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+Comparison	of	tree-based	ensemble	algorithms	for	merging	satellite	
+and	earth-observed	precipitation	data	at	the	daily	time	scale	
+Georgia Papacharalampous1, Hristos Tyralis2, Anastasios Doulamis3, Nikolaos Doulamis4 
+1 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, 
+National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece 
+(papacharalampous.georgia@gmail.com, https://orcid.org/0000-0001-5446-954X) 
+2 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, 
+National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece 
+(montchrister@gmail.com, 
+hristos@itia.ntua.gr, 
+https://orcid.org/0000-0002-8932-
+4997) 
+3 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, 
+National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece 
+(adoulam@cs.ntua.gr, https://orcid.org/0000-0002-0612-5889) 
+4 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, 
+National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece 
+(ndoulam@cs.ntua.gr, https://orcid.org/0000-0002-4064-8990) 
+Abstract: Merging satellite products and ground-based measurements is often required 
+for obtaining precipitation datasets that simultaneously cover large regions with high 
+density and are more accurate than pure satellite precipitation products. Machine and 
+statistical learning regression algorithms are regularly utilized in this endeavour. At the 
+same time, tree-based ensemble algorithms for regression are adopted in various fields 
+for solving algorithmic problems with high accuracy and low computational cost. The 
+latter can constitute a crucial factor for selecting algorithms for satellite precipitation 
+product correction at the daily and finer time scales, where the size of the datasets is 
+particularly large. Still, information on which tree-based ensemble algorithm to select in 
+such a case for the contiguous United States (US) is missing from the literature. In this 
+work, we conduct an extensive comparison between three tree-based ensemble 
+algorithms, specifically random forests, gradient boosting machines (gbm) and extreme 
+gradient boosting (XGBoost), in the context of interest. We use daily data from the 
+PERSIANN (Precipitation Estimation from Remotely Sensed Information using Artificial 
+Neural Networks) and the IMERG (Integrated Multi-satellitE Retrievals for GPM) gridded 
+
+2 
+ 
+datasets. We also use earth-observed precipitation data from the Global Historical 
+Climatology Network daily (GHCNd) database. The experiments refer to the entire 
+contiguous US and additionally include the application of the linear regression algorithm 
+for benchmarking purposes. The results suggest that XGBoost is the best-performing tree-
+based ensemble algorithm among those compared. They also suggest that IMERG is more 
+useful than PERSIANN in the context investigated. 
+Keywords: contiguous US; gradient boosting machines; IMERG; machine learning; 
+PERSIANN; random forests; remote sensing; satellite precipitation correction; spatial 
+interpolation; XGBoost 
+1. 
+Introduction	
+Machine and statistical learning algorithms (e.g., those documented in Hastie et al. 2009; 
+James et al. 2013; Efron and Hastie 2016) are increasingly adopted for solving a variety of 
+practical problems in hydrology (Dogulu et al. 2015; Xu et al. 2018; Quilty et al. 2019; 
+Curceac et al. 2020; Jehn et al. 2020; Quilty and Adamowski 2020; Rahman et al. 2020; 
+Althoff et al. 2021; Fischer and Schumann 2021; Papacharalampous and Tyralis 2022b) 
+and beyond (Ahmed et al. 2010; García-Gutiérrez et al. 2015; Goetz et al. 2015; Asri et al. 
+2016; Idowu et al. 2016; Bahl et al. 2018; Feng et al. 2020; Khanam and Foo 2021; Rustam 
+et al. 2021; Bamisile et al. 2022). Among the entire pool of such algorithms, the tree-based 
+ensemble ones (i.e., those combining decision trees under properly designed ensemble 
+learning strategies; Sagi and Rokach 2018) are of special interest for many practical 
+problems, as they can offer high predictive performance with low computational cost, 
+among their remaining benefits (Tyralis et al. 2019; Tyralis and Papacharalampous 2021). 
+Still, the known theoretical properties of the various tree-based ensemble algorithms 
+(including random forests, gradient boosting machines − gbm and extreme gradient 
+boosting – XGBoost; Breiman 2001, Friedman 2001, Chen and Guestrin 2016) cannot 
+support the selection of the most appropriate one among them for each practical problem. 
+Instead, such a selection could rely on attentively designed empirical comparisons. Thus, 
+such comparisons of tree-based ensemble algorithms are conducted with increasing 
+frequency in various scientific fields (Adler et al. 2011; Ahmad et al. 2018; Fan et al. 2018; 
+Besler et al. 2019; Ahmad and Zhang 2020; Ampomah et al. 2020; Liu et al. 2020; Rahaman 
+et al. 2021; Ziane et al. 2021; Khorrami et al. 2022; Mittendorf e al. 2022; Park and Kim 
+2022; Wei et al. 2022; Yao et al. 2022). 
+
+3 
+ 
+Tree-based ensemble algorithms are regularly applied and compared to other machine 
+and statistical learning algorithms for the task of merging satellite products and ground-
+based measurements. This task is the general focus of this work, together with the general 
+concept of tree-based ensemble algorithms, and is commonly executed in the literature in 
+the direction of obtaining precipitation datasets that cover large geographical regions 
+with high density and, simultaneously, are more accurate than pure satellite precipitation 
+products. The importance of this same task could be perceived through the inspection of 
+the major research topics appearing in the hydrological literature (see, e.g., those 
+discussed in Montanari et al. 2013, Blöschl et al. 2019). Relevant examples of applications 
+and comparisons are available in He et al. (2016), Meyer et al. (2016), Baez-Villanueva et 
+al. (2020), Chen et al. 2021, Nguyen et al. (2021), Shen and Yong (2021), Zhang et al. 
+(2021), Fernandez-Palomino et al. (2022), Lei et al. (2022), Lin et al. (2022), Zandi et al. 
+(2022) and Militino et al. (2023). 
+These examples refer to various temporal resolutions and many different geographical 
+regions around the globe (see also the reviews by Hu et al. 2019 and Abdollahipour et al. 
+2022), with the daily temporal resolution and the Unites States (US) being frequent cases. 
+Nonetheless, a relevant comparison of tree-based ensemble algorithms for the latter 
+temporal resolution and the latter geographical region is missing from the literature, with 
+the closest investigations at the moment being those available in the work by Lei et al. 
+(2022), which however focus on China. In this work, we fill this specific literature gap. 
+Notably, the selection of the most accurate regression algorithm from the tree-based 
+ensemble family could be particularly useful at the daily temporal scale, in which the size 
+of the datasets for large geographical areas might impose significant limitations on the 
+application of other accurate machine and statistical learning regression algorithms due 
+to their large computational cost. 
+The remainder of the paper is structured as follows: Section 2 describes the tree-based 
+ensemble algorithms compared in this work. It also describes a machine learning metric 
+that is utilized for ensuring some degree of explainability. Moreover, Section 3 presents 
+the data retrieved and utilized for the comparisons. The same section outlines the way in 
+which the tree-based ensemble algorithms are compared with each other. Furthermore, 
+Sections 4, 5 and 6 present the results, provide their discussion in light of the literature 
+and conclude the work, respectively. Lastly, Appendix A provides statistical software 
+information that assures the work’s reproducibility. 
+
+4 
+ 
+2. 
+Methods	
+Random forests, gradient boosting machines (gbm) and extreme gradient boosting 
+(XGBoost) were applied in a cross-validation setting (see Section 3.2) for conducting an 
+extensive comparison in the context of merging gridded satellite products and gauge-
+based measurements at the daily time scale. Additionally, the linear regression algorithm 
+was applied in the same setting for benchmarking purposes. In this section, we provide 
+brief descriptions of the four afore-mentioned algorithms. Extended descriptions are out 
+of the scope of this work, as they are widely available in the machine and statistical 
+learning literature (e.g., in Hastie et al. 2009; James et al. 2013; Efron and Hastie 2016). 
+2.1 Linear regression 
+The results of this work are reported in terms of relative scores (see Section 3.3). These 
+scores were computed with respect to the linear regression algorithm, which models the 
+dependent variable as a linear weighted sum of the predictor variables (Hastie et al. 2009, 
+pp 43–55). A squared error scoring function facilitates this algorithm’s fitting. 
+2.2 Random forests 
+Random forests (Breiman 2001) are the most commonly used algorithm in the context of 
+merging gridded satellite products and gauge-based measurements (see the examples in 
+Hengl et al. 2018). A detailed description of this algorithm can be found in Tyralis et al. 
+(2019b), along with a systematic review of its application in water resources. Notably, 
+random forests are an ensemble learning algorithm and, more precisely, an ensemble of 
+regression trees that is based on bagging (acronym for “bootstrap aggregation”) but with 
+an additional randomization procedure. The latter aims at reducing overfitting. In this 
+work, random forests were implemented with all their hyperparameters kept as default. 
+For instance, the number of trees was equal to 500. 
+2.3 Gradient boosting machines 
+Gradient boosting machines (Friedman 2001, Mayr et al. 2014) are also an ensemble 
+learning algorithm that is herein used with regression trees as base learners. The main 
+concept behind this ensemble algorithm and, more generally, behind all the boosting 
+algorithms (including the one described in Section 2.4) is the iterative training of new 
+base learners using the errors of previously trained base learners (Natekin and Knoll 
+2013, Tyralis and Papacharalampous 2021). For gradient boosting machines, a gradient 
+
+5 
+ 
+descent algorithm adapts the loss function for achieving optimal fitting. This loss function 
+is the squared error scoring function herein. Consistency with the respect to the 
+implementation of random forests is ensured by setting the number of trees equal to 500. 
+The remaining hyperparameters were kept as default. 
+2.4 Extreme gradient boosting 
+Extreme gradient boosting (XGBoost; Chen and Guestrin 2016) consists the third tree-
+based ensemble learning and the second boosting algorithm implemented in this work. In 
+the implementations of this work, all the hyperparameters were kept as default, except 
+for the maximum number of iterations that were set to 500. 
+Aside from applying XGBoost in a cross-validation setting for its comparison to the 
+remaining algorithms, we also utilized it with the same hyperparameter values for 
+assuring some degree of explainability in terms of variable importance under the more 
+general explainable machine learning culture (Linardatos et al. 2020, Roscher et al. 2020, 
+Belle and Papantonis 2021). Specifically, we computed the gain importance metric, which 
+is available in the XGBoost algorithm. This metric assesses the “fractional contribution of 
+each feature to the model based on the total gain of this feature’s splits”, with higher values 
+indicating more important features (Chen et al. 2022c). 
+3. 
+Data	and	application	
+3.1 Data 
+For our experiments, we retrieved and used daily earth-observed precipitation, gridded 
+satellite precipitation and elevation data for the gauged locations and grid points shown 
+in Figures 1 and 2. 
+
+6 
+ 
+ 
+Figure 1. Map of the geographical locations of the earth-located stations that offered data 
+for this work. 
+
+-120
+-100
+-80
+Longitude (°)3itude97 
+ 
+ 
+ 
+Figure 2. Maps of the geographical locations of the points composing the (a) PERSIANN 
+and (b) IMERG grids utilized in this work. 
+
+-120
+-100
+80
+Longitude (°)(b)8(a)&58 
+ 
+3.1.1 Earth-observed precipitation data 
+Daily precipitation totals from the Global Historical Climatology Network daily (GHCNd) 
+(Durre et al. 2008, 2010, Menne et al. 2012) were used for comparing the algorithms. 
+More precisely, data from 7 264 earth-located stations spanning across the contiguous 
+United States (see Figure 1) were extracted. This data cover the two-year time period 
+2014−2015. Data retrieval was made from the website of the NOAA National Climatic Data 
+Center (https://www1.ncdc.noaa.gov/pub/data/ghcn/daily; assessed on 2022-02-27). 
+3.1.2 Satellite precipitation data 
+For comparing the algorithms, we additionally used gridded satellite daily precipitation 
+data from the current operational PERSIANN (Precipitation Estimation from Remotely 
+Sensed Information using Artificial Neural Networks) system (see the geographical 
+locations of the extracted PERSIANN grid with a spatial resolution of 0.25 degree x 0.25 
+degree in Figure 2a) and the GPM IMERG (Integrated Multi-satellitE Retrievals) Late 
+Precipitation L3 1 day 0.1 degree x 0.1 degree V06 (see the geographical locations of the 
+extracted IMERG grid in Figure 2b). These two gridded satellite precipitation databases 
+were developed by the Centre for Hydrometeorology and Remote Sensing (CHRS) at the 
+University of California, Irvine (UCI) and the NASA Goddard Earth Sciences (GES) Data 
+and Information Services Center (DISC), respectively. More precisely, the PERSIANN data 
+were retrieved from the website of the Center for Hydrometeorology and Remote Sensing 
+(CHRS) (https://chrsdata.eng.uci.edu; assessed on 2022-03-07) and the IMERG data were 
+retrieved from the website of NASA (National Aeronautics and Space Administration) 
+Earth Data (https://doi.org/10.5067/GPM/IMERGDL/DAY/06; assessed on 2022-12-
+10). The extracted data cover the entire contiguous United States at the two-year time 
+period 2014−2015. 
+3.1.3 Elevation data 
+Elevation is a key predictor variable for many hydrological processes (Xiong et al. 2022). 
+Therefore, we estimated its value for all the geographical locations shown in Figures 1 
+and 2. For this estimation, we relied on the Amazon Web Services (AWS) Terrain Tiles 
+(https://registry.opendata.aws/terrain-tiles; assessed on 2022-09-25). 
+3.2 Validation setting and predictor variables 
+To formulate the regression settings of this work, we first defined earth-observed daily 
+
+9 
+ 
+total precipitation at a point of interest (which could be station 1 in Figure 3) as the 
+dependent variable. Then, we adopted procedures proposed in Papacharalampous et al. 
+(2023) to compute the observations of possible predictor variables. Separately for each 
+of the two satellite precipitation grids (see Figure 2), we determined the closest four grid 
+points to each ground-based station from those depicted in Figure 1. We also computed 
+the distances di, i = 1, 2, 3, 4 from these grid points and indexed the latter as Si, i = 1, 2, 3, 
+4 based on the following order: d1 < d2 <d3 < d4 (see Figure 3). Throughout this work, the 
+distances di, i = 1, 2, 3, 4 are also respectively called “PERSIANN distances 1−4” or “IMERG 
+distances 1−4” (depending on whether we refer to the PERSIANN grid or the IMERG grid) 
+and the daily precipitation values at the grid points 1−4 are called “PERSIANN values 1−4” 
+or “IMERG values 1−4” (depending on whether we refer to the PERSIANN grid or the 
+IMERG grid).  
+ 
+Figure 3. Setting of the regression problem. Note that the term “grid point” is used to 
+describe the geographical locations with satellite data, while the term “station” is used to 
+describe the geographical locations with ground-based measurements. Note also that, 
+throughout this work, the distances di, i = 1, 2, 3, 4 are also respectively called “PERSIANN 
+distances 1−4” or “IMERG distances 1−4” (depending on whether we refer to the 
+PERSIANN grid or the IMERG grid) and the daily precipitation values at the grid points 
+1−4 are called “PERSIANN values 1−4” or “IMERG values 1−4” (depending on whether we 
+refer to the PERSIANN grid or the IMERG grid). 
+Based on the above, the predictor variables for the technical problem of interest could 
+include the PERSIANN values 1−4, the IMERG values 1−4, the PERSIANN distances 1−4, 
+the IMERG distances 1−4 and the station’s elevation. We defined and examined three sets 
+
+grid point 2
+grid point 4d
+station 1
+dSatellite data grid
+Gauge station
+Distance, d, i = 1, 2, 3, 4
+d<d<d<d
+grid point 1
+grid point 310 
+ 
+of predictor variables (see Table 1). Each of them defines a different regression setting 
+that includes 4 833 007 samples. These samples were exploited under a two-fold cross-
+validation scheme for comparing the three tree-based ensemble algorithms outlined in 
+Section 2 in the context of merging gridded satellite precipitation products and gauge-
+based precipitation measurements at the daily temporal scale. The same samples were 
+explored by estimating the Spearman correlation (Spearman 1904) for the various pairs 
+of variables and by ranking the predictor variables based on their importance in the 
+regression. The latter methodological step was made by applying explainable machine 
+learning procedures offered by the XGBoost algorithm (see Section 2.4). 
+Table 1. Inclusion of predictor variables in the predictor sets examined in this work. 
+Predictor variable 
+Predictor set 1 
+Predictor set 2 
+Predictor set 3 
+PERSIANN value 1 
+✔ 
+× 
+✔ 
+PERSIANN value 2 
+✔ 
+× 
+✔ 
+PERSIANN value 3 
+✔ 
+× 
+✔ 
+PERSIANN value 4 
+✔ 
+× 
+✔ 
+IMERG value 1 
+× 
+✔ 
+✔ 
+IMERG value 2 
+× 
+✔ 
+✔ 
+IMERG value 3 
+× 
+✔ 
+✔ 
+IMERG value 4 
+× 
+✔ 
+✔ 
+PERSIANN distance 1 
+✔ 
+× 
+✔ 
+PERSIANN distance 2 
+✔ 
+× 
+✔ 
+PERSIANN distance 3 
+✔ 
+× 
+✔ 
+PERSIANN distance 4 
+✔ 
+× 
+✔ 
+IMERG distance 1 
+× 
+✔ 
+✔ 
+IMERG distance 2 
+× 
+✔ 
+✔ 
+IMERG distance 3 
+× 
+✔ 
+✔ 
+IMERG distance 4 
+× 
+✔ 
+✔ 
+Station elevation 
+✔ 
+✔ 
+✔ 
+3.3 Performance metrics and assessment 
+The performance assessment relied on procedures proposed by Papacharalampous et al. 
+(2023). These procedures are reported in what follows. First, we computed the median of 
+the squared error function, separately for each set {algorithm, predictor set, test fold}. 
+Note that the squared error scoring function can adequately support our performance 
+comparisons, as it is consistent for the mean functional of the predictive distributions 
+(Gneiting 2011). Subsequently, two relative scores (which are else referred to as “relative 
+improvements” throughout this work) were computed for each set {algorithm, predictor 
+set}. For that, the two median squared error (MedSE) values offered by each set 
+{algorithm, predictor set} (each corresponding to a different test fold) were utilized, 
+
+11 
+ 
+together with their corresponding MedSE values offered by the reference modelling 
+approach, which was defined as the linear regression when run with the same predictor 
+set as the modelling approach to which the relative score referred. More precisely, the 
+relative score was computed as the difference between the score of the set {algorithm, 
+predictor set} minus the score of the reference modelling approach, multiplied with 100 
+and divided by the score of the reference modelling approach. Then, mean relative scores 
+(which are else referred to as “mean relative improvements” throughout this work) were 
+computed by averaging, separately for each set {algorithm, predictor set}, the relative 
+scores. The procedures for computing the relative scores and the mean relative scores 
+were repeated by considering the set {linear regression, predictor set 1} as the reference 
+modelling approach for all the sets {algorithm, predictor set}. 
+Mean rankings of the machine and statistical learning algorithms were also computed. 
+For that, and separately for each set {case, predictor set, test fold}, we first ranked the four 
+algorithms based on their squared errors. Then, we averaged these rankings, separately 
+for each set {predictor set, test fold}. Lastly, we obtained the mean rankings reported by 
+averaging the two previously computed mean ranking values corresponding to the same 
+predictor set. We also computed the rankings collectively for all the predictor sets.  
+4. 
+Results	
+4.1 Regression setting exploration 
+Regression setting explorations can facilitate interpretations of the results of prediction 
+experiments, at least to some extent. Therefore, in Figure 4, we present the Spearman 
+correlation estimates for the various variable pairs appearing in the regression settings 
+examined in this work. As it could be expected, the magnitude of the relationships 
+between the predictand (i.e., the precipitation quantity observed at the earth-located 
+stations, which is referred to as “true value” in Figure 4) and the 17 predictor variables 
+seems to depend, to some extent, on the satellite rainfall product. Indeed, the Spearman 
+correlation estimates made for the relationships between the predictand and 
+precipitation quantities from the IMERG grid are equal to 0.45, while the corresponding 
+estimates for the case of the RERSIANN grid are equal to 0.40. The remaining relationships 
+between the predictand and predictor variables are far less intense, almost negligible, 
+based on the Spearman correlation statistic. Still, they could also provide information in 
+the regression settings. 
+
+12 
+ 
+ 
+Figure 4. Heatmap of the Spearman correlation estimates for the various variable pairs 
+appearing in the regression settings of this work. 
+The relationships between the predictor variables also exhibit various intensities. The 
+most intense among them, according to the Spearman correlation statistic, are the 
+relationships between the PERSIANN values, for which the estimates obtained are equal 
+to 0.90, 0.91, 0.92 and 0.93. The relationships between the IMERG values are also intense, 
+with the corresponding Spearman correlation estimates being equal to 0.80, 0.82, 0.83, 
+0.84 and 0.85. The Spearman correlation estimates referring to the relationships between 
+the distances, as well as those referring to the relationships between the distances and 
+the earth-located station’s elevation (with the latter being referred to as “station 
+elevation” in the visualizations), are either positive or negative and smaller (in absolute 
+terms) than the Spearman correlation estimates referring to the relationships between 
+the PERSIANN values and the relationships between the IMERG values. Still, some of them 
+are of similar magnitude as those referring to the relationships between the PERSIANN 
+and IMERG values. 
+
+sIP
+True
+G
+sip
+1sip
+IMERG
+IMERG
+Station e
+N
+PERSIANN
+PERSIANN
+PERSIANN
+Variable0.4
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+0.58
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+0.45
+0
+-0.01 -0.01 -0.01
+-0.01 -0.01 -0.01 -0.02
+-0.04
+PERSIANN
+ value
+T
+T
+T
+T
+人
+T
+T
+T
+T
+3
+3
+True
+?
+3
+、
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+-0.01
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+-0.02
+0.02 -0.02 -0.02 -0.02
+-0.02 -0.02 -0.02 -0.02
+0.52
+0.34
+-0.57
+-0.45
+-0.6
+-0.08
+0.79
+-0.1
+-0.01
+0.01 -0.01 -0.01 -0.01
+-0.01 -0.02 -0.01 -0.01
+0.56
+0.27
+-0.59-0.56
+-0.49
+9-0.34
+0.79
+-0.08
+distance 
+RG
+anceSpearman
+correlation -1.0
+-0.5
+0.0
+0.5
+1.013 
+ 
+Furthermore, Figure 5 presents the importance scores and rankings computed for the 
+17 predictor variables through the extreme gradient boosting (XGBoost) algorithm and 
+by considering all of these predictor variables in the regression setting. The four IMERG 
+values are found to be the most important predictor variables. Moreover, the fifth and 
+sixth most important predictors are PERSIANN value 1 and station elevation, respectively, 
+and PERSIANN values 2−4 follow in the line, while the eight distances are the eight least 
+important predictor variables. Notably, the fact that station elevation is more important 
+than three of the four PERSIANN values could not be expected by inspecting the Spearman 
+correlation estimates (see again Figure 4). 
+ 
+Figure 5. Barplot of the gain scores computed for the predictor variables by utilizing the 
+extreme gradient boosting algorithm. The predictor variables are presented from the 
+most to the least important ones (from top to bottom) based on the same scores. 
+4.2 Algorithm comparison 
+Figure 6 facilitates the comparison of the four machine and statistical learning algorithms 
+in terms of the square error function, separately for each predictor set. The mean relative 
+improvements (see Figure 6a) suggest that extreme gradient boosting (XGBoost) is the 
+best algorithm for all the predictor sets. For predictor set 1 (which incorporates, among 
+others, information from the PERSIANN gridded precipitation dataset; see Table 1), 
+random forests exhibits quite close performance to that of XGBoost. At the same time, for 
+predictor set 2 (which incorporates, among others, information from the IMERG gridded 
+precipitation dataset; see Table 1), gradient boosting machines (gbm) are those exhibiting 
+quite close performance to that of XGBoost. Also notably, the mean rankings (see Figure 
+
+PERSIANN distance 1
+TMERG distance 1
+IMERG distance 4
+0.0
+0.1
+0.2
+0.3
+0.4
+GainPredic
+PERSIANN distance 3
+TMERG distance 2
+PERSIANN distance 2
+PERSIANN distance 4Station elevation :
+tor variable
+PERSIANN value 2
+PERSIANN value 4 :
+PERSIANN value 3 :IMERG value 1
+IMERG value 2
+IMERG value 3
+IMERG value 4 :
+PERSAMNae :14 
+ 
+6b) of random forests and XGBoost are of similar magnitude. In terms of the same 
+criterion, gbm scores much closer to random forests and XGBoost than to linear 
+regression. 
+ 
+ 
+Figure 6. Heatmaps of the: (a) relative improvement (%) in terms of the median square 
+error metric, averaged across the two folds, as this improvement was provided by each 
+tree-based ensemble algorithm with respect to the linear regression algorithm; and (b) 
+mean ranking of each machine and statistical learning algorithm, averaged across the two 
+folds. The computations were made separately for each prediction set. The more reddish 
+the colour, the better the predictions on average. 
+Moreover, Figure 7 facilitates more detailed comparisons with respect the frequency 
+with which each algorithm appeared in the various positions from the first to the fourth 
+in the experiments. Here again, the comparisons can be made both across algorithms and 
+across predictor sets. The respective results are somewhat similar across predictor sets. 
+Indeed, linear regression is much more likely to be found at the fourth (i.e., the last) place 
+than at any other place. It is also more likely to be found at the first place than at the 
+second and third places. At the same time, random forests are more likely to be ranked 
+first than second, third and fourth, and gbm appears most often in the second and third 
+positions. The last position is the least likely for both gbm and XGBoost. The latter 
+algorithm is more likely to be ranked second; yet, the first and third positions are also far 
+more likely than the last. 
+
+Line
+regressic
+Rando
+fores
+Gradie
+boostir
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Mean ranking
+2.4
+2.6
+2.8
+3.0(b) Mean ranking
+Predictor set 1 
+2.97
+2.31
+2.45
+2.27
+Predictor set 2 
+3.05
+2.24
+2.43
+2.27
+Predictor set 3 -
+3.05
+2.27
+2.43
+2.26
+ms
+mgs
+emgLine
+regressic
+Rando
+fores
+Gradie
+boostir
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Mean relative
+improvement
+60
+40
+20
+0(a) Mean relative improvement
+set
+Predictor set 1 :
+0
+53.99
+34.72
+56.26
+Predictor s
+Predictor set 2 :
+0
+54.39
+62.61
+64.55
+Predictor set 3 :
+0
+37.57
+47.99
+52.66
+ms
+mgs
+eug15 
+ 
+  
+ 
+Figure 7. Heatmaps of the percentages (%) with which the four machine and statistical 
+learning algorithms were ranked from 1 to 4 for the predictor sets (a−c) 1−3. The rankings 
+summarized with this figure were computed separately for each pair {case, prediction 
+set}. The darker the colour, the higher the percentage. 
+Lastly, Figures 8 and 9 allow us to compare the degree of information that is offered by 
+the two gridded precipitation products within the context of our regression problem, 
+further than the comparisons already allowed by the variable importance explorations 
+using the gain metric incorporated into the XGBoost algorithm (see again Figure 5). 
+Overall, the IMERG dataset was proven to be far more information-rich than the 
+PERSIANN dataset, in terms of both mean relative improvement (see Figure 8a) and mean 
+ranking (see Figure 8b). Indeed, the relative improvements with respect to the linear 
+regression algorithm run with the predictor set 1 are much larger for the tree-based 
+algorithms when these algorithms are run with predictor set 2 than when they are run 
+with predictor set 1. Also, predictor set 3 (which contains information from both gridded 
+precipitation datasets) does not improve the performances notably in terms of mean 
+relative improvements with respect to predictor set 2, although it does in terms of mean 
+ranking. While the best modelling choice is {XGBoost, predictor set 3}, random forests 
+were ranked in the two first positions more often than any other algorithm for predictor 
+sets 2 and 3, when the ranking was made collectively for all the predictor sets (see Figure 
+9). Still, for the same predictor sets, XGBoost appeared in the last few positions much less 
+often and achieved the best performance in terms of mean ranking when run with 
+predictor set 3. 
+
+Line
+regressic
+Rando
+fores
+Gradie
+boostil
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Percentage
+10
+20
+30
+40
+50(c) Predictor set 3
+1
+21.66
+37.97
+16.6
+23.77
+2
+9.12
+19.66
+33.53
+37.7
+3
+12.18
+19.54
+40.54
+27.74
+4
+57.05
+22.83
+9.33
+10.79
+-
+ms
+mgs
+egLine
+regressic
+Rando
+fores
+Gradie
+boostil
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Percentage
+10
+20
+30
+40
+50(b) Predictor set 2
+1
+23.17
+39.16
+16.07
+21.6
+2
+7.44
+18.82
+34.28
+39.46
+3
+10.11
+20.86
+40.06
+28.97
+4
+59.27
+21.16
+9.59
+9.97
+T
+ms
+mgs
+emgLine
+regressic
+Rando
+fores
+Gradie
+boosti
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Percentage
+20
+30
+40
+50(a) Predictor set 1
+23.74
+31.44
+19.94
+24.88
+Ranking
+2
+10.71
+26.34
+28.74
+34.22
+3
+10.28
+22.06
+37.94
+29.72
+4
+55.27
+20.16
+13.38
+11.19
+ms
+gs
+eg16 
+ 
+ 
+ 
+Figure 8. Heatmaps of the: (a) relative improvement (%) in terms of the median square 
+error metric, averaged across the two folds, as this improvement was provided by each 
+tree-based ensemble algorithm with respect to the linear regression algorithm, with this 
+latter algorithm being run with the predictor set 1; and (b) mean ranking of each machine 
+and statistical learning algorithm, averaged across the two folds. The computations were 
+made collectively for all the predictor sets. The more reddish the colour, the better the 
+predictions on average. 
+ 
+ 
+Figure 9. Heatmaps of the percentages (%) with which the four machine and statistical 
+learning algorithms were ranked from 1 to 12 for the predictor sets (a−c) 1−3. The 
+rankings summarized with this figure were computed separately for each case and 
+collectively for all the predictor sets. 
+5. 
+Discussion	
+Overall, extreme gradient boosting (XGBoost) was proven to perform notably better than 
+random forests and gradient boosting machines (gbm) when merging gridded satellite 
+precipitation products and ground-based precipitation measurements for the contiguous 
+
+Linear
+regression
+Random
+forests
+Gradient
+boosting
+machines
+Extreme
+gradient
+boosting
+Algorithm
+Percentage
+5
+10 15
+20
+256
+7.07
+7.64
+15.59
+10.81
+7.
+9.86
+7.13
+13.62
+9.52
+8 -
+12.03
+7.57
+7.55
+7.85
+- 6
+13.78
+6.19
+4.94
+5.43
+10-
+26.84
+5.01
+3.48
+3.41
+11-
+2.32
+6.04
+2.59
+2.25
+12 -
+0.61
+5.36
+1.19
+1.85(c) Predictor set 3
+1:
+2.92
+20.83
+5.78
+10.02
+2 -
+5.03
+11.22
+7.11
+9.26
+3 -
+5.62
+7.78
+10.48
+13.19
+4 -
+5.71
+7.9
+12.71
+13.69
+5 -
+8.21
+7.34
+14.97
+12.7210 -
+6.67
+4.88
+5.42
+5.68
+11
+34.08
+6.13
+4.78
+5.33
+12
+7.3
+10.1
+3.92
+4.07
+Linear
+regression
+Random
+forests
+Gradient
+boosting
+machines
+Extreme
+gradient
+boosting
+Algorithm
+Percentage
+10
+20
+30(b) Predictor set 2
+1
+5.93
+14.38
+4.37
+7.54
+2
+5.33
+17.59
+8.02
+10.93
+3 -
+3.98
+8.35
+9.88
+11.22
+4 
+3.9
+7.09
+10.64
+11.05
+5 .
+4.38
+6.36
+10.22
+10.55
+- 9
+7.55
+6.3
+10.18
+9.6
+7 -
+7.92
+6.38
+11.32
+8.52
+8 -
+6.86
+6.43
+13.31
+8.1
+F 6
+6.1
+6.02
+7.92
+7.42Linear
+regression
+Random
+forests
+Gradient
+boosting
+machines
+Extreme
+gradient
+boosting
+Algorithm
+Percentage
+10
+20
+30Rankir
+2.74
+7.84
+6.47
+8.22
+7-
+2.97
+8.32
+7.12
+7.33
+8
+4.1
+8.2
+8.44
+9.56
+9
+9.82
+8.87
+11.77
+11.74
+10
+8.4
+6.4
+15.17
+8.64
+11 -
+11.36
+7
+8.52
+9.6
+12
+37.94
+12.98
+8.41
+6.27(a) Predictor set 1
+1
+7.14
+8.23
+5.67
+7.2
+2 
+3.67
+6.95
+6.66
+8.23
+4.25
+8.72
+8.32
+8.21
+4 -
+4.54
+8.57
+6.97
+7.24
+5 -
+3.07
+7.93
+6.48
+7.77
+gLine
+regressic
+Rando
+fores
+Gradie
+boosti
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Mean ranking
+6
+7
+8(b) Mean ranking
+Predictorset 1
+8.83
+6.7
+7.13
+6.66
+Predictor set 2
+8.06
+5.6
+6.16
+5.73
+Predictor set 3 -
+7.27
+5.28
+5.48
+5.11
+emgLine
+regressic
+Rando
+fores
+Gradie
+boostir
+machine
+Extrem
+gradie
+boostir
+Algorithm
+Mean relative
+improvement
+60
+40
+20
+0(a) Mean relative improvement
+Predictor set
+Predictor set 1 :
+0
+53.99
+34.72
+56.26
+Predictor set 2 :
+21.94
+64.4
+70.81
+72.33
+Predictor set 3 :
+43.14
+64.5
+70.43
+73.08
+ms
+mgs
+eug17 
+ 
+United States at the daily time scale. Also, the variable importance scores and the 
+predictive performance comparison across predictor sets indicate that the IMERG 
+product offers more useful predictors than the PERSIANN product for the same time scale. 
+In summary, when the former of these products is utilized (either alone or together with 
+the latter of them), random forests are far behind of both XGBoost and gbm in terms of 
+accuracy. On the other hand, when PERSIANN is utilized, without IMERG being utilized as 
+well, gbm is far behind of both XGBoost and random forests. 
+This latter result agrees, to some extent, with results obtained for the monthly time 
+scale in Papacharalampous et al. (2023), although the relative scores with respect to the 
+linear regression algorithm were found to be somewhat lower therein. Of course, the 
+comparison in this latter work relied on the PERSIANN satellite dataset only. Still, it is 
+accurate to deduce that the improvements in performance with respect to the linear 
+regression algorithm offered by XGBoost, gbm and random forests, when all these four 
+algorithms are run with the same predictor variables, are very large (i.e., from 
+approximately 25% to approximately 65%) for both the daily and monthly time scales. 
+Notably, even larger improvements could be achieved by combining predictions of 
+diverge algorithms in advanced or even simple ensemble learning frameworks, following 
+research efforts made in various fields (e.g., those by Wolpert 1992, Lichtendahl et al. 
+2013, Bogner et al. 2017, Sagi and Rokach 2018, Yao et al. 2018, Papacharalampous et al. 
+2019, Tyralis et al. 2019a, Kim et al. 2021, Lee and Ahn 2021, Tyralis et al. 2021, Granata 
+et al. 2022, Li and Yang 2022). 
+As the main concepts behind the boosting and random forest families of algorithms are 
+different (see Section 2, for brief summaries of these concepts), their combinations could 
+be investigated in the direction of achieving these further improvements with a low 
+computational cost. Moreover, their combination with the linear regression algorithm 
+could also be investigated. Indeed, in some contexts, even the least accurate algorithms 
+could benefit ensemble learning solutions (see, e.g., the relevant comparison outcome in 
+Papacharalampous and Tyralis 2020). In cases where the computational cost does not 
+constitute a limiting factor in algorithm selection, neural network (Cheng and Titterington 
+1994, Jain et al. 1996, Paliwal and Kumar 2009) and deep learning (LeCun et al. 2015, 
+Schmidhuber 2015) regression algorithms could be added to the ensembles. Lastly, 
+instead of aiming at providing accurate mean-value predictions, one could aim at 
+providing accurate median-value predictions coupled with useful uncertainty estimates. 
+
+18 
+ 
+This would require working on machine and statistical learning methods, such as those 
+summarized and popularized in the reviews by Papacharalampous and Tyralis (2022a) 
+and Tyralis and Papacharalampous (2022a). 
+6. 
+Conclusions	
+Precipitation datasets that simultaneously cover large regions with high density and are 
+more accurate than satellite precipitation products can be obtained by correcting such 
+products using earth-observed datasets together with machine and statistical learning 
+regression algorithms. Tree-based ensemble algorithms are adopted in various fields for 
+solving algorithmic problems with high accuracy and lower computational cost compared 
+to other algorithms. Still, information on which tree-based ensemble algorithm to select 
+when the merging is conducted for the contiguous United States (US) and at the daily time 
+scale, at which the computational requirements might constitute a crucial factor to 
+consider along with accuracy, is missing from the literature of satellite precipitation 
+product correction. 
+Herein, we work towards filling this methodological gap. We conduct an extensive 
+comparison between three tree-based ensemble algorithms, specifically random forests, 
+gradient boosting machines (gbm) and extreme gradient boosting (XGBoost). We exploit 
+daily information from the PERSIANN (Precipitation Estimation from Remotely Sensed 
+Information using Artificial Neural Networks) and the IMERG (Integrated Multi-satellitE 
+Retrievals for GPM) gridded datasets, and daily earth-observed information from the 
+Global Historical Climatology Network daily (GHCNd) database. The entire contiguous US 
+is examined and results that are generalizable are obtained. These results indicate that 
+XGBoost is more accurate than random forests and gbm. They also indicate that IMERG is 
+more useful than PERSIANN in the context investigated. 
+Conflicts	of	interest:	The authors declare no conflict of interest. 
+Author	contributions:	GP and HT conceptualized and designed the work with input from 
+AD and ND. GP and HT performed the analyses and visualizations, and wrote the first 
+draft, which was commented and enriched with new text, interpretations and discussions 
+by AD and ND. 
+Funding:	This work was conducted in the context of the research project BETTER RAIN 
+(BEnefiTTing from machine lEarning algoRithms and concepts for correcting satellite 
+RAINfall products). This research project was supported by the Hellenic Foundation for 
+
+19 
+ 
+Research and Innovation (H.F.R.I.) under the “3rd Call for H.F.R.I. Research Projects to 
+support Post-Doctoral Researchers” (Project Number: 7368). 
+Appendix	A 
+Statistical	software 
+We used the R programming language (R Core Team 2022) to implement the algorithms, 
+and to report and visualize the results. 
+For data processing and visualizations, we used the contributed R packages caret 
+(Kuhn 2022), data.table (Dowle and Srinivasan 2022), elevatr (Hollister 2022), 
+ncdf4 (Pierce 2021), rgdal (Bivand et al. 2022), sf (Pebesma 2018, 2022), spdep 
+(Bivand 2022, Bivand and Wong 2018, Bivand et al. 2013), tidyverse (Wickham et al. 
+2019, Wickham 2022). 
+The algorithms were implemented by using the contributed R packages gbm 
+(Greenwell et al. 2022), ranger (Wright 2022, Wright and Ziegler 2017), xgboost 
+(Chen et al. 2022c). 
+The performance metrics were computed by implementing the contributed R package 
+scoringfunctions (Tyralis and Papacharalampous 2022a, 2022b). 
+Reports were produced by using the contributed R packages devtools (Wickham et 
+al. 2022), knitr (Xie 2014, 2015, 2022), rmarkdown (Allaire et al. 2022, Xie et al. 2018, 
+2020). 
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+https://doi.org/10.1080/15435075.2021.1904945. 
+

1NAzT4oBgHgl3EQfDfoM/content/tmp_files/2301.00975v1.pdf.txt

@@ -0,0 +1,1995 @@
+1
+Surveillance Face Anti-spoofing
+Hao Fang, Ajian Liu, Jun Wan, Senior Member, IEEE, Sergio Escalera, Senior Member, IEEE
+Chenxu Zhao, Xu Zhang, Stan Z. Li, Fellow , IEEE, Zhen Lei, Senior Member, IEEE
+Abstract—Face Anti-spoofing (FAS) is essential to secure face
+recognition systems from various physical attacks. However,
+recent research generally focuses on short-distance applications
+(i.e., phone unlocking) while lacking consideration of long-
+distance scenes (i.e., surveillance security checks). In order to
+promote relevant research and fill this gap in the community,
+we collect a large-scale Surveillance High-Fidelity Mask (SuHi-
+FiMask) dataset captured under 40 surveillance scenes, which has
+101 subjects from different age groups with 232 3D attacks (high-
+fidelity masks), 200 2D attacks (posters, portraits, and screens),
+and 2 adversarial attacks. In this scene, low image resolution
+and noise interference are new challenges faced in surveillance
+FAS. Together with the SuHiFiMask dataset, we propose a
+Contrastive Quality-Invariance Learning (CQIL) network to
+alleviate the performance degradation caused by image quality
+from three aspects: (1) An Image Quality Variable module
+(IQV) is introduced to recover image information associated
+with discrimination by combining the super-resolution network.
+(2) Using generated sample pairs to simulate quality variance
+distributions to help contrastive learning strategies obtain robust
+feature representation under quality variation. (3) A Separate
+Quality Network (SQN) is designed to learn discriminative
+features independent of image quality. Finally, a large number
+of experiments verify the quality of the SuHiFiMask dataset and
+the superiority of the proposed CQIL.
+Index Terms—Face anti-spoofing, Dataset, Surveillance scenes.
+I. INTRODUCTION
+F
+ACE Presentation Attack Detection (PAD) technology is
+a crucial step to enhance the security of face recognition
+systems and plays an increasingly important role in resisting
+malicious attacks, such as print-attack [1], replay-attack [2], or
+face-mask [3]. Although current works [4]–[13] have achieved
+satisfactory performance in short-distance applications, such
+as phone unlocking, face payment, and access authentication,
+they are still sensitive to face quality and fail in long-distance
+applications, which hinders the expansion of FAS to surveil-
+lance scenarios.
+With the popularity of remote cameras and the improvement
+of surveillance networks, the development of smart cities
+Corresponding author: Jun Wan (e-mail: jun.wan@ia.ac.cn).
+Hao Fang, Ajian Liu, Jun Wan and Zhen Lei are with the National Laboratory
+of Pattern Recognition (NLPR), Institute of Automation Chinese Academy of
+Sciences (CASIA) and School of Artificial Intelligence, University of Chi-
+nese Academy of Sciences (UCAS), Beijing, China (e-mail: {fanghao2021,
+ajian.liu, jun.wan, zhen.lei}@ia.ac.cn).
+Xu Zhang is with Beijing Normal University School of Artificial Intelligence,
+Beijing, China (e-mail: xuzhang0908@mail.bnu.edu.cn).
+Chenxu Zhao is with the SailYond Technology, Beijing, China ( e-mail:
+zhaochenxu@sailyond.com).
+Sergio Escalera is with the Universitat de Barcelona (UB), Barcelona,
+Computer Vision Center (CVC), and Aalborg University (AAU) (e-mail:
+sergio@maia.ub.es).
+Stan
+Z.
+Li
+is
+with
+Westlake
+University,
+Hangzhou,
+China
+(e-mail:
+stan.zq.li@westlake.edu.cn).
+Fig. 1. Performance comparisons on the SiW, HiFiMask, OULU-NPU, and
+proposed SuHiFiMask dataset using the same ResNet18 network. It shows
+significant performance degradation under surveillance FAS.
+has put forward higher requirements for traditional visual
+technologies in surveillance. Benefited from the release of face
+recognition datasets [14]–[16] in the surveillance scene and
+driven by related algorithms [17]–[19], the face recognition
+system has gradually got rid of the constraint of verification
+distance, and can use the surveillance camera to complete
+real-time capture, self-service access control, and self-service
+supermarket payment. However, the FAS community is still
+stuck in the protection of the face recognition system under
+short-distance conditions, and cannot serve for the detection
+of spoofing faces under a long-distance natural behavior. We
+analyze two reasons that hinder the development of PAD tech-
+nologies: (1) Lack of a dataset that can truly simulate the
+attack in surveillance. The existing FAS datasets, whether 2D
+print or replay attacks [4], [20], [21], or 3D mask attacks [3],
+[22]–[25], require the subjects to face the acquisition device
+under distance constraints. However, diversified surveillance
+scenes, rich spoofing types, and natural human behavior are
+important assessment factors for the surveillance FAS dataset
+collection. (2) Low-quality faces in the surveillance scenar-
+ios cannot meet the requirements of fine-grained feature-
+based FAS tasks. The existing FAS algorithms, whether based
+on color-texture feature learning [26]–[29], face depth struc-
+ture fitting [4], [6], or remote photoplethysmography (rPPG)-
+based detection [30]–[32], require high-quality image details
+to ensure high performances. As illustrated in Fig. 1, the
+resolution of faces under long-distance surveillance is small
+and contains noise from motion blur, occlusion, bad weather,
+and other bad factors. These are new challenges for algorithm
+arXiv:2301.00975v1  [cs.CV]  3 Jan 2023
+
+Long Distance Scene
+Low Resolution Natural Weather Real Behavior
+ResNet18
+12.58
+ACER%
+12
+10
+8
+6
+4
+2.20
+2.23
+2
+0.21
+0
+Siw
+HiFiMask
+SuHiFiMask(Ours)
+OULU-NPU2
+design in surveillance FAS.
+In order to fill the gap in surveillance scenes of the FAS
+community, we target to solve two challenging problems an-
+alyzed above from two aspects: data collection and algorithm
+design. In Tab. I, we collect a large-scale FAS dataset based on
+surveillance scenes, namely SuHiFiMask. It has the following
+advantages: (a) Rich surveillance scenes. It includes 40 real
+surveillance scenes, such as movie theaters, security gates,
+and parking lots, which cover most face recognition scenes
+as much as possible. (b) Realistic distribution of human
+faces and natural behavior. It involves 101 participants of
+different ages, and genders distribution participating in the data
+collection. These subjects perform natural behaviors in daily
+life. (c) Rich spoofing Attacks. It has 232 high-fidelity masks
+(i.e. resin, plaster, silicone, headgear, head mold), 200 2D
+attacks (i.e. posters, portraits, and screens), and 2 adversarial
+attacks. (d) Realistic lighting and diverse weather. We collect
+data under real outdoor scenes with different weather (i.e.,
+sunny, snowy day) and light (i.e., day and night).
+For the algorithm design, the Contrastive Quality-Invariance
+Learning network (CQIL) is proposed in Fig. 5, which includes
+an Image Quality Variable (IQV) module and a two-stream
+framework consisting of a contrastive learning branch and a
+Separate Quality Network (SQN) branch. The IQV module is
+used to recover discriminative information related to FAS in
+the picture by super-resolution and deliver quality differences
+in contrast to the contrastive learning network backbone and
+SQN branch. The contrastive learning backbone [38] contains
+the online network and the target network. The online network
+continuously fits the target network during training, learning to
+approximate the same class with different quality distributions
+in the shared potential space. The SQN consists of a Quality-
+Invariance backbone network (CQI) (composed of a central
+differential convolution operator [7]), a quality discrimina-
+tor for separating quality, and the main classifier. CQI can
+effectively extract fine-grained features under environmental
+changes. The sample pairs generated by IQV are fed into CQI
+through adversarial learning, which allows CQI to focus on
+encoding features related to liveness while separating out the
+interference caused by quality. The main contributions of this
+paper are summarized below:
+• To the best of our knowledge, this is the first work
+to extend FAS to real surveillance scenes rather than
+mimicking low-resolution images and surveillance envi-
+ronments. We promote the development of this scenario
+through data collection and algorithm design.
+• We collect a large-scale surveillance FAS dataset, SuHi-
+FiMask, including 101 participants of different ages, 232
+masks and 200 2D attacks. A total of 10, 195 videos were
+collected by 7 mainstream cameras in 40 real scenes.
+• We
+propose
+a
+novel
+Contrastive
+Quality-Invariance
+Learning (CQIL) network to enhance the detection of face
+attacks in surveillance. Among them, an Image Quality
+Variable (IQV) module is designed to recover the FAS
+information in images and construct sample pairs to
+simulate face quality differences in realistic surveillance.
+A contrastive learning branch to obtain features robust
+to quality changes. And a Separate Quality Network
+(SQN) branch based on adversarial learning is introduced
+to further guide the model to learn quality-independent
+liveness features.
+• Extensive experiments are conducted on the SuHiFiMask
+and three other public datasets to demonstrate the chal-
+lenges of the SuHiFiMask and the effectiveness of the
+proposed method.
+II. RELATED WORK
+In this section, we review the current FAS works in con-
+strained environments and some preliminary attempts in the
+surveillance scenes.
+FAS under constrained Environments.
+Face spoofing (e.g., presentation attacks) is the typical
+physical attack to deceive the face recognition systems, where
+attackers present faces from spoof mediums, such as a pho-
+tograph, screen, or mask, instead of a living human. Ac-
+cording to the spoof mediums, we can roughly classify the
+existing attacks into 2D [4], [20], [21] and 3D attacks [3],
+[25]. Replay-Attack [2] and CASIA-FASD [1] are early FAS
+datasets, commonly used as benchmark for domain gener-
+alization evaluation. The spoof medium of the former is an
+electronic screen, while the latter introduces additional paper
+mediums based on different resolutions. With the advancement
+of acquisition equipment in mobile phones, there are also some
+high-resolution datasets recorded by replaying face video with
+a smartphone, such as Replay-Mobile [39], OULU-NPU [20],
+and SiW [4]. CelebA-Spoof [40] introduces rich attribute
+annotation information, which can be used as an auxiliary
+task to improve the generalization of the model in various
+attacks. Recently, with the cost reduction of multi-spectral
+sensors and the popularity of use scenes, some new sensors
+have been introduced to provide more possibilities for FAS
+methods. Holger et al. [23] use multi-spectral short wave
+infrared (SWIR) imaging to ensure the authenticity of a face
+even in the presence of partial disguises and masks. Zhang et
+al. [21] collect a CASIA-SURF dataset with 3 modalities (i.e.,
+RGB, Depth and NIR) using Intel RealSense SR300 camera,
+and propose a multi-modal multi-scale fusion method for FAS.
+Similarly, Liu et al. [41] introduce a CASIA-SURF CeFA
+dataset, covering 3 ethnicities, 1, 607 subjects, and 23, 538
+videos with 1280 × 720 resolution. As attack techniques are
+constantly upgraded, some new types of attacks have emerged,
+e.g., face mask [3], [25], [29]. Nesli et al. [3] provide a
+3DMAD which is recorded using the Microsoft Kinect sensor
+and consists of Depth and RGB modalities with 3D masks.
+George et al. [29] introduce a WMCA database with four
+channels, e.g., color, depth, near-infrared, and thermal, for
+face PAD which contains a wide variety of 2D and 3D
+presentation attacks, and propose MC-CNN method aiming
+to detect sophisticated attacks with multiple channels informa-
+tion. Heusch et al. [42] collect an HQ-WMCA database, which
+can be viewed as an extension of the WMCA [29] database via
+adding a new sensor acting in the shortwave infrared (SWIR)
+spectrum. A large-scale High-Fidelity Mask dataset, namely
+CASIA-SURF HiFiMask (briefly HiFiMask) was collected by
+
+3
+TABLE I
+COMPARISON OF PUBLIC FACE ANTI-SPOOFING DATASETS. * NOTES THAT SUHIFIMASK IS FOCUSED ON SURVEILLANCE SCENES WHERE BOTH REAL
+PEOPLE AND FAKE ATTACKS APPEAR AT THE SAME TIME. THEREFORE, THE NUMBER OF REAL VIDEOS AND THE NUMBER OF FAKE VIDEOS ARE BOTH
+10195.
+Dataset, Year
+#sub.
+Distance
+(Long/Short)
+Materials
+Scenes
+Light, Weather
+Attacks
+Devices
+#Videos
+(#Live/#Fake)
+3DMAD [33], 2013
+17
+Short
+Paper, Resin
+Constrained scenes
+Adjustment
+2D/2.5D image
+Kinect
+255(170/85)
+3DFS-DB [22], 2016
+26
+Short
+Plastic
+Office
+Adjustment
+2D/2.5D image, 3D Mask
+Kinect, Carmine 1.09
+520(260/260)
+BRSU [23], 2016
+137
+Short
+Silicone, Plastic,
+Resin, Latex
+Disguise,
+Counterfeiting
+Adjustment
+2D image
+SWIR, Color
+141(0/141)
+MARsV2 [34], 2016
+12
+Short
+ThatsMyFace,
+REAL-F
+Office
+Six directions of light
+3D Mask
+Logitech C920, Industrial Cam,
+EOS M3, Nexu 4, IPhone 6,
+Samsung S7, Sony Tablet S
+1008
+(504/504)
+SMAD [35], 2017
+Online
+Short
+Silicone
+-
+Varying light
+2D image, 3D Mask
+Varying Cam
+130(65/65)
+MLFP [36], 2017
+10
+Short
+Latex, Paper
+Indoor, Outdoor
+Daylight
+2D image
+Visible,
+Near infrared, Thermal
+1350
+(150/1200)
+ERPA [37], 2017
+5
+Short
+Resin, Silicone
+Indoor
+Room light
+3D Mask
+Xenic Gobi, Thermal Cam
+86
+WMCA [29], 2019
+72
+Short
+Plastic,
+Silicone, Paper
+Indoor
+Office/LED/Day light
+2D image, 3D Mask
+Intel RealSense SR 300,
+Seek Thermal, Compact PRO
+1670
+(347/1332)
+3DMask [26], 2020
+48
+Short
+Plaster
+Indoor, Outdoor
+Six directions of light
+2D image, 3D Mask
+Apple, Huawei, Samsung
+1152
+(288/864)
+HiFiMask [25], 2021
+75
+Short
+Transparent,
+Plaster, Resin
+White, Green,
+Tricolor, Sunshine,
+Shadow, Motion
+Six directions of light
+2D image, 3D Mask
+IPhone 11, IPhone X,
+MI10, P40, S20, Vivo, HJIM
+54,600
+(13,650/40,950)
+SuHiFiMask (ours), 2022
+101
+Long
+Resin, Plaster,
+Silicone, Paper
+Security check lane,
+Theater, Parking lot 1
+Day/Night light,
+Sunny/Windy/
+Cloudy/Snowy day
+2D image, Video replay,
+3D Mask
+Surveillance cameras2
+10,195*
+(10,195/10,195)
+1 40 real surveillance environments, including indoor as well as outdoor. Please see Fig.2 in Appendix for more details.
+2 dahua: DH-IPC-HFW4843M, DH-P80A1-SA; HIKVISION: DS-2CD3T87WD-L, DS-2CD3T86FWDV2-I3S; TP-LINK: TL-IPC586FP, TL-IPC586HP;
+ZHONGDUN: ZD5920-Gi4N (Brand name: Camera model) .
+Liu et al. [25]. Specifically, it consists of a total amount of
+54, 600 videos which are recorded from 75 subjects with 7
+kinds of sensors. Although the resolution and fidelity of these
+datasets are increasing high (i.e., resolution from 320×240 [2]
+to 1, 920 × 1, 080 [4], and spoofing types from print [1] to
+mask [25]), they are all oriented to FAS in a close constrained
+environment, ignoring the application requirements of remote
+surveillance scenes.
+The essence of FAS is a defensive measure for face recog-
+nition systems and has been studied for over a decade. Early
+works were mainly based on color texture [2], [43] and motion
+analysis [44]. The former is based on the consideration that
+the fake face is different from the live face in texture details,
+such as color distortions, and specular highlights, due to the
+intervention of spoofing mediums. However, these algorithms
+are not accurate enough because of the use of handcrafted
+features, such as LBP [2], HoG [45], and SURF [46]. The
+latter analyzes the attack samples as static or non-rigid motion
+compared with live faces from the perspective of motion.
+Unfortunately, these methods become vulnerable if someone
+presents a replay attack or a print attack with cut eye/mouth
+regions. Instead of using pre-defined features such as LBP
+and HOG, CNN-based methods [47], [48] design a unified
+framework of feature extraction and classification in an end-to-
+end manner. However, they treat FAS as a binary classification
+task, and will highly depend on the liveness-unrelated cues,
+such as color distortion, shape deformation, or background
+information. Intuitively, the live faces in any scene have con-
+sistent face-like geometry. Inspired by this, some works [4],
+[7], [49] leverage the physical-based depth information instead
+of binary classification loss as supervision, which are more
+faithful attack clues in any domain. Another works [8], [9],
+[50]–[52] treat FAS as a feature disentangled representation
+learning. Although these CNN-based methods achieve near-
+perfect performance under known attack clues, they still show
+poor generalization in the face of unknown attacks. To solve
+this problem, there are also some methods [53]–[56] that focus
+on improving the generalization of FAS in unknown domains.
+Examples are MADDG [5], SSDG [57], are SSAN [58], which
+aim to learn a generalized feature space via adversarial training
+and triplet loss strategies. In the case of FMeta [59], MT-
+FAS [60], D2AM [11], and SDA [61], they aim to find the
+generalized feature directions via meta-learning strategies.
+FAS in Surveillance Environments. The task of face recogni-
+tion in surveillance has been widely concerned by researchers,
+including data collection and algorithm design. SCface [16]
+was the first face recognition dataset released to simulate
+research in surveillance scenes, which contains 4, 160 still
+images captured by five different quality cameras. The QMUL-
+Survface dataset [14] further complements the low-resolution
+face recognition dataset by collecting 463, 507 face images
+from 15, 573 different identities in the real world using surveil-
+lance cameras. Then, IJB-C [15] aims to improve the repre-
+sentation of the global population by adding a list of names
+containing specific occupations such as artists, public speakers,
+and journalists from different countries to the surveillance
+scenario. In addition, based on these datasets, face recognition
+algorithms for surveillance scenes have been in full swing.
+Li et al. [18] introduce the adversarial generative networks
+and fully convolutional architectures to recognize ground-
+resolution faces in supervised discriminative learning. Consid-
+ering the incompleteness of these datasets, Zhong et al. [19]
+propose a sigmoid-constrained hypersphere loss (SFace) to
+reduce the intra-class distance of high-quality samples while
+preventing over-fitting label noise. Kim et al. [17] propose
+an adaptive marginal function to adjust the importance of
+different samples by emphasizing the role of clean samples
+in classification.
+In the FAS community, Chen et al. [62] explore the face
+anti-spoofing in surveillance scenes for the first time and
+proposed a dataset and benchmark. As for the dataset, they
+release the GREAT-FASD-S, which is first collected by two
+
+4
+multi-modal cameras, and then processed into low-quality
+images. And for the method, they propose the DAM-SE
+module to select the most informative channels and recover
+the image with the nearest neighbor interpolation algorithm.
+Aravena et al. [63] demonstrates that discarding a suitable
+percentage of low-quality samples can effectively improve
+the performance of the PAD algorithm. However, the nearest
+neighbor interpolation algorithm can not recover the original
+information by filling pixels with low-resolution images, and
+the method of directly discarding low-quality samples does
+not directly face the challenge of FAS in surveillance scenes.
+Fig. 2.
+An overview of some characteristics of SuHiFiMask. From top to
+bottom: forms of attack, diverse weather, and surveillance scenes.
+III. SUHIFIMASK
+In order to fill the gap in the face anti-spoofing dataset
+of surveillance scenes and promote the research of related
+algorithms, we collected the SuHiFiMask dataset that has the
+following advantages over existing datasets:
+Advantage 1: To the best of our knowledge, SuHiFiMask
+is the first dataset collected based on real surveillance scenes,
+rather than the low-quality datasets obtained by manual degra-
+dation, such as GREAT-FASD-S [62]. Compared to previous
+PAD datasets in controlled environments, the one we present
+inevitably introduces low-resolution face, pedestrian occlusion,
+changeable posture, motion blur, and other challenging situ-
+ations, which greatly increases the challenge of FAS tasks.
+In addition, as shown in the third column of the Tab. I, we
+define the dataset with the distance between the camera and
+the subject less than one meter as the short distance dataset,
+while the dataset with the distance between the camera and
+the subject greater than three meters is defined as the long-
+distance type. Advantage 2: SuHiFiMask considers the most
+comprehensive attack types, each of which contains diverse
+spoofing methods. As shown in Tab. I, 2D image, video
+replay and 3D mask all appear in SuHiFiMask to evaluate
+the algorithm’s perception of changes for paper color, screen
+moire and face structure in surveillance scenes. Different from
+the attack type under classical more constrained environments,
+as shown in Fig. 2, we introduce paper posters, humanoid
+stand-ups in 2D image, and headgear, head mold in the 3D
+mask to minimize the spoofing trace in the surveillance scenes.
+In order to effectively prevent criminals from hiding their
+identities through local occlusion during security inspection,
+we introduce two most effective adversarial attacks (ADV),
+instead of simply masking the face with paper classes [29]
+and partial paper [64]. Advantage 3:
+We designed 40 common real-world surveillance scenes,
+including daily life scenes (e.g., cafes, cinemas, and theaters)
+and security check scenes (e.g., security check lanes and park-
+ing lots) for deploying face recognition systems. In fact, the
+rich natural behaviors in different surveillance scenes greatly
+increase the difficulty of PAD due to pedestrian occlusion and
+non-frontal views. Advantage 4: We collect data in four types
+of weather (e.g., Sunny, Windy, Cloudy and Snowy days) and
+natural lighting (e.g., Day and Night lights) to fully simulate
+the complex and changeable surveillance scenes. Different
+weather and light bring diverse image style information and
+image artifacts, which will put forward higher requirements
+for the generalization of PAD technology.
+Based on the above acquisition advantages, our SuHiFiMask
+contains 10, 195 videos from 101 subjects of different age
+groups, which are collected by 7 mainstream surveillance
+cameras and see Fig.1 in Appendix for more details. In
+particular, as shown in the second and third rows of Fig. 2,
+SuHiFiMask is focused on surveillance scenes, and both real
+and fake attacks appear at the same time.
+As shown in the Fig. 3, the existing FAS dataset of a video
+contains only one real person or one type of attack. The subject
+faces the camera and remains stationary during the shooting
+to ensure the clarity of the collected data. In contrast, videos
+based on a surveillance scene contain multiple real people and
+multiple types of attacks. Subjects are not required to face the
+camera and move randomly in the scene while filming. This
+leads to low-resolution of face images, pedestrian occlusions,
+non-frontal poses, and other disturbances that affect the stabil-
+ity and generalization of the algorithm. Thus, the surveillance
+scene-based FAS dataset poses a greater challenge than the
+existing FAS dataset.
+Fig. 3. Comparison of the existing FAS dataset and the surveillance scene-
+based dataset. The left image is from MARsV2 and the right image is from
+the proposed SuHiFiMask.
+A. Acquisition Details of SuHiFiMask.
+Scenes and props. In order to cover real surveillance
+environments as much as possible, we carefully selected
+and rented 40 real-world scenarios that include daily places,
+such as cafes, yoga studios, and movie theaters, as well as
+security checkpoints, such as security lanes, parking lots, and
+entrance/exit gates. We provide 232 masks as the candidate
+
+Resin
+Plaster
+Silicone
+Headgear Head mold
+ADV
+Cardboard
+Poster
+Screen
+Attack
+Cloudy Day
+Snowy Day
+Windy Day
+Sunny Day
+Weather
+Sidewalk
+Cinema
+Corridor
+Theater
+Security lane
+SceneExisting FAS dataset
+Surveillance scene based FAS dataset
+Number of Subject: One
+Number of Subject: : Many
+Number of real or attack : Multiple real
+Number of real or attack : One
+people and multiple types of attacks
+Face Resolution : Low resolution
+Face Resolution : High resolution
+Posture and movement: Not facing the
+Posture and movement: Face the camera
+and remain still
+camera and moving freely5
+TABLE II
+STATISTICAL INFORMATION FOR EACH PROTOCOL OF THE PROPOSED SUHIFIMASK DATASET. NOTE THAT 1, 2, 3 IN THE FOURTH COLUMN MEAN RESIN,
+SILICONE, AND PLASTER. 4 REPRESENTS HEADGEAR AND HEAD MOLD.
+Pro.
+Subset
+#Subject
+Mask
+Quality score
+#Live
+#Mask
+#Other attack
+#All
+1
+Train
+40
+1&2&3&4
+[0, 1]
+118,520
+60,715
+22,333
+201,568
+Dev
+10
+1&2&3&4
+[0, 1]
+23,304
+11,856
+5663
+40,823
+Test
+51
+1&2&3&4
+[0, 1]
+69,878
+42,569
+19,743
+132,190
+2.1
+Train
+101
+1&2&3
+[0, 1]
+100,990
+40,454
+0
+141,444
+Dev
+101
+1&2&3
+[0, 1]
+20,521
+19,608
+0
+40,129
+Test
+101
+4
+[0, 1]
+42,539
+21,199
+0
+63,738
+2.2
+Train
+101
+1&2&4
+[0, 1]
+78,961
+36,829
+0
+115,790
+Dev
+101
+1&2&4
+[0, 1]
+20,505
+19,052
+0
+39,557
+Test
+101
+3
+[0, 1]
+42,521
+28,366
+0
+70,887
+2.3
+Train
+101
+1&3&4
+[0, 1]
+77,952
+28,994
+0
+106,946
+Dev
+101
+1&3&4
+[0, 1]
+20,594
+17,714
+0
+38,308
+Test
+101
+2
+[0, 1]
+42,498
+44,104
+0
+86,602
+2.4
+Train
+101
+2&3&4
+[0, 1]
+79,102
+29,087
+0
+108,189
+Dev
+101
+2&3&4
+[0, 1]
+20,627
+18,068
+0
+38,695
+Test
+101
+1
+[0, 1]
+42,513
+42,887
+0
+85,400
+3
+Train
+101
+1&2&3&4
+[0.4, 1]
+64,276
+35,898
+58,889
+159,063
+Dev
+101
+1&2&3&4
+[0.3, 0.4)
+37,990
+24,031
+27,255
+89,276
+Test
+101
+1&2&3&4
+[0, 0.3)
+84,368
+43,820
+36,369
+164,557
+pool for selection according to the scene requirements. Among
+them, some high-fidelity plaster and resin masks are from
+HiFiMask [25], and silicone material headgear and head mold
+masks are new additions to reduce the forgery traces exposed
+in the monitoring perspective, where the numbers of plaster
+masks, resin masks, silicone masks, headgear, and head mold
+were 93, 93, 23, 10, and 13, respectively. In addition to mask
+attacks, we printed 2D images of 50 subject in the form
+of humanoid upright cards and posters and provided video
+attacks by displaying images on a movable TV. In particular, in
+order to effectively prevent criminals from hiding their identity
+information during security checks in surveillance scenes, we
+crafted adversarial mask [65] and adversarial hat [66] that can
+induce face recognition systems to categorize the registered
+identity as unknown identity, aiming to increase the challenge
+to algorithm stability.
+Data collection and processing rules. To ensure the quality
+and challenge of data, we implemented the following criteria
+before each shot: a) Device adjustment. We adjusted the
+positions and angles of each camera to ensure that the entire
+scene is captured. b) Sample balance. We arranged a consistent
+number of live and fake subjects to ensure sample balance
+in SuHiFiMask. c) Static 2D attacks. We deployed posters
+and humanoid upright-card and electronic screen-based photos
+with the same identity as the subjects at random locations.
+We also considered the following criteria during each shot:
+a) We designed specific movement routes for each subject
+to ensure adequate pedestrian occlusion, versatile posture
+and comprehensive perspective. b) We requested subjects in
+different scenes to perform scene-related behaviors, such as
+eating and chatting in daily scenes, and self-service check-in
+security check scenes.
+After data collection, we performed the following pre-
+processing: a) Face detection. We used RetinaFace [67] to
+detect the face in each frame of the original video, and
+discarded those frames where the face could not be detected.
+b) Face tracking. We use face similarity to track the position
+of faces in consecutive frames and name each of the different
+face tracking boxes. c) Video sampling. We sample each video
+in 10-frame intervals and store the cropped face image in the
+corresponding face tracking box folder. d) Dataset naming
+rules. We named the folder of this video according to the
+following rule: Group Scene Camera Epoch Time.
+Ethical and legal considerations. Since we collected our
+filming scenes from real-world environments, we have a
+responsibility to maintain the public environment and protect
+pedestrian safety. We commissioned two companies to legally
+authorize the scenes for data collection. SuHiFiMask is a
+dataset consisting of videos taken from subjects of different
+age groups, and although this is not a subject explicitly
+modeled for human behavior, the relevant challenge factors
+are related to humans. Based on the consideration of the
+protection of human rights and legal interests, our collection
+process follows a strictly ethical procedure. We commission
+a data acquisition company to develop strict standards and
+obtain the signature authorization of all human subjects. The
+collected images and videos will be used to develop, train
+and optimize face anti-spoofing technologies to the extent
+permitted by Chinese laws. The dataset is balanced in terms of
+gender and age, that is, there is no hazard in terms of ethics.
+B. Evaluation protocol and Statistics.
+We define three protocols for SuHiFiMask to fully evaluate
+the performance in surveillance environments: Protocol 1-ID,
+Protocol 2-Mask, and Protocol 3-quality.
+Protocol 1-ID. Protocol 1 aims to evaluate the comprehen-
+sive performance of the algorithm being migrated to long-
+distance surveillance scenes. Compared with the classical
+constrained environment datasets, protocol 1 includes various
+unique factors in surveillance scenes, such as low resolution,
+pedestrian occlusion, changeable posture, motion blur, and
+other complex weather, which pose greater challenges to
+algorithm design. As shown in Tab. II, we divide the training
+set, development set, and testing set according to the identity
+information, including 40, 10, and 50 subjects, respectively.
+Protocol 2-Mask. Protocol 2 evaluates the generalization
+of the algorithm for the ‘unseen’ 3D facial mask type. The
+diversity and unpredictability of mask materials are important
+characteristics of spoofing means and are easily interfered with
+
+6
+by other liveness-unrelated factors. Thus, the generalization
+to mask materials is an important evaluation index. In this
+work, we divide protocol 2 into four sub-protocols by using the
+‘leave-one-type-out testing’ method, in which one unknown
+3D mask material is divided into the testing set for each sub-
+protocol. As shown in Tab. II, ‘1’, ‘2’, ‘3’, and ‘4’ in the fourth
+column indicate that the 3D mask material is headgear/head
+mold, resin, silicone, and plaster, respectively.
+Fig. 4.
+Some samples from protocol 3. The number below the image
+represents the quality score of the faces. Samples with different score intervals
+are proportionally categorized as the training set, development set, and testing
+set.
+Protocol 3-Quality. Protocol 3 evaluates the robustness of
+the algorithm to image quality degradation. Variable quality
+and disturbances are factors that affect the stability of the
+algorithm. Therefore, the robustness of the algorithm to quality
+degradation is an important metric to be evaluated. In this
+work, as shown in Fig. 4, we use the SER-FIQ [68] algorithm
+to calculate the image quality score which ranges from 0 to
+1. As shown in the fifth column of Tab. II, we assign images
+with scores [0.4, 1] as the training set, scores [0.3, 0.4) as the
+development set, and scores [0, 0.3) as the testing set.
+IV. METHODOLOGY
+In this section, we present a Contrastive Quality-Invariance
+Learning (CQIL) network for FAS tasks based on long-
+distance surveillance scenes. As shown in Fig. 5, CQIL
+contains an Image Quality Variable module (IQV) and a dual-
+stream framework with a contrastive learning branch and a
+Separate Quality Network (SQN) branch. IQV processes low-
+quality images into high-quality images by super-resolution
+and sends them to the contrastive learning branch and the SQN
+branch. The contrastive learning branch trains the network
+by using high-quality and low-quality images as input to
+the online network and the target network, respectively. The
+SQN branch makes the features extracted by the encoder
+independent of quality by adversarial learning. In addition,
+CQI uses high-quality images after super-resolution as input
+to extract richer discriminative features.
+Image Quality Variable Module (IQV). In contrast to the
+classical constrained environment, the difficulty of the FAS
+task based on surveillance scenes is the low resolution and
+variable quality of the images, which leads to insufficient
+information contained in the images and severely interferes
+with the extraction of robust features. To solve this problem,
+a possible solution is to increase the resolution of the image
+and extract robust invariant features. Inspired by CSRI [69],
+we introduce the Image Quality Variable (IQV) module to
+improve the image resolution of SuHiFiMask and recover
+information relevant to the FAS task. In addition, IQV tags
+the images processed by the SR network with label 0 and
+the original images with label 1. Then IQV sends them to
+the contrastive learning branch and the SQN branch. Since
+SuHiFiMask is the first unconstrained PAD dataset, there is
+no high-quality image as ground truth to optimize the super-
+resolution network. Thus, we use the existing high-definition
+PAD dataset to train the super-resolution network. As shown in
+Fig. 5, this process can be expressed as follows: 1) We degrade
+the high-fidelity dataset OULU-NPU [20] into a low-quality
+dataset using pre-processing methods such as interpolation and
+gaussian blurring. 2) We feed degraded low-resolution images
+into an SR network and use its original data for supervision
+to train the SR network. 3) We use the SR network with
+shared parameters to process SuHiFiMask’s images into high-
+quality images. Unlike the standalone super-resolution tasks,
+we combine the SR tasks with the FAS tasks by integrating
+the IQV module into the framework with the following two
+advantages below:
+• Training the FAS network with SR network-boosted
+resolution images can improve the performance of the
+FAS network.
+• The improved performance of other networks in CQIL
+can better guide the SR network to recover information
+related to the FAS task in the image.
+Finally, MSE loss is used to constrain the super-resolution
+network:
+Lmse = 1
+n
+�
+(ˆyi − yi)2
+(1)
+where n represents the number of pixels in the image, ˆyi, yi
+denote the pixel value of the image after super-resolution and
+the pixel value of ground truth respectively.
+Contrastive Learning Branch. To improve the robustness of
+FAS networks in a quality-variant surveillance environment,
+we propose a branch based on contrastive learning. Inspired
+by the BYOL [38], this branch obtains robustness to quality
+variations by fitting the distribution of potential features for
+different quality pictures. Specifically, during the training
+process, due to the constraints of Eq. 2 and Eq. 3, the online
+network will gradually fit the target network by closing the
+same class in the potential feature space for pairs of images
+of different quality, which makes it to obtain a powerful
+
+Train
+0.72
+0.63
+0.54
+Dev
+0.39
+0.34
+0.31
+Test
+0.22
+0.12
+0.037
+Fig. 5. Contrastive Quality-Invariance Learning (CQIL) network. IQV recovers information from the images and constructs sample pairs of different qualities.
+Sample pairs are sent to the contrastive learning branch and the SQN branch. The contrastive learning branch consists of an online network (encoder, projector,
+predictor) and a target network (encoder, projector). Above the image the SQN branch is shown, which contains the discriminator, CQI, GRL, and the main
+classifier.
+feature representation while ignoring the negative impact from
+different quality distributions.
+Lθ,ξ ≜
+��qθ (zθ) − ¯z′
+ξ
+��2
+2 = 2 − 2 ·
+�
+qθ (zθ) , z′
+ξ
+�
+∥qθ (zθ)∥2 ·
+���z′
+ξ
+���
+2
+(2)
+Lcontra = Lθ,ξ + �Lθ,ξ
+(3)
+where qθ (zθ) is the prediction of the online network output
+and z′
+ξ is the projection of the target network output, then we
+use ℓ2-normalize to turn qθ (zθ) and z′
+ξ into qθ (zθ) and ¯z′
+ξ.
+In addition, �Lθ,ξ is the result of Lθ,ξ symmetrization.
+As shown in Fig. 5, image pairs of different quality gen-
+erated by IQV are sent to the online and target networks.
+The online network is composed of an encoder network
+(Interchangeable backbone networks), a projector (Projection
+of extracted features into the latent space), and a predictor
+(with the same multi-layer perceptron structure). Similarly, the
+target network has an encoder and a projector with different
+weights from the online network. Unlike the weight update of
+the online network, the parameters of the target network are
+not updated in gradient descent [38], and the process can be
+expressed as follows:
+ξ←τξ + (1 − τ)θ
+(4)
+The parameters ξ and θ represent the parameters to be updated
+for the target network and the online network, respectively.
+The parameters θ of the online network are updated by the
+optimization of the loss function, the parameters ξ of the target
+network are updated by perceiving an exponential moving-
+average [70] of the online parameters and we perform the
+moving-average after each step by target decay rate τ.
+Separate Quality Network (SQN). For FAS data in surveil-
+lance scenes, which contains many variations (e.g., environ-
+ment, light, weather), we need operators that are more robust
+to variations to describe the required fine-grained information.
+Inspired by central differential convolution (CDC) [7], we use
+CDC to form a quality-independent backbone network (CQI)
+in the second branch, exploiting its powerful representation
+ability to extract fine-grained features under environmental
+variations. In addition, we use cross-entropy loss as a supervi-
+sion of CQI, so that this network can capture the cues related
+to liveness more robustly.
+The sample pairs generated by the IQV module have the
+following characteristics: 1) Both the super-resolution network
+processed images and the original images contain the object of
+the face (live or attack) in the center of their images, so even
+samples with very different quality share the same semantic
+feature space. 2) Although the quality of each image is differ-
+ent, they all contain discriminative information. Therefore, we
+make the discriminative features extracted by CQI independent
+of quality by adversarial learning. Specifically, we use the
+adversarial loss to optimize the backbone network CQI. And
+the gradient reversal layer (GRL) [71] allows the parameters
+of the quality discriminator to be optimized in the reverse
+direction. This process can be formulated as follows:
+min
+D max
+C Ladv(C, D) =
+− E(x.y)∼(X,YQ)
+N
+�
+i=1
+1[i = y]logD(C(x))
+(5)
+where YQ is the set of quality labels, N is the number of
+images of different quality, C stands for the CQI network
+backbone where we extracted the liveness-related information,
+and D represents the quality discriminator. Finally, we con-
+catenate the features extracted by CQI with those extracted by
+the contrastive learning branch and input them to the classifier
+for classification.
+
+SQN
+Contrastive learning
+cdc
+L
+cls
+Feature
+GRL
+Discriminator
+CQI
+Classifier
+ady
+concat
+Online network
+IQV
+ SuHiFiMask LR
+qe(z)
+face images
+Encoder
+Projector
+Predictor
+Resize
+Moving
+average
+contra
+i Shared
+Degraded LR face
+images
+GT
+Encoder
+Projector
+sg
+mse
+Target network8
+Algorithm 1 Contrastive Quality-Invariance Learning (CQIL)
+Input: image set X, label set Y , HD image set H.
+1: Initialize: encoder fθ, projector gθ, predictor qθ
+2: Initialize: encoder f ′
+ξ, projector g′
+ξ
+3: Initialize: network n of IQV, encoder c of CQI
+4: while not end of training do
+5:
+sample batch A ← {xi ∼ X}N
+i=1, B ← {yi ∼ Y }N
+i=1
+6:
+sample batch H ← {hi ∼ H}N
+i=1
+7:
+for xi ∈ A do
+8:
+li ← hi
+▷ Degradation into lq images
+9:
+compute Lmse, see Eq. 1
+10:
+xi1 ← n(xi), xi2 ← xi
+▷ Generate image pairs
+11:
+yi1, yi2 ← xi1, xi2
+▷ Generate quality labels
+12:
+z1 ← gθ (fθ (xi1)), z2 ← gθ (fθ (xi2))
+13:
+z′
+1 ← gθ (fθ (xi2)), z2′ ← gθ (fθ (xi1))
+14:
+compute Lcontra, see Eq. 2, Eq. 3
+15:
+X ← [xi1, xi2], Y ← [yi1, yi2], Z ← c(X)
+16:
+compute Ladv, by Eq. 5
+17:
+z3 ← c(xi), z4 ← fθ(xi)
+18:
+compute Lcls, Lcdc, Ltotal, by Eq. 6
+19:
+end for
+20: end while
+21: ∆θ = backward (Ltotal)
+22: θ ← θ−learningrate · ∆θ
+23: update ξ by Eq. 4
+24: update network n, encoder c
+Overall Loss. As mentioned, CQI is used to extract quality-
+independent discriminative features, and these features are
+concatenated with the robust features extracted from the con-
+trastive learning branch and fed to the main classifier. There-
+fore, the cross-entropy loss Lcdc and Lcls is well constrained
+for both CQI and the main classifier. In summary, the overall
+loss function Ltotal for stable and reliable training can be
+formulated as follows:
+Ltotal = λ1 · Lcls + λ2 · Lcontrast + λ3 · Ladv
++λ4 · Lcdc + λ5 · Lmse
+(6)
+whereλ1, λ2, λ3, λ4 and λ5 are five hyper-parameters to
+balance the proportion of the different loss functions.
+V. EXPERIMENTS
+A. Experiments Settings
+Dataset and Protocols. In experiments, a total of five
+datasets were used: OULU-NPU [20], CASIA-MFSD [1],
+RepalyAttack [2], MARsV2 [34] and the SuHiFiMask dataset.
+First, we conducted ablation experiments on three protocols of
+the proposed SuHiFiMask to demonstrate the effectiveness of
+each component of the proposed CQIL. Second, we present
+the respective baselines for the different protocols for the
+proposed dataset. Finally, we design several different cross-
+testing experiments to demonstrate the importance of the
+proposed dataset and the effectiveness of the method.
+Training Setting. Our proposed method is implemented
+with Pytorch. In the training stage, models are trained with
+Adam optimizer and the initial learning rate is 2e − 4. The
+batch size is set to 6 for CQIL. The epoch of the intra-testing
+is set to 10, and the lr decreases by 0.2 times per epoch. The
+epoch of the inter-testing is 300, and lr decreases by 0.2 times
+per 50 epochs. λ1, λ2, λ3, λ4 and λ5 are set to 2, 1.5, 0.5,
+1.5, 0.5 respectively.
+Performance Metrics and Implementation Details. We
+accept the Attack Presentation Classification Error Rate
+(APCER), Bonafide Presentation Classification Error Rate
+(BPCER), and ACER [72] as the evaluation metrics in our
+experiments. The ACER on each testing set is determined by
+the threshold value of the performance on the development
+set. In cross-testing experiments, we use Half Total Error
+Rate (HTER) [73] and Area Under Curve (AUC) as evalu-
+ation metrics. We use the ResNet18 [48], ViT [74], and the
+CDCN [7] network as the backbone, and report their results
+in experiments.
+B. Ablation Study.
+Here we conduct ablation experiments to verify the con-
+tribution of each module of the proposed CQIL on the three
+protocols of the SuHiFiMask dataset.
+TABLE III
+THE ABLATION STUDY OF DIFFERENT COMPONENTS. THE EVALUATION
+METRIC IS ACER (%).
+Method
+Prot.1
+Prot.2
+Prot.3
+ResNet18
+12.58
+16.55±51.71
+17.64
+CQIL-Model-1
+11.97
+16.01±50.23
+17.45
+CQIL-Model-2
+11.75
+15.67±48.12
+16.54
+CQIL-Model-3
+10.90
+15.14±46.66
+16.13
+CQIL-Model-4
+10.69
+14.90±45.92
+15.98
+Advantage of the proposed architecture. We compare four
+architectures with ResNet18 to demonstrate the advantages of
+each module of the proposed method. The CQIL-model-1 is a
+contrastive learning network with ResNet18 as its backbone.
+Since the training of the contrastive learning network requires
+the output of the IQV module, we use images processed by cu-
+bic interpolation and nearest-neighbor interpolation to mimic
+samples of different quality to eliminate the impact of the IQV
+module on performance. In addition, we additionally supervise
+the training of the online encoder using cross-entropy loss. In
+the testing phase, we use the features extracted by the online
+encoder for classification. As shown in Tab. III, CQIL-model-
+1 has a significant improvement in performance on all three
+protocols compared to ResNet18, which demonstrates that the
+contrastive learning branch using quality change as a contrast
+improves the robustness of the network in surveillance scenes.
+Advantage of SQN branch. Our proposed SQN branch
+takes sample pairs of different qualities generated by the
+IQV module as input and lets the discriminative features
+extracted by the encoder CQI be independent of the quality by
+adversarial learning. CQIL-model-2 extends the SQN branch
+on the basis of CQIL-model-1. In the testing phase, we
+concatenate the features extracted by the contrastive learning
+branch with the features extracted from CQI in the SQN
+
+9
+branch for classification. As shown in Tab. III, the performance
+of CQIL-model-2 is significantly improved on all three proto-
+cols, and the performance improvement is especially obvious
+in protocol 3, which verifies that SQN trained with samples
+of different quality have the ability to extract discriminative
+features independent of quality.
+Advantage of IQV module. CQIL-model-3 extends the
+complete IQV module based on CQIL-model-2 but uses low-
+quality original images to train the CQI encoder. CQIL-model-
+4 extends CQIL-model-3 by training CQI encoders using
+high-quality images generated by SR networks. The improved
+performance of CQIL-model-3 in Tab. III demonstrates that
+the sample pairs constructed by IQV more closely match
+the quality variation in the surveillance scene and IQV can
+effectively improve the performance of the SQN branch and
+contrastive learning branch. The improved performance of
+CQIL-Model-4 further validates the two advantages of IQV
+modules: 1) The SR network processed images can be used
+for CQI encoder training, thus improving the performance
+of the FAS task. 2) The performance-improved FAS network
+can better guide the SR network to recover the discriminative
+information of the images.
+C. Intra-Testing.
+Here, we conduct experiments on three different protocols
+of SuHiFiMask, showing that SuHiFiMask poses a challenge
+to existing FAS studies while also testing the performance of
+our proposed CQIL method in different data distributions.
+TABLE IV
+THE RESULTS OF INTRA-TESTING ON THREE PROTOCOLS OF
+SUHIFIMASK.
+Prot.
+Method
+APCER%
+BPCER%
+ACER%
+1
+ResNet18
+13.59
+11.57
+12.58
+ViT
+13.45
+9.89
+11.67
+CDCN
+20.46
+18.95
+20.41
+CQIL (ours)
+11.09
+10.29
+10.69
+2
+ResNet18
+20.46±184.60
+12.74±1.43
+16.60±51.05
+ViT
+19.56±181.71
+12.25±0.42
+15.89±45.01
+CDCN
+24.88±55.77
+24.44±12.51
+24.66±16.46
+CQIL (ours)
+18.83±169.37
+10.88±0.34
+14.86±46.04
+3
+ResNet18
+21.04
+13.64
+17.64
+ViT
+19.61
+13.95
+16.78
+CDCN
+28.70
+25.89
+27.30
+CQIL (ours)
+19.14
+12.82
+15.98
+Experiments on Protocol 1-ID. In protocol 1, the data
+distribution is similar for different sets. The training set,
+development set, and testing set contain all attack types,
+and also contain data for all quality scores. The protocol is
+appropriate to evaluate the performance of the FAS algorithm
+in long-distance surveillance scenes. As shown in Tab. IV,
+the proposed CQIL ranks first for three performance metrics
+(11.09%, 10.29%, 10.69%, respectively) compared to the
+generic network backbone ResNet, ViT, and the FAS task
+network CDCN with robust feature representation on the
+Protocol 1, showing that the proposed method performs well in
+the FAS task based on surveillance scene with low resolution
+and many interferences.
+Experiments on Protocol 2-Mask. We verify the algo-
+rithm’s ability to discriminate between different types of masks
+by protocol 2. As shown in Tab. V, our proposed CQIL
+achieved good results except for the APCER on protocol 2.1
+and protocol 2.2 which was not the highest performance,
+which proves that our method can extract discriminative
+features in low-quality mask images. It is worth mentioning
+that the testing set of protocol 2.1 is composed of headgear
+and head mold. These two types of masks are very similar
+to the human head structure, so the algorithm can no longer
+use features such as mask contours as a basis for prediction.
+Thus, the performance of CQIL on protocol 2.1 demonstrates
+the importance of CQI encoders that can extract fine-grained
+features.
+TABLE V
+THE RESULTS OF INTRA-TESTING ON FOUR SUB-PROTOCOLS OF
+SUHIFIMASK PROTOCOL 2.
+Prot.
+Method
+APCER%
+BPCER%
+ACER%
+2.1
+ResNet18
+43.33
+13.96
+28.65
+ViT
+42.54
+12.05
+27.29
+CDCN
+35.47
+20.69
+28.08
+CQIL (Ours)
+41.18
+11.81
+26.49
+2.2
+ResNet18
+8.50
+11.58
+10.04
+ViT
+8.56
+11.40
+9.98
+CDCN
+14.72
+21.41
+18.06
+CQIL (Ours)
+8.88
+10.26
+9.57
+2.3
+ResNet18
+17.52
+11.51
+14.52
+ViT
+13.70
+12.33
+13.02
+CDCN
+26.62
+29.20
+27.91
+CQIL (Ours)
+13.61
+10.92
+12.27
+2.4
+ResNet18
+12.48
+13.90
+13.19
+ViT
+13.33
+13.20
+13.26
+CDCN
+22.72
+26.47
+24.59
+CQIL (Ours)
+11.64
+10.54
+11.09
+TABLE VI
+THE RESULTS OF CROSS-DATASET TESTING FOR CASIA-MFSD,
+REPLAY-ATTACK AND SUHIFIMASK. THE EVALUATION METRIC IS
+HTER(%).
+Method
+Train
+CASIA-MFSD
+ReplayAttack
+Test
+Replay-
+Attack
+SuHiFi-
+Mask (Ours)
+CASIA-
+MFSD
+SuHiFi-
+Mask(Ours)
+ResNet18
+36.3
+44.5
+50.9
+42.1
+ViT
+34.9
+42.8
+44.8
+45.9
+CDCN
+15.6
+45.9
+32.6
+41.4
+AUX.(Depth)
+27.6
+43.8
+28.4
+39.6
+TABLE VII
+CROSS-TESTING RESULTS ON THE MARSV2 DEGRADED WITH DIFFERENT
+SIZE OF THE GAUSSIAN KERNEL WHEN TRAINED ON THE PROPOSED
+SUHIFIMASK.
+Method
+Train
+SuHiFiMask (ours)
+Test
+MARsV2
+MARsV2-3×3
+MARsV2-5×5
+Metric HTER(%)↓ AUC(%)↑ HTER (%)↓ AUC (%)↑ HTER(%)↓ AUC (%)↑
+ResNet18
+27.2
+79.6
+29.5
+79.2
+32.5
+75.5
+CDCN
+37.6
+66.8
+41.6
+61.7
+51.2
+52.7
+AUX.(Depth)
+26.8
+79.4
+41.1
+63.7
+48.7
+54.5
+CQIL (ours)
+21.8
+87.5
+26.2
+81.4
+30.9
+74.0
+Experiments on Protocol 3-Quality. Protocol 3 evaluates
+the stability of the algorithm to image quality degradation.
+Since the training, development, and testing sets of this pro-
+tocol differ only in quality, the algorithm is needed to learn a
+general feature extraction method on data with different quality
+
+10
+distributions. As shown in Tab. IV, our algorithm ranks first on
+protocol 3 (APCER, BPCER, and ACER are 19.14%, 12.82%,
+15.98%, respectively), which proves that our algorithm is
+effective in extracting discriminative features independent of
+quality.
+D. Inter-Testing.
+To evaluate the difficulty of surveillance-based FAS tasks
+and the effectiveness of CQIL working on low-quality datasets,
+we design a number of cross-testing experiments.
+Cross-dataset. To further evaluate the difficulty of the long-
+distance PAD task based on surveillance scenes, we design
+two cross-dataset experiments. (1) We train the model on the
+CASIA-MFSD dataset and perform the cross-test evaluation
+on the proposed SuHiFiMask and ReplayAttack datasets. (2)
+We train the model on the ReplayAttack dataset and evaluate
+it on the SuHiFiMask and CASIA-MFSD datasets for cross-
+testing. As shown in Tab. VI, the performance of the model
+tested on the proposed SuHiFiMask is significantly degraded
+relative to the performance testing on ReplayAttack or CASIA-
+MFSD. For example, the HTER (%) of the CDCN trained
+on CASIA-MFSD was increased by 30.3% for the test on
+the proposed dataset compared to the test on ReplayAttack.
+This shows the performance of existing algorithms degrades
+significantly when they encounter negative factors such as low
+resolution, motion blur, and occlusion. In particular, CQIL is
+a PAD method based on low-quality data and requires low-
+resolution images as input. So the generality of the method
+will be evaluated in the next subsection.
+Cross-quality. To demonstrate the generality of our method
+to low-quality datasets, we design a series of experiments
+across the quality. We train the different methods on the
+proposed SuHiFiMask and test them on MARsV2 after the
+degradation of gaussian kernels of different sizes. Specifically,
+since no existing work has provided available low-quality
+PAD datasets, we simulate low-quality datasets with different
+degrees of degradation by means of a gaussian kernel to verify
+the generality of different methods on low-quality datasets.
+Fig. 6 shows several samples of MARsV2 after treatment with
+gaussian kernels of different sizes. As shown in Tab. VII, our
+CQIL achieves good performance on the MARsV2 dataset at
+all degradation degrees. This demonstrates that our method can
+encode quality-independent discriminative features. However,
+there is a domain gap between the manually degraded low-
+quality dataset and the dataset based on the surveillance
+scenes. This results in CQIL not being able to take full
+advantage of encoders trained on low-quality data in real
+surveillance scenes.
+Fig. 6. MARsV2 with different sizes of the gaussian kernel processing.
+E. Visualization Analysis
+In this section, we further visualize the difficulties that low-
+quality data poses to FAS work and the performance of CQIL
+in surveillance scenes. First, we compare the features learned
+by ResNet18 on protocol 1 of HiFiMask, a dataset for the
+constrained environment, and on protocol 1 of SuHiFiMask, a
+proposed surveillance scene-based dataset. As shown in Fig 7,
+the performance of the algorithm degrades significantly on
+SuHiFiMask, which indicates that the low-quality data in the
+surveillance scenes add difficulties to the FAS work. Next,
+we compare the features learned by CQIL and ResNet18
+on protocol 3 of the proposed SuHiFiMask. Compared with
+ResNet18, the proposed CQIL is able to better distinguish
+between real faces and attacks, which demonstrates the better
+discriminative representation capacity of the proposed CQIL
+in surveillance scenes.
+Fig. 7. Feature distribution comparison on HiFiMask and SuHiFiMask using
+t-SNE [75]. The points with different colors denote features from different
+classes (blue: real faces; red: attack samples).
+VI. CONCLUSION
+In this paper, we release the first large-scale FAS dataset
+based on surveillance scenes, SuHiFiMask, with three chal-
+lenging protocols. We hope that this will fill the gap in FAS
+research in long-distance surveillance scenes. In addition, we
+propose a Contrastive Quality-Invariance Learning (CQIL)
+network to recover image information using super-resolution
+and enhance the robustness of the algorithm to quality vari-
+ations by fitting the quality variance distribution. Finally, we
+conduct comprehensive experiments on SuHiFiMask and three
+other datasets to verify the importance of the datasets for the
+FAS task and the effectiveness of the proposed method.
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+13
+Appendix
+A. Sample of faces
+As shown in the Fig. 8, we have listed some samples of
+pre-processed face images. The figure contains six sections,
+which are listed as close-up samples of real people, resin
+masks, silicone masks, plaster masks, headgear, head molds
+and other forms of attacks. Each column in the first five
+sections of the figure represents a mainstream surveillance
+camera, where C1 to C7 represents DS-2CD3T87WD-L, DS-
+2CD3T86FWDV2-I3S, TL-IPC586HP, TL-IPC586FP, DH-
+IPC-HFW4843M, DH-P80A1-SA, and ZD5920-Gi4N cam-
+eras. Each row in the first five sections of the figure represents
+a weather or shooting time, with samples taken on sunny days,
+cloudy days, windy days, snowy days, and nights, respectively.
+The sixth section of the figure lists head molds and other forms
+of attack, from left to right, in each column are adversarial
+masks, adversarial hats, replay attacks in electronic screens,
+posters, cardboards, and head molds.
+B. Sample of scenes
+As shown in the Fig. 9, we have listed all 40 scenes included
+in the SuHiFiMask, which include daily life scenes (e.g., cafes,
+cinemas, and theaters) and security check scenes (e.g., security
+check lanes and parking lots) for deploying face recognition
+systems. On the left side of the figure is the number of each
+scene in the row, which is the basis for naming the videos in
+the dataset. In addition, we need to increase the relevance of
+the data content and surveillance scenes by asking the subjects
+do scene-related behaviors in the scenes, such as asking the
+subjects sit around a coffee table and drink coffee in the coffee
+shop scenes. It is worth mentioning that some scenes in real
+life are vulnerable to attack in both day and night, so we
+identify the day and night of this scene as two different scenes,
+such as parking lot (day) and parking lot (night).
+
+14
+Fig. 8. We show some samples taken on snowy, cloudy, windy, sunny and night time days. Among them, C1 to C7 represents the surveillance cameras with
+DS-2CD3T87WD-L, DS-2CD3T86FWDV2-I3S, TL-IPC586HP, TL-IPC586FP, DH-IPC-HFW4843M, DH-P80A1-SA, and ZD5920-Gi4N, respectively.
+
+Live
+Resin
+C2
+C3
+C5
+C4
+C4
+C6
+C2
+C1
+Sunny
+ Cloudy
+Windy
+Snowy
+Night
+Silicone
+Plaster
+C5
+C2
+C1
+C3
+C4
+C6
+C5
+C4
+C6
+C7
+G
+Cloudy
+Windy
+Snowy
+Night
+Headgear
+Head mold and other attacks
+C1
+C2
+C3
+C4
+C5
+C6
+C7
+Cardboard
+ADV1
+ADV2
+Screen
+Poster
+Head mold
+Sunny
+Cloudy
+Windy
+Snowy
+Nighi15
+Fig. 9. We show the 40 surveillance scenes included in SuHiFiMask, S1-S40 on the left side of the figure are the numbers of each scene.
+
+Surveillance scenes
+S1-S5
+Gate(outdoor)
+Side Hall(day)
+Side Hall(night)
+Yoga Room
+Gate(indoor)
+S6-S10
+Café
+Rotating staircase
+Security check lanes
+Screening Room
+Game Room
+ S11-S15
+ Sidewalk(day)
+Road1( night)
+Sidewalk(night)
+Road1(day)
+Road2( day)
+S16-S20
+Road2(night)
+Restaurant
+Corridor
+Hall
+Escape routes
+福
+S21-S25
+Bedroom
+Balcony(outdoor)
+Balcony(indoor)
+Theater
+Lounge
+0
+S26-S30
+Malls
+Florist
+Washroom
+Escalator entrance
+Mall Lobby
+S31-S35
+Construction Sites
+Ticketing area
+Billboard
+Cinemas
+Commercial Street
+S36-S40
+Parking lot(day)
+Parking lot(night)
+Stairway entrance
+Neighborhood(night)
+Neighborhood2(night)

