read ∂ f/∂ t+ ·∂ f/∂+ (+×)·∂ f/∂=0 , (1-d_e^2∇_⊥^2) =- × + ×/n -p_e/n , ∂ /∂ t =- × ,×=where f(,,t) is the proton distribution function, d_e^2 = m_e is the electron skin depth, quasi-neutrality n_p≃ n_e≡ n is assumed,and the displacement current is neglected in the Ampére's law.In the generalized Ohm's law, the leading electron inertia termd_e^2∇^2 ≃ d_e^2∇_⊥^2 has been included (assuming k_^2 ≪ k_⊥^2 and a naturally anisotropic cascade). An isothermal closure for the electron pressure, p_e=nT_0,e, is adopted, and number density, n,and proton mean velocity, , are computed as v-space moments of f. We initialize the simulations with a Maxwellianproton distribution function with isotropic temperature T_0,p and an electron fluid with T_0,e=T_0,p,embedded in a uniform background magnetic field _0=B_0_z with B_0=1.We further impose initial random large-scale 3Disotropic magnetic perturbations, =_0+δ, with wave numbers 0.1≤ kd_i≤0.5 and δ B^ rms≃0.23. We use 384^2 grid points in the perpendicular xy-plane and 64 points in the parallel z direction, uniformly distributed to discretize a periodic simulation box with L_⊥ = 10 π d_p and L_=2L_⊥=20 π d_p, corresponding to a perpendicular resolution Δ x=Δ y ≃ 0.08 d_p = 0.8 d_eand Δ z≃ d_p.This corresponds to a spectral domain that spans more than two decadesin perpendicular wave numbers, 0.2≤ k_⊥ d_p≤38.4,and more that one decade in its parallel counterpart, 0.1≤ k_ d_p≤3.2.We apply (weak) spectral filters during the simulation in order to prevent spurious numerical effects at the smallest scales <cit.>, thus determining a cut-off in the turbulent energy spectrafor k_⊥ d_p≳20 and for k_d_p≳2.The velocity domain is limited in each directionby v_ max = ± 5 v_ th,pfor the β_p=1 case and by v_ max = ± 8 v_ th,p for β_p=0.2,with 51^3 and 61^3 uniformly distributed velocity grid points, respectively. § ANISOTROPY AND INTERMITTENCY OF KINETIC TURBULENCE Within a few outer-scale nonlinear times the initial condition freely-decaysinto a fully-developed turbulent state at t= t_*. Such time is identified by a peak in theroot-mean-square current density, J^ rms. In order to increase the statistics, the spectral analysisof turbulent fluctuations presented here includesa short time average over Δ t=10 Ω_p^-1≪ t_*, starting from t_*. Before discussing the spectral properties, a difference between the β_p=1 and β_p=0.2 regimes is first pointed out at the level of the spatial structures emergingin the fully-developed turbulent state. This is shown in Fig. <ref> where we draw thethree-dimensional contours of the magnetic field magnitudeat t=t_* in the two distinct regimes (top and bottom panelfor β_p=1 and β_p=0.2, respectively). As expected, starting with the same initially isotropic condition,in both cases the fluctuations gradually cascades into strongly anisotropic turbulence. However, while the β_p=1 regime exhibits perpendicular small-scale structures and very elongated fluctuations along _0 that are typical of Alfvénic turbulence,the β_p=0.2 case presents shorter parallel structuresthat are instead reminiscent of magnetosonic fluctuations. The spectral anisotropy of the turbulent fluctuations is shown in Fig. <ref>,where we draw the two-dimensional energy spectrum of the total magnetic fluctuations, δ B (top panels),and of the parallel electric fluctuations, δ E_ (bottom panels), for both regimes(left and right column for β_p=1 and β_p=0.2, respectively). Anisotropy is observed also at kρ_p<1, although this region contains few k points and is thus less relevant. At smaller scales, k_⊥ρ_p>1, the anisotropy is instead evident:the turbulent cascade is mainly perpendicular to _0and the fluctuations seem to follow a k_∼ k_⊥^2/3 pattern. This is more pronounced in the β_p=1 case, where the available sub-proton-scalerange is larger than in the low-β_p counterpart.Such pattern reveals a weaker anisotropy than the k_∼ k_⊥^1/3 scaling phenomenologically expected for both KAW and whistlerturbulence <cit.>,and it is rather in agreement with the one predicted for turbulence mainly concentrated within 2D sheet-like structures <cit.>. The spectra of fluctuations in the other quantities show the same behavior (not shown here).A classical intermittency analysis has been performed on both simulationsat about the peak of the nonlinear activity.In order to define the large scale limit of the inertial range, we evaluated theperpendicular and parallel auto-correlation functions, respectively defined asC(r_⊥) = ⟨δ(+ r_⊥)·δ() ⟩ andC(r_) = ⟨δ(+ r_)·δ() ⟩ <cit.>.We assumed isotropy in the perpendicular xy-plane, with the parallel direction along _0, i.e. along z.The e-folding length gives approximately the integral scale which is about λ_⊥∼ 3 d_pin the perpendicular direction (corresponding to k_⊥ d_p∼2), for both regimes.The situation is different in the parallel direction, where the parallel correlation length is λ_∼8 d_p for β_p=0.2,while is λ_∼12 d_p for β_p=1 (corresponding to k_d_p∼0.8 and ∼0.5, respectively).This is in qualitative agreement with the features spotted in Fig. <ref>,and, quantitatively, with the corresponding spectra (see Fig. <ref>),indicating differences already in the large-scale properties of the fluctuationspossibly due to a different decorrelation mechanism along the mean field.The level of intermittency can be better quantified by the PDFsof the magnetic field increments at a given scale r, defined asΔ b_r ≡[δ(+ r)-δ()]·r̂. We show here the statistics of the perpendicular increments, namely r≡ r_⊥,spanning this increment from lengths larger than the correlation scale λ_⊥, down to the smallest available scale (Δ x∼0.08 d_p).These distributions are reported in Fig. <ref>-(a) for the β_p=0.2 regimefor three cases, namely r/d_p=21, 0.6 and 0.08.These PDFs, as expected, become increasingly intermittent going towards smaller scales. In order to compare among cases, and among different times, we measured the scale-depended kurtosis χ– the fourth-order moment of the increments in Eq. (<ref>) – that can be measured as χ = ⟨Δ b_r^4 ⟩/⟨Δ b_r^2⟩^2. This quantity is reported in Fig. <ref>-(b), as a function of the perpendicular scale r,for the two values of β, at two distinct times. At large scale, for r>5 d_p, the distribution becomes Gaussian, where χ∼3,in agreement with the computation of the correlation lengths.At small scales, in the inertial range of turbulence, there is an enhancement due to theintermittent nature of the cascade, due to the presence of coherent structures and non-linear waves.At the smallest scales, a saturation of the multifractality is observed, in agreement with observations in the solar wind.In fact the study of high-order structure functions up to the 6^th moment and of their exponents,shows deviation from monofractality (not shown here). Here this process of saturation might also be slightly affected by the presence of artificial dissipation.It is important to notice, that at scales in the inertial-dispersive range,the case with β_p=0.2 is more intermittent than the β_p=1 regime,indicating a higher degree of coherency in the small-scale fluctuations.§ SPECTRAL FEATURES OF KINETIC-SCALE FLUCTUATIONSIn Fig. <ref>, we show the one-dimensional magnetic energy spectrum for both regimes (green and blue line for β_p=1 and β_p=0.2, respectively): the k_-averaged spectrum versus k_⊥, ⟨ E_B(k_⊥)⟩_k (top frame),and the k_⊥-averaged counterpart versus k_, ⟨ E_B(k_)⟩_k⊥ (bottom frame).The average procedure, e.g. ⟨ E(k_⊥)⟩_k, here is defined as the summationof E(k_,i,k_⊥,j) over the points of the {k_,i}_i=1,…,N_ grid, divided by those number of points, N_.Such procedure, when specified, can be restricted to a k_⊥-dependent sub-set of points, n_(k_⊥),of the entire k_ grid (see later in this Section). At large perpendicular scales, 0.4≲ k_⊥ρ_p≲2, a nearly -5/3 power law is visible in both cases, although the MHD range is too limited to draw conclusions. At small perpendicular scales, k_⊥ρ_p≳2, the β_p=1 regime exhibitsa power law very consistent with a -8/3 slope(this has been verified through compensated spectra), while the β_p=0.2 case shows a steeper spectrum, close to k_⊥^-3. For small parallel wave numbers, roughly k_ρ_p≲0.5,an excess of magnetic energy is present for β_p=0.2 and no clear power laws can be drawn for both regimes. For k_ρ_p≳0.5, instead, a -7/2 slope is observed at β_p=1, whereas at lower β it is again steeper (roughly between k_⊥^-9/2 and k_⊥^-5). Note that the kinetic-range cascade, expected to take place at kρ_p>1,in the parallel wave numbers already starts at k_ρ_p∼0.5 due to theanisotropic nature of the turbulent cascade itself (cf. Fig. <ref>). In particular, consistently with the spectral anisotropy and the intermittency analysis,the observed power laws for the magnetic spectrum at β_p=1,i.e., ∝ k_⊥^-8/3 and ∝ k_^-7/2,are in agreement with those predicted in <cit.>.A useful tool for the investigation of turbulent fluctuations properties are the spectral ratios of different quantities <cit.>. Here, in order to highlight the characteristicbehavior of small-scale fluctuations in the two different regimes, we consider the following quantities:C_p ≡ β_p^2 δ n^2/δ B_^2 , C_A ≡ δ E_⊥^2/δ B_⊥^2 , C_ ≡ δ B_^2/δ B^2 ,where τ≡ T_0,e/T_0,i=1 has been already assumed in normalizing C_p. Let us relate them to the characteristic signatures that themain oblique modes would leave on the above ratios <cit.>. Since we are interested in the oblique fluctuations and given the anisotropic behavior of the turbulent energy cascade shown in Fig. <ref>,the ratios defined above will be averaged over parallel wave numberssuch that k_≤ k_⊥^2/3ρ_p^-1/3. The resulting ratios are thus function of k_⊥ only,highlighting the properties of the main turbulent fluctuations and their connection with previous 2D numericalstudies <cit.>. We first consider C_p (Fig. <ref>, top panel):the normalized ratio between density and parallel magnetic fluctuationsis expected to be unity, C_p≈1, for low-frequency Alfvénic/KAW fluctuations,whereas higher frequency modes such as MS, WWs and IB should leave this ratio much smaller, namely C_p≪1. For β_p=1, the C_p ratio is about unity in nearly all the k_⊥ range, which is a signature of turbulence dominated by low-frequency Alfvénic/KAW-like fluctuations. In the β_p=0.2 case, instead, we obtain C_p≪1 at large scales, k_⊥ρ_p<1,and it then increases for k_⊥ρ_p>1, reaching values similar to those observed at β_p=1. In both regimes the behavior of C_p at the smallest scales, k_⊥ d_p≫1,is most likely due to a combined effect of k_⊥ d_e terms <cit.>and by the enhanced coupling of the MS, WWs and KAWs with the ion Bernstein branches in the oblique electromagnetic case <cit.>.Second, we consider C_A (Fig. <ref>, middle panel):at k_⊥ρ_p<1, this ratio is expected to be unity for Alfvénic fluctuations, C_A≈1, and to increase as C_A ≃ 1/2 β_p/1+β_p (k_⊥ρ_i)^2 for k_⊥ρ_p>1, i.e., in the KAW regime this ratio strongly depends on β_p.In the WWs regime, instead, this ratio does not depend on the beta and also increases as k_⊥^2: C_A≃2(k_⊥ρ_i)^2.Qualitatively, the relation C_A^WW≳C_A^KAW(β=1)≳ C_A^KAW(β=0.2) holds. From Fig. <ref> (middle panel), the behavior of C_A at β_p=1 is again consistent with predominantly Alfvénic/KAW-like fluctuations, whereas at β_p=0.2 the large-scale behavior is consistent with MS/WW-like fluctuations. Nevertheless, due to the fact the the above qualitative relation C_A^(β=1)≳ C_A^(β=0.2)is recovered at high k_⊥ and, in the same range, the C_p ratio for the β=0.2 case increases towards unity, a partial contribution of KAW type of fluctuations - but not dominant, as highlighted by the C_ ratio, below -cannot be excluded in the low-β regime. Note that the decrease of C_A at k_⊥ d_p≫1 is also consistent with acoupling with IB modes in both regimes <cit.>.Finally, let us consider the magnetic compressibility, C_ (Fig. <ref>, bottom panel):Alfvénic fluctuations would have small magnetic compressibility for k_⊥ρ_p≪1 that increases as one goes to smaller and smaller scales and, in the KAW regime, eventually settlesto a β-dependent value of C_≃β_p/(1+2β_p) at k_⊥ρ_p>1(represented in the bottom panel of Fig. <ref> by the green and blue horizontal dotted linesfor β_p=1 and β_p=0.2, respectively). Conversely, MS fluctuations have generally higher magnetic compressibilitythan the Alfvénic counterpart at k_⊥ρ_p<1 and, in the whistler regime,should settle to a β-independent value ofC_=k_⊥/2k≲1/2 at k_⊥ ρ_p > 1.From Fig. <ref> (bottom frame) we see that the magnetic compressibility is consistent with Alfvénic/KAW-like fluctuations at β_p=1, i.e. it is small at k_⊥ρ_p<1 and then it increases to the nearly constant value of C_≃β_i/(1+2β_i)=1/3expected for KAWs at k_⊥ρ_p>1. The β_i=0.2 regime instead exhibits a magnetic compressibility which is higher than that expected for Alfvénic/KAW fluctuations throughout the whole k_⊥ range, consistent with a mixture of MS, WWs and IB type of fluctuations <cit.>. Note that k_⊥ d_e effects can also enhance the compressibility ofKAWs <cit.>, so, consistently with the previous ratios,there could be a non-negligible contribution of KAW-like fluctuationsat k_⊥ρ_p≫1 also in this low-β regime. All these results are qualitatively in agreement with previous analysis performed in 2D fully-kinetic and hybrid-kineticsimulations <cit.>. § CONCLUSIONS We presented the first high-resolution simulations of 3D3V hybrid-kinetic turbulence including electron inertia effects (with m_p/m_e=100), ranging fromMHD scales to (perpendicular) scales well below the ion gyroradius.Two plasma beta parameters have been investigated: an “intermediate” β_p=1 regime and a “low” β_p=0.2 case.In both regimes, the spectral properties of the sub-proton turbulent cascade,such as its power laws and spectral anisotropy, and the intermittent behaviorof the fluctuations are in good agreement with solar-wind observations andwith the picture of turbulence mainly concentrated within 2D sheet-like structurespresented in <cit.>. In particular, all the turbulent fluctuations show a sub-proton-scale anisotropy pattern of the type k_∼ k_⊥^2/3 and, correspondingly, the magnetic energy spectrumexhibits power-laws in perpendicular and parallel wave numbers that arek_⊥^-8/3 and k_^-7/2 at β_p=1 (being slightly steeper in k_⊥ and much more steeper in k_for the low-β case, roughly going as k_⊥^-3 and k_^-5). This scenario has been supported also by intermittent analysis, which revealeddeviations from monofractality and a strongly intermittent behavior at the kinetic scales (the β_p=0.2 regime being slightly more intermittent than the intermediate-β case).Moreover, we find that the turbulent cascade is dominated by Alfvénic/KAWtype of fluctuations at β_p=1, whereas the low-β case presents a more complex scenario suggesting the simultaneous presence ofdifferent types of fluctuations, including magnetosonic and whistler-like ones. This picture seems to be supported also by the differences in the parallel correlation length of the magnetic fluctuations between the two regimes,thus possibly indicating a different decorrelation mechanism along the mean field.Nevertheless, signatures that may be interpreted asion Bernstein modes emerge in both regimes,although further focused investigations are needed in order to clarify this point. The presence of IB fluctuations would indeed point to a link between kinetic turbulence, dissipation andreconnection <cit.>, as suggested also bythe spectral properties <cit.>.The results presented here are in qualitative agreement with previous two-dimensional studies performed with fully-kinetic and hybrid-kineticsimulations <cit.>,although we stress that this scenario needs to include other important effects,such as the role of magnetic reconnection and the coupling withcoherent structures <cit.>. While the hybrid-kinetic model does not include all the electron kinetic physics and larger resolutions would be needed to better separate the electrons and protons kinetic scales, i.e., with a realistic mass ratio, the results presented herehave a far-reaching implications in the context of solar-wind turbulence,from a possible dependence of the kinetic-scale cascade on the plasma β parameter to the understanding of the fundamental processes at play in collisionless kinetic plasma turbulence.The authors acknowledge valuable discussions withF. Pegoraro, L. Franci, S. Landi, E. Papini, D. Groselj and C. H. K. Chen. S.S.C. and F.C. thank C. Cavazzoni (CINECA, Italy)for his essential contribution to the HVM code parallelization and performances. 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I. 1973, Soviet Journal of Experimental and Theoretical Physics, 37, 73 | http://arxiv.org/abs/1707.08429v2 | {
"authors": [
"S. S. Cerri",
"S. Servidio",
"F. Califano"
],
"categories": [
"physics.plasm-ph",
"astro-ph.SR",
"physics.space-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170726132643",
"title": "Kinetic cascade in solar-wind turbulence: 3D3V hybrid-kinetic simulations with electron inertia"
} |
http://arxiv.org/abs/1707.08953v2 | {
"authors": [
"Jose Beltran Jimenez",
"Lavinia Heisenberg",
"Gonzalo J. Olmo",
"Diego Rubiera-Garcia"
],
"categories": [
"hep-th",
"astro-ph.CO",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170727175746",
"title": "On gravitational waves in Born-Infeld inspired non-singular cosmologies"
} |
|
The Caccioppoli UltrafunctionsVieri Benci, Luigi Carlo Bersellicorresponding author, and Carlo Romano Grisanti====================================================================================Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, ITALY Ultrafunctions are a particular class of functions defined on a hyperreal field ℝ^∗⊃ℝ. They have been introduced and studied in some previous works (<cit.>,<cit.>,<cit.>). In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to PDE's and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.Keywords. Ultrafunctions, Non Archimedean Mathematics, Non Standard Analysis, Delta function, distributions. § INTRODUCTION The Caccioppoli ultrafunctions can be considered as a kind generalized functions. In many circumstances, the notion of real function is not sufficient to the needs of a theory and it is necessary to extend it. Among people working in partial differential equations, the theory of distributions of L. Schwartz is the most commonly used, but other notions of generalized functions have been introduced by J.F. Colombeau <cit.> and M. Sato <cit.>. This paper deals with a new kind of generalized functions, called “ultrafunctions”, which have been introduced recently in <cit.> and developed in <cit.>. They provide generalized solutions to certain equations which do not have any solution, not even among the distributions. Actually, the ultrafunctions are pointwise defined on a subset of (ℝ^∗)^N, where ℝ^∗ is the field of hyperreal numbes, namely the numerical field on which nonstandard analysis (NSA in the sequel) is based. We refer to Keisler <cit.> for a very clear exposition of NSA and in the following, starred quantities are the natural extensions of the corresponding classical quantities.The main novelty of this paper is that we introduce the space of Caccioppoli ultrafunctions V_Λ(Ω). They satisfy special properties which are very powerful in applications to Partial Differential Equations and Calculus of Variations. The construction of this space is rather technical, but contains some relevant improvements with respect to the previous notions present in the literature (see e.g. <cit.>). The main peculiarities of the ultrafunctions in V_Λ(Ω) are the following: there exist a generalized partial derivative D_i and a generalized integral(called poinwise integral) such that* the generalized derivative is a local operator namely, if supp(u)⊂ E^* (where E is an open set), then supp(D_iu)⊂ E^*.* ∀ u,v∈ V_Λ(Ω), D_iuv dx=- uD_iv dx; * the “generalized” Gauss theorem holds for any measurable set A (see Theorem <ref>)_AD·ϕ dx=_∂ Aϕ·𝐧_A dS; * to any distribution T∈𝒟^'(Ω) we can associate an equivalence class of ultrafunctions [u] such that, ∀ v∈[u], ∀φ∈𝒟(Ω),st( vφ° dx)=⟨ T,φ⟩ ,where st(·) denotes the standard part of an hyperreal number. The most relevant point, which is not present in the previous approaches to ultrafunctions, is that we are able the extend the notion of partial derivative so that it is a local operator and it satisfies the usual formula valid when integrating by parts, at the price of a suitable extension of the integral as well. In the proof of this fact, the Caccioppoli sets play a fundamental role.It is interesting to compare the result about the Caccioppoli ultrafunctions with the well-known Schwartz impossibility theorem: “there does not exist a differential algebra (𝔄,+,⊗,D) in which the distributions can be embedded, where D is a linear operator that extends the distributional derivative and satisfies the Leibniz rule (namely D(u⊗ v)=Du⊗ v+u⊗ Dv) and ⊗ is an extension of the pointwise product on 𝒞(ℝ).” The ultrafunctions extend the space of distributions; they do not violate the Schwartz theorem since the Leibniz rule, in general, does not hold (see Remark <ref>). Nevertheless, we can prove the integration by parts rule (<ref>) and the Gauss' divergence theorem (with the appropriate extensionof the usual integral), which are the main tools used in the applications. These results are a development of the theory previously introduced in <cit.> and <cit.>.The theory of ultrafunctions makes deep use of the techniques of NSA presented via the notion of Λ-limit. This presentation has the advantage that a reader, which does not know NSA, is able to follow most of the arguments. In the last section we present some very simple examples to show that the ultrafunctions can be used to perform a precise mathematical analysis of problems which are not tractable via the distributions. §.§ Plan of the paper In section <ref>, we present a summary of the theory of Λ-limits and their role in the development of the ultrafunctions using nonstandard methods, especially in the context of transferring as much as possible the language of classical analysis. In Section <ref>, we define the notion of ultrafunctions, with emphasis on the pointwise integral. In Section <ref>, we define the most relevant notion, namely the generalized derivative, and its connections with the pointwise integral, together with comparison with the classical and distributional derivative. In Section <ref>, we show how to construct a space satisfying all the properties of the generalized derivative and integrals. This Section is the most technical and can be skipped in a first reading. Finally, in Section <ref>, we present a general result and two very simple variational problem. In particular, the second problem is very elementary but without solutions in the standard H^1-setting. Nevertheless it has a natural and explicit candidate as solution. We show how this can be described by means of the language of ultrafunctions.§.§ Notations If X is a set and Ω is a subset of ℝ^N, then* 𝒫(X) denotes the power set of X and 𝒫_fin(X) denotes the family of finite subsets of X; * 𝔉(X,Y) denotes the set of all functions from X to Y and 𝔉(Ω)=𝔉(Ω,ℝ); * 𝒞(Ω) denotes the set of continuous functions defined on Ω; * 𝒞^k(Ω) denotes the set of functions defined on Ω which have continuous derivatives up to the order k;* H^k,p(Ω) denotes the usual Sobolev space of functions defined on Ω; * if W(Ω) is any function space, then W_c(Ω) will denote de function space of functions in W(Ω) having compact support; * 𝒞_0(Ω∪Ξ),Ξ⊆∂Ω, denotes the set of continuous functions in 𝒞(Ω∪Ξ) which vanish for x∈Ξ;* 𝒟(Ω) denotes the set of the infinitely differentiable functions with compact support defined on Ω; 𝒟^'(Ω) denotes the topological dual of 𝒟(Ω), namely the set of distributions on Ω; * for any ξ∈(ℝ^N)^∗,ρ∈ℝ^∗, we set 𝔅_ρ(ξ)={ x∈(ℝ^N)^∗: |x-ξ|<ρ}; * 𝔰𝔲𝔭𝔭(f)=(supp(f))^* where supp is the usual notion of support of a function or a distribution;* 𝔪𝔬𝔫(x)={y∈(ℝ^N)^∗:x∼ y} where x∼ y means that x-y is infinitesimal;* 𝔤𝔞𝔩(x)={y∈(ℝ^N)^∗:x-y is finite} ;* if W is a generic function space, its topological dual will be denoted by W^' and the pairing by ⟨·,·⟩ _W* we denote by χ_X the indicator (or characteristic) function of X, namelyχ_X(x)= 1 if x∈ X0 if x∉ X ; * |X| will denote the cardinality of X. § Λ-THEORY In this section we present the basic notions of Non Archimedean Mathematics and of Nonstandard Analysis, following a method inspired by <cit.> (see also <cit.> and <cit.>). §.§ Non Archimedean Fields Here, we recall the basic definitions and facts regarding non-Archimedean fields. In the following, 𝕂 will denote an ordered field. We recall that such a field contains (a copy of) the rational numbers. Its elements will be called numbers. Let 𝕂 be an ordered field and ξ∈𝕂. We say that: * ξ is infinitesimal if, for all positive n∈ℕ, |ξ|<1/n; * ξ is finite if there exists n∈ℕ such as |ξ|<n; * ξ is infinite if, for all n∈ℕ, |ξ|>n (equivalently, if ξ is not finite).An ordered field 𝕂 is called Non-Archimedean if it contains an infinitesimal ξ≠0. It is easily seen that all infinitesimal are finite, that the inverse of an infinite number is a nonzero infinitesimal number, and that the inverse of a nonzero infinitesimal number is infinite. A superreal field is an ordered field 𝕂 that properly extends ℝ. It is easy to show, due to the completeness of ℝ, that there are nonzero infinitesimal numbers and infinite numbers in any superreal field. Infinitesimal numbers can be used to formalize a new notion of closeness:We say that two numbers ξ,ζ∈𝕂 are infinitely close if ξ-ζ is infinitesimal. In this case, we write ξ∼ζ. Clearly, the relation ∼ of infinite closeness is an equivalence relation and we have the following theoremIf 𝕂 is a superreal field, every finite number ξ∈𝕂 is infinitely close to a unique real number r∼ξ, called the standard part of ξ. Given a finite number ξ, we denote its standard part by st(ξ), and we put st(ξ)=±∞ if ξ∈𝕂 is a positive (negative) infinite number. Let 𝕂 be a superreal field, and ξ∈𝕂 a number. The monad of ξ is the set of all numbers that are infinitely close to it: 𝔪𝔬𝔫(ξ)={ζ∈𝕂:ξ∼ζ},and the galaxy of ξ is the set of all numbers that are finitely close to it: 𝔤𝔞𝔩(ξ)={ζ∈𝕂:ξ-ζ is finite} By definition, it follows that the set of infinitesimal numbers is 𝔪𝔬𝔫(0) and that the set of finite numbers is 𝔤𝔞𝔩(0). §.§ The Λ-limit In this section we introduce a particular non-Archimedean field by means of Λ-theory [Readers expert in nonstandard analysis will recognize that Λ-theory is equivalent to the superstructure constructions of Keisler (see <cit.> for a presentation of the original constructions of Keisler). ] (for complete proofs and further informations the reader is referred to <cit.>, <cit.> and <cit.>). To recall the basics of Λ-theory we have to recall the notion of superstructure on a set (see also <cit.>): Let E be an infinite set. The superstructure on E is the setV_∞(E)=⋃_n∈ℕV_n(E),where the sets V_n(E) are defined by induction setting V_0(E)=Eand, for every n∈ℕ, V_n+1(E)=V_n(E)∪𝒫(V_n(E)). Here 𝒫(E) denotes the power set of E. Identifying the couples with the Kuratowski pairs and the functions and the relations with their graphs, it follows that V_∞(E) contains almost every usual mathematical object that can be constructed starting with E; in particular, V_∞(ℝ), which is the superstructure that we will consider in the following, contains almost every usual mathematical object of analysis.Throughout this paper we let 𝔏=𝒫_fin(V_∞(ℝ))and we order 𝔏 via inclusion. Notice that (𝔏,⊆) is a directed set. We add to 𝔏 a point at infinity Λ∉𝔏, and we define the following family of neighborhoods of Λ: {{Λ}∪ Q| Q∈𝒰},where 𝒰 is a fine ultrafilter on 𝔏, namely a filter such that* for every A,B⊆𝔏, if A∪ B=𝔏 then A∈𝒰 or B∈𝒰; * for every λ∈𝔏 the set Q(λ):={μ∈𝔏|λ⊆μ}∈𝒰. In particular, we will refer to the elements of 𝒰 as qualified sets and we will write Λ=Λ(𝒰) when we want to highlight the choice of the ultrafilter. A function φ: 𝔏→ E will be called net (with values in E). If φ(λ) is a real net, we have that lim_λ→Λφ(λ)=Lif and only if∀ε>0, ∃ Q𝒰,suchthat∀λQ, |φ(λ)-L|<ε. As usual, if a property P(λ) is satisfied by any λ in a neighborhood of Λ, we will say that it is eventually satisfied.Notice that the Λ-topology satisfies these interesting properties:If the net φ(λ) takes values in a compact set K, then it is a converging net.Suppose that the net φ(λ) has a subnet converging to L∈ℝ. We fix ε>0 arbitrarily and we have to prove that Q_ε∈𝒰 whereQ_ε={λ∈𝔏 | |φ(λ)-L| <ε} .We argue indirectly and we assume that Q_ε∉𝒰 Then, by the definition of ultrafilter, N=𝔏\ Q_ε∈𝒰 and hence∀λ∈ N, |φ(λ)-L|≥ε.This contradicts the fact that φ(λ) has a subnet which converges to L. Assume that φ: 𝔏→ E, where E is a first countable topological space; then if lim_λ→Λφ(λ)=x_0,there exists a sequence {λ_n} in 𝔏 such thatlim_n→∞φ(λ_n)=x_0We refer to the sequence φ_n:=φ(λ_n) as a subnet of φ(λ).It follows easily from the definitions. Let φ: 𝔏→ V be a net with value in bounded set of a reflexive Banach space equipped with the weak topology; then v:=lim_λ→Λφ(λ),is uniquely defined and there exists a sequence n↦φ(λ_n) which converges to v.The set of the hyperreal numbers ℝ^∗⊃ℝ is a set equipped with a topology τ such that * every net φ: 𝔏→ℝ has a unique limit in ℝ^∗, if 𝔏 and ℝ^∗ are equipped with the Λ and the τ topology respectively;* ℝ^∗ is the closure of ℝ with respect to the topology τ;* τ is the coarsest topology which satisfies the first property.The existence of such ℝ^∗ is a well known fact in NSA. The limit ξ∈ℝ^∗ of a net φ: 𝔏→ℝ with respect to the τ topology, following <cit.>, is called the Λ-limit of φ and the following notation will be used:ξ=lim_λ↑Λφ(λ)namely, we shall use the up-arrow “↑” to remind that the target space is equipped with the topology τ.Given ξ:=lim_λ↑Λφ(λ)andη:=lim_λ↑Λψ(λ),we setξ+η:=lim_λ↑Λ(φ(λ)+ψ(λ)),and ξ·η:=lim_λ↑Λ(φ(λ)·ψ(λ)). Then the following well known theorem holds:The definitions (<ref>) and (<ref>) are well posed and ℝ^*, equipped with these operations, is a non-Archimedean field. We observe that the field of hyperreal numbers is defined as a sort of completion of the real numbers. In fact ℝ^* is isomorphic to the ultrapowerℝ^𝔏/ℑwhereℑ={φ:𝔏→ℝ | φ(λ)=0 eventually}The isomorphism resembles the classical one between the real numbers and the equivalence classes of Cauchy sequences. This method is well known for the construction of real numbers starting from rationals. §.§ Natural extension of sets and functions For our purposes it is very important that the notion of Λ-limit can be extended to sets and functions (but also to differential and integral operators) in order to have a much wider set of objects to deal with, to enlarge the notion of variational problem and of variational solution. So we will define the Λ-limit of any bounded net of mathematical objects in V_∞(ℝ) (a net φ: 𝔏→ V_∞(ℝ) is called bounded if there exists n∈ℕ such that, ∀λ∈𝔏,φ(λ)∈ V_n(ℝ)). To do this, let us consider a netφ:𝔏→ V_n(ℝ).We will define lim_λ↑Λφ(λ) by induction on n. For n=0, lim_λ↑Λφ(λ) is defined by (<ref>). By induction we may assume that the limit is defined for n-1 and we define it for the net (<ref>) as follows:lim_λ↑φ(λ)={lim_λ↑ψ(λ) | ψ:𝔏→ V_n-1(ℝ), ∀λ∈𝔏, ψ(λ)∈φ(λ)} . A mathematical entity (number, set, function or relation) which is the Λ-limit of a net is called internal.If ∀λ∈𝔏, E_λ=E∈ V_∞(ℝ), we set lim_λ↑Λ E_λ=E^∗,namelyE^∗:={lim_λ↑Λψ(λ) | ψ(λ)∈ E} .E^∗ is called the natural extension of E. Notice that, while the Λ-limit of a constant sequence of numbers gives this number itself, a constant sequence of sets gives a larger set, namely E^∗. In general, the inclusion E⊆ E^∗ is proper.Given any set E, we can associate to it two sets: its natural extension E^∗ and the set E^σ, whereE^σ={ X^∗ | X∈ E} . Clearly E^σ is a copy of E, however it might be different as set since, in general, X^∗≠ X. If φ: 𝔏→ X is a net with values in a topological space we have the usual limit lim_λ→Λφ(λ),which, by Proposition <ref>, always exists in the Alexandrov compactification X∪{∞}. Moreover we have that the Λ-limit always exists and it is an element of X^*. In addition, the Λ-limit of a net is in X^σ if and only if φ is eventually constant. If X=ℝ and both limits exist, thenlim_λ→Λφ(λ)=st(lim_λ↑Λφ(λ)). The above equation suggests the following definition.If X is a topological space equipped with a Hausdorff topology, and ξ∈ X^* we setSt_X(ξ)=lim_λ→Λφ(λ),if there is a net φ: 𝔏→ X converging in the topology of X and such thatξ=lim_λ↑Λφ(λ),andSt_X(ξ)=∞otherwise.By the above definition we have that lim_λ→Λφ(λ)=St_X(lim_λ↑Λφ(λ)).Let f_λ: E_λ→ℝ,λ∈𝔏,be a net of functions. We define a functionf:(lim_λ↑Λ E_λ)→ℝ^∗as follows: for every ξ∈(lim_λ↑Λ E_λ) we setf(ξ):=lim_λ↑Λ f_λ(ψ(λ)),where ψ(λ) is a net of numbers such that ψ(λ)∈ E_λandlim_λ↑Λψ(λ)=ξ.A function which is a Λ-limit is called internal. In particular if, ∀λ∈𝔏, f_λ=f,f: E→ℝ,we set f^∗=lim_λ↑Λ f_λ.f^∗:E^∗→ℝ^∗ is called the natural extension of f. If we identify f with its graph, then f^∗ is the graph of its natural extension.§.§ Hyperfinite sets and hyperfinite sums An internal set is called hyperfinite if it is the Λ-limit of a net φ:𝔏→𝔉 where 𝔉 is a family of finite sets. For example, if E∈ V_∞(ℝ), the setE=lim_λ↑Λ(λ∩ E)is hyperfinite. Notice thatE^σ⊂E⊂ E^*so, we can say that every set is contained in a hyperfinite set. It is possible to add the elements of an hyperfinite set of numbers (or vectors) as follows: letA:= lim_λ↑ΛA_λ,be an hyperfinite set of numbers (or vectors); then the hyperfinite sum of the elements of A is defined in the following way: ∑_a∈ Aa= lim_λ↑Λ∑_a∈ A_λa.In particular, if A_λ={ a_1(λ),...,a_β(λ)(λ)}with β(λ)∈ℕ,then setting β= lim_λ↑Λ β(λ)∈ℕ^∗,we use the notation∑_j=1^βa_j= lim_λ↑Λ∑_j=1^β(λ)a_j(λ).§ ULTRAFUNCTIONS§.§ Caccioppoli spaces of ultrafunctions Let Ω be an open bounded set in ℝ^N, and let W(Ω) be a (real or complex) vector space such that 𝒟(Ω)⊆ W(Ω)⊆ L^1(Ω). A space of ultrafunctions modeled over the space W(Ω) is given byW_Λ(Ω):=lim_λ↑ΛW_λ(Ω)={lim_λ↑Λf_λ|f_λ∈ W_λ(Ω)} ,where W_λ(Ω)⊂ W(Ω) is an increasing net of finite dimensional spaces such thatW_λ(Ω)⊇ Span(W(Ω)∩λ). So, given any vector space of functions W(Ω), the space of ultrafunction generated by { W_λ(Ω)} is a vector space of hyperfinite dimension that includes W(Ω)^σ, as well as other functions in W(Ω)^*. Hence the ultrafunctions are particular internal functionsu: Ω^*→ℝ^∗.Given a space of ultrafunctions W_Λ(Ω), a σ-basis is an internal set of ultrafunctions {σ_a(x)} _a∈Γ such that, Ω⊂Γ⊂Ω^* and ∀ u∈ W_Λ(Ω), we can writeu(x)=∑_a∈Γu(a)σ_a(x).It is possible to prove (see e.g. <cit.>) that every space of ultrafunctions has a σ-basis. Clearly, if a,b∈Γ then σ_a(b)=δ_ab where δ_ab denotes the Kronecker delta. Now we will introduce a class of spaces of ultrafunctions suitable for most applications. To do this, we need to recall the notion of Caccioppoli set:A Caccioppoli set E is a Borel set such that χ_E∈ BV, namely such that ∇(χ_E) (the distributional gradient of the characteristic function of E) is a finite Radon measure concentrated on ∂ E. The numberp(E):=⟨ |∇(χ_E)|, 1⟩is called Caccioppoli perimeter of E. From now on, with some abuse of notation, the above expression will be written as follows:∫|∇(χ_E)|dx;this expression makes sense since “|∇(χ_E)|dx” is a measure.If E⊂Ω is a measurable set, we define the density function of E as follows:θ_E(x)=st(m(B_η(x)∩ E^*)/m(B_η(x)∩(Ω)^*)),where η is a fixed infinitesimal and m is the Lebesgue measure.Clearly θ_E(x) is a function whose value is 1 in int(E) and 0 in ℝ^N∖E; moreover, it is easy to prove that θ_E(x) is a measurable function and we have that∫θ_E(x)dx=m(E);also, if E is a bounded Caccioppoli set,∫|∇θ_E|dx=p(E).A set E is called special Caccioppoli set if it is open, bounded and m(∂ E)=0. The family of special Caccioppoli sets will be denoted by ℭ(Ω).Now we can define a space V(Ω) suitable for our aims: A function f∈ V(Ω) if and only iff(x)=∑_k=1^nf_k(x)θ_E_k(x)where f_k∈𝒞(ℝ^N), E_k∈ℭ(), and n is a number which depends on f. Such a function, will be called Caccioppoli function.Notice that V(Ω) is a module over the ring 𝒞(Ω) and that, ∀ f∈ V(Ω),(∫|f(x)| dx=0)⇒(∀ x∈ℝ^N, f(x)=0).Hence, in particular, (∫|f(x)|^2dx=0)^1/2,is a norm (and not a seminorm). V_Λ(Ω) is called Caccioppoli space of ultrafunctions if it satisfies the following properties:* V_Λ(Ω) is modeled on the space V(Ω) ;* V_Λ(Ω) has a σ-basis {σ_a(x)} _a∈Γ, Γ⊂(ℝ^N)^*, such that ∀ a∈Γ the support of σ_a is contained in 𝔪𝔬𝔫(a). Notice The existence of a Caccioppoli space of ultrafunctions will be proved in Section <ref>.Usually in the study of PDE's, the function space where to work depends on the problem or equation which we want to study. The same fact is true in the world of ultrafunctions. However, the Caccioppoli space V_Λ(Ω) have a special position since it satisfies the properties required by a large class of problems. First of all V_Λ(Ω)⊂(L^1(Ω))^*. This fact allows to define the pointwise integral (see next sub-section) for all the ultrafunctions. This integral turns out to be a very good tool. However, the space L^1 is not a good space for modeling ultrafunctions, since they are defined pointwise while the functions in L^1 are defined a.e. Thus, we are lead to the space L^1(Ω)∩𝒞(Ω), but this space does not contain functions such as f(x)θ_E(x) which are important in many situations; for example the Gauss' divergence theorem can be formulated as follows∫∇· F(x)θ_E(x)dx=∫_∂ E𝐧· F(x)dS,whenever the vector field F and E are sufficiently smooth. Thus the space V_Λ(Ω) seems to be the right space for a large class of problems.§.§ The pointwise integral From now on we will denote by V_Λ(Ω) a fixed Caccioppoli space of ultrafunctions and by {σ_a(x)} _a∈Γ a fixed σ-basis as in Definition <ref>. If u∈ V_Λ(Ω), we have that∫^*u(x)dx=∑_a∈Γu(a)η_a,whereη_a:=∫^*σ_a(x) dx. The equality (<ref>) suggests the following definition:For any internal function g : Ω^*→ℝ^*, we set g(x)dx:=∑_q∈Γg(q)η_q .In the sequel we will refer toas to the pointwise integral. From Definition <ref>, we have that ∀ u∈ V_Λ(Ω), ∫^*u(x)dx= u(x)dx,and in particular, ∀ f∈ V(Ω), ∫ f(x)dx= f^*(x)dx.But in general these equalities are not true for L^1 functions. For example iff(x)= 1 if x=x_0∈Ω,0 if x≠ x_0 ,we have that ∫^*f^*(x)dx=∫ f(x)dx=0, while f^*(x)dx=η_x_0>0. However, for any set E∈ℭ(Ω) and any function f∈𝒞(Ω)f^*(x)θ_E(x) dx=∫_Ef(x) dx,in factf^*(x)θ_E(x) dx=∫^*f^*(x)θ_E(x) dx=∫ f(x)θ_E(x) dx=∫_Ef(x) dx.Then, if f(x)≥0 and E is a bounded open set, we have thatf^*(x)χ_Edx< f^*(x)θ_E(x) dx< f^*(x)χ_Edx.sinceχ_E<θ_E<χ_E. As we will see in the following part of this paper, in many cases, it is more convenient to work with the pointwise integralrather than with the natural extension of the Lebesgue integral ∫^*. If ∂ E is smooth, we have that ∀ x∈∂ E, θ_E(x)=1/2 and hence, if E is open,f^*(x)χ_E(x) dx = f^*(x)θ_E(x) dx-1/2 f^*(x)χ_∂ E(x) dx =∫_Ef(x) dx-1/2 f^*(x)χ_∂ E(x) dx,and similarlyf^*(x)χ_E(x) dx=∫_Ef(x) dx+1/2 f^*(x)χ_∂ E(x) dx;of course, the term 1/2 f^*(x)χ_∂ E(x) dx is an infinitesimal number and it is relevant only in some particular problems.The pointwise integral allows us to define the following scalar product:u(x)v(x)dx=∑_q∈Γu(q)v(q)η_q. From now on, the norm of an ultrafunction will be given by ‖ u‖ =(|u(x)|^2 dx)^1/2.Notice that u(x)v(x)dx=∫^*u(x)v(x)dx⇔ uv∈ V_Λ(Ω).If {σ_a(x)} _a∈Γ is a σ, then{σ_a(x)/√(η_a)} _a∈Γis a orthonormal basis with respect to the scalar product (<ref>). Hence for every u∈ V_Λ(Ω),u(x)=∑_q∈Γ1/η_q( u(ξ)σ_q(ξ)dξ)σ_q(x).Moreover, we have that By (<ref>), we have thatσ_a(x)σ_b(x)dx=∑_q∈Γσ_a(q)σ_b(q)η_q=∑_q∈Γδ_aqδ_bqη_q=δ_abη_a,then the result. By the above equality, taking b=a we get (<ref>). §.§ The δ-bases Next, we will define the delta ultrafunctions:Given a point q∈Ω^∗, we denote by δ_q(x) an ultrafunction in V_Λ(Ω) such that ∀ v∈ V_Λ(Ω),v(x)δ_q(x) dx=v(q),and δ_q(x) is called delta (or the Dirac) ultrafunction concentrated in q. Let us see the main properties of the delta ultrafunctions: The delta ultrafunction satisfies the following properties:* For every q∈Ω^∗ there exists an unique delta ultrafunction concentrated in q;* for every a, b∈Ω^*, δ_a(b)=δ_b(a);* ‖δ_q‖ ^2=δ_q(q). 1. Let { e_j} _j=1^β be an orthonormal real basis of V_Λ(Ω), and set δ_q(x)=∑_j=1^βe_j(q)e_j(x).Let us prove that δ_q(x) actually satisfies (<ref>). Let v(x)=∑_j=1^βv_je_j(x) be any ultrafunction. Thenv(x)δ_q(x)dx =(∑_j=1^βv_je_j(x))(∑_k=1^βe_k(q)e_k(x))dx= =∑_j=1^β∑_k=1^βv_je_k(q) e_j(x)e_k(x)dx= =∑_j=1^β∑_k=1^βv_je_k(q)δ_jk=∑_j=1^βv_ke_k(q)=v(q).So δ_q(x) is a delta ultrafunction concentrated in q. It is unique: infact, if γ_q(x) is another delta ultrafunction concentrated in q, then for every y∈Ω^* we have:δ_q(y)-γ_q(y)=(δ_q(x)-γ_q(x))δ_y(x)dx=δ_y(q)-δ_y(q)=0,and hence δ_q(y)=γ_q(y) for every y∈Ω^∗.2.δ_a(b)=δ_a(x)δ_b(x) dx=δ_b(a).3. ‖δ_q‖ ^2=δ_q(x)δ_q(x) dx=δ_q(q). By the definition of Γ, ∀ a,b∈Γ, we have thatδ_a(x)σ_b(x)dx=σ_a(b)=δ_ab.From this it follows readily the following result The set {δ_a(x)} _a∈Γ (Γ⊂Ω^∗) is the dual basis of the sigma-basis; it will be called the δ-basis of V_Λ(Ω). Let us examine the main properties of the δ-basis:The δ-basis, satisfies the following properties:* u(x)=∑_q∈Γ[σ_q(ξ)u(ξ)dξ]δ_q(x);* * (i) is an immediate consequence of the definition of δ-basis. (ii) By Theorem <ref>, it follows that:δ_a(x)=∑_q∈Γ1/η_q(δ_a(ξ)σ_q(ξ)dξ)σ_q(x)=∑_q∈Γ1/η_qδ_aqσ_q(x)=1/η_aσ_a(x). (iii) Is an immediate consequence of (ii).§.§ The canonical extension of functions We have seen that every function f: Ω→ℝ has a natural extension f^*: Ω^*→ℝ^*. However, in general, f^* is not an ultrafunction; in fact, it is not difficult to prove that the natural extension f^∗ of a function f, is an ultrafunction if and only if f∈ V(Ω). So it is useful to define an ultrafunction f°∈ V_Λ(Ω) which approximates f^*. More in general, for any internal function u: Ω^*→ℝ^*, we will define an ultrafunction u° as follows:If u: Ω^*→ℝ^* is an internal function, we define u°∈ V_Λ(Ω) by the following formula:u°(x)=∑_q∈Γu(q)σ_q(x);if f: Ω→ℝ, with some abuse of notation, we setf°(x)=(f^*)°(x)=∑_q∈Γf^*(q)σ_q(x). Since Ω⊂Γ, for any internal function u, we have that ∀ x∈Ω,u(x)=u°(x)and∀ x∈Ω^*,u(x)=u°(x)⟺ u∈ V_Λ(Ω). Notice that P°: 𝔉(Ω)^*→ V_Λ(Ω)defined by P°(u)=u° is noting else but the orthogonal projection of u∈𝔉(Ω)^* with respect to the semidefinite bilinear form u(x)h(x)dx.If f∈𝒞(ℝ^N), and E∈ℭ(), then fθ_E∈ V(Ω) and hence(fθ_E)^°=f^*θ_E^* If a function f is not defined on a set S:=Ω∖Θ, by convention, we definef°(x)=∑_q∈Γ∩Θ^*f^*(q)σ_q(x).By the definition above, ∀ x∈Γ, we have that (1/|x|)^°=1/|x|if x≠00 if x=0. If f∈𝒞(Ω), then f°≠ f^* unless f∈ V_Λ(Ω). Let examine what f° looks like. Let f: Ω→ℝ beinFix x_0∈ A. Since A is bounded, there exists a set E∈ℭ(Ω) such that 𝔪𝔬𝔫(x_0)⊂ E^*⊂ A^*.We have that (see Example <ref>)f°(x) =∑_a∈Γf^*(a)σ_a(x) =∑_a∈Γf^*(a)θ_E^*(a)σ_a(x)+∑_a∈Γf^*(a)(1-θ_E^*(a))σ_a(x) =f^*(x)θ_E^*(x)+∑_a∈Γ∖ E^*f^*(a)(1-θ_E^*(a))σ_a(x).Since x_0∈ E^*, θ_E^*(x_0)=1; moreover, since 𝔪𝔬𝔫(x_0)⊂ E^*, by definition <ref>, (ii), ∀ a∈Γ∖ E^*, σ_a(x_0)=σ_x_0(a)=0. Thenf°(x_0)=f^*(x_0).If f∈𝒞(Ω), then, for any x∈Ω^*, such that |x| is finite, we get f°(x)=f^*(x). §.§ Canonical splitting of an ultrafunction In many applications, it is useful to split an ultrafunction u in a part w° which is the canonical extension of a standard function w and a part ψ which is not directly related to any classical object.If u∈ V_Λ(Ω), we setΞ={ x∈Ω | u(x)is infinite}andw(x)= st(u(x)) if x∈Ω∖Ξ0 if x∈Ξ.For every ultrafunction u consider the splittingu=w°+ψwhere* w=w_|Ω∖Ξ and w° which is defined by Definition <ref>, is called the functional part of u;* ψ:=u-w° is called the singular part of u.We will refer to S:={ x∈Ω^* | ψ(x)0} as to the singular set of the ultrafunction u.Notice that w°, the functional part of u, may assume infinite values, but they are determined by the values of w, which is a standard function defined on Ω∖Ξ.Take ε∼0, and u(x)=1/x^2+ε^2.In this case * w(x)=1/x^2,* ψ(x)= -ε^2/x^2(x^2+ε^2)if x≠0 1/ε^2if x=0,* S:={ x∈ℝ^* | ψ(x)0}⊂𝔪𝔬𝔫(0). We conclude this section with the following trivial propositions which, nevertheless, are very useful in applications: Let W be a Banach space such that 𝒟(Ω)⊂ W⊆ L_loc^1(Ω) and assume that u=w°+ψ is the canonical splittingu:=lim_λ↑Λ u_λ, lim_n→∞u_n=wweaklyinWand ∀ v∈ W,ψ v dx∼0.lim_n→∞‖ u_n-w‖ _W=0thenIt is an immediate consequence of Proposition <ref>. If we use the notation introduced in Definition <ref>, the above proposition can be reformulated as follows:If u:=lim_λ↑Λ u_λ, w=St_W_weak(u).If w=St_W(u). An immediate consequence of Proposition <ref> is the following:If w∈ L^1(Ω) thenw°(x)dx∼ w(x)dx.Since V_Λ(Ω) is dense in L^1(Ω) there is a sequence u_n∈ V_Λ(Ω) which converges strongly to w in L^1(Ω). Now set u:=lim_λ↑Λ u_|λ|.By Proposition <ref>, we have that u=w°+ψwith ‖ψ‖ _L^1*∼0. Since u and w° are in V_Λ(Ω), then also ψ∈ V_Λ(Ω), so that ψ dx=∫^*ψ dx∼0. Thenu(x)dx w°(x)dx.On the other hand, u(x)dx=∫^*u(x)dx=lim_λ↑Λ ∫ u_|λ|dx∼lim_λ→Λ ∫ u_|λ|dx=lim_n→∞ ∫_Ωu_ndx= w(x)dx.§ DIFFERENTIAL CALCULUS FOR ULTRAFUNCTIONS In this section, we will equip the Caccioppoli space of ultrafunctions V_Λ(Ω) with a suitable notion of derivative which generalizes the distributional derivative. Moreover we will extend the Gauss' divergence theorem to the environment of ultrafunctions and finally we will show the relationship between ultrafunctions and distributions. §.§ The generalized derivative If u∈ V_Λ(Ω)∩[C^1(Ω)]^*, then, ∂_i^*u is well defined and hence, using Definition <ref>, we can define an operatorD_i : V_Λ(Ω)∩[C^1(Ω)]^*→V_Λ(Ω)as followsD_iu°=(∂_i^*u)°.However it would be useful to extend the operator D_i to all the ultrafunctions in V_Λ(Ω) to include in the theory of ultrafunctions also the weak derivative. Moreover such an extension allows to compare ultrafunctions withdistributions. In this section we will define the properties that a generalized derivative must have (Definition <ref>) and in section <ref>, we will show that these properties are consistent; we will do that by a construction of the generalized derivative.The generalized derivative D_i: V_Λ(Ω)→ V_Λ(Ω)is an operator defined on a Caccioppoli ultrafunction space V_Λ(Ω) which satisfies the following properties:* V_Λ has σ-basis {σ_a(x)} _a∈Γ, such that ∀ a∈Γ the support of D_iσ_a is contained in 𝔪𝔬𝔫(a);* if u∈ V_Λ(Ω)∩[C^1(Ω)]^*, then, D_iu°=(∂_i^*u)°; * ∀ u,v∈ V_Λ(Ω), D_iuv dx=- uD_iv dx * if E∈ℭ(Ω), then ∀ v∈ V_Λ(Ω),D_iθ_Ev dx=-∫_∂ E^*v (𝐞_i·𝐧_E) dSwhere 𝐧_E is the measure theoretic unit outer normal, integrated on the reduced boundary of E with respect to the (n-1)-Hausdorff measure dS (see e.g. <cit.>) and (𝐞_1,....,𝐞_N) is the canonical basis of ℝ^N. We remark that, in the framework of the theory of Caccioppoli sets, the classical formula corresponding to IV is the following: ∀ v∈𝒞(Ω),∫∂_iθ_Ev dx=-∫_∂ Ev (𝐞_i·𝐧_E) dS.The existence of a generalized derivative will be proved in section <ref>.Now let us define some differential operators:* ∇=(∂_1,...,∂_N) will denote the usual gradient of standard functions;* ∇^*=(∂_1^*,...,∂_N^*) will denote the natural extension of the gradient (in the sense of NSA);* D=(D_1,...,D_N) will denote the canonical extension of the gradient in the sense of the ultrafunctions (Definition <ref>).Next let us consider the divergence: * ∇·ϕ=∂_1ϕ_1+...+∂_Nϕ_N will denote the usual divergence of standard vector fields ϕ∈[𝒞^1(Ω)]^N;* ∇^*·ϕ=∂_1^*ϕ_1+...+∂_N^*ϕ_N will denote the divergence of internal vector fields ϕ∈[𝒞^1(Ω)^*]^N;* D·ϕ will denote the unique ultrafunction D·ϕ∈ V_Λ(Ω) such that, ∀ v∈ V_Λ(Ω),D·ϕ vdx=-ϕ(x)· Dv dx. And finally, we can define the Laplace operator: * △° or D^2 will denote the Laplace operator defined by D∘ D.§.§ The Gauss' divergence theoremBy Definition <ref>, IV, for any set E∈ℭ_Λ(Ω) and v∈ V_Λ(Ω),D_iθ_Ev dx=-∫_∂ E^*v (𝐞_i·𝐧_E) dS,and by Definition <ref>, III, D_iv θ_Edx=∫_∂ E^*v (𝐞_i·𝐧_E) dS.Now, if we take a vector field ϕ=(v_1,...,v_N)∈[V_Λ(Ω)]^N, by the above identity, we getD·ϕ θ_E dx=∫_∂ E^*ϕ·𝐧_E dS.Now, if ϕ∈𝒞^1(Ω) and ∂ E is smooth, we get the divergence Gauss' theorem:∫_E∇·ϕ dx=∫_∂ Eϕ·𝐧_E dS. Then, (<ref>) is a generalization of the Gauss' theorem which makes sense for any set E∈ℭ_Λ(Ω). Next, we want to generalize Gauss' theorem to any subset A⊂Ω. First of all we need to generalize the notion of Caccioppoli perimeter p(E) to any arbitrary set. As we have seen in Section <ref>, if E∈ℭ(Ω) is a special Caccioppoli set, we have that p(E)=∫|∇θ_E| dx,and it is possible to define a (n-1)-dimensional measure dS as follows∫_∂ Ev(x) dS:=∫|∇θ_E| v(x) dx.In particular, if the reduced boundary of E coincides with ∂ E, we have that (see <cit.>)∫_∂ Ev(x) dS=∫_∂ Ev(x) dℋ^N-1. Then, the following definition is a natural generalization:If A is a measurable subset of Ω, we setp(A):=|Dθ_A^°| dxand ∀ v∈ V_Λ(Ω),_∂ Av(x) dS:= v(x) |Dθ_A^°| dx.Notice that_∂ Av(x) dS≠ v(x)χ_∂ A^°(x) dx.In fact the left hand term has been defined as follows:_∂ Av(x) dS=∑_x∈Γv(x) |Dθ_A^°(x)| η_xwhile the right hand term isv(x)χ_∂ A^°(x) dx=∑_x∈Γv(x)χ_∂ A^°(x) η_x,in particular if ∂ A is smooth and v(x) is bounded, ∑__x∈Γv(x)χ_∂ A^°(x) η_x is an infinitesimal number. If A is an arbitrary measurable subset of Ω, we have thatD·ϕ θ_A^° dx=_∂ Aϕ·𝐧_A^°(x) dS,where𝐧_A^°(x)= -Dθ_A^°(x)/|Dθ_A^°(x)|if Dθ_A^°(x)≠00 if Dθ_A^°(x)=0By Theorem <ref>, III,D·ϕ θ_A^°dx=-ϕ· Dθ_A^°dx,then, using the definition of 𝐧_A^°(x) and (<ref>), the above formula can be written as follows:D·ϕ θ_A^°dx=ϕ·𝐧_A^° |Dθ_A^°| dx=_∂ Aϕ·𝐧_A^° dS. Clearly, if E∈ℭ_Λ(Ω), then_∂ Eϕ·𝐧_E^° dS=∫_∂ Eϕ·𝐧_E dS.If A is the Koch snowflake, then the usual Gauss' theorem makes no sense since p(A)=+∞; on the other hand equation (<ref>) holds true. Moreover, the perimeter in the sense of ultrafunction is an infinite number given by Definition <ref>. In general, if ∂ A is a d-dimensional fractal set, it is an interesting open problem to investigate the relation between its Hausdorff measure and the ultrafunction “measure” dS=|Dθ_A^°| dx.§.§ Ultrafunctions and distributions One of the most important properties of the ultrafunctions is that they can be seen (in some sense that we will make precise in this section) as generalizations of the distributions.The space of generalized distribution on Ω is defined as follows: 𝒟_G^'(Ω)=V_Λ(Ω)/N,where N={τ∈ V_Λ(Ω) | ∀φ∈𝒟(Ω), ∫τφ dx∼0} . The equivalence class of u in V_Λ(Ω) will be denoted by [u]_𝒟.Let [u]_𝒟 be a generalized distribution. We say that [u]_𝒟 is a bounded generalized distribution if, ∀φ∈𝒟(Ω), ∫ uφ^* dxis finite. We will denote by 𝒟_GB^'(Ω) the set of the bounded generalized distributions.We have the following result. There is a linear isomorphism Φ:𝒟_GB^'(Ω)→𝒟^'(Ω),defined by ⟨Φ([u]_𝒟),φ⟩ _𝒟(Ω)=st( u φ^∗ dx).For the proof see e.g. <cit.>.From now on we will identify the spaces 𝒟_GB^'(Ω) and 𝒟^'(Ω); so, we will identify [u]_𝒟 with Φ([u]_𝒟) and we will write [u]_𝒟∈𝒟^'(Ω) and ⟨[u]_𝒟,φ⟩ _𝒟(Ω):=⟨Φ[u]_𝒟,φ⟩_𝒟(Ω)=st( u φ^∗ dx). Moreover, with some abuse of notation, we will write also that [u]_𝒟∈ L^2(Ω), [u]_𝒟∈ V(Ω), etc. meaning that the distribution [u]_𝒟 can be identified with a function f in L^2(Ω), V(Ω), etc. By our construction, this is equivalent to say that f^∗∈[u]_𝒟. So, in this case, we have that ∀φ∈𝒟(Ω),⟨[u]_𝒟,φ⟩ _𝒟(Ω)=st(∫^∗u φ^∗ dx)=st(∫^∗f^∗φ^∗dx)=∫ f φ dx.Since an ultrafunction u: Ω^*→ℝ^* is univocally determined by its value in Γ, we may think of the ultrafunction as being defined only on Γ and to denote them by V_Λ(Γ); the set V_Λ(Γ) is an algebra which extends the algebra of continuous functions 𝒞(Ω) if it is equipped with the pointwise product. Moreover, we recall that, by a well known theorem of Schwartz, any tempered distribution can be represented as ∂^αf, where α is a multi-index and f is a continuous function. If we identify T=∂^αf with the ultrafunction D^αf°, we have that the set of tempered distributions 𝒮' is contained in V_Λ(Γ). However the Schwartz impossibility theorem (see introduction) is not violated since (V_Λ(Γ),+, · , D) is not a differential algebra, since the Leibnitz rule does not hold for some couple of ultrafunctions. § CONSTRUCTION OF THE CACCIOPPOLI SPACE OF ULTRAFUNCTIONS In this section we will prove the existence of Caccioppoli spaces of ultrafunctions (see Definition <ref>) by an explicit construction. §.§ Construction of the space V_Λ(Ω) In this section we will construct a space of ultrafunctions V_Λ(Ω) and in the next section we will equip it with a σ-basis in such a way that V_Λ(Ω) becomes a Caccioppoli space of ultrafunctions according to Definition <ref>.Given a family of open sets ℜ_0 , we say that a family of open sets 𝔅={ E_k} _k∈ K is a basis for ℜ_0 if* ∀ k≠ h, E_k∩ E_h=∅; * ∀ A∈ℜ_0, there is a set of indices K_E⊂ K such thatA=int(⋃_k∈ K_EE_k);* 𝔅 is the smallest family of sets which satisfies the above properties.We we will refer to the family ℜ of all the open sets which can be written by the espression (<ref>) as to the family generated by ℜ_0. Let us verify that For any finite family of special Caccioppoli sets ℭ_0, there exists a basis 𝔅 whose elements are special Caccioppoli sets. Moreover also the set ℭ generated by ℭ_0 consists of special Caccioppoli sets. For any x∈Ω, we set E_x=⋂{ A∈ℭ_0 | x∈ A} .We claim that { E_x} _x∈Ω is a basis. Since ℭ_0 is a finite family, then also { E_x} _x∈Ω is a finite family and hence there is a finite set of indices K such that 𝔅={ E_k} _k∈ K. Now it is easy to prove that 𝔅 is a basis and it consists of special Caccioppoli sets. Also the last statement is trivial.We set ℭ_0,λ(Ω):=λ∩ℭ(Ω),and we denote by 𝔅_λ(Ω) and ℭ_λ(Ω) the relative basis and the generated family which exist by the previous lemma. Now setℭ_Λ(Ω)=lim_λ↑Λℭ_λ(Ω), 𝔅_Λ(Ω)=lim_λ↑Λ𝔅_λ(Ω).The following properties hold true* ℭ_Λ(Ω) and 𝔅_Λ(Ω) are hyperfinite;* ℭ(Ω)^σ⊂ℭ_Λ(Ω)⊂ℭ(Ω)^*;* if E∈ℭ_Λ(Ω), thenθ_E=∑_Q∈ K(E)θ_Q(x), ℭ_Λ(Ω)^* ℭ_Λ(Ω)It follows trivially by the construction.The next lemma is a basic step for the construction of the space V_Λ(Ω).For any Q∈𝔅_Λ(Ω) there exists a setand a family of functions * Ω⊂Ξ⊂Ω^*;* if Q,R∈𝔅_Λ(Ω) and Q≠ R, then * if a∈Ξ(Q), the* * ζ_a≥0;* ζ_a(a)=1. We setr(λ)=1/3min{ d(x,y) | x,y∈λ∩Ω} ,and we denote by ρ a smooth bell shaped function having support in B_1(0); then the functions ρ(x-a_λ/r(λ)), a_λ∈λ∩Ω have disjoint support. We setΞ:={lim_λ↑Λa_λ | a_λ∈λ∩Ω} ,so that Ω⊂Ξ⊂Ω^* and we divide all points a∈Ξ, among sets Ξ(Q), Q∈𝔅_Λ, in such a way that- if a∈ Q then a∈Ξ(Q);- if a∈∂ Q_1∩...∩∂ Q_l there exists a unique Q_j (j≤ l) such that a∈Ξ(Q_j).With this construction, claims 1. and 2. are trivially satisfied. Now, for any a∈Ξ(Q), set ρ_a(x):=lim_λ↑Λρ(x-a_λ/r(λ)),andζ_a(x):=ρ_a(x)θ_Q(x)/ρ_a(a)θ_Q(a).It is easy to check that the functions ζ_a satisfy 3.,4.,5. and 6. We setV_Λ^1(Ω)=span({ζ_a} _a∈Ξ)+lim_λ↑Λ(λ∩𝒞^1(Ω)),andV_Λ^1(Q)={ uθ_Q | u∈ V_Λ^1(Ω)} ;so we have that, for any a∈Ξ(Q), ζ_a∈ V_Λ^1(Q). Also, we set V_Λ^0(Ω)=Span({ f, ∂_if, fg, g∂_if|f,g∈ V_Λ^1(Ω),i=1,...,N} +lim_λ↑Λ(λ∩𝒞(Ω)))andV_Λ^0(Q)={ uθ_Q | u∈ V_Λ^0(Ω)} . Finally, we can define the V_Λ(Ω) as follows:V_Λ(Ω)=Q∈𝔅_Λ(Ω)⊕V_Λ^0(Q).Namely, if u∈ V_Λ(Ω), thenu(x)=∑_Q∈𝔅_Λ(Ω)u_Q(x)θ_Q(x),with u_Q∈ V_Λ^0(Ω). §.§ The σ-basisIn this section, we will introduce a σ-basis in such a way that V_Λ(Ω) becomes a Caccioppoli space of ultrafunctions, according to Definition <ref>. There exists a σ-basis V_Λ(Ω), {σ_a(x)} _a∈Γ, such that* Ω⊂Γ⊂Ω^*;* Γ=⋃__Q∈𝔅_Λ(Ω)Q_Γ, where Q∩Γ⊂ Q_Γ⊂Q∩Γ, and Q_Γ∩ R_Γ=∅forQ≠ R; * {σ_a(x)} _a∈ Q_Γ is a σ-basis V_Λ^0(Q).First we introduce in V_Λ(Ω) the following scalar product:⟨ u,v⟩ =∫^*uv dx. For any Q∈𝔅_Λ(Ω) we setZ(Q)={∑_a∈Ξ(Q)γ_aζ_a(x) | γ_a∈ℝ^*} ,where Ξ(Q) and the functions{ζ_a} _a∈Ξ are defined in Lemma <ref>.If we set 𝔡_a(x)=ζ_a(x)/∫^*|ζ_a(x)|^2 dxwe have that{𝔡_a(x)} _a∈Ξ(Q)is a δ-basis for Z(Q)⊂ V_Λ^0(Q) (with respect to the scalar product (<ref>)). In fact, if u∈ Z(Q), then u(x)=∑_b∈Ξ(Q)u(b)ζ_b(x), and hence, by Lemma <ref>, it follows that∫^*u(x)𝔡_a(x) dx =∫^*∑_b∈Ξ(Q)u(b)ζ_b(x)𝔡_a(x) dx =∑_b∈Ξ(Q)u(b)∫^*ζ_b(x)𝔡_a(x) dx =∑_b∈Ξ(Q)u(b)∫^*ζ_b(x)(ζ_a(x)/∫^*|ζ_a(x)|^2 dx )dx =∑_b∈Ξ(Q)u(b)δ_ab=u(a). Next, we want to complete this basis and to get a δ-basis for V_Λ^0(Q). To this aim, we take an orthonormal basis { e_k(x)} of Z(Q)^⊥ where Z(Q)^⊥ is the orthogonal complement of Z(Q) in V_Λ^0(Q) (with respect to the scalar product (<ref>)). For every a∈ Q\Ξ, set𝔡_a(x)=∑_ke_k(a)e_k(x);notice that this definition is not in contradiction with (<ref>) since in the latter a∈Ξ.For every v∈ Z(Q)^⊥, we have that∫^*v(x)𝔡_a(x) dx=v(a);in fact∫^*v(x)𝔡_a(x) dx =∫^*(∑_kv_ke_k(x))(∑_he_h(a)e_h(x)) dx=∑_k,hv_ke_h(a)∫^*e_k(x)e_h(x) dx =∑_k,hv_ke_h(a)δ_hk=∑_kv_ke_k(a)=v(a).It is not difficult to realize that {𝔡_a(x)} _a∈ Q\Ξ generates all Z(Q)^⊥ and hence we can select a set Ξ^⋆(Q)⊂ Q\Ξ such that {𝔡_a(x)} _a∈Ξ^⋆(Q) is a basis for Z(Q)^⊥. Taking Q_Γ=Ξ^⋆(Q)∪Ξ(Q),we have that {𝔡_a(x)} _a∈ Q_Γ is a basis for V_Λ^0(Q).Now let {σ_a(x)} _a∈ Q_Γ denote the dual basis of {𝔡_a(x)} _a∈ Q_Γ namely a basis such that, ∀ a,b∈ Q_Γ,∫^*σ_a(x)𝔡_b(x)dx=δ_ab. Clearly it is a σ-basis for V_Λ^0(Q). In fact, if u∈ V_Λ^0(Q), we have thatu(x)=∑_a∈ Q_Γ[∫^*u(t)𝔡_a(t) dt]σ_a(x)=∑_a∈ Q_Γu(a)σ_a(x).Notice that if a∈Ξ(Q), then σ_a(x)=ζ_a(x). The conclusion follows taking Γ:=⋃_Q∈𝔅_Λ(Ω)Q_Γ. By the above theorem, the following corollary follows straightforward. Caccioppoli space of ultrafunctionsIf E∈ℭ_Λ(Ω) (see (<ref>)), we setE_Γ=⋃_Q∈𝔅_Λ(Ω),Q⊂ EQ_ΓIf, for any internal set A, we define_Au(x)dx=∑_a∈Γ∩ Au(a)η_athen, we have the following result:If uθ_E∈ V_Λ(Ω) and E∈ℭ_Λ(Ω), then_E_Γu(x)dx=∫_E^*u(x)dx= u(x)θ_E(x)dx.We have that_E_Γu(x)dx =∑_a∈ E_Γu(a)η_a=∫^*∑_a∈ E_Γu(a)σ_a(x)dx =∫^*∑_Q⊂ E ∑_a∈ Q_Γu(a)σ_a(x)dxSince uθ_E∈ V_Λ(Ω), then, by (<ref>), we can write u(x)θ_E(x)=∑_Q⊂ Eu_Q(x)θ_Q(x). By Th. <ref>, 3., u_Q(x)θ_Q(x)=∑_a∈ Q_Γu(a)σ_a(x)∈ V_Λ^0(Q). Then by (<ref>) _E_Γu(x)dx =∫^*∑_Q⊂ E u_Q(x)θ_Q(x)dx=∑_Q⊂ E ∫^*u_Q(x)θ_Q(x)dx.By this equation and the fact that ∫^*u_Q(x)θ_Q(x)dx=∫_Q^*u(x)dx, it follows that _E_Γu(x)dx=∑_Q⊂ E ∫_Q^*u(x)dx=∫_E^*u(x)dx.Moreover, since u_Qθ_Q∈ V_Λ^0(Q)⊂ V_Λ(Ω),∫^*u_Q(x)θ_Q(x)dx= u_Q(x)θ_Q(x)dx,by (<ref>) we have_E_Γu(x)dx=∑_Q⊂ E u_Q(x)θ_Q(x)dx=∑_Q⊂ Eu_Q(x)θ_Q(x)dx= u(x)θ_E(x)dx. §.§ Construction of the generalized derivative Next we construct a generalized derivative on V_Λ(Ω).We setU_Λ^1=Q∈𝔅_Λ(Ω)⊕V_Λ^1(Q), and U_Λ^0=(U_Λ^1)^⊥,will denote the the orthogonal complement of U_Λ^1 in V_Λ(Ω). According to this decomposition, V_Λ(Ω)=U_Λ^1⊕ U_Λ^0 and we can define the following orthogonal projectorsP_i : V_Λ(Ω)→ U_Λ^i,i=0,1,hence, any ultrafunction u∈ V_Λ(Ω) has the following orthogonal splitting: u=u_1+u_0 where u_i=P_iu.Now we are able to define the generalized partial derivative for u∈ V_Λ^1(Ω).We define the generalized partial derivativeD_i:U_Λ^1(Ω)→ V_Λ(Ω), as follows: D_iuv dx=∑_Q∈𝔅_Λ(Ω)∂_i^*u_Qv_Qθ_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*(u_Q-u_R)v_Q (𝐞_i·𝐧_Q) dS,where 𝔜(Q)={ R∈𝔅_Λ(Ω)∪{ Q_∞} | Q≠ R,∂ Q∩∂ R≠∅}with Q_∞=Ω^*∖⋃_Q∈𝔅_Λ(Ω)Q,Moreover, if u=u_1+u_0∈ U_Λ^1⊕ U_Λ^0=V_Λ(Ω), we setD_iu=D_iu_1-(D_iP_1)^†u_0,where, for any linear operator L, L^† denotes the adjoint operator. Notice that, if u,v∈ U_Λ^1, by Th. <ref>,2., we haveD_iuv dx=∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*u_Qv_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*(u_Q-u_R)v_Q (𝐞_i·𝐧_Q) dS.In fact, if u, v∈ U_Λ^1, then u_Q, v_Q∈ V_Λ^1(Ω) and hence, by (<ref>), ∂_i^*u_Qv_Q∈ V_Λ^0(Ω) and so∂_i^*u_Qv_Qθ_Q dx=∫_Q^*∂_i^*u_Qv_Q dx.The operator D_i: V_Λ(Ω)→ V_Λ(Ω), given by Definition <ref>, satisfies the requests I, II, and III of Definition <ref>.Let us prove property I. If uθ_Q,vθ_R∈ V_Λ(Ω) and Q∩R=∅, by Definition <ref>,D_i(uθ_Q)vθ_R dx=0. Setδ:=max{ diam(Q) | Q∈𝔅_Λ(Ω)} .If q∈ Q and r∈ R, then|q-r|>2δ⇒Q∩R=∅,so, if σ_q∈ V_Λ^0(Q), and r∈ R, then|q-r|>2δ⇒Q∩R=∅,and hence, if we set ε_0>3δ,⋃{ R∈𝔅_Λ(Ω) | Q∩R≠∅}⊂ B_ε_0(q).Since σ_q∈ V_Λ^0(Q),𝔰𝔲𝔭𝔭(D_iσ_q)⊂⋃{ R∈𝔅_Λ(Ω) | Q∩R≠∅}⊂ B_ε_0(q). We prove property II. If u∈[C^1(Ω)]^*∩ V_Λ(Ω), then u=∑__Q∈𝔅_Λ(Ω)uθ_Q, and hence ∀ x∈∂ Q∩∂ R, u_Q(x)-u_R(x)=u(x)-u(x)=0. Then, by (<ref>), we have that, ∀ v∈ V_Λ(Ω)D_iuv dx =∑_Q∈𝔅_Λ(Ω)∂_i^*uv_Qθ_Q dx=∂_i^*u(∑_Q∈𝔅_Λ(Ω)v_Qθ_Q) dx =∂_i^*uv dx.The conclusion follows from the arbitrariness of v.Next let us prove property III. First we prove this property if u,v∈ U_Λ^1. By (<ref>), we have thatD_iuv dx =∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*uv dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*u_Qv_Q (𝐞_i·𝐧_Q) dS +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*u_Rv_Q (𝐞_i·𝐧_Q) dS =∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*uv dx-1/2∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*u_Rv_Q (𝐞_i·𝐧_Q) dS.Next we will compute uD_iv dx and we will show that it is equal to - D_iuv dx. So we replace u with v, in the above equality and we get∑_Q∈𝔅_Λ(Ω)∫_Q^*uD_iv dx =∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*v_Qu_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS.Now, we compute ∑__Q∈𝔅_Λ(Ω)__Q^*∂_i^*v_Qu_Q dx and the last term of the above expression separately. We have that ∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*v_Qu_Qdx =-∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*u_Qv_Qdx+∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS.Moreover, the last term in (<ref>), changing the order on which the terms are added, becomes∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS=∑_R∈𝔅_Λ(Ω)∑_Q∈𝔜(R)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS.In the right hand side we can change the name of the variables Q and R:∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Ru_Q (𝐞_i·𝐧_Q) dS =∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Qu_R (𝐞_i·𝐧_R) dS =-∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Qu_R (𝐞_i·𝐧_Q) dS.In the last step we have used the fact that x∈∂ Q∩∂ R⇒𝐧_R(x)=-𝐧_Q(x). Replacing (<ref>) and (<ref>) in (<ref>) we get ∑_Q∈𝔅_Λ(Ω)∫_Q^*uD_iv dx =-∑_Q∈𝔅_Λ(Ω)∫_Q^*∂_i^*u_Qv_Qdx+1/2∑_Q∈𝔅_Λ(Ω)∫_∂ Q^*u_Qv_Q (𝐞_i·𝐧_Q) dS -1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*v_Qu_R (𝐞_i·𝐧_Q) dS. Comparing (<ref>) and the above equation, we get that ∀ u,v∈ U_Λ^1, D_iuv dx=- uD_iv dx. Let us prove property III in the general case. We have that D_iu=D_iu_1-(D_iP_1)^†u_0hence D_iuv dx = D_iu_1v dx-(D_iP_1)^†u_0v dx = D_iu_1v_1 dx+ D_iu_1v_0 dx- u_0D_iP_1v dx = D_iu_1v_1 dx+ D_iu_1v_0 dx- u_0D_iv_1 dx.Now, replacing u with v and applying property (<ref>) for u_1,v_1∈ U_Λ^1, we get uD_iv dx = u_1D_iv_1 dx- D_iu_1v_0 dx+ u_0D_iv_1 dx =- D_iu_1v_1 dx- D_iu_1v_0 dx+ u_0D_iv_1 dx.Comparing the above equation with (<ref>) we get thatD_iuv dx=- uD_iv dx. Before proving property IV we need the following lemma:The following identity holds true: ∀ E∈ℭ_Λ(Ω), ∀ u∈ V_Λ(Ω)∩[C^1(Ω)]^* and ∀ v∈ V_Λ(Ω), D_i(uθ_E)v dx=∂_iuvθ_E dx-∫_∂ E^*uv (𝐞_i·𝐧_E) dS.We can writeuθ_E=∑_Q∈𝔅_Λ(Ω)h_Quθ_Q,whereh_Q= 1 if Q⊂ E0 if Q⊈ EThen, we have that h_Qu-h_Ru=u, if and only if,R∈𝔜_Q,E^+:={ R∈𝔅_Λ(Ω)∪{ Q_∞} | ∂ R∩∂ E≠∅, Q⊂ E, R⊂Ω∖ E} . Moreover, we have that,h_Qu-h_Ru=-u,if and only if,R∈𝔜_Q,E^-:={ R∈𝔅_Λ(Ω)∪{ Q_∞} | ∂ R∩∂ E≠∅, Q⊂Ω∖ E, R⊂ E} .Otherwise, we have that h_Qu-h_Ru=0or∂ R∩∂ E=∅.Then, by Theorem <ref>,D_i(u_Eθ_E)v dx =∑_Q∈𝔅_Λ(Ω)∂_i^*(h_Qu)vθ_Q dx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜(Q)∫_∂ Q∩∂ R^*(h_Qu-h_Ru)v (𝐞_i·𝐧_Q) dS =∑_Q∈𝔅_Λ(Ω)h_Q_Q_Γ∂_i^*uvdx-1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜_Q,E^+∫_∂ Q∩∂ R^*uv (𝐞_i·𝐧_Q) dS +1/2∑_Q∈𝔅_Λ(Ω)∑_R∈𝔜_Q,E^-∫_∂ Q∩∂ R^*uv (𝐞_i·𝐧_Q) dS =_E_Γ∂_i^*uv dx-1/2∑_Q∈𝔅_Λ(Ω),Q⊂ E∫_∂ Q∩∂ E^*uv (𝐞_i·𝐧_E) dS +1/2∑_Q∈𝔅_Λ(Ω), Q⊂Ω∖ E∫_∂ Q∩∂ E^*uv (𝐞_i·(-𝐧_E)) dS =∂_i^*uvθ_E dx-1/2∫_∂ E^*uv (𝐞_i·𝐧_E) dS+1/2∫_∂ E^*uv (𝐞_i·(-𝐧_E)) dS =∂_i^*uvθ_E dx-∫_∂ E^*uv (𝐞_i·𝐧_E) dS. Then (<ref>) holds true. The operator D_i: V_Λ(Ω)→ V_Λ(Ω) given by Definition <ref> satisfies the request IV of Definition <ref>.The result follows straighforward from (<ref>) just taking u=1. § SOME EXAMPLES We present a general minimization result and two very basic examples which can be analyzed in the framework of ultrafunctions. We have chosen these examples for their simplicity and also because we can give explicit solutions.§.§ A minimization result In this section we will consider a minimization problem. Let Ω be an open bounded set in ℝ^N and let Ξ⊂∂Ω be any nonempty portion of the boundary. We consider the following problem: minimize J(u)=∫_Ω[1/2a(u)|∇ u(x)|^p+f(x,u)]dx,p>1in the set𝒞^1(Ω)∩𝒞_0(Ω∪Ξ).We make the following assumptions: * a(u)≥0 and a(u)≥ k>0 for u sufficiently large;* a(u) is lower semicontinuous;* f(x,u) is a lower semicontinuous function in u, measurable in x, such that |f(x,u)|≤ M|u|^q, with 0<q<p and M∈ℝ^+.Clearly, the above assumptions are not sufficient to guarantee the existence of a solution, not even in a Sobolev space. We refer to <cit.> for a survey of this problem in the framework of Sobolev spaces. On the other hand, we have selected this problem since it can be solved in the framework of the ultrafunctions. More exactly, this problem becomes: find an ultrafunction u∈ V_Λ(Ω) which vanishes on Ξ^* and minimizesJ°(u):=_Ω[1/2a^*(u) |Du(x)|^p-f^*(x,u)]dx, p>1. We have the following result:If assumptions 1,2,3 are satisfied, then the functional J°(u) has a minimizer in the space{ v∈ V_Λ(Ω) | ∀ x∈Ξ^*, v(x)=0 } .Moreover, if J(u) has a minimizer w in V_Λ(Ω), then u=w°. The proof is based on a standard approximation by finite dimensional spaces. Let us observe that, for each finite dimensional space V_λ, we can consider the approximate problem: find u_λ∈ V_λ such that J°(u_λ)=min_v_λ∈ V_λJ°(v_λ).The above minimization problem has a solution, being the functional coercive, due to the hypotheses on a(·) and the fact that p>q. If we take a miminizing sequence u_λ^n∈ V_λ, then we can extract a subsequence weakly converging to some u_λ∈ V_λ. By observing that in finite dimensional spaces all norms are equivalent, it follows also that u_λ^n→ u_λ pointwise. Then, by the lower-semicontinuity of a and f, it follows that the pointwise limit satisfies J°(u_λ)≤lim inf J°(u_λ^n).Next, we use the very general properties of Λ-limits, as introduced in Section <ref>. We setu:=lim_λ↑Λu_λ.Then, taking a generic v:=lim_λ↑Λv_λ, from the inequality J°(u_λ)≤ J°(v_λ), we getJ°(u)≤ J°(v)∀ v∈ V_Λ(Ω).The last statement is trivial.Clearly, under this generality, the solution u could be very wild; however, we can state a regularization result which allows the comparison with variational and classical solutions.Let the assumptions of the above theorem hold. Ifℋ^N-1(Ξ)>0and there exists ν∈ℝ such thata(u)≥ν>0,then, the minimizer has the following formu=w^∘+ψ,where w∈ H^1,p(Ω) and ψ is null in the sense of distributions, namely∀φ∈𝒟(Ω),ψφ^*dx∼0.In this case J°(u)∼v∈ V(Ω)infJ°(v)with V(Ω) as in Definition <ref>. Moreover, if in addition a(u)<M, with M∈ℝ, we have that ‖ψ‖ _H^1,p(Ω)∼0and J°(u)∼ J(w). Finally, if u=w^∘ and J°(u)=J(w).Under the above hypotheses the minimization problem has an additional a priori estimates in H^1,p(Ω), due to the fact that a(·) is bounded away from zero. Moreover, the fact that the function vanishes on a non trascurable (N-1)-dimensional part of the boundary, shows that the generalized Poincar� inequality holds true. Hence, by Proposition <ref>, the approximating net { u_λ} has a subnet { u_n} such thatu_n→ uweakly in H^1,p(Ω).This proves the first statement, since obviously, ψ:=u-w^∘ vanishes in the sense of distributions. In this case, in general the minimum is not achieved in V(Ω) and hence J°(w^∘+ψ)<J(w).Next, if a(·) is bounded also from above, by classical results of semicontinuity of De Giorgi (see Boccardo <cit.> Section 9, Thm. 9.3) J is weakly l.s.c. Thus u is a minimizer and, by well known results, u_n→ u strongly in H^1,p(Ω). This implies, by Proposition <ref>, that u∼ w^∘, and hence ψ is infinitesimal in H^1,p(Ω), proving the second part. Finally, if the minimizer is a function w∈ H^1,p(Ω)∩𝒞(Ω)⊂ V_Λ(Ω), we have that u_λ=w° eventually; then‖ψ‖ _H^1,p(Ω)=0. §.§ The Poisson problem in ℝ^2 Now we cosider this very classical problem:- u=φ(x),φ∈𝒟(ℝ^N). If N≥3, the solution is given by φ(x)∗|x|^-N+2/(N-2)ω_Nand it can be characterized in several ways. First of all, it is the only solution the Schwartz space 𝒮' of tempered distributions obtained via the equation u(ξ)=φ(ξ)/|ξ|^2where T denotes the Fourier tranform of T.Moreover, it is the minimizer of the Dirichlet integralJ(u)=∫[1/2|∇ u(x)|^2-φ(x)u(x)]dxin the space 𝒟^1,2(ℝ^N) which is defined as the completion of 𝒞^1(ℝ^N) with respect to the Dirichlet norm‖ u‖ =√(∫|∇ u(x)|^2dx). Each of these characterizations provides a different method to prove its existence.The situation is completely different when N=2. In this case, it is well known that the fundamental class of solutions is given by2π·φ(x)∗ log|x|, however none of the previus characterization makes sense. In fact, we cannot use equation (<ref>), since 1/|ξ|^2∉ L_loc^1(ℝ^2) and hence 1/|ξ|^2 does not define a tempered distribution. Also, the space 𝒟^1,2(ℝ^2) is not an Hilbert space and the functional J(u) is not bounded from below in 𝒟^1,2(ℝ^2). On the contrary, using the theory of ultrafunctions, we can treat equation (<ref>) independently of the dimension. First of all, we recall that in equation (<ref>) with N≥3, the boundary conditions are replaced by the condition u∈𝒟^1,2(ℝ^N). This is a sort of Dirichlet boundary condition. In the theory of ultrafunctions it is not necessary to replace the Dirichlet boundary condition with such a trick. In fact we can reformulate the problem in the following way: find u∈ V_Λ(B_R) such that -°u =φ°(x) inB_Ru =0 on∂ B_Rwhere ° is the “generalized” Laplacian defined in Section <ref> and R is an infinite number such that χ__B_R∈ V_Λ(ℝ^N). [Such an R exists by overspilling (see e.g. <cit.>); in fact for any r∈ℝ, χ__B_r∈ V_Λ(ℝ^N). ]Clearly, the solutions of the above problem are the minimizers of the Dirichlet integralJ°(u)=[1/2|Du(x)|^2-φ°(x)u(x)]dxin the space u∈ V_Λ(B_R), with the Dirichlet boundary condition. Notice that, in the case of ultrafunctions, the problem has the same structure independently of N. In order to prove the existence, we can use Theorem <ref>[The fact that Ω is a standard set while B_R is an internal set does not change the proof.]. The fact that J°(u) may assume infinite values does not change the structure of the problem and shows the utility of the use of infinite quantities. The relation between the classical solution w and the ultrafunction u is given by u=w°+ψwithSt_𝒟'ψ=0. Some people might be disappointed that u depends on R and it is not a standard function; if this is the case it is sufficient to takew=St_𝒟'uand call w the standard solution of the Poisson problem with Dirichlet boundary condition at ∞. In this way we get the usual fundamental class of solutions and they can be characterized in the usual way also in the case N=2. Concluding, in the framework of ultrafunctions, the Poisson problem with Dirichlet boundary condition is the same problem independently of the space dimension and and it is very similar to the same problem when R is finite. This fact proves that the use of infinite numbers is an advantage which people should not ignore. §.§ An explicit example If the assumptions of Theorem <ref> do not hold true, the solution could not be related to any standard object. For example, if ℋ^N-1(Ξ)=0 and f(x,u)>k|u|^s,(p<N,k>0,0<s<q), the generalized solution u(x) takes infinite values for every x∈Ω. However, there are cases in which u(x) can be identified with a standard and meaningful function, but the minimization problem makes no sense in the usual mathematics. In the example which we will present here, we deal with a functional which might very well represent a physical model, even if the explicit solution cannot be interpreted in a standard world, since it involves the square of a measure (namely δ^2).Let us consider, for γ>0, the one dimensional variational problem of finding the minimum of the functional J(u)=∫_0^11/2a(u)|u'(x)|^2-γ u(x) dxamong the functions such that u(0)=0. In particular we are interested in the case in which a is the following degenerate functiona(s)={1 if s∈(-∞,1)∪(2,+∞), 0 if s∈[1,2]. .Formally, the Euler equation, if u∉[1,2], is u”(x)=-γ.We recall that, by standard arguments, u(1)≠1⇒ u'(1)=0.Hence, if γ<2, the solution is explicitly computed u(x)=γ/2(2x-x^2),since it turns out that 0≤ u(x)<1 for all x∈(0,1) and then the degeneracy does not take place.If γ>2, we see that the solution does not live in H^1(0,1), hence the problem has not a “classical” weak solution. More exactly we have the following result: If u(x)={ 1/2(2γ x-γ x^2)0<x<ξ1/2(-γ x^2+2γ x+2) ξ<x<1 .where Figure 2). First, we show that the generalized solution has at most one discontinuity. In fact, for γ>2 the solution satisfies u(ξ)=1, for some 0<ξ<1, and at that point the classical Euler equations are not anymore valid. On the other hand, where u>2, the solution satisfies a regular problem, hence we are in the situation of having at least the following possible candidate as solution with a jump at ξ=γ-√(γ^2-2γ)/γ=1-√(1-2/γ) and a discontinuity of derivatives at some ξ<η<1. In the specific case, we have (see Figure 1)u(x)={ 1/2(2γ x-γ x^2)0<x<ξ 2 ξ<x<ηγη^2/2-γη-γ x^2/2+γ x+2 η<x<1. .We now show that this is not possible because the functional takes a lower value on the solution with only a jump at x=ξ. In fact, if we consider the function ũ(x) defined as followsũ(x)={ 1/2(2γ x-γ x^2)0<x<ξ1/2(-γ x^2+2γ x+2) ξ<x<1 .we observe that u=ũ in [0,ξ], while u'=ũ'=γ(1-x) for all x∉[ξ,η] and, by explicit computations, we haveJ(ũ)-J(u)=γ^2[-η/2+η^2/2-η^3/6+ξ/2-ξ^2/2+ξ^3/6]=γ^2(Φ(η)-Φ(ξ))<0,ξ<ηwhere Φ(s)=-s/2+s^2/2-s^3/6 is strictly decreasing, since Φ'(s)=-1/2(s-1)^2≤0.Actually, the solution is the one shown in Figure 2. Next we show that there exists a unique point ξ such that the minimum is attained. We write the functional J(u), on a generic solution with a single jump from the value u=1 to the value u=2 at the point 0<ξ<1 and such that the Euler equation is satisfied before and after ξ. We obtain the following value for the functional (in terms of the point ξ) J(u)=F(ξ)=γ^2ξ^3/8-γ^2ξ^2/2+γ^2ξ/2-γ^2/6+3γξ/2-2γ+1/2ξ.We observe that, ∀ γ>2 F(0^+)=+∞and F(1)=-γ^2/24-γ/2+1/2<0.To study the behavior of F(ξ) one has to solve some fourth order equations (this could be possible in an explicit but cumbersome way), so we prefer to make a qualitative study. We evaluate F'(ξ) =3γ^2ξ^2/8-γ^2ξ+γ^2/2+3γ/2-1/2ξ^2,F”(ξ) =3γ^2ξ/4-γ^2+1/ξ^3,F”'(ξ) =3γ^2/4-3/ξ^4,hence we have that F”'(ξ)<0 if and only if 0<ξ<√(2/γ)<1.Consequently the function F”(ξ), which nevertheless satisfies ∀ γ>2 F”(0^+)=+∞, F”(1)=1-γ^2/4<0,has a unique negative minimum at the point √(2/γ). From this, we deduce that there exists one and only one point 0<ξ_0<√(2/γ) such that F”(ξ)>00<ξ<ξ_0 F”(ξ)<0ξ_0<ξ≤1.From the sign of F” we get that F' is strictly increasing in (0,ξ_0) and decreasing in (ξ_0,1). Next ∀ γ>2F'(0^+)=-∞, F'(1)=-γ^2/8+3γ/2-1/2hence, in the case that -γ^2/8+3γ/2-1/2>0 that is γ<2(3+2√(2))∼11.656...,then F' has a single zero ξ_1∈(0,ξ_0) and, being a change of sign, ξ_1 is a point of absolute minimum for F(ξ).If γ≥2(3+2√(2)) the above argument fails.In this case we can observe that F'(1/γ)=γ/2+3/8>0,hence F'(1), which is negative at ξ=1 and near ξ=0 vanishes exactly two times, at the point ξ_1, which is a point of local minimum and at another point ξ_2>ξ_1, which is a point of local maximum. Hence, to find the absolute minimum, we have to compare the value of F(ξ_1) with that of F(1).In particular, we have that ξ_1<√(2/γ) hence, we can show that the minimum is not at ξ=1 simply by observing that we can find at least a point where F(ξ)<F(1) and this point is √(2/γ). In fact M(γ):=F(√(2/γ))-F(1)=γ^3/2/√(2)-γ^2/8-5γ/2+2√(2)√(γ)-1/2≤0.In particular M(2)=0 and M'(γ)=1/4(-γ+3√(2)√(γ)+4√(2)/√(γ)-10)<0.This follows since by the substituting √(γ)→χ, we have to control the sign of the cubic M̃(χ)=-χ^3+3√(2)χ^2-10χ+4√(2)which is negative for all χ≥1, since M̃(1)=-11+7√(2), while M̃'(χ)=-3χ^2+6√(2)χ-10,is a parabola with negative minimum. It could be interesting to study this problem in dimension bigger that one, namely, to minimizeJ°(u)=_Ω(1/2a(u)|Du(x)|^2-γ u(x)) dxin the set { v∈ V_Λ(Ω) | ∀ x∈Ξ^*, u(x)=0 }and in particular to investigate the structure of the singular set of u, both in the general case and in some particular situations in which it is possible to find explicit solutions (e.g. Ω=B_R(0)).10 benci99 Benci V., An algebraic approach to nonstandard analysis, in: Calculus of Variations and Partial differential equations, (G.Buttazzo, et al., eds.), Springer, Berlin (1999), 285-326.ultra Benci V., Ultrafunctions and generalized solutions, Adv. Nonlinear Stud. 13, (2013), 461–486, arXiv:1206.2257.BGG V. Benci, S. Galatolo, M. Ghimenti, An elementary approach to Stochastic Differential Equations using the infinitesimals, in Contemporary Mathematics 530, Ultrafilters across Mathematics, American Mathematical Society, (2010), p. 1-22.BDN2003 Benci V., Di Nasso M., Alpha-theory: an elementary axiomatic for nonstandard analysis, Expo. Math. 21, (2003), 355-386.belu2012 Benci V., Luperi Baglini L., A model problem for ultrafunctions, in: Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems. Electron. J. Diff. Eqns., Conference 21 (2014), 11-21.belu2013 Benci V., Luperi Baglini L., Basic Properties of ultrafunctions, in: Analysis and Topology in Nonlinear Dierential Equations (D. G. Figuereido, J. M. do O, C. Tomei eds.), Progress in Nonlinear Dierential Equations and their Applications, 85 (2014), 61-86.milano Benci V., Luperi Baglini L., Ultrafunctions and applications, DCDS-S, Vol. 7, No. 4, (2014), 593-616. arXiv:1405.4152.algebra Benci V., Luperi Baglini L., A non archimedean algebra and the Schwartz impossibility theorem,, Monatsh. Math. (2014), 503-520.beyond Benci V., Luperi Baglini L., Generalized functions beyond distributions, AJOM 4, (2014), arXiv:1401.5270.gauss Benci V., Luperi Baglini L., A generalization of Gauss' divergence theorem, in: Recent Advances in Partial Dierential Equations and Applications, Proceedings of the International Conference on Recent Advances in PDEs and Applications (V. D. Radulescu, A. Sequeira, V. A. Solonnikov eds.), Contemporary Mathematics (2016), 69-84.nap V. Benci, L. Horsten, S. Wenmackers - Non-Archimedean probability, Milan J. Math. 81 (2013), 121-151. arXiv:1106.1524.boccardo Boccardo L., Croce, G. - Elliptic partial differential equations, De Gruyter, (2013)col85Colombeau, J.-F. Elementary introduction to new generalized functions. North-Holland Mathematics Studies, 113. Notes on Pure Mathematics, 103. North-Holland Publishing Co., Amsterdam, 1985eva-garEvans L.C., Gariepy R.F. - Measure theory and fine properties of functions - Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.keisler76 Keisler H. J., Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston, (1976).nelsonNelson, E. - Internal Set Theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc., 83 (1977), 1165–1198.rob Robinson A., Non-standard Analysis,Proceedings of the Royal Academy of Sciences, Amsterdam (Series A) 64, (1961), 432-440.sa59Sato, M. , Theory of hyperfunctions. II. J. Fac. Sci. Univ. Tokyo Sect. I 8 (1959) 139-193.sa60Sato, M., Theory of hyperfunctions. II. J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 387–437.squaSquassina M., Exstence, multiplicity, perturbation and concentration results for a class of quasi linear elliptic problems, Electronic Journal of Differential Equations, Monograph 07, 2006, (213 pages). ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu | http://arxiv.org/abs/1707.09265v1 | {
"authors": [
"Vieri Benci",
"Luigi Carlo Berselli",
"Carlo Romano Grisanti"
],
"categories": [
"math.AP",
"35D05, 46F30, 03H05, 26E30"
],
"primary_category": "math.AP",
"published": "20170727161020",
"title": "The Caccioppoli Ultrafunctions"
} |
[email protected] for High Technology Materials, University of New Mexico, 1313 Goddard SE, Albuquerque, NM, 87106, USADepartment of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10065, USA Donostia International Physics Center (DIPC), P de Manuel Lardizabal, 4, 20018 San Sebastian, Basque Country, SpainAir Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, NM 87117, USA Center for High Technology Materials, University of New Mexico, 1313 Goddard SE, Albuquerque, NM, 87106, USACenter for High Technology Materials, University of New Mexico, 1313 Goddard se, Albuquerque, NM, 87106, USA We have derived algebraic, analytic expressions for the chemical potential without any restriction on temperature for all types of doped, or extrinsic, gapped Dirac cone materials including gapped graphene, silicene, germaneneand single-layer transition metal dichalcogenides. As an important intermediate step of our derivations,we have established a reliable piecewise-linear model for calculating the density-of-states in molybdenum disulfide,showing good agreement with previously obtained numerical results. For the spin- and valley-resolvedband structure, we obtain an additional decrease of the chemical potential due to thermally induced dopingof the upper subband at finite temperature. It has been demonstrated that since the symmetry between the electron and hole states in MoS_2 is broken, the chemical potential could cross the zero-energy level at sufficiently hightemperature. These results allow us to investigate the collective properties, polarizability, plasmons and theirdamping.Emphasis is placed on low temperatures, when initial electron doping plays a crucial role. We clearly demonstrated the contribution of the initial doping to the finite-temperature collective propertiesof the considered materials. 73.21.-b, 73.63.-b, 71.45.Gm, 73.20.MfFinite-temperature Coulomb Excitations in Extrinsic Dirac StructuresGanesh Balakrishnan December 30, 2023 ====================================================================== § INTRODUCTIONDespite the fact that the microscopic properties of various low-dimensional materials have beenmeticulously examined over a fairly long period of time,<cit.> only successful fabrication of graphenein 2004<cit.> stimulated an intriguingly newresearch effort devoted to the study of atomically thintwo-dimensional (2D) materials. In particular, it was by virtue of its unique, yet unexpected massless Dirac electronic properties that led to high mobility (200,000 cm^2/V·s) and ballistic transport properties. <cit.> In the corners of the Brillouin zone, referred to as K and K' points,there is no energy band gap and the dispersions represent a linear Dirac cone structure. Due to the existence of such an energy spectrum, opening a sufficiently largeand tunable energy gap ingraphene has become an important issue in order to enable electron confinement. Researchers tried toachieve this by adjoininga variety ofinsulatingsubstrates<cit.> or evenexpos graphene to circularly-polarized radiation.<cit.>In finite-width nanoribbons, their energy band structure and gap are modified by the type of insulating `cousin' that is introduced.<cit.> In order to create a truly tunable band gap, one must use a material with large spin-orbit coupling or a buckled structure. In this regard, silicene, a 2D silicon structure was deemed a goodcandidate. Single-monolayer Si possesses a buckled structuresimply because of the larger ionic size of silicon compared to carbon. This results in a large spin–orbit band gap 1.55 meV and the possibility to modify its energy spectrum by applying an external perpendicular electric field<cit.>. These propertiesmake itdisplay an experimentally realizable Kane–Mele type of quantum spin Hall effect, or a topological insulator state,because of the existence of time-reversal symmetry.<cit.> Unlike graphene, the band structure ofsilicene and its nanoribbons <cit.>directly depends on spin and valley indices which give lead to plenty of nanoelectronic, valleytronic and spintronic applications. Germanene, the most recently discovered and fabricated member of atomically thin buckled 2D honeycomb lattices, <cit.> demonstrates substantially larger Fermi velocities and a band gap of 20 - 90 meV. Grown by molecular beam epitaxy<cit.> andinvestigated with x-ray absorption spectroscopy, Ge layers demonstrated satisfactory agreement between the experimentally obtained and theoretically predicted results for its inter-atomic distance. Another important class of innovative 2D materials is represented by direct-gap transition metaldichalchogenides, or TMDC's. Its chemical makeup consists of a transition metal atom M, such as molybdenum or tungsten, and two identical chalcogens C, i.e., sulfur, selenium or tellurium. Schematically, TMDC's aredescribed as MC_2. In our consideration, we are mostly going to focus on MoS_2, as their most studied representative. This material exhibits a semiconductor energy band structure with a very largedirect gap 1.78 eV, in contrast to its bulk states with indirect gap 1.3 eV, and substantial spin-orbit coupling.<cit.> Strictly speaking, MoS_2 is not a Dirac material since the mass terms play a crucialrole in its energy dispersions, however its low-energy Hamiltonian contains a t_0 a_0Σ· k term, corresponding to the linear Dirac cone dispersion.An effective two-band continuum model and lattice Hamiltonian, <cit.> based on the tight-binding model, accounts for the hybridization of the d orbitals in Mo and the p orbitals of sulfur atoms. It gives an adequate description of its low-energy band structure and predicts large spin splitting. <cit.> Due to the breaking of inversion symmetry and spin-orbit coupling, spin and valley physics is observed in all group-IV dichalcogenides, including MoS_2. <cit.>The low-energy states of such systems are no longer massive Dirac fermions since there is a a difference between electron and hole masses as well astrigonal warping effects.<cit.> Strain engineering, used to tune optical and electronic properties of conventional semiconductors, has also beenapplied to molybdenum disulfide, and its modified band structure has been theoretically calculated.<cit.> These unique electronic properties of a single-layerMoS_2were later used to create high-performance transistors operating at room temperature.<cit.>These electronic models and effective Hamiltonian have been widely used to investigate the collective propertiesof TMDC's<cit.> and their influence on the gap transition.<cit.> In optoelectronics, theband structure, spin and valley properties of molybdenum disulfide could be successfully controlled byoff-resonant dressing field.<cit.> Current many-body and quantum field theory methods in condensed matter physics <cit.> have providedhelpful ways to understand the electronic and transport properties of low-dimensional solids, includingdiverse bucked honeycomb materials.<cit.>In mostof these theories, we find the dynamical polarization function, or polarizability, to be the mainstay,fundamental quantity, describing the screening of an external potential by interacting electrons.<cit.> Also, the dynamic polarization function plays a key role in calculating the plasmon excitations, due tothe chargedensity oscillations, which occur in metals and semiconductors. Specifically,the plasmon dispersion relationalong with their lifetimeshave been theoretically investigated for a wide range of 2DDirac systems.<cit.>The interest in graphene plasmons is due in part to the fact thatthese excitations have no classical counterpart.<cit.> There has also been a considerable experimental effortfor investigating graphene plasmons,gate-tuning, infrared nano-imaging andconfinement<cit.> Graphene plasmonic resonances and instability at various wavelengths could be used in photodetectorsin the Terahertz range.<cit.> All these techniques could be successfully applied to the recently fabricated materials, discussed in the present work.Plasmonic applications have been widely based on nanoscale hybrid systems, in which grapheneplasmons are coupled to a surface plasmon excitations in metals. Technology has now gone a long way in combininggraphene with prefabricated plasmonic nanoarrays and metamaterials in order to produce plasmonics-based tunable hybrid optical devices.<cit.> Therefore, accurate knowledge of plasmon mode dispersions in graphene interfacedwith metallic substrates is crucial. Graphene-metal contacts are important components for all such devices.Consequently, exploration of plasmon modes at these metallic interfaces is a mandatory step toward fabricating the devices.High-resolution electron energy loss spectroscopy (EELS)has been employed to investigate those excitations at the surface ofBi_2 Se_3 to disclose the interplay between surface and Dirac plasmons in topological insulators<cit.>. Plasmons,their behavior, dispersions, quenching and environmental effects have been thoroughly studied in epitaxial graphene, in air-exposed graphene-Ru contacts, Graphene on Pt_3Ni (1 1 1), graphene grown on Cu (1 1 1)foils.<cit.>In all cases under consideration, we need to distinguish between extrinsic, ora sample initiallydoped at T=0, and intrinsic materials, with zero Fermi energy and completely empty conduction band.In the latter case, both the plasmon excitations and the electrical conductivity are completely suppressedat zero temperature due to the absence of free charge carriers. However, atfinite temperature, the conduction band would receive thermally induced doping in both cases.<cit.> In graphene,with zero energy band gap, this density is enhanced as n ⋍ T^2 and the plasmon dispersionbehaves like Ω_p^2 ⋍ q T. The properties of intrinsic finite-temperature plasmon excitations have been systematically examinedfor various materials including silicene.<cit.> In contrast, extrinsic ordoped structure at finite temperature is associated with a difficulty to obtain a reliableand accuratevalue for temperature-dependent chemical potential μ(T).Generally, it is known that μ(T) is decreased as the temperature is increased, and its value could be foundbased on carrier density conservation<cit.> In this work, our main objective is to obtain a setof non-integral, trancendental equations for a wide class of Dirac gapped materials with lineardensity-of-states (DOS), i.e., gapped graphene, silicene, germanene and transition metal dichalcogenides at arbitrary temperature.However, the range of our considered temperatures is limited by validity of linear or gapped Dirac coneapproximation for the energies, which recieve noticeable doping at those temperatures. Certain deviations start to build up at about 0.5 eV, <cit.> leading to various effect on the plasmons, suchas anisotropy, splitting and existence of additional acoustic plasmon branch.<cit.> Such energies are extremely far away from our range k_B T ∽ E_F^(0). In all our calculations, the energy is measured in the units of a typical Fermi energyE_F^(0) = 5.22meV, corresponding to electron density n^(0) = 1.0 · 10^15 m^-2.Once the chemical potential is known, one can obtain the finite-temperature dynamical polarizationfunction by Eq. (<ref>), which is a key component for all relevant many-body calculations.These includeoptical absorption, electronic transport, plasmonexcitations as well as electronexchange and correlation energies.<cit.> Here, we pay close attention tofinite-temperature plasmons, demonstrating how much initial doping contributes to each branch location at intermediate temperatures. Once the temperature becomes very high, k_B T ≫ E_F, thermally-induced doping dominates and thecontribution from the initial Fermi energy fades away.The rest of the paper is organized as follows. First, we derive the implicit analytic equations for thefinite-temperature chemical potential for all types of Dirac structures with linear DOS in Sec. <ref>. In Sec. <ref>, we calculate the dynamic polarization function which includes the single particle excitation mode frequencies. The single-particle modes combine with a charge cloud toproduce weakly interacting quasiparticles that vibrate collectively at the characteristic plasma frequencies.Emphasis has been placed on rather simple cases of gapped graphene and silicene at small, but finitetemperatures, when the zero temperature carrier doping plays a crucial role. We also briefly examinenon-local, hybrid plasmons in an open system of a semi-infinite conductor, Coulomb-coupled to a2D layer in the presence of finite doping doping of the electrons in the layer. Our concludingremarks and a concise discussion are presented in Sec. <ref>. We also provide two appendices with detailedderivations of the DOS for silicene and MoS_2 - in Appendix <ref>, and of thetemperature-dependent chemical potential in Appendix <ref>.§ CHEMICAL POTENTIALIn this section, we discuss our analytical derivations for the electron chemical potential as a function oftemperature. Being equivalent to the Fermi energy at T=0, the chemical potential normally decreases withincreasing temperature. Its specific value depends on multiple material parameters such as energy band gaps, Fermi velocities and the DOS of the of the electrons as well as the holes below the zero energy level.Thus, for a conventional 2D electron gas (2DEG) with no holes, the chemical potential could becomenegative at a certain temperature, which is not possible for a Dirac system with symmetry between the electron and hole states. Here, we are going to provide closed-form trancendental equations for thefinite-temperature μ(T) for a number of Dirac systems: graphene, buckled honeycomb lattices and transitionmetal dischacogenides.§.§ Buckled honeycomb lattices One of the most outstanding features of silicene and other buckled honeycomb lattices is the existence of two generally double degenerate pairs of energy subbands and two inequivalent band gaps. These gaps There is a fixed intrinsicspin-orbit gap 2 Δ_SO and a tunable sublattice-asymmetry gap Δ_zwhich is induced by and proportionalto an applied perpendicular electric field E_z. For small fields, Δ_z = E_z d_, whered_ is the out-of-plane displacement of a buckled lattice. The low-energy model Hamiltonian of a buckled honeycomb lattice has been found to be<cit.>Ĥ_ξ,σ = ħ v_F ( ξ k_x τ̂_x + k_y τ̂_y) ⊗Î_2 × 2 - ξΔ_SOΣ̂_z ⊗τ̂_z + Δ_z τ̂_z ⊗Î_2 × 2,where the Fermi velocity v_F = 0.5 · 10^6 m/s is half that for graphene, ξ =± 1 is the K/K' valley index, τ_x,y,z and Σ_x,y,z are Pauli matrices in two different spaces, attributed to pseudospin and real spin of the considered structure.Introducing spin index σ = ± 1, we can rewrite Eq. (<ref>) in a block-diagonal matrix form Ĥ_ξ,σ = ( [- ξσΔ_SO + Δ_z ħ v_F (k_x - i k_y); ħ v_F (k_x + i k_y)ξσΔ_SO - Δ_z ]).The energy dispersions are ε_ξ,σ^γ(k) = γ√((ξσΔ_z - Δ_SO)^2 +(ħ v_F k )^2 ),where γ = ± 1 determines the electron or hole state similar to graphene, with a finite or zero band gap. These dispersions, given by Eq. (<ref>), represent two pairs of spin-dependent energy subbands in a chosen valley with the two generally different band gaps |Δ_SO - ξσΔ_z |, which will be later referred to as Δ_<,> = |Δ_SO∓Δ_z |.Clearly, both energy gaps depend on the perpendicular electrostatic field and the two subbands corresponding to the ξσ = ± 1 indices switch their locations with increasing field strength. Small or zero E_z is relatedto a topological insulator (TI) states with Δ_z < Δ_SO. Once the two gaps become equal, we observe ametallic gapless state with Δ_<=0 and a finite Δ_>, defined as valley-spin polarized metal (VSPM). For larger fields, Δ_z ∽E_z would always exceed the constant intrinsic spin-orbit gap Δ_SO,which corresponds to the standard band insulator (BI) state. The DOS which is in general defined by ρ_d(E)= ∫d^2 k/(2 π)^2∑_γ =± 1 ∑_ξ,σ= ± 1 δ[ E - ε_ξ,σ^γ(k) ] ,is immediately obtained for silicene (see Appendix <ref>) as ρ_d(E) = 1/π ∑_γ = ± 1E/ħ^2 v_F^2∑_i = <,>Θ[E/γ - Δ_i] ,in terms of the unit step function Θ(x).We note that for systems sharing the same Dirac cone characteristics with arbitrary energy gap,the DOS is linear analogoustographene. Experimentally obtained linear V-shaped DOS was usedto verify the Dirac cone dispersion for germanene. <cit.> However, ρ_d (E) has afinite value only above the energy gap since only for this energy range electronic states existand thais how the band gap plays an importantrole.Finite-temperature chemical potential for an electronic system is obtained using conservationof carrier density nof electrons (n^(e)) and holes n^(h) concentrations at alltemperatures, including T=0. In this regard, we have n = n^(e) + (-1) n^(h) = ∫_0^∞ d E ρ_d(E) f_γ = 1(E,T) - ∫_-∞^0 d E ρ_d( E ){ 1 - f_γ = 1(E,T) }. At zero temperature, the density n is related to the Fermi energy E_F in a straightforward way. If only one subband is occupied, we obtainn = 1/2 π E_F^2 - Δ_<^2/ħ^2 v_F^2.Alternatively, if the doping density is such that both subbands are populate, then E_F is obtained from n = 1/π 1/ħ^2 v_F^2 [ E_F^2 - 1/2 ( Δ_<^2 + Δ_>^2) ] , and the critical density required to start filling the upper subband is n_c = 2 Δ_SOΔ_z / πħ^2 v_F^2.We prove in Appendix <ref> thatthe finite-temperature chemical potentail could be obtained from the following equationn( ħ v_F/k_B T)^2 =∑_γ = ± 1 γ/π ∑_i = <,> - Li_ 2{ - exp[γμ(T) - Δ_i/k_B T] }+ Δ_i/k_B T ln{ 1 + exp[γμ(T) - Δ_i/k_B T] },where Li_ 2 (x) is a polylogarithm function. Connecting the doping density n with the Fermi energy through eitherEq. (<ref>) or Eq. (<ref>) depending on how many subbands are doped at zero temperature, we derive thechemical potential with a value equal to E_F at T = 0, for all accessibletemperatures. Although this equation is transcendental and cannot be resolved algebraically, a quasi-analytic orone-step numerical solutions could be easily provided for any finite temperature without having to perform an integration. Clearly, the chemical potential for silicene depends on the energy band gaps Δ_i, i = < ,>. Our approach, discussedin Appendix <ref>, is valid for a variety of materials with linear energy dependence for the DOS, including MoS_2. Specifically, Eq. (<ref>) also describes μ(T) for gapped graphene with two degenerate subbands, or Δ_< =Δ_> = Δ_0. For gapless pristine graphene Δ_0 = 0 and π n = [ E_F/ (ħ v_F) ]^2, we have1/2 ( k_B T )^2E_F^2= - ∑_γ = ± 1γ Li_ 2{ - exp[γ μ(T)/k_B T] }.If the temperature is low with k_B T ≪ E_F, Eq. (<ref>) is reduced to the expressions, derived in Refs. [SDSS, SDSLi] and [myT]. All our considered materials could be effectively classified by the existing or broken symmetries of certain kinds,and, consequently, by the degeneracy of their energy subbands, which may be generally different for electronsand holes. In this respect, graphene represents the simplest case with a fourfold spin and valley degeneracy of±√((ħ v_F k)^2 + Δ_0^2) states. Silicene and germanene dispersions, yet showing complete electron/hole symmetry, exhibit spin- and valley-dependent pairs of subbands, each being double degenerate. Finally, MoS_2 demonstrates broken symmetry between its electrons and holes, and a finite energy separation between two non-equivalent holes subbands. Typical energy dispersions and DOS for silicene and molybdenum disulfide are presented in Fig. <ref>. For both materials, we consider the K valley with ξ = 1 so that the upper electron and lower hole subbands correspond to σ = 1 spin. Every time a new subband, or their degenerate manifold, begins to be doped, we see an immediate increase. or discontinuity, of the DOS, as schematically shown for silicene in the insets of Fig. <ref> (b). It is importantto observe that for both silicene and gapped graphene, ρ_d (E) is directly proportional to energy E, i.e., the DOS for Dirac materials with finite and zero gap are exactly the same as long as we measure it above theenery band gap. With no electronic states inside the gap region, we have ρ_d( E < 0) = 0.In Fig. <ref>, we display our results for the finite-temperature chemical potential for graphene and buckledhoneycomb lattices. First, we show how dissimilar this temperature dependence might be for various electronicsystems. In Fig. <ref> (a), this situation is described for a 2DEG with a parabolic energy band and constant DOS, graphene with ρ_s(E) ∽E, as well as two model structures - graphenewith no hole states, with doubly prevailing electrons ρ_d(E > 0) = 2 ρ_d ((E < 0) and doubly prevailinghole DOS. At lowtemperatures, all the curves, except the one for the 2DEG, are nearly identical since theholes do net play any important role. The hole distribution function is complimentary to that for electrons, i.e.,1-f_e(ϵ) → 0, and, therefore, inconsequential.However, when the temperature becomes comparable with the Fermi energy, the hole thermal excitations become crucial,causing an opposite effect compared with that for electrons. They mitigate the reduction of the chemical potentialand eventually prevent μ(T)fromcrossing the zero energy level. This is seen particularly well for ahole-dominating system (> h^(+)), for which the chemical potential starts to increase and ultimately exceeds theinitial E_F value.We conclude that only total electron/hole symmetry, but not the energy gap, keeps the chemicalpotential positive for arbitrary high temperatures. For the remaining plots, we consider the behavior of μ(T) for graphene and silicene with different gaps. At T=0, we maykeep the Fermi energy fixed so that the actual electron density n differs, in which casethe states with a largergap receive much smaller amount. Alternatively,we can dope the sample andμ(T) shows much stronger reduction, or we can fix the carrier density nso that the Fermi energyincreases inthe case with larger gap (see Eq. (<ref>)). The former case is shown in panels (b), (c) and (d), whereas the latter at (e) and (f). In general, the carrier density nis accepted to be the most meaningful parameter,determining the Fermi energy for each specific system.Plots (b) and (c) are designed to demonstrate complete reflection symmetry of the chemical potential between electronand hole doping for all energy band gaps. This is a manifestation of γ = ± 1 symmetry of these states persisting at arbitrary temperatures. Silicene with both filled subbands exhibits qualitatively similar finite-temperature behavior asgapped graphene. More generally, only the actual gaps are relevant, but not the exact type of states (such as topologicalinsulator or regular band insulator), or the way in which such a state has been achieved.We now pay attention to the following situation in silicene. At T=0, the Fermi level is chosen so that only the lower subband is filled. In contrast, the upper one Δ_> is located so closeto the Fermi level E_F, that it startsgetting populated at all, even very small temperatures. Thermally-induced doping, received by the Δ_>-subband, results in a much stronger reduction rate of μ(T), as shown in Fig. <ref> (e). Such situation suggests very specialthermal properties of silicene with closely located energy subbands, and shares this behavior with graphene having an additional spin and valley degeneracy. We investigate this phenomenon even further by adjusting the upper subband in the vicinity of the Fermi level, as shown in Fig. <ref> (f). Now, each μ(T) curve demonstrates a significantly pronounced decrease whenever theupper subband filling becomes essential. These two different types of μ(T) behavior in silicene could be used to achieveits additional tunability by introducing the upper energy subband Δ_>. §.§ Transition metal dichalcogenides The low-energy electronic states in monolayer molybdenum disulfide (MoS_2 , ML-MDS), a prototype transition metaldichalcogenide, could be effectively described by a two-band model Hamiltonian<cit.> Ĥ_d^ξ,σ = (1/2 ξσ λ_0 + ħ^2 k^2/4 m_eα) Î_2× 2 + ( Δ/2 - 1/2 ξσ λ_0 + ħ^2 k^2/4 m_eβ) Σ̂_z +t_0 a_0 Σ̂_ξ· k,whose important feature is a majorgap parameter Δ = 1.9 eV which results in the actualband gap ⋍ 1.7 eV,large compared to that for silicene. The spin-orbit couplingparameter λ_0=0.042 Δ represents asmaller, but essential correction, to the single-particle excitation spectrum and the band gap. The energy subbands are now spin- and valley- dependent since the corresponding degeneracy is lifted. The electron hopping parameter t_0 = 0.884 Δ and a_0=1.843Å shapethe Dirac cone term in the Hamiltonian Eq .(<ref>) as t_0 a_0= 4.95× 10^-29J· m,counting up to ≈ 0.47of ħ v_F value in graphene.Next, we includethe ∽ k^2 mass terms with α =2.21 = 5.140 β in which m_e is thefree electron mass. Our considered values of Fermi momentum at zero temperature are determinedby the experimentally allowed electron and hole doping densities n =10^14÷ 10^16 m^-2 as k_F = √(π n)⋍ 10^8÷ 10^9 m^-1.[Expression k_F = √(π n) holds true only for systems with a fourfold spin andvalley degeneracy, such as graphene,i.e., g = g_s g_v = 4. The general equation for the Fermi momentum in a2D material reads(2 π)^2 n = gπ k_F.] Anisotropic trigonal warping term t_1 a_0^2 (Σ̂_ξ· k) σ̂_x (Σ̂_ξ· k) is clearlybeyond out consideration since∽ t_1 = 0.1 eV = 0.053 Δ term does not cause any effect on the electron dispersions.The energy dispersion relations, corresponding to the HamiltonianEq. (<ref>) ε^ξ,σ_γ(k) = ϵ_0^ξ, σ(k) + γ√([ Δ_0^ξ,σ(k) ]^2 + (t_0 a_0 k)^2) ,formally represent gapped graphene witha k-dependent gap termΔ_0^ξ, σ(k) = ħ^2 k^2 β / (4 m_e)+ Δ / 2 - ξσ λ_0 / 2, and a band shift ϵ_0^ξ,σ(k) = ħ^2 k^2 α /(4 m_e) + ξσ λ_0 / 2.<cit.>Neglecting only the O(k^4) terms leads us toε^ξ,σ_γ(k)⋍1/2 ξσλ_0 + αħ^2/4 m_e k^2 +γ/2{[ (2 t_0 a_0)^2+( Δ - ξσ λ_0)βħ^2 / m_e ] k^2 + ( Δ - ξσ λ_0 )^2 }^1/2.This is the principal model which we will use to describe the energy dispersions of MoS_2 in our work.The ⋍ k^4 terms, trigonal warping and anisotropy are considered to be non-essential, even thoughas we will later see, cause certain discrepancies in the DOS.[Due to a largegap parameter Δ = 1.9 eV, the electronic states corresponding to largewave vectors are often involved and at k ≈ 5.0 k_F^(0) correction ⋍Δ β k^4 is no longer negligible.] We also show (see Appendix <ref>) that the curvature of the energy subbands in TMDC's is so small thateven the highestpossible doping density 10^17 m^-2 results in the Fermi energies∽λ_0. Thus, at zero or low temperatures, we do not need to consider any high-energycorrections to Eq. (<ref>)On the other hand, inclusion of higherorder terms into the dispersions, would enormously complicate the DOS calculation. At high temperatures,theelectronic states far from the Dirac point would also receivesubstantial temperature-induced doping density due to the so-called Fermi tail.<cit.> In that case, our model Hamiltonian in Eq. (<ref>) and especially, simplified dispersionrelation (<ref>) would no longer provide a satisfactory approximation. Consequently, our primary focus is on small but finite temperatures for which the initial doping density and E_F still play animportant role, beyond the O(T^2/T_F^2 ) approximation discussed in Ref. [SDSLi]. Here we note that for MoS_2, similar to the buckled honeycomb lattices, spin and valley indices alwaysappear together as a product, so that taking into account a × 2 degeneracy, a single composite index ν = ξσ could be effectively introduced. We are going to use only the index ν for the rest of ourconsideration.A valley- and spin-resolved gapped graphene approximation of dispersions (<ref>) arisesonce we also neglect themass terms in Eq. (<ref>)<cit.> ε^ν_γ(k)⋍νλ_0/2+γ√((t_0 a_0)^2 k^2+(Δ-νλ_0)^2/4).This approximation has a few important advantagesincluding simplicity and its formal resemblancewith gapped graphene so that all the crucial quantities such as the DOS, wave function,polarizabilityand many others are already known. Furthermore, it gives a quite a suitable description of the energyband structure of MoS_2, taking into account a large gap parameter and ν-dependent splitting of the two hole subbands. Nevertheless, the mass terms must be taken into account for a proper evaluation of the DOS and most of thetemperature-dependent properties of TMDC's. As we demonstrated in Appendix <ref>, even in the simplest possible parabolic subbands approximation, valid only for small wave vectors k → 0, the mass terms make a contribution, comparable with the Dirac cone and band gap parts of Hamiltonian (<ref>). Very small curvature of the energy subbands,mentioned above results in a tremendous ρ_d ∽ m_eff, so that even ⋍α/(4 m_e) correction, hardly noticeable on the electron band structure, becomes significant. It basically discards the DOS results obtainedfrom the gapped graphene model, even at zero or small wave vectors. Importance of the mass and even higher order terms for the plasmon calculation was discussed in Ref. [Scholz1].Taking into account all the required terms in (<ref>), rigorous numerical calculations giveaccurate results for the electron DOS for MoS_2. In Fig. <ref> (d), we present all three possible outcomes.Based on the gapped graphene model, the DOS is nearly twice as large as its numerical values .<cit.> In contrast, our ∽ k^2 model (<ref>) demonstrates quite a good match,especially in the low-energy range. We also note that the numerically obtained dependence is clearly linearfor a wide range of energies, much exceeding our considered diapason. In summary, we consider a piecewise linear approximationρ_d (E) relatively close (δE⋍λ_0 ) for each of the three non-degenerate subbands ρ_d(E) = c_0^(i) + c_1^(i) E and ρ_d(E) =0 in the gap region -Δ/2 + λ_0 < E < Δ/2. The expansion coefficientsare obtained as c_0^(1) =2.837 E_F^(0)/(ħ v_F)^2 = 0.043t_0^-1 a_0^-2, c_1^(1) = -1.397 (ħ v_F)^-2 = -0.308(t_0 a_0)^-2for γ = -1, ν = -1 and ≤ -Δ/2 - λ_0 ≥E≤-Δ/2 - λ_0. Finally, when γ = -1, but ν = 1 and E≤ -Δ/2 + λ_0, the hole DOS coefficients are c_0^(2) = 1.132 E_F^(0)/(ħ v_F)^2 = 0.0174t_0^-1 a_0^-2 c_1^(2) = -0.767 (ħ v_F)^-2 = -0.169(t_0 a_0)^-2. For electrons with γ=+1 atE= Δ/2 + δϵ, the two quasi-degenerate subbands lead to the DOSequal to c_0^(3) = 5.110 E_F^(0)/(ħ v_F)^2 = 0.078 t_0^-1 a_0^-2 and c_1^(3) =0.815 (ħ v_F)^-2 = 0.179(t_0 a_0)^-2.We adopt these values for ρ_d(E), which arise from the numerical calculations in orderto achieve the highest possible precision and credibility for our finite-temperature derivations. However, our effective model, presented in Appendix <ref>, gives the DOS results which show good agreement withthese numerical values and could be used for decisive estimates for various collective calculations for MoS_2.Once the DOS is known, we are in aposition to calculate the Fermi energy for a given doping density n_(0)^e. The new point here is that ρ_d (E) is not directly proportional to the energy so that E_F is determined by n_(0)^e = (E_F - Δ/2)[ c_0^(3) + c_1^(3)/2 ( Δ/2 + E_F ) ] ,orE_F^(e) = 1/2 c_1^(3) {-2c_0^(3) + [ ( 2 c_0^(3) + c_1^(3) Δ)^2 + 8 n_(0)^e c_1^(3)]^1/2}.For hole doping, the Fermi energy differs from the previously considered electron doping case, i.e.,E_F^(h) =1/c_1^(2) { - c_0^(2) + { -2 c_1^(2)n_(0)^h + [c_0^(2) - c_1^(2) (Δ/2 - λ_0)]^2 }^1/2 }Numerically, our results for theFermi energy for electrons and holes are presented in insets(i1) and (i2)of Fig. <ref>. Here, both linear and quadratic terms in the doping densityequations are present (see Eq. (<ref>)), and, most importantly, there is no symmetry between the electron and hole states. Unlike graphene, the linear dependence here dominates for bothcases due to the large energy band gap. Each curve starts from the corresponding bandgap - Δ/2 = 8.13 E_F^(0) for electrons and -Δ/2 + λ_0 = - 7.45 E_F^0.The corresponding well-known result for gapped graphene E_F^2 - Δ_0^2 =2 πn(ħ v_F )^2 are verified by putting c_1^(i)→ 0 and λ_0 → 0.The finite-temperature chemical potential for MoS_2 is obtained in a similar way, as we havedone for the buckled honeycomb lattices, except that we need to evaluate four differentterms related to the two separate hole subbands (see Eq. (<ref>)).The correspondingnumerical results are described in Fig. <ref>. As discussed above, the most specialproperty of TMDC's is the broken electron/hole symmetry. Consequently, the chemical potential forthe electron doping becomes negative at T ≈ 2.5 E_F, while μ(T) for hole doping does not ever change its sign. Broken electron/holesymmetry leads to two substantiallydifferent types of temperature dependence of the chemical potential. Thus, MoS_2represents a special material with unique symmetry properties and chemical potential dependence,so far encountered only in certain types of semiconductors but not in Dirac materials.