59AyT4oBgHgl3EQfpfhu/content/tmp_files/2301.00526v1.pdf.txt

@@ -0,0 +1,1227 @@
+ 
+1
+ Unusual acceleration and size effects in grain boundary migration with shear coupling 
+ 
+Liang Yanga, Xinyuan Songb, Tingting Yuc, Dahai Liua*, Chuang Dengb,* 
+a School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang 
+330063, China 
+b Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada 
+c School of Aviation and Mechanical Engineering, Changzhou Institute of Technology, Changzhou, 
+Jiangsu 213032, China. 
+* Corresponding author: dhliu@nchu.edu.cn(D. Liu), Chuang.Deng@umanitoba.ca (C. Deng) 
+ 
+ 
+Graphical abstract 
+ 
+
+acceleration and size effect
+mechanism
+50
+80
+30
+Lx- Ly Lz
+25
+2Lx
+Lx- 2 0L, 2 0Lz
+40
+20
+2L-3L,-3Lz
+60
+8Lx
+20
+aDisplacement (n)
+30
+(eV)
+40
+20
+20
+1010
+Original
+-10
+Fitted
+-20
+-15
+:2
+0
+4
+6
+10
+12
+0
+200
+400
+0
+50
+100
+Time (ps)
+Time (ps)
+Displacement (nm)150
+12
+140
+topfullyfixed
+第0ps
+120
+120-
+nm)
+100e
+90
+bottomfree
+80
+er
+E
+ien
+60.
+0g
+2L
+40
+30
+20x
+0
+0
+0
+20
+40
+60
+80
+0
+200
+400
+600
+800
+Time (ps)
+Time (ps)
+attempt to alleviate acceleration and size effect 
+2
+Abstract 
+Grain boundary (GB) migration is widely believed to maintain a linear relation between its 
+displacement and time under a constant driving force. In this study, we investigated the migration 
+behaviors of a set of GBs in Ni by applying the synthetic driving force and shear stress via atomistic 
+simulations. It was found that the displacements of some shear-coupling GBs do not exhibit a linear or 
+approximately linear relation with the time, as widely assumed, but evidently exhibit an acceleration 
+tendency. Moreover, the boundary velocity significantly decreases when increasing the bicrystal size 
+perpendicular to the GB plane. These behaviors were verified to be independent of the magnitude and 
+type of driving force but closely related to the temperature and revealed to be unique to 
+shear-coupling GBs exhibiting a rise in the kinetic energy component along the shear direction. 
+Moreover, after many attempts, we found that the acceleration in migration and size effect can be 
+largely alleviated by adopting one specific kind of boundary condition. Nevertheless, the continuous 
+rise of kinetic energy still exists and leads to the true driving force for GB migration lower than the 
+nominally applied value. For that reason, a technique is proposed to extract the true driving force 
+based on a quantitative analysis of the work-energy relation in the bicrystal system. Accordingly, the 
+calculated true mobility shows that the recently proposed mobility tensor may not be symmetric at 
+relatively large driving forces.   
+Keywords: grain boundary migration; shear-coupling; size effect; mobility; atomistic simulation. 
+ 
+1. Introduction 
+Grain boundary (GB) migration is crucial to a variety of behaviors (e.g., grain growth, 
+recrystallization, and plastic deformation) and mechanical properties (e.g., strength and ductility) in 
+polycrystalline materials [1]. Up to now abundant and deep insights have been gained into the 
+characters and underlying mechanisms of GB migration based on theoretical and experimental 
+investigations [2-9]. Thereinto, dramatic attentions were paid to GB mobility M, which can be defined 
+as the coefficient relating to migration velocity v and driving force P (i.e., M = v/P) and is considered 
+an intrinsic GB property (i.e., only depending on material parameters, temperature, and boundary 
+
+ 
+3
+crystallography) [10-12].  
+Nevertheless, computational studies [13-15] have revealed that the magnitude of driving force 
+can leave significant influences on mobility, owing to the force-induced variation of boundary 
+structure and/or migration mechanism. For example, Deng and Schuh [13] found that for both 
+symmetrical and inclined Ni Σ5 100 tilt GBs, their mobilities agree well with the intrinsic values 
+obtained by the thermal fluctuation method [16] only when the applied driving force is sufficiently 
+low; increasing the driving force will lead to diffusive-to-ballistic transition in the migration 
+mechanism and enlarge the discrepancy between the extracted and intrinsic mobility values. Moreover, 
+for shear-coupling migration GBs (i.e., simultaneous translation in GB plane during the migration 
+along the boundary normal direction), Han and coworkers [7,17,18] demonstrated that both GB 
+mobility and shear-coupling factor (ratio of GB sliding and migration rates) do not only strongly 
+depend on the magnitude but also the source of driving force (stress or a jump in chemical potential 
+across the boundary). They further revealed that the mobility traditionally defined as a scalar should 
+be a symmetrical second-rank tensor [18]. The tensor components can be extracted by applying 
+driving forces in the directions perpendicular and tangent to the boundary plane, respectively [18]. 
+In addition to the driving force, GB motion may also strongly depend on the size of simulation 
+cell. Zhou et al. [19] reported that the mobility of a Ni 100 tilt GB decreased monotonically with 
+decreasing the cell thickness (the size along the tilt axis), due to the interference between the free 
+surface and the collective rearrangement of atoms during boundary motion driven by an external 
+stress. A similar size-dependency of mobility was also observed in Ref. [20]. Race et al. [21] revealed 
+that the boundary area of a flat 111 tilt GB should reach the meso-scale or a large-enough value to 
+yield a converged migration velocity under the synthetic driving force (SDF). Meanwhile, simulations 
+for stress-driven [22,23] and SDF-driven [24] migration further discovered that the energy barrier for 
+disconnection nucleation or the driving force for GB migration would converge when the boundary 
+area was large enough for shear-coupling GBs. The energy barrier was also found to firstly decrease 
+and then keep steady with the increase of cell size in GB normal direction for 53.1º Σ5 100 tilt GB 
+[24]. Existing studies concerning the size effect on GB migration overall reached a consensus that the 
+
+ 
+4
+system size should be large enough to yield physically reliable results and conclusions. This agrees 
+well with the general understanding related to the size effect in modeling and simulation.  
+Although GB migration has been reported to suffer influences from various factors (e.g., 
+crystallography [11,25], temperature [13,26], driving force [10,14], pressure [27,28] and impurity 
+[29,30]), the mobility values extracted from M = v/P in these studies were all based on a basic premise 
+that the boundary velocity will maintain constant or approximately constant during the whole 
+migration process under a fixed driving force, i.e., the boundary displacement exhibiting a linear or 
+approximately linear relation with the migration time. This character has been widely observed in 
+existing research concerning GB migration (e.g., Refs. [3,31-34]). Nevertheless, in this study, we 
+found that velocities of some GBs did not keep constant during migration but exhibited an unusual 
+acceleration feature, i.e., the velocity varying significantly with GB relative position in the simulation 
+cell. This signifies a strong dependency of migration on the cell size in the direction perpendicular to 
+GB plane, which has not yet been discovered before.  
+The first effort of the present work was therefore to investigate the underlying mechanisms for 
+the acceleration in GB migration and effects of model size in GB normal direction on migration, 
+based on atomistic simulations of several GBs driven by the external stress and SDF. After the 
+corresponding conditions and mechanisms for these two phenomena were clarified, attentions were 
+paid to effectively alleviate the size effect and to extract the true driving force and mobility in the 
+presence of acceleration.  
+2. Methodology 
+In this study, the acceleration in GB migration, size effect and other related contents were 
+investigated based on atomistic simulations of several GBs in Ni. First, we simulated the migration of 
+Ni Σ5 100{310}, Σ29 100 {10 4 0} and Σ55 211 {952} symmetrical tilt GBs, which correspond 
+to P1, P148 and P233 GBs in the 388 GBs dataset constructed by Olmsted et al. [11], to reveal the 
+phenomena of acceleration and dependency of migration on the cell size in the direction perpendicular 
+to GB plane. These simulations were performed at 500 K and an SDF of 0.06 eV when increasing the 
+normal cell size for each GB. Additional simulations were carried out at lower SDF (i.e., 0.025 eV for 
+
+ 
+5
+P1 and 0.003–0.006 eV for P233) and under the external shear stress τext (τext = 250 MPa for P1 and τext 
+= 50 MPa for P233) to test whether the above phenomena are affected by the magnitude and type of 
+driving force, respectively. Second, we simulated the migration behaviors for Ni Σ3 110 twist GB 
+(i.e., P5) at 500 – 1000 K and 0.006 – 0.06 eV, Σ21 210 general GB (i.e., P81) at 1000 K and 0.001 – 
+0.06 eV, P1 at 1000 K and 0.06 – 0.003 eV. These simulations were aimed to explore the physical 
+causes for the acceleration and size-dependency from the aspects of shear coupling and work-energy 
+relation in the bicrystal system. Third, we chose the P1 GB as the representative example to attempt 
+approaches for inhibiting or alleviating the size effect by performing simulations adopting various 
+boundary conditions (BCs) and/or manipulating the internal stress in the system. The corresponding 
+results were compared with those reported in existing studies whenever possible. Finally, we still 
+chose P1 as the example to discuss how to extract the true driving force for GBs exhibiting 
+acceleration when the true driving force was not equal to the value nominally applied through the 
+SDF method or the external shear stress. 
+All simulations stated above were performed using the Large-scale Atomic/Molecular Massively 
+Parallel Simulator (LAMMPS) software package [35] with the embedded atom method potential 
+developed for Ni [36]. As shown in Fig. 1(a), a bicrystal simulation cell was used to construct a flat 
+GB, which was in y-z plane with y and z directions being periodic. Nevertheless, the boundary 
+conditions in x direction (parallel to the boundary normal direction) might be quite different, 
+depending on the simulation tests. For the above first and second groups of simulations, in order to 
+avoid translation of the whole bicrystal in GB normal direction, a slab of atoms (1nm thickness) near 
+the bottom surface were partially fixed (i.e., only the velocity and force components along the x 
+direction being set as zero) while the top surface was set free. For the third group of simulations, the 
+boundary conditions in x direction might be periodic, fully fixed, free, or one surface free while the 
+other fixed. When exploring the size effect on boundary migration, the cell size in the GB normal 
+direction (i.e., the grain size) or in the boundary plane (i.e., the boundary area) was varied accordingly. 
+The minimum cell size for each above GB was the same as those constructed by Olmsted in Ref. [11]. 
+For example, the minimum cell size for P1 was Lx = 18.3 nm, Ly = 3.2 nm, and Lz = 3.3 nm.  
+
+ 
+6
+ When the bicrystal system of each GB was constructed, its energy was minimized at 0 K 
+following the scheme introduced in Ref. [11]. Subsequently, the system was elevated to and 
+sufficiently relaxed at test temperatures (500 K or 1000 K) under the isothermal-isobaric ensemble 
+(NPT) for about 0.15–7 ns (depending on the temperature, GB, and cell size) with a default time step 
+of 5 fs. After the system was fully equilibrated, the boundary was driven to migrate under a jump in 
+chemical potential across the boundary or a shear stress. Thereinto, the former was realized by the 
+CROP-SDF method [14] while the latter by applying a shear force to individual atoms near the 
+bottom surface (1 nm thickness in x direction) in Fig. 1(a). The GB displacement under the SDF and 
+stress was computed by tracking the overall change of the potential energy artificially added to the 
+bicrystal system and by tracking atoms with the centro-symmetry parameters close to the maximum 
+value in the system, respectively. For the above four groups of simulations, the NPT ensemble was 
+also used during the boundary migration for each case, if not otherwise specified, to control the 
+internal normal stress components as close to 0 GPa as possible. For some specific simulations, the 
+internal shear stress along the shear direction also needed to be controlled at 0 GPa. The structure of 
+bicrystal model, if needed, was visualized by Ovito package [37]. Note that some results concerning 
+this study were presented in the Supplementary Material.  
+3. Results and discussion 
+3.1 Acceleration in migration and size effect 
+   Fig. 1(b-e) represents the migration data of Σ5 100 {310} tilt GB (i.e., P1 GB) when increasing 
+the cell size along different directions while under a constant external driving force. Two features can 
+be readily observed. First, the boundary velocities (slope of displacement curve) under various cell 
+sizes all gradually increase with the proceeding of boundary migration, suggesting a clear dependency 
+of migration velocity on the relative GB position along the boundary normal direction in the 
+simulation cell. In contrast, the boundary displacement was widely observed and commonly assumed 
+to exhibit a linear or approximately linear relation with the migration time (e.g., Refs. [3,31]). After a 
+detailed survey of existing studies, the acceleration in migration was only found in the work by 
+Coleman et al. [38], who simulated the migration of Ni Σ37 100 symmetrical tilt by applying the 
+
+ 
+7
+synthetic driving force and shear strain at 300 and 400 K. Unfortunately, the focus in Ref. [38] was 
+the atomic mechanisms of migration and no attention was paid to the acceleration.  
+ 
+Fig. 1 (a) Schematic of the bicrystal simulation cell. GB displacement vs. time for P1 GB simulated at 500 K 
+and under a synthetic force of 0.06 eV, when increasing the cell size along (b) x, (c) y, (d) z and (e) all three 
+directions. To avoid translation of the whole bicrystal along the x direction, the bottom surface was partially 
+fixed (i.e., setting the velocity and force components along x for atoms near the surface as zero) while the top 
+surface was set free. 
+ 
+Second, the velocity is independent of the cell size in GB plane (i.e., the boundary area or lateral 
+cell size) (see Fig. 1c-e) but strongly and negatively related to the size in the boundary normal 
+direction (x direction in this study) (see Fig. 1b and e). Nevertheless, the velocity of flat boundary has 
+been previously reported to show a strong and complex dependence on the boundary area [21]. In 
+addition to the velocity, the boundary area has also been reported to cause significant influence on GB 
+mobility [19,20] and the energy barrier of disconnection nucleation for GB migration [22-24]. 
+Moreover, these properties concerning GB migration exhibited a consistency in their dependency on 
+the boundary area, i.e., the boundary area should be sufficiently large to yield a converged property 
+value. The negative dependency of velocity on the cell size along the boundary normal (i.e., vertical 
+cell size) here is partially similar to the trend regarding the threshold driving force of disconnection 
+nucleation for Cu Σ5 100 {210} tilt GB (i.e., P6 GB in Ref. [11]) at 10 K revealed by Deng and 
+Deng [24], who found the threshold driving force, which in practice can be qualitatively regarded as 
+the reverse of GB mobility [9], overall declines when increasing the vertical cell size. Therefore, the 
+size effects regarding the boundary area are different between the present and existing studies, but a 
+
+ 
+8
+similarity appears in the dependency on the vertical cell size.  
+As a typical of low-period and high-angle CSL boundary, the migration behaviors of P1 GB have 
+been widely studied through atomistic simulations [6,10,11,13,14,17,34,39-41], but why the above 
+two features were not reported before? This can be attributed to multiple factors. First of all, there has 
+been no research adopting various vertical cell sizes for this GB up to now, and accordingly no insight 
+into the size effect was obtained. In studies adopting fixed vertical size [10,11,14,17,34,39], the 
+acceleration might also exist, though the displacement-time data was not directly provided in these 
+studies. Nevertheless, we deem that the acceleration might have been disregarded on the grounds that 
+the main attentions and efforts were focusing on exploring the intended objectives of individual 
+studies, as in our previous work [14,34]. Meanwhile, the disappearance of acceleration can also be 
+attributed to the relatively high temperatures (e.g., 1000, 1200 and 1400 K) tested in Refs. [10,11,39] 
+(see discussion in Section 3.2). In addition, the periodic boundary condition imposed along the 
+boundary normal direction will prevent the presence of acceleration in work [13,40]. This boundary 
+condition has been confirmed to inhibit the shear-coupling migration [21,41], which is a necessary but 
+not a sufficient condition for acceleration migration (see following discussion concerning Figs. 2 and 
+3). When exploring the shear coupling migration of Cu P1 GB, Cahn et al. [6] directly presented a 
+displacement-time data up to 1 nm, simulated by applying a constant shear strain 1 m/s at 800 K (see 
+Fig. 6 in Ref. [6]). To our understanding, 1 nm data may not be sufficiently long to evidently illustrate 
+the acceleration feature, in comparison with the displacement data in Fig. 1. Another 
+displacement-time data (up to 6 nm) was provided by Schartt and Mohles [41], who simulated the 
+migration of Ni P1 under 300–1000 K with free end boundary conditions and a synthetic force of 0.06 
+eV imposed through the ECO-SDF method. Nevertheless, the acceleration feature was still not 
+observed in Ref. [41] though it shows up in our re-tests of simulations in Fig. 1 by using the 
+ECO-SDF method. Therefore, the discrepancy concerning the acceleration should not be attributed to 
+the different versions of SDF method (i.e., CROP-SDF [14] or ECO-SDF [41]) utilized in the present 
+study and Ref. [41]. Since the temperature dependency of migration velocity and shear coupling 
+factor  in the range of 300-700 K in [41] also appears different from those previously reported 
+[6,9,13,17], we deem that the discrepancy may be resulted from the difference in the metastable 
+
+ 
+9
+structures for P1 GB adopted for various studies. 
+To evaluate whether the acceleration in migration or the corresponding size effect is unique to P1 
+GB or not, Fig. 2 show the results simulated for some other GBs or under simulation settings different 
+from Fig. 1. As shown in Fig. 2(a) and (b), the two features can also be observed for P148 and P233 
+GBs when adopting the same settings as for P1 in Fig. 1. Since the driving force applied in Figs. 1, 
+2(a) and (b) is a relatively high value (i.e., 0.06 eV ≈ 0.87 GPa) in comparison with experimentally 
+applied values, we tested lower forces for P1 and P233 GBs. It can be seen from in Fig. 2(c) and (d) 
+that these features still hold on for P1 GB at 0.025 eV (lower than KT = 0.043 eV, K Boltzmann 
+constant and T temperature) and for P233 GB at 0.003 eV which approaches typical experimental 
+values. Note that lower forces have also been tried for P1 but failed to yield continuous boundary 
+migration, agreeing with the threshold driving force of boundary migration determined for this GB at 
+500 K by Yu et al. [9]. It is important to note from Fig. 2(e) and (f) that the acceleration and size effect 
+still show up for P1 and P233 GBs when applying the external shear stress to drive the GB migration. 
+Therefore, while the shear coupling mode may be strongly influenced by both the magnitude and type 
+of the driving force [17], the acceleration and size effect does not exhibit such dependency.  
+The above analysis suggests that the acceleration in migration and negative dependency on the 
+vertical cell size are relatively common features for force-driven GB migration. The following content 
+will further reveal that the latter feature is resulted from the former one. These two features extend our 
+current understandings of size-effect on GB migration, which almost all focused on the size in the 
+boundary plane [19-23]. They also remind us that attentions should be taken for GBs exhibiting such 
+features when extracting the boundary velocity or mobility under a constant driving force, during 
+which the migration displacement and time were almost always assumed to keep a linear relation.  
+
+ 
+10
+ 
+Fig. 2 Other results simulated at 500 K supporting the acceleration in migration and size effect. Displacement 
+data in (a-d) were all simulated under the applied synthetic force while (e, f) under the external shear stress. 
+The shear stress was applied to a slab of atoms (1 nm thickness) near the top surface, as illustrated in Fig. 1, 
+while a slab of atoms near the bottom surface was set as a grid body. 
+3.2 Underlying mechanism for acceleration and size effects 
+The acceleration in boundary migration and vertical size effect have been revealed in the above 
+section, then for what types of GB or under what kinds of condition that such phenomenon will occur? 
+A preliminary analysis of the three GBs tested in Figs. 1 and 2 indicates that they are all 
+shear-coupling migration GBs and with  > 0.5. Fig. 3(a) chooses P5 (Ni Σ3 110 twist) GB as an 
+example to show the displacement-time (S-t) data and size-dependency of GBs without shear-coupling. 
+It can be seen that the displacement is linearly related to the migration time and the corresponding 
+velocities are the same under different vertical sizes, irrespective of temperature and magnitude of 
+driving force. These results seemingly indicate that only shear-coupling GBs will exhibit acceleration 
+
+ 
+11
+migration and negative size-dependency. However, as shown in Fig. 3(b) for P81 GB, the linear S-t 
+relation and constant v under different sizes can be observed also for shear-coupling GBs at various 
+temperatures and driving forces. Furthermore, the acceleration migration and size effect observed at 
+500 K for P1 GB (Fig. 1) unexpectedly transfers into uniform migration when raising the temperature 
+to 1000 K at which the shear coupling still exists, regardless of the magnitude of driving force (see 
+Fig. 3(c)). This kind of transition induced by the temperature also occurs for other shear-coupling GBs 
+(see Fig. s1 in the Supplementary file). To our understanding, the transition can be attributed to the 
+temperature-induced variation of disconnections mediated for GB migration, which has been widely 
+observed [7,23]. Based on these analyses, we conclude that the shear coupling is a necessary but not a 
+sufficient condition for the acceleration in migration and therefore the size effect, which may suffer 
+strong influence from the temperature. 
+ 
+Fig. 3 Examples of uniform migration for GBs with or without shear-coupling: (a) P5; (b) P81; (c) P1. (d) and 
+(e) presents kinetic energy E for P1 with cell size Lx, simulated at 500 K and 1000 K, respectively. 
+ 
+To further explore the fundamental mechanisms for acceleration, Fig. 3(d) compares the relative 
+
+ 
+12
+variation of kinetic energy (Ei, i = x, y or z) to the initial state for P1 GB at 500 and 1000 K, at which 
+the boundary exhibits accelerated (Fig. 1(b)) and uniform (Fig. 3(c)) migration, respectively. At 1000 
+K, all three components of the kinetic energy remain almost unchanged (i.e., Ex = Ey = Ez ≈ 0) 
+with the proceeding of migration. In contrast, at 500 K, although Ex and Ey still remain unchanged, 
+Ez firstly increases and then decreases (the final Ez is still much higher than zero). Note that the 
+shear movement is parallel to z direction. The comparison suggests that the work (Wext) done by the 
+external driving force (Pext) only contributes to shear-coupling migration at 1000 K, but to both 
+shear-coupling migration and a rise in the shear kinetic energy (Ez) at 500 K. This difference reminds 
+us that the accelerated and uniform migration can be qualitatively justified from the aspect of true 
+driving force (Ptrue) for boundary migration, which can be influenced by Wext and Ez.  
+During the process of boundary migration, the work and kinetic energy for the bicrystal system 
+meet the relation of Wext = Wtrue + E. Wtrue is the work done by Ptrue, i.e., Wtrue = PtrueꞏSꞏAGB, S and AGB 
+stand for the GB displacement and area, respectively. Considering Ex = Ey ≈ 0 in the case of both 
+accelerated and uniform migration, the relation can be given as Wext = Wtrue + Ez. At one specific 
+moment of migration, the work-energy relation can be further described as dPextꞏdS = dPtrueꞏdS + 
+d(Ez)/AGB, and the instant true driving force is dPtrue = dPext – (d(Ez)/dS)/AGB, where dPext is a fixed 
+value for the SDF method. Therefore, dPtrue will be constant when Ez keeps unchanged (e.g., 1000 K 
+at Fig. 3(d)), i.e., dPtrue = dPext. In such case, the boundary will accordingly exhibit uniform migration 
+(i.e., a linear S-t relation) and thus consistent velocities when adopting distinct vertical sizes but the 
+same Pext (e.g., Fig. 3(c)). Nevertheless, in the case of accelerated migration (i.e., Ez ≠ 0, see Fig. 
+3(d)), dPtrue depends on both dPext and d(Ez)/dS. From the d(Ez)/dS vs. S curve (the red curve) at 
+500 K for P1 GB shown in Fig. 4, we can observe that d(Ez)/dS continuously descends with the 
+boundary migration, suggesting a continuous rise in dPtrue and thus in migration velocity. Moreover, it 
+is conceivable that when applying the same dPext to bicrystal systems with different vertical sizes (e.g., 
+Fig. 1(b)), dPtrue will be lower (i.e., lower velocities) for larger systems due to higher d(Ez), which 
+can be further attributed to more atoms involving shear movement for larger systems. When applying 
+this interpretation to justify the size effect for systems with different sizes in the boundary plane (e.g., 
+
+ 
+13
+Fig. 1(c)), the contribution of GB area to dPtrue must also be considered. In summary, the above 
+analysis suggests that the acceleration in migration and negative dependency of velocity on the 
+vertical size are unique to shear-coupling GBs exhibiting a rise in the kinetic energy component along 
+the shear direction and can be justified from the aspect of true driving force based on the work-energy 
+relation in the bicrystal system. 
+80
+30
+ 
+Fig. 4 Variation of d(Ez)/dS with the boundary displacement for P1 GB, calculated based on the Ez-S curve 
+(blue curve) obtained by the least-square fitting of the original data at 500 K and 0.06 eV, given in Fig. 3(d). 
+3.3 Attempts to alleviate acceleration  
+Although the acceleration in migration has been demonstrated as a relatively common feature for 
+force-driven migration of flat GBs in Section 3.1, it is undesired if the purpose is to compute a GB 
+mobility by assuming v = MP. Then, is it possible to inhibit or alleviate the acceleration? For this 
+purpose, we have carried out a series of simulations by manipulating the boundary conditions and 
+internal stress in each bicrystal system (see Figs. 5-7).  
+Firstly, we tried to adopt periodic boundary condition in the GB normal direction (x direction in 
+Fig. 1(a)), which is one kind of boundary conditions widely used in previous studies [13,31,41]. It can 
+be seen from Fig. 5(a) that the boundary displacements under various vertical sizes all nearly exhibit a 
+linear relation with the time; the velocity nevertheless keeps increasing when enlarging the vertical 
+size, in contrast to a negative size dependency of velocity in Fig. 1. Meanwhile, in comparison with 
+0.06 and 0.025 eV applied for the boundary condition adopted in Figs. 1 and 2c, much larger driving 
+force (i.e., 0.15eV) must be applied to initiate the boundary movement for all cell sizes, suggesting a 
+
+ 
+14
+significant effect of boundary condition. Under the present boundary condition, the internal shear 
+stress sharply increases with the initiation of GB migration, and then experiences a short descending 
+and finally fluctuates around a very high value (see Fig. 5(b)). Furthermore, the shear stress is 
+obviously lower under larger cell size. The higher velocity under larger size in Fig. 5(a) is therefore 
+resulted from this tendency of shear stress, which can be attributed to more elastic energy released 
+with the boundary migration due to larger space along the GB normal. As shown in Fig. 5(c), the 
+periodic surface strongly inhibits the overall relative shear movement between two grains, as revealed 
+in Refs. [13,41]), but only enables local shear movement which is more evident for larger cells. This 
+precisely accounts for the linear S-t relation under various sizes in Fig. 5(a), according to the 
+discussion concerning mechanism for acceleration in Section 3.2. Evidently, the periodic boundary 
+can effectively eliminate the acceleration but not the size effect by significantly constraining the shear 
+movement.   
+  
+Fig. 5 Migration results vs. vertical size for P1 GB, simulated at 500 K and 0.15 eV by adopting periodic 
+boundary conditions along the GB normal direction: (a) displacement data; (b) shear stress; (c) snapshot of 
+boundary migration. Red and white arrows in (c) illustrate the migration and shear directions, respectively. 
+Atoms are colored by atom type to visualize the shear-coupling migration using the Ovito software [37]. 
+ 
+Secondly, we tried to set the top and bottom surfaces to be fully fixed. It can be seen from Fig. 6 
+that most of the results are similar to those under periodic boundary in Fig. 5, e.g., much larger 
+driving force, linear S-t and very high shear stress. Additionally, displacements are nearly independent 
+of the system size (Fig. 6(a)), though the normal stress continues to rise with the boundary movement 
+(see the example shown for cell size of Lx in Fig. 6(b)). The corresponding velocity is much lower 
+than that at 0.06 eV in Fig. 1 while close to that at 0.15 eV and Lx in Fig. 5. In consistency with the 
+
+Initialnitia]
+2L
+8L
+20LC
+GBGB 
+15
+periodic boundary, the boundary condition of fixed ends also inhibits the global shear movement and 
+enables only local shear. Fig. 6(c) shows that the local shear-coupling mode may change even switch 
+under fixed ends, as already observed in Ref. [42].  
+0
+50
+100
+150
+200
+250
+0
+5
+10
+15
+20
+Time (ps)
+Displacement (nm)
+Lx
+2Lx
+8Lx
+20Lx
+500 K, 0 15 eV
+.
+25
+(a)
+Lx
+  
+ 
+Fig. 6 Migration results for P1 GB simulated at 500 K and 0.15 eV while setting the top and bottom surfaces 
+to be fully fixed: (a) displacement data; (b) shear stress and normal stress under cell size Lx; (c) snapshot of 
+boundary migration. The red and yellow atoms colored in (c) are aimed to visualize the local shear movement 
+and shear-switching.  
+ 
+Thirdly, considering the impeding effect of the high internal shear stress on shear movement, we 
+performed simulations that controlling the stress as close to 0 GPa as possible under periodic 
+boundary (see Fig. 7). Although shear stresses are well controlled especially for larger systems (Fig. 
+7(b)), acceleration and size effect still exist (Fig. 7(a)). Moreover, the boundary stagnates long before 
+reaching the other end, and the final displacement value under each size is nearly half of the feasibly 
+maximum value (compare Figs. 5(a) and 7(a)). The inclined line displayed by the yellow atoms in 
+Figs. 7(c) illustrates that the shear movement is inhomogeneous along the GB normal direction, and 
+the top and bottom parts of the blue grain make shear along two opposite directions (see the white 
+arrows). These results should be resulted from the cell inclination when controlling shear stress (Fig. 
+7(c)), which leads to the variation of crystallographic orientation and thus erroneous exertion of the 
+orientation-dependent driving force for shear-coupling migration. We have also tried to control shear 
+stress for fixed and free boundary conditions but as well obtained cell inclination and other results 
+similar to those by using periodic boundary. Evidently, the above three attempts all failed to achieve 
+our anticipated objectives of effectively alleviating the acceleration and size effect.  
+ 
+
+CGB
+200ps:
+150ps
+150psInitiaInitial
+27
+8L
+201 
+16
+  
+ 
+Fig. 7 Migration results vs. vertical cell size for P1 GB, simulated at 500 K and 0.06 eV by adopting periodic 
+boundary while controlling shear stress close to 0 GPa: (a) displacement data; (b) shear stress; (c) snapshot of 
+boundary migration.  
+Finally, considering the inhibition of overall shear movement by periodic and fixed boundaries, 
+we performed simulations adopting two free boundaries or setting one boundary as free while the 
+other as fixed. It can be seen from Fig. 8(a) that the displacement data under two free boundaries and 
+one free while another partially fixed are consistent and exhibiting an acceleration tendency (as 
+observed in Fig. 1(b-e)), because the two grains across the GB are free to shear under these two 
+boundary conditions. When the bottom surface is fully fixed (i.e., the grain near this surface is not 
+allowed to shear), significant acceleration is observed through the whole migration process (the blue 
+curve in Fig. 8(a)). Nevertheless, in the case of fully fixed top surface, acceleration is only significant 
+at the early migration stage and gradually turns into uniform migration at the later stage (see the green 
+curve and black dashed line in Fig. 8(a)), suggesting a gradual weakening of acceleration. This 
+tendency can be justified from the similarity between the variations of Ez–t and S-t curves in Fig. 8(b) 
+and from the relation dPtrue = dPext – (d(Ez)/dS)/AGB. Moreover, as illustrated by the inset snapshots 
+in Fig. 8(b), the overall shear and coupling factor are not influenced by this boundary condition. 
+Furthermore, Fig. 8(c) presents the comparison of the linear S-t segments extracted from the complete 
+displacement data under various sizes. Interestingly, velocities are consistent for different cell sizes. 
+Therefore, the size effect for shear-coupling GB can be considered as being eliminated if only 
+focusing on the uniform migration stage. With these attempts, we may conclude that setting the top 
+surface (i.e., in the forward direction of GB migration) of the bicrystal as fully fixed while the bottom 
+surface (the backward direction of migration) as free is a relatively effective way to largely alleviate 
+
+0 ps
+5 ps
+15 ps
+25 ps 
+17
+acceleration migration and thus size-dependency.  
+ 
+   
+0
+200
+400
+600
+800
+0
+20
+40
+60
+80
+100
+120
+140
+Time (ps)
+Displacement (nm)
+ 
+ 
+Lx
+2Lx
+8Lx
+20Lx
+(c)
+ 
+Fig. 8 Attempts to alleviate acceleration by adopting one or two free boundaries for P1 GB with normal size 
+Lx, simulated at 500 K and 0.06 eV. (a) Comparison of displacement data under different boundary conditions. 
+(b) Variation of displacement, kinetic energy and snapshot with the time when setting the bottom surface as 
+free while the top one as fully fixed. (c) Comparison of the linear S-t segments extracted from the complete 
+displacement data under various cell sizes, simulated by adopting the same boundary condition as in (b). 
+When the surface is partially (as in Fig. 1) or fully (setting all force and velocity components for atoms near 
+the surface as zero) fixed, the grain near this surface can or can not make overall shear movement. The 
+complete displacement data for (c) can be found in Fig. s2 in the Supplementary file.  
+3.4 Extraction of true driving force and mobility 
+Although the accelerated migration and size effect for shear-coupling GBs can be effectively 
+weaken by adopting one special boundary condition, the kinetic energy of the system Ez still continues 
+to rise during the boundary migration (see Fig. 8(b)), and thus the true driving force Ptrue does not 
+equal to the externally applied value Pext and depends on the variation of Ez (see discussion in Section 
+3.2). Therefore, efforts should be paid to extract Ptrue and the corresponding true mobility Mtrue. As 
+already discussed in Section 3.2, Ptrue can be determined based on a quantitative analysis of the 
+
+(b) 150
+12
+top fullv fixedps
+120Isp
+58 ps90
+acen
+bottom freee
+60nm300
+20
+00
+80
+Time (ps) 
+18
+work-energy relation in the bicrystal system. Meanwhile, considering the continuous rise of Ez with 
+boundary migration, we may not obtain a constant but a time dependent Ptrue(t), and therefore the 
+quantitative analysis should be carried out for individual steps of GB migration. The artificial energy 
+added to the system by the SDF method in time interval dt can be written as: 
+dE = eꞏdn                                (1) 
+where e denotes the maximum potential energy added to a single atom (e.g., 0.06 eV in Fig. 1) and 
+dn indicates the number of atoms whose corresponding crystallography changed as the boundary 
+migrates. dn can be calculated by dn = (N/Lx)ꞏv(t)ꞏdt. Here, N and v(t) represent the total number of 
+atoms in the system and the instant migration velocity perpendicular to GB plane, respectively. 
+However, due to the continuous rise of Ez, the actual energy to drive boundary movement is  
+dE' = dE – dEz = dE – czꞏdt                           (2) 
+where cz is variation rate of dEz with respect to time. We can thus deduce the true maximum energy 
+imposed on per atom e'(t) as: 
+ e'(t) = (dE – dEz)/dn = e – cz/(vꞏN/Lx)                   (3) 
+For the SDF method, e is normally considered as Pext [11,13,41] and thus e'(t) can also be treated as 
+Ptrue(t), which then can be further used to calculate Mtrue.  
+From the internal stress data in Fig. 9(a), one can observe that the normal stress fluctuates around 
+0 GPa while the internal shear stress τxz roughly keeps a positive value and gradually declines when 
+adopting the boundary condition of a fully fixed top surface. This means that the normal stress leaves 
+no influence on GB migration, but one part of dE may be used to overcome the impeding effect of τxz 
+on migration, which can be quantitatively described in the form of shear strain energy Ess = 
+0.5Vꞏ
+2
+xz
+
+/G. V and G stand for the volume and shear modulus for bicrystal system, respectively. 
+However, Ess is essentially a type of elastic energy and will be dynamically stored and released with 
+the continuous migration of the GB, as supported by Fig. 9(b) which indicates that this energy 
+increment dEss also fluctuates around 0 eV (i.e., the long-time average of dEss equaling zero). 
+Therefore, Ess should make no contribution to the overall work-energy relation in the system and the 
+potential influence of τxz on GB migration does not need to be considered. Accordingly, Eq. (3) still 
+
+ 
+19
+holds for extracting Ptrue. 
+ 
+Fig. 9 Variation of (a) the internal stress and (b) dEss with the migration time for P1 GB simulated in Fig. 8(b) 
+Table 1 presents the calculated Ptrue and Mtrue for P1 GB at 500 K according to Eq. (3). In contrast 
+to the early migration stage, the velocity, driving force and mobility at the later stage (e.g., t > 50 ps 
+for Lx and t > 400 ps for 8Lx) all only rise slightly and are nearly consistent under Lx and 8Lx systems. 
+If we calculate the average value for v, Ptrue and Mtrue at the later stage, we can get the average v, Ptrue 
+and Mtrue for Lx system as 0.163 nm/ps, 0.0486 eV and 3.341nm/(ps eV) while v = 0.162 nm/ps, Ptrue = 
+0.0485 eV and Mtrue = 3.353 nm/(ps eV) for 8Lx. The relative differences of these three data between 
+the two systems are all lower than 1%. These results and comparisons again emphasize that the 
+acceleration and size-effect in the GB migration velocity have been nearly eliminated through 
+applying one special boundary condition. More importantly, they also signify that we can obtain 
+consistent true mobility values for systems with distinct vertical sizes if further considering the 
+correction of true driving force.   
+It should be noted that the above principle of correcting the driving force should also be 
+applicable to the shear-coupled migration driven by an external shear stress τext (i.e., applicable to Fig. 
+2(e) and (f)). As shown in the inset of Fig. 10(a) for τext-driven migration, Ex and Ey remain 
+unchanged while Ez overall increases linearly with time, in consistency with the tendency shown for 
+the SDF-driven migration in Fig. 8(b). The corresponding correction equation of Ptrue can be given as:   
+z
+true
+ext
+z
+GB
+c
+P
+v
+A
+
+
+
+
+                                (4) 
+where vz stands for the shear velocity of GB along the z direction.  
+
+ 
+20
+Table 1 True driving forces and mobility values extracted based on Eq. (3) for P1 GB, simulated at 500 K and 
+0.06 eV when setting the bottom surface as free while the top one as fully fixed. Both cz and v were obtained by 
+the least-square fitting into discrete Ek and S data. 
+Lx 
+8Lx 
+t 
+(ps) 
+v 
+(nm ps‒1) 
+Ptrue 
+(eV) 
+Mtrue 
+(nm ps‒1 eV‒1)
+t  
+(ps) 
+v 
+(nm ps‒1) 
+Ptrue 
+(eV) 
+Mtrue 
+(nm ps‒1 eV‒1)
+5* 
+0.030  
+-0.0016  -18.906  
+40* 
+0.035  
+0.0068  
+5.187  
+10 
+0.061  
+0.0299  
+2.054  
+80 
+0.065  
+0.0313  
+2.090  
+15 
+0.086  
+0.0386  
+2.237  
+120 
+0.089  
+0.0390  
+2.292  
+20 
+0.106  
+0.0426  
+2.489  
+160 
+0.108  
+0.0427  
+2.538  
+25 
+0.121  
+0.0447  
+2.707  
+200 
+0.123  
+0.0447  
+2.747  
+30 
+0.133  
+0.0461  
+2.879  
+240 
+0.134  
+0.0460  
+2.914  
+35 
+0.141  
+0.0469  
+3.010  
+280 
+0.142  
+0.0468  
+3.041  
+40 
+0.148  
+0.0475  
+3.109  
+320 
+0.149  
+0.0474  
+3.137  
+45 
+0.152  
+0.0479  
+3.182  
+360 
+0.153  
+0.0477  
+3.208  
+50 
+0.156  
+0.0481  
+3.237  
+400 
+0.157  
+0.0480  
+3.261  
+55 
+0.159  
+0.0483  
+3.279  
+440 
+0.159  
+0.0482  
+3.301  
+60 
+0.161  
+0.0485  
+3.314  
+480 
+0.161  
+0.0484  
+3.334  
+65 
+0.163  
+0.0486  
+3.345  
+520 
+0.163  
+0.0485  
+3.361  
+70 
+0.165  
+0.0488  
+3.375  
+560 
+0.164  
+0.0486  
+3.385  
+75 
+0.167  
+0.0489  
+3.405  
+600 
+0.166  
+0.0487  
+3.406  
+80 
+0.168  
+0.0490  
+3.435  
+640 
+0.167  
+0.0487  
+3.423  
+* Extracted Ptrue and Mtrue are erroneous due to the low velocity and kinetic energy at the very early stage of 
+migration  
+ 
+To compute the true mobility under an external shear stress, Fig. 10 still chooses the P1 GB as an 
+example and the shear stress has been carefully tuned so that the GB migrated at the same velocity as 
+that under the SDF, as shown in Fig. 10(a). Firstly, in comparison with Fig. 2(e), the acceleration in 
+migration is only significant at the early migration stage and turns into uniform migration at the later 
+stage for the current simulation (see blue curve in Fig. 10(a)). Secondly, the true mobility calculated 
+based on the corrected Ptrue according to Eq. (4) experiences a continuous rise and then becomes 
+nearly stable (see the blue square data in Fig. 10(b)). These results do reveal that the accelerated 
+migration under external shear stress can also be largely alleviated by the boundary condition as 
+utilized in Fig. 8(b) for the SDF-driven migration, and that the principle of correcting Ptrue is also 
+applicable to the case of τext-driven migration. 
+What is more, although the displacements or velocities are nearly the same under the SDF and 
+shear stress, the corresponding mobility values under the two types of driving force are significantly 
+
+ 
+21
+different (see Fig. 10(b)). Mtrue by SDF is on average 57.6% lower than that by τext. According to the 
+theory recently proposed by Chen et al. [18], the mobility values extracted by applying the driving 
+forces perpendicular and parallel to GB plane can be unified in a mobility tensor as following: 
+                        
+y
+  
+  
+  
+  
+  
+  
+xx
+xy
+xz
+x
+y
+x
+yy
+yz
+y
+z
+zx
+zy
+zz
+z
+M
+M
+M
+v
+v
+M
+M
+M
+v
+M
+M
+M
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                           (5) 
+Here x is perpendicular to GB plane while y and z are parallel to GB plane, φ is the driving force applied 
+along GB normal (e.g., the synthetic driving force [14,41]), and τy and τz are shear stresses. Moreover, 
+the GB mobility tensor should be symmetric according to the Onsager relation [18], i.e., Mxz = Mzx.  
+  
+ 
+Fig. 10 Migration results for P1 GB at 500 K when driven under τext : (a) displacement along GB normal; (b) 
+true mobility calculated based on the corrected Ptrue. The inset in (a) presents the variation of kinetic energy 
+Ei (i = x, y, z). To provide a better comparison of true mobility values between SDF-driven and τext-driven 
+migration, the magnitude of τext was chosen deliberately to ensure the corresponding displacement data as 
+close to that by SDF in Fig. 8(b) as possible. The shear stress was applied by adding a shear force 0.0037 
+eV/Å to a slab of atoms (1 nm thickness) near the bottom surface along z direction, while the top surface was 
+set as fully fixed.  
+ 
+As shown in Fig. 10(a),  equals 1.0 under both SDF and shear stress, indicating that vx = vz holds 
+under both types of driving force. In such case, we can deduce Mxx = Mzx for SDF and Mzz = Mxz for 
+shear stress. Nevertheless, the mobility data presented in Fig. 10(b) reveals Mzz ≈ 1.6Mxx, i.e., Mxz ≈ 
+1.6Mzx. Therefore, the symmetry of mobility tensor does not hold in the present study, seemingly 
+contradicting to the conclusion in Ref. [18]. Recently, we have systematically investigated the GB 
+mobility tensor and the effects of temperature and external driving force on its symmetry based on 
+
+ 
+22
+atomistic simulations of force-driven and force-free migration for the twist Ni Σ15 (2 1 1) (P14) GB 
+[43]. It is found that the symmetry holds at low driving force limit while fails at high driving forces. 
+Therefore, the non-equal Mxz and Mzx as shown in Fig. 10(b) is due to the relatively large driving 
+forces that have been applied in the current study, which is also consistent with the previous studies 
+that large driving forces can significantly change the underlying mechanisms for GB migration [13].   
+4. Conclusions 
+In this work, we investigated the migration behaviors of several GBs in Ni by atomistic 
+simulation. Based on that, the acceleration in GB migration and vertical size effects were revealed. 
+Then attentions were paid to reveal the concerning mechanisms for these phenomena and to 
+effectively alleviate them through manipulating the boundary conditions and internal stress in the 
+bicrystal system. At last, a method was proposed to extract the true driving force and true mobility, 
+based on which the symmetry of mobility tensor was discussed. The following conclusions can be 
+drawn based on this study: 
+(1) The migration displacements of some GBs driven under a constant external driving force are not 
+linearly related to the migration time, as widely assumed. The corresponding velocity nevertheless 
+gradually increases with the proceeding of boundary migration and is overall negatively related to 
+the cell size perpendicular to GB plane, irrespective of the magnitude and type of driving force. 
+These tendencies suggest that some previously published migration results for such GBs without 
+considering the acceleration and size effect may need to be calibrated. 
+(2) The acceleration and vertical size effect in migration are unique to shear-coupling GBs exhibiting 
+a rise in the kinetic energy component along the shear direction. It is precisely the rise of kinetic 
+energy that results in the true driving force for GB migration being lower than the nominally 
+applied value but continuing to increase, and thus leads to the accelerated migration. With the 
+increase of tested temperatures, the acceleration will transfer into uniform migration and the size 
+effect will accordingly disappear.  
+(3) Among the various attempts to eliminate or alleviate the acceleration and size-dependency in 
+migration, setting the cell surface in the forward direction of GB migration as being fully fixed 
+
+ 
+23
+while the surface in the backward direction as free is a relatively sample but effective way, which 
+is applicable to both shear stress and SDF-driven migration and does not change the 
+shear-coupling behaviors and migration mechanism.  
+(4) Based on a quantitative analysis of the work-energy relation in the bicrystal system, the true 
+driving force can be determined for GBs exhibiting accelerated migration. Under the coupling 
+effects of adopting one specific kind of boundary condition and correcting the true driving force, 
+we can obtain consistent true mobility values for such GBs with distinct vertical sizes.  
+Furthermore, the calculated true mobility suggests that the symmetry of the mobility tensor may 
+fail when the driving forces are too large. 
+ 
+Acknowledgment 
+The authors thank Dr. David L Olmsted for sharing the 388 Ni GB structure database. This 
+research was supported by the National Natural Science Foundation of China (Grant No. 
+52065045) and NSERC Discovery Grant (RGPIN-2019-05834), Canada, and the use of 
+computing resources provided by Compute/Calcul Canada.  
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+[21] C.P. Race, J. von Pezold, J. Neugebauer, Role of the mesoscale in migration kinetics of flat grain 
+boundaries, Phys. Rev. B 899 (2014) 214110. 
+[22] A. Rajabzadeh, F. Mompiou, M. Legros, N. Combe, Elementary mechanisms of shear-coupled 
+grain boundary migration, Phys. Rev. Lett. 110 (2013) 265507. 
+[23] N. Combe, F. Mompiou, M. Legros, Disconnections kinks and competing modes in 
+shear-coupled grain boundary migration, Phys. Rev. B. 93 (2016) 024109. 
+[24] Y. Deng, C. Deng, Size and rate dependent grain boundary motion mediated by disconnection 
+nucleation, Acta Mater. 131 (2017) 400–409. 
+
+ 
+25
+[25] T. Gorkaya, D.A. Molodov, G. Gottstein, Stress-driven migration of symmetrical <100> tilt grain 
+boundaries in Al bicrystals, Acta Mater. 57(2009) 5396–5405. 
+[26] G. Gottstein, L.S. Shvindlerman, The compensation effect in thermally activated interface 
+processes, Interface Sci. 6 (1998) 267–278. 
+[27] H. Hahn, H. Gleiter, The effect of pressure on grain growth and boundary mobility, Scr. Metall. 
+13(1979) 3–6. 
+[28] D.A. Molodov, B.B. Straumal, L.S. Shvindlerman, The effect of pressure on migration of <001> 
+tilt grain boundaries in tin bicrystals, Scr. Metall. 18(1984) 207–211. 
+[29] J.W. Cahn, The impurity-drag effect in grain boundary motion, Acta Metall. 10(1962) 789–798. 
+[30] G. Gottstein, L.S. Shvindlerman, On the orientation dependence of grain boundary migration, Scr. 
+Metall. Mater. 27 (1992) 1515–1520. 
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+simulations, Acta Mater. 103 (2016) 424–432. 
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+of a flat grain boundary, Acta Mater. 52 (2004) 2569–2576. 
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+Phys. Rev. Lett. 98 (2007) 165502. 
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+mobility in Al based on atomistic simulations, Mater. Lett. 263 (2020) 127293. 
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+(1995) 1–19. 
+[36] S.M. Foiles, J.J. Hoyt, Computation of grain boundary stiffness and mobility from boundary 
+fluctuations, Acta Mater. 54 (2006) 3351–3357. 
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+visualization tool, Model. Simul. Mater. Sci. Eng. 18 (2009) 015012. 
+[38] S.P. Coleman, D.E. Spearot, S.M. Foiles, The effect of synthetic driving force on the atomic 
+mechanisms associated with grain boundary motion below the interface roughening temperature, 
+Comput. Mater. Sci. 86 (2014) 38–42. 
+[39] E.R. Homer, S.M. Foiles, E.A. Holm, D.L. Olmsted, Phenomenology of shear-coupled grain 
+boundary motion in symmetric tilt and general grain boundaries, Acta Mater. 61 (2013) 
+1048–1060. 
+[40] C. Deng, C.A. Schuh, Atomistic simulation of slow grain boundary motion, Phys. Rev. Lett. 106 
+(2011) 045503. 
+[41] A.A. Schratt, V. Mohles, Efficient calculation of the ECO driving force for atomistic simulations 
+of grain boundary motion, Comput. Mater. Sci. 182 (2020) 109774. 
+[42] S.L. Thomas, K. Chen, J. Han, P.K. Purohit, D.J. Srolovitz, Reconciling grain growth and 
+shear-coupled grain boundary migration, Nat. Commun. 8 (2017) 1764. 
+
+ 
+26
+[43] X.Y. Song, L. Yang, C. Deng, Computing the intrinsic grain boundary mobility tensor, 
+https://doi.org/10.48550/arXiv.2212.11462.  
+
+ 
+1
+Supplemental Material:  
+Unusual acceleration and size effects in grain boundary migration with shear coupling 
+ 
+Liang Yanga, Xinyuan Songb, Tingting Yuc, Dahai Liua*, Chuang Dengb,* 
+a School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang 
+330063, China 
+b Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada 
+c School of Aviation and Mechanical Engineering, Changzhou Institute of Technology, Changzhou, 
+Jiangsu 213032, China. 
+* Corresponding author: dhliu@nchu.edu.cn(D. Liu), Chuang.Deng@umanitoba.ca (C. Deng) 
+ 
+1. Supporting results for the transition of acceleration migration into uniform migration  
+In Fig. 1 of the main text, we can readily observe the accelerated migration and negative 
+dependency of velocity on the vertical cell size for P1 GB at 500 K. Nevertheless, when raising the 
+temperature to 1000 K, both the acceleration and size effect disappear though the shear coupling still 
+exists. The results in Fig. s1 for P148 GB suggest that the transition from accelerated into uniform 
+migration, induced by the temperature, does also exist for other shear-coupling GBs, regardless of the 
+magnitude of driving force. 
+0
+100
+200
+300
+0
+10
+20
+30
+40
+50
+(a)
+Time (ps)
+Displacement (nm)
+ 
+ 
+Lx
+2Lx
+8Lx
+500 K
+0.060 eV
+β = 0.86
+0
+200
+400
+0
+10
+20
+30
+40
+50
+60
+(b)
+Time (ps)
+ 
+ 
+1000 K
+0.060 eV
+β = 0.22
+0
+2000
+4000
+6000
+8000
+0
+10
+20
+30
+40
+50
+60
+(c)
+Time (ps)
+ 
+ 
+1000 K
+0.006 eV
+β = 0.22
+ 
+Fig. s1 Variation of GB displacement with the vertical size for P148 GB, simulated under 
+distinct temperatures and driving forces. Concerning simulation settings were the same as 
+Fig. 1 in the main text. 
+
+ 
+2
+2. Supporting results for the effective alleviation of acceleration in migration  
+When adopting one special kind of boundary condition for GBs exhibiting acceleration, i.e., 
+setting the bottom surface as free while the top one as fully fixed, the acceleration will only be 
+significant at the early migration stage and gradually turn into uniform migration at the later stage. In 
+such case, both the acceleration in migration and size effect can be effectively alleviated. Nevertheless, 
+we only present the complete displacement and kinetic energy data for Lx system due to the limited 
+scope of the main text. Here Fig. s2 shows the complete data for all four distinct systems considered 
+in the main text to support the above conclusion. 
+ 
+  
+ 
+  
+ 
+Fig. s2 GB displacement and shear kinetic energy data for P1 GB with different vertical sizes, 
+simulated under 500 K and 0.06 eV when setting the bottom surface as free while the top one 
+as fully fixed.  
+ 
+
+(a) 150
+12(b) 300
+25ZL
+250-
+x
+20
+Lsp)
+200-15
+ace10100-
+(nm50-0
+80
+120
+160
+Time (ps)(c) 1200
+100
+81x
+120-OL
+1000-
+x
+80
+D1spl
+800-60
+cem
+600N
+40007200-
+200
+0
+0
+200
+400
+600
+800100
+Time (ps)(d) 3500
+2013000
+xispl2500
+1200
+spl
+lac150
+eme
+e
+len
+500N
+E
+100
+nm
+100050
+5000
+00
+800
+1200
+1600
+2000
+Time (ps)90
+ace
+e
+en60.(nm300
+20
+60
+80
+Time (ps)

69FAT4oBgHgl3EQfnx3V/content/tmp_files/2301.08631v1.pdf.txt

@@ -0,0 +1,830 @@
+Superconductivity in type II layered Weyl semi-metals
+B. Rosenstein1 and B. Ya.
+Shapiro2
+1Department of Electrohysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan, R.O.C.
+2Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel.
+Novel ”quasi two dimensional” typically layered (semi) metals offer a unique opportunity to
+control the density and even the topology of the electronic matter. In intercalated MoTe2 type
+II Weyl semi - metal the tilt of the dispersion relation cones is so large that topologically of the
+Fermi surface is distinct from a more conventional type I. Superconductivity observed recently in
+this compound [Zhang et al, 2D Materials 9, 045027 (2022)] demonstrated two puzzling phenomena:
+the gate voltage has no impact on critical temperature, Tc, in wide range of density, while it is very
+sensitive to the inter - layer distance. The phonon theory of pairing in a layered Weyl material
+including the effects of Coulomb repulsion is constructed and explains the above two features in
+MoTe2.
+The first feature turns out to be a general one for any type II topological material, while
+the second reflects properties of the intercalated materials affecting the Coulomb screening.
+arXiv:2301.08631v1  [cond-mat.supr-con]  20 Jan 2023
+
+2
+INTRODUCTION.
+The 3D and 2D topological quantum materials, such as topological insulators and Weyl semi - metals (WSM),
+attracted much interests due to their rich physics and promising prospects for application in electronic and spinotronic
+devices. The band structure in the so called type I WSM like graphene[1], is characterized by appearance linear
+dispersion relation (cones around several Dirac points) due to the band inversion. This is qualitatively distinct from
+conventional metals, semi - metals or semiconductors, in which bands are typically parabolic. In type-II WSM [2], the
+cones have such a strong tilt, κ, so that they exhibit a nearly flat band and the Fermi surface ”encircles” the Brillouin
+zone, Fig.1b, Fig.1c. It is topologically distinct from conventional ”pockets”, see Fig.1a. This in turn leads to exotic
+electronic properties different from both the those in both the conventional and in the type I WSM. Examples include
+the collapse of the Landau level spectrum in magnetoresistance [3], and novel quantum oscillations [4].
+The type II topology of the Fermi surface was achieved in particular in transition metal dichalcogenides [5]. Very
+recentlyMoTe2 layers intercalated by ionic liquid cations were studied[6]. The tilt value was estimated to as high
+as κ = 1.3 that places it firmly within the type II WSM class. The measurements included the Hall effect and the
+resistivity at low temperatures demonstrating appearance of superconductivity. They discovered two intriguing facts
+that are currently under discussion. First changing the gate voltage (chemical potential) surprisingly has no impact
+on critical temperature, Tc, in wide range of density of the electron gas. Second Tc turned out to be very sensitive to
+the inter - layer distance d: it increases from 10.5A to 11.7A, while the critical temperature jumps from 4.2K to 7K.
+In the present paper we propose a theoretical explanation of these observations based on appropriate generalization
+of the conventional superconductivity theory applied to these materials.
+Although early on unconventional mechanisms of superconductivity in WSM have been considered, accumulated
+experimental evidence points towards the conventional phonon mediated one [7–9]. In the previous paper[11] and a
+related work[10] a continuum theory of conventional superconductivity in WSM was developed. Magnetic response
+in the superconducting state was calculated[10][12]. The model was too ”mesoscopic” to describe the type II phase
+since the global topology of the Brillouin zone was beyond the scope of the continuum approach. Therefore we go
+beyond the continuum model in the present paper by modeling a type II layered WSM using a tight binding approach.
+The in-plane electron liquid model is similarl to that of graphene oxide[13] and other 2D WSM. It possesses a chiral
+symmetry between two Brave sublattices for all values of the tilt parameter κ, but lacks hexagonal symmetry. The
+second necessary additional feature is inclusion of Coulomb repulsion.
+It turns out that the screened Coulomb repulsion significantly opposes the phonon mediated pairing. Consequently
+a detailed RPA theory of screening in a layered material[14] is applied. We calculate the superconducting critical
+temperature taking into consideration the modification of the Coulomb interaction due to the dielectric constant of
+intercalator material and the inter-layered spacing d. The Gorkov equations for the two sublattices system are solved
+without resorting to the mesoscopic approach. Moreover since screening of Coulomb repulsion plays a much more
+profound role in quasi 2D materials the pseudo-potential simplification developed by McMillan[15] is not valid.
+Rest of the paper is organized as follows. In Section II the microscopic model of the layered WSM is described. The
+RPA calculation of both the intra- and inter - layer screening is presented. In Section III the Gorkov equations for the
+optical phonon mediated intra- layer pairing for a multiband system including the Coulomb repulsion is derived and
+solved numerically. In Section IV the phonon theory of pairing including the Coulomb repulsion for a layered material
+is applied to recent extensive experiments on MoTe2. The effect of intercalation and density on superconductivity is
+studied. This explains the both remarkable features of Tc observed[6] in MoTe2. The last Section contains conclusions
+and discussion.
+A ”GENERIC” LATTICE MODEL OF LAYERED WEYL SEMI-METALS
+Intra- layer hopping
+A great variety of tight binding models were used to describe Weyl (Dirac) semimetals in 2D. Historically the
+first was graphene (type I, κ = 0) , in which electrons hope between the neighboring cites of the honeycomb lattice.
+We restrict the discussion to systems with the minimal two cones of opposite chirality and negligible spin orbit
+coupling. The two Dirac cones appear in graphene at K and K′ crystallographic points in BZ. Upon modification
+(more complicated molecules like graphene oxide, stress, intercalation) the hexagonal symmetry is lost, however a
+discrete chiral symmetry between two sublattices, denoted by I = A, B, ensures the WSM. The tilted type I and even
+type II (for which typically κ > 1) crystals can be described by the same Hamiltonian with the tilt term added. This
+
+3
+FIG. 1. Two distict topologies of the Fermi surface in 2D. Topology of the 2D Brillouin zone is that of the surface of 3D
+torroid. On the left the “conventional” type I pocket is shown. In the ceter and on the right the type II topology is shown
+schematically. The filled states are in blue and envelop the torus. Despite the large difference in density of the two the Fermi
+surface properties like density of states are the same.
+2D model is extended to a layered system with inter - layer distance d. Physically the 2D WSM layers are separated
+by a dielectric material with inter - layer hopping neglected, so that they are coupled electromagnetically only[14].
+The lateral atomic coordinates are still considered on the honeycomb lattice are rn = n1a1 + n2a2, where lattice
+vectors are:
+a1 = a
+2
+�
+1,
+√
+3
+�
+; a2 = a
+2
+�
+1, −
+√
+3
+�
+,
+(1)
+despite the fact that hopping energies are different for jumps between nearest neighbors. Each site has three neighbors
+separated by δ1 = 1
+3 (a1 − a2) , δ2 = − 1
+3 (2a1 + a2) and δ3 = 1
+3 (a1 + 2a2), in different directions. The length of the
+lattice vectors a will be taken as the length unit and we set ℏ = 1. The hopping Hamiltonian including the tilt term
+is[13, 16]:
+K =
+√
+3
+4
+�
+nl
+�
+γ
+�
+ψsA†
+nl ψsB
+rn+δ1,l + ψsA†
+nl ψsB
+rn+δ2,l + tψsA†
+nl ψsB
+rn+δ3,l
+�
++ h.c. − κψsI†
+nl ψsI
+rn+a1,l − µnn,l
+�
+.
+(2)
+Here an integer l labels the layers. Operator ψsA†
+nl
+is the creation operators with spin s =↑, ↓, while the density
+operator is defined as nnl = ψsI†
+nl ψsI
+nl. The chemical potential is µ, while γ is the hopping energy for two neighbors
+at δ1, δ2 . Since the the system does not possesses hexagonal symmetry (only the chiral one), the third jump has the
+different hopping[13] tγ. Dimensionless parameter κ determines the tilt of the Dirac cones along the a1direction[16].
+In the 2D Fourier space, ψsA
+nl = N −2
+s
+�
+k ψsA
+kl e−ik·rn, one obtains for Hamiltonian (for finite discrete reciprocal lattice
+Ns × Ns):
+K = N −2
+s
+�
+kl ψs†
+klMkψs
+kl.
+(3)
+Here k = k1
+Ns b1 + k2
+Ns b2 (reciprocal lattice vectors are given in Appendix A) and matrix Mk = dxσx + dyσy + d0I in
+terms of Pauli matrices has components:
+dx = 2t
+√
+3 cos
+� 2π
+3Ns
+(k1 − k2)
+�
++ 4
+√
+3 cos
+� π
+Ns
+(k1 + k2)
+�
+cos
+�
+− π
+3Ns
+(k1 − k2)
+�
+;
+(4)
+dy = − 2t
+√
+3 sin
+� 2π
+3Ns
+(k1 − k2)
+�
++ 4
+√
+3 cos
+� π
+Ns
+(k1 + k2)
+�
+sin
+� π
+3Ns
+(k1 − k2)
+�
+;
+d0 =
+2
+√
+3
+�
+−κ cos
+� 2π
+Ns
+k1
+�
+− µ
+�
+.
+
+4
+Using γ as our energy unit from now on, the free electrons part of the Matsubara action for Grassmanian fields
+ψ∗sI
+kln is:
+Se = 1
+T
+�
+kln ψ∗sI
+kln
+��
+−iωn + d0
+k
+�
+δIJ + σIJ
+i di
+k
+�
+ψsJ
+kln.
+(5)
+where ωn = πT (2n + 1) is the Matsubara frequency. The Green Function of free electrons has the matrix form
+gkn =
+��
+−iωn + d0
+k
+�
+I + σidi
+k
+�−1 =
+�
+−iωn + d0
+k
+�
+I − σidi
+k
+(iωn − d0
+k)2 − dx2
+k − dy2
+k
+.
+(6)
+Now we turn to the spectrum of this model.
+FIG. 2. The topological phase diagram of the Weyl semimetal at large tilt parameter (κ = 1.3). Chemical potential (in units
+of γ = 500 meV) is marked on each contour. The electron type I topology at low values of µ undergoes transition to the type
+II at µ = µ1
+c = 0.8 meV. At yet larger µ > µ2
+c = 1.35. the Fermi surface becomes again type I. This time the excitations are
+hole rather than electrons.
+The range of the topological type II phase at large κ
+The spectrum of Hamiltonian of Eqs.(4) consists of two branches. The upper branch for µ = 0.9eV is given in Fig.
+2. The lower branch for a reasonable choice of parameters appropriate to MoTe2 is significantly below the Fermi
+surface and is not plotted. Blue regions represent the filled electron states. One observes a ”river” from one boundary
+to the other of the Brillouin zone (in coordinates k1 and k2, in terms of the original kx, ky it is a rhomb) characteristic
+to type II Fermi surface. Topologically this is akin to Fig.1b.
+In Fig. 3 the Fermi surfaces in a wide range of densities n = 7.×1013−4.5×1014cm−2 are given. Topologically they
+separate into three phases. At chemical potentials below µ1
+c = 0.796 eV , corresponding to densities n < n1
+c = 8.×1013
+cm−2 , the Fermi surface consists of one compact electron pocket similar to Fig.1a, so that the electronic matter
+is of the (”customary”) topological type I. The density is determined from the (nearly linear) relation between the
+
+5
+0.5
+0.6975
+0.6975
+0.9
+0.9
+1.2
+1.2
+1.2
+1.2
+1.35
+1.35
+1.35
+1.35
+1.5
+1.5
+1.5
+1.5
+I (electron)
+II
+II
+II
+I (hole)
+k1 (π/a)
+k2 (π/a)
+FIG. 3. Dispersion relation of WSM with κ = 1.3. The blue plane corresponds to chemical potentia lµ = 0.8 eV so that the
+Fermi surface has the type II topology.
+chemical potential and density given in Fig. 4 (blue line, scale on the right). In the range µ1
+c < µ < µ2
+c = 1.35eV
+the Fermi surface consists of two banks of a ”river” (blue color represents filled electron states) in Fig.2 and can be
+viewed topologically as in Fig.1b and Fig1c. The second critical density is n2
+c = 3.6 × 1014 cm−2. In this range the
+shape of both pieces of the Fermi surface largely does not depend on the density that is proportional to the area of
+the blue part of the surface.
+To make this purely topological observation quantitative, we present in Fig. 4 (green line, scale on the left) the
+density of states (DOS) as a function of chemical potential. One observes that it nearly constant away from the two
+topological I to II transitions where it peaks.
+Coulomb repulsion
+The electron-electron repulsion in the layered WSM can be presented in the form,
+V = e2
+2
+�
+nln′l′ nnlvC
+n−n′,l−l′nn′l′ =
+e2
+2N 2s
+�
+qll′ nqln−ql′vC
+q,l−l′,
+(7)
+where vC
+n−n′,l−l′ is the ”bare” Coulomb interaction between electrons with Fourier transform vC
+q,l−l′ = v2D
+q e−dq|l−l′|,
+v2D
+q
+= 2πe2/qϵ. Here ϵ is the dielectric constant of the intercalator material
+The long range Coulomb interaction is effectively taken into account using the RPA approximation.
+SCREENING IN LAYERED WSM.
+The screening in the layered system can be conveniently partitioned into the screening within each layer described
+by the polarization function Πqn and electrostatic coupling to carriers in other layers. We start with the former.
+
+6
+I
+II
+I
+0.6
+0.8
+1.0
+1.2
+1.4
+0
+1
+2
+3
+4
+chemical potential (ev)
+DOS(2 1014cm-2 ev-1)
+density (1014cm-2)
+FIG. 4. Electron density and DOS as function of the chemical potential µ.of WSM with κ = 1.3. DOS has cusps at both I to
+II transitions. Between the transitions it is nearly constant in the range of densities from 1.1 × 1014/cm2 to 4. × 1014/cm2.
+Polarization function of the electron gas in Layered WSM
+In a simple Fermi theory of the electron gas in normal state with Coulomb interaction between the electrons in
+RPA approximation the Matsubara polarization is calculated as a simple minus ”fish” diagram [14] in the form:
+Πqn = −
+�
+−2TTr
+�
+pm gpmgp+q,m+n
+�
+.
+(8)
+Using the GF of Eq.(6), one obtain:
+Πqn = 4T
+�
+pm
+(iωm + A) (iωm + B) + C
+�
+(iωm + A)2 − α2
+� �
+(iωm + B)2 − β2
+�,
+(9)
+where
+A = −d0
+p; B = iωn − d0
+p+q;
+C = dx
+pdx
+p+q + dy
+pdy
+p+q;
+(10)
+α2 = dx2
+p + dy2
+p ;
+β2 = dx2
+p+q + dy2
+p+q.
+Performing summation over m, one obtains:
+Πqn = −
+�
+p
+�
+�
+�
+α2−α(A−B)+C
+α[(A−B−α)2−β2] tanh α−A
+2T
++
+a2+α(A−B)+C
+α[(A−B+α)2−β2] tanh α+A
+2T
++ β2+β(A−B)+C
+β[(A−B+β)2−α2] tanh β−B
+2T
++
+β2−β(A−B)+C
+β[(A−B−β)2−α2] tanh β+B
+2T
+�
+�
+� .
+(11)
+Now we turn to screening due to other layers.
+Screening in a layered system
+Coulomb repulsion between electrons in different layers l and l′ within the RPA approximation is determined by
+the following integral equation:
+
+7
+V RP A
+q,l−l′,n = vC
+q,l−l′ + Πqn
+�
+l′′ vC
+q,l−l′′V RP A
+q,l′′−l′,n.
+(12)
+The polarization function Πqn in 2D was calculated in the previous subsection. This set of equations is decoupled by
+the Fourier transform in the z direction,
+V RP A
+q,qz,n =
+vC
+q,qz
+1 − ΠqnvC
+q,qz
+,
+(13)
+where
+vC
+q,qz =
+�
+l v2D
+q eiqzl−qd|l| = v2D
+q
+sinh [qd]
+cosh [qd] − cos [dqz].
+(14)
+The screened interaction in a single layer therefore is is given by the inverse Fourier transform [14]:
+V RP A
+q,l−l′,n = d
+2π
+� π/d
+qz=−π/d
+eiqzd(l−l′)
+vC
+q,qz
+1 − ΠqnvC
+q,qz
+.
+(15)
+Considering screened Coulomb potential at the same layer l = l′, the integration gives,
+V RP A
+qn
+= v2D
+q
+sinh [qd]
+�
+b2qn − 1
+,
+(16)
+where bqn = cosh [dq] − v2D
+q Πqn sinh [dq]. This formula is reliable only away from plasmon region bqn > 1. It turns
+out that to properly describe superconductivity, one can simplify the calculation at low temperature by considering
+the static limit Πqn ≃ Πq0. Consequently the potential becomes static: V RP A
+q
+≡ V RP A
+q,n=0.
+SUPERCONDUCTIVITY
+Superconductivity in WSM is caused by a conventional phonon pairing. The leading mode is an optical phonon
+mode assumed to be dispersionless. with energy Ω. The effective electron-electron attraction due to the electron -
+phonon attraction opposed by Coulomb repulsion (pseudo - potential) mechanism creates pairing below Tc. Further
+we assume the singlet s-channel electron-phonon interaction and neglect the inter-layers electrons pairing.
+In order
+to describe superconductivity, one should ”integrate out” the phonon and the spin fluctuations degrees of freedom to
+calculate the effective electron - electron interaction. We start with the phonons. The Matsubara action for effective
+electron-electron interaction via in-plane phonons and direct Coulomb repulsion calculated in the previous Section.
+It important to note that unlike in metal superconductors where a simplified pseudo - potential approach due to
+McMillan and other [15], in 2D and layered WSM, one have to resort to a more microscopic approach.
+Effective attraction due to phonon exchange opposed by the effective Coulomb repulsion
+The free and the interaction parts of the effective electron action (”integrating phonons”+RPA Coulomb interaction)
+in the quasi - momentum - Matzubara frequency representation, S = Se + Sint,
+Sint = 1
+2T
+�
+qll′mm′ nqln
+�
+δll′V ph
+q,m−m′ + V RP A
+q,l−l′
+�
+n−q,−l′,−n′.
+(17)
+Here nqln = �
+p ψ∗sI
+plnψsI
+q−p,l,n the Fourier transform of the electron density and Se was defined in Eq.(5). The effective
+electron - electron coupling due to phonons is:
+V ph
+qm = −
+�√
+3
+2
+�2
+g2Ω
+ωb2
+m + Ω2 ,
+(18)
+where the bosonic frequencies are ωb
+m = 2πmT.
+
+8
+Gorkov Green’s functions and the s-wave gap equations
+Normal and anomalous (Matsubara) intra - layer Gorkov Green’s functions are defined by expectation value of the
+fields,
+�
+ψIs
+knlψ∗s′J
+knl
+�
+= δss′GIJ
+kn and
+�
+ψIs
+knlψJs′
+−k,−n,l
+�
+= εss′F IJ
+kn, while the gap function is
+∆IJ
+qn =
+�
+pm Vq−p,n−mF IJ
+pm,
+(19)
+where Vqn = V ph
+qn + V RP A
+qn
+is a sublattice scalar. The gap equations in the sublattice matrix form are derived from
+Gorkov equations in Appendix B:
+∆qn = −
+�
+pm Vq−p,n−mgpm
+�
+I + ∆pmgt
+−p,−m∆∗
+−p,−mgpm
+�−1 ∆pmgt
+−p,−m.
+(20)
+In numerical simulation the gap equation was solved iteratively. Relatively large space cutoff Ns = 256 is required.
+The frequency cutoff Nt = 128 was required due to low temperatures approached. Typically 15 − 25 iterations were
+required. The parameters used were Ω = 16meV . The electron - phonon coupling g = 20meV . Now we turn to
+results concentrating on two puzzling experimental results of ref.[6].
+Independence of Tc on density in topological type II phase
+In Fig.5 the critical temperature for various values of density are plotted. The blue points are for dielectric
+constant[6], ε = 16, describing the intercalated imidazole cations [C2MIm] [17].
+The inter - layer distance was
+kept at d = 10.5A.
+ϵ=50
+ϵ=∞
+ϵ=16
+0.8
+0.9
+1.0
+1.1
+1.2
+0
+5
+10
+15
+chemical potential (ev)
+Tc (K)
+FIG. 5. Critical temperature of transition to superconducting state in type II layered WSM is shown as function of chemical
+potential (can be translated into carrier density via Fig.4). Three values of dielectric constant of the intercalant for fixed
+interlayer distance are shown. Parameters of the electron gas are the same as in previous figures.
+The significance and generatiozation of the observation are discussed below.
+Increase of Tc with dielectric constant of intercalator materials
+The main idea of the paper is that the difference in Tc between different intercalators is attributed not to small
+variations in the inter - layer spacing d, but rather to large differences in the dielectric constant of the intercalating
+materials due to its effect on the screening. In experiment of ref.[6] the imidazole cations [C2MIm]+ (1- ethyl - 3 -
+methyl - imidazolium) are short molecules[17] have ϵ = 16, while [C6MIm]+ (1- hexy l - 3 - methy l - imidazolium)
+are long molecules[18] with a larger value ϵ ≃ 50. The inter - layer distance d is slightly dependent intercalators
+
+9
+changing from 10.5˚
+A to 11.7˚
+A . The blue points in Fig 5 describe a material with dielectric constant ε = 16 should.
+This is contrasted[18] with the ε = 50 material, see the red point. Neglecting the Coulomb repulsion, see the green
+points, critical temperature (a much simpler calculation of Tc in this case similar to that in ref.[11] is needed in
+this case) becomes yet higher. This demonstrate the importance of the Coulomb repulsion in a quasi 2D system.
+Superconductivity is weaker for monolayer on substrate since both air and substrate have smaller dielectric constants
+and hence weaker screen the Coulomb repulsion.
+DISCUSSION AND CONCLUSION
+To summarize we have developed a theory of superconductivity in layered type II Weyl semi-metals that properly
+takes into account the Coulomb repulsion. The generalization goes beyond the simplistic pseudo - potential approach
+due to McMillan[15] and others and depends essentially on the intercalating material. The theory allows to explain
+the two puzzling phenomena observed recently in layered intercalated MoTe2 WSW compound [6]
+The first experimental observation is that the gate voltage (changes in the chemical potential or equivalently in
+density) has no impact on critical temperature Tc. For the 3D density range 8. × 1020cm−3 − 3.6 × 1021cm−3 the
+temperature changes within 5%. For the intercalating material [C2MIm]+ with inter - layer distance d = 10.5A the
+2D density range translates into 8.4×1013cm−2 −3.8×1014cm−2 wth slightly larger spacings d = 11.7A shown in Figs
+2-5. This feature is explained purely topologically, see schematic Fig.1. In the type II density range the shape of both
+pieces of the Fermi surface (the blue - yellow boundaries in Fig1b and Fig1c) largely does not depend on the density
+(that is proportional to the area of the blue part of the surface) leading, see Fig. 4 to approximate independence of
+the density of states (DOS) N (0) of chemical potential µ. This feature is akin to the DOS independence on µ for a
+parabolic (topologically type I like in Fig.1a) band in purely 2D materials, but has completely difficult origin.
+Using the somewhat naive BCS formula
+Tc ≃ Ω e−N(0)g2
+eff .
+(21)
+Here Ω is the phonon frequency and geff the effective electron - phonon coupling. Assuming that both Ω and g do
+not depend on the density one arrives at a conclusion that in the type II topological phase the critical temperature is
+density independent .
+The second experimental observation[6] was that Tc is in fact very sensitive to the intercalating material. For
+imidazole cations [C2MIm]+ the critical temperature is Tc = 4.2K, while for [C6MIm]+ the temperature jumps
+to Tc = 6.6K or 6.9K depending on the intercalation method. The inter - layer distance d is slightly dependent
+intercaltors increasing from 10.5˚
+A to 11.7˚
+A . Our calculation demonstrates that the difference in Tc between different
+intercalators cannot be attributed to small variations in the inter - layer spacing d. On the contrary there are large
+differences in the dielectric constant of the intercalating materials.
+While [C2MIm]+ have[17] a relatively small
+dielectric constant ϵ = 16, [C6MIm]+ is estimated[18] in the range ϵ = 40 − 60. Our theory accounts the difference
+in Tc due to changes in the screening of the Coulomb potential due to the inter - layer insulator.
+ACKNOWLEDGEMENTS.
+This work was supported by NSC of R.O.C. Grants No. 101-2112-M-009-014-MY3.
+APPENDIX A. DETAILS OF THE MODEL
+The system considered in the paper is fitted for the following values of the hipping and the tilt parameter. The
+dimensionless tilt parameter was taken from ref.[6] κ = 1.3. The hopping γ = 500 meV and t = 2. The calculations
+were performed on the discrete reciprocal lattice k1, k2 = 1, ...Ns with Ns = 256. Reciprocal lattice basis vectors are,
+b1 = 2π
+�
+1, 1
+√
+3
+�
+;
+b2 = 2π
+�
+1, − 1
+√
+3
+�
+,
+(22)
+
+10
+so that a convenient representation is k = k1
+Ns b1 + k2
+Ns b2 with
+kx = 2π
+Ns
+(k1 + k2) ,
+ky =
+2π
+√
+3Ns
+(k1 − k2) .
+(23)
+APPENDIX B. DERIVATION OF THE TWO SUBLATTICE GAP EQUATION
+Green’s functions and the s-wave Gorkov equations
+We derive the Gorkov’s equations (GE) within the functional integral approach[19] starting from the effective
+electron action for grassmanian fields ψ∗X, ψY .
+S = 1
+T
+�
+ψ∗X �
+G−1
+0
+�XY ψY + 1
+2ψ∗Y ψY V Y Xψ∗XψX
+�
+,
+(24)
+where X, Y denote space coordinate, sublattices (pseudospin) and spin of the electron. Finite temperature properties
+of the condensate are described at temperature T by the normal and the anomalous Matsubara Greens functions for
+spin singlet state.
+The GE in functional form are:
+�
+ψAψ∗B�
+δ
+δψ∗C
+� δS
+δψ∗B
+�
++
+�
+ψAψB�
+δ
+δψ∗C
+� δS
+δψB
+�
+= 0;
+(25)
+�
+ψAψ∗B�
+δ
+δψC
+� δS
+δψ∗B
+�
++
+�
+ψAψB�
+δ
+δψC
+� δS
+δψB
+�
+= δAC.
+(26)
+Performing the calculations and using the normal and anomalous Green functions in the form F AB =
+�
+ψAψB�
+; GAB =
+�
+ψAψ∗B�
+, one obtains:
+F AX ��
+G−1
+0
+�CX − vXCGCX + vCXGXX�
++ GAXvXCF XC = 0.
+(27)
+Skipping second and third terms in bracket in this expression and defining, superconducting gap ∆AB = vABF AB,
+one rewrites as a matrix products:
+�
+G−1
+0
+�CX F XA = GAX∆XC.
+(28)
+The first GE (multiplied from left by G0) is,
+F AB = −GAXGBY
+0
+∆XY ,
+(29)
+while the second GE similarly is:
+GAB − GAX∆XY GZY
+0
+∆∗ZUGUB
+0
+= GAB
+0
+.
+(30)
+Frequency-quasi-momentum and the spin-sublattice decomposition
+The generalized index A contains the space variables (space + Matsubara time, a), spin s and the sublattice I.
+After performing the Fourier series with combined quasi - momentum - frequency α:
+F s1s2IJ
+ab
+= ϵs1s2 �
+α eiα(a−b)F IJ
+α ;
+∆s1s2IJ
+ab
+= ϵs1s2 �
+α eiα(a−b)∆IJ
+α ;
+(31)
+Gs1s2IJ
+0ab
+= δs1s2 �
+α eiα(a−b)gIJ
+α ;
+V s1s2IJ
+ab
+=
+�
+α eiα(a−b)vα.
+
+11
+Substituting spins into Eq.(29,30), one obtains in the sublattice matrix form
+Fα = −Gα∆αgt
+−α;
+(32)
+Gα = G0α
+�
+I + ∆αgt
+−α∆∗
+−αG0α
+�−1 ,
+Convoluting the first GE by vν one obtains:
+∆ω = −
+�
+ν vω−νGν∆νgt
+−ν.
+(33)
+The solution of the second GE for G is:
+Gα = gα
+�
+I + ∆αgt
+−α∆∗
+−αgα
+�−1 .
+(34)
+Substituting into the first GE one obtain Eq.(20) in the text.
+[1] Katsnelson M.I. 2012 The Physics of Graphene, (Cambridge, Cambridge University Press).
+[2] Soluyanov A. A. , Gresch D. ,Wang Z. ,Wu Q. ,Troyer M. ,Dai X. & Bernevig B. A. 2015, Nature 527, 495.
+[3] Yu Z.-M. , Yao Y. , and Yang S. A. 2016 Phys. Rev. Lett. 117, 077202.
+[4] O’Brien T. E. ,Diez M. , and Beenakker C. W. J. , 2016 Phys.Rev. Lett. 116, 236401.
+[5] Wang C. et al. 2018 Nature, 555, 231; Lin, Z. et al. 2018 Nature, 562, 254; Huang H.,Zhou S. and DuanW., 2016 Phys.Rev.B
+94 121117; Yan M. et al, Nature Comm. 2017 8, 257; Furue Y. 2021 et al Phys. Rev. B 104,144510.
+[6] Zhang H. , Rousuli A. ,Zhang K. ,Zhong H. ,Wu Y. ,Yu P. ,Zhou S. 2022 2D Materials 9 045027.
+[7] Das Sarma S. andLi Q. 2013 Phys. Rev. B 88, 081404(R);Brydon P.M.R. ,Das Sarma S. ,Hui H.-Y. and Sau J. D. 2014
+Phys. Rev. B 90, 184512;Li D. ,Rosenstein B. ,Shapiro B. Ya. , andShapiro I. 2014 Phys. Rev. B 90 054517.
+[8] Fu L. and Berg E. 2010 Phys. Rev. Lett. 105, 097001.
+[9] Zhang J.-L. et al. 2012 Front. Phys., 7, 193.
+[10] Alidoust M. , Halterman K. , and Zyuzin A. A. 2017 Phys. Rev. B 95, 155124.
+[11] Li D. , Rosenstein B. , Shapiro B. Ya. , and Shapiro I. 2017 Phys. Rev. B 95, 094513.
+[12] Shapiro B Ya , Shapiro I , Li D. and Rosenstein B. 2018 J. Phys.:Condens. Matter 30 335403.
+[13] Wang S. T. et al. 2012 Appl. Phys. Let. 101, 183110.
+[14] Hawrylak P. , Eliasson G. , and Quinn J. J., 1988 Phys. Rev. B 37 10187.
+[15] Bilbro G. and McMillan L. 1976 Phys. Rev. B 14 1887.
+[16] Katayama S. ,Kobayashi A. ,Suzumura Y. 2006 J. Phys. Soc. Japan 75, 054705;Goerbig M. O. , Fuchs J. -N.,Montambaux
+G. , Pi´echon F. 2008 Phys. Rev. B 78, 045415; Hirata M. et al. 2016 Nature Commun. 7 12666.
+[17] Beal A R and Hughes H P 1979 J. Phys. C: Solid State Phys. 12 881 .
+[18] Yang L., Fishbine B. H. , Migliori A. and Pratt L.R., 2010 J. Chem. Phys. 132 044701.
+[19] Negele J.W. and Orlando H. , Quantum Many Particle Systems, 1998 Aspen, Advanced Book Classics, Westview Press.
+

69FAT4oBgHgl3EQfnx3V/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf,len=490
+page_content='Superconductivity in type II layered Weyl semi-metals  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Rosenstein1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Shapiro2  1Department of Electrohysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  2Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Novel ”quasi two dimensional” typically layered (semi) metals offer a unique opportunity to  control the density and even the topology of the electronic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In intercalated MoTe2 type  II Weyl semi - metal the tilt of the dispersion relation cones is so large that topologically of the  Fermi surface is distinct from a more conventional type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Superconductivity observed recently in  this compound [Zhang et al, 2D Materials 9, 045027 (2022)] demonstrated two puzzling phenomena:  the gate voltage has no impact on critical temperature, Tc, in wide range of density, while it is very  sensitive to the inter - layer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The phonon theory of pairing in a layered Weyl material  including the effects of Coulomb repulsion is constructed and explains the above two features in  MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The first feature turns out to be a general one for any type II topological material, while  the second reflects properties of the intercalated materials affecting the Coulomb screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='08631v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='supr-con]  20 Jan 2023  2  INTRODUCTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The 3D and 2D topological quantum materials, such as topological insulators and Weyl semi - metals (WSM),  attracted much interests due to their rich physics and promising prospects for application in electronic and spinotronic  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The band structure in the so called type I WSM like graphene[1], is characterized by appearance linear  dispersion relation (cones around several Dirac points) due to the band inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This is qualitatively distinct from  conventional metals, semi - metals or semiconductors, in which bands are typically parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In type-II WSM [2], the  cones have such a strong tilt, κ, so that they exhibit a nearly flat band and the Fermi surface ”encircles” the Brillouin  zone, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1b, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' It is topologically distinct from conventional ”pockets”, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This in turn leads to exotic  electronic properties different from both the those in both the conventional and in the type I WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Examples include  the collapse of the Landau level spectrum in magnetoresistance [3], and novel quantum oscillations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The type II topology of the Fermi surface was achieved in particular in transition metal dichalcogenides [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Very  recentlyMoTe2 layers intercalated by ionic liquid cations were studied[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The tilt value was estimated to as high  as κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='3 that places it firmly within the type II WSM class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The measurements included the Hall effect and the  resistivity at low temperatures demonstrating appearance of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' They discovered two intriguing facts  that are currently under discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' First changing the gate voltage (chemical potential) surprisingly has no impact  on critical temperature, Tc, in wide range of density of the electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Second Tc turned out to be very sensitive to  the inter - layer distance d: it increases from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5A to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='7A, while the critical temperature jumps from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2K to 7K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  In the present paper we propose a theoretical explanation of these observations based on appropriate generalization  of the conventional superconductivity theory applied to these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Although early on unconventional mechanisms of superconductivity in WSM have been considered, accumulated  experimental evidence points towards the conventional phonon mediated one [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In the previous paper[11] and a  related work[10] a continuum theory of conventional superconductivity in WSM was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Magnetic response  in the superconducting state was calculated[10][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The model was too ”mesoscopic” to describe the type II phase  since the global topology of the Brillouin zone was beyond the scope of the continuum approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Therefore we go  beyond the continuum model in the present paper by modeling a type II layered WSM using a tight binding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The in-plane electron liquid model is similarl to that of graphene oxide[13] and other 2D WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' It possesses a chiral  symmetry between two Brave sublattices for all values of the tilt parameter κ, but lacks hexagonal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The  second necessary additional feature is inclusion of Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  It turns out that the screened Coulomb repulsion significantly opposes the phonon mediated pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Consequently  a detailed RPA theory of screening in a layered material[14] is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' We calculate the superconducting critical  temperature taking into consideration the modification of the Coulomb interaction due to the dielectric constant of  intercalator material and the inter-layered spacing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The Gorkov equations for the two sublattices system are solved  without resorting to the mesoscopic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Moreover since screening of Coulomb repulsion plays a much more  profound role in quasi 2D materials the pseudo-potential simplification developed by McMillan[15] is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In Section II the microscopic model of the layered WSM is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The  RPA calculation of both the intra- and inter - layer screening is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In Section III the Gorkov equations for the  optical phonon mediated intra- layer pairing for a multiband system including the Coulomb repulsion is derived and  solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In Section IV the phonon theory of pairing including the Coulomb repulsion for a layered material  is applied to recent extensive experiments on MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The effect of intercalation and density on superconductivity is  studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This explains the both remarkable features of Tc observed[6] in MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The last Section contains conclusions  and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  A ”GENERIC” LATTICE MODEL OF LAYERED WEYL SEMI-METALS  Intra- layer hopping  A great variety of tight binding models were used to describe Weyl (Dirac) semimetals in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Historically the  first was graphene (type I, κ = 0) , in which electrons hope between the neighboring cites of the honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  We restrict the discussion to systems with the minimal two cones of opposite chirality and negligible spin orbit  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The two Dirac cones appear in graphene at K and K′ crystallographic points in BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Upon modification  (more complicated molecules like graphene oxide, stress, intercalation) the hexagonal symmetry is lost, however a  discrete chiral symmetry between two sublattices, denoted by I = A, B, ensures the WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The tilted type I and even  type II (for which typically κ > 1) crystals can be described by the same Hamiltonian with the tilt term added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Two distict topologies of the Fermi surface in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Topology of the 2D Brillouin zone is that of the surface of 3D  torroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' On the left the “conventional” type I pocket is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In the ceter and on the right the type II topology is shown  schematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The filled states are in blue and envelop the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Despite the large difference in density of the two the Fermi  surface properties like density of states are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  2D model is extended to a layered system with inter - layer distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Physically the 2D WSM layers are separated  by a dielectric material with inter - layer hopping neglected, so that they are coupled electromagnetically only[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The lateral atomic coordinates are still considered on the honeycomb lattice are rn = n1a1 + n2a2, where lattice  vectors are:  a1 = a  2  �  1,  √  3  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' a2 = a  2  �  1, −  √  3  �  ,  (1)  despite the fact that hopping energies are different for jumps between nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Each site has three neighbors  separated by δ1 = 1  3 (a1 − a2) , δ2 = − 1  3 (2a1 + a2) and δ3 = 1  3 (a1 + 2a2), in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The length of the  lattice vectors a will be taken as the length unit and we set ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The hopping Hamiltonian including the tilt term  is[13, 16]:  K =  √  3  4  �  nl  �  γ  �  ψsA†  nl ψsB  rn+δ1,l + ψsA†  nl ψsB  rn+δ2,l + tψsA†  nl ψsB  rn+δ3,l  �  + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' − κψsI†  nl ψsI  rn+a1,l − µnn,l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (2)  Here an integer l labels the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Operator ψsA†  nl  is the creation operators with spin s =↑, ↓, while the density  operator is defined as nnl = ψsI†  nl ψsI  nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The chemical potential is µ, while γ is the hopping energy for two neighbors  at δ1, δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Since the the system does not possesses hexagonal symmetry (only the chiral one), the third jump has the  different hopping[13] tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Dimensionless parameter κ determines the tilt of the Dirac cones along the a1direction[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  In the 2D Fourier space, ψsA  nl = N −2  s  �  k ψsA  kl e−ik·rn, one obtains for Hamiltonian (for finite discrete reciprocal lattice  Ns × Ns):  K = N −2  s  �  kl ψs†  klMkψs  kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (3)  Here k = k1  Ns b1 + k2  Ns b2 (reciprocal lattice vectors are given in Appendix A) and matrix Mk = dxσx + dyσy + d0I in  terms of Pauli matrices has components:  dx = 2t  √  3 cos  � 2π  3Ns  (k1 − k2)  �  + 4  √  3 cos  � π  Ns  (k1 + k2)  �  cos  �  − π  3Ns  (k1 − k2)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (4)  dy = − 2t  √  3 sin  � 2π  3Ns  (k1 − k2)  �  + 4  √  3 cos  � π  Ns  (k1 + k2)  �  sin  � π  3Ns  (k1 − k2)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  d0 =  2  √  3  �  −κ cos  � 2π  Ns  k1  �  − µ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  4  Using γ as our energy unit from now on, the free electrons part of the Matsubara action for Grassmanian fields  ψ∗sI  kln is:  Se = 1  T  �  kln ψ∗sI  kln  ��  −iωn + d0  k  �  δIJ + σIJ  i di  k  �  ψsJ  kln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (5)  where ωn = πT (2n + 1) is the Matsubara frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The Green Function of free electrons has the matrix form  gkn =  ��  −iωn + d0  k  �  I + σidi  k  �−1 =  �  −iωn + d0  k  �  I − σidi  k  (iωn − d0  k)2 − dx2  k − dy2  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (6)  Now we turn to the spectrum of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The topological phase diagram of the Weyl semimetal at large tilt parameter (κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Chemical potential (in units  of γ = 500 meV) is marked on each contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The electron type I topology at low values of µ undergoes transition to the type  II at µ = µ1  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' At yet larger µ > µ2  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' the Fermi surface becomes again type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This time the excitations are  hole rather than electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The range of the topological type II phase at large κ  The spectrum of Hamiltonian of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' (4) consists of two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The upper branch for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='9eV is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The lower branch for a reasonable choice of parameters appropriate to MoTe2 is significantly below the Fermi  surface and is not plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Blue regions represent the filled electron states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' One observes a ”river” from one boundary  to the other of the Brillouin zone (in coordinates k1 and k2, in terms of the original kx, ky it is a rhomb) characteristic  to type II Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Topologically this is akin to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 3 the Fermi surfaces in a wide range of densities n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='×1013−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5×1014cm−2 are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Topologically they  separate into three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' At chemical potentials below µ1  c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='796 eV , corresponding to densities n < n1  c = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='×1013  cm−2 , the Fermi surface consists of one compact electron pocket similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1a, so that the electronic matter  is of the (”customary”) topological type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The density is determined from the (nearly linear) relation between the  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='6975  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='6975  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5  I (electron)  II  II  II  I (hole)  k1 (π/a)  k2 (π/a)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Dispersion relation of WSM with κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The blue plane corresponds to chemical potentia lµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='8 eV so that the  Fermi surface has the type II topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  chemical potential and density given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 4 (blue line, scale on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In the range µ1  c < µ < µ2  c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='35eV  the Fermi surface consists of two banks of a ”river” (blue color represents filled electron states) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2 and can be  viewed topologically as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1b and Fig1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The second critical density is n2  c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='6 × 1014 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In this range the  shape of both pieces of the Fermi surface largely does not depend on the density that is proportional to the area of  the blue part of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  To make this purely topological observation quantitative, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 4 (green line, scale on the left) the  density of states (DOS) as a function of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' One observes that it nearly constant away from the two  topological I to II transitions where it peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Coulomb repulsion  The electron-electron repulsion in the layered WSM can be presented in the form,  V = e2  2  �  nln′l′ nnlvC  n−n′,l−l′nn′l′ =  e2  2N 2s  �  qll′ nqln−ql′vC  q,l−l′,  (7)  where vC  n−n′,l−l′ is the ”bare” Coulomb interaction between electrons with Fourier transform vC  q,l−l′ = v2D  q e−dq|l−l′|,  v2D  q  = 2πe2/qϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Here ϵ is the dielectric constant of the intercalator material  The long range Coulomb interaction is effectively taken into account using the RPA approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  SCREENING IN LAYERED WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The screening in the layered system can be conveniently partitioned into the screening within each layer described  by the polarization function Πqn and electrostatic coupling to carriers in other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' We start with the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  6  I  II  I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='4  0  1  2  3  4  chemical potential (ev)  DOS(2 1014cm-2 ev-1)  density (1014cm-2)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Electron density and DOS as function of the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='of WSM with κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' DOS has cusps at both I to  II transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Between the transitions it is nearly constant in the range of densities from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1 × 1014/cm2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' × 1014/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Polarization function of the electron gas in Layered WSM  In a simple Fermi theory of the electron gas in normal state with Coulomb interaction between the electrons in  RPA approximation the Matsubara polarization is calculated as a simple minus ”fish” diagram [14] in the form:  Πqn = −  �  −2TTr  �  pm gpmgp+q,m+n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (8)  Using the GF of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' (6), one obtain:  Πqn = 4T  �  pm  (iωm + A) (iωm + B) + C  �  (iωm + A)2 − α2  � �  (iωm + B)2 − β2  �,  (9)  where  A = −d0  p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' B = iωn − d0  p+q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  C = dx  pdx  p+q + dy  pdy  p+q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (10)  α2 = dx2  p + dy2  p ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  β2 = dx2  p+q + dy2  p+q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Performing summation over m, one obtains:  Πqn = −  �  p  �  �  �  α2−α(A−B)+C  α[(A−B−α)2−β2] tanh α−A  2T  +  a2+α(A−B)+C  α[(A−B+α)2−β2] tanh α+A  2T  + β2+β(A−B)+C  β[(A−B+β)2−α2] tanh β−B  2T  +  β2−β(A−B)+C  β[(A−B−β)2−α2] tanh β+B  2T  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (11)  Now we turn to screening due to other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Screening in a layered system  Coulomb repulsion between electrons in different layers l and l′ within the RPA approximation is determined by  the following integral equation:  7  V RP A  q,l−l′,n = vC  q,l−l′ + Πqn  �  l′′ vC  q,l−l′′V RP A  q,l′′−l′,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (12)  The polarization function Πqn in 2D was calculated in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This set of equations is decoupled by  the Fourier transform in the z direction,  V RP A  q,qz,n =  vC  q,qz  1 − ΠqnvC  q,qz  ,  (13)  where  vC  q,qz =  �  l v2D  q eiqzl−qd|l| = v2D  q  sinh [qd]  cosh [qd] − cos [dqz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (14)  The screened interaction in a single layer therefore is is given by the inverse Fourier transform [14]:  V RP A  q,l−l′,n = d  2π  � π/d  qz=−π/d  eiqzd(l−l′)  vC  q,qz  1 − ΠqnvC  q,qz  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (15)  Considering screened Coulomb potential at the same layer l = l′, the integration gives,  V RP A  qn  = v2D  q  sinh [qd]  �  b2qn − 1  ,  (16)  where bqn = cosh [dq] − v2D  q Πqn sinh [dq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This formula is reliable only away from plasmon region bqn > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' It turns  out that to properly describe superconductivity, one can simplify the calculation at low temperature by considering  the static limit Πqn ≃ Πq0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Consequently the potential becomes static: V RP A  q  ≡ V RP A  q,n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  SUPERCONDUCTIVITY  Superconductivity in WSM is caused by a conventional phonon pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The leading mode is an optical phonon  mode assumed to be dispersionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' with energy Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The effective electron-electron attraction due to the electron -  phonon attraction opposed by Coulomb repulsion (pseudo - potential) mechanism creates pairing below Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Further  we assume the singlet s-channel electron-phonon interaction and neglect the inter-layers electrons pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  In order  to describe superconductivity, one should ”integrate out” the phonon and the spin fluctuations degrees of freedom to  calculate the effective electron - electron interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' We start with the phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The Matsubara action for effective  electron-electron interaction via in-plane phonons and direct Coulomb repulsion calculated in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  It important to note that unlike in metal superconductors where a simplified pseudo - potential approach due to  McMillan and other [15], in 2D and layered WSM, one have to resort to a more microscopic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Effective attraction due to phonon exchange opposed by the effective Coulomb repulsion  The free and the interaction parts of the effective electron action (”integrating phonons”+RPA Coulomb interaction)  in the quasi - momentum - Matzubara frequency representation, S = Se + Sint,  Sint = 1  2T  �  qll′mm′ nqln  �  δll′V ph  q,m−m′ + V RP A  q,l−l′  �  n−q,−l′,−n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (17)  Here nqln = �  p ψ∗sI  plnψsI  q−p,l,n the Fourier transform of the electron density and Se was defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The effective  electron - electron coupling due to phonons is:  V ph  qm = −  �√  3  2  �2  g2Ω  ωb2  m + Ω2 ,  (18)  where the bosonic frequencies are ωb  m = 2πmT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  8  Gorkov Green’s functions and the s-wave gap equations  Normal and anomalous (Matsubara) intra - layer Gorkov Green’s functions are defined by expectation value of the  fields,  �  ψIs  knlψ∗s′J  knl  �  = δss′GIJ  kn and  �  ψIs  knlψJs′  −k,−n,l  �  = εss′F IJ  kn, while the gap function is  ∆IJ  qn =  �  pm Vq−p,n−mF IJ  pm,  (19)  where Vqn = V ph  qn + V RP A  qn  is a sublattice scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The gap equations in the sublattice matrix form are derived from  Gorkov equations in Appendix B:  ∆qn = −  �  pm Vq−p,n−mgpm  �  I + ∆pmgt  −p,−m∆∗  −p,−mgpm  �−1 ∆pmgt  −p,−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (20)  In numerical simulation the gap equation was solved iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Relatively large space cutoff Ns = 256 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The frequency cutoff Nt = 128 was required due to low temperatures approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Typically 15 − 25 iterations were  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The parameters used were Ω = 16meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The electron - phonon coupling g = 20meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Now we turn to  results concentrating on two puzzling experimental results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Independence of Tc on density in topological type II phase  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5 the critical temperature for various values of density are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The blue points are for dielectric  constant[6], ε = 16, describing the intercalated imidazole cations [C2MIm] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The inter - layer distance was  kept at d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  ϵ=50  ϵ=∞  ϵ=16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2  0  5  10  15  chemical potential (ev)  Tc (K)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Critical temperature of transition to superconducting state in type II layered WSM is shown as function of chemical  potential (can be translated into carrier density via Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Three values of dielectric constant of the intercalant for fixed  interlayer distance are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Parameters of the electron gas are the same as in previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The significance and generatiozation of the observation are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Increase of Tc with dielectric constant of intercalator materials  The main idea of the paper is that the difference in Tc between different intercalators is attributed not to small  variations in the inter - layer spacing d, but rather to large differences in the dielectric constant of the intercalating  materials due to its effect on the screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In experiment of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' [6] the imidazole cations [C2MIm]+ (1- ethyl - 3 -  methyl - imidazolium) are short molecules[17] have ϵ = 16, while [C6MIm]+ (1- hexy l - 3 - methy l - imidazolium)  are long molecules[18] with a larger value ϵ ≃ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The inter - layer distance d is slightly dependent intercalators  9  changing from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5˚  A to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='7˚  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The blue points in Fig 5 describe a material with dielectric constant ε = 16 should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  This is contrasted[18] with the ε = 50 material, see the red point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Neglecting the Coulomb repulsion, see the green  points, critical temperature (a much simpler calculation of Tc in this case similar to that in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' [11] is needed in  this case) becomes yet higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This demonstrate the importance of the Coulomb repulsion in a quasi 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Superconductivity is weaker for monolayer on substrate since both air and substrate have smaller dielectric constants  and hence weaker screen the Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSION  To summarize we have developed a theory of superconductivity in layered type II Weyl semi-metals that properly  takes into account the Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The generalization goes beyond the simplistic pseudo - potential approach  due to McMillan[15] and others and depends essentially on the intercalating material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The theory allows to explain  the two puzzling phenomena observed recently in layered intercalated MoTe2 WSW compound [6]  The first experimental observation is that the gate voltage (changes in the chemical potential or equivalently in  density) has no impact on critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' For the 3D density range 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' × 1020cm−3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='6 × 1021cm−3 the  temperature changes within 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' For the intercalating material [C2MIm]+ with inter - layer distance d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5A the  2D density range translates into 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='4×1013cm��2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='8×1014cm−2 wth slightly larger spacings d = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='7A shown in Figs  2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This feature is explained purely topologically, see schematic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' In the type II density range the shape of both  pieces of the Fermi surface (the blue - yellow boundaries in Fig1b and Fig1c) largely does not depend on the density  (that is proportional to the area of the blue part of the surface) leading, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 4 to approximate independence of  the density of states (DOS) N (0) of chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' This feature is akin to the DOS independence on µ for a  parabolic (topologically type I like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='1a) band in purely 2D materials, but has completely difficult origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  Using the somewhat naive BCS formula  Tc ≃ Ω e−N(0)g2  eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (21)  Here Ω is the phonon frequency and geff the effective electron - phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Assuming that both Ω and g do  not depend on the density one arrives at a conclusion that in the type II topological phase the critical temperature is  density independent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The second experimental observation[6] was that Tc is in fact very sensitive to the intercalating material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' For  imidazole cations [C2MIm]+ the critical temperature is Tc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='2K, while for [C6MIm]+ the temperature jumps  to Tc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='6K or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='9K depending on the intercalation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The inter - layer distance d is slightly dependent  intercaltors increasing from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='5˚  A to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='7˚  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Our calculation demonstrates that the difference in Tc between different  intercalators cannot be attributed to small variations in the inter - layer spacing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' On the contrary there are large  differences in the dielectric constant of the intercalating materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  While [C2MIm]+ have[17] a relatively small  dielectric constant ϵ = 16, [C6MIm]+ is estimated[18] in the range ϵ = 40 − 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Our theory accounts the difference  in Tc due to changes in the screening of the Coulomb potential due to the inter - layer insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  This work was supported by NSC of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' 101-2112-M-009-014-MY3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' DETAILS OF THE MODEL  The system considered in the paper is fitted for the following values of the hipping and the tilt parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The  dimensionless tilt parameter was taken from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' [6] κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The hopping γ = 500 meV and t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' The calculations  were performed on the discrete reciprocal lattice k1, k2 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='Ns with Ns = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Reciprocal lattice basis vectors are,  b1 = 2π  �  1, 1  √  3  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  b2 = 2π  �  1, − 1  √  3  �  ,  (22)  10  so that a convenient representation is k = k1  Ns b1 + k2  Ns b2 with  kx = 2π  Ns  (k1 + k2) ,  ky =  2π  √  3Ns  (k1 − k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (23)  APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' DERIVATION OF THE TWO SUBLATTICE GAP EQUATION  Green’s functions and the s-wave Gorkov equations  We derive the Gorkov’s equations (GE) within the functional integral approach[19] starting from the effective  electron action for grassmanian fields ψ∗X, ψY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  S = 1  T  �  ψ∗X �  G−1  0  �XY ψY + 1  2ψ∗Y ψY V Y Xψ∗XψX  �  ,  (24)  where X, Y denote space coordinate, sublattices (pseudospin) and spin of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' Finite temperature properties  of the condensate are described at temperature T by the normal and the anomalous Matsubara Greens functions for  spin singlet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  The GE in functional form are:  �  ψAψ∗B�  δ  δψ∗C  � δS  δψ∗B  �  +  �  ψAψB�  δ  δψ∗C  � δS  δψB  �  = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (25)  �  ψAψ∗B�  δ  δψC  � δS  δψ∗B  �  +  �  ψAψB�  δ  δψC  � δS  δψB  �  = δAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (26)  Performing the calculations and using the normal and anomalous Green functions in the form F AB =  �  ψAψB�  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' GAB =  �  ψAψ∗B�  , one obtains:  F AX ��  G−1  0  �CX − vXCGCX + vCXGXX�  + GAXvXCF XC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (27)  Skipping second and third terms in bracket in this expression and defining, superconducting gap ∆AB = vABF AB,  one rewrites as a matrix products:  �  G−1  0  �CX F XA = GAX∆XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (28)  The first GE (multiplied from left by G0) is,  F AB = −GAXGBY  0  ∆XY ,  (29)  while the second GE similarly is:  GAB − GAX∆XY GZY  0  ∆∗ZUGUB  0  = GAB  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (30)  Frequency-quasi-momentum and the spin-sublattice decomposition  The generalized index A contains the space variables (space + Matsubara time, a), spin s and the sublattice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  After performing the Fourier series with combined quasi - momentum - frequency α:  F s1s2IJ  ab  = ϵs1s2 �  α eiα(a−b)F IJ  α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  ∆s1s2IJ  ab  = ϵs1s2 �  α eiα(a−b)∆IJ  α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (31)  Gs1s2IJ  0ab  = δs1s2 �  α eiα(a−b)gIJ  α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  V s1s2IJ  ab  =  �  α eiα(a−b)vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  11  Substituting spins into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' (29,30), one obtains in the sublattice matrix form  Fα = −Gα∆αgt  −α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (32)  Gα = G0α  �  I + ∆αgt  −α∆∗  −αG0α  �−1 ,  Convoluting the first GE by vν one obtains:  ∆ω = −  �  ν vω−νGν∆νgt  −ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (33)  The solution of the second GE for G is:  Gα = gα  �  I + ∆αgt  −α∆∗  −αgα  �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content='  (34)  Substituting into the first GE one obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
+page_content=' (20) in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'}
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+1
+A Stochastic Multi-Objective Optimization
+Framework for Planning and Scheduling of
+Community Energy Storage Systems
+K.B.J. Anuradha, Student Member, IEEE, and Chathurika P. Mediwaththe, Member, IEEE
+Abstract—This paper explores a methodology to optimize the
+planning and the scheduling of a community energy storage
+(CES) considering the uncertainty of real power consumption
+and solar photovoltaic (SPV) generation of the customers in low
+voltage (LV) distribution networks. To this end, we develop a
+stochastic multi-objective optimization framework which mini-
+mizes the investment and the operation costs of the CES provider,
+and the social costs of the customers (i.e. cost of customers for
+trading energy with the grid and the CES). The uncertainty
+of SPV generation and real power consumption are modelled
+to follow the beta and normal distributions, respectively. Then,
+the roulette wheel mechanism (RWM) is exploited to formulate a
+scenario-based stochastic program. The initial scenarios obtained
+from the RWM, are then reduced by using the K-Means clus-
+tering algorithm, to keep the problem tractability. A case study
+highlights our model provides 10-21% more cumulative economic
+benefits for the customers and the CES provider, compared with
+the models that optimize only the CES scheduling. Also, the
+simulation results for different energy price schemes of the CES
+provider reflect, the customers change their power exchange with
+the CES and the grid significantly, to minimize their social costs.
+Index
+Terms—Community
+energy
+storage
+(CES),
+multi-
+objective optimization, planning and scheduling, power flow,
+roulette wheel mechanism (RWM), scenarios, uncertainty
+I. INTRODUCTION
+T
+HE integration of solar photovoltaic (SPV) systems in
+low voltage (LV) distribution networks, has undergone
+a rapid upsurge over the last few decades. However, the
+intermittent and non-dispatchable nature of SPV generation,
+may restrict their beneficiaries such as the customers from
+exploiting the merits of SPV fully. These issues can be
+efficiently alleviated by exploiting energy storage systems.
+Community energy storage (CES) devices are an emerging
+type of battery system, which is gaining increasing interest in
+the industry, as they can enable increased community access
+and network hosting capacity for renewable energy [1].
+An energy management framework which aims at optimiz-
+ing only the scheduling of a CES such as its charging and
+discharging pattern, may not deliver the expected rewards
+from a CES completely. Hence, it is imperative that the
+planning aspects including the location, the capacity and the
+rated power of a CES are optimized concurrently with its
+K. B. J. Anuradha is with The Australian National University, Canberra,
+ACT 0200, Australia (email: u7146121@anu.edu.au).
+Chathurika P. Mediwaththe is with The Australian National University,
+Canberra, ACT 0200, Australia, and also with the Commonwealth Scientific
+and Industrial Research Organisation, Canberra, ACT 2601, Australia (email:
+chathurika.mediwaththe@csiro.au).
+scheduling. Several studies have presented deterministic opti-
+mization frameworks to find the optimal CES planning and/or
+the scheduling, and thus achieve the objectives of different
+stakeholders [2]–[4]. Here, the authors have assumed both real
+power consumption and SPV generation of the customers are
+perfectly known ahead from their forecasts. However, due to
+the uncertainty of SPV generation and real power consumption
+of the customers, their forecast errors can be quite high at
+times. Eventually, this may result the optimization models de-
+scribed in [2]–[4] unable to achieve the objectives effectively.
+Also, different stakeholders have distinct objectives for them.
+Thus, a multi-objective optimization framework can reflect the
+trade-offs between those objectives comprehensively.
+In this paper, we examine the extent to which the optimal
+planning and scheduling of a CES benefit different stakehold-
+ers. To this end, we develop an energy management framework
+between the customers, the CES and the grid, by incorporating
+the uncertainty of real power consumption and SPV generation
+of the customers. Additionally, we leverage a linearized power
+flow model with our energy management framework to formu-
+late a mixed integer linear program (MILP). In summary, the
+main contributions of this paper can be highlighted as follows.
+• We develop a stochastic multi-objective optimization
+framework which optimizes both planning and scheduling
+of a CES for benefiting (i) the CES provider by minimiz-
+ing the investment and the operation costs of the CES,
+and (ii) the customers by minimizing their social costs.
+• The proposed optimization framework is capable of pro-
+viding significantly higher economic benefits for the CES
+provider and the customers than in the models which
+arbitrarily choose the CES connected node.
+• A case study compares our proposed stochastic model
+with its corresponding deterministic model for different
+energy price schemes of the CES provider. This study
+enables to understand how the economic benefits for
+the CES provider, and for the customers change due
+to the uncertainty of real power consumption and SPV
+generation of the customers, and the energy price scheme
+of the CES provider.
+Prior research work have presented optimization frame-
+works which focus on the scheduling of the CES [2], [3], and
+both planning and scheduling of CES [4]–[7]. For instance, the
+authors of [2] have presented a method for CES scheduling, to
+minimize the social costs of the customers while maximizing
+the revenue of the CES provider. In [3], an optimization
+arXiv:2301.01462v1  [eess.SY]  4 Jan 2023
+
+2
+framework is presented to schedule the CES, to minimize the
+real energy losses, and energy trading costs with the grid by
+the CES provider and the customers. The authors of [4] and
+[5] have presented models to optimize the CES planning and
+scheduling simultaneously, to enhance the hosting capacity of
+LV networks, and to mitigate the voltage excursions in three
+phase unbalanced LV networks, respectively. Also, analytical
+methods for optimizing the CES planning and scheduling have
+been discussed in [6], [7]. A common feature of [2]–[7] is
+that the authors have used deterministic models, assuming
+the SPV generation and the real power consumption of the
+customers are known ahead with no uncertainty. Hence, those
+models may not be efficient in providing realistic planning and
+operation decisions.
+The uncertainty of real power consumption and generation
+from SPV have been taken into account in [8], [9] for
+CES management problems. For instance, the authors of [8]
+have presented a method for optimizing both planning and
+scheduling of CES to accomplish multiple objectives. Here,
+the authors have used the normal distribution and the RWM
+to model the uncertainty of real power consumption and SPV
+generation. In [9], the authors of have investigated how the
+CES location impacts the voltage profile and real power losses
+in LV networks. For this, they have arbitrarily allocated the
+CES at different nodes. In contrast to [2]–[7], our paper opti-
+mizes both planning and scheduling aspects of the CES taking
+into account the uncertainty of real power consumption and
+SPV generation. Thus, our approach models the CES planning
+and its scheduling problem more realistically. Also, compared
+to [8], [9], we present a method to minimize the personal costs
+of the CES provider and the customers concurrently.
+This paper is structured as follows. Notations used in this
+paper are detailed in Section II. The system models of the CES,
+the customers and the network power flow are presented in
+Section III. The stochastic models of SPV generation and real
+power consumption of the customers are given in Section IV.
+The formulation of the multi-objective optimization framework
+is described in Section V. Section VI presents the validation of
+the results, and Section VII gives the conclusion of the paper.
+II. NOTATIONS
+A.
+Stochastic Model-related Notations
+To formulate a scenario-based stochastic program, the un-
+certainty of SPV generation and real power consumption of
+the customers are modelled using known probability density
+functions. Here, a scenario represents a possible combination
+of SPV generation and real power consumption of all the
+customers together with their corresponding probabilities at
+a given time. The initial set of scenarios is denoted by
+R, and r ∈ R. Due to the computational complexity of
+stochastic programs which use scenarios, it is imperative to
+use a scenario reduction approach to reduce the number of
+scenarios, and keep the problem tractability. Thus, a scenario
+reduction technique is used in this paper, and the set composed
+by the reduced scenarios is given by S, where s ∈ S.
+The implementation of the scenario-based stochastic program
+including the scenario reduction are discussed in Section IV.
+Fig. 1: Mutual power exchanges between the CES, the grid
+and a customer in scenario s at time t
+B. Network and Power Flow-related Notations
+In this paper, a distribution network with a radial topology
+is considered. It is described by the graph G = (V, E), where
+V = {0, 1, ..., N} is the set of all nodes, and E = {(i, j)} ⊂
+V×V is the set of all lines in the network. Node 0 (slack node)
+represents the secondary side of the distribution transformer.
+The resistance and the reactance of line (i, j) are rij (in Ω)
+and xij (in Ω), respectively. The set of customers at node j
+is represented by Cj, and c ∈ Cj ∀j ∈ V\ {0}. Also, t ∈ T ,
+where T is the set of time intervals, and ∆t is the difference
+between two adjacent time instances (in hours). The real and
+reactive power flow from i to j node in scenario s at time t
+are represented by Pij,s(t) (in kW) and Qij,s(t) (in kVAR),
+respectively. Also, ∀j ∈ V\ {0}, t∈ T , s∈ S, the real power
+absorption, reactive power absorption, voltage and squared
+voltage are given by pj,s(t) (in kW), qj,s(t) (in kVAR), Vj,s(t)
+(in V ) and Uj,s(t) (in V 2), respectively.
+III. SYSTEM MODELS
+This section first presents the network power flow model,
+followed by the power exchange model of the customers, and
+the CES model. We consider each customer, the CES and the
+grid can exchange power with each entity as shown in Fig. 1.
+A. Power Flow Model of the Network
+In this work, we consider the power absorption for the
+nodes as positive, and the power injections from the nodes as
+negative. Additionally, it is considered that there are multiple
+customers at each node. To model the network power flows,
+we use the linearized power flow equations (1) and (2) from
+the Distflow model in [10].
+Pij,s(t) = pj,s(t) +
+�
+k:j→k
+Pjk,s(t)
+∀(i, j) ∈ E, , t∈ T , s∈ S
+(1)
+Qij,s(t) = qj,s(t)+
+�
+k:j→k
+Qjk,s(t)
+∀(i, j) ∈ E, , t∈ T , s∈ S
+(2)
+
+External Grid
+Grid
+pCES,s(t) < 0
+pCES,s(t) ≥ 0
+0 > (0)sgd
+pejs(t) > 0
+pcS(t) > 0
+田田
+pCS (t) < 0
+cth Customer
+CES
+at node j3
+The real and reactive power absorbed by the node j at time t
+in scenario s, can be expressed as (3) and (4). The real power
+absorption for the CES installed node is governed by (3a),
+while for all the other nodes (except slack node), it is (3b).
+Th equation (4) handles the reactive power absorption for all
+nodes, and we assume the CES and the SPV devices operate
+at unity power factor. The real power consumption, reactive
+power consumption and SPV generation of the customer c at
+node j at time t in scenario s are given by pL
+cj,s(t) (in kW),
+qL
+cj,s(t) (in kVAR) and pP V
+cj,s(t) (in kW), respectively. For j ∈
+V\ {0} , t∈ T , s∈ S, the CES charging and discharging power
+are represented by pCES,ch
+j,s
+(t) and pCES,dis
+j,s
+(t), respectively.
+pj,s(t) =
+�
+c∈Cj
+pL
+cj,s(t)−
+�
+c∈Cj
+pP V
+cj,s(t)+pCES,ch
+j,s
+(t)−pCES,dis
+j,s
+(t)
+(3a)
+pj,s(t) =
+�
+c∈Cj
+pL
+cj,s(t)−
+�
+c∈Cj
+pP V
+cj,s(t)
+∀j ∈ V\ {0} , t∈ T , s∈ S
+(3b)
+qj,s(t) =
+�
+c∈Cj
+qL
+cj,s(t)
+∀j ∈ V\ {0} , t∈ T , s∈ S
+(4)
+The linearized Distflow equations described in (1)-(2) can
+be explicitly written as (5), where U0 = |V0|2 and U=|V(t)|2
+are the vectors of the squared voltage magnitude of the slack
+node, and the squared voltage magnitudes of all other nodes,
+respectively. 1 symbolizes a vector of all ones. Moreover, p
+and q are the vectors of real and reactive power absorption at
+each node. The matrices ˜R and ˜X ∈ RN×N have the elements
+Rij = 2 �
+(m,n)∈Li∩Lj rmn and Xij = 2 �
+(m,n)∈Li∩Lj xmn,
+respectively, where Li is the set of lines on the path which
+connects node 0 and i [2], [10]. Also, (6) ensures the squared
+voltage magnitude at each node is within its allowable voltage
+magnitude limits. Here, Umin =
+��V 2
+min
+�� 1 and Umax =
+��V 2
+max
+�� 1, where Vmin and Vmax are the allowable lower and
+upper bound of voltage, respectively.
+U = U01 − ˜Rp − ˜Xq
+∀t∈ T , s∈ S
+(5)
+Umin ≤ U ≤ Umax
+∀t∈ T , s∈ S
+(6)
+B. Power Exchange Model of the Customers
+According to Fig. 1, pG
+cj,s(t) and pCES
+cj,s (t) denote the real
+power exchange with the grid and the CES by the customer c at
+node j at time t in scenario s, respectively. When pG
+cj,s(t) > 0,
+it suggests a power import from the grid by a customer. On
+the contrary, when that customer exports power back to the
+grid, it will be pG
+cj,s(t) < 0. The same sign convention is used
+for customer power exchanges with the CES, and CES power
+exchange with the grid pG
+CES,s(t) (in kW).
+When a customer’s SPV generation is insufficient to fulfill
+its real power consumption, that deficit is attained by importing
+power from the grid and the CES. This is mathematically
+represented by (7a). Nevertheless, the imported power from
+each entity should be within the deficit quantity. This is
+ensured by (7b) and (7c). Besides, when that customer has
+excess SPV generation, it exports its surplus to the grid and
+the CES, which is mathematically interpreted by (8a). Similar
+to (7b) and (7c), the exported power to the CES and the grid
+should not exceed the mismatch of the SPV generation and
+the real power consumption. This is demonstrated by (8b) and
+(8c). Considering the ability of the CES to exchange power
+with the grid and the customers, pG
+CES,s(t) can be expressed
+in terms of pCES
+cj,s , pCES,ch
+j,s
+(t) and pCES,dis
+j,s
+(t) as in (9).
+If pL
+cj,s(t) ≥ pP V
+cj,s(t):
+0 ≤ pG
+cj,s(t) + pCES
+cj,s (t) = pL
+cj,s(t) − pP V
+cj,s(t)
+(7a)
+0 ≤ pG
+cj,s(t) ≤ pL
+cj,s(t) − pP V
+cj,s(t)
+(7b)
+0 ≤ pCES
+cj,s (t) ≤ pL
+cj,s(t) − pP V
+cj,s(t)
+(7c)
+Otherwise:
+pG
+cj,s(t) + pCES
+cj,s (t) = pL
+cj,s(t) − pP V
+cj,s(t) ≤ 0
+(8a)
+pL
+cj,s(t) − pP V
+cj,s(t) ≤ pG
+cj,s(t) ≤ 0
+(8b)
+pL
+cj,s(t) − pP V
+cj,s(t) ≤ pCES
+cj,s (t) ≤ 0
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T , s∈ S
+(8c)
+pG
+CES,s(t) = �N
+j=1
+��
+c∈Cj pCES
+cj,s (t) + pCES,ch
+j,s
+(t) − pCES,dis
+j,s
+(t)
+�
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T , s∈ S
+(9)
+C. Community Energy Storage Model
+In this paper, we assume the CES is owned by a third
+party, and we designate the owner as the CES provider. The
+planning constraints of the CES are given by (10)-(12), and the
+operation of the CES is mathematically modeled by (13)-(16).
+The equation (10) finds the optimal CES node, and we use
+a binary variable Lj, in which Lj = 1 suggests node j as
+the optimal CES node, and Lj = 0 means there is no CES at
+node j. Also, (10) guarantees only a single CES is installed in
+the network. The inequalities in (11) and (12) find the optimal
+CES capacity Ecap
+j
+(in kWh) and its rated power pRate
+j
+(in
+kW) at node j, respectively. Here, Ecap
+min (in kWh) and Ecap
+max
+(in kWh) are the minimum and maximum CES capacity limits,
+and pRate
+max is the maximum CES rated power limit.
+N
+�
+j=1
+Lj = 1
+∀j ∈ V\ {0} , Lj ∈ {0, 1}
+(10)
+
+4
+LjECap
+min ≤ ECap
+j
+≤ LjECap
+max
+∀j ∈ V\ {0} , Lj ∈ {0, 1}
+(11)
+0 ≤ pRate
+j
+≤ LjpRate
+max
+∀j ∈ V\ {0} , Lj ∈ {0, 1}
+(12)
+The inequality described in (13) avoids simultaneous charg-
+ing and discharging of the CES, while guaranteeing the CES
+charging and discharging power does not exceed its optimal
+rated power pRate
+j
+. For this, a binary variable Bj and two
+additional variables namely, y and z are used. Here, Bj ∈ [0, 1]
+and y, z ∈ R+ such that these variables also satisfy (13). When
+the CES is charging, Bj = 1. Hence, z = pCES,dis
+j,s
+(t) = 0 ac-
+cording to (13d) and (13e). Also, as stated in (13c), y = pRate
+j
+and thus, 0 ≤ pCES,ch
+j,s
+(t) ≤ y = pRate
+j
+≤ pRate
+max . When the
+CES discharges, Bj = 0, and hence, y = pCES,ch
+j,s
+(t) = 0
+according to (13a) and (13b). In this instance, z = pRate
+j
+,
+which generates the inequality 0 ≤ pCES,dis
+j,s
+(t) ≤ z =
+pRate
+j
+≤ pRate
+max . The equation (14) illustrates how the CES
+energy level changes with time t, where ECES
+j,s
+(t) (in kWh) is
+the energy level of the CES at node j at time t in scenario s.
+Additionally, ηch and ηdis are the charging and discharging
+efficiency of the CES, respectively. Furthermore, the CES
+energy level should be maintained with in the minimum and
+maximum state of charge (SoC) levels. This is regulated by
+(15), where ηmin and ηmax represent minimum and maximum
+percentage coefficients of the CES capacity, respectively. Also,
+it is required to guarantee the continuity of the CES operation
+over the next day, and thus, the CES energy level at the end
+of the day should be kept approximately same as the initial
+energy level at the start of the day . This is managed by (16)
+[2], [11]. Here, ε is a small positive number (in kWh), and
+tn ∈ TN where TN = {1, 2, ...., |T | /24}.
+0 ≤ pCES,ch
+j,s
+(t) ≤ y
+(13a)
+pCES,ch
+j,s
+(t) ≤ y ≤ pRate
+max Bj
+(13b)
+−pRate
+max (1 − Bj) ≤ y − pRate
+j
+≤ 0
+(13c)
+0 ≤ pCES,dis
+j,s
+(t) ≤ z
+(13d)
+pCES,dis
+j,s
+(t) ≤ z ≤ pRate
+max (1 − Bj)
+(13e)
+−pRate
+max Bj ≤ z − pRate
+j
+≤ 0
+∀j ∈ V\ {0} , t∈ T , s∈ S
+(13f)
+ECES
+j,s
+(t) = ECES
+j,s
+(t − 1) + (ηchpCES,ch
+j,s
+(t)
+− 1
+ηdis pCES,dis
+j,s
+(t))∆t
+∀j ∈ V\ {0} , t∈ T , s∈ S
+(14)
+ηminECap
+j
+≤ ECES
+j,s
+(t) ≤ ηmaxECap
+j
+∀j ∈ V\ {0} , t∈ T , s∈ S
+(15)
+��ECES
+j,s
+(24tn) − ECES
+j,s
+(0)
+�� ≤ ε
+∀j ∈ V\ {0} , tn∈ T N, s∈ S
+(16)
+IV. STOCHASTIC MODELS
+In this section, the uncertainty modeling of real power
+consumption and SPV generation are presented. Similar to
+[12], [13], the uncertainty of real power consumption and SPV
+generation are modelled by the probability density functions
+(PDFs) of normal and beta distributions, respectively.
+A. Uncertainty of the Real Power Consumption
+The uncertainty of real power consumption of the customers
+follows the probability density function of normal distribution
+PDFL(.) given in (17) [12], [14]. The forecasted real power
+consumption of the customer c at node j at time t is consid-
+ered as the mean real power consumption µL,t
+cj
+of PDFL(.)
+[12], [14]. The standard deviation and a sample real power
+consumption are denoted by σL,t
+cj
+and XL,t
+cj , respectively.
+PDFL(X) =
+1
+σL,t
+cj
+√
+2π
+e
+−0.5
+�
+XL,t
+cj
+−µL,t
+cj
+σL,t
+cj
+�2
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T
+(17)
+B. Uncertainty of the SPV Generation
+As mentioned in [12], [13], the uncertainty of SPV gener-
+ation of the customers mimic the probability density function
+of beta distribution PDFP V (.) as given in (18a). Also, the
+forecasted SPV generation of the customer c at node j at time t
+is taken as the mean SPV generation µP V,t
+cj
+of PDFP V (.). The
+equations (18a)-(18d) describe the relationship between the
+shape parameters αt
+cj, βt
+cj, a sample SPV generation XP V,t
+cj
+,
+SPV capacity of a customer PVcap,cj, the mean µP V,t
+cj
+and the
+standard deviation σP V,t
+cj
+of PDFP V (.) [12], [13]. Also, Γ(.)
+represents the gamma function.
+PDFP V (X) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Γ(α+β)
+Γ(α)Γ(β)
+�
+XP V,t
+cj
+�αt
+cj−1 �
+1 − XP V,t
+cj
+�βt
+cj−1
+∀
+0 < XP V,t
+cj
+< 1
+αt
+cj, βt
+cj ≥ 0
+0
+Otherwise
+(18a)
+µP V,t
+cj
+=
+αt
+cj
+αt
+cj + βt
+cj
+(18b)
+(σP V,t
+cj
+)2 =
+αt
+cjβt
+cj
+�
+αt
+cj + βt
+cj
+�2 �
+αt
+cj + βt
+cj + 1
+�
+(18c)
+σP V,t
+cj
+=
+0.2µP V,t
+cj
+PVcap,cj
++ 0.21
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T
+(18d)
+
+5
+C. Scenario-based Stochastic Program
+Since the normal and beta distributions are continuous
+PDFs, they represent infinite number of realizations of the
+random variables. Here, a realization refers to a sample real
+power consumption or SPV generation of a customer at a given
+time. A large number of realizations can model the uncertainty
+better at the expense of a large computational burden. But,
+continuous PDFs approximated as discrete functions by a
+finite number of realizations, as given in [15], can be used
+to eliminate the similar and less probable real power con-
+sumption or SPV generation values. Hence, the discretization
+of continuous PDFs, reduces the complexity of uncertainty
+modelling. Thus, both normal and beta PDFs are approximated
+as discrete functions by 7 realizations. Here, the approximated
+discrete functions are constructed to have 7 intervals, with
+every interval having a width of a standard deviation σ. The
+midpoint of an interval is a possible realization. For instance,
+when the forecasted real power consumption is µ, the intervals
+1-7 are centered around the 7 realizations µ, µ + σ, µ − σ,
+µ + 2σ, µ − 2σ, µ + 3σ and µ − 3σ as done in [16], [17]. The
+steps of scenario generation and reduction are given below.
+• Do Step 1 to Step 5 ∀t ∈ T , j ∈ V\ {0}, c ∈ Cj
+Step 1: Find the model parameters in (18) namely, σP V,t
+cj
+αt
+cj and βt
+cj, by using the forecasted SPV generation µP V,t
+cj
+.
+It is assumed that PVcap,cj ∀j ∈ V\ {0} , c ∈ Cj are known
+prior. The standard deviation σL,t
+cj
+in (17) is taken as 4% of
+the mean real power consumption µL,t
+cj
+of PDFL(.) [12].
+Step 2: Discretize the normal and beta distributions de-
+scribed in (17) and (18) into 7 intervals. For this, 7 possible
+realizations for each PDF are calculated as µ, µ + σ, µ − σ,
+µ + 2σ, µ − 2σ, µ + 3σ and µ − 3σ. Then, their respective
+probability densities are calculated from (17) and (18). Once
+the probability densities are available, the probability for the
+occurrence of each SPV generation and real power consump-
+tion is found by taking the product of probability density and
+the width of each discrete interval (i.e. σ).
+Step 3: Normalize the calculated probabilities of real power
+consumption and SPV generation. This is done by taking the
+sum of the probabilities, and dividing each probability by the
+sum [8], [16], [17]. This should be done for the 7 realizations
+obtained from each PDF separately. Since the continuous PDFs
+are approximated by discretization, sum of the 7 probabilities
+will only be close to unity but not exactly equal to 1. Hence,
+the normalization guarantees that the sum of the probabilities
+will be precisely equal to unity [8], [16], [17].
+Step 4: Use the roulette wheel mechanism (RWM) explained
+in [8], [16], [17] to construct two roulette wheels in the range
+[0,1], each having 7 intervals. For this, assign the normalized
+7 probabilities obtained from each PDF to [0,1] range. Hence,
+each interval has a width of the normalized probability of the
+respective real power consumption or SPV generation.
+Step 5: Generate Nr = |R| number of random numbers
+between 0 and 1, which follow the uniform distribution. Here,
+the random numbers are obtained from a uniform distribution,
+to guarantee they are generated without any bias.
+• Do Step 6 ∀t ∈ T , j ∈ V\ {0}, c ∈ Cj, r ∈ R
+Step 6: Assign each random number to the two roulette
+wheels according to their magnitudes. Select φL,t
+cj,r, φP V,t
+cj,r ,
+pL
+cj,r(t) and pP V
+cj,r(t) from the roulette wheels corresponding to
+the value of the random number, where φL,t
+cj,r and φP V,t
+cj,r are the
+normalized probabilities of pL
+cj,r(t) and pP V
+cj,r(t), respectively.
+In this way, the initial set of scenarios are obtained.
+• Do Step 7 ∀t ∈ T , j ∈ V\ {0}, c ∈ Cj
+Step 7: A scenario reduction approach is essential in sce-
+nario based stochastic programs to keep the problem tractabil-
+ity, while sustaining a fair approximation for the uncertainty.
+Thus, the initially generated scenarios in R, are then reduced
+to Ns = |S| number of scenarios to form a new scenario set
+S, by using the K-Means clustering algorithm [8], [18]. This
+will generate a new set of values for the probabilities and their
+realizations as φL,t
+cj,s, φP V,t
+cj,s , pL
+cj,s(t), pP V
+cj,s(t) ∀s ∈ S. The K-
+Means clustering method is illustrated in Algorithm 1.
+Algorithm 1 K-Means Clustering Algorithm
+1: Input φL,t
+cj,r ∀j ∈ V\ {0} , c ∈ Cj, t∈ T , r∈ R
+2: for each t in T do
+3:
+for each j in V\ {0} do
+4:
+for each c in Cj do
+5:
+Randomly initialize the centroids of K-Means clus-
+ters as Z =
+�
+φL,t
+cj,1, ..., φL,t
+cj,s, ..., φL,t
+cj,Ns
+�
+, where
+|Z| = Ns
+6:
+for each s in S do
+7:
+As ← ∅
+8:
+end for
+9:
+while centroids of clusters do not change do
+10:
+for each r in R do
+11:
+s∗ ← argmin
+s
+���φL,t
+cj,r − φL,t
+cj,s
+���
+12:
+As∗ ← As∗ ∪
+�
+φL,t
+cj,r
+�
+13:
+end for
+14:
+for each s in S do
+15:
+φL,t
+cj,s ←
+1
+|As|
+�
+φL,t
+cj,r∈As φL,t
+cj,r
+16:
+end for
+17:
+end while
+18:
+end for
+19:
+end for
+20: end for
+21: Repeat Step 1 to Step 20 for the inputs φP V,t
+cj,r , pL
+cj,r(t) and
+pP V
+cj,r(t), separately ∀j ∈ V\ {0} , c ∈ Cj, t∈ T , r∈ R
+22: Return φL,t
+cj,s, φP V,t
+cj,s , pL
+cj,s(t), pP V
+cj,s(t) ∀j ∈ V\ {0} , c ∈
+Cj, t∈ T , s∈ S
+• Do Step 8 ∀t ∈ T , s ∈ S
+Step 8: Calculate the overall probability ωs,t in (19), which
+gives the probability for the occurrence of scenario s at time
+t. The numerical values found for ωs,t, pL
+cj,s(t), and pP V
+cj,s(t)
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T , s∈ S are then fed into the system
+models and optimization framework in Section III and V.
+ωs,t =
+��N
+j=1
+��
+c∈Cj φL,t
+cj,sφP V,t
+cj,s
+��
+�Ns
+s=1
+��N
+j=1
+��
+c∈Cj φL,t
+cj,sφP V,t
+cj,s
+��
+(19)
+
+6
+V. MULTI-OBJECTIVE OPTIMIZATION FRAMEWORK
+In this paper, it is aimed to minimize the investment cost of
+the CES as a planning objective, and minimize the CES opera-
+tion cost and the social costs of the customers as the operation
+objectives. Thus, a multi-objective function is formulated by
+combining both planning and operation objectives.
+A. Objective Functions
+1) Minimizing the Investment Cost of the CES: The in-
+vestment cost for a CES can be expressed as (20) [8], [19].
+The first term of (20) relates the investment cost for the rated
+power of the CES, and latter for the capacity of the CES. Since
+minimizing the investment cost is a planning objective, it is not
+impacted by the uncertainty of the real power consumption and
+the SPV generation. Thus, (20) is independent of the scenarios.
+fInv,cost = ρCES(CCES,Inv
+Rate
+pRate
+j
++ CCES,Inv
+Cap
+ECap
+j
+) (20)
+Here, ρCES =
+d(1+d)τ
+(1+d)τ −1, where ρCES, d, and τ are the
+annual cost of the CES, discount rate and the CES life time
+(in years), respectively. Also, CCES,Inv
+Rate
+and CCES,Inv
+Cap
+are
+the CES investment cost per kW (in AUD/kW) and the CES
+investment cost per kWh (in AUD/kWh), respectively.
+2) Minimizing the Operation Cost of the CES: The cost for
+operating the CES is given by (21) [8]. Since (21) illustrates
+an operation objective, it is also a function of the scenarios.
+fop,cost =
+�
+t∈T
+� Ns
+�
+s=1
+ωs,t
+�
+CCES,oppCES
+j,s
+(t)
+�
+�
+(21)
+where pCES
+j,s
+(t) = ηchpCES,ch
+j,s
+(t) −
+1
+ηdis pCES,dis
+j,s
+(t) and
+CCES,op is the CES operation cost per kW (in AUD/kW).
+3) Minimizing the Social Costs of the Customers: Cus-
+tomers incur a cost or earn a revenue for trading energy with
+the CES and the grid, which is jointly named as the social
+costs as given in (22). Its first term denotes the energy trading
+cost with the grid, and latter for trading energy with the CES.
+Similar to (21), as the social costs of the customers is also an
+operation objective, fC,cost is a function of the scenarios.
+fC,cost =
+�
+t∈T
+Ns
+�
+s=1
+ωs,t
+�
+λG(t)
+N
+�
+j=1
+�
+c∈Cj
+pG
+cj,s(t)
++ λCES(t)
+N
+�
+j=1
+�
+c∈Cj
+pCES
+cj,s (t)
+�
+∆t
+(22)
+Here, λG(t) and λCES(t) are the grid energy price and CES
+provider’s energy price at time t, respectively. We adopt a one-
+for-one non-dispatchable energy buyback method for λG(t), to
+value energy imports and exports from/to the grid equally [20].
+B. Optimization Problem
+The three objective functions which determine the CES
+planning and its operation, are combined together to form
+a multi-objective function as given in (23). The objective
+functions are normalized by their corresponding nadir and
+utopia points to attain a Pareto optimal solution for each
+objective compatible with the weights assigned for them [21].
+x∗ = argmin
+x∈X
+w1
+�
+fInv,cost−f utopia
+Inv,cost
+f Nadir
+Inv,cost−f utopia
+Inv,cost
+�
++ w2
+�
+fop,cost−f utopia
+op,cost
+f Nadir
+op,cost−f utopia
+op,cost
+�
++w3
+�
+fC,cost−f utopia
+C,cost
+f Nadir
+C,cost−f utopia
+C,cost
+�
+(23)
+Here,
+x = (Lj, pRate
+j
+, ECap
+j
+, pCES,ch
+j
+, pCES,dis
+j
+, pCES
+cj
+, pG
+cj)
+(24)
+where x is the decision variable vector. Here, Lj, Ecap
+j
+and pRate
+j
+are the vectors of the optimal CES location, CES
+capacity and its rated power, respectively. Vectors of the CES
+charging power, CES discharging power, power exchange with
+the CES and the grid by the customers are given by pCES,ch
+j
+,
+pCES,dis
+j
+, pCES
+cj
+and pG
+cj, respectively. The feasible set is
+given by X, which is constrained by (1)-(16). Furthermore, the
+calculation of the utopia and nadir values for each objective
+function are done in line with the techniques mentioned in
+[21]. Also, w1, w2 and w3 are the weight coefficients of
+each objective function. The implementation of the overall
+optimization framework is succinctly given in Algorithm 2.
+Algorithm 2 Algorithm to Run the Stochastic Multi-Objective
+Optimization
+1: Input µL,t
+cj , µP V,t
+cj
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T
+2: Initialize the model parameters used.
+3: Execute Step 1 to Step 8 detailed in Section IV-C, in-
+cluding the Algorithm 1, to model the uncertainty of the
+real power consumption and the SPV generation of the
+customers.
+4: Return ωs,t, pL
+cj,s(t), and pP V
+cj,s(t) ∀j ∈ V\ {0} , c ∈
+Cj, t∈ T , s∈ S.
+5: Solve the multi-objective function in (23), subject to the
+set of constraints (1) - (16), as a MILP.
+VI. NUMERICAL AND SIMULATION RESULTS
+In the simulations, a radial distribution network with 7-
+nodes given in Fig. 2 was considered, and its line data
+can be found in [22]. The forecasted SPV generation and
+real power consumption data of 30 residential customers in
+an Australian community, for a period of 1 year, measured
+in 1-hour time intervals were used for simulations [23].
+Here, all the residential customers generate SPV power and
+consume real power. Nevertheless, due to the lack of real
+data on customers’ reactive power consumption, it was not
+considered for the simulations. Also, we randomly assigned
+the 30 customers for all the nodes except for the slack
+
+7
+Fig. 2: The 7-Node LV radial feeder with the number of
+customers marked at each node
+Fig. 3: Variation of the grid energy price λG(t) with time
+node (see Fig. 2). Therefore, �N
+j=1 |Cj| = 30. Additionally,
+|V0|
+=
+1p.u., |Vmin|
+=
+0.95p.u., |Vmax|
+=
+1.05p.u.,
+pRate
+max = 200kW, ECap
+min = 50kWh, ECap
+max = 1000kWh,
+ηch
+= 0.98, ηdis
+= 1.02, ηmin
+= 0.05, ηmax
+= 1,
+ε = 0.0001kWh, ∆t = 1h, d = 0.1, τ = 12.5 years,
+CCES,Inv
+Rate
+= 463AUD/kW, CCES,Inv
+Cap
+= 795AUD/kWh
+and CCES,op = 0.69AUD/kW. Moreover, the PV capacities
+of the customers (i.e. PVcap,cj) were obtained from [23]. Also,
+the values for CCES,Inv
+Rate
+, CCES,Inv
+Cap
+and CCES,op were taken
+assuming Li-ion as the CES technology [19].
+The three objectives of the multi-objective function in (23),
+were weighted according to their importance. For this, we
+used the analytic hierarchy process (AHP) illustrated in [24].
+We assigned an equal importance for fInv,cost and fop,cost,
+and a strong importance for fC,cost compared to fInv,cost and
+fop,cost. Therefore, the values of w1, w2, w3 were calculated
+as 1/7, 1/7, 5,7, respectively [24]. After computing the weight
+coefficients, the simulations were done over a period of 1 year
+(i.e. |T | = 8760) using the CPLEX solver in Python-Pyomo.
+A. Case Study I: Comparison of the Proposed Optimization
+Framework With its Corresponding Deterministic Model
+To understand the impact of uncertainty on the optimal
+planning and scheduling decisions of a CES, we compared
+the results of the proposed stochastic optimization framework
+and its corresponding deterministic model. The simulations
+were done for our stochastic model by considering 50 initial
+scenarios, which is then reduced to 10 (i.e. Ns = 10) by using
+the K-Means clustering algorithm. Simulations for the deter-
+ministic model were obtained by neglecting the uncertainty of
+real power consumption and SPV generation of the customers.
+The same set of constraints and the multi-objective function
+used for the proposed stochastic model (i.e. (1)-(16) and (23) ),
+were used for the deterministic model as well, while excluding
+the scenario dependency of (1)-(9), (13)-(16) and (23).
+In the simulations, a time-of-use (TOU) grid energy price
+λg(t) shown in Fig. 3 was used [25]. Due to the lack of
+accurate real data about the CES provider’s energy price
+λCES(t), we assumed three different energy price schemes
+for it as (i) λCES(t) = λG(t), (ii) λCES(t) = 0 and (iii)
+λCES(t) = λG,avg ,where λG,avg =
+�24
+t=1 λG(t)
+24
+. A summary
+of the numerical results obtained for the two types of the
+optimization models are given in Table I. For the three energy
+price schemes of λCES(t), in both deterministic and proposed
+stochastic models, the optimal CES location is node 7.
+The pRate
+j
+and ECap
+j
+in the stochastic model are higher than
+their values in the deterministic model. In the stochastic model,
+due to the impact of the higher values of the realizations with
+respect to µL,t
+cj , µP V,t
+cj
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T , both pRate
+j
+and ECap
+j
+will be greater than their values in the deterministic
+model. Moreover, in both models, the values of the planning
+decisions namely, Lj, pRate
+j
+and ECap
+j
+for their respective
+models have not changed irrespective of the CES provider’s
+energy price scheme. This has happened, as the planning
+decisions are independent from operation variables, and thus
+from λCES(t). Hence, the investment costs in the deterministic
+and stochastic models are AUD 32228 and 33695, respectively,
+irrespective of the CES provider’s energy price scheme.
+The operation objectives consider minimizing the CES
+operation cost fop,cost, and the social costs of the customers
+fC,cost. According to Table I, fop,cost is same for the deter-
+ministic model regardless of the CES provider’s energy price
+scheme. This is resulted as fop,cost is a function independent
+of λCES(t) (see (21). Besides, when λCES(t) = λG(t) and
+λCES(t) = λG,avg(t), fop,cost in the stochastic model is
+higher than the corresponding deterministic model values.
+Since the stochastic model takes into account the uncer-
+tainty of SPV generation and real power consumption of
+the customers, the costs in the stochastic model are higher
+than the ones in the deterministic model. This behaviour
+is seen for fC,cost as well when λCES(t) = λG(t) and
+λCES(t) = λG,avg(t). Additionally, as fC,cost < 0 when
+λCES(t) = 0, it implies that the customers earn a revenue. The
+customers minimize their social costs by importing power only
+from the CES as λCES(t) = 0. Also, the customers export
+power solely to the grid to maximize their social revenue. This
+is the intuition for fC,cost being negative when λCES(t) = 0
+for both deterministic and stochastic models. This is discussed
+in detail in Section VI-B-2. In the proposed stochastic model,
+fop,cost for the three energy price schemes of λCES(t) are
+different from each other. As the set of random numbers
+generated during stochastic modelling are unique and different
+for every execution of Algorithm 2, a unique set of values
+for ωs,t, pL
+cj,s(t), and pP V
+cj,s(t) ∀j ∈ V\ {0}, c ∈ Cj, t∈ T ,
+s∈ S are obtained. This results in getting different values for
+fop,cost, irrespective of fop,cost being independent of λCES(t).
+
+2
+6
+External
+Transformer
+IC2l = 4
+IC6l = 4
+Grid
+22/0.4 kvV
+0
+1
+3
+4
+L80
+ICil = 3
+IC3/ = 5
+IC4l = 6
+ICzl = 5
+5
+ICsl = 30.55
+0.50
+0.45
+0.40
+0.35
+0.30
+0.25
+T2
+T3
+T4 T5
+0.20
+0
+5
+10
+15
+20
+25
+Time Duration (24 Hours)8
+TABLE I: RESULTS OF THE DETERMINISTIC AND PROPOSED STOCHASTIC MODELS
+Optimal
+CES Node
+Lj
+Optimal CES
+Rated Power1
+pRate
+j
+(kW)
+Optimal CES
+Capacity1
+ECap
+j
+(kWh)
+CES Investment
+Cost1 (AUD)
+fInv,cost
+CES Operation
+Cost1(AUD)
+fop,cost
+Social
+Costs1 (AUD)
+fC,cost
+Deterministic
+λCES(t) = λG(t)
+7
+72
+240
+32228
+24674
+124830
+Model
+λCES(t) = 0
+7
+72
+240
+32228
+24674
+-94681
+λCES(t) = λG,avg
+7
+72
+240
+32228
+24674
+70810
+Proposed Stochastic
+Model (Ns = 10)
+λCES(t) = λG(t)
+7
+78 (7.69%)
+249 (3.61%)
+33695 (4.35%)
+27266 (9.51%)
+129429 (3.55%)
+λCES(t) = 0
+7
+78 (7.69%)
+249 (3.61%)
+33695 (4.35%)
+29712 (16.96%)
+-98201 (3.58%)
+λCES(t) = λG,avg
+7
+78 (7.69%)
+249 (3.61%)
+33695 (4.35%)
+29347 (15.92%)
+75263 (5.92%)
+1 Increment percentage values are computed with respect to their corresponding values found in the deterministic model
+Fig. 4: (a) Total power exchange with the grid and the CES by
+customers, (b) CES charging/discharging power and temporal
+variation of CES energy - When Ns = 10, λCES(t) = λG(t)
+B. Analysis of the Results of CES Scheduling and Mutual
+Power Exchanges Between the CES, the Grid and Customers
+The results obtained for the proposed stochastic optimiza-
+tion framework, for different energy price schemes of the CES
+provider are detailed next. Here, we do the analysis for a
+randomly selected a day, for a duration of 24 hours.
+1) When λCES(t)=λG(t): Fig. 4(a) depicts the total power
+exchange with the grid (brown plot) and the CES (orange plot)
+by the customers. As the customers do not have sufficient
+SPV generation during T1, T3 T4 and T5, they import power
+from the grid and the CES. Besides, the customers export their
+excess generation to the CES and the grid during T2. Since
+λCES(t)=λG(t), the customers do not have any preference
+whether to exchange power with the grid or the CES. The
+charging and discharging pattern of the CES (red plot), and the
+CES energy level variation (green plot) with time are shown
+in Fig. 4(b). During T1, the CES charges, and by the end of
+T1, it discharges completely. The CES is fully discharged by
+the end of T1 to exploit its full capacity to charge from the
+excess SPV generation during T2. This is evident as the CES
+energy level has reached its full capacity of 249 kWh during
+T2. During T3, the CES exports its power to the customers,
+and at the end of the day, CES reaches its initial energy level.
+Fig. 5: (a) Total power exchange with the grid and the CES by
+customers, (b) CES charging/discharging power and temporal
+variation of CES energy - When Ns = 10, λCES(t) = 0
+2) When λCES(t)=0: In this case, as λCES(t)=0, the
+customers neither incur a cost nor earn a revenue when trading
+energy with the CES. According to Fig. 5(a), during T1, the
+customers import power only from the CES, as the customers
+do not incur a cost for importing power from the CES. The
+same trend is followed by the customers during T3, T4 and T5.
+During T2, the customers export the excess SPV generation
+to the grid. Also, as λCES(t)=0, the customers do not export
+power to the CES as they cannot earn a revenue from the
+CES provider. Hence, when λCES(t) = 0, the customers do
+not incur a cost for all time. Instead, they earn a revenue from
+the grid which is shown in Table I as a negative value for
+fC,cost. Fig. 5(b) shows the charging and discharging pattern
+of the CES, including its energy level variation with time.
+3) When λCES(t)=λG,avg:
+In this paper, λG,avg
+=
+0.34180 AUD/kWh. Hence, during T3 λCES(t) < λG(t),
+and during T1, T2 T4, T5 λCES(t) > λG(t). As seen in Fig.
+6(a), during T1, the customers import power only from the
+grid. Since λCES(t) > λG(t) during this time period, it is not
+economically beneficial for the customers to import expensive
+power from the CES. During T2, in which the time period
+with high SPV generation, the customers export the excess
+SPV generation to the CES as λCES(t) > λG(t). Hence, the
+customers can earn a higher revenue from the CES provider.
+
+(a)
+100
+Power (kW)
+-100
+0
+10
+15
+oZ
+25
+Discharging Power (kW)
+(b)
+ CES Charging and,
+50
+CES Energy (kWh)
+200
+0
+100
+50
+0
+5
+10
+15
+25
+Time Duration (24 hours)
+ pa. (t)
+/= 1c E G
+5=1
+ pa5(t)
+/= 1cE G
+-(a)
+Power (kw)
+0
+-200
+0
+n
+10
+15
+20
+25
+Discharging Power (kW)
+(b)
+CES Charging and.
+ 50 
+ Energy (kWh)
+200
+0
+100
+50
+CES
+0
+-5
+10
+15
+20
+25
+Time Duration (24 hours)
+ pa. (t)
+5 = 1
+/= 1c E G
+5=1
+M
+Zpas(t)
+5 = 1
+/= 1cE G
+5 = 19
+TABLE II: RESULTS OF PROPOSED STOCHASTIC MODEL AND CASE I-IV - (WITH Ns = 10, λCES(t) = λG,avg)
+Case / CES Node
+Optimal CES
+Rated Power1
+pRate
+j
+(kW)
+Optimal CES
+Capacity1 ECap
+j
+(kWh)
+CES Investment
+Cost1 (AUD)
+fInv,cost
+CES Operation
+Cost1 (AUD)
+fop,cost
+Social
+Costs1 (AUD)
+fC,cost
+Cumulative Cost1
+(AUD)
+Proposed Model- 7 (optimal)
+78
+249
+33695
+29347
+75263
+138305
+Case I - 3 (chosen)
+124 (37.10%)
+380 (34.47%)
+51688 (34.81%)
+44900 (34.64%)
+77703 (3.14%)
+174291 (20.65%)
+Case II - 4 (chosen)
+98 (20.41%)
+303 (17.82%)
+41217 (18.25%)
+36723 (20.09%)
+76973 (2.22%)
+154913 (10.72%)
+Case III - 5 (chosen)
+127 (38.58%)
+366 (31.97%)
+50258 (32.96%)
+39093 (24.93%)
+77024 (2.29%)
+166375 (16.87%)
+Case IV - 6 (chosen)
+96 (18.75%)
+310 (19.68%)
+41837 (19.46%)
+35578 (17.51%)
+76491 (1.61%)
+153906 (10.14%)
+1 Increment percentage values are computed with respect to their corresponding values found in the proposed model
+Fig. 6: (a) Total power exchange with the grid and the CES by
+customers, (b) CES charging/discharging power and temporal
+variation of CES energy - When Ns = 10, λCES(t) = λG,avg
+Since λCES(t) < λG(t) during T3, the customers import
+power from the CES, so that they have to pay less for the
+imported power. During T4 and T5, the customers import
+power only from the grid as λCES(t) > λG(t). Fig. 6(b)
+shows the charging and discharging pattern, and the temporal
+variation of the CES energy level with time.
+C. Case Study II: Proposed Optimization Framework Vs Mod-
+els With Arbitrary CES Locations
+The merits of a CES may be fully exploited if its both
+planning and scheduling are optimized simultaneously. To
+test this, we compared our proposed model with four dif-
+ferent cases that randomly choose the CES location, with
+λCES(t) = λG,avg(t), Ns = 10, and taking into account the
+uncertainty of real power consumption and SPV generation.
+As given in Table II, the CES is allocated for nodes 3, 4,
+5 and 6 which represent Case I, II, III and IV, respectively.
+For Case I-IV, we considered the constraints (1)-(16), and
+the objective function in (23). Additionally, Lj = 1 where
+j ∈ {3, 4, 5, 6} in (10)-(12) for Case I-IV, respectively. In
+Case I-IV, pRate
+j
+and ECap
+j
+are significantly higher than in our
+proposed stochastic model. This has resulted in a substantial
+increase of the CES investment and operation costs. But,
+the social costs show only a minor increase for Case I-IV
+compared with the increase of the investment and operation
+costs of the CES. This can be explained in terms of the weight
+coefficients used for weighting the objective functions in (20)-
+(22). Since w1, w2 and w3 are 1/7, 1/7 and 5/7, respectively,
+the highest importance is given for minimizing the social costs.
+Hence, the optimization solver tries to maintain the social costs
+as much as close to fC,cost obtained in our proposed model.
+However, this comes at an expense as fInv,cost and fop,cost
+which have a less significance, increase significantly. But, the
+cumulative cost (i.e. sum of fInv,cost, fop,cost and fC,cost )
+is the least for the proposed model, while for Case I-IV, it is
+about 10-21% higher than the cost in our proposed model.
+D. Case Study III: Impact of Scenario Reduction Approaches
+In this section, we present a comparison of the results
+obtained for our optimization model utilizing two scenario re-
+duction methods namely, backward scenario reduction (BSR)
+method and K-Means clustering algorithm. In BSR method,
+the initial number of scenarios are reduced by minimizing the
+Monge-Kantorovich distance between the scenarios in both
+initial and reduced scenarios sets. Thereby, the initial scenarios
+are eliminated iteratively one by one, until the desired number
+of elements in the reduced scenario set is reached. Further
+explanation about the BSR method can be found in [12].
+In this case study, we considered λCES(t) = λG,avg, and
+the initial number of scenarios as 50. The proposed opti-
+mization framework was implemented under the two scenario
+reduction approaches by taking Ns = 10 and Ns = 30 for each
+method. The numerical results obtained for this case study are
+summarized in Table III. Note that, for all the cases, node 7
+was recorded as the optimal CES location.
+The costs for all the three objectives are the highest for the
+model which did not use a scenario reduction method. This is
+because, the model with Ns = 50 captures more uncertainty
+of the real power consumption and SPV generation, so that the
+optimal CES rated power and the capacity are higher than the
+models with a lesser number of scenarios. For both K-Means
+and BSR methods, even with a different number of reduced
+scenarios (i.e. Ns = 10 and Ns = 30), the CES planning
+aspects have not changed. This occurs as the planning aspects
+are independent of the number of scenarios. Nevertheless, as
+the operation decisions are scenario dependent, the operation
+cost of the CES, and the customers’ social costs have changed
+according to Ns. This trend is observed in the results obtained
+for the model which used the BSR method as well.
+
+(a)
+Power (kW)
+100
+0
+100
+-200
+0
+n
+10
+15
+20
+25
+Discharging Power (kW)
+(b)
+ CES Charging and,
+CES Energy (kWh)
+200
+0
+100
+50
+0
+5
+10
+15
+20
+Time Duration (24 hours)
+ pa. (t)
+5= 1
+/= 1c E G
+5=1
+MN
+ pa5(t)
+ws, EE5(t)
+/=1cE G
+-10
+TABLE III: RESULTS FOR DIFFERENT SCENARIO REDUCTION METHODS - (WITH λCES(t) = λG,avg)
+No. of
+scenarios
+Computational
+time1
+(min)
+Optimal CES
+Rated Power1
+pRate
+j
+(kW)
+Optimal CES
+Capacity1
+ECap
+j
+(kWh)
+CES Investment
+Cost1 (AUD)
+fInv,cost
+CES Operation
+Cost1 (AUD)
+fop,cost
+Social
+Costs1 (AUD)
+fC,cost
+Without Scenario
+Reduction
+Ns = 50
+91
+79
+257
+34573
+38434
+91321
+K-Means
+Clustering Algorithm
+Ns = 10
+38 (58.24%)
+78 (1.27%)
+249 (3.11%)
+33695 (2.54%)
+29347 (23.64%)
+75263 (17.58%)
+Ns = 30
+53 (41.76%)
+78 (1.27%)
+249 (3.11%)
+33695 (2.54%)
+30019 (21.89%)
+89425 (2.08%)
+Backward Scenario
+Reduction Method
+Ns = 10
+41 (54.95%)
+78 (1.27%)
+250 (2.72%)
+33735 (2.42%)
+29785 (22.50%)
+75709 (17.10%)
+Ns = 30
+57 (37.36%)
+78 (1.27%)
+250 2.72%)
+33735 (2.42%)
+30664 (20.22%)
+89981 (1.47%)
+1 Decrement percentage values are computed with respect to their corresponding values found without scenario reduction
+The models with reduced number of scenarios have con-
+verged for a solution in a lesser time than the case with
+Ns = 50. Scenario reduction methods like K-Means clustering
+algorithm and BSR method play a key role in reducing the
+computational time while maintaining the problem tractability.
+However, according to the results obtained for the models
+which used the K-Means and the BSR method, it is not
+conclusive to claim which scenario reduction method is better,
+as there is no any significant difference between the results.
+VII. CONCLUSION & FUTURE WORK
+In this paper, we have explored how the optimization of the
+planning and scheduling of a community energy storage (CES)
+benefit the CES provider by minimizing the CES investment
+and operation costs, and the customers by minimizing their
+social costs. The uncertainty of real power consumption and
+solar photovoltaic (SPV) generation of the customers have
+been accounted to formulate a scenario-based stochastic op-
+timization program. To reduce the computational burden of
+the stochastic model, we have used the K-Means clustering
+algorithm. It has been shown that, both the customers and the
+CES provider can significantly minimize their personal costs
+by optimizing both the CES planning and its scheduling.
+Future work includes extending the proposed model for
+unbalanced distribution networks, developing optimization
+models for networks with multiple CES, and considering the
+reactive power regulation capabilities of CES and SPV.
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+