§ PLASMON EXCITATIONS AT FINITE TEMPERATUREAs one of the most relevant applications of our finite-temperature chemical potential formalism,we briefly consider plasmons for an extrinsic, substantially doped at T=0, free-standing gapped Dirac cone material. The plasmon dispersion relation is calculated from the zeros of the dielectric function ϵ(q, ω),expressed in the random phase approximation (RPA) as ϵ(q, ω) = 1 - v(q)Π_T(q, ω | μ(T), T, Δ_i) = 0 ,where v(q) = 2 π e^2 / (ϵ_s q) is the Fourier-transformed two-dimensional Coulomb potential,and ϵ_s = 4 πϵ_0 ϵ_r ithϵ_b is the background dielectric constant in which the 2D material is embedded. At finite temperature, the dynamical polarization function Π_T is given as an integral transformation <cit.> of its T=0 counterpart Π_0, i.e.,Π_T(q, ω | μ(T), T, Δ_i) = 1/2 k_B T ∫_0^∞d ξ Π_0 (q,ω | ξ, Δ_i)/1 + cosh{ [ μ(T,E_F) - ξ] / (k_B T) }The evaluation of the zero-temperature polarizability is quite similar for buckled honeycomb lattices and MoS_2, since in both cases their low-energy bandstructure is represented by two generally inequivalent double-degenerate pairs of subbands which depend on the composite index ν = σ ξ. For any such pair, Π_0^(ν) is obtained in the one-loop approximation (g_0 = 2) as Π_0^(ν) (q,ω |E_F, Δ_ν) = g_0/8 π^2∫ d^2 k ∑_γ,γ' = ± 1(1 + γγ' k · ( k +q) + Δ_ν^2 /ε^ν (k) ε^ν(| k +q|) ) f[ E - ε^ν_γ (k)] - f[E - ε^ν_γ'(| k +q|)]/ε^ν_γ (k) - ε^ν_γ' (| k +q|),where f[E - γ E_ν(k)] is the Fermi-Dirac distribution function, showing electron and holeoccupation numbers for chosen energy E. At T=0, it is a Heaviside unit step functionΘ[ E - ε^ν_γ (k) ]. We note that the extra 1/2 comes from the form factor 1/2 [ 1 + γγ' ... which also depends on the absolute value of each energyε^ν (k) = 1 / γε^ν_γ (k).We also note that since any valleyor spin transitions are inadmissible and only one summation over the index ν is incuded, compared to theelectron/hole indices γ and γ'. Thus, for both types of materials, the dynamical polarization function is obtained as a sum of terms obtained fromEq.(<ref>) over ν <cit.>Π_0 (q,ω |E_F, Δ_i) = ∑_ν Π_0^(ν) (q,ω |E_F, Δ_ν) .Our results for the polarization functions and plasmon excitations are presented in Figs. <ref> and <ref>.First, we need to address the imaginary part of the polarizabilitysince it specifies the regions and intensity of the plasmon damping. We see that at a finite temperature the plasmon dissipation generally increases and thedamping-free regions are nearly absent. Once the temperature becomes high, the imaginary part of the polarization function is reduced as 1/T,<cit.> so there is no uniform temperature dependence of the plasmon damping.Doping and proportional increase of the energy band gaps, in contrast, increase the regions free of single-particleexcitation spectrum (see Fig. <ref> (b)), so the effect of a finite temperature and E_F on the plasmon dissipation coul be opposite. The real part of Π_0 (q,ω |E_F, Δ_i), shown in Fig. <ref> (a)-(d), designates thelocation and slope of the corresponding plasmon branches which are presented in panels (e) and (f). Here, the temperature and the doping produce a similar effects, shifting the location of the peaks to the right. The real part of the polarization function must be positive in order to enable a real solution of Eq. (<ref>)Plasmon branches are located at higher energies for a given wave vector due to either an initial increase of the Fermienergy or thermally-induced doping. §.§ Non-local plasmons in an open system Concluding our investigation of extrinsic 2D materials, we now turn to theplasmon excitations in so-called 2D open systems (2DMOS). Such a nanoscale hybrid arrangement being a part of graphene-based nanoscale devices, <cit.> consists of a 2D layer (it could be 2DEG, graphene, a buckled honeycomb layer, or MoS_2), which is Coulomb coupled (not chemically bound) with a semi-infinite conductor and its surface plasmon.<cit.> While a plasmon excitation in a closed system is determined by two-particle Green’s functions,in 2DMOS it more involved, depending on theCoulomb interaction with the environment.<cit.> The important feature of such a system is the screened Coulomb coupling between the electrons in graphene andthe conducting substrate.<cit.> Such a screened potential could be obtained using the nonlocal frequency-dependentinverse dielectric function<cit.>Consequently, two <cit.> ormore <cit.> linear plasmon branched have been obtained, confirming some previous experimental claims.<cit.> For finite temperature, their coupling to an external reservoir isreflected in existence of extraplasmon dissipation channels. <cit.> The plasmon branches in such system are obtained as zeros of a so-called structure factorS (q,ω | μ(T), Δ_i ), playing the role of the dielectric functionϵ (q, ω) in an isolated layer. It is obtained as<cit.> S (q,ω | μ(T), Δ_i) = 1 - v(q)Π^(0)(q,ω;μ) {1 - ϵ_B(ω)/1 + ϵ_B(ω) Exp(-2 a q) + 1},where a is the distance between the 2D layer and the surface and the bulk dielectric function is given in the local limit as ϵ_B = 1- Ω_p^2/ω^2. The bulk-plasma frequency, defined as Ω_p^2 = (n_m e^2)/(ϵ_o ϵ_S m^∗), depends on on the electron concentration n_m, its effective mass m^∗ and the substrate dielectric constant ϵ_S. This approximationstays valid for a large range of wave vectors q ≪ 2 × 10^9 m^-1 since the Fermiwavelength in metals is comparable with the inverse lattice constant. As a result the frequency of theupper plasmon branch in our system, equal to Ω_p/√(2) at q_∥→ 0, might range fromultra violet down to infrared or even terahertz, depending on the substate material and must stay commensurate with the energy band gap in the 2D layer. Previously, we reported that<cit.> in case of spin- and valey-dependent single-particle excitations in a 2D layer (which is true for buckled honeyomb lattices and MoS_2 ), such a hybrid structure could be effectively used to directly measure the dielectric properties or spin-orbit coupling parametersof such a layered material because the location of each plasmon branch, its damping rate and the signatures of theparticle-hole modes are independently determined by the material parameters of the 2D layer.Each of these properties is unique (for example, the two plasmon branches in silicene-based TDMOS depends onthe energy band gaps as Δ_i^1/2 and Δ_i^1/4, while the outermost PHMboundaries aredetermined solely by the lower gap Δ_<), so that an additional linear plasmon branch provides uswith the required earlier unknown piece of information about the specific material. This is not possiblein the case of a single plasmon branch in an isolated layer. In the present work, we additionally introduce finite doping and temperature into this consideration. Our numerical results for the non-local plasmons for a extrinsic systems are presented in Fig. <ref> We see that the locationand damping of both branches substantially changes in the presence ofan initial Fermi energy. At low temperature, this increase is especially apparent, similar to plasmons in an isolated layer. For a sample with stronger doping,E_F = 1.5 E_F^(0) at zero temperature, the upper plasmon branch is always located above the surface plasmon level Ω_p/√(2) and is never damped.§ CONCLUDING REMARKS We have carried out an extensive investigation of extrinsic, or doped, Dirac gappedmaterials at arbitrary finite temperatures and obtained a set ofalgebraic analytic equations determining the chemicalpotential. Our considered systems include graphene, with or without an energy band gap, buckled honeycomblattices with spin- and valley-dependent energy subbands and reduced degeneracy, as well as thetransition metaldichalcogenides having broken symmetry between the electron and hole states. Our results could also be usedto predict the finite-temperature chemical potential for parabolic or quasi-parabolic eigenstates insemiconductors with light or heavy holes.<cit.> In general, our model is limited only by the linear dependence of the DOS which stays valid over a wide energy range for all theabove mentioned materials. We have demonstrated that the chemical potential depends substantially on the energy band gap(s) of the consideredsystem since the DOS depends on the curvature of each subband. Specifically, we investigated structures with two non-degenerate, separated spin- and valley- resolved energy subbands in both valence and conduction bands,such as silicene. The upper subband would receive thermally-induced doping even if it is undoped at zero temperature. This is always reflected in a higher reduction rate of the chemical potential whenever the second subband doping starts playinga role. Consequently, one can tune the μ(T) dependence around any required temperature by bringing initial doping close to the higher-energy subband. The number of such separated subbands contributing to the DOS could be arbitrarily large foran electron in a quantum well or quasi-one-dimensional nanoribbons,<cit.> and all these cases could be effectively treated by our model.The behavior of the chemical potentialdepends on whether there is symmetry between the electron and hole states in the system. Ifthe DOS for the electrons and holes is equal or symmetric around the Dirac point, then the chemical potentail does notchange its sign even for high temperatures, i.e., it remains positive for electron doping or stays negative if E_F < 0.In fact, these two types of doping result in symmetric behavior, decreasing |μ(T) |, so that theelectron-hole symmetry persists at any finite temperature. Let us discuss the electron doping in more detail. Once thetemperature is sufficiently high, the hole states become thermally excited and it has an opposite effect to that of electron, decreasing the reduction of the chemical potential. At extremely high temperatures, the two processes have almost equivalent effects and μ(T) asymptotically tends to zero and never crosses the Dirac point. This situation changes if the DOS of the electron and holes differs and the two hole subbands are energeticallynot equivalent, as we observe for TMDC's. We have shown that for MoS_2 the chemical potential becomes negative, changing its sign at T ⋍ 2.5 E_F. Alternatively, μ(T) could never reach the zero energy line if there is hole doping at zero temperature and starts decreasing at sufficiently high temperatures.As a necessary intermediate step in our derivations, we obtained a piecewise-linear model for the DOSfor transition metal dichalcogenides, directly from the Hamiltonian parameters of the considered system.This model gives exact results for ρ_d(E) at the band edges (next to each energy gap) and afairly good approximation at higher energies. This model significantly improves the results obtainedfrom the spin- and valley-resolved gapped graphene approximation. Finally, we considered the way in which the initial doping affects the plasmon dispersions(ħΩ_p/ E_F)^2⋍Λq, and the effective length Λ depends on the doping.There could bean initial carrier density at T=0, and the thermally-induced one atfinite temperature. Thelatter doping type results in a finite polarization function and√(q T) plasmon dispersions even for intrinsic systems.<cit.> We demonstrate how much each type of doping contributes this effective length.Several many-body calculations or collective electronic models requires reliable knowledge ofthe chemical potential at finite temperature. In the absence of such information, much attention has been directed towards intrinsic, or undoped systems, or low temperatures, so that ⋍ (T/T_F)^2series expansions could have been applied. Our results are going to provide considerable assistance in transport studies, optical, thermally modulated conductivity for thematerials, discussed in our paper.Consequently, we expect our work to provide an important contribution to electronics,transport and plasmonics of these recently discovered structures, both theory and experiment. D.H. would like to thank the support from the Air ForceOffice of Scientific Research (AFOSR). § DENSITY-OF-STATESThe DOS for electrons and holes with energy dispersion ε^ ξ,σ_γ(k)is defined as ρ_d(E)= ∫d^2 k/(2 π)^2∑_γ = ± 1 ∑_ξ,σ = ± 1 δ[ E -ε^ ξ,σ_γ(k) ] ,where ξ = ± 1 and σ = ± 1 are valley and spin indices which in our considerations always appear as a product σξ, so that a single composite index ν = σξ = ± 1could be introduced. This leads to a double degeneracy of all considered dispersion relations anda transformation∑_ξ,σ = ± 1⟹ 2 ·∑_ν = ± 1, which we will use throughout this work.For a number of cases of 2D structures, Eq.(<ref>) could be immediately evaluatedusing the following property of a delta functionδ(f(x)) = ∑_iδ(x-x_i)/| df(x)/dx_ x=x_i|,where x_i are the roots of f(x), formally given as f(x) = E - ϵ_ν^γ(k) for variousenergy dispersions ϵ_ν^γ(k). In the case of silicene and germanene, the result isstraightforward with ρ_d(E) = 1/π ∑_γ = ± 1γ E/ħ^2 v_F^2∑_i = <,>Θ[ E/γ - Δ_i] ,i.e., the DOS for Dirac gapped systems is linear similar to graphene. However, it is finite only abovethe energy gaps. This result also covers the case for gapped grapheneif the two gaps areequal Δ_< = Δ_> = Δ_0. Furthermore, we arrive at well-known V-shaped∽E DOS for pristine gapless graphene if Δ_<,>→ 0. §.§ Molybdenum disulfide In our work, we discuss several effetive models of different complexity and accuracy, describing theenergy dispersion ε^ν_γ(k) for MoS_2.First, we consider a spin- and valley- resolved gapped graphene model given by Eq. (<ref>), in which we leave out all the mass ∽α and ∽β terms. It gives quite accurate resultsfor the energy eigenstates next to the corners of Brillouin zone, as shown in Fig. <ref>. However, it is straightforward to see that the corresponding DOS obtained asρ_d(E) = 1/π(t_0 a_0)^2∑_γ = ± 1 1/γ ∑_ν= ± 1(E - ν/2λ_0 )Θ[ γ(E -νλ_0/2) - 1/2( Δ - νλ_0 ) ] , is V-shaped and does not matchthe numerical results even near the band edges. Once we get into the "allowed“ energy rangesoutside the band gapε^ν=-1,1_γ = 1(k = 0) = Δ/2 for electrons, ε^ν=-1_γ = -1(k = 0) = -Δ/2 + λ_0 and ε^ν=1_γ= -1(k = 0) = -Δ/2 - λ_0 for the holes, the DOS experiences three different giant discontinuities due to each new contributing energy subband. These giant leaps could becalculated using the parabolic subbands approximation, obtained for k ≪ k_Fε_γ^ν(k)= 1/2[νλ_0 (1 - γ) + γ Δ] + [ ħ^2/4 m_e (α + γβ ) + γ(t_0 a_0)^2/Δ -νλ_0] k^2 .This result leads to the DOS given by<cit.> ρ_d(E) = 1/2πħ^2∑_γ, ν = ± 1| α+γβ/4 m_e + γ (t_0a_0)^2/ħ^2(Δ - νλ_0)|^-1Θ[ γ(E -νλ_0/2) - 1/2( Δ -νλ_0 ) ].The two terms in Eq. (<ref>) are of the same order of magnitude,consequently, each of them must be retained in our calculation. Physically it means that due to large gap Δ the curvature of each subband at k = 0 is so small that even the ⋍α/(4 m_e)correction is significant. It basically discards the DOS obtained for the gapped graphene model, even at k → 0. The fact that the mass and even higher order terms must be taken into accountfor the plasmon calculation was mentioned in Ref. [Scholz1].The principal model which yields quite reliable results for the DOSis derived by neglecting only non-essential O(k^4) terms with ε^ν_γ(k)⋍1/2 ν λ_0 + ħ^2 α/4 m_e k^2 +γ/2√((Δ - ν λ_0 )^2 + [ (2 t_0 a_0)^2+( Δ - ν λ_0) ħ^2β/m_e] k^2).Here, all ∽ k^2 terms are retained and the final expression for the DOS appears to be quite complicated.We use a general equation from our previous work<cit.> to come up with a linear approximationwhich, according to the most precise and generalized numerical results,<cit.> is validfor all experimentally allowable electron and hole doping densities. For the dispersions Eq. (<ref>),we use Eq. (<ref>) to obtain ρ_d(E) = 1/2π ∑_j ∑_γ, ν=± 1| α̃ + γ Ã_ν(Δ±λ_0, β |a_0 t_0) / 2 {E-ϵ̃_ν -α̃ ξ_ν^ (j) (E) }|^-1Θ[ γ(E -μλ_0/2) - 1/2( Δ - νλ_0 ) ],where ϵ̃_ν=νλ_0/2, Δ̃_ν=(Δ -νλ_0)/2, Ã_ν (Δ±λ_0, β |a_0 t_0)=(Δ -νλ_0)ħ^2 β /(4 m_e) + (t_0 a_0)^2, and α̃=ħ^2α/(4 m_e). ξ_ν^ (j)(E) are the roots off(E,ξ) = E - ϵ̃_ν - αξ - γ √(Ã_νξ +Δ̃_ν^2) = 0 .This euqation could be solved by expressing itin quadratic form. However, one must bear in mind that if bothparts of Eq. (<ref>) are squared, there might be additional non-physical solutions which must be disregarded. If this equation is written as (αξ)^2 + Bξ + C = 0, where B = Ã_ν + 2α and C = (E - ϵ̃_ν)^2 - Δ̃_ν^2, the only appropriatesolution is ξ^ (1)(E) = 1/(2 α)^2 ( B + √(B^2 - 4 α^2 C)).The other solution corresponds to theE - ϵ̃_ν - αξ= - γ √(Ã_νξ + Δ̃_ν^2) and is obviosly incorrect for α→ 0.We also note that the electron/hole index γ is no longer presentin this equation, so that the two ± solutions are not associated with electron or hole states.In order to illustrate the physics behind selecting the only appropriate solution, let us consider a simpleexample of gapless graphene with ε_γ(k) = ħ v_F k with additional small, not depending on γ, the mass term ∽α k^2, α≪ħ v_F/ k_F. The actual dispersion relation is nowϵ^γ(k) = γħ v_F | k | + α k^2, and the DOS is calculated as ρ_d(E) = 2/π ∑_γ = ± 1 ∑_jk^ (j)/|γ ħ v_F + 2 α k^ (j)|.The roots k^ (j) are the solutions of γħ v_F | k | + α k^2 = 0. Even thoughsuch a quadratic equation generally has two inequivalent solutions, only one of them± k^(1)⋍E/(γħ v_F) - α E^2/(ħ^3 v_F^3) satisfies the | k |-type equation. Consequently, we obtain the following expression ρ_d(E) ⋍2/π ∑_γ = ± 1 E/γ (ħ v_F)^2 -α (2 + γ)E^2/(ħ v_F)^4becoming equivalent to graphene DOS ρ_d(E) = 2/π E/ [γ(ħ v_F)^2 ]for α→ 0. For the holes with γ = -1, however, the linear and quadratic mass terms arecompeting, so that for k ≫ k_F, another solution is present. However such wave vectors are beyond the Diraccone model, and therefore, it is not physically acceptable. The obtained correction to the DOS is a small decrease,as it is expected to be for an energetically elevated location of the subband. However, in our model for MoS_2, the mass terms ∽α and ∽β are not small and represent afinite correction to the DOS. The small parameter which we used in our series expansions is the energyδϵ above each band gap.Now, we return to Eq. (<ref>) and present its solution as ξ^ (1) = 1/2α̃^2 {Ã_ν + 2α̃( E - ε̃_ν^(0))- [Ã_ν^2 + 4α̃^2 Δ̃_ν^2 + 4α̃ Ã_ν ( E -ε̃_ν^(0))]^1/2}.This solution is exact in the sense that no approximations have been made so far excpet forthe ∽ k^2 dispersions (<ref>). Substituted into Eq. (<ref>), it givesthe DOS for an arbitrary energy, for both electrons and holes. As the next step, we substitute this result for ξ^ (1) into Eq. (<ref>). We are interested in obtaining a linear approximation of the DOS next toeach subband edge. Let us first consider electrons withε^ν_1(k) =Δ/2 + δϵ, δϵ≪E. In this case, ξ_ν^ (1)⋍4 m_e/α ħ^2 { 1 - (a_0 t_0)^2 + ħ^2 β/(4 m_e)( Δ - νλ_0 )/(a_0 t_0)^2 + ħ^2 /(4 m_e)[(α + β) ( Δ - νλ_0 )]} δϵ,and the DOS is now approximately given by ρ_d (E) = 1/2 π ∑_ν=± 1Δ - νλ_0/(a_0 t_0)^2 + ħ^2/(4 m_e)(α + β)(Δ - νλ_0) +2δϵ[(a_0 t_0)^2 + ħ^2 β/(4 m_e)( Δ - νλ_0 )]^2/{ (a_0 t_0)^2 + ħ^2 /(4 m_e)[(α + β) ( Δ - νλ_0 )]}^3.The actual numerical resuts aredetermined from ρ_d (E) = c_0^(3) + c_1^(3)( E - Δ/2) , c_0^(3) = 0.180 1/t_0 a_0^2 = 11.74E_F^(0)/(ħ v_F)^2, c_1^(3) = 0.2681/(t_0 a_0)^2 = 1.2181/(ħ v_F)^2.In the valence band, we consider two separate hole subbands with ν = ± 1.If γ = -1, ν = 1 and E≤ -Δ/2 + λ_0, the DOS is ρ_d(E) = c_0^(2) + c_1^(2) [ E - ( Δ/2 - λ_0) ], and the expansion coefficients arec_0^(2) = 1/2 π Δ - λ_0/(a_0 t_0)^2 + (β - α) (Δ - λ_0), c_1^(2) = 1/π δϵ[(a_0 t_0)^2 + ħ^2 β/(4 m_e)( Δ - λ_0 )]^2/{ (a_0 t_0)^2 + ħ^2 /(4 m_e)[(β - α) ( Δ - λ_0 )]}^3 < 0 ,or c_0^(2) = 0.105 1/t_0 a_0^2 = 6.847E_F^(0)/(ħ v_F)^2, c_1^(2) = -0.2321/(t_0 a_0)^2 = - 1.0511/(ħ v_F)^2. Finally, for the lower hole subband with E⪅ - Δ/2 - λ_0,we obtain ρ_d(E) = c_0^(2) + c_1^(2) [ E - ( Δ/2 + λ_0) ] , c_0^(1) = 1/2 π ∑_ν = ± 1Δ -ν λ_0/(a_0 t_0)^2 + (β -α) (Δ - ν λ_0), c_1^(1) = 1/π∑_ν = ± 1δϵ[(a_0 t_0)^2 + ħ^2 β/(4 m_e)( Δ -ν λ_0 )]^2/{ (a_0 t_0)^2 + ħ^2 /(4 m_e)[(β - α) ( Δ - ν λ_0 )]}^3 < 0 ,andc_0^(1) = 0.233 1/t_0 a_0^2 = 15.17E_F^(0)/(ħ v_F)^2, c_1^(1) = -0.4581/(t_0 a_0)^2 = 2.0771/(ħ v_F)^2. It is straightforward to obtain our previous results for the gapped graphene model (<ref>) DOSif α = β→ 0. We also note that the slope of the DOS in the conductionband is negative, as it should be accoring to Fig. <ref> (d), and the summation over the ν index is present in all cases excpet the upper hole subband in Eqs. (<ref>).Our results(<ref>) - (<ref>) (here, we move from the conduction electrons to the valence band, i.e., from the right to left) represent a fairly good match with the previously obtained numerical values, specified in Sec. <ref> and later used for all our finite temperature calculations. Thecoefficients c_0^(i), i = 1, 2, 3 are equal to the giant discontinuities of the DOS δρ_d (E) at each subband edge or k = 0, except c_0^(1)⋍δρ_d^(2) (-Δ/2 + λ_0) + δρ_d^(1) (-Δ/2 - λ_0)), and, therefore, are accurate. The linear coefficients c_1^(i) , in fair agreement, are 20-25 % larger compared with thenumerical results since all the ∽ k^4 terms of our energy dispersions are neglected. Inclusionof these terms leads to the higher energies for chosen wave vector and a decrease of theDOS. This discrepancy is increased for higher energies, which is well seen for c_1^(1) for holes withE < -Δ/2 - λ_0. However,the subbands for such energy ranges do not receive any substantialdoping unless the temperature becomes very high. For most considered situations, we are limited for δϵ≈λ_0 within the band edges. In such a small range, c_1^(i) δϵ≪ c_0^(i), sothat the actual DOS values remain almost unaffected and our model yields accurate results. § CHEMICAL POTENTIAL Μ=Μ(T) AT A FINITE TEMPERATURE We now derive a set of algebraic equations for the finite-temperature chemical potential μ(T). At zero temperature, it is equal to the Fermi energy E_F = μ(T)|_T=0. Our derivation is based on the total carrier density conservation, which includes both electrons and holes, for zero and any finite temperatures<cit.> n = n^(e) + (-1) n^(h) = ∫_0^∞ d E ρ_d(E) f_γ = 1(E,T) - ∫_-∞^0 d E ρ_d( E ){ 1 - f_γ = 1(E,T) }.The electron and hole occupation probabilities are complimentary and for electron doping at T = 0 the hole states term has no effect on Eq. (<ref>) (for details seeRef. [proj]).We beginwith the relatively simple case for silicene with dispersions (<ref>). The DOSρ_d(E) for which buckled honeycomb lattices is given by Eq. (<ref>). The expression for theFermi energy E_F for fixed electron doping density n at zero temperature depends on whether either one orboth electron subbands are doped. The former case occurs for doping densitiesn ≤ n_c = 1/2 π Δ_>^2 - Δ_<^2/ħ^2 v_F^2 = 2/πħ^2 v_F^2 Δ_SOΔ_z ,and the Fermi energy is obtained formn = 1/2 π E_F^2 - Δ_<^2/ħ^2 v_F^2.Alternatively, if the doping density is sufficient to populate both subbands, E_F is determined byn = 1/π 1/ħ^2 v_F^2 [ E_F^2 - 1/2 ( Δ_<^2 + Δ_>^2) ] . Once the temperature is set finite, Eq. (<ref>) leads us ton = I^(e)(Δ_i,T) -I^(h)(Δ_i,T), with the two terms corresponding to electron and hole components ofthe total carrier density. These integrals are presented asI^(e)(Δ_i,T)=∫_Δ_<^∞ d E A (E,T) +∫_Δ_>^∞ d E A (E,T) ,where A(E,T) = 1/π E/ħ^2 v_F^2 {1+exp[ E - μ(T)/k_B T] }^-1.Each of the these integrals could be easily evaluated. Using a variable substitutionξ = (E - Δ_< )/k_B T, we obtain1/π 1/(ħ v_F )^2∫_Δ_<^∞ d E E {1+exp[ E - μ(T)/k_B T] }^-1 = k_B T/π(ħ v_F )^2∫_0^∞ d ξ(Δ_< + ξk_B T){ 1 + exp[ξ - μ(T) - Δ_</k_B T] }^-1With the help of the following notation R^(p) (T,X) = ∫_0^∞ d ξξ^p/ {1 + exp[ξ - X / (k_B T) ] },we obtain the final result of the integration as k_B TΔ_</π( ħ v_F)^2 R^(0) [T,μ(T) - Δ_<] + 1/π (k_B T/ħ v_F)^2R^(1) [T,μ(T)-Δ_< ] .For p=0 and 1, corresponding to the2DEG and gapless graphene, Eq. (<ref>) leads to<cit.> R^(0) (T, X) = ln{ 1 + exp[X/k_B T] },R^(1) (T, X) = - Li_ 2{ - exp[X/k_B T] },where Li_ 2 (z) is the second-order polylogarithm function ordilogarithm defined as Li_ p (z) = ∑_k=1^∞z^k/k^p,Li_ 2 (z) = - ∫_0^zln (1-t)/tdt .The second term of Eq.(<ref>), which only differs from the first one by its integration limits, is2 k_B TΔ_>/π( ħ v_F )^2 ln{ 1 +exp[μ(T) - Δ_>/k_B T] } - 1/π ( k_B T/ħ v_F)^2Li_ 2{ - exp[μ(T)-Δ_>/k_B T] }.The remaining term I^(h)(Δ_i,T), i = { <,> }, which describes the contribution from the holes,is also easily obtained I^(h)(Δ_i,T) = I^(h)_1 (Δ_i,T) + I^(h)_2 (Δ_i,T) 0.1in where ,I^(h)_1 (Δ_i,T) = 1/π k_B T /( ħ v_F )^2 ∑_i = <,>Δ_iR^(0){ T, - [μ(T) + Δ_i ] }0.1in and I^(h)_2 (Δ_i,T) = 1/π( k_B T/ħ v_F)^2∑_i = <,>R^(1){ T, - [μ(T) + Δ_i ] }.Now, the total carrier denisty from Eq. (<ref>) could be written as n= ( k_B T/ħ v_F)^2 ∑_γ = ± 1 γ/π ∑_i = <,>R^(1)[ T,γμ(T) - Δ_i] + Δ_i/k_B T R^(0)[ T,γμ(T) - Δ_i ]or, explicitly expressing the polylogarithm functions, we write n( ħ v_F/k_B T)^2 =∑_γ = ± 1 γ/π ∑_i = <,> - Li_ 2{ - exp[γμ(T) - Δ_i/k_B T] }+ Δ_i/k_B T ln{ 1 + exp[γμ(T) - Δ_i/k_B T] }. Using Eqs. (<ref>) or Eq. (<ref>) depending on whether only one or bothsubbands are filled at zero temperature,we obtain the equation which relates the finite-temperature chemical potential with its T = 0 value E_F.Energy band gap(s) obviously affects this result. The μ(T) for gapped graphene with two fourfold degenerate energy subbands is obtained if we substitute Δ_< = Δ_> = Δ_0 and ∑_i = <,>⟹× 2. For gapless graphene, Δ_0 = 0 and π n = [ E_F/ (ħ v_F) ]^2, so that we write 1/2 ( k_B T )^2E_F^2=∑_γ = ± 1γ R^(1)[T,γμ(T) ]= - ∑_γ = ± 1γ Li_ 2{ - exp[γ μ(T)/k_B T] }.If the temperature is kept low with k_B T ≪ E_F, this result is simplified as<cit.> R^1 (x) ⋍( x^2/2+ π^2/6)Θ(x) + xln(1 + e^- | x |) , Finalizing our derivations for silicene, we briefly address the case of hole doping with E_F < 0. The leftpart of Eq. (<ref>) is now modified as - n^(h) = - 2/π1/(ħ v_F)^2∑_i = < ,> ∫_- ∞^-Δ_idE |E|Θ(-E + E_F) .In analogy with electron doing, the Fermi energy depends on whether only the Δ_<-subband (whichis now the higher one) or they are bothdoped. The equations determining the Fermi energy for given holedoping density n are exactly similar to Eqs. (<ref>) and (<ref>), which confirms complete symmetrybetween the electron and hole states in silicene. The right part of Eq. (<ref>) remains unchangedexcept the chemical potential is negative μ < 0 for any finite temperature.§.§ μ(T) for transition metal dichalcogenidesIn cotrast to the previously considered buckled honeycomb lattices and graphene, the electron/hole symmetryin TMDC's (such as MoS_2) is clearly broken. Even in the simplest gapped graphene model given byEq. (<ref>), the two hole subbands are not degenerate and separated by λ_0 at k=0. For allreasonable doping densities n < 10^17 m^-2, the Fermi energy is such that the lower ε_ν= 1^γ = -1(k=0)= -Δ/2 - λ_0 subband could not be populated at zero temperature.This could be verified by rewriting Eq. (<ref>) as n_c = 2/πλ_0Δ/(t_0 a_0)^2= 1.0 · 10^18 m^-2. From here on, we are going to use a picewise-linear model for the DOS with theempirical coefficients, provided in Sec. <ref>. Our analytical model for the DOS,develop in the preceding Appendix in Eqs. (<ref>) - (<ref>) could also be employed here without losing much of precision. Let's us first consider electron doping with density n_(0)^e at zero temperature. The corresponding Fermi energy is determined by n_(0)^e = c_1^(3)/2( E_F^2 - Δ^2/4 ) + c_0^(3) ( E_F - Δ/2 ) ,orE_F^ e = 1/c_1^(3) {- c_0^(3) + [ ( c_0^(3) + c_1^(3) Δ/2)^2 + 2 n_(0)^e c_1^(3)]^1/2}.For hole doping the result is quite similar, except c_1^(2) < 0and the upper valenceband gap is -Δ/2 + λ_0: E_F^(h) =1/c_1^(2) { - c_0^(2) + { -2 c_1^(2)n_(0)^h + [c_0^(2) - c_1^(2) (Δ/2 - λ_0)]^2 }^1/2 }.Using this expression, we can further improve the result in Eq. (<ref>) for the critical hole doping density n_c needed to reach the lower subband at zero temperature. The required Fermi energy is E_F^(h)≤ - Δ/2 - λ_0, so that the corresponding hole density is n_c^h = - λ_0Δ c_1^(2) + 2 c_0^(2) λ_0 = 3.3 · 10^17m^-2.This critical density value is still far above the experimentally realizable values ⋍ 1.0 · 10^17 m^-2, and for all our calculations, the lower hole subband is never populated at T=0.Finally, we address the finite-temperature chemical potential for MoS_2. Once again, we start with the carrierdensity conservation equation (<ref>).The electron component is easily evaluated ask_B T(c_0^(3) + Δ/2)R^(0)[T,μ(T) - Δ/2] + c_1^(3)(k_B T)^2R^(1)[T,μ(T)-Δ/2] . The hole state integrals are written in the following form ∫_-∞^0 d E ρ_d( |E|){1 - f_1(E,T) }=∫_-∞^-Δ/2 + λ_0 d E [ - c_1^(2)E + c_0^(2)] {1+exp[ μ(T) - E/k_B T] }^-1 + + ∫_-∞^-Δ/2 - λ_0 d E [ - δ c_1^(1)E + δ c_0^(1)]{1+exp[ μ(T) - E/k_B T] }^-1,where δ c_1^(i) = c_1^(i) - c_2^(i), i = 1,2. They are evaluated as I^(h)(Δ,λ_0|T) = ∑_j = 1 ^4I^(h)_j0.1in where ,I^(h)_1 (Δ,λ_0|T) = k_B T ( Δ/2 - λ_0 + c_2^(0))R^(0){ T, - [μ(T) + Δ/2 - λ_0 ] }, I^(h)_2 (Δ,λ_0|T) = c_2^(1)( k_B T )^2R^(1){ T, - [μ(T) + Δ/2 - λ_0 ] }, I^(h)_3 (Δ,λ_0|T) = k_B T ( Δ/2 + λ_0 + δ c_1^(0))R^(0){ T, - [μ(T) + Δ/2 + λ_0 ] },I^(h)_2 (Δ,λ_0|T) = δ c_1^(1)( k_B T )^2R^(1){ T, - [μ(T) + Δ/2 + λ_0 ] }.Combined with the electron terms (<ref>), these hole integrals (<ref>) form the right sideof the carrier density conservation (<ref>). Its left side, corresponding to zero temperature, is given by Eq. (<ref>) for electron doping and byn_(0)^e = ( Δ/2 + E_F - λ_0) {c_2^(0) + c_2^(1)/2[Δ/2 - (E_F + λ_0)]},for hole doping. The symmetry between the electron and hole states is no longer present, which strongly affects the finite-temperature behavior of the chemical potential in TMDC's. | http://arxiv.org/abs/1707.08637v1 | {
"authors": [
"Andrii Iurov",
"Godfrey Gumbs",
"Danhong Huang",
"Ganesh Balakrishnan"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170726204813",
"title": "Finite-temperature Coulomb Excitations in Extrinsic Dirac Structures"
} |
Direct Observation of Orbital Hybridisation in a Cuprate SuperconductorJ. Chang December 30, 2023 ======================================================================== In many applications, and in systems/synthetic biology in particular, it is desirable to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point), or in other words to solve the switching problem. It was recently shown that, for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that this problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system. One of the main advantages of using the Koopman operator is the powerful computational tools developed for this framework. Besides the presence of numerical solutions, the switching/convergence problem can also serve as a building block for solving more complicated control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation. Potentially, our approach can be extended to problems with different parametrizations of control signals since the only fundamental limitation is the finite time application of the control signal. Monotone Systems, Koopman Operator, Isostables, Generalized Repressilator, Genetic Toggle Switch § INTRODUCTION Synthetic biology is an active field of research with applications in metabolic engineering, bioremediation and energy sector (<cit.>). One of the main goals of synthetic biology is to engineer biological functions in living cells (<cit.>) and control theory naturally plays an essential role toward that end. Control theoretic regulation of protein levels in microbes was achieved by <cit.>, <cit.>, and <cit.>. However, these works result in time-varying feedback control signals, which are affected by limitations due to physical constraints not only in sensing but also in actuation. Actuation limitations are observed with chemical induction, for instance. While the concentration of a chemical can be easily increased by adding this chemical to a culture, it is much more complicated to decrease its concentration (for instance, diluting is a labor-intensive process that is not suited to be performed repeatedly). Therefore, one of the challenges in synthetic biology is to derive control policies that not only achieve the desired objectives but are also simple enough to deal with the actuation limitations. In this context, an example of such control policies is based on temporal pulses, on which the present paper focuses.One of the basic but nevertheless important control problems is the problem of convergence to an equilibrium. In synthetic biology, it can, for instance, be motivated by the genetic toggle switch circuit, which is a major building block in applications. The genetic toggle switch by <cit.>, for example, consists of two interacting genes. Their design ensured that the concentration of proteins expressed by one gene is always much higher than the concentration of proteins expressed by the other gene: one gene is switched “on” while the other is switched “off”. In this case, the system is bistable and the control objective of the problem is to drive the state from one equilibrium (e.g. one gene switched “on”) to the other (e.g. the other gene switched “on”) in minimum time and given a fixed energy budget (e.g. to avoid cell death). In this paper, we propose to solve this convergence problem using a specific set of signals, namely temporal pulses u(t) with fixed length τ and magnitude μ: u(t) = μ h(t,τ) h(t,τ) = 1 0 ≤ t ≤τ ,0 t >τ .The main challenge in solving this convergence problem is the absence of well-developed theory and computational methods. This can be explained by the fact that classical optimal control methods rely on the semigroup property (of the value function and/or the dual variable), while our parametrization of control signals violates it. We, therefore, restrict our analysis to the class of monotone systems, properties of which allow simpler solutions. We note, however, that monotone systems play an important role in systems/synthetic biology and hence there exist many applications with this restriction. We solve the convergence problem by computing only one time-independent function, which we call the pulse control function and denote by r. This function links all the tunable parameters of the convergence problem so that finding a tradeoff between the convergence time and the energy budget becomes straightforward.In our solution, we use the Koopman operator (cf. <cit.>), which offers a linear infinite dimensional description of a nonlinear system and allows a spectral approach to the nonlinear system. In particular, the definition of the pulse control function involves the dominant eigenfunction of the Koopman operator of the unforced system (i.e., when the control signal is equal to zero). This allows to use computational methods developed in the Koopman operator framework to solve our problem. In particular, we show that the function r can be computed with Laplace averages (<cit.>).Our theoretical results generally do not apply to non-monotone systems. Let alone a solution using pulses may not exist. However, we can still use the function r in some cases to derive control policies for non-monotone systems. Furthermore, we can use our framework to solve more complicated control problems. We illustrate this idea on the problem ofsynchronization of cardiac cells modeled by non-monotone FitzHugh-Nagumo systems (<cit.>).Related Work. <cit.> proposed to solve the problem of switching between exponentially stable equilibria in monotone systems using temporal pulses. The authors derived a computational procedure to estimate the set of control signals allowing the switch. The control strategy proposed by <cit.> is open loop so that the control signal cannot be adjusted during the experiment. <cit.> considered a similar setting, but using the Koopman operator framework to estimate the set of all switching pulses and provide estimates for convergence time. In this paper, we present an optimization program for solving the convergence problem, which is a generalization of the switching problem considered by <cit.> and <cit.>. <cit.> considered the convergence/escape problem cast in the Koopman framework formalism. <cit.> did not restrict the class of systems but assumed full controllability, i.e. control signals affect all the states in an affine manner. <cit.> proposed to synchronize the cardiac cells by using geometric tools from Koopman operator framework and the techniques based on the Pontryagin's maximum principle. The authors did not parametrize the control signal, which led to complicated time-dependent optimal control signals. We will show that our scheme still achieves synchronization, but in the longer time than the optimal solution by <cit.>.To summarize, in this paper, we consider a convergence problem similarly to <cit.>, the target set is chosen similarly to <cit.>, while the class of systems and the control signals are restricted as in the works by <cit.> and <cit.>. We opted for restricting the class of systems in order to show optimality of our approach in this specific case, which indicates that systems' properties can be used to derive easy-to-implement and interpretable optimal solutions. Outline of the Paper. The rest of the paper is organized as follows. In Section <ref>, we cover basic definitions and properties of monotone systems and we introduce the Koopman operator framework. We formulate and discuss our convergence problem in Section <ref>, while also presenting the main result, which we prove in Section <ref>. In Section <ref>, we apply the theoretical results to the switching problem (open and closed loop control) and to the synchronization of cardiac cells.§ PRELIMINARIESConsider a system of the formẋ = f(x,u), x(0) = x_0, with f: ×→^n, u:→, and where ⊂^n, ⊂ are open and u belongs to the space _∞ of Lebesgue measurable functions with values from . We assume that f(x, u) is twice continuously differentiable (C^2) in (x, u) on ×.The flow map ϕ: ××_∞→^n induced by the system is such that ϕ(t, x_0, u) is a solution of (<ref>) with an initial condition x_0 and a control signal u. We denote the differential of a function g(x, y):^n×^m→^k with respect to x as ∂_x g(x, y).Let J(x) denote the Jacobian matrix of f(x,0) (i.e., J(x) = ∂_x f(x,0)/∂ x). For every stable fixed point x^∗ of the system, we assume that the eigenvectors of J(x^∗) are linearly independent (i.e., J(x^∗) is diagonalizable). The eigenvalues of J(x^∗) are denoted by λ_i with i = 1,…, n and are ordered by their real part, that is (λ_i) ≥(λ_j) for all i, j. We also denote the positive orthant by ^n = {x∈^n | x_i > 0,i = 1,…, n}, and the nonnegative orthant by ^n = {x∈^n | x_i ≥ 0,i = 1,…, n}. §.§ Koopman operator Autonomous nonlinear systems can be studied in the framework of the Koopman operator. A semigroup of Koopman operators acts on functions g:^n→ (also called observables) and is defined byU^t g(x) = g ∘ϕ(t,x,0),t ≥ 0where ∘ is the composition of functions. Provided that the vector field and observables are C^1, one can define the infinitesimal generator of the operator as L g(x) = (f(x, 0))^T ∇ g(x) on a compact set. The semigroup is linear (cf. <cit.>) and can be studied through its spectral properties. In this paper, we will limit our use of the Koopman operator to unforced systems (<ref>) on a basin of attraction of an exponentially stable equilibrium x^∗ (that is, the eigenvalues λ_j of J(x^∗) are such that (λ_j) < 0 for all j). The basin of attraction is defined by (x^∗) = {x∈^n | lim_t→∞ϕ(t,x,0) = x^∗}. In this case, the eigenvalues λ_j of the Jacobian matrix J(x^∗) are also the so-called Koopman eigenvalues, which are associated with the Koopman eigenfunctions s_j: → satisfyingU^t s_j(x) = s_j(ϕ(t, x,0)) = s_j(x) e^λ_j t,x∈,or equivalently∇ s_j(x)^T f(x, 0) = λ_j s_j(x).If the vector field f(·, 0) is a C^2 function and the Jacobian matrix J(x^∗) is diagonalizable, then the Koopman eigenfunctions s_j belong to C^1 (<cit.>).We refer to an eigenvalue λ_1 satisfying (λ_1) > (λ_j) for all λ_1 λ_j as the dominant eigenvalue. We assume that such an eigenvalue exists (it is the case for monotone systems) and we call the associated eigenfunction s_1 the dominant eigenfunction. If the dominant eigenvalue is such that (λ_1)<0, then the dominant eigenfunction s_1 can be computed through the Laplace averageg_λ^∗(x) = lim_t→∞1/T∫_0^T (g∘ϕ(t, x,0)) e^-λ t d t.For all g∈ C^1 that satisfy g(x^∗)=0 and (∇ g(x^∗))^T v_1 ≠ 0, where v_1 is the right eigenvector of J(x^∗) corresponding to λ_1, the Laplace average g_λ_1^∗ is equal to s_1(x) up to a multiplication with a scalar. If the algebraic and geometric multiplicities of λ_1 are equal to μ_1, then there are μ_1 independent eigenfunctions associated with λ_1 and they can be computed by choosing linearly independent right eigenvectors of J(x^∗) corresponding to λ_1. The eigenfunctions s_j(x) with j≥ 2 are generally harder to compute using Laplace averages, but can be obtained through other methods such as linear algebraic methods by <cit.>. The eigenfunctions can also be estimated from data by using the so-called dynamic mode decomposition (DMD) method (cf. <cit.>). The Koopman eigenfunctions capture important geometric properties of the system. In particular, the dominant Koopman eigenfunction s_1 is related to the notion of isostables.Suppose that s_1 ∈ C^1. An isostable ∂_α associated with the value α > 0 is the boundary of the set _α ={x∈^n||s_1(x)| ≤α}, that is∂_α ={x ∈^n||s_1(x)| = α}.A more rigorous definition based on the flow is due to <cit.>. Isostables are useful from a control perspective since they capture the dominant (or asymptotic) behavior of the unforced system. Indeed, it can be shown that trajectories with initial conditions on the same isostable ∂_α_1 converge synchronously toward the equilibrium, and reach other isostables ∂_α_2, with α_2<α_1, after a time= 1/|(λ_1)|ln(α_1/α_2) .In the case λ_1 ∈, for example, it can be shown that the trajectories starting from ∂_α share the same asymptotic evolutionϕ(t,x,0) → x^∗ + v_1αe^λ_1 t ,t→∞ .§.§ Monotone SystemsWe consider systems that preserve a partial order induced by a nonnegative orthant ^n, but our results can be extended to other conesin ^n.We define a partial order ≽ as follows: x≽ y if and only if x - y ∈^n (we write x⋡y if the relationx ≽ y does not hold). We will also write x≻ y if x≽ y and x y, and x≫ y if x- y ∈^n.Similarly, a partial order can be defined on the space of signals u∈_∞: u≽ v if u(t) - v(t) ∈^n for all t≥ 0.We also introduce concepts that are important for our subsequent discussion. Let [x, y] denote an interval in the order ≼, that is [x, y] = { z ∈^n | x≼ z ≼ y }. For a function W:^n →∪{-∞, +∞}, we refer tothe set (W) = {x∈^n | |W(x)| < ∞} as its effective domain. A function W:^n →∪{-∞, +∞} is called increasing if W(x) ≥ W(y) for all x≽ y on (W). Control systems in the form (<ref>) whose flows preserve a partial order relation ≽ are called monotone systems. The system ẋ = f(x,u) is called monotone if ϕ(t,x, u)≼ϕ(t,y, v) for all t≥ 0, and for all x≼ y, u≼ v.The unforced system ẋ = f(x, 0) is strongly monotoneif it is monotone and x≺ y implies that ϕ(t,x, 0)≪ϕ(t,y, 0) for all t> 0.A certificate for monotonicity is a condition on the vector field, for which we refer the reader to <cit.>. We finally consider the spectral properties of unforced monotone systems that are summarized in the following result.Consider that the system ẋ = f(x) with f∈ C^2() has an exponentially stable equilibrium x^∗ and let λ_j be the eigenvalues of J(x^∗) such that (λ_i)≥(λ_j) for all i ≤ j.(i)If the system is monotone with respect toon a set ⊆(x^∗), then λ_1 is real and negative, the right eigenvector v_1 of J(x^∗) can be chosen such that v_1 ≻ 0, while the eigenfunction s_1 can be chosen such that s_1(x) ≥ s_1(y) for all x, y∈ satisfying x≽ y. (ii) Furthermore, if the system is strongly monotone with respect toon a set ⊆(x^∗) then λ_1 is simple, real and negative, λ_1 > (λ_j) for all j ≥ 2, v_1 and s_1 can be chosen such that v_1≫ 0 and s_1(x)> s_1(y) for all x, y∈ satisfying x≻ y; The result is from <cit.>. Without loss of generality, we will assume that a dominant eigenfunction s_1 is increasing even if λ_1 is not simple. § CONVERGENCE TO AN ISOSTABLE PROBLEM§.§ Problem Formulation and DiscussionIn order to formulate the basic problem we want to address, consider the following assumptions: A1. The vector field f(x,u) in (<ref>) is twice continuously differentiable in (x, u) on ×. A2. The unforced system (<ref>) has an exponentially stable equilibrium x^∗ inwith a diagonalizable J(x^∗). A3. The system is monotone with respect to ^n× and forward-invariant on × in the sense that for all x∈, u∈_∞, the flow ϕ(t,x,u(·)) belongs to . A4. The eigenfunction s_1(x) is such that∇ s_1(x) ≫ 0 for all x∈(s_1). A5. f(x, μ_1) ≻ f(x, μ_2) for all x∈ and μ_1> μ_2≥ 0. A6. The space of control signals is limited to temporal pulses u(t) = μ h(t,τ), where h is defined in (<ref>). Assumption A1 guarantees existence and uniqueness of solutions, while Assumption A2 introduces a reference point x^∗. These assumptions are perhaps more restrictive than the ones usually met in control theory. That is, f(x,u) is usually assumed to be Lipschitz continuous in x for every fixed u, and the equilibria are asymptotically stable. Our assumptions are guided by our consequent use of the Koopman operator. Assumptions A1 and A2 guarantee the existence of continuously-differentiable eigenfunctions on the basin of attraction (x^∗) of x^∗. Monotonicity is crucial, but forward-invariance on × is a rather technical assumption on which our computational methods do not rely. Assumption A4 is well-posed since s_1∈ C^1 due to Assumption A1. If Assumptions A1 – A3 hold, then we have ∇ s_1(x) ≻ 0 and f(x,μ_1) ≽ f(x, μ_2) for μ_1 > μ_2 ≥0, hence Assumptions A4 and A5 serve as technical assumptions that guarantee uniqueness of solutions and a certain degree of regularity. We will comment throughout the paper on the case when Assumptions A4 – A5 do not hold. Assumption A6 is guided by many applications, where there is a need to parametrize in advance the control signal. In this paper, we choose the easiest parametrization, although the only fundamental limitation is to set u(t) to zero after some time τ. It is, therefore, possible to generalize our approach to more complicated control signals. We proceed by formulating a basic but fundamental problem.Problem 1. Converging to an isostable. Consider the system ẋ = f(x,u) satisfying assumptions A1–A6 and the initial state x_0. Compute a control signal u(t) = μ h(t,τ) such that the flow ϕ(t,x_0, u(·)) reaches the set _ε(x^∗) for some small ε>0 in minimum time units _ conv subject to the energy budget u_≤_ max.Our formulation based on the isostables is guided by our use of Koopman operator framework for computational purposes. However, there are other benefits in this formulation. One can view _ε(x^∗) as a ball in the (contracting) pseudometric d_K(x,y) = |s_1(x) - s_1(y)| on a basin of attraction (x^∗)(d_K(x, y) is a pseudometric, since d_K(x, y) can be equal to zero for some x y). By reformulating the standard convergence problem using a pseudometric defined through Koopman eigenfunctions, we take into account the dynamical properties of the unforced system. Furthermore, if ε is close to zero then the solution of Problem 1 can be used to solve a convergence-type problem. For example, the problem of switching between equilibria, which is considered in <cit.>, falls into this category. The main challenge in solving this problem is the parametrization of the control signal. Most of the control methods (such as dynamic programming, Pontryagin's maximum principle) are not tailored to deal with time parametrized control signals since they rely on the semigroup property of the value function or the dual variable. Hence it is not entirely clear how to systematically approach this problem through these methods. §.§ A Solution using a Static Optimization ProgramThe computational solution to our problem will be established by computing first a static function, which we call the pulse control function and define below. Let the function r:××→⋃{∞} such that r(x, μ,τ) = s_1(ϕ(τ, x, μ)), where s_1 is a dominant eigenfunction on the basin of attraction of x^∗, be called the pulse control function. By convention r(x, μ, τ) = ∞, if ϕ(τ, x, μ) ∉(x^∗) If s_1 is real-valued and increasing on (s_1) = (x^∗), then we assume it is extended to ^n so that s_1:^n→∪{-∞, +∞} is increasing on ^n. We note that the switching function proposed in <cit.> corresponds to r(x^∗, ·,·). The pulse control function r can be used in a context broader than the switching function by <cit.>, which we demonstrate in this paper. In particular, we will solve Problem 1 using the following result, which we will prove in what follows. Consider the system (<ref>), Problem 1 under Assumptions A1–A6 and the optimization program:γ^∗= min_μ≥ 0, τ≥ 0 1/|λ_1| ln(|r(x, μ,τ)|) + τ,subject to: r(x,μ,τ)≤ -ε, μ·τ≤_ max. If s_1(x_0)≤-ε, an optimal solution to (<ref>) is an optimal solution to Problem 1, if the former is feasible. Furthermore, the objective is nonincreasing in μ and τ and an optimal solution to (<ref>), if it exists, is achieved at the boundary of the admissible set to the constraint (<ref>) and/or the constraint (<ref>). If Assumptions A5 and A6 do not hold, then we can possibly have multiple minima including the points which do not activate the constraints. However, we can still compute a minimizing solution using the same program. Intuitively, the objective function is the convergence time (in fact, _ conv=γ^∗-1/|λ_1| ln(ε)), the constraint on r(x, μ, τ) ≤ - ε ensures that we stop when reaching _ε and the constraint μ·τ≤_ max is the energy budget. Since the optimum is attained when one of the constraints is active, the optimization program can be solved by a line search over μ (or τ) over the constraints curves, provided that r can be estimated at any given point. In our simulations we compute the function r for specific pairs of values (μ,τ) and take the minimum over these pairs. The two terms in the objective function of the static optimization problem show the tradeoff on the choice of the intermediate target isostable (which is to be reached after a time τ). For instance, choosing an isostable close to the equilibrium can lead to a large pulse duration τ (second term), but a small convergence time of the free motion (first term). Furthermore, the function r also allows to understand the tradeoff between the energy spent and the convergence time, which is not straightforward using standard optimal control theory. To summarize, we derived a static optimization problem, which has the same solution as the dynamic optimization problem (Problem 1). In order to compute the solution, one needs an efficient computational procedure for evaluating r at a given point. §.§ Computation of the Pulse Control FunctionThe eigenfunction s_1 can be estimated through Laplace averages (<ref>) and the function r is subsequently obtained since it is the composition of the eigenfunction s_1 with the flow. In particular, we can derive the following formula:r(x,μ,τ)= lim_t̅→∞1/t̅∫_0^t̅ g∘ϕ(t, ϕ(τ,x, μ),0) e^-λ_1 td t = lim_t̅→∞1/t̅∫_τ^t̅ g∘ϕ(t, x, μ h(·,τ)) e^-λ_1 (t-τ)d t ,where λ_1 is the dominant Koopman eigenvalue, g∈ C^1 satisfies g(x^∗)=0, v_1^T ∇ g(x^∗) ≠ 0, v_1 is the right eigenvector of J(x^∗) corresponding to λ_1 and h(t,τ) is the step function defined in (<ref>). In practice, we choose g(x)= w_1^T (x- x^∗), where w_1 is the dominant left eigenvector of J(x^∗). Since λ_1 is real according to Assumption A3 and Proposition <ref>), we have r(x, μ, τ)= lim_t̅→∞ w_1^T(ϕ(t̅, x, μ h(·,τ)) - x^∗) e^-λ_1 (t̅-τ)≈ w_1^T(ϕ(t̅, x, μ h(·,τ)) - x^∗) e^-λ_1 (t̅-τ)where the time t̅ should be chosen large enough.In this case the tolerance of the differential equation solver should be set to O(e^λ_1 (t̅-τ)). When only observed data are available, the eigenfunction — and therefore the function r — can be computed through dynamic mode decomposition methods (cf. <cit.>). This idea is illustrated in Appendix <ref>. §.§ Is the Control Space Rich Enough? Throughout the paper, we assume that the problem has a solution in the form of a temporal pulse u(t) = μ h(t, τ). We will argue that in the case of monotone systems, this is not a restrictive assumption. First of all, if the system is globally asymptotically stable, then clearly we can converge to x^∗ by using a temporal pulse.Assume now that the system is monotone with two exponentially stable equilibria x^∗ and x^∙ and basins of attraction (x^∗) and (x^∙), respectively. Let the system be defined on a forward-invariant set = (x^∗)⋃(x^∙). Assume also that x^∗≫ x^∙, which is typically fulfilled in many bistable monotone systems. If x_0∈(x^∗), then we can choose u = 0, which is a temporal pulse with τ =0. Consider now the case x_0 = x^∙. If there exists a control signal u^1∈_∞ driving the system from x^∙ to x^∗, then we have ϕ(t, x^∙, u^1) ≼ϕ(t, x^∙, μ), where u^1(t)≤μ for (almost) all t. At a time τ, the flow ϕ(τ, x^∙, u^1) will be in the vicinity of x^∗ and in the basin of attraction of (x^∗). The flow ϕ(τ, x^∙, μ) will also be in the basin of attraction of x^∗. Indeed, if ϕ(τ, x^∙, μ) ∈(x^∙), then [x^∙, ϕ(τ, x^∙, μ)]∈(x^∙), which contradicts that ϕ(τ, x^∙, u^1) ∈(x^∗) and ϕ(τ, x^∙, u^1) ∈[x^∙, ϕ(τ, x^∙, μ)] (cf. <cit.>). Hence if we can switch from x^∙ to x^∗ with a control signal u(t), then we can switch with a temporal pulse. Finally the case x_0 ∈(x^∙) is treated in a similar manner by first allowing the trajectory to converge to a neighborhood x^∙ with u^2 =0 and then applying the argument above.This discussion shows that using temporal pulses in the case of monotone systems does not restrict the space of feasible problems. However, we can strengthen the argument by showing that constant controls are optimal in the absence of energy constraints in Appendix <ref>.§ PROOF OF THE MAIN RESULT In order to prove the main result, we would need to use the properties of the pulse control function r, which we present in the following lemma. Let the system (<ref>) satisfy Assumptions A1–A5. Then r is a C^1 function on its effective domain (r). Furthermore, for all (x,μ,τ)∈(r)(i)∂_xr(x,μ,τ) ≫ 0, ∂_μ r(x,μ,τ) > 0and∂_τ r(x,μ,τ) > λ_1 r(x,μ,τ)(ii) If r(x,μ,τ)≤ 0, then ∂_τ r(x,μ,τ)> 0;(iii) If f(x, ν)≽ 0, then ∂_τ r(x,μ,τ)>0 for all finite τ> 0 and μ > ν.(o) First, we show that under the assumptions above for allt >0 and μ_1 > μ_2, we haveϕ(t, x, μ_1) ≻ϕ(t, x, μ_2), s_1(ϕ(t, x, μ_1)) > s_1(ϕ(t, x, μ_2)).Due to monotonicity, we have that ϕ(t,x, μ_1) ≽ϕ(t,x, μ_2) for allt >0andμ_1 > μ_2. All we need to show is that ϕ(t,x, μ_1) ϕ(t,x, μ_2) for all finite t>0. At t=0, the time derivatives of the flow are equal to f(x,μ_1) and f(x,μ_2). Since f(x,μ_1) ≻ f(x,μ_2) (Assumption A5), there exists a T>0 such that ϕ(t,x, μ_1) ≻ϕ(t,x, μ_2) for all t<T. If for some T we have that ϕ(T,x, μ_1) = ϕ(T,x, μ_2) and ϕ(t,x, μ_1) ≻ϕ(t, x, μ_2) for all t <T, then for some index i we haved ϕ_i(t,x, μ_1)/d t|_t =T <d ϕ_i(t,x, μ_2)/d t|_t =T.This implies that f_i(ϕ(T, x, μ_1),μ_1) < f_i(ϕ(T, x, μ_2), μ_2), which together with ϕ(T, x, μ_1) = ϕ(T,x, μ_2) contradicts Assumption A5. Therefore, ϕ(t,x, μ_1) ≻ϕ(t,x, μ_2) for all finite t> 0. Due to Assumption A4 we have that ∇ s_1(x )≫ 0, which in particular means that s_1(x) > s_1(y) for all x≻ y, and (<ref>) follows from (<ref>).(i)The flow is continuously-differentiable for constant control signals since f(x, u)∈ C^2 (Assumption A1), and hence r(x,μ,τ) = s_1(ϕ(τ,x,μ)) is a C^1 function. For x ≻ y, where(x,μ,τ) and (y,μ,τ)∈(r), we have s_1(ϕ(τ, x,μ)) > s_1(ϕ(τ, y, μ)) due to monotonicity (Assumption A3) and Assumption A4. Hence ∇_x r(x, μ,τ)≫ 0.For μ > ν, where(x,μ,τ) and (x, ν,τ)∈(r), we have s_1(ϕ(τ, x, μ)) > s_1(ϕ(τ, x, ν)) due to monotonicity (Assumption A3) and point (o). Hence ∂_μ r(x, μ,τ)> 0. Finally,∂_τ r(x, μ,τ)> λ_1 r(x,μ, τ) follows from:∂_τ r(x,μ, τ)=d s_1(ϕ(t,x,μ))dt|_t =τ =∇ s_1(ϕ( τ, x,μ))^T f(ϕ(τ, x,μ), μ) >∇ s_1(ϕ( τ, x,μ))^T f(ϕ( τ, x,μ), 0) =λ_1 s_1(ϕ(τ, x, μ))=λ_1 r(x,μ, τ),where the inequality is due to Assumption A4 and A5, and the following equality is due to (<ref>). (ii) This follows directly from point (i). (iii) This proof employs a fairly standard technique in monotone system theory. First note that f(x,ν)≽ 0 and Assumption A5 imply that f(x,μ)≻ 0. Consider a perturbed system ż = f(z,μ) +1/n, whereis a vector of ones and n is a positive integer. Let the flow of this system be ϕ_n(t,x,μ). We have that f(x,μ) +1/n ≫ f(x,μ)≻ 0. Since this is the derivative of the flow with respect to time around t=0, then ϕ_n(δ, x, μ)≫ x for a sufficiently small positive δ. Now for t > δ, we haveϕ_n(t, x, μ)= ϕ_n(t-δ, ϕ_n(δ, x, μ), μ) ≫ϕ_n(t-δ, x, μ), and hence ϕ_n(t, x, μ)≫ϕ_n(ξ, x, μ) for any t>ξ. This conclusion holds for all n>0. With n→∞, we have that ϕ_n(t,x,μ)→ϕ(t,x,μ). Therefore, ϕ(t,x, μ) ≽ϕ(ξ, x, μ) for all finite t>ξ and ∂_τϕ(τ, x, μ) ≽ 0.If the equality ∂_τϕ(τ, x, μ) = 0 is attained, then ϕ(τ, x, μ) is an equilibrium of the system ẋ = f(x, μ). This is impossible since an exponentially stable equilibrium cannot be reached in finite time τ due to uniqueness of solutions. Hence we have ϕ(t,x, μ) ≻ϕ(ξ, x, μ) for all finite t>ξ and, using Assumption A4, we obtain s_1(ϕ(t,x,μ)) > s_1(ϕ(ξ, x, μ)) for all finite t>ξ. The result follows. If Assumptions A4 and A5 do not hold, then all the inequalities in Lemma <ref> are not strict. For instance, we have that∂_x r(x,μ,τ) ≻ 0, ∂_μ r(x,μ,τ) ≥ 0and ∂_τ r(x,μ,τ) ≥λ_1 r(x,μ,τ) in point (i). We present additional properties of the function r in Appendix <ref>. Now we can present the proof of our main result.Theorem <ref> It is straightforward to verify that all the constraints and optimization objective are the same for Problem 1 and problem (<ref>). Hence, by construction the first part of the statement is fulfilled.According to the constraint (<ref>), we have that r(x_0,μ,τ)<0, which implies the following chain of inequalities∂_τ (ln(|r(x_0,μ,τ)| e^|λ_1| τ)) =∂_τ (|r(x_0,μ,τ)| e^|λ_1| τ)/|r(x_0,μ,τ)| e^|λ_1| τ =-∂_τ (r(x_0,μ,τ)) ·e^|λ_1| τ +|λ_1||r(x_0,μ,τ)| e^|λ_1| τ/|r(x_0,μ,τ)| e^|λ_1| τ <λ_1|r(x_0,μ,τ)| e^|λ_1| τ +|λ_1||r(x_0,μ,τ)| e^|λ_1| τ/|r(x_0,μ,τ)| e^|λ_1| τ = 0where the inequality follows from Lemma <ref>. Hence, the derivative of the objective function in (<ref>) with respect to τ is negative. Finally, ∂_μln(|r(x_0,μ,τ)| e^|λ_1| τ) is also negative according to Lemma <ref>. Hence, if there is a feasible point, the constraints (<ref>), (<ref>) are reached in order to minimize the objective.§ APPLICATIONS §.§ Implementation DetailsWe implemented our computational procedures both in python (using lsoda ordinary differential equation (ODE) solver) and Matlab (using ode15s ODE solver). We run our computational algorithm on a laptop equipped with a 4 core Intel i7 processor running at 2.4 GHz and 8 GB of RAM, however, we did not explicitly parallelize the computations. Computing one value of the function r is equivalent to computing one trajectory of the system, albeit with high precision (we set relative tolerance of the solvers to e^-10 – e^-14).§.§ Closed-Loop Switching in Generalized RepressilatorThe eight species generalized repressilator is an academic example, where each of the species represses another species in a ring topology (cf. <cit.>). The corresponding dynamic equations for a symmetric generalized repressilator are as follows: ẋ_1= p_1 1^0/1 + (x_8/p_1 2^0)^p_1 3^0 + p_1 4^0 - p_1 5^0 x_1 + u,ẋ_i= p_i 1^0/1 + (x_i-1/p_i 2^0)^p_i 3^0 + p_i 4^0 - p_i 5^0 x_i, ∀ i = 2,… 8,where p_i 1^0 = 40, p_i 2^0 = 1, p_i 3^0 = 2, p_i 4^0 = 1, and p_i 5^0 = 1. This system has two exponentially stable equilibria x^∗ and x^∙ and is monotone with respect to the cones _x = P_x ^8 and _u =, where P_x= ([1, -1, 1, -1, 1, -1, 1, -1]). We have also x^∙≼__x x^∗. It can be shown that the unforced system is strongly monotone in the interior of ^8 for all positive parameter values. We consider here the problem of switching the system from one equilibrium x^∙ to the other equilibrium x^∗ and we can verify that there exist pulse control signals that induce such a switch.In the left panel of Figure <ref>, we plot the level set of the function _ conv(x^∙, μ,τ, 10^-2) = 1/|λ_1|ln(r(x^∙, μ, τ)/10^-2) + τ, the level set _ max = 100 of the function μτ = _ max, and the level set r(x^∙,μ,τ) = 0. The last two are related to the constraints of the static optimization program (<ref>). Note that r is computed with the dominant eigenfunction associated with the target equilibrium x^∗.We also note that the function _ conv can escape to -∞ around the level set r(x^∙, μ,τ) ≈ 0. This is not a conflict with the interpretation ofthe function _ conv, since it represents the convergence time only if the value of |r(x^∙,μ,τ)| is larger than 10^-2. Otherwise the term 1/|λ_1|ln(|r(x^∙, μ, τ)|/10^-2) is negative, and the computational results are meaningless. This also explains why the level sets of _ conv appear to have the same asymptotics as the level set r(x^∙, μ,τ) ≈ 0 in Figure <ref>. Our goal is to compare the open-loop (proposed by <cit.>) and closed-loop solutions to the switching problem subject to perturbations of parameters p_i j^0. We consider two settings for the simulation. In both settings, we compute the control signals based on the nominal model (<ref>) with the parameter values p_i j^0, but the simulations are obtained with two sets of (exact) parameter values:Setting A.We set p_i 1^A = 50 for odd i. Setting B. We set p_i 1^B = 30 for odd i. The Euclidean distance between the nominal initial point and the actual initial point in Setting A and B is equal to 0.025 and 0.031, respectively. In order to compute an open-loop optimal control policy based on the nominal model (i.e. with parameter values p_i j^0), one can solve the static optimization program (<ref>). The plots in Figure <ref> also offer a graphical solution to the problem and a depiction of possible tradeoffs in the problem. In our case, the optimal solution lies at the intersection of the constraint curves (i.e. energy budget curve and level set r(x^∙,μ,τ) = 0). In our simulations, we pick a pair (μ^0, τ^0) lying near the zero level set of r below the level set _ max = 100. This is not an optimal solution for the energy budget _ max = 100, however, we pick a solution with a lower energy expenditure and a larger time τ (and hence larger convergence time) in order to have a possibility to react to the obtained measurements in the case of the closed-loop setting. We take τ^0 = 20 and compute μ^0 minimizing the time _ conv on a uniform grid of 100 points in [2, 10], which gives the value μ^0 = 3.53.For the closed-loop control, we take the same initial pair (τ^0, μ^0).In both setting A and B, we update the control signals every t_ samp = 2. For each update, we decrease the time of the pulse by t_ samp and we decrease the available energy budget by subtracting the energy already consumed. We then compute the values of the function r with a fixed τ = 20 - N t_ samp, where N is the number of previous updates and we choose the value μ on a uniform grid of 100 points in [2, 10], which minimizes _ conv with r(x^∙, μ, τ) < 0. The simulation results are depicted in the center and right panels of Figure (<ref>). In Simulation A, the system converges to the target equilibrium faster than the nominal one (i.e, with parameters p_i j^0) and the closed-loop solution saves energy and limits the overshoot in comparison with the open-loop solution. In Simulation B, the opposite occurs and all the energy budget is spent. In this case, the closed-loop solution allows the switch, while the open-loop (i.e. <cit.>) does not.§.§ Synchronization of Cardiac Cells Besides the switching problem, our approach can be used for more general problems. For example, a problem of importance in biology is to synchronize an ensemble of systems (e.g. cells) with a common control input. In this subsection, we propose to use the function r in order to compute a train of temporal pulses and solve the synchronization problem.Provided that the function r and the state x^j of each system are known, we design a pulse that drives every system toward isostables. The pair (μ^∗,τ^∗) is chosen in such a way that the corresponding pulse minimizes the maximum time delay between the different systems, and it follows from (<ref>) that this pair can be computed as(μ^∗,τ^∗) = _(μ,τ) ∈^2ln(max_j r(x_j,μ,τ)/min_j r(x_j,μ,τ)).From a practical point of view, we note that the values of the function r might be known only for some (x,μ,τ) (especially if it is computed from data). In this case, the value r at the different states x_j of the systems can be computed by interpolation, and the optimal pair (μ^∗,τ^∗) will be picked among the pairs (μ,τ) for which the value of r is known. We apply this closed-loop control (see Algorithm <ref>) to synchronize FitzHugh-Nagumo systems (cf. <cit.>), which have been proposed as simple models of excitable cardiac cells. This example is motivated by the synchronization of cardiac cells (i.e. defibrillation) and is directly inspired by the study by <cit.> proposing optimal defibrillation strategies. We consider here 100 FitzHugh-Nagumo cells described by the dynamics (see also <cit.>)V̇= 0.26 V(V-0.13)(1-V)-0.1 V wẇ= 0.013 (V-w)where V is the membrane potential and w is a recovery (gating) variable. The values of r were computed a priori on a 20 × 20 grid for (V,w) ∈[0,2]× [0,2] and on a 51 × 41 grid for (μ,τ) ∈ [0,0.5] × [10,50]. The time between two successive pulses is T_p=70 and the initial conditions are randomly distributed on [0,2]×[0,2]. The maximum time delay between the cells (computed with (<ref>)) after each pulse is shown in Figure <ref>(a) for the input obtained with the closed-loop control (optimal pairs (μ^∗,τ^∗)) and for periodic pulse trains. The best performance is obtained with the closed-loop control. As shown in Figure <ref>(b), the optimal pairs (μ^∗,τ^∗) are not identical at each iteration, since they depend on the states of the cells, which motivates the use of closed-loop control. The first pulses correspond to a maximum value τ^*=50, but μ^* takes intermediate values in the interval [0,0.5]. In particular, small values of μ are needed to obtain a fast convergence rate. We observe in Figure <ref>(a) that periodic pulse trains also synchronize the cells, but slower than our closed-loop approach. A periodic pulse train with maximum values (μ,τ)=(0.5,50) yields a slow rate of convergence (red curve), while smaller values (μ,τ) yield very large delays for the first iterations (green curve). Clearly, the optimal approach by <cit.> outperforms our method in terms of convergence and time delays, however, our optimal control policy is easier to implement. Furthermore, we can parametrize our control signal with different (non-constant) basis functions.The proposed closed-loop control could be easily adapted to incorporate additional constraints (e.g. maximum energy) and provides a solution to the synchronization problem. This solution is a compromise between a simple but not optimal periodic pulse train and the complex exact solution of the optimal control problem established by <cit.>. Future work could extend these preliminary results to more realistic cases, for instance where not all the states are observable.§ CONCLUSION In this paper, we studied a switching/convergence problem for monotone systems. Our solution reduces a dynamic optimization problem to the computation of the time-independent function r, which is defined using the Koopman operator. The properties of the function r lead to straightforward solutions to a tradeoff between the convergence time and the energy budget. This approach can potentially be extended beyond monotone systems and switching/convergence problems. In this paper, we illustrate the possible benefits of a closed-loop solution for the switching problem. We also apply our framework to the synchronization of cardiac cells represented by non-monotone FitzHugh-Nagumo models. In this paper, we have not addressed partial state observability and/or partial controllability issues. This constitutes one of the future work directions. § CONSTANT CONTROL SIGNALS ARE OPTIMAL FOR A MINIMUM-TIME PROBLEM Consider the following optimal control problem over bounded measurable control signals: V(z, μ, β)= inf_[ τ, u ∈_∞([0,μ]) ]τ, subject to (<ref>),x(0) = z, x(τ) ∈_β = {y∈^n | s_1(y) = β},wheres_1(x) is a C^1 increasing dominant eigenfunction defined on the basin of attraction of x^∗. Under our assumptions, the solution to this problem is surprisingly straightforward. Let the system (<ref>) satisfy Assumptions A1 – A3. Then (i) If s_1(z) < β, then the optimal solution to (<ref>), if it exists, is u(t) = μ for all t∈[0, τ]; (ii) If s_1(z) ≥β, then the optimal solution to (<ref>) is u(t) = 0 for all t≥0. (i) Let u^0(t) = μ for all t>0, and u^δ(t) be any admissible control signal, then u^0(t) ≽ u^δ(t) for all t∈[0, τ]. Then by monotonicity we have ϕ(t, z, u^0(·))≽ϕ(t, z, u^δ(·)), which leads to s_1(ϕ(t, z, u^0(·)))≥ s_1(ϕ(t, z, u^δ(·))) for all t≥ 0 due to Proposition <ref>.Hence β = s_1(ϕ(τ, z, u^0(·)))≥s_1(ϕ(τ, z, u^δ(·))), β > s_1(ϕ(t, z, u^0(·))) ≥s_1(ϕ(t, z, u^δ(·)))fort<τ, which implies that the target set _β^- is reached with u^0(·) at least as fast as with any other admissible control signal u^δ(·). Therefore, the control signal u^0(t) = μ is an optimal solution of the problem. (ii) The proof is similar to the point (i). This result justifies our use of temporal pulses to solve convergence problems for monotone systems. The problem (<ref>) has a direct relation to the function r. In particular, s_1(x)<β implies that r(x, μ, V(x, μ, β)) = β provided that the problem (<ref>) has a solution. On the other hand, there might be some values τ V(x, μ, β) such thatr(x, μ, τ) = β. In general, if s_1(x)<β, then V(x, μ, β) = min{τ∈ | r(x,μ,τ) = β},provided that the solution exists.In particular, if the premise of the points (ii) and (iii) in Lemma <ref> holds then V(x, μ, β) = τ if and only if r(x,μ,τ) = β, which again justifies our use of the pulse control function in these cases. § LEVEL SETS OF THE FUNCTION R If the premise of the points (ii) and (iii) in Lemma <ref> hold, then the level sets of r are the graphs of strictly decreasing functions, a result which simplifies their computations as discussed in <cit.>. The algorithms describing this procedure were developed by <cit.> and <cit.>. Let the system (<ref>) satisfy Assumptions A1, A4, A6, A7, and α > 0. (i) If s_1(x) < -α, then {(μ,τ) ∈^2| r(x, μ,τ) = -α} is a graph of a strictly decreasing function, i.e., this set does not contain pairs (μ_1, τ_1) (μ_2, τ_2) such that μ_1≤μ_2 and τ_1 ≤τ_2; (ii) If s_1(x) < α and f(x,η)≽ 0, then {(μ,τ) ∈^2 | r(x, μ,τ) = α} is a graph of a strictly decreasing function for μ>η. (i) First, we make sure that the function r(x,·,·) is increasing.This is guaranteed if ∂_μ r(x,μ,τ), ∂_τ r(x,μ,τ) are positive. Due to Lemma <ref> we have that ∂_τ r(x, μ, τ) >0 as long as r(x, μ, τ)≤ 0, and that ∂_μ r(x,μ,τ)>0. Now since ∂_τ r(x, μ, τ) >0 (for all (μ,τ) such that r(x,μ,τ)=-α<0), the implicit function theorem implies that there exists a function τ = g(μ, α) such that r(x, μ, g(μ, α)) = -α. Furthermore, the function g is C^1 in μ and ∂_μ g = - ∂_μ r(x, μ, τ) / ∂_τ r(x, μ, τ) < 0 in the neighborhood of the level set { (μ, τ)| r(x, μ, τ) =-α}. Therefore the level set {τ, μ | r(x, μ, τ) =-α} is a graph of a strictly decreasing function in μ. It also directly follows thatthe level set { (μ , τ) | r(x, μ, τ) =-α} is a graph of a strictly decreasing function in τ. (ii) We have that ∂_μ r(x,μ,τ) > 0. Now positivity of ∂_τ r(x,μ,τ) for positive r(x,μ,τ) follows from the point (iii) in Lemma <ref>. The rest of the proof follows on similar lines as in (i). § PURELY DATA-BASED CONTROL SETTINGOne can compute r from sampled data points (e.g. obtained from experiments). If data points z[k]=ϕ(τ+k T_s, x, μ h(·, τ))= ϕ(k T_s,ϕ(τ, x, μ),0) are given (with the sampling time T_s), then the DMD algorithm can be used to compute s_1(ϕ(τ, x, μ ))=r(x,μ,τ). We illustrate this in the following subsections.§.§ Dynamic mode decomposition algorithm The dynamic mode decomposition algorithm can be used to estimate the eigenfunctions of the Koopman operator from data. Assume that N snapshots of m trajectories of the (unforced) system are given, i.e.z^j[k] = g(ϕ(k T_s,x_0^j,0)) , j=1,…,m , k=1,…,Nwhere T_S is the sampling time, x_0^j ∈ℝ^n is the initial condition, and g is an observable (typically measuring one of the states). The algorithm yields the so-called DMD modes and eigenvalues, which are related to the eigenfunctions and eigenvalues of the Koopman operator. It is described in Algorithm <ref>. We will not elaborate further on the details of the method and refer the reader to <cit.>. We only note that the matrix Y X^†, where X^† is the pseudo-inverse of X, is a finite-dimensional approximation of the Koopman semigroup. Therefore the eigenvectors of Y X^† can be used to estimate the eigenfunctions of the Koopman operator.§.§ Application of the DMD Algorithm to Toxin-antitoxin SystemConsider the toxin-antitoxin system studied by <cit.>. We will use the model and parameter values as described in <cit.>. The system is bistable with two exponentially stable equilibria x^∙ and x^∗, but not monotone with respect to any orthant. However, it was established by <cit.> that it is eventually monotone, i.e. the flow satisfies the monotonicity property after some initial transient. The level sets of the function r(x^∙, μ, τ) (in the space of the parameters (μ, τ)) were computed by <cit.> and it appears that these sets are monotone curves although the system is not monotone. Since we cannot guarantee that the function r is increasing, we need to develop a different computational approach applicable to a broader class of systems. One possibility is to compute the function r from data. To do so, we used the DMD algorithm with 25 time series of the form (<ref>) (each of which corresponds to a different pair (μ,τ) ∈ [24,50] × [3,25]). For each time series, we used only 4 snapshots over the time interval [τ+12.5, τ+50] (i.e. k ∈{1,2,3,4} with T_s=12.5 in (<ref>)). The level sets of r computed with the DMD algorithm are accurate and similar to the ones obtained with Laplace averages (Figure <ref>). We stress that this example is only a proof-of-concept, and further research is required to automate the application of DMD to this problem. | http://arxiv.org/abs/1707.08462v2 | {
"authors": [
"Aivar Sootla",
"Alexandre Mauroy",
"Damien Ernst"
],
"categories": [
"math.OC",
"cs.SY"
],
"primary_category": "math.OC",
"published": "20170726142820",
"title": "An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator"
} |
We study the rectification of spin current in XXZ chains segmented in two parts, each with a different anisotropy parameter. Using exact diagonalization and a matrix product states algorithm we find that a large rectification (of the order of 10^4) is attainable even using a short chain of N=8 spins, when one half of the chain is gapless while the other has large enough anisotropy. We present evidence of diffusive transport when the current is driven in one direction and of a transition to an insulating behavior of the system when driven in the opposite direction, leading to a perfect diode in the thermodynamic limit. The above results are explained in terms of matching of spectrum of magnon excitations between the two halves of the chain.67.57.Lm, 03.65.Yz, 75.10.Jm, 67.80.−s Perfect diode in quantum spin chains Dario Poletti December 30, 2023 ====================================Introduction: A key challenging problem for modern physics is understanding and controlling transport properties of many-body quantum systems. Important results have been obtained for boundary driven systems, i.e. for systems coupled to baths at their extremities. Numerical and analytical results have shown that, by tuning interactions, disorder, noise, external fields and coupling to the baths one can modify the transport properties and access different regimes, including ballistic, diffusive, sub-diffusive, and insulating behavior <cit.>.Gaining a deeper understanding and control of transport properties at the nanoscale can lead to important technological advances. For instance, many-body nonlinear dynamics might be exploited to design nonlinear devices like heat diodes and transistors <cit.>. Quantum aspects in this quest are also attracting growing interest <cit.>. However, so far the relevance of phase transitions <cit.> for rectification has not been explored. In this work we study a spin chain segmented in two parts, each with different anisotropy. We couple the chain to two different magnetization baths at its edges and study the spin current in forward or reverse bias. Even for relatively small system sizes (N=8-10 spins), we find remarkable rectification (of the order of 10^4). We show that the rectification is due to a mismatch in the spectrum of magnon excitations of one partition of the chain compared to the other, a mismatch which only occurs in reverse bias. The mismatch is most prominent when one half of the chain is noninteracting (i.e. XX model), while the other has strong interactions (i.e. XXZ model in the gapped phase with large enough anisotropy). Our numerical and analytical results strongly indicate, in reverse bias, a transition to an insulating behavior beyond a critical value of the anisotropy, leading to perfect rectification in the thermodynamic limit.Model: We consider a bipartite spin-1/2 chain described by the Heisenberg XXZ Hamiltonian=∑_n=1^N-1[J_n(_n^x_n+1^x+_n^y_n+1^y)+Δ_n_n^z_n+1^z],where N (which we take to be even) is the total number of sites in the chain, _n^α, with α=x,y,z are the Pauli matrices for the n-th spin, J_n and Δ_n are the strengths of the XX tunneling and the ZZ coupling, the anisotropy parameter being Δ_n/J_n. The parameters J_n and Δ_n are chosen such that Δ_n=Δ_L and J_n=J_L when n < N/2 and Δ_n=Δ_R and J_n=J_R for n > N/2. At the junction between the two halves of the chain Δ_N/2=Δ_M and J_N/2=J_M.The chain is coupled to two differentbaths at its extremities, each tending to impose a particular magnetization. The evolution of the density operatorobeys the Lindblad master equation <cit.>d/d t=()=-/ħ[,]+∑_n=1,N_n(),where ħ is the (reduced) Planck constant,is the Lindbladian superoperator and the dissipator _n on sites n=1,N is given by_n()= γ[ _n (^+_n ^-_n - 1/2 {^-_n^+_n,}) . +.(1-_n) (^-_n ^+_n - 1/2 {^+_n^-_n,}) ].Here, ^+_n=(^-_n)^†=(^x_n+^y_n)/2, the parameter γ is the intensity of the coupling to the baths while the spin magnetization imposed by the baths is set by _n. We use _n∈ [0,0.5] and in most cases we focus on _1 and _N to be equal to 0 or 0.5, i.e. on one side of the chain the bath tends to set the spins to be pointing down (|⟩_n⟨| for _n=0) or to be in an equal mixture of up and down spins ([|⟩_n⟨| + |⟩_n⟨|]/2 for _n=0.5). The first case corresponds to the coupling to a magnet, acting as a reservoir for magnetization <cit.>, the second to a high temperature bath.The imbalance λ_1-λ_N imposed by the baths generates a spin current. For the `forward bias' _1>_N the spin current flows from left to right, while for the `reverse bias' _1<_N the current flows from right to left.The spin currentis defined via the continuity equation for the local observable _n^z which gives = tr(_n _ss), where_n=2J_n(^x_n^y_n+1-^y_n^x_n+1)/ħ and _ss is the steady state, (_ss)=0. We refer to the forward(reverse) spin current as _f(_r). To measure the spin current rectification, we calculate the rectification coefficient=-_f/_r.In absence of rectification =1 whiletends to infinity for a perfect diode.To observe a sharp signature of the rectification we also study the current contrast <cit.> =|_f+_r/_f-_r|.Note that the contrast =0 when there is no rectification and takes the value =1 for a perfect diode.Results: We study spin rectification in the above system by exact diagonalization for small chains and by a matrix product states algorithm for the evolution of the density matrices for larger systems <cit.>. In most of our simulations we set J_L=J_R, Δ_R=Δ_M=0 (no ZZ coupling on the right part of the chain and at the left-right interface) andvary the anisotropy on the left portion of the chain by changing the parameter Δ_L. We will then later discuss the case J_L J_R and Δ_R 0.In Fig. <ref>(a,b) we show the forward and reverse-bias currents, _f and _r. Both currents decrease as the interactions (anisotropy |Δ_L/J_L|) increase, however we can notice a marked difference in the magnitude of the forward and reverse currents at a large enough anisotropy. Hence we study both the rectification[Fig. <ref>(c)] and the contrast[Fig. <ref>(d)]. We notice that the rectification sharply increases for |Δ_L/J_L| ≈ 1+√(2) (this precise number will be justified later) and, already for chains of size N=8-10, it can reach values of 10^4 for an anisotropy |Δ_L/J_L|≈ 4. The increase in the rectification becomes more pronounced for longer chains, suggesting the possible occurrence of a transition in the thermodynamic limit. To visualize this effect more clearly we study the contrast , shown in Fig. <ref>(d). We observe that, as the interactions increases, around |Δ_L/J_L|≈ 1+√(2) the contrast changes from ≈ 0 to ≈ 1.It should be noted that in Fig. <ref>(a-d) the chains are not very long and that there is a large interface tunnelling J_M=J_L which broadens the transition. In Fig. <ref>(e,f) we show respectively the rectificationand the contrastas a function of the system size N for different values of the anisotropy Δ_L/J_L (and for J_M ≪ J_L). There is indeed a sharp contrast between Δ_L/J_L=1.5 and Δ_L/J_L=3; for the latter the rectification is orders of magnitude larger. To understand the change in the transport properties of the system with the increase in anisotropy, we first study the magnetization profile, ⟨^z_n⟩as a function of the position n. In Fig. <ref>(a) we focus on the forward bias case. In the right half of the chain the magnetization is fairly constant, however, for large Δ_L/J_L, a diffusive nature of transport is evidenced by the linear slope of the magnetization in the left half of the chain. For the reverse bias case the magnetization profile is very different, see Fig. <ref>(b). In this case, for large anisotropy the change in magnetization becomes more marked, switching rapidly at the interface of the two half-chains between -1 and 0. Such large interface resistance implies a low spin current and suggests insulating behavior. To confirm the insulating character of the system at large anisotropy and in reverse bias, we study the spin current as a function of the system size N. Fig. <ref>(c) indeed indicates an exponential drop of the spin current with N at Δ_L/J_L=3 and 5, while at lower Δ_L/J_L the current hardly varies (Δ_L/J_L=0.5) or decreases much more slowly (Δ_L/J_L=1.5). We also investigate the magnetization at site n=1. This site is directly coupled to a bath that tries to set the magnetization to -1. If the magnetization at site 1 is exactly -1, then there would be no effect of the bath on the chain, hence no transport (note in fact that in the forward bias scenario, the magnetization of the first and last spin does not match the one imposed by the baths). Hence we study in Fig. <ref>(d) the quantity δσ^z=1+^z_1, which we refer to as the `magnetization difference', versus the length of the spin chain, N, for different values of the anisotropy parameter. Here we notice that for Δ_L/J_L=3 and 5 the magnetization difference decreases exponentially as the system size increases, again a signature of an insulating regime, while, for smaller Δ_L/J_L, the decrease is markedly slower.The rectification mechanism: To explain the mechanism behind rectification and highlight the role of interactions, we study a small chain with a weak coupling between the two halves as well as with the baths, i.e. J_L=J_R, J_M=0.1 J_L and γ=0.1J_L/ħ. In Fig. <ref>(a) we find that the contrastversus the anisotropy has a resonant behavior, and, for certain values of Δ_L/J_L, the contrast drops significantly, indicating almost no rectification; for Δ_L/J_L ≳ 1+√(2), the contrast is close to 1, corresponding to strong rectification. Note that the use of small γ and J_M allows to clearly identify the resonances which was not possible for the parameters chosen in Fig. <ref>(a-d). The observed dips and the general rectification mechanism can be explained by the following considerations. Let us take two chains: one fully polarized, i.e. _ss,_j=0=⊗_n |⟩_n⟨|, while the other is at the infinite temperature state, i.e. _ss,_j=0.5=⊗_n (|⟩_n⟨|+|⟩_n⟨|)/2. As soon as these chains are coupled together, any magnon excitation generated at the interface propagates, provided there is no energy gap in the set-up made by the two half-chains. If instead there is a gap, the excitation is localized, and cannot propagate through the system. As a result the system becomes insulating.We consider first the case for which the half chain with non-zero anisotropy is polarized, i.e. _ss,_j=0 (reverse bias), and we compute the energy required to generate a localized excitation at the edge.The magnon excitation spectrum for this half chain is given by the eigenvalues of the following Toeplitz bordered matrix <cit.>.M(Δ)=[[ -2Δ_L2J_L 0 … … …;2J_L -4Δ_L2J_L 0 … …; 02J_L -4Δ_L2J_L 0 …; … … … … … …; … … 02J_L -4Δ_L2J_L; … … 0 02J_L -2Δ_L ]].We plot these eigenvalues E_ν,L in Fig. <ref>(b) as a function of Δ_L/J_L for a half chain of length N/2=5 with red continuous lines. For large enough anisotropy the eigenvalues are confined between -2Δ_L and -4Δ_L. This chain is coupled to another one with no anisotropy, Δ_R=0, prepared in the state _ss,_j=0.5. The energy spectrum E_ν,R for a single excitation in this chain is also given by the eigenvalues of (<ref>) after exchanging J_L with J_R and setting the anisotropy to 0. The energies E_ν,R are, for a large chain, within ± 4 J_R and for a half-chain of N/2=5 they are shown in Fig. <ref>(b) with blue dashed lines. The energetic cost of generating a magnon excitation is thus given by E_ν,L+E_ν',R.If E_ν,L+E_ν',R 0 for all ν,ν', then the generation of an excitation in the left and right halves of the chain requires to overcome an energy gap and transport will be hindered. The magnon excitation will be exponentially localized at the interface and will not be able to reach the baths. In Fig. <ref>(b) we use vertical black dashed lines to clearly show that when E_ν,L=-E_ν',R (note that E_ν',R is symmetric around 0) the contrast is significantly suppressed because of the absence of a gap.In the thermodynamic limit for the reverse bias, the presence of a gap for the magnon excitation results in a marked change of transport properties. In fact, for longer chains, the spectrum for the half-chain without anisotropy approaches a continuum between ± 4J_R, hence there will be no significant rectification as long as the largest of the E_ν,L is smaller than -4 J_R. This is why, for example, in Fig. <ref>(c,d) the rectification and contrast are very small for low interactions. For large interactions, Δ_L/J_L>1, and in the thermodynamic limit, the largest eigenvalue for the magnon excitation is E_ max,L=-2Δ_L+2J_L^2/Δ_L <cit.>. Hence, the critical anisotropy Δ_L,c, at which the rectification changes significantly, becomesΔ_L,c=J_R+√(J_R^2+J_L^2).It is thus clear that for J_L=J_R, the critical anisotropy in the thermodynamic limit is Δ_L,c/J_L=1+√(2). This is consistent with the numerical results shown in Fig. <ref>(c,d) which show that the current in reverse bias decreases exponentially with the system size. For completeness, in <cit.> we also show a characteristic time scale needed for a magnon excitation to propagate in the chain. Our numerics indicate that, for interaction strengths beyond the critical value Δ_L,c, the time scale goes to infinity as the system size increases.For the forward bias case instead, there will always be low energy excitations in the half-chain with non-zero anisotropy which can be matched by the quadratic chain, whose excitation spectrum does not change, resulting in good transport of excitations (for more details see <cit.>). Hence there is a very different non-equilibrium response in forward and reverse bias. Robustness of the rectification: It is important to study the robustness of the rectification to variations of the baths or Hamiltonian's parameters. First we consider the case for which the bath parameters λ_n are not set exactly to either 0 or 0.5. The continuous blue line in Fig. <ref>(a) shows the contrastwhen the parameter λ_n which should be 0 (i.e. λ_N in forward bias and λ_1 in reverse bias) is detuned by an amount δλ, while the other is kept at 0.5 (red continuous line). The contrast is close to 1 as long as δλ<10^-2, while for δλ=0.1 the contrast is ≈ 0.5 corresponding to a rectification coefficient ≈ 3. We now consider instead the case when the maximum value of λ_n is not 0.5 but 0.5-δλ (while the other is kept at 0). In this case the rectification is large and robust, as signalled by a contrastclose to 1 even at δλ=0.1 (blue dashed line). The rectification is thus very robust to changes of one of the bath parameters (the one that sets the local magnetization to 0), while more care is needed in ensuring that one of the baths is almost completely polarized.In Fig. <ref>(b) we show how the contrastchanges as we keep J_L=J_R and vary both Δ_L and Δ_R. The contrast is pronounced when the modulus of the anisotropies is different, with largest contrast when one of the two is 0. This is expected because a larger excitation gap is present when one half of the chain is quadratic. The symmetric behavior is due to the spectrum invariance for a change of all Δ_n→ -Δ_n. Conclusions: We have shown the possibility of large spin-current rectification even in short segmented spin chains. In the thermodynamic limit, our results are consistent with the prediction of an insulating behavior for reverse bias, leading to a perfect diode. The key ingredient for such effect is the presence of a magnon excitation gap between the two chains, which is possible because of large enough anisotropy in one half of the system. It is important to stress that in our set-up the rectification is maximum when half of the chain is quadratic (XX chain).Materials whose spin dynamics is well described by Heisenberg chains with Δ_L/J_L=1.2 or larger have been already studied <cit.>. Such anisotropy can suffice to obtain large rectification if such material is matched to one with lower tunnelling parameter J_R<J_L and negligible anisotropy (Δ_R=0) as shown from Eq. (<ref>). We also point out that recent proposals <cit.> and experiments with Rydberg atoms trapped with optical tweezers offer tremendous opportunities to engineer spin Hamiltonians with arbitrary values for the ratio Δ_L/J_L <cit.>. Moreover, cutting-edge technologies using adatoms on surface <cit.> or trapped ions <cit.> provide alternative frameworks for obtaining our proposed spin chain model.The rectification, here demonstrated for spin currents, is also expected for heat currents imposed by heat baths, due to the mismatch in spectral response between the two portions of the chain. More generally many-body interactions and phase transitions in the non-equilibrium scenarios are opening the door to the design of highly performing quantum devices, a research direction which should be further investigated. Acknowledgments: We acknowledge fruitful discussions with C. Guo and T. Prosen. D.P. and V. B. acknowledge support from SUTD-MIT IDC (Project No. IDG31600107), and Singapore Ministry of Education, Singapore Academic Research Fund Tier-I (project SUTDT12015005). E.P. was partially supported by CNPq (Brazil). G.B. acknowledges the financial support of the INFN through the project “QUANTUM”. 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However we can show that there are low frequencies modes that can be matched with the right half of the chain.We consider the two-time correlator S(t)=⟨σ^+_N/2(t)σ^-_N/2(0)⟩_ss=(σ^+_N/2 e^ tσ^-_N/2 ρ_ss), and its Fourier transform S(ω), where ρ̂_ss is the steady state. We consider a small (negligible) dissipation. The frequency response of S(ω) is given by the differences of the energies E_ν,L/R occupied by the steady state and excited by σ^-_N/2. Note that the spectrum of right half of the chain is unaffected by the bias (e.g. whether λ_j=0.5 to 0).As for the steady state of left half of chain in forward bias, it is given byρ̂_ss,λ_j=0.5=⊗_n=1^N/2(|⟩_n⟨|+|⟩_n⟨|)/2. The action of the operator σ^-_N/2 on the steady state results in =σ^-_N/2ρ̂_ss,λ_j=0.5=[⊗_n=1^N/2-1(|⟩_n⟨|+|⟩_n⟨|)/2]|⟩_N/2⟨|. Considering (i) the symmetry of the Hamiltonian to a global spin-flip, (ii) the fact that the Hamiltonian preserves the magnetization and (iii) focusing on the states near zero total magnetization (which are indeed present in ), it is easy to see that for half-chains of odd length there is a non-zero response of S(ω) at ω=0, while for even chains there be a finite response for a frequency which vanishes as the system size increases. For example let us consider =|⟩⟨| for a half-chain of length N/2=3, which is included in : For small dissipation, the Hamiltonian evolution, given by d/dt=()=-(/ħ)[,], connectsto other states whose total number of spins up (or down) in the bra and in the ket remains unchanged (for example '=|⟩⟨|). The evolution due to the anticommutation with the Hamiltonian is given by the difference of two phases. But since the Hamiltonian is symmetric to a global spin-flip, and since the total number of spins up minus spins down is opposite between the bra and the ket (in this example it is -1 for the ket and +1 for the bra), then there is a finite response for S(ω) at ω=0. §.§ Propagation time in the reverse bias scenario Here we give another indication of the presence of a transition to an insulating regime in the reverse bias scenario. Let us consider two chains each of length N/2: an XXZ chain (with anisotropy Δ_L and tunnelling J_L) prepared with spins all pointing down, and an XX chain (J_R=J_L) prepared in the infinite temperature state. We then connect these chains with an XX-type of coupling with small magnitude J_M and study the ensuing dynamics. Note that here no baths are used, but we just study Hamiltonian dynamics. We focus in particular on the spin of the XXZ chain closest to the interface between the two chains. At time t=0 this spin is pointing down, i.e. ⟨^z_N/2⟩=-1 and if the chain is not insulating, its polarization will change. We then compute the propagation time t^⋆ at which ⟨^z_N/2(t^*)⟩=-1+ϵ where ϵ is a small positive number which we take here to be 10^-2. In Fig. <ref> we show the inverse of the propagation time 1/t^⋆ as a function of the interaction Δ_L for various chain lengths, each indicated by a different symbol. Note that for the empty symbols the magnetization at site N/2 has not yet reached the value 1-ϵ despite the long time evolution (of the order of 1000 J/ħ); Hence we just plot the longest simulated time, which is a numerical lower bound to t^⋆. We see clearly that as the size of the system increases, the inverse of the propagation time goes quickly to 0 as soon as the anisotropy exceeds a critical value close to J_L(1+√(2)) as predicted in the main article. | http://arxiv.org/abs/1707.08823v3 | {
"authors": [
"Vinitha Balachandran",
"Giuliano Benenti",
"Emmanuel Pereira",
"Giulio Casati",
"Dario Poletti"
],
"categories": [
"cond-mat.stat-mech",
"cond-mat.quant-gas",
"quant-ph"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170727114639",
"title": "Perfect diode in quantum spin chains"
} |
empty arabic Department of Mathematics, Indiana University,Bloomington, IN 47408 [email protected] http://pages.iu.edu/ jlmeierDepartment of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588 [email protected] http://www.math.unl.edu/ azupan2 We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds.In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of the standard four-sphere that are unlikely to be standard.Finally, we give an analog of the Casson-Gordon Rectangle Condition for trisections that can be used to obstruct reducibility of a given trisection. Characterizing Dehn surgeries on links via trisections Alexander Zupan December 30, 2023 ======================================================§ OUTLINE The purpose of this note is to use both new and existing results to make clear the significant role of the trisection theory of smooth 4-manifolds in the classification of Dehn surgeries on links.The theory of Dehn surgery on knots has been thoroughly developed over the past forty years.In general, this research has focused on two major questions:First, which manifolds can be obtained by a surgery on a knot in a given manifold Y?Second, given a pair of manifolds Y and Y', for which knots K ⊂ Y does there exist a surgery to Y'?These two questions have contributed to the growth of powerful tools in low-dimensional topology, such as sutured manifold theory, the notion of thin position, and Heegaard Floer homology.For example, over the last 15 years, the Heegaard Floer homology theories of Ozsváth and Szabó have dramatically deepened our collective understanding of Dehn surgeries on knots (see, for instance, <cit.>).If we replace the word “knot" with “link" in the preceding paragraph, the situation changes significantly; for example, the classical Lickorish-Wallace Theorem asserts that every 3-manifold Y can be obtained by surgery on a link in S^3 <cit.>. For the second general question, concerning which links in a given 3-manifold Y yield a surgery to another given 3-manifold Y', we observe the following basic fact:Two framed links that are handleslide equivalent surger to the same 3-manifold <cit.>.Thus, surgery classification of links is necessarily considered up to handleslide equivalence, and tools which rely on the topology of a knot exterior S^3 ∖ν(K) are not nearly as useful, since handleslides can significantly alter this topology.Understanding link surgeries in particular 3-manifolds is intimately connected to smooth 4-manifold topology.Every smooth 4-manifold X can be described by a handle decomposition, characterized by the attaching link L for the 2-handles, which is contained in the boundary of the union of the 0- and 1-handles.In other words, X is associated to a framed link L ⊂#^k(S^1S^2) such that Dehn surgery on L yields #^k' (S^1S^2).Conversely, such a link L (which we will call admissible) describes a handle decomposition of a smooth 4-manifold, which we denote X_L.Thus, classifying all such surgeries would be equivalent to classifying all smooth 4-manifolds.Clearly, this is an insurmountable task, but to make the problem more tractable, we consider various restrictions placed on the parameters k, k', and n, where n represents the number of components of the link L.For example, let k = k' = 0.In the case that n=1, Gordon and Luecke proved that knots are determined by their complements, and thus the only knot in S^3 that admits an integral S^3 surgery (a cosmetic surgery) is a ± 1-framed unknot <cit.>. In this paper, we will describe the proof of the following theorem from <cit.>. If L ⊂ S^3 is a two-component link with tunnel number one with an integral surgery to S^3, then L is handleslide equivalent to a 0-framed Hopf link or ± 1-framed unlink. Another significant case occurs when k = 0 and k' = n.In other words, we wish to understand n-component links in S^3 with surgeries to #^n (S^1S^2).We call such a link L an R-link, noting that R-links correspond precisely to the collection of geometrically simply-connected homotopy 4-spheres, i.e. homotopy 4-spheres built without 1-handles.The Generalized Property R Conjecture (GPRC), Kirby Problem 1.82 <cit.>, contends that every R-link is handleslide equivalent to a 0-framed unlink.The conjecture is known to be true in the case n=1 via Gabai's proof of Property R <cit.>.In <cit.>, the authors, in collaboration with Trent Schirmer, proved a stable version of the GPRC for a class of links. If L ⊂ S^3 is an n-component R-link with tunnel number n, then the disjoint union of L with a 0-framed unknot is handleslide equivalent to a 0-framed unlink. As foreshadowed above, the proofs of these theorems are 4-dimensional in nature, utilizing a prominent new tool: trisections of smooth 4-manifolds.A trisection is a decomposition of a 4-manifold X into three simple pieces, a 4-dimensional version of a 3-dimensional Heegaard splitting.Elegantly connecting the two theories, Gay and Kirby proved that every smooth 4-manifold admits a trisection, and every pair of trisections for a given 4-manifold have a common stabilization <cit.>, mirroring the Reidemeister-Singer Theorem <cit.> in dimension three.Unlike Heegaard splittings, however, the stabilization operation of Gay and Kirby can be broken into three separate operations, called unbalanced stabilizations of types 1, 2, and 3 <cit.>.A trisection is said to be standard if it is an unbalanced stabilization of the genus zero trisection of S^4, and thus, every trisection of S^4 becomes standard after some number of Gay-Kirby stabilizations. In Section <ref>, we describe a process by which an R-link L paired with an admissible Heegaard surface Σ for its exterior is converted to a trisection (L,Σ) of the 4-manifold X_L.The new main result of this paper is a technical theorem that connects R-links to properties of these trisections.The terms {2}–standard and {2,3}–standard refer to trisections that become standard after allowing restricted types of unbalanced stabilizations; we will postpone the rigorous definitions for now. Suppose L is an R-link and Σ is any admissible surface for L. * If L satisfies the GPRC, then (L,Σ) is {2}–standard.* The link L satisfies the Stable GPRC if and only if (L,Σ) is {2,3}–standard.In Section <ref>, we analyze examples of Gompf-Schlarlemann-Thompson, the most prominent possible counterexamples to the GPRC.The first step in a program to disprove the GPRC via Theorem <ref> is to find low-genus admissible surfaces for these links, along with diagrams for their induced trisections.We outline this process; extensions of Section <ref> will appear in forthcoming work <cit.>.In Section <ref>, we introduce an analog of the Casson-Gordon Rectangle Condition <cit.> for trisection diagrams, giving a sufficient condition for a trisection diagram to correspond to an irreducible trisection. §.§ Acknowledgements The first author is supported by NSF grants DMS-1400543 and DMS-1664540, and the second author is supported by NSF grant DMS-1664578 and NSF-EPSCoR grant OIA-1557417. § TRISECTIONS AND ADMISSIBLE LINKS All manifolds are connected and orientable, unless otherwise stated.We will let ν( · ) refer to an open regular neighborhood in an ambient manifold that should be clear from context.The tunnel number of a link L ⊂ Y is the cardinality of the smallest collection of arcs a with the property that Y ∖ν(L ∪ a) is a handlebody.In this case, ν(L ∪ a) is a Heegaard surface cutting Y ∖ν(L) into a handlebody and a compression body.A framed link refers to a link with an integer framing on each component.Let L be a framed link in a 3-manifold Y, and let a be a framed arc connecting two distinct components of L, call them L_1 and L_2.The framings of L_1, L_2 and a induce an embedded surface S ⊂ Y, homeomorphic to a pair of pants, such that L_1 ∪ L_2 ∪ a is a core of S.Note that S has three boundary components, two of which are isotopic to L_1 and L_2.Let L_3 denote the third boundary component, with framing induced by S.If L' is the framed link (L ∖ L_1) ∪ L_3, we say that L' is obtained from L by a handleslide of L_1 over L_2 along a.If two links are related by a finite sequence of handleslides, we say they are handleslide equivalent.It is well-known that Dehn surgeries on handleslide equivalent framed links yield homeomorphic 3-manifolds <cit.>.Recall that an R-link is an n-component link in S^3 with a Dehn surgery to the manifold #^n(S^1S^2), which we henceforth denote by Y_n.Let U_n denote the n-component zero-framed unlink in S^3.If an R-link L is handleslide equivalent to U_n, we say that L has Property R.If the split union L ⊔ U_r is handleslide equivalent to U_m for some integers r and m, we say that L has Stable Property R.The following conjectures are well-known; the first is Kirby Problem 1.82 <cit.>. Every R-link has Property R.Every R-link has Stable Property R. In this section, we explore the relationship between R-links (and a more general family we call admissible links) and trisections of the smooth 4-manifolds that can be constructed from these links.Let X be a smooth, orientable, closed 4-manifold.A (g;k_1,k_2,k_3)–trisectionof X is a decomposition X = X_1∪ X_2∪ X_3 such that * Each X_i is a four-dimensional 1–handlebody, ♮^k_i(S^1× B^3); * If i≠j, then H_ij = X_i∩ X_j is a three-dimensional handlebody, ♮^g(S^1× D^2); and * The common intersection Σ = X_1∩ X_2∩ X_3 is a closed genus g surface. The surface Σ is called the trisection surface, and the parameter g is called the genus of the trisection.The trisectionis called balanced if k_1=k_2=k_3=k, in which case it is called a (g;k)–trisection; otherwise, it is called unbalanced.We call the union H_12∪ H_23∪ H_31 the spine of the trisection.In addition, we observe that ∂ X_i = Y_k_i = H_ij∪_Σ H_li is a genus g Heegaard splitting.Because there is a unique way to cap off Y_k_i with ♮^k_i(S^1× B^3) <cit.>,every trisection is uniquely determined by its spine.Like Heegaard splittings, trisections can be encoded with diagrams.A cut system for a genus g surface Σ is a collection of g pairwise disjoint simple closed curves that cut Σ into a 2g-punctured sphere.A cut system δ is said to define a handlebody H_δ if each curve in δ bounds a disk in H_δ.A triple (α,β,γ) of cut systems is called a (g;k_1,k_2,k_3)–trisection diagram forif α, β, and γ define the components H_α, H_β, and H_γ of the spine of .We set the convention that H_α = X_3∩ X_1, H_β = X_1∩ X_2, and H_γ = X_2∩ X_3.The careful reader will note that this convention differs slightly from <cit.>.With these conventions, (α,β), (β,γ), and (γ,α) are Heegaard diagrams for Y_k_1, Y_k_2, and Y_k_3, respectively.In <cit.>, Gay and Kirby prove that every smooth 4-manifold admits a trisection, and trisection diagrams, modulo handle slides within the three collections of curves, are in one-to-one correspondence with trisections. Trisections with genus at most two are well-understood. See Figure <ref>. * There is a unique genus zero trisection; the (0,0)–trisection describing S^4. * There are exactly six genus one trisections.Both ^2 and ^2 admit (1;0)–trisections; S^1× S^3 admits a (1;1)–trisection; and S^4 admits three unbalanced genus one trisections. * There is a unique irreducible (defined below) genus two trisection <cit.>, which describes S^2× S^2. Given trisectionsand ' for 4-manifolds X and X', we can obtain a trisection for X#X' by removing a neighborhood of a point in each trisection surface and gluing pairs of components ofand ' along the boundary of this neighborhood.The resulting trisection is uniquely determined in this manner; we denote it by #'.A trisectionis called reducible if = '#”, where neither ' nor ” is the genus zero trisection; otherwise, it is called irreducible.Equivalently,is reducible precisely when there exists a curve δ in Σ that bounds compressing disks in H_, H_, and H_.Such a curve δ represents the intersection of a decomposing 3-sphere with the trisection surface.In dimension three, stabilization of a Heegaard surface may be viewed as taking the connected sum with the genus one splitting of S^3, and a similar structure exists for trisections.Let _i denote the unique genus one trisection of S^4 satisfying k_i=1.Diagrams for these three trisections are shown in Figure <ref>.A trisectionis called i–stabilized if = '#_i, and is simply called stabilized if it is i–stabilized for some i=1,2,3.Two trisections ' and ” are called stably equivalent if there is a trisectionthat is a stabilization of both ' and ”.Gay and Kirby proved that any two trisections of a fixed 4-manifold are stably equivalent <cit.>.We say that a trisectionis standard ifcan be expressed as the connected sum of the trisections listed in Examples <ref>.Theorems in <cit.> classify trisections of genus two. Every trisectionwith genus g =2 is standard. Below, we see how this theorem implies Theorem <ref>, and for this purpose, we turn our attention to surgery on links.§.§ Admissible links and surfaces Recall that Y_k denotes #^k(S^1 × S^2), and let L be a framed n–component link in Y_k such that Dehn surgery on L yields Y_k'.We call such a link admissible.If L is an admissible link, L describes a closed 4-manifold X_L with a handle decomposition with k 1–handles, n 2–handles, and k' 3–handles.An admissible Heegaard surface Σ for L is a Heegaard surface cutting Y_k into two handlebodies H and H', such that a core of H contains L.As such, C = H ∖ν(L) is a compression body and Σ may be viewed as a Heegaard surface for the link exterior E(L) = Y_k ∖ν(L).Let H_L be the handlebody that results from Dehn filling C (or performing Dehn surgery on L in H) along the framing of the link L.An admissible pair consists of an admissible link together with an admissible Heegaard surface.For completeness, we will also allow the empty link, L =.An admissible surface Σ for the empty link is a (standard) genus g Heegaard surface for Y_k.A genus g Heegaard diagram (,) for Y_k is called standard if ∩ contains k curves, and the remaining g-k curves occur in pairs that intersect once and are disjoint from other pairs.A trisection diagram is called standard if each pair is a standard Heegaard diagram. Let L be an admissible n-component link in Y_k.Every admissible pair (L,Σ) gives rise to a trisection (L,Σ) with spine H' ∪ H ∪ H_L.If g(Σ) = g, then (L,Σ) is a (g;k,g-n,k')-trisection.Moreover, there is a trisection diagram (,,) for (L,Σ) such that * H_ = H', H_ = H, and H_ = H_L;* L is a sublink of , whereis viewed as a link framed by Σ in Y_k = H_∪ H_; and* (,) is a standard diagram for Y_g-n, where ∩ = ∖ L. This is proved (in slightly different formats) for L ≠ in both <cit.> and <cit.>.If L =, then it follows easily that X_L has a handle decomposition without 2-handles, H = H_L, and H' ∪ H ∪ H_L is the spine for the (g;k,g,k)-trisection (L,Σ) of X_L.In this case, there is a diagram such that =, the standard genus g diagram of Y_g. This machinery is enough to prove Theorem <ref>, classifying cosmetic surgeries on tunnel number one links in S^3.Note that the conventions H_ = H', H_ = H, and H_ = H_L agree with our earlier conventions identifying the union of the 0–handle and the 1–handles with X_1, the trace of the Dehn surgery on H_ along L with X_2, and the union of the 3–handles and the 4–handle with X_3. Suppose L ⊂ S^3 is a tunnel number one link with an integral Dehn surgery to S^3.Then there exists an admissible surface Σ⊂ S^3 and a genus two trisection (L,Σ) with a diagram (,,), where H_L = H_.By Lemma <ref>, (L,Σ) is a (2,0)-trisection, the two curves inare isotopic to the link L in S^3 = H_∪ H_, and the surface framing ofin Σ is the framing of L.By Theorem <ref>, the trisection (L,Σ) of X_L is standard, and (,,) is handleslide equivalent to a standard diagram (',',').Since (L,Σ) is (2,0)-trisection, X_L is diffeomorphic to either S^2S^2 or ±#±.In the first case,is handleslide equivalent to ', which is a zero-framed Hopf link in S^3 = H_∪ H_.In the second case,is handleslide equivalent to ', a 2-component unlink with framings ± 1 and ± 1, completing the proof. We now turn our attention to R-links.Note that if L is an R-link, then the smooth 4-manifold X_L has a handle decomposition with no 1-handles, n 2-handles, and n 3-handles; thus X_L is a simply connected 4-manifold with χ(X_L) = 2, so that X_L is a homotopy S^4.We describe an immediate connection between Stable Property R and trisections in the next lemma. Suppose L is an R-link with admissible surface Σ and (L,Σ) is a standard trisection of S^4.Then L has Stable Property R.By Lemma <ref>, the trisection (L,Σ) has a diagram (,,) such that (,) is the standard Heegaard diagram for Y_g-n.Viewingas a g-component link in S^3 = H_∪ H_, we have that (g-n) curves inbounds disks in H_, while the remaining n curves are isotopic to L (and are disjoint from the (g-n) disks).Thus, as a link in S^3, we have = L ⊔ U_g-n.In addition, the trisection (L,Σ) is a standard (g;0,g-n,n)-trisection of S^4 by hypothesis.As such, it must be a connected sum of g-n copies of _2 and n copies of _3, and it has a standard diagram, (',','), where g-n curves in ' are also curves in ', and the remaining n curves are also curves in '.Thus, in S^3 = H_'∪ H_', the curves ' comprise a g-component unlink, with surface framing equal to the zero framing on each component.Since (,,) and (',',') are trisection diagrams for the same trisection, we have thatis handleslide equivalent to ' via slides contained in Σ.Thus,and ' are handleslide equivalent links in S^3.We conclude that L has Stable Property R, as desired.Theorem <ref> can be quickly proved using this lemma and the following result from <cit.> as its main input. Every (g;0,1,g-1)-trisection is a standard trisection of S^4. Suppose L ⊂ S^3 is an n-component link with tunnel n with a Dehn surgery to #^n(S^1S^2).Then by Lemma <ref> there exists an admissible surface Σ⊂ S^3 and an (n+1;0,1,n)-trisection (L,Σ).By Theorem <ref>, the trisection (L,Σ) is standard, and by Lemma <ref>, L has Stable Property R.In fact, the proof of Lemma <ref> reveals that L ⊔ U_1 has Property R, as desired. § R-LINKS AND STABILIZATIONS In order to prove the third main theorem, we will further develop the connection between R-links, their induced trisections, and the various stabilization operations. If L = in Y_k and g(Σ) = g, then X_L = #^k(S^1 × S^3), and (,Σ) is the connected sum of k copies of the standard (1;1)-trisection of S^1 × S^3 and g-k copies of _2. By Waldhausen's Theorem <cit.>, Y_k has a standard Heegaard diagram, (,), and by Lemma <ref>, (,,) is a trisection diagram for (,Σ).The k curves in ∩ give rise to k summands of the standard genus one splitting of S^1 × S^3, and the remaining g-k pairs give rise to g-k copies of _2.In order to understand operations on an admissible link L and Heegaard surface Σ which will correspond to various stabilizations of (L,Σ), we introduce several additional definitions.Let (L_1,Σ_1) ⊂ Y_k_1 and (L_2,Σ_2) ⊂ Y_k_2 be any admissible pairs, and define the operation ∗ by(L_1,Σ_1) ∗ (L_2,Σ_2) = (L_1 ⊔ L_2, Σ_1 #Σ_2),where the connected sum is taken so that L_1 ⊔ L_2 is not separated by the surface Σ_1 #Σ_2.Note that (L_1,Σ_1) ∗ (L_2,Σ_2) ⊂ Y_k_1+k_2. If (L_1,Σ_1) and (L_2,Σ_2) are admissible pairs, then (L,Σ) = (L_1,Σ_1) ∗ (L_2,Σ_2) is an admissible pair, and (L,Σ) = (L_1,Σ_1) #(L_2,Σ_2). It is clear that the framed link L_1 ⊔ L_2 ⊂ Y_k_1+k_2 has the appropriate surgery.Suppose Σ_i bounds a handlebody H_i with core C_i containing L_i.Then there is a core C for H_1 ♮ H_2 such that L_1 ⊔ L_2 ⊂ C_1 ⊔ C_2 ⊂ C, and thus Σ_1 #Σ_2 is admissible as well.For the second claim, note that the curve δ arising from the connected sum Σ = Σ_1 #Σ_2 is a reducing curve for (L,Σ), splitting it into the trisections (L_1,Σ_1) and (L_2,Σ_2). Let U be a 0-framed unknot in S^3, and let Σ_U be the genus one splitting of S^3 such that one of the solid tori bounded by Σ_U contains U as a core.In addition, let W denote the knot S^1 ×{pt}⊂ S^1 × S^2 with framing given by the fibering, and let Σ_W be the genus one splitting of S^1 × S^2 such that one of the solid tori bounded by Σ_W contains W as a core.Note that both (U,Σ_U) and (W,Σ_W) are admissible pairs.Finally, let Σ_ be the genus one Heegaard surface for S^3, to be paired with the empty link. The links (W,Σ_W), (,Σ_), and (U,Σ_U) yield the following trisections: * (W,Σ_W) = _1.* (,Σ_) = _2.* (U,Σ_U) = _3.First, note that each trisection in question has genus one.Since framed surgery on W ⊂ Y_1 yields S^3, by Lemma <ref>, (W,Σ_W) is a (1;1,0,0)–trisection and must be _1. Similarly, (,Σ_) is a (1;0,1,0)–trisection and must be _2.Finally, framed surgery on U yields Y_1, so (U,Σ_U) is a (1;0,0,1)–trisection and must be _3. By combining Lemmas <ref>, <ref>, and <ref>, we obtainSuppose (L,Σ) is an admissible link, with = (L,Σ). * ((L,Σ) ∗ (W,Σ_W)) is the 1–stabilization of .* ((L,Σ) ∗ (,Σ_)) is the 2–stabilization of .* ((L,Σ) ∗ (U,Σ_U)) is the 3–stabilization of .In addition, if Σ_+ is the stabilization of Σ (as a Heegaard surface for Y_k), then (L,Σ_+) = (L,Σ) ∗ (,Σ_). We say that two trisections _1 and _2 of a 4-manifold X are 2–equivalent if there is a trisectionthat is the result of 2–stabilizations performed on both _1 and _2. If Σ_1 and Σ_2 are two distinct admissible surfaces for an admissible link L, then the trisections (L,Σ_1) and (L,Σ_2) are 2–equivalent. Since both Σ_1 and Σ_2 are Heegaard surfaces for E(L), they have a common stabilization Σ by the Reidemeister-Singer Theorem <cit.>.By Lemma <ref>, the surface Σ is admissible, and by Corollary <ref>, (L,Σ) can be obtained by 2–stabilizations of (L,Σ_i). Since 2–equivalence is an equivalence relation, Lemma <ref> implies that every admissible surface Σ for an admissible link L belongs to the same 2–equivalence class.Hence, L has a well-defined 2–equivalence class; namely, the 2–equivalence class of (L,Σ).If two admissible links L_1 and L_2 give rise to 2–equivalent trisections, we say that L_1 and L_2 are 2–equivalent.Suppose that L is an n-component admissible link with admissible surface Σ, so that Σ cuts Y_k into H ∪ H', and L is isotopic into a core C ⊂ H.As such, there is a collection of n compressing disks {D} with the property that each disk meets a unique component of L once and misses the other components.We call {D} a set of dualizing disks.Note that if (,,) is the trisection diagram for (L,Σ) guaranteed by Lemma <ref>, then the n disks bounded by the n curves inthat are not inare a set of dualizing disks for L. If admissible links L_1 and L_2 are related by a handleslide, then L_1 and L_2 are 2–equivalent. If L_i is an n–component link, then L_1 and L_2 have n-1 components in common and differ by a single component, L_1' ⊂ L_1 and L_2' ⊂ L_2, where a slide of L_1' over another component L' of L_1 along a framed arc a yields L_2'.Consider Γ = L_1 ∪ a, an embedded graph with n-1 components, and let Σ be a Heegaard surface cutting S^3 into H ∪ H', where Γ is contained in a core of H.Then L_1 is also contained in a core of H, and Σ is admissible (with respect to L_1).Let {D_1} be a set of dualizing disks for L_1.A priori, the arc a might meet some of the disks in {D_1}; however, if this is the case, we can perform a sequence of stabilizations on Σ, after which a avoids all of the disks {D_1}.Thus, we suppose without loss of generality that a ∩{D_1} = ∅. There is an isotopy taking Γ into Σ, preserving the intersections of L_i with the dualizing disks {D_1}, so that the framing of Γ agrees with its surface framing in Σ.As such, we can perform the handleslide of L_1' over L' along a within the surface Σ, so that the resulting link L_2 is also contained in Σ, with framing given by the surface framing.Let D_1' ∈{D_1} be the disk that meets L_1' once, and let D' ∈{D_1} be the disk that meets L' once.There is an arc a', isotopic in Σ to an arc in Γ, that connects D_1' to D'. See Figure <ref>. Let D_2' be the compressing disk obtained by banding D_1' to D' along a'.Then {D_2} = ({D_1}∖ D')∪ D_2' is a set of dualizing disks for L_2. Thus, by pushing L_2 back into H, we see that Σ is an admissible surface for L_2.Following Lemma <ref>, let H_i ∪ H_i' ∪ H_L_i be a spine for (L_i,Σ).By construction, H_1 = H_2 and H_1' = H_2'.Finally, since H_i is Dehn surgery on L_i in H_i , and L_1 and L_2 are related by a single handleslide, we have H_L_1 = H_L_2.It follows that (L_1,Σ) = (L_2,Σ), and we conclude that L_1 and L_2 are 2–equivalent. For the rest of the section, we will restrict our attention to admissible links in S^3.Let U_n denote the zero-framed, n–component unlink, so X_U_n = S^4.Recall that a standard trisection of S^4 is the connected sum of copies of _1, _2, and _3. Let Σ be any admissible surface for U_n, then (U_n,Σ) is standard. We induct on (n,g) with the dictionary ordering.If n=1, then E(U_1) is a solid torus.If g=1, then Σ = Σ_U, so that (U_1,Σ_U) = _3 by Lemma <ref>.If n=1 and g > 1, then Σ is stabilized <cit.> (see also <cit.>), which means that (U_1,Σ) is 2–stabilized by Corollary <ref>, and as such, (U_1,Σ) is standard by induction.In general, note that the Heegaard genus of an n–component unlink is n; thus g ≥ n for all possible pairs (n,g).For n>1, we have that E(U) is reducible, and so Haken's Lemma <cit.> implies that Σ is reducible, splitting into the connected sum of genus g_1 and g_2 surfaces Σ_1 and Σ_2, where Σ_i is a Heegaard surface for E(U_n_i).Then (U_n,Σ) = (U_n_1,Σ_1) #(U_n_2,Σ_2), where (n_i,g_i) < (n,g).Since both summands are standard trisections by induction, it follows that (U_n,Σ) is also standard, completing the proof. A trisectionis said to be 2–standard if it becomes standard after some number of 2-stabilizations.Similarly,is {2,3}–standard if it becomes standard after some number of 2- and 3-stabilizations. Suppose L has Property R. By Lemma <ref>, L and U_n are 2–equivalent links.Thus, (L,Σ) is 2–equivalent to some trisection coming from U_n, but all trisections induced by U_n are standard by Lemma <ref>, and thus (L,Σ) becomes standard after a finite sequence of 2–stabilizations. If L has Stable Property R, then L ⊔ U_n has Property R for some n, and thus ((L,Σ) ∗ (U,Σ_U) ∗…∗ (U,Σ_U)) is 2–standard by the above arguments.By Lemma <ref> and Corollary <ref>,((L,Σ) ∗ (U,Σ_U) ∗…∗ (U,Σ_U)) = (L,Σ) #_3 #…#_3;hence (L,Σ) is {2,3}–standard.Finally, if the trisection (L,Σ) is {2,3}–standard, then there exist integers s and t such that the connected sum of (L,Σ) with s copies of _2 and t copies of _3 is standard.Let (L_*,Σ_*) be the admissible pair given by(L_*,Σ_*) = (L,Σ) ∗(,Σ_) ∗…∗ (,Σ_)_s∗(U,Σ_U) ∗…∗ (U,Σ_U)_t.By assumption, (L_*,Σ_*) is standard, so by Lemma <ref>, the link L_* has Stable Property R.But by definition of ∗, we have L_* = L ⊔ U_t, and thus L also has Stable Property R, completing the proof. § TRISECTING THE GOMPF-SCHARLEMANN-THOMPSON EXAMPLESAlthough this view has changed in the past and may change in the future, it is the current view of the authors that the GPRC is likely false.In light of this opinion, we will outline the first steps one might take to employ Theorem <ref> to disprove the GRPC or the Stable GPRC.Let L be an R-link with admissible surface Σ.By Theorem <ref>, if (L,Σ) is not {2}–standard, then L fails to have Property R.Thus, we in this section we will show how to take the most promising potential counterexamples to the GPRC and construct admissible surfaces and their corresponding trisections.The possible counterexamples mentioned in the previous paragraph were produced by Gompf-Scharlemann-Thompson <cit.>, building on work of Akbulut-Kirby <cit.>. We will call this family the GST links.In order to describe the construction of the GST links, we need several preliminary details.Let Q denote the square knot, the connected sum of the right-handed and left-handed trefoil knots, and let F denote the genus two fiber surface for the square knot.In <cit.>, Scharlemann depicted an elegant way to think about the monodromy corresponding to the fibration of E(Q) by F:We may draw F as a topological annulus A, and such that * A disk D has been removed from A,* each component of A is split into six edges and six vertices, and* opposite inside edges and opposite outside edges of A are identified to form F. With respect to A, the monodromy φ is a 1/6th clockwise rotation of A, followed by an isotopy of D returning it to its original position.Let Y_Q be the closed 3-manifold obtained by 0-surgery on Q, so that Y_Q is a fibered 3-manifold with fiber surface F and monodromy φ, called the closed monodromy of Q. Note that the monodromy F is an honest 1/6th rotation of the annulus in Figure <ref>, since, in this case, the puncture has been filled in by the Dehn surgery.Details can be found in <cit.> and <cit.>, where the following lemma is first proved. For every rational number p/q with q odd, there is a family {V_p/q,V'_p/q,V”_p,q} of curves contained in F that are permuted by φ.We may subdivide A into six rectangular regions as shown in Figure <ref>.It is proved in <cit.> that F is a 3-fold branched cover of a 2-sphere S with four branch points.By naturally identifying S with 4-punctured sphere constructed by gluing two unit squares along their edges, there is a unique isotopy class of curve c_p/q with slope p/q in S.Let ρ:F → S denote the covering map.Scharlemann proves that ρ^-1(c_p,q) = {V_p/q,V'_p/q,V”_p,q}, and these curves are permuted by φ. Figure <ref> shows the three lifts, V_3/7, V_3/7', and V_3/7”, of the rational curve 3/7 to the fiber F of the square knot.Note that φ^6 is the identity map, and φ ^3 maps V_p/q to itself but with reversed orientation.Finally, we can define the GST links.The next lemma is also from <cit.>. The GST link L_n is handleslide equivalent to Q ∪ V_n/2n+1.The R-link L_n has Property R for n = 0,1,2 and is not known to have Property R for n ≥ 3. For ease of notation, let V_n = V_n/2n+1 and V_n' = V'_n/2n+1, so that L_n = Q ∪ V_n.Two links L and L' are said to be stably handleslide equivalent or just stably equivalent if there are integers n and n' so that L ⊔ U_n is handleslide equivalent to L' ⊔ U_n'.While we can find admissible surfaces for L_n, there is a simpler construction for a family of links L_n' stably equivalent to L_n for each n, and we note a link L has Stable Property R if and only if every link stably equivalent to L has Stable Property R. The link L_n = Q ∪ V_n is stably equivalent to L'_n = V_n ∪ V_n'. We will show that both links are stably equivalent to Q ∪ V_n ∪ V_n'.Since φ(V_n) = V_n', we have that V'_n is isotopic to V_n in Y_Q.Carrying this isotopy into S^3, we see that after some number of handleslides of V_n' over Q, the resulting curve C' is isotopic to V_n.Now C' can be slide over V_n to produce a split unknot U_1, and Q ∪ V_n ∪ V_n' is handleslide equivalent to L_n ⊔ U_1.On the other hand, V_n and V_n' are homologically independent in the genus two surface F.Thus, there is a sequence of slides of Q over V_n and V_n' converting it to a split unknot, so Q ∪ V_n ∪ V_n' is handleslide equivalent to L_n' ⊔ U_1 as well. Next, we will define an admissible surface for L_n'.Consider a collar neighborhood FI of F, and let N ⊂ S^3 denote the embedded 3-manifold obtained by crushing FI to a single curve.Letting Σ =N, we see that Σ is two copies of F, call them F_0 and F_1, glued along the curve Q. Consider L_n' embedded in F_0, and push L_n' slightly into N.Then Σ is an admissible surface for L_n'. First, FI is a genus four handlebody, as is N, since N is obtained by crushing the vertical boundary of FI.Moreover, since the exterior E(Q) is fibered with fiber F, we may view this fibering as an open book decomposition of S^3 with binding Q, and thus S^3 ∖ N is homeomorphic to N, so that Σ is a Heegaard surface for S^3.It remains to be seen that there is a core of N containing L_n', but it suffices to show that there is a pair D_n and D_n' of dualizing disks for L_n' in N.Note that for any properly embedded arc a ⊂ F_0, there is a compressing disk D(a) for N obtained by crushing the vertical boundary of the disk aI ⊂ FI.Let a_0 and a_0' be disjoint arcs embedded in F_0 such that a_0 meets V_n once and avoids V_n', and a_0' meets V_n' once and avoids V_n.Then D(a_0) and D(a_0') are dualizing disks for L_n', completing the proof. Lemma <ref> does more than simply prove Σ is admissible; it provides the key ingredients we need to construct a diagram for (L_n',Σ):Let a_1 and a_1' denote parallel copies of a_0 and a_0', respectively, in F_1, so that D(a_0) = a_0 ∪ a_1 and D(a_0') = a_0' ∪ a_1'.By Lemma <ref>, there is a genus four trisection diagram (,,) for (L_n',Σ) so that_1 =D(a_0) _2 =D(a_0') _1 = V_n _2 = V_n'.Noting that (,) defines a genus four splitting of Y_2, it follows that any curve disjoint from _1 ∪_2 ∪_1 ∪_2 that bounds a disk in either of H_ or H_ also bounds in the other handlebody.Let b_0 and b_0' denote non-isotopic disjoint arcs in F_0 that are disjoint from a_0 ∪ a_0' ∪ L_n'.Then b_0 ∪ b_1 and b_0' ∪ b_1' bound disks in N; thus letting_3 = _3 = b_0 ∪ b_1 _4 =_4 = b_0' ∪ b_1',we have that (,) is a standard diagram, corresponding to two of the cut systems in a diagram for (L_n,Σ).To find the curves in , let N' = S^3 ∖ N, and observe that N' also has the structure of FI crushed along its vertical boundary, and N' =N = F_0 ∪ F_1.One way to reconstruct S^3 from N and N', both of which are homeomorphic to crushed products FI, is to initially glue F_1 ⊂ N' to F_1 ⊂ N'.The result of this initial gluing is again homeomorphic to a crushed product FI.The second gluing then incorporates the monodromy, so that F_0 ⊂ N' is glued to F_0 ⊂ N via φ.The result of this gluing is that if a_1 is an arc in F_1 ⊂ N' and D'(a_1) is the corresponding product disk in N', then D'(a) = a_1 ∪φ(a_0), where a_0 is a parallel copy of a_1 in F_0 (using the product structure of N).Thus, in order to find curves in , we can choose any four arcs in F_1 cutting the surface into a planar component and construct their product disks.However, if we wish to a find a diagram with relatively little complication with respect to theandcurves we have already chosen, it makes sense to choose those four arcs to be a_1, a_1', b_1, and b_1'.Thus,_1 = a_1 ∪φ(a_0) _2 = a_1' ∪φ(a_0') _3 = b_1 ∪φ(b_0) _4 = b_1' ∪φ(b_0'). We have proved the following: The triple (,,) forms a (4;0,2,2)-trisection diagram for (L_n,Σ). § A RECTANGLE CONDITION FOR TRISECTION DIAGRAMS In the final section, we introduce a tool for potential future use.This tool is an adaptation of the Rectangle Condition for Heegaard diagrams of Casson and Gordon <cit.> to the setting of trisection diagrams.A collection of 3g-3 pairwise disjoint and nonisotopic curves in a genus g surface Σ is called a pants decomposition, as the curves cut Σ into 2g-2 thrice-punctured spheres, or pairs of pants.A pants decomposition defines a handlebody in the same way a cut system does, although a cut system is a minimal collection of curves defining a handlebody, whereas a pants decomposition necessarily contains superfluous curves.An extended Heegaard diagram is a pair of pants decompositions (,) determining a Heegaard splitting H_∪ H_.An extended trisection diagram is a triple of pants decompositions (,,) determining the spine H_∪ H_∪ H_ of a trisection.Suppose thatandare pants decompositions of Σ, and let P_ be a component of Σ∖ν() and P_ a component of Σ∖ν().Let a_1, a_2, and a_3 denote the boundary components of P_; b_1, b_2, and b_3 the boundary components of P_.We say that the pair (P_,P_) is saturated if for all i,j,k,l ∈{1,2,3}, i≠ j, k ≠ l, the intersection P_∩ P_ contains a rectangle R_i,j,k,l with boundary arcs contained in a_i, b_k, a_j, and b_l.We say that that pair of pants P_ is saturated with respect toif for every component P_ of Σ∖ν(), the pair (P_,P_) is saturated.An extended Heegaard diagram (,) satisfies the Rectangle Condition of Casson-Gordon if for every component P_ of Σ∖ν(), we have P_ is saturated with respect to .Casson and Gordon proved the following. Suppose that an extended Heegaard diagram (,) satisfies the Rectangle Condition.Then the induced Heegaard splitting H_∪ H_ is irreducible. Now, let (,,) be an extended trisection diagram.We say that (,,) satisfies the Rectangle Condition if for every component P_ of Σ∖ν(), we have that either P_ is saturated with respect toor P_ is saturated with respect to . Suppose that an extended trisection diagram satisfies the Rectangle Condition.Then the induced trisectionwith spine H_∪ H_∪ H_ is irreducible. Suppose by way of contradiction thatis reducible.Then there exists a curve δ⊂Σ =H_ that bounds disks D_1 ⊂ H_, D_2 ⊂ H_, and D_3 ⊂ H_.Let D_ denote the set of 3g-3 disks in H_ bounded by the curves , and define D_ and D_ similarly.There are several cases to consider.First, suppose that δ∈, so that D_1 ∈ D_, and let P_ be a component of Σ∖ν() that contains δ as a boundary component.Suppose without loss of generality that P_ is saturated with respect to .Then, for any curve b ∈, we have that b is the boundary of a component P_ of Σ∖ν(), where P_∩ P_ contains a rectangle with boundary arcs in δ and b.It follows that δ meets every curve b ∈, so δ∉.Suppose that D_2 and D_ have been isotoped to intersect minimally, so that these disks meet in arcs by a standard argument.There must be an outermost arc of intersection in D_2, which bounds a subdisk of D_2 with an arc δ' ⊂δ, and δ' is a wave (an arc with both endpoints on the same boundary curve) contained a single component P_ of Σ∖ν().Let b_1 and b_2 be the boundary components of P_ disjoint from δ'.Since P_ is saturated with respect to , there is rectangle R ⊂ P_∩ P_ with boundary arcs contained in b_1, δ, b_2, and some other curve in P_.Let δ” be the arc component of R contained in δ.Since the wave δ' separates b_1 from b_2 in P_, it follows that δ' ∩δ”≠, a contradiction.In the second case, suppose that δ is a curve in .Note that the Heegaard splitting determined by (,) is reducible, and thus by the contrapositive of Casson-Gordon's rectangle condition, there must be some pants decomposition P_ of Σ∖ν() such that P_ is not saturated with respect to , so that P_ is saturated with respect to β.Let P_ be a component of Σ∖ν() that contains δ as boundary component.By the above argument, δ∉, and if we intersect D_1 with D_ we can run an argument parallel to the one above to show that δ has a self-intersection, a contradiction.A parallel argument shows that δ∉.Finally, suppose that δ is not contained in any of , , or .By intersecting the disks D_1 and D_, we see that there is a wave δ' ⊂δ contained in some pants component P_ of Σ∖ν().Suppose without loss of generality that P_ is saturated with respect to .By intersecting D_2 with D_, we see that there is a wave δ”⊂δ contained in some pants component P_ of Σ∖ν().Let a_1 and a_2 be the components of P_ that avoid δ', and let b_1 and b_2 be the components of P_ that avoid δ”.By the Rectangle Condition, P_∩ P_ contains a rectangle R whose boundary is made of arcs in a_1, b_1, a_2, and b_2. As such, δ' ∩ R contains an arc connecting b_1 to b_2, while δ”∩ R contains an arc connecting a_1 to a_2, but this implies that δ' ∩δ”≠, a contradiction.We conclude that no such curve δ exists. Of course, at this time, the Rectangle Condition is a tool without an application, which elicits the following question: Is there an extended trisection diagram (,,) that satisfies the Rectangle Condition? Note that while it is easy to find three pants decompositions that satisfy the Rectangle Condition, the difficulty lies in finding three such pants decompositions which also determine a trisection; in pairs, they must be extended Heegaard diagrams for the 3-manifolds Y_k.amsalpha | http://arxiv.org/abs/1707.08955v1 | {
"authors": [
"Jeffrey Meier",
"Alexander Zupan"
],
"categories": [
"math.GT"
],
"primary_category": "math.GT",
"published": "20170727175815",
"title": "Characterizing Dehn surgeries on links via trisections"
} |
[ [ December 30, 2023 ===================== This paper deals with using word embedding models to trace the temporal dynamics of semantic relations between pairs of words. The set-up is similar to the well-known analogies task, but expanded with a time dimension. To this end, we apply incremental updating of the models with new training texts, including incremental vocabulary expansion, coupled with learned transformation matrices that let us map between members of the relation. The proposed approach is evaluated on the task of predicting insurgent armed groups based on geographical locations. The gold standard data for the time span 1994–2010 is extracted from the UCDP Armed Conflicts dataset. The results show that the method is feasible and outperforms the baselines, but also that important work still remains to be done.§ INTRODUCTION AND RELATED WORKIn this research, we make an attempt to model the dynamics of worldwide armed conflicts on the basis of English news texts. To this end, we employ the well-known framework of Continuous Bag-of-Words modeling <cit.> for training word embeddings on the English Gigaword news text corpus <cit.>. We learn linear projections from the embeddings of geographical locations where violent armed groups were active to the embeddings of these groups. These projections are then applied to the embeddings and gold standard data from the subsequent year, thus predicting what entities act as violent groups in the next time slice. To evaluate our approach, we adapt the UCDP Armed Conflict Dataset <cit.> (see Section <ref> for details). Here is a simplified example of the task: given that in 2003, the Kashmir Liberation Front and ULFA were involved in armed conflicts in India, and Lord's Resistance Army in Uganda, predict entities playing the same role in 2004 in Iraq (the correct answers are Ansar al-Islam, al-Mahdi Army and Islamic State). The nature of the task is conceptually similar to that of analogical reasoning, but with the added complexity of temporal change.Attempts to detect semantic change using unsupervised methods have a long history. Significant results have already been achieved in employing word embeddings to study diachronic language change. Among others, <cit.> show that the embedding of a given word for a given time period to a large extent is a linear combination of its embeddings for the previous time periods.<cit.> proposed an important distinction between cultural shifts and linguistic drifts. They proved that global embedding-based measures (comparing the similarities of words to all other words in the lexicon) are sensitive to regular processes of linguistic drift, while local measures (comparing nearest neighbors' lists) are a better fit for more irregular cultural shifts in word meaning. Our focus here is on cultural shifts: it is not the dictionary meanings of the names denoting locations and armed groups that change, but rather their `image' in the analyzed texts.Our measurement approach can also be defined as `local' to some extent: the linear projections that we learn are mostly based and evaluated on the nearest neighborhood data. However, this method is different in that its scope is not single words but pairs of typed entities (`location' and `armed group' in our case) and the semantic relations between them. §.§ ContributionsThe main contributions of this paper are: * We show that distributional semantic models, in particular word embeddings, can be used not only to trace diachronic semantic shifts in words, but also the temporal dynamics of semantic relations between pairs of words. * The necessary prerequisites for achieving decent performance in this task are incremental updating of the models with new textual data (instead of training from scratch each time new data is added) and some way of expanding the vocabulary of the models.§ GOLD STANDARD DATA ON ARMED CONFLICTSThe UCDP/PRIO Armed Conflict Dataset maintained by the Uppsala Conflict Data Program and the Peace Research Institute Oslo is a manually annotated geographical and temporal dataset with information on armed conflicts, in the time period from 1946 to the present <cit.>. It encodes conflicts, where at least one party is the government of a state. The Armed Conflict Dataset is widely used in statistical and macro-level conflict research; however, it was adapted and introduced to the NLP field only recently, starting with <cit.>. Whereas that work was focused on detecting the onset/endpoint of armed conflicts, the current paper further extends on this by using the dataset to evaluate the detection of changes in the semantic relation holding between participants of armed conflicts and their locations.Two essential notions in the UCDP data are those of event and armed conflict. Events can evolve into full-scale armed conflicts, defined as contested incompatibilities that concern government and/or territory where the use of armed force between two parties, of which at least one is the government of a state, results in at least 25 battle-related deaths <cit.>. The subset of the data that we employ is the UCDP Conflict Termination dataset <cit.>. It contains entries on starting and ending dates of about 2000 conflicts. We limit ourselves to the conflicts taking place between 1994 and 2010 (the Gigaword time span). Almost always, the first actor of the conflict (sideA) is the government of the corresponding location, and the second actor (sideB) is some insurgent armed group we are interested in. We omitted the conflicts where both sides were governments (about 2% of the entries) or where one of the sides was mentioned in the Gigaword less than 100 times (about 1% of the entries). In cases when the UCDP described the conflict as featuring several groups on the sideB, we created a separate entry for each. This resulted in a test set of 673 conflicts, with 137 unique Location–Insurgent pairs throughout the whole time span (many pairs appear several times in different years). In total, it mentions 52 locations (with India being the most frequent) and 128 armed insurgent groups (with ULFA or United Liberation Front of Assam being the most frequent). This test set is available for subsequent reuse (<http://ltr.uio.no/ andreku/armedconflicts/>).§ PREDICTING ARMED CONFLICT PARTICIPANTSIn this section, we provide a detailed description of our approach, starting with a synchronic example in <ref> and then moving on to a toy diachronic example on one pair of years in <ref>. In the next section <ref>, we conduct evaluation on the full test set. §.§ Synchronic modelingWe first conducted preliminary experiments to assess the hypothesis that the embeddings contain semantic relationships of the type `insurgent participant of an armed conflict in the location'. To this end, we trained a CBOW model on the full English Gigaword corpus (about 4.8 billion tokens in total), with a symmetric context window of 5 words, vector size 300, 10 negative samples and 5 iterations. Words with a frequency less than 100 were ignored during training. We used Gensim <cit.> for training, and in terms of corpus pre-processing we performed lemmatization, PoS-tagging and NER using Stanford CoreNLP <cit.>. Named entities were concatenated to one token (for example, United States became United::States_PROPN). Then, we used the 137 Location–Insurgent pairs derived in Section <ref> to learn a projection matrix from the embeddings for locations to the embeddings for insurgents. The idea and the theory behind this approach are extensively described in <cit.> and <cit.>, but essentially it involves training a linear regression which minimizes the error in transforming one set of vectors into another. Finding the optimal transformation matrix amounts to solving i normal equations (where i is the vector size in the embedding model being used), as shown in Equation <ref>:β_i = (X^⊺* X + λ * L)^-1 * X^⊺ * y_iwhere X is the matrix of 137 location word vectors (input), y_i is the array of the ith components of 137 corresponding insurgent word vectors (correct predictions), L is the identity matrix of the size i, with 0 at the top left cell, and λ is a real number used to tune the influence of regularization term (if λ = 0, there is no regularization). β_i is the array of i optimal coefficients which transform an arbitrary location vector into the ith component of the corresponding insurgent vector. After learning such an array for each vector component, we have a linear projection matrix which can `predict' an insurgent embedding from a location embedding.To evaluate the resulting projections, we employed leave-one-out cross-validation, i.e., measuring the average accuracy of predictions on each pair from the test set, after training the matrix on all the pairs except the one used for the testing. The transformation matrix was dot-multiplied by the location vector from the test pair. Then, we found n nearest neighbors in the word embedding model for this predicted vector. If the real insurgent in the test pair was present in these n neighbors, the accuracy for this pair was 1, otherwise 0. In Table <ref>, the average accuracies with different values of λ and n are reported.The relations of this kind are not symmetric: it is much easier to predict the location based on the insurgent than vice versa (see the right part of Table <ref>). Moreover, we find that the achieved results are roughly consistent with the performance of the same approach on the Google Analogies test set <cit.>. We converted the semantic sections in the Analogies test set containing only nouns (capitals–common, capitals–world, cities in states, currency and family) to sets of unique pairs. Then, linear projections with λ=1.0 were learned and evaluated for each of them. The average accuracies over these sections were 13.0@1, 48.77@5 and [email protected] results on predicting armed groups are still worse than on the Google Analogies, because of 3 factors: 1) one-to-many relationships in the UCDP dataset (multiple armed groups can be active in the same location) make learning the transformation matrix more difficult; 2) the frequency of words denoting armed groups is lower than any of the words in the Google Analogies data set, thus, embeddings for them are of lower quality; 3) training the matrix on the whole Gigaword model is suboptimal, as the majority of armed groups were not active throughout all its time span.All our experiments were also conducted using the very similar Continuous Skipgram models. However, as CBOW proved to consistently outperform Skipgram for our tasks, we only report results for CBOW, due to limited space.[It seems CBOW is often better than Skipgram with linear projections; cf. the same claim in <cit.>.]To sum up this section, many-to-one semantic relations between locations and insurgents do exist in the word embedding models. They are less expressed than one-to-one relations like those in the Google Analogies test set, but still can be found using linear projections. In the next section, we trace the dynamics of these relations as the models are updated with new data. §.§ Diachronic modelingOur approach to using learned transformation matrices to trace armed conflict dynamics through time consists of the following. We first train a CBOW model on the subsection of Gigaword texts belonging to the year 1994. Then, we incrementally update (train) this same model with new texts, saving a new model after each subsequent year. The size of the yearly subcorpora is about 250–320 million content words each. Importantly, we also use vocabulary expansion: new words are added to the vocabulary of the model if their frequency in the new yearly data satisfy our minimal threshold of 15.[We did not experiment with different thresholds. It was initially set to the value which produced a reasonable vocabulary size of several hundred thousand words.] Each yearly training session is performed in 5 iterations, with linearly decreasing learning rate.Note that we do not use any model alignment method (Procrustes, etc): our models are simply trained further with the new texts. A possible alternative to this can be incremental training of hierarchical softmax functions proposed in <cit.> or incremental negative sampling proposed in <cit.>; we leave it for future work. The experiment involves applying a learned transformation matrix across pairs of models. While in Section <ref> we evaluate the approach across the entire Gigaword time period, this section reports a preliminary example experiment for the transition from 2000 to 2001 alone. This means we will have one model saved after sequential training for the years up to 2000, and one saved after year 2001. Our aim is to find out whether the Location–Insurgent projection learned on the first model is able to reveal conflicts that appear in 2001. Thus, we extract from the UCDP dataset all the pairs related to the conflicts which took place between 1994 and 2000 (91 pairs total). The projection is trained on their embeddings from the first model (actually, on 79 pairs, as 12 armed group names were not present in the 2000 model and subsequently skipped). Then, this projection is applied to the second model embeddings of the 47 locations, which are subject to armed conflicts in the year 2001 (38 after skipping pairs with out-of-vocabulary elements). Table <ref> demonstrates the resulting performance (reflecting how close the predicted vectors are to the actual armed groups active in this or that location). Note that out of 38 pairs from 2001, 31 were already present in the previous data set (ongoing conflicts). This explains why the evaluation on all the pairs gives high results. However, even for the new conflicts, the projection performance is encouraging. Among others, it managed to precisely spot the 2001 insurgency of the members of the Kosovo Liberation Army in Macedonia, notwithstanding the fact that the initial set of training pairs did not mention Macedonia at all. Thus, it seems that the models at least partially `align' new data along the existing semantic axis trained before. In the next section, we systematically evaluate our approach on the whole set of UCDP conflicts in the Gigaword years (1994–2010).§ EVALUATION OF DIACHRONIC MODELSTo evaluate our approach on all the UCDP data, we again tested how good it is in predicting the future conflicts based on the projection matrices learned from the previous years. We did this for all the years between 1994 and 2010. The evaluation metrics are the same as in the Section <ref>: we calculated the ratio of correctly predicted armed groups names from the conflict pairs, for which the UCDP datasets stated that these conflicts were active in this particular year. As before, the models employed in the experiment were incrementally trained on each successive year with vocabulary expansion. Words present in the gold standard but absent from the models under analysis were skipped. At the worst case, 25% of pairs were skipped from the test set; on average, 13% were skipped each year (but see the note below about the incr. static baseline). At test time, all the entities were lowercased.We employ 3 baselines: 1) yearly models trained separately from scratch on the corpora containing texts from each year only (referred to as separate hereafter); 2) yearly models trained from scratch on all the texts from the particular year and the previous years (cumulative hereafter); 3) incrementally trained models without vocabulary expansion (incr. static hereafter).Initially, the linear projections for all models were trained on all the conflict pairs from the past and present years, similar to Section <ref> (dubbed up-to-now hereafter). However, the information about conflicts having ended several years before might not be strongly expressed in the model after it was incrementally updated with the data from all the subsequent years. For example, the 2005 model hardly contains much knowledge about the conflict relations between Mexico and the Popular Revolutionary Army (EPR) which stopped its activities after 1996.Thus, we additionally conducted a similar experiment, but this time the projections were learned only on the salient pairs (dubbed previous): that is, the pairs active in the last year up to which the model was trained. Table <ref> presents the results for these experiments, as well as baselines (averaged across 15 years). For the proposed incr. dynamic approach, the performance of the previous projections is comparable to that of the up-to-now projections on the accuracies @5 and @10, and is even higher on the accuracy @1 (statistically significant with t-test, p < 0.01). Thus, the single-year projections are somewhat more `focused', while taking much less time to learn, because of less training pairs. The fact that our models were incrementally updated, not trained from scratch, is crucial. The results of the separate baseline look more like random jitter. The cumulative baseline results are slightly better, probably simply because they are trained on more data. However, they still perform much worse thanthe models trained using incremental updates. This is because the former models are not connected to each other, and thus are initialized with a different layout of words in the vector space. This gives rise to formally different directions of semantic relations in each yearly model (the relations themselves are still there, but they are rotated and scaled differently).The results for the incr. static baseline, when tested only on the words present in the test model vocabulary (the left part of the table), seem better than those of the proposed incr. dynamic approach. This stems from the fact that incremental updating with static vocabulary means that we never add new words to the models; thus, they contain only the vocabulary learned from the 1994 texts. The result is that at test time we skip many more pairs than with the other approaches (about 62% in average). Subsequently, the projections are tested only on a minor part of the test sets.Of course, skipping large parts of the data would be a major drawback for any realistic application, so the incr. static baseline is not really plausible. For comparison, the right part of Table <ref> provides the accuracies for the setup in which all the pairs are evaluated (for pairs with OOV words the accuracy is always 0). Other tested approaches are not much affected by this change, but for incr. static the performance drops drastically. As a result, for the all pairs scenario, incremental updating with vocabulary expansion outperforms all the baselines (the differences are statistically significant with t-test, p < 0.01).§ CONCLUSIONWe have here shown how incrementally updated word embedding models with vocabulary expansion and linear projection matrices are able to trace the dynamics of subtle semantic relations over time. We applied this approach to the task of predicting armed groups active in particular geographical locations and showed that it significantly outperforms the baselines. However, it can be used for any kind of semantic relations. We believe that studying temporal shifts of such projections can lead to interesting findings far beyond the usual example of `king is to queen as man is to woman'.To our best knowledge, the behavior of semantic relations in updated word embedding models was not explored before. Our experiments show that the models do preserve these `directions' and that the learned projections not only hold for the word pairs known to the initial model, but can also be used to predict relations for the new words. In terms of future work, we plan to trace how quickly incremental updates to the model `dilute' the projections, rendering them useless with time. We observed this performance drop in our experiments, and it would be interesting to know more about the regularities governing this deterioration. Also, for the particular task of analyzing armed conflicts, we plan to research ways of improving accuracy in predicting completely new armed groups not present in the training data, and the methods of filtering out locations not involved in armed conflicts.emnlp_natbib | http://arxiv.org/abs/1707.08660v1 | {
"authors": [
"Andrey Kutuzov",
"Erik Velldal",
"Lilja Øvrelid"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170726230211",
"title": "Temporal dynamics of semantic relations in word embeddings: an application to predicting armed conflict participants"
} |
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland DGIST Research Institute, DGIST, 333 TechnoJungang, Hyeongpung, Daegu 42988, KoreaDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Quantronics Group, SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay, F-91191 Gif-sur-Yvette, FranceDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland We report the realization of a bonded-bridge on-chip superconducting coil and its use in impedance-matching a highly ohmic quantum dot (QD) to a 3 GHz measurement setup. The coil, modeled as a lumped-element LC resonator, is more compact and has a wider bandwidth than resonators based on coplanar transmission lines (e.g. λ/4 impedance transformers and stub tuners) at potentially better signal-to-noise ratios.In particular for measurements of radiation emitted by the device, such as shot noise, the 50× larger bandwidth reduces the time to acquire the spectral density. The resonance frequency, close to 3.25 GHz, is three times higher than that of the one previously reported wire-bonded coil.As a proof of principle, we fabricated an LC circuit that achieves impedance-matching to a 15 kΩ load and validate it with a load defined by a carbon nanotube QD of which we measure the shot noise in the Coulomb blockade regime. Measurements of a Quantum Dot with an Impedance-Matching On-Chip LC Resonator at GHz Frequencies C. Schönenberger December 30, 2023 ================================================================================================§ INTRODUCTIONSuperconducting qubit control and readout of very resistive quantum devices need microwave (MW) resonators operating at frequencies in the range of 1-10 GHz. To efficiently measure radiation emitted from a device, one attempts to balance the high device impedance to the Z_0 = 50 Ω characteristic impedance of the usual coaxial lines using an impedance-matching circuit. These circuits are often implemented with superconducting on-chip transmission lines. Examples include quarter-wavelength step transformer in the fluxonium qubit <cit.>, stub tuners for quantum point contacts <cit.> and QDs <cit.>. An alternative approach makes use of an LC resonator built either from a lumped element inductor <cit.> or form on-chip coils <cit.>. However, the resonance frequencies have typically been limited to a maximum of 1 GHz <cit.>. It is however important to improve LC circuits, since they are much less demanding in chip footprint as compared to transmission line circuits. While a transmission line circuit at 5 GHz needs up to 1 cm in length, an appropriate on-chip coil can be fabricated on a 100× smaller area. This is an important consideration for the future scaling of qubit networks for quantum processors. To understand the challenge, the key parameters of an LC matching circuit are its resonance frequency given by 1/(2π√(LC)) and its characteristic impedance Z_ c defined as √(L/C). In order to match to a high-impedance load, Z_ c should be as large as possible, meaning that one has to aim for a large inductance. At the same time, the resonance frequency should stay high, constraining the capacitance to as small as possible values. In fact, the limiting factor in achievable resonance frequency is the residual stray capacitance in the circuit, which we minimize in our approach.Addressing the compactness of MW resonators in this work, we fabricated a 200-μm-wide on-chip superconducting coil with a wire-bonded bridge (Fig. <ref>) and utilize it as a lumped LC matching circuit in a carbon nanotube (CNT) QD noise experiment at a working frequency close to 3.25 GHz. Compared to the one previously reported case of on-chip inductor with bonded bridge <cit.>,we achieved a threefold frequency increase with a similar footprint. Comparable compactness has been achieved only with Josephson junction arrays acting as quarter-wavelength resonators <cit.>.§ SAMPLE FABRICATION A planar spiral inductor raises the issue of connecting its inner end to the rest of the circuit. While it is possible to fabricate this bridge by lithography, we have realized that the close proximity of the bridge to the coil adds substantial capacitance, limiting the performance of the LC circuit. In addition, the microfabrication of air-bridges of this kind add several fabrication steps. That is why we developed a solution of bonding a wire between a geometry-restricted inner pad and an external bigger pad at the low-impedance end.The fabrication starts with preparing a region on the undoped Si / SiO_2 (500 μm / 170 nm) substrate where CNTs will be placed. Thus, we first evaporate Ti / Au (10 nm / 30 nm) for markers and CNT partial contacts in a square area (top-right corner in Fig. <ref>(a)). We then protect it with a PMMA/HSQ bilayer resist. Afterwards, we sputter 100 nm of Nb and lift the protection bilayer resist off. Subsequently, we e-beam-pattern bonding pads and the desired inductor in a new PMMA resist layer, then etch the Nb film with an Ar/Cl_2 inductively coupled plasma; the surrounding Nb becomes the ground plane. Next, we stamp CNTs <cit.> in the predefined region, locate them using a scanning electron microscope (SEM), and contact the chosen CNT in one Ti/Au evaporation step to the coil and to ground (Fig. <ref>(a)), utilizing the partial contacts. In the same lithography step, a side gate is created at a distance of 300 nm from the CNT (Fig. <ref>(c)). With the device glued onto a sample holder, we use Al wire to bond the remaining end of the coil to a neighboring bigger pad. Due to the relatively small size of the inner pad (70×70 μm^2, barely larger than the bonder wedge), the bonding operation is delicate (Fig <ref>(b)). Finally, we connect the latter pad to theMW line of the printed circuit board sample holder. Furthermore, the Nb ground plane of the sample is bonded with multiple wires along the wafer edge to the sample holder ground plane.The square spiral inductor used in this experiment (Fig. <ref>(a,b)) has an outer dimension of 210 μm, 14 turns with width w = 2 μ m and spacing s = 2 μ m. In the device investigated here two of the turns are shorted, lowering the effective inductance and thus shifting up the resonance frequency by several percents.§ HIGH-FREQUENCY SETUP AND RESONATOR CHARACTERIZATIONThe simplified schematic of the MW circuit is illustrated in Fig. <ref>(d). The impedance-matching circuit, contained in the light-blue dashed rectangle, is modeled with lumped elements: L is the inductance of the coil, C its capacitance to ground, and R_loss the effective loss resistance, accounting for the RF (radio frequency) loss of the superconducting material and the dielectric loss in the substrate. The input impedance Z_in (Fig. <ref>(d)) reads: Z_in = R_loss+ i ω L+ 1/G+ i ω C, with G=1/R, the load conductance. Z_in can be approximated at ω_0 = 1/√(LC) byZ_in(ω_0) ≈ R_loss+Z_ c^2/R.This approximation is valid for |Z_ c|≪ R. Full matching is achieved at ω_0 when Z_in(ω_0)=Z_0. Thus one obtains the condition Z_ c=√((Z_0-R_loss)R), meaning that the characteristic impedance of the LC circuit should be equal to the geometric mean of Z_0 and R, if we neglect R_loss. Typical values for Z_ c are between one and a few kΩ.In reflectance measurements, a continuous sinusoidal wave is applied to the LC circuit where it appears with amplitude V^+. Part of the incident signal is reflected back with amplitude V^- due to impedance mismatch. The reflection coefficient Γ is given by: Γ≡V^-/V^+= Z_in-Z_0 /Z_in + Z_0 . Themagnitude squared of the reflection coefficient is also referred as the reflectance.The complete measurement setup, depicted in Fig. <ref>(a), is the same as in Ref. <cit.>. The load of the matching circuit, the CNT device is tuned by V_SD and V_G—the DC source-drain and gate voltage, respectively. MW lines are used either in reflectometry mode (a microwave signal is sent to the sample and a reflected signal is read from it) or in noise mode where the MW output line conveys the noise signal produced by the sample. The two respective modes utilize a vector network analyzer (VNA) and a power spectrum analyzer (PSA).We characterize the impedance-matching LC resonator by analyzing reflectance measurements together with the simultaneously measured DC current I from which we derive the differential conductance . The LC parameters we thus obtain allow us to extract the QD noise from the measured noise power, as we will show.In practice, the measured reflection coefficient is not measured at the sample, but rather at the VNA, yielding Γ_VNA≡ V_o/V_i, where o and i refer to the VNA ports (see Fig. <ref>(a)). Γ_VNA is different from Γ by a frequency-dependent baseline b(f) produced by attenuations and amplifications in the setup:|Γ_VNA|^2 = b(f) · |Γ|^2 .If spurious reflections occur in components across the output line, then standing waves reside in sections of it and the baseline b(f) can exhibit a complicated pattern.In principle, we need to extract 5 parameters, L, C, b, R_loss, and R. There are two possible approaches: one can fit the frequency dependence of Γ_VNA for a fixed known value of G, for example for G=0, which is realized when the QD is deep in Coulomb blockade where the current is almost completely suppressed. Fig. <ref>(b) shows three measured curves for different G values. Unfortunately, the baseline function b(f) is also markedly frequency-dependent, shifting the minimum of the curves to different values. Due to this dependence, it is already difficult to extract the resonance frequency accurately.We therefore proceed with the second approach. We fix the frequency to a value close to resonance, in the following to f_m=3.23 GHz. We carry out simultaneous measurements of reflectance and DC current over several Coulomb diamonds. We obtain the reflectance |Γ_VNA|^2(V_G,V_SD) and the low-frequency (V_G,V_SD) maps. For each pair of (V_G,V_SD) this defines one point in the Γ_VNA- versus -scatter plot. To obtain the circuit parameters, we fit to this scatter plot the theoretical dependence Γ_VNA (G), thereby implicitly assuming that =G. We have shown in previous work that the real part of the CNT admittance at similar GHz frequencies is the same as dI/dV_SD <cit.>. The curve Γ_VNA(G) can be calculated using Eqs. (<ref>,<ref>,<ref>). It is displayed in Fig. <ref>(c) for a wide range of G values. At the resonance frequency, when sweeping the conductance domain from small to large G values, Γ_VNA(G) first decreases to reach zero reflectance at the matching point where G=G_match, then increases back for G > G_match. The fit together with the measured scatter plot is shown in Fig. <ref>(c).§ NOISE MEASUREMENT CALIBRATIONThe current noise produced by a QD can be modeled by a noise voltage source V_ noise as shown in Fig. <ref>(d,e). The power spectral density of the voltage source S_V is related to current spectral density S_I by S_V=R^2 S_I. The noise voltage needs to be transmitted through the LC circuit to be fed into the Z_0 - transmission line. The voltage V that appears at the interface is proportional to V_ noise, but contains in addition a frequency-dependent transmission function t_V(f) ≡ V/V_ noise, which can readily be derived from the LC circuit parameters: t_V(Ω) ≈Z_0 G/1 + iΩ(Z_0/Z_ c + R_loss/Z_c + Z_c/R) - Ω^2, where Ω=ω/ω_0 = 2π f√(LC) and Z_c=√(L/C) the characteristic impedance as introduced before. In this equation, the factor of i Ω shows that the total Q-factor has three terms, which can be grouped in an internal 1/Q_ int and an external loss part 1/Q_ ext. The two parts are respectively 1/Q_ int=R_ loss/Z_c and 1/Q_ ext=Z_0/Z_c + Z_c/R. At optimal impedance matching in the lossless picture, i.e. if the condition Z_c = √(R Z_0) is met, the external Q-factor is minimal and given by √(R/Z_0)/2 which yields values around 10-100.The transmitted voltage passes several elements, a directional coupler, a circulator and amplifiers at low and room temperature before being measured with the PSA. The overall gain along the chain is captured by g. It was determined with the method presented in Ref. <cit.> to be g=94.6 dB.In addition to the noise generated by the device, the amplifier does also add further current-independent noise. We split the total noise into two contributions ⟨δ P⟩ and ⟨δ P_0⟩, where the first term describes the noise power which is current-dependent and the second term contains all the rest, i.e. thermal and amplifier noise background. The different terms are related as follows <cit.>:⟨δ P⟩ = S_I R^2/Z_0 g∫_ BW|t_V (f)|^2df +⟨δ P_0⟩,where S_I refers to the excess current noise only (i.e. the thermal noise of the device is included in ⟨δ P_0 ⟩).§ EXPERIMENT By means of gate and bias voltage sweeps we measure the DC current and draw an initial charge stability diagram. Then we switch to MW reflectance measurements, where we restrict ourselves to a gate span covering a few Coulomb charge states. In this measurement mode, we simultaneously acquire the current I and reflectance |Γ_VNA|^2 as functions of V_ SD and V_ G. The VNA was set to the working frequency f_m=3.23 GHz with an output power of -25 dBm, attenuated by 87 dB before reaching the device. We deduce the parameters of the LC impedance-matching circuit from a comparison of the two measured maps (V_ G,V_ SD) (Fig. <ref>(a)) and |Γ_ VNA(V_ G,V_ SD)|^2 (Fig. <ref>(b)) as described in the previous section. In practice, the conductance domain of the resulted |Γ_ VNA(, f_m)|^2 scatter plot does not include values close to G_match, therefore we enrich the plot with a close-to-match point, |Γ_VNA ^near match(f_m)|^2, from the separately obtained near-match curve in Fig. <ref>(b).From complementary measurements we estimate an upper bound of R_loss≈ 1 Ω (see Supplemental material). Since R_loss≪ Z_0, the matching circuit can bemodeled as a lossless LC circuit for practical purposes. Thus we fix R_loss=0, and fit the curve |Γ_VNA|^2(G, f=f_m) (Fig. <ref>(c)), which yields L=37 nH, C=63 fF. We further deduce the resonance frequency f_0=3.28 GHz and the characteristic impedance Z_c =766 Ω; the resulting match conductance is the minimum of the calculated curve |Γ_VNA|^2(G, f=f_0): G_match=84 μS. For comparison, refitting with R_loss= 1 Ω, the calculated L, C, G_match vary by less than 1 %.Furthermore, for R_loss≤1 Ω we obtain Q_ ext≤ 15 and Q_ in≥ 766, yielding an empirical bandwidth f_0/Q ≳ 220MHz.Consequently, having fixed the parameters we then calculate a full map of the real part of the device admittance, i.e. G(V_G,V_ SD) (Fig. <ref>(c)). A comparison of cuts (Fig. <ref>(d)) demonstrates a very good agreement with , confirming the validity of the extracted parameters, at least at the frequency at which the reflectance map was acquired. The advantage of the reflectance measurement is, that G can be measured much faster and with less noise, once the circuit parameters are known.In the noise part of the experiment, we realize simultaneous current and noise measurements over the same few Coulomb diamond span as before. Each retained noise power value ⟨δ P ⟩ arises from averaging 500 identical-input measurements in a bandwidth BW=50 MHz and it is produced every800 ms. Fig. <ref>(b) shows the current noise S_I measured in the same gate range after applying Eq. <ref> for calibration. To compare with the Schottky noise, we show in Fig. <ref>(a) a calculated plot of 2 e | I |, where I represents the measured DC current. The Fano factor, F ≡ S_I / |2 e I |, is shown in Fig. <ref>(c). We observe values largely exceeding 1 within the Coulomb blockade, where sequential tunneling of single electrons is forbidden. It is seen in Fig. <ref>(d) that the Fano factor peaks at V_SD≈± 5 mV, where inelastic cotunneling sets in. Such a cotunneling event can excite the QD, enabling the sequential transfer of single electrons through the QD until the QD decays back to the ground state <cit.>. The respective train of charge bunches leads to the large Fano factor, which here reaches values as large as 8. Only super-Poissonian noise with F≤ 3 has been reported earlier in QDs <cit.>. At lower voltage bias, solely elastic cotunneling is present and the Fano factor should be 1 <cit.>, this being the case in those regions of the second Coulomb diamond closer to the inelastic cotunneling onset; at even lower bias, the calculation of F produces here divergent values, due to the division of "noisy" numbers by smallest current numbers, such that F>10 erroneously in the cyan domains of Fig. <ref>(c). § DISCUSSION AND CONCLUSIONWe have successfullydemonstrated the realization and application of a QD-impedance-matching on-chip superconducting coil working at 3.25 GHz. We have deduced the LC circuit parameters by using the dependence of the reflectance |Γ|^2 on the device resistance R at a fixed frequency. The alternative approach, where one fits the frequency dependence of |Γ|^2 to the expected functional form for a fixed value R, typically at G=1/R=0 deep in the Coulomb-blockade regime, has turned out to be less reliable, due to the large bandwidth offered by the LC circuit. Consequently, the circuit is affected by a frequency-dependent background b(f) due to, for example, spurious standing waves in parts of the MW lines. As compared to transmission line resonators used as impedance-matching circuits, the figure of merit g_SNR = SNR/SNR_0, where SNR_0 is the signal-to-noise ratio in the absence of any impedance-matching circuit <cit.>, has been shown to be the same for e.g. a stub tuner and an LC circuit, if one acquires the noise over the full bandwidths given by the full widths at half maxima (FWHM) of the respective transfer functions t_ V(f) <cit.>. The advantage of the LC circuit lies in its larger bandwidth 100 MHz as compared to a transmission line resonator. This is important in application where short interaction times are essential, i.e. for fast readout and qubit manipulation. The disadvantage is that one has to account for spurious resonances in the external system leading to a frequency-dependent background. This can be a problem if highly accurate pulses need to be transmitted for qubit manipulation. Residual and uncontrolled phase shifts may make it difficult to achieve high gate fidelities. 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Osborn, title title An analysis method for asymmetric resonator transmission applied to superconducting devices, 10.1063/1.3692073 journal journal Journal of Applied Physics volume 111 (year 2012), 10.1063/1.3692073, http://arxiv.org/abs/1108.3117 arXiv:1108.3117 NoStop§ SUPPLEMENTAL MATERIAL Loss characterization of a wire-bonded coilWe combined same-geometry coils and 50 Ω coplanar transmission lines in a “hanger" configuration (Fig. <ref>(a)). We then measured at 4 K the power transmission between ports 1 and 2 with P = -95 dBm (higher than the excitation power used in the experiment). We then fit the S_21 curve (Fig. <ref>(b)) with the formula <cit.>:S_21 = 2/2+Z_line/Z, Z_line = 50 Ω· e^ i ϕ.Here, Z=Z_in(G=0) from Eq. (<ref>) stands for a series R_lossLC cicuit; ϕ is a fitting parameter <cit.> that reproduces the asymmetry of the measured curve, caused by external spurious standing waves due to 50 Ω mismatches. The extracted parameters are: R_loss = 1.26 Ω, Z_ c = 954 Ω, f_ 0 = 3.35 GHz, ϕ = 0.56. The obtained loss resistance at 4 K is a comfortable upper limit of our 20 mK experiment, mainly because the Al bond wire becomes superconducting and Nb superconductivity gets reinforced in the mK range.The reduced characteristic impedance in the main experiment Z_ c= 766 Ω as compared to Z_ c= 954 Ω in the hanger is explained by a decreased inductance from the two-turn short in the coil. However, the frequency (and L· C) is very similar, suggesting that a short can also induce an increase in capacitance. The distributed structure of inductive and capacitive elements in a coil could indeed produce this effect.Fitting procedureWe extract the unknown circuit parameters L, C, R_loss, and b(f) by first fixing a frequency f_m close to the resonance frequency. We then plot the measured reflectance values |Γ_ VNA|^2 against the DC measuredfor the same values of V_G and V_ SD. This is shown in Fig. <ref> as green diamonds. We next assume thatis equal to G at GHz frequency. Then we can fit a theoretical dependence to the data points, based on Eqs. (<ref>,<ref>,<ref>). Both the dashed orange and full yellow trances in Fig. <ref> are candidate fitting curves. The reflectance values at G=0 and G →∞ yield the baseline b(f=f_ m) and R_loss; the lack of the latter G value explains why these two parameters cannot be reliably determined simultaneously. We therefore fix R_loss=0; this choice is also a reasonable in further noise extraction, as R_loss is in series with the much larger characteristic impedance Z_0=50 Ω of the setup output line.One could be convinced of the low influence of R_loss by observing the overlap of the lossless and lossy fits in Fig. <ref>. In spite of the stability of the extracted parameters in the two fit cases (less than 1 % for L and C), it is important to observe in Figs. <ref>(b), <ref> a 10 % deviation of the extracted G_match value from the measured one (77 μS). Therefore the accuracy of theextracted S_I is at most 10 %. In conclusion, working at a fixed frequency outputs effective values of L and C at that specific frequency, while the analysis of a whole frequency range, if possible, should provide more precise circuit parameters. | http://arxiv.org/abs/1707.09061v1 | {
"authors": [
"M. -C. Harabula",
"T. Hasler",
"G. Fülöp",
"M. Jung",
"V. Ranjan",
"C. Schönenberger"
],
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"physics.app-ph"
],
"primary_category": "quant-ph",
"published": "20170727222642",
"title": "Measurements of a Quantum Dot with an Impedance-Matching On-Chip LC Resonator at GHz Frequencies"
} |
0in 6.0in 8.0in [5]Journal of Computational and Applied Mathematics elsarticle-num numbers,sort compressTheoremTheorem ExampleExample LemmaLemma RemarkRemark CorollaryCorollary DefinitionDefinition AlgorithmAlgorithm AssumptionAssumption Proof[1][Proof.]#1 mymainaddress]Xiaoqiang Yuemymainaddress]Weiping Bumysecondaryaddress]Shi Shumycorrespondingauthor [mycorrespondingauthor]Corresponding author [email protected]]Menghuan Liumymainaddress]Shuai Wang[mymainaddress]School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P.R. China [mysecondaryaddress]Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, P.R. China The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions forone-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain Ω. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully FE approximation possesses the saturation error order under L^2(Ω) norm. Secondly, we theoretically prove the estimation 1+𝒪(τ^α h^-2β) on the condition number of the coefficient matrix, in which τ and h respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.Caputo-Riesz fractional diffusion equation, fully time-space FE scheme, condition number estimation, algorithmic complexity, adaptive AMG method [2010] 35R1165F1065F1565N55 § INTRODUCTION In recent years, there has been an explosion of research interest in numerical solutions for fractional differential equations, mainly due to the following two aspects: (i) the huge majority can't be solved analytically, (ii) the analytical solution (if luckily derived) always involve certain infinite series which sharply drives up the costs of its evaluation. Various numerical methods have been proposed to approximate more accurately and faster, such as finite difference (FD) method <cit.>, finite element (FE) method <cit.>, finite volume <cit.> method and spectral (element) method <cit.>. An essential challenge against standard differential equations lies in the presence of the fractional differential operator, which gives rise to nonlocality (space fractional, nearly dense or full coefficient matrix) or memory-requirement (time fractional, the entire time history of evaluations) issue, resulting in a vast computational cost.Preconditioned Krylov subspace methods are regarded as one of the potential solutions to the aforementioned challenge. Numerous preconditioners with various Krylov-subspace methods have been constructed respectively for one- and two-dimensional, linear and nonlinear space-fractional diffusion equations (SFDE) <cit.>. Multigrid method has been proven to be a superior solver and preconditioner for ill-conditioned Toeplitz systems as well as SFDE. Pang and Sun propose an efficient and robust geometric multigrid (GMG) with fast Fourier transform (FFT) for one-dimensional SFDE by an implicit FD scheme <cit.>. Bu et al. employ the GMG to one-dimensional multi-term time-fractional advection-diffusion equations via a fully discrete scheme by FD method in temporal and FE method in spatial directions <cit.>. Jiang and Xu construct optimal GMG for two-dimensional SFDE to get FE approximations <cit.>. Chen et al. make the first attempt to present an algebraic multigrid (AMG) method with line smoothers to the fractional Laplacian through localizing it into a nonuniform elliptic equation <cit.>. Zhao et al. invoke GMG for one-dimensional Riesz SFDE by an adaptive FE scheme using hierarchical matrices <cit.>. From the survey of references, in spite of quite a number of contributions to numerical methods and preconditioners, there are no calculations taking into account of fully discrete FE schemes and AMG methods for time-space Caputo-Riesz fractional diffusion equations.In this paper, we are concerned with the following time-space Caputo-Riesz fractional diffusion equation (CR-FDE)_0^CD^α_tu(x,t)=∂^2βu(x,t)/∂|x|^2β+f(x,t), t∈ I=(0,T], x∈Ω=(a,b) u(x,t)=0, t∈ I, x∈∂Ωu(x,0)=ψ_0(x), x∈Ωwith orders α∈(0,1) and β∈(1/2,1), the Caputo and Riesz fractional derivatives are respectively defined by_0^CD^α_tu=1/Γ(1-α)∫_0^t(t-s)^-α∂ u/∂ sds, ∂^2βu/∂|x|^2β=-1/2cos(βπ)(_xD_L^2βu + _xD_R^2βu),where_xD_L^2βu = 1/Γ(2-2β)∂^2/∂ x^2∫_a^x(x-s)^1-2βuds, _xD_R^2βu = 1/Γ(2-2β)∂^2/∂ x^2∫_x^b(s-x)^1-2βuds. The remainder of this paper proceeds as follows. A fully discrete FE method of (<ref>)-(<ref>) is developed in Section 2. Section 3 comes up with the theoretical estimation and verification experiments on the condition number of the coefficient matrix. The classical AMG method is introduced in Section 4 followed by its uniform convergence analysis and the construction of an adaptive AMG method. Section 5 reports and analyzes numerical results to show the benefits. We close in Section 6 with some concluding remarks.§ FULLY DISCRETE FINITE ELEMENT SCHEME FOR THE CR-FDE For simplicity, following <cit.>, we will use the symbols ≲, ≳ and ≃ throughout the paper. u_1≲ v_1 means u_1≤ C_1v_1, u_2≳ v_2 means u_2≥ c_2v_2 while u_3≃ v_3 means c_3v_3≤ u_3≤ C_3v_3, where C_1, c_2, c_3 and C_3 are generic positive constants independent of variables, time and space step sizes. §.§ Reminder about fractional calculus In this subsection, we briefly introduce some fractional derivative spaces and several auxiliary results. Here the L^2 inner product and norm are denoted by(u,v)_L^2(Ω)=∫_Ωuvdx, u_L^2(Ω)=(u,u)^1/2_L^2(Ω).(Left and right fractional derivative spaces) For constant μ>0, define normsu_J_L^μ(Ω) := (u^2_L^2(Ω)+_xD_L^μ u^2_L^2(Ω))^1/2, u_J_R^μ(Ω) := (u^2_L^2(Ω)+_xD_R^μ u^2_L^2(Ω))^1/2,and let J_L,0^μ(Ω) and J_R,0^μ(Ω) be closures of C_0^∞(Ω) under ·_J_L^μ(Ω) and ·_J_R^μ(Ω), respectively.(Fractional Sobolev space) For constant μ>0, define the normu_H^μ(Ω) := (u^2_L^2(Ω)+|ξ|^μũ^2_L^2(Ω_ξ))^1/2,and let H_0^μ(Ω) be the closure of C_0^∞(Ω) under ·_H^μ(Ω), where ũ is the Fourier transform of u.The equivalence between (<ref>) and the general definition of the norm has been established in <cit.>, which implies the reasonability of Definition <ref>.(see <cit.>, Proposition 1) If constant μ∈(0,1), u,v∈ J_L,0^2μ(Ω) (or J_R,0^2μ(Ω)), then(_xD_L^2μu,v)_L^2(Ω)=(_xD_L^μ u,_xD_R^μ v)_L^2(Ω), (_xD_R^2μu,v)_L^2(Ω)=(_xD_R^μ u,_xD_L^μ v)_L^2(Ω). (see <cit.>, Lemma 2.4) For constant μ > 0, we have(_xD_L^μ u, _xD_R^μ u)_L^2(Ω) = cos(πμ)_xD_L^μ u^2_L^2(Ω). (Fractional Poincaré-Friedrichs inequality, see <cit.>, Theorem 2.10) For u∈ J_L,0^μ(Ω), we haveu_L^2(Ω)≲_xD_L^μ u_L^2(Ω). §.§ Derivation of the fully discrete scheme By Lemma <ref>, we get the variational (weak) formulation of (<ref>)-(<ref>): given f∈ L^2(Ω,I), ϕ_0∈ L^2(Ω) and Q_t:=Ω×(0,t), to find u∈ℋ subject to u(x,0)=ψ_0(x) and(_0^CD^α_σ u,v)_Q_t+B^t_Ω(u,v)=(f,v)_Q_t, ∀ v∈ℋ^*,where ℋ:=H_0^β(Ω)× H^1(I), ℋ^*:=H_0^β(Ω)× L^2(I), and(_0^CD^α_σ u,v)_Q_t=∫_0^t(_0^CD^α_σ u,v)_L^2(Ω)dσ, (f,v)_Q_t=∫_0^t(f,v)_L^2(Ω)dσ, B^t_Ω(u,v)=∫_0^t1/2cos(βπ)[(_xD^β_Lu,_xD^β_Rv)_L^2(Ω) +(_xD^β_Ru,_xD^β_Lv)_L^2(Ω)]dσ.In order to acquire numerical solutions of u, we firstly make a (possibly nonuniform) temporal discretization by points 0=t_0<t_1<⋯<t_N=T, and a uniform spatial discretization by points x_i = a+ih (i=0,1,⋯,M), where h=(b-a)/M represents the space step size. LetI_j=(t_j-1,t_j), Ĩ_j=(0,t_j), j=1,2,⋯,N; Ω_h = {Ω_l:Ω_l=(x_l-1,x_l), l=1,2,⋯,M}. We observe that it is convenient to form the FE spaces in tensor products𝒱_n=𝒱^β_h(Ω_h)×𝒱_τ(Ĩ_n), 𝒱^*_n=𝒱^β_h(Ω_h)×𝒱^*_τ(I_n),where𝒱^β_h(Ω_h)={w_h∈ H_0^β(Ω)∩ C(Ω̅): w_h(x)|_Ω_l∈𝒫_1(Ω_l), l=1,⋯,M},𝒱_τ(Ĩ_n) = {v_τ∈𝒞(Ĩ_n): v_τ(0)=1, v_τ(t)|_I_j∈𝒫_1(I_j), j=1,⋯, n},𝒱^*_τ(I_n) = {v_τ∈ L^2(I_n): v_τ(t)|_I_n∈𝒫_0(I_n)},and 𝒫_k denotes the set of all polynomials of degree ≤ k. Apparently, for a given u_hτ(x,t)∈𝒱_n, we have ∂ u_hτ/∂ t∈𝒱^*_n, where ∂ u_hτ/∂ t is obtained by differentiating u_hτ with respect to t on each subinterval I_j (j=1,2,⋯,N). We obtain a fully discrete FE scheme in temporal and spatial directions of problem (<ref>): given Q_n:=Ω_h× I_n, to find u_hτ∈𝒱_n such that u_hτ(x,0)=ψ_0,I(x) and(_0^CD^α_t u_hτ,v_hτ)_Q_n+B^n_Ω(u_hτ,v_hτ)= (f,v_hτ)_Q_n, ∀ v_hτ∈𝒱^*_n,where ψ_0,I(x)∈𝒱_n satisfying ψ_0,I(x_i)=ψ_0(x_i) (i=0,1,⋯,M), and(_0^CD^α_tu_hτ,v_hτ)_Q_n=∫_t_n-1^t_n (_0^CD^α_t u_hτ, v_hτ)_L^2(Ω)dt, (f,v_hτ)_Q_n=∫_t_n-1^t_n(f,v_hτ)_L^2(Ω)dt, B^n_Ω(u_hτ,v_hτ) = ∫_t_n-1^t_n1/2cos(βπ)[(_xD^β_Lu_hτ,_xD^β_Rv_hτ)_L^2(Ω) +(_xD^β_Ru_hτ,_xD^β_Lv_hτ)_L^2(Ω)]dt. Letℒ_0(t)={ t_1-t/τ_1, t ∈ I_10, t∈Ĩ_n∖ I_1 ., ℒ̃_0(t)=1/Γ(1-α)∫_t_0^tdℒ_0(s)/(t-s)^α, ℒ̂_0(t)=1/Γ(1-α)∫_t_0^t_1dℒ_0(s)/(t-s)^α, ℒ_k(t)={ t_k+1-t/τ_k+1, t ∈ I_k+1t-t_k-1/τ_k, t ∈ I_k0, t∈Ĩ_n∖ (I_k ∪ I_k+1) ., ℒ̂_k(t)=1/Γ(1-α)∫_t_k-1^t_k+1dℒ_k(s)/(t-s)^α, k=1,⋯,n-1andℒ_n(t)={ t-t_n-1/τ_n, t ∈ I_n0, t∈Ĩ_n∖ I_n ., ℒ̃_n(t)=1/Γ(1-α)∫_t_n-1^tdℒ_n(s)/(t-s)^α.Note that𝒱^*_n={ϕ_l(x)× 1, l=1,⋯,M-1},where ϕ_l(x) is the shape function at x_l∈Ω_h. Usingu_hτ(x,t)=u_h^0(x)ℒ_0(t)+∑_k=1^n-1u_h^k(x)ℒ_k(t)+u_h^n(x)ℒ_n(t),we have(_0^CD^α_tu_hτ,ϕ_l× 1)_Q_1 =(u_h^0,ϕ_l)_L^2(Ω)(ℒ̃_0,1)_L^2(I_1) +(u_h^1,ϕ_l)_L^2(Ω)(ℒ̃_1,1)_L^2(I_1), (_0^CD^α_tu_hτ,ϕ_l× 1)_Q_n =(u_h^0,ϕ_l)_L^2(Ω)(ℒ̂_0,1)_L^2(I_n)+ ∑_k=1^n-1(u_h^k,ϕ_l)_L^2(Ω)(ℒ̂_k,1)_L^2(I_n) +(u_h^n,ϕ_l)_L^2(Ω)(ℒ̃_n,1)_L^2(I_n), n>1, ∫_t_n-1^t_n1×(_xD^β_Lu_hτ,_xD^β_Rϕ_l)_L^2(Ω)dt =(_xD^β_Lu_h^0,_xD^β_Rϕ_l)_L^2(Ω) (ℒ_0,1)_L^2(I_n)+∑_k=1^n-1(_xD^β_Lu_h^k,_xD^β_Rϕ_l)_L^2(Ω) (ℒ_k,1)_L^2(I_n) + (_xD^β_Lu_h^n,_xD^β_Rϕ_l)_L^2(Ω) (ℒ_n,1)_L^2(I_n), ∫_t_n-1^t_n1× (_xD^β_Ru_hτ,_xD^β_Lϕ_l)_L^2(Ω)dt =(_xD^β_Ru_h^0,_xD^β_Lϕ_l)_L^2(Ω) (ℒ_0,1)_L^2(I_n)+ ∑_k=1^n-1(_xD^β_Ru_h^k,_xD^β_Lϕ_l)_L^2(Ω) (ℒ_k,1)_L^2(I_n)+ (_xD^β_Ru_h^n,_xD^β_Lϕ_l)_L^2(Ω) (ℒ_n,1)_L^2(I_n). Substituting (<ref>)-(<ref>) into (<ref>), yieldsC^n_hτU^n_hτ = G^n_hτ,where the coefficient matrixC^n_hτ = M_h+Γ(3-α)/2τ^α_n A_h^β,the right-hand side vectorG^n_hτ=Γ(3-α)τ_n^α-1F^n_hτ+ [M_h-Γ(3-α)/2τ^α_n A_h^β]U^n-1_hτ - ∑_k=1^n-1τ_n^α-1×(t_n-t_k-1)^2-α-(t_n-1-t_k-1)^2-α -(t_n-t_k)^2-α+(t_n-1-t_k)^2-α/τ_k M_h (U^k_hτ - U^k-1_hτ),the mass matrixM_h = h/6([ 4 1; 1 4 1; ⋱ ⋱ ⋱; 1 4 1; 1 4 ])_(M-1)×(M-1),the stiffness matrix A_h^β = (a^h_i,j)_(M-1)×(M-1) with its entries{a^h_i,i = h^1-2β(2^4-2β-8)/2cos(βπ)Γ(4-2β), i=1,⋯,M-1 a^h_j,j+1=a^h_j+1,j=h^1-2β(3^3-2β-2^5-2β+7)/2cos(βπ)Γ(4-2β), j=1,⋯,M-2 a^h_k,k+l= a^h_k+l,k = h^1-2β/2cos(βπ)Γ(4-2β)[(l+2)^3-2β-4(l+1)^3-2β+6l^3-2β-4(l-1)^3-2β+(l-2)^3-2β], k=1,⋯,M-l-1 .,the vector F^n_hτ=(f^n_1,f^n_2,⋯,f^n_M-1)^T, f^n_l=(f,ϕ_l× 1)_Q_n, l=1,⋯,M-1and the fully FE approximationsU^k_hτ=(u^k_1,u^k_2,⋯,u^k_M-1)^T, u^0_j=ψ_0,I(x_j), u^k_j=u_h^k(x_j), k=1,⋯,n, j=1,⋯,M-1.(<ref>) is reduced via dividing both sides of (<ref>) by the factor τ^1-α_n / Γ(3-α), in case of the severe loss in convergence of the fully discrete FE scheme. Next, a number ofcharacterizations are established regarding A_h^β just defined by (<ref>). The stiffness matrix A_h^β is symmetric and satisfies * a^h_i,i > 0 for i=1,⋯,M-1;* a^h_i,j < 0 for i j, i,j=1,⋯,M-1;* ∑_j=1^M-1 a^h_i,j>0 for i=1,⋯,M-1;* The following relation holds for the particular case when h≤1/7∑_j=1^M-1 a^h_i,j≥{-h^1-2β(4-2^3-2β)/2cos(βπ)Γ(4-2β), i=1,M-1-2^2βh(2β-1)/cos(βπ)Γ(2-2β), i=2,⋯,M-2 .; * A_h^β is an M-matrix. The symmetric property of A_h^β is an obvious fact by (<ref>). Since β∈(1/2,1), then 4-2β<3 and cos(βπ)<0, which give immediately a^h_ii > 0, i=1,⋯,M-1. This proves the first part of the theorem. The second part is an immediate consequence of the facts that on the interval β∈(1/2,1), f(β)=3^3-2β-2^5-2β+7 is a strictly increasing function, and the bivariate functionf_β(l)=(l+2)^3-2β-4(l+1)^3-2β+6l^3-2β-4(l-1)^3-2β+(l-2)^3-2β > 0, 2≤ l≤ M-2.In fact, it is evident that1.5^2β > 1.5 > 3^3/2^5ln3/ln2⇒ f'(β)=-2ln3 · 3^3-2β + 2ln2 · 2^5-2β > 0,andf_β(l)=h^2β-3[g(x_l+2)-4g(x_l+1)+6g(x_l)-4g(x_l-1)+g(x_l-2)] > 0using Taylor's expansion with(l/l+1)^2+2β-(l/l-1)^2+2β > -30l/2β+1⇒ l^-1-2β+2β+1/30[(l+1)^-2-2β-(l-1)^-2-2β] > 0,where g(x)=(x-a)^3-2β and x_l=lh+a.To prove the third part, use(_xD^β_Lϕ_i,_xD^β_Rϕ_j)_L^2(Ω) = -(_xD_L^2β-1ϕ_i,dϕ_j/dx)_L^2(Ω)andϕ̃:=∑_j=1^M-1ϕ_j=1-ϕ_0-ϕ_Mto obtain the relation∑_j=1^M-1 a^h_i,j=-(_xD_L^2β-1ϕ̃,1)_Ω_i-(_xD_L^2β-1ϕ̃,1)_Ω_i+1 +(_xD_L^2β-1ϕ_i,1)_Ω_1-(_xD_L^2β-1ϕ_i,1)_Ω_M_h/2hcos(βπ),where_xD^2β-1_Lϕ̃={ (x-a)^2-2β/hΓ(3-2β), a < x < x_1 (x-a)^2-2β-(x-x_1)^2-2β/hΓ(3-2β), x_1 < x < x_M_h-1(x-a)^2-2β-(x-x_1)^2-2β-(x-x_M_h-1)^2-2β/hΓ(3-2β), x_M_h-1 < x < x_M_h.and_xD_L^2β-1ϕ_i(x)={0, x < x_i-1(x-x_i-1)^2-2β/hΓ(3-2β), x_i-1 < x < x_i (x-x_i-1)^2-2β-2(x-x_i)^2-2β/hΓ(3-2β), x_i < x < x_i+1(x-x_i-1)^2-2β-2(x-x_i)^2-2β+(x-x_i+1)^2-2β/hΓ(3-2β), x > x_i+1..Assume that Ω=(0,1) without loss of generality, one can easily derive∑_j=1^M-1 a^h_i,j=-(4-2^3-2β)h^3-2β-1+3(1-h)^3-2β -3(1-2h)^3-2β+(1-3h)^3-2β/2cos(βπ)h^2Γ(4-2β), i=1,M-1,∑_j=1^M-1 a^h_i,j=-3(ih)^3-2β-3[(i-1)h]^3-2β +[(i-2)h]^3-2β-[(i+1)h]^3-2β/2cos(βπ)h^2Γ(4-2β) - 3(1-ih)^3-2β-[1-(i-1)h]^3-2β-3[1-(i+1)h]^3-2β +[1-(i+2)h]^3-2β/2cos(βπ)h^2Γ(4-2β), i=2,⋯,M-2and deduce ∑_j=1^M-1 a^h_i,j>0 by Taylor's formula and β∈(1/2,1).Another step to do in the proof is the result 4, which follows fromh≤1/7⇒7-4(1-ξ)^-1-2β > -1/β h⇒-1+3(1-h)^3-2β-3(1-2h)^3-2β+(1-3h)^3-2β>0for all ξ∈(0,2h),(i-1/i)^3+2β > β+1/21(β+1)+30i⇒3(ih)^3-2β-3(ih-h)^3-2β+(ih-2h)^3-2β- (ih+h)^3-2β/h^3(3-2β)(2-2β)(2β-1)(ih)^-2β>1and3(1-ih)^3-2β-[1-(i-1)h]^3-2β-3[1-(i+1)h]^3-2β +[1-(i+2)h]^3-2β/h^3(3-2β)(2-2β)(2β-1)(1-ih)^-2β>1for i=2,⋯,M-2, together with the inequality (ih)^-2β+(1-ih)^-2β≥ 2^1+2β.Finally, according to properties 1 and 2, the result 5 will be proved by showing that (A_h^β)^-1 is nonnegative, which can be easily proved by contradiction with property 3. Observe from (<ref>)-(<ref>) that M_h and A_h^β are both symmetric Toeplitz matrices independent of any time terms. The under-mentioned corollaries are natural consequences of Theorem <ref>. The coefficient matrix C^n_hτ is a symmetric Toeplitz matrix. Furthermore, it will be independent of time level n if the temporal discretization is also uniform.The coefficient matrix C^n_hτ is an M-matrix, if and only ifτ_n^α/h^2β > -2cos(βπ)Γ(4-2β)/3Γ(3-α)(3^3-2β-2^5-2β+7). This result will follow from Theorem <ref>, if we can show thath/6+Γ(3-α)/2τ^α_n h^1-2β(3^3-2β-2^5-2β+7)/2cos(βπ)Γ(4-2β)<0,which is an immediate application of the condition (<ref>).§.§ Numerical experiments and the saturation error orderConsider (<ref>)-(<ref>) with Ω=(0,1), T=1, ψ_0(x)=0 andf(x,t) =Γ(3-α)/Γ(3-2α)t^2-2αx^2(1-x)^2 + t^2-α/cos(βπ)[x^2-2β+(1-x)^2-2β/Γ(3-2β)- 6x^3-2β+6(1-x)^3-2β/Γ(4-2β) + 12x^4-2β+12(1-x)^4-2β/Γ(5-2β)].The exact solution is u(x,t)=t^2-αx^2(1-x)^2. In the case of uniform temporal and spatial meshes, Tables <ref> and <ref> present errors e_0 := u(·,1)-u_hτ(·,1)_L^2(Ω) and convergence rates. From Tables <ref> and <ref>, we can obtain that the fully FE solution u_hτ achieves the saturation error order 𝒪(τ^2+h^2) under ·_0 norm.Fig. <ref> illustrates the comparisons of exact solutions and numerical solutions of α=0.2, 0.4 and β=0.6, 0.8 with t=1 and h=τ=1/32.§ CONDITION NUMBER ESTIMATION This section is devoted to deriving the condition number estimation on the coefficient matrix of (<ref>) in uniform temporal and and spatial discretizations. For the linear system (<ref>), we haveκ(C^n_hτ)= 1+𝒪(τ^α h^-2β). Let C_α = Γ(3-α)/2, we divide our proof in three steps. First, it is trivially true that C^n_hτ is spectrally equivalent to the matrix I+C_ατ^α M_h^-1/2A_h^β M_h^-1/2, i.e.κ(C^n_hτ)≃κ(I+C_ατ^α M_h^-1/2A_h^β M_h^-1/2). The next thing to do in the proof is to verifyλ_min(M_h^-1/2A_h^β M_h^-1/2)≳ 1, λ_max(M_h^-1/2A_h^β M_h^-1/2)≲ h^-2β,which is equivalent to(v⃗_h,v⃗_h)≲ (M_h^-1/2A_h^β M_h^-1/2v⃗_h,v⃗_h) ≲ h^-2β(v⃗_h,v⃗_h), ∀v⃗_h∈ℝ^M-1.Set u⃗_h = M_h^-1/2v⃗_h := (u^h_1,⋯,u^h_M-1)^T, rewrite (<ref>) as (M_hu⃗_h,u⃗_h)≲ (A_h^βu⃗_h,u⃗_h)≲ h^-2β(M_hu⃗_h,u⃗_h). It is sufficient to verify that (M_hu⃗_h,u⃗_h)≃ h(u⃗_h,u⃗_h). It follows by (<ref>) and the Cauchy-Schwarz inequality thath/3(u⃗_h, u⃗_h)≤(M_h u⃗_h, u⃗_h)=h[2/3∑_l=1^M-1(u^h_l)^2 +1/3∑_l=1^M-2u^h_lu^h_l+1]≤ h(u⃗_h, u⃗_h).Thus (<ref>) will follow if we can show that h(u⃗_h,u⃗_h)≲ (A_h^βu⃗_h,u⃗_h)≲ h^1-2β(u⃗_h,u⃗_h). We start by showing the second inequality. Utilizing Theorem <ref> and the Cauchy-Schwarz inequality, we arrive at(A_h^βu⃗_h, u⃗_h)≤ ∑_i=1^M-1 a^h_i,i (u^h_i)^2 -1/2∑_i=1^M-1∑_j i a^h_i,j [(u^h_i)^2 + (u^h_j)^2] = ∑_i=1^M-1 a^h_i,i (u^h_i)^2 - ∑_i=1^M-1∑_j=i+1^M-1 a^h_i,j (u^h_i)^2 - ∑_i=1^M-1∑_j=i+1^M-1 a^h_i,j (u^h_j)^2 = ∑_i=1^M-1 (u^h_i)^2 [a^h_i,i - ∑_j=i+1^M-1 a^h_i,j - ∑_j=1^i-1 a^h_i,j]≤2 a^h_1,1 (u⃗_h, u⃗_h) = (2^4-2β-8)/cos(βπ)Γ(4-2β) h^1-2β (u⃗_h, u⃗_h),which proves the second inequality. To prove the left inequality, set u_h:= Φ_hu⃗_h, we rewrite it as1/cos(βπ)(_xD^β_Lu_h,_xD^β_Ru_h)_L^2(Ω)=(A_h^βu⃗_h,u⃗_h) ≳ h(u⃗_h,u⃗_h)≃ (M_hu⃗_h,u⃗_h)= (u_h, u_h)_L^2(Ω)which can be deduced by (<ref>)-(<ref>), where Φ_h = (ϕ_1,⋯,ϕ_M-1).Finally, we have to show thatκ(I+C_ατ^α M_h^-1/2A_h^β M_h^-1/2) = λ_max(I+C_ατ^α M_h^-1/2A_h^β M_h^-1/2)/λ_min(I+C_ατ^α M_h^-1/2A_h^β M_h^-1/2)= 1+C_ατ^αλ_max(M_h^-1/2A_h^β M_h^-1/2)/1+C_ατ^αλ_min(M_h^-1/2A_h^β M_h^-1/2)≤1+C_α(2^4-2β-8)/cos(βπ)Γ(4-2β)τ^α h^-2β.This completes the proof based on the spectral equivalence relation (<ref>).The estimation (<ref>) is compatible with the correlative result 1+𝒪(τ h^-2) of integer order parabolic differential equations. An important particular case of Theorem <ref> is singled out in the following corollary. Let τ be proportional to h^μ with μα≥2β. Thenκ(C^n_hτ)=𝒪(1).In what follows, we examine the correctness of (<ref>) concerning Example <ref> with typical α and β for three specific cases: τ = h, τ = h^2 and τ is fixed (doesn't change along with h). In under-mentioned tables, λ_min and λ_max respectively indicate the smallest and largest eigenvalues, κ represents the condition number and ratio is the quotient of the condition number in fine grid divided by that in coarse grid.It is observed from Tables <ref>-<ref> that numerical results are in good agreement with our theoretical estimation.§ AMG'S CONVERGENCE ANALYSIS AND AN ADAPTIVE AMG METHOD Within the section, involving FFT to perform Toeplitz matrix-vector multiplications, we introduce the so-called Ruge-Stüben or classical AMG method <cit.> with low algorithmic complexity, fulfill its theoretical investigation, and then propose an adaptive AMG method through Corollary <ref>. The classical AMG method for the linear system (<ref>).Step 1 Perform the Setup phase to the coefficient matrix C^n_hτ.1.1 Set the strength-of-connection tolerance θ; 1.2 Build the ingredients required by a hierarchy of levels, coarsest to finest, including the grid transfer operator P. Step 2 Invoke the classical V(ϱ_1,ϱ_2)-cycle to solve (<ref>) until convergence. Below is the description of two-grid V(ϱ_1,ϱ_2)-cycle.2.1 Do ϱ_1 pre-smoothing steps on (<ref>); 2.2 Compute and restrict the residual: r^c=P^T(G^n_hτ - C^n_hτU^n_hτ); 2.3 Solve the residual equation on coarse level: (P^T C^n_hτ P) e^c = r^c; 2.4 Interpolation and correction: U^n_hτ=U^n_hτ+Pe^c; 2.5 Do ϱ_2 post-smoothing steps on (<ref>).In pre- and post-smoothing processes, damped-Jacobi iterative methods are favorable choices, which can maintain the low computational cost 𝒪(Mlog M) calculated by FFT. For theoretical investigations, we rewrite (<ref>) and the grid transfer operator P in block form regarding a given C/F splittingC^n_hτU^n_hτ = ( [ A_FF A_FC; A_CF A_CC ]) ( [ u_F; u_C ]) =( [ f_F; f_C ]) = G^n_hτ, P=( [ I_FC; I_CC ]),and introduce the following inner products(u_F,v_F)_0,F = (D_FFu_F,v_F), (u,v)_1 =(C^n_hτu,v), (u,v)_2 =(D^-1_hτC^n_hτu,C^n_hτv)with their associated norms ·_0,F=√((·,·)_0,F) and ·_i=√((·,·)_i) (i=1,2), where I_CC is the identity operator, D_FF= diag(A_FF) and D_hτ= diag(C^n_hτ).For simplicity, we here denote C^n_hτ=(c_ij)_(M-1)×(M-1), and only consider the two-grid V(0,1)-cycle, whose iteration matrix has the formM_h,H=S[I-P(P^T C^n_hτ P)^-1P^TC^n_hτ],where S is a relaxation operator usually chosen as damped-Jacobi or Gauss-Seidel iterative method.Combining Corollary <ref> and the two-level convergence theory in the work <cit.>, leads to the following lemmas and theorem. Under the condition (<ref>), for all e_h∈ℝ^M-1, damped-Jacobi and Gauss-Seidel relaxations satisfy the smoothing propertySe_h^2_1≤e_h^2_1-σ_1e_h^2_2with σ_1 independent of e_h and step sizes h and τ_n.On the strength of Theorem A.3.1 and A.3.2 in <cit.>, we produce that damped-Jacobi relaxation with parameter 0<ω<2/η satisfies (<ref>) withσ_1=ω(2-ωη),and Gauss-Seidel relaxation satisfies (<ref>) withσ_1=1/(1+γ_-)(1+γ_+),both independent of e_h, whereη≥ρ(D^-1_hτC^n_hτ), γ_-=max_i{1/w_ic_ii∑_j<iw_j|c_ij|}, γ_+=max_i{1/w_ic_ii∑_j>iw_j|c_ij|},and w=(w_i) is an arbitrary positive vector with C^n_hτw being also positive. By exploiting (<ref>)-(<ref>), the assumption (<ref>) and Corollary <ref>, we conclude that C^n_hτ is strictly diagonally dominant. Recall that C^n_hτw is a positive vector, yield γ_-<1, γ_+<1 andρ(D^-1_hτC^n_hτ)≤|D^-1_hτC^n_hτ|_w=max_i{1/w_i∑_jw_j|c_ij|/c_ii}<2,which implicitly mean that η, γ_- and γ_+ can be chosen to be independent of h and τ_n, and complete the proof.The inequality (<ref>) implies that there exists ϵ>0 such that ρ(D^-1_hτC^n_hτ)=2-3ϵ. Then η=2-2ϵ>ρ(D^-1_hτC^n_hτ) and hence the upper bound of parameter ω: 2/η=1/(1-ϵ)>1, which suggests that Jacobi relaxation with ω=1 is available in such a case.For all symmetric M-matrices, σ_1≤1/η<1 holds for damped-Jacobi relaxation, while σ_1∈(1/4,1) for Gauss-Seidel relaxation.Under the condition (<ref>) and a given C/F splitting, for all e_h=(e_F^T,e_C^T)^T∈ℝ^M-1, the direct interpolation I_FC satisfiese_F-I_FCe_C^2_0,F≤σ_2e_h^2_1with σ_2 independent of e_h, h and τ_n.According to Theorem A.4.3 in <cit.>, I_FC satisfies (<ref>) with σ_2 of the form regarding a given C/F splittingσ_2 ≥max_i∈ F{∑_j∈ N_ic_ij/∑_j∈ C_ic_ij}independent of e_h, where N_i={j i:c_ij0}, C_i is the subset of N_i whose values will be used to interpolate at F-point i. As a result of (<ref>) and the fact that c_ij (j∈ N_i) are all negative, the following relation holds: σ_2 > 1.Notice here that the classical Ruge-Stüben based coarsening strategy generates at least one of points i-1 and i+1 to be C-points and strongly influence i — viz. it retains i-1∈ C_i or i+1∈ C_i. Therefore, it can be seen that∑_j∈ N_ic_ij/∑_j∈ C_ic_ij<-c_ii/c_ii-1=-c_ii/c_ii+1 =-16cos(βπ)Γ(4-2β)+6Γ(3-α)τ^α_nh^-2β(2^4-2β- 8)/4cos(βπ)Γ(4-2β)+6Γ(3-α)τ^α_nh^-2β(3^3-2β-2^5-2β+7),indicating that σ_2 is independent of h and τ_n by plugging (<ref>), and thus prove the theorem.Let any C/F splitting be given. Under the condition (<ref>), there exist positive constants σ_1 and σ_2 independent of h and τ_n and satisfying σ_2>1>σ_1, such that a uniform two-grid convergence is achieved as followsM_h,H_1≤√(1-σ_1/σ_2). The proof of this result is straightforward and is based on Theorem A.4.1 and A.4.2 in <cit.>, Lemma <ref> and <ref>. Now observe from Theorem <ref> that, despite the independence of h and τ_n, σ_2 relies ruinously on θ in Step 1.1 of Algorithm <ref> due to the fact that C^n_hτ is nearly dense leading to a quite complicated adjacency graph. In addition, it is found that σ_2(θ) may be much larger than 1 as θ approaches zero, with that comes a sharp pullback in convergence rate. Hence, an appropriate θ is a critical component of Algorithm <ref> to handle fractional diffusion equations.We now turn to reveal a reference formula on θ. Note the heuristic that the distribution of ratios of off-diagonal elements relative to the maximum absolute off-diagonal element (namely the minor diagonal element for C^n_hτ) plays a major role in the choice of θ. Since C^n_hτ is a symmetric Toeplitz matrix from Corollary <ref>, its first row involving all off-diagonal elements of C^n_hτ is deserved to be the representative row. Taking β=0.8 as an example, Fig. <ref> shows the distribution of the ratios c_1j/c_12 (j≥2), which reminds us of the attenuation in off-diagonal elements, statesc_13/c_12≈ 0.160426, c_1j/c_12 < c_14/c_12≈ 0.034394, j=5,6,⋯,M-1,and suggests that c_1j (j≥4) should be viewed as weak couplings (wouldn't be used for interpolation) because they are less than 5% of c_12. Besides, for a better complexity and higher efficiency, only the nearest neighbors are potentially used to limit the interpolation matrix on each grid level to at most 3 coefficients per row, although c_13 reaches around 16% of c_12.It thus appears that the strength-of-connection tolerance θ should be of the formθ = c_13/c_12+ϵ_0,where ϵ_0 is some small number, which can be chosen to be 10^-5 in one-dimensional realistic problems.As is known, Algorithm <ref> is much more expensive for well-conditioned problems than basic iterative techniques, such as conjugate gradient (CG) or (plain) Jacobi iterative method. For the purpose of solving (<ref>) in an optimal way, an adaptive AMG method is proposed below by combining Algorithm <ref>, the reference formula (<ref>) and the condition number estimation (<ref>) in Corollary <ref> as the clear distinction to adaptively pick an appropriate solver. An adaptive AMG method S_ad for the linear system (<ref>).Step 1 If the condition (<ref>) is unsatisfied, then goto Step 2, else set S_ad as the CG or Jacobi iterative method; Step 2 Set S_ad as the classical AMG method described in Algorithm <ref>, with θ chosen via the reference formula (<ref>). § PERFORMANCE EVALUATION Let us illustrate the effectiveness of Algorithms <ref> and <ref>. Numerical experiments are performed in a 64 bit Fedora 18 platform, double precision arithmetic on Intel Xeon (W5590) with 24.0 GB RAM, 3.33 GHz, with an -O2 optimization parameter. In the following tables, dashed entries (-) indicate the solutions either diverge or fail to converge after 1000 iterations, Its is the number of iterations until the stopping criterion 10^-12 is reached, T_c represents the CPU time including both Setup and Solve phases with second as its unit, C_g and C_o respectively denote grid and operator complexities, which are defined as sums of the number of degrees of freedom and nonzero elements on all grid levels divided by those of the finest grid level, and used as measures for memory requirements, aritmetic operations and the execution time in Setup and Solve phases. Comparisons of the classical AMG over CG and Jacobi iterative methods for the case when (<ref>) is satisfied with two different fractional orders. As expected, the results in Table <ref> show that Jacobi, CG and AMG methods are robust with respect to the mesh size and fractional order, which indicates indirectly the correctness of (<ref>). In addition, CG method runs 3.28 and 3.03 times faster than Jacobi and AMG methods for M=4096 and α=β=0.6, respectively. Comparisons between the classical AMG method and CG method for the case when (<ref>) is unsatisfied. As shown in Table <ref>, AMG method converges robustly regarding to the mesh size and may be weakly dependent of β, while the number of iterations of CG method is quite unstable, and sometimes CG method even break down. Furthermore AMG method runs 12.24 times faster than CG method for M=4096 and β=0.8.Table <ref> shows the results of τ=h. Despite the advantage in computational cost and robustness over CG method, AMG method is nearly independent of α and β in this circumstance. Meanwhile, by an investigation in terms of number of iterations in Tables <ref> and <ref>, CG method converges faster because of the improvement in condition number from 𝒪(h^-2β) to 𝒪(h^α-2β). Comparisons of S_ad over the classical AMG and CG methods when the i-th time step size τ_i is chosen to beτ_i = {h^2, i=1,⋯,K_11/32, i=K_1+1,⋯,K_1+K_2 ..We can observe from Table <ref> that S_ad and AMG methods are fairly robust as to the mesh size, roughly 10 and 6 on the average. Yet the average number of iterations of CG method varies from 85 to 142. Moreover S_ad has a considerable advantage over others in CPU time, runs 1.72 and 6.09 times faster than AMG and CG methods for M=2048 and K_2=100. Analyze effects of the strength-of-connection tolerance θ on the performance of the classical AMG method. It is seen from Tables <ref> and <ref> that there is a unique threshold θ_0 independent of h which guarantees the robustness of the classical AMG method, and makes number of iterations of the classical AMG monotonically decreasing when θ<θ_0, or even the classical AMG possibly diverge when θ is small enough, e.g., θ_0=0.16043 and θ_0=0.00685 for cases β=0.8 and β=0.99. By direct calculations, we have c_13/c_12≈ 0.160426 and c_13/c_12≈ 0.006846. Utilizing the relation (<ref>) and ϵ_0=10^-5, the corresponding values of θ are respectively larger than those of θ_0. This confirms the reasonability of the reference formula (<ref>).§ CONCLUSION In this paper, we propose the variational formulation for a class of time-space Caputo-Riesz fractional diffusion equations, prove that the resulting matrix is a symmetric Toeplitz matrix, an M-matrix by appending a very weak constraint and its condition number is bounded by 1+𝒪(τ^α h^-2β), introduce the classical AMG method and prove rigorously that its convergence rate is independent of time and space step sizes, provide explicitly a reference formula of the strength-of-connection tolerance to guarantee the robustness and predictable behavior of AMG method in all cases, and develop an adaptive AMG method via our condition number estimation to decrease the computation cost. 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"authors": [
"Xiaoqiang Yue",
"Weiping Bu",
"Shi Shu",
"Menghuan Liu",
"Shuai Wang"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170726095542",
"title": "Fully Finite Element Adaptive Algebraic Multigrid Method for Time-Space Caputo-Riesz Fractional Diffusion Equations"
} |