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+arXiv:2301.11966v1  [quant-ph]  27 Jan 2023
+Generalized Uncertainty Principle for Entangled
+States of Two Identical Particles
+K. C. Lemos Filho∗1, B. B. Dilem†2, J. C. Fabris‡1,3, and J. A.Nogueira§1
+1Universidade Federal do Esp´ırito Santo – Ufes, Vit´oria, Esp´ırito Santo,
+29.075-910, Brasil
+2Instituto Federal do Esp´ırito Santo – Ifes, Alegre, Esp´ırito Santo,
+29.520-000, Brasil
+3National Research Nuclear University MEPhI, Kashirskoe sh. 31,
+Moscow 115409, Russia
+Abstract
+In this work we determine the consequences of the quantum en-
+tanglement of a system of two identical particles when the generalized
+uncertainty principle (GUP) is considered. GUP is usually associated
+with the existence of a minimal length. We focus on the main formu-
+lations of the GUP and then we determine the minimal uncertainties in
+position induced by those modified GUP’s. Our results point out that
+the minimal uncertainty is reduced by half of its usual value indepen-
+dently of the GUP employed. This implies that the minimal length is
+also reduced by half. On the other hand, it is generally expected that the
+minimal length must not depend on physical system. We overcome this
+apparent paradox by realizing that the entangled system is composed by
+two particles so that an effective parameter related to the minimal length
+must be employed.
+PACS numbers: 04.60.-m, 03.65.Ud, 03.65.Ta
+Keywords: Minimal length; generalized uncertainty principle; quantum entanglement.
+∗kim.vasco@gmail.com
+†bernardob@ifes.edu.br
+‡julio.fabris@cosmo-ufes.org
+§jose.nogueira@ufes.br
+1
+
+1
+Introduction
+Amongst the (many) new concepts introduced by the quantum mechanics, the quantum
+entanglement [1, 2] is one that, probably, more contradicts our common sense. Although,
+in the beginning, the quantum entanglement were only associated to theoretical aspects
+of quantum mechanics, specially those related to the non-locality or the complementarity
+(hidden variables) [3], nowadays it is a key component of the applications and experiments
+on quantum information, quantum computation and quantum teleportation [4, 5].
+The uncertainty principle is one of the fundamental cornerstones of the quantum me-
+chanics. Nevertheless, it is a principle: its correct form can not be proven. That opens
+the way to the possibility that its canonical form, described by the Heisenberg’s uncer-
+tainty principle (HUP), can be generalized. An example of the possible generalizations
+of HUP, whose origin can be traced back to quantum gravity, is given by introducing a
+non-zero minimal uncertainty in the measurement of position. That non-zero minimal
+uncertainty in position is, then, understood as a minimal-length scale, below which the
+necessary amount of energy to probe the position of a particle is so hight that it disturbs
+the space-time so that the concept of a length measurement loses its meaning. Hence, the-
+ories searching to describe a quantum approach for gravity lead generally to the existence
+of a minimal length. In fact, a minimal length actually appears in almost all proposed
+theories of the quantum gravity. For this reason theories formulated in a minimal-length
+scenario are considered to be effective theories of quantum gravity [6, 7, 8, 9].
+In 1999, Yoon-Ho Kim and Yanhua Shih conducted an experiment whose results ap-
+parently suggested a violation of the HUP [10]. However, G. Rigolin pointed out that
+in fact there is no violation of the HUP, because the HUP is derived for particles non-
+correlated (non-entangled). In the Kim and Shih’s experiment, the photons of the pair
+are correlated (entangled) when one of the physical slits is replaced by a virtual slit1:
+the canonical HUP is no longer applicable since the quantum entanglement modifies the
+canonical HUP [11, 12].
+An immediate question arising from the previous considerations is how the quantum
+entanglement modifies a generalized uncertainty principle (GUP), in others words, which
+is the effect of the quantum entanglement in the minimal-length scale. The answer for
+that question is important in order to know the role of the quantum entanglement at
+the minimal-length scale (maybe appearing in the Planck scale) or in the early Universe.
+Unfortunately, this issue has been little considered in the literature. In [13], G. Blado,
+F. Herrera and J. Erwin have studied the inseparability conditions with the most usual
+GUP correction, whereas D. Park has used a coupled harmonic oscillator in order to find
+the effects of the quantum entanglement with a linear GUP in [14].
+The purpose of this work is to answer that question considering the main proposals of
+generalization for the HUP which take into account the existence of a minimal length in
+nature. With this goal in mind, we will analyze the modifications in the HUP arising from
+the quantum entanglement of two identical particles determining the minimal uncertainty
+1The interaction of the photon with a physical slit destroys the correlation between the photons of the
+pair.
+2
+
+associated to them.
+The outline of this paper is as follows. In Section 2 we obtain an expression of the
+uncertainty principle for entangled states which is independent of the chosen GUP. In
+Section 3 we find the modified uncertainty principle for a pair of identical particles re-
+garding the main proposals of GUP’s: Kempf, Mangano and Mann GUP (KMM-GUP),
+Ali, Das and Vagenas GUP (ADV-GUP), Pedram GUP and exponential all orders GUP.
+In Section 4 we estimate an upper bound for the minimal-length value. We present our
+conclusions in the Section 5.
+2
+Uncertainty principle for entangled states
+The Hilbert space of the state vectors E of a system of N particles is given by the
+tensor product of the Hilbert spaces of the state vectors Ei of each particle [15, 16],
+E = E1 ⊗ · · · ⊗ EN.
+(1)
+The position and momentum linear operators of the i-th particle which act on the state
+vectors |ψ⟩ ∈ E are the extensions ˜Qi and ˜Pi defined as
+˜Qi = I1 ⊗ · · · ⊗ ˆxi ⊗ · · · ⊗ IN,
+(2)
+˜Pi = I1 ⊗ · · · ⊗ ˆpi ⊗ · · · ⊗ IN,
+(3)
+where Ii is the identity operator in Ei and ˆxi and ˆpi are the position and the momentum
+operators of the i-th particle acting on the state vectors |ψi⟩ ∈ Ei.
+The extensions ˜Qi and ˜Pi do not satisfy the canonical uncertainty principle (HUP),
+because ˜Qi and ˜Pi are not physical observables [11, 12, 15, 16].
+Physical observables
+are operators which commute with every permutation operators of the particles system.
+Hence, the operators ˜Q and ˜P, defined as
+˜Q :=
+N
+�
+i=1
+˜Qi
+(4)
+and
+˜P :=
+N
+�
+i=1
+˜Pi,
+(5)
+are physical observables and they satisfy the relation
+(∆Q)2 (∆P)2 ≥ 1
+4
+���⟨[ ˜Q, ˜P]⟩
+���
+2
+.
+(6)
+The relation (6) is general. It does not depend whether the system of particle is entangled
+or not.
+3
+
+As it was showed by G. Rigolin [11, 12], if the state of the particles system is entangled
+then the operators ˜Qi and ˜Pi do not satisfy the canonical Heisenberg uncertainty principle
+(HUP) - as previously stated.
+We briefly review the Rigolin’s result for a two particles system. From the definitions
+of ∆Q and ∆P we have2
+(∆ψQ)2 =
+�
+ψ| ˜Q2|ψ
+�
+−
+�
+ψ| ˜Q|ψ
+�2
+.
+(7)
+From now on we omit the subscript ψ for the sake of simplicity, whenever this does not
+cause any confusion. Thus,
+(∆Q)2 = (∆Q1)2 + (∆Q2)2 + 2
+��
+˜Q1 ˜Q2
+�
+−
+�
+˜Q1
+� �
+˜Q2
+��
+.
+(8)
+In the same way,
+(∆P)2 = (∆P1)2 + (∆P2)2 + 2
+��
+˜P1 ˜P2
+�
+−
+�
+˜P1
+� �
+˜P2
+��
+.
+(9)
+Using the results (8) and (9) into Eq. (6) we obtain
+�
+(∆Q1)2 + (∆Q2)2 + 2
+��
+˜Q1 ˜Q2
+�
+−
+�
+˜Q1
+� �
+˜Q2
+���
+×
+�
+(∆P1)2 + (∆P2)2 + 2
+��
+˜P1 ˜P2
+�
+−
+�
+˜P1
+� �
+˜P2
+���
+≥ 1
+4
+���⟨[ ˜Q, ˜P]⟩
+���
+2
+.
+(10)
+We now use the functions
+CQ(1, 2)
+:=
+�
+˜Q1 ˜Q2
+�
+−
+�
+˜Q1
+� �
+˜Q2
+�
+,
+(11)
+CP(1, 2)
+:=
+�
+˜P1 ˜P2
+�
+−
+�
+˜P1
+� �
+˜P2
+�
+,
+(12)
+which are called quantum covariance functions (QCF). By definition, QCF’s vanish if and
+only if the system is separable [17]. Therefore (11) and (12) are zero for any not entangled
+quantum system.
+Using the QCF’s (11) and (12) we have
+�
+(∆Q1)2 + (∆Q2)2 + 2CQ(1, 2)
+� �
+(∆P1)2 + (∆P2)2 + 2CP(1, 2)
+�
+≥ 1
+4
+���⟨[ ˜Q, ˜P]⟩
+���
+2
+,
+(13)
+or
+2
+�
+i,j=1
+CQ(i, j)
+2
+�
+k,l=1
+CP(k, l) ≥ 1
+4
+���⟨[ ˜Q, ˜P]⟩
+���
+2
+,
+(14)
+since CQ(i, i) = (∆Qi)2, CP(i, i) = (∆Pi)2, CQ(i, j) = CQ(j, i) and CP(i, j) = CP(j, i).
+2Note that [ ˜Q1, ˜Q2] = 0 and [ ˜P1, ˜P2] = 0.
+4
+
+In this work, we concern with the case of an entangled system of two identical particles,
+so we are going to handle Eq. (13) in order to express it in a more appropriate way. For
+this end, we define
+|ψ′⟩ :=
+�
+˜Q1 − ˜Q2
+�
+|ψ⟩,
+(15)
+with |ψ′⟩, |ψ⟩ ∈ E and ⟨ψ | ψ⟩ = 1. Therefore,
+⟨ψ′ | ψ′⟩ = (∆ψQ1)2 + (∆ψQ2)2 − 2
+�
+˜Q1 ˜Q2
+�
+ψ +
+�
+˜Q1
+�2
+ψ +
+�
+˜Q2
+�2
+ψ .
+(16)
+Now, using the Schwarz inequality, ⟨ψ | ψ⟩ ⟨ψ′ | ψ′⟩ ≥ ⟨ψ | ψ′⟩ ⟨ψ′ | ψ⟩, we have
+(∆ψQ1)2 + (∆ψQ2)2 ≥ 2
+��
+˜Q1 ˜Q2
+�
+ψ −
+�
+˜Q1
+�
+ψ
+�
+˜Q2
+�
+ψ
+�
+.
+(17)
+In the same way
+(∆ψP1)2 + (∆ψP2)2 ≥ 2
+��
+˜P1 ˜P2
+�
+ψ −
+�
+˜P1
+�
+ψ
+�
+˜P2
+�
+ψ
+�
+.
+(18)
+Finally, from inequalities (17), (18) and (13) we obtain
+�
+(∆Q1)2 + (∆Q2)2� �
+(∆P1)2 + (∆P2)2�
+≥ 1
+16
+���⟨[ ˜Q, ˜P]⟩
+���
+2
+.
+(19)
+In the case where (∆Q1)2 = (∆Q2)2 and (∆P1)2 = (∆P2)2 the inequality (19) becomes
+∆Qi∆Pi ≥ 1
+8
+���⟨[ ˜Q, ˜P]⟩
+��� .
+(20)
+It is worth noting that the expression of the inequality (20) is independent of the
+chosen uncertainty principle that does not take into account the quantum correlation.
+This uncertainty principle is related to the commutation relation [ ˜Q, ˜P].
+3
+Uncertainty principle for entangled states in different minimal-
+length scenarios
+In this section we consider a system of two entangled identical particles whose momenta
+have the same value but opposite directions, that is, ⃗p1 = −⃗p2, just as in the Kim
+and Shih’s experiment [10]. Therefore, in this case ⟨ˆp1⟩ + ⟨ˆp2⟩ = 0. Moreover, such a
+consideration also allows us to estimate, in the next section, an upper bound for the value
+of the minimal length based on the experimental results obtained by Kim and Shih.
+5
+
+3.1
+Heisenberg uncertainty principle
+Before we consider a minimal-length scenario it is appropriate to determine the change
+in the canonical HUP, that is, in a scenario in which effects of quantum gravity are not
+present. The canonical HUP for states of one simple-particle is
+∆x∆p ≥ ℏ
+2.
+(21)
+The commutation relation related to the HUP is
+[ˆx, ˆp] = iℏ.
+(22)
+Hence
+[ ˜Q, ˜P] = [ ˜Q1 + ˜Q2, ˜P1 + ˜P2] = 2iℏ.
+(23)
+Substituting Eq. (23) into Eq. (20) we get
+∆Qi∆Pi ≥ ℏ
+4.
+(24)
+The result (24) shows that for a system of two entangled identical particles the HUP is
+modified. Such an outcome is not new, it was already obtained by G. Rigolin in 2002 [11]
+and then in 2016 [12].
+From a quick glance at the result (24) and recalling that dimensionally (∆Q)min ∝ ℏ,
+we expect the minimal uncertainty in the position will be reduced by half for all GUP’s.
+3.2
+KMM GUP
+The GUP
+∆xi∆pi ≥ ℏ
+2
+�
+1 + β (∆pi)2 + β ⟨ˆpi⟩2�
+,
+(25)
+where β is a parameter related to the minimal length, has been proposed by A. Kempf,
+G. Mangano an R. B. Mann (KMM-GUP) [18] and it is the most used in the literature.
+The commutation relation related to it is given by
+[ˆxi, ˆpi] = iℏ
+�
+1 + βˆp2
+i
+�
+.
+(26)
+Hence
+[ ˜Q, ˜P] = iℏ
+�
+1 + β
+�
+˜P 2
+1 + ˜P 2
+2
+��
+= iℏ
+�
+1 + 2β ˜P 2
+i
+�
+.
+(27)
+Substituting Eq. (27) into Eq. (20) we get
+∆Qi∆Pi ≥ ℏ
+4
+�
+1 + β (∆Pi)2 + γ
+�
+,
+(28)
+where γ := β
+�
+˜Pi
+�2
+.
+6
+
+The modified KMM-GUP (28) induces the existence of a minimal uncertainty given
+by
+(∆Qi)min = ℏ
+2
+�
+β.
+(29)
+The result above shows that the non-zero minimal uncertainty in position induced by
+the KMM-GUP for two entangled identical particles is twice smaller than for a separable
+system of two identical particles (non-entangled).
+3.3
+ADV GUP
+A. Farag Ali, S. Das and E. C. Vagenas have proposed a GUP related to a commutation
+relation which has a linear and a quadratic term in the momentum operator [19],
+[ˆxi, ˆpi] = iℏ
+�
+1 − 2αˆpi + 4α2ˆp2
+i
+�
+,
+(30)
+where α is a parameter related to the minimal length. Besides the existence of a minimal
+length this linear approach induces a maximal uncertainty in the momentum, too. Then,
+from Eq. (30) we get
+[ ˜Q, ˜P] = 2iℏ
+�
+1 − α
+�
+˜P1 + ˜P2
+�
++ 2α2 �
+˜P 2
+1 + ˜P 2
+2
+��
+.
+(31)
+Therefore,
+⟨[ ˜Q, ˜P]⟩ = 2iℏ
+�
+1 − α
+��
+˜P1
+�
++
+�
+˜P2
+��
++ 2α2 ��
+˜P 2
+1
+�
++
+�
+˜P 2
+2
+���
+,
+(32)
+⟨[ ˜Q, ˜P]⟩ = 2iℏ
+�
+1 + 4α2 �
+˜P 2
+i
+��
+,
+(33)
+Substituting Eq. (33) into Eq. (20) we obtain
+∆Qi∆Pi ≥ ℏ
+4
+�
+1 + 4α2 (∆Pi)2 + γ′
+�
+,
+(34)
+where γ′ := 4α2 �
+˜Pi
+�2
+.
+The modified ADV-GUP (34) induces the existence of a non-zero minimal uncertainty
+given by
+(∆Qi)min = ℏα,
+(35)
+which once again is twice smaller than for a non-entangled system of two particles.
+It is important to note that the linear GUP (ADV-GUP) becomes non-linear in this
+case and consequently a maximal uncertainty in the momentum is no longer induced.
+7
+
+3.4
+Pedram GUP
+In order to overcome some problems arising from KMM-GUP and ADV-GUP - such
+as incorporation of a maximal momentum required in doubly special relativity (DSR)
+theories, commutative geometry and an approach valid for all order in the parameter
+related to the minimal length - P. Pedram has proposed a GUP [20, 21] based on the
+commutation relation given by
+[ˆxi, ˆpi] =
+iℏ
+1 − βˆp2
+i
+.
+(36)
+Hence
+[ ˜Q, ˜P] =
+iℏ
+1 − β ˜P 2
+1
++
+iℏ
+1 − β ˜P 2
+2
+.
+(37)
+Using the so-called Jensen’s Inequality [22] we have
+⟨[ ˜Q, ˜P]⟩ ≥
+iℏ
+1 − β
+�
+˜P 2
+1
+� +
+iℏ
+1 − β
+�
+˜P 2
+2
+�
+.
+(38)
+Substituting Eq. (38) into Eq. (20) we obtain
+∆Qi∆Pi ≥ ℏ
+4
+1
+�
+1 − β (∆Pi)2 − γ
+�.
+(39)
+Therefore, the modified Pedram-GUP (39) introduces a non-zero minimal uncertainty
+given by
+(∆Qi)min = 3ℏ
+8
+�
+3β,
+(40)
+which is also twice smaller than for a non-entangled system of a pair of identical particles.
+3.5
+Exponential all orders GUP
+In the canonical field theory in the context of non-commutative coherent states repre-
+sentation and field theory on non-anticommutative superspace the Feynman propagator
+displays an ultra-violet (UV) cut-off of the form e−βp2 [23, 24, 25, 26]. In consequence, K.
+Nouicer has proposed an exponential all orders GUP [27, 28] based on the commutation
+relation given by
+[ˆxi, ˆpi] = iℏeβˆp2
+i .
+(41)
+Hence
+[ ˜Q, ˜P] = iℏ
+�
+eβ ˜P 2
+1 + eβ ˜P 2
+2
+�
+.
+(42)
+Again, using the so-called Jensen’s Inequality [22] we have
+⟨[ ˜Q, ˜P]⟩ ≥ iℏ
+�
+eβ⟨ ˜P 2
+1⟩ + eβ⟨ ˜P 2
+2⟩�
+.
+(43)
+8
+
+Substituting Eq. (42) into Eq. (20) we get
+∆Qi∆Pi ≥ ℏ
+4eβ(∆Pi)2+γ.
+(44)
+Therefore, the modified exponential all order GUP (44) introduces a minimal uncer-
+tainty given by
+(∆Qi)min = ℏ
+2
+�
+eβ
+2 .
+(45)
+Finally, we note that (∆Qi)min is also twice smaller than for a non-entangled system of a
+pair of identical particles, as we have already said.
+3.6
+Minimal length
+If a non-zero minimal uncertainty in position can be interpreted as a minimal length then
+the previous results show that the minimal length for two entangled identical particles
+can be twice smaller than for a separable system. It is clear this statement must be taken
+with care because a minimal length should be a constant, that is, it must not depend on
+the physical system, it is a quantum gravitation effect. In fact, the minimal length should
+be an invariant as well as the light speed is. The answer for that apparent contradiction
+is possibly because the system is made up of two particles. In references [29], C. Quesne
+and V. M. Tkachuck have claimed that for a system composed by N particles the effective
+parameter related to the minimal length, β, is reduced by a factor
+1
+N2. Hence, ℏ
+2
+√β is not
+the correct minimal uncertainty in position (∆Qi)min for the modified KMM-GUP, but
+ℏ
+2
+√βi, where [29, 30]
+β = βi
+22.
+(46)
+Consequently, we find that (∆Qi)min = ℏ√β, therefore lmin = ℏ√β, as we expected3. In
+the same way for others GUP’s.
+The reader might want to claim that the minimal length is actually described by the
+minimal uncertainty of the entangled system. However, according to reference [12] we
+can presume that the minimal uncertainty for a system of N entangled identical particles
+is reduced by
+1
+N . Since, in principle, there is no limit for the number of particles for a
+entangled system, there would also be no limit for the minimal length.
+4
+Upper bound for the minimal-length value
+In the Kim and Shih’s experiment entangled identical pairs of photons were produced
+by spontaneous parametric down conversion (SPDC) with momentum conservation. A
+narrow physical slit was placed along the trajectory of one of the photons, whereas the
+3It is worth noting that if the minimal uncertainty was greater for the entangled system, we could sup-
+pose that the quantum entanglement decreases the accuracy of a position measurement thereby increasing
+the minimal uncertainty.
+9
+
+other photon of the pair (called 2) passed through a virtual slit. The ghost image exper-
+imental technique [31] ensured that the quantum correlation between the pair of photons
+was not destroyed. Then simultaneous detection of photons of the pairs were performed
+and data just for coincidence events were obtained in case when the photon 2 passed
+through a virtual slit (non-slit case) and in case when the photon 2 passed through a
+physical slit (slit case).
+From the experimental data obtained by Kim and Shih one gets that [12]
+∆P ns
+2
+∆P s
+2
+= 1.25
+2.15,
+(47)
+where ∆P ns
+2
+is uncertainty in momentum of the photon 2 in the non-slit case and ∆P s
+2 is
+uncertainty in momentum of the photon 2 in slit case. Since the width of the slit was 0.16
+mm (∆Q2 = 0.16 mm), the uncertainty in momentum in the slit case for KMM-GUP can
+be find from4
+0.16∆P s
+2 = ℏ
+2
+�
+1 + 4β (∆P s
+i )2 + γ
+�
+.
+(48)
+Eq. (48) has real roots only if β ≤ 0.162
+ℏ2η , where η := 1 + γ. Thus,
+∆P s
+2+ = 0.32
+ℏβ − ℏη
+0.32 − β ℏ3η2
+0.323
+(49)
+and
+∆P s
+2− = ℏη
+0.32 + β ℏ3η2
+0.323,
+(50)
+Now, using the above results we can estimate an upper bound for the minimal-length
+value induced by KMM-GUP. Hence, substituting the root (49) into Eq. (28) we have
+that
+β ≤ 3.58 × 10−2
+ℏ2η
+.
+(51)
+Therefore,
+lmin ≤ ℏ
+�
+3.58 × 10−2
+ℏ2η
+≤ ℏ
+�
+3.58 × 10−2
+ℏ2
+= 1.9 × 10−4m.
+(52)
+The substitution of the root (50) hold the inequality (28) for all β > 0.
+Consequently, the upper bound for the minimal-length value is order 10−4 m. There-
+fore, using the experiment described above a result with less restrictions than those re-
+ported in the literature is obtained [30, 32, 33, 34, 35, 36, 37, 38].
+4Note that in with-slit case the correlation between the photons of the pair was destroyed, therefore
+the photons were not entangled and the usual KMM-GUP, Eq. (25), is held.
+10
+
+5
+Conclusion
+In this work we find the non-zero minimal uncertainties induced by the main proposals
+of GUP’s (KMM, ADV, Pedram and exponential) which are modified due to the quantum
+entanglement of a system of two identical particles. In principle, our results have pointed
+out that the minimal uncertainties are reduced at half for a system of two entangled
+identical particles independently of the GUP. Hence, if a non-zero minimal uncertainty in
+position can be interpreted as a minimal length then the quantum entanglement reduces
+by half the minimal length. However, the minimal length must not depend on the phys-
+ical system. We overcome this apparent paradox by using the Quesne and Tkachuck’s
+proposal for a system composed. Consequently, despite the quantum entanglement to
+change the GUP, the minimal length does not change. Based on our results and using
+the reference [12] we can expect that the apparent minimal uncertainty for a system of
+N entangled identical particles is reduced by
+1
+N , nonetheless the minimal length does not
+change because the effective parameter β is also reduced by a factor
+1
+N2.
+Finally, we have estimated from the data obtained from the Kim and Shih’s experiment
+an upper bound value for the minimal length of the order of 10−4 m. Consequently, it is
+rather an inexpressive value (in the sense of leading to poor predictive power) as compared
+to ones have been found in the literature. This is due to the high imprecision of the
+experiment on the entangled system described above. However, we may expect that more
+refined version of the experiment may lead to more stringent bounds on the minimum
+length.
+Acknowledgements
+We would like to thank FAPES, CAPES and CNPq (Brazil) for financial support.
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+

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+1
+Diffusion Models For Stronger Face Morphing
+Attacks
+Zander Blasingame and Chen Liu
+Department of Electrical and Computer Engineering
+Clarkson University, Potsdam, New York, USA
+{blasinzw; cliu}@clarkson.edu
+Abstract—Face morphing attacks seek to deceive a Face Recognition (FR) system by presenting a morphed image consisting of the
+biometric qualities from two different identities with the aim of triggering a false acceptance with one of the two identities, thereby
+presenting a significant threat to biometric systems. The success of a morphing attack is dependent on the ability of the morphed
+image to represent the biometric characteristics of both identities that were used to create the image. We present a novel morphing
+attack that uses a Diffusion-based architecture to improve the visual fidelity of the image and improve the ability of the morphing attack
+to represent characteristics from both identities. We demonstrate the high fidelity of the proposed attack by evaluating its visual fidelity
+via the Fr´echet Inception Distance. Extensive experiments are conducted to measure the vulnerability of FR systems to the proposed
+attack. The proposed attack is compared to two state-of-the-art GAN-based morphing attacks along with two Landmark-based attacks.
+The ability of a morphing attack detector to detect the proposed attack is measured and compared against the other attacks.
+Additionally, a novel metric to measure the relative strength between morphing attacks is introduced and evaluated.
+Index Terms—Biometrics, Morphing Attack, GAN, Vulnerability Analysis, Face Recognition, Diffusion Models
+!
+1
+INTRODUCTION
+F
+ACE recognition (FR) systems have become one of the most
+common biometric modalities used for identity verification
+across a wide range of modern-day applications, from trivial tasks
+such as unlocking a smart phone to official businesses such as
+banking, e-commerce, and law enforcement. Unfortunately, while
+FR systems can often reach low false rejection and acceptance
+rates [1], they are especially vulnerable to a new class of emerging
+attacks, known as the face morphing attack [2], [3], [4], [5]. The
+face morphing attack aims to compromise a fundamental property
+of biometric security, i.e., the one-to-one mapping from biometric
+data to the associated identity. This compromise is achieved
+by creating a morphed face which contains biometric data of
+both identities in such a manner that presenting one morphed
+image triggers a match with two disjoint identities, violating the
+fundamental principle.
+This poses a significant threat towards FR systems, especially
+the applications of e-passports and border access. Notably the e-
+passport scenario, wherein the applicant submits a passport photo
+either in digital or printed form, is especially vulnerable to face
+morphing attack. Moreover, digital morphs can be easily generated
+hence offer a low-cost attack in the digital domain. It is not un-
+common for a digital image submitted to a document submission
+portal to not have its authenticity verified by a human agent [6].
+Critically, an adversary who is blacklisted from accessing a certain
+system can create a morph with a non-blacklisted individual to
+gain access. This is especially relevant for countries such as New
+Zealand, Estonia, and Ireland, where e-passports are used for both
+issuance and renewal of documents [7]. In 2018 a German activist
+was reported to have received a German passport with a photo of
+his faced morphed with an Italian politician [8].
+Due to the severity of face morphing attacks, an abundance of
+algorithms have been developed to identify these attacks [2], [9],
+[10], [11]. Methods for Morphing Attack Detection (MAD) can
+be broadly characterized into two classes based on the manner in
+which they obtain the features used for detection, i.e., handcrafted
+features or deep features. Handcrafted features are used in the
+so-called classical algorithms which seek to find evidence of the
+morphing attack in the pixel domain, whether that is evidence
+of degradation in image quality [10], residual noise left from
+the morphing attack [12], or local geometric features such as
+Local Binary Patterns (LBP) [13], Binarized Statisical Image
+Feature (BSIF) [14], and Local Phase Quantitization (LPQ) [4].
+Conversely, deep features are used with deep learning based algo-
+rithms. These features are often extracted by a deep Convolutional
+Neural Network (CNN) often from a pre-trained network such as
+ResNet150 [6], [15]. Generally, the best success has been found
+using deep CNN-based features in comparison to handcrafted
+features on both digital and print-scan face data [16].
+Comparatively, there has been less research on face mor-
+phing attack algorithms. Similar to the two classes of MAD
+algorithms, there exist two broad classes of face morphing at-
+tacks: Landmark-based attacks and deep learning-based attacks.
+Landmark-based morphing attacks use local features to create the
+morphed image by warping and aligning the landmarks within
+each face to then create a morphed face by pixel-wise compositing.
+Landmark-based attacks have been shown to be effective against
+FR systems [17]. Recent work has enhanced the effectiveness of
+Landmark-based attacks by using adversarial perturbation [18].
+In contrast, deep learning-based morphing attacks use a machine
+learning model to embed the original bona fide faces into a
+semantic representation which are then combined to produce a
+new representation which should contain information from both
+identities. This is new representation is then used to generate a
+morphed face.
+However, nearly all state-of-the-art deep learning based morph
+methods were based on the Generative Adversarial Network
+arXiv:2301.04218v1  [cs.CV]  10 Jan 2023
+
+2
+(a) Identity a
+(b) Morphed image
+(c) Identity b
+Fig. 1: Diffusion-based morphing attack. Samples are from FRLL dataset.
+(GAN) framework [19], [20] with the primary difference being
+architectural improvements such as using the StyleGAN2 [21]
+architecture over the vanilla GAN architecture or changes to
+the morph generation pipeline as seen in the Morphing through
+Identity Driven Prior GAN (MIPGAN) [22]. At the same time,
+there exists a handful of alternative state-of-the-art deep generative
+models which offer their own advantages in terms of visual
+fidelity, semantic representation capabilities, and inference speed.
+In particular, a class of generative models collectively known
+as “Diffusion models” haven been shown to possess high visual
+fidelity, beating even state-of-the-art GANs in visual fidelity [23],
+at the cost of increased inference time. As visual fidelity and
+semantic representation abilities are far more important for the
+potency of a potential morphing attack than inference speed, we
+present a novel methodology for generating strong face morphing
+attacks by leveraging Diffusion-based methods. Figure 1 shows a
+morphed image generated by the proposed attack constructed from
+two identities from the FRLL dataset [24]. We present a summary
+of contributions of the proposed work as shown below:
+•
+We propose a novel method for generating morphed faces
+by using a Diffusion-based model which calculates twin
+embeddings, one for the semantic details and one for
+the stochastic details, to generate images of high visual
+fidelity.
+•
+The proposed morphing attack is evaluated against four
+other morphing attacks by the Fr´echet Inception Distance,
+a quantitative measure of visual fidelity.
+•
+We evaluated our proposed attack against four other mor-
+phing attacks with extensive experiments assessing the
+vulnerability of two FR systems across three different
+datasets.
+•
+The proposed attack is further evaluated its ability to
+evade detection from MAD algorithms trained against
+other morphing attacks.
+•
+We introduce and evaluate a novel metric to measure the
+relative strength from one morphing attack to another.
+•
+A small study on the impact of the pre-processing in
+the FR pipeline on the vulnerability of FR systems to
+morphing attacks is presented.
+•
+Four variants of the proposed morphing attack are com-
+pared against each other, resulting in the creation of over
+10,000 new morphed images between them.
+2
+PRIOR WORK
+Several face morphing attacks have been developed by researchers
+with the morph generating process generally using face land-
+marks or deep learning. In particular we compare our proposed
+Diffusion-based attack against four state-of-the-art morphing at-
+tacks, two deep learning-based and two Landmark-based; namely,
+the OpenCV, FaceMorpher, StyleGAN2, and MIPGAN-II face
+morphing attacks. These were chosen as they represent several
+types of morphing attacks and provide a substantial baseline to
+measure the performance of the proposed Diffusion-based attack.
+To the best of our knowledge all deep learning-based attacks
+generate the morphed images via a type of GAN architecture [17],
+[19], [22], [25].
+2.1
+Landmark-Based Morphing Attacks
+The FaceMorpher and OpenCV attacks were chosen to represent
+the Landmark-based attacks as they are commonly used to repre-
+sent this class of attack [11], [17], [26].
+FaceMorpher is an open-source algorithm that uses the
+STASM landmark detector [27], [28]. From the landmarks on the
+images Delaunay triangles are formed, which are then warped and
+blended together. The areas outside the landmarks are averaged,
+typically introducing strong artifacts in the neck and hair regions
+of the image [29].
+OpenCV is a morphing attack which uses the open-source
+OpenCV library with a 68-point annotator from the Dlib li-
+brary [30]. The images and associated landmarks are used to form
+Delaunay triangles. Then, in a similar manner to FaceMorpher,
+the landmarks are warped and blended. In contrast to the approach
+of FaceMorpher, the areas outside the landmarks do not consist
+of an averaged image, but rather additional Delaunay triangles.
+However, these morphs also exhibit strong artifacts outside the
+facial area due to the missing landmarks [29].
+2.2
+GAN-Based Morphing Attacks
+As mentioned earlier, prior deep learning-based attacks have used
+a GAN architecture for the morph generation process [26], [31].
+GANs are a type of deep generative model which seeks to learn
+the sampling process for some data distribution Pdata on X , i.e.,
+given some simple distribution Z ∼ p(z) on Z, the generator G :
+Z → X is to learn G(Z) ∼ Pdata. A discriminator, sometimes
+
+3
+Fig. 2: Proposed architecture for Diffusion-based morphs, where the green traces indicate variables associated with identity a, likewise
+red traces denote identity b, and blue traces for the morphed identity ab.
+called the critic, D : X → [0, 1] is trained adversarially against
+the generator in a minimax game described by
+min
+G max
+D
+E
+x∼Pdata log D(x) +
+E
+z∼p(z) log(1 − (D ◦ G)(z)) (1)
+where the discriminator attempts to get better at distinguishing
+synthetic samples from genuine samples, while the generator tries
+to get better at deceiving the discriminator. For a GAN-based mor-
+phing attack it is necessary for an encoding algorithm E : X → Z
+to exist, which can embed images in the latent space such that the
+inversion has low distortion (G ◦ E)(x) ≈ x. The latent codes
+for two identities are then averaged to produce a new latent code
+representing the morphed face which is passed to the generator.
+Notably, there exists a trade-off between the inversion distortion
+and editability of the latent embeddings [32] Damer et al. proposed
+to use the GAN architecture for generating morphs by combining
+two latent codes encoded from two real identities to create a
+morphed code [19]. This proposed attack, known as MorGAN,
+was based on a modification to the vanilla GAN architecture
+by the addition of an explicitly defined encoder architecture that
+was trained jointly with the generator via a modified adversarial
+loss formulation. Since then the StyleGAN2 [17] and MIPGAN-
+II [22] attacks have improved upon the MorGAN formulation by
+improving the GAN architecture, loss formulation, or encoding
+algorithm.
+StyleGAN2 morphs offer promising results as shown in Damer
+et al. [19]. StyleGAN2 offers a host of improvements over the
+standard GAN implementation that enables the architecture to
+achieve state of the art image quality when generating high resolu-
+tion images. The StyleGAN2 model was pre-trained on the Flickr-
+Faces-HQ (FFHQ) dataset [33]. The faces were then cropped to
+possess the same landmark alignment as in the FFHQ dataset.
+Following the approach in [21], the images, xa, xb, are embedded
+by optimizing an initial latent code through stochastic gradient
+descent, minimizing the perceptual loss between the generated
+image and target image. After each embedded latent code, za, zb,
+is found, a morphed latent code is created by linearly interpolating
+between the two, zab = lerp(za, zb; 0.5). Lastly, the interpolated
+latent code is passed to the StyleGAN2 synthesis network to get
+the morphed image xab. The StyleGAN2 morphs are strong when
+used with images containing a uniform background, which makes
+them especially powerful when used in conjunction with the Face
+Research Lab London (FRLL) dataset [24].
+Fig. 3: The forward and reverse Diffusion processes.
+MIPGAN-II proposes an extension on StyleGAN2 by adding
+an optimization procedure for the latent vector used in creating
+the morphed image [22]. The StyleGAN2 portion of MIPGAN-
+II was pre-trained on the FFHQ dataset. The two bona fide
+images are embedded into the latent space using the StyleGAN2
+optimization procedure. The latent code is initially constructed
+as z0 = lerp(za, zb; 0.5). For n epochs the latent code is
+optimized to minimize a combination of perceptual loss, identity
+loss, identity difference loss, and Multi-Scale Structural Similarity
+loss, finding a fully optimized latent zn. The latent code zn is then
+passed to the StyleGAN2 synthesis network to create the morphed
+image. As MIPGAN-II presents a refinement on StyleGAN2 for
+the application of morphing attack, it possesses similar advantages
+and disadvantages that StyleGAN2 morphs offer.
+3
+DIFFUSION-BASED MORPHING ATTACK
+Unlike GANs which learn the sampling process for the data
+distribution through adversarial training between the generator
+and the critic, diffusion-based and score-based generative models
+learn the data distribution by learning a denoising process for
+varying noise levels. Diffusion-based models can achieve image
+fidelity superior to state-of-the-art generative models, matching
+even the acclaimed BigGAN-deep [34] model, while maintaining
+better coverage of the data distribution [23]. For these reasons we
+propose a morphing attack that uses Diffusion-based methods as
+the generative process.
+3.1
+Diffusion Models
+Given data distribution q(x0) on data space X , the goal is to
+learn a model pθ(x0) approximating q(x0) which can be easily
+sampled. Denoising Diffusion Probabilistic Models (DDPMs) [35]
+are latent variable models of the form
+pθ(x0) =
+�
+pθ(x0:T ) dx1:T
+(2)
+
+(x, x)
+lz(za, Zb)
+E(xo)
+q(XtXt-1, z)
+lx(x(), x)
+pe(Xt-1|xt, z)XT
+Xt
+Xt-1
+X04
+Algorithm 1 Diffusion Morphing Algorithm.
+Require: Bona fide images, x(a)
+0 , x(b)
+0
+∈ X , image space preprocessing function, ξ : X ×X → X , image space interpolation function,
+ℓX : X × X × [0, 1] → X , and latent space interpolation function, ℓZ : Z × Z × [0, 1] → Z.
+procedure DIFFUSIONMORPH(x(a)
+0 , x(b)
+0 )
+za ← E(x(a)
+0 )
+▷ Calculate semantic latent codes
+zb ← E(x(b)
+0 )
+x(a)
+0
+← ξ(x(a)
+0 , x(b)
+0 )
+▷ Preprocess images passed to stochastic encoder
+x(b)
+0
+← ξ(x(b)
+0 , x(a)
+0 )
+t ← 0
+while t < T do
+x(a)
+t+1 ← √αt+1f (t)
+θ (x(a)
+t
+, za) + √1 − αt+1ϵ(t)
+θ (x(a)
+t
+, za)
+▷ Forward pass of diffusion algorithm
+x(b)
+t+1 ← √αt+1f (t)
+θ (x(b)
+t , zb) + √1 − αt+1ϵ(t)
+θ (x(b)
+t , zb)
+t ← t + 1
+end while
+x(ab)
+T
+← ℓX (x(a)
+T , x(b)
+T ; 0.5)
+▷ Stochastic code interpolation
+zab ← ℓZ(za, zb; 0.5)
+▷ Semantic code interpolation
+t ← T
+while t > 0 do
+x(ab)
+t−1 ← √αt−1f (t)
+θ (x(ab)
+t
+, zab) + √1 − αt−1ϵ(t)
+θ (x(ab)
+t
+, zab)
+▷ Diffusion generative process
+t ← t − 1
+end while
+return x(ab)
+0
+end procedure
+where {xt}T
+t=1 ∈ X are latent variables and T ∈ N. The reverse
+process is a Markov chain starting at pθ(xT ) = N(0, I), i.e., a
+normal distribution with mean vector 0 and variance I (the identity
+matrix), with Gaussian transitions
+pθ(x0:T ) = pθ(x0)
+T
+�
+t=1
+pθ(xt−1|xt)
+(3)
+The diffusion (forward) process is fixed to a Markov chain that
+gradually adds Gaussian noise to the original sample x0 according
+to variance schedule {βt}T
+t=1, such that
+q(x1:T , x0) =
+T
+�
+t=1
+q(xt|xt−1)
+(4)
+and q(xt|xt−1) = N(√1 − βtxt−1, βtI). See Figure 3 for an
+illustration of this process. The transition probability p(xt−1|xt)
+is likely to be very complex, unless the gap between t and t − 1 is
+very small, i.e., T → ∞. In this case, p(xt−1|xt) can be modelled
+as N(µ(t)
+θ (xt), σt), where µ(t)
+θ
+: X → X is an estimator at step
+t parameterized by θ. Ho et al. [35] proposes to use the following
+form:
+µ(t)
+θ (xt) =
+1
+√αt
+�
+xt −
+βt
+√1 − ¯αt
+ϵ(t)
+θ (xt)
+�
+(5)
+where αt = 1 − βt, ¯αt = �t
+s=1 αs, and ϵ(t)
+θ
+: X → X is a
+function approximator parameterized by θ, which learns to predict
+the noise added to x0 to get xt. This is achieved by using a U-Net,
+a type of CNN consisting of several skip connections formed at
+each resolution size in the architecture to model ϵ(t)
+θ
+[36].
+Like with Variational Autoencoders, the model is trained by
+optimizing the variational bound on the negative log likelihood:
+E[− log pθ(x0)] ≤ E
+�
+− log pθ(xT ) −
+T
+�
+t=1
+log pθ(xt−1|xt)
+q(xt|xt−1)
+�
+Relaxing the constraint that the inference process has to be
+Markovian leads to another kind of diffusion model known as the
+Denoising Diffusion Implicit Model (DDIM) as proposed by Song
+et al. [37]. This model has the benefit of a deterministic generative
+process such that:
+xt−1 = √αt−1f (t)
+θ (xt) +
+�
+1 − αt−1ϵ(t)
+θ (xt)
+(6)
+where
+f (t)
+θ (xt) =
+1
+√αt
+(xt −
+√
+1 − αtϵ(t)
+θ (xt))
+(7)
+which allows for deterministic embedding of the image x0 into a
+latent representation xT . However, xT does not contain high-level
+semantics, which is a necessary property for deep learning-based
+morphing attacks.
+3.2
+Diffusion Autoencoders
+Preechakul et al. [38] proposed a Diffusion Autoencoder by
+employing a conditional DDIM. The DDIM is conditioned on a
+semantic representation z ∈ Z by modifying ϵ(t)
+θ
+to take z as an
+additional input. Training is done via a simplified loss [35] defined
+as:
+L =
+T
+�
+t=1
+E
+x0∼q(x0)
+ϵt∼N (0,I)
+||ϵ(t)
+θ (xt, z) − ϵt||2
+2
+(8)
+Therefore, Equation (6) functions as a decoder network from latent
+code (xT , z). Let E : X → Z be the semantic encoder network
+such that
+z = E(x0)
+which
+provides
+the
+information
+necessary
+for
+decoder
+pθ(xt−1|xt, z) to properly denoise images. Likewise, the stochas-
+tic encoder can be thought of as a reversal of Equation (6) such
+that
+xt+1 = √αt+1f (t)
+θ (xt, z) +
+�
+1 − αt+1ϵ(t)
+θ (xt, z)
+(9)
+
+5
+where xT is encouraged to encode only stochastic details left out
+from the semantic representation z.
+3.3
+Proposed Morphing Algorithm
+We propose a novel process for the creation of morphed images
+by employing both the stochastic and semantic encoders. In
+particular, let xa, xb ∈ X be two bona fide images of identities
+a, b, and let x(a)
+0
+= xa and x(b)
+0
+= b. Algorithm 1 outlines the
+structure of the proposed Diffusion-based attack, hereafter called
+the Diffusion attack for simplicity, with additional illustration
+provided in Figure 2
+Beyond the core components of the DDIM and semantic
+encoder, three additional functions are added to the architec-
+ture. Namely, these are the image space preprocessing func-
+tion, ξ : X × X → X , image space interpolation function,
+ℓX : X ×X ×[0, 1] → X , and latent space interpolation function,
+ℓZ : Z × Z × [0, 1] → Z.
+The interpolation functions are used to interpolate between
+the semantic and stochastic values by some factor γ ∈ [0, 1].
+The image space preprocessing function is used to prepare the
+image passed to the semantic encoder. The simplest form of
+the interpolation function is the linear interpolation function,
+lerp(a, b; γ) = γa+(1−γ)b, where linear interpolation was found
+to be the best choice for ℓZ. However, for ℓX Song et al. [37]
+suggests the usage of spherical linear interpolation. For a vector
+space V and two vectors u, v ∈ V the spherical interpolation by
+a factor of γ is given as
+slerp(u, v; γ) = sin((1 − γ)θ)
+sin θ
+u + sin(γθ)
+sin θ v
+(10)
+where θ = arccos(u·v)
+||u|| ||v|| .
+While the semantic code provides most of the fundamental
+information, such as positioning of facial features, the stochastic
+code is used to provide information on the details not explicitly
+associated with the identity, but are necessary for realism of the
+generated image. By altering the stochastic code, details such as
+direction of strands of hair, clothing, etc., are altered whilst the
+identity of the image is preserved [38]. Unlike the rather straight-
+forward nature of linearly interpolating between the semantic
+codes to produce an image with key identifying characteristics of
+both identities, the nature of the stochastic code can lead to images
+of low visual fidelity if the interpolation is not done carefully.
+In particular, linear interpolation between two stochastic codes
+is does guarantee a smooth interpolation between the stochastic
+details in the images. For this reason the preprocessing function,
+ξ, is used to prepare the image passed to the stochastic encoder.
+One strategy is to “pre-morph” the image when extracting the
+stochastic details, i.e., ξ performs an image space morph of the
+image with the goal of reducing the artifacts induced by stochastic
+interpolation.
+4
+EXPERIMENTAL SETUP
+To evaluate the effectiveness of the proposed morphing attack, the
+attack is evaluated on three datasets against two different state-of-
+the-art FR systems. All training, optimization, and evaluation was
+conducted on a system with dual Intel Xeon Silver 4114 CPUs
+and a NVIDIA Tesla V100 32GB GPU with CUDA version 10.1
+and CUDNN version 8.4. The proposed morphing attack, MAD
+algorithm, and the FR systems are implemented in PyTorch [39].
+4.1
+Face Recognition Systems
+In order to evaluate the strength of the proposed morphing attack,
+two publicly available FR systems are used, specifically, the
+FaceNet and VGGFace2 models. These models are widely used
+and representative recognition systems with state-of-the-art face
+verification performance [40], [41]. For both models the last fully
+connected layer is used to provid a rich feature representation
+of the input image. Then for a presented face, its feature vector
+is compared with that of the feature vector belonging to the
+target face. If the distance between these two representations is
+sufficiently “small”, the presented face is then said to have the
+same identity as the target face. The VGGFace2 model improves
+upon acclaimed VGGFace [42] by using an improved training
+dataset, also called VGGFace2. Cao et al. [40] present the SENet
+architecture—the Squeeze and Excitation Network (SENet) was
+introduced by Hu et al. [43]—as the optimal choice when used
+with the VGGFace2 dataset. Google’s FaceNet model consists of
+an Inception-ResNet V1 architecture which is pre-trained on the
+VGGFace2 dataset [41].
+Additionally, the two FR systems use a different pre-
+processing pipeline. As all datasets the images and generated
+morphs are cropped as to be appropriate for passport photos, a
+face extractor such as MTCNN [44], is omitted in the verification
+pipeline. The FaceNet model resizes the image such that the short
+side of the image is 180 pixels long and then the image is cropped
+to a 160 × 160 resolution. Lastly, the images are normalized to
+[−1, 1]. The VGGFace2 model resizes the image such that the
+short side of the image is 256 pixels long and then crops the
+image to 224 × 224 pixels. The mean RGB vector1 is subtracted
+from the cropped image to normalize the image.
+4.2
+Datasets
+In this work, the FERET [45], FRLL [24], and FRGC v2.0 [46]
+datasets were used to evaluate the proposed technique, as they
+are commonly used in MAD with a large number of different
+identities [17], [26]. Notably, the FRLL dataset consists of high
+quality close-up frontal images at a 1350 × 1350 resolution
+with 189 facial landmarks—a large number of landmarks. The
+StyleGAN2, MIPGAN-II, and diffusion models were all trained on
+the FFHQ dataset, which contains 70,000 images at a 1024×1024
+resolution [33]. Morphs using OpenCV, FaceMorpher, and Style-
+GAN2 were created by Sarkar et al. [17] on the FRLL, FERET,
+and FRGC datasets. Additionally, Zhang et al. [22] created morphs
+via MIPGAN-II on the three datasets.
+In order to create a morphed face, two component identities are
+needed. Naturally, if the two component identities are disparate,
+the resulting morph is likely to be very weak. To rectify this and
+for evaluation purposes the component identity pairs were selected
+by following the existing protocol offered by Sarkar et al. [17].
+These pairings resulted in 1222 unique morphs on FRLL, 964 on
+FRGC, and 529 on FERET.
+5
+RESULTS
+The proposed morphing attack is compared to state-of-the-art
+techniques drawing from both GAN-based and Landmark-based
+methods. The effectiveness of the proposed method is quantita-
+tively measured on three fronts, these are: the visual fidelity of
+1. The mean vector is specifically ⟨131.0912, 103.8827, 91.4953⟩ for the
+red, green, and blue channels.
+
+6
+(a) Identity a
+(b) OpenCV
+(c) StyleGAN2
+(d) Diffusion
+(e) MIPGAN-II
+(f) FaceMorpher
+(g) Identity b
+Fig. 4: Different generated morphs from two identities from the FRLL dataset.
+(a) Diffusion
+(b) MIPGAN-II
+Fig. 5: Comparison of Diffusion and MIPGAN-II morphed faces
+on FRGC. Images are resized to 256 × 256 and cropped to 224 ×
+224 to match VGGFace2 pre-processing pipeline.
+the generated morphed images, the vulnerability of state-of-the-art
+FR systems to the morphing attack, and the detection potential of
+the morphing attack, respectively. Furthermore, an exploration of
+interpolation techniques for the stochastic latent code is provided.
+5.1
+Evaluation of Visual Fidelity
+The visual fidelity of the Diffusion attack is compared against
+other morphing attacks. Whilst on first glance it may appear that
+the ability to deceive a FR system should imply a high level of
+visual fidelity, this is not a simple assertion. We posit two reasons
+for this discrepancy:
+1)
+The image pre-processing pipeline for an FR system may
+crop out a significant portion of artifacts in the original
+morphed image.
+2)
+The FR system is vulnerable to forms of adversarial
+attacks which can degrade the performance of the FR sys-
+tem while injecting noticeable artifacts into the morphed
+image.
+This can lead to a situation wherein the FR system is fooled by a
+morphed image; however, it would be trivial for a human agent to
+TABLE 1: FID across different morphing attacks. Lower is better.
+Morphing Attack
+FRLL
+FRGC
+FERET
+StyleGAN2
+45.19
+86.41
+41.91
+FaceMorpher
+91.97
+88.14
+79.58
+OpenCV
+85.71
+100.02
+91.94
+MIPGAN-II
+66.41
+115.96
+70.88
+Diffusion
+42.63
+64.16
+50.45
+notice the artifacts present in the image. Moreover, a deep learning
+system could be specifically trained to notice such artifacts, greatly
+reducing the potential of such an attack to go undetected.
+To quantitatively assess the visual fidelity of the generated
+images the Fr´echet Inception Distance (FID) is employed, as it has
+shown to correlate well with human assessment of fidelity [47].
+The FID is a measure of distance between the generated and
+target distributions, therefore, the lower the FID metric the more
+similar the generated distribution is to the target distribution which
+correlates well with visual fidelity. The metric is defined as the
+Fr´echet distance, or 2-Wasserstein metric2, between two Gaussian
+distributions each representing the activations the deepest layer of
+an Inception v3 network induced by images from the generated
+and target distributions.
+Table 1 shows the FID metric between the generated images
+from different morphing attacks and bona fide samples from
+the dataset the morphing attack is drawn from. The FID met-
+ric is calculated using pytorch-fid [48] Morphed images
+generated from the Diffusion attack generally had the lowest
+FID with the StyleGAN2-based attack following closely behind.
+Both Landmark-based morphs—OpenCV and FaceMorpher—had
+noticeably higher FIDs than the deep learning-based morphs.
+These results correlate well with visual inspection of the morphs as
+both Figure 4b and Figure 4f exhibit prominent artifacts outside
+the central face region. Likewise, the MIPGAN-II attack seems
+to struggle with some distortion outside the central face region,
+see Figure 4e. Interestingly, on the FRLL dataset the StyleGAN2
+morphing pipeline consistently darkens morphs relative to its com-
+ponent images; however, the visual fidelity is relatively high albeit
+with noticeable darkening. Importantly, stochastic details, such as
+hair, seem to be modelled well by the Diffusion attack whereas
+other attacks distort such details, the OpenCV, FaceMorpher, and
+MIPGAN-II attacks, or present details that have little similarity
+to both identities, the StyleGAN2 attack. While exhibiting the far
+less visual artifacts than other morphing techniques, the Diffusion
+attack does tend to slightly smooth out the skin texture. Overall,
+the Diffusion attack exhibits the highest consistent visual fidelity
+among all the presented attacks.
+Strangely, MIPGAN-II exhibited a much higher FID than the
+StyleGAN2 morphs. This is a surprising result as the MIPGAN-
+II attack was positioned as an improvement over the StyleGAN2
+attack. Figure 5 compares two morphed faces generated by the
+2. The 2-Wasserstein metric between two probability measures µ, ν with
+finite moments on Rn is defined as
+W2(µ, ν) =
+�
+inf
+π∈Π(µ,ν)
+�
+Rn×Rn ||x − y||2
+2 dπ(x, y)
+� 1
+2
+where Π(µ, ν) is the set of all distributions with marginals µ and ν.
+
+二二7
+TABLE 2: The APCER at specific BPCER values. Higher is better.
+Dataset
+FR System
+Morphing Attack
+APCER @ BPCER = 0.1%
+APCER @ BPCER = 1%
+APCER @ BPCER = 5%
+FRLL
+FaceNet
+StyleGAN2
+0.99
+0.05
+0
+FRLL
+FaceNet
+FaceMorpher
+2.25
+0.14
+0.05
+FRLL
+FaceNet
+OpenCV
+3.24
+0.33
+0
+FRLL
+FaceNet
+MIPGAN-II
+8.87
+0.47
+0.09
+FRLL
+FaceNet
+Diffusion
+8.83
+0.99
+0.23
+FRLL
+VGGFace2
+StyleGAN2
+0.05
+0.05
+0
+FRLL
+VGGFace2
+FaceMorpher
+1.36
+1.08
+0.23
+FRLL
+VGGFace2
+OpenCV
+2.35
+2.11
+0.28
+FRLL
+VGGFace2
+MIPGAN-II
+1.31
+0.99
+0.23
+FRLL
+VGGFace2
+Diffusion
+2.68
+2.07
+0.52
+FRGC
+FaceNet
+StyleGAN2
+74.04
+36.69
+17.73
+FRGC
+FaceNet
+FaceMorpher
+87.9
+38.85
+14.9
+FRGC
+FaceNet
+OpenCV
+84.1
+31.43
+11.36
+FRGC
+FaceNet
+MIPGAN-II
+96.54
+61.9
+33.48
+FRGC
+FaceNet
+Diffusion
+91.73
+48.86
+24.95
+FRGC
+VGGFace2
+StyleGAN2
+81.42
+46.22
+26.7
+FRGC
+VGGFace2
+FaceMorpher
+95.42
+63.65
+38.18
+FRGC
+VGGFace2
+OpenCV
+95.31
+64.62
+39.4
+FRGC
+VGGFace2
+MIPGAN-II
+91.92
+57.8
+30.84
+FRGC
+VGGFace2
+Diffusion
+93.71
+58.25
+32.18
+FERET
+FaceNet
+StyleGAN2
+15.65
+9.35
+2.49
+FERET
+FaceNet
+FaceMorpher
+10.71
+5.1
+0.91
+FERET
+FaceNet
+OpenCV
+8.79
+3.06
+0.17
+FERET
+FaceNet
+MIPGAN-II
+21.03
+10.77
+2.21
+FERET
+FaceNet
+Diffusion
+24.04
+13.95
+4.99
+FERET
+VGGFace2
+StyleGAN2
+54.08
+18.42
+5.73
+FERET
+VGGFace2
+FaceMorpher
+80.5
+32.65
+12.7
+FERET
+VGGFace2
+OpenCV
+81.01
+32.6
+12.87
+FERET
+VGGFace2
+MIPGAN-II
+66.5
+18.14
+5.84
+FERET
+VGGFace2
+Diffusion
+80.9
+35.2
+14.34
+Diffusion and MIPGAN-II attacks on the FRGC dataset. Numer-
+ous high frequency artifacts are present in Figure 5b, particularly,
+near the hairline and transition between hair and the background.
+Comparing Figure 5a and Figure 5b the hair generated by the
+MIPGAN-II attack looks unnatural with a strange texture as
+though an image sharpening filter has been applied to the image,
+greatly enhancing the magnitude of high frequency content, which
+aligns with the observation in Figure 4. Moreover, the MIPGAN-
+II images seem to be desaturated when compared to images
+produced by other attacks leading to a washed out appearance.
+Perhaps the low visual fidelity can be explained by the identity
+loss overpowering the perceptual quality loss leading to morphed
+images with low visual fidelity but high effectiveness against FR
+systems.
+5.2
+Vulnerability of FR Systems
+The strength of the proposed face morphing algorithm is further
+evaluated by measuring the ability of the morph to fool a FR
+system. The attack success is quantitatively verified against two
+state-of-the-art FR systems. To ensure a valid comparison between
+the five different morphing attacks the same pairs of component
+identities were used in evaluating every morphing attack, i.e., for
+every pair of component identities a morphed image was created
+for each of the five attacks. For both the FaceNet and VGGFace2
+FR systems the False Match Rate (FMR) is set at 0.1% following
+the guidelines of Frontex [49]. Additionally, the distance between
+faces is measured using the L2 distance between the outputs of
+the FR model.
+The vulnerability of FR systems to morphing attacks is as-
+sessed by comparing the error rates in detection, specifically,
+the Attack Presentation Classification Error Rate (APCER)3 is
+measured at specific Bona fide Presentation Classification Error
+Rate (BPCER)4 values. In Table 2 the APCER values for the five
+different morphing attacks is presented across all three datasets
+evaluated on three different BPCER values of 0.1%, 1%, and 5%.
+Due to a variety of factors—such as image quality and number
+of bona fide images per identity—the results vary between the
+different datasets; while there is some variance between both
+FR systems, they tend to agree more closely. Noticeably, all
+attacks performed rather poorly on the FRLL dataset, although
+the Diffusion-based morphing attack performed the best among
+them, which could be attributed to the limited number of bona
+fide images per identity; for in the FRLL dataset there are only
+two bona fide images per identity: a neutral face (used to create
+the morph) and a smiling face. Both FR systems on the FRLL
+dataset tend to give close relative rankings between the morphing
+attacks with the StyleGAN2-based attack being noticeably weaker
+than the rest. Both FR systems were much more vulnerable when
+evaluated on the FRGC dataset. The MIPGAN-II attack performed
+very well against FaceNet which makes sense as this technique
+was refined on the FRGC dataset in particular [22]. The attack was
+not as strong on VGGFace2 and instead that FR system was more
+vulnerable to OpenCV and FaceMorpher, this could possibly be
+attributed to the different pre-processing pipelines. The Diffusion-
+based generally performs close to the top performer on either FR
+system. As with FRGC, on the FERET dataset VGGFace2 is more
+3. APCER is the proportion of attack presentations incorrectly classified as
+bona fide presentations.
+4. BPCER is the proportion of bona fide presentations incorrectly classified
+as attack presentations.
+
+8
+TABLE 3: MMPMR at FMR = 0.1% across diffrent morphing attacks. Higher is better.
+FRLL
+FRGC
+FERET
+Morphing Attack
+FaceNet
+VGGFace2
+FaceNet
+VGGFace2
+FaceNet
+VGGFace2
+Geometric Mean
+StyleGAN2
+4.69
+6.05
+0.18
+0.85
+0.54
+0.76
+1.10
+FaceMorpher
+11.26
+36.4
+0.51
+9.15
+2.3
+10.78
+6.02
+OpenCV
+17.34
+40.93
+0.14
+12.16
+1.69
+11.12
+5.32
+MIPGAN-II
+30.96
+26.74
+3.12
+7.94
+6
+5.39
+9.34
+Diffusion
+28.14
+35.37
+2.68
+8.47
+6.47
+13.03
+11.13
+vulnerable to landmark-based attacks, OpenCV and FaceMorpher,
+than FaceNet. Diffusion-based morphs pose the greatest threat on
+FERET consistently having high APCER values. In general the
+following observations can be drawn from Table 2:
+•
+Among the five different attacks, FR systems are most vul-
+nerable to Diffusion attacks. Moreover, Diffusion attacks
+always rank in the top three in terms of performance.
+•
+FR systems are the least vulnerable to the StyleGAN2 at-
+tack. The StyleGAN2 attack is always outperformed by its
+successor, MIPGAN-II, and the other deep learning-based
+attack, Diffusion, while often falling behind landmark-
+based attacks.
+In addition to using the error rates to assess the vulnerability
+of FR systems the Mated Morphed Presentation Match Rate
+(MMPMR) [50] is used as a measure of vulnerability. Scherhag et
+al. [50] propose two variants of the MMPMR metric for the
+scenario in which there multiple bona fide images of an identity
+used in morph process, excluding the image used to create the
+morph, called the MinMax-MMPMR and ProdAvg-MMPMR.
+The MinMax-MMPMR metric is especially likely to increase the
+number of accepted morphs as the number of bona fide images
+per identity increases. Therefore, the ProdAvg-MMPMR is the
+specific MMPMR variant used to assess the vulnerability of FR
+systems, any mention hereafter to MMPMR refers specifically to
+ProdAvg-MMPMR unless stated otherwise.
+Let PM ∈ P(X) be the distribution of morphed images such
+that for some xab ∼ PM, xab denotes a morphed image made
+from identities a, b, where P(X) denotes the set of all probability
+measures on X . Let Pk ∈ P(X) denote the distribution of bona
+fide images of identity k. Then with abuse of notation Pk\xab is
+the distribution of bona fide images of identity k excluding those
+images used in creating the morph xab. The MMPMR metric for
+a particular threshold, γ > 0, equipped with FR system F : X →
+V is then defined as
+M(γ) =
+E
+xab∼PM
+�
+�
+k∈{a,b}
+E
+x∼Pk\xab
+�
+||F(xab)−F(x)||2 < γ
+��
+i.e., the expected success rate of the morphing attack to fool the
+FR system. The product term is the joint probability of successful
+verification of both identities
+Table 3 presents the MMPMR metric when the FMR is set
+at 0.1% for all datasets and FR systems. Interestingly, the FRLL
+dataset had the highest overall MMPMR metrics in contrast to
+the results from Table 2. This can likely be attributed to limited
+number of bona fide images per identity and FRLL in contrast
+with other datasets as the particular choice of MMPMR metric
+heavily punishes failed verifications in either identity, thus having
+only one possible image per identity could boost the metric for
+FRLL. On average the Diffusion attack greatly outperforms the
+TABLE 4: APCER at FMR = 0.1% across different margin sizes
+on the FaceNet FR system. Higher is Better.
+Margin Size
+Dataset
+Morphing Attack
+0
+20
+40
+80
+FRLL
+MIPGAN-II
+54.84
+56.53
+57.18
+58.03
+FRLL
+StyleGAN2
+15.12
+15.57
+17.14
+25.11
+FRLL
+FaceMorpher
+74.48
+75.26
+73.86
+47.91
+FRLL
+OpenCV
+76.21
+75.82
+74.96
+48.4
+FRLL
+Diffusion
+51.25
+54.47
+57.07
+59.02
+FERET
+MIPGAN-II
+19.33
+21.71
+22.11
+26.36
+FERET
+StyleGAN2
+14.12
+17.57
+18.14
+22.17
+FERET
+FaceMorpher
+36.11
+36.45
+36.28
+17.8
+FERET
+OpenCV
+36.11
+38.44
+37.64
+13.89
+FERET
+Diffusion
+23.24
+26.02
+25.91
+30.39
+FRGC
+MIPGAN-II
+12.33
+14.3
+16.2
+20.97
+FRGC
+StyleGAN2
+7.18
+8.71
+9.74
+14.42
+FRGC
+FaceMorpher
+17.5
+18.87
+20.39
+9.07
+FRGC
+OpenCV
+17.02
+18.47
+19.91
+6.6
+FRGC
+Diffusion
+9.7
+10.71
+12.27
+15.62
+other attacks; conversely, the Landmark-based attacks, on average,
+exhibit mediocre performance. In agreement with Table 2 the
+StyleGAN2 attack shows abysmal performance in comparison
+with the other attacks.
+5.2.1
+The Effect of Pre-processing on a FR System
+The impact of the pre-processing pipeline on the vulnerability of
+a FR system is examined, in particular the cropping process is
+further explored. To study this an additional margin size is added
+to the image after a initial face extraction and cropping performed
+by MTCNN, such that a margin size N adds back at most N pixels
+to the cropped image in both dimensions. Therefore, the larger N
+is the less tightly cropped the image passed to the FR system
+is. Table 4 shows the illustrates the impact of the margin size
+on the APCER metric on the FaceNet FR system. Generally, as
+the margin size increases the performance of the Landmark-based
+attacks decreases and the performance of the deep learning-based
+attacks increases. As illustrated in Figure 4 the Landmark-based
+attacks have noticeable artifacts outside the central face region;
+conversely, the deep learning-based morphs have less artifacts in
+the outside regions and generally look more realistic to a human
+observer. This observation aligns the visual fidelity results from
+Table 1. Therefore, a MAD algorithm or FR system which uses
+less tightly cropped faces would be more resilient against attacks
+with visual artifacts outside the core face region.
+5.2.2
+General Remarks on the Vulnerability Study
+The poor performance of the StyleGAN2 attack could be at-
+tributed to the darkening of images with light backgrounds, see
+Figure 4, and due to aliasing effects latent to the StyleGAN2
+generation pipeline which is addressed by Karras et al. [51].
+
+9
+TABLE 5: Ablation study on the impact morphing attack on validation accuracy.
+Training Attack
+Validation Attack
+Dataset
+Diffusion
+FaceMorpher
+MIPGAN-II
+OpenCV
+StyleGAN2
+Diffusion
+FaceMorpher
+MIPGAN-II
+OpenCV
+StyleGAN2
+FERET
+
+
+
+
+
+72.73
+99.23
+100
+99.95
+99.33
+FERET
+
+
+
+
+
+99.9
+76.39
+100
+99.85
+99.64
+FERET
+
+
+
+
+
+99.69
+99.38
+100
+99.95
+99.54
+FERET
+
+
+
+
+
+99.74
+99.48
+100
+99.74
+99.43
+FERET
+
+
+
+
+
+99.74
+98.56
+99.9
+99.74
+87.89
+FRGC
+
+
+
+
+
+75.89
+99.98
+99.97
+99.9
+99.93
+FRGC
+
+
+
+
+
+99.95
+99.48
+100
+99.9
+99.95
+FRGC
+
+
+
+
+
+99.83
+99.85
+99.82
+99.8
+99.85
+FRGC
+
+
+
+
+
+99.93
+100
+100
+99.23
+99.93
+FRGC
+
+
+
+
+
+99.93
+99.93
+99.94
+99.88
+97.83
+FRLL
+
+
+
+
+
+13.96
+99.58
+99.32
+99.65
+99.65
+FRLL
+
+
+
+
+
+99.23
+99.09
+98.91
+99.37
+99.44
+FRLL
+
+
+
+
+
+99.09
+98.95
+98.24
+99.02
+99.09
+FRLL
+
+
+
+
+
+99.51
+99.44
+99.19
+99.16
+99.58
+FRLL
+
+
+
+
+
+99.93
+99.86
+99.86
+99.93
+95.02
+Moreover, the structure of the StyleGAN2 latent space can make
+exploration in the space difficult which could possibly explain
+the poor performance in attacking the FR system compared to
+other attacks. MIPGAN-II, on the other hand, likely avoids these
+pitfalls due its explicit latent optimization process for fooling a FR
+system. The Diffusion attack utilizes an entirely different latent
+representation scheme which seems to yield an advantage in the
+task of generating morphed faces. The pre-processing pipeline of
+the FR system seems to mostly mitigate the artifacts latent to the
+Landmark-based attacks; however, such artifacts could easily be
+detected by a human observer.
+5.3
+Detectability of Morphing Attacks
+The performance of the proposed attack is further evaluated by the
+ability of Morphing Attack Detection (MAD) algorithms trained
+against other attacks to detect an unseen attack. To quantita-
+tively assess the detectability of a particular morphing attack a
+SE-ResNeXt101-32x4d model pre-trained on ImageNet [52] by
+NVIDIA is trained to detect morphing attacks. SE-ResNeXt101-
+32x4d is a state-of-the-art image recognition model based on the
+ResNeXt101-32x4d model [53] with the addition of the Squeeze-
+and-Excitation architecture [43]. For all experiments a 5-fold strat-
+ified k-fold cross validation strategy is employed, thus preserving
+the class balance between morphed and bona fide images in each
+fold. The model is fine-tuned on a collection of morphing attacks
+for 5 training epochs using exponential learning rate scheduler
+with differential learning rates in order to mitigate overfitting of
+the model.
+5.3.1
+Ablation Study
+To study the impact of a particular morphing attack on the ability
+of a MAD algorithm to detect morphing attacks an ablation
+study was conducted where the SE-ResNeXt101-32x4d model
+was trained on all the morphing attacks except for one holdout.
+Table 5 shows the validation accuracy of each morphing attack
+when different morphing attacks were withheld from the training
+process. Due to the similar natures between the OpenCV and
+FaceMorpher attacks the absence of one of these attacks does not
+greatly impact the validation accuracy. Interestingly, the absence
+of MIPGAN-II does not significantly change the validation accu-
+racy of the attacks; however, the omission of StyleGAN2 during
+training does decrease the performance of the StyleGAN2 during
+validation despite the presence of the MIPGAN-II. Notably, the
+Diffusion attack is very difficult to detect as a novel attack, which
+can be partially attributed to its unique morph generation process
+in contrast with the other morphing attacks.
+5.3.2
+A Metric For Relative Strength
+In this section we introduce a metric to measure the relative
+strength from one morph to another. We say a morph α is “strong”
+relative to a morph β if the following conditions are satisfied:
+1)
+It is easy to detect β when a detector is trained on α, i.e.,
+high transferability.
+2)
+It is hard to detect α when a detector is trained on β, i.e.,
+low detectability.
+Additionally, the relative strength metric, ∆(α||β), should be
+positive when α is stronger than β and negative when α is weaker.
+A relative strength of 0 would denote that the two morphing
+attacks are equally strong.
+As some of the morphing attacks are not deterministic but
+probabilistic, we chose to represent a morphing attack α by
+the random variable Xα : Ω → X such that P(Xα|xa, xb)
+denotes the distribution of morphs generated from images xa, xb.
+Moreover, we suppose there exists a detector f α : X → {0, 1}
+trained to distinguish between bona fide presentations and mor-
+phed presentations generated by α; wherein 0 denotes a bona fide
+presentation and 1 denotes a morphed presentation. The transfer-
+ability of a morphing attack α to β is defined as the probability the
+detector f α is able to detect the attack β given the probability f α
+detects α, i.e., T(α, β) = P(f α(Xβ) = 1|f α(Xα) = 1). This
+metric can be represented as a ratio of expectations taken over the
+pairs of component bona fide images:
+T(α, β) = P(f α(Xβ) = 1, f α(Xα) = 1)
+P(f α(Xα) = 1)
+= Exa,xb[P(f α(Xβ) = 1, f α(Xα) = 1|xa, xb)]
+Exa,xb[P(f α(Xα) = 1|xa, xb)]
+(11)
+Let {xα
+i }N
+i=1 denote a collection of N samples drawn from
+P(Xα|xa, xb) such that xα
+i denotes the morph generated from
+i-th pair of bona fide identities (ai, bi), and likewise for β. Then
+the metric in Equation (11) can be closely approximated by
+T(α, β) ≈
+�N
+i=1
+�
+f α(xβ
+i ) = 1 ∧ f α(xα
+i ) = 1
+�
+�N
+i=1
+�
+f α(xα
+i ) = 1
+�
+(12)
+
+10
+(a) RSM on FRGC
+(b) RSM on FERET
+(c) RSM on FRLL
+Fig. 6: Blue indicates higher strength and red indicates weak
+strength.
+i.e., the number of morphs from both α and β detected over the
+number of morphs detected from α.
+The relative strength metric (RSM) from α to β is defined
+as the log ratio of the transferability metrics between the two
+morphing attacks:
+∆(α||β) = log
+�T(α, β)
+T(β, α)
+�
+(13)
+The log of the ratio is chosen such that ∆(α||β) the RSM takes
+(a) Variant A
+(b) Variant B
+(c) Variant C
+(d) Variant D
+Fig. 7: Morphed image generated by different Diffusion attack
+variants on FRLL.
+positive values when α is “stronger” than β and negative values
+when weaker—with a value of zero denoting equal strength.
+Additionally, there is an antisymmetry such that ∆(α||β) =
+−∆(β||α).
+In contrast to the ablation study, the SE-ResNeXt101-32x4d
+model is only trained on a single attack per k-fold. The RSM is
+calculated between all attacks with the results shown in Figure 6.
+From Figure 6 it is observed that the RSM between the Landmark-
+based morphs and the RSM between the StyleGAN-based morphs
+is very small. As these attacks have similar morph generation
+pipelines it makes sense that the transferability between the attacks
+is near identical. In general, the Landmark-based attacks seem to
+be stronger than the StyleGAN-based attacks, in particular the
+FaceMorpher attack. The MIPGAN-II attack is generally weaker
+than the other attacks. Overall, the Diffusion attack is the least
+detectable among the attacks along with generally being the
+strongest attack across the three datasets.
+The results from Figure 6 corroborate with the results from
+Table 5 demonstrating the difficulty in detecting Diffusion attacks.
+From the perspective of training a MAD system including samples
+from the FaceMorpher, StyleGAN, and Diffusion attacks would
+greatly increase the ability for the system to detect unknown
+attacks. Additionally, Table 5 and Figure 6 demonstrates a par-
+ticular vulnerability existing MAD systems may have to the new
+Diffusion attack.
+5.4
+Study of the Diffusion-based Morphing Process
+The diffusion morphing algorithm leverages both a stochastic
+and semantic representation of an image. While the semantic
+representation contains many of the key “identifying” features,
+the stochastic representation represents many of the details nec-
+essary for high visual fidelity. Due to the importance of the
+stochastic code for high fidelity, we investigated several methods
+for finding the morphed stochastic latent code, x(ab)
+T
+. The first
+
+3.517
+StyleGAN2
+OpenCv
+Attack
+Training A
+MIPGAN-II
+0.000
+FaceMorpher
+Diffusion
+3.517
+OP
+eGAN2
+ValidationAttack2.734
+StyleGAN2
+OpenCv
+Attack
+Training A
+MIPGAN-II
+0.000
+FaceMorpher
+Diffusion
+2.734
+OR
+MIPGAN-II
+ValidationAttack18.2
+StyleGAN2
+Opencv
+Training Attack
+MIPGAN-II
+0.0
+FaceMorpher
+Diffusion
+18.2
+MIPGAN-II
+OR
+GAN
+Validation Attack11
+TABLE 6: MMPMR at FMR = 0.1% across different configurations. Higher is better. † indicates our default choices.
+FRLL
+FRGC
+FERET
+Variant
+ℓX
+ξ(x, y)
+FaceNet
+VGGFace2
+FaceNet
+VGGFace2
+FaceNet
+VGGFace2
+Geometric Mean
+A
+slerp
+x, y �→ x
+32.97
+34.71
+3.2
+9.59
+7.17
+11.54
+11.95
+B
+lerp
+x, y �→ x
+10.81
+11
+1.17
+2.17
+2.33
+4.69
+3.86
+C†
+slerp
+x, y �→ 1
+2 (x + y)
+28.14
+35.37
+2.68
+8.47
+6.47
+13.03
+11.13
+D
+slerp
+x, y �→ OpenCV(x, y)
+9.14
+9.34
+0
+1.37
+0.14
+1.42
+0
+TABLE 7: FID across different configurations. Lower is better.
+† indicates our default choices.
+Variant
+ℓX
+ξ(x, y)
+FRLL
+FRGC
+FERET
+A
+slerp
+x, y �→ x
+48.13
+52.97
+55.66
+B
+lerp
+x, y �→ x
+82.05
+119.33
+97.75
+C†
+slerp
+x, y �→ 1
+2 (x + y)
+42.63
+64.16
+50.45
+D
+slerp
+x, y �→ OpenCV(x, y)
+93.85
+84.51
+108.49
+variant, variant A, is the baseline implementation with ℓZ using
+linear interpolation, ℓX using spherical linear interpolation, and
+ξ does not perform any “pre-morphing”. Conversely, in variant B
+the stochastic codes are interpolated via linear interpolation. In
+variants C and D, instead of using the original image to calculate
+the stochastic code, the function ξ is used to construct the “pre-
+morph” passed to the stochastic encoder. Specifically, in variant
+C the two images are averaged pixel-wise and presented to the
+stochastic encoder; in contrast, in variant D the OpenCV morph is
+presented to the stochastic encoder.
+In Table 7 the FID is calculated between the generated morphs
+and the bona fide samples for each particular dataset. Variant
+C generally presents the lowest FID score closely followed by
+variant A. Both variants B and D exhibit clear degradation in
+performance when compared to variants A and C. Furthermore,
+the FID scores seems to correlate well to human assessment
+of the generated samples, see Figure 7. Noticeably, the linear
+interpolation in variant B results in an overly smoothed face and
+generally darker image, greatly degrading visual fidelity. Variant
+D has prominent visual artifacts, similar to the artifacts found in
+the OpenCV morphs. Moreover, the poor performance seems to
+be aided by an issue of differing alignment strategies between the
+OpenCV and diffusion pipeline.
+Notably, variant C often removes many of the high frequency
+artifacts found in variant A. This is likely due to the difficulty
+in smoothly interpolating between points in the stochastic latent
+space in contrast with the semantic latent space. As such, variant
+C which performs a pixel-wise average of the two source images
+before using the stochastic encoder seems to greatly improve
+the ability to smoothly interpolate between different stochastic
+representations. This appears to be the primary reason variant C
+has a generally lower FID when compared to variant A. Both
+Figure 7 and Table 7 demonstrate the large importance that the
+stochastic code plays in creating high fidelity morphed images.
+Due to the high fidelity exhibited by variant C, this particular
+diffusion process was used in evaluation against other morphing
+attacks.
+The MMPMR metric is calculated for each variant, see Ta-
+ble 6. Variant A is slightly stronger than variant C, with variants
+B and D falling far behind likely due to the high number of visual
+distortions. These results stand in contrast to the assessment of vi-
+sual fidelity wherein variant C outperforms variant A. This, again,
+illustrates a trade-off between visual fidelity and ability to fool
+the FR system; however, in this case the trade-off effectiveness
+against the FR system is relatively small in comparison to the
+gains in visual fidelity. Due to its excellent visual fidelity and
+strong MMPMR results variant C was chosen to be the default
+configuration for the Diffusion attack.
+6
+CONCLUSION
+By addressing some of the key limitations of prior deep-learning
+based morphing attacks, namely, the trade-off between visual
+fidelity and effectiveness against FR systems, we have proposed
+a novel morphing attack using Diffusion-based methods for the
+generative process. The proposed attack consistently generates
+realistic morphed images with high visual fidelity while also
+being able to strongly threaten FR systems. To evaluate the attack
+potential of the proposed attack, we evaluated the vulnerability
+of two FR systems over three distinct datasets and created over
+10,000 new morphs between the four variants, with the strongest
+variant achieving state-of-the-art performance. We conducted an
+additional study on the impact the pre-processing pipeline has on
+the vulnerability of an FR system to morphing attacks. A novel
+metric to assess the strength of morphing attacks relative to each
+other has been introduced. Moreover, the proposed attack was
+evaluated by its detection performance against a state-of-the-art
+MAD system. The Diffusion attack was shown to be very difficult
+to detect if not specifically trained against presenting showing
+the proposed attack can greatly threaten preexisting FR systems.
+The images generated by the Diffusion attack possess high visual
+fidelity, can fool state-of-the-art FR systems, and are difficult for
+MAD mechanisms to detect.
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+arXiv:2301.03557v1  [math.DS]  9 Jan 2023
+USING PREY ABUNDANCE TO SYNCHRONIZE TWO
+CHAOTIC GLV MODELS
+SHUBHANGI DWIVEDI
+E-MAIL: SHUBHANGI.DWIVEDI176@GMAIL.COM
+, NITU KUMARI
+E-MAIL: NITU@IITMANDI.AC.IN
+SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES, KAMAND, INDIAN
+INSTITUTE OF TECHNOLOGY, MANDI-HIMACHAL PRADESH, 175005, INDIA
+, AND RANJIT KUMAR UPADHYAY
+DEPARTMENT OF APPLIED MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY
+(ISM), DHANBAD-JHARKHAND, 826004 , INDIA
+E-MAIL: RANJIT.CHAOS@GMAIL.COM
+( Accepted: 02 November 2021 )
+Abstract.
+The concept of superfluous prey, or an excess of prey in certain areas within
+a patchy ecosystem, has significant implications for the synchronization of the predator
+population.
+These areas, known as ”hotspots,” have a higher density of prey compared
+to other areas and attract a higher concentration of predators.
+As a result, the preda-
+tor population becomes more stable and predictable, as they are less likely to migrate to
+other areas in search of food. This phenomenon can have important consequences for both
+the predators and their prey, as well as the overall functioning of the ecosystem.
+This
+work investigates the synchronization between two chaotic food webs using the general-
+ized Lotka-Volterra (GLV) model consisting of one prey and two predator populations.
+We, first, examine the impact of three functional responses (linear, Holling type II, and
+Holling type III) on system dynamics For the study, we consider the model with a linear
+functional response consisting of chaotic oscillations and apply controllers to stabilize its
+unstable fixed points.
+This research contributes to the understanding of how to apply
+chaotic ecological models to predict the population of competing species in one habitat
+using information about similar populations in another system. To do this, we configure a
+drive-response system where prey acts as the driving variable and both predators depend
+only on the prey. We use active and adaptive control methods to synchronize two coupled
+GLV models and verify the analytical results through numerical simulations.
+AMS Classification: —
+Keywords:
+Lyapunov exponents, slow manifold equation, chaos control, complete
+replacement synchronization, active and adaptive control techniques
+JOURNAL OF DYNAMICAL SYSTEMS & GEOMETRIC THEORIES
+©TARU PUBLICATIONS
+1
+
+2
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+1. Introduction
+In ancient philosophy and mythology, the word chaos meant the disordered state
+of unformed matter supposed to have existed before the ordered universe. In the
+course of its journey to truth, science has met with startling phenomena called chaos.
+Since the 1960s, with the discovery of chaotic systems, chaos has set a nonlinear
+dynamics research boom. Chaos theory is attributed to the work of Edward Lorenz.
+His 1963 paper, “Deterministic Non-periodic Flow” [1], is credited for laying the
+foundation for chaos theory. Hunt and Ott [2] reviewed the problem and proposed a
+computationally feasible entropy-based good definition of chaos. They define chaos
+as “ the existence of positive Expansion Entropy (EE) (equal to topological entropy
+for infinitely differentiable maps) on a given restraining region (bounded positive
+volume subset),” which confirms both the notions (COS as well as OS). Since EE
+enjoys the properties of simplicity, computability, and generality, so they call it a
+“ good ” definition of chaos. Chaotic systems are sensitive to initial conditions,
+topologically mixing and with dense periodic orbits [3], [4]. Because of slightest dif-
+ference, chaotic dynamical systems can lead to entirely different trajectories. The
+main characteristic of chaos is that the system does not repeat its past behaviour.
+Mathematically, chaotic dynamical systems are classified as non-linear dynamical
+systems having at least one positive Lyapunov exponent[5].
+Chaos theory is not just about chaos - it has two sides to it: chaos control and chaos
+synchronization. The study of chaos control and understanding chaotic model be-
+haviour has gained significant interest, with applications in various fields such as
+secure communications, biology, neural networks, finance, and more. Whereas chaos
+synchronization refers to the alignment in time of different chaotic processes. At
+first glance, chaotic systems may seem to defy synchronization, but it has been
+observed and studied in various contexts.
+In 1665, Dutch physicist Huygens observed the adjustment of rhythms via a cou-
+pling.
+He noticed that pendulum clocks suspended from the same beam would
+slowly adjust their phases. In 1984, Kuramoto set theory for the onset of sync, and
+Pecora and Carroll [6] reviewed the area of synchronization in chaotic systems and
+presented a more geometric view using synchronization manifold. In 1999, Blasius
+explained the theoretical analysis of seasonally synchronized chaotic population cy-
+cles [7]. All these contributions help the researcher think that synchronization is
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+3
+an essential phenomenon in physical and biological systems. In literature, this phe-
+nomenon has been nominated by various types, such as phase locking, frequency
+pulling, generalized synchrony, and complete locking, depending on the degree or
+type of synchronization.
+Synchronization of chaotic systems can be achieved by configuring drive and re-
+sponse systems, with the goal of using the output of the drive system to control
+the response system so that the output tracks the drive system asymptotically.
+However, creating identical chaotic synchronized ecological systems is questionable
+as it has a potential threat to biodiversity. As discovered in Pecora and Carroll’s
+pioneering work, another way of achieving complete synchronization between two
+systems is what is now called the technique of complete replacement. The complete
+replacement synchronization is helpful in a network of patchy ecosystems as it can
+help in identifying the underlying mechanism that derive the group co-ordination
+in the present of severely chaotic oscillations. In population biology, the chaotic
+dynamics may synchronize if populations are coupled through environmental or bi-
+ological interactions.
+From the ecological aspect, it is crucial to figure out the ecologically feasible cou-
+pling scheme that guarantee the permanence and global attractiveness of all species
+in a multi-patch ecosystem. Upadhyay and Rai [8] have demonstrated that the
+two non-identical chaotic ecological systems having different kinds of top-predators
+can be synchronized using an algorithm proposed by Lu and Cao[9]. The idea of
+this approach is that it takes care of the uncertainties involved in the parameter
+estimation. There are many methods in control theory for synchronizing chaotic
+systems, including the Adaptive Control Method, Back-stepping Method, Active
+Control Method, Time-Delay feedback approach, and others.
+Among above-listed methods, the linear active control technique for chaos syn-
+chronization is popular and effective for synchronizing identical and non-identical
+chaotic systems. In this work, we will use the active and adaptive control meth-
+ods for achieving synchronization of chaos within either identical or non-identical
+systems. The Active control method was first used for chaos synchronization by
+E.W. Bai and K.E. Lonngren [10], [11]. In this method, non-linear controllers are
+designed based on the Lyapunov stability theory to achieve synchronization in cou-
+pled systems using the known parameters of the drive and response systems.
+
+4
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+Since we will be dealing with an ecological model, the system’s parameters cannot be
+precisely known. The adaptive control is one of the popular technique for controlling
+and synchronizing non-linear systems with uncertain parameters [12]. The method
+allows the model to adapt data assimilation along the way that may be useful for
+predicting the real system’s future behaviour [13], [14]. In this method, control law
+and parameter update law are designed in such a way that the chaotic response
+system to behave like chaotic drive systems. This scheme maintains the consistent
+performance of a system in the presence of uncertainty, variations in parameters.
+Consequently, asymptotically global synchronization control of the chaotic system
+guarantees to converge the error dynamics to the equilibrium point.
+In this article, we investigate the three-dimensional chaotic generalized Lotka-Volterra
+system, a more general model than the competitive predator-prey examples of
+Lotka-Volterra types. We examine the properties of the model, including equilib-
+rium analysis, dissipative properties, the maximum Lyapunov exponent, and slow
+manifold analysis. We use linear feedback control to stabilize the model at its equi-
+librium points. Our goal is to synchronize two identical GLV models with the same
+drive variable and different initial conditions using complete replacement synchro-
+nization in an ecological context. Prey population is assumed to be in abundance in
+such a way that predators from nearby patches also feed upon it. To maintain syn-
+chronization indefinitely with only small adjustments within a two-patch system,
+we design the active control law(when system parameters are known) and adaptive
+control law (when system parameters are unknown) mathematically and validate
+the analytical results through numerical simulation.
+2.
+The Model
+The generalized Lotka-Volterra equations are autonomous and deterministic.
+The dynamics of the model in a more generalized form are defined as
+˙xi(t) = xi(t)(ri + fi(x1, x2, . . . , xn)),
+with initial condition
+xi(0) = x(i,0), for i ∈ {1, 2, . . ., n}.
+where n represents the number of species, xi(t) is the size of population i, ˙xi(t) is
+the time derivative of species i, t is the time variable, x(i,0) is the initial population
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+5
+of species i, ri is the self-growth of species i, and fi(x1, x2, x3, . . . , xn) is the nonlin-
+ear multi-variable function with intra and inter-species competition terms for each
+i. Although the populations are usually measured in integer numbers, xi(t) is real
+for each i and can be interpreted as density, biomass or some other measure which
+correlates with the number of species. Let Rn denotes the Euclidean space and the
+function fi is a continuous, smooth and real-valued function for each i. We make
+the following assumptions on fi for i ∈ {1, 2, . . ., n} [15].
+(i) fi, for each i ∈ {1, 2, . . ., n} is bounded on a domain D ⊂ Rn.
+(ii) There exist constant Ki > 0 for each i ∈ {1, 2, . . ., n} such that
+||fi(X) − fi(Y )|| ≤ Ki ||X − Y ||
+∀
+X, Y ∈ D ⊂ Rn.
+2.1. Model with linear functional response. The generalized Lotka Volterra
+(GLV) model and its variant have been studied by many authors [16],[17],[18]. Our
+main focus is on three-dimensional GLV chaotic system, which has been devised by
+Samardzija and Greller in 1988 [19]. We assume that the vector field for the model
+holds the above mentioned properties and takes the following form
+˙x1 = x1(1 − x2 + rx1 − px3x1),
+˙x2 = x2(−1 + x1),
+˙x3 = x3(−q + px2
+1).
+(1)
+where x1, x2, x3 are the state variables representing prey, middle predator and
+top predator populations respectively. Here p, q, r are positive parameters. Au-
+thors [18] found the system chaotic in particular parametric range p = 2.9851, q =
+3, r = 2 and shown interesting complex dynamical behaviour.
+For this set of
+parameter values, the orbit of all three states for two different initial conditions
+((1.0023, 1.0589, 0.6503) and (1.0023 + 10−3, 1.0589 + 10−3, 0.6503 + 10−3)) has sen-
+sitive dependence on initial conditions (SDIC) which is displayed in figure 2.1. Fig-
+ure 2.1 characterizes the SDIC in the system where the trajectories with initial
+condition (1.0023 + 10−3, 1.0589 + 10−3, 0.6503 + 10−3)) dominate over trajectories
+with initial condition ((1.0023, 1.0589, 0.6503) in long run.
+Further, we display the dynamics of GLV system for three different set of parame-
+ters to characterize its parameter- sensitivity. Different sets of parameters involved
+
+6
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Time
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+Prey
+IC: (1.0023, 1.0589, 0.6503)
+IC: (1.0033, 1.0599, 0.6513)
+(a)
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Time
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+Middle Predator
+IC: (1.0023, 1.0589, 0.6503)
+IC: (1.0033, 1.0599, 0.6513)
+(b)
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Time
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+Top Predator
+IC: (1.0023, 1.0589, 0.6503)
+IC: (1.0033, 1.0599, 0.6513)
+(c)
+Figure 1.
+(a), (b) and (c): Time series of x1, x2, and x3 for
+two nearby initial conditions ((1.0023, 1.0589, 0.6503) and (1.0023+
+10−3, 1.0589 + 10−3, 0.6503 + 10−3)).
+in the system are taken as
+(p, q, r) ∈ {(2.0451, 2.129, 2), (2.9851, 2.99, 2.1), (2.98098, 2.9799, 2)}.
+For simulation, we fix the initial condition at x1(0) = 1.0023, x2(0) = 1.0589, x3(0) =
+0.6503 for the GLV system. Figures 2, 3 and 4 show three dimensional attractor
+and two dimensional projections of the system on (x1, x2), (x1, x3) and (x2, x3)
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+7
+planes. From these figures, it can be inferred that the sensitivity of the system on
+parameters can help in restoring the hidden order out of its chaotic dynamics.
+(a)
+(b)
+(c)
+(d)
+Figure 2. (a): 3D attractor; (b), (c) and (d): 2D projections of
+the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively
+for (p, q, r) = (2.0451, 2.129, 2).
+Next subsections presents two other variants of GLV model with different func-
+tional responses.
+2.2. Model with Cyrtoid type (HT II) functional response. To elucidate
+the role of functional response, we replace the linear interaction term between prey
+and middle predator with Holling type II functional response ( (
+x1
+x1+d)x2).
+The
+ecological meaning of the non-linear interaction of prey with middle predator is
+that the prey’s contribution to the middle predator growth rate is x1x2
+x1+d. Using type
+II functional response, the dynamics of new GLV model are proposed as
+˙x1 = x1 − x1x2
+x1 + d + rx2
+1 − px1
+2x3,
+˙x2 = −x2 + ( x2x1
+x1 + d),
+˙x3 = −qx3 + px3x2
+1.
+(2)
+For simulation, we take the parametric values and initial condition as p = 2.514, q =
+2.9089, r = 2.1990507, d = .00198 and x1(0) = 1.78, x2(0) = 0.5020, x3(0) = 1.01
+
+(X2,x3) plane2D projection of the attractor on
+1.8
+1.6
+1.4.1
+1.2
+1.31.2
+3
+X
+1
+0.8
+0.6
+0.4
+0.7
+0.8
+0.9
+1(X,,x,) plane2D projection of the attractor on
+1.8
+1.6
+1.41.3
+1.4
+1.5
+1.61.2
+X
+1
+0.8
+0.6
+0.4
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2(x,X2) plane2D projection of the attractor on
+1.3
+1.21.3
+1.4
+1.5
+1.60.9
+0.8
+0.7
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2 AttractorThe Generalised Lotka Volterr1.6
+1.4
+1.2
+1
+X
+11.5
+3
+X
+0.5
+1.2
+1.1
+1
+0.9
+0.8
+0.8
+0.68
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+(a)
+(b)
+(c)
+(d)
+Figure 3. (a): 3D attractor ; (b), (c) and (d): 2D projections of
+the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively
+for (p, q, r) = (2.9851, 2.99, 2.1).
+respectively. Figure 5 displays the three-dimensional phase portrait of GLV system
+with HT II functional response which is a ‘stable focus’. Thus, a change in functional
+response in GLV system can lead to stable dynamics for a suitable set of parameter
+values and initial conditions.
+2.3.
+Model with Sigmoid type (HT III) functional response. Next, we
+change interaction term between prey and top predator population with Holling
+type III functional response. The inclusion of Holling type III functional response
+increases the search activity of top-predator for prey. With assumption of increasing
+prey density, the dynamics of new GLV model with Holling type III functional
+response are proposed as
+˙x1 = x1 − x1x2 + rx2
+1 − p x2
+1x3
+x2
+1 + d,
+˙x2 = −x2 + x1x2,
+˙x3 = −qx3 + p x2
+1x3
+x2
+1 + d.
+(3)
+
+(X2,X3) plane2D projection of the attractor on
+0.9
+0.81.04
+1.06
+1.08
+1.10.7
+0.6
+0.5
+0.9
+0.92
+0.94
+0.96
+0.98
+1
+1.02
+2(X,x) plane2D projection of the attractor on
+0.9
+0.81.1
+1.15
+1.20.6
+0.5
+0.85
+0.9
+0.95
+1.05
+X(X,X2) plane2D projection of the attractor on
+1.1
+1.051.1
+1.15
+1.20.95
+0.9
+0.85
+0.9
+0.95
+1.05AttractorThe Generalised Lotka Volterr
+0.9
+0.81.2
+1.1
+1
+X
+10.6 ~
+0.5
+1.1
+1.05
+1
+0.95
+0.9
+0.9
+0.8SYNCHRONIZATION OF CHAOS IN ECOLOGY
+9
+(a)
+(b)
+(c)
+(d)
+Figure 4. (a): 3D attractor; (b), (c) and (d): 2D projections of
+the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively
+for (p, q, r) = (2.98098, 2.9799, 2).
+(a)
+Figure 5. (a) Three-dimensional attractor of GLV equations with
+HT II functional response.
+For parameter values p = 7.34, q = 2.0, r = 0.507, d = 3.198, the system (3) has
+‘limit cycle’-like attractor which means model (3) has stable dynamics.
+For each variation, we have found different parameters values for which models (1),
+(2), and (3) show completely different dynamics. We infer that the GLV model’s
+unstable dynamics with linear function response can be turned into stable dynamics
+when linear functional response is altered by HT II or HT III. However, this may
+
+tional ResponseThe GL Attractor with HT II Fun2
+1.5
+X
+1X
+0.5
+0
+0.6
+0.4
+0.2
+0
+0.5(X2,X3) plane2D projection of the attractor on
+0.74
+0.72
+0.71
+1.05
+1.10.68
+3
+X
+0.66
+0.64
+0.62
+0.6
+0.8
+0.85
+0.9
+0.95(Xj,X3) plane2D projection of the attractor on
+0.74
+0.72
+0.7.02
+1.04
+1.060.68
+3
+X
+0.66
+0.64
+0.62
+0.6
+0.94
+0.96
+0.98
+1
+1(x,x,) plane2D projection of the attractor on
+1.1
+1.05.02
+1.04
+1.06x2 0.95
+0.9
+0.85
+0.8
+0.94
+0.96
+0.98
+1
+1 AttractorThe Generalised Lotka Volterr
+0.751.05
+1
+X
+10.7
+3
+0.65
+0.6
+1.1
+1
+0.9
+0.8
+0.9510
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+Figure
+6. Three-dimensional attractor of
+generalised Lotka
+Volterra equations with HT III functional response.
+not be very effective for arbitrarily given scenario. A case in point- predators usu-
+ally do not follow predation rate as HT II or HT III functional response when the
+prey population is in abundance. To overcome this problem, we control chaos in the
+model (1) through synchronization and achieve complete replacement synchroniza-
+tion in two coupled GLV models following linear functional response. Since GLV
+system has been shown to have a chaotic attractor for various set of parameters.
+Out of these sets, we pick the set of parameters p = 2.9851, q = 3, r = 2 as in [18]
+for further study.
+3.
+Mathematical Properties
+In this section, we discuss the dynamical and analytical properties of system
+(1) including positive Lyapunov exponent, equation of slow manifold, in-variance,
+dissipation, stability of feasible equilibrium points, and control of instability of
+unstable equilibrium points.
+3.1.
+Lyapunov exponents. For three-dimensional system, the local behaviour
+of the dynamics varies along three orthogonal directions in state space. In a given
+chaotic system , nearby initial conditions may be moving apart along one axis,
+and moving together along another. The Lyapunov exponent describes the average
+rate of separation between two nearby trajectories with different initial conditions
+subject to a flow [3]. Where a positive Lyapunov exponent confirms chaos in the
+system. For simulation, we take the parameter values of the system (1) as p =
+
+The Generalised Lotka Volterra Attractor with Holling Type Ill Functional Response
+1.4
+1.22.4
+2.2
+2
+1.8
+1.6
+1.4
+1.20.8~
+3
+0.6
+0.4 
+0.2 
+0>
+1.3
+1.2
+1.1
+1
+0.9
+0.8
+0.7
+1
+0.6
+0.8
+0.6
+X2
+0.5
+0.4SYNCHRONIZATION OF CHAOS IN ECOLOGY
+11
+2.0451, q = 2.129, r = 2. The dynamics of Lyapunov exponents are shown in figure
+7. The Lyapunov exponents of model (1) are as follows
+Figure 7. Dynamics of Lyapunov spectrum of system (1).
+L1 = 0.0138667 > 0, L2 = −0.275762 < 0, L3 = −0.293347 < 0.
+where L1 is the indicator of chaos in the system (1).
+3.2. Equation of slow manifold. The infusion of geometric and topological tech-
+niques in chaos theory motivates mathematicians to study the underlying geometric
+structures. In this line, expression of slow manifold permits to restore a part of the
+deterministic property of the system that was lost because of SDIC. To find an
+equation of slow manifold, we consider the system (1) as slow-fast autonomous dy-
+namical system (S-FADS). In S-FADS, variables are separated into two groups:, one
+is group of fast variable and other is of slow variables where slow variables are used
+to determine the behaviour of whole system. To get the equation, we consider that
+the slow manifold is locally defined by a plane orthogonal to tangent system’s left
+fast eigenvector. Under the set of parameter values p = 2.9851, q = 3, r = 2, the
+
+onents
+-L,=0.0138667
+-L,=-0.275762
+L.=-0.293347
+3Dynamics of Lyapunov Exp
+2
+nents800
+1000Lyapunov Expo
+0
+-3 
+0
+200
+400
+600
+Time12
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+equations of GLV model (1) can be given as
+(4)
+˙x1 = x1(1 − x2 + 2x1 − 2.9851x3x1),
+˙x2 = x2(−1 + x1),
+˙x3 = x3(−3 + 2.9851x2
+1).
+The Jacobian matrix J at point x = (x1, x2, x3)T is obtained as
+J =
+
+
+1 − x2 + 4x1 − 5.9702x1x3
+−x1
+−2.9851x2
+1
+x2
+−1 + x1
+0
+5.9702x1x3
+0
+−3 + 2.9851x2
+1
+
+
+Let λ1(x1, x2, x3) be a real, negative and dominant Eigen value ( i.e, fast Eigen
+value) for Jacobian matrix in a large part of attractor’s phase space domain.
+Furthermore, we assume that λ2(x1, x2, x3) and λ3(x1, x2, x3) be two slow Eigen
+values. Then Eigen vector ZT
+λ1 corresponding to fast Eigen value λ1 of JT (x1, x2, x3)
+is given by
+(5)
+|J − λ1I| Zλ1 = 0.
+where I is 3 × 3 identity matrix. Equation (6) gives,
+ZT
+λ1 =
+
+
+(−1 + x1 − λ1)(−3 + 2.9851x2
+1 − λ1)
+x1(−3 + 2.9851x1)2)
+2.9851x2
+1(−1 + x1 − λ1)
+
+ .
+On the attractive parts of phase space (where J(x) has a fast Eigen value λ1), the
+equation of the slow manifold is given by
+(6)
+˙x(t).ZT
+λ1 = 0.
+We use the equation (7) to define the equation of slow manifold. With the substi-
+tution of ˙x(t) and ZT
+λ1 in the equation (7), we write the equation of slow manifold
+as
+λ2
+1(x1 − x1x2 + 2x2
+1 − 2.9851x2
+1x3) + λ1(−4x1 + 7x2
+1 − 4.9851x3
+1 − 3x1x2 + 2.9851x3
+1x2−
+5.97020x4
+1 − 2.985100x2
+1x3 + 2.9851x3
+1x3) + (3x1 + x2
+1 − 8.985100x3
+1 − 2.9851x4
+1 + 5.970200x5
+1
++ 8.95530x1x3 − 17.910600x2
+1x3 − 0.044478x3
+1x3 + 17.821644x4
+1x3 − 8.91082x3
+1x3) = 0.
+(7)
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+13
+where λ1 is fast Eigen value of J(x). Because λ1(x1, x2, x3) is uncertain Eigen value,
+it is not easy to use this implicit equation to draw a slow manifold representation
+in the three dimensional phase space.
+3.3. Invariance property.
+Theorem 3.1. Let the system in vector notation is given as
+(8)
+˙x(t) = H(x(t)) =
+
+
+H1(x1, x2, x3)
+H2(x1, x2, x3)
+H3(x1, x2, x3)
+
+
+H1(x1, x2, x3) = x1(1 − x2 + rx1 − px3x1),
+H2(x1, x2, x3) = x2(−1 + x1),
+H3(x1, x2, x3) = x3(−q + px1
+2).
+where H1, H2 and H3 are continuously differentiable. Assume that H is locally
+Lipschitz and generates a flow φt(x). Let
+L : D ⊂ R3 → R3
+be a continuously differentiable function on a domain D ⊂ R3 such that ˙L(x) ≤ 0
+in D, then the largest invariant set Σ ⊂ D is the set; where ∇L.H(x) = 0 ∀x ∈ Σ.
+Proof. Consider
+L : D ⊂ R3 → R3
+be a continuously differentiable function on a domain D ⊂ R3 and defined as
+(9)
+L(x1, x2, x3) = x2
+1 + x2
+2 + x2
+3
+2
+.
+Equation (9) gives,
+(10)
+˙L(x1, x2, x3) = x1 ˙x1 + x2 ˙x2 + x3 ˙x3.
+The set D ⊂ R3 is said to be an invariant set under the flow φt if for any point
+x ∈ D
+φt(x) ∈ D ∀ t ∈ R.
+Let Σ be a smooth closed surface without boundary in D ⊂ R3 and suppose that
+n is a normal vector to the surface Σ at (x1, x2, x3). If we have
+(11)
+n. < ˙x1, ˙x2, ˙x3 >= 0 ∀ (x1, x2, x3) ∈ Σ.
+
+14
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+Let us consider Σ be the x1x2 plane i.e. x3 = 0. Note that the vector (0, 0, 1) is
+always normal to Σ and at the point (x1, x2, 0) ∈ Σ. So we have,
+( ˙x1, ˙x2, ˙x3) = (x1(1 − x2 + rx1 − px3x1), x2(−1 + x1), 0).
+Thus,
+⟨(0, 0, 1).(x1(1 − x2 + rx1 − px3x1), x2(−1 + x1), 0)⟩ = 0.
+Similar arguments can be verified for x1 and x2 planes which directs that each
+coordinate plane is an invariant subset. It implies that for any given positive initial
+condition, x1(t), x2(t) and x3(t) are positive for all t that is any trajectory starting
+in R3
++ can not cross the co-ordinate planes and it shows that R3
++ is an invariant set
+for the system.
+□
+3.4. Dissipation.
+Theorem 3.2. Consider the autonomous vector field
+˙x(t) = H(x) for x ∈ R3,
+and assume that it generates a flow φt(x).
+Let D0 is a domain in R3 which is
+supposed to have a volume V0, and φt(D0) is its evolution under the flow. If V (t)
+is the volume of Dt, then the time rate of change of volume is given as
+|dV
+dt |t=0 =
+�
+D0
+∇.Hdx.
+The system (1) is dissipative if its time-t map decreases volume for all t > 0.
+Proof. Dissipation in any dynamical system manifests itself as contraction of the
+phase volume on average. To check this, we express the volume V (t) in the following
+form using the definition of the Jacobian of transformation as
+(12)
+V (t) =
+�
+D0
+|dφt(x)
+dx
+|dx.
+Expanding φt(x) in the neighbourhood of t = 0.
+Since the vector field H(x) is
+smooth enough to have a tangent plane in each point on R3 so we can expand φt(x)
+by Taylor series expansion. Hence we get,
+(13)
+φt(x) = x + ˙xt + O(t2) for t → 0
+Since
+(14)
+˙x(t) = H(x),
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+15
+The equation (15) gives,
+(15)
+φt(x) = x + H(x)t + O(t2) for t → 0.
+It follows that
+∂φ
+∂x = I + ∂H
+∂x t + O(t2),
+(16)
+|∂φ
+∂x| = |I + ∂H
+∂x t| + O(t2).
+Here I is 3 × 3 identity matrix so det I will be equal to 1.
+By expanding the
+expression (17) by using expansion of determinant, we get the following
+(17)
+|∂φ
+∂x| = 1 + trace(∂H
+∂x )t + O(t2).
+Note that
+(18)
+trace(∂H
+∂x ) = ∇.H,
+therefore, we have
+(19)
+V (t) = V0 +
+�
+D0
+((∇.H)t + O(t2))dx.
+It gives
+(20)
+|dV
+dt |t=0 =
+�
+D0
+∇.Hdx,
+i.e. if the volume shrinks then divergence of vector field will be strictly negative [3].
+Now considering the equations of model (1) in vector notation and computing its
+Jacobian
+(21)
+J(x1, x2, x3) = ∂H
+∂x .
+The Jacobian J(x1, x2, x3) of the model is given by
+(22)
+
+
+1 − x2 + 2rx1 − 2px1x3
+−x1
+−px12
+x2
+−1 + x1
+0
+2px1x3
+0
+−q + px2
+1
+
+ ,
+we take the parameter values as
+(23)
+p = 2.9851,
+q = 3,
+r = 2.
+
+16
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+The above argument shows that the G.L.V dynamical system will be dissipative if
+the generalized divergence should be less than zero, i.e.
+(24)
+�
+i
+∂Hi
+∂xi
+< 0.
+where Einstein summation has been used. The divergence of vector field H on R3
+is as follows
+(25)
+∇.H = ∂H1
+∂x1 + ∂H2
+∂x2 + ∂H3
+∂x3 ,
+∇.H = −q − x2 + (2r + 1 + px1 − 2px3)x1.
+Hence system (1) will be dissipative if the following condition is satisfied,
+(26)
+(2r + 1 + px1 − 2px3)x1 < q + x2.
+□
+3.5. Existence and uniqueness of solution. Since we are dealing with a popula-
+tion dynamics model, hence, the existence of at least one solution is must. However,
+the uniqueness of existed solution will give more appropriate results. Here we men-
+tion two theorems for which solutions of the system (1) uniquely exist for all t > 0
+(complete detail of the proof can be seen in [18]).
+Theorem 3.3. If the functions f1, f2 and f3 satisfy assumptions (1) and (2),
+mentioned in section , then continuity of functions fi for i ∈ {1, 2, 3} assures that
+atleast one solution exists for the dynamics of system (1) in region D × I where
+I = (0, T ] and the spatial boundary of region D ⊂ R3 is defined as
+D = {x = (x1, x2, x3) : max|xi| ≤ M, for i ∈ {1, 2, 3}}
+where M > 0.
+Theorem 3.4. Let D be a closed subspace of complete normed linear space R3.
+Consider H : D ⊂ R3 → D is Lipschitz continuous so that there exist 0 < K < 1
+such that
+||H(χ) − H(ψ)|| < K||χ − ψ||
+with
+K = T.max(1 + 2M + 2rM + 4pM 2, 1 + 2M, q + 2pM 2)
+For 0 < K < 1, H(t) will be a contraction map. With the help of Banach fixed point
+theorem, it can be ensured that 0 < K < 1 is sufficient condition for uniqueness of
+solution of the system (1).
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+17
+3.6.
+Stability of feasible equilibrium points. The equilibrium points of system
+(1) are solutions of following algebraic equations
+xx1(1 − x2 + rx1 − px3x1) = 0,
+x2(−1 + x1) = 0,
+x3(−q + px2
+1) = 0.
+(27)
+We obtain five equilibrium points by solving the system (27),
+(28)
+X∗
+0 =
+
+
+0
+0
+0
+
+ ,
+X∗
+1 =
+
+
+1
+1 + r
+0
+
+ , X∗
+2 =
+
+
+�
+q
+p
+0
+1+r√ q
+p
+√pq
+
+ , X∗
+3 =
+
+
+− 1
+r
+0
+0
+
+ , X∗
+4 =
+
+
+−
+�
+q
+p
+0
+−1+r√ q
+p
+√pq
+
+ .
+From an ecological point of view, negative population density is not realistic as
+the population can not be negative, therefore, we take the vector x = (x1, x2, x3)
+as an element of R3
++. R3
++ is defined as
+(29)
+R3
++ = {X ∈ R3 : xi ≥ 0 for i ∈ {1, 2, 3}}.
+Since equilibrium points X∗
+0, X∗
+1 and X∗
+2 are elements of the set Int(R3
++), therefore,
+we study the local stability of ecologically feasible equilibrium points X∗
+0, X∗
+1 and
+X∗
+2.
+(I) Stability of Trivial Equilibrium Point X∗
+0.
+Theorem 3.5. Consider the dynamics of the model (1) in the following form
+(30)
+˙x = H(x) = Ax + f(x) for x ∈ R3.
+If following three conditions are satisfied
+(i) Constant matrix A3×3 has 3 Eigen-values with non-zero real part,
+(ii) f(x) is smooth and
+(iii) lim||x||→0
+||f(x)||
+||x||
+= 0,
+then in a neighbourhood of the critical point X∗
+0 = (0, 0, 0), there exists stable
+and unstable manifolds Ws and Wu with the same dimensions ns and nu as
+the stable and unstable manifolds Es and Eu of the system
+˙Z(t) = AZ.
+In x = 0, Es and Eu are tangent to Ws and Wu[20].
+
+18
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+Proof. For GLV system, X∗
+0 = (0, 0, 0) is trivial equilibrium point. Here we
+check all three mentioned conditions of theorem I.
+(i) Note that for model (1), the constant matrix A3×3 is
+A =
+
+
+
+
+1
+0
+0
+0
+−1
+0
+0
+0
+−q
+
+
+
+
+The determinant of the matrix A is non zero if q ̸= 0. Since, we have
+taken q = 3, therefore, all Eigen values of A have non zero real part.
+(ii) Since functions f1(x1, x2, x3), f2(x1, x2, x3) and f3(x1, x2, x3) for model
+(1) are considered as
+x1(−x2 + rx1 − px3x1)
+= f1(x1, x2, x3),
+x2x1
+= f2(x1, x2, x3),
+x3(px2
+1)
+= f3(x1, x2, x3).
+All three functions are continuous and have continuous partial derivative
+for all x ∈ R3 which implies that f(x) is smooth on R3. Hence, the
+second condition also holds.
+(iii) For any x ∈ R3, converting the Cartesian coordinates into spherical
+coordinates by making the following transformation
+
+
+x1 = r sin θ cos φ
+x2 = r sin θ sin φ
+x3 = r cos θ
+
+ ,
+where r ≥ 0, 0 ≤ θ ≤ π and 0 ≤ φ ≤ π.
+Using this transformation in model (1), we have
+lim
+||x||→0
+||f(x)||
+||x||
+= 0.
+Hence, the critical point (0, 0, 0) of system (1) is of the same type of
+critical point of the system
+˙Z(t) = AZ.
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+19
+The Eigen values of A are 1, −1 and −q which implies that (0, 0, 0) is
+saddle node for the system ˙Z(t) = AZ. Therefore, the trivial steady
+state X∗
+0 = (0, 0, 0) of the model (1) is a saddle point.
+□
+(II) Stability of Axial Equilibrium Point X∗
+1.
+The Jacobian matrix of model (1) for parameter values p = 2.9851, q =
+3, r = 2 is given as
+(31)
+J =
+
+
+1 − x2 + 4x1 − 5.9702x1x3
+−x1
+−2.9851x2
+1
+x2
+−1 + x1
+0
+5.9702x1x3
+0
+−3 + 2.9851x2
+1
+
+ .
+Jacobian matrix (31) of the model (1) about X∗
+1 = (1, 3, 0) yields the following
+Jacobian matrix
+(32)
+JX∗
+1 =
+
+
+2
+−1
+−2.9851
+3
+0
+0
+0
+0
+−.0149
+
+ .
+The characteristic equation |JX∗
+1 − λI| = 0 of matrix (32) is given as
+(33)
+λ3 − (1.9851)λ2 + (2.9702)λ + 0.0447 = 0.
+The characteristic equation (33) has the following Eigen values
+(34)
+λ1 = −0.014900, λ2 = 1 +
+√
+2ι, λ3 = 1 −
+√
+2ι.
+Since λ2 and λ3 have positive real parts, it implies that X∗
+1 = (1, 3, 0) is
+unstable equilibrium point.
+(III) Stability of Planer Equilibrium Point X∗
+2.
+The Jacobian matrix of model (1) for parameter values p = 2.9851, q =
+3, r = 2 is given as
+(35)
+J =
+
+
+1 − x2 + 4x1 − 5.9702x1x3
+−x1
+−2.9851x2
+1
+x2
+−1 + x1
+0
+5.9702x1x3
+0
+−3 + 2.9851x2
+1
+
+ .
+
+20
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+Jacobian matrix (35) of the model (1) about X∗
+2 = (1.002493, 0, 1.4159) yields
+the following Jacobian matrix
+(36)
+JX∗
+2 =
+
+
+3.004986
+−1.002493
+3.000002
+0
+.002493
+0
+6.00887
+0
+.000002
+
+ .
+The characteristic equation |JX∗
+2 − λI| = 0 of matrix (36) is given as
+(37)
+λ3 + (0.997507)λ2 + (18.027423)λ − 0.044949 = 0.
+The characteristic equation (37) has the following Eigen values
+(38)
+λ1 = −0.002493, λ2 = −0.5 + 4.216ι, λ3 = −0.5 − 4.216ι.
+Since all three Eigen-values of matrix (36) have negative real parts, it shows
+that X∗
+2 = (1002493, 0, 1.4159) is locally stable equilibrium point.
+It is clear that planer equilibrium point is stable whereas trivial and axial equilib-
+rium points are unstable equilibrium points. Since trivial equilibrium point refers
+the zero density of all three species, therefore, we neglect the instability of trivial
+equilibrium point. From an ecological point of view, we mainly focus on non-trivial
+unstable equilibrium point X∗
+1 and try to stabilize it by adding some external control
+inputs.
+3.7.
+Control of instability of axial equilibrium point. In order to suppress
+instability to X∗
+1 = (1, 3, 0), we consider the controlled GLV system in the following
+form
+(39)
+˙x1
+= x1(1 − x2 + rx1 − px3x1) + u1,
+˙x2
+= x2(−1 + x1) + u2,
+˙x3
+= x3(−q + px2
+1) + u3.
+We introduce the external control law
+(40)
+u1
+= −µ1(x1 − 1),
+u2
+= −µ2(x2 − 3),
+u3
+= −µ3(x3 − 0).
+with x1, x2, x3 as the feedback variable and µ1, µ2, µ3 as the positive feedback
+gains. We substitute control law (40) into (39) and hence, the controlled system
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+21
+(39) takes the following form
+(41)
+˙x1 = x1(1 − x2 + rx1 − px3x1)) − µ1(x1 − 1),
+˙x2 = x2(−1 + x1) − µ2(x2 − 3),
+˙x3 = x3(−q + px2
+1) − µ3(x3 − 0).
+Theorem 3.6. The equilibrium point X∗
+1 = (1, 3, 0) of the model (1) will be asymp-
+totically stable if positive gains µ1, µ2 and µ3 satisfy the following inequalities[21]
+(42)
+µ1
+> 2,
+µ1µ2
+> 1 + 2µ2,
+µ1µ2(µ3 + 0.0149)
+> µ2(2µ3 + 2.2528) + µ3 + 0.0149.
+Proof. The Jacobian matrix J of the system (41) is given by
+(43)
+
+
+1 − x2 + 2x1(r − px3) − µ1
+−x1
+−px2
+1
+x2
+−1 + x1 − µ2
+0
+2px1x3
+0
+−q + px2
+1 − µ3
+
+
+Let us consider that
+(44)
+e1
+= (x1 − 1),
+e2
+= (x2 − 3),
+e3
+= (x3 − 0).
+From (44), we get the error system as
+(45)
+˙e1 = (2 − µ1)e1 − e2 − 2.9851e3,
+˙e2 = 3e1 − µ2e2,
+˙e3 = −(0.014900 + µ3)e3.
+The system (1) with constant and known parameters, will be stabilized to steady
+state X∗
+1 = (1, 3, 0), if error system (45) stabilized to (0, 0, 0).
+To study the stability of equilibrium point (0, 0, 0) of error system, we consider the
+Lyapunov function L(e1, e2, e3) as:
+(46)
+L = 1
+2(e2
+1 + e2
+2 + e2
+3).
+The time derivative of L in the neighbourhood of (0, 0, 0) is given as
+(47)
+˙L = (2 − µ1)e2
+1 + 2e1e2 − µ2e2
+2 − 2.9851e1e3 − (0.0149 + µ2)e2
+3.
+
+22
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+The time derivative of Lyapunov function can be re-written in the following form
+(48)
+˙L = eT Me.
+where e = ((x1 − 1), (x2 − 3), (x3 − 0)) is the error vector in R3, eT is the transpose
+of error vector e and the matrix M is 3 × 3 is given as
+M =
+
+
+2 − µ1
+1
+−1.49250
+1
+−µ2
+0
+−1.49250
+0
+−(0.0149 + µ3)
+
+
+According to Lyapunov stability theory, the equilibrium point (0, 0, 0) of system
+(45) will be asymptotically stable if ˙L < 0. And ˙L < 0 if matrix M will be negative
+definite.
+Considering this, we find that mentioned condition will be fulfilled if
+positive feedback gains µ1, and µ2 satisfy the following inequalities,
+(49)
+µ1
+> 2,
+µ1µ2
+> 1 + 2µ2,
+µ1µ2(µ3 + 0.0149)
+> µ2(2µ3 + 2.2528) + µ3 + 0.0149.
+□
+4. Complete Replacement Synchronization
+To investigate complete replacement synchronization techniques, we consider two
+identical chaotic GLV systems having the same parameter but different initial con-
+ditions. Since the coupling between models is needed to maintain the synchronous
+state, we couple the states of both models with two controllers and drive the re-
+sponse system with prey species x1. For this, we remove prey from response system,
+and drive its counterpart. Here, we can think of prey species x1 as a driving vari-
+able for response system with an assumption that it is superfluous in the system of
+two coupled GLV models. This construction gives us a new five-dimensional drive-
+response system having drive and response variables as (x1d, x2d, x3d) and (x2r, x3r)
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+23
+respectively. The coupled chaotic system with x1d drive configuration is as follows:
+(50)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(Drive System)
+˙x1d
+= x1d(1 − x2d + rx1d − px3dx1d),
+˙x2d
+= x2d(−1 + x1d),
+˙x3d
+= x3d(−q + px2
+1d),
+( Response System)
+˙x2r
+= x2d(−1 + x1d) + u1,
+˙x3r
+= x3r(−q + px2
+1d) + u2.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+x2d(0) ̸= x2r(0) and x3d(0) ̸= x3r(0).
+4.1. Active control law for stability of synchronization manifold.
+Theorem 4.1. The identical synchronization manifold Ω = [x2d = x2r, x3d = x3r]
+is globally asymptotically stable for the coupling between drive and response system
+in equation (24) for positive µ1 and µ2, where µ1 and µ2 are large enough such that
+µ1 + 1 > x1d,
+µ2 + q > px2
+1d.
+Proof. We consider drive-response system given by equations (50) and add the uni-
+directional controllers to the response system through the linear positive constants
+µ1 and µ2. We choose two controllers for response system as
+(51)
+u1
+= −µ1(x2r(t) − x2d(t)),
+u2
+= −µ2(x3r(t) − x3d(t)).
+Existence of all forms of identical synchronization in any dynamical system
+(chaotic or not), are really manifestations of dynamical behaviour restricted to
+a flat hyper-plane in the phase space i.e. to say motion is continually confined to a
+hyper-plane which can be referred as synchronization manifold [22]. Therefore, we
+consider the identical synchronization manifold of the systems equation (50) as
+Ω = [x2d = x2r, x3d = x3r].
+Further, we consider the errors between states of drive and response systems of
+system (50) as
+(52)
+e2(t)
+= x2r(t) − x2d(t),
+e3(t)
+= x3r(t) − x3d(t).
+
+24
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+The dynamics of error system is as follows
+(53)
+˙e2
+= (−1 − µ1 + x1d)e2,
+˙e3
+= (−q − µ2 + px2
+1d)e3.
+where, we are now interested in the stability of origin. The Jacobian of right side
+of (53) is given by
+(54)
+J(e2, e3) =
+�
+−1 + µ1 + x1d
+0
+0
+−q + µ2 + px2
+1d
+�
+We treat the response system (x2d(t), x3r(t)) as a separate system driven by x1d,
+then the solutions of equation (54) convey us about convergence and divergence of
+two initially nearby trajectories of {x2r(t), x2d(t)} and {x3r(t), x3d(t)}.
+Next, We analyse the possibility of synchronization using the Lyapunov function
+construction method. We consider the Lyapunov function as
+(55)
+L(e2, e3) = 1
+2(e2
+2 + e2
+3).
+(56)
+dL
+dt = e2 ˙e2 + e3 ˙e3.
+Plugging dynamics of errors (53) into (55), we get
+(57)
+dL
+dt = −[(µ1 + 1 − x1d)e2
+2 + (µ2 + q − px2
+1d)e2
+3],
+which will be strictly negative for following conditions on µ1 and µ2
+(58)
+µ1 + 1 > x1d,
+µ2 + q > px2
+1d.
+for all t > 0. Condition (58) ensures that we consider the bounded density of prey
+species, then we can bound the positive feedback gains. Thus, if µ1 and µ2 satisfy
+(58), then it can be assured that dL
+dt < 0 for all t > 0 or in other words, the complete
+replacement synchronization follows as e2 and e3 → 0 as t → ∞.
+□
+4.1.1. Numerical Simulation. Since, a suitable coupling can influence both fre-
+quency as well as chaotic amplitude, therefore, the states coincide (or nearby co-
+incide) and regime of synchronization sets in. Thus, it is pre-arranged that the
+chosen coupling should assist the coupled states in coincidence without perturbing
+their chaotic rhythm. We numerically integrate the System (50) and display results
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+25
+in figure 4.1.1, where drive and response systems are shown to synchronize when
+considered positive feed-back gain are chosen as µ1 = 0.000024, µ2 = 1.345.
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Time
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+Population
+x1d
+x2d
+ x3d
+y2r
+y3r
+(a)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Time
+-0.14
+-0.12
+-0.1
+-0.08
+-0.06
+-0.04
+-0.02
+0
+Errors
+(e2)
+(e3)
+(b)
+x2d
+y2r
+x2d vs y2r
+(c)
+(d)
+Figure 8. (a): solutions of drive and response systems plotted
+over time, (b): errors between drive and response systems over
+time,(c): synchronization plot of {x2d, x2r}, (d): synchronization
+plot of {x3d, x3r}
+4.1.2. Lyapunov Spectrum For Drive-Response System. Since the necessary condi-
+tion for the stability of the synchronization manifold is the negative largest trans-
+verse Lyapunov exponent. In the case of complete replacement synchronization,
+the transverse Lyapunov exponents are also known as conditional Lyapunov expo-
+nents. It is because Lyapunov exponents for the new system depend on the coupling
+from the drive[23]. The Lyapunov spectrum is a global indicator of the system’s
+
+X
+VS
+3d3
+X
+3d26
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+state, aggregating over the behaviour of the entire system trajectory in phase space.
+The typical approach for observing these transitions is to see a change in sign of
+the Lyapunov exponents of the system, as obtained from ensemble averages of the
+eigenvalues of the Jacobian matrix of system (50) [24]. We solve equations of system
+(50) and get the Jacobian matrix as
+(59)
+J(x1d, x2d, x3d, x2d, x3r) = A5×5, where A = [aij].
+The entries of the matrix A are given as:
+a11 = 1 − x2d + 2(r − px3d)x1d,
+a12 = −x1d,
+a13 = −px2
+1d,
+a21 = x2d,
+a22 = −1 + x1d,
+a31 = 2px1dx3d,
+a33 = −q + px2
+1d,
+a41 = x2d,
+a42 = µ1,
+a44 = −1 + x1d − µ1x2d,
+a51 = 2px3rx1d,
+a53 = µ2,
+a55 = −q + px2
+1d − µ2,
+a14 =
+a15 =
+a23 =
+a24 =
+a25 =
+a32 = 0,
+a34 =
+a35 =
+a43 =
+a45 =
+a52 =
+a54 = 0.
+Averaging the eigenvalues of Jacobian J over all phase space configurations set-up
+by the chaotic trajectory, we get the five Lyapunov exponents of the system of two
+coupled GLV models.
+For set of parameter values {p, q, r} = {2.9851, 3, 2}, Lyapunov exponents of the
+drive-response system are obtained as
+(60)
+L1 = −0.011320,
+L2 = −0.174464,
+L3 = −0.22221,
+L4 = −5.011,
+L5 = −5.0059.
+Since, all Lyapunov exponents are negative which confirms the stable synchroniza-
+tion manifold, therefore, it can be concluded that the states of coupled GLV systems
+are synchronized.
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+27
+0
+20
+40
+60
+80
+100
+Time
+-6
+-5
+-4
+-3
+-2
+-1
+0
+Lyapunov exponents
+Dynamics of Lyapunov exponents
+L1=-0.01132
+L2=-0.17446
+L3=-0.22221
+L4=-5.011
+L5=-5.0059
+Figure 9. Lyapunov Exponents of two GLV models coupled with
+positive feedback gains µ1 = 0.000024 and µ2 = 1.345.
+4.2. Adaptive control law for stability of synchronization manifold. Us-
+ing the method [25], we design non-linear adaptive controller for global complete-
+replacement synchronization of two chaotic GLV systems with unknown parameters.
+We consider the drive system as
+(61)
+˙x1d
+= x1d(1 − x2d + rx1d − px3dx1d),
+˙x2d
+= x2d(−1 + x1d),
+˙x3d
+= x3d(−q + px2
+1d).
+The response system is given by controlled chaotic system
+(62)
+˙x2r
+= x2r(−1 + x1d) + u1,
+˙x3r
+= x3r(−q + px2
+1d) + u2.
+The synchronization error between drive and response systems is defined as
+(63)
+e2(t)
+= x2r(t) − x2d(t)
+e3(t)
+= x3r(t) − x3d(t).
+The error dynamics between drive and response systems is calculated as :
+(64)
+˙e2
+= −e2 + x1de2 + u1,
+˙e3
+= −qe3 + px2
+1de3 + u2.
+In (64), unknown parameters p and q are to be determined by using parameter
+estimates P(t) and Q(t) respectively. For this purpose, we consider adaptive control
+
+28
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+laws u1 and u2 with positive feedback gains µ1 and µ2 as
+(65)
+u1
+= e2 − x1de2 − µ1e2,
+u2
+= Q(t)e3 − P(t)x2
+1de3 − µ2e3.
+Using control law (65) into the error dynamics (64), we get
+(66)
+˙e2
+= −µ1e2,
+˙e3
+= −(q − Q(t))e3 + (p − P(t))x2
+1de3 − µ2e3.
+We define parameter estimation error as
+(67)
+ep(t) = (p − P(t)),
+eq(t) = (q − Q(t)).
+Using (67), we can simplify the error dynamics (66) as
+(68)
+˙e2 = −µ1e2,
+˙e3 = −eq(t)e3 + ep(t)x2
+1de3 − µ2e3.
+Differentiating (67) with respect to t, we get
+(69)
+˙ep
+= − ˙P(t),
+˙eq
+= − ˙Q(t)
+Theorem 4.2. The identical synchronization manifold Ω = [x2d = x2r, x3d = x3r]
+is globally asymptotically stable for the coupling between derive and response system
+in equation (50) for positive µ1 and µ2.
+Proof. The identical synchronization manifold for the systems equation (24) can be
+written as
+Ω = [x2d = x2r, x3d = x3r].
+Using change of coordinates
+(70)
+e =
+�
+e2
+e3
+�
+=
+�
+x2d − x2d
+x3r − x3d
+�
+,
+such that Ω can be written
+Ω = (0, 0).
+Next, we use Lyapunov stability theory for finding an update law for the parameter
+estimates. we consider the quadratic Lyapunov function as
+(71)
+L = 1
+2(e2
+2 + e2
+3 + e2
+p + e2
+q)
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+29
+Note that Lyapunov function L is positive definite on R4. Differentiating L along
+the trajectories of (66) and (69). We get,
+(72)
+dL
+dt
+= e2 ˙e2 + e3 ˙e3 + ep ˙ep + eq ˙eq,
+dL
+dt
+= −µ1e2
+2 − µ2e2
+3 + ep(t)(− ˙P(t) + x2
+1de3) + eq(t)(− ˙Q(t) − e2
+3).
+We want error system to be asymptotically stable i.e.
+(73)
+dL
+dt < 0.
+In view of (72), we take the parameter update law as
+(74)
+˙P(t) = x2
+1de3,
+˙Q(t) = −e2
+3.
+By substituting the parameter update law (74) into Lyapunov function, we obtain
+time derivative of L as
+(75)
+dL
+dt = −µ1e2
+2 − µ2e2
+3,
+From (75), it is clear that ˙L is negative semi-definite function on R4. Thus, we can
+conclude that the synchronization error vector e(t) and the parameter estimation
+error are globally bounded, i.e.
+(76)
+[e2, e3, ep, eq]T ∈ L∞.
+We define µ = min{µ1, µ2}, then it follows from (75) that
+(77)
+dL
+dt ≤ −µ||e||2.
+Integrating the inequality (77) with respect to τ from 0 to t. We get,
+(78)
+� t
+0
+µ||e(τ)||2dτ ≤ L(0) − L(t).
+From (78) it follows that e ∈ L2 and hence, ˙e(t) ∈ L∞.
+With the help of Barbalat’s lemma [26],[27], we conclude that e(t) → 0 exponentially
+as t → ∞ for all initial conditions e(0) ∈ R2.
+□
+
+30
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+4.2.1. Numerical Simulation. For numerical simulations, we use the classical fourth-
+order Runge-Kutta method to solve the GLV system. The initial value of parameter
+estimates are taken as P(0) = 3.9, Q(0) = 4. The initial values of states of drive and
+response systems are taken as (x1d(0), x2d(0), x3d(0)) = (4, 1.4, 1.41) and (x2r(0),
+x3r(0)) = (1, 1.414) respectively. The effectiveness of control law is verified through
+simulation results which are shown in figure 4.2.1. Figure shows the solutions of
+system (50).
+It is clear that although the initial values are different, the error
+dynamics approach to zero as time goes to ∞. Therefore, our numerical results
+confirm that the amplitude and frequency of state variables of response system
+become same with the drive system under the designed control law.
+5. Conclusion
+This work examines predator-prey systems in ecosystems to understand how
+they contribute to sustainable environment. Our focus is on the use of Generalized
+Lotka-Volterra (GLV) equations to model the competition and trophic relationships
+between various species. We consider three different forms of three-dimensional GLV
+models, each with different functional responses (linear, Holling type II, and Holling
+type III). We find that the model with the linear functional response exhibits un-
+stable dynamics, where alteration in functional response can stabilize the system
+dynamics for a particular scenario. To stabilize the dynamics in patchy ecosystem,
+we focus on the GLV model with the linear functional response for the remainder of
+the study. We investigate its fundamental properties and also examine the stability
+of equilibrium points and the suppression of instability at equilibrium. Through
+computation of Lyapunov exponent, we find that the model is chaotic due to one
+positive Lyapunov exponent and has two unstable equilibrium points for the con-
+stant parameters p = 2.9851, q = 3, r = 2.
+Further, we investigate the synchronization of two chaotic GLV models using two
+control schemes: the Active Control Technique and the Adaptive Control Technique.
+We consider a configuration in which the prey population in the drive system acts
+as a driving variable for the response system, allowing the other two predator popu-
+lations to depend only on the prey population. Using the Active Control Technique,
+we apply two simple linear controllers to synchronize the states of the GLV systems.
+
+SYNCHRONIZATION OF CHAOS IN ECOLOGY
+31
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Time
+-1
+0
+1
+2
+3
+4
+5
+Population
+x1d
+x2d
+ x3d
+y2r
+y3r
+(a)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Time
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+Errors
+(e2)
+(e3)
+(b)
+x2d
+0
+y2r
+x2d vs y2r
+(c)
+0
+x3d
+0
+y3r
+x3d vs y3r
+(d)
+Figure 10. (a): solutions of drive and response systems over time,
+(b):
+errors between drive and response systems over time, (c):
+synchronization plot for {x2d, x2r}, (d): synchronization plot for
+{x3d, x3r} for coupling strengths µ1 = 0.0038 and µ2 = 2.
+These controllers are easy to implement and more straightforward than previous re-
+sults. The stability of synchronization manifold is ensured through the transition
+of positive conditional Lyapunov exponent to negative one. We also examine the
+synchronization of two chaotic GLV systems with unknown parameters using the
+Adaptive Control Technique. We design two adaptive laws of parameters using the
+Lyapunov stability theory to ensure global and exponential synchronization of the
+systems. Our results show that both the Active and Adaptive Control Techniques
+are effective for achieving global synchronization in chaotic systems.
+
+32
+SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY
+Funding
+The second author’s research was funded by the Science and Engineering Re-
+search Board (SERB), under two separate grants with grant numbers MTR/2018/000727
+and EMR/2017/005203.
+Disclosure statement
+The authors declare that they have no conflict of interest.
+References
+[1] Lorenz EN. Deterministic nonperiodic flow. Journal of the atmospheric sciences. 1963;
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+[2] Hunt BR, Ott E. Defining chaos. Chaos: An Interdisciplinary Journal of Nonlinear Science.
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+[3] Solari HG, Natiello MA, Mindlin GB. Nonlinear dynamics: a two-way trip from physics to
+math. CRC Press; 1996.
+[4] Strogatz SH. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry,
+and engineering. CRC Press; 2018.
+[5] Janaki T, Rangarajan G. Lyapunov exponents for continuous-time dynamical systems [dis-
+sertation]. Msc, Thesis, Indian Institute of Science of education, India; 2003.
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+[7] Blasius B, Huppert A, Stone L. Complex dynamics and phase synchronization in spatially
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+[8] Upadhyay RK, Rai V. Complex dynamics and synchronization in two non-identical chaotic
+ecological systems. Chaos, Solitons & Fractals. 2009;40(5):2233–2241.
+[9] Lu J, Cao J. Adaptive complete synchronization of two identical or different chaotic (hy-
+perchaotic) systems with fully unknown parameters. Chaos: An Interdisciplinary Journal of
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+Solitons & Fractals. 1997;8(1):51–58.
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+prey model. International Journal of Stochastic Analysis. 1900;13(3):287–297.
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+lotka-volterra model. Bulletin of Mathematical Biology. 1988;50(5):465–491.
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+