.1mps*C[1]>p#1 =8pt [1] 1.1@widthequationsection ḍ d Tr Ø OD̅ḍ d Tr WD̅Φ̅ϕ̅R̅M̅F̅α̇β̇δ̇γ̇M_ plḍłℓ aḇH̋ GΓ^† C 0k̨qpsv̌xyzPŭn∇k_ NLf_ NL M A I J K [4] =15.5pt empty2028 Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals 1330 Marko Simonović,^ a Tobias Baldauf,^ b Matias Zaldarriaga,^ aJohn Joseph Carrasco^ c and Juna A. Kollmeier^ d 8pt ^ aSchool of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive,Princeton, NJ 08540, United States 2pt ^ bDAMTP, Center for Mathematical Sciences, Wilberforce Road,Cambridge, CB3 0WA, United Kingdom2pt ^ cInstitute of Theoretical Physics (IPhT), CEA-Saclay and University of Paris-SaclayF-91191 Gif-sur-Yvette cedex, France2pt ^ dThe Observatories of the Carnegie Institution for Science, 813 Santa Barbara St,Pasadena, CA 91101, United States 7pt AbstractWe present a new method for calculating loops in cosmological perturbation theory. This method is based on approximating a ΛCDM-like cosmology as a finite sum of complex power-law universes. Thedecomposition is naturally achieved using an FFTLog algorithm. For power-law cosmologies, all loop integrals are formally equivalent toloop integrals of massless quantum field theory. These integrals have analyticsolutions in terms of generalized hypergeometric functions. We provide explicit formulae for the one-loop and the two-loop power spectrum and the one-loop bispectrum. A chief advantage of our approach is that the difficult part of the calculation is cosmology independent, need be done only once, and can be recycled for any relevant predictions. Evaluation of standard loop diagrams then boils down to a simple matrix multiplication. We demonstrate the promise of this method for applications to higher multiplicity/loop correlation functions.10pt § INTRODUCTIONCosmological perturbation theory (PT) <cit.> offers a complementary approach to predicting and interrogating large-scale structure (LSS) observables in the weakly non-linear regime.While many investigations in the literature are aimed at understanding and exploiting the highly non-linear regime (e.g. galaxy and cluster formation), it is clear that upcoming experiments (e.g. CMB-S4, DESI, CHIME) will also provide voluminous datasets probingmatter distribution on very large scales. As we continue in this “era of precision cosmology", it is critical to exploit this investment of resources, to the fullest extent possible, in order to deliver on the promise of these upcoming surveys to measure cosmological parameters and potentially probe new physics.The PTapproach, when recognized as a classical effective field theory <cit.>, has a number of important advantages.It converges to the correct answer for clustering statistics on large scales as more orders are included.Its errors are parametrically controlled, so the typical size of the deviations from the correct answer can be easily estimated.For the scales of relevance, it is rapidly computable compared to the necessarily large and high-resolution cosmological simulations required to attempt adequate comparison of theory andlarge-scale data.Simulation boxes are computed with a single cosmology and thus many computations are required to investigate small changes in cosmological parameters, the effects of cosmic variance, and to perform consistency checks of scheme independence. PT works around these issues entirely.Indeed it can serve as a spectacular large-scale (IR) complement to simulations, allowing them to focus their power in the incredibly non-linear smaller-scale (UV) regimes where they excel. In the PT approach one treats dark matter and baryons as non-ideal self-gravitating fluids. At early times or on large scales these fluids are nearly homogeneous with small density fluctuations. This allows for the equations of motion to be solved perturbatively. The rigorous foundation of PT as an effective field theory oflarge scale structure (EFTofLSS) was recently made <cit.>, although many important results were known for a long time (for a review see <cit.>). One feature of these perturbative solutions is that they convolve initial density fields. Therefore observables, such as correlation functions of density-contrast, oroverdensity, are efficiently written as momentum integrals over a certain number of initial power spectra. These integrals are refereed to as loop integrals. They admit a graphical organization which is why the atomic units of loop contributions to correlation functions are often referred to as loop diagrams.Higher multiplicity/loop correlation calculations are critical not only to extending the scale of relevance of the analytic approach, but to break degeneracies and optimally extract cosmological parameters from the real data.While calculating loop integrals is a straightforward task in principle, the computational cost of exact solution becomes prohibitive for higher points (multiplicity) as well as higher loop order corrections. Higher multiplicity kernels quickly becomecomplicated and every loop brings additional three-dimensional integral. The linear power spectrum that appears in the integrand for real universe applications is known only as a numerical function which makes analytic solution of non-trivial integralsimpossible. Naive numerical integration, to desired precision, on the other hand, quickly becomes slow even with advanced Monte Carlo methods, due to the growth in dimensionality.This poses a direct challenge to our ability to interrogate large datasets and one that merely more and faster computers will not address. In order to simplify and speed up loop calculations we require new ideas, new strategies, to approach the problem. One inspiring idea, developed in <cit.> and <cit.>, is to use Fast Fourier Transform (FFT) for efficient evaluation of the one-loop power spectrum. After first “deconvolving” the lowest order PT solutions, and performing all angular integrals, the one-loop expressions reduce to a set of simple one-dimensional integrals that can be efficiently evaluated using FFT. Unfortunately, deconvolving higher order perturbative solutions and extending this approach to the one-loop bispectrum or the two-loop power spectrum proves to be challenging <cit.>.In this paper we build on ideas of <cit.> but choose a slightly different strategy which allows us to go beyond the one-loop power spectrum. Let us briefly sketch the main idea behind our proposal. Prior to doing any integrals, the linear power spectrum is expanded as a superposition of ideal self-similar power-law cosmologies. This is naturally accomplished using FFT in log k. Given some range of wavenumbers of interest, from k_ min to k_ max, the approximation for the linear power spectrum with N sampling points is <cit.>P̅_ lin(k_n) = ∑_m=-N/2^m=N/2 c_m k_n^ν+iη_m , where the coefficients c_m and the frequencies η_m are given byc_m = 1/N∑_l=0^N-1 P_ lin(k_l)k_l^-νk_ min^-iη_me^-2π i m l /N , η_m = 2π m/log (k_ max/k_ min) .Notice that the we denote the approximation for the linear power spectrum with P̅_ lin(k), while eq. (<ref>) uses the exact linear power spectrum P_ lin(k) to calculate the coefficients c_m. We will keep using the same notation throughout the paper. The parameter ν is an arbitrary real number. As we will see, the simplest choice ν=0 is insufficient in some applications, so we will use the more general form of the Fourier transform. In the terminology of <cit.> we call this ν parameterbias. Note that the powers in the power-law expansion are complex numbers. In practice, even a small number of power-laws, 𝒪(100), is enough to capture all features of the linear power spectrum including the BAO wiggles. One important thing to keep in mind is that the Fourier transform produces the power spectrum that is periodic in log k. Therefore, we will take care to choose k_ min and k_ max such that we cover the range of scales where we actually care about the value of the power spectrum. In other words we are choosing the momentum range where the loop integrals have the most of the support. However, one always has to be careful about possible contributions particularly from high k modes or short scales. Is this a limitation?Absolutely not.At the heart of the EFT understanding is the simple recognition that the PT idealized description of satisfying fluid-like equations of motion can only be valid at certain scales.This is much the same as the hydrodynamic description of liquid water is only valid at certain scales. Attempting to integrate this approximation over scales outside of its validity introduces non-parametrically controlled errors.Instead the information in the linear approximation must be supplemented by small-scale UV physics. This data is encoded in physical parameters like speed of sound or viscosity – potentially any dimensionally consistent operators. Such EFT parameters serve two roles.They must eat up any cutoff-dependence, by definition non-physical, and they must accurately represent the integrating out of any small-scale degrees of freedom.So baked into the framework that places PT on a rigorous footing is the realization that any integrals of the linear approximation should only be performed over a range of scales consistent with its validity. Notice that the decomposition (<ref>) reduces the evaluation of a loop diagram for an arbitrary cosmology to evaluation of the same diagram for a set of different power-law universes with numerical coefficients. Power-law momentum integralscan be done analytically. The final answer is a sum of familiar special functions which are straightforward to evaluate. In the simplest case of the one-loop power spectrum, the momentum integral for a power-law universe can be expressed entirely in terms of gamma functions <cit.>. Looking at higher order correlators an interesting pattern emerges. The perturbation theory kernels can always be written such that the general form of loop integrals in a power-law cosmology is formally identical to the one of a massless Quantum Field Theory (QFT) with cubic interactions.[More precisely, it is a QFT in three dimensions with the Euclidean signature.] This is just a formal relationship, but it should prove rather useful. Many results recently developed in the theory of scattering amplitudes can be applied to LSS correlation functions as well. Some steps in this direction have already been made for the one-loop bispectrum in <cit.>.In this work we derive formulas for the one-loop bispectrum and the two-loop power spectrum in power-law cosmologies which are suitable for effecient numerical evaluation. Generically, the higher multiplicity/loop correlation functions are expressed in terms of the generalized hypergeometric functions. One thing to keep in mind is that the powers ν+iη_m are complex and one has to be careful about the analytic continuation of all results to the entire complex plane. Before diving into the details, let us comment on one important virtue of our method. The decomposition (<ref>) is useful because it separates the cosmology dependent portion, encoded entirely in the coefficients c_m, from the loop calculations which have been reduced to that of much more tractable, ideal, cosmologies. This means that for the fixed value of bias ν, the momentum range (k_ min,k_ max) and the number of sampling points N the difficult part of the calculation which involves momentum integrals can be done only once, saved as a table of numbers and then used for any cosmology. As we will see, the evaluation of the contribution of loop diagrams reduces to a simple (small) matrix multiplication and it is very fast. This opens up the possibility of using our method in Markov chain Monte Carlo parameter estimation. In the rest of the paper we focus on three examples: the one-loop power spectrum, the one-loop bispectrum and the two-loop power spectrum. We will present our calculations in detail and for one-loop diagrams compare them with the standard numerical results. We leave a detailed comparison with the numerical two-loop power spectrum for future work. AMathematica notebook used to produce plots is available as an auxiliary file associated with the preprint of the paper onarXiv. § ONE-LOOP POWER SPECTRUMLet us first consider the simplest case—the one-loop power spectrum. In perturbation theory there are two different one-loop contributions. Using the usual approximation in which the time dependence is separated from k dependence (for a review see <cit.>), the one-loop power spectrum readsP_ 1-loop(k,τ) = D^4(τ)[P_22(k)+P_13(k)],where τis conformal time, D(τ) is the growth factor for matter fluctuations and the two terms in the square brackets are given byP_22(k) = 2 ∫_ F_2^2(, -̨) P_ lin(q) P_ lin( |-̨| ),P_13(k) = 6P_ lin(k) ∫_ F_3(, -, )̨ P_ lin(q),where ∫_≡∫d^3q/(2π)^3. Diagrammatic representation of these two contributions is shown in Fig. <ref>. The explicit form of kernels F_n can be calculated using well-known recursion relations <cit.>. One important point is that it is always possible to expand kernels in (<ref>) and (<ref>) in integer powers of k^2, q^2 and |-̨|^2. For example,F_2(,-̨) = 5/14 +3 k^2/28 q^2 +3 k^2/28 |-̨|^2-5 q^2/28 |-̨|^2-5 |-̨|^2/28 q^2 +k^4/14 |-̨|^2 q^2.A similar expression can be found for F_3(,-,)̨.[In the expansion of F_3(,-,)̨ some terms contain |+̨|^2. Given that the kernels are always integrated over , one is allowed to do the following change of coordinates →- and bring these terms to the same form as in (<ref>)] If we further decompose P_ lin(k) in power laws using (<ref>), the one-loop power spectrum becomes a sum of simple momentum integrals of the following form ∫_1/q^2ν_1|-̨|^2ν_2≡ k^3-2ν_12(ν_1,ν_2),where ν_1 and ν_2 are in general complex numbers.As we already mentioned, the form of the integral is identical to the one-loop massless two point function in QFT. The only difference is that in this case the powers of the “propagators" are complex numbers rather than integers. Still, the unknown dimensionless function (ν_1,ν_2) can be easily calculated using the standard technique with Feynman parameters. The result is a well known expression <cit.>(ν_1,ν_2) = 1/8π^3/2Γ(32 - ν_1) Γ(32 - ν_2) Γ(ν_12 - 32)/Γ(ν_1) Γ(ν_2) Γ(3 - ν_12) ,were ν_12=ν_1+ν_2 (throughout the paper we adopt the following notation ν_1...n≡ν_1+⋯ +ν_n). Notice that, thanks to the analytic continuation, (ν_1,ν_2) gives a finite answer even for the values of parameters for which the integral is formally divergent. In practice, breaking the loop calculation into many pieces can lead to some divergent terms. However, as long as thetotal sum is well defined and finite, for at least some power-law cosmology P_ lin(k)∼ k^ν, by analytic continuation it is guaranteed that eq. (<ref>) gives the correct answer.Sometimes the condition that the integral at hand is convergent for at least some power-law power spectrum cannot be met, and one has to use eq. (<ref>) with some care. For example, the function (ν_1,ν_2) vanishes if one of the arguments is zero (or a negative integer). Applying (<ref>) blindly would lead in these cases to paradoxical results. For instance, after power-law decomposition of the linear power spectrum, eq. (<ref>) would imply∫_0^∞ dq P_ lin(q) = 0,which is obviously the wrong answer. This is a consequence of the well known statement that in dimensional regularization all power-law divergences vanish: ∫_ q^ν = 0.[More precisely, this integral is related to a delta function <cit.>. Achange of coordinates relates ∫_1/q^3+2ν_1 = i/2π^2δ(ν_1).To get the consistent results one can use this equation. In practice, there is a much simpler way, as described in the main text.]Similar issues can appear in calculating loop diagrams. Luckily, for a ΛCDM-like cosmology, they can be always easily fixed. Let us imagine that the integral we are interested in is divergent for a given bias ν. Then, if the integral diverges in the UV(IR), one has to find the UV(IR) limit of the integrand. This can be easily done fixing all external momenta and sending the loop momentum to infinity(zero). This limit always has the form of eq. (<ref>) and it would be set to zero by dimensional regularization. Therefore, to get the correct answer, one simply has to add the UV(IR) contribution by hand. In the following sections we will give more details for each specific case we consider. Let us also point out that all UV divergences have a well defined momentum dependence. This momentum dependence is the same as for the counterterms in the EFTofLSS. Therefore, one can proceed without explicitly adding the UV-dependent terms to the loop calculation. The only effect of this choice is to change the usual values of the counterterms. In this sense we can say that eq. (<ref>) calculates only the “finite” part of the loop integral. As expected, the counterterms absorb all UV-dependent pieces. 10pt§.§ Symmetries of (ν_1,ν_2) and Recursion RelationsBefore moving on to applications, it is instructive to take a closer look at symmetries of the integral in (<ref>). For the one-loop power spectrum this is a straightforward exercise. We use it to introduce some notation and derive a couple of results that will be very useful in more complicated cases, such as the one-loop bispectrum or the two-loop power spectrum.The most obvious symmetry of the integral (<ref>) is invariance under the shift →-̨. Consequently, the function (ν_1,ν_2) is symmetric in ν_1 and ν_2(ν_1,ν_2) = (ν_2,ν_1).As we will see later, there are similar transformations for higher multiplicity/order diagrams and they always lead to some permutation of parameters ν_i. We are going to call these kind of identitiestranslation formulas, because they are derived using translations in momentum space.The integral (<ref>) preserves its form under rescaling of momenta, but this does not lead to any non-trivial condition on (ν_1,ν_2). However, a more complicated rescaling, such as inversion of momenta, does lead to interesting results. For simplicity, let us choose $̨ to be a unit vector:=̨̨̂,̨̂^2=1. Under an inversion→/q^2different factors in the integrand transform in the following wayd^3q → d^3q/q^6 ,q^2 → q^-2 ,|̨̂-|^2 → |̨̂-|^2/q^2 .Usingthese transformations we can write (ν_1,ν_2) = ∫_1/q^2ν_1|̨̂-|^2ν_2 = ∫_1/q^2(3-ν_12)|̨̂-|^2ν_2 ,which immediately implies theinversion formula (ν_1,ν_2) = (3-ν_12,ν_2) . There is one more method to find nontrivial identities for massless loop integrals which is based on the following relation between real and momentum space 1/q^2ν = Γ(32-ν)/Γ(ν)π^-3/2 2^-2ν∫_1/x^3-2ν e^-i· ,where∫_≡∫ d^3x. Let us illustrate the main idea behind this method. The starting point is to close all external lines in order to form additional loops. This is equivalent to integrating over all external momenta. For example, for the one-loop power spectrum we can start from ∫_1/k^3 (ν_1,ν_2) = ∫_,̨1/q^2ν_1|-̨|^2ν_2k^2(3-ν_12) .Notice that this expression has a form of a two-loop vacuum diagram. We have chosen to multiply(ν_1,ν_2)with the1/k^3factor such that the whole integral is dimensionless. On the l.h.s. the function(ν_1,ν_2)does not depend onkand the integral trivially reduces to ∫_1/k^3 (ν_1,ν_2) = (ν_1,ν_2) 1/(2π)^3∫d^3k/k^3 .On the r.h.s. we can use (<ref>) and integrate overand$̨. The momentum integrals lead to two delta functions which can be used to do two integrals in real space. At the end of the day, we are left with the following expression (ν_1,ν_2) 1/(2π)^3∫d^3k/k^3 = 1/64π^9/2Γ(32 - ν_1) Γ(32 - ν_2) Γ(ν_12 - 32)/Γ(ν_1) Γ(ν_2) Γ(3 - ν_12)∫d^3x/x^3 ,from which result (<ref>) immediately follows. As we can see, the one-loop two-point function is simple enough that relation (<ref>) is sufficient to fix its form. For higher multiplicity/loop correlation functions this is not the case. The reason is that in those cases the real space integrals are not trivial anymore. However, it turns out that they alway have the same structure as the original momentum integrals. The only difference is that the parameters are shifted: ν_i→ν̃_i≡ 32-ν_i. It follows that there is always an identity which relates two functions with parameters ν_i and ν̃_i. We will refer to these identities asreflection formulas. Finally, let us present recursion relations which connect functions whose parameters differ by an integer. These relations can be always derived using the fact that the integral of a total derivative vanishes. For instance,∫_∂/∂ q_i( q_i/q^2ν_1|-̨|^2ν_2) = 0.Expanding the derivative we find(3-2ν_1-ν_2)(ν_1,ν_2)+ν_2[(ν_1,ν_2+1)-(ν_1-1,ν_2+1)]=0,and a similar relation in which ν_1 and ν_2 are exchanged. The importance of these identities is that they relate different terms in the expansion of kernels, such as (<ref>). As we will see in the explicit calculations of P_22 and P_13 diagrams, thanks to the recursion relations all terms in the expansion of the kernels (for fixed ν_1 and ν_2) can be evaluated using a single function (ν_1,ν_2).10pt§.§ Numerical Evaluation of the One-loop Power SpectrumIn this section we will apply eq. (<ref>) to the calculation of the one-loop power spectrum. We will first separately discuss P_22 and P_13 diagrams (see Fig. <ref>). P_22 diagram.—Let us begin by reviewing some properties of the P_22 diagram in a cosmology with P_ lin(k)∼ k^ν. In particular, we are interested in finding the powers ν for which the integral is convergent. In order to do that we have to find the asymptotic form of the integrand in the UV and the IR regime. The behavior of the F_2 kernel in these two limits is F_2(,-̨) →k/q ,q→ 0,F_2(,-̨) →k^2/q^2 ,q→∞ .It follows that P_22 diagram is convergent if -1<ν<1/2. If we choose bias in FFT to be in this range, then the integral in P_22 is finite for each term in the sum (<ref>) and using (<ref>) we are guaranteed to get the same answer as with the usual numerical evaluation.Before turning to results, let us write the explicit formula for P_22 diagram. Using (<ref>) and (<ref>) we can write the approximation to the P_22 diagram in the following wayP̅_22(k) = 2 ∑_m_1,m_2 c_m_1 c_m_2∑_n_1,n_2 f_22(n_1,n_2) k^-2(n_1+n_2)∫_1/q^2ν_1-2n_1|-̨|^2ν_2-2n_2 .In this expression c_m_1 and c_m_2 are the coefficients in (<ref>) and n_1 and n_2 are integer powers of q^2 and |-̨|^2 in the expansion of F_2^2(,-̨). Corresponding rational coefficients in this expansion are labeled by f_22(n_1,n_2) and they can be read off from (<ref>). The complex numbers ν_1 and ν_2 are given by ν_1 = -12(ν+iη_m_1)and ν_2 = -12(ν+iη_m_2).Using the solution for the momentum integral, expression (<ref>) can be further simplified and written in the following way P̅_22(k) = k^3∑_m_1,m_2 c_m_1k^-2ν_1· M_22(ν_1,ν_2) · c_m_2k^-2ν_2 ,where the matrix M_22(ν_1,ν_2) is given byM_22(ν_1,ν_2)= (32 - ν_12) (12 - ν_12) [ν_1ν_2 (98ν_12^2 - 14ν_12+ 36 ) - 91 ν_12^2 + 3ν_12 + 58]/196 ν_1 (1+ν_1) (12 - ν_1) ν_2 (1+ν_2) (12-ν_2) (ν_1,ν_2).As we already pointed out, only a single function (ν_1,ν_2) is sufficient to calculate the full diagram. Thanks to the recursion relations (<ref>), all terms from the expansion of F_2 kernels are encoded in the ν-dependent prefactor in matrix M_22(ν_1,ν_2). One can use eq. (<ref>) to calculate the P_22 diagram. The result is shown in Fig. <ref>. As expected, the agreement with the usual numerical integration is excellent. An important thing to notice is that the only cosmology dependence in (<ref>) is in the coefficients c_m. For a given number of sampling points N, bias ν and k_ min and k_ max, the matrix M_22(ν_1,ν_2) is fixed. This means that it can be calculated only once and saved as a table of numbers. The evaluation of the P_22 diagram for an arbitrary cosmology then boils down to doing one FFT to determine coefficients c_m, calculating a vector c_mk^-2ν for each k and a simple matrix multiplication (<ref>). So far we restricted ourselves to biases in the range -1<ν< 12. Outside this range one has to be more careful because the integrals are not convergent anymore and eq. (<ref>) is not guaranteed to give the correct answer. For example, for biases ν<-1 the integrals are divergent in the IR. The leading piece of the P_22 diagram in this limit can be calculated by fixing k and sending → 0 in the integrand. The result isP_22^ IR (k) = P_ lin(k) k^2 σ_v^2,where σ_v^2≡16π^2∫_0^∞ dq P_ lin(q). The integral in σ_v^2 would be set to zero by eq. (<ref>) and missed in the final answer. Therefore, to get a correct result, one simply has to add P_22^ IR (k) to eq. (<ref>) at the end of the calculation. Notice that we have kept only the leading IR divergence, which is enough for biases in the range -3<ν<-1. If the bias was even smaller, one would have to keep track of sufficient number of subleading IR divergences. Similarly, when 12<ν<32, the leading UV divergence that has to be added on the r.h.s. of eq. (<ref>) to get the correct result is P_22^ UV (k) = 9/196π^2k^4 ∫_0^∞ dq P_ lin(q)/q^2 . However, for very high or low values of biases, the momentum range (k_ min,k_ max) has to be very wide for the integrals to converge to correct values. This implies a large number of frequencies and it is not practical. P_13 diagram.—The asymptotic behavior of the F_3 kernel in the UV and the IR limit is the same F_3(, -, )̨→k^2/q^2 ,q→ 0or q→∞ .Consequently, P_13 diagram is divergent in the UV for ν>-1 and divergent in the IR for ν<-1. In other words, P_13 diagram is never finite in a power-law cosmology. For ν>-1, the only possible mismatch between the true answer and eq. (<ref>) comes from the UV part of the integral. For fixed k and taking the limit →∞ P_13^ UV (k) = -61/105 P_ lin(q) k^2 σ_v^2 .On the other hand, for ν<-1, the possible error comes from the IR limit[Notice that P_13 diagram has two IR divergences →0 and →$̨, which are combined in a single expression. P_13^ IR (k) = -P_ lin(k) k^2 σ_v^2.As we already mentioned, in dimensional regularization this type of the integrals would be set to zero by (<ref>). This implies that to get the correct values forP_13, depending on the choice of bias, we have to add either the UV or IR term by hand. Notice that here we are writing down only the leading IR and UV parts of theP_13diagram. For biasesν<-3orν>1, one would have to include the corresponding subleading terms as well. < g r a p h i c s >< g r a p h i c s > Two contributions to the one-loop power spectrum calculated using direct numerical integration and eq. (<ref>) and eq. (<ref>). Both plots are produced using ν=-1.6, N=150, k_ min=3· 10^-4 h Mpc^-1 and k_ max=180 h Mpc^-1. For this value of bias both P_22 and P_13 are very different from their standard values.Let us see how the formulas above work in practice. With the same notation as for theP_22diagram, we can writeP̅_13(k) = 6P_ lin(k) ∑_m_1 c_m_1∑_n_1,n_2 f_13(n_1,n_2) k^-2(n_1+n_2)∫_1/q^2ν_1-2n_1|-̨|^-2n_2 .Solving the momentum integral, this expression can be further simplifiedP̅_13(k) = k^3 P_ lin(k) ∑_m_1 c_m_1k^-2ν_1· M_13(ν_1) ,where the vectorM_13(ν_1)is given byM_13(ν_1) = 1 + 9 ν_1/4tan(ν_1 π)/28 π (ν_1 + 1) ν_1 (ν_1 - 1) (ν_1 - 2) (ν_1 - 3) .Notice that to eq. (<ref>) one has to add the UV or the IR part of the integral. For example, forν>-1, we plot the result in Fig. <ref>. As expected, onceP_13^ UV(k)is added to eq. (<ref>), the agreement with the usual numerical result is excellent.The full one-loop power spectrum.—So far we were trying to reproduce the usual numerical results for separate pieces of the one-loop power spectrum. However, only their sum is a well defined observable. Thanks to the Equivalence Principle the IR divergences cancel and thetotal one-loop power spectrum is convergent for the range of power laws-3<ν<-1<cit.>. This means that with the choice of bias in this range, the formulas above should lead to the correct answer forP_ 1-loop(k), without having to deal with the IR divergences explicitly. In Fig. <ref> we plot the one-loop power spectrum calculated in this way and show that our method indeed agrees with the usual numerical result. As expected, the separate termsP_13andP_22are rather different from their usual values (see Fig. <ref>). However, the “mistake” that eq. (<ref>) makes in assigning some finite values to divergent integrals has to cancel between the two contributions in the same way the IR divergences cancel. Indeed, the IR limit of theP_22diagram is exactly the same as the IR limit ofP_13diagram, but with the opposite sign (see eq. (<ref>) and eq. (<ref>)).< g r a p h i c s >The full one-loop power spectrum calculated summing up contributions from Fig. <ref>. In conclusion, the one-loop power spectrum can be easily calculated using decomposition (<ref>). For practical applications, the most optimal choice of bias is in the range close to zero-0.5<ν<0because it requires the least number of frequencies to reproduce the linear power spectrum on relevant scales. For this range theP_22diagram can be evaluated directly applying our method. To get the correctP_13diagram, one has to addP_13^ UVterm to the r.h.s of eq. (<ref>). 10pt§.§ One-loop Power Spectrum of Biased TracersThe method described above can be also applied to the one-loop power spectrum of biased tracers <cit.> (for a review see <cit.>). In this section we give explicit formulas for all relevant one-loop contributions. We will follow the notation of <cit.>. The density contrast of biased tracers, such as the dark matter halosδ_h, is a local function of the underlying dark matter field. The functional dependence is expressed through all possible operators built from the tidal tensor∂_i∂_jΦ(and its derivatives), whereΦcan be either gravitational potentialΦ_gor velocity potentialΦ_v. These two potentials are the same at leading order in perturbation theory but starting from second order they differ. To calculate the one-loop power spectrum of biased tracers one has to keep in bias expansion all operators up to third order in perturbation theory[Notice that <cit.> is using b_δ_2=b_22 and b_δ_3=b_36.]δ_h = b_1 δ +b_2/2δ^2 +b__2_2 + b_3/6δ^3 + b__3_3 + b_(_2δ)_2δ + b_Γ_3Γ_3 .The operators_2,_3andΓ_3are defined as_2(Φ)≡ (∂_i∂_jΦ)^2-(∂^2Φ)^2,_3(Φ)≡ -∂_i∂_j Φ ∂_j∂_kΦ ∂_k ∂_i Φ -1/2 (∂^2Φ)^2 +3/2 (∂_i∂_jΦ)^2∂^2Φ ,Γ_3(Φ_g,Φ_v)≡_2(Φ_g) - _2(Φ_v) .However, only four renormalized operators contribute to the one-loop power spectrum. These areδ,[δ^2],[_2]and[Γ_3]. The final answer is given in terms of four corresponding renormalized bias parameters and six different momentum integrals <cit.> < g r a p h i c s >< g r a p h i c s >< g r a p h i c s >< g r a p h i c s > Four different contributions to the one-loop power spectrum of biased tracers. All plots are produced using ν=-1.6, N=150, k_ min=10^-5 h Mpc^-1 and k_ max=5 h Mpc^-1. For these values of parameters the difference with respect to the usual numerical calculation is less than 0.1% at all scales. P_h(k,τ) =b_1^2(P_ lin(k,τ)+ P_ 1-loop(k,τ)) + b_1 b_2ℐ_δ^2(k,τ) +2b_1 b__2 ℐ__2(k,τ) + ( 2b_1 b__2 + 4/5 b_1 b_Γ_3) ℱ__2(k,τ)+ 1/4 b_2^2ℐ_δ^2δ^2(k,τ) + b__2^2ℐ__2_2 (k,τ) + 1/2 b_2b__2 ℐ_δ_2_2(k,τ).In principle, at this order in perturbation theory one has to add higher derivative operators such as∂^2δ. However the contribution from this operator is trivial and it does not lead to a loop integral.The time dependence of all momentum integrals inP_h(k,τ)isD(τ)^4. The explicitk-dependences are ℐ_δ^2(k) = 2∫_ F_2(,-̨)P_ lin(q)P_ lin(|-̨|),(-3<ν<- 12)ℐ__2(k) = 2 ∫_σ^2(,-̨)F_2(,-̨) P_ lin(q)P_ lin(|-̨|),(-3<ν< 12)ℱ__2(k) = 4 P_ lin(k) ∫_σ^2(,-̨)F_2(,̨-) P_ lin(q),(-3<ν<-1)ℐ_δ^2δ^2(k)= 2 ∫_P_ lin(q)P_ lin(|-̨|),(-3<ν<- 32)ℐ__2_2 (k) = 2 ∫_ (σ^2(,-̨))^2 P_ lin(q)P_ lin(|-̨|),(-3<ν< 12)ℐ_δ_2_2(k)=2 ∫_σ^2(,-̨) P_ lin(q)P_ lin(|-̨|),(-3<ν<- 12 )whereσ^2(_̨1,_̨2)=(_̨1·_̨2/k_1k_2)^2-1. For each term we give a range of power laws for which the integral is convergent. Following the same steps as in the case of the one-loop power spectrum of matter fluctuations, we find that matrices analogous toM_22andM_13are given by M_ℐ_δ^2(ν_1,ν_2)= (3-2ν_12)(4-7ν_12)/17ν_1ν_2 (ν_1,ν_2) , M_ℐ__2(ν_1,ν_2)= -(3-2ν_12)(1-2ν_12)(6+7ν_12)/28ν_1(1+ν_1)ν_2(1+ν_2) (ν_1,ν_2) , M_ℱ__2(ν_1)= -15tan(ν_1π)/28π(ν_1+1)ν_1(ν_1-1)(ν_1-2)(ν_1-3) , M_ℐ_δ^2δ^2(ν_1,ν_2)= 2 (ν_1,ν_2) , M_ℐ__2_2(ν_1,ν_2)= (3-2ν_12)(1-2ν_12)/ν_1(1+ν_1)ν_2(1+ν_2) (ν_1,ν_2) , M_ℐ_δ^2_2(ν_1,ν_2)= 3-2ν_12/ν_1ν_2 (ν_1,ν_2) .In Fig. <ref> we plot some of the shapes and compare our method with the standard numerical evaluation.Notice that forℐ_δ^2δ^2(k)shape we subtract the constant shot-noise part and plot just the differenceℐ_δ^2δ^2(k)-ℐ_δ^2δ^2(0). This difference is convergent even forν>- 32.One important point to make is that the full one-loop power spectrum of biased tracers requires only a single function(ν_1,ν_2)with a single bias in the range-3<ν<- 32. This range can be extended to higher biases by adding the corresponding UV parts of the integrals in the same way as for the matter power spectrum. 10pt§ ONE-LOOP BISPECTRUMIn perturbation theory there are four different diagrams that contribute to the one-loop bispectrum and their sum can be schematically written like <cit.> B_ 1-loop(_̨1,_̨2,_̨3,τ) = D^4(τ)[B_222+B_321^I+B_321^II+B_411] .From translational invariance it follows that_̨1+_̨2+_̨3 = 0. The individual terms in square brackets are given by the following integralsB_222 = 8 ∫_ F_2 (,_̨1-) F_2(_̨1-,_̨2+) F_2(_̨2+,-) × P_ lin(q) P_ lin(|_̨1-|) P_ lin(|_̨2+|),B_321^I = 6 P_ lin(k_1) ∫_ F_3(,_̨2-,-_̨1) F_2(,_̨2-) P_ lin(q) P_ lin(|_̨2-|) + 5perms ,B_321^II = F_2(_̨1,_̨2) P_ lin(k_1) P_13(k_2) + 5perms ,B_411 = 12 P_ lin(k_1) P_ lin(k_2) ∫_ F_4(,-,-_̨1,-_̨2) P_ lin(q) + 2cyclicperms .The diagrammatic representation of all these contributions is shown in Fig. <ref>. < g r a p h i c s >Diagrammatic representation of four contributions to the one-loop bispectrum.To evaluate the one-loop bispectrum we can follow the same steps as for the one-loop power spectrum. After expanding the kernels and decomposing the linear power spectrum in power laws, all terms in the sums are proportional to the integral of the following form <cit.>∫_1/q^2ν_1|_̨1-|^2ν_2 |_̨2+|^2ν_3≡ k_1^3-2ν_123 (ν_1,ν_2,ν_3;x,y),wherex≡ k_3^2/k_1^2andy≡ k_2^2/k_1^2. Parametersν_1,ν_2andν_3have the same form as before (see (<ref>)). The overall scaling of the integral with momentum is fixed and here we choose to express that scaling in terms ofk_1. The rest defines a function(ν_1,ν_2,ν_3;x,y)which depends only on the ratiosxandy. Triangle inequality|k_2-k_3|≤ k_1≤ k_2+k_3implies that the physicallyallowed region in(x,y)plane is given by inequalities|√(x)-√(y)|≤ 1and√(x)+√(y)≥ 1and we will focus on evaluating the function in this region (see Fig. <ref>). Before giving the explicit expression suitable for numerical evaluation we present some important symmetry properties of(ν_1,ν_2,ν_3;x,y)which can be derived from its integral representation. < g r a p h i c s > Left panel: Domain of (ν_1,ν_2,ν_3;x,y) allowed by the triangle inequality. Six different regions correspond to six permutations of external momenta. All bispectrum configurations can be evaluated focusing on one of these regions. For example, in this paper we choose x≤ y≤ 1 which corresponds to k_3≤ k_2≤ k_1. Solid lines split the domain in three different parts that will be relevant for the evaluation of the two-loop power spectrum (see Section <ref>).Right panel: The same as left panel in more conventional variables k_3/k_1 and k_2/k_1. 10pt §.§ Symmetries of (ν_1,ν_2,ν_3;x,y) and Recursion RelationsAs in the case of the one-loop power spectrum, the simplest identities follow from shifts of the momentum. There are two basic translation formulas for function(ν_1,ν_2,ν_3;x,y). The first one follows from→_̨1-and reads(ν_1,ν_2,ν_3;x,y) = (ν_2,ν_1,ν_3;y,x).If we do a different shift,→ - _̨2, we get(ν_1,ν_2,ν_3;x,y)=x^3/2-ν_123(ν_3,ν_2,ν_1;1x,yx).These two formulas are sufficient to generate identities involving all six permutations of parametersν_1,ν_2andν_3. These are (ν_1,ν_2,ν_3; x,y)= (ν_2,ν_1,ν_3; y,x)= x^3/2-ν_123(ν_3,ν_2,ν_1; 1x,yx)=x^3/2-ν_123(ν_2,ν_3,ν_1; yx,1x)= y^3/2-ν_123(ν_3,ν_1,ν_2; 1y,xy)=y^3/2-ν_123(ν_1,ν_3,ν_2; xy,1y). An intuitive way to understand these symmetries is to realize that they map_̨1,_̨2and_̨3into each other, preserving the shape of the triangle. The six equations then correspond to nothing other but six possible permutations of three external momenta. From another point of view, for evaluation of the bispectrum we can always choose a small “corner” in the(x,y)plane (see Fig. <ref>). The choice that we make in this paper isx≤ y≤ 1which corresponds to the following ordering of momenta:k_3≤ k_2≤ k_1.Let us now derive the inversion formula for(ν_1,ν_2,ν_3;x,y). We can start from(ν_1,ν_2,ν_3;x,y) = ∫_1/q^2ν_1|̨̂_1-|^2ν_2 |√(y)̨̂_2+|^2ν_3 ,wherę̂_1and̨̂_2are unit vectors. Under inversion→/q^2, apart from transformations described in (<ref>), we also get |√(y)̨̂_2+|^2→y/q^2 |̨̂_2/√(y) +|^2.The whole integral then changes to(ν_1,ν_2,ν_3;x,y) = ∫_y^-ν_3/q^2(3-ν_123)|̨̂_1-|^2ν_2 |̨̂_2/√(y)+|^2ν_3 .It is easy to read off the inversion formula from this expression. One only has to keep in mind that, due to√(y)appearing in the denominator in the last term, the arguments of the function change to xyand 1y. We finally get (ν_1,ν_2,ν_3;x,y) =y^-ν_3(3-ν_123,ν_2,ν_3; xy, 1y ). Combining this inversion formula with translation formulas (<ref>), we get a set of three identities for functions evaluated at thesame point(x,y)(ν_1,ν_2,ν_3;x,y)= x^3/2-ν_23 (ν_3,3-ν_123,ν_1;x,y)= x^3/2-ν_23y^3/2-ν_13 (ν_2,ν_1,3-ν_123;x,y) = y^3/2-ν_13 (3-ν_123,ν_3,ν_2;x,y). Whenν_123=3the previous equation implies(ν_1,ν_2,3-ν_12) = x^3/2-ν_23y^3/2-ν_13 (ν_1,ν_2).This expression is sometimes referred to as thestar-triangle duality. The conditionν_123=3does not correspond to a generic situation but it can be used to derive another generic formula for(ν_1,ν_2,ν_3;x,y). The idea is to “split” one of the parameters using the one-loop integral, such thatν_123=3is satisfied. For example, we can choose to write1/|_̨2+|^2ν_3 = 1/(ν_123- 32, 3-ν_12)∫_1/s^2ν_123-3|_̨2+-|^6-2ν_12 .Now one can start with the integral representation of(ν_1,ν_2,ν_3;x,y)and use the previous formula. Notice that we have chosen parameters such that the integration incan be then easily done using (<ref>). The remaining integral inhas again the form of the one-loop bispectrum. Following these steps one derives the star-triangle formula(ν_1,ν_2,ν_3;x,y) = Γ(ν̃_1)Γ(ν_1)Γ(ν̃_2)Γ(ν_2)Γ(ν̃_3)Γ(ν_3)Γ(3-ν̃_123)Γ(3-ν_123) (ν̃_2,ν̃_1,3-ν̃_123 ;x,y),whereν̃_i = 32-ν_i. Finally, using the method described after eq. (<ref>), it is possible to derive the following reflection formula(ν_1,ν_2,ν_3;x,y) =Γ(ν̃_1)Γ(ν_1)Γ(ν̃_2)Γ(ν_2)Γ(ν̃_3)Γ(ν_3)Γ(3-ν̃_123)Γ(3-ν_123) x^3/2-ν_23 y^3/2-ν_13(ν̃_1,ν̃_2, ν̃_3 ; x,y). This is not an independent relation because it follows from eq. (<ref>) and eq. (<ref>).For arbitrary choice ofνeq. (<ref>) and eq. (<ref>) are not very useful, because they relate two functions with twodifferent biases. However, there are some special choices ofνfor which this is not the case. Let us remember that, up to an integer, the structure of parameters isν_i = -ν/2 - i η_m_i/2 ,which impliesν̃_i = 3/2 + ν/2 + i η_m_i/2 .Up to an integer, the transformationν→ν̃does not change the bias ifνis a odd integer multiple of- 12. For example, let us imagine thatν=-32. In this case ν_i = 3/4 - i η_m_i/2⇒ν̃_i = 3/4 + i η_m_i/2 .In other words, forν=-32,ν̃_iis just a complex conjugate ofν_i. For example, assumingν=-32, the reflection formula becomes(ν_1+n_1,ν_2+n_2,ν_3+n_3 ;x,y) =Γ(ν̃_1-n_1)Γ(ν_1+n_1)Γ(ν̃_2-n_2)Γ(ν_2+n_2)Γ(ν̃_3-n_3)Γ(ν_3+n_3)Γ(3-ν̃_123+n_123)Γ(3-ν_123-n_123)× x^3/2-ν_23-n_23 y^3/2-ν_13-n_13^*(ν_1-n_1,ν_2-n_2, ν_3-n_3 ; x,y),wheren_iare integers coming from the expansion of the kernels. This equation provides a simple relation between two functions in the same point and with the same parameters, but with the opposite sign of integer part ofν_i. For example, such pairs of functions do exist in the expansion of kernels inB_222and each of them can be calculated for the price of a single evaluation. Similar identity can be derived forν=- 12and the same conclusions apply to a set of formulas (<ref>).Let us conclude showing that the function(ν_1,ν_2,ν_3;x,y)satisfies a set of recursion relations <cit.>. As in the case of the one-loop power spectrum we can start from the following identity ∫_∂/∂ q_i( q_i/q^2ν_1|_̨1-|^2ν_2 |_̨2+|^2ν_3) = 0.After expanding the derivative in the integrand, the previous equation can be rewritten as ν_2(ν_1,ν_2+1,ν_3) + ν_3y (ν_1,ν_2,ν_3+1) =(ν_1+ν_123 -3) (ν_1,ν_2,ν_3) + ν_2(ν_1-1,ν_2+1,ν_3) +ν_3(ν_1-1,ν_2,ν_3+1) ,where we suppressed the(x,y)argument in all functions to avoid clutter. There are two other similar expressions that can be derived replacingq_iin the numerator of the integral in (<ref>) with(_̨1-)_ior(_̨2+)_i. These areν_1(ν_1+1,ν_2,ν_3) + ν_3x (ν_1,ν_2,ν_3+1) =(ν_2+ν_123 -3) (ν_1,ν_2,ν_3) + ν_1(ν_1+1,ν_2-1,ν_3) +ν_3(ν_1,ν_2-1,ν_3+1) , ν_1y(ν_1+1,ν_2,ν_3) + ν_2 x (ν_1,ν_2+1,ν_3) =(ν_3+ν_123 -3) (ν_1,ν_2,ν_3) + ν_1(ν_1+1,ν_2,ν_3-1) +ν_2(ν_1,ν_2+1,ν_3-1) .Notice that we wrote these equations such that the sum of the arguments in each function on the l.h.s. isν_123+1andthe sum of the arguments in each function on the r.h.s. isν_123. This splitting suggests the following interpretation of the recursion relations. They can be thought of as a system of three linear equations where three unknown functions are those in which one of the parameters is increased by 1. If we denote the r.h.s. of previous equations withA_1,A_2andA_3A_1≡ (ν_1+ν_123 -3) (ν_1,ν_2,ν_3) + ν_2(ν_1-1,ν_2+1,ν_3) +ν_3(ν_1-1,ν_2,ν_3+1), A_2≡ (ν_2+ν_123 -3) (ν_1,ν_2,ν_3) + ν_1(ν_1+1,ν_2-1,ν_3) +ν_3(ν_1,ν_2-1,ν_3+1) ,A_3≡ (ν_3+ν_123 -3) (ν_1,ν_2,ν_3) + ν_1(ν_1+1,ν_2,ν_3-1) +ν_2(ν_1,ν_2+1,ν_3-1) ,then the solution of the system is given by <cit.> (ν_1+1,ν_2,ν_3)= 1/2ν_1y(-A_1 x + A_2 y + A_3 ),(ν_1,ν_2+1,ν_3)= 1/2ν_2 x(A_1 x - A_2 y + A_3 ),(ν_1,ν_2,ν_3+1)= 1/2ν_3 x y(A_1 x + A_2 y - A_3 ).In other words, seven different functions whose parameters live on the planeν_123= const.determine three extra integrals on the planeν_123+1= const. These identities are very useful. For example, in the expansion of kernels inB_222they reduce the number of independent terms by roughly a factor of 2 (from 72 to 38). They are also very important for simplifying the two-loop calculation as we are going to see in the following sections. 10pt §.§ Evaluation of (ν_1,ν_2,ν_3;x,y)After making these general remarks based on the integral representation, let us turn to the explicit expression for(ν_1,ν_2,ν_3;x,y). Unlike(ν_1,ν_2), this function cannot be simply expressed in a closed form in terms of gamma functions. Starting from (<ref>) and using Feynman parameters we get (see Appendix <ref>)(ν_1,ν_2, ν_3 ;x,y) = 1/8π^3/2Γ(ν_123-3/2)/Γ(ν_1)Γ(ν_2)Γ(ν_3)×∫_0^1 du ∫_0^1 dv u^ν_1-1(1-u)^ν_2-1v^1/2-ν_3(1-v)^ν_3-1/( u v (1-u) + u(1-v) y + (1-u) (1-v) x )^ν_123-3/2 .The expression on the r.h.s. belongs to the class of hypergeometric functions of two variables. In particular,(ν_1,ν_2, ν_3 ;x,y)can be written as a linear combination of AppellF_4functions <cit.>. These special functions can be evaluated using their series representations. The region of convergence is given by√(x)+√(y)< 1, which unfortunately covers only the unphysical part of the(x,y)plane. As usual, this kind of problems can be circumvented by performing the analytic continuation. This can be done in several ways, depending on the region of parameter space that one wants to cover <cit.>. Although all results are formally equivalent andcan be related to each other, distinct expressions can be very different from the point of view of practical calculation. A series representation of(ν_1,ν_2, ν_3 ;x,y), optimized for numerical evaluation of the bispectrum, is given by the following formula (ν_1,ν_2,ν_3 ;x,y) = (πν_23)/8√(π)Γ(ν_1) Γ(ν_2) Γ(ν_3) Γ(3-ν_123)[ x^3/2-ν_23∑_n=0^∞a_n(ν_1,ν_2,ν_3)·x^n _2F_1(ν_1+n,32-ν_2+n, 3-ν_23+2n, 1-y ) . . -y^3/2-ν_13∑_n=0^∞ a_n(ν̃_1,ν̃_2,ν̃_3)·x^n _2F_1(ν̃_1+n,32-ν̃_2+n, 3-ν̃_23+2n, 1-y ) ],wherea_n(ν_1,ν_2,ν_3)=Γ(ν_1+ n)Γ(3 -ν_123 + n)/Γ(5/2-ν_23+ n)n!Γ(3/2-ν_3+n)Γ(3/2-ν_2+n)/Γ(3-ν_23+2n) .The functions_2F_1(…,1-y)that appear in the result are standard Gauss hypergeometric functions. We review their definition and some important properties in Appendix <ref>. The derivation of eq. (<ref>) is given in Appendix <ref>. Let us make some comments about expression (<ref>). The first thing to notice is that the series is always convergent if we restrict ourselves to the regionx≤ y≤ 1. The minimal allowed value of1-yin the given region is34, which corresponds to folded triangles. The smaller1-yis, the easier it gets to calculate the hypergeometric functions using their power series representation. For higher values ofyand smaller values ofxthe convergence is very fast. In the limits ofx→ 0andy→ 1,correspondingto squeezed triangles, only a few terms need be kept in the sum. The slowest convergence is for high values ofx. The limiting case isx=1andy=1, which corresponds to equilateral triangles. Even in this case only a relatively modest number of terms,∼𝒪(50), need be kept in the sum to reach satisfactory precision. Another important point to keep in mind is that dependences onxandyare explicitly separated in our formula. Furthermore, thex-dependence is trivial. This means that in practice, for a giveny, calculation for anyk_1andxcan be done evaluating the hypergeometric functions only once. This can speed up any full bispectrum calculation significantly.There are additional optimizations which can exploit many well-known properties of hypergeometric functions. One such property is that_2F_1(a+n_1,b+n_2,c+n_3,z)for any set of integers(n_1,n_2,n_3)can be always written as a linear combination of just two hypergeometric functions such as, for example,_2F_1(a,b,c,z)and_2F_1(a+1,b,c,z). Using this property one can prove the following recursion relation[Notice that in equation (10) of reference <cit.> there is a typo. The sign between the two terms in square brackets should be + instead of -.]<cit.> f_n-1 = f_n +c(1-a-b-2n)+2ab-2n(n-1)(c+2n-2)(c+2n) z f_n - (a+n)(b+n)(c-a+n)(c-b+n)(c+2n)^2((c+2n)^2-1)z^2 f_n+1 ,wheref_n=_2F_1(a+n,b+n,c+2n,z). This equations gives a way to recursively calculate all hypergeometric functions in the power series of (<ref>). In practice one should exercise some caution, as certain values of arguments in this recursion relation can be numerically unstable.In some special cases eq. (<ref>) further simplifies. For example, for the case of isosceles triangles,y=1and all hypergeometric functions are equal to one. As expected from the symmetry properties in (<ref>), the result in this case becomes symmetric inν_2andν_3. Finally, let us point out that if one of the parameters is a negative integer or zero, the sum in (<ref>) truncates. To see this explicitly, considerν_1=-N, withN≥0a non-negative integer. In the limitν_1→-N,1/Γ(ν_1)in the normalization goes to zero. If there were no terms in the sum which diverge in the same limit, the result would be zero. The hypergeometric functions are always regular. Therefore, we have to look at the coefficients. By inspection we see that alla_n(ν̃_1,ν̃_2,ν̃_3)coefficients are regular as well. The only divergence comes fromΓ(ν_1+n)in the coefficienta_n(ν_1,ν_2,ν_3), forn≤ N. Given thatΓ(-N+n)/Γ(-N) = (-1)^n Γ(N+1)/Γ(N-n+1) , we can rewrite the final answer in the following way(-N,ν_2,ν_3 ;x,y) = (-1)^N+1√(π)(πν_23)(πν_3)/8 Γ(ν_2) Γ(ν_3) Γ(3+N-ν_23)∑_n=0^N ∑_m=0^N-nN! (-1)^m+n/(N-m-n)!Γ( 32-ν_2+n+m)/Γ(5/2-ν_23+ n)Γ(ν_3 - N - 12+m)x^3/2-ν_23+n/n!y^m/m! .In conclusion, when one of the arguments of(ν_1,ν_2,ν_3 ;x,y)is a negative integer or zero, the function becomes a simple polynomial inxandyof degreeN. If two of the arguments are negative integers or zero, then the function vanishes.10pt §.§ Numerical Evaluation of the One-loop BispectrumLet us now turn to the numerical evaluation of the bispectrum. We will consider each term in (<ref>) separately.< g r a p h i c s >Equilateral B_222 diagram as a function of k. The calculation is done with bias ν=-0.25 and N=50 sampling points for the power spectrum. This is enough to reach the sub-percent precision.B_222 term.—We begin with theB_222contribution. We will fist find the range of biases for which the integral is convergent. Two of threeF_2kernels, have the same structure and asymptotic behavior as in theP_22diagram. The third kernel tends to𝒪(1)constant in the→ 0limit. In the UV limitF_2(_̨1-,_̨2+) →k^2/q^2 ,q→∞ .Combining all these limits it follows that the integral inB_222diagram is convergent for power laws in the range-1<ν<1. Therefore, choosing a bias close to zero, we expect our method to reproduce the results of the usual numerical integration. After expanding the kernels and linear power spectra in power laws, we can write the result as a matrix multiplication B̅_222(k_1,k_2,k_3) = k_1^3∑_m_i M_222(ν_1,ν_2,ν_3;x,y) · c_m_1k_1^-2ν_1· c_m_2k_1^-2ν_2· c_m_3k_1^-2ν_3 ,where the matrixM_222(ν_1,ν_2,ν_3;x,y)is given byM_222(ν_1,ν_2,ν_3;x,y) = 8∑_n_i f_222(n_1,n_2,n_3;x,y) (ν_1-n_1,ν_2-n_2,ν_3-n_3;x,y).As before,c_m_iare the coefficients of the expansion (<ref>) andn_1,n_2andn_3are integer powers ofq^2,|_̨1-|^2and|_̨2+|^2in the expansion of the three kernels in the integrand. The rational coefficients in this expansion are labeled byf_222(n_1,n_2,n_3;x,y). We give the explicit expression forM_222in Appendix <ref>.Let us make a couple of comments about the formulas above. As in the case of the one-loop power spectrum, the evaluation of the bispectrum boils down to a simple matrix multiplication. The matrixM_222is cosmology independent, so it has to be calculated only once. Notice thatM_222depends only on the shape of the triangle formed by the three external momenta, and not on its absolute size. It is clear from eq. (<ref>) that for fixedxandyone can calculate all triangles with differentk_1using the sameM_222. The size of this matrix isN^3, whereNis the number of sampling points of the linear power spectrum. In practice, in order to reach the sub-percent precision, it is enough to use𝒪(50)points. We plot theB_222diagram in Fig. <ref> and as expected it is in a very good agreement with the usual numerical evaluation.< g r a p h i c s >Equilateral B_321^I diagram as a function of k. The calculation is done with bias ν=-0.25 and N=50 sampling points for the power spectrum. This is enough to reach the sub-percent precision.B_321^I term.—Let us now turn toB_321^Idiagram. The asymptotic behavior of theF_3kernel in the integrand is the same in the UV and the IR F_3(,_̨2-,-_̨1) →k/q q→∞ and q→ 0.Therefore, the integral is convergent for power laws in the range-1<ν<0. For small negative bias we can use our method without dealing with the possible UV divergences. As in the previous case we can write B̅_321^I(k_1,k_2,k_3) = k_1^3 P_ lin(k_1) ∑_m_i M_321(ν_1,ν_2;x,y) · c_m_1k_1^-2ν_1· c_m_2k_1^-2ν_2 + 5perms ,where the matrixM_321(ν_1,ν_2;x,y)is given byM_321(ν_1,ν_2;x,y) = 6∑_n_i f_321(n_1,n_2,n_3;x,y) (⋯;x,y).We do not explicitly specify the argument becauseν_1andν_2can be at different positions in different terms. The explicit form of the matrix can be found in theMathematica notebook file associated with the preprint of the paper onarXiv. The same conclusions as in the previous case apply here as well. The matrixM_321is cosmology independent and hasN^2elements. This makes it numerically much less challengingthanM_222. In Fig. <ref> we compare our method forN=50sampling points in the power spectrum with the usual numerical evaluation and find an excellent agreement between the two. < g r a p h i c s >Equilateral B_411 diagram as a function of k. The calculation is done with bias ν=-0.25 and N=50 sampling points for the power spectrum. This is enough to each the sub-percent precision. B_411 term.—Unlike for previous diagrams, theB_411integral does not converge for any bias. The situation is similar toP_13diagram and the problem can be solved in a similar way. For small negative biases the integral is UV divergent. Therefore, to get the usual numerical result, one has to add the UV part of the loop integral which is given by <cit.> B_411^ UV =-P_ lin (k_2) P_ lin (k_3) σ_v^2/226380 k_2^2k_3^2( 12409 k_1^6 + 20085 k_1^4(k_2^2 + k_3^2) - k_1^2 (44518 k_2^4 - 76684 k_2^2k_3^2 +44518 k_3^4) + 12024 (k_2^2 - k_3^2)^2 (k_2^2 + k_3^2) )+ 2 perm.As in the case of the power spectrum, dimensional regularization would set these terms to zero. As expected, the structure of the UV part of theB_411diagram is such that it can be reabsorbed by the bispectrum counterterms in the EFT approach to LSS <cit.>. The regular terms can be organized in a vectorM_411which hasNelements and is cosmology independent. The approximation to theB_411diagram can be then written as B̅_411(k_1,k_2,k_3) = k_1^3 P_ lin(k_1) P_ lin(k_2) ∑_m M_411(ν;x,y) · c_mk_1^-2ν + 2perms ,whereM_411(ν;x,y) = 12∑_n_i f_411(n_1,n_2,n_3;x,y) (⋯;x,y).The explicit form of this vector can be found in theMathematica notebook file associated with the preprint of the paper onarXiv. In Fig. <ref> we find an excellent agreement of our method (includingB_411^ UVterms) with the usual numerical evaluation. < g r a p h i c s >Equilateral one-loop bispectrum as a function of k. The calculation is done with bias ν=-0.25 and N=50 sampling points for the power spectrum. With these parameters it is possible to reach sub-percent precision on all scales.The full one-loop bispectrum.—In Fig. <ref> we plot the equilateral bispectrum as a function ofkand compare our method with the usual numerical result. Given that typically there are no large cancellations between different diagrams, we achieve a similar precision for the full result as for each individual term in the sum.In summary, for each bispectrum shape given by ratiosxandy, one has to calculate three matricesM_222,M_321andM_411. These matrices haveN^3,N^2andNelements respectively whereN∼𝒪(50)is sufficient to achieve sub-percent precision on relevant scales. Computation of these matrices is relatively fast and it can be further optimized using properties of hypergeometric functions. The most practical approach for data analysis is to precompute all matrices and evaluate the one-loop bispectrum with different cosmological parameters as a simple matrix multiplication. § TWO-LOOP POWER SPECTRUMNow we can turn to the most complicated case of the two-loop power spectrum. There are four different contributions at this order in perturbation theory <cit.> (for the EFTofLSS treatment of the two-loop power spectrum see <cit.>)P_ 2-loop(k,τ) = D^4(τ)[P_33^I(k) + P_33^II(k) + P_24(k) + P_15(k) ].The explicit form of the four terms in the square brackets isP_33^I (k) = 9P_ lin(k) ∫_ F_3(,̨,-) P_ lin(q) ∫_ F_3(-,̨, -) P_ lin(p),P_33^II (k) = 6 ∫_∫_ F_3(,,-̨-) F_3(-,-,+-)̨ P_ lin(q) P_ lin(p) P_ lin(|-̨-|),P_24 (k) = 24 ∫_∫_ F_2(, -̨) F_4(,-,-,-)̨ P_ lin(q) P_ lin(p) P_ lin(|-̨|),P_15 (k) = 30 P_ lin(k) ∫_∫_ F_5(,̨,-,,-) P_ lin(q) P_ lin(p).The corresponding diagrams are shown in Fig. <ref>. In the first contribution,P_33^I (k), two integrals have the same structure as theP_13(k)part of the one-loop calculation. In other cases the integrals are not separable. After expanding kernels and linear power spectra in power laws, all terms in the sum can be written in the following form∫_1/q^2ν_4 |-̨|^2ν_5∫_1/p^2ν_1|-̨|^2ν_2 | - |^2ν_3≡ k^6-2ν_12345 (ν_1,…,ν_5).One important point to make is that at most three of five parametersν_1, …,ν_5are generic complex numbers. The reason for this is that there are at most three linear power spectra in the two-loop integrals. The other two parameters must be integers coming from the expansion of kernels. As we are going to see, this simplifies evaluation of some diagrams significantly. < g r a p h i c s >Diagrammatic representation of four contributions to the two-loop power spectrum. Before we move on, let us notice that the second integral in (<ref>) has identical structure as the one-loop bispectrum. Therefore, choosing the following change of coordinatesx=|-̨|^2/k^2andy=q^2/k^2, the function(ν_1,…,ν_5)that we are ultimately interested in can be written as follows(ν_1,…,ν_5) = 1/16π^2∫_x,y x^-ν_5 y^-ν_4(ν_1,ν_2,ν_3;x,y).We are going to use this equation and the series representation of(ν_1,ν_2,ν_3;x,y)to find the explicit formula for(ν_1,…,ν_5). One difficulty is that the region of integration is rather complicated:√(x)+√(y)≥ 1and|√(x)-√(y)|≤ 1. This reflects the constraints that physical momenta in the two-loop diagram have to satisfy. Before we discuss the explicit expression, we derive some symmetry properties of the function(ν_1,…,ν_5). These properties will prove useful in practical applications to the full two-loop integral. §.§ Symmetries of (ν_1,…,ν_5) and Recursion RelationsThe two-loop diagram of massless theory is known to have a lot of symmetries which translates to many symmetries of the function(ν_1,…,ν_5). All symmetry transformations were first derived in <cit.>. The full symmetry group isZ_2× S_6and it has2× 6! =1440elements <cit.>. We review here some of the relevant symmetry transformations. To derive them it is enough to use the integral representation(ν_1,…,ν_5) = ∫_1/q^2ν_4 |̨̂- |^2ν_5∫_1/p^2ν_1|̨̂- |^2ν_2 | - |^2ν_3 ,wherę̂is the unit vector. Two obvious symmetries are the following. First, exchanging integration variablesandleads to(ν_1,ν_2) ↔ (ν_4,ν_5), leavingν_3in the same position (ν_1, ν_2, ν_3, ν_4, ν_5) = (ν_4, ν_5, ν_3, ν_1, ν_2).Second, we can simultaneously shift both momenta→-̨and→-̨. Effectively, this produces the simultaneous exchange(ν_1 ↔ν_2)and(ν_4 ↔ν_5)(ν_1, ν_2, ν_3, ν_4, ν_5) = (ν_2, ν_1, ν_3, ν_5, ν_4).Additional translation formulas can be derived using translation formulas of(ν_1,ν_2,ν_3;x,y). Plugging the transformations (<ref>) into the integral representation (<ref>), it is straightforward to get the following extra identities (ν_1, ν_2, ν_3, ν_4, ν_5) = (ν_3,ν_1,ν_2, ν_5, ν_6), (ν_1, ν_2, ν_3, ν_4, ν_5) = (ν_3,ν_2,ν_1, ν_4, ν_6),whereν_6≡ 92-ν_12345. Transformations (<ref>) to (<ref>) form the symmetric group of degree fourS_4. This group has4!=24different elements. We recoverthese elements by starting from(ν_1, ν_2, ν_3, ν_4, ν_5)and successively applying identities (<ref>) to (<ref>). Invoking notation(1,2,3,4,5)≡(ν_1,…,ν_5), then the 24 equivalent functions are(1,2,3,4,5)(2,1,3,5,4)(4,5,3,1,2)(5,4,3,2,1)(3,1,2,5,6)(1,3,2,6,5)(5,6,2,3,1)(6,5,2,1,3)(3,2,1,4,6)(2,3,1,6,4)(4,6,1,3,2)(6,4,1,2,3)(3,5,4,1,6)(5,3,4,6,1)(1,6,4,3,5)(6,1,4,5,3)(3,4,5,2,6)(4,3,5,6,2)(2,6,5,3,4)(6,2,5,4,3)(5,2,6,4,1)(2,5,6,1,4)(4,1,6,5,2)(1,4,6,2,5). Applying inversion (<ref>) and star-triangle (<ref>) formulae for(ν_1,ν_2,ν_3;x,y)in the integral representation (<ref>), provides analogous results for(ν_1,…,ν_5): (ν_1,ν_2,ν_3,ν_4,ν_5) = (ν_2,ν_1,3-ν_123,ν_134- 32 ,ν_235 -32 ) , (ν_1,ν_2,ν_3,ν_4,ν_5) = Γ(ν̃_1)Γ(ν_1)Γ(ν̃_2)Γ(ν_2)Γ(ν̃_3)Γ(ν_3)Γ(3-ν̃_123)Γ(3-ν_123) (ν̃_2,ν̃_1,3-ν̃_123,ν_4,ν_5).Finally, combining these with translation leads to the following reflection formula(ν_1, ν_2,ν_3,ν_4,ν_5) = g (ν̃_1,ν̃_2,ν̃_3,ν̃_4,ν̃_5),where the prefactorgis given byg = Γ(ν̃_1)Γ(ν_1)Γ(ν̃_2)Γ(ν_2)Γ(ν̃_3)Γ(ν_3)Γ(ν̃_4)Γ(ν_4)Γ(ν̃_5)Γ(ν_5)Γ(ν̃_6)Γ(ν_6)Γ(3-ν̃_123)Γ(3-ν_123)Γ(3-ν̃_345)Γ(3-ν_345)Γ(ν̃_235-3/2)Γ(ν_235-3/2)Γ(ν̃_134-3/2)Γ(ν_134-3/2) . Applying successively all these transformations one can generate the entireZ_2× S_6group. However, almost all of these transformations fail to preserve the bias. Asin the case of the one-loop bipsectrum, onlyspecial choices ofνtrivially offer compact relations between functions with different integer parts of the parameters.Finally, let us write down an example of a recursion relation that(ν_1,…,ν_5)satisfies. Again, the simplest way to derive these identities is to use the analogous results for(ν_1,ν_2,ν_3;x,y). For instance, using recursion relation (<ref>) one can immediately write(3-ν_1-ν_123)(ν_1,ν_2, ν_3,ν_4,ν_5) + ν_2(ν_1,ν_2+1,ν_3,ν_4,ν_5) - ν_2 (ν_1-1,ν_2+1,ν_3,ν_4,ν_5)+ν_3(ν_1,ν_2,ν_3+1,ν_4-1,ν_5) - ν_3 (ν_1-1,ν_2,ν_3+1,ν_4,ν_5) = 0 .Other similar expressions can be found using the symmetry transformations discussed above. We will give some more details about the practical application of these equations in the following section. 10 pt §.§ Evaluation of (ν_1,…,ν_5)As we already mentioned, finding an explicit formula for(ν_1,…,ν_5)is not straightforward. In the integral (ν_1,…,ν_5) = 1/16π^2∫_x,y x^-ν_5 y^-ν_4(ν_1,ν_2,ν_3,x,y),the region of integration is complicated and we lack a simple representation of(ν_1,ν_2,ν_3;x,y)convergent in the entire domain. Our strategy to find the full solution is to first concentrate on a part of the region of integration and then use symmetries (<ref>) to find solutions elsewhere. One possibility is to start with the integralK(ν_1,…,ν_5)≡∫_𝒟_1 x^-ν_5 y^-ν_4 J(ν_1,ν_2,ν_3,x,y),over the following domain𝒟_1 ={ (x,y)| √(x)+√(y)≥ 1,x≤1,y≤ 1 }. Given that bothxandyare smaller than 1, we can use result (<ref>) to evaluate the integral. The details of this derivation are given in Appendix <ref>. Here we report only the final formula. We write the result in a form which resembles the series representation of(ν_1,ν_2,ν_3;x,y)K(ν_1, ν_2,ν_3,ν_4,ν_5)= (πν_23)/8√(π)Γ(ν_1) Γ(ν_2) Γ(ν_3) Γ(3-ν_123)×∑_n=0^∞[ a_n(ν_1,ν_2,ν_3)κ(32-ν_235+n ,-ν_4, ν_1+n ,32-ν_2+n, 3-ν_23+2n ) . . -a_n(ν̃_1,ν̃_2,ν̃_3)κ(-ν_5+n, 32-ν_134, ν̃_1+n,32 -ν̃_2+n, 3-ν̃_23+2n ) ]. The complicated part of the answer is the functionκ(⋯)which is given in terms of generalized hypergeometric functionsκ (α,β,a,b,c) = 11+α[ 11+β _3F_2([1,a,b;; 2+β,c ; ]1) . - 2·_4F_3([ a,b,1+β,3/2+β;; 1-d ,5/2+γ,3+γ ; ]1) . -2·_4F_3([ c-a,c-b,1+β+d,3/2+β+d;;1+d,d+γ+5/2,3+γ +d; ]1) ],whered≡ c-a-bandγ≡α+β. However, this is not the end of the story. We have found only one piece of the final answer, which corresponds to the integral over the region𝒟_1. If one splits the remaining part of domain of integration in (<ref>) in the following way (see Fig. <ref>)𝒟_2 ={ (x,y)| √(x)-√(y)≤ 1,x≥1,y≤ x } ,𝒟_3 ={ (x,y)| √(y)-√(x)≥ 1,y≥1,x≤ y } ,then we find that the integrals over𝒟_2and𝒟_3can be mapped to an integral of the form (<ref>). The proof is based on symmetries of(ν_1,ν_2,ν_3;x,y)given in eq. (<ref>). The full solution is then just a sum of three terms given by (<ref>), with slightly different parameters(ν_1,…,ν_5) =1/16π^2( K(ν_1,ν_2,ν_3,ν_4,ν_5) +K(ν_1,ν_3,ν_2, ν_6, ν_5) + K(ν_2,ν_3,ν_1, ν_6 ,ν_4) ).Obviously, the final result for(ν_1,…,ν_5)is very complicated and not very illuminating. Symmetries of the two-loop diagram that we discussed in the previous section are not manifest at all. It is actually quite remarkable that this messy formula satisfies all functional identities of(ν_1,…,ν_5). This is also an indication that there may exist a much simpler and elegant representation. However, for the time being, it remains elusive.The difficulties with the result (<ref>) are not only aesthetic but also practical. The biggest problem is that the sum on the r.h.s. of (<ref>) is not convergent for all values of parameters. Even when it is, sometimes there are big cancellations between different terms in the sum. Summing up many large numbers which eventually leads to a small answer can be numerically quite challenging. On the other hand, there is a large region of parameter space where the sum converges very rapidly. Using many of the symmetries of(ν_1,…,ν_5)such as (<ref>), it is always possible to evaluate the sum efficiently for any choice of parameters. Understanding the radius of convergence more quantitatively is very important for knowing ahead of time which symmetry transformation to use. Due to complexity of the final answer making some analytic progress is hard and we leave it for the future work. Another problem with eq. (<ref>) is the appearance of generalized hypergeometric functions whose argument is equal to 1. At this point the hypergeometric series which is usually used to calculate the function is either divergent or converges very slowly for a generic set of complex parameters. In order to calculate the hypergeometric functions efficiently one can use some of the functional identities such as recursion relations. In all examples when the integral in definition of(ν_1,…,ν_5)is convergent, the eq. (<ref>) agrees with the result of numerical integration. Our power series representation is typically several orders of magnitude faster.So far we considered the general case where the function(ν_1,…,ν_5)depends on five arbitrary complex numbers. However, as we already pointed out, at least two of(ν_1,…,ν_5)are integers which come from the expansion of perturbation theory kernels. In this more specialized case some of the formulas above simplify. There are two different situations that we meet in practice.One of integer parameters is zero or negative.—The simplest case is when one of integer parameters is zero. The integral (<ref>) becomes a product of two one-loop expressions and the result can be written in terms of gamma functions. For example, let us imagine thatν_1=0. It follows(0,ν_2,…,ν_5) = k^-6+2ν_2345∫_1/q^2ν_4 |-̨|^2ν_5∫_1/|-̨|^2ν_2 | - |^2ν_3 =k^-6+2ν_2345(ν_2,ν_3) ∫_ |-̨|^3 - 2 ν_23/q^2ν_4 |-̨|^2ν_5 =(ν_2,ν_3)(ν_4,ν_235-32).The next simplest case is when none of integer parameters is zero, but rather one of them is negative. Let us imagine thatν_1=-N, whereN>0. In this case the infinite sum in(ν_1,ν_2,ν_3;x,y)truncates (see eq. (<ref>)). The integral (<ref>) can again be expressed in terms of gamma functions only. It is straightforward to get(-N,ν_2, ν_3,ν_4,ν_5) = (-1)^N+1√(π)(πν_23)(πν_3)/8 Γ(ν_2) Γ(ν_3) Γ(3+N-ν_23)∑_n=0^N ∑_m=0^N-nN! (-1)^m+n/(N-m-n)! n! m!Γ( 32-ν_2+n+m)/Γ(5/2-ν_23+ n)Γ(ν_3 - N - 12+m) (ν_4-m, ν_235-32-n).Notice that forN=0this expression reduces to (<ref>). This is in agreement with results of <cit.> (see Appendix F of <cit.>). In practice, the sums always have at most a few terms. For all diagrams in the two-loop power spectrumN≤ 5. The cases in which other parameters are non-positive integers can be easily evaluated using translation formulas (<ref>).Let us point out that in the expansion of the perturbation theory kernels in theP_33,P_24andP_15diagrams, most of the terms do have a negative(zero) integer parameter. For example, the expansion ofF_5kernel in theP_15contribution has several thousand terms. Only ten of them have two positive integer parameters. In other words, the largest part of the two-loop result can be written in terms of gamma functions. Given that in dimensional regularization we do not expect different terms to have very different magnitudes, even neglecting contributions with two positive integer parameters may not affect the result considerably. Both integer parameters are positive.—Finally, let us discuss the option in which both integer parameters are positive. For the two-loop integral the only possibility is that both of these parameters are equal to one. This comes from the fact that we can have multiple inverse Laplacians in the perturbation theory kernels, but we never have a square of the inverse Laplacian. Under translation formulas (<ref>),all the apparently different cases reduce to the following two cases (1,ν_2,ν_3,ν_4,1)and(1,ν_2,ν_3,1,ν_5).Both of these terms can be calculated using eq. (<ref>). It is worth noting that improved numerical stability may be found by considering the related functions provided by the reflection formula (<ref>)(12,ν̃_2,ν̃_3,ν̃_4,12)and(12,ν̃_2,ν̃_3,12,ν̃_5). As in the case of the one-loop power spectrum and the one-loop bispectrum, from the expansion ofF_nkernels we get a lot of terms whereν_iparameters differ just by an integer. Many of those are related by recursion relations. Let us see how the recursion relations look like in the special case when two parameters are equal to one. For example, ifν_1=ν_4=1, then eq. (<ref>) becomes(1-ν_23)(1,ν_2, ν_3,1,ν_5) + ν_2 (1,ν_2+1,ν_3,1,ν_5) - ν_2 (0,ν_2+1,ν_3,1,ν_5) +ν_3(1,ν_2,ν_3+1,0,ν_5) - ν_3 (0,ν_2,ν_3+1,1,ν_5) = 0 .Notice that in three of the five terms one of the arguments is equal to zero. Therefore, they can be written in terms of gamma functions. In this way we get a simple functional identity which relates(1,ν_2,ν_3,1,ν_5)and(1,ν_2+1,ν_3,1,ν_5). Whenν_1=ν_5=1we get(1-ν_23)(1,ν_2, ν_3,ν_4,1) + ν_2 (1,ν_2+1,ν_3,ν_4,1) - ν_2 (0,ν_2+1,ν_3,ν_4,1) +ν_3(1,ν_2,ν_3+1,ν_4-1,1) - ν_3 (0,ν_2,ν_3+1,ν_4,1) = 0 .This equation is slightly more complicated because only two terms have one zero parameter, but it is still very useful. Similar recursion relations can be found exploiting symmetry properties of(ν_1,…,ν_5). In practice, these relations can reduce the number of terms that one has to evaluate by a factor of a few.The bottom line is that by using explicit expressions for(ν_1,…,ν_5), and its symmetry properties, it is possible to calculate all contributions to the two-loop power spectrum. As before, all information can be compressed in three cosmology independent matricesM_33,M_24andM_15which correspond toP^II_33,P_24andP_15diagrams. Evaluation of these matrices is not trivial because of the convergence properties of the series (<ref>). However, these matrices have to be calculated only once and once they are known, the evaluation of the two-loop power spectrum for any cosmology is just a simple matrix multiplication. These matrices have at mostN^3elements whereN∼𝒪(100). Therefore, evaluation of the two-loop power spectrum in onekbin is significantly faster than the usual numerical techniques. We leave the implementation and testing of our algorithm for the two-loop power spectrum for future work.§ CONCLUSIONS In this paper we demonstrate the path forward for the efficient computation of higher multiplicity/loop correlation functions in cosmological perturbation theory. Our starting point is similar to recent proposals for fast evaluation of the one-loop power spectrum <cit.> and it is based on representing the linear power spectrum as a sum of complex power laws. However, our implementation and generalization to higher order correlators is different. When comparison is possible, all methods agree.We mainly focus on deriving relevant analytic expressions for the one-loop and the two-loop power spectrum and the one-loop bispectrum. All one-loop diagrams evaluated using our method are in excellent agreement with the usual numerical results. We leave writing a dedicated code for the two-loop power spectrum (and possibly higher order correlation functions) for future work.Our method splits the computation of loop diagrams in two parts. The first, more “difficult” part is related to solving momentum integrals for power-law power spectra and it is cosmology independent. The second part is a simple matrix multiplication which evaluates the loops for aΛCDM-like cosmology. The matrices can be precomputed, they are cosmology independent and they are relatively small. For example, for the two-loop power spectrum, the largest matrix hasN^3elements, whereN∼𝒪(100). The number of operations needed for evaluation of the power spectrum or the bispectrum is significantly smaller than using direct numerical integration. Furthermore, the same building blocks used to calculate dark matter correlation functions can be also used for correlators of biased tracers. There are no fundamental obstacles in applying our method in redshift space as well.One interesting aspect of the method described in this paper is that it relies on evaluation of loop integrals that are formally identical to those of a massless QFT. This is a new bridge between cosmology and particle physics and the full potential of this connection is still to be explored. This remains the major direction for future work. One hope is that many developments in the theory of scattering amplitudes will prove useful for going beyond the lowest order statistics discussed in this paper. The first step in this direction is a practical analytic formula for the one-loop trispectrum. In principle, that would allow the calculation of the three-loop power spectrum or the two-loop bispectrum. In practice, following procedure described in this paper may turn out to be too difficult or impractical. After all, the trispectrum is a function of six variables, which makes it much more complicated than examples we considered so far.However, there are many alternative representations of loop integrals that may be more useful for higher loop diagrams. In this paper we have insisted on finding well-behaved power series representations for functions such as(ν_1,ν_2)or(ν_1,ν_2,ν_3;x,y). Alternative ways to evaluate these integrals include numerical integration using Mellin-Barnes representation of loop integrals (see for instance <cit.>), projecting onto a basis of other known higher-loop integrals as in <cit.>, or solving numerically partial differential equations that the loop integrals satisfy <cit.>. In some important situations things simplify. One such example is the one-loop covariance of the power spectrum. Given that there are only two independent vectors_̨1and_̨2, this special case of the one-loop four-point function depends only on three variables (before integrating over the angle between_̨1and_̨2). Furthermore, a set of diagrams in the one-loop covariance matrix which give the largest contribution to the final answer (see <cit.>) have the same structure as the one-loop bispectrum. These diagrams can be easily calculated using our function(ν_1,ν_2,ν_3;x,y). We leave application of our method to the covariance matrix and more generally one-loop four point function for future work. At the end, let us stress that the idea of representing theΛCDM-like cosmology as a set of power-law universes can be also very useful outside the context of PT. One can benefit form this decomposition whenever some numerically heavy integral has a simple solution for a power-law universe. One example of this kind is projection of the power spectrum or the bispectrum on the sky, which is difficult due to many integrals over highly oscillatory spherical Bessel functions. It was shown in <cit.> that decomposition (<ref>) can be used to find the solutions of these integrals very accurately and efficiently. It would be interesting to think of other similar applications in the future. 10 pt§ ACKNOWLEDGMENTSWe would like to thank Nima Arkani-Hamed, Valentin Assassi, Diego Blas, Jonathan Blazek, Paolo Creminelli, Guido D'Amico, Chris Hirata, Lam Hui, Mikhail Ivanov, David Kosower, Marcel Schmittfull, Roman Scoccimarro, Leonardo Senatore, Sergey Sibiryakov, Kris Sigurdson, Zachary Slepian, Gabriele Trevisan and Zvonimir Vlah for many useful discussions.J. J. M. C. is supported by the European Research Council under ERC-STG-639729,preQFT: Strategic Predictions for Quantum Field Theories. M.S. gratefully acknowledges support from the Institute for Advanced Study and the Raymond and Beverly Sackler Foundation. M.Z. is supported by NSF grants AST-1409709 and PHY-1521097 and by the Canadian Institute for Advanced Research (CIFAR) program on Gravity and the Extreme Universe.§ HYPERGEOMETRIC FUNCTIONSThe hypergeometric function _2F_1(a,b,c,z)is usually defined as a solution of Euler's hypergeometric equation:z(1-z) f”(z)+(c-(a+b+1)z) f'(z) -ab f(z) = 0 ,wherea,bandcare arbitrary complex numbers. The hypergeometric function has the power series representation:_2F_1(a,b,c,z)=Γ(c)/Γ(a)Γ(b)∑_n=0^∞Γ(a+n)Γ(b+n)/Γ(c+n)n! z^n,which is convergent inside the unit circle in the complex plane|z|<1. This power series can be used for numerical evaluation. The series is convergent at the pointz=1only when the parameters satisfy Re(c-a-b)>0. It should be stressed that the convergence sometimes may be slow or the series has large cancellations, particularly for parameters with large imaginary parts. In order to avoid such issues or evaluate the hypergeometric function outside the unit disc, one can use many functional identities. For example, one such identity is _2F_1(a,b,c,1-z) = Γ(c) Γ(c - a - b)/Γ(c - a) Γ(c - b) _2F_1(a, b, a + b - c + 1, z)+ Γ(c) Γ(a + b - c)/Γ(a) Γ(b) z^c - a - b _2F_1(c - a, c - b, 1 - a - b + c, z),which maps points close to|z|=1to a region aroundz=0where the series converges rapidly. Outside the unit disc the hypergeometric function can be calculated using_2F_1(a,b,c,1/z) = Γ(b - a) Γ(c)/Γ(b) Γ(c - a) (-z)^a _2F_1(a, a - c + 1, a - b + 1, z) + (a↔ b) .These two identities are sufficient to evaluate the hypergeometric function in the entire complex plane. It is possible to generalize the basic hypergeometric series (<ref>) and use it to define generalized hypergeometric functions _pF_q([ a_1 , a_2 ,…, a_p;b_1 , b_2 ,…,b_q ];z )≡ Γ(b_1)⋯Γ(b_q)/Γ(a_1)⋯Γ(a_p)∑_n=0^∞Γ(a_1+n)⋯Γ(a_p+n)/Γ(b_1+n)⋯Γ(b_q+n)z^n/n! ,wherepandqare positive integers. In this paper we use two generalized hypergeometric functions_4F_3and_3F_2. In these cases whenp=q+1the generalized hypergeometric series (<ref>) converges for|z|<1. A the point isz=1which is of special interest in evaluation of the two-loop power spectrum the series converges only when Re(b_1+⋯+b_q-a_1-⋯ a_q+1)>0. This condition is not always satisfied in practice. One simple way out is to use recursion relations which increase the real part of one ofb_iuntil the series becomes convergent. § EXPLICIT FORM OF THE M_222 MATRIXIn this appendix we give the explicit form of theM_222matrix. The starting point are threeF_2kernels in theB_222diagram8 F_2 (,_̨1-) F_2(_̨1-,_̨2+) F_2(_̨2+,-).Expanding this expression in powers ofq^2,|_̨1-|^2and|_̨2+|^2for a single set of parameters(ν_1,ν_2,ν_3)and factoring outk_1dependence we get a sum which can be rearranged in the following wayM_222 = 25x^21372_-2,2,2 -15 (y+1) x^21372_-1,2,2 -(10 y^2-9 y+10)x^21372_0,2,2 +3x^2y (y+1)686_1,2,2 +x^2y^2343_2,2,2 +75x2744_-2,1,2 +75x2744_-2,2,1-5 x(20 x+9 y+9)2744_-1,1,2 -5x(20x+9y+9)2744_-1,2,1 +3x(-10 y^2+9 y+10 x (2 y-1)-10)2744_0,1,2-3x(10 y^2-9 y+10 x (y-2)+10)2744_0,2,1 +x y (20 y x+9 x+9 y+9)1372_1,1,2 +x(9 y (y+1)+x (9 y+20))1372_1,2,1 + 3x(x+1) y^2686_2,1,2 +3xy(x+y)686_2,2,1 -1252744_-2,0,2 + 1251372_-2,1,1 - 1252744_-2,2,0 + 1251372_-1,-1,2 - 1251372_-1,0,1 -75 (2 x-y-1)2744_-1,0,2 -1251372_-1,1,0 -75 (2 x+y+1)1372_-1,1,1 + 1251372_-1,2,-1 - 75 (2 x-y-1)2744_-1,2,0 - 1252744_0,-2,2 - 1251372_0,-1,1 + 75(x-2y+1)2744_0,-1,2 + 3751372_0,-1,2 + 75(5x+5y-4)2744_0,0,1 + 5 (10 x^2+9 (2 y-1) x+10 y^2-9 y+10)2744_0,0,2 - 1251372_0,1,-1 + 75 (5 x-4 y+5)2744_0,1,0 + 5 (40 x^2+9 (y+1) x-20 y^2+18y-20)2744_0,1,1 -1252744_0,2,-2 + 75 (x+y-2)2744_0,2,-1 + 5(10 x^2-9 (y-2) x+10 y^2-9 y+10)2744_0,2,0 + 1251372_1,-2,1 + 75 y2744_1,-2,2 - 1251372_1,-1,0 - 75 (x+2 y+1)1372_1,-1,1 - 5y(9x+20y+9)2744_1,-1,2 -1251372_1,0,-1 - 75(4 x-5 (y+1))2744_1,0,0 - 5 (20x^2-9 (y+2) x-40 y^2-9 y+20)2744_1,0,1 + 3 y (-10 x^2+(20 y+9) x-10 (y+1))2744_1,0,2 + 1251372_1,1,-2 - 75(x+y+2)1372_1,1,-1 - 5(20 x^2-9 (2y+1) x+20 y^2-9 y-40)2744_1,1,0 + 3(10 (y+1) x^2+(10 y^2+9 y+10)x+10y (y+1))1372_1,1,1 + 752744_1,2,-2 - 5 (9 x+9 y+20)2744_1,2,-1 - 3(10 x^2-(9 y+20)x+10y(y+1))2744_1,2,0 - 1252744_2,-2,0 + 75y2744_2,-2,1 + 25 y^21372_2,-2,2 + 1251372_2,-1,-1 + 75 (x-2 y+1)2744_2,-1,0 - 5 y (9 x+20 y+9)2744_2,-1,1 - 15(x+1)y^21372_2,-1,2 - 1252744_2,0,-2 + 75 (x+y-2)2744_2,0,-1 + 5 (10 x^2-9 (y+1) x+2 (5 y^2+9y +5))2744_2,0,0 - 3 y (10 x^2+(10 y-9) x-20 y+10)2744_2,0,1 -(10 x^2-9 x+10) y^21372_2,0,2 + 752744_2,1,-2 - 5(9x+9y+20)2744_2,1,-1 - 3(10 x^2+(10-9 y) x+10 (y-2) y)2744_2,1,0 + y(9 x^2+9 (y+1) x+20 y)1372_2,1,1 + 251372_2,2,-2 - 15 (x+y)1372_2,2,-1 - (10 x^2-9 y x+10 y^2)1372_2,2,0 .We are using shorten notation in which_n_1,n_2,n_3≡(ν_1+n_1,ν_2+n_2,ν_3+n_3;x,y). Coefficientsf_222can be easily read off form this expression. Notice that there are 72 terms in the sum, but not all of them are independent. Using recursion relations (<ref>) one can further reduce this expression to a sum of 38 different_n_1,n_2,n_3functions. We do not write this sum explicitly because the coefficients multiplying_n_1,n_2,n_3functions become too cumbersome. Nevertheless, these new coefficients are still only rational functions that depend onx,y,ν_1,ν_2andν_3and application of recursion relations effectively reduces the cost of evaluating theB_222diagram by roughly a factor of 2. § DERIVATION OF (Ν_1,Ν_2, Ν_3 ;X,Y) Let us begin with the usual Feynman parametrization∫_1/q^2ν_1|_̨1-|^2ν_2 |_̨2+|^2ν_3 =Γ(ν_123)/Γ(ν_1) Γ(ν_2) Γ(ν_3)×∫_0^1 du_1 ∫_0^1 du_2 ∫_0^1 du_3 ∫_ u_1^ν_1-1u_2^ν_2-1u_3^ν_3-1δ^(D)(1-u_1-u_2-u_3)/(u_1q^2 +u_2|_̨1-|^2 + u_3 |_̨2+|^2 )^ν_123 = Γ(ν_123)/Γ(ν_1) Γ(ν_2) Γ(ν_3)∫_0^1 du_1 ∫_0^1-u_1 du_2 ∫_ u_1^ν_1-1u_2^ν_2-1(1-u_1-u_2)^ν_3-1/(u_1q^2 +u_2|_̨1-|^2 + (1-u_1-u_2) |_̨2+|^2 )^ν_123 .Next, we do the following change of variables:u_1=uvandu_2=(1-u)v. This transforms(1-u_1-u_2)intovand now both integrals inuandvhave the same boundaries[0,1]∫_1/q^2ν_1|_̨1-|^2ν_2 |_̨2+|^2ν_3 = Γ(ν_123)/Γ(ν_1)Γ(ν_2)Γ(ν_3)×∫_0^1 du ∫_0^1 dv ∫_ u^ν_1-1(1-u)^ν_2-1v^ν_12-1(1-v)^ν_3-1/( uvq^2 +(1-u)v|_̨1-|^2 +(1-v)|_̨2+|^2 )^ν_123 .At this point the momentum integral can be done easily. In the denominator we first complete the squareuvq^2 +(1-u)v|_̨1-|^2 +(1-v)|_̨2+|^2= (- (1-u) v_̨1 + (1-v)_̨2 )^2 + v ( u v (1-u) k_1^2 + u(1-v)k_2^2 + (1-u) (1-v) k_3^2),and use the following identity to do the integral in∫_1/(q^2 +m^2)^ν_123 = 1/8π^3/2Γ(ν_123-3/2)/Γ(ν_123)1/(m^2)^ν_123-3/2 .The expression for one-loop bispectrum simplifies and we are left with two integrals inuandv∫_1/q^2ν_1|_̨1-|^2ν_2 |_̨2+|^2ν_3 = k_1^3-2ν_123/8π^3/2Γ( ν_123-3/2)/Γ(ν_1)Γ(ν_2)Γ(ν_3)×∫_0^1 du ∫_0^1 dv u^ν_1-1(1-u)^ν_2-1v^1/2-ν_3(1-v)^ν_3-1/( u v (1-u) + u(1-v) y + (1-u) (1-v) x )^ν_123-3/2 ,from which we can read off(ν_1,ν_2, ν_3 ;x,y)(ν_1,ν_2, ν_3 ;x,y) = 1/8π^3/2Γ(ν_123-3/2)/Γ(ν_1)Γ(ν_2)Γ(ν_3)×∫_0^1 du ∫_0^1 dv u^ν_1-1(1-u)^ν_2-1v^1/2-ν_3(1-v)^ν_3-1/( u v (1-u) + u(1-v) y + (1-u) (1-v) x )^ν_123-3/2 .Notice that the denominator is linear invand that the integral invis nothing but the hypergeometric function(ν_1,ν_2,ν_3 ;x,y) = Γ(3/2-ν_3) Γ( ν_123-3/2)/4π^2Γ(ν_1) Γ(ν_2)∫_0^1 du u^ν_1-1(1-u)^ν_2-1× (x(1-u)+y u)^3/2-ν_123 _2F_1(3/2 - ν_3, ν_123-3/2, 3/2, 1- u(1-u)/x(1-u)+y u).At this point it is useful to transform this expression using (<ref>) (ν_1,ν_2,ν_3;x,y) = Γ(3/2-ν_3) Γ( ν_123-3/2)/4π^2Γ(ν_1) Γ(ν_2)∫_0^1 duu^ν_1-1(1-u)^ν_2-1(x(1-u)+y u)^3/2-ν_123[ Γ( 3/2) Γ(ν_12-3/2)/Γ( 3/2-ν_3 ) Γ( ν_123-3/2)u^3/2-ν_12(1-u)^3/2-ν_12/(x(1-u)+y u)^3/2-ν_12 _2F_1(ν_3, 3-ν_123, 5/2-ν_12, u(1-u)/x(1-u)+y u) . . + Γ( 3/2) Γ( 3/2- ν_12)/Γ(ν_3)Γ(3-ν_123) _2F_1(3/2 - ν_3, ν_123-3/2, ν_12- 1/2, u(1-u)/x(1-u)+y u)].The reason is that0≤ u(1-u)/(x(1-u)+y u)≤1for anyxandy, and one can use the power series representation of hypergeometric functions in order to solve the integral inu. Notice that this power series keeps the integral as simple aspossible, because only powers oru,(1-u)and(x(1-u)+y u)appear in the expression. Simplifying the gamma functions we get(ν_1,ν_2, ν_3;x,y) = (πν_12)/8√(π)Γ(ν_1) Γ(ν_2) Γ(ν_3)Γ(3-ν_123) [ ∑_n=0^∞Γ(ν_3+n) Γ(3-ν_123+n)/Γ(5/2-ν_12+n )n!∫_0^1 du u^1/2-ν_2+n(1-u)^1/2-ν_1+n/(x(1-u)+y u)^ν_3+n. . - ∑_n=0^∞Γ(3/2 - ν_3+n ) Γ( ν_123-3/2+n)/Γ( ν_12- 1/2+n )n!∫_0^1 du u^ν_1-1+n(1-u)^ν_2-1+n/(x(1-u)+y u)^ν_123-3/2+n].The integration inuleads to another hypergeometric function. The result can be written in the following way(ν_1,ν_2,ν_3;x,y) = (πν_12)/8√(π)Γ(ν_1) Γ(ν_2) Γ(ν_3)Γ(3-ν_123)[ ∑_n=0^∞ b_n(ν_1,ν_2,ν_3)· x^-ν_3-n _2F_1(3/2 - ν_2+n, ν_3+n, 3-ν_12+2n, 1-y/x) . . - ∑_n=0^∞ b_n(ν̃_1,ν̃_2,ν̃_3)·x^-ν_2-n y^3/2-ν_13 _2F_1(3/2-ν_3+n, ν_2+n, ν_12+2n, 1-y/x)],where the coefficientsb_nare given byb_n(ν_1,ν_2,ν_3) = Γ(ν_3+n) Γ(3-ν_123+n)/Γ(5/2-ν_12+n )n!Γ(3/2-ν_1+n ) Γ(3/2-ν_2+n )/Γ(3-ν_12+2n ) .One last step is to use the identity(ν_1,ν_2,ν_3;x,y)=x^3/2-ν_123(ν_3,ν_2,ν_1;1/x, y/x) ,in order to bring the result to its final form(ν_1,ν_2,ν_3;x,y) = (πν_23)/8√(π)Γ(ν_1) Γ(ν_2) Γ(ν_3) Γ(3-ν_123)∑_n=0^∞[ x^3/2-ν_23· b_n(ν_3,ν_2,ν_1) x^n _2F_1(ν_1+n, 3/2-ν_2+n, 3-ν_23+2n, 1-y ) . . -y^3/2-ν_13· b_n(ν̃_3,ν̃_2,ν̃_1) x^n _2F_1(ν_2+n, 3/2 -ν_1+n, ν_23+2n, 1-y) ].This precisely matches eq. (<ref>) wherea_n(ν_1,ν_2,ν_3) = b_n(ν_3,ν_2,ν_1).§ DERIVATION OF (Ν_1,…,Ν_5)We are interested in calculatingK(ν_1,…, ν_5) ≡∫_𝒟_1 x^-ν_5 y^-ν_4 J(ν_1,ν_2,ν_3,x,y), where the region of integration is given by 𝒟_1 ={ (x,y)| √(x)+√(y)≥ 1,x≤1,y≤ 1 } .In this domain the power series representation of(ν_1,ν_2,ν_3;x,y)is uniformly convergent, and we can use it to rewrite the integral in the following wayK(ν_1, …, ν_5) = (πν_23)/8√(π)Γ(ν_1) Γ(ν_2) Γ(ν_3) Γ(3-ν_123)∫_0^1dy∫_(1-√(y))^2^1 d x[ x^3/2-ν_23∑_n=0^∞a_n(ν_1,ν_2,ν_3)·x^n _2F_1(ν_1+n,32-ν_2+n, 3-ν_23+2n, 1-y ) . . -y^3/2-ν_13∑_n=0^∞ a_n(ν̃_1,ν̃_2,ν̃_3)·x^n _2F_1(ν̃_1+n,32-ν̃_2+n, 3-ν̃_23+2n, 1-y ) ].Therefore, the basic integral that we want to solve is the integral over the hypergeometric function. For simplicity, let us define κ(α,β,a,b,c) ≡∫_0^1 dy ∫_(1-√(y))^2^1 d xx^α y^β _2F_1(a,b,c,1-y).The integral inxis straightforward, leading toκ(α,β,a,b,c) = 1/1+α[∫_0^1 dy y^β _2F_1(a,b,c,1-y) . . - 2 ∫_0^1 dt t^2β+1 (1-t)^2+2α _2F_1(a,b,c,1-t^2) ],where in the second integral we did a change of variablesy=t^2. Both integrals can be expressed in terms of higher order hypergeometric functions. It is not difficult to findκ (α,β,a,b,c) = 1/1+α[ 1/1+β _3F_2([1,a,b;; 2+β,c ; ]1) . - 2·_4F_3([ a,b,1+β,3/2+β;; 1-d ,5/2+γ,3+γ ; ]1) . -2·_4F_3([ c-a,c-b,1+β+d,3/2+β+d;;1+d,d+γ+5/2,3+γ +d; ]1) ],whered=c-a-bandγ=α+β. The integration of_2F_1(⋯,1-y)function is straightforward. To integrate_2F_1(⋯,1-t^2)we first have to use (<ref>), expand the hypergeometric functions in power series, integrate in series, then resum the result.99 Lifshitz:1946a E. M. Lifshitz, “On the gravitational stability of the expanding universe," Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 587, 16 (1946) Fry:1983cjJ. N. 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Zaldarriaga, “Renormalized Halo Bias,” JCAP1408, 056 (2014)[arXiv:1402.5916 [astro-ph.CO]]. Mirbabayi:2014zcaM. Mirbabayi, F. Schmidt and M. Zaldarriaga, “Biased Tracers and Time Evolution,” JCAP1507, no. 07, 030 (2015)[arXiv:1412.5169 [astro-ph.CO]]. Desjacques:2016bnmV. Desjacques, D. Jeong and F. Schmidt, “Large-Scale Galaxy Bias,” arXiv:1611.09787 [astro-ph.CO]. Baldauf:2014qfaT. Baldauf, L. Mercolli, M. Mirbabayi and E. Pajer, “The Bispectrum in the Effective Field Theory of Large Scale Structure,” JCAP1505, no. 05, 007 (2015)[arXiv:1406.4135 [astro-ph.CO]]. Angulo:2014tfaR. E. Angulo, S. Foreman, M. Schmittfull and L. Senatore, “The One-Loop Matter Bispectrum in the Effective Field Theory of Large Scale Structures,” JCAP1510, no. 10, 039 (2015)[arXiv:1406.4143 [astro-ph.CO]]. Davydychev:1992xrA. I. Davydychev, “Recursive algorithm of evaluating vertex type Feynman integrals,” J. Phys. A25, 5587 (1992). Exton:1994abH. Exton, “On the system of partial differential equations associated with Appell's function F_4,” Journal of Physics A Mathematical General28, 631-641 (1995). Morita:1996abT. Morita, “Use of the Gauss Contiguous Relations in Computing the Hypergeometric Functions F(n+1/2,n+1/2,m;z),” Interdisciplinary Information Sciences2, 63-74 (1996). Carrasco:2013svaJ. J. M. Carrasco, S. Foreman, D. Green and L. Senatore, “The 2-loop matter power spectrum and the IR-safe integrand,” JCAP1407, 056 (2014)[arXiv:1304.4946 [astro-ph.CO]]. Carrasco:2013muaJ. J. M. Carrasco, S. Foreman, D. Green and L. Senatore, “The Effective Field Theory of Large Scale Structures at Two Loops,” JCAP1407, 057 (2014)[arXiv:1310.0464 [astro-ph.CO]]. Baldauf:2015ahaT. Baldauf, L. Mercolli and M. Zaldarriaga, “Effective field theory of large scale structure at two loops: The apparent scale dependence of the speed of sound,” Phys. Rev. D92, no. 12, 123007 (2015)[arXiv:1507.02256 [astro-ph.CO]]. Cataneo:2016suzM. Cataneo, S. Foreman and L. Senatore, “Efficient exploration of cosmology dependence in the EFT of LSS,” JCAP1704, no. 04, 026 (2017)[arXiv:1606.03633 [astro-ph.CO]]. Broadhurst:1986bxD. J. Broadhurst, “Exploiting the 1.440 Fold Symmetry of the Master Two Loop Diagram,” Z. Phys. C32, 249 (1986).Barfoot:1987kgD. T. Barfoot and D. J. Broadhurst, “Z(2) X S(6) Symmetry of the Two Loop Diagram,” Z. Phys. C41, 81 (1988).Czakon:2005rkM. Czakon, “Automatized analytic continuation of Mellin-Barnes integrals,” Comput. Phys. Commun.175, 559 (2006)[hep-ph/0511200]. Johansson:2012sfH. Johansson, D. A. Kosower and K. J. Larsen, “An Overview of Maximal Unitarity at Two Loops,” PoS LL2012, 066 (2012) [arXiv:1212.2132 [hep-th]]. Henn:2014qgaJ. M. Henn, “Lectures on differential equations for Feynman integrals,” J. Phys. A48, 153001 (2015)[arXiv:1412.2296 [hep-ph]]. Mohammed:2016sreI. Mohammed, U. Seljak and Z. Vlah, “Perturbative approach to covariance matrix of the matter power spectrum,” Mon. Not. Roy. Astron. Soc.466, no. 1, 780 (2017)[arXiv:1607.00043 [astro-ph.CO]]. Barreira:2017kxdA. Barreira and F. Schmidt, “Response Approach to the Matter Power Spectrum Covariance,” arXiv:1705.01092 [astro-ph.CO].Assassi:2017leaV. Assassi, M. Simonović and M. Zaldarriaga, “Efficient Evaluation of Cosmological Angular Statistics,” arXiv:1705.05022 [astro-ph.CO].] | http://arxiv.org/abs/1708.08130v1 | {
"authors": [
"Marko Simonović",
"Tobias Baldauf",
"Matias Zaldarriaga",
"John Joseph Carrasco",
"Juna A. Kollmeier"
],
"categories": [
"astro-ph.CO",
"hep-th"
],
"primary_category": "astro-ph.CO",
"published": "20170827195521",
"title": "Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals"
} |
B⃗E⃗v⃗x⃗k⃗k⃗^'r⃗k_∥k_⊥J_n(z)J_n^'(z)2J_n^2(z)V⃗f_a^(0)v_∥v_⊥Ωp_∥p_⊥p⃗p_⊥p_∥łαωω _RΩ _aΩ _eω _p,aω ^2_p,aω ^2_p,eΘ _pΘ _eΘu_aβ _∥ , aβ _∥w^2αδ̣ϵ _a,sδ_a,sϵ _nsrϵ _psrβ _c∂ _σaperiodic[ u_a√(1-E^-2)μ _aarctan 𝕀∋[ κP_e, critR_⊥R_∥PIC simulations and analytical predictionsInstitute for Physics and Astronomy, University of Potsdam, D-14476 Potsdam, GermanyDESY, Platanenallee 6, D-15738 Zeuthen, Germany Instytut Fizyki Ja̧drowej PAN, ul. Radzikowskiego 152, 31-342 Kraków, PolandPair beams produced by very high-energy radiation from TeV blazars emit gamma rays in the GeV band by inverse-Compton scattering of soft photons. The observed GeV-band signal is smaller than that expected from the full electromagnetic cascade. This means that the pair beams must be affected by other physical processes reducing their energy flux. One possible loss mechanism involves beam-plasma instabilities that we consider in the present work. For realistic parameters the pair beams can not be simulated by modern computers. Instead, we use a simple analytical model to find a range of the beam parameters that (i) provides a physical picture similar to that of realistic pair beams and (ii) at the same time can be handled by available computational resources. Afterwards, we performed corresponding 2D PIC simulations. We confirm that the beams experience only small changes in the relevant parameter regime, and other processes such as deflection in magnetic field must be at play. Plasma effects on relativistic pair beams from TeV blazars:I. Rafighi,[email protected], S. Vafin1, M. Pohl1,2, and J. Niemiec3 Accepted August 22, 2017==============================================================================§ INTRODUCTION Many blazars, a sub-class of active galactic nuclei, have been detected with gamma-ray telescopes such as HESS, VERITAS, Fermi and MAGIC as sources of gamma-rays with the energy E≥ 100 GeV <cit.>. These very-high energy photons interact with extragalactic background light (EBL) producing ultra-relativistic electron-positron pairs with the typical Lorentz factor 10^5<Γ<10^7<cit.>. The created pairs are subject of many investigations, as they can be affected by several physical processes: (i) inverse Compton scattering (ICS), (ii) deflection by the intergalactic magnetic field (IGMF), or (iii) collective plasma effects. The ICS would result in gamma-ray emission with characteristic energy in the GeV band. But, as indicated by Fermi-LAT data, the GeV gamma-ray emission is suppressed meaning that the ICS is not the fastest of the three processes. The effect of deflection by the IGMF has been well investigated <cit.> which led to constraints on the IGMF. However, these constraints are valid only under the assumptions that the multi-TeV gamma-ray emission persists on long timescales and that the pairs lose their energy only due to ICS. The last assumption is very crucial and remains debatable. The importance of collective plasma effects has been pointed out by several authors <cit.>. In fact, the pairs can induce electrostatic (two-stream, oblique) and electromagnetic (filamentation, Weibel) instabilities <cit.>. In this case, wave-particle interactions can reduce the energy of the pairs by 30-50 %<cit.>. Therefore, the collective plasma effects can also substantially suppress the GeV-band gamma-ray emission affecting as well the IGMF constraints. The pair beams constitute an extremely small fraction of the plasma density in the intergalactic medium (IGM), α=n_b/n≈ 10^-16-10^-18. This circumstance prohibits direct computer simulations of the beams due to insufficient computational power, and substantial adjustments in parameter values have been made in published simulation studies <cit.>.At the same time, an accurate analytical description of the non-linear evolution of the plasma system is also problematic. In this work, we combine numerical PIC simulations with a simple analytical model to determine physical parameters of the beam and plasma, so that (i) the problem can be treated with reasonable computational power, and (ii) the physical picture is adequate to realistic pair beams. The physical picture is determined by several aspects: (i) the ratio of the energy densities of the beam and background plasma, (ii) instabilities and their growth rates, and (iii) non-linear damping of plasma waves. Here, we are concerned only with the first two subjects, and the non-linear effects <cit.> will be analyzed in future papers.Such treatment is possible in the linear stage which we are interested in here. The created pairs are subject to the ICS and a full electromagnetic cascade that modifies their parameters. However, the goal of the present paper is to explore the potential dominance of plasma effects on the beam evolution. Therefore, we consider a pair beam created only by the initial TeV gamma-ray emission neglecting ICS. In this case, the typical parameters of the created beams depend on the distance from a blazar, and they are ⟨Γ⟩=10^5, Γ=10^3-10^8, n_b=10^-25-10^-19 cm^-3, Δθ≈1/⟨Γ⟩≈10^-5 (Δθ is the angular spread), whereas typical parameters of the IGM are T=10^4-10^7 K, n=10^-7 cm^-3<cit.>. Thus, the energy density ratio is ϵ=n_b⟨Γ⟩ m_e c^2/ (n k_B T)≈10^-10-10^-1 (k_B is the Boltzmann constant, m_e is the electron mass) indicating that the pair beam cannot considerably heat the IGM plasma. This point was realized by <cit.> who conducted simulations for ϵ=0.1. The parameters of the simulations by <cit.> are α=n_b/n≈ 10^-2, Γ≈10^2, and k_BT/(m_e c^2)≈ 10^-8, providing with ϵ≈ 10^8, a parameter regime that is not relevant for realistic pair beams. Moreover, such a high energy-density ratio causes anisotropic plasma heating that can eventually drive the Weibel instability as it will be shown below. Note that Kempf and Sironi have studied a beam distribution with ΔΓ≪⟨Γ⟩. We will also investigate this case in the present work, whereas a realistic distribution with ΔΓ≫⟨Γ⟩ will be studied in a separate paper. The pair beam can induce two unstable modes: electrostatic and electromagnetic. The growth rate of these instabilities sensitively depends on the momentum spread of the beam. If the momentum spread is small enough, then the instabilities evolve in the so-called reactive regime. In this case, the beam can be mathematically treated as a delta function <cit.> and the growth rates of the electrostatic and electromagnetic instabilities are maximal perpendicular to the direction of the beam propagation <cit.>. As the momentum spread increases, the electromagnetic instability becomes stabilized <cit.>, while the maximum growth rate of the electrostatic mode shifts to the direction parallel to the beam propagation <cit.>. This is the so-called kinetic regime. <cit.> have argued that the momentum spread of the realistic pair beam drastically reduces the growth rate of the electrostatic instability. Later, <cit.> have disputed this statement. <cit.> have demonstrated that the maximum growth rate occurs in the direction almost parallel to the beam (contrary to the reactive regime, when the maximum growth rate occurs in quasi-perpendicular direction to the beam). But <cit.> have assumed the parallel direction of the wave vector from the very beginning. In this case, the electrostatic growth rate, indeed, only weakly depends on the beam temperature <cit.>. Thus, we can conclude that the electrostatic instability for a blazar-induced beam evolves in the kinetic regime at all angles with the maximum growth rate parallel to the beam propagation. It can be shown (see below) that for the beam parameters used by <cit.> and <cit.> the electrostatic instability has evolved in the reactive regime. Thus, an adequate behaviour of the instability has not been simulated before.So far we have discussed only the electrostatic instability. Usually, the electromagnetic (Weibel) instability can be neglected due to its smaller growth rate, but that is not always the case. If we compare the growth rate of the parallel electrostatic instability with the maximum Weibel growth rate γ_W≈ (V_b/c)(α/Γ)^1/2, then γ_W/γ_react,∥≈α^1/6Γ^1/2=0.3-1 for α=10^-18-10^-15. Thus, the Weibel instability can be potentially competitive with the electrostatic one. Note that we have used γ_W assuming that the beam does not have any momentum spread, and the situation can be different for a beam with a finite temperature. <cit.> have shown that the Weibel instability is strongly suppressed by the non-relativistic perpendicular temperature of the beam. In this work, we will investigate the case of a relativistic temperature and demonstrate that the Weibel instability is suppressed in the case of a realistic blazar-induced beam. Additionally, we will demonstrate that for other conditions (relevant for PIC simulations) this mode can grow. Consequently, three criteria for a physically relevant simulation setup can be specified: (i) the energy density ratio, ϵ, must be much smaller than unity, (ii) the beam temperature must be high enough, so that the parallel electrostatic instability evolves in the kinetic regime at all angles, and (iii) the Weibel instability must be suppressed. The goal of the current work is to find parameters satisfying all these requirements and to model them using PIC simulations. In Sec. <ref>, we develop a simple analytical model of plasma instabilities. In Sec. <ref>, we evaluate a condition for the parallel electrostatic instability to be in the kinetic regime. In Sec. <ref>, we discuss our choice of physical parameters for PIC simulations. Sec. <ref> presents simulation results and their discussion. The final summary is given in Sec. <ref>.§ ANALYTICAL MODEL We already noted that the electrostatic instability evolves in the kinetic regime and has its maximum growth rate in the direction almost parallel to the beam propagation. At the same time, <cit.> demonstrated that the growth rate of the parallel electrostatic instability very weakly depends on the momentum spread of the beam. Therefore, we can use the well-known growth rate of the two-stream instability for a cold plasma,γ_TS= 3^1/2/2^4/3ω_pα^1/3Γ^-1. Thus, we need to investigate only the electromagnetic Weibel instability. It should be noted that the most unstable wave vector of the Weibel mode can be in transverse directionto the beam <cit.> as well as in the oblique direction <cit.>. Moreover, the work by <cit.> shows that for dilute beams the maximum growth rates of Weibel mode in the transverse and oblique directions can differ by a factor 2. Therefore, to make a rough estimation, we will study the Weibel instability only for wave vectors perpendicular to the beam. The PIC simulations described in the next section should include oblique modes as well. To derive analytical results, the beam-plasma system is modeled by a waterbag distribution <cit.>. Then, the distributions of the beam and the plasma, respectively, are f_b( p)= n_b 4p_⊥,b^2 (p_∥,b^+-p_∥,b^-)ł[ θł(p_z+p_⊥,b)̊-θł(p_z-p_⊥,b)̊]̊ ׳[ θł(p_y+p_⊥,b)̊-θł(p_y-p_⊥,b)̊]̊ł[ θł(p_x-p_∥,b^-)̊-θł(p_x-p_∥,b^+)̊]̊ ,f_p( p)= n 8p_⊥,p^2 p_∥,pł[ θł(p_z+p_⊥,p)̊-θł(p_z-p_⊥,p)̊]̊ ׳[ θł(p_y+p_⊥,p)̊-θł(p_y-p_⊥,p)̊]̊ł[ θł(p_x+p_∥,p)̊-θł(p_x-p_∥,p)̊]̊ ,where p_∥,b^±=p_0± p_∥,b; p_0 is the beam drift momentum; p_∥,b and p_⊥,b, respectively, the parallel and perpendicular momentum spreads of the beam; p_∥,p and p_⊥,p, respectively, the parallel and perpendicular momentum spreads of the background plasma; θ(x), the Heaviside step function. The beam and background plasma are assumed to be homogeneous with number densities, accordingly, n_b and n. It is useful to consider separately two cases:(i) p_⊥,b=p_∥,p=p_⊥,p=0 and (ii) p_∥,b=0. §.§ Case p_⊥,b=p_∥,p=p_⊥,p=0 We derive the dispersion equation for this case in Appendix A. It reads ł[ 1-ω_p^2ω -ω_b^2ω^2U_1 ]̊ł{1- ł(kcω)̊^2 - ω_p^2ω^2- .̊ ł. -ω_b^2ω^2ł[ ł(kcω)̊^2 U_1 +ł( 1- ł(kcω)̊^2)̊U_2]̊}̊- - ł(ω_b^2ω^2kcωU_3 )̊^2=0. Taking the limiting case p_∥,b≪ p_0, Eq. (<ref>) provides the classical text book result <cit.> ł[ 1-ω_p^2ω -ω_b^2Γω^2]̊ł{1- ł(kcω)̊^2 - ω_p^2ω^2- ω_b^2Γ^3ω^2- .̊ ł. - ω_b^2Γω^2ł(kV_oω)̊^2 }̊-ł(ω_b^2Γω^2kcω)̊^2=0,where V_0=p_0/(m_eΓ). Eq. (<ref>) predicts an instability with growth rate <cit.>:γ=ω_pβ_0 ł(αΓ)̊^1/2kcł(ω_p^2+ (kc)^2 )̊^1/2. Now, we will show that the solution (<ref>) is only slightly different for a large parallel momentum spread p_∥,b≫ p_0. Assuming p_∥,b^+≫ mc and p_∥,b^-≪ mc, we obtain U_1≈ U_3≈m_ec p_∥,b^+lnp_∥,b^+ m_ec,U_2≈m_ec p_∥,b^+. Neglecting unity in each bracket in Eq. (<ref>) results in the solutionγ= ω_p ł[ αm_e c p_∥,b^+lnp_∥,b^+ m_ec]̊^1/2kcł(ω_p^2+ (kc)^2 )̊^1/2. It is well seen from Eq. (<ref>) that even for p_∥,b^+=10^2 p_0, the difference between the solutions (<ref>) and (<ref>) is only a factor of 0.4. Thus, we can neglect the parallel momentum dispersion of the beam and use p_∥,b=0. §.§ Case p_∥,b=0 The dispersion equation for p_∥,b=0 is derived in Appendix B and has the following formł[ 1 - ω_p,p^2ω^2- ω_p,e^2 ω^2- (kv_⊥,p)^2 - ω_b,⊥^2ω^2 - (ku)^2]̊× ×ł[ 1 - (kc)^2+ω_p^2 + ω_b,∥^2 ω^2 -(kv_∥,p)^2 ω_p,e^2 3ω^2( ω^2 - (kv_⊥,p)^2 )- .̊ ł.- (kV_0)^2ω_b,⊥^2 ω^2 (ω^2 - (ku)^2)]̊ - ł[ kV_0ω_b,⊥^2 ω(ω^2- (ku)^2 )]̊^2=0,where ω_b,⊥^2=ω_b^2/Γ, ω_b,∥^2=ω_b^2/Γ^3. In principle, one can analyze Eq. (<ref>) analytically, but it is more useful and easier to treat two limiting cases of the cold background plasma and the cold beam.§.§.§ Cold background plasma v_∥,p=v_⊥,p=0 Neglecting unity in each bracket in Eq. (<ref>), we obtain that for p_⊥,b≥ p_0(α/Γ)^1/2 the solution is purely real (no instability can arise), whereas for p_⊥,b< p_0(α/Γ)^1/2 the Weibel mode is unstable for k<(ω_p/c)[(α/Γ)(p_0/p_⊥,b)^2-1]^1/2 with growth rateγ= ł( ω_p^2β_0^2αΓ(kc)^2(kc)^2+ω_p^2- (ku)^2 )̊^1/2 . Let us now assume that the beam obeys a relativistic Maxwellian distribution: f( p)= μ 4π (m_ec)^3Γ^2 K_2(μ/Γ)e^-μł[ ł( 1+p^2(m_ec)^2)̊^1/2- β_0 p_x m_ec]̊,where μ=Γμ_R =Γ m_ec^2/(k_BT_b), β_0=V_0/c. Here, T_b is the temperature of the beam in its rest frame. Then we can evaluate Δ p_⊥ (see Appendix C) and write the condition for the Weibel mode stability as α≤α_W= 2-π/2 Γμ_R=(2-π/2) k_BT_b/Γ m_ec^2. In the simulations by <cit.>, magnetic-field fluctuations grew at early times due to the Weibel instability driven by the beam, because α=10^-2 and α_W<5×10^-4 led to condition (<ref>) not being fulfilled. But in the simulations by <cit.>, the Weibel mode was suppressed, because α=2×10^-6<α_W=10^-5. For a realistic blazar-induced beam, the Weibel instability is also suppressed, since k_BT_b≈ m_ec^2 and α≪ 1/<Γ>.§.§.§ Cold beam p_⊥,b=0 Again neglecting unity in Eq. (<ref>), we can approximate it as Eω^4 + Fω^2 +G = 0.where E= (kc)^2+ω_p^2 , F= ł(kV_0ω_b,⊥)̊^2+ 13(kv_∥,pω_p,e)^2- (kv_⊥,p)^2((kc)^2+ω_p^2) ,G= - ł(kV_0)̊^2(kv_⊥,p)^2ω_b,⊥^2. The growth rate readsγ= ł[F+ (F^2-4EG)^1/2 2E]̊^1/2≈ł[ F+|F|2E- G|F|]̊^1/2.If F>0, Eq. (<ref>) describes the classical Weibel instability <cit.> with growth rateγ≈ł( F E)̊^1/2≈ł( (kv_∥,pω_p,e)^2/3- (kv_⊥,p)^2((kc)^2+ω_p^2)(kc)^2+ω_p^2)̊^1/2. Due to v_⊥,p≠0, the instability is stabilized at large wave vectors, but at small k the plasma is unstable for (v_∥,p/v_⊥,p)^2>3. These conditions were fulfilled in the simulations by <cit.>, where there was a growth of the magnetic-field fluctuations at later time around ω_p,et≈ 10^4. In the opposite case F<0 and assuming that v_⊥,p is large enough, Eq. (<ref>) reduces to Eq. (<ref>). § CONDITION FOR THE KINETIC REGIME The parallel electrostatic instability evolves in the kinetic regime <cit.>, ifł| v_∥,b c |̊≫α^1/3Γ^-1 which can be re-written as α≪α_kin= ł(Γł| v_∥,b/c|̊)̊^3. An analytical expression for α_kin is derived in Appendix C, and its functional behavior is illustrated in Fig. <ref>. For the simulation parameters used by <cit.>, μ_R=5×10^3 (T_b=10^6 K), Γ=10, and α=2.5×10^-6, we obtain α_kin≈2.8× 10^-9 and Eq. (<ref>) is not fulfilled.For the work of <cit.>, Γ=300, α=10^-2, μ>3, it results in α_kin≈ 7.1×10^-9, and Eq. (<ref>) is not satisfied again. Hence, both <cit.> and <cit.> did not simulate the electrostatic instability in the appropriate kinematic regime of pair cascades from AGN. § CHOICE OF PARAMETERS FOR PIC SIMULATIONS In the introduction, we have specified three criteria for a physically relevant setup for the beam-plasma system. First, the energy density ratio must satisfy ϵ=αΓ m_ec^2/(k_B T_p)≪1 yieldingα≪α_ϵ(Γ)= k_BT_pΓ m_ec^2≈10^-6-10^-3Γ.The behavior of α_ϵ(Γ) is shown in Figures <ref>, <ref>, and <ref> as red line. Second, the electrostatic instability should develop in the kinetic regime at all angles which is determined by Eq. (<ref>), for which we indicate α_kin by the green line in the figures. Lastly, the Weibel mode must be stable which requires satisfying Eq. (<ref>). Eq. (<ref>) is automatically fulfilled due to α_kin≪α_W for Γ>1 and μ_R>1. Fig. <ref> compares the functions α_kin(Γ) and α_ϵ(Γ). To satisfy Eqs. (<ref>) and (<ref>) for given values of Γ and μ, the value of α must be below both curves α_kin(Γ) and α_ϵ(Γ). We defined a simulation setup, henceforth referred to as run 1, that would satisfy all criteria. The main parameter values are Γ=5 and α=2×10^-4, and it is indicated in Figure <ref> by a black dot. In addition, we have specified three other setups (runs 2-4) that are listed inTable <ref>. The goal of these tests is to determine the impact of a violation of one of the criteria on the beam-plasma evolution. For run 2, the energy density ratio ϵ=2.5 is higher than unity, and one might expect a strong heating of the background plasma and subsequently the development of other instabilities. Run 3 considers the evolution of the electrostatic instability in the reactive regime (α>α_kin), and beam energy losses are expected to be larger. Finally, all the conditions are violated for run 4. The values of (α;Γ) for runs 2-4 are demonstrated by the black dots in Figs. <ref>-<ref>, respectively.§ THE SIMULATION CODEFor the simulation purposes we use EPOCH 2D, a multi-dimensional, fully electromagnetic, relativistic particle-in-cell code developed by the Collaborative Computational Plasma Physics (CCPP) consortium and funded by the Engineering and Physical Sciences Research Council (EPSRC). PIC codes solve Maxwell’s equations on a numerical (Eulerian) grid and follow charged computational particles (CP) as they move under the influence of the electromagnetic field and provide charge and current density <cit.>.The relevant equations are μ_0ϵ_0∂ E(x,t)/∂ t =∇× B(x,t) -μ_0 J(x,t)and∂ B(x,t)/∂ t =-∇× E(x,t),where the current density, J(x,t), is computed using the algorithm of <cit.>. Collisionless plasma is set up with a Maxwellian velocity distribution. For each CP the field pusher solves the relativistic equation of motion with a numerical approximation of Lorentz force equation. EPOCH isa refined version of the basic explicit PIC algorithm with higher-order weights and interpolation schemes <cit.>.Note that the 2D model can break down on the non-linear evolution stage, when 3D mode coupling becomes important <cit.>. As the electrostatic mode involves a narrow resonance, its modeling in a PIC simulation requires a very good wavenumber resolution of the numerical grid <cit.>. This implies a large number of grid points in any direction which we can establish only in 2D. Waves of arbitrary orientation will be included, albeit with only one linear polarization, as is nonlinear wave coupling, provided it does not build on the polarization out of the simulation plane. In the current study we are mainly interested in the linear growth of the instabilities, and so we accept these limitations. The simulation resolves the x–y plane with periodic boundary conditions. The simulation volume is filled with a beam of electrons and positrons and the background plasma of protons and electrons with real mass ratio. We performed a series of tests to verify the stability of the simulation against numerical artifacts. Of particular interest is avoiding artificial plasma heating arising from electric-field noise caused by the charge-density granularity in a particle simulation. We found that using 400 particles per cell and species is required to keep the plasma temperature as desired and the electric-field noise at a level significantly below the intensity of the electrostatic mode. The desired density ratio, α=n_b/n, is established with numerical weights. The simulation box contains 1024 ×1024 cells, each 1/8 of the skin length in size,λ_e=c/ω_pe=8 Δ_x. The timestep is chosen to satisfy the CFL condition and to resolve the plasma frequency, ω_pe=(n_0e^2/ϵ_0m_e)^1/2.Table <ref> lists the temperature of the IGM plasma, T_p, and of the beam in its rest frame, T_b. The IGM particles are initially at rest, while the beam is moving in x-direction with Lorentz factor Γ_b. For the IGM, EPOCH code generates a non relativistic thermal distribution using the method of <cit.>. However, we implemented the algorithm of <cit.> to set up the relativistic Maxwellian distribution for the beam.For the graphical presentationwe use the following normalization: distance and time are normalized to c/ω_p,e and ω_p,e^-1, and electric and magnetic fields are given in units of ω_p,ecm_e/e and ω_p,em_e/e, respectively. In order to reduce the well-known PIC-code phenomena of self heating and statistical noise, all simulations are performed with a high number of CPs (400 particles per species), a 6th-order field particle pusher, and a triangular-shaped cloud (TSC) shape function, with the peak of the triangle located at the position of the pseudoparticle. § DISCUSSION OF SIMULATION RESULTS§.§ Run 1 As mentioned above, for run 1 all relevant criteria for the beam are fulfilled. First of all, the beam/plasma energy density ratio ϵ=0.5 for run 1 is smaller than unity. Moreover, the beam is stable with respect to the Weibel instability, while the electrostatic mode grows as expected in the kinetic regime, i.e. at the parallel wave vector k_||≈ω_p/c to the beam.In Fig. <ref> we present the Fourier spectrum of the electric field, and it is evident that an electrostatic mode with 𝐄∥𝐤 dominates with peak intensity for wave vectors roughly aligned with the beam direction. The linear growth rate of the electric field is about γ≃ 4×10^-4ω_pe. The theoretically calculated maximum growth rate for parallel wave vectors is 5×10^-4ω_pe which approximately agrees with that derived numerically. Figs. <ref> and <ref> demonstrate that after 16,237ω_pe^-1, corresponding to about 8 growth times, the instability has saturated with negligible energy loss and heating of the beam. The latter is of interest because a widening of the lateral beam distribution would impose a temporal smearing of the ICS signal that would reduce the expected flux seen with Fermi-LAT. Our run 1 suggests that this effect is not efficient for realistic pair beams induced by gamma rays from AGN.Fig. <ref> illustrates the time evolution of the electric and magnetic field energy density. The electric field energy saturates after ∼ 7 growth times. It is clear that the beam transferred only a tiny fraction (∼ 10^-4%) of its initial kinetic energy into the electromagnetic fields. Accordingly, the change of the beam distribution is also very small (see Figs. <ref>-<ref>). This development of the beam-plasma interaction is caused by the initial momentum spread of the beam. It was also found by <cit.> that the beam momentum distribution does not relax to the plateau form when Δ p_⊥,b/m_e c∼1. The physical reason is that the electrostatic growth rate simply becomes much smaller than in the reactive regime. At the same time, the damping rates of the modulation instability and non-linear Landau damping depend on the resonant wave energy, and therefore they will stabilize the instability at smaller electric field energies.§.§ Run 2 In contrast to run 1, run 2 considers the beam/plasma energy-density ratio, ϵ=5, greater than 1. The only parameter changed compared to run 1 is the plasma temperature that became by an order of magnitude smaller. Due to the fact that the beam parameters remained the same, the Weibel mode is still stable. The electrostatic instability also evolves in kinetic regime with a growth rate around ≃ 5×10^-4ω_pe, and the time evolution of the Fourier spectrum (shown in Fig. <ref> at ω_pet=5036) is consistent with the value.Fig. <ref> shows that the electric field energy density saturates at nearly the same level as in run 1. Note that due to a smaller plasma temperature the initial electric noise level in run 2 is about by an order of magnitude smaller compared to run 1. Although the peak intensity of the electrostatic modes is now observed at a 10^∘ angle to the beam direction, the distribution function again did not evolve appreciably, in particular not to a plateau distribution, and the beam experienced only a tiny energy loss or widening. §.§ Run 3 With run 3, we explore the reactive regime of the electrostatic mode in contrast to runs 1 and 2, where the instability was kinetic. To do this, we have reduced the temperature of the beam and increased its gamma factor. Now, the electrostatic instability grows at an oblique direction (at about 30^∘) to the beam as is evident from the Fourier spectrum shown in Fig. <ref>. The growth rate for oblique propagation and the parameters of run 3 (assuming a cold beam <cit.>) is γ_TS= 3^1/2/2^4/3ω_peł(αΓ)̊^1/3ł(k_||^2 k^2γ^2+ k_⊥^2 k^2)̊^1/3≃ 9×10^-3ω_pe ,where the last equality applies for the parameters of run 3. The numerically determined growth rate is smaller than that by a factor 2-3. This difference may be explained by the fact that run 3 operates not very far from the condition α=α_kinł(Γ)̊ (see Fig. <ref>).The instability growth rate of run 3 is larger by an order of magnitude compared to runs 1 and 2. Therefore, we can expect a more substantial modification of the beam. Although the electric-field energy density remains small as shown in Fig. <ref>, we observe in Fig. <ref> a significant transverse widening of the beam that is not seen in runs 1 and 2. Fig. <ref> indicates that the width of the perpendicular momentum distribution of the beam increased by a factor of 3.§.§ Run 4 Finally, run 4 considers a situation in which all three constraints on the beam parameters are violated. Unlike runs 1 and 2, the fastest electrostatic mode develops for wave vectors that are quasi-perpendicular to the beam, as is well seen in Fig. <ref>. The numerical growth rate perfectly agrees with the analytical estimation for a cold beam, and it is about 2.2×10^-2ω_pe which is larger than in the runs 1, 2, and 3. Furthermore, Fig. <ref> demonstrates that the electric-field energy density assumes a considerably higher value than in three other runs on account of a higher growth rate. At the same time, the Weibel mode is destabilized resulting in a strong growth of magnetic field to a field strength even larger (see the red line in Fig. <ref>) than that of the electric field. Actually, the orange line in Fig. <ref> indicates that the dominant energy transfer is that to IGM electrons (∼ 0.5%), while the magnetic field receives only ∼ 10^-5%.This affects the momentum distribution of both the beam and IGM, as we present in Figs. <ref>-<ref>. In Fig. <ref> we also see a remarkable increase in the momentum spread of the beam and the IGM. This run 4 is similar to the simulations by <cit.> who observed a similar beam-plasma evolution.§ SUMMARY We have revisited the issue of plasma instabilities induced by electron-positron beams in the fully ionized intergalactic medium. This problem is related to pair beams produced by TeV radiation of blazars. The main objective of our study is to clarify the feedback of the beam-driven instabilities on the pairs. The largest difficulty is the impossibility to simulate realistic blazar-induced beams, even with modern computational resources. Therefore, parameters must be found that permit numerical modeling with similar physical properties. Two important criteria of the realistic pair beams have been noticed before: (i), the beam/IGM energy density ratio is much smaller than unity <cit.>, and (ii), the electrostatic mode evolves in the kinetic regime <cit.>. However, the simple estimation presented in the introduction shows that the Weibel mode can potentially compete with the kinetic electrostatic instability. To clarify this point, we have used a simple analytical model and demonstrated that the Weibel mode is actually stable for realistic parameters. This adds a third criterion for the pair beams. Previous PIC studies of the blazar-induced pair beams <cit.> considered only some of these requirements on the beam-plasma system. In contrast, we have performed a simulation (run 1), for which all of them are taken into account. Then, we have compared this case with three other simulations (runs 2-4), for which some criteria were violated.The results of run 1 indicate that the pair beam does not experience any significant modification. The electrostatic growth rate turns out to be quite small, and non-linear effects stabilize the beam very efficiently. However, once the electrostatic instability becomes reactive (runs 3-4), as is the case for the studies of <cit.> and <cit.>, the beam momentum distribution widens drastically in the transverse direction. A significant widening of the beam could in principle account for the observed low flux of cascade gamma rays in the GeV band on account of temporal smearing, but that requires a widening by a factor ≫ 10. In any case beam widening is only observed if the instability develops in the reactive regime, and that is not relevant for realistic pair beams arising from interactions of AGN gamma rays with extragalactic background light. Also, if the beam/IGM energy density ratio is high, then the beam effectively heats the IGM (run 4), as was seen in the simulations by <cit.>. To summarize, we have improved modeling of plasma instabilities for blazar-induced pair beams by including three relevant criteria for the beam. Our results suggest that such instabilities play a negligible role and cannot suppress the flux of cascade gamma rays in the GeV band. Thus, other suppression mechanisms of the energy flux from TeV blazars such magnetic-field deflection must be at play. The numerical simulations were performed with the EPOCH code that was in part funded by the UK EPSRC grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/ M022463/1. The numerical work was conducted on resources provided by The North-German Supercomputing Alliance (HLRN) under project bbp00003. M.P. acknowledges support through grant PO 1508/1-2 of the Deutsche Forschungsgemeinschaft. The work of J.N. is supported by Narodowe Centrum Nauki through research project DEC-2013/10/E/ST9/00662. § APPENDIX A: DERIVATION OF THE DISPERSION EQUATION FOR P_⊥,B=P_∥,P=P_⊥,P=0 AND K_∥=0 In the case p_⊥,b=p_∥,p=p_⊥,p=0, the beam and plasma distributions, respectively, reads f_b( p)= n_b (p_∥,b^+-p_∥,b^-)δ(p_z) δ(p_y) × ×ł[ θł(p_x-p_∥,b^-)̊-θł(p_x-p_∥,b^+)̊]̊ ,f_p( p)= n δ(p_x) δ(p_y)δ(p_z) ,where δ(x) is the Dirac delta function. The dielectric tensor is given by <cit.> and <cit.>: ϵ_i,j=δ_i,j+ ∑_a=p,b4π e^2ω^2× ×∫ d^3p ł( v_if_a( p) p_j- v_iv_jk_lk v-ω f_a( p)p_l)̊. Evaluating the dielectric tensor (<ref>) for the distribution functions (<ref>)-(<ref>) and for the wave vector k=(0,0,k) yieldsϵ_zy=ϵ_yz=ϵ_yx=ϵ_xy=0, ϵ_zz=ϵ_yy=1-ω_p^2ω -ω_b^2ω^2U_1 , ϵ_xx=1- ω_p^2ω^2-ω_b^2ω^2ł[ ł(kcω)̊^2 U_1 +ł( 1- ł(kcω)̊^2)̊U_2]̊, ϵ_xz=ϵ_zx= - ω_b^2ω^2kcωU_3, whereU_1= m_ec p_∥,b^+-p_∥,b^-lnł| p_∥,b^++ [ (p_∥,b^+ )^2 +m_e^2c^2]^1/2 p_∥,b^-+ [ (p_∥,b^-)^2 +m_e^2c^2]^1/2|̊,U_2= m_ec p_∥,b^+-p_∥,b^-ł( p_∥,b^+ [ (p_∥,b^+)^2 +m_e^2c^2]^1/2 - p_∥,b^- [ (p_∥,b^-)^2 +m_e^2c^2]^1/2)̊,U_3=m_ec 2(p_∥,b^+-p_∥,b^-)lnł| (p_∥,b^+)^2 +m_e^2c^2(p_∥,b^-)^2 +m_e^2c^2 |̊. Here, we have introduced ω_b^2=8π n_be^2/m_e. The dispersion equation reads: (Λ_i,j)=ł(ϵ_i,j+ k_ik_jc^2ω^2- ł(kcω)̊^2δ_i,j)̊= =Λ_yy(Λ_zzΛ_xx-Λ_zx^2)=0. Thus, the dispersion equation for electromagnetic fluctuations is ł[ 1-ω_p^2ω -ω_b^2ω^2U_1 ]̊ł{1- ł(kcω)̊^2 - ω_p^2ω^2- .̊ ł. -ω_b^2ω^2ł[ ł(kcω)̊^2 U_1 +ł( 1- ł(kcω)̊^2)̊U_2]̊}̊- - ł(ω_b^2ω^2kcωU_3 )̊^2=0. § APPENDIX B: DERIVATION OF THE DISPERSION EQUATION FOR P_∥,B=0 AND K_∥=0 For p_∥,b=0, the distribution function of the beam reads f_b( p)= n_b 4p_⊥,b^2 ł[ θł(p_z+p_⊥,b)̊-θł(p_z-p_⊥,b)̊]̊× ×ł[ θł(p_y+p_⊥,b)̊-θł(p_y-p_⊥,b)̊]̊δ(p_x-p_0).We will assume p_⊥,b≪ p_0. We will still model background protons with the distribution (<ref>), whereas the distribution function of the background electrons is given by Eq. (<ref>). Moreover, we will assume that the background electrons are non-relativistic, p_∥,p=m_e v_∥,p and p_⊥,p=m_e v_⊥,p. Now, it is easy to find that again ϵ_zy=ϵ_yz=ϵ_yx=ϵ_xy=0, but ϵ_zz= 1 - ω_p,p^2ω^2- ω_p,e^2 ω^2- (kv_⊥,p)^2 - ω_b^2/Γω^2 - (ku)^2,ϵ_yy= 1- ω_p^2+ω_b^2/Γω^2 - (kv_⊥,p)^2 ω_p,e^2 3ω^2( ω^2 - (kv_⊥,p)^2 )--(ku)^2ω_b^2/Γ 3ω^2 (ω^2 - (ku)^2 ) ,ϵ_xx= 1 - ω_p^2 + ω_b^2/Γ^3 ω^2 -(kv_∥,p)^2 ω_p,e^2 3ω^2( ω^2 - (kv_⊥,p)^2 )-- (kV_0)^2ω_b^2/Γω^2 (ω^2 - (ku)^2), ϵ_xz=ϵ_zx= - kV_0ω_b^2/Γω(ω^2- (ku)^2 ),where ω_p,e=(4π ne^2/m_e)^1/2, ω_p,p=(4π ne^2/m_p)^1/2 (m_p is the proton mass), u=V_0p_⊥,b/p_0.Finally, the Weibel instability is described by the equation ł[ 1 - ω_p,p^2ω^2- ω_p,e^2 ω^2- (kv_⊥,p)^2 - ω_b^2/Γω^2 - (ku)^2]̊× ×ł[ 1 - (kc)^2+ω_p^2 + ω_b^2/Γ^3 ω^2 -(kv_∥,p)^2 ω_p,e^2 3ω^2( ω^2 - (kv_⊥,p)^2 )- .̊ ł.- (kV_0)^2ω_b^2/Γω^2 (ω^2 - (ku)^2)]̊ - ł[ kV_0ω_b^2/Γω(ω^2- (ku)^2 )]̊^2=0. § APPENDIX C: APPROXIMATION FOR Α_KIN AT LARGE VALUES OF Μ_RFor μ_R≫1, we can use the series expansion c(p^2+m_e^2c^2)^1/2- V_0p_x ≈mc^2Γ+ (p_x-p_0)^2 2m_eΓ^3 + p_z^2+p_y^2 2m_eΓ .Then Eq. 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M., Bludman, S. A., and Rosenbluth, M. N. 1960, Phys. Fluids, 3, 741 [Yoon & Davidson(1987)]Yoon87Yoon, P. H., & Davidson, R. C. 1987, Phys. Rev. A, 35, 2718[Zenitani(2015)]Zenitani15Zenitani, S. 2015, Phys. Plasmas, 22, 042116 ] ] | http://arxiv.org/abs/1708.07797v1 | {
"authors": [
"Iman Rafighi",
"Sergei Vafin",
"Jacek Niemiec",
"Martin Pohl"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170825160731",
"title": "Plasma effects on relativistic pair beams from TeV blazars: PIC simulations and analytical predictions"
} |