CNAyT4oBgHgl3EQfePiT/content/tmp_files/2301.00318v1.pdf.txt

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+1
+Effect of Edge Roughness on resistance and
+switching voltage of Magnetic Tunnel Junctions
+Rachit R. Pandey1, Sutapa Dutta1, Heston A. Mendonca1, Ashwin A. Tulapurkar1
+1Solid State devices group, Department of Electrical Engineering, Indian Institute of Technology Bombay,
+Mumbai 400076,India
+Abstract—We investigate the impact of edge roughness on the
+electrical transport properties of magnetic tunnel junctions using
+non-equilibrium Green’s function formalism. We have modeled
+edge roughness as a stochastic variation in the cross-sectional
+profile of magnetic tunnel junction characterized by the stretched
+exponential decay of the correlation function. The stochastic
+variation in the shape and size changes the transverse energy
+mode profile and gives rise to the variations in the resistance
+and switching voltage of the magnetic tunnel junction. We find
+that the variations are larger as the magnetic tunnel junction
+size is scaled down due to the quantum confinement effect. A
+model is proposed for the efficient calculation of edge roughness
+effects by approximating the cross-sectional geometry to a circle
+with the same cross-sectional area. Further improvement can be
+obtained by approximating the cross-sectional area to an ellipse
+with an aspect ratio determined by the first transverse eigenvalue
+corresponding to the 2D cross section. These results would be
+useful for reliable design of the spin transfer torque- magnetic
+random access memory (STT-MRAM) with ultra-small magnetic
+tunnel junctions.
+Index Terms—Magnetic Tunnel Junction, spin transfer torque,
+circular edge roughness, non-equilibrium Green’s function
+I. INTRODUCTION
+Magnetic tunnel junction (MTJ) comprises two ferromag-
+netic layers (free layer and pinned layer) separated by a
+tunneling barrier. Binary information can be stored in MTJs
+corresponding to parallel (P) and anti-parallel (AP) configu-
+rations of the magnetizations. The information can be read
+by measuring the resistance which is low for P and high
+for AP configurations respectively. Spin transfer torque (STT)
+produced by application of large positive and negative voltages
+to free layer with respect to the fixed layer, stabilizes P and
+AP configurations respectively, and thus can be used for writ-
+ing the memory. As ferromagnetic layers with perpendicular
+magnetic anisotropy (PMA) have lower threshold switching
+voltage (Vc) with enhanced thermal stability, they are preferred
+over in-plane magnetized layers [1]. Reliability analysis of
+STT-MRAM (Magnetic Random Access Memory) in terms of
+write error, tunnel oxide breakdown, temperature variations,
+etc. has been carried out before [2], [3], [4], [5], [6], [7].
+In this paper, we have investigated the effect of lithographic
+imperfections on the performance of MTJ, which becomes
+more evident with the technology scaling. The circular edge
+roughness (CER) is defined as the straying of a pattern from
+its expected circular shape and is used to characterize the
+unwanted sidewall roughness emerging during fabrication pro-
+cesses [8], [9], [10]. The threshold voltages and resistances of
+the MTJ have been calculated using non-equilibrium Green’s
+function (NEGF) method, for 250 realizations of the sidewalls
+for fixed CER parameters. CER affects the area as well as
+the shape of MTJ, which in turn changes the transverse mode
+energies of the electrons tunneling across the barrier and thus
+gives rise to variations in the resistance and threshold voltage.
+We have calculated these variations for a range of CER
+parameters. The NEGF calculation needs transverse energy
+eigenvalues which were obtained by solving the Schrodinger
+equation for a 2d potential well with a random boundary
+corresponding to each realization.
+II. SIMULATION METHODOLOGY
+In the first step, charge current and spin current for a given
+applied voltage across MTJ is calculated using 1d NEGF
+formalism, as a function of transverse mode energy at 300 K
+temperature. The device Hamiltonian matrix is modeled using
+an effective mass tight binding approach. The transport of
+electrons across the device is assumed to be coherent. The
+effect of the contacts is taken into account as self-energy
+contributions to the Hamiltonian. Charge and spin currents
+are calculated from the energy-resolved electron correlation
+matrix [11], [12]. We used CoFeB as the ferromagnet for both
+the fixed and free layers with the Fermi energy EF =2.25 eV
+and the exchange splitting, ∆ = 2.15 eV. The barrier height
+from the Fermi level is taken as UB “ 0.76 eV. The effective
+mass of MgO (tunnelling barrier) and FM are taken as 0.16
+me and 0.38 me, respectively, where me is the free electron
+mass. The thickness of the oxide layer is set to 0.9 nm. The
+charge and parallel spin current (spin current along the fixed
+layer direction) as a function of the transverse mode energy
+are tabulated for a range of voltage values ranging from -0.6
+to 0.6 V for both P and AP configurations. In the second step,
+the transverse energy modes are found from the solution of
+the Schrodinger equation for 2d infinite well with a boundary
+given by the cross-section of the MTJ. If the cross-section is
+a perfect circle, the eigenvalues of Hamiltonian are known
+analytically. For an arbitrary cross-section, the eigenvalues
+can be found numerically using finite difference method by
+discretizing the area into a square grid. In the third step,
+the charge current and spin current for each transverse mode
+are summed up to get the net charge and spin current for a
+range of voltage values ranging from -0.6 to 0.6 V for both
+P and AP configurations. The resistance-area (RA) product
+arXiv:2301.00318v1  [cond-mat.mes-hall]  1 Jan 2023
+
+2
+calculated at 0.01 V, for MTJ with elliptical cross-section for
+different aspect ratios as a function of corresponding areas is
+shown in Fig. 1b. From this figure, we can see that as the area
+reduces, the RA product shows dependence on area as well as
+shape [13]. The critical spin current can be calculated from the
+Gilbert damping (αG) and the energy barrier between P and
+AP states (∆E), as Isc “ p4qαG{¯hq∆E. Further, the energy
+barrier is given by, ∆E
+“ p1{2qµ0MsAtF MHK, where
+Ms, A, tF M, HK denote the saturation magnetization, cross-
+sectional area, free layer thickness and effective perpendicular
+anisotropy respectively. The critical voltage can be found by
+interpolating spin current vs voltage data. If the radius of MTJ
+is 10 nm, assuming ∆E “ 40kBT (T=300 K), αG=0.08,
+tF M=2 nm and Ms “ 1.2 ˆ 106A{m, the HK comes out
+to be 3.5 ˆ 105A{m. The critical voltage for P to AP and AP
+to P switching as a function of area assuming circular cross-
+section and the same HK is shown by the magenta curve
+in Fig. 1c. Similar calculations for 8 nm and 6 nm radii are
+shown by green and blue curves respectively. Fig. 1d shows
+the critical voltage (assuming HK of 6 nm radius MTJ) for
+elliptical cross-section of different aspect ratios as a function
+of the area. We can see that as the area reduces, the threshold
+voltage shows dependence on area as well as shape.
+Fig. 1. (a) Schematic of MTJ without edge roughness. (b) RA
+product vs area for ellipse with different aspect ratios (AR=1
+corresponds to a circle).(c) Vc vs area for circle with energy
+barrier of 40kBT for radii=6 nm (blue), 8nm (green) and 10
+nm (magenta). (d) Vc vs area for ellipse with different aspect
+ratios for energy barrier of 40kBT for radius=6 nm.
+For incorporation of circular edge roughness into a circular
+cross-section of radius R0, we make a random line segment
+of length 2πR0 with auto-correlation function (R) given by
+the equation, Rpxq “ σ2e´pd{ξq2α, where the chord length
+d is given by d “ 2R0|sinpx{2R0q|. ξ, α, and σ denote the
+correlation length, roughness parameter and standard deviation
+respectively [14], [15] . A realization of random line segment
+is obtained as follows [16]: We numerically generate white
+noise series with unit power spectral density (PSD) and take
+its Fourier transform. This is then multiplied by the PSD of
+the correlation function. The inverse FT of the product gives
+us a random line segment. The random shape is constructed by
+Fig. 2. Coefficient of Variation plots for: (a) Resistance (P) (b)
+Resistance (AP) (c) Vc (P) (d) Vc (AP).
+Fig. 3. (a) Schematic of MTJ with edge roughness. (b)
+comparison of Vc (AP) for 20 trials obtained from detailed
+calculation(blue), circle approximation (red), ellipse approxi-
+mation (green).
+taking R0 `x as the radii distribution for angles from 0 to 2π.
+The coefficient of variation (CV=standard deviation/mean) for
+the quantities to be analyzed is obtained from 250 samples.
+III. RESULTS AND DISCUSSIONS
+Variation in the area and shape of MTJ cross-section due
+to the CER produces variation in the transverse energy mode
+profile. This in turn produces variation in the charge current
+and spin current flowing across the MTJ for a given applied
+voltage. The coefficient of variation of resistance and switch-
+ing voltage as a function of σ and ξ for α “ 0.5 and average
+radius 6 nm obtained from detailed calculation is shown as
+a 2D plot in Fig. 2. We can see that the variations become
+larger as σ and ξ increase. CV for different parameters at the
+centre of 2D plot (σ “ 0.67nm, ξ “ 15nm) are shown in the
+table I for different average radii of the cross-section under
+“detailed calculation” column heading. We can see that the
+variations increase as the MTJ size is scaled down. To find out
+the influence of area variation, for each of the 250 samples, we
+mapped the random shape to a perfect circle of the same area
+and found out the resistance and switching voltage (See Fig.
+1c). The CV obtained from this procedure is shown in table
+I under “circle approximation” coumn heading and it matches
+well with values obtained from detailed calculation.
+
+(a)
+(b)
+.AR=1
+(AP)
+RA product (2 um 
+-"AR=0.6 (AP)
+AR=0.4
+(AP)
+5.9
+CoFeB :0
+AR=1 (P)
+(free)
+AR=0.6 (P)
+1.2
+MgO
+AR=0.4 (P)
+1.1
+CoFeB
+(pinned)
+100
+300
+Area(nm2)
+(c)
+(d)
+..AR=1 (AP)
+0.16
+R=6 nm (AP)
+--AR=0.6 (AP)
+.AR=0.4 (AP)
+0.1
+M
+0.14
+M
+-0.22
+AR=1 (P)
+R=8 nm
+(P)
+-0.3
+R=10 nm
+AR=0.6 (P)
+-R=6 nm
+(P)
+-0.25
+AR=0.4 (P)
+100
+300
+100
+300
+Area(nm?)
+Area(nm?)25
+25
+(a)
+(b)
+0.5
+0.3
+15
+15
+0.3
+cS
+cS
+0.1
+0.1
+5
+5
+0.4
+1
+0.4
+1
+25
+25
+(c)
+(d)
+0.05
+0.05
+0.03
+15
+15
+0.03
+cS
+cS
+0.01
+0.01
+5
+5
+0.4
+1
+0.4
+0
+1(a)
+(b)
+detailed calculation
+ellipse approximation
+circle approximation
+0.159
+(v)
+CoFeB
+P
+(free)
+0.155
+U
+>
+MgO
+0.151
+CoFeB
+(pinned)
+5
+10
+15
+203
+TABLE I: % CV for α “ 0.5 σ “ 0.67nm ξ “ 15nm
+% CV of
+R0pnmq
+Detailed
+calculation
+Estimated
+from eq. 1
+Circle
+approx.
+6
+21.73
+20.5
+21.62
+RP
+8
+12.94
+13.86
+12.95
+10
+10.12
+10.20
+9.99
+6
+25.63
+23.32
+25.33
+RAP
+8
+14.26
+15.04
+14.15
+10
+10.89
+10.93
+10.69
+6
+2.18
+2.07
+1.99
+V cP
+8
+0.93
+0.92
+0.94
+10
+0.57
+0.57
+0.56
+6
+2.02
+1.83
+1.85
+V cAP
+8
+0.90
+0.90
+0.89
+10
+0.55
+0.53
+0.54
+The circle approximation is expected to work well when
+the ratio, (σ{R0) is small. Further, for the approximation
+to work well, the minimum normalized correlation function
+e´p2R0{ξq2α should be close to 1 i.e.p2R0{ξq2α should be
+small. If area variation due to CER plays a dominant role,
+we can estimate the variance in a quantity Q as,
+varpQq « pdQ{dAq2r2
+ż L
+0
+pL ´ xqRpxqdxs
+(1)
+where L “ 2πR0 is the average perimeter. The term in the
+square bracket in the above equation is the area variance. The
+CV of various parameters estimated with above equation is
+given under “estimated” column heading in table I. We can
+see that values estimated from area variation are fairly close to
+the numerically calculated values. These equations imply that
+the area variance is proportional to σ2 and it is an increasing
+function of ξ, which is consistent with trends seen in the 2d
+plots in Fig. 2. (area variance saturates at large values of ξ{L).
+To see if the circle approximation can be further improved,
+we mapped a given random shape to an ellipse. This is done as
+follows: We first note down the area. We calculate numerically
+the ground state energy of the 2d infinite well with boundary
+given by the random edge. We then compare ground state
+energy with the tabulated ground state energies of ellipses with
+the same area and different aspect ratios. An aspect ratio is
+assigned to the random figure by interpolation. Using tabulated
+data of Vc and resistance as a function of area for different
+aspect ratios (see Fig. 1), we can calculate the switching
+voltage and resistance of the random cross-section MTJ by
+interpolation. Fig. 3 b shows the Vc for AP to P state for 20
+different realizations (out of 250). The blue bar corresponds
+to Vc calculated by numerically “exact” way i.e. getting all
+the transverse energy modes to form the numerical solution of
+2d Schrodinger equation and summing up transverse currents
+for each mode. The green bar corresponds to the calculation
+by mapping the shape to an ellipse which needs only the
+ground state energy calculation and is hence faster. However,
+for large values of σ{R0 and R0{ξ, the contribution from the
+non-elliptical shape variation should be taken into account. It
+should be also noted that the area variation arising from CER
+gives rise to variation in the thermal stability as the energy
+barrier ∆E, depends on the area.
+IV. CONCLUSION
+We have demonstrated that edge roughness gives rise to
+variance in the area and shape of a magnetic tunnel junction.
+This in turn produces variance in the resistance and switching
+voltage. The variance becomes larger as the MTJ size is scaled
+down. These results would be useful for designing reliable
+MRAM cells.
+REFERENCES
+[1] Ikeda, S., Miura, H., Mizunuma, K., Gan, H. D., Endo, M., Kanai, S.,
+Hayakawa, J., Matsukura, F., and Ohno, H., “A perpendicular-anisotropy
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+724, 2010.
+[2] Nowak, J. J., Robertazzi, R. P., Sun, J. Z., Hu, G., Park, J.-H., Lee, J.,
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+Trouilloud, P. L., Kim, Y., and Worledge, D. C., “Dependence of voltage
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+access memory,” IEEE Magnetics Letters, vol. 7, pp. 1–4, 2016.
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+pert, C., and Mazoyer, P., “Failure and reliability analysis of stt-mram,”
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+Elsevier Science, 2000.
+

CNAyT4oBgHgl3EQfePiT/content/tmp_files/load_file.txt

@@ -0,0 +1,381 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf,len=380
+page_content='1  Effect of Edge Roughness on resistance and  switching voltage of Magnetic Tunnel Junctions  Rachit R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Pandey1, Sutapa Dutta1, Heston A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Mendonca1, Ashwin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Tulapurkar1  1Solid State devices group, Department of Electrical Engineering, Indian Institute of Technology Bombay,  Mumbai 400076,India  Abstract—We investigate the impact of edge roughness on the  electrical transport properties of magnetic tunnel junctions using  non-equilibrium Green’s function formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We have modeled  edge roughness as a stochastic variation in the cross-sectional  profile of magnetic tunnel junction characterized by the stretched  exponential decay of the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The stochastic  variation in the shape and size changes the transverse energy  mode profile and gives rise to the variations in the resistance  and switching voltage of the magnetic tunnel junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We find  that the variations are larger as the magnetic tunnel junction  size is scaled down due to the quantum confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' A  model is proposed for the efficient calculation of edge roughness  effects by approximating the cross-sectional geometry to a circle  with the same cross-sectional area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Further improvement can be  obtained by approximating the cross-sectional area to an ellipse  with an aspect ratio determined by the first transverse eigenvalue  corresponding to the 2D cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' These results would be  useful for reliable design of the spin transfer torque- magnetic  random access memory (STT-MRAM) with ultra-small magnetic  tunnel junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  Index Terms—Magnetic Tunnel Junction, spin transfer torque,  circular edge roughness, non-equilibrium Green’s function  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' INTRODUCTION  Magnetic tunnel junction (MTJ) comprises two ferromag-  netic layers (free layer and pinned layer) separated by a  tunneling barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Binary information can be stored in MTJs  corresponding to parallel (P) and anti-parallel (AP) configu-  rations of the magnetizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The information can be read  by measuring the resistance which is low for P and high  for AP configurations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Spin transfer torque (STT)  produced by application of large positive and negative voltages  to free layer with respect to the fixed layer, stabilizes P and  AP configurations respectively, and thus can be used for writ-  ing the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' As ferromagnetic layers with perpendicular  magnetic anisotropy (PMA) have lower threshold switching  voltage (Vc) with enhanced thermal stability, they are preferred  over in-plane magnetized layers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Reliability analysis of  STT-MRAM (Magnetic Random Access Memory) in terms of  write error, tunnel oxide breakdown, temperature variations,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' has been carried out before [2], [3], [4], [5], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  In this paper, we have investigated the effect of lithographic  imperfections on the performance of MTJ, which becomes  more evident with the technology scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The circular edge  roughness (CER) is defined as the straying of a pattern from  its expected circular shape and is used to characterize the  unwanted sidewall roughness emerging during fabrication pro-  cesses [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The threshold voltages and resistances of  the MTJ have been calculated using non-equilibrium Green’s  function (NEGF) method, for 250 realizations of the sidewalls  for fixed CER parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' CER affects the area as well as  the shape of MTJ, which in turn changes the transverse mode  energies of the electrons tunneling across the barrier and thus  gives rise to variations in the resistance and threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  We have calculated these variations for a range of CER  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The NEGF calculation needs transverse energy  eigenvalues which were obtained by solving the Schrodinger  equation for a 2d potential well with a random boundary  corresponding to each realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' SIMULATION METHODOLOGY  In the first step, charge current and spin current for a given  applied voltage across MTJ is calculated using 1d NEGF  formalism, as a function of transverse mode energy at 300 K  temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The device Hamiltonian matrix is modeled using  an effective mass tight binding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The transport of  electrons across the device is assumed to be coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The  effect of the contacts is taken into account as self-energy  contributions to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Charge and spin currents  are calculated from the energy-resolved electron correlation  matrix [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We used CoFeB as the ferromagnet for both  the fixed and free layers with the Fermi energy EF =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='25 eV  and the exchange splitting, ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='15 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The barrier height  from the Fermi level is taken as UB “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='76 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The effective  mass of MgO (tunnelling barrier) and FM are taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='16  me and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='38 me, respectively, where me is the free electron  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The thickness of the oxide layer is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The  charge and parallel spin current (spin current along the fixed  layer direction) as a function of the transverse mode energy  are tabulated for a range of voltage values ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 V for both P and AP configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' In the second step,  the transverse energy modes are found from the solution of  the Schrodinger equation for 2d infinite well with a boundary  given by the cross-section of the MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' If the cross-section is  a perfect circle, the eigenvalues of Hamiltonian are known  analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' For an arbitrary cross-section, the eigenvalues  can be found numerically using finite difference method by  discretizing the area into a square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' In the third step,  the charge current and spin current for each transverse mode  are summed up to get the net charge and spin current for a  range of voltage values ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 V for both  P and AP configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The resistance-area (RA) product  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='00318v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='mes-hall]  1 Jan 2023  2  calculated at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='01 V, for MTJ with elliptical cross-section for  different aspect ratios as a function of corresponding areas is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' From this figure, we can see that as the area  reduces, the RA product shows dependence on area as well as  shape [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The critical spin current can be calculated from the  Gilbert damping (αG) and the energy barrier between P and  AP states (∆E), as Isc “ p4qαG{¯hq∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Further, the energy  barrier is given by, ∆E  “ p1{2qµ0MsAtF MHK, where  Ms, A, tF M, HK denote the saturation magnetization, cross-  sectional area, free layer thickness and effective perpendicular  anisotropy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The critical voltage can be found by  interpolating spin current vs voltage data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' If the radius of MTJ  is 10 nm, assuming ∆E “ 40kBT (T=300 K), αG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='08,  tF M=2 nm and Ms “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='2 ˆ 106A{m, the HK comes out  to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='5 ˆ 105A{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The critical voltage for P to AP and AP  to P switching as a function of area assuming circular cross-  section and the same HK is shown by the magenta curve  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Similar calculations for 8 nm and 6 nm radii are  shown by green and blue curves respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 1d shows  the critical voltage (assuming HK of 6 nm radius MTJ) for  elliptical cross-section of different aspect ratios as a function  of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We can see that as the area reduces, the threshold  voltage shows dependence on area as well as shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (a) Schematic of MTJ without edge roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (b) RA  product vs area for ellipse with different aspect ratios (AR=1  corresponds to a circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (c) Vc vs area for circle with energy  barrier of 40kBT for radii=6 nm (blue), 8nm (green) and 10  nm (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (d) Vc vs area for ellipse with different aspect  ratios for energy barrier of 40kBT for radius=6 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  For incorporation of circular edge roughness into a circular  cross-section of radius R0, we make a random line segment  of length 2πR0 with auto-correlation function (R) given by  the equation, Rpxq “ σ2e´pd{ξq2α, where the chord length  d is given by d “ 2R0|sinpx{2R0q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' ξ, α, and σ denote the  correlation length, roughness parameter and standard deviation  respectively [14], [15] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' A realization of random line segment  is obtained as follows [16]: We numerically generate white  noise series with unit power spectral density (PSD) and take  its Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' This is then multiplied by the PSD of  the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The inverse FT of the product gives  us a random line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The random shape is constructed by  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Coefficient of Variation plots for: (a) Resistance (P) (b)  Resistance (AP) (c) Vc (P) (d) Vc (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (a) Schematic of MTJ with edge roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (b)  comparison of Vc (AP) for 20 trials obtained from detailed  calculation(blue), circle approximation (red), ellipse approxi-  mation (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  taking R0 `x as the radii distribution for angles from 0 to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  The coefficient of variation (CV=standard deviation/mean) for  the quantities to be analyzed is obtained from 250 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' RESULTS AND DISCUSSIONS  Variation in the area and shape of MTJ cross-section due  to the CER produces variation in the transverse energy mode  profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' This in turn produces variation in the charge current  and spin current flowing across the MTJ for a given applied  voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The coefficient of variation of resistance and switch-  ing voltage as a function of σ and ξ for α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='5 and average  radius 6 nm obtained from detailed calculation is shown as  a 2D plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We can see that the variations become  larger as σ and ξ increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' CV for different parameters at the  centre of 2D plot (σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='67nm, ξ “ 15nm) are shown in the  table I for different average radii of the cross-section under  “detailed calculation” column heading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We can see that the  variations increase as the MTJ size is scaled down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' To find out  the influence of area variation, for each of the 250 samples, we  mapped the random shape to a perfect circle of the same area  and found out the resistance and switching voltage (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The CV obtained from this procedure is shown in table  I under “circle approximation” coumn heading and it matches  well with values obtained from detailed calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  (a)  (b)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='AR=1  (AP)  RA product (2 um  "AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 (AP)  AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4  (AP)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='9  CoFeB :0  AR=1 (P)  (free)  AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 (P)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='2  MgO  AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4 (P)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='1  CoFeB  (pinned)  100  300  Area(nm2)  (c)  (d)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='.AR=1 (AP)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='16  R=6 nm (AP)  --AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 (AP)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4 (AP)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='1  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='14  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='22  AR=1 (P)  R=8 nm  (P)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='3  R=10 nm  AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='6 (P)  R=6 nm  (P)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='25  AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4 (P)  100  300  100  300  Area(nm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=')  Area(nm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' )25  25  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='3  15  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='3  cS  cS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='1  5  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4  1  25  25  (c)  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='03  15  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='03  cS  cS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='01  5  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='4  0  1(a)  (b)  detailed calculation  ellipse approximation  circle approximation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='159  (v)  CoFeB  P  (free)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='155  U  >  MgO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='151  CoFeB  (pinned)  5  10  15  203  TABLE I: % CV for α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='5 σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='67nm ξ “ 15nm  % CV of  R0pnmq  Detailed  calculation  Estimated  from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 1  Circle  approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  6  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='73  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
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+page_content='62  RP  8  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='94  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='86  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='95  10  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='12  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='20  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='99  6  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='63  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
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+page_content='33  RAP  8  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='26  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='04  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='15  10  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
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+page_content='99  V cP  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
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+page_content='57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='56  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='85  V cAP  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='89  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='54  The circle approximation is expected to work well when  the ratio, (σ{R0) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Further, for the approximation  to work well, the minimum normalized correlation function  e´p2R0{ξq2α should be close to 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='p2R0{ξq2α should be  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' If area variation due to CER plays a dominant role,  we can estimate the variance in a quantity Q as,  varpQq « pdQ{dAq2r2  ż L  0  pL ´ xqRpxqdxs  (1)  where L “ 2πR0 is the average perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The term in the  square bracket in the above equation is the area variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The  CV of various parameters estimated with above equation is  given under “estimated” column heading in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We can  see that values estimated from area variation are fairly close to  the numerically calculated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' These equations imply that  the area variance is proportional to σ2 and it is an increasing  function of ξ, which is consistent with trends seen in the 2d  plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' (area variance saturates at large values of ξ{L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  To see if the circle approximation can be further improved,  we mapped a given random shape to an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' This is done as  follows: We first note down the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We calculate numerically  the ground state energy of the 2d infinite well with boundary  given by the random edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' We then compare ground state  energy with the tabulated ground state energies of ellipses with  the same area and different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' An aspect ratio is  assigned to the random figure by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Using tabulated  data of Vc and resistance as a function of area for different  aspect ratios (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 1), we can calculate the switching  voltage and resistance of the random cross-section MTJ by  interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' 3 b shows the Vc for AP to P state for 20  different realizations (out of 250).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The blue bar corresponds  to Vc calculated by numerically “exact” way i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' getting all  the transverse energy modes to form the numerical solution of  2d Schrodinger equation and summing up transverse currents  for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The green bar corresponds to the calculation  by mapping the shape to an ellipse which needs only the  ground state energy calculation and is hence faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' However,  for large values of σ{R0 and R0{ξ, the contribution from the  non-elliptical shape variation should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' It  should be also noted that the area variation arising from CER  gives rise to variation in the thermal stability as the energy  barrier ∆E, depends on the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' CONCLUSION  We have demonstrated that edge roughness gives rise to  variance in the area and shape of a magnetic tunnel junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  This in turn produces variance in the resistance and switching  voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' The variance becomes larger as the MTJ size is scaled  down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content=' These results would be useful for designing reliable  MRAM cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
+page_content='  REFERENCES  [1] Ikeda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'}
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+arXiv:2301.11805v1  [math.LO]  27 Jan 2023
+A GAME FOR BAIRE’S GRAND THEOREM
+LORENZO NOTARO
+Abstract. Generalizing a result of Kiss, we provide a game that char-
+acterizes Baire class 1 functions between arbitrary separable metrizable
+spaces. We show that the determinacy of our game is equivalent to a
+generalization of Baire’s grand theorem, and that both these statements
+hold under AD and in Solovay’s model.
+1. Introduction
+A Polish space is a separable, completely metrizable topological space. Given
+two topological spaces X, Y , a function f : X → Y is said to be Baire class 1 if,
+for every open subset V of Y , the pre-image f −1(V ) is an Fσ subset of X, i.e. a
+countable union of closed subsets. If X is metrizable, then the open subsets of X
+are also Fσ subsets. All continuous functions with metrizable domain are Baire
+class 1.
+A classical result concerning this class of functions is the following theorem of
+Baire — known as Baire’s grand theorem — which provides a characterization of
+Baire class 1 functions from a Polish space to a separable metrizable space (e.g. see
+[5, Theorem 24.15]).
+Theorem (Baire). Let X be a Polish space, Y a separable metrizable space and
+f : X → Y . Then the following are equivalent:
+1) f is Baire class 1
+2) f↾K has a point of continuity for every compact K ⊆ X
+Actually, the separability hypothesis on X can be avoided [4, Corollary 1; 12, Ch.
+II, §31, X], but in this article we are interested in the separable case.
+We note that Baire class 1 functions have been, and still are, sometimes defined
+as pointwise limits of continuous functions — e.g. [16–18], Baire himself originally
+stated his grand theorem for pointwise limits of continuous real functions [14].
+This definition and ours are equivalent only under certain hypotheses — e.g. see
+[5, Theorem 24.10; 20].
+In this article, we study the generalization of Baire’s grand theorem in which
+the domain’s hypothesis is weakened from Polish to separable metrizable, and its
+relationship with the determinacy of a two-player game. The use of infinite two-
+player, perfect information games to characterize certain classes of functions has a
+2020 Mathematics Subject Classification. Primary 03E15, Secondary 26A21, 91A44.
+Key words and phrases. Baire class 1, Baire characterization theorem, topological game, re-
+duction game .
+The author would like to thank Rapha¨el Carroy for his careful readings and useful suggestions,
+which helped shaping this article. The author would like to acknowledge INDAM for the financial
+support. This research was also partially supported by the project PRIN 2017 “Mathematical
+Logic: models, sets, computability”, prot. 2017NWTM8R.
+1
+
+2
+LORENZO NOTARO
+long and established history — e.g. [1, 2, 6, 7, 13, 19, 21], see Motto Ros [11] for a
+detailed introduction on this subject.
+In Section 2 we define our game G(f), where f is a function between separable
+metrizable spaces. We prove that Player II has a winning strategy in G(f) if and
+only if f is Baire class 1 (Theorem 2.2). Then we show that Player I has a winning
+strategy in G(f) if and only if there is a compact K ⊆ X such that f↾K has no
+point of continuity (Theorem 2.3).
+In Section 3 we discuss the determinacy of our game. We start by observing
+that the determinacy of our game for every function is equivalent to GBT, the gen-
+eralization of Baire’s grand theorem in which the domain’s hypothesis is weakened
+from Polish to separable metrizable (Corollary 3.1). We note that AC and GBT are
+mutually inconsistent (Proposition 3.2). Then we show that GBT is equivalent to a
+separation property coming from Descriptive Set Theory (Theorem 3.4), and that
+both these statements hold under AD, the axiom of determinacy, and in Solovay’s
+model.
+2. The game
+Given X a topological space and (Un)n∈N a sequence of open subsets of X, we
+say that (Un)n∈N is convergent if it is decreasing with respect to ⊆, and if it is a
+local basis of some x ∈ X. In that case we write limn→∞ Un = x.
+Definition 2.1. Let X, Y be separable metrizable spaces and let f : X → Y . In
+our game G(f), at the nth round, Player I plays a nonempty open subset Un of X,
+and then Player II plays yn ∈ ran(f),
+I
+U0
+U1
+U2
+...
+II
+y0
+y1
+y2
+...
+with the rule: Un+1 ⊆ Un for each n ∈ N.
+At the end of a game run, Player
+I and II have produced a sequence (Un)n∈N of nonempty open subsets of X and
+a sequence (yn)n∈N in ran(f), respectively. Player II wins the run if either the
+sequence (Un)n∈N is not convergent or it converges to an x ∈ X and (yn)n∈N
+converges to f(x).
+This game is an elaboration of Kiss’ game [1] and a further generalization of
+Duparc’s eraser game [7].
+Since we use trees and operations over finite sequences throughout, we briefly
+recall their classical definition and notation. Let X be a nonempty set. We denote
+by X<N the set of finite sequences of elements of X, with ∅ being the empty
+sequence. Let u ∈ X<N and v ∈ X≤N. We write u ⊑ v when u is an initial segment
+of v and we say that v extends u. We call u⌢v the concatenation of u and v. For
+each n ≤ lenght(u), we let u↾n be the initial segment ⟨u(0), u(1), . . . , u(n − 1)⟩ if
+n > 0; otherwise, we let it be the empty sequence. Given an x ∈ X, we write
+u⌢x to denote the finite sequence u⌢⟨x⟩, where ⟨x⟩ is the sequence of length 1
+containing only x. Sometimes we use the notation ⃗x to denote an element of X<N.
+A tree T on X is a subset of X<N closed uder initial segments. A tree is said to
+be pruned if all its nodes have a proper extension. Given a tree T , we denote by
+[T ] the set {r ∈ XN | ∀n r↾n ∈ T }, whose elements are called branches of T . The
+family {Ns | s ∈ N<N} with Ns = {r ∈ NN | s ⊑ r}, is the standard basis of the
+Baire space.
+
+A GAME FOR BAIRE’S GRAND THEOREM
+3
+Kiss [1, Theorem 1] used his game to characterize Baire class 1 functions between
+separable complete metric spaces. The next theorem generalizes this result by pro-
+viding an analogous characterization for Baire class 1 functions between arbitrary
+separable metrizable spaces.
+Theorem 2.2. Let X, Y be separable metrizable spaces and f : X → Y . Then
+Player II has a winning strategy in G(f) if and only if f is Baire class 1.
+Proof. (⇐=): Assume that the function f is Baire class 1.
+As every separable
+metrizable space embeds into RN (i.e. the space of infinite sequences of real number
+with the product of the euclidean topology), we can assume Y ⊆ RN without loss
+of generality.
+By a theorem of Lebesgue, Hausdorff and Banach [5, Theorem 24.10] there exists
+a sequence (fn)n∈N of continuous functions from X to RN (with range not necessarily
+in Y ) pointwise converging to f. We fix such a sequence, we fix also a compatible
+metric d on RN and a sequence (qn)n∈N ⊂ ran(f) dense in ran(f).
+We now define by induction a map σ⋆ that maps partial plays of Player I into
+Y × N. Then πY ◦ σ⋆ is shown to be a winning strategy for Player II. We denote
+by σ⋆
+Y and σ⋆
+N the functions πY ◦ σ⋆ and πN ◦ σ⋆, respectively, where πY , πN are the
+projections.
+Here is the definition on σ⋆ by induction on the lengths of Player I’s partial
+plays: set σ⋆(U0) = (q0, 0) for each nonempty open subset U0 of X; fix k ∈ N,
+suppose that we have defined σ⋆ for all Player I’s partial plays of length up to k
+and consider a partial play ⃗U ⌢Uk of length k + 1, then
+1) if there is an n > σ⋆
+N(⃗U) such that diam(fn[Uk]) ≤ 2−n: fix an n satisfying
+the condition and an m such that d(qm, fn[Uk]) ≤ 2−n; set σ⋆(⃗U ⌢Uk) =
+(qm, n).
+2) otherwise: we set σ⋆(⃗U ⌢Uk) = σ⋆(⃗U).
+We now show that σ⋆
+Y is a winning strategy for Player II in G(f). Fix an infinite
+play (Uk)k∈N of Player I in G(f). If (Uk)k∈N is not convergent, then Player II wins.
+Assume that (Uk)k∈N converges to an x ∈ X, and set yk = σ⋆
+Y (U0, . . . , Uk), nk =
+σ⋆
+N(U0, . . . , Uk) for each k ∈ N. We now need to show limk→∞ yk = f(x).
+Claim 2.2.1. The sequence (nk)k∈N is nondecreasing and unbounded in N.
+Proof. The fact that (nk)k∈N is nondecreasing is a direct consequence of the defini-
+tion of σ⋆. Next note that, for all n ∈ N, the diameters of the sets in the sequence
+(fn[Uk])k∈N converge to 0, as fn is continuous and (Uk)k∈N is a local basis of x,
+decreasing with respect to ⊆.
+Fix a k ∈ N and an n > nk. By the previous observation, there exist a k′ > k such
+that diam(fn[Uk′]) ≤ 2−n. Fix one such k′, there are two cases: either nk′−1 > nk
+or nk′−1 = nk. In the latter case, the first condition in the inductive definition of
+σ⋆ happens at the k′-th round, hence nk′ > nk′−1 = nk. In either case, nk′ > nk.
+We just proved that for every k there is a k′ > k such that nk′ > nk, therefore
+(nk)k∈N is unbounded.
+□
+Let ¯k be the least k such that nk > 0.
+Claim 2.2.2. For all k ≥ ¯k, d(yk, fnk(x)) ≤ 21−nk.
+Proof. Fix a k ≥ ¯k and pick the smallest l ≤ k such that nl = nk. Note that
+yk = yl, as from the l-th round to the k-th σ⋆ does not change its response. From
+
+4
+LORENZO NOTARO
+the minimality of l it follows that the first condition of the inductive definition of σ⋆
+happens at the l-th round, therefore d(yl, fnl[Ul]) ≤ 2−nl and diam(fnl[Ul]) ≤ 2−nl.
+Since we assumed x = limn→∞ Un, x belongs to Ul and fnl(x) belongs to fnl[Ul],
+hence d(yl, fnl(x)) ≤ 21−nl. As nl = nk and yl = yk we are done.
+□
+Then, for each k ≥ ¯k,
+d(yk, f(x)) ≤ d(fnk(x), f(x)) + d(yk, fnk(x)) ≤ d(fnk(x), f(x)) + 21−nk.
+Since (nk)k∈N is unbounded and the fn’s pointwise converge to f, these inequalities
+imply that (yk)k∈N converges to f(x) and therefore σ⋆
+Y wins the run. As (Uk)k∈N
+was an arbitrary play of Player I, we have showed that σ⋆
+Y is a winning strategy for
+Player II in G(f).
+(=⇒): Suppose that Player II has a winning strategy in G(f), we show that the
+function f is Baire class 1.
+Fix a winning strategy σ for Player II in G(f) and fix a compatible metric d
+on X. As X is separable, there exists a scheme (Us)s∈N<N of open subsets of X
+satisfying the following properties:
+1) U∅ = X.
+2) For all s ∈ N<N, �
+n Us⌢n = Us.
+3) For all s ∈ N<N, diam(Us) ≤ 2−length(s).
+For each s ∈ N<N let
+ys = σ(Us↾0, Us↾1, Us↾2, . . . , Us).
+In other words, ys is the response of Player II following σ to the partial play
+(Us↾0, . . . , Us) of Player I. For every x ∈ X call Tx the tree {s ∈ N<N | x ∈ Us}.
+It follows directly from properties 1) and 2) of the scheme that all such trees are
+nonempty and pruned.
+Claim 2.2.3. For all x ∈ X and V open neighborhood of f(x), there is an s ∈ Tx
+such that for all t ∈ Tx if s ⊑ t then yt ∈ V .
+Proof. Assume by contradiction that there is an x ∈ X and a V open neighborhood
+of f(x) such that for all s ∈ Tx there is a t ∈ Tx that extends s and such that yt ̸∈ V .
+Then there exists a branch r ∈ [Tx] such that {n ∈ N | yr↾n ̸∈ V } is infinite. Fix
+one and note that Player I wins in G(f) by playing the sequence (Ur↾n)n∈N, as, by
+property 3) of the scheme, this sequence converges to x, whilst the corresponding
+play of Player II according to σ does not converge to f(x). Since we have assumed
+σ to be a winning strategy for Player II, we have reached a contradiction.
+□
+Fix V open subset of Y and a sequence (Vn)n∈N of open subsets of V such that
+V = �
+n∈N Vn = �
+n∈N Vn.
+Claim 2.2.4.
+f −1(V ) =
+�
+n∈N
+�
+s∈N<N
+�
+Us \
+�
+{Ut | t ⊒ s and yt ̸∈ Vn}
+�
+.
+Proof. Take an x in the set on the right-hand side. By definition, there exists an
+n ∈ N and an s ∈ N<N such that s ∈ Tx and for all t ∈ Tx extending s, yt ∈ Vn. Fix
+a branch r ∈ [Tx]∩Ns, then the sequence (yr↾k)k∈N is eventually in Vn, i.e. yr↾k ∈ Vn
+for all k greater than some m ∈ N. As (Ur↾k)k∈N converges to x by property 3) of
+
+A GAME FOR BAIRE’S GRAND THEOREM
+5
+the scheme, and σ is a winning strategy for Player II, we have limk→∞ yr↾k = f(x),
+and therefore f(x) ∈ Vn ⊆ V .
+Now pick an x in f −1(V ). There must be an n such that f(x) ∈ Vn. By Claim
+2.2.3, there exists an s ∈ N<N such that for all t ∈ Tx extending s, yt ∈ Vn. But
+this means that x belongs to the set on the right-hand side.
+□
+Since the set on right-hand is an Fσ subset of X and V was an arbitrary open
+subset of Y , we have showed that pre-images of open subsets of Y by f are Fσ sets.
+So f is Baire class 1.
+□
+Theorem 2.3. Let X, Y be separable metrizable spaces and f : X → Y . Then
+Player I has a winning strategy in G(f) if and only if there exists a compact set
+K ⊆ X such that f↾K has no point of continuity.
+To prove this theorem, we need the notion of pointwise oscillation of a function.
+Given X a topological space, Y a metric space and f : A → Y for some nonempty
+A ⊆ X, we define oscf(x) for each x ∈ X as
+oscf(x) = inf{diam(f(U ∩ A)) | U ⊆ X open neighborhood of x}.
+The function oscf is upper semi-continuous, i.e. for every ǫ > 0 the set {x ∈ X |
+oscf(x) ≥ ǫ} is closed.
+Lemma 2.4. Let X, Y be separable metric spaces, ǫ > 0 and f : X → Y such that
+oscf(x) ≥ ǫ for all x ∈ X. Then there is a countable Q ⊆ X such that oscf↾Q(x) ≥ ǫ
+for all x ∈ X.
+Proof. Let dY be the metric on Y and fix a sequence (yn)n∈N dense in Y . For each
+n, m ∈ N, let Qn,m be a countable and dense subset of f −1(B(yn, 2−m)). We claim
+that the countable set Q = �
+n,m Qn,m satisfies the wanted property.
+Fix x ∈ X, m ∈ N and an open neighborhood U of x. By assumption, there
+are x0, x1 ∈ U such that dY (f(x0), f(x1)) ≥ ǫ − 2−m. Let n0, n1 be such that
+f(xi) ∈ B(yni, 2−m) for i = 0, 1. In particular U ∩ f −1(B(yni, 2−m)) ̸= ∅, and
+therefore U ∩ Qni,m ̸= ∅ for i = 0, 1. Pick q0, q1 in U ∩ Qn0,m and U ∩ Qn1,m,
+respectively. Then,
+dY (f(q0), f(q1)) ≥ dY (f(x0), f(x1)) − dY (f(x0), f(q0)) − dY (f(x1), f(q1))
+≥ (ǫ − 2−m) − 21−m − 21−m = ǫ − 5 · 2−m.
+Indeed, for i = 0, 1, f(xi) and f(qi) both belong to B(yni, 2−m), and therefore their
+distance is less or equal to 21−m.
+We have showed that for each x ∈ X, for every open neighborhood U of x and for
+all m there are q0, q1 ∈ U ∩ Q such that dY (f(q0), f(q1)) is greater than ǫ − 5 · 2−m.
+In particular diam(f(U ∩ Q)) ≥ ǫ. Hence, for all x ∈ X, oscf↾Q(x) ≥ ǫ.
+□
+proof of Theorem 2.3. (⇐=) : Fix a compact set K ⊆ X such that f↾K has no
+continuity point. The winning strategy for Player I that we define is essentially the
+one defined by Kiss1 in [1, §2], the only difference being that we deal with a bit
+more care the amount of choice used in the construction (see Remark 2.5).
+1Kiss’ strategy, in turn, is based on the one defined by Carroy in [2, Theorem 4.1]
+
+6
+LORENZO NOTARO
+Fix a compatible metric dX on X and dY on Y . Since f↾K has no point of
+continuity, it follows that oscf↾K(x) > 0 for every x ∈ K. In particular, K =
+�
+n Kn, where
+Kn =
+�
+x ∈ K | oscf↾K(x) ≥ 1
+n
+�
+.
+By Baire’s category theorem, there are a nonempty open U ⊆ X and an n
+such that Kn ∩ U = K ∩ U. Let C be the closure of Kn ∩ U and ǫ = 1/n, then
+oscf↾C(x) ≥ ǫ for every x ∈ C.
+By Lemma 2.4, we know that there is a countable Q ⊆ C such that oscf↾Q(x) ≥ ǫ
+for every x ∈ Q. Let (qn)n∈N be an enumeration of Q. We now define a winning
+strategy τ for Player I by induction on the lengths of Player II’s partial plays. In
+particular, the map τ ranges among the open balls of X centered in Q, i.e. open
+sets of the form B(x, ρ) for some x ∈ Q and radius ρ > 0: first set τ(∅) = B(q0, 1)
+— we are setting the first move of Player I; fix k ∈ N, suppose that we have defined
+τ for all partial plays of Player II of lengths up to k and consider the partial play
+⃗y ⌢yk of length k + 1 with B(qnk, ρk) = τ(⃗y), then
+1) if dY (yk, f(qnk)) ≤ ǫ/8:
+let nk+1 be the least n such that qn ∈ B(qnk, ρk) and dY (f(qn), f(qnk)) ≥
+ǫ/3; let ρ be the greatest ρ ≤ ρk such that B(qnk+1, ρ) ⊆ B(qnk, ρk) and set
+τ(⃗y ⌢yk) = B(qnk+1, ρ/2).
+2) otherwise:
+τ(⃗y ⌢yk) = B(qnk, ρk/2).
+We now prove that τ is a winning strategy for Player I. Fix an infinite play (yk)k∈N
+of Player II and set Bk = B(xk, ρk) = τ(y0, . . . , yk) for every k. First we show that
+the sequence (Bk)k∈N converges to an x ∈ K. Indeed, it follows directly from τ’s
+inductive definition that �
+k Bk = �
+k Bk; the compactness of K guarantees that
+K ∩ �
+k Bk ̸= ∅; finally, the radii of (Bk)k∈N converge to 0, hence K ∩ �
+k Bk is a
+singleton {x} and (Bk)k∈N converges to x.
+So we are left to prove that the sequence (yk)k∈N does not converge to f(x).
+Suppose first that condition 1) of τ’s inductive definition happens only finitely
+many times during this game run. This means that there exists an n such that for
+all k ≥ n, xk = x, and therefore dY (yk, f(x)) > ǫ/8 for all k ≥ n. In this case
+(yk)k∈N certainly does not converge to f(x).
+Now suppose otherwise, and let the increasing sequence (kn)n∈N be such that
+condition 1) happens at the kn+1-th round for each n. More precisely, (kn)n∈N
+is the increasing sequence such that dY (yk, f(xk)) ≤ ǫ/8 if and only if k = kn for
+some (unique) n. For every n,
+dY (ykn, ykn+1) ≥ dY (f(xkn), f(xkn+1)) − dY (f(xkn), ykn) − dY (f(xkn+1), ykn+1)
+= dY (f(xkn), f(xkn+1)) − dY (f(xkn), ykn) − dY (f(xkn+1), ykn+1)
+≥ ǫ/3 − ǫ/8 − ǫ/8 = ǫ/12
+where the equality follows from xkn+1 = xkn+1, which holds because in the rounds
+between kn + 1 and kn+1 the strategy τ does not change the center of its balls; the
+last inequality follows directly from the definition of τ. Therefore, as (kn)n∈N is
+unbounded, the sequence (yk)k∈N does not converge.
+
+A GAME FOR BAIRE’S GRAND THEOREM
+7
+In either case (yk)k∈N does not converge to f(x), therefore τ wins the run. As
+(yk)k∈N was an arbitrary play of Player II, we have showed that τ is a winning
+strategy for Player I in G(f).
+(=⇒) : Suppose that Player I has a winning strategy in G(f), we want to prove
+that there exists a compact set K ⊆ X such that f↾K has no point of continuity.
+We show instead that there exists a compact K ⊆ X such that Player I has a
+winning strategy in G(f↾K). Indeed, if we do so, it would mean that the function
+f↾K is not Baire class 1, as otherwise Player II would have a winning strategy in
+G(f↾K) by Theorem 2.2. Then, by Baire’s grand theorem — which can be applied
+as K, being a compact separable metrizable space, is a Polish space — there would
+be a compact K′ ⊆ K such that f↾K′ has no point of continuity.
+Fix a winning strategy τ for Player I and fix also an enumeration (qn)n∈N of
+a countable dense subset of ran(f).
+Denote by S the tree {s ∈ N<N | s(n) ≤
+n for all n < length(s)}. Note that [S] is a compact subset of the Baire space.
+Consider the following map:
+ϕ : [S] −→ X
+r �−→ lim
+n→∞ τ(qr(0), qr(1), . . . , qr(n)).
+Since we are assuming τ winning for Player I, the limits in the definition always
+exist, and the map ϕ is well-defined. We now show that ϕ is continuous. Given
+an r ∈ [S] and V open neighborhood of ϕ(r), there exists an n ∈ N such that
+τ(qr(0), qr(1), . . . , qr(n−1)) ⊆ V , by definition of limit of sequences of open sets. But
+then the rules of the game force every t ∈ [S] ∩ Nr↾n to be mapped by ϕ into
+τ(qr(0), qr(1), . . . , qr(n−1)) ⊆ V . Therefore ϕ is continuous and K = ran(ϕ) is a
+compact subset of X.
+Next we show that Player I has a winning strategy in G(f↾K). Fix dY compatible
+metric on Y . For each y ∈ Y and k ∈ N, pick an n ≤ k such that dY (qn, y) =
+minm≤k dY (qm, y) and let q(y, k) = qn.
+We define the strategy τ ′ for Player I in G(f↾K) as follows: for each (y0, . . . , yk)
+partial play of Player II in G(f↾K), we let
+τ ′(y0, y1, . . . , yk) = τ(q(y0, 0), q(y1, 1), . . . , q(yk, k)) ∩ K.
+We claim that τ′ is a winning strategy for Player I in G(f↾K). Take an in-
+finite play (yk)k∈N of Player II. For each k, let nk be such that qnk = q(yk, k).
+Then (nk)k∈N belongs to [S], and the limit of the sequence (τ′(y0, . . . , yk))k∈N is
+ϕ((nk)k∈N) ∈ K, by definition of ϕ.
+If (yk)k∈N is not convergent then Player I wins the run. So suppose that (yk)k∈N
+converges to y ∈ ran(f). Then,
+dY (q(yk, k), y) ≤ dY (yk, y) + dY (q(yk, k), yk) = dY (yk, y) + min
+m≤k dY (qm, yk)
+≤ 2dY (yk, y) + min
+m≤k dY (qm, y).
+Since limk→∞ yk = y by assumption, and limk→∞ minm≤k dY (qm, y) = 0 by the
+density of (qn)n∈N in ran(f), it follows from the above inequalities that (q(yk, k))k∈N
+converges to y. Therefore,
+lim
+k→∞ yk = lim
+k→∞ q(yk, k) ̸= f( lim
+k→∞ τ(q(y0, 0), . . . , q(yk, k))) = f( lim
+k→∞ τ ′(y0, . . . yk))).
+
+8
+LORENZO NOTARO
+The inequality follows from having assumed τ winning strategy for Player I in G(f),
+and the last equality instead comes directly from having defined τ ′(y0, . . . , yk) as
+τ(q(y0, 0), . . . , q(yk, k)) ∩ K.
+Hence (yk)k∈N does not converge to f(limk→∞ τ ′(y0, . . . yk))), and τ′ wins the
+run. Since (yk)k∈N was an arbitrary play of Player II, τ ′ is a winning strategy for
+Player I in G(f↾K).
+□
+Remark 2.5. The careful reader may have noticed that in this section we didn’t use
+the axiom of choice, or even the axiom of dependent choice, in their full potential.
+Indeed, all the proofs contained or cited in this section go through assuming only
+ACω(R), the axiom of countable choice over the reals: “Every countable family of
+nonempty subsets of R has a choice function”.
+3. On the determinacy of G(f)
+Recall that a two-player game G is determined if either Player I or Player II
+has a winning strategy. Carroy [2, Theorem 4.1] proved that Duparc’s eraser game
+Ge(f), which characterizes Baire class 1 functions from and into NN, is determined
+for every function f, and used this determinacy result to give a new game-theoretic
+proof of Baire’s grand theorem restricted to functions between 0-dimensional Polish
+spaces [2, Theorem 4.6]. On the other hand, Kiss [1, §2] used Baire’s grand theorem
+to prove the determinacy of his game. Our game G(f) is a further generalization
+of both these games, and, again, a strong relationship between its determinacy and
+Baire’s grand theorem emerges as a direct corollary of the two main theorems of
+the previous section. Let us introduce the following statement, which is the same
+as Baire’s grand theorem with the hypothesis on the domain weakened from Polish
+to separable metrizable:
+(GBT)
+For all X, Y separable metrizable spaces and f : X → Y , f
+is Baire class 1 if and only if f↾K has a point of continuity
+for every compact K ⊆ X.
+The following is a direct corollary of Theorems 2.2 and 2.3.
+Corollary 3.1. The following are equivalent:
+1) G(f) is determined for every f.
+2) GBT
+But unlike Duparc’s and Kiss’ games, ours is not determined in general, as the
+next proposition shows.
+Proposition 3.2. (AC) GBT is false.
+Proof. Under the axiom of choice, there exists a Bernstein set — i.e. a set of reals
+with cardinality of the continuum which intersects every uncountable closed set but
+that contains none of them. Let X be a Bernstein set. Since the family of the Fσ
+subsets of a second countable space has at most the cardinality of the continuum,
+it follows from Cantor’s theorem that there must be a subset A ⊂ X which is not
+an Fσ subset of X.
+The function
+1A : X → 2, with
+1A(x) = 1 iff x ∈ A, is not Baire class 1, as A is
+not an Fσ subset of X. Nonetheless, we claim that
+1A↾K has a point of continuity
+for every compact K ⊆ X. Fix a compact K ⊆ X, then K needs to be countable,
+as we have assumed X to be a Bernstein set. But then K, being a countable and
+
+A GAME FOR BAIRE’S GRAND THEOREM
+9
+compact subset of R, has an isolated point, which is, in particular, a continuity
+point of
+1A↾K.
+□
+Note that the same argument shows that G(1A) is undetermined.
+A separable metrizable space is (absolutely) analytic precisely when it is the
+continuous image of a Polish space. Gerlits and Laczkovich [4, Theorem 2] showed
+that Baire’s grand theorem holds if the domain is assumed only to be an abso-
+lutely analytic metrizable space — actually, they stated this generalization for real
+functions, but their argument goes through assuming only separable metrizable
+codomains. From the theorems of the previous section it follows that the game
+G(f) is determined for every function f with analytic domain.
+We cannot hope to extend tout court this determinacy result to functions with
+co-analytic domains, where a separable metrizable space is said to be co-analytic if
+it is homeomorphic to the complement of an analytic set in a Polish space. In fact,
+the existence of a co-analytic Bernstein set is consistent with ZFC — in particular
+it follows from V = L, see [10, Theorem 13.12] — and the example defined in
+Proposition 3.2 would give us a function f with separable metrizable co-analytic
+domain witnessing the failure of GBT and the undeterminacy of G(f).
+We now focus on GBT, which, by Proposition 3.2, is inconsistent with AC. We
+first introduce a couple of statements coming from Descriptive Set Theory which
+are strictly related to GBT. We recall that, given three sets A, B, S we say that S
+separates A from B if A ⊆ S and B ∩ S = ∅.
+(HSP)
+For every disjoint A, B ⊆ NN such that there is no Fσ set
+separating A from B, there is a Cantor set C ⊆ A ∪ B with
+C ∩ B countable and dense in C.
+where HSP stands for Hurewicz’ Separation Property. The fact that the trace of
+B on C (i.e. B ∩ C) is countable and dense not only means that B ∩ C is Fσ in C,
+but also that it is Fσ-complete [5, Exercise 22.11]. HSP is known to hold under AD
+[3, Theorem 4.2; 5, §21.F].
+Fact 3.3. HSP is a strong statement, in the sense that HSP + DC is equiconsistent
+with the existence of an inaccessible cardinal. Indeed, HSP + DC implies PSP, the
+perfect set property for every subset of NN, and it is well-known that PSP + DC
+implies the consistency of an inaccessible cardinal [10, Propositions 11.4 and 11.5];
+on the other hand, Todorcevic and Di Prisco [8, Theorem 4.1] proved that HSP
+holds in Solovay’s model, hence the equiconsistency.
+Consider now this seemingly weaker statement:
+(WHSP)
+For every disjoint A, B ⊆ NN such that there is no Fσ set
+separating A from B, there is a Cantor set C ⊆ A ∪ B with
+C ∩ A dense and codense in C.
+This statement is clearly a consequence of HSP, but it doesn’t tell us anything
+about the definability of the trace of A or B on C.
+Theorem 3.4. The following are equivalent:
+1) GBT
+2) WHSP
+Proof. 1) =⇒ 2): let A, B ⊆ NN be disjoint subsets of the Baire space such that A
+cannot be separated from B by an Fσ set. Equivalently, A is not Fσ with respect
+
+10
+LORENZO NOTARO
+to the relative topology on A ∪ B. Therefore, the function f : A ∪ B → 2, with
+f(x) = 1 iff x ∈ A, is not Baire class 1, and by GBT there exists a compact set
+K ⊆ A ∪ B such that f↾K has not point of continuity. This means that A ∩ K
+is both dense and codense in K, as otherwise f would have a point of continuity.
+Finally notice that K, being a compact and perfect subset of NN, is actually a
+Cantor set [5, Theorem 7.4]. Hence WHSP holds.
+2) =⇒ 1): let X, Y be separable metrizable spaces and f : X → Y a function
+which is not Baire class 1. Every Polish space is the image of NN by a continuous
+and closed map [9]. As every separable metrizable space embeds into a Polish space,
+there exists a closed and continuous surjection g : X′ → X for some X′ ⊆ NN. Since
+the image of a closed set by a closed function is still closed by definition, also images
+of Fσ sets by a closed function remain Fσ. Therefore the function h = f◦g : X′ → Y
+is still not Baire class 1.
+As h is not Baire class 1, there is an open set V ⊆ Y such that h−1(V ) is not an
+Fσ set of X′. Fix such V and fix also a sequence of closed sets (Fn)n∈N such that
+V = �
+n Fn. It must be the case that, for some n, h−1(Fn) is not separable from
+h−1(Y \ V ) by an Fσ set, as otherwise h−1(V ) would be a countable union of Fσ
+sets, which is still Fσ. Fix such an n, then, by WHSP, there is a Cantor set C ⊆ X′
+with both h−1(Fn) ∩ C and h−1(Y \ V ) ∩ C dense in C.
+By continuity of g, the set g[C] is compact in X and f −1(Fn)∩g[C], f −1(Y \V )∩
+g[C] are both dense in g[C].
+We claim that the function f↾g[C] has no point of continuity. Take an x ∈ g[C],
+and fix two sequences (xk)k∈N ⊂ f −1(Fn) ∩ g[C], (x′
+k)k∈N ⊂ f −1(Y \ V ) ∩ g[C]
+converging to x. Such sequences exist because f −1(Fn)∩g[C] and f −1(Y \V )∩g[C]
+are both dense in g[C]. If the sequences (f(xk))k∈N and (f(x′
+k))k∈N converged in Y ,
+they would converge in Fn and in Y \ V , respectively, as both these sets are closed.
+Thus, even if their limits were to exist, they could not coincide. In particular x is
+not a point of continuity of f↾g[C]. Since x ∈ g[C] was arbitrary, we have that no
+x ∈ g[C] is a continuity point of f↾g[C].
+Given a function f : X → Y between separable metrizable spaces which is not
+Baire class 1, we have found a compact K ⊆ X such that f↾K has no point of
+continuity. On the other hand, if f : X → Y is Baire class 1, then the classical
+argument used in the proof of Baire’s grand theorem shows that f↾K has a point
+of continuity for every compact K ⊆ X (e.g. see [5, Theorem 24.15]), with no need
+to invoke WHSP. Hence GBT holds.
+□
+As HSP, and in particular WHSP, holds under AD [3, Theorem 4.2; 5, §21.F]
+and in Solovay’s model [8, Theorem 4.1], we can say the same of GBT and the full
+determinacy of our game, by Theorem 3.4 and Corollary 3.1. But the the precise
+consistency strength of these three statements (+DC) is still unknown.
+WHSP, compared to HSP, seems to be weak enough to be proved consistent
+relative to ZF, with no large cardinals needed (see Fact 3.3). Hence the following
+conjecture.
+Conjecture. GBT + DC is consistent relative to ZF.
+Lastly, notice that the definition of our game does not rely on the separability or
+the metrizability of the function’s domain and codomain, and it would make perfect
+sense to study it on wider classes of functions. Future research can shed light on
+
+A GAME FOR BAIRE’S GRAND THEOREM
+11
+how our game behaves on the class of functions with metrizable (not necessarily
+separable) domains and separable metrizable codomains.
+Would our results of
+Section 2 still hold? How much choice would be needed to prove them?
+References
+[1] Viktor Kiss, A game characterizing Baire class 1 functions, J. Symb. Log. 85 (2020), no. 1,
+456–466.
+[2] Rapha¨el Carroy, Playing in the first Baire class, MLQ Math. Log. Q. 60 (2014), no. 1-2,
+118–132.
+[3] Rapha¨el Carroy, Benjamin D. Miller, and D´aniel T. Soukup, The open dihypergraph di-
+chotomy and the second level of the Borel hierarchy, Trends in set theory, Contemp. Math.,
+vol. 752, Amer. Math. Soc., [Providence], RI, 2020, pp. 1–19.
+[4] J. Gerlits and M. Laczkovich, On barely continuous and Baire 1 functions, Topology, Vol.
+II (Proc. Fourth Colloq., Budapest, 1978), Colloq. Math. Soc. J´anos Bolyai, vol. 23, North-
+Holland, Amsterdam-New York, 1980, pp. 493–499.
+[5] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics,
+vol. 156, Springer-Verlag, New York, 1995.
+[6] William Wilfred Wadge, Reducibility and determinateness on the Baire space, ProQuest LLC,
+Ann Arbor, MI, 1983. Thesis (Ph.D.)–University of California, Berkeley.
+[7] J. Duparc, Wadge hierarchy and Veblen hierarchy. I. Borel sets of finite rank, J. Symbolic
+Logic 66 (2001), no. 1, 56–86.
+[8] Carlos Augusto Di Prisco and Stevo Todorcevic, Perfect-set properties in L(R)[U], Adv.
+Math. 139 (1998), no. 2, 240–259.
+[9] R. Engelking, On closed images of the space of irrationals, Proc. Amer. Math. Soc. 21 (1969),
+583–586.
+[10] Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics,
+Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings.
+[11] Luca Motto Ros, Game representations of classes of piecewise definable functions, MLQ
+Math. Log. Q. 57 (2011), no. 1, 95–112.
+[12] Kazimierz Kuratowski, Topologie, translated by Jan W. Jaworowski, Vol. 1, Accademic Press,
+New York-London, 1966.
+[13] Riccardo Camerlo and Jacques Duparc, Some remarks on Baire’s grand theorem, Arch. Math.
+Logic 57 (2018), no. 3-4, 195–201.
+[14] Ren´e Baire, Le¸cons sur les fonctions discontinues, Les Grands Classiques Gauthier-Villars.
+[Gauthier-Villars Great Classics], ´Editions Jacques Gabay, Sceaux, 1995 (French). Reprint of
+the 1905 original.
+[15] Olena Karlova, A generalization of a Baire theorem concerning barely continuous functions,
+Topology Appl. 258 (2019), 433–438.
+[16] Petr Poˇsta, Dirichlet problem and subclasses of Baire-one functions, Israel J. Math. 226
+(2018), no. 1, 177–188.
+[17] E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces
+containing l1, Israel J. Math. 20 (1975), no. 3-4, 375–384.
+[18] Felix Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962. Translated
+from the German by John R. Aumann et al.
+[19] Alessandro Andretta, More on Wadge determinacy, Ann. Pure Appl. Logic 144 (2006), no. 1-
+3, 2–32.
+[20] Olena Karlova, On Baire-one mappings with zero-dimensional domains, Colloq. Math. 146
+(2017), no. 1, 129–141.
+[21] M´arton Elekes, J´anos Flesch, Viktor Kiss, Don´at Nagy, M´ark Po´or, and Arkadi Predtetchin-
+ski, Games characterizing limsup functions and Baire class 1 functions, J. Symb. Log. 87
+(2022), no. 4, 1459–1473.
+Universit`a degli Studi di Torino, Dipartimento di Matematica “G. Peano”, Via Carlo
+Alberto 10, 10123 Torino, Italy
+Email address: lorenzo.notaro@unito.it
+

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+ASYNCHRONOUS DEEP DOUBLE DUELLING Q-LEARNING FOR
+TRADING-SIGNAL EXECUTION IN LIMIT ORDER BOOK
+MARKETS
+Peer Nagy
+Oxford-Man Institute of Quantitative Finance
+University of Oxford
+peer.nagy@eng.ox.ac.uk
+Jan-Peter Calliess
+Oxford-Man Institute of Quantitative Finance
+University of Oxford
+Stefan Zohren
+Oxford-Man Institute of Quantitative Finance
+University of Oxford
+January 23, 2023
+ABSTRACT
+We employ deep reinforcement learning (RL) to train an agent to successfully translate a high-
+frequency trading signal into a trading strategy that places individual limit orders. Based on the
+ABIDES limit order book simulator, we build a reinforcement learning OpenAI gym environment and
+utilise it to simulate a realistic trading environment for NASDAQ equities based on historic order book
+messages. To train a trading agent that learns to maximise its trading return in this environment, we
+use Deep Duelling Double Q-learning with the APEX (asynchronous prioritised experience replay)
+architecture. The agent observes the current limit order book state, its recent history, and a short-term
+directional forecast. To investigate the performance of RL for adaptive trading independently from a
+concrete forecasting algorithm, we study the performance of our approach utilising synthetic alpha
+signals obtained by perturbing forward-looking returns with varying levels of noise. Here, we find
+that the RL agent learns an effective trading strategy for inventory management and order placing
+that outperforms a heuristic benchmark trading strategy having access to the same signal.
+Keywords Limit Order Books · Quantitative Finance · Reinforcement Learning · LOBSTER ·
+1
+Introduction
+Successful quantitative trading strategies often work by generating trading signals, which exhibit a statistically significant
+correlation with future prices. These signals are then turned into actions, aiming to assume positions in order to gain
+from future price changes. The higher the signal frequency and strategy turnover, the more critical is the execution
+component of the strategy, which translates the signal into concrete orders that can be submitted to a market. Such
+markets are oftentimes organised as an order ledger represented by a limit order book (LOB).
+Limit order book prices have been shown to be predictable over short time periods, predicting a few successive ticks
+into the future with some accuracy. This has been done by either utilising the recent history of order book states
+[1, 2], order-flow data [3], or market-by-order (MBO) data directly as features [4]. However, given the short time
+horizons over which these predictions are performed, and correspondingly small price movements, predictability does
+not directly translate into trading profits. Transaction costs, strategy implementation details, and time delays add up to
+the challenging problem of translating high-frequency forecasts into a trading strategy, which determines when and
+which orders to send to the exchange. Moreover, different predictive signals have to be traded differently to achieve
+optimal results, depending on the forecast horizon, signal stability, and predictive power.
+arXiv:2301.08688v1  [q-fin.TR]  20 Jan 2023
+
+A PREPRINT - JANUARY 23, 2023
+In this paper, we use asynchronous off-policy reinforcement learning (RL), specifically Deep Duelling Double Q-
+learning with the APEX architecture [5], to learn an optimal trading strategy, given a noisy directional signal of
+short-term forward mid-quote returns. For this purpose, we developed an OpenAI gym [6] limit order book environment
+based on the ABIDES [7] market simulator, similar to [8]. We use this simulator to replay NASDAQ price-time priority
+limit order book markets using message data from the LOBSTER data set [9].
+We study the case of an artificial or synthetic signal, taking the future price as known and adding varying levels of noise,
+allowing us to investigate learning performance and to quantify the benefit of an RL-derived trading policy compared to
+a baseline strategy using the same noisy signal. This is not an unrealistic setup when choosing the correct level of noise.
+Practitioners often have dedicated teams researching and deriving alpha signals, often over many years, while other
+teams might work on translating those signals into profitable strategies. Our aim is to focus on the latter problem which
+becomes increasingly more difficult as signals become faster. It is thus interesting to see how an RL framework can be
+used to solve this problem. In particular, we show that the RL agent learns superior policies to the baselines, both in
+terms of strategy return and Sharpe ratio. Machine learning methods, such as RL, have become increasingly important
+to automate trade execution in the financial industry in recent years [10], underlining the practical use of research in
+this area.
+We make a number of contributions to the existing literature. By defining a novel action and state space in a LOB
+trading environment, we allow for the placement of limit orders at different prices. This allows the agent to learn
+a concrete high-frequency trading strategy for a given signal, trading either aggressively by crossing the spread, or
+conservatively, implicitly trading off execution probability and cost. In addition to the timing and level placement of
+limit orders, our RL agent also learns to use limit orders of single units of stock to manage its inventory as it holds
+variably sized long or short positions over time. More broadly, we demonstrate the practical use case of RL to translate
+predictive signals into limit order trading strategies, which is still usually a hand-crafted component of a trading system.
+We thus show that simulating limit order book markets and using RL to further automate the investment process is a
+promising direction for further research. To the best of our knowledge, this is also the first study applying the APEX [5]
+algorithm to limit order book environments.
+The remaining paper is structured as follows: Section 2 surveys related literature, section 3 explains the mechanics
+of limit order book markets and the APEX algorithm, section 4 details the construction of the artificial price signal,
+section 5 showcases our empirical results, and section 6 concludes our findings.
+2
+Related Work
+Reinforcement learning has been applied to learn different tasks in limit order book market environments, such as
+optimal trade execution [11, 12, 13, 14, 15], market making [16, 17], portfolio optimisation [18] or trading [19, 20, 21].
+The objective of optimal trade execution is to minimise the cost of trading a predetermined amount of shares over a
+given time frame. Trading direction and the number of shares is already pre-defined in the execution problem. Market
+makers, on the other hand, place limit orders on both sides of the order book and set out to maximise profits from
+capturing the spread, while minimising the risk of inventory accumulation and adverse selection. We summarise using
+the term “RL for trading” such tasks which maximise profit from taking directional bets in the market. This is a hard
+problem for RL to solve as the space of potential trading strategies is large, leading to potentially many local optima in
+the loss landscape, and actionable directional market forecasts are notoriously difficult due to arbitrage in the market.
+The work of [19] is an early study of RL for market microstructure tasks, including trade execution and predicting price
+movements. While the authors achieve some predictive power of directional price moves, forecasts are determined
+to be too erroneous for profitable trading. The most similar work to ours is [21] that provides the first end-to-end
+DRL framework for high-frequency trading, using PPO [22] to trade Intel stock. To model price impact, [21] use an
+approximation, moving prices proportionately to the square-root of traded volume. The action space is essentially
+limited to market orders, so there is no decision made on limit prices. The trained policy is able to produce a profitable
+trading strategy on the evaluated 20 test days. However, this is not compared to baseline strategies and the resulting
+performance is not statistically tested for significance. In contrast, we consider a larger action space, allowing for the
+placement of limit orders at different prices, thereby potentially lowering transaction costs of the learned HFT strategy.
+For a broader survey of deep RL (DRL) for trading, including portfolio optimisation, model-based and hierarchical RL
+approaches the reader is referred to [17].
+2
+
+A PREPRINT - JANUARY 23, 2023
+3
+Background
+3.1
+Limit Order Book Data
+Limit order books (LOBs) are one of the most popular financial market mechanisms used by exchanges around the
+world [23]. Market participants submit limit buy or sell orders, specifying a maximum (minimum) price at which they
+are willing to buy (sell), and the size of the order. The exchange’s limit order book then keeps track of unfilled limit
+orders on the buy side (bids) and the sell side (asks). If an incoming order is marketable, i.e. there are open orders on
+the opposing side of the order book at acceptable prices, the order is matched immediately, thereby removing liquidity
+from the book. The most popular matching prioritisation scheme is price-time priority. Here, limit orders are matched
+first based on price, starting with the most favourable price for the incoming order, and then based on arrival time,
+starting with the oldest resting limit order in the book, at each price level. For a more complete review of limit order
+book dynamics and pertaining models, we refer the reader to [23].
+In this paper, we consider equity limit order book data from the NASDAQ exchange [9], which also uses a price-time
+priority prioritisation. Our market simulator keeps track of the state of the LOB by replaying historical message data,
+consisting of new incoming limit orders, order cancellations or modifications. The RL agent can then inject new
+messages into the order flow and thereby, change the LOB state from its observed historical state.
+Our simulator reconstructs LOB dynamics from message data, so every marketable order takes liquidity from the book
+and thus has a direct price impact. Beyond that, we make no further assumptions on permanent market impact or
+reactions of other agents in the market, which we leave to future work.
+3.2
+Double DQN with Distributed Experience Replay
+We model the trader’s problem as a Markov Decision Process (MDP) [24, 25], described by the tuple ⟨S, A, T , r, γ⟩. S
+denotes a state space, A an action space, T a stochastic transition function, r a reward function and γ a discount factor.
+Observing the current environment state st ∈ S at time t, the trader takes action at ∈ A, which causes the environment
+to transition state according to the stochastic transition function T (st+1|st, at). After transitioning from st to st+1,
+the agent receives a reward rt+1 = r(st, at, st+1). We use Deep Double Q-learning [26] with a duelling network
+architecture [27] to approximate the optimal Q-function Q∗(s, a) = E[rt+1+γ maxa′ Q∗(st+1, a′)|at = a, st = s]. To
+speed up the learning process we employ the APEX training architecture [5], which combines asynchronous experience
+sampling using parallel environments with off-policy learning from experience replay buffers. Every episode i results
+in an experience trajectory τi = {st, at}T
+t=1, many of which are sampled from parallel environment instances and are
+then stored in the replay buffer. The environment sampling is done asynchronously using parallel processes running on
+CPUs. Experience data from the buffer is then sampled randomly and batched to perform a policy improvement step of
+the Q-network on the GPU. Prioritised sampling from the experience buffer has proven to degrade performance in our
+noisy problem setting, hence we are sampling uniformly from the buffer.1 After a sufficient number of training steps,
+the new policy is then copied to every CPU worker to update the behavioural policy.
+Double Q-learning [28, 26] stabilises the learning process by keeping separate Q-network weights for action selection
+(main network) and action validation (target network). The target network weights are then updated gradually in the
+direction of the main network’s weight every few iterations. The duelling network architecture [27] on the other hand
+uses two separate network branches (for both main and target Q-networks). One branch estimates the value function
+V (s) = maxa Q(s, a), while the other estimates the advantage function A(s, a) = Q(s, a) − V (s). The benefit of this
+architecture choice lies therein that the advantage of individual actions in some states might be irrelevant, and the state
+value, which can be learnt more easily, suffices for an action-value approximation.
+4
+Framework
+4.1
+Artificial Price Signal
+The artificial directional price signal dt ∈ ∆2 = {x ∈ R3 : x1 + x2 + x3 = 1, xi ≥ 0 for i = 1, 2, 3} the agent
+receives is modelled as a discrete probability distribution over 3 classes, corresponding to the averaged mid-quote price
+decreasing, remaining stable, or increasing over a fixed future time horizon of h ∈ N+ seconds. To achieve realistic
+levels of temporal stability of the signal process, dt is an exponentially weighted average, with persistence coefficient
+1In many application domains prioritised sampling, whereby we resample instances more frequently where the model initially
+performs poorly tends to aide learning. However, in our low signal-to-noise application domain, we noted poor performance.
+Investigating the matter, we found that prioritised sampling caused more frequent resampling of highly noisy instances where
+learning was particularly difficult, hence degrading performance.
+3
+
+A PREPRINT - JANUARY 23, 2023
+φ ∈ (0, 1), of Dirichlet random variables ϵt. The Dirichlet parameters α depend on the realised smoothed future return
+rt+h, specifically on whether the return lies within a neighbourhood of size k around zero, or above or below. Thus we
+have:
+dt = φdt−1 + (1 − φ)ϵt
+ϵt = Dirichlet (α(rt+h))
+rt+h = pt+h − pt
+pt
+where
+pt+h = 1
+h
+h
+�
+i=1
+pt+i
+(1)
+and prices pt refer to the mid-quote price at time t. The Dirichlet distribution is parametrised, so that, in expectation,
+the signal dt updates in the direction of future returns, where aH and aL determine the variance of the signal. The
+Dirichlet parameter vector is thus:
+α(rt+h) =
+�
+�
+�
+(aH, aL, aL)
+if rt+h < −k
+(aL, aH, aL)
+if − k ≤ rt+h < k
+(aL, aL, aH)
+if k ≤ rt+h.
+(2)
+4.2
+RL Problem Specification
+At each time step t, the agent receives a new state observation st. st consists of the time left in the current episode
+T − t given the episode’s duration of T, the agent’s cash balance Ct, stock inventory Xt, the directional signal dt ∈ ∆2,
+encoded as probabilities of prices decreasing, remaining approximately constant, or increasing; and price and volume
+quantities for the best bid and ask (level 1), including the agent’s own volume posted at bid and ask: ob,t and oa,t
+respectively. In addition to the most current observable variables at time t, the agent also observes a history of the
+previous l values, which are updated whenever there is an observed change in the LOB. Putting all this together, we
+obtain the following state observation:
+st =
+�
+�
+�
+�
+�
+T − u
+Cu
+Xu
+(d1
+u, d2
+u, d3
+u)′
+(pa,u, va,u, oa,u, pb,u, vb,u, ob,u)′
+�
+�
+�
+�
+�
+u={t−l,...,t}
+.
+After receiving the state observation, the agent then chooses an action at. It can place a buy or sell limit order of a
+single share at bid, mid-quote, or ask price; or do nothing and advance to the next time step. Actions, which would
+immediately result in positions outside the allowed inventory constraints [posmin, posmax] are disallowed and do not
+trigger an order. Whenever the execution of a resting limit order takes the inventory outside the allowed constraints, a
+market order in the opposing direction is triggered to reduce the position back to posmin for short positions or posmax
+for long positions. Hence, we define
+at ∈ A = ({−1, 1} × {−1, 0, 1}) ∪ {skip}
+so that in total there are 7 discrete actions available, three levels for both buy and sell orders, and a skip action.
+For the six actions besides the “skip” action, the first dimension encodes the trading direction (sell or buy) and the
+second dimension the price level (bid, mid-price, or ask). For example, a = (1, 0) describes the action to place a buy
+order at the mid price, and a = (−1, 1) a sell order at best ask. Rewards Rt+1 consist of a convex combination of a
+profit-and-loss-based reward Rpnl
+t+1 and a directional reward Rdir
+t+1. Rpnl
+t+1 is the log return of the agent’s mark-to-market
+portfolio value Mt, encompassing cash and the current inventory value, marked at the mid-price. The benefit of
+log-returns is that they are additive over time, rather than multiplicative like gross returns, so that, without discounting
+(γ = 1) the total profit-and-loss return �T
+s=t+1 Rpnl
+s
+= MT − Mt. The directional reward term Rdir
+t+1 incentivizes the
+agent to hold inventory in the direction of the signal and penalises the agent for inventory positions opposing the signal
+direction. The size of the directional reward can be scaled by the parameter κ > 0. Rdir
+t+1 is positive if the positive
+prediction has a higher score than the negative (dt,3 > dt,1) and the current inventory is positive; or if dt,3 < dt,1 and
+4
+
+A PREPRINT - JANUARY 23, 2023
+Xt < 0. Further, if the signal [−1, 0, 1] · dt has an opposite sign than inventory Xt, Rdir
+t+1 is negative. This can be
+summarised as follows:
+Mark-to-Market Value
+Mt = Ct + Xtpm
+t
+∆Mt = ∆Ct + Xt−1∆pm
+t + ∆xtpm
+t
+PnL Reward
+Rpnl
+t+1 = ln(Mt) − ln(Mt−1)
+Directional Reward
+Rdir
+t+1 = κ[−1, 0, 1] · dtXt
+Total Reward
+rt+1 = wdirRdir
+t+1 + (1 − wdir)Rpnl
+t+1
+(3)
+The weight on the directional reward wdir ∈ [0, 1) is reduced every learning step by a factor ψ ∈ (0, 1),
+wdir ← ψwdir
+so that initially the agent quickly learns to trade in the signal direction. Over the course of the learning process, Rpnl
+t
+becomes dominant and the agent maximises its mark-to-market profits.
+5
+Experimental Results
+We train all RL policies using the problem setup discussed in section 4.2 on 5.5 months of Apple (AAPL) limit order
+book data (2012-01-01 to 2012-05-16) and evaluate performance on 1.5 months of out-of-sample data (2012-05-17 to
+2012-06-31). We only use the first hour of every trading day (09:30 to 10:30) as the opening hour exhibits higher than
+average trading volume and price moves. Each hour of the data corresponds to a single RL episode.
+Our neural network architecture consists of 3 feed-forward layers, followed by an LSTM layer, for both the value- and
+advantage stream of the duelling architecture. The LSTM layer allows the agent to efficiently learn a memory-based
+policy with observations including 100 LOB states.
+We compare the resulting learned RL policies to a baseline trading algorithm, which receives the same artificially
+perturbed high-frequency signal of future mid-prices. The baseline policy trades aggressively by crossing the spread
+whenever the signal indicates a directional price move up or down until the inventory constraint is reached. The signal
+direction in the baseline algorithm is determined as the prediction class with the highest score (down, neutral, or up).
+When the signal changes from up or down to neutral, indicating no immediate expected price move, the baseline strategy
+places a passive order to slowly reduce position size until the inventory is cleared. This heuristic utilises the same action
+space as the RL agent and yielded better performance than trading using only passive orders (placed at the near touch),
+or only aggressive orders (at the far touch).
+Figure 1 plots a 17 second simulation window from the test period, comparing the simulated baseline strategy with
+the RL strategy. It can be seen that prices in the LOB are affected by the trading activity as both strategies inject new
+order flow into the market, in addition to the historical orders, thereby consuming or adding liquidity at the best bid and
+ask. During the plotted period, the baseline strategy incurs small losses due to the signal switching between predicting
+decreasing and increasing future prices. This causes the baseline strategy to trade aggressively, paying the spread with
+every trade. The RL strategy, on the other hand, navigates this difficult period better by trading more passively out of its
+long position, and again when building up a new position. Especially in the second half of the depicted time period, the
+RL strategy adds a large number of passive buy orders (green circles in the second panel of figure 1). This is shown
+by the green straight lines, which connect the orders to their execution or cancellation, some of which occur after the
+depicted period.
+5.1
+Oracle Signal
+The RL agent receives a noisy oracle signal of the mean return h = 10 seconds into the future (see equation 4.1). It
+chooses an action every 0.1s, allowing a sufficiently quick build-up of long or short positions using repeated limit
+orders of single stocks. The algorithm is constrained to keep the stock inventory within bounds of [posmin, posmax] =
+[−10, 10]. To change the amount of noise in the signal, we vary the aH parameter of the Dirichlet distribution, keeping
+aL = 1 constant in all cases. To keep the notation simple, we hence drop the H superscript and refer to the variable
+Dirichlet parameter aH simply as a. We consider three different noise levels, parametrising the Dirichlet distribution
+with a = 1.6 (low noise), a = 1.3 (mid noise), and a = 1.1 (high noise). A fixed return classification threshold
+k = 4 · 10−5 was chosen to achieve good performance of the baseline algorithm, placing around 85% of observations
+in the up or down category. The signal process persistence parameter is set to φ = 0.9.
+5
+
+A PREPRINT - JANUARY 23, 2023
+Figure 1: A short snapshot of simulation results (AAPL on 2012-06-14), comparing the RL policy (second panel) with
+the baseline (first panel). The first two panels plot the best bid, ask, and mid-price, overlaying trading events of buy
+orders (green) and sell orders (red). Circles mark new unmarketable limit orders entering the book. Crosses mark order
+executions (trades) and triangles order cancellations. Open orders are connected by lines to either cancellations or
+trades. Since we are simulating the entire LOB, trading activity can be seen to affect bid and ask prices. The third panel
+plots the evolution of the inventory position of both strategies, and the last panel the trading profits over the period in
+USD.
+Out-of-sample trading performance is visualised by the account curves in figure 2. The curves show the evolution of
+the portfolio value for a chronological evaluation of all test episodes. Every account curve shows the mean episodic
+log-return µ and corresponding Sharpe ratio S next to it. We show that all RL-derived policies are able to outperform
+their respective baseline strategies for the three noise levels investigated. Over the 31 test episodes, the cumulative RL
+algorithm out-performance over the baseline strategy ranges between 14.8 (a = 1.3) and 32.2 (a = 1.1) percentage
+points (and 20.7 for a = 1.6). In the case of the signal with the lowest signal-to-noise ratio (a=1.1), for which the
+baseline strategy incurs a loss for the test period, the RL agent has learned a trading strategy with an approximately
+zero mean return. Temporarily, the strategy even produces positive gains. Overall, it produces a sufficiently strong
+performance to not lose money while still trading actively and incurring transaction costs. Compared to a buy-and-hold
+strategy over the same time period, the noisy RL strategy similarly produces temporary out-performance, with both
+account curves ending up flat with a return around zero. Inspecting Sharpe ratios, we find that using RL to optimise the
+trading strategy is able to increase Sharpe ratios significantly. The increase in returns of the RL strategies is hence not
+simply explained by taking on more market risk.
+Figure 3a compares the mean return between the buy & hold, baseline, and RL policies for all out-of-sample episodes
+across the three noise levels. A single dashed grey line connects the return for a single test episode across the three
+trading strategies: buy & hold, baseline, and the RL policy. The solid blue lines representing the mean return across all
+episodes. Error bars represent the 95% bootstrapped confidence intervals for the means. Testing for the significance of
+the differences between RL and baseline returns across all episodes (t-test) is statistically significant (p ≪ 0.1) for all
+noise levels. Differences in Sharpe ratios are similarly significant. We can thus conclude that the high frequency trading
+strategies learned by RL outperform our baseline strategy for all levels of noise we have considered.
+6
+
+bid
+trade
+order
+ask
+- mid
+order
+V
+canc.
+X
+canc.
+trade
+O
+O
+V
+baseline
+570.8
+Pric
+570.4
+RL
+570.8
+Pric
+570.4
+Position
+10
+0
+baseline
+-10
+RL
+Profit
+0
+-10
+09:54:35
+09:54:40
+09:54:45
+09:54:50A PREPRINT - JANUARY 23, 2023
+Figure 2: Account curves, trading the noisy oracle signal in the test set, comparing the learned RL policies (solid lines)
+with the baseline trading strategy (dashed). The black line shows the performance of the buy & hold strategy over the
+same period. Different colours correspond to different signal noise levels. The RL policy is able to improve the trading
+performance across all signal noise levels.
+(a) Episodic mean strategy return of buy & hold, baseline, and
+RL strategies for high (a = 1.1), mid (a = 1.3), and low noise
+(a = 1.6) in 31 evaluation episodes. The grey dashed lines con-
+nect mean log-returns across strategies for all individual episodes.
+The blue line connects the mean of all episodes with 95% boot-
+strapped confidence intervals.
+(b) Turnover per episode: comparison between baseline and RL
+strategy. Lower noise results in a more persistent signal, decreas-
+ing baseline turnover, but a higher quality signal, resulting in the
+RL policy to increase trading activity and turnover.
+Figure 3: Mean return and turnover of the baseline and RL trading strategies.
+It is also informative to compare the amount of trading activity between the baseline and RL strategies (see figure
+3b). The baseline turnover decreases with an increasing signal-to-noise ratio (higher a), as the signal remains more
+stable over time, resulting in fewer trades. In contrast, the turnover of the RL trading agent increases with a higher
+signal-to-noise ratio, suggesting that the agent learns to trust the signal more and reflecting that higher transaction
+costs, resulting from the higher trading activity, can be sustained, given a higher quality signal. In the high noise
+case (a = 1.1), the RL agent learns to reduce trading activity relative to the other RL strategies, thereby essentially
+filtering the signal. The turnover is high in all cases due to the high frequency of the signal and the fact that we are only
+trading a small inventory. Nonetheless, performance is calculated net of spread-based transaction costs as our simulator
+adequately accounts for the execution of individual orders.
+Table 1 lists action statistics for all RL policies, including how often actions are skipped, and the price levels at which
+limit orders are placed, grouped by buy and sell orders. With the least informative signal, the strategy almost exclusively
+uses marketable limit orders, with buy orders being placed at the bid and sell orders at the ask price. With better signals
+being available (a = 1.3 and a = 1.6), buy orders are more often placed at the mid-quote price, thereby trading less
+aggressively and saving on transaction costs. Overall, the strategies trained on different signals all place the majority of
+sell orders at the best bid price, with the amount of skipped actions varying considerably across the signals.
+7
+
+(μ=0.25, S=19.30)
+1.2
+RL (a=1.1)
+baseline (a=l.1)
+(μ=0.21, S=14.69)
+1.0
+RL (a=1.3)
+baseline (a=1.3)
+0.8
+RL (a=1.6)
+(μ=0.14, S=11.28)
+baseline (a=1.6)
+Return
+0.6
+-(μ=0.11, S=7.34)
+Buy & Hold
+0.4
+0.2
+S=-%724)
+0.0
+=%.%0.
+-0.2
+ (μ=-0.07, S=-5.68)
+T
+T
+T
+T
+0
+5
+10
+15
+20
+25
+30
+35
+Hours Tradinga=1.1
+a=1.3
+a=1.6
+0.4-
+Return
+0.2
+0.0
+-0.2
+buy & hold
+buy & hold
+buy & hold
+baselin
+baseline
+baselinea=1.1
+a=1.3
+a=1.6
+800
+Turnover
+600
+400
+200
+T
+baseline
+RL
+baseline
+RL
+baseline
+RLA PREPRINT - JANUARY 23, 2023
+a=1.1
+a=1.3
+a=1.6
+action skipped [%]
+24.5
+43.8
+7.8
+sell levels (bid, mid, ask) [%]
+(95.4, 3.1, 1.5)
+(94.6, 2.8, 2.65)
+(97.2, 1.9, 0.9)
+buy levels (bid, mid, ask) [%]
+(1.1, 1.3, 97.5)
+(1.6, 52.9, 45.5)
+(1.7, 13.0, 85.3)
+Table 1: Actions taken by RL policy for the three different noise levels: the first row shows how often the policy
+chooses the “skip” action. Not choosing this action does however not necessarily result in an order being placed, e.g. if
+inventory constraints are binding. The last two rows show the relative proportion of limit order placement levels for sell
+orders, and buy orders, respectively.
+6
+Conclusions
+Using Deep Double Duelling Q-learning with asynchronous experience replay, a state-of-the-art off-policy reinforcement
+learning algorithm, we train a limit order trading strategy in an environment using historic market-by-order (MBO)
+exchange message data. For this purpose we develop an RL environment based on the ABIDES [7] market simulator,
+which reconstructs order book states dynamically from MBO data. Observing an artificial high-frequency signal of
+the mean return over the following 10 seconds, the RL policy successfully transforms a directional signal into a limit
+order trading strategy. The policies acquired by RL outperform our baseline trading algorithm, which places marketable
+limit orders to trade into positions and passive limit orders to exit positions, both in terms of mean return and Sharpe
+ratio. We investigate the effect of different levels of noise in the alpha signal on the RL performance. Unsurprisingly,
+more accurate signals lead to higher trading returns but we also find that RL provides a similar added benefit to trading
+performance across all noise levels investigated.
+The task of converting high-frequency forecasts into tradeable and profitable strategies is difficult to solve as transaction
+costs, due to high portfolio turnover, can have a prohibitively large impact on the bottom line profits. We suggest that
+RL can be a useful tool to perform this translational role and learn optimal strategies for a specific signal and market
+combination. We have shown that tailoring strategies in this way can significantly improve performance, and eliminates
+the need for manually fine-tuning execution strategies for different markets and signals. For practical applications,
+multiple different signals could even be combined into a single observation space. That way the problem of integrating
+different forecasts into a single coherent trading strategy could be directly integrated into the RL problem.
+While we here show an interesting use-case of RL in limit order book markets, we also want to motivate the need for
+further research in this area. There are many years of high-frequency market data available, which ought to be utilised
+to make further progress in LOB-based tasks and improve RL in noisy environments. This, together with the newest
+type of neural network architectures, such as attention-based transformers [29, 30], enables learning tasks in LOB
+environments directly from raw data with even better performance. For the task we have considered in this paper, future
+research could enlarge the action space, allowing for placement of limit orders deeper into the book and larger orders
+sizes. Allowing for larger sizes however would require a realistic model of market impact, considering the reaction of
+other market participants.
+8
+
+A PREPRINT - JANUARY 23, 2023
+References
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+9
+
+A PREPRINT - JANUARY 23, 2023
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+arXiv preprint arXiv:1904.10509, 2019.
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+Jordan, and Ion Stoica. Rllib: Abstractions for distributed reinforcement learning. In International Conference on
+Machine Learning, pages 3053–3062. PMLR, 2018.
+7
+Appendix
+We use the RLlib library [31] for a reference implementation of the APEX algorithm. Table 2 shows a selection of
+relevant parameters we used for RL training.
+Paramter
+Value
+timesteps_total
+300e6
+framework
+torch
+num_gpus
+1
+num_workers
+42
+batch_mode
+truncate_episode
+gamma
+.99
+lr_schedule
+[[0,2e-5], [1e6, 5e-6]]
+buffer_size
+2e6
+learning_starts
+5000
+train_batch_size
+50
+rollout_fragment_length
+50
+target_network_update_freq
+5000
+n_step
+3
+prioritized_replay
+False
+Table 2: Selected RL parameters for APEX algorithm using RLlib [31] library for training.
+Figure 4 shows confusion matrices interpreting the oracle signal scores as probabilities over the three classes: down,
+stationary, and up. The predicted class is thus the one with the highest score.
+Figure 4: Confusion matrices of the artificial oracle signal for three noise levels, from low to high noise.
+10
+
+a=1.1
+a=1.3
+a=1.6
+0.5
+0.23
+0.27
+0.69 0.11
+0.2
+0.77
+0.06 0.17
+down
+True label
+stationary
+0.350.31
+0.34
+0.34 0.340.32
+0.35 0.360.29
+up 0.29
+0.24
+0.48
+0.21
+0.11
+0.68
+0.2
+0.0580.74
+down
+stationary
+down
+stationary
+down
+stationary
+dn
+Tabe

M9FAT4oBgHgl3EQfxh78/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf,len=447
+page_content='ASYNCHRONOUS DEEP DOUBLE DUELLING Q-LEARNING FOR  TRADING-SIGNAL EXECUTION IN LIMIT ORDER BOOK  MARKETS  Peer Nagy  Oxford-Man Institute of Quantitative Finance  University of Oxford  peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='nagy@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='uk  Jan-Peter Calliess  Oxford-Man Institute of Quantitative Finance  University of Oxford  Stefan Zohren  Oxford-Man Institute of Quantitative Finance  University of Oxford  January 23, 2023  ABSTRACT  We employ deep reinforcement learning (RL) to train an agent to successfully translate a high-  frequency trading signal into a trading strategy that places individual limit orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Based on the  ABIDES limit order book simulator, we build a reinforcement learning OpenAI gym environment and  utilise it to simulate a realistic trading environment for NASDAQ equities based on historic order book  messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To train a trading agent that learns to maximise its trading return in this environment, we  use Deep Duelling Double Q-learning with the APEX (asynchronous prioritised experience replay)  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The agent observes the current limit order book state, its recent history, and a short-term  directional forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To investigate the performance of RL for adaptive trading independently from a  concrete forecasting algorithm, we study the performance of our approach utilising synthetic alpha  signals obtained by perturbing forward-looking returns with varying levels of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Here, we find  that the RL agent learns an effective trading strategy for inventory management and order placing  that outperforms a heuristic benchmark trading strategy having access to the same signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Keywords Limit Order Books · Quantitative Finance · Reinforcement Learning · LOBSTER ·  1  Introduction  Successful quantitative trading strategies often work by generating trading signals, which exhibit a statistically significant  correlation with future prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' These signals are then turned into actions, aiming to assume positions in order to gain  from future price changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The higher the signal frequency and strategy turnover, the more critical is the execution  component of the strategy, which translates the signal into concrete orders that can be submitted to a market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Such  markets are oftentimes organised as an order ledger represented by a limit order book (LOB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Limit order book prices have been shown to be predictable over short time periods, predicting a few successive ticks  into the future with some accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This has been done by either utilising the recent history of order book states  [1, 2], order-flow data [3], or market-by-order (MBO) data directly as features [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' However, given the short time  horizons over which these predictions are performed, and correspondingly small price movements, predictability does  not directly translate into trading profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Transaction costs, strategy implementation details, and time delays add up to  the challenging problem of translating high-frequency forecasts into a trading strategy, which determines when and  which orders to send to the exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Moreover, different predictive signals have to be traded differently to achieve  optimal results, depending on the forecast horizon, signal stability, and predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='08688v1  [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='TR]  20 Jan 2023  A PREPRINT - JANUARY 23, 2023  In this paper, we use asynchronous off-policy reinforcement learning (RL), specifically Deep Duelling Double Q-  learning with the APEX architecture [5], to learn an optimal trading strategy, given a noisy directional signal of  short-term forward mid-quote returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' For this purpose, we developed an OpenAI gym [6] limit order book environment  based on the ABIDES [7] market simulator, similar to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We use this simulator to replay NASDAQ price-time priority  limit order book markets using message data from the LOBSTER data set [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  We study the case of an artificial or synthetic signal, taking the future price as known and adding varying levels of noise,  allowing us to investigate learning performance and to quantify the benefit of an RL-derived trading policy compared to  a baseline strategy using the same noisy signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This is not an unrealistic setup when choosing the correct level of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Practitioners often have dedicated teams researching and deriving alpha signals, often over many years, while other  teams might work on translating those signals into profitable strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Our aim is to focus on the latter problem which  becomes increasingly more difficult as signals become faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' It is thus interesting to see how an RL framework can be  used to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In particular, we show that the RL agent learns superior policies to the baselines, both in  terms of strategy return and Sharpe ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Machine learning methods, such as RL, have become increasingly important  to automate trade execution in the financial industry in recent years [10], underlining the practical use of research in  this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  We make a number of contributions to the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' By defining a novel action and state space in a LOB  trading environment, we allow for the placement of limit orders at different prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This allows the agent to learn  a concrete high-frequency trading strategy for a given signal, trading either aggressively by crossing the spread, or  conservatively, implicitly trading off execution probability and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In addition to the timing and level placement of  limit orders, our RL agent also learns to use limit orders of single units of stock to manage its inventory as it holds  variably sized long or short positions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' More broadly, we demonstrate the practical use case of RL to translate  predictive signals into limit order trading strategies, which is still usually a hand-crafted component of a trading system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  We thus show that simulating limit order book markets and using RL to further automate the investment process is a  promising direction for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To the best of our knowledge, this is also the first study applying the APEX [5]  algorithm to limit order book environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  The remaining paper is structured as follows: Section 2 surveys related literature, section 3 explains the mechanics  of limit order book markets and the APEX algorithm, section 4 details the construction of the artificial price signal,  section 5 showcases our empirical results, and section 6 concludes our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  2  Related Work  Reinforcement learning has been applied to learn different tasks in limit order book market environments, such as  optimal trade execution [11, 12, 13, 14, 15], market making [16, 17], portfolio optimisation [18] or trading [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  The objective of optimal trade execution is to minimise the cost of trading a predetermined amount of shares over a  given time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Trading direction and the number of shares is already pre-defined in the execution problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Market  makers, on the other hand, place limit orders on both sides of the order book and set out to maximise profits from  capturing the spread, while minimising the risk of inventory accumulation and adverse selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We summarise using  the term “RL for trading” such tasks which maximise profit from taking directional bets in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This is a hard  problem for RL to solve as the space of potential trading strategies is large, leading to potentially many local optima in  the loss landscape, and actionable directional market forecasts are notoriously difficult due to arbitrage in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  The work of [19] is an early study of RL for market microstructure tasks, including trade execution and predicting price  movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' While the authors achieve some predictive power of directional price moves, forecasts are determined  to be too erroneous for profitable trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The most similar work to ours is [21] that provides the first end-to-end  DRL framework for high-frequency trading, using PPO [22] to trade Intel stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To model price impact, [21] use an  approximation, moving prices proportionately to the square-root of traded volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The action space is essentially  limited to market orders, so there is no decision made on limit prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The trained policy is able to produce a profitable  trading strategy on the evaluated 20 test days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' However, this is not compared to baseline strategies and the resulting  performance is not statistically tested for significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In contrast, we consider a larger action space, allowing for the  placement of limit orders at different prices, thereby potentially lowering transaction costs of the learned HFT strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  For a broader survey of deep RL (DRL) for trading, including portfolio optimisation, model-based and hierarchical RL  approaches the reader is referred to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  2  A PREPRINT - JANUARY 23, 2023  3  Background  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  Limit Order Book Data  Limit order books (LOBs) are one of the most popular financial market mechanisms used by exchanges around the  world [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Market participants submit limit buy or sell orders, specifying a maximum (minimum) price at which they  are willing to buy (sell), and the size of the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The exchange’s limit order book then keeps track of unfilled limit  orders on the buy side (bids) and the sell side (asks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' If an incoming order is marketable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' there are open orders on  the opposing side of the order book at acceptable prices, the order is matched immediately, thereby removing liquidity  from the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The most popular matching prioritisation scheme is price-time priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Here, limit orders are matched  first based on price, starting with the most favourable price for the incoming order, and then based on arrival time,  starting with the oldest resting limit order in the book, at each price level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' For a more complete review of limit order  book dynamics and pertaining models, we refer the reader to [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  In this paper, we consider equity limit order book data from the NASDAQ exchange [9], which also uses a price-time  priority prioritisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Our market simulator keeps track of the state of the LOB by replaying historical message data,  consisting of new incoming limit orders, order cancellations or modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The RL agent can then inject new  messages into the order flow and thereby, change the LOB state from its observed historical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Our simulator reconstructs LOB dynamics from message data, so every marketable order takes liquidity from the book  and thus has a direct price impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Beyond that, we make no further assumptions on permanent market impact or  reactions of other agents in the market, which we leave to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  Double DQN with Distributed Experience Replay  We model the trader’s problem as a Markov Decision Process (MDP) [24, 25], described by the tuple ⟨S, A, T , r, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' S  denotes a state space, A an action space, T a stochastic transition function, r a reward function and γ a discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Observing the current environment state st ∈ S at time t, the trader takes action at ∈ A, which causes the environment  to transition state according to the stochastic transition function T (st+1|st, at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' After transitioning from st to st+1,  the agent receives a reward rt+1 = r(st, at, st+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We use Deep Double Q-learning [26] with a duelling network  architecture [27] to approximate the optimal Q-function Q∗(s, a) = E[rt+1+γ maxa′ Q∗(st+1, a′)|at = a, st = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To  speed up the learning process we employ the APEX training architecture [5], which combines asynchronous experience  sampling using parallel environments with off-policy learning from experience replay buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Every episode i results  in an experience trajectory τi = {st, at}T  t=1, many of which are sampled from parallel environment instances and are  then stored in the replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The environment sampling is done asynchronously using parallel processes running on  CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Experience data from the buffer is then sampled randomly and batched to perform a policy improvement step of  the Q-network on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Prioritised sampling from the experience buffer has proven to degrade performance in our  noisy problem setting, hence we are sampling uniformly from the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1 After a sufficient number of training steps,  the new policy is then copied to every CPU worker to update the behavioural policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Double Q-learning [28, 26] stabilises the learning process by keeping separate Q-network weights for action selection  (main network) and action validation (target network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The target network weights are then updated gradually in the  direction of the main network’s weight every few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The duelling network architecture [27] on the other hand  uses two separate network branches (for both main and target Q-networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' One branch estimates the value function  V (s) = maxa Q(s, a), while the other estimates the advantage function A(s, a) = Q(s, a) − V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The benefit of this  architecture choice lies therein that the advantage of individual actions in some states might be irrelevant, and the state  value, which can be learnt more easily, suffices for an action-value approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  4  Framework  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  Artificial Price Signal  The artificial directional price signal dt ∈ ∆2 = {x ∈ R3 : x1 + x2 + x3 = 1, xi ≥ 0 for i = 1, 2, 3} the agent  receives is modelled as a discrete probability distribution over 3 classes, corresponding to the averaged mid-quote price  decreasing, remaining stable, or increasing over a fixed future time horizon of h ∈ N+ seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To achieve realistic  levels of temporal stability of the signal process, dt is an exponentially weighted average, with persistence coefficient  1In many application domains prioritised sampling, whereby we resample instances more frequently where the model initially  performs poorly tends to aide learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' However, in our low signal-to-noise application domain, we noted poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Investigating the matter, we found that prioritised sampling caused more frequent resampling of highly noisy instances where  learning was particularly difficult, hence degrading performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  3  A PREPRINT - JANUARY 23, 2023  φ ∈ (0, 1), of Dirichlet random variables ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The Dirichlet parameters α depend on the realised smoothed future return  rt+h, specifically on whether the return lies within a neighbourhood of size k around zero, or above or below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Thus we  have:  dt = φdt−1 + (1 − φ)ϵt  ϵt = Dirichlet (α(rt+h))  rt+h = pt+h − pt  pt  where  pt+h = 1  h  h  �  i=1  pt+i  (1)  and prices pt refer to the mid-quote price at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The Dirichlet distribution is parametrised, so that, in expectation,  the signal dt updates in the direction of future returns, where aH and aL determine the variance of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The  Dirichlet parameter vector is thus:  α(rt+h) =  �  �  �  (aH, aL, aL)  if rt+h < −k  (aL, aH, aL)  if − k ≤ rt+h < k  (aL, aL, aH)  if k ≤ rt+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  (2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  RL Problem Specification  At each time step t, the agent receives a new state observation st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' st consists of the time left in the current episode  T − t given the episode’s duration of T, the agent’s cash balance Ct, stock inventory Xt, the directional signal dt ∈ ∆2,  encoded as probabilities of prices decreasing, remaining approximately constant, or increasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' and price and volume  quantities for the best bid and ask (level 1), including the agent’s own volume posted at bid and ask: ob,t and oa,t  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In addition to the most current observable variables at time t, the agent also observes a history of the  previous l values, which are updated whenever there is an observed change in the LOB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Putting all this together, we  obtain the following state observation:  st =  �  �  �  �  �  T − u  Cu  Xu  (d1  u, d2  u, d3  u)′  (pa,u, va,u, oa,u, pb,u, vb,u, ob,u)′  �  �  �  �  �  u={t−l,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=',t}  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  After receiving the state observation, the agent then chooses an action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' It can place a buy or sell limit order of a  single share at bid, mid-quote, or ask price;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' or do nothing and advance to the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Actions, which would  immediately result in positions outside the allowed inventory constraints [posmin, posmax] are disallowed and do not  trigger an order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Whenever the execution of a resting limit order takes the inventory outside the allowed constraints, a  market order in the opposing direction is triggered to reduce the position back to posmin for short positions or posmax  for long positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Hence, we define  at ∈ A = ({−1, 1} × {−1, 0, 1}) ∪ {skip}  so that in total there are 7 discrete actions available, three levels for both buy and sell orders, and a skip action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  For the six actions besides the “skip” action, the first dimension encodes the trading direction (sell or buy) and the  second dimension the price level (bid, mid-price, or ask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' For example, a = (1, 0) describes the action to place a buy  order at the mid price, and a = (−1, 1) a sell order at best ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Rewards Rt+1 consist of a convex combination of a  profit-and-loss-based reward Rpnl  t+1 and a directional reward Rdir  t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Rpnl  t+1 is the log return of the agent’s mark-to-market  portfolio value Mt, encompassing cash and the current inventory value, marked at the mid-price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The benefit of  log-returns is that they are additive over time, rather than multiplicative like gross returns, so that, without discounting  (γ = 1) the total profit-and-loss return �T  s=t+1 Rpnl  s  = MT − Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The directional reward term Rdir  t+1 incentivizes the  agent to hold inventory in the direction of the signal and penalises the agent for inventory positions opposing the signal  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The size of the directional reward can be scaled by the parameter κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Rdir  t+1 is positive if the positive  prediction has a higher score than the negative (dt,3 > dt,1) and the current inventory is positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' or if dt,3 < dt,1 and  4  A PREPRINT - JANUARY 23, 2023  Xt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Further, if the signal [−1, 0, 1] · dt has an opposite sign than inventory Xt, Rdir  t+1 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This can be  summarised as follows:  Mark-to-Market Value  Mt = Ct + Xtpm  t  ∆Mt = ∆Ct + Xt−1∆pm  t + ∆xtpm  t  PnL Reward  Rpnl  t+1 = ln(Mt) − ln(Mt−1)  Directional Reward  Rdir  t+1 = κ[−1, 0, 1] · dtXt  Total Reward  rt+1 = wdirRdir  t+1 + (1 − wdir)Rpnl  t+1  (3)  The weight on the directional reward wdir ∈ [0, 1) is reduced every learning step by a factor ψ ∈ (0, 1),  wdir ← ψwdir  so that initially the agent quickly learns to trade in the signal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Over the course of the learning process, Rpnl  t  becomes dominant and the agent maximises its mark-to-market profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  5  Experimental Results  We train all RL policies using the problem setup discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2 on 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5 months of Apple (AAPL) limit order  book data (2012-01-01 to 2012-05-16) and evaluate performance on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5 months of out-of-sample data (2012-05-17 to  2012-06-31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We only use the first hour of every trading day (09:30 to 10:30) as the opening hour exhibits higher than  average trading volume and price moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Each hour of the data corresponds to a single RL episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Our neural network architecture consists of 3 feed-forward layers, followed by an LSTM layer, for both the value- and  advantage stream of the duelling architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The LSTM layer allows the agent to efficiently learn a memory-based  policy with observations including 100 LOB states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  We compare the resulting learned RL policies to a baseline trading algorithm, which receives the same artificially  perturbed high-frequency signal of future mid-prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The baseline policy trades aggressively by crossing the spread  whenever the signal indicates a directional price move up or down until the inventory constraint is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The signal  direction in the baseline algorithm is determined as the prediction class with the highest score (down, neutral, or up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  When the signal changes from up or down to neutral, indicating no immediate expected price move, the baseline strategy  places a passive order to slowly reduce position size until the inventory is cleared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This heuristic utilises the same action  space as the RL agent and yielded better performance than trading using only passive orders (placed at the near touch),  or only aggressive orders (at the far touch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Figure 1 plots a 17 second simulation window from the test period, comparing the simulated baseline strategy with  the RL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' It can be seen that prices in the LOB are affected by the trading activity as both strategies inject new  order flow into the market, in addition to the historical orders, thereby consuming or adding liquidity at the best bid and  ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' During the plotted period, the baseline strategy incurs small losses due to the signal switching between predicting  decreasing and increasing future prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This causes the baseline strategy to trade aggressively, paying the spread with  every trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The RL strategy, on the other hand, navigates this difficult period better by trading more passively out of its  long position, and again when building up a new position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Especially in the second half of the depicted time period, the  RL strategy adds a large number of passive buy orders (green circles in the second panel of figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This is shown  by the green straight lines, which connect the orders to their execution or cancellation, some of which occur after the  depicted period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  Oracle Signal  The RL agent receives a noisy oracle signal of the mean return h = 10 seconds into the future (see equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' It  chooses an action every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1s, allowing a sufficiently quick build-up of long or short positions using repeated limit  orders of single stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The algorithm is constrained to keep the stock inventory within bounds of [posmin, posmax] =  [−10, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To change the amount of noise in the signal, we vary the aH parameter of the Dirichlet distribution, keeping  aL = 1 constant in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' To keep the notation simple, we hence drop the H superscript and refer to the variable  Dirichlet parameter aH simply as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We consider three different noise levels, parametrising the Dirichlet distribution  with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6 (low noise), a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3 (mid noise), and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1 (high noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' A fixed return classification threshold  k = 4 · 10−5 was chosen to achieve good performance of the baseline algorithm, placing around 85% of observations  in the up or down category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The signal process persistence parameter is set to φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  5  A PREPRINT - JANUARY 23, 2023  Figure 1: A short snapshot of simulation results (AAPL on 2012-06-14), comparing the RL policy (second panel) with  the baseline (first panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The first two panels plot the best bid, ask, and mid-price, overlaying trading events of buy  orders (green) and sell orders (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Circles mark new unmarketable limit orders entering the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Crosses mark order  executions (trades) and triangles order cancellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Open orders are connected by lines to either cancellations or  trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Since we are simulating the entire LOB, trading activity can be seen to affect bid and ask prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The third panel  plots the evolution of the inventory position of both strategies, and the last panel the trading profits over the period in  USD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Out-of-sample trading performance is visualised by the account curves in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The curves show the evolution of  the portfolio value for a chronological evaluation of all test episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Every account curve shows the mean episodic  log-return µ and corresponding Sharpe ratio S next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We show that all RL-derived policies are able to outperform  their respective baseline strategies for the three noise levels investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Over the 31 test episodes, the cumulative RL  algorithm out-performance over the baseline strategy ranges between 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8 (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3) and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2 (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1) percentage  points (and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='7 for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In the case of the signal with the lowest signal-to-noise ratio (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1), for which the  baseline strategy incurs a loss for the test period, the RL agent has learned a trading strategy with an approximately  zero mean return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Temporarily, the strategy even produces positive gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Overall, it produces a sufficiently strong  performance to not lose money while still trading actively and incurring transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Compared to a buy-and-hold  strategy over the same time period, the noisy RL strategy similarly produces temporary out-performance, with both  account curves ending up flat with a return around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Inspecting Sharpe ratios, we find that using RL to optimise the  trading strategy is able to increase Sharpe ratios significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The increase in returns of the RL strategies is hence not  simply explained by taking on more market risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Figure 3a compares the mean return between the buy & hold, baseline, and RL policies for all out-of-sample episodes  across the three noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' A single dashed grey line connects the return for a single test episode across the three  trading strategies: buy & hold, baseline, and the RL policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The solid blue lines representing the mean return across all  episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Error bars represent the 95% bootstrapped confidence intervals for the means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Testing for the significance of  the differences between RL and baseline returns across all episodes (t-test) is statistically significant (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1) for all  noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Differences in Sharpe ratios are similarly significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We can thus conclude that the high frequency trading  strategies learned by RL outperform our baseline strategy for all levels of noise we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  6  bid  trade  order  ask  mid  order  V  canc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  X  canc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  trade  O  O  V  baseline  570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8  Pric  570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='4  RL  570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8  Pric  570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='4  Position  10  0  baseline  10  RL  Profit  0  10  09:54:35  09:54:40  09:54:45  09:54:50A PREPRINT - JANUARY 23, 2023  Figure 2: Account curves, trading the noisy oracle signal in the test set, comparing the learned RL policies (solid lines)  with the baseline trading strategy (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The black line shows the performance of the buy & hold strategy over the  same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Different colours correspond to different signal noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The RL policy is able to improve the trading  performance across all signal noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  (a) Episodic mean strategy return of buy & hold, baseline, and  RL strategies for high (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1), mid (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3), and low noise  (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6) in 31 evaluation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The grey dashed lines con-  nect mean log-returns across strategies for all individual episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  The blue line connects the mean of all episodes with 95% boot-  strapped confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  (b) Turnover per episode: comparison between baseline and RL  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Lower noise results in a more persistent signal, decreas-  ing baseline turnover, but a higher quality signal, resulting in the  RL policy to increase trading activity and turnover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Figure 3: Mean return and turnover of the baseline and RL trading strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  It is also informative to compare the amount of trading activity between the baseline and RL strategies (see figure  3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The baseline turnover decreases with an increasing signal-to-noise ratio (higher a), as the signal remains more  stable over time, resulting in fewer trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In contrast, the turnover of the RL trading agent increases with a higher  signal-to-noise ratio, suggesting that the agent learns to trust the signal more and reflecting that higher transaction  costs, resulting from the higher trading activity, can be sustained, given a higher quality signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In the high noise  case (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1), the RL agent learns to reduce trading activity relative to the other RL strategies, thereby essentially  filtering the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The turnover is high in all cases due to the high frequency of the signal and the fact that we are only  trading a small inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Nonetheless, performance is calculated net of spread-based transaction costs as our simulator  adequately accounts for the execution of individual orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Table 1 lists action statistics for all RL policies, including how often actions are skipped, and the price levels at which  limit orders are placed, grouped by buy and sell orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' With the least informative signal, the strategy almost exclusively  uses marketable limit orders, with buy orders being placed at the bid and sell orders at the ask price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' With better signals  being available (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6), buy orders are more often placed at the mid-quote price, thereby trading less  aggressively and saving on transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Overall, the strategies trained on different signals all place the majority of  sell orders at the best bid price, with the amount of skipped actions varying considerably across the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  7  (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='25, S=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='30)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  RL (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1)  baseline (a=l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1)  (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='21, S=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='69)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='0  RL (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3)  baseline (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8  RL (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6)  (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='14, S=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='28)  baseline (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6)  Return  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6  (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='11, S=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='34)  Buy & Hold  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  S=-%724)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='0  =%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='%0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  (μ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='07, S=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='68)  T  T  T  T  0  5  10  15  20  25  30  35  Hours Tradinga=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='4-  Return  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  buy & hold  buy & hold  buy & hold  baselin  baseline  baselinea=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6  800  Turnover  600  400  200  T  baseline  RL  baseline  RL  baseline  RLA PREPRINT - JANUARY 23, 2023  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6  action skipped [%]  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8  sell levels (bid, mid, ask) [%]  (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5)  (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='65)  (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='9)  buy levels (bid, mid, ask) [%]  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='9, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='7, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='0, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3)  Table 1: Actions taken by RL policy for the three different noise levels: the first row shows how often the policy  chooses the “skip” action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Not choosing this action does however not necessarily result in an order being placed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' if  inventory constraints are binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The last two rows show the relative proportion of limit order placement levels for sell  orders, and buy orders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  6  Conclusions  Using Deep Double Duelling Q-learning with asynchronous experience replay, a state-of-the-art off-policy reinforcement  learning algorithm, we train a limit order trading strategy in an environment using historic market-by-order (MBO)  exchange message data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' For this purpose we develop an RL environment based on the ABIDES [7] market simulator,  which reconstructs order book states dynamically from MBO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Observing an artificial high-frequency signal of  the mean return over the following 10 seconds, the RL policy successfully transforms a directional signal into a limit  order trading strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The policies acquired by RL outperform our baseline trading algorithm, which places marketable  limit orders to trade into positions and passive limit orders to exit positions, both in terms of mean return and Sharpe  ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We investigate the effect of different levels of noise in the alpha signal on the RL performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Unsurprisingly,  more accurate signals lead to higher trading returns but we also find that RL provides a similar added benefit to trading  performance across all noise levels investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  The task of converting high-frequency forecasts into tradeable and profitable strategies is difficult to solve as transaction  costs, due to high portfolio turnover, can have a prohibitively large impact on the bottom line profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We suggest that  RL can be a useful tool to perform this translational role and learn optimal strategies for a specific signal and market  combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' We have shown that tailoring strategies in this way can significantly improve performance, and eliminates  the need for manually fine-tuning execution strategies for different markets and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' For practical applications,  multiple different signals could even be combined into a single observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' That way the problem of integrating  different forecasts into a single coherent trading strategy could be directly integrated into the RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  While we here show an interesting use-case of RL in limit order book markets, we also want to motivate the need for  further research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' There are many years of high-frequency market data available, which ought to be utilised  to make further progress in LOB-based tasks and improve RL in noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' This, together with the newest  type of neural network architectures, such as attention-based transformers [29, 30], enables learning tasks in LOB  environments directly from raw data with even better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' For the task we have considered in this paper, future  research could enlarge the action space, allowing for placement of limit orders deeper into the book and larger orders  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Allowing for larger sizes however would require a realistic model of market impact, considering the reaction of  other market participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  8  A PREPRINT - JANUARY 23, 2023  References  [1] Zihao Zhang, Stefan Zohren, and Stephen Roberts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' DeepLOB: Deep convolutional neural networks for limit order  books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
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+page_content=' Multi-horizon forecasting for limit order books: Novel deep learning approaches  and hardware acceleration using intelligent processing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
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+page_content=' Double Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 23, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  [29] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and  Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  [30] Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Generating long sequences with sparse transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='10509, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  [31] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox, Ken Goldberg, Joseph Gonzalez, Michael  Jordan, and Ion Stoica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Rllib: Abstractions for distributed reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' In International Conference on  Machine Learning, pages 3053–3062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  7  Appendix  We use the RLlib library [31] for a reference implementation of the APEX algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' Table 2 shows a selection of  relevant parameters we used for RL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Paramter  Value  timesteps_total  300e6  framework  torch  num_gpus  1  num_workers  42  batch_mode  truncate_episode  gamma  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='99  lr_schedule  [[0,2e-5], [1e6, 5e-6]]  buffer_size  2e6  learning_starts  5000  train_batch_size  50  rollout_fragment_length  50  target_network_update_freq  5000  n_step  3  prioritized_replay  False  Table 2: Selected RL parameters for APEX algorithm using RLlib [31] library for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Figure 4 shows confusion matrices interpreting the oracle signal scores as probabilities over the three classes: down,  stationary, and up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content=' The predicted class is thus the one with the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  Figure 4: Confusion matrices of the artificial oracle signal for three noise levels, from low to high noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='  10  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='1  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='3  a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='17  down  True label  stationary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='29  up 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='0580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}
+page_content='74  down  stationary  down  stationary  down  stationary  dn  Tabe' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'}

MtFQT4oBgHgl3EQfVjao/content/tmp_files/2301.13301v1.pdf.txt

@@ -0,0 +1,2388 @@
+New experimental constraint on the 185W(n, γ)186W cross section
+A. C. Larsen,1, ∗ G. M. Tveten,1, 2, † T. Renstrøm,1, 2 H. Utsunomiya,3, 4 E. Algin,5 T. Ari-izumi,3
+K. O. Ay,6 F. L. Bello Garrote,1 L. Crespo Campo,1 F. Furmyr,1 S. Goriely,7 A. G¨orgen,1
+M. Guttormsen,1 V. W. Ingeberg,1 B. V. Kheswa,8, 9 I. K. B. Kullmann,10 T. Laplace,11 E. Lima,1
+M. Markova,1 J. E. Midtbø,1 S. Miyamoto,12 A. H. Mjøs,1 V. Modamio,1 M. Ozgur,6 F. Pogliano,1
+S. Riemer-Sørensen,13, 14 E. Sahin,1 S. Shen,13 S. Siem,1 A. Spyrou,15, 16, 17 and M. Wiedeking8, 18
+1Department of Physics, University of Oslo, N-0316 Oslo, Norway
+2Expert Analytics AS, N-0179 Oslo, Norway
+3Department of Physics, Konan University, Okamoto 8-9-1, Higashinada, Kobe 658-8501, Japan
+4Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China
+5Department of Metallurgical and Materials Engineering, Pamukkale University, 20160 Denizli, Turkey
+6Department of Physics, Eskisehir Osmangazi University, 26480 Eskisehir, Turkey
+7Institut d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles,
+Campus de la Plaine, CP-226, 1050 Brussels, Belgium
+8iThemba LABS, P.O. Box 722, 7129 Somerset West, South Africa
+9Department of Applied Physics and Engineering Mathematics,
+University of Johannesburg, Johannesburg, 2028, South Africa
+10Institute d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles, Belgium
+11Department of Nuclear Engineering, University of California, Berkeley, 94720, USA
+12Laboratory of Advanced Science and Technology for Industry,
+University of Hyogo, 3-1-2 Kouto, Kamigori, Ako-gun, Hyogo 678-1205, Japan
+13Institute of Theoretical Astrophysics, University of Oslo, N-0316 Oslo, Norway
+14Department of Mathematics and Cybernetics, SINTEF Digital, N-0314 Oslo, Norway
+15Physics Department, Michigan State University, East Lansing, Michigan 48824, USA
+16National Superconducting Cyclotron Laboratory,
+Michigan State University, East Lansing, Michigan 48824, USA
+17Joint Institute for Nuclear Astrophysics Center for the Evolution of the Elements,
+University of Notre Dame, Notre Dame, Indiana 46556, USA
+18School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa
+(Dated: February 1, 2023)
+In this work, we present new data on the 182,183,184W(γ, n) cross sections, utilizing a quasi-
+monochromatic photon beam produced at the NewSUBARU synchrotron radiation facility. Further,
+we have extracted the nuclear level density and γ-ray strength function of 186W from data on
+the 186W(α, α′γ)186W reaction measured at the Oslo Cyclotron Laboratory. Combining previous
+measurements on the 186W(γ, n) cross section with our new 182,183,184W(γ, n) and (α, α′γ)186W
+data sets, we have deduced the 186W γ-ray strength function in the range of 1 < Eγ < 6 MeV and
+7 < Eγ < 14 MeV.
+Our data are used to extract the level density and γ-ray strength functions needed as input to the
+nuclear-reaction code TALYS, providing an indirect, experimental constraint for the 185W(n, γ)186W
+cross section and reaction rate. Compared to the recommended Maxwellian-averaged cross section
+(MACS) in the KADoNiS-1.0 data base, our results are on average lower for the relevant energy
+range kBT ∈ [5, 100] keV, and we provide a smaller uncertainty for the MACS. The theoretical
+values of Bao et al.
+and the cross section experimentally constrained on photoneutron data of
+Sonnabend et al. are significantly higher than our result. The lower value by Mohr et al. is in very
+good agreement with our deduced MACS. Our new results could have implications for the s-process
+and in particular the predicted s-process production of 186,187Os nuclei.
+I.
+INTRODUCTION
+Neutron-capture reactions are known to be the main
+producers of elements heavier than iron in our Uni-
+verse [1, 2].
+The rapid (r) and the slow (s) neutron-
+capture processes are traditionally believed to account
+for almost 100% of the Solar-system heavy-element abun-
+∗ a.c.larsen@fys.uio.no
+† gry@xal.no
+dances [3, 4]. The r process takes place in an environment
+with an extremely high neutron density typicallly larger
+than 1024 neutrons/cm3, which produces very neutron-
+rich nuclei within a short time window (≈1s). In contrast,
+the s process is, as the name implies, a slow process;
+the neutron density is comparatively low (∼ 106 − 108
+neutrons/cm3 in asymptotic giant branch stars [5]) and
+it can take from days to thousands of years between each
+neutron-capture reaction.
+Consequently, the s-process
+“path” in the nuclear chart remains close to the valley of
+stability, as the β-decay rates are typically much faster
+than the (n, γ) rates when an unstable nucleus is reached.
+arXiv:2301.13301v1  [nucl-ex]  30 Jan 2023
+
+2
+108
+109
+110
+111
+112
+113
+114
+115
+116
+Neutron number
+73
+74
+75
+76
+77
+78
+Proton number
+W
+182
+W
+183
+W
+184
+W
+185
+W
+186
+W
+187
+Re
+185
+Re
+186
+Re
+187
+Re
+188
+Os
+186
+Os
+187
+Os
+188
+Os
+189
+Os
+190
+Os
+191
+Os
+192
+FIG. 1. (Color online) Schematic illustration of the nuclear chart in the W-Re-Os region. The black arrows indicate (n, γ)
+reactions on stable or near-stable isotopes, the blue dashed arrows show the possible (n, γ) branch on the long-lived W, Re and
+Os isotopes, while the pink arrows display the β− decay branch.
+However, this is not true for some particular nuclei
+along the s-process path. At the branch points [6] the
+β-decay rate is comparable to the (n, γ) rate, so that
+there is a non-negligible possibility for the nucleus to ei-
+ther undergo β-decay or capture another neutron. On
+the one hand, such branch points could complicate the
+s-process nucleosynthesis calculation significantly; on the
+other hand, they may provide valuable information about
+the neutron density and/or temperature at the astro-
+physical site for which the s process operates [7–9].
+In this work, we focus on the branch-point nucleus
+185W, with a laboratory half-life of 75.1(3) days [10].
+This nucleus is of interest for the Re/Os cosmochronol-
+ogy first discussed by Clayton [11]. The main idea behind
+the Re/Os cosmochronology is the following: the mat-
+ter from which the Solar system was formed, contained
+a given amount of 187Re and 187Os. Further, 187Re is
+usually assigned a pure r-process origin, while 187Os is
+produced only in the s process. As 187Re has a very long
+half-life of more than 4·1010 years [12], Clayton suggested
+to use the solar-system amount of 187Re and 187Os as a
+“clock”, which would display the time span for which
+nucleosynthesis events produced various elements up to
+the time of the formation of our Solar system. Provided
+that the 187Os amount stemming from the s process can
+be reliably calculated, the extra amount of 187Os origi-
+nates from the 187Re decay. Thus, at least in principle,
+the abundances of the parent/child pair 187Re/187Os can
+be used as a cosmochronometer, although not without
+complications [13–15]. As discussed in Refs. [7, 14–16],
+the branchings at 185W and 186Re (see Fig. 1) could well
+have a non-negligible impact on this cosmochronometer.
+Moreover, several authors [7–9, 17] have discussed the
+185W and 186Re branchings as a “neutron dosimeter” for
+the effective s-process neutron density; this application
+again depends on the radiative neutron-capture cross sec-
+tions of 185W and 186Re. No direct measurement of the
+neutron-capture cross section is possible on these target
+nuclei, and only constraints on the electromagnetic de-
+cay of the compound system have been obtained through
+photoneutron experiments at relatively high photon en-
+ergies [8, 9].
+Here
+we
+present
+new
+photoneutron
+data
+on
+182,183,184W that complete the (γ, n) measurements
+on the W isotopes (Sec. II). Moreover, in Sec. III,
+we present the
+186W(α, α′γ) data taken at the Oslo
+Cyclotron Laboratory, and the data analysis with the
+resulting level density and γ strength function of 186W.
+Using our new data to constrain the input to the
+nuclear reaction code TALYS-1.9 [18] we estimate the
+185W(n, γ)186W Maxwellian-averaged cross section and
+reaction rate, and compare to previous measurements
+and evaluations in Sec. IV B. Finally, we give a summary
+and outlook in Sec. V.
+II.
+THE (γ, n) EXPERIMENTS
+A.
+Experimental details
+The photo-neutron measurements on 182,183,184W took
+place at the NewSUBARU synchrotron radiation facil-
+ity.
+Figure 2 shows a schematic illustration of the γ-
+ray beam line and experimental setup. Beams of γ rays
+were produced through laser Compton scattering (LCS)
+of 1064 nm photons in head-on collisions with relativistic
+
+3
+Collision point                 Collision point
+P1 (Nd laser)                    P2 (CO2 laser)
+Collimator
+C1
+C2
+Lasers: Nd(w, 2w), CO2
+8.9 m                                                        7.5 m
+18.5 m
+GACKO
+g - ray
+Gamma hutch-2
+Gamma
+hutch-1
+FIG. 2. (Color online) A schematic illustration of the experimental set up at NewSUBARU used in the (γ, n) cross-section
+measurements.
+electrons at the most-efficient collision point P1. The γ
+beams were collimated using the Pb collimators C1 and
+C2, each 10 cm long, with 3 mm and 2 mm apertures,
+respectively. The beam profile on target nearly follows
+the geometrical aperture of the collimator C2 with re-
+spect to the collision point P1, thus avoiding any interac-
+tion between beam and other materials than the target.
+Throughout the experiment, the laser was periodically
+on for 80 ms and off for 20 ms, in order to measure
+background neutrons and γ-rays.
+In this experiment,
+the beams produced had an energy resolution ranging
+from 0.6 MeV to 0.9 MeV (full-width at half maximum,
+FWHM).
+The electrons were injected from a linear accelerator
+into the NewSUBARU storage ring with an initial energy
+of 974 MeV, then subsequently decelerated to nominal
+energies ranging from 608 MeV to 849 MeV, providing
+LCS γ-ray beams of energies up to 13 MeV and down to
+the neutron separation energies of the W isotopes (thus
+varied for each individual case).
+The maximum γ-ray
+energy of the beams was increased in steps of 0.25 MeV.
+The electron beam energy has been calibrated with the
+accuracy on the order of 10−5 [19].
+The energy is re-
+produced in every injection of an electron beam from a
+linear accelerator to the storage ring. The reproducibil-
+ity of the electron energy is assured in the deceleration
+down to 0.5 GeV by an automated control of the electron
+beam-optics parameters.
+The energy profiles of the produced γ-ray beams were
+measured with a 3.5in.×4.0in. LaBr3(Ce) (LaBr3) de-
+tector. The measured LaBr3 spectra were reproduced by
+a Geant4 code [20–23] that incorporated the kinematics
+of the LCS process, including the beam emittance and
+the interactions between the LCS beam and the LaBr3
+detector.
+In this way we were routinely able to sim-
+ulate the energy profile of the incoming γ beams with
+the maximum energies accurately determined by the cal-
+ibrated electron beam energy by best reproducing the
+LaBr3 spectra [24, 25].
+The W targets were made from isotopically enriched
+tungsten as metallic powder. The material was pressed
+together and enclosed in an aluminium cylinder with a
+thin cap. The targets had areal densities of 0.7421 g/cm2
+(182W), 0.754 g/cm2 (183W), and 1.7925 g/cm2 (184W).
+Due to the presence of the Al cap, we limited the γ-ray
+beam energy maximum 13 MeV to avoid getting contam-
+inant neutrons from 27Al (Sn=13.056 MeV).
+To measure the emitted neutrons, a high-efficiency
+4π detector was used, consisting of 20 3He proportional
+counters, arranged in three concentric rings and embed-
+ded in a 36 × 36 × 50 cm3 polyethylene neutron modera-
+tor [26]. The ring ratio technique, originally developed by
+Berman and Fultz [27], was used to determine the aver-
+age energy of the neutrons from the (γ, n) reactions. The
+efficiency of the neutron detector varies with the average
+neutron energy. The efficiency was measured with a cal-
+ibrated 252Cf source with the emission rate of 2.27 · 104
+s−1 with 2.2% uncertainty, and the energy dependence
+was determined by Monte Carlo simulations [28]. The
+efficiency of the neutron detector was simulated using
+isotropically distributed, mono-energetic neutrons. Once
+the neutron detection efficiency for a given beam energy
+has been determined, we were able to deduce the number
+of (γ, n) reactions that took place during each run.
+The LCS γ-ray flux was monitored by a 8in.×12in.
+NaI(Tl) (NaI) detector during neutron measurement runs
+with 100% detection efficiency for the beam energies used
+in this experiment.
+The number of incoming γ rays
+per measurement was determined using the pile-up and
+Poisson-fitting technique described in Refs. [29, 30].
+B.
+Analysis
+The measured photo-neutron cross section for an in-
+coming beam with maximum γ energy Emax is given by
+
+4
+FIG. 3. (Color online) The simulated energy profiles for the
+γ beams used. The distributions (integrated over all Eγ) are
+normalized to unity.
+the convoluted cross section,
+σEmax
+exp
+=
+� Emax
+Sn
+DEmax(Eγ)σ(Eγ)dEγ =
+Nn
+NtNγξϵng .
+(1)
+Here,
+DEmax
+is the normalized energy distribution
+(
+� Emax
+Sn
+DEmaxdEγ = 1) of the γ-ray beam obtained from
+Geant4 simulations. Examples of the simulated γ-beam
+profiles, DEmax, are shown in Fig. 3. Furthermore, σ(Eγ)
+is the true photo-neutron cross section as a function of
+energy. The quantity Nn represents the number of neu-
+trons detected, Nt gives the number of target nuclei per
+unit area, Nγ is the number of γ rays incident on tar-
+get, ϵn represents the neutron detection efficiency, and
+finally ξ = (1 − e−µt)/(µt) gives a correction factor for
+self-attenuation in the target. The factor g represents the
+fraction of the γ flux above Sn.
+We have determined the convoluted cross sections
+σEmax
+exp
+given by Eq. (1) for γ beams with maximum en-
+ergies in the range Sn ≤ Emax ≤ 13 MeV. The convo-
+luted cross sections σEmax
+exp
+are not connected to a specific
+Eγ, and we choose to plot them as a function of Emax.
+The convoluted cross sections of the three W isotopes,
+which are often called monochromatic cross sections, are
+shown in Fig. 4. The error bars in Fig. 4 represent the
+total uncertainty in the quantities comprising Eq. (1),
+and consists of ∼ 3.2% from the efficiency determination
+of the neutron detector, ∼ 1% from the pile-up method
+that gives the number of γ rays, and the statistical un-
+certainty in the number of detected neutrons [30]. The
+statistical uncertainty ranges between ∼ 5.0 % close to
+neutron threshold and 4.4 % for the highest maximum
+γ-ray beam energies. The systematic error is dominated
+by the uncertainty from the pile-up method and from the
+simulated efficiency of the neutron detector. For the to-
+tal uncertainty, we have added these uncorrelated errors
+in quadrature.
+By approximating the integral in Eq. (1) with a sum for
+FIG. 4.
+(Color online) Monochromatic cross sections of
+182,183,184W. The error bars contain statistical uncertainties
+from the number of detected neutrons, the uncertainty in the
+efficiency of the neutron detector and the uncertainly in the
+pile-up method used to determine the integrated γ-flux on
+target.
+each γ-beam profile, we are able to express the unfolding
+problem as a set of linear equations
+σf = Dσ,
+(2)
+where σf is the cross section folded with the beam pro-
+file D. The indexes i and j of the matrix element Dij
+corresponds to Emax and Eγ, respectively.
+The set of
+equations is given by
+�
+�
+�
+�
+�
+σ1
+σ2
+...
+σN
+�
+�
+�
+�
+�
+f
+=
+�
+�
+�
+�
+�
+D11
+D12
+· · · · · · D1M
+D21
+D22
+· · · · · · D2M
+...
+...
+...
+...
+...
+DN1 DN2 · · · · · · DNM
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+σ1
+σ2
+...
+...
+σM
+�
+�
+�
+�
+�
+�
+�
+�
+. (3)
+Each row of D corresponds to a Geant4 simulated γ beam
+profile belonging to a specific measurement characterized
+by Emax (see Fig. 3 for a visual representation of some of
+the rows in the response matrix D). It is clear that D is
+highly asymmetrical.
+The number of γ-ray beam energies used to study the
+cross section is much lower than the bin size (10 keV)
+of the simulated beam profiles above Sn. As the system
+of linear equations in Eq. (3) is under-determined, the
+true σ vector cannot be extracted by matrix inversion.
+In order to find σ, we utilize a folding iteration method.
+The main features of this method are as follows [31]:
+1) As a starting point, we choose for the 0th iteration,
+a constant trial function σ0. This initial vector is
+multiplied with D, and we get the 0th folded vector
+σ0
+f = Dσ0.
+2) The next trial input function, σ1, can be estab-
+lished by adding the difference of the experimen-
+
+0.025
+0.02
+Relative intensity
+0.015
+0.01
+0.005
+6
+8
+10
+12
+14
+E, [MeV]350
+W-182. before deconvolution
+300
+W-183. before deconvolution
+250
+W-184. before deconvolution
+[mb]
+200
+ouo
+150
+b
+100
+50
+8
+10
+11
+12
+13
+9
+x [MeV]5
+tally measured spectrum, σexp, and the folded spec-
+trum, σ0
+f , to σ0. In order to be able to add the
+folded and the input vector together, we first per-
+form a Piecewise Cubic Hermite Interpolating Poly-
+nomial (pchip) interpolation on the folded vector so
+that the two vectors have equal dimensions. Our
+new input vector is:
+σ1 = σ0 + (σexp − σ0
+f ).
+(4)
+3) The steps 1) and 2) are iterated i times giving
+σi
+f = Dσi
+(5)
+σi+1 = σi + (σexp − σi
+f)
+(6)
+until convergence is achieved.
+This means that
+σi+1
+f
+≈ σexp within the statistical errors. In order
+to quantitatively check convergence, we calculate
+the reduced χ2 of σi+1
+f
+and σexp after each iter-
+ation.
+Approximately four iterations are usually
+enough for convergence, which is defined when the
+reduced χ2 value approaches ≈ 1.
+We stopped iterating when the χ2 became lower than
+unity. In principle, the iteration could continue until the
+reduced χ2 approaches zero, but that results in large un-
+realistic fluctuations in σi due to over-fitting to the mea-
+sured points σexp.
+We estimate the total uncertainty in the unfolded cross
+sections by calculating an upper limit of the monochro-
+matic cross sections from Fig. 4 by adding and subtract-
+ing the errors to the measured cross section values. These
+upper and lower limits are then unfolded separately, re-
+sulting in the unfolded cross sections shown in Fig. 5.
+In Fig. 5, the unfolded cross sections for 182,183,184W
+are evaluated at the maximum energies of the incom-
+ing γ beams.
+The error bars represent the statistical
+errors and the systematic error due to the uncertainty in
+the absolute efficiency calibration of the neutron detec-
+tor. The results are compared to data on 182,184W from
+Goryachev et al. [32], and the agreement is overall quite
+reasonable although some local discrepancies can be ob-
+served.
+These discrepancies are sometimes not within
+the given uncertainties, and could be due to unknown
+systematic errors.
+III.
+THE OSLO EXPERIMENT
+A.
+Experimental details
+The 186W(α, α′γ) inelastic-scattering experiment was
+performed at the Oslo Cyclotron Laboratory. A fully-
+ionized 30-MeV α beam was delivered by the MC-35
+Scanditronix cyclotron and directed to the 186W target.
+The radio frequency was set to 23.76 MHz, giving a beam
+burst every 42.09 ns. The experiment was run for about
+FIG. 5. (Color online) Cross sections of 182,183,184W obtained
+after deconvolution. Also shown are cross sections of 182,184W
+from Goryachev et al. [32].
+eight days with typical beam intensities of 1.5 − 2.2 enA.
+The target was mounted on a 24-µm carbon backing, and
+the target thickness was 0.31 mg/cm2 with enrichment
+> 98% in 186W.
+To detect the outgoing charged particles, we used the
+Silicon Ring (SiRi) [33] placed in backward angles with
+respect to the beam direction. SiRi is a ∆E-E telescope
+array consisting of eight 1550-µm thick back (E) detec-
+tors, each of which has a 130-µm thick front (∆E) de-
+tector divided in eight strips. A 10.5-µm thick Al foil
+was placed in front of SiRi to reduce the amount δ elec-
+trons from the target. SiRi covers about 6% of 4π and
+the strips have an angular resolution of about 2◦, where
+the center of the strip is at 126 − 140◦ (in steps of 2◦);
+measured from the center of the front detector (at 133◦),
+the distance of SiRi from the center of the target was 5
+cm.
+The ∆E-E telescopes allow for separating different
+charged-particle species. Figure 6a shows the measured
+protons, deuterons, tritons, and α particles for a strip at
+130◦. To select the 186W(α, α′) events, a gate was set on
+the “banana” corresponding to the α particles. To cal-
+ibrate the SiRi front and back detectors, we used range
+calculations for our setup with the Qkinz code [34], see
+Fig. 6b.
+The resolution of the α particles was measured to
+be 330–360 keV FWHM for the peak of the elastically-
+scattered α particles. The relatively poor resolution is
+mainly due to a rather elongated beam spot on the tar-
+get (≈ 3–4 mm in diameter in the vertical direction, and
+≈ 1 mm in the horizontal direction). The master-gate
+signal for the data acquisition system was a logical signal
+of 2µs generated when an E detector gave a signal above
+threshold, which was set to ≈ 200 mV.
+Using the CACTUS array [35], we measured γ rays
+in coincidence with the inelastic scattered α particles.
+In the configuration used for this experiment, CAC-
+TUS consisted of 26 NaI(Tl) crystals of cylindrical shape
+
+500
+T
+W-182, NewSUBARU
+450
+W-183,NewSUBARU
+400
+W-184, NewSUBARU
+W-184, Goryachev et al.
+350
+W-182, Goryachev et al.
+Q
+[qw] (u'l)o
+300
+250
+200
+150
+100
+50
+0
+8
+10
+6
+12
+14
+16
+E, [MeV]6
+FIG. 6.
+(Color online) (a) Particle-identification spectrum for one of the front strips at 130◦ with its corresponding back
+detector (∆E–E banana plot); (b) a zoom on the α-particle banana with the Qkinz calculations used for calibration (crosses).
+(5in.× 5in.). All crystals were collimated with lead col-
+limators and had 2-mm thick Cu shields in front to at-
+tenuate X-rays. The NaI(Tl) detectors were mounted on
+the spherical CACTUS frame, so that the front end of
+each crystal was positioned 22 cm from the center of the
+target. The efficiency of CACTUS (for 26 NaI(Tl) detec-
+tors) is 14.1(2)% as measured with a 60Co source, and
+with a resolution of ≈ 6.8% FWHM for Eγ = 1.33 MeV.
+Using analog electronics, we obtained a lower threshold
+of about 350 keV for the NaI(Tl) detectors.
+The CACTUS detectors were calibrated in energy by
+gating on the protons in SiRi.
+As the target had a
+significant contamination of carbon (from the backing)
+and oxygen, we used peaks in the proton spectrum from
+the 12C(α, pγ)15N and 16O(α, pγ)19F reactions to further
+identify γ rays for calibration. In particular, we used the
+5.269-MeV transition from the 5/2+ first-excited level
+in 15N together with the 1.868-MeV transition from the
+13/2+ level at Ex = 4.648 MeV in 19F. Then we cross-
+checked the obtained calibration with the 1235-keV and
+2583-keV lines of 19F, in addition to the 511-keV γ ray
+from positron annihilation.
+To obtain α–γ coincident events, we applied a gate
+on the time-to-digital converter (TDC) spectra for
+the prompt peak, and subtracting randomly correlated
+events. The start of the TDCs is given by the master
+gate, and the stop signal is generated from the NaI(Tl)
+detectors (each NaI(Tl) has an individual TDC), with a
+built-in delay from the Mesytec shapers of ≈ 400 ns. The
+range of the TDCs was 1.2 µs. The gate on the prompt
+peak was set to ∆t = 0 ± 20 ns, while the gate for the
+background subtraction was set to ∆t = 135 ± 20 ns.
+Using the reaction kinematics, we determined the ini-
+tial excitation energy of the residual nucleus from the
+deposited energy of the α particles in SiRi. Applying the
+time gates for the γ rays, we obtained excitation-energy
+tagged, background-subtracted γ-ray spectra as shown in
+Fig. 7a.
+The γ-ray spectra needed to be corrected for the CAC-
+TUS detector response. For this purpose, we applied the
+iterative unfolding method of Ref. [36] available in the
+Oslo-method software package [37]. This method takes
+the raw γ-ray spectrum as a starting point for the un-
+folded (“true”) spectrum. This trial spectrum is folded
+with the known detector response, and then compared
+with the raw spectrum. By taking the difference between
+the folded spectrum and the raw spectrum, a new, im-
+proved trial spectrum is made. This process is repeated
+until the folded spectrum is approximately equal to the
+raw spectrum, within the experimental uncertainties. To
+preserve the experimental statistical fluctuations, and
+not introduce artificial, spurious ones, the Compton sub-
+traction method is also applied. This takes advantage of
+the fact that the Compton distribution is very smooth.
+For more details, see Ref. [36]. The unfolded γ-ray spec-
+tra for each Ex bin are shown in Fig. 7b.
+After unfolding, the first-generation γ rays were ex-
+tracted from the data by applying an iterative subtrac-
+tion method [38]. The first-generation γ rays are the ones
+that are emitted first in the decay cascades, and their dis-
+tribution represents the branching ratios for the various
+γ transitions at a given Ex bin. The principle behind
+the subtraction method is as follows. For a given Ex bin,
+say, at Ex = 4 MeV, the unfolded spectrum contains all
+the γ rays from all the possible decay cascades originat-
+ing from the levels populated in that Ex bin. If we now
+consider the Ex bins below Ex = 4 MeV, they will con-
+tain all the same γ rays as the Ex = 4 MeV bin, except
+the first-generation γs at Ex = 4 MeV. This is true if
+the Ex bins have the same decay cascades whether the
+levels in the bin were populated directly through the nu-
+clear reaction, or if they were populated from γ decay of
+
+104
+103
+14000
+TTTT
+[ke]
+9500
+(a) θ = 130°
+(b)
++
+Qkinz calculations
+detector [
+12000
+9000
+103
+二
+10000
+8500
+ deposited in △E
+102
+8000
+8000
+20
+102
+7500
+11
+6000
+7000
+Energy 
+10
+4000
+6500
+10
+6000
+2000
+5500
+0
+5000
+10000
+15000
+20000
+25000
+10000
+12000
+14000
+16000
+18000
+20000
+22000
+Energy deposited in E detector [keV]
+Energy deposited in E detector [keV]7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+1
+2
+3
+4
+5
+6
+7
+8
+ [MeV]
+x
+E
+(a)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+ [MeV]
+γ
+E
+(b)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+1
+10
+2
+10
+1
+10
+2
+10
+(c)
+max
+x
+E
+min
+x
+E
+min
+γ
+E
+FIG. 7. (Color online) Excitation-energy vs. γ-ray energy matrices of 186W. (a) Background-subtracted data; (b) unfolded
+γ-ray spectra; (c) first-generation γ-ray spectra. The lines indicate the limits set for the further analysis.
+above-lying levels. We refer the reader to Ref. [39] for a
+more in-depth discussion on the assumptions behind the
+first-generation method. The first-generation γ spectra
+are displayed in Fig. 7c.
+B.
+Extraction of level density and γ-ray
+transmission coefficient
+We now exploit the fact that the first-generation γ
+spectra represent the (relative) branching ratios for a
+given initial excitation-energy bin, and that we have
+many such branching ratios available for a large Ex re-
+gion. In the spirit of Fermi’s Golden Rule [40, 41], where
+the decay rate is proportional to the level density at the
+final excitation energy and the reduced transition proba-
+bility for decay between a given initial and final level, we
+use the following ansatz [42]:
+P(Eγ, Ex) ∝ ρ(Ex − Eγ) · T (Eγ),
+(7)
+where P(Eγ, Ex) is the matrix of first-generation γ rays
+(Fig. 7c), ρ(Ex − Eγ) is the level density at the excita-
+tion energy where the γ transition “lands” and T (Eγ) is
+the γ-ray transmission coefficient. Note that T (Eγ) is
+only a function of Eγ, which means that the Brink-Axel
+hypothesis [43, 44] is invoked. Brink stated that
+“...we assume that the energy dependence of
+the photo effect is independent of the detailed
+structure of the initial state so that, if it were
+possible to perform the photo effect on an ex-
+cited state, the cross section for absorption
+of a photon of energy E would still have an
+energy dependence given by (15).”
+where “(15)” is referring to the equation describing
+the Giant Dipole Resonance (GDR) with a Lorentzian
+function that only depends on the γ-transition energy.
+Brink’s original formulation (as well as Axel’s application
+of Brink’s hypothesis) concerned only E1 transitions, and
+there is a wealth of recent works in the literature dis-
+cussing the validity and/or violation of the hypothesis;
+see, e.g., Refs. [45–53].
+A necessary condition for the Oslo method is that the
+Brink hypothesis is at least approximately true for the
+specific excitation-energy region used for extracting the
+level density and γ-ray transmission coefficient. We have
+performed tests of this assumption for the application in
+the Oslo method in Ref. [39]. When the Brink hypothesis
+is applicable, we can fit the data of the first-generation γ
+rays to obtain a reliable estimate of the level density and
+the γ-ray transmission coefficient through an iterative
+optimization using a least-squares fit:
+χ2
+red =
+1
+Nfree
+Emax
+x�
+Ei=Emin
+x
+Ei
+�
+Eγ=Emin
+γ
+[P(Eγ, Ei) − Pth(Eγ, Ei)]2
+[∆P(Eγ, Ei)]2
+.
+(8)
+Here, P(Eγ, Ei) is the experimental matrix of first-
+generation γ rays where each row is normalized to unity:
+Ei
+�
+Ei=Emin
+γ
+P(Eγ, Ei) = 1,
+(9)
+and
+∆P(Eγ, Ei)
+is
+the
+uncertainties
+in
+the
+first-
+generation matrix (including statistical errors and an es-
+timate for systematic uncertainties due to unfolding and
+the first-generation method, see Ref. [42]).
+Moreover,
+Nfree is the number of degrees of freedom and Pth(Eγ, Ei)
+is the approximation for the theoretical first-generation
+matrix [42]:
+Pth(Eγ, Ei) =
+ρ(Ei − Eγ)T (Eγ)
+�Ei
+Eγ=Emin
+γ
+ρ(Ei − Eγ)T (Eγ)
+.
+(10)
+
+8
+0
+1
+2
+3
+4
+5
+6
+7
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+Probability distribution
+ = 4.14 MeV
+x
+E
+(a) 
+ first-gen. data 
+ 
+T
+ x 
+ρ
+ 
+0
+1
+2
+3
+4
+5
+6
+7
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+Probability distribution
+ = 5.26 MeV
+x
+E
+(d) 
+0
+1
+2
+3
+4
+5
+6
+7
+ = 4.59 MeV
+x
+E
+(b) 
+0
+1
+2
+3
+4
+5
+6
+7
+ [MeV]
+γ
+E
+ energy 
+γ
+Probability distribution
+ = 5.70 MeV
+x
+E
+(e) 
+0
+1
+2
+3
+4
+5
+6
+7
+ = 4.81 MeV
+x
+E
+(c) 
+0
+1
+2
+3
+4
+5
+6
+7
+ = 6.16 MeV
+x
+E
+(f) 
+FIG. 8. (Color online) Experimental first-generation spectra (black crosses) compared to the predicted ones using the extracted
+level density and γ-transmission coefficient (blue line) for various excitation-energy bins (224-keV wide).
+The number of degrees of freedom, Nfree, is given by
+Nfree = Nch(P) − Nch(ρ) − Nch(T ). For the present data
+set, we have used Emin
+γ
+= 0.90 MeV, Emin
+x
+= 4.0 MeV,
+and Emax
+x
+= 7.2 MeV as shown in Fig. 7c. Note that
+the neutron separation energy Sn of 186W is 7.1920(12)
+MeV [54], and as we have no way of discriminating
+against neutrons, the Oslo method is usually limited to a
+maximum excitation energy (close to) Sn. With bin size
+of 224 keV, and the limits applied as shown in Fig. 7c,
+we have the number of pixels in the first-generation ma-
+trix Nch(P) = 330, while the number of elements in the
+vectors of ρ and T is Nch(ρ) = Nch(T ) = 39, giving
+Nfree = 252. It is important to note that the number
+of data points in the first-generation matrix, Nch(P), is
+much bigger than the number of points to be estimated,
+which is 2 × 39 points; this is why the method usually
+converges very well. When convergence is reached, the
+extracted ρ(Ex − Eγ) and T (Eγ) are the ones that best
+describe the experimental P(Eγ, Ei) matrix.
+For this
+case, we obtain χ2
+red = 0.85 after 20 iterations.
+As a visual illustration of the fit, Fig. 8 shows some of
+the experimental first-generation spectra together with
+the spectra obtained for Pth. Overall, the agreement is
+quite good, although we remark that the experimental
+errors are rather large. Note that the fit is performed on
+all the first-generation spectra (for 15 excitation-energy
+bins), and so the fit is still well constrained.
+Schiller et al. showed [42] that the χ2 minimization
+obtains a unique solution for the relative variation of
+neighboring points in the functions ρ and T ; however,
+an equally good fit to the experimental P matrix is given
+by the transformation
+˜ρ(Ei − Eγ) = A exp[α(Ei − Eγ)] ρ(Ei − Eγ),
+(11)
+˜T (Eγ) = B exp(αEγ)T (Eγ).
+(12)
+Here, α is the common slope adjustment of ρ and T ,
+while A and B gives the absolute scaling of ρ and T ,
+respectively. These parameters must be determined from
+external data, as described in the following sections.
+C.
+Normalization of level density
+To normalize the level density by determining the α
+and A parameters, we make use of discrete levels [54]
+at low Ex and data on s-wave neutron resonance spac-
+ings [55] at the neutron separation energy Sn. The av-
+erage s-wave neutron resonance spacing D0 = 9.3(16)
+eV [55] represents the spacing of levels with Jπ = 1−, 2−
+as the target nucleus 185W has ground-state spin/parity
+Iπ
+t =
+3
+2
+−. To obtain the total level density at Sn, we
+need to apply a model for the spin distribution, in par-
+ticular the spin cutoff parameter σJ(Ex). Here, we use
+as a starting point the model of von Egidy and Bu-
+curescu [56, 57] employing the rigid-body moment of in-
+ertia.
+However, as shown by Uhrenholt et al. [58], at
+excitation energies around 7−8 MeV for heavy nuclei, a
+full rigid-body moment of inertia might not be reached
+yet: in Fig. 10 of Ref. [58], the effective moment of inertia
+is ≈ 85% of the rigid-body moment of inertia at Ex ≈ 8
+MeV. We take this as the reference value for which we
+will vary the spin cutoff parameter to obtain an estimate
+for the systematic uncertainty connected to the spin dis-
+
+9
+tribution, with the effective moment of inertia ranging
+from 70%−100% of the rigid-body moment of inertia:
+σ2
+J(Sn) = η 0.0146A5/3 1 +
+�
+1 + 4a(Sn − E1)
+2a
+,
+(13)
+where η is the reduction factor set to 0.85(15), A is the
+mass number of the nucleus (here 186), a is the level-
+density parameter and E1 is an excitation-energy shift
+taken from the global systematics of von Egidy and Bu-
+curescu [56, 57] calculated with the robin.c code in the
+Oslo-method software package (see Table I). This gives
+us a range of values for the estimated ρ(Sn), which is
+then calculated as [39, 42]
+ρ(Sn) =
+2σ2
+J
+D0
+�
+(It + 1)e−(It+1)2/2σ2
+J + Ite−I2
+t /2σ2
+J�,
+(14)
+assuming an equal parity distribution for all spins at
+the neutron separation energy. Uncertainties in the D0
+value and the spin cutoff parameter are propagated (for a
+derivation, see Appendix A). All the applied parameters
+are given in Table I.
+Moreover, due to the argument in the level density
+function being Ei − Eγ, we get an upper limit for the
+extracted level density given by Emax
+x
+−Emin
+γ
+. Therefore,
+we need to make an extrapolation from our data points
+up to ρ(Sn). Here, we use the constant-temperature (CT)
+model of Ericson [59]:
+ρCT(Ex) = 1
+T exp Ex − E0
+T
+,
+(15)
+where T denotes the nuclear “temperature” and E0 is a
+shift; both parameters are usually obtained from fits to
+discrete data and to neutron resonance spacings.
+The
+parameters used for 186W are shown in Table I.
+From the Oslo-method software, statistical uncertain-
+ties and an estimate of systematic errors due to the un-
+folding procedure and the first-generation method are
+calculated as described in Ref. [42].
+We also include
+systematic errors from the normalization procedure, ac-
+counting for the uncertainty in the experimental D0 value
+as well as the uncertainty in the moment of inertia and
+thus the spin cutoff parameter as described above. We es-
+timate the uncertainty (approximately one standard de-
+viation) including all these factors as
+δρ = ρrec
+��δD0
+D0
+�2
++
+�δσJ
+σJ
+�2
++
+�∆ρrec
+ρrec
+�2
+,
+(16)
+where ρrec is the central value (“recommended” nor-
+malization), and ∆ρrec represents statistical uncertain-
+ties and systematic errors from unfolding and the first-
+generation method. The resulting normalized level den-
+sity is shown in Fig. 9.
+D.
+Normalization of γ-ray strength
+Having the normalized level density at hand, we
+proceed to normalizing the γ-ray transmission coeffi-
+0
+1
+2
+3
+4
+5
+6
+7
+ [MeV]
+x
+E
+Excitation energy 
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+]
+-1
+) [MeV
+x
+E
+(
+ρ
+Level density 
+W 
+186
+ Oslo data, 
+ stat.+sys. 
+σ
+ 1
+ Known levels 
+ CT interpolation 
+) 
+n
+S
+(
+ρ
+ Estimated 
+FIG. 9. (Color online) Normalized level density of 186W. The
+discrete levels [54] are binned with the same bin size as our
+data (224 keV/channel). The dashed line shows the CT-model
+interpolation between our data and ρ(Sn). The black error
+bars represent statistical uncertainties from the experiment
+and systematic errors connected to the unfolding procedure
+and the first-generation method. The blue band includes also
+systematic errors from the normalization procedure (see text).
+0
+1
+2
+3
+4
+5
+6
+7
+8
+ [MeV]
+γ
+E
+-ray energy 
+γ
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+Transmission coeff. (arb. units)
+W Oslo data, this work 
+186
+ 
+ low-energy extrapolation 
+ high-energy extrapolation 
+FIG. 10. Gamma-ray transmission coefficient of 186W before
+normalization. The arrows indicate the fit regions used for de-
+termining the extrapolations (see text). The gray data points
+are not considered further in the analysis due to very low
+statistics in the first-generation matrix for these γ energies.
+cient T (Eγ) by determining the scaling parameter B in
+Eq. (12). Here we make use of the relation between the
+average, total radiative width ⟨Γγ0⟩ deduced from s-wave
+neutron resonances, the level density and the transmis-
+
+10
+TABLE I. Parameters used for the normalization of the level density and γ-ray transmission coefficient. Note that the E0
+parameter is adjusted to make ρCT(Sn) match with ρ(Sn) = 26.5·105 MeV−1. The uncertainty in ⟨Γγ0⟩ from Mughabghab [55]
+is given as 5 meV; however, this uncertainty seems too small based on the experimental errors in the radiative width for other
+W isotopes, and we have chosen a more conservative uncertainty in line with the experimental errors of 182,183,184,186W.
+Sn
+Iπ
+t
+D0
+σ2
+J(Sn)
+a
+E1
+ρ(Sn)
+T
+E0
+⟨Γγ0⟩
+σ2
+d
+Ed
+(MeV)
+(eV)
+(MeV−1) (MeV) 105 (MeV−1) (MeV) (MeV) (meV)
+(MeV)
+7.192 3/2− 9.3(16)
+47(8)
+19.38
+0.28
+26.5(64)
+0.51(1) -0.0077 60+13
+−9
+7.3(13) 0.86(19)
+sion coefficient [39, 60]:
+⟨Γγ0⟩ = BD0
+4π
+� Sn
+Eγ=0
+dEγT (Eγ)ρ(Sn − Eγ)×
+1
+�
+J=−1
+[g(Sn − Eγ, It − 1/2 + J) + g(Sn − Eγ, It + 1/2 + J)] ,
+(17)
+where g is the spin distribution [61, 62]:
+g(Ex, J) ≃ 2J + 1
+2σ2
+J
+exp
+�
+−(J + 1/2)2/2σ2
+J
+�
+.
+(18)
+As we need the spin distribution for the excitation-energy
+range Ex ∈ [0, Sn], we make use of the spin cutoff param-
+eter in the general form [63]
+σ2
+J(Ex) = σ2
+d + Ex − Ed
+Sn − Ed
+�
+σ2
+J(Sn) − σ2
+d
+�
+,
+(19)
+which is motivated also from microscopic calculations
+(e.g., shell-model calculations [64] and the work of Uhren-
+holt et al. [58]). Here, σ2
+d represents the spin cutoff pa-
+rameter at the low excitation energy Ed, where the lev-
+els are still resolved and with firm spin/parity assign-
+ments [54], see Table I.
+We need to estimate the γ-ray transmission coefficient
+for Eγ < Emin
+γ
+, i.e., where we do not have experimen-
+tal data, in order to calculate the integral in Eq. (17).
+Therefore, we extrapolate with a fit to the low-energy
+data points using the functional form E3
+γ exp(p1Eγ +p2),
+where p1 and p2 are free parameters1.
+Moreover, the
+statistics is very low at high γ-ray energies, and so we
+make use of an extrapolation here as well, here using a
+simple exponential, exp(p3Eγ + p4), where p3 and p4 are
+again free parameters. The fit regions and the extrapo-
+lation functions are shown in Fig. 10. The data points in
+gray color (Eγ > 6 MeV) are from a region in the first-
+generation matrix with very low statistics (see Fig. 7c),
+and we therefore choose to exclude those data points from
+the further analysis.
+To obtain the γ-ray strength function, we use the fact
+that γ decay at high excitation energies is largely domi-
+nated by dipole transitions (see, e.g., Refs. [67–69]). As
+1 This functional form is motivated by shell-model calculations of
+the low-energy γ strength, e.g. Refs. [65, 66].
+0
+1
+2
+3
+4
+5
+6
+ [MeV]
+γ
+-ray energy E
+γ
+8
+−
+10
+7
+−
+10
+6
+−
+10
+]
+-3
+) [MeV
+γ
+-ray strength function f(E
+γ
+W 
+186
+ Oslo data, 
+ stat.+sys. 
+σ
+ 1
+FIG. 11.
+(Color online) Gamma-ray strength function of
+186W. The black error bars represent statistical uncertain-
+ties from the experiment and systematic errors connected to
+the unfolding procedure and the first-generation method. The
+blue band includes also systematic errors from the normaliza-
+tion procedure (see text).
+our experimental data in principle contain transitions of
+both electric and magnetic character, we get the total
+dipole strength function f(Eγ) through
+f(Eγ) = T (Eγ)
+2πE3γ
+.
+(20)
+In accordance with the approach for the level density, we
+estimate the uncertainty in the γ-ray strength function
+through
+δf = frec
+��δD0
+D0
+�2
++
+�δσJ
+σJ
+�2
++
+�δΓγ0
+Γγ0
+�2
++
+�∆frec
+frec
+�2
+,
+(21)
+where ∆frec is again the central value (“recommended”
+normalization), and ∆frec represents statistical uncer-
+tainties and systematic errors from unfolding and the
+first-generation method. The resulting, normalized γ-ray
+strength function is shown in Fig. 11.
+
+11
+IV.
+RESULTS AND DISCUSSION
+A.
+Comparison to other data and models
+The level-density data are compared to various mod-
+els available in the TALYS-1.9 code [18], see Fig. 12.
+The models are: ldmodel 1, the composite formula of
+Gilbert and Cameron [70]; ldmodel 2, the back-shifted
+Fermi gas model [71]; ldmodel 3, the generalized super-
+fluid model [72]; ldmodel 4, calculated within the Hartree-
+Fock-BCS approach [73]; ldmodel 5, the combinatorial-
+plus-Hartree-Fock-Bogoliubov approach [74];
+and ld-
+model 6, the combinatorial model combined with a
+temperature-dependent Hartree-Fock-Bogoliubov calcu-
+lation [75].
+0
+1
+2
+3
+4
+5
+6
+7
+ [MeV]
+x
+E
+Excitation energy 
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+]
+-1
+) [MeV
+x
+E
+(
+ρ
+Level density 
+W 
+186
+ Oslo data, 
+ stat.+sys. 
+σ
+ 1
+ Known levels 
+ ldmodel 1 
+ ldmodel 2 
+ ldmodel 3 
+ ldmodel 4 
+ ldmodel 5 
+ ldmodel 6 
+FIG. 12. (Color online) Comparison of the level-density data
+from this work with models included in the TALYS code (see
+text).
+From a first look, none of the models seem to be in good
+agreement with the data, and we remark that the TALYS
+level densities have not been normalized to the D0 value
+from Ref. [55]. In adition, we take notice of two impor-
+tant issues: (i) the spin cutoff parameter we have used in
+our normalization procedure might not be representative
+of the corresponding spin distribution in the TALYS mod-
+els; (ii) our data can be re-normalized more coherently
+for each model by adopting its energy-dependence to ex-
+trapolate between the highest energy point and ρ(Sn), as
+was done e.g. in Ref. [76]. Nevertheless, it is clear that
+the overall shape of our data points are significantly dif-
+ferent from several of the level-density models. We also
+remark that the slope of our level-density data points is
+directly linked to the slope of the γ-strength function as
+given in Eq. (12). If we were to renormalize our level
+density to the TALYS models, this would inevitably lead
+to a change in slope in the γ-strength function as well.
+We now compare our γ-strength data from the (γ, n)
+measurements and the OCL experiment to external data
+found in the literature, as shown in Fig. 13a. We observe
+a good agreement with the E1 strength extracted from
+primary γ rays following neutron capture by Kopecky et
+al [78], which brings further support to the absolute nor-
+malization procedure.
+Moreover, we compare our new
+photoneutron data to several data sets found in the liter-
+ature, where the photoneutron cross section σγn is con-
+verted into dipole strength using the relation of Axel [79]:
+fγn(Eγ) =
+1
+3π2ℏ2c2
+σγn(Eγ)
+Eγ
+,
+(22)
+where σγn is in units of mb, Eγ in MeV, and the factor
+1/(3π2ℏ2c2) = 8.674 · 10−8 mb−1MeV−2. Overall, there
+is good agreement between the various data sets for the
+W isotopes.
+In Fig. 13a, we also compare the data with avail-
+able models in TALYS: strength 1, the Generalized
+Lorentzian [67]; strength 2, the Standard Lorentzian
+(Brink-Axel model) [43, 44];
+strength 3, the Quasi-
+Particle Random Phase Approximation (QRPA) on top
+of a Hartree-Fock-plus-BCS calculation [80]; strength 4,
+the QRPA on top of a Hartree-Fock-Bogoliubov (HFB)
+calculations [81];
+strength 5, the Hybrid model [82]
+with parameters from global systematics [18]; strength
+6, QRPA as in Ref. [81] but on top of a temperature-
+dependent HFB calculation [75]; and finally strength 7,
+a relativistic mean-field calculation plus a continuum
+QRPA calculation [83].
+Out of these models, strength
+4 and strength 6 match reasonably well the present Oslo
+data, but not the (γ, n) data. In general, the models are
+deviating significantly from each other and from either
+the Oslo data or the (γ, n) data.
+To obtain a model description that can reproduce our
+data reasonably well over the entire energy range, we
+take a pragmatic approach and exploit phenomenologi-
+cal models for the dipole strength. For the main part
+of the E1 strength which is dominated by the Giant
+Dipole Resonance (GDR), we apply the Hybrid model
+of Goriely [82]:
+f Hyb
+E1 (Eγ, Tf) =
+1
+3π2ℏ2c2
+EγσrΓrΓ(Eγ, Tf)
+(E2γ − E2r)2 + E2γΓrΓ(Eγ, Tf),
+(23)
+where σr is the peak cross section, Er the centroid, and
+Γr the width of the GDR. Further, the γ-energy and
+temperature dependent width Γ(Eγ, Tf) is given by
+Γ(Eγ, Tf) = 0.7 · Γr
+E2
+γ + 4π2T 2
+f
+EγEr
+.
+(24)
+The temperature of the final levels, Tf, is here considered
+as a constant, in line with the Brink-Axel hypothesis. We
+also include extra E1 strength (labeled “E1 pygmy” in
+Fig. 13b) to make a smooth connection between our data
+and the (γ, n) data.
+Finally, we also add a magnetic-
+dipole component (marked “M1 spin-flip” in Fig. 13b).
+
+12
+0
+2
+4
+6
+8
+10
+12
+14
+ [MeV]
+γ
+E
+-ray energy 
+γ
+8
+−
+10
+7
+−
+10
+6
+−
+10
+]
+-3
+) [MeV
+γ
+E
+(f
+-ray strength function 
+γ
+(a)
+W Oslo data, this work 
+186
+ 
+ stat.+sys. 
+σ
+ 1
+,n), this work 
+γ
+W(
+182
+ 
+,n), this work 
+γ
+W(
+183
+ 
+,n), this work 
+γ
+W(
+184
+ 
+,n), Berman et al. 
+γ
+W(
+186
+ 
+,n), Mohr et al. 
+γ
+W(
+186
+ 
+ 
+et al.
+, Kopecky 
+E1
+f
+W 
+184
+ 
+ 
+et al.
+, Kopecky 
+M1
+f
+W 
+184
+ 
+ strength1
+ strength2
+ strength3
+ strength4
+ strength5
+ strength6
+ strength7
+0
+2
+4
+6
+8
+10
+12
+14
+ [MeV]
+γ
+E
+-ray energy 
+γ
+8
+−
+10
+7
+−
+10
+6
+−
+10
+ E1 hybrid 
+ E1 pygmy 
+ M1 spin-flip 
+ total fit function 
+(b)
+FIG. 13. (Color online) (a) Comparison of γ-strength data from this work with data from the literature (Berman et al. [77],
+Mohr et al. [9], and Kopecky et al. [78]), and to models included in the TALYS code (see text); (b) Fit to the γ-ray strength
+function data of 186W and the 184W data of Kopecky et al. [78]) (see text).
+TABLE II. Parameters found from the model fits of ftot to the γ-strength data (see text). The uncertainties given are from
+the fit only. Note that EM1 and ΓM1 are fixed.
+Norm.
+Er
+Γr
+σr
+EPyg
+ΓPyg
+σPyg
+Tf
+EM1
+ΓM1
+σM1
+(MeV) (MeV) (mb)
+(MeV) (MeV) (mb)
+(MeV) (MeV) (MeV) (mb)
+Rec.
+12.9(1) 4.1(1) 382(2) 6.3(1) 2.6(2) 7.2(3) 0.43(3) 7.2
+2.5
+4.4(4)
+For both the E1 pygmy and the M1 spin-flip contribu-
+tions, we apply a resonance-like form using a Standard
+Lorentzian:
+fPyg,M1(Eγ) =
+1
+3π2ℏ2c2
+σPyg,M1Γ2
+Pyg,M1Eγ
+(E2γ − E2
+Pyg,M1)2 + Γ2
+Pyg,M1E2γ
+(25)
+where σPyg,M1, ΓPyg,M1, and EPyg,M1 are the peak cross
+section, width, and centroid for the pygmy (Pyg) and
+the spin-flip (M1) resonance, respectively. The total fit
+function is then given by
+ftot(Eγ) = f Hyb
+E1 (Eγ, Tf = const.) + fPyg(Eγ) + fM1(Eγ).
+(26)
+For the fit, we first constrain the Hybrid component by
+fitting only the Hybrid model to the GDR data (Mohr et
+al. [9] and Berman et al. [77]) in the range Eγ = 7.7−14.5
+MeV. We choose to fix the Tf parameter to the one used
+for the extrapolation of the level density (see Sec. III C)
+to ease the fit, as Tf is largely determined from the γ-
+strength function below neutron threshold. From this fit,
+we determine the GDR parameters σr, Er, and Γr, to be
+used as start values for the next fit including the data for
+γ energies below neutron threshold as well.
+For the spin-flip part, we use a fixed centroid EM1
+taken from systematics [63], and a fixed width of ΓM1 of
+2.5 MeV. The peak cross section σM1 is then found from a
+fit to the M1 data of 184W from Kopecky et al. [78]. Then
+we make a fit using the full energy range Eγ = 1.0 − 14.5
+MeV, with only the spin-flip parameters fixed, and with
+the first fit of the GDR data as starting values. In the fit,
+we include the present OCL data of 186W, the E1 data
+from Kopecky et al. [78] on 184W, and the GDR data
+from Mohr et al. [9] and Berman et al. [77]. The result-
+ing fit is shown in Fig. 13b, and the parameters are listed
+in Table II. As this model fit will be used to calculate the
+(n, γ) cross section and reactivity in the following sec-
+tion, we repeat the fit for all the different normalizations
+(varying D0, Γγ0, σJ and taking into account ∆f). All
+fits are performed within the ROOT software tool [84]
+using the Minuit package.
+The resulting fit function gives a reasonable description
+of the strength function data, although we note a poten-
+tial issue in that the region between Eγ = 6 − 8 MeV
+contains practically no data points for 186W. Moreover,
+the 184W data points from primary transitions following
+neutron capture typically have large fluctuations. Hence,
+it is very difficult to assess the actual parameters for the
+E1 pygmy, and the deduced parameters given in Table II
+
+13
+should be used with caution.
+We also remark that the data points at the lowest γ
+energies, Eγ < 1.5 MeV, might indicate some low-energy
+increase in the γ-strength function, as first observed in
+iron isotopes [85].
+However, in contrast to clear cases
+like 56Fe [68, 69, 85], it is hard to conclude here as there
+are only a few data points that might show an increas-
+ing trend. We therefore choose not to include an extra
+“upbend” component in the fit.
+B.
+Maxwellian-averaged cross section and reaction
+rate
+Using our level-density data and γ-strength function
+data, we now calculate the Maxwellian-averaged cross
+section (MACS) with the TALYS code, which is based on
+the statistical model of Wolfenstein [86] and Hauser and
+Feshbach [87]. The resulting MACS is shown in Fig. 14,
+where we also show the TALYS MACS with default in-
+puts (strength 1, ldmodel 1, a global optical-model po-
+tential, and no upbend), and the variation of the MACS
+as the different level-density and γ-strength models are
+used. We have tested using the semi-microscopic optical-
+model potential of Bauge et al. [88] for comparison with
+the one of Koning and Delaroche [89].
+As seen from
+Fig. 14 (dashed line versus dashed-dotted line), there
+is only a minor difference between the two for neutron
+energies around kBT = 30 keV, and overall the semi-
+microscopic potential gives a lower MACS. Nevertheless,
+the presented uncertainty band on our experimentally-
+constrained MACS includes the variation between the
+two different optical models in the lower uncertainty, in
+addition to uncertainties from D0, Γγ0, and σJ.
+In Fig. 14, we compare our result with the KADoNiS
+database [90], and find agreement within the error bars,
+although the KADoNiS values are overall larger than our
+central values. We remark that the KADoNiS values are
+from a weighted average of MACS constrained by pho-
+tonuclear data above Sn, while our results include infor-
+mation on both the level density as well as the γ-strength
+function below Sn.
+We have multiplied the KADoNiS
+MACS values with their corresponding stellar enhance-
+ment factor (SEF) as given in Ref. [90] for 185W(n, γ).
+Furthermore, our estimated uncertainty band is smaller
+than the KADoNiS uncertainties, Our result at kBT = 30
+keV, 508+76
+−106 mb, agrees well within error bars with the
+MACS from Mohr et al. [9], 553(60) mb. On the other
+hand, the evaluation of Bao et al. [91] of 703(113) mb,
+and the measurement of Sonnabend et al. [8], 687(110)
+mb, are both larger than our estimate, although still
+within the estimated uncertainties. We note that none
+of these values are directly measured, as Bao et al. gives
+a purely theoretical prediction, while the MACS value
+from Sonnabend et al. is constrained on (γ, n) data above
+Sn. In comparison with the TALYS estimates using the
+default input as well as the resulting MACS when vary-
+ing the level-density and γ-strength models, our deduced
+20
+40
+60
+80
+100
+ [keV]
+T
+B
+k
+Energy 
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+MACS [mb]
+ exp. constrained 
+ KADoNiS-1.0 
+ Bao et al. (2000) 
+ Sonnabend et al. (2003) 
+ Mohr et al. (2004) 
+ TALYS default 
+ Variations, NLD 
+SF 
+γ
+ Variations, 
+ Variation, n-OMP 
+FIG. 14.
+(Color online) Maxwellian-averaged cross section
+for the 185W(n, γ) reaction. The shaded band indicates the
+present data-constrained MACS. The thick, azure dashed-
+dotted line shows the TALYS result using default input, the
+thin, azure dashed lines show the TALYS MACS when vary-
+ing the level-density models, and the thin, cyan lines show
+the variation due to different γ-strength models.
+The dot-
+ted line shows the deviation from the default when using the
+optical-model potential of Bauge et al. [88].
+1
+−
+10
+1
+Temperature [GK]
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+6
+10
+×
+]
+-1
+ mol 
+-1
+ s
+3
+ [cm
+〉
+v
+σ
+〈
+A
+N
+exp. constrained 
+ KADoNiS-1.0 
+ TALYS default 
+ Variations, NLD 
+SF 
+γ
+ Variations, 
+ Variation, n-OMP 
+FIG. 15. (Color online) Reaction rate for the 185W(n, γ) reac-
+tion. The shaded band indicates the present data-constrained
+result. See also the caption of Fig. 14.
+MACS is in between the extremes.
+In Fig. 15, we show the corresponding reaction rate
+(stellar reactivity) deduced from our data compared to
+the KADoNiS rate, the TALYS default and the variations
+using different model inputs.
+Again we find that the
+KADoNiS values are overall higher than our estimated
+rate, in particular for temperatures below 0.3 GK.
+
+14
+To address possible implications for the s process
+and the Re/Os cosmochronometer in a reliable way, the
+branch points at 186Re and 191Os should also be consid-
+ered in realistic stellar models for thermally-pulsing AGB
+stars. The 191Os MACS has been estimated by a similar
+procedure as in this work by Kullmann et al. [92]. The
+186Re MACS remains to be experimentally constrained
+in the same way; the 186W(α, dγ)187Re data from this
+same experiment is currently being analyzed. With this
+experimentally-constrained MACS also at hand, we in-
+tend to perform a consistent study of the s process in
+this mass region.
+V.
+SUMMARY AND OUTLOOK
+In this work, we have performed photoneutron cross
+section measurements on the 182,183,184W isotopes. This
+completes the photoneutron measurements on the stable
+W isotopic chain. Furthermore, we have presented data
+on the 186W(α, α′γ) reaction, and used the extracted
+level density and γ-ray strength function to provide an
+experimentally constrained (n, γ) cross section for the
+branch-point nucleus 185W.
+In comparison with other data and the recommended
+MACS from the KADoNiS data base, we find that our
+estimated MACS and reaction rate are lower than most
+of the other available values, except for the result of Mohr
+et al. Our reaction rate could possibly impact the s pro-
+cess in this mass region, in particular the deduced neu-
+tron density and the calculation of the 186Os abundance.
+When the 186Re MACS also becomes available, we intend
+to perform a systematic study of the s-process conditions
+in the W-Re-Os region in the near future.
+ACKNOWLEDGMENTS
+The authors would like to thank J. C. M¨uller,
+P. A. Sobas, and J. C. Wikne at the Oslo Cyclotron Labo-
+ratory for operating the cyclotron and providing excellent
+experimental conditions. We sincerely thank T. W. Ha-
+gen, S. J. Rose and F. Zeiser for helping with the OCL
+experiment, Y.-W. Lui for helping with the NewSUB-
+ARU experiments, and S. N. Liddick for inspiring dis-
+cussions.
+A. C. L. gratefully acknowledges funding of
+this research by the European Research Council through
+ERC-STG-2014 under grant agreement no. 637686, and
+from the Research Council of Norway, project grant no.
+316116. S. G. acknowledges the support from the F.R.S.-
+FNRS. This work was supported in part by the National
+Science Foundation under Grant No.
+OISE-1927130
+(IReNA). The photoneutron cross section measurement
+was performed as part of the IAEA CRP on “Updating
+the Photonuclear Data Library and generating a Refer-
+ence Database for Photon Strength Functions” (F41032).
+A. G., V. W. I., and S. S. gratefully acknowledge fi-
+nancial support from the Research Council of Norway,
+project grant no. 325714. This work is in part based on
+the research supported partly by the National Research
+Foundation of South Africa (Grant Number: 118846).
+Appendix A: Uncertainty in ρ(Sn)
+To estimate the total NLD at the neutron separation
+energy using Eq. (14), we propagate errors from the D0
+value and the spin cutoff parameter σJ(Sn) assuming
+that they are independent variables, which is a justified
+assumption. Thus, we get that
+�δρ(Sn)
+ρ(Sn)
+�2
+=
+�δD0
+D0
+�2
++
+�δξ(σJ(Sn))
+ξ(σJ(Sn))
+�2
+,
+(A1)
+where ξ represents the function containing the depen-
+dency on the spin cutoff parameter σJ at the neutron
+separation energy Sn:
+ξ(σJ) =
+2σ2
+J
+Ite−I2
+t /2σ2
+J + (It + 1)e−(It+1)2/2σ2
+J .
+(A2)
+Now we take the derivative of ξ with respect to σJ and
+obtain:
+δξ
+δσJ
+=
+4σJ
+�
+Ite−I2
+t /2σ2
+J + (It + 1)e−(It+1)2/2σ2
+J
+�
+−
+2
+σJ
+�
+I3
+t e−I2
+t /2σ2
+J + (It + 1)3e−(It+1)2/2σ2
+J
+�
+�
+Ite−I2
+t /2σ2
+J + (It + 1)e−(It+1)2/2σ2
+J�2 .
+(A3)
+For convenience, we now define the auxilliary functions
+z1 ≡ I3
+t e−I2
+t /2σ2
+J + (It + 1)3e−(It+1)2/2σ2
+J,
+z2 ≡ Ite−I2
+t /2σ2
+J + (It + 1)e−(It+1)2/2σ2
+J.
+Using these and dividing Eq. (A3) on the function ξ(σJ),
+we get
+δξ
+ξδσJ
+= 2
+σJ
+−
+z1
+σ3
+Jz2
+= 2
+σJ
+�
+1 −
+1
+2σ2
+J
+z1
+z2
+�
+.
+(A4)
+Finally, we obtain
+�δξ
+ξ
+�2
+=
+�2δσJ
+σJ
+�2 �
+1 −
+1
+2σ2
+J
+z1
+z2
+�2
+.
+(A5)
+
+15
+This is what is implemented in the code d2rho in the Oslo
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@@ -0,0 +1,173 @@
+arXiv:2301.13124v1  [cs.CY]  30 Jan 2023
+MED1stMR: Mixed Reality to Enhance the Training of Medical First
+Responders for Challenging Contexts
+HELMUT SCHROM-FEIERTAG, GEORG REGAL, and MARKUS MURTINGER, AIT Austrian Insti-
+tute of Technology, Vienna
+1
+INTRODUCTION
+Mass-casualty incidents with a large number of injured persons caused by human-made or by natural disasters are
+increasing globally. In such situations, medical first responders (MFRs) need to perform diagnosis, basic life support,
+or other first aid to help stabilize victims and keep them alive to wait for the arrival of further support. Situational
+awareness and effective coping with acute stressors are essential [5] to enable first responders to take appropriate
+action that saves lives. Such tasks are particularly challenging for first responders, that lack the special training to act
+optimally in these situations. Increasingly severe consequences of natural disasters and terrorist threats will expand
+the occurrence probability of such stressful and demanding situations and require the development and deployment
+of innovative technological solutions adapted to the (cross-sectoral) needs of first responders.
+In that context, the adage “practice makes perfect” is well-fitting to situational training. Lectures, books, videos, etc.
+are no substitute for hands-on experiences, and humans often learn more from their mistakes than from their successes.
+Unfortunately, it is difficult to provide such training for large-scale emergency medicine or in dangerous conditions.
+Current training of medical first responders often happens through live exercises: medics practice stabilisation and
+wound care via moulage, a training exercise where live persons are given highly realistic “fake” wounds. The drawback
+of such training is the large effort needed to create such live training exercises (a large number of ‘victim actors’ needed,
+availability of infrastructure, etc.) and the lack of realistic treatments. Hence, such training or exercises are not executed
+very often and sometimes also fail to create “real” stressful and demanding environments.
+Virtual Reality (VR) has already been demonstrated in several domains to be a serious alternative, and in some areas
+also a significant improvement to conventional learning and training. Especially for the challenges in the training
+of MFRs, it can be highly useful for practising and learning domains where the context of the training is not easily
+available. VR training offers controlled, easy-to-create environments that can be created and trained repeatedly under
+the same conditions. This repetition makes it possible to master a new skill or process. Like in real-life training, trainees
+are transformed into active users who need to be physically and mentally engaged to evaluate the situation, take
+appropriate measures and act accordingly.
+There are two types of VR medical training systems realised up to date. Systems centred on teaching direct physical
+skills and procedures e.g. surgery (e.g. [7], [10], [14], [12]). Those usually employ highly sophisticated hardware user
+interfaces providing realistic haptic feedback or/and mimicking real devices. On the other end of the spectrum, there
+are VR systems helping the trainee to develop psychological skills required in real-world scenarios (e.g. [15], [11]) and
+in particular decision training.
+However, these approaches are currently rather disjunct and do not allow to train decision-making under stress
+together with physical skill scenarios. Also, other medical tasks or preclinical routines in patient care and treatment
+are not yet covered by haptic solutions. Training medical skills are about vision and haptics for tangible interaction,
+and if a simulation has only one of those two, it will provide only half of the experience.
+1
+
+Schrom-Feiertag, Regal and Murtinger
+2
+VISION OF THE PROJECT MED1STMR
+As an advanced alternative to VR, Mixed Reality (MR) environments have the potential to augment current VR training
+by providing a dynamic simulation of an environment and hands-on practice on injured victims.
+There are several interpretations of MR, one is that the real environment is augmented by digital objects, another
+is that physical objects are integrated into the VR. In our vision, MR is to be considered as the latter interpretation
+with a fully digital environment, where the user sees a fully digital environment without looking at the real world, but
+this digital environment is connected to real physical objects. This VR-based mixed reality is also called augmented
+virtuality (AV) according to [13].
+Building on this interpretation of MR, the main aim of MED1stMR is to develop a new generation of MR training
+with haptic feedback for enhanced realism. To this end, we will pursue the following pioneering concepts:
+2.1
+Integration of high-fidelity patient simulation manikins for enhanced realism
+Evidence shows that the use of VR is useful when the training domain is complex and difficult to master (cf. [9], [2])
+and when the audio-visual features assisted by haptic feedback of the training environment are crucial to the overall
+training success (cf. [1]). This makes virtual environments the solution for practising and learning domains where the
+context of the training is not easily available or replicable due to security and safety issues (e.g. [6]). It allows creating
+easily a diverse range of training scenarios tailored to the training goals and needs (single user vs. teams, from single
+to many people injured in large incidents, influence of psychological and contextual factors).
+Through the integration of high-fidelity patient simulation manikins and medical equipment into the MR experience,
+MED1stMR offers a much richer sensory experience. This MR training environment allows trainees to immerse into
+virtual scenarios and be able to feel and perceive actual movements of the limbs, head, and face through tactile and
+visual interaction as they are actuated. Furthermore, it enables systematic manipulation of a large set of potential
+influence factors in order to optimise training effects. This will bring virtual training closer to reality and enable both
+scenario training and medical training in the same MR training environment.
+2.2
+Biosignal feedback loop and smart scenario control to enhance effectiveness of MR training
+The wireless integration of wearables in MR training environments is emerging (e.g., [8]) and particularly in the train-
+ing of highly demanding skills such as piloting/aviation, medical surgeries (e.g., [4]) or first responders (e.g., [3]).
+In order to better support, assist and personalise MFRs training, we will integrate wearable technology for moni-
+toring trainees’ physiological data. Smart electronic devices can detect and transmit information regarding biosignals,
+informing on trainees’ physiological status. Monitoring these signals will provide the detection of physical and psy-
+chological strain and stress during training.
+This will provide information for the debriefing sessions and can be used for real-time scenario control through the
+trainer (manual control) or automatically by the training system through artificial intelligence-based adaptive smart
+scenarios. The data on trainee state and behaviour can then be used to constitute a feedback loop for personalising
+and adapting training to the trainees’ needs. Such a system can automatically adjust the scenarios according to the
+stress level of the trainee, for example, low stress increases the difficulty of the scenario and allows longer and more
+complex scenarios to be trained without the intervention of a trainer.
+2
+
+MED1stMR: Mixed Reality to Enhance Training of Medical First Responder
+3
+OUR MOTIVATION
+The development of such a training system for MFRs requires research, expertise, and knowledge in the areas of
+medical research, biosensors, and wearable technologies, human factors research, psychology, physiological research,
+technology experience, user research, VR/MR, and medical training simulation development to answer all the questions
+that arise in order to develop an optimal training system for MFRs:
+• How can haptic feedback for training medical skills on victims be provided for MFRs?
+• Which scenarios and use cases are most suitable for MR training and deliver the greatest benefits?
+• How can effective MR training scenarios be developed?
+• How effective are such training approaches, how good is the learning progress and how does it compare to
+real-world training?
+• How should a MR training curriculum be designed and merged with existing training curricula?
+• What about the costs for the training system and does the benefit justify the effort?
+4
+OUR CONTRIBUTION TO THE WORKSHOP
+MED1stMR develop a MR training system based on a combination of VR environment and the integration of VR-
+enabled manikins. With this new training environment, MED1stMR delivers a training platform for collaborative multi-
+user training to train the medical skills of MFRs as well the decision-making abilities in disaster situations. In the project,
+the training is designed for teams of up to four people to enable emergency teams to train together. The technological
+basis is the Refense trainings platform (www.refense.com) that allows up to 10 users on an area of 11 x 20 meters to
+immerse themselves into a realistic common shared scenario, see each other in real time with full-body VR tracking. A
+trainer can also be included as an invisible observer in the training. Every movement and voice spoken are recorded for
+debriefing of team collaboration and actions taken. The inclusion of manikins as tangible objects as learning support
+provides a more realistic experience and enables novel possibilities for hands-on tasks. The basis will build the ADAM-
+X manikin (https://medical-x.com/product/adam-x/) and will be advanced to a fully functional touch-enabled human
+manikin designed for practising skills in trauma emergency situation.
+The goal of the MED1stMR training solution is to train the situation awareness and the procedure in the first and
+second triage. The realisation of a biosignal feedback loop with body sensors allows to monitor trainee (stress, anxiety,
+etc.) states and behaviours of MFRs during training and will make this data available for scenario control. For this
+purpose, heart rate variability is measured as one of the most reliable indicators of stress. The trainer can adapt training
+to the personal needs of trainees and provides a new way of interaction between trainer and trainee and requires an
+appropriate user interface for support.
+A key point in MED1stMR is to examine the effectiveness of training for the different roles. To increase the effective-
+ness of the training, the simple repetition of the training scenarios as well as the recording of all movements, activities
+and communication during the training for the debriefing play an important role. The presentation of our project is in-
+tended to provide an insight into our approach and to open up an exchange of experiences with other people, projects,
+research, and developments and also to get an impression of how the topics are received, what others are doing in this
+field and where further and interesting research topics lie in this area.
+We can contribute existing knowledge to the workshop and discuss with the other participants’ challenges and help
+to set up a future research agenda for collaborative multi-user VR training.
+3
+
+Schrom-Feiertag, Regal and Murtinger
+ACKNOWLEDGEMENTS
+The project MED1stMR has received funding from the European Union’s Horizon 2020 Research and Innovation Pro-
+gramme under grant agreement No 101021775. The content reflects only the MED1stMR consortium’s view. Research
+Executive Agency and European Commission is not liable for any use that may be made of the information contained
+herein.
+REFERENCES
+[1] Andrea F Abate, Mariano Guida, Paolo Leoncini, Michele Nappi, and Stefano Ricciardi. 2009. A haptic-based approach to virtual training for
+aerospace industry. Journal of Visual Languages & Computing 20, 5 (2009), 318–325.
+[2] Cyril Bossard, Gilles Kermarrec, Cédric Buche, and Jacques Tisseau. 2008. Transfer of learning in virtual environments: a new challenge? Virtual
+Reality 12, 3 (2008), 151–161.
+[3] Meredith Carroll, Mitchell Ruble, Mark Dranias,Summer Rebensky, Maria Chaparro, Joanna Chiang, and Brent Winslow. 2020. Automatic detection
+of learner engagement using machine learning and wearable sensors. Journal of Behavioral and Brain Science 10, 3 (2020), 165–178.
+[4] Jonathan Currie, Raymond R Bond, Paul McCullagh, Pauline Black, Dewar D Finlay, Stephen Gallagher, Peter Kearney, Aaron Peace, Danail Stoy-
+anov, Colin D Bicknell, et al. 2019. Wearable technology-based metrics for predicting operator performance during cardiac catheterisation. Inter-
+national journal of computer assisted radiology and surgery 14, 4 (2019), 645–657.
+[5] Marie Ottilie Frenkel, Laura Giessing, Sebastian Egger-Lampl, Vana Hutter, Raoul RD Oudejans, Lisanne Kleygrewe, Emma Jaspaert, and Henning
+Plessner. 2021. The impact of the COVID-19 pandemic on European police officers: Stress, demands, and coping resources. Journal of Criminal
+justice 72 (2021), 101756.
+[6] Andrzej Grabowski and Jarosław Jankowski. 2015. Virtual reality-based pilot training for underground coal miners. Safety science 72 (2015),
+310–314.
+[7] Cuan M Harrington, Dara O Kavanagh, John F Quinlan, Donncha Ryan, Patrick Dicker, Dara O’Keeffe, Oscar Traynor, and Sean Tierney. 2018.
+Development and evaluation of a trauma decision-making simulator in Oculus virtual reality. The American Journal of Surgery 215, 1 (2018),
+42–47.
+[8] ByronHavardand Megan Podsiad. 2020. A meta-analysisof wearablesresearchin educational settings published 2016–2019. Educational Technology
+Research and Development 68, 4 (2020), 1829–1854.
+[9] Hsiu-Mei Huang, Ulrich Rauch, and Shu-Sheng Liaw. 2010. Investigating learners’ attitudes toward virtual reality learning environments: Based
+on a constructivist approach. Computers & Education 55, 3 (2010), 1171–1182.
+[10] Filip Jaskiewicz, Krystyna Frydrysiak, Katarzyna Starosta-Głowinska, and Dariusz Timler. 2019.
+The applicability of virtual reality in car-
+diopulmonary resuscitation training – opinion of medical professionals and students.
+Emergency Medical Service 6 (March 2019), 32–36.
+https://doi.org/10.36740/EmeMS201901103
+[11] Joshua Benjamin Kaplan, Aaron L Bergman, Michael Christopher, Sarah Bowen, and Matthew Hunsinger. 2017. Role of resilience in mindfulness
+training for first responders. Mindfulness 8, 5 (2017), 1373–1380.
+[12] Hannes Götz Kenngott, Anas Amin Preukschas, Martin Wagner, Felix Nickel, Michael Müller, Nadine Bellemann, Christian Stock, Markus Fangerau,
+Boris Radeleff, Hans-Ulrich Kauczor,et al. 2018. Mobile, real-time, and point-of-care augmented reality is robust, accurate,and feasible:a prospective
+pilot study. Surgical endoscopy 32, 6 (2018), 2958–2967.
+[13] Paul Milgram, Haruo Takemura, Akira Utsumi, and Fumio Kishino. 1995. Augmented reality: A class of displays on the reality-virtuality continuum.
+In Telemanipulator and telepresence technologies, Vol. 2351. International Society for Optics and Photonics, 282–292.
+[14] Felix Nickel, Julia A Brzoska, Matthias Gondan, Henriette M Rangnick, Jackson Chu, Hannes G Kenngott, Georg R Linke, Martina Kadmon, Lars
+Fischer, and Beat P Müller-Stich. 2015. Virtual reality training versus blended learning of laparoscopic cholecystectomy: a randomized controlled
+trial with laparoscopic novices. Medicine 94, 20 (2015).
+[15] Federica Pallavicini, Luca Argenton, Nicola Toniazzi, Luciana Aceti, and Fabrizia Mantovani. 2016. Virtual reality applications for stress manage-
+ment training in the military. Aerospace medicine and human performance 87, 12 (2016), 1021–1030.
+4
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='13124v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='CY]  30 Jan 2023  MED1stMR: Mixed Reality to Enhance the Training of Medical First  Responders for Challenging Contexts  HELMUT SCHROM-FEIERTAG, GEORG REGAL, and MARKUS MURTINGER, AIT Austrian Insti-  tute of Technology, Vienna  1  INTRODUCTION  Mass-casualty incidents with a large number of injured persons caused by human-made or by natural disasters are  increasing globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' In such situations, medical first responders (MFRs) need to perform diagnosis, basic life support,  or other first aid to help stabilize victims and keep them alive to wait for the arrival of further support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Situational  awareness and effective coping with acute stressors are essential [5] to enable first responders to take appropriate  action that saves lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Such tasks are particularly challenging for first responders, that lack the special training to act  optimally in these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Increasingly severe consequences of natural disasters and terrorist threats will expand  the occurrence probability of such stressful and demanding situations and require the development and deployment  of innovative technological solutions adapted to the (cross-sectoral) needs of first responders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  In that context, the adage “practice makes perfect” is well-fitting to situational training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Lectures, books, videos, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  are no substitute for hands-on experiences, and humans often learn more from their mistakes than from their successes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  Unfortunately, it is difficult to provide such training for large-scale emergency medicine or in dangerous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  Current training of medical first responders often happens through live exercises: medics practice stabilisation and  wound care via moulage, a training exercise where live persons are given highly realistic “fake” wounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The drawback  of such training is the large effort needed to create such live training exercises (a large number of ‘victim actors’ needed,  availability of infrastructure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=') and the lack of realistic treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Hence, such training or exercises are not executed  very often and sometimes also fail to create “real” stressful and demanding environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  Virtual Reality (VR) has already been demonstrated in several domains to be a serious alternative, and in some areas  also a significant improvement to conventional learning and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Especially for the challenges in the training  of MFRs, it can be highly useful for practising and learning domains where the context of the training is not easily  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' VR training offers controlled, easy-to-create environments that can be created and trained repeatedly under  the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' This repetition makes it possible to master a new skill or process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Like in real-life training, trainees  are transformed into active users who need to be physically and mentally engaged to evaluate the situation, take  appropriate measures and act accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  There are two types of VR medical training systems realised up to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Systems centred on teaching direct physical  skills and procedures e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' surgery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' [7], [10], [14], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Those usually employ highly sophisticated hardware user  interfaces providing realistic haptic feedback or/and mimicking real devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' On the other end of the spectrum, there  are VR systems helping the trainee to develop psychological skills required in real-world scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' [15], [11]) and  in particular decision training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  However, these approaches are currently rather disjunct and do not allow to train decision-making under stress  together with physical skill scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Also, other medical tasks or preclinical routines in patient care and treatment  are not yet covered by haptic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Training medical skills are about vision and haptics for tangible interaction,  and if a simulation has only one of those two, it will provide only half of the experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  1  Schrom-Feiertag, Regal and Murtinger  2  VISION OF THE PROJECT MED1STMR  As an advanced alternative to VR, Mixed Reality (MR) environments have the potential to augment current VR training  by providing a dynamic simulation of an environment and hands-on practice on injured victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  There are several interpretations of MR, one is that the real environment is augmented by digital objects, another  is that physical objects are integrated into the VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' In our vision, MR is to be considered as the latter interpretation  with a fully digital environment, where the user sees a fully digital environment without looking at the real world, but  this digital environment is connected to real physical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' This VR-based mixed reality is also called augmented  virtuality (AV) according to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  Building on this interpretation of MR, the main aim of MED1stMR is to develop a new generation of MR training  with haptic feedback for enhanced realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' To this end, we will pursue the following pioneering concepts:  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='1  Integration of high-fidelity patient simulation manikins for enhanced realism  Evidence shows that the use of VR is useful when the training domain is complex and difficult to master (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' [9], [2])  and when the audio-visual features assisted by haptic feedback of the training environment are crucial to the overall  training success (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' This makes virtual environments the solution for practising and learning domains where the  context of the training is not easily available or replicable due to security and safety issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' It allows creating  easily a diverse range of training scenarios tailored to the training goals and needs (single user vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' teams, from single  to many people injured in large incidents, influence of psychological and contextual factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  Through the integration of high-fidelity patient simulation manikins and medical equipment into the MR experience,  MED1stMR offers a much richer sensory experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' This MR training environment allows trainees to immerse into  virtual scenarios and be able to feel and perceive actual movements of the limbs, head, and face through tactile and  visual interaction as they are actuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Furthermore, it enables systematic manipulation of a large set of potential  influence factors in order to optimise training effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' This will bring virtual training closer to reality and enable both  scenario training and medical training in the same MR training environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='2  Biosignal feedback loop and smart scenario control to enhance effectiveness of MR training  The wireless integration of wearables in MR training environments is emerging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=', [8]) and particularly in the train-  ing of highly demanding skills such as piloting/aviation, medical surgeries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=', [4]) or first responders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=', [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  In order to better support, assist and personalise MFRs training, we will integrate wearable technology for moni-  toring trainees’ physiological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Smart electronic devices can detect and transmit information regarding biosignals,  informing on trainees’ physiological status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Monitoring these signals will provide the detection of physical and psy-  chological strain and stress during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  This will provide information for the debriefing sessions and can be used for real-time scenario control through the  trainer (manual control) or automatically by the training system through artificial intelligence-based adaptive smart  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The data on trainee state and behaviour can then be used to constitute a feedback loop for personalising  and adapting training to the trainees’ needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Such a system can automatically adjust the scenarios according to the  stress level of the trainee, for example, low stress increases the difficulty of the scenario and allows longer and more  complex scenarios to be trained without the intervention of a trainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  2  MED1stMR: Mixed Reality to Enhance Training of Medical First Responder  3  OUR MOTIVATION  The development of such a training system for MFRs requires research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' expertise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' and knowledge in the areas of  medical research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' biosensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' and wearable technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' human factors research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' physiological research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  technology experience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' user research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' VR/MR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' and medical training simulation development to answer all the questions  that arise in order to develop an optimal training system for MFRs:  How can haptic feedback for training medical skills on victims be provided for MFRs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  Which scenarios and use cases are most suitable for MR training and deliver the greatest benefits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  How can effective MR training scenarios be developed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  How effective are such training approaches, how good is the learning progress and how does it compare to  real-world training?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  How should a MR training curriculum be designed and merged with existing training curricula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  What about the costs for the training system and does the benefit justify the effort?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  4  OUR CONTRIBUTION TO THE WORKSHOP  MED1stMR develop a MR training system based on a combination of VR environment and the integration of VR-  enabled manikins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' With this new training environment, MED1stMR delivers a training platform for collaborative multi-  user training to train the medical skills of MFRs as well the decision-making abilities in disaster situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' In the project,  the training is designed for teams of up to four people to enable emergency teams to train together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The technological  basis is the Refense trainings platform (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='refense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='com) that allows up to 10 users on an area of 11 x 20 meters to  immerse themselves into a realistic common shared scenario, see each other in real time with full-body VR tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' A  trainer can also be included as an invisible observer in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Every movement and voice spoken are recorded for  debriefing of team collaboration and actions taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The inclusion of manikins as tangible objects as learning support  provides a more realistic experience and enables novel possibilities for hands-on tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The basis will build the ADAM-  X manikin (https://medical-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='com/product/adam-x/) and will be advanced to a fully functional touch-enabled human  manikin designed for practising skills in trauma emergency situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  The goal of the MED1stMR training solution is to train the situation awareness and the procedure in the first and  second triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The realisation of a biosignal feedback loop with body sensors allows to monitor trainee (stress, anxiety,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=') states and behaviours of MFRs during training and will make this data available for scenario control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' For this  purpose, heart rate variability is measured as one of the most reliable indicators of stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The trainer can adapt training  to the personal needs of trainees and provides a new way of interaction between trainer and trainee and requires an  appropriate user interface for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  A key point in MED1stMR is to examine the effectiveness of training for the different roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' To increase the effective-  ness of the training, the simple repetition of the training scenarios as well as the recording of all movements, activities  and communication during the training for the debriefing play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The presentation of our project is in-  tended to provide an insight into our approach and to open up an exchange of experiences with other people, projects,  research, and developments and also to get an impression of how the topics are received, what others are doing in this  field and where further and interesting research topics lie in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  We can contribute existing knowledge to the workshop and discuss with the other participants’ challenges and help  to set up a future research agenda for collaborative multi-user VR training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  3  Schrom-Feiertag, Regal and Murtinger  ACKNOWLEDGEMENTS  The project MED1stMR has received funding from the European Union’s Horizon 2020 Research and Innovation Pro-  gramme under grant agreement No 101021775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' The content reflects only the MED1stMR consortium’s view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Research  Executive Agency and European Commission is not liable for any use that may be made of the information contained  herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
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+page_content=' Medicine 94, 20 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  [15] Federica Pallavicini, Luca Argenton, Nicola Toniazzi, Luciana Aceti, and Fabrizia Mantovani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Virtual reality applications for stress manage-  ment training in the military.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content=' Aerospace medicine and human performance 87, 12 (2016), 1021–1030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}
+page_content='  4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'}