CloudScan - A configuration-free invoice analysis system using recurrent neural networksRasmus Berg Palm DTU Compute Technical University of Denmark [email protected] Winther DTU Compute Technical University of Denmark [email protected] Florian Laws Tradeshift Copenhagen, Denmark [email protected] 30, 2023 =============================================================================================================================================================================================================================== Computational fluid dynamics (CFD) studies are carried out on a two-bladed vertical axis wind turbine (VAWT) operating at a wind speed of 8 m/s for the tip speed ratios (λ) of 0.50 - 3.0. The blade consists of the NACA 0021 airfoil with the chord length of 0.265 m and a rotor radius of 1 m. Basic sensitivity studies for various time step sizes are carried out. The results are validated against available measurement data from literature. An excellent agreement is obtained for small λ up to optimum condition. For the higher tip speed ratios, the two dimensional CFD computations predict higher results than the wind tunnel experiment, but they are very similar to the field measurement data. Wake characteristics are presented in the present studies, showing that the wake becomes Gaussian at 5 times radius downsteam of the rotor. It is shown that complex flow phenomena occur due to dynamic stall onset especially for the smaller tip speed ratio.Keywords: Aerodynamics, CFD, Flow separation, VAWT § INTRODUCTIONThe growing demand for energy and the intensification of global climate change, the social development, and human survival are under a great threat due to shortage of fossil energy <cit.>. Alternatives for the energy sources need to be identified. Wind energy has received a great attention by renewable energy communities in the recent years due to its high potential in generating long term sustainable energies. This type of energy source has become the fastest growing industry for renewable energy with more than 30% annual growth rate <cit.>. Currently, there are two categories of modern wind turbines according to the axis of rotation: (1) Horizontal Axis Wind Turbines (HAWTs) and (2) Vertical Axis Wind Turbines (VAWTs), which are continuously developed. Nowadays, there is a strong interest from the wind energy community to harvest the energy within built environments at urban, sub-urban and remote areas in addition to large wind farms. HAWTs are not economically and socially suitable for such environments as they need large space to build, and a sophisticated yaw mechanism is required because wind direction often changes significantly <cit.>. On contrary, VAWT offers a simpler design than HAWT. The generator can be mounted on the ground, reducing the weight of the turbine that leads to a longer expected life time. Furthermore, the rotor has a small dependency towards wind direction so that no yaw mechanism is required. These advantages alleviate the manufacturing and maintenance costs of the rotor.Despite the mentioned superiorities, it is well known already that the flow field surrounding a VAWT rotor is complex involving a strong unsteadiness. During each rotor revolution, the blade sees varying local inflow velocity and angle of attack that can manifest as dynamic stall <cit.>. This effect has a huge influence to the aerodynamic performance of the rotor. It has been shown by Bangga et al. <cit.> that the generated power of a single bladed VAWT was negative for almost 22% of the azimuth angle range due to the dynamic stall effect. Laneville and Vitecoq <cit.> emphasized that dynamic stall presents for VAWT operating at tip speed ratio (λ) less or equal than 4, which is supported by the measurement carried out by Akins et al. <cit.>.Development of vertical axis wind turbines for the time being is focused on the small scale rotors. The rotor is usually proposed as an energy alternative in a region where electricity is not available like in the forest or where HAWT is hard to build like in the middle of cities. Several measurements, wind tunnel and field test, were carried out and documented in literature <cit.>. Kjellin et al. <cit.> performed a field test measurement of a 12 kW straight bladed vertical axis wind turbine. The results of the experiment was intended to be used as parameters in the control system of the turbine, in order to maintain the operation at optimum tip speed ratio. Li et al. <cit.> conducted both field and wind tunnel experiments focusing on the influences of pitch angle, Reynolds number and turbulence intensity on the power performance of a VAWT. The outcome shows various datasets of the power curve which are valuable for the validation tests. The effect of wind skew on VAWT was investigated in detail by Ferreira et al. <cit.>. It was shown that the tip vortices influence, generated in the downwind blade passage, on the vortices generated upwind is stronger than in the non-skewed case. Dabiri <cit.> suggested to design wind turbine arrays consisting of counter-rotating turbines to enhance the power production of the wind field. Despite the advantages of the experimental measurement, the technique cannot be widely used for its weakness of high cost, long cycle and poor flexibility <cit.>. There are a couple of low order simulation approaches for VAWT. The well known Blade Element Momentum (BEM) is one of them. This method discretizes the blade into several sections and the resulting forces are calculated based on the provided aerodynamic polars on the corresponding sections. The main drawback of the approach is that most flow physics are neglected such as the 3D effects, and thus the accuracy depends strongly upon the provided polars <cit.>. It has been documented by Li and Çalışal <cit.> that the momentum method <cit.> and the boundary-element method <cit.> are often used to predict power output from a turbine. Vortex methods are developed to deal with the issue of low order simulation approaches in the physical modelling by considering not only the 1D momentum theory but also the wake induced effects on the resulting rotor loads. In <cit.>, Li and Çalışal extended the numerical model to simulate twin-turbine system. Due to recent advances in high performance computations, computational fluid dynamics (CFD) approaches were applied in many engineering fields including VAWT with satisfactory results <cit.>. Ferreira et al. <cit.> presented 2D VAWT studies for various turbulence models, showing that Detached Eddy Simulations (DES) results are not only able to predict the generation and shedding of vorticity and it’s convection, but also show an acceptable sensitivity to grid refinement (both space and time) inferring that it can be used as a suitable comparison where validation data is limited or non existent. Song et al. <cit.> studied four meshing strategies for VAWT simulations. It has been shown that the required mesh for VAWT consists of 300 grid points on the airfoil surface with 15 cells within the boundary layer. The step size of T/240, where T denotes period of the rotor revolution, was observed to be reasonably small enough in capturing the loads. They obtained good results using 2D URANS simulations. Recently, Bangga et al. <cit.> conducted VAWT simulations using considerably finer mesh of 500 nodes on the airfoil surface in investigating the dynamic stall influence on a single bladed Darrieus rotor operating at λ = 2.0. The present studies focus on the 2D modelling of a two-bladed vertical axis wind turbine rotor using high fidelity CFD simulations, aiming to obtain deeper insights into the time accurate solutions of the rotor at various tip speed ratios. At first, preliminary studies into the time scale effect on the unsteady simulations will be carried out. Then, the results are validated against available measurement database from Li et al. <cit.>. Wake profiles are evaluated to quantify the momentum mixing of the flow in the wake area. The flow field characteristics and development of the shedding vortices surrounding the rotor will be presented and discussed in details.§ METHODOLOGY§.§ Studied Turbine and Test Case In exploring the power performance of a vertical axis wind turbine, very recently in 2016 Li et al. <cit.> performed wind tunnel and field test measurements of a straight bladed rotor under the influences of pitch angle, Reynolds number and wind speed. The wind tunnel experiments were carried out at the open test section of a circular type wind tunnel in Mie University, Japan <cit.>. The wind tunnel outlet diameter and length are 3.6 m and 4.5 m, respectively, with the maximum wind speed of 30 m/s. The studies were, however, carried out at the wind speed of 8.0 m/s with less than 0.5% turbulence intensity <cit.>. More detailed information of the wind tunnel was given in <cit.>. In the field test measurement, the rotor was mounted at a height h = 5.0 m from the ground.The rotor consists of two blades employing the NACA0021 airfoil cross-section rotating in clockwise direction. It shall be noted that the 2D simulations represent the mid-span section of the blade where the spanwise flow is not modelled. The turbine radius is 1.0 m, blade length is 1.2 m and the chord length is 0.265 m. In this study, a constant wind speed of 8 m/s and pitch angle of 6 were applied. A complete range of the power coefficient (C_power) curve was obtained by varying the rotational speed of the rotor. In wind turbine design, it is a common rule to specify a constant wind speed for various operating tip-speed-ratio (λ) to obtain a full power curve of the rotor. Usually, the employed wind speed is the designed rated condition that will be used in the experiment or depending on the field/wind farm situation. A similar technique was employed in obtaining the reference experimental data <cit.>. Since the measurement data from Li et al. <cit.> are used to validate the results, thus it is important to apply the same condition as in the experiment and the present test case was chosen based on this consideration. The simulations were carried out for eight different tip speed ratios, ranging from 0.5 up to 3.0.§.§ Computational Mesh and Numerical Setup Similar to preceding studies carried out by Castelli et al. <cit.>, Song et al. <cit.> and Bangga et al. <cit.>, some detailed rotor components like its main shaft, bolts, and member bars that connect blades with main shaft were omitted in the simulations. Two dimensional geometry was applied in the studies as the blade section is exactly the same along the blade length, neglecting the influence of tip vortices. The fully structured mesh technique was employed in the present studies. The mesh consists of three grid components, namely background, wake refinement and rotor meshes, as illustrated in Figure <ref>. The grid overlapping (Chimera) technique was applied in the studies, enabling high quality meshes to be built separately for each grid component. This approach greatly simplifies the mesh generation. The background mesh is discretized by 329 x 193 grid points in X and Y directions, respectively. The domain size is -25R x 120R in X direction and 50R wide in Y direction. A substantial mesh refinement was applied in the location where the rotor is located. Within this region, the wake refinement mesh consisting of an equi-distance cell size of Δ/R ≈ 0.12 extending from X/R = -5.3 up to 31.8 was introduced. This grid component consists of 257 x 141 grid points in X and Y directions, respectively. In the present studies, the resolution of this grid was increased up to 513 x 201 grid points (Δ/R ≈ 0.06), and basic sensitivity studies will be carried out.The rotor domain has a circular shape with the mesh component radius of 3R. The Chimera intersection area was defined near the outer radius for about 4 cells overlap. It is suggested not to apply the overlapping area too close to the blade wall to avoid unnecessary information loss from the data exchange between the fine mesh to the coarser meshes, especially when high velocity gradients need to be resolved such as flow involving massive separation. Figure <ref> presents detailed mesh around the blade/airfoil. The number of grid points on the airfoil surface is 321. In wall normal direction, 65 grid points were applied up to point p_1 and 129 grid points up to point p_2, see Figure <ref>. Among them, 32 cell layers are located within the boundary layer. These cell numbers are larger than the grid converged outcome of the study carried out by Song et al. <cit.>. To properly resolve the boundary layer, the first grid point adjacent to the wall was set to meet the non-dimensional wall distance, y^+, less than unity. The averaged value of y^+ is around 0.15 which is sufficiently small.To sum up, the mesh quality and quantity of the whole domains are described. Omitting the pole connector grid in the center of rotation of the rotor (origin), the maximum cell skewness is 0.75, which is small enough to avoid numerical errors due to high skewness cells influence. It shall be noted that the pole connector mesh was handled using the degenerate line boundary condition which allows a higher grid skewness on the pole boundary to be used. Several preceding works from the authors were carried out using this approach with good results <cit.>. The total number of grid points employed is of 327,524 divided into 92 different grid blocks. These grid blocks are further divided in the parallel computations of 24 CPUs carried out in the High Performance Computing Center Stuttgart (HLRS). The CFD simulations were carried out using a block-structured solver FLOWer from the German Aerospace Center (DLR) <cit.>. The code has been developed during the last years for wind turbine applications <cit.>. The time integration is carried out by an explicit hybrid 5-stage Runge-Kutta scheme. Dual time-stepping according to Jameson <cit.>, multigrid level 3 and implicit residual smoothing with variable coefficients were applied. The CFL numbers are 6.5 and 1.5 for the fine and coarse grid levels, respectively, in the multigrid scheme. The unsteady Reynolds-averaged Navier-Stokes (URANS) approach employing the Shear-Stress-Transport k-ω turbulence model according to Menter <cit.> was applied. The model has been widely used for industrial and academic purposes and is well known to be able to deliver good predictions for flows involving a strong adverse pressure gradient and separation. This has been also shown by previous works, for examples as in <cit.>. Three variants of time step sizes will be investigated, namely Δ t = T/360, T/720 and T/1440 which are equivalent to 1, 0.5 and 0.25 blade rotation per physical time step, respectively. The solutions for each time step were iterated within 70 sub-iterations. § RESULTS AND DISCUSSION In the present section, the results of the simulations for the two-bladed vertical axis wind turbine are presented and detailed discussions on the occurring physical phenomena are given. The discussions are divided into three main parts: Section <ref> presents the validation of the numerical results against available experimental data, and studies on temporal discretization sensitivity are presented. In Section <ref>, the wake characteristics downstream of the turbine are presented and discussed. At the end of the section, the dynamic stall phenomena at several tip speed ratios are evaluated in Section <ref> showing the unsteady development of the flow during the rotor revolution.§.§ Validation and Temporal Discretization Studies The simulations were carried out for 7 blade revolutions until the wake is fully developed at the time step size of T/360. Then the computations were restarted for 2 blade revolutions and smaller time step sizes (T/720 and T/1440) were applied as illustrated in Figure <ref>. This procedure was repeated for each simulated tip speed ratio. The power coefficient (C_power) was calculated based on the total moment coefficient around Z axis (M_Z) defined as C_power = -M_Z Ω/1/2ρ U_∞^3 Awhere Ω is the rotor rotational speed in [rad/s], ρ is the air density in [kg/m^3], U_∞ is the wind speed of 8 m/s and A is the cross section area of 2R. It shall be noted that positive power production is generated by negative driving moment for a rotor rotates in clockwise direction. In these studies, the results are time averaged only for the last rotor revolution.Figure <ref> presents the comparison between the CFD simulations to the measurement data, both field (grey line) and wind tunnel tests (purple line), obtained from Li et al. <cit.>. It can be seen that the wind tunnel and field test measurements exhibit very similar power curve at low tip speed ratios, approximately between λ = 1.0 to 2.0, but a closer look on the figure shows that the power generation in the field measurement is a bit larger. A noticeable deviation between these two results can be observed for the tip speed ratio larger than the optimum power coefficient position (λ≈ 2.19). It was documented in <cit.> that the fluctuation amplitudes obtained by field test show larger value than the results of wind tunnel at low and high tip speed ratios. Li et al. <cit.> argued that the deviation might occur as the result of the wind speed variation during the field test studies. It shall be noted that the same wind turbine was used in the wind tunnel and field measurements.In Figure <ref>, black line represents the CFD results employing the time step size of T/360, blue line of T/720 and red line of T/1440. The coarsest studied time scale results in an overestimation of the generated rotor power compared to measurement data. However, the general characteristics of the power curve is already reasonably captured. The prediction can be further improved by refining the time step size. As can be seen, the CFD predictions successively become more accurate as the time scale reduces to T/720 and T/1440. The computations even show excellent results for the time step size of T/1440 especially for the small tip speed ratio. The improvement of the C_power prediction by refining the time step size is a result from the variation of M_Z with Δ t depicted in Figure <ref>. It can be seen that M_Z becomes more positive and this causes reduction in the generated power production, alleviating the C_power overestimation. Physically, the use of the smaller time step size increases the ability of the CFD computations in resolving the trailed vortices that are highly unsteady. This causes a stronger induction effect that further reduces the local angle of attack seen by the blade section, which in turn reduces the local power generation. Furthermore, the finer time steps seems to be able to capture unsteady separation especially for the small tip speed ratio. It is worth noting that, in general, unsteady simulations require a certain time step to resolve unsteady effects accurately. The wind tunnel experiment is, however, still over estimated by the CFD predictions for the larger tip speed ratio, but their magnitude is similar to the field measurement data. The discrepancy between the CFD results to the wind tunnel experiment is expected to stem from two main factors, (1) a smaller time step size is required and (2) three dimensional effects due to tip loss occur. The latter becomes reasonable since the studied blade has no taper, and it is already well known that finite aspect ratio reduced the blade aerodynamic performance which may result in the reduction of rotor power in the wind tunnel test. §.§ Wake Characteristics The mean velocity (U/U_∞), streamwise variance (u'u'/U_∞^2) and shear stress (u'v'/U_∞^2) profiles at several downstream locations are shown in Figure <ref>. As already briefly mentioned in Section <ref>, two different resolutions of the wake refinement mesh were employed, namely 257 x 141 and 513 x 201 grid points (streamwise x crosswise) for the baseline (black-solid line) and fine (red-dashed line) meshes, respectively. It is shown that both the grid resolutions produce very similar magnitude of the wake profiles, inferring that the mesh is spatially converged. Figures <ref>-<ref> show that the streamwise velocity profile changes according to its distance to the rotor plane. A strong wake deficit is observed close to the rotor and it gradually becomes fuller with increasing streamwise distance. Flow acceleration occurs near the tip area especially at Y/R = 1.0. This seems to be related to the direction of the rotor rotation.The wake velocity profile becomes nearly Gaussian at X/R = 5.0, and this defines the near wake distance of the present case according to Sørensen et al.<cit.>. For the streamwise variance profile, strong peaks are observed for the whole rotor area at -1.0 < Y/R < 1.0. This behaviour is in contrast to the wake characteristics for HAWTs where the peaks are strong only near the tip and hub areas <cit.>, inferring that the tip vortices are the main contributors to the turbulence mixing in the wake. In case of VAWTs, the interactions of wake vortices occur very early close to the rotor because of the blade vortex interaction phenomena (BVI) <cit.>. As no turbulence inflow is considered in the present studies, both u'u'/U_∞^2 and u'v'/U_∞^2 are caused mainly by periodic motion of the blade passage. It is shown that the Reynolds stresses magnitudes reduce with increasing streamwise distance. This confirms preceding studies that only the stochastic part of u'v'/U_∞^2 enhances the momentum mixing in the wake area <cit.>. §.§ Dynamic Stall Onset In this section, the effect of dynamic stall on the aerodynamic performance of VAWT will be discussed. Three different tip speed ratios: below (λ = 1.50), near (λ = 2.13) and above (λ = 2.50) the optimum condition (λ = 2.19) are evaluated. Figure <ref> presents the normalized normal and tangential forces for these studied tip speed ratios. The forces direction is relative to the chord line. It shall be noted that the observation is made only for blade 1 employing the time step size of T/1440. It can be seen in Figure <ref> that F_N increases from θ = 0 until it reaches the maximum at θ = 90. Then, it drops gradually and a strong non-linearity of the curve is observed for θ > = 200 indicating the stall region. These characteristics occur at a smaller azimuth angle for F_T. It seems that F_N and F_T obtained from both the higher tip speed ratios (λ = 2.13 and 2.50) are very similar except in the stall area, but they differ strongly for the lower λ case. This confirms that the dynamic stall influence becomes stronger with decreasing tip speed ratio <cit.>.Figure <ref> presents the dimensionless vorticity field around the blade for the tip speed ratio of 1.50, which was calculated from the X- and Y-velocity components as: ω_Z = ∂ v/∂ x - ∂ u/∂ y.Blue and red colors represent the shedding vortices in clockwise (CW) and counter-clockwise (CCW) directions, respectively. The observation is focused only on the flow development for blade 1 at various azimuth angles during one rotor revolution. The employed time step size is T/1440. For clarity, the blade section is divided into two parts: the inner side is the side of airfoil where the normal vector its surface points towards the center of rotation and the outer side is otherwise. It shall be noted that positive normal force is when the force acts towards the center of rotation, and positive tangential force is towards the trailing edge.In Figure <ref>, at θ = 0, two main vortices shedding from the blade surfaces are observed, which are skewed towards the outer blade surface. This phenomenon stems from the pitch angle effect causing non-symmetrical flows on the inner and outer blade sides to occur. As a consequence, negative normal and tangential force coefficients are observed. The effect seems to be stronger for the larger operating tip speed ratios. With increasing azimuth angle, now the vortices are skewed towards the opposite direction because the angle of attack seen by the blade section increases. This causes the normal force augmentation up to maximum at θ = 72 (for λ = 1.50). Further increasing θ causes stronger trailing edge separation to occur on the inner side of the blade (Figure <ref>) indicated by a weak F_N drops at θ = 100. Slightly at a higher azimuth angle, at θ = 112, displacement effect by the vortical structure on the inner side of the blade causes a local augmentation of F_N. Despite that, F_T seems not too sensitive towards this effect where no local change at this azimuth position is observed in Figure <ref>. A sharp drop of the blade forces occurs for the higher azimuth angle up to θ = 132. The origin is quite obvious as the clockwise vortical structure start to detach from the blade surface, depicted in Figure <ref>. It has been documented by many authors that deep dynamic stall happens when the breakdown of the travelling vortex takes place <cit.>. The clockwise vortex breakdown (blue color) is caused by the flow insertion effect by the counter-clockwise vortex (red color) near the leading edge. This supports the numerical studies carried out for the other turbines in <cit.>. For clarity of the discussion, Figure <ref> presents the enlarged view of Figure <ref>, and flow streamlines surrounding the blade airfoil are plotted. The phenomenon starts when flow materials outside of the leading edge vortex (LEV) pocket are transported towards the leading edge in the near wall area creating a small but intense counter-clockwise vortex a bit downtsream of the leading edge, see green arrows in Figure <ref>. At the same time, a secondary clockwise vortex is formed near the leading edge, see yellow arrows in Figure <ref>. This causes the main LEV pocket to separate from the airfoil body, causing the sudden drop of the blade forces. In addition, the flow direction of the observed counter-rotating secondary LEVs is almost normal to the wall at point X (see Figure <ref>). This results in the increased pressure effect, as depicted in Figure <ref>, that has a significant contribution in the aerodynamic losses of the blade. The normal force gradually decreases up to θ = 270. On the other hand, the tangential force becomes more positive within these azimuth angles except at θ = 180 where a local drop is observed, which seems to be connected to the generation of the remarkable counter-clockwise vortex near the trailing edge in Figure <ref>. For the azimuth angles higher than 270, the flow field is dominated by shedding vortices from the inner and outer sides of the blade that cause loads fluctuation to occur.Generally, it can be seen that positive power generation (negative F_T) for blade-1 occurs within the azimuth angles of 0 < θ < 128 for λ = 1.50, 0 < θ < 234 for λ = 2.13 and almost the whole azimuth angle range (0 < θ < 360) for λ = 2.50. This indicates that the aerodynamic benefit is obtained for the larger operating tip speed ratio, even for λ greater than the optimum value. Furthermore, setting the operating tip speed ratio to a larger one may increase the life cycle of the turbine because the dynamic stall and load fluctuation effects are alleviated.In Figure <ref>, the distributions of pressure are presented for four different tip speed ratios and azimuth angles. The above discussions are further clarified as the shape of the pressure distributions seems to become more similar to the static airfoil conditions with increasing tip speed ratio. Furthermore, flow separation and variations of the pressure distribution shape between different azimuth angles are also alleviated. This is natural because the range of operating angle of attack reduces with λ, which in turn decreases the possibility of the blade to operate in the stall conditions.§ CONCLUSION AND OUTLOOK Numerical simulations using 2D computational fluid dynamics (CFD) approaches have been carried out in the present studies on a two-bladed vertical axis wind turbine (VAWT) for various tip speed ratios (0.50< λ < 3.0). The radius of the rotor is 1 m with the chord length of 0.265 m, and is operating at a constant wind speed of 8 m/s. The numerical results were compared to the experimental data by Li et al. <cit.> to assess the accuracy of the computations. Three time step sizes of T/360, T/720 and T/1440 were evaluated, where T represents the period of the rotor revolution. It was shown that the numerical results become more accurate for the smaller time step size. An excellent agreement with the experiment is obtained for the small tip speed ratio up to the optimum power coefficient at λ = 2.19 even though flow separation occurs. For the larger tip speed ratios, the numerical results overestimate the wind tunnel experiment, but they are very similar to the field measurement data. The discrepancy is expected to arise due to tip loss effects that are not captured in the 2D simulations and a smaller time step size may be required to model the small vortices. The characteristics of wake are presented in the present studies in terms of the steamwise velocity (U), streamwise variance (u'u') and Reynolds shear stress (u'v') profiles. Two wake refinement grid resolutions were investigated, showing that the grid is spatially converged. The wake profile becomes Gaussian at X/R = 5 indicating the location of the near wake distance for the studied case. It was shown that a remarkable flow acceleration occurs near the tip area especially at Y/R = 1.0 that is connected to the direction of the rotor revolution. Strong peaks of the Reynold stress components (u'u' and u'v') are observed within the rotor area at -1.0 < Y/R < 1.0 because noticeable momentum mixing occurs as a result from the blade vortex interaction. Additionally, it was shown that the strength of the Reynold stress components reduces with streamwise distance.The flow field surrounding the blade during one rotor revolution and its impact on blade loads were evaluated. Complex flow field phenomena were observed especially at the azimuth angle larger than 90, following the dynamic stall onset caused by the flow insertion effect. Power production for of the blade depends on the operating tip speed ratio. The azimuth range of the positive generated power from blade 1 increases with increasing λ. This indicates that designing VAWTs at higher tip speed ratios may be beneficial in term of power production, and this may can improve the life-time of the rotor since the dynamic stall effect is reduced. For the future works, several aspects can be suggested in the field of VAWT studies. Fully resolved 3D simulations may improve the accuracy of the results especially in capturing the tip loss effects at high tip speed ratios that are not captured in the present works. Furthermore, higher fidelity simulation approaches like Large Eddy Simulation (LES) or Detached-Eddy Simulation (DES) are recommended for this purpose. Additionally, the performance assessment of a multi-rotor system is suggested in the subsequent works. § ACKNOWLEDGEMENTSThe authors gratefully acknowledge these following institutions for the supports: Ministry of Research, Technology and Higher Education of Indonesia for the funding through Directorate General of Higher Education (DGHE) scholarship and the High Performance Computing Center Stuttgart (HLRS) for providing computational time in the CFD simulations. This paper is dedicated to F.F.S. Lesy who passed away on June 12, 2017 while accompanying the author G. Bangga in finishing the works. unsrt | http://arxiv.org/abs/1708.08365v2 | {
"authors": [
"Galih Bangga",
"Thorsten Lutz",
"Amgad Dessoky",
"Ewald Krämer"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170824133544",
"title": "Unsteady Navier-Stokes studies on loads, wake and dynamic stall characteristics of a two-bladed vertical axis wind turbine"
} |
A Study of Distributionally Robust Multistage Stochastic Optimization Jianqiu Huang, Kezhuo Zhou, and Yongpei Guan Department of Industrial and Systems Engineering University of Florida, Gainesville, FL 32611, USA Emails: [email protected], [email protected], and [email protected] ======================================================================================================================================================================================================================================= In this paper, we focus on a data-driven risk-averse multistage stochastic programming (RMSP) model considering distributional robustness. We optimize the RMSP over the worst-case distribution within an ambiguity set ofprobability distributions constructed directly from historical data samples. The proposed RMSP is intractable due to the multistage nested minimax structure in its objective function, so we reformulate it into a deterministic equivalent that contains a series of convex combination of expectation and conditional value at risk (CVaR), which can be solved by a customized stochastic dual dynamic programming (SDDP) algorithm in this paper. As the size of collected data samples increases to infinity, we show the consistency of the RMSP with distributional robustness to the traditional multistage stochastic programming. In addition,to test the computational performance of our proposed model and algorithm, we conduct numerical experiments for a risk-averse hydrothermal scheduling problem, the results of which demonstrate the effectiveness of our RMSP framework. Key words: multistage stochastic optimization; data-driven decision making; distributional robustness; hydrothermal scheduling § INTRODUCTIONThe multistage stochastic program (MSP) has been widely studied in literature for providing multi-period optimal decisionsunder uncertainty,since MSP can be naturally adopted to model various real-life applications with periodical decisions, hydrothermal scheduling <cit.>, power system operations <cit.>, transportation <cit.>, and supply chain planning <cit.>. For MSP models, the sequential decisions are made depending on the realization of stochastic parameters, which are assumed to follow some known probability distributions.Once a decision has been made for the previous period, an observation of the stochastic parameters for the current periodbecomes available and then the corresponding decision will be made consideringfuture uncertainty. The objective of MSP is to minimize the total expected costs incurred by the decisions over the planning horizon subject to a series of constraints, nonanticipativity constraints and other modeling constraints. Readers are referred to <cit.>, <cit.>, <cit.>, and <cit.> for more detailedMSP structures and properties. MSP models are generally computationally intractable, since each decision making depends on not only the past parameter realization, but also decisions made in each previous stage. To simplify the model, the stochastic parameters are assumed to be discrete random variables and modeled via scenario trees, leading to a deterministic equivalent model of MSP, where the corresponding expectations in the objective function convert to finite sums.Two types of decomposition algorithms are proposed to solve MSP, scenario-based methods where the sample of realizations is fixed, and sampling-based methods where the sample of realizations is obtained iteratively.Scenario-based methods use a small set of realizations from the complete sample space to approximately solve the original program, like diagonal quadratic approximation <cit.>, Lagrangian decomposition <cit.>, L-shaped methods <cit.>, and scenario aggregation methods <cit.>. On the other hand, sampling-based methods iteratively draw a subset of realizations from the complete sample space, where statistical bounds are utilized to create convergence criteria, stochastic dual dynamic programming (SDDP) – a Monte Carlo sampling-based method<cit.>,the stochastic decomposition with extension to the multistage case <cit.>,and progressive hedging <cit.>. These two types of methods are integrated together in <cit.>. Moreover, multistage distances are introduced and utilized to solve MSP models approximately <cit.>. Over the last few decades, risk-averse multistage stochastic programming (RMSP) has been attracting significant attentions, due to its advantages over MSP on modeling certain applications where low probability events have high impact. The classical MSP focuses on the average behavior, however, there are many real-life applications where low probability events have high impact.To handle this issue, various risk measures are designed to remedy the limitation of the traditional expectation operator in the MSP objective function, RMSP uses a convex combination of a expectation and a risk measure <cit.>. Seminal novel worksexploring the risk measure properties for RMSP include coherent risk measures <cit.>, time-consistent risk measures <cit.>, and regularity of risk measures <cit.>. Lagrangian relaxation is first proposed to solve the RMSP models <cit.>, and later come the advanced nested L-shaped decomposition algorithms <cit.>.Both MSP and RMSPrely onknown probability distributions, which is generally not practical due to the ambiguity of the probability distributions in real-life applications. Hence, ambiguity sets of probability distributions are proposed to model all possible probability distributions within certain range based on a series of historical data samples.Several types of ambiguity sets have been proposed, like moment-basedambiguity sets <cit.>, divergence-measure-based ambiguity sets <cit.>, and other metric-based ambiguity sets <cit.>.Seminal works also construct ambiguity sets based onthe relation between risk measures and robust optimization models <cit.>.Our contribution in this paper is thatwe first propose an RMSP formulation by integrating the general MSP model and the distributional ambiguity sets withL_∞-norm, which optimizes the total expected costs over the worst-case distribution within the ambiguity set. The ambiguity sets are constructed directly from the historical data samples. We then derive an equivalent reformulation of RMSP,where the objective function is replaced by a convex combination of an expected cost and a conditional value at risk (CVaR). A significant advantage of our reformulation is that it gets rid of the nested multistage minimax structure from the original objective function, leading to a tractable MSP based on the reference distributions constructed from historical data. We prove that optimal solutions and objective values of our RMSP with distributional robustness converge to those of risk-netural MSP as the size of data samples increases to infinity. In addition, we customize and analyze the SDDP algorithm to solve our proposed RMSP.Furthermore, we apply RMSP with distributional robustness to the hydrothermal scheduling problem, and implement corresponding computational experiments whose results verify the convergence of our method. The following sections of this paper are organized as follows. In Section <ref>, we present our RMSP model with distributional robustness and reformulate it into a tractable equivalent, and show the consistency of our RMSP model. In Section <ref>, we customize the SDDP algorithm to solve our RMSP, followed by a convergence analysis of our SDDP algorithm.Next, we apply our RMSP to a risk-averse hydrothermal scheduling problem in Section <ref>, and provide the corresponding computational experiment results. Finally we conclude our research in Section <ref>. § RISK-AVERSE MULTISTAGE STOCHASTIC PROGRAMIn this section, we present the reformulation of RMSP based on ambiguity sets with L_∞-norm ^t,∀ t = 2,…,T,and then provide the corresponding convergence analysis. At each stage t = 2, 3, ⋯, T, we denote the sample space of the stochastic parameters ξ_t as Ω_t = {ξ_t^1, ξ_t^2, ⋯, ξ_t^R}. Similarly to that of <cit.>, the ambiguity set ^t for possible probability distributions f = (f^1, f^2, ⋯, f^R) can be constructed in a data-driven way with two kinds of representations as follows:𝒟^t = { f≥ 0: ||f - f_0 ||_∞≤ d_t,∑_r=1^R f^r=1 },or𝒟^t = { f≥ 0: d_1^t ≤ ||f/f_0 ||_∞≤ d_2^t , ∑_r=1^R f^r=1 },where f_0 represents the reference distribution that can be established from the empirical distribution with historical data, d_t is a tolerance that decreases as the data size N_t increases, and d_1^t/ d_2^t are tolerance parameters that increase/decrease to 1 as the data size N_t grows to infinity.The decision rules of d_t, d_1^t, and d_2^t are beyond the scope of this paper and readers are referred to <cit.> for more details in this part.It is obvious that representations (<ref>) and (<ref>) can be transformed to each other by selecting proper parameters d_t, d_1^t and d_2^t. Therefore, we focus on the first approach to construct the distributional ambiguity set 𝒟^t in this paper. We denote the lower and upper bound of the ambiguity set as f_ℓ = f_0 -d_t and f_u = f_0 +d_t, respectively.At each stage t =2,…,T, we let _t^ℓ and _t^u-ℓ represent the probability measures induced by f_ℓ andf_u -f_ℓ, and we assume P_t^ℓ := ∑_r=1^Rf_ℓ^r ∈ (0,1) and P_t^u := ∑_r=1^Rf_u^r >1to avoid trivial cases.In the following, we reformulate the worst-case expected cost over ^t at each stage t into a convex combination of an expected cost and a CVaR. Finally, we show that both the set of optimal solutions and the objective value of RMSP converge to those of risk-neutral MSP as the data sample size grows to infinity, respectively.§.§ Equivalent ReformulationIn this subsection, we develop the equivalent reformulation of the following nested RMSP formulation:min_A_1x_1≥ b_1 c_1x_1 +sup__2 ∈𝒟^2__2[ min_A_2x_2 ≥ b_2 - B_2x_1 c_2x_2 + ⋯ + sup__T ∈𝒟^T__T[ min_A_Tx_T ≥ b_T -B_Tx_T-1 c_Tx_T]],where vectors c_t, b_t, and matrices A_t, B_t are assumed to be stagewise independent random variables forming the stochastic data process (c_t(ξ_t), b_t(ξ_t), A_t(ξ_t), B_t(ξ_t)) for t=2,…, T. The RMSP has compact feasible set and relatively complete recourse. In addition, the recourse function at each stage is bounded for each decision x. Due to the stagewise independence of the data process, formulation (<ref>) can reformulated as a series of dynamic programming equations. Starting from the last stage T, we define Q_T(x_T-1(ξ_T-1), ξ_T ) as the optimal value of the last stage program as follows:Q_T(x_T-1(ξ_T-1), ξ_T ) = min_x_Tc_T(ξ_T)x_T(ξ_T)s.t. A_T(ξ_T)x_T(ξ_T) ≥ b_T(ξ_T) - B_T(ξ_T)x_T-1(ξ_T-1).Backward to stage T-1, we have that Q_T-1(x_T-2(ξ_T-2), ξ_T-1 ) is equal to the optimal value of the programQ_T-1(x_T-2(ξ_T-2), ξ_T-1) = min_x_T-1 c_T-1(ξ_T-1)x_T-1(ξ_T-1) +sup__T∈𝒟^T_ξ_T∼_T[Q_T( x_T-1(ξ_T-1), ξ_T )] s.t. A_T-1(ξ_T-1)x_T-1(ξ_T-1) ≥ b_T-1(ξ_T-1) - B_T-1(ξ_T-1)x_T-2(ξ_T-2),and the cost-to-go function _T( x_T-1(ξ_T-1) ) for the worst-case expectation sup__T∈𝒟^T_ξ_T∼_T[Q_T( x_T-1(ξ_T-1),ξ_T )] can be defined and then reformulated as follows:eqn:Qe_T _T( x_T-1(ξ_T-1) )≡sup__T∈𝒟^T_ξ_T∼_T[Q_T( x_T-1(ξ_T-1), ξ_T )] =P_T^ℓ_ξ_T∼_T^ℓ[Q_T(x_T-1(ξ_T-1), ξ_T )] + (1-P_T^ℓ)_(P_T^u-1)/(P_T^u-P_T^ℓ)^ξ_T∼_T^u-ℓ[Q_T( x_T-1(ξ_T-1), ξ_T )] = P_T^ℓ_ξ_T∼_T^ℓ[Q_T(x_T-1(ξ_T-1), ξ_T )] + inf_ u_T-1∈{(1-P_T^ℓ)u_T-1(ξ_T-1) + (P_T^u-P_T^ℓ)_ξ_T∼_T^u-ℓ[Q_T(x_T-1(ξ_T-1), ξ_T) - u_T-1(ξ_T-1)]^+},where equality (<ref>) holds due to Theorem 6 in <cit.> and equality (<ref>) holds because of the CVaR definition.Thus, Q_T-1(x_T-2(ξ_T-2), ξ_T-1 ) can be reformulated by substituting Equation (<ref>) into (<ref>).min_x_T-1, u_T-1c_T-1(ξ_T-1)x_T-1(ξ_T-1) + (1-P_T^ℓ )u_T-1(ξ_T-1) + P_T^ℓ_ξ_T∼_T^ℓ[Q_T(x_T-1(ξ_T-1), ξ_T )] + (P_T^u-P_T^ℓ )_ξ_T∼_T^u-ℓ[Q_T(x_T-1(ξ_T-1), ξ_T) - u_T-1(ξ_T-1)]^+ s.t. A_T-1(ξ_T-1)x_T-1(ξ_T-1) ≥ b_T-1(ξ_T-1) - B_T-1(ξ_T-1)x_T-2(ξ_T-2). By repeating this process backward, we can write dynamic programming equations for each stage t=2,…,T asQ_t(x_t-1(ξ_t-1), ξ_t) = inf_x_t , u_t{c_t (ξ_t )x_t (ξ_t ) + (1-P_t+1^ℓ )u_t(ξ_t) + _t+1( x_t (ξ_t ), u_t(ξ_t ) ):A_t (ξ_t )x_t (ξ_t ) ≥ b_t (ξ_t ) - B_t (ξ_t )x_t-1(ξ_t-1) },where the cost-to-go function _t+1( x_t (ξ_t ), u_t(ξ_t ) ) is defined as _t+1( x_t (ξ_t ), u_t(ξ_t ) ) =P_t+1^ℓ_ξ_t+1∼_t+1^ℓ[Q_t+1(x_t (ξ_t ), ξ_t+1 )] + (P_t+1^u-P_t+1^ℓ )_ξ_t+1∼_t+1^u-ℓ[Q_t+1(x_t (ξ_t ), ξ_t+1) - u_t(ξ_t) ]^+ ,with _T+1(·) = 0 and P_T+1^u = P_T+1^ℓ= 1.Finally, the reformulated program at the first stage is described as follows:min_x_1, u_1c_1x_1+(1-P_2^ℓ )u_1+ _2( x_1 , u_1 ) s.t. A_1 x_1≥ b_1. §.§ Convergence AnalysisIn this section, we show the consistency of RMSP with distributional robustness by analyzing the convergence property of RMSP as the size of historical data samples increases to infinity. We find that when the size of historical data samples for constructing ambiguity sets at each stage goes to infinity, both the optimal objective value and the set of optimal solutions for RMSP converge to the counterparts of MSP under true while unknown distribution. We define the following notations for our proof. We let z(0) denote the optimal objective value and U(0) denote the set of optimal solutions for the MSP underthe true distribution. Similarly, we let ẑ(0) denote the optimal objective value and Û(0) denote the set of optimal solutions for the MSP under the reference distribution. We extend the notations to RMSP by denoting ẑ(d_2(N_2),…,d_T(N_T)) as the optimal objective value with data size N_t for each corresponding stage and Û(d_2(N_2),…,d_T(N_T)) as the optimal solutions under the same setting. For MSP, the recourse functions are overlapped in a nested structure and only in the last stage do we have a closed-form recourse function for each scenario. Moreover, the number of decision variables in stage t grows in an exponential rate of stage t. Thus, in order to better show our analysis of the convergence property, we denote x_t(ξ_t) as the decision in stage t for observation ξ_t, x_t as the set of decisions in stage t for all observations in stage t, and x as the set of decisions for all stages and scenarios. Besides, we let N=min{N_2,N_3,…,N_T} and d=max{d_2,d_3,…,d_T}. With the above notations, we provide our conclusion on the convergence analysis for RMSP. As the size of data sample N goes to ∞, ẑ(0)→ z(0). Furthermore, Û(0) converges to U(0), i.e., lim_N →∞sup_x∈Û(0)||x-U(0)||=0. Along with the proof in this paper, we will use N→∞ and d→ 0 interchangeably. Note that there exist underlying reference distributions along all stages for each ẑ(0), so we may use the notation ẑ__N(0) when we need to emphasize it. We follow the same notation rule for z(0) and throughout the proof in this paper. We also let ĥ(x,d_2,…,d_T) represent the objective value for RMSP corresponding to solution x and tolerances d_2,…,d_T, and similarly h(x) for MSP with true distribution under solution x. ẑ(0) = ĥ(x̂^*,0) ≤ĥ(x^*,0), where x̂^* represents an optimal solution to RMSP under corresponding tolerance, and x^* represents an optimal solution to MSP under true distribution. Taking upper limit with respect to the size of historical data on both sides of Inequality (<ref>), we have lim sup_N→∞ẑ(0) = lim sup_N→∞ĥ(x̂^*,0) ≤lim sup_N→∞ĥ(x^*,0) = lim_N→∞ĥ(x^*,0) = h(x^*)=z(0), limsupwhere the second equality holds because of the following reasons: once the decision variables x are fixed, the objective function is a polynomial of parameters f^r_t; the reference distribution for each stage t converges weakly to the true distribution, so the limit of the objective value exists and is consistent with that under true distribution. Next, we show that {ẑ__N(0)} converges. Under the Assumption <ref> it is easy to see that {ẑ__N(0)} belongs to a bounded set. So if {ẑ__N(0)} does not converge, we can find two subsequences that converge to different values, say z_1 and z_2, i.e., ẑ__N_t(0) → z_1,ẑ__N_s(0) → z_2 and z_1 ≠ z_2. Note that the corresponding optimal solutions{x̂_N_t^*} and {x̂_N_s^*} are bounded due to the assumption of compact feasible region, there exists subsequences of the two series of optimal solutions that converge respectively. For notation brevity, we still denote the two subsequences as{x̂_N_t^*} and {x̂_N_s^*}, and x̂_N_t^* → x_1, x̂_N_s^*→ x_2, where x_1 and x_2 are two feasible solution. Then we have z_1 = lim_t→∞ ẑ__N_t(0) = lim_t→∞ĥ(x̂_N_t^*,0) = lim_t→∞ĥ(x_1,0) = h(x_1) ≥z(0), z_2 = lim_s→∞ ẑ__N_s(0) = lim_s→∞ĥ(x̂_N_s^*,0) = lim_s→∞ĥ(x_2,0) = h(x_2) ≥z(0). But we have z_1 ≤ z(0) andz_2 ≤ z(0) due to (<ref>), so z_1 = z(0) =z_2, which is a contradiction. Thus, {ẑ__N(0)} converges and we have lim_N→∞ẑ(0) = lim sup_N→∞ẑ(0) = z(0). Finally, we prove the convergence property of Û(0) to U(0) by contradiction. Supposing that sup_x∈Û(0)||x-U(0)|| does not converge to zero as N grows to infinity, there exists a positive number ϵ_0 and a sequence of optimal solutions {x̂_N_k^*} such that ||x̂_N_k^*-U(0)|| > ϵ_0 for all k. Following the same idea as above, we have a subsequence of the optimal solutions {x̂_N_k^*} that converges. For notation brevity, we still denote the subsequence as{x̂_N_k^*} andx̂_N_k^* →x̅, where x̅ is a feasible solution. Then we have lim_k→∞ĥ(x̂_N_k^*,0) = lim_k→∞ĥ(x̅,0) = h(x̅). Since {x̂_N_k^*} is a sequence of optimal solutions, we conclude that h(x̅) = z(0) and accordingly x̅∈ U(0). However, as x̂_N_k^* →x̅ and ||x̂_N_k^*-U(0)|| > ϵ_0, we have ||x̅-U(0)|| ≥ϵ_0 > 0, which is a contradiction. This completes the proof. For all t≥ 2, as the size of historical data samples N_t increases to ∞, the distance tolerance d_t(N_t) → 0, max_ξ_t ∈Ω_t |f^t(ξ_t)-f_0^t(ξ_t)|→ 0, and ẑ(d_2(N_2),…,d_T(N_T))→ z(0). Furthermore, Û(d_2(N_2),…,d_T(N_T)) converges to U(0), i.e., lim_N_2,…,N_T →∞sup_x∈ U(d_2(N_2),…,d_T(N_T))||x-U(0)||=0. To prove that lim_N→∞ẑ(d_2(N_2),…,d_T(N_T)) = z(0), it is enough to show that lim sup_N→∞ẑ(d_2(N_2),…,d_T(N_T)) ≤ z(0), uppgoal and lim inf_N→∞ẑ(d_2(N_2),…,d_T(N_T)) ≥ z(0).lowgoal First, since ĥ(x̂_N,d^*,d_2(N_2),…,d_T(N_T)) ≥ĥ(x̂_N,d^*,0) ≥ĥ(x̂_N,0^*,0), where x̂_N,d^* and x̂_N,0^* represent the optimal solution to corresponding RMSP under tolerance (d_2(N_2),…,d_T(N_T)) and 0 respectively, by taking lower limit, we have lim inf_N→∞ẑ(d_2(N_2),…,d_T(N_T)) =lim inf_N→∞ĥ(x̂_N,d^*,d_2(N_2),…,d_T(N_T)) lowlim1 ≥lim inf_N→∞ ĥ(x̂_N,0^*,0) lowlim2 = lim_N→∞ ĥ(x̂_N,0^*,0) lowlim3 = lim_N→∞ ẑ(0)lowlim4 = z(0), lowlim5 where (<ref>) holds by definition, (<ref>) holds because of the inequality we just provide, and (<ref>) to (<ref>) hold because of Proposition <ref>. Thus, inequality (<ref>) is proved. Second, we have ĥ(x̂_N,d^*,d_2(N_2),…,d_T(N_T)) ≤ĥ(x,d_2(N_2),…,d_T(N_T)) for any feasible solution x. More specifically, it can be written briefly as minc_1x_1 +sup__2 ∈𝒟^2__2[ min c_2 x_2 + ⋯ + sup__T ∈𝒟^T__T[ minc_T x_T ]]≤c_1x_1 +sup__2 ∈𝒟^2__2[c_2 x_2 + ⋯ + sup__T ∈𝒟^T__T[c_T x_T ]] for all feasible x. Taking upper limit on both sides of inequality (<ref>), we can obtain that lim sup_N→∞ẑ(d_2(N_2),…,d_T(N_T)) = lim sup_N→∞ĥ(x̂_N,d^*,d_2(N_2),…,d_T(N_T)) upplim1 ≤lim sup_N→∞ { c_1x_1 +sup__2 ∈𝒟^2__2[c_2 x_2 + ⋯+ sup__T ∈𝒟^T__T[c_T x_T ]] } upplim2 = lim sup_d_T→0 ⋯lim sup_d_2→0{ c_1x_1 +sup__2 ∈𝒟^2__2[c_2 x_2 + ⋯+ sup__T ∈𝒟^T__T[c_T x_T ]] }upplim3 = lim sup_d_T→0 ⋯lim sup_d_3→0 { c_1x_1 + lim sup_d_2→0 sup__2 ∈𝒟^2__2[c_2 x_2 + ⋯+ sup__T ∈𝒟^T__T[c_T x_T ]] } upplim4 = lim sup_d_T→0 ⋯lim sup_d_3→0 { c_1x_1 + __2[c_2 x_2 + ⋯+ sup__T ∈𝒟^T__T[c_T x_T ] ] } upplim5 = ⋯upplim6 =c_1x_1 +__2[c_2 x_2 + ⋯+ __T[c_T x_T ]], upplim7 where (<ref>) holds by definition, (<ref>) holds because of inequality (<ref>), (<ref>) holds because N→∞ is equivalent to d_2(N_2),…,d_T(N_T) → 0, (<ref>) holds because here we only consider the limit when d_2 goes to zero, (<ref>) holds because for stage 2, the corresponding empirical distribution converges weakly to the true distribution and the recourse function for stage 2 is actually a linear expression of f_2^r, which leads to both the existence of the limit and the consistency,and (<ref>) to (<ref>) holds by taking the limit for stage 3,…, T. Since inequality (<ref>) holds for any feasible solution x, we conclude that lim sup_N→∞ẑ(d_2(N_2),…,d_T(N_T)) ≤ minc_1x_1 +__2[ minc_2 x_2 + ⋯+ __T[ minc_T x_T ]] = z(0). Thus, inequality (<ref>) is proved and lim_N→∞ẑ(d_2(N_2),…,d_T(N_T)) = z(0). The proof for convergence property of optimal solutions follows similar method to that in Proposition <ref> and is thus omitted here.§ SOLUTION APPROACHIn this section, we first utilize a scenario treeto model the data process ξ_2, …,ξ_T based on the historical data, and then customize the SDDP approach to solve our RMSP.We collect the historical data and then generate a finite scenario tree based on these data with two reference distributions ^ℓ and ^u-ℓ. At each stage t=2,…, T,we denoteN_t as the total number of different scenarios, and let p_ti^ℓ and p_ti^u-ℓ be the corresponding reference probabilities for scenario ξ_ti, i=1,…,N_t. Therefore, we can rewrite the dynamic programming equations in (<ref>) as follows:Q̂_ti(x̅_t-1) = inf_x_t , u_t{c_t (ξ_ti)x_t (ξ_ti) + (1-P_t+1^ℓ )u_t(ξ_ti) + _t+1( x_t (ξ_ti), u_t(ξ_ti) ):A_t(ξ_ti)x_t (ξ_ti) ≥ b_t (ξ_ti) - B_t (ξ_ti)x̅_t-1},for i=1,…, N_t, where x̅_t-1 is the current solution obtained from stage t-1, and the corresponding cost-to-go function can be written as follows:_t+1( x_t (ξ_tj ), u_t(ξ_tj ) ) = P_t+1^ℓ∑_i=1^N_t+1p_t+1i^ℓ[Q̂_t+1i(x_t (ξ_tj ))] + (P_t+1^u-P_t+1^ℓ )∑_i=1^N_t+1p_t+1i^u-ℓ[Q̂_t+1i(x_t (ξ_tj )) - u_t(ξ_tj) ]^+. At the first stage, we havemin_x_1, u_1c_1x_1+(1-P_2^ℓ )u_1+ _2( x_1 , u_1 ) s.t. A_1 x_1≥ b_1.In the following, we use variables x_t, u_t instead of x_t (ξ_tj ), u_t(ξ_tj ) for notation brevity, and we denote x̅_t , u̅_t as the current solutions for stage t. As the cost-to-go functions _t+1( x_t , u_t ) are convex, we can use supporting hyperplanes tomake lower approximations of the cost-to-go functions. We define 𝔔_t+1(x_t , u_t) as the current approximation of the cost-to-go function at stage t+1 for each t= 1,…,T-1. To generate a supporting hyperplane for _t+1( x_t , u_t ) at ( x̅_t , u̅_t ) in (<ref>), we first consider the subdifferential of function [Q̂_t+1i(x_t ) - u_t]^+ at point (x̅_t , u̅_t ), which is ∂ [Q̂_t+1i(x_t ) - u_t]^+= {[ [0,0] if Q̂_t+1i(x̅_t ) < u̅_t,; ⋃_g∈∂Q̂_t+1i(x̅_t )[g,-1]if Q̂_t+1i(x̅_t ) > u̅_t ,; ⋃_g∈∂Q̂_t+1i(x̅_t ) , ∈ [0,1] [ g, -] if Q̂_t+1i(x̅_t ) = u̅_t. ]. Applying the chain rule of subdifferentials, the subgradient of _t+1( x_t , u_t ) at ( x̅_t , u̅_t ) is [ P_t+1^ℓ∑_i=1^N_t+1p_t+1i^ℓ𝐠_t+1i + (P_t+1^u - P_t+1^ℓ)∑_i∈𝒥_t+1p_t+1i^u-ℓ𝐠_t+1i, -(P_t+1^u-P_t+1^ℓ )|𝒥_t+1|], where 𝐠_t+1i∈∂Q̂_t+1i(x̅_t ), i=1,…,N_t+1 and 𝒥_t+1:={j:Q̂_t+1i(x̅_t ) > u̅_t}. The supporting hyperplane of _t+1( x_t , u_t ) at ( x̅_t , u̅_t ) is constructed as follows. ℎ_t+1(x_t, u_t) :=_t+1(x̅_t, u̅_t) +[P_t+1^ℓ∑_i=1^N_t+1p_t+1i^ℓ𝐠_t+1i + (P_t+1^u-P_t+1^ℓ )∑_i=1^N_t+1p_t+1i^u-ℓ𝐠_t+1i](x_t - x̅_t)-(P_t+1^u-P_t+1^ℓ )|𝒥_t+1| (u_t -u̅_t).Therefore, we can update the cost-to-go functions by adding the new supporting hyperplane (<ref>), 𝔔_t+1(x_t , u_t) := max{𝔔_t+1(x_t , u_t), ℎ_t+1(x_t, u_t)}.§.§ Algorithm In this section, we adopt SDDP to solve our RMSP. SDDP was first proposed in <cit.> to solve MSP and later studied in <cit.> for MSP models where the objective function is a convex combination of expectation and CVaR. There are two parts in SDDP, a forward simulation that generates a simulated solution, and a backward recursion that improves the approximation of the cost-to-go functions at each stage. The detailed algorithm is provided in Algorithm <ref>. We describe the application of SDDP to our RMSP in the following section.In the backward recursion steps, we solve the optimization problem (<ref>) with the approximation𝔔_t+1(x_t , u_t) instead of the cost-to-go function _t+1( x_t , u_t ).Q̂_ti(x̅_t-1) = inf_x_t , u_t{c_t (ξ_ti)x_t (ξ_ti) + (1-P_t+1^ℓ )u_t(ξ_ti) + 𝔔_t+1( x_t (ξ_ti), u_t(ξ_ti) ):A_t(ξ_ti)x_t (ξ_ti) ≥ b_t (ξ_ti) - B_t (ξ_ti)x̅_t-1},for i=1,…, N_t, where the corresponding approximated expected cost-to-go functions can be updated by adding supporting hyperplanes (<ref>).At the first stage,min_x_1, u_1c_1x_1+(1-P_2^ℓ )u_1+ 𝔔_2( x_1 , u_1 ) s.t. A_1 x_1≥ b_1,whose optimal objective value is used as the lower bound of our problem. The forward simulation steps are performed by sampling independent scenarios from the historical data and computing the corresponding optimal value, which willbe used later to calculate the upper bound z̅ = c_1x̂_1 + (1-P_2^ℓ )û_1+ 𝔳̂_2,where𝔳̂_t =1M_t∑_i=1^M_t{P_t+1^ℓ (c_tix̂_ti + 𝔳̂_t+1 i)+ (1- P_t+1^ℓ )û_ti + (P_t+1^u- P_t+1^ℓ)[c_tix̂_ti + 𝔳̂_t+1 i -û_ti]^+}, t=2,…,T. §.§ Convergence PropertyIn this subsection, we provide the convergence of the proposed algorithm. When the basic optimal solutions are employed in the backward steps, the forward step procedure generates the optimal solution for the risk-averse multistage stochastic program w.p.1 after a sufficiently large number of backward and forward steps of the algorithm.Since the total number of scenarios generated from the historical data is finite and the forward steps generate the sample scenarios independently, the conditions in Proposition 3.1 in <cit.> hold. Thus, our proposition holds.§ RISK-AVERSE HYDROTHERMAL SCHEDULING PROBLEMIn this section, we investigate a risk-averse hydrothermal scheduling problem with distributional robustness by applying our proposed reformulation and algorithm in Sections <ref> – <ref>.The risk-averse hydrothermal scheduling problem is aiming to determine an optimal operational schedule for the hydrothermal system which minimizes the total expected costs including operation cost, fuel cost, and penalty cost for failing to satisfy the electricity load. The hydrothermal system is a combination of hydroelectric generators and other fuel-costing generators like thermal and nuclear generators.Hydroelectric generators utilize stored water in the system reservoirs to provide energy, whose generation is determined by the water inflow.The inflow is assumed to be the only uncertainty of this problem, which follows some unknown distribution and we only have historical data of the inflow volume at each stage. The risk-averse hydrothermal scheduling problem can be modeled as a stochastic dynamic program if the inflow volume of the current stage is predicted at the beginning of that stage. §.§ Problem FormulationTo describe the problem, we let T represent the planning horizon, andwe denote sets ofthermal generators and reservoirs as 𝒢 and ℛ, respectively. For each thermal generator g ∈𝒢, we label the maximum (minimum) generation as G^g(G^g), and denote the unit generation cost as c_t^g at time t. For each reservoir r∈ℛ, we denote the maximum (minimum) reservoir storage as R^r (R^r), and denote the inflow to reservoir r at time t as I^r_t. For convenience, we use energy units, MWh, as the units of R^r (R^r) and I^r_t, which can easily be done by multiplyingconstant coefficients of the true storage level and inflow amount, respectively. Furthermore, we use D_t to represent the total electricity load, anduse c^p_t to represent the penalty cost for each unit of unsatisfied load at time t. Next, we define five continuous decision variables using energy units MWh as follows. We definex^g_t as the thermal energy generated by generator g∈𝒢at time t, and x^p_t as the energy amount failed to satisfy at time t. For each reservoir r∈ℛ, we define the hydroelectric energy generation at time t as x^r_t, the water spillage as s_t^r, andthe storage level as v^r_t for reservoir r∈ℛ at time t, where the initial storage v^r_0 is given as a parameter. The generated hydroelectricity energy is a linear function of the water outflow, which can be represented by the outflow volume times a constant. Based on the above notation, we provide the formulation as follows.hydro-formulationmin_x_1(∑_g∈𝒢c_1^g x_1^g +c_1^p x_1^p) +sup__2 ∈𝒟^2_ξ_2∼_2[ min_ x_2 (c_2^g(ξ_2)x_2^g(ξ_2)+c_2^p(ξ_2)x_2^p(ξ_2))+ ⋯ + sup__T ∈𝒟^T_ξ_T∼_T[ min_ x_T (c_T^g(ξ_T)x_T^g(ξ_T)+c_T^p(ξ_T)x_T^p(ξ_T)) ]] s.t.∑_r∈ℛx_t^r + ∑_g∈𝒢x_t^g + x_t^p = D_t, ∀ t =1, …,T,v_t^r + x_t^r + s_t^r = v_t-1^r + I_t^r, ∀ t =1, …,T, ∀ r ∈ℛ, G^g ≤ x_t^g ≤G^g, ∀ t =1, …,T, ∀ g ∈𝒢, R^r ≤ v_t^r ≤R^r, ∀ t =1, …,T, ∀ r ∈ℛ,x_t^p, x_t^h, s_t ≥ 0, ∀ t =1, …,T,where the objective function (<ref>) is to minimize long-term expected thermal generation cost and penalty cost under the worst-case distribution within the ambiguity set. Constraints (<ref>) ensure the electricity load balance. Constraints (<ref>) restrict on the reservoir water balance, where the inflow amount equals to the sum of storage level difference, spillage and water outflow for hydroelectricity generation. Constraints (<ref>) and (<ref>) represent capacities of the thermal generation and the reservoir storage, respectively. Constraints (<ref>) are the nonnegative constraints of decision variables. §.§ Experiment Settings In the following, we perform numerical experiments on the risk-averse hydrothermal scheduling problem with distributional robustness by applying our proposed reformulation and algorithm in Sections <ref>-<ref>.We randomly generate four cases by assuming the inflow model follows four different classes of true probability distributions, lognormal, truncated normal, Weibull, and exponential distributions. For each case at each stage, we first create and collect N independent random samples from the true distribution to estimate the empirical distribution f_0, and then construct a scenario tree with S scenarios at each stage. Next, the ambiguity set is constructed as 𝒟 = {f≥ 0: f_0^r -d ≤ f^r ≤ f_0^r+d, ∀ r = 1, …, R, ∑_r=1^Rf^r =1 }, where d = max{z_α/2√(f^r_0(1-f_0^r))/√(N), ∀ r = 1,…,R } and z_α/2 is the α-level z-score (here we select α= 5%, and z_α/2 = 1.96). Finally, we compare our risk-averse optimal solutions with risk-neutral solutions under perfect information. All experiments were coded in C++ and implemented on a computer node with two AMD Opteron 2378 Quad Core Processors at 2.4GHz and 4GB memory. IBM ILOG CPLEX 12.3 is utilized as the linear programming solver. For the parameter setting of Algorithm <ref>, we set the optimality gap as 5%, the iteration limit K as 300, the sample number at each stage M_t, ∀ t =2,…,T as 6.For uncertainty set parameter settings, we create a scenario tree with 12 different scenarios at each stage, S=12. For the hydrothermal scheduling problemparameter settings, we consider a 52-week planning horizon for a single reservoir and a single thermal generator. We let the reservoir maximum capacity be R =10^6 and minimum capacity be R = 10^5. The initial storage is set as v_0 = 5.5 × 10^5. The thermal unit generation costs at each stage are randomly generated which vary from $ 45 / to $ 85 /, and the unit penalty cost is set as $ 1000 /.§.§ Computational ResultsIn the following, we compare our risk-averse solutions and their corresponding risk-neutral solutions with perfect information under various data sample sizes and true distribution settings in Figure <ref>.As indicated from Figure <ref>, our risk-averse solutions converges to their corresponding risk-neutral solution as the number collected data samples increases, and the convergence process evolves moderately quickly after collecting 2000 data samples, which numerically proves the convergence of our risk-averse multistage stochastic program with distributional ambiguity to the risk-neutral multistage stochastic program.We present the solution gaps between the risk-averse (RA) objective value and the risk-neutral (PI) objective value, and the algorithm iteration steps under various data sample size and distribution settingsin Table <ref>. In this table, the column “Gap" represents the gap between RA and PI, = (Z_a - Z_n)/Z_n, where Z_a is the risk-averse solution obtained from Algorithm <ref>, and Z_n is the risk-neutral objective value under the corresponding true distribution. The column “Step" represents the iteration steps when our algorithm stops, where 300 means that the corresponding case ceases due to the predefined iteration step limit.As indicated from Table <ref>, the solution gap decreases and the number of required iteration step becomes smaller as the number collected data samples increases, which is coincident with the convergence of the ambiguity set. That is, with more data samples collected, our ambiguity set size shrinks and thus it is faster to solve the risk-averse stochastic program over this ambiguity set. We provide the algorithm performance under various distribution settings with 9000 data samples in Figure <ref>. Figures (<ref>) - (<ref>) represent the upper bound and lower bound evolving processes under different true distributions lognormal, normal, exponential and Weibull, respectively. The horizontal axis is the number of performed iteration steps and the vertical axis is the total cost. Since the gap between upper bound and lower bound is significant in the first 30 iteration steps in the Figure <ref>, we provide a closer look at the convergent behavior of Algorithm <ref> in Figure <ref>, by using the log value of total cost in the vertical axis. The algorithm converges quickly in 35 steps and terminates with about 75 steps for each distribution setting. § CONCLUSIONIn this paper, we present an equivalent reformulation of RMSP, where we use a convex combination of expectation and CVaR to replace the worst-case expectation. The reformulation prevents repeating min-max patterns in the multistage program. As the size of collected data samples goes to infinity, we show that RMSP converges to the risk-neutral MSP, where the optimal objective value and the set of optimal solutions of RMSP converge to those of risk-neutral MSP. We adopt the SDDP algorithm to solve the reformulated RMSP and provide the convergence property for the algorithm. To test the RMSP computation performance, we implement numerical experiments for the risk-averse hydrothermal scheduling problem under different true distributions, which demonstrate the convergence of our RMSP to risk-neutral MSP as the collected data increase to infinity.=12pt plain | http://arxiv.org/abs/1708.07930v1 | {
"authors": [
"Jianqiu Huang",
"Kezhuo Zhou",
"Yongpei Guan"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170826045241",
"title": "A Study of Distributionally Robust Multistage Stochastic Optimization"
} |
eufm10 | http://arxiv.org/abs/1708.07527v2 | {
"authors": [
"Gabriel R. Bengochea",
"Gabriel León"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170824190410",
"title": "Novel vacuum conditions in inflationary collapse models"
} |
Leaf Counting with Deep Convolutional and Deconvolutional Networks Shubhra Aich and Ian Stavness Computer Science, University of SaskatchewanSaskatoon, [email protected], [email protected] ==================================================================================================================================================== In this paper, we investigate the problem of counting rosette leaves from an RGB image, an important task in plant phenotyping. We propose a data-driven approach for this task generalized over different plant species and imaging setups. To accomplish this task, we use state-of-the-art deep learning architectures: a deconvolutional network for initial segmentation and a convolutional network for leaf counting. Evaluation is performed on the leaf counting challenge dataset at CVPPP-2017. Despite the small number of training samples in this dataset, as compared to typical deep learning image sets, we obtain satisfactory performance on segmenting leaves from the background as a whole and counting the number of leaves using simple data augmentation strategies. Comparative analysis is provided against methods evaluated on the previous competition datasets. Our framework achieves mean and standard deviation of absolute count difference of 1.62 and 2.30 averaged over all five test datasets.§ INTRODUCTION Traditional plant phenotyping, which involves manual measurement of plant traits, is a slow, tedious and expensive task. In most cases, manual measurement techniques use sparse random sampling followed by the projection of those random measurements over the whole population which might incorporate measurement bias. Further, plant phenotyping has been identified as the current bottleneck in modern plant breeding and research programs <cit.>.Therefore, interest in image-based phenotyping techniques have expanded rapidly over the past 5 years. Automation of the estimation of these visual traits up to a satisfactory level of accuracy using suitable computer vision techniques can boost production speed and reduce costs since fewer field technicians would be required for manual measurement each year.In this paper, we work on estimating the number of leaves on a plant at the rosette stage, which is an indicator of plant health <cit.>.Our main objective is not only to develop a robust computer vision model, but also to generalize it so that the plant breeders can use this framework regardless of the plant species they are working on and of the quality of the image data they have acquired. Like one of the previous works <cit.>, we also pose this problem as a nonlinear regression problem, where given the images, our framework approximates the count directly without segmenting individual leaf instances. This regression hypothesis is useful for a couple of reasons. First, although this nonlinear regression problem appears to be very high dimensional, it is usually more efficient than counting by identifying the individual leaf instances. Second, from the perspective of supervised machine learning, collecting ground-truth leaf counts is much simpler than generating ground-truth segmented regions for each leaf in the color images. In section <ref> of this paper, we show that the performance of the systems developed under the regression hypothesis is comparable to the state-of-the-art counting by instance segmentation approaches. However, unlike <cit.>, we develop each of the components of our complete model in such a way that it can directly learn from the data without the need for manual heuristics or explicit knowledge on the plant species or other environmental factors. According to the state-of-the-art computer vision and machine learning literature, the best way to develop a generalized model without such prior knowledge is to use deep learning and therefore we adopt this paradigm in our work.Similar to <cit.>, we train a deep convolutional neural network to count leaves by regression. However, the focus of our present work is to develop a single network that can generalize across different rosette datasets, rather than separate networks each built and tuned to maximize performance on an individual dataset. We also develop a deep convolutional-deconvolutional neural network for automatic whole plant segmentation and explore the effect of using a binary segmentation mask as an additional input channel to the leaf counting network in order to improve generalized performance. We evaluate our method as part of the Leaf Counting Challenge 2017 (LCC-2017) and report performance across the five subsets of the competition dataset. Through this work, we hope to inaugurate the research and development of a useful and generalized system for plant breeders to study leaf development in individual plants and eventually to study crop emergence in the field.§ RELATED WORKWe classify the recent literature performing leaf counting either directly or via instance segmentation into three categories, i.e. Leaf Segmentation Challenge in CVPPP-2014 (LSC-2014), Leaf Counting Challenge in CVPPP-2015 (LCC-2015), and others. Below we provide a brief account of the methods under each of these categories.LSC-2014: In total, 4 methods evolve from this competition <cit.>. Although the training dataset for the competition included individual leaf instances indicated by different colors as the ground-truth, none of the 4 approaches use that ground truth to solve the instance segmentation problem. From that standpoint, they are all are eligible for the LCC-2017 competition also. The winner of this competition is IPK <cit.>. This method utilizes 3D histogram of the Lab color space of the training images to model both plant regions and background and test pixels are inferred non-parametrically using direct interpolation on the training data. Then, leaf centers are extracted using mathematical morphology of the distance map of the segmented foreground. These centers along with the foreground segmentation are processed byheuristics-based graph algorithms to generate final instance segmentation map. Next, comes the unsupervised Nottingham approach <cit.>, which segments the foreground using seeded region growing <cit.> over the superpixels <cit.> extracted from the Lab color map. For the subsets of the dataset containing non-overlapping images, empirical thresholds are used instead of the superpixel means as the initial seed. Like IPK <cit.>, they compute the distance map over the foreground pixels. Then, superpixels with centroids nearest to the local maxima in the distance map are chosen as the initial seeds with the assumption that they represent leaf centers the best for watershed based instance segmentation <cit.>. The MSU approach is adopted from the literature on multiple leaf alignment and tracking <cit.> and primarily based on template matching based on Chamfer Matching algorithm <cit.>. The authors use empirical threshold on the “a" plane of the Lab image to select foreground candidates on which template matching is performed. The main drawbacks of this approach are manual selection of both the threshold and the templates and exhaustive template matching with a large number of templates, i.e. 1080 templates for 2 subsets and 1920 for another. The last method submitted in LSC-2014 is Wageningen <cit.>. To segment the plant regions, the authors of this approach train a simple artificial neural network comprising one hidden layer of 10 units with six pixel-based features, i.e. red (R), green (G), blue (B), excessive green (2G-R-B), and variance and gradient magnitude of filtered green pixel values, and then post-process the network output using morphological operations with heuristically chosen parameters. After that, watershed transform <cit.> followed by empirical threshold based merging is performed to produce the instance segmentation result. A limitation of this method is the use of simple pixel features for foreground segmentation without using any contextual information in depth, resulting in the heavy usage of morphology to fine-tune the network output afterward.LCC-2015: Only the winning method of LCC-2015 competition, General Leaf Counting (GLC) <cit.>, is published in CVPPP-2015. To the best of our knowledge, this is the first approach posing the leaf counting problem as a nonlinear regression problem. The authors transform the original RGB image into a log-polar image <cit.> prior to further processing it to exploit the radial structure of the plants. Next, from the log-polar image, they extract patches based on the ground-truth foreground-background ratio in a sliding window fashion. These patch features are further vectorized with K-means <cit.> and triangle encoding <cit.>. Lastly, max-pooling over the patch features is performed to form the final feature vector for each image and a support vector regression network <cit.> is trained for the prediction task. A limitation of this system is that the authors use ground-truth plant segmentations in both training and testing phases of the counting module. While approximate plant segmentations could be generated by other methods <cit.>, the study used perfect segmentations and therefore it is not clear how robust their counting module is to noisy or imperfect segmentations that are typical of automatic segmentation procedures.Others: All the methods proposed since LCC-2015, addressing either the direct counting problem or counting by instance segmentationare found to be based on deep learning, which is not surprising given the resurgence of this subfield of machine learning in recent years. In the recurrent instance segmentation (RIS) approach <cit.>, the authors harness the power of sequential input processing of recurrent neural networks (RNN) <cit.> with the convolutional version of LSTM cells <cit.> to segment out one leaf instance at a time. Unlike the use of LSTM and RNN in natural language processing, the idea is to use convolutional LSTM instead of the original formulation to facilitate the training of the network by mitigating the computational complexity of fully connected layers as well as exploiting the semi-global statistical properties of images. To deal with the problem of possible ordering of individual instances in the image, the authors formulate the loss function based on the relaxed version of intersection over union (IoU) <cit.> and cross-entropy. The work done by Ren and Zemel <cit.> also use RNN similar to RIS <cit.>. However, their approach is primarily focused on extracting small patches each time to segment one instance using a similar idea of recurrent attention model <cit.> and then processing that small patch with LSTM <cit.> and a deconvolutional network <cit.> like architecture to segment a single instance. At the time of this writing, this work demonstrates the state-of-the-art performance for instance segmentation. Both this work and RIS use instance-level ground truth to train their networks and are therefore not directly comparable against ours. Nonetheless, we list their performance results in the Experiments section. Finally, the deep plant phenomics (DPP) approach <cit.> proposes a method addressing the problem of counting directly without both plant segmentation and instance segmentation. The authors customize their architectures as well as input dimensions to achieve state-of-the-art accuracy on different subsets of the LCC-2015 dataset. However, it is not known if the approach would generalize and if a single DPP network would provide consistent results across all datasets. Moreover, their training strategy relies on certain assumptions based on the nature of the images available in the LCC-2015 dataset; therefore, it is not clear how this approach would perform on the new types of images in the LCC-2017 competition. We will provide a detailed discussion about these issues while comparing our framework to DPP in section <ref>.§ OUR APPROACH The approach presented in this section is developed to participate in the LCC competition <cit.> at CVPPP 2017. The high-level design of our framework follows a traditional computer vision workflow where the segmentation module is followed by the counting module (Figure <ref>). Within each module, we incorporate task-specific convolutional architectures, which are trained without explicit knowledge of the plant species to develop a generalized framework able to learn only from the data. The architectures used for segmentation and counting are trained separately, but not independently since the binary mask generated by the segmentation model is used to train the counting model in conjunction with the RGB channels. In the following two subsections, we will describe the architectures along with the rationale behind their design. Training methodologies and data augmentation strategies for these models are described in the experiments section. §.§ SegmentationThe segmentation problem we address is that of differentiating the plant or foreground pixels, from the background. This kind of problem is also known as semantic segmentation where the semantics of the objects are utilized to accomplish the task. In recent years, many papers <cit.> have been published addressing the solution for semantic segmentation from RGB images. Some of these architectures belong to the class of neural networks called deconvolutional networks <cit.>. The main idea behind this kind of network is to construct a compact and informative set of feature maps or vectors from a set of input images, and then generate class-probability maps from the feature maps. Like other convolutional networks, construction of the feature set from the input data is done by a convolutional sub-network comprising multiple layers of convolution, pooling, and normalization operations. This convolutional sub-network is followed by a deconvolutional sub-network consisting of convolution-transpose, unpooling, and normalization operations to generate the desired probability maps. From the standpoint of semantic segmenatation, both height and width of the input and the output are the same. Hence, the deconvolutional part of the network is designed as a mirrored version of the convolutional part, except the input and the output layers, irrespective of the complexity of the problem and the dimensionality of the class-space. Usually the design of a deconvolutional network contains fully connected (FC) layers in the middle to generate the feature vector from the pooled feature maps <cit.>.The FC layers are used to extract features in the global context for segmentation, and are therefore important if global context is necessary for the segmentation task. However, we propose that features in the semi-global context should be sufficient to segment the leaf regions from the background in color images, and therefore the FC layers could be omitted for our application. An advantage of eliminating the FC layers is that it considerably reduces the number of trainable parameters without sacrificing performance. For these reasons, we adopt the SegNet architecture <cit.>, which omits FC layers and has shown promising results on SUN RGB-D <cit.> dataset comprising complicated indoor scenes and CamVid <cit.> video dataset of road scenes.The removal of FC layers in SegNet results in about 90% reduction of the number of trainable parameters as well as computational complexity. Figure <ref> depicts the segmentation network we employ. The front-end convolutional sub-structure of the network is the VGG architecture <cit.> with batch normalization followed by each convolutional layer. In the convolutional front-end of SegNet, there are five 2 × 2 pooling operations with zero overlapping following multiple convolution and rectification layers each time. Hence, the convolved feature maps are compressed 32 times before starting the decompression via the deconvolutional back-end. We hypothesize that such level of compression or semi-global consideration is qualitatively sufficient to solve a comparatively easier problem of whole plant segmentation (Figure <ref>) as compared to other domains of semantic segmentation. §.§ CountingAs shown in Figure <ref>, we use both the RGB image and the corresponding binary segmentation image to estimate the number of leaves after the segmentation is done. The rationale behind providing the counting module with the segmentation mask and the original RGB image instead of providing either the segmented region in the RGB image or the binary mask alone will be evident from Figure <ref>. Although the segmentation results generated by SegNet are sufficiently accurate for the counting phase for many images in the dataset, our network generates spurious segmentations for few of them. The poorly segmented images generally have lower average intensities and regions of leaves where the color and texture properties are washed out or blurred. We expect our network to do more or less accurate segmentation for these images by using semi-global contextual information, but we believe it fails due to the low number of available samples of that kind in the training dataset both in terms of absolute count and ratio of the samples of this particular kind to other kinds. The problem of this data scarcity is specific to the data-hungry approaches like deep learning, which requires a substantial number of training instances of a particular prototype to generate an accurate input-output mapping for that specific type.Therefore, providing both the segmentation and the original image as input, we hope to influence the network to recover the missed plant regions as well as reject the false detections from original image with the help of segmentation mask for counting. We call this four channel input as the SRGB (Segmentation + RGB) image. We also expect that providing the segmentation channel as input to the leaf counting network will help to suppress bias from features in the background of the training images, such as the soil, moss, pot/tray color, which will vary between datasets.The design of our leaf counting by regression network takes inspiration from the VGG architecture <cit.>, which reinforces the idea of deeper architectures with a long list of convolutional and rectification layers stacked one after another with several pooling layers in between and then the classification layer follows a couple of fully connected layers. Usually, this kind of convolutional networks use suitable amount of padding to maintain fixed height and width of the feature maps. Padding the input maps serves well when the network is trained with large-scale datasets containing samples in the order of millions. However, in our case, we have a small dataset of several hundred images to train, which is very difficult to augment beyond several thousand images. Hence, to retain the power of deeper architecture and to train the parameters without significant overfitting at the same time, we reduce the number of parameters effectively by using convolution without padding throughout the network. Moreover, we choose the filter size of the convolutional layers in such a way that before proceeding through the fully connected layers, the feature map turns into a vector. Thus, with zero padding and careful choice of filter size, we are able to reduce the number of parameters from 49M to 30M. Implementation details for both segmentation and regression networks are provided in the following section. § EXPERIMENTSIn this section, we provide a detailed account of our experimental setup. First, we describe the dataset used for evaluation. Next, the training strategies for both networks are specified. Finally, the performance of our framework is analyzed and compared against state-of-the-art literature from both quantitative and qualitative standpoints. §.§ DatasetThe dataset we use to evaluate our framework is provided to the teams registered for the Leaf Counting Challenge (LCC-2017). The objective of this challenge is to come up with the solutions able to count the number of leaves from plant images directly via learning algorithms without detecting individual leaf instances. All the RGB images in the dataset belong to either Tobacco or Arabidopsis plants. For the LCC competition, each RGB image is accompanied by a binary segmentation mask with 1 and 0 indicating plant and background pixels, respectively, and a center binary image with leaves centers denoted by single pixels.The training dataset is organized into 4 directories, namely A1, A2, A3, and A4. Directories A1 and A2 contain Arabidopsis images taken from growth chamber experiments with larger but different field of views covering many plants and then cropped to a single plant. Directory A3 enlists the Tobacco images with the field of view chosen to encompass a single plant. A4 is a subset of another public Arabidopsis dataset <cit.> collected using a time-lapse camera. In total, there are 27 Tobacco images in A3, and 783 Arabidopsis images in the rest of the directories. The organizers denote these directories along with the images as “SPLIT" images since they are split into separate folders according to the origin. In addition, all these directories contain CSV files including ground truth leaf counts under the same nomenclature.The “SPLIT" directory structure for the testing set is the same as training, except that it includes an extra directory denoted by A5, enlisting images from different sources of origin altogether with the objective to emulate a leaf counting task in the wild. Hence, the organizers represent A5 images under the nomenclature “WILD". §.§ Training and Implementation SegNet training: Unlike training in the original SegNet paper <cit.>, we trained our model from scratch without using any pretrained weights for initialization.Also in SegNet, the authors used different learning rates for different modules, whereas a fixed learning rate was used for all the layers in our training.We used an input and output image size of 224× 224 pixels in SegNet, whereas the original image size was approximately 500× 500 and 2000× 2500. While training deeper networks, the obvious advantage of using smaller input-output size than the original ones is data augmentation up to a considerable amount. We augmented the data and train the network in 3 stages. First, for each image, we extracted the union of top 20 object proposals <cit.>, flipped top-bottom and left-right, rotated them with an angular step size of 4 degree, cropped the largest square from the center position to avoid dark regions due to rotation, and created a couple of Gaussian blurred version and corresponding sharpened images. In this way, we generated about 0.8M augmented samples from 810 original images and trained the network for 5 epochs with randomly cropped 224× 224 subsamples. Second, we took the proposal images and their flipped versions and generated nearly 0.3M subsamples of size 224 × 224 deterministically with a fixed stride and train the network for another 8 epochs. Finally, we generated another 0.19M samples in a similar manner as in the second step, but this time from the original images instead of the proposals. Then, we fine-tuned the network with these 0.19M samples for 37 epochs. In all stages, SGD-momentum was used as the optimizer with initial learning rate, momentum and weight decay of 0.01, 0.9, and 0.0001, respectively and these parameters were changed later based on the training statistics. Spatial cross-entropy was used as the error criterion. The ratios of foreground to background weights in the cross-entropy calculation for the first stage was 2.0 and 1.2 for the later steps. In the test phase, we took dense 224× 224 samples deterministically with fixed stride from each of the test images and classified each pixel based on the aggregate probability over the samples. We initialized the convolutional weights with Xavier <cit.> prior to the start of training.Count network training: Training of the counting network is fairly straightforward compared to SegNet. In this phase, we used all the images as a whole without prior cropping or sampling operation for data augmentation to ensure that the ground truth leaf counts were valid for all augmented images. Also, while designing the network architecture, we experimented with adaptive operations to deal with variable sized images, but they did not seem to work better than resizing the images to a fixed size. Moreover, we had to be cautious in the choice of the size for resizing operation so that for bigger images with resolution like 2000 × 2500, properties of the small leaf regions did not deteriorate much. Considering this fact, we chose the modified image size to be 448 × 448 preserving the aspect ratio. Thus, the largest dimension was taken to be 448 and the smaller one was padded with zeros afterward.After the resize operation was performed, each of the images was augmented 8 times using intensity saturation, Gaussian blurring and sharpening, and additive Gaussian noise (Figure <ref>). Each image was also flipped top-bottom and left-right and rotated 180 along with similar augmentations. Thus, we generated 36 slightly different samples with the same ground truth from each original image, resulting in 29160 training instances for the regression network.After the data generation was done, the counting or regression network was trained for 40 epochs using Adam <cit.> with fixed learning rate and weight decay both set to 0.0001. Smooth-L_1 criterion was used as the loss function instead of simple L_1 criterion to prevent gradient explosions as described in <cit.>. At first, we started training the model with normalized FC layers of size 1024. However, based upon the training statistics and to reduce the risk of overfitting, we changed the size of FC layers to 512 and retrained the model with the already trained convolutional weights. Finally, the model trained until epoch 35 was used to generate the prediction for final submission.Implementation: We used Torch<cit.> as the deep learning framework for both models. All the convolutional filters of the segmentation network were of size 3× 3. For regression architecture, 9× 9 convolution was performed until the second max-pooling operation and 5×5 afterward. We used the convolutional stride of 1 throughout both networks. All the pooling operations were 2× 2 max-pooling with stride of 2. The dimension of all fully connected layers in the regression network was 512. Training was performed on a single NVIDIA Quadro P6000 Dell workstation. On this machine, training of SegNet took about 6-7 days, whereas the regression network was trained within a couple of days. Code is publicly available here. [<https://github.com/p2irc/leaf_count_ICCVW-2017>] §.§ Evaluation Evaluation of our complete framework was accomplished in three stages. First, we assessed the segmentation network in terms of the precision and recall (equation <ref>) of the plant pixels. Next, we performed a head-to-head comparison against the winner of the previous LCC competition. Finally, we compared our results to the state-of-the-art approaches. We also performed an ablation study by training our counting network with and without the segmentation channel as input, in order to cast some light on the issue regarding the need for foreground segmentation.Foreground segmentation: Even though the accuracy of binary segmentation is not a criterion for evaluation in the LCC competition <cit.>, we provide precision and recall (equation <ref>) of our segmentation model in Table <ref> to justify our assumption on the sufficiency of semi-global context for leaf segmentation. It is evident from Table <ref> that the segmentation results generated by SegNet using semi-global information are good enough to be used for the regression network. Performance of the segmentation network is comparatively lower for directory A3 (Table <ref>, red text) since there are only 27 Tobacco images in the A3 training set as compared to 783 Arabidopsis images in the rest of the directories. Precision = True Positive/True Positive+False Positive Recall = True Positive/True Positive+False Negative Comparison against the previous winner: Next, we provide comparisons in both Table <ref> and <ref>. Table <ref> provides comparisons against all the recent literature, whereas Table <ref> provides a head-to-head comparison against the LCC-2015 winner, which is more detailed due to the availability of the performance metrics for <cit.>. In Table <ref>, “CountDiff" refers to the mean and standard deviation (shown in parentheses) of the difference in count averaged over images. “AbsCountDiff" is the absolute of “CountDiff". The term “PercentAgreement" indicates the percentage of exact matches between the actual prediction and ground truth measurement for counts. “MSE" is the abbreviation for mean-squared error. From Table <ref>, it is evident that we achieve lower CountDiff and AbsCountDiff for directories A1 and A2. Lower CountDiff means that our model is less biased towards underestimation or overestimation than GLC <cit.>, whereas lower AbsCountDiff can be interpreted as the indicator of better average performance of the system. However, our framework performs poorly on directory A3 (Table <ref>, red text). The reason behind the failure is pretty straightforward. Note that, in the training set, there are in total 783 Arabidopsis images in A1, A2, and A4. On the other hand, there are only 27 Tobacco images in A3, which is scarce for the types of deep architectures we are using that contain millions of parameters. Hence, our regression network fails to model the distribution for leaf counting over the Tobacco images. This inadequacy is also reflected in the AbsCountDiff measure for the test directory A5, which is a mixture of Arabidopsis and Tobacco images altogether.For directory A2, although our CountDiff and AbsCountDiff are better than those of GLC, PercentAgreement of GLC is much better than ours. Apparently, it might seem to be a pitfall of our system. However, the combination of lower AbsCountDiff and lower PercentAgreement means that even though the number of exact predictions is low, all the predictions are pretty close to the original and the overall performance of the system is more or less uniform over the test images. On the contrary, comparatively higher values of AbsCountDiff and PercentAgreement, which belong to GLC for directory A2, refers to the situation where model performance is not uniform over the samples. In other words, predictions may be accurate for easier samples with no leaf overlap or moderate-sized leaves or both, but deteriorate for harder cases with smaller or overlapping leaves. In that sense, our generalized framework is capable of modeling and inferring leaf shapes under deformation and partial occlusion better than GLC given a few hundred images for a particular species. To facilitate this kind of comparative evaluation of our method by the readers, we enlist a set of combinations of the measures along with their possible interpretations in Table <ref>. Also, note that our average measurement (directory “All") is over 501 test images from 5 directories (A1-A5), whereas the average for GLC is taken over 98 test images from 3 directories (A1-A3).General comparison: Table <ref> shows that our method performs well as compared to all the LSC-2014 <cit.> and LCC-2015 <cit.>, except for the failure on directory A3 due to inadequate number of samples. Both RIS+CRF <cit.> and EERA <cit.> use instance-level ground truth. Hence, they are eligible for the segmentation competition (LSC), but not the counting competition (LCC). Nonetheless, we put them in the list to demonstrate our comparability to these state-of-the-art methods developed with instance segmentations that are more expensive in terms of training complexity/time and ground truth data requirements. DPP <cit.> is the only method close to ours in the style of approach, except that they use three shallow regression networks, each one highly customized over a single directory. Moreover, DPP uses random cropping from 10% - 25% for the purpose of data augmentation while training. This could result in mislabeled images if leaves are cropped out of certain images. The new rosette images in the LCC-2017 dataset include larger rosettes that cover more of the image frame (and extend outside the frame in certain cases, see rightmost image in Figure <ref>); therefore it is not clear how DPP would perform on the larger and more varied test images in the new competition dataset. Ablation study: To justify the inclusion of a segmentation network within our framework, we performed an ablation study by training our regression network using only RGB images as the input without foreground segmentation. We found slower convergence than that of using the segmentation images as input. However, counting results using only RGB images were comparable, which supports the approach proposed by DPP of using a regression network directly on RGB images and that the network learns relevant features directly without a priori segmentation. Nonetheless, we do expect that providing foreground segmentation as an additional input channel helps to push the regression architecture to train on localized features within the plant region in the image. This might help to suppress background features that could limit the generalizability of the counting model if provided images of rosettes grown in different backgrounds, e.g. in different pots, trays, or growth tables. The issue of localization of features in these types of regression networks requires additional attention as future work.§ ACKNOWLEDGMENTThis work was undertaken thanks in part to funding from the Canada First Research Excellence Fund and the Natural Sciences and Engineering Research Council of Canada.§ CONCLUSION AND FUTURE WORK In this paper, as a participant of the LCC-2017 competition, we provide a complete and generalized data-driven framework for leaf counting from RGB images directly without instance segmentation. We demonstrate that given a moderate amount of data on any species, our architectures are able to learn to estimate the number of leaves without prior knowledge on that particular species or surroundings of the plant. From the perspective of informed search strategies, we do plant segmentation prior to counting with the assumption that the additional foreground segmentation channel guides the regression model to extract necessary features only from the plant region and thus trains the model correctly. However, based upon other recent works and ours, the need for segmentation prior to counting by the deep networks is still an open question. As future work, we plan to investigate this issue in more detail, with the goal of achieving equivalent performance to that of instance segmentation architectures with much simpler and easier to train non-recurrent networks such as reported in the present study.ieee | http://arxiv.org/abs/1708.07570v2 | {
"authors": [
"Shubhra Aich",
"Ian Stavness"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170824223223",
"title": "Leaf Counting with Deep Convolutional and Deconvolutional Networks"
} |