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+BayesSpeech: A Bayesian Transformer Network for
+Automatic Speech Recognition
+Will Rieger
+Master of Science in Computer Science
+Department of Computer Science
+The University of Texas at Austin
+Abstract
+Recent developments using End-to-End Deep Learning models have
+been shown to have near or better performance than state of the art
+Recurrent Neural Networks (RNNs) on Automatic Speech Recognition
+tasks. These models tend to be lighter weight and require less training
+time than traditional RNN-based approaches. However, these models take
+frequentist approach to weight training. In theory, network weights are
+drawn from a latent, intractable probability distribution. We introduce
+BayesSpeech for end-to-end Automatic Speech Recognition. BayesSpeech
+is a Bayesian Transformer Network where these intractable posteriors are
+learned through variational inference and the local reparameterization
+trick without recurrence. We show how the introduction of variance in
+the weights leads to faster training time and near state-of-the-art perfor-
+mance on LibriSpeech-960.
+1. Introduction
+In the majority of neural networks, randomness is usually introduced through perturbation
+of the input or randomly removing nodes from the network (Hinton et al., 2012). There
+has been great success using these methods across a variety of domains including Automatic
+Speech Recognition (Park et al., 2019).
+Models continue to evolve.
+However and data
+augmentation methods rarely take large leaps in terms of the features they can help express.
+Newer models are generally larger and larger and require incredible amounts of compute to
+properly train. We especially see this in the field of Automatic Speech Recognition. Newer
+models such as Jasper (Li et al., 2019), the Conformer (Gulati et al., 2020), LAS (Chan et
+al., 2016), and the Transformer (Vaswani et al., 2017) all require training for multiple days
+across multiple GPUs. Creating deeper models can certainly help attain better performance
+on the domain task. But what if we approach the model differently and try to leverage their
+probabilistic nature?
+1
+arXiv:2301.11276v1  [eess.AS]  16 Jan 2023
+
+Most Neural Network models take a frequentist approach to model training. As you in-
+troduce non-linearities and apply gradient methods to solving these optimization problems,
+we become less and less likely to know we have reached a true minima.
+In theory, our
+true weights are drawn from an intractable prior distribution. If we approach the problem
+through a Bayesian lens, we can better contextualize our model’s output and weights on the
+input data. Using variational inference techniques, we can design a network whose weights
+are drawn from a learnable, tractable posterior. We present BayesSpeech; a Bayesian Trans-
+former Network for End-to-End Automatic Speech recognition where feed forward layers are
+contextualized with probability distributions.
+2. Background
+2.1 Automatic Speech Recognition
+Automatic Speech Recognition models have been evolving rapidly in recent years. Models
+can either be sub-domain specific and focus on speech representation (Mohamed et al., 2012)
+(Lee et al., 2009) (Conneau et al., 2020) (Devlin et al., 2018) (Schneider et al., 2019) (Chung
+et al., 2019) (Maas, 2013) (Baevski et al., 2020) attention (Vaswani et al., 2017) (Chorowski
+et al., 2015) (Conneau et al., 2020) or be end-to-end and incorporate the aforementioned
+components into one model and jointly train them.
+2.1.1 Connectionist Transporal Classification
+In order to jointly train end-to-end model including alignment, and encoding/decoding the
+input/output sequence, Connectionist Transporal (CTC) Loss can be used to better manage
+the alignments (Graves et al., 2006). Alignment of the input and output sequence becomes
+especially challenging in speech recognition tasks as the input sequence is generally longer
+than the output sequence. CTC Loss aids this process by penalizing models based on the
+joint probability of the current token in the sequence and all other tokens predicted.
+For decoders that only output set-length sequences, we can further augment the CTC loss
+by utilizing traditional Cross Entropy loss on the predictions (Hori et al., 2017). By enabling
+a joint CTC and Cross Entropy (CE) loss function we not only penalize characters based on
+their sequence but also on their absolute positioning in the output. This is covered further
+in Section 3.2.2.
+2.1.2 Models for End-to-End Speech Recognition
+End-to-End Speech recognition models combine all of the individual aspects of Automatic
+Speech Recognition into one model that is trained jointly. Traditional models rely on RNNs,
+2
+
+LSTMs, and generally recurrences for defining the output sequence (Liu et al., 2022)(Chan
+et al., 2016). These models are cumbersome to train and parallelize and create additional
+operational hurdles in properly tuning.
+Recently, new models involving convolutions and linear outputs have been used within
+Encoder-Decoder frameworks for end-to-end speech recognition tasks. These models such as
+Jasper (Li et al., 2019), Conformer (Gulati et al., 2020), and Transformer (Dong et al., 2018)
+are huge models with one billion or more parameters relying on feed-forward architectures.
+The three models were all trained for multiple days on multiple GPUs and required incred-
+ible compute power. The Speech-Transformer model (Dong et al., 2018) tried to address
+these issues by creating a thinner model to yield similar performance on the WSJ dataset.
+However, compared to its larger counterparts, it did not have the same performance charac-
+teristics. Although it did lend hope that smaller models could be trained to compete with
+their larger counterparts.
+2.2 Bayesian Methods
+Bayesian models have begun to show further promise in multiple fields such as Image Recog-
+nition (Blundell et al., 2015), attention mechanisms (Zhang et al., 2021) (Fan et al., 2020),
+and auto-encoders (Kingma & Welling, 2013). In a Bayesian approach, networks weights
+are samples from an intractable distribution which we can estimate over training iterations
+through Variational Inference.
+2.2.1 Variational Inference
+Variational Inference (VI) is the estimation of an intractable distribution through minimiz-
+ing the Kullback-Lieblier divergence (DKL) between a sample and some true distribution.
+Different works have shown that these estimation methods have value when applied to a
+Bayesian Neural Network (Graves, 2011) (Kingma & Welling, 2013). There are two different
+approaches to VI that largely depend on the what the true distribution is believed to be. If
+the true prior can be any distribution Monte-Carlo sampling is the only option for estimating
+the gradient from DKL. When using MC sampling, a network is sampled multiple times for
+the same input and the gradients are averaged together across the number of samples.
+If the prior is generally assumed to be Gaussian, the KL Divergence can be explicitly calcu-
+lated and the Local Reparameterization Trick (Kingma et al., 2015) can be used for finding
+the gradient with just one sample. This is discussed further in Section 3.1.4.
+3
+
+2.2.2 Bayes by Backprop
+Blundell et al.
+introduced the Bayes by Backprop algorithm for jointly learning the in-
+tractable distribution as well as a domain problem in a Bayesian Neural Network. They
+introduce the joint loss function in two parts:
+1. Weighting the KL Divergence of the model against the epoch
+2. Creating an Evidence Lower Bound (ELBO) on the loss function for the purpose of
+training
+We discuss the weighting of the KL Divergence loss term over time in Section 3.2.1. The
+introduction of ELBO loss serves as the method for tuning an efficient approximator for
+the Maximum Likelihood given an a-posteriori inference of the parameters. Because we are
+always sampling from an intractable distribution, the KL divergence term can be thought
+of as a regularization constant on the network. Over time, the KL divergence impact on
+loss will encourage the approximate posterior to be close to the true prior.
+While not
+readily apparent, ELBO loss is implicitly involved in the loss function described in Section
+3.2.3.
+3. Research & Methods
+As introduced above, sampling network weights (Bayesian approach) rather than explicitly
+defining them (frequentist approach) has been shown to have increased performance and
+faster convergence times. Our goal is to produce a network, leveraging Bayesian layers, to
+compete with state-of-the-art models and require less training time. BayesSpeech, is largely
+based on the no-recurrence model, the Transformer (Vaswani et al., 2017). While we leverage
+the model’s general architecture, we introduce modified Encoder and Decoder layers with
+Bayesian, Pointwise Feed Forward sub-layers. In this section, we explore the core components
+of the model, the model’s architecture, and a new training methodology for an ensemble loss
+function.
+3.1 Core Components
+3.1.1 Attention Mechanisms
+Part of Vaswani et al.’s Transformer (Vaswani et al., 2017) network was the introduction
+of Scaled Dot-Product attention and, further, Multi-Headed Attention. The goal of these
+mechanisms is to generate a temporal-rich representation of the inputs by attending to
+different positions within the input sequence. An attention function maps a query (or set
+of queries) in a matrix, Q, and a set of key-value pairs in matrices, K, V , to the input
+4
+
+sequence.
+Scaled Dot-Product Attention first takes the softmax of the matrix multiplication of Q, K.
+Then it matrix multiplies that value with V and normalizes by √dk (the dimension of the
+keys, K) (Equation 1). The normalization by the key size is used to prevent the softmax
+function from suffering the vanishing gradient problem.
+Attention(Q, K, V ) = Softmax(QKT
+√dk
+)V
+(1)
+Scaled Dot-Product attention only performs a single attention function at a time. Multi-Head
+Attention addresses this by linearly projecting the queries, keys, and values h times to each of
+the input dimensions (dq, dk, dv, respectively). Then Scaled Dot-Product attention is applied
+to these newly projected inputs in order to attend across the h different ”heads” (Equation
+2).
+Each headi is equal to Attention(QW Q
+i , KW K
+i , V W V
+i ).
+This way the model jointly
+attends different input representations across the different subspaces introduced through the
+projection.
+MultiHead(Q, K, V ) = Concat(head1, ..., headh)W O
+(2)
+3.1.2 Sinusoidal Positional Encoding
+Because the model does not have recurrence or convolutions, in order to make use of the
+temporal attentions from the Multi-Head modules, position information about the position of
+the tokens in the sequence must be introduced (Vaswani et al., 2017). Positional Encodings
+are used in the both the Encoder and Decoder modules to align each module’s outputs and
+allow them to be summed. The model leverages Sinusoidal embeddings (Equation 3) where
+the frequency corresponds to the token position (pos) and the dimension (i). Vaswani et al.
+hypothesize that using this encoding function will make it easy for the model to learn the
+attention weights for relative positions and adapt for longer sequences.
+PositionalEmbedding(pos,i) =
+�
+�
+�
+sin(pos/10000
+2i
+dmodel )
+if i < dmodel
+2
+cos(pos/10000
+2i
+dmodel )
+if i ≥ dmodel
+2
+(3)
+3.1.4 Proposal: Bayesian, Positionwise Feed Forward Layer
+Our primary proposal, is the Bayesian, Positionwise Feed Forward Layer (Figure 1). Rather
+than use two linear layers with dropout, we substitute the input layer with a Bayesian Linear
+Layer suplemented with the Local Reparameterization Trick (Bayes Linear LRT) (Kingma
+5
+
+et al., 2015) and remove the dropout. Other approaches to producing Bayesian components
+rely on sampling the same input sequence multiple times from a Monte-Carlo process in order
+to define an average gradient for modeling the intractable posterior (Blundell et al., 2015).
+In addition to being computationally expensive, this can also lead to high variance in the
+gradients increasing training time. The Local Reparameterization Trick addresses this by
+assuming that both the prior and posterior are Gaussian. Using only a single sample, the KL
+divergence between the estimators can be explicitly solved for. This new estimator is efficient
+(as it has less computational complexity) and reduces the variance in the gradient.
+For each Bayesian Layer in Figure 1, let din, dout be the input and output dimensions of the
+layer, respectiveley. We define matrices Wµ ∈ RdoutXdin and Wρ ∈ RdoutXdin representing the
+mean and variance scalar for each weight in the network. Similarly we have bias vectors for
+the output of Wµ and Wρ defined as bµ ∈ Rdout and bρ ∈ Rdout, respectively. Finally, at each
+iteration we sample a term from a standard Gaussian, ϵ ∼ N(0, 1).
+In order to calculate the output of the layer, we define a function for explicitly calculating
+the KL divergence when the prior and posterior are both Gaussian (Equation 4). The prior
+is represented by p and posterior represented by q. A sum is taken over all elements in the
+input matrices.
+KLD(µp, σp, µq, σq) = 1
+2
+�
+(2 log(σp
+σq
+) − (1 + (σp
+σq
+)2) + (µp − µq
+σp
+)2)
+(4)
+In the forward pass of the algorithm, we first must use our variance parameter ρ for estimating
+our standard deviation of weight and biases (Equation 5). The same applies for the bias
+Figure 1: Bayesian, Positionwise Feed Forward Layer Diagram
+6
+
+Activation (GELU
+Bayes Linear LRT
+Linearvector b (e.g. we arrive at a bσ using bρ).
+Wσ = log(1 + eWρ)
+(5)
+Next we sample from our Gaussian and introduce variance in the weights and biases for the
+input sequence X (Equation 6, Equation 7).
+Wout = XW T
+µ +
+�
+(W 2
+σ)TX ∗ ϵ
+(6)
+bout = XbT
+µ + bσ ∗ ϵ
+(7)
+Then we calculate the KL divergence between our estimated posteriors and true priors for
+the weights and biases: WKL = KLD(0, 1, Wµ, Wσ) and bKL = KLD(0, 0.1, bµ, bσ). Finally,
+the forward computation sets a global KL divergence term (KL = WKL + bKL) and returns
+Wout+bout. The KL divergence term is used in the join loss function for tuning our variational
+posterior.
+3.1.4 Encoder Feature Extraction
+Although the original Transformer architecture does not involve any convolutions, recent
+work in the Image Recognition domain ((Simonyan & Zisserman, 2014)) has lent itself useful
+for sequence-to-sequence ASR tasks (Hori et al., 2017). The modified VGG Network from
+(Hori et al., 2017) is used in the Encoder to further enhance in the input feature set drawing
+ideas from unsupervised speech representation tasks as seen in (Chung et al., 2019), (Lee et
+al., 2009), and (Mohamed et al., 2012). The output from this initial convolutional layer is
+passed to the encoder layers in the final network.
+3.1.5 BayesSpeech Model
+Putting this together, we arrive at our final model architecture (Figure 2). The model passes
+the input through an Encoder (Figure 2a) and then passes the encoder output through a
+Decoder (Figure 2b). In our model, we use 12 encoder block layers (de = 12) and 6 decoder
+block layers (dd = 6). These 18 inner layers contain a Mutli-Head attention block as well as
+a Bayesian Position-wise Feed Forward block. The model has a dimension of 512 and a feed
+forward dimension of 2148.
+3.2 Model Training
+In order to train our transformer model, we utilize a variation of the Bayes-By-Backprop
+algorithm (Blundell et al., 2015) with a joint Connectionist Temporal Classification and
+7
+
+(a) BayesSpeech Encoder
+(b) BayesSpeech Decoder
+Figure 2: BayesSpeech Encoder and Decoder Diagrams
+Cross Entropy loss function (Joint CTC, CrossEntropy Loss). The two-stage training is
+meant to:
+1. further tune the sampling mechanics for the variational posterior distribution the
+weights are drawn from
+2. and learn the temporal alignments and classification loss of the output tokens.
+We have found that trying to optimize each component separately leads to over-fitting in
+one of the domains of this problem. If we choose a large step size and seek to minimize
+the aggregate KL divergence across the Bayesian layers, we cannot further learn the align-
+ments. And if we choose a small step-size and learn the alignments, we introduce too much
+randomness in the output for our results to be meaningful. Therefore we introduce a scal-
+ing function, similar to the one in Bayes-by-Backprop for managing the tradeoff over epoch
+iterations (Minibatch Weighting). Our model was trained on the LibriSpeech-960 dataset
+(Panayotov et al., 2015). The utterances in the dataset were converted to Mel Spectrogram
+form with 80 channels a width of 20ms and a stride of 10ms.
+3.2.1 Tuning Variational Posterior
+The Bayesian part of our model tries to fit a variational posterior distribution (qθ) to a true
+intractable posterior (p) for each of the weights in the network. In order to do so, we reduce
+the problem of fitting qθ to that of a Minimum Description Length (MDL) problem (Hinton &
+van Camp, 1993a) (Rissanen, 1978) (Hinton & van Camp, 1993b). While we have introduced
+the explicit calculation of the Kullback-Leibler divergence between our variational posterior
+and true prior above (DKL(qθ||p)), it is important to conceptualize the divergence as the
+8
+
+VGG Extractor
+Linear
+Norm
+Sinusoidal Encoding
+Dropout
+Layer Norm
+Multi-Head Attention
+x de
+Encoder Layer(s)
+Bayesian Position-wise
+Feed Forward
+Norm
+LinearEmbedding
+Sinusoidal Encoding
+Layer Norm
+Multi-Head Attention
+Dropout
+Layer Norm
+C
+xdd
+Decoder Layer(s)
+Multi-Head Attention
+Layer Norm
+Norm
+Bayesian Position-wise
+Linear
+Feed ForwardMDL problem.
+The Minimum Description Length principal is that the best model for a given dataset bal-
+ances the tradeoff between describing the model and describing the misfit between the model
+and the data (Hinton & van Camp, 1993b). The KL divergence criteria we use has the goal
+of keeping weights simple by penalizing the amount of information they contain. Ultimately,
+this methodology will lead to a better separation between prediction accuracy and model
+complexity and is explicitly differentiable (Graves, 2011). The variational loss function used
+has two parts:
+1. Error Loss - the expected value of negative log probability in samples from qθ(β) (where
+β are the model’s parameters)
+2. Complexity Loss - the KL divergence between the tractable, variational posterior and
+the parameterized prior, DKL(qθ(β)||pα).
+In each batch, we seek to gently tune our model’s variational posterior (qθ) to continue
+random sampling but isolate different weights that have different levels of kurtosis. Due to
+the minibatch weighting, discussed in a later section, we see a consistent decline in the joint
+loss value dominated by the KL divergence term (blue, Figure 3b). Blundell et al. also show
+that using this relative kurtosis can create thinner models with an explicit scheme for weight
+pruning. Weights that are more leptokurtic are kept while platykurtic ones are discarded.
+While this is beyond the scope of this paper, it would present and interesting future research
+case for the model presented.
+(a) CTC Loss Over Training Iterations
+(b) Joint (Blue) and Scaled (Red) Loss Over Time
+Figure 3: Loss Functions over Training Iterations
+9
+
+CTCLossOverEpochs
+CTC LoSS
+CTCLoss(Smooth)
+4.0
+3.5
+3.0
+2.5
+2.0
+1.5
+0
+1000
+2000
+3000
+4000
+Training IterationComparisonofScaledandJointLoss
+1e7
+1e7
+7
+7.57
+6
+7.56
+Scaled Loss (Red)
+Joint Loss (Blue)
+5
+4
+7.55
+3
+7.54
+2
+1
+7.53
+0
+1000
+2000
+3000
+4000
+idx3.2.2 Joint CTC, CrossEntropy Loss
+In order to penalize the model for alignment of the input sequence to the output tokens, we
+utilize a joint Connectionist Transporal Classification (Graves et al., 2006) and Cross Entropy
+Loss function. The goal of this two term loss function is to manage a gradient through the
+alignment of tokens in the feature input (CTC) as well as the actual classification loss of
+the aligned output and the true tokens. The loss functions weights the two as so: L(X) =
+0.3 ∗ CTC(X) + 0.7 ∗ CE(X). We do this in order to help smooth out the gradient while
+maintaining the proper loss to back-propagate through the network. Due to the adversarial
+nature of the Bayesian outputs, we find that this joint loss descends rapidly then continues
+to descend without adjustment to the original learning rate (Figure 3a). In our training, we
+held the learning rate fixed at 10−6.
+3.2.3 Minibatch Weighting
+Blundell et al. found that earlier epochs have a greater importance on tuning of the varia-
+tional posterior than later ones. We adopt a similar methodology where we weight the KL
+divergence term according to the epoch (e) and number of epochs (ne) (Equation).
+MinibatchWeight(e, ne) = 2ne−e
+2ne − e
+(8)
+To better aid training over time, we choose an epoch indexer where the epoch index is
+integer divided by 10. When the training loop runs for multiple hours, this helps keep the
+KL divergence more heavily weighted at first. We then weight the KL divergence term by
+the minibatch weight term (Equation 9). The KLdiv term is the sum of all KL divergences
+over the Bayesian layers.
+L(X, e, ne) = MinibatchWeight(e, ne) ∗ KLdiv + 0.3 ∗ CTC(X) + 0.7 ∗ CE(X)
+(9)
+4. Results
+We split our model into two variants: one that outputs a character sequence and one that
+outputs tokenized word-pieces from a Sentencepiece language model with vocab size of 1000
+(Kudo & Richardson, 2018). We trained each model variant on a single A-100 GPU through
+Google Colab for 8 hours with a batchsize of 24.
+As shown in Table 1, our model performs nearly as well as the state of the art ASR models.
+Our Bayes speech model reaches respectable Word Error Rates with and without a language
+model on the LibriSpeech dataset. The Bayes Model as well was trained for just 8 hours on
+10
+
+Model
+WER (w/o LM)
+WER (w/ LM)
+test-clean
+test-other
+test-clean
+test-other
+LAS (Chan et al., 2016)
+2.89%
+6.98%
+2.33%
+5.17%
+Transformer (Vaswani et al., 2017)
+2.4%
+5.6%
+2.0%
+4.6%
+Conformer (Gulati et al., 2020)
+2.1%
+5.0%
+2.0%
+4.3%
+BayesSpeech
+4.5%
+6.5%
+4.0%
+5.7%
+Table 1: WER Results on LibriSpeech dataset
+a single GPU. For instance, the Conformer model was trained over the course of multiple
+days on multiple GPUs (8). During evaluation, we use beam search with a beam width of
+10 over the set of possible decoded sequences. This appears to be the standard decoding
+methodology giving the probabilistic output of the model’s decoder.
+When the input sequence passes through our Bayesian feed forward layers, we believe this
+creates an adversarial input stream. Rather than artificially augment the input Mel Spec-
+trogram inputs (Park et al., 2019), these layers produce a probabilistic feature encoding of
+the input. We believe that this general adversarial training technique allows our model to
+converge faster with less training time and resources. The randomness introduced in the
+model also helps better contextualize outputs. As we continue to tune the variational poste-
+rior over the weights, I imagine we would see a dramatic increase in performance. Because
+our model yielded reasonable results after 8 hours, we stopped training but future work may
+investigate if increased training could further improve our performance. There may also be a
+benefit to equally weighting the variational component and the CTC loss component of the
+global loss function. Similarly, in future work it may be useful to explore systematic model
+pruning as presented in Blundell et al.
+5. Conclusion
+Currently, best in class Automatic Speech Recognition solutions require multiple days of
+training on multiple GPUs. These models also take a frequentist approach to weight training.
+In this work, we present BayesSpeech; a Bayesian Transformer Network for learning an
+intractable posterior distribution over which weights are drawn in feed forward layers. We
+believe this probabilistic encoding of the input feature set creates a better representation of
+the input Mel Spectrogram. This mechanic in conjunction with a joint loss function yields
+near state-of-the-art results on the LibriSpeech dataset.
+11
+
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+

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+Weak solutions to the near-field reflector problem with
+spatial restrictions approached with generalized
+reflectors constructed from ellipsoids
+Dylanger S. Pittman
+400 Dowman Drive, Atlanta
+Abstract
+We motivate then formulate a novel variant of the near-field reflector problem
+and call it the near-field reflector problem with spatial restrictions. Let O be
+an anisotropic point source of light and assume that we are given a bounded
+open set U. Suppose that the light emitted from the source at O in directions
+defined by the aperture D ⊆ S2, of radiance g(m) for m ∈ D, is reflected off
+R ⊂ U, creating the irradiance f(x) for x ∈ T. The inverse problem consists of
+constructing the reflector R ⊆ U from the given position of the source O, the
+input aperture D, radiance g, ‘target’ set T, and irradiance f. We focus entirely
+on the case where the target set T is finite.
+Keywords:
+partial differential equations, geometric optics, geometry
+2020 MSC: 78A05, 35, 51, 53
+1. Introduction
+Let O be the origin of R3, and let S2 be the unit sphere centered at O. We
+treat points on S2 as unit vectors with initial points at O. Let an aperture be
+a subset of S2; in our work, the aperture will be an open set. Physically, it
+makes sense to consider O as the location of an anisotropic point source of light
+such that rays of light are emitted in a set of directions defined by an aperture
+D ⊆ S2.
+Email address: dpittm2@emory.edu (Dylanger S. Pittman)
+Preprint submitted to Constructive Approximation
+January 4, 2023
+arXiv:2301.00845v1  [math.AP]  2 Jan 2023
+
+Definition 1.1. Assume that we are given an aperture that is a connected open
+set D ⊆ S2, and a function ρ : D → (0, ∞) that is continuous and almost
+everywhere differentiable. Then a reflector is the set R = {mρ(m)|m ∈ D} ⊂
+R3.
+If ρ is a smooth function, we can call R a smooth reflector.
+Given an aperture, D, that is a connected open set, assume that we have a
+continuous, almost everywhere differentiable, positive function ρ : D → (0, ∞),
+and a corresponding reflector R = {mρ(m)|m ∈ D}. Suppose that a ray origi-
+nating from O in the direction m ∈ D is incident on the reflector R at the point
+mρ(m). If ρ is differentiable at m, there is a unit vector, n(m), normal to the
+reflector R at mρ(m). Therefore, by the reflection law of geometric optics, a
+ray from O of direction m reflects off the point mρ(m) in the direction
+y(m) = m − 2⟨m, n(m)⟩n(m)
+(1)
+where ⟨m, n(m)⟩ is the standard Euclidean inner product in R3 and n(m) is
+oriented such that ⟨m, n(m)⟩ > 0 [1].
+The reflector R is designed such that the ray described by the point mρ(m) ∈
+R and the direction y(m) corresponds to some element in a prespecified target
+set T. What one means by a ‘target set’ changes depending on the context, and
+the correspondence between y(m); also, an element of the target set can also
+vary depending on one’s needs. Hence a target set can represent many things.
+For example, if the target set T is a subset of S2, then a possible correspondence
+can be
+y(m)
+|y(m)| ∈ T; see [2]. Physically, in this case, T can be considered as a set
+of directions for rays of light. If T is a subset R3 \ {O}, then for an example of
+another possible correspondence, we can say that for every m ∈ D, there exists
+an a(m) > 0 such that a(m)y(m) + mρ(m) ∈ T; see [3] and [4]. Physically, in
+this case, T can be considered as a region that one wants to illuminate.
+Assume that g is an integrable and nonnegative function over an aperture D,
+and f is an integrable and nonnegative function over a target set T. Physically
+speaking, we say g(m) for m ∈ D is the radiance of the source at O in the
+directions m ∈ D, or that g is a radiance distribution over D. We also say f(x)
+2
+
+for x ∈ T is the irradiance of the target set at x ∈ T, or that f is an irradiance
+distribution over T.
+A reflector system comprises of an aperture D, O, a reflector R, an integrable
+and nonnegative function g over D, and a target set T with an integrable and
+nonnegative function f over T. From a physical perspective: light emitted from
+the source at O in directions defined by the aperture D, of radiance g(m) for
+m ∈ D, is reflected off R, creating the irradiance f(x) for x ∈ T. An example
+that can serve as an illustration is shown in Figure 1.
+A reflector problem is, in short, an inverse problem that seeks to complete
+a reflector system by creating a reflector that fits the other information given.
+Specifically, suppose we are given O, an aperture D, an integrable and non-
+negative function g over D, and a target set T with an integrable and nonneg-
+ative function f over T. The aim of a reflector problem is to find a continu-
+ous, almost everywhere differentiable, positive ρ over D such that the reflector
+R = {mρ(m)|m ∈ D} produces the specified in advance irradiance distribution
+f on T.
+Reflector problems have been well studied due to their utility in physics and
+engineering. Such problems have found numerous applications in the construc-
+tion of reflector antennas (see [5], [6]), mirror design [7], heat transfer [8], and
+beam shaping [9].
+We only consider in the high-frequency approximation of
+light, where the laws of geometric optics apply. We now proceed with a general
+description and motivation for the near-field reflector problem.
+2. The Near-Field Reflector Problem
+We discuss a reflector problem that we call the ‘near-field reflector prob-
+lem.’ In short, the near-field reflector problem aims to design a reflector that
+redistributes the light from the origin onto a set a finite distance away from the
+origin.
+In this part, when we say surface, we mean it in the differential geometric
+sense; see Definition 12.4 in [10]. Suppose that we are given a reflector system
+3
+
+O
+The plane reflector R
+The surface normal to R
+Light rays going to some target set T
+Figure 1: Here is the most basic example of a reflector system with a smooth reflector. Here R
+is a plane. Every point on R has a normal. Light originates from the point O with directions
+represented by points on the unit sphere S2 and travels according to some target set that is
+neither shown nor specified.
+4
+
+consisting of
+1. O,
+2. an aperture D ⊂ S2,
+3. a nonnegative g ∈ L1(D),
+4. a bounded Borel set T ⊂ R3 \ {O} (typically either a subset of a surface
+or a finite set),
+5. a nonnegative and integrable function f : T → [0, ∞),
+6. and a smooth function ρ : D → (0, ∞) with a smooth reflector R =
+{mρ(m)|m ∈ D}.
+From a physical perspective, this setup can be described as follows. The light
+is emitted from the source at O in directions defined by the aperture D. Each
+ray of direction m ∈ D has radiance g(m) and is reflected off R at the point
+mρ(m) in the direction y(m) as described by (1). For every m ∈ D, there exists
+an a(m) > 0 such that a(m)y(m) + mρ(m) ∈ T creating the irradiance f(x) for
+x ∈ T. A basic illustration of this situation is depicted in Figure 2. With this
+setup in mind, we proceed with a formulation of the near-field reflector problem
+Let u = (u1, u2) be smooth local coordinates on S2 such that D lies in one
+coordinate patch. The position vector of a point m ∈ D is m = m(u). We
+choose the coordinates u1, u2 so that ⟨m, m1 × m2⟩ = 1 in D; here, ⟨, ⟩ denotes
+the scalar product in R3 and mi = ∂m
+∂ui , i = 1, 2. Observe that this implies that
+⟨m, mi⟩ = 0, i = 1, 2. The first fundamental form of S2 is given by e = eijduiduj
+where eij = ⟨mi, mj⟩.
+Set r(m) = mρ(m), then r(m) defines a smooth surface R = {r(m)|m ∈ D}.
+Let g = gijduiduj be the first fundamental form of R where gij = ⟨ri, rj⟩ =
+ρiρj + ρ2eij, ri =
+∂r
+∂ui , and ρi =
+∂ρ
+∂ui .
+Let n(m) is the normal vector field on R such that ⟨n(m), m⟩ > 0 everywhere
+on R. Then
+n(m) = (ρ2 + | ˜∇ρ|2)−1/2(r − ˜∇ρ)
+(2)
+5
+
+O
+Reflector R
+Target set T with irradiance distribution f
+Surface normals to R
+Light Rays
+Figure 2: Here is an illustration of the near-field reflector problem in R3.
+The radiation
+intensity at the origin O is given by a nonnegative function g ∈ L1(D). We want to find a
+reflector R such that the reflected rays produce the prescribed irradiance distribution f on T.
+6
+
+where | ˜∇p|2 = ρiρjeij. This combined with equation (1) determines the direction
+a ray will go after reflecting off R [11].
+We can now track the path of each ray described by the direction m ∈ D
+to a point x(m) ∈ T. A ray, originating at O in direction m, hits the surface
+R at a point r(m). Then, said ray reflects off R at r(m) in the direction y(m)
+as defined by (1) and reaches T at some point x(m). Thus, from a physical
+perspective, an irradiance f(x(m)) is created by the rays reflected at x(m).
+This defines a mapping m → x that we call a reflector map; for convenience,
+we denote x(m) as the image of m under the reflector map. The reflector map
+x : D → T combined with equations (1) and (2) describes the ray tracing from
+D to T.
+If the reflector map is a diffeomorphism from D to T where T is a subset
+of a smooth surface, then one can introduce the first fundamental form of T as
+w = wijduiduj, where wij = ⟨xi, xj⟩, xi =
+∂x
+∂ui .
+According to the differential form of the energy conservation law [1],
+f(x(m))|J(x(m))| = g(m)
+(3)
+where J is the Jacobian determinant of the map x. Note that
+J(x(m)) = ±dν(x(m))
+dσ(m)
+= ±
+�
+det(wij)
+�
+det(eij)
+(4)
+where dσ is the surface area element on S2, and dν is the surface area element
+on T. We assign a ± sign to the Jacobian according to whether x preserves
+the orientation or reverses it. Therefore, by integration of (3), for all Borel sets
+ω ⊆ T,
+�
+x−1[ω]
+gdσ =
+�
+ω
+fdν
+(5)
+where x−1[ω] = {m ∈ D|x(m) ∈ ω} and
+�
+D gdσ =
+�
+T fdν.
+With this motivation, we can now state the near-field reflector problem.
+Assume that we are given O, an aperture D ⊂ S2 with a nonnegative function
+g ∈ L1(D), and a bounded Borel set T ⊂ R3\{O} with a nonnegative, integrable
+function f : T → [0, ∞). The goal is to find a smooth function ρ over D such
+that:
+7
+
+1. The ray originating from O in the direction m ∈ D reflects off the reflector
+R = {mρ(m)|m ∈ D} in accordance with equation (1) and reaches the
+target set T.
+2. g(m) on D is transformed by the reflector map into f on T; i.e. for all
+Borel subsets ω ⊆ T,
+�
+x−1[ω]
+gdσ =
+�
+ω
+fdν
+(6)
+where x : D → T the reflector map corresponding to the reflector R =
+{mρ(m)|m ∈ D}, x−1[ω] = {m ∈ D|x(m) ∈ ω}, dσ is the surface area
+element on S2, and dν is the area element on T (ν is typically some discrete
+or Lebesgue measure).
+3. The law of total energy conservation is obeyed:
+�
+D gdσ =
+�
+T fdν.
+The case where the reflector map is a diffeomorphism from D to T can be
+alternatively formulated as a PDE of Monge-Amp`ere type; specifically equation
+(4) from [12].
+There has been a lot of work done on the near-field reflector problem. In
+1972, Schruben [3] found that if the target set was a subset of a plane in R3, one
+can then derive an implicit integro-differential equation describing the reflector;
+the existence of a solution was not proved. Then in [13], Schruben considered
+the case where the target set was a small rotationally symmetric patch on the
+plane.
+In this case, when the radiance and the irradiance distributions are
+rotationally symmetric, the equation derived in [3] can be solved as an ODE.
+In 1989, Oliker [12] found a formulation of the near-field reflector problem in
+the form of a strongly non-linear PDE of Monge-Amp`ere type. The exploration
+of the said equation is difficult and in [12] was solved only for the rotationally
+symmetric case.
+In 1998, Kochengin and Oliker [4] introduced an alternative formulation to
+the near-field reflector problem, which was a geometric approach involving the
+analysis of the boundaries of convex sets generated by families of supporting
+ellipsoids. This approach can also be considered a weak solution to the PDE
+8
+
+introduced in [12]. The strategy was to assume that the target set was a finite
+set on a plane and constructively prove the existence of solutions for that case.
+Since the reflectors that were constructed were convex, one can use the Blaschke
+selection theorem (for more details, see [14]) to prove the existence of a solution
+with a continuous target set on the plane. This method was largely motivated
+by previous work done by Caffarelli and Oliker [2] which involved the analysis
+of the boundaries of convex sets generated by families of supporting paraboloids
+to solve a related problem.
+In [15] a provably convergent numerical algorithm was introduced that ex-
+plicitly finds the ellipsoids required to construct the reflectors described in [4].
+It was shown that this construction leads to infinitely many solutions; however,
+the algorithm has the benefit of converging to a unique solution if we fix an ini-
+tial point on the reflector. This algorithm and its variations have been explored
+extensively in various scenarios. For example, Fournier, Cassarly, and Rolland
+in [16] adapted the algorithm in [15], to situations where the light source is
+not a single point; specifically, a flat rotationally symmetric emitter. In [17] a
+method was proposed for smoothing out a reflector with a discrete irradiance
+distribution to a reflector with a continuous irradiance distribution. Optimal
+transport methods have also been studied [18].
+2.1. The Near-Field Reflector Problem with Spatial Restrictions
+In this paper, we study a novel variant of the near-field reflector problem
+where we have extreme limitations on where we can place and construct the
+reflectors. Specifically, we are given an open set U ⊂ R3 \{O}, and our reflector
+R must now be a subset of U.
+Definition 2.1. Given an x ∈ R3 \ {O} and a subset S ⊆ R3 \ {O}, then we
+define Proj(x) =
+x
+|x| as the projection of x onto S2 and Proj[S] = {Proj(x) ∈
+S2|x ∈ S} as the projection of S onto S2.
+Assume that we are given a positive, continuous, almost everywhere differen-
+tiable function ρ over Proj[U]. We have a reflector R = {mρ(m)|m ∈ Proj[U]}
+9
+
+which determines our reflector map x : Proj[U] → T which is determined by
+tracking the path of each ray described by the direction m ∈ Proj[U] to a point
+x(m) ∈ T. A ray, originating at O in direction m, hits the reflector R at a
+point mρ(m). Then, assuming ρ is differentiable at m, said ray reflects off R
+at mρ(m) in the direction y(m) as defined by (1) and reaches T at some point
+x(m). Thus, from a physical perspective, an irradiance f(x(m)) is created by
+the rays reflected at x(m). This defines a mapping m → x(m) that we call the
+reflector map; for convenience, we denote x(m) as the image of m under the
+reflector map.
+We can now formulate the near-field reflector problem with spatial restric-
+tions.
+Assume that we are given an open set U ⊂ R3 \ {O}, O, an aper-
+ture Proj[U] ⊂ S2, a nonnegative g ∈ L1(Proj[U]), and a bounded Borel set
+T ⊂ R3 \ {O} with an integrable function f : T → [0, ∞).
+The goal is to find a positive, continuous, almost everywhere differentiable
+function ρ over Proj[U] such that:
+1. R = {mρ(m)|m ∈ Proj[U]} ⊂ U.
+2. The ray originating from O in the direction m ∈ Proj[U] reflects off of R
+in accordance with equation (1) and reaches the target set T.
+3. g(m) on Proj[U] is transformed by the reflector map into f on T, i.e. for
+all Borel subsets ω ⊆ T,
+�
+x−1[ω]
+gdσ =
+�
+ω
+fdν
+(7)
+where x : Proj[U] → T is the reflector map, x−1[ω] = {m ∈ Proj[U]|x(m) ∈
+ω}, dσ is the surface area element on S2, and dν is the area element on T
+(ν is typically some discrete or Lebesgue measure).
+4. The law of total energy conservation is obeyed:
+�
+Proj[U] gdσ =
+�
+T fdν.
+This variation of the near-field reflector problem has clear applications to
+engineering; as often one has to grapple with restrictions of space in real-world
+designs. For example, in the construction of automotive headlights, there are
+10
+
+strict restrictions, guided purely by aesthetics, as to where a reflector can be
+placed and how a reflector must be shaped [19].
+However, to the author’s
+knowledge, no mathematical research has been done in this direction. We focus
+exclusively on the case where the target set is finite.
+3. Ellipsoids of Revolution
+We do all our work in R3. We denote S2 to be the unit sphere with the
+center at O and kx = x/|x| for all x ∈ R3 \ {O}. We borrow much of this
+geometric setup from [4] and [15]. Ellipsoids of revolution are of paramount
+importance when solving the near-field reflector problem due to their unique
+optical properties.
+Let x ∈ R3 \ {O} and d ∈ (0, ∞).
+We denote by Ed(x) an ellipsoid of
+revolution about the axis Ox and with foci at points O and x. The polar radius
+relative to O can be represented as:
+ψx,d(m) =
+d
+1 − ϵ⟨m, kx⟩, m ∈ S2
+(8)
+where ϵ is the eccentricity and
+ϵ =
+�
+1 + d2
+x2 − d
+|x|.
+(9)
+So in other words
+Ed(x) = {mψx,d(m)|m ∈ S2}.
+(10)
+From this point on, whenever we use the term ellipsoid we specifically refer to
+an ellipsoid of this kind with one of the foci always at O. Note that each Ed(x)
+is uniquely defined by the x ∈ R3 \ {O} and the d ∈ (0, ∞). In this paper, we
+define Ψx,d(m) = mψx,d(m).
+Note that for all possible values of d, we have that ϵ ∈ (0, 1). Also for a
+fixed x, as d → 0 the ellipsoid will degenerate into a line segment, i.e. Ed(x) →
+{tx + (1 − t)O|t ∈ [0, 1]}. Such an ellipsoid is called degenerate. Observe that as
+d → ∞, |ψx,d(m)| → ∞ for all m ∈ S2.
+An important property of ellipsoids can be described by the following propo-
+sition.
+11
+
+Proposition 3.1. Let c, d > 0. Then the ellipsoids Ecd(x) and Ed(x) have the
+same foci: O and x.
+From a physical perspective, the aforementioned property is important be-
+cause a reflector that is shaped like an ellipsoid Ed(x) will illuminate the focus
+x with the light emitted from O such that the total energy emitted from O is
+equal to the total energy reflected onto x. This property is still true no matter
+how large or small the ellipsoid is; all that matters is the location of the foci.
+4. Generalized Reflectors
+Before we proceed, we reiterate that the near-field reflector problem can
+be expressed analytically as a PDE of Monge Amp´ere Type. Specifically, the
+equation (4) from [12]. Therefore we will consider the following formulation of
+the near-field reflector problem with spatial restrictions as a weak formulation
+and its solutions, weak solutions. The following formulation only concerns the
+case where the target set is finite.
+4.1. Weak Solutions Using Generalized Reflectors
+Definition 4.1. Assume that we are given an aperture D ⊆ S2 that is an
+open set, and a function ρ : D → (0, ∞) that is not necessarily continuous
+and almost everywhere differentiable. Then a generalized reflector is the set
+R = {mρ(m)|m ∈ D} ⊂ R3.
+The upper half-space of R3 be represented as R3+ = {(x, y, z) ∈ R3|z > 0},
+and the lower half-space of R3 be represented as R3− = {(x, y, z) ∈ R3|z < 0}.
+Let σ denote the standard measure on S2. Consider an open set U ⊆ R3+, a
+corresponding aperture Proj[U], and a finite target set T ⊂ R3−.
+Let B be a countable family of open subsets of S2 such that σ(Proj[U] \
+�
+B∈B B) = 0, Proj[U] ⊆ �
+B∈B B, and σ(B∩B′) = 0 for all distinct B, B′ ∈ B.
+Let the set B(U) be the set of all such families.
+Since every ellipsoid requires foci and an eccentricity to be well defined, given
+a family B ∈ B(U), let UT (B) be the set of all functions B → T and V (B) be
+12
+
+the set of all functions B → (0, ∞). Thus we define
+ET (U) =
+� �
+B∈B
+Ψu(B),v(B)[B]
+����� B ∈ B(U), u ∈ UT (B), v ∈ V (B)
+�
+.
+(11)
+Assume we are given a Z ∈ ET (U). Let us define
+BZ =
+�
+Int(Proj[Ed(x) ∩ Z]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ Z]) ̸= 0
+�
+.
+(12)
+The geometry of the ellipsoid and the definition of BZ imply that there ex-
+ists unique u ∈ UT (B) and v ∈ V (B) such that Z = �
+B∈BZ Ψu(B),v(B)[B].
+Define uZ ∈ UT (BZ) and vZ ∈ V (BZ) be the unique functions such that
+Z = �
+B∈BZ ΨuZ(B),vZ(B)[B].
+Given some Z ∈ ET (U), let y1
+Z(m) = {B ∈
+BZ|m ∈ B} for m ∈ Proj[U]. Given a B ∈ B(U), let N (B) be the set of all
+injective functions s : B → N. For Z ∈ ET (U) and s ∈ N (BZ), define
+ρs
+Z(m) = ψuZ(s−1(min s[y1
+Z(m)])),vZ(s−1(min s[y1
+Z(m)]))(m), m ∈ Proj[U].
+(13)
+Observe that the function ρs
+Z is positive, not necessarily continuous, and almost
+everywhere differentiable. Let W(ρs
+Z) = {mρs
+Z(m)|m ∈ Proj[U]} and thus we
+describe a set of generalized reflectors
+RU
+1 (T) =
+�
+W(ρs
+Z)| Z ∈ ET (U) where Z ⊂ U, s ∈ N (BZ)
+�
+.
+(14)
+Assume we are given a generalized reflector R ∈ RU
+1 (T). Let us define
+BR =
+�
+Int(Proj[Ed(x) ∩ R]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ R]) ̸= 0
+�
+.
+(15)
+The geometry of the ellipsoid and the definition of BR imply that there exists
+an s ∈ N(BR), unique u ∈ UT (B) and unique v ∈ V (B) such that W(ρs
+Z) = R
+where Z = �
+B∈BR Ψu(B),v(B)[B].
+Therefore, for every generalized reflector
+R ∈ RU
+1 (T), we may define a unique BR ∈ B(U) such that for each B ∈ BR
+there are unique xB ∈ T and dB ∈ (0, ∞) such that, for some s ∈ N(BR),
+R = W (ρs
+Z) where Z = �
+B∈BR ΨxB,dB[B].
+Therefore, given a generalized reflector R ∈ RU
+1 (T), we obtain a correspond-
+ing BR; for each B ∈ BR we define unique xB and dB. We also obtain an
+sR ∈ N(BR) and a unique ZR = �
+B∈BR ΨxB,dB[B] such that R = W(ρsR
+ZR).
+13
+
+Given a generalized reflector R ∈ RU
+1 (T), for all m ∈ Proj[U] we define
+M(m) = xB ∈ T
+(16)
+where mρsR
+ZR(m) = ΨxB,dB(m). Let y2
+R(m) be the points of intersection between
+R \ {mρsR
+ZR(m)} and the line segment connecting mρsR
+ZR(m) to M(m).
+Given a generalized reflector R ∈ RU
+1 (T), the map α1 : Proj[U] → T ∪ R,
+α1(m) =
+�
+�
+�
+�
+�
+M(m)
+if y2
+R(m) = ∅
+y2
+R(m)
+if y2
+R(m) ̸= ∅
+(17)
+is called the generalized reflector map. Physically speaking, a ray of light of
+direction m originating from O can only reach the target set if y2
+R(m) is empty.
+Assume we are given a nonnegative g ∈ L1(S2). Let us define for all Borel
+X ⊆ S2
+µg(X) =
+�
+X
+g(m)dσ(m)
+(18)
+where σ denotes the standard measure on S2. Assume that g ≡ 0 outside of
+Proj[U]. Physically speaking, g is the radiance distribution of the source at O.
+In order to formulate and solve the generalized reflector problem (in the
+framework of weak solutions to be defined below), we need to define a mea-
+sure representing the energy generated by g and redistributed by a generalized
+reflector R ∈ RU
+1 (T).
+Given a generalized reflector R ∈ RU
+1 (T) and a set ω ⊆ T we define the
+visibility set of ω as
+V U
+1 (ω) =
+�
+A∈A
+A \ {m ∈ Proj[U]|α1(m) = y2
+R(m)}
+(19)
+where A = {B ∈ BR|xB ∈ ω}. We now need to show that V U
+1 (ω) is measurable.
+Note the following definition.
+Definition 4.2. For an element x ∈ R3 and a set A ⊂ R3, let the set Cx,A =
+{at + x(1 − t)|t ∈ [0, 1], a ∈ A} be the union of all line segments from x to A
+and Cx,A,∞ = {at + x(1 − t)|t ∈ [0, ∞), a ∈ A} be the union of all rays from x
+that intersect A.
+14
+
+We proceed with the following lemmas.
+Lemma 4.1. Let w : S2 → (0, ∞) be continuous and W(m) = mw(m) for all
+m ∈ S2. If B is a Borel set of S2, then CO,W [B] and CO,W [B],∞ are Borel sets
+of R3.
+Proof. Recall that all Borel sets can be formed from open sets through the
+operations of countable union, countable intersection, and relative complement.
+Let {Ei} be a countable collection of open sets of S2 such that through said
+operations, we obtain B. Then given the countable collection of open sets of R3,
+{Int(CO,W [Ei])}, through the same sequence of operations we used to obtain B
+from {Ei}, we obtain Int(CO,W [B]). Thus CO,W [B] is Borel. Therefore, assuming
+Wi ≡ (im)w(m) for all m ∈ S2 and i ∈ (0, ∞), �∞
+n=1 CO,Wn[B] = CO,W [B],∞ is
+Borel.
+Lemma 4.2. If B is a Borel set of S2, x ∈ R3 \ {O} and d ∈ (0, ∞), then
+Cx,Ψx,d[B] and Cx,Ψx,d[B],∞ are Borel sets of R3.
+Proof. Let S2
+x = {m+x|m ∈ S2} be the set of all unit vectors originating from x,
+i.e the unit sphere centered at x. Since x is another focus of the ellipsoid, there
+exists a continuous function w : S2 → (0, ∞) such that Ed(x) = {mw(m)+x|m ∈
+S2}. Let Wx(m) = mw(m) + x and let W(m) = mw(m). Note that since B is
+Borel in S2, Ψx,d[B] is Borel in Ed(x). Thus W −1
+x [Ψx,d[B]] is Borel in S2. By
+Lemma 4.1, CO,W [W −1
+x
+[Ψx,d[B]]] and CO,W [W −1
+x
+[Ψx,d[B]]],∞ are Borel sets of R3.
+Thus, by translation, Cx,Ψx,d[B] and Cx,Ψx,d[B],∞ are Borel sets of R3.
+We can now prove the following proposition.
+Proposition 4.1. Let R be a generalized reflector in RU
+1 (T). For any set ω ⊆ T
+the visibility set V U
+1 (ω) is Borel.
+Proof. We make use of the fact that sets formed from Borel sets through the
+operations of countable union, countable intersection, and relative complement
+are Borel. Recall that we obtain a sR ∈ N(BR). Note that by the definition
+of a generalized reflector in RU
+1 (T), R = �
+B∈BR ΨxB,dB [B′] where B′ = {m ∈
+15
+
+B|mρsR
+ZR = ΨxB,dB(m)} = B\�
+K∈K K where K = {A ∈ BR|sR(A) < sR(B)};
+note that B′ is clearly Borel and B ⊆ B′ ⊆ B.
+For B ∈ BR, we have that CxB,ΨxB,dB [B′] and CO,ΨxB,dB [B′] are Borel sets
+by Lemmas 4.2 and 4.1 respectively. Since BR is countable and functions of
+the form ΨxB,dB are continuous and bijective, R is Borel. Thus for all B ∈ BR,
+the set QB = CxB,ΨxB,dB [B′] ∩ (R \ ΨxB,dB[B′]) is Borel and therefore the set
+LB = CxB,QB,∞ ∩ ΨxB,dB[B′] is Borel. Thus Proj[LB] is Borel, as it is the
+preimage of LB under ΨxB,dB. Since
+{m ∈ Proj[U]|α1(m) = y2
+R(m)} =
+�
+B∈BR
+Proj[LB],
+(20)
+we have that {m ∈ Proj[U]|α1(m) = y2
+R(m)} is Borel and thus V U
+1 (ω) is Borel.
+Define for any generalized reflector R ∈ RU
+1 (T),
+G1(ω) = µg(V U
+1 (ω))
+(21)
+which we will deem the energy function of the generalized reflector problem.
+Let F be a nonnegative, finite measure on the finite set T. We say that a
+generalized reflector R ∈ RU
+1 (T) is a weak solution to the generalized reflector
+problem if the generalized reflector map α1 determined by R is such that
+F(ω) = G1(ω) for any Borel set ω ⊆ T.
+(22)
+It would be useful to point out the similarity of condition (22) and condition
+(6).
+4.2. Geometric Lemmas
+One thing that should be noted is that the definition of the generalized
+reflector map takes into account that there could potentially be a part of the
+generalized reflector that intercepts an already reflected ray before it can reach
+the target set. That fact inspires some key geometric lemmas.
+16
+
+Lemma 4.3. Let R ∈ RU
+1 (T) for some finite set T ⊂ R3− and open set U ⊆
+R3+. For all B ∈ BR, if m ∈ B, then α1(m) = xB if and only if y2
+R(m) = ∅.
+Proof. This follows directly from the definition of the reflector map of the gen-
+eralized reflector problem.
+Note the following definition.
+Definition 4.3. When we say that r is a ray in Cx,B,∞, then r = {at + x(1 −
+t)|t ∈ [0, ∞)} for some a ∈ B, similarly if we say r is a line segment in Cx,B,
+then r = {at + x(1 − t)|t ∈ [0, 1]} for some a ∈ B.
+Lemma 4.4. Let A, B ⊂ S2 be disjoint sets. Then for any x ∈ R3 \ {O} and
+a, b ∈ (0, ∞), Cx,Ψx,a[A] ∩ Ψx,b[B] = ∅ and Cx,Ψx,b[B] ∩ Ψx,a[A] = ∅ if and only
+if Cx,Ψx,a[A],∞ ∩ Ψx,b[B] = ∅.
+Proof. If Cx,Ψx,a[A],∞∩Ψx,b[B] ̸= ∅, then there exists a ray r in Cx,Ψx,a[A],∞ such
+that r intersects Ψx,b[B]. Thus, either there exists a line segment in Cx,Ψx,a[A]
+that intersects Ψx,b[B] and thus Cx,Ψx,a[A] ∩ Ψx,b[B] ̸= ∅, or there exists a line
+segment in Cx,Ψx,b[B] that intersects Ψx,a[A] and thus Cx,Ψx,b[B] ∩ Ψx,a[A] ̸= ∅.
+Conversely, if Cx,Ψx,a[A] ∩ Ψx,b[B] ̸= ∅, then there exists a line segment
+in Cx,Ψx,a[A] that intersects Ψx,b[B], said line segment coincides with a ray in
+Cx,Ψx,a[A],∞; thus Cx,Ψx,a[A],∞ ∩ Ψx,b[B] ̸= ∅. If Cx,Ψx,b[B] ∩ Ψx,a[A] ̸= ∅, then
+there exists a line segment in Cx,Ψx,b[B] that intersects Ψx,a[A], said line segment
+coincides with a ray in Cx,Ψx,a[A],∞; thus Cx,Ψx,a[A],∞ ∩ Ψx,b[B] ̸= ∅.
+These two lemmas give us the following result.
+Lemma 4.5. Assume that U is an open set in R3+, and T is a finite target
+set in R3−. Let R ∈ RU
+1 (T) be a generalized reflector and A, B ∈ BR such that
+A ̸= B and x = xA = xB. Then the following conditions are equivalent:
+1. for all m ∈ A and m′ ∈ B, α1(m) = α1(m′) = x,
+2. Cx,Ψx,dA[A] ∩ Ψx,dB[B] = ∅ and Cx,Ψx,dB [B] ∩ Ψx,dA[A] = ∅,
+3. Cx,Ψx,dA[A],∞ ∩ Ψx,dB[B] = ∅.
+17
+
+Proof. (2) and (3) are equivalent by Lemma 4.4. By Lemma 4.3, for all m ∈ A
+α1(m) = x if and only if y2
+R(m) = ∅. By definition, y2
+R(m) = ∅ if and only if
+the line segment between Ψx,dA(m) and x does not intersect R \ {Ψx,dA(m)}.
+Similarly, By Lemma 4.3, for all m′ ∈ B, α1(m′) = x if and only if y2
+R(m′) = ∅.
+By definition, y2
+R(m′) = ∅ if and only if the line segment between Ψx,dB(m′)
+and x does not intersect R\{Ψx,dB(m′)}. Therefore, statements (1) and (2) are
+equivalent.
+4.3. Generalized Reflectors Constructed in an Open Conical Cylinder of Arbi-
+trary Thickness
+Let S2
++ = {m ∈ S2|⟨m, (0, 0, 1)⟩ > 0} be the open hemisphere of the S2
+oriented towards the positive z−axis. Similarly, S2
+− = {m ∈ S2|⟨m, (0, 0, 1)⟩ <
+0} be the open hemisphere of the S2 oriented towards the negative z−axis.
+Given an open U ⊆ S2
++, and δ, z′ > 0, we then define an open conical cylinder
+of thickness δ as C δ
+U(z′) = CO,U,∞ ∩ {(x, y, z) ∈ R3|z′ + δ > z > z′}.
+In this paper, given a finite target set T ⊂ R3−, we aim to construct a
+generalized reflector R ∈ RC δ
+U(z′)
+1
+(T) that is a weak solution of the generalized
+reflector problem. This condition is very strict and the following strategies can
+potentially be applied to other kinds of open subsets in R3+.
+We first consider the case where the target set is a single point. We proceed
+with the following lemmas.
+Lemma 4.6. Let U be an open set in S2
++ and z′, δ > 0.
+Let {Si}i∈N be a
+countable collection of open subsets in U, {di}i∈N is a countable collection of
+distinct positive numbers, and x ∈ R3−. Assume that each Ψx,di[Si] ⊂ C δ
+U(z′)
+and denote Ψi = Ψx,di[Si]. Then we have that
+Proj
+�
+C δ
+U(z′) \
+�
+i∈N
+(CO,Ψi,∞ ∪ Cx,Ψi,∞)
+�
+= Proj
+�
+C δ
+U(z′) \
+�
+i∈N
+CO,Ψi,∞
+�
+. (23)
+Proof. Assume to the contrary that
+Proj
+�
+C δ
+U(z′) \
+�
+i∈N
+(CO,Ψi,∞ ∪ Cx,Ψi,∞)
+�
+̸= Proj
+�
+C δ
+U(z′) \
+�
+i∈N
+CO,Ψi,∞
+�
+. (24)
+18
+
+Then there exists a ray r in CO,U\�
+i∈N Si,∞ = CO,U,∞ \ �
+i∈N CO,Si,∞ such that
+r ∩ C δ
+U(z′) ⊂ �
+i∈N Cxi,Si,∞. Equivalently, one can say that there must be a ray
+of direction m ∈ U \ �
+i∈N Si originating from O that we denote as r such that
+r ∩ (C δ
+U(z′) \ �
+i∈N(CO,Ψi,∞ ∪ Cx,Ψi,∞)) = ∅.
+Consider the plane P(α) = {(x, y, z) ∈ R3|z = α}. Let m ∈ U \ �
+i∈N Si.
+Assume that there exists a set P(z′) ∩ �
+i∈N Cx,Ψi,∞ such that
+��
+P(z′) ∩
+�
+i∈N
+Cx,Ψi,∞
+�
+\
+�
+P(z′) ∩
+�
+i∈N
+CO,Ψi,∞
+��
+∩ CO,U,∞ ̸= ∅.
+(25)
+Otherwise there does not exist a ray r of direction m ∈ U \ �
+i∈N Si originating
+from O such that r ∩ (C δ
+U(z′) \ �
+i∈N(CO,Ψi,∞ ∪ Cx,Ψi,∞)) = ∅; a contradiction.
+Thus we assume such a ray exists r exists. Then m must be in a direction such
+that there exists a dmin > 0 where
+Ψx,dmin(m) ∈
+��
+P(z′) ∩
+�
+i∈N
+Cx,Ψi,∞
+�
+\
+�
+P(z′) ∩
+�
+i∈N
+CO,Ψi,∞
+��
+∩ CO,U,∞.
+(26)
+Since C δ
+U(z′) is bounded, there must also exist a dmax > 0 such that
+Ψx,dmax(m) ∈ P(z′ + δ) ∩ CO,U,∞.
+(27)
+Note that by our assumptions, for all d ∈ (dmin, dmax), there exists an α ∈ N
+such that the line segment between Ψx,d(m) and x is a subset of a line segment
+in Cx,Ψα. However, since all Ψi are closed, then for all d ∈ [dmin, dmax], there
+exists an α such that the line segment between Ψx,d(m) and x is a subset of a
+line segment in Cx,Ψα.
+Case 1. dmax > di for all i ∈ N.
+Recall that by our assumptions, Ψx,d(m) ∈ �
+i∈N Cx,Ψi,∞ for all d ∈ [dmin, dmax].
+However, since ψx,dmax(m) > ψx,di(m) for all i ∈ N, Ψx,dmax(m) cannot reside
+on the interior of any ellipsoid Ed′(x) where d′ ∈ {di}i∈N, thus Ψx,d(m) ̸∈
+�
+i∈N Cx,Ψi,∞. A contradiction.
+Case 2. There exists some α ∈ N such that dmax = dα.
+If there exists some α such that dmax = dα, then, since tΨx,dmax(m) + (1 −
+19
+
+t)x ̸∈ C δ
+U(z′) for all t > 1, Ψx,dmax(m) resides on the ellipsoid Edα(x). Therefore
+Ψx,dmax(m) ∈ Ψα ∩ P(z′ + δ) and thus m ∈ Sα. A contradiction.
+Case 3. There exists some α ∈ N such that dα > dmax.
+Assume that {di}i∈N is arranged such that di+1 ≥ di If there exists some α
+such that dα > dmax, then there exists a ray originating from x that intersects
+the point Ψx,dmax(m) that also intersects a point (xβ, yβ, zβ) ∈ Ψβ where dβ ≥
+dmax. The case where dβ = dmax has already been covered. When dβ > dmax:
+since x ∈ R3− and Ψx,dmax(m) ∈ P(z′ + δ), this implies that zβ > z′ + δ. A
+contradiction.
+Lemma 4.7. Recall that σ is the standard measure on S2. Let U be a Borel
+set in R3 \ {O} such that Int(U) ̸= ∅. Let x ∈ R3 \ {O}. Consider the set
+K(d) = Proj[Ed(x) ∩ U] and the corresponding function D(d) = σ(K(d)) for
+d ∈ (0∞). Then D(d) cannot be identically zero.
+Furthermore, if U is open, K(d) is open in S2 for all d ∈ (0, ∞).
+Proof. Clearly there exists a d′ ∈ (0, ∞) such that K(d′) ∩ Int(U) ̸= ∅. Then
+Ed′(x) ∩ Int(U) is open in Ed′(x) and thus Proj[Ed′(x) ∩ Int(U)] is open in S2.
+Therefore, D(d′) ≥ σ(Proj[K(d′) ∩ Int(U)]) > 0.
+Theorem 4.1. Let U be an open set in S2
++, δ, z′ > 0, and T = {x} ∈ R3−.
+Assume that we are given a nonnegative g ∈ L1(S2) where g ≡ 0 outside U.
+Then there exists a generalized reflector R ∈ RC δ
+U(z′)
+1
+(T) such that G1({x}) =
+µg(U).
+Proof. For convenience, label C∗ = C δ
+U(z′). Recall that σ is the standard mea-
+sure on S2.
+Consider the set K1(d) = Proj[Ed(x) ∩ C∗] and its corresponding function
+D1(d) = σ(K1(d)) where d ∈ (0, ∞). Note that since C∗ is bounded, D1(d) → 0
+as d → ∞ and D1(d) → 0 as d → 0. By construction, it is clear that D1 is
+bounded by 0 and σ(U). Therefore, Dmax
+1
+= sup{D1(d)|d ∈ (0, ∞)} exists and
+is finite, and by Lemma 4.7, Dmax
+1
+> 0.
+20
+
+Let ϵ1 ∈ [0, Dmax
+1
+). We define dmax1 to be a value such that D1(dmax1) =
+Dmax
+1
+− ϵ1 where ϵ1 = 0 if Dmax
+1
+∈ {D1(d)|d ∈ (0, ∞)}. We now eliminate the
+parts of U that had already been accounted for and the parts of C∗ that can no
+longer be used: let E1 = Proj[Edmax1 (x) ∩ C∗], Ψ1 = Ψx,dmax1 [E1],
+Q2 = C∗ \ (Cx,Ψ1,∞ ∪ CO,Ψ1,∞),
+(28)
+and U2 = U \ E1. Note by Lemma 4.6, U2 = Proj[Q2]. Let us define K2(d) =
+Proj[Ψx,d[U2] ∩ Q2] and D2(d) = σ(K2(d)).
+Note that since Q2 is bounded, D2(d) → 0 as d → ∞ and D2(d) → 0
+as d → 0. By construction, it is clear that D2 is bounded by 0 and σ(U2).
+Therefore, Dmax
+2
+= sup{D2(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma
+4.7, Dmax
+2
+> 0. Let ϵ2 ∈ [0, Dmax
+2
+). We define dmax2 to be a value such that
+D1(dmax1) ≥ D2(dmax2) = Dmax
+2
+− ϵ2.
+Given that U1 = U and Q1 = C∗, we can now recursively define a sequence
+of functions and sets for k ≥ 2:
+Ek−1 = Proj[Ψx,dmaxk−1 [Uk−1] ∩ Qk−1],
+(29)
+Ψk−1 = Ψx,dmaxk−1 [Ek−1],
+(30)
+Qk = Qk−1 \ (Cx,Ψk−1,∞ ∪ CO,Ψk−1,∞),
+(31)
+Uk = U \
+�
+�
+k−1
+�
+j=1
+Ej
+�
+� = Proj[Qk],
+(32)
+Kk(d) = Proj[Ed(x) ∩ Qk],
+(33)
+Dk(d) = σ(Kk(d)).
+(34)
+Also, note that since Qk is bounded, Dk(d) → 0 as d → ∞ and Dk(d) → 0
+as d → 0. By construction, it is clear that Dk is bounded by 0 and σ(Uk).
+Therefore, Dmax
+k
+= sup{Dk(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma
+4.7, Dmax
+k
+> 0. Let ϵk ∈ [0, Dmax
+k
+). We define dmaxk to be a value such that
+Dk−1(dmaxk−1) ≥ Dk(dmaxk) = Dmax
+k
+− ϵk.
+Observe that the set Kk(d) is open for all d > 0. We can therefore construct
+21
+
+a sequence
+�
+�
+�σ
+�
+�
+k�
+j=1
+Ej
+�
+�
+�
+�
+�
+∞
+k=1
+.
+(35)
+Claim 4.1. There exists {ϵi}i∈N such that
+�
+σ
+��k
+j=1 Ej
+��∞
+k=1 converges to
+σ(U).
+Proof. By construction, the sequence increases monotonically and is bounded
+between 0 and σ(U); thus it converges. Assume to the contrary that for every
+possible {ϵi}i∈N,
+�
+σ
+��k
+j=1 Ej
+��∞
+k=1 that converges to an L ∈ (0, σ(U)). Then
+σ
+�
+U \ �∞
+j=1 Ej
+�
+= σ(U) − L > 0.
+Consider the function
+D∗(d) = σ
+�
+Proj
+�
+Ed(x) ∩ lim
+j→∞ Qj
+��
+.
+(36)
+Observe that limj→∞ Qj = C∗\�∞
+i=1(Cx,Ψi,∞∪CO,Ψi,∞). Note that �∞
+i=1(Cx,Ψi,∞∪
+CO,Ψi,∞) ⊆ �∞
+i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞). Observe that for all i ∈ N, Int(Cx,Ψi,∞ ∪ CO,Ψi,∞) =
+Cx,Ψi,∞∪CO,Ψi,∞; thus �∞
+i=1 Int(Cx,Ψi,∞ ∪ CO,Ψi,∞) = �∞
+i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞).
+Thus limj→∞ Qj = C∗\�∞
+i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞) is open and thus Int(limj→∞ Qj) ̸=
+∅.
+Thus, by Lemma 4.7, there exists a d′ such that D∗(d′) > 0. By the def-
+inition of convergence, there exists an M such that for all m ≥ M, D∗(d′) >
+σ
+��∞
+j=m Ej
+�
+. Note that σ
+��∞
+j=m Ej
+�
+= �∞
+j=m σ(Ej) = �∞
+j=m Di(dmaxi) ≥
+Dm(dmaxm). For all k ∈ N, Dmax
+k
+is a limit point of {Dk(d)|d ∈ (0, ∞)}, there-
+fore as ϵk → 0, Dk(dmaxk) → Dmax
+k
+. Observe that Dm(d′) ≥ D∗(d′) because
+limj→∞ Qj ⊆ Qm and U \ �∞
+j=1 Ej ⊆ Um. Therefore there exists a sequence
+{ϵi}i∈N such that Dm(dmaxm) ≥ D∗(d′). For this sequence σ
+��∞
+j=m Ej
+�
+≥
+D∗(d′); a contradiction. Thus, there exists {ϵi}i∈N such that
+�
+σ
+��k
+j=1 Ej
+��∞
+k=1
+converges to σ(U).
+Let
+Z =
+∞
+�
+j=1
+Ψj.
+(37)
+22
+
+For some s ∈ N(BZ), consider the generalized reflector R = W(ρs
+Z) ∈ RC∗
+1 (T).
+By construction,
+Cx,Ψx,dmaxj [Int(Ej)],∞ ∩ Ψx,dmaxj′ [Int(Ej′)] = ∅
+(38)
+when j ̸= j′. Observe that if j′, j ∈ N where j′ > j, then dmaxj ̸= dmaxj′
+because otherwise Ej′ ⊆ Ej; thus BR = {Int(Ej) ⊂ S2|j ∈ N}.
+Thus, by
+Lemma 4.5, for all m ∈ B where B ∈ BR, we have α1(m) = x. Then, for any
+s ∈ N(BZ), the generalized reflector R = W(ρs
+Z) ∈ RC∗
+1 (T) is a weak solution
+to the generalized reflector problem such that G1({x}) = µg(U).
+We can now prove a result where our target set is made up of finitely many
+points. First, we prove the following lemma.
+Lemma 4.8. Assume that U is an open set in R3+, and T is a finite target
+set in R3−. Let R ∈ RU
+1 (T) be a generalized reflector and A, B ∈ BR such that
+A ̸= B. Then the following conditions are equivalent:
+1. for all m ∈ A and m′ ∈ B, α1(m) = xA and α1(m′) = xB,
+2. CxA,ΨxA,dA[A] ∩ ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B] ∩ ΨxA,dA[A] = ∅.
+Proof. For the case where xA = xB, we have Lemma 4.5. We now consider
+the case where xB ̸= xB. By Lemma 4.3, for all m ∈ A α1(m) = xA if and
+only if y2
+R(m) = ∅. By definition, y2
+R(m) = ∅ if and only if the line segment
+between ΨxA,dA(m) and xA does not intersect R \ {ΨxA,dA(m)}.
+Similarly,
+By Lemma 4.3, for all m′ ∈ B α1(m′) = xB if and only if y2
+R(m′) = ∅. By
+definition, y2
+R(m′) = ∅ if and only if the line segment between ΨxB,dB(m′) and
+x does not intersect R \ {ΨxB,dB(m′)}. Therefore, statements (1) and (2) are
+equivalent.
+Theorem 4.2. Let U be an open set in S2
++, δ, z′ > 0, and {x1, . . . , xk} ∈ R3−
+where k ≥ 2. Assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0
+23
+
+outside U. Let f1, f2, . . . , fk be nonnegative real numbers such that
+k
+�
+i=1
+fi = µg(U).
+(39)
+Assume that there exists n ≥ k disjoint open sets Bi in U where �
+i∈[n] Bi = U.
+Also assume that there exists a collection of k subsets of [n], {Ai}i∈[k], such that:
+At ∩At′ = ∅ where t ̸= t′, �
+i∈[k] Ai = [n], and µg(�
+i∈At Bi) = ft. Suppose that
+for all i ∈ [n] there exists ai, bi > 0 where z′ ≤ ai < ai + bi ≤ z′ + δ such that
+C bi
+Bi(ai) ∩ Cxj,C
+bj
+Bj (aj) = ∅ for all j ∈ [n] \ {i}.
+Then there exists a generalized reflector in R ∈ RC δ
+U(z′)
+1
+(T) such that G1({xi}) =
+fi for all i ∈ [k].
+Proof. For each i ∈ [n], C bi
+Bi(ai) is open and a generalized reflector Ri ∈
+R
+C
+bi
+Bi(ai)
+1
+({xi}) is constructed in the exact same way as Theorem 4.1. By our
+assumptions, since Ri ⊆ C bi
+Bi(ai), then
+(Ri ∩ C bi
+Bi(ai)) ∩ Cxj,Rj∩C
+bj
+Bj (aj) = ∅
+(40)
+for all j ∈ [n] \ {i}.
+Let F = �
+i∈[k] Ri and consider the generalized reflector R = W(ρF ) ∈
+RC δ
+U(z′)
+1
+(T). By construction, for an A, B ∈ BR such that A ̸= B, we have
+that CxA,ΨxA,dA[A] ∩ ΨxB,dB[B] = ∅ and CxB,Ψx,dB [B] ∩ ΨxA,dA[A] = ∅. Thus
+by Lemma 4.8, for any B ∈ BR, for all m ∈ B we have α1(m) = xB. Then,
+for any s ∈ N(BF ), the generalized reflector R = W(ρs
+F ) ∈ RC δ
+U (z′)
+1
+(T) is a
+weak solution to the generalized reflector problem such that G1({xi}) = fi for
+all i ∈ [k].
+We now will use Theorem 4.2 to construct a specific type of generalized
+reflector. Note the following definition.
+Definition 4.4. Let k ≥ 2, d > 0, ξ ∈ (−1, 0), and t ∈ R. Recall that, given a
+24
+
+point (x, y, z) ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], θ ∈ [0, 2π), such that
+x = r cos θ sin φ
+(41)
+y = r sin θ sin φ
+(42)
+z = r cos φ.
+(43)
+Define the set of points T ξ
+k,d(t) as
+��
+d cos
+�2πj
+k
++ t
+�
+sin (arccos(ξ)) , d sin
+�2πj
+k
++ t
+�
+sin (arccos(ξ)) , dξ
+�
+|j ∈ I
+�
+(44)
+where I = {0, 1, . . . , k − 1}.
+If we are additionally given an i ∈ {0, 1, . . . , k − 1}, we may define the set
+Pk,i(t) ⊂ S2, as
+��
+cos
+�
+θ + π(2i − 1)
+k
++ t
+�
+sin φ, sin
+�
+θ + π(2i − 1)
+k
++ t
+�
+sin φ, cos φ
+�
+����φ ∈ [0, π], θ ∈
+�
+0, 2π
+k
+��
+.
+(45)
+If k = 1, define the set of points T ξ
+1,d(t) = {(0, 0, −d)} and P1,0(t) = S2.
+It is good to observe that T ξ
+k,d(t) defines the points of a regular k-gon centered
+at the z-axis and that Pk,i(t) defines a spherical wedge.
+Theorem 4.3. Let δ, z′ > 0. Consider the open disk U = {m ∈ S2
++|⟨(0, 0, 1), m⟩ >
+c} where 0 < c < 1. Let d1, . . . , dn be a collection of not necessarily distinct
+positive numbers. Let k1, . . . , kn be a collection of not necessarily distinct pos-
+itive integers. Let ξ1, . . . , ξn be a collection of not necessarily distinct numbers
+such that ξi ∈ (−1, 0). Let t′
+1, . . . , t′
+n be a collection of not necessarily distinct
+elements of R. Let us denote Ti = T ξi
+ki,di(t′
+i) and let T = �n
+i=1 Ti.
+Assume that we a given a nonnegative g ∈ L1(S2
++) that is rotationally sym-
+metric about the z-axis such that g ≡ 0 outside U. Let f1, . . . , fn be a collection
+of positive numbers such that
+µg(U) =
+n
+�
+i=1
+fi.
+(46)
+25
+
+Then there exists a generalized reflector R ∈ RC δ
+U(z′)
+1
+(T) such that
+G1({x}) =
+�
+{j∈[n]|x∈Tj}
+fj
+kj
+(47)
+for all x ∈ T.
+Proof. By the intermediate value theorem, there exists a collection of numbers
+ζ1, . . . , ζn ∈ [c, 1) where ζn = c and µg
+�
+{m ∈ S2
++|⟨(0, 0, 1), m⟩ > ζi}
+�
+= �i
+j=1 fj.
+Define
+Bi = {m ∈ S2
++|⟨(0, 0, 1), m⟩ > ζi} \ {m ∈ S2
++|⟨(0, 0, 1), m⟩ ≥ ζi−1}
+(48)
+for all i ∈ {2, . . . , n} and B1 = {m ∈ S2
++|⟨(0, 0, 1), m⟩ > ζ1}. Thus µg(Bi) = fi.
+Consider the set Ti, let
+Ti(j) =
+�
+di cos
+�2πj
+ki
++ t′
+i
+�
+sin (arccos(ξi)) , di sin
+�2πj
+ki
++ t′
+i
+�
+sin (arccos(ξi)) , diξi
+�
+(49)
+where j ∈ {0, . . . , ki − 1} if k ≥ 2 and Ti(0) = (0, 0, −di).
+Let Pi(j) = Pki,j(t′
+i) where j ∈ {0, . . . , ki −1}, then by construction µg(Bi ∩
+Pi(j)) = fi
+ki . Let Ui(j) = C
+δ
+n
+Bi∩Int(Pi(j))
+�
+z′ + (i − 1) δ
+n
+�
+where j ∈ {0, . . . , ki −1}.
+Since Ti(j) ∈ R3−, any given line segment between a point in Ui(j) and the point
+Ti(j) will not intersect any set Ui′(j′) where i′ > i and j′ ∈ {0, . . . , ki′ − 1}.
+Therefore CTi(j),Ui(j) ∩ Ui′(j′) = ∅ where i′ > i and j′ ∈ {0, . . . , ki′ − 1}. Also,
+since CO,Pi(j),∞ is a convex set and Ui(j), {Ti(j)} are both subsets of CO,Pi(j),∞,
+we then have CTi(j),Ui(j) ⊂ CO,Pi(j),∞. Therefore, CTi(j),Ui(j) ∩ Ui(j′) = ∅
+where j′ ̸= j. Finally, if there exists a line segment between a point in Ui(j)
+and Ti(j) that intersects a Ui′(j′) where i > i′ and j′ ∈ {0, . . . , ki′ − 1}, then it
+must intersect CO,{m∈S2
++|⟨(0,0,1),m⟩>ζi−1},∞. However, CTi(j),Ui(j) is disjoint from
+CO,{m∈S2
++|⟨(0,0,1),m⟩>ζi−1},∞ and thus CTi(j),Ui(j) ∩ Ui′(j′) = ∅ where i > i′ and
+j′ ∈ {0, . . . , ki′ − 1}. Therefore, CTi(j),Ui(j) ∩ Ui′(j′) = ∅ when (i, j) ̸= (i′, j′).
+Therefore, by Theorem 4.2, there exists a generalized reflector R ∈ RCδ
+U(z′)
+1
+(T)
+such that
+G1({x}) =
+�
+{j∈[n]|x∈Tj}
+fj
+kj
+(50)
+26
+
+for all x ∈ T.
+5. Interpolated Reflectors
+The generalized reflector presented in the previous section might be impos-
+sible or, at best, very difficult to construct in the real world. Thus we introduce
+the following notion.
+Definition 5.1. Assume that we are given an aperture that is a connected open
+set D ⊆ S2 and a not necessarily continuous, almost everywhere differentiable
+function ρ : D → (0, ∞). Then an interpolated reflector is the set R =
+∂(CO,S) \ ∂(CO,S,∞) ⊂ R3 where S = {mρ(m)|m ∈ D}.
+It is interesting to note that, given an aperture that is a connected open set
+D ⊆ S2, a set is a reflector if and only if it is both a generalized reflector and
+an interpolated reflector.
+The type of interpolated reflector we construct below is a topological surface
+(see Chapter 4.36 in [20]) and thus consists of one connected component instead
+of countably many. In a practical sense, when designing an interpolated reflector
+as opposed to a generalized reflector, new challenges are introduced. Thus, we
+settle for finding a necessary and sufficient condition for the existence of an
+interpolated reflector.
+We will consider the following formulation of the near-field reflector problem
+as a weak formulation of equation (4) from [12] and its solutions, weak solutions.
+The following formulation only concerns the case where the target set is finite.
+5.1. Weak Solutions using Interpolated Reflectors
+Consider a connected open set U ⊆ R3+, a corresponding aperture Proj[U],
+and a finite target set T ⊆ R3−. Also, consider the set RU
+1 (T) as defined by
+(14). We then describe a set of interpolated reflectors
+RU
+2 (T) =
+�
+∂(CO,S) \ ∂(CO,S,∞)| S ∈ RU
+1 (T)
+�
+.
+(51)
+27
+
+It is interesting to note that the interpolated reflectors in RU
+2 (T) are all topo-
+logical surfaces.
+Assume we are given an interpolated reflector R ∈ RU
+2 (T). Let us define
+BR =
+�
+Int(Proj[Ed(x) ∩ R]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ R]) ̸= 0
+�
+.
+(52)
+The geometry of the ellipsoid and the definition of BR imply that there
+exists an s ∈ N(BR), unique u ∈ UT (B) and unique v ∈ V (B) such that
+∂(CO,W (ρs
+Z))\∂(CO,W (ρs
+Z),∞) = R where Z = �
+B∈BR Ψu(B),v(B)[B]. Therefore,
+for every interpolated reflector R ∈ RU
+2 (T), we may define a unique BR ∈ B(U)
+such that for each B ∈ BR there are unique xB ∈ T and dB ∈ (0, ∞) such
+that, for some s ∈ N(BR), R = ∂(CO,W (ρs
+Z)) \ ∂(CO,W (ρs
+Z),∞) where Z =
+�
+B∈BR ΨxB,dB[B].
+Therefore, given a interpolated reflector R ∈ RU
+2 (T), we obtain a cor-
+responding BR; for each B ∈ BR we define unique xB and dB.
+We also
+obtain an sR ∈ N(BR) and a unique ZR = �
+B∈BR ΨxB,dB[B] such that
+R = ∂(CO,S) \ ∂(CO,S,∞) where S = W(ρsR
+ZR).
+Given an interpolated reflector R ∈ RU
+2 (T), let
+M(m) = xB ∈ T
+(53)
+where mρsR
+ZR(m) = ΨxB,dB(m). Let y2
+R(m) be the points of intersection between
+R \ {mρsR
+ZR(m)} and the line segment connecting mρsR
+ZR(m) to M(m).
+Given an interpolated reflector R ∈ RU
+2 (T), the map α2 : Proj[U] → T ∪ R,
+α2(m) =
+�
+�
+�
+�
+�
+M(m)
+if y2
+R(m) = ∅
+y2
+R(m)
+if y2
+R(m) ̸= ∅
+(54)
+is called the interpolated reflector map. Physically speaking, a ray of light of
+direction m originating from O can only reach the target set if y2
+R(m) is empty.
+As before, we denote by g ∈ L1(S2) the energy density of the source O. Let
+us define for all Borel X ⊆ S2
+µg(X) =
+�
+X
+g(m)dσ(m)
+(55)
+28
+
+where σ denotes the standard measure on S2. Assume that g is a nonnegative
+function where g ≡ 0 outside of Proj[U]. Physically speaking, g is the radiance
+distribution of the source at O
+In order to formulate and solve the interpolated reflector problem (in the
+framework of weak solutions to be defined below), we need to define a measure
+representing the energy generated by g and redistributed by an interpolated
+reflector R ∈ RU
+2 (T).
+Given a interpolated reflector R ∈ RU
+2 (T) and a set ω ⊆ T we define the
+visibility set of ω as
+V U
+2 (ω) =
+�
+A∈A
+A \ {m ∈ Proj[U]|α2(m) = y2
+R(m)}
+(56)
+where A = {B ∈ BR|xB ∈ ω}. We now need to show that V U
+2 (ω) is measurable.
+Proposition 5.1. Let R be a interpolated reflector in RU
+2 (T).
+For any set
+ω ⊆ T, the visibility set V U
+2 (ω) is Borel.
+Proof. We make use of the fact that sets formed from Borel sets through the
+operations of countable union, countable intersection, and relative complement
+are Borel. Recall that we obtain a sR ∈ N(BR). Note that by the definition of
+an interpolated reflector in RU
+1 (T), R = �
+B∈BR ΨxB,dB [B′] ∪ SR where B′ =
+{m ∈ B|mρsR
+ZR = ΨxB,dB(m)} = B \ �
+K∈K K where K = {A ∈ BR|sR(A) <
+sR(B)} and SR = R \ �
+B∈BR ΨxB,dB[B′]. Note that B′ is clearly Borel and
+B ⊆ B′ ⊆ B.
+For B ∈ BR, we have that CxB,ΨxB,dB [B′] and CO,ΨxB,dB [B′] are Borel sets
+by Lemmas 4.2 and 4.1 respectively. Note that R is Borel as it is a boundary of
+an open set minus the boundary of another open set. Since BR is countable and
+functions of the form ΨxB,dB are continuous and bijective, �
+B∈BR ΨxB,dB [B′]
+is Borel.
+Therefore SR is also Borel.
+Thus for all B ∈ BR, the set QB =
+CxB,ΨxB,dB [B′]∩(R\ΨxB,dB[B′]) is Borel and therefore the set LB = CxB,QB,∞∩
+ΨxB,dB[B′] is Borel. Thus Proj[LB] is Borel, as it is the preimage of LB under
+29
+
+ΨxB,dB. Since
+{m ∈ Proj[U]|α2(m) = y2
+R(m)} =
+�
+B∈BR
+Proj[LB],
+(57)
+we have that {m ∈ Proj[U]|α2(m) = y2
+R(m)} is Borel and thus V U
+2 (ω) is Borel.
+Define for any interpolated reflector R ∈ RU
+2 (T),
+G2(ω) = µg(V U
+2 (ω))
+(58)
+which we will deem the energy function of interpolated reflector problem.
+Let F be a nonnegative, finite measure on the finite set T. We say that an
+interpolated reflector R ∈ RU
+2 (T) is a weak solution to the interpolated reflector
+problem if the interpolated reflector map α2 determined by R is such that
+F(ω) = G2(ω) for any Borel set ω ⊆ T.
+(59)
+5.2. Main Results
+Here we prove a necessary and sufficient condition for the existence of weak
+solutions to the interpolated reflector problem. We proceed with the following
+lemma.
+Lemma 5.1. Assume that U is an open set in R3+, and T is a finite target
+set in R3−. Assume we are given a nonnegative g ∈ L1(S2) such that g ≡ 0
+outside Proj[U] and g > 0 inside Proj[U]. Let R ∈ RU
+1 (T) be a generalized
+reflector, then we define the set Bx
+R = {B ∈ BR|x = xB} for x ∈ T. Then for
+any z, y ∈ T the following conditions are equivalent:
+1. for all m ∈ �
+A∈Bz
+R A and m′ ∈ �
+B∈By
+R B, α1(m) = z and α1(m′) = y,
+2. Cz,Ψz,dA[A] ∩Ψy,dB[B] = ∅ and Cy,Ψy,dB [B] ∩Ψz,dA[A] = ∅ for all A ∈ Bz
+R
+and B ∈ By
+R where A ̸= B,
+3. CxA,ΨxA,dA[A] ∩ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B] ∩ΨxA,dA[A] = ∅ for all
+A, B ∈ BR where A ̸= B,
+30
+
+4. G1({x}) = µg(�
+A∈Bx
+R A) and G1({y}) = µg(�
+B∈By
+R B).
+Proof. (1) ⇔ (2) by Lemma 4.8, clearly (3) ⇔ (2), and (1) trivially implies (4).
+We only need to prove that (4) implies (2); we prove the contrapositive.
+Assume that there exists an A ∈ Bx
+R and a B′ ∈ By
+R such that, without loss of
+generality, Cz,Ψz,dA[A] ∩ Ψy,dB′ [B′] ̸= ∅. Let Q = Cz,Ψz,dA[A] ∩ Ψy,dB′ [B′]. Note
+that Cz,Ψz,dA[A],∞ ∩ Cz,Ψy,dB′ [B′],∞ = Cz,Q,∞. Since Cz,Ψz,dA[A],∞ \ {O} and
+Cz,Ψy,dB′ [B′],∞\{O} are open, Cz,Q,∞\{O} is open; thus Cz,Q,∞∩EdB(y) is open
+in EdB(y). Thus Proj[Q] ⊆ B is open and, since g > 0 in U, µg(Proj[Q]) > 0,
+and thus G1({y}) ≤ µg(�
+B∈By
+R B) − µg(Proj[Q]).
+Note the following definition.
+Definition 5.2. Let K be a subset of Rn where n ≥ 2 such that K is compact.
+The complement U = Rn \ K is an open set. For sufficiently large R > 0,
+the set V = {x|R < |x|} is contained in U. Since V is connected, there exists
+a connected component of U that contains V . This is the unique unbounded
+connected component of U.
+We define the exterior boundary of K as the boundary of the unbounded
+connected component of Rn \ K. We denote this as ∂E(K).
+The following result gives a condition that is necessary and sufficient for the
+existence of weak solutions to the interpolated reflector problem.
+Theorem 5.1. Let U ⊂ R3+ be a simply connected open set such that U ⊂ R3+,
+and T ⊂ R3− be a finite set. Assume we are given a nonegative g ∈ L1(S2) such
+that g ≡ 0 outside Proj[U] and g > 0 inside Proj[U]. Let F be a measure over
+T such that
+F(T) = µg(Proj[U]).
+(60)
+Then there exists an interpolated reflector R2 ∈ RU
+2 (T) that is a weak solu-
+tion to the interpolated reflector problem as defined in (59) if and only if there
+exists a generalized reflector R1 ∈ RU
+1 (T) that is a weak solution to the gen-
+eralized reflector problem as defined in (22) where R1 is a subset of a simply
+31
+
+connected subset of
+U ∩ ∂E
+�
+�CO,R1 ∪
+�
+B∈BR1
+CxB,ΨxB,dB [B]
+�
+� .
+(61)
+Proof. It is clear that if R2 ∈ RU
+2 (T) that is a weak solution to the interpolated
+reflector problem as defined in (59), then, for any s ∈ N(BR2), R1 = W(ρs
+F )
+where F = �
+B∈BR2 ΨxB,dB[B] is a weak solution to the generalized reflector
+problem as defined in (22). Note that this implies that BR1 = BR2. Since
+g is positive in Proj[U], for our reflector R1, by Lemma 5.1, for all distinct
+A, B ∈ BR1, CxA,ΨxA,dA[A]∩ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B]∩ΨxA,dA[A] = ∅.
+Therefore, for all A ∈ BR1, ΨxA,dA[A]∩Int
+�
+CO,R1 ∪ �
+B∈BR1 CxB,ΨxB,dB [B]
+�
+=
+∅.
+Also observe that for all A ∈ BR1, (CO,ΨxA,dA[A],∞ \ CO,ΨxA,dA[A]) ∩
+Int
+�
+CO,R1 ∪ �
+B∈BR1 CxB,ΨxB,dB [B]
+�
+= ∅.
+Therefore, R1 is a subset of a simply connected subset of
+∂E
+�
+�CO,R1 ∪
+�
+B∈BR1
+CxB,ΨxB,dB [B]
+�
+� .
+(62)
+Since the interpolated reflector must be contained in U, we obtain (61).
+Conversely, if there exists a reflector R1 ∈ RU
+1 (T) that is a weak solution of
+the generalized reflector problem as defined in (22) where R1 ⊂ Q such that Q
+is a simply connected subset of (61), then R2 = ∂(CO,R1) \ ∂(CO,R1,∞) is also
+a subset of Q.
+6. Discussion
+In this note, with respect to the near-field reflector problem with spatial
+restrictions, we defined two different kinds of weak solutions. For the first weak
+solution, we proved, under certain assumptions, the existence of a generalized
+reflector where the target set is multiple points. A possible avenue for further
+research is to attempt to expand Theorem 4.3 for different target sets, apertures,
+and spatial restrictions. Another idea might be to try to come up with designs
+such that the generalized reflectors have finitely many connected components
+32
+
+instead of countably many. The author believes that the following statement is
+true.
+Conjecture 6.1. Let U be an open set in S2
++, δ, z′ > 0, and {x1, . . . , xk} ∈ R3−
+where k ≥ 2. Assume we are given a positive g ∈ L1(S2) where g ≡ 0 outside
+U. Let f1, f2, . . . , fk be nonnegative real numbers such that
+k
+�
+i=1
+fi = µg(U).
+(63)
+Then there exists a generalized reflector R ∈ RC δ
+U(z′)
+1
+(T) such that G1({xi}) = fi
+for all i ∈ [k].
+For the second weak solution, we proved a theorem that detailed a necessary
+and sufficient condition for the existence of an interpolated reflector. The ad-
+vantage of our interpolated reflectors, as opposed to our generalized reflectors,
+is that our interpolated reflector design is a topological surface; thus it is easier
+to construct from an engineering perspective. An obvious avenue for further
+work would be to create some practically useful interpolated reflectors; using
+Theorem 4.2 might be useful in this regard. In fact, it would be very useful if
+the following conjecture is true.
+Conjecture 6.2. Let U be a simply connected open set in S2
++, δ, z′ > 0, and
+{x1, . . . , xk} ∈ R3−. Assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0
+outside U. Let f1, f2, . . . , fk be nonnegative real numbers such that
+k
+�
+i=1
+fi = µg(U).
+(64)
+Then there exists an interpolated reflector in R ∈ RC δ
+U(z′)
+2
+(T) such that
+G2({xi}) = fi for all i ∈ [k].
+Another fruitful avenue of research might be to somehow expand these def-
+initions of weak solutions to account for cases where the target set is not finite.
+Then, proving the existence of generalized and interpolated reflectors with con-
+tinuous irradiance distributions.
+33
+
+As the reader might have noticed, we make no attempt to address the near-
+field reflector problem with spatial conditions with a reflector.
+Instead, we
+exclusively use generalized or interpolated reflectors. While it may be interesting
+to research reflectors in order to get a ‘stronger’ solution, it is the author’s
+view that, in general, it is not possible to construct a reflector under spatial
+conditions. For example, if the target set is a single point, the solution to the
+near-field reflector problem is an ellipsoid. However, if given spatial restrictions,
+a single ellipsoid, in general, cannot fit those restrictions; this is demonstrated
+in Theorem 4.1. If no reflector exists for a single point, the prospects for more
+complicated target sets and irradiance distributions appear limited.
+References
+[1] M. Born, E. Wolf, Principles of Optics: 60th Anniversary Edition, 7th
+Edition, Cambridge University Press, 2019.
+[2] L. Caffarelli, V. Oliker, Weak solutions of one inverse problem in geometric
+optics, Journal of Mathematical Sciences 154 (2008) 39–49.
+[3] J. Schruben, Formulation of relector-design problem for a lighting fixture,
+J. Opt. Soc. Am. 62 (12) (1972) 1498–1501.
+[4] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field
+scattering data, Inverse Problems 13 (2) (1997) 363–373.
+[5] X.-J. Wang, On design of reflector antenna, Inverse Problems 12 (1996)
+351–375.
+[6] B. Kinber, On two reflector antennas, Radio Eng. Electron. Phys. 7 (1962)
+973–979.
+[7] D. Burkhard, D. Shealy, Design of reflectors which will distribute sunlight
+in a specified manner, Solar Energy 17 (1975) 221–307.
+34
+
+[8] T. E. Horton, J. H. McDermit, Design of a specular aspheric surface to
+uniformly radiate a flat surface using a nonuniform collimated radiation
+source, J. Heat Transfer (1972) 453–458.
+[9] F. Fournier, A review of beam shaping strategies for LED lighting, in: T. E.
+Kidger, S. David (Eds.), Illumination Optics II, Vol. 8170, SPIE, 2011, pp.
+55 – 65.
+[10] A. Gray, Modern Differential Geometry of Curves and Surfaces with Math-
+ematica, 2nd Edition, CRC Press, Boca Raton, FL, 1997.
+[11] V. Oliker, E. Newman, The energy conservation equation in the reflector
+mapping problem, Applied Mathematics Letters 6 (1) (1993) 91–95.
+[12] V. Oliker, On reconstructing a reflecting surface from the scattering data in
+the geometric optics approximation, Inverse Problems 5 (1) (1989) 51–65.
+[13] J. Schruben, Analysis of rotationally symmetric reflectors for illuminating
+systems∗, J. Opt. Soc. Am. 64 (1) (1974) 55–58.
+[14] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd Edition,
+Encyclopedia of Mathematics and its Applications, Cambridge University
+Press, 2013.
+[15] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field
+scattering data ii. numerical solution ., Numer. Math. 79 (1998) 553–568.
+[16] F. Fournier, B. Cassarly, J. Rolland, Optimization of single reflectors for
+extended sources, Proc SPIE 7103 (09 2008).
+[17] F. Fournier, B. Cassarly, J. Rolland, Fast freeform reflector generation using
+source-target maps, Opt. Express 18 (5) (2010) 5295–5304.
+[18] T. Graf, V. Oliker, An optimal mass transport approach to the near-field
+reflector problem in optical design, Inverse Problems 28 (2012) 025001.
+35
+
+[19] F. R. Fournier, Freeform reflector design with extended sources, Ph.D.
+thesis, CREOL, the College of Optics and Photonics at the University of
+Central Florida (2010).
+[20] J. Munkres, Topology, 2nd Edition, Featured Titles for Topology, Prentice
+Hall, Incorporated, 2000.
+36
+

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+Density of states and spectral function of a superconductor out of a quantum-critical
+metal
+Shang-Shun Zhang1 and Andrey V. Chubukov1
+1School of Physics and Astronomy and William I. Fine Theoretical Physics Institute,
+University of Minnesota, Minneapolis, MN 55455, USA
+(Dated: February 1, 2023)
+We analyze the validity of a quasiparticle description of a superconducting state at a metallic
+quantum-critical point (QCP). A normal state at a QCP is a non-Fermi liquid with no coherent
+quasiparticles.
+A superconducting order gaps out low-energy excitations, except for a sliver of
+states for non-s-wave gap symmetry, and at a first glance, should restore a coherent quasiparticle
+behavior. We argue that this does not necessarily hold as in some cases the fermionic self-energy
+remains singular slightly above the gap edge.
+This singularity gives rise to markedly non-BCS
+behavior of the density of states and to broadening and eventual vanishing of the quasiparticle peak
+in the spectral function. We analyze the set of quantum-critical models with an effective dynamical
+4-fermion interaction, mediated by a gapless boson at a QCP, V (Ω) ∝ 1/Ωγ. We show that coherent
+quasiparticle behavior in a superconducting state holds for γ < 1/2, but breaks down for larger γ.
+We discuss signatures of quasiparticle breakdown and compare our results with the data.
+Introduction.
+Metals near a quantum critical
+point (QCP) display a number of non-Fermi liquid
+properties like linear-in-T resistivity, a broad peak in
+the spectral function near kF with linear-in-ω width,
+singular behavior of optical conductivity, etc [1–18].
+These properties are often thought to be caused by
+the coupling of fermions to near-gapless fluctuations of
+an order parameter, which condenses at a QCP [19–
+29].
+The same fermion-boson interaction gives rise to
+superconductivity near a QCP [30–42].
+A
+superconducting
+order
+gaps
+out
+low-energy
+excitations, leaving at most a tiny subset of gapless
+states for a non-s−wave order parameter.
+A general
+belief has been that this restores fermionic coherence. A
+frequently cited experimental evidence is the observed
+re-emergence
+of
+a
+quasiparticle
+peak
+below
+Tc
+in
+near-optimally doped cuprates (see e.g.,
+Ref. [43]).
+From theory side, the argument is that the fermionic
+self-energy in a superconductor has a conventional Fermi-
+liquid form Σ(ω) ∼ ω at the lowest ω, in distinction from
+a non-Fermi-liquid Σ(ω) ∝ ωa with a < 1 in the normal
+state
+[44–53].
+In this paper, we analyze theoretically
+whether fermions in a superconducting state at a QCP
+can be viewed as well-defined coherent quasiparticles.
+We argue that this is not necessarily the case as fermionic
+self-energy can still be singular on a real frequency axis
+immediately above the gap edge. This singularity gives
+rise to markedly non-BCS behavior of the density of
+states (DoS) and to broadening and eventual vanishing
+of the quasiparticle peak.
+For superconductivity away from a QCP, mediated by
+a massive boson, numerous earlier studies have found
+that the spectral function A(k, ω) at T
+= 0 has a
+δ-functional peak at ω = (∆2 + (ξk/Z)2)1/2, where
+ξk = vF (k − kF ) is a fermionic dispersion (vF is a
+Fermi velocity), ∆ is a superconducting gap, and Z is
+an inverse quasiparticle residue.
+A δ-functional peak
+FIG. 1.
+Three possible forms of the electronic spectral
+function
+A(k, ω)
+at
+T
+=
+0
+in
+a
+quantum
+critical
+superconductor at a small but finite k − kF and in the
+absence of impurity broadening.
+(a):
+A(k, ω) vanishes at
+|ω| = ∆ and has a well-defined peak at ω > ∆, (b): A(k, ω)
+diverges at |ω| = ∆, but it non-monotonic at larger ω. The
+peak in A(k, ω) at |ω| > ∆ broadens, but still exists. (c):
+A(k, ω) diverges at |ω| = ∆, and monotonically decreases at
+larger ω. In case (a) fermions can be viewed as well-defined
+quasiparticles, in case (c) the quasiparticle picture completely
+breaks down. The case (b) is the intermediate one between
+(a) and (c).
+holds for momenta near the Fermi surface, as long as
+ω < ∆ + ω0, where ω0 is a mass of a pairing boson
+in energy units. At larger ω, fermionic damping kicks
+in, and the peak broadens.
+The same physics leads
+to peak-dip-hump behavior of A(k, ω) as a function of
+ω, observed most spectacularly in near-optimally doped
+cuprate Bi2Sr2CaCu2O8+δ (see, e.g, Refs. [54, 55]). At
+a QCP, the pairing boson becomes massless and ω0
+vanishes. This creates a singular behavior near the gap
+edge at ω = ∆, which holds even when ξk is finite.
+A simple experimentation shows that there are three
+possible forms of A(k, ω), which we present in Fig. 1:
+it (i) either vanishes at ω = ∆ and has a well-defined
+peak at ω > ∆ whose width at small ξk is parametrically
+smaller than its energy; or (ii) diverges at ω = ∆,
+arXiv:2301.13679v1  [cond-mat.supr-con]  31 Jan 2023
+
+Three possible forms of A( k,w) in quantum-critical superconductor
+(a) Quasiparticle picture
+(b) Partial breakdown
+(c) Complete breakdown
+of quasiparticle
+of quasiparticle
+(k,w)
+A
+-△
+0
+-△
+0
+-△
+0
+3
+3
+32
+0
+0.5
+1
+2
+.
+0
+0.5
+1
+1.5
+2
+Leading exponent
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+Subleading exponent
+8
+c
+0
+0.2
+0.4
+0.6
+0.8
+!=7g
+0
+2
+4
+6
+8
+10
+D(!)
+. = 0:8; 8 ' 1:18
+gap edge
+0
+0.1
+0.2
+(" ! !)8
+0
+0.1
+0.2
+D(!) ! 1
+-10
+-5
+0
+5
+10
+!=7g
+0
+1
+2
+3
+4
+5
+DoS
+10!2
+10!1
+100
+(! ! ")=7g
+100
+101
+9 1=x0:5
+9 1=x0:59
+. = 0:35
+. = 0:8
+(a)
+(b)
+(c)
+FIG. 2.
+(a) Exponents ν and c for the leading and the subleading terms in the expansion D(ω) ≃ 1+α(∆−ω)ν +β(∆−ω)ν+c,
+where D(ω) = ∆(ω)/ω and the gap edge ∆ is the solution of D(ω = ∆) = 1. (b) Numerical result for D(ω) for γ = 0.8. Inset
+shows the power-law behavior near the gap edge with ν = 1.18, consistent with (a). (c) Fermionic DoS at T = 0 for γ = 0.35
+(thick green line) and γ = 0.8 (thin pink line). In both cases, the DoS vanishes below the gap edge ∆ and has a power-law
+singularity above it N(ω) ∝ 1/(ω − ∆)ν/2, but the exponent ν is different in the two cases, as we show in the right panel.
+but is non-monotonic at larger ω and displays a broad
+maximum at some ω > ∆, or (iii) diverges at ω = ∆ and
+monotonically decreases at larger ω.
+In the first case,
+fermions in a quantum-critical superconductor can be
+viewed as well-defined quasiparticles; in the last case the
+quasiparticle picture completely breaks down; the second
+case is the intermediate one between the other two. Our
+goal is to understand under what circumstances A(k, ω)
+of a quantum-critical superconductor has one of these
+forms.
+Model.
+For our study, we consider dispersion-
+full fermions, Yukawa-coupled to a massless boson. We
+assume, like in earlier works (see, e.g., Refs. [56]), that
+a boson is Landau overdamped, and its effective velocity
+is far smaller than vF . In this situation, the interaction
+that gives rise to non-Fermi liquid in the normal state
+and to superconductivity, is a purely dynamical V (Ω).
+The fermionic self-energy and the pairing gap, tuned into
+a proper spatial pairing channel, are then determined
+by two coupled equations in the frequency domain. At
+a QCP, V (Ω) is singular at vanishing Ω in spatial
+dimension D ≤ 3, and behaves as V (Ω) ∝ (¯g/Ω)γ,
+where ¯g is the effective fermion-boson coupling, and the
+exponent γ is determined by the underlying microscopic
+model.
+The most studied models of this kind are of
+fermions near an Ising-nematic or Ising/ferromagnetic
+QCP (γ = 1/3) and near an antiferromagnetic or charge
+density wave QCP (γ
+= 1/2).
+The same effective
+interaction emerges for dispersion-less fermions in a
+quantum dot coupled to Einstein bosons (the Yuakawa-
+SYK model) [57–60]. For this last case, the exponent γ is
+a continuous variable γ ∈ (0, 1), depending on the ratio of
+fermion and boson flavors. An extension of the Yukawa-
+SYK model to γ ∈ (1, 2) has recently been proposed [61].
+We follow these works and consider γ as a continuous
+variable. We note that the value of γ is generally larger
+deep in a superconducting state because of feedback
+from superconductivity on the bosonic polarization. For
+simplicity, we neglect potential in-gap states associated
+with non-s-wave pairing symmetry and focus on the
+spectral function of fermions away from the nodal points
+and on features in the density of states (DoS) above the
+gap edge. An extension to models with in-gap states is
+straightforward.
+In previous studies of the γ-model, we focused on
+the novel superconducting behavior at γ > 1, when the
+pairing interaction is attractive on the Matsubara axis,
+while on the real axis ReV (Ω) is repulsive [62, 63]. We
+argued that this dichotomy gives rise to phase slips of
+the gap function on the real axis.
+Here, we restrict
+ourselves to γ ≤ 1, when this physics is not present and,
+hence, does not interfere with the analysis of the validity
+of a quasiparticle description in a superconducting state.
+Pairing gap and quasiparticle residue.
+For
+superconductivity mediated by a dynamical interaction,
+the paring gap ∆(ω) and the inverse quasiparticle
+residue Z(ω) are functions of the running real fermionic
+frequency ω. We define the gap edge ∆ (often called the
+gap) from the condition ∆(ω) = ω at ω = ∆.
+For our purposes, it is convenient to introduce D(ω) =
+∆(ω)/ω. The gap edge is at |D| = 1. The equation for
+D(ω) that we need to solve is
+ωB(ω)D(ω) = A(ω) + C(ω),
+(1)
+where B(ω) and A(ω) are regular functions of ω (see
+[64, 65]). The C(ω) term depends on the running D(ω),
+C(ω) = ¯gγ sin πγ
+2
+� ω
+0
+dΩ
+Ωγ
+D(ω − Ω) − D(ω)
+�
+D2(ω − Ω) − 1
+.
+(2)
+Its presence makes Eq. (S3) an integral equation. The
+inverse residue Z(ω) is expressed via D(ω′) as
+Z(ω) = B(ω) + ¯gγ sin πγ
+2
+ω
+� ω
+0
+dΩ
+Ωγ
+1
+�
+D2(ω − Ω) − 1
+(3)
+
+3
+FIG. 3.
+Spectral function A(k, ω) at T = 0 for four representative γ. The broadening in the plots is intrinsic. (a-d): color-
+coded plot at negative ω, as measured by the ARPES intensity at T = 0. (e-f): constant-k cuts of A(k, ω) at ξk = 0 and at
+ξk = ±4¯g. For γ < 1/2, the spectral function has a sharp quasiparticle peak at ω + ∆ ∝ ξ2
+k. For γ > 1/2, the peak moves to
+ω + ∆ ∝ |ξk|1/(1−γ) and broadens up, which eventually disappears (see text).
+and is readily obtained once D(ω) is known.
+At γ
+= 0, which models a BCS superconductor,
+C(ω) = 0 and D(ω) = A(ω)/(ωB(ω)) is a regular
+function of frequency.
+Near the gap edge at ω > 0,
+D(ω)−1 ∼ ω−∆ and Z(ω) ≈ Z(∆) ≡ Z. We assume and
+then verify that D(ω) remains regular in some range of
+γ > 0. Substituting D(ω)−1 ∼ ω−∆ into (S7) for γ > 0,
+we obtain C(ω)−C(∆) ∼ (ω−∆)3/2−γ. We see that C(ω)
+is non-analytic near the gap edge, but for γ < 1/2, the
+exponent 3/2−γ is larger than one. In this situation, the
+non-analytic term in C(ω) generates a non-analytic term
+in D(ω) of order (ω − ∆)3/2−γ, which is smaller than the
+regular ω−∆ term. Evaluating the prefactors, we obtain
+slightly above the gap edge, at ω = ∆ + δ
+D′(∆ + δ) = 1 + αδ + A cos[π(3/2 − γ)]δ3/2−γ,
+D′′(∆ + δ) = −A sin[π(3/2 − γ)]δ3/2−γ,
+(4)
+where α ∼ 1/¯g, A = � α
+2
+¯gγ sin(πγ/2)
+∆B(∆)
+J(γ, 1) and J(γ, ν) is
+expressed via Beta functions:
+J(γ, ν) = B(1 − γ, γ − 1 − ν
+2) − B(1 − γ, γ − 1 + ν
+2). (5)
+For γ > 1/2, 3/2 − γ > 1, and the calculation of D(ω)
+has to be done differently. We find after straightforward
+analysis that the leading δ-dependent term in D(∆ + δ)
+is non-analytic and of order δν, where ν is the solution
+of J(γ, ν) = 0. The exponent ν ≈ 1 + 0.67(γ − 1/2) for
+γ ≈ 1/2 and ν ≈ 1.3 for γ = 1. The subleading term in
+D(∆ + δ) scales as δν+c, where c > 0 is approximately
+linear in γ − 1/2. In Fig. 2, we plot ν(γ) and c(γ) along
+with the numerical results of D(ω) for a representative
+γ = 0.8. The exponent ν extracted from this numerical
+D(ω) is 1.18, which matches perfectly with the analytical
+result. The behavior at γ = 1/2 is special, and we discuss
+it in Ref. [64].
+Substituting D(∆ + δ) into the formula for Z(ω), Eq.
+(S8), we obtain
+Z′(∆ + δ)=Z(∆)+B cos(π(γ + ν/2 − 1))δ1−γ−ν/2,(6)
+Z′′(∆ + δ)=B sin(π(γ + ν/2 − 1))δ1−γ−ν/2.
+(7)
+where B =
+¯gγ sin πγ
+2
+∆
+√
+2α B(1 − γ, ν
+2 + γ − 1). For γ < 1/2,
+Z(ω) = Z(∆) + O(δ1/2−γ) is approximately a constant
+near the gap edge.
+For γ > 1/2, the inverse residue
+diverges at the gap edge, indicating a qualitative change
+in the system behavior.
+Spectral function and DoS.
+The spectral function
+and the DoS per unit volume are given by
+A(k, ω) = − 1
+π ImGR(k, ω),
+N(ω) = 1
+V
+�
+k
+A(k, ω) = NF ωIm
+�
+1
+∆2(ω) − ω2 , (8)
+where
+the
+retarded
+Green’s
+function
+GR(k, ω)
+=
+−(ωZ(ω) + ξk)/(ξ2
+k + (∆2(ω) − ω2)Z2(ω)).
+ARPES
+intensity is proportional to A(k, ω)nF (ω), which at T = 0
+selects negative ω.
+At γ = 0 (BCS limit), N(ω) ∼
+1/(ω−∆)1/2, and the spectral function has a δ-functional
+peak at ω = (∆2 + (ξk/Z)2)1/2. In Fig. 2 (c,d), we show
+the DoS N(ω), obtained from the numerical solution of
+the full gap equation (S3) for representative γ = 0.35
+and 0.8. We see that in both cases the DoS describes a
+gapped continuum, but there is a qualitative difference in
+the behavior near the gap edge: for γ = 0.35, N(ω) has
+the same 1/δ1/2 singularity as for γ = 0, and for γ = 0.8
+
+(e) = 0.35
+(f) = 0.45
+1
+1
+A(k,w)
+Sk = 0
+0>
+......
+0<
+0.5
+0.5
+0
+0
+-10
+6-
+-8
+-7
+-6
+-5
+-4.5
+-4
+-3.5
+-3
+-2.5
+-2
+w/g
+w/g(g) = 0.65
+(h) = 0.8
+0.4
+0.4
+0.3
+0.3
+0.2
+0.2
+0.1
+0.1
+0
+0
+-2
+-1.5
+-1
+-0.5
+-1.5
+-1
+-0.5
+w/g
+w/g(a) = 0.35
+(b) = 0.45
+-2
+-1
+-4
+-2
+19
+-6
+3
+-8
+-3
+-10
+-4
+-12
+-10
+-5
+0
+5
+10
+-10
+-5
+0
+5
+10
+Sk/g
+Sk/g
+1.5.
+1.5.(c) = 0.65
+(d) = 0.8
+-0.5
+-0.5
+-1
+-1
+-1.5
+-1.5
+-2
+-10
+-5
+0
+5
+10
+-10
+-5
+0
+5
+10
+Sk/g
+Sk/g
+0.5
+0.5Max
+00.
+0.0.
+0.4
+the DOS behaves as 1/δ0.59, which perfectly matches the
+analytical form δ−ν/2, given that ν = 1.18 for γ = 0.8.
+The spectral function A(k, ω) is shown in Fig.
+(3).
+For comparison with ARPES, we set ω to be negative:
+ω = −(∆ + δ).
+For any γ, there is no frequency
+range, where A(k, ω) is a δ-function, simply because
+the bosonic mass vanishes at a QCP. Still, for γ < 1/2,
+D(−(∆+δ))−1 ∝ δ and Z(−(∆+δ)) ≈ Z(−∆) = Z(∆).
+In this situation, the spectral weight on the Fermi
+surface, integrated over an infinitesimally small range
+around ω
+=
+−∆ immediately above the real axis,
+is finite, like in BCS case.
+Away from the Fermi
+surface, the spectral function vanishes as |ω + ∆|1/2−γ
+at the gap edge and displays a quasiparticle peak at
+ω ≈ −(∆2 + (ξk/Z(∆))2)1/2. The peak is well defined
+at small δ as its width O(δ1/2−γ) is parametrically
+smaller than its frequency.
+This is the same behavior
+as in Fig. 1 (a).
+For γ
+>
+1/2, the situation is
+qualitatively different.
+Now Z(−∆ − δ) diverges at
+δ → 0 and D(−∆ − δ) − 1 ∼ |δ|ν ≪ |δ|.
+In this
+case, the integral of A(kF , ω) over an infinitesimally
+small range around ω = −∆ vanishes, which can be
+interpreted as a vanishing of a quasiparticle peak.
+At
+finite ξk, the spectral function diverges at the gap edge
+as 1/|ω + ∆|γ/2+γ−1. For γ slightly above 1/2, A(k, ω)
+is non-monotonic and possess a broad maximum at
+|ω + ∆| ∼ (ξk/¯gγ)
+1
+1−γ .
+This is the same behavior as
+in Fig. 1 (b).
+For larger γ, the maximum disappears,
+and A(k, ω) monotonically decreases at |ω| > ∆. This
+is the same behavior as in Fig. 1 (c).
+For small ξk,
+the maximum disappears at γ ∼ 0.9.
+For larger ξk,
+it disappears at smaller γ, first for positive ξk (see Fig. 4).
+Comparison with ARPES
+The behavior shown
+in Fig. 4 is our result in some range of γ > 1/2. For
+positive ξk (i.e., outside the Fermi surface), the spectral
+function has a single non-dispersing maximum at the
+gap edge, except for the smallest ξk, while for negative
+ξk, A(k, ω) has a kink at the gap edge ω = −∆ and
+a dispersing maximum at ω = −∆ − O
+�
+|ξk|1/(1−γ)�
+.
+This behavior is consistent with the ARPES data for
+Bi2201, Ref. [66].
+The data shows that the spectral
+function near the antinode, where our analysis is valid,
+displays an almost non-dispersing maximum at positive
+ξk, while for negative ξk it displays a non-dispersing
+kink at the same energy and a dispersing maximum
+at larger |ω|.
+We associate the non-dispersing feature
+at both positive and negative ξk with the gap edge
+∆, and associate the dispersing maximum, observed
+in [66] at ξk < 0, with the dispersing maximum in Fig. 4.
+Discussion and summary.
+In this work, we
+analyzed the applicability of quasiparticle description of
+a superconducting state which emerges out of a non-
+Fermi liquid at a metallic QCP. We considered the
+model with an effective dynamical 4-fermion interaction
+FIG. 4. (a) Spectral function A(k, ω) at positive and negative
+ξk = ±4¯g at γ = 0.6. To account for impurity scattering,
+we convoluted the spectral function with a Lorentzian of
+width ∼ 0.03¯g.
+(b) Spectral function at a set of discrete
+momenta. It displays a non-dispersing gap edge singularity
+(green dots) and a dispersing maximum (blue circles). This
+theoretical A(k, ω) is consistent with the ARPES data for
+Bi2201, Ref. [66] (see text).
+V (Ω) ∝ 1/Ωγ, mediated by a gapless boson at a QCP
+and analyzed the spectral function and the DoS for
+γ ∈ (0, 1). Interaction V (Ω) gives rise to a non-Fermi
+liquid in the normal state with self-energy Σ(ω) ∝ ω1−γ
+and to pairing below some finite Tc. A superconducting
+order gaps out low-energy excitations and, at a first
+glance, should restore fermionic coherence.
+We found,
+however, that this holds only for γ < 1/2. For larger γ
+the spectral function and the DoS exhibit qualitatively
+different behavior than that in a superconductor with
+coherent quasiparticles.
+(different power-laws).
+We
+argued that the quasiparticle peak broadens up and
+completely disappears for γ close to one.
+Away from a QCP, a pairing boson is massive and
+at the lowest energies a Fermi-liquid description holds
+already in the normal state and continue to hold in
+a superconductor.
+In particular, in the immediate
+vicinity of the gap edge, the system displays a BCS-
+like behavior for all γ. Still, the system behavior over
+a broad frequency range is governed by the physics at
+a QCP, as numerous experiments on the cuprates and
+other correlated systems indicate. We argued that our
+results are quite consistent with the ARPES data for
+Bi2201 [11, 66].
+We acknowledge with thanks useful conversations with
+a number of our colleagues. This work was supported by
+the U.S. Department of Energy, Office of Science, Basic
+Energy Sciences, under Award No. DE-SC0014402.
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+
+7
+Supplementary information for “Density of states and spectral function of a
+superconductor out of a quantum-critical metal”
+by Shang-Shun Zhang and Andrey V Chubukov
+GAP EQUATION ALONG THE REAL-FREQUENCY AXIS AND ITS SOLUTION
+We will use the approach pioneered by Marsiglio, Shossmann, and Carbotte [1]. In this approach, one first solves
+non-linear gap equation along the Matsubara axis, which can be done rather straightforwardly as the gap function
+∆(ωm) can be chosen to be real for all frequencies and is a regular function of ωm even when the pairing boson is
+massless. One then uses this ∆(ωm) as an input for the equation for complex ∆(ω) along the real frequency axis.
+The non-linear integral equation for D(ωm) = ωm∆(ωm) on the Matsubara axis, and the equation for the inverse
+quasiparticle residue Z(ωm) = 1 + Σ(ωm)/ωm (Σ(ωm) is the fermionic self-energy), have the form
+ωmD(ωm) = πT
+�
+ω′m
+(D(ω′
+m) − D(ωm)) sgn(ω′
+m)
+�
+1 + D2(ω′m)
+V (ω − ω′
+m),
+(S1)
+Z(ωm) = 1 + 1
+ωm
+πT
+�
+ω′m
+sgn(ω′
+m)
+�
+1 + D2(ω′m)
+V (ω − ω′
+m),
+(S2)
+where V (Ωm) = (¯g/|Ωm|)γ is the same as in the main text (¯g is an effective fermion-boson coupling, and γ depends
+on the underlying microscopic model).This set of equations has a non-zero solution D(ωm) below a finite pairing
+temperature Tp ∼ ¯g. Fig. S1 shows the numerical solution for ∆(ωm) at T = 10−6¯g ≪ Tp, for different γ. We see
+from the figure that ∆(ωm) approaches a constant value at small frequencies and decays as ω−γ
+m
+at high frequencies.
+This behavior holds for all γ and can be easily verified analytically.
+The gap equation along the real frequency axis is
+ωB(ω)D(ω) = A(ω) + C(ω),
+(S3)
+(Eqn. (1) in the main text). This equation is obtained by using the spectral representation of an analytic function
+on the upper frequency half-plane
+f(iωm) = 1
+π
+�
+dx Imf(x)
+x − iωm
+(S4)
+and, where possible, keeping D(ωm) as an input function. This approach was pioneered for electron-phonon interaction
+in Refs. [3–6] for the electron-phonon problem.
+FIG. S1. Numerical resulls for the gap function ∆(ωm) along the Matsubara axis. The calculation is performed at temperature
+T = 10−6¯g using the hybrid-frequency method [2].
+
+10
+10°
+6/(um)V
+1.0
+0.
+10
+102
+100
+104
+wm
+/g8
+20
+40
+60
+80
+100 120
+0
+0.5
+1
+1.5
+D(!)
+(a)
+. = 0:25
+#5
+D0
+D00
+20
+40
+60
+80
+100 120
+!=7g
+0
+0.5
+1
+1.5
+2
+Z(!)
+(b)
+. = 0:25
+Z0
+Z00
+1
+2
+3
+0
+0.5
+1
+1.5
+(c)
+. = 0:5
+1
+2
+3
+!=7g
+0
+2
+4
+6
+(d)
+. = 0:5
+0
+1
+2
+3
+4
+5
+0
+0.5
+1
+1.5
+(e)
+. = 0:8
+0
+1
+2
+3
+4
+5
+!=7g
+0
+2
+4
+6
+8
+10
+(f)
+. = 0:8
+FIG. S2. Numerical results for D(ω) and Z(ω) for γ = 0.25, γ = 0.5 and γ = 0.8.
+For our case, the functions A(ω) and B(ω) are directly expressed via D(ωm) along the Matsubara axis as
+A(ω) = 1
+2
+� ∞
+0
+dωm
+D(ωm)
+�
+1 + D2(ωm)
+×
+�
+¯gγ
+(ωm + iω)γ +
+¯gγ
+(ωm − iω)γ
+�
+,
+(S5)
+B(ω) = 1 + i
+2ω
+� ∞
+0
+dωm
+1
+�
+1 + D2(ωm)
+×
+�
+¯gγ
+(ωm + iω)γ −
+¯gγ
+(ωm − iω)γ
+�
+.
+(S6)
+and C(ω) is given by
+C(ω) = ¯gγ sin πγ
+2
+� ω
+0
+dΩ
+Ωγ
+D(ω − Ω) − D(ω)
+�
+D2(ω − Ω) − 1
+,
+(S7)
+(Eqn (3) in the main text). This function depends on the running D(ω − Ω), which makes Eq. (S3) an integral
+equation. The inverse residue Z(ω) is expressed via D(ω′) as
+Z(ω) = B(ω) + ¯gγ sin πγ
+2
+ω
+� ω
+0
+dΩ
+Ωγ
+1
+�
+D2(ω − Ω) − 1
+(S8)
+(Eqn (4) in the main text) and is readily obtained once D(ω) is known.
+The gap equation along the real-frequency axis has an iterative structure in the sense that D(ω) depends on D(ω′)
+at ω′ < ω. This allows us to solve this equation iteratively, using the low-frequency form D(ω) ≃ ∆(0)/ω as an input,
+with ∆(0) ≡ ∆(ωm = πT). In Fig. S2 we show the results for D(ω) and Z(ω) for three representative values of γ. In
+all cases, D(ω) and Z(ω) are real below the gap edge ω = ∆ and are complex above the gap edge, where ∆ is defined
+as ∆(ω) = 1 at ω = ∆. We see that for γ > 1/2, Z(ω) diverges at the gap edge. We use this fact in the main text in
+the analysis of the spectral function.
+
+9
+FIG. S3. The real and imaginary parts of the gap function near the gap edge ω = ∆ for γ = 1/2, obtained by solving the
+non-linear gap equation numerically. The leading term in D′(∆ − δ) − 1, shown in the inset of (a), is linear in δ∆ − ω, and the
+subleading scales as δ/| log |δ||, as is confirmed by the linear relation in panel (a), The imaginary part D′′ appears at negative
+δ above the gap edge. The numerical result in panel (b) clearly shows the scaling relation D′′ ∼ δ/ log2 (|δ|/¯g), expected from
+the Kramers-Kronig relation with D′.
+THE CASE OF γ = 1/2
+In the main text we argued that for γ < 1/2, the function D(∆ − δ) − 1 ∝ δ, where δ = ∆ − ω, and the correction
+scales as δ3/2−γ. More specifically, we found iteratively that
+D(∆ − δ) = 1 + δ
+∞
+�
+n=0
+αnδnϵ
+(S9)
+where ϵ = 1/2−γ and α0 = O(1/¯g). The expression for α1 is presented in the main text, after Eq. (4). It is proportional
+to J(γ, 1) = B(1 − γ, γ − 3/2) − B(1 − γ, γ − 1/2), where B(a, b) is a Beta function (B(a, b) = Γ(a)Γ(b)/Γ(a + b)).
+For small ϵ (i.e., for γ ≤ 1/2), α1 ∼ J(γ, 1) ∼ 1/ϵ. For the next term in (S9) we find α2 ∼ 1/ϵ2, and so on.
+We see that the perturbative expansion in δϵ in (S9) holds for (δ/¯g)ϵ/ϵ ≤ 1. Outside this range, all terms in Eq.
+(S9) are relevant. As γ approaches 1/2 from below and ϵ decreases, the perturbative regime shrinks to exponentially
+small δ < ¯g exp(−| log ϵ|/ϵ).
+To understand the form of D(ω) outside the perturbative regime, we express (δ/¯g)ϵ as eϵ log (δ/¯g) and expand (S10)
+in powers of log (δ/¯g). We obtain
+D(∆ − δ) = 1 + δ
+∞
+�
+n=0
+˜αn(log δ
+¯g )n
+(S10)
+where ˜α0 = α0 + α1 + α2 + .., ˜α1 = ϵα1 + 2ϵα2 + .., ˜α2 = 2ϵ2α2 + .... We see that each ˜αn is a series, in which the
+first term is independent on ϵ, and the others diverge as powers of 1/ϵ, because α1 ∼ 1/ϵ, α2 ∼ 1/ϵ2, and so on.
+We now argue that singular parts of ˜αn can be neglected. The argument is two-fold. First, in the calculations, the
+1/ϵ divergencies originate from the divergence of J(1/2 − ϵ, 1) ≈ 1/(2ϵ). This divergence is regularized by a finite
+boson mass, such that strictly at ϵ = 0, one has J(1/2, 1) = 0 instead of infinity. Second, if we assume that ˜α0
+in (S10) remains finite at ϵ = 0 and substitute the trial D(∆ − δ) = 1 + δ˜α0 in the gap equation at γ = 1/2 and
+compute iteratively the next term in D(∆−δ), we find it in the form ˜α1δ log (δ/¯g) with a finite ˜α1 = √¯g˜α0/(4∆B(∆)).
+Extending the iterative analysis, we find that all ˜αn are finite at γ = 1/2 (i.e., ϵ = 0), as we anticipated.
+We didn’t manage to sum up analytically the logarithmic series in (S10). The numerical solution for D(∆ − δ) for
+γ = 1/2 shows that D(∆ − δ) − 1 remains linear in δ (see Fig. S2 (c)), and the corrections scale as 1/| log (|δ|/¯g)|
+(see Fig. S3 (a)). By Kramers-Kronig relation, this implies that at negative δ, when ω > ∆ is above the gap edge,
+the imaginary part of D(ω) scales as D
+′′(∆ + |δ|) ∝ δ/ log2 (|δ|/¯g) (the same form is obtained by just noticing that
+log(−|δ|) = log(−(ω − ∆ + i0)) = log |δ| − iπ). This form of D
+′′(∆ + |δ|) is consistent with our numerical solution
+above the gap edge, Fig. S3 (b). The solution clearly shows that the ratio D
+′′(∆ + |δ|)/δ decreases at the smallest δ.
+We next use the result for D(ω) to obtain the inverse quasi-particle residue near the gap edge.
+Substituting
+D(∆ − δ) ≈ 1 + ˜α0δ into (S8), we obtain at γ = 1/2
+Z(∆ − δ) =
+1
+2∆
+� ¯g
+˜α0
+| log δ|.
+(S11)
+
+4
+3.5
+1
+1.2
+1.4
+1.6
+1.
+I log(8)|12
+000
+11.5
+GO
+8
+5
+10
+1og2 [8]15(a)
+5.5
+×10-4
+gap
+edge
+1
+5
+0
+1.843
+1.8435
+/
+3
+1
+4.5
+G0000(b) 13.5
+13
+12.510
+FIG. S4. (a) The spectral function A(k, ω) for γ = 1/2. (b) Constant ξk cuts along the blue lines in panel (a).
+Analytically continuing this function to negative δ, i.e., to ω above the threshold, we obtain
+Z(∆ + |δ|) =
+1
+2∆
+� ¯g
+˜α0
+(| log |δ|| + iπ) ,
+(S12)
+Note that the imaginary part of Z(ω) jumps to a finite value at ω infinitesimally above the threshold. This behavior
+is consistent with the numerical solution for Z(ω), see Fig. S2 (d).
+Finally, we use the results for D(ω) and Z(ω) and compute the spectral function near the gap edge. On the Fermi
+surface, the spectral function at negative ω and |ω| > ∆ takes the form
+A(kF , ω) ∝
+1
+|ω + ∆| log(¯g/|ω + ∆|).
+(S13)
+Slightly away from the Fermi surface, the spectral function has a peak at |ω| = ∆+δk where δk ∼ (ξ2
+k/¯g)/ log2(|¯g/ξk|).
+The peak width scales as δk/ log(|¯g/ξk|) and is logarithmically smaller than the energy variation |ω| − ∆. Also, for
+any non-zero ξk, the spectral function jumps at the gap edge to a finite value of order 1/ξ2
+k. In Fig. S4 we show the
+numerical result for the spectral function. It is consistent with the behavior we just described.
+UNIVERSAL FORM OF THE SPECTRAL FUNCTION AT 1/2 < γ < 1
+For frequencies ω near the gap edge and for momenta near the Fermi surface, when ξk is much smaller than |ωZ(ω)|,
+a straightforward calculation shows that for γ > 1/2, the spectral function can be expressed as a scaling function of
+ξk/|ω + ∆|1−γ¯gγ (we set ω < 0). Namely,
+A(k, ω) ∝
+1
+|ω + ∆|
+ν
+2 +1−γ Φ
+�
+ξk
+|ω + ∆|1−γ¯gγ
+�
+,
+(S14)
+where
+Φ(x) ≡
+x2 + Q2
+γ sin[π(ν − c)]/ sin(πc)
+�
+x2 + Q2γ cos(2πγ)
+�2 + Q4γ sin2(2πγ)
+(S15)
+with Qγ = sin(πγ/2)B(1 − γ, ν/2 + γ − 1). In Fig. S5, we plot the dimensionless function Φ(x) for different values of
+γ. At x ≫ 1, Φ(x) ∼ 1/x2; at x ≪ 1, Φ(x) ∼ const. For γ < γc ≃ 0.9, function Φ(x) contains a local maximum at
+x2
+∗ ∼
+�
+(u − v)2 + w2 − u,
+(S16)
+where u = Q2
+γ sin[π(ν − c)]/ sin(πc), v = Q2
+γ cos(2πγ), and w = Q2
+γ sin(2πγ). This maximum can be interpreted as an
+over-damped, but still existing quasi-particle peak. At γ > γc, the function Φ(x) monotonically decreases with x. In
+this case, the quasiparticle description breaks down completely.
+
+-2.5
+-3
+-3.5
+-10
+-5
+0
+5
+10
+Sk/g2
+1
+0
+-2.8
+-2.6
+-2.4
+-2.2
+-2
+-1.8
+-1.6
+w/gMin(a)
+0
+-0.5
+-1
+-1.5
+19
+3 
+-2(b)
+6
+5
+4
+ntensity
+3Max11
+100
+101
+102
+x
+10!4
+10!3
+10!2
+10!1
+)(x)
+0:5
+.
+1:0
+. = .c ' 0:9
+FIG. S5. The function Φ(x), Eqn (S15), for different values of γ.
+We emphasize that this behavior holds only for small enough ξk. For larger ξk, the spectral function does depend
+on the sign of ξk and as γ increases, the quasiparticle behavior gets completely destroyed first for positive ξk and
+then, at larger γ, for negative ξk.
+[1] F. Marsiglio, M. Schossmann, and J. P. Carbotte, Phys. Rev. B 37, 4965 (1988).
+[2] Y.-M. Wu, A. Abanov, Y. Wang, and A. V. Chubukov, Phys. Rev. B 102, 024525 (2020).
+[3] F. Marsiglio and J. P. Carbotte, Phys. Rev. B 43, 5355 (1991), for more recent results see F. Marsiglio and J.P. Carbotte,
+“Electron-Phonon Superconductivity”, in “The Physics of Conventional and Unconventional Superconductors”, Bennemann
+and Ketterson eds., Springer-Verlag, (2006) and references therein; F. Marsiglio, Annals of Physics 417, 168102-1-23 (2020).
+[4] A. Karakozov, E. Maksimov, and A. Mikhailovsky, Solid State Communications 79, 329 (1991).
+[5] R. Combescot, Phys. Rev. B 51, 11625 (1995).
+[6] Y.-M. Wu, A. Abanov, Y. Wang, and A. V. Chubukov, Phys. Rev. B 99, 144512 (2019).
+

VtE1T4oBgHgl3EQfIwPB/content/tmp_files/2301.02944v1.pdf.txt

@@ -0,0 +1,1226 @@
+Quantum Honest Byzantine Agreement as a
+Distributed Quantum Algorithm
+Marcus Edwards
+July 23, 2020
+1
+arXiv:2301.02944v1  [quant-ph]  7 Jan 2023
+
+Contents
+1
+Quantum Honest Byzantine Agreement
+3
+1.1
+Blockchain’s Resource Consumption and the Abundance of
+Quantum Resources
+. . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.2
+Coincidence-driven Consensus via Byzantine Agreement . . . . .
+5
+1.2.1
+Distribution of Correlated Lists . . . . . . . . . . . . . . .
+5
+1.2.2
+Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+1.3
+Casting to a Modular Quantum Neural Architecture . . . . . . .
+8
+1.3.1
+Restricted Boltzmann Machines . . . . . . . . . . . . . . .
+8
+1.3.2
+Associative Measuring Neurons . . . . . . . . . . . . . . .
+10
+1.3.3
+Training Via Quantum Binding Commitment . . . . . . .
+15
+1.3.4
+Commitment Schemes . . . . . . . . . . . . . . . . . . . .
+15
+1.3.5
+Collapsing Hash Functions
+. . . . . . . . . . . . . . . . .
+18
+1.3.6
+A New Type of Machine Learning Algorithm . . . . . . .
+19
+1.4
+Practical Implementation Limitations and Advantages . . . . . .
+20
+1.4.1
+Practical Constraints Today . . . . . . . . . . . . . . . . .
+20
+1.4.2
+Near-future Possibilities . . . . . . . . . . . . . . . . . . .
+27
+2
+References
+29
+2
+
+1
+Quantum Honest Byzantine Agreement
+Blockchain technology has three main components:
+the network, consensus
+algorithm and distributed data structure. Each of these brings with it particular
+issues of scalability and efficiency.
+By recasting the network and consensus
+algorithm components of blockchain to a quantum algorithm, we show that
+the efficiency and scalability of blockchain technology can be improved in the
+near-term without requiring powerful quantum computers to be available.
+1.1
+Blockchain’s Resource Consumption and the Abun-
+dance of Quantum Resources
+Crandall points out the abundance of resources in our world that could
+be harnessed for information processing tasks in his book “Nanotechnology:
+Molecular speculations on global abundance” [4].
+In light of the seemingly
+endless resources around us, it is difficult to stomach the irresponsible use of
+resources that we have encouraged through the design of our most prominent
+blockchain systems including Bitcoin.
+There are critical issues with the scaling properties and efficiency of these
+blockchain technologies which require solutions if any significant distributed
+ledgers are going to be possible to sustainably implement. The scaling properties
+of the immutable distributed data structures used in blockchain networks have
+been shown to cause demands on memory that are hard to justify. Blockchains
+that are based on proof-of-work consensus schemes like Bitcoin also encourage
+massively wasteful compute resource consumption.
+Competition in Bitcoin’s
+compute-intensive scheme coupled with the limitations of the blockchain data
+structure implementation by Bitcoin also causes issues with throughput of the
+system as a practical trading platform. The amount of transactions that can be
+processed by Bitcoin is less than seven per second. This is far from the reported
+3
+
+47,000 per second achieved by VISA [5]. These issues have motivated some push
+back against the spread of blockchain. China is seeking to stop Bitcoin mining
+in the country, for example [6].
+From a business perspective, blockchain technology is not expected to be
+viable for full adoption and practical use by mainstream banks for around
+another ten years [7]. Even so, banks are beginning to implement prototypes
+and blockchain applications of limited scale now. An IBM survey of 200 global
+banks [8] showed that 65% of these banks intended to roll out blockchain-based
+products between 2016 and 2019.
+The majority of blockchain applications that are being developed do not have
+solutions to the scalability and efficiency issues of their underlying cybersecurity
+schemes. They are also not prepared to face the challenges of attackers equipped
+with the quantum computers we expect to see developed within the next ten
+years.
+In the meantime, evidence is mounting that we will be capable of performing
+complex computing tasks using the world’s smallest resources within the next
+decade.
+Subatomic particles with properties that allow for us to use them
+in quantum computing applications include the spin- 1
+2 particles:
+electrons,
+protons, neutrons, neutrinos and quarks. These are especially convenient for
+use in computing applications because of how the limitation of 2 observables
+mimics the limitation of the 2 discrete levels used in digital logic and Boolean
+algebra.
+I explore the question of whether consensus networks like those in modern
+blockchains might be possible to improve upon using more efficient hybrid
+quantum-classical communication networks.
+4
+
+1.2
+Coincidence-driven Consensus via Byzantine Agree-
+ment
+Sun, Xin, et al. presented a voting protocol for blockchain in their 2019 paper
+[1].
+Their protocol makes use of a method for ensuring consensus between
+talliers from their previous work, called a Quantum Honest Success Byzantine
+Agreement Protocol (QHBA) [2]. This protocol is used in their voting scheme
+to identify dishonest ballot talliers.
+Definition 1 (Honest Success Byzantine Agreement Protocol (HBA))
+An honest success Byzantine agreement protocol involves n agents. One of the
+agents is the sender S, and holds an input value xs ∈ D, where D is a finite
+domain. A protocol achieves honest success Byzantine agreement if the protocol
+guarantees the following:
+1. If the sender is honest, then all honest agents agree on the same output
+value y = xs.
+2. If the sender is dishonest, then either all honest receivers abort the
+protocol, or all honest receivers decide on the same output value y ∈ D.
+The protocol is p-resilient if the protocol works when less than a fraction of p
+receivers are dishonest.
+The QHBA is
+m−2
+m -resilient.
+m is the number of receivers, and is more
+efficient than a classical HBA protocol when there are many dishonest receivers.
+1.2.1
+Distribution of Correlated Lists
+The first phase of the QHBA protocol is for correlated lists to be distributed
+among the agents using quantum secure direct communication.
+5
+
+Let the sender be S = P1. Each agent Pi ∈ {P n
+2 +1, ..., Pn} is tasked with
+distributing a list of numbers Li
+k to agent Pk ∈ {P1, ..., P n
+2 } such that:
+1. |Li
+k| = l ∀ k ∈ {1, ..., n/2}, where l is a multiple of 6.
+2. Li
+1 ∈ {0, 1, 2}l.
+l
+3 numbers on Li
+1 are 0.
+l
+3 are 1.
+l
+3 are 2.
+3. Li
+k ∈ {0, 1}l ∀ k ∈ {2, ..., n/2}
+4. ∀ j ∈ {1, ..., l}, if Li
+1[j] = 0, then Li
+2[j] = ... = Li
+n/2[j] = 0
+5. ∀ j ∈ {1, ..., l}, if Li
+1[j] = 1, then Li
+2[j] = ... = Li
+n/2[j] = 1
+6. ∀ j ∈ {1, ..., l}, if Li
+1[j] = 2, then ∀ k ∈ {2, ..., m} the probability that
+Li
+k[j] = 0 and that Li
+k[j] = 1 are equal.
+If the number of receivers that report non-compliant lists from a distributor
+passes a threshold, then that distributor is classified as dishonest.
+1.2.2
+Consensus
+Assuming h > n
+2 , the following procedure can be used to reach a consensus.
+First, the sender S sends a binary number B1k and a list of numbers ID1k
+to each receiver Pk. ID1k should indicate all the positions on L1 where B1k
+appears to Pk. An honest sender will send the same list to all receivers.
+Each Pk will compare the B1k and ID1k to their list Lk. If any honest Pk
+finds information that is not consistent, he/she sends ⊥ to the other receivers.
+Otherwise, he/she sends B1k and ID1k to the other receivers.
+After all these messages have been received, each honest Pk checks the
+following:
+1. If there were more than two agents who sent binary numbers and lists
+that were consistent with Lk but some had different binary numbers, Pk
+outputs ⊥.
+6
+
+2. If more than two agents sent the same binary numbers and lists which
+were consistent with Lk, these agents are considered to be honest. Pk
+outputs the binary number provided by these honest agents.
+3. If more than two agents sent the same binary numbers and lists which
+were consistent with Lk, any other agents are considered dishonest. If all
+of the dishonest agents sent ⊥ to Pk, then Pk sets vk to the binary value
+provided by the honest agents.
+4. In all other cases, Pk outputs ⊥.
+Consensus is achieved if at least n
+4 agents output the same bit value.
+Suppose Pj were a dishonest receiver, and j ≥ 2. Pj would want to send a
+binary number Bjk and list of numbers IDjk which was consistent with Lk.On
+Lj, there are
+l
+2 appearances of Bjk. On L1 there are only
+l
+3 appearances of
+Bjk. So, there are
+l
+6 positions of discord x, where L1[x] = 2. If Pj selects a
+discord position x then with probability 1
+2, Lk[x] ̸= Bjk. Pj has to avoid all
+discord positions in order to avoid being identified as dishonest. This has a
+( 2
+3)
+l
+3 probability of success which is very small when l is large. This is rationale
+behind the checks made by Pk listed above.
+In a nutshell, the receivers don’t know which instances of the false bit in
+their strings from the distributors are random, so they have to just echo what
+they were given by the sender. The final output is simply an agreed-upon bit.
+If the receivers should chose to transmit the false bit and say it should occur at
+a random index, they are recognized as dishonest when Bjk, IDjk are checked
+against the other receivers’ lists L.
+7
+
+1.3
+Casting to a Modular Quantum Neural Architecture
+I suggest that the QHBA protocol essentially reduces consensus to coincidence.
+The volume of coincidence is the input parameter which drives a receiver to echo
+its input. A lack of coincidence results in no useful output from a receiver, as
+per subsection 2.1.3. This is a similar mechanism to the learning mechanism in
+cognitive modular neural architectures like Haikonen’s architecture for artificial
+intelligence [3].
+1.3.1
+Restricted Boltzmann Machines
+Furthermore, the path of information between the participants in the QHBA
+protocol is suggestive of a fully-connected feed-forward neural network.
+The RBM-like connectivity graph of the Honest Success
+Byzantine Agreement
+At first glance, this appears to be a Restricted Boltzmann Machine (RBM).
+Restricted Boltzmann machines are an early machine learning neural network
+structure created by Geoffrey Hinton.
+A restricted Boltzmann machine is a
+shallow network with two layers. One is ”hidden”, one is ”visible”. The input
+layer’s nodes accept the system’s inputs, process these inputs, and pass their
+8
+
+outputs onto the nodes of the next layer. Each node is a McCulloch-Pitts neuron
+and has an activation function, typically of the following form.
+yj = T(
+�
+i
+wjixi + b)
+The input xi is multiplied by a weight wji and added to a bias b.
+The
+result is passed to an activation transfer function T, which produces a node’s
+output. This output is an amplification or suppression of the strength of the
+signal passing through it.
+In a fully-connected RBM, each node of the first layer outputs to each node of
+the second. Each combination of source and destination nodes ji has a different
+weight.
+The jth McCulloch-Pitts neuron in a network takes a set of weighted inputs
+xi and a bias signal, and outputs a signal yj which transforms its inputs
+according to a transfer function T.
+A simple instance of a McCulloch-Pitts neuron is the perceptron, which is
+a threshold binary classifier. Its output is always a 0 or 1. The perceptron will
+output a value of 1 if the following condition is satisfied.
+(
+�
+i
+wixi + b) > 0
+This type of neuron then plays the role of summarizing its inputs by putting
+them into a category. Artificial intelligence is most useful for the categorization
+of data which has very dense informational data points which share a structural
+pattern. A set of these neurons is typically arranged, and each neuron given the
+responsibility to categorize a different subset of a ”training data set”. A neural
+network’s output is often simply used to update a statistical model. This might
+mean updating a regression or more advanced statistical model of the training
+9
+
+data.
+As a thought experiment, let us consider the topology of the computer
+network implied by the QHBA protocol as a neural network. In this paradigm,
+the means of network communication between each protocol participant
+corresponds to a weighted I/O channel between neurons.
+The participants
+themselves correspond to neurons. The training data set is simply the sender’s
+list L1, and the output is the agreed-upon bit B1.
+1.3.2
+Associative Measuring Neurons
+If our network is to perform the QHBA protocol, the neurons must clearly be
+modified. Rather than the receiver neurons simply outputting a binary classifier
+when the perception threshold is reached, our version of honest receiver neurons
+must output its input, Bjk, IDjk when the amount of agreement or coincidence
+of its inputs’ values passes a threshold. This mechanism will actually suffice for
+the distribution neurons as well, with one small tweak. Rather than having a
+neuron simply output its input, we can have the neurons both measure and then
+output their inputs. This will not change the function of the reveiver neurons,
+but will allow the distributors to achieve the replacement of 2’s in the list L1
+provided to their inputs by the sender S with probabilistically distributed 1’s
+and 0’s.
+Definition 2 (Associative Measuring Neuron) An associative measuring
+neuron will conditionally propagate a quantum state from its quantum input
+to its output. Its output may be a classical channel or quantum channel that
+will accept only basis states. Let an associative measuring neuron k have the
+following output yk in terms of inputs x from n neurons, where |Xk > will
+be a superposition of basis vectors weighted appropriately to represent their
+multiplicities as inputs to the neuron, and the weight of |0⊗l > will represent the
+10
+
+neuron’s bias. Then the neuron achieves the non-linear effect of collapsing its
+output state to an that of an input which is sufficiently present so as to overcome
+the neuron’s bias.
+yk = M|Xk >
+(1)
+The behaviour of an associative measuring neuron is then to output the most
+recurring input with high likelihood, unless no input is repeated sufficiently
+enough, in which case the measurement will yield |0⊗l > with high probability.
+The input coincidence threshold is manifested by bk, which is the coefficient
+on the |0⊗l > basis state and should be trainable as well as the weights w. A
+simple way to achieve this functionality would be to implement an associative
+measuring neuron using a series of conditionally applied quantum gates.
+Associative Measuring Neuron Circuit
+In this circuit, D is the Grover Diffusion Operator.
+Grover Diffusion Operator
+Each Uki is a circuit composed of controlled Pauli-Z phase flip gates. Each
+of the Uki is a parameterized operator that takes a set of qubit indexes IDki, a
+boolean value Bki and a weight wki. All of these parameters are classical.
+11
+
+Uki
+Ukj
+Ukl
+Yk
+H
+D
+D
+DHH
+20><0-I
+H&If the boolean Bki is 1, then Uki is simply a multiply controlled CZ gate with
+all qubits in IDki included as controllers in x and targets in y. For example, if
+l = 6; a, c ∈ IDki and b, d, e, f, g /∈ IDki we would have the following.
+Parameterized Oracle Example with Bki = 1
+If the boolean Bki is 0, then a Pauli-X gate is first applied to the input xi.
+12
+
+Yka
+Z
+Ykb
+yke
+Z
+ykd
+ykg
+Lia
+Tib
+Lic
+Lid-
+Tie
+Ti
+LigYka
+Z
+ykb
+Yke
+Z
+Ykd
+ke
+Ykg
+Tia
+Tib-
+Tic
+Lid
+Lie-
+Lif
+TigParameterized Oracle Example with Bki = 0
+The weight wki dictates the number of times the operators Uki and D are
+repeated. The larger the weight with respect to other weights for other inputs
+to the same neuron, the more repeats. When there is only one weight and it
+is 1, then the operator should not be applied at all. Otherwise, Uki and D
+should be repeated a number of times proportional to the number of standard
+deviations Ni the weight wki is above the mean weight ¯
+wk. A negative N should
+be reflected as well, so we will always either subtract from or add to a default
+number of repetitions. This default will exactly be the bias bk and will be a
+learned parameter itself.
+Ni =
+wki − ¯
+wk
+�
+1
+l
+�l
+j=1(wkj − ¯
+wk)2
+(2)
+The number of repetitions total will be ⌊bk + Ni⌋.
+The function of the associative measuring neuron is comparable to a number
+of competing Grover searches [12] performed on the same quantum state. The
+algorithm is designed such that the effect of a search is proportional to the
+multiplicity of its corresponding input list Li
+k = xi, if that list matches the
+paramters IDki and Bk. In the case that IDki and Bk correspond correctly with
+xi, then the effect of Uki is to impose a negative phase on the bits corresponding
+to Bk in yk, which effectively ”tags” that state for amplitude amplification. In
+the case that xi does not correspond with IDki and Bk, the controlled Z gate is
+not applied since not all of its controlling qubits are 1’s. Hence, the bad value
+contributes nothing to the neuron’s final output state.
+S should encode a 2 into L1 by simply applying a Hadamard gate to
+13
+
+the corresponding qubits in a qubit register with a size equal to the length
+of L, preparing and sending this entire qubit register L to each distributor
+individually. With each message, S sends the parameters IDki and Bk. The
+effects of the associative measuring neuron’s operations will be to exactly
+replace any 2’s in the list L1 provided to their inputs by the sender S with
+probabilistically distributed 1’s and 0’s via measurement of the corresponding
+single qubit states which will be |0>+|1>
+√
+2
+or |0>−|1>
+√
+2
+depending on whether Bk
+is 1 or 0. The algorithm will leave the rest of the state unchanged and simply
+measure it.
+The receivers will also be associative measuring neurons and perform the
+same process. However, it will be assumed that they are more likely to have
+conflicting inputs, and that their weights will not all agree. Also, receivers will
+receive their inputs x from distributors and other receivers, but the parameters
+IDki and Bk will still be provided directly by S. The receivers should have a
+final state that approximates the following.
+|Xk >= bk|0⊗l > +
+�
+i
+�
+j̸=i
+(< xi|xj > wkiwkj)|xi >
+(3)
+The weights and biases will be naturally normalized.
+b2
+k +
+�
+i
+�
+j̸=i
+δ(xi, xj)(wkiwkj)2 = 1
+(4)
+While the machine begins with a nearly equal number of distributors and
+receivers, the neurons which do not receive consistent data do not output
+information and their input sources are considered unreliable. This decreases
+14
+
+the number of useful neurons in the network and may reduce the size of either
+of the two layers. This is how the QHBA selects trusted paths of information
+through the network. This is not dissimilar to the way that pathways between
+neurons are created through learning in Haikonen’s cognitive modular neural
+network.
+1.3.3
+Training Via Quantum Binding Commitment
+We can make use of an optimally secure mechanism called a Quantum Binding
+Commitment to define the training data for the system. In our network, we
+will want the consensus to result in agreement on a single bit. So, the measure
+of fidelity used in our training is trivial: a single bit (a sender’s vote) that is
+known initially only by the sender and the training code which can be a very
+simple, visible, immutable and infinitely running script.
+Quantum Bit Commitment Training
+1.3.4
+Commitment Schemes
+Computationally binding commitment schemes between two parties are
+composed of two phases.
+The Commitment Phase allows one party to send
+15
+
+Commitment
+ML Algorithmthe other party some information c related to a message m which does not give
+the receiver any information about m itself. However, the act of sending c binds
+the sender to provide the message m in the second stage, the Open Phase. In the
+Open Phase, the sender transmits m to the receiver and proves to the receiver
+that m does indeed correspond to c by providing a signature that ”opens c to
+m”.
+In our system, a successful opening will occur after a consensus is reached
+on a vote with well-tuned weights and biases. In the case that an opening is
+unsuccessful, this is used as negative feedback to motivate the ML to adjust its
+descent of the gradient.
+A classical definition of a computationally binding is the following from
+Unruh [21].
+Definition 3 (Classical-style binding) No algorithm A can output a
+commitment c and two signatures s, s’ that open c to two different messages
+m and m’.
+Computationally binding commitment schemes have been studied and
+defined in the quantum setting. Interestingly, when the algorithm A is allowed
+to be a quantum polynomial time algorithm, this definition was shown to be
+inadequate.
+While definition 3 holds for a particular classical-style binding
+commitment, Ambainis, Rosmanis, and Unruh showed that for this particular
+binding a quantum polynomial time algorithm A employed by an adversary
+could open c to any message that the adversary wished [22].
+Therefore Unruh was motivated to define a different type of binding that
+was useful in the quantum case. The new binding property is demonstrated by
+a pair of quantum games.
+Let A, B be algorithms and S, M, U be quantum registers.
+Vc is a
+16
+
+measurement which verifies that that U opens M.
+Mok measures m in the
+computational basis if ok = 1.
+The first game Game1 consists of four steps:
+(S, M, U, c) ← A(1γ)
+ok ← Vc(M, U)
+m ← Mok(M)
+b ← B(1γ, S, M, U)
+The second game Game2 omits the measurement in step three but is
+otherwise the same:
+(S, M, U, c) ← A(1γ)
+ok ← Vc(M, U)
+b ← B(1γ, S, M, U)
+A commitment scheme is ”collapse-binding” iff for any quantum polynomial
+time valid adversary, cAdv = |Pr[b = 1 : Game1] − Pr[b = 1 : Game2]| is
+negligible.
+This essentially expresses that if an adversary (A, B) provides a classical
+commitment c, there must be only one message he/she can open c to.
+A
+outputs a superposition of messages M and a superposition of corresponding
+opening signatures U. S is the adversary’s state. The assertion that |Pr[b = 1 :
+17
+
+Game1]−Pr[b = 1 : Game2]| is negligible limits the value of M to computational
+basis vectors for collapse-binding commitments. No quantum polynomial time
+algorithm B should be able to distinguish between the value of M whether M
+is measured in the computational basis or not.
+1.3.5
+Collapsing Hash Functions
+We will have to choose a specific collapse-binding commitment scheme for our
+system to use. However, the specific choice is relatively arbitrary as long as it
+satisfies the collapse-binding property.
+The games used to define the collapse-binding property of commitment
+schemes can also be applied to classify hash functions that are collapsing.
+Assume H is a one-to-one hash function.
+Definition 4 (Collapsing hash function - informal) H is a collapsing
+hash function iff no quantum polynomial time algorithm B can distinguish
+between Game1 and Game2.
+An adversary is valid if A outputs a classical
+value c and a register M where H(m) = c.
+This game based definition was clarified and made mathematical by Fehr in
+2018 [23].
+Definition 5 (Collapsing hash function - formal) A function H X → Y
+is ∈(q)-collapsing if
+cAdv[H](q) :=
+sup
+SMCU
+δq(M, M|CU) ≤∈ (q)
+for all q. The supremum is over all states SMCU = S H(M) CU with
+18
+
+complexity ≤ q.
+The collapsing property of a hash function is a counterpart of collision
+resistance. Unruh shows that the random quantum oracle is a collapsing hash
+[21] and so some hash function based commitment schemes are collapsing in the
+random oracle model. Unruh also showed that Merkle-Damgard hash functions
+are collapsing if their underlying compression algorithms are, which implies that
+SHA-2 is collapsing [24]. Czajkowski, et al. showed the same for Sponge hashes
+with certain conditions [25]. Sponge hash construction underlies SHA-3.
+1.3.6
+A New Type of Machine Learning Algorithm
+Our approach does not exactly fit into the category of supervised machine
+learning, since the idea here is not to train the neural network using a predefined
+data set until it reaches a fidelity threshold, and then to use the machine
+in production afterwards. Rather, our machine will be continuously training
+in production to achieve the effect of learning and accounting for the shifting
+behaviour of the participants in the network.
+The algorithm will train both weights and biases.
+As in any traditional
+RBM, each bias will effect all of the inputs to its neuron, while the weights
+will each be specific to channels between neurons. Hence, I expect that this
+approach will mitigate the impact of both dishonest individuals and groups of
+collaborating dishonest individuals.
+Another interesting paradigm shift has taken place. ML usually variationally
+combines well-defined, trivially simple and reliable elements (the neurons) to
+model uncertain, complex and large data sets (the training data). In our case,
+this is basically reversed. The training data is a single bit, for which we have an
+expectation value at the onset. The neurons are the uncertain elements. While
+we have defined the behaviour of the honest receiver neurons in the previous
+19
+
+subsection, we expect some participants to be dishonest and therefore deviate
+from this neuron model.
+The question that this algorithm is answering is fundamentally how to
+orchestrate unreliable and complex elements of a fair consensus system such
+that collaborative productivity is maximized. Since this new type of machine
+learning algorithm seems to be a suitable solution, there is the suggestion that
+a symbiosis of machine and human information processing can be useful for
+maximizing productivity.
+It would be interesting for future works to consider other applications where
+such a ”symbiotic” information processing approach could be useful.
+1.4
+Practical Implementation Limitations and Advantages
+1.4.1
+Practical Constraints Today
+It is an intended feature of the associative measuring neuron’s design that if
+the size of input lists and outputs is l = 6, its behaviour can be realized today
+using IBM’s commercially available Q System One or IBM Q 16 Melbourne
+system which is free to use for research purposes. Some restrictions must be
+applied to the neuron in order to ensure that it can be implemented using either
+system, since the superconducting Transmon qubit networks of these systems
+are not fully-connected.
+In order to make the associative measuring neuron
+compatible with the Melbourne, a sender simply must choose IDki and Bk that
+specify a controlled Z operation that is possible to implement. Many control
+configurations can be achieved using qubit swapping.
+20
+
+Y
+13
+12
+11
+10
+9
+8IBM Q 16 Melbourne Connectivity Graph
+The number of D and Uki operations applied, the less fidelity we will have
+in the neuron’s outputs due to decoherence.
+A minimum fidelity should be
+chosen and used to select the range of possible values that will be taken by the
+default repetition bias bk. This fidelity can be dynamically chosen based on
+the calibration parameters of the Melbourne, for example, which fluctuate but
+are available at a given time. The average T1 and T2 times for the IBM Q
+System One are reportedly 74µs and 69µs respectively [14]. The Melbourne’s
+decoherence times are similar but vary depending on the qubits involved, as
+evidenced by the figure. IBM reports that the average decoherence times for the
+Melbourne are T1 = 67.50µs and T2 = 22.40µs [16]. We only will consider the
+Melbourne’s limitations thoroughly, since it is less advanced and more limited
+than the IBM Q System One.
+IBM Q 16 Melbourne Calibration Details July 17th 2019
+The gate times of the Melbourne are updated continuously and published
+publicly [17], and the average amount of time required for a CX gate is around
+350ns.
+21
+
+qubit
+multi_qb_gate_error
+T1 (us)
+T2 (us)
+Frequency (GHz)
+readout error
+gate_error
+Q0
+73.32348273
+23.48828043
+5.100090141
+0.0215
+0.004031062
+Q1
+CX1 0:0.03, CX1 2:0.04
+63.23181621
+116.7289054
+5.238609742
+0.054
+0.012242205
+Q2
+CX2_3: 0.04
+46.13953307
+74.56571753
+5.032644087
+0.1864
+0.010450744
+Q3
+81.05055849
+74.78940464
+4.896205701
+0.047
+0.002494886
+Q4
+CX4 3:0.03, CX4 10:0.04
+55.43102145
+27.63898146
+5.028667392
+0.1226
+0.002551687
+Q5
+CX5 4:0.05,CX5 6:0.05,CX5 9:0.07
+27.79450766
+50.71953989
+5.06718735
+0.0568
+0.004714312
+Q6
+CX6 8:0.04
+56.16840169
+56.0630866
+4.923906934
+0.0478
+0.004816689
+Q7
+CX7 8:0.03
+32.50641909
+45.28966051
+4.974534967
+0.0598
+0.004438222
+Q8
+47.68062524
+71.45643335
+4.739563654
+0.0389
+0.004361702
+Q9
+CX9 8:0.04, CX9 10:0.05
+38.43726664
+79.71232612
+4.963421912
+0.0443
+0.006372041
+Q10
+56.99362705
+69.83941723
+4.945065458
+0.037
+0.003278348
+Q11
+CX11 3:0.05,CX11 10:0.05,CX11 12:0.06
+57.53451171
+71.43323367
+5.004981691
+0.0357
+0.0044898
+Q12
+CX12 2: 0.06
+78.13277541
+117.4664528
+4.760047973
+0.0918
+0.007732648
+Q13
+CX13_1:0.12, CX13_12:0.1
+21.39891833
+41.28178002
+4.968495889
+0.0498
+0.011006778IBM Q 16 Melbourne Gate Time Details August 7th 2019
+A CZ gate is realized in the IBM system using a CX and two single qubit
+Hadamard gates.
+Each Uki is a pair of controlled CCZ gates, and either 0
+or 6 X gates.
+A CCZ gate can be realized using CNOT, T †, and T gates
+via an optimal decomposition [20]. This requires six CX gates. The Grover
+diffusion operator can be realized using Hadamard gates surrounding a multiply
+controlled Z operation as well, as per [19].
+CCZ Gate Acheivable Using IBM Q
+Generally, the Grover diffusion operator would involve a multiply controlled
+Z gate with a number of controls equal to the size of the output register, minus
+one. We can get away with simplifying the Grover diffusion operator for our
+case by realizing that the operator will only ever be used to rotate the state
+22
+
+cX Gate
+GFGateTime (ns)
+CX1_0
+239
+CX1_2
+174
+CX2_3
+261
+CX4_3
+266
+CX5_4
+300
+CX5_6
+300
+CX7_8
+220
+CX9_8
+434
+CX9_10
+300
+CX11_10
+261
+CX11_12
+261
+CX13_12
+300
+CX13_1
+652
+CX12_2
+1043
+CX11_3
+286
+CX4_10
+261
+CX5_9
+348
+CX6_8
+348towards basis states with two non-zero qubit values. During each such rotation,
+the diffusion operator can be realized by a single CZ gate which involves the
+two qubits that correspond to the particular input xi’s corresponding indexes
+ID.
+This will rotate the high dimensional output state towards the target
+basis state in the relevant degrees of freedom, and leave the other degrees of
+freedom untouched. However, each input xi may adjust the overall output state
+in different degrees of freedom, and together rotate the state in any arbitrary
+direction. Using this approach, the Grover diffusion operator can be realized
+using six single qubit gates and one CX gate.
+The single qubit gates involved in the algorithm each have a time penalty
+as well. These time penalties can be understood by decomposing each unitary
+gates into its set of actual physical gates that are used to implement them in
+IBM’s system. IBM’s computers support three types of single qubit gates, the
+first two (u1, u2) are relevant for us:
+u1(λ) =
+�
+��
+1
+0
+0
+eλi
+�
+��
+u2(φ, λ) =
+1
+√
+2
+�
+��
+1
+−eλi
+eφi
+e(φi+λi)
+�
+��
+Any single qubit gate which has the form given by u1 is implemented using
+Frame Change (FC) operation, which does not physically take any time but
+actually influences the frame of the following operation and takes no time in
+and of itself. We can see that the T and T † gates do have the corresponding
+form.
+23
+
+T =
+�
+��
+1
+0
+0
+e
+π
+4 i
+�
+�� = u1(π
+4 )
+T † =
+�
+��
+1
+0
+0
+e− π
+4 i
+�
+�� = u1(−π
+4 )
+u1 Frame Change Physical Gate
+The Hadamard gate is also used in our algorithm, and matches the form
+of u2.
+Any gate which has the form of u2 is implemented using a physical
+Gaussian-Derivative (GD) pulse parameterized by two frame changes. A GD
+pulse takes 60ns itself, and invokes a 10ns buffer time.
+H =
+1
+√
+2
+�
+��
+1
+1
+1
+−1
+�
+�� = u2(2π, 3π)
+u2 Frame Change Physical Gate
+The final type of gate that is relevant for our work is the CX gate, which
+makes use of both FC and GD physical gates as well as Gaussian Flattop
+(GF) pulses. We have already addressed the time requirements for CX gates
+depending on the qubits involved.
+24
+
+U1
+FC
+bm
+(M)
+(-入)U2
+FC
+GD
+FC
+bm
+(9, 入)
+(-入)
+(π/2,π/2)
+(-Φ)CX Physical Gate
+Understanding this, we may say that the time requirement for Uki is at most
+equivalent to that of two CCZ gates and six X gates.
+T(Uki) ˙= 2 · 0ns + 6 · 350ns = 2100ns
+(5)
+Similarly, the time requirement for our simplified Grover diffusion operator
+is that of four Hadamard gates, two X gates and one CX gate.
+T(D) ˙= 2 · 0ns + 4 · 70ns + 1 · 350ns = 640ns
+(6)
+The overall time cost of a repetition of DUki is then given by equation (7).
+Trep = T(Uki) + T(D) = 2740ns
+(7)
+The time requirement for an associative measuring neuron’s operation in its
+entirety will then be given by equation (8).
+Tassoc =
+�
+i
+⌊bk + Ni⌋ · 2740ns
+(8)
+25
+
+FC
+GD
+GD
+I
+Control: wc
+(乙/μ)
+(/-")
+(t,0)
+GF
+GF
+CR: WT
+(π/4,0)
+(π/4, )
+-
++
+GD
+Target: WT
+(π/2,0)
+T
+-
+-To ensure this operation completes within an acceptable window, we simply
+enforce that Tassoc < T2. The most expensive associative measuring neuron
+operation will involve
+|P |
+2
+inputs xi.
+So, for example a system with ten
+participants would yield a maximal Tassoc time of max(Tassoc| |P|).
+max(Tassoc| |P|) =
+|P |
+2
+�
+i=0
+⌊bk + Ni⌋ · 2740ns
+In this scenario, Ni would be standard deviations of each |P |
+2 points. To keep
+an associative measuring neuron operation under the shortest time constraint,
+which is T2 = 22.40µs on the Melbourne, we must limit either the number of
+participants in the network |P|, or cap the number of standard deviations Ni at
+some maximum range. It is more appealing for the machine learning algorithm
+to take the number of participants as a parameter and adjust the range of the
+maximum considered standard deviation. So, we can define a maximum range
+max(Ni| |P|).
+T2 = max(Tassoc| |P|) =
+|P |
+2
+�
+i=0
+⌊bk + max(Ni)⌋ · 2740ns
+22.40µs =
+|P |
+2
+�
+i=0
+⌊bk + max(Ni)⌋ · 2740ns
+22.40µs
+2740ns =
+|P |
+2
+�
+i=0
+⌊bk + max(Ni)⌋
+26
+
+22.40µs
+2740ns −
+|P |
+2
+�
+i=0
+bk ˙=
+|P |
+2
+�
+i=0
+max(Ni)
+22.40µs
+2740ns −
+|P |
+2
+�
+i=0
+bk ˙= |P|
+2 max(Ni)
+max(Ni| |P|) ˙=
+2
+|P| · 22.40µs
+2740ns −
+|P |
+2
+�
+i=0
+bk
+In the worst case, � |P |
+2
+i=0 bk → |P |
+2
+since 0 ≤ bk ≤ 1.
+max(Ni| |P|) ˙=
+2
+|P| · 22.40µs
+2740ns − |P|
+2
+The system will become functionally useless when max(Ni| |P|) approaches
+0. Therefore we can conclude that the system will be able to handle only 6
+participants if our quantum associative measuring neurons were used at each
+node today.
+Also, if this system were implemented today, each participant would not
+have a local quantum computer to use for their associative measuring neuron
+operations. Rather, they would need to delegate their quantum computations
+to a central quantum computer. Today, the best option would be IBM’s system.
+The amount of time spent in the queue waiting for each others’ operations to
+complete would render the speedup from using Grover’s search pointless.
+1.4.2
+Near-future Possibilities
+Despite the conclusion that this system is not practical to implement today,
+the work we did in the last subsection gives us a method for predicting how
+27
+
+useful the system will be in the future, when we have access to better quantum
+computers.
+IBM claims that they intend to eventually improve their coherence (T2)
+times to 1-5 milliseconds, and suggest that they are exponentially approaching
+this goal according to a relationship similar to Moore’s law for integrated
+electronics [14]. Assuming this goal is reached within the next decade, which
+is generally considered to be feasible with at least a non-zero possibility, our
+scheme would be able to support roughly 85 participants in each vote.
+The true randomness of the probabilistic outcomes from measuring the
+states |0>+|1>
+√
+2
+and |0>−|1>
+√
+2
+is a valuable cryptographic asset when a quantum
+associative measuring neuron is used for a security protocol due to the outcome
+being truly random.
+Also, the Grover’s search algorithm provides a known
+quadratic speedup over equivalent classical methods, when the number of
+applied operations is compared to the number of classical records checked for
+the value searched for [13].
+However, it is important to point out that an associative measuring neuron
+with a limited repetition capacity can be easily classically simulated. So, the
+entire system described thus far could theoretically be replaced with a classical
+equivalent. This would mean that we do not gain the security and efficiency
+benefits of the quantum algorithms employed. However, it would mean that
+scaling the system to support any arbitrarily large number of users would be
+possible.
+An optimal network scheme would incorporate both quantum and classical
+elements to take advantage of as much quantum security and speedup as possible
+with the resources available whilst also supporting an arbitrary number of users.
+The
+machine
+learning
+inspired
+element
+of
+the
+consensus
+algorithm
+implemented at any scale would be beyond the capabilities of any quantum
+28
+
+computing technology that exists today. However, it would be well within the
+reach of modern classical technology. So, we will assume that it is purely classical
+for the forseeable future. However, it would be interesting for a future work to
+look into how quantum machine learning might increase the efficiency of this
+component as well.
+On the other hand, quantum-secure communication channels are already
+being established and demonstrated in the world by companies like NXM [18].
+We posit that quantum networks will also be available for practical use in the
+near-term, and we can expect to use these as a resource. The ability to perform
+a Hadamard gate, transmit and measure the resulting state is already quite
+feasible. So, we can assume that at least some of the channels used by the sender
+of any vote can benefit from the pure randomness of the quantum approach for
+encoding 2’s into the lists L1k.
+Our system does not make assumptions on the number of participants who
+will be interested in participating in any given vote. Therefore it is conceivable
+that votes involving small numbers of people (< 6 today, < 85 eventually) could
+occur, and benefit fully from the quadratic speedup of the Grover’s search.
+Larger votes could also occur, which would involve strictly classical neuron
+operations and simply trade efficiency for scalability.
+The ability to use a mixture of quantum and classical channels and neurons
+is an advantage.
+It enables this blockchain scheme to be viable throughout
+transitions in networking and computing technology.
+2
+References
+1. Sun, Xin, et al. “A Simple Voting Protocol on Quantum Blockchain.”
+International Journal of Theoretical Physics, vol. 58, no. 1, 2019, pp.
+275–281.
+29
+
+2. Sun, Xin, et al. “Quantum-Enhanced Logic-Based Blockchain I: Quantum
+Honest-Success Byzantine Agreement and Qulogicoin.” 2018.
+3. An Artificial Cognitive Neural System Based on a Novel Neuron Structure
+and a Reentrant Modular Architecture with Implications to Machine
+Consciousness
+4. Crandall, B. C. (2000). Nanotechnology: Molecular speculations on global
+abundance. Cambridge (Massachusetts): MIT Press.
+5. Vujicic, Dejan, et al. “Blockchain Technology, Bitcoin, and Ethereum:
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+32
+

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+Based on Springer Nature LATEX template
+Lyapunov Exponents for Temporal Networks
+Annalisa Caligiuri1, Victor M. Egu´ıluz1, Leonardo di
+Gaetano2, Tobias Galla1 and Lucas Lacasa1*
+1Institute for Cross-Disciplinary Physics and Complex Systems (IFISC),
+CSIC-UIB, Palma de Mallorca, Spain.
+2Department of Network and Data Science, Central European University,
+1100 Vienna, Austria.
+*Corresponding author(s). E-mail(s): lucas@ifisc.uib-csic.es;
+Abstract
+By interpreting a temporal network as a trajectory of a latent graph
+dynamical system, we introduce the concept of dynamical instabil-
+ity of a temporal network, and construct a measure to estimate the
+network Maximum Lyapunov Exponent (nMLE) of a temporal net-
+work trajectory. Extending conventional algorithmic methods from
+nonlinear time-series analysis to networks, we show how to quantify
+sensitive dependence on initial conditions, and estimate the nMLE
+directly from a single network trajectory. We validate our method
+for a range of synthetic generative network models displaying low
+and high dimensional chaos, and finally discuss potential applications.
+Keywords: Lyapunov exponent, temporal networks, chaos, complex systems
+1 Introduction
+Temporal networks (TNs) [1–3] are graphs whose topology changes in time.
+They are minimal mathematical models that encapsulate how the interaction
+architecture of elements in a complex system changes dynamically. TNs have
+been successfully used in a variety of areas ranging from epidemic spreading
+[4] or air transport [6] to neuroscience [7] to cite a few, and it has been shown
+that important dynamical processes running on networks (e.g. epidemics [4],
+synchronization, search [5]) display qualitatively different emergent patterns
+when the substrate is a TN, compared to the case of a static network. These
+1
+arXiv:2301.12966v1  [physics.data-an]  30 Jan 2023
+
+Based on Springer Nature LATEX template
+2
+Lyapunov Exponents for Temporal Networks
+effects are particularly relevant when the timescale of the dynamics running
+on the graph is comparable to that of the intrinsic evolution of the network,
+i.e., when there is no manifest separation of timescales. Relatively lesser
+work has, however, considered the intrinsic dynamics of the network from a
+principled point of view. Recently, a research programme has been proposed
+[8] in which TNs are to be interpreted as the trajectories of a latent ‘graph
+dynamical system’ (GDS). The GDS provides an explicit model for the time-
+evolution of the network. Similar to a conventional dynamical system (whose
+output is a time series of scalar or vector quantities), the output of a GDS is
+a time series of networks, i.e. a TN.
+The dynamics of TNs and GDS are indeed the objects of ongoing research.
+For instance, in [8] the authors consider how to extend the autocorrelation
+function of a signal to a graph-theoretical setting. They explore how TNs can
+oscillate and how harmonic modes, as well as decaying linear temporal correla-
+tions of various shapes, emerge. In a similar fashion, the memory of a temporal
+network has been studied from different angles, including the concept of mem-
+ory shape [9] as a multidimensional extension of memory (high order Markov
+chain theory) in conventional time series. In this work, we further pursue the
+abovementioned research framework programme, and consider the problem of
+dynamical instability and chaos quantification in TNs. Interpreting temporal
+networks as trajectories in graph space, we aim to generalise the concept
+of Lyapunov exponents as quantifiers of the sensitivity to initial conditions.
+Our objective is to define and measure Lyapunov exponents and sensitive
+dependence to initial conditions of the network as a whole. Our approach is
+therefore not to quantify chaos in the dynamics for example of every link, but
+rather to quantify chaos for the collective dynamics of the whole network.
+Since TNs in applications are frequently observed empirically, we focus our
+implementation and inference of network Lyapunov exponents solely on the
+observation of a single (long) TN trajectory, without the need to access the
+underlying GDS (but of course, the framework is also applicable if the GDS
+is explicitly accessible). Our algorithmic implementation can thus be seen as
+a conceptual network generalization of the classical algorithms by Wolf and
+Kantz [10–12, 14], originally proposed to quantify sensitive dependence on
+initial conditions directly from empirically observed time series (see also [13]
+for a similarly seminal work).
+Importantly, any new method needs to be validated. In our case, this is
+not trivial, since the notion of chaotic TNs is not common in existing litera-
+ture. A second objective of this work is thus to propose synthetic generative
+models of chaotic TNs, which can be used as templates to validate the meth-
+ods we develop for the quantification of chaos in TNs. These methods, once
+validated, can then be used in wider applications and further research.
+
+Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+3
+The rest of the paper is organised as follows. In Section 2 we introduce the
+theoretical background to our work, and we set the notation. We derive the
+network analog of the maximum Lyapunov exponent (MLE), and we outline
+an algorithmic implementation to estimate quantities such as the spectrum
+of local expansion rates, trajectory-averaged and volume-averaged expansion
+rates, and the network maximum Lyapunov exponent (nMLE). In Section 3
+we consider the relatively simple case of random network dynamics as a first
+example, and show how the method works in such scenario. Then, in Section 4
+we introduce a generative model of (low-dimensional) chaotic temporal net-
+works. This model provides us with ‘ground-truth’ access to the nMLEs of the
+network trajectories that the model generates. We show that the method we
+propose to infer nMLEs from a trajectory of networks correctly reconstructs
+this ‘true’ exponent. We also assess how the estimation of the nMLE is affected
+when the chaotic network trajectory is polluted with certain amounts of noise,
+and discuss at this point how to estimate negative nMLEs as well. In Section 5
+we introduce a different generative model of (high-dimensional) chaotic TNs.
+We demonstrate that the generated TNs show sensitive dependence to initial
+conditions, and that the nMLE varies as expected as a function of a network’s
+coupling parameter. In Section 6 we finally conclude and discuss potential
+applications of the method.
+2 Theory and method
+2.1 Lyapunov exponents for graph dynamical systems
+In nonlinear time series analysis, the maximum Lyapunov exponent λMLE of
+a dynamical system quantifies how two trajectories that are initially close
+separate over time. More precisely, one imagines two copies of the system,
+which are started from initial conditions at time t = 0 which are a distance d0
+apart. One then defines
+λMLE = lim
+t→∞ lim
+d0→0
+1
+t ln dt
+d0
+,
+(1)
+where dt is the distance between the two copies of the system at time t.
+In practice, the distance d0 is often small but finite, and the limit d0 → 0 may
+not be accessible. It then turns out that the long-time limit t → ∞ is not
+accessible either as the growth of dt is bounded by the size of the attractor
+of the system [16]. In such cases, the behaviour of dt usually undergoes a
+cross-over (at a time which we label τ), between an exponentially expanding
+phase (t < τ) to a saturated phase (t > τ). In the latter regime, dt fluctuates
+around the attractor’s size. We will call τ the saturation time.
+In the network setting, we assume there exists a (sometimes unknown)
+graph dynamical system that determines the evolution of a graph over time.
+We focus on discrete time. The GDS is then a map, which determines how
+
+Based on Springer Nature LATEX template
+4
+Lyapunov Exponents for Temporal Networks
+one network evolves in the next time step. For simplicity, we assume that
+the set of vertices is fixed, and that the vertices are distinguishable from
+one another and labelled. Thus, only the set of edges between these ver-
+tices evolves in time. A trajectory of the GDS then consists of a sequence
+of network snapshots. These trajectories define the TNs generated by the
+system [1, 3]. Each trajectory is given by the sequence of adjacency matrices
+S = (At)t≥0, At = {aij; t}; i, j = 1, 2, . . . , n where aij; t = 1 if the vertices i
+and j are connected at time t, and zero otherwise (this symmetric choice is
+for simple, undirected networks, but the method works essentially along the
+same lines for non-simple and directed networks, perhaps except for different
+normalization factors in the definition of the distance, see below). As such,
+we are considering labelled, unweighted networks of n nodes.
+It is not clear a priori if the specific choice of the distance used to quantify
+the deviation between two originally close network trajectories is critical. We
+conjecture that, as long as this distance is based on the full adjacency matrix
+–not on a projection of it–, results should hold independent of the specific met-
+ric. This is based on the fact that in dynamical systems, the MLE is invariant
+under different choices of the underlying norm ||.|| [14, 16]. There exist many
+graph distances [18]. For simplicity, we take an intuitive definition of such a
+distance which is based on the amount of edge overlap between two networks:
+given two networks with the same number of nodes n and adjacency matrices
+A and B,
+d(A, B) = 1
+2
+�
+i,j
+|aij − bij|,
+(2)
+where i and j take values from 1 to n1. In our setting, networks are simple
+and unweighted. In the particular case where A and B have the same number
+of edges, the distance defined in Eq. (2) is indeed a rewiring distance, i.e., it
+is given by the number of unique rewirings needed to transform A into B,
+and therefore d is a positive integer-valued function. One can then further
+normalize d as appropriate such that it is defined in [0, 1], as we will show
+later. Further details can be found in the Appendix, where we also introduce
+alternative distances.
+2.2 Inference of network Lyapunov exponents
+2.2.1 Local expansion rates
+We are interested in quantifying sensitive dependence on initial conditions (and
+in particular, the network version of λMLE, which we here call λnMLE) when
+the mechanics of the GDS is not known, and when we only have access to
+a (single) discrete-time network trajectory S = (A0, A1, A2, . . . ) of adjacency
+matrices. This is analogous to the case in which one would like to reconstruct
+the MLE of a conventional dynamical system from a single time series. The
+1The pre-factor is used to have a normalized distance when networks are simple, undirected
+and have the same number of links, otherwise a different normalization is needed.
+
+Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+5
+standard approach consists in using Wolf’s or Kantz’s algorithms [10–12, 14].
+The central idea is here to look for recurrences in the orbit, finding points in
+the time series which might be temporally separated but which are close in
+phase space. One then monitors the deviation of those points over time. Here,
+we extend this approach to the case of a time series of networks, i.e. a TN.
+We start by fixing an element At from S, where t is an arbitrary point in
+time. This adjacency matrix will be the ‘initial condition’ for our analysis. To
+extend Wolf’s algorithm we then proceed to look in S for another element At′
+at a different time t′, such that d(At, At′) < ϵ, where ϵ is a small threshold
+chosen before the analysis begins2. This recurrence in phase space allows us
+to use a single trajectory to explore how two close networks separate over
+time. We then set d0 := d(At, At′) as the initial distance. We then proceed to
+measure how distance evolves over time as we separately track the evolution
+of At+k and At′+k in S, where k = 1, 2, . . . . We write dk = ||At+k − At′+k||.
+Without loss of generality, we can always write the successive distances in
+terms of a sequence of local expansion rates ℓ1, ℓ2, . . . ,
+dk = dk−1 exp(ℓk).
+(3)
+Each of the ℓk can be positive (local expansion), negative (local contraction),
+or zero. Equation (3) generally models the case where two initially close tra-
+jectories (d0 < ϵ) deviate from each other over time. The ℓk can depend on k,
+since the expansion rates can vary as the trajectories pass through different
+parts of the attractor [14]. We then define a trajectory-averaged expansion rate
+ℓ as follows
+ℓ = 1
+τ
+τ
+�
+k=1
+ℓk = 1
+τ
+τ
+�
+k=1
+ln dk
+dk−1
+,
+(4)
+where τ is the saturation time defined earlier. Since we are considering a fixed
+trajectory (not an ensemble of trajectories), we thus have
+ℓ = 1
+τ ln dτ
+d0
+.
+(5)
+Provided the GDS is ergodic (i.e., that a single and long enough orbit ade-
+quately visits the whole graph phase space), ℓ converges to the network
+Maximum Lyapunov Exponent λnMLE in the limit of large τ, independent of
+the choice of the initial adjacency matrix At. However, in practice, τ will be
+finite, and thus we cannot readily assume that ℓ fully describes the long-term
+behaviour, or that it is independent of the initial condition At. It is thus inter-
+preted as a local Lyapunov exponent [15], and an average of this quantity over
+different initial conditions At will be required, as discussed further below.
+2In practice, At′ is selected as the closest network from At within the ball of radius ϵ.
+
+Based on Springer Nature LATEX template
+6
+Lyapunov Exponents for Temporal Networks
+Fig. 1 Illustration of a temporal network S = (G0, G1, G2, ...) as a trajectory of a latent
+graph dynamical system (GDS) and the methodology to compute its network Maximum
+Lyapunov exponent (Kantz version) λK
+nMLE. Each element of the trajectory is a network,
+i.e. a snapshot of the temporal network. λK
+nMLE is estimated directly from S (i.e., without
+accessing the GDS directly) by looking at recurrences in S and quantifying the average
+expansion around different network snapshots. In this illustration, a ball of radius ϵ around
+an arbitrary G0 is fixed, and four recurrences are found where d(G0, Gr) < ϵ. The initial
+distance d(G0, Gr) is averaged over the four recurrences and the average distance after one
+time step d(G1, Gr+1) is computed. λK
+nMLE is computed by averaging over time, volume and
+different initial conditions G0 (see the text for details).
+2.2.2 Wolf and Kantz methods of estimating maximum
+Lyapunov exponents for temporal networks
+The time it takes for two trajectories to reach a distance of the order of the
+attractor’s size depends on how close these two trajectories were initially. In
+other words, the saturation time τ depends on d0, and therefore on the choice
+
+Go
+G1
+S= (Go,G1, G2, ..., Grt, Gr+1, ..., Gr2, Gr+1, ...)
+Gr2+1
+Gr3
+Gra
+Go
+Gr2
+G1Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+7
+of the threshold ϵ. The limits d0 → 0 and τ → ∞ are thus related to one
+another. Conceptually, we would like the trajectories to be as close as possible
+initially, so that we can monitor the expansion rate for long times, allowing
+us to obtain a global MLE as opposed to a local Lyapunov exponent. To do
+this, we need to track the expansion for sufficiently long, but we also need to
+avoid the regime in which the distance is limited by the characteristic size of
+the attractor.
+Generalised Wolf approach to measuring the nMLE. We now construct
+the generalisation of Wolf’s approach, which will yield an estimation of the
+nMLE that we call λW
+nMLE. The aim is to compute ⟨ℓ⟩ = ⟨ 1
+τ ln dτ
+d0 ⟩, where the
+average is over choices of At and At′. In practice one considers a set of w
+initial choices of At, which we index i = 1, . . . , w, and for each of these choices,
+one additional point At′ on the trajectory such that d(At, At′) < ϵ. One then
+obtains
+λW
+nMLE = 1
+w
+w
+�
+i=1
+ℓ(i),
+(6)
+where ℓ(i) is the trajectory-averaged expansion rate ℓ computed for the i-th
+initial condition, obtained from Eq. (4).
+Algorithmically, this approach has the advantage that we do not need to fix
+the choice of τ, we are able to flexibly adjust τ for each initial condition,
+according to the specific d0 we are able to find in S. On the other hand, this
+approach is point-wise, in the sense that for each choice of At one only con-
+siders a single At′ nearby. As a consequence, this method does not necessarily
+capture the average expansion rate around each initial condition At.
+Generalised Kantz approach to measuring the NMLE. In order to cal-
+culate such an average expansion rate (for a given choice of At) one would, for
+a fixed At, have to average over the expansion rates for choices of At′ in an
+ϵ-ball about At. This volume-averaging is the basis of Kantz’s generalization
+[11, 12] of Wolf’s algorithm [10], see Fig.1 for an illustration. For a given ini-
+tial condition At Kantz’ method provides a trajectory and volume averaged
+expansion rate ⟨ 1
+τ ln dτ
+d0 ⟩volume. We will write Λ for the volume-averaged expan-
+sion rate. For fixed At, this could algorithmically be obtained as follows. One
+chooses N different At′ from the trajectory S, all within distance ϵ from At.
+We label these j = 1, . . . , N. For each At′ one then computes ℓ(j) via Eq. (4).
+Then one sets
+Λ(At) = 1
+N
+N
+�
+j=1
+ℓ(j),
+(7)
+where N is the number of initial conditions inside a ball of radius ϵ and cen-
+tered at At that we have found in the sequence S.
+In practice, Kantz algorithm proceeds slightly differently. Instead of first com-
+puting the ℓ(j), and then averaging the expansion rates, the average over the
+At′ is instead computed at the level of distances. That is to say, one makes N
+
+Based on Springer Nature LATEX template
+8
+Lyapunov Exponents for Temporal Networks
+choices of At′ as described above, and then obtains dk(j) for each j = 1, . . . , N
+and k = 0, 1, 2, . . . (k runs up to the relevant cut-off time). One then sets
+Λ(At) = 1
+τ ln
+N −1 �N
+j=1 dτ(j)
+N −1 �N
+j=1 d0(j)
+,
+(8)
+where τ is a priori fixed for all j. The numerator in the logarithm represents
+the volume average (over choices of At′ in a ball about At) of dτ, and the
+denominator is the volume average of d0.
+We stress that Eqs. (7) and (8) are mathematically different and do not
+necessarily lead to the same results. While Eq. (7) allows adapting the precise
+value τ (which depends on d0) for each trajectory, Eq. (8) instead requires
+us to use a uniform τ across all trajectories. The latter expression looks at
+the expansion in time of an initially small volume centered on At, and is
+closer in spirit to capturing the underlying nonlinear dynamics than Eqs. (7).
+Accordingly, Eq. (8) is the basis of the Kantz’ method.
+The quantity Λ in Eq. (8) is still a local quantity, in the sense that it
+was computed for a phase space volume around a fixed choice of At. In prin-
+ciple, the local volume-averaged expansion rate could vary across different
+regions in phase space. To capture the global long-term behaviour we there-
+fore additionally average over choices of At, and then finally obtain the global
+volume-averaged network maximum Lyapunov exponent
+λK
+nMLE =
+�1
+τ ln ⟨dτ⟩volume
+⟨d0⟩volume
+�
+At
+,
+(9)
+where we have written ⟨· · · ⟩At, for the average over initial conditions At. In
+practice this is carried out by averaging over a set of w choices of At, i.e.,
+λK
+nMLE = 1
+w
+w
+�
+j=1
+Λ(A(j)
+t )
+(10)
+where Λ(A(j)
+t ) stands for the expression in Eq. (8) for the fixed choice
+At = A(j)
+t .
+We note that the value of the saturation time τ or the radius of the ball
+ϵ will have to be selected after some numerical exploration. Indeed, a better
+estimate of the Lyapunov exponent is obtained when the cut-off time τ is
+large. This, in turn, is the case when the initial distance between At and At′ is
+small, hence favouring choosing a relatively small value of ϵ. However, a small
+value of ϵ complicates the task of finding points At′ that are at most a distance
+ϵ away from At on the given trajectory. In practice, a trade-off is to be struck.
+
+Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+9
+To summarise, from a given TN trajectory (i.e. a sequence of network
+snapshots) we first measure the local expansion rates {ℓk}τ
+k=1 via Eq. (3)
+for a fixed choice of At and At′. The set of ℓk obtained in this way provide
+information on the fluctuations of the local expansion rate (for fixed At and
+At′), and its trajectory-average ℓ. We can then proceed along two alternative
+routes. In the first approach (i) we average ℓ over different choices for the
+initial condition At [Eq. (6)], and obtain Wolf’s approximation to the nMLE
+λW
+nMLE. Alternatively, (ii) we can initially perform, for each initial condition
+At, an average over At′. This is done by computing the local expansion of a
+volume ⟨dk⟩volume and then averaging this over time [Eq. (8)]. This is then
+repeated for different choices of At, and once hence obtains a distribution
+P(Λ) describing the fluctuations across different points in the network phase
+space. Its mean provides Kantz’s approximation to the nMLE λK
+nMLE.
+In the following sections, we present a validation of this method for random,
+low-dimensional and high-dimensional chaotic temporal networks.
+3 The white-noise equivalent of a temporal
+network: independent and identically
+distributed random graphs
+Before addressing the case of chaotic dynamics, we briefly discuss the case of
+random network trajectories, with no correlations in time. One then expects
+no systematic expansion or compression in time, and the resulting Lyapunov
+exponent should hence vanish3. We here seek to verify that this is indeed the
+case for the procedures we have introduced to estimate the Lyapunov expo-
+nents of TNs. Studying this is of interest, among other reasons, because an
+empirically obtained time series may appear random. It is then important to
+be able to decide if the trajectory is consistent with an uncorrelated random
+trajectory in network space, or with a deterministic chaotic model.
+Here we study the simple case where S is an independently drawn sequence of
+Erd¨os-R´enyi graphs with n nodes and in which the probability that any two
+nodes are connected is p. This is an analog to white noise for TNs, i.e., a situ-
+ation in which the TN displays delta-distributed autocorrelation function [8].
+At odds with a deterministic GDS, the distances between different points on a
+network trajectory are now random variables. More precisely, since all elements
+of S are the adjacency matrices of Erd¨os-R´enyi graphs, the elements of these
+matrices are Bernoulli variables, taking values zero (with probability 1 − p) or
+one (with probability p). For independent adjacency matrices A and B, the
+possible values of |aij − bij| are then zero with probability p2 + (1 − p)2, and
+3By construction, d1 > d0 as we force d0 < ϵ, so in order to avoid spurious expansions at k = 1,
+in this section we don’t take into account d0 in the estimation of finite Lyapunov exponents, i.e.
+our starting time is k = 1.
+
+Based on Springer Nature LATEX template
+10
+Lyapunov Exponents for Temporal Networks
+one with probability 2p(1 − p). Thus we have
+d(A, B) ∼ Binomial[n(n − 1)/2, 2p(1 − p)].
+(11)
+Eq. (11) is numerically verified in panel (a) of Fig. 2.
+Fig. 2
+Panel (a): Probability distribution of the distance between consecutive networks in
+an i.i.d. sequence of |S| = 105 Erdos-Renyi graphs with n = 100 nodes and p = 0.01 (black
+line). The plot is in semi-log, where a Gaussian shape appears as an inverted parabola, so
+we can better appreciate the tails. Blue solid line is Eq. (11). Panel (b): P(ℓ) (black) and
+P(Λ) (red) associated with a sequence of |S| = 5·104 i.i.d. ER networks (n = 100, p = 0.01),
+for τ = 1 and w = O(104) initial conditions. The mean of both distributions (nMLE) is
+essentially zero for both methods, but the dispersion around the mean is larger in Wolf’s
+approach. The solid blue line is the theoretical prediction, i.e. a Gaussian distribution with
+mean 0 and variance as in Eq. (14) with τ = 1. The inset of panel (b) shows how the variance
+of ℓ shrinks as τ increases (resulting in a much lower uncertainty around a zero nMLE): dots
+are different numerical simulations with |S| = 104 and w = 103, for different τ; the blue line
+is Eq. (14).
+The quantity ℓ in Eq. (5) is given by ℓ = 1
+τ (ln dτ − ln d0). As we have just
+established, d0 and dτ are independent binomial random variables following
+the distribution in Eq. (11). For large networks (n ≫ 1) this distribution can
+be approximated as a Gaussian, with mean µ = qn(n − 1)/2 and variance
+σ2 = q(1 − q)n(n − 1)/2, where we have written q ≡ 2p(1 − p). Writing
+d0 = µ + σz0, with z0 a standard Gaussian random variable, we have
+ln d0 ≈ ln
+�
+µ(1 + σ
+µz0)
+�
+= ln µ + σ
+µz0 − 1
+2
+σ2
+µ2 z2
+0 + . . . ,
+(12)
+after an expansion in powers of σ/µ, where the latter quantity is of order
+O(1/n). The same expansion can be carried out for dτ, and we therefore find
+ℓ = 1
+τ
+�σ
+µ(zτ − z0) + 1
+2
+σ2
+µ2 (z2
+0 − z2
+τ)
+�
++ . . . .
+(13)
+
+(a)
+10-2
+P
+ simulation
+10
+— theory
+10-5
+60
+80
+100
+120
+140
+dsimulation
+(b)
+10-2
+Eq. (14)
+4 
+ P()
+- theory (Wolf)
+3
+L P(Λ)
+P(0), P(Λ)
+1
+10
+1
+2
+0
+-0.5
+0
+0.5
+l, ^Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+11
+We note that the second term in the bracket is of sub-leading order in 1/n.
+Hence, ℓ is to lowest order in 1/n approximately Gaussian, with mean zero
+and variance
+Var(ℓ) = 2
+τ 2
+σ2
+µ2 = 1
+τ 2
+4(1 − q)
+qn(n − 1)
+(14)
+This theory has been numerically verified, and in panel (b) of Fig.2 we plot
+P(ℓ) both for τ = 1 (outer panel) and Eq.14 for increasing values of τ (inset
+panel).
+The case of Λ (Kantz-version) should intuitively converge even faster than
+ℓ (Wolf-version) since in this case we are carrying out two averages instead
+of just one, i.e. P(Λ) should have a smaller variance than P(ℓ), for a given
+τ. This is confirmed in panel (b) of Fig. 2, where we also observe that both
+methods yield the same (correct) estimation of the nMLE, which in this case
+is approximately zero (both estimates are of the order of 10−6). Note that the
+main panels of Fig. 2) are for the case τ = 1, so it is a worst-case scenario:
+as τ increases Var(ℓ) shrinks [Eq. (14)] and the uncertainty around the null
+shrinks accordingly [see inset of Fig. 2(b)].
+We were not able to find a closed-form solution for P(Λ) as averages inside
+the ϵ-ball are random variables whose distribution explicitly depends on the
+specific initial condition At: this calculation is left as an open problem. In
+anycase, we conclude that an i.i.d. temporal network has a null MLE Lyapunov
+exponent and the methodology (in both variants) correctly estimates it.
+4 Low-dimensional chaotic networks
+4.1 Network generation: the dictionary trick
+To be able to validate the method in the context of chaotic dynamics, we
+ideally need to have access to chaotic network trajectories with a ground true
+nMLE. This is difficult as a general theory of chaotic GDS is not yet accessible.
+To circumvent this drawback, in this section we develop a method to construct
+(low-dimensional) chaotic network trajectories by symbolising in graph space-
+time series from low-dimensional chaotic maps. The method of graph-space
+symbolisation was first proposed as a so-called ‘dictionary trick’ in [8] and
+consists of the following steps:
+• We construct a network dictionary D. This is a set of networks that allows us
+to map a real-valued scalar x ∈ [0, 1]4 into a network, such that the distance
+between two scalars is preserved in graph space. The set D is therefore
+ordered and equipped with a metric, such that the distance between two real-
+valued scalars |x−x′| is preserved in the graph symbols. More concretely, the
+dictionary of networks D = (G1, G2, ..., GL) such that d(Gp, Gq) ∝ |p − q|
+(one can subsequently normalize d according to the length of the dictionary,
+such that we have d ∈ [0, 1]).
+4We choose the interval [0, 1] without loss of generality.
+
+Based on Springer Nature LATEX template
+12
+Lyapunov Exponents for Temporal Networks
+• Once such dictionary is built, any one-dimensional time series can be
+mapped into a sequence of networks. In particular, we can map chaotic
+time series with well-known MLEs into network trajectories, from which an
+independent estimate of the nMLE can be obtained.
+Algorithmically, the dictionary is generated sequentially with G1 ∼ ER(p)
+(an Erd¨os-R´enyi graph with parameter p) and then iteratively constructing
+Gk+1 from Gk by rewiring a link that (i) has not been rewired in any previous
+iteration of the algorithm, (ii) into a place that did not have a link in any pre-
+vious iteration of the algorithm. It is easy to see that such algorithm ensures
+that D provides a partition of [0, 1] of the form [0, 1] = ∪L−1
+k=0 [k/L, (k + 1)/L],
+where L is the number of networks in the dictionary. The dictionary is thus
+metrical, in the sense that the rewiring distance between any two elements
+in the dictionary is (for a sufficiently large refinement L) arbitrarily close to
+the associated real-valued scalars in the original interval. Once the dictionary
+is established, we can then generate synthetic temporal network trajectories
+as symbolizations of unit interval dynamics by matching points in the subin-
+terval [k/L, (k + 1)/L] to the symbol Gk+1. The resulting temporal network
+S inherits, by construction, the properties of the scalar time series, and in
+particular can be used to generate chaotic TNs.
+Fig. 3 Panel (a): Semi-log plot of the distance dk as a function of iteration index k, for
+two initially close network trajectories sampled from S. We can appreciate an initial expo-
+nentially expanding phase, followed by a saturation phase, although the local expansion
+rate strongly fluctuates. Panel (b) Volume-averaged distance ⟨dk⟩volume as a function of
+time k, for N = 17 initial graph conditions inside a volume centered at an initial graph of
+n = 500 nodes and m = 2000 edges. Network dynamics evolve according to a logistic map
+as described in the text, whose true Lyapunov exponent is ln 2 ≈ 0.693. We can see how the
+volume enclosing the graphs on average expands exponentially fast –with an exponent close
+to ln 2, as expected– until it reaches the attractor size, what happens at the saturation time
+τ ≈ 10.
+
+In2·k
+n = 500
+0.1
+m = 2000
+0.01
+(a)
+0
+10
+20
+30
+kn = 500
+m = 2000
+N = 17
+t (saturation time)
+0.01
+In2·k
+(b)
+0
+5
+10
+15
+20
+25
+30
+kSpringer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+13
+4.2 Results for the logistic map
+As a first validation, we consider the fully chaotic logistic map
+xt+1 = 4xt(1 − xt),
+xt ∈ [0, 1],
+(15)
+that generates chaotic trajectories with λMLE = ln 2 ≈ 0.693. Using the dic-
+tionary trick, from a signal extracted from Eq. (15) we generate a temporal
+network trajectory S of |S| = 3000 network snapshots. In this validation,
+networks have n = 500 nodes and m = 2000 edges.
+Fig. 4 Panel (a): Approximation to λW
+nMLE following Wolf’s approach (see text), computed
+by averaging ℓ over w randomly sampled initial conditions [Eq.6], as a function of w. We
+can see that the exponent converges to the ground true exponent ln 2 as w increases. Inset
+in (a): Probability distribution P(ℓ), sampled by estimating ℓ for w = 500 different initial
+graph conditions sampled randomly from S. The mean of this empirical distribution is
+λW
+nMLE = ⟨Λ⟩At ≈ 0.685, very close to the true exponent ln 2 ≈ 0.693. Panel (b): Same as
+panel (a), but using Kantz’s approach (see the text), where we compute the volume and
+trajectory averaged expansion rate Λ for w initial conditions. Convergence properties are
+similar in both cases.
+For illustration, in panel (a) of Fig. 3 we plot in semi-log scales the
+(properly normalized) distance dk as a function of the iteration index k, for
+two initially close network trajectories sampled from S. We can see an initial
+exponentially expanding phase (whose exponent is an estimation of ℓ) followed
+by a saturation, although the distance function shows strong fluctuations. To
+cope with these, in panel (b) of the same figure we plot the volume-averaged
+expansion ⟨dk⟩volume vs k for a ball of radius ϵ = 0.005 centered at a specific
+initial graph from S. We can now clearly see the initial exponential phase
+followed by a cross-over to a saturation phase. The cross-over marks the
+saturation time τ where the distance reaches the attractor size. Note that the
+slope of the exponential expansion (i.e. the estimate of Λ) is close to ln 2, the
+true MLE.
+
+1.2
+(a)
+2
+W
+0.685
+1.0
+P1
+0
+0.8
+0
+0.5
+1.0
+1.5
+In 2
+0.6
+0
+100
+200
+300
+400
+500
+w (# initial conditions)1.1
+3
+(b)
+XNMLE = 0.685
+1.0
+2
+P(Λ)
+0.9
+1
+0
+0.8
+-0.5
+0
+0.5
+1.0
+V
+- In 2
+0.7
+0
+100
+200
+300
+400
+500
+w (# initial conditions)Based on Springer Nature LATEX template
+14
+Lyapunov Exponents for Temporal Networks
+Figure 4 shows the estimated of the nMLE obtained both using Wolf’s
+approach [panel (a)] and Kantz’s approach [panel (b)]. These are from aver-
+aging ℓ (Wolf) and Λ (Kantz) over w = 500 initial graph conditions sampled
+from S. In both cases, the average quickly stabilises for w ≈ 100, and we
+obtain estimates λW
+nMLE ≈ λK
+nMLE ≈ 0.685, very close to the ground true
+λMLE = ln 2 ≈ 0.693.
+Fig. 5 Volume-averaged distance ⟨dk⟩volume as a function of time k, for a network dynamics
+evolving according to a chaotic logistic map xt+1 = 4xt(1 − xt), polluted with extrinsic
+Gaussian noise N(0, σ2) as described in the text, for four different noise intensities σ =
+0, 10−3, 10−2, 10−1. The exponential expansion phase –which systematically suggests the
+same exponent ln 2, as expected– is gradually erased as the noise intensity increases.
+4.3 Noisy chaotic networks
+To explore how noise contamination can complicate the estimation of the
+nMLE, we proceed to generate a temporal network S from Eq. (15) by using
+the dictionary trick, where before the network mapping, the original chaotic
+signal is contaminated by a certain amount of white Gaussian noise N(0, σ2)5.
+As we did in Section 3, we remove potential algorithmic biases by discarding
+⟨d0⟩ for the computation of Λ. Results are summarised in Fig. 5. The main
+observation is that noise pollution tends to reduce the extent of the exponen-
+tial phase (i.e., the saturation time τ decreases). For small amounts of noise,
+this phase is still observable, and the estimated nMLE continues to be con-
+sistent with that of the noise-free case. When the noise intensity is above a
+certain threshold, noise effectively hides the chaotic signal, and the exponential
+phase can no longer be identified, resulting in an apparent vanishing nMLE.
+These results are consistent with intuition and with the typical phenomenology
+observed in noisy chaotic time series [10], [11].
+5Note that we discard realizations of the noise that take the scalar variable outside the unit
+interval.
+
+10
+10°
+=0
+0=10-3
+0=10-2
+0=10-1
+exp(In2 k)
+10-4
+2
+4
+6
+8
+10
+12
+14
+kSpringer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+15
+4.4 Results for the parametric logistic map
+Here we consider the logistic map xt+1 = rxt(1 − xt). For each value of the
+parameter r > r∞, using the dictionary trick we generate a long sequence of
+networks Sr with the desired chaoticity properties, and proceed to estimate
+the network Lyapunov exponent using the method detailed in Section 2. In
+panel (a) of Fig.6 we plot λW
+nMLE vs λMLE of the map, for a range of values of
+the parameter r. The agreement is excellent in the region of parameters where
+the temporal network is chaotic.
+Fig. 6
+Panel (a): Scatterplot of λW
+nMLE, estimated from a temporal network Sr generated
+via the dictionary trick (see the text) from a logistic map xt+1 = rxt(1 − xt) for a range
+of values of r, vs the ground true λMLE. The solid line is the diagonal of perfect agreement
+y = x, highlighting the good agreement found in the chaotic region. The legend states
+the goodness of fit metric R2 of the fit of dk to an exponential function. The method is
+unable to capture negative Lyapunov exponents (observe that in those cases the R2 of the
+exponential fit is very bad), but these cases can easily be identified as periodic orbits using
+the autocorrelation function [8], see text for details. Panel (b): Estimate of the negative
+nMLE using two initially close temporal networks generated via the dictionary trick from the
+logistic map at r = 3.4 (period-2 orbit), where one initial condition belongs to the period-2
+attractor and the other is outside the attractor (see the text for details).
+4.5 A note on negative Lyapunov exponents
+The classical approach to estimate the MLE from a single trajectory displayed
+by Wolf and Kantz algorithms –based on recurrences of the trajectory– is, by
+construction, unable to capture negative MLEs. The reason is straightforward:
+once in the periodic attractor, the trajectory sequentially visits each element
+of the periodic orbit, and thus we won’t find recurrences that are close but
+away from the initial condition of interest. Accordingly, our method to esti-
+mate nMLE cannot work in that case for the same reasons, as confirmed in
+Fig. 6(a). This drawback can be solved using two alternative approaches.
+First, it is well known that a periodic time series has an autocorrelation
+function that peaks at the period of the time series. Interestingly, a recent
+work [8] has operationalised a way to estimate the autocorrelation function
+
+10-2
+r = 3.4
+入MLE = -0.137
+exp(-0.143k)
+10-3
+(b)
+0
+5
+10
+15
+20
+25
+k0.8
+R?
+< 0.2
+0.6
+E (0.2,0.7)
+E
+[0.7,0.8]
+0.4
+E [0.8,0.9]
+> 0.9
+0.2
+(a)
++
+0
+++
+-0.5
+0
+0.5
+ΛMLEBased on Springer Nature LATEX template
+16
+Lyapunov Exponents for Temporal Networks
+of temporal networks, whereby temporal networks that display periodicity
+are well characterised by a network version of the autocorrelation function.
+Accordingly, from a practical point of view, before attempting to estimate the
+nMLE of a given temporal network, it is sensible to apply the procedure of [8]
+and exclude that the temporal network is periodic –which would typically6
+mean a negative nMLE–. Once this test is done, it is sensible to conduct the
+nMLE analysis presented in this paper.
+Second, it is indeed possible to estimate negative nMLEs if one has access
+to the latent graph dynamical system (GDS), as in this case one does not need
+to undergo a Wolf/Kantz approach and one can generate through the GDS
+temporal networks from close initial graph conditions. To illustrate this, in
+panel (b) of Fig. 6 we plot the graph distance of two initially close networks
+evolving according to the logistic map for a value of the map’s parameter for
+which the orbit is periodic (the TNs are again generated via the dictionary
+trick). One initial condition is set at one of the orbit elements, whereas the
+other initial condition is a network close in graph space (but outside the peri-
+odic attractor). As we can see, there is an exponential shrinking of the initial
+distance, and the slope gives an estimate of the nMLE, which in this case is
+negative and in good agreement with the theoretical result.
+5 High-dimensional chaotic networks
+We now consider the case of high-dimensional chaotic dynamics for temporal
+networks. We first introduce a generative model, based on coupled Map Lat-
+tices (CML). These are high-dimensional dynamical systems with discrete time
+and continuous state variables, widely used to model complex spatio-temporal
+dynamics [20] in disparate contexts such as turbulence [21, 25], financial mar-
+kets [26], biological systems [27] or quantum field theories [28].
+Globally Coupled Maps (GCMs) [29] are a mean-field version of CMLS, where
+the diffusive coupling between the entities in a CML is replaced with an all-
+to-all coupling, mimicking the effect of a mean-field. We consider a globally
+coupled map of m entities, of the form
+xi(t + 1) = (1 − α)F[xi(t)] + α
+m
+m
+�
+j=1
+F[xj(t)], i = 1, 2, . . . , m,
+(16)
+where F(x) = 4x(1 − x), x ∈ [0, 1], where α ∈ [0, 1] is the strength of the
+mean-field coupling. In the uncoupled case α = 0, the system is composed of m
+independent fully chaotic dynamics. Its attractor is thus high-dimensional and,
+since there are m Lyapunov exponents all equal to ln 2, we have λMLE = ln 2.
+6Some pathological cases exist for which we can have seemingly periodic behavior but not a
+negative MLE, e.g. when we have a disconnected attractor composed by a number of bands and
+a trajectory that periodically visits the different chaotic bands
+
+Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+17
+At the other extreme, for complete coupling α = 1, the system is fully synchro-
+nized (i.e., for any time t we have xi(t) = xj(t) for all i, j), and the dynamics
+is reduced to the one-dimensional dynamics, again with λMLE = ln 2. We add
+that complete synchronization is in fact known to occur for α > 1/2 [29].
+For intermediate coupling the system shows a number of different macroscopic
+phases [29]. Among these one finds high-dimensional chaos for weak coupling
+α < 0.2. This is the so-called ‘turbulent state’. Interestingly, for CMLs with
+diffusive coupling, a scaling law has been established [22]
+λMLE = log 2 − βα1/p,
+(17)
+where p indicates the type of nonlinearity of F(x), i.e. p = 2 for the logistic
+map, p = 1 for tent maps, etc. Results for GCM are less clear. However, when
+the mean-field coupling can be considered ‘thermalized’ (i.e., independent of x)
+[23, 24] then Eq. (17) holds for β = 1. However such thermalization is known
+to be true only for tent maps (p = 1) and not logistic maps.
+Here we consider the range α ∈ [0, 0.2], i.e., the turbulent state of the
+GCM. We interpret the collection {xi}m
+i=1 as the (weighted) edge set of a fully
+connected undirected network backbone of n nodes and m = n(n−1)/2 edges.
+Once the time series of each edge {xi(t)}T
+t=1 has been computed from Eq. (16),
+we proceed to binarise each edge activity by using a two-symbol generating
+partition as follows: values xi(t) < 1/2 are mapped into the symbol 0, and
+xi(t) ≥ 1/2 onto the symbol 1[30]. Note that the use of a generating partition
+ensures that the symbolised (binary) series preserves the chaotic properties
+of the original signal [31–33]. Finally, we convert the (binary) evolution of
+the edges into a time-dependent adjacency matrix, thereby constructing a
+temporal network S. For values of α in the weak-coupling regime, we expect
+the temporal network to display sensitive dependence on initial conditions.
+In practice, the Wolf/Kantz methods of inferring the larges Lyapunov
+exponent proposed in the paper would require a very long sequence S for close
+enough recurrences to be observable in a system with large n. However, here
+we have access to the actual underlying GDS via Eq. (16). Given that the goal
+of this section is to show evidence that high-dimensional chaotic networks can
+be generated and their nMLE be estimated, we can use the GDS to generate
+the temporal network for any required initial condition. Accordingly, for a
+given initial condition {xi(0)}m
+i=1, we construct a perturbed copy {x′
+i(0)}m
+i=1
+(where |x′
+i(0) − xi(0)| < ϵ for some small choice of ϵ), generate temporal
+networks for both of these initial conditions, and track the network distance
+between the copies over time. We do this for 100 replicas to extract a volume-
+averaged distance, and then for 50 different initial conditions. Observe that,
+at odds with the model developed in the previous section, here the number of
+edges in each network snapshot is not fixed, and thus the network phase space
+is substantially larger. Similarly, the normalization factor of the distance
+function is now simply the total number of possible edges, n(n − 1)/2.
+
+Based on Springer Nature LATEX template
+18
+Lyapunov Exponents for Temporal Networks
+Fig. 7 Panel (a): Semi-log plot of the volume-averaged distance ⟨dk⟩volume as a function
+of the time step k, for a temporal network extracted from the GCM model with coupling
+constant α = 0, 0.05, 0.1. We observe an exponential phase, with different exponents for
+each value of the coupling constant. The solid lines are the best exponential fits. Panel (b):
+Estimate of the network maximum Lyapunov exponent λnMLE vs the coupling constant α
+for temporal networks generated from a GCM of logistic maps (blue circles) and tent maps
+(black squares). For each α, a total of 50 initial conditions were considered, and a ball of 100
+points for each initial condition was used. Error bars are standard deviation from the average
+over 50 different temporal network realizations. Blue lines report the theoretical predictions
+for logistic and tent CMLs [Eq. (17)], whereas the black line reports the theoretical prediction
+for GCM with thermalised mean-field, applicable for tent GCMs only.
+Results for a network of n = 100 nodes are shown in Fig. 7. In panel (a) we
+plot ⟨dk⟩volume vs time k, for three different coupling constants α = 0, 0.05, 0.1
+in the weak coupling regime. In every case we find a clear exponential phase.
+The exponent in the uncoupled phase α = 0 is indeed equal to ln 2, as expected,
+further validating the method. For increasing values of the coupling, interest-
+ingly, the nMLE seems to decrease and, as a byproduct, the saturation time τ
+increases. In Fig. 7(b) we plot, as blue dots, the estimated λnMLE as a function
+of the coupling α ∈ [0, 0.2], indeed showing a clear decrease. Such decrease
+might be induced by the fact that the m degrees of freedom are now coupled
+in some nontrivial way. Blue lines correspond to the theoretical predictions for
+logistic and tent CMLs obtained from Eq. (17). For completeness, we repeated
+the same analysis for network GCMs constructed from tent maps where λMLE
+is explicitly known (black line): F(x) = 1 − 2|x|, with x ∈ [−1, 1] and a sym-
+bolisation partition with x < 0 mapped to the symbol 0, and x ≥ 0 mapped
+to 1. Results for this case are plotted as black squares in Fig. 7(b)
+We conclude that (i) the TN thereby generated exhibits high-dimensional chaos
+and its nMLE, reconstructed with the methods we have developed, shows the
+expected behaviour, and (ii) this validation shows that the method works with
+TNs where not only the position but also the total number of edges itself
+fluctuates over time.
+
+(a)
+10
+10-
+2
+α=0
+exp(0.68 k) (R² = 0.99)
+α = 0.05
+exp(0.56 k) (R² = 0.99)
+α = 0.1
+10-3
+exp(0.44 k) (R2 = 0.99)
+0
+5
+10
+15
+k0.7
+(b)
+0.6
+0.5
+入nMLE
+E
+0.4
+logistic map network
+ log2 -βVa
+0.3
+.... log2 - βa
+ tent map network
+0.2
+ log2 + log(1-a)
+0
+0.1
+0.2
+aSpringer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+19
+6 Discussion
+In this work, we propose to look at temporal networks as trajectories of a
+latent Graph Dynamical System (GDS). This interpretation naturally leads
+us to explore whether these trajectories can show sensitive dependence on
+initial conditions, a fingerprint of chaotic behaviour. We have proposed a
+method to quantify this, and defined and computed the network Maximum
+Lyapunov Exponent (nMLE) for temporal network. Since the latent GDS
+is rarely available in practice, our algorithm exploits the recurrences of the
+temporal network in graph space. It generalizes the classical approaches of
+Wolf and Kantz to networks. We have validated the method by generating
+different synthetic GDS with known ground-truth nMLE.
+Conceptually speaking, quantifying chaos in the trajectory of structured
+objects (in our case, mathematical graphs) is somewhat close in spirit to quan-
+tifying the dynamical stability of (lattice) spin systems. Thus our approach
+shares some similarities with the damage-spreading [34] and self-overlap meth-
+ods [35] in statistical physics, and their applications to cellular automata [36]
+and random Boolean systems [37].
+Observe that we have focused on exponential expansion on nearby con-
+ditions –i.e., sensitive dependence–, since one of the goals of the paper is to
+conceptually postulate the existence of chaotic networks and to potentially
+operationalise a way to measure this deterministic fingerprint in observed TNs,
+without needs to having access to the underlying GDS. However, our approach
+can be straightforwardly extended to non-exponential divergence, e.g. alge-
+braic or otherwise, simply by suitably modifying the definition of expansion
+rates, thus yielding a way to quantify other types of dynamical instability.
+The rationale of this work is to consider graphs evolving over time as whole
+–yet not punctual– objects [8], and thus consider its evolution in graph space.
+It is however true that this approach might have a limitation for (large) real-
+world temporal networks, as it is often difficult to observe recurrences in high-
+dimensional space. A possible solution is to extract suitable scalar variables
+from the network, analyse sensitive dependence on initial conditions in each
+of them, and extract a consensus. We leave this approach for future work.
+Observe that throughout this work we have considered labelled networks.
+This choice was used for, convenience, illustration, and because TNs are usu-
+ally labelled, but we expect that a similar approach is possible for unlabelled
+TNs, i.e. graphs that evolve over time according to a certain graph dynamics.
+In this latter case, each network snapshot is no longer uniquely represented
+by a single adjacency matrix, in the sense that permutations of the rows and
+columns of the matrix lead to an equally valid description. It is then clear
+that one needs to use graph distances showing invariance under permutation
+of rows and columns in the adjacency matrices [38]. This could be, for exam-
+ple, distances based on the network spectrum, or graph kernels [39]. We leave
+this interesting extension as a question for further research, as well as the
+
+Based on Springer Nature LATEX template
+20
+Lyapunov Exponents for Temporal Networks
+quantification of the full Lyapunov spectrum beyond the maximum one.
+Finally we would like to add that the fact that the method does not rely
+on knowing the GDS and instead directly estimates the nMLE from tempo-
+ral network trajectories enables the investigation of these matters in empirical
+temporal networks. We foresee a range of potentially interesting applications
+in physical, biological, economic and social sciences –as indeed temporal net-
+works pervade these disciplines–. This approach is specially appealing in those
+systems where we don’t have access to the ‘equations of motion’ but it is sen-
+sible to expect some underlying deterministic dynamics, i.e. physical systems,
+but the approach is also extensible to systems with socially or biologically-
+mediated interactions, for instance: do flocks of birds [40–42] or crowd behavior
+[43], adequately modelled as temporal proximity networks, show chaos?
+Appendix: Graph distances
+Consider two adjacency matrices A, and B, each with binary entries (0 or
+1), describing two simple unweighted graphs with n nodes. The so-called edit
+distance [18] is a matrix distance defined as
+d(A, B) =
+n
+�
+i,j=1
+|aij − bij|.
+(18)
+The object d(A, B) counts the number of entries that are different in A and B.
+For simple undirected graphs (symmetric adjacency matrices), we need
+to account for the fact that the number of edges is only half the number of
+positive entries of the adjacency matrix, and therefore d(A, B)/2 measures
+the number of edges that exist in one graph but not on the other. We have
+d(A, B)/2 = 0 if and only if A = B. It is also easy to see that d(A, B)/2 only
+takes integer values for symmetric adjacency matrices A and B. If A ̸= B, then
+1 ≤ d(A, B)/2 ≤ n(n − 1)/2. We have d(A, B)/2 = 1 when the two graphs are
+identical except for one edge, which is present in one graph and absent in the
+other.
+One can directly use this unnormalized distance (as we do in Section 3) or
+subsequently normalize d(A, B) using different strategies, e.g. one can divide
+it over n(n − 1)/2 (as we do in Section 5), or just divide over the maximum
+possible distance, if further restrictions are imposed between A and B (as we
+do in Section 4).
+If we further impose that both graphs have the same number of edges, then
+the lower bound cannot be attained and 2 ≤ d(A, B)/2 when A ̸= B. This
+lower bound is reached when we only need a single edge rewiring to get from
+
+Springer Nature 2021 LATEX template
+Lyapunov Exponents for Temporal Networks
+21
+the first graph to the second. One can thus define the rewiring distance
+d(A, B) = 1
+4
+n
+�
+i,j=1
+|aij − bij|.
+(19)
+applicable for simple graphs (i.e. no self-links). This quanity measures the
+total number of rewirings needed to transform A into B when the associated
+graphs are simple, unweighted, undirected (symmetric adjacency matrices)
+and have the same number of nodes and edges.
+The rewiring distance above is based on the concept of non-overlapping
+edges, i.e., edges that are present in one graph, but not in the other. Thus,
+the edit and rewiring distances are based on |aij − bij| for the different edges,
+and hence assign the same importance to the presence or absence of an edge.
+One can instead construct measures of distance based on the number of links
+that are present in both networks. If the edge ij is present in both graphs
+then aijbij = 1, while this product is zero otherwise. One can prove that the
+following function is a distance [19]:
+d(A, B) = 1 −
+1
+2|E|
+n
+�
+i,j=1
+aijbij.
+(20)
+We replicated the analysis in Sec.
+4 for the distance defined above, and
+results (resulting nMLE) coincide.
+Acknowledgments We thank Federico Battiston for helpful comments on
+initial phases of this research, and Emilio Hern´andez-Fern´andez, Sandro Mel-
+oni, Lluis Arola-Fern´andez, Ernesto Estrada, Massimiliano Zanin, Diego Paz´o
+and Juan Manuel L´opez for helpful discussions around several aspects of the
+work. AC acknowledges funding by the Maria de Maeztu Programme (MDM-
+2017-0711) and the AEI under the FPI programme. LL acknowledges funding
+from project DYNDEEP (EUR2021-122007), and LL and VME acknowledge
+funding from project MISLAND (PID2020-114324GB-C22), both projects
+funded by the Spanish Ministry of Science and Innovation. This work has
+been partially supported by the Mar´ıa de Maeztu project CEX2021-001164-M
+funded by MCIN/AEI/10.13039/501100011033.
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+

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+1 
+ 
+Machine-learning-assisted environment-adaptive thermal 
+metamaterials 
+Peng Jin1, Liujun Xu2,3, Guoqiang Xu2, Jiaxin Li2, Cheng-Wei Qiu2,* and Jiping Huang1,* 
+ 
+1Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro 
+and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China 
+2Department of Electrical and Computer Engineering, National University of Singapore, 
+Singapore 117583, Singapore 
+3Graduate School of China Academy of Engineering Physics, Beijing 100193, China 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+*Corresponding authors. Emails: chengwei.qiu@nus.edu.sg; jphuang@fudan.edu.cn 
+
+2 
+ 
+Abstract 
+Adaptive metamaterials have prevailed recently owing to their extraordinary features like 
+dynamic response to external interference. However, highly complicated parameters, narrow 
+working ranges, and supervised manual intervention are still long-term and tricky obstacles to 
+the most advanced self-adaptive metamaterials. To surmount these barriers, we present 
+environment-adaptive thermal metamaterials driven by machine learning, which can 
+automatically sense ambient temperatures and regulate thermal functions promptly and 
+continuously. Thermal functions are robust when external thermal fields change their directions, 
+and simulations and experiments exhibit excellent performance. Based on this, we further 
+design two metadevices with on-demand adaptability, performing distinctive features with 
+isotropic materials, wide working temperatures, and spontaneous response. This work provides 
+a paradigm for intelligent diffusion metamaterial design and can be extended to other diffusion 
+fields, responding to more complex and variable environments. 
+ 
+ 
+
+3 
+ 
+Introduction 
+Metamaterials [1-5] have drawn intensive attraction due to their unprecedented ability to 
+manipulate physical fields. Profit from computer numerical control and three-dimensional 
+printing technology, metamaterials with novel functions are fabricated according to given 
+parameters and applied to laboratories or industries. Traditional metamaterials mainly focused 
+on static cases [6-15], lacking tunability for variable scenes. To tackle this issue, tunable 
+metamaterials with dynamic features have emerged, covering optics [16,17], acoustics [18], 
+and thermotics [19-21]. For example, many advanced thermal functions have been realized, 
+including macroscopic thermal diodes [19], tunable analog thermal materials [22], path-
+dependent thermal metadevices [23], and tunable hybrid thermal metamaterials [24]. On the 
+other hand, adaptive thermal metadevices are presented to maintain the robustness of functions 
+against environmental changes or switch functions depending on application scenes [20,25-30]. 
+However, the achievement of state-of-the-art self-adaptive thermal metamaterials is confronted 
+with three longstanding and strong barriers. Firstly, adaptive thermal metadevices with robust 
+functions usually require extremely complicated parameters [25-27], which are difficult to 
+prepare from natural bulk materials. Secondly, existing adaptive metadevices, especially 
+macroscopic thermal diodes [19] and energy-free temperature trapping [20], are limited to a 
+specific temperature range related to the phase change temperature of shape memory alloys. 
+Finally, most tunable or adaptive metamaterials [16-30] need to be adjusted through manual 
+control rather than automatically, lacking self-cognitive ability. 
+Recently, intelligent materials, involving interdisciplinary research and combining 
+intelligent algorithms with material design, have motivated applications in optics [31-33], 
+
+4 
+ 
+nanotechnology [34], theoretical physics [35], materials science [36], and thermal science [37]. 
+These advances inspire ideal self-adaptive thermal metamaterials with full embracement of 
+intelligence. In principle, ideal self-adaptive thermal metadevices should automatically 
+(without human aid) and timely adjust their dynamic components to keep function stable or 
+switch functions continuously in response to the broad range of ambient temperature change. 
+Developing such self-adaptive thermal metamaterials is highly desirable to be valuable in situ 
+scenes. However, the demanding technical performance requires an appropriate actuation 
+mechanism that integrates an algorithm-driven intelligent system with thermal metamaterial 
+design. Although these metamaterials have been used to design advanced self-adaptive optical 
+cloaks in wave systems [38], they fail in diffusion systems like heat transfer due to the lack of 
+controllable degrees of freedom. Existing machine-learning-based thermal metamaterials are 
+dictated by the inverse design method [39-42], which only calculates the parameters of 
+materials and sizes for desired functions. In addition, once such metamaterials are prepared, 
+their functions are not switchable, lacking response to various scenes. 
+Here, we introduce a machine-learning-assisted intelligent system and propose 
+environment-adaptive thermal metamaterials driven by big data. As a conceptual 
+implementation, we design an intelligent temperature gradient controller. We load the pre-
+trained artificial neural network into a hardware system and combine it with the bilayer 
+structure [43]. Depending on the sensing-feedback ambient temperatures, the thermal 
+conductivity of a spinning component could be adjusted to achieve a tunable temperature 
+gradient in a target region, verified by finite-element simulations and experiments. We then 
+propose two applications with on-demand adaptability. One is a thermal signal modulator with 
+
+5 
+ 
+functional robustness, making original thermal signals clearer. The other is an intelligent 
+thermoelectric generator with intelligent functional choice, which can automatically adjust the 
+electromotive force generated by thermoelectric materials [44] based on the ambient 
+temperature. The intelligent environment-adaptive thermal metamaterial features isotropic 
+materials, unlimited working temperatures, and cognitive responsiveness. A handy actuation 
+mechanism also integrates machine-learning-driven intelligent systems with diffusion 
+metamaterial design. Our work brings the design of self-adaptive diffusion metamaterials to a 
+new stage without human intervention. 
+Architecture 
+of 
+the 
+intelligent 
+temperature-gradient 
+controller 
+The architecture of the intelligent temperature-gradient controller is presented in Fig. 1. It 
+contains four main modules: a temperature acquisition module (micro infrared camera), a 
+computing system with a pre-trained artificial neural network (ANN), a stepper motor, and a 
+bilayer structure. We aim to manipulate the temperature gradient of the target region based on 
+the feedback of temperature information of its surroundings. Here, as a proof-of-concept 
+implementation, we consider a two-dimensional system with a bilayer metal structure. The 
+target region is the core region Ω!  consisting of poly-dimethylsiloxane (PDMS). The 
+component of the inner layer (Silicone pad) is approximately adiabatic for precise control of 
+thermal fields of Ω!, and the outer layer as a compensation layer (Magnesium alloy) is intended 
+not to disturb the thermal fields of the background (Inconel alloy). The thermal conductivity 
+from the inside out is 0.15, 1, 72.7, and 9.8 W m-1 K-1, respectively. After setting R1 = 30 mm 
+and R2 = 53 mm, R3 can be calculated as 60 mm [43]. 
+
+6 
+ 
+To characterize the temperature information of the bilayer structure’s surroundings, we 
+choose a series of discrete positions around the outer layer and extract their temperature using 
+a micro infrared camera. As shown in Fig. 1, the blue dashed circle is the chosen bilayer 
+structure’s surrounding, and the position marked 0◦ is the first position. Then, we take the 
+temperature of the amount of N equally spaced positions on the circumference in a 
+counterclockwise direction, serving as the input layer of ANN. 
+Thanks to the tunable analog thermal material [22], by spinning (angular velocity: 𝜔!) 
+the PDMS in the core region Ω!, the effective thermal conductivity of the spinning medium 
+can be tuned from near-zero (𝜔!	= 0) to near-infinity (larger 𝜔!). On the other hand, the 
+effective thermal conductivity of Ω! is the crucial factor affecting the temperature-gradient 
+distributions of Ω!; see lower part of Fig. 1 for an intuitive description. Color denotes 
+temperature profiles of the core region, and the white lines represent the isotherms from finite-
+element simulations. Here, sparser isotherms correspond to a smaller temperature gradient. 
+For the sake of “intelligence”, we utilize ANN to establish a mapping between extracted 
+temperature information (input data: 𝑻(#)) and the angular velocity of Ω! (output data: 𝜔!). 
+In Fig. 1, we show the structural component of the ANN, which is fully connected with four 
+hidden layers (50 neurons per layer). Activations of all neurons in the next layer are determined 
+by activations of those in the current layer (𝑯(%)), represented by 
+ 
+⎩
+⎪
+⎨
+⎪⎧𝑯(%&!) = ReLU0𝑾(%)𝑻(%) + 𝒃(%&!)4, 𝑖 = 0
+𝑯(%&!) = ReLU0𝑾(%)𝑯(%) + 𝒃(%&!)4, 0 < 𝑖 < 4
+𝜔! = ReLU0𝑾(%)𝑯(%) + 𝒃(%&!)4, 𝑖 = 4
+ 
+(1) 
+where ReLU	(a) = max(0, a) is the rectified linear unit function. W and b are weights and 
+biases for the neurons. i is the ordinal number of layers, and 𝑖	 = 	0 represents the input layer. 
+
+7 
+ 
+As the ANN is a data-driven-based model, we then prepare the dataset (see Supplementary Note 
+1). Finally, via the back propagation algorithm, the proposed ANN with selected 
+hyperparameters is well trained by the dataset (see Supplementary Note 1). 
+When the thermal field of the bilayer structure reaches equilibrium, the temperature 
+information (𝑇!, 𝑇', 𝑇(, ..., 𝑇)) at the circle with a radius of R3 = 60 mm is collected by micro 
+infrared camera and imported into the computing system as an array of input signals for the 
+pre-trained ANN. The computing system’s output signal is the stepper motor’s spinning angular 
+velocity 𝜔!. Considering a case where the output spinning angular velocity is 0 rad s-1, static 
+PDMS possesses thermal conductivity with 0.15 W m-1 K-1. At this time, the temperature 
+gradient in Ω! reaches its maximum value. 
+Intelligent response in omnidirectional simulations 
+Finite-element simulations are first used to demonstrate the performance of an intelligent 
+temperature-gradient controller. For proof-of-concept verification, we consider a two-
+dimensional bilayer structure whose components and sizes are the same as mentioned above. 
+The system’s left (right) end connects to a hot (cold) source. In our simulations, the cold source 
+(𝑇* = 283 K) is fixed, while the hot source (𝑇+) is changeable. We first extract the temperature 
+data (𝑇𝒂
+(#),	𝑇𝒃
+(#),	𝑇𝒄
+(#)) of N = 36 equally spaced positions in the white dashed circle in three 
+cases (𝑇+ = 293, 303, 313 K) of a static bilayer structure, as shown in Fig. 2a. For each case, 
+the first data is the temperature of the position marked 0◦ in the dashed circle. The order of 
+taking the temperature of these positions is counterclockwise, serving as the input layer of the 
+pre-trained ANN. Hence, via the pre-trained ANN, the spinning angular velocity 𝜔! of PDMS 
+is calculated individually as 0.10, 0.00067, and 0 rad s-1. After setting the above parameters 
+
+8 
+ 
+(𝑇+	and	𝜔!) in finite-element simulations, we show these three temperature profiles (color 
+distributions) of the bilayer structure; see Fig. 2b. No matter how the angular velocity 𝜔! of 
+PDMS changes, the temperature distributions of the background will not be disturbed (see 
+Supplementary Note 2). Finally, their corresponding temperature-gradient distributions in Ω! 
+are given in the right part of Fig. 2c. For comparison, we show the temperature-gradient 
+distributions in Ω! of pure background (size: 200 × 200 mm2) with three hot sources; see left 
+part of Fig. 2c. As anticipated, there is a mapping relationship between lowest/highest 𝑇+ − 𝑇* 
+and highest/lowest angular velocity 𝜔! (or say, lowest/highest |∇𝑇| in Ω!) in above scheme. 
+Further, the range of originally external temperature-gradient field [(𝑇+ − 𝑇*)/𝐿	: 50 to 150 K 
+m-1] can be adjusted to a wider range of temperature gradient (|∇𝑇|: 0 to 238.5 K m-1) in the 
+target region Ω!. 
+In the actual scene, we are unsure in which direction the external thermal field is exerted 
+on the bilayer structure. Therefore, we should ensure that the performance of the intelligent 
+temperature-gradient controller will not be affected by the change in the direction of the 
+external thermal field. In particular, we guarantee the hot and cold sources consistent with the 
+above and rotate them 30, 60, and 90◦ around the center of the bilayer structure. Apparently, 
+temperature distributions in the white dashed circle is different from each other when rotating 
+the same 𝑇+ and 𝑇*; see 𝑇(#), 𝑇/(#), 𝑇//(#), and 𝑇///(#) in Figs. 2a,d,g,j. Subsequently, we 
+calculate the angular velocity 𝜔! in cases with 𝑇+ = 293, 303, 313 K in the above four 
+directions using the pre-trained ANN and perform finite-element simulations. Finally, Figs. 
+2c,f,i,l verify that the performance of the intelligent temperature-gradient controller has good 
+robustness to the influence of external thermal flow’s direction. Such intelligent metadevice 
+
+9 
+ 
+also maintains functional stability under other cold sources and non-uniform external thermal 
+fields (see Supplementary Note 3). In addition, the calculated 𝜔! from pre-trained ANN is 
+almost consistent with the set 𝜔!,123456 when external hot and cold sources are given, as 
+shown in Figs. 2m-o. 
+Experimental realization and measurements 
+The intelligent temperature-gradient controller contains four parts: a micro infrared 
+camera, a computing system with pre-trained ANN, a stepper motor, and a bilayer structure; 
+see the real experimental setup in Supplementary Figure S7. A bilayer structure with a thickness 
+of 2 mm is connected to a hot and cold container on the two sides, serving as heat baths. Its 
+components and sizes are the same as those in simulations. The micro infrared camera is 
+controlled by the computing system. Every time the infrared camera is started, it measures the 
+temperature distribution of the bilayer structure and transmits the temperature data to the 
+computing system. The computing system consists of power, a microcomputer Raspberry Pi, a 
+power of motor driver, and a motor driver. A pre-trained ANN program runs in the Raspberry 
+Pi. The input data 𝑇(#) is from the temperature data measured by the micro infrared camera. 
+After the program processing, the computing system extracts the temperature data of the bilayer 
+structure’s surroundings, provided to the input layer 𝑇(#) of ANN. When reading the input 
+data 𝑇(#), the pre-trained ANN program in the computing system calculates the angular 
+velocity 𝜔! of PDMS in the core region Ω! of the bilayer structure. The corresponding signal 
+of controlling 𝜔! is transmitted to the stepper motor via the motor driver. Finally, the PDMS 
+spins around the center, driven by the stepper motor, and the temperature-gradient distribution 
+in Ω!  is regulated. To verify the performance of the intelligent device, we first set the 
+
+10 
+ 
+temperatures of the hot and cold baths to 293 K and 283 K. After starting this intelligent system, 
+the temperature data 𝑇𝒂
+(#) are measured and the angular velocity 𝜔! of PDMS is calculated 
+as 0.118 rad s-1 via pre-trained ANN, see Fig. 3a. Fig. 3b shows the temperature profile of the 
+bilayer structure recorded by infrared camera Fotric 430. Note that in the core region, the 
+temperature distribution is uniform, and in the background region, the temperature field is 
+nearly undistorted. Subsequently, we fixed the temperature of the cold bath to 283 K and 
+changed the temperature of the hot bath to 303 K. When the temperature field is stable, we start 
+the intelligent device again. The measured temperature data 𝑇𝒃
+(#) and the calculated 𝜔! =
+	0.0007 rad s-1 are shown in Fig. 3c. The temperature profile of the bilayer structure is shown 
+in Fig. 3d. At this time, the background temperature field is still undisturbed, and the uniformity 
+of the temperature distribution in the core region is slightly broken. Finally, with the same cold 
+bath, we further set the hot bath to 313 K. Fig. 3e displays the measured temperature data 𝑇𝒄
+(#). 
+As anticipated, the calculated 𝜔! is 0 rad s-1. Therefore, we get the temperature profile of the 
+bilayer structure, see Fig. 3f. We observe from Fig. 3f that the temperature distribution in the 
+core region has a maximal non-uniformity, also almost without disturbing the background 
+thermal field. Above temperature data 𝑇𝒂
+(#), 𝑇𝒃
+(#), and 𝑇𝒄
+(#), the relevant angular velocity 
+𝜔! , and their temperature profiles of the bilayer structure are consistent with the above 
+simulation results. For quantitative analysis, we use the central difference method to process 
+the discrete temperature data in Figs. 3b,d,f and obtain these temperature-gradient distributions 
+in the core region (see Fig. 3g), which is in good agreement with the simulation results in the 
+right part of Fig. 2c. As a result, we realize the manipulation of the core region’s temperature 
+gradient in the bilayer structure based on the feedback of temperature information 𝑻(#) of its 
+
+11 
+ 
+surroundings. 
+Potential applications 
+ 
+We realize a thermal signal modulator based on the intelligent temperature-gradient 
+controller for heat communications. The existing work adopted binary thermal spatial coding 
+to store information in heat communications (thermal signals) [45]. Binary 0 and 1 are 
+represented by encoding the temperature gradient in the working zone of the cloaking and 
+concentration with core-shell structures, where the core region is the working zone. For thermal 
+cloaking (concentration), there is a minimum (maximum) temperature-gradient distribution in 
+the working zone. Via the continuous arrangement of cloaking or concentration devices, 
+thermal signals could be stored and characterized by the temperature-gradient distributions in 
+the working zones of the arranged metadevices. Actually, binary encoding makes information 
+storage inefficient. Thermal signals should oscillate with space in a continuous mode to transmit 
+more encoding information simultaneously in heat communications. However, once the 
+continuous encoding is adopted, original thermal signals are easily disturbed and can only 
+oscillate with space in smaller amplitude due to thermal dissipation and thermal noise. Thanks 
+to the proposed thermal signal modulator, original disturbed thermal signals can be re-
+modulated to oscillate with space in a larger amplitude; see the schematic in Fig. 4a. We select 
+a square area (see the dashed box marked in Fig. 4a) in the working zone of devices as the 
+encoding zone. The area size is consistent with the size of the intelligent temperature-gradient 
+controller. To ensure the temperature field in the working zone of the device is not disturbed, 
+we make the effective thermal conductivity inside and outside the encoding zone consistent. 
+The original temperature gradient in the encoding zone is considered the external thermal field 
+
+12 
+ 
+of the intelligent temperature-gradient controller. In above simulations, external thermal fields 
+|∇𝑇|  vary from (𝑇+,789 − 𝑇*)/𝐿	 = 50 to (𝑇+,72: − 𝑇*)/𝐿	 = 150 K m-1. Through the 
+intelligent temperature-gradient controller, the average temperature gradient of the modulated 
+zone (Ω!;	see round dashed line marked in Fig. 4a) could be ranged from 0 to 238.5 K m-1. We 
+obtain a linear transformation relationship between the original temperature-gradient range and 
+modulated temperature-gradient range. Considering the oscillation of original thermal signals 
+driven by variable hot sources across the x direction in the sinusoidal law sin 2𝜋𝑥/𝜆, we apply 
+the linear transformation to original thermal signals and finally get the modulated thermal 
+signals, see Fig. 4b. We take 𝜆 as 10 m, and arrange 50 intelligent temperature-gradient 
+controllers (each size: 0.2 × 0.2 m2) in each range of 𝜆. Fig. 4c shows the temperature profiles 
+of several controllers and their original external thermal fields placed at x = 2.5, 17.5, 35, and 
+47.5 m (from finite-element simulations), respectively. Here, each x coordinate represents the 
+central position of each controller. We mark the original (modulated) temperature gradient 
+values in the encoding (modulated) zone of several controllers; see Fig. 4c. Such a thermal 
+signal modulator does not change the relative strength relationship between the original signals 
+but makes the difference between the original signals more obvious. 
+Conclusion and discussion 
+To sum up, we propose an environment-adaptive thermal metamaterial driven by a 
+machine-learning algorithm without manual intervention. As a conceptual verification, via pre-
+trained ANN (artificial neural network), we achieve the property of intelligent adjustment of 
+the temperature gradient of the spinning component PDMS (poly-dimethylsiloxane) of a bilayer 
+structure based on its ambient temperature. For algorithm implementation, we consider the 
+
+13 
+ 
+temperatures at a series of discrete positions on the outer contour of the bilayer structure 
+(ambient temperature) as the input data. Then, the pre-trained ANN outputs the PDMS’s 
+spinning angular velocity 𝜔!. 𝜔! can adjust the effective thermal conductivity of the spinning 
+PDMS and then adjust the temperature-gradient distribution in the core region. As for the 
+hardware implementation, we embed the pre-trained ANN into a microcomputer called 
+Raspberry Pi. One end is connected to the micro infrared camera to detect the ambient 
+temperature for the input data, and the other is connected to a stepper-motor driver to control 
+the 𝜔! of the motor, rotating the PDMS. Finite-element simulations and experiments have 
+confirmed this intelligent temperature-gradient controller. Meanwhile, the above thermal 
+manipulation is robust for the direction of external thermal fields. In addition, we design a 
+thermal signal modulator with functional robustness for the intelligent temperature-gradient 
+controller, which modulates original thermal signals clearer. 
+Incidentally, self-adaptive or intelligent devices involve two typical application scenarios. 
+The first is that the device displays stable function in a changeable environment like the thermal 
+signal modulator. The second is the device with intelligent functional choice responding to 
+environmental changes. We still design an application based on the intelligent temperature-
+gradient controller for the second typical scenario, an intelligent thermoelectric generator with 
+intelligent functional choice in a changeable environment. The adjustable temperature gradient 
+in the core region enables thermoelectric materials like Bi2Te3 to generate tunable electromotive 
+force, verified by finite-element simulations (see Supplementary Note 4). Finally, we make 
+metamaterials have the ability to perceive the environment. The proposed concept combines 
+diverse domains, such as artificial intelligence, metamaterials, energy utilization, heat 
+
+14 
+ 
+communications, and thermal management. We promise the interdisciplinary work to provide 
+new inspiration for progress in various areas, for example, intelligent thermal management in 
+chips. 
+ 
+Additional information 
+All study data are included in the article and/or Supplementary Information. 
+Acknowledgements 
+This work was supported by the National Natural Science Foundation of China to J.H. 
+(11725521 and 12035004), the Science and Technology Commission of Shanghai Municipality 
+to J.H. (20JC1414700), and the Singapore Ministry of Education to C.-W.Q. (R-263-000-E19-
+114). 
+Author contributions 
+P.J., C.-W.Q., and J.H. conceived of the idea. P.J. proposed the methodology, designed the 
+algorithm and hardware programs, and conducted the experiments. P.J., L.X., G.X., J.L., C.-
+W.Q., and J.H. made the visualization and wrote the manuscript. C.-W.Q. and J.H. supervised 
+the work. All authors contributed to the discussion and finalization of the manuscript. 
+Competing interests 
+The authors declared no competing interests. 
+ 
+ 
+ 
+ 
+
+15 
+ 
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+ 
+ 
+
+20 
+ 
+ 
+Fig. 1 The architecture of the intelligent temperature-gradient controller. The intelligent 
+controller comprises a micro infrared camera, a pre-trained artificial neural network, a stepper 
+motor, and a bilayer structure. The infrared camera measures the temperature data of the bilayer 
+structure’s surroundings. The measured temperature data is input into the pre-trained artificial 
+neural network, calculating the angular velocity 𝜔! of the spinning component in the core 
+region Ω!	of the bilayer structure. The stepper motor reads the angular velocity and drives the 
+spinning component to spin (through the spin disk). Finally, thermal functions in the core region 
+are regulated. 
+
+Input layer
+Hidden layers
+Output layer
+Ti 
+T2
+T3
+01
+2
+Spin disk
+Stepper motor
+n
+Artificialneuralnetwork(ANN)
+emperature
+Surroundings
+Angle
+Surroundings
+Variable
+Variable
+R.
+R
+Microinfraredcamera
+Bilayerstructure21 
+ 
+ 
+Fig. 2 Response of intelligent temperature-gradient controller in simulations. a N = 36 equally 
+spaced temperature data 𝑇𝒂
+(#) , 𝑇𝒃
+(#) , 𝑇𝒄
+(#)  in the white dashed circle in static bilayer 
+structures. The hot source is set as 293, 303, 313 K, respectively. The cold source is fixed at 
+283 K. The first data is the temperature of the position marked 0◦. Each temperature data is 
+taken every 10◦ in the counterclockwise direction. b Temperature profiles of the bilayer 
+
+a
+b
+d
+e
+90°
+.06
+120°
+60°
+120°
+60°
+T
+150°
+30°
+150°
+30°
+0.136
+30°
+Angle
+180°
+0°
+Angle
+180°
+0°
+@1= 0.0006
+@1= 0.00089
+313 K
+210°
+330°
+210°
+330°
+T(O)
+T (0)
+240°
+300°
+240°
+300°
+270°
+270°
+c
+f
+250
+250
+CE
+250
+250
+0=0
+@=0
+200
+200
+200
+Original
+150
+3 T. - 313 K
+150
+150
+Original
+2n T--313 K
+150
+，=0.00067
+Modt
+@1= 0.00089
+100
+c T. - 303 K
+100
+100
+T.-303K
+100 
+Modu
+50
+2 T. - 293 K
+50
+50
+A2 T--293 K
+@, = 0.10
+@, = 0.136
+0.02
+0.02
+0.059
+0.02
+0.02
+x (m)
+-0.02~0.02
+y (m)
+x (m)
+-0.02-0.02
+y (m)
+x (m)
+-0.02~0.02
+y (m)
+x (m)
+y(m)
+g
+h
+j
+k
+90°
+90°
+120°
+60°
+120°
+60°
+T "(0)
+150°
+30°
+=0.105
+150°
+30°
+@-0.115
+(0)..
+T"(0)
+.09
+.06
+283K
+4(0)
+Angle
+180°
+0
+Angle
+180°
+00
+01= 0.0008
++
+@=0.00085
+210°
+330°
+210°
+330°
+240°
+300°
+240°
+300°
+270°
+270°
+0
+00
+-
+250
+250
+0=0
+250
+250
+0m
+IVTI(K mr)
+200
+ated
+200
+200
+Original
+ted
+150
+3·T-313K
+150 
+150 
+C8 T, - 313 K
+150 
+Modu
+0=0.0008
+w1=0.00085
+100
+6T-303K
+100 
+100 -
+2 T, - 303 K
+100 
+50
+2·T=293K
+2 T, -293 K
+@, = 0.105
+50
+@, = 0.115
+0.02
+0.02
+0.02
+0.02
+0.001
+0.09
+0.02
+-0.020.02
+y(m)
+-0.020.02
+0
+y (m)
+0
+x (m)
+y (m)
+-0.02-0.02
+x (m)
+x (m)
+x (m)
+y(m)
+T= 293K T.=283K
+T. = 303 K T= 283 K
+T=313KT=283K
+0.14
+1 ×10-3
+m
+Wi.Target
+n
+0.12
+山
+W1,Target
+0.8
+.0.81
+山
+0.1
+0.08
+0.6
+Wi,Target
+(rad
+0F
+0.06
+0.4
+0.04
+0.2
+0.02
+0
+0
+0°
+30°
+60°
+90°
+0°
+30°
+60°
+90°
+0°
+30°
+60°
+90°
+Direction of external thermal field
+Direction of external thermal field
+Direction of external thermal field22 
+ 
+structure with spinning angular velocity 𝜔! = 0.10, 0.00067, 0 rad s-1, respectively. Left part 
+of c Temperature-gradient distributions |∇𝑇| in core region Ω! of static pure background 
+with three hot sources. Right part of c Temperature-gradient distributions |∇𝑇| in core region 
+Ω!  of the bilayer structure with 𝜔! = 0.10, 0.00067, 0 rad s-1, respectively. d-f Same 
+characterization with a-c when the external thermal field rotates 30◦ around the center of the 
+bilayer structure in the counterclockwise direction. g-i Same characterization with a-c when 
+the external thermal field rotates 60◦ around the center of the bilayer structure in the 
+counterclockwise direction. j-l Same characterization with a-c when the external thermal field 
+rotates 90◦ around the center of the bilayer structure in the counterclockwise direction. m-o 
+Comparison of calculated 𝜔! and targeted 𝜔!,123456 in above four directions of the external 
+thermal field in cases with 𝑇+ = 293, 303, 313 K, respectively. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+23 
+ 
+ 
+Fig. 3 Realization of the intelligent temperature-gradient controller. a,c,e Experimental 
+temperature data 𝑇𝒂
+(#), 𝑇𝒃
+(#), 𝑇𝒄
+(#) in the dashed circle with radius R3 = 60 mm, marked in 
+b,d,f, when hot bath is set to 293, 303, 313 K, respectively. The cold bath is fixed to 283 K. 
+b,d,f Measured temperature profile of the bilayer structure with spinning angular velocity 
+𝜔! = 0.118, 0.0007, 0 rad s-1, respectively. g Calculated temperature-gradient distributions 
+|∇𝑇|  in core region Ω!  of the bilayer structure with 𝜔! =  0.118, 0.0007, 0 rad s-1, 
+respectively. 
+ 
+ 
+ 
+Ta
+(0)(K)
+θ (∘)
+Tb
+(0)(K)
+θ (∘)
+θ (∘)
+Tc
+(0)(K)
+Exp.
+Simu.
+Exp.
+Simu.
+Exp.
+Simu.
+293 K
+283 K
+303 K
+283 K
+313 K
+283 K
+a
+b
+c
+d
+e
+f
+g
+ω1,Target
+ω1(rad s-1)
+0.118
+0.12
+ω1,Target
+ω1(rad s-1)
+0.0007
+0.001
+ω1,Target
+ω1(rad s-1)
+0
+0
+0∘
+θ∘
+0∘
+0∘
+Ta
+(0)
+Tb
+(0)
+Tc
+(0)
+R3= 60 mm
+0
+50
+-0.015
+100
+150
+200
+0.015
+250
+0
+0
+0.015
+-0.015
+y (m)
+x (m)
+|∇T| (K m-1)
+ω1 = 0
+ω1 = 0.0007
+ω1 = 0.118
+
+24 
+ 
+ 
+Fig. 4 Intelligent temperature-gradient controller for robust modulation of thermal signals. a 
+Schematic of the thermal signal modulator. b Comparison of modulated and original thermal 
+signals. Modulated (original) thermal signals are denoted by distributions of averaged 
+temperature gradients in the modulated zones of the controllers (only external thermal fields) 
+across the x direction. Each x coordinate represents the central position of each controller. c 
+Simulated temperature profiles of several controllers and their external thermal fields placed at 
+𝑥	 = 2.5, 17.5, 35, and 47.5 m, respectively. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Modulated zone
+Cold source (TC)
+Encoding zone
+x
+y
+z
+|∇T|
+|∇T′|
+Original 
+|∇T| = TH − TC
+L
+T(0)
+Designed 
+intelligent system
+ω1
+Modulated 
+|∇T′|
+|∇T|
+|∇T′|
+Original thermal signal
+Modulated thermal signal
+a
+238.5
+b
+ (K m-1)
+0
+10
+20
+30
+40
+50
+0
+50
+100
+150
+200
+250
+0
+50
+100
+150
+200
+x (m)
+Original
+Modulated
+c
+x = 2.5 m
+x = 17.5 m
+x = 35 m
+x = 47.5 m
+|∇T|
+Encoding zone
+TH
+TC
+T(0)
+ω1
+L
+x
+y
+Modulated 
+zone
+Ω1
+L = 0.2 m
+Original
+150 K m-1
+Modulated |∇T′| Ω1
+Unit: K
+283
+293
+303
+313
+|∇T|
+238.5 K m-1
+50 K m-1
+0 K m-1
+100 K m-1
+120 K m-1
+50 K m-1
+0 K m-1
+TC
+(fixed)
+TH
+(variable)
+Hot source (TH)
+