[email protected] Department of Physics, Chalmers University of Technology, Gothenburg, Sweden, 2 Department of Physics, University of Regensburg, 93040 Regensburg, Germany, 3 Institut für Theoretische Physik, Technische Universität Berlin, 10623 Berlin, GermanyTransition metal dichalcogenides (TMDs) exhibit a remarkable exciton physics including optically accessible (bright) as well as spin- and momentum-forbidden (dark) excitonic states.So far the dark exciton landscape has not been revealed leaving in particular the spectral position of momentum-forbidden dark states completely unclear.This has a significant impact on the technological application potential of TMDs, since the nature of the energetically lowest state determines, if the material is a direct-gap semiconductor. Here, we show how dark states can be experimentally revealed by probing the intra-excitonic 1s-2p transition. Distinguishing the optical response shortly after the excitation (< [100]fs) and after the exciton thermalization (> [1]ps) allows us to demonstrate the relative position of bright and dark excitons.We find both in theory and experiment a clear blue-shift in the optical response demonstrating for the first time the transition of bright exciton populations into lower lying momentum- and spin-forbidden dark excitonic states in monolayer WSe_2.Mapping of the dark exciton landscape in transition metal dichalcogenidesPhilipp Steinleitner^2, Philipp Merkl^2, Rupert Huber^2, Andreas Knorr^3, and Ermin Malic^1 December 30, 2023 ================================================================================================ Transition metal dichalcogenides (TMDs)are characterized by tightly Coulomb-bound electron-hole pairs that efficientlycouple to light and that can be spin-and valley-selectively excited <cit.>. Recent experimental and theoretical studies show that besides these bright excitonic states, dark optically inaccessible excitons <cit.> play a significant role for the optical response <cit.> and the non-equilibrium dynamics in TMDs <cit.>.In particular, the existence of energetically lower dark exciton states strongly limits the photoluminescence quantum efficiency of these materials <cit.>.We distinguish spin-forbidden dark states, where the Coulomb-bound electron and hole exhibit opposite spin, from momentum-forbidden dark states exhibiting a non-zero center-of-mass or angular momentum.Both cannot be accessed by light, since photons cannot provide the required spin or momentum to induce an interband transition into these states.Exciton populations occupying dark states can be optically addressed using mid-infrared spectroscopy <cit.>, however so far, the focus of these studies has been on revealing the radiative recombination time and the exciton formation dynamics.The exciton landscape and the microscopic origin of different types ofdark excitonic states has not been addressed yet. In particular, the spectral position of momentum-forbidden dark excitons has not been revealed, leaving the nature of the energetically lowest states unclear.This is of crucial importance for the technological application potential of TMDs, since it determines whether the material is a direct or an indirect-gap semiconductor.Here, we show how the entire exciton landscape including spin- and momentum-forbidden dark excitons can be identified. We combine a sophisticatedmicroscopic theory <cit.> with femtosecond infrared spectroscopy<cit.> probing the intra-excitonic transition between 1s and 2p states of different exciton types, cf. Figs. <ref>(a)-(b).While the 1s-2p transition is in the THz range in conventional semiconductors <cit.>, the excitonic binding energies in TMDs are one order of magnitude larger and the probing of exciton populations is therefore in the mid-infrared regime <cit.>.Each excitonic state is characterized by its binding energy, which also determines the 1s-2p transition energy. As a result, we can identify different excitonic states based on their linear response to an infrared probe pulse.Furthermore, distinguishing theresponse shortly after the optical excitation (coherent regime, < [100]fs) and after the exciton thermalization (incoherent regime, > [1]ps), we can track the exciton dynamics shifting the population from optically excited coherent excitons to incoherent excitonic states that are formed via exciton-phonon scattering<cit.>. After thermalization, the largest exciton population will be accumulated in the energetically lowest state according to the equilibrium Bose distribution. While the coherent response is always determined by the 1s-2p transition of the bright exciton with zero center-of-mass momentum Q, the incoherent response will be dominated by the energetically lowest exciton (Figs. <ref>(b)-(c)). The latter is characterized by alarger excitonic binding energy and thus a higher 1s-2p separation. As a result, we expect ablue-shift in the incoherent regime approximately [1-2]ps after the optical excitation for materials with dark excitons as the energetically lowest states. In this work, we demonstrate both in theory and experiment a clear blue-shift in the optical response (Fig. <ref>(d)) demonstrating for the first time the existence of energetically lower lying momentum-forbidden dark excitonic states in monolayer tungsten diselenide (WSe_2).To map the excitonic landscape in TMDs, we perform femtosecond infrared spectroscopy experiments and sophisticated calculations on microscopic footing based on the density matrix formalism <cit.>and TMD Bloch equations <cit.>.To obtain access to the optical response of TMD materials to an infrared pulse, we first solve the Heisenberg equation of motion for the microscopic polarization p_ij(t)=⟨ a_i^† a_j ⟩ (t) with the fermionic creation and annihilation operators a_i^† and a_j and the compund indices i,j containing all electronic quantum numbers, such as the band index λ and themomentum q. The microscopic polarization presents a measure for optical inter- and intra-band transitions.To account for the strong exciton physics in TMDs, this microscopic quantity is projected to an excitonic basis with p_𝐐^n=∑_𝐪p_𝐪,𝐐^cvϕ_𝐪^n <cit.>. The appearingeigenfunctions ϕ_𝐪^nare obtained by solvingthe Wannier equation, which is an eigenvalue equation for the exciton with the index n. Here, we take into account optically accessible brightexcitons (K-K excitons) that are locatedwithin the light cone characterized by the center-of-mass momentum Q≈ 0. Furthermore, we include momentum-forbidden intervalley dark excitonic states, where the Coulomb-bound hole and electron are located in different valleys. In particular, we include K-Λ^(') and K-K^(') (Γ-Λ^(') and Γ-K^(')) states with the hole at the K (Γ) point and the electron either in the Λ^(') or the K^(') valley. Here, we also take into account corresponding spin-forbidden excitonic states, where the Coulomb-bound electron and hole show opposite spin. The goal is to calculate the optical response of different monolayer TMDs to a weak infrared probe pulse after coherently pumping the 1s exciton in two limiting cases: (i) In the coherent limit shortlyafter the pump pulse (delay time t_pp < [100]fs), when the system is dominated by the optically generated excitonic coherence p_𝐐^n, cf. the orange line in Fig. <ref> (d).(ii) In the incoherent limit after exciton thermalization (delay time t_pp > [1]ps), when all interband coherences have decayed and the optical response is solely determined by the incoherent exciton density N_𝐐^n reflecting an equilibrium Bose distribution (cf. the blue line inFig. <ref> (d)). While the pump pulse excites coherent excitons, e.g. creates an excitonic polarization, the infrared probe pulse induces intra-excitonic coherences inducing transitions from1s to 2p excitonic states (Fig. <ref>(c)). To determine the infrared absorption of TMDs, wecalculate the macroscopic intraband current, which reads in the excitonic picture J (t)=∑_nm∑_𝐐j_nm( p^n*_𝐐(t)p^m_𝐐(t)+ N_𝐐^nm(t)).This equation contains both the infrared response in the coherent (first term) and incoherent limit (second term). Here,we have introduced the excitonic intraband current j_nm=∑_𝐪,λj_λ(𝐪) ϕ^n*_𝐪ϕ^m_𝐪 corresponding to the current j_λ( q)=-e_0/m_λ k weighted with excitonic wave functions. Here, m_λ corresponds to the effective mass of the considered electronic band λ. Furthermore,N_𝐐^nm describes the incoherent exciton density for n=m, while in the considered thermalized equilibrium situation the off-diagonal terms n≠ m vanish. The dynamics of the excitonic polarization p^n_𝐐(t) as a response to a weak probe pulse is given by the TMD Bloch equation, cf. Eq. (<ref>) in the appendix.The exciton distribution N^n_𝐐(t) can be obtained by taking into account the phonon-assisted formation of incoherent excitons as well as their thermalization towards an equilibrium distribution. While in a previous work <cit.> we have performed the full time- and momentum-dependent calculations, here we exploit the obtained thermalized Bose distribution of incoherent excitons. After solvingthe TMD Bloch equation for the excitonic polarization p_𝐐^n(t) in the coherent limit, we have access to the intra-band current fromEq.(<ref>).Then, we can determine the optical susceptibilityχ(ω)=J(ω)/ε_0 ω^2 A_0(ω), where ε_0 is the vacuum permittivity. Assuming an ultrashort and weak intraband probe pulse and furthermore neglecting terms higher than the 4th order in the field, the linear mid-infrared susceptibility reads in the coherent (χ^c) and incoherent limit (χ^ic)χ^c(ω) = 1/ε_0ω^2∑_nm(|j_1s,m| |p^n_0|^2 /Δε_m,n-iγ_0,c^mn-ħω+ c.c.), χ^ic(ω) = 1/ε_0ω^2∑_𝐐 nm j_nm∑_l( j_ml N_𝐐^nl-j_ln N_𝐐^lm)/ε^m_𝐐-ε^n_𝐐-iγ_𝐐,ic^mn-ħω.Here, we have introduced the quasi-static pump-induced excitonic polarization p^n_0 at the time of the probe pulse and a constant dephasing rate for the intra excitonic transition including γ^mn_0,c=γ^n_0,c+γ^m_0,c in the coherent andγ^mn_𝐐,ic in the incoherent regime. Since a microscopic description of the dephasing rates stemming from higher-order correlations is beyond the scope of this work, we assume a typical constant dephasing of 40 meV <cit.>.In bothregimes, we find that the absorption shows a Lorentzian shape. The spectral position of the Lorentzians is determined by the energy difference of the initial (n) and the final (m) excitonic state. The absorption intensity is given by the optically excited coherent or indirectly formed incoherent exciton populations in the involved states. While the population of the initial state enhances the absorption of the mid-infrared probe pulse, the population of the final state leads to negative contributions that could eventually result in gain <cit.>. Evaluating Eqs. (<ref>) and (<ref>), we have access to the optical response of TMD materials to an infrared probe pulse both in the coherent regime (t_pp<[100]fs) directly after the pump pulse and in the incoherent regime after the excitonic thermalization (t_pp>[1]ps). The corresponding probe absorption is shown in Fig. <ref> (d) for the exemplary material tungsten diselenide (WSe_2).In the coherent limit (yellow area), the initially pumped coherent 1s excitonsdominate the optical response to the infrared pulse. Here, the pronounced peak is located at [160]meV corresponding to the energy difference between the 1s and the 2p excitonic state of the bright K-K exciton.Notethat the2p exciton consists of the states 2p_+ and 2p_-, whichare not energetically degenerated due to a non-vanishing geometric phase in TMD monolayers <cit.>. As a result, we find that the infrared absorption is given by a a superposition of two Lorentzian peaks. For larger delay times between the pump and the probe pulse,incoherent exciton populations are formed <cit.> and have already thermalized into a Bose distribution. Here, predominantly lower energetic states are occupied, cf. Fig. <ref> (c). The excitonic landscape of WSe_2exhibits optically inaccessible dark states well below the initially pumped bright K-K exciton <cit.>.These momentum-forbidden intervalley excitons (K-Λ) are located at different high-symmetry points of the Brillouin zone. The involved conduction band at the Λ valley shows a significantly larger effective mass compared tothe K valley <cit.> resulting in a higher excitonic binding energy and therefore a larger 1s-2p separation. In the incoherent regime, we find that the infrared resonance is blue-shifted by approximately [40]meV, cf. the blue area in Fig. <ref>(d). The new peakat [200]meV corresponds to the1s-2p transition energy of the K-Λ exciton.Investigating in more detail the contribution of different excitonic states to the optical response in the incoherent limit, we find thatthe dark K-Λ^(') excitons clearly dominate followed by the contribution of K-K^(') excitons giving rise to a shoulder at energies around [180]meV, cf. Fig. <ref>(a).Our approach allows to further resolve the contribution of spin-unlike (spin-forbidden, ↑↓) and spin-like (spin-allowed, ↑↓) K-Λ^(') and K-K^(') states. Figures <ref>(b)-(c) show that by far the most pronounced optical response stems from the spin-unlike K-Λ' exciton, which is the energetically lowest state. In contrast, the spin-likeK-Λ exciton is far above the bright state due to the large spin-orbit splitting and therefore only marginally contributes (not shown). We also find a significant contribution from the spin-unlike (around [205]meV) and the spin-like (around [190] meV) K-Λ exciton followed by the spin-unlike (around [180] meV) K-K and the spin-like (around [170] meV) K-K' excitons (Fig. <ref> (c)).The energetic separation of these states can be explained by the different energy renormalization stemming from the Coulomb exchange coupling (X renormalization). The latter gives rise to a shift of spin-like 1s excitonic states, while it vanishes for spin-unlike states and for 2p excitons due to symmetry reasons <cit.>.In order to test the theoretical predictions, we perform a femtosecond near-infrared pump / mid-infrared probe experiment in a WSe_2 monolayer: A 90-fs laser pulse centered at a wavelength of [742]nm selectively injects bright K-K excitons with a vanishing center-of-mass momentum (Q ≈ 0). At a variable delay time t_pp, we subsequently probe the 1s-2p intra-excitonic transition with phase-locked few-cycle mid-infrared pulses covering the range of photon energies between [125]meV and [210]meV. Complete amplitude- and phase-sensitive electro-optic detection of the transmitted probe waveform allows us to extract the full complex-valued response function of the exciton ensemble <cit.>. The inset of Fig. <ref> depicts the real part of the pump-induced mid-infrared conductivity Δσ_1 (corresponding to optical absorption) for two characteristic delay times t_pp. The pump fluence Φ = [16]μ J/cm^2 is chosen such that a moderate exciton density of approximately [10^12]cm^-2 is maintained. For both delay times, a clear peak in Δσ_1 is observable corresponding to the absorptive intra-excitonic 1s-2p transition <cit.>. The peak energy of this transition exhibits a distinct blue-shift as the delay time increases from t_pp = [75]fs to t_pp = [5.1]ps. A more systematic study of the ultrafast evolution of the resonance energy extracted by fitting the experimental data with a Lorentz-oscillator is shown in the main Fig. <ref>. The resonance energy shows a strong blue shift from [162]meV at t_pp = [75]fs to [179]meV at t_pp = [275]fs. Within the next few picoseconds the resonance energy still slightly increases, leading to a saturation at a delay time t_pp = [5.1]ps and a total blue shift of [26]meV. This observation is in excellent agreement with the theoretical prediction of a blue-shift during the process of exciton thermalization resulting in a strong occupation ofenergetically lowest dark excitonic states. Once the thermalization is reached after approximately [2]ps <cit.>, the blue-shift is expected to saturate, as observed in the experiment. To make sure that the observed shift is not a result of a population-induced energy renormalization, we explicitly calculate the shift of the involved bright K-K as well as dark K-K', K-Λ, and K-Λ^' excitons as a function of the optically excited carrier occupation. The latter leads to an enhanced screening of the Coulomb potential and to Pauli blocking effects that reduce excitonic binding energies and consequently influence the 1s-2p transition, cf. Fig. <ref>(a). These effects are accounted for dynamically in the Γ term appearing inEq.(<ref>) in the appendix.In Fig. <ref>(b), we show thedependence of the relative shift of 1s and 2p states with the carrier population.Increasing the population results in a reduced excitonic binding energy, which leads to red-shift of the 1s-2p transition. In the modelled experimental situation, a pump pulse generates a coherent excitonic population that decays radiative and non-radiative. The decay reduces the population and therefore should lead to a blue-shift of the infrared resonance. One may therefore argue that the experimentally observed blue-shift at larger delay timescould be ascribed to this population-induced renormalization effect. However, in a moderate population regime of approximately [10^12]cm^-2 (well below the Mott transition <cit.>) as discussed in the experiment, we find only minor shifts of 1 or [2]meV for a population change of [10^12]cm^-2 confirming the trend measured in gating experiments <cit.> and in the pump intensity study of Ref. 2015_Steinleitner_NatMat. As a result, the population-induced energy renormalization cannot explain the observed blue shift in the range of [30]meV.The excitonic landscape of each TMD material is different and has been controversially discussed in literature. Although there is a huge number of investigations e.g. onMoS_2, it is still unclear, whether this material is a direct or indirect-gap semiconductor. While a temperature-dependent PL study suggests it to be a direct semiconductor <cit.> recent experimental data in an in-plane magnetic field imply that there should be alower lying spin-forbidden excitonic state <cit.>.Even theoretical calculations yield different ordering of dark and bright states in different TMD materials <cit.>. Our approach allows to directly address this question, since the optical infrared response significantly differs depending on the relative position of bright and dark excitonic states. While for direct semiconductors, only moderate population-induced blue shifts of the 1s-2p transition energyare expected,for indirect semiconductors these shifts will be in the range of a few tens of meV reflecting the relative position of dark excitons. In Fig. <ref>, we show a direct comparison of the infrared absorption of the four most studied TMD materials including (a) WSe_2, (b) MoSe_2, (c) WS_2 and (d) MoS_2.Based on DFT-input parameters on the electronic effective masses and band gap energies<cit.>, we find thatMoSe_2 is the only material that does not exhibit a blue-shift in the infrared response, cf. Fig<ref> (b). Surprisingly, we observe a small red-shift due to a minor population of spin-unlike K-K and spin-like K-K' excitons (dashed line Fig<ref> (b)). In contrast, the infrared absorption of all other studied TMDs exhibits pronounced blue-shifts in the range of [30-40]meV suggesting the existence of an energetically lowest dark state. The dark excitonic landscape of WS_2 results in a similar infrared absorption asalready discussed in the case ofWSe_2, however exhibiting a larger blue-shift and a clearer separation of the contribution of K-K and K-Λ excitons due to the larger effective masses of the Λ valleyin WS_2 <cit.>. For MoS_2, we reveal that the major contribution to the infrared absorption stems from the dark Γ-hole excitons, where the holes located at the Γ point have a large effective mass resulting in a strong excitonic binding energy and consequently a pronounced blue-shift of[40]meV of the 1s-2p transition energy in the incoherent regime. Here, the Coulomb exchange coupling lifts up spin-like states leaving spin-unlike Γ-K excitons as the lowest states. This has already been confirmed in a recent experiment <cit.>,where a brightening of these states has been observed in the presence of an in-plane magnetic field.In conclusion, we have shown how one can directly map the excitonic landscape in monolayer transition metal dichalcogenides by probing dark and bright exciton populations using infrared spectroscopy.We distinguish the optical response shortly afterexcitationstemming from coherent optically excited excitons and the response afterexciton thermalizationreflecting the population of energetically lowest incoherent excitons.This allows us to follow the exciton dynamics in different regimes and in particular to identify the energetically lowest state. We find thatMoSe_2 is the only direct-gap semiconductor, while MoS_2, WS_2, and WSe_2 exhibit dark excitons that lie below the optically accessible state. Our findings shed light to the fascinating dark excitonic landscape in TMDs and give new insights that will be relevant for the technological exploitation of atomically thin 2D materials. § APPENDIXThe applied theoretical approach is based on the density matrix formalism <cit.> taking into account all relevant many-particle interactions on microscopic footing.The considered many-particle Hamilton operator accounts for the Coulomb interaction, carrier-phonon, and carrier-light interaction. All occurring matrix elements are evaluated within an effective Hamiltonian tight-binding approach and explicitly include TMD characteristic symmetries and coupling strengths. The carrier-phonon interaction has been taken into account to calculate the excitonic linewidths as well as the phonon-assisted formation and thermalization of incoherent excitons <cit.>. The Coulomb interaction is considered within the Keldysh potential taking into account including the substrate-induced screening <cit.>. The carrier-light interaction is accounted for by the light-matter Hamiltonian in the p· A gauge <cit.>. It consists of an inter- and an intra-band contribution H_c-l=∑_λ(H_𝐀·𝐩^λλ+H_𝐀·𝐩^λλ̅). While the first is determined by the expectation value of the momentum operator and has been investigated in previous studies <cit.>, we focus here on the intra-band Hamiltonian H_𝐀·𝐩^λλ=∑_λj_λ(𝐤) A(t) a_λ𝐤^† a_λ𝐤. Here, j_λ(𝐤) isthe intra-band current projected to the polarization direction 𝐞_A of the incident light field𝐀(t)=A(t) 𝐞_A. The currentreads 𝐣_λ(𝐤)=-e_0/ħ∂ε_𝐤^λ/∂𝐤=- e_0ħ/m_λ 𝐤 with the effective mass m_λ of the considered band λ <cit.>. To determine the infrared absorption of TMDs, we calculate the macroscopic intraband current J(t) as defined in Eq. (<ref>). It includes the optical response in the coherent regime shortly after the optical excitation as well as in the incoherent regime after the exciton thermalization. The latter isdetermined by the thermalized incoherent exciton density N_𝐐^n that is approximated by Bose distribution. The coherent response is given by the excitonic microscopic polarization p^n_𝐐(t) describing the optical response to a weak probe pulse with the field amplitude A_0(t). It is given by the TMD Bloch equationiħṗ_𝐐^n(t) =(ε^n_𝐐-iγ_𝐐^n) p_𝐐^n(t)-∑_mj_nm A_0(t)p^m_𝐐(t)-∑_ qΩ^n_ q(t)(1-2∑_lmϕ_𝐪^mϕ_𝐪^l * p^l_𝐐(t)p^m*_𝐐(t))-Γ^n m_ij p^l_𝐐(t)p^m*_𝐐(t)p_𝐐^n(t). Here, we have introduced the excitonic interband Rabi frequencyΩ^n _ q(t)= ϕ_𝐪^n* 𝐌^vc_𝐪·𝐀(t) with the optical matrix elementM^vc_𝐪 <cit.>. Furthermore,it appears the excitonic dispersion ε^n_𝐐=E^n+ħ^2 Q^2/2M with the excitonic eigenenergy E^n that is obtained by solving the Wannier equation. To account for higher-order dephasing of the excitonic polarization, we have added a material and temperature dependent dephasing rate γ^n_𝐐 that has been obtained by calculating exciton-phonon scattering channels in these materials <cit.>. Finally, wehave included the Coulomb-induced non-linear coupling term Γ^n m_ij = ∑_ q q'ϕ_𝐪^n *( 2 ϕ_𝐪^i ϕ_𝐪^j *ϕ_𝐪' ^m V^cv 𝐪𝐪'_cv 𝐪'𝐪 - ϕ_𝐪' ^i ϕ_𝐪'^j *ϕ_𝐪^m ∑_λ V^λλ𝐪𝐪'_λλ𝐪'𝐪) that gives rise to the excitation-induced renormalization of the Rabi frequency and the electronic band structure.After solvingthe TMD Bloch equation (Eq.(<ref>)) for the excitonic polarization p_𝐐^n(t), we have access to the intra-band current fromEq.(<ref>) and can calculatethe optical susceptibilityχ(ω) determining the optical response to an infrared pulse in the coherent and incoherent regime. § ACKNOWLEDGMENTSWe acknowledge funding from the European Unions Horizon 2020 research and innovation programmunder grant agreement No 696656 (Graphene Flagship) and from the project 734690 (SONAR).Furthermore, the work was supported by the Swedish Research Council (VR) and the Deutsche Forschungsgemeinschaft (DFG) through Research Training Group GK1570 and collaborative research center SFB 1277 (project A05).Finally we thank P. 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"authors": [
"Gunnar Berghäuser",
"Philipp Steinleitner",
"Philipp Merkl",
"Rupert Huber",
"Andreas Knorr",
"Ermin Malic"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170825132206",
"title": "Mapping of the dark exciton landscape in transition metal dichalcogenides"
} |
Maximum A Posteriori Estimation of Distances Between Deep Features in Still-to-Video Face Recognition Andrey V. SavchenkoNational Research University Higher School of EconomicsLaboratory of Algorithms and Technologies for Network Analysis,36 Rodionova St., Nizhny Novgorod, [email protected] Natalya S. BelovaNational Research University Higher School of Economics20 Myasnitskaya St., Moscow, [email protected] 30, 2023 ================================================================================================================================================================================================================================================================================================================================================= This paper presents VEC-NBT, a variation on the unsupervised graph clustering technique VEC, which improves upon the performance of the original algorithm significantly for sparse graphs. VEC employs a novel application of the state-of-the-art word2vec model to embed a graph in Euclidean space via random walks on the nodes of the graph. In VEC-NBT, we modify the original algorithm to use a non-backtracking random walk instead of the normal backtracking random walk used in VEC. We introduce a modification to a non-backtracking random walk, which we call a begrudgingly-backtracking random walk, and show empirically that using this model of random walks for VEC-NBT requires shorter walks on the graph to obtain results with comparable or greater accuracy than VEC, especially for sparser graphs. § INTRODUCTION The word2vec algorithm <cit.> has become one of the most commonly used models for natural language processing, being both faster and more accurate than most other choices for embedding words into Euclidean space. In the word2vec model, an input corpus of sentences is used to create a co-occurrence matrix of words in the vocabulary, and the word vectors are optimized using a cost function related to the adjacencies of words to one another, a process known as Skip-Gram with Negative Sampling (SGNS). The resulting vectors can be used to perform various natural language processing tasks such as analogy prediction and sentiment analysis.This process is not limited to natural language processing. Identifying communities in a graph requires the definition of some measure of similarity between nodes of a graph. The VEC algorithm proposed by Ding et al. <cit.> quantifies this similarity by considering the nodes of a graph as words contained in sentences formed by random walks on the graph. Once this word-sentence representation is obtained, identifying graph communities is analogous to finding semantic or syntactic similarity between words in a language, since these similarities are often defined by the frequency of word pairs or groups in sentences. Starting at each node in the graph, VEC performs several random walks of fixed length. Using the skip-gram method, where two words are considered adjacent if they are within a certain distance of each other in one or more of the sentences, each sentence adds several node-pairs to the co-occurrence matrix of the list of nodes. Word2vec can be used to convert the random walks into node embeddings in Euclidean space, which is in effect a factorization of the co-occurrence matrix, as explained in <cit.>.The resulting vectors can then be clustered using simple techniques such as k-means clustering. The clusters identified by this method have been shown to improve considerably on those generated by classic algorithms such as spectral clustering and acyclic belief propagation <cit.>. However, this algorithm performs less effectively on extremely sparse graphs: if the average degree of the nodes is lower than 3, the algorithm cannot reliably determine the clusters. Unfortunately, many real-world graphs have this feature, such as protein-protein interaction graphs, our motivating example.We attempt to address these problems with the novel concept of using non-backtracking random walks to form the sentences, which encourages more homogeneous clusters. A non-backtracking random walk is a random walk which does not return to the node which it visited in the previous step. Intuitively, this idea is logical: backtracking does not add any new information to the algorithm, so reducing the number of repeated edges should produce better embeddings, which in turn give better clusters. Several groups have conducted theoretical explorations of non-backtracking random walks, which give a more formal justification to this claim <cit.>. We discuss some of the key ideas of these papers as well as show experimentally the clear improvement in performance which results from using non-backtracking random walks. § DESCRIPTION§.§ Notation Let G = (V,E) be a graph with vertex set V and edge set E, and |V| = n, |E| = m. The adjacency matrix A of G is defined as the n× n matrix with a_u,v = 1 if and only if (u,v)∈ E, and the degree matrix D of G is the n × n diagonal matrix indexed by v with each diagonal element equal to the degree of vertex v. A random walk on G is defined as a sequence of vertices (v_0,v_1,...,v_k), each connected by an edge in E, where at each step the next vertex is chosen randomly from those neighboring the current step with equal probability. A random walk on G is also a Markov process, where the transition probability matrix P = D^-1A, which gives x_kP = x_k+1 for starting distribution x_0 and k>0. This recurrence gives the closed form expression x_t = x_0P^t. We can also determine the stationary distribution π(v), defined as the distribution such that π P = π, as dv/vol(G), the degrees of the nodes divided by the total number of edges. For any non-bipartite connected graph G, the stationary distribution is the limit lim_t→∞x_0P^t <cit.>.The graph Laplacian L is defined as D - A, and is often represented as one of two normalized forms, L_sym = I - D^-1/2AD^-1/2 and L_rw = I - D^-1A: L_rw is closely related to P (it is in fact I-P), and L_sym is a symmetric matrix which is similar to L and so has the same eigenvalues. The eigenvalues of L_rw are also closely related to those of L <cit.>. The mixing rate of a graph, defined as ρ = lim sup_t→∞max_u,v|P^t(u,v)-π(v)|^1/t, defines how quickly a Markov chain with transition probabilities P reduces to the stationary distribution π(v), where a lower mixing rate indicates a faster mixing time. The mixing rate is intimately connected to the second eigenvalue of the graph Laplacian, as detailed in <cit.>. A Markov chain is irreducible if every node can be accessed from every other node; it is aperiodic if every node can have a cycle of any length (for instance, a bipartite graph is not aperiodic since all cycles for any node will have even length). A Markov chain is ergodic if it is aperiodic and irreducible. A non-backtracking random walk is defined as a random walk that chooses its next step from all neighbors except the one it visited in the previous step. A begrudgingly-backtracking random walk is the same, with the added condition that if the only choice of edge is the one visited previously, then it will resort to backtracking for that edge. Begrudgingly-backtracking random walks handle several problems caused by non-backtracking random walks especially on sparse graphs: in a non-backtracking random walk, walking to a dangling node (a node with only one connecting edge) forces the walk to end there, resulting in an artificially decreased length of the random walk. In addition, the dangling nodes will be weighted less heavily than they should be, since they will only ever be visited once per walk. §.§ Algorithm The graphs used to measure the performance of VEC-NBT are synthesized using the Stochastic Block Model (SBM), a canonical graph model used for community detection. SBM builds off of the classical Erdos-Renyi G(n,p) random graph model, where each edge of a graph with n nodes is formed with probability p. In SBM, specifically the planted partition model, G(n,p) is additionally given k clusters and each node is added to one of the clusters with equal probability. Edges are formed within the cluster with probability a while inter-cluster edges are formed with probability b, with b lower than a in order to form an assortative graph. We have elected to use graphs with constant scaling, b = λ a. In our model, the probability of an intra-cluster edge being formed is Q_n(k,k) = a = c/n while that of an inter-cluster edge being formed is Q_n(k,k^') = b = c(1-λ)/n. c is the average degree of nodes within the cluster and determines the sparsity of the graph. λ determines how connected the various clusters are: λ=1 would imply completely disjoint clusters, while λ=0 draws no distinction between clusters.Thus, our model can be represented as G(n,k,c,λ), which fully determines the graph.To generate a Euclidean embedding of the nodes of the graph, VEC performs r = 10 random walks per node, each with length l = 60. Since the graphs are unweighted, each neighbor is equally likely to be chosen. Each random walk is converted into a sentence by counting the number of pairwise co-occurrences for all node pairs: for the co-occurrence matrix W, W_i,j is the total number of times node j occurs within w=8 places of node i in all random walks (this defines a skip-gram with window size 8). Note that an isolated node (a node with a degree of 0) will not be part of any sentence, since the random walk will not reach it and random walks starting there will not be included in the corpus. This observation makes logical sense because a node with no connections cannot be said to be in any cluster, since there is no information about the node at all.Once the sentences have been formed, an existing implementation of a popular word embedding algorithm, namely, word2vec <cit.>, is used to convert the sentences into vectors. word2vec takes the corpus of random walks and embeds them into d-dimensional space by means of a stochastic gradient descent algorithm. An embedding dimension of d=50 is used in our tests to correspond to the original parameters used for VEC. Finally, the embeddings are clustered using a standard k-means clustering scheme.VEC is shown to have consistently performed better than standard community detection methods, such as spectral clustering and acyclic belief propagation, both in accuracy and robustness to random initialization of the graph <cit.>. However, accuracy is still lower than desirable for very sparse graphs. In addition, although 60 is not an excessively long random walk, reducing it would speed up the algorithm.We propose replacing the simple random walk in VEC with a begrudgingly-backtracking random walk, through which we find that both the accuracy and the runtime of VEC can be improved while using shorter random walks. By removing the possibility of revisiting an already encountered node, the begrudgingly-backtracking random walk diffuses over the nodes of the graph faster than a backtracking random walk due to its faster mixing rate <cit.>. Thus, a shorter begrudgingly-backtracking random walk can identify the community structure of sparse graphs better than a simple random walk of greater length. This is a classic example of exploration vs. exploitation in machine learning and statistics.§ THEORY VEC-NBT produces significantly better clustering than VEC consistently across a range of graph sparsity levels and number of nodes. This improvement in performance can be attributed to the faster mixing rate of a begrudgingly-backtracking random walk compared to a simple random walk.Consider a simple random walk on a non-bipartite graph Gwhere each node u has degree d_u. This is a first-order Markov chain with a transition probability matrix P as follows <cit.>.P(u,v) =1/d_u ifuv ∈ E 0otherwise As stated in Section 2.1, the stationary distribution for a random walk on G is given by π(v) = d_v/vol(G). However, a non-backtracking random walk on a graph G is a second-order Markov chain. We impose the additional constraint that G has a minimum degree of 2 for all nodes. To convert this random walk into a first-order Markov chain, the transition probability matrix P̃ is defined on the directed edge set of the graph instead of the vertex set <cit.>. P̃ is a 2m × 2m matrix with P̃((u,v), (x,y)) representing the transition probability between edge (u,v) to edge (x,y) such thatP̃((u,v), (x,y)) =1/d_v-1 ifv = x, y ≠ u0otherwise Note that P̃ is doubly stochastic. It is proven in <cit.> that since P̃ is irreducible and aperiodic, the non-backtracking random walk converges to the stationary distributionπ̃= 1/vol(G)where 1 is the unit vector of length 2m.As discussed in <cit.>, the mixing rate of the backtracking and the non-backtracking random walk is equal to the second largest eigenvalues of their respective transition probability matrices, P and P̃. Thus, if ρ and ρ̃ are the mixing times of the backtracking and non-backtracking random walks respectively and λ_2 is the second largest eigenvalue of the adjacency matrix of a d-regular graph G, we have ρ = λ_2/dandρ̃= λ_2 + √(λ_2^2 - 4(d-1))/2(d-1). Furthermore, <cit.> prove that the non-backtracking random walk has a faster mixing rate than a backtracking random walk, yielding the following bounds:For 2√(d-1)≤λ_2 ≤ d:d/2(d-1)≤ρ̃/ρ≤ 1 For λ_2 ≤ 2√(d-1) andd = n^o(n):ρ̃/ρ = d/2(d-1) + o(1)In VEC-NBT, we use a begrudgingly-backtracking random walk, a variation of the non-backtracking random walk. Let P̂ be a 2m × 2m matrix defining the transition probabilities of the begrudgingly-backtracking random walk on the edge set of a graph G such thatP̂((u,v), (x,y)) =1/d_v-1 ifv = x, y ≠ u1ifv = x, y = u, d_v = 1 0otherwiseP̂ is doubly stochastic.Since P̃ has been shown to be doubly stochastic, we only need consider those elements that would change from P̃ to P̂, and these are exactly the rows and columns corresponding to any dangling nodes. Let x be a node of G with degree 1, connected only to node y; then the single row going to x, (y,x), has only the element returning to y, (x,y), which has a weight of 1; similarly, the single column coming from x has only the element that sent it there.Hence, P̃ is the same as P̂, except in the rows and columns of degree-1 nodes: where P̃ has rows and columns of zeros, P̂ has a 1 where the edge can be included. Since P̃ is only defined on graphs where the minimal degree is greater than 1, P̃ is still doubly stochastic, and P̂ extends the definition to graphs with singlet connectivity. In this way we can relax the requirement that all elements of G have minimal degree 2.We can also show that P̂ is also irreducible and aperiodic, provided P̂ has a single connected component and at least one element with degree greater than 1: clearly if P̃ is irreducible, extending the graph to include single edges that can be traveled down and back will not make the chain reducible; and it can be shown that an irreducible Markov chain with at least one aperiodic node is aperiodic and P̃ is aperiodic, so adding more nodes will not change that result. Therefore, by the same argument made for π̃, we can see that the stationary distribution for the begrudgingly-backtracking random walk isπ̂= 1/vol(G) Thus, we can think of the begrudgingly-backtracking random walk as a variant of the non-backtracking random walk which only requires the graph G to have d_u ≥ 1 for each node u. Given the similarities between the two random walks, we hypothesize that the begrudgingly-backtracking random walk has a mixing rate that is equal or similar to that of the non-backtracking random walk, which explains the fast convergence and greater accuracy observed in our experiments. Furthermore, we suspect that even though the faster mixing rate of the non-backtracking random walk is proven only for a d-regular graph, it seems to hold in our experiments since the parameter c used to generate the SBM graphs ensures that each node has a constant average degree. These two points are yet to be fully explored and formalized. § NUMERICAL RESULTS We compare the performance of VEC and VEC-NBT on SBM graphs generated using the parameters given through two metrics: Correct Classification Rate (CCR) and Normalized Mutual Information (NMI). For random walks and embedding, VEC-NBT uses the same parameters used by VEC with the exception of the length of the random walk (l=5, 10) and the window size (w=5) and twice the number of random walks (r=20). Here, we show empirically that VEC-NBT consistently achieves better accuracy than VEC for sparser graphs (low values of c) and comparable accuracy to VEC at higher sparsity levels.CCR is defined as the number of correctly classified points divided by the total number of nodes. To ensure the calculated clustering matches the ground truth, we use a linear sum assignment to match the cluster assignments to the original labels. Because of this, CCR is defined (at least for 2 clusters) only between 0.5 and 1, since a measured CCR of 0 would indicate that every 1 was labeled as a 2 and vice versa - which is in fact a perfect clustering. In our graphs we have plotted CCR as a percentage between 1/K and 1 where K is the number of clusters.NMI is defined as the mutual information I(X;Y) normalized by the square root of the entropies H(X) and H(Y):NMI(X,Y) := I(X;Y)/√(H(X)H(Y)) = H(X)+H(Y)-H(X,Y)/√(H(X)H(Y)),a more technical metric measuring the information content of the resulting labels.Figures are shown with the original VEC algorithm (“BT") with solid lines and our new algorithm (“NBT") with dashed lines. CCR and NMI are shown for each algorithm on each plot. Note that red points correspond to CCR measurements, on a 50-100% scale, and blue points correspond to NMI measurements, between 0 and 1. NMI tends to be a more accurate indicator of performance.Unless otherwise specified, the x-axis is the sparsity of the graph, varying from 2 to 20; the number of clusters is 2; the graph has 10000 nodes; and the walks are 10 steps long.§ CONCLUSIONIn this paper, we presented VEC-NBT, which is a modification of VEC <cit.> using non-backtracking random walks instead of simple random walks. We show experimentally that VEC-NBT outperforms VEC for SBM model graphs across all ranges of parameters such as the number of nodes, number of clusters, and the length of the random walk, especially for sparser graphs. We discuss the theoretical basis for these results - the faster mixing rate of non-backtracking random walks compared to backtracking random walks. Finally, we analyze the connection between the begrudgingly-backtracking random walk we used and the non-backtracking random walk on which it was based. Future work will focus on formalizing our hypotheses about the behaviour of these two random walks, as well as exploring performance with other graph models which might better relate to real data.apalike | http://arxiv.org/abs/1708.07967v1 | {
"authors": [
"Brian Rappaport",
"Anuththari Gamage",
"Shuchin Aeron"
],
"categories": [
"stat.ML",
"cs.LG",
"cs.SI"
],
"primary_category": "stat.ML",
"published": "20170826134022",
"title": "Faster Clustering via Non-Backtracking Random Walks"
} |
Ultra-Dense Networks: A New Look at the Proportional Fair SchedulerMing Ding^, David López Pérez^†, Amir H. Jafari^∗, Guoqiang Mao^∦^, Zihuai Lin^^Data61, Australia, ^†Nokia Bell Labs, Ireland^∦School of Computing and Communication, University of Technology Sydney, Australia^∗Dept. of Electronic & Electrical Engineering, University of Sheffield, UK^The University of Sydney, AustraliaDecember 30, 2023 ============================================================================================================================================================================================================================================================================================================================================ In this paper, we theoretically study the proportional fair (PF) scheduler in the context of ultra-dense networks (UDNs). Analytical results are obtained for the coverage probability and the area spectral efficiency (ASE) performance of dense small cell networks (SCNs) with the PF scheduler employed at base stations (BSs). The key point of our analysis is that the typical user is no longer a random user as assumed in most studies in the literature. Instead, a user with the maximum PF metric is chosen by its serving BS as the typical user. By comparing the previous results of the round-robin (RR) scheduler with our new results of the PF scheduler, we quantify the loss of the multi-user diversity of the PF scheduler with the network densification, which casts a new look at the role of the PF scheduler in UDNs. Our conclusion is that the RR scheduler should be used in UDNs to simplify the radio resource management (RRM) [To appear in IEEE GLOBECOM2017. 1536-12762015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. Please find the final version in IEEE from the link: http://ieeexplore.ieee.org/document/xxxxxxx/. Digital Object Identifier: 10.1109/xxxxxxxxx], which is justified by the diminishing multi-user diversity of the PF scheduler in dense SCNs.For the case of sparse networks, we further derive an easy-to-compute upper-bound performance. In particular, for practical power models and 3GPP path loss functions, our studies reveal that compared with the baseline scheme with all BSs active, various IMC modes with different levels of power consumption can double or even triple the EE performance in dense SCNs, e.g., when the BS density is 10^3 BSs/km^2.The results derived from our analysis shed valuable new light on the deployment and the operation of future dense SCNs in 5G. ^Guoqiang Mao's research is supported by the Australian Research Council (ARC) Discovery projects DP110100538 and DP120102030 and the Chinese National Science Foundation project 61428102.and it can largely improve the performance of the 5th-generation (5G) networks Stochastic geometry, Line-of-sight (LoS), Non-line-of-sight (NLoS), Dense small cell networks (SCNs), Coverage probability, Area spectral efficiency, Energy efficiency. § INTRODUCTIONPlaceholder Network densification is envisioned to be the key solution to meet users' traffic demands in the 5th-generation (5G) networks <cit.>, since it has the potential to linearly improve the network capacity with the increasing number of deployed cells.Hence, dense small cell networks (SCNs) have attracted significant attention as one of the most promising approaches to rapidly increase network capacity and meet the ever-increasing data traffic demands. Indeed, the orthogonal deployment[The orthogonal deployment means that small cells and macrocells operate on different frequency spectrum, i.e., Small Cell Scenario #2a <cit.>).] of dense small cell networks (SCNs) within the existing macrocell networks <cit.> has been the workhorse for capacity enhancement in the 4th-generation (4G), developed by the 3rd Generation Partnership Project (3GPP), and this approach to capacity enhancement will continue in 5G with the adoption of ultra-dense networks (UDNs) <cit.>.This is due to its large spectrum reuse and its easy management; the latter one arising from its low interaction with the macrocell tier, e.g., no inter-tier interference. In this paper, we focus on the analysis of such orthogonal deployment of UDNs.Despite of its benefits, the SCN densification also opens up new research questions. In particular, scheduling has been conceived as an effective technique used at base stations (BSs) to efficiently use the available spectrum and improve the overall system throughput. In most cases, a proportional fair (PF) scheduler <cit.> serves as an appealing technique that offers a good trade-off between maximizing overall throughput and improving fairness among user equipments (UEs) with diverse channel conditions. However, the gains of the PF scheduler may be limited in UDNs, mostly because the number of UEs per active BS is considerably reduced, and partly because UEs may not experience very different channel conditions on different subcarriers due to the dominance of LOS considering the closeness of the UE and its serving BS. This gives rise to the question of whether the PF scheduling is as efficient for UDNs as it is for sparse networks, or whether it can be substituted by other ones of lower complexity, such as a round-robin (RR) one that semi-randomly selects UEs to serve. In this paper, we answer this fundamental question by theoretical analyses.This paper answers this fundamental question by comparing the performance of the PF scheduler and that of the RR scheduler as the SCN densifies. Network densification also brings questions about fundamental trade-offs. The smaller the cell size, the closer the UE to its serving BS – reducing the path loss, but the stronger the LOS – reducing the multi-user diversity. Furthermore, decreasing the cell size will not only diminish the multi-user diversity, but also will increase the interference due to the dominance of LOS. As a result, it is necessary to take a systematic view towards the tradeoffs of network densification. To the best of our knowledge, there has been no prior work on the theoretical study of the PF scheduler in the context of UDNs, the density of which could be as large as tens of thousands of BSs per km^2 <cit.>. Generally speaking, the existing work on PF schedulers does not scale well with the network densification. In more detail, the studies in the literature can be classified into 3 categories: (i) analysis of a scenario with merely one BS, e.g., based on the Gaussian approximation (GA) of the instantaneous data rate per user <cit.>, (ii) analysis of a scenario with a limited number of BSs <cit.>, which quickly becomes computationally infeasible for UDNs, and does not consider the reduction of the number of UEs per BS as the network densifies, and (iii) system-level simulations for large-scale networks <cit.>, which lacks analytical rigor.Placeholder Compared with the existing work, the main contributions of this paper are: * For the first time, we use stochastic geometry <cit.> to derive the analytical results of the coverage probability and the area spectral efficiency (ASE) performance for UDNs with the PF schedulers used at BSs. The key point of our analysis is that the typical user is no longer a random user as assumed in most studies of stochastic geometry <cit.>.Instead, a user with the maximum PF metric is chosen by its serving BS as the typical user. For the case of sparse networks, we further derive an easy-to-compute upper-bound performance. * By comparing the previous results of the RR scheduler and our new results of the PF scheduler, we quantify the loss of the multi-user diversity of the PF scheduler with the network densification, which leads to the conclusion that the RR scheduler should be used in UDNs to simplify the radio resource management (RRM). Placeholder The rest of this paper is structured as follows. Section <ref> describes the network scenario and the wireless system model considered in this paper. Section <ref> presents our theoretical results on the coverage probability and the ASE for UDNs while using PF scheduling. The numerical results are discussed in Section <ref>, with remarks casting a new look at the PF scheduler in the context of UDNs. The conclusions are drawn in Section <ref>.§ NETWORK SCENARIO AND SYSTEM MODELPlaceholder In this section, we present the network scenario, the wireless system model and the PF scheduler considered in this paper. §.§ Network ScenarioPlaceholder For a certain time-frequency resource block, we consider a downlink (DL) cellular network with BSs deployed on a plane according to a homogeneous Poisson point process (HPPP) Φ with a density of λ BSs/km^2. Active UEs are also Poisson distributed in the considered DL network with a density of ρ UEs/km^2. Here, we only consider active UEs in the network because non-active UEs do not trigger data transmission, and thus they are ignored in our analysis.Note that the total UE number in cellular networks should be much higher than the number of the active UEs, but at a certain time slot and on a certain frequency band, the active UEs with data traffic demands are not too many. As shown in <cit.>, a typical density of the active UEs in 5G is around 300UEs/km^2.In practice, a BS will enter an idle mode if there is no UE connected to it, which reduces the interference to neighboring UEs as well as the energy consumption of the network. The set of active BSs should be determined by a user association strategy (UAS). In this paper, we assume a practical UAS as in <cit.>, where each UE is connected to the BS having the maximum average received signal strength, which will be formally presented in Subsection <ref>.Note that in our previous work <cit.> and some other existing work, e.g., <cit.>, it was assumed that each UE should be associated with its closest BS. Such assumption is not appropriate for the considered path loss model in (<ref>), because in practice a UE should connect to a BS offering the largest received signal strength. Such BS does not necessarily have to be the nearest one to the UE, and it could be a farther one with a strong LoS path.Note that such BS idle mode operation is not trivial, which even changes the capacity scaling law <cit.>. Since UEs are randomly and uniformly distributed in the network, we assume that the active BSs also follow an HPPP distribution Φ̃ <cit.>, the density of which is denoted by λ̃ BSs/km^2. Note that λ̃≤λ and λ̃≤ρ, since one UE is served by at most one BS.Also note that a larger ρ requires more active BSs with a larger λ̃ to serve the active UEs.From <cit.>, λ̃ is given by it was shown that the formula proposed in <cit.> to calculate λ̃ is accurate for UDNs, which is given by λ̃=λ[1-1/(1+ρ/qλ)^q],where according to <cit.>, q depends on the path loss model, which will be presented in Subsection <ref>.According to <cit.>, Considering both active and inactive BSs,the per-BS coverage area size X can be approximately characterized by a Gamma distribution and the probability density function (PDF) of X can be expressed asf_X(x)=(qλ)^qx^q-1exp(qλ x)/(q),where (·) is the Gamma function <cit.>. The UE number per BS is denoted by a random variable (RV) K, and the probability mass function (PMF) of K can be calculated asf_K(k)=[K=k](a)= ∫_0^+∞(ρ x)^k/k!exp(-ρ x)f_X(x)dx(b)= Γ(k+q)/Γ(k+1)Γ(q)(ρ/ρ+qλ)^k(qλ/ρ+qλ)^q,where (a) is due to the HPPP distribution of UEs and (b) is obtained from (<ref>).Note that f_K(k) satisfies the normalization condition: ∑_k=0^+∞f_K(k)=1.It can be seen from (<ref>) that K follows a Negative Binomial distribution <cit.>, i.e., K∼NB(q,ρ/ρ+qλ).As discussed in Subsection <ref>, we assume that a BS with K=0 is not active, which will be ignored in our analysis due to its muted transmission. Thus, we focus on the active BSs and denote the UE number per active BS by a positive RV K̃. Considering (<ref>)and the fact that the only difference between K and K̃ lies in K̃≠0, we can conclude that K̃ follows a truncated Negative Binomial distribution, i.e., K̃∼truncNB(q,ρ/ρ+qλ), the PMF of which is denoted by f_K̃(k̃),k̃∈{ 1,2,…,+∞} and can be written asf_K̃(k̃)=[K̃=k̃]=f_K(k̃)/1-f_K(0). where the denominator (1-f_K(0)) represents the probability of a BS being active. Note that based on the definition of λ̃ in Subsection <ref>, we have that λ̃=(1-f_K(0))λ. Note that f_K̃(k̃) satisfies the normalization condition: ∑_k=1^+∞f_K̃(k̃)=1.Example: As an example, we plot the results of f_K̃(k̃) in Fig. <ref> with the following parameter values[With a 3GPP path loss model incorporating both line-of-sight (LoS) and non-LoS (NLoS) transmissions <cit.>, we have shown in <cit.> that q=4.05 when the UE density equals to ρ=300UEs/km^2. Note that our analytical results in this paper can work with any value of q.]: UE density ρ=300UEs/km^2, BS density λ∈{ 50,200,1000}BSs/km^2 and q=4.05.From this figure, we can draw the following observations: * The analytical results based on the truncated Negative Binomial distribution match well with the simulation results. More specifically, the maximum difference between the simulated PMF and the analytical PMF is shown to be less than 0.5 percentile.* For the active BSs, f_K̃(k̃) shows a more dominant peak at k̃=1 as λ, and in turn λ̃, increases. This is because the ratio of ρ to λ̃ gradually decreases toward 1 as λ increases, approaching the limit of one UE per active BS in ultra-dense SCNs. In particular, when λ=1000BSs/km^2, more than 80% of the active BSs will serve only one UE. Intuitively speaking, each of those BSs should dynamically engage all subframes for DL or UL transmissions based on the specific traffic demand of its served UE. In such way, the subframe resources can be fully utilized in dynamic TDD. Note that such operation is not available in static TDD due to its fixed TDD configuration. Furthermore, the cumulative mass function (CMF) of K̃ can be written asF_K̃(k̃)=∑_t=1^k̃f_K̃(t). §.§ Wireless System ModelPlaceholder Following <cit.>, we adopt a general path loss model, where the path loss ζ(r) is a multi-piece function of r written asζ(w)=ζ_1(w)=[ζ_1^L(w),; ζ_1^NL(w), ] [ with probability Pr_1^L(w); with probability (1-Pr_1^L(w)) ],when 0≤ w≤ d_1 ζ_2(w)=[ζ_2^L(w),; ζ_2^NL(w), ] [ with probability Pr_2^L(w); with probability (1-Pr_2^L(w)) ],when d_1<w≤ d_2 ⋮ ⋮ ζ_N(w)=[ζ_N^L(w),; ζ_N^NL(w), ] [ with probability Pr_N^L(w); with probability (1-Pr_N^L(w)) ],when w>d_N-1. ζ(r)=ζ_1(r),when 0≤ r≤ d_1 ζ_2(r),when d_1<r≤ d_2 ⋮ ⋮ ζ_N(r),when r>d_N-1,where each piece ζ_n(r),n∈{ 1,2,…,N} is modeled asζ_n(r)=[ζ_n^L(r)=A_n^Lr^-α_n^L,; ζ_n^NL(r)=A_n^NLr^-α_n^NL, ] [LoS Prob.: Pr_n^L(r); NLoS Prob.: 1-Pr_n^L(r) ],whereIn practice, A_n^L, A_n^NL, α_n^L and α_n^NL are constants obtainable from field tests <cit.>. * ζ_n^L(r) and ζ_n^NL(r),n∈{ 1,2,…,N} are the n-th piece path loss functions for the LoS transmission and the NLoS transmission, respectively,* A_n^L and A_n^NL are the path losses at a reference distance r=1 for the LoS and the NLoS cases, respectively,* α_n^L and α_n^NL are the path loss exponents for the LoS and the NLoS cases, respectively.Moreover, Pr_n^L(r) is the n-th piece LoS probability function that a transmitter and a receiver separated by a distance r has a LoS path, which is assumed to be a monotonically decreasing function with regard to r. Such assumption has been confirmed by existing measurement studies <cit.>. For convenience, {ζ_n^L(r)} and {ζ_n^NL(r)} are further stacked into piece-wise functions written asζ^Path(r)=ζ_1^Path(r),when 0≤ r≤ d_1 ζ_2^Path(r), when d_1<r≤ d_2 ⋮ ⋮ ζ_N^Path(r),when r>d_N-1,where the string variable Path takes the value of “L” and “NL” for the LoS and the NLoS cases, respectively. For convenience, {Pr_n^L(r)} is stacked into a piece-wise function asPr^L(r)=Pr_1^L(r),when 0≤ r≤ d_1 Pr_2^L(r), when d_1<r≤ d_2 ⋮ ⋮ Pr_N^L(r),when r>d_N-1.As a special case to show our analytical results, we consider the two-piece path loss and the exponential LoS probability functions defined by the 3GPP <cit.>. Specifically, we have N=2, ζ_1^L(w)=ζ_2^L(w)=A^Lw^-α^L, ζ_1^NL(w)=ζ_2^NL(w)=A^NLw^-α^NL, Pr_1^L(w)=1-5exp(-R_1/w), and Pr_2^L(w)=5exp(-w/R_2), where R_1=156 m, R_2=30 m, and d_1=R_1/ln10=67.75 m <cit.>. For clarity, this case is referred to as the 3GPP Case hereafter.As discussed in Subsection <ref>, we assume that each UE is connected to the BS having the maximum average received signal strength, which is equivalent to the BS with the largest ζ(r).Note that the generality and the practicality of the adopted path loss model (<ref>) have been well established in <cit.>. In more detail, this model is consistent with the ones adopted in the 3GPP <cit.>, <cit.>, and includes those models considered in <cit.> and <cit.> as its special cases. In <cit.>, only NLoS transmissions were considered, i.e., assuming Pr_n^L(r)=0,∀ n∈{ 1,2,…,N} in (<ref>). In <cit.>, only one piece of path loss function with one LoS path loss exponent and one NLoS path loss exponent was considered, i.e., assuming N=1 in (<ref>). Another important note is on the practical usage of the proposed path loss model, which is summarized in the following: * As have been addressed in <cit.>, the standard single-slope path loss function does not accurately capture the dependence of the path loss exponent on the link distance in many future networks, such as the multi-ray transmission environments, the dense/clustered small cells, the millimeter wave communications with blockage, etc. Therefore, a dual or more slope path loss function such as (<ref>) should be considered.* The realistic LoS probability functions usually take complicated mathematical forms, e.g., a sum of exponential functions in the 3GPP standards <cit.>, which will be addressed in more detail in Section . Therefore, to achieve both the analytical tractability and the accurate results, it is desirable to approximate the complicated LoS probability function as a few pieces of elementary functions such as the linear function. Such piece-wise LoS probability function is well captured by (<ref>) and can be studied in our framework.* With the justification of both (<ref>) and (<ref>), we can conclude that the proposed path loss model in(<ref>) is versatile to cope with various practical network scenarios.Finally, we assume that each BS/UE is equipped with an isotropic antenna, and that the multi-path fading between a BS and a UE is modeled as independently identical distributed (i.i.d.) Rayleigh fading <cit.>. In order to make the 3GPP Case even more practical, we will further consider distance-dependent Rician fading <cit.>and correlated shadow fading <cit.> in our simulations to show their minor impact on our conclusions. More specifically, we adopt the practical Rician fading defined in the 3GPP <cit.>, where the K factor in dB unit (the ratio between the power in the direct path and the power in the other scattered paths) is modeled as K[dB]=13-0.03r, where r is the distance in meter. We consider a practical correlated shadow fading defined in 3GPP <cit.>, where the shadow fading in dB is modeled as zero-mean Gaussian random variables, e.g., with a standard deviation of 10dB <cit.>. The correlation coefficient between the shadow fading values associated with two different BSs is denoted by τ, e.g., τ=0.5 in <cit.>. §.§ The PF SchedulerPlaceholder The original operation of the PF scheduler is as follows <cit.>, * First, the average throughput of each UE is tracked by an exponential moving average at the BS.* Second, each UE frequently feeds back its channel state information (CSI) to its serving BS, so that such BS can calculate the ratio of the instantaneous achievable rate to the average throughput for each user, which is defined as a PF metric for UE selection.* Finally, the UE with the maximum PF metric will be selected for DL transmission, which is formulated asu^*=u∈{ 1,2,…,k̃}max{R̃_u/R̅_u} ,where u, u^*, R̃_u and R̅_u denote the UE index, the selected UE index, the instantaneous achievable rate of UE u and the average throughput of UE u, respectively. Note that the distribution of k̃ has been discussed in (<ref>).according to our modeling of the UE number per active BS in (<ref>), we have that u∈{1,2,…,k̃}. Placeholder From a network performance analysis point of view, it is very difficult, if not impossible, to analyze the original PF scheduler given by (<ref>). This is because that the objective of a performance analysis is usually to derive the average user throughput R̅_u or aggregate inter-cell interference, but in this case it is part of the PF metric, i.e., R̃_u/R̅_u, and it should be known and plugged into the UE selection criterion of (<ref>) before the performance analysis of R̅_u is carried out. A widely adopted approach to tackle this dilemma is to use alternative measures of CSI in a PF metric, instead of R̃_u and R̅_u <cit.>.Fortunately, there is another way without approximation, but requiring modification of the definition of the PF metric. Here, we follow the framework developed in <cit.>, where the authors proposed to use the ratio of the instantaneous signal-to-noise ratio (SNR) to the average SNR as a PF metric instead of the original one. More specifically, the UE selection criterion of the PF scheduler proposed in <cit.> is given byu^*=u∈{ 1,2,…,k̃}max{Z̃_u/Z̅_u} ,where Z̃_u and Z̅_u denote the instantaneous SNR of UE u and the average SNR of UE u, respectively. Although this criterion of (<ref>) is not exactly the same as that of (<ref>), it captures the important characteristics of the PF scheduler: (i) allowing preference to UEs with relatively good instantaneous channels with respect to their average ones since R̃_u is a strictly monotonically increasing function of Z̃_u, and (ii) allocating the same portion of resource to each UE in the long term to enforce fairness, because the chance of Z̃_u≥Z̅_u is almost the same for all UEs. Since the accuracy and the practicality of (<ref>) have been well established in <cit.>, we will focus on studying the PF scheduler characterized by (<ref>)in the following sections.§ MAIN RESULTSPlaceholder In this section, we study the coverage probability and the ASE performanceof SCNs in terms of the coverage probability and the ASE by considering the performanceof a typical UE located at the origin o. §.§ The Coverage ProbabilityPlaceholder Considering the path loss model in (<ref>) and the PF scheduler model in (<ref>), we can derive p^cov(λ,γ) asp^cov(λ,γ)=∑_n=1^N(T_n^L+T_n^NL),where T_n^L=∫_d_n-1^d_n𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} f_R,n^L(r)dr, T_n^NL=∫_d_n-1^d_n𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}(r)dr, and d_0 and d_N are defined as 0 and +∞, respectively. Moreover, f_R,n^L(r) and f_R,n^NL(r) (d_n-1<r≤ d_n), are represented byf_R,n^L(r)=exp(-∫_0^r_1(1-Pr^L(u))2π uλ du)exp(-∫_0^rPr^L(u)2π uλ du)Pr_n^L(r)2π rλ,andf_R,n^NL(r)=exp(-∫_0^r_2Pr^L(u)2π uλ du)exp(-∫_0^r(1-Pr^L(u))2π uλ du)(1-Pr_n^L(r))2π rλ,where r_1=r_1{ζ^NL(r_1)=ζ_n^L(r)} and r_2=r_2{ζ^L(r_2)=ζ_n^NL(r)}. are given implicitly by the following equations asr_1=r_1{ζ^NL(r_1)=ζ_n^L(r)} ,andr_2=r_2{ζ^L(r_2)=ζ_n^NL(r)} .See Appendix A.Considering the truncated Negative Binomial distribution of the UE number per active BS, K̃, characterized in (<ref>), we can derive 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}, which will be used in Theorem <ref> as𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} =∑_k̃=1^K̃^max[1-∑_t=0^k̃([ k̃;t ])(-δ_n^L(r))^tℒ_I_agg^L(tγ/Pζ_n^L(r))]f_K̃(k̃),where K̃^max is a large enough integer that makes F_K̃(K̃^max) in (<ref>) close to one with a gap of a small value ϵ so that the expectation value in (<ref>) can be accurately evaluated over k̃, f_K̃(k̃) is obtained from (<ref>), δ_n^L(r) is expressed byδ_n^L(r)=exp(-γ P_N/Pζ_n^L(r)),and ℒ_I_agg^L(s) is the Laplace transform of I_agg for LoS signal transmission evaluated at s, which can be further written asℒ_I_agg^L(s)=exp(-2πλ̃∫_r^+∞Pr^L(u)u/1+(sPζ^L(u))^-1du)exp(-2πλ̃∫_r_1^+∞[1-Pr^L(u)]u/1+(sPζ^NL(u))^-1du). In a similar way, 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]} is computed by𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]} =∑_k̃=1^K̃^max[1-∑_t=0^k̃([ k̃;t ])(-δ_n^NL(r))^tℒ_I_agg^NL(tγ/Pζ_n^NL(r))]f_K̃(k̃),where δ_n^NL(r) is expressed byδ_n^NL(r)=exp(-γ P_N/Pζ_n^NL(r)),and ℒ_I_agg^NL(s) is the Laplace transform of I_agg for NLoS signal transmission evaluated at s, which can be further written asℒ_I_agg^NL(s)=exp(-2πλ̃∫_r_2^+∞Pr^L(u)u/1+(sPζ^L(u))^-1du)exp(-2πλ̃∫_r^+∞[1-Pr^L(u)]u/1+(sPζ^NL(u))^-1du).See Appendix B. First, we investigate the coverage probability that the typical UE's signal-to-interference-plus-noise ratio (SINR) is above a designated threshold γ:p^cov(λ,γ)=Pr[SINR>γ],where the typical UE's SINR is computed bySINR=Pζ(r)y(k̃)/I_agg+P_N,where y(k̃) is the channel gain on condition of the UE number k̃ in the typical UE's serving BS, P and P_N are the BS transmission power and the additive white Gaussian noise (AWGN) power at each UE, respectively, and I_agg is the cumulative interference given byI_agg=∑_i: b_i∈Φ̃∖ b_oPβ_ig_i,where b_o is the BS serving the typical UE, and b_i, β_i and g_i are the i-th interfering BS, the path loss from b_i to the typical UE and the multi-path fading channel gain associated with b_i (exponentially distributed), respectively. Note thatwhen all BSs are assumed to be active, the set of all BSs Φ should be used in the expression of I_agg <cit.>. Here,in (<ref>), only the BSs in Φ̃∖ b_o inject effective interference into the network, where Φ̃ denotes the set of the active BSs. In other words, the BSs in idle modes are not taken into account in the analysis of I_agg. It is very important to note that the distribution of y(k̃) should be derived according to (<ref>). More specifically, based on the variable definition in (<ref>), we can reformulate (<ref>) asu^*=u∈{ 1,2,…,k̃}max{Pζ(r)h_u/P_N/Pζ(r)×1/P_N} =u∈{ 1,2,…,k̃}max{ h_u} ,where h_u is an i.i.d. RV with a unit-mean exponential distribution due to our consideration of Rayleigh fading mentioned in Subsection <ref>. Thus, y(k̃) can be modeled as the maximum RV of k̃ i.i.d. exponential RVs. The complementary cumulative distribution function (CCDF) of y(k̃) is <cit.>F̅_Y(k̃)(y)=Pr[Y(k̃)>y]=1-(1-exp(-y))^k̃.It is easy to see that Pr[Y(k̃)>y] increases as k̃ grows, which in turn improves the typical UE's channel gain. Note that for the RR scheduler, the typical UE is randomly selected in the BS. Consequently, we have that k̃=1 in (<ref>), which leads to the degeneration of y(k̃) to h_u and the analytical results for RR have been derived in <cit.>. Based on the path loss model in (<ref>) and the PF scheduler model in (<ref>), we present our result of p^cov(λ,γ) in Theorem <ref>. shown on the next page Digging into Theorem <ref> and considering the truncated Negative Binomial distribution of the UE number per active BS, K̃ , we present our results on 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} and 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]} in Theorem <ref>.Plugging Theorem <ref> into Theorem <ref>, yields our theoretical results on p^cov(λ,γ). From Theorems <ref> and <ref>, we can draw an important and intuitive conclusion in Lemma <ref>.The p^cov(λ,γ) of the PF scheduler converges to that of the RR scheduler as .See Appendix C. Placeholder 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} and 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]} can be respectively upper bounded by𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}≤∑_k̃=1^K̃^max{ 1-[1-δ_n^L(r)ℒ_I_agg^L(γ/Pζ_n^L(r))]^k̃} f_K̃(k̃),and𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}≤∑_k̃=1^K̃^max{ 1-[1-δ_n^NL(r)ℒ_I_agg^NL(γ/Pζ_n^NL(r))]^k̃} f_K̃(k̃).See Appendix D. Although we have obtained the closed-form expressions of p^cov(λ,γ) for the PF scheduler in Theorems <ref> and <ref>, it is important to note that Theorem <ref> is computationally intensive for the case of sparse networks, where the maximum UE number per active BS K̃^max could be very large, leading to complex computations for ℒ_I_agg^L(tγ/Pζ_n^L(r)) and ℒ_I_agg^NL(tγ/Pζ_n^NL(r)),t∈{ 0,1,…,K̃^max} in (<ref>) and (<ref>), respectively. For example, when the UE density is ρ=300UEs/km^2 and the BS density is λ=10BSs/km^2,the average UE number per active BS is around 30UEs/BS, butK̃^max should be as large as 102 to make F_K̃(K̃^max) sufficiently close to one with a gap smaller than ϵ=0.001 with a gap of 0.001. As a result, we need to calculate the integrals in (<ref>) and (<ref>) at least 102 times for every possible value of r. The problem becomes even worse when the network is more sparse than λ=10BSs/km^2. The good news is that Theorem <ref> is very efficient for 5G UDNs, where K̃^max is expected to be less than 10 <cit.>. In the next subsection, we derive alternative and more efficient expressions for sparse networks.Based on the considered UAS, it is straightforward to derivePr[Pζ_n^L(r)g/I_agg+P_N>γ] =𝔼_[I_agg]{Pr[g>γ(I_agg+P_N)/Pζ_n^L(r)]} =𝔼_[I_agg]{F̅_G(γ(I_agg+P_N)/Pζ_n^L(r))} =𝔼_[I_agg]{𝔼_[k̃]{ 1-[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}} =𝔼_[k̃]{ 1-𝔼_[I_agg]{[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}} =𝔼_[k̃]{ 1-∑_t=0^k̃([ k̃;t ])(-1)^texp(tγ P_N/Pζ_n^L(r))𝔼_[I_agg]{exp(tγ I_agg/Pζ_n^L(r))}} =𝔼_[k̃]{ 1-∑_t=0^k̃([ k̃;t ])(-1)^t(δ_n^L(r))^tℒ_I_agg^L(tγ/Pζ_n^L(r))} =∑_k̃=1^K̃^max[1-∑_t=0^k̃([ k̃;t ])(-1)^t(δ_n^L(r))^tℒ_I_agg^L(tγ/Pζ_n^L(r))]f_K̃(k̃) §.§ A Low-Complexity Upper-Bound of p^cov(λ,γ)Placeholder We present upper bounds of 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} and 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]} in Theorem <ref>.The proposed upper bounds in Theorem <ref> require to calculate the integrals in (<ref>) and (<ref>) only once for every possible value of r, which makes the analysis of sparse networks very efficient.Based on the considered UAS, it is straightforward to derivePr[Pζ_n^L(r)g/I_agg+P_N>γ] =𝔼_[I_agg]{Pr[g>γ(I_agg+P_N)/Pζ_n^L(r)]} =𝔼_[I_agg]{F̅_G(γ(I_agg+P_N)/Pζ_n^L(r))} =𝔼_[I_agg]{𝔼_[k̃]{ 1-[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}} =𝔼_[k̃]{ 1-𝔼_[I_agg]{[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}}≤ 𝔼_[k̃]{ 1-[1-𝔼_[I_agg]{exp(γ(I_agg+P_N)/Pζ_n^L(r))}]^k̃} =𝔼_[k̃]{ 1-[1-δ_n^L(r)ℒ_I_agg^L(γ/Pζ_n^L(r))]^k̃} =∑_k̃=1^K̃^max{ 1-[1-δ_n^L(r)ℒ_I_agg^L(γ/Pζ_n^L(r))]^k̃} f_K̃(k̃) Consequently, plugging Theorem <ref> into Theorem <ref>, yields our theoretical results on an upper bound of p^cov(λ,γ), which is particularly useful for sparse networks. §.§ The Area Spectral EfficiencyPlaceholder We also investigate the area spectral efficiency (ASE) performance in bps/Hz/km^2, which is defined as <cit.>A^ASE(λ,γ_0)=λ̃∫_γ_0^+∞log_2(1+γ)f_Γ(λ,γ)dγ,where γ_0 is the minimum working SINR in a practical SCN, and f_Γ(λ,γ) is the PDF of the SINR γ observed at the typical UE for a particular value of λ. Based on the definition of p^cov(λ,γ) in (<ref>) and the partial integration theorem <cit.>, (<ref>) can be reformulated asBased on , which is the complementary cumulative distribution function (CCDF) of SINR, f_Γ(λ,γ) can be computed asf_Γ(λ,γ)=∂(1-p^cov(λ,γ))/∂γ.A^ASE(λ,γ_0)=λ̃/ln2∫_γ_0^+∞p^cov(λ,γ)/1+γdγ +λ̃log_2(1+γ_0)p^cov(λ,γ_0). Regarding A^ASE(λ,γ_0), it is important to note that: * Unlike <cit.>, in this work, λ̃ is used in the expression of A^ASE(λ,γ_0) because only the active BSs make an effective contribution to the ASE.* The ASE defined in this paper is different from that in <cit.>, where a constant rate based on γ_0 is assumed for the typical UE, no matter what the actual SINR value is. The definition of the ASE in (<ref>) can better capture the dependence of the transmission rate on SINR, but it is less tractable to analyze, as it requires one more fold of numerical integral compared with <cit.>.* Previously in Subsection <ref>, we have obtained a conclusion from Theorem <ref>: p^cov(λ,γ) with the BS IMC should be better than that with all BSs being active in dense SCNs due to λ̃≤λ. Here from (<ref>), we may arrive at an opposite conclusion for A^ASE(λ,γ_0) because it scales linearly with the active BS density λ̃, which caps at ρ in dense SCNs with the BS IMC. The takeaway message should not be that the IMC generates an inferior ASE in dense SCNs. Instead, since there is a finite number of active UEs in the network, some BSs are put to sleep and thus the spatial spectrum reuse in practice is fundamentally limited by ρ. The key advantage of the BS IMC is that the per-UE performance should increase with the network densification as discussed in Subsection <ref>.Regarding the computational complexity, in order to obtain p^cov(λ,γ) presented in Theorem <ref>, for a general case, three folds of integrals are respectively required for the calculation of { f_R,n^Path(r)}, {ℒ_I_agg(γ/Pζ_n^Path(r))}, and { T_n^Path}, where the string variable Path takes the value of “L” (for the LoS case) or “NL” (for the NLoS case). Note that an additional fold of integral is needed for the calculation of A^ASE(λ,γ_0) in (<ref>), making it a 4-fold integral computation. Placeholder§ SIMULATION AND DISCUSSIONPlaceholder In this section, we investigate network performance and use numerical results to validate the accuracy of our analysis. According to Tables A.1-3, A.1-4 and A.1-7 of <cit.> and <cit.>, we adopt the following parameters for the 3GPP Case: α^L=2.09, α^NL=3.75, A^L=10^-10.38, A^NL=10^-14.54An important footnote: Note that for the purpose of numerical calculation, all the distances should be converted to km because A^L and A^NL are defined for distances in km., BW=10 MHz,P=24 dBm, P_N=-95 dBm (including a noise figure of 9 dB at each UE). Besides, the UE density ρ is set to 300UEs/km^2, which leads to q=4.05 in (<ref>) and (<ref>) <cit.>. Parameters Description Valued_1 The cut-off point in the linear LoS probability function 300mα^L The path loss exponent for LoS 2.09α^NL The path loss exponent for NLoS 3.75A^L The constant value in the path loss function for LoS 10^-14.54A^NL The constant value in the path loss function for NLoS 10^-10.38P The BS transmission power 24dBmP_N The noise power -95dBm §.§ The Coverage Probability Performance of the 3GPP CasePlaceholder In Fig. <ref>, we plot the results of p^cov(λ,γ) with the PF scheduler for the 3GPP Case when ρ=300 UEs/km^2 and γ=0 dB. Note that our analytical results on the exact performance are obtained from Theorems <ref> and <ref>. In contrast, our analytical results on the upper-bound performance are obtained from Theorems <ref> and <ref>. As a benchmark, we also provide simulation results, and display the analytical results of the RR scheduler reported in <cit.>. Moreover, we show the ratio of the simulated p^cov(λ,γ) of the PF scheduler to that of the RR scheduler in Fig. <ref>.From these two figures, we can observe that: * As can be seen from Fig. <ref>, our analytical results well match the simulation results, which validates the accuracy of Theorems <ref> and <ref>. However, as discussed in Subsection <ref>, the evaluation of our analytical results is only efficient for dense and ultra-dense networks. Thus, in Fig. <ref>, we are only able to show the results of p^cov(λ,γ) for λ≥100 BSs/km^2. When λ<100 BSs/km^2, the proposed upper-bound results in Theorems <ref> and <ref> successfully capture the qualitative performance trend of the PF scheduler, with a maximum error of 0.04 in terms of p^cov(λ,γ) for sparse networks.* As predicted in Lemma <ref>, although the PF scheduler shows a better performance than the RR one for all BS densities, such performance gain diminishes as the network evolves into an UDN due to the loss of multi-user diversity. As can be seen from Fig. <ref>, the performance gain of the PF scheduler continuously decreases from around 175% (ratio=2.75) when λ=1 BSs/km^2 toward zero (ratio=1) in UDNs, e.g., λ=10^4 BSs/km^2.* The detailed explanation of the performance behavior in Fig. <ref> is provided as follows: * When λ∈[10^0,10^1] BSs/km^2, the network is noise-limited, and thus the coverage probabilities of both RR and PF increase with the BS density λ as the network is lightened up with more BSs.* When λ∈[10^1,10^2] BSs/km^2, p^cov(λ,γ) of the PF scheduler shows an interesting flat trail. This is because (i) the signal power is enhanced by LoS transmissions, as shown by the p^cov(λ,γ) of the RR scheduler in that BS density region; while (ii) the multi-user diversity decreases in that BS density region as exhibited in Fig. <ref>; and (iii) the above two factorsthe LoS signal power increase and the multi-user diversity decrease roughly cancel each other out.* When λ∈[10^2,10^3] BSs/km^2, the coverage probabilities of both RR and PF decrease with λ, as the network is pushed into the interference-limited region, and this performance degradation is due to the transition of a large number of interfering paths from NLoS to LoS, which accelerates the growth of the aggregate inter-cell interference <cit.>.* When λ>10^3 BSs/km^2, the coverage probabilities of both RR and PF continuously increase <cit.>.Such performance improvement is referred to as the Coverage Probability Takeoff in <cit.>. Such performance behavior can be attributed to the BS idle mode operations, i.e., (i) the signal power continues increasing with the network densification, and (ii) the interference power is controlled because not all BSs are turned on and emit interference., since only the BSs with active UE(s) eject interference.§.§ The Performance Impact of Rician FadingPlaceholder In this subsection, we investigate the performance for the 3GPP Case with distance-dependent Rician fading and correlated shadow fading, which has been introduced in Subsection <ref>. Due to its complex modeling, we conduct simulations to investigate this enhanced 3GPP Case, and the results are plotted in Fig. <ref>. As one can observe from this figure,All the conclusions in Subsections <ref> are qualitatively valid for Fig. <ref>, which shows the usefulness of our analysis and a less urgency to consider Rician fading in this performance analysis.that (i) our theoretical analysis is useful to predict the performance trend for the PF scheduler, and that (ii) it is not urgent to consider distance-dependent Rician fading in this performance analysis. Note that comparing Fig. <ref> with Fig. <ref>, we can see that Rician fading closes the performance gap between PF and RR more quickly than Rayleigh fading, due to its less variance in channel fluctuation for the PF scheduler to exploit. Only some quantitative deviations exists, which shows that (i) our theoretical analysis is useful to predict the performance trend for dense and ultra-dense SCNs of the PF scheduler, and that (ii) it is not urgent to consider distance-dependent Rician fading in this performance analysis. and correlated shadow fading §.§ The ASE Performance of the 3GPP CasePlaceholder In Fig. <ref>, we plot the results of A^ASE(λ,γ_0) when ρ=300 UEs/km^2 and γ_0=0 dB, with q^*=4.18 plugged into Proposition <ref>. From Fig. <ref>, we can draw the following conclusions: * For 3GPP Case 1, the ASE suffers from a slow growth or even a slight decrease when λ∈[20,200] BSs/km^2 because of the interference transition from NLoS to LoS <cit.>. Such performance degradation has also been confirmed in Fig. <ref>.* After such BS density region of interference transition, for both path loss models with ρ=300 UEs/km^2 and the BS IMC, the ASEs monotonically grow as λ increases in dense SCNs, but with noticeable performance gaps compared with those with ρ=+∞ UEs/km^2.* As discussed in Section <ref>, the takeaway message should not be that the IMC generates an inferior ASE in dense SCNs. Instead, since there is a finite number of the active UEs in the network, some BSs are put to sleep and thus the spatial spectrum reuse in practice is fundamentally limited by ρ. The key advantage of the BS IMC is that the per-UE performance should increase with the network densification as exhibited in Fig. <ref>.The intuition is that the randomness in channel fluctuation associated with Rayleigh fading is much stronger than that associated with Rician fading due to the large K factor in ultra-dense SCNs <cit.>, and the resulting stronger channel fluctuation implies a larger probability that the SINR could drop below the minimum working SINR γ_0 defined in (<ref>). The Impact of 3GPP Case 2 on the ASE PerformanceIn this subsection, we investigate the ASE performance for 3GPP Case 2, which has been introduced in Subsection <ref>. The parameters in the LoS probability function Pr^L(r) of 3GPP Case 2 are set to R_1=156 m and R_2=30 m <cit.>. Here, we directly apply the numerical integration in Theorem <ref> to evaluate the ASE performance for 3GPP Case 2.First, we study the value of q^* for the approximate λ̃ in 3GPP Case 2. The simulated results on the active BS density in case of various values of ρ are very similar to those shown in Fig. <ref>. For brevity, in Fig. <ref>, we only plot the average error of the active BS density for ρ=300UEs/km^2.Similar conclusions can be drawn for Fig. <ref> as discussed in Subsection <ref>, except for the small numerical difference of q^*=4.05 for ρ=300UEs/km^2 in 3GPP Case 2.In Fig. <ref>, we show the results of A^ASE(λ,γ_0) for 3GPP Case 2. As one can observe, all the conclusions in Subsection <ref> are qualitatively valid for Fig. <ref>. Only some quantitative deviations exists. Finally, we investigate the ASE performance, which is calculated from the results of p^cov(λ,γ) using (<ref>). Since the analytical results of p^cov(λ,γ) obtained from Theorems <ref> and <ref> have been validated in Fig. <ref>, we only plot the analytical results of A^ASE(λ,γ_0) for the PF and RR schedulers in Fig. <ref>with the 3GPP Case when ρ=300 UEs/km^2 and γ_0=0 dB. Note that we only investigate dense and ultra-dense SCNs with λ≥100 BSs/km^2 in Fig. <ref> because they are the focus of 5G and Theorem <ref> is computationally efficient to evaluate them.as explained at the end of Subsection <ref>, Theorem <ref> is computationally efficient for the BS density λ≥100 BSs/km^2, which allows us to obtain the ASE performance. As can be seen from this figure, the ASE performance of the PF scheduler quickly converges to that of the RR scheduler in UDNs, e.g., when λ=10^3 BSs/km^2, the ASE of the PF scheduler is around 590.1 bps/Hz/km^2, which is merely 4.52% higher than that of the RR scheduler around 564.6 bps/Hz/km^2. Such gain becomes practically zero as λ further increases.§ CONCLUSIONPlaceholder In this paper, we studied the network performance of the PF scheduler. Analytical results that are computationally efficient have been derived for dense and ultra-dense SCNs. For sparse networks, easy-to-compute analytical results of an upper-bound performance have been obtained to achieve a balance between accuracy and efficiency. Considering the negligible gain of the more complex channel-dependent PF scheduling in UDNs, which is justified by the diminishing multi-user diversity of the PF scheduler in dense SCNs, it is recommended to adopt simpler scheduling mechanisms such as the RR scheduler to simplify the RRM, and thus reduce network complexity for UDNs. As our future work, a non-full-buffer traffic model <cit.> will be studied.AcknowledgmentsGuoqiang Mao's research is supported by Australian Research Council (ARC) Discovery projects DP110100538 and DP120102030 and Chinese National Science Foundation project 61428102.§ APPENDIX A: PROOF OF THEOREM <REF> Placeholder Due to the page limit, we only provide the proof sketch of Theorem <ref> as follows. In (<ref>), T_n^L and T_n^NL are the components of the coverage probability for the case when the signal comes from the n-th piece LoS path and for the case when the signal comes from the n-th piece NLoS path, respectively. The calculation of T_n^L is based on (<ref>), in which f_R,n^L(r) characterizes the geometrical density function of the typical UE with no other LoS BS and no NLoS BS providing a better link to the typical UE than its serving BS (a BS with the n-th piece LoS path). Besides, 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} gives the expected coverage probability over all possible values of k̃ on condition of r. The logic of the calculation of T_n^NL is similar to that of T_n^L.§ APPENDIX B: PROOF OF THEOREM <REF> Placeholder Due to the page limit, here we only provides the key of the proof for Theorem <ref>. The derivation of (<ref>) is as follows, 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} =𝔼_[k̃]{Pr[y(k̃)>γ(I_agg+P_N)/Pζ_n^L(r)]}(a)= 𝔼_[k̃,I_agg]{F̅_Y(k̃)(γ(I_agg+P_N)/Pζ_n^L(r))}(b)= ∑_k̃=1^K̃^max𝔼_[I_agg]{1-∑_t=0^k̃([ k̃;t ])(-δ_n^L(r))^texp(-sI_agg)}f_K̃(k̃),where the step (a) of (<ref>) comes from (<ref>), and in the step (b) of (<ref>) s=tγ/Pζ_n^L(r) and 𝔼_[I_agg]{exp(-sI_agg)} =ℒ_I_agg^L(s) should be further plugged into (<ref>) to obtain (<ref>). The calculation of ℒ_I_agg^L(s) can be referred to <cit.>. The derivation of (<ref>) is very similar to (<ref>), which is omitted for brevity.§ APPENDIX C: PROOF OF LEMMA <REF> Placeholder The key of the proof for Lemma <ref> lies in (<ref>) and (<ref>) of Theorem <ref>. When λ→+∞, we have that K̃^max→1 and f_K̃(1)=1, because the resulting UDN is approach the limit of one UE per active BS. Hence, Theorem <ref> will degenerate to the results for the RR scheduler addressed in <cit.>.§ APPENDIX D: PROOF OF THEOREM <REF> Placeholder The key of the proof for Theorem <ref> lies in using Jensen's inequality as follows <cit.>which is because the function [1-(1-exp(-x))^k̃] defined in (<ref>) is a concave function with regard to exp(-x) when k̃≥1 and x∈[0,1]. Plugging (<ref>) into the step (a) of (<ref>) and with some mathematical manipulations, yields the proof. 𝔼_[I_agg]{ 1-(1-exp(-x))^k̃}≤1-(1-𝔼_[I_agg]{exp(-x)})^k̃. IEEEtran | http://arxiv.org/abs/1708.07961v2 | {
"authors": [
"Ming Ding",
"David Lopez Perez",
"Amir H. Jafari",
"Guoqiang Mao",
"Zihuai Lin"
],
"categories": [
"cs.NI",
"cs.IT",
"math.IT"
],
"primary_category": "cs.NI",
"published": "20170826120417",
"title": "Ultra-Dense Networks: A New Look at the Proportional Fair Scheduler"
} |
Backreaction in Axion Monodromy, 4-forms and the SwamplandAxion monodromy models can always be described in terms of an axion coupled to 3-form gauge fields with non-canonical kinetic terms. The presence of the saxions parametrising the kinetic metrics of the 3-form fields leads to backreaction effects in the inflationary dynamics. We review the case in which saxions backreact on the Kähler metric of the inflaton leading to a logarithmic scaling of the proper field distance at large field. This behaviour is universal in Type II string flux compactifications and consistent with a refinement of the Swampland Conjecture. The critical point at which this behaviour appears depends on the mass hierarchy between the inflaton and the saxions. However,in tractable compactifications, such a hierarchy cannot be realised without leaving the regime of validity of the effective theory, disfavouring transplanckian excursions in string theory.Corfu Summer Institute 2016 "School and Workshops on Elementary Particle Physics and Gravity" 31 August - 23 September, 2016 Corfu, Greece § INTRODUCTION Effective theories involving a transplanckian field range for a scalar have received a lot of attention in the recent years both from the phenomenology and the theory communities. The reason is twofold. First, they are important phenomenologically since a scalar field rolling down a potential for a transplanckian distance is the key feature of both large field inflationand the cosmological relaxation proposal for the EW hierarchy problem <cit.>. And second, because of the apparent aversion that string theory shows to transplanckian axions.Even if there is not any fundamental (only technical) problem with having a transplanckian axion from the point of view of quantum field theory, the embedding of these theories in string theory is full of control issues that always forces us to leave the regime of validity of the effective theory.Furthermore, recent constraints coming from the Weak Gravity Conjecture (WGC) <cit.> have led people think that these technical difficulties are in fact a sign of an underlying fundamental obstruction for realising a transplanckian scalar field range in a consistent theory of quantum gravity. If this is the case, a trans-planckian field range might constitute one of the few quantitative criteria known to distinguish between the string landscape and the swampland[Quantum field theories which cannot be embedded in a consistent theory of quantum gravity (and therefore cannot be derived from a consistent string theory compactification) are said to belong to the Swampland <cit.>.]. Here we will focus on the axion monodromy proposal <cit.>, which so far has been kept safe against quantum gravity arguments coming from the WGC constraining the axionic decay constants. Axion monodromy is a mechanism to induce a non-periodic potential (like a mass term) for an axion while preserving its discrete shift symmetry. This is achieved by inducing a multi-branched potential which converts the circle in field space into an helix. In this way, the field range can in principle be much bigger than the fundamental periodicity of the axion, while still enjoying the protection given by the shift symmetry against higher dimensional operators. The four-dimensional description of axion monodromy is given by coupling the axion to a Minkowski 4-form, which corresponds to the field strength of a 3-form gauge field living in the space-time dimensions. In the inflationary community this is known as the Kaloper-Sorbo description <cit.> although it was first analysed in detail by Dvali <cit.> in the context of the strong CP problem.As outlined, the embedding of these models in string theory leads to control issues when taking into account the backreaction of the inflaton into the other scalars of the compactification <cit.>. Therefore, the presence of a Kaloper-Sorbo coupling is not enough to guarantee a transplanckian field range. Here we will show how these backreaction issues appear in the four-dimensional description in terms of 4-forms, which in fact corresponds to a supersymmetric embedding ofthe Kaloper-Sorbo Lagrangian in N=1 supergravity. The above backreaction issues can then be naturally described by considering field-dependent kinetic metrics for the 3-form fields, without need of invoking unknown higher dimensional operators. Backreaction effects can lead to interesting features on the inflationary potential and even reduce the field range. However, so far, these problems seemed to be model dependent and not necessarily linked to the Planck mass. A universal feature, though, is the backreaction of the saxions into the Kähler metric of the inflaton, leading to a logarithmic scaling of the proper field distance at large field. In <cit.> Palti and Baume emphasised this logarithmic scaling and related it to one of the Swampland Conjectures of Ooguri-Vafa in <cit.>. The conjecture claims that there is always a tower of states whose mass decreases exponentially with the scalar vev for large values of the field. Therefore, the effective theory of a scalar is valid only up to a finite distance in field space. Based on the results obtained in Type IIA flux compactifications <cit.> and some hints coming from black hole physics <cit.>, the authors of these two papers defined a Refined Swampland Conjecture, stating that the logarithmic scaling appears necessarily close after crossing the Planck scale. If this is true, it would be the first model-independent argument constraining the field range of axion monodromy to a sub-planckian value.Our main motivation is to analyse in more detail these backreaction issues and discern if they necessarily appear around the Planck scale. We will recover the results of <cit.> regarding Type IIA flux compactifications by analysing the backreaction in the formulation in terms of 4-forms. Then we will show that the flux-independence of the critical field value at which the logarithmic scaling appears is a specific feature of these models and not a universal behaviour of string theory. The criterium is simple: everytime the inflaton potential can be set to zero without destabilising the rest of the scalars, the model enjoys the requirements to be able to delay the backreaction effects by generating a mass hierarchy between the inflaton and the other scalars <cit.>. However, a closer analysis reveals that, at least in flux toroidal compactifications, this mass hierarchy cannot be achieved without getting into trouble with the Kaluza-Klein scale and leaving the regime of validity of the effective theory <cit.>, supporting this way the Refined Swampland Conjecure. This proceedings article is based on <cit.> and the talk given at the “Workshop on Geometry and Physics” at Ringberg Castle in Germany. It is organised as follows. In section <ref> we review the Kaloper-Sorbo description and clarify the differences when embedded in a supergravity formulation. In section <ref> we discuss the backreaction on the Kähler metric of the inflaton leading to the aforementioned logarithmic scaling. We then analyse this effect in section <ref> in Type IIA flux compactifications and models including D7-brane position moduli in Type IIB. The latter are promising to provide a mass hierarchy that might delay backreaction effects in field space. We leave section <ref> for conclusions. § DESCRIPTION IN TERMS OF MINKOWSKI 4-FORMS Every four-dimensional scalar potential for an axion can always be rewritten in terms of a coupling to a 4-form field-strengthF_4. The simplest case of a mass term can be obtained from the following Lagrangian <cit.> (see also <cit.>), ℒ=-f^2(dϕ)^2- F_4∧ *F_4+2mF_4ϕ where f is the decay constant of the axion ϕ.Upon integrating out the Minkowski 3-form field, one gets a quadratic scalar potential *F_4=f_0+mϕ≡ρ(ϕ)→ V=(f_0+mϕ)^2with different branches labelled by f_0, a possible constant value of the 4-form field strength in the vacuum. The above scalar potential is indeed invariant under the combined discrete shift f_0→ f_0 + c ,ϕ→ϕ -c/mwhich identifies gauge equivalent branches when c/m=2π f. Tunneling between the different branches is mediated by nucleating a membrane electrically charged under the 3-form field, but the tunneling rate is not fast enough to constrain large field inflation significantly <cit.>. Invariance under the gauge symmetry of the 3-form field also constrains the form of the higher dimensional corrections, which should appear as functions of the field strength F_4 or ρ(ϕ) over the cut-off scale <cit.>. This guarantees in turn that the shift symmetry (<ref>) is also preserved. Upon integrating out the 3-form field, corrections depending on F_4 will give rise to corrections depending on the scalar potential itself, δℒ=∑_n c_n(F_4/M_p^4)^n→δ V =∑_n c_n (ρ(ϕ)/M^4_p)^n =∑_n c_n(V_0/M^4_p)^n so that they remain under control as long as the potential energy is subplanckian, even if ϕ takes transplanckian values. Finally, corrections going as powers of F_4ρ(ϕ) will give rise to higher derivative corrections for the axion <cit.>, suppressed during slow-roll inflation.The generalisation to multiple axions enjoying more elaborated scalar potentials is given by ℒ=K_ij(s,ϕ)dϕ^idϕ^j+K_ij(s,ϕ)ds^ids^j-Z_ab(s)F_4^a∧ *F_4^b +2F_4^aρ_a(ϕ)+V_loc(s)where we have included the possible presence of non-axionic scalars (saxions s^i). Since the saxions are not periodic, their scalar potential can receive contributions that cannot be written in terms of 4-form couplings and which we denote as V_loc(s^i). They will also appear parametrising the kinetic metric of the 3-form fields, which is indeed related to the Kähler metric of the scalar manifold if there is N=1 supersymmetry. Let us remark that all the dependence of the axions in the scalar potential only appears within the shift invariant functions ρ_a(ϕ) which couple to the 4-forms[One can also have periodic corrections that are shift invariant by themselves, like cosine-like terms. However, also these terms can be written in terms of a composite or effective 4-form with non-canonical kinetic term <cit.>.]. Upon integrating out the 3-form fields, one getsV=Z_ab(s)^-1ρ_a(ϕ)ρ_b(ϕ)+V_loc(s) .In <cit.> it was checked that the full four-dimensional closed string scalar potential of Type IIA/B flux orientifold compactifications on a Calabi-Yau three-fold can be written as (<ref>), where the 4-forms arise from dimensionally reducing the higher RR and NS p-form fields. The explicit form of Z(s) and ρ(ϕ) (which we display in section <ref>) was obtained by dimensionally reducing the 10d action and keeping track of all the 4-forms present in the compactification. A first analysis including open string fields has been performed in <cit.>.In the following we discuss the differences between the 4-form formulation (<ref>) and the Kaloper-Sorbo description (<ref>). These differences will be responsible for the control and backreaction issues which were not visible in (<ref>).*Non-linear couplingsThe shift invariant functions ρ(ϕ) can include mixing and higher order terms for the axions, leading to more generic scalar potentials beyond the usual quadratic term.*Multiple 3-forms The presence of multiple 3-form fields imply that the higher dimensional corrections will not appear as powers of the scalar potential itself, but of the different ρ-functions, δℒ=∑_n c_n(∏_aF_4^a/M_p^4)^n→δ V =∑_n c_n (∏_aρ_a(ϕ)/M^4_p)^n .However, they are still under control as long as all ρ(ϕ) remain subplanckian. *Non-canonical field-dependent metricsThe mixing between axions and saxions in the scalar potential arises from the presence of the saxions in the kinetic metric of the 3-form fields, as can be seen in (<ref>). This implies that, when displacing the inflaton from its minimum, the vacuum expectation values of the saxions will also change, backreacting on the effective theory of the inflaton. In particular, they can modify the field space metric leading to a reduction of the physical field range. If one would be able to integrate out all the saxions, one would recover the Kaloper-Sorbo description only in terms of 4-forms and axions. The higher order corrections would then be proportional to ρ(ϕ)/m_s^4 where the role of the cut-off scale is played by the saxion mass and not the Planck scale. However, as we will see, in many occasions the mass hierarchy between the inflaton and the saxion is not realisable in a consistent global compactification, invalidating such an effective theory. Therefore, we consider more appropriate to work in the 4-form formulation (<ref>) where the saxions appear explicitly parametrising the kinetic metrics.§ LOGARITHMIC SCALING OF THE PROPER FIELD DISTANCE In <cit.> Ooguri and Vafa conjectured that the proper field range for a non-periodic scalar (saxion) scales logarithmically with the field at large field values. In <cit.> it was argued that the same behaviour applies to axions when taking into account the backreaction of the saxions. We explain the latter in the following. The process of integrating out all the fields to obtain an effective theory involving only the inflaton is not feasible in general. In practice, one truncates the theory by freezing the saxions to their vevs at the minimum, and work with the remaining effective theory for the axion ϕ playing the role of the inflaton. However, this truncation is not valid when we displace ϕ away from the minimum, since the vevs of the saxions will generically depend on the inflaton vev as well. As a first approximation, one can then minimise the potential to obtain the leading order corrections of the saxions vevs as functions of ϕ, and plug them again into the effective theory to analyse the backreaction effects.From (<ref>) we can see that the minimum of the saxions will be given by some function involving the ρ(ϕ) functions. According to the conjecture, they should be such that the saxion scales at least linearly with the inflaton at large field. If this is so, when plugging this result into the Kähler metric of the inflaton, the proper field distance of the axion will scale at best logarithmically for large field,Δφ=∫ K_ϕ,ϕ̅^1/2 dϕ∼∫1/s(ϕ)dϕ∼∫1/ϕdϕ∼ log(ϕ) .Here we have used that K=-log(S+S^*) where S=s+iϕ for the sake of simplicity but it can be easily generalised to arbitrary Kähler potentials. Remarkably, we find this behaviour to appear universally in all F-term axion monodromy models constructed so far in the framework of flux string compactifications. Accompanying this logarithmic scaling, there is a tower of massive states which become exponentially light, M∼ s(ϕ)^-n∼ e^-nΔφ, and invalidates the effective theory, so that inflation cannot proceed in this regime. This leads to an upper bound on the physical field range available in F-term axion monodromy. The important question is, therefore, when this logarithmic behaviour appears. Is it always linked to the Planck scale or can it be delayed far out in field distance in a model-dependent way? To answer the question, let us be more concrete and split the saxion vev as follows, ⟨ s⟩ = s_0 +δ s(ϕ)where s_0 is the value at the minimum, δ s(ϕ)=λϕ for large field and λ is some proportional factor. The logarithmic behaviour will appear when δ s(ϕ_c)≈ s_0, ie. at ϕ_c=s_0/λ. The critical canonical field distance before backreaction effects dominate is therefore given byφ_c=∫^ϕ_c K_ϕ,ϕ̅^1/2 dϕ∼∫^ϕ_c1/sdϕ∼ϕ_c/s_0∼1/λ In the string theory examples studied in <cit.>, ϕ_c and s_0 scaled in the same way with the fluxes, leading to λ∼𝒪(1) in Planck units. This implied that the logarithmic scaling of the proper field distance appears necessarily around the Planck scale.However, this relation between ϕ_c and s_0 is an artefact of the models under consideration. The fact that, in those models, one cannot set the inflaton mass to zero without destabilising the saxion makes impossible to suppress the backreaction. A more promising avenue is to engineer a model with a flat direction, in which all scalars are stabilised but one, and then provide an additional source that stabilises the latter. If the limit of switching off this last source is well defined in the sense that we recover the old minimum with only one flat direction, then λ will depend on the mass hierarchy between the inflaton and the saxions. This provides a fundamental and not only a technical reason to look for single field inflationary models or at least, models in which the inflationary trajectory is not or barely mixed with non-axionic fields. Whether this is achievable in string theory, it is not clear yet. To sum up, let us remark that in general we find that λ∼(m_ϕ/m_s)^pwith p=0,1 depending on the model. In the next section, we will analyse some string theory models exemplifying both possibilities.§ BACKREACTION ISSUES IN STRING MODELSLet us now analyse these backreaction issues in concrete models of string theory. First, we will recover the results of <cit.> regarding Type IIA flux compactifications by using the formulation in terms of 4-forms. Then, we will propose some open string models which look promising to realise a flux-dependent λ, and therefore delay the backreaction effects to transplanckian values. §.§ Closed string sector of Type IIA The flux induced scalar potential of an orientifold Calabi-Yau compactification of Type IIA readsV=e^K_cs/s[1/2k|ρ_0|^2+g_ij/8kρ^iρ^j+2kg_ijρ̃^iρ̃^j+k|ρ_m|^2+1/kc_IJρ_h^Iρ_h^J]+V_loc where we have already integrated out the 4-forms. More details of the derivation can be found in <cit.>. From here one can read the metrics Z_ab(s) in (<ref>) which indeed depend on the saxions via volume factors labelled by k and the field metrics g_ij and c_IJ in the Kähler and complex structure moduli space respectively. The shift invariant ρ(ϕ) functions are given by ρ_0=e_0+ b^ie_i-m/6k_i jkb^ib^jb^k+k_i jk1/2 q_ib^jb^k-h_0c_3^0- h_ic_3^i ρ_i=e_i+k_ijkb^jq^k-m/2k_ijkb^jb^k ρ̃_i=q_i-mb_i ρ_m=mand ρ^I_h=h^I where e_0,e_i,q_i,m,h_I are internal fluxes while b^i,c_3^I are four dimensional axions coming from B_2 and C_3 respectively. Here i=1,…,h^1,1 and I=1,… h^2,1+1 zzz. Thanks to the separation between saxions and axions provided by the 4-forms, the minimisation procedure gets enormously simplified in this formalism. For simplicity, let us consider h^1,1=h^2,1=1. The vacuum expectation values of the saxions at the minimum of the potential reads_0∼ρ_i,0^3/2/ρ_h_0√(ρ_m) , u_0∼ρ_i,0^3/2/ρ_h_1√(ρ_m) , t_0∼ρ_i,0^1/2/√(ρ_m) where ρ_i,0=e_i+q^2/(2m) is the value of ρ_i at the minimum. These values are modified away from the minimum in the following way.Let us consider the inflaton to be the RR axion ϕ≡ c_3^0 (the same result can be obtained for c_3^i). It can be checked that for large field the dilaton depends linearly on ρ_0 which indeed is linear in c_3^0. Therefore,φ=∫ K_ϕϕ̅^1/2 dϕ= ∫1/s(ϕ)dϕ→∫1/ϕdϕ≃ log(ϕ)where we have used K=-log(s). The logarithmic behaviour appears at the critical value ϕ_c satisfying δ s(ϕ_c)∼ s_0. This occurs when Δρ_0≳ t_0ρ_i,0 implying ϕ_c∼ρ_i,0^3/2/(h_0 √(m))∼ s_0. Therefore λ∼𝒪(1) and the proper field distance before the logarithmic scaling appears φ_c=∫^ϕ_c K_ϕϕ̅^1/2 dϕ∼ϕ_c/s_0∼λ^-1∼𝒪(1)is flux independent and tied to the Planck scale. The same behaviour can be obtained if the inflaton is a NS axion ϕ≡ b^i. In that case, the saxion entering on the Kähler metric of the inflaton is the Kähler modulus t. The logarithmic behaviour will occur when Δρ_i≳ρ_i,0 implying a critical value given by b_c=√(ρ_i,0/m)∼ t_0. Therefore, the proper field distance before backreaction effects dominate is againφ_c=∫^ϕ_c K_ϕϕ̅^1/2 dϕ∼b_c/t_0∼λ^-1∼𝒪(1)flux independent and bounded by the Planck scale. The results do not change by including more Kähler or complex structure moduli.The underlying reason for having λ∼𝒪(1) gets manifest when computing the mass spectrum. The inflaton mass is proportional to h_0 for the the case of the RR axion or to m in the case of the NS axion, and therefore cannot be set to zero without destabilising the corresponding saxions (the vevs in (<ref>) go to infinity in that limit). §.§ D7-brane position moduli In order to get a flux-dependent tunable λ we need to engineer a model in which all scalars but one are stabilised, and identify this flat direction with the inflaton. Then we add an additional source that stabilises the flat direction in such a way that the new minimum can be understood as a deformation of the old minimum. If this is satisfied, then we can delay the critical field distance at which the backreaction becomes important without decreasing the Kähler metric of the inflaton (parametrised by the saxion vevs) in the same amount. The parameter λwill then depend on the mass hierarchy between the inflaton and the saxions andthus the proper field distance before the logarithmic scaling appears will not be necessarily linked to the Planck scale.Good candidates to realise these features are models in which a D7-brane position modulus plays the role of the inflaton<cit.>. In Type IIB toroidal orientifold compactifications the N=1 supergravity effective theory is given byK=-log((S+S̅)(U_3+U̅_3)-1/2 (Φ+ Φ̅)^2)-3log(T+T^*)-∑_i=1^2log(U_i+U_i^*)W=W_0(S,T,U_i)+μΦ^2where W_0(S,T,U_i) contains the necessary flux terms to stabilise all closed string moduli. Let us consider the inflaton to be parametrised by ϕ≡Φ-Φ̅. This effective theory contains two features which help to delay the backreaction effects:*The saxions parametrising (at the minimum) the Kähler metric of the inflaton do not belong to the same N=1 supermultiplet than the inflaton.*The structure of the superpotential allows us to set the inflaton mass to zero by choosing μ=0 without destabilising the rest of the scalars.If one minimises the potential with respect to all scalars but keeping the inflaton free, one gets that for large fieldK_ϕϕ̅^1/2=√(su_3)≃√(s_0u_0)+λϕwhere λ∼ m_ϕ/m_s up to numerical factors and m_s is the scale at which the dilaton and complex structure moduli are stabilised. Here m_ϕ is controlled by μ at leading order while m_s by the fluxes entering in W_0 (see <cit.> for concrete examples). Therefore the critical proper field distance before the logarithmic scaling dominates φ_c=∫^ϕ_c K_ϕϕ̅^1/2 dϕ∼λ^-1∼m_s/m_ϕ is flux dependent and in principle can be tuned larger than the Planck scale by generating a mass hierarchy between the inflaton and the saxions. This is a clear difference with respect to the previous models within the closed string sector of Type IIA, in which λ was flux-independent.The key question now is whether such a mass hierarchy can be realised in a full-fletched global compactification of string theory. Here we have two options, either decrease the inflaton mass by assuming μ≪ 1 or increase the saxion masses by choosing large fluxes in W_0. The first option is not possible in the toroidal compactification under consideration, since μ is a flux integer and cannot be tuned smaller than one. Therefore, we are left with the second option. However, if we embed the model in a KKLT moduli stabilisation scenario, large fluxes get in tension with having an exponentially small W_0. Furthermore, one also has to ensure that backreaction effects do not destabilise the Kähler modulus, which would make the KKLT minimum disappear.This in turn implies μ≪ W_0 which is not possible for μ being a flux integer. Embedding the model in the Large Volume Scenario does not help, because the condition required to suppress backreaction becomes μ≪ 1/𝒱, which is not possible for large volume 𝒱 either <cit.>. These difficulties can be traced back to the fact that we are trying to get the inflaton, which is stabilised at tree level by fluxes, lighter than the Kähler modulus, which is stabilised by non-perturbative effects. Therefore, a good alternative would be to consider an scenario in which all moduli are stabilised at tree level, which requires the presence of non-geometric fluxes in IIB or geometric fluxes in IIA. However, in that case, the required mass hierarchy cannot be realised either without getting the moduli masses heavier than the Kaluza-Klein scale <cit.>. Therefore,even if λ is flux dependent and in principle tunable in these models, in practice such a tuning is not possible in the global compactification and the critical proper field distance φ_c cannot be made larger than the Planck scale without leaving the regime where the employed effective supergravity theory is under control. This supports the Refined Swampland Conjecture of <cit.>. Notice, however, that a possible loophole comes from the quantisation of μ. If one can get an effective μ parameter depending not only on flux integers but vacuum expectation values of other fields, then in principle one might be able to get μ≪ 1 and delay the backreaction. However, without a concrete realisation of this fine-tuning in string theory, it is difficult to argue one way or the other, and could very well be that we are naively sweeping these backreaction issues under the carpet by appealing to the almighty string landscape. § CONCLUSIONS Axion monodromy models always admit a dual description in terms of coupling the axions to 4-form field strengths in four dimensions. This formulation is useful to make manifest the underlying shift symmetries of the system, study the tunneling transitions between different branches and constrain the form of the higher dimensional corrections.When embedded in N=1 supergravity, the kinetic metric of the 3-form fields will be parametrised by the saxions (non-periodic scalars) of the compactification, while all the dependence on the axions will appear only within the shift invariant functions ρ(ϕ), which couple linearly to the 4-forms. These field dependent metrics give rise to the backreaction issues obtained when realising axion monodromy in a concrete string theory model.The question addressed in this article is whether there is any universal behaviour appearing when crossing the Planck scale that forbids transplanckian excursions in axion monodromy. Interestingly, as emphasised in <cit.> saxions backreact on the Kähler metric of the inflaton such that at large field the proper field distance scales logarithmically with the field itself. This logarithmic scaling is consistent with the behaviour predicted by one of the Swampland Conjectures in <cit.>. We reproduce the results of <cit.>regarding flux compactifications of Type IIA by analysing the backreaction effects directly in the formulation in terms of 4-forms. It turns out that the critical field distance before the logarithmic behaviour appears is flux-independent and bounded by the Planck scale.We show, though, that it is possible to construct models in which this critical field distance is fiux-dependent and can be tuned larger than the Planck scale by realising a mass hierarchy between the inflaton and the saxions. This occurs whenever the inflaton mass can be set to zero without destabilising the rest of the scalars, so the true minimum is a deformation of the old minimum (the one obtained before stabilizing the inflaton). A good inflaton candidate to realise this feature is a D7-brane position modulus in Type IIB orientifold compactifications <cit.>. However, the mass hierarchy required to delay backreaction is not achievable in a toroidal compactification of Type IIB (or in F-theory on K3× K3) <cit.>. It remains to be seen if it can be realised in a more elaborated Calabi-Yau compactification involving mixing terms with many other moduli, in such a way that one can appeal to fine-tuning arguments in the landscape to reduce the inflaton mass <cit.>. For the moment, the multiple failing attempts and the lack of a concrete model realising this mass hierarchy, support the Refined Swampland Conjecture <cit.>, for which every effective theory of a scalar field breaks down upon a transplanckian field excursion in a consistent theory of quantum gravity. In the upcoming years, the clarification of this issue will be important in view of possible cosmological data, and will help deeping our understanding of the boundaries between the string landscape and the swampland.2em Acknowledgments:This article summarises a talk given at the “Workshop on Geometry and Physics”, which took place on November 20-25, 2016 at Ringberg Castle, Germany. The workshop was dedicated to the memory of Ioannis Bakas.utphys | http://arxiv.org/abs/1708.07456v1 | {
"authors": [
"Irene Valenzuela"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170824152701",
"title": "Backreaction in Axion Monodromy, 4-forms and the Swampland"
} |
[email protected] for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom [email protected] Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, United KingdomWe study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics—the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erdős–Rényi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erdős–Rényi phenomenology. 89.75.Hc, 64.60.aq Relaxation dynamics of maximally clustered networks Samuel Johnson December 30, 2023 ===================================================§ INTRODUCTIONNetwork science has enjoyed an unprecedented popularity in the last two decades as a paradigm for studying complex, interacting systems such as the Internet <cit.>, World Wide Web <cit.>, food webs <cit.>, scientific collaboration networks <cit.>, social and biological networks <cit.>, contact networks <cit.> and many others <cit.>. Many of these empirical networks exhibit a high degree of clustering or transitivity, i.e. a significant number of short, closed loops forming triangles <cit.>. This phenomenon is most commonly quantified by the clustering coefficient, defined as the proportion of connected triads that are also triangles in a network <cit.>.The classical random network models, the Erdős–Rényi random graph <cit.> and the configuration model <cit.>, both suffer from being unable to generate networks with significant values of the clustering coefficient thus making them unsuitable for modelling many real networks. High values of the clustering coefficient observed in empirical networks have lead to a surge of random network models that are capable of generating significant numbers of triangles <cit.>. The relationship between clustering and other network properties has also been studied extensively <cit.>. However, despite the large body of research, the inherent violation of edge independence in highly clustered networks has made it difficult to understand the full implications of clustering. Common issues encountered when dealing with highly clustered networks include difficulties of network sampling <cit.>, inability to use edge independence to derive accurate results <cit.> and potentially overstated inferences of causality <cit.>. This points to a need for more fundamental research in clustered networks.In this paper we explore simple dynamics of highly clustered networks relaxing to an unclustered equilibrium state. Specificaly, we study the evolution of the clustering coefficient under two edge rewiring schemes starting with fully clustered, degree-regular networks, i.e. networks in which all nodes have the same number of neighbours and a maximal number of triangles. We find that under both dynamics whose equilibrium distributions correspond to the Erdős–Rényi random graph and the configuration model respectively, a giant connected component emerges via a continuous phase transition. We provide an analytical prediction of the critical point for this transition as well as derive time evolution equations for various network properties.§ METHODS §.§ Network metricsWe consider undirected graphs with N nodes and L edges described by a symmetric N× N adjacency matrix 𝐀 with binary edge variables A_ij∈{ 0,1} for i,j∈{ 1,…,N} with A_ij=1, i≠ j indicating an edge between nodes i and j so that L=∑_i,jA_ij. The degree distribution of a network is defined as p_k=N_k/N, where N_k is the number of nodes with degree k. We denote the nth moment of the degree distribution by k^n.We define the multiplicity m_ij of an edge ij to be the number of triangles it participates in <cit.>. Similarly to the degree distribution, we define the edge multiplicity (or simply multiplicity) distribution as q_m=L_m/L, where L_m is the number of edges with multiplicity m. We denote the nth moment of the multiplicity distribution by m^n.The clustering coefficient of a network is defined as three times the number of triangles divided by the number of connected triples, i.e. C=3N_/N_∧ <cit.>. This measure of clustering is properly normalized so that C∈[0,1]. It also admits a probabilistic interpretation—it is the probability that a randomly chosen triple of nodes is closed.We can express the clustering coefficient in terms of the degree and multiplicity distributions. For any network we haveN_∧ = ∑_kk2N_k = N∑_kk2p_k = Nk^2-k/2and3N_ = ∑_mL_m = L∑_mmq_m = Lm.Putting the above results together and noting that in any network L=Nk/2, we obtain the following general expression for the clustering coefficient:C = km/k^2-k.§.§ Random network ensemblesWe study relaxation dynamics of k-regular networks under edge rewiring in two random network ensembles—the configuration model (CM) and the Erdős–Rényi random graph (ER).The CM <cit.> is defined by drawing a valid degree sequence 𝐤={ k_i}_i=1^N from a degree distribution p_k and producing a network realization uniformly at random from all possible networks with that degree sequence <cit.>. Provided the second moment of the degree distribution remains finite, it can be shown that the clustering coefficient scales as C∼ 1/N so that in the thermodynamic limit (N→∞) the resulting networks are tree-like <cit.>.The ER random graph <cit.> is defined by placing L edges uniformly at random between N nodes [Another common definition leading to a slightly different model is to place each of the possible N2 edges with equal probability p, but this does not enforce a fixed number of edges.]. If we require that the mean degree k=2L/N be fixed, the degree distribution of the ER model in the thermodynamic limit is Poisson with mean k <cit.>. The ER model is thus a special case of the CM and has the same scaling behaviour of the clustering coefficient.Given that both the CM and ER random graphs are asymptotically triangle-free, it is natural to consider them as equilibrium ensembles for relaxation dynamics of highly clustered networks into an unclustered state. To this end we describe two edge rewiring mechanisms that have the CM and the ER random graphs as equilibrium distributions (see <ref> for a graphical demonstration).*Double edge swap (CM). The double-edge swap <cit.> is defined by choosing two existing edges in the network at random and rewiring their ends to produce two new edges while deleting the original two. This is also known as degree-preserving randomization and so naturally produces network realizations in the CM ensemble with a fixed degree sequence. The double-edge swap defines a Markov chain whose equilibrium distribution is the CM <cit.>. *Edge replacement (ER). Alternatively, one can fully randomize a network by picking an edge at random and placing it anywhere in the network where there is no edge already <cit.>. In this scheme the number of edges is preserved but the degrees of the nodes are not. Edge replacement defines a Markov chain whose equilibrium distribution is the ER ensemble.A double-edge swap or an edge replacement constitutes an elementary rewiring step.§ RESULTSTo assess the evolution of network measures over time, we take into account the network size and the rewiring scheme (either CM or ER) to normalize the number of elementary rewiring steps per number of edges. If r_CM and r_ER are the number of elementary rewiring steps in the CM and ER ensembles respectively, we define the corresponding time variables ast_ER = r_ER/Lt_CM = 2r_CM/L.These definitions have the useful interpretation that when t_scheme=1, the rewiring scheme has, on average, modified each edge in the network. §.§ Multiplicity distributionThe multiplicity distribution evolves over time as edges are rewired and triangles are destroyed. The initial configuration of a k-regular network is a disjoint union of N/(k+1) cliques of size k+1 which ensures maximal clustering C=1. In other words, at time t=0, the multiplicity distribution isq_k-1 = 1 q_m=0if m≠k-1. Consider the smallest informative time step Δ t_CM=2/L or Δ t_ER=1/L corresponding to exactly one elementary rewiring step. At t=0 a clique of size k+1 has exactly k+12 edges all of which have maximal multiplicity k-1. Rewiring any single edge will destroy k-1 triangles leading to a decrease of 2(k-1)+1 edges with maximal multiplicity, one for the rewired edge and an additional two for each destroyed triangle. Assuming that no new triangles are created, the single rewired edge will have multiplicity zero. <Ref> shows the transition rates in the multiplicity distribution of a single clique. We now make the ansatz that this is the main way the multiplicity distribution changes over time—multiplicity is predominantly decreased by rewiring single edges from cliques and all such rewirings are independent. In this case, we can write down the full transition rate diagram between multiplicity classes as shown in <ref>. This gives the following time evolution equations for q_m:dq_m/dt = -(2m+1)q_m+2(m+1)q_m+1, [ for m=; k-1,…, 1 ] dq_0/dt = 3q_1+∑_m=2^k-1q_m. <Ref> shows the numerical solution of these ODEs which is in excellent agreement with simulation results. The calculations are valid both in the ER and the CM case. *Average multiplicity. Using the time evolution equations for the multiplicity distribution, we can derive exact expressions of its moments. Specifically we are interested in the average multiplicity m as it features in the expression for the clustering coefficient. We havedm/dt = ∑_m=1^k-1mdq_m/dt.Inserting <ref> we obtain the simple expressiondm/dt = -3m.Using the initial condition m(0)=k-1, this has solutionm = (k-1)e^-3t.<Ref> shows the analytic solution of the average multiplicity which is in perfect agreement with simulation results.§.§ Degree distributionIn the case of the ER model, the degree distribution is also changing over time. Consider the degree distribution p_k(t) as a function of time and a time step Δ t_ER. We can calculate the rate at which p_k(t) changes.An edge replacement event in the ER model consists of two steps. First, a random edge is selected. Second, a random pair of nodes that are not linked by an edge (let us call this pair a non-edge) is selected and the edge selected in the first step is deleted while the non-edge becomes an edge.When a random edge is selected, p_k can decrease if at least one end of the edge has degree k. Alternatively, p_k can increase if at least one end of the edge has degree k+1. The probability of reaching a node of degree k by following a randomly chosen edge is given by the so called excess degree distribution <cit.> which reads s_k=kp_k/k. Given this and the fact that a randomly chosen edge can have 0,1 or 2 nodes of degree k, we can calculate the expected number of nodes of degree k at the ends of a random edge:𝔼(k→ k-1)=2s_k^2+2s_k(1-s_k)=2s_k=2kp_k/k.This is the expected number of nodes whose degree would decrease from k to k-1 during a single edge selection step. Note that at the beginning of the process the degree distribution is regular so 𝔼(k→ k-1)=2 as expected.Similarly, the expected number of nodes whose degree would decrease from k+1 to k leading to an increase in p_k is:𝔼(k+1→ k)=2s_k+1=2(k+1)p_k+1/k. Now consider the second step in the edge replacement event, the selection of a non-edge. When a random non-edge is selected, p_k can also change in two ways. It can increase if at least one of the selected nodes has degree k-1 and it can decrease if at least one of the nodes has degree k. The calculation of the expected number of nodes changed as a result of this is similar to the previous case, but we must consider the distribution of non-degrees instead. To this end we study the graph complement of the original network defined as a network in which two nodes are linked if and only if they are not linked in the original network. From here on we denote by an overbar quantities in the graph complement.It is easy to see that the degrees of nodes in the complement are given by k=N-1-k where k is the degree of a node in the original network and we have p_k=p_k. Thus, the non-edges are selected proportionally to k not k as in the case of edge selection so we must work with the excess non-degree distribution given by s_k=kp_k/k. Note that the mean non-degree is given byk = ∑_kkp_k = N-1-k.As in the case of edge selection, the expected number of nodes whose degree would increase from k to k+1 thus reducing p_k during a single non-edge selection step is𝔼(k→ k+1)=2q_k=2kp_k/k = 2(N-1-k)/N-1-kp_k.When N is large we can approximate this by𝔼(k→ k+1)≃ 2p_k. Similarly, p_k can increase if we select a non-edge with at least one node with degree k-1. The expected number of such nodes in a single non-edge selection is𝔼(k-1→ k)≃ 2p_k-1. <Ref> describes pictorially the transition rates between degree classes as derived here. This allows us to write down the time evolution equations for p_k:dp_k/dt = 2p_k-1-2(1+k/k)p_k+2k+1/kp_k+1,for k=0,1,…. This system of ODEs is not closed, so in order to solve it numerically, we must truncate the system at some p_k^* setting p_k=0 for all k>k^*. The value of k^* should be set high enough so the probability mass unaccounted for is minimal for accurate predictions. We test our predictions by numerically solving the ODEs for a network with average degree k=2 and setting the cut-off k^*=8. The results are shown in <ref>. The numerical solution of the ODE system is in excellent agreement with simulation results. We also note that the cut-off is appropriate for this level of approximation as the total probability mass does not diverge from unity noticeably over the time period considered.*Second moment of the degree distribution. Using the time evolution equations for the degree distribution, we can derive exact expressions of its moments. Specifically, we are interested in the second moment k^2. We havedk^2/dt = ∑_kk^2dp_k/dt.Inserting <ref> we obtain the simple expressiondk^2/dt = -4k^2/k+4k+4.Using the initial condition k^2(0) = k^2 and recalling that the average degree k is constant, this has solutionk^2 = k(k+1-e^-4t/k).§.§ Clustering coefficientPutting together the results for the multiplicity and degree distributions, and using <ref>, we obtain exact expressions for the clustering coefficient as a function of time in both the CM and ER ensembles:C_CM = e^-3tC_ER = (k-1)e^-3t/k-e^-4t/k. We note that in the CM ensemble, the clustering coefficient has no dependence on the average degree while this is not the case for the ER ensemble. This is because the number of connected triples N_∧ in the CM ensemble is constant by virtue of having a fixed degree sequence while it is dependent on the evolving degree sequence in the ER ensemble. §.§ Giant connected componentWe find that under both rewiring schemes there is an emergence of global connectivity via the appearance of a giant connected component (GCC) at some critical time t^c (equivalently, critical clustering coefficient C^c). We confirm from simulation results that a GCC emerges in a continuous phase transition (<ref> for the CM and <ref> for the ER ensembles). Note that the large fluctuations in the 2-regular case is due to the fact that 2-regular networks are exactly at the poing of criticality in the unclustered CM case (C=0). This phenomenon has been studied in the context of reversible polymerization of rings <cit.>. We can calculate the critical point analytically by using the known result that a GCC in an ER random graph emerges when k=1 <cit.>. We conjecture that a GCC induced by edge rewiring emerges when the average number of external edges between the original N/(k+1) cliques of size k+1 exceeds one. If this is the case, the critical number of elementary rewiring steps isr^c = N/2(k+1).Expressing this in terms of the time variable, we obtain the critical time for both the CM and the ER rewiring schemes:t^c_CM = 2/k(k+1)t^c_ER = 1/k(k+1).Note that these differ by a factor of two. This is because in the CM rewiring scheme, even though every elementary rewiring step involves two edges, the two rewirings are not independent—during one rewiring step it is possible to connect at most two disconnected components.Expressed in terms of the clustering coefficient, the critical thresholds read:C^c_CM = e^-6/k(k+1)C^c_ER = (k-1)e^-3/k(k+1)/k-e^-4/k^2(k+1). <ref> confirm that these are in excellent agreement with simulations in the CM case and <ref> confirm a good agreement in the ER case which improves as the mean degree increases.What is the cause of the discrepancy of the analytical result for the critical point and the numerical simulations, particularly for low mean degree ER networks? We conjecture that this is due to some edges being rewired multiple times while others are not rewired at all. This would have the effect of increasing the critical time because we have to wait slightly longer until the average number of rewired edges discouting edges rewired multiple times reaches the point where long range connectedness emerges. <Ref> seems to confirm this to be the case. Let us calculate this revised critical time in the ER case.During an edge replacement step, the probability of any edge being chosen for rewiring is 1/L. So after r rewiring events the probability that a specific edge has not been rewired isℙ(not rewired) = (1-1/L)^r.Substituting r=Lt since we are in the ER case and taking the limit as L→∞, we getℙ(not rewired) = e^-t.The new revised time for ethe mergence of the GC, call it t^r, is then the time at which point this probability drops below a certain threshold. What is this threshold? It should be when the proportion of edges that have been rewired gives rise to a GCC which is precisely given by t^c. We can then writee^-t^r = 1-t^c.Note that by Taylor expansion we have t^r≃ t^c if this time is small as in the case when the average degree k→∞. This explains why the t^c value becomes a better predictor for the critical threshold as the mean degree increases as seen in <ref>.The revised critical point in the ER case is thust^r = -log(1-t^c) = log(k(k+1)/k(k+1)-1).<Ref> confirms that t^r is a better predictor of the location of the phase transition. The difference between t^c and t^r becomes negligible as the mean degree increases.Another aspect that could influence the position of the critical point is the possibility of having rewired more edges that needed to connect previously disconnected components. We show in the <ref> that this should have no bearing on the critical point in large networks.§ DISCUSSIONIn this paper we studied the evolution of highly clustered networks under random edge rewiring dynamics. Our main result is showing the existence of a phase transition in which a giant connected component emerges. <cit.> showed that equilibrium ensembles of degree-regular networks with prescribed clustering always admit a giant connected component. As a consequence, spreading processes such as infectious diseases in contact networks could always become endemic regardless of the level of clustering. By contrast, our work implies that this need not be the case in non-equilibrium systems. Depending on the precise mechanisms of time evolution of real networks and the level of clustering maintained, a giant connected component facilitating spreading processes may or may not exist. We have studied a model in which highly clustered populations undergo fully random connectivity changes and even in this simple scenario we observe two different modes of global connectivity.Another interesting aspect of our work is from the perspective of statistical mechanics. A maximally clustered network is essentially the lowest entropy microstate in the context of the random network ensembles studied here. This is because such a network, under relabelling of nodes, is unique and least likely to be produced by chance at equilibrium. By using this configuration as a starting state for network dynamics, we have shown that the emergence of global connectivity is effectively delayed. This raises several other research questions, for example, is random rewiring the most or least effective method of delaying the onset of global connectivity? It is probable that more sophisticated rewiring methods involving choice, such as those studied in explosive percolation <cit.>, would lead to different critical thresholds. We have also limited ourselves to studying rewiring that consistently destroys triangles, but what about rewiring with a view to increase the number of triangles? A number of greedy as well as equilibrium algorithms exist and are widely applied to model highly clustered networks <cit.>, but it is unclear how they cover the space of all networks and can lead to interesting behaviour such as hysteresis loops <cit.>. Indeed, clustering in networks still leaves much to be explored. J.K. was supported by the EPSRC under grant EP/IO1358X/1. S.J. is grateful for support from Spanish MINECO Grant No. FIS2013-43201-P (FEDER funds). * § EXTRANEOUS EDGESAnother mechanism that could change the location of the critical point t^c is the number of extraneous edges between already connected components. A GCC is formed when there are enough external edges between the initial cliques. Only one external edge is needed to connect two cliques, but there are multiple ways to do it and sometimes multiple edges end up linking together the same cliques. For example, we need only one edge two join two disconnected triangles, but there are a total of 9 ways to do it, moreover there is no guarantee that we will not end up with multiple edges between these triangles.More generally, let the average degree k be fixed, then at t=0 there are n=N/(k+1) cliques of size k+1. Any two cliques can therefore be connected in (k+1)^2 ways.Suppose we never want to make more than one external edge to connect disconnected components. Then at t=0 the number of choices for placing an external edge is given by(k+1)n2 = N(N-k-1)/2.After each rewiring event, the number of choices decreases by (k+1)^2, so after r-1 rewires, the probability of placing an extraneous edge on the next rewire, r, isℙ(extra edge on step r) = 2(k+1)^2r/N(N-k-1).Thus, the expected number of extraneous edges after r rewiring events is𝔼(extra edges by step r) = ∑_r^'=0^r2(k+1)^2r^'/N(N-k-1).In particular, setting r=r^c=N/2(k+1) we get 𝔼(extra edges by step r^c) =2(k+1)^2/N(N-k-1)·N/4(k+1)(N/2(k+1)+1) = N+2k+2/4(N-k-1). Taking the limit N→∞, we get𝔼(extra edges by step r_c) ≃1/4,which is fixed and independent of network size. Therefore, the formation of extraneous edges does not affect the location of the critical point in the large network limit. | http://arxiv.org/abs/1708.07739v1 | {
"authors": [
"Janis Klaise",
"Samuel Johnson"
],
"categories": [
"physics.soc-ph",
"cs.SI"
],
"primary_category": "physics.soc-ph",
"published": "20170825135316",
"title": "Relaxation dynamics of maximally clustered networks"
} |
Exotic Bifurcations Inspired by Walking Droplet Dynamics Aminur RahmanCorresponding Author, <[email protected]> Department of Mathematical Sciences, New Jersey Institute of Technology Current address: Department of Mathematics and Statistics, Texas Tech University , Denis Blackmore[2]===================================================================================================================================================================================================================================== We identify two rather novel types of (compound) dynamical bifurcations generated primarily by interactions of an invariant attracting submanifold with stable and unstable manifolds of hyperbolic fixed points. These bifurcation types - inspired by recent investigations of mathematical models for walking droplet (pilot-wave) phenomena - are introduced and illustrated. Some of the one-parameter bifurcation types are analyzed in detail and extended from the plane to higher-dimensional spaces. A few applications to walking droplet dynamics are analyzed. § INTRODUCTION Inspired by our recent research on the dynamical properties of mathematical models of walking droplet (pilot-wave) phenomena <cit.>, we shall describe and analyze what appear to be new types or classes of bifurcations. Owing largely to its potential for producing macroscopic analogs of certain quantum phenomena, walking droplet dynamics has become a very active area of research since the seminal work of Couder et al. <cit.>.In this study we focus on the dynamical systems models arising from walking droplets, interesting examples of which can be found in Gilet <cit.>, Milewski et al. <cit.>, Oza et al. <cit.>, Rahman and Blackmore <cit.>, and Shirokov <cit.>.Furthermore, a detailed summary of recent advancements in hydrodynamic pilot-waves can be found in <cit.>. Simulations of the solutions of some of the mathematical models for walkers are not only interesting for their quantum-like effects, they exhibit exotic bifurcations that are apt to attract the interest of dynamical systems researchers and enthusiasts. In particular, the two-parameter planar discrete dynamical systems model of Gilet <cit.> of the form G: ℝ^2→ℝ^2 defined as G(x,y;C;μ):=(x-CΨ^'(x)y,μ(y+Ψ(x))),where 0<C,μ<1 are parameters and Ψ is an odd, 2π-periodic function given byΨ(x):=1/√(π)(cosβsin3x+sinβsin5x),where β is usually chosen to be π/3 or π/6, exhibits not onlyNeimark–Sacker bifurcations, but more exotic chaotic bifurcations that have apparently not been analyzed in detail in the literature. Simulations of the dynamics of (<ref>) have shown that these exotic bifurcations are similar in certain respects if one of the parameters C,μ is varied and the other fixed, but also quite different in other ways. For example, in the C fixed case shown in Fig. <ref> , we see a progression of similarly shaped attractors ending in what appears to be a chaotic state. Actually, if μ is increased further (not shown), the chaotic attractor exhibits dramatic changes, which are apparently due to a series of dynamical crises (cf. Ott <cit.>).On the other hand, in Fig. <ref> we show a sequence in which μ is fixed and C varied from 0.45 to 0.528 at which point we see what appears to be a chaotic strange attractor. This chaotic strange attractor actually persists in shape up to around C=0.7 (not shown). Beyond 0.7 (also not shown) the attractor changes in shape more or less continuously until it finally breaks up into a chaotic splatter with just a ghost-like shadow of the prior shape - a last stage that is also indicative of dynamical crises. In this investigation, we concentrate on abstracting the bifurcation properties of the Gilet map associated with fixing μ and varying C.The bifurcations that we describe in the sequel are those for discrete dynamical systems comprising the iterates of differentiable, parameter-dependent self-maps of smooth finite-dimensional manifolds, and they are generated by interactions of closed (positively) invariant submanifolds with stable and unstable manifolds of saddle points as a single parameter is varied. In the interest of simplicity and clarity, we shall confine our attention to maps f∈ C^1(ℝ^m×ℝ ,ℝ^m), where m is a natural number (in ℕ) and, as usual,C^1(ℝ^m×ℝ,ℝ^m):={ f:ℝ^m×ℝ→ℝ^m:f is continuously differentiable}.Here, of course, ℝ^m and ℝ are, respectively, Euclidean m- and 1-space, which represent the phase and parameter spaces of the dynamical system and points in ℝ^m×ℝ to be denoted as (x,σ). An example of the kind of map we shall be investigating is that of the form of (<ref>) with μ a fixed constant in (0,1) and the other parameter C, which we denote as σ, varying in (0,1); namely,F(x,y;σ):=(x-σΨ^'(x)y,μ(y+Ψ(x))),In what follows, we assume a knowledge of the fundamentals of modern dynamical systems theory such as can be found in <cit.>. Our main results are detailed in the remainder of this paper, which is arranged as follows. In Section 2 and Section 3 we describe the planar forms of the bifurcations involving the interaction of an attracting invariant Jordan curve and stable and unstable manifolds of saddle points. These bifurcations share some features with the dynamical phenomena described in Aronson et al. <cit.> and Frouzakis et al. <cit.>, but they appear to be essentially new. More specifically, in Section 2, we introduce and analyze planar dynamical bifurcations generated by the interaction of an attracting Jordan curve and the stable manifold of a saddle point. In particular, the interaction first induces a bifurcation caused by a tangent homoclinic orbit, which is followed by a sequence of additional tangent homoclinic orbits interspersed with transverse homoclinic orbits as the parameter is increased. Ultimately, however, an increase in σ leads to a final tangent homoclinic orbit after which there is a parameter interval on which there a robust chaotic strange attractor, which is amenable to abstraction. As mentioned above, there are additional types of bifurcations for larger parameter values, which shall not be described in detail here. It should be noted that the bifurcations considered in this paper, which shall be designated as being of type 1, are directly related to those of the Gilet map (<ref>) where a slice-like region of non-injectivity of the map plays a key role. There is a second type - a variant of the Gilet map - where the map is a diffeomorphism that produces analogous bifurcations, which we shall analyze in a forthcoming paper.Section 3 is where we describe a modification of the bifurcation in the preceding section generated by the interaction of an attracting Jordan curve with a pair of stable manifolds, which can induce heteroclinic cycles that generate chaotic strange attractors followed by homoclinic bifurcations. Next, in Section 4, we discuss some higher-dimensional analogs of the planar bifurcations analyzed in the preceding sections. In Section 5, we describe several applications and examples of the bifurcations, with a focus on the phenomena and mathematical models that inspired our work on the bifurcations; namely, walking droplet dynamics. Finally, in Section 6, we summarize some of the conclusions reached in this research and adumbrate possible related future work, which includes analyzing the bifurcations in the dynamics of the map (<ref>) when C is fixed and μ varied.§ HOMOCLINIC BIFURCATIONS OF TYPE 1 IN THE PLANE In this and the next section, we restrict our attention to a C^1 map f:ℝ^2×ℝ→ℝ^2, with the tacit understanding that we could have just as well considered a 1-parameter dependent map on a simply-connected open subset of a smooth surface. The points of ℝ^2×ℝ shall be denoted by (x,σ)=((x,y),σ) and we use the standard notation f_σ:ℝ^2→ℝ^2 for the planar map with the parameter σ fixed at a particular value in (0,1).Let us set the stage for the homoclinic bifurcation of type 1 with more specificity. For this, we assume that p(σ):=(x̂(σ),ŷ(σ)) is a saddle point of f_σ, with one eigenvalue λ^u(σ)>1 and the other 0<λ^s(σ)<1 for all σ∈ a,b)⊂(0,1) such that x̂(a),ŷ(a)>0, where a<b . We may also assume that the linear unstable manifold is horizontal when σ=a; i.e., W_lin^u(p(a)) is the line y=ŷ(a). In addition, we assume that the saddle point, although it may move as the parameter is varied, remains within the open rectangle R_α,β:=(0,x̂(a)+α)×(0,ŷ(a)+β) for some α,β>0 and all a≤ν<b, and its stable manifold W^s (p(σ)) lies in the open vertical strip V_α :=(0,x̂(a)+α)×ℝ for all a≤σ<1.A principal feature of the bifurcation is an attracting invariant closed tubular neighborhood T(σ) of a C^1 closed Jordan curve C=C(σ), which we call an invariant tube with center C for the map. Then there is for some σ an associated compact invariant attracting setℭ(σ):= ⋂_n=1^∞ f_σ^n(T(σ)),which, for an appropriately chosen center, is equal to C(σ) for some initial subinterval of [a,b). It should be noted that such invariant circles and tubes rather frequently arise from Neimark–Sacker bifurcations of sinks,especially those of the spiral variety (cf. <cit.> and also <cit.>). We also assume that for the parameter values for which the invariant tubes exist, T(σ) is contained in the open rectangle ℛ_α,β:={(x,y):| x| <x̂(a)+α,| y| <ŷ(a)+β} , and even more; it is contained in the set Q(σ) comprising all points in ℛ _α,β to the left of W^s(p(σ)) and beneath W_lin^u(p(σ)), respectively. Moreover, we assume that f_σ has a positive (counterclockwise) rotation number in the sense that the iterates of any normal section of T(σ) completely traverse the tube in a counterclockwise manner.Finally, there is another important feature for maps of the type represented by Gilet's model. Namely, it is assumed that there is a a<σ_#<b such that for all σ∈ a,σ_#), the basin of attraction of ℭ(σ) contains all points (x,y) in ℛ _α,β to the left of W^s(p(σ)), except for points in a curvilinear slice Z(σ) below the saddle point with one edge and the image under f_σ of the other edge contained in W^s (p(σ)), having the the property that f_σ flips its interior across the stable manifold. This slice is a key agent in producing a chaotic strange attractor as the center of the tube expands for Gilet-type maps.Now that the contextual foundation has been established, we shall prove a result describing the homoclinic type bifurcations that we have in mind. Toward this end, it is useful to summarize the descriptions above in the form of a list of detailed but reasonably succinct properties. These attributes of the map f:ℝ^2×[a,b)→ℝ^2, illustrated in Fig. <ref>, are as follows:(A1) f∈ C^1:=C^1(ℝ^2×[ a,b),ℝ^2), where [a,b)⊂(0,1) and f_σ has a single saddle point p(σ):=(x̂(σ),ŷ(σ)), which is such that: (i) x̂(a),ŷ(a)>0; (ii) there are real constants κ_s, κ_u such that the eigenvalues λ^s(σ) and λ^u(σ) of f_σ^'(p(σ)) satisfy 0<λ ^s(σ)≤κ_s<1<κ_u≤λ^u(σ) for all a≤σ<b; (iii) the eigenvector corresponding to λ^u(a) is parallel to the x-axis; (iv) there is a vertical strip of the formV_α:={x:=(x,y)∈ ℝ^2:0<x<x̂ (a)+α}for some α>0 such that the stable manifold W^s(p (σ))⊂ S_α for all a≤σ<b and separates the plane into left and right components denoted as K_-(σ) and K_+(σ), respectively; and (v) there is a β>0 such that p(σ)∈ R_α,β:={(x,y)∈ℝ^2:0<x<x̂(a)+α,0<y<ŷ(a)+β}⊂ℛ_α,β:={(x,y):| x| <x̂(a)+α,| y| <ŷ(a)+β} for all σ∈ a,b). (A2) There exist a<σ_1≤σ_#<σ_t<σ _2<σ_3<b such that the following obtain: (i) there is a (positively) f_σ-invariant attracting tubular neighborhood T(σ) of a C^1 closed Jordan curve C(σ) for all σ∈ a,σ _2); (ii) f_σ restricted to T(σ) has a positive (counterclockwise) rotation number for all σ∈ a,σ_2); (iii) the closed curve can be chosen so that centerset ℭ(σ):= ⋂_n=1^∞ f_σ^n(T(σ))=C(σ) for all σ∈ a,σ_1); (iv) T(σ)⊂ K_- (σ)∩ℛ_α,β∩ H(σ), where H(σ) is the half-plane defined by y<ŷ(σ), for all σ∈ a,σ_2); (v) ℭ(σ) is a nonempty attracting set for all σ∈ a,σ_#)∪σ_2,σ_3) and has an open basin of attraction, denoted as 𝔅(σ), containing (K_-(σ)∩ℛ_α,β∩ℭ_ext(σ))∖ Z(σ) and ℭ(σ)∩ Z(σ)=∅for all σ∈ a,σ_#), where ℭ_ext(σ) is the exterior of the centerset and Z(σ) is a set defined as follows: There is for every σ∈ a,b) an orientation preserving C^1-diffeomorphism Φ_σ of an open neighborhood U_σ of a portion of W^s(p(σ)) below p(σ) such that Φ_σ(Z(σ)):={(ξ,η):-1≤ξ≤0,η≤φ(ξ)}, where φ:[-1,1]→(-1,0] is a C^1 function such that φ(0)=0 and φ^'(ξ)>0 when ξ<0. Moreover, the right-hand bounding curve c_r(σ):=Φ_σ^-1({ (0,η):φ(-1)≤η≤0}) and the f_σ image of left-hand bounding curve c_l(σ):=Φ_σ^-1( {(ξ,φ(ξ)):-1≤ξ≤0}) lies in W^s(p(σ)), while f_σ maps the interior Z(σ) into K_+(σ). We have now sufficiently prepared the way for the statement and proof of our first main result based on the above assumptions, which concerns the existence of what we call a homoclinic type 1 bifurcation. However, it is convenient to first introduce the following definition: The left unstable manifold is W_l^u(p(σ)):= ⋃_n=1^∞ f_σ^n({x∈ W^u(p (σ)):x∈{p(σ)}∪ K_-(σ) and d(x,p(σ))≤ν ∀ ν>0}). Let f:ℝ^2× a,b)→ℝ^2 satisfy (A1) and (A2) and the additional property:(A3) The distance Δ(σ):=dist( ℭ(σ),Z(σ)):=inf{|x-y| :(x,y )∈ℭ(σ)× Z(σ)} satisfiesΔ(a)>0 and is a nonincreasing function of σ on [a,σ_#)∪σ_2,σ_3) and lim _σ↑σ_3Δ(σ)=0. Then, there are a<σ_#≤σ_t≤σ_2 ≤σ_∗<σ_3<b_1<1, where σ_t and σ_∗ are the first and last, respectively, values of σ where there is a tangent intersection of the stable and unstable manifolds of p(σ) such that for all σ_∗<σ<b_1, f_σ has a chaotic strange attractor 𝔄(σ), which is the closure of the left unstable manifold: namely,𝔄(σ):=W_l^u(p(σ)).We begin by focusing on a tubular type strip Q=Q(σ;ν) for W^s(p(σ)) (excluding c_l (σ)) cut off just above p(σ) and below at the lower edge of R(α,β) as shown in Fig. <ref>, with the understanding that the dimensions can be taken to be as small as suits our purposes in what follows. In particular, we take the width of the strip to be 2ν and the top edge to be parallel to and at a distance of ν above W^u(p (σ)), where ν>0. In this context, we shall find it convenient to define W_ν^u(σ) to be the closed segment, with interior in K_-(σ), of the unstable manifold from p(σ) to the boundary of Q, and to use s_σ:=s_σ(x )=s(x) to denote the arclength from the fixed point to any point x∈ W_ν^u(σ) or any of its f_σ-iterates. We also define for every nonnegative integer n the endpoint e_σ^n, to be the boundary point of the C^1-submanifold (with boundary) 𝒲_σ^u(σ,n):=f_σ^n(W_ν ^u(σ)) for which s_σ is positive and actually increases without bound as n→∞ as long as σ<σ_#.The idea of the proof is illustrated rather simply in Figs. <ref> and <ref>, but some technical details are necessary. It follows from the hypotheses (A1)-(A3) that for any ν>0 there is a positive integer N=N_ν(σ) and a first σ=σ_t∈ a,σ_2) such that 𝒲_ν ^u(σ,N) is tangent to c_l(σ), which implies that 𝒲_ν^u(σ_t,N+1) is tangent to W^s(p (σ_t)). Then, as σ is increased, there must be a last value, σ_∗, with σ_t≤σ_∗<σ_2, such that 𝒲_ν^u(σ,N) crosses c_l(σ) into the interior of Z(σ), implying that Q_l,σ(ω,N+1):= f_σ ^N+1(Q_-(σ;ν)), where Q_-(σ;ν):=Q(σ;ν )∩K_-(σ), which contains a portion of 𝒲_ν^u(σ,N+1), actually crosses over Q(σ;ν) into K_+(σ) for all sufficiently small ν>0 whenever σ_∗<σ. As a consequence of this situation, called an ×-crossing, the desired result follows from the attracting horseshoe theorem in <cit.>, which incorporates the geometric chaos and fractal set arguments of Birkhoff–Moser–Smale theory (cf. <cit.>). In particular, there is an attracting horseshoe in f_σ^N+1(Q), which is depicted in Fig. <ref>, which persists as long as σ_∗ <σ<b_1, where σ=b_1 is the first parameter value where the unstable manifold has a tangent intersection with the stable manifold in Z(σ). The proof of Theorem 1 actually yields more information than provided in the statement; for example, it explains the “blinking effect” observed for the attractor as σ:=C increases. This is caused by a sequence of tangent intersections between the stable and unstable manifolds of the saddle point p(σ) - each a bifurcation value in its own right - as described by Newhouse <cit.>, which proves the following result. Let f be as in Theorem 1. Then the bifurcation value σ_∗ precipitating the creation of a robust stable chaotic strange attractor for σ_∗<σ<b_1, is proceeded by a sequence of bifurcation values σ_t<σ_t(1)<σ_t(2)<⋯<σ_∗ corresponding to successive tangent intersections between the stable and unstable manifolds of the saddle point as σ increases. § HETEROCLINIC-HOMOCLINIC BIFURCATIONS OF TYPE 1 IN THE PLANE Next, we consider a combined heteroclinic and homoclinic variant of the bifurcation in Section 2, illustrated in Fig. <ref>, that involves a pair of saddle points, which we set up first with the appropriate analogs of properties (A1) and (A2).(B1) f∈ C^1:=C^1(ℝ^2×[ a,b),ℝ^2) and f_σ has a pair of saddle points p(σ):=(x̂(σ),ŷ(σ)) and q(σ):=(x̆(σ),y̆(σ)), which are such that: (i) p(ω) is in the first quadrant and q(ω) is in the third quadrant of ℝ^2 for all a≤σ<b ; (ii) there real constants κ_s, κ_u such that the eigenvalues λ̂^s(σ) and λ̂^u (σ) of f_σ^'(p(σ)) and λ̃^s(σ) and λ̃^u(σ) of f_σ^'(q(ω)) satisfy 0<λ̂^s(σ),λ̃^s(σ)≤κ_s<1<κ_u≤λ̂^u(σ),λ̃^u(σ) for all a≤σ <b; (iii) the eigenvectors corresponding to λ̂^u(a) and λ̃^u(a) are parallel to the x-axis; (iv) there is a vertical strip of the form S_α:={x∈ ℝ^2:x̆(a)-α<| x| <x̂(a)+α} for some α>0 such that the stable manifolds W^s(p (σ)),W^s(q(σ))⊂ S_α for all a≤σ<b and separates the plane into left, middle and right components denoted as K_-(σ), K(σ) and K_+(σ), respectively; and(v) there is a β>0 such that p,q∈ R(α,β):={(x,y)∈ ℝ^2:x̆(a)-α<| x| <x̂(a)+α, y̆(a)-β<| y| <ŷ(a)+β} for all σ∈ a,b). (B2) There exist a<σ_1<σ_2<b such that the following obtain: (i) there is a (positively) f_σ-invariant attracting tubular neighborhood T(σ) of a C^1 closed Jordan curve C(σ) for all σ∈ a,σ_2); (ii) f_σ restricted to T(σ) has a positive (counterclockwise) rotation number for all σ∈ a,σ_2); (iii) the closed curve can be chosen so that centerset ℭ(σ):= ⋂_n=1^∞ f_ω^n(T(σ))=C(σ) for all σ∈ a,σ_1); (iv) T(σ)⊂ K(σ)∩ R(α,β) for all σ∈ a,σ_2); (v) T(σ) has an open basin of attraction, denoted as 𝔅_T(σ), containing (K(σ)∩ R(α,β)∩𝒞 _ext(σ))∖(Z(σ)∪Z̆(σ)) for all σ∈ a,σ_2), where 𝒞_ext(σ) is the exterior of the centerset and Z(σ) and Z̆(σ) are sets defined as follows: For Z(σ) there is for every σ∈ a,b) an orientation preserving C^1-diffeomorphism Φ_σ of an open neighborhood W_σ of a portion of W^s(p(σ)) below p(σ) such that Φ_σ(Z(σ)):={(ξ,η):-1≤ξ≤0,η≤φ(ξ)}, where φ:[-1,1]→(-1,0] is a C^1 function such that φ(0)=0 and φ^'(ξ)>0 when ξ<0. Moreover, the right-hand bounding curve c_r(σ):=Φ_σ^-1({ (0,η):φ(-1)≤η≤0}) and the f_σ image of left-hand bounding curve c_l(σ):=Φ_σ^-1( {(ξ,φ(ξ)):-1≤ξ≤0}) lie in W^s(p(σ)), while f_σ maps the interior of Z(σ) into K_+(σ). Analogously, there is a sectorial region Z̆(σ) (shown in Fig. <ref>) with vertex on W^s(q (σ)) above q(σ) and interior above the vertex, such that its (boundary) edges c̆_r(σ) and c̆_l(σ) lie in K(σ) and on W^s(q(σ)), respectively. Moreover, f_σ(c̆_r(σ))⊂ W^s(q(σ)) and f_σ maps the interior of Z̆(σ) into K_-(σ). We now have assembled the basic elements needed to formulate our basic result for heteroclinic-homoclinic bifurcations of type 1, of which there are several variations. In the interest of keeping things as simple as possible, we impose a symmetry requirement on the interaction of the invariant closed curve with the slice sets Z(σ) and Z̆(σ). For the result on the heteroclinic-homoclinic bifurcation, it is convenient to introduce the right unstable manifold. W_r^u(q(σ)):= ⋃_n=1^∞ f_σ^n({x∈ W^u(p (σ)):x∈{q(σ)}∪ K(σ) and d(x,q(σ))≤ν ∀ ν>0}), which is naturally the analog of the left unstable manifold of Theorem 1 and in the present context would be the generated by a portion the component of the unstable manifold of p(σ) in K(σ) (rather than in what was defined as K_-(σ) for Theorem 1). Let f:ℝ^2× a,b)→ℝ^2satisfy (B1) and (B2) and the additional property:(B3) Suppose the distance Δ(σ):=dist( ℭ(σ),Z(σ)):=inf{|x-y| :(x,y )∈ℭ(σ)× Z(σ)}=dist( ℭ(σ),Z̆(σ)):=inf{|x-y| :(x,y )∈ℭ(σ)×Z̆(σ)}, satisfies Δ(a)>0 and is a nonincreasing function of σon [a,σ_#)∪σ_2,σ_3) and lim _σ↑σ_3Δ(σ)=0. Then, there are a<σ_#≤σ_t^(1)≤σ_2≤σ_∗^(1)<σ_t^(2)<σ_∗^(2) <σ_3<b_1<1, where σ_t^(1)and σ _t^(2) are the first values of σ for which there are tangent intersections between W^u(p(σ)) and W^s(q(σ)) ( and W^u(q (σ)) and W^s(p(σ))) and W^u(p(σ)) and W^s(p(σ )) ( and W^u(p(σ)) andW^s(p(σ))), respectively. On the other hand, σ_∗^(1) and σ_∗^(2) are the last values of σ where there are tangent intersections between W^u(p(σ)) and W^s(q(σ )) (and W^u(q(σ)) andW^s(p(σ))) and W^u(p(σ)) and W^s(p(σ)) ( and W^u(p(σ)) and W^s(p(σ))), respectively. Then, for all σ_∗^(1)<σ<b_1, f_σ has a chaotic strange attractor 𝔄(σ ), which is the union of the closures of the left and right unstable manifolds: namely, 𝔄(σ):=W_l^u(p(σ)) ∪W_r^u(q(σ)).As in the proof of Theorem 1, we begin by focusing on a tubular type strip Q=Q(σ;ν) for W^s(p(σ)) (excluding c_l(σ)) cut off just above p(σ) and below at the lower edge of R(α,β) (as shown in Fig. <ref>), but we also introduce an analogous strip Q̃=Q̃(σ;ν) for W^s(q(σ)) with the understanding that the dimensions for both strips can be taken to be as small as suits our purposes in what follows. In particular, we take the width of the strips to be 2ν and the top edges to be parallel to and at a distance of ν above W^u(p (σ)) and below W^u(q(σ)), respectively, where ν>0. Again mimicking the proof of Theorem 1, but with twin objects for the saddle points p and q, it is convenient to define W_ν^u(σ) and W̃_ν^u(σ) to be the closed segments, with interiors in K(σ) as shown in Fig. <ref>, respectively, of the unstable manifold from p(σ) to the boundary of Q and the unstable manifold from q(σ) to the boundary of Q̃. In addition, we use s_σ:=s_σ(x)=s(x) and s̃_σ:=s̃_σ(x)=s̃(x) to denote the arclengths from the fixed point p to any point x ∈ W_ν^u(σ) and fixed point q to any point x∈W̃_ν^u(σ) or any of its f_σ-iterates. We also define for every nonnegative integer n the endpoints e_σ^n and ẽ_σ^n to be the boundary points of the C^1-submanifolds (with boundary) 𝒲 _σ^u(σ,n):=f_σ^n(W_ν^u(σ)) and 𝒲̃_σ^u(σ,n):=f_σ^n(W̃_ν^u(σ)) for which s_σ and s̃_σ are positive and actually increase without bound as n→∞ as long as σ<σ_#.As in the proof of Theorem 1, the details are best described and understood with the aid of figures illustrating the evolution of the dynamics and corresponding bifurcations such as in Figs. <ref> and <ref> for the homoclinic type bifurcations. Although we only present the basic geometry for the case at hand in Fig. <ref>, the analogous sequence of figures representing the evolution of the heteroclinic-homoclinic bifurcations can easily be envisaged by comparison with Figs. <ref> and <ref>, and we shall rely on this, along with an understanding of the proof of Theorem 1, in what follows.The hypotheses (B1)-(B3) imply that for any ν>0 there are positive integers N^(1)=N_ν^(1)(σ)<N^(2)=N_ν^(2)(σ) such that the following properties obtain:(i) There is a first σ=σ_t^(1)∈ a,σ_2) such that 𝒲_ν^u(σ,N^(1)) is tangent to c̆ _r(σ) and 𝒲̃_ν^u(σ,N^(1)) is tangent to c_l(σ). Consequently, it follows from the definition of the slice regions that 𝒲_ν^u(σ_t^(1),N^(1)+1) is tangent to W^s(q(σ_t^(1))) and 𝒲̃_ν ^u(σ_t^(1),N^(1)+1) is tangent to W^s(p (σ_t^(1))). (ii) Moreover, there is a first σ=σ_t^(2)∈ a,σ_2), with σ_t^(1)<σ_t^(2), such that 𝒲_ν^u(σ,N^(2)) is tangent to c_l(σ) and 𝒲̃_ν^u(σ,N^(2)) is tangent to c̆ _r(σ). Accordingly, the characterization of the slice regions then implies that 𝒲_ν^u(σ_t^(2),N^(2)+1) is tangent to W^s(p(σ_t^(2))) and 𝒲̃_ν ^u(σ_t^(2),N^(2)+1) is tangent to W^s(q (σ_t^(2))). Therefore, we conclude from (i) that as σ is increased, there must be a last value, σ_∗^(1), with σ_t^(1)≤σ_∗^(1)<σ_2, such that 𝒲_ν^u (σ,N^(1)) crosses c̆_r(σ) into the interior of Z̆(σ), implying that Q_l,σ(σ,N^(1)+1):= f_σ^N^(1)+1(Q_c(σ;ν)), where Q_c(σ ;ν):=Q(σ;ν)∩K(σ), which contains a portion of 𝒲_ν^u(σ,N^(1)+1), actually crosses over Q̃(σ;ν) into K_-(σ) for all sufficiently small ν>0 whenever σ_∗^(1)<σ. Furthermore, 𝒲̃_ν^u(σ,N^(1)) crosses c_l(σ) into the interior of Z(σ), implying that Q̃_l,σ(σ,N^(1)+1):= f_σ^N^(1)+1(Q̃_c(σ;ν)), where Q̃ _c(σ;ν):=Q̃(σ;ν)∩K(σ), which contains a portion of 𝒲̃_ν^u(σ,N^(1)+1), actually crosses over Q(σ;ν) into K_+(σ) for all sufficiently small ν>0 whenever σ_∗^(1)<σ. Consequently, for such values of the parameter σ we have a transverse heteroclinic 2-cycle connecting the saddle points p(σ) and q(σ), so it follows from the work of Bertozzi <cit.> (see also <cit.>) that one has a chaotic strange attractor of the form (<ref>) over the open parameter interval (σ_∗^(1),σ_t^(2)).When the parameter value increases to σ_t^(2), it follows from (ii) that we get a small Newhouse type bifurcation <cit.>, which amounts to a "blinking effect" followed by possibly more tangent intersections of W^u(p(σ)) and W^s(p(σ)) and W^u(q(σ)) and W^s(q(σ)). Moreover, there is a last such tangent intersection of these unstable and stable manifolds, beyond which we have the the geometric chaos and fractal set arguments of Birkhoff–Moser–Smale theory that imply the existence of the chaotic strange attractor defined by (<ref>), amounting to a two-fold version of the result for a single saddle point in Theorem 1. In particular, there is a two-fold manifestation of the attracting horseshoe in f_σ^N+1(Q) depicted in Fig. <ref>, with a copy at each of the saddle points. In a manner completely analogous to the homoclinic bifurcation, this (double) horseshoe bifiurcation persists as long as σ_∗<σ<b_1, where σ=b_1 is the first parameter value where the unstable manifolds W^u(p(σ)) and W^u(q(σ)) have tangent intersections with the stable manifolds W^s(p (σ)) and W^s(q(σ)) in Z(σ) and Z̆(σ), respectively. Finally, by combining the heteroclinic and homoclinic parts of the evolution of bifurcations, we obtain the desired result, thereby completing the proof.As in the case of Theorem 1, the proof of Theorem 2 yields more information than provided in the statement of the result. For example there are compound “blinking effects” observed for the attractor as σ:=C increases caused by sequences of tangent intersections: first the heteroclinic tangencies between W^u(p (σ)) and W^s(q(σ)) together with those between W^u(q(σ)) and W^s(p(σ)), followed by the homoclinic tangencies between the stable and unstable manifolds of p(σ) together with those of q(σ). Whence, the analysis in <cit.> together with the proof of Theorem 2 leads directly to a verification of the following analog of Corollary 1. Let f be as in Theorem 2. Then the bifurcation values σ_∗^(1) and σ_∗^(2) precipitating the creation of a robust stable chaotic strange attractors for σ_∗^(1)<σ<σ_t^(2) and σ_∗^(2)<σ<b_1 resulting from heteroclinic and homoclinic interactions, respectively, are proceeded by sequences of bifurcation values σ_t^(1)<σ_t(1)^(1) <σ_t(2)^(1)<⋯<σ_∗^(1) and σ_t ^(2)<σ_t(1)^(2)<σ_t(2)^(2)<⋯<σ_∗^(2) corresponding to successive tangent intersections between the stable and unstable manifolds W^s(p(σ)) and W^u(q(σ)) and W^s( q(σ)) and W^u(p (σ)), and W^s(p(σ)) and W^u(q(σ)) and W^s(q(σ)) and W^u( p(σ)) as σ increases. It is worth mentioning that the heteroclinic-homoclinic bifurcation is apt to experience dynamical crises for smaller parameter values than the homoclinic type because there are two unstable manifolds, rather than just one, capable of interacting with each stable manifold.§ SOME HIGHER-DIMENSIONAL GENERALIZATIONS There is an extensive array of possible generalizations of Theorems 1 and 2, some of which might be quite difficult to realize in terms of discrete dynamical systems in three or more dimensions. For example, suppose that instead of an invariant attracting closed curve, we have an attracting invariant torus in ℝ^3 on which the restricted dynamics is ergodic. In addition, assume there is a single saddle point with a 2-dimensional stable manifold and the analog of a slice set to which the torus tends to as the parameter of choice increases. Then one would expect an extremely interesting and complex analog of the 2-dimensional bifurcation described in Theorem 1. However, finding a relatively simple smooth map of ℝ^3 satisfying these properties is rather difficult, so we shall confine ourselves to a couple of much simpler examples. §.§ A simple 3-dimensional generalization The first generalization is a more or less trivial extension of Gilet's planar map; namely E(x,y,z;σ):=(x-σΨ^'(x)y,μ(y+Ψ(x)),0.8z) .To see this, we note that the fixed points of E are (x_∗,y_∗,0), where (x_∗,y_∗) are the fixed points of Gilet's planar map (<ref>) and that the x- and y-coordinate maps are independent of z. Therefore, the fixed points comprise denumerably many hyperbolic points of the form (x_k,μ(1-μ)^-1Ψ(x_k),0), with Ψ ^'(x_k)=0, each having a 2-dimensional stable and 1- dimensional unstable manifold, together with a denumerable set of hyperbolic fixed points (x̃_m,0,0), with Ψ(x_m)=0, each of which has a 3-dimensional stable manifold that bifurcates into a 1-dimensional stable manifold with a 2-dimensional unstable as the parameter σ increases. These later fixed points, associated with the Neimark–Sacker bifurcations of Gilet's map, are on invariant lines x=x_m,y=0, which bifurcate into invariant attracting cylinders of the form C(σ)×ℝ, where the curves C(σ) are as defined in Theorem 1. A simulation of E for increasing σ is shown in Fig. <ref>. §.§ A more complex 3-dimensional generalization We can create a more interesting extension of Gilet's map by adding a dependence of the z-coordinate map on the other variables; for example, as inÊ(x,y,z;σ):=(x-σΨ^'(x)y,μ(y+Ψ (x)),0.8z+0.1sin^2(z+Ψ^'(x)).It is easy to verify that the fixed points of (<ref>) are as follows: There are, as for (<ref>), denumerably many fixed points of the type ( x_k,μ(1-μ)^-1Ψ(x_k),0), with Ψ^'(x_k)=0, each having a 2-dimensional stable and 1-dimensional unstable manifold. There are also denumerably many fixed points of the form (x̃ _m,0,z_m), with Ψ(x_m)=0 and z_m the unique solution of 2z_m=sin^2(z_m+Ψ^'(x_m)). These latter fixed points start as sinks and bifurcate into hyperbolic fixed points each with a 2-dimensional unstable manifold and a 1-dimensional stable manifold parallel to the z-axis. Moreover, the cylinders C(σ)×ℝ are attracting and invariant for sufficiently large values of the parameter σ. The main difference between this extension and (<ref>) is that the unstable manifolds for the fixed points (x_k,μ(1-μ)^-1 Ψ(x_k),0) do not remain in the x,y-plane as they wrap around the cylinder C(σ)×ℝ, which leads to the more complex bifurcation behavior shown in Fig. <ref>.§ EXAMPLES Here we shall present examples of the bifurcation behavior described in Theorem 1 and Theorem 2, mainly through illustrations based on simulations and a bit of analysis. The examples, which comprise a homoclinic and heteroclinic-homoclinic type, are to be planar maps based upon Gilet's model and a minor modification thereof. §.§ Symmetric homoclinic bifurcations for Gilet's map It turns out that homoclinic bifurcations of the type described in Theorem 1 can be very effectively illustrated by considering a pair of contiguous symmetric cells (containing symmetric invariant closed curves with source point centers along the x-axis) for Gilet's map F(x,y;σ):=(x-σΨ^'(x)y,μ(y+Ψ(x))),where μ is fixed and σ=C is varied. The symmetric cells are chosen to be the ones symmetric about the stable manifold corresponding to the line x≃-1.57, with symmetric center sources (approximately) at ( -1.79,0) and (-1.35,0) as shown in Fig. <ref>. It should be noted that, in contrast to the description in Theorem 1, the saddle point on the stable manifold is below rather than above the symmetric invariant closed attracting curves, which means that the assumptions remain the same modulo a reflection in the x-axis.Now, it is a straightforward but rather tedious matter to verify that all of the hypotheses of Theorem 1 hold (modulo the reflection mentioned above), but all of the assumptions are illustrated quite clearly in Fig. <ref> save the existence of the orientation reversing slice regions. Consequently, we shall restrict our analysis to the identification of these regions for this example. A simple calculation shows that the saddle point of interest in this example is fixed at p=(x̂,μ(1-μ)^-1Ψ (x̂))≃(-1.57,μ(1-μ)^-1Ψ(-1.57)) ≃(-1.57,-0.206μ(1-μ)^-1). We also compute that F^'=μ[1-σ(yΨ^''(x)-(Ψ ^'(x))^2)], so it follows from the fact that Ψ^''(x) is positive ( ≃Ψ^''(x̂)) in a thin vertical strip centered at x=x̂. Hence, it follows that F^' changes from positive to negative along the stable manifold as y changes from negative to positive, which signals a reversal of orientation.The change in orientation also produces the slice regions Z described in (A2). To see this, it follows from symmetry that it is enough to describe the slice to the right of the stable manifold x=x̂. Of course, the left boundary curve for this slice is just a portion of the vertical line x=x̂ for y≥ỹ>0. The right-hand boundary curve for the slice is just y=x-x̂/σΨ^'(x) for x>x̂, which has a cusp at (x̂,ỹ), where ỹ=lim_x↓x̂(x-x̂/σΨ ^'(x)). Now it is straightforward to verify that this slice region has the properties described in (A2) and (A3) of Theorem1, as does the symmetric slice to the left of x=x̂. §.§ Heteroclinic-homoclinic bifurcations for modified Gilet's map To illustrate the bifurcations in Theorem 2, we consider the following modification of Gilet's map: It is the map F̃:ℝ ^2→ℝ^2 defined as follows:F̃(x,y;σ):={[ F(x,y;σ),| x|≤0.4; (x-[1+5(x+0.4)]σΨ̃^'(x)y,μ(y+Ψ̃(x))), -0.6<x<-0.4; (x-[1-5(x-0.4)]σΨ̃^'(x)y,μ(y+Ψ̃(x))), 0.4<x<0.6; (x,μ(y+Ψ̃(x))),| x|≥0.6 ].,where Ψ̃(x):={[ (10x+3)Ψ(0.4),-0.6<x<-0.4; (10x-3)Ψ(0.4),0.4<x<0.6;-3Ψ(0.4), x≤-0.6; 3Ψ(0.4), 0.6≤ x ].. We note that the map (<ref>) is continuous everywhere and smooth except along the lines x=±0.4,±0.6. It turns out that the vertical lines along which it fails to be C^∞ do not alter the qualitative nature of the bifurcation evolution as described in Theorem 2, which is evident from Fig. <ref>. Once again, we note that the hypotheses of Theorem 2 can be readily checked, but all save the existence of the Z slices are clearly shown in the simulations in Fig. <ref>. The existence of the orientation reversing slices Z(σ) and Z̆(σ) can be verified in a manner completely analogous to to that used in the previous example. § CONCLUDING REMARKS In our paper <cit.>, we proved that Gilet's walking droplet model <cit.> develops Neimark–Sacker bifurcations generating invariant attracting closed Jordan curves (topological circles) as either one of the parameters is increased. We also saw that the diameter of these circles increases with either increasing parameter, which ultimately gives rise to new types of bifurcations arising from the interactions of stable manifolds with unstable manifolds of saddle points winding around the expanding circles. The investigation in this paper comprises an in-depth analysis of two variants - one purely homoclinic and the other a combination of heteroclinic and homoclinic interactions of unstable and stable manifolds of saddle points - of these new bifurcations as the original interaction parameter C (identified with σ) is varied while the damping parameter μ is fixed. In addition to our analysis of these bifurcations, we showed by examples how these dynamical phenomena can be extended to higher dimensions.There are several research directions related to this work that we intend to pursue in the near future. First among these is a related new bifurcation somewhat like those studied here, but based on diffeomorphisms that do not include the orientation reversing, noninjective slice regions in Gilet's model. Naturally, we also intend to study the bifurcations in models of Gilet type with C fixed and μ varying, which, for example, appears to exhibit more striking dynamical crises behavior than the case studied here. Another line of research, which has strong connections with the quantum aspects of pilot waves, is the construction of invariant or approximately invariant measures for the dynamical models such as Gilet's and is something that we intend to investigate as part of our continuing investigation of the mathematical aspects of walking droplet phenomena.§ ACKNOWLEDGMENTS Discussions with Anatolij Prykarpatski were very helpful in writing this paper. 99ACHMD.G. Aronson, M.A. Chory, G.R Hall and R.P. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study, Commun. Math. Phys. 83 (1982), 303-354.ArrD. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour, Chapman and Hall, London, 1992.BertA. Bertozzi, Heteroclinic orbits and chaotic dynamics in planar fluid flow, SIAM J. Math. Anal. 19 (1998), 1271-1294.BushJ. Bush, Pilot-wave hydrodynamics, Ann. Rev. Fluid Mech. 49 (2015), 269-292.CPFB Y. Couder, S. Protiere, E. Fort, A. Boudaoud, Dynamical phenomena: Walking and orbiting droplets, Nature 437 (2005), 208.FKPC.E. Frouzakis, I.G. Kevrekidis and B.P. Peckham, A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle, Physica D 177 (2003), 101-121.GilT. Gilet, Dynamics and statistics of wave-particle interactions in a confined geometry, Phys. Rev. E 90 (2013), 05297.JBY. Joshi and D. Blackmore, Strange attractors for asymptotically zero maps, Chaos, Solitons & Fractals 68 (2014), 123-138.MGNBP. Milewski, C. Galeano-Rios, A. Nachbin and J. Bush, Faraday pilot-wave dynamics: modeling and computation, J. Fluid Mech. 778 (2015), 361-388.MoserJ. Moser, Stable and Random Motion in Dynamical Systems, Princeton University Press, Princeton, 1973.NeimJ. Neimark, On some cases of periodic motions depending on parameters, Dokl. Akad. Nauk SSSR 129 (1959), 736-739.NewS. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9-18.OttE. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993.OHRBA. Oza, D. Harris, R. Rosales and J. Bush, Pilot-wave dynamics in a rotating frame: the emergence of orbital quantification, J. Fluid Mech. 744 (2014), 404-429.PdMJ. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York, 1982.RB1A. Rahman and D. Blackmore, Neimark–Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers, Chaos, Solitons & Fractals 91 (2016), 339-349.RobC. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press Inc., Boca Raton, 1995.SackR. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, Rep. IMM-NYU 333 (1964), 1-62.ShirD. Shirokov, Bouncing droplets on a billiard table, Chaos 23 (2013), 013115.SmaleS. Smale, The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics, Springer-Verlag, New York, 1980.WY1Q. Wang and L-S. Young, Strange attractors with one dimension of instability, Commun. Math. Phys. 218 (2001), 1-97.WY2Q. Wang and L-S. Young, From invariant curves to strange attractors, Commun. Math. Phys. 225 (2002), 275-304.WY3Q. Wang and L-S. Young, Toward a theory of rank one attractors, Ann. of Math. (2) 167 (2008), 349-480.NDSS. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 2003. | http://arxiv.org/abs/1708.07593v1 | {
"authors": [
"Aminur Rahman",
"Denis Blackmore"
],
"categories": [
"math.DS"
],
"primary_category": "math.DS",
"published": "20170825011950",
"title": "Exotic Bifurcations Inspired by Walking Droplet Dynamics"
} |
=1L[1]>p#1OT1pzcmitfigures/C>c<=.08em =.05em =.03em =.65ex =0pt =.4ex =0pt ==.5em =.5em [email protected]@[email protected]@if.usp.brDepartamento de Física Matemática, Instituto de Física, Universidade de São Paulo,R. do Matão 1371, CEP. 05508-090, São Paulo, BrazilWe discuss the effect of Beyond the Standard Model charged current interactions on the detection of the Cosmic Neutrino Background by neutrino capture on tritium in a PTOLEMY-like detector. We show that the total capture rate can be substantially modified for Dirac neutrinos if scalar or tensor right-chiral currents, with strength consistent with current experimental bounds, are at play. We find that the total capture rate for Dirac neutrinos, Γ_ D^ BSM, can be between 0.3 to 2.2 ofwhat is expected for Dirac neutrinos in the Standard Model,Γ_ D^ SM, so that it can be made as large asthe rate expected for Majorana neutrinos with only Standard Modelinteractions. A non-negligible primordial abundance of right-handedneutrinos can only worsen the situation, increasingΓ_ D^ BSM by 30 to 90%. On the other hand, ifa much lower total rate is measured than what is expected forΓ_ D^ SM, it may be a sign of new physics.Impact of Beyond the Standard Model physics in the detection of the Cosmic Neutrino Background Renata Zukanovich Funchal December 30, 2023 ============================================================================================== § INTRODUCTION The accidental discovery of the Cosmic Microwave Background (CMB) radiation by Penzias and Wilson in 1965 laid the foundations for the enormous progress in our understanding of the evolution of the Universe. This is the oldest directly observed radiation in the Universe, dating from the epoch of recombination, and its precise study, carried out in the last decades by various cosmological probes, lead to the establishment of the standard model of cosmology. This model also predicts the existence of a Cosmic Neutrino Background (), a relic radiation that decoupled from matter when the Universe was merely a second old, which is expected to have played a crucial role in primordial nucleosynthesis and in large scale structures formation. The CMB anisotropies, an indirect imprint of the , have alreadyoffered two important constraints in connection to particle physics:a limit on the sum of neutrino masses and the effective number ofneutrino species.A confirmation of theby direct detectionusing experiments on Earth would not only represent a further triumphof modern cosmology, but it would also constitute an unique opportunityto probe neutrino properties. For a long time this was believed to bean impossible task since relic neutrinos are expected to be non-relativistictoday with an average momentum of about 10^-4 eV. Recent developmentshave allowed to revive the old suggestion by Weinberg <cit.>of capturing them on β-decaying nuclei, a process with no energy threshold. In fact, a real experimental proposal, the Princeton Tritium Observatory for Light, Early-Universe, Massive-Neutrino Yield (PTOLEMY) experiment <cit.> is currently assessing the prospects for using the process ν + ^3 H→ ^3 He+ e^-. The signature ofcapture would be a peak in the final electron spectrum at an energy 2 m_ν above the β-decay endpoint. This requires a very challenging energy resolution ≲ 0.1 eV for the final electrons to be distinguished from the β-decay background. This has triggered interest in the community to investigate what could potentially be learned in such experiment <cit.>.In particular, the authors of ref. <cit.> have shown how the direct measurement of thewould allow to discriminate Majorana from Dirac neutrinos, as the former would produce a capture rate twice as large as the latter.This is because for non-relativistic states chirality and helicity do not coincide, and it is helicity, not chirality, which is conserved by the . Their conclusions rely on the fact that only the neutrinos that interact weakly according to the Standard Model (SM) could be produced and kept in thermal equilibrium before decoupling, a feature that could be modified by new interactions or a different thermal history <cit.>.In this paper we try to answer the following question: if neutrinos have new Beyond the Standard Model (BSM) interactions, how would this affect the relic neutrino detection rate in PTOLEMY-like detectors? We implement these possible deviations using an effective lagrangian approach.We start in section <ref> by describing the gauge invariant operators that we will consider and computing the rate of neutrino capture on tritium. In section <ref> we introduce the experimental resolution and describe in detail when the signal from the electron produced in the capture can be distinguished from the electron produced by the β-decay background. In section <ref> we discuss the experimental bounds from β-decay on the BSMphysics coefficients, and we show how the capture rate is modified with respect to the standard case for various regions of the parameter space. In section <ref> we discuss how gravitational clusteringor a primordial abundance of right-handed neutrinos present in thetoday would affect our results. Finally our conclusions are drawn in<ref>. In appendix <ref> we discuss how the interplaybetween the experimental resolution and the neutrino mass ordering affectthe possibility of distinguishingthe electron peaks due to each neutrinomass eigenstate.§ EFFECTIVE LAGRANGIAN APPROACH FOR THE BSM NEUTRINO INTERACTIONS In the SM, the weak interactions have a purely V-A Lorentz structure. Sincethe simple fact that neutrinos have a non-zero mass constitutes already an evidence for BSM physics,we will allow here for other possibilities. This can be done in a model independentfashion using an effective field theory approach. We will consider dimension-sixoperators which are SU(2)_L×U(1)_Y invariant, but which also include right-handedneutrinos <cit.>. More precisely,we writeℒ_ BSM=ℒ_ SM^(4) + ℒ_m_ν+1/Λ^2∑_k=1^12 c_k^(6) Q_k^(6),where ℒ_ SM^(4) is the dimension-four SM lagrangian, ℒ_m_ν isthe neutrino mass lagrangian, which can either come from a dimension 4 operator involving right handedneutrinos or from the dimension 5 Weinberg operator; Λ is the maximum energy scale at which thetheory is still valid; and the c_k^(6) are dimensionless coupling constants. The set of operatorswith left- and right-handed neutrinos, Q_k^(6)={Q_k^(6)(ν_L),Q_k^(6)(ν_R)}, is givenin table <ref>. The terms relevant for our calculation of the BSM relic neutrino capture rate on β-decaying tritium can be obtained writing eq. (<ref>) in terms of mass eigenstatesℒ_ eff=-G_F/√(2) V_udU_ej {[e̅γ^μ(1-γ^5)ν_j][u̅γ_μ(1-γ^5)d] +∑_l,qϵ_lq [e̅𝒪_lν_j][u̅𝒪_q d]}+ h.c.,where a sum over the three neutrino mass eigenstates j=1,2,3 is implied. The couplings ϵ_lq,related to the dimensionless couplings c_k^(6) (see ref. <cit.>),parametrize the BSM physics effects, with l (q) labelling the Lorentz structure ofthe lepton (quark) current, as given by 𝒪_l (𝒪_q) in table <ref>.V_ud and U_ej correspond to the Cabibbo-Kobayashi-Maskawa (CKM) andPontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrices elements relevant to the process, respectively.Equation (<ref>) can be used to calculate the neutrino absorption on tritium ν_j + ^3 H→ ^3 He + e^-, in the presence of BSM interactions. To this end, we need to properly define the hadronic matrix elements involving the quark current in eq. (<ref>). Following ref. <cit.>, we have⟨ p(p_p)|u̅γ^μ(1±γ^5)d |n(p_n)⟩ =u_p(p_p)γ^μ[g_V(q^2)± g_A(q^2)γ^5]u_n(p_n), ⟨ p(p_p)|u̅d |n(p_n)⟩ =g_S(q^2) u_p(p_p) u_n(p_n), ⟨ p(p_p)|u̅γ^5 d |n(p_n)⟩ = g_P(q^2) u_p(p_p)γ^5 u_n(p_n), ⟨ p(p_p)|u̅σ^μν(1±γ^5)d |n(p_n)⟩ = g_T(q^2) u_p(p_p)σ^μν(1±γ^5)u_n(p_n).We have introduced the hadronic form factors g_h(q^2), with h=V,A,S,P,T corresponding to the vector, axial, scalar, pseudoscalar and tensor Lorentz structures, respectively.[Since it does not contribute to thecapture, we do not includethe weak magnetic term ⟨ p(p_p)|u̅γ_μ d |n(p_n)⟩_ WM = -ig_ WM/2 M_Nu_p (p_p)σ_μν(p_n-p_p)^ν u_n(p_n).] Although these form factors depend on the transferred momentum q^2=(p_n -p_p)^2, for the capture rate we are only interested in the q^2 ≃ 0 limit.In our numerical analysis we will use the values shown in table <ref><cit.>. Following the calculation of ref. <cit.>, the capture cross section for aneutrino mass eigenstate j, with helicity h_j=± 1 and velocity v_j including BSMeffects is given byσ_j^ BSM(h_j)v_j=G_F^2/2πV_ud^2U_ej^2 F_Z(E_e)m_ ^3He/m_ ^3H E_e p_e T_j(h_j,ϵ_lq),where m_ ^3 He and m_ ^3H are the helium and tritium masses, and E_e, m_e,p_e are the electron energy, mass and momentum, respectively. The T_j(h_j,ϵ_lq)function contains the dependence on the neutrino helicity and on the ϵ_lq parameters,T_j(h_j,ϵ_lq) = A(h_j)[g_V^2(ϵ_LL+ϵ_LR+1)^2+3 g_A^2(ϵ_LL-ϵ_LR+1)^2+g_S^2 ϵ_LS^2+48g_T^2 ϵ_LT^2..+2m_e/E_e [g_Sg_V ϵ_LS(ϵ_LL+ϵ_LR+1)-12g_Ag_T ϵ_LT(ϵ_LL-ϵ_LR+1)]] + A(-h_j)[g_V^2 (ϵ_RR+ϵ_RL)^2+3 g_A^2 (ϵ_RR-ϵ_RL)^2+g_S^2 ϵ_RS^2+48g_T^2 ϵ_RT^2..+2m_e/E_e [g_Sg_V ϵ_RS (ϵ_RR+ϵ_RL) -12g_Ag_Tϵ_RT (ϵ_RR-ϵ_RL)]] +2 m_j/E_j{g_Sg_V ϵ_RS (ϵ_LL+ϵ_LR+1)+ϵ_LS (ϵ_RR+ϵ_RL)). .-12g_Ag_T(ϵ_RT (ϵ_LL-ϵ_LR+1)+ϵ_LT (ϵ_RR-ϵ_RL))} +2 m_jm_e/E_jE_e{g_V^2 (ϵ_LL+ϵ_LR+1)(ϵ_RR+ϵ_RL)+3 g_A^2 (ϵ_LL-ϵ_LR+1)(ϵ_RR-ϵ_RL)..+g_S^2 ϵ_RS ϵ_LS+48g_T^2 ϵ_RT ϵ_LT},with m_j, E_j the mass and energy of the j-th neutrino mass eigenstate and A(h_j) = 1-2h_j v_j. Let us note that A(h_j) ≃ 1 for non-relativistic neutrinos, corresponding to the case on which we will focus in section <ref>. Furthermore,notice that the capture rate is independent of the pseudoscalar couplings ϵ_lP.The Fermi function F_Z(E_e), which takes cares of the enhancement of the cross section due to the Coulomb attraction between the proton and electron, is given byF_Z(E_e)=2π Z α E_e/p_e [1-e^-2π Z α E_e/p_e].Summing over all the neutrino mass eigenstates, one can calculate the total ^3H capture rateΓ_^ BSM = N_T ∑_j=1^3 Γ_^ BSM(j)= N_T∑_j=1^3 σ_j^ BSM (+1) v_jn_ν_+^j+σ_j^ BSM(-1) v_jn_ν_-^j,where N_T is the number of nuclei present in the sample and n_ν_±^j the number density at the present time of the helical state ν_±^j.§ DETECTION OF THEBY A PTOLEMY-LIKE DETECTOR A PTOLEMY-like experiment <cit.> aims to detect thethrough the neutrino capture by tritium, a reaction that has no energy threshold.We can safely assume thatneutrinos are non-relativistic today[As we know from oscillation experiments,only one neutrino can be massless.] as their root mean momentum is⟨ p ⟩≈ 0.6 meV≪ m_j <cit.>. This has two crucial consequences. First, the neutrino flavour eigenstateshave suffered decoherence into their mass eigenstates, so a detector would,in fact, measure the contribution of each neutrino mass eigenstate. Second,at the time of the creation of the , i.e. when neutrinos decoupledfrom the primordial plasma, they were ultrarelativistic, making chiral andhelical eigenstates effectively equal. However, as neutrinos evolved into anon-relativistic state due to the expansion of the Universe, chirality and helicity became different. Since neutrinos were free streaming, it was helicity,not chirality, that was conserved in the process.[If neutrinos underwent aclustering process, helicity would not be conserved either. We will comment more on this possibility in section <ref>.] This implies that the neutrino number density is n_ν_+^j=n_ν_-^j=n_0≈ 56 cm^-3in the Majorana case, while n_ν_-^j=n_0 and n_ν_+^j=0 in the Dirac case.If no BSM interactions are present, the function T_j(h_j,ϵ_lq) reduces toT_j(h_j,0) = A(h_j)[g_V^2+3 g_A^2],from which, using eq. (<ref>), we conclude that Γ^ M_ = 2Γ^ D_=85.7 [kg yr]^-1,where Γ^ M_ and Γ^ D_ are the Majorana and Dirac capture rates. We will consider in section <ref> the modifications to the neutrino abundance due to BSM physics.The signature of relic neutrinos in a PTOLEMY-like detector is given by the electron created in the capture process. Nonetheless, tritium can also undergo β-decay, giving rise to a continuous electron spectrum. As a consequence, one needs to discriminate the electrons produced by theneutrino capture from the electrons produced byβ-decays. Using kinematics, the electrons produced by the ν_j relic neutrinos capture will have a definite energy <cit.> E_e^, j≃ m_e+K_ end^0+2 m_j,where K_ end^0 corresponds to the β-decay endpoint energy.This implies that relic neutrinos could produce one or more peaks in theelectron energy spectrum at energies larger than the endpoint one.If so,and β-decay events can in principle be discriminatedfrom each other. It is clear that the finite energy resolution of the realdetector plays an essential role in establishing whether the two signalscan be separated or not. In order to estimate the signal in a more realisticway we will follow <cit.> and convolute thecapture rateof eq. (<ref>) and the β-decay background with a Gaussian functiondΓ_^ BSM/dE_e = 1/√(2πσ^2)∑_j=1^3∫_-∞^∞dE_e^' Γ_^ BSM(j)exp[-(E_e^'-E_e)^2/2σ^2] δ(E_e-E_e^, j),dΓ_β/dE_e = 1/√(2πσ^2)∫_-∞^∞dE_e^' dΓ_β/dE_e^' exp[-(E_e^'-E_e)^2/2σ^2], where σ is the expected experimental energy resolution. The complete expression for the β-decay rate dΓ_β/dE_e^' can be found in ref. <cit.>.In order to estimate the total number of events produced by theand β-decay in the region in which we expect asignal, we define the full width at half maximum (FWHM) of the Gaussian function as Δ = √(8ln 2) σ. With this definition, we haveN_^ BSM(Δ) =∫_E_e^-Δ/2^E_e^+Δ/2dE_e dΓ_^ BSM/dE_e,N_β(Δ) =∫_E_e^-Δ/2^E_e^+Δ/2dE_e dΓ_β/dE_e,which can be used to define the ratior_= N_^ BSM(Δ)/√( N_β(Δ)).We will consider that the signal can be discriminated from the backgroundwhen r_≥ 5. The future PTOLEMY experiment is expected to haveΔ = 0.15 eV <cit.> in such a way that a single peak is expected if the sum of the neutrino masses is about 0.1 eV. For smallermasses, a smaller value of Δ would be needed to discriminate the signalfrom the background. We study more in detail the interplay between Δ,neutrino masses and the position of the peaks observed at PTOLEMY-like detectors in Appendix <ref>. § ON THE CONTRIBUTIONS OF BSM PHYSICS TOCAPTURE RATE The BSM lagrangian of eq. (<ref>) generates not only new contributions tothe neutrino capture by tritium, but also modifies other low energy processes. To assessthe size of the modification to the neutrinocapture rate, we first need totake into account the experimental bounds on the ϵ_lq coefficients. Limitsfrom Cabbibo Universality <cit.>, radiative pion decay <cit.>and neutron decays <cit.> put bounds on the ϵ_Lq left-chiralcouplings; meanwhile, limits coming from the β-decay of several nuclei have been reviewed in ref. <cit.>. A complete compendium of the limits regardinglow energy decays is given in refs. <cit.>. For our purposes,we will consider the cases considered in ref. <cit.>, as they include couplingswith right-handed neutrinos. The constraints are given in terms of the following combinationsof couplings:C_V= g_V(1+ϵ_LL+ϵ_LR+ϵ_RL+ϵ_RR), C_V^' = g_V(1+ϵ_LL+ϵ_LR-ϵ_RL-ϵ_RR), C_A= -g_A(1+ϵ_LL-ϵ_LR-ϵ_RL+ϵ_RR), C_A^' = -g_A(1+ϵ_LL-ϵ_LR+ϵ_RL-ϵ_RR), C_S= g_S(ϵ_LS+ϵ_RS), C_S^' = g_S(ϵ_LS-ϵ_RS), C_T= 4 g_T(ϵ_LT+ϵ_RT), C_T^' = 4 g_T(ϵ_LT-ϵ_RT).Accordingly, we need to convert the bounds on the C_i^(') into bounds onϵ_lq at 3σ C.L. To this end, we have performed a scan over the ranges-10^-3≤ ϵ_LL≤ 10^-3, -10^-3≤ ϵ_LR≤ 10^-3, -2.8× 10^-3≤ ϵ_LS≤ 5× 10^-3, -2× 10^-3≤ ϵ_LT≤ 2.1× 10^-3,andϵ_Rq≤ 10^-1,keeping only the points consistent with each of the allowed regions of the C_h^(') in ref. <cit.>. Let us notice that,to translate the limits into contraints on the ϵ_lq parameters,we also scanned over the g_A(0)/g_V(0) value given in table <ref> sincesuch parameter is affected by the presence of BSM <cit.>.The ranges in which the scan is performed have been chosen to include the constraints ofrefs. <cit.> in the left-chiralcoefficients at the 3σ level. Although stronger limits can be imposedon right-handed couplings using pion decay <cit.>, we will notinclude them as they are strongly dependent on the flavour structure of themodel <cit.>. Finally, LHC bounds comingfrom pp → e + X + E_T have been studied in refs. <cit.>.However, the analysis is performed supposing the interactions of eq. (<ref>)remain pointlike up to the LHC energies, i.e. up to a few TeV. To allow forthe possibility that BSM physics appears just above the electroweak scale,in our analysis we will use only the bounds coming from low energy experiments. We found that the parameters ϵ_LL and ϵ_LR are unconstrainedby the experimental data as it has been previously noted in ref. <cit.>. For reference we summarize here the bounds without the correlations— which have been included in our numerical analysis — :*Only left-chiral couplings allowed in the fit (ϵ_Rq=0). The scalar and tensor terms have distinct dependence on the electron energy and mass, because of the different Lorentz structure. Computing the total capture rate Γ_^BSM using the points that pass the low energy experimental constraints, we find 0.985Γ_^ D≲Γ_^ BSM≲ 1.02Γ_^ D, where Γ_^D is the capture rate for Dirac neutrinos with only SM interactions. * Only vector-axial-vector couplings allowed in the fit (ϵ_LS=ϵ_RS=ϵ_LT=ϵ_RT=0): in this case we get |ϵ_RL|≲ 8 × 10^-2 and |ϵ_RR|≲ 5 × 10^-2 at 3σ level. Let us notice that the term linear in the right-handed couplings in eq. (<ref>) is proportional to m_j/E_j, so it would be negligible for an ultrarelativistic neutrino.This term comes from the interference of the SM contribution with the right-handed neutrino current. The terms proportional to (ϵ _RR±ϵ_RL)^2 come from the square of the right-handed currents, and are proportional to A(-h_j).Using the experimentally allowed range for ϵ_RR,RL, we find 0.89Γ_^ D≲Γ_^ BSM≲ 1.11Γ_^ D. *Only right-chiral scalar and tensor couplings allowed in the fit (ϵ_LS=ϵ_LT=ϵ_RL=ϵ_RR=0): in this case we get |ϵ_RS|≲ 1.1 × 10^-1 and |ϵ_RT|≲ 8 × 10^-2 at 3σ. Again the term proportional to the neutrino mass comes from the interference between SM and right-handed currents. Furthermore, we observe that this interference term does not depend on the neutrino helicity. This is due to the different Lorentz structures that appear in the BSM lagrangian. Considering the allowed parameter space, we find 0.61Γ_^ D≲Γ_^ BSM≲ 1.52Γ_^ D. Since in this case the parameter space is highly correlated due to the correlations coming from the β-decay bounds, we show in figure <ref> the rate between the BSM capture rate and the SM Dirac case in the (ϵ_RS, ϵ_RT) plane. *Five free couplings allowed in the fit: in this case we get |ϵ_RS|≲ 10^-1and |ϵ_RT|≲ 8 × 10^-2 at 3σ. Here the interference term proportional to the neutrino mass depends on the product between ϵ_LS,LT and ϵ_RS,RT.We show in figure <ref> the ratio between the BSM capture rate and the SM Dirac rate in the (ϵ_RS, ϵ_RT) plane, in which we find the strongest correlation between the couplings. We find that the ratio can be at the most 2.2 times the SM one, which is interesting as in this case Dirac neutrinos with BSM interactions can mimic Majorana neutrinos in the SM. However, there are regions in parameter space in which the rate is considerably lower than the SM one.Let us conclude stressing that pure Majorana neutrinos fall in the “only left-chiral couplings" category (case 1 above), with only a small modification of order 2% allowed in the capture rate. Dirac neutrinos have instead a much richer phenomenology, with all the above cases possible (depending on the gauge invariant operators of table <ref> generated in the UV theory). On the other hand, one could also worry about possible modifications of the tritiumβ-decay spectrum generated by BSM interactions, which could makethedetection more involved. Nevertheless, it has been shown inref. <cit.> that the endpoint of the β-decay spectrumis not significantly modified by BSM physics; thus, in principle, relicneutrino detection would be still possible in this case. § ON THE RELIC RIGHT-HANDED NEUTRINO ABUNDANCE As we have seen in section <ref>, without BSM contributionsthe neutrino number density today is expected to ben_ν_-^j =n_0, n_ν_+^j=n_0(Majorana), n_ν_-^j =n_0, n_ν_+^j=0(Dirac),with the capture rate in both cases given in eq. (<ref>).There are three ways in which this result can bemodified: (i) if neutrinos underwent a gravitational clustering process,(ii) if BSM interactions are present, and (iii) if an initial abundance of right-handed neutrinos was present in the early universe.Neutrino motion in the Dark Matter gravitational potential has the effect of modifying the direction of the neutrino momentum without affecting its spin <cit.>. The immediate consequence is that neutrinos undergo a process of gravitational clustering that tends to equilibrate the h_j=+1 and h_j=-1 populations. Since forMajorana neutrinos there is already equilibrium, eq. (<ref>)is still valid. The situation is different for Dirac neutrinos, for which we getn_ν_-^j =n_0/2, n_ν_+^j=n_0/2(Dirac, clustering).Nevertheless, eq. (<ref>) is still valid since the additional right-handed neutrino population in the Dirac case with clustering compensates for the loss in the left-handed neutrino population. Very recently, an N-body simulation has been consideredin ref. <cit.> to estimate the relic neutrino densityenhancement on Earth. The main result is that the clustering effectis negligible in the minimal Normal Ordering case while, for minimalInverted Ordering, the capture rate can be increased up to 20% forboth Dirac and Majorana neutrinos.We now turn to the case in which BSM interactions are present. Since BSM physics modify the electroweak rates, this could potentially affect the left-handed neutrino abundance. As we have seen in section <ref>, we must have at most ϵ_lq≲ 10^-1 to be compatible with β-decay and other low energy experimentalbounds (with many parameters much smaller). As such, the active neutrinos were maintained in equilibrium with the plasma mainly by SM interactions, and we do not expect a significant change in the left-handed neutrino number density n_ν_+^j.Let us finally consider the case in which an initial abundance of right-handed neutrinos is present. Such abundance can be either thermal or non-thermal. A thermal population can be achieved by non-standard interactions or in the presence of a tiny neutrino magneticmoment <cit.>.Following <cit.>, when the expansion of Universe becomes faster than the interaction rate, the right-handed neutrinos decouple as usual. At this freeze out temperature, T_R, the number densities of left- and right-handed neutrinos must be equaln_ν_R^j (T_R) = n_ν_L^j (T_R).Using entropy conservation, we can relate the right-handed neutrino abundance at late times with the left-handed abundance, obtaining <cit.> n_ν_R^j(T_ν)/n_ν_R^j(T_R)=g_*S(T_ν)/g_*S(T_R)(T_ν/T_R)^3,where g_*S(T) is the number of relativistic degree of freedom in entropy at the temperature T. Choosing T_ν in eq. (<ref>) to be the left-handed neutrino decoupling temperature, and using the definition of the effective number of thermal neutrino species N_ eff, one obtains <cit.> n_ν_R^j(T_ν) = (1/3Δ N_ eff)^3/4 n_ν_L^j(T_ν),where Δ N_ eff=N_ eff^ exp-3.046 and N_ eff=3.046 is the SM value with 3 left-handed neutrinos. The experimental determination of N_ eff by the Planck collaboration gives <cit.> N_ eff^ exp = 3.14^+0.44_-0.43He + Planck TT + low P + BAOat 95% C.L.Combining eq. (<ref>) with the experimental result, we get that the maximum density of right-handed neutrinos is <cit.> n_ν_+^j = n_(ν_-^j)^c = n_0^R ≃ 16cm^-3.The relic population of RH neutrinos modifies eq. (<ref>) even for vanishing non-standard interactions. In the pure SM case, since the capture rate is proportional to A(h_j)=1 for both left- and right-handed neutrinos, we can have an increase in Γ_^ D up to 28 % <cit.>. The difference is even larger if BSM interactions are turned on, although it depends crucially on the case considered. For instance, in the vector-axial-vector scenario, the capture rate is increased by roughly 30 %, while in the five parameter scenario the increase can be up to 70%. In this case, we have that therate can be as large as 2.8 Γ_^ D, reinforcing our results on the possibility of having Dirac neutrinos with a relic capture rate numerically similar to the Majorana one.The last possibility consists in having an initial non-thermal right-handedneutrino abundance. Following <cit.>, we will suppose that right-handed Dirac neutrinos initially form a degenerated Fermi gas, decoupled from the thermal bath. In this case, the right-handed neutrino density is related to the photon density n_γ byn_ν_R^j(T_γ)=1/6ζ(3)g_*S(T_γ)/g_*S(T_R)ϑn_γ,where ϑ=ε_F/T_R, ε_F the Fermi energy and T_R the freeze out temperature of the right-handed neutrinos. The experimental limit on ϑ obtained using Planck data is ϑ≲ 3.26, from which we get that the maximum right-handedneutrino density is <cit.> n_ν_+^j = n_(ν^j_-)^c≃ 36cm^-3.Since in this case we can have a larger right-handed neutrino population with respect to the thermal case, we expect larger modification in the capture rate. In the vector-axial-vector BSM case we find that the rate is increased between 40 and 90%, getting closer to the value expected for Majorana neutrinos in the SM. For the other three scenarios we found larger modifications. In the right-handed scalar-tensor case, the BSM capture rate has a maximum value of about 2.5Γ_^ D, while in the five-parameter case we obtain 3.5 Γ_^ D. We conclude noticing that, in all the cases in which a right-handed neutrino population (either thermal or non-thermal) is present, the increase in the number of neutrinos lead to an increase in the capture rate.§ CONCLUSIONS The detection of thewould be a milestone for both particlephysics and cosmology. Experiments using the neutrino capture in tritium are in development, so that the detection of themay become a reality in the near future.In this paper we have studied how the capture rate is modified if new interactions involving neutrinos are present. For definitiveness, we have focused on the interactions arising from generic BSM physics, including all the dimension-six operators that can modify the process ν + n → e + p. Once the experimental limits coming from low energy processes are considered, we have seen that for Majorana neutrinos the modifications to the capture rate are modest (of O(2 %)), while for Dirac neutrinos we can have much larger modifications, which can either increase or diminish the capture rate up to roughly a factor of two. Since in the SM case we expect the capture rate for Majorana neutrinos to be twice the one for Dirac neutrinos, we see that the measurement of the capture rate at future experiments will not be conclusive about the Majorana or Dirac nature of neutrinos. Another situation in which the observed neutrino capture rate can bedifferent from the standard one is the existence of a non negligiblecosmic population of right handed neutrinos. In this case the capturerate can either be left unaltered or increase (depending on the physicalorigin of the right handed population). This allows us to conclude that if a PTOLEMY-like experiment detects a capture rate smaller than the standard capture rate for Dirac neutrinos, it would unavoidably point to the presence of New Physics in the neutrino sector (since, as shown in section <ref>, the capture rate can be decreased in this case). If instead the measured capture rate is between the standard Dirac and Majorana case, or even abovethe standard Majorana case, the situation will not be clear, since the effect can be caused by Dirac neutrinos with either BSM interactions or an additional cosmological abundance of right-handed neutrinos. On the other hand, we have seen how important the right-chiral couplings are for the relic neutrino capture rate.Since the rate depends on ϵ_Rq when m_j/E_j is not negligible, a possible detection of thecan put stronger limits on the ϵ_Rq couplings that other low energy processes can not.Finally, we have also briefly discussed in appendix <ref> the problem of distinguishing the electron peaks generated by neutrino capture and β-decay. With an expected resolution of Δ=0.15 eV, the PTOLEMYexperiment will be able to detect only a single peak, corresponding to the captureof the three neutrino mass eigenstates. Assuming however two possible resolutions,Δ=0.01 eV (very aggressive) and Δ=0.001 eV (ultimate), we establisheda novel criteria to distinguish the electron peaks as a function of the separationbetween the experimental Gaussian distributions. The main result is that, given therange of neutrino parameters allowed by current oscillation experiments, the abilityto distinguishing the peaks depends crucially on the neutrino mass ordering, and evenfor the ultimatevalue Δ = 0.001 eV the three peaks could be only disentangledfor normal ordering. This result agrees with previous studies in theliterature <cit.>. This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Ciência e Tecnologia (CNPq). § BRIEF COMMENT ON THE NEUTRINO MASS ORDERING As we have already stressed, each neutrino mass eigenstate will produce an electron of energy given by eq. (<ref>) in a PTOLEMY-like experiment. A natural question is then whether each neutrino peak can be distinguished from the β-decay background and, if so, when each peak in the distribution can be distinguished from the peaks generated by the capture of the other neutrinos <cit.>. The answer depends cruciallynot only on the experimental resolution Δ, but also on the absolutevalue of the neutrino masses as well. In order to answer the above questions,we slightly modify eq. (<ref>) to consider the number of events due to the ν_j capture asN_^j(Δ) =∫_E_e^, j-Δ/2^E_e^, j+Δ/2dE_e dΓ_^ BSM(j)/dE_e,with E_e^, j given in eq. (<ref>). The criteria we use to distinguish the peaks from the background and between each other are the following: *we say that an electron peak due to neutrino capture can be distinguished from the β-decay background if r_^j ≡ N_^j(Δ)/√( N_β(Δ))≥ 5;*we count the number of distinguishable peaks according to the number of different values taken by the function Ξ^j_=∑_i=1^31-Θ(D_B(dΓ_^i/dE_e,dΓ_^j/dE_e)-4.5)Γ_^i , where D_B(p,q) is the Bhattacharya distance <cit.>, defined for two Gaussians distributions, p and q, as D_B(p,q)=1/4ln1/4σ_p^2/σ_q^2+σ_q^2/σ_p^2+2+1/4(μ_p-μ_q)^2/σ_p^2+σ_q^2. The value 4.5, which measures the separation between the peaks in the Θ function of eq. (<ref>), has been chosen because it corresponds to a distance of 6σ between the mean values of two Gaussians with σ_p=σ_q.The function Ξ^j_ of eq. (<ref>) has been constructed as follows: when the mass eigenstates are degenerate, the Bhattacharya distance vanishes and Ξ^j_ gives the total neutrino capture rate. Since Ξ^j_ takes a unique value for the three neutrino states, we have that only one peak will be seen experimentally. Meanwhile, if any eigenstate is separated enough to give a distance equal or larger than 6σ, the Ξ^j_ will correspond to the value of the capture rate for such mass eigenstate. Whether a PTOLEMY-like experiment will be able to distinguish between two or more neutrino capture peaks depends instead on the mass ordering and on the experimental resolution Δ. With the expected PTOLEMY resolution of Δ=0.15 eV, the Gaussian peaks for each electron will be too large to allow a distinction between the different contribution, so that a unique peak is expected. Nevertheless, we will try to understand how the electron peaks would look like for better experimental resolutions, which we take to be Δ=0.01 eV and Δ=0.001 eV. We show in figure <ref> how the Ξ^j_ function depend on the lightest neutrino mass m_0, for the mass eigenstates ν_1 (green), ν_2 (red) and ν_3 (blue). We consider both types of mass orderings and the two resolution already mentioned, Δ=0.01 eV and Δ=0.001 eV. We also scan over all the neutrino parameters at 3σ <cit.>. The gray points are those that can not be distinguished from the β-decay background. The upper left panel (Δ = 0.01 eV, normal ordering) should be interpreted as follows: for m_0 ≳ 3 × 10^-2 eV, the Ξ^j_ function takes only one value, so that only one peak would be measured, which corresponds to the capture of the three neutrinos. Since the peak is not gray, it can be distinguished from the β-decay background. For 8× 10^-3 eV ≲ m_0 ≲ 3× 10^-2 eV, two peaks could be measured, one due to the ν_3 capture (blue) and the other due to ν_1 and ν_2 (red/green). Finally, for m_0 ≲ 8× 10^-3 eV, only the ν_3 peak can be resolved, while the ν_1+ν_2 peak cannot be discriminated from the β-decay background. The other panels can be interpreted along the same reasoning. It is interesting to notice that there is only one situation in which the three peaks can be resolved, corresponding to the normal ordering for the extreme case Δ=0.001 eV. With the same resolution but inverted ordering, at most two peaks can be discriminated, since ν_1 and ν_2 tend to become degenerate as m_3 → 0.To better illustrate the interplay between the experimental resolution Δ and the importance of the neutrino mass ordering, we show in figure <ref> the expected spectra in a PTOLEMY-like experiment. In each plot we show normal (continuous line) and inverted (dashed line) ordering, for the two experimental resolutions we are discussing (a very agreessive Δ = 0.01 eV, upper panels, andan ultimate Δ = 0.001 eV, lower panels) and for some choices for the lightest neutrino mass. The gray line represents the β-decay background. This shows another potential problem in the peak detection; sinceΓ_^j∝U_ej^2,andU_ej^2 ≃{0.68, 0.3, 0.02 },the peak due to ν_3, although in principle distinguishable from the other peak(s), is much smaller, and will most probably be unresolved orunobservable in a real experiment.JHEP | http://arxiv.org/abs/1708.07841v2 | {
"authors": [
"Martín Arteaga",
"Enrico Bertuzzo",
"Yuber F. Perez-Gonzalez",
"Renata Zukanovich Funchal"
],
"categories": [
"hep-ph",
"astro-ph.CO"
],
"primary_category": "hep-ph",
"published": "20170825180003",
"title": "Impact of Beyond the Standard Model Physics in the Detection of the Cosmic Neutrino Background"
} |