_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/2301.02643v1.pdf.txt

@@ -0,0 +1,717 @@
+Auto-Assembly: a framework for automated robotic
+assembly directly from CAD.
+Fedor Chervinskii§, Sergei Zobov§, Aleksandr Rybnikov§, Danil Petrov§, Komal Vendidandi§
+Λ Γ Γ I V Λ L
+Abstract—In this work, we propose a framework called Auto-
+Assembly for automated robotic assembly from design files and
+demonstrate a practical implementation on modular parts joined
+by fastening using a robotic cell consisting of two robots. We show
+the flexibility of the approach by testing it on different input
+designs. Auto-Assembly consists of several parts: design analysis,
+assembly sequence generation, bill-of-process (BOP) generation,
+conversion of the BOP to control code, path planning, simulation,
+and execution of the control code to assemble parts in the physical
+environment.
+Index
+Terms—industry
+4.0,
+smart
+manufacturing,
+cyber-
+physical systems, smart factory, manufacturing automation, ma-
+nipulators, cellular manufacturing, digital twins, robotic assem-
+bly
+I. INTRODUCTION
+Assembly planning is one of the most laborious tasks
+when releasing a new product for manufacturing. Thus, many
+algorithms and methods around computer-aided design (CAD)
+and digital twins of the factories have emerged in recent
+years that help process engineers to prepare a new design
+for assembly (Computer-aided Assembly Process Planning
+techniques [1]). An emerging trend of Industry 4.0 [2] suggests
+that a digital, highly automated factory should be able to infer
+the process from the design. In practice, even for an automated
+factory, assembly planning has to be followed by an offline-
+programming of all the robots and devices to perform the
+assembly plan.
+Additive manufacturing technology (3D printing [3]) at the
+same time has achieved a much higher rate of process design
+automation. One can simply load a CAD file into a machine
+that will yield a part of the desired design. The main question
+we are trying to address in this paper is ”Could a robotic
+cell or even the whole factory work just as a 3D printer?”.
+When loaded with target assembly CAD design and given
+input parts in specified conditions (e.g. placed in special input
+jigs) - would it perform the required assembly?
+In this work, we show how this can be achieved under
+specific constraints, paving the road for future experiments
+towards a more general approach and wider applications.
+However, the framework we propose is general enough to
+accommodate more complex designs and conditions, like
+many types of tooling and different joining technologies.
+§Equal contribution.
+[chervinskii, zobov, rybnikov, danil.petrov, vendidandi]@arrival.com
+Fig. 1: Experimental setup: robotic cell with two UR5e manip-
+ulators. Left: UR5e with a screwdriver Likratec EH2 R1030-
+A and Right: UR5e with gripper Robotiq 2F-85 with custom
+designed gripper clamps. On the table: custom-designed 3D-
+printed jigs.
+II. RELATED WORK AND BACKGROUND
+An Assembly Planning for a given design typically starts
+from identifying the mating features or joints and suggesting
+a feasible Assembly Sequence, which could be automated as
+seen in [4], [5], [6].
+To proceed to the process planning, a virtual environment,
+also known as Digital Twin [7] is necessary. There are at-
+tempts to develop a common ontology, e.g. [8], [9] and unify
+interfaces between systems [10] to support process design
+automation.
+Sierla, Seppo, et al. [11] discuss the conceptual framework
+of automated assembly planning using a digital twin. It uses
+the XML-based AutomationML [12] data modeling frame-
+work. This framework aggregates different data exchange
+formats like CAEX for plant description, COLLADA for
+geometry and kinematics of 3D models, etc.
+There is still not sufficient work in joining together process
+planning, motion planning and execution using a common
+framework. In [13] authors used artificial intelligence to solve
+a Tooling Matching problem and developed an add-on for
+Octopuz [14] to do a Motion Planning and Robot Program
+Generation for disassembly, but not testing in physical cells.
+In another work, [15] a similar pipeline is described for an
+arXiv:2301.02643v1  [cs.RO]  6 Jan 2023
+
+Screwdriver
+Gripper
+Assembly
+Assembly Jig
+Parts Unload Jigs
+Screws Unload Jigarchitectural domain, mainly focusing on parametric design
+and modular assembly.
+We claim that Auto-Assembly is the first proposed frame-
+work that can generate and execute robotic assembly process
+for an arbitrary input CAD design.
+III. PROBLEM STATEMENT AND METHOD OVERVIEW
+The main objective of our work is to create a framework
+that enables a closed loop between design and robotic manu-
+facturing. A target framework should analyse the design and
+provide a simulation of assembly, executable programs (when
+possible) and other feedback. The primary aim of the feedback
+is to help in adapting the design and manufacturing to better
+correspond to each other.
+The feedback we should provide can be split into two
+categories:
+• Successful simulation and its’ artefacts can be directly
+used to decide on physical manufacturing. Users can
+choose between different processes to choose the one,
+based on the key performance indicators (KPI) they want
+to optimize: time, tooling price, energy consumption, etc.
+• In case of a failure, the system should provide all neces-
+sary feedback that helps to change the design, robot’s
+position, choose the robots with better parameters or
+different cell configuration. Such feedback can be: failed
+operations, missing appropriate tooling, parts or tools in
+collision, unreachable states.
+To achieve this, we implement a framework described in
+detail in Section IV. Section IV-A gives an overview of
+our system and its components. Section IV-B discusses 3D
+modelling of the assembly design files that form the base of
+our data extraction pipeline. Section IV-C reviews the usage
+of this extracted data to produce a set of possible assembly
+sequences.
+Each operation in the assembly sequence is enriched with
+tooling information as discussed in section IV-D. Finding a
+specific cell that contains all the resources like jigs, robots,
+and their tooling, etc to execute all the operations needed for
+an assembly is explained in section IV-E. Section IV-F tells
+about the generation of the control code that moves the robots
+to grasp, place and fasten parts in a cell.
+In section V, we test our framework on different assemblies
+and discuss the results. In section VI, we review our findings
+from the experiments and future work.
+IV. FRAMEWORK
+A. System architecture
+Auto-Assembly framework can be divided into two parts as
+shown in Fig. 2. The first part, called ”Artefacts generation”,
+works with CAD files provided by a design engineer. It is
+intended to run once on the input data and provide artefacts,
+which can then be stored and re-used to run the assembly
+process in the simulated and physical environments. This
+part includes Assembly Sequence Generation, Tool and Cell
+Matching, Bill-of-Process (BOP) Generation and Control Code
+generation.
+The second part can be seen as a deployed environment.
+It is represented as a system where we have many services,
+providing “abilities” which can be called from the domain-
+specific Process language (PL). An example of the PL script
+can be seen in a Listing 3. Here we describe the most important
+services and their respective abilities:
+• Robot Controller
+– Abilities to control the robots on a low level. As
+input, it takes a trajectory as a list of a robot’s joint
+states, and as output, interpolates the trajectory and
+moves the robot.
+– Abilities to control tooling connected to the robot,
+like grippers, screwdrivers, etc.
+• Motion Planner
+– Ability to plan a trajectory in the cell to move a
+robot to a target pose with cell objects taken as the
+collisions.
+• Jig Controller
+– Ability to return a pose of a part in a jig with respect
+to the jig origin.
+• Assembly Service
+– Ability to retrieve the information about fasteners
+and resulting parts’ pose with respect to the cell
+origin.
+• Transform Service
+– Ability to get the position of any object inside a cell
+with respect to any object in the cell.
+• 3D Simulator
+– Abilities to load objects from cell description and
+visualize cell state.
+• Database and Message Bus
+– Abilities to publish and retrieve JSON objects. This
+component is used as a message bus and data storage.
+All system parts exchange the data in a special format
+called Factory Control Model (FCM). It can be considered
+as a schema and also is a vital part of our system since it lets
+all the components speak the same language.
+B. CAD Data Preparation and Extraction
+For any given assembly, our framework needs two design
+files.
+• Design file containing part assembly with joints. Fasten-
+ers are labelled as separate joints in order to distinguish
+them from other parts.
+• Design file containing the jigs and gripper at different
+stages of assembly like grasping, placing, etc.
+The examples of these files for an assembly are depicted in
+Figs 6 and 7 and are created by us in Fusion 360. Our method
+is CAD-software agnostic as long as we can extract the CAD
+data using an API.
+From the design file in Fig. 6, we extract the joints and part
+occurrences information using Fusion 360 API [16]. Using
+this data, a joint register is created that maps every joint to its
+parts. The joint register follows the FCM schema.
+
+Fig. 2: System architecture
+Fig. 3: Listing of PL-code implementing high-level of robot
+control. Abilities get cell state and plan trajectory are imple-
+mented by Motion Planner and execute trajectory by Robot
+Controller
+From the design file in Fig. 7, we extract the pose of gripper
+occurrence relative to the part during grasping it from the jig
+and placing it at the assembly state using the Fusion 360 API.
+We call this data as recipes.
+C. Assembly Sequence Generation
+A CAD design contains a lot of important information about
+the part’s geometries, relations, and absolute poses. But what
+it lacks - the right assembling order - is the key information
+to move towards the assembled product. Assembly sequence
+encodes the order of operations needed to be performed on
+parts by the robotic cell. Although the operations can be
+executed sequentially, assembly sequences are represented by
+polytree (directed acyclic graph whose underlying undirected
+graph is a tree). Not any such tree represents a valid and
+feasible assembly sequence:
+• only directly joined parts should be neighbours;
+• the order of operations should take into account the
+geometrical limitations;
+• the number of generated assembly sequences should be
+reasonably limited. Naturally, it grows exponentially with
+the number of parts involved. This makes it hard to check
+all the generated sequences to pick the best one according
+to some criteria.
+The assembly sequence generation step aims to solve all three
+aforementioned issues, providing a limited number of valid
+sequences. The whole process can be divided into three steps:
+• a liaison graph generation;
+• assembly sequences generation based on the obtained
+liaison graph;
+
+Artefacts generation
+CAD file:
+parts geometries
+Assembly
+- joints
+Sequence
+- labelling
+Assembly
+CAD Data
+BOP
+Control Code
+Sequences
+Extraction
+generation
+Generation
+CAD file:
+Generation
+- tooling placing
+ 3D-model files
+absolute
+BOP
+- jigs placing
+positions and
+occurrencesof
+parts
+Tool and
+- joints
+- fastening
+Cell
+features
+Matching
+recipes
+Artefacts set
+Objects
+FCM
+PL script
+objects
+geometries
+Deploy
+graph
+Physical
+PL script
+Parts
+FCM objects
+Best Run
+environment
+geometries
+graph
+Selection
+Unsuccessful simulation runloop
+deploy
+Succesful run
+Databaseand
+artefacts set
+messagebus
+Services
+Simulation
+Robot
+Motion
+Assembly
+Transform
+3D
+controller
+planner
+Controller
+Service
+Service
+Simulator1-move_robot_to_position(cell, motion_group, manipulator_service,
+2
+planning_object,position,move_type,
+3 -
+ignored_collisions,ignored_collision_pairs)(
+4 -
+rules (
+5
+~ get_cell_state(cell = cell, out cell_state = cell_state)
+6 -
+~plan_trajectory(
+7
+position= position,
+8
+motion_group=motion_group
+9
+planning_object= planning_object
+10
+planning_socket_name="eef",
+11
+move_type=move_type,
+12
+ignoredcollision pairs = ignored collisionpairs
+13
+ignored_collisions = ignored_collisions,
+14
+cell_state = cell_state,
+15
+out result_motion_plan= trajectory
+16
+)
+17
+execute_trajectory(trajectory = trajectory)
+18
+} seq
+19 -
+constraints (
+20
+execute_trajectory.@provider_id.resource_id== @manipulator_service.id
+21
+22• geometry feasibility checking based on parts geometries
+This approach we used is described in [5]. Further, the high-
+level steps, important implementation details, and differences
+with the original paper are described.
+1) Liaison graph generation: The CAD file consists of the
+individual parts combined together with joints and fasteners.
+The information about the joints is crucial to accurately
+determine parts connectivity. Considering the parts as liaison
+graph nodes, connectivity information transfers into edges in
+this graph. We extracted the information about joints and
+fasteners from the design in CAD software to build up a liaison
+graph to further analyze it and generate assembly sequences.
+2) Sequences generation: An assembly sequence deter-
+mines the order of operations on parts. The liaison graph
+itself, being undirected, doesn’t set the order of operations
+in general. But the order should be based on the liaison graph
+since the latter contains the information about the connectivity
+in the resulting assembly. Usually, there are many sequences of
+operations. [5] describes the approach of extracting all possible
+assembly sequences from the liaison graph. We followed the
+suggested approach.
+3) Geometry feasibility checking: The geometrical feasibil-
+ity of an assembly process is the fundamental property, which
+should be checked first to eliminate irrelevant sequences.
+These irrelevant sequences could contain, for example, one
+part to be joined with another part, which is trapped already
+inside the sub-assembly. To prevent this, geometrical analysis
+of sub-assemblies is used. One sub-assembly is translated step-
+by-step w.r.t another sub-assembly in one of chosen directions
+until the bounding boxes of the sub-assemblies still intersect
+and the solid bodies’ intersection is checked. If the intersection
+represents a volume, it’s impossible to join the sub-assemblies
+in the chosen direction, and the remaining directions should
+be checked.
+Choosing the directions of translations alongside step size
+is important for the result. Due to the nature of assembly
+parts and their orientation alignment, directions along the
+main coordinate axes work well in the tested assemblies. In
+other cases, information from joints from the CAD file could
+be used to determine the potential directions. Step size is
+computed based on the minimal size of the part across both
+sub-assemblies. Precisely, the step size is computed as a 0.75
+ratio of the diagonal of the smallest bounding box part. The
+idea behind this value is to exclude the possibility of going
+completely through the smallest part with a single translation
+step.
+D. Tooling Matching
+The assembly sequence in itself doesn’t require specific
+tooling models, but this is information is required for the
+next steps in the assembling process. Given a graph of the
+assembly sequence from the previous section, we traverse
+this graph, considering the type of operation and parts used,
+assigning all the tooling models and adding recipes to process
+this operation.
+To archive this, we extract the following information from
+the CAD files:
+• For grippers:
+– Model of the part gripper can be applied.
+– List of positions for grasping the part, calculated with
+respect to the part origin. We use the information
+from “joints”, such as JointAxis, to extract the vec-
+tor of connection. Based on this vector poses are
+calculated.
+– States of digital inputs register to control the gripper.
+• For jigs:
+– Model of the part jig can hold.
+– Position of a part in a jig.
+• For screwdrivers:
+– Screw-picking requirements, such as type of screw-
+holder.
+We store this data in Tooling Database. In our approach
+Tooling Database is a storage with an API which allows
+adding, matching and visualizing of the tooling.
+By analyzing the dataset of the tooling used in physical
+world production in the automotive field, we concluded that
+the same information is stored in the tooling design files and
+propagated to the tooling integration in the physical cells. We
+decided to formalize the requirements and then store this data.
+For the cases where it can’t be calculated from the design files,
+we can manually put that information into the tooling database.
+E. Cell matching
+Cell description includes all the information representing
+an assembly cell. Cell description is used to deploy both
+environments (virtual and physical) and to choose a cell to
+execute Assembly Sequence. We topologically sort a graph of
+the Assembly Sequence and assign a level for every operation.
+The level is required to assign resources for the parallel
+operations when we should use different resources of the
+same model. Then by traversing each operation, we check the
+resources’ models required for this operation and find their
+representation in the cell. If there are no cells satisfying all
+the resource requirements for the Assembly Sequence, we fail,
+providing feedback with the exact operation and the resources
+model we were not able to assign. As a result of the execution
+of the described algorithm, we have an assembly sequence to
+be converted into a BOP.
+F. Control code generation
+For each operation in the BOP, we match a specific PL-
+script, which is self-containing to perform this type of oper-
+ation, and pass the operation, its resources and parts as the
+parameters, creating one PL-script, to assemble a product.
+An example of PL-script implementing unload operation is
+presented on the Listing 4
+V. EXPERIMENTS AND RESULTS
+The objectives of our experiments are:
+• To evaluate the framework.
+
+Fig. 4: Listing of PL-code for unload operation
+• To evaluate the assembly BOPs in the physical environ-
+ment to provide metrics and feedback on the assembly.
+The assembly we chose to test is shown in Fig. 6 and its
+tooling, and jig design are shown in Fig.7.
+• Data Preparation:
+– we extract the joint register and recipes as mentioned
+in section IV-B.
+– taking the joint register and part models files, as-
+sembly sequence generator produced 8 assembly
+sequences for this assembly. These sequences are
+mentioned in Fig. 5.
+– we enrich the assembly sequences using tool match-
+ing mentioned in section IV-D.
+– we convert the enriched assembly sequences to BOP
+using cell matching as mentioned in section IV-E.
+Fig. 5: A diagram of assembly sequences generation process
+for the design used in the experiments. Square blocks represent
+parts while circles represent (sub-)assemblies(D and E). The
+possible sequences are Left: ABDC, BADC, CABD, CBAD
+and Right: BCEA, CBEA, ABCE, ACBE.
+• Scene Preparation: Before starting the assembly, if it’s
+a simulated environment, the jigs are unloaded at the
+same poses as in the physical world in Fig. 1. If it’s
+the physical environment, the jigs and parts are placed in
+their respective poses.
+• Simulation
+Fig. 6: A simple assembly containing 3 parts. Profiles: A, C
+and connector: B
+Fig. 7: Assembly Design file showing the assembly jig,
+custom-designed gripper adapters, grasping, and insertion
+states of the gripper. Assembly state: Center of the table. Jig
+state: Top and bottom right of the table.
+1) Start simulation deployment with services and a
+database and message bus instance as mentioned in
+Fig. 2.
+2) We trigger execution of the PL code, which starts
+from running operations of type ”unload” on input
+parts. This operation effectively initializes part in-
+stances on respective positions in the input jigs, so
+that the parts are now represented in the digital twin
+of the cell, as active objects with poses, visible for
+the simulator as well as for the motion planner.
+3) The rest of the PL code is executed, sequentially
+reading necessary gripper positions, planning and
+executing trajectories, and triggering gripper control
+programs for grasping/releasing/fastening, all by
+calling respective PL functions that use abilities of
+underlying systems described in IV-A.
+4) The user can observe the execution of assembly
+in the 3D simulator. During the execution of the
+assembly process, the motion planner gives us direct
+feedback, on whether it can reach a certain pose in
+the assembly or not.
+For our example assembly, the initial results were the
+following:
+– Of all the possible 8 assembly sequences, only one
+assembly sequence (ABDC) passed through the cell
+matching, as the cell resource descriptions (in this
+case jigs) support this. Many sequences get filtered
+
+1- unload_operation(operation, cell, jig, out part_instance_id)(
+2- rules {
+3 ~
+~ read_fcm#find_part(
+4
+object_id = @operation,
+5
+level = 2,
+6
+format ="array"
+7
+object_filters=[{"path":"type","operation":"EQ","value":"part"}]
+8
+link_filters=[{"path":"type","operation":"EQ","value":"uses"}],
+9
+out id = part)
+10 -
+ get_part_pose_wrt_jig(
+11
+jig_id=@jig.id,
+12
+absoccurrenceid=@part.abs_occurrence_object,
+13
+out pose_from_jig)
+14 -
+~ unload_part(
+15
+part_object = @part,
+16
+cell_object = @cell,
+17
+pose = @pose_from_jig,
+18
+out part_instance_id)
+19
+ seq
+20
+;assembly
+assembly
+C
+D
+A
+E
+1
+I
+-
+A
+B
+B
+cC
+AQ0060000600000000000
+0000000060000000000
+Q0000000000000000
+e00000000000000
+Q00000000
+00000008
+9000000000000
+000000
+00000000000
+000000000
+0000000
+00000
+00000000
+QC0000000000000
+0000000000
+0000000
+0000000
+QDO0000000008based on the cell resources (jigs, robot tooling, etc.).
+All the sequences for any given assembly are fea-
+sible, if the cell has the resources to hold sub-
+assemblies, For example, the assembly sequence
+CABD is possible when the cell has the jig that
+supports moving part C first to the assembly pose,
+then creating a sub-assembly D by moving parts A
+and B in the same order. Now the question here is,
+how to make the decision on which resources in this
+case jigs are needed to be designed to hold the sub-
+assemblies. If there are cycles in the graph, multiple
+BOPs pass through the cell matching which needs
+the same cell resources, which enables us to simulate
+and select the best one based on the metrics.
+– We also noticed that our initial design failed due to
+fastening robot reachability, we took this feedback
+from the framework and changed the fastening posi-
+tion in the assembly to assemble a product.
+To adjust the design, we changed the positions of the
+screws in the assembly to other holes without losing the
+structure stability. After this design adjustment, we were
+able to successfully simulate the one feasible assembly
+sequence.
+This is one of the main features of the proposed frame-
+work - to get this kind of feedback about the prod-
+uct/tooling/cell design compatibility as soon as possible
+with minimal manual input.
+• Running assembly on physical robotic cell: Once we
+find an assembly sequence that passes in the simulation,
+we can proceed to the physical assembly process.
+This is achieved by running the same generated PL code
+as before but now in a physical robotic environment. The
+only thing that differs compared to the previous pipeline
+in simulation is the first step - deployment of the systems.
+For a physical assembly we deploy the robot and the
+gripper drivers to be connected to robots and devices,
+such that in parallel with updating the state of the digital
+twin, these controllers will be changing the states of the
+tools in the physical world, such as robots moving along
+precomputed trajectories, gripper opening/closing and the
+screwdriver fastening the screws.
+We evaluated this assumption on the physical robotic cell
+with two collaborative robots the layout of which can be
+seen in Fig. 1. The results of these experiments are two
+folds:
+1) On one side, we can see that as soon as the
+digital twin is accurate enough, all computed gripper
+positions allow performing most of the operations,
+such as picking a screw, grasping and releasing a
+part, and in some cases to fasten a screw.
+2) On another side, some operations show that an ac-
+cumulated tolerance stack of robot calibration, tool
+accuracy, and parts accuracy leads to the inability
+to perform the joint operation such as fastening
+successfully, and the screwing position requires cor-
+rection.
+The example of running assembly in the virtual and phys-
+ical environments can be seen in Fig. 8. The process of
+assembly of the provided CAD by running the generated
+PL code can be seen in the accompanying video.
+Fig. 8: Assembly process. Left: Physical environment and
+Right: Virtual environment.
+VI. CONCLUSION AND FUTURE SCOPE
+In this paper, we implemented and tested a framework to
+run a robotic assembly of a product by using only CAD
+files as input. We were able to use the feedback provided
+by the framework to change the original design and achieve
+successful assembly. We re-iterated the whole pipeline and
+transferred the assembly from the virtual to the physical
+world. We conclude that this transfer can be done only if the
+digital twin matches the physical cell precisely, which requires
+additional work, such as robots and cell calibration, but it’s out
+of the scope of this paper. The choice of design of our system
+proved its flexibility since we were able to analyze and change
+artefacts produced during the different steps of the execution.
+The system is general enough to support new products and
+cell configurations.
+The next step is to validate our framework on more complex
+assemblies, including new types of operations and operations
+which involve more than two parts.
+In our experiment, we relied only on the parts’ dimensional
+precision and the accuracy of the robots. While it could work
+for some parts, and partially worked in our case, it would
+likely fail on many other parts and materials. To address this
+problem, computer vision and other perception methods should
+be introduced into the framework to deal with variations in the
+real assembly process.
+In the section IV-C3 the constraint we chose could lead to
+some possible assembly sequences being rejected. To solve
+this issue, we plan to implement a geometrical feasibility
+check based on joints from the CAD files or other optimization
+algorithms.
+The method used in the section IV-E can lead to a sub-
+optimal configuration or even to setups where some robots
+can’t reach parts. This approach was chosen as the easiest to
+track and implement. In future works, we plan to implement a
+more sophisticated scheduling algorithm based on geometrical
+and utilization constraints.
+
+REFERENCES
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+[7] Grieves, Michael. ”Digital twin: manufacturing excellence through vir-
+tual factory replication.” White paper 1.2014 (2014): 1-7.
+[8] J¨arvenp¨a¨a, Eeva, et al. ”The development of an ontology for describ-
+ing the capabilities of manufacturing resources.” Journal of Intelligent
+Manufacturing 30.2 (2019): 959-978.
+[9] Lemaignan, Severin, et al. ”MASON: A proposal for an ontology
+of manufacturing domain.” IEEE Workshop on Distributed Intelligent
+Systems: Collective Intelligence and Its Applications (DIS’06). IEEE,
+2006.
+[10] Rust, Romana, et al. ”COMPAS FAB: Robotic fabrication package for
+the COMPAS Framework” Gramazio Kohler Research, ETH Zurich.
+https://github.com/compas-dev/compas fab (2018)
+[11] Sierla, Seppo, et al. ”Automatic assembly planning based on digital
+product descriptions.” Computers in Industry 97 (2018): 34-46
+[12] Drath, Rainer, ed. AutomationML: the industrial cookbook. Walter de
+Gruyter GmbH & Co KG, 2021.
+[13] Beck, Joshua, Alexander Neb, and Katharina Barbu. ”Towards a CAD-
+based Automated Robot Offline-Programming Approach for Disassem-
+bly.” Procedia CIRP 104 (2021): 1280-1285.
+[14] Octopuz® https://octopuz.com
+[15] Huang, Yijiang. Automated motion planning for robotic assembly of
+discrete architectural structures. Diss. Massachusetts Institute of Tech-
+nology, 2018.
+[16] Autodesk® Fusion 360® https://www.autodesk.com/
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf,len=375
+page_content='Auto-Assembly: a framework for automated robotic  assembly directly from CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Fedor Chervinskii§, Sergei Zobov§, Aleksandr Rybnikov§, Danil Petrov§, Komal Vendidandi§  Λ Γ Γ I V Λ L  Abstract—In this work, we propose a framework called Auto-  Assembly for automated robotic assembly from design files and  demonstrate a practical implementation on modular parts joined  by fastening using a robotic cell consisting of two robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We show  the flexibility of the approach by testing it on different input  designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Auto-Assembly consists of several parts: design analysis,  assembly sequence generation, bill-of-process (BOP) generation,  conversion of the BOP to control code, path planning, simulation,  and execution of the control code to assemble parts in the physical  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Index  Terms—industry  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='0,  smart  manufacturing,  cyber-  physical systems, smart factory, manufacturing automation, ma-  nipulators, cellular manufacturing, digital twins, robotic assem-  bly  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' INTRODUCTION  Assembly planning is one of the most laborious tasks  when releasing a new product for manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Thus, many  algorithms and methods around computer-aided design (CAD)  and digital twins of the factories have emerged in recent  years that help process engineers to prepare a new design  for assembly (Computer-aided Assembly Process Planning  techniques [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' An emerging trend of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='0 [2] suggests  that a digital, highly automated factory should be able to infer  the process from the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' In practice, even for an automated  factory, assembly planning has to be followed by an offline-  programming of all the robots and devices to perform the  assembly plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Additive manufacturing technology (3D printing [3]) at the  same time has achieved a much higher rate of process design  automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' One can simply load a CAD file into a machine  that will yield a part of the desired design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The main question  we are trying to address in this paper is ”Could a robotic  cell or even the whole factory work just as a 3D printer?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  When loaded with target assembly CAD design and given  input parts in specified conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' placed in special input  jigs) - would it perform the required assembly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  In this work, we show how this can be achieved under  specific constraints, paving the road for future experiments  towards a more general approach and wider applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  However, the framework we propose is general enough to  accommodate more complex designs and conditions, like  many types of tooling and different joining technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  §Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  [chervinskii, zobov, rybnikov, danil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='petrov, vendidandi]@arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='com  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 1: Experimental setup: robotic cell with two UR5e manip-  ulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Left: UR5e with a screwdriver Likratec EH2 R1030-  A and Right: UR5e with gripper Robotiq 2F-85 with custom  designed gripper clamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' On the table: custom-designed 3D-  printed jigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' RELATED WORK AND BACKGROUND  An Assembly Planning for a given design typically starts  from identifying the mating features or joints and suggesting  a feasible Assembly Sequence, which could be automated as  seen in [4], [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  To proceed to the process planning, a virtual environment,  also known as Digital Twin [7] is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' There are at-  tempts to develop a common ontology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' [8], [9] and unify  interfaces between systems [10] to support process design  automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Sierla, Seppo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' [11] discuss the conceptual framework  of automated assembly planning using a digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' It uses  the XML-based AutomationML [12] data modeling frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' This framework aggregates different data exchange  formats like CAEX for plant description, COLLADA for  geometry and kinematics of 3D models, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  There is still not sufficient work in joining together process  planning, motion planning and execution using a common  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' In [13] authors used artificial intelligence to solve  a Tooling Matching problem and developed an add-on for  Octopuz [14] to do a Motion Planning and Robot Program  Generation for disassembly, but not testing in physical cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  In another work, [15] a similar pipeline is described for an  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='02643v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='RO]  6 Jan 2023  Screwdriver  Gripper  Assembly  Assembly Jig  Parts Unload Jigs  Screws Unload Jigarchitectural domain, mainly focusing on parametric design  and modular assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  We claim that Auto-Assembly is the first proposed frame-  work that can generate and execute robotic assembly process  for an arbitrary input CAD design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' PROBLEM STATEMENT AND METHOD OVERVIEW  The main objective of our work is to create a framework  that enables a closed loop between design and robotic manu-  facturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' A target framework should analyse the design and  provide a simulation of assembly, executable programs (when  possible) and other feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The primary aim of the feedback  is to help in adapting the design and manufacturing to better  correspond to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The feedback we should provide can be split into two  categories:  Successful simulation and its’ artefacts can be directly  used to decide on physical manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Users can  choose between different processes to choose the one,  based on the key performance indicators (KPI) they want  to optimize: time, tooling price, energy consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  In case of a failure, the system should provide all neces-  sary feedback that helps to change the design, robot’s  position, choose the robots with better parameters or  different cell configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Such feedback can be: failed  operations, missing appropriate tooling, parts or tools in  collision, unreachable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  To achieve this, we implement a framework described in  detail in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Section IV-A gives an overview of  our system and its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Section IV-B discusses 3D  modelling of the assembly design files that form the base of  our data extraction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Section IV-C reviews the usage  of this extracted data to produce a set of possible assembly  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Each operation in the assembly sequence is enriched with  tooling information as discussed in section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Finding a  specific cell that contains all the resources like jigs, robots,  and their tooling, etc to execute all the operations needed for  an assembly is explained in section IV-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Section IV-F tells  about the generation of the control code that moves the robots  to grasp, place and fasten parts in a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  In section V, we test our framework on different assemblies  and discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' In section VI, we review our findings  from the experiments and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' FRAMEWORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' System architecture  Auto-Assembly framework can be divided into two parts as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The first part, called ”Artefacts generation”,  works with CAD files provided by a design engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' It is  intended to run once on the input data and provide artefacts,  which can then be stored and re-used to run the assembly  process in the simulated and physical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' This  part includes Assembly Sequence Generation, Tool and Cell  Matching, Bill-of-Process (BOP) Generation and Control Code  generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The second part can be seen as a deployed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  It is represented as a system where we have many services,  providing “abilities” which can be called from the domain-  specific Process language (PL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' An example of the PL script  can be seen in a Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Here we describe the most important  services and their respective abilities:  Robot Controller  – Abilities to control the robots on a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' As  input, it takes a trajectory as a list of a robot’s joint  states, and as output, interpolates the trajectory and  moves the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – Abilities to control tooling connected to the robot,  like grippers, screwdrivers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Motion Planner  – Ability to plan a trajectory in the cell to move a  robot to a target pose with cell objects taken as the  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Jig Controller  – Ability to return a pose of a part in a jig with respect  to the jig origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Assembly Service  – Ability to retrieve the information about fasteners  and resulting parts’ pose with respect to the cell  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Transform Service  – Ability to get the position of any object inside a cell  with respect to any object in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  3D Simulator  – Abilities to load objects from cell description and  visualize cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Database and Message Bus  – Abilities to publish and retrieve JSON objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' This  component is used as a message bus and data storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  All system parts exchange the data in a special format  called Factory Control Model (FCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' It can be considered  as a schema and also is a vital part of our system since it lets  all the components speak the same language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' CAD Data Preparation and Extraction  For any given assembly, our framework needs two design  files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Design file containing part assembly with joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Fasten-  ers are labelled as separate joints in order to distinguish  them from other parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Design file containing the jigs and gripper at different  stages of assembly like grasping, placing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The examples of these files for an assembly are depicted in  Figs 6 and 7 and are created by us in Fusion 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Our method  is CAD-software agnostic as long as we can extract the CAD  data using an API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  From the design file in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 6, we extract the joints and part  occurrences information using Fusion 360 API [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Using  this data, a joint register is created that maps every joint to its  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The joint register follows the FCM schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 2: System architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 3: Listing of PL-code implementing high-level of robot  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Abilities get cell state and plan trajectory are imple-  mented by Motion Planner and execute trajectory by Robot  Controller  From the design file in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 7, we extract the pose of gripper  occurrence relative to the part during grasping it from the jig  and placing it at the assembly state using the Fusion 360 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  We call this data as recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Assembly Sequence Generation  A CAD design contains a lot of important information about  the part’s geometries, relations, and absolute poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' But what  it lacks - the right assembling order - is the key information  to move towards the assembled product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Assembly sequence  encodes the order of operations needed to be performed on  parts by the robotic cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Although the operations can be  executed sequentially, assembly sequences are represented by  polytree (directed acyclic graph whose underlying undirected  graph is a tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Not any such tree represents a valid and  feasible assembly sequence:  only directly joined parts should be neighbours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  the order of operations should take into account the  geometrical limitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  the number of generated assembly sequences should be  reasonably limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Naturally, it grows exponentially with  the number of parts involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' This makes it hard to check  all the generated sequences to pick the best one according  to some criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The assembly sequence generation step aims to solve all three  aforementioned issues, providing a limited number of valid  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The whole process can be divided into three steps:  a liaison graph generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  assembly sequences generation based on the obtained  liaison graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Artefacts generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='CAD file:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='parts geometries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Assembly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='joints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='labelling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Assembly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='CAD Data  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='BOP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Control Code  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Sequences  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Extraction  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='CAD file:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Generation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='tooling placing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='3D-model files  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='absolute  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='BOP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='jigs placing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='positions and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='occurrencesof  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='parts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Tool and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='joints  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='fastening  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Cell  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='features  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Matching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='recipes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Artefacts set  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Objects  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='FCM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='PL script  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='objects  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='geometries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Deploy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='graph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Physical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='PL script  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Parts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='FCM objects  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Best Run  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='environment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='geometries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='graph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Selection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Unsuccessful simulation runloop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='deploy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Succesful run  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Databaseand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='artefacts set  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='messagebus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Services  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Simulation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Robot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Motion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Assembly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Transform  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='3D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='controller  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='planner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Controller  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Service  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Service  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='Simulator1-move_robot_to_position(cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' motion_group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' manipulator_service,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  2  planning_object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='move_type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  3 -  ignored_collisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='ignored_collision_pairs)(  4 -  rules (  5  ~ get_cell_state(cell = cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' out cell_state = cell_state)  6 -  ~plan_trajectory(  7  position= position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  8  motion_group=motion_group  9  planning_object= planning_object  10  planning_socket_name="eef",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  11  move_type=move_type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  12  ignoredcollision pairs = ignored collisionpairs  13  ignored_collisions = ignored_collisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  14  cell_state = cell_state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  15  out result_motion_plan= trajectory  16  )  17  execute_trajectory(trajectory = trajectory)  18  } seq  19 -  constraints (  20  execute_trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' @provider_id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='resource_id== @manipulator_service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='id  21  22• geometry feasibility checking based on parts geometries  This approach we used is described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Further, the high-  level steps, important implementation details, and differences  with the original paper are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  1) Liaison graph generation: The CAD file consists of the  individual parts combined together with joints and fasteners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The information about the joints is crucial to accurately  determine parts connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Considering the parts as liaison  graph nodes, connectivity information transfers into edges in  this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We extracted the information about joints and  fasteners from the design in CAD software to build up a liaison  graph to further analyze it and generate assembly sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  2) Sequences generation: An assembly sequence deter-  mines the order of operations on parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The liaison graph  itself, being undirected, doesn’t set the order of operations  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' But the order should be based on the liaison graph  since the latter contains the information about the connectivity  in the resulting assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Usually, there are many sequences of  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' [5] describes the approach of extracting all possible  assembly sequences from the liaison graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We followed the  suggested approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  3) Geometry feasibility checking: The geometrical feasibil-  ity of an assembly process is the fundamental property, which  should be checked first to eliminate irrelevant sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  These irrelevant sequences could contain, for example, one  part to be joined with another part, which is trapped already  inside the sub-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' To prevent this, geometrical analysis  of sub-assemblies is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' One sub-assembly is translated step-  by-step w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='t another sub-assembly in one of chosen directions  until the bounding boxes of the sub-assemblies still intersect  and the solid bodies’ intersection is checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' If the intersection  represents a volume, it’s impossible to join the sub-assemblies  in the chosen direction, and the remaining directions should  be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Choosing the directions of translations alongside step size  is important for the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Due to the nature of assembly  parts and their orientation alignment, directions along the  main coordinate axes work well in the tested assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' In  other cases, information from joints from the CAD file could  be used to determine the potential directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Step size is  computed based on the minimal size of the part across both  sub-assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Precisely, the step size is computed as a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='75  ratio of the diagonal of the smallest bounding box part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The  idea behind this value is to exclude the possibility of going  completely through the smallest part with a single translation  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Tooling Matching  The assembly sequence in itself doesn’t require specific  tooling models, but this is information is required for the  next steps in the assembling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Given a graph of the  assembly sequence from the previous section, we traverse  this graph, considering the type of operation and parts used,  assigning all the tooling models and adding recipes to process  this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  To archive this, we extract the following information from  the CAD files:  For grippers:  – Model of the part gripper can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – List of positions for grasping the part, calculated with  respect to the part origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We use the information  from “joints”, such as JointAxis, to extract the vec-  tor of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Based on this vector poses are  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – States of digital inputs register to control the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  For jigs:  – Model of the part jig can hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – Position of a part in a jig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  For screwdrivers:  – Screw-picking requirements, such as type of screw-  holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  We store this data in Tooling Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' In our approach  Tooling Database is a storage with an API which allows  adding, matching and visualizing of the tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  By analyzing the dataset of the tooling used in physical  world production in the automotive field, we concluded that  the same information is stored in the tooling design files and  propagated to the tooling integration in the physical cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We  decided to formalize the requirements and then store this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  For the cases where it can’t be calculated from the design files,  we can manually put that information into the tooling database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Cell matching  Cell description includes all the information representing  an assembly cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Cell description is used to deploy both  environments (virtual and physical) and to choose a cell to  execute Assembly Sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We topologically sort a graph of  the Assembly Sequence and assign a level for every operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The level is required to assign resources for the parallel  operations when we should use different resources of the  same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Then by traversing each operation, we check the  resources’ models required for this operation and find their  representation in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' If there are no cells satisfying all  the resource requirements for the Assembly Sequence, we fail,  providing feedback with the exact operation and the resources  model we were not able to assign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' As a result of the execution  of the described algorithm, we have an assembly sequence to  be converted into a BOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Control code generation  For each operation in the BOP, we match a specific PL-  script, which is self-containing to perform this type of oper-  ation, and pass the operation, its resources and parts as the  parameters, creating one PL-script, to assemble a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  An example of PL-script implementing unload operation is  presented on the Listing 4  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' EXPERIMENTS AND RESULTS  The objectives of our experiments are:  To evaluate the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 4: Listing of PL-code for unload operation  To evaluate the assembly BOPs in the physical environ-  ment to provide metrics and feedback on the assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The assembly we chose to test is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 6 and its  tooling, and jig design are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Data Preparation:  – we extract the joint register and recipes as mentioned  in section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – taking the joint register and part models files, as-  sembly sequence generator produced 8 assembly  sequences for this assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' These sequences are  mentioned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – we enrich the assembly sequences using tool match-  ing mentioned in section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – we convert the enriched assembly sequences to BOP  using cell matching as mentioned in section IV-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 5: A diagram of assembly sequences generation process  for the design used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Square blocks represent  parts while circles represent (sub-)assemblies(D and E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The  possible sequences are Left: ABDC, BADC, CABD, CBAD  and Right: BCEA, CBEA, ABCE, ACBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Scene Preparation: Before starting the assembly, if it’s  a simulated environment, the jigs are unloaded at the  same poses as in the physical world in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' If it’s  the physical environment, the jigs and parts are placed in  their respective poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Simulation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 6: A simple assembly containing 3 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Profiles: A, C  and connector: B  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 7: Assembly Design file showing the assembly jig,  custom-designed gripper adapters, grasping, and insertion  states of the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Assembly state: Center of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Jig  state: Top and bottom right of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  1) Start simulation deployment with services and a  database and message bus instance as mentioned in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  2) We trigger execution of the PL code, which starts  from running operations of type ”unload” on input  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' This operation effectively initializes part in-  stances on respective positions in the input jigs, so  that the parts are now represented in the digital twin  of the cell, as active objects with poses, visible for  the simulator as well as for the motion planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  3) The rest of the PL code is executed, sequentially  reading necessary gripper positions, planning and  executing trajectories, and triggering gripper control  programs for grasping/releasing/fastening, all by  calling respective PL functions that use abilities of  underlying systems described in IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  4) The user can observe the execution of assembly  in the 3D simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' During the execution of the  assembly process, the motion planner gives us direct  feedback, on whether it can reach a certain pose in  the assembly or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  For our example assembly, the initial results were the  following:  – Of all the possible 8 assembly sequences, only one  assembly sequence (ABDC) passed through the cell  matching, as the cell resource descriptions (in this  case jigs) support this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Many sequences get filtered  1- unload_operation(operation, cell, jig, out part_instance_id)(  2- rules {  3 ~  ~ read_fcm#find_part(  4  object_id = @operation,  5  level = 2,  6  format ="array"  7  object_filters=[{"path":"type","operation":"EQ","value":"part"}]  8  link_filters=[{"path":"type","operation":"EQ","value":"uses"}],  9  out id = part)  10 -  get_part_pose_wrt_jig(  11  jig_id=@jig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='id,  12  absoccurrenceid=@part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='abs_occurrence_object,  13  out pose_from_jig)  14 -  ~ unload_part(  15  part_object = @part,  16  cell_object = @cell,  17  pose = @pose_from_jig,  18  out part_instance_id)  19  seq  20  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='assembly  assembly  C  D  A  E  1  I A  B  B  cC  AQ0060000600000000000  0000000060000000000  Q0000000000000000  e00000000000000  Q00000000  00000008  9000000000000  000000  00000000000  000000000  0000000  00000  00000000  QC0000000000000  0000000000  0000000  0000000  QDO0000000008based on the cell resources (jigs, robot tooling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  All the sequences for any given assembly are fea-  sible, if the cell has the resources to hold sub-  assemblies, For example, the assembly sequence  CABD is possible when the cell has the jig that  supports moving part C first to the assembly pose,  then creating a sub-assembly D by moving parts A  and B in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Now the question here is,  how to make the decision on which resources in this  case jigs are needed to be designed to hold the sub-  assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' If there are cycles in the graph, multiple  BOPs pass through the cell matching which needs  the same cell resources, which enables us to simulate  and select the best one based on the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  – We also noticed that our initial design failed due to  fastening robot reachability, we took this feedback  from the framework and changed the fastening posi-  tion in the assembly to assemble a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  To adjust the design, we changed the positions of the  screws in the assembly to other holes without losing the  structure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' After this design adjustment, we were  able to successfully simulate the one feasible assembly  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  This is one of the main features of the proposed frame-  work - to get this kind of feedback about the prod-  uct/tooling/cell design compatibility as soon as possible  with minimal manual input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Running assembly on physical robotic cell: Once we  find an assembly sequence that passes in the simulation,  we can proceed to the physical assembly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  This is achieved by running the same generated PL code  as before but now in a physical robotic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The  only thing that differs compared to the previous pipeline  in simulation is the first step - deployment of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  For a physical assembly we deploy the robot and the  gripper drivers to be connected to robots and devices,  such that in parallel with updating the state of the digital  twin, these controllers will be changing the states of the  tools in the physical world, such as robots moving along  precomputed trajectories, gripper opening/closing and the  screwdriver fastening the screws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  We evaluated this assumption on the physical robotic cell  with two collaborative robots the layout of which can be  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The results of these experiments are two  folds:  1) On one side, we can see that as soon as the  digital twin is accurate enough, all computed gripper  positions allow performing most of the operations,  such as picking a screw, grasping and releasing a  part, and in some cases to fasten a screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  2) On another side, some operations show that an ac-  cumulated tolerance stack of robot calibration, tool  accuracy, and parts accuracy leads to the inability  to perform the joint operation such as fastening  successfully, and the screwing position requires cor-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The example of running assembly in the virtual and phys-  ical environments can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The process of  assembly of the provided CAD by running the generated  PL code can be seen in the accompanying video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' 8: Assembly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' Left: Physical environment and  Right: Virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' CONCLUSION AND FUTURE SCOPE  In this paper, we implemented and tested a framework to  run a robotic assembly of a product by using only CAD  files as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We were able to use the feedback provided  by the framework to change the original design and achieve  successful assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We re-iterated the whole pipeline and  transferred the assembly from the virtual to the physical  world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' We conclude that this transfer can be done only if the  digital twin matches the physical cell precisely, which requires  additional work, such as robots and cell calibration, but it’s out  of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' The choice of design of our system  proved its flexibility since we were able to analyze and change  artefacts produced during the different steps of the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The system is general enough to support new products and  cell configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The next step is to validate our framework on more complex  assemblies, including new types of operations and operations  which involve more than two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  In our experiment, we relied only on the parts’ dimensional  precision and the accuracy of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' While it could work  for some parts, and partially worked in our case, it would  likely fail on many other parts and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' To address this  problem, computer vision and other perception methods should  be introduced into the framework to deal with variations in the  real assembly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  In the section IV-C3 the constraint we chose could lead to  some possible assembly sequences being rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' To solve  this issue, we plan to implement a geometrical feasibility  check based on joints from the CAD files or other optimization  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content='  The method used in the section IV-E can lead to a sub-  optimal configuration or even to setups where some robots  can’t reach parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' This approach was chosen as the easiest to  track and implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
+page_content=' In future works, we plan to implement a  more sophisticated scheduling algorithm based on geometrical  and utilization constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'}
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_NE1T4oBgHgl3EQf8gXW/content/tmp_files/2301.03547v1.pdf.txt

@@ -0,0 +1,466 @@
+ON PERTURBATIONS RETAINING CONSERVATION
+LAWS OF DIFFTRENTIAL EQUATIONS
+ALEXEY SAMOKHIN
+Abstract. The paper deals with perturbations of the equation
+that have a number of conservation laws. When a small term is
+added to the equation its conserved quantities usually decay at in-
+dividual rates, a phenomenon known as a selective decay. These
+rates are described by the simple law using the conservation laws’
+generating functions and the added term. Yet some perturbation
+may retain a specific quantity(s), such as energy, momentum and
+other physically important characteristics of solutions. We intro-
+duce a procedure for finding such perturbations and demonstrate
+it by examples including the KdV-Burgers equation and a system
+from magnetodynamics. Some interesting properties of solutions
+of such perturbed equations are revealed and discussed.
+Keywords: conservation laws, perturbed equations, selective de-
+cay, traveling waves.
+MSC[2010]: 35Q53, 35B36.
+1. Introduction
+Many physical systems are modeled using equations that have a sig-
+nificant number of conservation laws. Yet when an additional (usually
+dissipative) term is added to the equation its conserved quantities decay
+at individual rates, which are connected to their generating functions
+[1]. The famous example is the KdV equation (it has infinitely many
+conservation laws) and the KdV-Burgers equation (with additional,
+with respect to KdV, dissipative term and only one conservation law).
+To be precise let E(u) = 0 be a system of equations describing an
+ideal (unperturbed) media state. A scalar H depending on u and its
+derivatives is a conservation law if for ⟨H⟩, the integral of H over some
+fixed spatial domain, ∂⟨H⟩
+∂t
+����
+E
+= 0.
+For the perturbed equation the quantity H is constant no more and
+∂⟨H⟩
+∂t
+̸= 0 is called the decay rate of H, cf. [2].
+A perturbed state usually satisfies the equation E(u) + LF(u) = 0,
+where L is a small parameter diagonal matrix diag(λi); for L = 0 we
+get the ideal state equation. The decay rate depends on the additional
+term L F(u). The connection between decay rate and LF(u) was called
+a ’balance law’ in [3].
+1
+arXiv:2301.03547v1  [nlin.PS]  9 Jan 2023
+
+2
+ALEXEY SAMOKHIN
+This law expresses ∂t⟨H⟩ in terms of scalar product of LF(u) and
+the generating function g of the conserved quantity H, [1]:
+∂⟨H⟩
+∂t
+= ⟨g · LF⟩
+(1)
+Remarks
+• The right-hand side of (1) is not unique: e.g, one can get a
+different but equivalent form integrating by parts.
+• In the case of the integrand in the right-hand side of (1) is null or
+an exact form we get the situation when the conserved quantity
+⟨H⟩ is conserved as well for the correspondent perturbed state.
+• Let us restrict considerations to R[u], the ring of differential
+polynoms of u. Then all perturbations F retaining the conser-
+vation law with the generating function g must satisfy
+g · LFdx1 . . . dxn ∈ Im(d),
+where d : Λn−1 → Λn and Λk are k-forms of spatial variables.
+Of course, the intersection of the principal ideal g · R[u] with
+Im(d) is huge.
+A considerable difference in decay rates leads to a simple method,
+first discovered by Taylor, [4], for finding quasi-stationary states of
+plasma which are of great practical importance.
+He studied the model where the decay of energy E is monotonic but
+those of momentum M and helicity are not necessarily so. Such an
+inequality in decay rates leads to a distinct physical phenomenon of
+’self–organization’ or quasi–stable states.
+There exist a very simple procedure for finding solutions of the above
+described behavior. It was suggested in [4], and is known as ’Taylor
+trick’. The procedure is as follows.
+Taking into consideration their comparative decay rates, minimize
+E with M as constrain. Put δ(E + λM = 0), M and presumed con-
+stant, λ being Lagrange multiplier. This Euler–Lagrange equation is
+not necessarily compatible with the initial equation but nevertheless it
+gives a way for good approximations of self-organization phenomena.
+There is a considerable number of publication in the field, see a recent
+paper [5] for recent developments.
+Another application of selective decay is given in [6]. The problem
+is the behavior of the soliton which, while moving in non-dissipative
+and dispersion-constant medium encounters a finite-width barrier with
+varying dissipation and/or dispersion; beyond the layer dispersion is
+constant (but not necessarily of the same value) and dissipation is null.
+The transmitted wave either retains the form of a soliton (though of
+different parameters) or scatters a into a number of them. Using the
+relative decay of the KdV conserved quantities a simple algorithm to
+predict the number and amplitudes of resulting solitons was obtained.
+
+ON PERTURBATIONS RETAINING CONSERVATION LAWS
+3
+In [7] the selective decay approach was applied to some well-known
+equations of mathematical physics (KdV and KdV-Burgers equation,
+BBM and its dissipative generalization, two-dimensional generalized
+shallow water wave equation). It have showed that the Taylor trick
+extremals are associated with first-order PDEs and travelling wave so-
+lutions.
+In this paper we search, for some popular equations, their low-order
+perturbations which retain a chosen conservation law (in a sense that
+the perturbed equation has the same conserved quantity as initial one).
+Examples include KdV and its conserved energy or momentum and the
+Kadomtsev-Pogutse system of equation from magnetohydrodynamics
+with its three known conserved quantities. Some interesting properties
+of solutions of such perturbed equations are revealed and discussed.
+2. KdV and KdV-Burgers
+The generalized KdV equation (KdV-Burgers equation) considered
+here is of the form
+ut = 2uux + uxxx + λuxx;
+(2)
+The classical KdV equation corresponds to λ = 0.
+The first three conserved quantities for KdV are
+m
+=
+� +∞
+−∞
+u(x, t) dx — mass,
+M
+=
+� +∞
+−∞
+u2(x, t) dx — momentum,
+E
+=
+� +∞
+−∞
+�
+2u3(x, t) − 3(ux(x, t))2�
+dx — energy,
+and there are infinite number of them.
+The generating functions for the above conservation laws of the KdV
+are, up to multiplication constants, 1, u and u2 + uxx correspondingly.
+As for the equation (2), it has a form of a conservation law, ut = Fx,
+the ”mass”
+� +∞
+−∞
+u dx is a conserved quantity. For a soliton this mass
+is equal to 12aγ.
+But the impulse ⟨u2⟩ =
+� +∞
+−∞
+u2 dx declines monotonically:
+Mt = 1
+2⟨u2⟩t = ⟨uut⟩ = ⟨u(u2 + uxx + λux)x⟩ = 2
+3u3��+∞
+−∞ − u2
+x|+∞
+−∞ − λ⟨u2
+x⟩
+=
+(3)
+By analogy, for the energy
+
+4
+ALEXEY SAMOKHIN
+Et = ⟨
+�
+2u3(x, t) − 3(ux(x, t))2�
+⟩t = 6λ⟨uxx(u2 + uxx)⟩
+(4)
+Thus the energy does not necessary declines.
+2.1. Transformations of KdV that retain momentum. Now let
+us find perturbations of the form F(u, ux, uxx) that retain momen-
+tum. Accordingly to the remark 2 above, the differential form λu ·
+F(u, ux, uxx)dx must be exact. Thus
+u · F(u, ux, uxx) = Dx(A(u, ux))
+(5)
+for some A(u, ux). Here
+Dx = ∂
+∂x +
+∞
+�
+n=0
+uxn+1
+∂
+∂uxn
+is the operator of the full differentiation with respect to x.
+Below we restrict the search to polynomials of u and its derivatives.
+Then in (5) the polynomial Dx(A(u, ux)) is divisible by u, so A(u, ux) =
+u2B(u, ux).
+On the other hand
+Dx(u2B(u, ux)) = 2uuxB(u, ux) + u2(ux
+∂B
+∂u + uxx
+∂B
+∂ux
+).
+Hence the second order retaining momentum perturbation is defined
+by
+F(u, ux, uxx) = 2uxB(u, ux) + u(ux
+∂B
+∂u + uxx
+∂B
+∂ux
+)
+for an arbitrary B. Note that F is linear in uxx.
+For instance, if B = ux the λ transformation of the KdV equation
+ut = 2uux + uxxx + λ(2u2
+x + uuxx)
+(6)
+retains ⟨u2⟩ as its conserved quantity.
+Remark 1. This construction can be generalized. If g is the gen-
+erating function for some conserved quantity Cl of an one-spational
+equation E, then F = g−1Dx(g2Φ) is the addendum to E which re-
+tains Cl, Φ being a arbitrary function of u and its derivatives.
+Remark 2. The equation (6) has travelling wave solutions, in par-
+ticular shock waves of the form
+3
+2λ
+�
+a tanh
+�a3λ2 + 3a
+λ2
+t + ax
+�
++ 1
+λ
+�
+.
+(7)
+This shock moves to the left. If require u|−∞ = 0 then (7) becomes
+the shock wave
+3
+2λ2
+�
+1 + tanh
+� 4
+λ2t + 1
+λx
+��
+
+ON PERTURBATIONS RETAINING CONSERVATION LAWS
+5
+with the velocity 4/λ, see figure 1, left.
+Figure 1. The travelling wave solution
+Left: for the equation (6). Right: For the equation (9),
+a = 1/2; λ = 1.
+Remark 4. The perturbed equation has only translations in x and
+t as its point symmetries, but a lot of conservation laws.
+2.2. Transformations of KdV that retain energy. Now for energy
+saving transformations of KdV. Since the generating function of energy
+is, up to a constant multiplier, u2 + uxx, one must solve
+(u2 + uxx) · F(u, ux, uxx, uxxx) = Dx(A(u, ux, uxx))
+(8)
+for some A(u, ux, uxx), to find an low-order F(u, ux, uxx), the suitable
+transformation term. By analogy to the momentum case, the one pos-
+sibility is A = (u2 + uxx)2B
+F(u, ux, uxx) = 2Dx(u2+uxx)B+(u2+uxx)(ux
+∂B
+∂u +uxx
+∂B
+∂ux
++uxxx
+∂B
+∂uxx
+),
+for an arbitrary B = B(u, ux, uxx). If B = u then F = 5u2ux+2uuxxx+
+uxuxx
+The corresponding transformed equation is
+ut = 2uux + uxxx + λ(5u2ux + 2uuxxx + uxuxx).
+(9)
+Its point symmetries are only translations in x and t.
+Remark 5. The equation (9) has travelling wave solutions, in par-
+ticular — solutons of the form of a vertically shifted soliton
+u(x, t) = −6a2 tanh2(a(4a4·λt+x))+4a2 = 6a2 sech2(a(4a4·λt+x))−2a2
+(10)
+found by Maple, with the velocity V = 4a4λ, see figure 1, right.
+
+6
+ALEXEY SAMOKHIN
+Yet it is not the whole answer. Computer experiments demonstrate
+that an arbitrary initial datum for this equation scatters into a number
+of solitary peaks of different but constant height and velocity and a ’tail’
+(see figures 2 and 3) — in a manner of the KdV itself, cf. [6].
+Figure 2. Left: Initial profile 1.5 sech2(0.5x) for the
+equation (9), λ = 1.
+Right: Resulting profile at t = 6: single soliton-like
+peak of a constant form and velocity and an oscillating
+tail moving in opposite direction
+Figure 3. Left: Initial profile sech2(0.1x) for the equa-
+tion (9), λ = 1.
+Right: Resulting profile at t = 40: multiple soliton-like
+peaks of a constant form and velocity and (seemingly)
+no tail.
+The analytical description of these peaks is so far unknown. The
+reason is that the equation on travelling waves, u = u(x + V t), here
+V u′ = 2uu′ + u′′ + λ(5u2u′ + 2uu′′′ + u′u′′
+can be readily integrated introducing the new dependent variable u′ =
+p(u) which leads to a linear first order ordinary differential equation on
+z(u) = p(u)p′(u),
+(2uλ + 1)z′ + λz = V − 5λu2 − 2u.
+
+ON PERTURBATIONS RETAINING CONSERVATION LAWS
+7
+But the resulting general solution looks hopelessly implicit. The likes
+of (10) arise in the case of a very special combination of the arbitrary
+constants entering this general solution, and such combinations are
+hard to discover.
+3. Two-dimensional MHD System
+Consider the Kadomtsev-Pogutse sysnem of equations
+�
+∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy
+=
+0
+vt + uxvy − uyvx
+=
+0
+(11)
+which describes quasi-stationary states of plasma. It has three conser-
+vation laws, that is there are three non–trivial conserved densities (two
+of them depending on arbitrary functions): the total energy E (mag-
+netic plus kinetic energy), generalized ’cross helicity’ Hc and mean
+magnetic potential A,
+E
+=
+1
+2⟨u2
+x + u2
+y + v2
+x + v2
+y⟩
+H
+=
+⟨f ′(v) · (uxvx + uyvy)⟩
+A
+=
+⟨Φ(v)⟩
+(12)
+Their generating functions are, respective order,
+�
+u
+∆v
+�
+,
+�
+f(v)
+f ′(v)∆u
+�
+,
+�
+0
+Φ′(v)
+�
+(13)
+where f and Φ are arbitrary functions.
+Let us seek transformations of (11) of the form
+�
+∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy
+=
+νF(u, v)
+vt + uxvy − uyvx
+=
+ηG(u, v)
+(14)
+Here F, G are functions of u(x, y, t), v(x, y, t) and their derivatives.
+3.1. Energy-retaining transformations. In this instance
+∂⟨E⟩/∂t = 0 implies
+(−νu · F − η∆v · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) =
+(DxA(u, v) + DyB(u, v))dx ∧ dy.
+(15)
+There are a lot of solutions to (15). We restrict ourselves to some
+low-order examples.
+3.1.1. Ortogonal transformations. One can always get zero right hand
+side in equation (15): just put F = η∆ and G = −νu. The vector
+(F, G) is orthogonal to the generating function so ∂⟨E⟩/∂t = 0. It
+works if the number of any system of equations is greater than one.
+
+8
+ALEXEY SAMOKHIN
+3.1.2. Splitted sum transformations. Another solution may be obtained
+assuming
+− νu · F(u, v) = DxA(u, v),
+η∆v · G(u, v) = DyB(u, v).
+(16)
+Here again A, B are functions of u(x, y, t), v(x, y, t) and their deriva-
+tives. This equations may be solved by analogy to the KdV case.
+One of numerous solutions here is A = νun,
+B = η(∆v)2, so F =
+−νnun−2ux, G = 2η∆vy
+3.1.3. {ν = η}—case transformations. Take A = Gvx, B = Gvy.
+Then uF = vxDxG + vyDyG. For instance, choose G = u2; it fol-
+lows that F = 2(uxvx + uyvy).
+3.2. Mean magnetic potential retaining transformations. Here
+∂⟨A⟩/∂t = 0 implies
+(−ν0 · F − ηΦ′(v) · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) =
+(DxA(u, v) + DyB(u, v))dx ∧ dy.
+(17)
+Thus F is an arbitrary function. Then one possible solution is
+−ηΦ′(v) · Φ(v)(αvx + βvy) = DxαΦ2 + DyβΦ2, α, β ∈ R.
+That is, to retain the mean magnetic potential of (11), its first equation
+may be transformed in arbitrary way and the second one by ηG =
+−ηΦ(v)(αvx + βvy) for all α, β ∈ R.
+3.3. Cross helicity retaining transformations. Here ∂⟨Hc⟩/∂t = 0
+implies
+−νf(v)·F(u, v)−ηf ′(v)∆(u)·G(u, v) = DxA(u, v)+DyB(u, v). (18)
+In the case ηη = ν it is not hard to find some suitable transformations
+(F, G). Namely, take
+A = −ηf 2(v)f ′(v)ux, B = −ηf 2(v)f ′(v)uy;
+It follows
+F = [2f ′2(v) + f(v)f ′′(v)](vxux + uyvy), G = f ′(v)f(v)∆u.
+For f(v) = v it comes to
+F = −2η(vxux + uyvy) G = −ηv∆u.
+
+ON PERTURBATIONS RETAINING CONSERVATION LAWS
+9
+Conclusion
+The paper deals with perturbations of the equation that have a num-
+ber of conservation laws. When a small term is added to the equation
+its conserved quantities usually decay at individual rates, a phenome-
+non known as a selective decay. These rates are described by the simple
+law using the conservation laws’ generating functions and the added
+term. Yet some perturbation may retain a specific quantity(s), such
+as energy, momentum and other physically important characteristics
+of solutions. We introduced a procedure for finding such perturbations
+and demonstrated it by examples including the KdV-Burgers equation
+and a system from magnetodynamics.
+Our worked out examples show that the perturbed equations retain-
+ing a specific conservation law frequently also retain additional alge-
+braic properties such as travelling wave solutions or a presence of other
+conservation laws.
+Thus the present paper as well as [5] and our previous research of
+the KdV solitons in nonhomogeneous media, [6], persuades that the
+selective decay approach is a valid and effective instrument to obtain
+qualitative approximations and estimates for behavior of solutions.
+The figures in this paper were generated numerically using Maple
+PDETools package.
+The mode of operation uses the default Euler
+method, which is a centered implicit scheme, and can be used to find
+solutions to PDEs that are first order in time, and arbitrary order in
+space, with no mixed partial derivatives.
+References
+[1] A. V. Samokhin, Decay velocity of conservation laws for nonevolution equa-
+tions,Acta Applicanda Math., v. 41 n. 1, 1–11 (1995)
+[2] A. C. Ting, M. H. Matthaeus, D. Montgomery, Turbulent relaxation processes
+in magnetohydrodynamics Phys. Fluids, v.29, 3261–3274 (1986)
+[3] E. van Groesen, F. Mainardi, Balance laws and centro velocity in dissipative
+systems, J. Math. Phys.v. 31 (11), 2136–2140 (1990)
+[4] J. B. Taylor. Relaxation of toroidal plasma and generation of reverse magnetic
+fields, Phys. Rev.Lett., v. 33, 1139–1141 (1974)
+[5] R. Brecht1, W. Bauer, A. Bihlo, F. Gay-Balmaz, S. MacLachlan. Selective decay
+for the rotating shallow-water equations with a structure-preserving discretiza-
+tion Phys. Fluids, v.33, 116604 (2021); https://doi.org/10.1063/5.0062573
+[6] A. V. Samokhin, The KdV soliton crosses a dissipative and dispersive border,
+Journal of Differential Geometry and its Applications.75, Part A, 11 pages(April
+2021) https://doi.org/10.1016/j.difgeo.2021.101723
+[7] A. V. Samokhin, Taylor Trick and Travelling Wave Solutions, Lobachevskii
+Journal
+of
+Mathematics,
+2022,
+43,
+n.
+10,
+2808—2815,
+(2022).
+DOI:
+10.1134/S1995080222130406
+Institute of Control Sciences of Russian Academy of Sciences 65
+Profsoyuznaya street, Moscow 117997, Russia
+Email address:
+samohinalexey@gmail.com
+

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@@ -0,0 +1,1425 @@
+Proceedings of Machine Learning Research – Under Review:1–20, 2023
+Full Paper – MIDL 2023 submission
+DBGDGM: Dynamic Brain Graph Deep Generative Model
+Alexander Campbell∗1,2
+ajrc4@cl.cam.ac.uk
+Simeon Spasov∗1
+ses88@cl.cam.ac.uk
+Nicola Toschi1
+Pietro Li`o3,4
+1 Department of Computer Science and Technology, University of Cambridge, United Kingdom
+2 The Alan Turing Institute, United Kingdom
+3 University of Rome Tor Vergata, Italy
+4 A.A. Martinos Center for Biomedical Imaging, Harvard Medical School, United States
+Editors: Under Review for MIDL 2023
+Abstract
+Graphs are a natural representation of brain activity derived from functional magnetic
+imaging (fMRI) data. It is well known that clusters of anatomical brain regions, known as
+functional connectivity networks (FCNs), encode temporal relationships which can serve
+as useful biomarkers for understanding brain function and dysfunction. Previous works,
+however, ignore the temporal dynamics of the brain and focus on static graphs. In this
+paper, we propose a dynamic brain graph deep generative model (DBGDGM) which simul-
+taneously clusters brain regions into temporally evolving communities and learns dynamic
+unsupervised node embeddings. Specifically, DBGDGM represents brain graph nodes as
+embeddings sampled from a distribution over communities that evolve over time.
+We
+parameterise this community distribution using neural networks that learn from subject
+and node embeddings as well as past community assignments. Experiments demonstrate
+DBGDGM outperforms baselines in graph generation, dynamic link prediction, and is com-
+parable for graph classification. Finally, an analysis of the learnt community distributions
+reveals overlap with known FCNs reported in neuroscience literature.
+Keywords: Dynamic graph, generative model, functional magnetic resonance imaging
+1. Introduction
+Functional magnetic resonance imaging (fMRI) is a non-invasive imaging technique pri-
+marily used to measure blood-oxygen level dependent (BOLD) signal in the brain (Huettel
+et al., 2004). A natural representation of fMRI data is as a discrete-time graph, henceforth
+referred to as a dynamic brain graph (DBG), consisting of a set of fixed nodes correspond-
+ing to anatomically separated brain regions and a set of time-varying edges determined by
+a measure of dynamic functional connectivity (dFC) (Calhoun et al., 2014). DBGs have
+been widely used in graph-based network analysis for understanding brain function (Hirsch
+and Wohlschlaeger, 2022; Raz et al., 2016) and dysfunction (Alonso Mart´ınez et al., 2020;
+Dautricourt et al., 2022; Yu et al., 2015).
+∗ Contributed equally
+© 2023 A. Campbell, S. Spasov, N. Toschi & P. Li`o.
+arXiv:2301.11408v1  [cs.LG]  26 Jan 2023
+
+Campbell Spasov Toschi Li`o
+Recently, there is growing interest in using deep learning-based methods for learning
+representations of graph-structured data (Goyal and Ferrara, 2018; Hamilton, 2020). A
+graph representation typically consists of a low-dimensional vector embedding of either the
+entire graph (Narayanan et al., 2017) or a part of it’s structure such as nodes (Grover and
+Leskovec, 2016), edges (Gao et al., 2019), or sub-graphs (Adhikari et al., 2017). Although
+originally formulated for static graphs (i.e. not time-varying), several existing methods have
+been extended (Mahdavi et al., 2018; Goyal et al., 2020), and new ones proposed (Zhou
+et al., 2018; Sankar et al., 2020), for dynamic graphs. The embeddings are usually learnt
+in either a supervised or unsupervised fashion and typically used in tasks such as node
+classification (Pareja et al., 2020) and dynamic link prediction (Goyal et al., 2018).
+To date, very few deep learning-based methods have been designed for, or existing
+methods applied to, representation learning of DBGs. Those that do, tend to use graph
+neural networks (GNNs) that are designed for learning node- and graph-level embeddings
+for use in graph classification (Kim et al., 2021; Dahan et al., 2021). Although node/graph-
+level embeddings are effective at representing local/global graph structure, they are less
+adept at representing topological structures in-between these two extremes such a clusters
+of nodes or communities (Wang et al., 2017). Recent methods that explicitly incorporate
+community embeddings alongside node embeddings have shown improved performance for
+static graph representation learning tasks (Sun et al., 2019; Cavallari et al., 2017). How to
+leverage the relatedness of graph, node, and community embeddings in a unified framework
+for DBG representation learning remains under-explored. We refer to Appendix A for a
+summary of related work.
+Contributions
+To address these shortcomings, we propose DBGDGM, a hierarchical
+deep generative model (DGM) designed for unsupervised representing learning on DBGs
+derived from multi-subject fMRI data. Specifically, DBGDGM represents nodes as embed-
+dings sampled from a distribution over communities that evolve over time. The community
+distribution is parameterized using neural networks (NNs) that learn from graph and node
+embeddings as well as past community assignments.
+We evaluate DBGDGM on multi-
+ple real-world fMRI datasets and show that it outperforms state-of-the-art baselines for
+graph reconstruction, dynamic link prediction, and achieves comparable results for graph
+classification.
+2. Problem formulation
+We consider a dataset of multi-subject DBGs derived from fMRI data D ≡ G(1:S, 1:T) =
+{G(s, t)}S, T
+s, t=1 that share a common set of nodes V = {v1, . . . , vN} over T ∈ N timepoints
+for S ∈ N subjects. Each G(s, t) ∈ G(1:S, 1:T) denotes a non-attributed, unweighted, and
+undirected brain graph snapshot for the s-th subject at the t-th timepoint. We define a
+brain graph snapshot as a tuple G(s, t) = (V, E(s, t)) where E(s, t) ⊆ V × V denotes an edge
+set. The i-th edge for the s-th subject at the t-th timepoint e(s, t)
+i
+∈ E(s, t) is defined e(s, t)
+i
+=
+(w(s,t)
+i
+, c(s,t)
+i
+) where w(s,t)
+i
+is a source node and c(s,t)
+i
+is a target node. We assume each node
+corresponds to a brain region making the number of nodes |V| = V ∈ N fixed over subjects
+and time. We also assume edges correspond to a measure of dFC allowing the number of
+edges |E(s, t)| = E(s, t) ∈ N vary over subjects and time. We further assume there exists
+2
+
+DBGDGM: Dynamic Brain Graph Deep Generative Model
+K ∈ N clusters of nodes, or communities, the membership of which dynamically changes
+over time for each subject. Let z(s, t)
+i
+∈ [1 : K] denote the latent community assignment of
+the i-th edge for the s-th subject at the t-th timepoint. For each subject’s DBG our aim
+is to learn, in an unsupervised fashion, graph α(s) ∈ RHα, node φ(s, t)
+1:N = [φ(s, t)
+n
+] ∈ RN×Hφ,
+and community ψ(s, t)
+1:K = [ψ(s, t)
+k
+] ∈ RK×Hψ representations of dimensions Hα, Hφ, Hψ ∈ N,
+respectively, for use in a variety of downstream tasks.
+3. Method
+Figure 1: Plate diagram for DBGDGM. La-
+tent and observed variables are denoted by
+white-and gray-shaded circles, respectively.
+Solid black squares denote non-linear map-
+pings parameterized by NNs.
+DBGDGM defines a hierarchical deep gen-
+erative model and inference network for
+the end-to-end learning of graph, node,
+and community embeddings from multi-
+subject DBG data. Specifically, DBGDGM
+treats the embeddings and edge commu-
+nity assignments as latent random vari-
+ables collectively denoted Ω(s, t) = {α(s),
+φ(s, t)
+1:N ,
+ψ(s, t)
+1:K , {z(s, t)
+i
+}E(s, t)
+i=1
+},
+which along
+with the observed DBGs, defines a proba-
+bilistic latent variable model with joint den-
+sity pθ(G1:S, 1:T , Ω1:S, 1:T ).
+3.1. Generative model
+Graph embeddings
+We begin the gen-
+erative process by sampling graph embed-
+dings from a prior α(s) ∼ pθα(α(s)) imple-
+mented as a normal distribution following
+pθα(α(s)) = Normal(0Hα, IHα)
+(1)
+where 0Hα is a matrix of zeros and IHα is a identity matrix. Each embedding is a vector
+α(s) ∈ RHα representing subject-specific information that remains fixed over time.
+Node and community embeddings
+Next, let φ(s, t)
+n
+∈ RHφ and ψ(s, t)
+k
+∈ RHψ denote
+the n-th node and the k-th community embedding, respectively. To incorporate tempo-
+ral dynamics, we assume node and community embeddings are related through Markov
+chains with prior transition distributions φ(s, t)
+n
+∼ pθφ(φ(s, t)
+n
+|φ(s, t−1)
+n
+, α(s)) and ψ(s, t)
+k
+∼
+pθψ(ψ(s, t)
+k
+|ψ(s, t−1)
+k
+, α(s)). We specify each prior to be a normal distribution following
+pθφ(φ(s, t)
+n
+|φ(s, t−1)
+n
+, α(s)) = Normal(φ(s, t−1)
+n
+, σφIHφ)
+(2)
+pθψ(ψ(s, t)
+k
+|ψ(s, t−1)
+k
+, α(s)) = Normal(ψ(s, t−1)
+k
+, σψIHψ)
+(3)
+where the graph embeddings are used for initializing the means, i.e., φ(s, 0)
+n
+= α(s), ψ(s, 0)
+k
+=
+α(s) and the standard deviations σφ, σψ ∈ R>0 are hyperparameters controlling how smoothly
+each embedding changes between consecutive timepoints.
+3
+
+Campbell Spasov Toschi Li`o
+Edge generation
+We next describe the edge generative process of a graph snapshot
+G(s, t) ∈ G(1:S, 1:T). Similar to Sun et al. (2019), for each edge e(s, t)
+i
+= (w(s, t)
+i
+, c(s, t)
+i
+) ∈ E(s, t)
+we first sample a latent community assignment z(s, t)
+i
+∈ [1 : K] from a conditional prior
+z(s, t)
+i
+∼ pθz(z(s, t)
+i
+|w(s, t)
+i
+) implemented as a categorical distribution
+pθz(z(s, t)
+i
+|w(s, t)
+i
+) = Categorical(π(s, t)
+θz
+),
+π(s, t)
+θz
+= MLPθz(φ(s, t)
+wi
+)
+(4)
+where MLPθz : RHφ → RK is a Lz-layered multilayered perception (MLP) that parame-
+terizes community probabilities using node embeddings indexed by w(s, t)
+i
+. In other words,
+each source node w(s, t)
+i
+is represented as a mixture of communities. A linked target node
+c(s, t)
+i
+∈ [1 : N] is then sampled from the conditional likelihood c(s, t)
+i
+∼ pθc(c(s, t)
+i
+|z(s, t)
+i
+) which
+is also implemented as a categorical distribution
+pθc(c(s, t)
+i
+|z(s, t)
+i
+) = Categorical(π(s, t)
+θc
+),
+π(s, t)
+θc
+= MLPθc(ψ(s, t)
+zi
+)
+(5)
+where MLPθc : RHψ → RN is a Lc-layered MLP that parameterizes node probabilities using
+community embeddings indexed by z(s, t)
+i
+. That is, each community assignment z(s, t)
+i
+is
+represented as a mixture of nodes. By integrating out the latent community assignment
+variable
+p(c(s, t)
+i
+|w(s, t)
+i
+) =
+�
+z(s, t)
+i
+∈[1:K]
+pθc(c(s, t)
+i
+|z(s, t)
+i
+)pθz(z(s, t)
+i
+|w(s, t)
+i
+)
+(6)
+we define the likelihood of node c(s, t)
+i
+being a linked neighbor of node w(s, t)
+i
+, in a given
+graph snapshot.
+Factorized generative model
+Given this model specification, the joint probability of
+the observed data and the latent variables can be factorized following
+pθ(G1:S 1:T , Ω1:S,1:T ) =
+S
+�
+s=1
+�
+pθα(α(s))
+T
+�
+t=1
+�
+V�
+n=1
+pθφ(φ(s, t)
+n
+|φ(s, t−1)
+n
+)
+K
+�
+k=1
+pθψ(ψ(s,t)
+k
+|ψ(s,t−1)
+k
+)
+E(s, t)
+�
+i=1
+pθz(z(s, t)
+i
+|φ(s, t)
+wi
+)pθc(c(s, t)
+i
+|ψ(s, t)
+zi
+)
+��
+(7)
+where θ = {θc , θz} is the set of generative model parameters, i.e., NN weights. The gener-
+ative model of DBGDGM summarized in Appendix B
+3.2. Inference network
+To learn the embeddings, we must infer the posterior distribution over all latent variables
+conditioned on the observed data pθ(Ω(1:S, 1:T)|G(1:S, 1:T)). However, exact inference is in-
+tractable due the log marginal likelihood requiring integrals that are hard to evaluate, i.e.,
+4
+
+DBGDGM: Dynamic Brain Graph Deep Generative Model
+log pθ(G(1:S, 1:T)) =
+�
+Ω log pθ(G(1:S, 1:T), Ω(1:S, 1:T))dΩ. As a result, we use variational infer-
+ence (Jordan et al., 1999) to approximate the true posterior with a variational distribution
+qλ(Ω(1:S,1:T)) with parameters λ. To do this, we maximize a lower bound on the log marginal
+likelihood of the DBGs, referred to as the ELBO (evidence lower bound), defined as
+LELBO(θ, λ) = Eqλ
+�
+log pθ(G1:S, 1:T , Ω1:S, 1:T )
+qλ(Ω(1:S, 1:T))
+�
+≤ log pθ(G(1:S, 1:T))
+(8)
+where Eqλ[·] denotes the expectation taken with respect to the variational distribution
+qλ(Ω(1:S, 1:T)). By maximizing the ELBO with respect to the generative and variational
+parameters θ and λ we train our generative model and perform Bayesian inference, respec-
+tively.
+Structured variational distribution
+To ensure a good approximation to true posterior,
+we retain the Markov properties of the node and community embeddings. This results in a
+structured variational distribution (Hoffman and Blei, 2015; Saul and Jordan, 1995) which
+factorizes following
+qλ(Ω(1:S, 1:T)) =
+S
+�
+s=1
+�
+qλα(α(s))
+T
+�
+t=1
+�
+V�
+n=1
+qλφ(φ(s, t)
+n
+| φ(s, t−1)
+n
+)
+K
+�
+k=1
+qλψ(ψ(s, t)
+k
+| ψ(s, t−1)
+k
+)
+E(s, t)
+�
+i=1
+qλz(z(s, t)
+i
+| φ(s, t)
+wi
+, φ(s, t)
+ci
+)
+��
+(9)
+where each distribution is specified to mimic the structure of the generative model so that
+qλα(α(s)) = Normal(µ(s)
+λα, σ(s)
+λα )
+(10)
+qλφ(φ(s, t)
+n
+|φ(s, t−1)
+n
+) = Normal(µ(s, t)
+λφ , σ(s, t)
+λφ
+)
+{µ(s, t)
+λφ , σ(s, t)
+λφ
+} = GRUλφ(φ(s, t−1)
+n
+) (11)
+qλψ(ψ(s, t)
+k
+|ψ(s, t−1)
+n
+) = Normal(µ(s, t)
+λψ , σ(s, t)
+λψ )
+{µ(s, t)
+λψ , σ(s, t)
+λψ } = GRUλψ(ψ(s, t−1)
+k
+) (12)
+qλz(z(s, t)
+i
+|φ(s, t)
+wi
+, φ(s, t)
+ci
+) = Categorical(π(s, t)
+λz
+)
+π(s, t)
+λz
+= MLPλz(φ(s, t)
+wi
+⊙ φ(s, t)
+ci
+)
+(13)
+where GRUλj : RHj → RHj is a Lj-layered GRU for each j ∈ {φ, ψ} and MLPλz :
+RHφ → RK is Lz-layered MLP. Furthermore, we use MLPs to initialize the GRUs with
+the graph embeddings such that φ(s, 0)
+n
+= MLPλφ(α(s)) and ψ(s, 0)
+k
+= MLPλψ(α(s)) where
+MLPλj : RNα → RNj. This allows for subject-specific variation to be incorporated in the
+temporal dynamics of the node and community embeddings. Another difference with the
+generative model is now the variational distribution of the community assignment qλz(·) in-
+cludes information from neighboring nodes via c(s, t)
+i
+. Finally, we use the same NN from the
+generative model to parameterize the variational distribution of the community assignment,
+i.e., λz = θz. This not only spares additional trainable parameters for the variational dis-
+tribution but also further links the variational parameters of qλ(·) to generative parameters
+of pθ(·) resulting in more robust learning (Farnoosh and Ostadabbas, 2021). The set of pa-
+rameters for the inference network is therefore λ = {λα = {µ(s)
+λα, σ(s)
+λα }S
+s=1, λφ, λψ, λz = θz}.
+5
+
+Campbell Spasov Toschi Li`o
+Model
+HCP
+UKB
+NLL (↓)
+MSE (↓)
+NLL (↓)
+MSE (↓)
+CMN
+5.999 ± 0.029 *
+0.050 ± 0.005 *
+5.861 ± 0.017 *
+0.050 ± 0.003 *
+VGAE
+5.857 ± 0.017 *
+0.051 ± 0.002 *
+5.851 ± 0.027 *
+0.061 ± 0.002 *
+OSBM
+5.808 ± 0.026 *
+0.051 ± 0.003 *
+5.726 ± 0.039 *
+0.052 ± 0.003 *
+VGRAPH
+5.569 ± 0.046 *
+0.022 ± 0.004 *
+5.716 ± 0.037 *
+0.020 ± 0.003 *
+VGRNN
+5.674 ± 0.034 *
+0.011 ± 0.003 *
+5.649 ± 0.035 *
+0.014 ± 0.002 *
+ELSM
+5.924 ± 0.040 *
+0.081 ± 0.002 *
+5.809 ± 0.024 *
+0.115 ± 0.003 *
+DBGDGM
+4.587 ± 0.045
+0.001 ± 0.002
+4.586 ± 0.084
+0.004 ± 0.003
+AUROC (↑)
+AP (↑)
+AUROC (↑)
+AP (↑)
+CMN
+0.665 ± 0.007 *
+0.654 ± 0.006 *
+0.678 ± 0.004 *
+0.668 ± 0.005 *
+VGAE
+0.661 ± 0.010 *
+0.674 ± 0.008 *
+0.688 ± 0.010 *
+0.607 ± 0.009 *
+OSBM
+0.655 ± 0.027 *
+0.675 ± 0.024 *
+0.678 ± 0.032 *
+0.682 ± 0.033 *
+VGRAPH
+0.689 ± 0.004 *
+0.682 ± 0.002 *
+0.664 ± 0.002 *
+0.621 ± 0.001 *
+VGRNN
+0.689 ± 0.007 *
+0.698 ± 0.006 *
+0.698 ± 0.009 *
+0.696 ± 0.007 *
+ELSM
+0.669 ± 0.004 *
+0.662 ± 0.002 *
+0.661 ± 0.001 *
+0.662 ± 0.002 *
+DBGDGM
+0.768 ± 0.026
+0.732 ± 0.032
+0.786 ± 0.040
+0.762 ± 0.038
+Table 1: Graph reconstruction (top) and dynamic link prediction (bottom) results (mean
+± standard deviation over 5 runs).
+First and second-best results shown in bold and
+underlined. Statistically significant difference from DBGDGM marked *.
+Training objective
+Substituting the variational distribution from (9) and the joint dis-
+tribution from (7) into the ELBO (8) gives the full training objective which can be optimized
+using stochastic gradient descent. We estimate all gradients using the reparameterization
+trick (Kingma and Welling, 2013) and the Gumbel-softmax trick (Jang et al., 2016; Mad-
+dison et al., 2016). We refer to Appendix B further details on the ELBO and learning the
+parameters.
+4. Experiments
+We evaluate DBGDGM against baseline models on the tasks of graph reconstruction, dy-
+namic link prediction, and graph classification. Each task is designed to evaluate the use-
+fulness of the learnt embeddings.
+Datasets
+We construct two multi-subject DBG datasets using publicly available fMRI
+scans from the Human Connectome Project (HCP) (Van Essen et al., 2013) and UK Biobank
+(UKB) (Sudlow et al., 2015). We randomly sample S = 300 subjects ensuring an even
+male/female split.
+To create DBGs, we parcellate each scan into V = 360 region-wise
+BOLD signals using the Glasser atlas (Glasser et al., 2016), apply sliding-window Pearson
+correlation (Calhoun et al., 2014) with a non-overlapping window of size and stride of 30,
+and threshold the top 5% values of the lower triangle of each correlation matrix as connected
+following Kim et al. (2021). The described procedure gives T = 16 graph snapshots for each
+subject. Biological sex is taken as graph-level labels. We refer to Appendix C for further
+details on each dataset.
+6
+
+DBGDGM: Dynamic Brain Graph Deep Generative Model
+Baselines
+We compare DBGDGM against a range of different unsupervised probabilistic
+baseline models. For static baselines, we include variational graph autoencoder (VGAE) (Kipf
+and Welling, 2016b), a deep generative version of the overlapping stochastic block model
+(OSBM) (Mehta et al., 2019), and vGraph (VGRAPH) (Sun et al., 2019). For dynamic
+baselines we include variational graph recurrent neural network (VGRNN) (Hajiramezanali
+et al., 2019) and evolving latent space model (ELSM) (Gupta et al., 2019). For the graph re-
+construction and link prediction tasks, we also include a heuristic baseline based on common
+neighbors between nodes at previous snapshots (CMN). Finally, for graph classification we
+include a support vector machine which takes as import static FC matrices (FCM) (Abra-
+ham et al., 2017). Further details about baseline model can be found in Appendix D.
+Implementation
+We split both datasets into 80/10/10% training/validation/test data
+along the time dimension. We train all models using the Adam optimizer (Kingma and
+Ba, 2014) with decoupled weight decay (Loshchilov and Hutter, 2017). All baseline hy-
+perparameters are set following their original implementations. For DBGDGM, choose the
+number of communities K based on validation NLL. Finally, we train all models 5 times
+using different random seeds. Implementation details can be found in Appendix E.
+Evaluation metrics
+For graph reconstruction, we evaluate the probability of the edges
+in the test dataset using negative log-likelihood (NLL). We also compare the mean-squared
+error (MSE) between actual and reconstructed node degree over all test snapshots. For dy-
+namic link prediction, we sample an equal number of positive and negative edges in the test
+dataset and measure performance using area under the receiver operator curve (AUROC)
+and average precision (AP). Finally, for graph classification we predict the biological sex
+for each subjects’ DBG and evaluate on accuracy. To predict graph labels, we average node
+embeddings per subject for the baselines and the community embeddings for DBGDGM
+before training a SVM using 10-fold cross-validation. For comparing models, we use the al-
+most stochastic order (ASO) test (Dror et al., 2019) with significance level 0.05 and correct
+for multiple comparisons (Bonferroni, 1936).
+5. Results
+Dynamic graph reconstruction and link prediction.
+We summarize the average test
+results of all models over 5 runs using optimally tuned hyperparameters. From Table 1,
+it is clear that DBGDGM outperforms baselines on both tasks. For graph reconstruction,
+DBGDGM shows an 18% and 30% relative improvement in NLL on HCP and UKB, re-
+spectively, compared to the second-best baselines. For dynamic link prediction, the relative
+improvement is > 11% in AUCROC and > 5% in AP compared to second-best baselines de-
+pending on dataset. We attribute these statistically significant gains to DBGDGM’s ability
+to learn dynamic brain connectivity more effectively.
+Graph classification
+For graph classification, DBGDGM achieves ∼ 75% accuracy for
+HCP and ∼ 73% for UKB (see Fig. 2). We outperform 4 baselines and show indiscernible
+performance to VGAE and OSBM. To show the interpretative power of DBGDGM, we re-
+run the graph classification experiment for HCP with the embeddings of each community
+separately. We find a community which comprises brain regions in the Cingulo-opercular
+7
+
+Campbell Spasov Toschi Li`o
+Figure 2: Graph classification results (5 runs). Statistical significance from DBGDGM marked *.
+Figure 3: Overlap between communities learned by DBGDGM and FCNs from Ji et al. (2019).
+(CON) and the Somatomotor (SMN) networks, which achieves 68% accuracy. This finding
+is in agreement with studies that show SMN is predictive of gender (Zhang et al., 2018).
+Interpretability analysis
+We use the learnt distributions over the nodes to calculate
+overlap between each community and known functional connectivity networks (FCNs) from Ji
+et al. (2019) (see Appendix F). Figure 3 shows that DBGDGM finds communities that sig-
+nificantly overlap with existing FCNs. In particular, nodes in community 1 almost fully
+corresponds to the visual network (VIS1 + VIS2), which is in keeping with the nature of
+the experiment (the resting state data was acquired with eyes open and cross-hair fixation).
+Remarkably, the second and third most homogeneous communities correspond to a large
+degree to the DMN, which is well known to dominate resting state activity as a whole
+(Yeshurun et al., 2021). The inspection of additional communities and respective predictive
+power, along with their evolution in time at the region-of-interest granularity, has the poten-
+tial to unveil the yet largely unexplored relationships between dynamic brain connectivity
+changes and, e.g. psychiatric or neurological disorders (Heitmann and Breakspear, 2017).
+8
+
+HCP
+100%
+AUD
+CON
+DAN
+80%
+DMN
+FPN
+LAN
+ORA
+60%
+PMM
+SMN
+VIS1
+VIS2
+40%
+VMM
+20%
+0%
+5
+4
+6
+7
+8
+9 10 11 12 13 14 15 16UKB
+100%
+AUD
+CON
+DAN
+80%
+DMN
+FPN
+LAN
+ORA
+60%
+PMM
+SMN
+VIS1
+VIS2
+40%
+VMM
+20%
+0%
+5
+3
+4
+6
+7
+8
+910 1112 13 14 15 16HCP
+UKB
+90%
+90%
+*
+*
+85%
+80%
+80%
+75%
+70%
+2
+Accura
+70%
+65%
+60%
+60%
+50% -
+55%
+50%
+VGAE
+OSBM
+ELSM
+VGRAPH
+FCM
+VGAE
+OSBM
+VGRNN
+ELSM
+DBGDGM
+VGRAPH
+DBGDGM
+VGRNN
+FCMDBGDGM: Dynamic Brain Graph Deep Generative Model
+6. Conclusion
+We propose DBGDGM, a hierarchical DGM designed for unsupervised representing learning
+of DBGs. Specifically, DBGDGM jointly learns graph-, community-, and node-level embed-
+dings that outperform baselines on classification, interpretability, and dynamic link predic-
+tion with statistical significance. Moreover, an analysis of the learnt dynamic community-
+node distributions shows significant overlap with existing FCNs from neuroscience literature
+further validating our method.
+Acknowledgments
+This work is supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.
+Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium
+(Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by
+the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Re-
+search; and by the McDonnell Center for Systems Neuroscience at Washington University.
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+DBGDGM: Dynamic Brain Graph Deep Generative Model
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+//www.sciencedirect.com/science/article/pii/S105381191401012X.
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+Appendix A. Related work
+Dynamic graph generative models
+Classic generative models for graph-structured
+data are designed for capturing a small set of specific properties (e.g., degree distribution,
+eigenvalues, modularity) of static graphs (Erdos et al., 1960; Barab´asi and Albert, 1999;
+Nowicki and Snijders, 2001). DGMs that exploit the learning capacity of NNs are able to
+learn more expressive graph distributions (Mehta et al., 2019; Kipf and Welling, 2016b;
+Sarkar et al., 2020). Recent DGMs for dynamic graphs are majority VAE-based (Kingma
+and Welling, 2013) and cannot learn community representations (Hajiramezanali et al.,
+2019; Gracious et al., 2021; Zhang et al., 2021). The few that do, are designed for static
+graphs (Sun et al., 2019; Khan et al., 2021; Cavallari et al., 2017).
+Learning representations of dynamic brain graphs
+Unsupervised representation
+learning methods for DBGs tend to focus on clustering DBGs into a finite number of con-
+nectivity patterns that recur over time (Allen et al., 2014; Spencer and Goodfellow, 2022).
+Community detection is another commonly used method but mainly applied to static brain
+graphs (Pavlovi´c et al., 2020; Esfahlani et al., 2021). Extensions to DBGs are typically not
+end-to-end trainable and do not scale to multi-subject datasets (Ting et al., 2020; Martinet
+et al., 2020). Recent deep learning-based methods are predominately GNN-based (Kim
+et al., 2021; Dahan et al., 2021). Unlike DBGDGM, these methods are supervised and
+focus on learning deterministic node- and graph-level representations.
+Appendix B. Method
+B.1. Generative model
+Algorithm 1 summarizes the generative model for DBGDGM.
+15
+
+Campbell Spasov Toschi Li`o
+Algorithm 1: DBGDGM generative model
+Input: {E(s, t)}S, T
+s, t=1
+Hyperparameters: K, Hα, Hψ, Hφ, Lψ, Lφ, Lz, σ2
+ψ, σ2
+φ,
+Initialize: D ← ∅
+for s ← 1 to S do
+α(s) ∼ p(α(s)) = Normal(0Hα, IHα)
+for t ← 1 to T do
+for k ← 1 to K do
+ψ(s,t)
+k
+∼ p(ψ(s, t)
+k
+|ψ(s, t−1)
+k
+) = Normal(ψ(s, t−1)
+k
+, σψIHψ)
+end
+for n ← 1 to V do
+φ(s,t)
+n
+∼ p(φ(s, t)
+n
+|φ(s, t−1)
+n
+) = Normal(φ(s, t−1)
+k
+, σφIHφ)
+end
+˜E(s, t) ← ∅
+for i ← 1 to |E(s, t)| do
+z(s, t)
+i
+∼ p(z(s, t)
+i
+|w(s, t)
+i
+) = Categorical(fθπ(φ(s, t)
+wi
+))
+c(s, t)
+i
+∼ p(c(s, t)
+i
+|z(s, t)
+i
+) = Categorical(fθπ(ψ(s, t)
+zi
+))
+˜E(s, t) ← ˜E(s, t) ∪ {(w(s, t)
+i
+, c(s, t)
+i
+)}
+end
+G(s, t) ← (V, ˜E(s, t))
+D ← D ∪ {G(s, t)}
+end
+end
+B.2. Training objective and learning the parameters
+Substituting the variational distribution from (9) and the joint distribution from (7) into
+the ELBO (8) gives the full training objective defined as
+LELBO(θ, λ) =
+S
+�
+s=1
+T
+�
+t=1
+E(s, t)
+�
+i=1
+�
+Eqλz qλψ
+�
+log pθ(c(s, t)
+i
+|w(s, t)
+i
+, ψ(s, t)
+zi
+)
+�
+− Eqλφ
+�
+DKL[qλz(z(s, t)
+i
+| φ(s, t)
+wi
+, φ(s, t)
+ci
+)||pθz(z(s, t)
+i
+| φ(s, t)
+wi
+)]
+��
+−
+S
+�
+s=1
+�
+DKL[qλα(α(s))||pθα(α(s))]
+T
+�
+t=1
+�
+(14)
+−
+V
+�
+n=1
+Eqλφ
+�
+DKL[qλφ(φ(s, t)
+n
+| φ(s, t−1)
+n
+)||pθφ(φ(s, t)
+n
+| φ(s, t−1)
+n
+)]
+�
+−
+K
+�
+k=1
+Eqλψ
+�
+DKL[qλψ(ψ(s, t)
+k
+| ψ(s, t−1)
+k
+)||pθψ(ψ(s, t)
+k
+| ψ(s, t−1)
+k
+)]
+���
+16
+
+DBGDGM: Dynamic Brain Graph Deep Generative Model
+where DKL[·||·] denotes the Kullback-Leibler (KL) divergence. By maximizing (14), the
+parameters (θ, λ) of the generative model and inference network can be jointly learnt.
+Learning the parameters
+In order to use efficient stochastic gradient-based optimiza-
+tion techniques (Robbins and Monro, 1951) for learning (θ, λ), the gradient of the ELBO
+has to be estimated. The main challenge of this is obtaining gradients of the variables under
+expectation, i.e., Eq∗[·], since they are sampled. To allow gradients to flow through these
+sampling steps, we use the reparameterization trick (Kingma and Welling, 2013; Rezende
+et al., 2014) for the normal distributions and the Gumbel-softmax trick (Jang et al., 2016;
+Maddison et al., 2016) for the categorical distributions. All gradients are now easily com-
+puted via back-propagation (Rumelhart et al., 1986) making DBGDGM end-to-end train-
+able. In addition, we analytically calculate the KL terms for both normal and categorical
+distributions, which leads to lower variance gradient estimates and faster training as com-
+pared to noisy Monte Carlo estimates.
+Appendix C. Datasets
+To create multi-subject DBG datasets, we use real fMRI scans from the UK Biobank (Sud-
+low et al., 2015) and Human Connectome Project (Van Essen et al., 2013). Both data
+sources represent well-characterized population cohorts that have undergone standardized
+neuroimaging and clinical assessments to ensure high quality.
+UK Biobank1 (UKB)
+The UKB dataset consists of S = 300 resting-rate fMRI scans
+(i.e. 3D image of the brain taken over consecutive timepoints) randomly sampled from the
+v1.3 January 2017 release ensuring an equal male/female split (i.e. sex balanced) with an
+age range of 44 − 57 years. The total number of images for each scan is 490 timepoints (6
+minutes duration with a repetition time of 0.74s). The dataset is minimally preprocessed
+following the pipeline described in Alfaro-Almagro et al. (2018).
+Human Connectome Project2 (HCP)
+The HCP dataset similarly consists of S = 300
+sex balanced resting-state fMRI scans randomly sampled from the S1200 release with an
+age range of 22 − 35 years. Only images from the first scanning-session using left-right
+phase encoding are used. The total number of images for each scan is 1, 200 timepoints (15
+minutes duration with a repetition time of 0.72s). The dataset is minimally preprocessed
+following the pipeline described in Glasser et al. (2013)
+Further preprocessing
+The fMRI scans from each dataset are further preprocessed to
+create DBGs. Firstly, each scan is transformed into a multivariate timeseries of BOLD
+signals using the Glasser atlas (Glasser et al., 2016) to average voxels within V = 360 brain
+regions. Next, to ensure comparability with UKB, we truncate the length of HCP timeseries
+to 490 timepoints. Following the commonly used sliding-window method (Calhoun et al.,
+2014), we use Pearson correlation to calculate FC matrices within non-overlapping windows
+of length 1 < W ≤ 490 along the temporal dimension. At every window, we create an
+edge set of a unweighted and undirected graph with no self-edges by thresholding the top
+1 ≤ ϵ < 100 percentile values of the lower triangle of the FC matrix (excluding the principal
+1. https://www.ukbiobank.ac.uk
+2. https://www.humanconnectome.org
+17
+
+Campbell Spasov Toschi Li`o
+diagonal) as connected following Kim et al. (2021). For both datasets, we choose W = 30
+and ϵ = 5 resulting in T = ⌊490/30⌋ = 16 graph snapshots each with E(s, t) = ⌊(360(360 −
+1)/2)(5/100)⌋ = 3, 231 edges.
+Appendix D. Baselines
+We compare DBGDGM against a range of static and dynamic unsupervised graph repre-
+sentation learning baseline models, all with publicly available code. In particular, we focus
+on baselines that are generative and can quantify uncertainty. We leave comparisons to
+popular deterministic baselines such as DynamicTriad (Zhou et al., 2018), DySAT (Sankar
+et al., 2020), and DynNode2Vec (Mahdavi et al., 2018) for future work. Furthermore, since
+all of the baselines were originally designed to model large single-graph datasets, we had to
+adapt each implementation to work with smaller multi-graph datasets.
+Variational graph auto encoder3 (VGAE) (Kipf and Welling, 2016b)
+An extension
+of the variational autoencoder (Kingma and Welling, 2013) (VAE) for graph structured
+data. Specifically, VGAE uses a graph convolutional network (GCN) (Kipf and Welling,
+2016a) to learn a distribution over node embeddings. Originally designed for static graphs,
+we train VGAE on each dynamic graph snapshot independently.
+Overlapping stochastic block model4 (OSBM) (Mehta et al., 2019)
+A deep gener-
+ative version of the overlapping stochastic block model (Miller et al., 2009). In particular,
+OSBM places a stick-breaking prior over the number of communities which allows the model
+to automatically infer the optimal number of communities from the data during training.
+Similar to VGAE, OSBM uses a GCN to parameterize the distribution over node embed-
+dings and is designed for static graphs.
+Variational graph RNN5 (VGRNN) (Hajiramezanali et al., 2019)
+An extension of
+VGAE for dynamic graphs. Using a modified graph RNN architecture, VGRNN is able
+to learn dependencies between and within changing graph topology over time.
+Similar
+to DBGDGM, the prior distribution over node embeddings is parameterized using hidden
+states from previous timepoints.
+Evolving latent space model6 (ELSM) (Gupta et al., 2019)
+A generative model for
+dynamic graphs that learns node embeddings and performs community detection. In par-
+ticular, node embeddings are initially sampled from a Gaussian mixture model over com-
+munities and then evolved over time using an LSTM. Unlike the previous baselines, ELSM
+does not use a GNNs to parameterize model distributions.
+vGraph7 (VGRAPH) (Sun et al., 2019)
+Similar to DBGDGM, VGRAPH simultane-
+ously learns node embeddings and community assignments by modeling nodes as being
+3. https://github.com/tkipf/gae
+4. https://github.com/nikhil-dce/SBM-meet-GNN
+5. https://github.com/VGraphRNN/VGRNN
+6. https://github.com/sh-gupta/ELSM
+7. https://github.com/fanyun-sun/vGraph
+18
+
+DBGDGM: Dynamic Brain Graph Deep Generative Model
+generated from a mixture of communities. The generative process of VGRAPH also re-
+lies on edge information. Since VGRAPH only models static graphs, we train it on each
+dynamic graph snapshot independently.
+Common neighbors (CMN)
+In light of recent work demonstrating that heuristic meth-
+ods are able to outperform deep-learning based models on dynamic link prediction tasks (Skard-
+ing et al., 2022; Poursafaei et al., 2022), we include our own heuristic-based generative model
+baseline. More formally, let π(t)
+vi ∈ [0, 1]V denote a vector of Jaccard index scores for node
+v(t)
+i
+∈ V with all other nodes v(t)
+j
+∈ V for i ̸= j. The Jaccard index between two nodes
+v(t)
+i , v(t)
+j
+∈ V is defined |Γ(v(t)
+i ) ∩ Γ(v(t)
+j )|/|Γ(v(t)
+i ) ∪ Γ(v(t)
+j )| where Γ(v(t)
+i ) denotes the set of
+neighbors of node v(t)
+i . We define the probability of node v(t)
+i
+having a linked neighbor v(t)
+j
+at snapshot t as
+p(v(t)
+j |v(t)
+i ) = Categorical(π(t−1)
+vi
+).
+(15)
+This simple generative model captures the intuition that nodes are more likely to form links
+if they had common neighbors in a previous snapshot.
+Appendix E. Implementation details
+Software and hardware
+All models are developed in Python 3.7 (Python Core Team,
+2019) using scikit-learn 1.1.1 (Pedregosa et al., 2011), PyTorch(Paszke et al., 2019), and
+numpy 1.1.1 (Harris et al., 2020). Statistical significance tests are carried out using deep-
+significance 1.1.1 (Ulmer et al., 2022). Experiments are performed on a Linux server (Debian
+5.10.113-1) with a NVIDIA RTX A6000 GPU with 48 GB memory and 16 CPUs.
+Training and testing
+All baselines are implemented as per the original paper and/or
+code repository given in Appendix D. For the static graph baselines VGAE, OSBM, VGRAPH
+we train on each snapshot independently and use the node and/or community embeddings
+at the last training snapshot to make predictions.
+Hyperparameter optimization
+We use model and training hyperparameter values de-
+scribed in the original implementation of each baseline as a starting point for tuning on
+the validation dataset. Since searching for optional values for each hyperparameter con-
+figuration was outside the scope of this paper, we focus mainly on tuning the dimensions
+of hidden layers. For DBGDGM, we use a learning rate of 1e-4 with a weight decay of
+0. We choose the number of communities K ∈ {3, 6, 8, 12, 16, 24} based on lowest average
+validation NLL (see Figure 4). In the generative model, we fix the temporal smoothness
+hyperparameters σφ = σψ = 0.01. In the inference network, we fix the number of layers
+for all NNs to Lφ = Lψ = Lz = 1. For the Gumbel-softmax reparameterization trick we
+anneal the softmax temperature parameter starting from a maximum of 1 to a minimum
+of 0.05 at a rate of 3e-4. Finally, we train all models for 1, 000 epochs using early-stopping
+with a patience of 15 based on the lowest validation NLL.
+Appendix F. Interpretability analysis
+Using DBGDGM, for each community we average the node distributions across subjects
+and timepoints and take the top 10% most probable nodes. We use these high probability
+19
+
+Campbell Spasov Toschi Li`o
+3
+6
+8
+12
+16
+24
+Number of communities
+4.45
+4.50
+4.55
+4.60
+4.65
+4.70
+4.75
+4.80
+Validation nll
+hcp
+3
+6
+8
+12
+16
+24
+Number of communities
+4.35
+4.40
+4.45
+4.50
+4.55
+4.60
+4.65
+4.70
+4.75
+Validation nll
+ukb
+Figure 4: Elbow plot for finding the optimal number of communities K.
+nodes to calculate overlap between each community and the brain regions that comprise
+each functional network from Ji et al. (2019). More specifically, the coloured proportions in
+Figure 3 represent the proportion of top nodes in each community, which belong to a given
+functional network.
+Abbreviation
+Functional network
+AUD
+Auditory network
+CON
+Cingulo-opercular network
+DAN
+Dorsal-attention network
+DMN
+Default mode network
+FPN
+Frontoparietal network
+LAN
+Language network
+ORA
+Orbito-affective network
+PMM
+Posterior-multimodal network
+SMN
+Somatomotor network
+VIS1
+Visual network 1
+VIS2
+Visual network 2
+VMM
+Ventral-multimodal network
+Table 2: Functional connectivity networks (FCNs) from Ji et al. (2019)
+20
+