a_HArg φθ_a ρ_dm (m_a/m)^2_max θ_A,ext θ^ str_A t_start k_start t_cut r_min r_max κ_min k_max z̅_2 α_WS a_H0 1/2 𝒪 ℒΛ_QCD#1Eq. (<ref>) ReIm n_axn_misalignatanInstitut für Kernphysik, Technische Universität DarmstadtSchlossgartenstraße 2, D-64289 Darmstadt, [email protected],[email protected] We evaluate the efficiency of axion production from spatially random initial conditions in the axion field, so a network of axionic strings is present.For the first time, we perform numerical simulations which fully account for the large short-distance contributions to the axionic string tension, and the resulting dense network of high-tension axionic strings.We find nevertheless that the total axion production is somewhat less efficient than in the angle-averaged misalignment case.Combining our results with a recent determination of the hot QCD topological susceptibility <cit.>, we find that if the axion makes up all of the dark matter, then the axion mass is m_a = 26.2± 3.4 μeV. The dark-matter axion mass Vincent B. Klaer, Guy D. Moore December 30, 2023 ==================================§ INTRODUCTION The QCD axion <cit.> is a hypothetical particle, predicted in models which solve the QCD theta problem <cit.> via the Peccei-Quinn mechanism <cit.>. In the simplest models <cit.> it is the angular mode of a complex scalar, √(2) φ = f e^i or =, which would be the Goldstone boson of a spontaneously broken U(1) symmetry, except that the symmetry is also anomalous (explicitly broken by QCD).Therefore QCD effects induce a small, temperature-dependent “tilt” in the potential, and therefore make the angular fluctuations massive, so the axion mass is m_a ≠ 0. The axion is a dark matter candidate <cit.> because early Universe dynamics generically generate large coherent oscillations in the axion field – essentially a Bose-Einstein condensate of axions at or near rest – which act as a pressureless fluid on scales longer than a few meters.In this paper we will predict the axion's mass, given the following hypotheses:*the axion exists;*PQ symmetry is restored either during or at some point after inflation, so that the axion field starts out “random,” meaning that its value at points out of post-inflationary causal contact are uncorrelated <cit.>;*The cosmological epoch where axions are produced – roughly, temperatures around 1 GeV– follows standard FRW behavior with the expected standard-model matter content;*the axion makes up 100% of the dark matter, so its current energy density is set by measurements of Ω_dm h^2; /s = 0.39eV with s the entropy density <cit.>.These assumptions give rise to a rich dynamics, with a network of axionic cosmic strings <cit.> which collapses once the axion mass becomes large in units of the system age, m_a t ≫ 1, through the action of axionic domain walls <cit.>, leaving a final state with small-amplitude axionic fluctuations which evolve adiabatically thereafter. Under our assumptions, a mass prediction should be possible, because the model has one principal free parameter, [ There are two other relevant parameters.There is the number N_a of minima around the U(1) circle, cos→cos N_a. But if N_a ≠ 1 then the model predicts stable domain walls which are a cosmological disaster <cit.>. Also there is the mass of the radial excitation in the complex scalar field, m.This must be heavy, and we find below that the results are quite weakly dependent on its exact value.] the axion “decay constant” (the vacuum expectation value breaking the U(1) symmetry) f_a (defined below in Lflat).This parameter determines the axion mass, see def_ma.And given our other assumptions, it also sets the dark matter density.By computing the relation between f_a and the dark matter density, we should then be able to predict f_a and therefore m_a.Such a prediction is valuable because it informs experiment, and because if the axion is then discovered at this mass, it will clarify its role as the dark matter.In the next section, we lay out our methodology for relating f_a to the axionic dark matter density.Section <ref> presents our numerical results.We end with a discussion.A few technical issues and numerical tests are postponed to an appendix. But for the impatient reader, we present our main results here. While there have been numerous previous studies of this problem <cit.>, ours is the first which includes the physics of the large tension associated with axionic strings.This large tension leads to a much higher density of strings which are more robust and survive longer than in previous simulations.However, this makes surprisingly little difference in the final axion number produced. Let us set as a baseline for axion production, the angle-averaged misalignment value of the axion density. This is the axion density value we would find if the axion field starts out uniform in space with value , averaged over ∈ [-π,π] (without the approximation, sometimes made, of replacing 1-cos→^2/2 in the potential) [ Specifically, we write the conformal-time axion mass squared as m_a^2=t^n+2/t_*^n+4 (t, n, and t_* defined in the next section) and evolve θ(t) according to d^2θ/dt^2+(2/t)dθ/dt = - m_a^2 sinθ to a time t > 4t_*.The axion number density at time t is = (m θ^2 + θ̇^2/m)/2.We average over starting values θ∈ [-π,π] and use the resultingaverage to normalize the result of a string simulation at the same t/t_* value.]. At a given f_a value, we find the axion number density produced in the inhomogeneous case is actually smaller than the misalignment value, by a factor of about 0.78.Sinceincreases with increasing f_a, this inefficiency must be compensated by a larger value of f_a, and hence a smaller value of m_a, than has generally been assumed; we find m_a ≃ 26.2 ± 3.4μeV.We postpone discussion of this result and its errors to the conclusions.§ METHODOLOGY Our approach will be as follows.The Lagrangian for the axion field is [ We use [-+++] metric convention, and standard complex-field normalization φ = (φ_r+iφ_i)/√(2).] - =g^μν∂_μφ^* ∂_νφ + m^2/8f_a^2( 2 φ^* φ - f_a^2 )^2 + χ(T) ( 1 - cos).The middle term is the symmetry breaking Lagrangian, and the last term is the “tilt” in the potential due to QCD effects.This tilt gives rise to an axion mass ofm_a^2(T) = χ(T)/f_a^2 ,m_a^2(T=0) =χ(T=0)/f_a^2 .In a radiation dominated FRW universe, in comoving coordinates and conformal time, the metric is g_μν = t^2 η_μν and the temperature is T ∝ t^-1, so-√(g) _conf = t^2 ( -φ̇^* φ̇+ ∇φ^* ·∇φ + t^2 m^2/8f_a^2( 2φ^*φ - f_a^2 )^2 + t^2 χ(t) ( 1 - cos) ) .Here χ(T(t)) is the topological susceptibility.Model calculations <cit.> and a recent lattice calculation <cit.> indicate that χ(T) is approximately power law between 1.5 GeV and 400 MeV, which, we will see, is wider than the relevant temperature range we need.Therefore we will treat χ(T) as a power law, χ(T) ∝ T^-n, so t^2 χ(t) = f_a^2 t^n+2/t_*^n+4, with t_* the natural scale where the susceptibility begins to influence the dynamics; t_* m_a,conf(t_*) = 1 where m_a,conf = √(t^2χ(t)) / f_a is the conformal-time axion mass.In terms of physical time, t_* is the moment when m_a H=1.In the following we will suppress the subscript and write m_a,conf=m_a, except in the discussion.That is, masses and times will always be expressed in conformal units.To initialize the network, we choose an independent random phase at every lattice site.We then evolve the fields for an initial time under strong damping (φ̈+2φ̇/t →φ̈+ φ̇/t for times t <) to prepare a string network relatively close to the scaling network density.The length and strength of damping is chosen such that the string network will roughly match on to the scaling network density; we will also study the dependence on the initial conditions below.The model has two sorts of metastable defects, strings and domain walls.A string is identified as a linear structure wherechanges by 2π in circling the string.A domain wall is a surface on which = π; each string has one such domain wall ending on it.The domain walls only become distinct structures once m_a t ≫ 1; the surface tension of an isolated domain wall is σ = 8m_a f_a^2, which grows rapidly with time.Therefore the domain walls straighten out and pull the strings together, annihilating both networks and leaving small fluctuations in thefield.We evolve all fields until this dynamics is complete and there are no strings left.Then we count the axion abundance by extractingand d/dt from the simulation and applying the method of <cit.> to determine the axion content.This determines the total density of axions from all sources – we make no attempt to distinguish which axions arise from strings, from walls, or from misalignment, as we do not believe such a distinction can be made unambiguously.We express the axion number produced as a ratio to the angle-averaged misalignment value and we determine f_a such that the dark matter abundance is correct.[ We implement the misalignment case in the same code by turning off the scalar gradient terms.We also implemented misalignment in a simple dedicated code as a cross-check.]The scale m for “radial” excitations in the φ field is may be as large as m ∼ f_a ∼ 10^11GeV and must be at least 10^3GeV (see Subsection <ref>), while the relevant length scale is H at the QCD epoch (around 1 GeV temperature), which is of order 10^-18GeV.We handle this huge scale hierarchy by observing that the only important physics it gives rise to is very thin, high-tension axionic cosmic strings. Specifically, the string tension should be T_str≃κπ f_a^2 with κ≡ln(m/H) ∈ [50,70]. We address this physics via the technique we recently introduced <cit.>.Specifically, we add abelian-Higgs degrees of freedom which are massive away from string cores but which induce a large string tension.In the Appendix <ref> we review the procedure and explain how we implement χ(T) (1-cos) into this method.The outcome is that the string tension is maintained by some extra, massive degrees of freedom, but the mass scale m for these degrees of freedom must be resolved by the lattice, ma ≲ 1 with a the lattice spacing.The correct physical limit involves this scale becoming heavy compared to the physics of IR fluctuations in the axion field.Since the correct physical picture arises when m is large, we will hold ma fixed, that is, we keep m fixed in lattice units, throughout a simulation.In our implementation, the extra degrees of freedom introduce one new parameter q_1, which determines the value added to κ; most of our results are labeled (q_1,q_2)=(4,3), which means that κ has been increased by 50 through the added degrees of freedom <cit.>.§ NUMERICAL RESULTS In Appendix <ref> we show that our results are in the large volume limit if we keep L/t_* ≥ 5, and we show that the axion number does not evolve after the string network is gone and can be measured as soon as no string is left, without concern that it will evolve further.Therefore box size and axion number measurement do not contribute to our error budget.Here we will instead focus on those effects which still do. §.§ Lattice spacing For axionic strings to evolve correctly, the string core must be resolved by our lattice spacing.We need to check that our lattice is fine enough, in the sense of ma the product of the heavy scale and the lattice spacing is sufficiently small.That said, numerical cost scales as (ma)^-4 and required RAM scales as (ma)^-3, so we want the largest value which we can get away with.To test the (ma) dependence, we fix all other parameters in terms of m, and we consider axion production at various ma values in Figure <ref>.Because we have used an improved action, the result should naively converge in the small ma limit with corrections vanishing as (ma)^4, motivating the axis choice in the left-hand plot.However, the right-hand plot shows that the data fit better assuming (ma)^2 dependence.Indeed the χ^2 for an (ma)^2 fit is about 1, while for an (ma)^4 fit, χ^2=15.Therefore we will assume that the errors are quadratic in spacing, despite our improved action.The fit indicates an upwards correction between (ma)=1 and the continuum limit (ma)=0 of 10± 1%.In the rest of this study we will use ma=1 and correct the final results upwards by 10%.It remains to explain why the axion production scales with (ma)^2 despite our improved action, which should give (ma)^4 convergence if fields are smooth.We believe this occurs because a small fraction of string has a velocity close to 1, and therefore a large Lorentz contraction factor.If the (energy-weighted) fraction of string with velocity-squared v^2 > v^2_0 only vanishes linearly in the v^2_0→ 1 limit, then the fraction of string with γ^-2 < ϵ would then scale linearly in ϵ. Such scaling is consistent with our measured string velocity distribution.It also makes sense from the string equations of motion. In flat space, with χ=0, and in the Nambu-Goto limit, labeling the string location as x_i(σ,t) with σ an affine parameter along the string, one can make a gauge choice such that ẋ_i x_i'=0 and such that √(x_i' x_i'/(1 - ẋ_j ẋ_j))=1. The equation of motion is thenẍ_i = x_i”which is solved by <cit.> x_i' = α_i(σ+t)+β_i(σ-t)/2 ,ẋ_i =α_i(σ+t)-β_i(σ-t)/2 ,α_i(r) α_i(r) = 1 = β_i(r) β_i(r) ∀ r.Here α,β are backwards and forwards propagating waves which take values on the unit sphere.Even in curved space we may satisfy the gauge choice instantaneously.The relevant question for the distribution of string velocities is the distribution of angles between α and β, since v^2=ẋ^2 = (1-α·β)/2.The measure of α·β values is uniform in [-1,1] because α,β take values on the unit sphere.While we do not expect the distribution of α·β values to be uniform in [-1,1], neither do we have a reason why it should avoid α·β=-1, so the probability distribution should not vanish at α·β=-1, and therefore the fraction of string with γ^-2 < ϵ should indeed vanish linearly in small ϵ.Consider ϵ = (ma)^2, corresponding to a gamma-factor of γ > 1/ma and therefore a Lorentz contracted string thickness [ In this parametric argument we are neglecting order-1 factors which make the string somewhat thicker than 1/m and mean that, for ma=1, most string is actually properly treated.] of γ^-1(1/m) < a.Our hypothesis for string velocities then states that an ((ma)^2) fraction of string should be Lorentz contracted to a thickness of less than 1 lattice spacing. Such string is mistreated regardless of how improved our update algorithm is.Therefore, even if typical string is treated correctly with ((ma)^4) errors, the fraction which is mistreated is of order (ma)^2.This allows (ma)^2 scaling corrections, regardless of the level of lattice action improvement. §.§ Initial network density We want the axion production from a string network which is initially in the scaling regime.But this cannot be exactly achieved; initial conditions will typically produce a network which is either denser or less dense than scaling.The network evolves towards scaling, and if mt_* is large enough then initial conditions should have little effect.But it would still be good to check how sensitive the final axion number is to the starting conditions.We address this in Figure <ref>.We introduce the scaled network densityξ = t^2 ∫_all stringγ dl/4V_space ,with γ the local gamma-factor of the string so that the integral represents the total invariant length of string (length scaled by a γ factor to account for the energy content), and with V_space the volume of the simulation.This combination should approach a fixed “scaling” value as t increases (for χ(T)=0, that is, in the absence of potential tilt).We measure ξ once early in an evolution and again later in the evolution, just before the walls start to influence the string network evolution.We perform several evolutions with initial conditions with more or less damping, leading to denser or rarer initial networks.The ratio of the starting to final ξ, ξ_init/ξ_late, then indicates whether the network started too thin or too dense, and therefore from which side it is approaching the scaling solution.We find a roughly linear correlation between this starting density and axion production, with more axions arising from denser starting networks.However the dependence is quite weak.Based on the figure, we will try to use initial conditions with this ξ ratio close to 1, and we will assign a 5% systematic error based on incomplete network scaling. §.§ Thin-core limit Next we must consider the effects of finite mt_*, meaning that the strings are of finite thickness.This is clearly an artifact because in the physical case there is a hierarchy of many orders of magnitude between string thickness and axion mass.We have incorporated the logarithmic sensitivity to this hierarchy by implementing auxiliary fields to give rise to the resulting high string tension.But there can still be effects suppressed by powers of 1/(mt_*), probably starting at first order.In particular, strings may lose energy via the radiation of unphysical massive modes.We only expect such radiation from short length-scale structures on the strings, which should generally get smoothed out by axion emission so long as mt_* ≫κandm_a/m ≪ 1. However, because m_a grows as a large power of conformal time, m_a ∝ t^1+n/2, the latter condition may not be maintained, given the persistence of high-tension strings. And if the axion mass m_a comes of order the heavy-mode mass m then one might expect that axion production is lost to heavy-mode production, and the simulation could result in an underestimate of axion production. [ In previous work <cit.> we showed that, for a theory of a pure scalar field, the axionic domain walls spontaneously decompose as soon as (m_a/m)^2 > 1/39.The added degrees of freedom in our string cores prevent this physics from occurring; the domain walls remain strongly metastable up to and past m_a/m = 1.]We can “fix” this problem by artificially capping the value of m_a, so that rather than growing with time at all times, t^2 χ(t) grows up to some value and then becomes constant.But this replaces one unphysical behavior with another, and it will introduce new artifacts.The axionic wall tension is proportional to m_a.These walls cause the network to collapse, and limiting their tension artificially extends the life of the network. We show this effect in Figure <ref>. The figure shows how the total length of strings, rescaled as in def_xi, and a similar rescaled wall area (without γ factor or the conventional factor of 4),ξ_wall≡t ∫_all wall d^2Σ/V_space ,evolve with time under the influence of various choices for a maximal m_a/m value.We see that the wall area starts to decline as the wall surface tension turns on around t=1.6 t_*, and later around t=2.8t_* the surface tension becomes large enough to influence the string network evolution, drawing together the strings and collapsing the network by t=4t_*. However, artificially limiting the axion mass slows down the collapse of the network; for the smallest value we considered, the last bits of string survive almost to t=5t_*.We also expect that the network with a maximal m_a/m value will produce more axions than without such a cutoff. [ Something special happens if we choose a maximum value for m_a which is very close to m/2.In this case, there is a resonant nonlinear mode-coupling process which converts mass m/2 axions into mass-m excitations, leading to a reduction in the axion production for values of (m_a/m)_max very close to 0.5. This effect is clearly an artifact, so we avoid this special value.] The reason is that, as m_a increases, the energy stored in the string network becomes less and less useful for producing axions.While increasing m_a increases the energy in axion fluctuations and in domain walls in proportion to m_a, it does not change the energy in strings. Therefore, as m_a increases, the capacity for strings to make axions is diluted; since making an axion costs energy m_a, an energy E can only produce E/m_a axions.Limiting m_a turns off this dilution, allowing the string energy to produce more axions, and could therefore result in overproduction of axions.The issue should disappear if we can reach a large enough value of mt_*.But it is useful to consider different cutoff values for m_a/m and take the large mt_* limit for each.If the continuum limits are the same, then it lends credence to the belief that we have achieved the continuum limit.According to Figure <ref>, The difference between different cutoff choices falls below 10% starting around mt_*=300 (third-from-leftmost points).On the two still-finer lattices, the choices =0.5 and =0.2 agree to within 3%.So these values can be close to the continuum limit.Note that the last point in the figure, with mt_*=625, was achieved by loosening ma from ma=1.0 to ma=1.25 and using the result of Figure <ref> to extrapolate it to the same value as the other points; it is also at a slightly smaller physical volume, Lt_*=4.1 rather than 5.1.The error bars shown in the figure are statistical only.However the statistical errors for points with the same mt_* value but differentvalues are strongly correlated, since they are calculated from simulations which are identical up to the point when m_a reaches the smaller upper-bound value. Therefore the determination of the difference between differentchoices has smaller errors.In particular this difference is not linear in 1/mt_*, but drops to a small value at a sufficient m_a,maxt_* value. That complicates the continuum limit.Here we will perform a linear extrapolation of the three smallest 1/mt_* data points, each for =0.2 and 0.5.We find 0.696(46) and 0.729(41) respectively.The fact that these answers are not the same indicates that our lattices are not yet abundantly fine.We assign a 10% error bar for the continuum-extrapolated value, to include these systematic issues, adopting / = 0.71(7). This is for n=7, κ=50 from extra degrees of freedom, and before performing the small ma extrapolation. §.§ String tension and temperature-dependent susceptibility Having discussed numerical artifacts, we now turn to actual physical parameters which are relevant but not completely known:the string tension κ and the strength of the temperature dependence n in χ(T) ∝ T^-n. The slope n is calculable in lattice QCD.Recently Borsanyi et al have presented <cit.> results up to and beyond the relevant temperature range.Using their results at 600 and 1200 MeV, we estimate n=7.6 ± 0.5.Most groups find results which agree with Borsanyi et al at lower temperatures <cit.>, although no group has reproduced these higher temperatures and even below 600 MeV there are some results which appear discrepant <cit.>. Therefore we will explore other n values but consider values near n=7.5 to be likely correct.We also feel that we gain some physical insight by considering different n values, especially much smaller values.We do this in Figure <ref>. The figure shows that small n values lead to more axions than in the misalignment mechanism, while large n values lead to less.But between n=7 and n=8 the dependence is not very strong.Therefore our choice to use n=7 elsewhere, which we made mostly for simplicity, does not appear to be very critical.We interpret the results of Figure <ref> as follows.The larger the n-value, the more rapidly the axion mass m_a turns on, and therefore the heavier the axion is when the string network breaks up and loses its energy.That means that for small n, the network can still produce relatively many of the relatively-light axions, but for large n the axions quickly become heavy and the string energy cannot produce a large number of them.This is consistent with what we saw whenwas small.Indeed, the results at n=2 had (m_a/m)^2=0.07 at the time the string network had completely disappeared, so walls broke up and axion production occurred when axions were still relatively light.Finally we consider the κ value.Above we define κ as κ = ln(m/H).For us H = 1/t the inverse system age. Therefore the contribution from axionic modes to κ is ln(mt), which we approximate to its value at t=3t_* since this is when the string network is breaking up.In addition there is a contribution from the extra massive degrees of freedom we have added, so our simulations haveκ = ln(3mt_*) + 2(q_1^2 + q_2^2),where the charges (q_1,q_2) are explained in Appendix <ref>.We do not know what the physical value of κ should be, because we don't know the model-dependent microscopic origin of the axion field.In the single complex-field case <cit.> we don't know the radial mass m; if the axion is a composite or arises from more complicated physics <cit.>, we do not know the compositeness scale and whether there is an extra contribution to the string tension from the microscopic physics giving rise to the axion field.We can reasonably guess that m < f_a ≃ 2× 10^11 GeV.Also the requirement that the radial excitations decay by the time the Universe reaches a temperature of 1 GeV, along with an estimate for their decay rate <cit.>, Γ_m ∼α_s^2/64π^3m^3/f_a^2, sets very roughly m > 10^3 GeV.These limits correspond to approximately κ∈ [48,67].For mt_*=300 and q_1=4 the κ value in the simulation is 50+ln(900)=57, which is in this range.By considering other values of (q_1,q_2), we achieve κ values larger and smaller than the physically interesting range.We show results for the axion production as a function of κ in Figure <ref>.The figure shows that higher tension strings give a significantly denser string network, with strings which break up later, but nevertheless produces only mildly more axions (note the false zero for the y-axis in the right plot). Therefore our ignorance of the physical value of m,κ is not very significant in bracketing the physical value of the axion.A simple linear fit to the left frame in Figure <ref>, and the range we quoted above for κ, gives a systematic error of ± 3% due to the unknown value of κ.Note that the chosen initial conditions for both our n dependence and our κ dependence studies produced somewhat underdense networks.One can see in the left frame of Figure <ref> that the underdensity is worse for the highest tensions, so the true κ dependence is somewhat underestimated.Also note that the larger κ values are farther from the large mt_* limit, leading to a slight overestimate in the produced axions due toeffects.§ DISCUSSION If we take the temperature dependence of the topological susceptibility to scale as χ(T) ∝ T^-n with n=7.6 in the relevant temperature range <cit.>, and assume that axionic string cores arise at a mass scale m ∼ 10^7 GeV so the extra string tension is κ = 58 ± 10, then our results indicate an axion production efficiency which is 0.78(12) times as efficient as in the angle-averaged misalignment mechanism. The indicated error is dominated by the extrapolation to the large mt_* limit, with the uncertainty due to κ added linearly (not in quadrature).Now we use this result to calculate the axion mass.There has been relatively little entropy production since the Universe was 1 GeV in temperature, so the ratio of axion number-density to entropy density is approximately the same at the end of the axion-production epoch as it is now.We can express our results by saying that the axion number density, determined at t=4t_* or T=T_*/4 and then back-extrapolated to the temperature T_* where m(T_*) H(T_*)=1, was(T=T_*) ≃ K H(T_*) f_a^2,where a numerical evaluation finds that the angle-averaged misalignment value for K is K=16.61, and our result is K=13.0 ± 2.0. By dividing this by the entropy density at that temperature, s=2g_* π^2 T_*^3/45, we get the modern axion-number to entropy ratio, which can be multiplied by the vacuum axion mass m_a = √(χ(T=0))/f_a to give the modern dark matter density to entropy density ratio. We combine this with the Planck result <cit.>,n_b/s ≃ 8.59× 10^-11 ,/s= Ω_dm h^2/Ω_b h^2m_p n_b/s≃0.1194/0.0221(938 MeV) (8.59× 10^-11) ≃ 0.39eV,thermal QCD results for the entropy density s and energy density ε of the thermal plasma from Borsanyi et al<cit.>,ε(T) = π^2 T^4 g_*/30 ,s(T) = 2π^2 T^3 g_*/45 ,g_*(1 GeV) ≃ 73,χ(T)≃( 1 GeV/T)^7.6 (1.02(35)× 10^-11 GeV^4),Hubble's law H^2 = 8π ε/(3 m_pl^2) with m_pl≃ 1.22× 10^19 GeV, the thermal value for the axion mass m_a^2(T) = χ(T)/f_a^2, and vacuum value χ(T=0)=(0.076 GeV)^4<cit.>, to obtainf_a = (2.21 ± 0.29)× 10^11 GeV, m_a = 26.2 ± 3.4μeV , T_* = 1.54 ± 0.05GeV .Taking the errors quoted in Ref.<cit.> at face value, the dominant error in f_a and hence in m_a is from our determination of K, while the error in T_* arises equally from the errors in K and in χ(T).fa-result, ma-result, and T*-result constitute the main results of our study.Our most striking result is that the axion production from random initial conditions, with the resulting dense and high-tension axionic string network, is actually smaller than the angle-averaged misalignment value.The deficit gets larger at large n, where thepotential tilts more abruptly; if it tilts more gradually then the axion production exceeds the misalignment value.Furthermore, although axion production is larger from high-tension strings than from strings with a lower tension, the dependence is quite weak; a factor of 10 increase in string tension between our results and the results of <cit.>, along with the resulting factor of 3 increase in the string network density, has led to less than a 30% increase in axion production.This clearly requires some explanation.The conventional wisdom has been (see for instance <cit.>) that axions are produced by misalignment in the space between walls, by walls, and by strings.Therefore the production is the sum of three terms, and must be larger than the misalignment contribution.We argue that this picture involves assumptions and commits double counting.It does not make sense to consider misalignment axions to be independent from walls.Within the misalignment mechanism, half of all axions emerge from the range of angles |(t=0)| ∈ [2.76,π].But it is precisely the regions with ∼π which become the domain walls.Much or most of the “misalignment” axion field energy becomes the domain walls; it is double counting to speak of both domain-wall axions and misalignment axions as independent contributions.Of course, since the axion field is initially very inhomogeneous, it is also not obvious that there are any spacetime regions where homogeneous misalignment is a useful description. Consider also what happens to the energy in domain walls.After the potential tilts and the domain walls become relatively thin and distinct, the wall surface tension induces forces on the strings.The walls lose their energy to accelerating the strings, which consumes the wall area (see Figure <ref>).Also in this epoch, it is not simple for walls or strings to emit axions.The axion frequency m_a increases with time, and any process involving time scales longer than m_a^-1 has a frequency-mismatch problem to produce massive axions.That is, long wavelength fluctuations of walls or strings are incapable of producing axions because they drive the axion field at frequencies below m_a. We saw this very clearly in our previous study of 2+1D axion production with massive strings <cit.>.What about the energy of the string network?The high string tension means that the network stores much more energy.But after the time scale t_*, the energy in domain walls and in axionic fluctuations increases with the axion mass as E ∝ m_a ∝ t^1+n/2, while the string energy does not increase as m_a increases.Therefore the string network's ability to produce axions dilutes with time.The network only annihilates when the walls are able to influence string dynamics, which occurs when the wall energy is comparable to the string energy.That is, the strings only fragment when their energy is comparable to the energy that was present in the wall network which caused them to fragment.And there is still the question of how efficiently the resulting small loops turn their energy into axions.To improve this analysis, we see a few directions which need to be pursued.First, we need simulations with more RAM, so that larger boxes, and therefore larger mt_* values, can be studied.We need to be more systematic in setting the initial network density and understanding the approach to network scaling.It should be straightforward to reduce statistical and extrapolation errors to the few percent level, with the dedication of more computer power.Also, we would like to investigate some of the late network evolution in more microscopic detail.The string network breaks up into loops which then annihilate in a way which somehow does not produce many axions. It should be possible to cut such loops out of a simulation and resolve them with a much finer lattice, which can then properly separate the m_a and m mass scales and follow the loop dynamics down to short scales.This could help explain why so few axions are produced (or determine whether our limited lattice spacing is causing a systematic neglect of some relevant but shorter-distance physics).§ ACKNOWLEDGMENTS We would like to thank Mark Hindmarsh for very useful conversations, and Daniel Robaina, Thomas Jahn, and Max Eller.We also thank the GSI Helmholtzzentrum and the TU Darmstadt and its Institut für Kernphysik for supporting this research.§ ALGORITHMIC DETAILS Here we explain in more detail how our numerics work.Following Ref. <cit.>, we embed the axion field as a global U(1) symmetry of a theory with two scalars φ_1, φ_2 and one U(1) gauge symmetry, so one linear combination of the U(1)×U(1) symmetry is gauged and one is global.Both are spontaneously broken by the scalar vacuum values:- (φ_1,φ_2,A_μ) = 1/4e^2 F_μν F^μν + | (∂_μ -i q_1 A_μ) φ_1 |^2 + | (∂_μ -i q_2 A_μ) φ_2 |^2 = + m_1^2/8 v_1^2( 2φ_1^* φ_1 - v_1^2 )^2 + m_2^2/8 v_2^2( 2φ_2^* φ_2 - v_2^2 )^2 .Here q_1, q_2 are the field charges with q_2=q_1-1. The axion is the angle= q_2 _1 - q_1 _2which is gauge-invariant. This procedure exactly reproduces the global symmetry and the way strings act as a source for the axion field; the dynamics are modified only by a large induced string tension, κ≃ 2(q_1^2+q_2^2), and heavy degrees of freedom which should decouple from the dynamics in the continuum limit (in the sense of mt ≫ 1, or mt_* ≫ 1 for our current purposes)<cit.>. We consider v_1=v_2 and m^2_1=m^2_2 = e^2(q_1^2 v_1^2 + q_2^2 v_2^2)=m_e^2 so all heavy fields have a common mass.The axion decay constant is f_a^2 = v_1^2 v_2^2/(v_1^2 q_1^2 + v_2^2 q_2^2).In our lattice units we normalize our fields such that v_1=1=v_2.The topological susceptibility part of the potential (which breaks the global U(1) symmetry, “tilting” the potential for the axion field), is implemented ast^2 χ(t) ( 1 - cos Arg )⇒f_a^2 t^n+2/t_*^n+4 F(2φ_1^*φ_1) F(2φ_2^* φ_2) ( 1 - cos( q_2 _1 - q_1 _2 ) )F(r)≡{[ 25/16 r ( 8/5 - r ) ,r< 4/5 ,; 1 , r > 4/5 . ]. The function F(r) is inserted to soften the behavior of the susceptibility term in string cores; without this term the introduced potential becomes violently nondifferentiable wherever φ^* φ∼ 0 for either field, which causes problems for space-discretized equations of motion.The modification only changes the tilted potential inside string cores, where its effect is very subdominant to the leading potential terms.But without this modification we do not get consistently stable evolution near string cores.Our results are insensitive to the specific form of F(r), provided F(1)=1, F'(1)=0, F(0)=0, and F'(r) is continuous, which motivated our choice.A similar modification is common in single-scalar simulations of axionic strings, where most authors <cit.> have made the substitution (1-cos) →√(2) Re φ, a substitution which is correct only for the angular dynamics and only where 2φ^* φ = 1.This replacement is justified because it is simpler, is nonsingular at 0, and is only of much influence outside string cores, where it is nearly equivalent to the correct form.We have explicitly checked that in the single-scalar model, axion production and string dynamics are nearly indistinguishable whether we use (1-cos) or √(2) Re φ as the “tilt” in the potential.Our numerical implementation uses a standard leapfrog algorithm and the noncompact formulation of U(1).The only novel feature is that we use an a^2-improved action for both the scalar and gauge parts, which requires a somewhat nontrivial treatment of electric fields in which the link's canonical momentum is not the same as the link's time derivative <cit.>.More details and tests are in <cit.>.§ OTHER NUMERICAL TESTS Here we detail some tests which have little bearing on the extrapolation to a final result, and which we have therefore not put in the main development.In the main text we spend some effort considering when to stop the growth of χ(T).But we do not discuss when to measure the axion number, arguing only that it is sufficient to measure after the string network is gone and only small fluctuations remain.Here we justify this claim.Figure <ref> shows what happens when we measure the axion number before the string network has finished collapsing.The figure shows the density of strings in blue, and the density of axions, as measured at the indicated time, in black.This measurement is somewhat ambiguous because it involves identifying the axion angle ∈ [-π,π] which is discontinuous across domain walls.Such a discontinuity leads to “ringing” in the Fourier spectrum and formally gives a logarithmically UV divergent particle number (cut off by lattice effects). We “fix” this problem by truncating the largestvalues, reflecting ∈ [π/2,π] to →π -, and similarly for ∈ [-π,-π/2]. Despite this “cap” on the maximum size of || < π/2, we nevertheless find a very large axion density if we measure axions before the network has decayed.However, we see that after the strings are gone, the axion number becomes completely independent of further time evolution.Our “cap” on largevalues has no effect on this final plateau, because || > π/2 virtually never happens and represents a tiny fraction of the axion number. In light of this result, we generally measureas soon as no strings remain, but when we evolve for longer and remeasure later, we get an answer which agrees at the 1% level. If we repeat this analysis for the misalignment scenario, we find that instead of becoming virtually t-independent at t=4t_*, the axion number becomes virtually t-independent already by t=2t_*.This difference reflects the absence of topological structures in the misalignment scenario.The other test which proves to play almost no role in the final axion density is the box volume.To test out to very large and quite small volumes, we used the rather small value of mt_*=200.Keeping everything else besides the volume unchanged, we find in Figure <ref> that the volume has less than a <2% effect on the axion abundance down to a box length of Lt_* = 2.Note that any box larger than Lt_*=8 should have essentially zero volume dependence, since the box periodicity is invisible for Lt>2 and the axion number becomes an adiabatic invariant by t=4t_*.The smallest volume shown, L=1.28t_*, shows a larger generated axion number, with larger statistical fluctuations (we used more simulations for smaller volumes so the product of simulations and volumes is about the same for each data point).The reason is that, in a small fraction of small-L simulations, after all strings annihilate, there remains a domain wall stretching across the whole box.This domain wall is metastable and lasts indefinitely, until it dominates the axion number.This is purely a small volume artifact; nothing of the sort ever occurs for the larger volumes.Because the volume dependence is so mild, it should be possible to study the axion production in boxes down to Lt_* =2 or 3. However, to be conservative, we have generally tried to keep Lt_* ≥ 5.Recall that there cannot be lattice volume dependence for L > 2t, and we need t=4t_*, so the box size dependence should be exactly zero for Lt_* > 8.The extremely weak box size sensitivity allows us to relax this value somewhat.unsrt | http://arxiv.org/abs/1708.07521v1 | {
"authors": [
"Vincent B. Klaer",
"Guy D. Moore"
],
"categories": [
"hep-ph",
"astro-ph.CO",
"hep-lat"
],
"primary_category": "hep-ph",
"published": "20170824182840",
"title": "The dark-matter axion mass"
} |
0012017 2017 XX 999 XX Special Astrophysical Obsevatory, Nizhnii Arkhyz, Russia Saint-Petersburg University, Saint-Petersburg, RussiaLeibniz-Institut für Astrophysik Potsdam (AIP), Potsdam, GermanyEuropean Southern Observatory, Garching, GermanyKazan Federal University, Kazan, RussiaMain (Pulkovo) Astronomical Observatory, Saint-Petersburg, Russiaevolution of magnetic field of OBA stars A.S.Medvedev et al. Special Astrophysical Observatory, Nizhnii Arkhyz, Russia Saint-Petersburg University, Russia Leibniz-Institut für Astrophysik Potsdam (AIP), Potsdam, Germany European Southern Observatory, Garching, GermanyKazan Federal University, Kazan, Russia Main (Pulkovo) Observatory, Saint-Petersburg, Russia 2017 2017 later We review the measurements of magnetic fields of OBA stars. Based on these data we confirm thatmagnetic fields are distributed according to a lognormal law with a mean logB=-0.5 (B in kG) with a standarddeviation σ=0.5. The shape of the magnetic field distribution is similar to that for neutronstars. This finding is in favor of the hypothesis that the magnetic field of a neutron star isdetermined mainly by the magnetic field of its predecessor, the massive OB star. Further, we modelthe evolution of an ensemble of magnetic massive stars in the Galaxy. We use our own populationsynthesis code to obtain the distribution of stellar radii, ages, masses, temperatures, effectivemagnetic fields, and magnetic fluxes from the pre-main sequence via zero age main sequence(ZAMS) up to the terminal age main sequence stages. A comparison of the obtained in our modelmagnetic field distribution (MFD) with thatobtained from the recent measurements of the stellarmagnetic field allows us to conclude that the evolution of magnetic fields of massive stars is slowif not absent. The shape of the real MFD shows no indications of the magnetic desert proposedpreviously. Based on this finding we argue that the observed fraction of magnetic stars is determined by physical conditions at the PMS stage of stellar evolution.Statistics of magnetic field measurements in OBA stars and the evolution of their magnetic fields A. S. Medvedev1A. S.Medvedev: [email protected]. F. Kholtygin2A. F.Kholtygin: [email protected]. Hubrig3M. Schöller4S. Fabrika1,5G. G. Valyavin1G. A. Chountonov1 Yu. V. Milanova2O. A. Tsiopa6V. A. Yakovleva2Received 22 May 2017; Accepted 3 August 2017 =================================================================================================================================================================================================================================================================== § INTRODUCTION Knowledge of the properties of the magnetic fields in massive stars is very important for our understanding the mechanisms oftheir formation and evolution as well as their impact on the stellar parameters and evolution. Accurate studies of the age, environment, and kinematic characteristics of magnetic stars are promising to give us new insightinto the origin of the magnetic fields (e.g. <cit.>, 2013; <cit.>).For a number of early B-type stars the magnetic fields were detected several tens of years ago (e.g. <cit.>). The first magnetic field detection in an O-type star was made in 2002 by <cit.>, even though the existence of magnetic O-type stars had been suspected for a long time.The recent systematic surveys MiMeS (The Magnetism in Massive Stars) and BOB (The B fields in OB stars) aiming at the detection and studying of the magnetic fields inmassive stars (<cit.>; <cit.>) strongly enhanced the number of known magnetic OBA stars incomparison with ∼500 OBA stars with confirmed magnetic field which were known before 2009(e.g. <cit.>). Because of the increase in the number of new detections it was also found that incidence of massive magnetic stars is about of 7% only (<cit.>).The detection rate obtained by the BOB collaboration is of 6± 4% (<cit.>) which is consistentwith that given by the MiMeS group.Measuring the magnetic fields of hundreds of OBA stars opens the possibility to study the magnetic fielddistribution for different types of stars (e.g., <cit.>). The understanding of the natureof the magnetic field of massive stars can be based on our knowledge of their MFD. Checking the real MFDshowed that some previous ideas about the magnetic field evolution can be incorrect. For example, the MFD predicted by <cit.> for massive stars on the main sequence in the mass range from 8 to 45 M_⊙based on the hypothesis that the net magnetic flux for a massive star and its descendant, the neutron star, is the same (see their Fig. 4) does not agree with the real distribution of the magneticfield (e.g. <cit.>). The disagreement between the predicted MFD and the one based on the real measurements is connected with the dissipation ofthe stellar magnetic flux during the evolution from massive stars to neutron stars(see Fig. 1 in the paper by <cit.> and subsection <ref> in the present paper).It is shown that the magnetic fluxes of neutron stars are about of three orders of magnitude lower than for theirprogenitors, the massive OB stars.It means that the studying how the magnetic fields and the magnetic fluxes change during the massive starevolution plays a decisive role in the understanding the nature of the magnetic field of this group of stars.In the present paper we consider the evolution of the magnetic fields and fluxes of OBA stars at the mainsequence stage. Our paper is organized in the following way: The accepted model of the evolution of magnetic OBA stars is describedin Sect. <ref>. In Sect. <ref>we review the empiricalmagnetic field distributions of OBA stars. In Sect. <ref> we discuss the magnetic desertproblem and give some constrains on the rate of dissipation of stellar magnetic fields.Finally we summarize our results and give some conclusions in Sect. <ref>.§ THE MODEL§.§ Population synthesisInformation about the intrinsic distribution of stellar magnetic fields can only be obtained from ananalysis of observations of certain sample of stars.Even the stars of the same spectral class and even subclass have significant diversity inmasses, ages andother characteristics. These characteristics influence the magnitude of alarge-scale magnetic field due, for example, the geometrical effects and the dissipation processes.Therefore, to simulate the magnetic field distribution for the model ensemble of stars, these variations in the stellar parameters have to be taken into account. Here we outline the main features of our populationsynthesis model realizing this aim.The first step of our modeling is a creation of the initial ensemble of stars. We suppose that a total numberof stars in the ensemble is N_tot and the stellar massesM ∈[M_min,M_max]. The mass distribution at the ZAMS is descibed by a powerlaw with an exponent of -2.3. This corresponds to the high-mass region of the stellar initial massfunction (<cit.>). The appearance time t_* of an each star at ZAMS is generated randomlyusing the uniformdistribution in the range t_* ∈[0,T], where Tis the total simulation time. We use the constant stellar birthrate λ. This means that approximately one star appears in the time interval Δ T=1/λ. The simulationtime T has to be at least three times longer than the main sequence lifetime τ_MS of a leastmassive star in the ensemble to be sure that the model star ensemble is stationary.Next steps include the computation of stellar parameters and selection of the stars remaining on the mainsequence at the moment of time T. Evolution of the stars in the ensemble is simulated using the rapidsingle-star evolution code SSE developed by <cit.>. This code is based on analyticalapproximations of evolutionary tracks computed by <cit.>. It can be used for simulations ofall stages of stellar evolution from the ZAMS to final remnants and it is valid for masses in the range0.1–100 M_⊙.Our population synthesis code is created on the base of the Astrophysical Multipurpose Software Environment (AMUSE)developed by <cit.>. The platform AMUSE uses pythonas an interface between different existing astrophysical codes (stellar dynamics, stellar evolution,hydrodynamics, etc.) and provides a framework in which these codes can becoupled[see http://www.amusecode.org for details].§.§ Evolution of stellar magnetic fields §.§.§ Basic definitionsIn our model the magnetic field of a star is defined via the net magnetic flux Φ at the stellar surface: Φ = ∫_S |B_r| dS.Here B_r is the radial projection of the B-field and dS is a surface element. This parametrization isvery useful for describing the magnetic properties of stellar populations because the assumption about of theconservation of magnetic flux in theabsence of dynamo or dissipation mechanismsis very good (e.g. <cit.>).In the case of a dipole field the relation between the net magnetic flux Φ_d and the polar fieldB_d can be easily derived: Φ_d = 4/3 π B_d R^2, where R is a radius of a star. However, current techniques used to measure stellar magnetic fields can give us only the longitudinal componentB_l of the field. Although B_l is directly connected to the polar field B_d, it also dependson the rotation phase ϕ, angle between the magnetic dipole and rotation axes β, and the inclinationof the rotation axis to the line of sight i (e.g. <cit.>). The later two parameters arecompletely random and have a wide range of possible values. Even the phase-averaged field⟨B_l⟩_ϕ is very sensitive to their variations.Because of these reasons, we use the root-mean-square (rms) field ℬ instead ofB_d as the main characteristic of stellar magnetic fields. If N is the total number of thefield measurements B_l^k (k=1 … N), then the rms-field is given by the formula(e.g. <cit.>): ℬ = √(1/N∑_k = 1^n (B_l^k)^2 ). The phase-averaged ratio ℬ / B_d and its asymptotical behavior at N →∞ wereinvestigated by Kholtygin et al. (2010a). They demonstrated that in the case of a dipole configuration therms-field ℬ weakly depends on random values of the rotational phase ϕ, inclinationi and the angle β. This conclusion also is valid for quadrupole and more complex field configurations.Following this paper we adopt that ℬ≈ 0.2 B_d. Then we can find the net magnetic flux using next relation: Φ = 4 π R^2 ℬ. It gives a good estimation of the net magnetic flux for any field configurations(<cit.>) . §.§.§ Magnetic field function at ZAMSIn order to simulate the evolution of stellar magnetic fields with the stellar age t it is necessary firstto define the initial magnetic field function for the stars at the ZAMS (t=0). We assume the lognormaldistribution of the net magnetic fluxes: f(Φ| t = 0) = A/Φσexp{ -1/2(logΦ - ⟨logΦ⟩/σ)^2 }, where Φ is the net magnetic flux, ⟨logΦ⟩ is the mean value of the logΦ,σ is the width (in dex) of the distribution and A = 1/√(2 π)ln10. We chose the lognormal distribution of the magnetic fluxes mainly due to similarity between lognormaland empirical distributions derived from real samples of magnetic stars (this similarity is discussed inSection <ref>). A lognormal magnetic flux distribution (magnetic flux function)may be obtained from the simple assumption that during the pre-main sequence evolution the magnetic field ofa star is experiencing a finite number of cycles of amplification/damping by some random factor. Under ratherwide conditions the resulting magnetic field function will coincide to a lognormal distribution even fora small number of the cycles (see <cit.> for details). §.§.§ Dissipation of magnetic fieldsObservational evidences imply that the magnetic fields of Ap stars decay. The rate ofdissipation depends strongly on stellar mass (e.g. <cit.>).According to <cit.> the dissipation of magnetic fields can be represented as theexponential function ∝ e^ - α_dτ, where τ is the age of a star expressedin terms of the main sequence lifetime of the star, and α_d∼ 2.0 is the dissipationfactor. We also assume exponential decay of stellar magnetic fields on the time-scale t_d whichis set to be proportional to the lifetime of a star on the main sequence t_MS. Introducing thedissipation parameter as τ_d =t_d/ t_MS, we obtain the following expression for the temporal evolution of the net magnetic flux: Φ(t) = Φ(0) exp{-1/τ_d(t/t_MS) }. The time dependence of the rms-field B(t) can be derived from this relation and by using the formula (<ref>). §.§ Random sampling and parameter estimation §.§.§ Generation of empirical distributionsTo derive an empirical magnetic fields distribution from a sample consisting of N_smplstars, the first step is to split the entire range of magnetic fields into a set of N_binsequally-sized intervals (bins) and then count how many stars fall into each bin. Obviously, large samples aremore preferable for obtaining empirical distributions which would accurately resemble the intrinsic magneticfield function. Unfortunately, the current number of known magnetic stars is still very limited,therefore it is important to understandi) how the intrinsic model magnetic field function would manifest itself as empirical distribution andii) how the number of stars in a sample affects the magnitude of possible variations in distribution functiondue to statistical effects. For these purposes we developed the random sampling method which allows toproduce empirical distributions from the model.The random sampling is used to generate a large set of equally-sized samples from the ensemble of magneticstars created by the population synthesis module (see Section <ref>). Thendistributions of magnetic fields in each of the samples are passing through the special procedure to obtainaverage distribution as well as confidence boundaries for the uncertainties in a counts number for each ofthe defined bins.If n_k(i) is the number of stars in i-th bin of k-th sample then the averagednumber of stars in the bin is⟨n(i)⟩ = 1/N_smpl∑_k=1^N_smpl n_k(i), where N_smpl is the number of generated samples. The confidence boundaries for a desired level ofsignificance are calculated using the Poisson statistics.§.§.§ Parameter estimationSince real samples of massive magnetic stars are small, the standard χ^2 statistics is no longer validfor the parameter estimation. For this reason, instead of χ^2 we use the C statistics in form C_q = 2 ∑_i=1^N_bins[ e_i - n_i + n_i ( lnn_i - lne_i ) ], where n_i is a number of stars in each of the bins of the empirical distribution, e_i is theexpected number of stars given by the model, and N_bins is a number of bins. The C statistics were introduced by <cit.> as replacement of χ^2that would be valid even for a very small samples. It is commonly used in X-ray andgamma-ray astronomy in cases when the number of photons received by a detector is low.The following method is used for generating confidence intervals for the model parameters. First, we find theminimum value (C_q)_min when the model is fitted to the data and q parameters are varied. Then,assuming the level of significance α, we determine the locus of points in parameter space for which C - (C_q)_min > Δ C(α). According to <cit.> the Δ C statistics can be represented as Δ C(α) = χ^2_q(α) + O( 1/√(N_smpl)), where N_smpl is the total number of stars in a samples. For N_smpl≫ 1 the Δ C statistics followsthe χ^2 distribution for q degrees of freedom. However, we generate the distribution of Δ Cfrom our model using the Monte Carlo simulations, because the term O(N_smpl^-1/2) cannot be neglectedwhen N_smpl≲ 10.The C statistics may also be easily modified for the purpose of simultaneous fitting of different datasets.§.§ Parameters of the modelA short summary describing the main parameters of the model is presented in Table <ref>.Note that two of the parameters such as the mass interval [M_min, M_max] and the samplesize N_smpl, may be determined directly from characteristics of empirical data. For example, anempirical magnetic fields distribution derived from a sample of eleven O-type stars suggests thatM ∈ [16, 60] M_⊙ and N_smpl = 11. The remaining three parameters however have to beestimated from fitting of the empirical data. Some examples of magnetic field functions produced by the model are presented in Fig. <ref>.Such parameters as ⟨logΦ⟩ and σ define the position and overall width ofthe magnetic fields function. The mass interval influences some minor asymmetry in the shape of the magneticfield function due to changes of stellar radii during evolution on the main sequence. The dissipationparameter τ_d has the most profound impact on a magnetic field function.In the case of no dissipation of magnetic fields, or if τ_d≫ 1, the magnetic field functionwouldbe a bell-shaped curve, with some minor asymmetry. However, if the magnetic field decay is fast(τ_d≪ 1),the model would produce a highly asymmetrical magnetic field function. Finally,the sample size parameter N_smpl defines possible manifestations of a magnetic field function asan empirical distribution derived form a finite number of magnetic fields measurements.§ ANALYSIS OF THE EMPIRICAL MAGNETIC FIELD DISTRIBUTIONS FOR OBA STARS§.§ Magnetic field measurementsIn order to compare our model with observations, we first divided all known magnetic stars into three groups:BA stars (this groups contains mainly Ap/Bp stars), O and early B type stars combined together (OB stars),and O type stars only. These groups correspond to the mass ranges 1.5–16, 3–16 and16–60 M_⊙.Magnetic field measurements for BA stars were taken from the catalogue by Bychkov et al. (2009) which containshundreds of magnetic stars. However only 288 stars with most reliable measurements were selected, based on statistical criteria described by Kholtygin et al. (2011a). The data for O- and early B-type stars werecompiled from multiple sources(<cit.>; <cit.>, 2015a, 2015b; <cit.>; <cit.>) with thetotal amount of 73 stars, including 11 O-type stars.If only dipolar magnetic field strengths B_d were given in the used sources, we converted them torms magnetic field ℬ via relation (<ref>).The empirical magnetic field distribution f(B) for the sample of stars with measured magneticfields was introduced by <cit.> and <cit.>. It equals tof(ℬ)≈N(ℬ,ℬ +Δℬ)/N_totΔℬ, where N(ℬ,ℬ+Δℬ) is the number of stars with the rms magneticfields ℬ in the interval (ℬ, ℬ+Δℬ) and N_totis the totalnumber of stars with measured rms magnetic field.§.§ BA stars The empirical distribution of magnetic fields derived from our sample of BA stars is presented inFig. <ref>. It is bell-shaped with a maximum located at ℬ≈ 300G, and may beeasily approximated by lognormal distribution. In Section <ref> we found that the shape ofthe model distribution was highly dependent on the value of the dissipation parameter (see alsoFig. <ref>). Therefore, one may expect that τ_d has to be large to produce thesymmetrical magnetic field functions which would be consistent with the empirical datainFig. <ref>. At the same time the maximal symmetry would be achievedonly if the dissipation parameter was infinite.Thus we decided to consider two models:i) the model ℳ_0 in which we assume no dissipation of stellar magnetic fields, i.e.τ_d→∞;ii) the model ℳ_1 with a dissipation and a finite value of τ_d. We find that the model ℳ_0 is in a good agreement with the empirical magnetic field distributionfor BA stars (see Fig. <ref>, top panel). The parameters of the best-fit and thecorrespondingconfidence intervals as well as the value of the C statistics are presented in Table <ref>.Applying the model ℳ_1 for the analysis of the empirical data we discovered that from astatistical point of view the results were equally good for any value of the dissipation parameter takenfrom the range τ_d≳ 0.5. Below this threshold the quality of fitting is rapidlydecreasing beyond acceptable limits. Thus we conclude that τ_d = 0.5 corresponds to thehighest rate of magnetic field decay allowed by the agreement with the empirical distribution of BA stars.We assume this value for the futher analysis using ℳ_1 model.The model ℳ_1 which provide the best fitof the empirical distribution of BA stars magneticfields is given in Fig. <ref> (bottom panel). Like the model ℳ_0 it also gives a goodagreement with the data but it also posess the higher and less scattered values of the net magnetic flux⟨logΦ⟩ for stars at the ZAMS as it seen in Table <ref>. §.§ OB starsThe empirical magnetic field distribution of O- and early B-type stars is very similar inappearance to that of BA stars. The resemblance becomes obviousfrom Fig. <ref> in which these distributions are presented together. This is why first we used thepreviously obtained for BA stars best-fitting parameters to calculate the modeldistribution of OB stars. Thus, we confirmed our hypothesis that both empirical distributions arelikely drawn from the same intrinsic magnetic field function (the corresponding p-values for two of themodels are p_null(ℳ_0) = 0.4 and p_null(ℳ_1) = 0.31).Both the model ℳ_0 and model ℳ_1 are in a good agreement with the empiricalmagnetic field distribution for OB stars (see Fig. <ref>). The parameters of the best-fitand thecorresponding confidence intervals as well as the value of the C statistics are presented inTable <ref>.We compared the proper best-fitting parameters for OB stars with those obtainedfor BA stars (Table <ref>). We also find that both models give agood agreement with the data (Fig. <ref>). Therefore we can conclude that the results of ouranalysis strongly imply a close similarity between the magnetic field distribution for BA and OB stars.§.§ O stars and OBA stars Up to date the number of known O-type magnetic stars is still very small. Consequently, it isvirtually impossible to make reliable estimates of the parameters describing the corresponding magneticfield function from the available data because of large uncertainties of counting statistics. Nevertheless,we applied the same procedure as before and found the best-fitting parameters for the modelsℳ_0 and ℳ_1. A comparison of the model and the empirical magnetic field distributionsfor O stars and for the models ℳ_0 and ℳ_1 is given in Fig. <ref>.We obtained significantly higher values for ⟨logΦ⟩in comparison to the other groups of stars, although the estimated 95% confidence intervals forthe model parameters are turned to be also much wider (Table <ref>).The null hypothesisthat the empirical magnetic field distribution of O-type stars is drawn from the same magnetic fieldfunction as the distributions of BA and OB stars gives the p-value of 0.1,which is quite low but still higher than the commonly accepted rejection value of 0.05.In order to test further the hypothesis that all of the empirical distributions could be drawn from a singlemagnetic field function, we applied the technique of simultaneous fitting, which was described above inSection <ref> to all sample of OBA stars including the individual subsamples of BA,OB and O stars considered above. The results of the fitting are listed in the last 2 rows ofTable <ref>. This result supports our hypothesis on the similarity of all of the consideredempirical distributions. It means that the magnetic field distribution for OBA stars obeys a commondistribution law. §.§ Magnetic Flux distributionIn this subsection we outline the evolution of magnetic field of the massive stars after the main sequence.In Fig. <ref> we present the magnetic fluxes distributions for OB stars, Wolf-Rayet (WR) stars anddifferent types of the neutron stars (NSs). For OB stars we took the value of rms magnetic fields from the catalogue by Bychkov et al. (2009).The radii of stars were taken either from the original papers for considered stars or from the CADARSCatalogue (<cit.>). All known galactic NSs were divided into three groups: normal pulsars, milliseconds pulsars, and magnetars.The data for neutron stars are taken from the ATNF Pulsar Catalogue (<cit.>) and<cit.>. We put for all NSs the standard radius R_*=10km.The distribution of the magnetic fluxes both for these groups of neutron stars and for OBstars are the same what was presented by Igoshev & Kholtygin (2011) in their Fig. 1.We add to their data the distribution of the upper limits of the magnetic fluxes for WR stars.The upper limits of magnetic fields for 11 WR stars was taken from the paper <cit.>. Theupper limit of the magnetic field for the star WR 136 is given by Kholtygin et al. (<cit.>) and forWR 6 by <cit.>. For all WR stars we use the standard radius R_* = 5R_⊙.We can see in Fig. <ref> that although the width of distributions for NSs are close to that for OB stars, the mean fluxes are very different. The youngest group of NS (magnetars) has the largest fluxes,while the magnetic fluxes of the oldest NS (millisecond pulsars) are on the average smaller by 7orders-of-magnitude. The mean fluxes of all remaining NSs (Normal pulsars) have intermediate values.We see that the mean fluxes of all types of NSs, excluding magnetars, are much lower than the meanfluxes of their progenitors, viz. the massive OB stars.On the other hand, the upper limits of the magnetic fluxes for WR stars are close to those for normal pulsars. It means that the main dissipation of the magnetic flux during the evolution from the main sequence massiveOB stars occurs between the ZAMS and the WR stage. At the same time, this can mean that the changes of the magnetic fluxes during of the supernova explosion are relatively small in agreement with the hypothesis by <cit.>. § DISCUSSION§.§ The “magnetic desert” problem <cit.> proposed a scenario whereby the magnetic dichotomy between BA and Vega-likemagnetism originates from the bifurcation between stable and unstable large scale magnetic configurationsin differentially rotating stars. This means that the number of BA (and possibly O) stars with the rms-magnetic fields in interval between ∼1 G and ∼300 G is small if notnegligible.At the same time the empirical magnetic field functions obtained by us for OB and BA stars reveal that the same regular shape can be fitted by a lognormal distribution. There are no visible peculiarities thatmight indicate the existence of a threshold magnetic field reported for BA and early B-type starsby Auriére et al. (2007) and by Kholtygin et al. (2010a). The cumulative distributions produced bytruncated magnetic field functions are too steep in comparison to the empirical ones(see Fig. <ref>).The same issue also has been discussed by Fossati et al. (2015a), who reported the detection of weak magneticfields (below 100 G) in two early B-type stars: β CMa (HD 44743) and ϵ CMa (HD 52089).The estimated upper limit for a dipolar field strength of the latter star is B_d ≳ 13G, which isfar beyond the threshold value B_d ≈ 300G obtained by Auriére et al. (2007), and B_d ≈ 1500G obtained byKholtygin et al. (2010a). Based on the presented considerations, we also conclude that there is no evidence for any `magnetic threshold' at least for BA and late B-typestars.On the other hand, Wade et al. (2015) reported preliminary results on a study where a modeling approach by<cit.> has been applied to infer upper limits on dipole magnetic fields in a sample of O-type stars from the MiMeS survey. The results of <cit.> implyupper limits on surface dipoles of about 40 G at 50% confidence, and 105 G at 80% confidence. The later values are more consistent with the empirical distributions(Fig. <ref>). Therefore, we cannot rule out the possibility that the critical value for a magneticfield may exist, but this value should be much lower than proposed earlier. §.§ Constrains on dissipation of stellar magnetic fieldsAn important result of our analysis of the empirical magnetic fields distributions is that the time scaleof the magnetic fields dissipation should be at least comparable with the lifetime of a star on the mainsequence. Using the technique of simultaneous fitting we found that the dissipation parameter has to be greaterthan 0.5 to be consistent with the empirical distributions. This result is in a good agreement with thevalue obtained by Kholtygin et al. (2010b). However, we cannot estimate the dissipation parameterτ_d more precisely with the current sample of magnetic stars. It means that models withmoderate dissipation and without the dissipation all are equally consistent with the empirical data.We would also like to note, that according to Landstreet et al. (2008), the mean magnetic fields of earlyA- and late B-type stars decrease by a factor of 3–4 in the range τ∈ (0.0–0.2) and thenremain almost constant. The dissipation parameter in that case would be lying within the range of0.1–0.2and, according to our simulations, would produce a highly asymmetrical shape of the magneticfield function, inconsistent with the empirical distributions. We believe that this discrepancy may beexplained by a limited size of the magnetic stars sample used in their analysis.§.§ Intrinsic magnetic field function for early-type starsAs we discussed earlier, the empirical distributions are very similar in their appearance. Moreover, it ispossible to successfully describe all three of our samples of magnetic stars with a single model. Thus,we consider the possibility that magnetic properties of early type massive stars may be explained by acommon magnetic field function that would be the same in a very wide range of masses(≳ 2 M_⊙).Another strong evidence supporting this hypothesis is that the occurrence of early type magnetic starsis remarkably constant and does not depend on a spectral type or mass of a star. The incidence of magneticBA stars – which is slightly less than ten per cent – has been known for some time(<cit.>; <cit.>). Recently similar magnetic field detection rates were reported for earlyB- and O-type stars by the BOB(<cit.>) and MiMeS collaborations <cit.>. It iscertainly became more evident now that magnetic stars across the upper main sequence share some commonphysics related to the phenomena of stellar magnetism (see also <cit.>, 2015).Given the fact that the incidence of magnetic Herbig Ae/Be stars is also estimated as 7%(<cit.>), we come to the conclusion that magnetic properties of massive stars areprobably determined by physical conditions at early stages of pre-main-sequence evolution, perhapseven during protostar formation. § CONCLUSIONS Our investigations of the measured magnetic fields and their evolution for an ensemble of galactic OBA stars show thatthe distribution of these magnetic field for all mentioned spectral types can be described by a lognormal law.We develop the population synthesis code to model the magnetic field evolution for OBA stars.After the analysis of the empirical magnetic field distributions using our model we came to the following conclusions: * The empirical magnetic field distribution for BA stars can be fitted by the lognormal distribution with the mean ⟨logB⟩≈ -0.5 and the standard deviationσ = 0.5. * Our model can be used to reproduce the empirical magnetic field and net magnetic flux distributions for OBA stars supposing that the net magnetic flux distributions at ZAMS is lognormal with parameters ⟨logΦ⟩≈ 26.5 for model without magnetic field dissipation and for a modelwith dissipation ⟨logΦ⟩≈ 26.9. * The dissipation parameter τ_d≳ 0.5 in an accordance with an estimation by <cit.>.* Our modeling shows that there is no magnetic desert in the distribution of the magnetic field of OBA star suggested in the past by some authors (e.g., <cit.>).ASM, AFK, SF, GGV and GAC thank the RFBR grant 16-02-00604 A for the support. GGV acknowledges the Russian Foundation for Basic Research (RFBR grant N15-02-05183 A) [Alecian et al. 2014]Alecian:2014crAlecian, E., Kochukhov, O., Petit, V. et al. 2014, , 567, A28 [Aurière et al. 2007]Auriere:2007dwAurière, M., Wade, G. A., Silvester, J. et al. 2007, , 475, 1053[Babcock 1947]Babcock:1947gfBabcock H. W., 1947, , 105, 105[Bohlender et al. 1993]Bohlender:1993vyBohlender, D. A., Landstreet, J. D., Thompson, I.B. 1993, , 269, 355[Braithwaite & Nordlund 2006]Braithwaite:2006daBraithwaite, J., Nordlund, A. 2006, , 450, 1077[Bychkov et al. 2009]Bychkov:2009jjBychkov, V. D., Bychkova, L.V.,Madej, J. 2009, , 394, 1338[Cash (1979)]Cash:1979bi Cash, W. 1979, , 228, 939[Castro 2015]Castro:2015bvCastro, N., Fossati, L., Hubrig, S. et al. 2015, , 581, A81[Chevrotiére et al. (2014)]Chevrotiere-2014de la Chevrotiére, A., St-Louis, N., Moffat, A. F. J. 2014, , 781, 73[Donati et al. (2002)]Donati-2002 Donati, J.-F., Babel, J. Harries, T. 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"authors": [
"A. S. Medvedev",
"A. F. Kholtygin",
"S. Hubrig",
"M. Schöeller",
"S. Fabrika",
"G. G. Valyavin",
"G. A. Chountonov",
"Yu. V. Milanova",
"O. A. Tsiopa",
"V. A. Yakovleva"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170825172819",
"title": "Statistics of magnetic field measurements and evolution of magnetic field of OBA stars"
} |
\begin \end [] \ ⇒ → | http://arxiv.org/abs/1708.07394v2 | {
"authors": [
"Ulrich Horst",
"Dörte Kreher"
],
"categories": [
"q-fin.MF",
"math.PR",
"q-fin.TR",
"60F17, 91G80"
],
"primary_category": "q-fin.MF",
"published": "20170824132727",
"title": "Second order approximations for limit order books"
} |
kimuniv-m]Yun-Hyok Kye kimuniv-m]Chol-Jun Yucor [email protected] kimuniv-m]Un-Gi Jong hk]Yue Chen icl]Aron Walsh[cor]Corresponding author[kimuniv-m]Department of Computational Materials Design, Faculty of Materials Science, Kim Il Sung University, Ryongnam-Dong, Taesong District, Pyongyang, Democratic People's Republic of Korea [hk]Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China [icl]Department of Materials, Imperial College London, London SW7 2AZ, United Kingdom The chemical stability of methylammonium lead iodide (MAPbI3) under humid conditions remains the primary challenge facing halide perovskite solar cells. We investigate defect processes in the water-intercalated iodide perovskite (MAPbI3_H2O) and monohydrated phase (MAPbI3·H2O) within a first-principles thermodynamic framework. We consider the formation energies of isolated and aggregated vacancy defects with different charge states under I-rich and I-poor conditions. It is found that a PbI2 (partial Schottky) vacancy complex can be formed readily, while the MAI vacancy complex is difficult to form in the hydrous compounds. Vacancies in the hydrous phases create deep charge transition levels, indicating the degradation of halide perovskite upon exposure to moisture. Electronic structure analysis supports a novel mechanism of water-mediated vacancy-pair formation. Critical role of water in defect aggregation and chemical degradation of perovskite solar cells [ Received: date / Accepted: date =============================================================================================== Low-cost perovskite solar cells (PSCs) based on methylammonium lead iodide (CH3NH3PbI3 or MAPbI3) are rapidly evolving, with a record power-conversion efficiency (PCE) from under 4% in 2009 <cit.> to over 22% in recent years <cit.>. However, PSCs have a critical problem of easy degradation by extrinsic as well as intrinsic factors, still preventing their outdoor installation <cit.>. In particular, the facile decomposition of MAPbI3 upon exposure to moisture has been recognized to be the major extrinsic factor of PSC degradation <cit.>. In fact, the PCE of MAPbI3 solar cells drops by nearly 90% in a few days under an ambient environment (T= 300 K, relative humidity (RH) = 30-50%) <cit.>, while MAPbI3 can be decomposed into MAI, PbI2, and HI in a few hours at high humidity conditions <cit.>. For a chemical explanation of this phenomenon, hydrolysis of MAPbI3 was initially suggested as the main mechanism, and based on the first-principles calculations, deprotonation of MA+ by H2O was proposed as the principal cause of the hydrolysis <cit.>. Soon afterwards, however, it was demonstrated that MAPbI3 readily transformed to the monohydrate phase MAPbI3·H2O at moderate humidity (RH ≤ 60%), while to the dihydrate phase (MA)4PbI6·2H2O at high humidity(RH ≥ 80%), at the initial stage of the MAPbI3 water-mediated decomposition process, which could be reversed by drying treatment <cit.>. This can be explained by the hydrogen bonding interaction between the lead iodide framework and the organic MA+ cations in the perovskite crystal being weakened upon its hydration <cit.>. MA+ can readily diffuse and separate from the PbI6 octahedra, resulting in a rapid decomposition of MAPbI3. The activation barrier for vacancy-mediated MA+ migration was confirmed to be reduced from 1.18 eV in MAPbI3 to 0.38 eV in water-intercalated and 1.14 eV in monohydrated phases <cit.>. Although there have been some theoretical studies of the intrinsic point defects in MAPbX3 (X = I, Br, Cl) <cit.>, those in the hydrate phases remain unexplored. In this Letter, we investigate the origin of perovskite decomposition through point defect processes in water-intercalated MAPbI3, denoted as MAPbI3_H2O hereafter, and monohydrated phase, MAPbI3·H2O. The water-intercalatedMAPbI3_H2O is suggested as an intermediate phase during the transition to the hydrated phases due to the relatively low activation energies for water insertion into the perovskite surface (0.27 eV <cit.> or 0.31 eV <cit.>), as well as for water molecular diffusion within the bulk crystal (0.28 eV <cit.>). A density functional theory (DFT) approach combined with ab initio thermodynamics is utilized to describe defect formation and interactions. Electrostatic stabilization by water is found to play a key role in defect clustering and ultimately in the stability of perovskites in humid environments.In the first stage, we performed structural optimizations of pristine MAPbI3,water-intercalated MAPbI3_H2O, and monohydrated MAPbI3·H2O. The lattice constant and bandgap of MAPbI3 were calculated to be 6.33 Å and 1.53 eV, which are in good agreement with the experimental values of 6.32-6.33 Å <cit.> and 1.50 eV <cit.>. For the case of MAPbI3_H2O, the unit cell containing a water molecule in the large interstitial space formed by a huge PbI6 framework became triclinic after optimization. The initial structure of MAPbI3·H2O with a monoclinic crystalline lattice and experimentally identified atomic positions <cit.> was also optimized, giving the lattice constants of a=10.46 Å, b=4.63 Å, c=11.10 Å and β=101.50^∘ agreed well with the experimental values <cit.>. The bandgaps were calculated to be 1.86 eV in MAPbI3_H2O, and 2.47 eV in MAPbI3·H2O, being comparable with the previous DFT value of 2.52 eV for the monohydrated phase <cit.> and experimental value <cit.>. In the second stage, using the optimized unit cells, we built (3×3×3) supercells for MAPbI3 (324 atoms) and MAPbI3_H2O (405 atoms), while the (2×3×2) supercell for MAPbI3·H2O (360 atoms), with and without vacancy defects, and performed atomic relaxations with the fixed lattice constants (see Figures S1-S3). Isolated vacancy point (V_I, V_MA, V_Pb) and pair defects (V_MAI, V_PbI2) were created. For each vacancy defect, various charge states were considered to identify the thermodynamic charge transition levels. Figure <ref> presents the defect formation energy diagrams at I-poor (Pb-rich and MA-rich) and I-rich (Pb-poor and MA-poor) conditions. These conditions correspond to an iodine precursor in orthorhombic solid form and a lead precursor in fcc solid form, respectively.Amongst the vacancy point defects, lead vacancies with a charge state of -2 (V^-2_Pb) in these three compounds, and additionally -1 charge state (V^-1_Pb) and neutral state (V^0_Pb) in the case of hydrous compounds, have the lowest formation energies in the whole range of Fermi level (E_F) at I-rich conditions. For I-poor growth, meanwhile, the iodine vacancies with a charge state of +1 (V^+1_I) have the lowest formation energies in the lower part of E_F, whereas the V^-2_Pb in the higher range of E_F. Note that for the case of MAPbI3 our results are consistent with the previous DFT results <cit.>, with some minor numerical differences due to the inclusion of dispersion corrections in this work.Under I-poor conditions, MA vacancies with a neutral state (V^0_MA) and a charge state of -1 (V^-1_MA) have typically higher formation energies than V_Pb and V_I in MAPbI3 and MAPbI3_H2O, but in-between values are found in the case of MAPbI3·H2O. For vacancy pair defects, which can be viewed as compensated partial Schottky-type aggregates, we considered various charge states.We calculated the binding energy defined as E_b=H_f[A]+H_f[B]-H_f[AB] <cit.>.Table <ref> summarizes the formation and binding energies of the neutralpairs of V^0_MAI and V^0_PbI2 (for E_b of charged pairs, see Table S1). For the case of MAPbI3, the formation energy of V^0_MAI is 0.23 eV, which is much lower than 1.80 eV reported by Kim et al. <cit.>. If we use the MAI molecule instead of MAI solid as they did, it becomes 1.98 eV in better agreement. V^0_PbI2 has the formation energy of -0.03 eV, being slightly lower than 0.03 eV reported by Kim et al. <cit.>, possibly due to a different crystal lattice. In general, the formation energies of these complex defects in the hydrous compounds are higher than those in the pristine MAPbI3.The formation of V^0_PbI2 in all the compounds is more favorable than the formation of the individual vacancy point defects V^+1_I and V^-2_Pb due to their positive binding energies (Table <ref>). Therefore, it is expected that V^+1_I and V^-2_Pb are formed first (they are dominant defects), and then the interaction between them leads to the formation of V^0_PbI2 independently of whether the hydrous compound or not. Water adsorption into the perovskite crystal reduces the activation barrier for vacancy-mediated I- ion migration <cit.>, resulting in an enhancement of V^0_PbI2 formation. As shown in Figure <ref>, the formation energy of V^+1_I (V^-2_Pb) at the I-rich condition is higher (lower) than at the I-poor condition (their concentrations have a reverse feature), and thus the reaction speed of V^0_PbI2 formation can be lower at the I-rich condition (Pb2+ ion migration is quite difficult). Experimentally, the I-rich condition can be realized by directly adding I_3^- in solution <cit.>. In indirect ways, increasing PbI2 concentration relative to MAI concentration as in many other experimental works <cit.> can effectively inhibit the formation of V^0_PbI2 from the decomposition of MAPbI3 from the viewpoint of chemical equilibrium of reaction, and furthermore, formation of a passivating MAPbI3/PbI2 interface <cit.> also provide PbI2 excess environment to prevent the forward reaction for V^0_PbI2 formation. Surprisingly, the binding energies of V^0_MAI in the hydrous compounds are negative, although it is positive in the pristine perovskite. This indicates that in the hydrous compounds other products such as HI, CH3NH2 and I2 rather than MAI can be formed during chemical decomposition.Next, we derived thermodynamic transition levels ε(q_1/q_2) between defects in different charge states q_1 and q_2. Figure <ref> shows the possible transition levels together with the relative band alignment of MAPbI3_H2O and MAPbI3·H2O with respect to MAPbI3. Water molecules inserted through the film surface extract electrons, resulting in the shift of the valence band maximum (VBM) towards lower values and the conduction band minimum (CBM) towards higher values, and thus the bandgap change from the pristine to the water-intercalated and to the monohydrated phase. It is apparent that in the case of MAPbI3 all of the vacancy defects have shallow transition levels, whereas in the case of hydrous compounds the vacancies exhibit deep trap behavior that can facilitate the recombination of charge carriers, resulting in the degradation of solar cell performance. Specifically, in the case of MAPbI3, V_I and V_MA are shallow donors and acceptors due to their transition levels ε(1+/0) and ε(0/1-) near the CBM and VBM, respectively. In the case of hydrous compounds, transition levels are located deep in bandgap region, although V_MA transition levels are not far away from the valence band. In particular, V_Pb has no transition level (ε(0/2-) inside the valence band) in MAPbI3, but two deep transition levels ε(0/1-) and ε(1-/2-) are possible in the hydrous compounds. Similar features are found for V_MAI (two transition levels) and for V_PbI2, which has two/four transition levels in the pristine/hydrous MAPbI3.The electronic density of states (DOS) in the pristine and vacancy-containing structures are shown in Figure <ref>. Only the neutral states are considered. To make clear the role of each species, the atom-projected DOS (PDOS) are presented in Figure S4-S6. In all the compounds, the lower conduction band is from Pb 6p, while the upper valence band is from mostly I 5p states and minor part of Pb 6s <cit.>. In the case of MAPbI3,V_I formation causes a donor level near the conduction band and the positive potential results results in a downshift of the local valence band, while V_Pb and V_MA form acceptor states and their negative potential result in a local upshift of the valence band. The DOS characteristics of V_PbI2 and V_MAI can be explained by combining the effects of individual point defects. Similar features are observed in the hydrous compounds. One distinction is that the n-type donor level created by V_Iis deeper in the band gap due to the presence of the water molecule. The electronic states of the water molecule overlap with I 5p at about -2 (-3) eV and with the MA states at about -4 (-5) eV in MAPbI3_H2O (MAPbI3·H2O), through the hydrogen bonding interaction between water and I atoms of PbI6 as well as MA moiety. When the vacancy defect V_I or V_Pb is formed, similar interactions are observed. To study charge transfer during the formation of a vacancy defect, we plot the electron density difference Δρ=ρ_V_D-ρ_perf+ρ_D in Figure <ref>. In the case of MAPbI3,charge is depleted around the V^+_I defect, while charge is accumulated around V^-_MA and V^-2_Pb, as expected.When the hydrous phases are formed, the extent of charge exchange is reduced because the water molecule can donate (for positive point defect) or accept (for negative ones) some electrons, indicating a stabilization by waterof the Schottky-type defects.In the absence of water, V_I, V_MA and V_Pb with the charge states of +1, -1 and -2 are the most stable over a large range of Fermi energy. In the case of hydrous compounds, however, these defects exchange the electrostatic stabilization by water opens up alternative charge states, leading to the creation of deep levels, which can enhance non-radiative processes and ultimately lead to the degradation of perovskite solar cells.In summary, we have investigated defect processes in MAPbI3 and its hydrous phases MAPbI3_H2O and MAPbI3·H2O in order to reveal the effect of water on the performance and stability of iodide perovskites. The formation energies of V_I, V_MA, V_Pb, V_MAI and V_PbI2 with various charge states, the binding energies of the complex defects (V_MAI and V_PbI2), the density of states, and electronic charge density differences were presented to draw the following conclusions.The formation of V_PbI2 from its individual vacancy point defects is spontaneous, and due to the greatly reduced kinetic barrier for I- ion migration when hydrated, the concentration of V_I should be reduced to prevent this formation, which can be realized by imposingI-rich conditions.In the hydrous compounds, the formation of individual point defects V_I and V_MA is more favorable than the formation of V_MAI, and thus I2 or CH3NH2 or HI can be formed rather than MAI during the decomposition. Unlike inbulkMAPbI3, all the vacancy defects create deep transition levels in the hydrous compounds arising from electrostatic interactions with water molecules.To overcome the negative effects of water on the performance and stability of halide perovskites, controlling the processing conditions such asthe halide chemical potential during growth and annealing will be important, in additional to the physical encapsulation of devices.§ COMPUTATIONAL METHODSThe formation enthalpy of a point defect with a charge state q is calculated using the grand canonical expression <cit.>, Δ H_f[D^q]≅{E[D^q]+E_corr[D^q]}-E_perf-n_iμ_i+qE_F where E[D^q] and E_perf are the total energies of the supercell including a defect D and the perfect crystal supercell, and n_i and μ_i are the number of removed (minus sign) or added (plus sign) i-type species and its chemical potential. E_corr[D^q] is a correction to the error in the total energy of charged supercell that can be calculated by E_corr=α q^2/ε L in the monopole approximation, where α is the Madelung constant, ε the static dielectric constant, and L the lattice constant, respectively <cit.>. Using the density functional perturbation theory, we computed the isotropic static dielectric constants as 23.55, 25.88, and 16.30 for MAPbI3, MAPbI3_H2O, and MAPbI3·H2O. E_F is the Fermi energy expressed as referenced to the valence band: E_F=ϵ_VBM+Δϵ_F+Δ V, where ϵ_VBM is the energy level of the VBM, Δϵ_F is the Fermi level with respect to the VBM and Δ V is the potential alignment. We ignored the band-filling correction for shallow defects due to its negligible value with large supercell size in this work. Self-interaction and spin-orbit coupling were not included, which are not expected to affect the comparison between the defect physics of pristine and hydrated MAPbI3 compounds and thereby draw the conclusion about the decomposition mechanism upon hydration, but they will be important for quantitative defect spectroscopy <cit.>.The chemical potentials depend on the growth conditions, which can fall betweenI-rich or I-poor conditions. The I-rich condition corresponds to the iodine precursor in orthorhombic solid form (space group Cmca), and thus the upper limit of the iodine chemical potential is that μ_I^rich=E_I(orth). The synthesis equations constrain the chemical potentials as follows, μ_MAI+μ_PbI2=μ_MAPbI3 μ_Pb+2μ_I=μ_PbI2 μ_MA+μ_I=μ_MAI Eqs. (<ref>) and (<ref>) correspond to the real synthetic reactions, but Eq. (<ref>) acts only as a theoretical reference.From Eq. (<ref>), we identified the iodine poor conditions, μ_I^poor≈1/2(E_PbI2-μ_Pb^rich), where μ_Pb^rich=E_Pb(fcc)referred to the bulk Pb in fcc phase and E_PbI2 is referred to the bulk PbI2 in rhombohedral phase (space group P3̄m1) <cit.>, and Pb-poor condition, μ_Pb^poor≈ E_PbI2-2μ_I^rich.Then, using Eqs. (<ref>) and (<ref>), the MA-poor and -rich conditions are established like μ_MA^poor≈ E_MAPbI3-E_PbI2-μ_I^rich and μ_MA^rich≈ E_MAPbI3-E_PbI2-μ_I^poor.For the vacancy pair defects, the chemical potentials were not affected by the iodine chemical potential; μ_MAI≈ E_MAPbI3-E_PbI2 and μ_PbI2≈ E_PbI2.Pseudo-cubic unit cells were adopted for MAPbI3, and (3×3×3) and (2×3×2) supercells were used for vacancy-containing MAPbI3, MAPbI3_H2O and MAPbI3·H2O, respectively. The DFT total energies were calculated using the Quantum ESPRESSO code <cit.> with the ultrasoft pseudopotentials provided in the code and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional <cit.> added by the van der Waals (vdW) energy in the flavor of vdW-DF-OB86 <cit.>. Scalar-relativistic effects are included. A plane-wave cutoff energy of 40 Ry and a Γ point for structural relaxation of vacancy-containing supercells, while 2×2×2 special k-points for DOS calculations, were used for all the configurations. All the atomic positions of each configuration were relaxed until the forces on atoms converge to 5× 10^-5 Ry/Bohr. § APPENDIX A. SUPPLEMENTARY DATASupplementary data related to this article can be found at URL.§ NOTESThe authors declare no competing financial interest.elsarticle-num-names | http://arxiv.org/abs/1708.07606v3 | {
"authors": [
"Yun-Hyok Kye",
"Chol-Jun Yu",
"Un-Gi Jong",
"Yue Chen",
"Aron Walsh"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170825035006",
"title": "Critical role of water in defect aggregation and chemical degradation of perovskite solar cells"
} |
Evaluation of Deep Learning on an Abstract Image Classification Dataset Sebastian [email protected] Rodríguez-Sá[email protected] University of InnsbruckTechnikerstrasse 21a, Innsbruck, Austria December 30, 2023 ====================================================================================================================================================================================================================empty200mm(.2,-7cm) Copyright IEEE. To be published at proceedings of MBCC/ICCV2017 Convolutional Neural Networks have become state of the art methods for image classification over the last couple of years. By now they perform better than human subjects on many of the image classification datasets. Most of these datasets are based on the notion of concrete classes (images are classified by the type of object in the image). In this paper we present a novel image classification dataset, using abstract classes, which should be easy to solve for humans, but variations of it are challenging for CNNs. The classification performance of popular CNN architectures is evaluated on this dataset and variations of the dataset that might be interesting for further research are identified. § INTRODUCTIONConvolutional Neural Networks have become the method of choice for image classification since the system by Krizhevsky <cit.> won the ImageNet competition in 2012 by a large margin. In 2015 Russakovsky <cit.> reported the classification accuracy of human subjects, on the same dataset, to be around 94.9% correctly classified images. In the same year, He <cit.> were the first to present a network that outperformed human subjects on ImageNet. Since then, image classification is often perceived as either solved, or in the process of being solved.Popular datasets used for image classification like MNIST<cit.>, ImageNet<cit.>, PASCAL<cit.>, and CIFAR10/100<cit.> all classify the images by the type of a prominent object or feature in the image. We will call such classes concrete classes. Concrete classes have in common that they can be identified by analyzing local features, or the distribution of multiple local features. In this paper, we present a dataset that consists of abstract classes. Abstract classes imply that images can not be classified by simply considering local features. In our case, the two types of classes are identity/non–identity and symmetry/non–symmetry.§ RELATED WORK Fleuret <cit.> have already presented a dataset with abstract classes, using very simple black and white line drawings. This dataset is somewhat reminiscent of the “Bongard problems”, presented by Bongard in 1970 <cit.> as a set of problems that, according to Bongard, neural networks would never be able to solve (though he did not have simple classification in mind, but a textual description of what separates the two classes). In previous work <cit.> <cit.>, we have tested different convolutional neural network architectures on the dataset by Fleuret and came to the conclusion that current CNN architectures have shortcomings when shape comparison is needed to distinguish two classes. As Dodge <cit.> argue, the Fleuret dataset is too simplistic and too far from natural images to draw any practical conclusions from it. Our goal is to present a more realistic dataset, with abstract classes, that is equally hard to classify for CNNs.§ DATASET.1 10mm The presented dataset consists of two separate tasks: * The symmetry–task: The system has to decide whether an arrangement of red pawns on a checkerboard is symmetric along one of the mid lines of the checker board, or not.* The identity–task: The system has to decide whether the arrangement of red pawns on two checkerboards is identical, or not.There are multiple reasons for selecting these specific tasks: * The use of checkerboards, with randomly positioned pawns, allows us to very easily generate random samples, without inadvertently introducing additional, unwanted clues to the dataset. As we could show <cit.> for the dataset by Fleuret <cit.>, these unintended clues can be used by CNNs to classify images, and might lead to wrong conclusions about what CNNs are able to learn.* Although the images have a random component (the position of the pawns on the board), the images are still semi-realistic and show a simplified representation of what might be observable in reality.* According to the gestalt principles <cit.>, symmetry is an important property for humans to understand and order the world. It therefore seemed like a good choice for one of the tasks. The identity–task was chosen since the tests on the dataset by Fleuret <cit.> showed that CNNs have specific weakness when it comes to detecting identity.Example images for both of these tasks can be seen in <ref>. The difficulty of both tasks can be controlled in multiple different ways: * The number of pawns, breaking the symmetry/identity, can be adjusted. It should be evident that detecting a single, out of place pawn is more difficult than detecting ten pawns that are out of place.* The task can be made more challenging by increasing the visual variability of the presented images. We are using three different levels of variability: * For the lowest amount of variability, the camera position as well as the board positions are fixed. See <ref> and <ref> for example images.* For more variability, the camera is randomly moved on a plane, resulting in different board positions for each of the images. Still, in the identity–task, the relative position of the two boards stays the same. See <ref> and <ref> for example images.* The highest variability is achieved by randomly positioning the camera on a sphere, with a random radius, around the checkerboard. This results in different view points, as well as different sizes of the boards. See <ref> and <ref> for example images. * For the symmetry–task, the two checkerboards can be arranged randomly. See <ref> for example images.For each of these variability–schemes, we test versions of the dataset with one, five, and ten pawns that are out of place.The images for the dataset were generated by an automated procedure, using the 3D modeling software Blender <cit.>. Thus, an arbitrary amount of training– and testing–images can be produced very quickly in arbitrary resolutions. This also opens the door to varying lighting conditions, added clutter, additional chess pieces, …, to make the dataset more challenging. The scripts to generate the dataset can be found online[<https://github.com/Paethon/chess_image_dataset>].§ EXPERIMENTSTo evaluate the dataset, we generated 20000 training images as well as 1000 testing images for each of the tasks and difficulty levels. A resolution of 224 × 224 pixels was used. We tested the dataset on the popular network architectures Alexnet <cit.>, VGG16 <cit.>, and GoogLeNet <cit.>.For AlexNet and GoogLeNet, the standard implementations provided with the nVidia DIGITS <cit.> deep learning framework version 5.0.0 using Caffe <cit.> version 0.15.13 as a back end were used. Since we were not able to train VGG16 on the presented dataset from scratch, we used the predefined network, pre–trained on ImageNet, from the DIGITS model store. All networks were trained using ADAM, with a base learning rate of 1 × 10^-5 for 120 epochs. The training was manually stopped in cases where further improvement was not to be expected (e.g. perfect accuracy was already achieved).For each task, the networks were trained in order of increasing difficulty, and the learned weights were used as initialization for the next, more difficult, task. A network was, for example, trained on the symmetry–task with a fixed camera position and ten out of place pawns. After successful training of this network, the weights were used to initialize the network to be trained on the same task with five out of place pawns. A variation of this approach was presented by Bengio <cit.> under the name of curriculum learning. This approach was absolutely critical for training some of the more difficult variations of the dataset. We were, for example, not able to achieve a classification accuracy above chance with GoogLeNet on the identity–task with random board positions (<ref>) without this curriculum learning approach, despite the fact that we could reach a good classification accuracy of 0.86 using curriculum learning. During training, each network was evaluated on the testing set after each epoch, and an accuracy measure was recorded. Accuracy is defined as |s_c|/|s| where |s| is the number of tested samples (the number of images to be classified) and |s_c| is the number of correctly classified samples. Since we have two possible classes for all our experiments, a purely random classifier would achieve an accuracy of ≈ 0.5. For each network and task, we report the highest achieved accuracy for all of the evaluations, after each of the 120 training epochs. We thus expect even a random classifier to get a maximum accuracy above 0.5. If we assume an equal probability for both classes, 1000 samples classified per test, and 120 tests, we expect a purely random classifier to achieve a mean maximum accuracy over all 120 evaluations of ≈ 0.54, with a standard deviation of ≈ 6.6 × 10^-3. These values were determined using simulation.§.§ Discussion#+ORGTBL: SEND resultstbl orgtbl-to-latex :booktabs t :splice t :skip 0 | Task| AlexNet | VGG16 | GoogLeNet | |———————————————————+———+——-+———–| |———————————————————+———+——-+———–| | *identity*| | | | | fixed position, 10 diff (Fig.<ref>)|1.00 |1.00 |0.99 | | fixed position, 5 diff|1.00 |0.99 |0.97 | | fixed position, 1 diff|0.99 |1.00 |1.00 | |———————————————————+———+——-+———–| | camera translation, 10 diff (Fig.<ref>)|0.99 |0.99 |0.99 | | camera translation, 5 diff|0.98 |0.99 |0.98 | | camera translation, 1 diff|0.90 |0.98 |0.96 | |———————————————————+———+——-+———–| | random board placement, 10 diff (Fig.<ref>) |0.80 |0.89 |0.95 | | random board placement, 5 diff|0.73 |0.88 |0.94 | | random board placement, 1 diff|0.54 |0.69 |0.86 | |———————————————————+———+——-+———–| | camera rotation, 10 diff (Fig.<ref>) |0.54 |0.64 |0.55 | | camera rotation, 5 diff |0.52 |0.63 |0.53 | | camera rotation, 1 diff |0.51 |0.54 |0.50 | |———————————————————+———+——-+———–| |———————————————————+———+——-+———–| | *symmetry*| | | | | fixed position, 10 diff (Fig.<ref>)|1.00 |1.00 |1.00 | | fixed position, 5 diff|1.00 |1.00 |1.00 | | fixed position, 1 diff|0.99 |1.00 |1.00 | |———————————————————+———+——-+———–| | camera translation, 10 diff (Fig.<ref>)|0.99 |1.00 |1.00 | | camera translation, 5 diff|0.98 |0.99 |0.98 | | camera translation, 1 diff|0.85 |0.99 |0.92 | |———————————————————+———+——-+———–| | camera rotation, 10 diff (Fig.<ref>) |0.75 |0.85 |0.79 | | camera rotation, 5 diff |0.59 |0.80 |0.78 | | camera rotation, 1 diff |0.52 |0.59 |0.63 | |———————————————————+———+——-+———–|<ref> shows the highest achieved accuracy during training. The identity–task with fixed camera position and camera translation (<ref>) was solved almost perfectly by all tested network architectures. This is not very surprising, since the same checker board positions will always be at the same pixel positions. Thus, the networks can learn a very direct mapping, to check for identity and symmetry.Somewhat more surprising is the almost perfect performance of all three networks on the dataset variation with random camera translation (<ref>). Especially, since the translation of the camera also imparts perspective effects on the images (if the checkerboard is rendered at the top of the image, it is smaller in comparison to being rendered at the bottom). Still, the relative position of all the checkerboard positions is constant in all the images, up to some scaling factor. This might explain the overall good performance of the networks. AlexNet does perform somewhat worse with only one pawn out of place, but it still reaches a good accuracy of 0.90.Random board placement and fixed camera angle (<ref>) is interesting, since the tested architectures perform very differently on this task. GoogLeNet performs very well, even solving one pawn out of place well above chance. AlexNet performs much worse, and does not solve the one pawn out of place variant at all. VGG16 lies somewhere in the middle. The less than perfect performance is interesting, since human subjects would very likely not consider this task more difficult than the variations with fixed camera position, or camera translation. It could be the case, that features have to be integrated on a more global scope than in the other tasks, which leads to diminished performance.The variant with camera rotation (<ref>) was not solved convincingly by any of the architectures. VGG16 performs slightly better than chance, with an accuracy of 0.64 and 0.63 for ten and five out of place pawns respectively, but it also completely fails with only one out of place pawn. The images that VGG16 can correctly classify predominantly show the checkerboard in a very favorably position (top-down with little rotation). AlexNet and GoogLeNet seem to be confused enough by the rest of the training set so that they are not even able to classify these easier images.The symmetry–task seems to be easier for the networks in general. This likely has two reasons. On one hand, only 64 board positions have to be compared in comparison with 128 positions for the identity–task. On the other hand, the positions to be compared are also spatially closer, especially for the more difficult variations of the dataset.The variation with fixed camera and board position (<ref>) is solved perfectly by all the networks. Added camera translation (<ref>) shows a similar pattern to what we have seen for the identity–task. All networks solve this problem more or less perfectly, except for AlexNet, which is only able to achieve an accuracy of 0.85 for one out of place pawn. This suggests that there seems to be a general flaw in the AlexNet architecture for these kinds of problems.Adding camera rotation (<ref>) leads to more variable results. None of the networks perform perfectly, but all of them perform significantly above chance for the variation with ten out of place pawns. VGG16 and GoogLenet even perform slightly above chance for one out of place pawn.The experiments reveal a few variations of the dataset that seem to be interesting for further research: * Symmetry–task with camera rotation: This variant seems to be at the border of being solvable by current architectures and the difficulty scales well with the number of out of place pawns.* Identity–task with random board placement: The network architecture seems to be especially relevant for this task.* Identity–task with camera rotation: None of the networks were able to solve any of the variants of this task convincingly, but the fact that VGG16 does perform slightly above chance indicates that it might be possible to create a network architecture that performs much better. It would be interesting to evaluate these variations of the dataset on additional network architectures, and to analyze how human subjects solve problems of this kind. Our hypothesis is that such problems are generally not solved in a pure feed forward manner by humans, and some attentional mechanisms and iterative processing of the images are required. Attention is defined by the Encyclopedia Britannica as “the concentration of awareness on some phenomenon to the exclusion of other stimuli”. Since brains do have capacity limitations, it is impossible to process all visual information at any given time, as shown by Tsotsos <cit.>. Therefore, an attentional mechanism has to assign the available resources to task relevant stimuli. We hypothesize that pawn positions are compared not as a whole, but by an iterative switching of attention between smaller areas of the board or boards. To substantiate this hypothesis, we propose to test the classification accuracy and classification speed of human subjects on the same dataset, while also collecting eye tracking data, to get a rough estimate of shifting attention. Processing of the images in this way would hint at the possibility that attention and iterative processes might be more efficient at, or even necessary, for solving the problem classes presented in our dataset.It would also be interesting to see whether the time humans need to correctly classify an image correlates with the classification performance of a CNN. A human might for example need less time to classify a pawn arrangement if a pawn is misplaced in one of the corners.It would also be interesting to see whether current CNN architectures that already incorporate some form of attention, as well as a form of iterative processing of images, would perform better on the dataset than the already tested standard architectures. Sermanet <cit.> have shown that incorporating attention and iterative refinement of class predictions can improve the performance of CNNs.§ CONCLUSIONWe presented a novel image classification dataset that should be trivial to classify for humans. Nonetheless, certain variations of it are poorly classified by the tested CNN architectures AlexNet, VGG16, and GoogLeNet. We identified three variations of the dataset that might be interesting for further research. Detecting symmetry of pawn positions of a checkerboard, together with camera rotation, is interesting, since it seems to be on the border of what current CNN architectures can solve. Depending on the number of pawns that break the symmetry, it can, or can not be solved. Detecting identity of pawn positions on two randomly positioned checkerboards, with fixed camera position, is the second interesting variation of the dataset. From our perspective, it seems like it should be an easy task for human subjects, but the tested architectures showed highly variable performance. Third, the identity–task, with camera rotation, was not convincingly solved by any of the architectures. We therefore proposed to do additional tests on these specific variations of the dataset. In addition, experiments involving human subjects might be interesting to determine under which circumstances and by which processes humans are able to classify this dataset. Our hypothesis is that humans use some form of attentional mechanism and iterative processing to solve problems of this kind. We further hypothesize that such an approach is therefore more efficient for the given task at hand, and incorporating these principles might benefit machine learning methods.We want to thank the reviewers for the helpful comments and proposing further research.ieee | http://arxiv.org/abs/1708.07770v1 | {
"authors": [
"Sebastian Stabinger",
"Antonio Rodriguez-Sanchez"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170825151022",
"title": "Evaluation of Deep Learning on an Abstract Image Classification Dataset"
} |
Inverse of a Special Matrix and Application Thuan Nguyen School of Electrical and Computer Engineering, Oregon State University, Corvallis, OR, 97331Email: [email protected] 30, 2023 =========================================================================================================================================================== The matrix inversion is an interesting topic in algebra mathematics. However, to determine an inverse matrix from a given matrix is required many computation tools and time resourceif the size of matrix is huge. In this paper, we have shown an inverse closed form for an interesting matrix which has much applications in communication system. Base on this inverse closed form, the channel capacity closed form of a communication system can be determined via the error rate parameter α. Keywords: Inverse matrix, convex optimization, channel capacity. § MATRIX CONSTRUCTION In Wireless communication system or Free Space Optical communication system, due to the shadow effect or the turbulent of environment, the channel conditions can be flipped from “good" to “bad" or “bad" to “good" state such as Markov model after the transmission time σ <cit.> <cit.>. For simple intuition, in “bad" channel, a signal will be transmitted incorrectly and in “good" channel, the signal is received perfectly. Suppose a system has total n channels, the “good" channel is noted as “1" and “bad" channel is “0", respectively, the transmission time between transmitter and receiver is σ, the probability the channel is flipped after the transmission time σ is α. We note that if the system using a binary code such as On-Off Keying in Free Space Optical communication, then the flipped probability α is equivalent to the error rate.Consider a simple case for n=2, suppose that at the beginning, both channel is “good" channel, the probability of system has both of channels are “good" after transmission time σ, for example, is (1-α)^2. Let call A_ij is the probability of system from the state has i-1 “good" channels and n-i+1 “bad" channels transfers to state has j-1 “good" and n-j+1 “bad" channels. Obviously that 1 ≤ i ≤ n+1 and 1 ≤ j ≤ n+1. For example, thetransition matrix A_2 and A_3 for n=2 and n=3 are constructed respectively as follows:[ A_2 = [ (1-α)^2 2α(1-α) α^2;α(1-α) (1-α)^2+α^2α(1-α); α^2 2α(1-α) (1-α)^2; ]. ][ A_3 = [ (1-α)^3 3(1-α)^2α 3α^2(1-α) α^3;(1-α)^2α 2(1-α)α^2+(1-α)^3 2(1-α)^2α+α^3(1-α)α^2;(1-α)α^2 2(1-α)^2α+α^3 2(1-α)α^2+(1-α)^3(1-α)^2α; α^3 3α^2(1-α) 3(1-α)^2α (1-α)^3; ]. ] These transition matrices are obviously size (n+1)×(n+1) since the number of “good" channels can achieve n+1 discrete values from 0,1,…,n. Moreover, these class matrices have several interesting properties: (1) all entries in matrix A_n can be determined by Proposition <ref>; (2) the inverse of matrix A_n is given by Proposition <ref>. Moreover, this matrices are obviously central symmetric matrix.For n channels system, the transition matrix A_n has size (n+1) × (n+1) and all entries A_n_ij in row i column j will be established byA_n_ij=∑_s=max(i-j,0)^s=min(n+1-j,i-1)j-i+s n+1-is i-1α^j-i+2s (1-α)^n-(j-i+2s) From the definition, A_n_ij is the probability from state has i-1 “good" channels or i-1 bit “1"transfer to statehas j-1 “good" channels or j-1 bit “1". Therefore, suppose s is the number channels in i-1 “good" channels that is flipped to “bad" channelsafter the transmission time σ and 0 ≤ s ≤ i-1. Thus, to maintain j-1 “good" channels after the time σ, the number of “bad" channels in n+1-i “bad" channels must be flipped to “good" channels is:(j-1)-((i-1)-s)=j-i+sTherefore, the total number of channels are flipped their state after transmission time σ is:s+(j-i+s)=j-i+2sand the total number of channels that preserves their state after transmission time σ isn-(j-i+2s). However, 0 ≤ s ≤ i-1. Similarly, the number of “bad" channels in n+1-i “bad" channels must be flipped to “good" channels should be in 0 ≤ j-i+s ≤ n+1-i. Hence:maxs = min (n+1-j;i-1 )mins = max (0;i-j ) Therefore, A_n_ij can be determined by below form: A_n_ij=∑_s=max(i-j,0)^s=min(n+1-j,i-1)j-i+s n+1-is i-1α^j-i+2s (1-α)^n-(j-i+2s) All the entries of inverse matrix A_n^-1 given in Proposition <ref> can be determined via original transition matrix A_n for ∀ α≠ 1/2.A_n_ij^-1=(-1)^i+j(1-2α)^nA_n_ij Due to the pages limitation, we will show the detailed proof at the end of this paper.To illustrate our result, an example of the inverse matrix A_2 are shown as follows:[ A_2^-1 =1(1-2α)^2[ (1-α)^2-2α(1-α) α^2; -α(1-α) (1-α)^2+α^2 -α(1-α); α^2-2α(1-α) (1-α)^2; ] ]. Next, base on the existence of inverse matrix closed form, we will show that a capacity closed form for adiscrete memory-less channel can be established. We note that in <cit.>, the authors said that haven't closed form for channel capacity problem. However, with our approach, the closed form can be established for a wide range of channel with error rate α is small.§ OPTIMIZE SYSTEM CAPACITY A discrete memoryless channel is characterized by a channel matrix A ∈𝐑^m × n with m and n representing the numbers of distinct input (transmitted) symbols x_i, i = 1, 2, …, m,and output (received) symbols y_j, j = 1, 2, …, n, respectively. The matrix entry A_ij represents the conditional probability that given a symbol x_i is transmitted, the symbol x_j is received.Let p = (p_1, p_2, …, p_m)^T be the input probability mass vector, where p_i denotes the probability of transmitting symbol x_i, then the probability mass vector of output symbols q = (q_1, q_2, …, q_n)^T = A^Tp, where q_i denotes the probability of receiving symbol y_i. For simplicity, we only consider the case n=m such that the number of transmitted input patterns is equal the number of received input patterns. The mutual information between input and output symbolsis:I(X;Y) = H(Y) - H(Y|X),whereH(Y)= -∑_j=1^j=nq_jlogq_jH(Y|X)= ∑_i=1^m∑_j=1^np_i A_ijlogA_ij.Thus, the mutual information function can be written as:I(X;Y) = -∑_j=1^j=n(A^Tp)_jlog(A^Tp)_j +∑_i=1^m∑_j=1^np_iA_ijlogA_ij,where (A^Tp)_j denotes the jth component of the vector q = (A^Tp).The capacity C of a discrete memoryless channel associated with a channel matrix A puts a theoretical maximum rate that information can be transmitted over the channel <cit.>.It is defined as: C = max_pI(X;Y).Therefore, finding the channel capacity is to find an optimal input probability mass vector p such that the mutual information between the input and output symbols is maximized. For a given channel matrix A, I(X;Y) is a concave function in p <cit.>.Therefore, maximizing I(X;Y) is equivalent to minimizing -I(X;Y), and the capacity problem can be cast as the following convex problem:Minimize:∑_j=1^n(A^Tp)_jlog(A^Tp)_j -∑_i=1^m∑_j=1^np_i A_ijlogA_ijSubject to:p_i ≽01^Tp = 1 Optimal numerical values of p^* can be found efficiently using various algorithms such as gradient methods <cit.> <cit.>.However, in this paper, we try to figure out the closed form for optimal distribution p via KKT condition. The KKT conditions state that for the following canonical optimization problem:ProblemMiminize: f(x) Subject to:g_i(x) ≤0, i = 1, 2, … n, h_j(x) = 0, j = 1, 2, …, m, construct the Lagrangian function:L(x,λ, ν) = f(x) + ∑_i=1^nλ_i g_i(x) + ∑_j=1^mν_j h_j(x), then for i = 1, 2, …, n, j = 1, 2, …, m, the optimal point x^* must satisfy: g_i(x^*) ≤ 0, h_j(x^*) = 0,dL(x, λ, ν)/dx|_x = x^*, λ = λ^*, ν = ν^* = 0,λ_i^*x_i^* = 0,λ_i^* ≥ 0. Our transition matrix that is already established in previous part can represent as a channel matrix.In the optical transmission, for example, the transmission bits are denoted by the different levels of energy, for example, in On-Off Keying code bit “1" and “0" is represented by high and low power level. This energy is received by a photo diode and converse directly to the voltage for example. However, these photo diode work base on the aggregate property when collecting all the incident energy, that said, if two channels transmit a bit “1" then the photo diode will receive the same energy “2" even though this energy comes from a different pair of channels. Therefore, the received signal is completely dependentto the number of bits “1" in transmission side. Hence, in receiver side, the photo diode will recognize n+1 states 0,1,2,…,n. From this property, the transition matrix A is the previous section is exactly the system channel matrix. The channel capacity of system, therefore, is determined as an optimization problem in (<ref>). Next, we will show that the above optimization problem can be solved efficiently by KKT condition. We note that our method can establish the closed form for general channel matrix and then the results are applied to special matrix A_n. First, we try to optimize directly with input distribution p, however, the KKT condition for input distribution is too complicated to construct the first derivation. On the other hand, base on the existence of inverse channel matrix, the output variable is more suitable to work with KKT condition since.Due to 0 ≤ q_j ≤ 1, the Lagrange function from (<ref>) with output variable q is: L(q_j,λ_j,ν_j)= I(X,Y) + ∑_j=1^j=nq_jλ_j + ν (∑_j=1^j=nq_j-1)Using KKT conditions, at optimal point q_j^*, λ_j^*, ν^*: q_j^*≥ 0 ∑_j=1^j=nq_j^* = 1 ν^* - λ_j^* - dI(X,Y)dq_j^*= 0 λ_j^*≥ 0 λ_j^* q_j^*=0 Because 0 ≤ p_i ≤ 1, i=1,…, (n) and ∑_i=1^np_i = 1, so always exist p_i> 0. From q_j = ∑_i=1^i=np_iA_ij with ∀ A_ij>0, we can see clearly that q_j > 0 with ∀ q_j or q_j^* > 0 with ∀ q_j^*.Therefore with fifth condition, λ_j^* = 0 with ∀λ_j^*. Then, we have simplified KKT conditions:∑_j=1^j=nq_j^* = 1 ν^* - dI(X,Y)dq_j^*= 0 The derivations are determined by: dI(X,Y)dq_j = ∑_i=1^i=nA_ji^-1∑_j=1^j=nA_ijlogA_ij - (1+logq_j) Let call: ∑_i=1^i=nA_ji^-1∑_j=1^j=nA_ijlogA_ij=K_j Next, using derivation of I(X,Y) at q_j=q_j^* and last KKT condition:ν^*=K_j -(1+logq_j^*) Hence:q_j^* = 2^K_j-ν^* -1 Next, using first KKT simplified condition, we have the sum of all output states is 1. ∑_j=1^j=n 2^K_j-ν^*-1 = 12^ν^* = ∑_j=1^j=n2^K_j-1 Therefore, ν^* can be figured out by:ν^*= log∑_j=1^j=n2^K_j-1 From the second KKT simplified condition, we can compute ∀ q_j^*:q_j^* = 2^K_j-ν^* -1 And finally:p^T^*=q^T^*A_ij^-1Due to the channel matrix is a closed form of α, the optimal input vector p and output vector q also is a function of α. However, we note that since the KKT condition works directly to the output variable q, the optimal input p can be invalid p_i> 1 or p_i <0. In next step, our simulations shown that forn ≤ 10 and α≤ 0.2, both output and input vector are valid. That said, our approach will be worked with a good system where the error probability α is small. In case of the invalid optimal input vector, the upper bound of channel capacity, of course, will be established.§ CONCLUSIONIn this paper, our contributions are twofold: (1) establish an inverse closed form for a class of channel matrix based on the error probability α; (2) figure out the closed form for channel matrix with small error rate α and determine the upper bound system capacity for a high error rate channel.unsrt Proof for Proposition <ref>.To simplify our notation, the “good" and “bad" channel are represented by bit “1" and “0", respectively. Next, we will use the definition to show that: A_nA_n^-1=IIf matrix A_n^* is constructed by A_n^*_ij=(-1)^i+jA_n_ij, then we need to show that: A_nA_n^* =B = (1-2α)^n I Firstly, we note that the A_n_ij and A_n^*_ij is only different by sign of the first index (-1)^i+j. Therefore, B_ij which is computed by product of row i in matrix A_n_ij and column j in matrixA_n^*_ij, can be computed by: B_ij= ∑_k=1^k=n+1A_n_ikA_n^*_kjNote that the A_n_ik is the probability from state i “good" channels (with i-1 bit “1" and n-i+1 bit “0") to medium state has k “good" channels (with k-1 bit “1" and n-k+1 bit “0"). Moreover, if the sign is ignored, thenA_n^*_kj also is the probability going from state k to state j, too. However, the state k includes Cn k-1 sub-states which have a same number of “good" and “bad" channels.For example with n=2, state k=2 includes two sub-states that contains one “good" and one “bad" channels are “10" an “01". Therefore, the total number of sub-states while k runs from 1 to n is ∑_k=1^k=n+1Cn k-1= 2^n sub-states.Let compute B_ij by divided into two subsets:Compute B_ij for i=j: This means that B_ii is the sum of the probability from state i-1 bit “1" go tostates has k-1bit “1"then come back to state has i-1 bit “1". In 2^n sub-states, we can divide back to n+1 categories by the number of different position between i and k.∙ If all the bit in i and k are the same, then the probability is:Cn 0(1-α)^n(1-α)^n=Cn 0(1-α)^2n ∙ If all the bit in i and k different at only one position, then the probability is:Cn 1(1-α)^2(n-1)(1-α)^2 ∙ If all the bit in i and k different at only two positions, then the probability is:Cn 2(1-α)^2(n-2)(1-α)^2×2 ∙ If all the bit in i and k different at all positions, then the probability is:Cn n(1-α)^2n Therefore, B_ii can be determined by the probability of all n+1 categories such as:B_ii=∑_t=0^t=nCn tα^2t (1-α)^2n-2t=((1-α)^2 -α^2)^n=(1-2α)^nCompute B_ij for i≠ j: Letdivide A_n^*_kj into two subsets: k+j is oddand A_n^*_kj < 0 or k+j is evenand A_n^*_kj > 0, respectively. Therefore, B_ij=∑_k=1^k=nA_n_ikA_n^*_kj also is distributed into positive or negative subsets. Next, we will show that the positive subset inB_ij is equal the negative subset then B_ij=0 for i ≠ j. Indeed, suppose that state i with i-1 bit “1" go to state k_1 and then to back to state j with j-1 bit “1" and B_ik_1 is positive value.Next, we will show that existence a state k_2 such that B_ik_2 is negative value and B_ik_1=-B_ik_2. Let call s is the number of positions where state i and j have a same bit. Obviously that s≤ n-1 due to i≠ j. For example if n=4 and i=1111 and j=0001, we have s=1 because i and j share a same bit “1" in the positions fourth. Supposethat an arbitrary state k_1 are picked, we will show how to chose the state k_2with B_ik_1=-B_ik_2. Consider two follows cases:∙ If (n-s) is odd. k_2 is constructed by maintain s position of k_1 where i and j have same bit and flip bit inthe n-s rest positions.∙ If (n-s) is even. k_2 is constructed by maintain s+1 position of k_1 where s position are i and j have a same bit and one position where i and j have a different bit, next n-s-1 rest positions will be flipped. Note that since s≤ n-1 then we are able to flip n-s-1 rest positions.We obviously can see that k_1 and k_2 satisfied the probability condition |B_ik_1|=|B_ik_2| due to the number of flipped bit between i and k_1 equals the number of flipped bit between k_2 and j andthe number of flipped bit between j and k_1 equals the number of flipped bit between k_2 and i.Next, we will prove that k_1 and k_2 make B_ik_1 and B_ik_2 in different subsets. Indeed, call number of bit “1" in k_1 is b_1, number of bit “1" in k_2 is b_2, number of bit “1" in s bit same of i and j is b_s, respectively. Therefore, the number of bit “1" of k_1 in (n-s) rest positions is (k_1-k_s), the number of bit “1" of k_2 in (n-s) rest positions is (k_2-k_s).∙ If (n-s) is odd. Since all bit in (n-s) rest positions of k_1 is flipped to create k_2, then total number of bit “1" inn-s bit of k_1 and k_2 is (k_1-k_s+k_2-k_s=n-s) is odd. So, (k_1+k_2) should be an odd number.That said (k_1-k_2) is odd or (k_1+j)-(k_2+j) is odd. Therefore, B_ik_1 and B_ik_2 bring the contradict sign.∙ If (n-s) is even. Because, we fix one more position to create k_2, then number of flippedbit (n-s-1) is odd number. If one more bit is fixed in k_1 is “0", we have a same result with case (n-s) is odd. If fixed bit is “1", similarly in first case (k_1-k_s-1)+(k_2-k_s-1)=n-s-1 is odd number, therefore (k1+k2) is odd number. That said (k_1-k_2) is odd or (k_1+j)-(k_2+j) is odd. Therefore, B_ik_1 and B_ik_2 bring the contradict sign.Therefore, the state k_2 always can be created from a random state k_1 and B_ik_1 and B_ik_2 bring a contradict sign. That said for i ≠ j, B_ij=0. Therefore: B=(1-2α)^n I The Proposition <ref>, therefore, are proven. | http://arxiv.org/abs/1708.07795v1 | {
"authors": [
"Thuan Nguyen"
],
"categories": [
"cs.DM",
"cs.IT",
"math.IT"
],
"primary_category": "cs.DM",
"published": "20170825160123",
"title": "Inverse of a Special Matrix and Application"
} |
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